University Calculus, Early Transcendentals, 2nd Edition

UNIVERSITY CALCULUS EARLY TRANSCENDENTALS Second Edition Joel Hass University of California, Davis Maurice D. Weir Na...

Author: Joel Hass | Maurice D. Weir | George B. Thomas

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UNIVERSITY

CALCULUS EARLY TRANSCENDENTALS Second Edition

Joel Hass University of California, Davis Maurice D. Weir Naval Postgraduate School George B. Thomas, Jr. Massachusetts Institute of Technology

Editor in Chief: Deirdre Lynch Senior Acquisitions Editor: William Hoffman Sponsoring Editor: Caroline Celano Senior Content Editor: Elizabeth Bernardi Editorial Assistant: Brandon Rawnsley Senior Managing Editor: Karen Wernholm Associate Managing Editor: Tamela Ambush Senior Production Project Manager: Sheila Spinney Digital Assets Manager: Marianne Groth Supplements Production Coordinator: Kerri McQueen Associate Media Producer: Stephanie Green Software Development: Kristina Evans and Marty Wright Executive Marketing Manager: Jeff Weidenaar Marketing Coordinator: Kendra Bassi Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Advisor: Michael Joyce Image Manager: Rachel Youdelman Manufacturing Manager: Evelyn Beaton Senior Manufacturing Buyer: Carol Melville Senior Media Buyer: Ginny Michaud Design Manager: Andrea Nix Production Coordination, Composition, and Illustrations: Nesbitt Graphics, Inc. Cover Design: Andrea Nix Cover Image: Black Shore III—Iceland, 2007. All content copyright © 2009 Josef Hoflehner For permission to use copyrighted material, grateful acknowledgment is made to the copyright holders on page C-1, which is hereby made part of this copyright page. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and Pearson Education was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Hass, Joel. University calculus: early transcendentals/Joel Hass, Maurice D. Weir, George B. Thomas, Jr.—2nd ed. p. cm. Rev. ed. of: University calculus. c2007. ISBN 978-0-321-71739-9 (alk. paper) 1. Calculus—Textbooks. I. Weir, Maurice D. II. Thomas, George B. (George Brinton), 1914–2006. III. Title. QA303.2.H373 2011 515—dc22

2010035141

Copyright © 2012, 2007, Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6—CRK—14 13 12 11

www.pearsonhighered.com

ISBN 13: 978-0-321-71739-9 ISBN 10: 0-321-71739-2

CONTENTS

1

Preface

ix

Functions

1 1.1 1.2 1.3 1.4 1.5 1.6

2

14

Limits and Continuity 2.1 2.2 2.3 2.4 2.5 2.6

3

Functions and Their Graphs 1 Combining Functions; Shifting and Scaling Graphs Trigonometric Functions 21 Graphing with Calculators and Computers 29 Exponential Functions 33 Inverse Functions and Logarithms 39

Rates of Change and Tangents to Curves 52 Limit of a Function and Limit Laws 59 The Precise Definition of a Limit 70 One-Sided Limits 79 Continuity 86 Limits Involving Infinity; Asymptotes of Graphs QUESTIONS TO GUIDE YOUR REVIEW 110 PRACTICE EXERCISES 111 ADDITIONAL AND ADVANCED EXERCISES 113

52

97

Differentiation 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

116 Tangents and the Derivative at a Point 116 The Derivative as a Function 120 Differentiation Rules 129 The Derivative as a Rate of Change 139 Derivatives of Trigonometric Functions 149 The Chain Rule 156 Implicit Differentiation 164 Derivatives of Inverse Functions and Logarithms Inverse Trigonometric Functions 180 Related Rates 186 Linearization and Differentials 195 QUESTIONS TO GUIDE YOUR REVIEW 206 PRACTICE EXERCISES 206 ADDITIONAL AND ADVANCED EXERCISES 211

170

iii

iv

Contents

4

Applications of Derivatives 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

5

289 Area and Estimating with Finite Sums 289 Sigma Notation and Limits of Finite Sums 299 The Definite Integral 305 The Fundamental Theorem of Calculus 317 Indefinite Integrals and the Substitution Method 328 Substitution and Area Between Curves 335 QUESTIONS TO GUIDE YOUR REVIEW 345 PRACTICE EXERCISES 345 ADDITIONAL AND ADVANCED EXERCISES 349

Applications of Definite Integrals 6.1 6.2 6.3 6.4 6.5 6.6

7

230

Integration 5.1 5.2 5.3 5.4 5.5 5.6

6

Extreme Values of Functions 214 The Mean Value Theorem 222 Monotonic Functions and the First Derivative Test Concavity and Curve Sketching 235 Indeterminate Forms and L’Hôpital’s Rule 246 Applied Optimization 255 Newton’s Method 266 Antiderivatives 271 QUESTIONS TO GUIDE YOUR REVIEW 281 PRACTICE EXERCISES 281 ADDITIONAL AND ADVANCED EXERCISES 285

214

353

Volumes Using Cross-Sections 353 Volumes Using Cylindrical Shells 364 Arc Length 372 Areas of Surfaces of Revolution 378 Work 383 Moments and Centers of Mass 389 QUESTIONS TO GUIDE YOUR REVIEW 397 PRACTICE EXERCISES 397 ADDITIONAL AND ADVANCED EXERCISES 399

Integrals and Transcendental Functions 7.1 7.2 7.3

The Logarithm Defined as an Integral 401 Exponential Change and Separable Differential Equations Hyperbolic Functions 420 QUESTIONS TO GUIDE YOUR REVIEW 428 PRACTICE EXERCISES 428 ADDITIONAL AND ADVANCED EXERCISES 429

401 411

Contents

8

Techniques of Integration 8.1 8.2 8.3 8.4 8.5 8.6 8.7

9

448

563

Parametrizations of Plane Curves 563 Calculus with Parametric Curves 570 Polar Coordinates 579 Graphing in Polar Coordinates 583 Areas and Lengths in Polar Coordinates 587 Conics in Polar Coordinates 591 QUESTIONS TO GUIDE YOUR REVIEW 598 PRACTICE EXERCISES 599 ADDITIONAL AND ADVANCED EXERCISES 600

Vectors and the Geometry of Space 11.1 11.2 11.3 11.4 11.5 11.6

486

Sequences 486 Infinite Series 498 The Integral Test 507 Comparison Tests 512 The Ratio and Root Tests 517 Alternating Series, Absolute and Conditional Convergence 522 Power Series 529 Taylor and Maclaurin Series 538 Convergence of Taylor Series 543 The Binomial Series and Applications of Taylor Series 550 QUESTIONS TO GUIDE YOUR REVIEW 559 PRACTICE EXERCISES 559 ADDITIONAL AND ADVANCED EXERCISES 561

Parametric Equations and Polar Coordinates 10.1 10.2 10.3 10.4 10.5 10.6

11

431

Infinite Sequences and Series 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10

10

Integration by Parts 432 Trigonometric Integrals 439 Trigonometric Substitutions 444 Integration of Rational Functions by Partial Fractions Integral Tables and Computer Algebra Systems 456 Numerical Integration 461 Improper Integrals 471 QUESTIONS TO GUIDE YOUR REVIEW 481 PRACTICE EXERCISES 481 ADDITIONAL AND ADVANCED EXERCISES 483

v

Three-Dimensional Coordinate Systems 602 Vectors 607 The Dot Product 616 The Cross Product 624 Lines and Planes in Space 630 Cylinders and Quadric Surfaces 638 QUESTIONS TO GUIDE YOUR REVIEW 643 PRACTICE EXERCISES 644 ADDITIONAL AND ADVANCED EXERCISES 646

602

vi

Contents

12

Vector-Valued Functions and Motion in Space 12.1 12.2 12.3 12.4 12.5 12.6

13

Curves in Space and Their Tangents 649 Integrals of Vector Functions; Projectile Motion 657 Arc Length in Space 664 Curvature and Normal Vectors of a Curve 668 Tangential and Normal Components of Acceleration 674 Velocity and Acceleration in Polar Coordinates 679 QUESTIONS TO GUIDE YOUR REVIEW 682 PRACTICE EXERCISES 683 ADDITIONAL AND ADVANCED EXERCISES 685

Partial Derivatives 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

14

649

686 Functions of Several Variables 686 Limits and Continuity in Higher Dimensions 694 Partial Derivatives 703 The Chain Rule 714 Directional Derivatives and Gradient Vectors 723 Tangent Planes and Differentials 730 Extreme Values and Saddle Points 740 Lagrange Multipliers 748 QUESTIONS TO GUIDE YOUR REVIEW 757 PRACTICE EXERCISES 758 ADDITIONAL AND ADVANCED EXERCISES 761

Multiple Integrals 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

763 Double and Iterated Integrals over Rectangles 763 Double Integrals over General Regions 768 Area by Double Integration 777 Double Integrals in Polar Form 780 Triple Integrals in Rectangular Coordinates 786 Moments and Centers of Mass 795 Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 814 QUESTIONS TO GUIDE YOUR REVIEW 823 PRACTICE EXERCISES 823 ADDITIONAL AND ADVANCED EXERCISES 825

802

Contents

15

Integration in Vector Fields 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8

16

17

828

Line Integrals 828 Vector Fields and Line Integrals: Work, Circulation, and Flux 834 Path Independence, Conservative Fields, and Potential Functions 847 Green’s Theorem in the Plane 858 Surfaces and Area 870 Surface Integrals 880 Stokes’ Theorem 889 The Divergence Theorem and a Unified Theory 900 QUESTIONS TO GUIDE YOUR REVIEW 911 PRACTICE EXERCISES 911 ADDITIONAL AND ADVANCED EXERCISES 914

First-Order Differential Equations 16.1 16.2 16.3 16.4 16.5

Online

Solutions, Slope Fields, and Euler’s Method 16-2 First-Order Linear Equations 16-10 Applications 16-16 Graphical Solutions of Autonomous Equations 16-22 Systems of Equations and Phase Planes 16-29

Second-Order Differential Equations 17.1 17.2 17.3 17.4 17.5

vii

Online

Second-Order Linear Equations 17-1 Nonhomogeneous Linear Equations 17-8 Applications 17-17 Euler Equations 17-23 Power Series Solutions 17-26

Appendices

AP-1 A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10 A.11

Real Numbers and the Real Line AP-1 Mathematical Induction AP-6 Lines, Circles, and Parabolas AP-10 Conic Sections AP-18 Proofs of Limit Theorems AP-26 Commonly Occurring Limits AP-29 Theory of the Real Numbers AP-31 Complex Numbers AP-33 The Distributive Law for Vector Cross Products AP-43 The Mixed Derivative Theorem and the Increment Theorem Taylor’s Formula for Two Variables AP-48

AP-44

Answers to Odd-Numbered Exercises

A-1

Index

I-1

Credits

C-1

A Brief Table of Integrals

T-1

This page intentionally left blank

PREFACE We have significantly revised this edition of University Calculus, Early Transcendentals to meet the changing needs of today’s instructors and students. The result is a book with more examples, more mid-level exercises, more figures, better conceptual flow, and increased clarity and precision. As with the previous edition, this new edition provides a briefer, modern introduction to calculus that supports conceptual understanding but retains the essential elements of a traditional course. These enhancements are closely tied to an expanded version of MyMathLab® for this text (discussed further on), providing additional support for students and flexibility for instructors. In this second edition, we introduce the basic transcendental functions in Chapter 1. After reviewing the basic trigonometric functions, we present the family of exponential functions using an algebraic and graphical approach, with the natural exponential described as a particular member of this family. Logarithms are then defined as the inverse functions of the exponentials, and we also discuss briefly the inverse trigonometric functions. We fully incorporate these functions throughout our developments of limits, derivatives, and integrals in the next five chapters of the book, including the examples and exercises. This approach gives students the opportunity to work early with exponential and logarithmic functions in combinations with polynomials, rational and algebraic functions, and trigonometric functions as they learn the concepts, operations, and applications of single-variable calculus. Later, in Chapter 7, we revisit the definition of transcendental functions, now giving a more rigorous presentation. Here we define the natural logarithm function as an integral with the natural exponential as its inverse. Today, an increasing number of students become familiar with the terminology and operational methods of calculus in high school. However, their conceptual understanding of calculus is often quite limited when they enter college. We have acknowledged this reality by concentrating on concepts and their applications throughout. We encourage students to think beyond memorizing formulas and to generalize concepts as they are introduced. Our hope is that after taking calculus, students will be confident in their problem-solving and reasoning abilities. Mastering a beautiful subject with practical applications to the world is its own reward, but the real gift is the ability to think and generalize. We intend this book to provide support and encouragement for both.

Changes for the Second Edition CONTENT In preparing this edition we have maintained the basic structure of the Table of Contents from the first edition, yet we have paid attention to requests by current users and reviewers to postpone the introduction of parametric equations until we present polar coordinates. We have made numerous revisions to most of the chapters, detailed as follows:

•

Functions We condensed this chapter to focus on reviewing function concepts and introducing the transcendental functions. Prerequisite material covering real numbers, intervals, increments, straight lines, distances, circles, and the conic sections is presented in Appendices 1–4.

ix

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• •

•

•

•

•

•

•

Limits To improve the flow of this chapter, we combined the ideas of limits involving infinity and their associations with asymptotes to the graphs of functions, placing them together in the final section of Chapter 2. Differentiation While we use rates of change and tangents to curves as motivation for studying the limit concept, we now merge the derivative concept into a single chapter. We reorganized and increased the number of related rates examples, and we added new examples and exercises on graphing rational functions. L’Hôpital’s Rule is presented as an application section, consistent with our early coverage of the transcendental functions. Antiderivatives and Integration We maintain the organization of the first edition in placing antiderivatives as the final topic of Chapter 4, covering applications of derivatives. Our focus is on “recovering a function from its derivative” as the solution to the simplest type of first-order differential equation. Integrals, as “limits of Riemann sums,” motivated primarily by the problem of finding the areas of general regions with curved boundaries, are a new topic forming the substance of Chapter 5. After carefully developing the integral concept, we turn our attention to its evaluation and connection to antiderivatives captured in the Fundamental Theorem of Calculus. The ensuing applications then define the various geometric ideas of area, volume, lengths of paths, and centroids, all as limits of Riemann sums giving definite integrals, which can be evaluated by finding an antiderivative of the integrand. Series We retain the organizational structure and content of the first edition for the topics of sequences and series. We have added several new figures and exercises to the various sections, and we revised some of the proofs related to convergence of power series in order to improve the accessibility of the material for students. The request stated by one of our users as, “anything you can do to make this material easier for students will be welcomed by our faculty,” drove our thinking for revisions to this chapter. Parametric Equations Several users requested that we move this topic into Chapter 10, where we also cover polar coordinates. We have done this, realizing that many departments choose to cover these topics at the beginning of Calculus III, in preparation for their coverage of vectors and multivariable calculus. Vector-Valued Functions We streamlined the topics in this chapter to place more emphasis on the conceptual ideas supporting the later material on partial derivatives, the gradient vector, and line integrals. We condensed the discussions of the Frenet frame and Kepler’s three laws of planetary motion. Multivariable Calculus We have further enhanced the art in these chapters, and we have added many new figures, examples, and exercises. As with the first edition, we continue to make the connections of multivariable ideas with their single-variable analogues studied earlier in the book. Vector Fields We devoted considerable effort to improving the clarity and mathematical precision of our treatment of vector integral calculus, including many additional examples, figures, and exercises. Important theorems and results are stated more clearly and completely, together with enhanced explanations of their hypotheses and mathematical consequences. The area of a surface is still organized into a single section, and surfaces defined implicitly or explicitly are treated as special cases of the more general parametric representation. Surface integrals and their applications then follow as a separate section. Stokes’ Theorem and the Divergence Theorem continue to be presented as generalizations of Green’s Theorem to three dimensions. A number of new examples and figures have been added illustrating these important themes.

EXERCISES AND EXAMPLES We know that the exercises and examples are critical components in learning calculus. Because of this importance, we have updated, improved, and increased the number of exercises in nearly every section of the book. There are over 750 new exercises in this edition. We continue our organization and grouping of exercises by

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topic as in earlier editions, progressing from computational problems to applied and theoretical problems. Over 70 examples have been added to clarify or deepen the meaning of the topic being discussed and to help students understand its mathematical consequences or applications to science and engineering. The new examples appear throughout the text, with emphasis on key concepts and results. ART Because of their importance to learning calculus, we have continued to improve existing figures in University Calculus, Early Transcendentals, and we have created a significant number of new ones. We continue to use color consistently and pedagogically to enhance the conceptual idea that is being illustrated. We have also taken a fresh look at all of the figure captions, paying considerable attention to clarity and precision in short statements.

y y 1x

No matter what positive number is, the graph enters this band at x 1 and stays. y

N – 1 0 y –

z

M 1

x

–

No matter what positive number is, the graph enters this band at x – 1 and stays.

FIGURE 2.50, page 98 The geometric explanation of a finite limit as x : ; q .

y x

FIGURE 15.9, page 835 A surface in a space occupied by a moving fluid.

MYMATHLAB AND MATHXL The increasing use of and demand for online homework systems has driven the changes to MyMathLab and MathXL® for University Calculus, Early Transcendentals. The MyMathLab and MathXL courses now include significantly more exercises of all types.

Continuing Features RIGOR The level of rigor is consistent with the first edition. We continue to distinguish between formal and informal discussions and to point out their differences. We think starting with a more intuitive, less formal, approach helps students understand a new or difficult concept so they can then appreciate its full mathematical precision and outcomes. We pay attention to defining ideas carefully and to proving theorems appropriate for calculus students, while mentioning deeper or subtler issues they would study in a more advanced course. Our organization and distinctions between informal and formal discussions give the instructor a degree of flexibility in the amount and depth of coverage of the various topics. For example, while we do not prove the Intermediate Value Theorem or the Extreme Value Theorem for continuous functions on a … x … b, we do state these theorems precisely, illustrate their meanings in numerous examples, and use them to prove other important results. Furthermore, for those instructors who desire greater depth of coverage, in Appendix 7 we discuss the reliance of the validity of these theorems on the completeness of the real numbers.

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WRITING EXERCISES Writing exercises placed throughout the text ask students to explore and explain a variety of calculus concepts and applications. In addition, the end of each chapter contains a list of questions for students to review and summarize what they have learned. Many of these exercises make good writing assignments. END-OF-CHAPTER REVIEWS In addition to problems appearing after each section, each chapter culminates with review questions, practice exercises covering the entire chapter, and a series of Additional and Advanced Exercises serving to include more challenging or synthesizing problems. WRITING AND APPLICATIONS As always, this text continues to be easy to read, conversational, and mathematically rich. Each new topic is motivated by clear, easy-to-understand examples and is then reinforced by its application to real-world problems of immediate interest to students. A hallmark of this book has been the application of calculus to science and engineering. These applied problems have been updated, improved, and extended continually over the last several editions. TECHNOLOGY Technology can be incorporated in a course using this text, according to the taste of the instructor. Each section contains exercises requiring the use of technology; these are marked with a T if suitable for calculator or computer use, or they are labeled Computer Explorations if a computer algebra system (CAS, such as Maple or Mathematica) is required.

Text Versions UNIVERSITY CALCULUS, EARLY TRANSCENDENTALS, Second Edition Complete (Chapters 1–15), ISBN 0-321-71739-2 | 978-0-321-71739-9 Single Variable Calculus (Chapters 1–10), ISBN 0-321-69459-7 | 978-0-321-69459-1 Multivariable Calculus (Chapters 9–15), ISBN 0-321-69460-0 | 978-0-321-69460-7 University Calculus, Early Transcendentals, introduces and integrates transcendental functions (such as inverse trigonometric, exponential, and logarithmic functions) into the exposition, examples, and exercises of the early chapters alongside the algebraic functions. Electronic versions of the text are available within MyMathLab (www.mymathlab.com) or at CourseSmart.com.

Instructor’s Edition for University Calculus, Early Transcendentals, Second Edition ISBN 0-321-71747-3 | 978-0-321-71747-4 In addition to including all of the answers present in the student editions, the Instructor’s Edition includes even-numbered answers for Chapters 1–11.

Print Supplements INSTRUCTOR’S SOLUTIONS MANUAL Single Variable Calculus (Chapters 1–10), ISBN 0-321-71748-1 | 978-0-321-71748-1 Multivariable Calculus (Chapters 9–15), ISBN 0-321-71749-X | 978-0-321-71749-8 The Instructor’s Solutions Manual by William Ardis, Collin County Community College, contains complete worked-out solutions to all of the exercises in University Calculus, Early Transcendentals.

STUDENT’S SOLUTIONS MANUAL Single Variable Calculus (Chapters 1–10), ISBN 0-321-69462-7 | 978-0-321-69462-1 Multivariable Calculus (Chapters 9–15), ISBN 0-321-69454-6 | 978-0-321-69454-6 The Student’s Solutions Manual by William Ardis, Collin County Community College, is designed for the student and contains carefully worked-out solutions to all of the oddnumbered exercises in University Calculus, Early Transcendentals.

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JUST-IN-TIME ALGEBRA AND TRIGONOMETRY FOR EARLY TRANSCENDENTALS CALCULUS, Third Edition ISBN 0-321-32050-6 | 978-0-321-32050-6 Sharp algebra and trigonometry skills are critical to mastering calculus, and Just-in-Time Algebra and Trigonometry for Early Transcendentals Calculus by Guntram Mueller and Ronald I. Brent is designed to bolster these skills while students study calculus. As students make their way through calculus, this text is with them every step of the way, showing them the necessary algebra or trigonometry topics and pointing out potential problem spots. The easy-to-use table of contents has algebra and trigonometry topics arranged in the order in which students will need them as they study calculus.

CALCULUS REVIEW CARDS The Calculus Review Cards (one for Single Variable and another for Multivariable) are a student resource containing important formulas, functions, definitions, and theorems that correspond precisely to the Thomas’ Calculus series. These cards can work as a reference for completing homework assignments or as an aid in studying, and are available bundled with a new text. Contact your Pearson sales representative for more information.

Media and Online Supplements TECHNOLOGY RESOURCE MANUALS Maple Manual by James Stapleton, North Carolina State University Mathematica Manual by Marie Vanisko, Carroll College TI-Graphing Calculator Manual by Elaine McDonald-Newman, Sonoma State University These manuals cover Maple 13, Mathematica 7, and the TI-83 Plus/TI-84 Plus and TI-89, respectively. Each manual provides detailed guidance for integrating a specific software package or graphing calculator throughout the course, including syntax and commands. These manuals are available to students and instructors through the University Calculus, Early Transcendentals Web site, www.pearsonhighered.com/thomas, and MyMathLab.

WEB SITE www.pearsonhighered.com/thomas The University Calculus, Early Transcendentals Web site contains the chapters on First-Order and Second-Order Differential Equations, including odd-numbered answers, and provides the expanded historical biographies and essays referenced in the text. Also available is a collection of Maple and Mathematica modules and the Technology Resource Manuals.

Video Lectures with Optional Captioning The Video Lectures with Optional Captioning feature an engaging team of mathematics instructors who present comprehensive coverage of topics in the text. The lecturers’ presentations include examples and exercises from the text and support an approach that emphasizes visualization and problem solving. Available only through MyMathLab and MathXL.

MyMathLab Online Course (access code required) MyMathLab is a text-specific, easily customizable online course that integrates interactive multimedia instruction with textbook content. MyMathLab gives you the tools you need to deliver all or a portion of your course online, whether your students are in a lab setting or working from home.

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Interactive homework exercises, correlated to your textbook at the objective level, are algorithmically generated for unlimited practice and mastery. Most exercises are freeresponse and provide guided solutions, sample problems, and tutorial learning aids for extra help.

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•

“Getting Ready” chapter includes hundreds of exercises that address prerequisite skills in algebra and trigonometry. Each student can receive remediation for just those skills he or she needs help with.

•

Personalized homework assignments that you can design to meet the needs of your class. MyMathLab tailors the assignment for each student based on his or her test or quiz scores. Each student receives a homework assignment that contains only the problems he or she still needs to master.

•

Personalized Study Plan, generated when students complete a test or quiz or homework, indicates which topics have been mastered and links to tutorial exercises for topics students have not mastered. You can customize the Study Plan so that the topics available match your course content.

•

Multimedia learning aids, such as video lectures and podcasts, Java applets, animations, and a complete multimedia textbook, help students independently improve their understanding and performance. You can assign these multimedia learning aids as homework to help your students grasp the concepts.

•

Homework and Test Manager lets you assign homework, quizzes, and tests that are automatically graded. Select just the right mix of questions from the MyMathLab exercise bank, instructor-created custom exercises, and/or TestGen® test items.

•

Gradebook, designed specifically for mathematics and statistics, automatically tracks students’ results, lets you stay on top of student performance, and gives you control over how to calculate final grades. You can also add offline (paper-and-pencil) grades to the gradebook.

•

MathXL Exercise Builder allows you to create static and algorithmic exercises for your online assignments. You can use the library of sample exercises as an easy starting point, or you can edit any course-related exercise.

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Pearson Tutor Center (www.pearsontutorservices.com) access is automatically included with MyMathLab. The Tutor Center is staffed by qualified math instructors who provide textbook-specific tutoring for students via toll-free phone, fax, email, and interactive Web sessions.

Students do their assignments in the Flash®-based MathXL Player, which is compatible with almost any browser (Firefox®, Safari™, or Internet Explorer®) on almost any platform (Macintosh® or Windows®). MyMathLab is powered by CourseCompass™, Pearson Education’s online teaching and learning environment, and by MathXL®, our online homework, tutorial, and assessment system. MyMathLab is available to qualified adopters. For more information, visit www.mymathlab.com or contact your Pearson representative.

MathXL Online Course (access code required) MathXL is an online homework, tutorial, and assessment system that accompanies Pearson’s textbooks in mathematics or statistics.

•

Interactive homework exercises, correlated to your textbook at the objective level, are algorithmically generated for unlimited practice and mastery. Most exercises are freeresponse and provide guided solutions, sample problems, and learning aids for extra help.

•

“Getting Ready” chapter includes hundreds of exercises that address prerequisite skills in algebra and trigonometry. Each student can receive remediation for just those skills he or she needs help with.

•

Personalized Homework assignments are designed by the instructor to meet the needs of the class, and then personalized for each student based on his or her test or quiz scores. As a result, each student receives a homework assignment in which the problems cover only the objectives for which he or she has not achieved mastery.

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•

Personalized Study Plan, generated when students complete a test or quiz or homework, indicates which topics have been mastered and links to tutorial exercises for topics students have not mastered. Instructors can customize the available topics in the study plan to match their course concepts.

•

Multimedia learning aids, such as video lectures, Java applets, and animations, help students independently improve their understanding and performance. These are assignable as homework, to further encourage their use.

•

Gradebook, designed specifically for mathematics and statistics, automatically tracks students’ results, lets you stay on top of student performance, and gives you control over how to calculate final grades.

•

MathXL Exercise Builder allows you to create static and algorithmic exercises for your online assignments. You can use the library of sample exercises as an easy starting point or use the Exercise Builder to edit any of the course-related exercises.

•

Homework and Test Manager lets you create online homework, quizzes, and tests that are automatically graded. Select just the right mix of questions from the MathXL exercise bank, instructor-created custom exercises, and/or TestGen test items.

The new, Flash-based MathXL Player is compatible with almost any browser (Firefox, Safari, or Internet Explorer) on almost any platform (Macintosh or Windows). MathXL is available to qualified adopters. For more information, visit our Web site at www .mathxl.com, or contact your Pearson representative.

TestGen TestGen (www.pearsonhighered.com/testgen) enables instructors to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text. TestGen is algorithmically based, allowing instructors to create multiple but equivalent versions of the same question or test with the click of a button. Instructors can also modify test bank questions or add new questions. Tests can be printed or administered online. The software and testbank are available for download from Pearson Education’s online catalog. (www.pearsonhighered.com)

PowerPoint® Lecture Slides These classroom presentation slides are geared specifically to the sequence and philosophy of the Thomas’ Calculus series. Key graphics from the book are included to help bring the concepts alive in the classroom.These files are available to qualified instructors through the Pearson Instructor Resource Center, www.pearsonhighered/irc, and MyMathLab.

Acknowledgments We would like to express our thanks to the people who made many valuable contributions to this edition as it developed through its various stages:

Accuracy Checkers Rhea Meyerholtz Tom Wegleitner Gary Williams

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Reviewers for the Second Edition Robert A. Beezer, University of Puget Sound Przemyslaw Bogacki, Old Dominion University Leonard Chastofsky, University of Georgia Meighan Dillon, Southern Polytechnic State University Anne Dougherty, University of Colorado Said Fariabi, San Antonio College Klaus Fischer, George Mason University Tim Flood, Pittsburg State University Rick Ford, California State University—Chico Robert Gardner, East Tennessee State University Christopher Heil, Georgia Institute of Technology David Hemmer, SUNY—Buffalo Joshua Brandon Holden, Rose-Hulman Institute of Technology Alexander Hulpke, Colorado State University Jacqueline Jensen, Sam Houston State University Jennifer M. Johnson, Princeton University Hideaki Kaneko, Old Dominion University Przemo Kranz, University of Mississippi John Kroll, Old Dominion University Glenn Ledder, University of Nebraska—Lincoln Matthew Leingang, New York University Xin Li, University of Central Florida Maura Mast, University of Massachusetts—Boston Val Mohanakumar, Hillsborough Community College—Dale Mabry Campus Aaron Montgomery, Central Washington University Yibiao Pan, University of Pittsburgh Christopher M. Pavone, California State University at Chico Cynthia Piez, University of Idaho Brooke Quinlan, Hillsborough Community College—Dale Mabry Campus Rebecca A. Segal, Virginia Commonwealth University Andrew V. Sills, Georgia Southern University Edward E. Slaminka, Auburn University Alex Smith, University of Wisconsin—Eau Claire Mark A. Smith, Miami University Donald Solomon, University of Wisconsin—Milwaukee John Sullivan, Black Hawk College Maria Terrell, Cornell University Blake Thornton, Washington University in St. Louis David Walnut, George Mason University Adrian Wilson, University of Montevallo Bobby Winters, Pittsburg State University Dennis Wortman, University of Massachusetts—Boston

FPO

1 FUNCTIONS OVERVIEW Functions are fundamental to the study of calculus. In this chapter we review what functions are and how they are pictured as graphs, how they are combined and transformed, and ways they can be classified. We review the trigonometric functions, and we discuss misrepresentations that can occur when using calculators and computers to obtain a function’s graph. We also discuss inverse, exponential, and logarithmic functions. The real number system, Cartesian coordinates, straight lines, circles, parabolas, ellipses, and hyperbolas are reviewed in the Appendices.

1.1

Functions and Their Graphs Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation, a graph, a numerical table, or a verbal description; we will use all four representations throughout this book. This section reviews these function ideas.

Functions; Domain and Range The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle. The distance an object travels at constant speed along a straight-line path depends on the elapsed time. In each case, the value of one variable quantity, say y, depends on the value of another variable quantity, which we might call x. We say that “y is a function of x” and write this symbolically as y = ƒ(x)

(“y equals ƒ of x”).

In this notation, the symbol ƒ represents the function, the letter x is the independent variable representing the input value of ƒ, and y is the dependent variable or output value of ƒ at x.

DEFINITION A function ƒ from a set D to a set Y is a rule that assigns a unique (single) element ƒsxd H Y to each element x H D. The set D of all possible input values is called the domain of the function. The set of all values of ƒ(x) as x varies throughout D is called the range of the function. The range may not include every element in the set Y. The domain and range of a function can be any sets of objects, but often in calculus they are sets of real numbers interpreted as points of a coordinate line. (In Chapters 12–15, we will encounter functions for which the elements of the sets are points in the coordinate plane or in space.)

1

2

x

Chapter 1: Functions

f

Input (domain)

Output (range)

f (x)

FIGURE 1.1 A diagram showing a function as a kind of machine.

x a D domain set

f (a)

f (x)

Y set containing the range

FIGURE 1.2 A function from a set D to a set Y assigns a unique element of Y to each element in D.

Often a function is given by a formula that describes how to calculate the output value from the input variable. For instance, the equation A = pr 2 is a rule that calculates the area A of a circle from its radius r (so r, interpreted as a length, can only be positive in this formula). When we define a function y = ƒsxd with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of real x-values for which the formula gives real y-values, the so-called natural domain. If we want to restrict the domain in some way, we must say so. The domain of y = x 2 is the entire set of real numbers. To restrict the domain of the function to, say, positive values of x, we would write “y = x 2, x 7 0.” Changing the domain to which we apply a formula usually changes the range as well. The range of y = x 2 is [0, q d. The range of y = x 2, x Ú 2, is the set of all numbers obtained by squaring numbers greater than or equal to 2. In set notation (see Appendix 1), the range is 5x 2 ƒ x Ú 26 or 5y ƒ y Ú 46 or [4, q d. When the range of a function is a set of real numbers, the function is said to be realvalued. The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed, or half open, and may be finite or infinite. The range of a function is not always easy to find. A function ƒ is like a machine that produces an output value ƒ(x) in its range whenever we feed it an input value x from its domain (Figure 1.1). The function keys on a calculator give an example of a function as a machine. For instance, the 2x key on a calculator gives an output value (the square root) whenever you enter a nonnegative number x and press the 2x key. A function can also be pictured as an arrow diagram (Figure 1.2). Each arrow associates an element of the domain D with a unique or single element in the set Y. In Figure 1.2, the arrows indicate that ƒ(a) is associated with a, ƒ(x) is associated with x, and so on. Notice that a function can have the same value at two different input elements in the domain (as occurs with ƒ(a) in Figure 1.2), but each input element x is assigned a single output value ƒ(x).

EXAMPLE 1 Let’s verify the natural domains and associated ranges of some simple functions. The domains in each case are the values of x for which the formula makes sense. Function y y y y y

= = = = =

x2 1>x 2x 24 - x 21 - x 2

Domain (x)

Range ( y)

s - q, q d s - q , 0d ´ s0, q d [0, q d s - q , 4] [-1, 1]

[0, q d s - q , 0d ´ s0, q d [0, q d [0, q d [0, 1]

The formula y = x 2 gives a real y-value for any real number x, so the domain is s - q , q d. The range of y = x 2 is [0, q d because the square of any real number is nonnegative and every nonnegative number y is the square of its own square root, y = A 2y B 2 for y Ú 0. The formula y = 1>x gives a real y-value for every x except x = 0. For consistency in the rules of arithmetic, we cannot divide any number by zero. The range of y = 1>x, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since y = 1>(1>y). That is, for y Z 0 the number x = 1>y is the input assigned to the output value y. The formula y = 1x gives a real y-value only if x Ú 0. The range of y = 1x is [0, q d because every nonnegative number is some number’s square root (namely, it is the square root of its own square). In y = 14 - x, the quantity 4 - x cannot be negative. That is, 4 - x Ú 0, or x … 4. The formula gives real y-values for all x … 4. The range of 14 - x is [0, q d, the set of all nonnegative numbers. Solution

1.1

3

Functions and Their Graphs

The formula y = 21 - x 2 gives a real y-value for every x in the closed interval from -1 to 1. Outside this domain, 1 - x 2 is negative and its square root is not a real number. The values of 1 - x 2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of 21 - x 2 is [0, 1].

Graphs of Functions If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ. In set notation, the graph is 5sx, ƒsxdd ƒ x H D6. The graph of the function ƒsxd = x + 2 is the set of points with coordinates (x, y) for which y = x + 2. Its graph is the straight line sketched in Figure 1.3. The graph of a function ƒ is a useful picture of its behavior. If (x, y) is a point on the graph, then y = ƒsxd is the height of the graph above the point x. The height may be positive or negative, depending on the sign of ƒsxd (Figure 1.4). y

f (1)

y

f (2) x

yx2

0

1

x

2 f(x)

2 (x, y) –2

x

y x2

-2 -1 0 1

4 1 0 1

3 2

9 4

2

4

x

0

FIGURE 1.3 The graph of ƒsxd = x + 2 is the set of points (x, y) for which y has the value x + 2 .

EXAMPLE 2

FIGURE 1.4 If (x, y) lies on the graph of ƒ, then the value y = ƒsxd is the height of the graph above the point x (or below x if ƒ(x) is negative).

Graph the function y = x 2 over the interval [-2, 2].

Make a table of xy-pairs that satisfy the equation y = x 2. Plot the points (x, y) whose coordinates appear in the table, and draw a smooth curve (labeled with its equation) through the plotted points (see Figure 1.5).

Solution

y

How do we know that the graph of y = x 2 doesn’t look like one of these curves? (–2, 4)

(2, 4)

4

y

y

y x2 3 ⎛3 , 9⎛ ⎝2 4⎝

2 (–1, 1)

1

–2

0

–1

1

2

y x 2?

y x 2?

(1, 1) x

FIGURE 1.5 Graph of the function in Example 2.

x

x

4

Chapter 1: Functions

To find out, we could plot more points. But how would we then connect them? The basic question still remains: How do we know for sure what the graph looks like between the points we plot? Calculus answers this question, as we will see in Chapter 4. Meanwhile, we will have to settle for plotting points and connecting them as best we can.

Representing a Function Numerically We have seen how a function may be represented algebraically by a formula (the area function) and visually by a graph (Example 2). Another way to represent a function is numerically, through a table of values. Numerical representations are often used by engineers and scientists. From an appropriate table of values, a graph of the function can be obtained using the method illustrated in Example 2, possibly with the aid of a computer. The graph consisting of only the points in the table is called a scatterplot.

EXAMPLE 3 Musical notes are pressure waves in the air. The data in Table 1.1 give recorded pressure displacement versus time in seconds of a musical note produced by a tuning fork. The table provides a representation of the pressure function over time. If we first make a scatterplot and then connect approximately the data points (t, p) from the table, we obtain the graph shown in Figure 1.6. p (pressure)

TABLE 1.1 Tuning fork data

Time

Pressure

Time

0.00091 0.00108 0.00125 0.00144 0.00162 0.00180 0.00198 0.00216 0.00234 0.00253 0.00271 0.00289 0.00307 0.00325 0.00344

-0.080 0.200 0.480 0.693 0.816 0.844 0.771 0.603 0.368 0.099 -0.141 -0.309 -0.348 -0.248 -0.041

0.00362 0.00379 0.00398 0.00416 0.00435 0.00453 0.00471 0.00489 0.00507 0.00525 0.00543 0.00562 0.00579 0.00598

Pressure 0.217 0.480 0.681 0.810 0.827 0.749 0.581 0.346 0.077 -0.164 -0.320 -0.354 -0.248 -0.035

1.0 0.8 0.6 0.4 0.2 –0.2 –0.4 –0.6

Data

0.001 0.002 0.003 0.004 0.005 0.006

t (sec)

FIGURE 1.6 A smooth curve through the plotted points gives a graph of the pressure function represented by Table 1.1 (Example 3).

The Vertical Line Test for a Function Not every curve in the coordinate plane can be the graph of a function. A function ƒ can have only one value ƒ(x) for each x in its domain, so no vertical line can intersect the graph of a function more than once. If a is in the domain of the function ƒ, then the vertical line x = a will intersect the graph of ƒ at the single point (a, ƒ(a)). A circle cannot be the graph of a function since some vertical lines intersect the circle twice. The circle in Figure 1.7a, however, does contain the graphs of two functions of x: the upper semicircle defined by the function ƒ(x) = 21 - x 2 and the lower semicircle defined by the function g (x) = - 21 - x 2 (Figures 1.7b and 1.7c).

1.1 y

y

y

–1 –1

0

1

x

–1

yx

0

1

2

x

3

x, -x,

x Ú 0 x 6 0,

First formula Second formula

y f (x) 2 y1

1 –1

(c) y –1 x 2

whose graph is given in Figure 1.8. The right-hand side of the equation means that the function equals x if x Ú 0, and equals -x if x 6 0. Piecewise-defined functions often arise when real-world data are modeled. Here are some other examples.

y

–2

x

Sometimes a function is described by using different formulas on different parts of its domain. One example is the absolute value function ƒxƒ = e

FIGURE 1.8 The absolute value function has domain s - q , q d and range [0, q d .

y –x

1 0

Piecewise-Defined Functions

1 –1

x

y x

3 2

–3 –2

1

FIGURE 1.7 (a) The circle is not the graph of a function; it fails the vertical line test. (b) The upper semicircle is the graph of a function ƒsxd = 21 - x 2 . (c) The lower semicircle is the graph of a function gsxd = - 21 - x 2 .

y y –x

0

(b) y 1 x 2

(a) x 2 y 2 1

5

Functions and Their Graphs

EXAMPLE 4

y x2

0

1

The function -x, ƒsxd = • x 2, 1,

x

2

FIGURE 1.9 To graph the function y = ƒsxd shown here, we apply different formulas to different parts of its domain (Example 4).

x 6 0 0 … x … 1 x 7 1

First formula Second formula Third formula

is defined on the entire real line but has values given by different formulas, depending on the position of x. The values of ƒ are given by y = -x when x 6 0, y = x 2 when 0 … x … 1, and y = 1 when x 7 1. The function, however, is just one function whose domain is the entire set of real numbers (Figure 1.9).

y yx

3

EXAMPLE 5

The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted :x; . Figure 1.10 shows the graph. Observe that

2 y ⎣x⎦

1 –2 –1

1

2

3

x

:2.4; = 2, :2; = 2,

:1.9; = 1, :0.2; = 0,

:0; = 0, : -0.3; = -1

: -1.2; = -2, : -2; = -2.

–2

FIGURE 1.10 The graph of the greatest integer function y = :x; lies on or below the line y = x , so it provides an integer floor for x (Example 5).

EXAMPLE 6 The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted < x = . Figure 1.11 shows the graph. For positive values of x, this function might represent, for example, the cost of parking x hours in a parking lot which charges $1 for each hour or part of an hour.

6

Chapter 1: Functions

Increasing and Decreasing Functions

y yx

3

If the graph of a function climbs or rises as you move from left to right, we say that the function is increasing. If the graph descends or falls as you move from left to right, the function is decreasing.

2 y ⎡x⎤

1 –2 –1

1

2

x

3

DEFINITIONS Let ƒ be a function defined on an interval I and let x1 and x2 be any two points in I.

–1

1. If ƒsx2) 7 ƒsx1 d whenever x1 6 x2 , then ƒ is said to be increasing on I. 2. If ƒsx2 d 6 ƒsx1 d whenever x1 6 x2 , then ƒ is said to be decreasing on I.

–2

FIGURE 1.11 The graph of the least integer function y = < x = lies on or above the line y = x , so it provides an integer ceiling for x (Example 6).

It is important to realize that the definitions of increasing and decreasing functions must be satisfied for every pair of points x1 and x2 in I with x1 6 x2 . Because we use the inequality 6 to compare the function values, instead of … , it is sometimes said that ƒ is strictly increasing or decreasing on I. The interval I may be finite (also called bounded) or infinite (unbounded) and by definition never consists of a single point (Appendix 1). The function graphed in Figure 1.9 is decreasing on s - q , 0] and increasing on [0, 1]. The function is neither increasing nor decreasing on the interval [1, q d because of the strict inequalities used to compare the function values in the definitions.

EXAMPLE 7

Even Functions and Odd Functions: Symmetry The graphs of even and odd functions have characteristic symmetry properties.

DEFINITIONS

A function y = ƒsxd is an even function of x if ƒs -xd = ƒsxd, odd function of x if ƒs -xd = -ƒsxd,

for every x in the function’s domain. y y x2 (x, y)

(–x, y)

x

0 (a) y y x3

0

(x, y) x

(–x, –y)

The names even and odd come from powers of x. If y is an even power of x, as in y = x 2 or y = x 4 , it is an even function of x because s -xd2 = x 2 and s -xd4 = x 4 . If y is an odd power of x, as in y = x or y = x 3 , it is an odd function of x because s -xd1 = -x and s -xd3 = -x 3 . The graph of an even function is symmetric about the y-axis. Since ƒs -xd = ƒsxd, a point (x, y) lies on the graph if and only if the point s -x, yd lies on the graph (Figure 1.12a). A reflection across the y-axis leaves the graph unchanged. The graph of an odd function is symmetric about the origin. Since ƒs -xd = -ƒsxd, a point (x, y) lies on the graph if and only if the point s -x, -yd lies on the graph (Figure 1.12b). Equivalently, a graph is symmetric about the origin if a rotation of 180° about the origin leaves the graph unchanged. Notice that the definitions imply that both x and -x must be in the domain of ƒ.

EXAMPLE 8 (b)

FIGURE 1.12 (a) The graph of y = x 2 (an even function) is symmetric about the y-axis. (b) The graph of y = x 3 (an odd function) is symmetric about the origin.

ƒsxd = x 2

Even function: s -xd2 = x 2 for all x; symmetry about y-axis.

ƒsxd = x 2 + 1

Even function: s -xd2 + 1 = x 2 + 1 for all x; symmetry about y-axis (Figure 1.13a).

ƒsxd = x

Odd function: s -xd = -x for all x; symmetry about the origin.

ƒsxd = x + 1

Not odd: ƒs -xd = -x + 1 , but -ƒsxd = -x - 1 . The two are not equal. Not even: s -xd + 1 Z x + 1 for all x Z 0 (Figure 1.13b).

1.1

y

Functions and Their Graphs

7

y

y x2 1

yx1

y x2

yx 1

1 x

0

(a)

–1

x

0

(b)

FIGURE 1.13 (a) When we add the constant term 1 to the function y = x 2 , the resulting function y = x 2 + 1 is still even and its graph is still symmetric about the y-axis. (b) When we add the constant term 1 to the function y = x , the resulting function y = x + 1 is no longer odd. The symmetry about the origin is lost (Example 8).

Common Functions A variety of important types of functions are frequently encountered in calculus. We identify and briefly describe them here. Linear Functions A function of the form ƒsxd = mx + b , for constants m and b, is called a linear function. Figure 1.14a shows an array of lines ƒsxd = mx where b = 0, so these lines pass through the origin. The function ƒsxd = x where m = 1 and b = 0 is called the identity function. Constant functions result when the slope m = 0 (Figure 1.14b). A linear function with positive slope whose graph passes through the origin is called a proportionality relationship.

m –3

y m2 y 2x

y –3x m –1

m1

y

yx

1 m 2 1 y x 2 x

y –x 0

2 1 0

(a)

y3 2

1

2

x

(b)

FIGURE 1.14 (a) Lines through the origin with slope m. (b) A constant function with slope m = 0.

DEFINITION Two variables y and x are proportional (to one another) if one is always a constant multiple of the other; that is, if y = kx for some nonzero constant k.

If the variable y is proportional to the reciprocal 1>x, then sometimes it is said that y is inversely proportional to x (because 1>x is the multiplicative inverse of x). Power Functions A function ƒsxd = x a , where a is a constant, is called a power function. There are several important cases to consider.

8

Chapter 1: Functions

(a) a = n, a positive integer. The graphs of ƒsxd = x n , for n = 1, 2, 3, 4, 5, are displayed in Figure 1.15. These functions are defined for all real values of x. Notice that as the power n gets larger, the curves tend to flatten toward the x-axis on the interval s -1, 1d, and to rise more steeply for ƒ x ƒ 7 1. Each curve passes through the point (1, 1) and through the origin. The graphs of functions with even powers are symmetric about the y-axis; those with odd powers are symmetric about the origin. The even-powered functions are decreasing on the interval s - q , 0] and increasing on [0, q d; the odd-powered functions are increasing over the entire real line s - q , q ). y

y

yx

1 –1

y

y x2

1 0

–1

FIGURE 1.15

1

x

–1

y

y x3

1

0

1

x

–1

–1

0

y y x5

y x4

1 x

1

–1

–1

1

0

1

x

–1

–1

0

1

x

–1

Graphs of ƒsxd = x n, n = 1, 2, 3, 4, 5, defined for - q 6 x 6 q .

(b) a = -1

or a = -2 .

The graphs of the functions ƒsxd = x -1 = 1>x and gsxd = x -2 = 1>x 2 are shown in Figure 1.16. Both functions are defined for all x Z 0 (you can never divide by zero). The graph of y = 1>x is the hyperbola xy = 1, which approaches the coordinate axes far from the origin. The graph of y = 1>x 2 also approaches the coordinate axes. The graph of the function ƒ is symmetric about the origin; ƒ is decreasing on the intervals s - q , 0) and s0, q ). The graph of the function g is symmetric about the y-axis; g is increasing on s - q , 0) and decreasing on s0, q ). y y y 12 x

y 1x 1 0

1

x

Domain: x 0 Range: y 0

(a)

1 0

x 1 Domain: x 0 Range: y 0 (b)

FIGURE 1.16 Graphs of the power functions ƒsxd = x a for part (a) a = -1 and for part (b) a = -2.

(c) a =

1 1 3 2 , , , and . 2 3 2 3

3 The functions ƒsxd = x 1>2 = 2x and gsxd = x 1>3 = 2 x are the square root and cube root functions, respectively. The domain of the square root function is [0, q d, but the cube root function is defined for all real x. Their graphs are displayed in Figure 1.17, along with the graphs of y = x 3>2 and y = x 2>3 . (Recall that x 3>2 = sx 1>2 d3 and x 2>3 = sx 1>3 d2 .)

Polynomials A function p is a polynomial if psxd = an x n + an - 1x n - 1 + Á + a1 x + a0 where n is a nonnegative integer and the numbers a0 , a1 , a2 , Á , an are real constants (called the coefficients of the polynomial). All polynomials have domain s - q , q d. If the

1.1

9

Functions and Their Graphs

y y

y

y

y

y x

y x 23

3

y x

1 1 0

x 32

1 Domain: 0 x Range: 0 y

x

0

1 Domain: – x Range: – y

Graphs of the power functions ƒsxd = x a for a =

FIGURE 1.17

1

1 x

0

x

x

0 1 Domain: – x Range: 0 y

1 Domain: 0 x Range: 0 y

2 1 1 3 , , , and . 2 3 2 3

leading coefficient an Z 0 and n 7 0, then n is called the degree of the polynomial. Linear functions with m Z 0 are polynomials of degree 1. Polynomials of degree 2, usually written as psxd = ax 2 + bx + c, are called quadratic functions. Likewise, cubic functions are polynomials psxd = ax 3 + bx 2 + cx + d of degree 3. Figure 1.18 shows the graphs of three polynomials. Techniques to graph polynomials are studied in Chapter 4. 3 2 y x x 2x 1 3 2 3 y

4

y y

2

–2

16

2 –1

–4

y (x 2)4(x 1)3(x 1)

y 8x 4 14x 3 9x 2 11x 1

0

2

x

4

1

–2

x

2

–4

–1

–6 –2

1

x

2

–8 –10

–4

–16

–12 (a)

FIGURE 1.18

0

(c)

(b)

Graphs of three polynomial functions.

Rational Functions A rational function is a quotient or ratio ƒ(x) = p(x)>q(x), where p and q are polynomials. The domain of a rational function is the set of all real x for which qsxd Z 0. The graphs of several rational functions are shown in Figure 1.19. y y

8 2 y 5x 2 8x 3 3x 2

y

4 2 2 y 2x 3 2 7x 4

–4

2

4

x

–5

0

5

2 10

–1

–2

y 11x3 2 2x 1

4 Line y 5 3

1

–2

6

x

–4

–2 0 –2

2

4

6

x

–4

–2

NOT TO SCALE

–4

–6 –8 (a)

FIGURE 1.19 of the graph.

(b)

(c)

Graphs of three rational functions. The straight red lines are called asymptotes and are not part

10

Chapter 1: Functions

Algebraic Functions Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots) lies within the class of algebraic functions. All rational functions are algebraic, but also included are more complicated functions (such as those satisfying an equation like y 3 - 9xy + x 3 = 0, studied in Section 3.7). Figure 1.20 displays the graphs of three algebraic functions. y x 1/3(x 4)

y

y x(1 x)2/5

y y 3 (x 2 1) 2/3 4 y

4 3 2 1

1

–1 –1 –2 –3

x

4

x

0

0

5 7

x

1

–1

(c)

(b)

(a)

FIGURE 1.20 Graphs of three algebraic functions.

Trigonometric Functions The six basic trigonometric functions are reviewed in Section 1.3. The graphs of the sine and cosine functions are shown in Figure 1.21. y

y

1

3 0

–

2

1

– 2 x

0 –1

–1 (a) f (x) sin x

3 2

5 2

x

2

(b) f(x) cos x

FIGURE 1.21 Graphs of the sine and cosine functions.

Exponential Functions Functions of the form ƒsxd = a x , where the base a 7 0 is a positive constant and a Z 1, are called exponential functions. All exponential functions have domain s - q , q d and range s0, q d, so an exponential function never assumes the value 0. We discuss exponential functions in Section 1.5. The graphs of some exponential functions are shown in Figure 1.22. y

y y 10 x

y 10 –x

12

12

10

10

8

8 y

6 y 3x

4 2 –1

–0.5

FIGURE 1.22

0 (a)

1

6 4

y 2x 0.5

3 –x

2

y 2 –x x

Graphs of exponential functions.

–1

–0.5

0 (b)

0.5

1

x

1.1

11

Functions and Their Graphs

Logarithmic Functions These are the functions ƒsxd = loga x, where the base a Z 1 is a positive constant. They are the inverse functions of the exponential functions, and we discuss these functions in Section 1.6. Figure 1.23 shows the graphs of four logarithmic functions with various bases. In each case the domain is s0, q d and the range is s - q , q d. y

y

y log 2 x y log 3 x

1

0

1

–1

x y log5 x

1

y log10 x –1

FIGURE 1.23 Graphs of four logarithmic functions.

0

1

x

FIGURE 1.24 Graph of a catenary or hanging cable. (The Latin word catena means “chain.”)

Transcendental Functions These are functions that are not algebraic. They include the trigonometric, inverse trigonometric, exponential, and logarithmic functions, and many other functions as well. A particular example of a transcendental function is a catenary. Its graph has the shape of a cable, like a telephone line or electric cable, strung from one support to another and hanging freely under its own weight (Figure 1.24). The function defining the graph is discussed in Section 7.3.

Exercises 1.1 Functions In Exercises 1–6, find the domain and range of each function. 1. ƒsxd = 1 + x 2

2. ƒsxd = 1 - 2x

3. F(x) = 25x + 10 4 5. ƒstd = 3 - t

4. g(x) = 2x 2 - 3x 2 6. G(t) = 2 t - 16

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers. 7. a.

b.

y

y

8. a.

0

y

b.

x

0

x

Finding Formulas for Functions 9. Express the area and perimeter of an equilateral triangle as a function of the triangle’s side length x.

y

10. Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.

0

x

0

x

11. Express the edge length of a cube as a function of the cube’s diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length.

12

Chapter 1: Functions

12. A point P in the first quadrant lies on the graph of the function ƒsxd = 2x . Express the coordinates of P as functions of the slope of the line joining P to the origin.

y

31. a.

b.

(–1, 1) (1, 1) 1

2

13. Consider the point (x, y) lying on the graph of the line 2x + 4y = 5. Let L be the distance from the point (x, y) to the origin (0, 0). Write L as a function of x. 14. Consider the point (x, y) lying on the graph of y = 2x - 3. Let L be the distance between the points (x, y) and (4, 0). Write L as a function of y.

15. ƒsxd = 5 - 2x

16. ƒsxd = 1 - 2x - x 2

17. g sxd = 2ƒ x ƒ

18. g sxd = 2 -x

19. F std = t> ƒ t ƒ

20. G std = 1> ƒ t ƒ

21. Find the domain of y =

x + 3 4 - 2x 2 - 9

22. Find the range of y = 2 +

.

27. F sxd = e

4 - x , x 2 + 2x ,

28. G sxd = e

1>x , x,

x 6 0 0 … x

0

2

0

1

2

3

4

y 3

(2, 1) 5

x Ú 0 x 6 0.

38. y = -

1 39. y = - x

40. y =

1 x2

1 ƒxƒ 42. y = 2 -x 44. y = -42x 46. y = s -xd2>3

Even and Odd Functions In Exercises 47–58, say whether the function is even, odd, or neither. Give reasons for your answer.

b.

2

:x;, <x=,

37. y = -x 3

48. ƒsxd = x -5

47. ƒsxd = 3

x

y

30. a.

ƒsxd = e

45. y = -x 3>2

2

2

36. Graph the function

y

b. (1, 1)

1

35. Does < -x = = - :x ; for all real x? Give reasons for your answer.

43. y = x 3>8

y

b. < x = = 0 ?

34. What real numbers x satisfy the equation :x; = < x = ?

41. y = 2ƒ x ƒ

Find a formula for each function graphed in Exercises 29–32. 29. a.

t

Increasing and Decreasing Functions Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

x … 1 x 7 1

2

–A

x

T 3T 2T 2

Why is ƒ(x) called the integer part of x?

Piecewise-Defined Functions Graph the functions in Exercises 25–28.

0 … x … 1 1 6 x … 2

T

a. :x; = 0 ?

b. ƒ x + y ƒ = 1

1 - x, 2 - x,

T 2

T 2

The Greatest and Least Integer Functions 33. For what values of x is

b. y 2 = x 2 a. ƒ y ƒ = x 24. Graph the following equations and explain why they are not graphs of functions of x.

26. g sxd = e

y

b. (T, 1)

0

23. Graph the following equations and explain why they are not graphs of functions of x.

0 … x … 1 1 6 x … 2

y

32. a.

0

x . x2 + 4

x, 25. ƒsxd = e 2 - x,

(–2, –1)

x 1 (1, –1) (3, –1)

A

2

a. ƒ x ƒ + ƒ y ƒ = 1

x

3

1

Functions and Graphs Find the domain and graph the functions in Exercises 15–20.

y

2

x

1 –1

–1 –2 –3

1

x 2 (2, –1)

t

49. ƒsxd = x + 1

50. ƒsxd = x 2 + x

51. gsxd = x 3 + x

52. gsxd = x 4 + 3x 2 - 1

2

x x2 - 1

53. gsxd =

1 x2 - 1

54. gsxd =

55. hstd =

1 t - 1

56. hstd = ƒ t 3 ƒ

57. hstd = 2t + 1

58. hstd = 2 ƒ t ƒ + 1

Theory and Examples 59. The variable s is proportional to t, and s = 25 when t = 75. Determine t when s = 60.

1.1 60. Kinetic energy The kinetic energy K of a mass is proportional to the square of its velocity y. If K = 12,960 joules when y = 18 m>sec, what is K when y = 10 m>sec?

66. a. y = 5x

Functions and Their Graphs

b. y = 5x

c. y = x 5

y g

61. The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10. 62. Boyle’s Law Boyle’s Law says that the volume V of a gas at constant temperature increases whenever the pressure P decreases, so that V and P are inversely proportional. If P = 14.7 lb>in2 when V = 1000 in3, then what is V when P = 23.4 lb>in2?

h

x x

14 x

x x

b. Confirm your findings in part (a) algebraically.

x

64. The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long. a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

y

70. Three hundred books sell for $40 each, resulting in a revenue of (300)($40) = $12,000. For each $5 increase in the price, 25 fewer books are sold. Write the revenue R as a function of the number x of $5 increases.

B

P(x, ?)

A –1

0

x

1

x

In Exercises 65 and 66, match each equation with its graph. Do not use a graphing device, and give reasons for your answer. b. y = x 7

T 68. a. Graph the functions ƒsxd = 3>sx - 1d and g sxd = 2>sx + 1d together to identify the values of x for which 3 2 6 . x - 1 x + 1 b. Confirm your findings in part (a) algebraically. 69. For a curve to be symmetric about the x-axis, the point (x, y) must lie on the curve if and only if the point sx, -yd lies on the curve. Explain why a curve that is symmetric about the x-axis is not the graph of a function, unless the function is y = 0 .

b. Express the area of the rectangle in terms of x.

65. a. y = x 4

f

T 67. a. Graph the functions ƒsxd = x>2 and g sxd = 1 + s4>xd together to identify the values of x for which x 4 7 1 + x. 2

22 x

x

0

63. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.

x

13

71. A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs $5/ft for the legs and $10/ft for the hypotenuse, write the total cost C of construction as a function of h. 72. Industrial costs A power plant sits next to a river where the river is 800 ft wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs $180 per foot across the river and $100 per foot along the land.

c. y = x 10

2 mi P

x

Q

City

y 800 ft

g h

Power plant NOT TO SCALE

0 f

x

a. Suppose that the cable goes from the plant to a point Q on the opposite side that is x ft from the point P directly opposite the plant. Write a function C(x) that gives the cost of laying the cable in terms of the distance x. b. Generate a table of values to determine if the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

14

Chapter 1: Functions

1.2

Combining Functions; Shifting and Scaling Graphs In this section we look at the main ways functions are combined or transformed to form new functions.

Sums, Differences, Products, and Quotients Like numbers, functions can be added, subtracted, multiplied, and divided (except where the denominator is zero) to produce new functions. If ƒ and g are functions, then for every x that belongs to the domains of both ƒ and g (that is, for x H Dsƒd ¨ Dsgd), we define functions ƒ + g, ƒ - g , and ƒg by the formulas sƒ + gdsxd = ƒsxd + gsxd. sƒ - gdsxd = ƒsxd - gsxd. sƒgdsxd = ƒsxdgsxd. Notice that the + sign on the left-hand side of the first equation represents the operation of addition of functions, whereas the + on the right-hand side of the equation means addition of the real numbers ƒ(x) and g(x). At any point of Dsƒd ¨ Dsgd at which gsxd Z 0, we can also define the function ƒ>g by the formula ƒ ƒsxd a g bsxd = swhere gsxd Z 0d. gsxd Functions can also be multiplied by constants: If c is a real number, then the function cƒ is defined for all x in the domain of ƒ by scƒdsxd = cƒsxd.

EXAMPLE 1

The functions defined by the formulas ƒsxd = 2x

and

g sxd = 21 - x

have domains Dsƒd = [0, q d and Dsgd = s - q , 1]. The points common to these domains are the points [0, q d ¨ s - q , 1] = [0, 1]. The following table summarizes the formulas and domains for the various algebraic combinations of the two functions. We also write ƒ # g for the product function ƒg. Function

Formula

Domain

ƒ + g ƒ - g g - ƒ ƒ#g

sƒ + gdsxd = 2x + 21 - x sƒ - gdsxd = 2x - 21 - x sg - ƒdsxd = 21 - x - 2x sƒ # gdsxd = ƒsxdgsxd = 2xs1 - xd ƒ ƒsxd x g sxd = gsxd = A 1 - x gsxd g 1 - x sxd = = ƒ ƒsxd A x

[0, 1] = Dsƒd ¨ Dsgd [0, 1] [0, 1] [0, 1]

ƒ>g g>ƒ

[0, 1) sx = 1 excludedd (0, 1] sx = 0 excludedd

The graph of the function ƒ + g is obtained from the graphs of ƒ and g by adding the corresponding y-coordinates ƒ(x) and g(x) at each point x H Dsƒd ¨ Dsgd, as in Figure 1.25. The graphs of ƒ + g and ƒ # g from Example 1 are shown in Figure 1.26.

1.2

y

y

yfg

g(x) 1 x

8 6

15

Combining Functions; Shifting and Scaling Graphs

y ( f g)(x) y g(x)

4 2

g(a)

y f (x)

1 2

f (a) g(a)

yf•g

f (a) x

a

0

f(x) x

1

FIGURE 1.25 Graphical addition of two functions.

0

1 5

2 5

3 5

4 5

1

x

FIGURE 1.26 The domain of the function ƒ + g is the intersection of the domains of ƒ and g, the interval [0, 1] on the x-axis where these domains overlap. This interval is also the domain of the function ƒ # g (Example 1).

Composite Functions Composition is another method for combining functions.

DEFINITION If ƒ and g are functions, the composite function ƒ ⴰ g (“ƒ composed with g”) is defined by sƒ ⴰ gdsxd = ƒsgsxdd. The domain of ƒ ⴰ g consists of the numbers x in the domain of g for which g(x) lies in the domain of ƒ. The definition implies that ƒ ⴰ g can be formed when the range of g lies in the domain of ƒ. To find sƒ ⴰ gdsxd, first find g(x) and second find ƒ(g(x)). Figure 1.27 pictures ƒ ⴰ g as a machine diagram and Figure 1.28 shows the composite as an arrow diagram. f g f (g(x)) x g

x

g

g(x)

f

f (g(x))

FIGURE 1.27 A composite function ƒ ⴰ g uses the output g(x) of the first function g as the input for the second function f.

f

g(x)

FIGURE 1.28 Arrow diagram for ƒ ⴰ g . If x lies in the domain of g and g(x) lies in the domain of f, then the functions f and g can be composed to form (ƒ ⴰ g)(x).

To evaluate the composite function g ⴰ ƒ (when defined), we find ƒ(x) first and then g(ƒ(x)). The domain of g ⴰ ƒ is the set of numbers x in the domain of ƒ such that ƒ(x) lies in the domain of g. The functions ƒ ⴰ g and g ⴰ ƒ are usually quite different.

16

Chapter 1: Functions

EXAMPLE 2

If ƒsxd = 2x and gsxd = x + 1, find

(a) sƒ ⴰ gdsxd

(b) sg ⴰ ƒdsxd

(c) sƒ ⴰ ƒdsxd

(d) sg ⴰ gdsxd.

Solution

Composite

Domain [-1, q d

(a) sƒ ⴰ gdsxd = ƒsg sxdd = 2g sxd = 2x + 1 (b) sg ⴰ ƒdsxd = g sƒsxdd = ƒsxd + 1 = 2x + 1 (c) sƒ ⴰ ƒdsxd = ƒsƒsxdd = 2ƒsxd = 21x = x

1>4

(d) sg ⴰ gdsxd = g sg sxdd = g sxd + 1 = sx + 1d + 1 = x + 2

[0, q d [0, q d s - q, q d

To see why the domain of ƒ ⴰ g is [-1, q d, notice that gsxd = x + 1 is defined for all real x but belongs to the domain of ƒ only if x + 1 Ú 0, that is to say, when x Ú -1. Notice that if ƒsxd = x 2 and gsxd = 2x, then sƒ ⴰ gdsxd = A 2x B 2 = x. However, the domain of ƒ ⴰ g is [0, q d, not s - q , q d, since 2x requires x Ú 0.

Shifting a Graph of a Function A common way to obtain a new function from an existing one is by adding a constant to each output of the existing function, or to its input variable. The graph of the new function is the graph of the original function shifted vertically or horizontally, as follows.

Shift Formulas Vertical Shifts y

y x2 2 y x2 1 y x2

y = ƒsxd + k

Shifts the graph of ƒ up k units if k 7 0 Shifts it down ƒ k ƒ units if k 6 0

Horizontal Shifts

y = ƒsx + hd

y x2 2

Shifts the graph of ƒ left h units if h 7 0 Shifts it right ƒ h ƒ units if h 6 0

2 1 unit

EXAMPLE 3

1 –2

0 –1

2

x

2 units

–2

FIGURE 1.29 To shift the graph of ƒsxd = x 2 up (or down), we add positive (or negative) constants to the formula for ƒ (Examples 3a and b).

(a) Adding 1 to the right-hand side of the formula y = x 2 to get y = x 2 + 1 shifts the graph up 1 unit (Figure 1.29). (b) Adding -2 to the right-hand side of the formula y = x 2 to get y = x 2 - 2 shifts the graph down 2 units (Figure 1.29). (c) Adding 3 to x in y = x 2 to get y = sx + 3d2 shifts the graph 3 units to the left, while adding -2 shifts the graph 2 units to the right (Figure 1.30). (d) Adding -2 to x in y = ƒ x ƒ , and then adding -1 to the result, gives y = ƒ x - 2 ƒ - 1 and shifts the graph 2 units to the right and 1 unit down (Figure 1.31).

Scaling and Reflecting a Graph of a Function To scale the graph of a function y = ƒsxd is to stretch or compress it, vertically or horizontally. This is accomplished by multiplying the function ƒ, or the independent variable x, by an appropriate constant c. Reflections across the coordinate axes are special cases where c = -1.

1.2 Add a positive constant to x.

Add a negative constant to x.

y

y y x – 2 – 1

4 y (x 3) 2

y x2

17

Combining Functions; Shifting and Scaling Graphs

y (x 2) 2 1

1 –3

–4 1

0

–2

2

–1

4

6

x

x

2

FIGURE 1.30 To shift the graph of y = x 2 to the left, we add a positive constant to x (Example 3c). To shift the graph to the right, we add a negative constant to x.

FIGURE 1.31 Shifting the graph of y = ƒ x ƒ 2 units to the right and 1 unit down (Example 3d).

Vertical and Horizontal Scaling and Reflecting Formulas For c 7 1, the graph is scaled:

y = cƒsxd

Stretches the graph of ƒ vertically by a factor of c.

1 y = c ƒsxd

Compresses the graph of ƒ vertically by a factor of c.

y = ƒscxd y = ƒsx>cd

Compresses the graph of ƒ horizontally by a factor of c. Stretches the graph of ƒ horizontally by a factor of c.

For c = -1, the graph is reflected:

y = -ƒsxd y = ƒs -xd

EXAMPLE 4

Reflects the graph of ƒ across the x-axis. Reflects the graph of ƒ across the y-axis. Here we scale and reflect the graph of y = 2x.

(a) Vertical: Multiplying the right-hand side of y = 2x by 3 to get y = 32x stretches the graph vertically by a factor of 3, whereas multiplying by 1>3 compresses the graph by a factor of 3 (Figure 1.32). (b) Horizontal: The graph of y = 23x is a horizontal compression of the graph of y = 2x by a factor of 3, and y = 2x>3 is a horizontal stretching by a factor of 3 (Figure 1.33). Note that y = 23x = 232x so a horizontal compression may correspond to a vertical stretching by a different scaling factor. Likewise, a horizontal stretching may correspond to a vertical compression by a different scaling factor. (c) Reflection: The graph of y = - 2x is a reflection of y = 2x across the x-axis, and y = 2 -x is a reflection across the y-axis (Figure 1.34). y

y y 3x

5

compress

y x

stretch

2

2

1

compress

1 0

y 3 x

3

3

–1

y x

4

4

1

2

3

y 3 x 4

x

FIGURE 1.32 Vertically stretching and compressing the graph y = 1x by a factor of 3 (Example 4a).

stretch

1 –1

0

y

y –x

1

2

3

4

y x y x3 x

FIGURE 1.33 Horizontally stretching and compressing the graph y = 1x by a factor of 3 (Example 4b).

1

–3

–2

–1

1

2

3

x

–1 y –x

FIGURE 1.34 Reflections of the graph y = 1x across the coordinate axes (Example 4c).

18

Chapter 1: Functions

Given the function ƒsxd = x 4 - 4x 3 + 10 (Figure 1.35a), find formulas to

EXAMPLE 5

(a) compress the graph horizontally by a factor of 2 followed by a reflection across the y-axis (Figure 1.35b). (b) compress the graph vertically by a factor of 2 followed by a reflection across the x-axis (Figure 1.35c). y 20

y

y 16x 4 32x 3 10 y f (x)

x4

4x 3

10

y – 12 x 4 2x 3 5

20 10

10 –1

0 –10

10 1

2

3

4

x

–2

–1

0 –10

x

1

–1

0

1

2

3

4

x

–10 –20

–20 (b)

(a)

(c)

FIGURE 1.35 (a) The original graph of f. (b) The horizontal compression of y = ƒsxd in part (a) by a factor of 2, followed by a reflection across the y-axis. (c) The vertical compression of y = ƒsxd in part (a) by a factor of 2, followed by a reflection across the x-axis (Example 5). Solution

(a) We multiply x by 2 to get the horizontal compression, and by -1 to give reflection across the y-axis. The formula is obtained by substituting -2x for x in the right-hand side of the equation for ƒ: y = ƒs -2xd = s -2xd4 - 4s -2xd3 + 10 = 16x 4 + 32x 3 + 10 . (b) The formula is y = -

1 1 ƒsxd = - x 4 + 2x 3 - 5. 2 2

Exercises 1.2 Algebraic Combinations In Exercises 1 and 2, find the domains and ranges of ƒ, g, ƒ + g , and ƒ # g. 1. ƒsxd = x,

g sxd = 2x - 1

2. ƒsxd = 2x + 1,

g sxd = 2x - 1

In Exercises 3 and 4, find the domains and ranges of ƒ, g, ƒ>g, and g >ƒ. 3. ƒsxd = 2,

g sxd = x 2 + 1

4. ƒsxd = 1,

g sxd = 1 + 2x

Composites of Functions 5. If ƒsxd = x + 5 and g sxd = x 2 - 3 , find the following. a. ƒ( g (0)) c. ƒ( g (x))

b. g(ƒ(0)) d. g(ƒ(x))

e. ƒ(ƒ(-5))

f. g (g (2))

g. ƒ(ƒ(x))

h. g (g (x))

6. If ƒsxd = x - 1 and gsxd = 1>sx + 1d , find the following. a. ƒ(g (1>2))

b. g (ƒ(1>2))

c. ƒ(g (x))

d. g (ƒ(x))

e. ƒ(ƒ(2))

f. g (g (2))

g. ƒ(ƒ(x))

h. g (g (x))

In Exercises 7–10, write a formula for ƒ ⴰ g ⴰ h. 7. ƒ(x) = x + 1,

gsxd = 3x , hsxd = 4 - x

8. ƒ(x) = 3x + 4, 9. ƒsxd = 2x + 1, 10. ƒsxd =

x + 2 , 3 - x

gsxd = 2x - 1, hsxd = x 2 1 , x + 4

gsxd = gsxd =

x2 , x + 1 2

1 hsxd = x hsxd = 22 - x

1.2

Combining Functions; Shifting and Scaling Graphs

x . Find a function y = g(x) so that x - 2 (ƒ ⴰ g)(x) = x.

Let ƒsxd = x - 3, g sxd = 2x , hsxd = x 3 , and jsxd = 2x . Express each of the functions in Exercises 11 and 12 as a composite involving one or more of ƒ, g, h, and j.

19. Let ƒ(x) =

11. a. y = 2x - 3

20. Let ƒ(x) = 2x 3 - 4. Find a function y = g(x) so that (ƒ ⴰ g)(x) = x + 2.

b. y = 2 2x

c. y = x 1>4

d. y = 4x f. y = s2x - 6d3

e. y = 2sx - 3d

3

12. a. y = 2x - 3

Shifting Graphs 21. The accompanying figure shows the graph of y = -x 2 shifted to two new positions. Write equations for the new graphs.

b. y = x 3>2

c. y = x 9

d. y = x - 6

e. y = 22x - 3

f. y = 2x 3 - 3

19

y

13. Copy and complete the following table. g(x)

ƒ(x)

(ƒ ⴰ g)(x)

a. x - 7

2x

?

b. x + 2 c.

?

3x

?

2x - 5

2x 2 - 5

d.

x x - 1

x x - 1

?

e.

?

1 1 + x

x

f.

1 x

?

x

–7

Position (a)

0

4

y –x 2

x

Position (b)

22. The accompanying figure shows the graph of y = x 2 shifted to two new positions. Write equations for the new graphs. y Position (a)

14. Copy and complete the following table. g(x)

ƒ(x)

(ƒ ⴰ g)(x)

ƒxƒ

?

a.

1 x - 1

b.

?

x - 1 x

x x + 1

c.

?

2x

ƒxƒ ƒxƒ

d. 2x

?

y x2 3

x

0 Position (b)

15. Evaluate each expression using the given table of values: –5

-2

-1

0

1

2

ƒ(x)

1

0

-2

1

2

g(x)

2

1

0

-1

0

x

a. ƒsgs -1dd

b. gsƒs0dd

c. ƒsƒs -1dd

d. gsgs2dd

e. gsƒs -2dd

f. ƒsgs1dd

23. Match the equations listed in parts (a)–(d) to the graphs in the accompanying figure. a. y = sx - 1d2 - 4

b. y = sx - 2d2 + 2

c. y = sx + 2d2 + 2

d. y = sx + 3d2 - 2 y

Position 2

Position 1

16. Evaluate each expression using the functions ƒ(x) = 2 - x,

g(x) = b

-x, x - 1,

-2 … x 6 0 0 … x … 2.

a. ƒsgs0dd

b. gsƒs3dd

c. gsgs -1dd

d. ƒsƒs2dd

e. gsƒs0dd

f. ƒsgs1>2dd

In Exercises 17 and 18, (a) write formulas for ƒ ⴰ g and g ⴰ ƒ and find the (b) domain and (c) range of each.

3 (–2, 2) Position 3

2 1

–4 –3 –2 –1 0

(2, 2) 1 2 3 Position 4

(–3, –2)

1 17. ƒ(x) = 2x + 1, g (x) = x 18. ƒ(x) = x 2, g (x) = 1 - 2x

(1, –4)

x

20

Chapter 1: Functions

24. The accompanying figure shows the graph of y = -x 2 shifted to four new positions. Write an equation for each new graph.

55. The accompanying figure shows the graph of a function ƒ(x) with domain [0, 2] and range [0, 1]. Find the domains and ranges of the following functions, and sketch their graphs.

y (1, 4)

y

(–2, 3) (b)

(a) (2, 0)

1

y f (x)

0

2

x

(–4, –1)

(c)

(d)

Exercises 25–34 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. 25. x 2 + y 2 = 49

Down 3, left 2

26. x 2 + y 2 = 25

Up 3, left 4

27. y = x

3

Right 1, down 1

29. y = 2x

Left 0.81

30. y = - 2x

Right 3

31. y = 2x - 7

Up 7

1 32. y = sx + 1d + 5 2

e. ƒsx + 2d

f. ƒsx - 1d

g. ƒs -xd

h. -ƒsx + 1d + 1

56. The accompanying figure shows the graph of a function g(t) with domain [-4, 0] and range [-3, 0] . Find the domains and ranges of the following functions, and sketch their graphs.

–2

y g(t)

Down 5, right 1

0

t

–3

a. g s -td

b. -g std

c. g std + 3

d. 1 - g std

e. g s -t + 2d

f. g st - 2d

g. g s1 - td

h. -g st - 4d

Left 2, down 1

35. y = 2x + 4

36. y = 29 - x

37. y = ƒ x - 2 ƒ

38. y = ƒ 1 - x ƒ - 1

39. y = 1 + 2x - 1

40. y = 1 - 2x

41. y = sx + 1d

42. y = sx - 8d2>3

43. y = 1 - x 2>3

44. y + 4 = x 2>3

3 x - 1 - 1 45. y = 2

46. y = sx + 2d3>2 + 1

2>3

1 x - 2

1 49. y = x + 2

53. y =

d. -ƒsxd

–4

Graph the functions in Exercises 35–54.

51. y =

c. 2ƒ(x)

y

Up 1, right 1

34. y = 1>x 2

47. y =

b. ƒsxd - 1

Left 1, down 1

28. y = x 2>3

33. y = 1>x

a. ƒsxd + 2

x

1 sx - 1d2 1 + 1 x2

1 48. y = x - 2 50. y = 52. y = 54. y =

1 x + 2 1 - 1 x2 1 sx + 1d2

Vertical and Horizontal Scaling Exercises 57–66 tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph. 57. y = x 2 - 1,

stretched vertically by a factor of 3

58. y = x 2 - 1,

compressed horizontally by a factor of 2

59. y = 1 +

1 , x2

compressed vertically by a factor of 2

1 , stretched horizontally by a factor of 3 x2 61. y = 2x + 1, compressed horizontally by a factor of 4 60. y = 1 +

62. y = 2x + 1,

stretched vertically by a factor of 3

63. y = 24 - x 2,

stretched horizontally by a factor of 2

64. y = 24 - x ,

compressed vertically by a factor of 3

2

65. y = 1 - x ,

compressed horizontally by a factor of 3

66. y = 1 - x ,

stretched horizontally by a factor of 2

3 3

1.3 Graphing In Exercises 67–74, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation. 68. y =

69. y = sx - 1d3 + 2

70. y = s1 - xd3 + 2

1 71. y = - 1 2x

2 72. y = 2 + 1 x 74. y = s -2xd2>3

3 73. y = - 2 x

A

1 -

Combining Functions 77. Assume that ƒ is an even function, g is an odd function, and both ƒ and g are defined on the entire real line ⺢ . Which of the following (where defined) are even? odd?

x 2

67. y = - 22x + 1

21

Trigonometric Functions

a. ƒg

b. ƒ>g

c. g >ƒ

d. ƒ 2 = ƒƒ

e. g 2 = gg

f. ƒ ⴰ g

g. g ⴰ ƒ

h. ƒ ⴰ ƒ

i. g ⴰ g

78. Can a function be both even and odd? Give reasons for your answer. T 79. (Continuation of Example 1.) Graph the functions ƒsxd = 2x and g sxd = 21 - x together with their (a) sum, (b) product, (c) two differences, (d) two quotients.

75. Graph the function y = ƒ x 2 - 1 ƒ . 76. Graph the function y = 2ƒ x ƒ .

2 T 80. Let ƒsxd = x - 7 and g sxd = x . Graph ƒ and g together with ƒ ⴰ g and g ⴰ ƒ .

Trigonometric Functions

1.3

This section reviews radian measure and the basic trigonometric functions.

Angles

B'

Angles are measured in degrees or radians. The number of radians in the central angle A¿CB¿ within a circle of radius r is defined as the number of “radius units” contained in the arc s subtended by that central angle. If we denote this central angle by u when measured in radians, this means that u = s>r (Figure 1.36), or

s B 1 C

C ir

A' A r

it cir cl

e

Un

θ

cle of ra diu

sr

s = ru

FIGURE 1.36 The radian measure of the central angle A¿CB¿ is the number u = s>r. For a unit circle of radius r = 1, u is the length of arc AB that central angle ACB cuts from the unit circle.

(u in radians).

(1)

If the circle is a unit circle having radius r = 1, then from Figure 1.36 and Equation (1), we see that the central angle u measured in radians is just the length of the arc that the angle cuts from the unit circle. Since one complete revolution of the unit circle is 360° or 2p radians, we have p radians = 180°

(2)

and 180 1 radian = p ( L 57.3) degrees

1 degree =

or

p ( L 0.017) radians. 180

Table 1.2 shows the equivalence between degree and radian measures for some basic angles.

TABLE 1.2

Angles measured in degrees and radians

Degrees

180

135

90

45

0

30

45

60

90

120

135

150

180

270

360

U (radians)

p

3p 4

p 2

p 4

0

p 6

p 4

p 3

p 2

2p 3

3p 4

5p 6

p

3p 2

2p

22

Chapter 1: Functions

An angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies along the positive x-axis (Figure 1.37). Angles measured counterclockwise from the positive x-axis are assigned positive measures; angles measured clockwise are assigned negative measures. y

y Terminal ray Initial ray x Positive measure

Initial ray

Negative measure

Terminal ray

x

FIGURE 1.37 Angles in standard position in the xy-plane.

Angles describing counterclockwise rotations can go arbitrarily far beyond 2p radians or 360°. Similarly, angles describing clockwise rotations can have negative measures of all sizes (Figure 1.38). y

y

y

y

3 x

– 5 2 x

9 4 hypotenuse

opposite

adjacent opp hyp adj cos hyp opp tan adj

hyp opp hyp sec adj adj cot opp csc

y

y r x

O

r

x

Nonzero radian measures can be positive or negative and can go beyond 2p.

The Six Basic Trigonometric Functions

FIGURE 1.39 Trigonometric ratios of an acute angle.

P(x, y)

x

Angle Convention: Use Radians From now on, in this book it is assumed that all angles are measured in radians unless degrees or some other unit is stated explicitly. When we talk about the angle p>3, we mean p>3 radians (which is 60°), not p>3 degrees. We use radians because it simplifies many of the operations in calculus, and some results we will obtain involving the trigonometric functions are not true when angles are measured in degrees.

sin

FIGURE 1.38

– 3 4

x

You are probably familiar with defining the trigonometric functions of an acute angle in terms of the sides of a right triangle (Figure 1.39). We extend this definition to obtuse and negative angles by first placing the angle in standard position in a circle of radius r. We then define the trigonometric functions in terms of the coordinates of the point P(x, y) where the angle’s terminal ray intersects the circle (Figure 1.40). y sine: sin u = r x cosine: cos u = r y tangent: tan u = x

r cosecant: csc u = y r secant: sec u = x x cotangent: cot u = y

These extended definitions agree with the right-triangle definitions when the angle is acute. Notice also that whenever the quotients are defined,

FIGURE 1.40 The trigonometric functions of a general angle u are defined in terms of x, y, and r.

sin u cos u 1 sec u = cos u tan u =

1 tan u 1 csc u = sin u cot u =

1.3

2

3

2 1

4

As you can see, tan u and sec u are not defined if x = cos u = 0. This means they are not defined if u is ;p>2, ;3p>2, Á . Similarly, cot u and csc u are not defined for values of u for which y = 0, namely u = 0, ;p, ;2p, Á . The exact values of these trigonometric ratios for some angles can be read from the triangles in Figure 1.41. For instance,

6

4

3

2

1

2

p 1 = 4 22 p 1 cos = 4 22 p tan = 1 4 sin

1

sin

23 2p = , 3 2

p 1 = 6 2

sin

23 p = 3 2

cos

2p 1 = - , 3 2

cos

tan

2p = - 23. 3

⎛cos 2 , sin 2 ⎛ ⎛– 1 , 3 ⎛ ⎝ 3 3⎝ ⎝ 2 2 ⎝ y

y

A all pos

P 1

3 2

x T tan pos

sin

23 p p 1 = cos = 6 2 3 2 p p 1 tan = tan = 23 6 3 23 The CAST rule (Figure 1.42) is useful for remembering when the basic trigonometric functions are positive or negative. For instance, from the triangle in Figure 1.43, we see that

FIGURE 1.41 Radian angles and side lengths of two common triangles.

S sin pos

23

Trigonometric Functions

C cos pos

2 3 x

1 2

FIGURE 1.42 The CAST rule, remembered by the statement “Calculus Activates Student Thinking,” tells which trigonometric functions are positive in each quadrant.

FIGURE 1.43 The triangle for calculating the sine and cosine of 2p>3 radians. The side lengths come from the geometry of right triangles.

Using a similar method we determined the values of sin u, cos u, and tan u shown in Table 1.3. TABLE 1.3

Values of sin u, cos u, and tan u for selected values of u

Degrees

180

u (radians)

p

135 3p 4

90 p 2

45 p 4

0 0

30 p 6

45 p 4

60 p 3

90 p 2

120 2p 3

135 3p 4

150 5p 6

22 2

1 2

- 22 2 -1

sin u

0

- 22 2

-1

- 22 2

0

1 2

22 2

23 2

1

23 2

cos u

-1

- 22 2

0

22 2

1

23 2

22 2

1 2

0

-

tan u

0

-1

0

23 3

1

23

1

1 2

- 23

180

270 3p 2

360

0

-1

0

- 23 2

-1

0

1

- 23 3

0

p

2p

0

24

Chapter 1: Functions

Periodicity and Graphs of the Trigonometric Functions

Periods of Trigonometric Functions Period P : tan sx + pd = tan x cot sx + pd = cot x Period 2P :

sin sx + 2pd = sin x cos sx + 2pd = cos x sec sx + 2pd = sec x csc sx + 2pd = csc x

When an angle of measure u and an angle of measure u + 2p are in standard position, their terminal rays coincide. The two angles therefore have the same trigonometric function values: sin(u + 2p) = sin u, tan(u + 2p) = tan u, and so on. Similarly, cos(u - 2p) = cos u, sin(u - 2p) = sin u, and so on. We describe this repeating behavior by saying that the six basic trigonometric functions are periodic.

DEFINITION A function ƒ(x) is periodic if there is a positive number p such that ƒ(x + p) = ƒ(x) for every value of x. The smallest such value of p is the period of ƒ. When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable by x instead of u. Figure 1.44 shows that the tangent and cotangent functions have period p = p, and the other four functions have period 2p. Also, the symmetries in these graphs reveal that the cosine and secant functions are even and the other four functions are odd (although this does not prove those results). y y

y y cos x

– – 2

2

0

y sin x

3 2 2

x

y

coss -xd = cos x secs -xd = sec x

3 2 2

2

0

y

y sec x

1 – 3 – – 0 2 2

– – 2

3 2 2

x

y sinx Domain: – x Range: –1 y 1 Period: 2 (b)

Domain: – x Range: –1 y 1 Period: 2 (a)

Even

y tan x

x

0 3 2 2

x

Domain: x , 3 , . . . 2 2 Range: – y Period: (c) y

y csc x

1 – – 0 2

– 3 – – 2 2

y cot x

1 2

3 2 2

x

– – 0 2

2

3 2 2

x

Odd

sin s -xd tan s -xd cscs -xd cots -xd

= = = =

Domain: x , 3 , . . . 2 2 Range: y –1 or y 1 Period: 2 (d)

-sin x -tan x -csc x -cot x

Domain: x 0, , 2, . . . Range: y –1 or y 1 Period: 2

Domain: x 0, , 2, . . . Range: – y Period: (f)

(e)

FIGURE 1.44 Graphs of the six basic trigonometric functions using radian measure. The shading for each trigonometric function indicates its periodicity.

y P(cos , sin )

Trigonometric Identities

x 2 y2 1

The coordinates of any point P(x, y) in the plane can be expressed in terms of the point’s distance r from the origin and the angle u that ray OP makes with the positive x-axis (Figure 1.40). Since x>r = cos u and y>r = sin u, we have

sin cos O

x 1

x = r cos u,

y = r sin u.

When r = 1 we can apply the Pythagorean theorem to the reference right triangle in Figure 1.45 and obtain the equation

FIGURE 1.45 The reference triangle for a general angle u .

cos2 u + sin2 u = 1.

(3)

1.3

25

Trigonometric Functions

This equation, true for all values of u, is the most frequently used identity in trigonometry. Dividing this identity in turn by cos2 u and sin2 u gives

1 + tan2 u = sec2 u 1 + cot2 u = csc2 u

The following formulas hold for all angles A and B (Exercise 58).

Addition Formulas cossA + Bd = cos A cos B - sin A sin B sinsA + Bd = sin A cos B + cos A sin B

(4)

There are similar formulas for cos sA - Bd and sin sA - Bd (Exercises 35 and 36). All the trigonometric identities needed in this book derive from Equations (3) and (4). For example, substituting u for both A and B in the addition formulas gives

Double-Angle Formulas cos 2u = cos2 u - sin2 u sin 2u = 2 sin u cos u

(5)

Additional formulas come from combining the equations cos2 u + sin2 u = 1,

cos2 u - sin2 u = cos 2u.

We add the two equations to get 2 cos2 u = 1 + cos 2u and subtract the second from the first to get 2 sin2 u = 1 - cos 2u. This results in the following identities, which are useful in integral calculus.

Half-Angle Formulas cos2 u =

1 + cos 2u 2

(6)

sin2 u =

1 - cos 2u 2

(7)

The Law of Cosines If a, b, and c are sides of a triangle ABC and if u is the angle opposite c, then

c 2 = a 2 + b 2 - 2ab cos u.

This equation is called the law of cosines.

(8)

26

Chapter 1: Functions

We can see why the law holds if we introduce coordinate axes with the origin at C and the positive x-axis along one side of the triangle, as in Figure 1.46. The coordinates of A are (b, 0); the coordinates of B are sa cos u, a sin ud. The square of the distance between A and B is therefore

y B(a cos , a sin )

c 2 = sa cos u - bd2 + sa sin ud2

c

a

= a 2scos2 u + sin2 ud + b 2 - 2ab cos u ('')''* 1

C

b

A(b, 0)

x

= a 2 + b 2 - 2ab cos u. FIGURE 1.46 The square of the distance between A and B gives the law of cosines.

The law of cosines generalizes the Pythagorean theorem. If u = p>2, then cos u = 0 and c 2 = a 2 + b 2.

Transformations of Trigonometric Graphs The rules for shifting, stretching, compressing, and reflecting the graph of a function summarized in the following diagram apply to the trigonometric functions we have discussed in this section. Vertical stretch or compression; reflection about y = d if negative

Vertical shift

y = aƒ(bsx + cdd + d Horizontal shift

Horizontal stretch or compression; reflection about x = -c if negative

The transformation rules applied to the sine function give the general sine function or sinusoid formula ƒ(x) = A sin a

2p (x - C)b + D, B

where ƒ A ƒ is the amplitude, ƒ B ƒ is the period, C is the horizontal shift, and D is the vertical shift. A graphical interpretation of the various terms is given below. y

(

Horizontal shift (C)

Amplitude (A)

This axis is the line y D.

D

DA

)

y A sin 2 (x C) D B

DA

Vertical shift (D) This distance is the period (B).

0

x

Two Special Inequalities For any angle u measured in radians, - ƒ u ƒ … sin u … ƒ u ƒ

and

- ƒ u ƒ … 1 - cos u … ƒ u ƒ .

1.3

To establish these inequalities, we picture u as a nonzero angle in standard position (Figure 1.47). The circle in the figure is a unit circle, so ƒ u ƒ equals the length of the circular arc AP. The length of line segment AP is therefore less than ƒ u ƒ . Triangle APQ is a right triangle with sides of length

y P

O cos

sin

1

Q

27

Trigonometric Functions

QP = ƒ sin u ƒ ,

AQ = 1 - cos u.

From the Pythagorean theorem and the fact that AP 6 ƒ u ƒ , we get

x A(1, 0)

sin2 u + (1 - cos u) 2 = (AP) 2 … u2.

1 – cos

(9)

The terms on the left-hand side of Equation (9) are both positive, so each is smaller than their sum and hence is less than or equal to u2: sin2 u … u2

FIGURE 1.47 From the geometry of this figure, drawn for u 7 0, we get the inequality sin2 u + (1 - cos u) 2 … u2.

(1 - cos u) 2 … u2.

and

By taking square roots, this is equivalent to saying that ƒ sin u ƒ … ƒ u ƒ

and

ƒ 1 - cos u ƒ … ƒ u ƒ ,

- ƒ u ƒ … sin u … ƒ u ƒ

and

- ƒ u ƒ … 1 - cos u … ƒ u ƒ .

so

These inequalities will be useful in the next chapter.

Exercises 1.3 Radians and Degrees 1. On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4p>5 radians? (b) 110°? 2. A central angle in a circle of radius 8 is subtended by an arc of length 10p . Find the angle’s radian and degree measures. 3. You want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of the arc to the disk’s center. To the nearest tenth of an inch, how long should the arc be? 4. If you roll a 1-m-diameter wheel forward 30 cm over level ground, through what angle will the wheel turn? Answer in radians (to the nearest tenth) and degrees (to the nearest degree). Evaluating Trigonometric Functions 5. Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables. U

P

2P>3

0

P>2

3P>2

U

6. Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables.

P>6

P>4

5P>6

sin u cos u tan u cot u sec u csc u In Exercises 7–12, one of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 7. sin x =

3 , 5

p x H c , pd 2

9. cos x =

1 , 3

x H c-

p , 0d 2

10. cos x = -

11. tan x =

1 , 2

x H cp,

3p d 2

1 12. sin x = - , 2

3P>4

sin u cos u tan u cot u sec u csc u

P>3

x H c0,

8. tan x = 2,

5 , 13

p d 2

p x H c , pd 2 x H cp,

3p d 2

Graphing Trigonometric Functions Graph the functions in Exercises 13–22. What is the period of each function? 13. sin 2x

14. sin ( x>2)

15. cos px

16. cos

17. -sin

px 3

19. cos ax -

px 2

18. -cos 2px p b 2

20. sin ax +

p b 6

28

Chapter 1: Functions

21. sin ax -

p b + 1 4

22. cos ax +

2p b - 2 3

Graph the functions in Exercises 23–26 in the ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have? 23. s = cot 2t

24. s = -tan pt

pt 25. s = sec a b 2

t 26. s = csc a b 2

T 27. a. Graph y = cos x and y = sec x together for -3p>2 … x … 3p>2 . Comment on the behavior of sec x in relation to the signs and values of cos x. b. Graph y = sin x and y = csc x together for -p … x … 2p . Comment on the behavior of csc x in relation to the signs and values of sin x. T 28. Graph y = tan x and y = cot x together for -7 … x … 7 . Comment on the behavior of cot x in relation to the signs and values of tan x.

Solving Trigonometric Equations For Exercises 51–54, solve for the angle u, where 0 … u … 2p. 3 51. sin2 u = 52. sin2 u = cos2 u 4 53. sin 2u - cos u = 0

54. cos 2u + cos u = 0

Theory and Examples 55. The tangent sum formula The standard formula for the tangent of the sum of two angles is

Derive the formula. 56. (Continuation of Exercise 55.) Derive a formula for tan sA - Bd . 57. Apply the law of cosines to the triangle in the accompanying figure to derive the formula for cos sA - Bd . y

29. Graph y = sin x and y = :sin x; together. What are the domain and range of :sin x; ?

1

30. Graph y = sin x and y = < sin x = together. What are the domain and range of < sin x = ?

A

Using the Addition Formulas Use the addition formulas to derive the identities in Exercises 31–36. 31. cos ax -

p b = sin x 2

32. cos ax +

p b = -sin x 2

33. sin ax +

p b = cos x 2

34. sin ax -

p b = -cos x 2

35. cos sA - Bd = cos A cos B + sin A sin B (Exercise 57 provides a different derivation.) 36. sin sA - Bd = sin A cos B - cos A sin B 37. What happens if you take B = A in the trigonometric identity cos sA - Bd = cos A cos B + sin A sin B ? Does the result agree with something you already know? 38. What happens if you take B = 2p in the addition formulas? Do the results agree with something you already know? In Exercises 39–42, express the given quantity in terms of sin x and cos x. 39. cos sp + xd

40. sin s2p - xd

41. sin a

42. cos a

3p - xb 2

43. Evaluate sin

7p p p as sin a + b . 12 4 3

44. Evaluate cos

11p p 2p b. as cos a + 12 4 3

p 45. Evaluate cos . 12

p 12

0

x

1

58. a. Apply the formula for cos sA - Bd to the identity sin u = cos a

p - u b to obtain the addition formula for sin sA + Bd . 2

b. Derive the formula for cos sA + Bd by substituting -B for B in the formula for cos sA - Bd from Exercise 35. 59. A triangle has sides a = 2 and b = 3 and angle C = 60° . Find the length of side c. 60. A triangle has sides a = 2 and b = 3 and angle C = 40° . Find the length of side c. 61. The law of sines The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then sin C sin A sin B a = b = c . Use the accompanying figures and the identity sin sp - ud = sin u , if required, to derive the law. A

5p . 46. Evaluate sin 12

50. sin2

B

1

3p + xb 2

Using the Double-Angle Formulas Find the function values in Exercises 47–50. 5p p 47. cos2 48. cos2 8 12 49. sin2

tan A + tan B . 1 - tan A tan B

tansA + Bd =

3p 8

c

B

h

a

A

c

b

b C

B

a

h

C

62. A triangle has sides a = 2 and b = 3 and angle C = 60° (as in Exercise 59). Find the sine of angle B using the law of sines.

1.4 63. A triangle has side c = 2 and angles A = p>4 and B = p>3 . Find the length a of the side opposite A. T 64. The approximation sin x L x It is often useful to know that, when x is measured in radians, sin x L x for numerically small values of x. In Section 3.11, we will see why the approximation holds. The approximation error is less than 1 in 5000 if ƒ x ƒ 6 0.1 . a. With your grapher in radian mode, graph y = sin x and y = x together in a viewing window about the origin. What do you see happening as x nears the origin? b. With your grapher in degree mode, graph y = sin x and y = x together about the origin again. How is the picture different from the one obtained with radian mode?

Graphing with Calculators and Computers

69. The period B

29

Set the constants A = 3, C = D = 0 .

a. Plot ƒ(x) for the values B = 1, 3, 2p, 5p over the interval -4p … x … 4p . Describe what happens to the graph of the general sine function as the period increases. b. What happens to the graph for negative values of B? Try it with B = -3 and B = -2p . 70. The horizontal shift C Set the constants A = 3, B = 6, D = 0 . a. Plot ƒ(x) for the values C = 0, 1 , and 2 over the interval -4p … x … 4p . Describe what happens to the graph of the general sine function as C increases through positive values. b. What happens to the graph for negative values of C ? c. What smallest positive value should be assigned to C so the graph exhibits no horizontal shift? Confirm your answer with a plot.

General Sine Curves For ƒsxd = A sin a

2p sx - Cdb + D, B

identify A, B, C, and D for the sine functions in Exercises 65–68 and sketch their graphs. 1 1 65. y = 2 sin sx + pd - 1 66. y = sin spx - pd + 2 2 2pt p 2 1 L 67. y = - p sin a tb + p 68. y = sin , L 7 0 L 2 2p COMPUTER EXPLORATIONS In Exercises 69–72, you will explore graphically the general sine function ƒsxd = A sin a

71. The vertical shift D Set the constants A = 3, B = 6, C = 0 . a. Plot ƒ(x) for the values D = 0, 1 , and 3 over the interval -4p … x … 4p . Describe what happens to the graph of the general sine function as D increases through positive values. b. What happens to the graph for negative values of D? 72. The amplitude A

Set the constants B = 6, C = D = 0 .

a. Describe what happens to the graph of the general sine function as A increases through positive values. Confirm your answer by plotting ƒ(x) for the values A = 1, 5 , and 9. b. What happens to the graph for negative values of A?

2p sx - Cdb + D B

as you change the values of the constants A, B, C, and D. Use a CAS or computer grapher to perform the steps in the exercises.

1.4

Graphing with Calculators and Computers A graphing calculator or a computer with graphing software enables us to graph very complicated functions with high precision. Many of these functions could not otherwise be easily graphed. However, care must be taken when using such devices for graphing purposes, and in this section we address some of the issues involved. In Chapter 4 we will see how calculus helps us determine that we are accurately viewing all the important features of a function’s graph.

Graphing Windows When using a graphing calculator or computer as a graphing tool, a portion of the graph is displayed in a rectangular display or viewing window. Often the default window gives an incomplete or misleading picture of the graph. We use the term square window when the units or scales on both axes are the same. This term does not mean that the display window itself is square (usually it is rectangular), but instead it means that the x-unit is the same as the y-unit. When a graph is displayed in the default window, the x-unit may differ from the y-unit of scaling in order to fit the graph in the window. The viewing window is set by specifying an interval [a, b] for the x-values and an interval [c, d] for the y-values. The machine selects equally spaced x-values in [a, b] and then plots the points (x, ƒ(x)). A point is plotted if and

30

Chapter 1: Functions

only if x lies in the domain of the function and ƒ(x) lies within the interval [c, d]. A short line segment is then drawn between each plotted point and its next neighboring point. We now give illustrative examples of some common problems that may occur with this procedure. Graph the function ƒsxd = x 3 - 7x 2 + 28 in each of the following display or viewing windows:

EXAMPLE 1

(a) [-10, 10] by [-10, 10]

(b) [-4, 4] by [-50, 10]

(c) [-4, 10] by [-60, 60]

Solution

(a) We select a = -10, b = 10, c = -10, and d = 10 to specify the interval of x-values and the range of y-values for the window. The resulting graph is shown in Figure 1.48a. It appears that the window is cutting off the bottom part of the graph and that the interval of x-values is too large. Let’s try the next window.

10

10 –4 –10

60 4 –4

10

–10

–50

(a)

(b)

10

–60 (c)

FIGURE 1.48 The graph of ƒsxd = x 3 - 7x 2 + 28 in different viewing windows. Selecting a window that gives a clear picture of a graph is often a trial-and-error process (Example 1).

(b) We see some new features of the graph (Figure 1.48b), but the top is missing and we need to view more to the right of x = 4 as well. The next window should help. (c) Figure 1.48c shows the graph in this new viewing window. Observe that we get a more complete picture of the graph in this window, and it is a reasonable graph of a third-degree polynomial.

EXAMPLE 2

When a graph is displayed, the x-unit may differ from the y-unit, as in the graphs shown in Figures 1.48b and 1.48c. The result is distortion in the picture, which may be misleading. The display window can be made square by compressing or stretching the units on one axis to match the scale on the other, giving the true graph. Many systems have built-in functions to make the window “square.” If yours does not, you will have to do some calculations and set the window size manually to get a square window, or bring to your viewing some foreknowledge of the true picture. Figure 1.49a shows the graphs of the perpendicular lines y = x and y = -x + 3 22, together with the semicircle y = 29 - x 2 , in a nonsquare [-4, 4] by [-6, 8] display window. Notice the distortion. The lines do not appear to be perpendicular, and the semicircle appears to be elliptical in shape. Figure 1.49b shows the graphs of the same functions in a square window in which the x-units are scaled to be the same as the y-units. Notice that the scaling on the x-axis for Figure 1.49a has been compressed in Figure 1.49b to make the window square. Figure 1.49c gives an enlarged view of Figure 1.49b with a square [-3, 3] by [0, 4] window.

If the denominator of a rational function is zero at some x-value within the viewing window, a calculator or graphing computer software may produce a steep near-vertical line segment from the top to the bottom of the window. Example 3 illustrates this situation.

1.4 8

Graphing with Calculators and Computers 4

4

–4

–6

4

31

6

–3 –6 (a)

–4 (b)

3 0 (c)

FIGURE 1.49 Graphs of the perpendicular lines y = x and y = -x + 322 , and the semicircle y = 29 - x 2 appear distorted (a) in a nonsquare window, but clear (b) and (c) in square windows (Example 2).

1 . 2 - x Solution Figure 1.50a shows the graph in the [-10, 10] by [-10, 10] default square window with our computer graphing software. Notice the near-vertical line segment at x = 2. It is not truly a part of the graph and x = 2 does not belong to the domain of the function. By trial and error we can eliminate the line by changing the viewing window to the smaller [-6, 6] by [-4, 4] view, revealing a better graph (Figure 1.50b).

EXAMPLE 3

Graph the function y =

10

4

–10

10

–6

6

–10 (a)

–4 (b)

1 . A vertical line may appear 2 - x without a careful choice of the viewing window (Example 3). FIGURE 1.50 Graphs of the function y =

Sometimes the graph of a trigonometric function oscillates very rapidly. When a calculator or computer software plots the points of the graph and connects them, many of the maximum and minimum points are actually missed. The resulting graph is then very misleading.

EXAMPLE 4

Graph the function ƒsxd = sin 100x.

Solution Figure 1.51a shows the graph of ƒ in the viewing window [-12, 12] by [-1, 1]. We see that the graph looks very strange because the sine curve should oscillate periodically between -1 and 1. This behavior is not exhibited in Figure 1.51a. We might 1

–12

1

12

–1 (a)

–6

1

6

–1 (b)

–0.1

0.1

–1 (c)

FIGURE 1.51 Graphs of the function y = sin 100x in three viewing windows. Because the period is 2p>100 L 0.063 , the smaller window in (c) best displays the true aspects of this rapidly oscillating function (Example 4).

32

Chapter 1: Functions

experiment with a smaller viewing window, say [-6, 6] by [-1, 1], but the graph is not better (Figure 1.51b). The difficulty is that the period of the trigonometric function y = sin 100x is very small s2p>100 L 0.063d. If we choose the much smaller viewing window [-0.1, 0.1] by [-1, 1] we get the graph shown in Figure 1.51c. This graph reveals the expected oscillations of a sine curve. 1 sin 50x. 50 Solution In the viewing window [-6, 6] by [-1, 1] the graph appears much like the cosine function with some small sharp wiggles on it (Figure 1.52a). We get a better look when we significantly reduce the window to [-0.6, 0.6] by [0.8, 1.02], obtaining the graph in Figure 1.52b. We now see the small but rapid oscillations of the second term, s1>50d sin 50x, added to the comparatively larger values of the cosine curve.

EXAMPLE 5

Graph the function y = cos x +

1.02

1

–6

6

–0.6

0.6 0.8 (b)

–1 (a)

FIGURE 1.52 In (b) we see a close-up view of the function 1 sin 50x graphed in (a). The term cos x clearly dominates the y = cos x + 50 1 second term, sin 50x , which produces the rapid oscillations along the 50 cosine curve. Both views are needed for a clear idea of the graph (Example 5).

Obtaining a Complete Graph Some graphing devices will not display the portion of a graph for ƒ(x) when x 6 0. Usually that happens because of the procedure the device is using to calculate the function values. Sometimes we can obtain the complete graph by defining the formula for the function in a different way.

EXAMPLE 6

Graph the function y = x 1>3 .

Some graphing devices display the graph shown in Figure 1.53a. When we 3 compare it with the graph of y = x 1>3 = 2 x in Figure 1.17, we see that the left branch for Solution

2

–3

2

3

–2 (a)

–3

3

–2 (b)

FIGURE 1.53 The graph of y = x 1>3 is missing the left branch in (a). In x # 1>3 (b) we graph the function ƒsxd = ƒ x ƒ , obtaining both branches. (See ƒxƒ Example 6.)

1.5

Exponential Functions

33

x 6 0 is missing. The reason the graphs differ is that many calculators and computer software programs calculate x 1>3 as e s1>3dln x . Since the logarithmic function is not defined for negative values of x, the computing device can produce only the right branch, where x 7 0. (Logarithmic and exponential functions are introduced in the next two sections.) To obtain the full picture showing both branches, we can graph the function ƒsxd =

x # 1>3 ƒxƒ . ƒxƒ

This function equals x 1>3 except at x = 0 (where ƒ is undefined, although 0 1>3 = 0). The graph of ƒ is shown in Figure 1.53b.

Exercises 1.4 Choosing a Viewing Window T In Exercises 1–4, use a graphing calculator or computer to determine which of the given viewing windows displays the most appropriate graph of the specified function. 1. ƒsxd = x 4 - 7x 2 + 6x a. [-1, 1] by [-1, 1]

b. [-2, 2] by [-5, 5]

c. [-10, 10] by [-10, 10] d. [-5, 5] by [-25, 15] 2. ƒsxd = x 3 - 4x 2 - 4x + 16 a. [-1, 1] by [-5, 5]

b. [-3, 3] by [-10, 10]

c. [-5, 5] by [-10, 20]

d. [-20, 20] by [-100, 100]

3. ƒsxd = 5 + 12x - x 3 a. [-1, 1] by [-1, 1]

b. [-5, 5] by [-10, 10]

c. [-4, 4] by [-20, 20]

d. [-4, 5] by [-15, 25]

19. ƒsxd =

x2 + 2 x2 + 1

20. ƒsxd =

x2 - 1 x2 + 1

21. ƒsxd =

x - 1 x2 - x - 6

22. ƒsxd =

8 x2 - 9

24. ƒsxd =

x2 - 3 x - 2

6x 2 - 15x + 6 4x 2 - 10x 25. y = sin 250x 23. ƒsxd =

27. y = cos a 29. y = x +

26. y = 3 cos 60x

x b 50

28. y =

x 1 sin a b 10 10

1 sin 30x 10

30. y = x 2 +

1 cos 100x 50

31. Graph the lower half of the circle defined by the equation x 2 + 2x = 4 + 4y - y 2 . 32. Graph the upper branch of the hyperbola y 2 - 16x 2 = 1 .

4. ƒsxd = 25 + 4x - x 2 a. [-2, 2] by [-2, 2]

b. [-2, 6] by [-1, 4]

33. Graph four periods of the function ƒsxd = - tan 2x .

c. [-3, 7] by [0, 10]

d. [-10, 10] by [-10, 10]

34. Graph two periods of the function ƒsxd = 3 cot

Finding a Viewing Window T In Exercises 5–30, find an appropriate viewing window for the given function and use it to display its graph. x3 x2 - 2x + 1 5. ƒsxd = x 4 - 4x 3 + 15 6. ƒsxd = 3 2 5 4 3 4 7. ƒsxd = x - 5x + 10 8. ƒsxd = 4x - x 9. ƒsxd = x29 - x 2

10. ƒsxd = x 2s6 - x 3 d

11. y = 2x - 3x 2>3

12. y = x 1>3sx 2 - 8d

13. y = 5x

14. y = x 2>3s5 - xd

2>5

- 2x

15. y = ƒ x 2 - 1 ƒ 17. y =

x + 3 x + 2

1.5

16. y = ƒ x 2 - x ƒ 18. y = 1 -

1 x + 3

x + 1. 2

35. Graph the function ƒsxd = sin 2x + cos 3x . 36. Graph the function ƒsxd = sin3 x . Graphing in Dot Mode T Another way to avoid incorrect connections when using a graphing device is through the use of a “dot mode,” which plots only the points. If your graphing utility allows that mode, use it to plot the functions in Exercises 37–40. 1 1 37. y = 38. y = sin x x - 3 x3 - 1 39. y = x :x; 40. y = 2 x - 1

Exponential Functions Exponential functions are among the most important in mathematics and occur in a wide variety of applications, including interest rates, radioactive decay, population growth, the spread of a disease, consumption of natural resources, the earth’s atmospheric pressure, temperature change of a heated object placed in a cooler environment, and the dating of

34

Chapter 1: Functions

fossils. In this section we introduce these functions informally, using an intuitive approach. We give a rigorous development of them in Chapter 7, based on important calculus ideas and results.

Exponential Behavior

Don’t confuse the exponential function 2x with the power function x 2. In the exponential function, the variable x is in the exponent, whereas the variable x is the base in the power function.

When a positive quantity P doubles, it increases by a factor of 2 and the quantity becomes 2P. If it doubles again, it becomes 2(2P) = 22P, and a third doubling gives 2(22P) = 23P. Continuing to double in this fashion leads us to the consideration of the function ƒ(x) = 2x. We call this an exponential function because the variable x appears in the exponent of 2x. Functions such as g(x) = 10 x and h(x) = (1>2) x are other examples of exponential functions. In general, if a Z 1 is a positive constant, the function ƒ(x) = a x,

a0

is the exponential function with base a.

EXAMPLE 1 In 2010, $100 is invested in a savings account, where it grows by accruing interest that is compounded annually (once a year) at an interest rate of 5.5%. Assuming no additional funds are deposited to the account and no money is withdrawn, give a formula for a function describing the amount A in the account after x years have elapsed. If P = 100, at the end of the first year the amount in the account is the original amount plus the interest accrued, or Solution

P + a

5.5 bP = (1 + 0.055)P = (1.055)P. 100

At the end of the second year the account earns interest again and grows to (1 + 0.055) # (1.055P) = (1.055) 2P = 100 # (1.055) 2.

P = 100

Continuing this process, after x years the value of the account is A = 100 # (1.055) x. This is a multiple of the exponential function with base 1.055. Table 1.4 shows the amounts accrued over the first four years. Notice that the amount in the account each year is always 1.055 times its value in the previous year.

TABLE 1.4

Year 2010 2011 2012 2013 2014

Savings account growth Amount (dollars) 100 100(1.055) 100(1.055) 2 100(1.055) 3 100(1.055) 4

= = = =

105.50 111.30 117.42 123.88

Increase (dollars)

5.50 5.80 6.12 6.46

In general, the amount after x years is given by P(1 + r) x, where r is the interest rate (expressed as a decimal).

1.5

y 10 x

a n = 1442443 a # a # Á # a.

12 10

n factors

If x = 0, then a 0 = 1, and if x = -n for some positive integer n, then

8 6 y 3x

4 2

a -n =

y 2x

–1 –0.5 0 0.5 1 (a) y 2 x, y 3 x, y 10 x

n

a 1>n = 2a, which is the positive number that when multiplied by itself n times gives a. If x = p>q is any rational number, then

y 10 –x

q

a p>q = 2a p =

12 10 8 6 4 2

y 2 –x

n

1 1 = aa b . an

If x = 1>n for some positive integer n, then

x

y

y

35

For integer and rational exponents, the value of an exponential function ƒ(x) = a x is obtained arithmetically as follows. If x = n is a positive integer, the number a n is given by multiplying a by itself n times:

y

3 –x

Exponential Functions

–1 –0.5 0 0.5 1 (b) y 2 –x, y 3 –x, y 10 –x

x

FIGURE 1.54 Graphs of exponential functions.

A 2a B . q

p

If x is irrational, the meaning of a x is not so clear, but its value can be defined by considering values for rational numbers that get closer and closer to x. This informal approach is based on the graph of the exponential function. In Chapter 7 we define the meaning in a rigorous way. We displayed the graphs of several exponential functions in Section 1.1, and show them again here in Figure 1.54. These graphs describe the values of the exponential functions for all real inputs x. The value at an irrational number x is chosen so that the graph of a x has no “holes” or “jumps.” Of course, these words are not mathematical terms, but they do convey the informal idea. We mean that the value of a x, when x is irrational, is chosen so that the function ƒ(x) = a x is continuous, a notion that will be carefully explored in the next chapter. This choice ensures the graph retains its increasing behavior when a 7 1, or decreasing behavior when 0 6 a 6 1 (see Figure 1.54). Arithmetically, the graphical idea can be described in the following way, using the exponential function ƒ(x) = 2x as an illustration. Any particular irrational number, say x = 23, has a decimal expansion 23 = 1.732050808 . . . . We then consider the list of numbers, given as follows in the order of taking more and more digits in the decimal expansion,

TABLE 1.5 Values of 213 for

rational r closer and closer to 23 r

2r

1.0 1.7 1.73 1.732 1.7320 1.73205 1.732050 1.7320508 1.73205080 1.732050808

2.000000000 3.249009585 3.317278183 3.321880096 3.321880096 3.321995226 3.321995226 3.321997068 3.321997068 3.321997086

21, 21.7, 21.73, 21.732, 21.7320, 21.73205, . . . .

(1)

We know the meaning of each number in list (1) because the successive decimal approximations to 23 given by 1, 1.7, 1.73, 1.732, and so on, are all rational numbers. As these decimal approximations get closer and closer to 23, it seems reasonable that the list of numbers in (1) gets closer and closer to some fixed number, which we specify to be 223. Table 1.5 illustrates how taking better approximations to 23 gives better approximations to the number 213 L 3.321997086. It is the completeness property of the real numbers (discussed briefly in Appendix 7) which guarantees that this procedure gives a single number we define to be 213 (although it is beyond the scope of this text to give a proof ). In a similar way, we can identify the number 2x (or a x, a 7 0) for any irrational x. By identifying the number a x for both rational and irrational x, we eliminate any “holes” or “gaps” in the graph of a x. In practice you can use a calculator to find the number a x for irrational x, taking successive decimal approximations to x and creating a table similar to Table 1.5. Exponential functions obey the familiar rules of exponents listed on the next page. It is easy to check these rules using algebra when the exponents are integers or rational numbers. We prove them for all real exponents in Chapters 4 and 7.

36

Chapter 1: Functions

Rules for Exponents If a 7 0 and b 7 0, the following rules hold true for all real numbers x and y. ax 1. a x # a y = a x + y 2. y = a x - y a 3. (a x ) y = (a y ) x = a xy 4. a x # b x = (ab) x 5.

a ax = a b b bx

EXAMPLE 2 1. 2. 3. 4. 5.

x

We illustrate using the rules for exponents to simplify numerical expressions.

31.1 # 30.7 = 31.1 + 0.7 = 31.8

A 210 B 210

3

=

A 210 B

A 522 B 22 = 522 p

7

#8

p

4 a b 9

= (56)

1>2

=

3-1

# 22

=

2 A 210 B = 10

= 52 = 25

p

41>2 2 = 1>2 3 9

The Natural Exponential Function e x The most important exponential function used for modeling natural, physical, and economic phenomena is the natural exponential function, whose base is the special number e. The number e is irrational, and its value is 2.718281828 to nine decimal places. (In Section 3.8 we will see a way to calculate the value of e.) It might seem strange that we would use this number for a base rather than a simple number like 2 or 10. The advantage in using e as a base is that it simplifies many of the calculations in calculus. If you look at Figure 1.54a you can see that the graphs of the exponential functions y = a x get steeper as the base a gets larger. This idea of steepness is conveyed by the slope of the tangent line to the graph at a point. Tangent lines to graphs of functions are defined precisely in the next chapter, but intuitively the tangent line to the graph at a point is a line that just touches the graph at the point, like a tangent to a circle. Figure 1.55 shows the slope of the graph of y = a x as it crosses the y-axis for several values of a. Notice that the slope is exactly equal to 1 when a equals the number e. The slope is smaller than 1 if a 6 e, and larger than 1 if a 7 e. This is the property that makes the number e so useful in calculus: The graph of y e x has slope 1 when it crosses the y-axis.

y

y

y 2x m 0.7

y

y ex

m 1.1

m1

1

1 0 (a)

x

y 3x

1 0

(b)

x

0

x

(c)

FIGURE 1.55 Among the exponential functions, the graph of y = e x has the property that the slope m of the tangent line to the graph is exactly 1 when it crosses the y-axis. The slope is smaller for a base less than e, such as 2x, and larger for a base greater than e, such as 3x.

1.5

37

Exponential Functions

Exponential Growth and Decay The exponential functions y = e kx, where k is a nonzero constant, are frequently used for modeling exponential growth or decay. The function y = y0 e kx is a model for exponential growth if k 7 0 and a model for exponential decay if k 6 0. Here y0 represents a constant. An example of exponential growth occurs when computing interest compounded continuously modeled by y = P # e rt, where P is the initial monetary investment, r is the interest rate as a decimal, and t is time in units consistent with r. An example of exponential -4 decay is the model y = A # e -1.2 * 10 t, which represents how the radioactive isotope carbon-14 decays over time. Here A is the original amount of carbon-14 and t is the time in years. Carbon-14 decay is used to date the remains of dead organisms such as shells, seeds, and wooden artifacts. Figure 1.56 shows graphs of exponential growth and exponential decay. y y 20 1.4

15

1

10

y e1.5x

y e –1.2 x

0.6

5 0.2 –1

– 0.5

0

0.5

1

1.5

2

x

– 0.5

0

0.5

1

1.5

2

2.5

3

x

(b)

(a)

FIGURE 1.56 Graphs of (a) exponential growth, k = 1.5 7 0, and (b) exponential decay, k = - 1.2 6 0.

Investment companies often use the model y = Pe rt in calculating the growth of an investment. Use this model to track the growth of $100 invested in 2000 at an annual interest rate of 5.5%.

EXAMPLE 3

Let t = 0 represent 2010, t = 1 represent 2011, and so on. Then the exponential growth model is y(t) = Pe rt, where P = 100 (the initial investment), r = 0.055 (the annual interest rate expressed as a decimal), and t is time in years. To predict the amount in the account in 2014, after four years have elapsed, we take t = 4 and calculate

Solution

y(4) = 100e 0.055(4) = 100e 0.22 = 124.61.

Nearest cent using calculator

This compares with $123.88 in the account when the interest is compounded annually from Example 1.

EXAMPLE 4

Laboratory experiments indicate that some atoms emit a part of their mass as radiation, with the remainder of the atom re-forming to make an atom of some new element. For example, radioactive carbon-14 decays into nitrogen; radium eventually decays into lead. If y0 is the number of radioactive nuclei present at time zero, the number still present at any later time t will be y = y0 e -rt,

r 7 0.

38

Chapter 1: Functions

The number r is called the decay rate of the radioactive substance. (We will see how this formula is obtained in Section 7.2.) For carbon-14, the decay rate has been determined experimentally to be about r = 1.2 * 10 -4 when t is measured in years. Predict the percent of carbon-14 present after 866 years have elapsed. If we start with an amount y0 of carbon-14 nuclei, after 866 years we are left with the amount Solution

y(866) = y0 e (-1.2 * 10 L (0.901)y0.

-4

)(866)

Calculator evaluation

That is, after 866 years, we are left with about 90% of the original amount of carbon-14, so about 10% of the original nuclei have decayed. In Example 7 in the next section, you will see how to find the number of years required for half of the radioactive nuclei present in a sample to decay (called the half-life of the substance). You may wonder why we use the family of functions y = e kx for different values of the constant k instead of the general exponential functions y = a x. In the next section, we show that the exponential function a x is equal to e kx for an appropriate value of k. So the formula y = e kx covers the entire range of possibilities, and we will see that it is easier to use.

Exercises 1.5 Sketching Exponential Curves In Exercises 1–6, sketch the given curves together in the appropriate coordinate plane and label each curve with its equation. 1. y = 2x, y = 4x, y = 3-x, y = (1>5) x 2. y = 3x, y = 8x, y = 2-x, y = (1>4) x 3. y = 2-t and y = -2t

4. y = 3-t and y = -3t

5. y = e x and y = 1>e x

6. y = -e x and y = -e -x

In each of Exercises 7–10, sketch the shifted exponential curves. 7. y = 2x - 1 and y = 2-x - 1 8. y = 3x + 2 and y = 3-x + 2 9. y = 1 - e x and y = 1 - e -x 10. y = -1 - e x and y = -1 - e -x Applying the Laws of Exponents Use the laws of exponents to simplify the expressions in Exercises 11–20. 11. 162 # 16-1.75 44.2 43.7 4 15. A 251>8 B 13.

17. 2

23

19. a

# 723

2 22

b

4

12. 91>3 # 91>6

35>3 32>3 16. A 1322 B 22>2 14.

18.

1>2 A 23 B # A 212 B

20. a

1>2

26 2 b 3

Composites Involving Exponential Functions Find the domain and range for each of the functions in Exercises 21–24. 1 21. ƒ(x) = 22. g(t) = cos (e -t ) 2 + ex 3 23. g(t) = 21 + 3-t 24. ƒ(x) = 1 - e 2x Applications T In Exercises 25–28, use graphs to find approximate solutions. 25. 2x = 5

26. e x = 4

27. 3x - 0.5 = 0

28. 3 - 2-x = 0

T In Exercises 29–36, use an exponential model and a graphing calculator to estimate the answer in each problem. 29. Population growth The population of Knoxville is 500,000 and is increasing at the rate of 3.75% each year. Approximately when will the population reach 1 million? 30. Population growth The population of Silver Run in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year. a. Estimate the population in 1915 and 1940. b. Approximately when did the population reach 50,000? 31. Radioactive decay The half-life of phosphorus-32 is about 14 days. There are 6.6 grams present initially. a. Express the amount of phosphorus-32 remaining as a function of time t. b. When will there be 1 gram remaining?

1.6

Inverse Functions and Logarithms

39

32. If John invests $2300 in a savings account with a 6% interest rate compounded annually, how long will it take until John’s account has a balance of $4150?

35. Cholera bacteria Suppose that a colony of bacteria starts with 1 bacterium and doubles in number every half hour. How many bacteria will the colony contain at the end of 24 hr?

33. Doubling your money Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.25% compounded annually.

36. Eliminating a disease Suppose that in any given year the number of cases of a disease is reduced by 20%. If there are 10,000 cases today, how many years will it take

34. Tripling your money Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded continuously.

b. to eliminate the disease; that is, to reduce the number of cases to less than 1?

a. to reduce the number of cases to 1000?

Inverse Functions and Logarithms

1.6

A function that undoes, or inverts, the effect of a function ƒ is called the inverse of ƒ. Many common functions, though not all, are paired with an inverse. In this section we present the natural logarithmic function y = ln x as the inverse of the exponential function y = e x, and we also give examples of several inverse trigonometric functions.

One-to-One Functions A function is a rule that assigns a value from its range to each element in its domain. Some functions assign the same range value to more than one element in the domain. The function ƒsxd = x 2 assigns the same value, 1, to both of the numbers -1 and +1; the sines of p>3 and 2p>3 are both 13>2. Other functions assume each value in their range no more than once. The square roots and cubes of different numbers are always different. A function that has distinct values at distinct elements in its domain is called one-to-one. These functions take on any one value in their range exactly once. y

y y x

y x3 x

0

x

0

DEFINITION A function ƒ(x) is one-to-one on a domain D if ƒsx1 d Z ƒsx2 d whenever x1 Z x2 in D.

EXAMPLE 1

Some functions are one-to-one on their entire natural domain. Other functions are not one-to-one on their entire domain, but by restricting the function to a smaller domain we can create a function that is one-to-one. The original and restricted functions are not the same functions, because they have different domains. However, the two functions have the same values on the smaller domain, so the original function is an extension of the restricted function from its smaller domain to the larger domain.

(a) One-to-one: Graph meets each horizontal line at most once. y

y x2

Same y-value y Same y-value

1 –1 0

0.5 1

x

6

5 6

x

y sin x (b) Not one-to-one: Graph meets one or more horizontal lines more than once.

FIGURE 1.57 (a) y = x 3 and y = 1x are one-to-one on their domains s - q , q d and [0, q d. (b) y = x 2 and y = sin x are not one-to-one on their domains s - q , q d .

(a) ƒsxd = 1x is one-to-one on any domain of nonnegative numbers because 1x1 Z 1x2 whenever x1 Z x2 . (b) gsxd = sin x is not one-to-one on the interval [0, p] because sin sp>6d = sin s5p>6d. In fact, for each element x1 in the subinterval [0, p>2d there is a corresponding element x2 in the subinterval sp>2, p] satisfying sin x1 = sin x2, so distinct elements in the domain are assigned to the same value in the range. The sine function is one-toone on [0, p>2], however, because it is an increasing function on [0, p>2] giving distinct outputs for distinct inputs. The graph of a one-to-one function y = ƒsxd can intersect a given horizontal line at most once. If the function intersects the line more than once, it assumes the same y-value for at least two different x-values and is therefore not one-to-one (Figure 1.57).

40

Chapter 1: Functions

The Horizontal Line Test for One-to-One Functions A function y = ƒsxd is one-to-one if and only if its graph intersects each horizontal line at most once.

Inverse Functions Since each output of a one-to-one function comes from just one input, the effect of the function can be inverted to send an output back to the input from which it came.

DEFINITION Suppose that ƒ is a one-to-one function on a domain D with range R. The inverse function ƒ -1 is defined by ƒ -1sbd = a if ƒsad = b. The domain of ƒ -1 is R and the range of ƒ -1 is D.

The symbol ƒ -1 for the inverse of ƒ is read “ƒ inverse.” The “ -1” in ƒ -1 is not an exponent; ƒ -1sxd does not mean 1> ƒ(x). Notice that the domains and ranges of ƒ and ƒ -1 are interchanged. Suppose a one-to-one function y = ƒsxd is given by a table of values

EXAMPLE 2 x

1

2

3

4

5

6

7

8

ƒ(x)

3

4.5

7

10.5

15

20.5

27

34.5

A table for the values of x = ƒ -1s yd can then be obtained by simply interchanging the values in the columns (or rows) of the table for ƒ: y -1

ƒ s yd

3

4.5

7

10.5

15

20.5

27

34.5

1

2

3

4

5

6

7

8

If we apply ƒ to send an input x to the output ƒ(x) and follow by applying ƒ -1 to ƒ(x), we get right back to x, just where we started. Similarly, if we take some number y in the range of ƒ, apply ƒ -1 to it, and then apply ƒ to the resulting value ƒ -1syd, we get back the value y with which we began. Composing a function and its inverse has the same effect as doing nothing. sƒ -1 ⴰ ƒdsxd = x,

for all x in the domain of ƒ

sƒ ⴰ ƒ -1 dsyd = y,

for all y in the domain of ƒ -1 sor range of ƒd

Only a one-to-one function can have an inverse. The reason is that if ƒsx1 d = y and ƒsx2 d = y for two distinct inputs x1 and x2 , then there is no way to assign a value to ƒ -1syd that satisfies both ƒ -1sƒsx1 dd = x1 and ƒ -1sƒsx2 dd = x2 . A function that is increasing on an interval so it satisfies the inequality ƒsx2 d 7 ƒsx1 d when x2 7 x1 is one-to-one and has an inverse. Decreasing functions also have an inverse. Functions that are neither increasing nor decreasing may still be one-to-one and have an

1.6

41

Inverse Functions and Logarithms

inverse, as with the function ƒ(x) = 1>x for x Z 0 and ƒ(0) = 0, defined on (- q , q ) and passing the horizontal line test.

Finding Inverses The graphs of a function and its inverse are closely related. To read the value of a function from its graph, we start at a point x on the x-axis, go vertically to the graph, and then move horizontally to the y-axis to read the value of y. The inverse function can be read from the graph by reversing this process. Start with a point y on the y-axis, go horizontally to the graph of y = ƒsxd, and then move vertically to the x-axis to read the value of x = ƒ -1syd (Figure 1.58).

y

DOMAIN OF

y 5 f (x)

f RANGE OF

f –1

y

y

0

x

x DOMAIN OF

x 5 f –1(y) y

0

f

RANGE OF

f

–1

–1 is

(a) To find the value of f at x, we start at x, go up to the curve, and then over to the y-axis.

(b) The graph of f the graph of f, but with x and y interchanged. To find the x that gave y, we start at y and go over to the curve and down to the x-axis. The domain of f –1 is the range of f. The range of f –1 is the domain of f.

y

f –1

x y5x x5f

–1(y)

(b, a) 0

y DOMAIN OF

(c) To draw the graph of f –1 in the more usual way, we reflect the system across the line y 5 x.

f –1

y 5 f –1(x)

f –1

(a, b)

RANGE OF

RANGE OF

x

x

x

0 DOMAIN OF

f –1

(d) Then we interchange the letters x and y. We now have a normal-looking graph of f –1 as a function of x.

FIGURE 1.58 The graph of y = ƒ -1(x) is obtained by reflecting the graph of y = ƒ(x) about the line y = x.

We want to set up the graph of ƒ -1 so that its input values lie along the x-axis, as is usually done for functions, rather than on the y-axis. To achieve this we interchange the x- and y-axes by reflecting across the 45° line y = x. After this reflection we have a new graph that represents ƒ -1 . The value of ƒ -1sxd can now be read from the graph in the usual way, by starting with a point x on the x-axis, going vertically to the graph, and then horizontally

42

Chapter 1: Functions

to the y-axis to get the value of ƒ -1sxd. Figure 1.58 indicates the relationship between the graphs of ƒ and ƒ -1 . The graphs are interchanged by reflection through the line y = x. The process of passing from ƒ to ƒ -1 can be summarized as a two-step procedure. 1. 2.

y

y 2x 2

yx

Solve the equation y = ƒsxd for x. This gives a formula x = ƒ -1s yd where x is expressed as a function of y. Interchange x and y, obtaining a formula y = ƒ -1sxd where ƒ -1 is expressed in the conventional format with x as the independent variable and y as the dependent variable.

EXAMPLE 3

1 x + 1, expressed as a function of x. 2

Find the inverse of y =

Solution y 1x1 2

1.

Solve for x in terms of y:

1 –2

1 x + 1 2 2y = x + 2

x

1

FIGURE 1.59 Graphing ƒsxd = s1>2dx + 1 and ƒ -1sxd = 2x - 2 together shows the graphs’ symmetry with respect to the line y = x (Example 3).

The graph is a straight line satisfying the horizontal line test (Fig. 1.59).

x = 2y - 2. 2.

–2

y =

Interchange x and y:

y = 2x - 2.

The inverse of the function ƒsxd = s1>2dx + 1 is the function ƒ -1sxd = 2x - 2. (See Figure 1.59.) To check, we verify that both composites give the identity function: 1 ƒ -1sƒsxdd = 2 a x + 1b - 2 = x + 2 - 2 = x 2 ƒsƒ -1sxdd =

1 s2x - 2d + 1 = x - 1 + 1 = x. 2

y

EXAMPLE 4

y x 2, x 0

Find the inverse of the function y = x 2, x Ú 0, expressed as a function

of x. yx

For x Ú 0, the graph satisfies the horizontal line test, so the function is one-toone and has an inverse. To find the inverse, we first solve for x in terms of y: Solution

y x

y = x2 2y = 2x 2 = ƒ x ƒ = x 0

x

FIGURE 1.60 The functions y = 1x and y = x 2, x Ú 0 , are inverses of one another (Example 4).

ƒ x ƒ = x because x Ú 0

We then interchange x and y, obtaining y = 1x. The inverse of the function y = x , x Ú 0, is the function y = 1x (Figure 1.60). Notice that the function y = x 2, x Ú 0, with domain restricted to the nonnegative real numbers, is one-to-one (Figure 1.60) and has an inverse. On the other hand, the function y = x 2, with no domain restrictions, is not one-to-one (Figure 1.57b) and therefore has no inverse. 2

Logarithmic Functions If a is any positive real number other than 1, the base a exponential function ƒ(x) = a x is one-to-one. It therefore has an inverse. Its inverse is called the logarithm function with base a.

DEFINITION The logarithm function with base a, y = loga x, is the inverse of the base a exponential function y = a x (a 7 0, a Z 1).

1.6 y

y 2x yx

y log 2 x

2 1 0

x

1 2

Inverse Functions and Logarithms

43

The domain of loga x is (0, q ), the range of a x. The range of loga x is (- q , q ), the domain of a x. Figure 1.23 in Section 1.1 shows the graphs of four logarithmic functions with a 7 1. Figure 1.61a shows the graph of y = log2 x. The graph of y = a x, a 7 1, increases rapidly for x 7 0, so its inverse, y = loga x, increases slowly for x 7 1. Because we have no technique yet for solving the equation y = a x for x in terms of y, we do not have an explicit formula for computing the logarithm at a given value of x. Nevertheless, we can obtain the graph of y = loga x by reflecting the graph of the exponential y = a x across the line y = x. Figure 1.61 shows the graphs for a = 2 and a = e. Logarithms with base 2 are commonly used in computer science. Logarithms with base e and base 10 are so important in applications that calculators have special keys for them. They also have their own special notation and names:

(a)

loge x is written as ln x. log10 x is written as log x.

y 8

y ex

The function y = ln x is called the natural logarithm function, and y = log x is often called the common logarithm function. For the natural logarithm,

7 6

ln x = y 3 e y = x.

5 4

In particular, if we set x = e, we obtain e

(1, e)

2

y ln x

ln e = 1

1 –2

–1

0

1

2

e

4

x

because e 1 = e.

Properties of Logarithms (b) x

FIGURE 1.61 (a) The graph of 2 and its inverse, log2 x. (b) The graph of e x and its inverse, ln x.

HISTORICAL BIOGRAPHY* John Napier (1550–1617)

Logarithms, invented by John Napier, were the single most important improvement in arithmetic calculation before the modern electronic computer. What made them so useful is that the properties of logarithms reduce multiplication of positive numbers to addition of their logarithms, division of positive numbers to subtraction of their logarithms, and exponentiation of a number to multiplying its logarithm by the exponent. We summarize these properties for the natural logarithm as a series of rules that we prove in Chapter 3. Although here we state the Power Rule for all real powers r, the case when r is an irrational number cannot be dealt with properly until Chapter 4. We also establish the validity of the rules for logarithmic functions with any base a in Chapter 7.

THEOREM 1—Algebraic Properties of the Natural Logarithm For any numbers b 7 0 and x 7 0, the natural logarithm satisfies the following rules: 1. Product Rule:

ln bx = ln b + ln x

2. Quotient Rule:

b ln x = ln b - ln x

3. Reciprocal Rule:

1 ln x = -ln x

4. Power Rule:

ln x r = r ln x

Rule 2 with b = 1

*To learn more about the historical figures mentioned in the text and the development of many major elements and topics of calculus, visit www.aw.com/thomas.

44

Chapter 1: Functions

EXAMPLE 5

Here we use the properties in Theorem 1 to simplify three expressions.

(a) ln 4 + ln sin x = ln (4 sin x)

Product Rule

x + 1 = ln (x + 1) - ln (2x - 3) (b) ln 2x - 3 1 (c) ln = -ln 8 8

Quotient Rule Reciprocal Rule

= -ln 23 = -3 ln 2

Power Rule

Because a x and loga x are inverses, composing them in either order gives the identity function. Inverse Properties for ax and loga x 1. Base a: a loga x = x, 2. Base e: e

ln x

= x,

loga a x = x,

a 7 0, a Z 1, x 7 0

ln e = x,

x 7 0

x

Substituting a x for x in the equation x = e ln x enables us to rewrite a x as a power of e: x

a x = e ln (a )

Substitute a x for x in x = e ln x.

= e = e (ln a) x. x ln a

Power Rule for logs Exponent rearranged

Thus, the exponential function a is the same as e kx for k = ln a. x

Every exponential function is a power of the natural exponential function. a x = e x ln a That is, a x is the same as e x raised to the power ln a: a x = e kx for k = ln a. For example, 2x = e (ln 2) x = e x ln 2,

and

5 - 3x = e (ln 5) ( - 3x) = e - 3x ln 5.

Returning once more to the properties of a x and loga x, we have ln x = ln (a loga x) = (loga x)(ln a).

Inverse Property for a x and loga x Power Rule for logarithms, with r = loga x

Rewriting this equation as loga x = (ln x)>(ln a) shows that every logarithmic function is a constant multiple of the natural logarithm ln x. This allows us to extend the algebraic properties for ln x to loga x. For instance, loga bx = loga b + loga x. Change of Base Formula Every logarithmic function is a constant multiple of the natural logarithm. ln x loga x = (a 7 0, a Z 1) ln a

Applications In Section 1.5 we looked at examples of exponential growth and decay problems. Here we use properties of logarithms to answer more questions concerning such problems.

EXAMPLE 6 If $1000 is invested in an account that earns 5.25% interest compounded annually, how long will it take the account to reach $2500?

1.6

Inverse Functions and Logarithms

45

From Example 1, Section 1.5 with P = 1000 and r = 0.0525, the amount in the account at any time t in years is 1000(1.0525) t, so to find the time t when the account reaches $2500 we need to solve the equation

Solution

1000(1.0525) t = 2500. Thus we have (1.0525) t = 2.5

Divide by 1000.

ln (1.0525) = ln 2.5 t

Take logarithms of both sides.

t ln 1.0525 = ln 2.5 ln 2.5 L 17.9 t = ln 1.0525

Power Rule Values obtained by calculator

The amount in the account will reach $2500 in 18 years, when the annual interest payment is deposited for that year.

EXAMPLE 7

The half-life of a radioactive element is the time required for half of the radioactive nuclei present in a sample to decay. It is a remarkable fact that the half-life is a constant that does not depend on the number of radioactive nuclei initially present in the sample, but only on the radioactive substance. To see why, let y0 be the number of radioactive nuclei initially present in the sample. Then the number y present at any later time t will be y = y0 e -kt . We seek the value of t at which the number of radioactive nuclei present equals half the original number: 1 y 2 0 1 e -kt = 2 1 -kt = ln = -ln 2 2 ln 2 t = . k

y0 e -kt =

Reciprocal Rule for logarithms

(1)

This value of t is the half-life of the element. It depends only on the value of k; the number y0 does not have any effect. The effective radioactive lifetime of polonium-210 is so short that we measure it in days rather than years. The number of radioactive atoms remaining after t days in a sample that starts with y0 radioactive atoms is

Amount present y0 y y0 e–5 10

-3

–3t

y = y0 e -5 * 10 t.

1y 2 0 1y 4 0

The element’s half-life is ln 2 k ln 2 = 5 * 10 -3 L 139 days.

Half-life =

0

139

278

t (days)

Half-life

FIGURE 1.62 Amount of polonium-210 present at time t, where y0 represents the number of radioactive atoms initially present (Example 7).

Eq. (1) The k from polonium’s decay equation

This means that after 139 days, 1/2 of y0 radioactive atoms remain; after another 139 days (or 278 days altogether) half of those remain, or 1/4 of y0 radioactive atoms remain, and so on (see Figure 1.62).

Inverse Trigonometric Functions The six basic trigonometric functions of a general radian angle x were reviewed in Section 1.3. These functions are not one-to-one (their values repeat periodically). However, we can restrict their domains to intervals on which they are one-to-one. The sine function

46

Chapter 1: Functions

increases from -1 at x = -p>2 to +1 at x = p>2. By restricting its domain to the interval [-p>2, p>2] we make it one-to-one, so that it has an inverse sin-1 x (Figure 1.63). Similar domain restrictions can be applied to all six trigonometric functions.

y x sin y y sin –1x Domain: –1 x 1 Range: –/ 2 y / 2

2

–1

Domain restrictions that make the trigonometric functions one-to-one

x

1

– 2

– 2

sin x 1

1 0

–1

FIGURE 1.63 The graph of y = sin-1 x.

2

x

tan x

cos x 2

0 –1

y = sin x Domain: [-p>2, p>2] Range: [- 1, 1] y

y

y

y

x

– 2

2

x

y = tan x Domain: s -p>2, p>2d Range: s - q , q d

y = cos x Domain: [0, p] Range: [-1, 1] y

sec x

0

y

csc x

cot x 1 0

y y sin x, – 2 x 2 Domain: [–/2, /2] Range: [–1, 1] 1 – 2

x

2

0 –1 (a) y

x sin y y sin –1x Domain: [–1, 1] Range: [–/2, /2]

2

1 2

x

– 2

2

0

x 0

y = csc x Domain: [-p>2, 0d ´ s0, p>2] Range: s - q , -1] ´ [1, q d

y = sec x Domain: [0, p>2d ´ sp>2, p] Range: s - q , -1] ´ [1, q d

2

x

y = cot x Domain: s0, pd Range: s - q , q d

Since these restricted functions are now one-to-one, they have inverses, which we denote by y y y y y y

= = = = = =

sin-1 x cos-1 x tan-1 x sec-1 x csc-1 x cot-1 x

or or or or or or

y y y y y y

= = = = = =

arcsin x arccos x arctan x arcsec x arccsc x arccot x

These equations are read “y equals the arcsine of x” or “y equals arcsin x” and so on. –1

0

1

x

– 2

(b)

FIGURE 1.64 The graphs of (a) y = sin x, -p>2 … x … p>2 , and (b) its inverse, y = sin-1 x . The graph of sin-1 x , obtained by reflection across the line y = x , is a portion of the curve x = sin y .

Caution The -1 in the expressions for the inverse means “inverse.” It does not mean reciprocal. For example, the reciprocal of sin x is ssin xd-1 = 1>sin x = csc x. The graphs of the six inverse trigonometric functions are obtained by reflecting the graphs of the restricted trigonometric functions through the line y = x. Figure 1.64b shows the graph of y = sin-1 x and Figure 1.65 shows the graphs of all six functions. We now take a closer look at two of these functions.

The Arcsine and Arccosine Functions We define the arcsine and arccosine as functions whose values are angles (measured in radians) that belong to restricted domains of the sine and cosine functions.

1.6 Domain: –1 x 1 Range: – y 2 2 y 2

y

–1

x

1

2

y cos –1x 2

– 2

–1

–2

–1

x

1

2

y sec –1x

2

–2 –1

1

2

–1

Domain: – ∞ x ∞ Range: 0 y y

y csc–1x 1

2

2

x

– 2

x

(d)

FIGURE 1.65

(c)

Domain: x –1 or x 1 Range: – y , y 0 2 2 y

Domain: x –1 or x 1 Range: 0 y , y 2 y

y tan –1x x 1 2

– 2

(b)

(a)

–2

Domain: – ∞ x ∞ Range: – y 2 2 y

Domain: –1 x 1 0 y Range:

y sin –1x

47

Inverse Functions and Logarithms

–2

–1

(e)

y cot –1x

1

2

x

(f )

Graphs of the six basic inverse trigonometric functions.

DEFINITION y sin1 x is the number in [-p>2, p>2] for which sin y = x. y cos1 x is the number in [0, p] for which cos y = x. The “Arc” in Arcsine and Arccosine For a unit circle and radian angles, the arc length equation s = r u becomes s = u , so central angles and the arcs they subtend have the same measure. If x = sin y , then, in addition to being the angle whose sine is x, y is also the length of arc on the unit circle that subtends an angle whose sine is x. So we call y “the arc whose sine is x.”

The graph of y = sin-1 x (Figure 1.64b) is symmetric about the origin (it lies along the graph of x = sin y). The arcsine is therefore an odd function: sin-1s -xd = -sin-1 x.

(2)

The graph of y = cos-1 x (Figure 1.66b) has no such symmetry.

EXAMPLE 8

Evaluate (a) sin-1 a

23 1 b and (b) cos-1 a- b. 2 2

Solution

y

(a) We see that x2 1 y2 5 1

sin-1 a

Arc whose sine is x

Arc whose cosine is x

Angle whose sine is x 0

x Angle whose cosine is x

1

x

23 p b = 2 3

because sin (p>3) = 23>2 and p>3 belongs to the range [-p>2, p>2] of the arcsine function. See Figure 1.67a. (b) We have 2p 1 cos-1 a- b = 2 3 because cos (2p>3) = -1>2 and 2p>3 belongs to the range [0, p] of the arccosine function. See Figure 1.67b.

48

Chapter 1: Functions

y

Using the same procedure illustrated in Example 8, we can create the following table of common values for the arcsine and arccosine functions.

y cos x, 0 x Domain: [0, ] Range: [–1, 1]

1

2

0 –1

x

23>2 22>2 1> 2 -1>2 - 22>2 - 23>2

(a) y x cos y

y cos –1 x Domain: [–1, 1] Range: [0, ]

2 –1 0

sin x -1 x

x

cos-1 x

p>3 23>2 p>4 22>2 p>6 1> 2 -p>6 -1>2 -p>4 - 22>2 -p>3 - 23>2

p>6 p>4 p>3 2p>3 3p>4 5p>6

x

1

y

(b)

2 3 0 1

FIGURE 1.66 The graphs of (a) y = cos x, 0 … x … p , and (b) its inverse, y = cos-1 x . The graph of cos-1 x , obtained by reflection across the line y = x , is a portion of the curve x = cos y .

y

sin –1 3 2 3

cos–1 ⎛– 1⎛ 2 p 3 ⎝ 2⎝ 2

3

3 x

sin 3 2 3 (a)

FIGURE 1.67 (Example 8).

Chicago

–1 0

2p 3 x

cos ⎛ 2 p⎛ – 1 ⎝3 ⎝ 2 (b)

Values of the arcsine and arccosine functions

179 Springfield

180

61

12 62 b

St. Louis

Plane

EXAMPLE 9

During an airplane flight from Chicago to St. Louis, the navigator determines that the plane is 12 mi off course, as shown in Figure 1.68. Find the angle a for a course parallel to the original correct course, the angle b, and the drift correction angle c = a + b.

a

c

From Figure 1.68 and elementary geometry, we see that 180 sin a = 12 and 62 sin b = 12, so 12 a = sin-1 L 0.067 radian L 3.8° 180 12 b = sin-1 L 0.195 radian L 11.2° 62

Solution FIGURE 1.68 Diagram for drift correction (Example 9), with distances rounded to the nearest mile (drawing not to scale).

c = a + b L 15°.

y –1(–x)

cos

cos–1x –1 –x

0

x

1

x

FIGURE 1.69 cos-1 x and cos-1s -xd are supplementary angles (so their sum is p).

Identities Involving Arcsine and Arccosine As we can see from Figure 1.69, the arccosine of x satisfies the identity cos-1 x + cos-1s -xd = p, or cos-1 s -xd = p - cos-1 x. Also, we can see from the triangle in Figure 1.70 that for x 7 0, sin-1 x + cos-1 x = p>2.

(3) (4)

(5)

1.6

cos–1x

1

x

sin–1x

FIGURE 1.70 sin-1 x and cos-1 x are complementary angles (so their sum is p>2).

49

Inverse Functions and Logarithms

Equation (5) holds for the other values of x in [-1, 1] as well, but we cannot conclude this from the triangle in Figure 1.70. It is, however, a consequence of Equations (2) and (4) (Exercise 76). The arctangent, arccotangent, arcsecant, and arccosecant functions are defined in Section 3.9. There we develop additional properties of the inverse trigonometric functions in a calculus setting using the identities discussed here.

Exercises 1.6 Identifying One-to-One Functions Graphically Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not? 1.

2.

y

y

y 3x 3

Graphing Inverse Functions Each of Exercises 11–16 shows the graph of a function y = ƒsxd . Copy the graph and draw in the line y = x . Then use symmetry with respect to the line y = x to add the graph of ƒ -1 to your sketch. (It is not necessary to find a formula for ƒ -1 .) Identify the domain and range of ƒ -1 . 11.

x

0

–1

0 1 y x4 x2

12. y

y

x

y f (x) 2 1 , x 0 x 1 1

3.

4.

y

1 0

y f (x) 1 1x , x 0 x

1

y y int x 0

x

1

y 2x

13.

14. y

y

x

y f (x) sin x, –x 1 2 2

5.

6.

y y 1x

y y x1/3

– 2

2

0

– 2

x

15.

16. y

6

8. ƒsxd = e

2x + 6, x + 4,

10. ƒsxd = e

x … -3 x 7 -3

x , 2

x … 0

x , x + 2

x 7 0

2 - x 2, x 2,

x … 1 x 7 1

1 9. ƒsxd = d

x 6 0 x Ú 0

f (x) f (x) 6 2x, 0x3

In Exercises 7–10, determine from its graph if the function is one-to-one. 3 - x, 3,

x

–1

y

7. ƒsxd = e

2

0

x

x

0

y f (x) tan x, –x 2 2

0

3

x

1 –1 0

x 1, 1 x 0 2 x, 0 x 3 3

2 3

x

–2

17. a. Graph the function ƒsxd = 21 - x 2, 0 … x … 1 . What symmetry does the graph have? b. Show that ƒ is its own inverse. (Remember that 2x 2 = x if x Ú 0 .) 18. a. Graph the function ƒsxd = 1>x . What symmetry does the graph have? b. Show that ƒ is its own inverse.

50

Chapter 1: Functions

Formulas for Inverse Functions Each of Exercises 19–24 gives a formula for a function y = ƒsxd and shows the graphs of ƒ and ƒ -1 . Find a formula for ƒ -1 in each case. 19. ƒsxd = x 2 + 1,

x Ú 0

20. ƒsxd = x 2,

x … 0

y

y

y f (x)

1

y f –1(x)

0

0

21. ƒsxd = x - 1

x

1

x

1

2

x Ú 1

y

y y f (x)

1 x

1

–1

0

40. a. Find the inverse of ƒsxd = - x + 1 . Graph the line y = -x + 1 together with the line y = x . At what angle do the lines intersect?

c. What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y = x ?

y f (x)

1

c. What can you conclude about the inverses of functions whose graphs are lines parallel to the line y = x ?

b. Find the inverse of ƒsxd = -x + b (b constant). What angle does the line y = -x + b make with the line y = x ?

y f –1(x)

y f –1(x)

39. a. Find the inverse of ƒsxd = x + 1 . Graph ƒ and its inverse together. Add the line y = x to your sketch, drawing it with dashes or dots for contrast. b. Find the inverse of ƒsxd = x + b (b constant). How is the graph of ƒ -1 related to the graph of ƒ?

y f –1(x)

22. ƒsxd = x - 2x + 1,

3

–1

b. What can you conclude about the inverse of a function y = ƒsxd whose graph is a line through the origin with a nonzero slope m? 38. Show that the graph of the inverse of ƒsxd = mx + b , where m and b are constants and m Z 0 , is a line with slope 1> m and y-intercept -b>m .

y f (x)

1

Inverses of Lines 37. a. Find the inverse of the function ƒsxd = mx , where m is a constant different from zero.

x

1

Logarithms and Exponentials 41. Express the following logarithms in terms of ln 2 and ln 3. b. ln (4> 9)

a. ln 0.75 23. ƒsxd = sx + 1d2,

x Ú -1 24. ƒsxd = x 2>3,

y

1

y f (x) 1

–1 0

x

1

Each of Exercises 25–36 gives a formula for a function y = ƒsxd . In each case, find ƒ -1sxd and identify the domain and range of ƒ -1 . As a check, show that ƒsƒ -1sxdd = ƒ -1sƒsxdd = x . 25. ƒsxd = x 5

26. ƒsxd = x 4,

27. ƒsxd = x + 1

28. ƒsxd = s1>2dx - 7>2

3

29. ƒsxd = 1>x 2, 31. ƒsxd =

x 7 0

x + 3 x - 2

33. ƒsxd = x 2 - 2x,

x Ú 0

30. ƒsxd = 1>x 3, 32. ƒsxd =

x … 1

f. ln 213.5

e. ln 322

y f –1(x)

x

3 d. ln 2 9

42. Express the following logarithms in terms of ln 5 and ln 7.

y f –1(x)

1 0

x Ú 0

y y f (x)

–1

c. ln (1> 2)

x Z 0

2x 2x - 3

34. ƒsxd = s2x 3 + 1d1>5

(Hint: Complete the square.) x + b , b 7 -2 and constant 35. ƒsxd = x - 2 36. ƒsxd = x2 - 2bx, b 7 0 and constant, x … b

a. ln (1> 125)

b. ln 9.8

c. ln 727

d. ln 1225

e. ln 0.056

f. sln 35 + ln s1>7dd>sln 25d

Use the properties of logarithms to write the expressions in Exercises 43 and 44 as a single term. sin u 1 b 43. a. ln sin u - ln a b. ln s3x 2 - 9xd + ln a b 5 3x 1 c. ln s4t 4 d - ln b 2 44. a. ln sec u + ln cos u b. ln s8x + 4d - 2 ln c 3 2 t - 1 - ln st + 1d c. 3 ln2

Find simpler expressions for the quantities in Exercises 45–48. 45. a. e ln 7.2 46. a. e

ln sx 2 + y 2d

b. e -ln x b. e

c. e ln x - ln y

2

-ln 0.3

c. e ln px - ln 2

47. a. 2 ln 2e

b. ln sln e e d

48. a. ln se sec u d

b. ln se se d d

x

c. ln se -x

2

- y2

d

c. ln se 2 ln x d

In Exercises 49–54, solve for y in terms of t or x, as appropriate. 49. ln y = 2t + 4

50. ln y = -t + 5

51. ln s y - bd = 5t

52. ln sc - 2yd = t

53. ln s y - 1d - ln 2 = x + ln x 54. ln s y 2 - 1d - ln s y + 1d = ln ssin xd

1.6 In Exercises 55 and 56, solve for k. 55. a. e 2k = 4 b. 100e 10k = 200 1 56. a. e 5k = b. 80e k = 1 4 In Exercises 57–60, solve for t. 1 57. a. e -0.3t = 27 b. e kt = 2 1 58. a. e -0.01t = 1000 b. e kt = 10 59. e 2t = x 2

c. e k>1000 = a c. e sln 0.8dk = 0.8 c. e sln 0.2dt = 0.4 c. e sln 2dt =

1 2

60. e sx de s2x + 1d = e t 2

Simplify the expressions in Exercises 61–64. 61. a. 5log5 7 d. log4 16 62. a. 2log2 3 d. log11 121 63. a. 2log4 x log5 s3x 2d

64. a. 25

b. 8log822 e. log3 23

c. 1.3log1.3 75 1 f. log4 a b 4

b. 10 log10 s1>2d

c. plogp 7

e. log121 11

1 f. log3 a b 9

x

b. log e se d

69. a. arccos (-1) 70. a. arcsin (-1)

c. log4 s2e

c. shifting left 1, up 3 units.

x

sin x

d

23 -1 b c. cos-1 a b 2 22 b. arccos (0) b. arcsin a-

75. Find a formula for the inverse function ƒ -1 and verify that (ƒ ⴰ ƒ -1)(x) = (ƒ -1 ⴰ ƒ )(x) = x. 50 100 a. ƒ(x) = b. ƒ(x) = 1 + 2-x 1 + 1.1-x 76. The identity sin-1 x + cos-1 x = P>2 Figure 1.70 establishes the identity for 0 6 x 6 1 . To establish it for the rest of [-1, 1] , verify by direct calculation that it holds for x = 1 , 0, and -1 . Then, for values of x in s -1, 0d , let x = -a, a 7 0 , and apply Eqs. (3) and (5) to the sum sin-1s -ad + cos-1s -ad .

a. shifting down 3 units.

Arcsine and Arccosine In Exercises 67–70, find the exact value of each expression. - 23 -1 1 b c. sin-1 a b 67. a. sin-1 a b b. sin-1 a 2 2 22 b. cos-1 a

74. If a composite ƒ ⴰ g is one-to-one, must g be one-to-one? Give reasons for your answer.

b. shifting right 1 unit.

Express the ratios in Exercises 65 and 66 as ratios of natural logarithms and simplify. log x a log 2 x log 2 x 65. a. b. c. log 3 x log 8 x log x2 a log210 x log 9 x log a b 66. a. b. c. log 3 x log b a log22 x

1 68. a. cos-1 a b 2

51

77. Start with the graph of y = ln x. Find an equation of the graph that results from

c. log2 se sln 2dssin xd d

b. 9log3 x

Inverse Functions and Logarithms

1 22

b

Theory and Examples 71. If ƒ(x) is one-to-one, can anything be said about gsxd = -ƒsxd ? Is it also one-to-one? Give reasons for your answer. 72. If ƒ(x) is one-to-one and ƒ(x) is never zero, can anything be said about hsxd = 1>ƒsxd ? Is it also one-to-one? Give reasons for your answer. 73. Suppose that the range of g lies in the domain of ƒ so that the composite ƒ ⴰ g is defined. If ƒ and g are one-to-one, can anything be said about ƒ ⴰ g ? Give reasons for your answer.

d. shifting down 4, right 2 units. e. reflecting about the y-axis. f. reflecting about the line y = x. 78. Start with the graph of y = ln x. Find an equation of the graph that results from a. vertical stretching by a factor of 2. b. horizontal stretching by a factor of 3. c. vertical compression by a factor of 4. d. horizontal compression by a factor of 2. T 79. The equation x 2 = 2x has three solutions: x = 2, x = 4, and one other. Estimate the third solution as accurately as you can by graphing. T 80. Could x ln 2 possibly be the same as 2ln x for x 7 0? Graph the two functions and explain what you see. 81. Radioactive decay The half-life of a certain radioactive substance is 12 hours. There are 8 grams present initially. a. Express the amount of substance remaining as a function of time t. b. When will there be 1 gram remaining? 82. Doubling your money Determine how much time is required for a $500 investment to double in value if interest is earned at the rate of 4.75% compounded annually. 83. Population growth The population of Glenbrook is 375,000 and is increasing at the rate of 2.25% per year. Predict when the population will be 1 million. 84. Radon-222 The decay equation for radon-222 gas is known to be y = y0 e -0.18t , with t in days. About how long will it take the radon in a sealed sample of air to fall to 90% of its original value?

2 LIMITS AND CONTINUITY OVERVIEW Mathematicians of the seventeenth century were keenly interested in the study of motion for objects on or near the earth and the motion of planets and stars. This study involved both the speed of the object and its direction of motion at any instant, and they knew the direction at a given instant was along a line tangent to the path of motion. The concept of a limit is fundamental to finding the velocity of a moving object and the tangent to a curve. In this chapter we develop the limit, first intuitively and then formally. We use limits to describe the way a function varies. Some functions vary continuously; small changes in x produce only small changes in ƒ(x). Other functions can have values that jump, vary erratically, or tend to increase or decrease without bound. The notion of limit gives a precise way to distinguish between these behaviors.

2.1

Rates of Change and Tangents to Curves Calculus is a tool that helps us understand how a change in one quantity is related to a change in another. How does the speed of a falling object change as a function of time? How does the level of water in a barrel change as a function of the amount of liquid poured into it? In this section we introduce the ideas of average and instantaneous rates of change, and show that they are closely related to the slope of a curve at a point P on the curve. We give precise developments of these important concepts in the next chapter, but for now we use an informal approach so you will see how they lead naturally to the main idea of the chapter, the limit. The idea of a limit plays a foundational role throughout calculus.

Average and Instantaneous Speed HISTORICAL BIOGRAPHY Galileo Galilei (1564–1642)

In the late sixteenth century, Galileo discovered that a solid object dropped from rest (not moving) near the surface of the earth and allowed to fall freely will fall a distance proportional to the square of the time it has been falling. This type of motion is called free fall. It assumes negligible air resistance to slow the object down, and that gravity is the only force acting on the falling body. If y denotes the distance fallen in feet after t seconds, then Galileo’s law is y = 16t 2, where 16 is the (approximate) constant of proportionality. (If y is measured in meters, the constant is 4.9.) A moving body’s average speed during an interval of time is found by dividing the distance covered by the time elapsed. The unit of measure is length per unit time: kilometers per hour, feet (or meters) per second, or whatever is appropriate to the problem at hand.

52

2.1

EXAMPLE 1

Rates of Change and Tangents to Curves

53

A rock breaks loose from the top of a tall cliff. What is its average speed

(a) during the first 2 sec of fall? (b) during the 1-sec interval between second 1 and second 2? The average speed of the rock during a given time interval is the change in distance, ¢y, divided by the length of the time interval, ¢t. (Increments like ¢y and ¢t are reviewed in Appendix 3.) Measuring distance in feet and time in seconds, we have the following calculations:

Solution

(a) For the first 2 sec:

¢y 16s2d2 - 16s0d2 ft = = 32 sec 2 - 0 ¢t

(b) From sec 1 to sec 2:

¢y 16s2d2 - 16s1d2 ft = = 48 sec 2 - 1 ¢t

We want a way to determine the speed of a falling object at a single instant t0, instead of using its average speed over an interval of time. To do this, we examine what happens when we calculate the average speed over shorter and shorter time intervals starting at t0. The next example illustrates this process. Our discussion is informal here, but it will be made precise in Chapter 3.

EXAMPLE 2

Find the speed of the falling rock in Example 1 at t = 1 and t = 2 sec.

We can calculate the average speed of the rock over a time interval [t0 , t0 + h], having length ¢t = h, as

Solution

¢y 16st0 + hd2 - 16t0 2 . = h ¢t

(1)

We cannot use this formula to calculate the “instantaneous” speed at the exact moment t0 by simply substituting h = 0, because we cannot divide by zero. But we can use it to calculate average speeds over increasingly short time intervals starting at t0 = 1 and t0 = 2. When we do so, we see a pattern (Table 2.1).

TABLE 2.1

Average speeds over short time intervals [t0, t0 + h] Average speed:

¢y 16st0 + hd2 - 16t0 2 = h ¢t

Length of time interval h

Average speed over interval of length h starting at t0 1

Average speed over interval of length h starting at t0 2

1 0.1 0.01 0.001 0.0001

48 33.6 32.16 32.016 32.0016

80 65.6 64.16 64.016 64.0016

The average speed on intervals starting at t0 = 1 seems to approach a limiting value of 32 as the length of the interval decreases. This suggests that the rock is falling at a speed of 32 ft> sec at t0 = 1 sec. Let’s confirm this algebraically.

54

Chapter 2: Limits and Continuity

If we set t0 = 1 and then expand the numerator in Equation (1) and simplify, we find that ¢y 16s1 + 2h + h 2 d - 16 16s1 + hd2 - 16s1d2 = = h h ¢t =

32h + 16h 2 = 32 + 16h. h

For values of h different from 0, the expressions on the right and left are equivalent and the average speed is 32 + 16h ft>sec. We can now see why the average speed has the limiting value 32 + 16s0d = 32 ft>sec as h approaches 0. Similarly, setting t0 = 2 in Equation (1), the procedure yields ¢y = 64 + 16h ¢t for values of h different from 0. As h gets closer and closer to 0, the average speed has the limiting value 64 ft >sec when t0 = 2 sec, as suggested by Table 2.1. The average speed of a falling object is an example of a more general idea which we discuss next.

Average Rates of Change and Secant Lines Given an arbitrary function y = ƒsxd, we calculate the average rate of change of y with respect to x over the interval [x1 , x2] by dividing the change in the value of y, ¢y = ƒsx2 d - ƒsx1 d, by the length ¢x = x2 - x1 = h of the interval over which the change occurs. (We use the symbol h for ¢x to simplify the notation here and later on.)

y y f (x) Q(x 2, f(x 2 ))

Secant y

P(x1, f(x1))

DEFINITION The average rate of change of y = ƒsxd with respect to x over the interval [x1 , x2] is

x h 0

x2

x1

FIGURE 2.1 A secant to the graph y = ƒsxd . Its slope is ¢y>¢x , the average rate of change of ƒ over the interval [x1 , x2] .

x

ƒsx2 d - ƒsx1 d ¢y ƒsx1 + hd - ƒsx1 d = = , x2 - x1 h ¢x

h Z 0.

Geometrically, the rate of change of ƒ over [x1, x2] is the slope of the line through the points Psx1, ƒsx1 dd and Qsx2 , ƒsx2 dd (Figure 2.1). In geometry, a line joining two points of a curve is a secant to the curve. Thus, the average rate of change of ƒ from x1 to x2 is identical with the slope of secant PQ. Let’s consider what happens as the point Q approaches the point P along the curve, so the length h of the interval over which the change occurs approaches zero. We will see that this procedure leads to defining the slope of a curve at a point.

Defining the Slope of a Curve P L O

FIGURE 2.2 L is tangent to the circle at P if it passes through P perpendicular to radius OP.

We know what is meant by the slope of a straight line, which tells us the rate at which it rises or falls—its rate of change as a linear function. But what is meant by the slope of a curve at a point P on the curve? If there is a tangent line to the curve at P—a line that just touches the curve like the tangent to a circle—it would be reasonable to identify the slope of the tangent as the slope of the curve at P. So we need a precise meaning for the tangent at a point on a curve. For circles, tangency is straightforward. A line L is tangent to a circle at a point P if L passes through P perpendicular to the radius at P (Figure 2.2). Such a line just touches the circle. But what does it mean to say that a line L is tangent to some other curve C at a point P?

2.1

Rates of Change and Tangents to Curves

55

To define tangency for general curves, we need an approach that takes into account the behavior of the secants through P and nearby points Q as Q moves toward P along the curve (Figure 2.3). Here is the idea: 1. 2. 3.

HISTORICAL BIOGRAPHY

Start with what we can calculate, namely the slope of the secant PQ. Investigate the limiting value of the secant slope as Q approaches P along the curve. (We clarify the limit idea in the next section.) If the limit exists, take it to be the slope of the curve at P and define the tangent to the curve at P to be the line through P with this slope.

This procedure is what we were doing in the falling-rock problem discussed in Example 2. The next example illustrates the geometric idea for the tangent to a curve.

Pierre de Fermat (1601–1665) Secants

Tangent

P

P Q

Tangent

Secants

Q

FIGURE 2.3 The tangent to the curve at P is the line through P whose slope is the limit of the secant slopes as Q : P from either side.

Find the slope of the parabola y = x 2 at the point P(2, 4). Write an equation for the tangent to the parabola at this point.

EXAMPLE 3

We begin with a secant line through P(2, 4) and Qs2 + h, s2 + hd2 d nearby. We then write an expression for the slope of the secant PQ and investigate what happens to the slope as Q approaches P along the curve: Solution

Secant slope =

¢y s2 + hd2 - 22 h 2 + 4h + 4 - 4 = = h h ¢x =

h 2 + 4h = h + 4. h

If h 7 0, then Q lies above and to the right of P, as in Figure 2.4. If h 6 0, then Q lies to the left of P (not shown). In either case, as Q approaches P along the curve, h approaches zero and the secant slope h + 4 approaches 4. We take 4 to be the parabola’s slope at P. y y x2

Secant slope is

(2 h) 2 4 h 4. h

Q(2 h, (2 h) 2) Tangent slope 4 Δy (2 h)2 4 P(2, 4) Δx h 0

2

2h

x

NOT TO SCALE

FIGURE 2.4 Finding the slope of the parabola y = x 2 at the point P(2, 4) as the limit of secant slopes (Example 3).

Chapter 2: Limits and Continuity

The tangent to the parabola at P is the line through P with slope 4: y = 4 + 4sx - 2d

Point-slope equation

y = 4x - 4.

Instantaneous Rates of Change and Tangent Lines The rates at which the rock in Example 2 was falling at the instants t = 1 and t = 2 are called instantaneous rates of change. Instantaneous rates and slopes of tangent lines are intimately connected, as we will now see in the following examples.

EXAMPLE 4

Figure 2.5 shows how a population p of fruit flies (Drosophila) grew in a 50-day experiment. The number of flies was counted at regular intervals, the counted values plotted with respect to time t, and the points joined by a smooth curve (colored blue in Figure 2.5). Find the average growth rate from day 23 to day 45. There were 150 flies on day 23 and 340 flies on day 45. Thus the number of flies increased by 340 - 150 = 190 in 45 - 23 = 22 days. The average rate of change of the population from day 23 to day 45 was Solution

Average rate of change:

¢p 340 - 150 190 = = L 8.6 flies>day. 45 - 23 22 ¢t

p 350 Q(45, 340) Number of flies

56

300 p 190

250 200 P(23, 150)

150

p 8.6 flies/day t t 22

100 50 0

10

20 30 Time (days)

40

50

t

FIGURE 2.5 Growth of a fruit fly population in a controlled experiment. The average rate of change over 22 days is the slope ¢p>¢t of the secant line (Example 4).

This average is the slope of the secant through the points P and Q on the graph in Figure 2.5. The average rate of change from day 23 to day 45 calculated in Example 4 does not tell us how fast the population was changing on day 23 itself. For that we need to examine time intervals closer to the day in question.

EXAMPLE 5

How fast was the number of flies in the population of Example 4 growing

on day 23? To answer this question, we examine the average rates of change over increasingly short time intervals starting at day 23. In geometric terms, we find these rates by calculating the slopes of secants from P to Q, for a sequence of points Q approaching P along the curve (Figure 2.6). Solution

2.1

57

Rates of Change and Tangents to Curves

p

Slope of PQ ≤p/≤t (flies / day) 340 45 330 40 310 35 265 30

(45, 340) (40, 330) (35, 310) (30, 265) FIGURE 2.6

-

150 23 150 23 150 23 150 23

L 8.6 L 10.6 L 13.3 L 16.4

Q(45, 340)

300 Number of flies

Q

B(35, 350)

350

250 200 P(23, 150)

150 100 50 0

10 20 30 A(14, 0) Time (days)

40

50

t

The positions and slopes of four secants through the point P on the fruit fly graph (Example 5).

The values in the table show that the secant slopes rise from 8.6 to 16.4 as the t-coordinate of Q decreases from 45 to 30, and we would expect the slopes to rise slightly higher as t continued on toward 23. Geometrically, the secants rotate about P and seem to approach the red tangent line in the figure. Since the line appears to pass through the points (14, 0) and (35, 350), it has slope 350 - 0 = 16.7 flies>day (approximately). 35 - 14 On day 23 the population was increasing at a rate of about 16.7 flies>day. The instantaneous rates in Example 2 were found to be the values of the average speeds, or average rates of change, as the time interval of length h approached 0. That is, the instantaneous rate is the value the average rate approaches as the length h of the interval over which the change occurs approaches zero. The average rate of change corresponds to the slope of a secant line; the instantaneous rate corresponds to the slope of the tangent line as the independent variable approaches a fixed value. In Example 2, the independent variable t approached the values t = 1 and t = 2. In Example 3, the independent variable x approached the value x = 2. So we see that instantaneous rates and slopes of tangent lines are closely connected. We investigate this connection thoroughly in the next chapter, but to do so we need the concept of a limit.

Exercises 2.1 Average Rates of Change In Exercises 1–6, find the average rate of change of the function over the given interval or intervals. 1. ƒsxd = x 3 + 1 b. [-1, 1]

a. [2, 3] 2. g sxd = x

2

a. [-1, 1]

b. [-2, 0]

3. hstd = cot t a. [p>4, 3p>4]

b. [p>6, p>2]

[1, 2]

Slope of a Curve at a Point In Exercises 7–14, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P. 7. y = x 2 - 3,

P(2, 1)

8. y = 5 - x ,

P(1, 4)

9. y = x 2 - 2x - 3, 10. y = x - 4x, 2

b. [-p, p]

[0, 2]

6. Psud = u3 - 4 u2 + 5u;

2

4. g std = 2 + cos t a. [0, p]

5. Rsud = 24u + 1;

11. y = x 3,

P(2, -3)

P(1, -3)

P(2, 8)

58

Chapter 2: Limits and Continuity b. What is the average rate of increase of the profits between 2002 and 2004?

P(1, 1)

13. y = x - 12x,

P(1, -11)

3

14. y = x 3 - 3x 2 + 4,

c. Use your graph to estimate the rate at which the profits were changing in 2002.

P(2, 0)

Instantaneous Rates of Change 15. Speed of a car The accompanying figure shows the timeto-distance graph for a sports car accelerating from a standstill.

a. Find the average rate of change of F(x) over the intervals [1, x] for each x Z 1 in your table.

s P

650 600

Distance (m)

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1 .

Q4 Q3

500 400

T 19. Let g sxd = 2x for x Ú 0 .

Q2

a. Find the average rate of change of g(x) with respect to x over the intervals [1, 2], [1, 1.5] and [1, 1 + h] .

300 200

b. Make a table of values of the average rate of change of g with respect to x over the interval [1, 1 + h] for some values of h approaching zero, say h = 0.1, 0.01, 0.001, 0.0001, 0.00001 , and 0.000001.

Q1

100 0

5 10 15 20 Elapsed time (sec)

t

c. What does your table indicate is the rate of change of g(x) with respect to x at x = 1 ?

a. Estimate the slopes of secants PQ1 , PQ2 , PQ3 , and PQ4 , arranging them in order in a table like the one in Figure 2.6. What are the appropriate units for these slopes? b. Then estimate the car’s speed at time t = 20 sec . 16. The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon. a. Estimate the slopes of the secants PQ1 , PQ2 , PQ3 , and PQ4 , arranging them in a table like the one in Figure 2.6. b. About how fast was the object going when it hit the surface?

Distance fallen (m)

80

T 20. Let ƒstd = 1>t for t Z 0 . a. Find the average rate of change of ƒ with respect to t over the intervals (i) from t = 2 to t = 3 , and (ii) from t = 2 to t = T . b. Make a table of values of the average rate of change of ƒ with respect to t over the interval [2, T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001 , and 2.000001. c. What does your table indicate is the rate of change of ƒ with respect to t at t = 2 ?

21. The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

Q3

60 Q2

40

0

P

Q4

d. Calculate the limit as h approaches zero of the average rate of change of g(x) with respect to x over the interval [1, 1 + h] .

d. Calculate the limit as T approaches 2 of the average rate of change of ƒ with respect to t over the interval from 2 to T. You will have to do some algebra before you can substitute T = 2 .

y

20

T 18. Make a table of values for the function Fsxd = sx + 2d>sx - 2d at the points x = 1.2, x = 11>10, x = 101>100, x = 1001>1000, x = 10001>10000 , and x = 1 .

s

Q1

5 Elapsed time (sec)

10

t

T 17. The profits of a small company for each of the first five years of its operation are given in the following table: Year

Profit in $1000s

2000 2001 2002 2003 2004

6 27 62 111 174

a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can.

Distance traveled (mi)

12. y = 2 - x 3,

40 30 20 10 0

1

2 3 Elapsed time (hr)

4

t

a. Estimate the bicyclist’s average speed over the time intervals [0, 1], [1, 2.5], and [2.5, 3.5]. b. Estimate the bicyclist’s instantaneous speed at the times t = 12, t = 2, and t = 3. c. Estimate the bicyclist’s maximum speed and the specific time at which it occurs.

2.2 22. The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after being driven for t days.

59

a. Estimate the average rate of gasoline consumption over the time intervals [0, 3], [0, 5], and [7, 10]. b. Estimate the instantaneous rate of gasoline consumption at the times t = 1, t = 4, and t = 8.

A Remaining amount (gal)

Limit of a Function and Limit Laws

c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

16 12 8 4 0

1

2

3 4 5 6 7 8 Elapsed time (days)

9 10

t

Limit of a Function and Limit Laws

2.2

In Section 2.1 we saw that limits arise when finding the instantaneous rate of change of a function or the tangent to a curve. Here we begin with an informal definition of limit and show how we can calculate the values of limits. A precise definition is presented in the next section.

Limits of Function Values

HISTORICAL ESSAY

Frequently when studying a function y = ƒ(x), we find ourselves interested in the function’s behavior near a particular point c, but not at c. This might be the case, for instance, if c is an irrational number, like p or 22, whose values can only be approximated by “close” rational numbers at which we actually evaluate the function instead. Another situation occurs when trying to evaluate a function at c leads to division by zero, which is undefined. We encountered this last circumstance when seeking the instantaneous rate of change in y by considering the quotient function ¢y>h for h closer and closer to zero. Here’s a specific example where we explore numerically how a function behaves near a particular point at which we cannot directly evaluate the function.

Limits y

2 2 y f (x) x 1 x 1

1

–1

0

1

x

EXAMPLE 1

How does the function ƒsxd =

y

x2 - 1 x - 1

behave near x = 1? 2

The given formula defines ƒ for all real numbers x except x = 1 (we cannot divide by zero). For any x Z 1, we can simplify the formula by factoring the numerator and canceling common factors:

Solution

yx1 1

–1

0

1

FIGURE 2.7 The graph of ƒ is identical with the line y = x + 1 except at x = 1 , where ƒ is not defined (Example 1).

x

ƒsxd =

sx - 1dsx + 1d = x + 1 x - 1

for

x Z 1.

The graph of ƒ is the line y = x + 1 with the point (1, 2) removed. This removed point is shown as a “hole” in Figure 2.7. Even though ƒ(1) is not defined, it is clear that we can make the value of ƒ(x) as close as we want to 2 by choosing x close enough to 1 (Table 2.2).

60

Chapter 2: Limits and Continuity

TABLE 2.2 The closer x gets to 1, the closer ƒ(x) = (x 2 - 1)>(x - 1)

seems to get to 2 x2 1 x 1, x1

Values of x below and above 1

ƒ(x)

0.9 1.1 0.99 1.01 0.999 1.001 0.999999 1.000001

1.9 2.1 1.99 2.01 1.999 2.001 1.999999 2.000001

x1

Let’s generalize the idea illustrated in Example 1. Suppose ƒ(x) is defined on an open interval about c, except possibly at c itself. If ƒ(x) is arbitrarily close to L (as close to L as we like) for all x sufficiently close to c, we say that ƒ approaches the limit L as x approaches c, and write lim ƒsxd = L,

x:c

which is read “the limit of ƒ(x) as x approaches c is L.” For instance, in Example 1 we would say that ƒ(x) approaches the limit 2 as x approaches 1, and write lim ƒsxd = 2,

x:1

or

x2 - 1 = 2. x:1 x - 1 lim

Essentially, the definition says that the values of ƒ(x) are close to the number L whenever x is close to c (on either side of c). This definition is “informal” because phrases like arbitrarily close and sufficiently close are imprecise; their meaning depends on the context. (To a machinist manufacturing a piston, close may mean within a few thousandths of an inch. To an astronomer studying distant galaxies, close may mean within a few thousand light-years.) Nevertheless, the definition is clear enough to enable us to recognize and evaluate limits of specific functions. We will need the precise definition of Section 2.3, however, when we set out to prove theorems about limits. Here are several more examples exploring the idea of limits.

EXAMPLE 2

The limiting value of a function does not depend on how the function is defined at the point being approached. Consider the three functions in Figure 2.8. The function ƒ has limit 2 as x : 1 even though ƒ is not defined at x = 1. The function g has y

–1

y

y

2

2

2

1

1

1

0

2 (a) f (x) x 1 x 1

1

x

–1

0

1

x

⎧ x2 1 , x 1 ⎪ (b) g(x) ⎨ x 1 ⎪ 1, x1 ⎩

–1

0

1

x

(c) h(x) x 1

FIGURE 2.8 The limits of ƒ(x), g(x), and h(x) all equal 2 as x approaches 1. However, only h(x) has the same function value as its limit at x = 1 (Example 2).

2.2

61

limit 2 as x : 1 even though 2 Z gs1d. The function h is the only one of the three functions in Figure 2.8 whose limit as x : 1 equals its value at x = 1. For h, we have limx:1 hsxd = hs1d. This equality of limit and function value is significant, and we return to it in Section 2.5.

y yx c

EXAMPLE 3

x

c

Limit of a Function and Limit Laws

(a) If ƒ is the identity function ƒsxd = x, then for any value of c (Figure 2.9a), lim ƒsxd = lim x = c.

(a) Identity function

x:c

y

(b) If ƒ is the constant function ƒsxd = k (function with the constant value k), then for any value of c (Figure 2.9b),

yk

k

x:c

lim ƒsxd = lim k = k.

x:c

c

0

x:c

For instances of each of these rules we have

x

lim x = 3

x:3

(b) Constant function

lim s4d = lim s4d = 4.

and

x: -7

x:2

We prove these rules in Example 3 in Section 2.3.

FIGURE 2.9 The functions in Example 3 have limits at all points c.

Some ways that limits can fail to exist are illustrated in Figure 2.10 and described in the next example.

y

1

y

0

y ⎧1 ⎪ , x0 y ⎨x ⎪ 0, x 0 ⎩

⎧ 0, x 0 y⎨ ⎩ 1, x 0

x

1

x

0

x

0

⎧ x0 ⎪ 0, y⎨ 1 ⎪ sin x , x 0 ⎩

–1 (a) Unit step function U(x)

(b) g(x)

(c) f (x)

FIGURE 2.10 None of these functions has a limit as x approaches 0 (Example 4).

Discuss the behavior of the following functions as x : 0.

EXAMPLE 4 (a) Usxd = e

(b) gsxd = L (c) ƒsxd = •

0, 1,

x 6 0 x Ú 0

1 x,

x Z 0

0,

x = 0

0,

x … 0

1 sin x ,

x 7 0

62

Chapter 2: Limits and Continuity Solution

(a) It jumps: The unit step function U(x) has no limit as x : 0 because its values jump at x = 0. For negative values of x arbitrarily close to zero, Usxd = 0. For positive values of x arbitrarily close to zero, Usxd = 1. There is no single value L approached by U(x) as x : 0 (Figure 2.10a). (b) It grows too “large” to have a limit: g(x) has no limit as x : 0 because the values of g grow arbitrarily large in absolute value as x : 0 and do not stay close to any fixed real number (Figure 2.10b). (c) It oscillates too much to have a limit: ƒ(x) has no limit as x : 0 because the function’s values oscillate between +1 and -1 in every open interval containing 0. The values do not stay close to any one number as x : 0 (Figure 2.10c).

The Limit Laws To calculate limits of functions that are arithmetic combinations of functions having known limits, we can use several easy rules.

THEOREM 1—Limit Laws lim ƒsxd = L

x:c

1. Sum Rule: 2. Difference Rule: 3. Constant Multiple Rule: 4. Product Rule: 5. Quotient Rule: 6. Power Rule: 7. Root Rule:

If L, M, c, and k are real numbers and lim gsxd = M,

and

x:c

then

lim sƒsxd + gsxdd = L + M

x:c

lim sƒsxd - gsxdd = L - M

x:c

lim sk # ƒsxdd = k # L

x:c

lim sƒsxd # gsxdd = L # M

x:c

lim

x:c

ƒsxd L = , M gsxd

M Z 0

lim [ƒ(x)]n = L n, n a positive integer

x:c

n

n

lim 2ƒ(x) = 2L = L 1>n, n a positive integer

x:c

(If n is even, we assume that lim ƒ(x) = L 7 0.) x:c

In words, the Sum Rule says that the limit of a sum is the sum of the limits. Similarly, the next rules say that the limit of a difference is the difference of the limits; the limit of a constant times a function is the constant times the limit of the function; the limit of a product is the product of the limits; the limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0); the limit of a positive integer power (or root) of a function is the integer power (or root) of the limit (provided that the root of the limit is a real number). It is reasonable that the properties in Theorem 1 are true (although these intuitive arguments do not constitute proofs). If x is sufficiently close to c, then ƒ(x) is close to L and g (x) is close to M, from our informal definition of a limit. It is then reasonable that ƒsxd + gsxd is close to L + M; ƒsxd - gsxd is close to L - M; kƒ(x) is close to kL; ƒ(x)g(x) is close to LM; and ƒ(x)>g(x) is close to L>M if M is not zero. We prove the Sum Rule in Section 2.3, based on a precise definition of limit. Rules 2–5 are proved in Appendix 4. Rule 6 is obtained by applying Rule 4 repeatedly. Rule 7 is proved in more

2.2

Limit of a Function and Limit Laws

63

advanced texts. The sum, difference, and product rules can be extended to any number of functions, not just two. Use the observations limx:c k = k and limx:c x = c (Example 3) and the properties of limits to find the following limits.

EXAMPLE 5

x4 + x2 - 1 x:c x2 + 5

(a) lim sx 3 + 4x 2 - 3d

(b) lim

x:c

(c) lim 24x 2 - 3 x: -2

Solution

(a) lim sx 3 + 4x 2 - 3d = lim x 3 + lim 4x 2 - lim 3 x:c

x:c

x:c

x:c

= c 3 + 4c 2 - 3 x4 + x2 - 1 = x:c x2 + 5

(b) lim

=

(c)

Power and Multiple Rules

lim sx + x - 1d 4

2

x:c

Quotient Rule

lim sx 2 + 5d

x:c

lim x 4 + lim x 2 - lim 1

x:c

x:c

x:c

lim x 2 + lim 5

x:c

=

Sum and Difference Rules

x:c

c + c - 1 c2 + 5 4

2

Power or Product Rule

lim 24x 2 - 3 = 2 lim s4x 2 - 3d

x: -2

Sum and Difference Rules

x: -2

Root Rule with n = 2

= 2 lim 4x 2 - lim 3

Difference Rule

= 24s -2d2 - 3

Product and Multiple Rules

x: -2

x: -2

= 216 - 3 = 213 Two consequences of Theorem 1 further simplify the task of calculating limits of polynomials and rational functions. To evaluate the limit of a polynomial function as x approaches c, merely substitute c for x in the formula for the function. To evaluate the limit of a rational function as x approaches a point c at which the denominator is not zero, substitute c for x in the formula for the function. (See Examples 5a and 5b.) We state these results formally as theorems.

THEOREM 2—Limits of Polynomials If Psxd = an x n + an - 1 x n - 1 + Á + a0 , then lim Psxd = Pscd = an c n + an - 1 c n - 1 + Á + a 0 .

x:c

THEOREM 3—Limits of Rational Functions If P(x) and Q(x) are polynomials and Qscd Z 0, then lim

x:c

Psxd Pscd = . Qsxd Qscd

64

Chapter 2: Limits and Continuity

EXAMPLE 6

The following calculation illustrates Theorems 2 and 3: s -1d3 + 4s -1d2 - 3 x 3 + 4x 2 - 3 0 = = = 0 2 2 6 x: -1 x + 5 s -1d + 5 lim

Identifying Common Factors It can be shown that if Q(x) is a polynomial and Qscd = 0 , then sx - cd is a factor of Q(x). Thus, if the numerator and denominator of a rational function of x are both zero at x = c , they have sx - cd as a common factor.

Eliminating Zero Denominators Algebraically Theorem 3 applies only if the denominator of the rational function is not zero at the limit point c. If the denominator is zero, canceling common factors in the numerator and denominator may reduce the fraction to one whose denominator is no longer zero at c. If this happens, we can find the limit by substitution in the simplified fraction.

EXAMPLE 7

Evaluate

y 2 x2 y x x2 x (1, 3)

3

x2 + x - 2 . x:1 x2 - x lim

We cannot substitute x = 1 because it makes the denominator zero. We test the numerator to see if it, too, is zero at x = 1. It is, so it has a factor of sx - 1d in common with the denominator. Canceling the sx - 1d’s gives a simpler fraction with the same values as the original for x Z 1: Solution

–2

0

x

1

sx - 1dsx + 2d x + 2 x2 + x - 2 = = x , 2 xsx 1d x - x

(a) y yx2 x (1, 3)

3

if x Z 1.

Using the simpler fraction, we find the limit of these values as x : 1 by substitution: x2 + x - 2 x + 2 1 + 2 = lim x = = 3. 2 1 x:1 x:1 x - x lim

–2

0

1

x

See Figure 2.11.

(b)

FIGURE 2.11 The graph of ƒsxd = sx 2 + x - 2d>sx 2 - xd in part (a) is the same as the graph of g sxd = sx + 2d>x in part (b) except at x = 1, where ƒ is undefined. The functions have the same limit as x : 1 (Example 7).

Using Calculators and Computers to Estimate Limits When we cannot use the Quotient Rule in Theorem 1 because the limit of the denominator is zero, we can try using a calculator or computer to guess the limit numerically as x gets closer and closer to c. We used this approach in Example 1, but calculators and computers can sometimes give false values and misleading impressions for functions that are undefined at a point or fail to have a limit there, as we now illustrate.

EXAMPLE 8

Estimate the value of lim

x:0

2x 2 + 100 - 10 . x2

Solution Table 2.3 lists values of the function for several values near x = 0. As x approaches 0 through the values ;1, ;0.5, ;0.10, and ;0.01, the function seems to approach the number 0.05. As we take even smaller values of x, ;0.0005, ;0.0001, ;0.00001, and ;0.000001, the function appears to approach the value 0. Is the answer 0.05 or 0, or some other value? We resolve this question in the next example.

2.2

Limit of a Function and Limit Laws

TABLE 2.3 Computer values of ƒ(x) =

65

2x 2 + 100 - 10 near x = 0 x2

x

ƒ(x)

;1 ;0.5 ;0.1 ;0.01

0.049876 0.049969 t approaches 0.05? 0.049999 0.050000

;0.0005 ;0.0001 ;0.00001 ;0.000001

0.050000 0.000000 t approaches 0? 0.000000 0.000000

Using a computer or calculator may give ambiguous results, as in the last example. The calculator does not keep track of enough digits to avoid rounding errors in computing the values of f (x) when x is very small. We cannot substitute x = 0 in the problem, and the numerator and denominator have no obvious common factors (as they did in Example 7). Sometimes, however, we can create a common factor algebraically.

EXAMPLE 9

Evaluate

2x 2 + 100 - 10 . x:0 x2 Solution This is the limit we considered in Example 8. We can create a common factor by multiplying both numerator and denominator by the conjugate radical expression 2x 2 + 100 + 10 (obtained by changing the sign after the square root). The preliminary algebra rationalizes the numerator: lim

2x 2 + 100 - 10 2x 2 + 100 - 10 # 2x 2 + 100 + 10 = 2 x x2 2x 2 + 100 + 10 2 x + 100 - 100 = 2 x A 2x 2 + 100 + 10 B

Therefore, lim

x:0

x2

=

x 2 A 2x 2 + 100 + 10 B

=

1 . 2x + 100 + 10 2

Common factor x2

Cancel x2 for x Z 0.

2x 2 + 100 - 10 1 = lim 2 2 x:0 x 2x + 100 + 10 =

1 20 + 100 + 10

=

1 = 0.05. 20

2

Denominator not 0 at x = 0; substitute.

This calculation provides the correct answer, in contrast to the ambiguous computer results in Example 8. We cannot always algebraically resolve the problem of finding the limit of a quotient where the denominator becomes zero. In some cases the limit might then be found with the

66

Chapter 2: Limits and Continuity

aid of some geometry applied to the problem (see the proof of Theorem 7 in Section 2.4), or through methods of calculus (illustrated in Section 4.5). The next theorem is also useful.

y h f L

The Sandwich Theorem g x

c

0

The following theorem enables us to calculate a variety of limits. It is called the Sandwich Theorem because it refers to a function ƒ whose values are sandwiched between the values of two other functions g and h that have the same limit L at a point c. Being trapped between the values of two functions that approach L, the values of ƒ must also approach L (Figure 2.12). You will find a proof in Appendix 5.

FIGURE 2.12 The graph of ƒ is sandwiched between the graphs of g and h.

THEOREM 4—The Sandwich Theorem Suppose that gsxd … ƒsxd … hsxd for all x in some open interval containing c, except possibly at x = c itself. Suppose also that lim gsxd = lim hsxd = L.

y

2

The Sandwich Theorem is also called the Squeeze Theorem or the Pinching Theorem.

2 y1 x 4

0

x:c

Then limx:c ƒsxd = L.

y u(x)

1

–1

x:c

2 y1 x 2

EXAMPLE 10

Given that

x

1

FIGURE 2.13 Any function u(x) whose graph lies in the region between y = 1 + sx 2>2d and y = 1 - sx 2>4d has limit 1 as x : 0 (Example 10).

1 -

x2 x2 … usxd … 1 + 4 2

find limx:0 usxd, no matter how complicated u is. Solution

Since lim s1 - sx 2>4dd = 1

y ⎢ ⎢

EXAMPLE 11

y sin

–

u:0

(c) For any function ƒ, lim ƒ ƒ(x) ƒ = 0 implies lim ƒ(x) = 0. x:c

x:c

(a) In Section 1.3 we established that - ƒ u ƒ … sin u … ƒ u ƒ for all u (see Figure 2.14a). Since limu:0 s - ƒ u ƒ d = limu:0 ƒ u ƒ = 0, we have

y

lim sin u = 0 .

y ⎢ ⎢

1 0

(b) lim cos u = 1

u:0

Solution

(a)

–1

The Sandwich Theorem helps us establish several important limit rules:

(a) lim sin u = 0

y – ⎢ ⎢

–1

–2

x:0

the Sandwich Theorem implies that limx:0 usxd = 1 (Figure 2.13).

1

2

lim s1 + sx 2>2dd = 1,

and

x:0

y

for all x Z 0,

u:0

y 1 cos 1

2

(b)

FIGURE 2.14 The Sandwich Theorem confirms the limits in Example 11.

(b) From Section 1.3, 0 … 1 - cos u … ƒ u ƒ for all u (see Figure 2.14b), and we have limu:0 s1 - cos ud = 0 or lim cos u = 1 .

u:0

(c) Since - ƒ ƒsxd ƒ … ƒsxd … ƒ ƒsxd ƒ and - ƒ ƒsxd ƒ and ƒ ƒsxd ƒ have limit 0 as x : c, it follows that lim x:c ƒ(x) = 0 .

2.2

Limit of a Function and Limit Laws

67

Another important property of limits is given by the next theorem. A proof is given in the next section.

THEOREM 5 If ƒsxd … gsxd for all x in some open interval containing c, except possibly at x = c itself, and the limits of ƒ and g both exist as x approaches c, then lim ƒsxd … lim gsxd.

x:c

x:c

The assertion resulting from replacing the less than or equal to (…) inequality by the strict less than (6) inequality in Theorem 5 is false. Figure 2.14a shows that for u Z 0, - ƒ u ƒ 6 sin u 6 ƒ u ƒ , but in the limit as u : 0, equality holds.

Exercises 2.2 Limits from Graphs 1. For the function g(x) graphed here, find the following limits or explain why they do not exist. a. lim g sxd x: 1

b. lim g sxd

c. lim g sxd

x: 2

y y f (x)

1

d. lim g sxd

x: 3

x:2.5

–1

y

1

2

x

–1 y g(x) 1

1

x

3

2

4. Which of the following statements about the function y = ƒsxd graphed here are true, and which are false? a. lim ƒsxd does not exist. x:2

2. For the function ƒ(t) graphed here, find the following limits or explain why they do not exist. a. lim ƒstd t: -2

b. lim ƒstd t: -1

c. lim ƒstd

d.

t: 0

lim ƒstd

t: -0.5

b. lim ƒsxd = 2 x:2

c. lim ƒsxd does not exist. x:1

d. lim ƒsxd exists at every point c in s -1, 1d . x:c

e. lim ƒsxd exists at every point c in (1, 3).

s

x:c

y s f (t)

1

–1

0

–2

y f (x)

1 1

t –1

–1

1

2

3

x

–1 –2

3. Which of the following statements about the function y = ƒsxd graphed here are true, and which are false? a. lim ƒsxd exists. x: 0

b. lim ƒsxd = 0 x: 0

c. lim ƒsxd = 1 x: 0

d. lim ƒsxd = 1 x: 1

e. lim ƒsxd = 0 x: 1

f. lim ƒsxd exists at every point c in s -1, 1d . x: c

g. lim ƒsxd does not exist. x: 1

Existence of Limits In Exercises 5 and 6, explain why the limits do not exist. x 1 6. lim x:1 x - 1 ƒxƒ 7. Suppose that a function ƒ(x) is defined for all real values of x except x = c . Can anything be said about the existence of limx:c ƒsxd ? Give reasons for your answer. 5. lim

x:0

8. Suppose that a function ƒ(x) is defined for all x in [-1, 1] . Can anything be said about the existence of limx:0 ƒsxd ? Give reasons for your answer.

68

Chapter 2: Limits and Continuity

9. If limx:1 ƒsxd = 5 , must ƒ be defined at x = 1 ? If it is, must ƒs1d = 5 ? Can we conclude anything about the values of ƒ at x = 1 ? Explain. 10. If ƒs1d = 5 , must limx:1 ƒsxd exist? If it does, then must limx:1 ƒsxd = 5 ? Can we conclude anything about limx:1 ƒsxd ? Explain.

Using Limit Rules 51. Suppose limx:0 ƒsxd = 1 and limx:0 g sxd = -5 . Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation. lim

x:0

2ƒsxd - g sxd sƒsxd + 7d2>3

Calculating Limits Find the limits in Exercises 11–22. 12. lim s -x + 5x - 2d

13. lim 8st - 5dst - 7d

14. lim sx 3 - 2x 2 + 4x + 8d

t:6

15. lim

x:2

x + 3 x + 6

19. lim s5 - yd4>3 y: -3

h: 0

=

x: -2

3 23h + 1 + 1

=

y 2 + 5y + 6 20. lim s2z - 8d1>3

x:5

x - 5 x 2 - 25

x 2 + 3x - 10 25. lim x + 5 x: -5

z: 0

22. lim

h :0

25h + 4 - 2 h

x: -3

lim

-2x - 4 29. lim 3 x: -2 x + 2x 2

30. lim

1 x

- 1

x - 1

x:1

u - 1 u3 - 1

u: 1

2x - 3 35. lim x:9 x - 9 37. lim

x:1

x - 1 2x + 3 - 2

2x 2 + 12 - 4 39. lim x - 2 x:2 41. lim

x: -3

2 - 2x 2 - 5 x + 3

t: -1

y: 0

1 x - 1

32. lim

+

y: 2

36. lim

x: 4

x + 2

x: -2

x: 4

2x 2 + 5 - 3 4 - x

5 - 2x 2 + 9

lim sin2 x

=

7 4

(a)

lim 5hsxd 4x:1

A lim p(x) B A lim A 4 - r(x) B B lim hsxd 45x:1

A lim p(x) B A lim 4 - lim r (x) B x:1

x:1

2s5ds5d 5 = 2 s1ds4 - 2d

a. lim ƒsxdg sxd

b. lim 2ƒsxdg sxd

c. lim sƒsxd + 3g sxdd

d. lim

x:c

x:c

ƒsxd ƒsxd - g sxd

54. Suppose limx:4 ƒsxd = 0 and limx:4 g sxd = -3 . Find a. lim sg sxd + 3d

b. lim xƒsxd

c. lim sg sxdd2

d. lim

x:4

x:4

x:4

x:4

g sxd ƒsxd - 1

55. Suppose limx:b ƒsxd = 7 and limx:b g sxd = -3 . Find a. lim sƒsxd + g sxdd

b. lim ƒsxd # g sxd

c. lim 4g sxd

d. lim ƒsxd>g sxd

x:b x:b

lim tan x

x: p>3

48. lim (x 2 - 1)(2 - cos x)

x:b

x: 0

49. lim 2x + 4 cos (x + p) 50. lim 27 + sec x

a. lim spsxd + r sxd + ssxdd x: -2

b.

lim psxd # r sxd # ssxd

x: -2

c. lim s -4psxd + 5r sxdd>ssxd x: -2

(b)

x:1

x:b

56. Suppose that limx:-2 psxd = 4, limx:-2 r sxd = 0 , and limx:-2 ssxd = -3 . Find

x: p>4

x: 0

=

x:c

2

x: -p

s1 + 7d

2>3

x:1

x:c

2 - 2x

40. lim 42. lim

=

4x - x 2

2x 2 + 8 - 3 38. lim x + 1 x: -1

46.

x:0

(c)

x:0

s2ds1d - s -5d

53. Suppose limx:c ƒsxd = 5 and limx:c g sxd = -2 . Find

y - 8 y4 - 16

34. lim

45. lim sec x 1 + x + sin x 3 cos x

x:0

x:1

1 x + 1

3

44.

47. lim

=

x

x: 0

43. lim (2 sin x - 1) x:0

x:0

A lim ƒ(x) + lim 7 B 2>3

x:1

3y 4 - 16y 2

Limits with trigonometric functions Find the limits in Exercises 43–50. x:0

x:1

t 2 + 3t + 2 t2 - t - 2 5y 3 + 8y 2

4

33. lim

2 lim ƒsxd - lim g sxd

lim 25hsxd 25hsxd x:1 = psxds4 - rsxdd lim spsxds4 - rsxddd

x 2 - 7x + 10 26. lim x - 2 x: 2 28. lim

31. lim

(b)

52. Let limx:1 hsxd = 5, limx:1 psxd = 1 , and limx:1 r sxd = 2 . Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

x + 3 x 2 + 4x + 3

24. lim

t2 + t - 2 t:1 t2 - 1

27. lim

x:0

A lim A ƒsxd + 7 B B 2>3

y: 2

Limits of quotients Find the limits in Exercises 23–42. 23. lim

lim 2ƒsxd - lim g sxd

x:0

x:0

y + 2

18. lim

(a)

lim sƒsxd + 7d2>3

x:0

x: 2

s: 2>3

x: -1

21. lim

2

16. lim 3ss2s - 1d

17. lim 3s2x - 1d2

lim s2ƒsxd - g sxdd

x:0

x:0

=

11. lim s2x + 5d x: -7

=

(c)

2.2 Limits of Average Rates of Change Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form lim

h :0

ƒsx + hd - ƒsxd h

59. ƒsxd = 3x - 4, 61. ƒsxd = 2x,

x = 2

x = 7

x = -2

60. ƒsxd = 1>x,

x = -2

62. ƒsxd = 23x + 1,

x = 0

Using the Sandwich Theorem 63. If 25 - 2x 2 … ƒsxd … 25 - x 2 for limx:0 ƒsxd .

-1 … x … 1, find

64. If 2 - x 2 … g sxd … 2 cos x for all x, find limx:0 g sxd . 65. a. It can be shown that the inequalities x sin x x2 6 6 1 6 2 - 2 cos x hold for all values of x close to zero. What, if anything, does this tell you about 1 -

x sin x ? 2 - 2 cos x Give reasons for your answer. lim

x: 0

T b. Graph y = 1 - sx 2>6d, y = sx sin xd>s2 - 2 cos xd, and y = 1 together for -2 … x … 2 . Comment on the behavior of the graphs as x : 0 . 66. a. Suppose that the inequalities 1 - cos x x2 1 1 6 6 2 24 2 x2 hold for values of x close to zero. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about lim

x: 0

b. Support your conclusion in part (a) by graphing g near c = 22 and using Zoom and Trace to estimate y-values on the graph as x : 22 . 69. Let Gsxd = sx + 6d>sx 2 + 4x - 12d .

58. ƒsxd = x 2,

x = 1

69

c. Find limx:22 g sxd algebraically.

occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function ƒ. 57. ƒsxd = x 2,

Limit of a Function and Limit Laws

1 - cos x ? x2

Give reasons for your answer.

T b. Graph the equations y = s1>2d - sx 2>24d, y = s1 - cos xd>x 2 , and y = 1>2 together for -2 … x … 2 . Comment on the behavior of the graphs as x : 0 . Estimating Limits T You will find a graphing calculator useful for Exercises 67–76. 67. Let ƒsxd = sx 2 - 9d>sx + 3d . a. Make a table of the values of ƒ at the points x = -3.1, -3.01, -3.001 , and so on as far as your calculator can go. Then estimate limx:-3 ƒsxd . What estimate do you arrive at if you evaluate ƒ at x = -2.9, -2.99, -2.999, Á instead? b. Support your conclusions in part (a) by graphing ƒ near c = -3 and using Zoom and Trace to estimate y-values on the graph as x : -3 . c. Find limx:-3 ƒsxd algebraically, as in Example 7. 68. Let g sxd = sx 2 - 2d>(x - 22). a. Make a table of the values of g at the points x = 1.4, 1.41, 1.414 , and so on through successive decimal approximations of 22 . Estimate limx:22 g sxd .

a. Make a table of the values of G at x = -5.9, -5.99, - 5.999, and so on. Then estimate limx:-6 Gsxd . What estimate do you arrive at if you evaluate G at x = - 6.1, -6.01, -6.001, Á instead? b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x : -6 . c. Find limx:-6 Gsxd algebraically. 70. Let hsxd = sx 2 - 2x - 3d>sx 2 - 4x + 3d . a. Make a table of the values of h at x = 2.9, 2.99, 2.999, and so on. Then estimate limx:3 hsxd . What estimate do you arrive at if you evaluate h at x = 3.1, 3.01, 3.001, Á instead? b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x : 3. c. Find limx:3 hsxd algebraically. 71. Let ƒsxd = sx 2 - 1d>s ƒ x ƒ - 1d . a. Make tables of the values of ƒ at values of x that approach c = -1 from above and below. Then estimate limx:-1 ƒsxd . b. Support your conclusion in part (a) by graphing ƒ near c = -1 and using Zoom and Trace to estimate y-values on the graph as x : -1 . c. Find limx:-1 ƒsxd algebraically. 72. Let Fsxd = sx 2 + 3x + 2d>s2 - ƒ x ƒ d . a. Make tables of values of F at values of x that approach c = -2 from above and below. Then estimate limx:-2 Fsxd . b. Support your conclusion in part (a) by graphing F near c = -2 and using Zoom and Trace to estimate y-values on the graph as x : -2 . c. Find limx:-2 Fsxd algebraically. 73. Let g sud = ssin ud>u . a. Make a table of the values of g at values of u that approach u0 = 0 from above and below. Then estimate lim u : 0 g sud . b. Support your conclusion in part (a) by graphing g near u0 = 0 . 74. Let Gstd = s1 - cos td>t 2 . a. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate lim t:0 Gstd . b. Support your conclusion in part (a) by graphing G near t0 = 0 . 75. Let ƒsxd = x 1>s1 - xd . a. Make tables of values of ƒ at values of x that approach c = 1 from above and below. Does ƒ appear to have a limit as x : 1 ? If so, what is it? If not, why not? b. Support your conclusions in part (a) by graphing ƒ near c = 1.

70

Chapter 2: Limits and Continuity

76. Let ƒsxd = s3x - 1d>x . a. Make tables of values of ƒ at values of x that approach c = 0 from above and below. Does ƒ appear to have a limit as x : 0 ? If so, what is it? If not, why not?

b. If lim

x:2

82. If lim

Theory and Examples 77. If x 4 … ƒsxd … x 2 for x in [-1, 1] and x 2 … ƒsxd … x 4 for x 6 -1 and x 7 1 , at what points c do you automatically know limx:c ƒsxd ? What can you say about the value of the limit at these points? 78. Suppose that g sxd … ƒsxd … hsxd for all x Z 2 and suppose that

ƒsxd x2

x: -2

= 1 , find b. lim

x: -2

x:0

b. Confirm your estimate in part (a) with a proof. T 84. a. Graph hsxd = x 2 cos s1>x 3 d to estimate limx:0 hsxd , zooming in on the origin as necessary. b. Confirm your estimate in part (a) with a proof. COMPUTER EXPLORATIONS Graphical Estimates of Limits In Exercises 85–90, use a CAS to perform the following steps: a. Plot the function near the point c being approached. b. From your plot guess the value of the limit.

ƒsxd - 5 = 3 , find lim ƒsxd . x: 2 x - 2 x: 2

x:2

x 4 - 16 x - 2

86. lim

x: -1

3 2 1 + x - 1 87. lim x x:0

81. a. If lim

89. lim

x:0

2.3

ƒsxd x:0 x

b. lim

T 83. a. Graph g sxd = x sin s1>xd to estimate limx:0 g sxd , zooming in on the origin as necessary.

85. lim ƒsxd x: -2 x

a. lim ƒsxd

= 1 , find

x: 2

Can we conclude anything about the values of ƒ, g, and h at x = 2 ? Could ƒs2d = 0 ? Could limx:2 ƒsxd = 0 ? Give reasons for your answers. ƒsxd - 5 = 1 , find lim ƒsxd . 79. If lim x:4 x - 2 x: 4 80. If lim

x2

a. lim ƒsxd

lim g sxd = lim hsxd = -5 .

x:2

ƒsxd

x:0

b. Support your conclusions in part (a) by graphing ƒ near c = 0

ƒsxd - 5 = 4 , find lim ƒsxd . x - 2 x:2

1 - cos x x sin x

88. lim

x 3 - x 2 - 5x - 3 sx + 1d2 x2 - 9

2x 2 + 7 - 4 2x 2 90. lim x:0 3 - 3 cos x x:3

The Precise Definition of a Limit We now turn our attention to the precise definition of a limit. We replace vague phrases like “gets arbitrarily close to” in the informal definition with specific conditions that can be applied to any particular example. With a precise definition, we can prove the limit properties given in the preceding section and establish many important limits. To show that the limit of ƒ(x) as x : c equals the number L, we need to show that the gap between ƒ(x) and L can be made “as small as we choose” if x is kept “close enough” to c. Let us see what this would require if we specified the size of the gap between ƒ(x) and L.

y

Upper bound: y9

⎧9 To satisfy ⎪ ⎨7 this ⎪ ⎩5

Lower bound: y5

3 4 5

x

⎧ ⎨ ⎩

0

Consider the function y = 2x - 1 near x = 4. Intuitively it appears that y is close to 7 when x is close to 4, so limx:4 s2x - 1d = 7. However, how close to x = 4 does x have to be so that y = 2x - 1 differs from 7 by, say, less than 2 units?

EXAMPLE 1

y 2x 1

Restrict to this

FIGURE 2.15 Keeping x within 1 unit of x = 4 will keep y within 2 units of y = 7 (Example 1).

We are asked: For what values of x is ƒ y - 7 ƒ 6 2? To find the answer we first express ƒ y - 7 ƒ in terms of x: Solution

ƒ y - 7 ƒ = ƒ s2x - 1d - 7 ƒ = ƒ 2x - 8 ƒ . The question then becomes: what values of x satisfy the inequality ƒ 2x - 8 ƒ 6 2? To find out, we solve the inequality: ƒ 2x - 8 ƒ -2 6 3

6 6 6 6

2 2x - 8 6 2 2x 6 10 x 6 5

-1 6 x - 4 6 1.

Solve for x. Solve for x 4.

Keeping x within 1 unit of x = 4 will keep y within 2 units of y = 7 (Figure 2.15).

2.3

1 10

f (x) f(x) lies in here

L L

1 10

Definition of Limit for all x c in here

x

c

0

c

c

x

FIGURE 2.16 How should we define d 7 0 so that keeping x within the interval sc - d, c + dd will keep ƒ(x) within the 1 1 ,L + b? interval aL 10 10

y

L L

71

In the previous example we determined how close x must be to a particular value c to ensure that the outputs ƒ(x) of some function lie within a prescribed interval about a limit value L. To show that the limit of ƒ(x) as x : c actually equals L, we must be able to show that the gap between ƒ(x) and L can be made less than any prescribed error, no matter how small, by holding x close enough to c.

y

L

The Precise Definition of a Limit

f(x)

f(x) lies in here

Suppose we are watching the values of a function ƒ(x) as x approaches c (without taking on the value of c itself ). Certainly we want to be able to say that ƒ(x) stays within onetenth of a unit from L as soon as x stays within some distance d of c (Figure 2.16). But that in itself is not enough, because as x continues on its course toward c, what is to prevent ƒ(x) from jittering about within the interval from L - (1>10) to L + (1>10) without tending toward L? We can be told that the error can be no more than 1>100 or 1>1000 or 1>100,000. Each time, we find a new d-interval about c so that keeping x within that interval satisfies the new error tolerance. And each time the possibility exists that ƒ(x) jitters away from L at some stage. The figures on the next page illustrate the problem. You can think of this as a quarrel between a skeptic and a scholar. The skeptic presents P-challenges to prove that the limit does not exist or, more precisely, that there is room for doubt. The scholar answers every challenge with a d-interval around c that keeps the function values within P of L. How do we stop this seemingly endless series of challenges and responses? By proving that for every error tolerance P that the challenger can produce, we can find, calculate, or conjure a matching distance d that keeps x “close enough” to c to keep ƒ(x) within that tolerance of L (Figure 2.17). This leads us to the precise definition of a limit.

L

DEFINITION Let ƒ(x) be defined on an open interval about c, except possibly at c itself. We say that the limit of ƒ(x) as x approaches c is the number L, and write

for all x c in here x 0

c

lim ƒsxd = L,

x c

c

FIGURE 2.17 The relation of d and P in the definition of limit.

x:c

if, for every number P 7 0, there exists a corresponding number d 7 0 such that for all x, 0 6 ƒx - cƒ 6 d

Q

ƒ ƒsxd - L ƒ 6 P.

One way to think about the definition is to suppose we are machining a generator shaft to a close tolerance. We may try for diameter L, but since nothing is perfect, we must be satisfied with a diameter ƒ(x) somewhere between L - P and L + P. The d is the measure of how accurate our control setting for x must be to guarantee this degree of accuracy in the diameter of the shaft. Notice that as the tolerance for error becomes stricter, we may have to adjust d. That is, the value of d, how tight our control setting must be, depends on the value of P, the error tolerance.

Examples: Testing the Definition The formal definition of limit does not tell how to find the limit of a function, but it enables us to verify that a suspected limit is correct. The following examples show how the definition can be used to verify limit statements for specific functions. However, the real purpose of the definition is not to do calculations like this, but rather to prove general theorems so that the calculation of specific limits can be simplified.

72

Chapter 2: Limits and Continuity

y

L

y f(x)

1 10

L

L

c c 1/10 c 1/10 Response: x c 1/10 (a number) 0

The challenge: Make f(x) – L 1 10

1 100 L 1 L 100

1 100

L

L 1 L 100

1 10

x

c

0

x

New challenge: Make f (x) – L 1 100

y f (x)

y f (x) L

1 1000

L

L

1 1000

L

1 1000

x

c

0

x

c

0

New challenge: 1 1000

Response: x c 1/1000

y

y y f (x) 1 L 100,000

L

L

1 100,000

L

c New challenge: 1 100,000

y y f (x)

1 L 100,000

0

0

y

1 L 1000

L

x c c 1/100 c 1/100 Response: x c 1/100

x

c

0

y

L

y f(x)

y f (x)

L

1 10

y

y f (x)

1 L 10

L L

y

y

y f (x) L L L

1 100,000

x 0

c

x

Response: x c 1/100,000

EXAMPLE 2

c

0

x

New challenge: ...

Show that lim s5x - 3d = 2.

x:1

Solution Set c = 1, ƒsxd = 5x - 3, and L = 2 in the definition of limit. For any given P 7 0, we have to find a suitable d 7 0 so that if x Z 1 and x is within distance d of c = 1, that is, whenever

0 6 ƒ x - 1 ƒ 6 d, it is true that ƒ(x) is within distance P of L = 2, so ƒ ƒsxd - 2 ƒ 6 P.

2.3

y

The Precise Definition of a Limit

73

We find d by working backward from the P-inequality:

y 5x 3

ƒ s5x - 3d - 2 ƒ = ƒ 5x - 5 ƒ 6 P

2

5ƒx - 1ƒ 6 P

2

ƒ x - 1 ƒ 6 P>5.

2

Thus, we can take d = P>5 (Figure 2.18). If 0 6 ƒ x - 1 ƒ 6 d = P>5, then 1 1 1 5 5

0

ƒ s5x - 3d - 2 ƒ = ƒ 5x - 5 ƒ = 5 ƒ x - 1 ƒ 6 5sP>5d = P,

x

which proves that limx:1s5x - 3d = 2. The value of d = P>5 is not the only value that will make 0 6 ƒ x - 1 ƒ 6 d imply 5x - 5 ƒ 6 P. Any smaller positive d will do as well. The definition does not ask for a ƒ “best” positive d, just one that will work.

–3

EXAMPLE 3

NOT TO SCALE

(a) lim x = c

FIGURE 2.18 If ƒsxd = 5x - 3 , then 0 6 ƒ x - 1 ƒ 6 P>5 guarantees that ƒ ƒsxd - 2 ƒ 6 P (Example 2).

x:c

Prove the following results presented graphically in Section 2.2. (b) lim k = k

(k constant)

x:c

Solution

(a) Let P 7 0 be given. We must find d 7 0 such that for all x 0 6 ƒx - cƒ 6 d

implies

ƒ x - c ƒ 6 P.

The implication will hold if d equals P or any smaller positive number (Figure 2.19). This proves that limx:c x = c. (b) Let P 7 0 be given. We must find d 7 0 such that for all x

y yx c c c c

0 6 ƒx - cƒ 6 d

implies

ƒ k - k ƒ 6 P.

Since k - k = 0, we can use any positive number for d and the implication will hold (Figure 2.20). This proves that limx:c k = k.

c

c c c

0

x

FIGURE 2.19 For the function ƒsxd = x , we find that 0 6 ƒ x - c ƒ 6 d will guarantee ƒ ƒsxd - c ƒ 6 P whenever d … P (Example 3a).

Finding Deltas Algebraically for Given Epsilons In Examples 2 and 3, the interval of values about c for which ƒ ƒsxd - L ƒ was less than P was symmetric about c and we could take d to be half the length of that interval. When such symmetry is absent, as it usually is, we can take d to be the distance from c to the interval’s nearer endpoint.

EXAMPLE 4 For the limit limx:5 2x - 1 = 2, find a d 7 0 that works for P = 1. That is, find a d 7 0 such that for all x 0 6 ƒx - 5ƒ 6 d

y yk

k k k

Solution

1.

Q

ƒ 2x - 1 - 2 ƒ 6 1.

We organize the search into two steps.

Solve the inequality ƒ 2x - 1 - 2 ƒ 6 1 to find an interval containing x = 5 on which the inequality holds for all x Z 5. ƒ 2x - 1 - 2 ƒ 6 1

0

c

c

c

x

FIGURE 2.20 For the function ƒsxd = k , we find that ƒ ƒsxd - k ƒ 6 P for any positive d (Example 3b).

-1 6 2x - 1 - 2 6 1 1 6 2x - 1 6 3 1 6 x - 1 6 9 2 6 x 6 10

74 (

Chapter 2: Limits and Continuity

3

3

2

( 10

8

5

x

2. FIGURE 2.21 An open interval of radius 3 about x = 5 will lie inside the open interval (2, 10).

The inequality holds for all x in the open interval (2, 10), so it holds for all x Z 5 in this interval as well. Find a value of d 7 0 to place the centered interval 5 - d 6 x 6 5 + d (centered at x = 5) inside the interval (2, 10). The distance from 5 to the nearer endpoint of (2, 10) is 3 (Figure 2.21). If we take d = 3 or any smaller positive number, then the inequality 0 6 ƒ x - 5 ƒ 6 d will automatically place x between 2 and 10 to make ƒ 2x - 1 - 2 ƒ 6 1 (Figure 2.22): 0 6 ƒx - 5ƒ 6 3

y

Q

ƒ 2x - 1 - 2 ƒ 6 1.

y x 1 3

How to Find Algebraically a D for a Given ƒ, L, c, and P>0 The process of finding a d 7 0 such that for all x

2

0 6 ƒx - cƒ 6 d

ƒ ƒsxd - L ƒ 6 P

can be accomplished in two steps.

1 3 0

Q

1 2

1. Solve the inequality ƒ ƒsxd - L ƒ 6 P to find an open interval (a, b) containing c on which the inequality holds for all x Z c.

3 5

8

x

10

2. Find a value of d 7 0 that places the open interval sc - d, c + dd centered at c inside the interval (a, b). The inequality ƒ ƒsxd - L ƒ 6 P will hold for all x Z c in this d-interval.

NOT TO SCALE

FIGURE 2.22 The function and intervals in Example 4.

EXAMPLE 5

Prove that limx:2 ƒsxd = 4 if ƒsxd = e

Solution

1.

y x2 4

x Z 2 x = 2.

Our task is to show that given P 7 0 there exists a d 7 0 such that for all x 0 6 ƒx - 2ƒ 6 d

y

x 2, 1,

Q

ƒ ƒsxd - 4 ƒ 6 P.

Solve the inequality ƒ ƒsxd - 4 ƒ 6 P to find an open interval containing x = 2 on which the inequality holds for all x Z 2. For x Z c = 2, we have ƒsxd = x 2 , and the inequality to solve is ƒ x 2 - 4 ƒ 6 P: ƒ x2 - 4 ƒ 6 P

(2, 4)

4

-P 6 x 2 - 4 6 P 4

4 - P 6 x2 6 4 + P 24 - P 6 ƒ x ƒ 6 24 + P

(2, 1) 0

4

2

24 - P 6 x 6 24 + P.

x

The inequality ƒ ƒsxd - 4 ƒ 6 P holds for all x Z 2 in the open interval A 24 - P, 24 + P B (Figure 2.23).

4

FIGURE 2.23 An interval containing x = 2 so that the function in Example 5 satisfies ƒ ƒsxd - 4 ƒ 6 P .

Assumes P 6 4 ; see below. An open interval about x = 2 that solves the inequality

2.

Find a value of d 7 0 that places the centered interval s2 - d, 2 + dd inside the interval A 24 - P, 24 + P B . Take d to be the distance from x = 2 to the nearer endpoint of A 24 - P, 24 + P B .

In other words, take d = min E 2 - 24 - P, 24 + P - 2 F , the minimum (the

2.3

The Precise Definition of a Limit

75

smaller) of the two numbers 2 - 24 - P and 24 + P - 2. If d has this or any smaller positive value, the inequality 0 6 ƒ x - 2 ƒ 6 d will automatically place x between 24 - P and 24 + P to make ƒ ƒsxd - 4 ƒ 6 P. For all x, 0 6 ƒx - 2ƒ 6 d

Q

ƒ ƒsxd - 4 ƒ 6 P.

This completes the proof for P 6 4. If P Ú 4, then we take d to be the distance from x = 2 to the nearer endpoint of the interval A 0, 24 + P B . In other words, take d = min E 2, 24 + P - 2 F . (See Figure 2.23.)

Using the Definition to Prove Theorems We do not usually rely on the formal definition of limit to verify specific limits such as those in the preceding examples. Rather, we appeal to general theorems about limits, in particular the theorems of Section 2.2. The definition is used to prove these theorems (Appendix 5). As an example, we prove part 1 of Theorem 1, the Sum Rule.

EXAMPLE 6

Given that limx:c ƒsxd = L and limx:c gsxd = M, prove that lim sƒsxd + gsxdd = L + M.

x:c

Solution

Let P 7 0 be given. We want to find a positive number d such that for all x 0 6 ƒx - cƒ 6 d

Q

ƒ ƒsxd + gsxd - sL + Md ƒ 6 P.

Regrouping terms, we get ƒ ƒsxd + gsxd - sL + Md ƒ = ƒ sƒsxd - Ld + sgsxd - Md ƒ … ƒ ƒsxd - L ƒ + ƒ gsxd - M ƒ .

Triangle Inequality: ƒa + bƒ … ƒaƒ + ƒbƒ

Since limx:c ƒsxd = L, there exists a number d1 7 0 such that for all x 0 6 ƒ x - c ƒ 6 d1

Q

ƒ ƒsxd - L ƒ 6 P>2.

Similarly, since limx:c gsxd = M, there exists a number d2 7 0 such that for all x 0 6 ƒ x - c ƒ 6 d2

Q

ƒ gsxd - M ƒ 6 P>2.

Let d = min 5d1, d26, the smaller of d1 and d2 . If 0 6 ƒ x - c ƒ 6 d then ƒ x - c ƒ 6 d1 , so ƒ ƒsxd - L ƒ 6 P>2, and ƒ x - c ƒ 6 d2 , so ƒ gsxd - M ƒ 6 P>2. Therefore P P ƒ ƒsxd + gsxd - sL + Md ƒ 6 2 + 2 = P. This shows that limx:c sƒsxd + gsxdd = L + M. Next we prove Theorem 5 of Section 2.2.

EXAMPLE 7 Given that limx:c ƒsxd = L and limx:c gsxd = M, and that ƒsxd … gsxd for all x in an open interval containing c (except possibly c itself), prove that L … M. We use the method of proof by contradiction. Suppose, on the contrary, that L 7 M. Then by the limit of a difference property in Theorem 1,

Solution

lim s gsxd - ƒsxdd = M - L.

x:c

76

Chapter 2: Limits and Continuity

Therefore, for any P 7 0, there exists d 7 0 such that ƒ sgsxd - ƒsxdd - sM - Ld ƒ 6 P

0 6 ƒ x - c ƒ 6 d.

whenever

Since L - M 7 0 by hypothesis, we take P = L - M in particular and we have a number d 7 0 such that ƒ sg sxd - ƒsxdd - sM - Ld ƒ 6 L - M

whenever

0 6 ƒ x - c ƒ 6 d.

whenever

0 6 ƒx - cƒ 6 d

Since a … ƒ a ƒ for any number a, we have sgsxd - ƒsxdd - sM - Ld 6 L - M which simplifies to gsxd 6 ƒsxd

whenever

0 6 ƒ x - c ƒ 6 d.

But this contradicts ƒsxd … gsxd. Thus the inequality L 7 M must be false. Therefore L … M.

Exercises 2.3 Centering Intervals About a Point In Exercises 1–6, sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of d 7 0 such that for all x, 0 6 ƒ x - c ƒ 6 d Q a 6 x 6 b . 1. a = 1,

b = 7,

c = 5

2. a = 1,

b = 7,

c = 2

3. a = -7>2, 4. a = -7>2,

b = -1>2,

c = -3

b = -1>2,

c = -3>2

b = 4>7,

5. a = 4>9,

y 5 4 1 3 4

10. f (x) x c1 L1 1 y x 4

c = 3

0

y f (x) 2x 1 c3 L4 0.2 y 2x 1

4.2 4 3.8

c = 1>2

b = 3.2391,

6. a = 2.7591,

9.

1

9 16

x

25 16

2

–1 0

Finding Deltas Graphically In Exercises 7–14, use the graphs to find a d 7 0 such that for all x 0 6 ƒ x - c ƒ 6 d Q ƒ ƒsxd - L ƒ 6 P . 7.

12.

y 2x 4 f(x) 2x 4 c5 L6 0.2

6.2 6 5.8

0

f (x) 4 x 2 c –1 L3 0.25

f (x) x 2 c2 L4 1 y x2

y –3 x 3 2

y 4 x2

5 4

7.65 7.5 7.35

x

5 4.9

y

y

y

f (x) – 3 x 3 2 c –3 L 7.5 0.15

x

NOT TO SCALE

11.

8.

y

2.61 3 3.41

3.25 3 2.75

3 0

5.1

3

2

x 5

NOT TO SCALE NOT TO SCALE

–3.1

–3

–2.9

0

NOT TO SCALE

x –

5 –1 3 – 2 2 NOT TO SCALE

0

x

2.3 13.

y

2 –x c –1 L2 0.5

f(x)

y

x:c

f (x) 1x c1 2 L2 0.01

2.01

2 –x

that for all x 0 6 ƒx - cƒ 6 d 31. ƒsxd = 3 - 2 x,

2 2.5

1.99 y 1x

1.5

x

0

x

1 1 1 2 2.01 1.99

0

33. ƒsxd =

x2 - 4 , x - 2

34. ƒsxd =

x2 + 6x + 5 , x + 5

NOT TO SCALE

36. ƒsxd = 4>x,

Each of Exercises 15–30 gives a function ƒ(x) and numbers L, c, and P 7 0 . In each case, find an open interval about c on which the inequality ƒ ƒsxd - L ƒ 6 P holds. Then give a value for d 7 0 such that for all x satisfying 0 6 ƒ x - c ƒ 6 d the inequality ƒ ƒsxd - L ƒ 6 P holds.

16. ƒsxd = 2x - 2,

L = -6,

17. ƒsxd = 2x + 1, 18. ƒsxd = 2x,

L = 1, L = 1>2,

19. ƒsxd = 219 - x, 20. ƒsxd = 2x - 7, 21. ƒsxd = 1>x,

L = 3, L = 4,

L = 1>4,

P = 0.1

c = 1>4,

P = 0.1

c = 10,

c = -3,

c = 2,

L = 3,

c = 23,

23. ƒsxd = x2,

L = 4,

c = -2,

25. ƒsxd = x2 - 5, 26. ƒsxd = 120>x,

L = 11, L = 5,

c = -1, c = 4, c = 24,

P = 0.4

39. lim 2x - 5 = 2

40. lim 24 - x = 2

P = 1 P = 0.05

x:1

42. lim ƒsxd = 4 x: -2

x:3

x:0

ƒsxd = e

if

x Z -2 x = -2

x 2, 1,

44.

x2 - 9 = -6 x + 3

x Z 1 x = 1

x 2, 2,

ƒsxd = e

if

1 43. lim x = 1 x:1

lim

x: 23

46. lim

x:1

47. lim ƒsxd = 2

if

ƒsxd = e

4 - 2x, 6x - 4,

48. lim ƒsxd = 0

if

ƒsxd = e

2x, x>2,

x:0

x 6 1 x Ú 1 x 6 0 x Ú 0

x:0

P = 0.5

y

P = 0.1 P = 1 P = 1 c = 2,

P = 0.03

28. ƒsxd = mx,

m 7 0,

L = 3m,

c = 3,

P = c 7 0

L = sm>2d + b, L = m + b,

c = 1,

– 1 2 1 –

y x sin 1x

1 2 1

1 1 = 3 x2

x2 - 1 = 2 x - 1

1 49. lim x sin x = 0

P = 0.1

L = 2m,

m 7 0,

41. lim ƒsxd = 1

x:1

m 7 0,

30. ƒsxd = mx + b, P = 0.05

P = 0.5

38. lim s3x - 7d = 2

x: -3

27. ƒsxd = mx,

m 7 0, 29. ƒsxd = mx + b, c = 1>2, P = c 7 0

P = 0.05

37. lim s9 - xd = 5

45. lim

P = 1

c = 23, c = 4,

L = -1,

P = 0.02

c = 0,

22. ƒsxd = x2,

24. ƒsxd = 1>x,

P = 0.01

c = -2,

P = 0.05

c = -5,

x:9

c = 4,

P = 0.03

Prove the limit statements in Exercises 37–50.

Finding Deltas Algebraically

L = 5,

P = 0.02

c = 2,

x:4

15. ƒsxd = x + 1,

ƒ ƒsxd - L ƒ 6 P .

c = -1,

35. ƒsxd = 21 - 5x, –1 – 16 25

Q

c = 3,

32. ƒsxd = -3x - 2,

2

–16 9

77

Using the Formal Definition Each of Exercises 31–36 gives a function ƒ(x), a point c, and a positive number P . Find L = lim ƒsxd. Then find a number d 7 0 such

14. y

The Precise Definition of a Limit

x

78

Chapter 2: Limits and Continuity When Is a Number L Not the Limit of ƒ(x) as x : c?

1 50. lim x 2 sin x = 0

Showing L is not a limit We can prove that limx:c ƒsxd Z L by providing an P 7 0 such that no possible d 7 0 satisfies the condition

x:0

y y x2

1

for all x,

0 6 ƒx - cƒ 6 d

Q

ƒ ƒsxd - L ƒ 6 P .

We accomplish this for our candidate P by showing that for each d 7 0 there exists a value of x such that y x 2 sin 1x –1

0

2 –

2

0 6 ƒx - cƒ 6 d

and

ƒ ƒsxd - L ƒ Ú P .

x

1

y y f (x) L

–1

y –x 2

L L

Theory and Examples

f (x)

51. Define what it means to say that lim g sxd = k . x: 0

52. Prove that lim ƒsxd = L if and only if lim ƒsh + cd = L . h :0

x:c

0

53. A wrong statement about limits Show by example that the following statement is wrong.

57. Let ƒsxd = e

y

The number L is the limit of ƒ(x) as x approaches c if, given any P 7 0 , there exists a value of x for which ƒ ƒsxd - L ƒ 6 P.

yx1

Explain why the function in your example does not have the given value of L as a limit as x : c .

2

T 55. Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9 in2 , you need to know how much deviation from the ideal cylinder diameter of c = 3.385 in. you can allow and still have the area come within 0.01 in2 of the required 9 in2 . To find out, you let A = psx>2d2 and look for the interval in which you must hold x to make ƒ A - 9 ƒ … 0.01 . What interval do you find?

1

y f (x)

I

R

x

1 yx

a. Let P = 1>2 . Show that no possible d 7 0 satisfies the following condition: For all x,

0 6 ƒx - 1ƒ 6 d

Q

ƒ ƒsxd - 2 ƒ 6 1>2.

That is, for each d 7 0 show that there is a value of x such that 0 6 ƒx - 1ƒ 6 d

V

x

c

x, x 6 1 x + 1, x 7 1.

54. Another wrong statement about limits Show by example that the following statement is wrong.

56. Manufacturing electrical resistors Ohm’s law for electrical circuits like the one shown in the accompanying figure states that V = RI . In this equation, V is a constant voltage, I is the current in amperes, and R is the resistance in ohms. Your firm has been asked to supply the resistors for a circuit in which V will be 120 volts and I is to be 5 ; 0.1 amp . In what interval does R have to lie for I to be within 0.1 amp of the value I0 = 5 ?

c

a value of x for which 0 x c and f (x) L

The number L is the limit of ƒ(x) as x approaches c if ƒ(x) gets closer to L as x approaches c. Explain why the function in your example does not have the given value of L as a limit as x : c .

c

and

ƒ ƒsxd - 2 ƒ Ú 1>2.

This will show that limx:1 ƒsxd Z 2 . b. Show that limx:1 ƒsxd Z 1 . c. Show that limx:1 ƒsxd Z 1.5 .

2.4 x 6 2 x = 2 x 7 2.

x 2, 58. Let hsxd = • 3, 2,

One-Sided Limits

79

y

2

y y g(x) 1

y h(x)

4 3

y2

2 1

y x2

0

2

–1

x

0

x

COMPUTER EXPLORATIONS In Exercises 61–66, you will further explore finding deltas graphically. Use a CAS to perform the following steps: a. Plot the function y = ƒsxd near the point c being approached.

Show that

b. Guess the value of the limit L and then evaluate the limit symbolically to see if you guessed correctly.

a. lim hsxd Z 4 x: 2

b. lim hsxd Z 3

c. Using the value P = 0.2 , graph the banding lines y1 = L - P and y2 = L + P together with the function ƒ near c.

x: 2

c. lim hsxd Z 2 x: 2

d. From your graph in part (c), estimate a d 7 0 such that for all x

59. For the function graphed here, explain why

0 6 ƒx - cƒ 6 d Q ƒ ƒsxd - L ƒ 6 P . Test your estimate by plotting ƒ, y1 , and y2 over the interval 0 6 ƒ x - c ƒ 6 d . For your viewing window use c - 2d … x … c + 2d and L - 2P … y … L + 2P . If any function values lie outside the interval [L - P, L + P] , your choice of d was too large. Try again with a smaller estimate.

a. lim ƒsxd Z 4 x: 3

b. lim ƒsxd Z 4.8 x: 3

c. lim ƒsxd Z 3 x: 3

y

4.8

61.

4

y f (x)

3

62. 63.

0

x

3

60. a. For the function graphed here, show that limx : -1 g sxd Z 2 . b. Does limx : -1 g sxd appear to exist? If so, what is the value of the limit? If not, why not?

2.4

64.

e. Repeat parts (c) and (d) successively for P = 0.1, 0.05 , and 0.001. x4 - 81 ƒsxd = , c = 3 x - 3 5x3 + 9x2 ƒsxd = 5 , c = 0 2x + 3x2 sin 2x ƒsxd = , c = 0 3x xs1 - cos xd ƒsxd = , c = 0 x - sin x

65. ƒsxd =

3 x - 1 2 , x - 1

66. ƒsxd =

3x - s7x + 1d2x + 5 , x - 1

c = 1

2

c = 1

One-Sided Limits In this section we extend the limit concept to one-sided limits, which are limits as x approaches the number c from the left-hand side (where x 6 c) or the right-hand side sx 7 cd only.

Approaching a Limit from One Side To have a limit L as x approaches c, a function ƒ must be defined on both sides of c and its values ƒ(x) must approach L as x approaches c from either side. Because of this, ordinary limits are called two-sided.

80

Chapter 2: Limits and Continuity

y y x x 1

x

0

–1

If ƒ fails to have a two-sided limit at c, it may still have a one-sided limit, that is, a limit if the approach is only from one side. If the approach is from the right, the limit is a right-hand limit. From the left, it is a left-hand limit. The function ƒsxd = x> ƒ x ƒ (Figure 2.24) has limit 1 as x approaches 0 from the right, and limit -1 as x approaches 0 from the left. Since these one-sided limit values are not the same, there is no single number that ƒ(x) approaches as x approaches 0. So ƒ(x) does not have a (two-sided) limit at 0. Intuitively, if ƒ(x) is defined on an interval (c, b), where c 6 b, and approaches arbitrarily close to L as x approaches c from within that interval, then ƒ has right-hand limit L at c. We write lim ƒsxd = L.

x:c +

c+ ”

FIGURE 2.24 Different right-hand and left-hand limits at the origin.

The symbol “x : means that we consider only values of x greater than c. Similarly, if ƒ(x) is defined on an interval (a, c), where a 6 c and approaches arbitrarily close to M as x approaches c from within that interval, then ƒ has left-hand limit M at c. We write lim ƒsxd = M.

x:c -

The symbol “x : c - ” means that we consider only x-values less than c. These informal definitions of one-sided limits are illustrated in Figure 2.25. For the function ƒsxd = x> ƒ x ƒ in Figure 2.24 we have lim ƒsxd = 1

and

x:0 +

lim ƒsxd = -1.

x:0 -

y

y

f (x)

L 0

c

M

f (x) x

x

x

0

(a) lim+ f (x) L

(b)

x: c

c

x

lim _ f (x) M

x: c

FIGURE 2.25 (a) Right-hand limit as x approaches c. (b) Left-hand limit as x approaches c. y

The domain of ƒsxd = 24 - x 2 is [-2, 2]; its graph is the semicircle in Figure 2.26. We have

EXAMPLE 1 y 4 x 2

lim 24 - x 2 = 0

x: -2 +

–2

0

2

x

FIGURE 2.26 The function f (x) = 24 - x2 has right-hand limit 0 at x = -2 and left-hand limit 0 at x = 2 (Example 1).

and

lim 24 - x 2 = 0.

x:2 -

The function does not have a left-hand limit at x = -2 or a right-hand limit at x = 2. It does not have ordinary two-sided limits at either -2 or 2. One-sided limits have all the properties listed in Theorem 1 in Section 2.2. The righthand limit of the sum of two functions is the sum of their right-hand limits, and so on. The theorems for limits of polynomials and rational functions hold with one-sided limits, as do the Sandwich Theorem and Theorem 5. One-sided limits are related to limits in the following way. THEOREM 6 A function ƒ(x) has a limit as x approaches c if and only if it has left-hand and right-hand limits there and these one-sided limits are equal: lim ƒsxd = L

x:c

3

lim ƒsxd = L

x:c -

and

lim ƒsxd = L.

x:c +

2.4 y

EXAMPLE 2

1

limx:0+ ƒsxd = 1, limx:0- ƒsxd and limx:0 ƒsxd do not exist. The function is not defined to the left of x = 0. limx:1- ƒsxd = 0 even though ƒs1d = 1, limx:1+ ƒsxd = 1, limx:1 ƒsxd does not exist. The right- and left-hand limits are not equal. limx:2- ƒsxd = 1, limx:2+ ƒsxd = 1, limx:2 ƒsxd = 1 even though ƒs2d = 2. limx:3- ƒsxd = limx:3+ ƒsxd = limx:3 ƒsxd = ƒs3d = 2. limx:4- ƒsxd = 1 even though ƒs4d Z 1, limx:4+ ƒsxd and limx:4 ƒsxd do not exist. The function is not defined to the right of x = 4.

At x = 1:

0

1

2

3

x

4

FIGURE 2.27 Graph of the function in Example 2.

At x = 2:

At x = 3: At x = 4:

y

81

For the function graphed in Figure 2.27,

At x = 0:

y f (x)

2

One-Sided Limits

At every other point c in [0, 4], ƒ(x) has limit ƒ(c).

L

f(x)

Precise Definitions of One-Sided Limits

f(x) lies in here

L

The formal definition of the limit in Section 2.3 is readily modified for one-sided limits.

L for all x c in here

DEFINITIONS

x 0

c

c

We say that ƒ(x) has right-hand limit L at c, and write lim ƒsxd = L

x

(see Figure 2.28)

x:c +

if for every number P 7 0 there exists a corresponding number d 7 0 such that for all x

FIGURE 2.28 Intervals associated with the definition of right-hand limit.

c 6 x 6 c + d

Q

ƒ ƒsxd - L ƒ 6 P.

We say that ƒ has left-hand limit L at c, and write y

lim ƒsxd = L

x:c -

L L

if for every number P 7 0 there exists a corresponding number d 7 0 such that for all x c - d 6 x 6 c Q ƒ ƒsxd - L ƒ 6 P.

f(x) f(x) lies in here

EXAMPLE 3

Prove that

L

lim 2x = 0.

x:0 +

for all x c in here

Let P 7 0 be given. Here c = 0 and L = 0, so we want to find a d 7 0 such that for all x

Solution

x 0

(see Figure 2.29)

c

x

0 6 x 6 d

c

FIGURE 2.29 Intervals associated with the definition of left-hand limit.

Q

ƒ 2x - 0 ƒ 6 P,

or 0 6 x 6 d

Q

2x 6 P.

82

Chapter 2: Limits and Continuity

Squaring both sides of this last inequality gives

y

x 6 P2

f(x) x

If we choose d = P2 we have

Q

0 6 x 6 d = P2

f(x)

L0

0 6 x 6 d.

if

x

FIGURE 2.30

2

x

lim 1x = 0 in Example 3.

x: 0 +

2x 6 P,

or 0 6 x 6 P2

Q

ƒ 2x - 0 ƒ 6 P.

According to the definition, this shows that limx:0+ 2x = 0 (Figure 2.30). The functions examined so far have had some kind of limit at each point of interest. In general, that need not be the case. Show that y = sin s1>xd has no limit as x approaches zero from either side (Figure 2.31).

EXAMPLE 4

y 1

x

0

y sin 1x –1

FIGURE 2.31 The function y = sin s1>xd has neither a righthand nor a left-hand limit as x approaches zero (Example 4). The graph here omits values very near the y-axis. Solution As x approaches zero, its reciprocal, 1>x, grows without bound and the values of sin (1>x) cycle repeatedly from -1 to 1. There is no single number L that the function’s values stay increasingly close to as x approaches zero. This is true even if we restrict x to positive values or to negative values. The function has neither a right-hand limit nor a lefthand limit at x = 0.

Limits Involving (sin U)/U A central fact about ssin ud>u is that in radian measure its limit as u : 0 is 1. We can see this in Figure 2.32 and confirm it algebraically using the Sandwich Theorem. You will see the importance of this limit in Section 3.5, where instantaneous rates of change of the trigonometric functions are studied. y 1

–3

–2

y sin (radians)

–

2

3

NOT TO SCALE

FIGURE 2.32 The graph of ƒsud = ssin ud>u suggests that the rightand left-hand limits as u approaches 0 are both 1.

2.4

83

One-Sided Limits

y

THEOREM 7—Limit of the ratio sin U>U as U : 0 sin u lim = 1 su in radiansd u:0 u

T 1

(1)

P

tan

Proof The plan is to show that the right-hand and left-hand limits are both 1. Then we will know that the two-sided limit is 1 as well. To show that the right-hand limit is 1, we begin with positive values of u less than p>2 (Figure 2.33). Notice that

1 sin

cos Q

A(1, 0)

Area ¢OAP 6 area sector OAP 6 area ¢OAT.

x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

O 1

We can express these areas in terms of u as follows:

FIGURE 2.33 The figure for the proof of Theorem 7. By definition, TA>OA = tan u , but OA = 1 , so TA = tan u .

1 1 1 base * height = s1dssin ud = sin u 2 2 2 u 1 2 1 Area sector OAP = r u = s1d2u = 2 2 2 1 1 1 Area ¢OAT = base * height = s1dstan ud = tan u. 2 2 2 Area ¢OAP =

(2)

Thus, Equation (2) is where radian measure comes in: The area of sector OAP is u>2 only if u is measured in radians.

1 1 1 sin u 6 u 6 tan u. 2 2 2 This last inequality goes the same way if we divide all three terms by the number (1>2) sin u , which is positive since 0 6 u 6 p>2: 1 6

u 1 6 . cos u sin u

Taking reciprocals reverses the inequalities: 1 7

sin u 7 cos u. u

Since limu:0+ cos u = 1 (Example 11b, Section 2.2), the Sandwich Theorem gives lim

u:0 +

sin u = 1. u

Recall that sin u and u are both odd functions (Section 1.1). Therefore, ƒsud = ssin ud>u is an even function, with a graph symmetric about the y-axis (see Figure 2.32). This symmetry implies that the left-hand limit at 0 exists and has the same value as the right-hand limit: lim

u:0 -

sin u sin u = 1 = lim+ , u u u:0

so limu:0 ssin ud>u = 1 by Theorem 6.

EXAMPLE 5

Show that (a) lim

h:0

cos h - 1 = 0 h

and

(b) lim

x:0

sin 2x 2 = . 5x 5

84

Chapter 2: Limits and Continuity Solution (a) Using the half-angle formula cos h = 1 - 2 sin2 sh>2d, we calculate

2 sin2 sh>2d cos h - 1 = lim h h h:0 h:0 lim

= - lim

u:0

sin u sin u u

Let u = h>2 . Eq. (1) and Example 11a in Section 2.2

= -s1ds0d = 0.

(b) Equation (1) does not apply to the original fraction. We need a 2x in the denominator, not a 5x. We produce it by multiplying numerator and denominator by 2>5: s2>5d # sin 2x sin 2x = lim x:0 5x x:0 s2>5d # 5x lim

EXAMPLE 6

Find lim

t:0

=

sin 2x 2 lim 5 x:0 2x

=

2 2 s1d = 5 5

Now, Eq. (1) applies with u = 2x.

tan t sec 2t . 3t

From the definition of tan t and sec 2t, we have

Solution

tan t sec 2t sin t 1 1 1 = lim t # cos t # 3t 3 cos 2t t:0 t:0 lim

=

1 1 (1)(1)(1) = . 3 3

Eq. (1) and Example 11b in Section 2.2

Exercises 2.4 Finding Limits Graphically 1. Which of the following statements about the function y = ƒsxd graphed here are true, and which are false?

y y f (x) 2

y y f (x)

1

1

–1

0

2

x

–1

0

lim ƒsxd = 1

1

2

3

x

b. lim- ƒsxd = 0

a.

c. lim- ƒsxd = 1

d. lim- ƒsxd = lim+ ƒsxd

c. lim ƒsxd = 2

d. lim- ƒsxd = 2

e. lim ƒsxd exists.

f. lim ƒsxd = 0

f. lim ƒsxd does not exist.

g. lim ƒsxd = 1

x: 0

e. lim+ ƒsxd = 1

h. lim ƒsxd = 1

i. lim ƒsxd = 0

j. lim- ƒsxd = 2

x: 1

g. lim+ ƒsxd = lim- ƒsxd

x: 2

h. lim ƒsxd exists at every c in the open interval s -1, 1d .

l. lim+ ƒsxd = 0

i. lim ƒsxd exists at every c in the open interval (1, 3).

a.

lim ƒsxd = 1

1

x: -1 + x:0 x:0 x:0 x:1

k.

lim ƒsxd does not exist .

x: -1 -

x: 0 x: 0

x: 2

x: 0

2. Which of the following statements about the function y = ƒsxd graphed here are true, and which are false?

x: -1 + x:2 x:1 x:0

b. lim ƒsxd does not exist. x:2 x:1 x:1

x:0

x:c x:c

j.

lim ƒsxd = 0

x: -1 -

k. lim+ ƒsxd does not exist. x:3

2.4 3 - x, 3. Let ƒsxd = • x + 1, 2

x 6 2

One-Sided Limits

85

6. Let g sxd = 2x sins1>xd .

x 7 2. y 1

y

y x

y3x 3

y x sin 1x

y x1 2

2

0

1 2 0

x

4

1

2

1

x

a. Find limx:2+ ƒsxd and limx:2- ƒsxd . b. Does limx:2 ƒsxd exist? If so, what is it? If not, why not? y –x

–1

c. Find limx:4- ƒsxd and limx:4+ ƒsxd . d. Does limx:4 ƒsxd exist? If so, what is it? If not, why not?

4. Let ƒsxd = d

a. Does limx:0+ g sxd exist? If so, what is it? If not, why not?

3 - x, 2,

x 6 2 x = 2

x , 2

x 7 2.

b. Does limx:0- g sxd exist? If so, what is it? If not, why not? c. Does limx:0 g sxd exist? If so, what is it? If not, why not? 7. a. Graph ƒsxd = e

y

b. Find limx:1- ƒsxd and limx:1+ ƒsxd . c. Does limx:1 ƒsxd exist? If so, what is it? If not, why not?

y3x

8. a. Graph ƒsxd = e

3 y x 2 –2

0

2

x

c. Does limx:1 ƒsxd exist? If so, what is it? If not, why not? Graph the functions in Exercises 9 and 10. Then answer these questions. a. What are the domain and range of ƒ?

b. Does limx:2 ƒsxd exist? If so, what is it? If not, why not?

b. At what points c, if any, does limx:c ƒsxd exist?

c. Find limx:-1- ƒsxd and limx:-1+ ƒsxd .

c. At what points does only the left-hand limit exist?

d. Does limx:-1 ƒsxd exist? If so, what is it? If not, why not? 0,

x … 0

1 sin x ,

x 7 0.

d. At what points does only the right-hand limit exist? 21 - x 2, 9. ƒsxd = • 1, 2, x, 10. ƒsxd = • 1, 0,

y 1

0

x Z 1 x = 1.

1 - x 2, 2,

b. Find limx:1+ ƒsxd and limx:1- ƒsxd .

a. Find limx:2+ ƒsxd, limx:2- ƒsxd , and ƒ(2).

5. Let ƒsxd = •

x Z 1 x = 1.

x 3, 0,

x ⎧ x0 ⎪0, y⎨ 1 ⎪ sin x , x 0 ⎩

–1

a. Does limx:0+ ƒsxd exist? If so, what is it? If not, why not? b. Does limx:0- ƒsxd exist? If so, what is it? If not, why not? c. Does limx:0 ƒsxd exist? If so, what is it? If not, why not?

0 … x 6 1 1 … x 6 2 x = 2

-1 … x 6 0, or 0 6 x … 1 x = 0 x 6 -1 or x 7 1

Finding One-Sided Limits Algebraically Find the limits in Exercises 11–18. 11. 13.

x + 2 + 1

lim

x: -0.5 - A x

lim a

x: -2 +

14. lim- a x:1

15. lim+ h:0

12. lim+

2x + 5 x b ba 2 x + 1 x + x

x + 6 3 - x 1 b ba x ba 7 x + 1

2h 2 + 4h + 5 - 25 h

x:1

x - 1 Ax + 2

86

Chapter 2: Limits and Continuity

26 - 25h 2 + 11h + 6 h ƒx + 2ƒ 17. a. lim +sx + 3d b. x + 2 x: -2

41. lim

16. lim-

u :0

h: 0

18. a. lim+ x: 1

22x sx - 1d ƒx - 1ƒ

lim -sx + 3d

x: -2

b. limx: 1

ƒx + 2ƒ x + 2

22x sx - 1d ƒx - 1ƒ

Use the graph of the greatest integer function y = :x; , Figure 1.10 in Section 1.1, to help you find the limits in Exercises 19 and 20. :u; :u; 19. a. lim+ b. limu u u: 3 u: 3 20. a. lim+st - : t; d

b. lim-st - : t; d

t:4

t: 4

sin U 1 U Find the limits in Exercises 21–42. Using lim

U:0

21. lim

sin 22u

22. lim

22u sin 3y 23. lim y:0 4y tan 2x 25. lim x x:0 u: 0

t: 0

sin kt t

sk constantd

h h :0 sin 3h 2t 26. lim t: 0 tan t 24. lim-

x csc 2x x:0 cos 5x

27. lim

28. lim 6x 2scot xdscsc 2xd

x + x cos x 29. lim x:0 sin x cos x

30. lim

x: 0

x 2 - x + sin x 2x x: 0

1 - cos u 31. lim u: 0 sin 2u sin s1 - cos td 1 - cos t sin u 35. lim u: 0 sin 2u

x - x cos x 32. lim x: 0 sin2 3x sin ssin hd 34. lim sin h h :0 sin 5x 36. lim x: 0 sin 4x

37. lim u cos u

38. lim sin u cot 2u

tan 3x 39. lim x:0 sin 8x

40. lim

33. lim

t:0

u: 0

2.5

u: 0

y: 0

sin 3y cot 5y y cot 4y

tan u u2 cot 3u

42. lim

u :0

u cot 4u sin2 u cot2 2u

Theory and Examples 43. Once you know limx:a+ ƒsxd and limx:a- ƒsxd at an interior point of the domain of ƒ, do you then know limx:a ƒsxd ? Give reasons for your answer. 44. If you know that limx:c ƒsxd exists, can you find its value by calculating limx:c+ ƒsxd ? Give reasons for your answer. 45. Suppose that ƒ is an odd function of x. Does knowing that limx:0+ ƒsxd = 3 tell you anything about limx:0- ƒsxd ? Give reasons for your answer. 46. Suppose that ƒ is an even function of x. Does knowing that limx:2- ƒsxd = 7 tell you anything about either limx: -2- ƒsxd or limx:-2+ ƒsxd ? Give reasons for your answer. Formal Definitions of One-Sided Limits 47. Given P 7 0 , find an interval I = s5, 5 + dd, d 7 0 , such that if x lies in I, then 2x - 5 6 P . What limit is being verified and what is its value? 48. Given P 7 0 , find an interval I = s4 - d, 4d, d 7 0 , such that if x lies in I, then 24 - x 6 P . What limit is being verified and what is its value? Use the definitions of right-hand and left-hand limits to prove the limit statements in Exercises 49 and 50. x x - 2 49. lim50. lim+ = -1 = 1 x:0 ƒ x ƒ x:2 ƒ x - 2 ƒ

51. Greatest integer function Find (a) limx:400+ :x; and (b) limx:400- :x; ; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx:400 :x; ? Give reasons for your answer. 52. One-sided limits

Let ƒsxd = e

x 2 sin s1>xd, x 6 0 x 7 0. 2x,

Find (a) limx:0+ ƒsxd and (b) limx:0- ƒsxd ; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx:0 ƒsxd ? Give reasons for your answer.

Continuity When we plot function values generated in a laboratory or collected in the field, we often connect the plotted points with an unbroken curve to show what the function’s values are likely to have been at the times we did not measure (Figure 2.34). In doing so, we are assuming that we are working with a continuous function, so its outputs vary continuously with the inputs and do not jump from one value to another without taking on the values in between. The limit of a continuous function as x approaches c can be found simply by calculating the value of the function at c. (We found this to be true for polynomials in Theorem 2.) Intuitively, any function y = ƒsxd whose graph can be sketched over its domain in one continuous motion without lifting the pencil is an example of a continuous function. In this section we investigate more precisely what it means for a function to be continuous.

2.5

87

We also study the properties of continuous functions, and see that many of the function types presented in Section 1.1 are continuous.

y 500 Distance fallen (m)

Continuity

Q4 Q3

375

Continuity at a Point

Q2

250

To understand continuity, it helps to consider a function like that in Figure 2.35, whose limits we investigated in Example 2 in the last section.

Q1

125 0

5 Elapsed time (sec)

10

FIGURE 2.34 Connecting plotted points by an unbroken curve from experimental data Q1 , Q2 , Q3 , Á for a falling object.

t

EXAMPLE 1 Find the points at which the function ƒ in Figure 2.35 is continuous and the points at which ƒ is not continuous. The function ƒ is continuous at every point in its domain [0, 4] except at x = 1, x = 2, and x = 4. At these points, there are breaks in the graph. Note the relationship between the limit of ƒ and the value of ƒ at each point of the function’s domain.

Solution

Points at which ƒ is continuous:

At x = 0, At x = 3, y

At 0 6 c 6 4, c Z 1, 2, y f (x)

2

At x = 1, 1

2

3

At x = 2,

x

4

lim ƒsxd = ƒs3d.

x:3

lim ƒsxd = ƒscd.

x:c

Points at which ƒ is not continuous:

1

0

lim ƒsxd = ƒs0d.

x:0 +

FIGURE 2.35 The function is continuous on [0, 4] except at x = 1, x = 2 , and x = 4 (Example 1).

At x = 4, At c 6 0, c 7 4,

lim ƒsxd does not exist.

x:1

lim ƒsxd = 1, but 1 Z ƒs2d.

x:2

lim ƒsxd = 1, but 1 Z ƒs4d.

x:4 -

these points are not in the domain of ƒ.

To define continuity at a point in a function’s domain, we need to define continuity at an interior point (which involves a two-sided limit) and continuity at an endpoint (which involves a one-sided limit) (Figure 2.36).

Continuity from the right

Two-sided continuity

Continuity from the left

DEFINITION Interior point: A function y = ƒsxd is continuous at an interior point c of its domain if lim ƒsxd = ƒscd.

x:c

y f (x) a

c

b

x

FIGURE 2.36 Continuity at points a, b, and c.

Endpoint: A function y = ƒsxd is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if lim ƒsxd = ƒsad

x:a +

or

lim ƒsxd = ƒsbd,

x:b -

respectively.

If a function ƒ is not continuous at a point c, we say that ƒ is discontinuous at c and that c is a point of discontinuity of ƒ. Note that c need not be in the domain of ƒ. A function ƒ is right-continuous (continuous from the right) at a point x = c in its domain if limx:c+ ƒsxd = ƒscd. It is left-continuous (continuous from the left) at c if limx:c- ƒsxd = ƒscd. Thus, a function is continuous at a left endpoint a of its domain if it

88

Chapter 2: Limits and Continuity

is right-continuous at a and continuous at a right endpoint b of its domain if it is leftcontinuous at b. A function is continuous at an interior point c of its domain if and only if it is both right-continuous and left-continuous at c (Figure 2.36).

y y 4

2

–2

0

x2

The function ƒsxd = 24 - x 2 is continuous at every point of its domain [-2, 2] (Figure 2.37), including x = -2, where ƒ is right-continuous, and x = 2, where ƒ is left-continuous.

EXAMPLE 2

x

2

FIGURE 2.37 A function that is continuous at every domain point (Example 2).

EXAMPLE 3

The unit step function U(x), graphed in Figure 2.38, is right-continuous at x = 0, but is neither left-continuous nor continuous there. It has a jump discontinuity at x = 0.

y y U(x)

1

We summarize continuity at a point in the form of a test.

x

0

Continuity Test A function ƒ(x) is continuous at an interior point x = c of its domain if and only if it meets the following three conditions.

FIGURE 2.38 A function that has a jump discontinuity at the origin (Example 3).

1. 2. 3.

(c lies in the domain of ƒ). (ƒ has a limit as x : c). (the limit equals the function value).

When we say a function is continuous at c, we are asserting that all three conditions hold. For one-sided continuity and continuity at an endpoint, the limits in parts 2 and 3 of the test should be replaced by the appropriate one-sided limits.

y 4 3

The function y = :x; introduced in Section 1.1 is graphed in Figure 2.39. It is discontinuous at every integer because the left-hand and right-hand limits are not equal as x : n:

y ⎣x⎦

EXAMPLE 4

2 1 x –1

ƒ(c) exists limx:c ƒsxd exists limx:c ƒsxd = ƒscd

1

2

3

4

–2

FIGURE 2.39 The greatest integer function is continuous at every noninteger point. It is right-continuous, but not left-continuous, at every integer point (Example 4).

lim :x; = n - 1

x:n -

and

lim :x; = n.

x:n +

Since :n ; = n, the greatest integer function is right-continuous at every integer n (but not left-continuous). The greatest integer function is continuous at every real number other than the integers. For example, lim :x; = 1 = :1.5; .

x:1.5

In general, if n - 1 6 c 6 n, n an integer, then

lim :x; = n - 1 = :c; .

x:c

Figure 2.40 displays several common types of discontinuities. The function in Figure 2.40a is continuous at x = 0. The function in Figure 2.40b would be continuous if it had ƒs0d = 1. The function in Figure 2.40c would be continuous if ƒ(0) were 1 instead of 2. The discontinuities in Figure 2.40b and c are removable. Each function has a limit as x : 0, and we can remove the discontinuity by setting ƒ(0) equal to this limit. The discontinuities in Figure 2.40d through f are more serious: limx:0 ƒsxd does not exist, and there is no way to improve the situation by changing ƒ at 0. The step function in Figure 2.40d has a jump discontinuity: The one-sided limits exist but have different values. The function ƒsxd = 1>x 2 in Figure 2.40e has an infinite discontinuity. The function in Figure 2.40f has an oscillating discontinuity: It oscillates too much to have a limit as x : 0.

2.5

y f (x)

0

2

y f (x)

1

1 x

x

0

(b)

(c) y

y y f (x) 12 x

1

1 0

y f (x)

1

0

(a)

y

y

y

y

0

89

Continuity

1 x

y f (x)

0

x

(d) y sin 1x

x

x –1

(e)

(f)

FIGURE 2.40 The function in (a) is continuous at x = 0 ; the functions in (b) through (f ) are not.

Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. For example, the semicircle function graphed in Figure 2.37 is continuous on the interval [-2, 2], which is its domain. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval. y

EXAMPLE 5 y 1x

0

x

(a) The function y = 1>x (Figure 2.41) is a continuous function because it is continuous at every point of its domain. It has a point of discontinuity at x = 0, however, because it is not defined there; that is, it is discontinuous on any interval containing x = 0. (b) The identity function ƒsxd = x and constant functions are continuous everywhere by Example 3, Section 2.3. Algebraic combinations of continuous functions are continuous wherever they are defined.

FIGURE 2.41 The function y = 1>x is continuous at every value of x except x = 0 . It has a point of discontinuity at x = 0 (Example 5).

THEOREM 8—Properties of Continuous Functions If the functions ƒ and g are continuous at x = c, then the following combinations are continuous at x = c. 1. 2. 3. 4. 5. 6. 7.

Sums: Differences: Constant multiples: Products: Quotients: Powers: Roots:

ƒ + g ƒ - g k # ƒ, for any number k ƒ#g ƒ>g, provided gscd Z 0 ƒ n, n a positive integer n 2ƒ, provided the root is defined on an open interval containing c, where n is a positive integer

90

Chapter 2: Limits and Continuity

Most of the results in Theorem 8 follow from the limit rules in Theorem 1, Section 2.2. For instance, to prove the sum property we have lim sƒ + gdsxd = lim sƒsxd + gsxdd

x:c

x:c

= lim ƒsxd + lim gsxd,

Sum Rule, Theorem 1

= ƒscd + gscd = sƒ + gdscd.

Continuity of ƒ, g at c

x:c

x:c

This shows that ƒ + g is continuous.

EXAMPLE 6 (a) Every polynomial Psxd = an x n + an - 1x n - 1 + Á + a0 is continuous because lim Psxd = Pscd by Theorem 2, Section 2.2. x:c

(b) If P(x) and Q(x) are polynomials, then the rational function Psxd>Qsxd is continuous wherever it is defined sQscd Z 0d by Theorem 3, Section 2.2. The function ƒsxd = ƒ x ƒ is continuous at every value of x. If x 7 0, we have ƒsxd = x, a polynomial. If x 6 0, we have ƒsxd = -x, another polynomial. Finally, at the origin, limx:0 ƒ x ƒ = 0 = ƒ 0 ƒ .

EXAMPLE 7

The functions y = sin x and y = cos x are continuous at x = 0 by Example 11 of Section 2.2. Both functions are, in fact, continuous everywhere (see Exercise 70). It follows from Theorem 8 that all six trigonometric functions are then continuous wherever they are defined. For example, y = tan x is continuous on Á ´ s -p>2, p>2d ´ sp>2, 3p>2d ´ Á .

Inverse Functions and Continuity The inverse function of any function continuous on an interval is continuous over its domain. This result is suggested from the observation that the graph of ƒ-1, being the reflection of the graph of ƒ across the line y = x, cannot have any breaks in it when the graph of ƒ has no breaks. A rigorous proof that ƒ-1 is continuous whenever ƒ is continuous on an interval is given in more advanced texts. It follows that the inverse trigonometric functions are all continuous over their domains. We defined the exponential function y = a x in Section 1.5 informally by its graph. Recall that the graph was obtained from the graph of y = a x for x a rational number by filling in the holes at the irrational points x, so the function y = a x was defined to be continuous over the entire real line. The inverse function y = loga x is also continuous. In particular, the natural exponential function y = e x and the natural logarithm function y = ln x are both continuous over their domains.

Composites All composites of continuous functions are continuous. The idea is that if ƒ(x) is continuous at x = c and g(x) is continuous at x = ƒscd, then g ⴰ ƒ is continuous at x = c (Figure 2.42). In this case, the limit as x : c is g(ƒ(c)). g˚ f Continuous at c

c

f

g

Continuous at c

Continuous at f (c) f (c)

FIGURE 2.42 Composites of continuous functions are continuous.

g( f (c))

2.5

Continuity

91

THEOREM 9—Composite of Continuous Functions If ƒ is continuous at c and g is continuous at ƒ(c), then the composite g ⴰ ƒ is continuous at c. Intuitively, Theorem 9 is reasonable because if x is close to c, then ƒ(x) is close to ƒ(c), and since g is continuous at ƒ(c), it follows that g(ƒ(x)) is close to g(ƒ(c)). The continuity of composites holds for any finite number of functions. The only requirement is that each function be continuous where it is applied. For an outline of a proof of Theorem 9, see Exercise 6 in Appendix 5.

EXAMPLE 8

Show that the following functions are continuous everywhere on their respective domains. x 2>3 1 + x4

(a) y = 2x 2 - 2x - 5

(b) y =

(c) y = `

(d) y = `

x - 2 ` x2 - 2

x sin x ` x2 + 2

Solution y 0.4 0.3 0.2 0.1 –2

–

0

2

FIGURE 2.43 The graph suggests that y = ƒ sx sin xd>sx 2 + 2d ƒ is continuous (Example 8d).

x

(a) The square root function is continuous on [0, q d because it is a root of the continuous identity function ƒsxd = x (Part 7, Theorem 8). The given function is then the composite of the polynomial ƒsxd = x 2 - 2x - 5 with the square root function gstd = 2t , and is continuous on its domain. (b) The numerator is the cube root of the identity function squared; the denominator is an everywhere-positive polynomial. Therefore, the quotient is continuous. (c) The quotient sx - 2d>sx 2 - 2d is continuous for all x Z ; 22 , and the function is the composition of this quotient with the continuous absolute value function (Example 7). (d) Because the sine function is everywhere-continuous (Exercise 70), the numerator term x sin x is the product of continuous functions, and the denominator term x 2 + 2 is an everywhere-positive polynomial. The given function is the composite of a quotient of continuous functions with the continuous absolute value function (Figure 2.43). Theorem 9 is actually a consequence of a more general result which we now state and prove.

THEOREM 10—Limits of Continuous Functions b and limx:c ƒ(x) = b, then

If g is continuous at the point

limx:c g(ƒ(x)) = g(b) = g(limx:c ƒ(x)). Proof

Let P 7 0 be given. Since g is continuous at b, there exists a number d1 7 0 such that

ƒ g( y) - g(b) ƒ 6 P whenever 0 6 ƒ y - b ƒ 6 d1. Since limx:c ƒ(x) = b, there exists a d 7 0 such that ƒ ƒ(x) - b ƒ 6 d1 If we let y = ƒ(x), we then have that

whenever

0 6 ƒ x - c ƒ 6 d.

ƒ y - b ƒ 6 d1 whenever 0 6 ƒ x - c ƒ 6 d, which implies from the first statement that ƒ g( y) - g(b) ƒ = ƒ g(ƒ(x)) - g(b) ƒ 6 P whenever 0 6 ƒ x - c ƒ 6 d. From the definition of limit, this proves that limx:c g(ƒ(x)) = g(b).

92

Chapter 2: Limits and Continuity

EXAMPLE 9 (a)

lim cos a2x + sin a

x:p/2

(b) lim sin-1 a x:1

We sometimes denote e u by exp u when u is a complicated mathematical expression.

As an application of Theorem 10, we have the following calculations. 3p 3p + xb b = cos a lim 2x + lim sin a + xb b 2 2 x:p/2 x:p/2 = cos (p + sin 2p) = cos p = -1.

1 - x 1 - x b = sin-1 a lim b 2 x:1 1 - x 2 1 - x 1 = sin-1 a lim b 1 + x x:1 p 1 = sin-1 = 2 6

Arcsine is continuous. Cancel common factor (1 - x).

(c) lim 2x + 1 e tan x = lim 2x + 1 # exp a lim tan xb x:0

x:0

Exponential is continuous.

x:0

= 1 # e0 = 1

Intermediate Value Theorem for Continuous Functions Functions that are continuous on intervals have properties that make them particularly useful in mathematics and its applications. One of these is the Intermediate Value Property. A function is said to have the Intermediate Value Property if whenever it takes on two values, it also takes on all the values in between.

THEOREM 11—The Intermediate Value Theorem for Continuous Functions If ƒ is a continuous function on a closed interval [a, b], and if y0 is any value between ƒ(a) and ƒ(b), then y0 = ƒscd for some c in [a, b]. y y f (x) f (b)

y0 f (a)

0

y

a

c

b

x

3 2 1

0

1

2

3

4

x

FIGURE 2.44 The function 2x - 2, 1 … x 6 2 ƒsxd = e 3, 2 … x … 4 does not take on all values between ƒs1d = 0 and ƒs4d = 3 ; it misses all the values between 2 and 3.

Theorem 11 says that continuous functions over finite closed intervals have the Intermediate Value Property. Geometrically, the Intermediate Value Theorem says that any horizontal line y = y0 crossing the y-axis between the numbers ƒ(a) and ƒ(b) will cross the curve y = ƒsxd at least once over the interval [a, b]. The proof of the Intermediate Value Theorem depends on the completeness property of the real number system (Appendix 7) and can be found in more advanced texts. The continuity of ƒ on the interval is essential to Theorem 11. If ƒ is discontinuous at even one point of the interval, the theorem’s conclusion may fail, as it does for the function graphed in Figure 2.44 (choose y0 as any number between 2 and 3). A Consequence for Graphing: Connectedness Theorem 11 implies that the graph of a function continuous on an interval cannot have any breaks over the interval. It will be connected—a single, unbroken curve. It will not have jumps like the graph of the greatest integer function (Figure 2.39), or separate branches like the graph of 1>x (Figure 2.41).

2.5

93

Continuity

A Consequence for Root Finding We call a solution of the equation ƒsxd = 0 a root of the equation or zero of the function ƒ. The Intermediate Value Theorem tells us that if ƒ is continuous, then any interval on which ƒ changes sign contains a zero of the function. In practical terms, when we see the graph of a continuous function cross the horizontal axis on a computer screen, we know it is not stepping across. There really is a point where the function’s value is zero.

EXAMPLE 10

Show that there is a root of the equation x 3 - x - 1 = 0 between 1 and 2.

Let ƒ(x) = x 3 - x - 1. Since ƒ(1) = 1 - 1 - 1 = -1 6 0 and ƒ(2) = 2 - 2 - 1 = 5 7 0, we see that y0 = 0 is a value between ƒ(1) and ƒ(2). Since ƒ is continuous, the Intermediate Value Theorem says there is a zero of ƒ between 1 and 2. Figure 2.45 shows the result of zooming in to locate the root near x = 1.32. Solution 3

5

1

1 –1

1.6

2

–2

–1 (a)

(b)

0.02

0.003

1.320

1.330

–0.02

1.3240

1.3248

–0.003 (c)

(d)

FIGURE 2.45 Zooming in on a zero of the function ƒsxd = x 3 - x - 1 . The zero is near x = 1.3247 (Example 10).

EXAMPLE 11

y 4

y542 x

Use the Intermediate Value Theorem to prove that the equation 22x + 5 = 4 - x 2

2

3

has a solution (Figure 2.46).

2

Solution

22x + 5 + x 2 = 4,

y 5 2x 1 5 1 0

We rewrite the equation as

c

2

x

FIGURE 2.46 The curves y = 22x + 5 and y = 4 - x 2 have the same value at x = c where 22x + 5 = 4 - x 2 (Example 11).

and set ƒ(x) = 22x + 5 + x 2. Now g(x) = 22x + 5 is continuous on the interval [-5>2, q) since it is the composite of the square root function with the nonnegative linear function y = 2x + 5. Then ƒ is the sum of the function g and the quadratic function y = x 2, and the quadratic function is continuous for all values of x. It follows that ƒ(x) = 22x + 5 + x 2 is continuous on the interval [-5>2, q). By trial and error, we find the function values ƒ(0) = 25 L 2.24 and ƒ(2) = 29 + 4 = 7, and note that ƒ is also continuous on the finite closed interval [0, 2] ( [-5>2, q). Since the value y0 = 4 is between the numbers 2.24 and 7, by the Intermediate Value Theorem there is a number c H [0, 2] such that ƒ(c) = 4. That is, the number c solves the original equation.

94

Chapter 2: Limits and Continuity

Continuous Extension to a Point Sometimes the formula that describes a function f does not make sense at a point x = c. It might nevertheless be possible to extend the domain of f, to include x = c, creating a new function that is continuous at x = c. For example, the function y = ƒ(x) = ssin xd>x is continuous at every point except x = 0. In this it is like the function y = 1>x. But y = ssin xd>x is different from y = 1>x in that it has a finite limit as x : 0 (Theorem 7). It is therefore possible to extend the function’s domain to include the point x = 0 in such a way that the extended function is continuous at x = 0. We define a new function Fsxd =

L

sin x x ,

x Z 0

1,

x = 0.

The function F(x) is continuous at x = 0 because sin x = Fs0d lim x:0 x (Figure 2.47). y

y

(0, 1) ⎛– , 2⎛ ⎝ 2 ⎝ – 2

(0, 1)

f (x) ⎛ , 2⎛ ⎝ 2 ⎝

⎛– , 2⎛ ⎝ 2 ⎝ 2

0

x

– 2

⎛ , 2⎛ ⎝ 2 ⎝ 0

(a)

F(x)

2

x

(b)

FIGURE 2.47 The graph (a) of ƒsxd = ssin xd>x for -p>2 … x … p>2 does not include the point (0, 1) because the function is not defined at x = 0 . (b) We can remove the discontinuity from the graph by defining the new function F(x) with Fs0d = 1 and Fsxd = ƒsxd everywhere else. Note that Fs0d = limx:0 ƒsxd .

More generally, a function (such as a rational function) may have a limit even at a point where it is not defined. If ƒ(c) is not defined, but limx:c ƒsxd = L exists, we can define a new function F(x) by the rule y 2

y

Fsxd = e

x2 x 6 x2 4

1 –1

0

1

2

3

4

x

(a)

EXAMPLE 12

y

5 4 –1

2

x 3 y x2

Show that ƒsxd =

1 0

ƒsxd, if x is in the domain of ƒ L, if x = c. The function F is continuous at x = c. It is called the continuous extension of ƒ to x = c. For rational functions ƒ, continuous extensions are usually found by canceling common factors.

x2 + x - 6 , x2 - 4

x Z 2

has a continuous extension to x = 2, and find that extension. 1

2

3

4

x

(b)

FIGURE 2.48 (a) The graph of ƒ(x) and (b) the graph of its continuous extension F(x) (Example 12).

Solution

Although ƒ(2) is not defined, if x Z 2 we have sx - 2dsx + 3d x2 + x - 6 x + 3 ƒsxd = = = . 2 x + 2 sx - 2dsx + 2d x - 4

The new function Fsxd =

x + 3 x + 2

2.5

Continuity

95

is equal to ƒ(x) for x Z 2, but is continuous at x = 2, having there the value of 5>4. Thus F is the continuous extension of ƒ to x = 2, and x2 + x - 6 5 = lim ƒsxd = . 4 x:2 x:2 x2 - 4 The graph of ƒ is shown in Figure 2.48. The continuous extension F has the same graph except with no hole at (2, 5>4). Effectively, F is the function ƒ with its point of discontinuity at x = 2 removed. lim

Exercises 2.5 Continuity from Graphs In Exercises 1–4, say whether the function graphed is continuous on [-1, 3] . If not, where does it fail to be continuous and why? 1.

2.

y g(x)

–1

c. Does limx:1 ƒsxd = ƒs1d ? d. Is ƒ continuous at x = 1 ?

1

0

1

2

x

3

–1

3.

0

1

2

3

x

y h(x) 2

1

1 1

2

x

3

b. Is ƒ continuous at x = 2 ? 9. What value should be assigned to ƒ(2) to make the extended function continuous at x = 2 ?

y

2

0

7. a. Is ƒ defined at x = 2 ? (Look at the definition of ƒ.) 8. At what values of x is ƒ continuous?

4. y

–1

b. Does limx:1 ƒsxd exist?

2

1

c. Does limx:-1+ ƒsxd = ƒs -1d ? 6. a. Does ƒ(1) exist?

y f(x) 2

b. Does limx: -1+ ƒsxd exist? d. Is ƒ continuous at x = -1 ?

y

y

5. a. Does ƒs -1d exist?

–1

y k(x)

1

0

2

Exercises 5–10 refer to the function x 2 - 1, 2x, ƒsxd = e 1, -2x + 4, 0,

-1 0 x 1 2

… 6 = 6 6

x x 1 x x

10. To what new value should ƒ(1) be changed to remove the discontinuity?

3

x

Applying the Continuity Test At which points do the functions in Exercises 11 and 12 fail to be continuous? At which points, if any, are the discontinuities removable? Not removable? Give reasons for your answers.

6 0 6 1

11. Exercise 1, Section 2.4

6 2 6 3

13. y =

1 - 3x x - 2

14. y =

1 + 4 sx + 2d2

15. y =

x + 1 x 2 - 4x + 3

16. y =

x + 3 x 2 - 3x - 10

18. y =

x2 1 2 ƒxƒ + 1

At what points are the functions in Exercises 13–30 continuous?

graphed in the accompanying figure.

17. y = ƒ x - 1 ƒ + sin x

y y f (x) 2

19. y =

(1, 2)

y 2x

y –2x 4

cos x x

21. y = csc 2x

(1, 1) –1 y x2 1

0

1

2

–1

The graph for Exercises 5–10.

3

12. Exercise 2, Section 2.4

x

23. y =

x tan x x2 + 1

x + 2 20. y = cos x px 22. y = tan 2 24. y =

2x 4 + 1 1 + sin2 x 4

25. y = 22x + 3

26. y = 23x - 1

27. y = s2x - 1d1>3

28. y = s2 - xd1>5

96

Chapter 2: Limits and Continuity

x2 - x - 6 , 29. gsxd = • x - 3 5, x3 - 8 , x2 - 4 30. ƒsxd = d 3, 4,

47. For what values of a and b is

x Z 3

-2, ƒsxd = • ax - b, 3,

x = 3

x Z 2, x Z -2

continuous at every x?

x = 2 x = -2

48. For what values of a and b is

Limits Involving Trigonometric Functions Find the limits in Exercises 31–38. Are the functions continuous at the point being approached? p 31. lim sin sx - sin xd 32. lim sin a cos stan tdb 2 x:p t: 0 33. lim sec s y sec y - tan y - 1d 2

2

y:1

34. lim tan a x:0

p cos ssin x 1>3 db 4

35. lim cos a t:0

p 219 - 3 sec 2t

37. lim+ sin a x:0

p 2x e b 2

x … -1 -1 6 x 6 1 x Ú 1

b 36.

lim 2csc2 x + 5 13 tan x

x: p/6

38. lim cos-1 (ln 2x) x: 1

Continuous Extensions 39. Define g(3) in a way that extends g sxd = sx 2 - 9d>sx - 3d to be continuous at x = 3 . 40. Define h(2) in a way that extends hstd = st 2 + 3t - 10d>st - 2d to be continuous at t = 2 .

ax + 2b, gsxd = • x 2 + 3a - b, 3x - 5,

x … 0 0 6 x … 2 x 7 2

continuous at every x? T In Exercises 49–52, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0 . If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be? 10 ƒ x ƒ - 1 x

49. ƒsxd =

10 x - 1 x

50. ƒsxd =

51. ƒsxd =

sin x ƒxƒ

52. ƒsxd = s1 + 2xd1>x

Theory and Examples 53. A continuous function y = ƒsxd is known to be negative at x = 0 and positive at x = 1 . Why does the equation ƒsxd = 0 have at least one solution between x = 0 and x = 1? Illustrate with a sketch.

41. Define ƒ(1) in a way that extends ƒssd = ss 3 - 1d>ss 2 - 1d to be continuous at s = 1 .

54. Explain why the equation cos x = x has at least one solution.

42. Define g(4) in a way that extends

56. A function value Show that the function Fsxd = sx - ad2 # sx - bd2 + x takes on the value sa + bd>2 for some value of x.

g sxd = sx 2 - 16d> sx 2 - 3x - 4d to be continuous at x = 4 . 43. For what value of a is ƒsxd = e

x 2 - 1, 2ax,

x 6 3 x Ú 3

continuous at every x? x 6 -2 x Ú -2

continuous at every x?

58. Explain why the following five statements ask for the same information.

a 2x - 2a, ƒsxd = b 12,

b. Find the x-coordinates of the points where the curve y = x 3 crosses the line y = 3x + 1 . c. Find all the values of x for which x 3 - 3x = 1 . d. Find the x-coordinates of the points where the cubic curve y = x 3 - 3x crosses the line y = 1 .

45. For what values of a is x Ú 2 x 6 2

continuous at every x? 46. For what value of b is x - b , x 6 0 gsxd = • b + 1 x 2 + b, x 7 0 continuous at every x?

57. Solving an equation If ƒsxd = x 3 - 8x + 10 , show that there are values c for which ƒ(c) equals (a) p ; (b) - 23 ; (c) 5,000,000.

a. Find the roots of ƒsxd = x 3 - 3x - 1 .

44. For what value of b is x, g sxd = e 2 bx ,

55. Roots of a cubic Show that the equation x 3 - 15x + 1 = 0 has three solutions in the interval [-4, 4] .

e. Solve the equation x 3 - 3x - 1 = 0 . 59. Removable discontinuity Give an example of a function ƒ(x) that is continuous for all values of x except x = 2 , where it has a removable discontinuity. Explain how you know that ƒ is discontinuous at x = 2 , and how you know the discontinuity is removable. 60. Nonremovable discontinuity Give an example of a function g(x) that is continuous for all values of x except x = -1 , where it has a nonremovable discontinuity. Explain how you know that g is discontinuous there and why the discontinuity is not removable.

2.6 61. A function discontinuous at every point a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the function ƒsxd = e

1, if x is rational 0, if x is irrational

Limits Involving Infinity; Asymptotes of Graphs

97

68. The sign-preserving property of continuous functions Let ƒ be defined on an interval (a, b) and suppose that ƒscd Z 0 at some c where ƒ is continuous. Show that there is an interval sc - d, c + dd about c where ƒ has the same sign as ƒ(c). 69. Prove that ƒ is continuous at c if and only if lim ƒsc + hd = ƒscd .

h:0

is discontinuous at every point. b. Is ƒ right-continuous or left-continuous at any point?

70. Use Exercise 69 together with the identities

62. If functions ƒ(x) and g(x) are continuous for 0 … x … 1 , could ƒ(x)>g (x) possibly be discontinuous at a point of [0, 1]? Give reasons for your answer.

sin sh + cd = sin h cos c + cos h sin c , cos sh + cd = cos h cos c - sin h sin c

63. If the product function hsxd = ƒsxd # g sxd is continuous at x = 0 , must ƒ(x) and g(x) be continuous at x = 0 ? Give reasons for your answer.

64. Discontinuous composite of continuous functions Give an example of functions ƒ and g, both continuous at x = 0 , for which the composite ƒ ⴰ g is discontinuous at x = 0 . Does this contradict Theorem 9? Give reasons for your answer. 65. Never-zero continuous functions Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer. 66. Stretching a rubber band Is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? Give reasons for your answer. 67. A fixed point theorem Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0 … ƒsxd … 1 for every x in [0, 1]. Show that there must exist a number c in [0, 1] such that ƒscd = c (c is called a fixed point of ƒ).

to prove that both ƒsxd = sin x and g sxd = cos x are continuous at every point x = c . Solving Equations Graphically T Use the Intermediate Value Theorem in Exercises 71–78 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. 71. x 3 - 3x - 1 = 0 72. 2x 3 - 2x 2 - 2x + 1 = 0 73. xsx - 1d2 = 1

sone rootd

74. x = 2 x

75. 2x + 21 + x = 4 76. x 3 - 15x + 1 = 0 77. cos x = x 78. 2 sin x = x mode.

sthree rootsd

sone rootd . Make sure you are using radian mode. sthree rootsd . Make sure you are using radian

Limits Involving Infinity; Asymptotes of Graphs

2.6

In this section we investigate the behavior of a function when the magnitude of the independent variable x becomes increasingly large, or x : ; q . We further extend the concept of limit to infinite limits, which are not limits as before, but rather a new use of the term limit. Infinite limits provide useful symbols and language for describing the behavior of functions whose values become arbitrarily large in magnitude. We use these limit ideas to analyze the graphs of functions having horizontal or vertical asymptotes.

y 4 3 y 1x

2 1 –1 0 –1

1

2

3

4

FIGURE 2.49 The graph of y = 1>x approaches 0 as x : q or x : - q .

x

Finite Limits as x : —ˆ The symbol for infinity s q d does not represent a real number. We use q to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, the function ƒsxd = 1>x is defined for all x Z 0 (Figure 2.49). When x is positive and becomes increasingly large, 1>x becomes increasingly small. When x is negative and its magnitude becomes increasingly large, 1>x again becomes small. We summarize these observations by saying that ƒsxd = 1>x has limit 0 as x : q or x : - q , or that 0 is a limit of ƒsxd = 1>x at infinity and negative infinity. Here are precise definitions.

98

Chapter 2: Limits and Continuity

DEFINITIONS 1. We say that ƒ(x) has the limit L as x approaches infinity and write lim ƒsxd = L

x: q

if, for every number P 7 0, there exists a corresponding number M such that for all x x 7 M

Q

ƒ ƒsxd - L ƒ 6 P. 2. We say that ƒ(x) has the limit L as x approaches minus infinity and write lim ƒsxd = L

x: - q

if, for every number P 7 0, there exists a corresponding number N such that for all x x 6 N

y y 1x

M 1 –

and

lim

x: ; q

1 x = 0.

(1)

We prove the second result in Example 1, and leave the first to Exercises 87 and 88.

EXAMPLE 1 0

y –

lim k = k

x: ; q

y

N – 1

ƒ ƒsxd - L ƒ 6 P.

Intuitively, limx: q ƒsxd = L if, as x moves increasingly far from the origin in the positive direction, ƒ(x) gets arbitrarily close to L. Similarly, limx:- q ƒsxd = L if, as x moves increasingly far from the origin in the negative direction, ƒ(x) gets arbitrarily close to L. The strategy for calculating limits of functions as x : ; q is similar to the one for finite limits in Section 2.2. There we first found the limits of the constant and identity functions y = k and y = x. We then extended these results to other functions by applying Theorem 1 on limits of algebraic combinations. Here we do the same thing, except that the starting functions are y = k and y = 1>x instead of y = k and y = x. The basic facts to be verified by applying the formal definition are

No matter what positive number is, the graph enters this band at x 1 and stays.

Q

x

Show that

1 (a) lim x = 0 x: q

(b)

lim

x: - q

1 x = 0.

Solution No matter what positive number is, the graph enters this band at x – 1 and stays.

(a) Let P 7 0 be given. We must find a number M such that for all x

FIGURE 2.50 The geometry behind the argument in Example 1.

The implication will hold if M = 1>P or any larger positive number (Figure 2.50). This proves limx: q s1>xd = 0. (b) Let P 7 0 be given. We must find a number N such that for all x

x 7 M

x 6 N

Q

Q

1 1 ` x - 0 ` = ` x ` 6 P.

1 1 ` x - 0 ` = ` x ` 6 P.

The implication will hold if N = -1>P or any number less than -1>P (Figure 2.50). This proves limx:- q s1>xd = 0. Limits at infinity have properties similar to those of finite limits.

THEOREM 12 All the limit laws in Theorem 1 are true when we replace limx:c by limx: q or limx:- q . That is, the variable x may approach a finite number c or ; q .

2.6

Limits Involving Infinity; Asymptotes of Graphs

99

EXAMPLE 2 The properties in Theorem 12 are used to calculate limits in the same way as when x approaches a finite number c. 1 1 (a) lim a5 + x b = lim 5 + lim x x: q x: q x: q

Sum Rule

= 5 + 0 = 5 (b)

lim

x: - q

p23 = x2 =

Known limits

1 1 lim p23 # x # x

x: - q

1 lim p23 # lim x x: - q

x: - q

#

1 lim x

x: - q

= p23 # 0 # 0 = 0

2

–5

1

Line y 5 3

0

5

10

To determine the limit of a rational function as x : ; q , we first divide the numerator and denominator by the highest power of x in the denominator. The result then depends on the degrees of the polynomials involved. x

EXAMPLE 3

These examples illustrate what happens when the degree of the numerator is less than or equal to the degree of the denominator. 5 + s8>xd - s3>x 2 d 5x 2 + 8x - 3 = lim x: q x: q 3x 2 + 2 3 + s2>x 2 d

–1

(a) lim –2

NOT TO SCALE

= FIGURE 2.51 The graph of the function in Example 3a. The graph approaches the line y = 5>3 as ƒ x ƒ increases.

8 y 6

lim

x: - q

See Fig. 2.51.

s11>x 2 d + s2>x 3 d

Divide numerator and denominator by x3.

2 - s1>x d 3

0 + 0 = 0 2 - 0

See Fig. 2.52.

Horizontal Asymptotes

2

0

11x + 2 = x: - q 2x 3 - 1 lim

5 + 0 - 0 5 = 3 + 0 3

Divide numerator and denominator by x2.

A case for which the degree of the numerator is greater than the degree of the denominator is illustrated in Example 10.

11x 2 2x 3 1

4

–2

(b)

=

y

–4

Known limits

Limits at Infinity of Rational Functions

2 y 5x 2 8x 3 3x 2

y

Product Rule

2

4

6

–2

x

If the distance between the graph of a function and some fixed line approaches zero as a point on the graph moves increasingly far from the origin, we say that the graph approaches the line asymptotically and that the line is an asymptote of the graph. Looking at ƒsxd = 1>x (see Figure 2.49), we observe that the x-axis is an asymptote of the curve on the right because 1 lim x = 0

x: q

–4 –6

and on the left because 1 lim x = 0.

x: - q

–8

FIGURE 2.52 The graph of the function in Example 3b. The graph approaches the x-axis as ƒ x ƒ increases.

We say that the x-axis is a horizontal asymptote of the graph of ƒsxd = 1>x.

DEFINITION A line y = b is a horizontal asymptote of the graph of a function y = ƒsxd if either lim ƒsxd = b

x: q

or

lim ƒsxd = b.

x: - q

100

Chapter 2: Limits and Continuity

The graph of the function 5x 2 + 8x - 3 3x 2 + 2

ƒsxd =

sketched in Figure 2.51 (Example 3a) has the line y = 5>3 as a horizontal asymptote on both the right and the left because 5 3

lim ƒsxd =

x: q

EXAMPLE 4

ƒsxd =

2 y1

y –1

f(x) –2

For x Ú 0:

x3 – 2 x3 + 1

For x 6 0:

FIGURE 2.53 The graph of the function in Example 4 has two horizontal asymptotes.

5 . 3

x3 - 2 ƒ x ƒ 3 + 1.

We calculate the limits as x : ; q .

x

0

x: - q

Find the horizontal asymptotes of the graph of

y

Solution

lim ƒsxd =

and

lim

x: q

lim

x: - q

1 - (2>x 3) x3 - 2 x3 - 2 = lim = lim = 1. x: q x 3 + 1 x: q 1 + (1>x 3) ƒxƒ3 + 1 x3 - 2 = ƒxƒ3 + 1

lim

x: - q

x3 - 2 = (-x) 3 + 1

lim

x: - q

1 - (2>x 3) -1 + (1>x 3)

= -1.

The horizontal asymptotes are y = -1 and y = 1. The graph is displayed in Figure 2.53. Notice that the graph crosses the horizontal asymptote y = -1 for a positive value of x.

EXAMPLE 5 y = e x because

The x-axis (the line y = 0) is a horizontal asymptote of the graph of

y

lim e x = 0.

x: - q

y ex

To see this, we use the definition of a limit as x approaches - q. So let P 7 0 be given, but arbitrary. We must find a constant N such that for all x,

1

x 6 N N ln

x

FIGURE 2.54 The graph of y = e x approaches the x-axis as x : - q (Example 5).

Q

ƒex

- 0 ƒ 6 P.

Now ƒ e x - 0 ƒ = e x, so the condition that needs to be satisfied whenever x 6 N is e x 6 P. Let x = N be the number where e x = P. Since e x is an increasing function, if x 6 N , then e x 6 P. We find N by taking the natural logarithm of both sides of the equation e N = P, so N = ln P (see Figure 2.54). With this value of N the condition is satisfied, and we conclude that limx:- q e x = 0.

EXAMPLE 6

Find (a) lim sin s1>xd and x: q

(b)

lim x sin s1>xd.

x: ; q

Solution (a) We introduce the new variable t = 1>x. From Example 1, we know that t : 0 + as

x : q (see Figure 2.49). Therefore,

1 lim sin x = lim+ sin t = 0. t:0

x: q

2.6

Limits Involving Infinity; Asymptotes of Graphs

101

(b) We calculate the limits as x : q and x : - q :

y

sin t 1 lim x sin x = lim+ t = 1 q x: t:0

1

sin t 1 lim x sin x = lim- t = 1. q x: t:0

The graph is shown in Figure 2.55, and we see that the line y = 1 is a horizontal asymptote.

1 y x sin x –1

and

x

1

Likewise, we can investigate the behavior of y = ƒ(1>x) as x : 0 by investigating y = ƒ(t) as t : ; q , where t = 1>x.

FIGURE 2.55 The line y = 1 is a horizontal asymptote of the function graphed here (Example 6b).

EXAMPLE 7

Find lim-e 1>x. x:0

We let t = 1>x. From Figure 2.49, we can see that t : - q as x : 0 -. (We make this idea more precise further on.) Therefore, Solution

y

y

–3

–2

lim e 1>x =

1 0.8 0.6 0.4 0.2

e1 ⁄x

–1

0

x:0 -

lim e t = 0

(Figure 2.56). The Sandwich Theorem also holds for limits as x : ; q . You must be sure, though, that the function whose limit you are trying to find stays between the bounding functions at very large values of x in magnitude consistent with whether x : q or x : - q .

x

FIGURE 2.56 The graph of y = e 1>x for x 6 0 shows limx:0- e 1>x = 0 (Example 7).

EXAMPLE 8

Using the Sandwich Theorem, find the horizontal asymptote of the curve y = 2 +

Solution

sin x 1 0 … ` x ` … `x`

y 2 sinx x

and limx: ; q ƒ 1>x ƒ = 0, we have limx:; q ssin xd>x = 0 by the Sandwich Theorem. Hence,

2 1 0

2

3

sin x x .

We are interested in the behavior as x : ; q. Since

y

–3 –2 –

Example 5

t: - q

lim a2 +

x: ; q

x

FIGURE 2.57 A curve may cross one of its asymptotes infinitely often (Example 8).

sin x x b = 2 + 0 = 2,

and the line y = 2 is a horizontal asymptote of the curve on both left and right (Figure 2.57). This example illustrates that a curve may cross one of its horizontal asymptotes many times.

EXAMPLE 9

Find lim A x - 2x 2 + 16 B . x: q

Both of the terms x and 2x 2 + 16 approach infinity as x : q , so what happens to the difference in the limit is unclear (we cannot subtract q from q because the symbol does not represent a real number). In this situation we can multiply the numerator and the denominator by the conjugate radical expression to obtain an equivalent algebraic result: Solution

lim A x - 2x 2 + 16 B = lim A x - 2x 2 + 16 B

x: q

x: q

= lim

x: q

x 2 - (x 2 + 16) x + 2x + 16 2

x + 2x 2 + 16 x + 2x 2 + 16

= lim

x: q

-16 x + 2x 2 + 16

.

102

Chapter 2: Limits and Continuity

As x : q , the denominator in this last expression becomes arbitrarily large, so we see that the limit is 0. We can also obtain this result by a direct calculation using the Limit Laws: 16 - x 0 lim = lim = = 0. x: q x + 2x 2 + 16 x: q 16 x2 1 + 21 + 0 1 + + 2 A x2 x -16

Oblique Asymptotes 2 y5 x 235x111 1 2x 2 4 2 2x 2 4

y

The vertical distance between curve and line goes to zero as x → `

6 5 4 3

y5 x 11 2

Find the oblique asymptote of the graph of

2

3

4

ƒsxd =

x2 - 3 2x - 4

in Figure 2.58.

1 1

EXAMPLE 10

Oblique asymptote

x52

2

–1 0 –1

If the degree of the numerator of a rational function is 1 greater than the degree of the denominator, the graph has an oblique or slant line asymptote. We find an equation for the asymptote by dividing numerator by denominator to express ƒ as a linear function plus a remainder that goes to zero as x : ; q .

x

x

Solution

sx - 3d: 2

We are interested in the behavior as x : ; q . We divide s2x - 4d into

–2

x + 1 2

–3

2x - 4 x 2 - 3 x 2 - 2x 2x - 3 2x - 4 1

FIGURE 2.58 The graph of the function in Example 10 has an oblique asymptote.

This tells us that ƒsxd =

x2 - 3 x 1 = ¢ + 1≤ + ¢ ≤. 2x - 4 2 2x - 4 123 linear g(x)

y

B

You can get as high as you want by taking x close enough to 0. No matter how high B is, the graph goes higher. y 1x

x 0

x

x No matter how low –B is, the graph goes lower.

You can get as low as you want by taking x close enough to 0.

14243 remainder

As x : ; q , the remainder, whose magnitude gives the vertical distance between the graphs of ƒ and g, goes to zero, making the slanted line x + 1 2 an asymptote of the graph of ƒ (Figure 2.58). The line y = g(x) is an asymptote both to the right and to the left. The next subsection will confirm that the function ƒ(x) grows arbitrarily large in absolute value as x : 2 (where the denominator is zero), as shown in the graph. g(x) =

Notice in Example 10 that if the degree of the numerator in a rational function is greater than the degree of the denominator, then the limit as ƒ x ƒ becomes large is + q or - q , depending on the signs assumed by the numerator and denominator.

–B

FIGURE 2.59 One-sided infinite limits: 1 1 lim = q and lim- x = - q . x:0 + x x: 0

Infinite Limits Let us look again at the function ƒsxd = 1>x. As x : 0 + , the values of ƒ grow without bound, eventually reaching and surpassing every positive real number. That is, given any positive real number B, however large, the values of ƒ become larger still (Figure 2.59).

2.6

Limits Involving Infinity; Asymptotes of Graphs

103

Thus, ƒ has no limit as x : 0 + . It is nevertheless convenient to describe the behavior of ƒ by saying that ƒ(x) approaches q as x : 0 + . We write 1 lim ƒsxd = lim+ x = q . x:0

x:0 +

In writing this equation, we are not saying that the limit exists. Nor are we saying that there is a real number q , for there is no such number. Rather, we are saying that limx:0+ s1>xd does not exist because 1>x becomes arbitrarily large and positive as x : 0 + . As x : 0 - , the values of ƒsxd = 1>x become arbitrarily large and negative. Given any negative real number -B, the values of ƒ eventually lie below -B. (See Figure 2.59.) We write 1 lim ƒsxd = lim- x = - q . x:0

x:0 -

y

Again, we are not saying that the limit exists and equals the number - q . There is no real number - q . We are describing the behavior of a function whose limit as x : 0 - does not exist because its values become arbitrarily large and negative.

y 1 x1 1

EXAMPLE 11 –1

0

1

2

x

3

Find lim+ x:1

1 x - 1

and

lim

x:1 -

1 . x - 1

Geometric Solution The graph of y = 1>sx - 1d is the graph of y = 1>x shifted 1 unit to the right (Figure 2.60). Therefore, y = 1>sx - 1d behaves near 1 exactly the way y = 1>x behaves near 0: FIGURE 2.60 Near x = 1 , the function y = 1>sx - 1d behaves the way the function y = 1>x behaves near x = 0 . Its graph is the graph of y = 1>x shifted 1 unit to the right (Example 11).

lim

x:1 +

1 = q x - 1

lim

x:1 -

1 = -q. x - 1

Think about the number x - 1 and its reciprocal. As x : 1+ , we have sx - 1d : 0 and 1>sx - 1d : q . As x : 1- , we have sx - 1d : 0 - and 1>sx - 1d : -q. Analytic Solution +

EXAMPLE 12

Discuss the behavior of ƒsxd =

1 x2

as

x : 0.

As x approaches zero from either side, the values of 1>x 2 are positive and become arbitrarily large (Figure 2.61). This means that

Solution

y No matter how high B is, the graph goes higher.

B

lim ƒsxd = lim

x:0

f(x) 12 x

x 0

and

x

x

FIGURE 2.61 The graph of ƒ (x) in Example 12 approaches infinity as x : 0.

x:0

1 = q. x2

The function y = 1>x shows no consistent behavior as x : 0. We have 1>x : q if x : 0 + , but 1>x : - q if x : 0 - . All we can say about limx:0 s1>xd is that it does not exist. The function y = 1>x 2 is different. Its values approach infinity as x approaches zero from either side, so we can say that limx:0 s1>x 2 d = q .

EXAMPLE 13

These examples illustrate that rational functions can behave in various ways near zeros of the denominator. (a) lim

sx - 2d2 sx - 2d2 x - 2 = lim = lim = 0 2 x:2 sx - 2dsx + 2d x:2 x + 2 x - 4

(b) lim

x - 2 x - 2 1 1 = lim = lim = 2 x + 2 4 sx 2dsx + 2d x:2 x:2 x - 4

x:2

x:2

104

Chapter 2: Limits and Continuity

(c)

lim

x:2 +

(d) limx:2

x - 3 x - 3 = lim+ = -q x:2 sx - 2dsx + 2d x2 - 4

The values are negative for x 7 2, x near 2.

x - 3 x - 3 = lim= q x:2 sx - 2dsx + 2d x2 - 4

The values are positive for x 6 2, x near 2.

x - 3 x - 3 = lim does not exist. 2 sx 2dsx + 2d x:2 x - 4 -sx - 2d 2 - x -1 (f) lim = lim = lim = -q 3 x:2 sx - 2d x:2 sx - 2d3 x:2 sx - 2d2 (e) lim

See parts (c) and (d).

x:2

y

In parts (a) and (b) the effect of the zero in the denominator at x = 2 is canceled because the numerator is zero there also. Thus a finite limit exists. This is not true in part (f ), where cancellation still leaves a zero factor in the denominator.

y f (x)

Precise Definitions of Infinite Limits

B

0

c

c

x c

FIGURE 2.62 For c - d 6 x 6 c + d, the graph of ƒ(x) lies above the line y = B.

y c

c

Instead of requiring ƒ(x) to lie arbitrarily close to a finite number L for all x sufficiently close to c, the definitions of infinite limits require ƒ(x) to lie arbitrarily far from zero. Except for this change, the language is very similar to what we have seen before. Figures 2.62 and 2.63 accompany these definitions.

c

DEFINITIONS 1. We say that ƒ(x) approaches infinity as x approaches c, and write lim ƒsxd = q , x:c

if for every positive real number B there exists a corresponding d 7 0 such that for all x 0 6 ƒx - cƒ 6 d Q ƒsxd 7 B. 2. We say that ƒ(x) approaches minus infinity as x approaches c, and write lim ƒsxd = - q , x:c

x

0

if for every negative real number -B there exists a corresponding d 7 0 such that for all x Q ƒsxd 6 -B. 0 6 ƒx - cƒ 6 d

–B

The precise definitions of one-sided infinite limits at c are similar and are stated in the exercises. y f (x)

EXAMPLE 14 FIGURE 2.63 For c - d 6 x 6 c + d, the graph of ƒ(x) lies below the line y = -B.

Solution

Prove that lim

x:0

1 = q. x2

Given B 7 0, we want to find d 7 0 such that 0 6 ƒx - 0ƒ 6 d

implies

1 7 B. x2

Now, 1 7 B x2

if and only if x 2 6

or, equivalently, ƒxƒ 6

1 . 2B

1 B

2.6

Limits Involving Infinity; Asymptotes of Graphs

105

Thus, choosing d = 1> 2B (or any smaller positive number), we see that ƒxƒ 6 d

implies

1 1 7 2 Ú B. x2 d

Therefore, by definition, 1 = q. x2

lim

x:0

Vertical Asymptotes

y

Notice that the distance between a point on the graph of ƒsxd = 1>x and the y-axis approaches zero as the point moves vertically along the graph and away from the origin (Figure 2.64). The function ƒ(x) = 1>x is unbounded as x approaches 0 because

Vertical asymptote

Horizontal asymptote

y 1x

1 lim x = q

0

x

1

and

x:0 +

1 Horizontal asymptote, y0

Vertical asymptote, x0

We say that the line x = 0 (the y-axis) is a vertical asymptote of the graph of ƒ(x) = 1>x. Observe that the denominator is zero at x = 0 and the function is undefined there.

DEFINITION A line x = a is a vertical asymptote of the graph of a function y = ƒsxd if either

FIGURE 2.64 The coordinate axes are asymptotes of both branches of the hyperbola y = 1>x .

lim ƒsxd = ; q

or

x:a +

EXAMPLE 15

We are interested in the behavior as x : ; q and the behavior as x : -2, where the denominator is zero. The asymptotes are quickly revealed if we recast the rational function as a polynomial with a remainder, by dividing sx + 2d into sx + 3d:

6 4

Horizontal asymptote, y1

3

y

x3 x2

1

1 x2

1

3

1 x + 2 x + 3 x + 2 1

2 1

–5 –4 –3 –2 –1 0 –1

x + 3 . x + 2

Solution

y

5

lim ƒsxd = ; q .

x:a -

Find the horizontal and vertical asymptotes of the curve y =

Vertical asymptote, x –2

1 lim x = - q .

x:0 -

2

–2 –3 –4

FIGURE 2.65 The lines y = 1 and x = - 2 are asymptotes of the curve in Example 15.

x

This result enables us to rewrite y as: y = 1 +

1 . x + 2

As x : ; q , the curve approaches the horizontal asymptote y = 1; as x : -2, the curve approaches the vertical asymptote x = -2. We see that the curve in question is the graph of ƒ(x) = 1>x shifted 1 unit up and 2 units left (Figure 2.65). The asymptotes, instead of being the coordinate axes, are now the lines y = 1 and x = -2.

106

Chapter 2: Limits and Continuity y

EXAMPLE 16

8 7 6 5 4 3 2 1

Vertical asymptote, x –2

Find the horizontal and vertical asymptotes of the graph of

y – 28 x 4

ƒsxd = -

Vertical asymptote, x 2 Horizontal asymptote, y 0

–4 –3 –2 –1 0

1 2 3 4

x

8 . x2 - 4

Solution We are interested in the behavior as x : ; q and as x : ;2, where the denominator is zero. Notice that ƒ is an even function of x, so its graph is symmetric with respect to the y-axis. (a) The behavior as x : ; q . Since limx: q ƒsxd = 0, the line y = 0 is a horizontal asymptote of the graph to the right. By symmetry it is an asymptote to the left as well (Figure 2.66). Notice that the curve approaches the x-axis from only the negative side (or from below). Also, ƒs0d = 2.

(b) The behavior as x : ;2. Since lim ƒsxd = - q

x:2 +

FIGURE 2.66 Graph of the function in Example 16. Notice that the curve approaches the x-axis from only one side. Asymptotes do not have to be two-sided.

y

lim ƒsxd = q ,

x:2 -

the line x = 2 is a vertical asymptote both from the right and from the left. By symmetry, the line x = -2 is also a vertical asymptote. There are no other asymptotes because ƒ has a finite limit at every other point.

EXAMPLE 17 The graph of the natural logarithm function has the y-axis (the line x = 0) as a vertical asymptote. We see this from the graph sketched in Figure 2.67 (which is the reflection of the graph of the natural exponential function across the line y = x) and the fact that the x-axis is a horizontal asymptote of y = e x (Example 5). Thus,

y ex

4

lim ln x = - q .

3

x:0 +

2

y ln x

The same result is true for y = loga x whenever a 7 1.

1 –1

and

1

2

3

4

x

–1

FIGURE 2.67 The line x = 0 is a vertical asymptote of the natural logarithm function (Example 17).

EXAMPLE 18

The curves 1 y = sec x = cos x

and

sin x y = tan x = cos x

both have vertical asymptotes at odd-integer multiples of p>2, where cos x = 0 (Figure 2.68). y

y

y sec x

1 – 3 – – 2 2

0

y tan x

1 2

3 2

x

0 – 3 – – –1 2 2 2

3 2

FIGURE 2.68 The graphs of sec x and tan x have infinitely many vertical asymptotes (Example 18).

Dominant Terms In Example 10 we saw that by long division we could rewrite the function ƒ(x) =

x2 - 3 2x - 4

x

2.6

Limits Involving Infinity; Asymptotes of Graphs

107

as a linear function plus a remainder term: ƒ(x) = a

x 1 + 1b + a b. 2 2x - 4

This tells us immediately that y 20 15 10 f (x) 5 –2

–1

0

g(x) 3x 4 1

2

x

ƒsxd L

x + 1 2

For |x| large,

ƒsxd L

1 2x - 4

For x near 2, this term is very large.

1 is near 0. 2x - 4

If we want to know how ƒ behaves, this is the way to find out. It behaves like y = sx>2d + 1 when |x| is large and the contribution of 1>s2x - 4d to the total value of ƒ is insignificant. It behaves like 1>s2x - 4d when x is so close to 2 that 1>s2x - 4d makes the dominant contribution. We say that sx>2d + 1 dominates when x is numerically large, and we say that 1>s2x - 4d dominates when x is near 2. Dominant terms like these help us predict a function’s behavior.

–5

Let ƒsxd = 3x 4 - 2x 3 + 3x 2 - 5x + 6 and g sxd = 3x 4 . Show that although ƒ and g are quite different for numerically small values of x, they are virtually identical for ƒ x ƒ very large, in the sense that their ratios approach 1 as x : q or x : - q.

EXAMPLE 19

(a) y 500,000

The graphs of ƒ and g behave quite differently near the origin (Figure 2.69a), but appear as virtually identical on a larger scale (Figure 2.69b). We can test that the term 3x 4 in ƒ, represented graphically by g, dominates the polynomial ƒ for numerically large values of x by examining the ratio of the two functions as x : ; q . We find that

Solution 300,000

100,000 –20

–10

0

10

20

x

ƒsxd = x: ; q gsxd lim

–100,000 (b)

FIGURE 2.69 The graphs of ƒ and g are (a) distinct for ƒ x ƒ small, and (b) nearly identical for ƒ x ƒ large (Example 19).

=

3x 4 - 2x 3 + 3x 2 - 5x + 6 x: ; q 3x 4 lim

lim a1 -

x: ; q

5 2 2 1 + 4b + 2 3 3x x 3x x

= 1, which means that ƒ and g appear nearly identical when ƒ x ƒ is large.

Summary In this chapter we presented several important calculus ideas that are made meaningful and precise by the concept of the limit. These include the three ideas of the exact rate of change of a function, the slope of the graph of a function at a point, and the continuity of a function. The primary methods used for calculating limits of many functions are captured in the algebraic limit laws of Theorem 1 and in the Sandwich Theorem, all of which are proved from the precise definition of the limit. We saw that these computational rules also apply to one-sided limits and to limits at infinity. Moreover, we can sometimes apply these rules when calculating limits of simple transcendental functions, as illustrated by our examples or in cases like the following: lim

x:0

ex - 1 ex - 1 1 1 1 = lim x = = . = lim x x 2x 1 + 1 2 (e 1)(e + 1) e + 1 x:0 x:0 e - 1

108

Chapter 2: Limits and Continuity

However, calculating more complicated limits involving transcendental functions such as x

x ln x 1 and lim a1 + x b , , lim x:0 e 2x - 1 x:0 x x:0 requires more than simple algebraic techniques. The derivative is exactly the tool we need to calculate limits such as these (see Section 4.5), and this notion is the main subject of our next chapter. lim

Exercises 2.6 Finding Limits 1. For the function ƒ whose graph is given, determine the following limits. b.

d. lim ƒ(x)

e. lim+ ƒ(x)

f. lim- ƒ(x)

g. lim ƒ(x)

h. lim ƒ(x)

i.

x: 2

x: -3

x:0

lim + ƒ(x)

c.

x: -3 x: 0

x: -3 x: 0

x: q

sin 2x x 2 - t + sin t 11. lim t + cos t t: - q 9. lim

x: q

3 2

f

1 –6 –5 –4 –3 –2 –1 –1

1

2

3

4 5

6

x

–2 –3

2. For the function ƒ whose graph is given, determine the following limits. c. lim- ƒ(x)

a. lim ƒ(x)

b. lim+ ƒ(x)

d. lim ƒ(x)

e.

g. lim ƒ(x)

h. lim+ ƒ(x)

i. lim- ƒ(x)

j. lim ƒ(x)

k. lim ƒ(x)

l.

x: 2

x: -3 x: 0

x: 2

x: 2

lim + ƒ(x)

f.

x: -3 x: 0

2

3

2x + 3 5x + 7

14. ƒsxd =

2x 3 + 7 x - x2 + x + 7

15. ƒsxd =

x + 1 x2 + 3

16. ƒsxd =

3x + 7 x2 - 2

17. hsxd =

7x 3 x - 3x 2 + 6x

18. g sxd =

1 x 3 - 4x + 1

19. g sxd =

10x 5 + x 4 + 31 x6

20. hsxd =

9x 4 + x 2x + 5x 2 - x + 6

21. hsxd =

-2x 3 - 2x + 3 3x 3 + 3x 2 - 5x

22. hsxd =

-x 4 x 4 - 7x 3 + 7x 2 + 9

4 5

6

x

23. lim

x: q

–2

25.

–3

3

8x 2 - 3 A 2x 2 + x

lim ¢

x: - q

x: q

In Exercises 3–8, find the limit of each function (a) as x : q and (b) as x : - q . (You may wish to visualize your answer with a graphing calculator or computer.) 4. ƒsxd = p -

2 x2

1 - x3 ≤ x 2 + 7x

lim

x: - q

31. lim

x: q

5

3

4

5

3 2 x + 2x

2x 5>3 - x 1>3 + 7 x

8>5

lim ¢

x: - q

+ 3x + 2x

x2 + x - 1 ≤ 8x 2 - 3

x 2 - 5x x: q A x + x - 2

26. lim 28. lim

x: q

3

2 + 2x 2 - 2x

5

2x - 2x 3

29.

24.

22x + x -1 3x - 7

27. lim

2 3. ƒsxd = x - 3

lim

u: -q

Limits as x : ˆ or x : ˆ The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36.

1 1

cos u 3u r + sin r 12. lim r: q 2r + 7 - 5 sin r 10.

lim ƒ(x)

x: - q

f

2

4 + (22>x 2)

13. ƒsxd =

lim ƒ(x)

x: -3 -

y 3

3 - s2>xd

8. hsxd =

Limits of Rational Functions In Exercises 13–22, find the limit of each rational function (a) as x : q and (b) as x : - q .

x: 0

x: q

–6 –5 –4 –3 –2 –1 –1

3 - s1>x 2 d

1 8 - s5>x 2 d

6. g sxd =

Find the limits in Exercises 9–12.

lim ƒ(x)

x: - q

y

x: 4

-5 + s7>xd

7. hsxd =

lim - ƒ(x)

a. lim ƒ(x)

1 2 + s1>xd

5. g sxd =

30. lim

x: q

32.

lim

x: - q

x -1 + x -4 x -2 - x -3 3 2 x - 5x + 3 2x + x 2>3 - 4

1>3

2.6

33. lim

x: q

35. lim

x: q

2x 2 + 1 x + 1

34.

x - 3

36.

24x 2 + 25

2x 2 + 1 x + 1

lim

x: - q

4 - 3x 3

lim

x: - q

2x 6 + 9

Infinite Limits Find the limits in Exercises 37–48. 5 1 37. lim+ 38. limx: 0 3x x: 0 2x 3 1 39. lim40. lim+ x: 2 x - 2 x: 3 x - 3 3x 2x 41. lim + 42. lim x: -8 x + 8 x: -5 2x + 10 4 sx - 7d2 2 45. a. lim+ 1>3 x: 0 3x 2 46. a. lim+ 1>5 x: 0 x 4 47. lim 2>5 x: 0 x 43. lim

44. lim

x: 7

x: 0

-1 x 2sx + 1d

51.

lim

x: sp>2d

60. lim a

1 + 7b as t 3>5 a. t : 0 +

tan x -

48. lim

x: 0

lim s1 + csc ud

u :0 -

50.

1 x 2>3

lim

x: s-p>2d

sec x +

52. lim s2 - cot ud u: 0

Find the limits in Exercises 53–58. 1 as 53. lim 2 x - 4 a. x : 2+ b. c. x : -2+ d. x as 54. lim 2 x - 1 a. x : 1+ b. c. x : -1+ d.

61. lim a

1 2 + b as x 2>3 sx - 1d2>3 a. x : 0 + b. x : 0 c. x : 1+

x : 2-

x : -2-

x : 1x : -1-

x 1 55. lim a - x b as 2

1 1 b as x 1>3 sx - 1d4>3 a. x : 0 + b. x : 0 d. x : 1-

Graphing Simple Rational Functions Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms. 1 1 63. y = 64. y = x - 1 x + 1 65. y =

1 2x + 4

66. y =

-3 x - 3

67. y =

x + 3 x + 2

68. y =

2x x + 1

Inventing Graphs and Functions In Exercises 69–72, sketch the graph of a function y = ƒsxd that satisfies the given conditions. No formulas are required—just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) 69. ƒs0d = 0, ƒs1d = 2, ƒs -1d = -2, lim ƒsxd = -1, and x: - q lim ƒsxd = 1 x: q

70. ƒs0d = 0, lim ƒsxd = 0, lim+ ƒsxd = 2, and x: ; q

x:0

lim- ƒsxd = -2

x:0

b. x : 0 d. x : -1

71. ƒs0d = 0, lim ƒsxd = 0, lim- ƒsxd = lim + ƒsxd = q , x: ; q

x: -1

x:1

lim+ ƒsxd = - q , and lim - ƒsxd = - q x: -1

x:1

72. ƒs2d = 1, ƒs -1d = 0, lim ƒsxd = 0, lim+ ƒsxd = q ,

x - 1 as 2x + 4 + a. x : -2 c. x : 1+

b. x : -2 d. x : 0 -

x 2 - 3x + 2 as 57. lim x 3 - 2x 2 + a. x : 0 c. x : 2-

b. x : 2+ d. x : 2

2

x: q

56. lim

e. What, if anything, can be said about the limit as x : 0 ?

x: - q

x:0

In Exercises 73–76, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) 73. lim ƒsxd = 0, lim- ƒsxd = q , and lim+ ƒsxd = q x: ; q

74.

58. lim

b. x : -2+ d. x : 1+

x:0

lim- ƒsxd = - q , and lim ƒsxd = 1

-

e. What, if anything, can be said about the limit as x : 0 ? x 2 - 3x + 2 as x 3 - 4x a. x : 2+ c. x : 0 -

d. x : 1-

62. lim a

2

a. x : 0 + 3 2 c. x : 2

b. t : 0 -

c. x : 1+

2 x: 0 3x 1>3 2 b. lim- 1>5 x: 0 x

109

Find the limits in Exercises 59–62. 3 59. lim a2 - 1>3 b as t a. t : 0 + b. t : 0 -

b. lim-

Find the limits in Exercises 49–52. 49.

Limits Involving Infinity; Asymptotes of Graphs

75.

x:2

x:2

lim g sxd = 0, lim- g sxd = - q , and lim+ g sxd = q

x: ; q

x:3

x: 3

lim hsxd = -1, lim hsxd = 1, lim- hsxd = -1, and

x: - q

x: q

x:0

lim+ hsxd = 1

x:0

76.

lim k sxd = 1, lim- k sxd = q , and lim+ k sxd = - q

x: ; q

x:1

x:1

110

Chapter 2: Limits and Continuity

77. Suppose that ƒ(x) and g(x) are polynomials in x and that limx: q sƒsxd>g sxdd = 2 . Can you conclude anything about limx: - q sƒsxd>g sxdd ? Give reasons for your answer.

Modify the definition to cover the following cases. a. lim- ƒsxd = q x:c

b. lim+ ƒsxd = - q

78. Suppose that ƒ(x) and g(x) are polynomials in x. Can the graph of ƒ(x)>g (x) have an asymptote if g(x) is never zero? Give reasons for your answer. 79. How many horizontal asymptotes can the graph of a given rational function have? Give reasons for your answer. Finding Limits of Differences when x : ; ˆ Find the limits in Exercises 80–86. 80. lim

x: q

81. lim

x: q

82. 83.

A 2x + 9 - 2x + 4 B

lim

x: - q

lim

x: - q

84. lim

x: q

85. lim

x: q

c. lim- ƒsxd = - q x:c

Use the formal definitions from Exercise 93 to prove the limit statements in Exercises 94–98. 1 1 94. lim+ x = q 95. lim- x = - q x:0 x:0 1 = -q x - 2 1 = q 98. limx:1 1 - x 2 x:2

2

A 2x 2 + 3 + x B

A 2x + 24x 2 + 3x - 2 B

A 29x - x - 3x B

A 2x 2 + 3x - 2x 2 - 2x B

86. lim A 2x 2 + x - 2x 2 - x B x: q

x2 - 4 x - 1 x2 - 1 103. y = x

87. If ƒ has the constant value ƒsxd = k , then lim ƒsxd = k . x: q

88. If ƒ has the constant value ƒsxd = k , then lim ƒsxd = k . x: - q

Use formal definitions to prove the limit statements in Exercises 89–92. -1 1 = q 89. lim 2 = - q 90. lim x:0 x x: 0 ƒ x ƒ -2 1 = -q = q 91. lim 92. lim x:3 sx - 3d2 x: -5 sx + 5d2 93. Here is the definition of infinite right-hand limit.

x2 - 1 2x + 4 x3 + 1 104. y = x2

Additional Graphing Exercises T Graph the curves in Exercises 105–108. Explain the relationship between the curve’s formula and what you see. 105. y =

ƒsxd 7 B.

106. y =

-1

24 - x 2 p 108. y = sin a 2 b x + 1

T Graph the functions in Exercises 109 and 110. Then answer the following questions. a. How does the graph behave as x : 0 +? b. How does the graph behave as x : ; q ? c. How does the graph behave near x = 1 and x = -1?

x: c

if, for every positive real number B, there exists a corresponding number d 7 0 such that for all x

x

24 - x 2 1 107. y = x 2>3 + 1>3 x

We say that ƒ(x) approaches infinity as x approaches c from the right, and write lim+ ƒsxd = q ,

Q

1 = q x - 2

102. y =

101. y =

Using the Formal Definitions Use the formal definitions of limits as x : ; q to establish the limits in Exercises 87 and 88.

Chapter 2

x:2

Oblique Asymptotes Graph the rational functions in Exercises 99–104. Include the graphs and equations of the asymptotes. x2 + 1 x2 99. y = 100. y = x - 1 x - 1

2

c 6 x 6 c + d

97. lim+

96. lim-

A 2x + 25 - 2x - 1 B 2

x:c

Give reasons for your answers. 109. y =

3 1 ax - x b 2

2>3

110. y =

3 x a b 2 x - 1

Questions to Guide Your Review

1. What is the average rate of change of the function g(t) over the interval from t = a to t = b ? How is it related to a secant line? 2. What limit must be calculated to find the rate of change of a function g(t) at t = t0 ?

3. Give an informal or intuitive definition of the limit lim ƒsxd = L.

x:c

Why is the definition “informal”? Give examples.

2>3

Chapter 2

Practice Exercises

111

4. Does the existence and value of the limit of a function ƒ(x) as x approaches c ever depend on what happens at x = c ? Explain and give examples.

13. What does it mean for a function to be right-continuous at a point? Left-continuous? How are continuity and one-sided continuity related?

5. What function behaviors might occur for which the limit may fail to exist? Give examples.

14. What does it mean for a function to be continuous on an interval? Give examples to illustrate the fact that a function that is not continuous on its entire domain may still be continuous on selected intervals within the domain.

6. What theorems are available for calculating limits? Give examples of how the theorems are used. 7. How are one-sided limits related to limits? How can this relationship sometimes be used to calculate a limit or prove it does not exist? Give examples. 8. What is the value of lim u:0 sssin ud>ud ? Does it matter whether u is measured in degrees or radians? Explain. 9. What exactly does limx:c ƒsxd = L mean? Give an example in which you find a d 7 0 for a given ƒ, L, c , and P 7 0 in the precise definition of limit. 10. Give precise definitions of the following statements. a. limx:2- ƒsxd = 5 c. limx:2 ƒsxd = q

b. limx:2+ ƒsxd = 5 d. limx:2 ƒsxd = - q

11. What conditions must be satisfied by a function if it is to be continuous at an interior point of its domain? At an endpoint? 12. How can looking at the graph of a function help you tell where the function is continuous?

Chapter 2

1, -x, ƒsxd = e 1, -x, 1,

17. Under what circumstances can you extend a function ƒ(x) to be continuous at a point x = c ? Give an example. 18. What exactly do limx: q ƒsxd = L and limx:- q ƒsxd = L mean? Give examples. 19. What are limx:; q k (k a constant) and limx:; q s1>xd ? How do you extend these results to other functions? Give examples. 20. How do you find the limit of a rational function as x : ; q ? Give examples. 21. What are horizontal and vertical asymptotes? Give examples.

e. cos (g(t)) g. ƒstd + g std x -1 x 0 x

… 6 = 6 Ú

-1 x 6 0 0 x 6 1 1.

Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = -1, 0 , and 1. Are any of the discontinuities removable? Explain. 2. Repeat the instructions of Exercise 1 for 0, 1>x, ƒsxd = d 0, 1,

x 0 x x

… 6 = 7

-1 ƒxƒ 6 1 1 1.

3. Suppose that ƒ(t) and g(t) are defined for all t and that limt:t0 ƒstd = -7 and limt:t0 g std = 0 . Find the limit as t : t0 of the following functions. c. ƒstd # g std

16. What does it mean for a function to have the Intermediate Value Property? What conditions guarantee that a function has this property over an interval? What are the consequences for graphing and solving the equation ƒsxd = 0 ?

Practice Exercises

Limits and Continuity 1. Graph the function

a. 3ƒ(t)

15. What are the basic types of discontinuity? Give an example of each. What is a removable discontinuity? Give an example.

b. sƒstdd2 ƒstd d. g std - 7

f. ƒ ƒstd ƒ h. 1>ƒ(t)

4. Suppose the functions ƒ(x) and g(x) are defined for all x and that limx:0 ƒsxd = 1>2 and limx:0 g sxd = 22 . Find the limits as x : 0 of the following functions. a. -g sxd c. ƒsxd + g sxd e. x + ƒsxd

b. g sxd # ƒsxd

d. 1>ƒ(x) ƒsxd # cos x f. x - 1

In Exercises 5 and 6, find the value that limx:0 g sxd must have if the given limit statements hold. 4 - g sxd b = 1 5. lim a x x:0 6. lim ax lim g sxdb = 2 x: -4

x:0

7. On what intervals are the following functions continuous? a. ƒsxd = x 1>3

b. g sxd = x 3>4

c. hsxd = x -2>3

d. ksxd = x -1>6

8. On what intervals are the following functions continuous? a. ƒsxd = tan x

b. g sxd = csc x

cos x c. hsxd = x - p

d. ksxd =

sin x x

112

Chapter 2: Limits and Continuity

Finding Limits In Exercises 9–28, find the limit or explain why it does not exist. x 2 - 4x + 4 x + 5x 2 - 14x a. as x : 0

9. lim

3

x2 + x 10. lim 5 x + 2x 4 + x 3 a. as x : 0 11. lim

x:1

1 - 2x 1 - x

x:0

s2 + xd3 - 8 x x: 0

16. lim

x 2>3 - 16

18. lim

2x - 1 tan (2x) 19. lim x:0 tan (px)

23. lim

x - a x4 - a4 sx + hd2 - x 2 14. lim h x: 0 2

x: 64

2x - 8

19 - 1b A 27

1>3

- a

19 + 1b A 27

1>3

.

Evaluate this exact answer and compare it with the value you found in part (b).

x: a

x:1

x:p

a

b. as x : -1 12. lim

x 1>3 - 1

21. lim sin a

c. It can be shown that the exact value of the solution in part (b) is

2

sx + hd2 - x 2 13. lim h h :0 1 1 2 + x 2 15. lim x x:0 17. lim

b. as x : 2

3 T 34. Let ƒsud = u - 2u + 2 . a. Use the Intermediate Value Theorem to show that ƒ has a zero between -2 and 0. b. Solve the equation ƒsud = 0 graphically with an error of magnitude at most 10 -4 .

Continuous Extension 35. Can ƒsxd = xsx 2 - 1d> ƒ x 2 - 1 ƒ be extended to be continuous at x = 1 or -1 ? Give reasons for your answers. (Graph the function—you will find the graph interesting.) 36. Explain why the function ƒsxd = sin s1>xd has no continuous extension to x = 0 .

20. lim csc x x: p -

x + sin x b 2

8x 3 sin x - x

22. lim cos2 sx - tan xd x: p

24. lim

x: 0

cos 2x - 1 sin x

25. lim+ ln (t - 3)

26. lim t 2 ln A 2 - 2t B

27. lim+ 2u ecos (p>u)

28. lim+

t:3

t: 1

u: 0

z: 0

2e1>z 1>z e + 1

In Exercises 29–32, find the limit of g(x) as x approaches the indicated value. 29. lim+ s4g sxdd1>3 = 2

T In Exercises 37–40, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be? x - 1 , a = 1 4 x - 2x 5 cos u 38. g sud = , a = p>2 4u - 2p 37. ƒsxd =

39. hstd = s1 + ƒ t ƒ d1>t, a = 0 x 40. k sxd = , a = 0 1 - 2ƒ x ƒ

x:0

30.

lim

x: 25

31. lim

x:1

32. lim

1 = 2 x + g sxd

3x 2 + 1 = q g sxd

x: -2

5 - x2 2g sxd

= 0

Limits at Infinity Find the limits in Exercises 41–54. 2x + 3 41. lim 42. x: q 5x + 7 43. 45.

T Roots 33. Let ƒsxd = x 3 - x - 1 . a. Use the Intermediate Value Theorem to show that ƒ has a zero between -1 and 2. b. Solve the equation ƒsxd = 0 graphically with an error of magnitude at most 10 -8 . c. It can be shown that the exact value of the solution in part (b) is a

269 1>3 269 1>3 1 1 + a + b b . 2 18 2 18

Evaluate this exact answer and compare it with the value you found in part (b).

lim

x: - q

x 2 - 4x + 8 3x 3

x 2 - 7x x: - q x + 1 lim

47. lim

x: q

sin x :x;

cos u - 1 u u: q

48. lim

lim

x: - q

44. lim

x: q

2x2 + 3 5x 2 + 7 1 x 2 - 7x + 1

x4 + x3 x: q 12x 3 + 128 sIf you have a grapher, try graphing the function for -5 … x … 5.d 46. lim

sIf you have a grapher, try graphing ƒsxd = xscos s1>xd - 1d near the origin to “see” the limit at infinity.d

x + sin x + 22x x + sin x x: q

x 2>3 + x -1 x 2>3 + cos2 x

49. lim

50. lim

1 51. lim e1>x cos x x: q

1 52. lim ln a1 + t b t: q

53.

lim tan-1 x

x: - q

x: q

54.

1 lim e3t sin-1 t

t: - q

Chapter 2 Horizontal and Vertical Asymptotes 55. Use limits to determine the equations for all vertical asymptotes. x2 + 4 a. y = x - 3

x2 - x - 2 b. ƒ(x) = 2 x - 2x + 1

x2 + x - 6 c. y = 2 x + 2x - 8

Chapter 2 T

113

56. Use limits to determine the equations for all horizontal asymptotes. a. y =

1 - x2 x2 + 1

c. g(x) =

b. ƒ(x) =

2x + 4 x 2

d. y =

2x + 4 2x + 4

x2 + 9 A 9x 2 + 1

Additional and Advanced Exercises

1. Assigning a value to 00 The rules of exponents tell us that a 0 = 1 if a is any number different from zero. They also tell us that 0 n = 0 if n is any positive number. If we tried to extend these rules to include the case 0 0 , we would get conflicting results. The first rule would say 0 0 = 1 , whereas the second would say 0 0 = 0 . We are not dealing with a question of right or wrong here. Neither rule applies as it stands, so there is no contradiction. We could, in fact, define 0 0 to have any value we wanted as long as we could persuade others to agree. What value would you like 0 0 to have? Here is an example that might help you to decide. (See Exercise 2 below for another example.) a. Calculate x x for x = 0.1 , 0.01, 0.001, and so on as far as your calculator can go. Record the values you get. What pattern do you see? b. Graph the function y = x x for 0 6 x … 1 . Even though the function is not defined for x … 0 , the graph will approach the y-axis from the right. Toward what y-value does it seem to be headed? Zoom in to further support your idea.

T

Additional and Advanced Exercises

2. A reason you might want 00 to be something other than 0 or 1 As the number x increases through positive values, the numbers 1>x and 1> (ln x) both approach zero. What happens to the number 1 ƒsxd = a x b

1>sln xd

as x increases? Here are two ways to find out. a. Evaluate ƒ for x = 10 , 100, 1000, and so on as far as your calculator can reasonably go. What pattern do you see? b. Graph ƒ in a variety of graphing windows, including windows that contain the origin. What do you see? Trace the y-values along the graph. What do you find? 3. Lorentz contraction In relativity theory, the length of an object, say a rocket, appears to an observer to depend on the speed at which the object is traveling with respect to the observer. If the observer measures the rocket’s length as L0 at rest, then at speed y the length will appear to be L = L0

1 -

B

y2 . c2

This equation is the Lorentz contraction formula. Here, c is the speed of light in a vacuum, about 3 * 10 8 m>sec . What happens to L as y increases? Find limy:c- L . Why was the left-hand limit needed?

4. Controlling the flow from a draining tank Torricelli’s law says that if you drain a tank like the one in the figure shown, the rate y at which water runs out is a constant times the square root of the water’s depth x. The constant depends on the size and shape of the exit valve.

x Exit rate y ft 3min

Suppose that y = 2x>2 for a certain tank. You are trying to maintain a fairly constant exit rate by adding water to the tank with a hose from time to time. How deep must you keep the water if you want to maintain the exit rate a. within 0.2 ft3>min of the rate y0 = 1 ft3>min ?

b. within 0.1 ft3>min of the rate y0 = 1 ft3>min ? 5. Thermal expansion in precise equipment As you may know, most metals expand when heated and contract when cooled. The dimensions of a piece of laboratory equipment are sometimes so critical that the shop where the equipment is made must be held at the same temperature as the laboratory where the equipment is to be used. A typical aluminum bar that is 10 cm wide at 70°F will be y = 10 + st - 70d * 10 -4 centimeters wide at a nearby temperature t. Suppose that you are using a bar like this in a gravity wave detector, where its width must stay within 0.0005 cm of the ideal 10 cm. How close to t0 = 70°F must you maintain the temperature to ensure that this tolerance is not exceeded? 6. Stripes on a measuring cup The interior of a typical 1-L measuring cup is a right circular cylinder of radius 6 cm (see accompanying figure). The volume of water we put in the cup is therefore a function of the level h to which the cup is filled, the formula being V = p62h = 36ph . How closely must we measure h to measure out 1 L of water s1000 cm3 d with an error of no more than 1% s10 cm3 d ?

114

Chapter 2: Limits and Continuity 17. A function continuous at only one point ƒsxd = e

Let

x, if x is rational 0, if x is irrational.

a. Show that ƒ is continuous at x = 0 . Stripes about 1 mm wide

b. Use the fact that every nonempty open interval of real numbers contains both rational and irrational numbers to show that ƒ is not continuous at any nonzero value of x. 18. The Dirichlet ruler function If x is a rational number, then x can be written in a unique way as a quotient of integers m>n where n 7 0 and m and n have no common factors greater than 1. (We say that such a fraction is in lowest terms. For example, 6>4 written in lowest terms is 3>2.) Let ƒ(x) be defined for all x in the interval [0, 1] by

(a) r 6 cm

ƒsxd = e Liquid volume V 36h

h

1>n, 0,

if x = m>n is a rational number in lowest terms if x is irrational.

For instance, ƒs0d = ƒs1d = 1, ƒs1>2d = 1>2, ƒs1>3d = ƒ(2>3) = 1>3, ƒs1>4d = ƒs3>4d = 1>4 , and so on. a. Show that ƒ is discontinuous at every rational number in [0, 1].

(b)

A 1-L measuring cup (a), modeled as a right circular cylinder (b) of radius r = 6 cm Precise Definition of Limit In Exercises 7–10, use the formal definition of limit to prove that the function is continuous at c. 7. ƒsxd = x2 - 7,

8. g sxd = 1>s2xd,

c = 1

9. hsxd = 22x - 3,

c = 1>4

c = 2 10. Fsxd = 29 - x,

c = 5

11. Uniqueness of limits Show that a function cannot have two different limits at the same point. That is, if limx:c ƒsxd = L1 and limx:c ƒsxd = L2 , then L1 = L2 . 12. Prove the limit Constant Multiple Rule:

13. One-sided limits find

x: c

If limx:0 ƒsxd = A and limx:0 ƒsxd = B , +

-

a. limx:0+ ƒsx 3 - xd

b. limx:0- ƒsx 3 - xd

c. limx:0+ ƒsx 2 - x 4 d

d. limx:0- ƒsx 2 - x 4 d

a. If limx:c ƒsxd exists but limx:c g sxd does not exist, then limx:csƒsxd + g sxdd does not exist. b. If neither limx:c ƒsxd nor limx:c g sxd exists, then limx:c sƒsxd + g sxdd does not exist. c. If ƒ is continuous at x, then so is ƒ ƒ ƒ . d. If ƒ ƒ ƒ is continuous at c, then so is ƒ. In Exercises 15 and 16, use the formal definition of limit to prove that the function has a continuous extension to the given value of x. x2 - 1 , x + 1

x = -1

16. g sxd =

19. Antipodal points Is there any reason to believe that there is always a pair of antipodal (diametrically opposite) points on Earth’s equator where the temperatures are the same? Explain. 20. If lim sƒsxd + g sxdd = 3 and lim sƒsxd - g sxdd = -1 , find x:c

x 2 - 2x - 3 , 2x - 6

x:c

lim ƒsxdg sxd .

x:c

21. Roots of a quadratic equation that is almost linear The equation ax 2 + 2x - 1 = 0 , where a is a constant, has two roots if a 7 -1 and a Z 0 , one positive and one negative: -1 + 21 + a , a

r-sad =

-1 - 21 + a . a

a. What happens to r+sad as a : 0 ? As a : -1+ ? b. What happens to r-sad as a : 0 ? As a : -1+ ? c. Support your conclusions by graphing r+sad and r-sad as functions of a. Describe what you see.

14. Limits and continuity Which of the following statements are true, and which are false? If true, say why; if false, give a counterexample (that is, an example confirming the falsehood).

15. ƒsxd =

c. Sketch the graph of ƒ. Why do you think ƒ is called the “ruler function”?

r+sad =

lim kƒsxd = k lim ƒsxd for any constant k .

x:c

b. Show that ƒ is continuous at every irrational number in [0, 1]. (Hint: If P is a given positive number, show that there are only finitely many rational numbers r in [0, 1] such that ƒsrd Ú P .)

x = 3

d. For added support, graph ƒsxd = ax 2 + 2x - 1 simultaneously for a = 1, 0.5, 0.2, 0.1, and 0.05. 22. Root of an equation Show that the equation x + 2 cos x = 0 has at least one solution. 23. Bounded functions A real-valued function ƒ is bounded from above on a set D if there exists a number N such that ƒsxd … N for all x in D. We call N, when it exists, an upper bound for ƒ on D and say that ƒ is bounded from above by N. In a similar manner, we say that ƒ is bounded from below on D if there exists a number M such that ƒsxd Ú M for all x in D. We call M, when it exists, a lower bound for ƒ on D and say that ƒ is bounded from below by M. We say that ƒ is bounded on D if it is bounded from both above and below. a. Show that ƒ is bounded on D if and only if there exists a number B such that ƒ ƒsxd ƒ … B for all x in D.

Chapter 2 b. Suppose that ƒ is bounded from above by N. Show that if limx:c ƒsxd = L , then L … N . c. Suppose that ƒ is bounded from below by M. Show that if limx:c ƒsxd = L , then L Ú M .

24. Max 5a, b6 and min 5a, b6 a. Show that the expression

ƒa - bƒ a + b + 2 2 equals a if a Ú b and equals b if b Ú a . In other words, max {a, b} gives the larger of the two numbers a and b. max 5a, b6 =

b. Find a similar expression for min 5a, b6, the smaller of a and b.

sin U U The formula limu:0ssin ud>u = 1 can be generalized. If limx:c ƒsxd = 0 and ƒ(x) is never zero in an open interval containing the point x = c, except possibly c itself, then sin ƒsxd lim = 1. x: c ƒsxd Here are several examples.

sin sx 2 - x - 2d sin sx 2 - x - 2d # = lim x + 1 x: -1 x: -1 sx 2 - x - 2d sx 2 - x - 2d sx + 1dsx - 2d = 1 # lim = -3 x + 1 x + 1 x: -1 x: -1 lim

d. lim

sin A 1 - 2x B x - 1

x:1

1 # lim

x:1

x:1

sin A 1 - 2x B 1 - 2x = x - 1 1 - 2x = lim

sx - 1d A 1 + 2x B

x:1

1 - x

sx - 1d A 1 + 2x B

Find the limits in Exercises 25–30. 25. lim

sin s1 - cos xd x

sin ssin xd 27. lim x x:0 sin sx 2 - 4d x - 2 x:2

29. lim

26. lim+ x:0

sin x sin 2x

sin sx 2 + xd 28. lim x x:0 30. lim

x:9

sin A 2x - 3 B x - 9

Oblique Asymptotes Find all possible oblique asymptotes in Exercises 31–34. 31. y =

sin x 2 sin x 2 x2 lim x = 1 # 0 = 0 = lim x 2 x: 0 x: 0 x x: 0

33. y = 2x 2 + 1

b. lim

= lim

A 1 - 2x B A 1 + 2x B

sin x 2 = 1 x: 0 x 2

a. lim

115

c. lim

x:0

Generalized Limits Involving

Additional and Advanced Exercises

2x 3>2 + 2x - 3 2x + 1

1 32. y = x + x sin x 34. y = 2x 2 + 2x

=-

1 2

3 DIFFERENTIATION OVERVIEW In the beginning of Chapter 2, we discussed how to determine the slope of a curve at a point and how to measure the rate at which a function changes. Now that we have studied limits, we can define these ideas precisely and see that both are interpretations of the derivative of a function at a point. We then extend this concept from a single point to the derivative function, and we develop rules for finding this derivative function easily, without having to calculate any limits directly. These rules are used to find derivatives of most of the common functions reviewed in Chapter 1, as well as various combinations of them. The derivative is one of the key ideas in calculus, and we use it to solve a wide range of problems involving tangents and rates of change.

Tangents and the Derivative at a Point

3.1

In this section we define the slope and tangent to a curve at a point, and the derivative of a function at a point. We will see that the derivative gives a way to find both the slope of a graph and the instantaneous rate of change of a function.

Finding a Tangent to the Graph of a Function

y y f(x) Q(x 0 h, f(x 0 h)) f (x 0 h) f (x 0)

To find a tangent to an arbitrary curve y = ƒ(x) at a point Psx0 , ƒsx0 dd, we use the procedure introduced in Section 2.1. We calculate the slope of the secant through P and a nearby point Qsx0 + h, ƒsx0 + hdd. We then investigate the limit of the slope as h : 0 (Figure 3.1). If the limit exists, we call it the slope of the curve at P and define the tangent at P to be the line through P having this slope.

P(x 0, f(x 0)) h 0

x0

x0 h

FIGURE 3.1 The slope of the tangent ƒsx0 + hd - ƒsx0 d line at P is lim . h h:0

x

DEFINITIONS number

The slope of the curve y = ƒsxd at the point Psx0 , ƒsx0 dd is the ƒsx0 + hd - ƒsx0 d h h:0

m = lim

(provided the limit exists).

The tangent line to the curve at P is the line through P with this slope.

In Section 2.1, Example 3, we applied these definitions to find the slope of the parabola ƒ(x) = x 2 at the point P(2, 4) and the tangent line to the parabola at P. Let’s look at another example.

116

3.1

117

EXAMPLE 1

y

(a) Find the slope of the curve y = 1>x at any point x = a Z 0. What is the slope at the point x = -1? (b) Where does the slope equal -1>4? (c) What happens to the tangent to the curve at the point (a, 1>a) as a changes?

y 5 1x slope is – 12 a

0

Tangents and the Derivative at a Point

x

a

Solution

(a) Here ƒsxd = 1>x. The slope at (a, 1>a) is slope is –1 at x 5 –1

FIGURE 3.2 The tangent slopes, steep near the origin, become more gradual as the point of tangency moves away (Example 1). y y 1x slope is – 1 4

1 1 - a ƒsa + hd - ƒsad a + h 1 a - sa + hd lim = lim = lim h h h:0 h:0 h:0 h asa + hd = lim

h:0

Notice how we had to keep writing “limh:0” before each fraction until the stage where we could evaluate the limit by substituting h = 0. The number a may be positive or negative, but not 0. When a = -1, the slope is -1>(-1) 2 = -1 (Figure 3.2). (b) The slope of y = 1>x at the point where x = a is -1>a 2. It will be -1>4 provided that

⎛2, 1 ⎛ ⎝ 2⎝

x

⎛–2, – 1 ⎛ ⎝ 2⎝

slope is – 1 4

FIGURE 3.3 The two tangent lines to y = 1>x having slope -1>4 (Example 1).

-h 1 -1 = lim = - 2. hasa + hd h:0 asa + hd a

1 1 = - . 4 a2

This equation is equivalent to a 2 = 4, so a = 2 or a = -2. The curve has slope -1>4 at the two points (2, 1>2) and s -2, -1>2d (Figure 3.3). (c) The slope -1>a 2 is always negative if a Z 0. As a : 0 +, the slope approaches - q and the tangent becomes increasingly steep (Figure 3.2). We see this situation again as a : 0 - . As a moves away from the origin in either direction, the slope approaches 0 and the tangent levels off to become horizontal.

Rates of Change: Derivative at a Point The expression ƒsx0 + hd - ƒsx0 d , h

h Z 0

is called the difference quotient of ƒ at x0 with increment h. If the difference quotient has a limit as h approaches zero, that limit is given a special name and notation.

DEFINITION

The derivative of a function ƒ at a point x0, denoted ƒ¿(x0), is ƒ¿(x0) = lim

h:0

ƒ(x0 + h) - ƒ(x0) h

provided this limit exists.

If we interpret the difference quotient as the slope of a secant line, then the derivative gives the slope of the curve y = ƒ(x) at the point P(x0, ƒ(x0)). Exercise 31 shows

118

Chapter 3: Differentiation

that the derivative of the linear function ƒ(x) = mx + b at any point x0 is simply the slope of the line, so ƒ¿(x0) = m, which is consistent with our definition of slope. If we interpret the difference quotient as an average rate of change (Section 2.1), the derivative gives the function’s instantaneous rate of change with respect to x at the point x = x0 . We study this interpretation in Section 3.4.

EXAMPLE 2

In Examples 1 and 2 in Section 2.1, we studied the speed of a rock falling freely from rest near the surface of the earth. We knew that the rock fell y = 16t 2 feet during the first t sec, and we used a sequence of average rates over increasingly short intervals to estimate the rock’s speed at the instant t = 1. What was the rock’s exact speed at this time?

We let ƒstd = 16t 2 . The average speed of the rock over the interval between t = 1 and t = 1 + h seconds, for h 7 0, was found to be Solution

ƒs1 + hd - ƒs1d 16s1 + hd2 - 16s1d2 16sh 2 + 2hd = = = 16sh + 2d. h h h The rock’s speed at the instant t = 1 is then lim 16sh + 2d = 16s0 + 2d = 32 ft>sec.

h:0

Our original estimate of 32 ft > sec in Section 2.1 was right.

Summary We have been discussing slopes of curves, lines tangent to a curve, the rate of change of a function, and the derivative of a function at a point. All of these ideas refer to the same limit.

The following are all interpretations for the limit of the difference quotient, ƒsx0 + hd - ƒsx0 d . h h:0 lim

1. 2. 3. 4.

The slope of the graph of y = ƒsxd at x = x0 The slope of the tangent to the curve y = ƒsxd at x = x0 The rate of change of ƒ(x) with respect to x at x = x0 The derivative ƒ¿(x0) at a point

In the next sections, we allow the point x0 to vary across the domain of the function ƒ.

3.1

Tangents and the Derivative at a Point

119

Exercises 3.1 Slopes and Tangent Lines In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P1 and P2 . 1.

2.

y

2

23. ƒsxd = x 2 + 4x - 1

24. g sxd = x 3 - 3x

25. Find equations of all lines having slope -1 that are tangent to the curve y = 1>sx - 1d .

y P2

2

Tangent Lines with Specified Slopes At what points do the graphs of the functions in Exercises 23 and 24 have horizontal tangents?

26. Find an equation of the straight line having slope 1>4 that is tangent to the curve y = 2x .

P2 1

1

–2

1

0

2

x

P1 –1

P1 0

3.

–1

x

1

–2

4.

y

28. Speed of a rocket At t sec after liftoff, the height of a rocket is 3t 2 ft. How fast is the rocket climbing 10 sec after liftoff ? 29. Circle’s changing area What is the rate of change of the area of a circle sA = pr 2 d with respect to the radius when the radius is r = 3?

y

3

30. Ball’s changing volume What is the rate of change of the volume of a ball sV = s4>3dpr 3 d with respect to the radius when the radius is r = 2 ?

2

31. Show that the line y = mx + b is its own tangent line at any point (x0, mx0 + b).

4 2

P2

P1

1

P2

P1

32. Find the slope of the tangent to the curve y = 1> 2x at the point where x = 4.

1 0

1

2

x

–2

–1

Rates of Change 27. Object dropped from a tower An object is dropped from the top of a 100-m-high tower. Its height above ground after t sec is 100 - 4.9t 2 m. How fast is it falling 2 sec after it is dropped?

0

1

2

x

Testing for Tangents 33. Does the graph of

In Exercises 5–10, find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. 5. y = 4 - x 2, 7. y = 2 2x,

s -1, 3d s1, 2d

s -2, -8d

9. y = x 3,

6. y = sx - 1d2 + 1, 1 8. y = 2 , x 10. y =

1 , x3

s1, 1d

s2, 5d

12. ƒsxd = x - 2x 2,

x , x - 2

s3, 3d

14. g sxd =

15. hstd = t 3,

s2, 8d

17. ƒsxd = 2x,

s4, 2d

g sxd = e

21. y =

x = -1

1 , x - 1

x = 3

s8, 3d

20. y = 1 - x 2,

x = 2

x - 1 , x + 1

x = 0

22. y =

x Z 0 x = 0

have a tangent at the origin? Give reasons for your answer. Vertical Tangents We say that a continuous curve y = ƒsxd has a vertical tangent at the point where x = x0 if lim h:0 sƒsx0 + hd - ƒsx0 dd>h = q or - q . For example, y = x 1>3 has a vertical tangent at x = 0 (see accompanying figure):

In Exercises 19–22, find the slope of the curve at the point indicated. 19. y = 5x 2,

x sin s1>xd, 0,

s1, -1d

8 , s2, 2d x2 16. hstd = t 3 + 3t, s1, 4d 18. ƒsxd = 2x + 1,

x Z 0 x = 0

34. Does the graph of

1 a-2, - b 8

11. ƒsxd = x 2 + 1,

x 2 sin s1>xd, 0,

have a tangent at the origin? Give reasons for your answer.

s -1, 1d

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.

13. g sxd =

ƒsxd = e

lim

h:0

ƒs0 + hd - ƒs0d h 1>3 - 0 = lim h h h:0 1 = lim 2>3 = q . h:0 h

120

Chapter 3: Differentiation 36. Does the graph of

y y f (x)

0, x 6 0 1, x Ú 0 have a vertical tangent at the point (0, 1)? Give reasons for your answer. Usxd = e

x 13

x

0

T Graph the curves in Exercises 37–46. a. Where do the graphs appear to have vertical tangents? b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 35 and 36.

VERTICAL TANGENT AT ORIGIN

However, y = x 2>3 has no vertical tangent at x = 0 (see next figure): g s0 + hd - g s0d h 2>3 - 0 lim = lim h h h:0 h :0 1 = lim 1>3 h :0 h does not exist, because the limit is q from the right and - q from the left.

37. y = x 2>5

38. y = x 4>5

39. y = x 1>5

40. y = x 3>5

41. y = 4x

2>5

- 2x

42. y = x 5>3 - 5x 2>3

45. y = e

- 2ƒ x ƒ , x … 0 x 7 0 2x,

y g(x)

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises 47–50: b. Holding x0 fixed, the difference quotient

x 23

qshd =

0

ƒsx0 + hd - ƒsx0 d h

at x0 becomes a function of the step size h. Enter this function into your CAS workspace.

x

c. Find the limit of q as h : 0 .

NO VERTICAL TANGENT AT ORIGIN

d. Define the secant lines y = ƒsx0 d + q # sx - x0 d for h = 3, 2 , and 1. Graph them together with ƒ and the tangent line over the interval in part (a). 5 47. ƒsxd = x 3 + 2x, x0 = 0 48. ƒsxd = x + x , x0 = 1 49. ƒsxd = x + sin s2xd, x0 = p>2

35. Does the graph of -1, x 6 0 0, x = 0 1, x 7 0

have a vertical tangent at the origin? Give reasons for your answer.

3.2

46. y = 2 ƒ 4 - x ƒ

a. Plot y = ƒsxd over the interval sx0 - 1>2d … x … sx0 + 3d .

y

ƒsxd = •

44. y = x 1>3 + sx - 1d1>3

43. y = x 2>3 - sx - 1d1>3

50. ƒsxd = cos x + 4 sin s2xd,

x0 = p

The Derivative as a Function In the last section we defined the derivative of y = ƒsxd at the point x = x0 to be the limit

HISTORICAL ESSAY

ƒsx0 + hd - ƒsx0 d . h h:0

ƒ¿sx0 d = lim

The Derivative

We now investigate the derivative as a function derived from ƒ by considering the limit at each point x in the domain of ƒ.

DEFINITION The derivative of the function ƒ(x) with respect to the variable x is the function ƒ¿ whose value at x is ƒ¿sxd = lim

h:0

provided the limit exists.

ƒsx + hd - ƒsxd , h

3.2 y f (x) Secant slope is f (z) f (x) zx

Q(z, f(z))

f (z) f (x)

P(x, f(x)) hzx

The Derivative as a Function

121

We use the notation ƒ(x) in the definition to emphasize the independent variable x with respect to which the derivative function ƒ¿(x) is being defined. The domain of ƒ¿ is the set of points in the domain of ƒ for which the limit exists, which means that the domain may be the same as or smaller than the domain of ƒ. If ƒ¿ exists at a particular x, we say that ƒ is differentiable (has a derivative) at x. If ƒ¿ exists at every point in the domain of ƒ, we call ƒ differentiable. If we write z = x + h, then h = z - x and h approaches 0 if and only if z approaches x. Therefore, an equivalent definition of the derivative is as follows (see Figure 3.4). This formula is sometimes more convenient to use when finding a derivative function.

zxh

x

Derivative of f at x is f(x h) f (x) f '(x) lim h h→0 lim

z→x

Alternative Formula for the Derivative ƒszd - ƒsxd z - x z:x

ƒ¿sxd = lim

f(z) f (x) zx

FIGURE 3.4 Two forms for the difference quotient.

Calculating Derivatives from the Definition The process of calculating a derivative is called differentiation. To emphasize the idea that differentiation is an operation performed on a function y = ƒsxd, we use the notation d ƒsxd dx

Derivative of the Reciprocal Function d 1 1 a b = - 2, dx x x

as another way to denote the derivative ƒ¿sxd. Example 1 of Section 3.1 illustrated the differentiation process for the function y = 1>x when x = a. For x representing any point in the domain, we get the formula d 1 1 a b = - 2. dx x x

x Z 0

Here are two more examples in which we allow x to be any point in the domain of ƒ.

EXAMPLE 1

Differentiate ƒsxd =

x . x - 1

We use the definition of derivative, which requires us to calculate ƒ(x + h) and then subtract ƒ(x) to obtain the numerator in the difference quotient. We have

Solution

ƒsxd =

x x - 1

and ƒsx + hd =

sx + hd , so sx + hd - 1

ƒsx + hd - ƒsxd h h:0

ƒ¿sxd = lim

Definition

x + h x x - 1 x + h - 1 = lim h h:0 = lim

1 # sx + hdsx - 1d - xsx + h - 1d h sx + h - 1dsx - 1d

a c ad - cb - = b d bd

= lim

-h 1 # h sx + h - 1dsx - 1d

Simplify.

= lim

-1 -1 = . sx + h - 1dsx - 1d sx - 1d2

Cancel h Z 0.

h:0

h:0

h:0

122

Chapter 3: Differentiation

EXAMPLE 2 (a) Find the derivative of ƒsxd = 1x for x 7 0. (b) Find the tangent line to the curve y = 1x at x = 4. Solution Derivative of the Square Root Function d 1 2x = , dx 2 2x

(a) We use the alternative formula to calculate ƒ¿ : ƒ¿sxd = lim

x 7 0

z:x

= lim

z:x

ƒszd - ƒsxd z - x 1z - 1x z - x

= lim

y y 1x1 4

(4, 2)

y x

ƒ¿s4d =

1 0

1z - 1x

A 1z - 1x B A 1z + 1x B 1 1 = lim = . z:x 1z + 1x 21x (b) The slope of the curve at x = 4 is z:x

4

x

FIGURE 3.5 The curve y = 1x and its tangent at (4, 2). The tangent’s slope is found by evaluating the derivative at x = 4 (Example 2).

1 1 = . 4 224

The tangent is the line through the point (4, 2) with slope 1>4 (Figure 3.5): 1 y = 2 + sx - 4d 4 1 y = x + 1. 4

Notations There are many ways to denote the derivative of a function y = ƒsxd, where the independent variable is x and the dependent variable is y. Some common alternative notations for the derivative are ƒ¿sxd = y¿ =

dy dƒ d = = ƒsxd = Dsƒdsxd = Dx ƒsxd. dx dx dx

The symbols d>dx and D indicate the operation of differentiation. We read dy>dx as “the derivative of y with respect to x,” and dƒ>dx and (d>dx)ƒ(x) as “the derivative of ƒ with respect to x.” The “prime” notations y¿ and ƒ¿ come from notations that Newton used for derivatives. The d>dx notations are similar to those used by Leibniz. The symbol dy>dx should not be regarded as a ratio (until we introduce the idea of “differentials” in Section 3.11). To indicate the value of a derivative at a specified number x = a, we use the notation ƒ¿sad =

dy df d ` = ` = ƒsxd ` . dx x = a dx x = a dx x=a

For instance, in Example 2 ƒ¿s4d =

d 1 1 1 1x ` = ` = = . 4 dx 21x x = 4 x=4 224

Graphing the Derivative We can often make a reasonable plot of the derivative of y = ƒsxd by estimating the slopes on the graph of ƒ. That is, we plot the points sx, ƒ¿sxdd in the xy-plane and connect them with a smooth curve, which represents y = ƒ¿sxd.

3.2

EXAMPLE 3

y

123

Graph the derivative of the function y = ƒsxd in Figure 3.6a.

We sketch the tangents to the graph of ƒ at frequent intervals and use their slopes to estimate the values of ƒ¿sxd at these points. We plot the corresponding sx, ƒ¿sxdd pairs and connect them with a smooth curve as sketched in Figure 3.6b.

Solution

Slope 0 A 10

The Derivative as a Function

Slope –1 B C

y f (x) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Slope – 4 3 E D Slope 0

⎧ ⎪ ⎨ ⎪ ⎩

5

What can we learn from the graph of y = ƒ¿sxd? At a glance we can see

8

0

4 x-units 10 15

5

x

1. 2. 3.

where the rate of change of ƒ is positive, negative, or zero; the rough size of the growth rate at any x and its size in relation to the size of ƒ(x); where the rate of change itself is increasing or decreasing.

(a) Slope

Differentiable on an Interval; One-Sided Derivatives A function y = ƒsxd is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval. It is differentiable on a closed interval [a, b] if it is differentiable on the interior (a, b) and if the limits

4 y f '(x)

3

E'

2 1 A' –1 –2

D' 10

5 C' B' Vertical coordinate –1

15

x

h:0 +

lim

ƒsa + hd - ƒsad h

Right-hand derivative at a

lim-

ƒsb + hd - ƒsbd h

Left-hand derivative at b

h:0

(b)

FIGURE 3.6 We made the graph of y = ƒ¿sxd in (b) by plotting slopes from the graph of y = ƒsxd in (a). The vertical coordinate of B¿ is the slope at B and so on. The slope at E is approximately 8>4 = 2. In (b) we see that the rate of change of ƒ is negative for x between A¿ and D¿; the rate of change is positive for x to the right of D¿.

exist at the endpoints (Figure 3.7). Right-hand and left-hand derivatives may be defined at any point of a function’s domain. Because of Theorem 6, Section 2.4, a function has a derivative at a point if and only if it has left-hand and right-hand derivatives there, and these one-sided derivatives are equal.

EXAMPLE 4 Show that the function y = ƒ x ƒ is differentiable on s - q , 0d and s0, q d but has no derivative at x = 0. Solution From Section 3.1, the derivative of y = mx + b is the slope m. Thus, to the right of the origin,

d d d sxd = sxd = s1 # xd = 1. dx ƒ ƒ dx dx

d smx + bd = m, ƒ x ƒ = x dx

d d d sxd = s -xd = s -1 # xd = -1 dx ƒ ƒ dx dx

ƒ x ƒ = -x

To the left,

Slope f(a h) f(a) lim h h→0

(Figure 3.8). There is no derivative at the origin because the one-sided derivatives differ there:

Slope f (b h) f (b) lim h h→0

Right-hand derivative of ƒ x ƒ at zero = lim+ h:0

= lim+ h:0

y f (x)

ƒ0 + hƒ - ƒ0ƒ ƒhƒ = lim+ h h:0 h h h

ƒ h ƒ = h when h 7 0

= lim+1 = 1 h:0

a

ah h0

bh h0

b

x

Left-hand derivative of ƒ x ƒ at zero = limh:0

= limh:0

FIGURE 3.7 Derivatives at endpoints are one-sided limits.

ƒ0 + hƒ - ƒ0ƒ ƒhƒ = limh h:0 h -h h

ƒ h ƒ = -h when h 6 0

= lim- -1 = -1. h:0

124

Chapter 3: Differentiation

EXAMPLE 5

y

In Example 2 we found that for x 7 0,

y ⏐x⏐ y' –1

d 1 . 1x = dx 21x

y' 1

0

x

We apply the definition to examine if the derivative exists at x = 0:

y' not defined at x 0: right-hand derivative left-hand derivative

FIGURE 3.8 The function y = ƒ x ƒ is not differentiable at the origin where the graph has a “corner” (Example 4).

lim

h:0 +

20 + h - 20 1 = lim+ = q. h h:0 1h

Since the (right-hand) limit is not finite, there is no derivative at x = 0. Since the slopes of the secant lines joining the origin to the points (h, 1h) on a graph of y = 1x approach q , the graph has a vertical tangent at the origin. (See Figure 1.17 on page 9).

When Does a Function Not Have a Derivative at a Point? A function has a derivative at a point x0 if the slopes of the secant lines through Psx0, ƒsx0 dd and a nearby point Q on the graph approach a finite limit as Q approaches P. Whenever the secants fail to take up a limiting position or become vertical as Q approaches P, the derivative does not exist. Thus differentiability is a “smoothness” condition on the graph of ƒ. A function can fail to have a derivative at a point for many reasons, including the existence of points where the graph has

P

P

Q Q

Q

Q

1. a corner, where the one-sided

2. a cusp, where the slope of PQ approaches q from one side and - q from the other.

derivatives differ.

P

Q

P

Q P Q

Q

Q

3. a vertical tangent, where the slope of PQ approaches q from both sides or approaches - q from both sides (here, - q ).

4. a discontinuity (two examples shown).

Q

3.2

The Derivative as a Function

125

Another case in which the derivative may fail to exist occurs when the function’s slope is oscillating rapidly near P, as with ƒ(x) = sin (1>x) near the origin, where it is discontinuous (see Figure 2.31).

Differentiable Functions Are Continuous A function is continuous at every point where it has a derivative.

THEOREM 1—Differentiability Implies Continuity x = c, then ƒ is continuous at x = c.

If ƒ has a derivative at

Proof Given that ƒ¿scd exists, we must show that limx:c ƒsxd = ƒscd, or equivalently, that limh:0 ƒsc + hd = ƒscd. If h Z 0, then ƒsc + hd = ƒscd + sƒsc + hd - ƒscdd = ƒscd +

ƒsc + hd - ƒscd # h. h

Now take limits as h : 0. By Theorem 1 of Section 2.2, lim ƒsc + hd = lim ƒscd + lim

h:0

h:0

h:0

ƒsc + hd - ƒscd h

# lim h

= ƒscd + ƒ¿scd # 0

h: 0

= ƒscd + 0 = ƒscd. Similar arguments with one-sided limits show that if ƒ has a derivative from one side (right or left) at x = c, then ƒ is continuous from that side at x = c. Theorem 1 says that if a function has a discontinuity at a point (for instance, a jump discontinuity), then it cannot be differentiable there. The greatest integer function y = :x; fails to be differentiable at every integer x = n (Example 4, Section 2.5). Caution The converse of Theorem 1 is false. A function need not have a derivative at a point where it is continuous, as we saw in Example 4.

Exercises 3.2 Finding Derivative Functions and Values Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified. 1. ƒsxd = 4 - x 2;

ƒ¿s - 3d, ƒ¿s0d, ƒ¿s1d

2. Fsxd = sx - 1d2 + 1;

F¿s -1d, F¿s0d, F¿s2d

1 ; g¿s - 1d, g¿s2d, g¿ A 23 B t2 1 - z ; k¿s -1d, k¿s1d, k¿ A 22 B 4. k szd = 2z 3. g std =

5. psud = 23u ;

p¿s1d, p¿s3d, p¿s2>3d

6. r ssd = 22s + 1 ;

r¿s0d, r¿s1d, r¿s1>2d

In Exercises 7–12, find the indicated derivatives. dy dr 7. if y = 2x 3 8. if r = s 3 - 2s 2 + 3 dx ds 9.

ds dt

if

s =

11.

dp dq

if

p =

t 2t + 1 1 2q + 1

10.

dy dt

if

1 y = t - t

12.

dz dw

if

z =

1 23w - 2

126

Chapter 3: Differentiation

Slopes and Tangent Lines In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable. 9 13. ƒsxd = x + x ,

27.

14. k sxd =

x = -3

8 2x - 2

,

sx, yd = s6, 4d

18. w = g szd = 1 + 24 - z,

y

y f 2 (x)

y f1(x)

1 , x = 2 2 + x x + 3 , x = -2 15. s = t 3 - t 2, t = -1 16. y = 1 - x In Exercises 17–18, differentiate the functions. Then find an equation of the tangent line at the indicated point on the graph of the function. 17. y = ƒsxd =

28.

y

x

0

29.

30.

y

sz, wd = s3, 2d

x

0

y

y f3(x)

y f4(x)

In Exercises 19–22, find the values of the derivatives. 19.

ds ` dt t = -1

20.

dy ` dx x = 23

21.

dr ` du u = 0

if

r =

dw ` 22. dz z = 4

if

w = z + 1z

if

x

0

s = 1 - 3t 2

x

0

1 y = 1 - x

if

31. a. The graph in the accompanying figure is made of line segments joined end to end. At which points of the interval [-4, 6] is ƒ¿ not defined? Give reasons for your answer.

2 24 - u

y (6, 2)

(0, 2)

Using the Alternative Formula for Derivatives Use the formula

y f (x)

ƒszd - ƒsxd z - x z: x

ƒ¿sxd = lim

0

(– 4, 0)

1

x

6

to find the derivative of the functions in Exercises 23–26. (1, –2)

1 x + 2 x 25. g sxd = x - 1 23. ƒsxd =

(4, –2)

24. ƒsxd = x 2 - 3x + 4 26. g sxd = 1 + 1x

b. Graph the derivative of ƒ. The graph should show a step function.

Graphs Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)– (d). y'

32. Recovering a function from its derivative a. Use the following information to graph the function ƒ over the closed interval [-2, 5] . i) The graph of ƒ is made of closed line segments joined end to end.

y'

ii) The graph starts at the point s -2, 3d . iii) The derivative of ƒ is the step function in the figure shown here.

x

0

y'

x

0 (a)

(b)

y'

y'

y' f '(x) 1 –2

0

(c)

x

0

(d)

x

0

1

3

5

x

–2

b. Repeat part (a), assuming that the graph starts at s -2, 0d instead of s - 2, 3d .

3.2 33. Growth in the economy The graph in the accompanying figure shows the average annual percentage change y = ƒstd in the U.S. gross national product (GNP) for the years 1983–1988. Graph dy>dt (where defined). 7%

127

The Derivative as a Function

36. Weight loss Jared Fogle, also known as the “Subway Sandwich Guy,” weighed 425 lb in 1997 before losing more than 240 lb in 12 months (http://en.wikipedia.org/wiki/Jared_Fogle). A chart showing his possible dramatic weight loss is given in the accompanying figure.

6 W

5 4

500

3

425

1 0 1983

1984

1985

1986

1987

Weight (lbs)

2

1988

t

1 2 3 4 5 6 7 8 9 10 11 12 Time (months)

0

a. Estimate Jared’s rate of weight loss when i) t = 1

a. Use the graphical technique of Example 3 to graph the derivative of the fruit fly population. The graph of the population is reproduced here.

ii) t = 4

iii) t = 11

b. When does Jared lose weight most rapidly and what is this rate of weight loss? c. Use the graphical technique of Example 3 to graph the derivative of weight W.

p

Number of flies

200 100

34. Fruit flies (Continuation of Example 4, Section 2.1.) Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment.

350 300 250 200 150 100 50 0

300

One-Sided Derivatives Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P. 37. 10

20 30 Time (days)

40

y

38.

y

y f (x)

t

50

y 2x

y f (x)

y x2

y2 2

b. During what days does the population seem to be increasing fastest? Slowest?

P(1, 2)

yx

35. Temperature The given graph shows the temperature T in °F at Davis, CA, on April 18, 2008, between 6 A.M. and 6 P.M.

1 x

P(0, 0)

0

1

2

x

T

Temperature (°F)

80

y

39.

70

40.

y y f (x)

y f (x)

P(1, 1)

y 2x 1

60

1

50 1

40 0 6 a.m.

3

6

9

9 a.m. 12 noon 3 p.m. Time (hrs)

12

t

ii) 9 A.M.

iii) 2 P.M.

x

1

yx

1

x

6 p.m.

a. Estimate the rate of temperature change at the times i) 7 A.M.

0

P(1, 1) y x

y 1x

iv) 4 P.M.

b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times? c. Use the graphical technique of Example 3 to graph the derivative of temperature T versus time t.

In Exercises 41 and 42, determine if the piecewise-defined function is differentiable at the origin. 41. ƒsxd = e

2x - 1, x 2 + 2x + 7,

42. gsxd = e

x 2>3, x 1>3,

x Ú 0 x 6 0

x Ú 0 x 6 0

128

Chapter 3: Differentiation

Differentiability and Continuity on an Interval Each figure in Exercises 43–48 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

55. Derivative of ƒ Does knowing that a function ƒ(x) is differentiable at x = x0 tell you anything about the differentiability of the function -ƒ at x = x0 ? Give reasons for your answer.

a. differentiable? b. continuous but not differentiable? c. neither continuous nor differentiable?

56. Derivative of multiples Does knowing that a function g(t) is differentiable at t = 7 tell you anything about the differentiability of the function 3g at t = 7 ? Give reasons for your answer.

Give reasons for your answers. 43.

44. y

y

y f (x) D: –3 x 2 2

2

1

1

–3

–2

–1

0 1

2

x

–2

y f (x) D: –2 x 3

–1

0

–1

–1

–2

–2

45.

54. Tangent to y 1x Does any tangent to the curve y = 1x cross the x-axis at x = -1 ? If so, find an equation for the line and the point of tangency. If not, why not?

1

2

3

x

57. Limit of a quotient Suppose that functions g(t) and h(t) are defined for all values of t and g s0d = hs0d = 0 . Can limt:0 sg stdd>shstdd exist? If it does exist, must it equal zero? Give reasons for your answers. 58. a. Let ƒ(x) be a function satisfying ƒ ƒsxd ƒ … x 2 for -1 … x … 1 . Show that ƒ is differentiable at x = 0 and find ƒ¿s0d . b. Show that

ƒsxd = L

46. y

1 x 2 sin x ,

x Z 0

0,

x = 0

is differentiable at x = 0 and find ƒ¿s0d .

y y f (x) D: –3 x 3

T 59. Graph y = 1> A 21x B in a window that has 0 … x … 2 . Then, on the same screen, graph

y f (x) D: –2 x 3 3

y =

1 –3 –2 –1 0 –1

1

2

3

2

x

1

–2 –2

47.

–1

1

0

3

x

48. y

y y f (x) D: –1 x 2

4

y f (x) D: –3 x 3

2

1

–1

2

0

1

2

x

–3 –2 –1 0

1 2

3

x

Theory and Examples In Exercises 49–52, a. Find the derivative ƒ¿sxd of the given function y = ƒsxd . b. Graph y = ƒsxd and y = ƒ¿sxd side by side using separate sets of coordinate axes, and answer the following questions.

1x + h - 1x h

for h = 1, 0.5, 0.1 . Then try h = -1, -0.5, -0.1 . Explain what is going on. T 60. Graph y = 3x 2 in a window that has -2 … x … 2, 0 … y … 3 . Then, on the same screen, graph y =

sx + hd3 - x 3 h

for h = 2, 1, 0.2 . Then try h = -2, -1, -0.2 . Explain what is going on. 61. Derivative of y x Graph the derivative of ƒsxd = ƒ x ƒ . Then graph y = s ƒ x ƒ - 0d>sx - 0d = ƒ x ƒ >x . What can you conclude? T 62. Weierstrass’s nowhere differentiable continuous function The sum of the first eight terms of the Weierstrass function ƒ(x) = g nq= 0 s2>3dn cos s9npxd is g sxd = cos spxd + s2>3d1 cos s9pxd + s2>3d2 cos s92pxd + s2>3d3 cos s93pxd + Á + s2>3d7 cos s97pxd . Graph this sum. Zoom in several times. How wiggly and bumpy is this graph? Specify a viewing window in which the displayed portion of the graph is smooth.

c. For what values of x, if any, is ƒ¿ positive? Zero? Negative? d. Over what intervals of x-values, if any, does the function y = ƒsxd increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.) 49. y = -x

2

51. y = x 3>3

50. y = -1>x 52. y = x 4>4

53. Tangent to a parabola Does the parabola y = 2x 2 - 13x + 5 have a tangent whose slope is -1 ? If so, find an equation for the line and the point of tangency. If not, why not?

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises 63–68. a. Plot y = ƒsxd to see that function’s global behavior. b. Define the difference quotient q at a general point x, with general step size h. c. Take the limit as h : 0 . What formula does this give? d. Substitute the value x = x0 and plot the function y = ƒsxd together with its tangent line at that point.

3.3 e. Substitute various values for x larger and smaller than x0 into the formula obtained in part (c). Do the numbers make sense with your picture? f. Graph the formula obtained in part (c). What does it mean when its values are negative? Zero? Positive? Does this make sense with your plot from part (a)? Give reasons for your answer. 63. ƒsxd = x 3 + x 2 - x,

x0 = 1

64. ƒsxd = x 1>3 + x 2>3, 65. ƒsxd =

4x , x2 + 1

Differentiation Rules

129

x0 = 1

x0 = 2

x - 1 , x0 = -1 3x 2 + 1 67. ƒsxd = sin 2x, x0 = p>2 66. ƒsxd =

68. ƒsxd = x 2 cos x,

x0 = p>4

Differentiation Rules

3.3

This section introduces several rules that allow us to differentiate constant functions, power functions, polynomials, exponential functions, rational functions, and certain combinations of them, simply and directly, without having to take limits each time.

Powers, Multiples, Sums, and Differences A simple rule of differentiation is that the derivative of every constant function is zero. y c

(x h, c)

(x, c)

yc

dƒ d scd = 0. = dx dx

h 0

x

Derivative of a Constant Function If ƒ has the constant value ƒsxd = c, then

xh

FIGURE 3.9 The rule sd>dxdscd = 0 is another way to say that the values of constant functions never change and that the slope of a horizontal line is zero at every point.

x

Proof We apply the definition of the derivative to ƒsxd = c, the function whose outputs have the constant value c (Figure 3.9). At every value of x, we find that ƒsx + hd - ƒsxd c - c = lim = lim 0 = 0. h h h:0 h:0 h:0

ƒ¿sxd = lim

From Section 3.1, we know that d 1 1 a b = - 2, dx x x

or

d -1 A x B = -x - 2. dx

From Example 2 of the last section we also know that d A 2x B = 1 , dx 22x

or

d 1>2 A x B = 12 x - 1>2. dx

These two examples illustrate a general rule for differentiating a power x n. We first prove the rule when n is a positive integer.

Power Rule for Positive Integers: If n is a positive integer, then d n x = nx n - 1 . dx

130

Chapter 3: Differentiation

HISTORICAL BIOGRAPHY

Proof of the Positive Integer Power Rule

The formula

z n - x n = sz - xdsz n - 1 + z n - 2 x + Á + zx n - 2 + x n - 1 d

Richard Courant (1888–1972)

can be verified by multiplying out the right-hand side. Then from the alternative formula for the definition of the derivative, ƒszd - ƒsxd z n - xn = lim z - x z x z:x z:x

ƒ¿sxd = lim

= lim sz n - 1 + z n - 2x + Á + zx n - 2 + x n - 1 d z:x

n terms

= nx n - 1. The Power Rule is actually valid for all real numbers n. The number n could be a negative integer or a fractional power or an irrational number. To apply the Power Rule, we subtract 1 from the original exponent n and multiply the result by n. Here we state the general version of the rule, but postpone its proof until Section 3.8.

Power Rule (General Version) If n is any real number, then d n x = nx n - 1 dx for all x where the powers x n and x n - 1 are defined.

EXAMPLE 1 (a) x 3

Differentiate the following powers of x.

(b) x 2/3

(c) x 22

(d)

1 x4

(e) x -4>3

(f) 2x 2 + p

Solution

(a)

d 3 (x ) = 3x 3 - 1 = 3x 2 dx

(b)

d 2>3 2 2 (x ) = x (2>3) - 1 = x -1>3 3 3 dx

(c)

d 22 A x B = 22x 22 - 1 dx

(d)

d 1 d -4 4 a b = (x ) = -4x -4 - 1 = -4x -5 = - 5 dx x 4 dx x

(e)

d -4>3 4 4 (x ) = - x -(4>3) - 1 = - x -7>3 3 3 dx

(f)

d d p A 2x 2 + p B = dx A x 1 + (p>2) B = a1 + 2 bx 1 + (p>2) - 1 = 12 (2 + p)2x p dx

The next rule says that when a differentiable function is multiplied by a constant, its derivative is multiplied by the same constant.

Derivative Constant Multiple Rule If u is a differentiable function of x, and c is a constant, then du d scud = c . dx dx

In particular, if n is any real number, then d scx n d = cnx n - 1 . dx

3.3

131

Proof

y y 3x 2

3

Differentiation Rules

cusx + hd - cusxd d cu = lim dx h h:0 usx + hd - usxd h h:0

= c lim

Slope 3(2x) 6x Slope 6(1) 6

(1, 3)

= c

du dx

Derivative definition with ƒsxd = cusxd Constant Multiple Limit Property

u is differentiable.

y x2

EXAMPLE 2

2

(a) The derivative formula 1

0

Slope 2x Slope 2(1) 2 (1, 1)

1

2

x

FIGURE 3.10 The graphs of y = x 2 and y = 3x 2 . Tripling the y-coordinate triples the slope (Example 2).

Denoting Functions by u and Y The functions we are working with when we need a differentiation formula are likely to be denoted by letters like ƒ and g. We do not want to use these same letters when stating general differentiation rules, so instead we use letters like u and y that are not likely to be already in use.

d s3x 2 d = 3 # 2x = 6x dx says that if we rescale the graph of y = x 2 by multiplying each y-coordinate by 3, then we multiply the slope at each point by 3 (Figure 3.10). (b) Negative of a function The derivative of the negative of a differentiable function u is the negative of the function’s derivative. The Constant Multiple Rule with c = -1 gives d d d du s -ud = s -1 # ud = -1 # sud = - . dx dx dx dx The next rule says that the derivative of the sum of two differentiable functions is the sum of their derivatives.

Derivative Sum Rule If u and y are differentiable functions of x, then their sum u + y is differentiable at every point where u and y are both differentiable. At such points, du dy d su + yd = + . dx dx dx

For example, if y = x 4 + 12x, then y is the sum of u(x) = x 4 and y(x) = 12x. We then have dy d 4 d = (x ) + (12x) = 4x 3 + 12. dx dx dx Proof We apply the definition of the derivative to ƒsxd = usxd + ysxd: [usx + hd + ysx + hd] - [usxd + ysxd] d [usxd + ysxd] = lim dx h h:0 = lim c h:0

= lim

h:0

usx + hd - usxd ysx + hd - ysxd + d h h

usx + hd - usxd ysx + hd - ysxd du dy + lim = + . h h dx dx h:0

Combining the Sum Rule with the Constant Multiple Rule gives the Difference Rule, which says that the derivative of a difference of differentiable functions is the difference of their derivatives: d d du dy du dy su - yd = [u + s -1dy] = + s -1d = . dx dx dx dx dx dx

132

Chapter 3: Differentiation

The Sum Rule also extends to finite sums of more than two functions. If u1 , u2 , Á , un are differentiable at x, then so is u1 + u2 + Á + un , and dun du2 du1 d su + u2 + Á + un d = + + Á + . dx 1 dx dx dx For instance, to see that the rule holds for three functions we compute du3 du3 du1 du2 d d d su + u2 + u3 d = ssu1 + u2 d + u3 d = su + u2 d + = + + . dx 1 dx dx 1 dx dx dx dx

A proof by mathematical induction for any finite number of terms is given in Appendix 2.

EXAMPLE 3 Solution

Find the derivative of the polynomial y = x 3 +

dy d 4 2 d d 3 d = x + a x b s5xd + s1d dx dx dx 3 dx dx = 3x 2 +

4 2 x - 5x + 1. 3

Sum and Difference Rules

8 4# 2x - 5 + 0 = 3x 2 + x - 5 3 3

We can differentiate any polynomial term by term, the way we differentiated the polynomial in Example 3. All polynomials are differentiable at all values of x.

EXAMPLE 4

Does the curve y = x 4 - 2x 2 + 2 have any horizontal tangents? If so,

where? y

Solution

y x 4 2x 2 2

dy d 4 = sx - 2x 2 + 2d = 4x 3 - 4x. dx dx dy = 0 for x: Now solve the equation dx

(0, 2)

(–1, 1)

–1

1

0

The horizontal tangents, if any, occur where the slope dy>dx is zero. We have

(1, 1)

1

x

FIGURE 3.11 The curve in Example 4 and its horizontal tangents.

4x 3 - 4x = 0 4xsx 2 - 1d = 0 x = 0, 1, -1. The curve y = x 4 - 2x 2 + 2 has horizontal tangents at x = 0, 1, and -1. The corresponding points on the curve are (0, 2), (1, 1), and s -1, 1d. See Figure 3.11. We will see in Chapter 4 that finding the values of x where the derivative of a function is equal to zero is an important and useful procedure.

Derivatives of Exponential Functions We briefly reviewed exponential functions in Section 1.5. When we apply the definition of the derivative to ƒ(x) a x, we get d x a x+h - ax (a ) = lim dx h h:0

a x # ah - a x h h:0

= lim

Derivative definition

#

a xh = ax ah

= lim a x #

ah - 1 h

Factoring out ax

= a x # lim

ah - 1 h

a x is constant as h : 0.

h:0

h:0

= ¢ lim

h:0

ah - 1 # x ≤ a. h

(++)++* a fixed numberL

(1)

3.3

y

a 3 a e a 2.5

133

Thus we see that the derivative of a x is a constant multiple L of a x. The constant L is a limit unlike any we have encountered before. Note, however, that it equals the derivative of ƒ(x) = a x at x = 0: ah - a0 ah - 1 = lim = L. h h h:0 h:0

a2

L 1.0 1.1

Differentiation Rules

ƒ¿(0) = lim

0.92 h y a 1,a0 h

0.69

h

0

FIGURE 3.12 The position of the curve y = (a h - 1)>h, a 7 0, varies continuously with a. The limit L of y as h : 0 changes with different values of a. The number e between a = 2 and a = 3 is the number for which the limit equals 1 as h : 0.

The limit L is therefore the slope of the graph of ƒ(x) = a x where it crosses the y-axis. In Chapter 7, where we carefully develop the logarithmic and exponential functions, we prove that the limit L exists and has the value ln a. For now we investigate values of L by graphing the function y = (a h - 1)>h and studying its behavior as h approaches 0. Figure 3.12 shows the graphs of y = (a h - 1)>h for four different values of a. The limit L is approximately 0.69 if a = 2, about 0.92 if a = 2.5, and about 1.1 if a = 3. It appears that the value of L is 1 at some number a chosen between 2.5 and 3. That number is given by a = e L 2.718281828. With this choice of base we obtain the natural exponential function ƒ(x) = e x as in Section 1.5, and see that it satisfies the property eh - 1 = 1. h h:0

ƒ¿(0) = lim

(2)

That the limit is 1 implies an important relationship between the natural exponential function e x and its derivative: d x eh - 1 # x (e ) = lim ¢ ≤ e dx h h:0 = 1 # e x = e x.

Eq. (1) with a = e Eq. (2)

Therefore the natural exponential function is its own derivative.

Derivative of the Natural Exponential Function d x (e ) = e x dx

Find an equation for a line that is tangent to the graph of y = e x and goes through the origin.

EXAMPLE 5

Since the line passes through the origin, its equation is of the form y = mx, where m is the slope. If it is tangent to the graph at the point (a, e a), the slope is m = (e a - 0)>(a - 0). The slope of the natural exponential at x = a is e a. Because these slopes are the same, we then have that e a = e a>a. It follows that a = 1 and m = e, so the equation of the tangent line is y = ex . See Figure 3.13. Solution

y 6

y ex

4 (a,

2 –1

a

ea) x

FIGURE 3.13 The line through the origin is tangent to the graph of y = e x when a = 1 (Example 5).

We might ask if there are functions other than the natural exponential function that are their own derivatives. The answer is that the only functions that satisfy the property that ƒ¿(x) = ƒ(x) are functions that are constant multiples of the natural exponential function, ƒ(x) = c # e x, c any constant. We prove this fact in Section 7.2. Note from the Constant Multiple Rule that indeed d d x (c # e x ) = c # (e ) = c # e x. dx dx

134

Chapter 3: Differentiation

Products and Quotients While the derivative of the sum of two functions is the sum of their derivatives, the derivative of the product of two functions is not the product of their derivatives. For instance, d 2 d sx # xd = sx d = 2x, dx dx

while

d d sxd # sxd = 1 # 1 = 1. dx dx

The derivative of a product of two functions is the sum of two products, as we now explain. Derivative Product Rule If u and y are differentiable at x, then so is their product uy, and d dy du suyd = u + y . dx dx dx The derivative of the product uy is u times the derivative of y plus y times the derivative of u. In prime notation, suyd¿ = uy¿ + yu¿ . In function notation, d [ƒsxdgsxd] = ƒsxdg¿sxd + gsxdƒ¿sxd. dx

EXAMPLE 6

1 Find the derivative of (a) y = x A x 2 + e x B ,

(b) y = e 2x.

Solution

(a) We apply the Product Rule with u = 1>x and y = x 2 + e x : d 1 2 1 1 c A x + e x B d = x A 2x + e x B + A x 2 + e x B a- 2 b dx x x ex ex = 2 + x - 1 - 2 x Picturing the Product Rule Suppose u(x) and y(x) are positive and increase when x increases, and h 7 0.

(b)

y(x h) y

= 1 + (x - 1)

d dy du suyd = u + y , and dx dx dx d 1 1 a b = - 2 dx x x

ex . x2

d 2x d x# x d x d x (e ) = (e e ) = e x # (e ) + e x # (e ) = 2e x # e x = 2e 2x dx dx dx dx

u(x h) y

Proof of the Derivative Product Rule

y(x) u(x)y(x)

0

usx + hdysx + hd - usxdysxd d suyd = lim dx h h:0

y(x) u

u(x)

u u(x h)

Then the change in the product uy is the difference in areas of the larger and smaller “squares,” which is the sum of the upper and right-hand reddish-shaded rectangles. That is, ¢(uy) = u(x + h)y(x + h) - u(x)y(x) = u(x + h)¢y + y(x)¢u.

Division by h gives ¢(uy) ¢u ¢y = u(x + h) + y(x) . h h h

The limit as h : 0 + gives the Product Rule.

To change this fraction into an equivalent one that contains difference quotients for the derivatives of u and y, we subtract and add usx + hdysxd in the numerator: usx + hdysx + hd - usx + hdysxd + usx + hdysxd - usxdysxd d suyd = lim dx h h:0 = lim cusx + hd h:0

ysx + hd - ysxd usx + hd - usxd + ysxd d h h

ysx + hd - ysxd usx + hd - usxd + ysxd # lim . h h h:0 h:0

= lim usx + hd # lim h:0

As h approaches zero, usx + hd approaches u(x) because u, being differentiable at x, is continuous at x. The two fractions approach the values of dy>dx at x and du>dx at x. In short, d dy du suyd = u + y . dx dx dx

3.3

EXAMPLE 7

Differentiation Rules

135

Find the derivative of y = sx 2 + 1dsx 3 + 3d.

Solution

(a) From the Product Rule with u = x 2 + 1 and y = x 3 + 3, we find d C sx 2 + 1dsx 3 + 3d D = sx 2 + 1ds3x 2 d + sx 3 + 3ds2xd dx

d dy du suyd = u + y dx dx dx

= 3x 4 + 3x 2 + 2x 4 + 6x = 5x 4 + 3x 2 + 6x. (b) This particular product can be differentiated as well (perhaps better) by multiplying out the original expression for y and differentiating the resulting polynomial: y = sx 2 + 1dsx 3 + 3d = x 5 + x 3 + 3x 2 + 3 dy = 5x 4 + 3x 2 + 6x. dx This is in agreement with our first calculation. The derivative of the quotient of two functions is given by the Quotient Rule.

Derivative Quotient Rule If u and y are differentiable at x and if ysxd Z 0, then the quotient u>y is differentiable at x, and d u a b = dx y

y

du dy - u dx dx . 2 y

In function notation, gsxdƒ¿sxd - ƒsxdg¿sxd d ƒsxd d = . c dx gsxd g 2sxd

EXAMPLE 8

Find the derivative of (a) y =

t2 - 1 , (b) y = e -x. t3 + 1

Solution

(a) We apply the Quotient Rule with u = t 2 - 1 and y = t 3 + 1: dy st 3 + 1d # 2t - st 2 - 1d # 3t 2 = dt st 3 + 1d2

(b)

=

2t 4 + 2t - 3t 4 + 3t 2 st 3 + 1d2

=

-t 4 + 3t 2 + 2t . st 3 + 1d2

ex # 0 - 1 # ex d 1 d -x -1 = x = -e - x (e ) = a xb = dx dx e e (e x) 2

ysdu>dtd - usdy>dtd d u a b = dt y y2

136

Chapter 3: Differentiation

Proof of the Derivative Quotient Rule usxd usx + hd ysx + hd ysxd d u a b = lim dx y h h:0 ysxdusx + hd - usxdysx + hd = lim hysx + hdysxd h:0 To change the last fraction into an equivalent one that contains the difference quotients for the derivatives of u and y, we subtract and add y(x)u(x) in the numerator. We then get ysxdusx + hd - ysxdusxd + ysxdusxd - usxdysx + hd d u a b = lim dx y hysx + hdysxd h:0 usx + hd - usxd ysx + hd - ysxd ysxd - usxd h h = lim . ysx + hdysxd h:0 Taking the limits in the numerator and denominator now gives the Quotient Rule. Exercise 74 outlines another proof. The choice of which rules to use in solving a differentiation problem can make a difference in how much work you have to do. Here is an example.

EXAMPLE 9

Find the derivative of y =

sx - 1dsx2 - 2xd . x4

Using the Quotient Rule here will result in a complicated expression with many terms. Instead, use some algebra to simplify the expression. First expand the numerator and divide by x 4 : Solution

y =

sx - 1dsx 2 - 2xd x 3 - 3x 2 + 2x = = x -1 - 3x -2 + 2x -3 . x4 x4

Then use the Sum and Power Rules: dy = -x -2 - 3s -2dx -3 + 2s -3dx -4 dx 6 6 1 = - 2 + 3 - 4. x x x

Second- and Higher-Order Derivatives If y = ƒsxd is a differentiable function, then its derivative ƒ¿sxd is also a function. If ƒ¿ is also differentiable, then we can differentiate ƒ¿ to get a new function of x denoted by ƒ–. So ƒ– = sƒ¿d¿ . The function ƒ– is called the second derivative of ƒ because it is the derivative of the first derivative. It is written in several ways: ƒ–sxd =

d 2y dx 2

=

dy¿ d dy a b = = y– = D 2sƒdsxd = D x2 ƒsxd. dx dx dx

The symbol D 2 means the operation of differentiation is performed twice. If y = x 6 , then y¿ = 6x 5 and we have y– = Thus D 2(x 6) = 30x 4 .

dy¿ d = (6x 5) = 30x 4 . dx dx

3.3

How to Read the Symbols for Derivatives “y prime” y¿ “y double prime” y– d 2y “d squared y dx squared” dx 2 y‡ “y triple prime” y snd “y super n” d ny “d to the n of y by dx to the n” dx n D n “D to the n”

Differentiation Rules

137

If y– is differentiable, its derivative, y‡ = dy–>dx = d 3y>dx 3, is the third derivative of y with respect to x. The names continue as you imagine, with y snd =

d ny d sn - 1d y = = D ny dx dx n

denoting the nth derivative of y with respect to x for any positive integer n. We can interpret the second derivative as the rate of change of the slope of the tangent to the graph of y = ƒsxd at each point. You will see in the next chapter that the second derivative reveals whether the graph bends upward or downward from the tangent line as we move off the point of tangency. In the next section, we interpret both the second and third derivatives in terms of motion along a straight line.

EXAMPLE 10

The first four derivatives of y = x 3 - 3x 2 + 2 are First derivative:

y¿ = 3x 2 - 6x

Second derivative:

y– = 6x - 6

Third derivative:

y‡ = 6

Fourth derivative:

y s4d = 0.

The function has derivatives of all orders, the fifth and later derivatives all being zero.

Exercises 3.3 Derivative Calculations In Exercises 1–12, find the first and second derivatives. 1. y = - x 2 + 3

2. y = x 2 + x + 8

3. s = 5t - 3t

4. w = 3z 7 - 7z 3 + 21z 2

3

5

4x 3 - x + 2e x 3 1 7. w = 3z -2 - z 5. y =

9. y = 6x 2 - 10x - 5x -2 5 1 11. r = 2 2s 3s

x3 x2 x + + 3 2 4 4 8. s = -2t -1 + 2 t 10. y = 4 - 2x - x -3 6. y =

12 1 4 - 3 + 4 12. r = u u u

27. y =

sx + 1dsx + 2d 1 28. y = 2 sx - 1dsx - 2d sx - 1dsx + x + 1d 2

29. y = 2e -x + e 3x 31. y = x 3e x 33. y = x 9>4 + e -2x 35. s = 2t 3>2 + 3e 2

x 2 + 3e x 2e x - x 32. w = re -r 30. y =

34. y = x -3>5 + p3>2 p 1 36. w = 1.4 + z 2z

7 2 37. y = 2x - xe

3 9.6 38. y = 2 x + 2e 1.3

es 39. r = s

40. r = e u a

1 + u-p>2 b u2

In Exercises 13–16, find y¿ (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate.

Find the derivatives of all orders of the functions in Exercises 41–44.

13. y = s3 - x 2 dsx 3 - x + 1d 14. y = s2x + 3ds5x 2 - 4xd

41. y =

1 15. y = sx 2 + 1d ax + 5 + x b 16. y = s1 + x 2 dsx 3>4 - x -3 d

43. y = sx - 1dsx 2 + 3x - 5d 44. y = s4x 3 + 3xds2 - xd

Find the derivatives of the functions in Exercises 17–40. 2x + 5 17. y = 3x - 2 x2 - 4 19. g sxd = x + 0.5 2 -1

21. y = s1 - tds1 + t d 1s - 1 23. ƒssd = 1s + 1 1 + x - 4 1x 25. y = x

4 - 3x 18. z = 3x 2 + x t2 - 1 20. ƒstd = 2 t + t - 2 22. w = s2x - 7d-1sx + 5d 5x + 1 24. u = 2 1x 26. r = 2 a

1 2u

3 x4 - x2 - x 2 2

x5 120

Find the first and second derivatives of the functions in Exercises 45–52. 45. y = 47. r =

x3 + 7 x

46. s =

su - 1dsu 2 + u + 1d

49. w = a + 2ub

42. y =

u3 1 + 3z bs3 - zd 3z

51. w = 3z 2e 2z

48. u =

t 2 + 5t - 1 t2 2 sx + xdsx 2 - x + 1d x4 q + 3 2

50. p =

sq - 1d3 + sq + 1d3

52. w = e zsz - 1dsz 2 + 1d

138

Chapter 3: Differentiation

53. Suppose u and y are functions of x that are differentiable at x = 0 and that us0d = 5,

u¿s0d = -3,

ys0d = -1,

y¿s0d = 2 .

Find the values of the following derivatives at x = 0 . a.

d suyd dx

b.

d u a b dx y

c.

d y a b dx u

d.

d s7y - 2ud dx

62. Find all points (x, y) on the graph of gsxd = 13 x 3 tangent lines parallel to the line 8x - 2y = 1.

u¿s1d = 0,

ys1d = 5,

64. Find all points (x, y) on the graph of ƒsxd = x 2 with tangent lines passing through the point (3, 8). y 10

y¿s1d = -1 .

f (x) 5 x 2 (3, 8)

Find the values of the following derivatives at x = 1 . a.

d suyd dx

b.

d u a b dx y

c.

d y a b dx u

d.

x 2 + 1 with

63. Find all points (x, y) on the graph of y = x>(x - 2) with tangent lines perpendicular to the line y = 2x + 3.

54. Suppose u and y are differentiable functions of x and that us1d = 2,

3 2

6

d s7y - 2ud dx

(x, y) 2

Slopes and Tangents 55. a. Normal to a curve Find an equation for the line perpendicular to the tangent to the curve y = x 3 - 4x + 1 at the point (2, 1).

–2

2

4

x

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope? c. Tangents having specified slope Find equations for the tangents to the curve at the points where the slope of the curve is 8. 56. a. Horizontal tangents Find equations for the horizontal tangents to the curve y = x 3 - 3x - 2 . Also find equations for the lines that are perpendicular to these tangents at the points of tangency. b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope? Find an equation for the line that is perpendicular to the curve’s tangent at this point. 57. Find the tangents to Newton’s serpentine (graphed here) at the origin and the point (1, 2). y y 24x x 1

(1, 2)

1 2 3 4

0

8 x2 4

T b. Graph the curve and tangent together. The tangent intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates.

x 50 - 1 x:1 x - 1

67. lim

68.

x 2>9 - 1 x: -1 x + 1 lim

69. Find the value of a that makes the following function differentiable for all x-values.

(2, 1) 1 2 3

66. a. Find an equation for the line that is tangent to the curve y = x 3 - 6x 2 + 5x at the origin.

Theory and Examples For Exercises 67 and 68 evaluate each limit by first converting each to a derivative at a particular x-value.

y

2 1

T c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent simultaneously (Solver key).

x

58. Find the tangent to the Witch of Agnesi (graphed here) at the point (2, 1). y

T b. Graph the curve and tangent line together. The tangent intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates.

T c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent simultaneously (Solver key).

2 1 0

65. a. Find an equation for the line that is tangent to the curve y = x 3 - x at the point s -1, 0d .

x

59. Quadratic tangent to identity function The curve y = ax 2 + bx + c passes through the point (1, 2) and is tangent to the line y = x at the origin. Find a, b, and c. 60. Quadratics having a common tangent The curves y = x 2 + ax + b and y = cx - x 2 have a common tangent line at the point (1, 0). Find a, b, and c. 61. Find all points (x, y) on the graph of ƒsxd = 3x 2 - 4x with tangent lines parallel to the line y = 8x + 5.

gsxd = e

ax, x 2 - 3x,

if x 6 0 if x Ú 0

70. Find the values of a and b that make the following function differentiable for all x-values. ƒsxd = e

ax + b, bx 2 - 3,

x 7 -1 x … -1

71. The general polynomial of degree n has the form Psxd = an x n + an - 1 x n - 1 + Á + a2 x 2 + a1 x + a0 where an Z 0 . Find P¿sxd .

3.4 72. The body’s reaction to medicine The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form R = M2 a

The Derivative as a Rate of Change

139

c. What is the formula for the derivative of a product u1 u2 u3 Á un of a finite number n of differentiable functions of x? 76. Power Rule for negative integers Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is,

C M - b, 2 3

where C is a positive constant and M is the amount of medicine absorbed in the blood. If the reaction is a change in blood pressure, R is measured in millimeters of mercury. If the reaction is a change in temperature, R is measured in degrees, and so on. Find dR>dM . This derivative, as a function of M, is called the sensitivity of the body to the medicine. In Section 4.5, we will see how to find the amount of medicine to which the body is most sensitive.

d -m (x ) = -mx -m - 1 dx where m is a positive integer. 77. Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form nRT an 2 - 2, V - nb V in which a, b, n, and R are constants. Find dP>dV . (See accompanying figure.) P =

73. Suppose that the function y in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule? 74. The Reciprocal Rule a. The Reciprocal Rule says that at any point where the function y(x) is differentiable and different from zero, d 1 1 dy a b = - 2 . dx y y dx Show that the Reciprocal Rule is a special case of the Derivative Quotient Rule. b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule. 75. Generalizing the Product Rule The Derivative Product Rule gives the formula d dy du suyd = u + y dx dx dx

78. The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is hq km Asqd = q + cm + , 2

for the derivative of the product uy of two differentiable functions of x. a. What is the analogous formula for the derivative of the product uyw of three differentiable functions of x? b. What is the formula for the derivative of the product u1 u2 u3 u4 of four differentiable functions of x?

3.4

where q is the quantity you order when things run low (shoes, radios, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). Find dA>dq and d 2A>dq 2 .

The Derivative as a Rate of Change In Section 2.1 we introduced average and instantaneous rates of change. In this section we study further applications in which derivatives model the rates at which things change. It is natural to think of a quantity changing with respect to time, but other variables can be treated in the same way. For example, an economist may want to study how the cost of producing steel varies with the number of tons produced, or an engineer may want to know how the power output of a generator varies with its temperature.

Instantaneous Rates of Change If we interpret the difference quotient sƒsx + hd - ƒsxdd>h as the average rate of change in ƒ over the interval from x to x + h, we can interpret its limit as h : 0 as the rate at which ƒ is changing at the point x.

140

Chapter 3: Differentiation

DEFINITION

The instantaneous rate of change of ƒ with respect to x at x0 is

the derivative ƒ¿sx0 d = lim

h:0

ƒsx0 + hd - ƒsx0 d , h

provided the limit exists. Thus, instantaneous rates are limits of average rates. It is conventional to use the word instantaneous even when x does not represent time. The word is, however, frequently omitted. When we say rate of change, we mean instantaneous rate of change.

EXAMPLE 1

The area A of a circle is related to its diameter by the equation A =

p 2 D . 4

How fast does the area change with respect to the diameter when the diameter is 10 m? Solution

The rate of change of the area with respect to the diameter is dA p pD = # 2D = . 4 2 dD

When D = 10 m, the area is changing with respect to the diameter at the rate of sp>2d10 = 5p m2>m L 15.71 m2>m.

Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk Suppose that an object is moving along a coordinate line (an s-axis), usually horizontal or vertical, so that we know its position s on that line as a function of time t: s = ƒstd. Position at time t … s f(t)

Δs

The displacement of the object over the time interval from t to t + ¢t (Figure 3.14) is

and at time t Δ t s Δs f (t Δt)

FIGURE 3.14 The positions of a body moving along a coordinate line at time t and shortly later at time t + ¢t . Here the coordinate line is horizontal.

¢s = ƒst + ¢td - ƒstd,

s

and the average velocity of the object over that time interval is yay =

ƒst + ¢td - ƒstd displacement ¢s = = . travel time ¢t ¢t

To find the body’s velocity at the exact instant t, we take the limit of the average velocity over the interval from t to t + ¢t as ¢t shrinks to zero. This limit is the derivative of ƒ with respect to t.

DEFINITION Velocity (instantaneous velocity) is the derivative of position with respect to time. If a body’s position at time t is s = ƒstd, then the body’s velocity at time t is ystd =

ƒst + ¢td - ƒstd ds = lim . dt ¢t ¢t:0

3.4

s s f (t) ds 0 dt

t

0 (a) s increasing: positive slope so moving upward

141

Besides telling how fast an object is moving along the horizontal line in Figure 3.14, its velocity tells the direction of motion. When the object is moving forward (s increasing), the velocity is positive; when the object is moving backward (s decreasing), the velocity is negative. If the coordinate line is vertical, the object moves upward for positive velocity and downward for negative velocity. The blue curves in Figure 3.15 represent position along the line over time; they do not portray the path of motion, which lies along the s-axis. If we drive to a friend’s house and back at 30 mph, say, the speedometer will show 30 on the way over but it will not show -30 on the way back, even though our distance from home is decreasing. The speedometer always shows speed, which is the absolute value of velocity. Speed measures the rate of progress regardless of direction.

DEFINITION

s

The Derivative as a Rate of Change

Speed is the absolute value of velocity. ds Speed = ƒ ystd ƒ = ` ` dt

s f (t) ds 0 dt

Figure 3.16 shows the graph of the velocity y = ƒ¿std of a particle moving along a horizontal line (as opposed to showing a position function s = ƒstd such as in Figure 3.15). In the graph of the velocity function, it’s not the slope of the curve that tells us if the particle is moving forward or backward along the line (which is not shown in the figure), but rather the sign of the velocity. Looking at Figure 3.16, we see that the particle moves forward for the first 3 sec (when the velocity is positive), moves backward for the next 2 sec (the velocity is negative), stands motionless for a full second, and then moves forward again. The particle is speeding up when its positive velocity increases during the first second, moves at a steady speed during the next second, and then slows down as the velocity decreases to zero during the third second. It stops for an instant at t = 3 sec (when the velocity is zero) and reverses direction as the velocity starts to become negative. The particle is now moving backward and gaining in speed until t = 4 sec, at which time it achieves its greatest speed during its backward motion. Continuing its backward motion at time t = 4, the particle starts to slow down again until it finally stops at time t = 5 (when the velocity is once again zero). The particle now remains motionless for one full second, and then moves forward again at t = 6 sec, speeding up during the final second of the forward motion indicated in the velocity graph.

EXAMPLE 2 t

0 (b) s decreasing: negative slope so moving downward

FIGURE 3.15 For motion s = ƒstd along a straight line (the vertical axis), y = ds>dt is (a) positive when s increases and (b) negative when s decreases.

HISTORICAL BIOGRAPHY Bernard Bolzano (1781–1848)

The rate at which a body’s velocity changes is the body’s acceleration. The acceleration measures how quickly the body picks up or loses speed. In Chapter 12 we will study motion in the plane and in space, where acceleration of an object may also lead to a change in direction. A sudden change in acceleration is called a jerk. When a ride in a car or a bus is jerky, it is not that the accelerations involved are necessarily large but that the changes in acceleration are abrupt.

DEFINITIONS Acceleration is the derivative of velocity with respect to time. If a body’s position at time t is s = ƒstd, then the body’s acceleration at time t is dy d 2s astd = = 2. dt dt Jerk is the derivative of acceleration with respect to time: da d 3s jstd = = 3. dt dt Near the surface of the Earth all bodies fall with the same constant acceleration. Galileo’s experiments with free fall (see Section 2.1) lead to the equation s =

1 2 gt , 2

142

Chapter 3: Differentiation y MOVES FORWARD

FORWARD AGAIN

(y 0)

(y 0)

Velocity y f '(t) Speeds up

Steady

Slows down

(y const)

Speeds up Stands still (y 0)

0

1

2

3

4

5

6

7

t (sec)

Greatest speed

Speeds up

Slows down

MOVES BACKWARD

(y 0)

FIGURE 3.16 The velocity graph of a particle moving along a horizontal line, discussed in Example 2.

where s is the distance fallen and g is the acceleration due to Earth’s gravity. This equation holds in a vacuum, where there is no air resistance, and closely models the fall of dense, heavy objects, such as rocks or steel tools, for the first few seconds of their fall, before the effects of air resistance are significant. The value of g in the equation s = s1>2dgt 2 depends on the units used to measure t and s. With t in seconds (the usual unit), the value of g determined by measurement at sea level is approximately 32 ft>sec2 (feet per second squared) in English units, and g = 9.8 m>sec2 (meters per second squared) in metric units. (These gravitational constants depend on the distance from Earth’s center of mass, and are slightly lower on top of Mt. Everest, for example.) The jerk associated with the constant acceleration of gravity sg = 32 ft>sec2 d is zero: j = t (seconds) t0

s (meters)

t1

5

0

10 15 t2

20 25

d sgd = 0. dt

An object does not exhibit jerkiness during free fall.

EXAMPLE 3

Figure 3.17 shows the free fall of a heavy ball bearing released from rest at time t = 0 sec. (a) How many meters does the ball fall in the first 3 sec? (b) What is its velocity, speed, and acceleration when t = 3?

30 35 40 t3

45

Solution

(a) The metric free-fall equation is s = 4.9t 2 . During the first 3 sec, the ball falls ss3d = 4.9s3d2 = 44.1 m. (b) At any time t, velocity is the derivative of position:

FIGURE 3.17 A ball bearing falling from rest (Example 3).

ystd = s¿std =

d s4.9t 2 d = 9.8t. dt

3.4

The Derivative as a Rate of Change

143

At t = 3, the velocity is ys3d = 29.4 m>sec in the downward (increasing s) direction. The speed at t = 3 is speed = ƒ ys3d ƒ = 29.4 m>sec. The acceleration at any time t is astd = y¿std = s–std = 9.8 m>sec2 . At t = 3, the acceleration is 9.8 m>sec2 .

EXAMPLE 4

A dynamite blast blows a heavy rock straight up with a launch velocity of 160 ft> sec (about 109 mph) (Figure 3.18a). It reaches a height of s = 160t - 16t 2 ft after t sec.

s y0

Height (ft)

smax

(a) How high does the rock go? (b) What are the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? (c) What is the acceleration of the rock at any time t during its flight (after the blast)? (d) When does the rock hit the ground again?

t?

256

Solution

(a) In the coordinate system we have chosen, s measures height from the ground up, so the velocity is positive on the way up and negative on the way down. The instant the rock is at its highest point is the one instant during the flight when the velocity is 0. To find the maximum height, all we need to do is to find when y = 0 and evaluate s at this time. At any time t during the rock’s motion, its velocity is

s0 (a) s, y 400

y =

s 160t 16t 2

ds d = s160t - 16t 2 d = 160 - 32t ft>sec. dt dt

The velocity is zero when 160

0 –160

160 - 32t = 0 5

10

t

y ds 160 32t dt (b)

FIGURE 3.18 (a) The rock in Example 4. (b) The graphs of s and y as functions of time; s is largest when y = ds>dt = 0 . The graph of s is not the path of the rock: It is a plot of height versus time. The slope of the plot is the rock’s velocity, graphed here as a straight line.

or

t = 5 sec.

The rock’s height at t = 5 sec is smax = ss5d = 160s5d - 16s5d2 = 800 - 400 = 400 ft. See Figure 3.18b. (b) To find the rock’s velocity at 256 ft on the way up and again on the way down, we first find the two values of t for which sstd = 160t - 16t 2 = 256. To solve this equation, we write 16t 2 - 160t + 256 = 0 16st 2 - 10t + 16d = 0 st - 2dst - 8d = 0 t = 2 sec, t = 8 sec. The rock is 256 ft above the ground 2 sec after the explosion and again 8 sec after the explosion. The rock’s velocities at these times are ys2d = 160 - 32s2d = 160 - 64 = 96 ft>sec. ys8d = 160 - 32s8d = 160 - 256 = -96 ft>sec.

144

Chapter 3: Differentiation

At both instants, the rock’s speed is 96 ft> sec. Since ys2d 7 0, the rock is moving upward (s is increasing) at t = 2 sec; it is moving downward (s is decreasing) at t = 8 because ys8d 6 0. (c) At any time during its flight following the explosion, the rock’s acceleration is a constant a =

dy d = s160 - 32td = -32 ft>sec2 . dt dt

The acceleration is always downward. As the rock rises, it slows down; as it falls, it speeds up. (d) The rock hits the ground at the positive time t for which s = 0. The equation 160t - 16t 2 = 0 factors to give 16ts10 - td = 0, so it has solutions t = 0 and t = 10. At t = 0, the blast occurred and the rock was thrown upward. It returned to the ground 10 sec later.

Derivatives in Economics Cost y (dollars) Slope marginal cost

y c (x)

xh x Production (tons/week)

0

x

FIGURE 3.19 Weekly steel production: c(x) is the cost of producing x tons per week. The cost of producing an additional h tons is csx + hd - csxd .

Engineers use the terms velocity and acceleration to refer to the derivatives of functions describing motion. Economists, too, have a specialized vocabulary for rates of change and derivatives. They call them marginals. In a manufacturing operation, the cost of production c(x) is a function of x, the number of units produced. The marginal cost of production is the rate of change of cost with respect to level of production, so it is dc>dx. Suppose that c(x) represents the dollars needed to produce x tons of steel in one week. It costs more to produce x + h tons per week, and the cost difference, divided by h, is the average cost of producing each additional ton: csx + hd - csxd average cost of each of the additional = h tons of steel produced. h The limit of this ratio as h : 0 is the marginal cost of producing more steel per week when the current weekly production is x tons (Figure 3.19): csx + hd - csxd dc = lim = marginal cost of production. dx h h:0 Sometimes the marginal cost of production is loosely defined to be the extra cost of producing one additional unit:

y

csx + 1d - csxd ¢c = , 1 ¢x which is approximated by the value of dc>dx at x. This approximation is acceptable if the slope of the graph of c does not change quickly near x. Then the difference quotient will be close to its limit dc>dx, which is the rise in the tangent line if ¢x = 1 (Figure 3.20). The approximation works best for large values of x. Economists often represent a total cost function by a cubic polynomial

y c(x)

x 1

⎧ ⎪ ⎨ c ⎪ ⎩

dc dx

csxd = ax 3 + bx 2 + gx + d

0

x

x1

x

FIGURE 3.20 The marginal cost dc>dx is approximately the extra cost ¢c of producing ¢ x = 1 more unit.

where d represents fixed costs, such as rent, heat, equipment capitalization, and management costs. The other terms represent variable costs, such as the costs of raw materials, taxes, and labor. Fixed costs are independent of the number of units produced, whereas variable costs depend on the quantity produced. A cubic polynomial is usually adequate to capture the cost behavior on a realistic quantity interval.

EXAMPLE 5

Suppose that it costs csxd = x 3 - 6x 2 + 15x

3.4

The Derivative as a Rate of Change

145

dollars to produce x radiators when 8 to 30 radiators are produced and that rsxd = x 3 - 3x 2 + 12x gives the dollar revenue from selling x radiators. Your shop currently produces 10 radiators a day. About how much extra will it cost to produce one more radiator a day, and what is your estimated increase in revenue for selling 11 radiators a day? Solution

The cost of producing one more radiator a day when 10 are produced is about

c¿s10d: c¿sxd =

d 3 A x - 6x 2 + 15x B = 3x 2 - 12x + 15 dx

c¿s10d = 3s100d - 12s10d + 15 = 195. The additional cost will be about $195. The marginal revenue is d 3 (x - 3x 2 + 12x) = 3x 2 - 6x + 12. dx The marginal revenue function estimates the increase in revenue that will result from selling one additional unit. If you currently sell 10 radiators a day, you can expect your revenue to increase by about r¿sxd =

r¿s10d = 3s100d - 6s10d + 12 = $252 if you increase sales to 11 radiators a day.

EXAMPLE 6 To get some feel for the language of marginal rates, consider marginal tax rates. If your marginal income tax rate is 28% and your income increases by $1000, you can expect to pay an extra $280 in taxes. This does not mean that you pay 28% of your entire income in taxes. It just means that at your current income level I, the rate of increase of taxes T with respect to income is dT>dI = 0.28. You will pay $0.28 in taxes out of every extra dollar you earn. Of course, if you earn a lot more, you may land in a higher tax bracket and your marginal rate will increase.

y 1

Sensitivity to Change

y 2p p 2 0

1

p

(a)

When a small change in x produces a large change in the value of a function ƒ(x), we say that the function is relatively sensitive to changes in x. The derivative ƒ¿sxd is a measure of this sensitivity.

EXAMPLE 7

dy /dp

Genetic Data and Sensitivity to Change

The Austrian monk Gregor Johann Mendel (1822–1884), working with garden peas and other plants, provided the first scientific explanation of hybridization. His careful records showed that if p (a number between 0 and 1) is the frequency of the gene for smooth skin in peas (dominant) and s1 - pd is the frequency of the gene for wrinkled skin in peas, then the proportion of smooth-skinned peas in the next generation will be

2

dy 2 2p dp

y = 2ps1 - pd + p 2 = 2p - p 2 . 0

1

p

(b)

FIGURE 3.21 (a) The graph of y = 2p - p 2 , describing the proportion of smooth-skinned peas in the next generation. (b) The graph of dy>dp (Example 7).

The graph of y versus p in Figure 3.21a suggests that the value of y is more sensitive to a change in p when p is small than when p is large. Indeed, this fact is borne out by the derivative graph in Figure 3.21b, which shows that dy>dp is close to 2 when p is near 0 and close to 0 when p is near 1. The implication for genetics is that introducing a few more smooth skin genes into a population where the frequency of wrinkled skin peas is large will have a more dramatic effect on later generations than will a similar increase when the population has a large proportion of smooth skin peas.

146

Chapter 3: Differentiation

Exercises 3.4 Motion Along a Coordinate Line Exercises 1–6 give the positions s = ƒstd of a body moving on a coordinate line, with s in meters and t in seconds. a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? 1. s = t 2 - 3t + 2, 2. s = 6t - t , 2

0 … t … 2

0 … t … 6

3. s = -t 3 + 3t 2 - 3t,

0 … t … 3

4. s = st >4d - t + t , 0 … t … 3 5 25 5. s = 2 - t , 1 … t … 5 t 25 6. s = , -4 … t … 0 t + 5 4

3

2

7. Particle motion At time t, the position of a body moving along the s-axis is s = t 3 - 6t 2 + 9t m . a. Find the body’s acceleration each time the velocity is zero. b. Find the body’s speed each time the acceleration is zero. c. Find the total distance traveled by the body from t = 0 to t = 2 . 8. Particle motion At time t Ú 0 , the velocity of a body moving along the horizontal s-axis is y = t 2 - 4t + 3 . a. Find the body’s acceleration each time the velocity is zero.

12. Speeding bullet A 45-caliber bullet shot straight up from the surface of the moon would reach a height of s = 832t - 2.6t 2 ft after t sec. On Earth, in the absence of air, its height would be s = 832t - 16t 2 ft after t sec. How long will the bullet be aloft in each case? How high will the bullet go? 13. Free fall from the Tower of Pisa Had Galileo dropped a cannonball from the Tower of Pisa, 179 ft above the ground, the ball’s height above the ground t sec into the fall would have been s = 179 - 16t 2 . a. What would have been the ball’s velocity, speed, and acceleration at time t ? b. About how long would it have taken the ball to hit the ground? c. What would have been the ball’s velocity at the moment of impact? 14. Galileo’s free-fall formula Galileo developed a formula for a body’s velocity during free fall by rolling balls from rest down increasingly steep inclined planks and looking for a limiting formula that would predict a ball’s behavior when the plank was vertical and the ball fell freely; see part (a) of the accompanying figure. He found that, for any given angle of the plank, the ball’s velocity t sec into motion was a constant multiple of t. That is, the velocity was given by a formula of the form y = kt . The value of the constant k depended on the inclination of the plank. In modern notation—part (b) of the figure—with distance in meters and time in seconds, what Galileo determined by experiment was that, for any given angle u , the ball’s velocity t sec into the roll was y = 9.8ssin udt m>sec .

b. When is the body moving forward? Backward?

Free-fall position

c. When is the body’s velocity increasing? Decreasing? Free-Fall Applications 9. Free fall on Mars and Jupiter The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t in seconds) are s = 1.86t 2 on Mars and s = 11.44t 2 on Jupiter. How long does it take a rock falling from rest to reach a velocity of 27.8 m> sec (about 100 km> h) on each planet? 10. Lunar projectile motion A rock thrown vertically upward from the surface of the moon at a velocity of 24 m> sec (about 86 km> h) reaches a height of s = 24t - 0.8t 2 m in t sec. a. Find the rock’s velocity and acceleration at time t. (The acceleration in this case is the acceleration of gravity on the moon.) b. How long does it take the rock to reach its highest point? c. How high does the rock go?

?

θ

a. What is the equation for the ball’s velocity during free fall? b. Building on your work in part (a), what constant acceleration does a freely falling body experience near the surface of Earth? Understanding Motion from Graphs 15. The accompanying figure shows the velocity y = ds>dt = ƒstd (m> sec) of a body moving along a coordinate line.

d. How long does it take the rock to reach half its maximum height? e. How long is the rock aloft? 11. Finding g on a small airless planet Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of 15 m> sec. Because the acceleration of gravity at the planet’s surface was gs m>sec2 , the explorers expected the ball bearing to reach a height of s = 15t - s1>2dgs t 2 m t sec later. The ball bearing reached its maximum height 20 sec after being launched. What was the value of gs ?

(b)

(a)

y (m/sec) y f (t)

3 0

2

4

6

8 10

t (sec)

–3

a. When does the body reverse direction? b. When (approximately) is the body moving at a constant speed?

3.4 c. Graph the body’s speed for 0 … t … 10 .

The Derivative as a Rate of Change

147

18. The accompanying figure shows the velocity y = ƒstd of a particle moving on a horizontal coordinate line.

d. Graph the acceleration, where defined. 16. A particle P moves on the number line shown in part (a) of the accompanying figure. Part (b) shows the position of P as a function of time t.

y

y f(t)

P

s (cm)

0

0

1 2 3 4 5 6 7 8 9

t (sec)

(a) s (cm) s f (t)

2 0

1

2

3

4

a. When does the particle move forward? Move backward? Speed up? Slow down? 5

6

t (sec)

–2

c. When does the particle move at its greatest speed?

(6, 4)

–4

b. When is the particle’s acceleration positive? Negative? Zero? d. When does the particle stand still for more than an instant?

(b)

a. When is P moving to the left? Moving to the right? Standing still? b. Graph the particle’s velocity and speed (where defined).

19. Two falling balls The multiflash photograph in the accompanying figure shows two balls falling from rest. The vertical rulers are marked in centimeters. Use the equation s = 490t 2 (the freefall equation for s in centimeters and t in seconds) to answer the following questions.

17. Launching a rocket When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it lands. The figure here shows velocity data from the flight of the model rocket. Use the data to answer the following.

a. How fast was the rocket climbing when the engine stopped? b. For how many seconds did the engine burn? 200

Velocity (ft /sec)

150 100 50 0 –50 –100 0

2

4 6 8 10 Time after launch (sec)

12

c. When did the rocket reach its highest point? What was its velocity then? d. When did the parachute pop out? How fast was the rocket falling then? e. How long did the rocket fall before the parachute opened? f. When was the rocket’s acceleration greatest? g. When was the acceleration constant? What was its value then (to the nearest integer)?

a. How long did it take the balls to fall the first 160 cm? What was their average velocity for the period? b. How fast were the balls falling when they reached the 160-cm mark? What was their acceleration then? c. About how fast was the light flashing (flashes per second)?

148

Chapter 3: Differentiation

20. A traveling truck The accompanying graph shows the position s of a truck traveling on a highway. The truck starts at t = 0 and returns 15 h later at t = 15 . a. Use the technique described in Section 3.2, Example 3, to graph the truck’s velocity y = ds>dt for 0 … t … 15 . Then repeat the process, with the velocity curve, to graph the truck’s acceleration dy>dt. b. Suppose that s = 15t 2 - t 3 . Graph ds>dt and d 2s>dt 2 and compare your graphs with those in part (a).

Position, s (km)

500

Economics 23. Marginal cost Suppose that the dollar cost of producing x washing machines is csxd = 2000 + 100x - 0.1x 2 . a. Find the average cost per machine of producing the first 100 washing machines. b. Find the marginal cost when 100 washing machines are produced. c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

400

24. Marginal revenue Suppose that the revenue from selling x washing machines is

300

1 rsxd = 20,000 a1 - x b dollars.

200

a. Find the marginal revenue when 100 machines are produced. 100 0

5 10 Elapsed time, t (hr)

15

21. The graphs in the accompanying figure show the position s, velocity y = ds>dt , and acceleration a = d 2s>dt 2 of a body moving along a coordinate line as functions of time t. Which graph is which? Give reasons for your answers. y A

B

b. Use the function r¿sxd to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week. c. Find the limit of r¿sxd as x : q . How would you interpret this number? Additional Applications 25. Bacterium population When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10 6 + 10 4t - 10 3t 2 . Find the growth rates at a. t = 0 hours .

C

b. t = 5 hours . c. t = 10 hours . t

0

22. The graphs in the accompanying figure show the position s, the velocity y = ds>dt , and the acceleration a = d 2s>dt 2 of a body moving along the coordinate line as functions of time t. Which graph is which? Give reasons for your answers. y A

26. Draining a tank The number of gallons of water in a tank t minutes after the tank has started to drain is Qstd = 200s30 - td2 . How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min? T 27. Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula y = 6 a1 -

2

t b m. 12

a. Find the rate dy>dt (m> h) at which the tank is draining at time t. b. When is the fluid level in the tank falling fastest? Slowest? What are the values of dy>dt at these times?

0

t B

c. Graph y and dy>dt together and discuss the behavior of y in relation to the signs and values of dy>dt. 28. Inflating a balloon The volume V = s4>3dpr 3 of a spherical balloon changes with the radius. a. At what rate sft3>ftd does the volume change with respect to the radius when r = 2 ft ?

C

b. By approximately how much does the volume increase when the radius changes from 2 to 2.2 ft?

3.5

Derivatives of Trigonometric Functions

149

29. Airplane takeoff Suppose that the distance an aircraft travels along a runway before takeoff is given by D = s10>9dt 2 , where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 200 km> h. How long will it take to become airborne, and what distance will it travel in that time?

velocity function ystd = ds>dt = ƒ¿std and the acceleration function astd = d 2s>dt 2 = ƒ–std . Comment on the object’s behavior in relation to the signs and values of y and a. Include in your commentary such topics as the following:

30. Volcanic lava fountains Although the November 1959 Kilauea Iki eruption on the island of Hawaii began with a line of fountains along the wall of the crater, activity was later confined to a single vent in the crater’s floor, which at one point shot lava 1900 ft straight into the air (a Hawaiian record). What was the lava’s exit velocity in feet per second? In miles per hour? (Hint: If y0 is the exit velocity of a particle of lava, its height t sec later will be s = y0 t - 16t 2 ft . Begin by finding the time at which ds>dt = 0 . Neglect air resistance.)

b. When does it move to the left (down) or to the right (up)?

Analyzing Motion Using Graphs T Exercises 31–34 give the position function s = ƒstd of an object moving along the s-axis as a function of time t. Graph ƒ together with the

3.5

a. When is the object momentarily at rest? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? 31. s = 200t - 16t 2, 0 … t … 12.5 (a heavy object fired straight up from Earth’s surface at 200 ft> sec) 32. s = t 2 - 3t + 2,

0 … t … 5

33. s = t 3 - 6t 2 + 7t,

0 … t … 4

34. s = 4 - 7t + 6t - t 3, 2

0 … t … 4

Derivatives of Trigonometric Functions Many phenomena of nature are approximately periodic (electromagnetic fields, heart rhythms, tides, weather). The derivatives of sines and cosines play a key role in describing periodic changes. This section shows how to differentiate the six basic trigonometric functions.

Derivative of the Sine Function To calculate the derivative of ƒsxd = sin x, for x measured in radians, we combine the limits in Example 5a and Theorem 7 in Section 2.4 with the angle sum identity for the sine function: sin sx + hd = sin x cos h + cos x sin h. If ƒsxd = sin x, then ƒ¿sxd = lim

h:0

= lim

h:0

ƒsx + hd - ƒsxd sin sx + hd - sin x = lim h h h:0

Derivative definition

ssin x cos h + cos x sin hd - sin x sin x scos h - 1d + cos x sin h = lim h h h:0

= lim asin x # h:0

= sin x # lim

h:0

cos h - 1 sin h b + lim acos x # b h h h:0

cos h - 1 sin h + cos x # lim = sin x # 0 + cos x # 1 = cos x. h h:0 h

(+++)+++* limit 0

(+)+* limit 1

The derivative of the sine function is the cosine function: d ssin xd = cos x. dx

Example 5a and Theorem 7, Section 2.4

150

Chapter 3: Differentiation

EXAMPLE 1

We find derivatives of the sine function involving differences, products,

and quotients. dy d = 2x (sin x) dx dx

(a) y = x 2 - sin x:

Difference Rule

= 2x - cos x dy d d x = ex (sin x) + (e ) sin x dx dx dx = e x cos x + e x sin x = e x (cos x + sin x)

x (b) y = e sin x:

d x# (sin x) - sin x # 1 dy dx = dx x2

sin x (c) y = x :

=

Product Rule

Quotient Rule

x cos x - sin x x2

Derivative of the Cosine Function With the help of the angle sum formula for the cosine function, cos sx + hd = cos x cos h - sin x sin h, we can compute the limit of the difference quotient: cossx + hd - cos x d scos xd = lim dx h h:0 = lim

h:0

scos x cos h - sin x sin hd - cos x h

Derivative definition

Cosine angle sum identity

cos xscos h - 1d - sin x sin h h h:0

= lim

= lim cos x # h:0

cos h - 1 sin h - lim sin x # h h h:0

= cos x # lim

cos h - 1 sin h - sin x # lim h h:0 h:0 h

y y cos x

1 –

0 –1 y'

= cos x # 0 - sin x # 1

Example 5a and Theorem 7, Section 2.4

x

= -sin x. y' –sin x

1 –

0 –1

FIGURE 3.22 The curve y¿ = -sin x as the graph of the slopes of the tangents to the curve y = cos x .

x

The derivative of the cosine function is the negative of the sine function: d scos xd = -sin x. dx

3.5

Derivatives of Trigonometric Functions

151

Figure 3.22 shows a way to visualize this result in the same way we did for graphing derivatives in Section 3.2, Figure 3.6.

EXAMPLE 2

We find derivatives of the cosine function in combinations with other

functions. (a) y = 5e x + cos x: dy d d = s5e x d + (cos x) dx dx dx

Sum Rule

= 5e x - sin x (b) y = sin x cos x: dy d d = sin x (cos x) + cos x (sin x) dx dx dx

Product Rule

= sin xs -sin xd + cos xscos xd = cos2 x - sin2 x (c) y =

cos x : 1 - sin x d d (1 - sin x) (cos x) - cos x (1 - sin x) dy dx dx = dx s1 - sin xd2 =

s1 - sin xds -sin xd - cos xs0 - cos xd s1 - sin xd2

=

1 - sin x s1 - sin xd2

=

1 1 - sin x

Quotient Rule

sin2 x + cos2 x = 1

Simple Harmonic Motion The motion of an object or weight bobbing freely up and down with no resistance on the end of a spring is an example of simple harmonic motion. The motion is periodic and repeats indefinitely, so we represent it using trigonometric functions. The next example describes a case in which there are no opposing forces such as friction to slow the motion.

EXAMPLE 3

A weight hanging from a spring (Figure 3.23) is stretched down 5 units beyond its rest position and released at time t = 0 to bob up and down. Its position at any later time t is

–5

s = 5 cos t. 0

5

Rest position Position at t0

s

FIGURE 3.23 A weight hanging from a vertical spring and then displaced oscillates above and below its rest position (Example 3).

What are its velocity and acceleration at time t ? We have s = 5 cos t Position:

Solution

Velocity:

y =

ds d = s5 cos td = -5 sin t dt dt

Acceleration:

a =

dy d = s -5 sin td = -5 cos t. dt dt

152

Chapter 3: Differentiation

Notice how much we can learn from these equations:

s, y 5

y –5 sin t

1.

s 5 cos t

2. 0

2

3 2

2 5 2

t

–5

3.

FIGURE 3.24 The graphs of the position and velocity of the weight in Example 3.

4.

As time passes, the weight moves down and up between s = -5 and s = 5 on the s-axis. The amplitude of the motion is 5. The period of the motion is 2p, the period of the cosine function. The velocity y = -5 sin t attains its greatest magnitude, 5, when cos t = 0, as the graphs show in Figure 3.24. Hence, the speed of the weight, ƒ y ƒ = 5 ƒ sin t ƒ , is greatest when cos t = 0, that is, when s = 0 (the rest position). The speed of the weight is zero when sin t = 0. This occurs when s = 5 cos t = ;5, at the endpoints of the interval of motion. The weight is acted on by the spring and by gravity. When the weight is below the rest position, the combined forces pull it up, and when it is above the rest position, they pull it down. The weight’s acceleration is always proportional to the negative of its displacement. This property of springs is called Hooke’s Law, and is studied further in Section 6.5. The acceleration, a = -5 cos t, is zero only at the rest position, where cos t = 0 and the force of gravity and the force from the spring balance each other. When the weight is anywhere else, the two forces are unequal and acceleration is nonzero. The acceleration is greatest in magnitude at the points farthest from the rest position, where cos t = ;1.

EXAMPLE 4

The jerk associated with the simple harmonic motion in Example 3 is j =

da d = s -5 cos td = 5 sin t. dt dt

It has its greatest magnitude when sin t = ;1, not at the extremes of the displacement but at the rest position, where the acceleration changes direction and sign.

Derivatives of the Other Basic Trigonometric Functions Because sin x and cos x are differentiable functions of x, the related functions sin x tan x = cos x ,

cot x =

cos x , sin x

1 sec x = cos x ,

and

csc x =

1 sin x

are differentiable at every value of x at which they are defined. Their derivatives, calculated from the Quotient Rule, are given by the following formulas. Notice the negative signs in the derivative formulas for the cofunctions.

The derivatives of the other trigonometric functions: d stan xd = sec2 x dx

d scot xd = -csc2 x dx

d ssec xd = sec x tan x dx

d scsc xd = -csc x cot x dx

To show a typical calculation, we find the derivative of the tangent function. The other derivations are left to Exercise 60.

3.5

EXAMPLE 5 Solution

Derivatives of Trigonometric Functions

Find d(tan x)> dx.

We use the Derivative Quotient Rule to calculate the derivative: d d ssin xd - sin x scos xd dx dx cos2 x cos x cos x - sin x s -sin xd = cos2 x

d sin x d b = stan xd = a dx dx cos x

EXAMPLE 6 Solution

153

cos x

=

cos2 x + sin2 x cos2 x

=

1 = sec2 x. cos2 x

Quotient Rule

Find y– if y = sec x.

Finding the second derivative involves a combination of trigonometric deriva-

tives. y = sec x y¿ = sec x tan x y– =

Derivative rule for secant function

d ssec x tan xd dx

= sec x

d d (tan x) + tan x (sec x) dx dx

= sec xssec2 xd + tan xssec x tan xd = sec3 x + sec x tan2 x

Derivative Product Rule Derivative rules

The differentiability of the trigonometric functions throughout their domains gives another proof of their continuity at every point in their domains (Theorem 1, Section 3.2). So we can calculate limits of algebraic combinations and composites of trigonometric functions by direct substitution.

EXAMPLE 7

We can use direct substitution in computing limits provided there is no division by zero, which is algebraically undefined. 22 + sec x 22 + sec 0 22 + 1 23 = - 23 = = = -1 cossp tan xd cossp tan 0d cossp 0d x:0 lim

Exercises 3.5 Derivatives In Exercises 1–18, find dy>dx.

5. y = csc x - 41x + 7

1. y = -10x + 3 cos x

3 2. y = x + 5 sin x

3. y = x cos x

4. y = 2x sec x + 3

2

7. ƒsxd = sin x tan x

1 x2 8. gsxd = csc x cot x 6. y = x 2 cot x -

9. y = ssec x + tan xdssec x - tan xd 10. y = ssin x + cos xd sec x

154 11. y =

Chapter 3: Differentiation cot x 1 + cot x

4 1 13. y = cos x + tan x

12. y =

cos x 1 + sin x

14. y =

cos x x x + cos x

44. Find all points on the curve y = cot x, 0 6 x 6 p , where the tangent line is parallel to the line y = -x . Sketch the curve and tangent(s) together, labeling each with its equation. In Exercises 45 and 46, find an equation for (a) the tangent to the curve at P and (b) the horizontal tangent to the curve at Q.

15. y = x sin x + 2x cos x - 2 sin x 2

16. y = x 2 cos x - 2x sin x - 2 cos x 17. ƒsxd = x sin x cos x 3

45.

18. gsxd = s2 - xd tan x

21. s =

y Q P ⎛ , 2⎛ ⎝2 ⎝

2

In Exercises 19–22, find ds>dt. 19. s = tan t - e-t

46.

y

2

20. s = t 2 - sec t + 5e t

1 + csc t 1 - csc t

22. s =

sin t 1 - cos t

1

P ⎛ , 4⎛ ⎝4 ⎝

4

In Exercises 23–26, find dr>du . 23. r = 4 - u 2 sin u

24. r = u sin u + cos u

25. r = sec u csc u

26. r = s1 + sec ud sin u

0

2 2 y 4 cot x 2csc x 1

x Q

In Exercises 27–32, find dp>dq. 27. p = 5 + 29. p = 31. p =

1 cot q

28. p = s1 + csc qd cos q

sin q + cos q cos q q sin q

q - 1 33. Find y– if 2

30. p =

tan q 1 + tan q

32. p =

3q + tan q q sec q

a. y = csc x . a. y = -2 sin x .

b. y = 9 cos x .

49.

-3p>2 … x … 2p

36. y = tan x,

-p>2 6 x 6 p>2

x = -p>3, 0, p>3 37. y = sec x,

-p>2 6 x 6 p>2

x = -p>3, p>4 38. y = 1 + cos x,

-3p>2 … x … 2p

x = -p>3, 3p>2 T Do the graphs of the functions in Exercises 39–42 have any horizontal tangents in the interval 0 … x … 2p ? If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher. 39. y = x + sin x 40. y = 2x + sin x 41. y = x - cot x 42. y = x + 2 cos x 43. Find all points on the curve y = tan x, -p>2 6 x 6 p>2 , where the tangent line is parallel to the line y = 2x . Sketch the curve and tangent(s) together, labeling each with its equation.

x

1 1 47. lim sin a x - b 2 x:2 48.

x = -p, 0, 3p>2

3

Trigonometric Limits Find the limits in Exercises 47–54.

b. y = sec x .

Tangent Lines In Exercises 35–38, graph the curves over the given intervals, together with their tangents at the given values of x. Label each curve and tangent with its equation.

2

y 1 2 csc x cot x

lim

x: - p>6

34. Find y s4d = d 4 y>dx 4 if

35. y = sin x,

1 4

0

lim

u :p>6

21 + cos sp csc xd

sin u u -

1 2

51. lim sec ce x + p tan a x:0

52. lim sin a x:0

u :p>4

tan u - 1 u - p4

p b - 1d 4 sec x

p + tan x b tan x - 2 sec x

53. lim tan a1 t: 0

lim

50.

p 6

sin t t b

54. lim cos a u :0

pu b sin u

Theory and Examples The equations in Exercises 55 and 56 give the position s = ƒstd of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = p>4 sec . 55. s = 2 - 2 sin t

56. s = sin t + cos t

57. Is there a value of c that will make sin2 3x , x2 ƒsxd = L c,

x Z 0 x = 0

continuous at x = 0 ? Give reasons for your answer. 58. Is there a value of b that will make g sxd = e

x + b, cos x,

x 6 0 x Ú 0

continuous at x = 0 ? Differentiable at x = 0 ? Give reasons for your answers.

3.5

Derivatives of Trigonometric Functions

59. By computing the first few derivatives and looking for a pattern, find d 999>dx 999 scos xd .

See the accompanying figure.

60. Derive the formula for the derivative with respect to x of

y

a. sec x.

b. csc x.

155

Slope f '(x)

c. cot x.

61. A weight is attached to a spring and reaches its equilibrium position sx = 0d. It is then set in motion resulting in a displacement of

Slope

C

x = 10 cos t,

B

f(x h) f(x) h

A

where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Slope

f (x h) f(x h) 2h

y f (x) h 0

xh

h xh

x

x

–10

0

Equilibrium position at x 5 0

10

a. To see how rapidly the centered difference quotient for ƒsxd = sin x converges to ƒ¿sxd = cos x , graph y = cos x together with y =

sin sx + hd - sin sx - hd 2h

x

a. Find the spring’s displacement when t = 0, t = p>3, and t = 3p>4. b. Find the spring’s velocity when t = 0, t = p>3, and t = 3p>4. 62. Assume that a particle’s position on the x-axis is given by x = 3 cos t + 4 sin t,

over the interval [-p, 2p] for h = 1, 0.5 , and 0.3. Compare the results with those obtained in Exercise 63 for the same values of h. b. To see how rapidly the centered difference quotient for ƒsxd = cos x converges to ƒ¿sxd = -sin x , graph y = -sin x together with

where x is measured in feet and t is measured in seconds. a. Find the particle’s position when t = 0, t = p>2, and t = p.

y =

b. Find the particle’s velocity when t = 0, t = p>2, and t = p. T 63. Graph y = cos x for -p … x … 2p . On the same screen, graph y =

sin sx + hd - sin x h

for h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try h = -1, -0.5 , and -0.3 . What happens as h : 0 + ? As h : 0 - ? What phenomenon is being illustrated here? T 64. Graph y = -sin x for -p … x … 2p . On the same screen, graph cos sx + hd - cos x y = h for h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try h = -1, -0.5 , and -0.3 . What happens as h : 0 + ? As h : 0 - ? What phenomenon is being illustrated here? 65. Centered difference quotients The centered difference quotient T

ƒsx + hd - ƒsx - hd 2h is used to approximate ƒ¿sxd in numerical work because (1) its limit as h : 0 equals ƒ¿sxd when ƒ¿sxd exists, and (2) it usually gives a better approximation of ƒ¿sxd for a given value of h than the difference quotient ƒsx + hd - ƒsxd . h

cos sx + hd - cos sx - hd 2h

over the interval [-p, 2p] for h = 1, 0.5 , and 0.3. Compare the results with those obtained in Exercise 64 for the same values of h. 66. A caution about centered difference quotients (Continuation of Exercise 65.) The quotient ƒsx + hd - ƒsx - hd 2h may have a limit as h : 0 when ƒ has no derivative at x. As a case in point, take ƒsxd = ƒ x ƒ and calculate lim

h:0

ƒ0 + hƒ - ƒ0 - hƒ . 2h

As you will see, the limit exists even though ƒsxd = ƒ x ƒ has no derivative at x = 0 . Moral: Before using a centered difference quotient, be sure the derivative exists. T 67. Slopes on the graph of the tangent function Graph y = tan x and its derivative together on s -p>2, p>2d . Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers.

156

Chapter 3: Differentiation

T 68. Slopes on the graph of the cotangent function Graph y = cot x and its derivative together for 0 6 x 6 p . Does the graph of the cotangent function appear to have a smallest slope? A largest slope? Is the slope ever positive? Give reasons for your answers.

b. With your grapher still in degree mode, estimate lim

h:0

T 69. Exploring (sin kx) / x Graph y = ssin xd>x , y = ssin 2xd>x , and y = ssin 4xd>x together over the interval -2 … x … 2 . Where does each graph appear to cross the y-axis? Do the graphs really intersect the axis? What would you expect the graphs of y = ssin 5xd>x and y = ssin s -3xdd>x to do as x : 0 ? Why? What about the graph of y = ssin kxd>x for other values of k ? Give reasons for your answers.

c. Now go back to the derivation of the formula for the derivative of sin x in the text and carry out the steps of the derivation using degree-mode limits. What formula do you obtain for the derivative? d. Work through the derivation of the formula for the derivative of cos x using degree-mode limits. What formula do you obtain for the derivative?

T 70. Radians versus degrees: degree mode derivatives What happens to the derivatives of sin x and cos x if x is measured in degrees instead of radians? To find out, take the following steps.

e. The disadvantages of the degree-mode formulas become apparent as you start taking derivatives of higher order. Try it. What are the second and third degree-mode derivatives of sin x and cos x?

a. With your graphing calculator or computer grapher in degree mode, graph ƒshd =

cos h - 1 . h

sin h h

and estimate limh:0 ƒshd . Compare your estimate with p>180 . Is there any reason to believe the limit should be p>180 ?

3.6

The Chain Rule How do we differentiate F(x) = sin (x 2 - 4)? This function is the composite ƒ ⴰ g of two functions y = ƒ(u) = sin u and u = gsxd = x 2 - 4 that we know how to differentiate. The answer, given by the Chain Rule, says that the derivative is the product of the derivatives of ƒ and g. We develop the rule in this section.

Derivative of a Composite Function The function y =

3 1 1 x = s3xd is the composite of the functions y = u and u = 3x. 2 2 2

We have dy 3 = , 2 dx 2

3 1

C: y turns B: u turns A: x turns

FIGURE 3.25 When gear A makes x turns, gear B makes u turns and gear C makes y turns. By comparing circumferences or counting teeth, we see that y = u>2 (C turns one-half turn for each B turn) and u = 3x (B turns three times for A’s one), so y = 3x>2 . Thus, dy>dx = 3>2 = s1>2ds3d = sdy>dudsdu>dxd .

Since

dy 1 = , 2 du

and

du = 3. dx

3 1 = # 3, we see in this case that 2 2 dy dy du # . = dx du dx

If we think of the derivative as a rate of change, our intuition allows us to see that this relationship is reasonable. If y = ƒsud changes half as fast as u and u = gsxd changes three times as fast as x, then we expect y to change 3>2 times as fast as x. This effect is much like that of a multiple gear train (Figure 3.25). Let’s look at another example.

EXAMPLE 1

The function y = s3x 2 + 1d2

3.6

The Chain Rule

157

is the composite of y = ƒ(u) = u 2 and u = g(x) = 3x 2 + 1. Calculating derivatives, we see that dy du # = 2u # 6x du dx = 2s3x 2 + 1d # 6x = 36x 3 + 12x. Calculating the derivative from the expanded formula (3x 2 + 1) 2 = 9x 4 + 6x 2 + 1 gives the same result: dy d = (9x 4 + 6x 2 + 1) dx dx = 36x 3 + 12x. The derivative of the composite function ƒ(g(x)) at x is the derivative of ƒ at g(x) times the derivative of g at x. This is known as the Chain Rule (Figure 3.26). Composite f g

˚

Rate of change at x is f '(g(x)) • g'(x).

x

g

f

Rate of change at x is g'(x).

Rate of change at g(x) is f '( g(x)).

u g(x)

y f (u) f(g(x))

FIGURE 3.26 Rates of change multiply: The derivative of ƒ ⴰ g at x is the derivative of ƒ at g(x) times the derivative of g at x.

THEOREM 2—The Chain Rule If ƒ(u) is differentiable at the point u = gsxd and g(x) is differentiable at x, then the composite function sƒ ⴰ gdsxd = ƒsgsxdd is differentiable at x, and sƒ ⴰ gd¿sxd = ƒ¿sgsxdd # g¿sxd. In Leibniz’s notation, if y = ƒsud and u = gsxd, then dy du dy # , = dx du dx where dy>du is evaluated at u = gsxd.

A Proof of One Case of the Chain Rule: Let ¢u be the change in u when x changes by ¢x, so that ¢u = gsx + ¢xd - gsxd. Then the corresponding change in y is ¢y = ƒsu + ¢ud - ƒsud. If ¢u Z 0, we can write the fraction ¢y>¢x as the product ¢y ¢y ¢u # = ¢x ¢u ¢x

(1)

158

Chapter 3: Differentiation

and take the limit as ¢x : 0: dy ¢y = lim dx ¢x:0 ¢x ¢y ¢u # = lim ¢x:0 ¢u ¢x ¢y # lim ¢u = lim ¢x:0 ¢u ¢x:0 ¢x ¢y # lim ¢u = lim ¢u:0 ¢u ¢x:0 ¢x dy du # . = du dx

(Note that ¢u : 0 as ¢x : 0 since g is continuous.)

The problem with this argument is that if the function g(x) oscillates rapidly near x, then ¢u can be zero even when ¢x Z 0, so the cancellation of ¢u in Equation (1) would be invalid. A complete proof requires a different approach that avoids this problem, and we give one such proof in Section 3.11.

EXAMPLE 2 An object moves along the x-axis so that its position at any time t Ú 0 is given by xstd = cosst 2 + 1d. Find the velocity of the object as a function of t. We know that the velocity is dx>dt. In this instance, x is a composite function: x = cossud and u = t 2 + 1. We have

Solution

dx = -sin sud du du = 2t. dt

x = cossud u = t2 + 1

By the Chain Rule, dx dx # du = dt du dt = -sin sud # 2t = -sin st 2 + 1d # 2t = -2t sinst 2 + 1d.

dx evaluated at u du

“Outside-Inside” Rule A difficulty with the Leibniz notation is that it doesn’t state specifically where the derivatives in the Chain Rule are supposed to be evaluated. So it sometimes helps to think about the Chain Rule using functional notation. If y = ƒ(g(x)), then dy = ƒ¿sgsxdd # g¿sxd. dx In words, differentiate the “outside” function ƒ and evaluate it at the “inside” function g(x) left alone; then multiply by the derivative of the “inside function.”

EXAMPLE 3 Solution

Differentiate sin sx 2 + e x d with respect to x.

We apply the Chain Rule directly and find d sin (x 2 + e x ) = cos (x 2 + e x ) # (2x + e x). (+)+* (+)+* (+)+* dx inside

inside left alone

derivative of the inside

3.6

EXAMPLE 4

The Chain Rule

159

Differentiate y = e cos x.

Here the inside function is u = g (x) = cos x and the outside function is the exponential function ƒ(x) = e x. Applying the Chain Rule, we get

Solution

dy d cos x d = (e ) = e cos x (cos x) = e cos x (-sin x) = -e cos x sin x. dx dx dx Generalizing Example 4, we see that the Chain Rule gives the formula d u du e = eu . dx dx For example, d d kx (e ) = e kx # (kx) = ke kx, dx dx

for any constant k

and 2 2 d d x2 A e B = e x # dx (x 2) = 2xe x . dx

Repeated Use of the Chain Rule We sometimes have to use the Chain Rule two or more times to find a derivative. HISTORICAL BIOGRAPHY

EXAMPLE 5

Johann Bernoulli (1667–1748)

Solution

Find the derivative of gstd = tan s5 - sin 2td.

Notice here that the tangent is a function of 5 - sin 2t, whereas the sine is a function of 2t, which is itself a function of t. Therefore, by the Chain Rule, d (tan (5 - sin 2t)) dt d = sec2 s5 - sin 2td # (5 - sin 2t) dt

Derivative of tan u with u = 5 - sin 2t

= sec2 s5 - sin 2td # a0 - cos 2t #

Derivative of 5 - sin u with u = 2t

g¿std =

= sec2 s5 - sin 2td # s -cos 2td # 2 = -2scos 2td sec2 s5 - sin 2td.

d (2t)b dt

The Chain Rule with Powers of a Function If ƒ is a differentiable function of u and if u is a differentiable function of x, then substituting y = ƒsud into the Chain Rule formula dy dy du # = dx du dx leads to the formula d du ƒsud = ƒ¿sud . dx dx If n is any real number and ƒ is a power function, ƒsud = u n , the Power Rule tells us that ƒ¿sud = nu n - 1 . If u is a differentiable function of x, then we can use the Chain Rule to extend this to the Power Chain Rule: d n du su d = nu n - 1 . dx dx

d A u n B = nu n - 1 du

160

Chapter 3: Differentiation

EXAMPLE 6

The Power Chain Rule simplifies computing the derivative of a power of

an expression. (a)

d d s5x 3 - x 4 d7 = 7s5x 3 - x 4 d6 (5x 3 - x 4) dx dx

= 7s5x 3 - x 4 d6s5 # 3x 2 - 4x 3 d

Power Chain Rule with u = 5x 3 - x 4, n = 7

= 7s5x 3 - x 4 d6s15x 2 - 4x 3 d (b)

d d 1 a b = s3x - 2d-1 dx 3x - 2 dx d s3x - 2d dx = -1s3x - 2d-2s3d 3 = s3x - 2d2 = -1s3x - 2d-2

Power Chain Rule with u = 3x - 2, n = - 1

In part (b) we could also find the derivative with the Derivative Quotient Rule. (c)

d d (sin5 x) = 5 sin4 x # sin x dx dx 4 = 5 sin x cos x

(d)

d d 23x + 1 Ae B = e 23x + 1 # dx A 23x + 1 B dx = e 23x + 1 # =

Power Chain Rule with u = sin x, n = 5, because sinn x means ssin xdn, n Z -1 .

1 (3x + 1) -1>2 # 3 2

3 223x + 1

Power Chain Rule with u = 3x + 1, n = 1>2

e 23x + 1

In Section 3.2, we saw that the absolute value function y = ƒ x ƒ is not differentiable at x 0. However, the function is differentiable at all other real numbers, as we now show. Since ƒ x ƒ = 2x 2, we can derive the following formula:

EXAMPLE 7

d d ( x ) = 2x 2 dx ƒ ƒ dx 1 # d 2 = (x ) 22x 2 dx 1 # = 2x 2ƒxƒ x = , x Z 0. ƒxƒ

Derivative of the Absolute Value Function d x , (ƒxƒ) = dx ƒxƒ

x Z 0

EXAMPLE 8

Power Chain Rule with u = x 2, n = 1>2, x Z 0 2x 2 = ƒ x ƒ

Show that the slope of every line tangent to the curve y = 1>s1 - 2xd3 is

positive. We find the derivative:

Solution

dy d = s1 - 2xd-3 dx dx = -3s1 - 2xd-4 #

d s1 - 2xd dx

= -3s1 - 2xd-4 # s -2d =

6 . s1 - 2xd4

Power Chain Rule with u = s1 - 2xd, n = -3

3.6

161

The Chain Rule

At any point (x, y) on the curve, x Z 1>2 and the slope of the tangent line is dy 6 , = dx s1 - 2xd4 the quotient of two positive numbers.

EXAMPLE 9

The formulas for the derivatives of both sin x and cos x were obtained under the assumption that x is measured in radians, not degrees. The Chain Rule gives us new insight into the difference between the two. Since 180° = p radians, x° = px>180 radians where x° is the size of the angle measured in degrees. By the Chain Rule, d px d px p p b = cos a b = cossx°d. sin sx°d = sin a 180 180 180 180 dx dx See Figure 3.27. Similarly, the derivative of cossx°d is -sp>180d sin sx°d. The factor p>180 would compound with repeated differentiation. We see here the advantage for the use of radian measure in computations. y sin(x°) sin x 180

y 1

x

180

y sin x

FIGURE 3.27 The function sin sx°d oscillates only p>180 times as often as sin x oscillates. Its maximum slope is p>180 at x = 0 (Example 9).

Exercises 3.6 Derivative Calculations In Exercises 1–8, given y = ƒsud and u = gsxd , find dy>dx = ƒ¿sgsxddg¿sxd . 1. y = 6u - 9,

u = s1>2dx 4

2. y = 2u 3,

u = 8x - 1

3. y = sin u,

u = 3x + 1

4. y = cos u,

u = -x>3

5. y = cos u,

u = sin x

6. y = sin u,

u = x - cos x

7. y = tan u,

u = 10x - 5

8. y = -sec u,

u = x 2 + 7x

In Exercises 9–22, write the function in the form y = ƒsud and u = gsxd . Then find dy>dx as a function of x. 9. y = s2x + 1d5 11. y = a1 13. y = a

x b 7

10. y = s4 - 3xd9

-7

x2 1 + x - xb 8

12. y = a

x - 1b 2

-10

4

19. y = e-5x

20. y = e 2x>3

22. y = e A42x + x B

21. y = e 5 - 7x

2

Find the derivatives of the functions in Exercises 23–50. 23. p = 23 - t 25. s =

3 2r - r 2 24. q = 2

4 4 sin 3t + cos 5t 5p 3p

26. s = sin a

3pt 3pt b + cos a b 2 2

27. r = scsc u + cot ud-1

28. r = 6ssec u - tan ud3>2 x 1 29. y = x 2 sin4 x + x cos-2 x 30. y = x sin-5 x - cos3 x 3 -1 1 1 b s3x - 2d7 + a4 31. y = 2 21 2x 32. y = s5 - 2xd-3 +

1 2 a + 1b 8 x

4

14. y = 23x 2 - 4x + 6

33. y = s4x + 3d4sx + 1d-3

34. y = s2x - 5d-1sx 2 - 5xd6

35. y = xe -x + e 3x

36. y = (1 + 2x) e-2x

15. y = sec stan xd

1 16. y = cot ap - x b

37. y = (x - 2x + 2) e

17. y = sin3 x

18. y = 5 cos-4 x

2

5x>2

39. hsxd = x tan A 21x B + 7

38. y = (9x 2 - 6x + 2) e x 1 40. k sxd = x 2 sec a x b

3

162

Chapter 3: Differentiation

41. ƒsxd = 27 + x sec x 43. ƒsud = a

sin u b 1 + cos u

42. g sxd =

2

44. g std = a

t 2t + 1

49. y = cos A e- u

2

1 + sin 3t b 3 - 2t

87. Suppose that functions ƒ and g and their derivatives with respect to x have the following values at x = 2 and x = 3 . -1

1 46. r = sec2u tan a b u

45. r = sin su2 d cos s2ud 47. q = sin a

tan 3x sx + 7d4

b

sin t 48. q = cot a t b

B

50. y = u3e-2u cos 5u

51. y = sin2 spt - 2d

52. y = sec2 pt

53. y = s1 + cos 2td-4

54. y = s1 + cot st>2dd-2

55. y = st tan td10

56. y = st -3>4 sin td4>3

2

59. y = a

58. y = A e sin (t>2) B

(pt - 1)

t2 b t - 4t

3

60. y = a

3

t bb 12

-5

t 62. y = cos a5 sin a b b 3

61. y = sin scos s2t - 5dd 63. y = a1 + tan4 a

3

3t - 4 b 5t + 2

3

64. y =

1 A 1 + cos2 s7td B 3 6

65. y = 21 + cos st 2 d

66. y = 4 sin A 21 + 1t B

67. y = tan2 ssin3 td

68. y = cos4 ssec2 3td

69. y = 3t s2t 2 - 5d4

70. y = 43t + 32 + 21 - t

-1

1 73. y = cot s3x - 1d 9

x 74. y = 9 tan a b 3

75. y = x s2x + 1d4

76. y = x 2 sx 3 - 1d5

77. y = e

x2

+ 5x

78. y = sin (x 2e x )

Finding Derivative Values In Exercises 79–84, find the value of sƒ ⴰ gd¿ at the given value of x. 79. ƒsud = u 5 + 1,

u = g sxd = 1x,

x = 1

1 1 , x = -1 80. ƒsud = 1 - u , u = g sxd = 1 - x pu , u = g sxd = 5 1x, x = 1 81. ƒsud = cot 10 1 , u = g sxd = px, x = 1>4 82. ƒsud = u + cos2 u 2u , u = g sxd = 10x 2 + x + 1, x = 0 83. ƒsud = 2 u + 1 84. ƒsud = a

2

u - 1 b , u + 1

ƒ(x)

g(x)

2 3

8 3

2 -4

1>3 2p

-3 5

a. 2ƒsxd,

b. ƒsxd + g sxd,

x = 2

c. ƒsxd # g sxd,

x = 3

x = 3

x = 2

d. ƒsxd>g sxd,

e. ƒsg sxdd,

x = 2

f. 2ƒsxd,

g. 1>g 2sxd,

x = 3

h. 2ƒ2sxd + g 2sxd,

x = 2 x = 2

88. Suppose that the functions ƒ and g and their derivatives with respect to x have the following values at x = 0 and x = 1 . x

ƒ(x)

g(x)

ƒ(x)

g(x)

0 1

1 3

1 -4

5 -1>3

1>3 -8>3

Find the derivatives with respect to x of the following combinations at the given value of x. a. 5ƒsxd - g sxd, x = 1 ƒsxd c. , x = 1 g sxd + 1 e. g sƒsxdd,

x = 0

b. ƒsxdg 3sxd, d. ƒsg sxdd,

x = 0

f. sx 11 + ƒsxdd-2,

x = 0

x = 1

x = 0

89. Find ds>dt when u = 3p>2 if s = cos u and d u>dt = 5 . 72. y = A 1 - 1x B

3

g(x)

g. ƒsx + g sxdd,

Second Derivatives Find y– in Exercises 71–78. 1 71. y = a1 + x b

ƒ(x)

Find the derivatives with respect to x of the following combinations at the given value of x.

In Exercises 51–70, find dy>dt.

57. y = ecos

x

u = g sxd =

1 - 1, x2

x = -1

85. Assume that ƒ¿s3d = -1, g¿s2d = 5, gs2d = 3, and y = ƒsgsxdd. What is y¿ at x = 2? 86. If r = sin sƒstdd, ƒs0d = p>3, and ƒ¿s0d = 4, then what is dr>dt at t = 0?

90. Find dy>dt when x = 1 if y = x 2 + 7x - 5 and dx>dt = 1>3 . Theory and Examples What happens if you can write a function as a composite in different ways? Do you get the same derivative each time? The Chain Rule says you should. Try it with the functions in Exercises 91 and 92. 91. Find dy>dx if y = x by using the Chain Rule with y as a composite of a. y = su>5d + 7

and

u = 5x - 35

b. y = 1 + s1>ud

and

u = 1>sx - 1d .

92. Find dy>dx if y = x 3>2 by using the Chain Rule with y as a composite of a. y = u 3 b. y = 1u

and and

u = 1x u = x3 .

93. Find the tangent to y = ssx - 1d>sx + 1dd2 at x = 0. 94. Find the tangent to y = 2x 2 - x + 7 at x = 2. 95. a. Find the tangent to the curve y = 2 tan spx>4d at x = 1 . b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval -2 6 x 6 2 ? Give reasons for your answer. 96. Slopes on sine curves a. Find equations for the tangents to the curves y = sin 2x and y = -sin sx>2d at the origin. Is there anything special about how the tangents are related? Give reasons for your answer.

3.6

The Chain Rule

163

b. Can anything be said about the tangents to the curves y = sin mx and y = -sin sx>md at the origin sm a constant Z 0d ? Give reasons for your answer.

102. Particle acceleration A particle moves along the x-axis with velocity dx>dt = ƒsxd . Show that the particle’s acceleration is ƒsxdƒ¿sxd .

c. For a given m, what are the largest values the slopes of the curves y = sin mx and y = - sin sx>md can ever have? Give reasons for your answer.

103. Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation

d. The function y = sin x completes one period on the interval [0, 2p] , the function y = sin 2x completes two periods, the function y = sin sx>2d completes half a period, and so on. Is there any relation between the number of periods y = sin mx completes on [0, 2p] and the slope of the curve y = sin mx at the origin? Give reasons for your answer. 97. Running machinery too fast Suppose that a piston is moving straight up and down and that its position at time t sec is s = A cos s2pbtd , with A and b positive. The value of A is the amplitude of the motion, and b is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston’s velocity, acceleration, and jerk? (Once you find out, you will know why some machinery breaks when you run it too fast.) 98. Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is y = 37 sin c

2p sx - 101d d + 25 365

L , Ag

T = 2p

where g is the constant acceleration of gravity at the pendulum’s location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportionality constant, dL = kL . du Assuming this to be the case, show that the rate at which the period changes with respect to temperature is kT>2. Suppose that ƒsxd = x 2 and g sxd = ƒ x ƒ . Then the

104. Chain Rule composites

sƒ ⴰ gdsxd = ƒ x ƒ 2 = x 2

T 105. The derivative of sin 2x Graph the function y = 2 cos 2x for -2 … x … 3.5 . Then, on the same screen, graph y =

a. On what day is the temperature increasing the fastest? b. About how many degrees per day is the temperature increasing when it is increasing at its fastest? y

40 20 0 ...... ..

for h = 1.0, 0.5 , and 0.2. Experiment with other values of h, including negative values. What do you see happening as h : 0 ? Explain this behavior.

y = . ... .... ............ ..... .

. .... ....

x

Ja

–20

. .. .. .. ... .... ....

sin 2sx + hd - sin 2x h

106. The derivative of cos sx 2 d Graph y = -2x sin sx 2 d for -2 … x … 3 . Then, on the same screen, graph

n Fe b M ar A pr M ay Ju n Ju l A ug Se p O ct N ov D ec Ja n Fe b M ar

Temperature (˚F)

60

sg ⴰ ƒdsxd = ƒ x 2 ƒ = x 2

are both differentiable at x = 0 even though g itself is not differentiable at x = 0 . Does this contradict the Chain Rule? Explain.

and is graphed in the accompanying figure.

....... .... . ........ .. . .... .... ... . .... . ... ... ... ... ... ... ..

and

for h = 1.0, 0.7, and 0.3 . Experiment with other values of h. What do you see happening as h : 0 ? Explain this behavior. Using the Chain Rule, show that the Power Rule sd>dxdx n = nx n - 1 holds for the functions xn in Exercises 107 and 108. 107. x1>4 = 2 1x

99. Particle motion The position of a particle moving along a coordinate line is s = 21 + 4t , with s in meters and t in seconds. Find the particle’s velocity and acceleration at t = 6 sec. 100. Constant acceleration Suppose that the velocity of a falling body is y = k1s m>sec (k a constant) at the instant the body has fallen s m from its starting point. Show that the body’s acceleration is constant. 101. Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to 1s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s 2 .

cos ssx + hd2 d - cos sx 2 d h

108. x3>4 = 2x1x

COMPUTER EXPLORATIONS Trigonometric Polynomials 109. As the accompanying figure shows, the trigonometric “polynomial” s = ƒstd = 0.78540 - 0.63662 cos 2t - 0.07074 cos 6t - 0.02546 cos 10t - 0.01299 cos 14t gives a good approximation of the sawtooth function s = g std on the interval [-p, p] . How well does the derivative of ƒ approximate the derivative of g at the points where dg>dt is defined? To find out, carry out the following steps.

164

Chapter 3: Differentiation s = hstd = 1.2732 sin 2t + 0.4244 sin 6t + 0.25465 sin 10t

a. Graph dg>dt (where defined) over [-p, p] .

+ 0.18189 sin 14t + 0.14147 sin 18t

b. Find dƒ>dt. c. Graph dƒ>dt. Where does the approximation of dg>dt by dƒ>dt seem to be best? Least good? Approximations by trigonometric polynomials are important in the theories of heat and oscillation, but we must not expect too much of them, as we see in the next exercise. s

graphed in the accompanying figure approximates the step function s = kstd shown there. Yet the derivative of h is nothing like the derivative of k. s s k(t)

s g(t) s f (t)

2

– –

0

s h(t)

1

t

– 2

2

0

t

–1

110. (Continuation of Exercise 109.) In Exercise 109, the trigonometric polynomial ƒ(t) that approximated the sawtooth function g(t) on [-p, p] had a derivative that approximated the derivative of the sawtooth function. It is possible, however, for a trigonometric polynomial to approximate a function in a reasonable way without its derivative approximating the function’s derivative at all well. As a case in point, the “polynomial”

a. Graph dk>dt (where defined) over [-p, p] . b. Find dh>dt. c. Graph dh>dt to see how badly the graph fits the graph of dk>dt. Comment on what you see.

Implicit Differentiation

3.7

Most of the functions we have dealt with so far have been described by an equation of the form y = ƒsxd that expresses y explicitly in terms of the variable x. We have learned rules for differentiating functions defined in this way. Another situation occurs when we encounter equations like x 3 + y 3 - 9xy = 0,

y

5

(x 0, y 1) A

x 3 y 3 9xy 0

x 2 + y 2 - 25 = 0.

y f2(x)

(x 0, y 2) x0

(x 0, y3)

or

(See Figures 3.28, 3.29, and 3.30.) These equations define an implicit relation between the variables x and y. In some cases we may be able to solve such an equation for y as an explicit function (or even several functions) of x. When we cannot put an equation Fsx, yd = 0 in the form y = ƒsxd to differentiate it in the usual way, we may still be able to find dy>dx by implicit differentiation. This section describes the technique.

y f1(x)

0

y 2 - x = 0,

5

x

y f3 (x)

FIGURE 3.28 The curve x 3 + y 3 - 9xy = 0 is not the graph of any one function of x. The curve can, however, be divided into separate arcs that are the graphs of functions of x. This particular curve, called a folium, dates to Descartes in 1638.

Implicitly Defined Functions We begin with examples involving familiar equations that we can solve for y as a function of x to calculate dy>dx in the usual way. Then we differentiate the equations implicitly, and find the derivative to compare the two methods. Following the examples, we summarize the steps involved in the new method. In the examples and exercises, it is always assumed that the given equation determines y implicitly as a differentiable function of x so that dy>dx exists.

EXAMPLE 1

Find dy>dx if y 2 = x.

The equation y 2 = x defines two differentiable functions of x that we can actually find, namely y1 = 1x and y2 = - 1x (Figure 3.29). We know how to calculate the derivative of each of these for x 7 0:

Solution

dy1 1 = dx 21x

and

dy2 1 = . dx 21x

3.7

y2 x

y

Slope 1 1 2y1 2x

y1 x

P(x, x )

165

But suppose that we knew only that the equation y 2 = x defined y as one or more differentiable functions of x for x 7 0 without knowing exactly what these functions were. Could we still find dy>dx? The answer is yes. To find dy>dx, we simply differentiate both sides of the equation y 2 = x with respect to x, treating y = ƒsxd as a differentiable function of x:

x

0

Implicit Differentiation

y2 = x dy = 1 2y dx

Q(x, x ) y 2 x Slope 1 1 2y 2 2x

The Chain Rule gives

d 2 Ay B = dx

dy d 2 [ƒsxd] = 2ƒsxdƒ¿sxd = 2y . dx dx

dy 1 . = 2y dx

FIGURE 3.29 The equation y 2 - x = 0 , or y 2 = x as it is usually written, defines two differentiable functions of x on the interval x 7 0. Example 1 shows how to find the derivatives of these functions without solving the equation y 2 = x for y.

This one formula gives the derivatives we calculated for both explicit solutions y1 = 1x and y2 = - 1x: dy1 1 1 = = 2y1 dx 21x

y y1 25

EXAMPLE 2 x2

and

dy2 1 1 1 = = . = 2y2 dx 21x 2 A - 1x B

Find the slope of the circle x 2 + y 2 = 25 at the point s3, -4d.

The circle is not the graph of a single function of x. Rather, it is the combined graphs of two differentiable functions, y1 = 225 - x 2 and y2 = - 225 - x 2 (Figure 3.30). The point s3, -4d lies on the graph of y2 , so we can find the slope by calculating the derivative directly, using the Power Chain Rule:

Solution

–5

0

5

x

(3, – 4) y2 –25 x 2

Slope – xy 3 4

dy2 -6 3 -2x ` = ` = = . 2 x=3 4 dx x = 3 2225 - x 2225 - 9

d a-(25 - x2)1>2 b = dx 1 - (25 - x 2) -1>2(-2x) 2

We can solve this problem more easily by differentiating the given equation of the circle implicitly with respect to x:

FIGURE 3.30 The circle combines the graphs of two functions. The graph of y2 is the lower semicircle and passes through s3, -4d .

d 2 d 2 d (x ) + (y ) = (25) dx dx dx 2x + 2y

dy = 0 dx dy x = -y. dx

x The slope at s3, -4d is - y `

s3, -4d

= -

3 3 = . -4 4

Notice that unlike the slope formula for dy2>dx, which applies only to points below the x-axis, the formula dy>dx = -x>y applies everywhere the circle has a slope. Notice also that the derivative involves both variables x and y, not just the independent variable x. To calculate the derivatives of other implicitly defined functions, we proceed as in Examples 1 and 2: We treat y as a differentiable implicit function of x and apply the usual rules to differentiate both sides of the defining equation.

166

Chapter 3: Differentiation

Implicit Differentiation 1. Differentiate both sides of the equation with respect to x, treating y as a differentiable function of x. 2. Collect the terms with dy>dx on one side of the equation and solve for dy>dx.

EXAMPLE 3

y 4

y2 x2 sin xy

Solution

We differentiate the equation implicitly. y 2 = x 2 + sin xy

2

–4

–2

0

Find dy>dx if y 2 = x 2 + sin xy (Figure 3.31).

2

4

d 2 d d A y B = dx A x 2 B + dx A sin xy B dx

x

–2 –4

FIGURE 3.31 The graph of y 2 = x 2 + sin xy in Example 3.

2y

Differentiate both sides with respect to x Á

2y

dy d = 2x + scos xyd A xy B dx dx

Á treating y as a function of x and using the Chain Rule.

2y

dy dy = 2x + scos xyd ay + x b dx dx

Treat xy as a product.

dy dy - scos xyd ax b = 2x + scos xydy dx dx s2y - x cos xyd

Collect terms with dy>dx.

dy = 2x + y cos xy dx dy 2x + y cos xy = 2y - x cos xy dx

Solve for dy>dx.

Notice that the formula for dy>dx applies everywhere that the implicitly defined curve has a slope. Notice again that the derivative involves both variables x and y, not just the independent variable x.

Derivatives of Higher Order Implicit differentiation can also be used to find higher derivatives.

EXAMPLE 4

Find d 2y>dx 2 if 2x 3 - 3y 2 = 8.

To start, we differentiate both sides of the equation with respect to x in order to find y¿ = dy>dx.

Solution

d d (2x 3 - 3y 2) = s8d dx dx 6x 2 - 6yy¿ = 0

Treat y as a function of x. 2

x y¿ = y ,

when y Z 0

Solve for y¿.

We now apply the Quotient Rule to find y– . y– =

2xy - x 2y¿ d x2 2x x2 ay b = = y - 2 # y¿ 2 dx y y

Finally, we substitute y¿ = x 2>y to express y– in terms of x and y. 2x x2 x2 2x x4 y– = y - 2 a y b = y - 3 , y y

when y Z 0

3.7

Curve of lens surface

Normal line

167

Lenses, Tangents, and Normal Lines

Tangent

Light ray

Implicit Differentiation

A Point of entry P

In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry (angles A and B in Figure 3.32). This line is called the normal to the surface at the point of entry. In a profile view of a lens like the one in Figure 3.32, the normal is the line perpendicular to the tangent of the profile curve at the point of entry.

B

Show that the point (2, 4) lies on the curve x 3 + y 3 - 9xy = 0. Then find the tangent and normal to the curve there (Figure 3.33).

EXAMPLE 5

FIGURE 3.32 The profile of a lens, showing the bending (refraction) of a ray of light as it passes through the lens surface.

The point (2, 4) lies on the curve because its coordinates satisfy the equation given for the curve: 23 + 43 - 9s2ds4d = 8 + 64 - 72 = 0. To find the slope of the curve at (2, 4), we first use implicit differentiation to find a formula for dy>dx:

Solution

y

x 3 + y 3 - 9xy = 0

nt

ge

n Ta

d 3 d 3 d d (x ) + (y ) (9xy) = (0) dx dx dx dx

4 No al

rm

x 3 y 3 9xy 0

0

2

3x 2 + 3y 2 x

FIGURE 3.33 Example 5 shows how to find equations for the tangent and normal to the folium of Descartes at (2, 4).

dy dy dx - 9 ax + y b = 0 dx dx dx

s3y 2 - 9xd

dy + 3x 2 - 9y = 0 dx 3sy 2 - 3xd

Differentiate both sides with respect to x. Treat xy as a product and y as a function of x.

dy = 9y - 3x 2 dx dy 3y - x 2 = 2 . dx y - 3x

Solve for dy>dx.

We then evaluate the derivative at sx, yd = s2, 4d: dy 3y - x 2 3s4d - 22 8 4 = = . ` = 2 ` = 2 5 10 dx s2, 4d y - 3x s2, 4d 4 - 3s2d The tangent at (2, 4) is the line through (2, 4) with slope 4>5: y = 4 + y =

4 sx - 2d 5

4 12 x + . 5 5

The normal to the curve at (2, 4) is the line perpendicular to the tangent there, the line through (2, 4) with slope -5>4: y = 4 y = -

5 sx - 2d 4

5 13 x + . 4 2

168

Chapter 3: Differentiation

Exercise 3.7 Differentiating Implicitly Use implicit differentiation to find dy>dx in Exercises 1–16. 1. x 2y + xy 2 = 6

2. x 3 + y 3 = 18xy

3. 2xy + y = x + y

4. x 3 - xy + y 3 = 1

5. x 2sx - yd2 = x 2 - y 2

6. s3xy + 7d2 = 6y 2x - y 8. x 3 = x + 3y

2

7. y 2 =

x - 1 x + 1

9. x = tan y

41. Parallel tangents Find the two points where the curve x 2 + xy + y 2 = 7 crosses the x-axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents? 42. Normals parallel to a line Find the normals to the curve xy + 2x - y = 0 that are parallel to the line 2x + y = 0 . 43. The eight curve Find the slopes of the curve y 4 = y 2 - x 2 at the two points shown here.

10. xy = cot sxyd

11. x + tan (xy) = 0

12. x 4 + sin y = x 3y 2

1 13. y sin a y b = 1 - xy

14. x cos s2x + 3yd = y sin x

15. e 2x = sin (x + 3y)

16. e x

2

y

y

⎛3 , 3 ⎛ ⎝ 4 2 ⎝

1

⎛3 , 1 ⎛ ⎝ 4 2⎝

= 2x + 2y

Find dr>du in Exercises 17–20. + r

x

17. u

1>2

Second Derivatives In Exercises 21–26, use implicit differentiation to find dy>dx and then d 2y>dx 2 . 21. x + y = 1 2

2

23. y = e

x2

2

0

3 4 18. r - 2 2u = u2>3 + u3>4 2 3 20. cos r + cot u = e ru

= 1 1 19. sin sr ud = 2 1>2

22. x

+ 2x

2>3

+ y

2>3

= 1

24. y - 2x = 1 - 2y

y4 5 y2 2 x2 –1

44. The cissoid of Diocles (from about 200 B.C.) Find equations for the tangent and normal to the cissoid of Diocles y 2s2 - xd = x 3 at (1, 1).

2

y

26. xy + y 2 = 1

25. 2 1y = x - y

y 2(2 2 x) 5 x 3

27. If x + y = 16 , find the value of d y>dx at the point (2, 2). 3

3

2

2

28. If xy + y 2 = 1 , find the value of d 2y>dx 2 at the point s0, -1d . 29. y 2 + x 2 = y 4 - 2x

s -2, 1d and s -2, -1d

at

30. sx 2 + y 2 d2 = sx - yd2

s1, 0d and s1, -1d

at

Slopes, Tangents, and Normals In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. 31. x 2 + xy - y 2 = 1, 32. x + y = 25, 2

2

33. x 2y 2 = 9,

s2, 3d

0

1

x

45. The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y 4 - 4y 2 = x 4 - 9x 2 at the four indicated points. y

s3, -4d

s -1, 3d

y 4 2 4y 2 5 x 4 2 9x 2

34. y 2 - 2x - 4y - 1 = 0,

s -2, 1d

35. 6x 2 + 3xy + 2y 2 + 17y - 6 = 0, 36. x - 23xy + 2y = 5, 2

(1, 1)

1

In Exercises 29 and 30, find the slope of the curve at the given points.

2

37. 2xy + p sin y = 2p,

A 23, 2 B

s1, p>2d

38. x sin 2y = y cos 2x,

sp>4, p>2d

39. y = 2 sin spx - yd,

s1, 0d

40. x 2 cos2 y - sin y = 0,

s0, pd

s -1, 0d

(–3, 2)

2

–3

(–3, –2)

–2

(3, 2) x 3 (3, –2)

3.7 46. The folium of Descartes (See Figure 3.28.) a. Find the slope of the folium of Descartes x + y - 9xy = 0 at the points (4, 2) and (2, 4). 3

3

Implicit Differentiation

169

52. The graph of y 2 = x 3 is called a semicubical parabola and is shown in the accompanying figure. Determine the constant b so that the line y = - 13 x + b meets this graph orthogonally.

b. At what point other than the origin does the folium have a horizontal tangent?

y

y2 5 x3

c. Find the coordinates of the point A in Figure 3.28, where the folium has a vertical tangent. Theory and Examples 47. Intersecting normal The line that is normal to the curve x 2 + 2xy - 3y 2 = 0 at (1, 1) intersects the curve at what other point?

y 5 21 x 1 b 3 x

0

48. Power rule for rational exponents Let p and q be integers with q 7 0. If y = x p>q, differentiate the equivalent equation y q = x p implicitly and show that, for y Z 0, d p>q p (p>q) -1 . x = qx dx 49. Normals to a parabola Show that if it is possible to draw three normals from the point (a, 0) to the parabola x = y 2 shown in the accompanying diagram, then a must be greater than 1>2. One of the normals is the x-axis. For what value of a are the other two normals perpendicular?

T In Exercises 53 and 54, find both dy>dx (treating y as a differentiable function of x) and dx>dy (treating x as a differentiable function of y). How do dy>dx and dx>dy seem to be related? Explain the relationship geometrically in terms of the graphs. 53. xy 3 + x 2y = 6 54. x 3 + y 2 = sin2 y

y x y2

x

(a, 0)

0

COMPUTER EXPLORATIONS Use a CAS to perform the following steps in Exercises 55–62.

b. Using implicit differentiation, find a formula for the derivative dy>dx and evaluate it at the given point P.

50. Is there anything special about the tangents to the curves y 2 = x 3 and 2x 2 + 3y 2 = 5 at the points s1, ;1d ? Give reasons for your answer. y

3y 2

5 (1, 1) x

0 (1, –1)

51. Verify that the following pairs of curves meet orthogonally. a. x 2 + y 2 = 4, b. x = 1 - y 2,

x 2 = 3y 2 x =

1 2 y 3

c. Use the slope found in part (b) to find an equation for the tangent line to the curve at P. Then plot the implicit curve and tangent line together on a single graph. 55. x 3 - xy + y 3 = 7,

Ps2, 1d

56. x 5 + y 3x + yx 2 + y 4 = 4, 2 + x , 1 - x

Ps0, 1d

58. y 3 + cos xy = x 2,

Ps1, 0d

y 59. x + tan a x b = 2,

P a1,

57. y 2 + y =

y2 x3 2x 2

a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.

Ps1, 1d

p b 4

60. xy 3 + tan (x + yd = 1,

p P a , 0b 4

61. 2y 2 + sxyd1>3 = x 2 + 2,

Ps1, 1d

62. x21 + 2y + y = x 2,

P s1, 0d

170

Chapter 3: Differentiation

Derivatives of Inverse Functions and Logarithms

3.8

In Section 1.6 we saw how the inverse of a function undoes, or inverts, the effect of that function. We defined there the natural logarithm function ƒ -1(x) = ln x as the inverse of the natural exponential function ƒ(x) = e x. This is one of the most important functioninverse pairs in mathematics and science. We learned how to differentiate the exponential function in Section 3.3. Here we learn a rule for differentiating the inverse of a differentiable function and we apply the rule to find the derivative of the natural logarithm function. y

y 2x 2

Derivatives of Inverses of Differentiable Functions yx

y 1x1 2

We calculated the inverse of the function ƒsxd = s1>2dx + 1 as ƒ -1sxd = 2x - 2 in Example 3 of Section 1.6. Figure 3.34 shows again the graphs of both functions. If we calculate their derivatives, we see that d d 1 1 ƒsxd = a x + 1b = 2 dx dx 2

1 –2

1

x

d d -1 ƒ sxd = s2x - 2d = 2. dx dx

–2

FIGURE 3.34 Graphing a line and its inverse together shows the graphs’ symmetry with respect to the line y = x . The slopes are reciprocals of each other.

The derivatives are reciprocals of one another, so the slope of one line is the reciprocal of the slope of its inverse line. (See Figure 3.34.) This is not a special case. Reflecting any nonhorizontal or nonvertical line across the line y = x always inverts the line’s slope. If the original line has slope m Z 0, the reflected line has slope 1> m. y

y y = f(x)

b = f (a)

(a, b)

a = f –1(b)

0

a

x

(b, a)

0

The slopes are reciprocal: ( f –1)'(b) =

y = f –1(x) b

x

1 or ( f –1) (b) 1 ' = f '(a) f '( f –1(b))

FIGURE 3.35 The graphs of inverse functions have reciprocal slopes at corresponding points.

The reciprocal relationship between the slopes of ƒ and ƒ -1 holds for other functions as well, but we must be careful to compare slopes at corresponding points. If the slope of y = ƒsxd at the point (a, ƒ(a)) is ƒ¿sad and ƒ¿sad Z 0, then the slope of y = ƒ -1sxd at the point (ƒ(a), a) is the reciprocal 1>ƒ¿sad (Figure 3.35). If we set b = ƒsad, then sƒ -1 d¿sbd =

1 1 . = ƒ¿sad ƒ¿sƒ -1sbdd

If y = ƒsxd has a horizontal tangent line at (a, ƒ(a)), then the inverse function ƒ -1 has a vertical tangent line at (ƒ(a), a), and this infinite slope implies that ƒ -1 is not differentiable at ƒ(a). Theorem 3 gives the conditions under which ƒ -1 is differentiable in its domain (which is the same as the range of ƒ).

3.8

171

Derivatives of Inverse Functions and Logarithms

THEOREM 3—The Derivative Rule for Inverses If ƒ has an interval I as domain and ƒ¿sxd exists and is never zero on I, then ƒ -1 is differentiable at every point in its domain (the range of ƒ). The value of sƒ -1 d¿ at a point b in the domain of ƒ -1 is the reciprocal of the value of ƒ¿ at the point a = ƒ -1sbd: sƒ -1 d¿sbd =

1 ƒ¿sƒ -1sbdd

(1)

or dƒ -1 1 . ` = dx x = b dƒ ` dx x = ƒ -1sbd

Theorem 3 makes two assertions. The first of these has to do with the conditions under which ƒ -1 is differentiable; the second assertion is a formula for the derivative of ƒ -1 when it exists. While we omit the proof of the first assertion, the second one is proved in the following way: ƒsƒ -1sxdd = x

Inverse function relationship

d ƒsƒ -1sxdd = 1 dx ƒ¿sƒ -1sxdd #

Differentiating both sides

d -1 ƒ sxd = 1 dx

Chain Rule

d -1 1 ƒ sxd = . dx ƒ¿sƒ -1sxdd

Solving for the derivative

The function ƒsxd = x 2, x Ú 0 and its inverse ƒ -1sxd = 1x have derivatives ƒ¿sxd = 2x and sƒ -1 d¿sxd = 1> A 21x B . Let’s verify that Theorem 3 gives the same formula for the derivative of ƒ -1sxd:

EXAMPLE 1

1 ƒ¿sƒ -1sxdd 1 = 2sƒ -1sxdd 1 = . 2s 1xd

sƒ -1 d¿sxd = y y x 2, x 0

4

Theorem 3 gives a derivative that agrees with the known derivative of the square root function. Let’s examine Theorem 3 at a specific point. We pick x = 2 (the number a) and ƒs2d = 4 (the value b). Theorem 3 says that the derivative of ƒ at 2, ƒ¿s2d = 4, and the derivative of ƒ -1 at ƒ(2), sƒ -1 d¿s4d, are reciprocals. It states that

Slope 4 (2, 4)

3

Slope 1– 4

2

(4, 2)

y x

1 0

sƒ -1 d¿s4d = 1

2

3

4

ƒ¿sxd = 2x with x replaced by ƒ -1sxd

x

1 1 1 1 = = ` = . 2x x = 2 4 ƒ¿s2d ƒ¿sƒ -1s4dd

See Figure 3.36. FIGURE 3.36 The derivative of ƒ -1sxd = 1x at the point (4, 2) is the reciprocal of the derivative of ƒsxd = x 2 at (2, 4) (Example 1).

We will use the procedure illustrated in Example 1 to calculate formulas for the derivatives of many inverse functions throughout this chapter. Equation (1) sometimes enables us to find specific values of dƒ -1>dx without knowing a formula for ƒ -1 .

172

Chapter 3: Differentiation y 6

(2, 6)

y x3 2 Slope 3x 2 3(2)2 12

Let ƒsxd = x 3 - 2. Find the value of dƒ -1>dx at x = 6 = ƒs2d without finding a formula for ƒ -1sxd.

EXAMPLE 2

Solution

We apply Theorem 3 to obtain the value of the derivative of ƒ -1 at x = 6:

Reciprocal slope: 1 12 (6, 2)

–2

0

6

dƒ ` = 3x 2 ` = 12 dx x = 2 x=2 dƒ -1 1 1 ` = = . 12 dx x = ƒs2d dƒ ` dx x = 2

x

Eq. (1)

–2

See Figure 3.37. FIGURE 3.37 The derivative of ƒsxd = x 3 - 2 at x = 2 tells us the derivative of ƒ -1 at x = 6 (Example 2).

Derivative of the Natural Logarithm Function Since we know the exponential function ƒ(x) = e x is differentiable everywhere, we can apply Theorem 3 to find the derivative of its inverse ƒ -1(x) = ln x: (ƒ -1)¿(x) = = =

1 ƒ¿(ƒ -1(x)) 1 e

ƒ -1(x)

Theorem 3

ƒ¿(u) = e u

1 e ln x

1 = x.

Inverse function relationship

Alternate Derivation Instead of applying Theorem 3 directly, we can find the derivative of y = ln x using implicit differentiation, as follows: y = ln x ey = x d y d (e ) = (x) dx dx ey

Inverse function relationship Differentiate implicitly.

dy = 1 dx

Chain Rule

dy 1 1 = y = x. dx e

ey = x

No matter which derivation we use, the derivative of y = ln x with respect to x is d 1 (ln x) = x , dx

x 7 0.

The Chain Rule extends this formula for positive functions usxd:

d 1 du ln u = , u dx dx

u 7 0.

(2)

3.8

EXAMPLE 3 (a)

Derivatives of Inverse Functions and Logarithms

173

We use Equation (2) to find derivatives.

d 1 1 d 1 s2d = x , ln 2x = s2xd = 2x dx 2x dx

x 7 0

(b) Equation (2) with u = x 2 + 3 gives 2x d 1 # d 2 1 # 2x = 2 . ln sx 2 + 3d = 2 sx + 3d = 2 dx x + 3 dx x + 3 x + 3 (c) Equation (2) with u = ƒ x ƒ gives an important derivative: d d du ln x = ln u # dx ƒ ƒ du dx 1# x = u ƒxƒ 1 ƒxƒ x = 2 x 1 = x.

= Derivative of ln |x| d 1 ln ƒ x ƒ = x , x Z 0 dx

#

x ƒxƒ

u = ƒxƒ,x Z 0 x d A ƒxƒ B = dx ƒxƒ Substitute for u.

So 1>x is the derivative of ln x on the domain x 7 0, and the derivative of ln (-x) on the domain x 6 0. Notice from Example 3a that the function y = ln 2x has the same derivative as the function y = ln x. This is true of y = ln bx for any constant b, provided that bx 7 0: d 1 # d 1 1 ln bx = sbxd = sbd = x . dx bx dx bx

(3)

EXAMPLE 4

A line with slope m passes through the origin and is tangent to the graph of y = ln x. What is the value of m?

Suppose the point of tangency occurs at the unknown point x = a 7 0. Then we know that the point (a, ln a) lies on the graph and that the tangent line at that point has slope m = 1>a (Figure 3.38). Since the tangent line passes through the origin, its slope is Solution

y

2 1

0

(a, ln a)

1 2 y ln x

ln a - 0 ln a = a . a - 0

m = 1 Slope a 3

4

Setting these two formulas for m equal to each other, we have 5

x

ln a 1 a = a ln a = 1

FIGURE 3.38 The tangent line intersects the curve at some point (a, ln a), where the slope of the curve is 1>a (Example 4).

e ln a = e 1 a = e 1 m = e.

The Derivatives of au and loga u We start with the equation a x = e ln (a

x

)

= e x ln a , which was seen in Section 1.6:

d x d x ln a d a = e = e x ln a # sx ln ad dx dx dx = a x ln a.

d u du e = eu dx dx

174

Chapter 3: Differentiation

If a 7 0, then d x a = a x ln a. dx

(4)

This equation shows why e x is the exponential function preferred in calculus. If a = e, then ln a = 1 and the derivative of a x simplifies to d x e = e x ln e = e x . dx With the Chain Rule, we get a more general form for the derivative of a general exponential function.

If a 7 0 and u is a differentiable function of x, then a u is a differentiable function of x and d u du a = a u ln a . dx dx

EXAMPLE 5

(5)

Here are some derivatives of general exponential functions.

(a)

d x 3 = 3x ln 3 dx

Eq. (5) with a = 3, u = x

(b)

d -x d 3 = 3-x sln 3d s -xd = -3-x ln 3 dx dx

Eq. (5) with a = 3, u = -x

(c)

d sin x d 3 = 3sin x sln 3d ssin xd = 3sin x sln 3d cos x dx dx

Á , u = sin x

In Section 3.3 we looked at the derivative ƒ¿(0) for the exponential functions f(x) = a x at various values of the base a. The number ƒ¿(0) is the limit, limh:0 (a h - 1)>h, and gives the slope of the graph of a x when it crosses the y-axis at the point (0, 1). We now see from Equation (4) that the value of this slope is lim

h:0

ah - 1 = ln a. h

(6)

In particular, when a = e we obtain lim

h:0

eh - 1 = ln e = 1. h

However, we have not fully justified that these limits actually exist. While all of the arguments given in deriving the derivatives of the exponential and logarithmic functions are correct, they do assume the existence of these limits. In Chapter 7 we will give another development of the theory of logarithmic and exponential functions which fully justifies that both limits do in fact exist and have the values derived above. To find the derivative of loga u for an arbitrary base (a 7 0, a Z 1), we start with the change-of-base formula for logarithms (reviewed in Section 1.6) and express loga u in terms of natural logarithms, loga x =

ln x . ln a

3.8

Derivatives of Inverse Functions and Logarithms

175

Taking derivatives, we have d d ln x loga x = a b dx dx ln a =

1 # d ln x ln a dx

=

1 #1 ln a x

=

1 . x ln a

ln a is a constant.

If u is a differentiable function of x and u 7 0, the Chain Rule gives the following formula.

For a 7 0 and a Z 1, d 1 du loga u = . dx u ln a dx

(7)

Logarithmic Differentiation The derivatives of positive functions given by formulas that involve products, quotients, and powers can often be found more quickly if we take the natural logarithm of both sides before differentiating. This enables us to use the laws of logarithms to simplify the formulas before differentiating. The process, called logarithmic differentiation, is illustrated in the next example.

EXAMPLE 6

Find dy> dx if y =

sx 2 + 1dsx + 3d1>2 , x - 1

x 7 1.

We take the natural logarithm of both sides and simplify the result with the algebraic properties of logarithms from Theorem 1 in Section 1.6:

Solution

ln y = ln

sx 2 + 1dsx + 3d1>2 x - 1

= ln ssx 2 + 1dsx + 3d1>2 d - ln sx - 1d

Rule 2

= ln sx 2 + 1d + ln sx + 3d1>2 - ln sx - 1d

Rule 1

1 ln sx + 3d - ln sx - 1d. 2

Rule 4

= ln sx 2 + 1d +

We then take derivatives of both sides with respect to x, using Equation (2) on the left: 1 dy 1 1 # y dx = x 2 + 1 2x + 2

#

1 1 . x + 3 x - 1

Next we solve for dy> dx:

dy 2x 1 1 + b. = ya 2 2x + 6 x - 1 dx x + 1

176

Chapter 3: Differentiation

Finally, we substitute for y: dy sx 2 + 1dsx + 3d1>2 2x 1 1 = a 2 b. + x - 1 2x + 6 x - 1 dx x + 1

Irrational Exponents and the Power Rule (General Version) The definition of the general exponential function enables us to raise any positive number to any real power n, rational or irrational. That is, we can define the power function y = x n for any exponent n.

DEFINITION

For any x 7 0 and for any real number n, x n = e n ln x.

Because the logarithm and exponential functions are inverses of each other, the definition gives ln x n = n ln x,

for all real numbers n.

That is, the rule for taking the natural logarithm of any power holds for all real exponents n, not just for rational exponents. The definition of the power function also enables us to establish the derivative Power Rule for any real power n, as stated in Section 3.3.

General Power Rule for Derivatives For x 7 0 and any real number n, d n x = nx n - 1. dx If x … 0, then the formula holds whenever the derivative, x n, and x n - 1 all exist.

Proof Differentiating x n with respect to x gives d n d n ln x x = e dx dx = e n ln x #

d sn ln xd dx

Definition of x n, x 7 0 Chain Rule for e u

n = xn # x

Definition and derivative of ln x

= nx n - 1 .

x n # x -1 = x n - 1

In short, whenever x 7 0, d n x = nx n - 1 . dx For x 6 0, if y = x n, y¿ , and x n - 1 all exist, then ln ƒ y ƒ = ln ƒ x ƒ n = n ln ƒ x ƒ .

3.8

Derivatives of Inverse Functions and Logarithms

177

Using implicit differentiation (which assumes the existence of the derivative y¿ ) and Example 3(c), we have y¿ n y = x. Solving for the derivative, y xn y¿ = n x = n x = nx n - 1.

y = xn

It can be shown directly from the definition of the derivative that the derivative equals 0 when x = 0 and n Ú 1. This completes the proof of the general version of the Power Rule for all values of x.

EXAMPLE 7 Solution

Differentiate ƒ(x) = x x, x 7 0.

We note that ƒ(x) = x x = e x ln x, so differentiation gives ƒ¿(x) =

d x ln x (e ) dx

= e x ln x

d (x ln x) dx

d dx

e u, u = x ln x

1 = e x ln x aln x + x # x b = x x (ln x + 1).

x 7 0

The Number e Expressed as a Limit In Section 1.5 we defined the number e as the base value for which the exponential function y = a x has slope 1 when it crosses the y-axis at (0, 1). Thus e is the constant that satisfies the equation lim

h:0

eh - 1 = ln e = 1. h

Slope equals ln e from Eq. (6).

We now prove that e can be calculated as a certain limit.

THEOREM 4—The Number e as a Limit limit

The number e can be calculated as the

e = lim s1 + xd1>x . x:0

y e 3 2

Proof If ƒsxd = ln x, then ƒ¿sxd = 1>x, so ƒ¿s1d = 1. But, by the definition of derivative,

y (1 x)1/x

ƒs1 + hd - ƒs1d ƒs1 + xd - ƒs1d = lim x h h:0 x:0

ƒ¿s1d = lim

1

0

x

FIGURE 3.39 The number e is the limit of the function graphed here as x : 0.

ln s1 + xd - ln 1 1 = lim x ln s1 + xd x x:0 x:0

= lim

ln 1 = 0

= lim ln s1 + xd1>x = ln c lim s1 + xd1>x d.

ln is continuous, Theorem 10 in Chapter 2.

x:0

x:0

178

Chapter 3: Differentiation

Because ƒ¿s1d = 1, we have

ln c lim s1 + xd1>x d = 1. x:0

Therefore, exponentiating both sides we get lim s1 + xd1>x = e.

x:0

See Figure 3.39 on the previous page. Approximating the limit in Theorem 4 by taking x very small gives approximations to e. Its value is e L 2.718281828459045 to 15 decimal places.

Exercises 3.8 Derivatives of Inverse Functions In Exercises 1–4: a. Find ƒ -1sxd . b. Graph ƒ and ƒ -1 together.

c. Evaluate dƒ> dx at x = a and dƒ -1>dx at x = ƒsad to show that at these points dƒ -1>dx = 1>sdƒ>dxd .

1. ƒsxd = 2x + 3,

a = -1

2. ƒsxd = s1>5dx + 7,

3. ƒsxd = 5 - 4x,

a = 1>2

4. ƒsxd = 2x2,

x Ú 0,

a = -1 a = 5

3 x are inverses of one an5. a. Show that ƒsxd = x3 and g sxd = 1 other.

b. Graph ƒ and g over an x-interval large enough to show the graphs intersecting at (1, 1) and s -1, -1d . Be sure the picture shows the required symmetry about the line y = x . c. Find the slopes of the tangents to the graphs of ƒ and g at (1, 1) and s - 1, -1d (four tangents in all). d. What lines are tangent to the curves at the origin?

6. a. Show that hsxd = x3>4 and ksxd = s4xd1>3 are inverses of one another. b. Graph h and k over an x-interval large enough to show the graphs intersecting at (2, 2) and s -2, -2d . Be sure the picture shows the required symmetry about the line y = x . c. Find the slopes of the tangents to the graphs at h and k at (2, 2) and s -2, -2d . d. What lines are tangent to the curves at the origin?

7. Let ƒsxd = x3 - 3x2 - 1, x Ú 2 . Find the value of dƒ -1>dx at the point x = -1 = ƒs3d .

8. Let ƒsxd = x - 4x - 5, x 7 2 . Find the value of dƒ >dx at the point x = 0 = ƒs5d . 2

-1

9. Suppose that the differentiable function y = ƒsxd has an inverse and that the graph of ƒ passes through the point (2, 4) and has a slope of 1> 3 there. Find the value of dƒ -1>dx at x = 4 .

13. y = ln st 2 d 3 15. y = ln x 17. y = ln su + 1d

14. y = ln st 3>2 d 10 16. y = ln x 18. y = ln s2u + 2d

19. y = ln x3

20. y = sln xd3

21. y = t sln td2 4

22. y = t2ln t 4

x x ln x 4 16 ln t 25. y = t ln x 27. y = 1 + ln x

24. y = (x 2 ln x) 4

29. y = ln sln xd

30. y = ln sln sln xdd

23. y =

1 + ln t t x ln x 28. y = 1 + ln x 26. y =

31. y = ussin sln ud + cos sln udd 32. y = ln ssec u + tan ud 1 33. y = ln x2x + 1 1 + ln t 35. y = 1 - ln t

sx2 + 1d5 21 - x

1 1 + x ln 2 1 - x

36. y = 2ln 1t 38. y = ln a

37. y = ln ssec sln udd 39. y = ln a

34. y =

b

40. y = ln

2sin u cos u b 1 + 2 ln u sx + 1d5

C sx + 2d20

Logarithmic Differentiation In Exercises 41–54, use logarithmic differentiation to find the derivative of y with respect to the given independent variable. 41. y = 2xsx + 1d 43. y =

t At + 1

42. y = 2sx2 + 1dsx - 1d2 44. y =

1 A t st + 1d

10. Suppose that the differentiable function y = gsxd has an inverse and that the graph of g passes through the origin with slope 2. Find the slope of the graph of g-1 at the origin.

45. y = (sin u)2u + 3

46. y = stan ud22u + 1

47. y = t st + 1dst + 2d

48. y =

Derivatives of Logarithms In Exercises 11–40, find the derivative of y with respect to x, t, or u , as appropriate.

49. y =

u + 5 u cos u

50. y =

11. y = ln 3x

51. y =

x2x2 + 1 sx + 1d2>3

52. y =

12. y = ln k x, k constant

1 t st + 1dst + 2d u sin u 2sec u sx + 1d10

C s2x + 1d5

3.8

53. y =

3

xsx - 2d

54. y =

Bx + 1 2

3

xsx + 1dsx - 2d

Bsx + 1ds2x + 3d 2

Finding Derivatives In Exercises 55–62, find the derivative of y with respect to x, t, or u , as appropriate. 55. y = ln (cos2 u)

56. y = ln s3ue-u d

57. y = ln s3te-t d

58. y = ln s2e-t sin td

59. y = ln a

60. y = ln a

eu b 1 + eu

61. y = escos t + ln td

In Exercises 63–66, find dy> dx.

2u 1 + 2u

64. ln xy = e x + y

65. x y = y x

66. tan y = e x + ln x

70. y = 2ss

2

x 2y– + xy¿ + y = 0. 100. Using mathematical induction, show that

d

72. y = t

73. y = log 2 5u

74. y = log 3 s1 + u ln 3d

75. y = log 4 x + log 4 x 2

76. y = log 25 e x - log 5 1x

77. y = log 2 r # log 4 r

x + 1 79. y = log 3 a a b x - 1

ln 3

78. y = log 3 r # log 9 r b

7x 80. y = log 5 a b 3x + 2 B

ln 5

81. y = u sin slog 7 ud

sin u cos u 82. y = log 7 a b eu 2u

83. y = log 5 e x

84. y = log 2 a

x 2e 2

2 2x + 1 86. y = 3 log 8 slog 2 td

85. y = 3

log 2 t

b

88. y = t log 3 A essin tdsln 3d B

87. y = log 2 s8t ln 2 d

Logarithmic Differentiation with Exponentials In Exercises 89–96, use logarithmic differentiation to find the derivative of y with respect to the given independent variable. sx + 1d

89. y = sx + 1d

90. y = x

91. y = s 1tdt

92. y = t2t

93. y = ssin xdx

94. y = x sin x

95. y = x

96. y = sln xdln x

x

ln x

COMPUTER EXPLORATIONS In Exercises 101–108, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function y = ƒsxd together with its derivative over the given interval. Explain why you know that ƒ is one-to-one over the interval.

1-e

71. y = x

p

n

x 98. Show that lim n: q a1 + n b = e x for any x 7 0.

b

In Exercises 67–88, find the derivative of y with respect to the given independent variable. 69. y = 52s

sides of this equation with respect to x, using the Chain Rule to express sg ⴰ ƒd¿sxd as a product of derivatives of g and ƒ. What do you find? (This is not a proof of Theorem 3 because we assume here the theorem’s conclusion that g = ƒ -1 is differentiable.)

(n - 1)! dn ln x = (-1) n - 1 . dx n xn

63. ln y = e y sin x

68. y = 3-x

b. Solve the equation y = ƒsxd for x as a function of y, and name the resulting inverse function g. c. Find the equation for the tangent line to ƒ at the specified point sx0 , ƒsx0 dd . d. Find the equation for the tangent line to g at the point sƒsx0 d, x0 d located symmetrically across the 45° line y = x (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions ƒ and g, the identity, the two tangent lines, and the line segment joining the points sx0 , ƒsx0 dd and sƒsx0 d, x0 d . Discuss the symmetries you see across the main diagonal. 101. y = 23x - 2,

2 … x … 4, 3

3x + 2 , -2 … x … 2, x0 = 1>2 2x - 11 4x 103. y = 2 , -1 … x … 1, x0 = 1>2 x + 1 3 x 104. y = 2 , -1 … x … 1, x0 = 1>2 x + 1 105. y = x 3 - 3x 2 - 1,

97. If we write g(x) for ƒ -1sxd , Equation (1) can be written as

106. y = 2 - x - x3,

g¿sƒsadd # ƒ¿sad = 1 .

If we then write x for a, we get g¿sƒsxdd # ƒ¿sxd = 1 . The latter equation may remind you of the Chain Rule, and indeed there is a connection. Assume that ƒ and g are differentiable functions that are inverses of one another, so that sg ⴰ ƒdsxd = x . Differentiate both

x0 = 3

102. y =

Theory and Applications 1 g¿sƒsadd = , or ƒ¿sad

179

99. If y = A sin (ln x) + B cos (ln x), where A and B are constants, show that

62. y = e sin t sln t 2 + 1d

67. y = 2x

Derivatives of Inverse Functions and Logarithms

2 … x … 5, -2 … x … 2,

27 10 3 x0 = 2 x0 =

-3 … x … 5, x0 = 1 p p 108. y = sin x, - … x … , x0 = 1 2 2 107. y = e , x

In Exercises 109 and 110, repeat the steps above to solve for the functions y = ƒsxd and x = ƒ -1s yd defined implicitly by the given equations over the interval. 109. y1>3 - 1 = sx + 2d3, 110. cos y = x , 1>5

-5 … x … 5,

0 … x … 1,

x0 = 1>2

x0 = -3>2

180

Chapter 3: Differentiation

Inverse Trigonometric Functions

3.9

We introduced the six basic inverse trigonometric functions in Section 1.6, but focused there on the arcsine and arccosine functions. Here we complete the study of how all six inverse trigonometric functions are defined, graphed, and evaluated, and how their derivatives are computed.

Inverses of tan x, cot x, sec x, and csc x The graphs of these four basic inverse trigonometric functions are shown again in Figure 3.40. We obtain these graphs by reflecting the graphs of the restricted trigonometric functions (as discussed in Section 1.6) through the line y = x. Let’s take a closer look at the arctangent, arccotangent, arcsecant, and arccosecant functions. Domain: – ∞ x ∞ Range: – y 2 2 y 2 –2

–1

– 2

Domain: – ∞ x ∞ Range: 0 y y

y tan –1x x 1 2

(a)

2

2

–2

Domain: x –1 or x 1 Range: – y , y 0 2 2 y

Domain: x –1 or x 1 Range: 0 y , y 2 y

–1

y

2

y sec

–1x

–2 1

(b)

cot –1x

2

x

–2

–1

1

2

y csc–1x

–1

1

x

– 2

x

(c)

2

(d)

FIGURE 3.40 Graphs of the arctangent, arccotangent, arcsecant, and arccosecant functions.

The arctangent of x is a radian angle whose tangent is x. The arccotangent of x is an angle whose cotangent is x, and so forth. The angles belong to the restricted domains of the tangent, cotangent, secant, and cosecant functions.

DEFINITION y tan1 x is the number in s -p>2, p>2d for which tan y = x. y cot1 x is the number in s0, pd for which cot y = x. y sec1 x is the number in [0, p/2) ´ (p/2, p] for which sec y = x. y csc1 x is the number in [-p/2, 0) ´ (0, p/2] for which csc y = x.

We use open or half-open intervals when describing the ranges to avoid values where the tangent, cotangent, secant, and cosecant functions are undefined. (See Figure 3.40.) The graph of y = tan-1 x is symmetric about the origin because it is a branch of the graph x = tan y that is symmetric about the origin (Figure 3.40a). Algebraically this means that tan-1 s -xd = -tan-1 x; the arctangent is an odd function. The graph of y = cot-1 x has no such symmetry (Figure 3.40b). Notice from Figure 3.40a that the graph of the arctangent function has two horizontal asymptotes; one at y = p>2 and the other at y = -p>2.

3.9 Domain: x 1 Range: 0 y , y 2 y

B

y

2

sec –1x

–1

0

x

1

Caution There is no general agreement about how to define sec-1 x for negative values of x. We chose angles in the second quadrant between p>2 and p. This choice makes sec-1 x = cos-1 s1>xd. It also makes sec-1 x an increasing function on each interval of its domain. Some tables choose sec-1 x to lie in [-p, -p>2d for x 6 0 and some texts choose it to lie in [p, 3p>2d (Figure 3.41). These choices simplify the formula for the derivative (our formula needs absolute value signs) but fail to satisfy the computational equation sec-1 x = cos-1 s1>xd. From this, we can derive the identity

– 2

p 1 1 sec-1 x = cos-1 a x b = - sin-1 a x b 2

C – – 3 2

FIGURE 3.41 There are several logical choices for the left-hand branch of y = sec-1 x . With choice A, sec-1 x = cos-1 s1>xd , a useful identity employed by many calculators.

EXAMPLE 1

The accompanying figures show two values of tan1 x. y

tan–1 1 tan–1 3 3 6 3 6 2 1 x 0 3

y

tan–1 ⎛⎝–3⎛⎝ – 3

1 0

x

y sin–1x Domain: –1 x 1 Range: – /2 y /2

–1

1

– 3 x –3

tan ⎛– ⎛ –3 ⎝ 3⎝

tan 1 6 3

y

(1)

by applying Equation (5) in Section 1.6.

2

2

181

The inverses of the restricted forms of sec x and csc x are chosen to be the functions graphed in Figures 3.40c and 3.40d. 3 2

A

Inverse Trigonometric Functions

tan-1 x

23 1

p>3 p>4

23>3 - 23>3 -1 - 23

p>6 -p>6 -p>4 -p>3

The angles come from the first and fourth quadrants because the range of tan-1 x is s -p>2, p>2d.

x

– 2

FIGURE 3.42 The graph of y = sin-1 x has vertical tangents at x = -1 and x = 1.

The Derivative of y = sin-1 u We know that the function x = sin y is differentiable in the interval -p>2 6 y 6 p>2 and that its derivative, the cosine, is positive there. Theorem 3 in Section 3.8 therefore assures us that the inverse function y = sin-1 x is differentiable throughout the interval -1 6 x 6 1. We cannot expect it to be differentiable at x = 1 or x = -1 because the tangents to the graph are vertical at these points (see Figure 3.42).

182

Chapter 3: Differentiation

We find the derivative of y = sin-1 x by applying Theorem 3 with ƒsxd = sin x and ƒ sxd = sin-1 x: -1

1 ƒ¿sƒ -1sxdd

Theorem 3

=

1 cos ssin-1 xd

ƒ¿sud = cos u

=

1 21 - sin2 ssin-1 xd

cos u = 21 - sin2 u

=

1 . 21 - x 2

sin ssin-1 xd = x

sƒ -1 d¿sxd =

If u is a differentiable function of x with ƒ u ƒ 6 1, we apply the Chain Rule to get

d du 1 ssin-1 ud = , 2 dx dx 21 - u

EXAMPLE 2

ƒ u ƒ 6 1.

Using the Chain Rule, we calculate the derivative d 1 ssin-1 x 2 d = dx 21 - sx 2 d2

#

d 2 2x sx d = . dx 21 - x 4

The Derivative of y = tan-1 u We find the derivative of y = tan-1 x by applying Theorem 3 with ƒsxd = tan x and ƒ -1sxd = tan-1 x. Theorem 3 can be applied because the derivative of tan x is positive for -p>2 6 x 6 p>2: 1 ƒ¿sƒ -1sxdd

Theorem 3

=

1 sec2 stan-1 xd

ƒ¿sud = sec2 u

=

1 1 + tan2 stan-1 xd

sec2 u = 1 + tan2 u

=

1 . 1 + x2

tan stan-1 xd = x

sƒ -1 d¿sxd =

The derivative is defined for all real numbers. If u is a differentiable function of x, we get the Chain Rule form:

d du 1 . stan-1 ud = dx 1 + u 2 dx

The Derivative of y = sec-1 u Since the derivative of sec x is positive for 0 6 x 6 p/2 and p>2 6 x 6 p, Theorem 3 says that the inverse function y = sec-1 x is differentiable. Instead of applying the formula

3.9

Inverse Trigonometric Functions

183

in Theorem 3 directly, we find the derivative of y = sec-1 x, ƒ x ƒ 7 1, using implicit differentiation and the Chain Rule as follows: y = sec-1 x sec y = x

Inverse function relationship

d d ssec yd = x dx dx sec y tan y

Differentiate both sides.

dy = 1 dx

Chain Rule

dy 1 = sec y tan y . dx

Since ƒ x ƒ 7 1, y lies in s0, p>2d ´ sp>2, pd and sec y tan y Z 0 .

To express the result in terms of x, we use the relationships sec y = x

and

tan y = ; 2sec2 y - 1 = ; 2x 2 - 1

to get y

dy 1 . = ; dx x2x 2 - 1

y sec –1x

2 –1

0

1

Can we do anything about the ; sign? A glance at Figure 3.43 shows that the slope of the graph y = sec-1 x is always positive. Thus,

x

FIGURE 3.43 The slope of the curve y = sec-1 x is positive for both x 6 -1 and x 7 1 .

d sec-1 x = d dx

1 x2x 2 - 1 1 x2x 2 - 1 +

if x 7 1 if x 6 -1.

With the absolute value symbol, we can write a single expression that eliminates the “;” ambiguity: d 1 sec-1 x = . dx ƒ x ƒ 2x 2 - 1 If u is a differentiable function of x with ƒ u ƒ 7 1, we have the formula

d du 1 ssec-1 ud = , 2 dx dx u 2u 1 ƒ ƒ

EXAMPLE 3

ƒ u ƒ 7 1.

Using the Chain Rule and derivative of the arcsecant function, we find d d 1 sec-1 s5x 4 d = s5x 4 d 4 4 2 dx dx 5x 2s5x d 1 ƒ ƒ =

1 s20x 3 d 5x 225x 8 - 1

=

4 . x225x 8 - 1

4

5x 4 7 1 7 0

184

Chapter 3: Differentiation

Derivatives of the Other Three Inverse Trigonometric Functions We could use the same techniques to find the derivatives of the other three inverse trigonometric functions—arccosine, arccotangent, and arccosecant—but there is an easier way, thanks to the following identities.

Inverse Function–Inverse Cofunction Identities cos-1 x = p>2 - sin-1 x cot-1 x = p>2 - tan-1 x csc-1 x = p>2 - sec-1 x

We saw the first of these identities in Equation (5) of Section 1.6. The others are derived in a similar way. It follows easily that the derivatives of the inverse cofunctions are the negatives of the derivatives of the corresponding inverse functions. For example, the derivative of cos-1 x is calculated as follows: d d p (cos-1 x) = a - sin-1 xb dx dx 2 = -

d (sin-1 x) dx

= -

1 . 21 - x 2

Identity

Derivative of arcsine

The derivatives of the inverse trigonometric functions are summarized in Table 3.1.

TABLE 3.1

Derivatives of the inverse trigonometric functions

1.

dssin-1 ud du 1 = , 2 dx dx 21 - u

2.

dscos-1 ud du 1 = , 2 dx dx 21 - u

3.

dstan-1 ud du 1 = dx 1 + u 2 dx

4.

dscot-1 ud du 1 = 2 dx dx 1 + u

5.

dssec-1 ud du 1 = , 2 dx dx u 2u 1 ƒ ƒ

6.

dscsc-1 ud du 1 = , 2 dx ƒ u ƒ 2u - 1 dx

ƒuƒ 6 1 ƒuƒ 6 1

ƒuƒ 7 1 ƒuƒ 7 1

3.9

Inverse Trigonometric Functions

185

Exercises 3.9 Common Values Use reference triangles in an appropriate quadrant, as in Example 1, to find the angles in Exercises 1–8. 1 1. a. tan-1 1 b. tan-1 A - 23 B c. tan-1 a b 23 -1 2. a. tan-1s -1d b. tan-1 23 c. tan-1 a b 23

4. a. sin

b. sin-1 a

-1 b 2

1 a b 2

-1

b. sin

-1

a

1 22 -1

b

b 22 -1 a b 22 -2 a b 23 2 a b 23

1 5. a. cos-1 a b 2

b. cos-1

6. a. csc-1 22

b. csc-1

7. a. sec-1 A - 22 B

b. sec-1

8. a. cot-1 s -1d

b. cot-1 A 23 B

c. sin-1 a c. sin

-1

- 23 b 2

23 a b 2

c. cos-1 a

23 b 2

22 bb 2

1 11. tan asin-1 a- b b 2

x: 1

15. lim tan

-1

-1 23

12. cot asin-1 a-

14.

17. lim sec-1 x

18.

19. lim csc x: q

-1

x

20.

lim tan

43. You are sitting in a classroom next to the wall looking at the blackboard at the front of the room. The blackboard is 12 ft long and starts 3 ft from the wall you are sitting next to. Show that your viewing angle is

3'

23 bb 2

lim csc

You

Wall x

44. Find the angle a .

65°

x

x: - q x: - q

x x - cot-1 15 3

b

lim sec-1 x -1

41. y = x sin-1 x + 21 - x 2

Theory and Examples

lim cos-1 x

x: - q

x 7 1

x 42. y = ln sx 2 + 4d - x tan-1 a b 2

x: -1 +

-1

x,

if you are x ft from the front wall.

c. sec-1s -2d c. cot-1 a

38. y = 2s 2 - 1 - sec-1 s -1

1 40. y = cot-1 x - tan-1 x

1 10. sec acos-1 b 2

16.

x: q

39. y = tan 2x - 1 + csc 2

12'

x

x: q

-1

a = cot-1

Limits Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.) 13. lim- sin-1 x

37. y = s 21 - s 2 + cos-1 s

c. csc-1 2

Evaluations Find the values in Exercises 9–12. 9. sin acos-1 a

36. y = cos-1 se -t d

Blackboard

3. a. sin-1 a

35. y = csc-1 se t d

x

Finding Derivatives In Exercises 21–42, find the derivative of y with respect to the appropriate variable. 21. y = cos-1 sx 2 d

22. y = cos-1 s1>xd

23. y = sin-1 22 t

24. y = sin-1 s1 - td

25. y = sec-1 s2s + 1d

26. y = sec-1 5s

27. y = csc-1 sx 2 + 1d, x 7 0 x 28. y = csc-1 2 3 1 29. y = sec-1 t , 0 6 t 6 1 30. y = sin-1 2 t 31. y = cot-1 2t

32. y = cot-1 2t - 1

33. y = ln stan-1 xd

34. y = tan-1 sln xd

21 50

45. Here is an informal proof that tan-1 1 + tan-1 2 + tan-1 3 = p . Explain what is going on.

186

Chapter 3: Differentiation

46. Two derivations of the identity sec-1 s xd = P sec1 x

55. What is special about the functions

-1

a. (Geometric) Here is a pictorial proof that sec s -xd = p - sec-1 x . See if you can tell what is going on. y

y sec –1x

ƒsxd = sin-1

0

1

x

x

cos-1 s -xd = p - cos-1 x sec

Eq. (4), Section 1.6

-1

x = cos s1>xd

48. a. csc 49. a. sec 50. a. cot

-1 -1

-1

b. cos-1 2 b. csc-1 2

(1>2)

b. sin 22 -1

s -1>2d

1 2x + 1 2

and

1 g sxd = tan-1 x ?

T 57. Find the values of a. sec-1 1.5

b. csc-1 s -1.5d

c. cot-1 2

T 58. Find the values of a. sec-1s -3d

b. csc-1 1.7

c. cot-1 s -2d

T In Exercises 59–61, find the domain and range of each composite function. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. 59. a. y = tan-1 stan xd

-1

0

g sxd = 2 tan-1 1x ?

Eq. (1)

Which of the expressions in Exercises 47–50 are defined, and which are not? Give reasons for your answers. 47. a. tan-1 2

and

Explain.

b. (Algebraic) Derive the identity sec-1 s -xd = p - sec-1 x by combining the following two equations from the text: -1

x Ú 0,

56. What is special about the functions

2 –1

x - 1 , x + 1

Explain.

–x

ƒsxd = sin-1

b. cos s -5d

-1

b. y = tan stan-1 xd

60. a. y = sin ssin xd

b. y = sin ssin-1 xd

61. a. y = cos-1 scos xd

b. y = cos scos-1 xd

51. Use the identity csc-1 u =

p - sec-1 u 2

to derive the formula for the derivative of csc-1 u in Table 3.1 from the formula for the derivative of sec-1 u . 52. Derive the formula dy 1 = dx 1 + x2 -1

for the derivative of y = tan x by differentiating both sides of the equivalent equation tan y = x . 53. Use the Derivative Rule in Section 3.8, Theorem 3, to derive d 1 sec-1 x = , ƒ x ƒ 7 1. 2 dx x 2x - 1 ƒ ƒ 54. Use the identity p cot-1 u = - tan-1 u 2 to derive the formula for the derivative of cot-1 u in Table 3.1 from the formula for the derivative of tan-1 u .

3.10

T Use your graphing utility for Exercises 62–66. 62. Graph y = sec ssec-1 xd = sec scos-1s1>xdd . Explain what you see. 63. Newton’s serpentine Graph Newton’s serpentine, y = 4x>sx2 + 1d . Then graph y = 2 sin s2 tan-1 xd in the same graphing window. What do you see? Explain. 64. Graph the rational function y = s2 - x 2 d>x 2 . Then graph y = cos s2 sec-1 xd in the same graphing window. What do you see? Explain. 65. Graph ƒsxd = sin-1 x together with its first two derivatives. Comment on the behavior of ƒ and the shape of its graph in relation to the signs and values of ƒ¿ and ƒ–. 66. Graph ƒsxd = tan-1 x together with its first two derivatives. Comment on the behavior of ƒ and the shape of its graph in relation to the signs and values of ƒ¿ and ƒ–.

Related Rates In this section we look at problems that ask for the rate at which some variable changes when it is known how the rate of some other related variable (or perhaps several variables) changes. The problem of finding a rate of change from other known rates of change is called a related rates problem.

3.10

Related Rates

187

Related Rates Equations Suppose we are pumping air into a spherical balloon. Both the volume and radius of the balloon are increasing over time. If V is the volume and r is the radius of the balloon at an instant of time, then V =

4 3 pr . 3

Using the Chain Rule, we differentiate both sides with respect to t to find an equation relating the rates of change of V and r, dV dr dV dr = = 4pr 2 . dt dr dt dt So if we know the radius r of the balloon and the rate dV>dt at which the volume is increasing at a given instant of time, then we can solve this last equation for dr>dt to find how fast the radius is increasing at that instant. Note that it is easier to directly measure the rate of increase of the volume (the rate at which air is being pumped into the balloon) than it is to measure the increase in the radius. The related rates equation allows us to calculate dr>dt from dV>dt. Very often the key to relating the variables in a related rates problem is drawing a picture that shows the geometric relations between them, as illustrated in the following example. Water runs into a conical tank at the rate of 9 ft3>min. The tank stands point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep?

EXAMPLE 1

dV 9 ft 3/min dt 5 ft

Solution

Figure 3.44 shows a partially filled conical tank. The variables in the problem are V = volume sft3 d of the water in the tank at time t smind

x dy ? dt when y 6 ft

x = radius sftd of the surface of the water at time t

10 ft y

y = depth sftd of the water in the tank at time t. We assume that V, x, and y are differentiable functions of t. The constants are the dimensions of the tank. We are asked for dy>dt when

FIGURE 3.44 The geometry of the conical tank and the rate at which water fills the tank determine how fast the water level rises (Example 1).

y = 6 ft

dV = 9 ft3>min. dt

and

The water forms a cone with volume V =

1 2 px y. 3

This equation involves x as well as V and y. Because no information is given about x and dx>dt at the time in question, we need to eliminate x. The similar triangles in Figure 3.44 give us a way to express x in terms of y: 5 x y = 10

or

x =

y . 2

Therefore, find V =

y 2 p 3 1 pa b y = y 3 2 12

to give the derivative dV p # 2 dy p dy 3y = = y2 . 12 4 dt dt dt

188

Chapter 3: Differentiation

Finally, use y = 6 and dV>dt = 9 to solve for dy>dt. dy p s6d2 4 dt dy 1 = p L 0.32 dt 9 =

At the moment in question, the water level is rising at about 0.32 ft> min.

Related Rates Problem Strategy 1. Draw a picture and name the variables and constants. Use t for time. Assume that all variables are differentiable functions of t. 2. Write down the numerical information (in terms of the symbols you have chosen). 3. Write down what you are asked to find (usually a rate, expressed as a derivative). 4. Write an equation that relates the variables. You may have to combine two or more equations to get a single equation that relates the variable whose rate you want to the variables whose rates you know. 5. Differentiate with respect to t. Then express the rate you want in terms of the rates and variables whose values you know. 6. Evaluate. Use known values to find the unknown rate.

EXAMPLE 2

A hot air balloon rising straight up from a level field is tracked by a range finder 150 m from the liftoff point. At the moment the range finder’s elevation angle is p>4, the angle is increasing at the rate of 0.14 rad> min. How fast is the balloon rising at that moment?

Balloon

d 0.14 rad/min dt when /4

Range finder

dy ? y dt when /4

150 m

FIGURE 3.45 The rate of change of the balloon’s height is related to the rate of change of the angle the range finder makes with the ground (Example 2).

We answer the question in six steps. 1. Draw a picture and name the variables and constants (Figure 3.45). The variables in the picture are u = the angle in radians the range finder makes with the ground. y = the height in feet of the balloon. We let t represent time in minutes and assume that u and y are differentiable functions of t. The one constant in the picture is the distance from the range finder to the liftoff point (150 m). There is no need to give it a special symbol. 2. Write down the additional numerical information. Solution

3. 4.

du p = 0.14 rad>min when u = 4 dt Write down what we are to find. We want dy>dt when u = p>4. Write an equation that relates the variables y and u. y = tan u or y = 150 tan u 150

5.

Differentiate with respect to t using the Chain Rule. The result tells how dy>dt (which we want) is related to du>dt (which we know). dy du = 150 ssec2 ud dt dt

6.

Evaluate with u = p>4 and du>dt = 0.14 to find dy>dt. dy = 150 A 22 B 2s0.14d = 42 dt

sec

p = 22 4

At the moment in question, the balloon is rising at the rate of 42 m > min.

3.10

Related Rates

189

EXAMPLE 3

y

A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car?

Situation when x 0.8, y 0.6 y dy –60 dt

ds 20 dt

We picture the car and cruiser in the coordinate plane, using the positive x-axis as the eastbound highway and the positive y-axis as the southbound highway (Figure 3.46). We let t represent time and set

Solution

0

dx ? dt

x

x

FIGURE 3.46 The speed of the car is related to the speed of the police cruiser and the rate of change of the distance between them (Example 3).

x = position of car at time t y = position of cruiser at time t s = distance between car and cruiser at time t. We assume that x, y, and s are differentiable functions of t. We want to find dx>dt when x = 0.8 mi,

dy = -60 mph, dt

y = 0.6 mi,

ds = 20 mph. dt

Note that dy>dt is negative because y is decreasing. We differentiate the distance equation s2 = x2 + y2 (we could also use s = 2x 2 + y 2), and obtain 2s

dy ds dx = 2x + 2y dt dt dt dy ds dx 1 = s ax + y b dt dt dt 1

=

2x 2 + y 2

ax

dy dx + y b. dt dt

Finally, we use x = 0.8, y = 0.6, dy>dt = -60, ds>dt = 20, and solve for dx>dt. 20 =

1 2s0.8d + s0.6d 2

2

a0.8

dx + (0.6)(-60)b dt

202s0.8d2 + s0.6d2 + s0.6ds60d dx = = 70 0.8 dt At the moment in question, the car’s speed is 70 mph.

EXAMPLE 4

y P 10 u 0

Q (x, 0)

x

A particle P moves clockwise at a constant rate along a circle of radius 10 m centered at the origin. The particle’s initial position is (0, 10) on the y-axis, and its final destination is the point (10, 0) on the x-axis. Once the particle is in motion, the tangent line at P intersects the x-axis at a point Q (which moves over time). If it takes the particle 30 sec to travel from start to finish, how fast is the point Q moving along the x-axis when it is 20 m from the center of the circle? We picture the situation in the coordinate plane with the circle centered at the origin (see Figure 3.47). We let t represent time and let u denote the angle from the x-axis to the radial line joining the origin to P. Since the particle travels from start to finish in 30 sec, it is traveling along the circle at a constant rate of p>2 radians in 1>2 min, or p rad>min. In other words, du>dt = -p, with t being measured in minutes. The negative sign appears because u is decreasing over time. Solution

FIGURE 3.47 The particle P travels clockwise along the circle (Example 4).

190

Chapter 3: Differentiation

Setting xstd to be the distance at time t from the point Q to the origin, we want to find dx>dt when x = 20 m

du = -p rad>min. dt

and

To relate the variables x and u, we see from Figure 3.47 that x cos u = 10, or x = 10 sec u. Differentiation of this last equation gives du dx = 10 sec u tan u = -10p sec u tan u. dt dt Note that dx>dt is negative because x is decreasing (Q is moving toward the origin). When x = 20, cos u = 1>2 and sec u = 2. Also, tan u = 2sec2 u - 1 = 23. It follows that dx = s -10pds2d A 23 B = -20 23p. dt At the moment in question, the point Q is moving toward the origin at the speed of 2023p L 108.8 m>min. A

12,000 u R

FIGURE 3.48 Jet airliner A traveling at constant altitude toward radar station R (Example 5).

x

EXAMPLE 5 A jet airliner is flying at a constant altitude of 12,000 ft above sea level as it approaches a Pacific island. The aircraft comes within the direct line of sight of a radar station located on the island, and the radar indicates the initial angle between sea level and its line of sight to the aircraft is 30°. How fast (in miles per hour) is the aircraft approaching the island when first detected by the radar instrument if it is turning upward (counterclockwise) at the rate of 2>3 deg>sec in order to keep the aircraft within its direct line of sight? Solution The aircraft A and radar station R are pictured in the coordinate plane, using the positive x-axis as the horizontal distance at sea level from R to A, and the positive y-axis as the vertical altitude above sea level. We let t represent time and observe that y = 12,000 is a constant. The general situation and line-of-sight angle u are depicted in Figure 3.48. We want to find dx>dt when u = p>6 rad and du>dt = 2>3 deg>sec. From Figure 3.48, we see that

12,000 = tan u x

x = 12,000 cot u.

or

Using miles instead of feet for our distance units, the last equation translates to x =

12,000 cot u. 5280

Differentiation with respect to t gives du 1200 dx = csc2 u . 528 dt dt When u = p>6, sin2 u = 1>4, so csc2 u = 4. Converting du>dt = 2>3 deg>sec to radians per hour, we find du 2 p = a bs3600d rad>hr. 3 180 dt

1 hr = 3600 sec, 1 deg = p>180 rad

Substitution into the equation for dx>dt then gives 1200 dx p 2 = abs4d a b a bs3600d L -380. 528 3 180 dt The negative sign appears because the distance x is decreasing, so the aircraft is approaching the island at a speed of approximately 380 mi>hr when first detected by the radar.

3.10

Related Rates

191

EXAMPLE 6

Figure 3.49a shows a rope running through a pulley at P and bearing a weight W at one end. The other end is held 5 ft above the ground in the hand M of a worker. Suppose the pulley is 25 ft above ground, the rope is 45 ft long, and the worker is walking rapidly away from the vertical line PW at the rate of 6 ft>sec. How fast is the weight being raised when the worker’s hand is 21 ft away from PW ?

P

W M 6 ft/sec

O

We let OM be the horizontal line of length x ft from a point O directly below the pulley to the worker’s hand M at any instant of time (Figure 3.49). Let h be the height of the weight W above O, and let z denote the length of rope from the pulley P to the worker’s hand. We want to know dh>dt when x = 21 given that dx>dt = 6. Note that the height of P above O is 20 ft because O is 5 ft above the ground. We assume the angle at O is a right angle. At any instant of time t we have the following relationships (see Figure 3.49b): Solution

5 ft

x (a)

P

20 - h + z = 45 20 2 + x 2 = z 2.

z

20 ft W h O

x

M

(b)

FIGURE 3.49 A worker at M walks to the right, pulling the weight W upward as the rope moves through the pulley P (Example 6).

Total length of rope is 45 ft. Angle at O is a right angle.

If we solve for z = 25 + h in the first equation, and substitute into the second equation, we have 20 2 + x 2 = s25 + hd2.

(1)

Differentiating both sides with respect to t gives 2x

dx dh = 2s25 + hd , dt dt

and solving this last equation for dh>dt we find x dx dh = . dt 25 + h dt

(2)

Since we know dx>dt, it remains only to find 25 + h at the instant when x = 21. From Equation (1), 20 2 + 212 = s25 + hd2 so that s25 + hd2 = 841,

or

25 + h = 29.

Equation (2) now gives 126 dh 21 # = 6 = L 4.3 ft>sec 29 29 dt as the rate at which the weight is being raised when x = 21 ft.

Exercises 3.10 1. Area Suppose that the radius r and area A = pr 2 of a circle are differentiable functions of t. Write an equation that relates dA>dt to dr>dt.

6. If x = y 3 - y and dy>dt = 5, then what is dx>dt when y = 2?

2. Surface area Suppose that the radius r and surface area S = 4pr 2 of a sphere are differentiable functions of t. Write an equation that relates dS>dt to dr>dt.

8. If x 2y 3 = 4>27 and dy>dt = 1>2, then what is dx>dt when x = 2?

3. Assume that y = 5x and dx>dt = 2. Find dy>dt. 4. Assume that 2x + 3y = 12 and dy>dt = -2. Find dx>dt. 5. If y = x 2 and dx>dt = 3, then what is dy>dt when x = -1?

7. If x 2 + y 2 = 25 and dx>dt = - 2, then what is dy>dt when x = 3 and y = -4? 9. If L = 2x 2 + y 2, dx>dt = -1, and dy>dt = 3, find dL>dt when x = 5 and y = 12. 10. If r + s 2 + y3 = 12, dr>dt = 4, and ds>dt = -3, find dy>dt when r = 3 and s = 1.

192

Chapter 3: Differentiation

11. If the original 24 m edge length x of a cube decreases at the rate of 5 m>min, when x = 3 m at what rate does the cube’s

a. Assuming that x, y, and z are differentiable functions of t, how is ds>dt related to dx>dt, dy>dt, and dz>dt?

a. surface area change?

b. How is ds>dt related to dy>dt and dz>dt if x is constant?

b. volume change?

c. How are dx>dt, dy>dt, and dz>dt related if s is constant?

12. A cube’s surface area increases at the rate of 72 in2>sec. At what rate is the cube’s volume changing when the edge length is x = 3 in? 13. Volume The radius r and height h of a right circular cylinder are related to the cylinder’s volume V by the formula V = pr 2h . a. How is dV>dt related to dh>dt if r is constant? b. How is dV>dt related to dr>dt if h is constant? c. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant? 14. Volume The radius r and height h of a right circular cone are related to the cone’s volume V by the equation V = s1>3dpr 2h . a. How is dV>dt related to dh>dt if r is constant? b. How is dV>dt related to dr>dt if h is constant? c. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant? 15. Changing voltage The voltage V (volts), current I (amperes), and resistance R (ohms) of an electric circuit like the one shown here are related by the equation V = IR . Suppose that V is increasing at the rate of 1 volt> sec while I is decreasing at the rate of 1> 3 amp> sec. Let t denote time in seconds. V

19. Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure u is A =

1 ab sin u . 2

a. How is dA>dt related to du>dt if a and b are constant? b. How is dA>dt related to du>dt and da>dt if only b is constant? c. How is dA>dt related to du>dt, da>dt , and db>dt if none of a, b, and u are constant? 20. Heating a plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm> min. At what rate is the plate’s area increasing when the radius is 50 cm? 21. Changing dimensions in a rectangle The length l of a rectangle is decreasing at the rate of 2 cm> sec while the width w is increasing at the rate of 2 cm> sec. When l = 12 cm and w = 5 cm , find the rates of change of (a) the area, (b) the perimeter, and (c) the lengths of the diagonals of the rectangle. Which of these quantities are decreasing, and which are increasing? 22. Changing dimensions in a rectangular box Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates: dx = 1 m>sec, dt

dy = -2 m>sec, dt

dz = 1 m>sec . dt

Find the rates at which the box’s (a) volume, (b) surface area, and (c) diagonal length s = 2x 2 + y 2 + z 2 are changing at the instant when x = 4, y = 3 , and z = 2 .

I

R

a. What is the value of dV>dt?

23. A sliding ladder A 13-ft ladder is leaning against a house when its base starts to slide away (see accompanying figure). By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft> sec.

b. What is the value of dI>dt?

a. How fast is the top of the ladder sliding down the wall then?

c. What equation relates dR>dt to dV>dt and dI>dt?

b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?

d. Find the rate at which R is changing when V = 12 volts and I = 2 amp. Is R increasing, or decreasing? 16. Electrical power The power P (watts) of an electric circuit is related to the circuit’s resistance R (ohms) and current I (amperes) by the equation P = RI 2 . a. How are dP>dt, dR>dt, and dI>dt related if none of P, R, and I are constant?

c. At what rate is the angle u between the ladder and the ground changing then? y

y(t)

b. How is dR>dt related to dI>dt if P is constant?

13-ft ladder

17. Distance Let x and y be differentiable functions of t and let s = 2x 2 + y 2 be the distance between the points (x, 0) and (0, y) in the xy-plane. a. How is ds>dt related to dx>dt if y is constant? b. How is ds>dt related to dx>dt and dy>dt if neither x nor y is constant? c. How is dx>dt related to dy>dt if s is constant? 18. Diagonals If x, y, and z are lengths of the edges of a rectangular box, the common length of the box’s diagonals is s = 2x 2 + y 2 + z 2 .

0

x(t)

x

24. Commercial air traffic Two commercial airplanes are flying at an altitude of 40,000 ft along straight-line courses that intersect at right angles. Plane A is approaching the intersection point at a speed of 442 knots (nautical miles per hour; a nautical mile is 2000 yd). Plane B is approaching the intersection at 481 knots. At what rate is the distance between the planes changing when A is 5

3.10 nautical miles from the intersection point and B is 12 nautical miles from the intersection point?

27. A growing sand pile Sand falls from a conveyor belt at the rate of 10 m3>min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high? Answer in centimeters per minute.

193

Ring at edge of dock

25. Flying a kite A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft> sec. How fast must she let out the string when the kite is 500 ft away from her? 26. Boring a cylinder The mechanics at Lincoln Automotive are reboring a 6-in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one thousandth of an inch every 3 min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.800 in.?

Related Rates

6'

33. A balloon and a bicycle A balloon is rising vertically above a level, straight road at a constant rate of 1 ft> sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft> sec passes under it. How fast is the distance s(t) between the bicycle and balloon increasing 3 sec later? y

28. A draining conical reservoir Water is flowing at the rate of 50 m3>min from a shallow concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. a. How fast (centimeters per minute) is the water level falling when the water is 5 m deep? b. How fast is the radius of the water’s surface changing then? Answer in centimeters per minute.

y(t)

29. A draining hemispherical reservoir Water is flowing at the rate of 6 m3>min from a reservoir shaped like a hemispherical bowl of radius 13 m, shown here in profile. Answer the following questions, given that the volume of water in a hemispherical bowl of radius R is V = sp>3dy 2s3R - yd when the water is y meters deep.

s(t)

Center of sphere 13 Water level y

a. At what rate is the water level changing when the water is 8 m deep?

x(t)

0

r

x

34. Making coffee Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in3>min. a. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? b. How fast is the level in the cone falling then?

b. What is the radius r of the water’s surface when the water is y m deep? 6"

c. At what rate is the radius r changing when the water is 8 m deep? 30. A growing raindrop Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop’s radius increases at a constant rate.

6" How fast is this level falling?

31. The radius of an inflating balloon A spherical balloon is inflated with helium at the rate of 100p ft3>min. How fast is the balloon’s radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing? 32. Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft> sec.

How fast is this level rising?

a. How fast is the boat approaching the dock when 10 ft of rope are out? b. At what rate is the angle u changing at this instant (see the figure)?

6"

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Chapter 3: Differentiation

35. Cardiac output In the late 1860s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Würzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about 7 L> min. At rest it is likely to be a bit under 6 L> min. If you are a trained marathon runner running a marathon, your cardiac output can be as high as 30 L> min. Your cardiac output can be calculated with the formula

Light Ball at time t 0 1/2 sec later 50-ft pole

Q y = , D where Q is the number of milliliters of CO2 you exhale in a minute and D is the difference between the CO2 concentration (ml> L) in the blood pumped to the lungs and the CO2 concentration in the blood returning from the lungs. With Q = 233 ml>min and D = 97 - 56 = 41 ml>L , y =

away from the light. (See accompanying figure.) How fast is the shadow of the ball moving along the ground 1>2 sec later? (Assume the ball falls a distance s = 16t 2 ft in t sec .)

233 ml>min L 5.68 L>min , 41 ml>L

fairly close to the 6 L> min that most people have at basal (resting) conditions. (Data courtesy of J. Kenneth Herd, M.D., Quillan College of Medicine, East Tennessee State University.) Suppose that when Q = 233 and D = 41 , we also know that D is decreasing at the rate of 2 units a minute but that Q remains unchanged. What is happening to the cardiac output?

Shadow 0

x(t)

30

x

NOT TO SCALE

40. A building’s shadow On a morning of a day when the sun will pass directly overhead, the shadow of an 80-ft building on level ground is 60 ft long. At the moment in question, the angle u the sun makes with the ground is increasing at the rate of 0.27°> min. At what rate is the shadow decreasing? (Remember to use radians. Express your answer in inches per minute, to the nearest tenth.)

36. Moving along a parabola A particle moves along the parabola y = x 2 in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 10 m> sec. How fast is the angle of inclination u of the line joining the particle to the origin changing when x = 3 m ? 37. Motion in the plane The coordinates of a particle in the metric xy-plane are differentiable functions of time t with dx>dt = -1 m>sec and dy>dt = -5 m>sec . How fast is the particle’s distance from the origin changing as it passes through the point (5, 12)? 38. Videotaping a moving car You are videotaping a race from a stand 132 ft from the track, following a car that is moving at 180 mi> h (264 ft> sec), as shown in the accompanying figure. How fast will your camera angle u be changing when the car is right in front of you? A half second later? Camera

132'

80'

41. A melting ice layer A spherical iron ball 8 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 10 in3>min , how fast is the thickness of the ice decreasing when it is 2 in. thick? How fast is the outer surface area of ice decreasing? 42. Highway patrol A highway patrol plane flies 3 mi above a level, straight road at a steady 120 mi> h. The pilot sees an oncoming car and with radar determines that at the instant the line-ofsight distance from plane to car is 5 mi, the line-of-sight distance is decreasing at the rate of 160 mi> h. Find the car’s speed along the highway. 43. Baseball players A baseball diamond is a square 90 ft on a side. A player runs from first base to second at a rate of 16 ft> sec.

Car

39. A moving shadow A light shines from the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft

a. At what rate is the player’s distance from third base changing when the player is 30 ft from first base? b. At what rates are angles u1 and u2 (see the figure) changing at that time?

3.11 c. The player slides into second base at the rate of 15 ft> sec. At what rates are angles u1 and u2 changing as the player touches base?

Third base

1

2

195

44. Ships Two ships are steaming straight away from a point O along routes that make a 120° angle. Ship A moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yd). Ship B moves at 21 knots. How fast are the ships moving apart when OA = 5 and OB = 3 nautical miles?

Second base 90'

Linearization and Differentials

Player 30'

First base

Home

3.11

Linearization and Differentials Sometimes we can approximate complicated functions with simpler ones that give the accuracy we want for specific applications and are easier to work with. The approximating functions discussed in this section are called linearizations, and they are based on tangent lines. Other approximating functions, such as polynomials, are discussed in Chapter 9. We introduce new variables dx and dy, called differentials, and define them in a way that makes Leibniz’s notation for the derivative dy>dx a true ratio. We use dy to estimate error in measurement, which then provides for a precise proof of the Chain Rule (Section 3.6).

Linearization As you can see in Figure 3.50, the tangent to the curve y = x 2 lies close to the curve near the point of tangency. For a brief interval to either side, the y-values along the tangent line 4

2 y x2

y x2

y 2x 1

y 2x 1

(1, 1)

(1, 1) –1

3 0

0

2 0

y x 2 and its tangent y 2x 1 at (1, 1). 1.2

Tangent and curve very close near (1, 1). 1.003

y x2

y x2 y 2x 1

(1, 1)

(1, 1)

y 2x 1 0.8

1.2 0.8

Tangent and curve very close throughout entire x-interval shown.

0.997 0.997

1.003

Tangent and curve closer still. Computer screen cannot distinguish tangent from curve on this x-interval.

FIGURE 3.50 The more we magnify the graph of a function near a point where the function is differentiable, the flatter the graph becomes and the more it resembles its tangent.

196

Chapter 3: Differentiation

y

give good approximations to the y-values on the curve. We observe this phenomenon by zooming in on the two graphs at the point of tangency or by looking at tables of values for the difference between ƒ(x) and its tangent line near the x-coordinate of the point of tangency. The phenomenon is true not just for parabolas; every differentiable curve behaves locally like its tangent line. In general, the tangent to y = ƒsxd at a point x = a , where ƒ is differentiable (Figure 3.51), passes through the point (a, ƒ(a)), so its point-slope equation is

y f(x) Slope f '(a) (a, f(a))

0

y L(x)

x

a

y = ƒsad + ƒ¿sadsx - ad.

FIGURE 3.51 The tangent to the curve y = ƒsxd at x = a is the line Lsxd = ƒsad + ƒ¿sadsx - ad .

Thus, this tangent line is the graph of the linear function Lsxd = ƒsad + ƒ¿sadsx - ad. For as long as this line remains close to the graph of ƒ, L(x) gives a good approximation to ƒ(x). If ƒ is differentiable at x = a, then the approximating function

DEFINITIONS

Lsxd = ƒsad + ƒ¿sadsx - ad is the linearization of ƒ at a. The approximation ƒsxd L Lsxd of ƒ by L is the standard linear approximation of ƒ at a. The point x = a is the center of the approximation.

Find the linearization of ƒsxd = 21 + x at x = 0 (Figure 3.52).

EXAMPLE 1

y

y 5 x 4 4

y1 x 2

y1 x 2

1.1

2 y 1 x

y 1 x

1.0

1

–1

0

1

2

3

x

4

FIGURE 3.52 The graph of y = 21 + x and its linearizations at x = 0 and x = 3 . Figure 3.53 shows a magnified view of the small window about 1 on the y-axis. Solution

0.9 –0.1

0

0.1

0.2

FIGURE 3.53 Magnified view of the window in Figure 3.52.

Since ƒ¿sxd =

1 s1 + xd-1>2 , 2

we have ƒs0d = 1 and ƒ¿s0d = 1>2, giving the linearization Lsxd = ƒsad + ƒ¿sadsx - ad = 1 +

x 1 sx - 0d = 1 + . 2 2

See Figure 3.53. The following table shows how accurate the approximation 21 + x L 1 + sx>2d from Example 1 is for some values of x near 0. As we move away from zero, we lose

3.11

Linearization and Differentials

197

accuracy. For example, for x = 2, the linearization gives 2 as the approximation for 23 , which is not even accurate to one decimal place. Approximation

True value

ƒ True value approximation ƒ

21.2 L 1 +

0.2 2

= 1.10

1.095445

0.004555 6 10-2

21.05 L 1 +

0.05 2

= 1.025

1.024695

0.000305 6 10-3

1.002497

0.000003 6 10-5

21.005 L 1 +

0.005 = 1.00250 2

Do not be misled by the preceding calculations into thinking that whatever we do with a linearization is better done with a calculator. In practice, we would never use a linearization to find a particular square root. The utility of a linearization is its ability to replace a complicated formula by a simpler one over an entire interval of values. If we have to work with 21 + x for x close to 0 and can tolerate the small amount of error involved, we can work with 1 + sx>2d instead. Of course, we then need to know how much error there is. We further examine the estimation of error in Chapter 9. A linear approximation normally loses accuracy away from its center. As Figure 3.52 suggests, the approximation 21 + x L 1 + sx>2d will probably be too crude to be useful near x = 3. There, we need the linearization at x = 3.

EXAMPLE 2 Solution

Find the linearization of ƒsxd = 21 + x at x = 3.

We evaluate the equation defining Lsxd at a = 3. With ƒs3d = 2,

ƒ¿s3d =

1 1 = , s1 + xd-1>2 ` 2 4 x=3

we have Lsxd = 2 +

5 x 1 (x - 3) = + . 4 4 4

At x = 3.2, the linearization in Example 2 gives 21 + x = 21 + 3.2 L

5 3.2 + = 1.250 + 0.800 = 2.050, 4 4

which differs from the true value 24.2 L 2.04939 by less than one one-thousandth. The linearization in Example 1 gives 21 + x = 21 + 3.2 L 1 +

y

3.2 = 1 + 1.6 = 2.6, 2

a result that is off by more than 25%.

EXAMPLE 3 0

2

Find the linearization of ƒsxd = cos x at x = p>2 (Figure 3.54).

x y cos x

y –x 2

FIGURE 3.54 The graph of ƒsxd = cos x and its linearization at x = p>2 . Near x = p>2, cos x L -x + sp>2d (Example 3).

Since ƒsp>2d = cossp>2d = 0, ƒ¿sxd = -sin x, and ƒ¿sp>2d = -sin sp>2d = -1, we find the linearization at a = p>2 to be

Solution

Lsxd = ƒsad + ƒ¿sadsx - ad = 0 + s -1d ax = -x +

p . 2

p b 2

198

Chapter 3: Differentiation

An important linear approximation for roots and powers is s1 + xdk L 1 + kx

sx near 0; any number kd

(Exercise 15). This approximation, good for values of x sufficiently close to zero, has broad application. For example, when x is small, 21 + x L 1 +

1 x 2

k = 1>2

1 = s1 - xd-1 L 1 + s -1ds -xd = 1 + x 1 - x 3 2 1 + 5x 4 = s1 + 5x 4 d1>3 L 1 +

k = -1; replace x by -x .

5 1 s5x 4 d = 1 + x 4 3 3

k = 1>3; replace x by 5x 4 .

1 1 1 = s1 - x 2 d-1>2 L 1 + a- bs -x 2 d = 1 + x 2 2 2 2 21 - x

k = - 1>2; replace x by -x 2 .

Differentials We sometimes use the Leibniz notation dy>dx to represent the derivative of y with respect to x. Contrary to its appearance, it is not a ratio. We now introduce two new variables dx and dy with the property that when their ratio exists, it is equal to the derivative.

DEFINITION Let y = ƒsxd be a differentiable function. The differential dx is an independent variable. The differential dy is dy = ƒ¿sxd dx.

Unlike the independent variable dx, the variable dy is always a dependent variable. It depends on both x and dx. If dx is given a specific value and x is a particular number in the domain of the function ƒ, then these values determine the numerical value of dy.

EXAMPLE 4 (a) Find dy if y = x 5 + 37x. (b) Find the value of dy when x = 1 and dx = 0.2. Solution

(a) dy = s5x 4 + 37d dx (b) Substituting x = 1 and dx = 0.2 in the expression for dy, we have dy = s5 # 14 + 37d0.2 = 8.4. The geometric meaning of differentials is shown in Figure 3.55. Let x = a and set dx = ¢x. The corresponding change in y = ƒsxd is ¢y = ƒsa + dxd - ƒsad. The corresponding change in the tangent line L is ¢L = L(a + dx) - L(a) = ƒ(a) + ƒ¿(a)[(a + dx) - a] - ƒ(a) (++++++)++++++* L(a dx)

= ƒ¿(a) dx.

()* L(a)

3.11 y

Linearization and Differentials

199

y f (x) (a dx, f (a dx)) y f(a dx) f (a) L f '(a)dx (a, f (a)) dx x When dx is a small change in x, the corresponding change in the linearization is precisely dy.

Tangent line 0

x

a dx

a

FIGURE 3.55 Geometrically, the differential dy is the change ¢L in the linearization of ƒ when x = a changes by an amount dx = ¢x .

That is, the change in the linearization of ƒ is precisely the value of the differential dy when x = a and dx = ¢x. Therefore, dy represents the amount the tangent line rises or falls when x changes by an amount dx = ¢x. If dx Z 0, then the quotient of the differential dy by the differential dx is equal to the derivative ƒ¿sxd because dy ƒ¿sxd dx = ƒ¿sxd = . dx dx

dy , dx = We sometimes write

dƒ = ƒ¿sxd dx in place of dy = ƒ¿sxd dx, calling dƒ the differential of ƒ. For instance, if ƒsxd = 3x 2 - 6, then dƒ = ds3x 2 - 6d = 6x dx. Every differentiation formula like dsu + yd du dy = + dx dx dx

or

dssin ud du = cos u dx dx

or

dssin ud = cos u du.

has a corresponding differential form like dsu + yd = du + dy

EXAMPLE 5 We can use the Chain Rule and other differentiation rules to find differentials of functions. (a) dstan 2xd = sec2s2xd ds2xd = 2 sec2 2x dx sx + 1d dx - x dsx + 1d x x dx + dx - x dx dx (b) d a = = b = x + 1 sx + 1d2 sx + 1d2 sx + 1d2

Estimating with Differentials Suppose we know the value of a differentiable function ƒ(x) at a point a and want to estimate how much this value will change if we move to a nearby point a + dx. If dx ¢x is small, then we can see from Figure 3.55 that ¢y is approximately equal to the differential dy. Since ƒsa + dxd = ƒsad + ¢y,

¢x = dx

200

Chapter 3: Differentiation

the differential approximation gives ƒsa + dxd L ƒsad + dy when dx = ¢x. Thus the approximation ¢y L dy can be used to estimate ƒsa + dxd when ƒ(a) is known and dx is small. dr 0.1

a 10

EXAMPLE 6 The radius r of a circle increases from a = 10 m to 10.1 m (Figure 3.56). Use dA to estimate the increase in the circle’s area A. Estimate the area of the enlarged circle and compare your estimate to the true area found by direct calculation. Solution

Since A = pr 2 , the estimated increase is dA = A¿sad dr = 2pa dr = 2ps10ds0.1d = 2p m2 .

A ≈ dA 2 a dr

FIGURE 3.56 When dr is small compared with a, the differential dA gives the estimate Asa + drd = pa 2 + dA (Example 6).

Thus, since Asr + ¢rd L Asrd + dA, we have As10 + 0.1d L As10d + 2p = ps10d2 + 2p = 102p. The area of a circle of radius 10.1 m is approximately 102p m2 . The true area is As10.1d = ps10.1d2 = 102.01p m2 . The error in our estimate is 0.01p m2 , which is the difference ¢A - dA.

EXAMPLE 7

Use differentials to estimate

(a) 7.971>3 (b) sin (p>6 + 0.01). Solution

(a) The differential associated with the cube root function y = x1>3 is dy =

1 dx. 3x2>3

We set a = 8, the closest number near 7.97 where we can easily compute f (a) and f ¿(a). To arrange that a + dx = 7.97, we choose dx = -0.03. Approximating with the differential gives f (7.97) = f (a + dx) L f (a) + dy 1 (-0.03) = 81>3 + 3(8)2>3 1 = 2 + (-0.03) = 1.9975 12 This gives an approximation to the true value of 7.971>3, which is 1.997497 to 6 decimals. (b) The differential associated with y = sin x is dy = cos x dx.

3.11

Linearization and Differentials

201

To estimate sin (p>6 + 0.01), we set a = p>6 and dx = 0.01. Then f (p>6 + 0.01) = f (a + dx) L f (a) + dy = sin =

p p + acos b (0.01) 6 6

23 1 + (0.01) L 0.5087 2 2

For comparison, the true value of sin (p>6 + 0.01) to 6 decimals is 0.508635. The method in part (b) of Example 7 is used by some calculator and computer algorithms to give values of trigonometric functions. The algorithms store a large table of sine and cosine values between 0 and p>4. Values between these stored values are computed using differentials as in Example 7b. Values outside of [0, p>4] are computed from values in this interval using trigonometric identities.

Error in Differential Approximation Let ƒ(x) be differentiable at x = a and suppose that dx = ¢x is an increment of x. We have two ways to describe the change in ƒ as x changes from a to a + ¢x: The true change:

¢ƒ = ƒsa + ¢xd - ƒsad

The differential estimate:

dƒ = ƒ¿sad ¢x.

How well does dƒ approximate ¢ƒ? We measure the approximation error by subtracting dƒ from ¢ƒ: Approximation error = ¢ƒ - dƒ = ¢ƒ - ƒ¿sad¢x = ƒ(a + ¢x) - ƒ(a) - ƒ¿(a)¢x (++++)++++* ƒ

= a

ƒ(a + ¢x) - ƒ(a) - ƒ¿(a)b # ¢x ¢x

(+++++++)+++++++* Call this part P.

= P # ¢x. As ¢x : 0, the difference quotient

ƒsa + ¢xd - ƒsad ¢x approaches ƒ¿sad (remember the definition of ƒ¿sad), so the quantity in parentheses becomes a very small number (which is why we called it P). In fact, P : 0 as ¢x : 0. When ¢x is small, the approximation error P ¢x is smaller still. ¢ƒ = ƒ¿(a)¢x + P ¢x

()* true change

(+)+* estimated change

()* error

Although we do not know the exact size of the error, it is the product P # ¢x of two small quantities that both approach zero as ¢x : 0. For many common functions, whenever ¢x is small, the error is still smaller.

202

Chapter 3: Differentiation

Change in y ƒsxd near x a If y = ƒsxd is differentiable at x = a and x changes from a to a + ¢x, the change ¢y in ƒ is given by ¢y = ƒ¿sad ¢x + P ¢x

(1)

in which P : 0 as ¢x : 0.

In Example 6 we found that 2 ¢A = p(10.1) 2 - p(10) 2 = (102.01 - 100)p = (2p + 0.01p) ()* ()* m

dA

error

so the approximation error is ¢A - dA = P¢r = 0.01p and P = 0.01p>¢r = 0.01p>0.1 = 0.1p m.

Proof of the Chain Rule Equation (1) enables us to prove the Chain Rule correctly. Our goal is to show that if ƒ(u) is a differentiable function of u and u = gsxd is a differentiable function of x, then the composite y = ƒsgsxdd is a differentiable function of x. Since a function is differentiable if and only if it has a derivative at each point in its domain, we must show that whenever g is differentiable at x0 and ƒ is differentiable at gsx0 d, then the composite is differentiable at x0 and the derivative of the composite satisfies the equation dy ` = ƒ¿s gsx0 dd # g¿sx0 d. dx x = x0 Let ¢x be an increment in x and let ¢u and ¢y be the corresponding increments in u and y. Applying Equation (1) we have ¢u = g¿sx0 d¢x + P1 ¢x = sg¿sx0 d + P1 d¢x, where P1 : 0 as ¢x : 0. Similarly, ¢y = ƒ¿su0 d¢u + P2 ¢u = sƒ¿su0 d + P2 d¢u, where P2 : 0 as ¢u : 0. Notice also that ¢u : 0 as ¢x : 0. Combining the equations for ¢u and ¢y gives ¢y = sƒ¿su0 d + P2 dsg¿sx0 d + P1 d¢x, so ¢y = ƒ¿su0 dg¿sx0 d + P2 g¿sx0 d + ƒ¿su0 dP1 + P2P1 . ¢x Since P1 and P2 go to zero as ¢x goes to zero, three of the four terms on the right vanish in the limit, leaving dy ¢y ` = ƒ¿su0 dg¿sx0 d = ƒ¿sgsx0 dd # g¿sx0 d. = lim dx x = x0 ¢x:0 ¢x

3.11

Linearization and Differentials

203

Sensitivity to Change The equation df = ƒ¿sxd dx tells how sensitive the output of ƒ is to a change in input at different values of x. The larger the value of ƒ¿ at x, the greater the effect of a given change dx. As we move from a to a nearby point a + dx, we can describe the change in ƒ in three ways:

Absolute change Relative change Percentage change

True

Estimated

¢ƒ = ƒsa + dxd - ƒsad ¢ƒ ƒsad ¢ƒ * 100 ƒsad

dƒ = ƒ¿sad dx dƒ ƒsad dƒ * 100 ƒsad

You want to calculate the depth of a well from the equation s = 16t 2 by timing how long it takes a heavy stone you drop to splash into the water below. How sensitive will your calculations be to a 0.1-sec error in measuring the time?

EXAMPLE 8

The size of ds in the equation ds = 32t dt depends on how big t is. If t = 2 sec, the change caused by dt = 0.1 is about ds = 32s2ds0.1d = 6.4 ft. Three seconds later at t = 5 sec, the change caused by the same dt is ds = 32s5ds0.1d = 16 ft. For a fixed error in the time measurement, the error in using ds to estimate the depth is larger when it takes a longer time before the stone splashes into the water. Solution

Exercises 3.11 Finding Linearizations In Exercises 1–5, find the linearization L(x) of ƒ(x) at x = a . 1. ƒsxd = x 3 - 2x + 3,

a = 2

14. ƒ(x) = sin-1 x,

2. ƒsxd = 2x + 9, a = -4 1 3. ƒsxd = x + x , a = 1 2

3 x, 4. ƒsxd = 2

a = -8

5. ƒ(x) = tan x,

a = p

(b) cos x

16. Use the linear approximation s1 + xdk L 1 + kx to find an approximation for the function ƒ(x) for values of x near zero.

(c) tan x

(d) e x

(e) ln (1 + x)

Linearization for Approximation In Exercises 7–14, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate. 7. ƒsxd = x + 2x, 2

-1

8. ƒsxd = x ,

a = 0.1

a = 0.9

9. ƒsxd = 2x + 3x - 3, 2

10. ƒsxd = 1 + x, 3 x, 11. ƒsxd = 2

a = 8.1 a = 8.5

a = p>12

15. Show that the linearization of ƒsxd = s1 + xdk at x = 0 is Lsxd = 1 + kx .

6. Common linear approximations at x = 0 Find the linearizations of the following functions at x = 0. (a) sin x

x , a = 1.3 x + 1 -x 13. ƒ(x) = e , a = -0.1 12. ƒsxd =

a = -0.9

a. ƒsxd = s1 - xd6 c. ƒsxd =

1 21 + x

e. ƒsxd = s4 + 3xd1>3

b. ƒsxd =

2 1 - x

d. ƒsxd = 22 + x 2 f. ƒsxd =

3

B

a1 -

1 b 2 + x

2

17. Faster than a calculator Use the approximation s1 + xdk L 1 + kx to estimate the following. a. s1.0002d50

3 1.009 b. 2

18. Find the linearization of ƒsxd = 2x + 1 + sin x at x = 0 . How is it related to the individual linearizations of 2x + 1 and sin x at x = 0 ?

204

Chapter 3: Differentiation

Derivatives in Differential Form In Exercises 19–38, find dy.

49. The change in the volume V = pr 2h of a right circular cylinder when the radius changes from r0 to r0 + dr and the height does not change

19. y = x 3 - 3 1x

20. y = x21 - x 2

2x 1 + x2 23. 2y 3>2 + xy - x = 0

22. y =

2 1x 3s1 + 1xd 24. xy 2 - 4x 3>2 - y = 0

50. The change in the lateral surface area S = 2prh of a right circular cylinder when the height changes from h0 to h0 + dh and the radius does not change

25. y = sin s5 1xd

26. y = cos sx 2 d

Applications 51. The radius of a circle is increased from 2.00 to 2.02 m.

21. y =

27. y = 4 tan sx >3d

28. y = sec sx 2 - 1d

3

29. y = 3 csc s1 - 21xd

30. y = 2 cot a

31. y = e2x

32. y = xe-x

33. y = ln (1 + x2)

34. y = ln a

1 b 1x

b 2x - 1 1 36. y = cot-1 a 2 b + cos-1 2x x

35. y = tan-1 (e x ) 2

37. y = sec-1 (e-x )

38. y = etan

-1

x + 1

a. Estimate the resulting change in area. b. Express the estimate as a percentage of the circle’s original area. 52. The diameter of a tree was 10 in. During the following year, the circumference increased 2 in. About how much did the tree’s diameter increase? The tree’s cross-section area? 53. Estimating volume Estimate the volume of material in a cylindrical shell with length 30 in., radius 6 in., and shell thickness 0.5 in.

2x 2 + 1

0.5 in. 6 in. 30 in.

Approximation Error In Exercises 39–44, each function ƒ(x) changes value when x changes from x0 to x0 + dx . Find a. the change ¢ƒ = ƒsx0 + dxd - ƒsx0 d ; b. the value of the estimate dƒ = ƒ¿sx0 d dx ; and c. the approximation error ƒ ¢ƒ - dƒ ƒ . y

54. Estimating height of a building A surveyor, standing 30 ft from the base of a building, measures the angle of elevation to the top of the building to be 75°. How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%? 55. Tolerance The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s

y f (x)

a. circumference? f f (x 0 dx) f (x 0) df f '(x 0 ) dx

(x 0, f(x 0 )) dx 0

x0 = 1,

40. ƒsxd = 2x + 4x - 3, 42. ƒsxd = x 4, 43. ƒsxd = x -1,

x0 = 1,

x0 = 1,

dx = 0.1

dx = 0.1 dx = 0.1

x0 = 2,

b. volume?

57. Tolerance The height and radius of a right circular cylinder are equal, so the cylinder’s volume is V = ph 3 . The volume is to be calculated with an error of no more than 1% of the true value. Find approximately the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h. 58. Tolerance

dx = 0.1

x0 = 0.5,

44. ƒsxd = x 3 - 2x + 3,

dx = 0.1

x0 = -1,

2

41. ƒsxd = x 3 - x,

x

x 0 dx

x0

39. ƒsxd = x 2 + 2x,

56. Tolerance The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s a. surface area?

Tangent

b. area?

dx = 0.1

Differential Estimates of Change In Exercises 45–50, write a differential formula that estimates the given change in volume or surface area.

a. About how accurately must the interior diameter of a 10-mhigh cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value? b. About how accurately must the tank’s exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount?

45. The change in the volume V = s4>3dpr of a sphere when the radius changes from r0 to r0 + dr

59. The diameter of a sphere is measured as 100 ; 1 cm and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation.

46. The change in the volume V = x 3 of a cube when the edge lengths change from x0 to x0 + dx

60. Estimate the allowable percentage error in measuring the diameter D of a sphere if the volume is to be calculated correctly to within 3%.

47. The change in the surface area S = 6x 2 of a cube when the edge lengths change from x0 to x0 + dx

61. The effect of flight maneuvers on the heart The amount of work done by the heart’s main pumping chamber, the left ventricle, is given by the equation

3

48. The change in the lateral surface area S = pr2r 2 + h 2 of a right circular cone when the radius changes from r0 to r0 + dr and the height does not change

W = PV +

Vdy2 , 2g

3.11 where W is the work per unit time, P is the average blood pressure, V is the volume of blood pumped out during the unit of time, d (“delta”) is the weight density of the blood, y is the average velocity of the exiting blood, and g is the acceleration of gravity. When P, V, d , and y remain constant, W becomes a function of g, and the equation takes the simplified form b W = a + g sa, b constantd . As a member of NASA’s medical team, you want to know how sensitive W is to apparent changes in g caused by flight maneuvers, and this depends on the initial value of g. As part of your investigation, you decide to compare the effect on W of a given change dg on the moon, where g = 5.2 ft>sec2 , with the effect the same change dg would have on Earth, where g = 32 ft>sec2 . Use the simplified equation above to find the ratio of dWmoon to dWEarth . 62. Measuring acceleration of gravity When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on the earth’s surface, depending on the change in g. By keeping track of ¢T , we can estimate the variation in g from the equation T = 2psL>gd1>2 that relates T, g, and L. a. With L held constant and g as the independent variable, calculate dT and use it to answer parts (b) and (c). b. If g increases, will T increase or decrease? Will a pendulum clock speed up or slow down? Explain. c. A clock with a 100-cm pendulum is moved from a location where g = 980 cm>sec2 to a new location. This increases the period by dT = 0.001 sec . Find dg and estimate the value of g at the new location. 63. The linearization is the best linear approximation Suppose that y = ƒsxd is differentiable at x = a and that g sxd = msx - ad + c is a linear function in which m and c are constants. If the error Esxd = ƒsxd - g sxd were small enough near x = a , we might think of using g as a linear approximation of ƒ instead of the linearization Lsxd = ƒsad + ƒ¿sadsx - ad . Show that if we impose on g the conditions 1. Esad = 0 Esxd 2. lim x - a = 0 x: a

The approximation error is zero at x = a . The error is negligible when compared with x - a .

then g sxd = ƒsad + ƒ¿sadsx - ad . Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x = a and negligible in comparison with x - a . The linearization, L(x): y f(a) f '(a)(x a)

Some other linear approximation, g(x): y m(x a) c y f (x)

a

x

64. Quadratic approximations a. Let Qsxd = b0 + b1sx - ad + b2sx - ad2 be a quadratic approximation to ƒ(x) at x = a with the properties: i) Qsad = ƒsad ii) Q¿sad = ƒ¿sad iii) Q–sad = ƒ–sad.

205

Determine the coefficients b0 , b1 , and b2 . b. Find the quadratic approximation to ƒsxd = 1>s1 - xd at x = 0. T c. Graph ƒsxd = 1>s1 - xd and its quadratic approximation at x = 0 . Then zoom in on the two graphs at the point (0, 1). Comment on what you see. T d. Find the quadratic approximation to gsxd = 1>x at x = 1 . Graph g and its quadratic approximation together. Comment on what you see. T e. Find the quadratic approximation to hsxd = 21 + x at x = 0 . Graph h and its quadratic approximation together. Comment on what you see. f. What are the linearizations of ƒ, g, and h at the respective points in parts (b), (d), and (e)? 65. The linearization of 2x a. Find the linearization of ƒsxd = 2x at x = 0 . Then round its coefficients to two decimal places. T b. Graph the linearization and function together for -3 … x … 3 and -1 … x … 1 . 66. The linearization of log3 x a. Find the linearization of ƒsxd = log 3 x at x = 3 . Then round its coefficients to two decimal places. T b. Graph the linearization and function together in the window 0 … x … 8 and 2 … x … 4 .

COMPUTER EXPLORATIONS In Exercises 67–72, use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval I. Perform the following steps: a. Plot the function ƒ over I. b. Find the linearization L of the function at the point a. c. Plot ƒ and L together on a single graph. d. Plot the absolute error ƒ ƒsxd - Lsxd ƒ over I and find its maximum value. e. From your graph in part (d), estimate as large a d 7 0 as you can, satisfying ƒx - aƒ 6 d

Q

ƒ ƒsxd - Lsxd ƒ 6 P

for P = 0.5, 0.1, and 0.01. Then check graphically to see if your d-estimate holds true. 67. ƒsxd = x 3 + x 2 - 2x, 68. ƒsxd =

(a, f (a))

Linearization and Differentials

x - 1 , 4x 2 + 1

a = 1

[-1, 2],

3 1 c- , 1 d, a = 4 2

69. ƒsxd = x 2>3sx - 2d,

[-2, 3],

a = 2

70. ƒsxd = 1x - sin x,

[0, 2p],

a = 2

71. ƒ(x) = x2 ,

a = 1

x

[0, 2],

72. ƒ(x) = 2x sin-1 x, [0, 1],

a =

1 2

206

Chapter 3: Differentiation

Chapter 3

Questions to Guide Your Review

1. What is the derivative of a function ƒ ? How is its domain related to the domain of ƒ? Give examples. 2. What role does the derivative play in defining slopes, tangents, and rates of change? 3. How can you sometimes graph the derivative of a function when all you have is a table of the function’s values? 4. What does it mean for a function to be differentiable on an open interval? On a closed interval? 5. How are derivatives and one-sided derivatives related? 6. Describe geometrically when a function typically does not have a derivative at a point. 7. How is a function’s differentiability at a point related to its continuity there, if at all? 8. What rules do you know for calculating derivatives? Give some examples. 9. Explain how the three formulas d n sx d = nx n - 1 a. dx d du scud = c b. dx dx du1 du2 Á dun d su + u2 + Á + un d = + + + c. dx 1 dx dx dx enable us to differentiate any polynomial.

20. What is the rule for calculating the derivative of a composite of two differentiable functions? How is such a derivative evaluated? Give examples. 21. If u is a differentiable function of x, how do you find sd>dxdsu n d if n is an integer? If n is a real number? Give examples. 22. What is implicit differentiation? When do you need it? Give examples. 23. What is the derivative of the natural logarithm function ln x? How does the domain of the derivative compare with the domain of the function? 24. What is the derivative of the exponential function ax, a 7 0 and a Z 1? What is the geometric significance of the limit of (ah - 1)>h as h : 0? What is the limit when a is the number e? 26. What is logarithmic differentiation? Give an example. 27. How can you write any real power of x as a power of e? Are there any restrictions on x? How does this lead to the Power Rule for differentiating arbitrary real powers?

11. What is a second derivative? A third derivative? How many derivatives do the functions you know have? Give examples. x

12. What is the derivative of the exponential function e ? How does the domain of the derivative compare with the domain of the function? 13. What is the relationship between a function’s average and instantaneous rates of change? Give an example. 14. How do derivatives arise in the study of motion? What can you learn about a body’s motion along a line by examining the derivatives of the body’s position function? Give examples. 15. How can derivatives arise in economics? 16. Give examples of still other applications of derivatives. 17. What do the limits limh:0 sssin hd>hd and limh:0 sscos h - 1d>hd have to do with the derivatives of the sine and cosine functions? What are the derivatives of these functions?

28. What is one way of expressing the special number e as a limit? What is an approximate numerical value of e correct to 7 decimal places? 29. What are the derivatives of the inverse trigonometric functions? How do the domains of the derivatives compare with the domains of the functions? 30. How do related rates problems arise? Give examples. 31. Outline a strategy for solving related rates problems. Illustrate with an example. 32. What is the linearization L(x) of a function ƒ(x) at a point x = a ? What is required of ƒ at a for the linearization to exist? How are linearizations used? Give examples. 33. If x moves from a to a nearby value a + dx , how do you estimate the corresponding change in the value of a differentiable function ƒ(x)? How do you estimate the relative change? The percentage change? Give an example.

Practice Exercises

Derivatives of Functions Find the derivatives of the functions in Exercises 1– 64. 1. y = x - 0.125x + 0.25x 5

19. At what points are the six basic trigonometric functions continuous? How do you know?

25. What is the derivative of loga x? Are there any restrictions on a?

10. What formula do we need, in addition to the three listed in Question 9, to differentiate rational functions?

Chapter 3

18. Once you know the derivatives of sin x and cos x, how can you find the derivatives of tan x, cot x, sec x, and csc x? What are the derivatives of these functions?

2

3. y = x 3 - 3sx 2 + p2 d 5. y = sx + 1d2sx 2 + 2xd

2. y = 3 - 0.7x + 0.3x 1 4. y = x 7 + 27x p + 1 3

7

6. y = s2x - 5ds4 - xd-1

7. y = su2 + sec u + 1d3 9. s =

1t 1 + 1t

11. y = 2 tan2 x - sec2 x

8. y = a-1 -

csc u u2 - b 2 4

1 1t - 1 1 2 12. y = sin x sin2 x 10. s =

2

Chapter 3

13. s = cos4 s1 - 2td

2 14. s = cot3 a t b

71. y 2 =

15. s = ssec t + tan td5

16. s = csc5 s1 - t + 3t 2 d

73. e x + 2y = 1

74. y 2 = 2e -1>x

17. r = 22u sin u

18. r = 2u 2cos u

75. ln (x>y) = 1

76. x sin-1 y = 1 + x 2

20. r = sin A u + 2u + 1 B

19. r = sin 22u 1 2 2 x csc x 2 23. y = x -1>2 sec s2xd2

22. y = 2 1x sin 1x

25. y = 5 cot x

26. y = x cot 5x

21. y =

2

-2

81. r cos 2s + sin2 s = p

-1 30. s = 15s15t - 1d3

x + x 33. y = B x2

34. y = 4x2x + 1x

2

sin u b cos u - 1

2

36. r = a

1 + sin u b 1 - cos u

38. y = 20s3x - 4d s3x - 4d

3 s5x 2 + sin 2xd3>2 41. y = 10e-x>5

40. y = s3 + cos3 3xd-1>3

1>4

42. y = 22e22x

1 4x 1 4x xe e 4 16

44. y = x2e-2>x 46. y = ln ssec2 ud

45. y = ln ssin2 ud

47. y = log2 sx >2d

48. y = log5 s3x - 7d

2

-t

50. y = 92t

51. y = 5x3.6

52. y = 22x-22

53. y = sx + 2dx + 2

54. y = 2sln xdx>2

-1

55. y = sin 21 - u ,

0 6 u 6 1

2

56. y = sin-1 a

1

2y 57. y = ln cos-1 x

b,

2 b. y 2 = 1 - x

a. x 3 + y 3 = 1 84. a. By differentiating dy>dx = x>y .

62. y = 22x - 1 sec 63. y = csc

-1

-1>5

64. y = s1 + x2 detan

-1

that

Numerical Values of Derivatives 85. Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1 . x

ƒ(x)

g (x)

ƒ(x)

g(x)

0 1

1 3

1 5

-3 1>2

1>2 -4

Find the first derivatives of the following combinations at the given value of x. a. 6ƒsxd - g sxd, x = 1 ƒsxd , x = 1 c. g sxd + 1 e. g sƒsxdd,

b. ƒsxdg 2sxd, d. ƒsg sxdd,

x = 0

g. ƒsx + g sxdd,

z 7 1

1x

x = 0 x = 0

f. sx + ƒsxdd3>2,

x = 1

x = 0

x

ƒ(x)

ƒ(x)

0 1

9 -3

-2 1>5

Find the first derivatives of the following combinations at the given value of x. a. 1x ƒsxd,

0 6 u 6 p>2

ssec ud,

show

b. Then show that d 2y>dx 2 = -1>y 3 .

58. y = z cos-1 z - 21 - z2 1 59. y = t tan-1 t - ln t 2 60. y = s1 + t 2 d cot-1 2t -1

implicitly,

86. Suppose that the function ƒ(x) and its first derivative have the following values at x = 0 and x = 1 .

y 7 1

61. y = z sec-1 z - 2z2 - 1,

x2 - y2 = 1

2

37. y = s2x + 1d22x + 1 39. y =

82. 2rs - r - s + s 2 = -3

83. Find d 2y>dx 2 by implicit differentiation:

2 2 1x b 32. y = a 2 1x + 1

49. y = 8

80. q = s5p 2 + 2pd-3>2

In Exercises 81 and 82, find dr>ds.

28. y = x -2 sin2 sx 3 d

2

1x 2 b 31. y = a 1 + x

43. y =

78. x y = 22

2

4t b 29. s = a t + 1

35. r = a

= 2

79. p 3 + 4pq - 3q 2 = 2

24. y = 1x csc sx + 1d

2

x

1 + x A1 - x

72. y 2 =

In Exercises 79 and 80, find dp>dq. 3

27. y = x sin s2x d 2

77. ye tan

-1

x x + 1

207

Practice Exercises

x = 1

c. ƒs 1xd, x = 1 ƒsxd , x = 0 e. 2 + cos x

x

Implicit Differentiation In Exercises 65–78, find dy>dx by implicit differentiation. 65. xy + 2x + 3y = 1

66. x 2 + xy + y 2 - 5x = 2

67. x 3 + 4xy - 3y 4>3 = 2x

68. 5x 4>5 + 10y 6>5 = 15

69. 1xy = 1

70. x y = 1 2 2

b. 2ƒsxd,

x = 0

d. ƒs1 - 5 tan xd,

x = 0

px f. 10 sin a b ƒ 2sxd, 2

x=1

87. Find the value of dy>dt at t = 0 if y = 3 sin 2x and x = t 2 + p . 88. Find the value of ds>du at u = 2 if s = t 2 + 5t and t = su 2 + 2ud1>3 .

89. Find the value of dw>ds at s = 0 if w = sin A e 1r B and r = 3 sin ss + p>6d .

208

Chapter 3: Differentiation

90. Find the value of dr>dt at t = 0 if r = su2 + 7d1>3 and u2t + u = 1 . 91. If y 3 + y = 2 cos x , find the value of d 2y>dx 2 at the point (0, 1). 92. If x 1>3 + y 1>3 = 4 , find d 2y>dx 2 at the point (8, 8). Applying the Derivative Definition In Exercises 93 and 94, find the derivative using the definition. 93. ƒstd =

1 2t + 1

94. g sxd = 2x 2 + 1 95. a. Graph the function ƒsxd = e

x 2, -x 2,

-1 … x 6 0 0 … x … 1.

b. Is ƒ continuous at x = 0 ? c. Is ƒ differentiable at x = 0 ? Give reasons for your answers. 96. a. Graph the function x, ƒsxd = e tan x, b. Is ƒ continuous at x = 0 ?

-1 … x 6 0 0 … x … p>4.

c. Is ƒ differentiable at x = 0 ? Give reasons for your answers. 97. a. Graph the function x, 0 … x … 1 ƒsxd = e 2 - x, 1 6 x … 2. b. Is ƒ continuous at x = 1 ? c. Is ƒ differentiable at x = 1 ? Give reasons for your answers. 98. For what value or values of the constant m, if any, is sin 2x, x … 0 ƒsxd = e mx, x 7 0 a. continuous at x = 0 ?

105. Normals parallel to a line Find the points on the curve y = tan x, -p>2 6 x 6 p>2 , where the normal is parallel to the line y = -x>2 . Sketch the curve and normals together, labeling each with its equation. 106. Tangent and normal lines Find equations for the tangent and normal to the curve y = 1 + cos x at the point sp>2, 1d . Sketch the curve, tangent, and normal together, labeling each with its equation. 107. Tangent parabola The parabola y = x 2 + C is to be tangent to the line y = x . Find C. 108. Slope of tangent Show that the tangent to the curve y = x 3 at any point sa, a 3 d meets the curve again at a point where the slope is four times the slope at sa, a 3 d . 109. Tangent curve For what value of c is the curve y = c>sx + 1d tangent to the line through the points s0, 3d and s5, -2d ? 110. Normal to a circle Show that the normal line at any point of the circle x 2 + y 2 = a 2 passes through the origin. In Exercises 111–116, find equations for the lines that are tangent and normal to the curve at the given point. 111. x 2 + 2y 2 = 9, s1, 2d 112. e x + y 2 = 2, s0, 1d 113. xy + 2x - 5y = 2, s3, 2d 114. s y - xd2 = 2x + 4, s6, 2d 115. x + 1xy = 6, s4, 1d 116. x 3>2 + 2y 3>2 = 17, s1, 4d 117. Find the slope of the curve x 3y 3 + y 2 = x + y at the points (1, 1) and s1, -1d . 118. The graph shown suggests that the curve y = sin sx - sin xd might have horizontal tangents at the x-axis. Does it? Give reasons for your answer. y

y sin (x sin x)

1

b. differentiable at x = 0 ?

–2

0

–

x

2

Give reasons for your answers. Slopes, Tangents, and Normals 99. Tangents with specified slope Are there any points on the curve y = sx>2d + 1>s2x - 4d where the slope is -3>2 ? If so, find them. 100. Tangents with specified slope Are there any points on the curve y = x - e -x where the slope is 2? If so, find them. 101. Horizontal tangents Find the points on the curve y = 2x 3 - 3x 2 - 12x + 20 where the tangent is parallel to the x-axis.

–1

Analyzing Graphs Each of the figures in Exercises 119 and 120 shows two graphs, the graph of a function y = ƒsxd together with the graph of its derivative ƒ¿sxd . Which graph is which? How do you know? y

119. A

A

2

1

B

1 –1

0

a. perpendicular to the line y = 1 - sx>24d . b. parallel to the line y = 22 - 12x . 104. Intersecting tangents Show that the tangents to the curve y = sp sin xd>x at x = p and x = -p intersect at right angles.

4

2

3

102. Tangent intercepts Find the x- and y-intercepts of the line that is tangent to the curve y = x 3 at the point s -2, -8d . 103. Tangents perpendicular or parallel to lines Find the points on the curve y = 2x 3 - 3x 2 - 12x + 20 where the tangent is

y

120.

–1 B –2

1

x

0

1

2

x

Chapter 3 121. Use the following information to graph the function y = ƒsxd for -1 … x … 6 . i) The graph of ƒ is made of line segments joined end to end. ii) The graph starts at the point s -1, 2d . iii) The derivative of ƒ, where defined, agrees with the step function shown here. y

–1

1

2

3

4

5

6

x

u :sp>2d

–2

122. Repeat Exercise 121, supposing that the graph starts at s -1, 0d instead of s -1, 2d . Exercises 123 and 124 are about the accompanying graphs. The graphs in part (a) show the numbers of rabbits and foxes in a small arctic population. They are plotted as functions of time for 200 days. The number of rabbits increases at first, as the rabbits reproduce. But the foxes prey on rabbits and, as the number of foxes increases, the rabbit population levels off and then drops. Part (b) shows the graph of the derivative of the rabbit population, made by plotting slopes. 123. a. What is the value of the derivative of the rabbit population when the number of rabbits is largest? Smallest? b. What is the size of the rabbit population when its derivative is largest? Smallest (negative value)? 124. In what units should the slopes of the rabbit and fox population curves be measured? Number of rabbits Initial no. rabbits 1000 Initial no. foxes 40

-

4 tan2 u + tan u + 1 tan2 u + 5

130. lim+

1 - 2 cot2 u 5 cot u - 7 cot u - 8

131. lim

x sin x 2 - 2 cos x

x:0

–1

2000

lim

u :0

1

209

Trigonometric Limits Find the limits in Exercises 125–132. sin x 3x - tan 7x 125. lim 126. lim 2x x:0 2x 2 - x x:0 sin ssin ud sin r 127. lim 128. lim u tan 2r r: 0 u :0 129.

y f '(x)

Practice Exercises

2

132. lim

u :0

1 - cos u u2

Show how to extend the functions in Exercises 133 and 134 to be continuous at the origin. 133. g sxd =

tan stan xd tan x

134. ƒsxd =

tan stan xd sin ssin xd

Logarithmic Differentiation In Exercises 135–140, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable. 135. y =

2sx2 + 1d

2cos 2x st + 1dst - 1d 5 137. y = a b , st - 2dst + 3d 2u2u 138. y = 2u2 + 1 139. y = ssin ud2u

136. y =

3x + 4 A 2x - 4

10

t 7 2

140. y = sln xd1>sln xd

Related Rates 141. Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2pr 2 + 2prh .

(20, 1700) 1000

a. How is dS>dt related to dr>dt if h is constant? b. How is dS>dt related to dh>dt if r is constant?

Number of foxes 0

50

100 Time (days) (a)

150

200

c. How is dS>dt related to dr>dt and dh>dt if neither r nor h is constant? d. How is dr>dt related to dh>dt if S is constant? 142. Right circular cone The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation S = pr2r 2 + h 2 .

100

a. How is dS>dt related to dr>dt if h is constant? 50

b. How is dS>dt related to dh>dt if r is constant? (20, 40)

c. How is dS>dt related to dr>dt and dh>dt if neither r nor h is constant?

0

143. Circle’s changing area The radius of a circle is changing at the rate of -2>p m>sec. At what rate is the circle’s area changing when r = 10 m ?

–50 –100

0

50

100 150 Time (days) Derivative of the rabbit population (b)

200

144. Cube’s changing edges The volume of a cube is increasing at the rate of 1200 cm3>min at the instant its edges are 20 cm long. At what rate are the lengths of the edges changing at that instant?

210

Chapter 3: Differentiation

145. Resistors connected in parallel If two resistors of R1 and R2 ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation 1 1 1 + . = R R1 R2 1.2'

R1

R2 R

If R1 is decreasing at the rate of 1 ohm> sec and R2 is increasing at the rate of 0.5 ohm> sec, at what rate is R changing when R1 = 75 ohms and R2 = 50 ohms ?

151. Moving searchlight beam The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, du>dt = -0.6 rad/sec.

146. Impedance in a series circuit The impedance Z (ohms) in a series circuit is related to the resistance R (ohms) and reactance X (ohms) by the equation Z = 2R 2 + X 2 . If R is increasing at 3 ohms> sec and X is decreasing at 2 ohms> sec, at what rate is Z changing when R = 10 ohms and X = 20 ohms ?

a. How fast is the light moving along the shore when it reaches point A?

b. How many revolutions per minute is 0.6 rad> sec?

147. Speed of moving particle The coordinates of a particle moving in the metric xy-plane are differentiable functions of time t with dx>dt = 10 m>sec and dy>dt = 5 m>sec . How fast is the particle moving away from the origin as it passes through the point s3, -4d ? 148. Motion of a particle A particle moves along the curve y = x 3>2 in the first quadrant in such a way that its distance from the origin increases at the rate of 11 units per second. Find dx>dt when x = 3 . 149. Draining a tank Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft3>min . a. What is the relation between the variables h and r in the figure? b. How fast is the water level dropping when h = 6 ft ?

4'

x

A 1 km

152. Points moving on coordinate axes Points A and B move along the x- and y-axes, respectively, in such a way that the distance r (meters) along the perpendicular from the origin to the line AB remains constant. How fast is OA changing, and is it increasing, or decreasing, when OB = 2r and B is moving toward O at the rate of 0.3r m> sec? Linearization 153. Find the linearizations of a. tan x at x = -p>4

r

b. sec x at x = -p>4 .

Graph the curves and linearizations together. 10'

154. We can obtain a useful linear approximation of the function ƒsxd = 1>s1 + tan xd at x = 0 by combining the approximations

h Exit rate: 5 ft 3/min

150. Rotating spool As television cable is pulled from a large spool to be strung from the telephone poles along a street, it unwinds from the spool in layers of constant radius (see accompanying figure). If the truck pulling the cable moves at a steady 6 ft> sec (a touch over 4 mph), use the equation s = r u to find how fast (radians per second) the spool is turning when the layer of radius 1.2 ft is being unwound.

1 L 1 - x 1 + x

and

tan x L x

to get 1 L 1 - x. 1 + tan x Show that this result is the standard linear approximation of 1>s1 + tan xd at x = 0 . 155. Find the linearization of ƒsxd = 21 + x + sin x - 0.5 at x = 0 . 156. Find the linearization of ƒsxd = 2>s1 - xd + 21 + x - 3.1 at x = 0 .

Chapter 3 Differential Estimates of Change 157. Surface area of a cone Write a formula that estimates the change that occurs in the lateral surface area of a right circular cone when the height changes from h0 to h0 + dh and the radius does not change.

Additional and Advanced Exercises

211

159. Compounding error The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of a. the radius. b. the surface area. c. the volume.

h

160. Finding height To find the height of a lamppost (see accompanying figure), you stand a 6 ft pole 20 ft from the lamp and measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.

r V 5 1 pr 2h 3 S 5 pr r 2 1 h 2 (Lateral surface area)

158. Controlling error a. How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

h

b. Suppose that the edge is measured with the accuracy required in part (a). About how accurately can the cube’s volume be calculated from the edge measurement? To find out, estimate the percentage error in the volume calculation that might result from using the edge measurement.

Chapter 3

6 ft 20 ft a

Additional and Advanced Exercises

1. An equation like sin2 u + cos2 u = 1 is called an identity because it holds for all values of u . An equation like sin u = 0.5 is not an identity because it holds only for selected values of u , not all. If you differentiate both sides of a trigonometric identity in u with respect to u , the resulting new equation will also be an identity. Differentiate the following to show that the resulting equations hold for all u . a. sin 2u = 2 sin u cos u

satisfy the conditions ƒs0d = g s0d

3. a. Find values for the constants a, b, and c that will make ƒsxd = cos x

and

g sxd = a + bx + cx 2

satisfy the conditions ƒs0d = g s0d,

ƒ¿s0d = g¿s0d,

and

ƒ–s0d = g–s0d .

b. Find values for b and c that will make ƒsxd = sin sx + ad

and

g sxd = b sin x + c cos x

ƒ¿s0d = g¿s0d .

c. For the determined values of a, b, and c, what happens for the third and fourth derivatives of ƒ and g in each of parts (a) and (b)? 4. Solutions to differential equations a. Show that y = sin x, y = cos x , and y = a cos x + b sin x (a and b constants) all satisfy the equation

b. cos 2u = cos2 u - sin2 u 2. If the identity sin sx + ad = sin x cos a + cos x sin a is differentiated with respect to x, is the resulting equation also an identity? Does this principle apply to the equation x 2 - 2x - 8 = 0 ? Explain.

and

y– + y = 0 . b. How would you modify the functions in part (a) to satisfy the equation y– + 4y = 0 ? Generalize this result. 5. An osculating circle Find the values of h, k, and a that make the circle sx - hd2 + s y - kd2 = a 2 tangent to the parabola y = x 2 + 1 at the point (1, 2) and that also make the second derivatives d 2y>dx 2 have the same value on both curves there. Circles like this one that are tangent to a curve and have the same second derivative as the curve at the point of tangency are called osculating circles (from the Latin osculari, meaning “to kiss”). We encounter them again in Chapter 12.

212

Chapter 3: Differentiation

6. Marginal revenue A bus will hold 60 people. The number x of people per trip who use the bus is related to the fare charged ( p dollars) by the law p = [3 - sx>40d]2 . Write an expression for the total revenue r(x) per trip received by the bus company. What number of people per trip will make the marginal revenue dr>dx equal to zero? What is the corresponding fare? (This fare is the one that maximizes the revenue, so the bus company should probably rethink its fare policy.)

c. Find the particle’s velocity and acceleration at the points in part (b). d. When does the particle first reach the origin? What are its velocity, speed, and acceleration then? 11. Shooting a paper clip On Earth, you can easily shoot a paper clip 64 ft straight up into the air with a rubber band. In t sec after firing, the paper clip is s = 64t - 16t 2 ft above your hand. a. How long does it take the paper clip to reach its maximum height? With what velocity does it leave your hand?

7. Industrial production a. Economists often use the expression “rate of growth” in relative rather than absolute terms. For example, let u = ƒstd be the number of people in the labor force at time t in a given industry. (We treat this function as though it were differentiable even though it is an integer-valued step function.) Let y = g std be the average production per person in the labor force at time t. The total production is then y = uy . If the labor force is growing at the rate of 4% per year sdu>dt = 0.04ud and the production per worker is growing at the rate of 5% per year sdy>dt = 0.05yd , find the rate of growth of the total production, y. b. Suppose that the labor force in part (a) is decreasing at the rate of 2% per year while the production per person is increasing at the rate of 3% per year. Is the total production increasing, or is it decreasing, and at what rate? 8. Designing a gondola The designer of a 30-ft-diameter spherical hot air balloon wants to suspend the gondola 8 ft below the bottom of the balloon with cables tangent to the surface of the balloon, as shown. Two of the cables are shown running from the top edges of the gondola to their points of tangency, s -12, - 9d and s12, -9d . How wide should the gondola be? y x 2 y 2 225

b. On the moon, the same acceleration will send the paper clip to a height of s = 64t - 2.6t 2 ft in t sec. About how long will it take the paper clip to reach its maximum height, and how high will it go? 12. Velocities of two particles At time t sec, the positions of two particles on a coordinate line are s1 = 3t 3 - 12t 2 + 18t + 5 m and s2 = -t 3 + 9t 2 - 12t m . When do the particles have the same velocities? 13. Velocity of a particle A particle of constant mass m moves along the x-axis. Its velocity y and position x satisfy the equation 1 1 msy 2 - y0 2 d = k sx0 2 - x 2 d , 2 2 where k, y0 , and x0 are constants. Show that whenever y Z 0 , m

dy = -kx . dt

14. Average and instantaneous velocity a. Show that if the position x of a moving point is given by a quadratic function of t, x = At 2 + Bt + C , then the average velocity over any time interval [t1, t2] is equal to the instantaneous velocity at the midpoint of the time interval. b. What is the geometric significance of the result in part (a)? 15. Find all values of the constants m and b for which the function y = e

x

0 (–12, –9)

(12, –9)

Suspension cables Gondola

sin x, x 6 p mx + b, x Ú p

15 ft

is

8 ft

b. differentiable at x = p .

a. continuous at x = p . 16. Does the function

Width NOT TO SCALE

9. Pisa by parachute On August 5, 1988, Mike McCarthy of London jumped from the top of the Tower of Pisa. He then opened his parachute in what he said was a world record low-level parachute jump of 179 ft. Make a rough sketch to show the shape of the graph of his speed during the jump. (Source: Boston Globe, Aug. 6, 1988.) 10. Motion of a particle The position at time t Ú 0 of a particle moving along a coordinate line is s = 10 cos st + p>4d . a. What is the particle’s starting position st = 0d ? b. What are the points farthest to the left and right of the origin reached by the particle?

ƒsxd =

L

1 - cos x , x Z 0 x 0,

x = 0

have a derivative at x = 0 ? Explain. 17. a. For what values of a and b will ƒsxd = e

ax, ax 2 - bx + 3,

x 6 2 x Ú 2

be differentiable for all values of x? b. Discuss the geometry of the resulting graph of ƒ.

Chapter 3 18. a. For what values of a and b will g sxd = e

ax + b, ax 3 + x + 2b,

x … -1 x 7 -1

a. b.

be differentiable for all values of x? b. Discuss the geometry of the resulting graph of g. 19. Odd differentiable functions Is there anything special about the derivative of an odd differentiable function of x? Give reasons for your answer. 20. Even differentiable functions Is there anything special about the derivative of an even differentiable function of x? Give reasons for your answer. 21. Suppose that the functions ƒ and g are defined throughout an open interval containing the point x0 , that ƒ is differentiable at x0 , that ƒsx0 d = 0 , and that g is continuous at x0 . Show that the product ƒg is differentiable at x0 . This process shows, for example, that although ƒ x ƒ is not differentiable at x = 0 , the product x ƒ x ƒ is differentiable at x = 0 . 22. (Continuation of Exercise 21.) Use the result of Exercise 21 to show that the following functions are differentiable at x = 0 . a. ƒ x ƒ sin x d. hsxd = e

b. x 2>3 sin x x 2 sin s1>xd, 0,

3 c. 2 x s1 - cos xd

x Z 0 x = 0 x 2 sin s1>xd, 0,

dx

2

d 3suyd dx

3

=

d 2u du dy d 2y y + 2 + u 2. 2 dx dx dx dx

=

d 3u d 2u dy d 3y du d 2y y + 3 2 + u 3. + 3 3 2 dx dx dx dx dx dx

213

d suyd d nu d n - 1u dy Á = + n n y + n dx dx dx n - 1 dx nsn - 1d Á sn - k + 1d d n - ku d ky + k! dx n - k dx k n d y + Á + u n. dx The equations in parts (a) and (b) are special cases of the equation in part (c). Derive the equation in part (c) by mathematical induction, using n

c.

m m m! m! a b + a b = + . k!sm - kd! sk + 1d!sm - k - 1d! k k + 1 27. The period of a clock pendulum The period T of a clock pendulum (time for one full swing and back) is given by the formula T 2 = 4p2L>g , where T is measured in seconds, g = 32.2 ft>sec2 , and L, the length of the pendulum, is measured in feet. Find approximately a. the length of a clock pendulum whose period is T = 1 sec . b. the change dT in T if the pendulum in part (a) is lengthened 0.01 ft.

23. Is the derivative of hsxd = e

d 2suyd

Additional and Advanced Exercises

x Z 0 x = 0

continuous at x = 0 ? How about the derivative of k sxd = xhsxd ? Give reasons for your answers. 24. Suppose that a function ƒ satisfies the following conditions for all real values of x and y: i) ƒsx + yd = ƒsxd # ƒs yd .

ii) ƒsxd = 1 + xg sxd , where limx:0 g sxd = 1 . Show that the derivative ƒ¿sxd exists at every value of x and that ƒ¿sxd = ƒsxd . 25. The generalized product rule Use mathematical induction to prove that if y = u1 u2 Á un is a finite product of differentiable functions, then y is differentiable on their common domain and dy dun du2 Á du1 Á un + u1 = u un + Á + u1 u2 Á un - 1 . dx dx 2 dx dx 26. Leibniz’s rule for higher-order derivatives of products Leibniz’s rule for higher-order derivatives of products of differentiable functions says that

c. the amount the clock gains or loses in a day as a result of the period’s changing by the amount dT found in part (b). 28. The melting ice cube Assume that an ice cube retains its cubical shape as it melts. If we call its edge length s, its volume is V = s 3 and its surface area is 6s 2 . We assume that V and s are differentiable functions of time t. We assume also that the cube’s volume decreases at a rate that is proportional to its surface area. (This latter assumption seems reasonable enough when we think that the melting takes place at the surface: Changing the amount of surface changes the amount of ice exposed to melt.) In mathematical terms, dV = -k s6s 2 d, dt

k 7 0.

The minus sign indicates that the volume is decreasing. We assume that the proportionality factor k is constant. (It probably depends on many things, such as the relative humidity of the surrounding air, the air temperature, and the incidence or absence of sunlight, to name only a few.) Assume a particular set of conditions in which the cube lost 1> 4 of its volume during the first hour, and that the volume is V0 when t = 0 . How long will it take the ice cube to melt?

4 APPLICATIONS OF DERIVATIVES OVERVIEW In this chapter we use derivatives to find extreme values of functions, to determine and analyze the shapes of graphs, and to solve equations numerically. We also introduce the idea of recovering a function from its derivative. The key to many of these applications is the Mean Value Theorem, which paves the way to integral calculus in Chapter 5.

Extreme Values of Functions

4.1

This section shows how to locate and identify extreme (maximum or minimum) values of a function from its derivative. Once we can do this, we can solve a variety of problems in which we find the optimal (best) way to do something in a given situation (see Section 4.6). Finding maximum and minimum values is one of the most important applications of the derivative.

DEFINITIONS Let ƒ be a function with domain D. Then ƒ has an absolute maximum value on D at a point c if ƒsxd … ƒscd

and an absolute minimum value on D at c if

y 1

ƒsxd Ú ƒscd

y sin x

y cos x

– 2

0

2

for all x in D.

x

–1

FIGURE 4.1 Absolute extrema for the sine and cosine functions on [-p>2, p>2] . These values can depend on the domain of a function.

214

for all x in D

Maximum and minimum values are called extreme values of the function ƒ. Absolute maxima or minima are also referred to as global maxima or minima. For example, on the closed interval [-p>2, p>2] the function ƒsxd = cos x takes on an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On the same interval, the function gsxd = sin x takes on a maximum value of 1 and a minimum value of -1 (Figure 4.1). Functions with the same defining rule or formula can have different extrema (maximum or minimum values), depending on the domain. We see this in the following example.

4.1

Extreme Values of Functions

215

EXAMPLE 1 The absolute extrema of the following functions on their domains can be seen in Figure 4.2. Each function has the same defining equation, y = x2, but the domains vary. Notice that a function might not have a maximum or minimum if the domain is unbounded or fails to contain an endpoint.

y x2

Function rule

Domain D

Absolute extrema on D

(a) y = x 2

s - q, q d

(b) y = x 2

[0, 2]

(c) y = x 2

(0, 2]

No absolute maximum Absolute minimum of 0 at x = 0 Absolute maximum of 4 at x = 2 Absolute minimum of 0 at x = 0 Absolute maximum of 4 at x = 2 No absolute minimum

(d) y = x 2

(0, 2)

y x2

y

D (– , )

FIGURE 4.2

HISTORICAL BIOGRAPHY Daniel Bernoulli (1700–1789)

y x2

y

x

2 (b) abs max and min

y x2

y

D (0, 2]

D [0, 2]

2 (a) abs min only

No absolute extrema

x

D (0, 2)

2 (c) abs max only

y

x

2 (d) no max or min

x

Graphs for Example 1.

Some of the functions in Example 1 did not have a maximum or a minimum value. The following theorem asserts that a function which is continuous at every point of a finite closed interval [a, b] has an absolute maximum and an absolute minimum value on the interval. We look for these extreme values when we graph a function.

THEOREM 1—The Extreme Value Theorem If ƒ is continuous on a closed interval [a, b], then ƒ attains both an absolute maximum value M and an absolute minimum value m in [a, b]. That is, there are numbers x1 and x2 in [a, b] with ƒsx1 d = m, ƒsx2 d = M, and m … ƒsxd … M for every other x in [a, b].

The proof of the Extreme Value Theorem requires a detailed knowledge of the real number system (see Appendix 7) and we will not give it here. Figure 4.3 illustrates possible locations for the absolute extrema of a continuous function on a closed interval [a, b]. As we observed for the function y = cos x, it is possible that an absolute minimum (or absolute maximum) may occur at two or more different points of the interval. The requirements in Theorem 1 that the interval be closed and finite, and that the function be continuous, are key ingredients. Without them, the conclusion of the theorem

216

Chapter 4: Applications of Derivatives

(x2, M) y f (x)

y f (x)

M

M x1

a

x2

b

m

m

x

a

b

x

Maximum and minimum at endpoints

(x1, m) Maximum and minimum at interior points

y f (x) y f (x)

M

m

m a

M

x2

b

x

Maximum at interior point, minimum at endpoint

a

x1

b

x

Minimum at interior point, maximum at endpoint

FIGURE 4.3 Some possibilities for a continuous function’s maximum and minimum on a closed interval [a, b].

need not hold. Example 1 shows that an absolute extreme value may not exist if the interval fails to be both closed and finite. Figure 4.4 shows that the continuity requirement cannot be omitted.

y No largest value 1 yx 0 x1 0

1 Smallest value

Local (Relative) Extreme Values x

FIGURE 4.4 Even a single point of discontinuity can keep a function from having either a maximum or minimum value on a closed interval. The function x, 0 … x 6 1 0, x = 1 is continuous at every point of [0, 1] except x = 1 , yet its graph over [0, 1] does not have a highest point. y = e

Figure 4.5 shows a graph with five points where a function has extreme values on its domain [a, b]. The function’s absolute minimum occurs at a even though at e the function’s value is smaller than at any other point nearby. The curve rises to the left and falls to the right around c, making ƒ(c) a maximum locally. The function attains its absolute maximum at d. We now define what we mean by local extrema.

DEFINITIONS A function ƒ has a local maximum value at a point c within its domain D if ƒsxd … ƒscd for all x H D lying in some open interval containing c. A function ƒ has a local minimum value at a point c within its domain D if ƒsxd Ú ƒscd for all x H D lying in some open interval containing c.

If the domain of ƒ is the closed interval [a, b], then ƒ has a local maximum at the endpoint x = a, if ƒ(x) … ƒ(a) for all x in some half-open interval [a, a + d), d 7 0. Likewise, ƒ has a local maximum at an interior point x = c if ƒ(x) … ƒ(c) for all x in some open interval (c - d, c + d), d 7 0, and a local maximum at the endpoint x = b if ƒ(x) … ƒ(b) for all x in some half-open interval (b - d, b], d 7 0. The inequalities are reversed for local minimum values. In Figure 4.5, the function ƒ has local maxima at c and d and local minima at a, e, and b. Local extrema are also called relative extrema. Some functions can have infinitely many local extrema, even over a finite interval. One example is the function ƒ(x) = sin (1>x) on the interval (0, 1]. (We graphed this function in Figure 2.40.)

4.1

217

Extreme Values of Functions

Absolute maximum No greater value of f anywhere. Also a local maximum.

Local maximum No greater value of f nearby.

Local minimum No smaller value of f nearby.

y f (x)

Absolute minimum No smaller value of f anywhere. Also a local minimum.

Local minimum No smaller value of f nearby. a

c

e

d

b

x

FIGURE 4.5 How to identify types of maxima and minima for a function with domain a … x … b.

An absolute maximum is also a local maximum. Being the largest value overall, it is also the largest value in its immediate neighborhood. Hence, a list of all local maxima will automatically include the absolute maximum if there is one. Similarly, a list of all local minima will include the absolute minimum if there is one.

Finding Extrema Local maximum value

The next theorem explains why we usually need to investigate only a few values to find a function’s extrema. y f (x)

THEOREM 2—The First Derivative Theorem for Local Extreme Values If ƒ has a local maximum or minimum value at an interior point c of its domain, and if ƒ¿ is defined at c, then ƒ¿scd = 0. Secant slopes 0 (never negative)

x

Secant slopes 0 (never positive)

c

x

x

FIGURE 4.6 A curve with a local maximum value. The slope at c, simultaneously the limit of nonpositive numbers and nonnegative numbers, is zero.

Proof To prove that ƒ¿scd is zero at a local extremum, we show first that ƒ¿scd cannot be positive and second that ƒ¿scd cannot be negative. The only number that is neither positive nor negative is zero, so that is what ƒ¿scd must be. To begin, suppose that ƒ has a local maximum value at x = c (Figure 4.6) so that ƒsxd - ƒscd … 0 for all values of x near enough to c. Since c is an interior point of ƒ’s domain, ƒ¿scd is defined by the two-sided limit lim

x:c

ƒsxd - ƒscd x - c .

This means that the right-hand and left-hand limits both exist at x = c and equal ƒ¿scd. When we examine these limits separately, we find that ƒ¿scd = lim+

ƒsxd - ƒscd … 0. x - c

ƒ¿scd = lim-

ƒsxd - ƒscd Ú 0. x - c

x:c

Because sx - cd 7 0 and ƒsxd … ƒscd

(1)

Similarly, x:c

Because sx - cd 6 0 and ƒsxd … ƒscd

(2)

Together, Equations (1) and (2) imply ƒ¿scd = 0. This proves the theorem for local maximum values. To prove it for local minimum values, we simply use ƒsxd Ú ƒscd, which reverses the inequalities in Equations (1) and (2).

218

Chapter 4: Applications of Derivatives

Theorem 2 says that a function’s first derivative is always zero at an interior point where the function has a local extreme value and the derivative is defined. Hence the only places where a function ƒ can possibly have an extreme value (local or global) are

y y x3 1

–1

0

1

x

1. 2. 3.

interior points where ƒ¿ = 0, interior points where ƒ¿ is undefined, endpoints of the domain of ƒ.

At x = c and x = e in Fig. 4.5 At x = d in Fig. 4.5 At x = a and x = b in Fig. 4.5

The following definition helps us to summarize.

–1 (a)

DEFINITION An interior point of the domain of a function ƒ where ƒ¿ is zero or undefined is a critical point of ƒ.

y 1 y x1/3 –1

0

1

x

–1 (b)

FIGURE 4.7 Critical points without extreme values. (a) y¿ = 3x 2 is 0 at x = 0 , but y = x 3 has no extremum there. (b) y¿ = s1>3dx -2>3 is undefined at x = 0 , but y = x 1>3 has no extremum there.

Thus the only domain points where a function can assume extreme values are critical points and endpoints. However, be careful not to misinterpret what is being said here. A function may have a critical point at x = c without having a local extreme value there. For instance, both of the functions y = x 3 and y = x 1>3 have critical points at the origin, but neither function has a local extreme value at the origin. Instead, each function has a point of inflection there (see Figure 4.7). We define and explore inflection points in Section 4.4. Most problems that ask for extreme values call for finding the absolute extrema of a continuous function on a closed and finite interval. Theorem 1 assures us that such values exist; Theorem 2 tells us that they are taken on only at critical points and endpoints. Often we can simply list these points and calculate the corresponding function values to find what the largest and smallest values are, and where they are located. Of course, if the interval is not closed or not finite (such as a 6 x 6 b or a 6 x 6 q), we have seen that absolute extrema need not exist. If an absolute maximum or minimum value does exist, it must occur at a critical point or at an included right- or left-hand endpoint of the interval.

How to Find the Absolute Extrema of a Continuous Function ƒ on a Finite Closed Interval 1. Evaluate ƒ at all critical points and endpoints. 2. Take the largest and smallest of these values.

EXAMPLE 2

Find the absolute maximum and minimum values of ƒsxd = x 2 on

[-2, 1]. The function is differentiable over its entire domain, so the only critical point is where ƒ¿sxd = 2x = 0, namely x = 0. We need to check the function’s values at x = 0 and at the endpoints x = -2 and x = 1: Solution

Critical point value: ƒs0d = 0 Endpoint values:

ƒs -2d = 4 ƒs1d = 1.

The function has an absolute maximum value of 4 at x = -2 and an absolute minimum value of 0 at x = 0.

EXAMPLE 3 Find the absolute maximum and minimum values of ƒ(x) = 10x (2 - ln x) on the interval [1, e 2].

4.1

Figure 4.8 suggests that ƒ has its absolute maximum value near x = 3 and its absolute minimum value of 0 at x = e 2. Let’s verify this observation. We evaluate the function at the critical points and endpoints and take the largest and smallest of the resulting values. The first derivative is

Solution

y 30

(e, 10e)

25 20

219

Extreme Values of Functions

(1, 20)

15

1 ƒ¿(x) = 10(2 - ln x) - 10x a x b = 10(1 - ln x).

10 5

(e 2, 0) 1

0

2

3

4

5

6

7

x

8

The only critical point in the domain [1, e 2] is the point x = e, where ln x = 1. The values of ƒ at this one critical point and at the endpoints are

FIGURE 4.8 The extreme values of ƒ(x) = 10x(2 - ln x) on [1, e 2] occur at x = e and x = e 2 (Example 3).

Critical point value:

ƒ(e) = 10e

Endpoint values:

ƒ(1) = 10(2 - ln 1) = 20 ƒ(e 2) = 10e 2(2 - 2 ln e) = 0.

We can see from this list that the function’s absolute maximum value is 10e L 27.2; it occurs at the critical interior point x = e. The absolute minimum value is 0 and occurs at the right endpoint x = e 2. Find the absolute maximum and minimum values of ƒsxd = x 2>3 on the

EXAMPLE 4 interval [-2, 3].

We evaluate the function at the critical points and endpoints and take the largest and smallest of the resulting values. The first derivative

Solution y

y x 2/3, –2 ≤ x ≤ 3

ƒ¿sxd =

Absolute maximum; also a local maximum

Local maximum 2

has no zeros but is undefined at the interior point x = 0. The values of ƒ at this one critical point and at the endpoints are

1 –2

–1

0

2 -1>3 2 = x 3 3 32 x

Critical point value: ƒs0d = 0 1 2 3 Absolute minimum; also a local minimum

x

Endpoint values:

3 ƒs -2d = s -2d2>3 = 2 4 3 ƒs3d = s3d2>3 = 2 9.

FIGURE 4.9 The extreme values of ƒsxd = x 2>3 on [-2, 3] occur at x = 0 and x = 3 (Example 4).

3 9 L 2.08, and it We can see from this list that the function’s absolute maximum value is 2 occurs at the right endpoint x = 3. The absolute minimum value is 0, and it occurs at the interior point x = 0 where the graph has a cusp (Figure 4.9).

Exercises 4.1 Finding Extrema from Graphs In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1. 1.

2.

y

3.

y f (x) y h(x)

y f (x) 0

a

c1

c2

b

x

y

y

y h(x)

0

4.

y

0

a

c

b

x

a

c

b

x 0

a

c

b

x

220

Chapter 4: Applications of Derivatives

5.

6.

y

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

y

y g(x)

y g(x)

15. ƒ(x) = ƒ x ƒ , -1 6 x 6 2 6 , -1 6 x 6 1 16. y = 2 x + 2 0

a

c

b

x

a

0

c

x

b

17. g(x) = e

-x, 0 … x 6 1 x - 1, 1 … x … 2

In Exercises 7–10, find the absolute extreme values and where they occur. 7.

8.

y

18. h(x) =

y 2

x

1

9.

0

–2

–1

10.

y

x

2

(1, 2)

2

–3

x

2 –1

x

ƒ (x)

a b c

0 0 5

x + 1,

L cos x,

12.

14.

x

ƒ(x)

a b c

does not exist 0 2

x

ƒ (x)

a b c

0 0 5

a b c

24. ƒsxd = 4 - x 2,

-3 … x … 1

1 , 0.5 … x … 2 x2 1 26. Fsxd = - x , -2 … x … -1

29. g sxd = 24 - x , 30. g sxd = - 25 - x ,

c

(b)

b (c)

c

a (d)

-

36. ƒstd = ƒ t - 5 ƒ ,

b

c

- 25 … x … 0

5p p … u … 2 6 p p ƒsud = tan u, - … u … 3 4 p 2p g sxd = csc x, … x … 3 3 p p g sxd = sec x, - … x … 3 6 ƒstd = 2 - ƒ t ƒ , -1 … t … 3

31. ƒsud = sin u,

37. g(x) = xe -x,

a

-2 … x … 1 2

35. (a)

-1 … x … 1 2

34. b

-1 … x … 8

28. hsxd = -3x 2>3,

does not exist does not exist 1.7

a

-4 … x … 1

25. Fsxd = -

ƒ (x)

x

-2 … x … 3

-1 … x … 2

33. b c

p 2

23. ƒsxd = x 2 - 1,

32.

a

0 … x …

Absolute Extrema on Finite Closed Intervals In Exercises 21–40, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

3 x, 27. hsxd = 2

13.

-1 … x 6 0

2 x - 5, 3 22. ƒsxd = - x - 4,

In Exercises 11–14, match the table with a graph. 11.

0 6 x 6 2p

21. ƒsxd =

x

2

0 … x … 4

y

5

0

20. ƒ(x) =

-1 … x 6 0

2x,

19. y = 3 sin x,

1 –1

L

1 x,

4 … t … 7

-1 … x … 1

38. h(x) = ln (x + 1),

0 … x … 3

1 39. ƒ(x) = x + ln x,

0.5 … x … 4

40. g(x) = e -x , 2

-2 … x … 1

4.1 In Exercises 41–44, find the function’s absolute maximum and minimum values and say where they are assumed. 41. ƒsxd = x 4>3,

-1 … x … 8

42. ƒsxd = x 5>3,

-1 … x … 8

43. g(ud = u ,

-32 … u … 1

3>5

44. hsud = 3u2>3,

Extreme Values of Functions

221

Theory and Examples 79. A minimum with no derivative The function ƒsxd = ƒ x ƒ has an absolute minimum value at x = 0 even though ƒ is not differentiable at x = 0 . Is this consistent with Theorem 2? Give reasons for your answer. 80. Even functions If an even function ƒ(x) has a local maximum value at x = c , can anything be said about the value of ƒ at x = -c ? Give reasons for your answer.

-27 … u … 8

Finding Critical Points In Exercises 45–52, determine all critical points for each function. 45. y = x 2 - 6x + 7

46. ƒ(x) = 6x 2 - x 3

47. ƒ(x) = x(4 - x) 3 2 49. y = x 2 + x

48. g(x) = (x - 1) 2(x - 3) 2 x2 50. ƒ(x) = x - 2

51. y = x 2 - 32 2x

52. g(x) = 22x - x 2

81. Odd functions If an odd function g(x) has a local minimum value at x = c , can anything be said about the value of g at x = -c ? Give reasons for your answer. 82. We know how to find the extreme values of a continuous function ƒ(x) by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers. 83. The function

Finding Extreme Values In Exercises 53–68, find the extreme values (absolute and local) of the function and where they occur.

V sxd = xs10 - 2xds16 - 2xd,

0 6 x 6 5,

models the volume of a box.

54. y = x 3 - 2x + 4

a. Find the extreme values of V.

55. y = x 3 + x 2 - 8x + 5

56. y = x 3(x - 5) 2

b. Interpret any values found in part (a) in terms of the volume of the box.

57. y = 2x 2 - 1 1 59. y = 3 2 1 - x2 x 61. y = 2 x + 1 63. y = e x + e -x

58. y = x - 4 2x

65. y = x ln x

66. y = x 2 ln x

67. y = cos-1 (x 2)

68. y = sin-1 (e x )

53. y = 2x 2 - 8x + 9

60. y = 23 + 2x - x

84. Cubic functions Consider the cubic function 2

x + 1 x 2 + 2x + 2 64. y = e x - e -x 62. y =

Local Extrema and Critical Points In Exercises 69–76, find the critical points, domain endpoints, and extreme values (absolute and local) for each function. 69. y = x

sx + 2d

70. y = x

2>3

4 - 2x, x + 1,

2

72. y = x 2 23 - x

71. y = x24 - x 2 73. y = e

sx - 4d

2>3

x … 1 x 7 1

-x 2 - 2x + 4, x … 75. y = e 2 -x + 6x - 4, x 7 15 1 1 - x2 - x + , 4 2 4 y = • 76. x 3 - 6x 2 + 8x,

74. y = e

3 - x, x 6 0 3 + 2x - x 2, x Ú 0

1 1 x … 1 x 7 1

In Exercises 77 and 78, give reasons for your answers. 77. Let ƒsxd = sx - 2d2>3 . a. Does ƒ¿s2d exist? b. Show that the only local extreme value of ƒ occurs at x = 2 . c. Does the result in part (b) contradict the Extreme Value Theorem? d. Repeat parts (a) and (b) for ƒsxd = sx - ad2>3 , replacing 2 by a. 78. Let ƒsxd = ƒ x 3 - 9x ƒ . a. Does ƒ¿s0d exist? c. Does ƒ¿s -3d exist?

ƒsxd = ax 3 + bx 2 + cx + d . a. Show that ƒ can have 0, 1, or 2 critical points. Give examples and graphs to support your argument. b. How many local extreme values can ƒ have? 85. Maximum height of a vertically moving body body moving vertically is given by s = -

1 2 gt + y0 t + s0, 2

The height of a

g 7 0,

with s in meters and t in seconds. Find the body’s maximum height. 86. Peak alternating current Suppose that at any given time t (in seconds) the current i (in amperes) in an alternating current circuit is i = 2 cos t + 2 sin t . What is the peak current for this circuit (largest magnitude)? T Graph the functions in Exercises 87–90. Then find the extreme values of the function on the interval and say where they occur. 87. ƒsxd = ƒ x - 2 ƒ + ƒ x + 3 ƒ , 88. gsxd = ƒ x - 1 ƒ - ƒ x - 5 ƒ , 89. hsxd = ƒ x + 2 ƒ - ƒ x - 3 ƒ , 90. ksxd = ƒ x + 1 ƒ + ƒ x - 3 ƒ ,

-5 … x … 5 -2 … x … 7 -q 6 x 6 q -q 6 x 6 q

COMPUTER EXPLORATIONS In Exercises 91–98, you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. a. Plot the function over the interval to see its general behavior there.

b. Does ƒ¿s3d exist?

b. Find the interior points where ƒ¿ = 0 . (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot ƒ¿ as well.

d. Determine all extrema of ƒ.

c. Find the interior points where ƒ¿ does not exist.

222

Chapter 4: Applications of Derivatives

d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function’s absolute extreme values on the interval and identify where they occur. 91. ƒsxd = x 4 - 8x 2 + 4x + 2,

[-20>25, 64>25]

92. ƒsxd = -x + 4x - 4x + 1, 4

3

93. ƒsxd = x 2>3s3 - xd,

[1, 15]

We know that constant functions have zero derivatives, but could there be a more complicated function whose derivative is always zero? If two functions have identical derivatives over an interval, how are the functions related? We answer these and other questions in this chapter by applying the Mean Value Theorem. First we introduce a special case, known as Rolle’s Theorem, which is used to prove the Mean Value Theorem. f '(c) 0

Rolle’s Theorem y f (x)

a

c

As suggested by its graph, if a differentiable function crosses a horizontal line at two different points, there is at least one point between them where the tangent to the graph is horizontal and the derivative is zero (Figure 4.10). We now state and prove this result.

x

b

(a)

THEOREM 3—Rolle’s Theorem Suppose that y = ƒsxd is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). If ƒsad = ƒsbd, then there is at least one number c in (a, b) at which ƒ¿scd = 0.

y f '(c1 ) 0

a

[0, 2p] 1 96. ƒsxd = x - sin x + , [0, 2p] 2 97. ƒ(x) = px 2e - 3x>2, [0, 5] 3>4

[-2, 2]

y

0

95. ƒsxd = 2x + cos x,

98. ƒ(x) = ln (2x + x sin x),

[-3>4, 3]

[-1, 10>3]

The Mean Value Theorem

4.2

0

94. ƒsxd = 2 + 2x - 3x 2>3,

c1

f '(c3 ) 0 y f (x)

f '(c2 ) 0

c2

c3

b

x

(b)

FIGURE 4.10 Rolle’s Theorem says that a differentiable curve has at least one horizontal tangent between any two points where it crosses a horizontal line. It may have just one (a), or it may have more (b).

HISTORICAL BIOGRAPHY Michel Rolle (1652–1719)

Proof Being continuous, ƒ assumes absolute maximum and minimum values on [a, b] by Theorem 1. These can occur only 1. 2. 3.

at interior points where ƒ¿ is zero, at interior points where ƒ¿ does not exist, at the endpoints of the function’s domain, in this case a and b.

By hypothesis, ƒ has a derivative at every interior point. That rules out possibility (2), leaving us with interior points where ƒ¿ = 0 and with the two endpoints a and b. If either the maximum or the minimum occurs at a point c between a and b, then ƒ¿scd = 0 by Theorem 2 in Section 4.1, and we have found a point for Rolle’s Theorem. If both the absolute maximum and the absolute minimum occur at the endpoints, then because ƒsad = ƒsbd it must be the case that ƒ is a constant function with ƒsxd = ƒsad = ƒsbd for every x H [a, b]. Therefore ƒ¿sxd = 0 and the point c can be taken anywhere in the interior (a, b). The hypotheses of Theorem 3 are essential. If they fail at even one point, the graph may not have a horizontal tangent (Figure 4.11). Rolle’s Theorem may be combined with the Intermediate Value Theorem to show when there is only one real solution of an equation ƒsxd = 0, as we illustrate in the next example.

EXAMPLE 1

Show that the equation x 3 + 3x + 1 = 0

has exactly one real solution.

4.2

y

y

y

y f (x)

a

y f (x)

b

x

a

(a) Discontinuous at an endpoint of [a, b]

y

223

The Mean Value Theorem

x0 b

y f(x)

x

(b) Discontinuous at an interior point of [a, b]

(1, 5)

a

x0

x

b

(c) Continuous on [a, b] but not differentiable at an interior point

FIGURE 4.11 There may be no horizontal tangent if the hypotheses of Rolle’s Theorem do not hold.

–1

Solution

y x 3 3x 1

1 0

1

We define the continuous function ƒsxd = x 3 + 3x + 1.

x

Since ƒ(-1) = -3 and ƒ(0) = 1, the Intermediate Value Theorem tells us that the graph of ƒ crosses the x-axis somewhere in the open interval (-1, 0). (See Figure 4.12.) The derivative

(–1, –3)

FIGURE 4.12 The only real zero of the polynomial y = x 3 + 3x + 1 is the one shown here where the curve crosses the x-axis between -1 and 0 (Example 1).

ƒ¿sxd = 3x 2 + 3 is never zero (because it is always positive). Now, if there were even two points x = a and x = b where ƒ(x) was zero, Rolle’s Theorem would guarantee the existence of a point x = c in between them where ƒ¿ was zero. Therefore, ƒ has no more than one zero. Our main use of Rolle’s Theorem is in proving the Mean Value Theorem.

The Mean Value Theorem Tangent parallel to chord

y Slope f'(c)

B Slope

f (b) f (a) ba

A 0

a y f(x)

The Mean Value Theorem, which was first stated by Joseph-Louis Lagrange, is a slanted version of Rolle’s Theorem (Figure 4.13). The Mean Value Theorem guarantees that there is a point where the tangent line is parallel to the chord AB.

c

b

x

FIGURE 4.13 Geometrically, the Mean Value Theorem says that somewhere between a and b the curve has at least one tangent parallel to chord AB.

THEOREM 4—The Mean Value Theorem Suppose y = ƒsxd is continuous on a closed interval [a, b] and differentiable on the interval’s interior (a, b). Then there is at least one point c in (a, b) at which ƒsbd - ƒsad = ƒ¿scd. b - a

(1)

Proof We picture the graph of ƒ and draw a line through the points A(a, ƒ(a)) and B(b, ƒ(b)). (See Figure 4.14.) The line is the graph of the function gsxd = ƒsad +

ƒsbd - ƒsad sx - ad b - a

(2)

(point-slope equation). The vertical difference between the graphs of ƒ and g at x is HISTORICAL BIOGRAPHY Joseph-Louis Lagrange (1736–1813)

hsxd = ƒsxd - gsxd = ƒsxd - ƒsad -

ƒsbd - ƒsad sx - ad. b - a

Figure 4.15 shows the graphs of ƒ, g, and h together.

(3)

224

Chapter 4: Applications of Derivatives

B(b, f (b))

y f (x)

y f (x)

B h(x)

A(a, f (a)) y g(x)

A

h(x) f (x) g(x)

a

y y 1 x 2, –1 x 1 1

–1

0

FIGURE 4.14 The graph of ƒ and the chord AB over the interval [a, b].

x

b

x

FIGURE 4.15 The chord AB is the graph of the function g(x). The function hsxd = ƒsxd - g sxd gives the vertical distance between the graphs of ƒ and g at x.

The function h satisfies the hypotheses of Rolle’s Theorem on [a, b]. It is continuous on [a, b] and differentiable on (a, b) because both ƒ and g are. Also, hsad = hsbd = 0 because the graphs of ƒ and g both pass through A and B. Therefore h¿scd = 0 at some point c H sa, bd. This is the point we want for Equation (1). To verify Equation (1), we differentiate both sides of Equation (3) with respect to x and then set x = c: h¿sxd = ƒ¿sxd -

ƒsbd - ƒsad b - a

Derivative of Eq. (3) . . .

h¿scd = ƒ¿scd -

ƒsbd - ƒsad b - a

. . . with x = c

0 = ƒ¿scd -

ƒsbd - ƒsad b - a

h¿scd = 0

y B(2, 4)

3 y x2 2

ƒ¿scd = 1

a

x

1

FIGURE 4.16 The function ƒsxd = 21 - x 2 satisfies the hypotheses (and conclusion) of the Mean Value Theorem on [1, 1] even though ƒ is not differentiable at -1 and 1.

4

x

b

ƒsbd - ƒsad , b - a

Rearranged

(1, 1)

which is what we set out to prove. 1

A(0, 0)

x

2

FIGURE 4.17 As we find in Example 2, c = 1 is where the tangent is parallel to the chord. s

Distance (ft)

400

s f (t)

A Physical Interpretation

240

80 0

The function ƒsxd = x 2 (Figure 4.17) is continuous for 0 … x … 2 and differentiable for 0 6 x 6 2. Since ƒs0d = 0 and ƒs2d = 4, the Mean Value Theorem says that at some point c in the interval, the derivative ƒ¿sxd = 2x must have the value s4 - 0d>s2 - 0d = 2. In this case we can identify c by solving the equation 2c = 2 to get c = 1. However, it is not always easy to find c algebraically, even though we know it always exists.

EXAMPLE 2

(8, 352)

320

160

The hypotheses of the Mean Value Theorem do not require ƒ to be differentiable at either a or b. Continuity at a and b is enough (Figure 4.16).

At this point, the car’s speed was 30 mph. t

5 Time (sec)

FIGURE 4.18 Distance versus elapsed time for the car in Example 3.

We can think of the number sƒsbd - ƒsadd>sb - ad as the average change in ƒ over [a, b] and ƒ¿scd as an instantaneous change. Then the Mean Value Theorem says that at some interior point the instantaneous change must equal the average change over the entire interval.

EXAMPLE 3 If a car accelerating from zero takes 8 sec to go 352 ft, its average velocity for the 8-sec interval is 352>8 = 44 ft>sec. The Mean Value Theorem says that at some point during the acceleration the speedometer must read exactly 30 mph (44 ft>sec) (Figure 4.18).

4.2

The Mean Value Theorem

225

Mathematical Consequences At the beginning of the section, we asked what kind of function has a zero derivative over an interval. The first corollary of the Mean Value Theorem provides the answer that only constant functions have zero derivatives.

COROLLARY 1 If ƒ¿sxd = 0 at each point x of an open interval (a, b), then ƒsxd = C for all x H sa, bd, where C is a constant.

Proof We want to show that ƒ has a constant value on the interval (a, b). We do so by showing that if x1 and x2 are any two points in (a, b) with x1 6 x2, then ƒsx1 d = ƒsx2 d. Now ƒ satisfies the hypotheses of the Mean Value Theorem on [x1 , x2]: It is differentiable at every point of [x1, x2] and hence continuous at every point as well. Therefore, ƒsx2 d - ƒsx1 d = ƒ¿scd x2 - x1 at some point c between x1 and x2. Since ƒ¿ = 0 throughout (a, b), this equation implies successively that ƒsx2 d - ƒsx1 d = 0, x2 - x1

ƒsx2 d - ƒsx1 d = 0,

and

ƒsx1 d = ƒsx2 d.

At the beginning of this section, we also asked about the relationship between two functions that have identical derivatives over an interval. The next corollary tells us that their values on the interval have a constant difference.

y

y 5 x2 1 C

COROLLARY 2 If ƒ¿sxd = g¿sxd at each point x in an open interval (a, b), then there exists a constant C such that ƒsxd = gsxd + C for all x H sa, bd. That is, ƒ - g is a constant function on (a, b).

C52 C51

Proof At each point x H sa, bd the derivative of the difference function h = ƒ - g is

C50

2

C 5 –1

h¿sxd = ƒ¿sxd - g¿sxd = 0.

C 5 –2

Thus, hsxd = C on (a, b) by Corollary 1. That is, ƒsxd - gsxd = C on (a, b), so ƒsxd = gsxd + C.

1 0

x

–1 –2

FIGURE 4.19 From a geometric point of view, Corollary 2 of the Mean Value Theorem says that the graphs of functions with identical derivatives on an interval can differ only by a vertical shift there. The graphs of the functions with derivative 2x are the parabolas y = x 2 + C , shown here for selected values of C.

Corollaries 1 and 2 are also true if the open interval (a, b) fails to be finite. That is, they remain true if the interval is sa, q d, s - q , bd, or s - q , q d. Corollary 2 plays an important role when we discuss antiderivatives in Section 4.8. It tells us, for instance, that since the derivative of ƒsxd = x 2 on s - q , q d is 2x, any other function with derivative 2x on s - q , q d must have the formula x 2 + C for some value of C (Figure 4.19).

EXAMPLE 4 Find the function ƒ(x) whose derivative is sin x and whose graph passes through the point (0, 2). Since the derivative of gsxd = -cos x is g¿(x) = sin x, we see that ƒ and g have the same derivative. Corollary 2 then says that ƒsxd = -cos x + C for some

Solution

226

Chapter 4: Applications of Derivatives

constant C. Since the graph of ƒ passes through the point (0, 2), the value of C is determined from the condition that ƒs0d = 2: ƒs0d = -cos s0d + C = 2,

so

C = 3.

The function is ƒsxd = -cos x + 3.

Finding Velocity and Position from Acceleration We can use Corollary 2 to find the velocity and position functions of an object moving along a vertical line. Assume the object or body is falling freely from rest with acceleration 9.8 m>sec2. We assume the position s(t) of the body is measured positive downward from the rest position (so the vertical coordinate line points downward, in the direction of the motion, with the rest position at 0). We know that the velocity y(t) is some function whose derivative is 9.8. We also know that the derivative of gstd = 9.8t is 9.8. By Corollary 2, ystd = 9.8t + C for some constant C. Since the body falls from rest, ys0d = 0. Thus 9.8s0d + C = 0,

and

C = 0.

The velocity function must be ystd = 9.8t. What about the position function s(t)? We know that s(t) is some function whose derivative is 9.8t. We also know that the derivative of ƒstd = 4.9t 2 is 9.8t. By Corollary 2, sstd = 4.9t 2 + C for some constant C. Since ss0d = 0, 4.9s0d2 + C = 0,

and

C = 0.

The position function is sstd = 4.9t 2 until the body hits the ground. The ability to find functions from their rates of change is one of the very powerful tools of calculus. As we will see, it lies at the heart of the mathematical developments in Chapter 5.

Proofs of the Laws of Logarithms The algebraic properties of logarithms were stated in Section 1.6. We can prove those properties by applying Corollary 2 of the Mean Value Theorem to each of them. The steps in the proofs are similar to those used in solving problems involving logarithms. Proof that ln bx ln b ln x have the same derivative:

The argument starts by observing that ln bx and ln x

d b d 1 ln (bx) = = x = ln x. dx bx dx According to Corollary 2 of the Mean Value Theorem, then, the functions must differ by a constant, which means that ln bx = ln x + C for some C. Since this last equation holds for all positive values of x, it must hold for x = 1. Hence, ln (b # 1) = ln 1 + C ln b = 0 + C C = ln b.

ln 1 = 0

4.2

The Mean Value Theorem

227

By substituting, we conclude ln bx = ln b + ln x. Proof that ln x r r ln x values of x,

We use the same-derivative argument again. For all positive d 1 d r ln x r = r (x ) dx x dx 1 = r rx r - 1 x d 1 = r# x = (r ln x). dx

Chain Rule Derivative Power Rule

Since ln x r and r ln x have the same derivative, ln x r = r ln x + C for some constant C. Taking x to be 1 identifies C as zero, and we’re done. You are asked to prove the Quotient Rule for logarithms, b ln a x b = ln b - ln x, in Exercise 75. The Reciprocal Rule, ln (1>x) = -ln x, is a special case of the Quotient Rule, obtained by taking b = 1 and noting that ln 1 = 0.

Laws of Exponents The laws of exponents for the natural exponential e x are consequences of the algebraic properties of ln x. They follow from the inverse relationship between these functions.

Laws of Exponents for e x For all numbers x, x1, and x2, the natural exponential e x obeys the following laws: 1 1. e x1 # e x2 = e x1 + x2 2. e -x = x e e x1 3. x2 = e x1 - x2 4. (e x1) x2 = e x1x2 = (e x2) x1 e

Proof of Law 1 Let y1 = e x1

and

x1 = ln y1

and

y2 = e x2.

(4)

Then x2 = ln y2

Take logs of both sides of Eqs. (4).

x1 + x2 = ln y1 + ln y2 e

x1 + x2

= ln y1 y2

Product Rule for logarithms

= e = y1 y2 = e x1e x2.

Exponentiate. e ln u = u

ln y1 y2

The proof of Law 4 is similar. Laws 2 and 3 follow from Law 1 (Exercises 77 and 78).

228

Chapter 4: Applications of Derivatives

Exercises 4.2 Checking the Mean Value Theorem Find the value or values of c that satisfy the equation ƒsbd - ƒsad = ƒ¿scd b - a in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–8. 1. ƒsxd = x 2 + 2x - 1, 2. ƒsxd = x 2>3,

1 c , 2d 2

4. ƒsxd = 2x - 1, 5. ƒsxd = sin-1 x, 7. ƒsxd = x - x , 2

x 3, 8. g(x) = e 2 x ,

[-1, 2]

20. Show that a cubic polynomial can have at most three real zeros.

-2 … x … 0 0 6 x … 2

Show that the functions in Exercises 21–28 have exactly one zero in the given interval.

Which of the functions in Exercises 9–14 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. ,

[-1, 8]

10. ƒsxd = x

4>5

,

[0, 1]

11. ƒsxd = 2xs1 - xd, 12. ƒsxd =

L

sin x x , 0,

iv) y = x 3 - 33x 2 + 216x = xsx - 9dsx - 24d

19. Show that if ƒ– 7 0 throughout an interval [a, b], then ƒ¿ has at most one zero in [a, b]. What if ƒ– 6 0 throughout [a, b] instead?

[2, 4]

9. ƒsxd = x

iii) y = x 3 - 3x 2 + 4 = sx + 1dsx - 2d2

18. Suppose that ƒ– is continuous on [a, b] and that ƒ has three zeros in the interval. Show that ƒ– has at least one zero in (a, b). Generalize this result.

[-1, 1]

2>3

ii) y = x 2 + 8x + 15

nx n - 1 + sn - 1dan - 1x n - 2 + Á + a1.

[1, 3]

6. ƒsxd = ln (x - 1),

i) y = x 2 - 4

b. Use Rolle’s Theorem to prove that between every two zeros of x n + an - 1x n - 1 + Á + a1 x + a0 there lies a zero of

[0, 1]

1 3. ƒsxd = x + x ,

3

[0, 1]

Roots (Zeros) 17. a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

21. ƒsxd = x 4 + 3x + 1,

[-2, -1]

4 + 7, x2

s - q , 0d

22. ƒsxd = x 3 +

s0, q d

23. g std = 2t + 21 + t - 4, 24. g std =

[0, 1]

u 25. r sud = u + sin2 a b - 8, 3

-p … x 6 0 x = 0

13. ƒ(x) = e

x - x, 2x 2 - 3x - 3,

14. ƒ(x) = e

2x - 3, 6x - x 2 - 7,

15. The function x, 0 … x 6 1 0, x = 1

is zero at x = 0 and x = 1 and differentiable on (0, 1), but its derivative on (0, 1) is never zero. How can this be? Doesn’t Rolle’s Theorem say the derivative has to be zero somewhere in (0, 1)? Give reasons for your answer. 16. For what values of a, m, and b does the function 3, ƒsxd = • -x 2 + 3x + a, mx + b,

1 + 5, u3

28. r sud = tan u - cot u - u,

0 … x … 2 2 6 x … 3

ƒsxd = e

27. r sud = sec u -

x = 0 0 6 x 6 1 1 … x … 2

satisfy the hypotheses of the Mean Value Theorem on the interval [0, 2]?

s -1, 1d

s - q, q d s - q, q d

26. r sud = 2u - cos2 u + 22,

-2 … x … - 1 -1 6 x … 0

2

1 + 21 + t - 3.1, 1 - t

s0, p>2d s0, p>2d

Finding Functions from Derivatives 29. Suppose that ƒs -1d = 3 and that ƒ¿sxd = 0 for all x. Must ƒsxd = 3 for all x? Give reasons for your answer. 30. Suppose that ƒs0d = 5 and that ƒ¿sxd = 2 for all x. Must ƒsxd = 2x + 5 for all x? Give reasons for your answer. 31. Suppose that ƒ¿sxd = 2x for all x. Find ƒ(2) if a. ƒs0d = 0

b. ƒs1d = 0

c. ƒs -2d = 3.

32. What can be said about functions whose derivatives are constant? Give reasons for your answer. In Exercises 33–38, find all possible functions with the given derivative. 33. a. y¿ = x

b. y¿ = x 2

c. y¿ = x 3

34. a. y¿ = 2x

b. y¿ = 2x - 1

c. y¿ = 3x 2 + 2x - 1

35. a. y¿ = -

1 x2

b. y¿ = 1 -

1 x2

c. y¿ = 5 +

1 x2

4.2

36. a. y¿ =

1

b. y¿ =

2 2x

1 2x t 2

37. a. y¿ = sin 2t

b. y¿ = cos

38. a. y¿ = sec2 u

b. y¿ = 2u

c. y¿ = 4x -

1 2x

c. y¿ = sin 2t + cos

t 2

c. y¿ = 2u - sec2 u

In Exercises 39–42, find the function with the given derivative whose graph passes through the point P. 39. ƒ¿sxd = 2x - 1, 40. g¿(x) =

Ps0, 0d

1 + 2x, x2

41. ƒ¿(x) = e ,

Ps0, 0d

43. y = 9.8t + 5,

ss0d = 10

44. y = 32t - 2,

ss0.5d = 4 ss0d = 0 s(p2) = 1

48. a = 9.8,

y(0) = 20, ys0d = -3,

s(0) = 5 ss0d = 0

49. a = -4 sin 2t,

ys0d = 2,

ss0d = -3

9 3t cos p , p2

ys0d = 0,

ss0d = -1

50. a =

What does the graph do? Why does the function behave this way? Give reasons for your answers. 60. Rolle’s Theorem a. Construct a polynomial ƒ(x) that has zeros at x = -2, -1, 0, 1, and 2 . b. Graph ƒ and its derivative ƒ¿ together. How is what you see related to Rolle’s Theorem? c. Do gsxd = sin x and its derivative g¿ illustrate the same phenomenon as ƒ and ƒ¿?

Exercises 47–50 give the acceleration a = d 2s>dt 2 , initial velocity, and initial position of a body moving on a coordinate line. Find the body’s position at time t. 47. a = e t,

58. The arithmetic mean of a and b The arithmetic mean of two numbers a and b is the number sa + bd>2 . Show that the value of c in the conclusion of the Mean Value Theorem for ƒsxd = x 2 on any interval [a, b] is c = sa + bd>2 .

ƒsxd = sin x sin sx + 2d - sin2 sx + 1d.

Finding Position from Velocity or Acceleration Exercises 43–46 give the velocity y = ds>dt and initial position of a body moving along a coordinate line. Find the body’s position at time t.

2t 2 46. y = p cos p ,

Theory and Examples 57. The geometric mean of a and b The geometric mean of two positive numbers a and b is the number 2ab . Show that the value of c in the conclusion of the Mean Value Theorem for ƒsxd = 1>x on an interval of positive numbers [a, b] is c = 2ab .

T 59. Graph the function

42. r¿std = sec t tan t - 1,

45. y = sin pt,

229

P(- 1, 1)

3 P a0, b 2

2x

The Mean Value Theorem

Applications 51. Temperature change It took 14 sec for a mercury thermometer to rise from -19°C to 100 C when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the rate of 8.5 C>sec. 52. A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?

61. Unique solution Assume that ƒ is continuous on [a, b] and differentiable on (a, b). Also assume that ƒ(a) and ƒ(b) have opposite signs and that ƒ¿ Z 0 between a and b. Show that ƒsxd = 0 exactly once between a and b. 62. Parallel tangents Assume that ƒ and g are differentiable on [a, b] and that ƒsad = g sad and ƒsbd = g sbd . Show that there is at least one point between a and b where the tangents to the graphs of ƒ and g are parallel or the same line. Illustrate with a sketch. 63. Suppose that ƒ¿(x) … 1 for 1 … x … 4. Show that ƒ(4) ƒ(1) … 3. 64. Suppose that 0 6 ƒ¿(x) 6 1>2 for all x-values. Show that ƒ(-1) 6 ƒ(1) 6 2 + ƒ(-1). 65. Show that ƒ cos x - 1 ƒ … ƒ x ƒ for all x-values. (Hint: Consider ƒ(t) = cos t on [0, x].) 66. Show that for any numbers a and b, the sine inequality ƒ sin b - sin a ƒ … ƒ b - a ƒ is true. 67. If the graphs of two differentiable functions ƒ(x) and g(x) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer.

53. Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme’s speed exceeded 7.5 knots (sea miles per hour).

68. If ƒ ƒ(w) - ƒ(x) ƒ … ƒ w - x ƒ for all values w and x and ƒ is a differentiable function, show that -1 … ƒ¿(x) … 1 for all x-values.

54. A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.

70. Let ƒ be a function defined on an interval [a, b]. What conditions could you place on ƒ to guarantee that

55. Show that at some instant during a 2-hour automobile trip the car’s speedometer reading will equal the average speed for the trip. 56. Free fall on the moon On our moon, the acceleration of gravity is 1.6 m>sec2 . If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later?

69. Assume that ƒ is differentiable on a … x … b and that ƒsbd 6 ƒsad. Show that ƒ¿ is negative at some point between a and b.

min ƒ¿ …

ƒsbd - ƒsad … max ƒ¿, b - a

where min ƒ¿ and max ƒ¿ refer to the minimum and maximum values of ƒ¿ on [a, b]? Give reasons for your answers.

230

Chapter 4: Applications of Derivatives

T 71. Use the inequalities in Exercise 70 to estimate ƒs0.1d if ƒ¿sxd = 1>s1 + x 4 cos xd for 0 … x … 0.1 and ƒs0d = 1 . T 72. Use the inequalities in Exercise 70 to estimate ƒs0.1d if ƒ¿sxd = 1>s1 - x 4 d for 0 … x … 0.1 and ƒs0d = 2 . 73. Let ƒ be differentiable at every value of x and suppose that ƒs1d = 1 , that ƒ¿ 6 0 on s - q , 1d , and that ƒ¿ 7 0 on s1, q d. a. Show that ƒsxd Ú 1 for all x. b. Must ƒ¿s1d = 0 ? Explain. 74. Let ƒsxd = px 2 + qx + r be a quadratic function defined on a closed interval [a, b]. Show that there is exactly one point c in (a, b) at which ƒ satisfies the conclusion of the Mean Value Theorem.

4.3

75. Use the same-derivative argument, as was done to prove the Product and Power Rules for logarithms, to prove the Quotient Rule property. 76. Use the same-derivative argument to prove the identities a. tan-1 x + cot-1 x =

p 2

b. sec-1 x + csc-1 x =

p 2

77. Starting with the equation e x1e x2 = e x1 + x2, derived in the text, show that e -x = 1>e x for any real number x. Then show that e x1>e x2 = e x1 - x2 for any numbers x1 and x2. 78. Show that (e x1) x2 = e x1 x2 = (e x2) x1 for any numbers x1 and x2.

Monotonic Functions and the First Derivative Test In sketching the graph of a differentiable function, it is useful to know where it increases (rises from left to right) and where it decreases (falls from left to right) over an interval. This section gives a test to determine where it increases and where it decreases. We also show how to test the critical points of a function to identify whether local extreme values are present.

Increasing Functions and Decreasing Functions As another corollary to the Mean Value Theorem, we show that functions with positive derivatives are increasing functions and functions with negative derivatives are decreasing functions. A function that is increasing or decreasing on an interval is said to be monotonic on the interval.

COROLLARY 3

Suppose that ƒ is continuous on [a, b] and differentiable on

(a, b). If ƒ¿sxd 7 0 at each point x H sa, bd, then ƒ is increasing on [a, b]. If ƒ¿sxd 6 0 at each point x H sa, bd, then ƒ is decreasing on [a, b]. Proof Let x1 and x2 be any two points in [a, b] with x1 6 x2 . The Mean Value Theorem applied to ƒ on [x1, x2] says that ƒsx2 d - ƒsx1 d = ƒ¿scdsx2 - x1 d for some c between x1 and x2 . The sign of the right-hand side of this equation is the same as the sign of ƒ ¿scd because x2 - x1 is positive. Therefore, ƒsx2 d 7 ƒsx1 d if ƒ¿ is positive on (a, b) and ƒsx2 d 6 ƒsx1 d if ƒ¿ is negative on (a, b). Corollary 3 tells us that f (x) = 1x is increasing on the interval [0, b] for any b 7 0 because f ¿(x) = 1> 1x is positive on (0, b). The derivative does not exist at x = 0, but Corollary 3 still applies. The corollary is valid for infinite as well as finite intervals, so f (x) = 1x is increasing on [0, q ). To find the intervals where a function ƒ is increasing or decreasing, we first find all of the critical points of ƒ. If a 6 b are two critical points for ƒ, and if the derivative ƒ¿ is continuous but never zero on the interval (a, b), then by the Intermediate Value Theorem applied to ƒ¿ , the derivative must be everywhere positive on (a, b), or everywhere negative there. One way we can determine the sign of ƒ¿ on (a, b) is simply by evaluating the derivative at a single point c in (a, b). If ƒ¿(c) 7 0, then ƒ¿(x) 7 0 for all x in (a, b) so ƒ is increasing on [a, b] by Corollary 3; if ƒ¿(c) 6 0, then ƒ is decreasing on [a, b]. The next example illustrates how we use this procedure.

4.3 y y x3 – 12x – 5

Monotonic Functions and the First Derivative Test

231

20

Find the critical points of ƒsxd = x 3 - 12x - 5 and identify the intervals on which ƒ is increasing and on which ƒ is decreasing.

10

Solution

EXAMPLE 1

(–2, 11)

–4 –3 –2 –1 0

1

2

3

The function ƒ is everywhere continuous and differentiable. The first derivative ƒ¿sxd = 3x 2 - 12 = 3sx 2 - 4d = 3sx + 2dsx - 2d

x

4

is zero at x = -2 and x = 2. These critical points subdivide the domain of ƒ to create nonoverlapping open intervals s - q , -2d, s -2, 2d, and s2, q d on which ƒ¿ is either positive or negative. We determine the sign of ƒ¿ by evaluating ƒ¿ at a convenient point in each subinterval. The behavior of ƒ is determined by then applying Corollary 3 to each subinterval. The results are summarized in the following table, and the graph of ƒ is given in Figure 4.20.

–10

–20 (2, –21)

FIGURE 4.20 The function ƒsxd = x 3 - 12x - 5 is monotonic on three separate intervals (Example 1).

- q 6 x 6 -2 ƒ¿s -3d = 15 + increasing

Interval ƒ œ evaluated Sign of ƒ œ Behavior of ƒ

2 6 x 6 q ƒ¿s3d = 15 + increasing

-2 6 x 6 2 ƒ¿s0d = -12 decreasing

x –3

–2

–1

0

1

2

3

We used “strict” less-than inequalities to specify the intervals in the summary table for Example 1. Corollary 3 says that we could use … inequalities as well. That is, the function ƒ in the example is increasing on - q 6 x … -2, decreasing on -2 … x … 2, and increasing on 2 … x 6 q . We do not talk about whether a function is increasing or decreasing at a single point.

HISTORICAL BIOGRAPHY

First Derivative Test for Local Extrema

Edmund Halley (1656–1742)

In Figure 4.21, at the points where ƒ has a minimum value, ƒ¿ 6 0 immediately to the left and ƒ¿ 7 0 immediately to the right. (If the point is an endpoint, there is only one side to consider.) Thus, the function is decreasing on the left of the minimum value and it is increasing on its right. Similarly, at the points where ƒ has a maximum value, ƒ¿ 7 0 immediately to the left and ƒ¿ 6 0 immediately to the right. Thus, the function is increasing on the left of the maximum value and decreasing on its right. In summary, at a local extreme point, the sign of ƒ¿sxd changes. Absolute max f ' undefined Local max f'0 No extremum f'0 f'0

y f(x)

No extremum f'0

f'0

f'0

f'0

f'0 Local min

Local min f'0

f'0 Absolute min a

c1

c2

c3

c4

c5

b

x

FIGURE 4.21 The critical points of a function locate where it is increasing and where it is decreasing. The first derivative changes sign at a critical point where a local extremum occurs.

These observations lead to a test for the presence and nature of local extreme values of differentiable functions.

232

Chapter 4: Applications of Derivatives

First Derivative Test for Local Extrema Suppose that c is a critical point of a continuous function ƒ, and that ƒ is differentiable at every point in some interval containing c except possibly at c itself. Moving across this interval from left to right, 1. if ƒ¿ changes from negative to positive at c, then ƒ has a local minimum at c; 2. if ƒ¿ changes from positive to negative at c, then ƒ has a local maximum at c; 3. if ƒ¿ does not change sign at c (that is, ƒ¿ is positive on both sides of c or negative on both sides), then ƒ has no local extremum at c.

The test for local extrema at endpoints is similar, but there is only one side to consider. Proof of the First Derivative Test Part (1). Since the sign of ƒ¿ changes from negative to positive at c, there are numbers a and b such that a 6 c 6 b, ƒ¿ 6 0 on (a, c), and ƒ¿ 7 0 on (c, b). If x H sa, cd, then ƒscd 6 ƒsxd because ƒ¿ 6 0 implies that ƒ is decreasing on [a, c]. If x H sc, bd, then ƒscd 6 ƒsxd because ƒ¿ 7 0 implies that ƒ is increasing on [c, b]. Therefore, ƒsxd Ú ƒscd for every x H sa, bd. By definition, ƒ has a local minimum at c. Parts (2) and (3) are proved similarly.

EXAMPLE 2

Find the critical points of ƒsxd = x 1>3sx - 4d = x 4>3 - 4x 1>3.

Identify the intervals on which ƒ is increasing and decreasing. Find the function’s local and absolute extreme values. Solution The function ƒ is continuous at all x since it is the product of two continuous functions, x 1>3 and sx - 4d. The first derivative

d 4>3 4 4 - 4x 1>3R = x 1>3 - x -2>3 x 3 3 dx Q 4sx - 1d 4 = x -2>3 Qx - 1R = 3 3x 2>3

ƒ¿sxd =

is zero at x = 1 and undefined at x = 0. There are no endpoints in the domain, so the critical points x = 0 and x = 1 are the only places where ƒ might have an extreme value. The critical points partition the x-axis into intervals on which ƒ¿ is either positive or negative. The sign pattern of ƒ¿ reveals the behavior of ƒ between and at the critical points, as summarized in the following table. Interval Sign of ƒ œ

y 4

Behavior of ƒ y x1/3(x 4)

2 1 –1 0 –1

1

2

3

4

x

–2 –3

(1, 3)

FIGURE 4.22 The function ƒsxd = x 1>3 sx - 4d decreases when x 6 1 and increases when x 7 1 (Example 2).

x 6 0 decreasing

0 6 x 6 1 decreasing

x 7 1 + increasing x

–1

0

1

2

Corollary 3 to the Mean Value Theorem tells us that ƒ decreases on s - q , 0], decreases on [0, 1], and increases on [1, q d. The First Derivative Test for Local Extrema tells us that ƒ does not have an extreme value at x = 0 (ƒ¿ does not change sign) and that ƒ has a local minimum at x = 1 (ƒ¿ changes from negative to positive). The value of the local minimum is ƒs1d = 11>3s1 - 4d = -3. This is also an absolute minimum since ƒ is decreasing on s - q , 1] and increasing on [1, q d. Figure 4.22 shows this value in relation to the function’s graph. Note that limx:0 ƒ¿sxd = - q , so the graph of ƒ has a vertical tangent at the origin.

4.3

233

Monotonic Functions and the First Derivative Test

EXAMPLE 3

Find the critical points of ƒ(x) = (x 2 - 3)e x. Identify the intervals on which ƒ is increasing and decreasing. Find the function’s local and absolute extreme values. The function ƒ is continuous and differentiable for all real numbers, so the critical points occur only at the zeros of ƒ¿.

Solution

Using the Derivative Product Rule, we find the derivative d x d 2 e + (x - 3) # e x ƒ¿(x) = (x 2 - 3) # dx dx = (x 2 - 3) # e x + (2x) # e x = (x 2 + 2x - 3)e x. Since e x is never zero, the first derivative is zero if and only if x 2 + 2x - 3 = 0 (x + 3)(x - 1) = 0. y

y (x 2 3)e x

The zeros x = -3 and x = 1 partition the x-axis into intervals as follows.

4 3

Interval Sign of ƒ œ

2 1 –5 –4 –3 –2 –1

–1

1

2

3

x

Behavior of ƒ

x 6 -3 + increasing

1 6 x + increasing x

–4

–2

-3 6 x 6 1 decreasing

–3

–2

–1

0

1

2

3

–3 –4

We can see from the table that there is a local maximum (about 0.299) at x = -3 and a local minimum (about -5.437) at x = 1. The local minimum value is also an absolute minimum because ƒ(x) 7 0 for ƒ x ƒ 7 23. There is no absolute maximum. The function increases on (- q , -3) and (1, q ) and decreases on (-3, 1). Figure 4.23 shows the graph.

–5 –6

FIGURE 4.23 The graph of ƒ(x) = (x 2 - 3) e x (Example 3).

Exercises 4.3 Analyzing Functions from Derivatives Answer the following questions about the functions whose derivatives are given in Exercises 1–14: a. What are the critical points of ƒ? b. On what intervals is ƒ increasing or decreasing? c. At what points, if any, does ƒ assume local maximum and minimum values? 1. ƒ¿sxd = xsx - 1d

2. ƒ¿sxd = sx - 1dsx + 2d

3. ƒ¿sxd = sx - 1d2sx + 2d

4. ƒ¿sxd = sx - 1d2sx + 2d2

5. ƒ¿(x) = (x - 1)e -x 6. ƒ¿sxd = sx - 7dsx + 1dsx + 5d x 2(x - 1) , x Z -2 7. ƒ¿(x) = x + 2 (x - 2)(x + 4) , x Z -1, 3 8. ƒ¿(x) = (x + 1)(x - 3) 6 4 , 9. ƒ¿(x) = 1 - 2 , x Z 0 10. ƒ¿(x) = 3 x 2x

11. ƒ¿sxd = x -1>3sx + 2d

13. ƒ¿(x) = (sin x - 1)(2 cos x + 1), 0 … x … 2p 14. ƒ¿(x) = (sin x + cos x)(sin x - cos x), 0 … x … 2p Identifying Extrema In Exercises 15–44: a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function’s local and absolute extreme values, if any, saying where they occur. 15.

16.

y 2 1

–3 –2 –1 –1

x Z 0

12. ƒ¿sxd = x -1>2sx - 3d

–2

2

y 5 f (x) 1

2

y

1 3

x

–3 –2 –1 –1 –2

y 5 f (x) 1

2

3

x

234

Chapter 4: Applications of Derivatives

17.

18.

y

57. ƒsxd = sin 2x,

y

0 … x … p

58. ƒsxd = sin x - cos x, 2

y 5 f (x)

1 –3 –2 –1 –1

1

2

2

y 5 f (x) x

3

1

–3 –2 –1 –1

1

2

3

–2

–2

19. g std = -t 2 - 3t + 3

20. g std = -3t 2 + 9t + 5

x

0 … x … 2p

59. ƒsxd = 23 cos x + sin x, -p 60. ƒsxd = -2x + tan x, 2 x x 61. ƒsxd = - 2 sin , 0 … 2 2 62. ƒsxd = -2 cos x - cos2 x, 63. ƒsxd = csc x - 2 cot x, 2

0 … x … 2p p 6 x 6 2 x … 2p -p … x … p

0 6 x 6 p -p p 6 x 6 2 2

21. hsxd = -x 3 + 2x 2

22. hsxd = 2x 3 - 18x

64. ƒsxd = sec x - 2 tan x,

23. ƒsud = 3u2 - 4u3

24. ƒsud = 6u - u3

25. ƒsrd = 3r + 16r

26. hsrd = sr + 7d3

27. ƒsxd = x 4 - 8x 2 + 16 3 29. Hstd = t 4 - t 6 2

28. g sxd = x 4 - 4x 3 + 4x 2

Theory and Examples Show that the functions in Exercises 65 and 66 have local extreme values at the given values of u , and say which kind of local extreme the function has.

31. ƒsxd = x - 6 2x - 1

32. g sxd = 4 2x - x + 3

33. g sxd = x28 - x

2

34. g sxd = x 2 25 - x

x Z 2

36. ƒsxd =

3

x2 - 3 , x - 2

30. Kstd = 15t 3 - t 5

37. ƒsxd = x 1>3sx + 8d 39. hsxd = x

40. k sxd = x

41. ƒ(x) = e

1>3

2x

sx - 4d 2

+ e

-x

42. ƒ(x) = e

2

sx - 4d

2>3

2

2x

44. ƒ(x) = x ln x

43. ƒ(x) = x ln x

2

a. ƒ¿sxd 7 0 for x 6 1 and ƒ¿sxd 6 0 for x 7 1; b. ƒ¿sxd 6 0 for x 6 1 and ƒ¿sxd 7 0 for x 7 1; c. ƒ¿sxd 7 0 for x Z 1; d. ƒ¿sxd 6 0 for x Z 1. 68. Sketch the graph of a differentiable function y = ƒsxd that has

In Exercises 45–56: a. Identify the function’s local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? T c. Support your findings with a graphing calculator or computer grapher. 45. ƒsxd = 2x - x 2, - q 6 x … 2 -q 6 x … 0

46. ƒsxd = sx + 1d , 2

1 … x 6 q

47. g sxd = x 2 - 4x + 4,

-4 … x 6 q -3 … t 6 q -q 6 t … 3

48. g sxd = -x - 6x - 9, 2

49. ƒstd = 12t - t , 3

50. ƒstd = t 3 - 3t 2, 51. hsxd =

u , 0 … u … 2p, at u = 0 and u = 2p 2 u 66. hsud = 5 sin , 0 … u … p, at u = 0 and u = p 2 67. Sketch the graph of a differentiable function y = ƒsxd through the point (1, 1) if ƒ¿s1d = 0 and 65. hsud = 3 cos

2

x3 3x + 1 38. g sxd = x 2>3sx + 5d

35. ƒsxd =

2

x3 - 2x 2 + 4x, 3

0 … x 6 q

52. k sxd = x 3 + 3x 2 + 3x + 1, 53. ƒsxd = 225 - x 2,

-q 6 x … 0

-5 … x … 5

54. ƒsxd = 2x 2 - 2x - 3, 3 … x 6 q x - 2 , 0 … x 6 1 55. g sxd = 2 x - 1 x2 , -2 6 x … 1 56. g sxd = 4 - x2 In Exercises 57–64: a. Find the local extrema of each function on the given interval, and say where they occur. T b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ¿.

a. a local minimum at (1, 1) and a local maximum at (3, 3); b. a local maximum at (1, 1) and a local minimum at (3, 3); c. local maxima at (1, 1) and (3, 3); d. local minima at (1, 1) and (3, 3). 69. Sketch the graph of a continuous function y = g sxd such that a. g s2d = 2, 0 6 g¿ 6 1 for x 6 2, g¿sxd : 1- as x : 2-, -1 6 g¿ 6 0 for x 7 2, and g¿sxd : -1+ as x : 2+ ; b. g s2d = 2, g¿ 6 0 for x 6 2, g¿sxd : - q as x : 2-, g¿ 7 0 for x 7 2, and g¿sxd : q as x : 2+ . 70. Sketch the graph of a continuous function y = hsxd such that a. hs0d = 0, -2 … hsxd … 2 for all x, h¿sxd : q as x : 0 -, and h¿sxd : q as x : 0 + ; b. hs0d = 0, -2 … hsxd … 0 for all x, h¿sxd : q as x : 0 -, and h¿sxd : - q as x : 0 + . 71. Discuss the extreme-value behavior of the function ƒ(x) = x sin (1>x), x Z 0. How many critical points does this function have? Where are they located on the x-axis? Does ƒ have an absolute minimum? An absolute maximum? (See Exercise 49 in Section 2.3.) 72. Find the intervals on which the function ƒsxd = ax 2 + bx + c, a Z 0 , is increasing and decreasing. Describe the reasoning behind your answer. 73. Determine the values of constants a and b so that ƒ(x) = ax 2 + bx has an absolute maximum at the point (1, 2). 74. Determine the values of constants a, b, c, and d so that ƒ(x) = ax 3 + bx 2 + cx + d has a local maximum at the point (0, 0) and a local minimum at the point (1, -1).

4.4

Concavity and Curve Sketching

235

a. ln (cos x) on [-p>4, p>3],

79. Find the absolute maximum value of ƒsxd = x 2 ln s1>xd and say where it is assumed.

b. cos (ln x) on [1>2, 2].

80. a. Prove that e x Ú 1 + x if x Ú 0.

75. Locate and identify the absolute extreme values of

76. a. Prove that ƒ(x) = x - ln x is increasing for x 7 1.

b. Use the result in part (a) to show that

b. Using part (a), show that ln x 6 x if x 7 1.

ex Ú 1 + x +

77. Find the absolute maximum and minimum values of ƒsxd = e x - 2x on [0, 1].

81. Show that increasing functions and decreasing functions are oneto-one. That is, show that for any x1 and x2 in I, x2 Z x1 implies ƒsx2 d Z ƒsx1 d.

78. Where does the periodic function ƒsxd = 2e sin sx>2d take on its extreme values and what are these values?

Use the results of Exercise 81 to show that the functions in Exercises 82–86 have inverses over their domains. Find a formula for dƒ -1>dx using Theorem 3, Section 3.8.

y y 2e sin (x/2)

x

0

y

CA

VE

UP

y x3

CO

DO

N

f ' increases x

0

CO NC AV E

N W

82. ƒsxd = s1>3dx + s5>6d

83. ƒsxd = 27x 3

84. ƒsxd = 1 - 8x 3

85. ƒsxd = s1 - xd3

86. ƒsxd = x 5>3

Concavity and Curve Sketching

4.4

f ' decreases

1 2 x . 2

We have seen how the first derivative tells us where a function is increasing, where it is decreasing, and whether a local maximum or local minimum occurs at a critical point. In this section we see that the second derivative gives us information about how the graph of a differentiable function bends or turns. With this knowledge about the first and second derivatives, coupled with our previous understanding of symmetry and asymptotic behavior studied in Sections 1.1 and 2.6, we can now draw an accurate graph of a function. By organizing all of these ideas into a coherent procedure, we give a method for sketching graphs and revealing visually the key features of functions. Identifying and knowing the locations of these features is of major importance in mathematics and its applications to science and engineering, especially in the graphical analysis and interpretation of data.

Concavity FIGURE 4.24 The graph of ƒsxd = x is concave down on s - q , 0d and concave up on s0, q d (Example 1a). 3

As you can see in Figure 4.24, the curve y = x 3 rises as x increases, but the portions defined on the intervals s - q , 0d and s0, q d turn in different ways. As we approach the origin from the left along the curve, the curve turns to our right and falls below its tangents. The slopes of the tangents are decreasing on the interval s - q , 0d. As we move away from the origin along the curve to the right, the curve turns to our left and rises above its tangents. The slopes of the tangents are increasing on the interval s0, q d. This turning or bending behavior defines the concavity of the curve.

DEFINITION

The graph of a differentiable function y = ƒsxd is

(a) concave up on an open interval I if ƒ¿ is increasing on I; (b) concave down on an open interval I if ƒ¿ is decreasing on I.

If y = ƒsxd has a second derivative, we can apply Corollary 3 of the Mean Value Theorem to the first derivative function. We conclude that ƒ¿ increases if ƒ– 7 0 on I, and decreases if ƒ– 6 0.

236

Chapter 4: Applications of Derivatives

The Second Derivative Test for Concavity Let y = ƒsxd be twice-differentiable on an interval I.

y 4

1. If ƒ– 7 0 on I, the graph of ƒ over I is concave up. 2. If ƒ– 6 0 on I, the graph of ƒ over I is concave down.

y x2

3 EU

P

CAV CON

NC

P

–2

–1

CO

EU

y'' 0

If y = ƒsxd is twice-differentiable, we will use the notations ƒ– and y– interchangeably when denoting the second derivative.

AV

2 1

y'' 0

0

1

EXAMPLE 1

x

2

(a) The curve y = x 3 (Figure 4.24) is concave down on s - q , 0d where y– = 6x 6 0

and concave up on s0, q d where y– = 6x 7 0. (b) The curve y = x 2 (Figure 4.25) is concave up on s - q , q d because its second derivative y– = 2 is always positive.

FIGURE 4.25 The graph of ƒsxd = x 2 is concave up on every interval (Example 1b).

EXAMPLE 2

Determine the concavity of y = 3 + sin x on [0, 2p].

The first derivative of y = 3 + sin x is y¿ = cos x, and the second derivative is y– = -sin x. The graph of y = 3 + sin x is concave down on s0, pd, where y– = -sin x is negative. It is concave up on sp, 2pd, where y– = -sin x is positive (Figure 4.26). Solution

Points of Inflection The curve y = 3 + sin x in Example 2 changes concavity at the point sp, 3d. Since the first derivative y¿ = cos x exists for all x, we see that the curve has a tangent line of slope -1 at the point sp, 3d. This point is called a point of inflection of the curve. Notice from Figure 4.26 that the graph crosses its tangent line at this point and that the second derivative y– = -sin x has value 0 when x = p. In general, we have the following definition.

y 4 3

y 5 3 1 sin x (p, 3)

2 1 0 –1

p

2p

x

y'' 5 – sin x

DEFINITION A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.

FIGURE 4.26 Using the sign of y– to determine the concavity of y (Example 2).

We observed that the second derivative of ƒ(x) = 3 + sin x is equal to zero at the inflection point sp, 3d. Generally, if the second derivative exists at a point of inflection (c, ƒ(c)), then ƒ–(c) = 0. This follows immediately from the Intermediate Value Theorem whenever ƒ– is continuous over an interval containing x = c because the second derivative changes sign moving across this interval. Even if the continuity assumption is dropped, it is still true that ƒ–(c) = 0, provided the second derivative exists (although a more advanced agrument is required in this noncontinuous case). Since a tangent line must exist at the point of inflection, either the first derivative ƒ¿(c) exists (is finite) or a vertical tangent exists at the point. At a vertical tangent neither the first nor second derivative exists. In summary, we conclude the following result.

At a point of inflection (c, ƒ(c)), either ƒ–(c) = 0 or ƒ–(c) fails to exist.

The next example illustrates a function having a point of inflection where the first derivative exists, but the second derivative fails to exist.

4.4

The graph of ƒ(x) = x 5>3 has a horizontal tangent at the origin because ƒ¿(x) = (5>3)x = 0 when x = 0. However, the second derivative 2>3

y x5/3

2

ƒ–(x) =

1 1 Point of inflection

0 –1

237

EXAMPLE 3

y

–1

Concavity and Curve Sketching

10 -1>3 d 5 2>3 a x b = x 9 dx 3

x

fails to exist at x = 0. Nevertheless, ƒ–(x) 6 0 for x 6 0 and ƒ–(x) 7 0 for x 7 0, so the second derivative changes sign at x = 0 and there is a point of inflection at the origin. The graph is shown in Figure 4.27.

–2

FIGURE 4.27 The graph of ƒ(x) = x 5>3 has a horizontal tangent at the origin where the concavity changes, although ƒ– does not exist at x = 0 (Example 3).

Here is an example showing that an inflection point need not occur even though both derivatives exist and ƒ– = 0. The curve y = x 4 has no inflection point at x = 0 (Figure 4.28). Even though the second derivative y– = 12x 2 is zero there, it does not change sign.

EXAMPLE 4

As our final illustration, we show a situation in which a point of inflection occurs at a vertical tangent to the curve where neither the first nor the second derivative exists.

y y x4 2

The graph of y = x 1>3 has a point of inflection at the origin because the second derivative is positive for x 6 0 and negative for x 7 0:

EXAMPLE 5

1 y'' 0 –1

1

0

y– =

x

FIGURE 4.28 The graph of y = x 4 has no inflection point at the origin, even though y– = 0 there (Example 4).

However, both y¿ = x -2>3>3 and y– fail to exist at x = 0, and there is a vertical tangent there. See Figure 4.29. To study the motion of an object moving along a line as a function of time, we often are interested in knowing when the object’s acceleration, given by the second derivative, is positive or negative. The points of inflection on the graph of the object’s position function reveal where the acceleration changes sign.

y Point of inflection

d2 d 1 -2>3 2 ax 1>3 b = a x b = - x -5>3 . 9 dx 3 dx 2

y 5 x1/3 0

x

EXAMPLE 6 A particle is moving along a horizontal coordinate line (positive to the right) with position function sstd = 2t 3 - 14t 2 + 22t - 5,

t Ú 0.

Find the velocity and acceleration, and describe the motion of the particle. FIGURE 4.29 A point of inflection where y¿ and y– fail to exist (Example 5).

Solution

The velocity is ystd = s¿std = 6t 2 - 28t + 22 = 2st - 1ds3t - 11d,

and the acceleration is astd = y¿std = s–std = 12t - 28 = 4s3t - 7d. When the function s(t) is increasing, the particle is moving to the right; when s(t) is decreasing, the particle is moving to the left. Notice that the first derivative sy = s¿d is zero at the critical points t = 1 and t = 11>3. Interval Sign of Y s œ Behavior of s Particle motion

0 6 t 6 1 + increasing right

1 6 t 6 11>3 decreasing left

11>3 6 t + increasing right

238

Chapter 4: Applications of Derivatives

The particle is moving to the right in the time intervals [0, 1) and s11>3, q d, and moving to the left in (1, 11>3). It is momentarily stationary (at rest) at t = 1 and t = 11>3. The acceleration astd = s–std = 4s3t - 7d is zero when t = 7>3. Interval Sign of a s ﬂ Graph of s

0 6 t 6 7>3 concave down

7>3 6 t + concave up

The particle starts out moving to the right while slowing down, and then reverses and begins moving to the left at t = 1 under the influence of the leftward acceleration over the time interval [0, 7>3). The acceleration then changes direction at t = 7>3 but the particle continues moving leftward, while slowing down under the rightward acceleration. At t = 11>3 the particle reverses direction again: moving to the right in the same direction as the acceleration.

Second Derivative Test for Local Extrema Instead of looking for sign changes in ƒ¿ at critical points, we can sometimes use the following test to determine the presence and nature of local extrema.

THEOREM 5—Second Derivative Test for Local Extrema

Suppose ƒ– is continuous

on an open interval that contains x = c. 1. If ƒ¿scd = 0 and ƒ–scd 6 0, then ƒ has a local maximum at x = c. 2. If ƒ¿scd = 0 and ƒ–scd 7 0, then ƒ has a local minimum at x = c. 3. If ƒ¿scd = 0 and ƒ–scd = 0, then the test fails. The function ƒ may have a local maximum, a local minimum, or neither.

f ' 5 0, f '' , 0 ⇒ local max

f ' 5 0, f '' . 0 ⇒ local min

Proof Part (1). If ƒ–scd 6 0, then ƒ–sxd 6 0 on some open interval I containing the point c, since ƒ– is continuous. Therefore, ƒ¿ is decreasing on I. Since ƒ¿scd = 0, the sign of ƒ¿ changes from positive to negative at c so ƒ has a local maximum at c by the First Derivative Test. The proof of Part (2) is similar. For Part (3), consider the three functions y = x 4, y = -x 4 , and y = x 3 . For each function, the first and second derivatives are zero at x = 0. Yet the function y = x 4 has a local minimum there, y = -x 4 has a local maximum, and y = x 3 is increasing in any open interval containing x = 0 (having neither a maximum nor a minimum there). Thus the test fails. This test requires us to know ƒ– only at c itself and not in an interval about c. This makes the test easy to apply. That’s the good news. The bad news is that the test is inconclusive if ƒ– = 0 or if ƒ– does not exist at x = c. When this happens, use the First Derivative Test for local extreme values. Together ƒ¿ and ƒ– tell us the shape of the function’s graph—that is, where the critical points are located and what happens at a critical point, where the function is increasing and where it is decreasing, and how the curve is turning or bending as defined by its concavity. We use this information to sketch a graph of the function that captures its key features.

EXAMPLE 7

Sketch a graph of the function ƒsxd = x 4 - 4x 3 + 10

using the following steps. (a) Identify where the extrema of ƒ occur. (b) Find the intervals on which ƒ is increasing and the intervals on which ƒ is decreasing.

4.4

Concavity and Curve Sketching

239

(c) Find where the graph of ƒ is concave up and where it is concave down. (d) Sketch the general shape of the graph for ƒ. (e) Plot some specific points, such as local maximum and minimum points, points of in-

flection, and intercepts. Then sketch the curve. The function ƒ is continuous since ƒ¿sxd = 4x 3 - 12x 2 exists. The domain of ƒ is s - q , q d, and the domain of ƒ¿ is also s - q , q d. Thus, the critical points of ƒ occur only at the zeros of ƒ¿ . Since

Solution

ƒ¿sxd = 4x 3 - 12x 2 = 4x 2sx - 3d, the first derivative is zero at x = 0 and x = 3. We use these critical points to define intervals where ƒ is increasing or decreasing. x 6 0 decreasing

Interval Sign of ƒ œ Behavior of ƒ

0 6 x 6 3 decreasing

3 6 x + increasing

(a) Using the First Derivative Test for local extrema and the table above, we see that there is no extremum at x = 0 and a local minimum at x = 3. (b) Using the table above, we see that ƒ is decreasing on s - q , 0] and [0, 3], and increasing on [3, q d. (c) ƒ–sxd = 12x 2 - 24x = 12xsx - 2d is zero at x = 0 and x = 2. We use these points to define intervals where ƒ is concave up or concave down. x 6 0 + concave up

Interval Sign of ƒ ﬂ Behavior of ƒ

0 6 x 6 2 concave down

2 6 x + concave up

We see that ƒ is concave up on the intervals s - q , 0d and s2, q d, and concave down on (0, 2). (d) Summarizing the information in the last two tables, we obtain the following.

y y x 4 4x 3 10

x<0

0<x<2

2<x<3

3<x

decreasing concave up

decreasing concave down

decreasing concave up

increasing concave up

20 15

The general shape of the curve is shown in the accompanying figure.

(0, 10) Inflection 10 point 5 –1

0 –5 –10

1 Inflection point

2

3 (2, – 6)

–15 –20

(3, –17) Local minimum

FIGURE 4.30 The graph of ƒsxd = x 4 - 4x 3 + 10 (Example 7).

4

x

decr

decr

decr

incr

conc up

conc down

conc up

conc up

0

2

3

infl point

infl point

local min

General shape

(e) Plot the curve’s intercepts (if possible) and the points where y¿ and y– are zero. Indicate any local extreme values and inflection points. Use the general shape as a guide to sketch the curve. (Plot additional points as needed.) Figure 4.30 shows the graph of ƒ.

240

Chapter 4: Applications of Derivatives

The steps in Example 7 give a procedure for graphing the key features of a function.

Procedure for Graphing y ƒ(x) 1. Identify the domain of ƒ and any symmetries the curve may have. 2. Find the derivatives y¿ and y– . 3. Find the critical points of ƒ, if any, and identify the function’s behavior at each one. 4. Find where the curve is increasing and where it is decreasing. 5. Find the points of inflection, if any occur, and determine the concavity of the curve. 6. Identify any asymptotes that may exist (see Section 2.6). 7. Plot key points, such as the intercepts and the points found in Steps 3–5, and sketch the curve together with any asymptotes that exist.

EXAMPLE 8

Sketch the graph of ƒsxd =

sx + 1d2 . 1 + x2

Solution

1. 2.

The domain of ƒ is s - q , q d and there are no symmetries about either axis or the origin (Section 1.1). Find ƒ¿ and ƒ– . ƒsxd =

ƒ¿sxd =

=

ƒ–sxd =

=

3.

4.

x-intercept at x = -1, y-intercept sy = 1d at x = 0

sx + 1d2 1 + x2 s1 + x 2 d # 2sx + 1d - sx + 1d2 # 2x s1 + x 2 d2 2s1 - x 2 d s1 + x 2 d2

Critical points: x = - 1, x = 1

s1 + x 2 d2 # 2s -2xd - 2s1 - x 2 d[2s1 + x 2 d # 2x] s1 + x 2 d4 4xsx 2 - 3d s1 + x 2 d3

After some algebra

Behavior at critical points. The critical points occur only at x = ;1 where ƒ¿sxd = 0 (Step 2) since ƒ¿ exists everywhere over the domain of ƒ. At x = -1, ƒ–(-1) = 1 7 0, yielding a relative minimum by the Second Derivative Test. At x = 1, f –(1) = -1 6 0, yielding a relative maximum by the Second Derivative test. Increasing and decreasing. We see that on the interval s - q , -1d the derivative ƒ¿sxd 6 0, and the curve is decreasing. On the interval s -1, 1d, ƒ¿sxd 7 0 and the curve is increasing; it is decreasing on s1, q d where ƒ¿sxd 6 0 again.

4.4

Inflection points. Notice that the denominator of the second derivative (Step 2) is always positive. The second derivative ƒ– is zero when x = - 23, 0, and 23. The second derivative changes sign at each of these points: negative on A - q , - 23 B , positive on A - 23, 0 B , negative on A 0, 23 B , and positive again on A 23, q B . Thus each point is a point of inflection. The curve is concave down on the interval A - q , - 23 B , concave up on A - 23, 0 B , concave down on A 0, 23 B , and concave up again on A 23, q B .

6.

Asymptotes. Expanding the numerator of ƒ(x) and then dividing both numerator and denominator by x 2 gives

=

(1, 2)

Point of inflection where x 3

2 y1

1 Horizontal asymptote –1 Point of inflection where x 3

1

FIGURE 4.31 The graph of y =

x

sx + 1d2

241

5.

ƒsxd =

y

Concavity and Curve Sketching

7.

1 + x2

(Example 8).

sx + 1d2 x 2 + 2x + 1 = 1 + x2 1 + x2 1 + s2>xd + s1>x 2 d s1>x 2 d + 1

.

Expanding numerator

Dividing by x 2

We see that ƒsxd : 1+ as x : q and that ƒsxd : 1- as x : - q . Thus, the line y = 1 is a horizontal asymptote. Since ƒ decreases on s - q , -1d and then increases on s -1, 1d, we know that ƒs -1d = 0 is a local minimum. Although ƒ decreases on s1, q d, it never crosses the horizontal asymptote y = 1 on that interval (it approaches the asymptote from above). So the graph never becomes negative, and ƒs -1d = 0 is an absolute minimum as well. Likewise, ƒs1d = 2 is an absolute maximum because the graph never crosses the asymptote y = 1 on the interval s - q , -1d, approaching it from below. Therefore, there are no vertical asymptotes (the range of ƒ is 0 … y … 2). The graph of ƒ is sketched in Figure 4.31. Notice how the graph is concave down as it approaches the horizontal asymptote y = 1 as x : - q , and concave up in its approach to y = 1 as x : q .

EXAMPLE 9

Sketch the graph of ƒ(x) =

x2 + 4 . 2x

Solution

1.

2.

3.

The domain of ƒ is all nonzero real numbers. There are no intercepts because neither x nor ƒ(x) can be zero. Since ƒ(-x) = -ƒ(x), we note that ƒ is an odd function, so the graph of ƒ is symmetric about the origin. We calculate the derivatives of the function, but first rewrite it in order to simplify our computations: ƒ(x) =

x2 + 4 x 2 = + x 2x 2

Function simplified for differentiation

ƒ¿(x) =

x2 - 4 1 2 - 2 = 2 x 2x 2

Combine fractions to solve easily ƒ¿(x) = 0.

ƒ–(x) =

4 x3

Exists throughout the entire domain of ƒ

The critical points occur at x = ;2 where ƒ¿(x) = 0. Since ƒ–(-2) 6 0 and ƒ–(2) 7 0, we see from the Second Derivative Test that a relative maximum occurs at x = -2 with ƒ(-2) = -2, and a relative minimum occurs at x = 2 with ƒ(2) = 2.

242

Chapter 4: Applications of Derivatives

4.

5. y 4

2 y5 x 14 2x

(2, 2) 2 –4

–2 (–2, –2)

6.

y5 x 2 2

0

4

On the interval (- q , -2) the derivative ƒ¿ is positive because x 2 - 4 7 0 so the graph is increasing; on the interval (-2, 0) the derivative is negative and the graph is decreasing. Similarly, the graph is decreasing on the interval (0, 2) and increasing on (2, q ). There are no points of inflection because ƒ–(x) 6 0 whenever x 6 0, ƒ–(x) 7 0 whenever x 7 0, and ƒ– exists everywhere and is never zero throughout the domain of ƒ. The graph is concave down on the interval (- q , 0) and concave up on the interval (0, q ). From the rewritten formula for ƒ(x), we see that

x

lim+ a

x:0

–2 –4

FIGURE 4.32 The graph of y =

x2 + 4 2x

7.

x 2 + xb = +q 2

and

lim- a

x:0

x 2 + x b = - q, 2

so the y-axis is a vertical asymptote. Also, as x : q or as x : - q , the graph of ƒ(x) approaches the line y = x>2. Thus y = x>2 is an oblique asymptote. The graph of ƒ is sketched in Figure 4.32.

(Example 9).

EXAMPLE 10

Sketch the graph of ƒ(x) = e 2>x.

The domain of ƒ is (- q , 0) h (0, q ) and there are no symmetries about either axis or the origin. The derivatives of ƒ are Solution

ƒ¿(x) = e 2>x a-

2e 2>x 2 b = - 2 2 x x

and ƒ–(x) =

x 2(2e 2>x)(-2>x 2) - 2e 2>x(2x) x4

=

4e 2>x(1 + x) x4

.

y y e 2x

5 4 3 Inflection 2 point 1 –2

–1

y1 0

1

2

3

x

FIGURE 4.33 The graph of y = e 2>x has a point of inflection at (-1, e -2). The line y = 1 is a horizontal asymptote and x = 0 is a vertical asymptote (Example 10).

Both derivatives exist everywhere over the domain of ƒ. Moreover, since e 2>x and x 2 are both positive for all x Z 0, we see that ƒ¿ 6 0 everywhere over the domain and the graph is everywhere decreasing. Examining the second derivative, we see that ƒ–(x) = 0 at x = -1. Since e 2>x 7 0 and x 4 7 0, we have ƒ– 6 0 for x 6 -1 and ƒ– 7 0 for x 7 -1, x Z 0. Therefore, the point (-1, e -2) is a point of inflection. The curve is concave down on the interval (- q , -1) and concave up over (-1, 0) h (0, q ). From Example 7, Section 2.6, we see that limx:0- ƒ(x) = 0. As x : 0 +, we see that 2>x : q , so limx:0+ ƒ(x) = q and the y-axis is a vertical asymptote. Also, as x : - q , 2>x : 0 - and so limx:- q ƒ(x) = e 0 = 1. Therefore, y = 1 is a horizontal asymptote. There are no absolute extrema since ƒ never takes on the value 0. The graph of ƒ is sketched in Figure 4.33.

Graphical Behavior of Functions from Derivatives As we saw in Examples 7–10, we can learn much about a twice-differentiable function y = ƒsxd by examining its first derivative. We can find where the function’s graph rises and falls and where any local extrema are located. We can differentiate y¿ to learn how the graph bends as it passes over the intervals of rise and fall. We can determine the shape of the function’s graph. Information we cannot get from the derivative is how to place the graph in the xy-plane. But, as we discovered in Section 4.2, the only additional information we need to position the graph is the value of ƒ at one point. Information about the asymptotes is found using limits (Section 2.6). The following

4.4

243

Concavity and Curve Sketching

figure summarizes how the derivative and second derivative affect the shape of a graph.

y f (x)

y f (x)

Differentiable ⇒ smooth, connected; graph may rise and fall

y' 0 ⇒ rises from left to right; may be wavy

or

y f (x)

y' 0 ⇒ falls from left to right; may be wavy

or

y'' 0 ⇒ concave up throughout; no waves; graph may rise or fall

y'' 0 ⇒ concave down throughout; no waves; graph may rise or fall

y'' changes sign at an inflection point

y' 0 and y'' 0 at a point; graph has local maximum

y' 0 and y'' 0 at a point; graph has local minimum

or y' changes sign ⇒ graph has local maximum or local minimum

Exercises 4.4 Analyzing Functions from Graphs Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the intervals on which the functions are concave up and concave down. 1.

2.

3 2 y x x 2x 1 3 2 3 y

4 y x 2x2 4 4 y

0

2 3

7. y sin x , – 2 x 2 y

0

4.

y 3 (x 2 1)2/3 4 y

0

x

y

x

2

2

x

0

8.

3 y 2 cos x 2 x, – x 2 y

x

0

3.

6. y tan x 4x, – x

y x sin 2x, – 2 x 2 3 3 y

– 2 3

x

0

5.

y 9 x1/3(x 2 7) 14 y

0

x

–

0

3 2

x

NOT TO SCALE

x

Graphing Equations Use the steps of the graphing procedure on page 240 to graph the equations in Exercises 9–58. Include the coordinates of any local and absolute extreme points and inflection points. 9. y = x 2 - 4x + 3 11. y = x - 3x + 3 3

10. y = 6 - 2x - x 2 12. y = xs6 - 2xd2

244

Chapter 4: Applications of Derivatives 14. y = 1 - 9x - 6x 2 - x 3

13. y = -2x 3 + 6x 2 - 3 15. y = sx - 2d3 + 1 16. y = 1 - sx + 1d3 17. y = x 4 - 2x 2 = x 2sx 2 - 2d

18. y = -x 4 + 6x 2 - 4 = x 2s6 - x 2 d - 4 19. y = 4x 3 - x 4 = x 3s4 - xd 20. y = x 4 + 2x 3 = x 3sx + 2d 21. y = x 5 - 5x 4 = x 4sx - 5d 22. y = x a

x - 5b 2

4

23. y = x + sin x,

0 … x … 2p

24. y = x - sin x,

0 … x … 2p

29. y = x 1>5 31. y =

0 … x … 2p

2x 2 + 1 33. y = 2x - 3x 2>3 35. y = x 2>3 a

5 - xb 2

37. y = x28 - x

2

39. y = 216 - x 2 x2 - 3 41. y = x - 2 8x 43. y = 2 x + 4 45. y = ƒ x 2 - 1 ƒ 2 -x, 47. y = 2 ƒ x ƒ = e 2x,

32. y =

0 … t … 2p

74. y¿ = sin t,

0 … t … 2p

21 - x 2 2x + 1

34. y = 5x 2>5 - 2x 36. y = x 2>3(x - 5)

78. y¿ = x -4>5sx + 1d

Sketching y from Graphs of y œ and y ﬂ Each of Exercises 81–84 shows the graphs of the first and second derivatives of a function y = ƒsxd . Copy the picture and add to it a sketch of the approximate graph of ƒ, given that the graph passes through the point P. 81.

82.

y

y

y f '(x)

y f'(x)

P

x

x

42. y = 2x + 1 3

P

y f ''(x)

83. P

y f '(x)

x

0

ex 50. y = x

51. y = ln (3 - x 2)

52. y = x (ln x) 2

53. y = e x - 2e -x - 3x

54. y = xe -x

55. y = ln (cos x)

56. y =

y f ''(x)

y

x 6 0 x Ú 0

49. y = xe 1>x

1 1 + e -x

-2x, x … 0 2x, x 7 0

-x 2, x … 0 x 7 0 x 2,

38. y = (2 - x ) 2 40. y = x 2 + x

5 44. y = 4 x + 5 46. y = ƒ x 2 - 2 x ƒ

0 6 u 6 2p

77. y¿ = x -2>3sx - 1d

48. y = 2ƒ x - 4 ƒ

57. y =

u 70. y¿ = csc2 , 2 p 6 2 p

76. y¿ = sx - 2d-1>3

2 3>2

3

66. y¿ = sx 2 - 2xdsx - 5d2

75. y¿ = sx + 1d-2>3

80. y¿ = e

30. y = x 2>5 x

73. y¿ = cos t,

79. y¿ = 2 ƒ x ƒ = e

25. y = 23x - 2 cos x, 0 … x … 2p -p p 4 26. y = x - tan x, 6 x 6 3 2 2 27. y = sin x cos x, 0 … x … p 28. y = cos x + 23 sin x,

65. y¿ = s8x - 5x 2 d(4 - x) 2 p p 67. y¿ = sec2 x, - 6 x 6 2 2 p p 68. y¿ = tan x, - 6 x 6 2 2 u 69. y¿ = cot , 0 6 u 6 2p 2 p 71. y¿ = tan2 u - 1, - 6 u 2 72. y¿ = 1 - cot2 u, 0 6 u 6

y f ''(x)

84.

y y f '(x)

ln x

2x ex 58. y = 1 + ex

Sketching the General Shape, Knowing y œ Each of Exercises 59–80 gives the first derivative of a continuous function y = ƒsxd . Find y– and then use steps 2–4 of the graphing procedure on page 240 to sketch the general shape of the graph of ƒ. 59. y¿ = 2 + x - x 2

60. y¿ = x 2 - x - 6

61. y¿ = xsx - 3d2

62. y¿ = x 2s2 - xd

63. y¿ = xsx 2 - 12d

64. y¿ = sx - 1d2s2x + 3d

x

0 y f ''(x) P

Graphing Rational Functions Graph the rational functions in Exercises 85–102. 2x 2 + x - 1 x2 - 1 4 x + 1 87. y = x2 1 89. y = 2 x - 1 85. y =

x 2 - 49 x + 5x - 14 x2 - 4 88. y = 2x x2 90. y = 2 x - 1 86. y =

2

4.4

95. y = 97. y = 99. y = 101. y = 102. y =

y S

y f (x) R

P

107.

s

s f(t)

0

Theory and Examples 103. The accompanying figure shows a portion of the graph of a twicedifferentiable function y = ƒsxd . At each of the five labeled points, classify y¿ and y– as positive, negative, or zero.

108.

5

5

ƒ¿s2d = ƒ¿s -2d = 0,

ƒs0d = 4,

ƒ¿sxd 6 0

for

ƒ x ƒ 6 2,

ƒs2d = 0,

ƒ–sxd 6 0

for

x 6 0,

ƒ–sxd 7 0

for

x 7 0.

ƒ x ƒ 7 2,

x 6 2 2 6 x 4 4 6 x 6 x 7

y

Derivatives

2

y¿ y¿ y¿ y¿ y¿ y¿ y¿

1 6 4 4 6 6 7 6

6 = 7 7 7 = 6

0, 0, 0, 0, 0, 0, 0,

y– y– y– y– y– y– y–

7 7 7 = 6 6 6

0 0 0 0 0 0 0

106. Sketch the graph of a twice-differentiable function y = ƒsxd that passes through the points s -2, 2d, s -1, 1d, s0, 0d, s1, 1d, and (2, 2) and whose first two derivatives have the following sign patterns. y¿:

+

y–:

-

-2 -1

-

+ 0

+

2 -

1

-

15

t

c c f (x)

105. Sketch the graph of a twice-differentiable function y = ƒsxd with the following properties. Label coordinates where possible.

x

10 Time (sec)

Cost

ƒs - 2d = 8,

for

t

109. Marginal cost The accompanying graph shows the hypothetical cost c = ƒsxd of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

x

104. Sketch a smooth connected curve y = ƒsxd with

ƒ¿sxd 7 0

15

s f (t)

Q

0

10 Time (sec)

s

0

T

245

Motion Along a Line The graphs in Exercises 107 and 108 show the position s = ƒstd of an object moving up and down on a coordinate line. (a) When is the object moving away from the origin? Toward the origin? At approximately what times is the (b) velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration positive? Negative?

Displacement

93. y =

x2 - 4 x2 - 2 x2 - 4 x + 1 x2 - x + 1 x - 1 x3 + x - 2 x - x2 x - 1 x 2(x - 2)

Displacement

x2 - 2 92. y = x2 - 1 x2 94. y = x + 1 x2 - x + 1 96. y = x - 1 3 2 x - 3x + 3x - 1 98. y = x2 + x - 2 x 100. y = x2 - 1 8 (Agnesi's witch) x2 + 4 4x (Newton's serpentine) x2 + 4

91. y = -

Concavity and Curve Sketching

20 40 60 80 100 120 Thousands of units produced

x

110. The accompanying graph shows the monthly revenue of the Widget Corporation for the last 12 years. During approximately what time intervals was the marginal revenue increasing? Decreasing? y y r(t)

0

5

10

t

111. Suppose the derivative of the function y = ƒsxd is y¿ = sx - 1d2sx - 2d . At what points, if any, does the graph of ƒ have a local minimum, local maximum, or point of inflection? (Hint: Draw the sign pattern for y¿ .)

246

Chapter 4: Applications of Derivatives

112. Suppose the derivative of the function y = ƒsxd is

120. Suppose that the second derivative of the function y = ƒsxd is y– = x 2(x - 2) 3(x + 3).

y¿ = sx - 1d2sx - 2dsx - 4d .

For what x-values does the graph of ƒ have an inflection point? At what points, if any, does the graph of ƒ have a local minimum, local maximum, or point of inflection? 113. For x 7 0 , sketch a curve y = ƒsxd that has ƒs1d = 0 and ƒ¿sxd = 1>x . Can anything be said about the concavity of such a curve? Give reasons for your answer. 114. Can anything be said about the graph of a function y = ƒsxd that has a continuous second derivative that is never zero? Give reasons for your answer. 115. If b, c, and d are constants, for what value of b will the curve y = x 3 + bx 2 + cx + d have a point of inflection at x = 1 ? Give reasons for your answer. 116. Parabolas a. Find the coordinates of the vertex of the parabola y = ax 2 + bx + c, a Z 0 . b. When is the parabola concave up? Concave down? Give reasons for your answers. 117. Quadratic curves What can you say about the inflection points of a quadratic curve y = ax 2 + bx + c, a Z 0 ? Give reasons for your answer. 118. Cubic curves What can you say about the inflection points of a cubic curve y = ax 3 + bx 2 + cx + d, a Z 0 ? Give reasons for your answer. 119. Suppose that the second derivative of the function y = ƒsxd is y– = (x + 1)(x - 2). For what x-values does the graph of ƒ have an inflection point?

4.5

121. Find the values of constants a, b, and c so that the graph of y = ax 3 + bx 2 + cx has a local maximum at x = 3, local minimum at x = -1, and inflection point at (1, 11). 122. Find the values of constants a, b, and c so that the graph of y = (x 2 + a)>(bx + c) has a local minimum at x = 3 and a local maximum at (-1, -2). COMPUTER EXPLORATIONS In Exercises 123–126, find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function’s first and second derivatives. How are the values at which these graphs intersect the x-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function? 123. y = x 5 - 5x 4 - 240

124. y = x 3 - 12x 2

4 5 x + 16x 2 - 25 5 x4 x3 126. y = - 4x 2 + 12x + 20 4 3 125. y =

127. Graph ƒsxd = 2x 4 - 4x 2 + 1 and its first two derivatives together. Comment on the behavior of ƒ in relation to the signs and values of ƒ¿ and ƒ– . 128. Graph ƒsxd = x cos x and its second derivative together for 0 … x … 2p . Comment on the behavior of the graph of ƒ in relation to the signs and values of ƒ– .

Indeterminate Forms and L’Hôpital’s Rule

HISTORICAL BIOGRAPHY Guillaume François Antoine de l’Hôpital (1661–1704) Johann Bernoulli (1667–1748)

John (Johann) Bernoulli discovered a rule using derivatives to calculate limits of fractions whose numerators and denominators both approach zero or + q . The rule is known today as l’Hôpital’s Rule, after Guillaume de l’Hôpital. He was a French nobleman who wrote the first introductory differential calculus text, where the rule first appeared in print. Limits involving transcendental functions often require some use of the rule for their calculation.

Indeterminate Form 0/0 If we want to know how the function Fsxd =

x - sin x x3

behaves near x = 0 (where it is undefined), we can examine the limit of Fsxd as x : 0. We cannot apply the Quotient Rule for limits (Theorem 1 of Chapter 2) because the limit of the denominator is 0. Moreover, in this case, both the numerator and denominator approach 0, and 0>0 is undefined. Such limits may or may not exist in general, but the limit does exist for the function Fsxd under discussion by applying l’Hôpital’s Rule, as we will see in Example 1d.

4.5

Indeterminate Forms and L’Hôpital’s Rule

247

If the continuous functions ƒ(x) and g(x) are both zero at x = a, then lim

x:a

ƒsxd gsxd

cannot be found by substituting x = a. The substitution produces 0>0, a meaningless expression, which we cannot evaluate. We use 0>0 as a notation for an expression known as an indeterminate form. Other meaningless expressions often occur, such as q > q , q # 0, q - q , 0 0, and 1q, which cannot be evaluated in a consistent way; these are called indeterminate forms as well. Sometimes, but not always, limits that lead to indeterminate forms may be found by cancellation, rearrangement of terms, or other algebraic manipulations. This was our experience in Chapter 2. It took considerable analysis in Section 2.4 to find limx:0 ssin xd>x. But we have had success with the limit ƒsxd - ƒsad , x - a x:a

ƒ¿sad = lim

from which we calculate derivatives and which produces the indeterminant form 0>0 when we substitute x = a. L’Hôpital’s Rule enables us to draw on our success with derivatives to evaluate limits that otherwise lead to indeterminate forms.

THEOREM 6— L’Hôpital’s Rule Suppose that ƒsad = gsad = 0, that ƒ and g are differentiable on an open interval I containing a, and that g¿sxd Z 0 on I if x Z a. Then ƒsxd ƒ¿sxd = lim , lim x:a gsxd x:a g¿sxd assuming that the limit on the right side of this equation exists.

We give a proof of Theorem 6 at the end of this section.

Caution To apply l’Hôpital’s Rule to ƒ>g, divide the derivative of ƒ by the derivative of g. Do not fall into the trap of taking the derivative of ƒ>g. The quotient to use is ƒ¿>g¿, not sƒ>gd¿ .

EXAMPLE 1 The following limits involve 0>0 indeterminate forms, so we apply l’Hôpital’s Rule. In some cases, it must be applied repeatedly. (a) lim

x:0

3x - sin x 3 - cos x 3 - cos x = lim = ` = 2 x 1 1 x:0 x=0

1 21 + x - 1 221 + x 1 = lim = (b) lim x 2 x:0 x:0 1 (c) lim

x:0

21 + x - 1 - x>2 x

0 ; apply l’Hôpital’s Rule. 0

2

s1>2ds1 + xd-1>2 - 1>2 2x x:0

= lim = lim

x:0

-s1>4ds1 + xd-3>2 1 = 2 8

0 Still ; apply l’Hôpital’s Rule again. 0 Not

0 ; limit is found. 0

248

Chapter 4: Applications of Derivatives

(d) lim

x:0

x - sin x x3

0 ; apply l’Hôpital’s Rule. 0

= lim

1 - cos x 3x 2

0 Still ; apply l’Hôpital’s Rule again. 0

= lim

sin x 6x

0 Still ; apply l’Hôpital’s Rule again. 0

= lim

cos x 1 = 6 6

0 Not ; limit is found. 0

x:0

x:0

x:0

Here is a summary of the procedure we followed in Example 1.

Using L’Hôpital’s Rule To find lim

x:a

ƒsxd gsxd

by l’Hôpital’s Rule, continue to differentiate ƒ and g, so long as we still get the form 0>0 at x = a. But as soon as one or the other of these derivatives is different from zero at x = a we stop differentiating. L’Hôpital’s Rule does not apply when either the numerator or denominator has a finite nonzero limit.

EXAMPLE 2

Be careful to apply l’Hôpital’s Rule correctly: lim

x:0

1 - cos x x + x2

= lim

x:0

sin x 1 + 2x

0 0 Not

0 0

It is tempting to try to apply l’Hôpital’s Rule again, which would result in lim

x:0

cos x 1 = , 2 2

but this is not the correct limit. l’Hôpital’s Rule can be applied only to limits that give indeterminate forms, and limx:0 (sin x)>(1 + 2x) does not give an indeterminate form. Instead, this limit is 0>1 = 0, and the correct answer for the original limit is 0. L’Hôpital’s Rule applies to one-sided limits as well.

EXAMPLE 3 (a) lim+ x:0

Recall that q and + q mean the same thing.

In this example the one-sided limits are different.

sin x x2 = lim+ x:0

(b) limx:0

0 0

cos x = q 2x

sin x x2 = limx:0

Positive for x 7 0 0 0

cos x = -q 2x

Negative for x 6 0

Indeterminate Forms ˆ / ˆ , ˆ # 0, ˆ ˆ Sometimes when we try to evaluate a limit as x : a by substituting x = a we get an indeterminant form like q > q , q # 0, or q - q , instead of 0>0. We first consider the form q> q.

4.5

Indeterminate Forms and L’Hôpital’s Rule

249

In more advanced treatments of calculus, it is proved that l’Hôpital’s Rule applies to the indeterminate form q > q as well as to 0>0. If ƒsxd : ; q and gsxd : ; q as x : a, then lim

x:a

ƒsxd ƒ¿sxd = lim gsxd x:a g¿sxd

provided the limit on the right exists. In the notation x : a, a may be either finite or infinite. Moreover, x : a may be replaced by the one-sided limits x : a + or x : a -. Find the limits of these q > q forms:

EXAMPLE 4 (a)

lim

x:p>2

sec x 1 + tan x

ln x

(b) lim

x: q

(c) lim

x: q

22x

ex . x2

Solution

(a) The numerator and denominator are discontinuous at x = p>2, so we investigate the one-sided limits there. To apply l’Hôpital’s Rule, we can choose I to be any open interval with x = p>2 as an endpoint. lim

x:sp>2d -

q q from the left so we apply l’Hôpital’s Rule.

sec x 1 + tan x =

lim

x:sp>2d -

sec x tan x = sec2 x

lim

x:sp>2d -

sin x = 1

The right-hand limit is 1 also, with s - q d>s - q d as the indeterminate form. Therefore, the two-sided limit is equal to 1. (b) lim

x: q

ln x 22x

x: q

x

(c)

lim

x: q

1>x

= lim

1> 2x

= lim

x: q

1 = 0 2x

1>x 1> 2x

=

2x x =

1 2x

x

e e ex = lim = lim = q 2 x: q 2x x: q 2 x

Next we turn our attention to the indeterminate forms q # 0 and q - q . Sometimes these forms can be handled by using algebra to convert them to a 0>0 or q > q form. Here again we do not mean to suggest that q # 0 or q - q is a number. They are only notations for functional behaviors when considering limits. Here are examples of how we might work with these indeterminate forms.

EXAMPLE 5

Find the limits of these q # 0 forms:

1 (a) lim ax sin x b x: q

(b) lim+ 2x ln x x:0

Solution

(a)

sin h 1 1 lim ax sin x b = lim+ a sin hb = lim+ = 1 h h:0 h h:0

x: q

(b) lim+ 2x ln x = lim+ x:0

x:0

= lim+ x:0

ln x 1> 2x

q # 0 converted to q > q

1>x -1>2x 3>2

= lim+ A -22x B = 0 x:0

q # 0; let h = 1>x.

l’Hôpital’s Rule applied

250

Chapter 4: Applications of Derivatives

EXAMPLE 6

Find the limit of this q - q form: lim a

x:0

Solution

1 1 - xb. sin x

If x : 0 + , then sin x : 0 + and 1 1 - x : q - q. sin x

Similarly, if x : 0 - , then sin x : 0 - and 1 1 - x : - q - s- qd = - q + q . sin x Neither form reveals what happens in the limit. To find out, we first combine the fractions: x - sin x 1 1 - x = sin x x sin x

Common denominator is x sin x.

Then we apply l’Hôpital’s Rule to the result: lim a

x:0

x - sin x 1 1 - x b = lim sin x x:0 x sin x = lim

1 - cos x sin x + x cos x

= lim

sin x 0 = = 0. 2 2 cos x - x sin x

x:0

x:0

0 0 Still

0 0

Indeterminate Powers Limits that lead to the indeterminate forms 1q, 0 0, and q 0 can sometimes be handled by first taking the logarithm of the function. We use l’Hôpital’s Rule to find the limit of the logarithm expression and then exponentiate the result to find the original function limit. This procedure is justified by the continuity of the exponential function and Theorem 10 in Section 2.5, and it is formulated as follows. (The formula is also valid for one-sided limits.)

If limx:a ln ƒ(x) = L, then lim ƒ(x) = lim e ln ƒ(x) = e L.

x:a

x:a

Here a may be either finite or infinite.

EXAMPLE 7

Apply l’Hôpital’s Rule to show that limx:0+ (1 + x) 1>x = e.

The limit leads to the indeterminate form 1q. We let ƒ(x) = (1 + x) 1>x and find limx:0+ ln ƒ(x). Since Solution

1 ln ƒ(x) = ln (1 + x) 1>x = x ln (1 + x),

4.5

Indeterminate Forms and L’Hôpital’s Rule

251

l’Hôpital’s Rule now applies to give lim+ ln ƒ(x) = lim+

x:0

x:0

ln (1 + x) x

1 1 + x = lim+ 1 x:0 =

0 0

l’Hôpital’s Rule applied

1 = 1. 1

Therefore, lim+ (1 + x) 1>x = lim+ ƒ(x) = lim+ e ln ƒ(x) = e 1 = e. x:0

EXAMPLE 8

x:0

x:0

Find limx: q x 1>x.

The limit leads to the indeterminate form q 0. We let ƒ(x) = x 1>x and find limx: q ln ƒ(x). Since Solution

ln ƒ(x) = ln x 1>x =

ln x x ,

l’Hôpital’s Rule gives lim ln ƒ(x) = lim

ln x x

q q

= lim

1>x 1

l’Hôpital’s Rule applied

x: q

x: q

x: q

=

0 = 0. 1

Therefore lim x 1>x = lim ƒ(x) = lim e ln ƒ(x) = e 0 = 1. x: q

x: q

x: q

Proof of L’Hôpital’s Rule HISTORICAL BIOGRAPHY Augustin-Louis Cauchy (1789–1857)

When gsxd = x, Theorem 7 is the Mean Value Theorem.

The proof of l’Hôpital’s Rule is based on Cauchy’s Mean Value Theorem, an extension of the Mean Value Theorem that involves two functions instead of one. We prove Cauchy’s Theorem first and then show how it leads to l’Hôpital’s Rule.

THEOREM 7—Cauchy’s Mean Value Theorem Suppose functions ƒ and g are continuous on [a, b] and differentiable throughout (a, b) and also suppose g¿sxd Z 0 throughout (a, b). Then there exists a number c in (a, b) at which ƒ¿scd ƒsbd - ƒsad = . g¿scd gsbd - gsad

Proof We apply the Mean Value Theorem of Section 4.2 twice. First we use it to show that gsad Z gsbd. For if gsbd did equal gsad, then the Mean Value Theorem would give g¿scd =

gsbd - gsad = 0 b - a

for some c between a and b, which cannot happen because g¿sxd Z 0 in (a, b).

252

Chapter 4: Applications of Derivatives

We next apply the Mean Value Theorem to the function Fsxd = ƒsxd - ƒsad -

ƒsbd - ƒsad [ gsxd - gsad]. gsbd - gsad

This function is continuous and differentiable where ƒ and g are, and Fsbd = Fsad = 0. Therefore, there is a number c between a and b for which F¿scd = 0. When expressed in terms of ƒ and g, this equation becomes F¿scd = ƒ¿scd -

ƒsbd - ƒsad [ g¿scd] = 0 gsbd - gsad

so that ƒ¿scd ƒsbd - ƒsad = . g¿scd gsbd - gsad

y

slope 5

f '(c) g'(c) B (g(b), f (b))

P

slope 5

A 0

f (b) 2 f (a) g(b) 2 g(a)

Cauchy’s Mean Value Theorem has a geometric interpretation for a general winding curve C in the plane joining the two points A = sgsad, ƒsadd and B = sgsbd, ƒsbdd. In Chapter 11 you will learn how the curve C can be formulated so that there is at least one point P on the curve for which the tangent to the curve at P is parallel to the secant line joining the points A and B. The slope of that tangent line turns out to be the quotient ƒ¿>g¿ evaluated at the number c in the interval sa, bd, which is the left-hand side of the equation in Theorem 7. Because the slope of the secant line joining A and B is

(g(a), f (a))

ƒsbd - ƒsad , gsbd - gsad

x

FIGURE 4.34 There is at least one point P on the curve C for which the slope of the tangent to the curve at P is the same as the slope of the secant line joining the points A(g(a), ƒ(a)) and B(g(b), ƒ(b)).

the equation in Cauchy’s Mean Value Theorem says that the slope of the tangent line equals the slope of the secant line. This geometric interpretation is shown in Figure 4.34. Notice from the figure that it is possible for more than one point on the curve C to have a tangent line that is parallel to the secant line joining A and B. Proof of l’Hôpital’s Rule We first establish the limit equation for the case x : a + . The method needs almost no change to apply to x : a -, and the combination of these two cases establishes the result. Suppose that x lies to the right of a. Then g¿sxd Z 0, and we can apply Cauchy’s Mean Value Theorem to the closed interval from a to x. This step produces a number c between a and x such that ƒ¿scd ƒsxd - ƒsad = . g¿scd gsxd - gsad But ƒsad = gsad = 0, so ƒ¿scd ƒsxd = . g¿scd gsxd As x approaches a, c approaches a because it always lies between a and x. Therefore, lim+

x:a

ƒsxd ƒ¿scd ƒ¿sxd = lim+ = lim+ , gsxd c:a g¿scd x:a g¿sxd

which establishes l’Hôpital’s Rule for the case where x approaches a from above. The case where x approaches a from below is proved by applying Cauchy’s Mean Value Theorem to the closed interval [x, a], x 6 a.

4.5

Indeterminate Forms and L’Hôpital’s Rule

253

Exercises 4.5 Finding Limits in Two Ways In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

cos u - 1 u :0 e u - u - 1

43. lim

x + 2 1. lim 2 x: -2 x - 4

sin 5x 2. lim x x: 0

45. lim

5x 2 - 3x 3. lim x: q 7x 2 + 1

x3 - 1 4. lim 3 x: 1 4x - x - 3

47. lim

1 - cos x 5. lim x: 0 x2

2x 2 + 3x 6. lim 3 q x: x + x + 1

49. lim

t: q

t - 4t + 15 t: -3 t 2 - t - 12

9. lim

5x 3 - 2x 11. lim x: q 7x 3 + 3 sin t t t:0

8x x: 0 cos x - 1

3t + 3 t: -1 4t3 - t + 3

17. 19.

lim

u :p>2

2u - p cos (2p - u)

1 - sin u u :p>2 1 + cos 2u lim

21. lim

x: 0

x2 ln (sec x)

t (1 - cos t) 23. lim t:0 t - sin t 25.

lim

x: (p>2) -

sin x - x x: 0 x3

61. lim a

ax -

p b sec x 2

26.

lim

30. lim

a

32. lim

log2 x log3 (x + 3)

33. lim+

ln (x 2 + 2x) ln x

34. lim+

ln (e x - 1) ln x

25y + 25 - 5 y

36. lim

x: 0

35. lim

y: 0

37. lim (ln 2x - ln (x + 1)) x: q

39. lim+ x: 0

sln x) 2 ln ssin x)

41. lim+ a x: 1

1 1 b x - 1 ln x

x: q

x: 0

y: 0

2ay + a 2 - a , y

x + 2 b x - 1

x

lim x 2 ln x

64.

x:0 +

lim x tan a

x:0 +

62.

p - xb 2

66.

lim a

x: q

x

x2 + 1 b x + 2

1>x

lim x sln xd2

x:0 +

lim sin x # ln x

x:0 +

29x + 1

2x + 1 sec x 69. lim x:sp>2d - tan x 71.

2x - 3x x x x: q 3 + 4

73.

ex x q xe x:

lim

68. lim+

lim

2x

2sin x cot x 70. lim+ csc x x:0 x:0

72.

2x + 4x x x x: - q 5 - 2 lim

74.

lim

x:0 +

x e -1>x

75. Which one is correct, and which one is wrong? Give reasons for your answers. 0 x - 3 x - 3 1 1 b. lim 2 = lim = = 0 = 6 6 x:3 2x x:3 x - 3 x2 - 3 76. Which one is correct, and which one is wrong? Give reasons for your answers. a. lim

x:3

a 7 0

x 2 - 2x 2x - 2 = lim x:0 x 2 - sin x x:0 2x - cos x 2 2 = lim = = 1 2 + 0 x:0 2 + sin x

a. lim

3x + 1 1 b x sin x

42. lim+ (csc x - cot x + cos x) x: 0

x:0

2

x: 0

x: 0

58. lim (e x + x) 1>x 1 60. lim+ a1 + x b x:0

x: q

38. lim+ (ln x - ln sin x) 40. lim+ a

x: q

x:0

67. lim

3x - 1 x x: 0 2 - 1

ln (x + 1) log2 x

56. lim x 1>ln x

Theory and Applications L’Hôpital’s Rule does not help with the limits in Exercises 67–74. Try it—you just keep on cycling. Find the limits some other way.

2

p - xb tan x 2

31. lim

x: q

65.

(1>2) u - 1 u u: 0

x2x x: 0 2 - 1 x

(x - (p>2))

x: (p>2) -

28. lim

29. lim

63.

t sin t 24. lim t: 0 1 - cos t

3sin u - 1 u u :0

27. lim

3u + p sin (u + (p>3))

ln (csc x)

lim

x:e

lim+ x x

x: q

x - 1 x: 1 ln x - sin px x: p>2

54. lim+ (ln x) 1>(x - e)

x: q

59.

lim

x:1

1>x

-1>ln x

sin 5t t: 0 2t

u: -p>3

sin 3x - 3x + x 2 sin x sin 2x x:0

52. lim+ x 1>(x - 1)

57. lim (1 + 2x) 1>(2 ln x)

20. lim 22.

50. lim

x:0

x - 8x 2 12. lim x: q 12x 2 + 5x

18.

u - sin u cos u tan u - u u :0

55. lim+ x

16. lim

15. lim

48. lim

x: q

10. lim

2

se x - 1d2 x:0 x sin x

x - sin x x:0 x tan x

53. lim (ln x)

14. lim

13. lim

x: q

x:1

3

2

h2

h:0

46. lim x 2e -x

51. lim+ x 1>(1 - x)

x 2 - 25 8. lim x: - 5 x + 5

3

e h - (1 + h)

Indeterminate Powers and Products Find the limits in Exercise 51–66.

Applying l’Hôpital’s Rule Use l’Hôpital’s rule to find the limits in Exercises 7– 50. x - 2 7. lim 2 x: 2 x - 4

et + t2 et - t

44. lim

b.

x 2 - 2x 2x - 2 -2 = = 2 = lim 0 - 1 x:0 x 2 - sin x x:0 2x - cos x lim

254

Chapter 4: Applications of Derivatives

77. Only one of these calculations is correct. Which one? Why are the others wrong? Give reasons for your answers. a. lim+ x ln x = 0 # (- q ) = 0

a. Use l’Hôpital’s Rule to show that x

1 lim a1 + x b = e. x: q

x: 0

b. lim+ x ln x = 0 # (- q ) = - q x: 0

ln x -q = q = -1 (1>x) ln x d. lim+ x ln x = lim+ x: 0 x:0 (1>x) (1>x) = lim+ (-x) = 0 = lim+ x: 0 (-1>x 2) x: 0 78. Find all values of c that satisfy the conclusion of Cauchy’s Mean Value Theorem for the given functions and interval. c. lim+ x ln x = lim+ x: 0

T b. Graph

x:0

a. ƒsxd = x,

g sxd = x ,

sa, bd = s -2, 0d

b. ƒsxd = x,

g sxd = x ,

sa, bd arbitrary

2 2

c. ƒsxd = x 3>3 - 4x,

g sxd = x 2,

ƒ(x) = a1 +

9x - 3 sin 3x , 5x 3 c,

x = 0

continuous at x = 0 . Explain why your value of c works.

T 81. ˆ ˆ Form

85. Show that lim a1 +

k: q

lim A x - 2x 2 + x B

x: q

by graphing ƒsxd = x - 2x 2 + x over a suitably large interval of x-values. b. Now confirm your estimate by finding the limit with l’Hôpital’s Rule. As the first step, multiply ƒ(x) by the fraction sx + 2x 2 + xd>sx + 2x 2 + xd and simplify the new numerator.

A 2x 2 + 1 - 2x B .

Estimate the value of 2x 2 - s3x + 1d2x + 2 x - 1 x:1 lim

by graphing. Then confirm your estimate with l’Hôpital’s Rule. 84. This exercise explores the difference between the limit lim a1 +

x: q

k

r b = e r. k

86. Given that x 7 0, find the maximum value, if any, of a. x 1>x 2

b. x 1>x

n

c. x 1>x (n a positive integer) 87. Use limits to find horizontal asymptotes for each function. 1 a. y = x tan a x b 88. Find ƒ¿s0d for ƒsxd = e

a. Estimate the value of

T 83. 0/0 Form

x

n

tan 2x sin bx a lim a 3 + 2 + x b = 0? x:0 x x

x: q

1 g(x) = a1 + x b

d. Show that limx: q x 1>x = 1 for every positive integer n.

80. For what values of a and b is

82. Find lim

and

c. Confirm your estimate of limx: q ƒ(x) by calculating it with l’Hôpital’s Rule.

sa, bd = s0, 3d

x Z 0

x

together for x Ú 0. How does the behavior of ƒ compare with that of g? Estimate the value of limx: q ƒ(x).

79. Continuous extension Find a value of c that makes the function ƒsxd = •

1 b x2

1 b x2

x

and the limit x

1 lim a1 + x b = e. x: q

b. y = e -1/x , 0, 2

3x + e 2x 2x + e 3x

x Z 0 x = 0.

T 89. The continuous extension of (sin x)x to [0, p] a. Graph ƒ(x) = (sin x) x on the interval 0 … x … p. What value would you assign to ƒ to make it continuous at x = 0? b. Verify your conclusion in part (a) by finding limx:0+ ƒ(x) with l’Hôpital’s Rule. c. Returning to the graph, estimate the maximum value of ƒ on [0, p]. About where is max ƒ taken on? d. Sharpen your estimate in part (c) by graphing ƒ¿ in the same window to see where its graph crosses the x-axis. To simplify your work, you might want to delete the exponential factor from the expression for ƒ¿ and graph just the factor that has a zero. T 90. The function (sin x)tan x (Continuation of Exercise 89.) a. Graph ƒ(x) = (sin x) tan x on the interval -7 … x … 7. How do you account for the gaps in the graph? How wide are the gaps? b. Now graph ƒ on the interval 0 … x … p. The function is not defined at x = p>2, but the graph has no break at this point. What is going on? What value does the graph appear to give for ƒ at x = p>2? (Hint: Use l’Hôpital’s Rule to find lim ƒ as x : (p>2) - and x : (p>2) +.) c. Continuing with the graphs in part (b), find max ƒ and min ƒ as accurately as you can and estimate the values of x at which they are taken on.

4.6

Applied Optimization

255

Applied Optimization

4.6

What are the dimensions of a rectangle with fixed perimeter having maximum area? What are the dimensions for the least expensive cylindrical can of a given volume? How many items should be produced for the most profitable production run? Each of these questions asks for the best, or optimal, value of a given function. In this section we use derivatives to solve a variety of optimization problems in business, mathematics, physics, and economics.

x

Solving Applied Optimization Problems 1. Read the problem. Read the problem until you understand it. What is given? What is the unknown quantity to be optimized? 2. Draw a picture. Label any part that may be important to the problem. 3. Introduce variables. List every relation in the picture and in the problem as an equation or algebraic expression, and identify the unknown variable. 4. Write an equation for the unknown quantity. If you can, express the unknown as a function of a single variable or in two equations in two unknowns. This may require considerable manipulation. 5. Test the critical points and endpoints in the domain of the unknown. Use what you know about the shape of the function’s graph. Use the first and second derivatives to identify and classify the function’s critical points.

12

x x

x

12 (a)

x

12 2x 12 12 2x

x

EXAMPLE 1

An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-in. sheet of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible?

x

(b)

FIGURE 4.35 An open box made by cutting the corners from a square sheet of tin. What size corners maximize the box’s volume (Example 1)?

We start with a picture (Figure 4.35). In the figure, the corner squares are x in. on a side. The volume of the box is a function of this variable: Solution

Vsxd = xs12 - 2xd2 = 144x - 48x 2 + 4x 3.

Since the sides of the sheet of tin are only 12 in. long, x … 6 and the domain of V is the interval 0 … x … 6. A graph of V (Figure 4.36) suggests a minimum value of 0 at x = 0 and x = 6 and a maximum near x = 2. To learn more, we examine the first derivative of V with respect to x:

Maximum y

Volume

y x(12 – 2x)2, 0x6

0

dV = 144 - 96x + 12x 2 = 12s12 - 8x + x 2 d = 12s2 - xds6 - xd. dx

min

min 2

6

V = hlw

x

Of the two zeros, x = 2 and x = 6, only x = 2 lies in the interior of the function’s domain and makes the critical-point list. The values of V at this one critical point and two endpoints are

NOT TO SCALE

FIGURE 4.36 The volume of the box in Figure 4.35 graphed as a function of x.

Critical-point value:

Vs2d = 128

Endpoint values:

Vs0d = 0,

Vs6d = 0.

The maximum volume is 128 in3 . The cutout squares should be 2 in. on a side.

256

Chapter 4: Applications of Derivatives

EXAMPLE 2

You have been asked to design a one-liter can shaped like a right circular cylinder (Figure 4.37). What dimensions will use the least material?

2r

h

Solution Volume of can: If r and h are measured in centimeters, then the volume of the can in cubic centimeters is

pr 2h = 1000. Surface area of can: FIGURE 4.37 This one-liter can uses the least material when h = 2r (Example 2).

1 liter = 1000 cm3

A = ()* 2pr 2 + 2prh ()*

circular cylindrical ends wall

How can we interpret the phrase “least material”? For a first approximation we can ignore the thickness of the material and the waste in manufacturing. Then we ask for dimensions r and h that make the total surface area as small as possible while satisfying the constraint pr 2h = 1000. To express the surface area as a function of one variable, we solve for one of the variables in pr 2h = 1000 and substitute that expression into the surface area formula. Solving for h is easier: h =

1000 . pr 2

Thus, A = 2pr 2 + 2prh = 2pr 2 + 2pr a = 2pr 2 +

1000 b pr 2

2000 r .

Our goal is to find a value of r 7 0 that minimizes the value of A. Figure 4.38 suggests that such a value exists.

A Tall and thin can Short and wide can —— , r . 0 A 5 2pr 2 1 2000 r

Tall and thin

min

0 Short and wide

r 3

500 p

FIGURE 4.38 The graph of A = 2pr 2 + 2000>r is concave up.

Notice from the graph that for small r (a tall, thin cylindrical container), the term 2000>r dominates (see Section 2.6) and A is large. For large r (a short, wide cylindrical container), the term 2pr 2 dominates and A again is large.

4.6

Applied Optimization

257

Since A is differentiable on r 7 0, an interval with no endpoints, it can have a minimum value only where its first derivative is zero. 2000 dA = 4pr dr r2 2000 0 = 4pr r2 4pr 3 = 2000 r =

Set dA>dr = 0 . Multiply by r 2.

500 L 5.42 A p 3

Solve for r.

3 What happens at r = 2 500>p? The second derivative

4000 d 2A = 4p + dr 2 r3 is positive throughout the domain of A. The graph is therefore everywhere concave up and 3 500>p is an absolute minimum. the value of A at r = 2 The corresponding value of h (after a little algebra) is h =

1000 500 = 2 3 p = 2r. A pr 2

The one-liter can that uses the least material has height equal to twice the radius, here with r L 5.42 cm and h L 10.84 cm.

Examples from Mathematics and Physics EXAMPLE 3 A rectangle is to be inscribed in a semicircle of radius 2. What is the largest area the rectangle can have, and what are its dimensions?

y x 2 1 y2 5 4

Let sx, 24 - x 2 d be the coordinates of the corner of the rectangle obtained by placing the circle and rectangle in the coordinate plane (Figure 4.39). The length, height, and area of the rectangle can then be expressed in terms of the position x of the lower right-hand corner:

⎛x, 4 2 x 2⎛ Solution ⎝ ⎝ 2 –2 –x

0

x 2

FIGURE 4.39 The rectangle inscribed in the semicircle in Example 3.

x

Length: 2x,

Height: 24 - x 2,

Area: 2x24 - x 2 .

Notice that the values of x are to be found in the interval 0 … x … 2, where the selected corner of the rectangle lies. Our goal is to find the absolute maximum value of the function Asxd = 2x24 - x 2 on the domain [0, 2]. The derivative dA -2x 2 = + 224 - x 2 2 dx 24 - x is not defined when x = 2 and is equal to zero when -2x 2

+ 224 - x 2 = 0 24 - x 2 -2x 2 + 2s4 - x 2 d = 0 8 - 4x 2 = 0 x 2 = 2 or x = ; 22.

258

Chapter 4: Applications of Derivatives

Of the two zeros, x = 22 and x = - 22, only x = 22 lies in the interior of A’s domain and makes the critical-point list. The values of A at the endpoints and at this one critical point are Critical-point value: Endpoint values:

A A 22 B = 22224 - 2 = 4 As0d = 0,

As2d = 0.

The area has a maximum value of 4 when the rectangle is 24 - x 2 = 22 units high and 2x = 222 units long.

EXAMPLE 4

The speed of light depends on the medium through which it travels, and is generally slower in denser media. Fermat’s principle in optics states that light travels from one point to another along a path for which the time of travel is a minimum. Describe the path that a ray of light will follow in going from a point A in a medium where the speed of light is c1 to a point B in a second medium where its speed is c2 .

HISTORICAL BIOGRAPHY Willebrord Snell van Royen (1580–1626)

y A a

u1

Angle of incidence u1 P

0

x Medium 2

Since light traveling from A to B follows the quickest route, we look for a path that will minimize the travel time. We assume that A and B lie in the xy-plane and that the line separating the two media is the x-axis (Figure 4.40). In a uniform medium, where the speed of light remains constant, “shortest time” means “shortest path,” and the ray of light will follow a straight line. Thus the path from A to B will consist of a line segment from A to a boundary point P, followed by another line segment from P to B. Distance traveled equals rate times time, so Solution

Medium 1

b

u2 d2x

d Angle of refraction

x

B

Time =

FIGURE 4.40 A light ray refracted (deflected from its path) as it passes from one medium to a denser medium (Example 4).

distance rate .

From Figure 4.40, the time required for light to travel from A to P is 2a 2 + x 2 AP . t1 = c1 = c1 From P to B, the time is 2b 2 + sd - xd2 PB t2 = c2 = . c2 The time from A to B is the sum of these: t = t1 + t2 =

2b 2 + sd - xd2 2a 2 + x 2 + . c1 c2

This equation expresses t as a differentiable function of x whose domain is [0, d]. We want to find the absolute minimum value of t on this closed interval. We find the derivative dt x d - x = 2 2 2 dx c1 2a + x c2 2b + sd - xd2 dt/dx negative

0

dt/dx zero

x0

and observe that it is continuous. In terms of the angles u1 and u2 in Figure 4.40,

dt/dx positive x d

FIGURE 4.41 The sign pattern of dt>dx in Example 4.

sin u1 sin u2 dt = c1 - c2 . dx The function t has a negative derivative at x = 0 and a positive derivative at x = d. Since dt>dx is continuous over the interval [0, d], by the Intermediate Value Theorem for continuous functions (Section 2.5), there is a point x0 H [0, d ] where dt>dx = 0 (Figure 4.41).

4.6

Applied Optimization

259

There is only one such point because dt>dx is an increasing function of x (Exercise 62). At this unique point we then have sin u1 sin u2 c1 = c2 . This equation is Snell’s Law or the Law of Refraction, and is an important principle in the theory of optics. It describes the path the ray of light follows.

Examples from Economics Suppose that rsxd = the revenue from selling x items csxd = the cost of producing the x items psxd = rsxd - csxd = the profit from producing and selling x items. Although x is usually an integer in many applications, we can learn about the behavior of these functions by defining them for all nonzero real numbers and by assuming they are differentiable functions. Economists use the terms marginal revenue, marginal cost, and marginal profit to name the derivatives r¿(x), c¿(x), and p¿(x) of the revenue, cost, and profit functions. Let’s consider the relationship of the profit p to these derivatives. If r(x) and c(x) are differentiable for x in some interval of production possibilities, and if psxd = rsxd - csxd has a maximum value there, it occurs at a critical point of p(x) or at an endpoint of the interval. If it occurs at a critical point, then p¿sxd = r¿sxd c¿sxd = 0 and we see that r¿(x) = c¿(x). In economic terms, this last equation means that

At a production level yielding maximum profit, marginal revenue equals marginal cost (Figure 4.42).

y

Dollars

Cost c(x)

Revenue r(x) Break-even point B

0

Maximum profit, c'(x) r'(x)

Local maximum for loss (minimum profit), c'(x) r'(x) x Items produced

FIGURE 4.42 The graph of a typical cost function starts concave down and later turns concave up. It crosses the revenue curve at the break-even point B. To the left of B, the company operates at a loss. To the right, the company operates at a profit, with the maximum profit occurring where c¿sxd = r¿sxd . Farther to the right, cost exceeds revenue (perhaps because of a combination of rising labor and material costs and market saturation) and production levels become unprofitable again.

260

Chapter 4: Applications of Derivatives

Suppose that rsxd = 9x and csxd = x 3 - 6x 2 + 15x, where x represents millions of MP3 players produced. Is there a production level that maximizes profit? If so, what is it?

EXAMPLE 5

y

Notice that r¿sxd = 9 and c¿sxd = 3x 2 - 12x + 15.

Solution

3x 2 - 12x + 15 = 9 3x 2 - 12x + 6 = 0

c(x) x 3 6x2 15x

Set c¿sxd = r¿sxd.

The two solutions of the quadratic equation are r(x) 9x Maximum for profit

Local maximum for loss 0 2 2

x

2 2

2

NOT TO SCALE

FIGURE 4.43 The cost and revenue curves for Example 5.

x1 =

12 - 272 = 2 - 22 L 0.586 6

x2 =

12 + 272 = 2 + 22 L 3.414. 6

and

The possible production levels for maximum profit are x L 0.586 million MP3 players or x L 3.414 million. The second derivative of psxd = rsxd - csxd is p–sxd = -c–sxd since r–sxd is everywhere zero. Thus, p–(x) = 6(2 - x), which is negative at x = 2 + 22 and positive at x = 2 - 22. By the Second Derivative Test, a maximum profit occurs at about x = 3.414 (where revenue exceeds costs) and maximum loss occurs at about x = 0.586. The graphs of r(x) and c(x) are shown in Figure 4.43.

Exercises 4.6 Mathematical Applications Whenever you are maximizing or minimizing a function of a single variable, we urge you to graph it over the domain that is appropriate to the problem you are solving. The graph will provide insight before you calculate and will furnish a visual context for understanding your answer. 1. Minimizing perimeter What is the smallest perimeter possible for a rectangle whose area is 16 in2 , and what are its dimensions? 2. Show that among all rectangles with an 8-m perimeter, the one with largest area is a square. 3. The figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long. a. Express the y-coordinate of P in terms of x. (Hint: Write an equation for the line AB.) b. Express the area of the rectangle in terms of x. c. What is the largest area the rectangle can have, and what are its dimensions?

B

P(x, ?)

A 0

5. You are planning to make an open rectangular box from an 8-in.by-15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume? 6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0, b). Show that the area of the triangle enclosed by the segment is largest when a = b. 7. The best fencing plan A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? 8. The shortest fence A 216 m2 rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed?

y

–1

4. A rectangle has its base on the x-axis and its upper two vertices on the parabola y = 12 - x 2 . What is the largest area the rectangle can have, and what are its dimensions?

x

1

x

9. Designing a tank Your iron works has contracted to design and build a 500 ft3 , square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible.

4.6 a. What dimensions do you tell the shop to use?

x NOT TO SCALE

b. Briefly describe how you took weight into account. 3

10. Catching rainwater A 1125 ft open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy.

261

Applied Optimization

x

x

x

10"

Base

Lid

x

x x

x 15"

a. If the total cost is c = 5sx 2 + 4xyd + 10xy, what values of x and y will minimize it? b. Give a possible scenario for the cost function in part (a). 11. Designing a poster You are designing a rectangular poster to contain 50 in2 of printing with a 4-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used? 12. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.

a. Write a formula V(x) for the volume of the box. b. Find the domain of V for the problem situation and graph V over this domain. c. Use a graphical method to find the maximum volume and the value of x that gives it. d. Confirm your result in part (c) analytically. T 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid. a. Write a formula V(x) for the volume of the box. b. Find the domain of V for the problem situation and graph V over this domain.

3

3

y

c. Use a graphical method to find the maximum volume and the value of x that gives it. d. Confirm your result in part (c) analytically.

x

e. Find a value of x that yields a volume of 1120 in3 . f. Write a paragraph describing the issues that arise in part (b).

13. Two sides of a triangle have lengths a and b, and the angle between them is u . What value of u will maximize the triangle’s area? (Hint: A = s1>2dab sin u .) 14. Designing a can What are the dimensions of the lightest open-top right circular cylindrical can that will hold a volume of 1000 cm3 ? Compare the result here with the result in Example 2. 15. Designing a can You are designing a 1000 cm3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure 2r units on a side. The total amount of aluminum used up by the can will therefore be A = 8r 2 + 2prh rather than the A = 2pr 2 + 2prh in Example 2. In Example 2, the ratio of h to r for the most economical can was 2 to 1. What is the ratio now? T 16. Designing a box with a lid A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.

x

x

x

x

24"

24" x

x x

36"

x 18"

The sheet is then unfolded.

24"

Base

36"

18. A rectangle is to be inscribed under the arch of the curve y = 4 cos s0.5xd from x = -p to x = p . What are the dimensions of the rectangle with largest area, and what is the largest area?

262

Chapter 4: Applications of Derivatives

19. Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume?

cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.

20. a. The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume?

24. The trough in the figure is to be made to the dimensions shown. Only the angle u can be varied. What value of u will maximize the trough’s volume?

Girth distance around here

1'

1' 1'

Length

Square end

T b. Graph the volume of a 108-in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a). 21. (Continuation of Exercise 20.)

20'

25. Paper folding A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it with paper. a. Show that L 2 = 2x 3>s2x - 8.5d .

b. What value of x minimizes L 2 ?

a. Suppose that instead of having a box with square ends you have a box with square sides so that its dimensions are h by h by w and the girth is 2h + 2w . What dimensions will give the box its largest volume now?

c. What is the minimum value of L? D

C

R

Girth

L2 x 2

L

Crease

Q (originally at A)

h

x x A

w h

T b. Graph the volume as a function of h and compare what you see with your answer in part (a). 22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.

P

B

26. Constructing cylinders Compare the answers to the following two construction problems. a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume? b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?

y x Circumference 5 x

x y

y

23. A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the

(a)

(b)

4.6 27. Constructing cones A right triangle whose hypotenuse is 23 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume that can be made this way.

h

3

Applied Optimization

263

39. Shortest beam The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

Beam

Building

r

y x 28. Find the point on the line a + = 1 that is closest to the origin. b 29. Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible. 30. Find a postitive number for which the sum of its reciprocal and four times its square is the smallest possible. 31. A wire b m long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part? 32. Answer Exercise 31 if one piece is bent into a square and the other into a circle. 33. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.

w

5

4

h 3

34. Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 3. (See accompanying figure.)

w

8' wall 27'

40. Motion on a line The positions of two particles on the s-axis are s1 = sin t and s2 = sin st + p>3d , with s1 and s2 in meters and t in seconds. a. At what time(s) in the interval 0 … t … 2p do the particles meet? b. What is the farthest apart that the particles ever get? c. When in the interval 0 … t … 2p is the distance between the particles changing the fastest? 41. The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least? 42. Projectile motion The range R of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity y0 at an angle a with the horizontal, then in Chapter 12 we find that y02 R = g sin 2a,

h r5 3

35. What value of a makes ƒsxd = x 2 + sa>xd have a. a local minimum at x = 2 ? b. a point of inflection at x = 1 ? 36. What values of a and b make ƒsxd = x 3 + ax 2 + bx have a. a local maximum at x = -1 and a local minimum at x = 3 ? b. a local minimum at x = 4 and a point of inflection at x = 1 ? Physical Applications 37. Vertical motion The height above ground of an object moving vertically is given by s = -16t 2 + 96t + 112 ,

where g is the downward acceleration due to gravity. Find the angle a for which the range R is the largest possible. T 43. Strength of a beam The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. (See the accompanying figure.) a. Find the dimensions of the strongest beam that can be cut from a 12-in.-diameter cylindrical log. b. Graph S as a function of the beam’s width w, assuming the proportionality constant to be k = 1. Reconcile what you see with your answer in part (a). c. On the same screen, graph S as a function of the beam’s depth d, again taking k = 1 . Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of k? Try it.

with s in feet and t in seconds. Find a. the object’s velocity when t = 0; b. its maximum height and when it occurs; c. its velocity when s = 0 . 38. Quickest route Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time?

12" d

w

264

Chapter 4: Applications of Derivatives

T 44. Stiffness of a beam The stiffness S of a rectangular beam is proportional to its width times the cube of its depth. a. Find the dimensions of the stiffest beam that can be cut from a 12-in.-diameter cylindrical log. b. Graph S as a function of the beam’s width w, assuming the proportionality constant to be k = 1 . Reconcile what you see with your answer in part (a). c. On the same screen, graph S as a function of the beam’s depth d, again taking k = 1 . Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of k? Try it. 45. Frictionless cart A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 10 cos pt . a. What is the cart’s maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then?

c. The visibility that day was 5 nautical miles. Did the ships ever sight each other? T d. Graph s and ds>dt together as functions of t for -1 … t … 3 , using different colors if possible. Compare the graphs and reconcile what you see with your answers in parts (b) and (c). e. The graph of ds>dt looks as if it might have a horizontal asymptote in the first quadrant. This in turn suggests that ds>dt approaches a limiting value as t : q . What is this value? What is its relation to the ships’ individual speeds? 48. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.) Normal Light receiver

b. Where is the cart when the magnitude of the acceleration is greatest? What is the cart’s speed then?

0

10

Light source A

a. At what times in the interval 0 6 t do the masses pass each other? (Hint: sin 2t = 2 sin t cos t .) b. When in the interval 0 … t … 2p is the vertical distance between the masses the greatest? What is this distance? (Hint: cos 2t = 2 cos2 t - 1.)

s1 0 s2

m2

1

Angle of reflection 2

B

Plane mirror

s

46. Two masses hanging side by side from springs have positions s1 = 2 sin t and s2 = sin 2t , respectively.

m1

Angle of incidence

49. Tin pest When metallic tin is kept below 13.2°C, it slowly becomes brittle and crumbles to a gray powder. Tin objects eventually crumble to this gray powder spontaneously if kept in a cold climate for years. The Europeans who saw tin organ pipes in their churches crumble away years ago called the change tin pest because it seemed to be contagious, and indeed it was, for the gray powder is a catalyst for its own formation. A catalyst for a chemical reaction is a substance that controls the rate of reaction without undergoing any permanent change in itself. An autocatalytic reaction is one whose product is a catalyst for its own formation. Such a reaction may proceed slowly at first if the amount of catalyst present is small and slowly again at the end, when most of the original substance is used up. But in between, when both the substance and its catalyst product are abundant, the reaction proceeds at a faster pace. In some cases, it is reasonable to assume that the rate y = dx>dt of the reaction is proportional both to the amount of the original substance present and to the amount of product. That is, y may be considered to be a function of x alone, and y = k xsa - xd = kax - k x 2,

s

47. Distance between two ships At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yd) and continued to do so all day. Ship B was sailing east at 8 knots and continued to do so all day. a. Start counting time with t = 0 at noon and express the distance s between the ships as a function of t. b. How rapidly was the distance between the ships changing at noon? One hour later?

where x = the amount of product a = the amount of substance at the beginning k = a positive constant . At what value of x does the rate y have a maximum? What is the maximum value of y? 50. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the figure. Assume that the

4.6

Applied Optimization

265

landing path of the airplane is the graph of a cubic polynomial function y = ax 3 + bx 2 + cx + d, where y s -Ld = H and y s0d = 0 .

54. Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.

a. What is dy>dx at x = 0 ?

55. Show that if rsxd = 6x and csxd = x 3 - 6x 2 + 15x are your revenue and cost functions, then the best you can do is break even (have revenue equal cost).

b. What is dy>dx at x = -L ? c. Use the values for dy>dx at x = 0 and x = -L together with y s0d = 0 and y s -Ld = H to show that 3

2

x x y sxd = H c2 a b + 3 a b d. L L

57. You are to construct an open rectangular box with a square base and a volume of 48 ft3. If material for the bottom costs $6>ft2 and material for the sides costs $4>ft2, what dimensions will result in the least expensive box? What is the minimum cost?

y

Landing path

56. Production level Suppose that csxd = x 3 - 20x 2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items.

58. The 800-room Mega Motel chain is filled to capacity when the room charge is $50 per night. For each $10 increase in room charge, 40 fewer rooms are filled each night. What charge per room will result in the maximum revenue per night?

H = Cruising altitude Airport x L

Business and Economics 51. It costs you c dollars each to manufacture and distribute backpacks. If the backpacks sell at x dollars each, the number sold is given by

Biology 59. Sensitivity to medicine (Continuation of Exercise 72, Section 3.3.) Find the amount of medicine to which the body is most sensitive by finding the value of M that maximizes the derivative dR>dM , where R = M2 a

a n = x - c + bs100 - xd , where a and b are positive constants. What selling price will bring a maximum profit? 52. You operate a tour service that offers the following rates: $200 per person if 50 people (the minimum number to book the tour) go on the tour. For each additional person, up to a maximum of 80 people total, the rate per person is reduced by $2. It costs $6000 (a fixed cost) plus $32 per person to conduct the tour. How many people does it take to maximize your profit? 53. Wilson lot size formula One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is hq km Asqd = q + cm + , 2 where q is the quantity you order when things run low (shoes, radios, brooms, or whatever the item might be), k is the cost of placing an order (the same, no matter how often you order), c is the cost of one item (a constant), m is the number of items sold each week (a constant), and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). a. Your job, as the inventory manager for your store, is to find the quantity that will minimize A(q). What is it? (The formula you get for the answer is called the Wilson lot size formula.) b. Shipping costs sometimes depend on order size. When they do, it is more realistic to replace k by k + bq , the sum of k and a constant multiple of q. What is the most economical quantity to order now?

C M - b 2 3

and C is a constant. 60. How we cough a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity y can be modeled by the equation y = csr0 - rdr 2 cm>sec,

r0 … r … r0 , 2

where r0 is the rest radius of the trachea in centimeters and c is a positive constant whose value depends in part on the length of the trachea. Show that y is greatest when r = s2>3dr0; that is, when the trachea is about 33% contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough. T b. Take r0 to be 0.5 and c to be 1 and graph y over the interval 0 … r … 0.5 . Compare what you see with the claim that y is at a maximum when r = s2>3dr0 . Theory and Examples 61. An inequality for positive integers Show that if a, b, c, and d are positive integers, then sa 2 + 1dsb 2 + 1dsc 2 + 1dsd 2 + 1d Ú 16 . abcd

266

Chapter 4: Applications of Derivatives a. Explain why you need to consider values of x only in the interval [0, 2p] .

62. The derivative dt>dx in Example 4 a. Show that ƒsxd =

b. Is ƒ ever negative? Explain.

x

65. a. The function y = cot x - 22 csc x has an absolute maximum value on the interval 0 6 x 6 p . Find it.

2a 2 + x 2

is an increasing function of x.

T b. Graph the function and compare what you see with your answer in part (a).

b. Show that g sxd =

66. a. The function y = tan x + 3 cot x has an absolute minimum value on the interval 0 6 x 6 p>2 . Find it.

d - x 2b + sd - xd 2

2

T b. Graph the function and compare what you see with your answer in part (a).

is a decreasing function of x. c. Show that dt d - x x = 2 2 2 dx c1 2a + x c2 2b + sd - xd2 is an increasing function of x.

67. a. How close does the curve y = 2x come to the point (3>2, 0)? (Hint: If you minimize the square of the distance, you can avoid square roots.) T b. Graph the distance function D(x) and y = 2x together and reconcile what you see with your answer in part (a).

63. Let ƒ(x) and g(x) be the differentiable functions graphed here. Point c is the point where the vertical distance between the curves is the greatest. Is there anything special about the tangents to the two curves at c? Give reasons for your answer.

y

(x, x)

y x

y f (x) y g(x)

a

c

b

0

x

64. You have been asked to determine whether the function ƒsxd = 3 + 4 cos x + cos 2x is ever negative.

4.7

⎛ 3 , 0⎛ ⎝2 ⎝

x

68. a. How close does the semicircle y = 216 - x 2 come to the point A 1, 23 B ? T b. Graph the distance function and y = 216 - x 2 together and reconcile what you see with your answer in part (a).

Newton’s Method In this section we study a numerical method, called Newton’s method or the Newton–Raphson method, which is a technique to approximate the solution to an equation ƒsxd = 0. Essentially it uses tangent lines in place of the graph of y = ƒsxd near the points where ƒ is zero. (A value of x where ƒ is zero is a root of the function ƒ and a solution of the equation ƒsxd = 0.)

Procedure for Newton’s Method The goal of Newton’s method for estimating a solution of an equation ƒsxd = 0 is to produce a sequence of approximations that approach the solution. We pick the first number x0 of the sequence. Then, under favorable circumstances, the method does the rest by moving step by step toward a point where the graph of ƒ crosses the x-axis (Figure 4.44). At each step the method approximates a zero of ƒ with a zero of one of its linearizations. Here is how it works. The initial estimate, x0 , may be found by graphing or just plain guessing. The method then uses the tangent to the curve y = ƒsxd at sx0, ƒsx0 dd to approximate the curve, calling

4.7

y f (x) (x0, f (x0))

y = ƒsxn d + ƒ¿sxn dsx - xn d.

(x1, f (x1))

We can find where it crosses the x-axis by setting y = 0 (Figure 4.45):

(x2, f(x2 ))

0 = ƒsxn d + ƒ¿sxn dsx - xn d

Root sought

-

x x3 Fourth

x2 Third

x1 Second

x0 First

ƒsxn d = x - xn ƒ¿sxn d

APPROXIMATIONS

x = xn -

FIGURE 4.44 Newton’s method starts with an initial guess x0 and (under favorable circumstances) improves the guess one step at a time.

If ƒ¿sxn d Z 0

Newton’s Method 1. Guess a first approximation to a solution of the equation ƒsxd = 0. A graph of y = ƒsxd may help. 2. Use the first approximation to get a second, the second to get a third, and so on, using the formula

y f (x) Point: (xn, f (xn )) Slope: f'(xn ) Tangent line equation: y f (xn ) f '(xn )(x xn ) (xn, f (xn))

ƒsxn d ƒ¿sxn d

This value of x is the next approximation xn + 1 . Here is a summary of Newton’s method.

y

xn + 1 = xn -

Tangent line (graph of linearization of f at xn )

Root sought 0

267

the point x1 where the tangent meets the x-axis (Figure 4.44). The number x1 is usually a better approximation to the solution than is x0 . The point x2 where the tangent to the curve at sx1, ƒsx1 dd crosses the x-axis is the next approximation in the sequence. We continue on, using each approximation to generate the next, until we are close enough to the root to stop. We can derive a formula for generating the successive approximations in the following way. Given the approximation xn , the point-slope equation for the tangent to the curve at sxn, ƒsxn dd is

y

0

Newton’s Method

ƒsxn d , ƒ¿sxn d

if ƒ¿sxn d Z 0.

(1)

Applying Newton’s Method xn

f (xn ) xn1 xn f'(xn )

FIGURE 4.45 The geometry of the successive steps of Newton’s method. From xn we go up to the curve and follow the tangent line down to find xn + 1 .

x

Applications of Newton’s method generally involve many numerical computations, making them well suited for computers or calculators. Nevertheless, even when the calculations are done by hand (which may be very tedious), they give a powerful way to find solutions of equations. In our first example, we find decimal approximations to 22 by estimating the positive root of the equation ƒsxd = x 2 - 2 = 0.

EXAMPLE 1

Find the positive root of the equation ƒsxd = x 2 - 2 = 0.

Solution

With ƒsxd = x 2 - 2 and ƒ¿sxd = 2x, Equation (1) becomes xn + 1 = xn -

xn 2 - 2 2xn

= xn -

xn 1 + xn 2

=

xn 1 + xn . 2

268

Chapter 4: Applications of Derivatives y

The equation

20

xn + 1 =

y x3 x 1

xn 1 + xn 2

enables us to go from each approximation to the next with just a few keystrokes. With the starting value x0 = 1, we get the results in the first column of the following table. (To five decimal places, 22 = 1.41421.)

15 10

Error

Number of correct digits

-0.41421 0.08579 0.00246 0.00001

1 1 3 5

5

–1

0

2

1

x

3

x0 x1 x2 x3

FIGURE 4.46 The graph of ƒsxd = x 3 - x - 1 crosses the x-axis once; this is the root we want to find (Example 2).

= = = =

1 1.5 1.41667 1.41422

Newton’s method is the method used by most calculators to calculate roots because it converges so fast (more about this later). If the arithmetic in the table in Example 1 had been carried to 13 decimal places instead of 5, then going one step further would have given 22 correctly to more than 10 decimal places.

y x3 x 1 (1.5, 0.875)

Find the x-coordinate of the point where the curve y = x 3 - x crosses the horizontal line y = 1.

EXAMPLE 2

Root sought x1

x2

x0

x 1

1.5

The curve crosses the line when x 3 - x = 1 or x 3 - x - 1 = 0. When does ƒsxd = x - x - 1 equal zero? Since ƒs1d = -1 and ƒs2d = 5, we know by the Intermediate Value Theorem there is a root in the interval (1, 2) (Figure 4.46). We apply Newton’s method to ƒ with the starting value x0 = 1. The results are displayed in Table 4.1 and Figure 4.47. At n = 5, we come to the result x6 = x5 = 1.3247 17957. When xn + 1 = xn , Equation (1) shows that ƒsxn d = 0. We have found a solution of ƒsxd = 0 to nine decimals. Solution

3

1.3478 (1, –1)

FIGURE 4.47 The first three x-values in Table 4.1 (four decimal places).

TABLE 4.1 The result of applying Newton’s method to ƒsxd = x3 - x - 1

with x0 = 1

y 25

n

xn

ƒ(xn)

ƒ(xn)

xn1 xn

0 1 2 3 4 5

1 1.5 1.3478 26087 1.3252 00399 1.3247 18174 1.3247 17957

-1 0.875 0.1006 82173 0.0020 58362 0.0000 00924 -1.8672E-13

2 5.75 4.4499 05482 4.2684 68292 4.2646 34722 4.2646 32999

1.5 1.3478 26087 1.3252 00399 1.3247 18174 1.3247 17957 1.3247 17957

B0(3, 23) 20 y x3 x 1 15

10 B1(2.12, 6.35) 5 –13 –1

0

ƒsxn d ƒ¿sxn d

Root sought 13

x2 x1 1

1.6 2.12

x0

x

3

FIGURE 4.48 Any starting value x0 to the right of x = 1> 23 will lead to the root.

In Figure 4.48 we have indicated that the process in Example 2 might have started at the point B0s3, 23d on the curve, with x0 = 3. Point B0 is quite far from the x-axis, but the tangent at B0 crosses the x-axis at about (2.12, 0), so x1 is still an improvement over x0 . If we use Equation (1) repeatedly as before, with ƒsxd = x 3 - x - 1 and ƒ¿sxd = 3x 2 - 1, we obtain the nine-place solution x7 = x6 = 1.3247 17957 in seven steps.

4.7

Newton’s Method

269

Convergence of the Approximations

y y f (x)

r 0

x0

x1

x

In Chapter 9 we define precisely the idea of convergence for the approximations xn in Newton’s method. Intuitively, we mean that as the number n of approximations increases without bound, the values xn get arbitrarily close to the desired root r. (This notion is similar to the idea of the limit of a function g(t) as t approaches infinity, as defined in Section 2.6.) In practice, Newton’s method usually gives convergence with impressive speed, but this is not guaranteed. One way to test convergence is to begin by graphing the function to estimate a good starting value for x0 . You can test that you are getting closer to a zero of the function by evaluating ƒ ƒsxn d ƒ , and check that the approximations are converging by evaluating ƒ xn - xn + 1 ƒ . Newton’s method does not always converge. For instance, if ƒsxd = e

FIGURE 4.49 Newton’s method fails to converge. You go from x0 to x1 and back to x0 , never getting any closer to r.

- 2r - x, 2x - r,

x 6 r x Ú r,

the graph will be like the one in Figure 4.49. If we begin with x0 = r - h, we get x1 = r + h, and successive approximations go back and forth between these two values. No amount of iteration brings us closer to the root than our first guess. If Newton’s method does converge, it converges to a root. Be careful, however. There are situations in which the method appears to converge but there is no root there. Fortunately, such situations are rare. When Newton’s method converges to a root, it may not be the root you have in mind. Figure 4.50 shows two ways this can happen. y f (x) Starting point Root sought

FIGURE 4.50

x0

Root found

y f (x) x1

x

x2

x1

Root sought

Root found

x

x0

Starting point

If you start too far away, Newton’s method may miss the root you want.

Exercises 4.7 Root Finding 1. Use Newton’s method to estimate the solutions of the equation x 2 + x - 1 = 0 . Start with x0 = -1 for the left-hand solution and with x0 = 1 for the solution on the right. Then, in each case, find x2 .

6. Use Newton’s method to find the negative fourth root of 2 by solving the equation x 4 - 2 = 0 . Start with x0 = -1 and find x2 .

2. Use Newton’s method to estimate the one real solution of x 3 + 3x + 1 = 0 . Start with x0 = 0 and then find x2 .

8. Estimating pi You plan to estimate p>2 to five decimal places by using Newton’s method to solve the equation cos x = 0 . Does it matter what your starting value is? Give reasons for your answer.

3. Use Newton’s method to estimate the two zeros of the function ƒsxd = x 4 + x - 3 . Start with x0 = -1 for the left-hand zero and with x0 = 1 for the zero on the right. Then, in each case, find x2 . 4. Use Newton’s method to estimate the two zeros of the function ƒsxd = 2x - x 2 + 1 . Start with x0 = 0 for the left-hand zero and with x0 = 2 for the zero on the right. Then, in each case, find x2 . 5. Use Newton’s method to find the positive fourth root of 2 by solving the equation x 4 - 2 = 0 . Start with x0 = 1 and find x2 .

7. Guessing a root Suppose that your first guess is lucky, in the sense that x0 is a root of ƒsxd = 0 . Assuming that ƒ¿sx0 d is defined and not 0, what happens to x1 and later approximations?

Theory and Examples 9. Oscillation Show that if h 7 0 , applying Newton’s method to ƒsxd = •

2x,

x Ú 0

2 -x, x 6 0

leads to x1 = -h if x0 = h and to x1 = h if x0 = -h . Draw a picture that shows what is going on.

270

Chapter 4: Applications of Derivatives

10. Approximations that get worse and worse Apply Newton’s method to ƒsxd = x 1>3 with x0 = 1 and calculate x1 , x2 , x3 , and x4 . Find a formula for ƒ xn ƒ . What happens to ƒ xn ƒ as n : q ? Draw a picture that shows what is going on.

23. Intersection of curves At 2 e -x = x 2 - x + 1?

what

value(s)

of

x

does

24. Intersection of curves At ln (1 - x 2) = x - 1?

what

value(s)

of

x

does

11. Explain why the following four statements ask for the same information: iii) Find the roots of ƒsxd = x 3 - 3x - 1.

25. Use the Intermediate Value Theorem from Section 2.5 to show that ƒsxd = x 3 + 2x - 4 has a root between x = 1 and x = 2. Then find the root to five decimal places.

iii) Find the x-coordinates of the intersections of the curve y = x 3 with the line y = 3x + 1.

26. Factoring a quartic Find the approximate values of r1 through r4 in the factorization

iii) Find the x-coordinates of the points where the curve y = x 3 - 3x crosses the horizontal line y = 1 .

8x 4 - 14x 3 - 9x 2 + 11x - 1 = 8sx - r1 dsx - r2 dsx - r3 dsx - r4 d.

iv) Find the values of x where the derivative of g sxd = s1>4dx 4 - s3>2dx 2 - x + 5 equals zero. 12. Locating a planet To calculate a planet’s space coordinates, we have to solve equations like x = 1 + 0.5 sin x . Graphing the function ƒsxd = x - 1 - 0.5 sin x suggests that the function has a root near x = 1.5 . Use one application of Newton’s method to improve this estimate. That is, start with x0 = 1.5 and find x1 . (The value of the root is 1.49870 to five decimal places.) Remember to use radians. T 13. Intersecting curves The curve y = tan x crosses the line y = 2x between x = 0 and x = p>2 . Use Newton’s method to find where. T 14. Real solutions of a quartic Use Newton’s method to find the two real solutions of the equation x 4 - 2x 3 - x 2 - 2x + 2 = 0 . T 15. a. How many solutions does the equation sin 3x = 0.99 - x 2 have? b. Use Newton’s method to find them.

y

y 8x 4 14x 3 9x 2 11x 1

2 –1

1

–2 –4 –6 –8 –10 –12

T 27. Converging to different zeros Use Newton’s method to find the zeros of ƒsxd = 4x 4 - 4x 2 using the given starting values. a. x0 = -2 and x0 = -0.8 , lying in A - q , - 22>2 B b. x0 = -0.5 and x0 = 0.25 , lying in A - 221>7, 221>7 B c. x0 = 0.8 and x0 = 2 , lying in A 22>2, q B

d. x0 = - 221>7 and x0 = 221>7

16. Intersection of curves a. Does cos 3x ever equal x? Give reasons for your answer. b. Use Newton’s method to find where. 17. Find the four real zeros of the function ƒsxd = 2x 4 - 4x 2 + 1. T 18. Estimating pi Estimate p to as many decimal places as your calculator will display by using Newton’s method to solve the equation tan x = 0 with x0 = 3. 19. Intersection of curves At what value(s) of x does cos x = 2x? 20. Intersection of curves At what value(s) of x does cos x = -x?

28. The sonobuoy problem In submarine location problems, it is often necessary to find a submarine’s closest point of approach (CPA) to a sonobuoy (sound detector) in the water. Suppose that the submarine travels on the parabolic path y = x 2 and that the buoy is located at the point s2, -1>2d . a. Show that the value of x that minimizes the distance between the submarine and the buoy is a solution of the equation x = 1>sx 2 + 1d . b. Solve the equation x = 1>sx 2 + 1d with Newton’s method. y

21. The graphs of y = x 2(x + 1) and y = 1>x (x 7 0) intersect at one point x = r. Use Newton’s method to estimate the value of r to four decimal places. y

x

2

y 5 x 2(x 1 1)

y x2 Submarine track in two dimensions

1 CPA

3

1 –1

0

0

⎛r, 1 ⎛ ⎝ r⎝ y 5 1x

2

1

2

1

2

x

1 Sonobuoy ⎛⎝2, – ⎛⎝ 2 x

22. The graphs of y = 2x and y = 3 - x 2 intersect at one point x = r. Use Newton’s method to estimate the value of r to four decimal places.

T 29. Curves that are nearly flat at the root Some curves are so flat that, in practice, Newton’s method stops too far from the root to give a useful estimate. Try Newton’s method on ƒsxd = sx - 1d40 with a starting value of x0 = 2 to see how close your machine comes to the root x = 1 . See the accompanying graph.

4.8

Antiderivatives

30. The accompanying figure shows a circle of radius r with a chord of length 2 and an arc s of length 3. Use Newton’s method to solve for r and u (radians) to four decimal places. Assume 0 6 u 6 p.

y

s5 3

r u y (x 1) 40 Slope 40

1

(2, 1)

Nearly flat

4.8

2

r

Slope –40

0

271

1

x

2

Antiderivatives We have studied how to find the derivative of a function. However, many problems require that we recover a function from its known derivative (from its known rate of change). For instance, the laws of physics tell us the acceleration of an object falling from an initial height and we can use this to compute its velocity and its height at any time. More generally, starting with a function ƒ, we want to find a function F whose derivative is ƒ. If such a function F exists, it is called an antiderivative of ƒ. We will see in the next chapter that antiderivatives are the link connecting the two major elements of calculus: derivatives and definite integrals.

Finding Antiderivatives DEFINITION A function F is an antiderivative of ƒ on an interval I if F¿sxd = ƒsxd for all x in I.

The process of recovering a function F(x) from its derivative ƒ(x) is called antidifferentiation. We use capital letters such as F to represent an antiderivative of a function ƒ, G to represent an antiderivative of g, and so forth.

EXAMPLE 1

Find an antiderivative for each of the following functions. 1 (a) ƒsxd = 2x (b) gsxd = cos x (c) h(x) = x + 2e 2x We need to think backward here: What function do we know has a derivative equal to the given function?

Solution

(a) Fsxd = x 2

(b) Gsxd = sin x

(c) H(x) = ln ƒ x ƒ + e 2x

Each answer can be checked by differentiating. The derivative of Fsxd = x 2 is 2x. The derivative of Gsxd = sin x is cos x, and the derivative of H(x) = ln ƒ x ƒ + e 2x is (1>x) + 2e 2x.

272

Chapter 4: Applications of Derivatives

The function Fsxd = x 2 is not the only function whose derivative is 2x. The function x + 1 has the same derivative. So does x 2 + C for any constant C. Are there others? Corollary 2 of the Mean Value Theorem in Section 4.2 gives the answer: Any two antiderivatives of a function differ by a constant. So the functions x 2 + C, where C is an arbitrary constant, form all the antiderivatives of ƒsxd = 2x. More generally, we have the following result. 2

THEOREM 8 If F is an antiderivative of ƒ on an interval I, then the most general antiderivative of ƒ on I is Fsxd + C where C is an arbitrary constant.

Thus the most general antiderivative of ƒ on I is a family of functions Fsxd + C whose graphs are vertical translations of one another. We can select a particular antiderivative from this family by assigning a specific value to C. Here is an example showing how such an assignment might be made.

EXAMPLE 2

y

y

x3

C 2 1

C2 C1 C0 C –1 C –2 x

0 –1

Solution

Find an antiderivative of ƒsxd = 3x 2 that satisfies Fs1d = -1.

Since the derivative of x 3 is 3x 2, the general antiderivative Fsxd = x 3 + C

gives all the antiderivatives of ƒ(x). The condition Fs1d = -1 determines a specific value for C. Substituting x = 1 into Fsxd = x 3 + C gives Fs1d = (1) 3 + C = 1 + C.

(1, –1)

–2

Since Fs1d = -1, solving 1 + C = -1 for C gives C = -2. So Fsxd = x 3 - 2

FIGURE 4.51 The curves y = x 3 + C fill the coordinate plane without overlapping. In Example 2, we identify the curve y = x 3 - 2 as the one that passes through the given point s1, - 1d .

is the antiderivative satisfying Fs1d = -1. Notice that this assignment for C selects the particular curve from the family of curves y = x 3 + C that passes through the point (1, -1) in the plane (Figure 4.51). By working backward from assorted differentiation rules, we can derive formulas and rules for antiderivatives. In each case there is an arbitrary constant C in the general expression representing all antiderivatives of a given function. Table 4.2 gives antiderivative formulas for a number of important functions. The rules in Table 4.2 are easily verified by differentiating the general antiderivative formula to obtain the function to its left. For example, the derivative of (tan kx)>k + C is sec2 kx, whatever the value of the constants C or k Z 0, and this establishes Formula 4 for the most general antiderivative of sec2 kx.

EXAMPLE 3

Find the general antiderivative of each of the following functions.

(a) ƒsxd = x 5 (d) isxd = cos

1 2x (e) j(x) = e -3x

(b) gsxd =

x 2

(c) hsxd = sin 2x (f ) k(x) = 2x

4.8

Antiderivatives

273

TABLE 4.2 Antiderivative formulas, k a nonzero constant

Function

General antiderivative

1.

xn

1 x n + 1 + C, n + 1

2.

sin kx

3.

cos kx

4.

sec2 kx

5.

csc2 kx

1 - cos kx + C k 1 sin kx + C k 1 tan kx + C k 1 - cot kx + C k

6.

sec kx tan kx

7.

csc kx cot kx

Function

n Z -1

8.

e kx

1 kx e + C k

9.

1 x

ln ƒ x ƒ + C,

1 21 - k 2x 2

1 -1 sin kx + C k

1 1 + k 2x 2 1 x2k 2x 2 - 1

1 tan-1 kx + C k

10. 11. 12.

1 sec kx + C k 1 - csc kx + C k

General antiderivative

13.

sec-1 kx + C, kx 7 1 a

a kx

x Z 0

1 b a kx + C, a 7 0, a Z 1 k ln a

In each case, we can use one of the formulas listed in Table 4.2.

Solution (a) Fsxd =

x6 + C 6

Formula 1 with n = 5

(b) gsxd = x -1>2 , so

Gsxd = (c) Hsxd = (d) Isxd =

x 1>2 + C = 22x + C 1>2

Formula 1 with n = -1>2

-cos 2x + C 2

Formula 2 with k = 2

sin sx>2d x + C = 2 sin + C 2 1>2

(e) J(x) = -

1 -3x + C e 3

(f) K(x) = a

Formula 3 with k = 1>2 Formula 8 with k = - 3

1 b 2x + C ln 2

Formula 13 with a = 2, k = 1

Other derivative rules also lead to corresponding antiderivative rules. We can add and subtract antiderivatives and multiply them by constants. TABLE 4.3 Antiderivative linearity rules

1. 2. 3.

Constant Multiple Rule: Negative Rule: Sum or Difference Rule:

Function

General antiderivative

kƒ(x) -ƒsxd ƒsxd ; gsxd

kFsxd + C, k a constant -Fsxd + C Fsxd ; Gsxd + C

The formulas in Table 4.3 are easily proved by differentiating the antiderivatives and verifying that the result agrees with the original function. Formula 2 is the special case k = -1 in Formula 1.

274

Chapter 4: Applications of Derivatives

EXAMPLE 4

Find the general antiderivative of ƒsxd =

3 2x

+ sin 2x.

Solution We have that ƒsxd = 3gsxd + hsxd for the functions g and h in Example 3. Since Gsxd = 22x is an antiderivative of g(x) from Example 3b, it follows from the Constant Multiple Rule for antiderivatives that 3Gsxd = 3 # 22x = 62x is an antiderivative of 3gsxd = 3> 2x. Likewise, from Example 3c we know that Hsxd = s -1>2d cos 2x is an antiderivative of hsxd = sin 2x. From the Sum Rule for antiderivatives, we then get that

Fsxd = 3Gsxd + Hsxd + C = 62x -

1 cos 2x + C 2

is the general antiderivative formula for ƒ(x), where C is an arbitrary constant.

Initial Value Problems and Differential Equations Antiderivatives play several important roles in mathematics and its applications. Methods and techniques for finding them are a major part of calculus, and we take up that study in Chapter 8. Finding an antiderivative for a function ƒ(x) is the same problem as finding a function y(x) that satisfies the equation dy = ƒsxd. dx This is called a differential equation, since it is an equation involving an unknown function y that is being differentiated. To solve it, we need a function y(x) that satisfies the equation. This function is found by taking the antiderivative of ƒ(x). We fix the arbitrary constant arising in the antidifferentiation process by specifying an initial condition ysx0 d = y0 . This condition means the function y(x) has the value y0 when x = x0 . The combination of a differential equation and an initial condition is called an initial value problem. Such problems play important roles in all branches of science. The most general antiderivative Fsxd + C (such as x 3 + C in Example 2) of the function ƒ(x) gives the general solution y = Fsxd + C of the differential equation dy>dx = ƒsxd. The general solution gives all the solutions of the equation (there are infinitely many, one for each value of C). We solve the differential equation by finding its general solution. We then solve the initial value problem by finding the particular solution that satisfies the initial condition ysx0 d = y0 . In Example 2, the function y = x 3 - 2 is the particular solution of the differential equation dy>dx = 3x 2 satisfying the initial condition y(1) = -1.

Antiderivatives and Motion We have seen that the derivative of the position function of an object gives its velocity, and the derivative of its velocity function gives its acceleration. If we know an object’s acceleration, then by finding an antiderivative we can recover the velocity, and from an antiderivative of the velocity we can recover its position function. This procedure was used as an application of Corollary 2 in Section 4.2. Now that we have a terminology and conceptual framework in terms of antiderivatives, we revisit the problem from the point of view of differential equations.

EXAMPLE 5

A hot-air balloon ascending at the rate of 12 ft>sec is at a height 80 ft above the ground when a package is dropped. How long does it take the package to reach the ground?

4.8

Antiderivatives

275

Solution Let y(t) denote the velocity of the package at time t, and let s(t) denote its height above the ground. The acceleration of gravity near the surface of the earth is 32 ft>sec2. Assuming no other forces act on the dropped package, we have

dy = -32. dt

Negative because gravity acts in the direction of decreasing s

This leads to the following initial value problem (Figure 4.52):

s v(0) 5 12

dy = -32 dt ys0d = 12.

Differential equation: Initial condition:

Balloon initially rising

This is our mathematical model for the package’s motion. We solve the initial value problem to obtain the velocity of the package. 1.

Solve the differential equation: The general formula for an antiderivative of -32 is y = -32t + C.

dv –32 5 dt s(t)

2.

Having found the general solution of the differential equation, we use the initial condition to find the particular solution that solves our problem. Evaluate C: 12 = -32s0d + C

0

ground

Initial condition ys0d = 12

C = 12.

FIGURE 4.52 A package dropped from a rising hot-air balloon (Example 5).

The solution of the initial value problem is y = -32t + 12. Since velocity is the derivative of height, and the height of the package is 80 ft at time t = 0 when it is dropped, we now have a second initial value problem. Differential equation: Initial condition:

ds = -32t + 12 dt ss0d = 80

Set y = ds>dt in the previous equation.

We solve this initial value problem to find the height as a function of t. 1.

Solve the differential equation: Finding the general antiderivative of -32t + 12 gives s = -16t 2 + 12t + C.

2.

Evaluate C: 80 = -16s0d2 + 12s0d + C C = 80.

Initial condition ss0d = 80

The package’s height above ground at time t is s = -16t 2 + 12t + 80. Use the solution: To find how long it takes the package to reach the ground, we set s equal to 0 and solve for t: -16t 2 + 12t + 80 = 0 -4t 2 + 3t + 20 = 0 -3 ; 2329 -8 t L -1.89, t L 2.64. t =

Quadratic formula

The package hits the ground about 2.64 sec after it is dropped from the balloon. (The negative root has no physical meaning.)

Chapter 4: Applications of Derivatives

Indefinite Integrals A special symbol is used to denote the collection of all antiderivatives of a function ƒ.

DEFINITION The collection of all antiderivatives of ƒ is called the indefinite integral of ƒ with respect to x, and is denoted by L

ƒsxd dx.

The symbol 1 is an integral sign. The function ƒ is the integrand of the integral, and x is the variable of integration.

After the integral sign in the notation we just defined, the integrand function is always followed by a differential to indicate the variable of integration. We will have more to say about why this is important in Chapter 5. Using this notation, we restate the solutions of Example 1, as follows: 2x dx = x 2 + C,

L L

cos x dx = sin x + C, 1 a x + 2e 2x b dx = ln ƒ x ƒ + e 2x + C.

L

This notation is related to the main application of antiderivatives, which will be explored in Chapter 5. Antiderivatives play a key role in computing limits of certain infinite sums, an unexpected and wonderfully useful role that is described in a central result of Chapter 5, called the Fundamental Theorem of Calculus.

EXAMPLE 6

Evaluate L

sx 2 - 2x + 5d dx.

If we recognize that sx 3>3d - x 2 + 5x is an antiderivative of x 2 - 2x + 5, we can evaluate the integral as Solution

antiderivative $++%++&

L

(x 2 - 2x + 5) dx =

x3 - x 2 + 5x + C. 3 #

276

arbitrary constant

If we do not recognize the antiderivative right away, we can generate it term-by-term with the Sum, Difference, and Constant Multiple Rules: L

sx 2 - 2x + 5d dx = =

L L

= a =

x 2 dx -

L

x 2 dx - 2

2x dx +

L

L

x dx + 5

5 dx

L

1 dx

x2 x3 + C1 b - 2 a + C2 b + 5sx + C3 d 3 2

x3 + C1 - x 2 - 2C2 + 5x + 5C3 . 3

4.8

Antiderivatives

277

This formula is more complicated than it needs to be. If we combine C1, -2C2 , and 5C3 into a single arbitrary constant C = C1 - 2C2 + 5C3 , the formula simplifies to x3 - x 2 + 5x + C 3 and still gives all the possible antiderivatives there are. For this reason, we recommend that you go right to the final form even if you elect to integrate term-by-term. Write L

sx 2 - 2x + 5d dx =

x 2 dx 2x dx + 5 dx L L L x3 = - x 2 + 5x + C. 3

Find the simplest antiderivative you can for each part and add the arbitrary constant of integration at the end.

Exercises 4.8 Finding Antiderivatives In Exercises 1–24, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

18. a. sec x tan x

b. 4 sec 3x tan 3x

c. sec

19. a. e 3x

b. e -x

c. e x>2

-2x

1. a. 2x

b. x

c. x - 2x + 1

20. a. e

2. a. 6x

b. x 7

c. x 7 - 6x + 8

21. a. 3x

b. 2-x

3. a. -3x -4

b. x -4

c. x -4 + 2x + 3

22. a. x 23

b. x p

2

-3

b.

x + x2 2

c. -x -3 + x - 1

b.

5 x2

c. 2 -

2 x3

b.

1 2x 3

c. x 3 -

7. a.

3 2x 2

b.

8. a.

4 3 2x 3

b.

9. a.

2 -1>3 x 3

10. a.

1 -1>2 x 2

4. a. 2x -3 5. a.

1 x2

6. a. -

1 2 2x 1

3 32 x 1 -2>3 b. x 3

b. -

1 -3>2 x 2

7 b. x

1 11. a. x 1 3x

5 x2 1 x3

c. 2x + 3 x + c. 2

1 2x 1

2 5x

27.

c. -

3 -5>2 x 2

29. 31.

b. 3 sin x

14. a. p cos px

px p cos b. 2 2

px + p cos x c. cos 2

15. a. sec x

x 2 b. sec2 3 3

3x c. -sec 2

16. a. csc2 x

b. -

17. a. csc x cot x

b. - csc 5x cot 5x

2

3x 3 csc2 2 2

25.

1 -4>3 x 3

13. a. -p sin px

b.

c. 1 +

2

c. 1 - 8 csc2 2x c. -p csc

px px cot 2 2

b.

c. x 22 - 1 1 c. 1 + 4x 2

1 2(x 2 + 1)

c. p x - x -1

b. x 2 + 2x

Finding Indefinite Integrals In Exercises 25–70, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

c. -

5 c. 1 - x

2

21 - x 2 x 1 24. a. x - a b 2

3 2 x

4 1 - 2 3x x c. sin px - 3 sin 3x

12. a.

23. a.

b. e

c. e -x>5 x 5 c. a b 3

4x>3

2

px px tan 2 2

33. 35. 37. 39. 41.

L L

sx + 1d dx

26.

a3t 2 +

28.

t b dt 2

s2x 3 - 5x + 7d dx

30.

1 1 a 2 - x 2 - b dx 3 L x

32.

L

L L L L L

x -1>3 dx

34.

3 x B dx A 2x + 2

36.

a8y -

38.

2 b dy y 1>4

2xs1 - x -3 d dx

40.

t2t + 2t dt t2

42.

L

s5 - 6xd dx a

t2 + 4t 3 b dt L 2 L

s1 - x 2 - 3x 5 d dx

2 1 a - 3 + 2xb dx x L 5 L

x -5>4 dx

2x 2 b dx + 2 L 2x a

a

1 1 - 5>4 b dy 7 y L L L

x -3sx + 1d dx 4 + 2t dt t3

278 43. 45. 47. 49. 51. 53. 55. 56. 57. 59. 61. 63. 65.

67.

69.

L L L

Chapter 4: Applications of Derivatives

s -2 cos td dt

44.

u du 3

46.

s -3 csc2 xd dx

48.

7 sin

csc u cot u du 2 L L L L

50.

se 3x + 5e -x d dx

52.

(e -x + 4x ) dx

54.

L L L

s -5 sin td dt

78.

3 cos 5u du

79.

a-

80.

sec2 x b dx 3

2 sec u tan u du L5 L L

s2e x - 3e -2x d dx s1.3dx dx

s4 sec x tan x - 2 sec2 xd dx

1 scsc2 x - csc x cot xd dx L2 58.

L

1 + cos 4t dt 2

60.

L

1 ax -

L

3x

23

5 b dx x2 + 1 dx

62. 64.

L L L L

s2 cos 2x - 3 sin 3xd dx 1 - cos 6t dt 2 a x

2 21 - y 2

22 - 1

-

dx

cot2 x dx 68. s1 - cot2 xd dx L L 2 2 (Hint: 1 + cot x = csc x) cos u stan u + sec ud du 70.

csc u du csc u - sin u L

L

s7x - 2d3 dx =

s7x - 2d4 + C 28

s3x + 5d-1 s3x + 5d-2 dx = + C 72. 3 L 73. 74.

L L

sec2 s5x - 1d dx = csc2 a

1 tan s5x - 1d + C 5

x - 1 x - 1 b dx = -3 cot a b + C 3 3

dx L 2a - x 2

1 dx = ln (x + 1) + C, Lx + 1

2

x = sin-1 a a b + C

tan-1 x tan-1 x 1 dx = ln x - ln s1 + x 2 d + C x 2 2 L x

ssin-1 xd2 dx = xssin-1 xd2 - 2x + 221 - x 2 sin-1 x + C L 83. Right, or wrong? Say which for each formula and give a brief reason for each answer. 82.

L L

x sin x dx =

x2 sin x + C 2

x sin x dx = -x cos x + C

x sin x dx = -x cos x + sin x + C L 84. Right, or wrong? Say which for each formula and give a brief reason for each answer. c.

a. b.

L L

c.

tan u sec2 u du =

sec3 u + C 3

tan u sec2 u du =

1 2 tan u + C 2

tan u sec2 u du =

1 sec2 u + C 2

L 85. Right, or wrong? Say which for each formula and give a brief reason for each answer. s2x + 1d3 s2x + 1d2 dx = + C a. 3 L b.

L

3s2x + 1d2 dx = s2x + 1d3 + C

6s2x + 1d2 dx = s2x + 1d3 + C L 86. Right, or wrong? Say which for each formula and give a brief reason for each answer. c.

a. b.

L L

22x + 1 dx = 2x 2 + x + C 22x + 1 dx = 2x 2 + x + C

1 22x + 1 dx = A 22x + 1 B 3 + C 3 L 87. Right, or wrong? Give a brief reason why. c.

-15(x + 3) 2

1 1 dx = + C 2 x + 1 sx + 1d L x 1 dx = + C 76. 2 x + 1 L sx + 1d 75.

77.

1 b dy y 1>4

s2 + tan2 ud du s1 + tan2 ud du 66. L L (Hint: 1 + tan2 u = sec2 u)

L

x dx 1 = a tan-1 a a b + C 2 2 a + x L

b.

ssin 2x - csc2 xd dx

L

xe x dx = xe x - e x + C

a.

Checking Antiderivative Formulas Verify the formulas in Exercises 71–82 by differentiation. 71.

81.

L

L

(x - 2) 4

dx = a

3

x + 3 b + C x - 2

88. Right, or wrong? Give a brief reason why. x 7 -1

x cos (x 2) - sin (x 2) L

x2

dx =

sin (x 2) + C x

4.8 Initial Value Problems 89. Which of the following graphs shows the solution of the initial value problem dy = 2x, dx

y = 4 when x = 1 ?

y

y

y

100.

dr = cos pu, du

101.

dy 1 = sec t tan t, 2 dt

y s0d = 1

102.

dy = 8t + csc2 t, dt

p y a b = -7 2

279

Antiderivatives

r s0d = 1

3 dy , t 7 1, y(2) = 0 = 2 dt t2t - 1 8 dy + sec2 t, y(0) = 1 = 104. dt 1 + t2 d 2y = 2 - 6x; y¿s0d = 4, y s0d = 1 105. dx 2 d 2y = 0; y¿s0d = 2, y s0d = 0 106. dx 2 103.

4

4

(1, 4)

3

4

(1, 4)

3

2

3 2

1

1

1

–1 0

x

1

–1 0

(a)

(1, 4)

2

1

x

–1 0

(b)

1

x

107.

d 2r 2 = 3; dt 2 t

dr = 1, ` dt t = 1

r s1d = 1

108.

3t d 2s = ; 8 dt 2

ds = 3, ` dt t = 4

s s4d = 4

(c)

Give reasons for your answer. 90. Which of the following graphs shows the solution of the initial value problem dy = -x, dx

y = 1 when x = -1 ?

y

111. y s4d = -sin t + cos t ;

y

y

d 3y

= 6; y–s0d = -8, y¿s0d = 0, y s0d = 5 dx 3 d3 u 1 = 0; u–s0d = -2, u¿s0d = - , us0d = 22 110. 2 dt 3 109.

y‡s0d = 7, (–1, 1) (–1, 1) 0

x

x

(b)

(a)

Give reasons for your answer.

0 (c)

y‡s0d = 0,

x

dy = 10 - x, dx

dy 1 = 2 + x, x 7 0; y s2d = 1 dx x dy = 9x 2 - 4x + 5, y s -1d = 0 94. dx dy = 3x -2>3, 95. dx

i)

dy 1 , y s4d = 0 = dx 2 2x ds = 1 + cos t, s s0d = 4 97. dt 98.

ds = cos t + sin t, dt

99.

dr = -p sin pu, du

d 2y dx 2

= 6x

ii) Its graph passes through the point (0, 1), and has a horizontal tangent there. b. How many curves like this are there? How do you know?

Solution (Integral) Curves Exercises 115–118 show solution curves of differential equations. In each exercise, find an equation for the curve through the labeled point. 115.

116.

y s -1d = -5

96.

y s0d = 3

114. a. Find a curve y = ƒsxd with the following properties:

y s0d = -1

93.

y–s0d = y¿s0d = 1,

113. Find the curve y = ƒsxd in the xy-plane that passes through the point (9, 4) and whose slope at each point is 32x .

Solve the initial value problems in Exercises 91–112. dy = 2x - 7, y s2d = 0 91. dx 92.

y s0d = 0

112. y s4d = -cos x + 8 sin 2x ;

(–1, 1) 0

y–s0d = y¿s0d = -1,

y dy x 1 dx

dy 1 4 x1/3 3 dx

y

2 1 (1, 0.5)

0

s spd = 1

1

2

x

(–1, 1) –1

1

0

–1

r s0d = 0

–1

1

2

x

280

Chapter 4: Applications of Derivatives

117.

123. Motion along a coordinate line A particle moves on a coordinate line with acceleration a = d 2s>dt 2 = 152t - A 3> 2t B , subject to the conditions that ds>dt = 4 and s = 0 when t = 1 . Find

118. dy sin x cos x dx y

y

dy 1 sin x dx 2x

6

a. the velocity y = ds>dt in terms of t

1 0

2

x

(–, –1)

b. the position s in terms of t.

4 (1, 2)

2

0

1

2

3

x

–2

Applications 119. Finding displacement from an antiderivative of velocity

T 124. The hammer and the feather When Apollo 15 astronaut David Scott dropped a hammer and a feather on the moon to demonstrate that in a vacuum all bodies fall with the same (constant) acceleration, he dropped them from about 4 ft above the ground. The television footage of the event shows the hammer and the feather falling more slowly than on Earth, where, in a vacuum, they would have taken only half a second to fall the 4 ft. How long did it take the hammer and feather to fall 4 ft on the moon? To find out, solve the following initial value problem for s as a function of t. Then find the value of t that makes s equal to 0. Differential equation:

a. Suppose that the velocity of a body moving along the s-axis is ds = y = 9.8t - 3 . dt iii) Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0 . iii) Find the body’s displacement from t = 1 to t = 3 given that s = -2 when t = 0 . iii) Now find the body’s displacement from t = 1 to t = 3 given that s = s0 when t = 0 . b. Suppose that the position s of a body moving along a coordinate line is a differentiable function of time t. Is it true that once you know an antiderivative of the velocity function ds>dt you can find the body’s displacement from t = a to t = b even if you do not know the body’s exact position at either of those times? Give reasons for your answer. 120. Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m>sec2 . How fast will the rocket be going 1 min later? 121. Stopping a car in time You are driving along a highway at a steady 60 mph (88 ft>sec) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in 242 ft? To find out, carry out the following steps.

Initial conditions:

125. Motion with constant acceleration The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line is s =

Initial conditions:

d 2s = -k sk constantd dt 2 ds = 88 and s = 0 when t = 0 . dt Measuring time and distance from when the brakes are applied

2. Find the value of t that makes ds>dt = 0 . (The answer will involve k.) 3. Find the value of k that makes s = 242 for the value of t you found in Step 2. 122. Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft>sec) to 0 in 45 ft. What constant deceleration does it take to do that?

a 2 t + y0 t + s0 , 2

(1)

where y0 and s0 are the body’s velocity and position at time t = 0 . Derive this equation by solving the initial value problem Differential equation: Initial conditions:

d 2s = a dt 2 ds = y0 and s = s0 when t = 0 . dt

126. Free fall near the surface of a planet For free fall near the surface of a planet where the acceleration due to gravity has a constant magnitude of g length-units>sec2 , Equation (1) in Exercise 125 takes the form s = -

1. Solve the initial value problem Differential equation:

d 2s = -5.2 ft>sec2 dt 2 ds = 0 and s = 4 when t = 0 dt

1 2 gt + y0 t + s0 , 2

(2)

where s is the body’s height above the surface. The equation has a minus sign because the acceleration acts downward, in the direction of decreasing s. The velocity y0 is positive if the object is rising at time t = 0 and negative if the object is falling. Instead of using the result of Exercise 125, you can derive Equation (2) directly by solving an appropriate initial value problem. What initial value problem? Solve it to be sure you have the right one, explaining the solution steps as you go along. 127. Suppose that ƒsxd =

d A 1 - 2x B dx

and

g sxd =

d sx + 2d . dx

Find: a.

L

ƒsxd dx

b.

L

g sxd dx

Chapter 4

c.

L

[-ƒsxd] dx

d.

L

[-g sxd] dx

[ƒsxd + g sxd] dx [ƒsxd - g sxd] dx f. L L 128. Uniqueness of solutions If differentiable functions y = Fsxd and y = Gsxd both solve the initial value problem e.

dy = ƒsxd, dx

y sx0 d = y0 ,

on an interval I, must Fsxd = Gsxd for every x in I? Give reasons for your answer.

Chapter 4

Practice Exercises

281

COMPUTER EXPLORATIONS Use a CAS to solve the initial value problems in Exercises 129–132. Plot the solution curves. 129. y¿ = cos2 x + sin x,

y spd = 1

1 130. y¿ = x + x, y s1d = -1 1 , y s0d = 2 131. y¿ = 24 - x 2 2 132. y– = x + 2x, y s1d = 0, y¿s1d = 0

Questions to Guide Your Review

1. What can be said about the extreme values of a function that is continuous on a closed interval? 2. What does it mean for a function to have a local extreme value on its domain? An absolute extreme value? How are local and absolute extreme values related, if at all? Give examples.

14. What is a cusp? Give examples. 15. List the steps you would take to graph a rational function. Illustrate with an example. 16. Outline a general strategy for solving max-min problems. Give examples.

3. How do you find the absolute extrema of a continuous function on a closed interval? Give examples.

17. Describe l’Hôpital’s Rule. How do you know when to use the rule and when to stop? Give an example.

4. What are the hypotheses and conclusion of Rolle’s Theorem? Are the hypotheses really necessary? Explain.

18. How can you sometimes handle limits that lead to indeterminate forms q > q , q # 0, and q - q ? Give examples.

5. What are the hypotheses and conclusion of the Mean Value Theorem? What physical interpretations might the theorem have?

19. How can you sometimes handle limits that lead to indeterminate q q forms 1 , 00, and q ? Give examples.

6. State the Mean Value Theorem’s three corollaries. 7. How can you sometimes identify a function ƒ(x) by knowing ƒ¿ and knowing the value of ƒ at a point x = x0 ? Give an example.

20. Describe Newton’s method for solving equations. Give an example. What is the theory behind the method? What are some of the things to watch out for when you use the method?

8. What is the First Derivative Test for Local Extreme Values? Give examples of how it is applied.

21. Can a function have more than one antiderivative? If so, how are the antiderivatives related? Explain.

9. How do you test a twice-differentiable function to determine where its graph is concave up or concave down? Give examples.

22. What is an indefinite integral? How do you evaluate one? What general formulas do you know for finding indefinite integrals?

10. What is an inflection point? Give an example. What physical significance do inflection points sometimes have? 11. What is the Second Derivative Test for Local Extreme Values? Give examples of how it is applied. 12. What do the derivatives of a function tell you about the shape of its graph? 13. List the steps you would take to graph a polynomial function. Illustrate with an example.

Chapter 4

23. How can you sometimes solve a differential equation of the form dy>dx = ƒsxd ? 24. What is an initial value problem? How do you solve one? Give an example. 25. If you know the acceleration of a body moving along a coordinate line as a function of time, what more do you need to know to find the body’s position function? Give an example.

Practice Exercises

Extreme Values 1. Does ƒsxd = x3 + 2x + tan x have any local maximum or minimum values? Give reasons for your answer. 2. Does g sxd = csc x + 2 cot x have any local maximum values? Give reasons for your answer.

3. Does ƒsxd = s7 + xds11 - 3xd1>3 have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ.

282

Chapter 4: Applications of Derivatives

4. Find values of a and b such that the function ƒsxd =

7

b. Show that ƒ has a local maximum value at x = 25 L 1.2585 3 and a local minimum value at x = 2 2 L 1.2599 .

ax + b x2 - 1

has a local extreme value of 1 at x = 3 . Is this extreme value a local maximum, or a local minimum? Give reasons for your answer. 5. Does g(x) = e x - x have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of g.

6. Does ƒ(x) = 2e x>(1 + x 2) have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ.

In Exercises 7 and 8, find the absolute maximum and absolute minimum values of ƒ over the interval. 7. ƒ(x) = x - 2 ln x,

1 … x … 3

8. ƒ(x) = (4>x) + ln x2,

1 … x … 4

9. The greatest integer function ƒsxd = :x; , defined for all values of x, assumes a local maximum value of 0 at each point of [0, 1). Could any of these local maximum values also be local minimum values of ƒ? Give reasons for your answer. 10. a. Give an example of a differentiable function ƒ whose first derivative is zero at some point c even though ƒ has neither a local maximum nor a local minimum at c. b. How is this consistent with Theorem 2 in Section 4.1? Give reasons for your answer. 11. The function y = 1>x does not take on either a maximum or a minimum on the interval 0 6 x 6 1 even though the function is continuous on this interval. Does this contradict the Extreme Value Theorem for continuous functions? Why? 12. What are the maximum and minimum values of the function y = ƒ x ƒ on the interval -1 … x 6 1 ? Notice that the interval is not closed. Is this consistent with the Extreme Value Theorem for continuous functions? Why? T 13. A graph that is large enough to show a function’s global behavior may fail to reveal important local features. The graph of ƒsxd = sx8>8d - sx6>2d - x5 + 5x3 is a case in point. a. Graph ƒ over the interval -2.5 … x … 2.5 . Where does the graph appear to have local extreme values or points of inflection? b. Now factor ƒ¿sxd and show that ƒ has a local maximum at x = 3 2 5 L 1.70998 and local minima at x = ; 23 L ;1.73205 . c. Zoom in on the graph to find a viewing window that shows 3 5 and x = 23 . the presence of the extreme values at x = 2 The moral here is that without calculus the existence of two of the three extreme values would probably have gone unnoticed. On any normal graph of the function, the values would lie close enough together to fall within the dimensions of a single pixel on the screen. (Source: Uses of Technology in the Mathematics Curriculum, by Benny Evans and Jerry Johnson, Oklahoma State University, published in 1990 under National Science Foundation Grant USE-8950044.)

c. Zoom in to find a viewing window that shows the presence of 7 3 the extreme values at x = 25 and x = 2 2. The Mean Value Theorem 15. a. Show that g std = sin2 t - 3t decreases on every interval in its domain. b. How many solutions does the equation sin2 t - 3t = 5 have? Give reasons for your answer. 16. a. Show that y = tan u increases on every interval in its domain. b. If the conclusion in part (a) is really correct, how do you explain the fact that tan p = 0 is less than tan sp>4d = 1 ? 17. a. Show that the equation x4 + 2x2 - 2 = 0 has exactly one solution on [0, 1]. T b. Find the solution to as many decimal places as you can. 18. a. Show that ƒsxd = x>sx + 1d increases on every interval in its domain. b. Show that ƒsxd = x3 + 2x has no local maximum or minimum values. 19. Water in a reservoir As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal>min. (An acre-foot is 43,560 ft3 , the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.) 20. The formula Fsxd = 3x + C gives a different function for each value of C. All of these functions, however, have the same derivative with respect to x, namely F¿sxd = 3 . Are these the only differentiable functions whose derivative is 3? Could there be any others? Give reasons for your answers. 21. Show that x d d 1 a ab = b x + 1 dx x + 1 dx even though x 1 Z . x + 1 x + 1 Doesn’t this contradict Corollary 2 of the Mean Value Theorem? Give reasons for your answer.

22. Calculate the first derivatives of ƒsxd = x2>sx2 + 1d and g sxd = -1>sx2 + 1d . What can you conclude about the graphs of these functions? Analyzing Graphs In Exercises 23 and 24, use the graph to answer the questions.

23. Identify any global extreme values of ƒ and the values of x at which they occur. y y f (x) (1, 1)

T 14. (Continuation of Exercise 13.)

a. Graph ƒsxd = sx >8d - s2>5dx - 5x - s5>x d + 11 over the interval -2 … x … 2 . Where does the graph appear to have local extreme values or points of inflection? 8

5

2

0

⎛2, 1⎛ ⎝ 2⎝ x

Chapter 4 24. Estimate the intervals on which the function y = ƒsxd is a. increasing.

Practice Exercises

In Exercises 49–52, graph each function. Then use the function’s first derivative to explain what you see.

b. decreasing.

49. y = x2>3 + sx - 1d1>3

50. y = x2>3 + sx - 1d2>3

c. Use the given graph of ƒ¿ to indicate where any local extreme values of the function occur, and whether each extreme is a relative maximum or minimum.

51. y = x1>3 + sx - 1d1>3

52. y = x2>3 - sx - 1d1>3

Sketch the graphs of the rational functions in Exercises 53–60.

y

53. y =

x + 1 x - 3

54. y =

2x x + 5

55. y =

x2 + 1 x

56. y =

x2 - x + 1 x

57. y =

x3 + 2 2x

58. y =

x4 - 1 x2

59. y =

x2 - 4 x2 - 3

60. y =

x2 x2 - 4

(2, 3) y f ' (x) (–3, 1) x –1

283

–2

Using L’Hôpital’s Rule Use l’Hôpital’s Rule to find the limits in Exercises 61–72. Each of the graphs in Exercises 25 and 26 is the graph of the position function s = ƒstd of an object moving on a coordinate line (t represents time). At approximately what times (if any) is each object’s (a) velocity equal to zero? (b) Acceleration equal to zero? During approximately what time intervals does the object move (c) forward? (d) Backward? 25.

tan x x

63. lim

x:p

65. lim s f (t) 3

6

9

12 14

t

x:p>2

69. lim scsc x - cot xd x:0

26.

s

4

6

8

t

tan x x + sin x

66. lim

sin mx sin nx

x:0

68. lim+ 2x sec x x:0

70. lim a x:0

1 1 - 2b x4 x

x3 x3 - 2 b x - 1 x + 1 2

Find the limits in Exercises 73–84.

Graphs and Graphing Graph the curves in Exercises 27–42. 27. y = x2 - sx3>6d

64. lim

x:1

x: q x: q

2

xa - 1 xb - 1

71. lim A 2x2 + x + 1 - 2x2 - x B

s f (t)

72. lim a 0

62. lim

x:0

sin2 x x:0 tan sx2 d 67. lim - sec 7x cos 3x

s

0

x2 + 3x - 4 x - 1 x:1

61. lim

73. lim

x:0

28. y = x3 - 3x2 + 3

29. y = -x3 + 6x2 - 9x + 3

10x - 1 x

74. lim

u :0

2-sin x - 1 x x:0 e - 1

2sin x - 1 x x:0 e - 1

76. lim

5 - 5 cos x x x:0 e - x - 1

78. lim

75. lim

77. lim

30. y = s1>8dsx3 + 3x2 - 9x - 27d 31. y = x3s8 - xd

32. y = x2s2x2 - 9d

33. y = x - 3x2>3

34. y = x1>3sx - 4d

35. y = x23 - x

36. y = x24 - x2

37. y = (x - 3)2 ex

38. y = xe-x

39. y = ln (x2 - 4x + 3)

40. y = ln (sin x)

1 41. y = sin-1 a x b

1 42. y = tan-1 a x b

2

Each of Exercises 43–48 gives the first derivative of a function y = ƒsxd . (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph. 43. y¿ = 16 - x2

44. y¿ = x2 - x - 6

45. y¿ = 6xsx + 1dsx - 2d

46. y¿ = x2s6 - 4xd

47. y¿ = x4 - 2x2

48. y¿ = 4x2 - x4

79. lim+

t - ln s1 + 2td

e 1 81. lim+ a t - t b t: 0 t

b 83. lim a1 + x b x: q

4 - 4e x xe x x:0

80. lim

sin2 spxd

x:4 ex - 4

t2

t: 0

3u - 1 u

kx

+ 3 - x

82. lim+ e-1>y ln y y:0

7 2 84. lim a1 + x + 2 b x: q x

Optimization 85. The sum of two nonnegative numbers is 36. Find the numbers if a. the difference of their square roots is to be as large as possible. b. the sum of their square roots is to be as large as possible. 86. The sum of two nonnegative numbers is 20. Find the numbers a. if the product of one number and the square root of the other is to be as large as possible. b. if one number plus the square root of the other is to be as large as possible.

284

Chapter 4: Applications of Derivatives

87. An isosceles triangle has its vertex at the origin and its base parallel to the x-axis with the vertices above the axis on the curve y = 27 - x2 . Find the largest area the triangle can have.

Newton’s Method 95. Let ƒsxd = 3x - x3 . Show that the equation ƒsxd = -4 has a solution in the interval [2, 3] and use Newton’s method to find it.

88. A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft3 , to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?

96. Let ƒsxd = x4 - x3 . Show that the equation ƒsxd = 75 has a solution in the interval [3, 4] and use Newton’s method to find it.

89. Find the height and radius of the largest right circular cylinder that can be put in a sphere of radius 23 . 90. The figure here shows two right circular cones, one upside down inside the other. The two bases are parallel, and the vertex of the smaller cone lies at the center of the larger cone’s base. What values of r and h will give the smaller cone the largest possible volume?

Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives) in Exercises 97–120. You may need to try a solution and then adjust your guess. Check your answers by differentiation. 97.

L

sx3 + 5x - 7d dx

4 99. a3 2t + 2 b dt t L dr 101. 2 L sr + 5d 103.

L

12'

105.

r

107.

h

L L

3u2u2 + 1 du x3s1 + x4 d-1>4 dx sec2

s ds 10

98.

L

100.

a8t3 -

t2 + tb dt 2

a

-

1

L 22t 6 dr 102. L A r - 22 B 3 u 104. du L 27 + u2 106. 108.

L L

s2 - xd3>5 dx csc2 ps ds

6'

109. 91. Manufacturing tires Your company can manufacture x hundred grade A tires and y hundred grade B tires a day, where 0 … x … 4 and 40 - 10x y = . 5 - x

111. 112. 113.

Your profit on a grade A tire is twice your profit on a grade B tire. What is the most profitable number of each kind to make?

115.

92. Particle motion The positions of two particles on the s-axis are s1 = cos t and s2 = cos st + p>4d .

117.

a. What is the farthest apart the particles ever get? 119.

b. When do the particles collide? T 93. Open-top box An open-top rectangular box is constructed from a 10-in.-by-16-in. piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Find analytically the dimensions of the box of largest volume and the maximum volume. Support your answers graphically. 94. The ladder problem What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot. y

(8, 6)

6 0

8

x

L

csc 22u cot 22u du

3 b dt t4

sec

110.

u u tan du 3 3

L 1 - cos 2u x 2 b sin dx aHint: sin u = 4 2 L x cos2 dx 2 L 3 5 2 114. a x - xb dx a 2 + 2 b dx x + 1 L L x 1 116. a et - e-t b dt (5s + s5) ds L 2 L 2

L

u1 - p du

3 dx L 2x2x2 - 1

118. 120.

L

2p + r dr

du L 216 - u2

Initial Value Problems Solve the initial value problems in Exercises 121–124. dy x2 + 1 121. , y s1d = -1 = dx x2 2 dy 1 122. = ax + x b , y s1d = 1 dx 3 d 2r = 152t + ; r¿s1d = 8, r s1d = 0 dt 2 2t d3r 124. 3 = -cos t; r–s0d = r¿s0d = 0, r s0d = -1 dt 123.

Applications and Examples 125. Can the integrations in (a) and (b) both be correct? Explain. dx a. = sin-1 x + C L 21 - x2 dx dx b. = - = -cos-1 x + C 2 L 21 - x L 21 - x2

Chapter 4 126. Can the integrations in (a) and (b) both be correct? Explain. a. b.

dx

= -

L 21 - x2

L

-

dx 21 - x2

= -cos-1 x + C

dx -du = L 21 - x2 L 21 - s - ud2 =

x = -u dx = -du

L 21 - u

2

130. y = 10xs2 - ln xd,

y

u = -x

132. g(x) = e23 - 2x - x

T 133. Graph the following functions and use what you see to locate and estimate the extreme values, identify the coordinates of the inflection points, and identify the intervals on which the graphs are concave up and concave down. Then confirm your estimates by working with the functions’ derivatives.

128. The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex on the curve y = sln xd>x2 . What dimensions give the rectangle its largest area, and what is that area?

1

c. y = s1 + x) e-x

136. A round underwater transmission cable consists of a core of copper wires surrounded by nonconducting insulation. If x denotes the ratio of the radius of the core to the thickness of the insulation, it is known that the speed of the transmission signal is given by the equation y = x2 ln s1>xd . If the radius of the core is 1 cm, what insulation thickness h will allow the greatest transmission speed? Insulation x r h

y ln2x x

0.2 0.1

2

T 135. Graph ƒsxd = ssin xdsin x over [0, 3p] . Explain what you see.

x

y

b. y = e-x

T 134. Graph ƒsxd = x ln x . Does the function appear to have an absolute minimum value? Confirm your answer with calculus.

2 e –x

0

s0, e2]

+1

a. y = sln xd> 1x

1

1 e , d 2e 2

2

y

Chapter 4

c

129. y = x ln 2x - x,

4

127. The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex 2 on the curve y = e-x . What dimensions give the rectangle its largest area, and what is that area?

0

In Exercises 129 and 130, find the absolute maximum and minimum values of each function on the given interval.

131. ƒ(x) = ex>2x

= cos-1 u + C = cos s -xd + C

285

In Exercises 131 and 132, find the absolute maxima and minima of the functions and say where they are assumed.

-du

-1

Additional and Advanced Exercises

Core

h r

x

Additional and Advanced Exercises

Functions and Derivatives 1. What can you say about a function whose maximum and minimum values on an interval are equal? Give reasons for your answer. 2. Is it true that a discontinuous function cannot have both an absolute maximum and an absolute minimum value on a closed interval? Give reasons for your answer. 3. Can you conclude anything about the extreme values of a continuous function on an open interval? On a half-open interval? Give reasons for your answer. 4. Local extrema Use the sign pattern for the derivative dƒ = 6sx - 1dsx - 2d2sx - 3d3sx - 4d4 dx to identify the points where ƒ has local maximum and minimum values.

5. Local extrema a. Suppose that the first derivative of y = ƒsxd is y¿ = 6sx + 1dsx - 2d2 . At what points, if any, does the graph of ƒ have a local maximum, local minimum, or point of inflection? b. Suppose that the first derivative of y = ƒsxd is y¿ = 6x sx + 1dsx - 2d . At what points, if any, does the graph of ƒ have a local maximum, local minimum, or point of inflection? 6. If ƒ¿sxd … 2 for all x, what is the most the values of ƒ can increase on [0, 6]? Give reasons for your answer. 7. Bounding a function Suppose that ƒ is continuous on [a, b] and that c is an interior point of the interval. Show that if ƒ¿sxd … 0 on [a, c) and ƒ¿sxd Ú 0 on (c, b], then ƒ(x) is never less than ƒ(c) on [a, b].

286

Chapter 4: Applications of Derivatives

8. An inequality a. Show that -1>2 … x>s1 + x2 d … 1>2 for every value of x. b. Suppose that ƒ is a function whose derivative is ƒ¿sxd = x>s1 + x2 d . Use the result in part (a) to show that 1 ƒ ƒsbd - ƒsad ƒ … 2 ƒ b - a ƒ for any a and b.

drill the hole near the bottom, the water will exit at a higher velocity but have only a short time to fall. Where is the best place, if any, for the hole? (Hint: How long will it take an exiting particle of water to fall from height y to the ground?) Tank kept full, top open

y

9. The derivative of ƒsxd = x2 is zero at x = 0 , but ƒ is not a constant function. Doesn’t this contradict the corollary of the Mean Value Theorem that says that functions with zero derivatives are constant? Give reasons for your answer.

h Exit velocity 64(h y) y

10. Extrema and inflection points Let h = ƒg be the product of two differentiable functions of x.

b. If the graphs of ƒ and g have inflection points at x = a , does the graph of h have an inflection point at a? In either case, if the answer is yes, give a proof. If the answer is no, give a counterexample. 11. Finding a function Use the following information to find the values of a, b, and c in the formula ƒsxd = sx + ad> sbx2 + cx + 2d .

Ground

0

a. If ƒ and g are positive, with local maxima at x = a , and if ƒ¿ and g¿ change sign at a, does h have a local maximum at a?

x

Range

16. Kicking a field goal An American football player wants to kick a field goal with the ball being on a right hash mark. Assume that the goal posts are b feet apart and that the hash mark line is a distance a 7 0 feet from the right goal post. (See the accompanying figure.) Find the distance h from the goal post line that gives the kicker his largest angle b . Assume that the football field is flat. Goal posts b

i) The values of a, b, and c are either 0 or 1.

Goal post line

a

ii) The graph of ƒ passes through the point s -1, 0d . iii) The line y = 1 is an asymptote of the graph of ƒ. 12. Horizontal tangent For what value or values of the constant k will the curve y = x3 + kx2 + 3x - 4 have exactly one horizontal tangent?

h

Optimization 13. Largest inscribed triangle Points A and B lie at the ends of a diameter of a unit circle and point C lies on the circumference. Is it true that the area of triangle ABC is largest when the triangle is isosceles? How do you know? 14. Proving the second derivative test The Second Derivative Test for Local Maxima and Minima (Section 4.4) says: a. ƒ has a local maximum value at x = c if ƒ¿scd = 0 and ƒ–scd 6 0 b. ƒ has a local minimum value at x = c if ƒ¿scd = 0 and ƒ–scd 7 0 . To prove statement (a), let P = s1>2d ƒ ƒ–scd ƒ . Then use the fact that ƒ–scd = lim

h:0

Football

17. A max-min problem with a variable answer Sometimes the solution of a max-min problem depends on the proportions of the shapes involved. As a case in point, suppose that a right circular cylinder of radius r and height h is inscribed in a right circular cone of radius R and height H, as shown here. Find the value of r (in terms of R and H) that maximizes the total surface area of the cylinder (including top and bottom). As you will see, the solution depends on whether H … 2R or H 7 2R .

ƒ¿sc + hd - ƒ¿scd ƒ¿sc + hd = lim h h h :0

to conclude that for some d 7 0 , 0 6 ƒhƒ 6 d

Q

r

ƒ¿sc + hd 6 ƒ–scd + P 6 0 . h

Thus, ƒ¿sc + hd is positive for -d 6 h 6 0 and negative for 0 6 h 6 d . Prove statement (b) in a similar way. 15. Hole in a water tank You want to bore a hole in the side of the tank shown here at a height that will make the stream of water coming out hit the ground as far from the tank as possible. If you drill the hole near the top, where the pressure is low, the water will exit slowly but spend a relatively long time in the air. If you

H

h

R

Chapter 4 18. Minimizing a parameter Find the smallest value of the positive constant m that will make mx - 1 + s1>xd greater than or equal to zero for all positive values of x. Limits 19. Evaluate the following limits. a. lim

x: 0

2 sin 5x 3x

b. lim sin 5x cot 3x x: 0

c. lim x csc 22x 2

x: 0

d.

x - sin x x - tan x

e. lim

x: 0

lim ssec x - tan xd

x: p>2

f. lim

x: 0

sin x2 x sin x

sec x - 1 x3 - 8 g. lim h. lim 2 x: 0 x: 2 x - 4 x2 20. L’Hôpital’s Rule does not help with the following limits. Find them some other way. a. lim

2x + 5

x: q

b. lim

x: q

2x + 5

2x x + 7 2x

Theory and Examples 21. Suppose that it costs a company y = a + bx dollars to produce x units per week. It can sell x units per week at a price of P = c - ex dollars per unit. Each of a, b, c, and e represents a positive constant. (a) What production level maximizes the profit? (b) What is the corresponding price? (c) What is the weekly profit at this level of production? (d) At what price should each item be sold to maximize profits if the government imposes a tax of t dollars per item sold? Comment on the difference between this price and the price before the tax. 22. Estimating reciprocals without division You can estimate the value of the reciprocal of a number a without ever dividing by a if you apply Newton’s method to the function ƒsxd = s1>xd - a . For example, if a = 3 , the function involved is ƒsxd = s1>xd - 3 . a. Graph y = s1>xd - 3 . Where does the graph cross the x-axis? b. Show that the recursion formula in this case is xn + 1 = xns2 - 3xn d , so there is no need for division. q

a q-1 b, x0

287

25. Free fall in the fourteenth century In the middle of the fourteenth century, Albert of Saxony (1316–1390) proposed a model of free fall that assumed that the velocity of a falling body was proportional to the distance fallen. It seemed reasonable to think that a body that had fallen 20 ft might be moving twice as fast as a body that had fallen 10 ft. And besides, none of the instruments in use at the time were accurate enough to prove otherwise. Today we can see just how far off Albert of Saxony’s model was by solving the initial value problem implicit in his model. Solve the problem and compare your solution graphically with the equation s = 16t2 . You will see that it describes a motion that starts too slowly at first and then becomes too fast too soon to be realistic. T 26. Group blood testing During World War II it was necessary to administer blood tests to large numbers of recruits. There are two standard ways to administer a blood test to N people. In method 1, each person is tested separately. In method 2, the blood samples of x people are pooled and tested as one large sample. If the test is negative, this one test is enough for all x people. If the test is positive, then each of the x people is tested separately, requiring a total of x + 1 tests. Using the second method and some probability theory it can be shown that, on the average, the total number of tests y will be 1 y = N a1 - qx + x b . With q = 0.99 and N = 1000 , find the integer value of x that minimizes y. Also find the integer value of x that maximizes y. (This second result is not important to the real-life situation.) The group testing method was used in World War II with a savings of 80% over the individual testing method, but not with the given value of q. 27. Assume that the brakes of an automobile produce a constant deceleration of k ft>sec2 . (a) Determine what k must be to bring an automobile traveling 60 mi>hr (88 ft>sec) to rest in a distance of 100 ft from the point where the brakes are applied. (b) With the same k, how far would a car traveling 30 mi>hr travel before being brought to a stop? 28. Let ƒ(x), g(x) be two continuously differentiable functions satisfying the relationships ƒ¿sxd = g sxd and ƒ–sxd = -ƒsxd . Let hsxd = ƒ 2sxd + g 2sxd . If hs0d = 5 , find h(10).

23. To find x = 2a , we apply Newton’s method to ƒsxd = xq - a . Here we assume that a is a positive real number and q is a positive integer. Show that x1 is a “weighted average” of x0 and a>x0q - 1 , and find the coefficients m0, m1 such that x1 = m0 x0 + m1 a

Additional and Advanced Exercises

m0 7 0, m1 7 0, m0 + m1 = 1. q-1

What conclusion would you reach if x0 and a>x0 What would be the value of x1 in that case?

were equal?

24. The family of straight lines y = ax + b (a, b arbitrary constants) can be characterized by the relation y– = 0 . Find a similar relation satisfied by the family of all circles sx - hd2 + sy - hd2 = r2 , where h and r are arbitrary constants. (Hint: Eliminate h and r from the set of three equations including the given one and two obtained by successive differentiation.)

29. Can there be a curve satisfying the following conditions? d 2y>dx 2 is everywhere equal to zero and, when x = 0, y = 0 and dy>dx = 1 . Give a reason for your answer. 30. Find the equation for the curve in the xy-plane that passes through the point s1, -1d if its slope at x is always 3x2 + 2 . 31. A particle moves along the x-axis. Its acceleration is a = -t 2 . At t = 0 , the particle is at the origin. In the course of its motion, it reaches the point x = b , where b 7 0 , but no point beyond b. Determine its velocity at t = 0 . 32. A particle moves with acceleration a = 2t - (1> 2t) . Assuming that the velocity y = 4>3 and the position s = - 4>15 when t = 0 , find a. the velocity y in terms of t. b. the position s in terms of t. 33. Given ƒsxd = ax2 + 2bx + c with a 7 0 . By considering the minimum, prove that ƒsxd Ú 0 for all real x if and only if b2 - ac … 0 .

288

Chapter 4: Applications of Derivatives

34. Schwarz’s inequality B

a. In Exercise 33, let ƒsxd = sa1 x + b1 d2 + sa2 x + b2 d2 + Á + san x + bn d2,

d2

and deduce Schwarz’s inequality:

b. Show that equality holds in Schwarz’s inequality only if there exists a real number x that makes ai x equal -bi for every value of i from 1 to n. 35. The best branching angles for blood vessels and pipes When a smaller pipe branches off from a larger one in a flow system, we may want it to run off at an angle that is best from some energysaving point of view. We might require, for instance, that energy loss due to friction be minimized along the section AOB shown in the accompanying figure. In this diagram, B is a given point to be reached by the smaller pipe, A is a point in the larger pipe upstream from B, and O is the point where the branching occurs. A law due to Poiseuille states that the loss of energy due to friction in nonturbulent flow is proportional to the length of the path and inversely proportional to the fourth power of the radius. Thus, the loss along AO is skd1 d>R4 and along OB is skd2 d>r4 , where k is a constant, d1 is the length of AO, d2 is the length of OB, R is the radius of the larger pipe, and r is the radius of the smaller pipe. The angle u is to be chosen to minimize the sum of these two losses: L = k

d1 4

R

+ k

A

sa1 b1 + a2 b2 + Á + an bn d2 … A a1 2 + a2 2 + Á + an 2 B A b1 2 + b 2 2 + Á + bn 2 B .

d2 4

r

.

b d 2 sin

O

d1

d 2 cos

C

a

In our model, we assume that AC = a and BC = b are fixed. Thus we have the relations d1 + d2 cos u = a

d2 sin u = b ,

so that d2 = b csc u , d1 = a - d2 cos u = a - b cot u . We can express the total loss L as a function of u : L = ka

a - b cot u b csc u + b. R4 r4

a. Show that the critical value of u for which dL>du equals zero is uc = cos-1

r4 . R4

b. If the ratio of the pipe radii is r>R = 5>6 , estimate to the nearest degree the optimal branching angle given in part (a).

5 INTEGRATION OVERVIEW A great achievement of classical geometry was obtaining formulas for the areas and volumes of triangles, spheres, and cones. In this chapter we develop a method to calculate the areas and volumes of very general shapes. This method, called integration, is a tool for calculating much more than areas and volumes. The integral is of fundamental importance in statistics, the sciences, and engineering. As with the derivative, the integral also arises as a limit, this time of increasingly fine approximations. We use it to calculate quantities ranging from probabilities and averages to energy consumption and the forces against a dam’s floodgates. We study a variety of these applications in the next chapter, but in this chapter we focus on the integral concept and its use in computing areas of various regions with curved boundaries.

Area and Estimating with Finite Sums

5.1

The definite integral is the key tool in calculus for defining and calculating quantities important to mathematics and science, such as areas, volumes, lengths of curved paths, probabilities, and the weights of various objects, just to mention a few. The idea behind the integral is that we can effectively compute such quantities by breaking them into small pieces and then summing the contributions from each piece. We then consider what happens when more and more, smaller and smaller pieces are taken in the summation process. Finally, if the number of terms contributing to the sum approaches infinity and we take the limit of these sums in the way described in Section 5.3, the result is a definite integral. We prove in Section 5.4 that integrals are connected to antiderivatives, a connection that is one of the most important relationships in calculus. The basis for formulating definite integrals is the construction of appropriate finite sums. Although we need to define precisely what we mean by the area of a general region in the plane, or the average value of a function over a closed interval, we do have intuitive ideas of what these notions mean. So in this section we begin our approach to integration by approximating these quantities with finite sums. We also consider what happens when we take more and more terms in the summation process. In subsequent sections we look at taking the limit of these sums as the number of terms goes to infinity, which then leads to precise definitions of the quantities being approximated here.

y

1 y 1 x2

0.5 R

0

0.5

1

FIGURE 5.1 The area of the region R cannot be found by a simple formula.

x

Area Suppose we want to find the area of the shaded region R that lies above the x-axis, below the graph of y = 1 - x 2 , and between the vertical lines x = 0 and x = 1 (Figure 5.1). Unfortunately, there is no simple geometric formula for calculating the areas of general shapes having curved boundaries like the region R. How, then, can we find the area of R? While we do not yet have a method for determining the exact area of R, we can approximate it in a simple way. Figure 5.2a shows two rectangles that together contain the region R. Each rectangle has width 1>2 and they have heights 1 and 3>4, moving from left to right. The height of each rectangle is the maximum value of the function ƒ,

289

290

Chapter 5: Integration

y

1

y y 1 x2

(0, 1)

1

(0, 1)

y 1 x2

⎛ 1 , 15 ⎛ ⎝ 4 16 ⎝

⎛1 , 3⎛ ⎝2 4⎝

⎛1 , 3⎛ ⎝2 4⎝

0.5

⎛3 , 7 ⎛ ⎝ 4 16 ⎝

0.5 R

0

R

0.5

x

1

0

0.25

0.5

(a)

0.75

x

1

(b)

FIGURE 5.2 (a) We get an upper estimate of the area of R by using two rectangles containing R. (b) Four rectangles give a better upper estimate. Both estimates overshoot the true value for the area by the amount shaded in light red.

obtained by evaluating ƒ at the left endpoint of the subinterval of [0, 1] forming the base of the rectangle. The total area of the two rectangles approximates the area A of the region R, A L 1#

3 1 + 2 4

#

7 1 = = 0.875. 2 8

This estimate is larger than the true area A since the two rectangles contain R. We say that 0.875 is an upper sum because it is obtained by taking the height of each rectangle as the maximum (uppermost) value of ƒ(x) for a point x in the base interval of the rectangle. In Figure 5.2b, we improve our estimate by using four thinner rectangles, each of width 1>4, which taken together contain the region R. These four rectangles give the approximation A L 1#

15 1 + 4 16

#

#

3 1 + 4 4

7 1 + 4 16

#

25 1 = = 0.78125, 4 32

which is still greater than A since the four rectangles contain R. Suppose instead we use four rectangles contained inside the region R to estimate the area, as in Figure 5.3a. Each rectangle has width 1>4 as before, but the rectangles are y 1

y y 1 x2

⎛ 1 , 15 ⎛ ⎝ 4 16 ⎝

1

⎛ 1 , 63 ⎛ ⎝ 8 64 ⎝ ⎛ 3 , 55 ⎛ ⎝ 8 64 ⎝

y 1 x2

⎛1 , 3⎛ ⎝2 4 ⎝ ⎛ 5 , 39 ⎛ ⎝ 8 64 ⎝ ⎛3 , 7 ⎛ ⎝ 4 16 ⎝

0.5

0.5 ⎛ 7 , 15 ⎛ ⎝ 8 64 ⎝

0

0.25

0.5 (a)

0.75

1

x

0

0.25 0.5 0.75 1 0.125 0.375 0.625 0.875

x

(b)

FIGURE 5.3 (a) Rectangles contained in R give an estimate for the area that undershoots the true value by the amount shaded in light blue. (b) The midpoint rule uses rectangles whose height is the value of y = ƒsxd at the midpoints of their bases. The estimate appears closer to the true value of the area because the light red overshoot areas roughly balance the light blue undershoot areas.

5.1

Area and Estimating with Finite Sums

291

shorter and lie entirely beneath the graph of ƒ. The function ƒsxd = 1 - x 2 is decreasing on [0, 1], so the height of each of these rectangles is given by the value of ƒ at the right endpoint of the subinterval forming its base. The fourth rectangle has zero height and therefore contributes no area. Summing these rectangles with heights equal to the minimum value of ƒ(x) for a point x in each base subinterval gives a lower sum approximation to the area, 15 16

A L y

#

3 1 + 4 4

#

7 1 + 4 16

#

17 1 1 + 0# = = 0.53125. 4 4 32

This estimate is smaller than the area A since the rectangles all lie inside of the region R. The true value of A lies somewhere between these lower and upper sums:

1

0.53125 6 A 6 0.78125.

y 1 x2

1

0

x

(a)

By considering both lower and upper sum approximations, we get not only estimates for the area, but also a bound on the size of the possible error in these estimates since the true value of the area lies somewhere between them. Here the error cannot be greater than the difference 0.78125 - 0.53125 = 0.25. Yet another estimate can be obtained by using rectangles whose heights are the values of ƒ at the midpoints of their bases (Figure 5.3b). This method of estimation is called the midpoint rule for approximating the area. The midpoint rule gives an estimate that is between a lower sum and an upper sum, but it is not quite so clear whether it overestimates or underestimates the true area. With four rectangles of width 1>4 as before, the midpoint rule estimates the area of R to be A L

y 1

63 64

#

55 1 + 4 64

#

39 1 + 4 64

#

15 1 + 4 64

#

172 1 = 4 64

#

1 = 0.671875. 4

In each of our computed sums, the interval [a, b] over which the function ƒ is defined was subdivided into n subintervals of equal width (also called length) ¢x = sb - ad>n, and ƒ was evaluated at a point in each subinterval: c1 in the first subinterval, c2 in the second subinterval, and so on. The finite sums then all take the form

y 1 x2

ƒsc1 d ¢x + ƒsc2 d ¢x + ƒsc3 d ¢x + Á + ƒscn d ¢x.

1

0

x

(b)

FIGURE 5.4 (a) A lower sum using 16 rectangles of equal width ¢x = 1>16. (b) An upper sum using 16 rectangles.

By taking more and more rectangles, with each rectangle thinner than before, it appears that these finite sums give better and better approximations to the true area of the region R. Figure 5.4a shows a lower sum approximation for the area of R using 16 rectangles of equal width. The sum of their areas is 0.634765625, which appears close to the true area, but is still smaller since the rectangles lie inside R. Figure 5.4b shows an upper sum approximation using 16 rectangles of equal width. The sum of their areas is 0.697265625, which is somewhat larger than the true area because the rectangles taken together contain R. The midpoint rule for 16 rectangles gives a total area approximation of 0.6669921875, but it is not immediately clear whether this estimate is larger or smaller than the true area.

EXAMPLE 1

Table 5.1 shows the values of upper and lower sum approximations to the area of R, using up to 1000 rectangles. In Section 5.2 we will see how to get an exact value of the areas of regions such as R by taking a limit as the base width of each rectangle goes to zero and the number of rectangles goes to infinity. With the techniques developed there, we will be able to show that the area of R is exactly 2>3.

Distance Traveled Suppose we know the velocity function y(t) of a car moving down a highway, without changing direction, and want to know how far it traveled between times t = a and t = b . The position function s(t) of the car has derivative y(t). If we can find an antiderivative F(t) of y(t) then

292

Chapter 5: Integration

TABLE 5.1 Finite approximations for the area of R

Number of subintervals

Lower sum

Midpoint rule

Upper sum

2 4 16 50 100 1000

0.375 0.53125 0.634765625 0.6566 0.66165 0.6661665

0.6875 0.671875 0.6669921875 0.6667 0.666675 0.66666675

0.875 0.78125 0.697265625 0.6766 0.67165 0.6671665

we can find the car’s position function s(t) by setting sstd = Fstd + C. The distance traveled can then be found by calculating the change in position, ssbd - ssad = F(b) - F(a). If the velocity function is known only by the readings at various times of a speedometer on the car, then we have no formula from which to obtain an antiderivative function for velocity. So what do we do in this situation? When we don’t know an antiderivative for the velocity function y(t), we can approximate the distance traveled with finite sums in a way similar to our estimates for area discussed before. We subdivide the interval [a, b] into short time intervals on each of which the velocity is considered to be fairly constant. Then we approximate the distance traveled on each time subinterval with the usual distance formula distance = velocity * time and add the results across [a, b]. Suppose the subdivided interval looks like t a

t1

t

t

t2

t3

b

t (sec)

with the subintervals all of equal length ¢t. Pick a number t1 in the first interval. If ¢t is so small that the velocity barely changes over a short time interval of duration ¢t, then the distance traveled in the first time interval is about yst1 d ¢t. If t2 is a number in the second interval, the distance traveled in the second time interval is about yst2 d ¢t. The sum of the distances traveled over all the time intervals is D L yst1 d ¢t + yst2 d ¢t + Á + ystn d ¢t, where n is the total number of subintervals.

EXAMPLE 2

The velocity function of a projectile fired straight into the air is ƒstd = 160 - 9.8t m>sec. Use the summation technique just described to estimate how far the projectile rises during the first 3 sec. How close do the sums come to the exact value of 435.9 m? We explore the results for different numbers of intervals and different choices of evaluation points. Notice that ƒ(t) is decreasing, so choosing left endpoints gives an upper sum estimate; choosing right endpoints gives a lower sum estimate. Solution

(a) Three subintervals of length 1, with ƒ evaluated at left endpoints giving an upper sum: t1

t2

t3

0

1

2

t

3

t

5.1

Area and Estimating with Finite Sums

293

With ƒ evaluated at t = 0, 1, and 2, we have D L ƒst1 d ¢t + ƒst2 d ¢t + ƒst3 d ¢t = [160 - 9.8s0d]s1d + [160 - 9.8s1d]s1d + [160 - 9.8s2d]s1d = 450.6. (b) Three subintervals of length 1, with ƒ evaluated at right endpoints giving a lower sum:

0

t1

t2

t3

1

2

3

t

t

With ƒ evaluated at t = 1, 2, and 3, we have D L ƒst1 d ¢t + ƒst2 d ¢t + ƒst3 d ¢t = [160 - 9.8s1d]s1d + [160 - 9.8s2d]s1d + [160 - 9.8s3d]s1d = 421.2. (c) With six subintervals of length 1>2, we get t1 t 2 t 3 t 4 t 5 t 6 0

1

2

t

t1 t 2 t 3 t 4 t 5 t 6 3

t

0

1

2

3

t

t

These estimates give an upper sum using left endpoints: D L 443.25; and a lower sum using right endpoints: D L 428.55. These six-interval estimates are somewhat closer than the three-interval estimates. The results improve as the subintervals get shorter. As we can see in Table 5.2, the left-endpoint upper sums approach the true value 435.9 from above, whereas the right-endpoint lower sums approach it from below. The true value lies between these upper and lower sums. The magnitude of the error in the closest entries is 0.23, a small percentage of the true value. Error magnitude = ƒ true value - calculated value ƒ = ƒ 435.9 - 435.67 ƒ = 0.23. Error percentage =

0.23 L 0.05%. 435.9

It would be reasonable to conclude from the table’s last entries that the projectile rose about 436 m during its first 3 sec of flight. TABLE 5.2 Travel-distance estimates

Number of subintervals

Length of each subinterval

Upper sum

Lower sum

3 6 12 24 48 96 192

1 1>2 1>4 1>8 1>16 1>32 1>64

450.6 443.25 439.58 437.74 436.82 436.36 436.13

421.2 428.55 432.23 434.06 434.98 435.44 435.67

294

Chapter 5: Integration

Displacement Versus Distance Traveled If an object with position function s(t) moves along a coordinate line without changing direction, we can calculate the total distance it travels from t = a to t = b by summing the distance traveled over small intervals, as in Example 2. If the object reverses direction one or more times during the trip, then we need to use the object’s speed ƒ ystd ƒ , which is the absolute value of its velocity function, y(t), to find the total distance traveled. Using the velocity itself, as in Example 2, gives instead an estimate to the object’s displacement, ssbd - ssad, the difference between its initial and final positions. To see why using the velocity function in the summation process gives an estimate to the displacement, partition the time interval [a, b] into small enough equal subintervals ¢t so that the object’s velocity does not change very much from time tk - 1 to tk . Then ystk d gives a good approximation of the velocity throughout the interval. Accordingly, the change in the object’s position coordinate during the time interval is about

s s(5)

Height (ft)

400

256

()

()

s(2)

s(8)

144

ystk d ¢t. The change is positive if ystk d is positive and negative if ystk d is negative. In either case, the distance traveled by the object during the subinterval is about ƒ ystk d ƒ ¢t.

s0

The total distance traveled is approximately the sum ƒ yst1 d ƒ ¢t + ƒ yst2 d ƒ ¢t + Á + ƒ ystn d ƒ ¢t.

s(0)

FIGURE 5.5 The rock in Example 3. The height s = 256 ft is reached at t = 2 and t = 8 sec. The rock falls 144 ft from its maximum height when t = 8.

TABLE 5.3 Velocity Function

t

Y(t)

t

Y(t)

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

160 144 128 112 96 80 64 48 32

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

16 0 -16 -32 -48 -64 -80 -96

We revisit these ideas in Section 5.4.

EXAMPLE 3

In Example 4 in Section 3.4, we analyzed the motion of a heavy rock blown straight up by a dynamite blast. In that example, we found the velocity of the rock at any time during its motion to be ystd = 160 - 32t ft>sec. The rock was 256 ft above the ground 2 sec after the explosion, continued upward to reach a maximum height of 400 ft at 5 sec after the explosion, and then fell back down to reach the height of 256 ft again at t = 8 sec after the explosion. (See Figure 5.5.) If we follow a procedure like that presented in Example 2, and use the velocity function ystd in the summation process over the time interval [0, 8], we will obtain an estimate to 256 ft, the rock’s height above the ground at t = 8. The positive upward motion (which yields a positive distance change of 144 ft from the height of 256 ft to the maximum height) is canceled by the negative downward motion (giving a negative change of 144 ft from the maximum height down to 256 ft again), so the displacement or height above the ground is being estimated from the velocity function. On the other hand, if the absolute value ƒ ystd ƒ is used in the summation process, we will obtain an estimate to the total distance the rock has traveled: the maximum height reached of 400 ft plus the additional distance of 144 ft it has fallen back down from that maximum when it again reaches the height of 256 ft at t = 8 sec. That is, using the absolute value of the velocity function in the summation process over the time interval [0, 8], we obtain an estimate to 544 ft, the total distance up and down that the rock has traveled in 8 sec. There is no cancelation of distance changes due to sign changes in the velocity function, so we estimate distance traveled rather than displacement when we use the absolute value of the velocity function (that is, the speed of the rock). As an illustration of our discussion, we subdivide the interval [0, 8] into sixteen subintervals of length ¢t = 1>2 and take the right endpoint of each subinterval in our calculations. Table 5.3 shows the values of the velocity function at these endpoints. Using ystd in the summation process, we estimate the displacement at t = 8: s144 + 128 + 112 + 96 + 80 + 64 + 48 + 32 + 16 1 + 0 - 16 - 32 - 48 - 64 - 80 - 96d # = 192 2 Error magnitude = 256 - 192 = 64

5.1

Area and Estimating with Finite Sums

295

Using ƒ ystd ƒ in the summation process, we estimate the total distance traveled over the time interval [0, 8]: s144 + 128 + 112 + 96 + 80 + 64 + 48 + 32 + 16 1 + 0 + 16 + 32 + 48 + 64 + 80 + 96d # = 528 2 Error magnitude = 544 - 528 = 16 If we take more and more subintervals of [0, 8] in our calculations, the estimates to 256 ft and 544 ft improve, approaching their true values.

Average Value of a Nonnegative Continuous Function The average value of a collection of n numbers x1, x2 , Á , xn is obtained by adding them together and dividing by n. But what is the average value of a continuous function ƒ on an interval [a, b]? Such a function can assume infinitely many values. For example, the temperature at a certain location in a town is a continuous function that goes up and down each day. What does it mean to say that the average temperature in the town over the course of a day is 73 degrees? When a function is constant, this question is easy to answer. A function with constant value c on an interval [a, b] has average value c. When c is positive, its graph over [a, b] gives a rectangle of height c. The average value of the function can then be interpreted geometrically as the area of this rectangle divided by its width b - a (Figure 5.6a). y

y

0

y g(x)

yc

c

a

(a)

b

c

x

0

a

(b)

b

x

FIGURE 5.6 (a) The average value of ƒsxd = c on [a, b] is the area of the rectangle divided by b - a . (b) The average value of g(x) on [a, b] is the area beneath its graph divided by b - a .

What if we want to find the average value of a nonconstant function, such as the function g in Figure 5.6b? We can think of this graph as a snapshot of the height of some water that is sloshing around in a tank between enclosing walls at x = a and x = b. As the water moves, its height over each point changes, but its average height remains the same. To get the average height of the water, we let it settle down until it is level and its height is constant. The resulting height c equals the area under the graph of g divided by b - a. We are led to define the average value of a nonnegative function on an interval [a, b] to be the area under its graph divided by b - a. For this definition to be valid, we need a precise understanding of what is meant by the area under a graph. This will be obtained in Section 5.3, but for now we look at an example.

y f(x) sin x 1

0

2

FIGURE 5.7 Approximating the area under ƒsxd = sin x between 0 and p to compute the average value of sin x over [0, p] , using eight rectangles (Example 4).

x

EXAMPLE 4

Estimate the average value of the function ƒsxd = sin x on the interval

[0, p]. Looking at the graph of sin x between 0 and p in Figure 5.7, we can see that its average height is somewhere between 0 and 1. To find the average we need to calculate the area A under the graph and then divide this area by the length of the interval, p - 0 = p. We do not have a simple way to determine the area, so we approximate it with finite sums. To get an upper sum approximation, we add the areas of eight rectangles of equal

Solution

296

Chapter 5: Integration

width p>8 that together contain the region beneath the graph of y = sin x and above the x-axis on [0, p]. We choose the heights of the rectangles to be the largest value of sin x on each subinterval. Over a particular subinterval, this largest value may occur at the left endpoint, the right endpoint, or somewhere between them. We evaluate sin x at this point to get the height of the rectangle for an upper sum. The sum of the rectangle areas then estimates the total area (Figure 5.7): A L asin

3p 5p 3p 7p # p p p p p + sin + sin + sin + sin + sin + sin + sin b 8 4 8 2 2 8 4 8 8

L s.38 + .71 + .92 + 1 + 1 + .92 + .71 + .38d #

p p = s6.02d # L 2.365. 8 8

To estimate the average value of sin x we divide the estimated area by p and obtain the approximation 2.365>p L 0.753. Since we used an upper sum to approximate the area, this estimate is greater than the actual average value of sin x over [0, p]. If we use more and more rectangles, with each rectangle getting thinner and thinner, we get closer and closer to the true average value. Using the techniques covered in Section 5.3, we will show that the true average value is 2>p L 0.64. As before, we could just as well have used rectangles lying under the graph of y = sin x and calculated a lower sum approximation, or we could have used the midpoint rule. In Section 5.3 we will see that in each case, the approximations are close to the true area if all the rectangles are sufficiently thin.

Summary The area under the graph of a positive function, the distance traveled by a moving object that doesn’t change direction, and the average value of a nonnegative function over an interval can all be approximated by finite sums. First we subdivide the interval into subintervals, treating the appropriate function ƒ as if it were constant over each particular subinterval. Then we multiply the width of each subinterval by the value of ƒ at some point within it, and add these products together. If the interval [a, b] is subdivided into n subintervals of equal widths ¢x = sb - ad>n, and if ƒsck d is the value of ƒ at the chosen point ck in the kth subinterval, this process gives a finite sum of the form ƒsc1 d ¢x + ƒsc2 d ¢x + ƒsc3 d ¢x + Á + ƒscn d ¢x. The choices for the ck could maximize or minimize the value of ƒ in the kth subinterval, or give some value in between. The true value lies somewhere between the approximations given by upper sums and lower sums. The finite sum approximations we looked at improved as we took more subintervals of thinner width.

Exercises 5.1 Area In Exercises 1–4, use finite approximations to estimate the area under the graph of the function using a. a lower sum with two rectangles of equal width. b. a lower sum with four rectangles of equal width. c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width.

3. ƒsxd = 1>x between x = 1 and x = 5. 4. ƒsxd = 4 - x 2 between x = -2 and x = 2. Using rectangles whose height is given by the value of the function at the midpoint of the rectangle’s base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles. 5. ƒsxd = x 2 between x = 0 and x = 1. 6. ƒsxd = x 3 between x = 0 and x = 1.

1. ƒsxd = x 2 between x = 0 and x = 1.

7. ƒsxd = 1>x between x = 1 and x = 5.

2. ƒsxd = x 3 between x = 0 and x = 1.

8. ƒsxd = 4 - x 2 between x = -2 and x = 2.

5.1 Distance 9. Distance traveled The accompanying table shows the velocity of a model train engine moving along a track for 10 sec. Estimate the distance traveled by the engine using 10 subintervals of length 1 with b. right-endpoint values. Velocity (in. / sec)

Time (sec)

Velocity (in. / sec)

0 1 2 3 4 5

0 12 22 10 5 13

6 7 8 9 10

11 6 2 6 0

297

12. Distance from velocity data The accompanying table gives data for the velocity of a vintage sports car accelerating from 0 to 142 mi> h in 36 sec (10 thousandths of an hour).

a. left-endpoint values.

Time (sec)

Area and Estimating with Finite Sums

Time (h)

Velocity (mi / h)

Time (h)

Velocity (mi / h)

0.0 0.001 0.002 0.003 0.004 0.005

0 40 62 82 96 108

0.006 0.007 0.008 0.009 0.010

116 125 132 137 142

mi/hr 160

10. Distance traveled upstream You are sitting on the bank of a tidal river watching the incoming tide carry a bottle upstream. You record the velocity of the flow every 5 minutes for an hour, with the results shown in the accompanying table. About how far upstream did the bottle travel during that hour? Find an estimate using 12 subintervals of length 5 with

140 120 100

a. left-endpoint values.

80

b. right-endpoint values.

60

Time (min)

Velocity (m / sec)

Time (min)

Velocity (m / sec)

0 5 10 15 20 25 30

1 1.2 1.7 2.0 1.8 1.6 1.4

35 40 45 50 55 60

1.2 1.0 1.8 1.5 1.2 0

11. Length of a road You and a companion are about to drive a twisty stretch of dirt road in a car whose speedometer works but whose odometer (mileage counter) is broken. To find out how long this particular stretch of road is, you record the car’s velocity at 10-sec intervals, with the results shown in the accompanying table. Estimate the length of the road using a. left-endpoint values.

40 20 0

0.002 0.004 0.006 0.008 0.01

hours

a. Use rectangles to estimate how far the car traveled during the 36 sec it took to reach 142 mi> h. b. Roughly how many seconds did it take the car to reach the halfway point? About how fast was the car going then? 13. Free fall with air resistance An object is dropped straight down from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in ft>sec2 and recorded every second after the drop for 5 sec, as shown: t

0

1

2

3

4

5

a

32.00

19.41

11.77

7.14

4.33

2.63

b. right-endpoint values.

Time (sec)

Velocity (converted to ft / sec) (30 mi / h 44 ft / sec)

0 10 20 30 40 50 60

0 44 15 35 30 44 35

Time (sec)

Velocity (converted to ft / sec) (30 mi / h 44 ft / sec)

70 80 90 100 110 120

15 22 35 44 30 35

a. Find an upper estimate for the speed when t = 5. b. Find a lower estimate for the speed when t = 5. c. Find an upper estimate for the distance fallen when t = 3. 14. Distance traveled by a projectile An object is shot straight upward from sea level with an initial velocity of 400 ft> sec. a. Assuming that gravity is the only force acting on the object, give an upper estimate for its velocity after 5 sec have elapsed. Use g = 32 ft>sec2 for the gravitational acceleration. b. Find a lower estimate for the height attained after 5 sec.

298

Chapter 5: Integration

Average Value of a Function In Exercises 15–18, use a finite sum to estimate the average value of ƒ on the given interval by partitioning the interval into four subintervals of equal length and evaluating ƒ at the subinterval midpoints. 15. ƒsxd = x 3 on

16. ƒsxd = 1>x on

[0, 2]

17. ƒstd = s1>2d + sin2 pt

on

[1, 9]

Month

Jan

Feb

Mar

Apr

May

Jun

Pollutant release rate (tons> day)

0.20

0.25

0.27

0.34

0.45

0.52

Month

Jul

Aug

Sep

Oct

Nov

Dec

Pollutant release rate (tons> day)

0.63

0.70

0.81

0.85

0.89

0.95

[0, 2]

y y 1 sin 2 t 2

1.5

Measurements are taken at the end of each month determining the rate at which pollutants are released into the atmosphere, recorded as follows.

1 0.5 0

18. ƒstd = 1 - acos

1

pt b 4

t

2

4

on

y

[0, 4] a. Assuming a 30-day month and that new scrubbers allow only 0.05 ton> day to be released, give an upper estimate of the total tonnage of pollutants released by the end of June. What is a lower estimate?

4 y 1 ⎛cos t ⎛ ⎝ 4⎝

1

b. In the best case, approximately when will a total of 125 tons of pollutants have been released into the atmosphere? 0

1

2

3

4

t

21. Inscribe a regular n-sided polygon inside a circle of radius 1 and compute the area of the polygon for the following values of n: a. 4 (square)

Examples of Estimations 19. Water pollution Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in the following table. Time (h)

0

1

2

3

4

Leakage (gal / h)

50

70

97

136

190

Time (h)

5

6

7

8

265

369

516

720

Leakage (gal / h)

a. Give an upper and a lower estimate of the total quantity of oil that has escaped after 5 hours. b. Repeat part (a) for the quantity of oil that has escaped after 8 hours.

c. The tanker continues to leak 720 gal> h after the first 8 hours. If the tanker originally contained 25,000 gal of oil, approximately how many more hours will elapse in the worst case before all the oil has spilled? In the best case?

20. Air pollution A power plant generates electricity by burning oil. Pollutants produced as a result of the burning process are removed by scrubbers in the smokestacks. Over time, the scrubbers become less efficient and eventually they must be replaced when the amount of pollution released exceeds government standards.

b. 8 (octagon)

c. 16

d. Compare the areas in parts (a), (b), and (c) with the area of the circle. 22. (Continuation of Exercise 21. ) a. Inscribe a regular n-sided polygon inside a circle of radius 1 and compute the area of one of the n congruent triangles formed by drawing radii to the vertices of the polygon. b. Compute the limit of the area of the inscribed polygon as n: q. c. Repeat the computations in parts (a) and (b) for a circle of radius r. COMPUTER EXPLORATIONS In Exercises 23–26, use a CAS to perform the following steps. a. Plot the functions over the given interval. b. Subdivide the interval into n = 100 , 200, and 1000 subintervals of equal length and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation ƒsxd = saverage valued for x using the average value calculated in part (c) for the n = 1000 partitioning. 23. ƒsxd = sin x on [0, p]

24. ƒsxd = sin2 x on [0, p]

p 1 25. ƒsxd = x sin x on c , p d 4

p 1 26. ƒsxd = x sin2 x on c , p d 4

5.2

5.2

Sigma Notation and Limits of Finite Sums

299

Sigma Notation and Limits of Finite Sums In estimating with finite sums in Section 5.1, we encountered sums with many terms (up to 1000 in Table 5.1, for instance). In this section we introduce a more convenient notation for sums with a large number of terms. After describing the notation and stating several of its properties, we look at what happens to a finite sum approximation as the number of terms approaches infinity.

Finite Sums and Sigma Notation Sigma notation enables us to write a sum with many terms in the compact form n

Á + an - 1 + an . a ak = a1 + a2 + a3 +

k=1

The Greek letter © (capital sigma, corresponding to our letter S), stands for “sum.” The index of summation k tells us where the sum begins (at the number below the © symbol) and where it ends (at the number above © ). Any letter can be used to denote the index, but the letters i, j, and k are customary. The index k ends at k 5 n.

n The summation symbol (Greek letter sigma)

ak a k is a formula for the kth term.

k51

The index k starts at k 5 1.

Thus we can write 11

12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92 + 10 2 + 112 = a k 2, k=1

and 100

ƒs1d + ƒs2d + ƒs3d + Á + ƒs100d = a ƒsid. i=1

The lower limit of summation does not have to be 1; it can be any integer.

EXAMPLE 1 A sum in sigma notation

The sum written out, one term for each value of k

The value of the sum

ak

1 + 2 + 3 + 4 + 5

15

k a s -1d k

s -1d1s1d + s -1d2s2d + s -1d3s3d

-1 + 2 - 3 = -2

k a k + 1

1 2 + 1 + 1 2 + 1

7 1 2 + = 2 3 6

k2 a k - 1 k=4

52 42 + 4 - 1 5 - 1

16 25 139 + = 3 4 12

5 k=1 3 k=1 2 k=1 5

300

Chapter 5: Integration

EXAMPLE 2

Express the sum 1 + 3 + 5 + 7 + 9 in sigma notation.

The formula generating the terms changes with the lower limit of summation, but the terms generated remain the same. It is often simplest to start with k = 0 or k = 1, but we can start with any integer. Solution

4

Starting with k = 0:

1 + 3 + 5 + 7 + 9 = a s2k + 1d

Starting with k = 1:

1 + 3 + 5 + 7 + 9 = a s2k - 1d

Starting with k = 2:

1 + 3 + 5 + 7 + 9 = a s2k - 3d

Starting with k = -3:

1 + 3 + 5 + 7 + 9 = a s2k + 7d

k=0 5 k=1 6 k=2 1

k = -3

When we have a sum such as 3 2 a sk + k d

k=1

we can rearrange its terms, 3 2 2 2 2 a sk + k d = s1 + 1 d + s2 + 2 d + s3 + 3 d

k=1

= s1 + 2 + 3d + s12 + 22 + 32 d 3

Regroup terms.

3

= a k + a k 2. k=1

k=1

This illustrates a general rule for finite sums: n

n

n

a sak + bk d = a ak + a bk .

k=1

k=1

k=1

Four such rules are given below. A proof that they are valid can be obtained using mathematical induction (see Appendix 2).

Algebra Rules for Finite Sums n

1.

2.

n

n

a (ak + bk) = a ak + a bk

Sum Rule:

k=1

k=1

k=1

n

n

n

a (ak - bk) = a ak - a bk

Difference Rule:

k=1

3.

Constant Multiple Rule:

4.

Constant Value Rule:

k=1

k=1

# a cak = c a ak

(Any number c)

# ac = n c

(c is any constant value.)

n

n

k=1

k=1

n

EXAMPLE 3

k=1

We demonstrate the use of the algebra rules.

n

n

n

Difference Rule and Constant Multiple Rule

(a) a s3k - k 2 d = 3 a k - a k 2 k=1 n

k=1

k=1

n

n

n

k=1

k=1

k=1

k=1

(b) a s -ak d = a s -1d # ak = -1 # a ak = - a ak

Constant Multiple Rule

5.2 3

3

Sigma Notation and Limits of Finite Sums

3

(c) a sk + 4d = a k + a 4 k=1

k=1

Sum Rule

k=1

= s1 + 2 + 3d + s3 # 4d = 6 + 12 = 18

Constant Value Rule

1 1 (d) a n = n # n = 1 n

Constant Value Rule (1>n is constant)

k=1

HISTORICAL BIOGRAPHY Carl Friedrich Gauss (1777–1855)

301

Over the years people have discovered a variety of formulas for the values of finite sums. The most famous of these are the formula for the sum of the first n integers (Gauss is said to have discovered it at age 8) and the formulas for the sums of the squares and cubes of the first n integers.

EXAMPLE 4

Show that the sum of the first n integers is n

ak =

k=1

Solution

nsn + 1d . 2

The formula tells us that the sum of the first 4 integers is s4ds5d = 10. 2

Addition verifies this prediction: 1 + 2 + 3 + 4 = 10. To prove the formula in general, we write out the terms in the sum twice, once forward and once backward. 1 n

+ +

2 sn - 1d

+ +

+ +

3 sn - 2d

Á Á

+ +

n 1

If we add the two terms in the first column we get 1 + n = n + 1. Similarly, if we add the two terms in the second column we get 2 + sn - 1d = n + 1. The two terms in any column sum to n + 1. When we add the n columns together we get n terms, each equal to n + 1, for a total of nsn + 1d. Since this is twice the desired quantity, the sum of the first n integers is sndsn + 1d>2. Formulas for the sums of the squares and cubes of the first n integers are proved using mathematical induction (see Appendix 2). We state them here. n

The first n squares: The first n cubes:

2 ak =

k=1 n

nsn + 1ds2n + 1d 6

3 ak = a

k=1

nsn + 1d 2 b 2

Limits of Finite Sums The finite sum approximations we considered in Section 5.1 became more accurate as the number of terms increased and the subinterval widths (lengths) narrowed. The next example shows how to calculate a limiting value as the widths of the subintervals go to zero and their number grows to infinity.

EXAMPLE 5 Find the limiting value of lower sum approximations to the area of the region R below the graph of y = 1 - x 2 and above the interval [0, 1] on the x-axis using equal-width rectangles whose widths approach zero and whose number approaches infinity. (See Figure 5.4a.)

302

Chapter 5: Integration

We compute a lower sum approximation using n rectangles of equal width ¢x = s1 - 0d>n, and then we see what happens as n : q . We start by subdividing [0, 1] into n equal width subintervals

Solution

n - 1 n 1 2 1 c0, n d, c n , n d, Á , c n , n d . Each subinterval has width 1>n. The function 1 - x 2 is decreasing on [0, 1], and its smallest value in a subinterval occurs at the subinterval’s right endpoint. So a lower sum is constructed with rectangles whose height over the subinterval [sk - 1d>n, k>n] is ƒsk>nd = 1 - sk>nd2 , giving the sum k n 1 2 1 1 1 1 cƒ a n b d a n b + cƒ a n b d a n b + Á + cƒ a n b d a n b + Á + cƒ a n b d a n b . We write this in sigma notation and simplify, n

n

k=1

k=1 n

2

k k 1 1 a ƒ a n b a n b = a a1 - a n b b a n b k2 1 = a an - 3 b n k=1 n

n

k2 1 = a n - a 3 k=1 k=1 n 1 1 = n # n - 3 ak 2 n k=1 1 sndsn + 1ds2n + 1d = 1 - a 3b 6 n

Difference Rule

n

= 1 -

2n 3 + 3n 2 + n . 6n 3

Constant Value and Constant Multiple Rules Sum of the First n Squares

Numerator expanded

We have obtained an expression for the lower sum that holds for any n. Taking the limit of this expression as n : q , we see that the lower sums converge as the number of subintervals increases and the subinterval widths approach zero: lim a1 -

n: q

2n 3 + 3n 2 + n 2 2 b = 1 - = . 3 6 3 6n

The lower sum approximations converge to 2>3. A similar calculation shows that the upper sum approximations also converge to 2>3. Any finite sum approximation g nk = 1 ƒsck ds1>nd also converges to the same value, 2>3. This is because it is possible to show that any finite sum approximation is trapped between the lower and upper sum approximations. For this reason we are led to define the area of the region R as this limiting value. In Section 5.3 we study the limits of such finite approximations in a general setting.

Riemann Sums HISTORICAL BIOGRAPHY Georg Friedrich Bernhard Riemann (1826–1866)

The theory of limits of finite approximations was made precise by the German mathematician Bernhard Riemann. We now introduce the notion of a Riemann sum, which underlies the theory of the definite integral studied in the next section. We begin with an arbitrary bounded function ƒ defined on a closed interval [a, b]. Like the function pictured in Figure 5.8, ƒ may have negative as well as positive values. We subdivide the interval [a, b] into subintervals, not necessarily of equal widths (or lengths), and form sums in the same way as for the finite approximations in Section 5.1. To do so, we choose n - 1 points 5x1, x2 , x3 , Á , xn - 16 between a and b and satisfying a 6 x1 6 x2 6 Á 6 xn - 1 6 b.

5.2

To make the notation consistent, we denote a by x0 and b by xn , so that a = x0 6 x1 6 x2 6 Á 6 xn - 1 6 xn = b.

y

y 5 f(x)

0 a

303

Sigma Notation and Limits of Finite Sums

The set

b

x

P = 5x0 , x1, x2 , Á , xn - 1, xn6

is called a partition of [a, b]. The partition P divides [a, b] into n closed subintervals [x0 , x1], [x1, x2], Á , [xn - 1, xn].

FIGURE 5.8 A typical continuous function y = ƒsxd over a closed interval [a, b].

The first of these subintervals is [x0 , x1], the second is [x1, x2], and the kth subinterval of P is [xk - 1, xk], for k an integer between 1 and n. kth subinterval x0 a

x1

•••

x2

x k1

x xk

•••

xn b

x n1

The width of the first subinterval [x0 , x1] is denoted ¢x1 , the width of the second [x1, x2] is denoted ¢x2 , and the width of the kth subinterval is ¢xk = xk - xk - 1 . If all n subintervals have equal width, then the common width ¢x is equal to sb - ad>n . D x1

D x2

D xk

D xn x

x0 5 a

{{{

x2

x1

x k21

xk

{{{

xn21

xn 5 b

In each subinterval we select some point. The point chosen in the kth subinterval [xk - 1, xk] is called ck . Then on each subinterval we stand a vertical rectangle that stretches from the x-axis to touch the curve at sck , ƒsck dd. These rectangles can be above or below the x-axis, depending on whether ƒsck d is positive or negative, or on the x-axis if ƒsck d = 0 (Figure 5.9). On each subinterval we form the product ƒsck d # ¢xk . This product is positive, negative, or zero, depending on the sign of ƒsck d. When ƒsck d 7 0, the product ƒsck d # ¢xk is the area of a rectangle with height ƒsck d and width ¢xk . When ƒsck d 6 0, the product ƒsck d # ¢xk is a negative number, the negative of the area of a rectangle of width ¢xk that drops from the x-axis to the negative number ƒsck d. y

y f (x)

(cn, f (cn ))

(ck, f (ck )) kth rectangle

c1 0 x0 a

c2 x1

x2

x k1

ck xk

cn x n1

xn b

x

(c1, f (c1))

(c 2, f (c 2))

FIGURE 5.9 The rectangles approximate the region between the graph of the function y = ƒsxd and the x-axis. Figure 5.8 has been enlarged to enhance the partition of [a, b] and selection of points ck that produce the rectangles.

304

Chapter 5: Integration

Finally we sum all these products to get n

SP = a ƒsck d ¢xk . k=1

The sum SP is called a Riemann sum for ƒ on the interval [a, b]. There are many such sums, depending on the partition P we choose, and the choices of the points ck in the subintervals. For instance, we could choose n subintervals all having equal width ¢x = sb - ad>n to partition [a, b], and then choose the point ck to be the right-hand endpoint of each subinterval when forming the Riemann sum (as we did in Example 5). This choice leads to the Riemann sum formula n

Sn = a ƒ aa + k

y

k=1

y f(x)

0 a

b

x

b - a # b - a n b a n b.

Similar formulas can be obtained if instead we choose ck to be the left-hand endpoint, or the midpoint, of each subinterval. In the cases in which the subintervals all have equal width ¢x = sb - ad>n, we can make them thinner by simply increasing their number n. When a partition has subintervals of varying widths, we can ensure they are all thin by controlling the width of a widest (longest) subinterval. We define the norm of a partition P, written 7P7 , to be the largest of all the subinterval widths. If 7P7 is a small number, then all of the subintervals in the partition P have a small width. Let’s look at an example of these ideas.

(a)

EXAMPLE 6 The set P = {0, 0.2, 0.6, 1, 1.5, 2} is a partition of [0, 2]. There are five subintervals of P: [0, 0.2], [0.2, 0.6], [0.6, 1], [1, 1.5], and [1.5, 2]:

y

y f(x)

x1 0

0 a

b

x 2 0.2

x3 0.6

x5

x4 1

1.5

2

x

x

(b)

FIGURE 5.10 The curve of Figure 5.9 with rectangles from finer partitions of [a, b]. Finer partitions create collections of rectangles with thinner bases that approximate the region between the graph of ƒ and the x-axis with increasing accuracy.

The lengths of the subintervals are ¢x1 = 0.2, ¢x2 = 0.4, ¢x3 = 0.4, ¢x4 = 0.5, and ¢x5 = 0.5. The longest subinterval length is 0.5, so the norm of the partition is 7P7 = 0.5. In this example, there are two subintervals of this length. Any Riemann sum associated with a partition of a closed interval [a, b] defines rectangles that approximate the region between the graph of a continuous function ƒ and the x-axis. Partitions with norm approaching zero lead to collections of rectangles that approximate this region with increasing accuracy, as suggested by Figure 5.10. We will see in the next section that if the function ƒ is continuous over the closed interval [a, b], then no matter how we choose the partition P and the points ck in its subintervals to construct a Riemann sum, a single limiting value is approached as the subinterval widths, controlled by the norm of the partition, approach zero.

Exercises 5.2 Sigma Notation Write the sums in Exercises 1–6 without sigma notation. Then evaluate them. 2

6k 1. a k + 1 k=1 4

3

k - 1 2. a k k=1 5

3. a cos kp

4. a sin kp

p 5. a s -1dk + 1 sin k k=1

6. a s -1dk cos kp

k=1 3

k=1 4 k=1

7. Which of the following express 1 + 2 + 4 + 8 + 16 + 32 in sigma notation? 6

a. a 2k - 1 k=1

5

b. a 2k k=0

4

c. a 2k + 1 k = -1

8. Which of the following express 1 - 2 + 4 - 8 + 16 - 32 in sigma notation? 6

a. a s -2dk - 1 k=1

5

b. a s -1dk 2k k=0

3

c. a s -1dk + 1 2k + 2 k = -2

5.3 9. Which formula is not equivalent to the other two? 4 s -1dk - 1 a. a k=2 k - 1

2 s -1dk b. a k=0 k + 1

1 s -1dk c. a k = -1 k + 2

10. Which formula is not equivalent to the other two? 4

-1

3

a. a sk - 1d2 k=1

b. a sk + 1d2 k = -1

c. a k 2 k = -3

Express the sums in Exercises 11–16 in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. 11. 1 + 2 + 3 + 4 + 5 + 6 13.

12. 1 + 4 + 9 + 16

1 1 1 1 + + + 2 4 8 16

15. 1 -

14. 2 + 4 + 6 + 8 + 10

1 1 1 1 + - + 5 2 3 4

3 5 2 4 1 + - + 5 5 5 5 5

16. -

Values of Finite Sums n

n

17. Suppose that a ak = -5 and a bk = 6. Find the values of k=1

n

k=1

n

n

bk b. a k=1 6

a. a 3ak k=1 n

c. a sak + bk d k=1

n

d. a sak - bk d k=1

n

18. Suppose that a ak = 0 and a bk = 1. Find the values of k=1

k=1

n

n

a. a 8ak

b. a 250bk

c. a sak + 1d

d. a sbk - 1d

k=1 n

k=1 n

k=1

k=1

Evaluate the sums in Exercises 19–32. 10

19. a. a k k=1 13

20. a. a k k=1

7

10

10

b. a k 2

c. a k 3

k=1 13

k=1 13

b. a k 2

c. a k 3

k=1

k=1

5

21. a s -2kd

pk 22. a k = 1 15

23. a s3 - k 2 d

24. a sk 2 - 5d

k=1 6 k=1

5.3

7

k=1 5

26. a ks2k + 1d 5

k3 + a a kb 27. a k = 1 225 k=1

k=1 7

3

2

7

k3 28. a a kb - a k=1 k=1 4

7

500

264

29. a. a 3

b. a 7

c. a 10

30. a. a k

b. a k 2

c. a ksk - 1d

31. a. a 4

b. a c

c. a sk - 1d

1 32. a. a a n + 2nb

c b. a n

k c. a 2 k=1 n

k=1 36

k=1 17

k=9 n

k=3 n

k=1 n k=1

k=1 n k=1

k=3 71

k = 18 n k=1 n

Riemann Sums In Exercises 33–36, graph each function ƒ(x) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum © 4k = 1ƒsck d ¢xk , given that ck is the (a) left-hand endpoint, (b) righthand endpoint, (c) midpoint of the kth subinterval. (Make a separate sketch for each set of rectangles.) 33. ƒsxd = x 2 - 1,

[0, 2]

[-p, p]

34. ƒsxd = -x 2,

[0, 1]

36. ƒsxd = sin x + 1,

[-p, p]

37. Find the norm of the partition P = 50, 1.2, 1.5, 2.3, 2.6, 36.

k=1

n

5

25. a ks3k + 5d

35. ƒsxd = sin x,

e. a sbk - 2ak d

305

The Definite Integral

38. Find the norm of the partition P = 5-2, -1.6, -0.5, 0, 0.8, 16. Limits of Riemann Sums For the functions in Exercises 39–46, find a formula for the Riemann sum obtained by dividing the interval [a, b] into n equal subintervals and using the right-hand endpoint for each ck. Then take a limit of these sums as n : q to calculate the area under the curve over [a, b]. 39. ƒsxd = 1 - x 2 over the interval [0, 1]. 40. ƒsxd = 2x over the interval [0, 3]. 41. ƒsxd = x 2 + 1 over the interval [0, 3]. 42. ƒsxd = 3x 2 over the interval [0, 1]. 43. ƒsxd = x + x 2 over the interval [0, 1]. 44. ƒsxd = 3x + 2x 2 over the interval [0, 1].

6 k=1

45. ƒsxd = 2x 3 over the interval [0, 1]. 46. ƒsxd = x 2 - x 3 over the interval [1, 0].

The Definite Integral In Section 5.2 we investigated the limit of a finite sum for a function defined over a closed interval [a, b] using n subintervals of equal width (or length), sb - ad>n. In this section we consider the limit of more general Riemann sums as the norm of the partitions of [a, b] approaches zero. For general Riemann sums the subintervals of the partitions need not have equal widths. The limiting process then leads to the definition of the definite integral of a function over a closed interval [a, b].

Definition of the Definite Integral The definition of the definite integral is based on the idea that for certain functions, as the norm of the partitions of [a, b] approaches zero, the values of the corresponding Riemann

Chapter 5: Integration

sums approach a limiting value J. What we mean by this limit is that a Riemann sum will be close to the number J provided that the norm of its partition is sufficiently small (so that all of its subintervals have thin enough widths). We introduce the symbol P as a small positive number that specifies how close to J the Riemann sum must be, and the symbol d as a second small positive number that specifies how small the norm of a partition must be in order for convergence to happen. We now define this limit precisely.

DEFINITION Let ƒ(x) be a function defined on a closed interval [a, b]. We say that a number J is the definite integral of ƒ over [a, b] and that J is the limit of the Riemann sums g nk = 1 ƒsck d ¢xk if the following condition is satisfied: Given any number P 7 0 there is a corresponding number d 7 0 such that for every partition P = 5x0 , x1, Á , xn6 of [a, b] with 7P7 6 d and any choice of ck in [xk - 1, xk], we have n

` a ƒsck d ¢xk - J ` 6 P. k=1

The definition involves a limiting process in which the norm of the partition goes to zero. In the cases where the subintervals all have equal width ¢x = sb - ad>n, we can form each Riemann sum as n

n

k=1

k=1

Sn = a ƒsck d ¢xk = a ƒsck d a

b - a n b,

¢xk = ¢x = sb - ad>n for all k

where ck is chosen in the subinterval ¢xk. If the limit of these Riemann sums as n : q exists and is equal to J, then J is the definite integral of ƒ over [a, b], so n

J = lim a ƒsck d a n: q k=1

n

b - a lim a ƒsck d ¢x. n b = n: q k=1

¢x = sb - ad>n

Leibniz introduced a notation for the definite integral that captures its construction as a limit of Riemann sums. He envisioned the finite sums g nk = 1 ƒsck d ¢xk becoming an infinite sum of function values ƒ(x) multiplied by “infinitesimal” subinterval widths dx. The sum symbol a is replaced in the limit by the integral symbol 1, whose origin is in the letter “S.” The function values ƒsck d are replaced by a continuous selection of function values ƒ(x). The subinterval widths ¢xk become the differential dx. It is as if we are summing all products of the form ƒsxd # dx as x goes from a to b. While this notation captures the process of constructing an integral, it is Riemann’s definition that gives a precise meaning to the definite integral. The symbol for the number J in the definition of the definite integral is b

La

ƒsxd dx,

which is read as “the integral from a to b of ƒ of x dee x” or sometimes as “the integral from a to b of ƒ of x with respect to x.” The component parts in the integral symbol also have names: The function is the integrand. Upper limit of integration Integral sign

Lower limit of integration

⌠b ⎮ ⎮ ⌡a

x is the variable of integration.

f(x) dx

⎧⎪ ⎪ ⎨ ⎪ ⎩

306

Integral of f from a to b

When you find the value of the integral, you have evaluated the integral.

5.3

The Definite Integral

307

When the condition in the definition is satisfied, we say that the Riemann sums of ƒ on [a, b] b converge to the definite integral J = 1a ƒsxd dx and that ƒ is integrable over [a, b]. We have many choices for a partition P with norm going to zero, and many choices of points ck for each partition. The definite integral exists when we always get the same limit J, no matter what choices are made. When the limit exists we write it as the definite integral n

b

lim a ƒsck d ¢xk = J = ƒ ƒ P ƒ ƒ :0

ƒsxd dx. La The limit of any Riemann sum is always taken as the norm of the partitions approaches zero and the number of subintervals goes to infinity. If we choose the partitions with n equal subintervals of width ¢x = (b - a)>n, and we pick the point ck at the right endpoint of the k th subinterval, so ck = a + k ¢x = a + k(b - a)>n, then the formula for the definite integral becomes k=1

b

n

f (x) dx = lim a f aa + k n: q

(b - a) b - a n ba n b

(1) k=1 La Equation (1) gives an explicit formula that can be used to compute definite integrals. Other choices of partitions and locations of points ck result in the same value for the definite integral when we take the limit as n : q provided that the norm of the partition approaches zero. The value of the definite integral of a function over any particular interval depends on the function, not on the letter we choose to represent its independent variable. If we decide to use t or u instead of x, we simply write the integral as b

b

ƒstd dt

or

b

ƒsud du

instead of

ƒsxd dx. La La La No matter how we write the integral, it is still the same number that is defined as a limit of Riemann sums. Since it does not matter what letter we use, the variable of integration is called a dummy variable.

Integrable and Nonintegrable Functions Not every function defined over the closed interval [a, b] is integrable there, even if the function is bounded. That is, the Riemann sums for some functions may not converge to the same limiting value, or to any value at all. A full development of exactly which functions defined over [a, b] are integrable requires advanced mathematical analysis, but fortunately most functions that commonly occur in applications are integrable. In particular, every continuous function over [a, b] is integrable over this interval, and so is every function having no more than a finite number of jump discontinuities on [a, b]. (See Figures 1.9 and 1.10. The latter functions are called piecewise-continuous functions, and they are defined in Additional Exercises 11–18 at the end of this chapter.) The following theorem, which is proved in more advanced courses, establishes these results.

THEOREM 1—Integrability of Continuous Functions If a function ƒ is continuous over the interval [a, b], or if ƒ has at most finitely many jump discontinuities b there, then the definite integral 1a ƒsxd dx exists and ƒ is integrable over [a, b]. The idea behind Theorem 1 for continuous functions is given in Exercises 86 and 87. Briefly, when ƒ is continuous we can choose each ck so that ƒsck d gives the maximum value of ƒ on the subinterval [xk - 1, xk], resulting in an upper sum. Likewise, we can choose ck to give the minimum value of ƒ on [xk - 1, xk] to obtain a lower sum. The upper and lower sums can be shown to converge to the same limiting value as the norm of the partition P tends to zero. Moreover, every Riemann sum is trapped between the values of the upper and lower sums, so every Riemann sum converges to the same limit as well. Therefore, the number J in the definition of the definite integral exists, and the continuous function ƒ is integrable over [a, b].

308

Chapter 5: Integration

For integrability to fail, a function needs to be sufficiently discontinuous that the region between its graph and the x-axis cannot be approximated well by increasingly thin rectangles. The next example shows a function that is not integrable over a closed interval.

EXAMPLE 1

The function ƒsxd = e

1, if x is rational 0, if x is irrational has no Riemann integral over [0, 1]. Underlying this is the fact that between any two numbers there is both a rational number and an irrational number. Thus the function jumps up and down too erratically over [0, 1] to allow the region beneath its graph and above the x-axis to be approximated by rectangles, no matter how thin they are. We show, in fact, that upper sum approximations and lower sum approximations converge to different limiting values. If we pick a partition P of [0, 1] and choose ck to be the point giving the maximum value for ƒ on [xk - 1, xk] then the corresponding Riemann sum is n

n

U = a ƒsck d ¢xk = a s1d ¢xk = 1, k=1

k=1

n

n

since each subinterval [xk - 1, xk] contains a rational number where ƒsck d = 1. Note that the lengths of the intervals in the partition sum to 1, g nk = 1 ¢xk = 1. So each such Riemann sum equals 1, and a limit of Riemann sums using these choices equals 1. On the other hand, if we pick ck to be the point giving the minimum value for ƒ on [xk - 1, xk], then the Riemann sum is L = a ƒsck d ¢xk = a s0d ¢xk = 0, k=1

k=1

since each subinterval [xk - 1, xk] contains an irrational number ck where ƒsck d = 0. The limit of Riemann sums using these choices equals zero. Since the limit depends on the choices of ck , the function ƒ is not integrable. Theorem 1 says nothing about how to calculate definite integrals. A method of calculation will be developed in Section 5.4, through a connection to the process of taking antiderivatives.

Properties of Definite Integrals b

In defining 1a ƒsxd dx as a limit of sums g nk = 1ƒsck d ¢xk , we moved from left to right across the interval [a, b]. What would happen if we instead move right to left, starting with x0 = b and ending at xn = a? Each ¢xk in the Riemann sum would change its sign, with xk - xk - 1 now negative instead of positive. With the same choices of ck in each subinterval, the sign of a any Riemann sum would change, as would the sign of the limit, the integral 1b ƒsxd dx. Since we have not previously given a meaning to integrating backward, we are led to define a

b

ƒsxd dx = -

ƒsxd dx. Lb La Although we have only defined the integral over an interval [a, b] when a 6 b, it is convenient to have a definition for the integral over [a, b] when a = b, that is, for the integral over an interval of zero width. Since a = b gives ¢x = 0, whenever ƒ(a) exists we define a

ƒsxd dx = 0. La Theorem 2 states basic properties of integrals, given as rules that they satisfy, including the two just discussed. These rules, listed in Table 5.4, become very useful in the process of computing integrals. We will refer to them repeatedly to simplify our calculations. Rules 2 through 7 have geometric interpretations, shown in Figure 5.11. The graphs in these figures are of positive functions, but the rules apply to general integrable functions.

THEOREM 2 When ƒ and g are integrable over the interval [a, b], the definite integral satisfies the rules in Table 5.4.

5.3

309

The Definite Integral

TABLE 5.4 Rules satisfied by definite integrals a

1.

Order of Integration:

2.

Zero Width Interval:

3.

Constant Multiple:

4.

Sum and Difference:

5.

Additivity:

b

ƒsxd dx = -

Lb

ƒsxd dx

La

A definition

a

A definition when ƒ(a) exists

ƒsxd dx = 0

La b

b

kƒsxd dx = k ƒsxd dx La La b

Any constant k

b

sƒsxd ; gsxdd dx =

La b

b

ƒsxd dx ;

La

c

ƒsxd dx +

La

gsxd dx

c

ƒsxd dx =

ƒsxd dx La Lb La Max-Min Inequality: If ƒ has maximum value max ƒ and minimum value min ƒ on [a, b], then

6.

min ƒ # sb - ad …

ƒsxd dx … max ƒ # sb - ad.

b

La

b

ƒsxd Ú gsxd on [a, b] Q

Domination:

7.

La

b

ƒsxd dx Ú

La

gsxd dx

b

ƒsxd Ú 0 on [a, b] Q

y

y

La

ƒsxd dx Ú 0

(Special case)

y

y 2 f (x)

y f (x) g(x)

y f (x)

y g(x) y f (x) y f (x)

0

x

a

0

a

a

b

ƒsxd dx = 0

b

La

y y f (x)

max f

b b L

a L

0

a

f (x) dx

y f (x)

y g(x) c

(d) Additivity for definite integrals: c

ƒsxd dx +

Lb

ƒsxd dx =

La

b

ƒsxd dx +

y f (x)

min f

b

b

b

sƒsxd + gsxdd dx =

y

c

f (x) dx

x

b

(c) Sum: (areas add)

b

kƒsxd dx = k ƒsxd dx La La

y

La

0 a

(b) Constant Multiple: (k = 2)

(a) Zero Width Interval: La

b

x

c

La

ƒsxd dx

x

0 a

b

x

0 a

b

(e) Max-Min Inequality:

(f ) Domination:

min ƒ # sb - ad …

ƒsxd Ú gsxd on [a, b]

FIGURE 5.11 Geometric interpretations of Rules 2–7 in Table 5.4.

b

ƒsxd dx La … max ƒ # sb - ad

b

Q

La

x

b

ƒsxd dx Ú

La

gsxd dx

La

gsxd dx

310

Chapter 5: Integration

While Rules 1 and 2 are definitions, Rules 3 to 7 of Table 5.4 must be proved. The following is a proof of Rule 6. Similar proofs can be given to verify the other properties in Table 5.4. Proof of Rule 6 Rule 6 says that the integral of ƒ over [a, b] is never smaller than the minimum value of ƒ times the length of the interval and never larger than the maximum value of ƒ times the length of the interval. The reason is that for every partition of [a, b] and for every choice of the points ck , min ƒ # sb - ad = min ƒ # a ¢xk n

n

a ¢xk = b - a

k=1

k=1

= a min ƒ # ¢xk n

Constant Multiple Rule

k=1 n

… a ƒsck d ¢xk

min ƒ … ƒsck d

… a max ƒ # ¢xk

ƒsck d … max ƒ

= max ƒ # a ¢xk

Constant Multiple Rule

k=1 n

k=1

n

k=1

= max ƒ # sb - ad. In short, all Riemann sums for ƒ on [a, b] satisfy the inequality min ƒ # sb - ad … a ƒsck d ¢xk … max ƒ # sb - ad. n

k=1

Hence their limit, the integral, does too.

EXAMPLE 2

To illustrate some of the rules, we suppose that 1

L-1

4

ƒsxd dx = 5,

1

L1

ƒsxd dx = -2,

and

L-1

hsxd dx = 7.

Then 1

1.

4

ƒsxd dx = -

L4

L1

ƒsxd dx = -s -2d = 2

1

2.

1

L-1

[2ƒsxd + 3hsxd] dx = 2

L-1

1

ƒsxd dx + 3

L-1 L-1 = 2s5d + 3s7d = 31

4

3.

Rule 1

1

ƒsxd dx =

EXAMPLE 3

L-1

hsxd dx

Rules 3 and 4

4

ƒsxd dx +

L1

ƒsxd dx = 5 + s -2d = 3

Rule 5

1

Show that the value of 10 21 + cos x dx is less than or equal to 22.

The Max-Min Inequality for definite integrals (Rule 6) says that min ƒ # sb - ad b is a lower bound for the value of 1a ƒsxd dx and that max ƒ # sb - ad is an upper bound.

Solution

The maximum value of 21 + cos x on [0, 1] is 21 + 1 = 22, so 1

L0

21 + cos x dx … 22 # s1 - 0d = 22 .

5.3

The Definite Integral

311

Area Under the Graph of a Nonnegative Function We now return to the problem that started this chapter, that of defining what we mean by the area of a region having a curved boundary. In Section 5.1 we approximated the area under the graph of a nonnegative continuous function using several types of finite sums of areas of rectangles capturing the region—upper sums, lower sums, and sums using the midpoints of each subinterval—all being cases of Riemann sums constructed in special ways. Theorem 1 guarantees that all of these Riemann sums converge to a single definite integral as the norm of the partitions approaches zero and the number of subintervals goes to infinity. As a result, we can now define the area under the graph of a nonnegative integrable function to be the value of that definite integral.

DEFINITION If y = ƒsxd is nonnegative and integrable over a closed interval [a, b], then the area under the curve y ƒ(x) over [a, b] is the integral of ƒ from a to b, b

A =

La

ƒsxd dx.

For the first time we have a rigorous definition for the area of a region whose boundary is the graph of any continuous function. We now apply this to a simple example, the area under a straight line, where we can verify that our new definition agrees with our previous notion of area.

EXAMPLE 4

y

b

Compute 10 x dx and find the area A under y = x over the interval

[0, b], b 7 0.

b yx

Solution b

0

b

x

FIGURE 5.12 The region in Example 4 is a triangle.

The region of interest is a triangle (Figure 5.12). We compute the area in two ways.

(a) To compute the definite integral as the limit of Riemann sums, we calculate lim ƒ ƒ P ƒ ƒ :0 g nk = 1 ƒsck d ¢xk for partitions whose norms go to zero. Theorem 1 tells us that it does not matter how we choose the partitions or the points ck as long as the norms approach zero. All choices give the exact same limit. So we consider the partition P that subdivides the interval [0, b] into n subintervals of equal width ¢x = sb - 0d>n = b>n, and we choose ck to be the right endpoint in each subinterval. The partition is b 2b 3b nb kb P = e 0, n , n , n , Á , n f and ck = n . So kb # b a ƒsck d ¢x = a n n n

n

k=1

k=1

ƒsck d = ck

n

kb 2 = a 2 k=1 n n

=

b2 k n 2 ka =1

=

b2 n2

=

b2 1 a1 + n b. 2

#

nsn + 1d 2

Constant Multiple Rule

Sum of First n Integers

312

Chapter 5: Integration

As n : q and 7P7 : 0, this last expression on the right has the limit b 2>2. Therefore,

y

b

b

x dx =

yx

L0 (b) Since the area equals the definite integral for a nonnegative function, we can quickly derive the definite integral by using the formula for the area of a triangle having base length b and height y = b. The area is A = s1>2d b # b = b 2>2. Again we conclude b that 10 x dx = b 2>2.

b a a 0

a

x

b

ba

b2 . 2

Example 4 can be generalized to integrate ƒsxd = x over any closed interval [a, b], 0 6 a 6 b.

(a)

b

y

0

x dx =

La

La

b

x dx +

L0

x dx

a

= a

x

0

b

x dx +

L0 L0 b2 a2 + . = 2 2

b

y5x

Rule 5

x dx

Rule 1 Example 4

In conclusion, we have the following rule for integrating f(x) = x:

(b) b

y

x dx =

La

b2 a2 - , 2 2

a 6 b

(2)

y5x a 0

b

x

(c)

FIGURE 5.13 (a) The area of this trapezoidal region is A = (b 2 - a 2)>2. (b) The definite integral in Equation (2) gives the negative of the area of this trapezoidal region. (c) The definite integral in Equation (2) gives the area of the blue triangular region added to the negative of the area of the gold triangular region.

This computation gives the area of a trapezoid (Figure 5.13a). Equation (2) remains valid when a and b are negative. When a 6 b 6 0, the definite integral value sb 2 - a 2 d>2 is a negative number, the negative of the area of a trapezoid dropping down to the line y = x below the x-axis (Figure 5.13b). When a 6 0 and b 7 0, Equation (2) is still valid and the definite integral gives the difference between two areas, the area under the graph and above [0, b] minus the area below [a, 0] and over the graph (Figure 5.13c). The following results can also be established using a Riemann sum calculation similar to that in Example 4 (Exercises 63 and 65).

b

La

c dx = csb - ad, b

La

x 2 dx =

c any constant

b3 a3 - , 3 3

a 6 b

(3)

(4)

Average Value of a Continuous Function Revisited In Section 5.1 we introduced informally the average value of a nonnegative continuous function ƒ over an interval [a, b], leading us to define this average as the area under the graph of y = ƒsxd divided by b - a. In integral notation we write this as b

1 ƒsxd dx. b - a La We can use this formula to give a precise definition of the average value of any continuous (or integrable) function, whether positive, negative, or both. Alternatively, we can use the following reasoning. We start with the idea from arithmetic that the average of n numbers is their sum divided by n. A continuous function ƒ on [a, b] may have infinitely many values, but we can still sample them in an orderly way. Average =

5.3

The Definite Integral

313

We divide [a, b] into n subintervals of equal width ¢x = sb - ad>n and evaluate ƒ at a point ck in each (Figure 5.14). The average of the n sampled values is

y y f (x) (ck, f(ck ))

n ƒsc1 d + ƒsc2 d + Á + ƒscn d 1 = n n a ƒsck d

x1

k=1

0 x0 a

n

x

ck

xn b

=

¢x ƒsck d b - a ka =1

=

1 ƒsck d ¢x. b - a ka =1

¢x =

b-a ¢x 1 n , so n = b - a

n

FIGURE 5.14 A sample of values of a function on an interval [a, b].

Constant Multiple Rule

The average is obtained by dividing a Riemann sum for ƒ on [a, b] by sb - ad. As we increase the size of the sample and let the norm of the partition approach zero, the b average approaches (1>(b - a)) 1a ƒsxd dx. Both points of view lead us to the following definition.

DEFINITION If ƒ is integrable on [a, b], then its average value on [a, b], also called its mean, is y

b

avsƒd =

2 2 f(x) 4 x

y 2

1

EXAMPLE 5 –2

–1

1

2

x

1 ƒsxd dx. b - a La

Find the average value of ƒsxd = 24 - x 2 on [-2, 2].

We recognize ƒsxd = 24 - x 2 as a function whose graph is the upper semicircle of radius 2 centered at the origin (Figure 5.15). Since we know the area inside a circle, we do not need to take the limit of Riemann sums. The area between the semicircle and the x-axis from -2 to 2 can be computed using the geometry formula

Solution FIGURE 5.15 The average value of ƒsxd = 24 - x 2 on [-2, 2] is p>2 (Example 5). The area of the rectangle shown here is 4 # (p>2) = 2p, which is also the area of the semicircle.

Area =

1 2

#

pr 2 =

1 2

#

ps2d2 = 2p.

Because ƒ is nonnegative, the area is also the value of the integral of ƒ from -2 to 2, 2

L-2

24 - x 2 dx = 2p.

Therefore, the average value of ƒ is 2

avsƒd =

p 1 1 24 - x 2 dx = s2pd = . 4 2 2 - s -2d L-2

Notice that the average value of ƒ over [-2, 2] is the same as the height of a rectangle over [-2, 2] whose area equals the area of the upper semicircle (see Figure 5.15).

Exercises 5.3 Interpreting Limits as Integrals Express the limits in Exercises 1–8 as definite integrals.

n

3.

n

¢xk, where P is a partition of [0, 2]

1.

lim a ck2 ƒ ƒ P ƒ ƒ :0 k = 1 n

2.

lim a 2ck3 ¢xk, where P is a partition of [-1, 0] ƒ ƒ P ƒ ƒ :0 k=1

4.

lim a sck2 - 3ck d ¢xk, where P is a partition of [-7, 5] ƒ ƒPƒ ƒ :0 k=1 n

1 lim a a ck b ¢xk, where P is a partition of [1, 4]

ƒ ƒPƒ ƒ :0 k = 1 n

5.

1 ¢xk, where P is a partition of [2, 3] lim a ƒ ƒ P ƒ ƒ : 0 k = 1 1 - ck

314

Chapter 5: Integration n

6. 7.

lim a 24 - ck2 ¢xk, where P is a partition of [0, 1] ƒ ƒPƒ ƒ :0 k=1 n

lim a ssec ck d ¢xk, where P is a partition of [-p>4, 0]

ƒ ƒPƒ ƒ :0 k = 1

Using Known Areas to Find Integrals In Exercises 15–22, graph the integrands and use areas to evaluate the integrals. 4

15.

n

8.

lim a stan ck d ¢xk, where P is a partition of [0, p>4]

ƒ ƒPƒ ƒ :0 k = 1

a

3>2

16.

29 - x 2 dx

18.

ƒ x ƒ dx

20.

s2 - ƒ x ƒ d dx

22.

L-2

x + 3b dx 2

3

17.

L-3

0

19.

L-2

1

5

ƒsxd dx = -4,

L1

5

ƒsxd dx = 6,

L1

g sxd dx = 8.

L1

2

g sxd dx

b.

L1

d.

ƒsxd dx

L2 5

[ƒsxd - g sxd] dx

L1

f.

[4ƒsxd - g sxd] dx

L1

10. Suppose that ƒ and h are integrable and that 9

L1

9

ƒsxd dx = -1,

L7

9

ƒsxd dx = 5,

L7

hsxd dx = 4.

9

L1

b.

c.

[2ƒsxd - 3hsxd] dx

d.

7

e.

ƒsxd dx

L1

f.

2 11 ƒsxd

L1

b.

L2

d.

b.

0

c.

L-3

[-g sxd] dx

d.

L22

L1

L0

37.

L0

2a

3

2b

39.

L0

x dx

La

3b

x 2 dx

40.

L0

x 2 dx

2

7 dx

L3

42.

22

2

g sud du g srd

L-3 22

dr

s2t - 3d dt

L0 1

45.

L2

a1 +

z b dz 2

44.

L0

1

3u 2 du

L1 L0

s2z - 3d dz

L3

48.

L1>2

2

49.

A t - 22 B dt

0

46.

2

47.

5x dx

L0

24u 2 du

0

s3x 2 + x - 5d dx

50.

L1

s3x 2 + x - 5d dx

3

ƒszd dz

L3

b.

L4

ƒstd dt 1

14. Suppose that h is integrable and that 1-1 hsrd dr = 0 and 3 1-1 hsrd dr = 6. Find 3

L1

u2 du

s 2 ds

L0

p>2

x dx

La

3

a.

36.

34.

L0

23a

43.

L-3 0

4

t 2 dt

0.3

x 2 dx

1

13. Suppose that ƒ is integrable and that 10 ƒszd dz = 3 and 4 10 ƒszd dz = 7. Find a.

33.

1>2

35.

0

g std dt

3

r dr

u du

Lp

27

522

32.

31.

41.

0

L0

L0.5

2p

x dx

[-ƒsxd] dx

12. Suppose that 1-3 g std dt = 22. Find a.

L1

30.

Use the rules in Table 5.4 and Equations (2)–(4) to evaluate the integrals in Exercises 41–50.

L1

-3

2.5

x dx

23ƒszd dz 2

ƒstd dt

26.

Evaluating Definite Integrals Use the results of Equations (2) and (4) to evaluate the integrals in Exercises 29–40.

38. 2

ƒsud du 1

c.

on a. [-1, 0], b. [-1, 1]

dx = 5. Find

2

b

0 6 a 6 b

2s ds,

[hsxd - ƒsxd] dx

L9

b 7 0

28. ƒsxd = 3x + 21 - x 2

7

11. Suppose that a.

ƒsxd dx

L9

4x dx,

L0

3t dt, 0 6 a 6 b La on a. [-2, 2], b. [0, 2]

1

L7

24.

22

[ƒsxd + hsxd] dx

L7

9

b 7 0

La 27. ƒsxd = 24 - x 2

9

-2ƒsxd dx

b

x dx, 2 L0

25.

29.

Use the rules in Table 5.4 to find a.

23.

b

5

3ƒsxd dx 5

e.

g sxd dx

L5

2

c.

A 1 + 21 - x 2 B dx

L-1

b

1

L2

L-1

1

Use areas to evaluate the integrals in Exercises 23–28.

Use the rules in Table 5.4 to find a.

21.

s1 - ƒ x ƒ d dx

L-1

1 2

216 - x 2 dx

L-4

1

Using the Definite Integral Rules 9. Suppose that ƒ and g are integrable and that

s -2x + 4d dx

L1>2

1

hsrd dr

b. -

L3

hsud du

Finding Area by Definite Integrals In Exercises 51–54, use a definite integral to find the area of the region between the given curve and the x-axis on the interval [0, b]. 51. y = 3x 2

52. y = px 2

53. y = 2x

54. y =

x + 1 2

5.3 Finding Average Value In Exercises 55–62, graph the function and find its average value over the given interval. 55. ƒsxd = x 2 - 1

C 0, 23 D

on

x2 56. ƒsxd = on 2

57. ƒsxd = -3x - 1 on 2

[0, 3]

58. ƒsxd = 3x 2 - 3 on

[0, 1]

59. ƒstd = st - 1d2 on

[0, 3]

60. ƒstd = t 2 - t on

[0, 1]

[-2, 1]

on a. [-1, 0] , b. [0, 1], and c. [-1, 1]

b

2

c dx

La

64.

0

x 2 dx,

La

a 6 b

66.

L-1

1

s3x 2 - 2x + 1d dx

68.

La

a 6 b

70.

on

[a, b]

Q

ƒsxd dx Ú 0.

La

78. Integrals of nonpositive functions Show that if ƒ is integrable then b

ƒsxd … 0

on

[a, b]

Q

La

ƒsxd dx … 0.

79. Use the inequality sin x … x, which holds for x Ú 0, to find an 1 upper bound for the value of 10 sin x dx.

80. The inequality sec x Ú 1 + sx 2>2d holds on s -p>2, p>2d . Use it 1 to find a lower bound for the value of 10 sec x dx. 81. If av(ƒ) really is a typical value of the integrable function ƒ(x) on [a, b], then the constant function av(ƒ) should have the same integral over [a, b] as ƒ. Does it? That is, does b

1

x 3 dx,

77. Integrals of nonnegative functions Use the Max-Min Inequality to show that if ƒ is integrable then

x 3 dx

L-1

b

69.

sx - x 2 d dx

L-1

2

67.

s2x + 1d dx

L0

b

65.

1

b

Definite Integrals as Limits Use the method of Example 4a or Equation (1) to evaluate the definite integrals in Exercises 63–70. 63.

315

75. Show that the value of 10 sin sx 2 d dx cannot possibly be 2. 1 76. Show that the value of 10 2x + 8 dx lies between 222 L 2.8 and 3.

ƒsxd Ú 0

61. g sxd = ƒ x ƒ - 1 on a. [-1, 1] , b. [1, 3], and c. [- 1, 3] 62. hsxd = - ƒ x ƒ

The Definite Integral

s3x - x 3 d dx

L0

La

b

avsƒd dx =

La

ƒsxd dx?

Give reasons for your answer. Theory and Examples 71. What values of a and b maximize the value of b

sx - x d dx? 2

La

72. What values of a and b minimize the value of b

sx 4 - 2x 2 d dx?

73. Use the Max-Min Inequality to find upper and lower bounds for the value of 1

1 dx. 1 + x2 L0 74. (Continuation of Exercise 73. ) Use the Max-Min Inequality to find upper and lower bounds for 0.5

L0

a. avsƒ + gd = avsƒd + avsgd b. avskƒd = k avsƒd c. avsƒd … avsgd

(Hint: Where is the integrand positive?)

La

82. It would be nice if average values of integrable functions obeyed the following rules on an interval [a, b].

1 dx 1 + x2

1

and

1 dx. 1 + x2 L0.5

Add these to arrive at an improved estimate of 1

1 dx. 2 L0 1 + x

if

sany number kd ƒsxd … g sxd

on

[a, b].

Do these rules ever hold? Give reasons for your answers. 83. Upper and lower sums for increasing functions a. Suppose the graph of a continuous function ƒ(x) rises steadily as x moves from left to right across an interval [a, b]. Let P be a partition of [a, b] into n subintervals of length ¢x = sb - ad>n. Show by referring to the accompanying figure that the difference between the upper and lower sums for ƒ on this partition can be represented graphically as the area of a rectangle R whose dimensions are [ƒsbd - ƒsad] by ¢x. (Hint: The difference U - L is the sum of areas of rectangles whose diagonals Q0 Q1, Q1 Q2 , Á , Qn - 1Qn lie along the curve. There is no overlapping when these rectangles are shifted horizontally onto R.) b. Suppose that instead of being equal, the lengths ¢xk of the subintervals of the partition of [a, b] vary in size. Show that U - L … ƒ ƒsbd - ƒsad ƒ ¢xmax, where ¢xmax is the norm of P, and hence that lim ƒ ƒ P ƒ ƒ :0 sU - Ld = 0.

316

Chapter 5: Integration y

y

y f (x) y f(x) f (b) f (a) Q3 Q1

R

Q2 Δx

0 x 0 a x1 x 2

xn b

0 x

a

x1

x2

x3

x k1 x k

x n1

b

x

y

84. Upper and lower sums for decreasing functions (Continuation of Exercise 83.) a. Draw a figure like the one in Exercise 83 for a continuous function ƒ(x) whose values decrease steadily as x moves from left to right across the interval [a, b]. Let P be a partition of [a, b] into subintervals of equal length. Find an expression for U - L that is analogous to the one you found for U - L in Exercise 83a. b. Suppose that instead of being equal, the lengths ¢xk of the subintervals of P vary in size. Show that the inequality

0

U - L … ƒ ƒsbd - ƒsad ƒ ¢xmax

a

xk

x k1

b

a

xk

x k1

b

x

y

of Exercise 83b still holds and hence that lim ƒ ƒ P ƒ ƒ :0 sU - Ld = 0. 85. Use the formula sin h + sin 2h + sin 3h + Á + sin mh =

cos sh>2d - cos ssm + s1>2ddhd 2 sin sh>2d

to find the area under the curve y = sin x from x = 0 to x = p>2 in two steps:

0

a. Partition the interval [0, p>2] into n subintervals of equal length and calculate the corresponding upper sum U; then b. Find the limit of U as n : q and ¢x = sb - ad>n : 0. 86. Suppose that ƒ is continuous and nonnegative over [a, b], as in the accompanying figure. By inserting points x1, x2 , Á , xk - 1, xk , Á , xn - 1 as shown, divide [a, b] into n subintervals of lengths ¢x1 = x1 - a, ¢x2 = x2 - x1, Á , ¢xn = b - xn - 1, which need not be equal.

a. If mk = min 5ƒsxd for x in the k th subinterval6, explain the connection between the lower sum L = m1 ¢x1 + m2 ¢x2 + Á + mn ¢xn and the shaded regions in the first part of the figure.

b. If Mk = max 5ƒsxd for x in the k th subinterval6, explain the connection between the upper sum U = M1 ¢x1 + M2 ¢x2 + Á + Mn ¢xn and the shaded regions in the second part of the figure. c. Explain the connection between U - L and the shaded regions along the curve in the third part of the figure.

x

ba

87. We say ƒ is uniformly continuous on [a, b] if given any P 7 0, there is a d 7 0 such that if x1, x2 are in [a, b] and ƒ x1 - x2 ƒ 6 d, then ƒ ƒsx1 d - ƒsx2 d ƒ 6 P . It can be shown that a continuous function on [a, b] is uniformly continuous. Use this and the figure for Exercise 86 to show that if ƒ is continuous and P 7 0 is given, it is possible to make U - L … P # sb - ad by making the largest of the ¢xk’s sufficiently small.

88. If you average 30 mi> h on a 150-mi trip and then return over the same 150 mi at the rate of 50 mi> h, what is your average speed for the trip? Give reasons for your answer. COMPUTER EXPLORATIONS If your CAS can draw rectangles associated with Riemann sums, use it to draw rectangles associated with Riemann sums that converge to the integrals in Exercises 89–94. Use n = 4 , 10, 20, and 50 subintervals of equal length in each case. 1

89.

L0

s1 - xd dx =

1 2

5.4 1

90.

L0

p

4 3

sx 2 + 1d dx =

91.

L-p

p>4

92.

2

94.

L1

1

sec2 x dx = 1

L0

cos x dx = 0

93.

L-1

ƒ x ƒ dx = 1

1 x dx (The integral’s value is about 0.693.)

The Fundamental Theorem of Calculus

d. Solve the equation ƒsxd = saverage valued for x using the average value calculated in part (c) for the n = 1000 partitioning. 95. ƒsxd = sin x

a. Plot the functions over the given interval.

96. ƒsxd = sin x

b. Partition the interval into n = 100 , 200, and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b).

on

102. ƒ(x) =

p c , pd 4

on

on

[0, 1]

on

[0, 1]

on

[2, 5]

1 21 - x

on

2

1 c0, d 2

In this section we present the Fundamental Theorem of Calculus, which is the central theorem of integral calculus. It connects integration and differentiation, enabling us to compute integrals using an antiderivative of the integrand function rather than by taking limits of Riemann sums as we did in Section 5.3. Leibniz and Newton exploited this relationship and started mathematical developments that fueled the scientific revolution for the next 200 years. Along the way, we present an integral version of the Mean Value Theorem, which is another important theorem of integral calculus and is used to prove the Fundamental Theorem.

HISTORICAL BIOGRAPHY Sir Isaac Newton (1642–1727) y y f (x)

f (c), average height a

100. ƒ(x) = e ln x 101. ƒ(x) = x

p c , pd 4

The Fundamental Theorem of Calculus

5.4

0

-x2

[0, p]

on

1 97. ƒsxd = x sin x

99. ƒ(x) = xe -x

[0, p]

on

2

1 98. ƒsxd = x sin2 x In Exercises 95–102, use a CAS to perform the following steps:

317

c ba

b

x

FIGURE 5.16 The value ƒ(c) in the Mean Value Theorem is, in a sense, the average (or mean) height of ƒ on [a, b]. When ƒ Ú 0 , the area of the rectangle is the area under the graph of ƒ from a to b, b

ƒscdsb - ad =

La

ƒsxd dx.

Mean Value Theorem for Definite Integrals In the previous section we defined the average value of a continuous function over a b closed interval [a, b] as the definite integral 1a ƒsxd dx divided by the length or width b - a of the interval. The Mean Value Theorem for Definite Integrals asserts that this average value is always taken on at least once by the function ƒ in the interval. The graph in Figure 5.16 shows a positive continuous function y = ƒsxd defined over the interval [a, b]. Geometrically, the Mean Value Theorem says that there is a number c in [a, b] such that the rectangle with height equal to the average value ƒ(c) of the function and base width b - a has exactly the same area as the region beneath the graph of ƒ from a to b.

THEOREM 3—The Mean Value Theorem for Definite Integrals ous on [a, b], then at some point c in [a, b],

If ƒ is continu-

b

ƒscd =

1 ƒsxd dx. b - a La

Proof If we divide both sides of the Max-Min Inequality (Table 5.4, Rule 6) by sb - ad, we obtain b

min ƒ …

1 ƒsxd dx … max ƒ. b - a La

318

Chapter 5: Integration

y y f(x)

1

Average value 1/2 not assumed

1 2 0

1

x

2

Since ƒ is continuous, the Intermediate Value Theorem for Continuous Functions (Section 2.5) says that ƒ must assume every value between min ƒ and max ƒ. It must therefore b assume the value s1>sb - add 1a ƒsxd dx at some point c in [a, b]. The continuity of ƒ is important here. It is possible that a discontinuous function never equals its average value (Figure 5.17).

EXAMPLE 1

Show that if ƒ is continuous on [a, b], a Z b, and if b

FIGURE 5.17 A discontinuous function need not assume its average value.

La

ƒsxd dx = 0,

then ƒsxd = 0 at least once in [a, b]. Solution

The average value of ƒ on [a, b] is b 1 1 avsƒd = ƒsxd dx = b - a La b - a

#

0 = 0.

By the Mean Value Theorem, ƒ assumes this value at some point c H [a, b].

Fundamental Theorem, Part 1 It can be very difficult to compute definite integrals by taking the limit of Riemann sums. We now develop a powerful new method for evaluating definite integrals, based on using antiderivatives. This method combines the two strands of calculus. One strand involves the idea of taking the limits of finite sums to obtain a definite integral, and the other strand contains derivatives and antiderivatives. They come together in the Fundamental Theorem of Calculus. We begin by considering how to differentiate a certain type of function that is described as an integral. If ƒ(t) is an integrable function over a finite interval I, then the integral from any fixed number a H I to another number x H I defines a new function F whose value at x is

area F(x)

y

y f (t)

0

a

x

b

t

FIGURE 5.18 The function F(x) defined by Equation (1) gives the area under the graph of ƒ from a to x when ƒ is nonnegative and x 7 a.

y y f (t)

f(x) 0

a

x xh

b

FIGURE 5.19 In Equation (1), F(x) is the area to the left of x. Also, Fsx + hd is the area to the left of x + h . The difference quotient [Fsx + hd - Fsxd]>h is then approximately equal to ƒ(x), the height of the rectangle shown here.

x

Fsxd =

t

ƒstd dt. (1) La For example, if ƒ is nonnegative and x lies to the right of a, then F(x) is the area under the graph from a to x (Figure 5.18). The variable x is the upper limit of integration of an integral, but F is just like any other real-valued function of a real variable. For each value of the input x, there is a well-defined numerical output, in this case the definite integral of ƒ from a to x. Equation (1) gives a way to define new functions (as we will see in Section 7.1), but its importance now is the connection it makes between integrals and derivatives. If ƒ is any continuous function, then the Fundamental Theorem asserts that F is a differentiable function of x whose derivative is ƒ itself. At every value of x, it asserts that d Fsxd = ƒsxd. dx To gain some insight into why this result holds, we look at the geometry behind it. If ƒ Ú 0 on [a, b], then the computation of F¿sxd from the definition of the derivative means taking the limit as h : 0 of the difference quotient Fsx + hd - Fsxd . h For h 7 0, the numerator is obtained by subtracting two areas, so it is the area under the graph of ƒ from x to x + h (Figure 5.19). If h is small, this area is approximately equal to the area of the rectangle of height ƒ(x) and width h, which can be seen from Figure 5.19. That is, Fsx + hd - Fsxd L hƒsxd. Dividing both sides of this approximation by h and letting h : 0, it is reasonable to expect that Fsx + hd - Fsxd = ƒsxd. F¿sxd = lim h h:0 This result is true even if the function ƒ is not positive, and it forms the first part of the Fundamental Theorem of Calculus.

5.4

The Fundamental Theorem of Calculus

319

THEOREM 4—The Fundamental Theorem of Calculus, Part 1 If ƒ is continuous x on [a, b], then Fsxd = 1a ƒstd dt is continuous on [a, b] and differentiable on (a, b) and its derivative is ƒsxd: x

F ¿sxd =

d ƒstd dt = ƒsxd. dx La

(2)

Before proving Theorem 4, we look at several examples to gain a better understanding of what it says. In each example, notice that the independent variable appears in a limit of integration, possibly in a formula.

EXAMPLE 2

Use the Fundamental Theorem to find dy>dx if

x

(a) y =

La

5

st 3 + 1d dt

(b) y =

x2

(c) y =

L1

3t sin t dt

Lx 4

cos t dt

(d) y =

1 dt 2 + et

L1 + 3x

2

We calculate the derivatives with respect to the independent variable x.

Solution

x dy d (a) Eq. (2) with ƒ(t) = t 3 + 1 st 3 + 1d dt = x 3 + 1 = dx dx La 5 x dy d d (b) Table 5.4, Rule 1 3t sin t dt = = a- 3t sin t dtb dx dx Lx dx L5 x d = 3t sin t dt dx L5 Eq. (2) with ƒ(t) 3t sin t = -3x sin x 2 (c) The upper limit of integration is not x but x . This makes y a composite of the two functions, u

y =

cos t dt

L1

u = x 2.

and

We must therefore apply the Chain Rule when finding dy>dx. dy dy = dx du = a

#

du dx u

d cos t dtb du L1

= cos u

#

#

du dx

du dx

= cossx 2 d # 2x = 2x cos x 2 4

(d)

d d 1 dt = adx L1 + 3x2 2 + e t dx L4 d = dx L4

1 + 3x2

1 + 3x2

1 dtb 2 + et

1 dt 2 + et

d 1 (1 + 3x 2) (1 + 3x2) dx 2 + e 6x = 2 2 + e (1 + 3x ) = -

Rule 1

Eq. (2) and the Chain Rule

320

Chapter 5: Integration

Proof of Theorem 4 We prove the Fundamental Theorem, Part 1, by applying the definition of the derivative directly to the function F(x), when x and x + h are in (a, b). This means writing out the difference quotient Fsx + hd - Fsxd (3) h and showing that its limit as h : 0 is the number ƒ(x) for each x in (a, b). Thus, F(x + h) - F(x) h h:0

F ¿(x) = lim

x+h

x

1 c ƒstd dt ƒstd dt d h:0 h L a La x+h 1 = lim ƒstd dt Table 5.4, Rule 5 h:0 h L x According to the Mean Value Theorem for Definite Integrals, the value before taking the limit in the last expression is one of the values taken on by ƒ in the interval between x and x + h. That is, for some number c in this interval, = lim

1 h Lx

x+h

ƒstd dt = ƒscd.

(4)

As h : 0, x + h approaches x, forcing c to approach x also (because c is trapped between x and x + h). Since ƒ is continuous at x, ƒ(c) approaches ƒ(x): lim ƒscd = ƒsxd.

(5)

h:0

In conclusion, we have 1 h:0 h L x

F¿(x) = lim

x+h

ƒstd dt

= lim ƒscd

Eq. (4)

= ƒsxd.

Eq. (5)

h:0

If x = a or b, then the limit of Equation (3) is interpreted as a one-sided limit with h : 0 + or h : 0 -, respectively. Then Theorem 1 in Section 3.2 shows that F is continuous for every point in [a, b]. This concludes the proof.

Fundamental Theorem, Part 2 (The Evaluation Theorem) We now come to the second part of the Fundamental Theorem of Calculus. This part describes how to evaluate definite integrals without having to calculate limits of Riemann sums. Instead we find and evaluate an antiderivative at the upper and lower limits of integration.

THEOREM 4 (Continued)—The Fundamental Theorem of Calculus, Part 2 If ƒ is continuous at every point in [a, b] and F is any antiderivative of ƒ on [a, b], then b

La

ƒsxd dx = Fsbd - Fsad.

Proof Part 1 of the Fundamental Theorem tells us that an antiderivative of ƒ exists, namely x

Gsxd =

ƒstd dt. La Thus, if F is any antiderivative of ƒ, then Fsxd = Gsxd + C for some constant C for a 6 x 6 b (by Corollary 2 of the Mean Value Theorem for Derivatives, Section 4.2).

5.4

The Fundamental Theorem of Calculus

321

Since both F and G are continuous on [a, b], we see that F(x) = G(x) + C also holds when x = a and x = b by taking one-sided limits (as x : a + and x : b - d. Evaluating Fsbd - Fsad, we have Fsbd - Fsad = [Gsbd + C] - [Gsad + C] = Gsbd - Gsad b

=

a

ƒstd dt -

La

La

ƒstd dt

b

=

ƒstd dt - 0

La b

=

La

ƒstd dt.

The Evaluation Theorem is important because it says that to calculate the definite integral of ƒ over an interval [a, b] we need do only two things: 1. 2.

Find an antiderivative F of ƒ, and b Calculate the number Fsbd - Fsad, which is equal to 1a ƒsxd dx.

This process is much easier than using a Riemann sum computation. The power of the theorem follows from the realization that the definite integral, which is defined by a complicated process involving all of the values of the function ƒ over [a, b], can be found by knowing the values of any antiderivative F at only the two endpoints a and b. The usual notation for the difference Fsbd - Fsad is Fsxd d

b

b

or

cFsxd d ,

a

a

depending on whether F has one or more terms.

EXAMPLE 3

We calculate several definite integrals using the Evaluation Theorem, rather than by taking limits of Riemann sums. p

(a)

L0

cos x dx = sin x d

p

d sin x = cos x dx

0

= sin p - sin 0 = 0 - 0 = 0 0

(b)

L-p>4

sec x tan x dx = sec x d

0

d sec x = sec x tan x dx

-p>4

= sec 0 - sec a4

4

(c)

p b = 1 - 22 4

3 4 4 a 1x - 2 b dx = cx 3>2 + x d 2 x 1 1 L = cs4d3>2 +

3 d 4 4 ax 3>2 + x b = x 1>2 - 2 2 dx x

4 4 d - cs1d3>2 + d 4 1

= [8 + 1] - [5] = 4 1

1

(d)

dx = ln ƒ x + 1 ƒ d x + 1 L0 0 = ln 2 - ln 1 = ln 2 1

(e)

d 1 ln ƒ x + 1 ƒ = x + 1 dx

1

dx = tan-1 x d 2 0 L0 x + 1 = tan-1 1 - tan-1 0 =

d 1 tan-1 x = = 2 dx x + 1

p p - 0 = . 4 4

322

Chapter 5: Integration

Exercise 82 offers another proof of the Evaluation Theorem, bringing together the ideas of Riemann sums, the Mean Value Theorem, and the definition of the definite integral.

The Integral of a Rate We can interpret Part 2 of the Fundamental Theorem in another way. If F is any antiderivative of ƒ, then F¿ = ƒ. The equation in the theorem can then be rewritten as b

La

F¿sxd dx = Fsbd - Fsad.

Now F¿sxd represents the rate of change of the function Fsxd with respect to x, so the last equation asserts that the integral of F¿ is just the net change in F as x changes from a to b. Formally, we have the following result.

THEOREM 5—The Net Change Theorem The net change in a differentiable function Fsxd over an interval a … x … b is the integral of its rate of change: b

Fsbd - Fsad =

EXAMPLE 4

F¿sxd dx.

La

(6)

Here are several interpretations of the Net Change Theorem.

(a) If csxd is the cost of producing x units of a certain commodity, then c¿sxd is the marginal cost (Section 3.4). From Theorem 5, x2

Lx1

c¿sxd dx = csx2 d - csx1 d,

which is the cost of increasing production from x1 units to x2 units. (b) If an object with position function sstd moves along a coordinate line, its velocity is ystd = s¿std. Theorem 5 says that t2

Lt1

ystd dt = sst2 d - sst1 d,

so the integral of velocity is the displacement over the time interval t1 … t … t2. On the other hand, the integral of the speed ƒ ystd ƒ is the total distance traveled over the time interval. This is consistent with our discussion in Section 5.1. If we rearrange Equation (6) as b

Fsbd = Fsad +

La

F¿sxd dx,

we see that the Net Change Theorem also says that the final value of a function Fsxd over an interval [a, b] equals its initial value Fsad plus its net change over the interval. So if ystd represents the velocity function of an object moving along a coordinate line, this means that the object’s final position sst2 d over a time interval t1 … t … t2 is its initial position sst1 d plus its net change in position along the line (see Example 4b).

EXAMPLE 5

Consider again our analysis of a heavy rock blown straight up from the ground by a dynamite blast (Example 3, Section 5.1). The velocity of the rock at any time t during its motion was given as ystd = 160 - 32t ft>sec. (a) Find the displacement of the rock during the time period 0 … t … 8. (b) Find the total distance traveled during this time period.

5.4

The Fundamental Theorem of Calculus

323

Solution

(a) From Example 4b, the displacement is the integral s160 - 32td dt = C 160t - 16t 2 D 0 L0 = s160ds8d - s16ds64d = 256.

8

L0

8

8

ystd dt =

This means that the height of the rock is 256 ft above the ground 8 sec after the explosion, which agrees with our conclusion in Example 3, Section 5.1. (b) As we noted in Table 5.3, the velocity function ystd is positive over the time interval [0, 5] and negative over the interval [5, 8]. Therefore, from Example 4b, the total distance traveled is the integral 8

L0

5

ƒ ystd ƒ dt =

8

ƒ ystd ƒ dt +

L0

L5

ƒ ystd ƒ dt

5

=

L0

8

s160 - 32td dt -

L5

s160 - 32td dt

= C 160t - 16t 2 D 0 - C 160t - 16t 2 D 5 5

8

= [s160ds5d - s16ds25d] - [s160ds8d - s16ds64d - ss160ds5d - s16ds25dd] = 400 - s -144d = 544. Again, this calculation agrees with our conclusion in Example 3, Section 5.1. That is, the total distance of 544 ft traveled by the rock during the time period 0 … t … 8 is (i) the maximum height of 400 ft it reached over the time interval [0, 5] plus (ii) the additional distance of 144 ft the rock fell over the time interval [5, 8].

The Relationship Between Integration and Differentiation The conclusions of the Fundamental Theorem tell us several things. Equation (2) can be rewritten as

y –2

–1

0

1

2

x

x

d ƒstd dt = ƒsxd, dx La

–1

which says that if you first integrate the function ƒ and then differentiate the result, you get the function ƒ back again. Likewise, replacing b by x and x by t in Equation (6) gives

–2 f (x) 5 x 2 2 4

–3 –4 y 4

g(x) 5 4 2 x 2

3

x

F¿std dt = Fsxd - Fsad, La so that if you first differentiate the function F and then integrate the result, you get the function F back (adjusted by an integration constant). In a sense, the processes of integration and differentiation are “inverses” of each other. The Fundamental Theorem also says that every continuous function ƒ has an antiderivative F. It shows the importance of finding antiderivatives in order to evaluate definite integrals easily. Furthermore, it says that the differential equation dy>dx = ƒsxd has a solution (namely, any of the functions y = F(x) + C ) for every continuous function ƒ.

2

Total Area

1 –2

–1

0

1

2

x

FIGURE 5.20 These graphs enclose the same amount of area with the x-axis, but the definite integrals of the two functions over [-2, 2] differ in sign (Example 6).

The Riemann sum contains terms such as ƒsck d ¢xk that give the area of a rectangle when ƒsck d is positive. When ƒsck d is negative, then the product ƒsck d ¢xk is the negative of the rectangle’s area. When we add up such terms for a negative function, we get the negative of the area between the curve and the x-axis. If we then take the absolute value, we obtain the correct positive area. Figure 5.20 shows the graph of ƒsxd = x 2 - 4 and its mirror image gsxd = 4 - x reflected across the x-axis. For each function, compute

EXAMPLE 6

2

324

Chapter 5: Integration

(a) the definite integral over the interval [-2, 2], and (b) the area between the graph and the x-axis over [-2, 2]. Solution 2

(a)

L-2 and

ƒsxd dx = c

2

8 32 8 x3 - 4x d = a - 8b - a- + 8b = - , 3 3 3 3 -2 2

L-2

gsxd dx = c4x -

2

32 x3 d = . 3 -2 3

(b) In both cases, the area between the curve and the x-axis over [-2, 2] is 32>3 units. Although the definite integral of ƒsxd is negative, the area is still positive. To compute the area of the region bounded by the graph of a function y = ƒsxd and the x-axis when the function takes on both positive and negative values, we must be careful to break up the interval [a, b] into subintervals on which the function doesn’t change sign. Otherwise we might get cancellation between positive and negative signed areas, leading to an incorrect total. The correct total area is obtained by adding the absolute value of the definite integral over each subinterval where ƒ(x) does not change sign. The term “area” will be taken to mean this total area.

EXAMPLE 7 Figure 5.21 shows the graph of the function ƒsxd = sin x between x = 0 and x = 2p. Compute

y 1

y sin x

(a) the definite integral of ƒ(x) over [0, 2p]. (b) the area between the graph of ƒ(x) and the x-axis over [0, 2p].

Area 2 0

Area –2 2

2

x

Solution

2p

–1

FIGURE 5.21 The total area between y = sin x and the x-axis for 0 … x … 2p is the sum of the absolute values of two integrals (Example 7).

The definite integral for ƒsxd = sin x is given by L0

sin x dx = -cos x d

2p

= -[cos 2p - cos 0] = -[1 - 1] = 0. 0

The definite integral is zero because the portions of the graph above and below the x-axis make canceling contributions. The area between the graph of ƒ(x) and the x-axis over [0, 2p] is calculated by breaking up the domain of sin x into two pieces: the interval [0, p] over which it is nonnegative and the interval [p, 2p] over which it is nonpositive. p

L0 2p

Lp

sin x dx = -cos x d

p

= -[cos p - cos 0] = -[-1 - 1] = 2 0

sin x dx = -cos x d

2p p

= -[cos 2p - cos p] = -[1 - s -1d] = -2

The second integral gives a negative value. The area between the graph and the axis is obtained by adding the absolute values Area = ƒ 2 ƒ + ƒ -2 ƒ = 4.

Summary: To find the area between the graph of y = ƒsxd and the x-axis over the interval [a, b]: 1. Subdivide [a, b] at the zeros of ƒ. 2. Integrate ƒ over each subinterval. 3. Add the absolute values of the integrals.

5.4

Find the area of the region between the x-axis and the graph of ƒ(x) = x 3 - x 2 - 2x, -1 … x … 2.

y x 3 x 2 2x

First find the zeros of ƒ. Since

Solution –1

325

EXAMPLE 8

y Area 5 12

The Fundamental Theorem of Calculus

0

8 Area ⎢⎢– ⎢⎢ 3 8 3

x

2

ƒsxd = x 3 - x 2 - 2x = xsx 2 - x - 2d = xsx + 1dsx - 2d, the zeros are x = 0, -1, and 2 (Figure 5.22). The zeros subdivide [-1, 2] into two subintervals: [-1, 0], on which ƒ Ú 0, and [0, 2], on which ƒ … 0. We integrate ƒ over each subinterval and add the absolute values of the calculated integrals. 0

L-1

sx 3 - x 2 - 2xd dx = c

FIGURE 5.22 The region between the curve y = x 3 - x 2 - 2x and the x-axis (Example 8).

2

sx 3 - x 2 - 2xd dx = c

L0

0

5 x3 x4 1 1 - x2 d = 0 - c + - 1 d = 4 3 4 3 12 -1 2

8 8 x3 x4 - x 2 d = c4 - - 4 d - 0 = 4 3 3 3 0

The total enclosed area is obtained by adding the absolute values of the calculated integrals. Total enclosed area =

5 8 37 + `- ` = 12 3 12

Exercises 5.4 Evaluating Integrals Evaluate the integrals in Exercises 1–34. 0

1.

4

s2x + 5d dx

L-2

2.

L0 L0 1

7.

a3x -

x3 b dx 4

A x + 1x B dx 2

L0

4.

L-2

27.

sx 3 - 2x + 3d dx

29.

Lp>2

8.

L0

10.

Lp>4

12.

1 + cos 2t dt 2 Lp>2

x

L1

dx

33.

s1 + cos xd dx

L0

4 sec u tan u du

L0 L-p>3

1 - cos 2t dt 2

15.

p>6

tan2 x dx

L0

16.

L0 -1

19.

L1 1

21.

L22

L0

L-4

sin 2x dx

18.

sr + 1d2 dr

20.

a

22.

u7 1 - 5 b du 2 u

28.

L-p>3 23

L-23 -1

L-3

a4 sec2 t +

p b dt t2

st + 1dst 2 + 4d dt y 5 - 2y

1 scos x + ƒ cos x ƒ d dx L0 2

L0 L0

30.

x p - 1 dx

L2

y3

1 a x - e - x b dx

1>13

4 21 - x

L1

2

dx

32.

L0 0

34.

L-1

dx 1 + 4x 2

p x - 1 dx

In Exercises 35–38, guess an antiderivative for the integrand function. Validate your guess by differentiation and then evaluate the given definite integral. (Hint: Keep in mind the Chain Rule in guessing an antiderivative. You will learn how to find such antiderivatives in the next section.) 35. 37.

2

xe x dx

L0

x dx

L2 21 + x 2

2

36.

L1

ln x x dx

p>3

38.

L0

sin2 x cos x dx

Derivatives of Integrals Find the derivatives in Exercises 39–44. a. by evaluating the integral and differentiating the result.

dy

dx

scos x + sec xd2 dx

L0

2

e 3x dx

5 -p>4

p>8

17.

ssec x + tan xd2 dx

x 1>3

p

ƒ x ƒ dx

1 p>4

sx 1>3 + 1ds2 - x 2>3 d

L1

4

p>3

14.

26.

4

31.

p>3

csc u cot u du

8

24.

p>3

sin 2x dx 2 sin x

1>2 -6>5

p

2 sec2 x dx

s 2 + 2s ds s2

ln 2

32

0

13.

sx 2 - 2x + 3d dx

p

2

6.

3p>4

11.

25.

L-1

p>3

9.

x b dx 2

L1

1

xsx - 3d dx 4

5.

a5 -

L-3

2

3.

22

23.

b. by differentiating the integral directly.

326 39.

Chapter 5: Integration

d dx L0

1x

cos t dt

40.

t4

41.

d 1u du dt L0

42.

x3

43.

d e - t dt dx L0

44.

d dx L1

sin x

d du L0

tan u

d dt L0

63.

3t 2 dt

x

L0

46. y =

2t

ax 4 +

y sec tan

3 21 - x

2

b dx

– 4

L1x

1 dt, L1 t

48. y = x

x

49. y =

x

dt

L0

21 - t 2 0

Ltan x

65. e

dt 1 + t2

53. y =

L0

2t dt

x1>p

56. y =

55. y =

3

L2x L-1

1

dt

sin-1 x

1

54. y =

x2

2t

L0

cos t dt

sin-1 t dt

Area In Exercises 57–60, find the total area between the region and the x-axis. 57. y = -x 2 - 2x, -3 … x … 2 -2 … x … 2

59. y = x 3 - 3x 2 + 2x, 60. y = x

1>3

- x,

0 … x … 2

-1 … x … 8

Find the areas of the shaded regions in Exercises 61–64. 61.

y

y 1 cos x

62.

dy 1 = x, dx

x

yspd = -3 ys0d = 4

d. y =

1 dt - 3 Lp t

66. y¿ = sec x, 1 68. y¿ = x ,

ys -1d = 4

ys1d = - 3

Express the solutions of the initial value problems in Exercises 69 and 70 in terms of integrals. dy = sec x, ys2d = 3 69. dx dy = 21 + x 2, dx

ys1d = -2

Theory and Examples 71. Archimedes’ area formula for parabolic arches Archimedes (287–212 B.C.), inventor, military engineer, physicist, and the greatest mathematician of classical times in the Western world, discovered that the area under a parabolic arch is two-thirds the base times the height. Sketch the parabolic arch y = h - s4h>b 2 dx 2, -b>2 … x … b>2 , assuming that h and b are positive. Then use calculus to find the area of the region enclosed between the arch and the x-axis. 72. Show that if k is a positive constant, then the area between the x-axis and one arch of the curve y = sin k x is 2>k .

74. Revenue from marginal revenue Suppose that a company’s marginal revenue from the manufacture and sale of eggbeaters is dr = 2 - 2>sx + 1d2 , dx

y sin x

6

L-1

sec t dt + 4

L0

dollars. Find cs100d - cs1d , the cost of printing posters 2–100.

x

y 1

t

dc 1 = dx 21x

x

0

1

73. Cost from marginal cost The marginal cost of printing a poster when x posters have been printed is

y2

2

b. y =

sec t dt + 4

67. y¿ = sec x,

70.

58. y = 3x 2 - 3,

0

x

1 dt - 3 L1 t x

c. y =

p ƒxƒ 6 2

,

y 1 t2

– 4

x

a. y =

st 3 + 1d10 dtb

L0

1 y sec 2 t

Initial Value Problems Each of the following functions solves one of the initial value problems in Exercises 65–68. Which function solves which problem? Give brief reasons for your answers.

3

sin x

52. y =

sin st 3 d dt

t2 t2 dt dt 2 L-1 t + 4 L3 t + 4

50. y = a 51. y =

L2

2

x

4

x 7 0

x2

sin st 2 d dt

0

–2

0

47. y =

2

sec2 y dy

x

21 + t 2 dt

y

2

Find dy>dx in Exercises 45–56. 45. y =

64.

y

5 6

x

where r is measured in thousands of dollars and x in thousands of units. How much money should the company expect from a production run of x = 3 thousand eggbeaters? To find out, integrate the marginal revenue from x = 0 to x = 3 .

5.4 75. The temperature T s°Fd of a room at time t minutes is given by T = 85 - 3 225 - t

0 … t … 25.

for

a. Find the room’s temperature when t = 0, t = 16, and t = 25.

The Fundamental Theorem of Calculus

83. Suppose that ƒ is the differentiable function shown in the accompanying graph and that the position at time t (sec) of a particle moving along a coordinate axis is t

s =

b. Find the room’s average temperature for 0 … t … 25. 76. The height H sftd of a palm tree after growing for t years is given by H = 2t + 1 + 5t 1>3

for

0 … t … 8.

4 (3, 3) 3 (2, 2) 2 1 (1, 1)

x

77. Suppose that 11 ƒstd dt = x 2 - 2x + 1. Find ƒ(x). x 78. Find ƒ(4) if 10 ƒstd dt = x cos px. 79. Find the linearization of

L2

ƒsxd dx

y

b. Find the tree’s average height for 0 … t … 8.

x+1

L0

meters. Use the graph to answer the following questions. Give reasons for your answers.

a. Find the tree’s height when t = 0, t = 4, and t = 8.

ƒsxd = 2 -

327

(5, 2)

1 2 3 4 5 6 7 8 9

0 –1 –2

9 dt 1 + t

y f (x)

x

a. What is the particle’s velocity at time t = 5?

at x = 1.

b. Is the acceleration of the particle at time t = 5 positive, or negative?

80. Find the linearization of x2

g sxd = 3 +

sec st - 1d dt

L1

c. What is the particle’s position at time t = 3? d. At what time during the first 9 sec does s have its largest value?

at x = -1.

e. Approximately when is the acceleration zero?

81. Suppose that ƒ has a positive derivative for all values of x and that ƒs1d = 0. Which of the following statements must be true of the function x

g sxd =

L0

ƒstd dt ?

f. When is the particle moving toward the origin? Away from the origin? g. On which side of the origin does the particle lie at time t = 9? 84. Find lim

x: q

Give reasons for your answers.

1

x

dt

2x L1 2t

.

a. g is a differentiable function of x. b. g is a continuous function of x. c. The graph of g has a horizontal tangent at x = 1. d. g has a local maximum at x = 1.

COMPUTER EXPLORATIONS x In Exercises 85–88, let Fsxd = 1a ƒstd dt for the specified function ƒ and interval [a, b]. Use a CAS to perform the following steps and answer the questions posed.

e. g has a local minimum at x = 1.

a. Plot the functions ƒ and F together over [a, b].

f. The graph of g has an inflection point at x = 1.

b. Solve the equation F¿sxd = 0. What can you see to be true about the graphs of ƒ and F at points where F¿sxd = 0? Is your observation borne out by Part 1 of the Fundamental Theorem coupled with information provided by the first derivative? Explain your answer.

g. The graph of dg>dx crosses the x-axis at x = 1. 82. Another proof of the Evaluation Theorem a. Let a = x0 6 x1 6 x2 Á 6 xn = b be any partition of [a, b], and let F be any antiderivative of ƒ. Show that n

F(b) - F(a) = a [F(xi d - F(xi-1 d] . i=1

b. Apply the Mean Value Theorem to each term to show that F(xi) - F(xi-1) = ƒ (ci)(xi - xi-1) for some ci in the interval (xi-1, xi). Then show that F(b) - F(a) is a Riemann sum for ƒ on [a, b]. c. From part sbd and the definition of the definite integral, show that b

F sbd - F sad =

La

ƒsxd dx.

c. Over what intervals (approximately) is the function F increasing and decreasing? What is true about ƒ over those intervals? d. Calculate the derivative ƒ¿ and plot it together with F. What can you see to be true about the graph of F at points where ƒ¿sxd = 0? Is your observation borne out by Part 1 of the Fundamental Theorem? Explain your answer. 85. ƒsxd = x 3 - 4x 2 + 3x,

[0, 4]

86. ƒsxd = 2x 4 - 17x 3 + 46x 2 - 43x + 12, x 87. ƒsxd = sin 2x cos , 3 88. ƒsxd = x cos px,

[0, 2p]

[0, 2p]

9 c0, d 2

328

Chapter 5: Integration u(x)

In Exercises 89–92, let Fsxd = 1a ƒstd dt for the specified a, u, and ƒ. Use a CAS to perform the following steps and answer the questions posed. a. Find the domain of F. b. Calculate F¿sxd and determine its zeros. For what points in its domain is F increasing? Decreasing? c. Calculate F–sxd and determine its zero. Identify the local extrema and the points of inflection of F. d. Using the information from parts (a)– (c), draw a rough handsketch of y = Fsxd over its domain. Then graph F(x) on your CAS to support your sketch.

5.5

89. a = 1,

usxd = x 2,

ƒsxd = 21 - x 2

90. a = 0,

usxd = x ,

ƒsxd = 21 - x 2

91. a = 0,

usxd = 1 - x,

92. a = 0,

usxd = 1 - x ,

2

2

ƒsxd = x 2 - 2x - 3 ƒsxd = x 2 - 2x - 3

In Exercises 93 and 94, assume that ƒ is continuous and u(x) is twicedifferentiable. usxd

93. Calculate

d dx La

94. Calculate

d2 dx 2 La

ƒstd dt and check your answer using a CAS. usxd

ƒstd dt and check your answer using a CAS.

Indefinite Integrals and the Substitution Method The Fundamental Theorem of Calculus says that a definite integral of a continuous function can be computed directly if we can find an antiderivative of the function. In Section 4.8 we defined the indefinite integral of the function ƒ with respect to x as the set of all antiderivatives of ƒ, symbolized by

L

ƒsxd dx.

Since any two antiderivatives of ƒ differ by a constant, the indefinite integral 1 notation means that for any antiderivative F of ƒ,

L

ƒsxd dx = Fsxd + C,

where C is any arbitrary constant. The connection between antiderivatives and the definite integral stated in the Fundamental Theorem now explains this notation. When finding the indefinite integral of a function ƒ, remember that it always includes an arbitrary constant C. We must distinguish carefully between definite and indefinite integrals. A definite b integral 1a ƒsxd dx is a number. An indefinite integral 1 ƒsxd dx is a function plus an arbitrary constant C. So far, we have only been able to find antiderivatives of functions that are clearly recognizable as derivatives. In this section we begin to develop more general techniques for finding antiderivatives.

Substitution: Running the Chain Rule Backwards If u is a differentiable function of x and n is any number different from -1, the Chain Rule tells us that un+1 d du a b = un . dx n + 1 dx From another point of view, this same equation says that u n + 1>sn + 1d is one of the antiderivatives of the function u nsdu>dxd. Therefore, L

un

du un+1 + C. dx = n + 1 dx

(1)

5.5

Indefinite Integrals and the Substitution Method

329

The integral in Equation (1) is equal to the simpler integral L

u n du =

un+1 + C, n + 1

which suggests that the simpler expression du can be substituted for sdu>dxd dx when computing an integral. Leibniz, one of the founders of calculus, had the insight that indeed this substitution could be done, leading to the substitution method for computing integrals. As with differentials, when computing integrals we have du =

EXAMPLE 1

Solution

Find the integral

L

du dx. dx

sx 3 + xd5s3x 2 + 1d dx.

We set u = x 3 + x. Then du =

du dx = s3x 2 + 1d dx, dx

so that by substitution we have L

sx 3 + xd5s3x 2 + 1d dx =

Let u = x 3 + x, du = s3x 2 + 1d dx.

L u6 = + C 6 =

EXAMPLE 2 Solution

u 5 du

Find

L

Integrate with respect to u.

sx 3 + xd6 + C 6

Substitute x 3 + x for u.

22x + 1 dx.

The integral does not fit the formula L

u n du,

with u = 2x + 1 and n = 1>2, because du =

du dx = 2 dx dx

is not precisely dx. The constant factor 2 is missing from the integral. However, we can introduce this factor after the integral sign if we compensate for it by a factor of 1>2 in front of the integral sign. So we write L

22x + 1 dx =

1 22x + 1 # 2 dx 2 L (1)1* ()* u

du

1 = u 1>2 du 2L

Let u = 2x + 1, du = 2 dx.

=

1 u 3>2 + C 2 3>2

Integrate with respect to u.

=

1 s2x + 1d3>2 + C 3

Substitute 2x + 1 for u.

The substitutions in Examples 1 and 2 are instances of the following general rule.

330

Chapter 5: Integration

THEOREM 6—The Substitution Rule If u = gsxd is a differentiable function whose range is an interval I, and ƒ is continuous on I, then L

ƒsgsxddg¿sxd dx =

L

ƒsud du.

Proof By the Chain Rule, F(g(x)) is an antiderivative of ƒsgsxdd # g¿sxd whenever F is an antiderivative of ƒ: d Fsgsxdd = F¿sgsxdd # g¿sxd dx = ƒsgsxdd # g¿sxd.

Chain Rule F¿ = ƒ

If we make the substitution u = gsxd, then

L

d Fsgsxdd dx dx L = Fsgsxdd + C = Fsud + C

ƒsgsxddg¿sxd dx =

= =

L L

Fundamental Theorem u = gsxd

F¿sud du

Fundamental Theorem

ƒsud du

F¿ = ƒ

The Substitution Rule provides the following substitution method to evaluate the integral

L

ƒsgsxddg¿sxd dx,

when ƒ and g¿ are continuous functions: 1.

Substitute u = gsxd and du = sdu>dxd dx = g¿sxd dx to obtain the integral L

2. 3.

ƒsud du.

Integrate with respect to u. Replace u by g(x) in the result.

EXAMPLE 3 Solution

Find

L

sec2 s5t + 1d # 5 dt.

We substitute u = 5t + 1 and du = 5 dt. Then, L

sec2 s5t + 1d # 5 dt =

EXAMPLE 4

Find

L

sec2 u du L = tan u + C

Let u = 5t + 1, du = 5 dt.

= tan s5t + 1d + C

Substitute 5t + 1 for u.

cos s7u + 3d du.

d tan u = sec2 u du

5.5

Indefinite Integrals and the Substitution Method

331

We let u = 7u + 3 so that du = 7 du. The constant factor 7 is missing from the du term in the integral. We can compensate for it by multiplying and dividing by 7, using the same procedure as in Example 2. Then,

Solution

L

cos s7u + 3d du = =

1 cos s7u + 3d # 7 du 7L

Place factor 1>7 in front of integral.

1 cos u du 7L

Let u = 7u + 3, du = 7 du.

1 sin u + C 7 1 = sin s7u + 3d + C 7

=

Integrate. Substitute 7u + 3 for u.

There is another approach to this problem. With u = 7u + 3 and du = 7 du as before, we solve for du to obtain du = s1>7d du. Then the integral becomes L

cos u #

cos s7u + 3d du =

1 du 7

L 1 = sin u + C 7 1 = sin s7u + 3d + C 7

Let u = 7u + 3, du = 7 du, and du = s1>7d du. Integrate. Substitute 7u + 3 for u.

We can verify this solution by differentiating and checking that we obtain the original function cos s7u + 3d.

EXAMPLE 5 Sometimes we observe that a power of x appears in the integrand that is one less than the power of x appearing in the argument of a function we want to integrate. This observation immediately suggests we try a substitution for the higher power of x. This situation occurs in the following integration. 3

L

x2ex dx =

L

=

eu

#

x2 dx 1 du 3

L 1 = eu du 3L 1 = eu + C 3 1 3 = ex + C 3

HISTORICAL BIOGRAPHY George David Birkhoff (1884–1944)

#

3

ex

Let u = x3, du = 3x2 dx, (1>3) du = x2 dx.

Integrate with respect to u. Replace u by x 3 .

EXAMPLE 6

An integrand may require some algebraic manipulation before the substitution method can be applied. This example gives two integrals obtained by multiplying the integrand by an algebraic form equal to 1, leading to an appropriate substitution. (a)

dx e x dx -x = 2x Le + e Le + 1 du = 2 Lu + 1 = tan-1 u + C = tan-1 se x d + C x

Multiply by se x>e x d = 1. Let u = e x, u 2 = e 2x, du = e x dx. Integrate with respect to u. Replace u by e x.

332

Chapter 5: Integration

(b)

sec x dx =

L

L

(sec x)(1) dx =

sec x #

L sec x + sec x tan x = dx L sec x + tan x

sec x + tan x dx sec x + tan x

sec x + tan x is a form of 1 sec x + tan x

2

u = tan x + sec x, du = (sec2 x + sec x tan x) dx

du u

=

L = ln ƒ u ƒ + C = ln ƒ sec x + tan x ƒ + C.

EXAMPLE 7 Sometimes we can use trigonometric identities to transform integrals we do not know how to evaluate into ones we can evaluate using the Substitution Rule. (a)

(b)

L

L

sin2 x dx =

1 - cos 2x dx 2 L

=

1 s1 - cos 2xd dx 2L

=

sin 2x x 1 1 sin 2x x + C = + C 2 2 2 2 4

cos2 x dx =

sin 2x 1 + cos 2x x dx = + + C 2 2 4 L

sin2 x =

1 - cos 2x 2

cos2 x =

1 + cos 2x 2

It may happen that an extra factor of x appears in the integrand when we try a substitution u = gsxd. In that case, it may be possible to solve the equation u = gsxd for x in terms of u. Replacing the extra factor of x with that expression may then allow for an integral we can evaluate. Here’s an example of this situation. x22x + 1 dx. L Solution Our previous integration in Example 2 suggests the substitution u = 2x + 1 with du = 2 dx. Then,

EXAMPLE 8

Evaluate

22x + 1 dx =

1 2u du. 2

However, in this case the integrand contains an extra factor of x multiplying the term 12x + 1. To adjust for this, we solve the substitution equation u = 2x + 1 to obtain x = (u - 1)>2, and find that x22x + 1 dx =

1 1 su - 1d # 2u du. 2 2

The integration now becomes

L

1 1 su - 1d2u du = su - 1du 1>2 du 4L 4L 1 = su 3>2 - u 1>2 d du 4L 1 2 2 = a u 5>2 - u 3>2 b + C 4 5 3

x22x + 1 dx =

=

1 1 s2x + 1d5>2 - s2x + 1d3>2 + C. 10 6

Substitute.

Multiply terms.

Integrate. Replace u by 2x + 1.

5.5

Indefinite Integrals and the Substitution Method

333

The success of the substitution method depends on finding a substitution that changes an integral we cannot evaluate directly into one that we can. If the first substitution fails, try to simplify the integrand further with additional substitutions (see Exercises 67 and 68).

EXAMPLE 9

Evaluate

2z dz L 2z 2 + 1 3

.

Solution We can use the substitution method of integration as an exploratory tool: Substitute for the most troublesome part of the integrand and see how things work out. For the integral here, we might try u = z 2 + 1 or we might even press our luck and take u to be the entire cube root. Here is what happens in each case. Solution 1: Substitute u = z 2 + 1.

2z dz L 2z + 1 3

2

du L u 1>3

= =

L

Let u = z 2 + 1, du = 2z dz .

u -1>3 du

In the form 1 u n du

u 2>3 + C 2>3

=

Integrate.

3 2>3 u + C 2 3 = sz 2 + 1d2>3 + C 2 =

Replace u by z 2 + 1 .

3 Solution 2: Substitute u = 2z 2 + 1 instead.

2z dz L 2z + 1 3

2

=

L

= 3

3u 2 du u

L

= 3# =

3 2 Let u = 2 z + 1, u 3 = z 2 + 1, 3u 2 du = 2z dz.

u du

u2 + C 2

Integrate.

3 2 sz + 1d2>3 + C 2

Replace u by sz 2 + 1d1>3 .

Exercises 5.5 Evaluating Indefinite Integrals Evaluate the indefinite integrals in Exercises 1–16 by using the given substitutions to reduce the integrals to standard form. 1. 2. 3.

L L L

4.

2s2x + 4d5 dx,

u = 2x + 4

5.

727x - 1 dx,

u = 7x - 1

6.

2xsx 2 + 5d-4 dx,

u = x2 + 5

7.

4x 3 dx, 2 L sx + 1d 4

L

L L

u = x4 + 1

s3x + 2ds3x 2 + 4xd4 dx,

A 1 + 2x B 1>3 2x sin 3x dx,

dx,

u = 3x

u = 3x 2 + 4x

u = 1 + 2x

334 8. 9.

L L

Chapter 5: Integration

x sin s2x 2 d dx,

u = 2x 2

41.

u = 2t

43.

sec 2t tan 2t dt, 2

10. 11. 12. 13. 14. 15.

16.

t t a1 - cos b sin dt, 2 2 L 9r 2 dr L 21 - r 3 L L

,

u = 1 - cos

t 2

45.

u = 1 - r3

47.

12s y 4 + 4y 2 + 1d2s y 3 + 2yd dy, 1x sin2 sx 3>2 - 1d dx,

1 1 cos2 a x b dx, 2 Lx

u = y 4 + 4y 2 + 1

u = x 3>2 - 1

51.

1 u = -x

csc2 2u cot 2u du L a. Using u = cot 2u

b. Using u = csc 2u

19.

L 25x + 8 a. Using u = 5x + 8

23 - 2s ds 4

L

u 21 - u2 du

1 dx 21. 2 L 1x s1 + 1xd 23.

L

sec2 s3x + 2d dx

x x sin cos dx 25. 3 3 L 5 r3 r2 a - 1 b dr 27. 18 L 5

29. 30.

L L

sin s2t + 1d dt 2 cos s2t + 1d L 1 1 cos a t - 1b dt 33. 2 Lt 35.

1 1 1 sin cos du 2 u u Lu

37.

t s1 + t d dt

39.

L

44.

sx + 1d2s1 - xd5 dx

46.

x 3 2x 2 + 1 dx

48.

x dx 2 3 L sx - 4d L

50.

(cos x) e sin x dx

b. Using u = 25x + 8

18. 20. 22. 24.

1 ds L 25s + 4 L L L

61.

3y27 - 3y 2 dy 63. cos s3z + 4d dz 65. tan2 x sec2 x dx

x x tan sec2 dx 26. 2 2 L 3 r5 r 4 a7 b dr 28. 10 L 7

4 3

1 1 2 - x dx 2 Lx A

36.

L

sx + 5dsx - 5d1>3 dx 3x 5 2x 3 + 1 dx

x dx 3 L sx - 4d L

(sin 2u) e sin

2

u

du

e sin

-1

x

56.

L

ln 2t t dt dx

58.

L x2x 4 - 1 1 du 60. L 2e 2u - 1

dx

L 21 - x 2 ssin-1 xd2 dx L 21 - x 2 dy -1 2 L stan yds1 + y d

e cos

-1

x

dx L 21 - x 2 2tan-1 x dx 64. 2 L 1 + x dy 66. -1 L ssin yd21 - y 2 62.

If you do not know what substitution to make, try reducing the integral step by step, using a trial substitution to simplify the integral a bit and then another to simplify it some more. You will see what we mean if you try the sequences of substitutions in Exercises 67 and 68. 67.

18 tan2 x sec2 x dx 3 2 L s2 + tan xd a. u = tan x , followed by y = u 3 , then by w = 2 + y c. u = 2 + tan3 x

sec z tan z

cos 2u L 2u sin 2u 2

68.

1 x2 - 1 dx 3 2 Lx A x

21 + sin2 sx - 1d sin sx - 1d cos sx - 1d dx L a. u = x - 1 , followed by y = sin u , then by w = 1 + y2 b. u = sin sx - 1d , followed by y = 1 + u 2 c. u = 1 + sin2 sx - 1d

du

x - 1 dx 38. 5 LA x 40.

L

x24 - x dx

b. u = tan3 x , followed by y = 2 + u

dz L 2sec z 1 cos s 1t + 3d dt 34. L 1t 32.

52.

L

1

dx L x ln x dz 57. 1 + ez L 5 dr 59. 2 L 9 + 4r

y - p y - p b cot a b dy 2 2

31.

3

L

xsx - 1d10 dx

x4 dx L Ax - 1 3

sec2 (e 2x + 1) dx L 2xe - 2x 1 1>x e sec (1 + e 1>x ) tan (1 + e 1>x ) dx 54. 2 Lx

x 1>2 sin sx 3>2 + 1d dx csc a

L

42.

55.

dx

L

L

53.

Evaluate the integrals in Exercises 17–66. 17.

49.

x3 - 3 dx 11 LA x

Evaluate the integrals in Exercises 69 and 70. s2r - 1d cos 23s2r - 1d2 + 6 dr 69. L 23s2r - 1d2 + 6 70.

sin 2u L 2u cos3 1u

du

5.6

ds p = 8 sin2 at + b, 12 dt

dr p 74. = 3 cos2 a - u b, 4 du

5.6

335

d 2s p = -4 sin a2t - b, s¿s0d = 100, s s0d = 0 2 dt 2 d 2y = 4 sec2 2x tan 2x, y¿s0d = 4, y s0d = -1 76. dx 2 77. The velocity of a particle moving back and forth on a line is y = ds>dt = 6 sin 2t m>sec for all t. If s = 0 when t = 0 , find the value of s when t = p>2 sec .

Initial Value Problems Solve the initial value problems in Exercises 71–76. ds 71. = 12t s3t 2 - 1d3, s s1d = 3 dt dy 72. = 4x sx 2 + 8d-1>3, y s0d = 0 dx 73.

Substitution and Area Between Curves

75.

s s0d = 8

78. The acceleration of a particle moving back and forth on a line is a = d 2s>dt 2 = p2 cos pt m>sec2 for all t. If s = 0 and y = 8 m/sec when t = 0 , find s when t = 1 sec .

p r s0d = 8

Substitution and Area Between Curves There are two methods for evaluating a definite integral by substitution. One method is to find an antiderivative using substitution and then to evaluate the definite integral by applying the Evaluation Theorem. The other method extends the process of substitution directly to definite integrals by changing the limits of integration. We apply the new formula introduced here to the problem of computing the area between two curves.

The Substitution Formula The following formula shows how the limits of integration change when the variable of integration is changed by substitution.

THEOREM 7—Substitution in Definite Integrals If g¿ is continuous on the interval [a, b] and ƒ is continuous on the range of gsxd = u, then ƒsgsxdd # g¿sxd dx =

b

La

gsbd

Lgsad

ƒsud du.

Proof Let F denote any antiderivative of ƒ. Then, ƒsgsxdd # g¿sxd dx = Fsgsxdd d

b

La

x=b x=a

d Fsgsxdd dx = F¿sgsxddg¿sxd = ƒsgsxddg¿sxd

= Fsgsbdd - Fsg sadd = Fsud d

u = gsbd u = gsad

gsbd

=

Lgsad

ƒsud du.

Fundamental Theorem, Part 2

To use the formula, make the same u-substitution u = gsxd and du = g¿sxd dx you would use to evaluate the corresponding indefinite integral. Then integrate the transformed integral with respect to u from the value g(a) (the value of u at x = a) to the value g(b) (the value of u at x = b).

336

Chapter 5: Integration 1

EXAMPLE 1 Solution

Evaluate

3x 2 2x 3 + 1 dx.

L-1

We have two choices.

Method 1: Transform the integral and evaluate the transformed integral with the trans-

formed limits given in Theorem 7. 1

L-1

3x 2 2x 3 + 1 dx

Let u = x 3 + 1, du = 3x 2 dx . When x = - 1, u = s -1d3 + 1 = 0 . When x = 1, u = s1d3 + 1 = 2 .

2

=

1u du

L0

2

=

2 3>2 u d 3 0

=

422 2 3>2 C 2 - 0 3>2 D = 23 C 222 D = 3 3

Evaluate the new definite integral.

Method 2: Transform the integral as an indefinite integral, integrate, change back to x, and use the original x-limits.

3x 2 2x 3 + 1 dx =

1u du L 2 = u 3>2 + C 3

L

=

Integrate with respect to u.

2 3 sx + 1d3>2 + C 3 1

1

L-1

Let u = x 3 + 1, du = 3x 2 dx .

3x 2 2x 3 + 1 dx =

2 3 sx + 1d3>2 d 3 -1

Replace u by x 3 + 1 . Use the integral just found, with limits of integration for x.

=

2 C ss1d3 + 1d3>2 - ss -1d3 + 1d3>2 D 3

=

422 2 3>2 C 2 - 0 3>2 D = 23 C 222 D = 3 3

Which method is better—evaluating the transformed definite integral with transformed limits using Theorem 7, or transforming the integral, integrating, and transforming back to use the original limits of integration? In Example 1, the first method seems easier, but that is not always the case. Generally, it is best to know both methods and to use whichever one seems better at the time.

EXAMPLE 2

We use the method of transforming the limits of integration.

p>2

(a)

Lp>4

u # s -dud

0

cot u csc2 u du =

L1

0

= -

L1

u du 0

= -c

u2 d 2 1

= -c

s1d2 s0d2 1 d = 2 2 2

Let u = cot u, du -du When u = p>4, u When u = p>2, u

= - csc2 u du, = csc2 u du. = cot (p>4) = 1. = cot (p>2) = 0.

5.6 p>4

(b)

L-p>4

p>4

tan x dx =

L-p>4

= -

Substitution and Area Between Curves

337

sin x cos x dx

22>2

L22>2

Let u = cos x, du = - sin x dx.

du u

When x = - p>4, u = 22>2. When x = p>4, u = 22>2.

= -ln ƒ u ƒ d

22>2

= 0

22>2

Integrate, zero-width interval

Definite Integrals of Symmetric Functions

y

The Substitution Formula in Theorem 7 simplifies the calculation of definite integrals of even and odd functions (Section 1.1) over a symmetric interval [-a, a] (Figure 5.23).

THEOREM 8 –a

0

a

a

(a) If ƒ is even, then

(a)

a

L-a

ƒsxd dx = 2

L0

ƒsxd dx.

a

y

(b) If ƒ is odd, then

0 –a

Let ƒ be continuous on the symmetric interval [-a, a].

x

a

x

L-a

ƒ(x) dx = 0.

Proof of Part (a) a

L-a

(b)

0

ƒsxd dx =

FIGURE 5.23 (a) For ƒ an even function, the integral from -a to a is twice the integral from 0 to a. (b) For ƒ an odd function, the integral from -a to a equals 0.

L-a

= -

a

ƒsxd dx +

ƒsxd dx

L0

-a

a

ƒsxd dx +

L0

L0

ƒsxd dx

Order of Integration Rule

a

Let u = -x, du = -dx . When x = 0, u = 0 . When x = -a, u = a .

a

= -

ƒs -uds -dud +

L0 a

=

ƒs -ud du +

L0 L0

L0

ƒsxd dx

a

L0

a

=

Additivity Rule for Definite Integrals

ƒsxd dx

a

ƒsud du +

L0

ƒsxd dx

ƒ is even, so ƒs -ud = ƒsud .

a

= 2

L0

ƒsxd dx

The proof of part (b) is entirely similar and you are asked to give it in Exercise 114. The assertions of Theorem 8 remain true when ƒ is an integrable function (rather than having the stronger property of being continuous).

2

EXAMPLE 3

Evaluate

L-2

sx 4 - 4x 2 + 6d dx.

338

Chapter 5: Integration

Since ƒsxd = x 4 - 4x 2 + 6 satisfies ƒs -xd = ƒsxd, it is even on the symmetric interval [-2, 2], so Solution

2

L-2

2

sx 4 - 4x 2 + 6d dx = 2

L0

= 2 c

sx 4 - 4x 2 + 6d dx 2

x5 4 - x 3 + 6x d 5 3 0

= 2 a

32 32 232 + 12b = . 5 3 15

Areas Between Curves

y

Suppose we want to find the area of a region that is bounded above by the curve y = ƒsxd, below by the curve y = gsxd, and on the left and right by the lines x = a and x = b (Figure 5.24). The region might accidentally have a shape whose area we could find with geometry, but if ƒ and g are arbitrary continuous functions, we usually have to find the area with an integral. To see what the integral should be, we first approximate the region with n vertical rectangles based on a partition P = 5x0 , x1, Á , xn6 of [a, b] (Figure 5.25). The area of the kth rectangle (Figure 5.26) is

Upper curve y f(x)

a b

x

Lower curve y g(x)

¢Ak = height * width = [ƒsck d - gsck d] ¢xk .

FIGURE 5.24 The region between the curves y = ƒsxd and y = gsxd and the lines x = a and x = b .

We then approximate the area of the region by adding the areas of the n rectangles: n

n

A L a ¢Ak = a [ƒsck d - gsck d] ¢xk .

y

k=1

y f(x)

Riemann sum

k=1

As 7P7 : 0, the sums on the right approach the limit 1a [ƒsxd - gsxd] dx because ƒ and g are continuous. We take the area of the region to be the value of this integral. That is, b

a x 0 x1

x n1 x2

x b xn

n

b

A = lim a [ƒsck d - gsck d] ¢xk = ƒ ƒ P ƒ ƒ :0

y g(x)

k=1

FIGURE 5.25 We approximate the region with rectangles perpendicular to the x-axis.

La

[ƒsxd - gsxd] dx.

DEFINITION If ƒ and g are continuous with ƒsxd Ú gsxd throughout [a, b], then the area of the region between the curves y f (x) and y g(x) from a to b is the integral of ( f - g) from a to b:

y

b

A =

(ck , f(ck ))

La

[ƒsxd - gsxd] dx.

f (ck ) g(ck ) a

Ak

ck b (ck , g(ck )) xk

FIGURE 5.26 The area ¢Ak of the kth rectangle is the product of its height, ƒsck d - g sck d , and its width, ¢xk .

x

When applying this definition it is helpful to graph the curves. The graph reveals which curve is the upper curve ƒ and which is the lower curve g. It also helps you find the limits of integration if they are not given. You may need to find where the curves intersect to determine the limits of integration, and this may involve solving the equation ƒsxd = gsxd for values of x. Then you can integrate the function ƒ - g for the area between the intersections. Find the area of the region bounded above by the curve y = 2e -x + x, below by the curve y = e x>2 , on the left by x = 0, and on the right by x = 1.

EXAMPLE 4

5.6 y

Substitution and Area Between Curves

339

Figure 5.27 displays the graphs of the curves and the region whose area we want to find. The area between the curves over the interval 0 … x … 1 is given by

Solution (x, f(x))

2

y 2e –x x

1

A =

0.5

L0

1

c(2e -x + x) -

1 x 1 1 e ddx = c-2e -x + x 2 - e x d 2 2 2 0

y 1 ex 2

(x, g(x))

= a-2e -1 +

0

1

x

FIGURE 5.27 The region in Example 4 with a typical approximating rectangle.

1 1 1 - eb - a-2 + 0 - b 2 2 2

e 2 = 3 - e - L 0.9051. 2

EXAMPLE 5 line y = -x.

Find the area of the region enclosed by the parabola y = 2 - x 2 and the

y

First we sketch the two curves (Figure 5.28). The limits of integration are found by solving y = 2 - x 2 and y = -x simultaneously for x.

Solution

(x, f (x)) y 2 x2

(–1, 1)

2 - x2 x2 - x - 2 sx + 1dsx - 2d x = -1, x

x

–1

0

1

x

2

(x, g(x)) y –x

b

A =

Richard Dedekind (1831–1916)

0 L

(4, 2)

yx2 A y0 2

0

y x (x, f (x)) B

1

Solve.

L-1

[s2 - x2 d - s -xd] dx

L-1

s2 + x - x2 d dx = c2x +

2

x2 x3 - d 2 3 -1

8 9 4 1 1 - b - a-2 + + b = . 2 3 2 3 2

Find the area of the region in the first quadrant that is bounded above by y = 1x and below by the x-axis and the line y = x - 2.

(x x 2) dx

(x, f(x))

Factor.

EXAMPLE 6

L 2

2

Rewrite.

If the formula for a bounding curve changes at one or more points, we subdivide the region into subregions that correspond to the formula changes and apply the formula for the area between curves to each subregion.

4

Area x dx

La

= a4 +

HISTORICAL BIOGRAPHY

2

Equate ƒ(x) and g(x).

2

[ƒsxd - gsxd] dx = 2

=

Area

-x 0 0 2.

The region runs from x = -1 to x = 2. The limits of integration are a = -1, b = 2. The area between the curves is

(2, –2)

FIGURE 5.28 The region in Example 5 with a typical approximating rectangle.

y

= = = =

(x, g(x)) 4

x

Solution The sketch (Figure 5.29) shows that the region’s upper boundary is the graph of ƒsxd = 1x. The lower boundary changes from gsxd = 0 for 0 … x … 2 to gsxd = x - 2 for 2 … x … 4 (both formulas agree at x = 2). We subdivide the region at x = 2 into subregions A and B, shown in Figure 5.29. The limits of integration for region A are a = 0 and b = 2. The left-hand limit for region B is a = 2. To find the right-hand limit, we solve the equations y = 1x and y = x - 2 simultaneously for x:

(x, g(x))

FIGURE 5.29 When the formula for a bounding curve changes, the area integral changes to become the sum of integrals to match, one integral for each of the shaded regions shown here for Example 6.

1x = x - 2 x x - 5x + 4 sx - 1dsx - 4d x 2

= = = =

Equate ƒ(x) and g(x).

sx - 2d = x - 4x + 4 0 0 1, x = 4. 2

2

Square both sides. Rewrite. Factor. Solve.

340

Chapter 5: Integration

Only the value x = 4 satisfies the equation 1x = x - 2. The value x = 1 is an extraneous root introduced by squaring. The right-hand limit is b = 4. For 0 … x … 2: For 2 … x … 4:

ƒsxd - gsxd = 1x - 0 = 1x ƒsxd - gsxd = 1x - sx - 2d = 1x - x + 2

We add the areas of subregions A and B to find the total area: 2

Total area =

4

1x dx + s 1x - x + 2d dx 0 L L2 (')'* ('''')''''* area of A

area of B

2

4

x2 2 2 = c x 3>2 d + c x 3>2 + 2x d 3 3 2 0 2 =

2 2 2 s2d3>2 - 0 + a s4d3>2 - 8 + 8b - a s2d3>2 - 2 + 4b 3 3 3

=

10 2 s8d - 2 = . 3 3

Integration with Respect to y If a region’s bounding curves are described by functions of y, the approximating rectangles are horizontal instead of vertical and the basic formula has y in place of x. For regions like these: y

y

d

y

d

x f(y)

x f (y)

d x g(y) Δ(y)

Δ(y) x g(y)

Δ(y)

c x

x

0

c

c 0

x g(y)

x f(y)

x

0

use the formula d

A =

[ƒs yd - gs yd] dy. Lc In this equation ƒ always denotes the right-hand curve and g the left-hand curve, so ƒs yd - gs yd is nonnegative.

EXAMPLE 7

Find the area of the region in Example 6 by integrating with respect to y.

y

We first sketch the region and a typical horizontal rectangle based on a partition of an interval of y-values (Figure 5.30). The region’s right-hand boundary is the line x = y + 2, so ƒs yd = y + 2. The left-hand boundary is the curve x = y 2 , so gs yd = y 2 . The lower limit of integration is y = 0. We find the upper limit by solving x = y + 2 and x = y 2 simultaneously for y: Solution

2 1

0

(g(y), y)

x

xy2

y f(y) g(y) y0

(4, 2)

y2

2

( f(y), y) 4

x

FIGURE 5.30 It takes two integrations to find the area of this region if we integrate with respect to x. It takes only one if we integrate with respect to y (Example 7).

y + 2 2 y - y - 2 s y + 1ds y - 2d y = -1, y

= = = =

y2 0 0 2

Equate ƒs yd = y + 2 and gs yd = y 2. Rewrite. Factor. Solve.

The upper limit of integration is b = 2. (The value y = -1 gives a point of intersection below the x-axis.)

5.6

Substitution and Area Between Curves

341

The area of the region is d

A =

Lc

2

[ƒs yd - gs yd] dy =

[y + 2 - y 2] dy

L0 2

=

L0

[2 + y - y 2] dy

= c2y + = 4 +

y2 y3 2 d 2 3 0

8 10 4 - = . 2 3 3

This is the result of Example 6, found with less work.

Exercises 5.6 Evaluating Definite Integrals Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises 1–46. 3

1. a.

0

2y + 1 dy

L0

b.

2. a.

L0

b.

3. a.

tan x sec x dx

b.

4. a.

tan x sec x dx

L-p>4

3 cos x sin x dx

b.

5. a.

t 3s1 + t 4 d3 dt

b.

27

6. a.

t st 2 + 1d1>3 dt

L0

b.

1

7. a.

b.

1

8. a.

23

9. a.

L0

x3

L0 2x 4 + 9

b.

dx

b.

dx

b.

24 + 3 sin z

sin w dw 14. a. 2 L-p>2 s3 + 2 cos wd

b.

Lp>6

31.

b.

L0 4

2t 5 + 2t s5t 4 + 2d dt

16.

26.

L-1

sin t dt L0 2 - cos t L1

Lp>4

dx

Lp>2

s1 - cos 3td sin 3t dt

34.

2 cot

u du 3

36.

2 cos u du 1 + ssin ud2

38.

ln 23

cos z

dz

sin w dw s3 + 2 cos wd2 dy

L1 2 1y s1 + 1yd

2

39.

L0 1

41.

2

43.

L22 L-1

cot t dt 6 tan 3x dx

L0 p>4

Lp>6 ep>4

40.

L1

csc2 x dx 1 + scot xd2 4 dt ts1 + ln2 td

322>4

4 ds

42.

2

sec2 ssec-1 xd dx x2x 2 - 1

-22>2

45.

Lp>4

p>12

e x dx 1 + e 2x

L0 24 - s

4 sin u du 1 - 4 cos u

p>2

x dx 2

L-p>2

L0

dx L2 x ln x 16 dx 32. L2 2x2ln x

tan

p>2

37.

p>3

28. 30.

dx xsln xd2 L2

p

35.

s1 + e cot u d csc2 u du

4

2 ln x x dx

L0

1 t -2 sin2 a1 + t b dt

p>2

p>2

33.

dx

L-p 24 + 3 sin z p>2

1

L0

x3

L-1 2x 4 + 9

0

15.

4x

L-23 2x 2 + 1

p

dz

s1 + e tan u d sec2 u du

4

10 1y dy s1 + y3>2 d2 1 L

t t t t a2 + tan b sec2 dt b. a2 + tan b sec2 dt 2 2 2 2 L-p>2 L-p>2 L0

L0

2

p>2

cos z

27. 29.

p>3

b.

24.

p

t st 2 + 1d1>3 dt

5r dr 2 2 L0 s4 + r d

0

s1 - cos 3td sin 3t dt

L0

2p

13. a.

L0

-1>2

2u cos2 su3>2 d du

p>4

L-27

0

12. a.

25.

L-1 0

p>6

11. a.

t 3s1 + t 4 d3 dt

3 cos x sin x dx

23

4x 2x 2 + 1

1

10. a.

23.

4

10 1y dy s1 + y 3>2 d2 0 L

s y 3 + 6y 2 - 12y + 9d-1>2 s y 2 + 4y - 4d dy

L0

1

5r dr 2 2 L-1 s4 + r d

s4y - y 2 + 4y 3 + 1d-2>3 s12y 2 - 2y + 4d dy

L0

1

L0

s1 - sin 2td3>2 cos 2t dt

L0

3

2

L2p

1

L0

u u cot5 a b sec2 a b du 6 6

p>4

5s5 - 4 cos td1>4 sin t dt 20.

2p2

3p 2

L0

22.

2

p

Lp

1

0 2

L0

21.

r21 - r 2 dr

L-1

p>4

3p>2

18.

1

1

r21 - r 2 dr

L0

cos-3 2u sin 2u du

p

19.

2y + 1 dy

L-1

1

p>6

17.

dy y24y - 1 2

44.

L0 2

cos ssec-1 xd dx

L2>23

x2x 2 - 1

-22>3

46.

ds 29 - 4s 2

L-2>3

dy y29y 2 - 1

342

Chapter 5: Integration

Area Find the total areas of the shaded regions in Exercises 47–62. 47.

55.

y 1

x 12y 2 12y 3

48. y

x 2y 2 2y

y

0

y (1 cos x) sin x

y x4 x 2

1

56. –2

0

x

2

57.

x

0

y

y

yx

1 y x2

49.

50.

–

–2

–1

x

0

–1

0

1

y1 2 y x 4

x

1

y (cos x)(sin( sin x)) 2 y

y

x

0

1

x

2

y –2x 4

1

–1

–2 – –1 2

–

–2

0 –1

x

58.

y

–3 1

y 3(sin x)1 cos x

51.

52.

0

y

xy2

y x2 1

x

2

y y1

1

2 y

59.

60.

1

cos 2 x x

2

0

y 1 sec2 t 2

– 3

0

y 3 y

t

(–3, 5)

5

y x2 4

– 4 sin2 t

–4 –3

53.

y

y y –x 2 3x (2, 2) 2

0

–2 –1

x

2

y 2x 3 x 2 5x

x 1 y –x 2 2x

(–2, –10) –10

(1, –3)

(–3, –3)

1

–4 (–2, 8)

8

(2, 8) y 2x 2 y x 4 2x 2

61.

4

(–2, 4) –2 –1 –1

1

x

2

54.

1 2 3

1

x y3

(3, 6)

3 y x x 3

y x 3

x y –x 2

y –5

y 6

y 4 x2

2 –2 –1

NOT TO SCALE

62.

y

(3, –5)

(3, 1) –2

0

3

x

⎛–2, – 2 ⎛ ⎝ 3⎝

(1, 1)

x y2

0

1

x

Find the areas of the regions enclosed by the lines and curves in Exercises 63–72. 63. y = x 2 - 2 65. y = x

4

and

and

y = 2

y = 8x

64. y = 2x - x 2

and

y = -3

66. y = x - 2x

and

y = x

2

5.6 67. y = x 2

y = -x 2 + 4x

and

68. y = 7 - 2x

2

y = x + 4

69. y = x 4 - 4x 2 + 4

and

y = x2

70. y = x2a 2 - x 2,

a 7 0,

and

71. y = 2 ƒ x ƒ are there?)

5y = x + 6 (How many intersection points

and

72. y = ƒ x 2 - 4 ƒ

y = 0

y = sx 2>2d + 4

and

Find the areas of the regions enclosed by the lines and curves in Exercises 73–80. x = 0,

73. x = 2y 2, 74. x = y 2

y = 3

and

x = y + 2

and

75. y 2 - 4x = 4

4x - y = 16

and

76. x - y = 0

and

x + 2y 2 = 3

77. x + y = 0

and

x + 3y 2 = 2

2 2

and

79. x = y - 1

x = ƒ y ƒ 21 - y and x = 2y

80. x = y - y 3

x + y4 = 2

and 2

82. x 3 - y = 0 83. x + 4y 2 = 4 84. x + y = 3 2

3x 2 - y = 4

and

x + y 4 = 1,

and

for

x Ú 0

4x + y = 0 2

and

Find the areas of the regions enclosed by the lines and curves in Exercises 85–92. 85. y = 2 sin x 86. y = 8 cos x

y = sin 2x,

0 … x … p

and

y = sec x,

-p>3 … x … p>3

87. y = cos spx>2d

and

y = 1 - x2

88. y = sin spx>2d

and

y = x

89. y = sec x, 2

90. x = tan2 y

y = tan x,

x = -p>4,

2

and

x = -tan2 y,

91. x = 3 sin y 2cos y 92. y = sec2 spx>3d

and

and

a. Sketch the region and draw a line y = c across it that looks about right. In terms of c, what are the coordinates of the points where the line and parabola intersect? Add them to your figure.

106. Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the curve x = 21y , above left by the curve x = sy - 1d2, and above right by the line x = 3 - y.

x 2y

96. Find the area of the “triangular” region in the first quadrant bounded on the left by the y-axis and on the right by the curves y = sin x and y = cos x . 97. Find the area between the curves y = ln x and y = ln 2x from x = 1 to x = 5. 98. Find the area between the curve y = tan x and the x-axis from x = -p>4 to x = p>3. 99. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e 2x, below by the curve y = e x, and on the right by the line x = ln 3.

2

x

107. The figure here shows triangle AOC inscribed in the region cut from the parabola y = x 2 by the line y = a 2 . Find the limit of the ratio of the area of the triangle to the area of the parabolic region as a approaches zero. y

94. Find the area of the propeller-shaped region enclosed by the curves x - y 1>3 = 0 and x - y 1>5 = 0 . 95. Find the area of the region in the first quadrant bounded by the line y = x , the line x = 2 , the curve y = 1>x 2 , and the x-axis.

1

0

0 … y … p>2

Area Between Curves 93. Find the area of the propeller-shaped region enclosed by the curve x - y 3 = 0 and the line x - y = 0 .

x3y

1

x = p>4

-1 … x … 1

x ( y 1)2

2

-p>4 … y … p>4

x = 0,

y = x 1>3,

and

105. Find the area of the region in the first quadrant bounded on the left by the y-axis, below by the line y = x>4 , above left by the curve y = 1 + 1x , and above right by the curve y = 2> 1x .

y

and

2

103. The region bounded below by the parabola y = x 2 and above by the line y = 4 is to be partitioned into two subsections of equal area by cutting across it with the horizontal line y = c.

104. Find the area of the region between the curve y = 3 - x 2 and the line y = - 1 by integrating with respect to a. x, b. y.

x4 - y = 1

and

102. Find the area of the region between the curve y = 21 - x and the interval -1 … x … 1 of the x-axis.

c. Find c by integrating with respect to x. (This puts c into the integrand as well.)

2

Find the areas of the regions enclosed by the curves in Exercises 81–84. 81. 4x 2 + y = 4

101. Find the area of the region between the curve y = 2x>s1 + x 2 d and the interval -2 … x … 2 of the x-axis.

b. Find c by integrating with respect to y. (This puts c in the limits of integration.)

78. x - y 2>3 = 0 2

343

100. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e x>2, below by the curve y = e -x>2, and on the right by the line x = 2 ln 2.

2

and

Substitution and Area Between Curves

y x2 C y a2 (a, a 2)

A (–a,

a 2)

–a

O

a

x

108. Suppose the area of the region between the graph of a positive continuous function ƒ and the x-axis from x = a to x = b is 4 square units. Find the area between the curves y = ƒsxd and y = 2ƒsxd from x = a to x = b .

344

Chapter 5: Integration

109. Which of the following integrals, if either, calculates the area of the shaded region shown here? Give reasons for your answer. 1

a.

sx - s -xdd dx =

L-1 L-1

xy

1

L-1

1

b.

116. By using a substitution, prove that for all positive numbers x and y,

2x dx

Lx

1

s -x - sxdd dx =

L-1

The Shift Property for Definite Integrals A basic property of definite integrals is their invariance under translation, as expressed by the equation

-2x dx y

y –x

b

yx

1

y

1 1 t dt = L1 t dt .

La

–1

1

x

ƒsxd dx =

ƒsx + cd dx .

La - c

(1)

The equation holds whenever ƒ is integrable and defined for the necessary values of x. For example in the accompanying figure, show that -1

L-2

–1

110. True, sometimes true, or never true? The area of the region between the graphs of the continuous functions y = ƒsxd and y = g sxd and the vertical lines x = a and x = b sa 6 bd is

b-c

1

sx + 2d3 dx =

L0

x 3 dx

because the areas of the shaded regions are congruent. y

b

[ƒsxd - g sxd] dx . La Give reasons for your answer.

y ( x 2)3

y x3

Theory and Examples 111. Suppose that F(x) is an antiderivative of ƒsxd = ssin xd>x, x 7 0 . Express 3

L1

sin 2x x dx

–2

–1

0

1

x

in terms of F. 112. Show that if ƒ is continuous, then 1

L0

1

ƒsxd dx =

L0

ƒs1 - xd dx .

113. Suppose that

118. For each of the following functions, graph ƒ(x) over [a, b] and ƒsx + cd over [a - c, b - c] to convince yourself that Equation (1) is reasonable. a. ƒsxd = x 2,

1

ƒsxd dx = 3.

L0 Find

0

L-1 if a. ƒ is odd,

117. Use a substitution to verify Equation (1).

ƒsxd dx

b. ƒ is even.

114. a. Show that if ƒ is odd on [-a, a] , then a

L-a

ƒsxd dx = 0.

b. Test the result in part (a) with ƒsxd = sin x and a = p>2 . 115. If ƒ is a continuous function, find the value of the integral a

I =

ƒsxd dx ƒsxd + ƒsa - xd L0

by making the substitution u = a - x and adding the resulting integral to I.

b. ƒsxd = sin x,

a = 0, a = 0,

c. ƒsxd = 2x - 4,

b = 1,

c = 1

b = p,

a = 4,

c = p>2

b = 8,

c = 5

COMPUTER EXPLORATIONS In Exercises 119–122, you will find the area between curves in the plane when you cannot find their points of intersection using simple algebra. Use a CAS to perform the following steps: a. Plot the curves together to see what they look like and how many points of intersection they have. b. Use the numerical equation solver in your CAS to find all the points of intersection. c. Integrate ƒ ƒsxd - g sxd ƒ over consecutive pairs of intersection values. d. Sum together the integrals found in part (c). x3 x2 1 - 2x + , g sxd = x - 1 3 2 3 x4 - 3x 3 + 10, g sxd = 8 - 12x 120. ƒsxd = 2 121. ƒsxd = x + sin s2xd, g sxd = x 3 119. ƒsxd =

122. ƒsxd = x 2 cos x,

g sxd = x 3 - x

Chapter 5

Chapter 5

Practice Exercises

345

Questions to Guide Your Review

1. How can you sometimes estimate quantities like distance traveled, area, and average value with finite sums? Why might you want to do so? 2. What is sigma notation? What advantage does it offer? Give examples. 3. What is a Riemann sum? Why might you want to consider such a sum? 4. What is the norm of a partition of a closed interval? 5. What is the definite integral of a function ƒ over a closed interval [a, b]? When can you be sure it exists?

9. What is the Fundamental Theorem of Calculus? Why is it so important? Illustrate each part of the theorem with an example. 10. What is the Net Change Theorem? What does it say about the integral of velocity? The integral of marginal cost? 11. Discuss how the processes of integration and differentiation can be considered as “inverses” of each other. 12. How does the Fundamental Theorem provide a solution to the initial value problem dy>dx = ƒsxd, ysx0 d = y0 , when ƒ is continuous? 13. How is integration by substitution related to the Chain Rule?

6. What is the relation between definite integrals and area? Describe some other interpretations of definite integrals.

14. How can you sometimes evaluate indefinite integrals by substitution? Give examples.

7. What is the average value of an integrable function over a closed interval? Must the function assume its average value? Explain.

15. How does the method of substitution work for definite integrals? Give examples.

8. Describe the rules for working with definite integrals (Table 5.4). Give examples.

16. How do you define and calculate the area of the region between the graphs of two continuous functions? Give an example.

Chapter 5

Practice Exercises

Finite Sums and Estimates 1. The accompanying figure shows the graph of the velocity (ft> sec) of a model rocket for the first 8 sec after launch. The rocket accelerated straight up for the first 2 sec and then coasted to reach its maximum height at t = 8 sec .

5 Velocity (m/sec)

200 Velocity (ft/sec)

4 3 2

150

1

100

0

50 0

2

4 6 Time (sec)

10

8

10

10

3. Suppose that a ak = -2 and a bk = 25 . Find the value of 2

4

6

8

Time after launch (sec)

a. Assuming that the rocket was launched from ground level, about how high did it go? (This is the rocket in Section 3.3, Exercise 17, but you do not need to do Exercise 17 to do the exercise here.) b. Sketch a graph of the rocket’s height aboveground as a function of time for 0 … t … 8 . 2. a. The accompanying figure shows the velocity (m> sec) of a body moving along the s-axis during the time interval from t = 0 to t = 10 sec. About how far did the body travel during those 10 sec? b. Sketch a graph of s as a function of t for 0 … t … 10 assuming ss0d = 0 .

k=1

k=1

10

10

ak a. a k=1 4

b. a sbk - 3ak d k=1 10

10

5 d. a a - bk b k=1 2

c. a sak + bk - 1d k=1

20

20

4. Suppose that a ak = 0 and a bk = 7 . Find the values of k=1

20

k=1

20

a. a 3ak

b. a sak + bk d

2bk 1 b c. a a 7 2 k=1

d. a sak - 2d

k=1 20

k=1 20 k=1

346

Chapter 5: Integration

Definite Integrals In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval and the numbers ck are chosen from the subintervals of P.

18. x 3 + 1y = 1,

x = 0,

y = 0,

for

0 … x … 1

y x 3 y 1, 0 x 1

1

n

5. 6.

lim a s2ck - 1d-1>2 ¢xk , where P is a partition of [1, 5] ƒ ƒPƒ ƒ :0 k=1 n

lim a cksck 2 - 1d1>3 ¢xk , where P is a partition of [1, 3]

ƒ ƒPƒ ƒ :0 k = 1

0

n

ck 7. lim a acos a bb ¢xk , where P is a partition of [-p, 0] 2 ƒ ƒPƒ ƒ :0 k = 1 n

8.

lim a ssin ck dscos ck d ¢xk , where P is a partition of [0, p>2]

ƒ ƒPƒ ƒ :0 k = 1 2 1-2

5 1-2 ƒsxd

5 1-2

3ƒsxd dx = 12, dx = 6 , and 9. If find the values of the following. 2

a.

5

ƒsxd dx

L-2

b.

5

e.

5

g sxd dx

L5

a

L-2

ƒsxd dx

L2

-2

c.

g sxd dx = 2 ,

d.

s -pg sxdd dx

L-2

ƒsxd + g sxd b dx 5

2

2

1

2

2

g sxd dx

L0

b.

g sxd dx

L1

0

c.

2

ƒsxd dx

L2

d.

L0

22 ƒsxd dx

L0

s g sxd - 3ƒsxdd dx

Area In Exercises 11–14, find the total area of the region between the graph of ƒ and the x-axis. 11. ƒsxd = x 2 - 4x + 3, 12. ƒsxd = 1 - sx 2>4d,

0 … x … 3 -2 … x … 3

13. ƒsxd = 5 - 5x 2>3,

0 … x … 4

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26. 15. y = x,

y = 1>x ,

16. y = x,

y = 1> 1x,

x = 2

2

17. 1x + 1y = 1,

x = 2

x = 0,

y = 4x - 2

x = 0

y = 4x - 16

y = x,

0 … x … p>4

24. y = ƒ sin x ƒ , y = 1, -p>2 … x … p>2 25. y = 2 sin x, y = sin 2x, 0 … x … p 26. y = 8 cos x,

y = sec2 x,

-p>3 … x … p>3

27. Find the area of the “triangular” region bounded on the left by x + y = 2 , on the right by y = x 2 , and above by y = 2 .

29. Find the extreme values of ƒsxd = x 3 - 3x 2 and find the area of the region enclosed by the graph of ƒ and the x-axis. 30. Find the area of the region cut from the first quadrant by the curve x 1>2 + y 1>2 = a 1>2. 31. Find the total area of the region enclosed by the curve x = y 2>3 and the lines x = y and y = -1 .

33. Area Find the area between the curve y = 2sln xd>x and the x-axis from x = 1 to x = e . 34. a. Show that the area between the curve y = 1>x and the x-axis from x = 10 to x = 20 is the same as the area between the curve and the x-axis from x = 1 to x = 2 .

Initial Value Problems x

35. Show that y = x 2 + d2 y

y = 0

dx 2

= 2 -

1 dt solves the initial value problem L1 t 1 ; x2

y¿s1d = 3,

y s1d = 1.

36. Show that y = 10 A 1 + 22sec t B dt solves the initial value problem x

y 1

d 2y dx 2 x y 1

0

20. x = 4 - y 2,

b. Show that the area between the curve y = 1>x and the x-axis from ka to kb is the same as the area between the curve and the x-axis from x = a to x = b s0 6 a 6 b, k 7 0d .

-1 … x … 8

14. ƒsxd = 1 - 1x,

21. y 2 = 4x, 23. y = sin x,

y = 3

32. Find the total area of the region between the curves y = sin x and y = cos x for 0 … x … 3p>2.

2

e.

x = 0,

28. Find the area of the “triangular” region bounded on the left by y = 1x , on the right by y = 6 - x , and below by y = 1.

10. If 10 ƒsxd dx = p, 10 7g sxd dx = 7 , and 10 g sxd dx = 2 , find the values of the following. a.

19. x = 2y 2,

22. y 2 = 4x + 4,

x

1

1

x

= 2sec x tan x ;

y¿s0d = 3,

y s0d = 0.

Express the solutions of the initial value problems in Exercises 37 and 38 in terms of integrals. dy sin x 37. = x , y s5d = -3 dx

Chapter 5

38.

dy = 22 - sin2 x , dx

Evaluating Definite Integrals Evaluate the integrals in Exercises 73–112.

y s -1d = 2

Solve the initial value problems in Exercises 39–42. dy 1 , y s0d = 0 = 39. dx 21 - x 2 dy 1 = 2 - 1, y s0d = 1 40. dx x + 1 dy 1 , x 7 1; y s2d = p = 41. dx x2x 2 - 1 42.

dy 2 1 , = dx 1 + x2 21 - x 2

45.

L L

L

stan xd-3>2 sec2 x dx

51. 52. 53.

L L

(sec u tan u) 21 + sec u du

L

L-1

56.

97.

2t dt 2 L0 t - 25 sln xd-3 dx 59. x L 61. 63.

58. 60.

2

L

x3x dx

62.

3 dr L 21 - 4sr - 1d2

64.

L1

2ln x x dx

L

tan sln yd dy y

103.

L

1 2 r csc s1 + ln rd dr

105.

L

2tan x sec2 x dx

6 dr L 24 - sr + 1d2

dx dx 66. 2 2 L 2 + sx - 1d L 1 + s3x + 1d dx 67. L s2x - 1d2s2x - 1d2 - 4 65.

68.

69. 71.

dx e sin

2x

dx

L 2 2x - x 2 dy L 2tan - 1 y s1 + y 2 d

88.

sec x tan x dx

90.

5ssin xd3>2 cos x dx

92.

a

3 sin x cos x

L1

-sx + 1d

70. 72.

2sin

-1

x dx

L 21 - x 2 stan-1 xd2 dx L

1 + x

2

96.

csc2 x dx

Lp>4

tan2

L0

u du 3

csc z cot z dz

Lp>4

98.

p>4

sec2 x dx s1 + 7 tan xd2>3

8

8 2 - 2 b dx 3x x

L0 L1

a

102.

log4 u du u

L-ln 2

e 2w dw

ln 9

1 -1>3 dx x s1 + 7 ln xd

e use u - 1d1>2 du

L0

sln sy + 1dd2 dy y + 1 L1 e 8 ln 3 log u 3 du 104. u L1 3

1>5

6 dx

L-3>4 29 - 4x 2 3 dt 107. 2 4 L-2 + 3t dy 109. L y24y 2 - 1 2>3

15 sin4 3x cos 3x dx

L-p>2

0

dx

100.

3>4

111.

94.

e rs3e r + 1d-3>2 dr

L0

8

dx

x 1 + b dx 8 2x

e

L-2

L1

p b dt 4

p>2

21 + 3 sin2 x

e

101.

cos2 a4t -

L0

p

2

dy

L22>3 ƒ y ƒ 29y - 1 2

6 dx L-1>5 24 - 25x 2 3 dt 108. 3 + t2 L23 24 dy 110. L y2y 2 - 16 106.

-26>25

112.

L-2>25

Average Values 113. Find the average value of ƒsxd = mx + b a. over [-1, 1]

L sx + 3d2sx + 3d - 25 2

-1

x dx 6

L0

ln 5

99.

x 3s1 + 9x 4 d-3>2 dx

3p>4

L0

(csc2 x) e cot x dx

L

4

57.

86.

L-p>3

L1

2(7 - 5r) 2

3p>4

sec2 u du

p>2

93.

dr

p>4

-1

e

dx 3x - 4

84.

Lp

du

3

L0

p>2

91.

95.

54.

sin2 5r dr

cot2

4

ssec2 xd e tan x dx

L0

0

89.

1u

L1

1>2

82.

L0

x

e y csc se y + 1d cot se y + 1d dy

1

55.

2

L0

A 1 + 1u B 1>2

1

80.

x -1>3s1 - x 2>3 d3>2 dx

3p

e sec se - 7d dx x

L

50.

4

78.

p>3

85. 87.

1 + 2 sec2 s2u - pdb du a L 22u - p st + 1d2 - 1 2 2 at - t b at + t b dt dt 47. 48. t4 L L 1t sin s2t 3>2 d dt

L1>8

x -4>3 dx

L1

p

83.

s2u + 1 + 2 cos s2u + 1dd du

L

76.

36 dx s2x + 1d3 0 L 1

s8s 3 - 12s 2 + 5d ds

L0

27

4 dy 2 L1 y 4 dt 77. L1 t1t

81.

44.

74.

2

75.

y s0d = 2

2scos xd-1>2 sin x dx

L-1

1

s3x 2 - 4x + 7d dx

1

46.

49.

1

73.

79.

Evaluating Indefinite Integrals Evaluate the integrals in Exercises 43–72. 43.

Practice Exercises

b. over [-k, k] 114. Find the average value of a. y = 23x over [0, 3] b. y = 2ax over [0, a]

dy ƒ y ƒ 25y 2 - 3

347

348

Chapter 5: Integration

115. Let ƒ be a function that is differentiable on [a, b]. In Chapter 2 we defined the average rate of change of ƒ over [a, b] to be ƒsbd - ƒsad b - a and the instantaneous rate of change of ƒ at x to be ƒ¿sxd . In this chapter we defined the average value of a function. For the new definition of average to be consistent with the old one, we should have ƒsbd - ƒsad = average value of ƒ¿ on [a, b] . b - a Is this the case? Give reasons for your answer. 116. Is it true that the average value of an integrable function over an interval of length 2 is half the function’s integral over the interval? Give reasons for your answer. 117. a. Verify that 1 ln x dx = x ln x - x + C. b. Find the average value of ln x over [1, e]. 118. Find the average value of ƒsxd = 1>x on [1, 2].

Theory and Examples 129. Is it true that every function y = ƒsxd that is differentiable on [a, b] is itself the derivative of some function on [a, b]? Give reasons for your answer. 130. Suppose that F(x) is an antiderivative of ƒsxd = 21 + x 4 . Ex1

press 10 21 + x 4 dx in terms of F and give a reason for your answer. 1 131. Find dy>dx if y = 1x 21 + t 2 dt . Explain the main steps in your calculation. 0 132. Find dy>dx if y = 1cos x s1>s1 - t 2 dd dt . Explain the main steps in your calculation. 133. A new parking lot To meet the demand for parking, your town has allocated the area shown here. As the town engineer, you have been asked by the town council to find out if the lot can be built for $10,000. The cost to clear the land will be $0.10 a square foot, and the lot will cost $2.00 a square foot to pave. Can the job be done for $10,000? Use a lower sum estimate to see. (Answers may vary slightly, depending on the estimate used.)

T 119. Compute the average value of the temperature function

0 ft

2p ƒsxd = 37 sin a sx - 101db + 25 365

36 ft 54 ft

for a 365-day year. (See Exercise 98, Section 3.6.) This is one way to estimate the annual mean air temperature in Fairbanks, Alaska. The National Weather Service’s official figure, a numerical average of the daily normal mean air temperatures for the year, is 25.7ºF, which is slightly higher than the average value of ƒ(x).

51 ft 49.5 ft Vertical spacing 15 ft

T 120. Specific heat of a gas Specific heat Cy is the amount of heat required to raise the temperature of one mole (gram molecule) of a gas with constant volume by 1ºC. The specific heat of oxygen depends on its temperature T and satisfies the formula

64.4 ft 67.5 ft 42 ft

Cy = 8.27 + 10 -5 s26T - 1.87T 2 d .

Ignored

Find the average value of Cy for 20° … T … 675°C and the temperature at which it is attained. Differentiating Integrals In Exercises 121–128, find dy>dx. 7x2

x

121. y =

22 + cos3 t dt

L2

122. y =

1

123. y =

6 dt 4 Lx 3 + t

2

124. y =

1 dt 2 Lsec x t + 1 e2x

0

125. y =

Lln x2

e

cos t

sin-1 x

127. y =

L0

22 + cos3 t dt

L2

126. y =

dt dt

21 - 2t 2

ln (t + 1) dt 2

L1 Ltan-1 x

134. Skydivers A and B are in a helicopter hovering at 6400 ft. Skydiver A jumps and descends for 4 sec before opening her parachute. The helicopter then climbs to 7000 ft and hovers there. Forty-five seconds after A leaves the aircraft, B jumps and descends for 13 sec before opening his parachute. Both skydivers descend at 16 ft> sec with parachutes open. Assume that the skydivers fall freely (no effective air resistance) before their parachutes open. a. At what altitude does A’s parachute open? b. At what altitude does B’s parachute open? c. Which skydiver lands first?

p>4

128. y =

54 ft

e

2t

dt

Chapter 5

Chapter 5 1

1

7ƒsxd dx = 7, does

L0

L0

ƒsxd dx = 1?

ƒsxd dx = 4 and ƒsxd Ú 0, does

L0

1

L0

2ƒsxd dx = 24 = 2 ?

Give reasons for your answers. 2

5

ƒsxd dx = 4,

5

g sxd dx = 2 . L-2 L2 L-2 Which, if any, of the following statements are true?

2. Suppose

ƒsxd dx = 3,

2

3. Initial value problem Show that 1 y = a

x

L0

ƒstd sin asx - td dt

solves the initial value problem

dx

2

lim ƒsxd

1 - x, ƒsxd = • x 2, -1,

(Hint: sin sax - atd = sin ax cos at - cos ax sin at .) 4. Proportionality Suppose that x and y are related by the equation y

x = 2

L0 21 + 4t 2

dt.

2

Show that d y/dx is proportional to y and find the constant of proportionality. 5. Find ƒ(4) if x2

L-1

0

ƒsxd dx =

L-1

2

s1 - xd dx +

= cx =

L0

0

3

x 2 dx +

2

L2 3

x2 x3 d + c d + c-x d 2 -1 3 0 2

3 8 19 + - 1 = . 2 3 6

ƒsxd

t 2 dt = x cos px . L0 L0 6. Find ƒsp/2d from the following information. a.

-1 … x 6 0 0 … x 6 2 2 … x … 3

(Figure 5.32) over [-1, 3] is 3

1

lim ƒsxd

x:c +

exist and are finite at every interior point of I, and the appropriate onesided limits exist and are finite at the endpoints of I. All piecewise continuous functions are integrable. The points of discontinuity subdivide I into open and half-open subintervals on which ƒ is continuous, and the limit criteria above guarantee that ƒ has a continuous extension to the closure of each subinterval. To integrate a piecewise continuous function, we integrate the individual extensions and add the results. The integral of

dy = 0 and y = 0 when x = 0 . dx

+ a 2y = ƒsxd,

and

x:c -

sƒsxd + g sxdd = 9

b.

L5 L-2 c. ƒsxd … g sxd on the interval -2 … x … 5

d 2y

Piecewise Continuous Functions Although we are mainly interested in continuous functions, many functions in applications are piecewise continuous. A function ƒ(x) is piecewise continuous on a closed interval I if ƒ has only finitely many discontinuities in I, the limits

5

ƒsxd dx = -3

a.

9. Finding a curve Find the equation for the curve in the xy-plane that passes through the point s1, -1d if its slope at x is always 3x 2 + 2 . 10. Shoveling dirt You sling a shovelful of dirt up from the bottom of a hole with an initial velocity of 32 ft> sec. The dirt must rise 17 ft above the release point to clear the edge of the hole. Is that enough speed to get the dirt out, or had you better duck?

1

b. If

ƒstd dt = x cos px

b.

y

4

i) ƒ is positive and continuous. ii) The area under the curve y = ƒsxd from x = 0 to x = a is a a p + sin a + cos a . 2 2 2 7. The area of the region in the xy-plane enclosed by the x-axis, the curve y = ƒsxd, ƒsxd Ú 0 , and the lines x = 1 and x = b is equal to 2b 2 + 1 - 22 for all b 7 1 . Find ƒ(x). 8. Prove that

L0

2 y1x

–1

a

u

L0

x

ƒstd dtb du =

L0

ƒsudsx - ud du .

(Hint: Express the integral on the right-hand side as the difference of two integrals. Then show that both sides of the equation have the same derivative with respect to x.)

y x2

3

2

x

349

Additional and Advanced Exercises

Theory and Examples 1. a. If

Additional and Advanced Exercises

1

0 –1

1

2 3 y –1

FIGURE 5.32 Piecewise continuous functions like this are integrated piece by piece.

x

s -1d dx

350

Chapter 5: Integration

The Fundamental Theorem applies to piecewise continuous funcx tions with the restriction that sd>dxd 1a ƒstd dt is expected to equal ƒ(x) only at values of x at which ƒ is continuous. There is a similar restriction on Leibniz’s Rule (see Exercises 31–38). Graph the functions in Exercises 11–16 and integrate them over their domains.

For example, let’s estimate the sum of the square roots of the first n positive integers, 21 + 22 + Á + 2n . The integral 1

L0

11. ƒsxd = e

x , -4,

12. ƒsxd = e

2 -x, x 2 - 4,

13. g std = e

t, sin pt,

14. hszd = e

21 - z, s7z - 6d-1>3,

2 3>2 2 x d = 3 3 0

is the limit of the upper sums

-8 … x 6 0 0 … x … 3

2>3

1

1x dx =

Sn =

-4 … x 6 0 0 … x … 3

=

0 … t 6 1 1 … t … 2

1 An

#

1 2 n + An

#

n 1 Á + n n + A

#

1 n

21 + 22 + Á + 2n . n 3>2

y

0 … z 6 1 1 … z … 2

1, 15. ƒsxd = • 1 - x 2, 2,

-2 … x 6 -1 -1 … x 6 1 1 … x … 2

r, 16. hsrd = • 1 - r 2, 1,

-1 … r 6 0 0 … r 6 1 1 … r … 2

y x

0

17. Find the average value of the function graphed in the accompanying figure.

1 n

n1 1 n

2 n

x

Therefore, when n is large, Sn will be close to 2>3 and we will have 2 Root sum = 21 + 22 + Á + 2n = Sn # n 3>2 L n 3>2 . 3

y 1

The following table shows how good the approximation can be. x 0

1

2

18. Find the average value of the function graphed in the accompanying figure. y

n

Root sum

s2>3dn3>2

Relative error

10 50 100 1000

22.468 239.04 671.46 21,097

21.082 235.70 666.67 21,082

1.386>22.468 L 6% 1.4% 0.7% 0.07%

1

23. Evaluate x 0

1

2

n: q

19. limb: 1

L0 21 - x

21. lim a n: q

dx 2

1 20. lim x x: q

15 + 25 + 35 + Á + n 5 n6

by showing that the limit is

Limits Find the limits in Exercises 19–22. b

lim

3

1 x

L0

tan-1 t dt

1 1 1 + + Á + b n + 1 n + 2 2n

1 22. lim n A e 1>n + e 2>n + Á + e sn - 1d>n + e n>n B n: q

L0

x 5 dx

and evaluating the integral. 24. See Exercise 23. Evaluate lim

n: q

1 3 s1 + 23 + 33 + Á + n 3 d . n4

25. Let ƒ(x) be a continuous function. Express Approximating Finite Sums with Integrals In many applications of calculus, integrals are used to approximate finite sums—the reverse of the usual procedure of using finite sums to approximate integrals.

n 1 1 2 lim n cƒ a n b + ƒ a n b + Á + ƒ a n b d

n: q

as a definite integral.

Chapter 5 26. Use the result of Exercise 25 to evaluate

f. Find the x-coordinate of each point of inflection of the graph of g on the open interval s -3, 4d .

1 a. lim 2 s2 + 4 + 6 + Á + 2nd , n: q n 1 b. lim 16 s115 + 215 + 315 + Á + n 15 d , n: q n

g. Find the range of g.

3p np p 2p 1 c. lim n asin n + sin n + sin n + Á + sin n b . n: q What can be said about the following limits? 1 d. lim 17 s115 + 215 + 315 + Á + n 15 d n: q n 1 e. lim 15 s115 + 215 + 315 + Á + n 15 d n: q n 27. a. Show that the area An of an n-sided regular polygon in a circle of radius r is An =

351

Additional and Advanced Exercises

nr 2 2p sin n . 2

30. A differential equation Show that both of the following condip tions are satisfied by y = sin x + 1x cos 2t dt + 1: i) y– = - sin x + 2 sin 2x ii) y = 1 and y¿ = -2 when x = p . In applications, we sometimes encounter functions

Leibniz’s Rule like x2

ƒsxd =

21x

s1 + td dt

Lsin x

g sxd =

and

L1x

sin t 2 dt ,

defined by integrals that have variable upper limits of integration and variable lower limits of integration at the same time. The first integral can be evaluated directly, but the second cannot. We may find the derivative of either integral, however, by a formula called Leibniz’s Rule.

b. Find the limit of An as n : q . Is this answer consistent with what you know about the area of a circle? 28. Let

Leibniz’s Rule (n - 1) 12 22 + 3 + Á + . 3 n n n3

If ƒ is continuous on [a, b] and if u(x) and y(x) are differentiable functions of x whose values lie in [a, b], then

2

Sn =

ysxd

d dy du ƒstd dt = ƒsysxdd - ƒsusxdd . dx Lusxd dx dx

To calculate limn: q Sn, show that 2

2

2

n - 1 1 1 2 Sn = n c a n b + a n b + Á + a n b d and interpret Sn as an approximating sum of the integral 1

x 2 dx.

L0

(Hint: Partition [0, 1] into n intervals of equal length and write out the approximating sum for inscribed rectangles.) Defining Functions Using the Fundamental Theorem 29. A function defined by an integral The graph of a function ƒ consists of a semicircle and two line segments as shown. Let x g sxd = 11 ƒstd dt .

Figure 5.33 gives a geometric interpretation of Leibniz’s Rule. It shows a carpet of variable width ƒ(t) that is being rolled up at the left at the same time x as it is being unrolled at the right. (In this interpretation, time is x, not t.) At time x, the floor is covered from u(x) to y(x). The rate du>dx at which the carpet is being rolled up need not be the same as the rate dy>dx at which the carpet is being laid down. At any given time x, the area covered by carpet is ysxd

Asxd =

Lusxd

ƒstd dt .

y y 3

Uncovering

y f(x)

f (u(x))

1 –3

–1

–1

1

3

y f (t)

x

Covering f(y(x))

0 u(x)

a. Find g(1). b. Find g(3). c. Find g s -1d . d. Find all values of x on the open interval s -3, 4d at which g has a relative maximum. e. Write an equation for the line tangent to the graph of g at x = -1 .

y(x)

A(x)

f (t) dt

u(x) L

y(x) t

FIGURE 5.33 Rolling and unrolling a carpet gives a geometric interpretation of Leibniz’s Rule: dA dy du = ƒsysxdd - ƒsusxdd . dx dx dx

352

Chapter 5: Integration

At what rate is the covered area changing? At the instant x, A(x) is increasing by the width ƒ(y(x)) of the unrolling carpet times the rate dy>dx at which the carpet is being unrolled. That is, A(x) is being increased at the rate ƒsysxdd

x

40. For what x 7 0 does x s x d = sx x dx ? Give reasons for your answer. 41. Find the areas between the curves y = 2slog2 xd>x and y = 2slog4 xd>x and the x-axis from x = 1 to x = e . What is the ratio of the larger area to the smaller? 42. a. Find df> dx if

dy . dx

ex

At the same time, A is being decreased at the rate ƒsusxdd

ƒsxd =

du , dx

L1

2 ln t t dt .

b. Find ƒ(0).

the width at the end that is being rolled up times the rate du>dx. The net rate of change in A is

c. What can you conclude about the graph of ƒ? Give reasons for your answer. x

t dt . 1 + t4 L2 44. Use the accompanying figure to show that 43. Find ƒ¿s2d if ƒsxd = e gsxd and g sxd =

dA dy du = ƒsysxdd - ƒsusxdd , dx dx dx which is precisely Leibniz’s Rule. To prove the rule, let F be an antiderivative of ƒ on [a, b]. Then

p>2

L0

ysxd

Lusxd

ƒstd dt = Fsysxdd - Fsusxdd .

1

sin x dx =

p sin-1 x dx . 2 L0

y 2

Differentiating both sides of this equation with respect to x gives the equation we want: ysxd

d d ƒstd dt = cFsysxdd - Fsusxdd d dx Lusxd dx = F¿sysxdd

y sin–1 x

1

dy du - F¿susxdd dx dx

y sin x

Chain Rule 0

du dy - ƒsusxdd . = ƒsysxdd dx dx

x

2

1

45. Napier’s inequality Here are two pictorial proofs that Use Leibniz’s Rule to find the derivatives of the functions in Exercises 31–38.

b 7 a 7 0

x

1 31. ƒsxd = dt L1>x t sin x

32. ƒsxd =

Lcos x

ln b - ln a 1 1 6 6 a. b b - a

Q

Explain what is going on in each case.

1 dt 1 - t2

a.

y

L3

21y

33. g s yd =

L1y y2

34. g(y) =

L2y

y ln x

sin t 2 dt 35. y =

Lx2>2

3

36. y =

2x

L2x

L2

x2

et t dt

ln 2t dt

37. y =

ln t dt

Le

a

x

b

L1

ln x t

L0

sin e dt

e 2x

38. y =

0

b.

y

ln t dt

y 1x

41x

Theory and Examples 39. Use Leibniz’s Rule to find the value of x that maximizes the value of the integral x+3

Lx

t s5 - td dt .

0

a

b

x

(Source: Roger B. Nelson, College Mathematics Journal, Vol. 24, No. 2, March 1993, p. 165.)

6 APPLICATIONS OF DEFINITE INTEGRALS OVERVIEW In Chapter 5 we saw that a continuous function over a closed interval has a definite integral, which is the limit of any Riemann sum for the function. We proved that we could evaluate definite integrals using the Fundamental Theorem of Calculus. We also found that the area under a curve and the area between two curves could be computed as definite integrals. In this chapter we extend the applications of definite integrals to finding volumes, lengths of plane curves, and areas of surfaces of revolution. We also use integrals to solve physical problems involving the work done by a force, and to find the location of an object’s center of mass. Each application comes from a process leading to an approximation by a Riemann sum, and then taking a limit to obtain an appropriate definite integral.

Volumes Using Cross-Sections

6.1 y

Px

Cross-section S(x) with area A(x)

Volume = area * height = A # h.

S

0

In this section we define volumes of solids using the areas of their cross-sections. A crosssection of a solid S is the plane region formed by intersecting S with a plane (Figure 6.1). We present three different methods for obtaining the cross-sections appropriate to finding the volume of a particular solid: the method of slicing, the disk method, and the washer method. Suppose we want to find the volume of a solid S like the one in Figure 6.1. We begin by extending the definition of a cylinder from classical geometry to cylindrical solids with arbitrary bases (Figure 6.2). If the cylindrical solid has a known base area A and height h, then the volume of the cylindrical solid is

This equation forms the basis for defining the volumes of many solids that are not cylinders, like the one in Figure 6.1. If the cross-section of the solid S at each point x in the interval [a, b] is a region S(x) of area A(x), and A is a continuous function of x, we can define and calculate the volume of the solid S as the definite integral of A(x). We now show how this integral is obtained by the method of slicing.

a x b

x

FIGURE 6.1 A cross-section S(x) of the solid S formed by intersecting S with a plane Px perpendicular to the x-axis through the point x in the interval [a, b]. A base area Plane region whose area we know

h height Cylindrical solid based on region Volume base area × height Ah

FIGURE 6.2 The volume of a cylindrical solid is always defined to be its base area times its height.

353

354

Chapter 6: Applications of Definite Integrals

Slicing by Parallel Planes

y

We partition [a, b] into subintervals of width (length) ¢xk and slice the solid, as we would a loaf of bread, by planes perpendicular to the x-axis at the partition points a = x0 6 x1 6 Á 6 xn = b. The planes Pxk , perpendicular to the x-axis at the partition points, slice S into thin “slabs” (like thin slices of a loaf of bread). A typical slab is shown in Figure 6.3. We approximate the slab between the plane at xk - 1 and the plane at xk by a cylindrical solid with base area Asxk d and height ¢xk = xk - xk - 1 (Figure 6.4). The volume Vk of this cylindrical solid is Asxk d # ¢xk , which is approximately the same volume as that of the slab:

S

0

a xk21

Volume of the k th slab L Vk = Asxk d ¢xk .

xk b

x

The volume V of the entire solid S is therefore approximated by the sum of these cylindrical volumes,

FIGURE 6.3 A typical thin slab in the solid S.

n

n

V L a Vk = a Asxk d ¢xk . k=1

y Plane at xk21

k=1

This is a Riemann sum for the function A(x) on [a, b]. We expect the approximations from these sums to improve as the norm of the partition of [a, b] goes to zero. Taking a partition of [a, b] into n subintervals with 7P7 : 0 gives

Approximating cylinder based on S(xk ) has height Δxk 5 xk 2 xk21

n

b

lim a Asxk d ¢xk = n: q k=1

La

Asxddx.

So we define the limiting definite integral of the Riemann sum to be the volume of the solid S. 0 xk21

DEFINITION The volume of a solid of integrable cross-sectional area A(x) from x = a to x = b is the integral of A from a to b,

Plane at xk xk

The cylinder’s base is the region S(xk ) with area A(xk )

x

b

V =

La

Asxd dx.

NOT TO SCALE

FIGURE 6.4 The solid thin slab in Figure 6.3 is shown enlarged here. It is approximated by the cylindrical solid with base Ssxk d having area Asxk d and height ¢xk = xk - xk - 1 .

This definition applies whenever A(x) is integrable, and in particular when it is continuous. To apply the definition to calculate the volume of a solid using cross-sections perpendicular to the x-axis, take the following steps:

Calculating the Volume of a Solid 1. Sketch the solid and a typical cross-section. 2. Find a formula for A(x), the area of a typical cross-section. 3. Find the limits of integration. 4. Integrate A(x) to find the volume.

EXAMPLE 1

A pyramid 3 m high has a square base that is 3 m on a side. The crosssection of the pyramid perpendicular to the altitude x m down from the vertex is a square x m on a side. Find the volume of the pyramid. Solution

1.

A sketch. We draw the pyramid with its altitude along the x-axis and its vertex at the origin and include a typical cross-section (Figure 6.5).

6.1

2.

y

Volumes Using Cross-Sections

355

A formula for A(x). The cross-section at x is a square x meters on a side, so its area is Asxd = x 2.

Typical cross-section

3. 4.

x 0

The limits of integration. The squares lie on the planes from x = 0 to x = 3. Integrate to find the volume:

3 x

3

V =

x

L0

3

3

Asxd dx =

L0

x 2 dx =

x3 d = 9 m3. 3 0

3 x (m)

3

FIGURE 6.5 The cross-sections of the pyramid in Example 1 are squares.

29 2 x 2

EXAMPLE 2 A curved wedge is cut from a circular cylinder of radius 3 by two planes. One plane is perpendicular to the axis of the cylinder. The second plane crosses the first plane at a 45° angle at the center of the cylinder. Find the volume of the wedge. Solution We draw the wedge and sketch a typical cross-section perpendicular to the x-axis (Figure 6.6). The base of the wedge in the figure is the semicircle with x Ú 0 that is cut from the circle x 2 + y 2 = 9 by the 45° plane when it intersects the y-axis. For any x in the interval [0, 3], the y-values in this semicircular base vary from y = - 29 - x 2 to y = 29 - x 2. When we slice through the wedge by a plane perpendicular to the x-axis, we obtain a cross-section at x which is a rectangle of height x whose width extends across the semicircular base. The area of this cross-section is

Asxd = sheightdswidthd = sxd A 229 - x 2 B

y

= 2x29 - x 2 . x

x

0

45° x

3

The rectangles run from x = 0 to x = 3, so we have b

V = –3

⎛x, –9 x 2 ⎛ 2 ⎝ ⎝

La

3

Asxd dx =

L0

2x29 - x 2 dx 3

2 = - s9 - x 2 d3>2 d 3 0 2 3>2 = 0 + s9d 3 = 18.

FIGURE 6.6 The wedge of Example 2, sliced perpendicular to the x-axis. The cross-sections are rectangles.

Let u = 9 - x 2, du = -2x dx , integrate, and substitute back.

EXAMPLE 3 Cavalieri’s principle says that solids with equal altitudes and identical cross-sectional areas at each height have the same volume (Figure 6.7). This follows immediately from the definition of volume, because the cross-sectional area function A(x) and the interval [a, b] are the same for both solids. b

Same volume

HISTORICAL BIOGRAPHY Bonaventura Cavalieri (1598–1647)

a

Same cross-section area at every level

FIGURE 6.7 Cavalieri’s principle: These solids have the same volume, which can be illustrated with stacks of coins.

356

Chapter 6: Applications of Definite Integrals

Solids of Revolution: The Disk Method

y y x R(x) x 0

x

x

4

The solid generated by rotating (or revolving) a plane region about an axis in its plane is called a solid of revolution. To find the volume of a solid like the one shown in Figure 6.8, we need only observe that the cross-sectional area A(x) is the area of a disk of radius R(x), the distance of the planar region’s boundary from the axis of revolution. The area is then Asxd = psradiusd2 = p[Rsxd]2.

(a)

So the definition of volume in this case gives y y x

Volume by Disks for Rotation About the x-axis

R(x) x

b

V =

0

La

b

Asxd dx =

La

p[Rsxd]2 dx.

x 4

x

This method for calculating the volume of a solid of revolution is often called the disk method because a cross-section is a circular disk of radius R(x).

Disk

(b)

FIGURE 6.8 The region (a) and solid of revolution (b) in Example 4.

The region between the curve y = 2x, 0 … x … 4, and the x-axis is revolved about the x-axis to generate a solid. Find its volume.

EXAMPLE 4

We draw figures showing the region, a typical radius, and the generated solid (Figure 6.8). The volume is Solution

b

V =

p[Rsxd]2 dx

La 4

=

L0

p C 2x D dx

Radius Rsxd = 2x for rotation around x-axis

2

4 4 s4d2 x2 = p x dx = p d = p = 8p. 2 0 2 L0

EXAMPLE 5

The circle x2 + y2 = a2

is rotated about the x-axis to generate a sphere. Find its volume. Solution We imagine the sphere cut into thin slices by planes perpendicular to the x-axis (Figure 6.9). The cross-sectional area at a typical point x between -a and a is

Asxd = py 2 = psa 2 - x 2 d.

R(x) = 2a 2 - x 2 for rotation around x-axis

Therefore, the volume is a

V =

L-a

a

Asxd dx =

L-a

psa 2 - x 2 d dx = p ca 2x -

a

x3 4 d = pa 3. 3 -a 3

The axis of revolution in the next example is not the x-axis, but the rule for calculating the volume is the same: Integrate psradiusd2 between appropriate limits.

EXAMPLE 6

Find the volume of the solid generated by revolving the region bounded by y = 2x and the lines y = 1, x = 4 about the line y = 1.

6.1 y x2

1

y2

5

357

Volumes Using Cross-Sections

(x, y)

a2 A(x) 5 p(a2 2 x2) x2 1 y2 5 a2

–a x

x a x

Δx

FIGURE 6.9 The sphere generated by rotating the circle x 2 + y 2 = a 2 about the x-axis. The radius is Rsxd = y = 2a 2 - x 2 (Example 5). Solution We draw figures showing the region, a typical radius, and the generated solid (Figure 6.10). The volume is 4

V =

p[Rsxd]2 dx

L1 4

=

L1

p C 2x - 1 D dx 4

= p L1 = pc

Radius R(x) = 2x - 1 for rotation around y = 1

2

C x - 22x + 1 D dx

Expand integrand.

7p x2 2 - 2 # x 3>2 + x d = . 2 3 6 1 4

Integrate.

y R(x) x 1 (x, x)

y y x

R(x) x 1

y1

1

0

0

y x

1

1

x

4

(x, 1)

1

y1 x

x

4

x

(b)

(a)

FIGURE 6.10 The region (a) and solid of revolution (b) in Example 6.

To find the volume of a solid generated by revolving a region between the y-axis and a curve x = Rs yd, c … y … d, about the y-axis, we use the same method with x replaced by y. In this case, the circular cross-section is As yd = p[radius]2 = p[Rs yd]2, and the definition of volume gives

358

Chapter 6: Applications of Definite Integrals

Volume by Disks for Rotation About the y-axis d

V =

y

d

A( y) dy =

Lc

p[Rs yd]2 dy.

Lc

EXAMPLE 7

Find the volume of the solid generated by revolving the region between the y-axis and the curve x = 2>y, 1 … y … 4, about the y-axis.

We draw figures showing the region, a typical radius, and the generated solid (Figure 6.11). The volume is Solution

4 x 2y

4

V =

y 1

4

R( y) 2y

=

x

2

0

p[Rs yd]2 dy

L1

L1

2 Radius R( y) = y for rotation around y-axis

2

2 p a y b dy 4

4

3 4 1 = p dy = 4p c- y d = 4p c d = 3p. 2 4 y 1 1 L

(a) y

EXAMPLE 8 4

Find the volume of the solid generated by revolving the region between the parabola x = y 2 + 1 and the line x = 3 about the line x = 3.

x 2y

We draw figures showing the region, a typical radius, and the generated solid (Figure 6.12). Note that the cross-sections are perpendicular to the line x = 3 and have y-coordinates from y = - 22 to y = 22. The volume is Solution

⎛ 2 , y⎛ ⎝y ⎝ y

V =

1 R( y) 2y

0 2

=

x

22

L-22 22

L-22

(b)

p[Rs yd]2 dy

y = ; 22 when x = 3 Radius R( y) = 3 - (y 2 + 1) for rotation around axis x = 3

p[2 - y 2]2 dy

22

= p [4 - 4y 2 + y 4] dy L-22

FIGURE 6.11 The region (a) and part of the solid of revolution (b) in Example 7.

= p c4y = y

y

R( y) 2 y 2 x3

(3, 2 )

2

y

–2

Integrate.

64p22 . 15

R( y) 3 ( y 2 1) 2 y2

2

0

y 5 22 4 3 d y + 5 -22 3

Expand integrand.

y 1

3

x y2 1

5

(3, –2 ) (a)

x

0

–2

3

1 x y2 1

(b)

FIGURE 6.12 The region (a) and solid of revolution (b) in Example 8.

5

x

6.1

359

Volumes Using Cross-Sections

y

y ( x, R(x))

y

(x, r(x))

0

0

y R(x)

0 a

x

x

y r(x)

x b

x

x

x Washer

FIGURE 6.13 The cross-sections of the solid of revolution generated here are washers, not disks, so the integral b 1a Asxd dx leads to a slightly different formula.

Solids of Revolution: The Washer Method If the region we revolve to generate a solid does not border on or cross the axis of revolution, the solid has a hole in it (Figure 6.13). The cross-sections perpendicular to the axis of revolution are washers (the purplish circular surface in Figure 6.13) instead of disks. The dimensions of a typical washer are y

Outer radius:

Rsxd

Inner radius:

rsxd

(–2, 5) R(x) –x 3

y –x 3

r(x) x 2 1 –2 Interval of integration

The washer’s area is Asxd = p[Rsxd]2 - p[rsxd]2 = ps[Rsxd]2 - [rsxd]2 d.

(1, 2) y x2 1 x 0

1

Consequently, the definition of volume in this case gives

x

Volume by Washers for Rotation About the x-axis

(a)

b

V =

La

b

A(x) dx =

La

p([Rsxd]2 - [r(x)]2) dx.

y (–2, 5) R(x) –x 3 (1, 2)

r (x) x 2 1

This method for calculating the volume of a solid of revolution is called the washer method because a thin slab of the solid resembles a circular washer of outer radius R(x) and inner radius r(x). The region bounded by the curve y = x 2 + 1 and the line y = -x + 3 is revolved about the x-axis to generate a solid. Find the volume of the solid.

EXAMPLE 9 x

x

We use the four steps for calculating the volume of a solid as discussed early in this section.

Solution Washer cross-section Outer radius: R(x) –x 3 Inner radius: r (x) x 2 1 (b)

FIGURE 6.14 (a) The region in Example 9 spanned by a line segment perpendicular to the axis of revolution. (b) When the region is revolved about the x-axis, the line segment generates a washer.

1. 2.

Draw the region and sketch a line segment across it perpendicular to the axis of revolution (the red segment in Figure 6.14a). Find the outer and inner radii of the washer that would be swept out by the line segment if it were revolved about the x-axis along with the region. These radii are the distances of the ends of the line segment from the axis of revolution (Figure 6.14). Outer radius:

Rsxd = -x + 3

Inner radius:

r sxd = x 2 + 1

360

Chapter 6: Applications of Definite Integrals

3.

Find the limits of integration by finding the x-coordinates of the intersection points of the curve and line in Figure 6.14a. x 2 + 1 = -x + 3 x2 + x - 2 = 0 sx + 2dsx - 1d = 0 x = -2,

4.

b

V = R(y) y

4 r (y)

y 2

y x 2 or x y x

2 (a)

y

L-2

pss -x + 3d2 - sx 2 + 1d2 d dx

Values from Steps 2 and 3

1

y 2x or y x 2

y 2

Rotation around x-axis

= p s8 - 6x - x 2 - x 4 d dx L-2

y

0

ps[Rsxd]2 - [rsxd]2 d dx

La 1

=

(2, 4)

Interval of integration

Limits of integration

Evaluate the volume integral.

y

r(y)

x = 1

= p c8x - 3x 2 -

Simplify algebraically.

1

117p x3 x5 d = 5 -2 5 3

To find the volume of a solid formed by revolving a region about the y-axis, we use the same procedure as in Example 9, but integrate with respect to y instead of x. In this situation the line segment sweeping out a typical washer is perpendicular to the y-axis (the axis of revolution), and the outer and inner radii of the washer are functions of y.

R(y) y

The region bounded by the parabola y = x 2 and the line y = 2x in the first quadrant is revolved about the y-axis to generate a solid. Find the volume of the solid.

EXAMPLE 10

4

y

First we sketch the region and draw a line segment across it perpendicular to the axis of revolution (the y-axis). See Figure 6.15a. The radii of the washer swept out by the line segment are Rs yd = 2y, rs yd = y>2 (Figure 6.15). The line and parabola intersect at y = 0 and y = 4, so the limits of integration are c = 0 and d = 4. We integrate to find the volume: Solution

x

y 2

x y 0 2

x

(b)

FIGURE 6.15 (a) The region being rotated about the y-axis, the washer radii, and limits of integration in Example 10. (b) The washer swept out by the line segment in part (a).

d

V =

ps[Rs yd]2 - [rs yd]2 d dy

Rotation around y-axis

2 y 2 p a c2y d - c d b dy 2 L0

Substitute for radii and limits of integration.

Lc 4

=

4 y2 y2 y3 4 8 = p ay b dy = p c d = p. 4 2 12 3 0 L0

6.1

Volumes Using Cross-Sections

361

Exercises 6.1 Volumes by Slicing Find the volumes of the solids in Exercises 1–10. 1. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 4. The cross-sections perpendicular to the axis on the interval 0 … x … 4 are squares whose diagonals run from the parabola y = - 2x to the parabola y = 2x. 2. The solid lies between planes perpendicular to the x-axis at x = -1 and x = 1. The cross-sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = x 2 to the parabola y = 2 - x 2. y

8. The base of a solid is the region bounded by the graphs of y = 2x and y = x>2. The cross-sections perpendicular to the x-axis are a. isosceles triangles of height 6. b. semi-circles with diameters running across the base of the solid. 9. The solid lies between planes perpendicular to the y-axis at y = 0 and y = 2. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola x = 25y 2. 10. The base of the solid is the disk x 2 + y 2 … 1. The cross-sections by planes perpendicular to the y-axis between y = - 1 and y = 1 are isosceles right triangles with one leg in the disk.

2

y

y x2

0 y 2 x2

x

3. The solid lies between planes perpendicular to the x-axis at x = -1 and x = 1. The cross-sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle y = - 21 - x 2 to the semicircle y = 21 - x 2. 4. The solid lies between planes perpendicular to the x-axis at x = - 1 and x = 1. The cross-sections perpendicular to the x-axis between these planes are squares whose diagonals run from the semicircle y = - 21 - x 2 to the semicircle y = 21 - x 2. 5. The base of a solid is the region between the curve y = 2 2sin x and the interval [0, p] on the x-axis. The cross-sections perpendicular to the x-axis are a. equilateral triangles with bases running from the x-axis to the curve as shown in the accompanying figure.

0 x2 1 y2 5 1

1

x

11. Find the volume of the given tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.)

3 4 5

y

12. Find the volume of the given pyramid, which has a square base of area 9 and height 5.

y 5 2sin x 0 p

x

b. squares with bases running from the x-axis to the curve.

5

6. The solid lies between planes perpendicular to the x-axis at x = -p>3 and x = p>3. The cross-sections perpendicular to the x-axis are a. circular disks with diameters running from the curve y = tan x to the curve y = sec x. b. squares whose bases run from the curve y = tan x to the curve y = sec x. 7. The base of a solid is the region bounded by the graphs of y = 3x, y = 6, and x = 0. The cross-sections perpendicular to the x-axis are a. rectangles of height 10. b. rectangles of perimeter 20.

3 3

13. A twisted solid A square of side length s lies in a plane perpendicular to a line L. One vertex of the square lies on L. As this square moves a distance h along L, the square turns one revolution about L to generate a corkscrew-like column with square cross-sections. a. Find the volume of the column. b. What will the volume be if the square turns twice instead of once? Give reasons for your answer.

362

Chapter 6: Applications of Definite Integrals

14. Cavalieri’s principle A solid lies between planes perpendicular to the x-axis at x = 0 and x = 12. The cross-sections by planes perpendicular to the x-axis are circular disks whose diameters run from the line y = x>2 to the line y = x as shown in the accompanying figure. Explain why the solid has the same volume as a right circular cone with base radius 3 and height 12. y

36. x = 22y>s y 2 + 1d, x

12

Volumes by the Disk Method In Exercises 15–18, find the volume of the solid generated by revolving the shaded region about the given axis.

2 x

2

x

17. About the y-axis

x tan ⎛ y⎛ ⎝4 ⎝

0

2

0

y = 0,

21. y = 29 - x 2, 23. y = 2cos x, 24. y = sec x, -x

25. y = e ,

x = 2 y = 0

20. y = x 3,

22. y = x - x 2,

0 … x … p>2,

y = 0, y = 0,

y = 0,

x = -p>4, x = 0,

y = 0,

x

0

39. y = x,

x

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises 19–28 about the x-axis. 19. y = x 2,

y = 1

4

y1

y = 1,

40. y = 22x,

2

-2 … y … 0,

1

x

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises 39–44 about the x-axis.

y sin x cos x

x

x = 0

y

x = 2

y = 0

x = 0

x = p>4

x = 1

x = 0

y = 2,

41. y = x 2 + 1, 0

0 … y … p>2,

38. The y-axis

x

18. About the x-axis 1 2

37. The x-axis

– 2

y

1

y = 1

x tan y

3y 2

3

0

y

y = -1,

y = 2

Volumes by the Washer Method Find the volumes of the solids generated by revolving the shaded regions in Exercises 37 and 38 about the indicated axes.

y cos x

y

x 2y 2

x = 0,

y

16. About the y-axis

0

x = 0,

x = 0,

34. The region enclosed by x = 2cos spy>4d, x = 0 35. x = 2> 2y + 1, x = 0, y = 0, y = 3

0

1

31. The region enclosed by x = 25 y 2,

33. The region enclosed by x = 22 sin 2y,

y5 x 2

y

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises 31–36 about the y-axis. 32. The region enclosed by x = y 3>2,

y5x

15. About the x-axis

30. The region in the first quadrant bounded above by the line y = 2 , below by the curve y = 2 sin x, 0 … x … p>2 , and on the left by the y-axis, about the line y = 2

x = 0

y = x + 3

42. y = 4 - x 2,

y = 2 - x

43. y = sec x,

y = 22,

-p>4 … x … p>4

44. y = sec x,

y = tan x,

x = 0,

x = 1

In Exercises 45–48, find the volume of the solid generated by revolving each region about the y-axis. 45. The region enclosed by the triangle with vertices (1, 0), (2, 1), and (1, 1) 46. The region enclosed by the triangle with vertices (0, 1), (1, 0), and (1, 1)

26. The region between the curve y = 2cot x and the x-axis from x = p>6 to x = p>2.

47. The region in the first quadrant bounded above by the parabola y = x 2, below by the x-axis, and on the right by the line x = 2

27. The region between the curve y = 1>(2 2x) and the x-axis from x = 1>4 to x = 4.

48. The region in the first quadrant bounded on the left by the circle x 2 + y 2 = 3, on the right by the line x = 23, and above by the

28. y = e x - 1,

y = 0,

x = 1,

x = 3

line y = 23

In Exercises 29 and 30, find the volume of the solid generated by revolving the region about the given line.

In Exercises 49 and 50, find the volume of the solid generated by revolving each region about the given axis.

29. The region in the first quadrant bounded above by the line y = 22 , below by the curve y = sec x tan x, and on the left by the y-axis, about the line y = 22

49. The region in the first quadrant bounded above by the curve y = x 2, below by the x-axis, and on the right by the line x = 1, about the line x = -1

6.1 50. The region in the second quadrant bounded above by the curve y = -x 3, below by the x-axis, and on the left by the line x = -1, about the line x = -2 Volumes of Solids of Revolution 51. Find the volume of the solid generated by revolving the region bounded by y = 2x and the lines y = 2 and x = 0 about a. the x-axis.

b. the y-axis.

c. the line y = 2.

d. the line x = 4.

52. Find the volume of the solid generated by revolving the triangular region bounded by the lines y = 2x, y = 0, and x = 1 about a. the line x = 1.

60. Designing a plumb bob Having been asked to design a brass plumb bob that will weigh in the neighborhood of 190 g, you decide to shape it like the solid of revolution shown here. Find the plumb bob’s volume. If you specify a brass that weighs 8.5 g>cm3 , how much will the plumb bob weigh (to the nearest gram)? y (cm)

y 5 x 36 2 x 2 12

0 6

b. the line x = 2.

x (cm)

53. Find the volume of the solid generated by revolving the region bounded by the parabola y = x 2 and the line y = 1 about a. the line y = 1.

b. the line y = 2.

c. the line y = -1. 54. By integration, find the volume of the solid generated by revolving the triangular region with vertices (0, 0), (b, 0), (0, h) about a. the x-axis.

363

Volumes Using Cross-Sections

b. the y-axis.

Theory and Applications 55. The volume of a torus The disk x 2 + y 2 … a 2 is revolved about the line x = b sb 7 ad to generate a solid shaped like a doughnut

61. Designing a wok You are designing a wok frying pan that will be shaped like a spherical bowl with handles. A bit of experimentation at home persuades you that you can get one that holds about 3 L if you make it 9 cm deep and give the sphere a radius of 16 cm. To be sure, you picture the wok as a solid of revolution, as shown here, and calculate its volume with an integral. To the nearest cubic centimeter, what volume do you really get? s1 L = 1000 cm3.d y (cm)

a

and called a torus. Find its volume. (Hint: 1-a 2a 2 - y 2 dy = pa 2>2 , since it is the area of a semicircle of radius a.)

x 2 y 2 16 2 256

56. Volume of a bowl A bowl has a shape that can be generated by revolving the graph of y = x 2>2 between y = 0 and y = 5 about the y-axis.

0

a. Find the volume of the bowl.

–7

b. Related rates If we fill the bowl with water at a constant rate of 3 cubic units per second, how fast will the water level in the bowl be rising when the water is 4 units deep?

x (cm) 9 cm deep

–16

57. Volume of a bowl a. A hemispherical bowl of radius a contains water to a depth h. Find the volume of water in the bowl. b. Related rates Water runs into a sunken concrete hemispherical bowl of radius 5 m at the rate of 0.2 m3>sec . How fast is the water level in the bowl rising when the water is 4 m deep? 58. Explain how you could estimate the volume of a solid of revolution by measuring the shadow cast on a table parallel to its axis of revolution by a light shining directly above it. 59. Volume of a hemisphere Derive the formula V = s2>3dpR 3 for the volume of a hemisphere of radius R by comparing its cross-sections with the cross-sections of a solid right circular cylinder of radius R and height R from which a solid right circular cone of base radius R and height R has been removed, as suggested by the accompanying figure. R 2 2 h 2

62. Max-min The arch y = sin x, 0 … x … p , is revolved about the line y = c, 0 … c … 1 , to generate the solid in the accompanying figure. a. Find the value of c that minimizes the volume of the solid. What is the minimum volume? b. What value of c in [0, 1] maximizes the volume of the solid? T c. Graph the solid’s volume as a function of c, first for 0 … c … 1 and then on a larger domain. What happens to the volume of the solid as c moves away from [0, 1]? Does this make sense physically? Give reasons for your answers. y y 5 sin x c

h h

0 R

h

y5c

R p

x

364

Chapter 6: Applications of Definite Integrals

63. Consider the region R bounded by the graphs of y = ƒ(x) 7 0, x = a 7 0, x = b 7 a, and y = 0 (see accomanying figure). If the volume of the solid formed by revolving R about the x-axis is 4p, and the volume of the solid formed by revolving R about the line y = -1 is 8p, find the area of R.

64. Consider the region R given in Exercise 63. If the volume of the solid formed by revolving R around the x-axis is 6p, and the volume of the solid formed by revolving R around the line y = -2 is 10p, find the area of R.

y y 5 f (x) R 0

6.2

a

b

x

Volumes Using Cylindrical Shells b

In Section 6.1 we defined the volume of a solid as the definite integral V = 1a Asxd dx, where A(x) is an integrable cross-sectional area of the solid from x = a to x = b. The area A(x) was obtained by slicing through the solid with a plane perpendicular to the x-axis. However, this method of slicing is sometimes awkward to apply, as we will illustrate in our first example. To overcome this difficulty, we use the same integral definition for volume, but obtain the area by slicing through the solid in a different way.

Slicing with Cylinders Suppose we slice through the solid using circular cylinders of increasing radii, like cookie cutters. We slice straight down through the solid so that the axis of each cylinder is parallel to the y-axis. The vertical axis of each cylinder is the same line, but the radii of the cylinders increase with each slice. In this way the solid is sliced up into thin cylindrical shells of constant thickness that grow outward from their common axis, like circular tree rings. Unrolling a cylindrical shell shows that its volume is approximately that of a rectangular slab with area A(x) and thickness ¢x. This slab interpretation allows us to apply the same integral definition for volume as before. The following example provides some insight before we derive the general method. The region enclosed by the x-axis and the parabola y = ƒsxd = 3x - x 2 is revolved about the vertical line x = -1 to generate a solid (Figure 6.16). Find the volume of the solid.

EXAMPLE 1

Solution Using the washer method from Section 6.1 would be awkward here because we would need to express the x-values of the left and right sides of the parabola in Figure 6.16a in terms of y. (These x-values are the inner and outer radii for a typical washer, requiring us to solve y = 3x - x 2 for x, which leads to complicated formulas.) Instead of rotating a horizontal strip of thickness ¢y, we rotate a vertical strip of thickness ¢x. This rotation produces a cylindrical shell of height yk above a point xk within the base of the vertical strip and of thickness ¢x. An example of a cylindrical shell is shown as the orange-shaded region in Figure 6.17. We can think of the cylindrical shell shown in the figure as approximating a slice of the solid obtained by cutting straight down through it, parallel to the axis of revolution, all the way around close to the inside hole. We then cut another cylindrical slice around the enlarged hole, then another, and so on, obtaining n cylinders. The radii of the cylinders gradually increase, and the heights of

6.2

365

Volumes Using Cylindrical Shells

y

y y 3x

2

x2

1 –2

–1

0

1

2

x

3

0

3

x

–1 Axis of revolution x –1

Axis of revolution x –1

–2 (a)

(b)

FIGURE 6.16 (a) The graph of the region in Example 1, before revolution. (b) The solid formed when the region in part (a) is revolved about the axis of revolution x = - 1.

y

yk 23

0

xk

3

x 5 –1

FIGURE 6.17 A cylindrical shell of height yk obtained by rotating a vertical strip of thickness ¢xk about the line x = -1 . The outer radius of the cylinder occurs at xk , where the height of the parabola is yk = 3xk - x k2 (Example 1).

x

the cylinders follow the contour of the parabola: shorter to taller, then back to shorter (Figure 6.16a). Each slice is sitting over a subinterval of the x-axis of length (width) ¢xk . Its radius is approximately s1 + xk d, and its height is approximately 3xk - xk 2. If we unroll the cylinder at xk and flatten it out, it becomes (approximately) a rectangular slab with thickness ¢xk (Figure 6.18). The outer circumference of the kth cylinder is 2p # radius = 2ps1 + xk d, and this is the length of the rolled-out rectangular slab. Its volume is approximated by that of a rectangular solid, ¢Vk = circumference * height * thickness = 2ps1 + xk d # A 3xk - xk2B

# ¢xk.

Summing together the volumes ¢Vk of the individual cylindrical shells over the interval [0, 3] gives the Riemann sum 2 a ¢Vk = a 2psxk + 1d A 3xk - xk B ¢xk. n

n

k=1

k=1

Δ xk

Outer circumference 5 2p ⋅ radius 5 2p (1 1 x k ) Radius 5 1 1 x k

(3x k 2 x k 2 )

h 5 (3x k 2 x k 2 )

Δ x k 5 thickness l 5 2p(1 1 x k )

FIGURE 6.18 Cutting and unrolling a cylindrical shell gives a nearly rectangular solid (Example 1).

366

Chapter 6: Applications of Definite Integrals

Taking the limit as the thickness ¢xk : 0 and n : q gives the volume integral V = lim a 2psxk + 1d A 3xk - x k2 B ¢xk n: q n

k=1

3

=

2psx + 1ds3x - x 2 d dx

L0 3

=

L0

2ps3x 2 + 3x - x 3 - x 2 d dx 3

s2x 2 + 3x - x 3 d dx = 2p L0 3

3 45p 2 1 = 2p c x 3 + x 2 - x 4 d = . 3 2 4 2 0 We now generalize the procedure used in Example 1.

The Shell Method

The volume of a cylindrical shell of height h with inner radius r and outer radius R is pR2h - pr2h = 2p a

R + r b (h)(R - r). 2

Suppose the region bounded by the graph of a nonnegative continuous function y = ƒsxd and the x-axis over the finite closed interval [a, b] lies to the right of the vertical line x = L (Figure 6.19a). We assume a Ú L, so the vertical line may touch the region, but not pass through it. We generate a solid S by rotating this region about the vertical line L. Let P be a partition of the interval [a, b] by the points a = x0 6 x1 6 Á 6 xn = b, and let ck be the midpoint of the kth subinterval [xk - 1, xk]. We approximate the region in Figure 6.19a with rectangles based on this partition of [a, b]. A typical approximating rectangle has height ƒsck d and width ¢xk = xk - xk - 1 . If this rectangle is rotated about the vertical line x = L, then a shell is swept out, as in Figure 6.19b. A formula from geometry tells us that the volume of the shell swept out by the rectangle is ¢Vk = 2p * average shell radius * shell height * thickness = 2p # sck - Ld # ƒsck d # ¢xk .

Vertical axis of revolution Vertical axis of revolution

y 5 f (x)

y 5 f (x) Δ xk

ck

a a x5L

xk21

ck

(a)

b xk

x

x k21

b

xk

Rectangle height 5 f (ck )

Δ xk (b)

FIGURE 6.19 When the region shown in (a) is revolved about the vertical line x = L, a solid is produced which can be sliced into cylindrical shells. A typical shell is shown in (b).

x

6.2

Volumes Using Cylindrical Shells

367

We approximate the volume of the solid S by summing the volumes of the shells swept out by the n rectangles based on P: n

V L a ¢Vk. k=1

The limit of this Riemann sum as each ¢xk : 0 and n : q gives the volume of the solid as a definite integral: n

b

V = lim a ¢Vk = q n:

k=1

2psshell radiusdsshell heightd dx.

La b

=

La

2psx - Ldƒsxd dx.

We refer to the variable of integration, here x, as the thickness variable. We use the first integral, rather than the second containing a formula for the integrand, to emphasize the process of the shell method. This will allow for rotations about a horizontal line L as well.

Shell Formula for Revolution About a Vertical Line The volume of the solid generated by revolving the region between the x-axis and the graph of a continuous function y = ƒsxd Ú 0, L … a … x … b, about a vertical line x = L is b

V =

shell shell 2p a ba b dx. radius height La

EXAMPLE 2 The region bounded by the curve y = 2x, the x-axis, and the line x = 4 is revolved about the y-axis to generate a solid. Find the volume of the solid. Solution Sketch the region and draw a line segment across it parallel to the axis of revolution (Figure 6.20a). Label the segment’s height (shell height) and distance from the axis of revolution (shell radius). (We drew the shell in Figure 6.20b, but you need not do that.)

y

Shell radius y x

y x

Shell radius

2

2

y x

4 x 0

4

x

x = Shell height x

Shell height

x f (x) x

0

(4, 2)

Interval of integration

x –4

Interval of integration (a)

(b)

FIGURE 6.20 (a) The region, shell dimensions, and interval of integration in Example 2. (b) The shell swept out by the vertical segment in part (a) with a width ¢ x.

Chapter 6: Applications of Definite Integrals

The shell thickness variable is x, so the limits of integration for the shell formula are a = 0 and b = 4 (Figure 6.20). The volume is then V = =

b

shell shell 2p a ba b dx radius height

4

2psxd A 2x B dx

La L0

4

4

128p 2 . = 2p x 3>2 dx = 2p c x 5>2 d = 5 5 0 0 L So far, we have used vertical axes of revolution. For horizontal axes, we replace the x’s with y’s. The region bounded by the curve y = 2x, the x-axis, and the line x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid by the shell method.

EXAMPLE 3

This is the solid whose volume was found by the disk method in Example 4 of Section 6.1. Now we find its volume by the shell method. First, sketch the region and draw a line segment across it parallel to the axis of revolution (Figure 6.21a). Label the segment’s length (shell height) and distance from the axis of revolution (shell radius). (We drew the shell in Figure 6.21b, but you need not do that.) In this case, the shell thickness variable is y, so the limits of integration for the shell formula method are a = 0 and b = 2 (along the y-axis in Figure 6.21). The volume of the solid is Solution

b

V =

La

shell shell 2p a ba b dy radius height

2

=

L0

2ps yds4 - y 2 d dy 2

= 2p s4y - y 3 d dy L0 y4 2 = 2p c2y 2 d = 8p. 4 0 y Shell height 2 y

y

4 y2 (4, 2)

y x

4 y2 Shell height 0 2 Interval of integration

368

(4, 2)

x y2

y 4

y y Shell radius 0

4 (a)

x Shell radius

x (b)

FIGURE 6.21 (a) The region, shell dimensions, and interval of integration in Example 3. (b) The shell swept out by the horizontal segment in part (a) with a width ¢y.

6.2

369

Volumes Using Cylindrical Shells

Summary of the Shell Method Regardless of the position of the axis of revolution (horizontal or vertical), the steps for implementing the shell method are these. 1. Draw the region and sketch a line segment across it parallel to the axis of revolution. Label the segment’s height or length (shell height) and distance from the axis of revolution (shell radius). 2. Find the limits of integration for the thickness variable. 3. Integrate the product 2p (shell radius) (shell height) with respect to the thickness variable (x or y) to find the volume.

The shell method gives the same answer as the washer method when both are used to calculate the volume of a region. We do not prove that result here, but it is illustrated in Exercises 37 and 38. (Exercise 45 outlines a proof.) Both volume formulas are actually special cases of a general volume formula we will look at when studying double and triple integrals in Chapter 14. That general formula also allows for computing volumes of solids other than those swept out by regions of revolution.

Exercises 6.2 Revolution About the Axes In Exercises 1–6, use the shell method to find the volumes of the solids generated by revolving the shaded region about the indicated axis. 1.

5. The y-axis y

y

2

y x 2 1 x 3

1 2 y2 x 4

2

0

x

2

0

3.

2

x

y y 2

3

x y2

y 3

x 3 y2 2

x

0

3

0

3

x

Revolution About the y-Axis Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines in Exercises 7–12 about the y-axis.

4. y

x

3

0

1

0

9x

x 3 9

y 2 y1 x 4

2

y

5

2.

y

6. The y-axis

x

7. y = x,

y = -x>2,

x = 2

8. y = 2x,

y = x>2,

x = 1

9. y = x 2,

y = 2 - x,

x = 0,

for x Ú 0

10. y = 2 - x 2,

y = x 2,

11. y = 2x - 1,

y = 2x,

x = 0

y = 0,

x = 1,

12. y = 3> A 22x B ,

x = 0 x = 4

370

Chapter 6: Applications of Definite Integrals

13. Let ƒsxd = e

0 6 x … p x = 0

ssin xd>x, 1,

26. y = x 4,

a. The line x = 1

a. Show that xƒsxd = sin x, 0 … x … p. b. Find the volume of the solid generated by revolving the shaded region about the y-axis in the accompanying figure. y

0

14. Let g sxd = e

stan xd2>x, 0,

c. The x-axis

In Exercises 27 and 28, use the shell method to find the volumes of the solids generated by revolving the shaded regions about the indicated axes. b. The line y = 1

27. a. The x-axis c. The line y = 8>5

⎧ sin x , 0 x y⎨ x x0 ⎩ 1,

1

y = 4 - 3x 2

d. The line y = -2>5

y

1

x

x 12(y 2 y 3)

0 6 x … p>4 x = 0

a. Show that xg sxd = stan xd2, 0 … x … p>4. b. Find the volume of the solid generated by revolving the shaded region about the y-axis in the accompanying figure. y

2 ⎧ tan x , 0 x 4 y⎨ x x0 ⎩ 0,

4

0

x

1

b. The line y = 2

28. a. The x-axis c. The line y = 5

d. The line y = -5>8

y x

4

0

2

Revolution About the x-Axis Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines in Exercises 15–22 about the x-axis. 15. x = 2y, 16. x = y , 2

x = -y, x = -y,

17. x = 2y - y 2, 19. y = ƒ x ƒ ,

y = 2,

x = 0

18. x = 2y - y 2,

y = 0,

y = x - 2

22. y = 2x,

y = 0,

y = 2 - x

x = y

y = 2x,

y = 2

Revolution About Horizontal and Vertical Lines In Exercises 23–26, use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines. x = 2

a. The y-axis c. The line x = -1 e. The line y = 7 24. y = x 3,

y = 8,

25. y = x + 2,

b. The line x = 4 d. The x-axis f. The line y = -2

x = 0

a. The y-axis c. The line x = -2 e. The line y = 8

y2 2

1

2

x

Choosing the Washer Method or Shell Method For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the y-axis, for example, and washers are used, we must integrate with respect to y. It may not be possible, however, to express the integrand in terms of y. In such a case, the shell method allows us to integrate with respect to x instead. Exercises 29 and 30 provide some insight. 29. Compute the volume of the solid generated by revolving the region bounded by y = x and y = x 2 about each coordinate axis using a. the shell method.

b. the washer method.

30. Compute the volume of the solid generated by revolving the triangular region bounded by the lines 2y = x + 4, y = x, and x = 0 about a. the x-axis using the washer method.

b. The line x = 3 d. The x-axis f. The line y = -1

y = x2

a. The line x = 2 c. The x-axis

(2, 2)

x

0

y Ú 0 20. y = x,

y = 1

y = 0,

y4 y2 4 2

y = 2

21. y = 2x,

23. y = 3x,

x

b. The line x = -1 d. The line y = 4

b. the y-axis using the shell method. c. the line x = 4 using the shell method. d. the line y = 8 using the washer method. In Exercises 31–36, find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so.

6.2 31. The triangle with vertices (1, 1), (1, 2), and (2, 2) about a. the x-axis

b. the y-axis

c. the line x = 10>3

d. the line y = 1

32. The region bounded by y = 2x, y = 2, x = 0 about a. the x-axis

b. the y-axis

c. the line x = 4

d. the line y = 2

33. The region in the first quadrant bounded by the curve x = y - y 3 and the y-axis about b. the line y = 1

a. the x-axis

34. The region in the first quadrant bounded by x = y - y 3, x = 1, and y = 1 about a. the x-axis

b. the y-axis

c. the line x = 1

d. the line y = 1

a. the x-axis

b. the y-axis

35. The region bounded by y = 2x and y = x 2>8 about 36. The region bounded by y = 2x - x 2 and y = x about b. the line x = 1

a. the y-axis

37. The region in the first quadrant that is bounded above by the curve y = 1>x 1>4 , on the left by the line x = 1>16 , and below by the line y = 1 is revolved about the x-axis to generate a solid. Find the volume of the solid by a. the washer method.

b. the shell method.

38. The region in the first quadrant that is bounded above by the curve y = 1> 2x, on the left by the line x = 1>4, and below by the line y = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid by a. the washer method.

b. the shell method.

Theory and Examples 39. The region shown here is to be revolved about the x-axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? Explain. y x 3y2 2

(1, 1)

1

x y2 0

–2

1

x

40. The region shown here is to be revolved about the y-axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? Give reasons for your answers.

Volumes Using Cylindrical Shells

371

41. A bead is formed from a sphere of radius 5 by drilling through a diameter of the sphere with a drill bit of radius 3. a. Find the volume of the bead. b. Find the volume of the removed portion of the sphere. 42. A Bundt cake, well known for having a ringed shape, is formed by revolving around the y-axis the region bounded by the graph of y = sin (x 2 - 1) and the x-axis over the interval 1 … x … 11 + p. Find the volume of the cake. 43. Derive the formula for the volume of a right circular cone of height h and radius r using an appropriate solid of revolution. 44. Derive the equation for the volume of a sphere of radius r using the shell method. 45. Equivalence of the washer and shell methods for finding volume Let ƒ be differentiable and increasing on the interval a … x … b, with a 7 0, and suppose that ƒ has a differentiable inverse, ƒ -1 . Revolve about the y-axis the region bounded by the graph of ƒ and the lines x = a and y = ƒsbd to generate a solid. Then the values of the integrals given by the washer and shell methods for the volume have identical values: ƒsbd

Lƒsad

p ssƒ -1s ydd2 - a 2 d dy =

b

La

2pxsƒsbd - ƒsxdd dx.

To prove this equality, define ƒstd

Wstd =

Lƒsad

pssƒ -1s ydd2 - a 2 d dy

t

Sstd =

La

2pxsƒstd - ƒsxdd dx.

Then show that the functions W and S agree at a point of [a, b] and have identical derivatives on [a, b]. As you saw in Section 4.8, Exercise 128, this will guarantee Wstd = Sstd for all t in [a, b]. In particular, Wsbd = Ssbd. (Source: “Disks and Shells Revisited,” by Walter Carlip, American Mathematical Monthly, Vol. 98, No. 2, Feb. 1991, pp. 154–156.) 46. The region between the curve y = sec-1 x and the x-axis from x = 1 to x = 2 (shown here) is revolved about the y-axis to generate a solid. Find the volume of the solid. y

3

y sec –1 x

y 0

1

1

2

x

y x2

0

1 y –x 4

–1

x

47. Find the volume of the solid generated by revolving the region en2 closed by the graphs of y = e - x , y = 0, x = 0, and x = 1 about the y-axis. 48. Find the volume of the solid generated by revolving the region enclosed by the graphs of y = e x>2, y = 1, and x = ln 3 about the x-axis.

372

Chapter 6: Applications of Definite Integrals

Arc Length

6.3

We know what is meant by the length of a straight line segment, but without calculus, we have no precise definition of the length of a general winding curve. If the curve is the graph of a continuous function defined over an interval, then we can find the length of the curve using a procedure similar to that we used for defining the area between the curve and the x-axis. This procedure results in a division of the curve from point A to point B into many pieces and joining successive points of division by straight line segments. We then sum the lengths of all these line segments and define the length of the curve to be the limiting value of this sum as the number of segments goes to infinity.

Length of a Curve y = ƒ(x) Suppose the curve whose length we want to find is the graph of the function y = ƒ(x) from x = a to x = b. In order to derive an integral formula for the length of the curve, we assume that ƒ has a continuous derivative at every point of [a, b]. Such a function is called smooth, and its graph is a smooth curve because it does not have any breaks, corners, or cusps. y

Pk21 Pk B 5 Pn y 5 f (x)

x0 5 a

x1

x2

xk21

xk

b 5 xn

x

P0 5 A P1 P2

FIGURE 6.22 The length of the polygonal path P0P1P2 Á Pn approximates the length of the curve y = ƒ(x) from point A to point B.

y Pk21 Δyk

y 5 f (x) Lk

Pk

Δx k

0

xk21

xk

x

FIGURE 6.23 The arc Pk - 1Pk of the curve y = ƒ(x) is approximated by the straight line segment shown here, which has length Lk = 2(¢xk) 2 + (¢yk) 2.

We partition the interval [a, b] into n subintervals with a = x0 6 x1 6 x2 6 Á 6 xn = b. If yk = ƒ(xk), then the corresponding point Pk (xk, yk) lies on the curve. Next we connect successive points Pk - 1 and Pk with straight line segments that, taken together, form a polygonal path whose length approximates the length of the curve (Figure 6.22). If ¢xk = xk - xk - 1 and ¢yk = yk - yk - 1, then a representative line segment in the path has length (see Figure 6.23) Lk = 2(¢xk) 2 + (¢yk) 2, so the length of the curve is approximated by the sum n

n

2 2 a Lk = a 2(¢xk) + (¢yk) .

k=1

k=1

(1)

We expect the approximation to improve as the partition of [a, b] becomes finer. Now, by the Mean Value Theorem, there is a point ck, with xk - 1 6 ck 6 xk, such that ¢yk = ƒ¿(ck) ¢xk.

6.3

373

Arc Length

With this substitution for ¢yk , the sums in Equation (1) take the form n

n

n

2 2 2 a Lk = a 2(¢xk) + (ƒ¿(ck)¢xk) = a 21 + [ƒ¿(ck)] ¢xk.

k=1

k=1

(2)

k=1

Because 21 + [ƒ¿(x)]2 is continuous on [a, b], the limit of the Riemann sum on the righthand side of Equation (2) exists as the norm of the partition goes to zero, giving n

n

b

lim a Lk = lim a 21 + [ƒ¿(ck)]2 ¢xk = n: q n: q

21 + [ƒ¿(x)]2 dx. La We define the value of this limiting integral to be the length of the curve. k=1

k=1

DEFINITION If ƒ¿ is continuous on [a, b], then the length (arc length) of the curve y = ƒ(x) from the point A = (a, ƒ(a)) to the point B = (b, ƒ(b)) is the value of the integral b

L =

EXAMPLE 1

y

La

1

0 –1

y =

B

Solution

422 3>2 x - 1, 3

y =

a

3 1>2 x = 222x 1>2 2

dy 2 2 b = A 222x 1>2 B = 8x. dx

1

=

y

1 + a

L0 B 2 3

#

1 dy 2 b dx = 21 + 8x dx dx L0 1

13 1 s1 + 8xd3>2 d = L 2.17. 8 6 0

joining the points A = (0, -1) and B = A 1, 422>3 - 1 B on the curve (see Figure 6.24): 2.17 7 212 + (1.89) 2 L 2.14

EXAMPLE 2

A 4

Eq. (3) with a = 0, b = 1 Let u = 1 + 8x, integrate, and replace u by 1 + 8x.

Notice that the length of the curve is slightly larger than the length of the straight-line segment

B

1

#

x = 1, y L 0.89

The length of the curve over x = 0 to x = 1 is L =

0

(3)

0 … x … 1.

422 3>2 x - 1 3

dy 422 = 3 dx

A

3 y 5 x 1 1x 12

dy 2 b dx. dx

We use Equation (3) with a = 0, b = 1, and

x

FIGURE 6.24 The length of the curve is slightly larger than the length of the line segment joining points A and B (Example 1).

La B

1 + a

Find the length of the curve (Figure 6.24)

y 5 42 x 3/2 2 1 3 (1, 0.89)

b

21 + [ƒ¿(x)]2 dx =

x

FIGURE 6.25 The curve in Example 2, where A = (1, 13>12) and B = (4, 67>12).

Find the length of the graph of ƒ(x) =

Solution

Decimal approximations

x3 1 + x, 12

1 … x … 4.

A graph of the function is shown in Figure 6.25. To use Equation (3), we find ƒ¿(x) =

x2 1 - 2 4 x

374

Chapter 6: Applications of Definite Integrals

so 2

1 + [ƒ¿(x)]2 = 1 + a

x2 x4 1 1 1 - 2b = 1 + a - + 4b 4 16 2 x x 2 x4 x2 1 1 1 = + + 4 = a + 2b . 16 2 4 x x

The length of the graph over [1, 4] is 4

L =

21 + [ƒ¿(x)] dx =

L1

= c

EXAMPLE 3

4 2

x2 1 + 2 bdx 4 x

4

72 64 x3 1 1 1 - xd = a - b - a - 1b = = 6. 12 12 4 12 12 1

Find the length of the curve 1 x (e + e -x ), 2

y = Solution

L1

a

0 … x … 2.

We use Equation (3) with a = 0, b = 2, and y =

1 x (e + e -x ) 2

dy 1 = (e x - e -x ) 2 dx a 1 + a

dy 2 1 b = (e 2x - 2 + e -2x ) 4 dx

2 dy 2 1 1 b = (e 2x + 2 + e -2x ) = c (e x + e -x) d . 4 2 dx

The length of the curve from x = 0 to x = 2 is 2

L =

1 + a

L0 B

2 dy 2 1 x b dx = (e + e -x ) dx 2 dx 0 L

Eq. (3) with a = 0, b = 2

2

=

1 x 1 ce - e -x d = (e 2 - e -2) L 3.63. 2 2 0

Dealing with Discontinuities in dy/dx At a point on a curve where dy> dx fails to exist, dx> dy may exist. In this case, we may be able to find the curve’s length by expressing x as a function of y and applying the following analogue of Equation (3):

Formula for the Length of x = gs yd, c … y … d If g ¿ is continuous on [c, d], the length of the curve x = g s yd from A = (g(c), c) to B = (g(d ), d ) is d

L =

1 + a

Lc B

2

d

dx b dy = 21 + [g¿s yd]2 dy. dy Lc

(4)

6.3

EXAMPLE 4 Solution

0

375

Find the length of the curve y = sx>2d2>3 from x = 0 to x = 2.

The derivative -1>3 1>3 dy 1 2 x 1 2 a b = ax b = a b 3 2 2 3 dx

y

1

Arc Length

x 2/3 y5 ⎛ ⎛ ⎝ 2⎝

1

is not defined at x = 0, so we cannot find the curve’s length with Equation (3). We therefore rewrite the equation to express x in terms of y: (2, 1)

2

FIGURE 6.26 The graph of y = (x>2) 2>3 from x = 0 to x = 2 is also the graph of x = 2y 3>2 from y = 0 to y = 1 (Example 4).

x y = a b 2 x y 3>2 = 2

x

2>3

Raise both sides to the power 3/2.

x = 2y 3>2.

Solve for x.

From this we see that the curve whose length we want is also the graph of x = 2y 3>2 from y = 0 to y = 1 (Figure 6.26). The derivative 3 dx = 2 a b y 1>2 = 3y 1>2 2 dy is continuous on [0, 1]. We may therefore use Equation (4) to find the curve’s length: d

L =

1 + a

Lc B

2

1

dx b dy = 21 + 9y dy dy L0

1#2 s1 + 9yd3>2 d 9 3 0 1

= =

2 A 10210 - 1 B L 2.27. 27

Eq. (4) with c = 0, d = 1. Let u = 1 + 9y, du>9 = dy, integrate, and substitute back.

The Differential Formula for Arc Length If y = ƒ(x) and if ƒ¿ is continuous on [a, b], then by the Fundamental Theorem of Calculus we can define a new function x

s(x) =

21 + [ƒ¿(t)]2 dt. (5) La From Equation (3) and Figure 6.22, we see that this function s(x) is continuous and measures the length along the curve y = ƒ(x) from the initial point P0(a, ƒ(a)) to the point Q(x, ƒ(x)) for each x H [a, b]. The function s is called the arc length function for y = ƒ(x). From the Fundamental Theorem, the function s is differentiable on (a, b) and dy 2 ds = 21 + [ƒ¿(x)]2 = 1 + a b . dx dx B Then the differential of arc length is ds =

B

1 + a

dy 2 b dx. dx

(6)

A useful way to remember Equation (6) is to write ds = 2dx 2 + dy 2,

(7)

which can be integrated between appropriate limits to give the total length of a curve. From this point of view, all the arc length formulas are simply different expressions for the equation

376

Chapter 6: Applications of Definite Integrals

L = 1 ds. Figure 6.27a gives the exact interpretation of ds corresponding to Equation (7). Figure 6.27b is not strictly accurate, but is to be thought of as a simplified approximation of Figure 6.27a. That is, ds L ¢s.

y

ds

EXAMPLE 5

Find the arc length function for the curve in Example 2 taking A = (1, 13>12) as the starting point (see Figure 6.25).

dy

dx

Solution

In the solution to Example 2, we found that

x

0

1 + [ƒ¿(x)]2 = a

(a) y

2

x2 1 + 2b . 4 x

Therefore the arc length function is given by x

s(x) = ds

L1

= c

dy

x

L1

a

t2 1 + 2 b dt 4 t

t x 11 1 1 - td = - x + . 12 12 12 1 3

3

To compute the arc length along the curve from A = (1, 13>12) to B = (4, 67>12), for instance, we simply calculate

dx x

0

x

21 + [ƒ¿(t)]2 dt =

(b)

1 11 43 - + = 6. 12 4 12

s(4) =

FIGURE 6.27 Diagrams for remembering the equation ds = 2dx 2 + dy 2 .

This is the same result we obtained in Example 2.

Exercises 6.3 Finding Lengths of Curves Find the lengths of the curves in Exercises 1–14. If you have a grapher, you may want to graph these curves to see what they look like. 1. y = s1>3dsx 2 + 2d3>2 2. y = x 3>2

from

4. x = sy 3>2>3d - y 1>2

5. x = sy >4d + 1>s8y d

from

6. x = sy 3>6d + 1>s2yd

from from

y = 1 to y = 2 y = 2 to y = 3

7. y = s3>4dx 4>3 - s3>8dx 2>3 + 5,

1 … x … 8

8. y = sx 3>3d + x 2 + x + 1>s4x + 4d,

0 … x … 2

2

x from x = 1 to x = 2 8 x2 ln x 10. y = from x = 1 to x = 3 2 4 3 x 1 11. y = + , 1 … x … 3 3 4x 5 x 1 1 12. y = , + … x … 1 5 12x3 2 y

13. x =

2sec4 t - 1 dt,

L0

-p>4 … y … p>4

15. 16. 17. 18. 19. 20.

c. Use your grapher’s or computer’s integral evaluator to find the curve’s length numerically. y = x 2, -1 … x … 2 y = tan x, -p>3 … x … 0 x = sin y, 0 … y … p x = 21 - y 2, -1>2 … y … 1>2 y 2 + 2y = 2x + 1 from s -1, -1d to s7, 3d y = sin x - x cos x, 0 … x … p x

21. y =

tan t dt,

L0

0 … x … p>6

y

22. x =

L0

2sec2 t - 1 dt,

-p>3 … y … p>4

Theory and Examples 23. a. Find a curve through the point (1, 1) whose length integral (Equation 3) is 4

x

14. y =

b. Graph the curve to see what it looks like.

y = 1 to y = 3 y = 1 to y = 9

from

2

9. y = ln x -

a. Set up an integral for the length of the curve.

x = 0 to x = 3

x = 0 to x = 4

3. x = sy 3>3d + 1>s4yd 4

from

T Finding Integrals for Lengths of Curves In Exercises 15–22, do the following.

23t - 1 dt, 4

L-2

-2 … x … -1

L =

L1

A

1 +

1 dx . 4x

b. How many such curves are there? Give reasons for your answer.

6.3 24. a. Find a curve through the point (0, 1) whose length integral (Equation 4) is 2 1 1 + 4 dy . L = A y 1 L b. How many such curves are there? Give reasons for your answer.

377

Arc Length

which is the length L of the curve y = ƒsxd from a to b. y f (x)

25. Find the length of the curve Tangent fin with slope f'(xk–1)

x

y =

L0

2cos 2t dt

(xk–1, f (xk–1)) xk

from x = 0 to x = p>4. + 26. The length of an astroid The graph of the equation x y 2>3 = 1 is one of a family of curves called astroids (not “asteroids”) because of their starlike appearance (see the accompanying figure). Find the length of this particular astroid by finding the length of half the first-quadrant portion, y = (1 - x 2>3) 3>2, 12>4 … x … 1, and multiplying by 8. 2>3

xk–1

xk

x

33. Approximate the arc length of one-quarter of the unit circle (which is p>2) by computing the length of the polygonal approximation with n = 4 segments (see accompanying figure). y

y 1 x 2/3 1 y 2/3 5 1

–1

0

1

x 0

–1

27. Length of a line segment Use the arc length formula (Equation 3) to find the length of the line segment y = 3 - 2x, 0 … x … 2. Check your answer by finding the length of the segment as the hypotenuse of a right triangle. 28. Circumference of a circle Set up an integral to find the circumference of a circle of radius r centered at the origin. You will learn how to evaluate the integral in Section 8.3. 29. If 9x 2 = y(y - 3) 2, show that ds 2 = 30. If 4x 2 - y 2 = 64, show that 4 ds = 2 A 5x 2 - 16 B dx 2. y 31. Is there a smooth (continuously differentiable) curve y = ƒsxd 2

whose length over the interval 0 … x … a is always 22a ? Give reasons for your answer. 32. Using tangent fins to derive the length formula for curves Assume that ƒ is smooth on [a, b] and partition the interval [a, b] in the usual way. In each subinterval [xk - 1, xk] , construct the tangent fin at the point sxk - 1, ƒsxk - 1 dd , as shown in the accompanying figure. a. Show that the length of the kth tangent fin over the interval [xk - 1, xk] equals 2s¢xk d2 + sƒ¿sxk - 1 d ¢xk d2 . b. Show that n k=1

b

La

x

34. Distance between two points Assume that the two points (x1, y1) and (x2, y2) lie on the graph of the straight line y = mx + b. Use the arc length formula (Equation 3) to find the distance between the two points. 35. Find the arc length function for the graph of ƒ(x) = 2x 3>2 using (0, 0) as the starting point. What is the length of the curve from (0, 0) to (1, 2)? 36. Find the arc length function for the curve in Exercise 8, using (0, 1>4) as the starting point. What is the length of the curve from (0, 1>4) to (1, 59>24)? COMPUTER EXPLORATIONS In Exercises 37–42, use a CAS to perform the following steps for the given graph of the function over the closed interval.

(y + 1) 2 2 dy . 4y

lim a slength of k th tangent find = n: q

0.25 0.5 0.75 1

21 + sƒ¿sxdd2 dx ,

a. Plot the curve together with the polygonal path approximations for n = 2, 4, 8 partition points over the interval. (See Figure 6.22.) b. Find the corresponding approximation to the length of the curve by summing the lengths of the line segments. c. Evaluate the length of the curve using an integral. Compare your approximations for n = 2, 4, 8 with the actual length given by the integral. How does the actual length compare with the approximations as n increases? Explain your answer. 37. ƒsxd = 21 - x 2,

-1 … x … 1

38. ƒsxd = x 1>3 + x 2>3,

0 … x … 2

39. ƒsxd = sin spx d, 2

0 … x … 22

40. ƒsxd = x 2 cos x, 0 … x … p x - 1 1 , - … x … 1 41. ƒsxd = 2 4x 2 + 1 42. ƒsxd = x 3 - x 2, -1 … x … 1

378

Chapter 6: Applications of Definite Integrals

Areas of Surfaces of Revolution

6.4

When you jump rope, the rope sweeps out a surface in the space around you similar to what is called a surface of revolution. The surface surrounds a volume of revolution, and many applications require that we know the area of the surface rather than the volume it encloses. In this section we define areas of surfaces of revolution. More general surfaces are treated in Chapter 15.

Defining Surface Area

x y x A

B

2y

y x

0

x NOT TO SCALE

(a)

(b)

FIGURE 6.28 (a) A cylindrical surface generated by rotating the horizontal line segment AB of length ¢x about the x-axis has area 2py¢x . (b) The cut and rolled-out cylindrical surface as a rectangle.

If you revolve a region in the plane that is bounded by the graph of a function over an interval, it sweeps out a solid of revolution, as we saw earlier in the chapter. However, if you revolve only the bounding curve itself, it does not sweep out any interior volume but rather a surface that surrounds the solid and forms part of its boundary. Just as we were interested in defining and finding the length of a curve in the last section, we are now interested in defining and finding the area of a surface generated by revolving a curve about an axis. Before considering general curves, we begin by rotating horizontal and slanted line segments about the x-axis. If we rotate the horizontal line segment AB having length ¢x about the x-axis (Figure 6.28a), we generate a cylinder with surface area 2py¢x. This area is the same as that of a rectangle with side lengths ¢x and 2py (Figure 6.28b). The length 2py is the circumference of the circle of radius y generated by rotating the point (x, y) on the line AB about the x-axis. Suppose the line segment AB has length L and is slanted rather than horizontal. Now when AB is rotated about the x-axis, it generates a frustum of a cone (Figure 6.29a). From classical geometry, the surface area of this frustum is 2py * L, where y * = s y1 + y2 d>2 is the average height of the slanted segment AB above the x-axis. This surface area is the same as that of a rectangle with side lengths L and 2py * (Figure 6.29b). Let’s build on these geometric principles to define the area of a surface swept out by revolving more general curves about the x-axis. Suppose we want to find the area of the surface swept out by revolving the graph of a nonnegative continuous function y = ƒsxd, a … x … b, about the x-axis. We partition the closed interval [a, b] in the usual way and use the points in the partition to subdivide the graph into short arcs. Figure 6.30 shows a typical arc PQ and the band it sweeps out as part of the graph of ƒ.

L y

L

A y*

y1

B

2py * y2 x

0

NOT TO SCALE

(a)

(b)

FIGURE 6.29 (a) The frustum of a cone generated by rotating the slanted line segment AB of length L about the x-axis has area y1 + y2 2py * L . (b) The area of the rectangle for y * = , the average 2 height of AB above the x-axis.

6.4

y

y 5 f (x)

PQ

0 a xk21 x k

b

FIGURE 6.30 The surface generated by revolving the graph of a nonnegative function y = ƒsxd, a … x … b, about the x-axis. The surface is a union of bands like the one swept out by the arc PQ.

x

Areas of Surfaces of Revolution

379

As the arc PQ revolves about the x-axis, the line segment joining P and Q sweeps out a frustum of a cone whose axis lies along the x-axis (Figure 6.31). The surface area of this frustum approximates the surface area of the band swept out by the arc PQ. The surface area of the frustum of the cone shown in Figure 6.31 is 2py *L, where y * is the average height of the line segment joining P and Q, and L is its length ( just as before). Since ƒ Ú 0, from Figure 6.32 we see that the average height of the line segment is y * = sƒsxk - 1 d + ƒsxk dd>2, and the slant length is L = 2s¢xk d2 + s¢yk d2. Therefore, Frustum surface area = 2p #

ƒsxk - 1 d + ƒsxk d 2

# 2s¢xk d2

+ s¢yk d2

= psƒsxk - 1 d + ƒsxk dd2s¢xk d2 + s¢yk d2. The area of the original surface, being the sum of the areas of the bands swept out by arcs like arc PQ, is approximated by the frustum area sum n 2 2 a psƒsxk - 1 d + ƒsxk dd2s¢xk d + s¢yk d .

(1)

k=1

We expect the approximation to improve as the partition of [a, b] becomes finer. Moreover, if the function ƒ is differentiable, then by the Mean Value Theorem, there is a point sck , ƒsck dd on the curve between P and Q where the tangent is parallel to the segment PQ (Figure 6.33). At this point, ¢yk ƒ¿sck d = , ¢xk

P Q

x k1 xk

x

¢yk = ƒ¿sck d ¢xk . With this substitution for ¢yk , the sums in Equation (1) take the form

FIGURE 6.31 The line segment joining P and Q sweeps out a frustum of a cone.

n 2 2 a psƒsxk - 1 d + ƒsxk dd2s¢xk d + sƒ¿sck d ¢xk d

k=1

n

P

Segment length: L (xk )2 (yk )2 y f (x)

yk

Q

= a psƒsxk - 1 d + ƒsxk dd21 + sƒ¿sck dd2 ¢xk .

(2)

k=1

These sums are not the Riemann sums of any function because the points xk - 1, xk , and ck are not the same. However, it can be proved that as the norm of the partition of [a, b] goes to zero, the sums in Equation (2) converge to the integral b

r1 f(xk – 1) r2 f (xk ) xk – 1

xk

xk

FIGURE 6.32 Dimensions associated with the arc and line segment PQ.

La

2pƒsxd21 + sƒ¿sxdd2 dx.

We therefore define this integral to be the area of the surface swept out by the graph of ƒ from a to b.

DEFINITION If the function ƒsxd Ú 0 is continuously differentiable on [a, b], the area of the surface generated by revolving the graph of y = ƒsxd about the x-axis is b

S =

La

2py

1 + a

B

b dy 2 b dx = 2pƒsxd21 + sƒ¿sxdd2 dx. dx La

(3)

The square root in Equation (3) is the same one that appears in the formula for the arc length differential of the generating curve in Equation (6) of Section 6.3. Find the area of the surface generated by revolving the curve y = 22x, 1 … x … 2, about the x-axis (Figure 6.34).

EXAMPLE 1

380

Chapter 6: Applications of Definite Integrals Solution

(ck , f(ck )) yk

We evaluate the formula

Tangent parallel to chord

P

b

S =

La

Q

ck xk

1 + a

B

dy 2 b dx dx

Eq. (3)

with

y f (x) xk – 1

2py

a = 1,

xk

FIGURE 6.33 If ƒ is smooth, the Mean Value Theorem guarantees the existence of a point ck where the tangent is parallel to segment PQ.

b = 2,

y = 22x,

dy 1 = . dx 2x

First, we perform some algebraic manipulation on the radical in the integrand to transform it into an expression that is easier to integrate. 1 + a

B

2 dy 2 1 b = 1 + a b dx B 2x

y

=

y 5 2x (2, 22)

A

2x + 1 x + 1 1 = . 1 + x = A x 2x

(1, 2)

With these substitutions, we have 2

S =

0 1

2

x

L1

2p # 22x

2x + 1 2x

2

dx = 4p 2x + 1 dx L1

8p 2 sx + 1d3>2 d = A 323 - 222 B . 3 3 1

= 4p #

2

Revolution About the y-Axis

FIGURE 6.34 In Example 1 we calculate the area of this surface.

For revolution about the y-axis, we interchange x and y in Equation (3).

Surface Area for Revolution About the y-Axis If x = gsyd Ú 0 is continuously differentiable on [c, d], the area of the surface generated by revolving the graph of x = gsyd about the y-axis is d

S =

y

A (0, 1)

Lc

2px

1 + a

B

2

d

dx b dy = 2pgsyd21 + sg¿s ydd2 dy. dy Lc

(4)

The line segment x = 1 - y, 0 … y … 1, is revolved about the y-axis to generate the cone in Figure 6.35. Find its lateral surface area (which excludes the base area).

EXAMPLE 2

xy1

Solution

Here we have a calculation we can check with a formula from geometry:

0

Lateral surface area =

B (1, 0) x

base circumference * slant height = p22 . 2

To see how Equation (4) gives the same result, we take FIGURE 6.35 Revolving line segment AB about the y-axis generates a cone whose lateral surface area we can now calculate in two different ways (Example 2).

c = 0,

d = 1,

B

1 + a

x = 1 - y, 2

dx = -1, dy

dx b = 21 + s -1d2 = 22 dy

6.4

Areas of Surfaces of Revolution

381

and calculate d

S =

Lc

2px

1 + a

B

= 2p22 c y -

2

1

dx b dy = 2ps1 - yd22 dy dy L0

y2 1 1 d = 2p22 a1 - b 2 0 2

= p22. The results agree, as they should.

Exercises 6.4 Finding Integrals for Surface Area In Exercises 1–8: a. Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis.

Find the areas of the surfaces generated by revolving the curves in Exercises 13–23 about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. 13. y = x 3>9,

0 … x … 2;

T b. Graph the curve to see what it looks like. If you can, graph the surface too.

14. y = 2x,

3>4 … x … 15>4;

T c. Use your grapher’s or computer’s integral evaluator to find the surface’s area numerically.

16. y = 2x + 1,

1. y = tan x,

0 … x … p>4;

x-axis

2. y = x 2,

0 … x … 2;

x-axis

3. xy = 1,

1 … y … 2;

y-axis

4. x = sin y, 5. x

1>2

+ y

1>2

0 … y … p; = 3 from

6. y + 22y = x,

15. y = 22x - x ,

0 … y … 1;

y-axis

0 … y … 15>4;

y-axis

y-axis

⎛ 15 ⎛ ⎝1, 4 ⎝

15 4

(4, 1) to (1, 4); x-axis

x-axis

x-axis

1 … y … 3; y

y-axis

1 … y … 2;

1 … x … 5;

18. x = s1>3dy 3>2 - y 1>2, 19. x = 224 - y,

x-axis

0.5 … x … 1.5;

2

17. x = y 3>3,

x-axis

x 24 y

y-axis

y

7. x =

tan t dt,

L0

0 … y … p>3;

y-axis 0

x

8. y =

2t - 1 dt, 2

L1

1 … x … 25;

Finding Surface Area 9. Find the lateral (side) surface area of the cone generated by revolving the line segment y = x>2, 0 … x … 4 , about the x-axis. Check your answer with the geometry formula Lateral surface area =

Lateral surface area =

20. x = 22y - 1,

5>8 … y … 1;

1 * base circumference * slant height. 2

11. Find the surface area of the cone frustum generated by revolving the line segment y = sx>2d + s1>2d, 1 … x … 3 , about the x-axis. Check your result with the geometry formula Frustum surface area = psr1 + r2 d * slant height. 12. Find the surface area of the cone frustum generated by revolving the line segment y = sx>2d + s1>2d, 1 … x … 3, about the y-axis. Check your result with the geometry formula Frustum surface area = psr1 + r2 d * slant height.

y-axis

y

1 * base circumference * slant height . 2

10. Find the lateral surface area of the cone generated by revolving the line segment y = x>2, 0 … x … 4, about the y-axis. Check your answer with the geometry formula

x

4

x-axis

1

5 8

21. x = se y + e -y d>2,

(1, 1)

⎛1 , ⎝2 0

1 2

x 2y 1

5⎛ 8⎝ 1

x

0 … y … ln 2; y-axis y

x

e y e –y 2

ln 2

0 1

x

382

Chapter 6: Applications of Definite Integrals

22. y = s1>3dsx 2 + 2d3>2, ds = 2dx + dy 2

2

0 … x … 22;

y-axis (Hint: Express

in terms of dx, and evaluate the integral

S = 1 2px ds with appropriate limits.)

23. x = s y 4>4d + 1>s8y 2 d,

1 … y … 2;

29. The shaded band shown here is cut from a sphere of radius R by parallel planes h units apart. Show that the surface area of the band is 2pRh .

x-axis (Hint: Express

ds = 2dx 2 + dy 2 in terms of dy, and evaluate the integral S = 1 2py ds with appropriate limits.) h

24. Write an integral for the area of the surface generated by revolving the curve y = cos x, -p>2 … x … p>2, about the x-axis. In Section 8.4 we will see how to evaluate such integrals.

26. Testing the new definition The lateral (side) surface area of a cone of height h and base radius r should be pr 2r 2 + h 2, the semiperimeter of the base times the slant height. Show that this is still the case by finding the area of the surface generated by revolving the line segment y = sr>hdx, 0 … x … h, about the x-axis.

30. Here is a schematic drawing of the 90-ft dome used by the U.S. National Weather Service to house radar in Bozeman, Montana. a. How much outside surface is there to paint (not counting the bottom)? T b. Express the answer to the nearest square foot. Axis

25. Testing the new definition Show that the surface area of a sphere of radius a is still 4pa 2 by using Equation (3) to find the area of the surface generated by revolving the curve y = 2a 2 - x 2, -a … x … a, about the x-axis.

R

T 27. Enameling woks Your company decided to put out a deluxe version of a wok you designed. The plan is to coat it inside with white enamel and outside with blue enamel. Each enamel will be sprayed on 0.5 mm thick before baking. (See accompanying figure.) Your manufacturing department wants to know how much enamel to have on hand for a production run of 5000 woks. What do you tell them? (Neglect waste and unused material and give your answer in liters. Remember that 1 cm3 = 1 mL , so 1 L = 1000 cm3 .)

45 ft

Center Radius 45 ft

22.5 ft

y (cm) x 2 y 2 16 2 256

0 –7

31. An alternative derivation of the surface area formula Assume ƒ is smooth on [a, b] and partition [a, b] in the usual way. In the kth subinterval [xk - 1, xk], construct the tangent line to the curve at the midpoint mk = sxk - 1 + xk d>2 , as in the accompanying figure.

x (cm)

a. Show that 9 cm deep

r1 = ƒsmk d - ƒ¿smk d

–16

28. Slicing bread Did you know that if you cut a spherical loaf of bread into slices of equal width, each slice will have the same amount of crust? To see why, suppose the semicircle y = 2r 2 - x 2 shown here is revolved about the x-axis to generate a sphere. Let AB be an arc of the semicircle that lies above an interval of length h on the x-axis. Show that the area swept out by AB does not depend on the location of the interval. (It does depend on the length of the interval.)

¢xk 2

and

r2 = ƒsmk d + ƒ¿smk d

b. Show that the length Lk of the tangent line segment in the kth subinterval is Lk = 2s¢xk d2 + sƒ¿smk d ¢xk d2. y f (x)

y y r 2 x 2

A

B

r2 r1 r

–r

0

ah

a h

x

¢xk . 2

xk – 1

mk xk

xk

x

6.5

383

(Hint: Revolve the first-quadrant portion y = s1 - x 2>3 d3>2, 0 … x … 1, about the x-axis and double your result.)

c. Show that the lateral surface area of the frustum of the cone swept out by the tangent line segment as it revolves about the x-axis is 2pƒsmk d21 + sƒ¿smk dd2 ¢xk.

y

d. Show that the area of the surface generated by revolving y = ƒsxd about the x-axis over [a, b] is

1

n b lateral surface area b = 2pƒsxd21 + sƒ¿sxdd2 dx. lim a a n: q k = 1 of k th frustum La

x 2/3 y 2/3 1

–1

32. The surface of an astroid Find the area of the surface generated by revolving about the x-axis the portion of the astroid x 2>3 + y 2>3 = 1 shown in the accompanying figure.

6.5

Work

0

1

x

Work In everyday life, work means an activity that requires muscular or mental effort. In science, the term refers specifically to a force acting on a body (or object) and the body’s subsequent displacement. This section shows how to calculate work. The applications run from compressing railroad car springs and emptying subterranean tanks to forcing subatomic particles to collide and lifting satellites into orbit.

Work Done by a Constant Force When a body moves a distance d along a straight line as a result of being acted on by a force of constant magnitude F in the direction of motion, we define the work W done by the force on the body with the formula W = Fd

sConstant-force formula for workd.

(1)

From Equation (1) we see that the unit of work in any system is the unit of force multiplied by the unit of distance. In SI units (SI stands for Système International, or International System), the unit of force is a newton, the unit of distance is a meter, and the unit of work is a newton-meter sN # md. This combination appears so often it has a special name, the joule. In the British system, the unit of work is the foot-pound, a unit frequently used by engineers. Joules The joule, abbreviated J, is named after the English physicist James Prescott Joule (1818–1889). The defining equation is 1 joule = s1 newtonds1 meterd .

In symbols, 1 J = 1 N # m .

EXAMPLE 1 Suppose you jack up the side of a 2000-lb car 1.25 ft to change a tire. The jack applies a constant vertical force of about 1000 lb in lifting the side of the car (but because of the mechanical advantage of the jack, the force you apply to the jack itself is only about 30 lb). The total work performed by the jack on the car is 1000 * 1.25 = 1250 ft-lb. In SI units, the jack has applied a force of 4448 N through a distance of 0.381 m to do 4448 * 0.381 L 1695 J of work. Work Done by a Variable Force Along a Line If the force you apply varies along the way, as it will if you are stretching or compressing a spring, the formula W = Fd has to be replaced by an integral formula that takes the variation in F into account. Suppose that the force performing the work acts on an object moving along a straight line, which we take to be the x-axis. We assume that the magnitude of the force is a continuous function F of the object’s position x. We want to find the work done over the interval from x = a to x = b. We partition [a, b] in the usual way and choose an arbitrary point ck in each subinterval [xk - 1, xk]. If the subinterval is short enough, the continuous function F

384

Chapter 6: Applications of Definite Integrals

will not vary much from xk - 1 to xk . The amount of work done across the interval will be about Fsck d times the distance ¢xk , the same as it would be if F were constant and we could apply Equation (1). The total work done from a to b is therefore approximated by the Riemann sum n

Work L a Fsck d ¢xk. k=1

We expect the approximation to improve as the norm of the partition goes to zero, so we define the work done by the force from a to b to be the integral of F from a to b: b

n

lim a Fsck d ¢xk = n: q k=1

La

Fsxd dx.

DEFINITION The work done by a variable force F(x) in the direction of motion along the x-axis from x = a to x = b is b

W =

La

Fsxd dx.

(2)

The units of the integral are joules if F is in newtons and x is in meters, and foot-pounds if F is in pounds and x is in feet. So the work done by a force of Fsxd = 1>x 2 newtons in moving an object along the x-axis from x = 1 m to x = 10 m is 10

W =

L1

10

1 1 1 + 1 = 0.9 J. dx = - x d = 10 x2 1

Hooke’s Law for Springs: F = kx Hooke’s Law says that the force required to hold a stretched or compressed spring x units from its natural (unstressed) length is proportional to x. In symbols, F = kx.

Compressed

(3)

F x 0

Uncompressed

1

x

The constant k, measured in force units per unit length, is a characteristic of the spring, called the force constant (or spring constant) of the spring. Hooke’s Law, Equation (3), gives good results as long as the force doesn’t distort the metal in the spring. We assume that the forces in this section are too small to do that.

(a) F Force (lb)

EXAMPLE 2 Find the work required to compress a spring from its natural length of 1 ft to a length of 0.75 ft if the force constant is k = 16 lb>ft. F 16x

4

0

Work done by F from x 0 to x 0.25 0.25 Amount compressed

x (ft)

(b)

FIGURE 6.36 The force F needed to hold a spring under compression increases linearly as the spring is compressed (Example 2).

Solution We picture the uncompressed spring laid out along the x-axis with its movable end at the origin and its fixed end at x = 1 ft (Figure 6.36). This enables us to describe the force required to compress the spring from 0 to x with the formula F = 16x. To compress the spring from 0 to 0.25 ft, the force must increase from

Fs0d = 16 # 0 = 0 lb

to

Fs0.25d = 16 # 0.25 = 4 lb.

The work done by F over this interval is 0.25

W =

L0

16x dx = 8x 2 d

0.25

= 0.5 ft-lb. 0

Eq. (2) with a = 0, b = 0.25, Fsxd = 16x

6.5

Work

385

EXAMPLE 3

A spring has a natural length of 1 m. A force of 24 N holds the spring stretched to a total length of 1.8 m. (a) Find the force constant k. (b) How much work will it take to stretch the spring 2 m beyond its natural length? (c) How far will a 45-N force stretch the spring?

x0

Solution 0.8 1

24 N

(a) The force constant. We find the force constant from Equation (3). A force of 24 N maintains the spring at a position where it is stretched 0.8 m from its natural length, so 24 = ks0.8d k = 24>0.8 = 30 N>m.

x (m)

FIGURE 6.37 A 24-N weight stretches this spring 0.8 m beyond its unstressed length (Example 3).

Eq. (3) with F = 24, x = 0.8

(b) The work to stretch the spring 2 m. We imagine the unstressed spring hanging along the x-axis with its free end at x = 0 (Figure 6.37). The force required to stretch the spring x m beyond its natural length is the force required to hold the free end of the spring x units from the origin. Hooke’s Law with k = 30 says that this force is Fsxd = 30x. The work done by F on the spring from x = 0 m to x = 2 m is 2

W =

2

30x dx = 15x 2 d = 60 J.

0 L0 (c) How far will a 45-N force stretch the spring? We substitute F = 45 in the equation F = 30x to find 45 = 30x, or x = 1.5 m. A 45-N force will keep the spring stretched 1.5 m beyond its natural length.

The work integral is useful to calculate the work done in lifting objects whose weights vary with their elevation. x 20

EXAMPLE 4

A 5-lb bucket is lifted from the ground into the air by pulling in 20 ft of rope at a constant speed (Figure 6.38). The rope weighs 0.08 lb> ft. How much work was spent lifting the bucket and rope?

The bucket has constant weight, so the work done lifting it alone is weight * distance = 5 # 20 = 100 ft-lb. The weight of the rope varies with the bucket’s elevation, because less of it is freely hanging. When the bucket is x ft off the ground, the remaining proportion of the rope still being lifted weighs s0.08d # s20 - xd lb. So the work in lifting the rope is Solution

0

20

Work on rope = FIGURE 6.38 Lifting the bucket in Example 4.

L0

20

s0.08ds20 - xd dx =

L0

s1.6 - 0.08xd dx

= C 1.6x - 0.04x 2 D 0 = 32 - 16 = 16 ft-lb. 20

The total work for the bucket and rope combined is 100 + 16 = 116 ft-lb.

Pumping Liquids from Containers How much work does it take to pump all or part of the liquid from a container? Engineers often need to know the answer in order to design or choose the right pump to transport water or some other liquid from one place to another. To find out how much work is required to pump the liquid, we imagine lifting the liquid out one thin horizontal slab at a time and applying the equation W = Fd to each slab. We then evaluate the integral this leads to as the slabs become thinner and more numerous. The integral we get each time depends on the weight of the liquid and the dimensions of the container, but the way we find the integral is always the same. The next example shows what to do.

386

Chapter 6: Applications of Definite Integrals y

y 2x or x 1 y 2

8

(5, 10) 1y 2

y

We imagine the oil divided into thin slabs by planes perpendicular to the y-axis at the points of a partition of the interval [0, 8]. The typical slab between the planes at y and y + ¢y has a volume of about 2

p 1 ¢V = psradiusd2sthicknessd = p a yb ¢y = y 2 ¢y ft3. 2 4

Δy

The force F( y) required to lift this slab is equal to its weight,

0 5

FIGURE 6.39 Example 5.

The conical tank in Figure 6.39 is filled to within 2 ft of the top with olive oil weighing 57 lb>ft3 . How much work does it take to pump the oil to the rim of the tank? Solution

10

10 y

EXAMPLE 5

x

Fs yd = 57 ¢V =

The olive oil and tank in

57p 2 y ¢y lb. 4

Weight = (weight per unit volume) * volume

The distance through which F( y) must act to lift this slab to the level of the rim of the cone is about s10 - yd ft, so the work done lifting the slab is about ¢W =

57p s10 - ydy 2 ¢y ft-lb. 4

Assuming there are n slabs associated with the partition of [0, 8], and that y = yk denotes the plane associated with the kth slab of thickness ¢yk , we can approximate the work done lifting all of the slabs with the Riemann sum n

W L a

k=1

57p s10 - yk dyk 2 ¢yk ft-lb. 4

The work of pumping the oil to the rim is the limit of these sums as the norm of the partition goes to zero and the number of slabs tends to infinity: n

W = lim a n: q k=1

8

57p 57p (10 - yk)yk 2 ¢yk = (10 - y)y 2 dy 4 L0 4 8

=

57p s10y 2 - y 3 d dy 4 L0

=

3 y4 8 57p 10y c d L 30,561 ft-lb. 4 3 4 0

Exercises 6.5 Springs 1. Spring constant It took 1800 J of work to stretch a spring from its natural length of 2 m to a length of 5 m. Find the spring’s force constant.

3. Stretching a rubber band A force of 2 N will stretch a rubber band 2 cm (0.02 m). Assuming that Hooke’s Law applies, how far will a 4-N force stretch the rubber band? How much work does it take to stretch the rubber band this far?

2. Stretching a spring A spring has a natural length of 10 in. An 800-lb force stretches the spring to 14 in.

4. Stretching a spring If a force of 90 N stretches a spring 1 m beyond its natural length, how much work does it take to stretch the spring 5 m beyond its natural length? 5. Subway car springs It takes a force of 21,714 lb to compress a coil spring assembly on a New York City Transit Authority subway car from its free height of 8 in. to its fully compressed height of 5 in. a. What is the assembly’s force constant? b. How much work does it take to compress the assembly the first half inch? the second half inch? Answer to the nearest in.-lb.

a. Find the force constant. b. How much work is done in stretching the spring from 10 in. to 12 in.? c. How far beyond its natural length will a 1600-lb force stretch the spring?

6.5 6. Bathroom scale A bathroom scale is compressed 1> 16 in. when a 150-lb person stands on it. Assuming that the scale behaves like a spring that obeys Hooke’s Law, how much does someone who compresses the scale 1> 8 in. weigh? How much work is done compressing the scale 1> 8 in.?

10 ft Ground level 0

Work

387

12 ft

y Δy

Work Done by a Variable Force 7. Lifting a rope A mountain climber is about to haul up a 50-m length of hanging rope. How much work will it take if the rope weighs 0.624 N> m?

20 y

8. Leaky sandbag A bag of sand originally weighing 144 lb was lifted at a constant rate. As it rose, sand also leaked out at a constant rate. The sand was half gone by the time the bag had been lifted to 18 ft. How much work was done lifting the sand this far? (Neglect the weight of the bag and lifting equipment.)

14. Emptying a cistern The rectangular cistern (storage tank for rainwater) shown has its top 10 ft below ground level. The cistern, currently full, is to be emptied for inspection by pumping its contents to ground level.

9. Lifting an elevator cable An electric elevator with a motor at the top has a multistrand cable weighing 4.5 lb> ft. When the car is at the first floor, 180 ft of cable are paid out, and effectively 0 ft are out when the car is at the top floor. How much work does the motor do just lifting the cable when it takes the car from the first floor to the top?

b. How long will it take a 1> 2-hp pump, rated at 275 ft-lb> sec, to pump the tank dry?

10. Force of attraction When a particle of mass m is at (x, 0), it is attracted toward the origin with a force whose magnitude is k>x 2. If the particle starts from rest at x = b and is acted on by no other forces, find the work done on it by the time it reaches x = a, 0 6 a 6 b. 11. Leaky bucket Assume the bucket in Example 4 is leaking. It starts with 2 gal of water (16 lb) and leaks at a constant rate. It finishes draining just as it reaches the top. How much work was spent lifting the water alone? (Hint: Do not include the rope and bucket, and find the proportion of water left at elevation x ft.) 12. (Continuation of Exercise 11.) The workers in Example 4 and Exercise 11 changed to a larger bucket that held 5 gal (40 lb) of water, but the new bucket had an even larger leak so that it, too, was empty by the time it reached the top. Assuming that the water leaked out at a steady rate, how much work was done lifting the water alone? (Do not include the rope and bucket.)

Pumping Liquids from Containers 13. Pumping water The rectangular tank shown here, with its top at ground level, is used to catch runoff water. Assume that the water weighs 62.4 lb>ft3 . a. How much work does it take to empty the tank by pumping the water back to ground level once the tank is full?

b. If the water is pumped to ground level with a (5> 11)horsepower (hp) motor (work output 250 ft-lb> sec), how long will it take to empty the full tank (to the nearest minute)? c. Show that the pump in part (b) will lower the water level 10 ft (halfway) during the first 25 min of pumping. d. The weight of water What are the answers to parts (a) and (b) in a location where water weighs 62.26 lb>ft3 ? 62.59 lb>ft3 ?

a. How much work will it take to empty the cistern?

c. How long will it take the pump in part (b) to empty the tank halfway? (It will be less than half the time required to empty the tank completely.) d. The weight of water What are the answers to parts (a) through (c) in a location where water weighs 62.26 lb>ft3 ? 62.59 lb>ft3 ? Ground level

0 10

10 ft 20 y

20 ft

12 ft

15. Pumping oil How much work would it take to pump oil from the tank in Example 5 to the level of the top of the tank if the tank were completely full? 16. Pumping a half-full tank Suppose that, instead of being full, the tank in Example 5 is only half full. How much work does it take to pump the remaining oil to a level 4 ft above the top of the tank? 17. Emptying a tank A vertical right-circular cylindrical tank measures 30 ft high and 20 ft in diameter. It is full of kerosene weighing 51.2 lb>ft3 . How much work does it take to pump the kerosene to the level of the top of the tank? 18. a. Pumping milk Suppose that the conical container in Example 5 contains milk (weighing 64.5 lb>ft3) instead of olive oil. How much work will it take to pump the contents to the rim? b. Pumping oil How much work will it take to pump the oil in Example 5 to a level 3 ft above the cone’s rim? 19. The graph of y = x 2 on 0 … x … 2 is revolved about the y-axis to form a tank that is then filled with salt water from the Dead Sea (weighing approximately 73 lb/ft3). How much work does it take to pump all of the water to the top of the tank? 20. A right-circular cylindrical tank of height 10 ft and radius 5 ft is lying horizontally and is full of diesel fuel weighing 53 lb/ft3. How much work is required to pump all of the fuel to a point 15 ft above the top of the tank?

388

Chapter 6: Applications of Definite Integrals

21. Emptying a water reservoir We model pumping from spherical containers the way we do from other containers, with the axis of integration along the vertical axis of the sphere. Use the figure here to find how much work it takes to empty a full hemispherical water reservoir of radius 5 m by pumping the water to a height of 4 m above the top of the reservoir. Water weighs 9800 N>m3. y

5

0

26. Golf A 1.6-oz golf ball is driven off the tee at a speed of 280 ft> sec (about 191 mph). How many foot-pounds of work are done on the ball getting it into the air?

27. On June 11, 2004, in a tennis match between Andy Roddick and Paradorn Srichaphan at the Stella Artois tournament in London, England, Roddick hit a serve measured at 153 mi> h. How much work was required by Andy to serve a 2-oz tennis ball at that speed? 28. Softball How much work has to be performed on a 6.5-oz softball to pitch it 132 ft> sec (90 mph)?

4m

y x y –y

Δy

25

25. Baseball How many foot-pounds of work does it take to throw a baseball 90 mph? A baseball weighs 5 oz, or 0.3125 lb.

y2

22. You are in charge of the evacuation and repair of the storage tank shown here. The tank is a hemisphere of radius 10 ft and is full of benzene weighing 56 lb>ft3 . A firm you contacted says it can empty the tank for 1> 2¢ per foot-pound of work. Find the work required to empty the tank by pumping the benzene to an outlet 2 ft above the top of the tank. If you have $5000 budgeted for the job, can you afford to hire the firm?

29. Drinking a milkshake The truncated conical container shown here is full of strawberry milkshake that weighs 4>9 oz>in3. As you can see, the container is 7 in. deep, 2.5 in. across at the base, and 3.5 in. across at the top (a standard size at Brigham’s in Boston). The straw sticks up an inch above the top. About how much work does it take to suck up the milkshake through the straw (neglecting friction)? Answer in inch-ounces. y 8 7

8y

y

(1.75, 7) y 17.5 14

y Δy

Outlet pipe

x 2 y 2 z 2 100 10

y 14x 17.5

2 ft

0

0 10

x

z

Work and Kinetic Energy 23. Kinetic energy If a variable force of magnitude F(x) moves a body of mass m along the x-axis from x1 to x2 , the body’s velocity y can be written as dx> dt (where t represents time). Use Newton’s second law of motion F = msdy>dtd and the Chain Rule dy dy dy dx = = y dt dx dt dx to show that the net work done by the force in moving the body from x1 to x2 is

x 1.25 Dimensions in inches

30. Water tower Your town has decided to drill a well to increase its water supply. As the town engineer, you have determined that a water tower will be necessary to provide the pressure needed for distribution, and you have designed the system shown here. The water is to be pumped from a 300-ft well through a vertical 4-in. pipe into the base of a cylindrical tank 20 ft in diameter and 25 ft high. The base of the tank will be 60 ft above ground. The pump is a 3-hp pump, rated at 1650 ft # lb>sec . To the nearest hour, how long will it take to fill the tank the first time? (Include the time it takes to fill the pipe.) Assume that water weighs 62.4 lb>ft3 . 10 ft

x2

1 1 W = Fsxd dx = my2 2 - my1 2, 2 2 Lx1 where y1 and y2 are the body’s velocities at x1 and x2 . In physics, the expression s1>2dmy2 is called the kinetic energy of a body of mass m moving with velocity y. Therefore, the work done by the force equals the change in the body’s kinetic energy, and we can find the work by calculating this change. In Exercises 24–28, use the result of Exercise 23.

24. Tennis A 2-oz tennis ball was served at 160 ft> sec (about 109 mph). How much work was done on the ball to make it go this fast? (To find the ball’s mass from its weight, express the weight in pounds and divide by 32 ft>sec2 , the acceleration of gravity.)

25 ft

Ground

60 ft

4 in. 300 ft Water surface

Submersible pump NOT TO SCALE

6.6

32. Forcing electrons together Two electrons r meters apart repel each other with a force of

mMG . r2

Here, M = 5.975 * 10 24 kg is Earth’s mass, G = 6.6720 * 10 -11 N # m2 kg-2 is the universal gravitational constant, and r is measured in meters. The work it takes to lift a 1000-kg satellite from Earth’s surface to a circular orbit 35,780 km above Earth’s center is therefore given by the integral 35,780,000

Work =

6.6

L6,370,000

389

Evaluate the integral. The lower limit of integration is Earth’s radius in meters at the launch site. (This calculation does not take into account energy spent lifting the launch vehicle or energy spent bringing the satellite to orbit velocity.)

31. Putting a satellite in orbit The strength of Earth’s gravitational field varies with the distance r from Earth’s center, and the magnitude of the gravitational force experienced by a satellite of mass m during and after launch is Fsrd =

Moments and Centers of Mass

F =

23 * 10 -29 newtons. r2

a. Suppose one electron is held fixed at the point (1, 0) on the x-axis (units in meters). How much work does it take to move a second electron along the x-axis from the point s -1, 0d to the origin? b. Suppose an electron is held fixed at each of the points s -1, 0d and (1, 0). How much work does it take to move a third electron along the x-axis from (5, 0) to (3, 0)?

1000MG dr joules. r2

Moments and Centers of Mass Many structures and mechanical systems behave as if their masses were concentrated at a single point, called the center of mass (Figure 6.40). It is important to know how to locate this point, and doing so is basically a mathematical enterprise. Here we consider masses distributed along a line or region in the plane. Masses distributed across a region or curve in three-dimensional space are treated in Chapters 14 and 15.

Masses Along a Line We develop our mathematical model in stages. The first stage is to imagine masses m1, m2 , and m3 on a rigid x-axis supported by a fulcrum at the origin. x1 m1

x2

x3

m2

m3

0

x

Fulcrum at origin

The resulting system might balance, or it might not, depending on how large the masses are and how they are arranged along the x-axis. Each mass mk exerts a downward force mk g (the weight of mk ) equal to the magnitude of the mass times the acceleration due to gravity. Note that gravitational acceleration is downward, hence negative. Each of these forces has a tendency to turn the x-axis about the origin, the way a child turns a seesaw. This turning effect, called a torque, is measured by multiplying the force mk g by the signed distance xk from the point of application to the origin. By convention, a positive torque induces a counterclockwise turn. Masses to the left of the origin exert positive (counterclockwise) torque. Masses to the right of the origin exert negative (clockwise) torque. The sum of the torques measures the tendency of a system to rotate about the origin. This sum is called the system torque. System torque = m1 gx1 + m2 gx2 + m3 gx3 The system will balance if and only if its torque is zero.

(1)

390

Chapter 6: Applications of Definite Integrals

If we factor out the g in Equation (1), we see that the system torque is g "

#

sm 1 x1 + m2 x2 + m3 x3 d. (++++)++++* a feature of the system

a feature of the environment

Thus, the torque is the product of the gravitational acceleration g, which is a feature of the environment in which the system happens to reside, and the number sm1 x1 + m2 x2 + m3 x3 d, which is a feature of the system itself, a constant that stays the same no matter where the system is placed. The number sm1 x1 + m2 x2 + m3 x3 d is called the moment of the system about the origin. It is the sum of the moments m1 x1, m2 x2, m3 x3 of the individual masses. M0 = Moment of system about origin = a mk xk (We shift to sigma notation here to allow for sums with more terms.) We usually want to know where to place the fulcrum to make the system balance, that is, at what point x to place it to make the torques add to zero. x1

x2

0

x

x3

m2

m1

m3

x

Special location for balance

The torque of each mass about the fulcrum in this special location is signed distance downward Torque of mk about x = a ba b of mk from x force = sxk - xdmk g. When we write the equation that says that the sum of these torques is zero, we get an equation we can solve for x: a sxk - xdmk g = 0 x =

a mk xk . a mk

Sum of the torques equals zero.

Solved for x

This last equation tells us to find x by dividing the system’s moment about the origin by the system’s total mass: FIGURE 6.40 A wrench gliding on ice turning about its center of mass as the center glides in a vertical line.

x =

system moment about origin a mk xk = . system mass a mk

(2)

The point x is called the system’s center of mass.

Masses Distributed over a Plane Region y yk

xk

Suppose that we have a finite collection of masses located in the plane, with mass mk at the point sxk , yk d (see Figure 6.41). The mass of the system is

mk (xk, yk )

0

yk xk

System mass:

M = a mk .

Each mass mk has a moment about each axis. Its moment about the x-axis is mk yk , and its moment about the y-axis is mk xk . The moments of the entire system about the two axes are x

FIGURE 6.41 Each mass mk has a moment about each axis.

Moment about x-axis:

Mx = a mk yk,

Moment about y-axis:

My = a mk xk.

6.6

Moments and Centers of Mass

391

The x-coordinate of the system’s center of mass is defined to be Ba x la nc x el in e

y

x =

My a mk xk = . M a mk

(3)

y c.m. yy Balan ce line

0

With this choice of x, as in the one-dimensional case, the system balances about the line x = x (Figure 6.42). The y-coordinate of the system’s center of mass is defined to be

x

y =

x

FIGURE 6.42 A two-dimensional array of masses balances on its center of mass.

Mx a mk yk = . M a mk

(4)

With this choice of y, the system balances about the line y = y as well. The torques exerted by the masses about the line y = y cancel out. Thus, as far as balance is concerned, the system behaves as if all its mass were at the single point sx, yd. We call this point the system’s center of mass.

Thin, Flat Plates

y Strip of mass m

~y

Strip c.m. ~ x

~ ~y) (x,

x =

~ y

0

In many applications, we need to find the center of mass of a thin, flat plate: a disk of aluminum, say, or a triangular sheet of steel. In such cases, we assume the distribution of mass to be continuous, and the formulas we use to calculate x and y contain integrals instead of finite sums. The integrals arise in the following way. Imagine that the plate occupying a region in the xy-plane is cut into thin strips parallel to one of the axes (in Figure 6.43, the y-axis). The center of mass of a typical strip is ' ' ' ' s x , y d. We treat the strip’s mass ¢m as if it were concentrated at s x , y d. The moment of the ' ' strip about the y-axis is then x ¢m. The moment of the strip about the x-axis is y ¢m. Equations (3) and (4) then become

~ x

x

FIGURE 6.43 A plate cut into thin strips parallel to the y-axis. The moment exerted by a typical strip about each axis is the moment its mass ¢m would exert if concentrated at the strip’s center of mass ' ' s x , y d.

' My a x ¢m = , M a ¢m

y =

' Mx a y ¢m = . M a ¢m

The sums are Riemann sums for integrals and approach these integrals as limiting values as the strips into which the plate is cut become narrower and narrower. We write these integrals symbolically as ' 1 x dm x = 1 dm

and

' 1 y dm y = . 1 dm

Moments, Mass, and Center of Mass of a Thin Plate Covering a Region in the xy-Plane ' Moment about the x-axis: Mx = y dm L Moment about the y-axis: Mass: Center of mass:

My = M =

L

' x dm

L My x = , M

(5)

dm y =

Mx M

392

Chapter 6: Applications of Definite Integrals

Density A material’s density is its mass per unit area. For wires, rods, and narrow strips, we use mass per unit length.

y (cm) (1, 2)

2

The differential dm is the mass of the strip. Assuming the density d of the plate to be a continuous function, the mass differential dm equals the product d dA (mass per unit area times area). Here dA represents the area of the strip. To evaluate the integrals in Equations (5), we picture the plate in the coordinate plane and sketch a strip of mass parallel to one of the coordinate axes. We then express the strip’s ' ' mass dm and the coordinates s x , y d of the strip’s center of mass in terms of x or y. Finally, ' ' we integrate y dm, x dm, and dm between limits of integration determined by the plate’s location in the plane.

EXAMPLE 1

The triangular plate shown in Figure 6.44 has a constant density of d = 3 g>cm2. Find

y 2x

(a) the plate’s moment My about the y-axis. (b) the plate’s mass M. (c) the x-coordinate of the plate’s center of mass (c.m.).

x1

Solution y0

0

x (cm)

1

FIGURE 6.44 The plate in Example 1.

y (1, 2)

2

Method 1: Vertical Strips (Figure 6.45)

(a) The moment My : The typical vertical strip has the following relevant data. ' ' center of mass (c.m.): s x , y d = sx, xd length: 2x width: dx area: dA = 2x dx mass: dm = d dA = 3 # 2x dx = 6x dx ' distance of c.m. from y-axis: x = x The moment of the strip about the y-axis is ' x dm = x # 6x dx = 6x 2 dx.

y 2x (x, 2x) Strip c.m. is halfway. ~ ~y) (x, x) (x,

x

The moment of the plate about the y-axis is therefore

2x

0

My = x

1

dx

L

' x dm =

L

dm =

y (cm)

Strip c.m. is halfway. ⎛ y 2 , y⎛ ~~ (x, y) ⎝ ⎝ 4

y

dy 1

y 2

(1, y)

2 0

1

L0

1

6x dx = 3x 2 d = 3 g. 0

My 2 g # cm 2 = = cm. M 3g 3

By a similar computation, we could find Mx and y = Mx>M.

(1, 2)

y 2

0

(c) The x-coordinate of the plate’s center of mass: x =

1

1

1

FIGURE 6.45 Modeling the plate in Example 1 with vertical strips.

y x 2 ⎛ y , y⎛ ⎝2 ⎝

L0

6x 2 dx = 2x 3 d = 2 g # cm.

(b) The plate’s mass: M =

Units in centimeters

2

1

x (cm)

FIGURE 6.46 Modeling the plate in Example 1 with horizontal strips.

Method 2: Horizontal Strips

(Figure 6.46)

(a) The moment My: The y-coordinate of the center of mass of a typical horizontal strip is y (see the figure), so ' y = y. The x-coordinate is the x-coordinate of the point halfway across the triangle. This makes it the average of y> 2 (the strip’s left-hand x-value) and 1 (the strip’s right-hand x-value): s y>2d + 1 y y + 2 ' 1 x = = + = . 2 4 2 4

6.6

Moments and Centers of Mass

393

We also have length: 1 -

y 2 - y = 2 2

width: dy area: dA = mass: distance of c.m. to y-axis:

2 - y dy 2

dm = d dA = 3 #

2 - y dy 2

y + 2 ' x = . 4

The moment of the strip about the y-axis is y + 2 ' # 3 # 2 - y dy = 3 s4 - y 2 d dy. x dm = 4 2 8 The moment of the plate about the y-axis is My =

L

' x dm =

y3 2 3 3 3 16 s4 - y 2 d dy = c4y d = a b = 2 g # cm. 8 3 0 8 3 L0 8 2

(b) The plate’s mass: y2 2 3 3 3 (2 - y) dy = c2y d = (4 - 2) = 3 g. 2 2 2 2 0 0 L 2

M =

L

dm =

(c) The x-coordinate of the plate’s center of mass:

My 2 g # cm 2 = = cm. M 3g 3

x =

By a similar computation, we could find Mx and y. If the distribution of mass in a thin, flat plate has an axis of symmetry, the center of mass will lie on this axis. If there are two axes of symmetry, the center of mass will lie at their intersection. These facts often help to simplify our work. y 4 y 4 x2 Center of mass 2 ⎛ 4x ⎛ ~~ (x, y) ⎝ x, 2 ⎝

4 x2

y 2 –2

0

x dx

2

x

FIGURE 6.47 Modeling the plate in Example 2 with vertical strips.

EXAMPLE 2 Find the center of mass of a thin plate covering the region bounded above by the parabola y = 4 - x 2 and below by the x-axis (Figure 6.47). Assume the density of the plate at the point (x, y) is d = 2x 2, which is twice the square of the distance from the point to the y-axis. The mass distribution is symmetric about the y-axis, so x = 0. We model the distribution of mass with vertical strips since the density is given as a function of the varible x. The typical vertical strip (see Figure 6.47) has the following relevant data.

Solution

4 - x2 ' ' b center of mass (c.m.): s x , y d = ax, 2 length: width: area: mass: distance from c.m. to x-axis:

4 - x2 dx dA = s4 - x 2 d dx dm = d dA = ds4 - x 2 d dx 4 - x2 ' y = 2

The moment of the strip about the x-axis is 4 - x2 ' y dm = 2

#

ds4 - x 2 d dx =

d s4 - x 2 d2 dx. 2

394

Chapter 6: Applications of Definite Integrals

The moment of the plate about the x-axis is Mx =

L

' y dm =

2

2

d s4 - x 2 d2 dx = x 2s4 - x 2 d2 dx 2 -2 -2 L L

2

=

L-2

s16x 2 - 8x 4 + x 6 d dx =

2048 105

2

M =

dm =

L

2

ds4 - x 2 d dx =

L-2

2

=

L-2

s8x 2 - 2x 4 d dx =

L-2

2x 2s4 - x 2 d dx

256 . 15

Therefore, Mx 2048 = M 105

y =

#

15 8 = . 7 256

The plate’s center of mass is 8 sx, yd = a0, b. 7

Plates Bounded by Two Curves

y y f (x)

Suppose a plate covers a region that lies between two curves y = g(x) and y = ƒ(x), where ƒ(x) Ú g(x) and a … x … b. The typical vertical strip (see Figure 6.48) has

~~ (x, y) y g(x) 0

a

dx

b

x

FIGURE 6.48 Modeling the plate bounded by two curves with vertical strips. The strip 1 ' c.m. is halfway, so y = [ƒ(x) + g(x)]. 2

center of mass (c.m.): length: width: area: mass:

' ' (x , y ) = (x, 12 [ƒ(x) + g(x)]) ƒ(x) - g(x) dx dA = [ƒ(x) - g(x)] dx dm = d dA = d[ƒ(x) - g(x)] dx.

The moment of the plate about the y-axis is b

My =

L

x dm =

La

xd[ƒ(x) - g(x)] dx,

and the moment about the x-axis is 1 [ƒ(x) + g(x)] # d[ƒ(x) - g(x)] dx La 2 b

Mx =

L

y dm = b

=

d 2 [ƒ (x) - g 2(x)] dx. La 2

These moments give the formulas

b

x =

1 dx [ƒ(x) - g(x)] dx M La

y =

d 2 1 [ƒ (x) - g 2(x)] dx M La 2

(6)

b

(7)

6.6

Moments and Centers of Mass

395

EXAMPLE 3 Find the center of mass for the thin plate bounded by the curves g(x) = x>2 and ƒ(x) = 2x, 0 … x … 1, (Figure 6.49) using Equations (6) and (7) with the density function d(x) = x 2. Solution We first compute the mass of the plate, where dm = d[ƒ(x) - g(x)] dx: 1

M =

L0

x 2 a2x -

1

1

9 x x3 2 1 b dx = b dx = c x 7>2 - x 4 d = . ax5>2 7 2 2 8 56 0 L0

Then from Equations (6) and (7) we get 1

x =

y

56 x x 2 # x a2x - b dx 9 L0 2 1

1 f (x) 5 x

=

56 x4 ax 7>2 b dx 9 L0 2

=

56 2 9>2 308 1 5 c x x d = , 9 9 10 405 0

1

c.m.

1

g(x) 5 x 2

and x

1

0

y =

56 x2 x2 ax b dx 9 L0 2 4 1

FIGURE 6.49 The region in Example 3.

=

28 x4 ax 3 b dx 9 L0 4

=

252 28 1 4 1 5 c x x d = . 9 4 20 405 0

1

The center of mass is shown in Figure 6.49.

y A typical small segment of wire has dm ds a du.

y a2 x 2

Centroids

~~ (x, y) (a cos u, a sin u)

du

u –a

0

a

x

(a)

The center of mass in Example 3 is not located at the geometric center of the region. This is due to the region’s non-uniform density. When the density function is constant, it cancels out of the numerator and denominator of the formulas for x and y. Thus, when the density is constant, the location of the center of mass is a feature of the geometry of the object and not of the material from which it is made. In such cases, engineers may call the center of mass the centroid of the shape, as in “Find the centroid of a triangle or a solid cone.” To do so, just set d equal to 1 and proceed to find x and y as before, by dividing moments by masses.

y

EXAMPLE 4 Find the center of mass (centroid) of a thin wire of constant density d shaped like a semicircle of radius a. a

–a

0

We model the wire with the semicircle y = 2a 2 - x 2 (Figure 6.50). The distribution of mass is symmetric about the y-axis, so x = 0. To find y, we imagine ' ' the wire divided into short subarc segments. If ( x , y ) is the center of mass of a subarc ' ' and u is the angle between the x-axis and the radial line joining the origin to ( x , y ), ' then y = a sin u is a function of the angle u measured in radians (see Figure 6.50a). ' ' The length ds of the subarc containing ( x , y ) subtends an angle of du radians, so ds = a du. Thus a typical subarc segment has these relevant data for calculating y: Solution

c.m. ⎛ 2 ⎞ ⎝0, a⎠

a

x

(b)

FIGURE 6.50 The semicircular wire in Example 4. (a) The dimensions and variables used in finding the center of mass. (b) The center of mass does not lie on the wire.

length: mass: distance of c.m. to x-axis:

ds = a du dm = d ds = da du ' y = a sin u.

Mass per unit length times length

396

Chapter 6: Applications of Definite Integrals

Hence,

p p ' # da 2 C -cos u D 0 10 a sin u da du 1 y dm 2 = p a. y = = = p dap 1 dm 10 da du The center of mass lies on the axis of symmetry at the point s0, 2a>pd, about two-thirds of the way up from the origin (Figure 6.50b). Notice how d cancels in the equation for y, so we could have set d = 1 everywhere and obtained the same value for y.

In Example 4 we found the center of mass of a thin wire lying along the graph of a differentiable function in the xy-plane. In Chapter 15 we will learn how to find the center of mass of wires lying along more general smooth curves in the plane (or in space).

Exercises 6.6 Thin Plates with Constant Density In Exercises 1–14, find the center of mass of a thin plate of constant density d covering the given region. 1. The region bounded by the parabola y = x 2 and the line y = 4 2. The region bounded by the parabola y = 25 - x 2 and the x-axis 3. The region bounded by the parabola y = x - x 2 and the line y = -x 4. The region enclosed by the parabolas y = x 2 - 3 and y = -2x 2 5. The region bounded by the y-axis and the curve x = y - y 3, 0 … y … 1

a. Find the volume of the solid. b. Find the center of mass of a thin plate covering the region if the plate’s density at the point (x, y) is dsxd = 1>x. c. Sketch the plate and show the center of mass in your sketch. 18. The region between the curve y = 2>x and the x-axis from x = 1 to x = 4 is revolved about the x-axis to generate a solid. a. Find the volume of the solid.

6. The region bounded by the parabola x = y 2 - y and the line y = x

b. Find the center of mass of a thin plate covering the region if the plate’s density at the point (x, y) is dsxd = 2x .

7. The region bounded by the x-axis and the curve y = cos x, -p>2 … x … p>2

c. Sketch the plate and show the center of mass in your sketch.

8. The region between the curve y = sec2 x, -p>4 … x … p>4 and the x-axis T

17. The region bounded by the curves y = ;4> 2x and the lines x = 1 and x = 4 is revolved about the y-axis to generate a solid.

9. The region between the curve y = 1>x and the x-axis from x = 1 to x = 2. Give the coordinates to two decimal places. 10. a. The region cut from the first quadrant by the circle x 2 + y 2 = 9 b. The region bounded by the x-axis and the semicircle y = 29 - x 2 Compare your answer in part (b) with the answer in part (a). 11. The region in the first and fourth quadrants enclosed by the curves y = 1>(1 + x 2) and y = -1>(1 + x 2) and by the lines x = 0 and x = 1 12. The region bounded by the parabolas y = 2x 2 - 4x and y = 2x - x 2 13. The region between the curve y = 1> 1x and the x-axis from x = 1 to x = 16 14. The region bounded above by the curve y = 1>x 3 , below by the curve y = -1>x 3, and on the left and right by the lines x = 1 and x = a 7 1 . Also, find lim a: q x . Thin Plates with Varying Density 15. Find the center of mass of a thin plate covering the region between the x-axis and the curve y = 2>x 2, 1 … x … 2, if the plate’s density at the point (x, y) is dsxd = x 2. 16. Find the center of mass of a thin plate covering the region bounded below by the parabola y = x 2 and above by the line y = x if the plate’s density at the point (x, y) is dsxd = 12x.

Centroids of Triangles 19. The centroid of a triangle lies at the intersection of the triangle’s medians You may recall that the point inside a triangle that lies one-third of the way from each side toward the opposite vertex is the point where the triangle’s three medians intersect. Show that the centroid lies at the intersection of the medians by showing that it too lies one-third of the way from each side toward the opposite vertex. To do so, take the following steps. i) Stand one side of the triangle on the x-axis as in part (b) of the accompanying figure. Express dm in terms of L and dy. ii) Use similar triangles to show that L = sb>hdsh - yd . Substitute this expression for L in your formula for dm. iii) Show that y = h>3. iv) Extend the argument to the other sides. y

h2y

dy h L

y

Centroid

x

0 b (a)

(b)

Chapter 6 Use the result in Exercise 19 to find the centroids of the triangles whose vertices appear in Exercises 20–24. Assume a, b 7 0 . 20. s -1, 0d , (1, 0), (0, 3) 21. (0, 0), (1, 0), (0, 1) 22. (0, 0), (a, 0), (0, a) 23. (0, 0), (a, 0), (0, b) 24. (0, 0), (a, 0), (a> 2, b) Thin Wires 25. Constant density Find the moment about the x-axis of a wire of constant density that lies along the curve y = 2x from x = 0 to x = 2. 26. Constant density Find the moment about the x-axis of a wire of constant density that lies along the curve y = x 3 from x = 0 to x = 1. 27. Variable density Suppose that the density of the wire in Example 4 is d = k sin u (k constant). Find the center of mass. 28. Variable density Suppose that the density of the wire in Example 4 is d = 1 + k ƒ cos u ƒ (k constant). Find the center of mass. Plates Bounded by Two Curves In Exercises 29–32, find the centroid of the thin plate bounded by the graphs of the given functions. Use Equations (6) and (7) with d = 1 and M = area of the region covered by the plate. 29. g(x) = x 2 and ƒ(x) = x + 6 30. g(x) = x 2 (x + 1), ƒ(x) = 2, and x = 0 31. g(x) = x 2(x - 1) and ƒ(x) = x 2 32. g(x) = 0, ƒ(x) = 2 + sin x, x = 0, and x = 2p (Hint:

L

397

Theory and Examples Verify the statements and formulas in Exercises 33 and 34. 33. The coordinates of the centroid of a differentiable plane curve are x =

1 x ds , length

y =

1 y ds . length

y

ds

x

y

x

0

34. Whatever the value of p 7 0 in the equation y = x 2>s4pd, the y-coordinate of the centroid of the parabolic segment shown here is y = s3>5da. y 2 y5 x 4p

a y53a 5

x sin x dx = sin x - x cos x + C.)

Chapter 6

Practice Exercises

0

x

Questions to Guide Your Review

1. How do you define and calculate the volumes of solids by the method of slicing? Give an example. 2. How are the disk and washer methods for calculating volumes derived from the method of slicing? Give examples of volume calculations by these methods.

3. Describe the method of cylindrical shells. Give an example. 4. How do you find the length of the graph of a smooth function over a closed interval? Give an example. What about functions that do not have continuous first derivatives?

5. How do you define and calculate the area of the surface swept out

6. How do you define and calculate the work done by a variable force directed along a portion of the x-axis? How do you calculate the work it takes to pump a liquid from a tank? Give examples. 7. How do you calculate the force exerted by a liquid against a portion of a flat vertical wall? Give an example. 8. What is a center of mass? a centroid? 9. How do you locate the center of mass of a thin flat plate of material? Give an example. 10. How do you locate the center of mass of a thin plate bounded by two curves y = ƒ(x) and y = g(x) over a … x … b?

by revolving the graph of a smooth function y = ƒsxd, a … x … b , about the x-axis? Give an example.

Chapter 6

Practice Exercises

Volumes Find the volumes of the solids in Exercises 1–16. 1. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 1 . The cross-sections perpendicular to the x-axis between these planes are circular disks whose diameters run from the parabola y = x 2 to the parabola y = 2x.

2. The base of the solid is the region in the first quadrant between the line y = x and the parabola y = 2 2x . The cross-sections of the solid perpendicular to the x-axis are equilateral triangles whose bases stretch from the line to the curve. 3. The solid lies between planes perpendicular to the x-axis at x = p>4 and x = 5p>4 . The cross-sections between these planes are circular

398

Chapter 6: Applications of Definite Integrals

disks whose diameters run from the curve y = 2 cos x to the curve y = 2 sin x. 4. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 6 . The cross-sections between these planes are squares whose bases run from the x-axis up to the curve x 1>2 + y 1>2 = 26 .

15. Volume of a solid sphere hole A round hole of radius 23 ft is bored through the center of a solid sphere of a radius 2 ft. Find the volume of material removed from the sphere. 16. Volume of a football The profile of a football resembles the ellipse shown here. Find the football’s volume to the nearest cubic inch. y

y

y2 4x 2 1 121 12

6 – 11 2

x1/2 y1/2 6

6

x

6. The base of the solid is the region bounded by the parabola y 2 = 4x and the line x = 1 in the xy-plane. Each cross-section perpendicular to the x-axis is an equilateral triangle with one edge in the plane. (The triangles all lie on the same side of the plane.) 7. Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve y = 3x 4, and the lines x = 1 and x = -1 about (a) the x-axis; (b) the y-axis; (c) the line x = 1 ; (d) the line y = 3. 8. Find the volume of the solid generated by revolving the “triangular” region bounded by the curve y = 4>x 3 and the lines x = 1 and y = 1>2 about (a) the x-axis; (b) the y-axis; (c) the line x = 2 ; (d) the line y = 4. 9. Find the volume of the solid generated by revolving the region bounded on the left by the parabola x = y 2 + 1 and on the right by the line x = 5 about (a) the x-axis; (b) the y-axis; (c) the line x = 5. 10. Find the volume of the solid generated by revolving the region bounded by the parabola y 2 = 4x and the line y = x about (a) the x-axis; (b) the y-axis; (c) the line x = 4 ; (d) the line y = 4. 11. Find the volume of the solid generated by revolving the “triangular” region bounded by the x-axis, the line x = p>3 , and the curve y = tan x in the first quadrant about the x-axis. 12. Find the volume of the solid generated by revolving the region bounded by the curve y = sin x and the lines x = 0, x = p , and y = 2 about the line y = 2.

x

Lengths of Curves Find the lengths of the curves in Exercises 17–20. 17. y = x 1>2 - s1>3dx 3>2,

5. The solid lies between planes perpendicular to the x-axis at x = 0 and x = 4 . The cross-sections of the solid perpendicular to the x-axis between these planes are circular disks whose diameters run from the curve x 2 = 4y to the curve y 2 = 4x.

11 2

0

18. x = y

2>3

,

1 … x … 4

1 … y … 8

19. y = x 2 - (ln x)>8,

1 … x … 2

20. x = sy >12d + s1>yd, 3

1 … y … 2

Areas of Surfaces of Revolution In Exercises 21–24, find the areas of the surfaces generated by revolving the curves about the given axes. 21. y = 22x + 1, 22. y = x >3, 3

0 … x … 1;

23. x = 24y - y 2, 24. x = 2y,

0 … x … 3;

x-axis

1 … y … 2;

2 … y … 6;

x-axis y-axis

y-axis

Work 25. Lifting equipment A rock climber is about to haul up 100 N (about 22.5 lb) of equipment that has been hanging beneath her on 40 m of rope that weighs 0.8 newton per meter. How much work will it take? (Hint: Solve for the rope and equipment separately, then add.) 26. Leaky tank truck You drove an 800-gal tank truck of water from the base of Mt. Washington to the summit and discovered on arrival that the tank was only half full. You started with a full tank, climbed at a steady rate, and accomplished the 4750-ft elevation change in 50 min. Assuming that the water leaked out at a steady rate, how much work was spent in carrying water to the top? Do not count the work done in getting yourself and the truck there. Water weighs 8 lb> U.S. gal. 27. Stretching a spring If a force of 20 lb is required to hold a spring 1 ft beyond its unstressed length, how much work does it take to stretch the spring this far? An additional foot?

13. Find the volume of the solid generated by revolving the region 2 bounded by the curve x = e y and the lines y = 0, x = 0, and y = 1 about the x-axis.

28. Garage door spring A force of 200 N will stretch a garage door spring 0.8 m beyond its unstressed length. How far will a 300-N force stretch the spring? How much work does it take to stretch the spring this far from its unstressed length?

14. Find the volume of the solid generated by revolving about the x-axis the region bounded by y = 2 tan x, y = 0, x = -p>4 , and x = p>4 . (The region lies in the first and third quadrants and resembles a skewed bowtie.)

29. Pumping a reservoir A reservoir shaped like a right-circular cone, point down, 20 ft across the top and 8 ft deep, is full of water. How much work does it take to pump the water to a level 6 ft above the top?

Chapter 6 30. Pumping a reservoir (Continuation of Exercise 29.) The reservoir is filled to a depth of 5 ft, and the water is to be pumped to the same level as the top. How much work does it take? 31. Pumping a conical tank A right-circular conical tank, point down, with top radius 5 ft and height 10 ft is filled with a liquid whose weight-density is 60 lb>ft3 . How much work does it take to pump the liquid to a point 2 ft above the tank? If the pump is driven by a motor rated at 275 ft-lb> sec (1> 2 hp), how long will it take to empty the tank? 32. Pumping a cylindrical tank A storage tank is a right-circular cylinder 20 ft long and 8 ft in diameter with its axis horizontal. If the tank is half full of olive oil weighing 57 lb>ft3 , find the work done in emptying it through a pipe that runs from the bottom of the tank to an outlet that is 6 ft above the top of the tank. Centers of Mass and Centroids 33. Find the centroid of a thin, flat plate covering the region enclosed by the parabolas y = 2x 2 and y = 3 - x 2.

Chapter 6

Additional and Advanced Exercises

399

34. Find the centroid of a thin, flat plate covering the region enclosed by the x-axis, the lines x = 2 and x = -2 , and the parabola y = x 2. 35. Find the centroid of a thin, flat plate covering the “triangular” region in the first quadrant bounded by the y-axis, the parabola y = x 2>4 , and the line y = 4. 36. Find the centroid of a thin, flat plate covering the region enclosed by the parabola y 2 = x and the line x = 2y. 37. Find the center of mass of a thin, flat plate covering the region enclosed by the parabola y 2 = x and the line x = 2y if the density function is dsyd = 1 + y . (Use horizontal strips.) 38. a. Find the center of mass of a thin plate of constant density covering the region between the curve y = 3>x 3>2 and the x-axis from x = 1 to x = 9. b. Find the plate’s center of mass if, instead of being constant, the density is dsxd = x . (Use vertical strips.)

Additional and Advanced Exercises

Volume and Length 1. A solid is generated by revolving about the x-axis the region bounded by the graph of the positive continuous function y = ƒsxd , the x-axis, and the fixed line x = a and the variable line x = b, b 7 a . Its volume, for all b, is b 2 - ab . Find ƒ(x).

45° wedge

2. A solid is generated by revolving about the x-axis the region bounded by the graph of the positive continuous function y = ƒsxd , the x-axis, and the lines x = 0 and x = a . Its volume, for all a 7 0 , is a 2 + a . Find ƒ(x). 3. Suppose that the increasing function ƒ(x) is smooth for x Ú 0 and that ƒs0d = a . Let s(x) denote the length of the graph of ƒ from (0, a) to sx, ƒsxdd, x 7 0 . Find ƒ(x) if ssxd = Cx for some constant C. What are the allowable values for C? 4. a. Show that for 0 6 a … p>2 , a

21 + cos2 u du 7 2a2 + sin2 a . L0 b. Generalize the result in part (a).

r5

1 2

Surface Area 7. At points on the curve y = 22x , line segments of length h = y are drawn perpendicular to the xy-plane. (See accompanying figure.) Find the area of the surface formed by these perpendiculars from (0, 0) to s3, 223d . y

5. Find the volume of the solid formed by revolving the region bounded by the graphs of y = x and y = x 2 about the line y = x. 6. Consider a right-circular cylinder of diameter 1. Form a wedge by making one slice parallel to the base of the cylinder completely through the cylinder, and another slice at an angle of 45° to the first slice and intersecting the first slice at the opposite edge of the cylinder (see accompanying diagram). Find the volume of the wedge.

23 2x

(3, 23)

y 2x 0 x

3

x

400

Chapter 6: Applications of Definite Integrals

8. At points on a circle of radius a, line segments are drawn perpendicular to the plane of the circle, the perpendicular at each point P being of length ks, where s is the length of the arc of the circle measured counterclockwise from (a, 0) to P and k is a positive constant, as shown here. Find the area of the surface formed by the perpendiculars along the arc beginning at (a, 0) and extending once around the circle.

12. If you haul a telephone pole on a two-wheeled carriage behind a truck, you want the wheels to be 3 ft or so behind the pole’s center of mass to provide an adequate “tongue” weight. The 40-ft wooden telephone poles used by Verizon have a 27-in. circumference at the top and a 43.5-in. circumference at the base. About how far from the top is the center of mass? 13. Suppose that a thin metal plate of area A and constant density d occupies a region R in the xy-plane, and let My be the plate’s moment about the y-axis. Show that the plate’s moment about the line x = b is

0 a a

Centers of Mass 11. Find the centroid of the region bounded below by the x-axis and above by the curve y = 1 - x n , n an even positive integer. What is the limiting position of the centroid as n : q ?

y

x

Work 9. A particle of mass m starts from rest at time t = 0 and is moved along the x-axis with constant acceleration a from x = 0 to x = h against a variable force of magnitude Fstd = t 2. Find the work done. 10. Work and kinetic energy Suppose a 1.6-oz golf ball is placed on a vertical spring with force constant k = 2 lb>in. The spring is compressed 6 in. and released. About how high does the ball go (measured from the spring’s rest position)?

a. My - bdA if the plate lies to the right of the line, and b. bdA - My if the plate lies to the left of the line. 14. Find the center of mass of a thin plate covering the region bounded by the curve y 2 = 4ax and the line x = a, a = positive constant , if the density at (x, y) is directly proportional to (a) x, (b) ƒ y ƒ . 15. a. Find the centroid of the region in the first quadrant bounded by two concentric circles and the coordinate axes, if the circles have radii a and b, 0 6 a 6 b , and their centers are at the origin. b. Find the limits of the coordinates of the centroid as a approaches b and discuss the meaning of the result. 16. A triangular corner is cut from a square 1 ft on a side. The area of the triangle removed is 36 in2 . If the centroid of the remaining region is 7 in. from one side of the original square, how far is it from the remaining sides?

7 INTEGRALS AND TRANSCENDENTAL FUNCTIONS OVERVIEW Our treatment of the logarithmic and exponential functions has been rather informal until now, appealing to intuition and graphs to describe what they mean and to explain some of their characteristics. In this chapter, we give a rigorous approach to the definitions and properties of these functions, and we study a wide range of applied problems in which they play a role. We also introduce the hyperbolic functions and their inverses, with their applications to integration and hanging cables.

7.1

The Logarithm Defined as an Integral In Chapter 1, we introduced the natural logarithm function ln x as the inverse of the exponential function e x. The function e x was chosen as that function in the family of general exponential functions a x, a 7 0, whose graph has slope 1 as it crosses the y-axis. The function a x was presented intuitively, however, based on its graph at rational values of x. In this section we recreate the theory of logarithmic and exponential functions from an entirely different point of view. Here we define these functions analytically and recover their behaviors. To begin, we use the Fundamental Theorem of Calculus to define the natural logarithm function ln x as an integral. We quickly develop its properties, including the algebraic, geometric, and analytic properties as seen before. Next we introduce the function e x as the inverse function of ln x, and establish its previously seen properties. Defining ln x as an integral and e x as its inverse is an indirect approach. While it may at first seem strange, it gives an elegant and powerful way to obtain the key properties of logarithmic and exponential functions.

Definition of the Natural Logarithm Function The natural logarithm of a positive number x, written as ln x, is the value of an integral.

DEFINITION

The natural logarithm is the function given by x

ln x =

1 dt, L1 t

x 7 0.

From the Fundamental Theorem of Calculus, ln x is a continuous function. Geometrically, if x 7 1, then ln x is the area under the curve y = 1>t from t = 1 to t = x (Figure 7.1). For 0 6 x 6 1, ln x gives the negative of the area under the curve from x to 1.

401

402

Chapter 7: Integrals and Transcendental Functions

The function is not defined for x … 0. From the Zero Width Interval Rule for definite integrals, we also have 1

ln 1 =

1 dt = 0. L1 t

y

x

If 0 x 1, then ln x

L 1

1

1 dt t

x L

1 dt t

gives the negative of this area. x

If x 1, then ln x

1 L

gives this area.

1 dt t y ln x

1 y 1x 0

x

1

x

x 1

If x 1, then ln x

L 1

1 dt 0. t

y ln x

FIGURE 7.1 The graph of y = ln x and its relation to the function y = 1>x, x 7 0 . The graph of the logarithm rises above the x-axis as x moves from 1 to the right, and it falls below the axis as x moves from 1 to the left.

Notice that we show the graph of y = 1>x in Figure 7.1 but use y = 1>t in the integral. Using x for everything would have us writing x

TABLE 7.1 Typical 2-place

ln x =

values of ln x x

ln x

0 0.05 0.5 1 2 3 4 10

undefined -3.00 -0.69 0 0.69 1.10 1.39 2.30

L1

1 x dx,

with x meaning two different things. So we change the variable of integration to t. By using rectangles to obtain finite approximations of the area under the graph of y = 1>t and over the interval between t = 1 and t = x, as in Section 5.1, we can approximate the values of the function ln x. Several values are given in Table 7.1. There is an important number between x = 2 and x = 3 whose natural logarithm equals 1. This number, which we now define, exists because ln x is a continuous function and therefore satisfies the Intermediate Value Theorem on [2, 3].

DEFINITION The number e is that number in the domain of the natural logarithm satisfying e

ln (e) =

1 dt = 1. L1 t

Interpreted geometrically, the number e corresponds to the point on the x-axis for which the area under the graph of y = 1>t and above the interval [1, e] equals the area of the unit square. That is, the area of the region shaded blue in Figure 7.1 is 1 sq unit when x = e. We will see further on that this is the same number e L 2.718281828 we have encountered before.

7.1

The Logarithm Defined as an Integral

403

The Derivative of y ln x By the first part of the Fundamental Theorem of Calculus (Section 5.4), x

d d 1 1 ln x = dt = x . dx dx L1 t For every positive value of x, we have d 1 ln x = x . dx

(1)

Therefore, the function y = ln x is a solution to the initial value problem dy>dx = 1>x, x 7 0, with y s1d = 0. Notice that the derivative is always positive. If u is a differentiable function of x whose values are positive, so that ln u is defined, then applying the Chain Rule we obtain

d 1 du ln u = , u dx dx

u 7 0.

(2)

As established in Example 3(c) of Section 3.8, we also have

d 1 ln ƒ x ƒ = x , dx

x Z 0.

(3)

y y 5 1x

Moreover, if b is any constant with bx 7 0

1

d 1 # d 1 1 ln bx = sbxd = sbd = x . dx bx dx bx

1 2

1 (a)

0

x

2

The derivative dsln xd>dx = 1>x is positive for x 7 0, so ln x is an increasing function of x. The second derivative, -1>x 2 , is negative, so the graph of ln x is concave down. (See Figure 7.2.) The function ln x has the following familiar algebraic properties, which we stated in Section 1.6. In Section 4.2 we showed these properties are a consequence of Corollary 2 of the Mean Value Theorem.

y y 5 ln x

0

The Graph and Range of ln x

x

(1, 0)

(b)

FIGURE 7.2 (a) The rectangle of height y = 1>2 fits beneath the graph of y = 1>x for the interval 1 … x … 2 . (b) The graph of the natural logarithm.

1. ln bx = ln b + ln x 1 3. ln x = -ln x

b 2. ln x = ln b - ln x 4. ln xr = r ln x, r rational

We can estimate the value of ln 2 by considering the area under the graph of y = 1>2 and above the interval [1, 2]. In Figure 7.2(a) a rectangle of height 1> 2 over the interval [1, 2]

404

Chapter 7: Integrals and Transcendental Functions

fits under the graph. Therefore, the area under the graph, which is ln 2, is greater than the area, 1> 2, of the rectangle. So ln 2 7 1>2. Knowing this we have n 1 ln 2n = n ln 2 7 n a b = . 2 2 This result shows that ln (2n) : q as n : q . Since ln x is an increasing function, we get that lim ln x = q . x: q

We also have lim ln x = lim ln t -1 = lim (-ln t) = - q . t: q

x:0 +

t: q

x = 1>t = t -1

We defined ln x for x 7 0, so the domain of ln x is the set of positive real numbers. The above discussion and the Intermediate Value Theorem show that its range is the entire real line, giving the graph of y = ln x shown in Figure 7.2(b).

The Integral 1 s1/ud du Equation (3) leads to the following integral formula.

If u is a differentiable function that is never zero,

L

1 u du = ln ƒ u ƒ + C.

(4)

Equation (4) applies anywhere on the domain of 1>u, the points where u Z 0. It says that integrals of a certain form lead to logarithms. If u = ƒsxd, then du = ƒ¿sxd dx and ƒ¿sxd dx = ln ƒ ƒsxd ƒ + C L ƒsxd whenever ƒ(x) is a differentiable function that is never zero.

EXAMPLE 1

Here we recognize an integral of the form p>2

L-p>2

5

du u.

u = 3 + 2 sin u, du = 2 cos u du, us -p>2d = 1, usp>2d = 5

4 cos u 2 du = u du 3 + 2 sin u L1 = 2 ln ƒ u ƒ d

L

5 1

= 2 ln ƒ 5 ƒ - 2 ln ƒ 1 ƒ = 2 ln 5 Note that u = 3 + 2 sin u is always positive on [-p>2, p>2], so Equation (4) applies.

The Integrals of tan x, cot x, sec x, and csc x Equation (4) tells us how to integrate these trigonometric functions. L

tan x dx =

sin x cos x dx =

-du u

L L = -ln ƒ u ƒ + C = -ln ƒ cos x ƒ + C 1 = ln + C = ln ƒ sec x ƒ + C ƒ cos x ƒ

u = cos x 7 0 on s -p>2, p>2d, du = -sin x dx

Reciprocal Rule

7.1

The Logarithm Defined as an Integral

405

For the cotangent, L

cot x dx =

u = sin x, du = cos x dx

cos x dx du = u sin x L L

= ln ƒ u ƒ + C = ln ƒ sin x ƒ + C = -ln ƒ csc x ƒ + C. To integrate sec x, we multiply and divide by (sec x + tan x). L =

sec x dx =

L

L

sec x

(sec x + tan x) sec2 x + sec x tan x dx = dx sec x + tan x (sec x + tan x) L u = sec x + tan x, du = (sec x tan x + sec2 x) dx

du u = ln ƒ u ƒ + C = ln ƒ sec x + tan x ƒ + C

For csc x, we multiply and divide by (csc x + cot x). L =

csc x dx =

L

L

csc x

(csc x + cot x) csc2 x + csc x cot x dx = dx csc x + cot x (csc x + cot x) L

-du u = -ln ƒ u ƒ + C = -ln ƒ csc x + cot x ƒ + C

u = csc x + cot x, du = (-csc x cot x - csc2 x) dx

Integrals of the tangent, cotangent, secant, and cosecant functions L L

tan u du = ln ƒ sec u ƒ + C

L

cot u du = ln ƒ sin u ƒ + C

L

sec u du = ln ƒ sec u + tan u ƒ + C csc u du = -ln ƒ csc u + cot u ƒ + C

The Inverse of ln x and the Number e The function ln x, being an increasing function of x with domain s0, q d and range s - q , q d, has an inverse ln-1 x with domain s - q , q d and range s0, q d. The graph of ln-1 x is the graph of ln x reflected across the line y = x. As you can see in Figure 7.3,

y 8

y ln–1x or x ln y

7

lim ln-1 x = q

x: q

6 5 4 e

(1, e)

2

y ln x

1 –2

–1

0

1

2

e

4

FIGURE 7.3 The graphs of y = ln x and y = ln-1 x = exp x . The number e is ln-1 1 = exp s1d .

x

and

lim ln-1 x = 0.

x: - q

The function ln-1 x is also denoted by exp x. We now show that ln-1 x = exp x is an exponential function with base e. The number e was defined to satisfy the equation ln (e) = 1, so e = exp (1). We can raise the number e to a rational power r using algebra: 1 3 2 e 2 = e # e, e -2 = 2 , e 1>2 = 2e, e 2>3 = 2 e , e and so on. Since e is positive, e r is positive too. Thus, e r has a logarithm. When we take the logarithm, we find that for r rational ln e r = r ln e = r # 1 = r. Then applying the function ln-1 to both sides of the equation ln e r = r, we find that e r = exp r for r rational. exp is ln-1. (5) We have not yet found a way to give an exact meaning to e x for x irrational. But ln-1 x has meaning for any x, rational or irrational. So Equation (5) provides a way to extend the definition of e x to irrational values of x. The function exp x is defined for all x, so we use it to assign a value to e x at every point.

406

Chapter 7: Integrals and Transcendental Functions

Typical values of ex x

e x (rounded)

-1 0 1 2 10 100

0.37 1 2.72 7.39 22026 2.6881 * 1043

DEFINITION For every real number x, we define the natural exponential function to be e x = exp x.

For the first time we have a precise meaning for a number raised to an irrational power. Usually the exponential function is denoted by e x rather than exp x. Since ln x and e x are inverses of one another, we have

Inverse Equations for e x and ln x e ln x = x

sall x 7 0d

ln se x d = x

sall xd

The Derivative and Integral of e x Transcendental Numbers and Transcendental Functions Numbers that are solutions of polynomial equations with rational coefficients are called algebraic: -2 is algebraic because it satisfies the equation x + 2 = 0, and 23 is algebraic because it satisfies the equation x 2 - 3 = 0. Numbers such as e and p that are not algebraic are called transcendental. We call a function y = ƒsxd algebraic if it satisfies an equation of the form

The exponential function is differentiable because it is the inverse of a differentiable function whose derivative is never zero. We calculate its derivative using Theorem 3 of Section 3.8 and our knowledge of the derivative of ln x. Let ƒsxd = ln x Then, dy d -1 d x se d = ln x = dx dx dx

Pn y n + Á + P1 y + P0 = 0 in which the P’s are polynomials in x with rational coefficients. The function y = 1> 2x + 1 is algebraic because it satisfies the equation sx + 1dy 2 - 1 = 0. Here the polynomials are P2 = x + 1, P1 = 0, and P0 = -1. Functions that are not algebraic are called transcendental.

y = e x = ln-1 x = ƒ -1sxd.

and

=

d -1 ƒ sxd dx

=

1 ƒ¿sƒ -1sxdd

Theorem 3, Section 3.8

=

1 ƒ¿se x d

ƒ -1sxd = e x

=

1 1 a xb e

1 ƒ¿szd = z with z = e x

= ex. That is, for y = e x, we find that dy>dx = e x so the natural exponential function e x is its own derivative, just as we claimed in Section 3.3. We will see in the next section that the only functions that behave this way are constant multiples of e x. The Chain Rule extends the derivative result in the usual way to a more general form.

If u is any differentiable function of x, then d u du e = eu . dx dx

(6)

Since e x 7 0, its derivative is also everywhere positive, so it is an increasing and continuous function for all x, having limits lim e x = 0

x: - q

and

lim e x = q .

x: q

7.1

The Logarithm Defined as an Integral

407

It follows that the x-axis (y = 0) is a horizontal asymptote of the graph y = e x (see Figure 7.3). Equation (6) also tells us the antiderivative of eu.

L

eu du = eu + C

If ƒ(x) = e x, then from Equation (6), ƒ¿(0) = e 0 = 1. That is, the exponential function e x has slope 1 as it crosses the y-axis at x = 0. This agrees with our assertion for the natural exponential in Section 3.3.

Laws of Exponents Even though e x is defined in a seemingly roundabout way as ln-1 x, it obeys the familiar laws of exponents from algebra. Theorem 1 shows us that these laws are consequences of the definitions of ln x and e x . We proved the laws in Section 4.2, and they are still valid because of the inverse relationship between ln x and e x .

THEOREM 1—Laws of Exponents for e x For all numbers x, x1, and x2 , the natural exponential e x obeys the following laws: 1 1. e x1 # e x2 = e x1 + x2 2. e -x = x e e x1 3. x2 = e x1 - x2 4. se x1 dx2 = e x1x2 = (e x2) x1 e

The General Exponential Function a x Since a = e ln a for any positive number a, we can think of a x as se ln a dx = e x ln a. We therefore make the following definition, consistent with what we stated in Section 1.6.

DEFINITION

For any numbers a 7 0 and x, the exponential function with base

a is given by a x = e x ln a. The General Power Function xr is the function e r ln x

#

When a = e, the definition gives a x = e x ln a = e x ln e = e x 1 = e x. Similarly, the power function f (x) = xr is defined to be xr = er ln x for any real number r, rational or irrational. Theorem 1 is also valid for a x, the exponential function with base a. For example, a x1 # a x2 = e x1 ln a # e x2 ln a = e x1 ln a + x2 ln a

Definition of a x Law 1

= e

sx1 + x2dln a

Factor ln a

= a

x1 + x2

Definition of a x

.

Starting with the definition a x = e x ln a, a 7 0, we get the derivative d x d x ln a a = e = (ln a) e x ln a = (ln a) a x, dx dx so

d x a = a x ln a. dx

408

Chapter 7: Integrals and Transcendental Functions

Alternatively, we get the same derivative rule by applying logarithmic differentiation: y = ax ln y = x ln a 1 dy y dx = ln a dy = y ln a = a x ln a. dx

Taking logarithms Differentiating with respect to x

With the Chain Rule, we get a more general form, as in Section 3.8. If a 7 0 and u is a differentiable function of x, then a u is a differentiable function of x and d u du . a = a u ln a dx dx The integral equivalent of this last result is

L

au + C. ln a

Logarithms with Base a

y

y 2x

If a is any positive number other than 1, the function a x is one-to-one and has a nonzero derivative at every point. It therefore has a differentiable inverse.

yx

DEFINITION For any positive number a Z 1, the logarithm of x with base a, denoted by log a x, is the inverse function of a x.

y log 2 x

2 1 0

a u du =

1 2

FIGURE 7.4 The graph of 2x and its inverse, log 2 x .

x

The graph of y = loga x can be obtained by reflecting the graph of y = a x across the 45° line y = x (Figure 7.4). When a = e, we have log e x = inverse of e x = ln x. Since log a x and a x are inverses of one another, composing them in either order gives the identity function. Inverse Equations for a x and log a x a loga x = x loga sa x d = x

sx 7 0d sall xd

As stated in Section 1.6, the function loga x is just a numerical multiple of ln x. We see this from the following derivation: y = loga x Defining equation for y y a = x Equivalent equation y ln a = ln x Natural log of both sides y ln a = ln x Algebra Rule 4 for natural log ln x ln a ln x loga x = ln a y =

Solve for y. Substitute for y.

7.1

TABLE 7.2

409

It then follows easily that the arithmetic rules satisfied by log a x are the same as the ones for ln x. These rules, given in Table 7.2, can be proved by dividing the corresponding rules for the natural logarithm function by ln a. For example,

Rules for base a

logarithms For any numbers x 7 0 and y 7 0,

ln xy = ln x + ln y ln y ln xy ln x = + ln a ln a ln a loga xy = loga x + loga y.

Product Rule: loga xy = loga x + loga y

1.

The Logarithm Defined as an Integral

Rule 1 for natural logarithms Á Á divided by ln a Á Á gives Rule 1 for base a logarithms.

Quotient Rule:

2.

x loga y = loga x - loga y

Derivatives and Integrals Involving loga x To find derivatives or integrals involving base a logarithms, we convert them to natural logarithms. If u is a positive differentiable function of x, then

Reciprocal Rule:

3.

d d ln u 1 d 1 # 1 du b = . sloga ud = a sln ud = dx dx ln a ln a dx ln a u dx

1 loga y = -loga y Power Rule: loga x y = y loga x

4.

d 1 # 1 du sloga ud = dx ln a u dx

EXAMPLE 2

We illustrate the derivative and integral results.

#

3 d 1 s3x + 1d = 3x + 1 dx sln 10ds3x + 1d

(a)

d 1 log10 s3x + 1d = dx ln 10

(b)

log2 x ln x 1 x dx = ln 2 x dx L L

log2 x =

ln x ln 2

=

1 u du ln 2 L

=

2 sln xd2 1 sln xd 1 u2 + C + C = + C = ln 2 2 ln 2 2 2 ln 2

u = ln x,

1 du = x dx

Summary In this section we used calculus to give precise definitions of the logarithmic and exponential functions. This approach is somewhat different from our earlier treatments of the polynomial, rational, and trigonometric functions. There we first defined the function and then we studied its derivatives and integrals. Here we started with an integral from which the functions of interest were obtained. The motivation behind this approach was to avoid mathematical difficulties that arise when we attempt to define functions such as a x for any real number x, rational or irrational. Defining ln x as the integral of the function 1>t from t = 1 to t = x enabled us to define all of the exponential and logarithmic functions, and then derive their key algebraic and analytic properties.

Exercises 7.1 Integration Evaluate the integrals in Exercises 1–46. -2

1.

L-3

dx x

3. 0

2.

L-1

3 dx 3x - 2

5.

2y dy L y - 25 2

3 sec2 t dt 6 L + 3 tan t

8r dr 2 L 4r - 5 sec y tan y dy 6. L 2 + sec y 4.

410 7.

Chapter 7: Integrals and Transcendental Functions

dx 21x + 2x L

8.

sec x dx L 2ln ssec x + tan xd

ln 3

9.

e x dx

Lln 2

(ln x) 3 dx 2x L1

10.

4

11.

12.

8e sx + 1d dx

L

49.

e x>2 dx

Lln 4 e 2r dr 15. L 2r 17. 19. 21. 22.

L

2t e

-t 2

ln (ln x) dx L x ln x

L

dt

2e y cos e y dy

Lln sp>6d er 25. r dr L1 + e 1

L0

2-u du

22

L1

2

x2sx d dx

p>2

31.

7cos t sin t dt

L0 4

33.

e

dr L 2r ln x dx 18. L x2ln2 x + 1

x 2xs1 + ln xd dx

L2

26.

37.

s 12 + 1dx 12 dx

L0 L

log10 x x dx 4

39.

L1

ln 2 log 2 x dx x

log 2 sx + 2d dx 41. x + 2 L0 9 2 log sx + 1d 10 dx 43. x + 1 L0 dx x log 10 x L

5-u du

21x dx L1 1x p>4 tan t 1 a b sec2 t dt 32. 3 L0 2 ln x 2 34. x dx L1 e

36. 38. 40.

L1

dy = e -t sec2 spe -t d, dt

= sec2 x,

y s0d = 0

y¿s0d = 1

and

55. y = sx 2>8d - ln x,

4 … x … 8

56. x = sy>4d2 - 2 ln sy>4d,

4 … y … 12

T 57. The linearization of ln s1 + xd at x = 0 Instead of approximating ln x near x = 1, we approximate ln s1 + xd near x = 0. We get a simpler formula this way. a. Derive the linearization ln s1 + xd L x at x = 0. b. Estimate to five decimal places the error involved in replacing ln s1 + xd by x on the interval [0, 0.1]. c. Graph ln s1 + xd and x together for 0 … x … 0.5. Use different colors, if available. At what points does the approximation of ln s1 + xd seem best? Least good? By reading coordinates from the graphs, find as good an upper bound for the error as your grapher will allow. 58. The linearization of e x at x = 0 a. Derive the linear approximation e x L 1 + x at x = 0.

log 2 x x dx

T b. Estimate to five decimal places the magnitude of the error involved in replacing e x by 1 + x on the interval [0, 0.2].

e

2 ln 10 log 10 x dx x

T c. Graph e x and 1 + x together for -2 … x … 2. Use different colors, if available. On what intervals does the approximation appear to overestimate e x ? Underestimate e x ?

10

42.

log 10 s10xd dx x

L1>10 3 2 log sx - 1d 2 dx 44. x - 1 L2 46.

dx 2

y¿s1d = 0

4

L1 L1

x sln 2d - 1 dx

dx 2 xslog L 8 xd

Initial Value Problems Solve the initial value problems in Exercises 47–52. dy = e t sin se t - 2d, y sln 2d = 0 47. dt 48.

2

4

2

45.

2

2xe x cos se x d dx

30.

3

35.

d 2y

and

54. In Section 6.2, Exercise 6, we revolved about the y-axis the region between the curve y = 9x N 2x 3 + 9 and the x-axis from x = 0 to x = 3 to generate a solid of volume 36p. What volume do you get if you revolve the region about the x-axis instead? (See Section 6.2, Exercise 6, for a graph.)

dx x L1 + e L-2

d 2y

Find the lengths of the curves in Exercises 55 and 56.

L0

0

28.

52.

y¿s0d = 0

and

53. The region between the curve y = 1>x 2 and the x-axis from x = 1>2 to x = 2 is revolved about the y-axis to generate a solid. Find the volume of the solid.

e -1>x dx 3 L x

2ln p

24.

y s0d = 1

Theory and Applications

2

20.

= 2e -x,

-2r

e csc sp + td csc sp + td cot sp + td dt

23.

29.

L

e sec pt sec pt tan pt dt

ln sp>2d

27.

tan x ln (cos x) dx

16.

e 1>x dx 2 L x L

14.

dx 2

= 1 - e 2t, y s1d = -1 dt 2 dy 1 = 1 + x , y s1d = 3 51. dx 50.

ln 9

13.

d 2y

y sln 4d = 2>p

59. Show that for any number a 7 1 a

L1

ln a

ln x dx +

L0

e y dy = a ln a.

(See accompanying figure.) y

y ln x

ln a

0

1

a

x

7.2 60. The geometric, logarithmic, and arithmetic mean inequality x

a. Show that the graph of e is concave up over every interval of x-values. b. Show, by reference to the accompanying figure, that if 0 6 a 6 b then e sln a + ln bd>2 # sln b - ln ad 6

ln b

Lln a

e x dx 6

e ln a + e ln b # sln b - ln ad. 2 y ex

Exponential Change and Separable Differential Equations

411

T 66. Could x ln 2 possibly be the same as 2ln x for x 7 0? Graph the two functions and explain what you see. T 67. Which is bigger, pe or eP? Calculators have taken some of the mystery out of this once-challenging question. (Go ahead and check; you will see that it is a surprisingly close call.) You can answer the question without a calculator, though. a. Find an equation for the line through the origin tangent to the graph of y = ln x.

F C E B A ln a

M [–3, 6] by [–3, 3] ln a ln b 2

D ln b

x

b. Give an argument based on the graphs of y = ln x and the tangent line to explain why ln x 6 x>e for all positive x Z e. c. Show that ln sx e d 6 x for all positive x Z e.

NOT TO SCALE

c. Use the inequality in part (b) to conclude that b - a a + b 2ab 6 6 . 2 ln b - ln a This inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in turn is less than their arithmetic mean. (For more about this inequality, see “The Geometric, Logarithmic, and Arithmetic Mean Inequality” by Frank Burk, American Mathematical Monthly, Vol. 94, No. 6, June–July 1987, pp. 527–528.)

d. Conclude that x e 6 e x for all positive x Z e. e. So which is bigger, pe or e p ? T 68. A decimal representation of e Find e to as many decimal places as your calculator allows by solving the equation ln x = 1 using Newton’s method in Section 4.7. Calculations with Other Bases T 69. Most scientific calculators have keys for log10 x and ln x. To find logarithms to other bases, we use the equation log a x = sln xd>sln ad. Find the following logarithms to five decimal places.

Grapher Explorations 61. Graph ln x, ln 2x, ln 4x, ln 8x, and ln 16x (as many as you can) together for 0 6 x … 10. What is going on? Explain.

a. log 3 8

62. Graph y = ln ƒ sin x ƒ in the window 0 … x … 22, -2 … y … 0. Explain what you see. How could you change the formula to turn the arches upside down?

g. ln x, given that log 2 x = -1.5

63. a. Graph y = sin x and the curves y = ln sa + sin xd for a = 2, 4, 8, 20, and 50 together for 0 … x … 23. b. Why do the curves flatten as a increases? (Hint: Find an a-dependent upper bound for ƒ y¿ ƒ .) 64. Does the graph of y = 1x - ln x, x 7 0, have an inflection point? Try to answer the question (a) by graphing, (b) by using calculus. T 65. The equation x 2 = 2x has three solutions: x = 2, x = 4, and one other. Estimate the third solution as accurately as you can by graphing.

7.2

b. log 7 0.5

c. log 20 17

d. log 0.5 7

e. ln x, given that log 10 x = 2.3 f. ln x, given that log 2 x = 1.4 h. ln x, given that log 10 x = -0.7 70. Conversion factors a. Show that the equation for converting base 10 logarithms to base 2 logarithms is ln 10 log10 x. ln 2 b. Show that the equation for converting base a logarithms to base b logarithms is log 2 x =

log b x =

ln a log a x. ln b

Exponential Change and Separable Differential Equations Exponential functions increase or decrease very rapidly with changes in the independent variable. They describe growth or decay in many natural and industrial situations. The variety of models based on these functions partly accounts for their importance. We now investigate the basic proportionality assumption that leads to such exponential change.

412

Chapter 7: Integrals and Transcendental Functions

Exponential Change In modeling many real-world situations, a quantity y increases or decreases at a rate proportional to its size at a given time t. Examples of such quantities include the amount of a decaying radioactive material, the size of a population, and the temperature difference between a hot object and its surrounding medium. Such quantities are said to undergo exponential change. If the amount present at time t = 0 is called y0 , then we can find y as a function of t by solving the following initial value problem: dy = ky dt

Initial condition:

y = y0

(1a)

when

t = 0.

(1b)

If y is positive and increasing, then k is positive, and we use Equation (1a) to say that the rate of growth is proportional to what has already been accumulated. If y is positive and decreasing, then k is negative, and we use Equation (1a) to say that the rate of decay is proportional to the amount still left. We see right away that the constant function y = 0 is a solution of Equation (1a) if y0 = 0. To find the nonzero solutions, we divide Equation (1a) by y:

y k 1.3

k1

1 y

k 0.6 y0

Differential equation:

x (a) y

#

dy = k dt

yZ0

1 dy dt = k dt y L L dt ln ƒ y ƒ = kt + C ƒ y ƒ = e kt + C ƒ y ƒ = e C # e kt y = ;e e y = Ae kt .

C kt

y0 k 0.5

Integrate with respect to t; 1 s1>ud du = ln ƒ u ƒ + C . Exponentiate.

ea+b = ea # eb

If ƒ y ƒ = r, then y = ;r. A is a shorter name for ;e C.

By allowing A to take on the value 0 in addition to all possible values ;e C , we can include the solution y = 0 in the formula. We find the value of A for the initial value problem by solving for A when y = y0 and t = 0:

k 1 k 1.3 x (b)

FIGURE 7.5 Graphs of (a) exponential growth and (b) exponential decay. As |k| increases, the growth (k 7 0) or decay (k 6 0) intensifies.

y0 = Ae k

#0

= A.

The solution of the initial value problem in Equations (1a) and (1b) is therefore y = y0 e kt .

(2)

Quantities changing in this way are said to undergo exponential growth if k 7 0 and exponential decay if k 6 0. The number k is called the rate constant of the change. (See Figure 7.5.) The derivation of Equation (2) shows also that the only functions that are their own derivatives are constant multiples of the exponential function. Before presenting several examples of exponential change, let’s consider the process we used to derive it.

Separable Differential Equations Exponential change is modeled by a differential equation of the form dy>dx = ky for some nonzero constant k. More generally, suppose we have a differential equation of the form dy = ƒ(x, y), dx

(3)

7.2

Exponential Change and Separable Differential Equations

413

where ƒ is a function of both the independent and dependent variables. A solution of the equation is a differentiable function y = y(x) defined on an interval of x-values (perhaps infinite) such that d y(x) = ƒ(x, y (x)) dx on that interval. That is, when y(x) and its derivative y¿(x) are substituted into the differential equation, the resulting equation is true for all x in the solution interval. The general solution is a solution y(x) that contains all possible solutions and it always contains an arbitrary constant. Equation (3) is separable if ƒ can be expressed as a product of a function of x and a function of y. The differential equation then has the form dy = gsxdHs yd. dx

g is a function of x; H is a function of y.

When we rewrite this equation in the form dy gsxd = , dx hs yd

H( y) =

1 h( y)

its differential form allows us to collect all y terms with dy and all x terms with dx: hs yd dy = gsxd dx. Now we simply integrate both sides of this equation: hs yd dy = gsxd dx. (4) L L After completing the integrations we obtain the solution y defined implicitly as a function of x. The justification that we can simply integrate both sides in Equation (4) is based on the Substitution Rule (Section 5.5): L

hs yd dy = = =

EXAMPLE 1

L L L

hs ysxdd

dy dx dx

hs ysxdd

gsxd dx hs ysxdd

g(x) dy = dx h( y)

gsxd dx.

Solve the differential equation dy = s1 + yde x, dx

y 7 -1.

Since 1 + y is never zero for y 7 -1, we can solve the equation by separating the variables. Solution

dy = s1 + yde x dx dy = s1 + yde x dx dy = e x dx 1 + y dy = e x dx L1 + y L ln s1 + yd = e x + C The last equation gives y as an implicit function of x.

Treat dy/dx as a quotient of differentials and multiply both sides by dx. Divide by (1 + y). Integrate both sides. C represents the combined constants of integration.

414

Chapter 7: Integrals and Transcendental Functions

EXAMPLE 2 Solution

Solve the equation ysx + 1d

dy = xs y 2 + 1d. dx

We change to differential form, separate the variables, and integrate: ysx + 1d dy = xs y 2 + 1d dx y dy x dx = x + 1 y2 + 1 y dy L1 + y

=

2

L

a1 -

x Z -1

1 b dx x + 1

Divide x by x + 1.

1 ln s1 + y 2 d = x - ln ƒ x + 1 ƒ + C. 2 The last equation gives the solution y as an implicit function of x. The initial value problem dy = ky, dt

ys0d = y0

involves a separable differential equation, and the solution y = y0 e kt expresses exponential change. We now present several examples of such change.

Unlimited Population Growth

y Yeast biomass (mg)

500 400 300 200 100 0 0

5 Time (hr)

10

FIGURE 7.6 Graph of the growth of a yeast population over a 10-hour period, based on the data in Table 7.3.

TABLE 7.3

Time (hr) 0 1 2 3 4 5 6 7 8 9 10

Population of Yeast Yeast biomass (mg) 9.6 18.3 29.0 47.2 71.1 119.1 174.6 257.3 350.7 441.0 513.3

t

Strictly speaking, the number of individuals in a population (of people, plants, animals, or bacteria, for example) is a discontinuous function of time because it takes on discrete values. However, when the number of individuals becomes large enough, the population can be approximated by a continuous function. Differentiability of the approximating function is another reasonable hypothesis in many settings, allowing for the use of calculus to model and predict population sizes. If we assume that the proportion of reproducing individuals remains constant and assume a constant fertility, then at any instant t the birth rate is proportional to the number y(t) of individuals present. Let’s assume, too, that the death rate of the population is stable and proportional to y(t). If, further, we neglect departures and arrivals, the growth rate dy> dt is the birth rate minus the death rate, which is the difference of the two proportionalities under our assumptions. In other words, dy>dt = ky so that y = y0 e kt , where y0 is the size of the population at time t = 0. As with all kinds of growth, there may be limitations imposed by the surrounding environment, but we will not go into these here. When k is positive, the proportionality dy>dt = ky models unlimited population growth. (See Figure 7.6.)

EXAMPLE 3

The biomass of a yeast culture in an experiment is initially 12 grams. After 10 minutes the mass is 15 grams. Assuming that the equation for unlimited population growth gives a good model for the growth of the yeast when the mass is below 100 grams, how long will it take for the mass to double from its initial value?

Let y(t) be the yeast biomass after t minutes. We use the exponential growth model dy>dt = ky for unlimited population growth, with solution y = y0ekt. We have y0 = y(0) = 12. We are also told that, Solution

y(10) = 12ek(10) = 15. Solving this equation for k, we find ek(10) =

15 12

5 10k = ln a b 4 k =

5 1 ln a b L 0.022314. 10 4

7.2

Exponential Change and Separable Differential Equations

415

Then the mass of the yeast in grams after t minutes is given by the equation y = 12e(0.022314)t. To solve the problem we find the time t for which y(t) = 24, which is twice the initial amout: 12e(0.022314)t = 24 (0.022314)t = ln a

24 b 12 ln 2 t = L 31.06 0.022314 It takes about 31 minutes for the yeast population to double. In the next example we model the number of people within a given population who are infected by a disease which is being eradicated from the population. Here the constant of proportionality k is negative, and the model describes an exponentially decaying number of infected individuals.

EXAMPLE 4

One model for the way diseases die out when properly treated assumes that the rate dy> dt at which the number of infected people changes is proportional to the number y. The number of people cured is proportional to the number y that are infected with the disease. Suppose that in the course of any given year the number of cases of a disease is reduced by 20%. If there are 10,000 cases today, how many years will it take to reduce the number to 1000? We use the equation y = y0 e kt . There are three things to find: the value of y0 , the value of k, and the time t when y = 1000. The value of y0 . We are free to count time beginning anywhere we want. If we count from today, then y = 10,000 when t = 0, so y0 = 10,000. Our equation is now Solution

y = 10,000e kt .

(5)

The value of k. When t = 1 year, the number of cases will be 80% of its present value, or 8000. Hence, 8000 = 10,000e ks1d e = 0.8 ln se k d = ln 0.8 k = ln 0.8 6 0. k

y

ln 0.8 L - 0.223

y = 10,000e sln 0.8dt . y

5

(6)

The value of t that makes y = 1000. We set y equal to 1000 in Equation (6) and solve for t: 1000 = 10,000e sln 0.8dt

10,000e(ln 0.8)t

e sln 0.8dt = 0.1 sln 0.8dt = ln 0.1 ln 0.1 t = L 10.32 years. ln 0.8

1,000 0

Logs of both sides

At any given time t,

10,000

5,000

Eq. (5) with t = 1 and y = 8000

10

FIGURE 7.7 A graph of the number of people infected by a disease exhibits exponential decay (Example 4).

t

Logs of both sides

It will take a little more than 10 years to reduce the number of cases to 1000. (See Figure 7.7)

Radioactivity Some atoms are unstable and can spontaneously emit mass or radiation. This process is called radioactive decay, and an element whose atoms go spontaneously through this process is called radioactive. Sometimes when an atom emits some of its mass through this process of radioactivity, the remainder of the atom re-forms to make an atom of some

416

Chapter 7: Integrals and Transcendental Functions

For radon-222 gas, t is measured in days and k = 0.18 . For radium-226, which used to be painted on watch dials to make them glow at night (a dangerous practice), t is measured in years and k = 4.3 * 10 -4 .

new element. For example, radioactive carbon-14 decays into nitrogen; radium, through a number of intermediate radioactive steps, decays into lead. Experiments have shown that at any given time the rate at which a radioactive element decays (as measured by the number of nuclei that change per unit time) is approximately proportional to the number of radioactive nuclei present. Thus, the decay of a radioactive element is described by the equation dy>dt = -ky, k 7 0. It is conventional to use -k, with k 7 0, to emphasize that y is decreasing. If y0 is the number of radioactive nuclei present at time zero, the number still present at any later time t will be y = y0 e -kt,

k 7 0.

In Section 1.6, we defined the half-life of a radioactive element to be the time required for half of the radioactive nuclei present in a sample to decay. It is an interesting fact that the half-life is a constant that does not depend on the number of radioactive nuclei initially present in the sample, but only on the radioactive substance. We found the half-life is given by Half-life =

ln 2 k

(7)

For example, the half-life for radon-222 is half-life =

ln 2 L 3.9 days. 0.18

EXAMPLE 5

The decay of radioactive elements can sometimes be used to date events from Earth’s past. In a living organism, the ratio of radioactive carbon, carbon-14, to ordinary carbon stays fairly constant during the lifetime of the organism, being approximately equal to the ratio in the organism’s atmosphere at the time. After the organism’s death, however, no new carbon is ingested, and the proportion of carbon-14 in the organism’s remains decreases as the carbon-14 decays. Scientists who do carbon-14 dating use a figure of 5730 years for its half-life. Find the age of a sample in which 10% of the radioactive nuclei originally present have decayed. We use the decay equation y = y0 e -kt . There are two things to find: the value of k and the value of t when y is 0.9y0 (90% of the radioactive nuclei are still present). That is, find t when y0 e -kt = 0.9y0 , or e -kt = 0.9. Solution

The value of k. We use the half-life Equation (7): ln 2 ln 2 = sabout 1.2 * 10-4 d. k = 5730 half-life The value of t that makes e -kt = 0.9. e-kt = 0.9 e-sln 2>5730dt = 0.9 ln 2 t = ln 0.9 5730 5730 ln 0.9 t = L 871 years ln 2

Logs of both sides

The sample is about 871 years old.

Heat Transfer: Newton’s Law of Cooling Hot soup left in a tin cup cools to the temperature of the surrounding air. A hot silver bar immersed in a large tub of water cools to the temperature of the surrounding water. In situations like these, the rate at which an object’s temperature is changing at any given time is roughly proportional to the difference between its temperature and the temperature of the

7.2

Exponential Change and Separable Differential Equations

417

surrounding medium. This observation is called Newton’s Law of Cooling, although it applies to warming as well. If H is the temperature of the object at time t and HS is the constant surrounding temperature, then the differential equation is dH = -ksH - HS d. dt

(8)

If we substitute y for sH - HS d, then dy d d dH = sH - HS d = sH d dt dt dt dt S dH - 0 = dt dH = dt = -ksH - HS d = -ky.

HS is a constant.

Eq. (8) H - HS = y

Now we know that the solution of dy>dt = -ky is y = y0 e tuting sH - HS d for y, this says that

-kt

, where ys0d = y0 . Substi-

H - HS = sH0 - HS de -kt ,

(9)

where H0 is the temperature at t = 0. This equation is the solution to Newton’s Law of Cooling.

EXAMPLE 6

A hard-boiled egg at 98°C is put in a sink of 18°C water. After 5 min, the egg’s temperature is 38°C. Assuming that the water has not warmed appreciably, how much longer will it take the egg to reach 20°C? Solution We find how long it would take the egg to cool from 98°C to 20°C and subtract the 5 min that have already elapsed. Using Equation (9) with HS = 18 and H0 = 98, the egg’s temperature t min after it is put in the sink is

H = 18 + s98 - 18de -kt = 18 + 80e -kt . To find k, we use the information that H = 38 when t = 5: 38 = 18 + 80e -5k 1 e -5k = 4 1 -5k = ln = -ln 4 4 k =

1 ln 4 = 0.2 ln 4 5

sabout 0.28d.

The egg’s temperature at time t is H = 18 + 80e -s0.2 ln 4dt . Now find the time t when H = 20: 20 = 18 + 80e -s0.2 ln 4dt 80e -s0.2 ln 4dt = 2 1 e -s0.2 ln 4dt = 40 -s0.2 ln 4dt = ln t =

1 = -ln 40 40

ln 40 L 13 min. 0.2 ln 4

The egg’s temperature will reach 20°C about 13 min after it is put in the water to cool. Since it took 5 min to reach 38°C, it will take about 8 min more to reach 20°C.

418

Chapter 7: Integrals and Transcendental Functions

Exercises 7.2 Verifying Solutions In Exercises 1–4, show that each function y = ƒsxd is a solution of the accompanying differential equation. 1. 2y¿ + 3y = e -x a. y = e -x c. y = e 2. y¿ = y 2

-x

+ Ce

b. y = e -x + e -s3>2dx -s3>2dx

1 1 1 a. y = - x b. y = c. y = x + 3 x + C x t e 1 dt, x 2y¿ + xy = e x 3. y = x L1 t x 2x 3 1 21 + t 4 dt, y¿ + y = 1 4. y = 4L 1 + x4 21 + x 1 Initial Value Problems In Exercises 5–8, show that each function is a solution of the given initial value problem. Differential Initial Solution equation condition candidate 5. y¿ + y = 6. y¿ = e -x

2

2 1 + 4e 2x - 2xy

7. xy¿ + y = -sin x, x 7 0 8. x 2y¿ = xy - y 2, x 7 1

ys -ln 2d =

p 2

y = e -x tan-1 s2e x d

ys2d = 0 p ya b = 0 2

y = sx - 2de -x cos x y = x

ysed = e

y =

2

x ln x

Separable Differential Equations Solve the differential equation in Exercises 9–22. dy dy 9. 2 2xy = 1, x, y 7 0 10. = x 2 2y, y 7 0 dx dx dy dy 11. 12. = ex-y = 3x 2 e -y dx dx dy dy 13. 14. 22xy = 2y cos2 2y = 1 dx dx dy dy 15. 2x = e y + 2x, x 7 0 16. ssec xd = e y + sin x dx dx dy 17. = 2x21 - y 2, -1 6 y 6 1 dx dy e 2x - y 18. = x+y dx e dy dy 2 19. y 20. = 3x 2y 3 - 6x 2 = xy + 3x - 2y - 6 dx dx dy 2 2 1 dy 21. x 22. = ye x + 2 2y e x = e x - y + e x + e -y + 1 dx dx Applications and Examples The answers to most of the following exercises are in terms of logarithms and exponentials. A calculator can be helpful, enabling you to express the answers in decimal form. 23. Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process did not come to a halt some 30,000 years ago as many scientists contend. In

northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years. a. If t represents time in years and y represents tooth size, use the condition that y = 0.99y0 when t = 1000 to find the value of k in the equation y = y0 e kt . Then use this value of k to answer the following questions. b. In about how many years will human teeth be 90% of their present size? c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)? 24. Atmospheric pressure The earth’s atmospheric pressure p is often modeled by assuming that the rate dp> dh at which p changes with the altitude h above sea level is proportional to p. Suppose that the pressure at sea level is 1013 millibars (about 14.7 pounds per square inch) and that the pressure at an altitude of 20 km is 90 millibars. a. Solve the initial value problem Differential equation: dp>dh = kp sk a constantd Initial condition: p = p0 when h = 0 to express p in terms of h. Determine the values of p0 and k from the given altitude-pressure data. b. What is the atmospheric pressure at h = 50 km ? c. At what altitude does the pressure equal 900 millibars? 25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of d-glucono lactone into gluconic acid, for example, dy = -0.6y dt when t is measured in hours. If there are 100 grams of d-glucono lactone present when t = 0, how many grams will be left after the first hour? 26. The inversion of sugar The processing of raw sugar has a step called “inversion” that changes the sugar’s molecular structure. Once the process has begun, the rate of change of the amount of raw sugar is proportional to the amount of raw sugar remaining. If 1000 kg of raw sugar reduces to 800 kg of raw sugar during the first 10 hours, how much raw sugar will remain after another 14 hours? 27. Working underwater The intensity L(x) of light x feet beneath the surface of the ocean satisfies the differential equation dL = -kL . dx As a diver, you know from experience that diving to 18 ft in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below one-tenth of the surface value. About how deep can you expect to work without artificial light? 28. Voltage in a discharging capacitor Suppose that electricity is draining from a capacitor at a rate that is proportional to the voltage V across its terminals and that, if t is measured in seconds, dV 1 = V. 40 dt Solve this equation for V, using V0 to denote the value of V when t = 0. How long will it take the voltage to drop to 10% of its original value?

7.2 29. Cholera bacteria Suppose that the bacteria in a colony can grow unchecked, by the law of exponential change. The colony starts with 1 bacterium and doubles every half-hour. How many bacteria will the colony contain at the end of 24 hours? (Under favorable laboratory conditions, the number of cholera bacteria can double every 30 min. In an infected person, many bacteria are destroyed, but this example helps explain why a person who feels well in the morning may be dangerously ill by evening.) 30. Growth of bacteria A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000. How many bacteria were present initially? 31. The incidence of a disease (Continuation of Example 3.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%. a. How long will it take to reduce the number of cases to 1000? b. How long will it take to eradicate the disease, that is, reduce the number of cases to less than 1? 32. The U.S. population The U.S. Census Bureau keeps a running clock totaling the U.S. population. On March 26, 2008, the total was increasing at the rate of 1 person every 13 sec. The population figure for 2:31 P.M. EST on that day was 303,714,725. a. Assuming exponential growth at a constant rate, find the rate constant for the population’s growth (people per 365-day year). b. At this rate, what will the U.S. population be at 2:31 P.M. EST on March 26, 2015? 33. Oil depletion Suppose the amount of oil pumped from one of the canyon wells in Whittier, California, decreases at the continuous rate of 10% per year. When will the well’s output fall to one-fifth of its present value? 34. Continuous price discounting To encourage buyers to place 100-unit orders, your firm’s sales department applies a continuous discount that makes the unit price a function p(x) of the number of units x ordered. The discount decreases the price at the rate of $0.01 per unit ordered. The price per unit for a 100-unit order is ps100d = $20.09 . a. Find p(x) by solving the following initial value problem: dp 1 Differential equation: = p 100 dx Initial condition:

ps100d = 20.09.

b. Find the unit price p(10) for a 10-unit order and the unit price p(90) for a 90-unit order. c. The sales department has asked you to find out if it is discounting so much that the firm’s revenue, rsxd = x # psxd , will actually be less for a 100-unit order than, say, for a 90-unit order. Reassure them by showing that r has its maximum value at x = 100 . d. Graph the revenue function rsxd = xpsxd for 0 … x … 200 . 35. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay? 36. Polonium-210 The half-life of polonium is 139 days, but your sample will not be useful to you after 95% of the radioactive nuclei present on the day the sample arrives has disintegrated. For about how many days after the sample arrives will you be able to use the polonium?

Exponential Change and Separable Differential Equations

419

37. The mean life of a radioactive nucleus Physicists using the radioactivity equation y = y0 e -kt call the number 1> k the mean life of a radioactive nucleus. The mean life of a radon nucleus is about 1>0.18 = 5.6 days . The mean life of a carbon-14 nucleus is more than 8000 years. Show that 95% of the radioactive nuclei originally present in a sample will disintegrate within three mean lifetimes, i.e., by time t = 3>k . Thus, the mean life of a nucleus gives a quick way to estimate how long the radioactivity of a sample will last. 38. Californium-252 What costs $27 million per gram and can be used to treat brain cancer, analyze coal for its sulfur content, and detect explosives in luggage? The answer is californium-252, a radioactive isotope so rare that only 8 g of it have been made in the Western world since its discovery by Glenn Seaborg in 1950. The half-life of the isotope is 2.645 years—long enough for a useful service life and short enough to have a high radioactivity per unit mass. One microgram of the isotope releases 170 million neutrons per minute. a. What is the value of k in the decay equation for this isotope? b. What is the isotope’s mean life? (See Exercise 37.) c. How long will it take 95% of a sample’s radioactive nuclei to disintegrate? 39. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room whose temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions. a. How much longer would it take the soup to cool to 35°C? b. Instead of being left to stand in the room, the cup of 90°C soup is put in a freezer whose temperature is -15°C . How long will it take the soup to cool from 90°C to 35°C? 40. A beam of unknown temperature An aluminum beam was brought from the outside cold into a machine shop where the temperature was held at 65°F. After 10 min, the beam warmed to 35°F and after another 10 min it was 50°F. Use Newton’s Law of Cooling to estimate the beam’s initial temperature. 41. Surrounding medium of unknown temperature A pan of warm water (46°C) was put in a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 min after that, it was 33°C. Use Newton’s Law of Cooling to estimate how cold the refrigerator was. 42. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be a. 15 min from now?

b. 2 hours from now?

c. When will the silver be 10°C above room temperature? 43. The age of Crater Lake The charcoal from a tree killed in the volcanic eruption that formed Crater Lake in Oregon contained 44.5% of the carbon-14 found in living matter. About how old is Crater Lake? 44. The sensitivity of carbon-14 dating to measurement To see the effect of a relatively small error in the estimate of the amount of carbon-14 in a sample being dated, consider this hypothetical situation: a. A bone fragment found in central Illinois in the year 2000 contains 17% of its original carbon-14 content. Estimate the year the animal died. b. Repeat part (a) assuming 18% instead of 17%. c. Repeat part (a) assuming 16% instead of 17%.

420

Chapter 7: Integrals and Transcendental Functions

45. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the

7.3

melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery? 46. Art forgery A painting attributed to Vermeer (1632–1675), which should contain no more than 96.2% of its original carbon14, contains 99.5% instead. About how old is the forgery?

Hyperbolic Functions The hyperbolic functions are formed by taking combinations of the two exponential functions e x and e - x. The hyperbolic functions simplify many mathematical expressions and occur frequently in mathematical and engineering applications. In this section we give a brief introduction to these functions, their graphs, their derivatives, their integrals, and their inverse functions.

Definitions and Identities The hyperbolic sine and hyperbolic cosine functions are defined by the equations e x - e -x e x + e -x sinh x = and cosh x = . 2 2 We pronounce sinh x as “cinch x,” rhyming with “pinch x,” and cosh x as “kosh x,” rhyming with “gosh x.” From this basic pair, we define the hyperbolic tangent, cotangent, secant, and cosecant functions. The defining equations and graphs of these functions are shown in Table 7.4. We will see that the hyperbolic functions bear many similarities to the trigonometric functions after which they are named. The six basic hyperbolic functions

TABLE 7.4

y 3 y⫽ e 2 2 1

y ⫽ sinh x 1 2 3 x –x y⫽–e 2

–3 –2 –1 –1 –2 –3

3 2 1

–x y⫽ e 2

1 2 3

y⫽1 x

(b)

Hyperbolic cosine: e x + e -x cosh x = 2

x 1 2 y ⫽ sech x

(d)

Hyperbolic secant: 1 2 = x sech x = cosh x e + e -x

–2 –1

2 1 –2 –1 –1

y ⫽ coth x y ⫽ tanh x x 1 2 y ⫽ –1

(c)

Hyperbolic tangent: sinh x e x - e -x tanh x = = x cosh x e + e -x Hyperbolic cotangent: cosh x e x + e -x coth x = = x sinh x e - e -x

y y⫽1

2

–2 y ⫽ coth x

y

–2 –1 0

x y⫽ e 2

–3 –2 –1

(a)

Hyperbolic sine: e x - e -x sinh x = 2

2

y

y y ⫽ cosh x

x

x 1 2 y ⫽ csch x

(e)

Hyperbolic cosecant: 1 2 csch x = = x sinh x e - e -x

7.3

Identities for hyperbolic functions TABLE 7.5

cosh2 x - sinh2 x = 1 sinh 2x = 2 sinh x cosh x cosh 2x = cosh2 x + sinh2 x cosh2 x =

cosh 2x + 1 2

sinh2 x =

cosh 2x - 1 2

tanh2 x = 1 - sech2 x coth2 x = 1 + csch2 x

Hyperbolic Functions

421

Hyperbolic functions satisfy the identities in Table 7.5. Except for differences in sign, these resemble identities we know for the trigonometric functions. The identities are proved directly from the definitions, as we show here for the second one: 2 sinh x cosh x = 2 a

e x - e -x e x + e -x ba b 2 2

e 2x - e -2x 2 = sinh 2x.

=

The other identities are obtained similarly, by substituting in the definitions of the hyperbolic functions and using algebra. Like many standard functions, hyperbolic functions and their inverses are easily evaluated with calculators, which often have special keys for that purpose. For any real number u, we know the point with coordinates (cos u, sin u) lies on the unit circle x 2 + y 2 = 1. So the trigonometric functions are sometimes called the circular functions. Because of the first identity

Derivatives of hyperbolic functions

cosh2 u - sinh2 u = 1,

d du ssinh ud = cosh u dx dx

with u substituted for x in Table 7.5, the point having coordinates (cosh u, sinh u) lies on the right-hand branch of the hyperbola x 2 - y 2 = 1. This is where the hyperbolic functions get their names (see Exercise 86).

TABLE 7.6

d du scosh ud = sinh u dx dx d du stanh ud = sech2 u dx dx d du scoth ud = -csch2 u dx dx d du ssech ud = -sech u tanh u dx dx d du scsch ud = -csch u coth u dx dx

Derivatives and Integrals of Hyperbolic Functions The six hyperbolic functions, being rational combinations of the differentiable functions e x and e -x , have derivatives at every point at which they are defined (Table 7.6). Again, there are similarities with trigonometric functions. The derivative formulas are derived from the derivative of e u : d e u - e -u d ssinh ud = a b 2 dx dx e u du>dx + e -u du>dx 2 du = cosh u . dx

=

Integral formulas for hyperbolic functions TABLE 7.7

L L L L L L

sinh u du = cosh u + C

Definition of sinh u

Derivative of e u Definition of cosh u

This gives the first derivative formula. From the definition, we can calculate the derivative of the hyperbolic cosecant function, as follows:

cosh u du = sinh u + C

d d 1 scsch ud = a b dx dx sinh u

sech2 u du = tanh u + C

= -

cosh u du sinh2 u dx

Quotient Rule

csch2 u du = -coth u + C

= -

1 cosh u du sinh u sinh u dx

Rearrange terms.

sech u tanh u du = -sech u + C

= -csch u coth u

csch u coth u du = -csch u + C

Definition of csch u

du dx

Definitions of csch u and coth u

The other formulas in Table 7.6 are obtained similarly. The derivative formulas lead to the integral formulas in Table 7.7.

422

Chapter 7: Integrals and Transcendental Functions

EXAMPLE 1 (a)

(b)

d d A tanh 21 + t 2 B = sech2 21 + t 2 # dt A 21 + t 2 B dt t sech2 21 + t 2 = 2 21 + t L

1

(c)

L0

1

sinh2 x dx =

L0

cosh 2x - 1 dx 2

Table 7.5

1

L0

1

=

1 1 sinh 2x scosh 2x - 1d dx = c - xd 2 L0 2 2 0

=

sinh 2 1 - L 0.40672 4 2

ln 2

(d)

u = sinh 5x , du = 5 cosh 5x dx

cosh 5x 1 du dx = 5L u sinh 5x L 1 1 = ln ƒ u ƒ + C = ln ƒ sinh 5x ƒ + C 5 5

coth 5x dx =

ln 2

4e x sinh x dx =

L0

4e x

e x - e -x dx = 2 L0

= C e 2x - 2x D 0

ln 2

Evaluate with a calculator. ln 2

s2e 2x - 2d dx

= se 2 ln 2 - 2 ln 2d - s1 - 0d

= 4 - 2 ln 2 - 1 L 1.6137

Inverse Hyperbolic Functions The inverses of the six basic hyperbolic functions are very useful in integration (see Chapter 8). Since dssinh xd>dx = cosh x 7 0, the hyperbolic sine is an increasing function of x. We denote its inverse by y = sinh-1 x. For every value of x in the interval - q 6 x 6 q , the value of y = sinh-1 x is the number whose hyperbolic sine is x. The graphs of y = sinh x and y = sinh-1 x are shown in Figure 7.8a. The function y = cosh x is not one-to-one because its graph in Table 7.4 does not pass the horizontal line test. The restricted function y = cosh x, x Ú 0, however, is oneto-one and therefore has an inverse, denoted by y = cosh-1 x. For every value of x Ú 1, y = cosh-1 x is the number in the interval 0 … y 6 q whose hyperbolic cosine is x. The graphs of y = cosh x, x Ú 0, and y = cosh-1 x are shown in Figure 7.8b. Like y = cosh x, the function y = sech x = 1>cosh x fails to be one-to-one, but its restriction to nonnegative values of x does have an inverse, denoted by y = sech-1 x. For every value of x in the interval s0, 1], y = sech-1 x is the nonnegative number whose hyperbolic secant is x. The graphs of y = sech x, x Ú 0, and y = sech-1 x are shown in Figure 7.8c. The hyperbolic tangent, cotangent, and cosecant are one-to-one on their domains and therefore have inverses, denoted by y = tanh-1 x,

y = coth-1 x,

These functions are graphed in Figure 7.9.

y = csch-1 x.

7.3 y ⫽ cosh x, y⫽x x0

y y

y ⫽ sinh x y ⫽ x

8 7 6 5 4 3 2 1

y ⫽ sinh –1 x (x ⫽ sinh y) 2 1 –6 –4 –2

2

4

6

x

0

423

Hyperbolic Functions

y

3

y⫽x

y ⫽ sech –1 x (x ⫽ sech y, y 0)

2 y ⫽ cosh –1 x (x ⫽ cosh y, y 0) x 1 2 3 4 5 6 7 8 (b)

1 0

1

2 (c)

y ⫽ sech x x0 x 3

(a)

FIGURE 7.8 The graphs of the inverse hyperbolic sine, cosine, and secant of x. Notice the symmetries about the line y = x. y

y x ⫽ tanh y y ⫽ tanh –1 x

–1

y

x ⫽ coth y y ⫽ coth–1 x

0

1

x

–1

(a)

x ⫽ csch y y ⫽ csch–1 x

0

1

x

0

(b)

x

(c)

FIGURE 7.9 The graphs of the inverse hyperbolic tangent, cotangent, and cosecant of x.

Useful Identities

Identities for inverse hyperbolic functions TABLE 7.8

sech

-1

x = cosh

-1

1 x

1 csch-1 x = sinh-1 x 1 coth-1 x = tanh-1 x

We use the identities in Table 7.8 to calculate the values of sech-1 x, csch-1 x, and coth-1 x on calculators that give only cosh-1 x, sinh-1 x, and tanh-1 x. These identities are direct consequences of the definitions. For example, if 0 6 x … 1, then 1 sech acosh-1 a x b b =

1 1 cosh acosh-1 a x b b

=

1 = x. 1 ax b

We also know that sech ssech-1 xd = x, so because the hyperbolic secant is one-to-one on s0, 1], we have 1 cosh-1 a x b = sech-1 x.

Derivatives of Inverse Hyperbolic Functions An important use of inverse hyperbolic functions lies in antiderivatives that reverse the derivative formulas in Table 7.9. The restrictions ƒ u ƒ 6 1 and ƒ u ƒ 7 1 on the derivative formulas for tanh-1 u and -1 coth u come from the natural restrictions on the values of these functions. (See Figure 7.9a and b.) The distinction between ƒ u ƒ 6 1 and ƒ u ƒ 7 1 becomes important when we convert the derivative formulas into integral formulas. We illustrate how the derivatives of the inverse hyperbolic functions are found in Example 2, where we calculate dscosh-1 ud>dx. The other derivatives are obtained by similar calculations.

424

Chapter 7: Integrals and Transcendental Functions

TABLE 7.9

Derivatives of inverse hyperbolic functions

dssinh-1 ud du 1 = 2 dx dx 21 + u dscosh-1 ud du 1 = , 2 dx 2u - 1 dx

u 7 1

dstanh-1 ud du 1 = , 2 dx dx 1 - u

ƒuƒ 6 1

dscoth-1 ud du 1 = , dx 1 - u 2 dx

ƒuƒ 7 1

dssech-1 ud du 1 = , 2 dx dx u21 - u

0 6 u 6 1

dscsch-1 ud du 1 = , u Z 0 2 dx dx ƒ u ƒ 21 + u

EXAMPLE 2

Show that if u is a differentiable function of x whose values are greater

than 1, then d du 1 scosh-1 ud = . 2 dx 2u - 1 dx First we find the derivative of y = cosh-1 x for x 7 1 by applying Theorem 3 of Section 3.8 with ƒsxd = cosh x and ƒ -1sxd = cosh-1 x. Theorem 3 can be applied because the derivative of cosh x is positive for 0 6 x. Solution

1

sƒ -1 d¿sxd =

ƒ¿sƒ

-1

Theorem 3, Section 3.8

sxdd

1 sinh scosh-1 xd

=

ƒ¿sud = sinh u

1 2 2cosh scosh-1 xd - 1 1 = 2 2x - 1 =

HISTORICAL BIOGRAPHY

cosh2 u - sinh2 u = 1, sinh u = 2cosh2 u - 1 cosh scosh-1 xd = x

The Chain Rule gives the final result:

Sonya Kovalevsky (1850–1891)

d du 1 scosh-1 ud = . 2 dx 2u - 1 dx With appropriate substitutions, the derivative formulas in Table 7.9 lead to the integration formulas in Table 7.10. Each of the formulas in Table 7.10 can be verified by differentiating the expression on the right-hand side.

EXAMPLE 3

Evaluate 1

2 dx

L0 23 + 4x 2

.

7.3

2.

u = sinh-1 a a b + C,

du L 2a + u 2

2

a 7 0

du u = cosh-1 a a b + C, L 2u 2 - a 2

u 7 a 7 0

1 -1 u a tanh a a b + C,

3.

4.

5.

u2 6 a2

du = d 2 2 1 La - u -1 u a coth a a b + C,

u2 7 a2

u du 1 = - a sech-1 a a b + C, L u2a 2 - u 2 du L u2a + u 2

Solution

425

Integrals leading to inverse hyperbolic functions

TABLE 7.10

1.

Hyperbolic Functions

2

u 1 = - a csch-1 ƒ a ƒ + C,

0 6 u 6 a u Z 0 and a 7 0

The indefinite integral is 2 dx du = L 23 + 4x 2 L 2a 2 + u 2 u = sinh-1 a a b + C = sinh-1 a

2x 23

u = 2x,

du = 2 dx,

a = 23

Formula from Table 7.10

b + C.

Therefore, 1

2 dx

L0 23 + 4x 2

= sinh-1 a

2x 23

1

b d = sinh-1 a 0

= sinh-1

2 b - sinh-1 s0d 23 2 a b - 0 L 0.98665. 23

Exercises 7.3 Values and Identities Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh2 x - sinh2 x = 1 to find the values of the remaining five hyperbolic functions. 3 4 17 , 3. cosh x = 15

4 3 13 , 4. cosh x = 5

x 7 0

a. sinh 2x = 2 sinh x cosh x. x 7 0

Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can. 5. 7. 9. 10.

sinh sx + yd = sinh x cosh y + cosh x sinh y, cosh sx + yd = cosh x cosh y + sinh x sinh y. Then use them to show that

2. sinh x =

1. sinh x = -

11. Prove the identities

2 cosh (ln x) 6. sinh (2 ln x) 8. cosh 3x - sinh 3x cosh 5x + sinh 5x ssinh x + cosh xd4 ln scosh x + sinh xd + ln scosh x - sinh xd

b. cosh 2x = cosh2 x + sinh2 x . 12. Use the definitions of cosh x and sinh x to show that cosh2 x - sinh2 x = 1 . Finding Derivatives In Exercises 13–24, find the derivative of y with respect to the appropriate variable. x 1 13. y = 6 sinh 14. y = sinh s2x + 1d 3 2

426

Chapter 7: Integrals and Transcendental Functions

17. y = ln ssinh zd

1 16. y = t 2 tanh t 18. y = ln scosh zd

19. y = sech us1 - ln sech ud

20. y = csch us1 - ln csch ud

15. y = 2 2t tanh 2t

1 tanh2 y 2

21. y = ln cosh y -

22. y = ln sinh y -

p>4

55.

23. y = sx + 1d sech sln xd (Hint: Before differentiating, express in terms of exponentials and simplify.) 24. y = s4x 2 - 1d csch sln 2xd In Exercises 25–36, find the derivative of y with respect to the appropriate variable. 25. y = sinh-1 1x 29. y = s1 - td coth

-1

2t

31. y = cos-1 x - x sech-1 x 33. y = csch

-1

1 a b 2

30. y = s1 - t d coth

-1

34. y = csch

-1

cosh

u

2

b. 38.

L

L

sech x dx = tan-1 ssinh xd + C

x sech-1 x dx =

x2 1 sech-1 x - 21 - x 2 + C 2 2

x2 - 1 x x coth-1 x dx = coth-1 x + + C 39. 2 2 L 1 tanh-1 x dx = x tanh-1 x + ln s1 - x 2 d + C 40. 2 L

-1

L

x 6 cosh a - ln 3 b dx 43. 2 L x tanh dx 45. 7 L 1 sech ax - b dx 47. 2 L 2

49.

sech 2t tanh 2t dt L

2t

42.

L

46. 48. 50.

L L

Lln 2

-ln 2

53.

L-ln 4

52. 54.

x Z 0 ƒxƒ 7 1

65. sech-1 s3>5d

66. csch-1 s -1> 13d

Evaluate the integrals in Exercises 67–74 in terms of a. inverse hyperbolic functions. b. natural logarithms. L0

1>3

dx

68.

24 + x 2

L0

2

sinh

x dx 5

69.

4 cosh s3x - ln 2d dx u

coth

L

23

71.

L1>5

73.

70.

6 dx 21 + 9x 2

1>2

dx 2 L5>4 1 - x

p

du

csch s5 - xd dx

csch sln td coth sln td dt t L tanh 2x dx

L0 ln 2

2e u cosh u du

0 6 x … 1

64. coth-1 s5>4d

L0

dx 1 - x2

2

dx x21 - 16x

2

72.

dx L1 x24 + x 2 e

cos x dx

L0 21 + sin x 2

74.

dx L1 x21 + sln xd2

Applications and Examples 75. Show that if a function ƒ is defined on an interval symmetric about the origin (so that ƒ is defined at -x whenever it is defined at x), then

ln 2

coth x dx

ƒxƒ 6 1

2

ln 4

51.

x Ú 1

63. tanh-1 s -1>2d

3>13

44.

-q 6 x 6 q

62. cosh-1 s5>3d

223

sinh 2x dx

2

61. sinh-1 s -5>12d

67.

Evaluating Integrals Evaluate the integrals in Exercises 41–60. 41.

x 4 sinh2 a b dx 2

Use the formulas in the box here to express the numbers in Exercises 61–66 in terms of natural logarithms.

sech x dx = sin-1 stanh xd + C

L

L0

21 + x 2 1 csch-1 x = ln a x + b, ƒxƒ 1 x + 1 , coth-1 x = ln 2 x - 1

0 6 x 6 p>2

Integration Formulas Verify the integration formulas in Exercises 37–40. 37. a.

60.

-1

35. y = sinh-1 stan xd 36. y = cosh-1 ssec xd,

8 cosh 1x dx 1x

ln 10

x cosh2 a b dx 2 L-ln 2

x = ln A x + 2x - 1 B , 1 1 + x , tanh x = ln 2 1 - x 1 + 21 - x 2 sech-1 x = ln a b, x

t

32. y = ln x + 21 - x 2 sech-1 x

u

L1

sinh-1 x = ln A x + 2x 2 + 1 B ,

28. y = su2 + 2ud tanh-1 su + 1d 2

58.

Inverse Hyperbolic Functions and Integrals When hyperbolic function keys are not available on a calculator, it is still possible to evaluate the inverse hyperbolic functions by expressing them as logarithms, as shown here.

26. y = cosh-1 2 2x + 1

27. y = s1 - ud tanh-1 u

4

cosh sln td dt t L1 0

59.

2 sinh ssin ud cos u du

L0

2

57.

1 coth2 y 2

2

L-p>4

p>2

cosh stan ud sec2 u du 56.

L0

4e -u sinh u du

ƒsxd =

ƒsxd - ƒs -xd ƒsxd + ƒs -xd + . 2 2

Then show that sƒsxd + ƒs -xdd>2 is even and that sƒsxd - ƒs -xdd>2 is odd.

(1)

7.3 76. Derive the formula sinh-1 x = ln A x + 2x 2 + 1 B for all real x. Explain in your derivation why the plus sign is used with the square root instead of the minus sign. 77. Skydiving If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body’s velocity t sec into the fall satisfies the differential equation m

427

choose a coordinate system for the plane of the cable in which the x-axis is horizontal, the force of gravity is straight down, the positive y-axis points straight up, and the lowest point of the cable lies at the point y = H>w on the y-axis (see accompanying figure), then it can be shown that the cable lies along the graph of the hyperbolic cosine w H y = w cosh x . H

dy = mg - ky 2 , dt

where k is a constant that depends on the body’s aerodynamic properties and the density of the air. (We assume that the fall is short enough so that the variation in the air’s density will not affect the outcome significantly.)

Hyperbolic Functions

y

y ⫽ H cosh w x w H

Hanging cable

a. Show that

satisfies the differential equation and the initial condition that y = 0 when t = 0 . b. Find the body’s limiting velocity, limt: q y . c. For a 160-lb skydiver smg = 160d , with time in seconds and distance in feet, a typical value for k is 0.005. What is the diver’s limiting velocity? 78. Accelerations whose magnitudes are proportional to displacement Suppose that the position of a body moving along a coordinate line at time t is

H w

H 0

gk mg tanh a m tb y = A A k

x

Such a curve is sometimes called a chain curve or a catenary, the latter deriving from the Latin catena, meaning “chain.” a. Let P(x, y) denote an arbitrary point on the cable. The next accompanying figure displays the tension at P as a vector of length (magnitude) T, as well as the tension H at the lowest point A. Show that the cable’s slope at P is dy w tan f = = sinh x . H dx y

a. s = a cos kt + b sin kt.

y ⫽ H cosh w x w H

b. s = a cosh kt + b sinh kt . Show in both cases that the acceleration d 2s>dt 2 is proportional to s but that in the first case it is directed toward the origin, whereas in the second case it is directed away from the origin. 79. Volume A region in the first quadrant is bounded above by the curve y = cosh x , below by the curve y = sinh x , and on the left and right by the y-axis and the line x = 2 , respectively. Find the volume of the solid generated by revolving the region about the x-axis. 80. Volume The region enclosed by the curve y = sech x , the x-axis, and the lines x = ; ln 23 is revolved about the x-axis to generate a solid. Find the volume of the solid. 81. Arc length Find the length of the graph of y = s1>2d cosh 2x from x = 0 to x = ln 25 . 82. Use the definitions of the hyperbolic functions to find each of the following limits. lim tanh x

(a) lim tanh x

(b)

(c) lim sinh x

(d)

(e) lim sech x

(f ) lim coth x

(g) lim+coth x

(h) lim-coth x

x: q

x: q x: q x: 0

(i)

x: - q

lim sinh x

x: - q x: q x: 0

lim csch x

x: - q

83. Hanging cables Imagine a cable, like a telephone line or TV cable, strung from one support to another and hanging freely. The cable’s weight per unit length is a constant w and the horizontal tension at its lowest point is a vector of length H. If we

T ⎛ ⎛ A ⎝0, H w⎝

P(x, y)

T cos

H x

0

b. Using the result from part (a) and the fact that the horizontal tension at P must equal H (the cable is not moving), show that T = wy . Hence, the magnitude of the tension at P(x, y) is exactly equal to the weight of y units of cable. 84. (Continuation of Exercise 83.) The length of arc AP in the Exercise 83 figure is s = s1>ad sinh ax , where a = w>H . Show that the coordinates of P may be expressed in terms of s as 1 x = a sinh-1 as,

y =

A

s2 +

1 . a2

85. Area Show that the area of the region in the first quadrant enclosed by the curve y = s1>ad cosh ax , the coordinate axes, and the line x = b is the same as the area of a rectangle of height 1> a and length s, where s is the length of the curve from x = 0 to x = b . Draw a figure illustrating this result. 86. The hyperbolic in hyperbolic functions Just as x = cos u and y = sin u are identified with points (x, y) on the unit circle, the functions x = cosh u and y = sinh u are identified with

428

Chapter 7: Integrals and Transcendental Functions b. Differentiate both sides of the equation in part (a) with respect to u to show that

points sx, yd on the right-hand branch of the unit hyperbola, x2 - y2 = 1.

A¿sud =

1

0

u→

c. Solve this last equation for A(u). What is the value of A(0)? What is the value of the constant of integration C in your solution? With C determined, what does your solution say about the relationship of u to A(u)?

P(cosh u, sinh u)

u⫽0

x

1 u→

–1

∞

y

1 . 2

x2 ⫺ y 2 ⫽ 1

y sy m pt ot e

−∞

x 2 ⫺ y2 ⫽ 1

y 2

x ⫹

y2

⫽1

P(cos u, sin u)

A

Since cosh2 u - sinh2 u = 1 , the point (cosh u, sinh u) lies on the right-hand branch of the hyperbola x 2 - y 2 = 1 for every value of u (Exercise 86).

P(cosh u, sinh u) u is twice the area of sector AOP. x u⫽0

A

O

A

x u⫽0

e

ot

pt

m

sy

A

Another analogy between hyperbolic and circular functions is that the variable u in the coordinates (cosh u, sinh u) for the points of the right-hand branch of the hyperbola x 2 - y 2 = 1 is twice the area of the sector AOP pictured in the accompanying figure. To see why this is so, carry out the following steps.

O u is twice the area of sector AOP.

a. Show that the area A(u) of sector AOP is Asud =

1 cosh u sinh u 2 L1

Chapter 7

cosh u

2x 2 - 1 dx .

Questions to Guide Your Review

1. How is the natural logarithm function defined as an integral? What are its domain, range, and derivative? What arithmetic properties does it have? Comment on its graph. 2. What integrals lead to logarithms? Give examples. 3. What are the integrals of tan x and cot x? sec x and csc x? 4. How is the exponential function e x defined? What are its domain, range, and derivative? What laws of exponents does it obey? Comment on its graph. 5. How are the functions a x and loga x defined? Are there any restrictions on a? How is the graph of loga x related to the graph of ln x? What truth is there in the statement that there is really only one exponential function and one logarithmic function? 6. How do you solve separable first-order differential equations?

Chapter 7

L

e x sin se x d dx

7. What is the law of exponential change? How can it be derived from an initial value problem? What are some of the applications of the law? 8. What are the six basic hyperbolic functions? Comment on their domains, ranges, and graphs. What are some of the identities relating them? 9. What are the derivatives of the six basic hyperbolic functions? What are the corresponding integral formulas? What similarities do you see here with the six basic trigonometric functions? 10. How are the inverse hyperbolic functions defined? Comment on their domains, ranges, and graphs. How can you find values of sech-1 x, csch-1 x , and coth-1 x using a calculator’s keys for cosh-1 x, sinh-1 x , and tanh-1 x ? 11. What integrals lead naturally to inverse hyperbolic functions?

Practice Exercises

Integration Evaluate the integrals in Exercises 1–12. 1.

One of the analogies between hyperbolic and circular functions is revealed by these two diagrams (Exercise 86).

2.

L

p

3. e t cos s3e t - 2d dt

L0

tan

x dx 3

1>4

4.

L1>6

2 cot px dx

p>6

5.

cos t dt L-p>2 1 - sin t

6.

ex sec ex dx L

Chapter 7

7.

ln sx - 5d dx L x - 5 7

9.

L1

1

Le x2ln x

L

10.

L1

24. a. If sln xd>x = sln 2d>2 , must x = 2 ?

1 dx 5x

b. If sln xd>x = -2 ln 2 , must x = 1>2 ? Give reasons for your answers.

4

dx

12.

L2

s1 + ln tdt ln t dt

25. The quotient slog4 xd>slog2 xd has a constant value. What value? Give reasons for your answer. T 26. log x s2d vs. log2 sxd How does ƒsxd = logx s2d compare with gsxd = log2 sxd ? Here is one way to find out.

Solving Equations with Logarithmic or Exponential Terms In Exercises 13–18, solve for y. 13. 3y = 2y + 1

14. 4-y = 3y + 2

15. 9e 2y = x 2

16. 3y = 3 ln x

17. ln (y - 1) = x + ln y

18. ln (10 ln y) = ln 5x

a. Use the equation loga b = sln bd>sln ad to express ƒ(x) and g(x) in terms of natural logarithms. b. Graph ƒ and g together. Comment on the behavior of ƒ in relation to the signs and values of g. In Exercises 27–30, solve the differential equation. dy 3ysx + 1d2 = 2y cos2 2y 27. 28. y¿ = y - 1 dx

Theory and Applications 19. The function ƒsxd = e x + x , being differentiable and one-to-one, has a differentiable inverse ƒ -1sxd . Find the value of dƒ -1>dx at the point ƒ(ln 2).

29. yy¿ = sec y 2 sec2 x

dƒ -1 1 ` = . dx ƒsxd ƒ¿sxd

33. x dy - A y + 2y B dx = 0, ys1d = 1

21. A particle is traveling upward and to the right along the curve y = ln x . Its x-coordinate is increasing at the rate sdx>dtd = 1x m>sec . At what rate is the y-coordinate changing at the point se 2, 2d ?

dx ex , ys0d = 1 = 2x dy e + 1 35. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed? 34. y -2

22. A girl is sliding down a slide shaped like the curve y = 9e -x>3 . Her y-coordinate is changing at the rate dy>dt = s -1>4d29 - y ft>sec . At approximately what rate is her x-coordinate changing when she reaches the bottom of the slide at x = 9 ft ? (Take e 3 to be 20 and round your answer to the nearest ft> sec.)

t: q

36. Cooling a pie A deep-dish apple pie, whose internal temperature was 220°F when removed from the oven, was set out on a breezy 40°F porch to cool. Fifteen minutes later, the pie’s internal temperature was 180°F. How long did it take the pie to cool from there to 70°F?

Additional and Advanced Exercises

1. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y = e -x , and the vertical line x = t, t 7 0 . Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits. a. lim Astd

30. y cos2 x dy + sin x dx = 0

In Exercises 31–34, solve the initial value problem. dy = e -x - y - 2, ys0d = - 2 31. dx dy y ln y , ys0d = e 2 = 32. dx 1 + x2

20. Find the inverse of the function ƒsxd = 1 + s1>xd, x Z 0 . Then show that ƒ -1sƒsxdd = ƒsƒ -1sxdd = x and that

Chapter 7

429

23. The functions ƒsxd = ln 5x and gsxd = ln 3x differ by a constant. What constant? Give reasons for your answer.

cos s1 - ln yd dy y 32

3 x dx

e2

11.

8.

Additional and Advanced Exercises

b. lim Vstd>Astd t: q

c. lim+ Vstd>Astd t: 0

2. Varying a logarithm’s base a. Find lim loga 2 as a : 0 +, 1-, 1+ , and q . T b. Graph y = loga 2 as a function of a over the interval 0 6 a … 4. T 3. Graph ƒsxd = tan-1 x + tan-1s1>xd for -5 … x … 5 . Then use calculus to explain what you see. How would you expect ƒ to behave beyond the interval [-5, 5] ? Give reasons for your answer.

T

4. Graph ƒsxd = ssin xdsin x over [0, 3p] . Explain what you see. 5. Even-odd decompositions a. Suppose that g is an even function of x and h is an odd function of x. Show that if g sxd + h sxd = 0 for all x then g sxd = 0 for all x and hsxd = 0 for all x. b. Use the result in part (a) to show that if ƒsxd = ƒE sxd + ƒO sxd is the sum of an even function ƒE sxd and an odd function ƒO sxd , then ƒE sxd = sƒsxd + ƒs -xdd>2

and

ƒO sxd = sƒsxd - ƒs -xdd>2 .

c. What is the significance of the result in part (b)?

430

Chapter 7: Integrals and Transcendental Functions

6. Let g be a function that is differentiable throughout an open interval containing the origin. Suppose g has the following properties: g sxd + g s yd for all real numbers x, y, and 1 - g sxdg s yd x + y in the domain of g.

i. g sx + yd =

ii. lim g shd = 0 h:0

g shd = 1 h:0 h

iii. lim

a. Show that gs0d = 0 . b. Show that g¿sxd = 1 + [g sxd]2 . c. Find g (x) by solving the differential equation in part (b). 7. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y = 1>s1 + x 2 d and y = -1>s1 + x 2 d and by the lines x = 0 and x = 1 . 8. Solid of revolution The region between the curve y = 1>s2 1xd and the x-axis from x = 1>4 to x = 4 is revolved about the x-axis to generate a solid.

9. The Rule of 70 If you use the approximation ln 2 L 0.70 (in place of 0.69314 Á ), you can derive a rule of thumb that says, “To estimate how many years it will take an amount of money to double when invested at r percent compounded continuously, divide r into 70.” For instance, an amount of money invested at 5% will double in about 70>5 = 14 years . If you want it to double in 10 years instead, you have to invest it at 70>10 = 7% . Show how the Rule of 70 is derived. (A similar “Rule of 72” uses 72 instead of 70, because 72 has more integer factors.) T 10. Urban gardening A vegetable garden 50 ft wide is to be grown between two buildings, which are 500 ft apart along an east-west line. If the buildings are 200 ft and 350 ft tall, where should the garden be placed in order to receive the maximum number of hours of sunlight exposure? (Hint: Determine the value of x in the accompanying figure that maximizes sunlight exposure for the garden.)

350 ft tall 200 ft tall

a. Find the volume of the solid. b. Find the centroid of the region.

u2

West x

50

u1 450 2 x

East

8 TECHNIQUES OF INTEGRATION OVERVIEW The Fundamental Theorem tells us how to evaluate a definite integral once we have an antiderivative for the integrand function. Table 8.1 summarizes the forms of antiderivatives for many of the functions we have studied so far, and the substitution method helps us use the table to evaluate more complicated functions involving these basic ones. In this chapter we study a number of other important techniques for finding antiderivatives (or indefinite integrals) for many combinations of functions whose antiderivatives cannot be found using the methods presented before.

TABLE 8.1

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

L L L L L L L L L L L

Basic integration formulas

k dx = kx + C x n dx =

sany number kd

xn+1 + C n + 1

sn Z -1d

12. 13.

dx x = ln ƒ x ƒ + C

14.

e x dx = e x + C

15.

a x dx =

ax + C ln a

sa 7 0, a Z 1d

16.

sin x dx = -cos x + C

17.

cos x dx = sin x + C

18.

sec2 x dx = tan x + C

19.

csc2 x dx = -cot x + C

20.

sec x tan x dx = sec x + C csc x cot x dx = -csc x + C

21. 22.

L L L L L L

tan x dx = ln ƒ sec x ƒ + C cot x dx = ln ƒ sin x ƒ + C sec x dx = ln ƒ sec x + tan x ƒ + C csc x dx = -ln ƒ csc x + cot x ƒ + C sinh x dx = cosh x + C cosh x dx = sinh x + C x = sin-1 a a b + C

dx L 2a - x 2

2

dx x 1 = a tan-1 a a b + C 2 2 La + x dx L x2x - a 2

dx L 2a + x 2

2

2

x 1 = a sec-1 ` a ` + C x = sinh-1 a a b + C

dx x = cosh-1 a a b + C L 2x 2 - a 2

sa 7 0d sx 7 a 7 0d

431

432

Chapter 8: Techniques of Integration

8.1

Integration by Parts Integration by parts is a technique for simplifying integrals of the form

L

ƒsxdgsxd dx.

It is useful when ƒ can be differentiated repeatedly and g can be integrated repeatedly without difficulty. The integrals

L

x cos x dx

and

L

x 2e x dx

are such integrals because ƒsxd = x or ƒ(x) = x 2 can be differentiated repeatedly to become zero, and g(x) = cos x or gsxd = e x can be integrated repeatedly without difficulty. Integration by parts also applies to integrals like L

ln x dx

and

L

e x cos x dx.

In the first case, ƒ(x) = ln x is easy to differentiate and g(x) = 1 easily integrates to x. In the second case, each part of the integrand appears again after repeated differentiation or integration.

Product Rule in Integral Form If ƒ and g are differentiable functions of x, the Product Rule says that d [ƒsxdgsxd] = ƒ¿sxdgsxd + ƒsxdg¿sxd. dx In terms of indefinite integrals, this equation becomes d [ƒsxdgsxd] dx = [ƒ¿sxdgsxd + ƒsxdg¿sxd] dx dx L L or d [ f sxdgsxd] dx = ƒ¿sxdgsxd dx + ƒ(x)g¿(x) dx. L dx L L Rearranging the terms of this last equation, we get

L

ƒsxdg¿sxd dx =

d [ƒsxdgsxd] dx ƒ¿(x)g(x) dx, L dx L

leading to the integration by parts formula

L

ƒsxdg¿sxd dx = ƒsxdgsxd -

L

ƒ¿sxdgsxd dx

(1)

Sometimes it is easier to remember the formula if we write it in differential form. Let u = ƒsxd and y = gsxd. Then du = ƒ¿sxd dx and dy = g¿sxd dx. Using the Substitution Rule, the integration by parts formula becomes

8.1

Integration by Parts

433

Integration by Parts Formula L

u dy = uy -

L

y du

(2)

This formula expresses one integral, 1 u dy, in terms of a second integral, 1 y du. With a proper choice of u and y, the second integral may be easier to evaluate than the first. In using the formula, various choices may be available for u and dy. The next examples illustrate the technique. To avoid mistakes, we always list our choices for u and dy, then we add to the list our calculated new terms du and y, and finally we apply the formula in Equation (2).

EXAMPLE 1

Find L

Solution

We use the formula u = x, du = dx,

L

x cos x dx.

u dy = uy -

dy = cos x dx, y = sin x.

L

y du with

Simplest antiderivative of cos x

Then L

x cos x dx = x sin x -

L

sin x dx = x sin x + cos x + C.

There are four choices available for u and dy in Example 1: 1. Let u = 1 and dy = x cos x dx. 3. Let u = x cos x and dy = dx.

2. Let u = x and dy = cos x dx. 4. Let u = cos x and dy = x dx.

Choice 2 was used in Example 1. The other three choices lead to integrals we don’t know how to integrate. For instance, Choice 3 leads to the integral L

sx cos x - x 2 sin xd dx.

The goal of integration by parts is to go from an integral 1 u dy that we don’t see how to evaluate to an integral 1 y du that we can evaluate. Generally, you choose dy first to be as much of the integrand, including dx, as you can readily integrate; u is the leftover part. When finding y from dy, any antiderivative will work and we usually pick the simplest one; no arbitrary constant of integration is needed in y because it would simply cancel out of the right-hand side of Equation (2).

EXAMPLE 2

Find L

ln x dx.

Since 1 ln x dx can be written as 1 ln x # 1 dx, we use the formula 1 u dy = uy - 1 y du with

Solution

u = ln x 1 du = x dx,

Simplifies when differentiated

dy = dx y = x.

Easy to integrate Simplest antiderivative

434

Chapter 8: Techniques of Integration

Then from Equation (2), L

ln x dx = x ln x -

L

1 x # x dx = x ln x -

L

dx = x ln x - x + C.

Sometimes we have to use integration by parts more than once.

EXAMPLE 3

Evaluate

L Solution

x 2e x dx.

With u = x 2, dy = e x dx, du = 2x dx, and y = e x , we have L

x 2e x dx = x 2e x - 2

L

xe x dx.

The new integral is less complicated than the original because the exponent on x is reduced by one. To evaluate the integral on the right, we integrate by parts again with u = x, dy = e x dx. Then du = dx, y = e x , and L

xe x dx = xe x -

L

e x dx = xe x - e x + C.

Using this last evaluation, we then obtain L

x 2e x dx = x 2e x - 2

L

xe x dx

= x 2e x - 2xe x + 2e x + C. The technique of Example 3 works for any integral 1 x ne x dx in which n is a positive integer, because differentiating x n will eventually lead to zero and integrating e x is easy. Integrals like the one in the next example occur in electrical engineering. Their evaluation requires two integrations by parts, followed by solving for the unknown integral.

EXAMPLE 4

Evaluate L

Solution

e x cos x dx.

Let u = e x and dy = cos x dx. Then du = e x dx, y = sin x, and L

e x cos x dx = e x sin x -

L

e x sin x dx.

The second integral is like the first except that it has sin x in place of cos x. To evaluate it, we use integration by parts with u = e x,

dy = sin x dx,

y = -cos x,

du = e x dx.

Then L

e x cos x dx = e x sin x - a-e x cos x = e x sin x + e x cos x -

L

L

s -cos xdse x dxdb

e x cos x dx.

8.1

Integration by Parts

435

The unknown integral now appears on both sides of the equation. Adding the integral to both sides and adding the constant of integration give 2

L

e x cos x dx = e x sin x + e x cos x + C1 .

Dividing by 2 and renaming the constant of integration give L

EXAMPLE 5

e x cos x dx =

e x sin x + e x cos x + C. 2

Obtain a formula that expresses the integral

cosn x dx L in terms of an integral of a lower power of cos x. We may think of cosn x as cosn - 1 x

Solution

u = cosn - 1 x

#

cos x. Then we let

and

dy = cos x dx,

so that du = sn - 1d cosn - 2 x s -sin x dxd

and

y = sin x.

Integration by parts then gives

L

cosn x dx = cosn - 1 x sin x + sn - 1d sin2 x cosn - 2 x dx L = cosn - 1 x sin x + sn - 1d s1 - cos2 xd cosn - 2 x dx L = cosn - 1 x sin x + sn - 1d cosn - 2 x dx - sn - 1d cosn x dx. L L

If we add sn - 1d cosn x dx L to both sides of this equation, we obtain n

L

cosn x dx = cosn - 1 x sin x + sn - 1d cosn - 2 x dx. L

We then divide through by n, and the final result is

L

cosn x dx =

cosn - 1 x sin x n - 1 + n n

L

cosn - 2 x dx.

The formula found in Example 5 is called a reduction formula because it replaces an integral containing some power of a function with an integral of the same form having the power reduced. When n is a positive integer, we may apply the formula repeatedly until the remaining integral is easy to evaluate. For example, the result in Example 5 tells us that

L

cos2 x sin x 2 + cos x dx 3 3L 2 1 = cos2 x sin x + sin x + C. 3 3

cos3 x dx =

436

Chapter 8: Techniques of Integration

Evaluating Definite Integrals by Parts The integration by parts formula in Equation (1) can be combined with Part 2 of the Fundamental Theorem in order to evaluate definite integrals by parts. Assuming that both ƒ¿ and g¿ are continuous over the interval [a, b], Part 2 of the Fundamental Theorem gives Integration by Parts Formula for Definite Integrals ƒsxdg¿sxd dx = ƒsxdgsxd D a -

b

La

b

b

La

ƒ¿sxdgsxd dx

(3)

In applying Equation (3), we normally use the u and y notation from Equation (2) because it is easier to remember. Here is an example. Find the area of the region bounded by the curve y = xe -x and the x-axis from x = 0 to x = 4.

y

EXAMPLE 6

1

–1

Solution

y xe –x

0.5

1

0

2

3

The region is shaded in Figure 8.1. Its area is 4

L0

x

4

xe -x dx.

Let u = x, dy = e -x dx, y = -e -x , and du = dx. Then,

–0.5

4

L0

–1

xe -x dx = -xe -x D 0 -

4

4

L0

s -e -x d dx

= [-4e -4 - s0d] +

FIGURE 8.1 The region in Example 6.

= -4e -4 - e -x D 0

4

L0

e -x dx

4

= -4e -4 - e -4 - s -e 0 d = 1 - 5e -4 L 0.91.

Exercises 8.1 Integration by Parts Evaluate the integrals in Exercises 1–24 using integration by parts. 1. 3.

L L

x dx 2

2.

t 2 cos t dt

4.

x sin

L L

2

5. 7. 9. 11. 13.

L1 L L L L

15.

u cos pu du

17.

x 2 sin x dx

19.

L L L

x 3e x dx

16.

sx 2 - 5xde x dx

18.

x 5e x dx

20.

e u sin u du

22.

e 2x cos 3x dx

24.

L L L

e

x ln x dx xe x dx

6. 8.

x 2 e - x dx

10.

tan-1 y dy

12.

x sec2 x dx

14.

L1 L L L L

x 3 ln x dx xe 3x dx (x 2 - 2x + 1) e 2x dx

21. 23.

L L

L L

p 4e -p dp sr 2 + r + 1de r dr t 2e 4t dt e -y cos y dy e -2x sin 2x dx

sin-1 y dy

Using Substitution Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

4x sec2 2x dx

25.

L

e 23s + 9 ds

1

26.

L0

x21 - x dx

8.1 p>3

27. 29.

x tan2 x dx

L0 L

28.

sin sln xd dx

30.

L L

ln sx + x 2 d dx

31. 33. 35. 37. 39. 41. 43. 45.

L L

x sec x dx x (ln x) 2 dx

ln x dx 2 L x 4

L L L

x 3 e x dx

38.

x 3 2x 2 + 1 dx

40.

sin 3x cos 2x dx

42.

x

L L

x

e sin e dx

44.

cos 2x dx

46.

3

L L L

L0 2

49.

L2>23

x 5 e x dx x 2 sin x 3 dx sin 2x cos 4x dx

e 1x dx L 1x L

p>2

47.

a

2x e 1x dx p>2

u 2 sin 2u du

48.

t sec-1 t dt

50.

x 3 cos 2x dx

L0

1>22

L0

2n - 1 2n + 1 bp … x … a bp, 2 2

n an arbitrary positive integer? Give reasons for your answer. y 10

cos 1x dx 32. L 1x 1 dx 34. x (ln x) 2 L (ln x) 3 36. x dx L

2

437

d. What pattern do you see? What is the area between the curve and the x-axis for

zsln zd2 dz

Evaluating Integrals Evaluate the integrals in Exercises 31–50. Some integrals do not require integration by parts.

Integration by Parts

0

y x cos x

2

3 5 2 2

7 2

x

–10

53. Finding volume Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e x , and the line x = ln 2 about the line x = ln 2 . 54. Finding volume Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e -x , and the line x = 1 a. about the y-axis. b. about the line x = 1 . 55. Finding volume Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos x, 0 … x … p>2 , about a. the y-axis.

2x sin-1 sx 2 d dx

Theory and Examples 51. Finding area Find the area of the region enclosed by the curve y = x sin x and the x-axis (see the accompanying figure) for

b. the line x = p>2 . 56. Finding volume Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin x, 0 … x … p , about a. the y-axis. b. the line x = p .

a. 0 … x … p. b. p … x … 2p.

(See Exercise 51 for a graph.)

c. 2p … x … 3p . d. What pattern do you see here? What is the area between the curve and the x-axis for np … x … sn + 1dp, n an arbitrary nonnegative integer? Give reasons for your answer. y

d. Find the centroid of the region.

5 0

2

3

x

–5

52. Finding area Find the area of the region enclosed by the curve y = x cos x and the x-axis (see the accompanying figure) for a. p>2 … x … 3p>2. b. 3p>2 … x … 5p>2. c. 5p>2 … x … 7p>2.

a. Find the area of the region. b. Find the volume of the solid formed by revolving this region about the x-axis. c. Find the volume of the solid formed by revolving this region about the line x = -2.

y x sin x

10

57. Consider the region bounded by the graphs of y = ln x, y = 0, and x = e.

58. Consider the region bounded by the graphs of y = tan-1 x, y = 0, and x = 1. a. Find the area of the region. b. Find the volume of the solid formed by revolving this region about the y-axis. 59. Average value A retarding force, symbolized by the dashpot in the accompanying figure, slows the motion of the weighted spring so that the mass’s position at time t is y = 2e -t cos t,

t Ú 0.

438

Chapter 8: Techniques of Integration

Find the average value of y over the interval 0 … t … 2p. y

The idea is to take the most complicated part of the integral, in this case ƒ -1sxd , and simplify it first. For the integral of ln x, we get L

ln x dx =

y = ln x, x = e y dx = e y dy

ye y dy

L = ye y - e y + C

= x ln x - x + C . 0

For the integral of cos-1 x we get

y

Mass

L

cos-1 x dx = x cos-1 x -

L

y = cos-1 x

cos y dy

= x cos-1 x - sin y + C

Dashpot

= x cos-1 x - sin scos-1 xd + C . Use the formula 60. Average value In a mass-spring-dashpot system like the one in Exercise 59, the mass’s position at time t is y = 4e -t ssin t - cos td,

t Ú 0.

Find the average value of y over the interval 0 … t … 2p. Reduction Formulas In Exercises 61–64, use integration by parts to establish the reduction formula. 61. 62. 63.

L L L

x n cos x dx = x n sin x - n

L

x n - 1 sin x dx

x n sin x dx = -x n cos x + n

L

x ne ax dx =

n ax

x e a

n - a

a Z 0

sln xdn dx = xsln xdn - n sln xdn - 1 dx L L 65. Show that La

a

b

Lx

67. 69.

L L

La

21 - x 2 dx =

a. b.

L

L

a.

L

= xƒ -1sxd -

ƒs yd dy

L

b.

y = ƒ-1sxd, x = ƒs yd dx = ƒ¿s yd dy

yƒ¿s yd dy

= yƒs yd -

sin-1 x dx

68.

sec-1 x dx

70.

ƒs yd dy

Integration by parts with u = y, dy = ƒ¿s yd dy

L L

tan-1 x dx log2 x dx

ƒ -1sxd dx = xƒ -1sxd -

L

xa

d -1 ƒ sxdb dx. dx

(5)

L

cos-1 x dx = x cos-1 x - sin scos-1 xd + C

cos-1 x dx = x cos-1 x - 21 - x 2 + C L Can both integrations be correct? Explain.

Eq. (4)

Eq. (5)

72. Equations (4) and (5) lead to different formulas for the integral of tan-1 x :

Integrating Inverses of Functions Integration by parts leads to a rule for integrating inverses that usually gives good results: ƒ -1sxd dx =

(4)

71. Equations (4) and (5) give different formulas for the integral of cos-1 x :

sx - adƒ(x) dx.

1 1 1 x 21 - x 2 + dx. 2 2 L 21 - x 2

y = ƒ -1sxd

Exercises 71 and 72 compare the results of using Equations (4) and (5).

66. Use integration by parts to obtain the formula L

ƒs yd dy

Another way to integrate ƒ -1sxd (when ƒ -1 is integrable, of course) is to use integration by parts with u = ƒ -1sxd and dy = dx to rewrite the integral of ƒ -1 as

b

ƒ(t) dtb dx =

L

to evaluate the integrals in Exercises 67–70. Express your answers in terms of x.

L

64.

b

ƒ -1sxd dx = xƒ -1sxd -

x n - 1 cos x dx

x n - 1e ax dx,

L

L

L

tan-1 x dx = x tan-1 x - ln sec stan-1 xd + C

tan-1 x dx = x tan-1 x - ln 21 + x 2 + C L Can both integrations be correct? Explain.

Eq. (4) Eq. (5)

Evaluate the integrals in Exercises 73 and 74 with (a) Eq. (4) and (b) Eq. (5). In each case, check your work by differentiating your answer with respect to x. 73.

L

sinh-1 x dx

74.

L

tanh-1 x dx

8.2

8.2

Trigonometric Integrals

439

Trigonometric Integrals Trigonometric integrals involve algebraic combinations of the six basic trigonometric functions. In principle, we can always express such integrals in terms of sines and cosines, but it is often simpler to work with other functions, as in the integral

L

sec2 x dx = tan x + C.

The general idea is to use identities to transform the integrals we have to find into integrals that are easier to work with.

Products of Powers of Sines and Cosines We begin with integrals of the form:

L

sinm x cosn x dx,

where m and n are nonnegative integers (positive or zero). We can divide the appropriate substitution into three cases according to m and n being odd or even.

Case 1 If m is odd, we write m as 2k + 1 and use the identity sin2 x = 1 - cos2 x to obtain sinm x = sin2k + 1 x = ssin2 xdk sin x = s1 - cos2 xdk sin x.

(1)

Then we combine the single sin x with dx in the integral and set sin x dx equal to -dscos xd. If m is even and n is odd in 1 sinm x cosn x dx, we write n as 2k + 1 and use the identity cos2 x = 1 - sin2 x to obtain Case 2

cosn x = cos2k + 1 x = scos2 xdk cos x = s1 - sin2 xdk cos x. We then combine the single cos x with dx and set cos x dx equal to d(sin x). Case 3

If both m and n are even in 1 sinm x cosn x dx, we substitute sin2 x =

1 - cos 2x , 2

cos2 x =

1 + cos 2x 2

to reduce the integrand to one in lower powers of cos 2x.

Here are some examples illustrating each case.

EXAMPLE 1

Evaluate

L

sin3 x cos2 x dx.

(2)

440

Chapter 8: Techniques of Integration Solution

This is an example of Case 1. L

sin3 x cos2 x dx = = = =

L L L L

=

EXAMPLE 2

sin2 x cos2 x sin x dx

m is odd.

s1 - cos2 xd cos2 x s -d scos xdd

sin x dx = -d(cos x)

s1 - u 2 dsu 2 ds -dud

u = cos x

su 4 - u 2 d du

Multiply terms.

cos5 x cos3 x u5 u3 + C = + C 5 5 3 3

Evaluate L

Solution

L

This is an example of Case 2, where m = 0 is even and n = 5 is odd.

cos5 x dx = = =

L L L

cos4 x cos x dx =

EXAMPLE 3

L

s1 - sin2 xd2 dssin xd

cos x dx = d(sin x)

s1 - u 2 d2 du

u = sin x

s1 - 2u 2 + u 4 d du

Square 1 - u 2.

= u -

2 3 1 2 1 u + u5 + C = sin x - sin3 x + sin5 x + C 5 5 3 3

Evaluate L

Solution

cos5 x dx.

sin2 x cos4 x dx.

This is an example of Case 3. L

sin2 x cos4 x dx =

L

a

2

1 - cos 2x 1 + cos 2x ba b dx 2 2

m and n both even

=

1 s1 - cos 2xds1 + 2 cos 2x + cos2 2xd dx 8L

=

1 s1 + cos 2x - cos2 2x - cos3 2xd dx 8L

=

1 1 cx + sin 2x (cos2 2x + cos3 2x) dx d 8 2 L

For the term involving cos2 2x, we use L

cos2 2x dx = =

1 s1 + cos 4xd dx 2L 1 1 ax + sin 4xb . 2 4

Omitting the constant of integration until the final result

8.2

Trigonometric Integrals

441

For the cos3 2x term, we have s1 - sin2 2xd cos 2x dx L 1 1 1 = s1 - u 2 d du = asin 2x - sin3 2xb . 2L 2 3 Combining everything and simplifying, we get L

cos3 2x dx =

L

u = sin 2x, du = 2 cos 2x dx Again omitting C

1 1 1 ax - sin 4x + sin3 2xb + C . 16 4 3

sin2 x cos4 x dx =

Eliminating Square Roots In the next example, we use the identity cos2 u = s1 + cos 2ud>2 to eliminate a square root.

EXAMPLE 4

Evaluate

p>4

21 + cos 4x dx. L0 Solution To eliminate the square root, we use the identity 1 + cos 2u or 1 + cos 2u = 2 cos2 u. cos2 u = 2 With u = 2x, this becomes 1 + cos 4x = 2 cos2 2x. Therefore, p>4

L0

p>4

21 + cos 4x dx =

p>4

22 cos2 2x dx =

L0

222cos2 2x dx

L0

p>4

= 22

p>4

ƒ cos 2x ƒ dx = 22

cos 2x dx

L0 L0 p>4 22 22 sin 2x = 22 c d = [1 - 0] = . 2 0 2 2

cos 2x Ú 0 on [0, p>4]

Integrals of Powers of tan x and sec x We know how to integrate the tangent and secant and their squares. To integrate higher powers, we use the identities tan2 x = sec2 x - 1 and sec2 x = tan2 x + 1, and integrate by parts when necessary to reduce the higher powers to lower powers.

EXAMPLE 5

Evaluate L

Solution

L

tan4 x dx = = = =

L L L

tan2 x # tan2 x dx = tan2 x sec2 x dx tan2 x sec2 x dx tan x sec x dx 2

L

tan4 x dx.

L L L

2

L

tan2 x # ssec2 x - 1d dx

tan2 x dx ssec2 x - 1d dx sec2 x dx +

In the first integral, we let u = tan x,

du = sec2 x dx

and have L

u 2 du =

1 3 u + C1 . 3

L

dx

442

Chapter 8: Techniques of Integration

The remaining integrals are standard forms, so L

EXAMPLE 6

1 3 tan x - tan x + x + C. 3

tan4 x dx =

Evaluate L

Solution

sec3 x dx.

We integrate by parts using u = sec x,

dy = sec2 x dx,

y = tan x,

du = sec x tan x dx.

Then L

sec3 x dx = sec x tan x = sec x tan x -

L L

stan xdssec x tan x dxd ssec2 x - 1d sec x dx

tan2 x = sec2 x - 1

= sec x tan x +

sec x dx sec3 x dx. L L Combining the two secant-cubed integrals gives 2

L

sec3 x dx = sec x tan x +

L

sec x dx

and L

EXAMPLE 7

sec3 x dx =

1 1 sec x tan x + ln ƒ sec x + tan x ƒ + C. 2 2

Evaluate L

tan4 x sec4 x dx.

Solution

L

(tan4 x)(sec4 x) dx = = =

L L

(tan4 x)(1 + tan2 x)(sec2 x) dx (tan4 x + tan6 x) (sec2 x) dx (tan4 x)(sec2 x) dx +

(tan6 x) (sec2 x) dx L u5 u7 = u4 du + u6 du = + + C 5 7 L L tan5 x tan7 x = + + C 5 7 L

sec2 x = 1 + tan2 x

sec2 x = 1 + tan2 x

Products of Sines and Cosines The integrals sin mx sin nx dx, sin mx cos nx dx, and cos mx cos nx dx L L L arise in many applications involving periodic functions. We can evaluate these integrals through integration by parts, but two such integrations are required in each case. It is simpler to use the identities 1 sin mx sin nx = [cos sm - ndx - cos sm + ndx], (3) 2 1 (4) sin mx cos nx = [sin sm - ndx + sin sm + ndx], 2 1 (5) cos mx cos nx = [cos sm - ndx + cos sm + ndx]. 2

8.2

Trigonometric Integrals

443

These identities come from the angle sum formulas for the sine and cosine functions (Section 1.3). They give functions whose antiderivatives are easily found.

EXAMPLE 8

Evaluate L

Solution

sin 3x cos 5x dx.

From Equation (4) with m = 3 and n = 5, we get L

sin 3x cos 5x dx = =

1 [sin s -2xd + sin 8x] dx 2L 1 ssin 8x - sin 2xd dx 2L

= -

cos 8x cos 2x + + C. 16 4

Exercises 8.2 Powers of Sines and Cosines Evaluate the integrals in Exercises 1–22.

p>2

27. p

1. 3. 5.

L L L

cos 2x dx

2.

cos3 x sin x dx

4.

sin3 x dx

6.

3 sin

L0

7.

L

sin x dx

x 8. sin dx 2 L0

11.

L L

cos3 x dx

10.

sin3 x cos3 x dx

12.

cos2 x dx

14.

3 cos5 3x dx

L0

cos3 2x sin5 2x dx

L

21 + sin x dx

L0

cos4 x

L5p>6 21 - sin x

30.

21 - sin 2x dx

Lp>2 p

u21 - cos 2u du

L0

1 - sin x .b B 1 - sin x

3p>4

dx

p>2

5

p>6

9.

p

29.

31.

p

5

28.

aHint: Multiply by

cos3 4x dx

L

Lp>3 21 - cos x

dx

x dx 3

sin4 2x cos 2x dx

L

p/6

sin2 x

32.

L-p

s1 - cos2 td3>2 dt

Powers of Tangents and Secants Evaluate the integrals in Exercises 33–50. 33.

L

sec2 x tan x dx

34.

sec3 x tan x dx

36.

sec2 x tan2 x dx

38.

L

sec x tan2 x dx

p>2

13.

L

sin2 x dx

L0

35.

L

L

sec3 x tan3 x dx

p>2

15.

sin7 y dy

16.

8 sin4 x dx

18.

L0

L

7 cos7 t dt

37.

8 cos4 2px dx

39.

L

p

17.

L0

L

L

16 sin2 x cos2 x dx

20.

8 sin4 y cos2 y dy

L0

41.

p>2

21.

L

8 cos3 2u sin 2u du

22.

2p

L0 L0

sec4 u du

40. 42.

L L

e x sec3 e x dx 3 sec4 3x dx

Lp>4

csc4 u du

44.

L

sec6 x dx p>4

45.

L

4 tan3 x dx

46.

tan5 x dx

48.

L-p>4

6 tan4 x dx

p

1 - cos x dx 2 A

24.

21 - sin2 t dt

26.

L0

p

25.

L

2 sec3 x dx

p>2

43.

Integrating Square Roots Evaluate the integrals in Exercises 23–32. 23.

L-p>3

sin2 2u cos3 2u du

L0

sec4 x tan2 x dx

0

p

19.

L

21 - cos 2x dx

47.

21 - cos2 u du

49.

p

L0

L

L

cot6 2x dx

p>3

Lp>6

cot3 x dx

50.

L

8 cot4 t dt

444

Chapter 8: Techniques of Integration

Products of Sines and Cosines Evaluate the integrals in Exercises 51–56. 51.

L

sin 3x cos 2x dx

52.

L

p

53.

L-p

65. sin 2x cos 3x dx

67.

L

54.

cos 3x cos 4x dx

59. 61.

2

L L L

68.

L

x cos3 x dx

Applications

56.

L-p>2

69. Arc length Find the length of the curve

cos x cos 7x dx

sin u cos 3u du

58.

cos3 u sin 2u du

60.

sin u cos u cos 3u du

62.

L L L

3

sec x dx L tan x

cos 2u sin u du sin3 u cos 2u du

71. Volume Find the volume generated by revolving one arch of the curve y = sin x about the x-axis. 72. Area Find the area between the x-axis and the curve y = 21 + cos 4x, 0 … x … p .

sin u sin 2u sin 3u du

73. Centroid Find the centroid of the region bounded by the graphs of y = x + cos x and y = 0 for 0 … x … 2p.

8.3

74. Volume Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = p>3 about the x-axis.

3

64.

0 … x … p>4 .

y = ln ssec xd,

70. Center of gravity Find the center of gravity of the region bounded by the x-axis, the curve y = sec x , and the lines x = -p>4, x = p>4 .

2

Assorted Integrations Use any method to evaluate the integrals in Exercises 63–68. 63.

x sin2 x dx

cot x dx 2 cos x L

sin x cos x dx

L0

Exercises 57–62 require the use of various trigonometric identities before you evaluate the integrals. 57.

L

66.

p>2

sin 3x sin 3x dx

p>2

55.

L

tan2 x csc x dx

sin x dx 4 L cos x

Trigonometric Substitutions Trigonometric substitutions occur when we replace the variable of integration by a trigonometric function. The most common substitutions are x = a tan u, x = a sin u, and x = a sec u. These substitutions are effective in transforming integrals involving 2a 2 + x 2 , 2a 2 - x 2, and 2x 2 - a 2 into integrals we can evaluate directly since they come from the reference right triangles in Figure 8.2. With x = a tan u, a 2 + x 2 = a 2 + a 2 tan2 u = a 2s1 + tan2 ud = a 2 sec2 u. With x = a sin u, a 2 - x 2 = a 2 - a 2 sin2 u = a 2s1 - sin2 ud = a 2 cos2 u.

a 2 x 2

a

x

a 2 x 2

x

x

a x a tan

x a sin

a x a sec

a 2 x 2 asec

a 2 x 2 acos

x 2 a2 atan

FIGURE 8.2 Reference triangles for the three basic substitutions identifying the sides labeled x and a for each substitution.

x 2 a2

8.3

Trigonometric Substitutions

445

With x = a sec u, x 2 - a 2 = a 2 sec2 u - a 2 = a 2ssec2 u - 1d = a 2 tan2 u. 2 tan–1 ax

x a

0 – 2

We want any substitution we use in an integration to be reversible so that we can change back to the original variable afterward. For example, if x = a tan u, we want to be able to set u = tan-1 sx>ad after the integration takes place. If x = a sin u, we want to be able to set u = sin-1 sx>ad when we’re done, and similarly for x = a sec u. As we know from Section 1.6, the functions in these substitutions have inverses only for selected values of u (Figure 8.3). For reversibility, x = a tan u

requires

x u = tan-1 a a b

with

-

p p 6 u 6 , 2 2

x = a sin u

requires

x u = sin-1 a a b

with

-

p p … u … , 2 2

requires

x aa b

sin–1 ax

2

–1

0

1

x a

x = a sec u

u = sec-1

with

– 2

–1

sec–1 ax

0 1

if

x a Ú 1,

d

p 6 u … p 2

if

x a … -1.

To simplify calculations with the substitution x = a sec u, we will restrict its use to integrals in which x>a Ú 1. This will place u in [0, p>2d and make tan u Ú 0. We will then have 2x 2 - a 2 = 2a 2 tan2 u = ƒ a tan u ƒ = a tan u, free of absolute values, provided a 7 0.

p 2

0 … u 6

2 x a

FIGURE 8.3 The arctangent, arcsine, and arcsecant of x> a, graphed as functions of x> a.

Procedure For a Trigonometric Substitution 1. Write down the substitution for x, calculate the differential dx, and specify the selected values of u for the substitution. 2. Substitute the trigonometric expression and the calculated differential into the integrand, and then simplify the results algebraically. 3. Integrate the trigonometric integral, keeping in mind the restrictions on the angle u for reversibility. 4. Draw an appropriate reference triangle to reverse the substitution in the integration result and convert it back to the original variable x.

EXAMPLE 1

Evaluate dx L 24 + x 2

Solution

.

We set x = 2 tan u,

dx = 2 sec2 u du,

-

p p 6 u 6 , 2 2

4 + x 2 = 4 + 4 tan2 u = 4s1 + tan2 ud = 4 sec2 u.

446

Chapter 8: Techniques of Integration

4 x 2

Then x

dx

L 24 + x 2

2

24 + x 2 . 2

2 sec2 u du L 24 sec2 u

=

sec2 u du L ƒ sec u ƒ

2sec2 u = ƒ sec u ƒ

=

sec u 7 0 for -

sec u du L = ln ƒ sec u + tan u ƒ + C

FIGURE 8.4 Reference triangle for x = 2 tan u (Example 1): x tan u = 2 and sec u =

=

= ln `

24 + x 2 x + ` + C. 2 2

p p 6 u 6 2 2

From Fig. 8.4

Notice how we expressed ln ƒ sec u + tan u ƒ in terms of x: We drew a reference triangle for the original substitution x = 2 tan u (Figure 8.4) and read the ratios from the triangle.

EXAMPLE 2

Evaluate x 2 dx L 29 - x 2

Solution

.

We set x = 3 sin u,

dx = 3 cos u du,

-

p p 6 u 6 2 2

9 - x 2 = 9 - 9 sin2 u = 9s1 - sin2 ud = 9 cos2 u. Then x 2 dx L 29 - x 2 3

=

9 sin2 u # 3 cos u du ƒ 3 cos u ƒ L

= 9

x

L

cos u 7 0 for -

p p 6 u 6 2 2

1 - cos 2u du 2 L

= 9

9 x 2 FIGURE 8.5 Reference triangle for x = 3 sin u (Example 2): x sin u = 3 and cos u =

sin2 u du

29 - x 2 . 3

EXAMPLE 3

=

sin 2u 9 au b + C 2 2

=

9 su - sin u cos ud + C 2

sin 2u = 2 sin u cos u

=

9 x x 29 - x 2 asin-1 - # b + C 2 3 3 3

From Fig. 8.5

=

9 -1 x x sin - 29 - x 2 + C. 2 3 2

Evaluate dx L 225x - 4 2

Solution

,

x 7

2 . 5

We first rewrite the radical as 225x 2 - 4 =

4 b 25 ax 2 25 B

= 5

2 x2 - a b 5 C

2

8.3

Trigonometric Substitutions

447

to put the radicand in the form x 2 - a 2 . We then substitute x =

2 sec u, 5

dx =

2 sec u tan u du, 5

0 6 u 6

p 2

2

2 4 4 x2 - a b = sec2 u 5 25 25 =

4 4 ssec2 u - 1d = tan2 u 25 25

2

5x

tan u 7 0 for 0 6 u 6 p>2

2 2 2 x 2 - a b = ƒ tan u ƒ = tan u. 5 5 5 C

25x 2 4

With these substitutions, we have

dx 2

L 225x - 4 2

FIGURE 8.6 If x = s2>5dsec u, 0 6 u 6 p>2 , then u = sec-1 s5x>2d , and we can read the values of the other trigonometric functions of u from this right triangle (Example 3).

=

dx L 5 2x - s4>25d 2

=

s2>5d sec u tan u du # L 5 s2>5d tan u

=

1 1 sec u du = ln ƒ sec u + tan u ƒ + C 5L 5

=

225x 2 - 4 5x 1 + ln ` ` + C. 5 2 2

From Fig. 8.6

EXERCISES 8.3 Using Trigonometric Substitutions Evaluate the integrals in Exercises 1–14. 1.

dx

2.

L 29 + x 2 2

3.

7. 9.

L0 L

3 dx

4.

dx 2 L0 8 + 2x 1>222

dx

6.

29 - x 2 225 - t 2 dt dx

L 24x 2 - 49

,

2y - 49 dy, y

8. 7 x 7 2

10.

L

13.

L

dx , L x 2 2x 2 - 1

21 - 4x 2

25.

27. 5 dx

L 225x 2 - 9 2y - 25

,

3 x 7 5

x 7 1

12. 14.

L

y3

dy,

2 dx , L x 3 2x 2 - 1

15. 17.

x L 29 - x 2 x 3 dx L 2x 2 + 4

dx

16. 18.

100 dx 2 L 36 + 25x L0

x2 dx 2 L4 + x dx L x 2 2x 2 + 1

4x 2 dx s1 - x 2 d3>2

dx , L sx - 1d3>2 2

s1 - x 2 d3>2 x6

L

22.

dx

x 7 1

29 - w 2 dw w2 L L

x 2x 2 - 4 dx 1

24.

dx L0 s4 - x 2 d3>2

26.

x 2 dx , L sx - 1d5>2

28.

2

s1 - x 2 d1>2 x4

L

29.

8 dx 2 2 L s4x + 1d

30.

6 dt 2 2 L s9t + 1d

31.

x 3 dx 2 Lx - 1

32.

x dx 2 L 25 + 4x

33.

y 2 dy L s1 - y 2 d5>2

34.

y 7 5 x 7 1

Assorted Integrations Use any method to evaluate the integrals in Exercises 15–34. Most will require trigonometric substitutions, but some can be evaluated by other methods.

20.

L w 2 24 - w 2

21 - 9t 2 dt

2

y 7 7

8 dw

23>2

23.

2 dx

L0

2

11.

21.

L 21 + 9x 2 2

dx 2 L-2 4 + x 3>2

5.

19.

s1 - r 2 d5>2 r8

L

x 7 1

dx

dr

In Exercises 35–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals. ln 4

35.

L0

2e 2t + 9 1>4

37.

e t dt

L1>12

2 dt 1t + 4t1t

ln s4>3d

36.

Lln s3>4d e

38.

e t dt s1 + e 2t d3>2 dy

L1 y21 + sln yd2

448

Chapter 8: Techniques of Integration

dx

39.

L x2x - 1 x dx 41. L 2x 2 - 1 43. 45.

47.

x dx

44.

L 21 + x 4 4 - x x dx L B (Hint: Let x = u 2.) L

56. Consider the region bounded by the graphs of y = 2x tan-1 x and y = 0 for 0 … x … 1. Find the volume of the solid formed by revolving this region about the x-axis (see accompanying figure).

dx 1 + x2 L dx 42. L 21 - x 2 40.

2

46.

2x 21 - x dx

48.

y

21 - (ln x) 2 dx x ln x L

y 5 x tan21 x

x dx 3 L B1 - x (Hint: Let u = x 3>2.) 2x - 2 L 2x - 1

dx

Initial Value Problems Solve the initial value problems in Exercises 49–52 for y as a function of x. dy = 2x 2 - 4, x Ú 2, ys2d = 0 49. x dx dy = 1, x 7 3, ys5d = ln 3 50. 2x 2 - 9 dx dy = 3, ys2d = 0 51. sx 2 + 4d dx dy 52. sx 2 + 1d2 = 2x 2 + 1, ys0d = 1 dx

1

0

57. Evaluate 1 x 3 21 - x 2 dx using a. integration by parts. b. a u-substitution. c. a trigonometric substitution. 58. Path of a water skier Suppose that a boat is positioned at the origin with a water skier tethered to the boat at the point (30, 0) on a rope 30 ft long. As the boat travels along the positive y-axis, the skier is pulled behind the boat along an unknown path y = ƒ(x), as shown in the accompanying figure. a. Show that ƒ¿(x) =

- 2900 - x 2 . x

(Hint: Assume that the skier is always pointed directly at the boat and the rope is on a line tangent to the path y = ƒ(x).)

Applications and Examples 53. Area Find the area of the region in the first quadrant that is enclosed by the coordinate axes and the curve y = 29 - x 2>3 .

b. Solve the equation in part (a) for ƒ(x), using ƒ(30) = 0. y y 5 f (x) path of skier

54. Area Find the area enclosed by the ellipse 2

y x + 2 = 1. a2 b 55. Consider the region bounded by the graphs of y = sin-1 x, y = 0, and x = 1>2. 2

x

boat

30 ft rope (x, f (x)) skier

f (x)

a. Find the area of the region. b. Find the centroid of the region.

0

x

(30, 0)

x

NOT TO SCALE

8.4

Integration of Rational Functions by Partial Fractions This section shows how to express a rational function (a quotient of polynomials) as a sum of simpler fractions, called partial fractions, which are easily integrated. For instance, the rational function (5x - 3)>(x 2 - 2x - 3) can be rewritten as 5x - 3 3 2 = + . x + 1 x 3 x - 2x - 3 2

You can verify this equation algebraically by placing the fractions on the right side over a common denominator sx + 1dsx - 3d. The skill acquired in writing rational functions as such a sum is useful in other settings as well (for instance, when using certain transform methods to solve differential equations). To integrate the rational function

8.4

Integration of Rational Functions by Partial Fractions

449

(5x - 3)>(x 2 - 2x - 3) on the left side of our previous expression, we simply sum the integrals of the fractions on the right side: 5x - 3 3 2 dx = dx + dx x + 1 x 3 sx + 1dsx 3d L L L = 2 ln ƒ x + 1 ƒ + 3 ln ƒ x - 3 ƒ + C. The method for rewriting rational functions as a sum of simpler fractions is called the method of partial fractions. In the case of the preceding example, it consists of finding constants A and B such that 5x - 3 A B = + . x + 1 x - 3 x 2 - 2x - 3

(1)

(Pretend for a moment that we do not know that A = 2 and B = 3 will work.) We call the fractions A>sx + 1d and B>sx - 3d partial fractions because their denominators are only part of the original denominator x 2 - 2x - 3. We call A and B undetermined coefficients until proper values for them have been found. To find A and B, we first clear Equation (1) of fractions and regroup in powers of x, obtaining 5x - 3 = Asx - 3d + Bsx + 1d = sA + Bdx - 3A + B. This will be an identity in x if and only if the coefficients of like powers of x on the two sides are equal: A + B = 5,

-3A + B = -3.

Solving these equations simultaneously gives A = 2 and B = 3.

General Description of the Method Success in writing a rational function ƒ(x)> g(x) as a sum of partial fractions depends on two things:

• •

The degree of ƒ(x) must be less than the degree of g(x). That is, the fraction must be proper. If it isn’t, divide ƒ(x) by g(x) and work with the remainder term. See Example 3 of this section. We must know the factors of g(x). In theory, any polynomial with real coefficients can be written as a product of real linear factors and real quadratic factors. In practice, the factors may be hard to find.

Here is how we find the partial fractions of a proper fraction ƒ(x)> g(x) when the factors of g are known. A quadratic polynomial (or factor) is irreducible if it cannot be written as the product of two linear factors with real coefficients. That is, the polynomial has no real roots. Method of Partial Fractions (ƒ(x)> g(x) Proper) 1. Let x - r be a linear factor of g(x). Suppose that sx - rdm is the highest

power of x - r that divides g(x). Then, to this factor, assign the sum of the m partial fractions: Am A2 A1 + + Á + . 2 (x - r) sx - rdm sx - rd Do this for each distinct linear factor of g(x). continued

450

Chapter 8: Techniques of Integration

2. Let x 2 + px + q be an irreducible quadratic factor of g(x) so that x 2 + px + q has no real roots. Suppose that sx 2 + px + qdn is the highest power of this factor that divides g(x). Then, to this factor, assign the sum of the n partial fractions: Bn x + Cn B1 x + C1 B2 x + C2 + 2 + Á + 2 . 2 (x + px + q) sx + px + qd sx + px + qdn 2

Do this for each distinct quadratic factor of g(x). 3. Set the original fraction ƒ(x)> g(x) equal to the sum of all these partial fractions. Clear the resulting equation of fractions and arrange the terms in decreasing powers of x. 4. Equate the coefficients of corresponding powers of x and solve the resulting equations for the undetermined coefficients.

EXAMPLE 1

Use partial fractions to evaluate x 2 + 4x + 1 dx. L sx - 1dsx + 1dsx + 3d

Solution

The partial fraction decomposition has the form C x 2 + 4x + 1 B A + + . = x - 1 x + 1 x + 3 sx - 1dsx + 1dsx + 3d

To find the values of the undetermined coefficients A, B, and C, we clear fractions and get x 2 + 4x + 1 = Asx + 1dsx + 3d + Bsx - 1dsx + 3d + Csx - 1dsx + 1d = A(x 2 + 4x + 3) + B(x 2 + 2x - 3) + C(x 2 - 1) = sA + B + Cdx 2 + s4A + 2Bdx + s3A - 3B - Cd. The polynomials on both sides of the above equation are identical, so we equate coefficients of like powers of x, obtaining Coefficient of x 2: Coefficient of x 1: Coefficient of x 0:

A + B + C = 1 4A + 2B = 4 3A - 3B - C = 1

There are several ways of solving such a system of linear equations for the unknowns A, B, and C, including elimination of variables or the use of a calculator or computer. Whatever method is used, the solution is A = 3>4, B = 1>2, and C = -1>4. Hence we have 3 1 x 2 + 4x + 1 1 1 1 1 dx = c + d dx 2x + 1 4x + 3 L sx - 1dsx + 1dsx + 3d L 4x - 1 =

3 1 1 ln x - 1 ƒ + ln ƒ x + 1 ƒ - ln ƒ x + 3 ƒ + K, 4 ƒ 2 4

where K is the arbitrary constant of integration (to avoid confusion with the undetermined coefficient we labeled as C).

EXAMPLE 2

Use partial fractions to evaluate 6x + 7 dx. 2 L sx + 2d

8.4

Integration of Rational Functions by Partial Fractions

451

First we express the integrand as a sum of partial fractions with undetermined coefficients.

Solution

6x + 7 B A + = 2 x + 2 sx + 2d sx + 2d2 6x + 7 = Asx + 2d + B

Multiply both sides by sx + 2d2 .

= Ax + s2A + Bd Equating coefficients of corresponding powers of x gives A = 6

and

2A + B = 12 + B = 7,

or

A = 6

and

B = -5.

Therefore, 6x + 7 5 6 dx = a b dx 2 x + 2 sx + 2d2 L sx + 2d L = 6

dx - 5 sx + 2d-2 dx x L L + 2

= 6 ln ƒ x + 2 ƒ + 5sx + 2d-1 + C.

EXAMPLE 3

Use partial fractions to evaluate 2x 3 - 4x 2 - x - 3 dx. x 2 - 2x - 3 L

First we divide the denominator into the numerator to get a polynomial plus a proper fraction. 2x 2 x - 2x - 3 2x 3 - 4x 2 - x - 3 2x 3 - 4x 2 - 6x 5x - 3

Solution

Then we write the improper fraction as a polynomial plus a proper fraction. 2x 3 - 4x 2 - x - 3 5x - 3 = 2x + 2 2 x - 2x - 3 x - 2x - 3 We found the partial fraction decomposition of the fraction on the right in the opening example, so 2x 3 - 4x 2 - x - 3 5x - 3 dx = 2x dx + dx 2 2 x - 2x - 3 L L L x - 2x - 3 3 2 dx + dx x + 1 x 3 L L L = x 2 + 2 ln ƒ x + 1 ƒ + 3 ln ƒ x - 3 ƒ + C. =

EXAMPLE 4

2x dx +

Use partial fractions to evaluate -2x + 4 dx. 2 L sx + 1dsx - 1d 2

The denominator has an irreducible quadratic factor as well as a repeated linear factor, so we write

Solution

C -2x + 4 Ax + B D = 2 + . + x - 1 sx 2 + 1dsx - 1d2 x + 1 sx - 1d2

(2)

452

Chapter 8: Techniques of Integration

Clearing the equation of fractions gives -2x + 4 = sAx + Bdsx - 1d2 + Csx - 1dsx 2 + 1d + Dsx 2 + 1d = sA + Cdx 3 + s -2A + B - C + Ddx 2 + sA - 2B + Cdx + sB - C + Dd. Equating coefficients of like terms gives Coefficients of x 3 : Coefficients of x 2 : Coefficients of x 1 : Coefficients of x 0 :

0 0 -2 4

= = = =

A + C - 2A + B - C + D A - 2B + C B - C + D

We solve these equations simultaneously to find the values of A, B, C, and D: -4 = -2A,

A = 2

Subtract fourth equation from second.

C = -A = -2

From the first equation

B = (A + C + 2)>2 = 1

From the third equation and C = -A

D = 4 - B + C = 1.

From the fourth equation

We substitute these values into Equation (2), obtaining -2x + 4 2x + 1 2 1 = 2 . + 2 x 1 sx + 1dsx - 1d x + 1 sx - 1d2 2

Finally, using the expansion above we can integrate: -2x + 4 2x + 1 1 2 dx = a 2 b dx + 2 2 x 1 sx + 1dsx 1d x + 1 sx 1d2 L L =

L

a

2x 1 2 1 b dx + 2 + x - 1 x2 + 1 x + 1 sx - 1d2

1 = ln sx 2 + 1d + tan-1 x - 2 ln ƒ x - 1 ƒ + C. x - 1

EXAMPLE 5

Use partial fractions to evaluate dx . 2 L xsx + 1d 2

Solution

The form of the partial fraction decomposition is Bx + C Dx + E . 1 A = x + 2 + 2 2 xsx + 1d x + 1 sx + 1d2 2

Multiplying by xsx 2 + 1d2 , we have 1 = Asx2 + 1d2 + sBx + Cdxsx2 + 1d + sDx + Edx = Asx4 + 2x2 + 1d + Bsx4 + x2 d + Csx3 + xd + Dx2 + Ex = sA + Bdx4 + Cx3 + s2A + B + Ddx2 + sC + Edx + A. If we equate coefficients, we get the system A + B = 0,

C = 0,

2A + B + D = 0,

C + E = 0,

A = 1.

8.4

Integration of Rational Functions by Partial Fractions

453

Solving this system gives A = 1, B = -1, C = 0, D = -1, and E = 0. Thus, -x -x dx 1 + 2 = cx + 2 d dx 2 2 xsx + 1d x + 1 sx + 1d2 L L = =

L

dx x -

x dx x dx 2 2 x + 1 sx + 1d2 L L

dx 1 du 1 du x - 2 u - 2 u2 L L L

u = x 2 + 1, du = 2x dx

1 1 + K = ln ƒ x ƒ - ln ƒ u ƒ + 2 2u 1 1 + K = ln ƒ x ƒ - ln sx 2 + 1d + 2 2sx 2 + 1d = ln

ƒxƒ 2x + 1 2

+

1 + K. 2sx 2 + 1d

Another Way to Determine the Coefficients EXAMPLE 6

Find A, B, and C in the equation C x - 1 A B = + + x + 1 sx + 1d3 sx + 1d2 sx + 1d3

by clearing fractions, differentiating the result, and substituting x = -1. Solution

We first clear fractions: x - 1 = Asx + 1d2 + Bsx + 1d + C.

Substituting x = -1 shows C = -2. We then differentiate both sides with respect to x, obtaining 1 = 2Asx + 1d + B. Substituting x = -1 shows B = 1. We differentiate again to get 0 = 2A, which shows A = 0. Hence, x - 1 1 2 = . 3 2 sx + 1d sx + 1d sx + 1d3 In some problems, assigning appropriate small values to x, such as x = 0, ;1, ;2, to get equations in A, B, and C provides an alternative method to finding the constants that appear in partial fractions.

EXAMPLE 7

Find A, B, and C in the expression C x2 + 1 A B = + + x - 1 x - 2 x - 3 sx - 1dsx - 2dsx - 3d

by assigning numerical values to x.

454

Chapter 8: Techniques of Integration

Clear fractions to get

Solution

x 2 + 1 = Asx - 2dsx - 3d + Bsx - 1dsx - 3d + Csx - 1dsx - 2d. Then let x = 1, 2, 3 successively to find A, B, and C: x = 1:

s1d2 + 1 2 A s2d2 + 1 5 B s3d2 + 1 10 C

x = 2:

x = 3:

= = = = = = = = =

As -1ds -2d + Bs0d + Cs0d 2A 1 As0d + Bs1ds -1d + Cs0d -B -5 As0d + Bs0d + Cs2ds1d 2C 5.

Conclusion: 5 5 x2 + 1 1 + . = x - 1 x - 2 x - 3 sx - 1dsx - 2dsx - 3d

Exercises 8.4 Expanding Quotients into Partial Fractions Expand the quotients in Exercises 1–8 by partial fractions. 1.

5x - 13 sx - 3dsx - 2d

2.

5x - 7 x 2 - 3x + 2

3.

x + 4 sx + 1d2

4.

2x + 2 x 2 - 2x + 1

z + 1 5. 2 z sz - 1d

z 6. 3 z - z 2 - 6z

t2 + 8 7. 2 t - 5t + 6

t4 + 9 8. 4 t + 9t 2

Nonrepeated Linear Factors In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals. dx 9. 2 L1 - x 11. 13. 15.

x + 4 dx 2 L x + 5x - 6 8 y dy 2 L4 y - 2y - 3

dt 3 2 L t + t - 2t

dx 10. 2 L x + 2x 2x + 1 dx 2 L x - 7x + 12 1 y + 4 dy 14. 2 L1>2 y + y 16.

x + 3 dx 3 L 2x - 8x

1

x 3 dx 2 L0 x + 2x + 1

x 3 dx L-1 x - 2x + 1 2

20.

x 2 dx 2 L sx - 1dsx + 2x + 1d

23

1

21.

dx 2 L0 sx + 1dsx + 1d y + 2y + 1

22.

2

23.

L

sy + 1d 2

2

dy

L1

3t 2 + t + 4 dt t3 + t

24.

8x 2 + 8x + 2 dx 2 2 L s4x + 1d

25.

2s + 2 ds ss + 1dss - 1d3 L

26.

s 4 + 81 ds 2 2 L sss + 9d

27.

x2 - x + 2 dx 3 L x - 1

28.

1 dx 4 Lx + x

29.

x2 dx Lx - 1

30.

x2 + x dx 2 L x - 3x - 4

31.

2u3 + 5u2 + 8u + 4 du 2 2 L su + 2u + 2d

32.

u4 - 4u3 + 2u2 - 3u + 1 du su2 + 1d3 L

2

4

4

Improper Fractions In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.

0

18.

dx 2 L sx - 1d 2

Irreducible Quadratic Factors In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

12.

Repeated Linear Factors In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals. 17.

19.

33.

2x 3 - 2x 2 + 1 dx x2 - x L

34.

x4 dx 2 x - 1 L

8.4

35.

9x 3 - 3x + 1 dx x3 - x2 L y + y - 1 4

37.

y + y

dy

38.

2y

y y

4

2 Ly - y + y - 1 3

455

56. The y-axis

16x 3 dx L 4x - 4x + 1 2

2

3

L

36.

Integration of Rational Functions by Partial Fractions

2 (x 1)(2 x)

1

dy

Evaluating Integrals Evaluate the integrals in Exercises 39–50. 39. 41. 43. 44.

Le

2t

e 4t + 2e 2t - e t dt e 2t + 1 L sin u du 42. 2 L cos u + cos u - 2

e t dt + 3e t + 2 cos y dy

2 L sin y + sin y - 6

sx - 2d2 tan-1 s2xd - 12x 3 - 3x s4x 2 + 1dsx - 2d2

L

sx + 1d2 tan-1 s3xd + 9x 3 + x s9x 2 + 1dsx + 1d2

L

45.

1 dx L x 3>2 - 1x

46.

47.

1x + 1 dx x

48.

49.

L (Hint: Let x + 1 = u 2.) 1 dx 4 L x(x + 1)

50.

aHint: Multiply by

y

dx

(3, 1.83)

1 dx L (x 1>3 - 1) 1x (Hint: Let x = u 6.) 1 dx x1x + 9 L 1 dx 6 5 L x (x + 4)

x3 .b x3

dx = 1 dt

52. s3t 4 + 4t 2 + 1d 53. st 2 + 2td 54. st + 1d

st 7 2d,

dx = 2 23, dt

dx = 2x + 2 dt

dx = x2 + 1 dt

xs3d = 0

xs1d = -p 23>4 xs1d = 1

st 7 -1d,

xs0d = 0

55. The x-axis y (0.5, 2.68)

(5, 0.98)

0

3

5

x

T 59. Social diffusion Sociologists sometimes use the phrase “social diffusion” to describe the way information spreads through a population. The information might be a rumor, a cultural fad, or news about a technical innovation. In a sufficiently large population, the number of people x who have the information is treated as a differentiable function of time t, and the rate of diffusion, dx> dt, is assumed to be proportional to the number of people who have the information times the number of people who do not. This leads to the equation dx = kxsN - xd , dt where N is the number of people in the population. Suppose t is in days, k = 1>250 , and two people start a rumor at time t = 0 in a population of N = 1000 people. b. When will half the population have heard the rumor? (This is when the rumor will be spreading the fastest.) T 60. Second-order chemical reactions Many chemical reactions are the result of the interaction of two molecules that undergo a change to produce a new product. The rate of the reaction typically depends on the concentrations of the two kinds of molecules. If a is the amount of substance A and b is the amount of substance B at time t = 0 , and if x is the amount of product at time t, then the rate of formation of x may be given by the differential equation

3

dx = ksa - xdsb - xd , dt

3x x 2 (2.5, 2.68)

or dx 1 = k, sa - xdsb - xd dt

2

0

2 9 y 4x3 13x x 2x 2 3x

a. Find x as a function of t.

st, x 7 0d,

Applications and Examples In Exercises 55 and 56, find the volume of the solid generated by revolving the shaded region about the indicated axis.

y

T 57. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = tan-1 x , and the line x = 23 . T 58. Find the x-coordinate of the centroid of this region to two decimal places.

dx

Initial Value Problems Solve the initial value problems in Exercises 51–54 for x as a function of t. 51. st 2 - 3t + 2d

x

1

0

40.

0.5

2.5

x

where k is a constant for the reaction. Integrate both sides of this equation to obtain a relation between x and t (a) if a = b , and (b) if a Z b . Assume in each case that x = 0 when t = 0 .

456

Chapter 8: Techniques of Integration

8.5

Integral Tables and Computer Algebra Systems In this section we discuss how to use tables and computer algebra systems to evaluate integrals.

Integral Tables A Brief Table of Integrals is provided at the back of the book, after the index. (More extensive tables appear in compilations such as CRC Mathematical Tables, which contain thousands of integrals.) The integration formulas are stated in terms of constants a, b, c, m, n, and so on. These constants can usually assume any real value and need not be integers. Occasional limitations on their values are stated with the formulas. Formula 21 requires n Z -1, for example, and Formula 27 requires n Z -2. The formulas also assume that the constants do not take on values that require dividing by zero or taking even roots of negative numbers. For example, Formula 24 assumes that a Z 0, and Formulas 29a and 29b cannot be used unless b is positive.

EXAMPLE 1

Find L

Solution

xs2x + 5d-1 dx.

We use Formula 24 at the back of the book (not 22, which requires n Z -1): x b xsax + bd-1 dx = a - 2 ln ƒ ax + b ƒ + C. a L

With a = 2 and b = 5, we have L

EXAMPLE 2

xs2x + 5d-1 dx =

5 x - ln ƒ 2x + 5 ƒ + C. 2 4

Find dx L x22x - 4

Solution

.

We use Formula 29b: dx ax - b 2 = tan-1 + C. A b 1b L x2ax - b

With a = 2 and b = 4, we have dx 2x - 4 x - 2 2 = tan-1 + C = tan-1 + C. A 4 A 2 L x22x - 4 24

EXAMPLE 3

Find L

Solution

L

x sin-1 x dx.

We begin by using Formula 106: x n sin-1 ax dx =

xn+1 a x n + 1 dx , sin-1 ax n + 1 n + 1 L 21 - a 2x 2

n Z -1.

8.5

Integral Tables and Computer Algebra Systems

457

With n = 1 and a = 1, we have L

x sin-1 x dx =

x 2 -1 x 2 dx 1 sin x . 2 2 L 21 - x 2

Next we use Formula 49 to find the integral on the right: x2 L 2a 2 - x 2

a 2 -1 x 1 sin a a b - x2a 2 - x 2 + C. 2 2

dx =

With a = 1, x 2 dx L 21 - x

2

=

1 -1 1 sin x - x21 - x 2 + C. 2 2

The combined result is L

x sin-1 x dx =

x 2 -1 1 1 1 sin x - a sin-1 x - x21 - x 2 + Cb 2 2 2 2

= a

x2 1 1 - bsin-1 x + x21 - x 2 + C¿ . 2 4 4

Reduction Formulas The time required for repeated integrations by parts can sometimes be shortened by applying reduction formulas like L

tann x dx =

L L

sinn x cosm x dx = -

1 tann - 2 x dx tann - 1 x n - 1 L

sln xdn dx = xsln xdn - n

L

(1)

sln xdn - 1 dx

sinn - 1 x cosm + 1 x n - 1 + sinn - 2 x cosm x dx m + n m + nL

(2)

sn Z -md. (3)

By applying such a formula repeatedly, we can eventually express the original integral in terms of a power low enough to be evaluated directly. The next example illustrates this procedure.

EXAMPLE 4

Solution

Find

tan5 x dx. L We apply Equation (1) with n = 5 to get L

tan5 x dx =

1 4 tan x tan3 x dx. 4 L

We then apply Equation (1) again, with n = 3, to evaluate the remaining integral: L

tan3 x dx =

1 2 1 tan x tan x dx = tan2 x + ln ƒ cos x ƒ + C. 2 2 L

The combined result is L

tan5 x dx =

1 4 1 tan x - tan2 x - ln ƒ cos x ƒ + C¿ . 4 2

As their form suggests, reduction formulas are derived using integration by parts. (See Example 5 in Section 8.1.)

458

Chapter 8: Techniques of Integration

Integration with a CAS A powerful capability of computer algebra systems is their ability to integrate symbolically. This is performed with the integrate command specified by the particular system (for example, int in Maple, Integrate in Mathematica).

EXAMPLE 5

Suppose that you want to evaluate the indefinite integral of the function ƒsxd = x 2 2a 2 + x 2 .

Using Maple, you first define or name the function: 7 ƒ:= x ¿2 * sqrt sa ¿2 + x ¿2d; Then you use the integrate command on ƒ, identifying the variable of integration: 7 intsƒ, xd; Maple returns the answer 1 1 1 xsa 2 + x 2 d3>2 - a 2x2a 2 + x 2 - a 4 ln A x + 2a 2 + x 2 B . 4 8 8 If you want to see if the answer can be simplified, enter 7 simplifys%d; Maple returns 1 1 1 2 a x2a 2 + x 2 + x 3 2a 2 + x 2 - a 4 ln A x + 2a 2 + x 2 B . 8 4 8 If you want the definite integral for 0 … x … p>2, you can use the format 7 intsƒ, x = 0..Pi>2d; Maple will return the expression 1 1 2 1 ps4a 2 + p2 ds3>2d a p24a 2 + p2 + a 4 ln s2d 64 32 8 1 1 4 - a 4 ln A p + 24a 2 + p2 B + a ln sa 2 d. 8 16 You can also find the definite integral for a particular value of the constant a: 7 a:= 1; 7 intsƒ, x = 0..1d; Maple returns the numerical answer 3 1 22 + ln A 22 - 1 B . 8 8

EXAMPLE 6

Use a CAS to find L

Solution

sin2 x cos3 x dx.

With Maple, we have the entry 7 int sssin ¿2dsxd * scos ¿3dsxd, xd;

with the immediate return -

1 1 2 sinsxd cossxd4 + cossxd2 sinsxd + sinsxd. 5 15 15

Computer algebra systems vary in how they process integrations. We used Maple in Examples 5 and 6. Mathematica would have returned somewhat different results:

8.5

1.

Integral Tables and Computer Algebra Systems

459

In Example 5, given In [1]:= Integrate [x ¿2 * Sqrt [a ¿2 + x ¿2], x] Mathematica returns Out [1]= 2a 2 + x 2 a

a2 x x3 1 + b - a 4 Log C x + 2a 2 + x 2 D 8 4 8

without having to simplify an intermediate result. The answer is close to Formula 22 in the integral tables. 2.

The Mathematica answer to the integral In [2]:= Integrate [Sin [x]¿2 * Cos [x]¿3, x] in Example 6 is Out [2]=

Sin [x] 1 1 Sin [3 x] Sin [5 x] 8 48 80

differing from the Maple answer. Both answers are correct. Although a CAS is very powerful and can aid us in solving difficult problems, each CAS has its own limitations. There are even situations where a CAS may further complicate a problem (in the sense of producing an answer that is extremely difficult to use or interpret). Note, too, that neither Maple nor Mathematica returns an arbitrary constant +C. On the other hand, a little mathematical thinking on your part may reduce the problem to one that is quite easy to handle. We provide an example in Exercise 67.

Nonelementary Integrals The development of computers and calculators that find antiderivatives by symbolic manipulation has led to a renewed interest in determining which antiderivatives can be expressed as finite combinations of elementary functions (the functions we have been studying) and which cannot. Integrals of functions that do not have elementary antiderivatives are called nonelementary integrals. These integrals can sometimes be expressed with infinite series (Chapter 9) or approximated using numerical methods for their evaluation (Section 8.6). Examples of nonelementary integrals include the error function (which measures the probability of random errors) erf sxd =

2 2p L0

x

e -t dt 2

and integrals such as sin x 2 dx

21 + x 4 dx

and

L L that arise in engineering and physics. These and a number of others, such as

L

ex x dx,

x

L

e se d dx,

1 dx, L ln x

L

ln sln xd dx,

L

sin x x dx,

0 6 k 6 1, 21 - k 2 sin2 x dx, L look so easy they tempt us to try them just to see how they turn out. It can be proved, however, that there is no way to express these integrals as finite combinations of elementary functions. The same applies to integrals that can be changed into these by substitution. The integrands all have antiderivatives, as a consequence of the Fundamental Theorem of Calculus, Part 1, because they are continuous. However, none of the antiderivatives are elementary. None of the integrals you are asked to evaluate in the present chapter fall into this category, but you may encounter nonelementary integrals in your other work.

460

Chapter 8: Techniques of Integration

Exercises 8.5 Using Integral Tables Use the table of integrals at the back of the book to evaluate the integrals in Exercises 1–26. dx

1.

L x2x - 3 x dx 3. L 2x - 2 5. 7. 9. 11. 13. 15. 17. 19. 21.

L L L

L L L L L

6.

29 - 4x dx x2

8.

x24x - x 2 dx

10.

L

12.

24 - x 2 dx x

14.

e 2t cos 3t dt

16.

x cos-1 x dx

18.

x 2 tan-1 x dx sin 3x cos 2x dx

t 8 sin 4t sin dt 23. 2 L 25.

L x2x + 4 x dx 4. L s2x + 3d3>2

x22x - 3 dx

dx L x27 + x 2

cos

u u cos du 3 4

40.

dx

2.

L

xs7x + 5d3>2 dx

dx L x 2 24x - 9 L

2x - x 2 dx x

dx L x27 - x 2 L L L

2x 2 - 4 dx x e -3t sin 4t dt

22.

L L

tan x dx x2

L

cos

u cos 7u du 2

-1

cos-1 1x dx 30. 1x L

31. 33. 34. 36.

1x dx

1x L 21 - x L

dx

cot t21 - sin2 t dt, dt

L tan t24 - sin2 t L

tan-1 1y dy

32.

L

22 - x dx 1x

0 6 t 6 p>2 35. 37.

L

dy L y23 + sln yd2 1 dx L 2x 2 + 2x + 5 (Hint: Complete the square.)

dx

39.

L

25 - 4x - x 2 dx

x 2 22x - x 2 dx

45. 47. 49.

L L L L L

sin5 2x dx

42.

sin2 2u cos3 2u du

44.

4 tan3 2x dx

46.

2 sec3 px dx

48.

csc5 x dx

50.

L L L L L

8 cos4 2pt dt 2 sin2 t sec4 t dt 8 cot4 t dt 3 sec4 3x dx 16x 3sln xd2 dx

Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula. L

e t sec3 se t - 1d dt

52.

csc3 2u L

L0

55.

L1

2u 23>2

1

53.

2

x 2 + 6x dx 28. 2 2 L sx + 3d

sin

43.

sin 2x cos 3x dx

x3 + x + 1 dx 27. 2 2 L sx + 1d L

41.

51.

t t sin sin dt 24. 3 6 L 26.

L 2x - 4x + 5

Using Reduction Formulas Use reduction formulas to evaluate the integrals in Exercises 41–50.

-1

20.

x2 2

x tan-1 x dx

Substitution and Integral Tables In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

29.

38.

22x 2 + 1 dx

54.

sr 2 - 1d3>2 dr r

56.

dy s1 - y 2 d5>2

L0 1>23

L0

du

dt st 2 + 1d7>2

Applications 57. Surface area Find the area of the surface generated by revolving the curve y = 2x 2 + 2, 0 … x … 22 , about the x-axis. 58. Arc length Find the length of the curve y = x 2, 0 … x … 23>2 . 59. Centroid Find the centroid of the region cut from the first quadrant by the curve y = 1> 2x + 1 and the line x = 3 . 60. Moment about y-axis A thin plate of constant density d = 1 occupies the region enclosed by the curve y = 36>s2x + 3d and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis. T 61. Use the integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve y = x 2, -1 … x … 1 , about the x-axis. 62. Volume The head of your firm’s accounting department has asked you to find a formula she can use in a computer program to calculate the year-end inventory of gasoline in the company’s tanks. A typical tank is shaped like a right circular cylinder of radius r and length L, mounted horizontally, as shown in the accompanying figure. The data come to the accounting office as depth measurements taken with a vertical measuring stick marked in centimeters.

8.6 a. Show, in the notation of the figure, that the volume of gasoline that fills the tank to a depth d is -r + d

V = 2L L-r b. Evaluate the integral.

2r 2 - y 2 dy.

Numerical Integration

461

d. What pattern do you see? Predict the formula for 1 x 4 ln x dx and then see if you are correct by evaluating it with a CAS. e. What is the formula for 1 x n ln x dx, n Ú 1 ? Check your answer using a CAS. 66. Evaluate the integrals a.

y

ln x dx 2 L x

b.

ln x dx 3 L x

c.

ln x dx. 4 L x

d. What pattern do you see? Predict the formula for Measuring stick r

ln x dx 5 L x and then see if you are correct by evaluating it with a CAS.

0 d Depth of gasoline

–r

e. What is the formula for ln x n dx, L x

L

Check your answer using a CAS.

63. What is the largest value

67. a. Use a CAS to evaluate

b

La

2x - x 2 dx

p>2

L0

can have for any a and b? Give reasons for your answer.

b

b. In succession, find the integral when n = 1, 2, 3, 5, and 7. Comment on the complexity of the results.

x22x - x 2 dx

can have for any a and b? Give reasons for your answer.

c. Now substitute x = sp>2d - u and add the new and old integrals. What is the value of

COMPUTER EXPLORATIONS In Exercises 65 and 66, use a CAS to perform the integrations. 65. Evaluate the integrals a.

L

x ln x dx

8.6

b.

sinn x dx sinn x + cosn x

where n is an arbitrary positive integer. Does your CAS find the result?

64. What is the largest value

La

n Ú 2?

2

L

x ln x dx

c.

3

L

x ln x dx.

p>2

L0

sinn x dx? sin x + cosn x n

This exercise illustrates how a little mathematical ingenuity solves a problem not immediately amenable to solution by a CAS.

Numerical Integration The antiderivatives of some functions, like sin (x 2), 1> ln x, and 21 + x 4, have no elementary formulas. When we cannot find a workable antiderivative for a function ƒ that we have to integrate, we can partition the interval of integration, replace ƒ by a closely fitting polynomial on each subinterval, integrate the polynomials, and add the results to approximate the integral of ƒ. This procedure is an example of numerical integration. In this section we study two such methods, the Trapezoidal Rule and Simpson’s Rule. In our presentation we assume that ƒ is positive, but the only requirement is for it to be continuous over the interval of integration [a, b].

Trapezoidal Approximations The Trapezoidal Rule for the value of a definite integral is based on approximating the region between a curve and the x-axis with trapezoids instead of rectangles, as in

462

Chapter 8: Techniques of Integration y 5 f (x) Trapezoid area 1 ( y 1 y 2 )Dx 2 1

y1

x0 5 a

x1

y2

yn21

x2

yn

x n21 x n 5 b

Dx

x

FIGURE 8.7 The Trapezoidal Rule approximates short stretches of the curve y = ƒsxd with line segments. To approximate the integral of ƒ from a to b, we add the areas of the trapezoids made by joining the ends of the segments to the x-axis.

Figure 8.7. It is not necessary for the subdivision points x0, x1, x2, Á , xn in the figure to be evenly spaced, but the resulting formula is simpler if they are. We therefore assume that the length of each subinterval is ¢x =

b - a n .

The length ¢x = sb - ad>n is called the step size or mesh size. The area of the trapezoid that lies above the ith subinterval is ¢x a

yi - 1 + yi ¢x b = s yi - 1 + yi d, 2 2

where yi - 1 = ƒsxi - 1 d and yi = ƒsxi d. This area is the length ¢x of the trapezoid’s horizontal “altitude” times the average of its two vertical “bases.” (See Figure 8.7.) The area below the curve y = ƒsxd and above the x-axis is then approximated by adding the areas of all the trapezoids: T =

1 1 s y + y1 d¢x + s y1 + y2 d¢x + Á 2 0 2 +

1 1 sy + yn - 1 d¢x + s yn - 1 + yn d¢x 2 n-2 2

1 1 = ¢x a y0 + y1 + y2 + Á + yn - 1 + yn b 2 2 =

¢x s y0 + 2y1 + 2y2 + Á + 2yn - 1 + yn d, 2

where y0 = ƒsad,

y1 = ƒsx1 d,

...,

yn - 1 = ƒsxn - 1 d,

yn = ƒsbd.

The Trapezoidal Rule says: Use T to estimate the integral of ƒ from a to b. It is equivalent to the midpoint rule discussed in Section 5.1.

8.6

Numerical Integration

463

y

The Trapezoidal Rule b

To approximate 1a ƒsxd dx, use y5

T =

x2

¢x ay0 + 2y1 + 2y2 + Á + 2yn - 1 + yn b . 2

The y’s are the values of ƒ at the partition points x0 = a, x1 = a + ¢x, x2 = a + 2¢x, Á , xn - 1 = a + sn - 1d¢x, xn = b, 4

where ¢x = sb - ad>n. 49 16 36 16

1 0

25 16

1

2

Use the Trapezoidal Rule with n = 4 to estimate 11 x 2 dx. Compare the estimate with the exact value.

EXAMPLE 1

5 4

6 4

7 4

x

2

FIGURE 8.8 The trapezoidal approximation of the area under the graph of y = x 2 from x = 1 to x = 2 is a slight overestimate (Example 1).

Partition [1, 2] into four subintervals of equal length (Figure 8.8). Then evaluate y = x 2 at each partition point (Table 8.2). Using these y values, n = 4, and ¢x = s2 - 1d>4 = 1>4 in the Trapezoidal Rule, we have

Solution

T = TABLE 8.2

x

y = x

1 5 4 6 4 7 4 2

1 25 16 36 16 49 16 4

¢x ay0 + 2y1 + 2y2 + 2y3 + y4 b 2

=

49 25 36 1 a1 + 2 a b + 2 a b + 2 a b + 4b 8 16 16 16

=

75 = 2.34375. 32

2

Since the parabola is concave up, the approximating segments lie above the curve, giving each trapezoid slightly more area than the corresponding strip under the curve. The exact value of the integral is 2

L1

2

x 2 dx =

8 7 x3 1 d = - = . 3 1 3 3 3

The T approximation overestimates the integral by about half a percent of its true value of 7> 3. The percentage error is s2.34375 - 7>3d>s7>3d L 0.00446, or 0.446%. y

Simpson’s Rule: Approximations Using Parabolas Parabola yn21 yn

y0

y1 y2 h h 0 a 5 x0 x1 x2

y 5 f (x)

h

xn21 xn5 b

FIGURE 8.9 Simpson’s Rule approximates short stretches of the curve with parabolas.

x

Another rule for approximating the definite integral of a continuous function results from using parabolas instead of the straight line segments that produced trapezoids. As before, we partition the interval [a, b] into n subintervals of equal length h = ¢x = sb - ad>n, but this time we require that n be an even number. On each consecutive pair of intervals we approximate the curve y = ƒsxd Ú 0 by a parabola, as shown in Figure 8.9. A typical parabola passes through three consecutive points sxi - 1, yi - 1 d, sxi , yi d, and sxi + 1, yi + 1 d on the curve. Let’s calculate the shaded area beneath a parabola passing through three consecutive points. To simplify our calculations, we first take the case where x0 = -h, x1 = 0, and

464

Chapter 8: Techniques of Integration

x2 = h (Figure 8.10), where h = ¢x = sb - ad>n. The area under the parabola will be the same if we shift the y-axis to the left or right. The parabola has an equation of the form

y (0, y1) (–h, y0)

y0

y1

(h, y2 ) y 5 Ax 2 1 Bx 1 C

y = Ax 2 + Bx + C, so the area under it from x = -h to x = h is

y2

h

Ap =

L-h

sAx 2 + Bx + Cd dx h

–h

0

h

x

FIGURE 8.10 By integrating from -h to h, we find the shaded area to be h s y + 4y1 + y2 d. 3 0

=

Ax 3 Bx 2 + + Cx d 3 2 -h

=

2Ah 3 h + 2Ch = s2Ah 2 + 6Cd. 3 3

Since the curve passes through the three points s -h, y0 d, s0, y1 d, and sh, y2 d, we also have y0 = Ah 2 - Bh + C,

y1 = C,

y2 = Ah 2 + Bh + C,

from which we obtain C = y1, Ah - Bh = y0 - y1, 2

Ah 2 + Bh = y2 - y1, 2Ah 2 = y0 + y2 - 2y1 . Hence, expressing the area Ap in terms of the ordinates y0, y1 , and y2 , we have h h h s2Ah 2 + 6Cd = ss y0 + y2 - 2y1 d + 6y1 d = s y0 + 4y1 + y2 d. 3 3 3

Ap =

Now shifting the parabola horizontally to its shaded position in Figure 8.9 does not change the area under it. Thus the area under the parabola through sx0 , y0 d, sx1, y1 d, and sx2 , y2 d in Figure 8.9 is still h s y + 4y1 + y2 d. 3 0 Similarly, the area under the parabola through the points sx2 , y2 d, sx3 , y3 d, and sx4 , y4 d is h s y + 4y3 + y4 d. 3 2 Computing the areas under all the parabolas and adding the results gives the approximation b

La

ƒsxd dx L

h h s y + 4y1 + y2 d + s y2 + 4y3 + y4 d + Á 3 0 3 +

= HISTORICAL BIOGRAPHY Thomas Simpson (1720–1761)

h sy + 4yn - 1 + yn d 3 n-2

h s y + 4y1 + 2y2 + 4y3 + 2y4 + Á + 2yn - 2 + 4yn - 1 + yn d. 3 0

The result is known as Simpson’s Rule. The function need not be positive, as in our derivation, but the number n of subintervals must be even to apply the rule because each parabolic arc uses two subintervals.

8.6

Numerical Integration

465

Simpson’s Rule b To approximate 1a ƒsxd dx, use S =

¢x s y0 + 4y1 + 2y2 + 4y3 + Á + 2yn - 2 + 4yn - 1 + yn d. 3

The y’s are the values of ƒ at the partition points x0 = a, x1 = a + ¢x, x2 = a + 2¢x, Á , xn - 1 = a + sn - 1d¢x, xn = b. The number n is even, and ¢x = sb - ad>n. Note the pattern of the coefficients in the above rule: 1, 4, 2, 4, 2, 4, 2, Á , 4, 1.

EXAMPLE 2

2

Use Simpson’s Rule with n = 4 to approximate 10 5x 4 dx.

Partition [0, 2] into four subintervals and evaluate y = 5x 4 at the partition points (Table 8.3). Then apply Simpson’s Rule with n = 4 and ¢x = 1>2:

Solution

TABLE 8.3

x

y = 5x 4

0 1 2 1 3 2 2

0 5 16 5 405 16 80

S = =

¢x ay0 + 4y1 + 2y2 + 4y3 + y4 b 3 405 5 1 b + 80b a0 + 4 a b + 2s5d + 4 a 6 16 16

= 32

1 . 12

This estimate differs from the exact value (32) by only 1> 12, a percentage error of less than three-tenths of one percent, and this was with just four subintervals.

Error Analysis Whenever we use an approximation technique, the issue arises as to how accurate the approximation might be. The following theorem gives formulas for estimating the errors when using the Trapezoidal Rule and Simpson’s Rule. The error is the difference between the apb proximation obtained by the rule and the actual value of the definite integral 1a ƒ(x) dx. THEOREM 1—Error Estimates in the Trapezoidal and Simpson’s Rules If ƒ– is continuous and M is any upper bound for the values of ƒ ƒ– ƒ on [a, b], then the error ET in the trapezoidal approximation of the integral of ƒ from a to b for n steps satisfies the inequality ƒ ET ƒ …

Msb - ad3 . 12n 2

Trapezoidal Rule

If ƒs4d is continuous and M is any upper bound for the values of ƒ ƒs4d ƒ on [a, b], then the error ES in the Simpson’s Rule approximation of the integral of ƒ from a to b for n steps satisfies the inequality ƒ ES ƒ …

Msb - ad5 . 180n 4

Simpson’s Rule

To see why Theorem 1 is true in the case of the Trapezoidal Rule, we begin with a result from advanced calculus, which says that if ƒ – is continuous on the interval [a, b], then b

La

ƒsxd dx = T -

b - a # ƒ–scds¢xd2 12

466

Chapter 8: Techniques of Integration

for some number c between a and b. Thus, as ¢x approaches zero, the error defined by ET = -

b - a # ƒ–scds¢xd2 12

approaches zero as the square of ¢x. The inequality ƒ ET ƒ …

b - a max ƒ ƒ–sxd ƒ s¢xd2 12

where max refers to the interval [a, b], gives an upper bound for the magnitude of the error. In practice, we usually cannot find the exact value of max ƒ ƒ–sxd ƒ and have to estimate an upper bound or “worst case” value for it instead. If M is any upper bound for the values of ƒ ƒ–sxd ƒ on [a, b], so that ƒ ƒ–sxd ƒ … M on [a, b], then ƒ ET ƒ …

b - a Ms¢xd2 . 12

If we substitute sb - ad>n for ¢x, we get ƒ ET ƒ …

Msb - ad3 . 12n 2

To estimate the error in Simpson’s Rule, we start with a result from advanced calculus that says that if the fourth derivative ƒ(4) is continuous, then b

La

ƒsxd dx = S -

b - a # s4d ƒ scds¢xd4 180

for some point c between a and b. Thus, as ¢x approaches zero, the error, ES = -

b - a # s4d ƒ scds¢xd4 , 180

approaches zero as the fourth power of ¢x. (This helps to explain why Simpson’s Rule is likely to give better results than the Trapezoidal Rule.) The inequality b - a ƒ ES ƒ … 180 max ƒ ƒs4dsxd ƒ s¢xd4, where max refers to the interval [a, b], gives an upper bound for the magnitude of the error. As with max ƒ ƒ– ƒ in the error formula for the Trapezoidal Rule, we usually cannot find the exact value of max ƒ ƒs4dsxd ƒ and have to replace it with an upper bound. If M is any upper bound for the values of ƒ ƒs4d ƒ on [a, b], then b - a 4 ƒ ES ƒ … 180 Ms¢xd . Substituting sb - ad>n for ¢x in this last expression gives ƒ ES ƒ …

Msb - ad5 . 180n 4 2

Find an upper bound for the error in estimating 10 5x 4 dx using Simpson’s Rule with n = 4 (Example 2).

EXAMPLE 3

To estimate the error, we first find an upper bound M for the magnitude of the fourth derivative of ƒsxd = 5x 4 on the interval 0 … x … 2. Since the fourth derivative has

Solution

8.6

Numerical Integration

467

the constant value ƒs4dsxd = 120, we take M = 120. With b - a = 2 and n = 4, the error estimate for Simpson’s Rule gives ƒ ES ƒ …

Msb - ad5 120s2d5 1 = = . 4 4 # 12 180n 180 4

This estimate is consistent with the result of Example 2. Theorem 1 can also be used to estimate the number of subintervals required when using the Trapezoidal or Simpson’s Rule if we specify a certain tolerance for the error.

EXAMPLE 4

Estimate the minimum number of subintervals needed to approximate the integral in Example 3 using Simpson’s Rule with an error of magnitude less than 10 -4. Solution

Using the inequality in Theorem 1, if we choose the number of subintervals n

to satisfy Msb - ad5 6 10 -4, 180n 4 then the error ES in Simpson’s Rule satisfies ƒ ES ƒ 6 10-4 as required. From the solution in Example 3, we have M = 120 and b - a = 2, so we want n to satisfy 120(2) 5 180n 4 or, equivalently, n4 7

1 10 4

6

64 # 10 4 . 3

It follows that n 7 10 a

64 b 3

1>4

L 21.5.

Since n must be even in Simpson’s Rule, we estimate the minimum number of subintervals required for the error tolerance to be n = 22.

EXAMPLE 5

As we saw in Chapter 7, the value of ln 2 can be calculated from the

integral 2

ln 2 =

L1

1 x dx. 2

Table 8.4 shows T and S values for approximations of 11 s1>xd dx using various values of n. Notice how Simpson’s Rule dramatically improves over the Trapezoidal Rule. TABLE 8.4 Trapezoidal Rule approximations sTn d and Simpson’s Rule 2 approximations sSn d of ln 2 = 11 s1>xd dx

n 10 20 30 40 50 100

Tn

ƒ Error ƒ less than Á

Sn

ƒ Error ƒ less than Á

0.6937714032 0.6933033818 0.6932166154 0.6931862400 0.6931721793 0.6931534305

0.0006242227 0.0001562013 0.0000694349 0.0000390595 0.0000249988 0.0000062500

0.6931502307 0.6931473747 0.6931472190 0.6931471927 0.6931471856 0.6931471809

0.0000030502 0.0000001942 0.0000000385 0.0000000122 0.0000000050 0.0000000004

468

Chapter 8: Techniques of Integration

In particular, notice that when we double the value of n (thereby halving the value of h = ¢x), the T error is divided by 2 squared, whereas the S error is divided by 2 to the fourth. This has a dramatic effect as ¢x = s2 - 1d>n gets very small. The Simpson approximation for n = 50 rounds accurately to seven places and for n = 100 agrees to nine decimal places (billionths)! If ƒ(x) is a polynomial of degree less than four, then its fourth derivative is zero, and ES = -

b - a (4) b - a ƒ scds¢xd4 = s0ds¢xd4 = 0. 180 180

Thus, there will be no error in the Simpson approximation of any integral of ƒ. In other words, if ƒ is a constant, a linear function, or a quadratic or cubic polynomial, Simpson’s Rule will give the value of any integral of ƒ exactly, whatever the number of subdivisions. Similarly, if ƒ is a constant or a linear function, then its second derivative is zero, and 146 ft 122 ft 76 ft 54 ft 40 ft

Vertical spacing 20 ft

30 ft 13 ft Ignored

FIGURE 8.11 The dimensions of the swamp in Example 6.

ET = -

b - a b - a ƒ–scds¢xd2 = s0ds¢xd2 = 0. 12 12

The Trapezoidal Rule will therefore give the exact value of any integral of ƒ. This is no surprise, for the trapezoids fit the graph perfectly. Although decreasing the step size ¢x reduces the error in the Simpson and Trapezoidal approximations in theory, it may fail to do so in practice. When ¢x is very small, say ¢x = 10 -5 , computer or calculator round-off errors in the arithmetic required to evaluate S and T may accumulate to such an extent that the error formulas no longer describe what is going on. Shrinking ¢x below a certain size can actually make things worse. You should consult a text on numerical analysis for more sophisticated methods if you are having problems with round-off error using the rules discussed in this section.

EXAMPLE 6

A town wants to drain and fill a small polluted swamp (Figure 8.11). The swamp averages 5 ft deep. About how many cubic yards of dirt will it take to fill the area after the swamp is drained?

To calculate the volume of the swamp, we estimate the surface area and multiply by 5. To estimate the area, we use Simpson’s Rule with ¢x = 20 ft and the y’s equal to the distances measured across the swamp, as shown in Figure 8.11. Solution

¢x s y0 + 4y1 + 2y2 + 4y3 + 2y4 + 4y5 + y6 d 3 20 = s146 + 488 + 152 + 216 + 80 + 120 + 13d = 8100 3

S =

The volume is about s8100ds5d = 40,500 ft3 or 1500 yd3 .

Exercises 8.6 Estimating Integrals The instructions for the integrals in Exercises 1–10 have two parts, one for the Trapezoidal Rule and one for Simpson’s Rule.

II. Using Simpson’s Rule a. Estimate the integral with n = 4 steps and find an upper bound for ƒ ES ƒ .

I. Using the Trapezoidal Rule

b. Evaluate the integral directly and find ƒ ES ƒ .

a. Estimate the integral with n = 4 steps and find an upper bound for ƒ ET ƒ .

c. Use the formula s ƒ ES ƒ >strue valuedd * 100 to express ƒ ES ƒ as a percentage of the integral’s true value.

b. Evaluate the integral directly and find ƒ ET ƒ .

c. Use the formula s ƒ ET ƒ >strue valuedd * 100 to express ƒ ET ƒ as a percentage of the integral’s true value.

2

1.

L1

3

x dx

2.

L1

s2x - 1d dx

8.6 1

3.

L-1

0

sx 2 + 1d dx

4.

st 3 + td dt

6.

L-2

2

5.

L0

L-1

8.

1 ds ss 1d2 L2 1

sin t dt

L0

st 3 + 1d dt

4

1 ds 2 s L1 p

9.

sx 2 - 1d dx

1

2

7.

10.

sin pt dt

L0

Estimating the Number of Subintervals In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10 -4 by (a) the Trapezoidal Rule and (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.) 2

11.

L1

3

x dx

12.

sx 2 + 1d dx

14.

L-1

16.

2

17.

L-2

1 ds 2 L1 s L0

L-1

18.

L0

2.2 3.2 4.5 5.9 7.8 10.2 12.7 16.0 20.6 26.2 37.1

25. Wing design The design of a new airplane requires a gasoline tank of constant cross-sectional area in each wing. A scale drawing of a cross-section is shown here. The tank must hold 5000 lb of gasoline, which has a density of 42 lb>ft3 . Estimate the length of the tank by Simpson’s Rule. y0

st 3 + 1d dt

y1

y2

y3

y4

y5

y6

y0 1.5 ft, y1 1.6 ft, y2 1.8 ft, y3 1.9 ft, y4 2.0 ft, y5 y6 2.1 ft Horizontal spacing 1 ft

1 ds 2 L2 ss - 1d 3

2x + 1 dx

20.

sin sx + 1d dx

22.

1 dx L0 2x + 1

2

21.

Zero to 30 mph 40 mph 50 mph 60 mph 70 mph 80 mph 90 mph 100 mph 110 mph 120 mph 130 mph

sx 2 - 1d dx

4

3

19.

Time (sec)

1

st 3 + td dt

L0

Speed change

0

2

15.

s2x - 1d dx

L1

1

13.

469

Numerical Integration

1

L-1

cos sx + pd dx

Estimates with Numerical Data 23. Volume of water in a swimming pool A rectangular swimming pool is 30 ft wide and 50 ft long. The accompanying table shows the depth h(x) of the water at 5-ft intervals from one end of the pool to the other. Estimate the volume of water in the pool using the Trapezoidal Rule with n = 10 applied to the integral 50

V =

L0

30 # hsxd dx .

Position (ft) x

Depth (ft) h(x)

Position (ft) x

Depth (ft) h(x)

0 5 10 15 20 25

6.0 8.2 9.1 9.9 10.5 11.0

30 35 40 45 50

11.5 11.9 12.3 12.7 13.0

24. Distance traveled The accompanying table shows time-tospeed data for a sports car accelerating from rest to 130 mph. How far had the car traveled by the time it reached this speed? (Use trapezoids to estimate the area under the velocity curve, but be careful: The time intervals vary in length.)

26. Oil consumption on Pathfinder Island A diesel generator runs continuously, consuming oil at a gradually increasing rate until it must be temporarily shut down to have the filters replaced. Use the Trapezoidal Rule to estimate the amount of oil consumed by the generator during that week.

Day

Oil consumption rate (liters> h)

Sun Mon Tue Wed Thu Fri Sat Sun

0.019 0.020 0.021 0.023 0.025 0.028 0.031 0.035

Theory and Examples 27. Usable values of the sine-integral function The sine-integral function, x sin t “Sine integral of x” Si sxd = dt , L0 t is one of the many functions in engineering whose formulas cannot be simplified. There is no elementary formula for the antiderivative of (sin t)> t. The values of Si(x), however, are readily estimated by numerical integration. Although the notation does not show it explicitly, the function being integrated is ƒstd = •

sin t t ,

t Z 0

1,

t = 0,

470

Chapter 8: Techniques of Integration

the continuous extension of (sin t)> t to the interval [0, x]. The function has derivatives of all orders at every point of its domain. Its graph is smooth, and you can expect good results from Simpson’s Rule.

Applications T 32. The length of one arch of the curve y = sin x is given by p

L =

y

Estimate L by Simpson’s Rule with n = 8 .

x

sin t y t

Si (x)

1

0

L0

sin t dt t

x

2

t

T 33. Your metal fabrication company is bidding for a contract to make sheets of corrugated iron roofing like the one shown here. The cross-sections of the corrugated sheets are to conform to the curve y = sin

a. Use the fact that ƒ ƒ ƒ … 1 on [0, p>2] to give an upper bound for the error that will occur if s4d

p Si a b = 2 L0

p>2

21 + cos2 x dx .

L0

sin t t dt

is estimated by Simpson’s Rule with n = 4 .

3p x, 20

0 … x … 20 in .

If the roofing is to be stamped from flat sheets by a process that does not stretch the material, how wide should the original material be? To find out, use numerical integration to approximate the length of the sine curve to two decimal places. Original sheet

y

Corrugated sheet

b. Estimate Si sp>2d by Simpson’s Rule with n = 4 . c. Express the error bound you found in part (a) as a percentage of the value you found in part (b). 20 in.

28. The error function The error function, erf sxd =

x

2 2p L0

0

e -t dt , 2

important in probability and in the theories of heat flow and signal transmission, must be evaluated numerically because there is 2 no elementary expression for the antiderivative of e -t . a. Use Simpson’s Rule with n = 10 to estimate erf (1). b. In [0, 1],

`

20

y sin 3 x 20

2 d4 ae -t b ` … 12 . dt 4

x (in.)

T 34. Your engineering firm is bidding for the contract to construct the tunnel shown here. The tunnel is 300 ft long and 50 ft wide at the base. The cross-section is shaped like one arch of the curve y = 25 cos spx>50d . Upon completion, the tunnel’s inside surface (excluding the roadway) will be treated with a waterproof sealer that costs $1.75 per square foot to apply. How much will it cost to apply the sealer? (Hint: Use numerical integration to find the length of the cosine curve.)

Give an upper bound for the magnitude of the error of the estimate in part (a).

y 25 cos ( x/50)

y

b 1a ƒsxd dx

29. Prove that the sum T in the Trapezoidal Rule for is a Riemann sum for ƒ continuous on [a, b]. (Hint: Use the Intermediate Value Theorem to show the existence of ck in the subinterval [xk - 1, xk] satisfying ƒsck d = sƒsxk - 1 d + ƒsxk dd>2 .)

–25

b

30. Prove that the sum S in Simpson’s Rule for 1a ƒsxd dx is a Riemann sum for ƒ continuous on [a, b]. (See Exercise 29.)

0

300 ft 25

T 31. Elliptic integrals The length of the ellipse y x + 2 = 1 a2 b turns out to be p>2

Length = 4a

L0

21 - e 2 cos2 t dt ,

where e = 2a - b >a is the ellipse’s eccentricity. The integral in this formula, called an elliptic integral, is nonelementary except when e = 0 or 1. 2

x (ft) NOT TO SCALE

2

2

2

a. Use the Trapezoidal Rule with n = 10 to estimate the length of the ellipse when a = 1 and e = 1>2 . b. Use the fact that the absolute value of the second derivative of ƒstd = 21 - e 2 cos2 t is less than 1 to find an upper bound for the error in the estimate you obtained in part (a).

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis. 35. y = sin x, 36. y = x 2>4,

0 … x … p 0 … x … 2

37. Use numerical integration to estimate the value of sin-1 0.6 =

0.6

L0

dx 21 - x 2

.

For reference, sin-1 0.6 = 0.64350 to five decimal places. 38. Use numerical integration to estimate the value of 1

1 dx . 2 L0 1 + x

p = 4

8.7

y y ln2x x

0.1

0

1

471

Improper Integrals

8.7 0.2

Improper Integrals

2

3

4

5

x

6

(a) y

Up to now, we have required definite integrals to have two properties. First, that the domain of integration [a, b] be finite. Second, that the range of the integrand be finite on this domain. In practice, we may encounter problems that fail to meet one or both of these conditions. The integral for the area under the curve y = sln xd>x 2 from x = 1 to x = q is an example for which the domain is infinite (Figure 8.12a). The integral for the area under the curve of y = 1> 1x between x = 0 and x = 1 is an example for which the range of the integrand is infinite (Figure 8.12b). In either case, the integrals are said to be improper and are calculated as limits. We will see in Chapter 9 that improper integrals play an important role when investigating the convergence of certain infinite series.

Infinite Limits of Integration y 1 x

Consider the infinite region that lies under the curve y = e -x>2 in the first quadrant (Figure 8.13a). You might think this region has infinite area, but we will see that the value is finite. We assign a value to the area in the following way. First find the area A(b) of the portion of the region that is bounded on the right by x = b (Figure 8.13b).

1

0

x

1

b

Asbd =

L0

(b)

b

e -x>2 dx = -2e -x>2 d = -2e -b>2 + 2 0

Then find the limit of A(b) as b : q

FIGURE 8.12 Are the areas under these infinite curves finite? We will see that the answer is yes for both curves.

lim Asbd = lim s -2e -b>2 + 2d = 2.

b: q

b: q

The value we assign to the area under the curve from 0 to q is q

L0

e -x>2 dx = lim

b: q L 0

b

e -x>2 dx = 2.

y

DEFINITION Integrals with infinite limits of integration are improper integrals of Type I.

Area 2

1. If ƒ(x) is continuous on [a, q d, then q

x

La

(a) y

b

ƒsxd dx = lim

ƒsxd dx.

b: q L a

2. If ƒ(x) is continuous on s - q , b], then b

Area 2e –b/2 2

L- q

b

ƒsxd dx =

lim

a: - q L a

ƒsxd dx.

3. If ƒ(x) is continuous on s - q , q d, then b (b)

FIGURE 8.13 (a) The area in the first quadrant under the curve y = e-x>2. (b) The area is an improper integral of the first type.

x q

L- q

q

c

ƒsxd dx =

L- q

ƒsxd dx +

Lc

ƒsxd dx,

where c is any real number. In each case, if the limit is finite we say that the improper integral converges and that the limit is the value of the improper integral. If the limit fails to exist, the improper integral diverges.

472

Chapter 8: Techniques of Integration

It can be shown that the choice of c in Part 3 of the definition is unimportant. We can q evaluate or determine the convergence or divergence of 1- q ƒsxd dx with any convenient choice. Any of the integrals in the above definition can be interpreted as an area if ƒ Ú 0 on the interval of integration. For instance, we interpreted the improper integral in Figure 8.13 as an area. In that case, the area has the finite value 2. If ƒ Ú 0 and the improper integral diverges, we say the area under the curve is infinite. Is the area under the curve y = sln xd>x 2 from x = 1 to x = q finite? If so, what is its value?

EXAMPLE 1

We find the area under the curve from x = 1 to x = b and examine the limit as b : q . If the limit is finite, we take it to be the area under the curve (Figure 8.14). The area from 1 to b is Solution

y y ln2x x

0.2

b

b

0.1

0

Integration by parts with u = ln x, dy = dx>x 2, du = dx>x, y = - 1>x

b

ln x 1 1 1 dx = csln xd a- x b d a- x b a x b dx 2 1 L1 x L1 b

1

b

= -

ln b 1 - cx d b 1

= -

ln b 1 - + 1. b b

x

FIGURE 8.14 The area under this curve is an improper integral (Example 1).

The limit of the area as b : q is q

L1

b

ln x ln x dx = lim dx 2 q b: L x x2 1 = lim c-

ln b 1 - + 1d b b

= - c lim

ln b d - 0 + 1 b

b: q

b: q

= - c lim

1>b d + 1 = 0 + 1 = 1. b: q 1

l’Hôpital’s Rule

Thus, the improper integral converges and the area has finite value 1.

EXAMPLE 2

Evaluate q

L- q Solution

According to the definition (Part 3), we can choose c = 0 and write q

HISTORICAL BIOGRAPHY Lejeune Dirichlet (1805–1859)

dx . 1 + x2

L- q

0

dx dx = + 2 2 q 1 + x L- 1 + x L0

q

dx . 1 + x2

Next we evaluate each improper integral on the right side of the equation above. 0

dx = 2 L- q 1 + x = =

0

lim

a: - q L a

dx 1 + x2

lim tan-1 x d

a: - q

0 a

lim stan-1 0 - tan-1 ad = 0 - a-

a: - q

p p b = 2 2

8.7 q

L0

b

b: q

y

b 0

= lim stan-1 b - tan-1 0d = b: q

1 1 x2

p p - 0 = 2 2

Thus,

Area

q

0

473

dx dx = lim 2 b: q L 1 + x2 0 1 + x = lim tan-1 x d

y

Improper Integrals

L- q

x

NOT TO SCALE

FIGURE 8.15 The area under this curve is finite (Example 2).

dx p p = + = p. 2 2 2 1 + x

Since 1>s1 + x 2 d 7 0, the improper integral can be interpreted as the (finite) area beneath the curve and above the x-axis (Figure 8.15).

ˆ

The Integral

L1

dx xp

The function y = 1>x is the boundary between the convergent and divergent improper integrals with integrands of the form y = 1>x p . As the next example shows, the improper integral converges if p 7 1 and diverges if p … 1. q

For what values of p does the integral 11 dx>x p converge? When the integral does converge, what is its value?

EXAMPLE 3

Solution

If p Z 1, x -p + 1 dx 1 1 1 d = sb -p + 1 - 1d = a p - 1 - 1b . p = -p + 1 1 p 1 p x b 1 1 L b

b

Thus, q

L1

b

dx dx = lim p xp b: q L 1 x = lim c b: q

1 , 1 1 a p - 1 - 1b d = • p - 1 1 - p b q,

p 7 1 p 6 1

because

lim

b: q

0, 1 = e q, bp-1

p 7 1 p 6 1.

Therefore, the integral converges to the value 1>s p - 1d if p 7 1 and it diverges if p 6 1.

474

Chapter 8: Techniques of Integration

If p = 1, the integral also diverges: q

dx = xp L1

L1

q

dx x b

= lim

b: q L 1

dx x

= lim ln x D 1 b

b: q

= lim sln b - ln 1d = q . b: q

Integrands with Vertical Asymptotes

y

Another type of improper integral arises when the integrand has a vertical asymptote—an infinite discontinuity—at a limit of integration or at some point between the limits of integration. If the integrand ƒ is positive over the interval of integration, we can again interpret the improper integral as the area under the graph of ƒ and above the x-axis between the limits of integration. Consider the region in the first quadrant that lies under the curve y = 1> 1x from x = 0 to x = 1 (Figure 8.12b). First we find the area of the portion from a to 1 (Figure 8.16).

y 1 x

Area 2 2a

1

0

1

1

dx = 21x d = 2 - 21a. 1x a La a

1

x

Then we find the limit of this area as a : 0 +: dx = lim+ A 2 - 21a B = 2. a:0 1x La 1

lim+

a:0

FIGURE 8.16 The area under this curve is an example of an improper integral of the second kind.

Therefore the area under the curve from 0 to 1 is finite and is defined to be 1

1

dx dx = lim+ = 2. a:0 L a 1x L0 1x

DEFINITION Integrals of functions that become infinite at a point within the interval of integration are improper integrals of Type II. 1. If ƒ(x) is continuous on (a, b] and discontinuous at a, then b

b

ƒsxd dx = lim+

La

c:a

ƒsxd dx.

Lc

2. If ƒ(x) is continuous on [a, b) and discontinuous at b, then b

La

c

ƒsxd dx = limc:b

La

ƒsxd dx.

3. If ƒ(x) is discontinuous at c, where a 6 c 6 b, and continuous on [a, cd ´ sc, b], then b

c

ƒsxd dx =

b

ƒsxd dx +

ƒsxd dx. La La Lc In each case, if the limit is finite we say the improper integral converges and that the limit is the value of the improper integral. If the limit does not exist, the integral diverges.

8.7

475

In Part 3 of the definition, the integral on the left side of the equation converges if both integrals on the right side converge; otherwise it diverges.

y y

Improper Integrals

1 1x

EXAMPLE 4

Investigate the convergence of 1

1 dx. 1 x L0 The integrand ƒsxd = 1>s1 - xd is continuous on [0, 1) but is discontinuous at x = 1 and becomes infinite as x : 1- (Figure 8.17). We evaluate the integral as

Solution

1

b

b

0

lim-

b:1

x

1

L0

b 1 dx = lim- C -ln ƒ 1 - x ƒ D 0 1 - x b:1

= lim- [-ln s1 - bd + 0] = q . b:1

FIGURE 8.17 The area beneath the curve and above the x-axis for [0, 1) is not a real number (Example 4).

The limit is infinite, so the integral diverges.

EXAMPLE 5

Evaluate 3

dx . L0 sx - 1d2>3 The integrand has a vertical asymptote at x = 1 and is continuous on [0, 1) and (1, 3] (Figure 8.18). Thus, by Part 3 of the definition above,

Solution

y

y

3

1

3

dx dx dx = + . 2>3 2>3 L0 sx - 1d L0 sx - 1d L1 sx - 1d2>3

1 (x 1)2/3

Next, we evaluate each improper integral on the right-hand side of this equation. 1

b

dx dx = lim2>3 b:1 sx 1d sx 1d2>3 0 0 L L = lim- 3sx - 1d1>3 D 0

1

b

b:1

0

c

b

3

= lim- [3sb - 1d1>3 + 3] = 3

x

1

FIGURE 8.18 Example 5 shows that the area under the curve exists (so it is a real number).

b:1

3

3

dx dx = lim+ 2>3 c:1 L1 sx - 1d Lc sx - 1d2>3 = lim+ 3sx - 1d1>3 D c

3

c:1

3 = lim+ C 3s3 - 1d1>3 - 3sc - 1d1>3 D = 32 2

c:1

We conclude that 3

L0

dx 3 = 3 + 32 2. sx - 1d2>3

Improper Integrals with a CAS Computer algebra systems can evaluate many convergent improper integrals. To evaluate the integral q

L2

x + 3 dx sx - 1d(x 2 + 1)

476

Chapter 8: Techniques of Integration

(which converges) using Maple, enter 7 ƒ:= sx + 3d>ssx - 1d * sx ¿2 + 1dd; Then use the integration command 7 intsƒ, x = 2..infinityd; Maple returns the answer 1 p + ln s5d + arctan s2d. 2

-

To obtain a numerical result, use the evaluation command evalf and specify the number of digits as follows: 7 evalfs%, 6d; The symbol % instructs the computer to evaluate the last expression on the screen, in this case s -1>2dp + ln s5d + arctan s2d. Maple returns 1.14579. Using Mathematica, entering In [1]:= Integrate [sx + 3d>ssx - 1dsx ¿2 + 1dd, 5x, 2, Infinity6] returns Out [1]=

-p + ArcTan [2] + Log [5]. 2

To obtain a numerical result with six digits, use the command “N[%, 6]”; it also yields 1.14579.

Tests for Convergence and Divergence When we cannot evaluate an improper integral directly, we try to determine whether it converges or diverges. If the integral diverges, that’s the end of the story. If it converges, we can use numerical methods to approximate its value. The principal tests for convergence or divergence are the Direct Comparison Test and the Limit Comparison Test.

y 1 y e –x

2

EXAMPLE 6 Solution

q

Does the integral 11 e -x dx converge?

By definition, q

(1, e –1) y 0

1

x

FIGURE 8.19 The graph of e -x lies below the graph of e -x for x 7 1 (Example 6). 2

e -x dx = lim 2

b

b: q L 1

L1

e –x

b

2

e -x dx. 2

We cannot evaluate this integral directly because it is nonelementary. But we can show 2 b that its limit as b : q is finite. We know that 11 e -x dx is an increasing function of b. Therefore either it becomes infinite as b : q or it has a finite limit as b : q . It does 2 not become infinite: For every value of x Ú 1, we have e -x … e -x (Figure 8.19) so that b

L1

e -x dx …

b

2

L1

e -x dx = -e -b + e -1 6 e -1 L 0.36788.

Hence, q

L1

e -x dx = lim 2

b: q L 1

b

e -x dx 2

converges to some definite finite value. We do not know exactly what the value is except that it is something positive and less than 0.37. Here we are relying on the completeness property of the real numbers, discussed in Appendix 7.

8.7

Improper Integrals

477

The comparison of e -x and e -x in Example 6 is a special case of the following test. 2

THEOREM 2—Direct Comparison Test Let ƒ and g be continuous on [a, q d with 0 … ƒsxd … gsxd for all x Ú a . Then

HISTORICAL BIOGRAPHY Karl Weierstrass (1815–1897)

q

1.

La

q

ƒsxd dx

converges if

gsxd dx

diverges if

q

2.

La

gsxd dx

La

converges.

q

ƒsxd dx

La

diverges.

Proof The reasoning behind the argument establishing Theorem 2 is similar to that in Example 6. If 0 … ƒsxd … gsxd for x Ú a, then from Rule 7 in Theorem 2 of Section 5.3 we have b

b

ƒsxd dx …

La

b 7 a.

gsxd dx,

La

From this it can be argued, as in Example 6, that q

q

ƒsxd dx

La

converges if

La

gsxd dx

converges.

ƒsxd dx

diverges.

Turning this around says that q

La

EXAMPLE 7 q

(a)

L1 0 … q

(b)

L1

q

gsxd dx

diverges if

La

These examples illustrate how we use Theorem 2.

sin2 x dx x2 sin2 x 1 … 2 x2 x

converges because

on

1 dx 2x - 0.1

q

and

L1

1 dx x2

converges.

Example 3

1 x dx

diverges.

Example 3

diverges because

2

1 1 Ú x 2 2x - 0.1

[1, q d

on

[1, q d

q

and

L1

THEOREM 3—Limit Comparison Test tinuous on [a, q d, and if lim

x: q

If the positive functions ƒ and g are con-

ƒsxd = L, gsxd

0 6 L 6 q,

then q

q

ƒsxd dx

La both converge or both diverge.

and

La

gsxd dx

478

Chapter 8: Techniques of Integration

We omit the more advanced proof of Theorem 3. Although the improper integrals of two functions from a to q may both converge, this does not mean that their integrals necessarily have the same value, as the next example shows.

EXAMPLE 8

Show that q

L1 y

dx 1 + x2

q

converges by comparison with 11 s1>x 2 d dx. Find and compare the two integral values.

y 12 x

1

The functions ƒsxd = 1>x 2 and gsxd = 1>s1 + x 2 d are positive and continuous on [1, q d. Also, Solution

lim

y 1 2 1x

x: q

1>x 2 ƒsxd 1 + x2 = lim = lim 2 gsxd x: q 1>s1 + x d x: q x2 = lim a

1

0

2

3

x: q

x

1 + 1b = 0 + 1 = 1, x2 q

FIGURE 8.20 The functions in Example 8.

a positive finite limit (Figure 8.20). Therefore,

dx converges because 1 + x2 L1

L1 converges. The integrals converge to different values, however: q

L1

dx 1 = = 1 2 2 1 x

q

dx x2

Example 3

and q

L1

b

dx dx = lim 2 2 q b: 1 + x L1 1 + x = lim [tan-1 b - tan-1 1] = b: q

q

EXAMPLE 9

Investigate the convergence of

TABLE 8.5 b

b 2 5 10 100 1000 10000 100000

L1

1 ex dx x

0.5226637569 1.3912002736 2.0832053156 4.3857862516 6.6883713446 8.9909564376 11.2935415306

L1

p p p = . 2 4 4

1 - e -x dx. x

The integrand suggests a comparison of ƒsxd = s1 - e -x d>x with gsxd = 1>x. However, we cannot use the Direct Comparison Test because ƒsxd … gsxd and the integral of gsxd diverges. On the other hand, using the Limit Comparison Test we find that Solution

lim

x: q

ƒsxd x 1 - e -x b a b = lim s1 - e -x d = 1, = lim a x 1 q gsxd x: x: q q

1 - e -x dx diverges because x

q

dx x diL1 L1 verges. Approximations to the improper integral are given in Table 8.5. Note that the values which is a positive finite limit. Therefore,

do not appear to approach any fixed limiting value as b : q .

8.7

Improper Integrals

479

Exercises 8.7 Evaluating Improper Integrals The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables. q

1.

L0

q

dx 2 x + 1

2.

L1

1

1

1

L0 21 - x 2 -2 2 dx 9. 2 L- q x - 1 11.

L2 q

13.

10. 12.

2x dx sx + 1d2

14.

51.

L2

2 dt t2 - 1

du

L0 2u + 2u q dx 17. L0 s1 + xd1x

x dx sx + 4d3>2

53.

s + 1

16.

ds

L0 24 - s q 1 18. dx L1 x2x 2 - 1 q q 16 tan-1 x dy 19. 20. dx 2 -1 1 + x2 L0 s1 + y ds1 + tan yd L0 2

2

q

0

21.

ue u du

L- q 0

23.

e

L- q

- ƒxƒ

22.

L0

24.

2

26.

L0 24 - s 2 ds 29. L1 s2s 2 - 1 4 dx 31. L-1 2 ƒ x ƒ 2

q

L-1

-x2

dx

s -ln xd dx

L0

du u + 5u + 6 2

4r dr L0 21 - r 4 4 dt 30. L2 t2t 2 - 4 2 dx 32. L0 2 ƒ x - 1 ƒ 28.

q

34.

L0

p>2

dx sx + 1dsx 2 + 1d

p>2

tan u du

L0 p

37.

36.

L0

x e

cos u du L-p>2 sp - 2ud1>3 1 -1x e 40. dx L0 1x 38.

L0 2p - u -2 -1>x

cot u du

L0 p>2

sin u du

ln 2

39.

L0 L1 Lp q

L4 q

59.

L1

63.

50.

2y - 1 dx

L1

54. 56.

2 dt t 3>2 - 1

58.

ex x dx

60.

2e - x dx

L2 L2 q

2 + cos x dx x

x

L0

q

2x + 1 dx x2

1

L4

q

52.

2x + 1

q

61.

-x ln ƒ x ƒ dx

q

dy

6

q

Testing for Convergence In Exercises 35–64, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. 35.

L2

q

57.

48.

Lp q

L2

dx 1x - 1 du 1 + eu dx 2x 2 - 1 x dx 2x 4 - 1 1 + sin x dx x2 1 dx ln x

q

ln sln xd dx

Lee q

dx

62.

L1

1 dx e x - 2x

q

64.

L- q 2x 4 + 1

dx x -x L- q e + e

Theory and Examples 65. Find the values of p for which each integral converges.

1

ds

27.

33.

L1

L-1

q

dx x + 1 3

q

55.

46.

1

x ln x dx

L0

2xe

L- q

1

25.

2e -u sin u du

q

dx

L-1

q

2

L- q

dx L0 1 - x 1

q

L- q

2

u + 1

15.

0.999

49.

q

44.

ln ƒ x ƒ dx

q

47.

2 dx x2 + 4

q

2 dy y2 - y

1

L0 r

2

dx 2 L0 1 - x 1

45.

dr

2

2

L- q

43.

L0 24 - x 1 dx 6. L-8 x 1>3 8.

dt L0 1t + sin t 1 dt 42. (Hint: t Ú sin t for t Ú 0) L0 t - sin t 2

dx

4.

dx

q

x 4

dx 3. L0 1x 1 dx 5. L-1 x 2>3 7.

dx 1.001

p

41.

dx

q

2

a.

dx p L1 xsln xd

b.

L2

dx xsln xd p

ˆ

b

66. 1 ˆ ƒ(x) dx may not equal lim 1-b ƒ(x) dx Show that b: ˆ q

L0

2x dx x2 + 1

diverges and hence that q

2x dx 2 L- q x + 1 diverges. Then show that b

lim

b: q

2x dx = 0. 2 L-b x + 1

Exercises 67–70 are about the infinite region in the first quadrant between the curve y = e -x and the x-axis. 67. Find the area of the region. 68. Find the centroid of the region. 69. Find the volume of the solid generated by revolving the region about the y-axis.

480

Chapter 8: Techniques of Integration

70. Find the volume of the solid generated by revolving the region about the x-axis. 71. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = p>2 . 72. The region in Exercise 71 is revolved about the x-axis to generate a solid. a. Find the volume of the solid. b. Show that the inner and outer surfaces of the solid have infinite area. 73. Estimating the value of a convergent improper integral whose domain is infinite a. Show that q

L3

1 -9 e 6 0.000042 , 3

e -3x dx = q

and hence that 13 e -x dx 6 0.000042 . Explain why this q 2 2 3 means that 10 e -x dx can be replaced by 10 e -x dx without introducing an error of magnitude greater than 0.000042. 2

3 -x 2 10 e dx

numerically. T b. Evaluate 74. The infinite paint can or Gabriel’s horn As Example 3 shows, q the integral 11 sdx>xd diverges. This means that the integral q

1 2p x

1 dx , 4 A x 1 L which measures the surface area of the solid of revolution traced out by revolving the curve y = 1>x, 1 … x, about the x-axis, diverges also. By comparing the two integrals, we see that, for every finite value b 7 1 , b

L1

1 2p x

1 +

b

A

1 1 dx 7 2p x dx . x4 L1

1 +

T

a. Plot the integrand ssin td>t for t 7 0 . Is the sine-integral function everywhere increasing or decreasing? Do you think Si sxd = 0 for x 7 0 ? Check your answers by graphing the function Si (x) for 0 … x … 25 . b. Explore the convergence of q

sin t t dt .

L0 If it converges, what is its value? The function

76. Error function

x

erf sxd =

2e -t

2

dt , L0 2p called the error function, has important applications in probability and statistics. T a. Plot the error function for 0 … x … 25 . b. Explore the convergence of q

2e -t

2

dt . L0 2p If it converges, what appears to be its value? You will see how to confirm your estimate in Section 14.4, Exercise 41. 77. Normal probability distribution The function ƒsxd =

1

e-2 A

B

1 x-m 2 s

s 22p

is called the normal probability density function with mean m and standard deviation s. The number m tells where the distribution is centered, and s measures the “scatter” around the mean. From the theory of probability, it is known that q

y

L- q

y 1x

ƒsxd dx = 1 .

In what follows, let m = 0 and s = 1 . T a. Draw the graph of ƒ. Find the intervals on which ƒ is increasing, the intervals on which ƒ is decreasing, and any local extreme values and where they occur.

0 1 b

b. Evaluate n

x

However, the integral q

2

1 p a x b dx

L-n for n = 1, 2, and 3.

c. Give a convincing argument that

L1 for the volume of the solid converges. a. Calculate it.

b. This solid of revolution is sometimes described as a can that does not hold enough paint to cover its own interior. Think about that for a moment. It is common sense that a finite amount of paint cannot cover an infinite surface. But if we fill the horn with paint (a finite amount), then we will have covered an infinite surface. Explain the apparent contradiction. 75. Sine-integral function The integral x

Si sxd =

sin t dt , L0 t

called the sine-integral function, has important applications in optics.

ƒsxd dx

q

L- q

ƒsxd dx = 1 .

(Hint: Show that 0 6 ƒsxd 6 e -x>2 for x 7 1, and for b 7 1, q

Lb

e -x>2 dx : 0

b : q .)

as

78. Show that if ƒ(x) is integrable on every interval of real numbers and a and b are real numbers with a 6 b , then a

q

b

q

a. 1- q ƒsxd dx and 1a ƒsxd dx both converge if and only if 1- q ƒsxd dx and 1b ƒsxd dx both converge. a

q

b

q

b. 1- q ƒsxd dx + 1a ƒsxd dx = 1- q ƒsxd dx + 1b ƒsxd dx when the integrals involved converge.

Chapter 8 COMPUTER EXPLORATIONS In Exercises 79–82, use a CAS to explore the integrals for various values of p (include noninteger values). For what values of p does the integral converge? What is the value of the integral when it does converge? Plot the integrand for various values of p.

Chapter 8

2. When applying the formula for integration by parts, how do you choose the u and dy? How can you apply integration by parts to an integral of the form 1ƒsxd dx ? 3. If an integrand is a product of the form sinn x cosm x , where m and n are nonnegative integers, how do you evaluate the integral? Give a specific example of each case. 4. What substitutions are made to evaluate integrals of sin mx sin nx, sin mx cos nx, and cos mx cos nx? Give an example of each case. 5. What substitutions are sometimes used to transform integrals involving 2a 2 - x 2, 2a 2 + x 2 , and 2x 2 - a 2 into integrals that can be evaluated directly? Give an example of each case. 6. What restrictions can you place on the variables involved in the three basic trigonometric substitutions to make sure the substitutions are reversible (have inverses)? 7. What is the goal of the method of partial fractions?

3. 5. 7.

L L L L

ln sx + 1d dx

2.

tan-1 3x dx

4.

sx + 1d2e x dx

6.

e x cos 2x dx

8.

x dx L x - 3x + 2 dx 11. 2 L xsx + 1d 2

80.

q

81.

L0

x p ln x dx

Le q

x p ln x dx

82.

L- q

x p ln ƒ x ƒ dx

8. When the degree of a polynomial ƒ(x) is less than the degree of a polynomial g(x), how do you write ƒ(x)> g(x) as a sum of partial fractions if g(x) a. is a product of distinct linear factors? b. consists of a repeated linear factor? c. contains an irreducible quadratic factor? What do you do if the degree of ƒ is not less than the degree of g? 9. How are integral tables typically used? What do you do if a particular integral you want to evaluate is not listed in the table? 10. What is a reduction formula? How are reduction formulas used? Give an example. 11. How would you compare the relative merits of Simpson’s Rule and the Trapezoidal Rule? 12. What is an improper integral of Type I? Type II? How are the values of various types of improper integrals defined? Give examples. 13. What tests are available for determining the convergence and divergence of improper integrals that cannot be evaluated directly? Give examples of their use.

L L L L

sin u du 2 L cos u + cos u - 2

14.

x 2 ln x dx

15.

16.

x cos-1 a b dx 2

3x 2 + 4x + 4 dx x3 + x L

17.

y + 3 dy 3 L 2y - 8y

19.

dt 4 2 L t + 4t + 3

20.

t dt 4 2 Lt - t - 2

21.

x3 + x2 dx Lx + x - 2

22.

x3 + 1 dx 3 Lx - x

23.

x 3 + 4x 2 dx L x + 4x + 3

24.

2x 3 + x 2 - 21x + 24 dx x 2 + 2x - 8 L

25.

dx L x A 32x + 1 B

26.

dx 3 L xA1 + 2 xB

27.

ds s Le - 1

28.

ds L 2e s + 1

x 2 sin s1 - xd dx e -2x sin 3x dx

x dx L x + 4x + 3 x + 1 dx 12. 2 L x sx - 1d 10.

cos u du 2 L sin u + sin u - 6

13.

Partial Fractions Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first. 9.

L0

x p ln x dx

Practice Exercises

Integration by Parts Evaluate the integrals in Exercises 1–8 using integration by parts. 1.

q

e

79.

481

Questions to Guide Your Review

1. What is the formula for integration by parts? Where does it come from? Why might you want to use it?

Chapter 8

Practice Exercises

2

2

2

4x dx 3 L x + 4x s3y - 7d dy 18. L sy - 1dsy - 2dsy - 3d

482

Chapter 8: Techniques of Integration

Trigonometric Substitutions Evaluate the integrals in Exercises 29–32 (a) without using a trigonometric substitution, (b) using a trigonometric substitution. y dy x dx 29. 30. L 216 - y 2 L 24 + x 2 31.

x dx 2 L4 - x

32.

t dt L 24t 2 - 1

Evaluate the integrals in Exercises 33–36. x dx 2 L9 - x dx 35. 2 L9 - x

50. Heat capacity of a gas Heat capacity Cy is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal> deg-mol (calories per degree gram molecular weight). The heat capacity of oxygen depends on its temperature T and satisfies the formula

dx 2 L xs9 - x d dx 36. L 29 - x 2

33.

34.

Cy = 8.27 + 10 -5 s26T - 1.87T 2 d .

Trigonometric Integrals Evaluate the integrals in Exercises 37–44. 37. 39. 41. 43.

L L L L

sin3 x cos4 x dx

38.

tan4 x sec2 x dx

40.

sin 5u cos 6u du

42.

21 + cos st>2d dt

44.

L L L

Use Simpson’s Rule to find the average value of Cy and the temperature at which it is attained for 20° … T … 675°C.

cos5 x sin5 x dx tan3 x sec3 x dx

e 2tan e + 1 dt 2

t

Numerical Integration 45. According to the error-bound formula for Simpson’s Rule, how many subintervals should you use to be sure of estimating the value of 3

ln 3 =

L1

51. Fuel efficiency An automobile computer gives a digital readout of fuel consumption in gallons per hour. During a trip, a passenger recorded the fuel consumption every 5 min for a full hour of travel.

Time

Gal> h

0 5 10 15 20 25 30

2.5 2.4 2.3 2.4 2.4 2.5 2.6

cos 3u cos 3u du t

L

T 49. Mean temperature Use Simpson’s Rule to approximate the average value of the temperature function 2p ƒsxd = 37 sin a sx - 101db + 25 365 for a 365-day year. This is one way to estimate the annual mean air temperature in Fairbanks, Alaska. The National Weather Service’s official figure, a numerical average of the daily normal mean air temperatures for the year, is 25.7°F, which is slightly higher than the average value of ƒ(x).

1 x dx

by Simpson’s Rule with an error of no more than 10 -4 in absolute value? (Remember that for Simpson’s Rule, the number of subintervals has to be even.) 46. A brief calculation shows that if 0 … x … 1 , then the second derivative of ƒsxd = 21 + x 4 lies between 0 and 8. Based on this, about how many subdivisions would you need to estimate the integral of ƒ from 0 to 1 with an error no greater than 10 -3 in absolute value using the Trapezoidal Rule?

0 ft 36 ft 54 ft

How close do you come to this value by using the Trapezoidal Rule with n = 6 ? Simpson’s Rule with n = 6 ? Try them and find out.

51 ft 49.5 ft Vertical spacing 15 ft

54 ft

2

ƒsxd dx

with an error magnitude less than 10 -5 . You have determined that s4d ƒ ƒ sxd ƒ … 3 throughout the interval of integration. How many subintervals should you use to assure the required accuracy? (Remember that for Simpson’s Rule the number has to be even.)

2.5 2.4 2.3 2.4 2.4 2.3

52. A new parking lot To meet the demand for parking, your town has allocated the area shown here. As the town engineer, you have been asked by the town council to find out if the lot can be built for $11,000. The cost to clear the land will be $0.10 a square foot, and the lot will cost $2.00 a square foot to pave. Use Simpson’s Rule to find out if the job can be done for $11,000.

2 sin2 x dx = p .

L1

35 40 45 50 55 60

b. If the automobile covered 60 mi in the hour, what was its fuel efficiency (in miles per gallon) for that portion of the trip?

p

48. You are planning to use Simpson’s Rule to estimate the value of the integral

Gal> h

a. Use the Trapezoidal Rule to approximate the total fuel consumption during the hour.

47. A direct calculation shows that

L0

Time

64.4 ft 67.5 ft 42 ft Ignored

Chapter 8 Improper Integrals Evaluate the improper integrals in Exercises 53–62. 3

53. 55.

L3 L0 q

61.

L- q

ln x dx

L0

du 56. L-2 su + 1d3>5 q

2 du u 2 - 2u

58.

x 2e -x dx

60.

dx 4x 2 + 9

62.

L1

63.

L6

65.

L1 q

67.

L- q

89.

xe 3x dx

91.

L- q q

L- q

q

du

64.

L0 q

ln z z dz

66.

2 dx e + e -x

68.

4 dx x 2 + 16

93.

L1

e -u cos u du -t

e dt 1t

q

x

69.

x dx L 1 + 1x

70.

71.

dx 2 2 L xsx + 1d

72.

73.

2 - cos x + sin x dx sin2 x L

74.

75.

9 dy 4 L 81 - y

76.

79. 81.

L

u cos s2u + 1d du

sin 2u du s1 + cos 2ud2 L x dx L 22 - x

Chapter 8

dx x 2s1 + e x d

L- q

x3 + 2 dx 2 L4 - x dx sin2 u du 2 L cos u q

78.

L2

L

ssin-1 xd2 dx

dx sx - 1d2

t dt L 29 - 4t Le

82.

L1 L

86.

Lp>4

21 - y 2 dy y2 L

e t dt + 3e t + 2

ln y

92.

e ln1x dx

94. 96.

x3 dx 2 L1 + x 1 + x2 dx 101. 3 L1 + x

105.

L

L 28 - 2x 2 - x 4 2

L

x 3e sx d dx

tan-1 x dx 2 L x L

tan3 t dt

cot y dy L ln sin y L

e u 23 + 4e u du dy

L 2e

2y

- 1

x2 dx 3 L1 + x 1 + x2 dx 102. 3 L (1 + x)

2x # 21 + 1x dx

104.

L

21 + 11 + x dx 1>2

L 2x21 + x

107.

ln x dx L x + x ln x

109.

x ln x ln x dx x

111.

x dx

100.

1

L

dx

1 dx L x21 - x 4 a

106.

L0

31 + 21 - x 2 dx

1 dx # # L x ln x ln (ln x) ln (ln x) 1 (ln x) ln x c x + d dx 110. x L 21 - x dx 112. x L 108.

a

113. a. Show that 10 ƒ(x) dx = 10 ƒ(a - x) dx. b. Use part (a) to evaluate p>2

L0

sin x dx. sin x + cos x

sin x dx sin x + cos x L 1 - cos x dx 116. L 1 + cos x 114.

115.

sin2 x dx 2 L 1 + sin x

Additional and Advanced Exercises 2. 3.

dx L xsx + 1dsx + 2d Á sx + md L

x sin-1 x dx

483

4x 3 - 20x dx 98. 4 2 L x - 10x + 9

99.

103.

90.

dy

y3

dr 97. 1 + 1r L

x 3 dx L x - 2x + 1 21 + cos 4x dx

2t

88.

2

sin 5t dt 2 L 1 + scos 5td

2

p>2

80.

95.

L 2 -2x - x 2

Evaluating Integrals Evaluate the integrals in Exercises 1–6. 1.

z + 1 dz 2 L z sz + 4d

q

Assorted Integrations Evaluate the integrals in Exercises 69–116. The integrals are listed in random order so you need to decide which integration technique to use.

77.

84.

2

0

2u2 + 1 q

87.

3y - 1 dy 4y 3 - y 2

Which of the improper integrals in Exercises 63–68 converge and which diverge? q

85.

dy 2 L y - 2y + 2

0

dy

L0 (y - 1) 2>3

q

59.

54.

L0 29 - x 2

q

57.

1

dx

2

83.

Additional and Advanced Exercises

4.

L

sin-1 1y dy

484 5.

Chapter 8: Techniques of Integration

dt L t - 21 - t

6.

2

20. Finding volume The infinite region bounded by the coordinate axes and the curve y = -ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

dx 4 x + 4 L

Evaluate the limits in Exercises 7 and 8. x

7. lim

x: q L -x

1

sin t dt

8. lim+ x x: 0 L x

cos t dt t2

Evaluate the limits in Exercises 9 and 10 by identifying them with definite integrals and evaluating the integrals. n-1

n

k n 9. lim a ln 1 + n n: q k=1 A Applications 11. Finding arc length

10. lim a n: q

k=0

1 2n 2 - k 2

Find the length of the curve

x

y =

L0

2cos 2t dt,

0 … x … p>4 .

21. Centroid of a region Find the centroid of the region in the first quadrant that is bounded below by the x-axis, above by the curve y = ln x , and on the right by the line x = e . 22. Centroid of a region Find the centroid of the region in the plane enclosed by the curves y = ;s1 - x 2 d-1>2 and the lines x = 0 and x = 1 . Find the length of the curve y = ln x from

23. Length of a curve x = 1 to x = e .

24. Finding surface area Find the area of the surface generated by revolving the curve in Exercise 23 about the y-axis. 25. The surface generated by an astroid The graph of the equation x 2>3 + y 2>3 = 1 is an astroid (see accompanying figure). Find the area of the surface generated by revolving the curve about the x-axis.

12. Finding arc length Find the length of the graph of the function y = ln s1 - x 2 d, 0 … x … 1>2 .

y 1

13. Finding volume The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x21 - x is revolved about the y-axis to generate a solid. Find the volume of the solid. 14. Finding volume The region in the first quadrant that is enclosed

by the x-axis, the curve y = 5> A x25 - x B , and the lines x = 1 and x = 4 is revolved about the x-axis to generate a solid. Find the volume of the solid.

15. Finding volume The region in the first quadrant enclosed by the coordinate axes, the curve y = e x , and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid. 16. Finding volume The region in the first quadrant that is bounded above by the curve y = e x - 1 , below by the x-axis, and on the right by the line x = ln 2 is revolved about the line x = ln 2 to generate a solid. Find the volume of the solid. 17. Finding volume Let R be the “triangular” region in the first quadrant that is bounded above by the line y = 1 , below by the curve y = ln x , and on the left by the line x = 1 . Find the volume of the solid generated by revolving R about a. the x-axis. b. the line y = 1 . 18. Finding volume (Continuation of Exercise 17. ) Find the volume of the solid generated by revolving the region R about a. the y-axis. b. the line x = 1 . 19. Finding volume

The region between the x-axis and the curve

y = ƒsxd = e

0, x ln x,

x = 0 0 6 x … 2

is revolved about the x-axis to generate the solid shown here. a. Show that ƒ is continuous at x = 0 .

–1

0

y 5 x ln x

1

26. Length of a curve Find the length of the curve x

32t - 1 dt , L1 27. For what value or values of a does y =

q

1 … x … 16.

a

ax 1 b dx 2x x2 + 1 converge? Evaluate the corresponding integral(s). L1

q

28. For each x 7 0 , let Gsxd = 10 e -xt dt . Prove that xGsxd = 1 for each x 7 0 . 29. Infinite area and finite volume What values of p have the following property: The area of the region between the curve y = x -p, 1 … x 6 q , and the x-axis is infinite but the volume of the solid generated by revolving the region about the x-axis is finite. 30. Infinite area and finite volume What values of p have the following property: The area of the region in the first quadrant enclosed by the curve y = x -p , the y-axis, the line x = 1 , and the interval [0, 1] on the x-axis is infinite but the volume of the solid generated by revolving the region about one of the coordinate axes is finite.

q

2

x

–1

t x - 1e -t dt, x 7 0 . L0 For each positive x, the number ≠sxd is the integral of t x - 1e -t with respect to t from 0 to q . Figure 8.21 shows the graph of ≠ near the origin. You will see how to calculate ≠s1>2d if you do Additional Exercise 23 in Chapter 14. ≠sxd =

0

1

The Gamma Function and Stirling’s Formula Euler’s gamma function ≠sxd (“gamma of x”; ≠ is a Greek capital g) uses an integral to extend the factorial function from the nonnegative integers to other real values. The formula is

b. Find the volume of the solid. y

x 2/3 y 2/3 1

x

Chapter 8 y

Additional and Advanced Exercises

32. Stirling’s formula Scottish mathematician James Stirling (1692–1770) showed that x

3

x e lim a b ≠sxd = 1 , x: q x A 2p

y (x)

2

so, for large x,

1

x

–3

–2

–1

0

485

1

2

3

x 2p ≠sxd = a e b s1 + Psxdd, A x

x

–1

Psxd : 0 as x : q .

(2)

Dropping Psxd leads to the approximation

–2

x

x 2p ≠sxd L a e b A x

–3

(Stirling’s formula) .

(3)

a. Stirling’s approximation for n! Use Equation (3) and the fact that n! = n≠snd to show that FIGURE 8.21 Euler’s gamma function ≠sxd is a continuous function of x whose value at each positive integer n + 1 is n!. The defining integral formula for ≠ is valid only for x 7 0 , but we can extend ≠ to negative noninteger values of x with the formula ≠sxd = s≠sx + 1dd>x , which is the subject of Exercise 31.

n

n n! L a e b 22np

n n 2n! L e .

(5)

T b. Compare your calculator’s value for n! with the value given by Stirling’s approximation for n = 10, 20, 30, Á , as far as your calculator can go.

a. Show that ≠s1d = 1 . b. Then apply integration by parts to the integral for ≠sx + 1d to show that ≠sx + 1d = x≠sxd. This gives

T c. A refinement of Equation (2) gives x x 2p 1>s12xd e s1 + Psxdd ≠sxd = a e b A x or

≠s2d = 1≠s1d = 1 ≠s3d = 2≠s2d = 2 ≠s4d = 3≠s3d = 6

x ≠sxd L a e b

x

2p 1>s12xd e , A x

which tells us that

o

n

(1)

c. Use mathematical induction to verify Equation (1) for every nonnegative integer n.

(4)

As you will see if you do Exercise 104 in Section 10.1, Equation (4) leads to the approximation

31. If n is a nonnegative integer, Ω(n + 1) n!

≠sn + 1d = n ≠snd = n!

(Stirling’s approximation) .

n n! L a e b 22np e 1>s12nd . Compare the values given for 10! by your calculator, Stirling’s approximation, and Equation (6).

(6)

9 INFINITE SEQUENCES AND SERIES OVERVIEW Everyone knows how to add two numbers together, or even several. But how do you add infinitely many numbers together? In this chapter we answer this question, which is part of the theory of infinite sequences and series. An important application of this theory is a method for representing a known differentiable function ƒ(x) as an infinite sum of powers of x, so it looks like a “polynomial with infinitely many terms.” Moreover, the method extends our knowledge of how to evaluate, differentiate, and integrate polynomials, so we can work with even more general functions than those encountered so far. These new functions are often solutions to important problems in science and engineering.

9.1 HISTORICAL ESSAY Sequences and Series

Sequences Sequences are fundamental to the study of infinite series and many applications of mathematics. We have already seen an example of a sequence when we studied Newton’s Method in Section 4.7. There we produced a sequence of approximations xn that became closer and closer to the root of a differentiable function. Now we will explore general sequences of numbers and the conditions under which they converge to a finite number.

Representing Sequences A sequence is a list of numbers a1, a2 , a3 , Á , an , Á in a given order. Each of a1, a2, a3 and so on represents a number. These are the terms of the sequence. For example, the sequence 2, 4, 6, 8, 10, 12, Á , 2n, Á has first term a1 = 2, second term a2 = 4, and nth term an = 2n. The integer n is called the index of an , and indicates where an occurs in the list. Order is important. The sequence 2, 4, 6, 8 Á is not the same as the sequence 4, 2, 6, 8 Á . We can think of the sequence a1, a2 , a3 , Á , an , Á as a function that sends 1 to a1 , 2 to a2 , 3 to a3 , and in general sends the positive integer n to the nth term an . More precisely, an infinite sequence of numbers is a function whose domain is the set of positive integers. The function associated with the sequence 2, 4, 6, 8, 10, 12, Á , 2n, Á sends 1 to a1 = 2, 2 to a2 = 4, and so on. The general behavior of this sequence is described by the formula an = 2n.

486

9.1

Sequences

487

We can equally well make the domain the integers larger than a given number n0 , and we allow sequences of this type also. For example, the sequence 12, 14, 16, 18, 20, 22 Á is described by the formula an = 10 + 2n. It can also be described by the simpler formula bn = 2n , where the index n starts at 6 and increases. To allow such simpler formulas, we let the first index of the sequence be any integer. In the sequence above, 5an6 starts with a1 while 5bn6 starts with b6 . Sequences can be described by writing rules that specify their terms, such as n - 1 1 an = 2n, bn = s -1dn + 1 n , cn = n , dn = s -1dn + 1, or by listing terms: 5an6 =

E 21, 22, 23, Á , 2n, Á F

1 1 1 1 5bn6 = e 1, - , , - , Á , s -1dn + 1 n , Á f 2 3 4 n - 1 1 2 3 4 5cn6 = e 0, , , , , Á , n , Á f 2 3 4 5

5dn6 = 51, -1, 1, -1, 1, -1, Á , s -1dn + 1, Á 6. We also sometimes write 5an6 =

E 2n F n = 1. q

Figure 9.1 shows two ways to represent sequences graphically. The first marks the first few points from a1, a2 , a3 , Á , an , Á on the real axis. The second method shows the graph of the function defining the sequence. The function is defined only on integer inputs, and the graph consists of some points in the xy-plane located at s1, a1 d, s2, a2 d, Á , sn, an d, Á . an a1

a2 a3 a4 a5

1

0

2 an 5 n

a3 a 2

0

n

1 2 3 4 5

an

a1

0

3 2 1

1

1 1 an 5 n

0

1

2

3

4

5

n

an a2 a4

a5 a3

a1

0

1

1 an 5 (21) n11 1n

0

n

FIGURE 9.1 Sequences can be represented as points on the real line or as points in the plane where the horizontal axis n is the index number of the term and the vertical axis an is its value.

Convergence and Divergence Sometimes the numbers in a sequence approach a single value as the index n increases. This happens in the sequence 1 1 1 1 e 1, , , , Á , n , Á f 2 3 4

488

Chapter 9: Infinite Sequences and Series

whose terms approach 0 as n gets large, and in the sequence 1 2 3 4 1 e 0, , , , , Á , 1 - n , Á f 2 3 4 5 whose terms approach 1. On the other hand, sequences like

E 21, 22, 23, Á , 2n, Á F

L2e 0

a2 a3

aN

a1

L L1e

L1e (n, an) L2e (N, aN)

0

1 2 3

N

n

n

FIGURE 9.2 In the representation of a sequence as points in the plane, an : L if y = L is a horizontal asymptote of the sequence of points 5sn, an d6 . In this figure, all the an’s after aN lie within P of L.

HISTORICAL BIOGRAPHY Nicole Oresme (ca. 1320–1382)

51, -1, 1, -1, 1, -1, Á , s -1dn + 1, Á 6

an

an

L

have terms that get larger than any number as n increases, and sequences like bounce back and forth between 1 and -1, never converging to a single value. The following definition captures the meaning of having a sequence converge to a limiting value. It says that if we go far enough out in the sequence, by taking the index n to be larger than some value N, the difference between an and the limit of the sequence becomes less than any preselected number P 7 0.

DEFINITIONS The sequence 5an6 converges to the number L if for every positive number P there corresponds an integer N such that for all n, n 7 N

Q

ƒ an - L ƒ 6 P.

If no such number L exists, we say that 5an6 diverges. If 5an6 converges to L, we write limn: q an = L, or simply an : L, and call L the limit of the sequence (Figure 9.2). The definition is very similar to the definition of the limit of a function ƒ(x) as x tends to q (limx: q ƒsxd in Section 2.6). We will exploit this connection to calculate limits of sequences.

EXAMPLE 1 1 (a) lim n = 0 n: q

Show that (b) lim k = k n: q

sany constant kd

Solution

(a) Let P 7 0 be given. We must show that there exists an integer N such that for all n, n 7 N

Q

1 ` n - 0 ` 6 P.

This implication will hold if s1>nd 6 P or n 7 1>P. If N is any integer greater than 1>P, the implication will hold for all n 7 N. This proves that limn: q s1>nd = 0. (b) Let P 7 0 be given. We must show that there exists an integer N such that for all n, n 7 N

ƒ k - k ƒ 6 P. Since k - k = 0, we can use any positive integer for N and the implication will hold. This proves that limn: q k = k for any constant k.

EXAMPLE 2

Q

Show that the sequence 51, -1, 1, -1, 1, -1, Á , s -1dn + 1, Á 6 diverges.

Solution Suppose the sequence converges to some number L. By choosing P = 1>2 in the definition of the limit, all terms an of the sequence with index n larger than some N must lie within P = 1>2 of L. Since the number 1 appears repeatedly as every other term of the sequence, we must have that the number 1 lies within the distance P = 1>2 of L.

9.1 an

M

0 12 3

N

n

Sequences

489

It follows that ƒ L - 1 ƒ 6 1>2, or equivalently, 1>2 6 L 6 3>2. Likewise, the number -1 appears repeatedly in the sequence with arbitrarily high index. So we must also have that ƒ L - s -1d ƒ 6 1>2, or equivalently, -3>2 6 L 6 -1>2. But the number L cannot lie in both of the intervals (1> 2, 3> 2) and s -3>2, -1>2d because they have no overlap. Therefore, no such limit L exists and so the sequence diverges. Note that the same argument works for any positive number P smaller than 1, not just 1> 2. The sequence {1n} also diverges, but for a different reason. As n increases, its terms become larger than any fixed number. We describe the behavior of this sequence by writing

(a) an

lim 2n = q .

n: q

0 12 3

N

n

m

(b)

FIGURE 9.3 (a) The sequence diverges to q because no matter what number M is chosen, the terms of the sequence after some index N all lie in the yellow band above M. (b) The sequence diverges to - q because all terms after some index N lie below any chosen number m.

In writing infinity as the limit of a sequence, we are not saying that the differences between the terms an and q become small as n increases. Nor are we asserting that there is some number infinity that the sequence approaches. We are merely using a notation that captures the idea that an eventually gets and stays larger than any fixed number as n gets large (see Figure 9.3a). The terms of a sequence might also decrease to negative infinity, as in Figure 9.3b.

DEFINITION The sequence 5an6 diverges to infinity if for every number M there is an integer N such that for all n larger than N, an 7 M. If this condition holds we write lim an = q or an : q . n: q

Similarly, if for every number m there is an integer N such that for all n 7 N we have an 6 m, then we say 5an6 diverges to negative infinity and write lim an = - q or an : - q . n: q

A sequence may diverge without diverging to infinity or negative infinity, as we saw in Example 2. The sequences 51, -2, 3, -4, 5, -6, 7, -8, Á 6 and 51, 0, 2, 0, 3, 0, Á 6 are also examples of such divergence.The convergence or divergence of a sequence is not affected by the values of any number of its initial terms (whether we omit or change the first 10, 1000, or even the first million terms does not matter). From Figure 9.2, we can see that only the part of the sequence that remains after discarding some initial number of terms determines whether the sequence has a limit and the value of that limit when it does exist.

Calculating Limits of Sequences Since sequences are functions with domain restricted to the positive integers, it is not surprising that the theorems on limits of functions given in Chapter 2 have versions for sequences.

THEOREM 1 Let 5an6 and 5bn6 be sequences of real numbers, and let A and B be real numbers. The following rules hold if limn: q an = A and limn: q bn = B. 1. 2. 3. 4.

Sum Rule: Difference Rule: Constant Multiple Rule: Product Rule:

5. Quotient Rule:

limn: q san + bn d = A + B limn: q san - bn d = A - B limn: q sk # bn d = k # B sany number kd limn: q san # bn d = A # B an A limn: q = if B Z 0 B bn

The proof is similar to that of Theorem 1 of Section 2.2 and is omitted.

490

Chapter 9: Infinite Sequences and Series

EXAMPLE 3

By combining Theorem 1 with the limits of Example 1, we have:

1 1 (a) lim a- n b = -1 # lim n = -1 # 0 = 0 n: q n: q (b) lim a n: q

(c)

lim

n: q

Constant Multiple Rule and Example la

n - 1 1 1 lim a1 - n b = lim 1 - lim n = 1 - 0 = 1 n b = n: q n: q n: q

5 1 = 5 # lim n n: q n2

#

1 lim n = 5 # 0 # 0 = 0

Difference Rule and Example la

Product Rule

n: q

s4>n 6 d - 7 4 - 7n 6 0 - 7 = -7. = lim = 1 + 0 n: q n 6 + 3 n: q 1 + s3>n 6 d

(d) lim

Sum and Quotient Rules

Be cautious in applying Theorem 1. It does not say, for example, that each of the sequences 5an6 and 5bn6 have limits if their sum 5an + bn6 has a limit. For instance, 5an6 = 51, 2, 3, Á 6 and 5bn6 = 5-1, -2, -3, Á 6 both diverge, but their sum 5an + bn6 = 50, 0, 0, Á 6 clearly converges to 0. One consequence of Theorem 1 is that every nonzero multiple of a divergent sequence 5an6 diverges. For suppose, to the contrary, that 5can6 converges for some number c Z 0. Then, by taking k = 1>c in the Constant Multiple Rule in Theorem 1, we see that the sequence cn

1 e c # can f = 5an6

L bn an 0

n

FIGURE 9.4 The terms of sequence {bn} are sandwiched between those of {an} and {cn}, forcing them to the same common limit L.

converges. Thus, 5can6 cannot converge unless 5an6 also converges. If 5an6 does not converge, then 5can6 does not converge. The next theorem is the sequence version of the Sandwich Theorem in Section 2.2. You are asked to prove the theorem in Exercise 109. (See Figure 9.4.)

THEOREM 2—The Sandwich Theorem for Sequences Let 5an6, 5bn6, and 5cn6 be sequences of real numbers. If an … bn … cn holds for all n beyond some index N, and if limn: q an = limn: q cn = L, then limn: q bn = L also. An immediate consequence of Theorem 2 is that, if ƒ bn ƒ … cn and cn : 0, then bn : 0 because -cn … bn … cn . We use this fact in the next example.

EXAMPLE 4

Since 1>n : 0, we know that

(a)

cos n n :0

because

cos n 1 1 - n … n … n;

(b)

1 :0 2n

because

0 …

1 (c) s -1dn n : 0

because

1 1 1 - n … s -1dn n … n .

1 1 … n; 2n

The application of Theorems 1 and 2 is broadened by a theorem stating that applying a continuous function to a convergent sequence produces a convergent sequence. We state the theorem, leaving the proof as an exercise (Exercise 110).

THEOREM 3—The Continuous Function Theorem for Sequences Let 5an6 be a sequence of real numbers. If an : L and if ƒ is a function that is continuous at L and defined at all an , then ƒsan d : ƒsLd.

9.1

y

EXAMPLE 5

y 2x

2

(1, 2)

491

Show that 2sn + 1d>n : 1.

We know that sn + 1d>n : 1. Taking ƒsxd = 1x and L = 1 in Theorem 3 gives 1sn + 1d>n : 11 = 1.

Solution

⎛ 1 , 21/2⎛ ⎝2 ⎝

The sequence 51>n6 converges to 0. By taking an = 1>n, ƒsxd = 2x , and L = 0 in Theorem 3, we see that 21>n = ƒs1>nd : ƒsLd = 20 = 1. The sequence 521>n6 converges to 1 (Figure 9.5).

EXAMPLE 6

⎛ 1 , 21/3⎛ ⎝3 ⎝

1

Sequences

Using L’Hôpital’s Rule

0

1 3

1 2

1

x

FIGURE 9.5 As n : q , 1>n : 0 and 21>n : 20 (Example 6). The terms of {1>n} are shown on the x-axis; the terms of {21>n} are shown as the y-values on the graph of ƒ(x) = 2x.

The next theorem formalizes the connection between limn: q an and limx: q ƒsxd. It enables us to use l’Hôpital’s Rule to find the limits of some sequences.

THEOREM 4 Suppose that ƒ(x) is a function defined for all x Ú n0 and that 5an6 is a sequence of real numbers such that an = ƒsnd for n Ú n0 . Then lim ƒsxd = L

Q

x: q

lim an = L.

n: q

Proof Suppose that limx: q ƒsxd = L. Then for each positive number P there is a number M such that for all x, x 7 M

Q

ƒ ƒsxd - L ƒ 6 P.

Let N be an integer greater than M and greater than or equal to n0 . Then n 7 N

EXAMPLE 7

Q

an = ƒsnd

and

ƒ an - L ƒ = ƒ ƒsnd - L ƒ 6 P.

Show that lim

n: q

ln n n = 0.

The function sln xd>x is defined for all x Ú 1 and agrees with the given sequence at positive integers. Therefore, by Theorem 4, limn: q sln nd>n will equal limx: q sln xd>x if the latter exists. A single application of l’Hôpital’s Rule shows that Solution

lim

x: q

1>x 0 ln x lim = = 0. x = x: 1 q 1

We conclude that limn: q sln nd>n = 0. When we use l’Hôpital’s Rule to find the limit of a sequence, we often treat n as a continuous real variable and differentiate directly with respect to n. This saves us from having to rewrite the formula for an as we did in Example 7.

EXAMPLE 8

Does the sequence whose nth term is an = a

converge? If so, find limn: q an .

n + 1 b n - 1

n

492

Chapter 9: Infinite Sequences and Series

The limit leads to the indeterminate form 1q . We can apply l’Hôpital’s Rule if we first change the form to q # 0 by taking the natural logarithm of an :

Solution

ln an = ln a

n + 1 b n - 1

= n ln a

n

n + 1 b. n - 1

Then, lim ln an = lim n ln a

n: q

n: q

ln a = lim

n: q

= lim

n + 1 b n - 1

q # 0 form

n + 1 b n - 1 1>n

0 form 0

-2>sn 2 - 1d -1>n

n: q

L’Hôpital’s Rule: differentiate numerator and denominator.

2

2n 2 = 2. n: q n - 1

= lim

2

Since ln an : 2 and ƒsxd = e x is continuous, Theorem 4 tells us that The sequence 5an6 converges to e 2 .

an = e ln an : e 2 .

Commonly Occurring Limits The next theorem gives some limits that arise frequently.

THEOREM 5 1. lim

n: q

The following six sequences converge to the limits listed below:

ln n n = 0

n

2. lim 2n = 1 n: q

3. lim x 1>n = 1

sx 7 0d

n: q

4. lim x n = 0 n: q

n

x 5. lim a1 + n b = e x q n:

sany xd

xn = 0 n: q n!

6. lim

s ƒ x ƒ 6 1d sany xd

In Formulas (3) through (6), x remains fixed as n : q . Proof The first limit was computed in Example 7. The next two can be proved by taking logarithms and applying Theorem 4 (Exercises 107 and 108). The remaining proofs are given in Appendix 6.

EXAMPLE 9 (a)

These are examples of the limits in Theorem 5.

ln sn 2 d 2 ln n = n :2#0 = 0 n n

Formula 1

(b) 2n 2 = n 2>n = sn 1/n d2 : s1d2 = 1

Formula 2

(c) 23n = 31>nsn 1/n d : 1 # 1 = 1

Formula 3 with x = 3 and Formula 2

n

n

1 (d) a- b : 0 2

Formula 4 with x = -

1 2

9.1

Factorial Notation The notation n! (“n factorial”) means the product 1 # 2 # 3 Á n of the integers from 1 to n. Notice that sn + 1d! = sn + 1d # n! . Thus, 4! = 1 # 2 # 3 # 4 = 24 and 5! = 1 # 2 # 3 # 4 # 5 = 5 # 4! = 120 . We define 0! to be 1. Factorials grow even faster than exponentials, as the table suggests. The values in the table are rounded.

n

en

n!

1 5 10 20

3 148 22,026 4.9 * 10 8

1 120 3,628,800 2.4 * 10 18

(e) a (f)

n

Sequences

493

n

n - 2 -2 -2 n b = a1 + n b : e

100 n :0 n!

Formula 5 with x = -2 Formula 6 with x = 100

Recursive Definitions So far, we have calculated each an directly from the value of n. But sequences are often defined recursively by giving 1. 2.

The value(s) of the initial term or terms, and A rule, called a recursion formula, for calculating any later term from terms that precede it.

EXAMPLE 10 (a) The statements a1 = 1 and an = an - 1 + 1 for n 7 1 define the sequence 1, 2, 3, Á , n, Á of positive integers. With a1 = 1, we have a2 = a1 + 1 = 2, a3 = a2 + 1 = 3, and so on. (b) The statements a1 = 1 and an = n # an - 1 for n 7 1 define the sequence 1, 2, 6, 24, Á , n!, Á of factorials. With a1 = 1, we have a2 = 2 # a1 = 2, a3 = 3 # a2 = 6, a4 = 4 # a3 = 24, and so on. (c) The statements a1 = 1, a2 = 1, and an + 1 = an + an - 1 for n 7 2 define the sequence 1, 1, 2, 3, 5, Á of Fibonacci numbers. With a1 = 1 and a2 = 1, we have a3 = 1 + 1 = 2, a4 = 2 + 1 = 3, a5 = 3 + 2 = 5, and so on. (d) As we can see by applying Newton’s method (see Exercise 133), the statements x0 = 1 and xn + 1 = xn - [ssin xn - x n2 d>scos xn - 2xn d] for n 7 0 define a sequence that, when it converges, gives a solution to the equation sin x - x 2 = 0.

Bounded Monotonic Sequences Two concepts that play a key role in determining the convergence of a sequence are those of a bounded sequence and a monotonic sequence.

DEFINITIONS A sequence 5an6 is bounded from above if there exists a number M such that an … M for all n. The number M is an upper bound for 5an6. If M is an upper bound for 5an6 but no number less than M is an upper bound for 5an6, then M is the least upper bound for 5an6. A sequence {an} is bounded from below if there exists a number m such that an Ú m for all n. The number m is a lower bound for {an}. If m is a lower bound for {an} but no number greater than m is a lower bound for {an}, then m is the greatest lower bound for {an}. If {an} is bounded from above and below, the {an} is bounded. If {an} is not bounded, then we say that {an} is an unbounded sequence.

EXAMPLE 11 (a) The sequence 1, 2, 3, Á , n, Á has no upper bound since it eventually surpasses every number M. However, it is bounded below by every real number less than or equal to 1. The number m = 1 is the greatest lower bound of the sequence. n 1 2 3 (b) The sequence , , , Á , , Á is bounded above by every real number greater 2 3 4 n + 1 than or equal to 1. The upper bound M = 1 is the least upper bound (Exercise 125). 1 The sequence is also bounded below by every number less than or equal to , which is 2 its greatest lower bound.

494

Chapter 9: Infinite Sequences and Series

If a sequence {an} converges to the number L, then by definition there is a number N such that ƒ an - L ƒ 6 1 if n 7 N. That is,

Convergent sequences are bounded

L - 1 6 an 6 L + 1

for n 7 N.

If M is a number larger than L + 1 and all of the finitely many numbers a1, a2, Á , aN, then for every index n we have an … M so that {an} is bounded from above. Similarly, if m is a number smaller than L - 1 and all of the numbers a1, a2, Á , aN, then m is a lower bound of the sequence. Therefore, all convergent sequences are bounded. Although it is true that every convergent sequence is bounded, there are bounded sequences that fail to converge. One example is the bounded sequence {s -1dn + 1} discussed in Example 2. The problem here is that some bounded sequences bounce around in the band determined by any lower bound m and any upper bound M (Figure 9.6). An important type of sequence that does not behave that way is one for which each term is at least as large, or at least as small, as its predecessor.

an M

n

0 123

DEFINITION A sequence 5an6 is nondecreasing if an … an + 1 for all n. That is, a1 … a2 … a3 … Á . The sequence is nonincreasing if an Ú an + 1 for all n. The sequence 5an6 is monotonic if it is either nondecreasing or nonincreasing.

m

FIGURE 9.6 Some bounded sequences bounce around between their bounds and fail to converge to any limiting value.

EXAMPLE 12 (a) The sequence 1, 2, 3, Á , n, Á is nondecreasing. n 1 2 3 , Á is nondecreasing. (b) The sequence , , , Á , 2 3 4 n + 1 1 1 1 1 (c) The sequence 1, , , , Á , n , Á is nonincreasing. 2 4 8 2 (d) The constant sequence 3, 3, 3, Á , 3, Á is both nondecreasing and nonincreasing. (e) The sequence 1, -1, 1, -1, 1, -1, Á is not monotonic. A nondecreasing sequence that is bounded from above always has a least upper bound. Likewise, a nonincreasing sequence bounded from below always has a greatest lower bound. These results are based on the completeness property of the real numbers, discussed in Appendix 7. We now prove that if L is the least upper bound of a nondecreasing sequence then the sequence converges to L, and that if L is the greatest lower bound of a nonincreasing sequence then the sequence converges to L.

y

M L

y⫽M y⫽L

THEOREM 6—The Monotonic Sequence Theorem If a sequence {an} is both bounded and monotonic, then the sequence converges.

L⫺⑀

0

FIGURE 9.7 If the terms of a nondecreasing sequence have an upper bound M, they have a limit L … M .

x

Proof Suppose {an} is nondecreasing, L is its least upper bound, and we plot the points s1, a1 d, s2, a2 d, Á , sn, an d, Á in the xy-plane. If M is an upper bound of the sequence, all these points will lie on or below the line y = M (Figure 9.7). The line y = L is the lowest such line. None of the points sn, an d lies above y = L, but some do lie above any lower line y = L - P , if P is a positive number. The sequence converges to L because (a) an … L for all values of n, and (b) given any P 7 0, there exists at least one integer N for which aN 7 L - P . The fact that 5an6 is nondecreasing tells us further that an Ú aN 7 L - P

for all n Ú N.

Thus, all the numbers an beyond the Nth number lie within P of L. This is precisely the condition for L to be the limit of the sequence {an}. The proof for nonincreasing sequences bounded from below is similar.

9.1

Sequences

495

It is important to realize that Theorem 6 does not say that convergent sequences are monotonic. The sequence {s -1dn + 1>n} converges and is bounded, but it is not monotonic since it alternates between positive and negative values as it tends toward zero. What the theorem does say is that a nondecreasing sequence converges when it is bounded from above, but it diverges to infinity otherwise.

Exercises 9.1 Finding Terms of a Sequence Each of Exercises 1–6 gives a formula for the nth term an of a sequence 5an6 . Find the values of a1, a2 , a3 , and a4 . 1 - n n2 s -1dn + 1 3. an = 2n - 1 1. an =

5. an =

2. an =

1 n!

4. an = 2 + s -1dn

n

2 2n + 1

6. an =

2 - 1 2n n

Each of Exercises 7–12 gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence. 7. a1 = 1, 8. a1 = 1, 9. a1 = 2, 10. a1 = -2,

an + 1 = an + s1>2n d

12. a1 = 2,

an + 1 = s -1dn + 1an>2

an + 1 = nan>sn + 1d an + 2 = an + 1>an

Finding a Sequence’s Formula In Exercises 13–26, find a formula for the nth term of the sequence. 13. The sequence 1, -1, 1, -1, 1, Á

1’s with alternating signs

14. The sequence -1, 1, -1, 1, -1, Á

1’s with alternating signs

15. The sequence 1, -4, 9, -16, 25, Á

Squares of the positive integers, with alternating signs Reciprocals of squares of the positive integers, with alternating signs Powers of 2 divided by multiples of 3

1 1 1 1 16. The sequence 1, - , , - , , Á 4 9 16 25 17.

1 2 22 23 24 , , , , ,Á 9 12 15 18 21

Alternating 1’s and 0’s

26. The sequence 0, 1, 1, 2, 2, 3, 3, 4, Á

Each positive integer repeated

Convergence and Divergence Which of the sequences 5an6 in Exercises 27–90 converge, and which diverge? Find the limit of each convergent sequence. n + s - 1dn 27. an = 2 + s0.1dn 28. an = n 29. an =

1 - 2n 1 + 2n

1 - n3 70 - 4n 2

38. an = a2 -

1 1 b a3 + n b 2n 2 n

n + 1 1 b a1 - n b 2n

39. an =

s - 1dn + 1 2n - 1

1 40. an = a- b 2

41. an =

2n An + 1

42. an =

43. an = sin a

p 1 + nb 2

1 s0.9dn

44. an = np cos snpd

45. an =

sin n n

46. an =

sin2 n 2n

47. an =

n 2n

48. an =

3n n3

50. an =

ln n ln 2n

ln sn + 1d

55. an = 210n

22. The sequence 2, 6, 10, 14, 18, Á

Every other even positive integer Integers differing by 3 divided by factorials

5 8 11 14 17 , , , , ,Á 1 2 6 24 120

34. an =

37. an = a

Every other odd positive integer

23.

1 - 32n n + 3 32. an = 2 n + 5n + 6

1 36. an = s - 1dn a1 - n b

21. The sequence 1, 5, 9, 13, 17, Á

20. The sequence -3, -2, -1, 0, 1, Á

2n + 1

35. an = 1 + s -1dn

49. a n =

19. The sequence 0, 3, 8, 15, 24, Á

30. an =

n 2 - 2n + 1 n - 1

Integers differing by 2 divided by products of consecutive integers Squares of the positive integers diminished by 1 Integers, beginning with -3

3 1 1 3 5 18. - , - , , , , Á 2 6 12 20 30

Cubes of positive integers divided by powers of 5

25. The sequence 1, 0, 1, 0, 1, Á

33. an =

an + 2 = an + 1 + an

a2 = -1,

125 1 8 27 64 , , , , ,Á 25 125 625 3125 15,625

1 - 5n 4 31. an = 4 n + 8n 3

an + 1 = an>sn + 1d

11. a1 = a2 = 1,

24.

2n

51. an = 81>n

52. an = s0.03d1>n

7 53. an = a1 + n b n

3 57. an = a n b 59. an =

ln n n 1>n

1>n

n

1 54. an = a1 - n b

n

n

56. an = 2n 2 58. an = sn + 4d1>sn + 4d 60. an = ln n - ln sn + 1d

496

Chapter 9: Infinite Sequences and Series n

n

62. an = 232n + 1

61. an = 24nn 63. an =

n! (Hint: Compare with 1> n.) nn

64. an =

s -4dn n!

65. an =

66. an =

n! 2 # 3n

1 67. an = a n b

n

1 68. an = ln a1 + n b 70. an = a

n b n + 1

n

71. an = a n

72. an = a1 -

1 b n2 s10>11dn 74. an = s9/10dn + s11/12dn

73. an =

,

x 7 0

3 # 6n 2-n # n! n

2

n

n

a a + 2b 1 3 7 17 ,Á. , , , ,Á, , 1 2 5 12 b a + b

1>n

1 1 83. an = a b + 3 22n sln nd200 85. an = n

tan-1 n

84. an = 2n 2 + n sln nd5 86. an = 2n 88. an =

100. A sequence of rational numbers is described as follows:

n

n 1 sin n 2n - 1 1 79. an = 2n sin 2n 81. an = tan-1 n

80. an = s3n + 5n d1>n 2n

x b 2n + 1 n

77. an =

1 78. an = n a1 - cos n b

1

Write out enough early terms of the sequence to deduce a general formula for xn that holds for n Ú 2 .

1>sln nd

75. an = tanh n

76. an = sinh sln nd

82. an =

xn + 1 = x1 + x2 + Á + xn .

n! 10 6n

3n + 1 b 69. an = a 3n - 1

n

Theory and Examples 99. The first term of a sequence is x1 = 1 . Each succeeding term is the sum of all those that come before it:

Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xn and yn be, respectively, the numerator and the denominator of the nth fraction rn = xn>yn . a. Verify that x 12 - 2y 12 = -1, x 22 - 2y 22 = +1 and, more generally, that if a 2 - 2b 2 = -1 or +1 , then sa + 2bd2 - 2sa + bd2 = +1 respectively.

101. Newton’s method The following sequences come from the recursion formula for Newton’s method, xn + 1 = xn -

1 n

L1

90. an =

1 p dx, L1 x

p 7 1

Recursively Defined Sequences In Exercises 91–98, assume that each sequence converges and find its limit. 72 91. a1 = 2, an + 1 = 1 + an an + 6 an + 2

92. a1 = -1,

an + 1 =

93. a1 = -4,

an + 1 = 28 + 2an

94. a1 = 0,

an + 1 = 28 + 2an

95. a1 = 5,

an + 1 = 25an

96. a1 = 3,

an + 1 = 12 - 2an

97. 2, 2 +

1 ,2 + 2

1 1 2 + 2

,2 +

xn + 1 = xn -

b. x0 = 1,

xn + 1 = xn -

c. x0 = 1,

xn + 1 = xn - 1

tan xn - 1 sec2 xn

102. a. Suppose that ƒ(x) is differentiable for all x in [0, 1] and that ƒs0d = 0 . Define sequence 5an6 by the rule an = nƒs1>nd . Show that lim n: q an = ƒ¿s0d. Use the result in part (a) to find the limits of the following sequences 5an6 . c. an = nse 1>n - 1d

2 d. an = n ln a1 + n b ,Á

1 2 +

1 2

98. 21, 31 + 21, 41 + 31 + 21, 51 + 41 + 31 + 21, Á

x n2 - 2 xn 1 = + xn 2xn 2

a. x0 = 1,

1 b. an = n tan-1 n

1 2 +

ƒsxn d . ƒ¿sxn d

Do the sequences converge? If so, to what value? In each case, begin by identifying the function ƒ that generates the sequence.

n

1 x dx

-1 ,

b. The fractions rn = xn>yn approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that r n2 - 2 = ;s1>yn d2 and that yn is not less than n.)

87. an = n - 2n 2 - n

2n 2 - 1 - 2n 2 + n

1 89. an = n

or

103. Pythagorean triples A triple of positive integers a, b, and c is called a Pythagorean triple if a 2 + b 2 = c 2 . Let a be an odd positive integer and let b = j

a2 k 2

and

c = l

a2 m 2

be, respectively, the integer floor and ceiling for a 2>2 .

9.1

117. an =

2n - 1 2n

118. an =

119. an = ss -1dn + 1d a ⎡a 2⎡ ⎢ ⎢ ⎢2⎢

⎢a 2 ⎢ ⎢ ⎢ ⎣2⎣

Sequences

497

2n - 1 3n

n + 1 n b

120. The first term of a sequence is x1 = cos s1d . The next terms are x2 = x1 or cos (2), whichever is larger; and x3 = x2 or cos (3), whichever is larger (farther to the right). In general, xn + 1 = max 5xn , cos sn + 1d6 .

121. an =

a

1 + 22n

a. Show that a 2 + b 2 = c 2 . (Hint: Let a = 2n + 1 and express b and c in terms of n.)

123. an =

b. By direct calculation, or by appealing to the accompanying figure, find

124. a1 = 1,

lim

a: q

j

a2 k 2

l

a2 m 2

.

104. The nth root of n! a. Show that limn: q s2npd1>s2nd = 1 and hence, using Stirling’s approximation (Chapter 8, Additional Exercise 32a), that n n 2n! L e for large values of n. T b. Test the approximation in part (a) for n = 40, 50, 60, Á , as far as your calculator will allow. 105. a. Assuming that limn: q s1>n c d = 0 if c is any positive constant, show that ln n lim c = 0 n: q n if c is any positive constant. b. Prove that limn: q s1>n c d = 0 if c is any positive constant. (Hint: If P = 0.001 and c = 0.04 , how large should N be to ensure that ƒ 1>n c - 0 ƒ 6 P if n 7 N ?) 106. The zipper theorem Prove the “zipper theorem” for sequences: If 5an6 and 5bn6 both converge to L, then the sequence a1, b1, a2 , b2 , Á , an , bn , Á converges to L. n

107. Prove that limn: q 2n = 1 . 108. Prove that limn: q x 109. Prove Theorem 2.

1>n

= 1, sx 7 0d . 110. Prove Theorem 3.

In Exercises 111–114, determine if the sequence is monotonic and if it is bounded. s2n + 3d! 3n + 1 111. an = 112. an = n + 1 sn + 1d! 2n3n 113. an = n!

2 1 114. an = 2 - n - n 2

Which of the sequences in Exercises 115–124 converge, and which diverge? Give reasons for your answers. 1 1 115. an = 1 - n 116. an = n - n

122. an =

2n

n + 1 n

4n + 1 + 3n 4n an + 1 = 2an - 3

125. The sequence {n/ (n + 1)} has a least upper bound of 1 Show that if M is a number less than 1, then the terms of 5n>sn + 1d6 eventually exceed M. That is, if M 6 1 there is an integer N such that n>sn + 1d 7 M whenever n 7 N . Since n>sn + 1d 6 1 for every n, this proves that 1 is a least upper bound for 5n>sn + 1d6 . 126. Uniqueness of least upper bounds Show that if M1 and M2 are least upper bounds for the sequence 5an6 , then M1 = M2 . That is, a sequence cannot have two different least upper bounds.

127. Is it true that a sequence 5an6 of positive numbers must converge if it is bounded from above? Give reasons for your answer.

128. Prove that if 5an6 is a convergent sequence, then to every positive number P there corresponds an integer N such that for all m and n, m 7 N

and

n 7 N

Q

ƒ am - an ƒ 6 P .

129. Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L1 and L2 are numbers such that an : L1 and an : L2 , then L1 = L2 . 130. Limits and subsequences If the terms of one sequence appear in another sequence in their given order, we call the first sequence a subsequence of the second. Prove that if two subsequences of a sequence 5an6 have different limits L1 Z L2 , then 5an6 diverges. 131. For a sequence 5an6 the terms of even index are denoted by a2k and the terms of odd index by a2k + 1 . Prove that if a2k : L and a2k + 1 : L , then an : L . 132. Prove that a sequence 5an6 converges to 0 if and only if the sequence of absolute values 5ƒ an ƒ6 converges to 0.

133. Sequences generated by Newton’s method Newton’s method, applied to a differentiable function ƒ(x), begins with a starting value x0 and constructs from it a sequence of numbers 5xn6 that under favorable circumstances converges to a zero of ƒ. The recursion formula for the sequence is xn + 1 = xn -

ƒsxn d . ƒ¿sxn d

a. Show that the recursion formula for ƒsxd = x 2 - a, a 7 0 , can be written as xn + 1 = sxn + a>xn d>2 . T b. Starting with x0 = 1 and a = 3 , calculate successive terms of the sequence until the display begins to repeat. What number is being approximated? Explain.

498

Chapter 9: Infinite Sequences and Series

T 134. A recursive definition of P/ 2 If you start with x1 = 1 and define the subsequent terms of 5xn6 by the rule xn = xn - 1 + cos xn - 1 , you generate a sequence that converges rapidly to p>2 . (a) Try it. (b) Use the accompanying figure to explain why the convergence is so rapid. y

1

b. If the sequence converges, find an integer N such that ƒ an - L ƒ … 0.01 for n Ú N . How far in the sequence do you have to get for the terms to lie within 0.0001 of L?

xn 1 1

an + 1 = an +

138. a1 = 1,

an + 1 = an + s - 2dn

139. an = sin n

x

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the sequences in Exercises 135–146.

sin n n

143. an = s0.9999dn 145. an =

8n n!

n

1 5n

137. a1 = 1,

141. an =

9.2

0.5 136. an = a1 + n b

n

135. an = 2n

cos xn 1 xn 1

0

a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit L?

1 140. an = n sin n ln n 142. an = n 144. an = (123456) 1>n 146. an =

n 41 19n

Infinite Series An infinite series is the sum of an infinite sequence of numbers a1 + a2 + a3 + Á + an + Á The goal of this section is to understand the meaning of such an infinite sum and to develop methods to calculate it. Since there are infinitely many terms to add in an infinite series, we cannot just keep adding to see what comes out. Instead we look at the result of summing the first n terms of the sequence and stopping. The sum of the first n terms sn = a1 + a2 + a3 + Á + an is an ordinary finite sum and can be calculated by normal addition. It is called the nth partial sum. As n gets larger, we expect the partial sums to get closer and closer to a limiting value in the same sense that the terms of a sequence approach a limit, as discussed in Section 9.1. For example, to assign meaning to an expression like 1 +

1 1 1 1 + + + + Á 2 4 8 16

we add the terms one at a time from the beginning and look for a pattern in how these partial sums grow.

Partial sum First: Second: Third: o nth:

s1 = 1 1 s2 = 1 + 2 1 1 s3 = 1 + + 2 4 o 1 1 1 sn = 1 + + + Á + n - 1 2 4 2

Value

Suggestive expression for partial sum

1 3 2 7 4 o n 2 - 1 2n - 1

2 - 1 1 2 2 1 2 4 o 1 2 - n-1 2

9.2

Infinite Series

499

Indeed there is a pattern. The partial sums form a sequence whose nth term is sn = 2 -

1 . 2n - 1

This sequence of partial sums converges to 2 because limn: q s1>2n - 1 d = 0. We say “the sum of the infinite series 1 +

1 1 1 + + Á + n - 1 + Á is 2.” 2 4 2

Is the sum of any finite number of terms in this series equal to 2? No. Can we actually add an infinite number of terms one by one? No. But we can still define their sum by defining it to be the limit of the sequence of partial sums as n : q , in this case 2 (Figure 9.8). Our knowledge of sequences and limits enables us to break away from the confines of finite sums.

1

⎧ ⎪ ⎨ ⎪ ⎩

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1/4

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

⎧ ⎨ ⎩

1/2

1/8

0

2

FIGURE 9.8 As the lengths 1, 12 , 14 , 18 , Á are added one by one, the sum approaches 2.

HISTORICAL BIOGRAPHY Blaise Pascal (1623–1662)

DEFINITIONS

Given a sequence of numbers 5an6, an expression of the form a1 + a2 + a3 + Á + an + Á

is an infinite series. The number an is the nth term of the series. The sequence 5sn6 defined by s1 = a1 s2 = a1 + a2 o

n

sn = a1 + a2 + Á + an = a ak k=1

o is the sequence of partial sums of the series, the number sn being the nth partial sum. If the sequence of partial sums converges to a limit L, we say that the series converges and that its sum is L. In this case, we also write q

a1 + a2 + Á + an + Á = a an = L. n=1

If the sequence of partial sums of the series does not converge, we say that the series diverges.

When we begin to study a given series a1 + a2 + Á + an + Á , we might not know whether it converges or diverges. In either case, it is convenient to use sigma notation to write the series as q

a an ,

n=1

q

a ak ,

k=1

or

a an

A useful shorthand when summation from 1 to q is understood

500

Chapter 9: Infinite Sequences and Series

Geometric Series Geometric series are series of the form q

a + ar + ar 2 + Á + ar n - 1 + Á = a ar n - 1 n=1

in which a and r are fixed real numbers and a Z 0. The series can also be written as q g n = 0 ar n . The ratio r can be positive, as in 1 +

1 1 1 + + Á + a b 2 4 2

n-1

+ Á,

r = 1>2, a = 1

or negative, as in n-1

1 1 1 + - Á + a- b + Á. 3 9 3 If r = 1, the nth partial sum of the geometric series is 1 -

r = -1>3, a = 1

sn = a + as1d + as1d2 + Á + as1dn - 1 = na, and the series diverges because limn: q sn = ; q , depending on the sign of a. If r = -1, the series diverges because the nth partial sums alternate between a and 0. If ƒ r ƒ Z 1, we can determine the convergence or divergence of the series in the following way: sn = a + ar + ar 2 + Á + ar n - 1 rsn = ar + ar 2 + Á + ar n - 1 + ar n sn - rsn = a - ar n sns1 - rd = as1 - r n d sn =

as1 - r n d , 1 - r

sr Z 1d.

Multiply sn by r. Subtract rsn from sn . Most of the terms on the right cancel. Factor. We can solve for sn if r Z 1 .

If ƒ r ƒ 6 1, then r n : 0 as n : q (as in Section 9.1) and sn : a>s1 - rd. If ƒ r ƒ 7 1, then ƒ r n ƒ : q and the series diverges. If ƒ r ƒ 6 1, the geometric series a + ar + ar 2 + Á + ar n - 1 + Á converges to a>s1 - rd: q

a n-1 = , a ar 1 - r

n=1

ƒ r ƒ 6 1.

If ƒ r ƒ Ú 1, the series diverges. We have determined when a geometric series converges or diverges, and to what value. Often we can determine that a series converges without knowing the value to which it converges, as we will see in the next several sections. The formula a>s1 - rd for the sum of a geometric series applies only when the summation index begins with n = 1 in the exq q pression g n = 1 ar n - 1 (or with the index n = 0 if we write the series as g n = 0 ar n).

EXAMPLE 1

The geometric series with a = 1>9 and r = 1>3 is q

1 1 1 1 1 + + + Á = a a b 9 27 81 n=1 9 3

EXAMPLE 2

n-1

=

1>9 1 = . 6 1 - s1>3d

The series q

s -1dn5 5 5 5 = 5 - + + Á n 4 16 64 4 n=0 a

9.2

Infinite Series

501

is a geometric series with a = 5 and r = -1>4. It converges to

a

5 a = 4. = 1 - r 1 + s1>4d

EXAMPLE 3

You drop a ball from a meters above a flat surface. Each time the ball hits the surface after falling a distance h, it rebounds a distance rh, where r is positive but less than 1. Find the total distance the ball travels up and down (Figure 9.9).

ar

ar 2

Solution

The total distance is

ar 3

2ar 1 + r = a . s = a + 2ar + 2ar 2 + 2ar 3 + Á = a + 1 - r 1 - r (''''')'''''* This sum is 2ar>s1 - rd.

If a = 6 m and r = 2>3, for instance, the distance is (a)

s = 6

EXAMPLE 4 Solution

1 + s2>3d 5>3 = 6a b = 30 m. 1>3 1 - s2>3d

Express the repeating decimal 5.232323 Á as the ratio of two integers.

From the definition of a decimal number, we get a geometric series 23 23 23 + 5.232323 Á = 5 + + + Á 100 s100d2 s100d3 2

= 5 +

23 1 1 a1 + + a b + Áb 100 100 100

a = 1, r = 1>100

('''''''')''''''''* 1>s1 - 0.01d

= 5 +

23 518 23 1 a b = 5 + = 100 0.99 99 99

Unfortunately, formulas like the one for the sum of a convergent geometric series are rare and we usually have to settle for an estimate of a series’ sum (more about this later). The next example, however, is another case in which we can find the sum exactly. q

(b)

FIGURE 9.9 (a) Example 3 shows how to use a geometric series to calculate the total vertical distance traveled by a bouncing ball if the height of each rebound is reduced by the factor r. (b) A stroboscopic photo of a bouncing ball.

EXAMPLE 5

1 Find the sum of the “telescoping” series a . nsn + 1d n=1

We look for a pattern in the sequence of partial sums that might lead to a formula for sk . The key observation is the partial fraction decomposition

Solution

1 1 1 = n , n + 1 nsn + 1d so k

k

1 1 1 = a an b a n + 1 n = 1 nsn + 1d n=1 and sk = a

1 1 1 1 1 1 1 1 - b + a - b + a - b + Á + a b. 1 2 2 3 3 4 k k + 1

Removing parentheses and canceling adjacent terms of opposite sign collapses the sum to 1 sk = 1 . k + 1 We now see that sk : 1 as k : q . The series converges, and its sum is 1: q

1 a nsn + 1d = 1.

n=1

502

Chapter 9: Infinite Sequences and Series

The nth-Term Test for a Divergent Series One reason that a series may fail to converge is that its terms don’t become small.

EXAMPLE 6

The series q

3 n + 1 2 4 Á + n + 1 + Á n = 1 + 2 + 3 + n n=1 a

diverges because the partial sums eventually outgrow every preassigned number. Each term is greater than 1, so the sum of n terms is greater than n. q

Notice that limn: q an must equal zero if the series g n = 1 an converges. To see why, let S represent the series’ sum and sn = a1 + a2 + Á + an the nth partial sum. When n is large, both sn and sn - 1 are close to S, so their difference, an , is close to zero. More formally, an = sn - sn - 1

:

S - S = 0.

Difference Rule for sequences

This establishes the following theorem. Caution q Theorem 7 does not say that g n = 1 an converges if an : 0 . It is possible for a series to diverge when an : 0 .

q

THEOREM 7

If a an converges, then an : 0. n=1

Theorem 7 leads to a test for detecting the kind of divergence that occurred in Example 6.

The nth-Term Test for Divergence q

lim an fails to exist or is different from zero. a an diverges if n: q

n=1

EXAMPLE 7

The following are all examples of divergent series.

q

(a) a n 2 diverges because n 2 : q . n=1 q

(b) a

n=1 q

n + 1 n + 1 n diverges because n : 1.

limn: q an Z 0

(c) a s -1dn + 1 diverges because limn: q s -1dn + 1 does not exist. n=1 q

-n -n 1 (d) a diverges because limn: q = - Z 0. 2n + 5 2 n = 1 2n + 5

EXAMPLE 8 1 +

The series 1 1 1 1 1 1 1 1 1 + + + + + + Á + n + n + Á + n + Á 2 2 4 4 4 4 2 2 2

(')'* 2 terms

('''')''''* 4 terms

(''''')'''''* 2n terms

diverges because the terms can be grouped into infinitely many clusters each of which adds to 1, so the partial sums increase without bound. However, the terms of the series form a sequence that converges to 0. Example 1 of Section 9.3 shows that the harmonic series also behaves in this manner.

9.2

Infinite Series

503

Combining Series Whenever we have two convergent series, we can add them term by term, subtract them term by term, or multiply them by constants to make new convergent series.

THEOREM 8

If gan = A and gbn = B are convergent series, then gsan + bn d = gan + gbn = A + B gsan - bn d = gan - gbn = A - B gkan = kgan = kA sany number kd.

1. Sum Rule:

2. Difference Rule: 3. Constant Multiple Rule:

Proof The three rules for series follow from the analogous rules for sequences in Theorem 1, Section 9.1. To prove the Sum Rule for series, let An = a1 + a2 + Á + an ,

Bn = b1 + b2 + Á + bn .

Then the partial sums of gsan + bn d are sn = sa1 + b1 d + sa2 + b2 d + Á + san + bn d = sa1 + Á + an d + sb1 + Á + bn d = An + Bn . Since An : A and Bn : B, we have sn : A + B by the Sum Rule for sequences. The proof of the Difference Rule is similar. To prove the Constant Multiple Rule for series, observe that the partial sums of gkan form the sequence sn = ka1 + ka2 + Á + kan = ksa1 + a2 + Á + an d = kAn , which converges to kA by the Constant Multiple Rule for sequences. As corollaries of Theorem 8, we have the following results. We omit the proofs.

1. Every nonzero constant multiple of a divergent series diverges. 2. If gan converges and gbn diverges, then gsan + bn d and gsan - bn d both diverge.

Caution Remember that gsan + bn d can converge when gan and gbn both diverge. For example, gan = 1 + 1 + 1 + Á and gbn = s -1d + s -1d + s -1d + Á diverge, whereas gsan + bn d = 0 + 0 + 0 + Á converges to 0.

EXAMPLE 9 q

Find the sums of the following series. q

3n - 1 - 1 1 1 = a a n-1 - n-1 b 6n - 1 6 n=1 n=1 2

(a) a

q

q

1 1 = a n-1 - a n-1 n=1 2 n=1 6 1 1 = 1 - s1>2d 1 - s1>6d = 2 -

6 4 = 5 5

Difference Rule Geometric series with a = 1 and r = 1>2, 1>6

504

Chapter 9: Infinite Sequences and Series q

q

4 1 (b) a n = 4 a n n=0 2 n=0 2 = 4a

Constant Multiple Rule

1 b 1 - s1>2d

Geometric series with a = 1, r = 1>2

= 8

Adding or Deleting Terms We can add a finite number of terms to a series or delete a finite number of terms without altering the series’ convergence or divergence, although in the case of convergence q q this will usually change the sum. If g n = 1 an converges, then g n = k an converges for any k 7 1 and q

q

Á + ak - 1 + a an = a1 + a2 + a an .

n=1

Conversely, if

q g n = k an

n=k

converges for any k 7 1, then q

q g n = 1 an

converges. Thus,

q

1 1 1 1 1 a 5n = 5 + 25 + 125 + a 5n n=1 n=4 and q

q

1 1 1 1 1 a n = a a 5n b - 5 - 25 - 125 . n=4 5 n=1 The convergence or divergence of a series is not affected by its first few terms. Only the “tail” of the series, the part that remains when we sum beyond some finite number of initial terms, influences whether it converges or diverges. HISTORICAL BIOGRAPHY

Reindexing

Richard Dedekind (1831–1916)

As long as we preserve the order of its terms, we can reindex any series without altering its convergence. To raise the starting value of the index h units, replace the n in the formula for an by n - h: q

q

a an =

n=1

Á. a an - h = a1 + a2 + a3 +

n=1+h

To lower the starting value of the index h units, replace the n in the formula for an by n + h: q

q

a an =

n=1

Á. a an + h = a1 + a2 + a3 +

n=1-h

We saw this reindexing in starting a geometric series with the index n = 0 instead of the index n = 1, but we can use any other starting index value as well. We usually give preference to indexings that lead to simple expressions.

EXAMPLE 10

We can write the geometric series q

1 1 1 Á a 2n - 1 = 1 + 2 + 4 + n=1 as q

1 a 2n , n=0

q

1 a 2n - 5 , n=5

q

or even

1 a 2n + 4 . n = -4

The partial sums remain the same no matter what indexing we choose.

9.2

Infinite Series

505

Exercises 9.2 Finding nth Partial Sums In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.

Using the nth-Term Test In Exercises 27–34, use the nth-Term Test for divergence to show that the series is divergent, or state that the test is inconclusive. q

2 2 2 2 + + + Á + n-1 + Á 3 9 27 3

n 27. a n = 1 n + 10

9 9 9 9 + + Á + + Á + 100 100 n 100 2 100 3

1 29. a n=0 n + 4

1 1 1 1 + - + Á + s -1dn - 1 n - 1 + Á 2 4 8 2 4. 1 - 2 + 4 - 8 + Á + s -1dn - 1 2n - 1 + Á

1 31. a cos n

1. 2 + 2.

q

q

1

5 5 5 5 + # + # + Á + + Á 1#2 2 3 3 4 nsn + 1d

Series with Geometric Terms In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges. q

7. a

n=0 q

q

s -1d 4n

n

1 8. a n n=2 4

5 1 11. a a n + n b 2 3 n=0

5 1 12. a a n - n b 2 3 n=0

s -1dn 1 b 13. a a n + 5n n=0 2

2n + 1 14. a a n b 5 n=0

4

2

3

4

q

37. a A ln 2n + 1 - ln 2n B

q

q

6 42. a n = 1 s2n - 1ds2n + 1d

4 41. a n = 1 s4n - 3ds4n + 1d q

q

45. a a

5

5

Find the sum of each series in Exercises 41–48.

40n 43. a 2 2 n = 1 s2n - 1d s2n + 1d

1 1 1 1 1 17. a b + a b + a b + a b + a b + Á 8 8 8 8 8 18. a

q

3 3 36. a a 2 b sn + 1d2 n=1 n

n=1

4

3

q

1 1 35. a a n b n + 1 n=1

q

2 2 2 2 15. 1 + a b + a b + a b + a b + Á 5 5 5 5 16. 1 + s -3d + s -3d2 + s -3d3 + s -3d4 + Á 2

Telescoping Series In Exercises 35–40, find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum.

40. a A 2n + 4 - 2n + 3 B

In Exercises 15–18, determine if the geometric series converges or diverges. If a series converges, find its sum. 3

n=0

1 1 b - cos-1 a bb 39. a acos-1 a n + 1 n + 2 n=1

q

2

n=1

n=1 q

n=0 q

q

34. a cos np

38. a stan snd - tan sn - 1dd

5 10. a s -1dn n 4

q

q

1 33. a ln n

n=1 q

q

7 9. a a1 - n b 4 n=1

q

en 32. a n n=0 e + n

n=1

1 + Á 5. # + # + # + Á + 2 3 3 4 4 5 sn + 1dsn + 2d 6.

1

q

n 30. a 2 n=1 n + 3

q

3. 1 -

1

q nsn + 1d 28. a sn + 2dsn + 3d n=1

6

-2 -2 -2 -2 -2 b + a b + a b + a b + a b + Á 3 3 3 3 3

n=1 q

1

-

2n

1 2n + 1

b

q

2n + 1 44. a 2 2 n = 1 n sn + 1d q

46. a a n=1

1 1 - 1>sn + 1d b 21>n 2

1 1 b 47. a a ln sn + 2d ln sn + 1d n=1 q

Repeating Decimals Express each of the numbers in Exercises 19–26 as the ratio of two integers. 19. 0.23 = 0.23 23 23 Á 20. 0.234 = 0.234 234 234 Á 21. 0.7 = 0.7777 Á 22. 0.d = 0.dddd Á ,

where d is a digit

23. 0.06 = 0.06666 Á 24. 1.414 = 1.414 414 414 Á 25. 1.24123 = 1.24 123 123 123 Á 26. 3.142857 = 3.142857 142857 Á

48. a stan-1 snd - tan-1 sn + 1dd n=1

Convergence or Divergence Which series in Exercises 49–68 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. q

49. a a n=0 q

1 22

b

n

q

50. a A 22 B

n

n=0 q

3 51. a s -1dn + 1 n 2

52. a s -1dn + 1n

53. a cos np

54. a

n=1 q n=0

n=1 q n=0

cos np 5n

506

Chapter 9: Infinite Sequences and Series

q

1 56. a ln n 3 n=1

2 57. a n n = 1 10

1 58. a n , n=0 x

n=0 q

q

q

q

n=1 q

nn 62. a n = 1 n! q

2 + 3 4n

n

n

87. What happens if you add a finite number of terms to a divergent series or delete a finite number of terms from a divergent series? Give reasons for your answer.

q

n 65. a ln a b n + 1 n=1 e 67. a a p b

86. If gan converges and an 7 0 for all n, can anything be said about gs1>an d ? Give reasons for your answer.

2 + 4 64. a n n n=1 3 + 4

n

q

85. Show by example that gsan>bn d may converge to something other than A> B even when A = gan, B = gbn Z 0 , and no bn equals 0.

n

n=1 q

n! 61. a n n = 0 1000 n

ƒxƒ 7 1

1 60. a a1 - n b

2n - 1 3n n=0

63. a

84. Find convergent geometric series A = gan and B = gbn that illustrate the fact that gan bn may converge without being equal to AB.

q

59. a

q

83. Show by example that gsan>bn d may diverge even though gan and gbn converge and no bn equals 0.

q

55. a e -2n

n 66. a ln a b 2n + 1 n=1

88. If gan converges and gbn diverges, can anything be said about their term-by-term sum gsan + bn d ? Give reasons for your answer.

q

n

e np 68. a ne n=0 p

n=0

Geometric Series with a Variable x In each of the geometric series in Exercises 69–72, write out the first few terms of the series to find a and r, and find the sum of the series. Then express the inequality ƒ r ƒ 6 1 in terms of x and find the values of x for which the inequality holds and the series converges. q

q

69. a s -1dnx n n=0 q

71. a 3 a n=0

x - 1 b 2

70. a s -1dnx 2n n=0 q

s -1d 1 a b 2 3 + sin x n=0

n

n

In Exercises 73–78, find the values of x for which the given geometric series converges. Also, find the sum of the series (as a function of x) for those values of x. q

73. a 2nx n

74. a s -1dnx -2n

75. a s -1dnsx + 1dn

1 76. a a- b sx - 3dn 2 n=0

77. a sinn x

78. a sln xdn

n=0 q

n=0 q

n=0 q

a. a = 2

b. a = 13>2 .

90. Find the value of b for which 1 + e b + e 2b + e 3b + Á = 9 . 91. For what values of r does the infinite series

n

72. a

q

89. Make up a geometric series gar n - 1 that converges to the number 5 if

n

1 + 2r + r 2 + 2r 3 + r 4 + 2r 5 + r 6 + Á converge? Find the sum of the series when it converges. 92. Show that the error sL - sn d obtained by replacing a convergent geometric series with one of its partial sums sn is ar n>s1 - rd . 93. The accompanying figure shows the first five of a sequence of squares. The outermost square has an area of 4 m2 . Each of the other squares is obtained by joining the midpoints of the sides of the squares before it. Find the sum of the areas of all the squares.

q

n=0

n=0

Theory and Examples 79. The series in Exercise 5 can also be written as q

1 a n = 1 sn + 1dsn + 2d

q

1 . a n = -1 sn + 3dsn + 4d

and

Write it as a sum beginning with (a) n = -2 , (b) n = 0 , (c) n = 5 . 80. The series in Exercise 6 can also be written as q

5 a n = 1 nsn + 1d

q

and

5 . a n = 0 sn + 1dsn + 2d

Write it as a sum beginning with (a) n = -1 , (b) n = 3 , (c) n = 20 .

94. Helga von Koch’s snowflake curve Helga von Koch’s snowflake is a curve of infinite length that encloses a region of finite area. To see why this is so, suppose the curve is generated by starting with an equilateral triangle whose sides have length 1. a. Find the length Ln of the nth curve Cn and show that limn: q Ln = q . b. Find the area An of the region enclosed by Cn and show that limn: q An = (8>5) A1 .

81. Make up an infinite series of nonzero terms whose sum is a. 1

b. -3

c. 0.

82. (Continuation of Exercise 81. ) Can you make an infinite series of nonzero terms that converges to any number you want? Explain.

C1

C2

C3

C4

9.3

9.3

The Integral Test

507

The Integral Test Given a series, the most basic question we can ask is whether it converges or not. In this section and the next two, we study this question, starting with series that have nonnegative terms. Such a series converges if its sequence of partial sums is bounded. If we establish that a given series does converge, we generally do not have a formula available for its sum, so we investigate methods to approximate the sum instead.

Nondecreasing Partial Sums q

Suppose that g n = 1 an is an infinite series with an Ú 0 for all n. Then each partial sum is greater than or equal to its predecessor because sn + 1 = sn + an : s1 … s2 … s3 … Á … sn … sn + 1 … Á . Since the partial sums form a nondecreasing sequence, the Monotonic Sequence Theorem (Theorem 6, Section 9.1) gives the following result.

q

Corollary of Theorem 6 A series g n = 1 an of nonnegative terms converges if and only if its partial sums are bounded from above.

EXAMPLE 1

The series q

1 1 1 Á + 1 + Á a n = 1 + 2 + 3 + n

n=1

is called the harmonic series. The harmonic series is divergent, but this doesn’t follow from the nth-Term Test. The nth term 1> n does go to zero, but the series still diverges. The reason it diverges is because there is no upper bound for its partial sums. To see why, group the terms of the series in the following way: 1 +

1 1 1 1 1 1 1 1 1 1 + a + b + a + + + b + a + + Á + b + Á. 5 7 2 3 4 6 8 9 10 16 ('')''* 7 24 = 12

(''''')'''''* 7 48 = 12

('''''')''''''* 8 7 16 = 12

The sum of the first two terms is 1.5. The sum of the next two terms is 1>3 + 1>4, which is greater than 1>4 + 1>4 = 1>2. The sum of the next four terms is 1>5 + 1>6 + 1>7 + 1>8, which is greater than 1>8 + 1>8 + 1>8 + 1>8 = 1>2. The sum of the next eight terms is 1>9 + 1>10 + 1>11 + 1>12 + 1>13 + 1>14 + 1>15 + 1>16, which is greater than 8>16 = 1>2. The sum of the next 16 terms is greater than 16>32 = 1>2, and so on. In general, the sum of 2n terms ending with 1>2n + 1 is greater than 2n>2n + 1 = 1>2. The sequence of partial sums is not bounded from above: If n = 2k , the partial sum sn is greater than k >2. The harmonic series diverges.

The Integral Test We now introduce the Integral Test with a series that is related to the harmonic series, but whose nth term is 1>n 2 instead of 1> n.

EXAMPLE 2

Does the following series converge? q

1 1 1 1 Á + 1 + Á a n 2 = 1 + 4 + 9 + 16 + n2 n=1

508

Chapter 9: Infinite Sequences and Series

y

q 2 11 s1>x d

(1, f(1)) Graph of f(x) 12 x 1 12

1 1 1 1 + 2 + 2 + Á + 2 2 1 2 3 n = ƒs1d + ƒs2d + ƒs3d + Á + ƒsnd

sn =

(2, f(2)) 1 32 (3, f(3))

1 42

1 22 0

q

We determine the convergence of g n = 1s1>n 2 d by comparing it with dx. To carry out the comparison, we think of the terms of the series as values of the function ƒsxd = 1>x 2 and interpret these values as the areas of rectangles under the curve y = 1>x 2 . As Figure 9.10 shows, Solution

1

3

2

4 …

1 n2

n

(n, f(n))

n1 n…

6 ƒs1d +

x

1 dx 2 L1 x

Rectangle areas sum to less than area under graph.

q

1 dx 2 L1 x 6 1 + 1 = 2. 6 1 +

FIGURE 9.10 The sum of the areas of the rectangles under the graph of ƒ(x) = 1>x 2 is less than the area under the graph (Example 2).

Caution The series and integral need not have the same value in the convergent case. As we q noted in Example 2, g n = 1 s1>n 2 d = q 2 2 p >6 while 11 s1>x d dx = 1 .

q

As in Section 8.7, Example 3, q 2 11 s1>x d dx = 1 .

q

Thus the partial sums of g n = 1 (1>n 2) are bounded from above (by 2) and the series converges. The sum of the series is known to be p2>6 L 1.64493.

THEOREM 9—The Integral Test Let 5an6 be a sequence of positive terms. Suppose that an = ƒsnd, where ƒ is a continuous, positive, decreasing function of q x for all x Ú N (N a positive integer). Then the series g n = N an and the integral q 1N ƒsxd dx both converge or both diverge.

Proof We establish the test for the case N = 1. The proof for general N is similar. We start with the assumption that ƒ is a decreasing function with ƒsnd = an for every n. This leads us to observe that the rectangles in Figure 9.11a, which have areas a1, a2 , Á , an , collectively enclose more area than that under the curve y = ƒsxd from x = 1 to x = n + 1. That is,

y

y f (x)

a1

n+1

a2 an

0

1

2

n

3

n1

L1

ƒsxd dx … a1 + a2 + Á + an .

x

In Figure 9.11b the rectangles have been faced to the left instead of to the right. If we momentarily disregard the first rectangle of area a1 , we see that

(a) y

a2 + a3 + Á + an … y f (x)

a1

0

n

2 2 11 s1>x d dx 6 11 s1>x d dx

a2 1

n

L1

ƒsxd dx.

If we include a1 , we have

a3 2

an n1 n

3

a1 + a2 + Á + an … a1 +

x

(b)

FIGURE 9.11 Subject to the conditions of q the Integral Test, the series g n = 1 an and the q integral 11 sxd dx both converge or both diverge.

n

L1

ƒsxd dx.

Combining these results gives n+1

L1

ƒsxd dx … a1 + a2 + Á + an … a1 +

n

L1

ƒsxd dx.

These inequalities hold for each n, and continue to hold as n : q . q If 11 ƒsxd dx is finite, the right-hand inequality shows that gan is finite. If q 11 ƒsxd dx is infinite, the left-hand inequality shows that gan is infinite. Hence the series and the integral are both finite or both infinite.

9.3

The Integral Test

509

EXAMPLE 3

Show that the p-series q 1 1 1 1 Á + 1p + Á a p = 1p + 2p + 3p + n n=1 n ( p a real constant) converges if p 7 1, and diverges if p … 1. Solution

If p 7 1, then ƒsxd = 1>x p is a positive decreasing function of x. Since q

L1

ˆ 1 The p-series a p n1 n converges if p 7 1, diverges if p … 1.

q

x -p + 1 1 d x -p dx = lim c p dx = x b: q -p + 1 1 L1 1 1 = lim a - 1b 1 - p b: q b p - 1 1 1 = s0 - 1d = , 1 - p p - 1 b

b p - 1 : q as b : q because p - 1 7 0.

the series converges by the Integral Test. We emphasize that the sum of the p-series is not 1>s p - 1d. The series converges, but we don’t know the value it converges to. If p 6 1, then 1 - p 7 0 and q

L1

1 1 dx = lim sb 1 - p - 1d = q . 1 - p b: q xp

The series diverges by the Integral Test. If p = 1, we have the (divergent) harmonic series 1 +

1 1 1 + + Á + n + Á. 2 3

We have convergence for p 7 1 but divergence for all other values of p. The p-series with p = 1 is the harmonic series (Example 1). The p-Series Test shows that the harmonic series is just barely divergent; if we increase p to 1.000000001, for instance, the series converges! The slowness with which the partial sums of the harmonic series approach infinity is impressive. For instance, it takes more than 178 million terms of the harmonic series to move the partial sums beyond 20. (See also Exercise 43b.) q

The series g n = 1 s1>sn 2 + 1dd is not a p-series, but it converges by the Integral Test. The function ƒsxd = 1>sx 2 + 1d is positive, continuous, and decreasing for x Ú 1, and

EXAMPLE 4

q

L1

1 b dx = lim C arctan x D 1 q b: x + 1 = lim [arctan b - arctan 1] b: q p p p = = . 2 4 4 2

The series converges, but p>4 is not the sum of the series.

Error Estimation For some convergent series, such as the geometric series or the telescoping series in Example 5 of Section 9.2, we can actually find the total sum of the series. That is, we can find the limiting value S of the sequence of partial sums. For most convergent series, however, we cannot easily find the total sum. Nevertheless, we can estimate the sum by adding the first n terms to get sn, but then we need to know how far off sn is from the total sum S. Suppose that a series ©an is shown to be convergent by the Integral Test, and we want to estimate the size of the remainder Rn between the total sum S of the series and its nth partial sum sn. That is, we wish to estimate Rn = S - sn = an + 1 + an + 2 + an + 3 + Á .

510

Chapter 9: Infinite Sequences and Series

To get a lower bound for the remainder, we compare the sum of the areas of the rectangles with the area under the curve y = ƒ(x) for x Ú n (see Figure 9.11a). We see that q

Rn = an + 1 + an + 2 + an + 3 + Á Ú

Ln + 1

ƒ(x) dx.

Similarly, from Figure 9.11b, we find an upper bound with Rn = an + 1 + an + 2 + an + 3 + Á …

q

Ln

ƒ(x) dx.

These comparisons prove the following result, giving bounds on the size of the remainder.

Bounds for the Remainder in the Integral Test Suppose {ak} is a sequence of positive terms with ak = ƒ(k), where ƒ is a continuous positive decreasing function of x for all x Ú n, and that ©an converges to S. Then the remainder Rn = S - sn satisfies the inequalities q

Ln + 1

q

ƒ(x) dx … Rn …

Ln

ƒ(x) dx.

(1)

If we add the partial sum sn to each side of the inequalities in (1), we get q

sn +

Ln + 1

q

ƒ(x) dx … S … sn +

Ln

ƒ(x) dx

(2)

since sn + Rn = S . The inequalities in (2) are useful for estimating the error in approximating the sum of a series known to converge by the Integral Test. The error can be no larger than the length of the interval containing S, as given by (2).

EXAMPLE 5 n = 10. Solution

Estimate the sum of the series ©s1>n 2 d using the inequalities in (2) and

We have that q

Ln

b

1 1 1 1 1 dx = lim c- x d = lim a- + n b = n . b b: q b: q x2 n

Using this result with the inequalities in (2), we get s10 +

1 1 … S … s10 + . 11 10

Taking s10 = 1 + (1>4) + (1>9) + (1>16) + Á + (1>100) L 1.54977, these last inequalities give 1.64068 … S … 1.64997. If we approximate the sum S by the midpoint of this interval, we find that q

1 a n 2 L 1.6453. n=1 The error in this approximation is less than half the length of the interval, so the error is less than 0.005.

9.3

The Integral Test

511

Exercises 9.3 Applying the Integral Test Use the Integral Test to determine if the series in Exercises 1–10 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

Theory and Examples For what values of a, if any, do the series in Exercises 41 and 42 converge? q

q

1 1. a 2 n=1 n

q

1 2. a 0.2 n=1 n

a 1 41. a a b n + 4 n=1 n + 2

q

q

43. a. Draw illustrations like those in Figures 9.7 and 9.8 to show that the partial sums of the harmonic series satisfy the inequalities

q

1 3. a 2 n + 4 n=1

1 4. a n + 4 n=1

q

q

n=1 q

ln (n 2) 8. a n q

n - 4 10. a 2 n = 2 n - 2n + 1

n n>3 n=1 e

9. a

Determining Convergence or Divergence Which of the series in Exercises 11–40 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.) q

q

1 11. a n n = 1 10

n=1

q

q

5 14. a n=1 n + 1

15. a

n=1

q

q n=2

1

2n 27. a n = 2 ln n

q

1 29. a n n = 1 sln 2d n=3

s1>nd sln nd2ln2 n - 1

q

1 33. a n sin n n=1 q

en 35. a 2n n=1 1 + e q

8 tan-1 n 37. a 2 n=1 1 + n q

39. a sech n n=1

a. Use the accompanying graph to show that the partial sum s50 = g n = 1 A 1> 2n + 1 B satisfies 50

n

1 28. a a1 + n b n=1

q

44. Are there any values of x for which g n = 1s1>nxd converges? Give reasons for your answer.

ˆ

2 25. a n=1 n + 1 q

T b. There is absolutely no empirical evidence for the divergence of the harmonic series even though we know it diverges. The partial sums just grow too slowly. To see what we mean, suppose you had started with s1 = 1 the day the universe was formed, 13 billion years ago, and added a new term every second. About how large would the partial sum sn be today, assuming a 365-day year?

47. g n 1 A 1/ 2n 1 B diverges

q

q

1 x dx = 1 + ln n .

46. (Continuation of Exercise 45.) Is there a “largest” convergent series of positive numbers? Explain.

5n 22. a n 4 + 3 n=1

1 24. a n = 1 2n - 1

2n A 2n + 1 B

-2 n2n

n=2

q

L1

q

ln n 19. a n

q

1 1 Á + 1 x dx … 1 + 2 + n

45. Is it true that if g n = 1 an is a divergent series of positive numbers, q then there is also a divergent series g n = 1 bn of positive numbers with bn 6 an for every n? Is there a “smallest” divergent series of positive numbers? Give reasons for your answers.

q

q

2n

q

n=1

2n 21. a n n=1 3

ln n

q

31. a

2n

n=1

-2 23. a n=0 n + 1 n=1

16. a

-8 18. a n

q

26. a

q

3

q

1 17. a - n 8 n=1 20. a

q

n 13. a n=1 n + 1

12. a e -n

L1

… 1 +

n=2

2

n+1

n

q

n 7. a 2 n=1 n + 4 q

ln sn + 1d =

1 6. a 2 n = 2 nsln nd

5. a e -2n

2a 1 42. a a b n + 1 n=3 n - 1

51 n

L1

50

1 2x + 1

dx 6 s50 6

L0

1 2x + 1

dx.

Conclude that 11.5 6 s50 6 12.3.

q

1 30. a n n = 1 sln 3d

y 1

q

1 32. a 2 n = 1 ns1 + ln nd q

f (x) 5

1 34. a n tan n n=1 q

1 x 1 1

···

2 36. a n n=1 1 + e

0

1 2 3 4 5 ··· 48 49 50 51

x

q

n 38. a 2 n=1 n + 1 q

40. a sech2 n n=1

b. What should n be in order that the partial sum sn = g i = 1 A 1> 2i + 1 B satisfy sn 7 1000? n

512

Chapter 9: Infinite Sequences and Series ˆ

q

48. g n 1 s1>n4 d converges a. Use the accompanying graph to determine the error if s30 = q 30 g n = 1 s1>n 4 d is used to estimate the value of g n = 1 s1>n 4 d. y

1 +

f (x) 5 14 x

26

2310

q

1 1 c. a d. a 3 3 n = 2 n lnsn d n = 2 nsln nd 57. Euler’s constant Graphs like those in Figure 9.11 suggest that as n increases there is little change in the difference between the sum 1 1 + Á + n 2

and the integral n

29

30

31

··· 33

32

x

n

b. Find n so that the partial sum sn = g i = 1 s1>i 4 d estimates the q value of g n = 1 s1>n 4 d with an error of at most 0.000001.

1 x dx . L1 To explore this idea, carry out the following steps. ln n =

a. By taking ƒsxd = 1>x in the proof of Theorem 9, show that ln sn + 1d … 1 +

q

49. Estimate the value of g n = 1 s1>n 3 d to within 0.01 of its exact value. q

50. Estimate the value of g n = 2 s1>sn 2 + 4dd to within 0.1 of its exact value. q

51. How many terms of the convergent series g n = 1 s1>n 1.1 d should be used to estimate its value with error at most 0.00001? q

52. How many terms of the convergent series g n = 4 s1>nsln nd3 d should be used to estimate its value with error at most 0.01? 53. The Cauchy condensation test The Cauchy condensation test says: Let 5an6 be a nonincreasing sequence (an Ú an + 1 for all n) of positive terms that converges to 0. Then gan converges if and only if g2na2n converges. For example, gs1>nd diverges because g2n # s1>2n d = g1 diverges. Show why the test works. 54. Use the Cauchy condensation test from Exercise 53 to show that q

1 a. a diverges; n ln n n=2 q

1 b. a p converges if p 7 1 and diverges if p … 1 . n=1 n

or 0 6 ln sn + 1d - ln n … 1 +

q

dx xsln xd p

s p a positive constantd L2 converges if and only if p 7 1 . b. What implications does the fact in part (a) have for the convergence of the series q

1 a nsln nd p ?

1 1 + Á + n - ln n 2 is bounded from below and from above. an = 1 +

b. Show that 1 6 n + 1 Ln

Give reasons for your answer. 56. (Continuation of Exercise 55.) Use the result in Exercise 55 to determine which of the following series converge and which diverge. Support your answer in each case. 1 a. a n = 2 nsln nd

9.4

n+1

1 x dx = ln sn + 1d - ln n ,

and use this result to show that the sequence 5an6 in part (a) is decreasing. Since a decreasing sequence that is bounded from below converges, the numbers an defined in part (a) converge: 1 +

1 1 + Á + n - ln n : g . 2

The number g , whose value is 0.5772 Á , is called Euler’s constant. 58. Use the Integral Test to show that the series q

ae

-n2

n=0

converges. 59. a. For the series g(1>n 3), use the inequalities in Equation (2) with n = 10 to find an interval containing the sum S.

n=2

q

1 1 + Á + n - ln n … 1 . 2

Thus, the sequence

55. Logarithmic p-series a. Show that the improper integral

1 1 + Á + n … 1 + ln n 2

q

b. As in Example 5, use the midpoint of the interval found in part (a) to approximate the sum of the series. What is the maximum error for your approximation? 60. Repeat Exercise 59 using the series g(1>n 4).

1 b. a 1.01 n = 2 nsln nd

Comparison Tests We have seen how to determine the convergence of geometric series, p-series, and a few others. We can test the convergence of many more series by comparing their terms to those of a series whose convergence is known.

9.4

Comparison Tests

513

THEOREM 10—The Comparison Test Let gan, gcn, and gdn be series with nonnegative terms. Suppose that for some integer N dn … an … cn

n 7 N.

for all

(a) If gcn converges, then gan also converges. (b) If gdn diverges, then gan also diverges.

Proof In Part (a), the partial sums of gan are bounded above by

y c1

M = a1 + a2 + Á + aN +

c2 c3 a1

a3

a2 1

2

c4 c 5

cn1 c n a a4 a5 n 3 4 5 ··· n1 n

n

FIGURE 9.12 If the total area gcn of the taller cn rectangles is finite, then so is the total area gan of the shorter an rectangles.

q

a cn .

n=N+1

They therefore form a nondecreasing sequence with a limit L … M. That is, if gcn converges, then so does gan. Figure 9.12 depicts this result, where each term of each series is interpreted as the area of a rectangle (just like we did for the integral test in Figure 9.11). In Part (b), the partial sums of gan are not bounded from above. If they were, the partial sums for gdn would be bounded by M * = d1 + d2 + Á + dN +

q

a an

n=N+1

and gdn would have to converge instead of diverge.

EXAMPLE 1

We apply Theorem 10 to several series.

(a) The series q

5 a 5n - 1

n=1

diverges because its nth term 5 = 5n - 1

HISTORICAL BIOGRAPHY

1

1 7 n 1 n 5 is greater than the nth term of the divergent harmonic series. (b) The series q

1 1 1 1 Á a n! = 1 + 1! + 2! + 3! + n=0

Albert of Saxony (ca. 1316–1390)

converges because its terms are all positive and less than or equal to the corresponding terms of q

1 1 1 1 + a n = 1 + 1 + + 2 + Á. 2 2 n=0 2 The geometric series on the left converges and we have q

1 1 1 + a n = 1 + = 3. 2 1 s1>2d n=0

q

The fact that 3 is an upper bound for the partial sums of g n = 0 s1>n!d does not mean that the series converges to 3. As we will see in Section 9.9, the series converges to e. (c) The series 5 +

2 1 1 1 1 1 + + 1 + + + + Á + + Á n 7 3 2 + 21 4 + 22 8 + 23 2 + 2n

converges. To see this, we ignore the first three terms and compare the remaining terms q with those of the convergent geometric series g n = 0 s1>2n d. The term 1> A 2n + 2n B of

514

Chapter 9: Infinite Sequences and Series

the truncated sequence is less than the corresponding term 1>2n of the geometric series. We see that term by term we have the comparison 1 +

1 2 + 21

+

1 4 + 22

+

1

1 1 1 + Á … 1 + + + + Á. 2 4 8 8 + 23

So the truncated series and the original series converge by an application of the Comparison Test.

The Limit Comparison Test We now introduce a comparison test that is particularly useful for series in which an is a rational function of n.

THEOREM 11—Limit Comparison Test all n Ú N (N an integer). 1. If lim

n: q

Suppose that an 7 0 and bn 7 0 for

an = c 7 0, then gan and gbn both converge or both diverge. bn

an = 0 and gbn converges, then gan converges. n: q bn

2. If lim 3. If lim

n: q

an = q and gbn diverges, then gan diverges. bn

Proof We will prove Part 1. Parts 2 and 3 are left as Exercises 55a and b. Since c>2 7 0, there exists an integer N such that for all n n 7 N Q `

an c - c` 6 . 2 bn

Limit definition with P = c>2, L = c , and an replaced by an>bn

Thus, for n 7 N, -

an c c - c 6 , 6 2 2 bn an 3c c 6 6 , 2 2 bn

3c c a bbn 6 an 6 a bbn . 2 2 If gbn converges, then gs3c>2dbn converges and gan converges by the Direct Comparison Test. If gbn diverges, then gsc>2dbn diverges and gan diverges by the Direct Comparison Test.

EXAMPLE 2

Which of the following series converge, and which diverge? q

q

(a)

3 5 7 9 2n + 1 2n + 1 + + + + Á = a = a 2 2 4 9 16 25 n = 1 sn + 1d n = 1 n + 2n + 1

(b)

1 1 1 1 1 + + + + Á = a n 7 1 3 15 2 - 1 n=1

(c)

1 + 3 ln 3 1 + n ln n 1 + 2 ln 2 1 + 4 ln 4 + + + Á = a 2 9 14 21 n=2 n + 5

q

q

9.4 Solution

Comparison Tests

515

We apply the Limit Comparison Test to each series.

(a) Let an = s2n + 1d>sn 2 + 2n + 1d. For large n, we expect an to behave like 2n>n 2 = 2>n since the leading terms dominate for large n, so we let bn = 1>n. Since q

q

n=1

n=1

1 a bn = a n diverges

and an 2n 2 + n = lim 2 = 2, n: q bn n: q n + 2n + 1 lim

gan diverges by Part 1 of the Limit Comparison Test. We could just as well have taken bn = 2>n, but 1> n is simpler. (b) Let an = 1>s2n - 1d. For large n, we expect an to behave like 1>2n , so we let bn = 1>2n . Since q q 1 a bn = a 2n converges n=1 n=1 and an 2n lim = lim n n: q bn n: q 2 - 1 1 = lim n n: q 1 - s1>2 d = 1, gan converges by Part 1 of the Limit Comparison Test. (c) Let an = s1 + n ln nd>sn 2 + 5d. For large n, we expect an to behave like sn ln nd>n 2 = sln nd>n, which is greater than 1> n for n Ú 3, so we let bn = 1>n. Since q

q

n=2

n=2

1 a bn = a n diverges

and an n + n 2 ln n = lim n: q bn n: q n2 + 5 = q, lim

gan diverges by Part 3 of the Limit Comparison Test. q

EXAMPLE 3

Does a

n=1

ln n converge? n 3>2

Because ln n grows more slowly than n c for any positive constant c (Section 9.1, Exercise 105), we can compare the series to a convergent p-series. To get the p-series, we see that Solution

ln n n 1>4 1 6 3>2 = 5>4 3>2 n n n for n sufficiently large. Then taking an = sln nd>n 3>2 and bn = 1>n 5>4 , we have an ln n lim = lim 1>4 b q q n: n: n n = lim

n: q

1>n s1>4dn -3>4

l’Hôpital’s Rule

4 = 0. n 1>4 Since gbn = gs1>n 5>4 d is a p-series with p 7 1, it converges, so gan converges by Part 2 of the Limit Comparison Test. = lim

n: q

516

Chapter 9: Infinite Sequences and Series

Exercises 9.4 Comparison Test In Exercises 1–8, use the Comparison Test to determine if each series converges or diverges. q

q

n - 1 2. a 4 n=1 n + 2

1 1. a 2 n = 1 n + 30 q

3. a

n + 2 4. a 2 n=2 n - n

2n - 1 cos2 n 5. a 3>2 n=1 n n=2 q

q

8. a

n=1

2n 2 + 3

q

q

13. a

2n 4 1 15. a ln n n=2 n=1 q

n

3 2 2n + 2 n 1 + cos n 20. a n2 n=1

n=1 q

q

n + 2n 2n 21. a n = 1 3n - 1 n=1 q

10n + 1 23. a nsn + 1dsn + 2d n=1 q

n b 25. a a n = 1 3n + 1 q

28. a

sln nd2 n3

n

3

q

26. a

n=1 q

29. a

2n 3 + 2 1

1 31. a n = 1 1 + ln n

2n ln n ln sn + 1d 32. a n + 1 n=2

2n 34. a 2 n=1 n + 1

1 - n 35. a n n = 1 n2

n=1 q

q

n=2 q

q

51. a

n=1

sec n n 1.3 1 n

n2n

q

49. a

n=1 q

coth n n2 n

2n 52. a 2 n=1 n

1 54. a 2 2 2 n=1 1 + 2 + 3 + Á + n

n: q

60. Suppose that an 7 0 and lim n 2an = 0. Prove that gan conn: q verges.

q

sin2 n n n=1 2

19. a q

22. a

n + 1

n 2 2n q 3 5n - 3n 24. a 2 n sn - 2dsn 2 + 5d n=3 1

n=1 q

59. Suppose that an 7 0 and lim an = q . Prove that gan diverges.

Determining Convergence or Divergence Which of the series in Exercises 17–54 converge, and which diverge? Use any method, and give reasons for your answers. q

48. a

58. Prove that if gan is a convergent series of nonnegative terms, then ga n2 converges.

1 16. a ln a1 + 2 b n n=1 q (Hint: Limit Comparison with g n = 1 s1>n 2 dd

18. a

tan n n 1.1

n=1 q

-1

57. Suppose that an 7 0 and bn 7 0 for n Ú N (N an integer). If lim n: q san>bn d = q and gan converges, can anything be said about g bn ? Give reasons for your answer.

q

1

n=1 q

n=1 q

-1

q

q

q

47. a

q

1 46. a tan n

56. If g n = 1 an is a convergent series of nonnegative numbers, can q anything be said about g n = 1san>nd ? Explain.

(Hint: Limit Comparison with g n = 2 s1>ndd

17. a

q

q

1 45. a sin n

Theory and Examples 55. Prove (a) Part 2 and (b) Part 3 of the Limit Comparison Test.

n

2n + 3 b 14. a a n = 1 5n + 4

n

q sn - 1d! 44. a n = 1 sn + 2d!

1 53. a Á + n n=1 1 + 2 + 3 +

2 12. a n n=1 3 + 4

5n

q

q

q

q

n b 42. a ln a n + 1 n=1

tanh n n2 n=1

A Hint: Limit Comparison with g n = 1 A 1> 2n B B

nsn + 1d 11. a 2 n = 2 sn + 1dsn - 1d

q

50. a

q

q

q

(Hint: First show that s1>n!d … s1>nsn - 1dd for n Ú 2.)

2n + 1

n - 2 9. a 3 2 n=1 n - n + 3 q (Hint: Limit Comparison with g n = 1 s1>n 2 dd

q

2n + 3n 40. a n n n=1 3 + 4

q

Limit Comparison Test In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.

n + 1 10. a 2 n=1 A n + 2

q

n + 1 # 1 39. a 2 n = 1 n + 3n 5n

1 43. a n! n=2

1 6. a n n = 1 n3

q

3n - 1 + 1 3n n=1

41. a

q

n + 4 7. a A n4 + 4 n=1

q

38. a

2n - n n n = 1 n2

q

1

q

1 37. a n - 1 + 1 n=1 3

n=1

q

1 27. a n = 3 ln sln nd q

30. a

n=1 q

33. a

sln nd2 n 3>2 1

n 2n 2 - 1 n + 2n 36. a 2 n n=1 n 2 n=2 q

61. Show that g n = 2 ssln ndq>n p d converges for - q 6 q 6 q and p 7 1. q

q

(Hint: Limit Comparison with g n = 2 1>n r for 1 6 r 6 p.)

62. (Continuation of Exercise 61.) Show that g n = 2 ssln ndq>n p d diverges for - q 6 q 6 q and 0 6 p … 1. q

(Hint: Limit Comparison with an appropriate p-series.)

In Exercises 63–68, use the results of Exercises 61 and 62 to determine if each series converges or diverges. q

63. a

n=2 q

65. a

n=2 q

sln nd3 n

4

sln nd1000 n 1.001

1 67. a 1.1 3 n = 2 n sln nd

q

ln n 64. a n n=2 A q

66. a

n=2 q

68. a

n=2

sln nd1>5 n 0.99 1

2n # ln n

9.5

The Ratio and Root Tests

517

70. a. Use Theorem 8 to show that

COMPUTER EXPLORATIONS 69. It is not yet known whether the series

q

q

1 1 1 + a a 2 b S = a nsn + 1d n = 1 nsn + 1d n=1 n

q

1 a n 3 sin2 n

n=1

q

converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps.

where S = g n = 1 s1>n 2 d, the sum of a convergent p-series. b. From Example 5, Section 9.2, show that q

a. Define the sequence of partial sums

1 S = 1 + a 2 . n = 1 n sn + 1d

k

1 sk = a 3 2 . n sin n n=1 What happens when you try to find the limit of sk as k : q ? Does your CAS find a closed form answer for this limit? b. Plot the first 100 points sk, sk d for the sequence of partial sums. Do they appear to converge? What would you estimate the limit to be? c. Next plot the first 200 points sk, sk d . Discuss the behavior in your own words.

c. Explain why taking the first M terms in the series in part (b) gives a better approximation to S than taking the first M terms q in the original series g n = 1 s1>n 2 d.

d. The exact value of S is known to be p2>6. Which of the sums 1000000

1 a 2 n=1 n

1000

or

1 1 + a 2 n = 1 n sn + 1d

gives a better approximation to S?

d. Plot the first 400 points sk, sk d . What happens when k = 355 ? Calculate the number 355> 113. Explain from your calculation what happened at k = 355 . For what values of k would you guess this behavior might occur again?

9.5

The Ratio and Root Tests The Ratio Test measures the rate of growth (or decline) of a series by examining the ratio an + 1>an . For a geometric series gar n , this rate is a constant ssar n + 1 d>sar n d = rd, and the series converges if and only if its ratio is less than 1 in absolute value. The Ratio Test is a powerful rule extending that result.

THEOREM 12—The Ratio Test suppose that

Let gan be a series with positive terms and

lim

n: q

an + 1 an = r.

Then (a) the series converges if r 6 1, (b) the series diverges if r 7 1 or r is infinite, (c) the test is inconclusive if r = 1.

Proof (a) R<1. Let r be a number between r and 1. Then the number P = r - r is positive. Since an + 1 an : r, an + 1>an must lie within P of r when n is large enough, say for all n Ú N. In particular, an + 1 an 6 r + P = r,

when n Ú N.

518

Chapter 9: Infinite Sequences and Series

That is, aN + 1 6 raN , aN + 2 6 raN + 1 6 r 2aN , aN + 3 6 raN + 2 6 r 3aN , o aN + m 6 raN + m - 1 6 r maN . These inequalities show that the terms of our series, after the Nth term, approach zero more rapidly than the terms in a geometric series with ratio r 6 1. More precisely, consider the series gcn , where cn = an for n = 1, 2, Á , N and cN + 1 = raN , cN + 2 = r 2aN , Á , cN + m = r maN , Á . Now an … cn for all n, and q

Á + aN - 1 + aN + raN + r 2aN + Á a cn = a1 + a2 + = a1 + a2 + Á + aN - 1 + aN s1 + r + r 2 + Á d.

n=1

The geometric series 1 + r + r 2 + Á converges because ƒ r ƒ 6 1, so gcn converges. Since an … cn , gan also converges. (b) 1
1 a n

q

1 a n2 n=1 n=1 show that some other test for convergence must be used when r = 1. q

and

1>sn + 1d an + 1 n : 1. = an = n + 1 1>n

1 For a n : n=1 q

2 1>sn + 1d2 an + 1 n = = a b : 12 = 1. an n + 1 1>n 2

1 For a 2 : n=1 n

In both cases, r = 1, yet the first series diverges, whereas the second converges. The Ratio Test is often effective when the terms of a series contain factorials of expressions involving n or expressions raised to a power involving n.

EXAMPLE 1 q

2n + 5 3n n=0

(a) a

Solution

Investigate the convergence of the following series. q s2nd! (b) a n = 1 n!n!

q

4nn!n! (c) a n = 1 s2nd!

We apply the Ratio Test to each series. q

(a) For the series g n = 0 s2n + 5d>3n , s2n + 1 + 5d>3n + 1 an + 1 1 = = an 3 s2n + 5d>3n

#

2n + 1 + 5 1 = 3 2n + 5

1 # # a 2 + 5 ## 2-n -n b : 1 + 5 2

3

2 2 = . 1 3

The series converges because r = 2>3 is less than 1. This does not mean that 2> 3 is the sum of the series. In fact, q

q

n

q

2n + 5 5 5 2 1 21 a 3n = a a 3 b + a 3n = 1 - s2>3d + 1 - s1>3d = 2 . n=0 n=0 n=0

9.5

(b) If an =

The Ratio and Root Tests

519

s2nd! s2n + 2d! , then an + 1 = and n!n! sn + 1d!sn + 1d! n!n!s2n + 2ds2n + 1ds2nd! an + 1 an = sn + 1d!sn + 1d!s2nd! =

s2n + 2ds2n + 1d 4n + 2 = : 4. n + 1 sn + 1dsn + 1d

The series diverges because r = 4 is greater than 1. (c) If an = 4nn!n!>s2nd!, then 4n + 1sn + 1d!sn + 1d! # s2nd! an + 1 = an s2n + 2ds2n + 1ds2nd! 4nn!n! =

4sn + 1dsn + 1d 2sn + 1d = : 1. 2n + 1 s2n + 2ds2n + 1d

Because the limit is r = 1, we cannot decide from the Ratio Test whether the series converges. When we notice that an + 1>an = s2n + 2d>s2n + 1d, we conclude that an + 1 is always greater than an because s2n + 2d>s2n + 1d is always greater than 1. Therefore, all terms are greater than or equal to a1 = 2, and the nth term does not approach zero as n : q . The series diverges.

The Root Test The convergence tests we have so far for gan work best when the formula for an is relatively simple. However, consider the series with the terms an = e

n>2n, 1>2n,

n odd n even.

To investigate convergence we write out several terms of the series: q

3 5 7 1 1 1 1 Á a an = 21 + 22 + 23 + 24 + 25 + 26 + 27 + n=1 3 5 7 1 1 1 1 + + + + + + + Á. 2 4 8 16 32 64 128 Clearly, this is not a geometric series. The nth term approaches zero as n : q , so the nth-Term Test does not tell us if the series diverges. The Integral Test does not look promising. The Ratio Test produces =

1 , 2n an + 1 an = d n + 1 , 2

n odd n even.

As n : q , the ratio is alternately small and large and has no limit. However, we will see that the following test establishes that the series converges.

THEOREM 13—The Root Test

Let gan be a series with an Ú 0 for n Ú N ,

and suppose that n

lim 2an = r.

n: q

Then (a) the series converges if r 6 1, (b) the series diverges if r 7 1 or r is infinite, (c) the test is inconclusive if r = 1.

520

Chapter 9: Infinite Sequences and Series

Proof n

n

(a) R<1. Choose an P 7 0 so small that r + P 6 1. Since 2an : r, the terms 2an eventually get closer than P to r. In other words, there exists an index M Ú N such that n

2an 6 r + P

when n Ú M.

Then it is also true that an 6 sr + Pdn q gn=M

for n Ú M.

Now, sr + Pd , a geometric series with ratio sr + Pd 6 1, converges. By q comparison, g n = M an converges, from which it follows that n

q

q

n=1

n=M

Á + aM - 1 + a an = a1 + a an

converges. n (b) 1nd and g n = 1 s1>n 2 d show that the test is not conclusive when r = 1. The first series diverges and the second converges, but in both cases n 2an : 1.

EXAMPLE 2

Consider again the series with terms an = e

n>2n, 1>2n,

n odd n even.

Does gan converge? Solution

We apply the Root Test, finding that 2an = e n

n

2n>2, 1>2,

n odd n even.

Therefore, n

2n n 1 … 2an … . 2 2 n

n

Since 2n : 1 (Section 9.1, Theorem 5), we have limn: q 2an = 1>2 by the Sandwich Theorem. The limit is less than 1, so the series converges by the Root Test.

EXAMPLE 3 q

n2 (a) a n n=1 2 Solution

Which of the following series converge, and which diverge? q

2n (b) a 3 n=1 n

q

1 (c) a a b n=1 1 + n

We apply the Root Test to each series.

q

n

2 2n 2 n2 n n (a) a n converges because n = n n = B2 n=1 2 22 q

n 2n n 2 (b) a 3 diverges because = A n3 n=1 n q

n

n

n nB A2

2

2

:

12 6 1. 2

2 2 : 3 7 1. n 3 1 2 n A B n

1 1 1 n (c) a a b converges because a b = : 0 6 1. 1 + n B 1 + n n=1 1 + n

9.5

The Ratio and Root Tests

521

Exercises 9.5 Using the Ratio Test In Exercises 1–8, use the Ratio Test to determine if each series converges or diverges. q

q

n

2 1. a n = 1 n! q

2. a

n=1 q

sn - 1d!

q

4

q

7. a

n5n 8. a s2n + 3d ln sn + 1d n=1

n! 32n

n=1

n! 37. a s2n + 1d! n=1

n! 38. a n n n=1

n 39. a n n = 2 sln nd

40. a

n2nsn + 1d! 3nn! n=1

q

q

q

n=2 q

q

q

n 2sn + 2d!

q

Using the Root Test In Exercises 9–16, use the Root Test to determine if each series converges or diverges.

q sn!d2 43. a n = 1 s2nd!

q

Recursively Defined Terms Which of the series g n = 1 an defined by the formulas in Exercises 45–54 converge, and which diverge? Give reasons for your answers. 45. a1 = 2,

an + 1 =

1 + sin n an n

46. a1 = 1,

an + 1 =

1 + tan-1 n an n

1 , 3

an + 1 =

3n - 1 a 2n + 5 n

48. a1 = 3,

an + 1 =

n a n + 1 n

49. a1 = 2,

2 an + 1 = n an

16. a 1 + n n=2 n

50. a1 = 5,

an + 1 =

2n a 2 n

Determining Convergence or Divergence In Exercises 17–44, use any method to determine if the series converges or diverges. Give reasons for your answer.

51. a1 = 1,

an + 1 =

1 + ln n an n n + ln n a n + 10 n

q

q

4n 10. a n n = 1 s3nd

7 9. a s2n + 5dn n=1 q

4n + 3 b 11. a a n = 1 3n - 5

n

q

8 13. a 2n n = 1 (3 + (1>n)) q

1 15. a a1 - n b

q

1 12. a aln ae + n b b 2

n=1 q

14. a sinn a n=1

1 2n

n+1

b

n2

n=1

(Hint: lim s1 + x>ndn = e x) n: q

q

1

q

n 22 17. a n n=1 2 q

19. a n!e

52. a1 =

1 , 2

an + 1 =

n=1 q

53. a1 =

1 , 3

an + 1 = 2an

54. a1 =

1 , 2

an + 1 = san dn + 1

q

n 10 21. a n 10 n=1

n - 2 22. a a n b

2 + s - 1dn 23. a 1.25n n=1

s -2dn 24. a 3n n=1

q

q

n

n=1 q

ln n 27. a 3 n=1 n q

n

n=1 q

q

1 b 26. a a1 3n n=1 q

q

ln n 31. a n

n ln n 32. a n n=1 2

sn + 1dsn + 2d n! n=1

33. a

q

q

34. a e -nsn 3 d n=1

n

n

Convergence or Divergence Which of the series in Exercises 55–62 converge, and which diverge? Give reasons for your answers. q q s3nd! 2nn!n! 55. a 56. a s2nd! n!sn + 1d!sn + 2d! n=1 n=1 q sn!dn 57. a n 2 n = 1 sn d

sln ndn nn n=1

1 1 30. a a n - 2 b n n=1

n=1 q

n

28. a

1 1 29. a a n - 2 b n n=1 q

n

q

n! 20. a n n = 1 10

n=1 q

3 25. a a1 - n b

47. a1 =

18. a n 2e -n

-n

n sln ndsn>2d

3n 42. a 3 n n=1 n 2 q s2n + 3ds2n + 3d 44. a 3n + 2 n=1

n! ln n 41. a n = 1 nsn + 2d!

n+2

3 6. a n = 2 ln n

n 5. a n n=1 4

36. a

q

2n + 1 4. a n - 1 n = 1 n3

3. a 2 n = 1 sn + 1d q

n + 2 3n

q sn + 3d! 35. a n n = 1 3!n!3

q

59. a

n=1 q

nn sn2d

q

58. a

n=1 q

sn!dn 2

n sn d

nn 60. a n 2 n = 1 s2 d

2 1 # 3 # Á # s2n - 1d 61. a 4n2nn! n=1 q 1 # 3 # Á # s2n - 1d 62. a # # Á # s2nd]s3n + 1d n = 1 [2 4

522

Chapter 9: Infinite Sequences and Series 65. Let an = e

Theory and Examples 63. Neither the Ratio Test nor the Root Test helps with p-series. Try them on

n>2n, 1>2n,

if n is a prime number otherwise.

Does gan converge? Give reasons for your answer.

q

66. Show that g n = 1 2(n )>n! diverges. Recall from the Laws of Expo2 nents that 2(n ) = s2n dn. q

1 a p n=1 n

2

and show that both tests fail to provide information about convergence. 64. Show that neither the Ratio Test nor the Root Test provides information about the convergence of q

1 a sln nd p n=2

9.6

s p constantd .

Alternating Series, Absolute and Conditional Convergence A series in which the terms are alternately positive and negative is an alternating series. Here are three examples: s -1dn + 1 1 1 1 1 + Á + - + - Á + n 5 2 3 4 s -1dn4 1 1 1 -2 + 1 - + - + Á + + Á 2 4 8 2n 1 - 2 + 3 - 4 + 5 - 6 + Á + s -1dn + 1n + Á 1 -

(1) (2) (3)

We see from these examples that the nth term of an alternating series is of the form an = s -1dn + 1un

or

an = s -1dnun

where un = ƒ an ƒ is a positive number. Series (1), called the alternating harmonic series, converges, as we will see in a moment. Series (2), a geometric series with ratio r = -1>2, converges to -2>[1 + s1>2d] = -4>3. Series (3) diverges because the nth term does not approach zero. We prove the convergence of the alternating harmonic series by applying the Alternating Series Test. This test is for convergence of an alternating series and cannot be used to conclude that such a series diverges. The test is also valid for the alternating series -u1 + u2 - u3 + Á , like the one in Series (2) given above.

THEOREM 14—The Alternating Series Test q

n+1

a s -1d

n=1

The series

un = u1 - u2 + u3 - u4 + Á

converges if all three of the following conditions are satisfied: 1. The un’s are all positive. 2. The positive un’s are (eventually) nonincreasing: un Ú un + 1 for all n Ú N, for some integer N. 3. un : 0. Proof Assume N = 1. If n is an even integer, say n = 2m, then the sum of the first n terms is s2m = su1 - u2 d + su3 - u4 d + Á + su2m - 1 - u2m d = u1 - su2 - u3 d - su4 - u5 d - Á - su2m - 2 - u2m - 1 d - u2m .

9.6

Alternating Series, Absolute and Conditional Convergence

523

The first equality shows that s2m is the sum of m nonnegative terms since each term in parentheses is positive or zero. Hence s2m + 2 Ú s2m , and the sequence 5s2m6 is nondecreasing. The second equality shows that s2m … u1 . Since 5s2m6 is nondecreasing and bounded from above, it has a limit, say lim s2m = L.

m: q

(4)

If n is an odd integer, say n = 2m + 1, then the sum of the first n terms is s2m + 1 = s2m + u2m + 1 . Since un : 0, lim u2m + 1 = 0

m: q

and, as m : q , s2m + 1 = s2m + u2m + 1 : L + 0 = L.

(5)

Combining the results of Equations (4) and (5) gives limn: q sn = L (Section 9.1, Exercise 131).

EXAMPLE 1

The alternating harmonic series q

n+1

a s -1d

n=1

1 1 1 1 Á n = 1 - 2 + 3 - 4 +

clearly satisfies the three requirements of Theorem 14 with N = 1; it therefore converges. Rather than directly verifying the definition un Ú un + 1, a second way to show that the sequence {un} is nonincreasing is to define a differentiable function ƒsxd satisfying ƒsnd = un. That is, the values of ƒ match the values of the sequence at every positive integer n. If ƒ¿sxd … 0 for all x greater than or equal to some positive integer N, then ƒsxd is nonincreasing for x Ú N. It follows that ƒsnd Ú ƒsn + 1d, or un Ú un + 1, for n Ú N. Consider the sequence where un = 10n>sn 2 + 16d. Define ƒsxd = 10x>sx + 16d. Then from the Derivative Quotient Rule,

EXAMPLE 2 2

ƒ¿sxd =

10s16 - x 2 d … 0 sx 2 + 16d2

whenever x Ú 4.

It follows that un Ú un + 1 for n Ú 4. That is, the sequence {un} is nonincreasing for n Ú 4. u1 u2 u3 u4

0

s2

s4

L

s3

FIGURE 9.13 The partial sums of an alternating series that satisfies the hypotheses of Theorem 14 for N = 1 straddle the limit from the beginning.

s1

x

A graphical interpretation of the partial sums (Figure 9.13) shows how an alternating series converges to its limit L when the three conditions of Theorem 14 are satisfied with N = 1. Starting from the origin of the x-axis, we lay off the positive distance s1 = u1 . To find the point corresponding to s2 = u1 - u2 , we back up a distance equal to u2 . Since u2 … u1 , we do not back up any farther than the origin. We continue in this seesaw fashion, backing up or going forward as the signs in the series demand. But for n Ú N, each forward or backward step is shorter than (or at most the same size as) the preceding step because un + 1 … un . And since the nth term approaches zero as n increases, the size of step we take forward or backward gets smaller and smaller. We oscillate across the limit L, and the amplitude of oscillation approaches zero. The limit L lies between any two successive sums sn and sn + 1 and hence differs from sn by an amount less than un + 1 . Because ƒ L - sn ƒ 6 un + 1

for n Ú N,

we can make useful estimates of the sums of convergent alternating series.

Chapter 9: Infinite Sequences and Series

THEOREM 15—The Alternating Series Estimation Theorem If the alternating q series g n = 1 s -1dn + 1un satisfies the three conditions of Theorem 14, then for n Ú N, sn = u1 - u2 + Á + s -1dn + 1un approximates the sum L of the series with an error whose absolute value is less than un + 1 , the absolute value of the first unused term. Furthermore, the sum L lies between any two successive partial sums sn and sn + 1, and the remainder, L - sn , has the same sign as the first unused term. We leave the verification of the sign of the remainder for Exercise 61.

EXAMPLE 3

We try Theorem 15 on a series whose sum we know:

q

1 1 1 1 1 1 1 n 1 a s -1d 2n = 1 - 2 + 4 - 8 + 16 - 32 + 64 - 128 n=0

------

524

+

1 - Á. 256

The theorem says that if we truncate the series after the eighth term, we throw away a total that is positive and less than 1> 256. The sum of the first eight terms is s8 = 0.6640625 and the sum of the first nine terms is s9 = 0.66796875. The sum of the geometric series is 1 1 2 = = , 3 3>2 1 - s -1>2d and we note that 0.6640625 6 (2>3) 6 0.66796875. The difference, s2>3d - 0.6640625 = 0.0026041666 Á , is positive and is less than s1>256d = 0.00390625.

Absolute and Conditional Convergence We can apply the tests for convergence studied before to the series of absolute values of a series with both positive and negative terms.

DEFINITION A series gan converges absolutely (is absolutely convergent) if the corresponding series of absolute values, g ƒ an ƒ , converges. The geometric series in Example 3 converges absolutely because the corresponding series of absolute values q

1 1 1 1 Á a 2n = 1 + 2 + 4 + 8 + n=0 converges. The alternating harmonic series does not converge absolutely because the corresponding series of absolute values is the (divergent) harmonic series.

DEFINITION A series that converges but does not converge absolutely converges conditionally.

The alternating harmonic series converges conditionally. Absolute convergence is important for two reasons. First, we have good tests for convergence of series of positive terms. Second, if a series converges absolutely, then it converges, as we now prove.

9.6

Alternating Series, Absolute and Conditional Convergence

525

q

THEOREM 16—The Absolute Convergence Test q

If a ƒ an ƒ converges, then n=1

a an converges.

n=1

Proof For each n, - ƒ an ƒ … an … ƒ an ƒ , q gn=1

0 … an + ƒ an ƒ … 2 ƒ an ƒ .

so

q gn=1

If 2 ƒ an ƒ converges and, by the Direct Comparison Test, ƒ an ƒ converges, then q the nonnegative series g n = 1 san + ƒ an ƒ d converges. The equality an = san + ƒ an ƒ d - ƒ an ƒ q now lets us express g n = 1 an as the difference of two convergent series: q

q

q

q

a an = a san + ƒ an ƒ - ƒ an ƒ d = a san + ƒ an ƒ d - a ƒ an ƒ .

Therefore,

n=1

n=1

q g n = 1 an

n=1

converges.

n=1

Caution We can rephrase Theorem 16 to say that every absolutely convergent series converges. However, the converse statement is false: Many convergent series do not converge absolutely (such as the alternating harmonic series in Example 1).

EXAMPLE 4

This example gives two series that converge absolutely.

q

1 1 1 1 (a) For a s -1dn + 1 2 = 1 - + + Á , the corresponding series of absolute 4 9 16 n n=1 values is the convergent series q

1 1 1 1 Á. a n 2 = 1 + 4 + 9 + 16 + n=1 The original series converges because it converges absolutely. q

sin n sin 1 sin 2 sin 3 (b) For a 2 = + + + Á , which contains both positive and nega1 4 9 n=1 n tive terms, the corresponding series of absolute values is q

ƒ sin 1 ƒ ƒ sin 2 ƒ sin n + + Á, a ` n2 ` = 1 4 n=1 q

which converges by comparison with g n = 1 s1>n 2 d because ƒ sin n ƒ … 1 for every n. The original series converges absolutely; therefore it converges. If p is a positive constant, the sequence 51>n p6 is a decreasing sequence with limit zero. Therefore the alternating p-series

EXAMPLE 5

q

s -1dn - 1 1 1 1 = 1 - p + p - p + Á, a np 2 3 4 n=1

p 7 0

converges. If p 7 1, the series converges absolutely. If 0 6 p … 1, the series converges conditionally. Conditional convergence: Absolute convergence:

1 1 1 + + Á 22 23 24 1 1 1 1 - 3>2 + 3>2 - 3>2 + Á 2 3 4 1 -

526

Chapter 9: Infinite Sequences and Series

Rearranging Series We can always rearrange the terms of a finite sum. The same result is true for an infinite series that is absolutely convergent (see Exercise 68 for an outline of the proof ).

THEOREM 17—The Rearrangement Theorem for Absolutely Convergent Series If q g n = 1 an converges absolutely, and b1, b2 , Á , bn , Á is any arrangement of the sequence 5an6, then gbn converges absolutely and q

q

a bn = a an .

n=1

n=1

If we rearrange the terms of a conditionally convergent series, we get different results. In fact, it can be proved that for any real number r, a given conditionally convergent series can be rearranged so its sum is equal to r. (We omit the proof of this fact.) Here’s an example of summing the terms of a conditionally convergent series with different orderings, with each ordering giving a different value for the sum. We know that the alternating harmonic series g n = 1 s -1dn + 1>n converges to some number L. Moreover, by Theorem 15, L lies between the successive partial sums s2 = 1>2 and s3 = 5>6, so L Z 0. If we multiply the series by 2 we obtain q

EXAMPLE 6

q

2L = 2 a

n=1

s -1dn + 1 1 1 1 1 1 1 1 1 1 1 + - Áb = 2 a1 - + - + - + - + n 2 3 4 5 6 7 8 9 10 11 = 2 - 1 +

2 1 2 2 2 1 1 1 2 - + - + - + - + - Á. 5 7 5 3 2 3 4 9 11

Now we change the order of this last sum by grouping each pair of terms with the same odd denominator, but leaving the negative terms with the even denominators as they are placed (so the denominators are the positive integers in their natural order). This rearrangement gives s2 - 1d -

1 2 1 1 2 2 1 1 1 1 + a - b - + a - b - + a - b - + Á 5 5 7 7 2 3 3 4 6 8 = a1 q

= a

n=1

1 1 1 1 1 1 1 1 1 1 + - + - + - + + - Áb 5 7 2 3 4 6 8 9 10 11

s -1dn + 1 = L. n

So by rearranging the terms of the conditionally convergent series g n = 1 2s -1dn + 1>n, the q series becomes g n = 1 s -1dn + 1>n, which is the alternating harmonic series itself. If the two series are the same, it would imply that 2L = L, which is clearly false since L Z 0. q

Example 6 shows that we cannot rearrange the terms of a conditionally convergent series and expect the new series to be the same as the original one. When we are using a conditionally convergent series, the terms must be added together in the order they are given to obtain a correct result. On the other hand, Theorem 17 guarantees that the terms of an absolutely convergent series can be summed in any order without affecting the result.

Summary of Tests We have developed a variety of tests to determine convergence or divergence for an infinite series of constants. There are other tests we have not presented which are sometimes given in more advanced courses. Here is a summary of the tests we have considered.

9.6

1. 2. 3. 4.

Alternating Series, Absolute and Conditional Convergence

527

The nth-Term Test: Unless an : 0, the series diverges. Geometric series: gar n converges if ƒ r ƒ 6 1; otherwise it diverges. p-series: g1>n p converges if p 7 1; otherwise it diverges.

Series with nonnegative terms: Try the Integral Test, Ratio Test, or Root Test. Try comparing to a known series with the Comparison Test or the Limit Comparison Test. 5. Series with some negative terms: Does g ƒ an ƒ converge? If yes, so does gan since absolute convergence implies convergence. 6. Alternating series: gan converges if the series satisfies the conditions of the Alternating Series Test.

Exercises 9.6 Determining Convergence or Divergence In Exercises 1–14, determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. q

q

1

1. a s -1dn + 1

2. a s -1dn + 1

2n 1 3. a s -1dn + 1 n n3 n=1 n=1 q

n=1 q

1 n 3>2

4 4. a s -1dn sln nd2 n=2 q

q

n 5. a s -1dn 2 n + 1 n=1

n2 + 5 6. a s -1dn + 1 2 n + 4 n=1

2n 7. a s -1dn + 1 2 n n=1

10 n 8. a s -1dn sn + 1d! n=1

q

q

n 9. a s -1dn + 1 a b 10 n=1 q

q

n

q

1 10. a s -1dn + 1 ln n n=2 q

11. a s -1dn + 1

ln n n

1 12. a s -1dn ln a1 + n b

13. a s -1dn + 1

2n + 1 n + 1

14. a s -1dn + 1

n=1 q n=1

n=1 q n=1

3 2n + 1 2n + 1

Absolute and Conditional Convergence Which of the series in Exercises 15– 48 converge absolutely, which converge, and which diverge? Give reasons for your answers. q q s0.1dn 15. a s -1dn + 1s0.1dn 16. a s -1dn + 1 n n=1 q

17. a s -1dn n=1 q

1 2n

n+1

19. a s -1d n=1 q

n n3 + 1

n=1 q

18. a

s -1dn

1 + 2n n! 20. a s -1dn + 1 n 2 n=1 q

sin n n2

22. a s -1dn

3 + n 23. a s -1dn + 1 5 + n n=1

s -2dn + 1 24. a n n=1 n + 5

q

q

25. a s -1dn + 1 n=1

1 + n n2

n=1 q

q

26. a s -1dn + 1 A 210 B n=1

n

q

n=1 q

-1

1 28. a s -1dn + 1 n ln n n=2 q

tan n 29. a s -1dn 2 n + 1 n=1

ln n 30. a s -1dn n - ln n n=1

n 31. a s -1dn n + 1 n=1

32. a s -5d-n

s -100dn 33. a n! n=1

s -1dn - 1 34. a 2 n = 1 n + 2n + 1

q

q

q

35. a

cos np

n=1 q

n2n s -1dnsn + 1dn 37. a s2ndn n=1 q s2nd! 39. a s -1dn n 2 n!n n=1 q

q

n=1 q

q

36. a

n=1 q

cos np n

s -1dn + 1sn!d2 s2nd! n=1

38. a

q sn!d2 3n 40. a s -1dn s2n + 1d! n=1 q

41. a s -1dn A 2n + 1 - 2n B 42. a s -1dn A 2n 2 + n - n B n=1 q

n=1

43. a s -1dn A 2n + 1n - 2n B n=1 q

44. a

n=1 q

s -1dn 2n + 2n + 1

q

45. a s -1dn sech n n=1

46. a s -1dn csch n n=1

47.

n=1 q

1 21. a s -1dn n + 3 n=1

q

27. a s -1dnn 2s2>3dn

1 1 1 1 1 1 - + + + Á 4 6 8 10 12 14

48. 1 +

1 1 1 1 1 1 1 - + + + Á 4 9 16 25 36 49 64

Error Estimation In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. q

1 49. a s -1dn + 1 n n=1

q

1 50. a s -1dn + 1 n 10 n=1

528

Chapter 9: Infinite Sequences and Series

q

51. a s -1dn + 1 n=1

s0.01dn n

As you will see in Section 9.7, the sum is ln (1.01).

is the same as the sum of the first n terms of the series 1 1 1 1 1 + # + # + # + # + Á. 1#2 2 3 3 4 4 5 5 6

q

52.

1 = a s -1dnt n, 1 + t n=0

0 6 t 6 1

Do these series converge? What is the sum of the first 2n + 1 terms of the first series? If the series converge, what is their sum?

In Exercises 53–56, determine how many terms should be used to estimate the sum of the entire series with an error of less than 0.001. q

q

q

n=1

q

q

A n + 3 2n B 3

1 56. a s -1dn ln sln sn + 2dd n=1

q

` a an ` … a ƒ an ƒ . n=1

q

1

q

64. Show that if g n = 1 an converges absolutely, then

n 54. a s -1dn + 1 2 n + 1 n=1

1 53. a s -1dn 2 n + 3 n=1 55. a s -1dn + 1

q

63. Show that if g n = 1 an diverges, then g n = 1 ƒ an ƒ diverges.

n=1

q g n = 1 an

q gn=1

65. Show that if and so do the following. q

T Approximate the sums in Exercises 57 and 58 with an error of magnitude less than 5 * 10 -6 . q

1 57. a s -1dn s2nd! n=0 q

As you will see in Section 9.9, the sum is cos 1, the cosine of 1 radian. As you will see in Section 9.9, the sum is e -1 .

1 58. a s -1dn n! n=0

Theory and Examples 59. a. The series 1 1 1 1 1 1 1 1 - + - + - + Á + n - n + Á 3 2 9 4 27 8 3 2 does not meet one of the conditions of Theorem 14. Which one? b. Use Theorem 17 to find the sum of the series in part (a). T 60. The limit L of an alternating series that satisfies the conditions of Theorem 14 lies between the values of any two consecutive partial sums. This suggests using the average sn + sn + 1 1 = sn + s -1dn + 2an + 1 2 2

s20 +

1 2

#

1 21

as an approximation to the sum of the alternating harmonic series. The exact sum is ln 2 = 0.69314718 . Á 61. The sign of the remainder of an alternating series that satisfies the conditions of Theorem 14 Prove the assertion in Theorem 15 that whenever an alternating series satisfying the conditions of Theorem 14 is approximated with one of its partial sums, then the remainder (sum of the unused terms) has the same sign as the first unused term. (Hint: Group the remainder’s terms in consecutive pairs.) 62. Show that the sum of the first 2n terms of the series 1 -

q

a. a san + bn d

b. a san - bn d

n=1 q

n=1

c. a kan (k any number) n=1

q

1 1 1 1 1 1 1 1 1 + - + - + - + - + Á 5 5 2 2 3 3 4 4 6

q

66. Show by example that g n = 1 an bn may diverge even if g n = 1 an q and g n = 1 bn both converge. T 67. In the alternating harmonic series, suppose the goal is to arrange the terms to get a new series that converges to -1>2 . Start the new arrangement with the first negative term, which is -1>2 . Whenever you have a sum that is less than or equal to -1>2, start introducing positive terms, taken in order, until the new total is greater than -1>2 . Then add negative terms until the total is less than or equal to -1>2 again. Continue this process until your partial sums have been above the target at least three times and finish at or below it. If sn is the sum of the first n terms of your new series, plot the points sn, sn d to illustrate how the sums are behaving. 68. Outline of the proof of the Rearrangement Theorem (Theorem 17) q

a. Let P be a positive real number, let L = g n = 1 an , and let k

sk = g n = 1 an . Show that for some index N1 and for some index N2 Ú N1 , q

to estimate L. Compute

bn both converge absolutely, then

P a ƒ an ƒ 6 2 n = N1

and

P ƒ sN2 - L ƒ 6 2 .

Since all the terms a1, a2 , Á , aN 2 appear somewhere in the sequence 5bn6 , there is an index N3 Ú N2 such that if n n Ú N3 , then A g k = 1 bk B - sN2 is at most a sum of terms am with m Ú N1 . Therefore, if n Ú N3 , n

n

k=1

k=1 q

` a bk - L ` … ` a bk - sN2 ` + ƒ sN2 - L ƒ … a ƒ ak ƒ + ƒ sN2 - L ƒ 6 P. k = N1

q

b. The argument in part (a) shows that if g n = 1 an converges q q q absolutely then g n = 1 bn converges and g n = 1 bn = g n = 1 an . q q Now show that because g n = 1 ƒ an ƒ converges, g n = 1 ƒ bn ƒ q converges to g n = 1 ƒ an ƒ .

9.7

Power Series

529

Power Series

9.7

Now that we can test many infinite series of numbers for convergence, we can study sums that look like “infinite polynomials.” We call these sums power series because they are defined as infinite series of powers of some variable, in our case x. Like polynomials, power series can be added, subtracted, multiplied, differentiated, and integrated to give new power series. With power series we can extend the methods of calculus we have developed to a vast array of functions making the techniques of calculus applicable in a much wider setting.

Power Series and Convergence We begin with the formal definition, which specifies the notation and terminology used for power series.

A power series about x 0 is a series of the form

DEFINITIONS q

n 2 Á + cn x n + Á . a cn x = c0 + c1 x + c2 x +

n=0

(1)

A power series about x a is a series of the form q

n 2 Á + cnsx - adn + Á a cnsx - ad = c0 + c1sx - ad + c2sx - ad +

n=0

(2)

in which the center a and the coefficients c0, c1, c2, Á , cn, Á are constants.

Equation (1) is the special case obtained by taking a = 0 in Equation (2). We will see that a power series defines a function ƒsxd on a certain interval where it converges. Moreover, this function will be shown to be continuous and differentiable over the interior of that interval.

EXAMPLE 1

Taking all the coefficients to be 1 in Equation (1) gives the geometric

power series q

n 2 Á + xn + Á . ax = 1 + x + x +

n=0

This is the geometric series with first term 1 and ratio x. It converges to 1>s1 - xd for ƒ x ƒ 6 1. We express this fact by writing 1 = 1 + x + x2 + Á + xn + Á , 1 - x

Reciprocal Power Series q

1 = a x n, 1 - x n=0

ƒxƒ 6 1

-1 6 x 6 1.

(3)

Up to now, we have used Equation (3) as a formula for the sum of the series on the right. We now change the focus: We think of the partial sums of the series on the right as polynomials Pnsxd that approximate the function on the left. For values of x near zero, we need take only a few terms of the series to get a good approximation. As we move toward x = 1, or -1, we must take more terms. Figure 9.14 shows the graphs of ƒsxd = 1>s1 - xd and the approximating polynomials yn = Pnsxd for n = 0, 1, 2, and 8. The function ƒsxd = 1>s1 - xd is not continuous on intervals containing x = 1, where it has a vertical asymptote. The approximations do not apply when x Ú 1.

530

Chapter 9: Infinite Sequences and Series y y

9

1 1x

8 7 y8 1 x x 2 x 3 x 4 x 5 x 6 x7 x 8 5 4 y2 1 x x2

3

y1 1 x

2 1 –1

y0 1

0

1

x

FIGURE 9.14 The graphs of ƒsxd = 1>s1 - xd in Example 1 and four of its polynomial approximations.

EXAMPLE 2

The power series n

1 -

1 1 1 sx - 2d + sx - 2d2 + Á + a- b sx - 2dn + Á 2 4 2

(4)

matches Equation (2) with a = 2, c0 = 1, c1 = -1>2, c2 = 1>4, Á , cn = s -1>2dn . This x - 2 . The series converges for is a geometric series with first term 1 and ratio r = 2 x - 2 ` ` 6 1 or 0 6 x 6 4. The sum is 2 1 = 1 - r

1 2 = x, x - 2 1 + 2

so y n sx - 2d sx - 2d2 1 2 + - Á + a- b sx - 2dn + Á , x = 1 2 4 2

2 1

y0 1

0

1

2 y2 3 3x x 2 4 (2, 1) y 2x y1 2 x 2 x 3 2

FIGURE 9.15 The graphs of ƒsxd = 2>x and its first three polynomial approximations (Example 2).

0 6 x 6 4.

Series (4) generates useful polynomial approximations of ƒsxd = 2>x for values of x near 2: P0sxd = 1 P1sxd = 1 -

x 1 sx - 2d = 2 2 2

P2sxd = 1 -

3x x2 1 1 sx - 2d + sx - 2d2 = 3 + , 2 4 2 4

and so on (Figure 9.15). The following example illustrates how we test a power series for convergence by using the Ratio Test to see where it converges and diverges.

EXAMPLE 3 q

For what values of x do the following power series converge?

xn x3 x2 (a) a s -1dn - 1 n = x + - Á 2 3 n=1

9.7

Power Series

531

q

x 2n - 1 x3 x5 (b) a s -1dn - 1 - Á = x + 5 2n - 1 3 n=1 q

xn x2 x3 (c) a = 1 + x + + + Á 2! 3! n = 0 n! q

(d) a n!x n = 1 + x + 2!x 2 + 3!x 3 + Á n=0

Apply the Ratio Test to the series g ƒ un ƒ , where un is the nth term of the power series in question. (Recall that the Ratio Test applies to series with nonnegative terms.)

Solution

un + 1 xn+1 # n n ` = (a) ` un ` = ` x : x. n + 1 x n + 1ƒ ƒ ƒ ƒ The series converges absolutely for ƒ x ƒ 6 1. It diverges if ƒ x ƒ 7 1 because the nth term does not converge to zero. At x = 1, we get the alternating harmonic series 1 - 1>2 + 1>3 - 1>4 + Á , which converges. At x = -1, we get -1 - 1>2 1>3 - 1>4 - Á , the negative of the harmonic series; it diverges. Series (a) converges for -1 6 x … 1 and diverges elsewhere. –1

0

1

x

un + 1 x 2n + 1 # 2n - 1 2n - 1 2 x : x2. (b) ` un ` = ` ` = 2n + 1 x 2n - 1 2n + 1

2sn + 1d - 1 = 2n + 1

The series converges absolutely for x 2 6 1. It diverges for x 2 7 1 because the nth term does not converge to zero. At x = 1 the series becomes 1 - 1>3 + 1>5 - 1>7 + Á , which converges by the Alternating Series Theorem. It also converges at x = -1 because it is again an alternating series that satisfies the conditions for convergence. The value at x = -1 is the negative of the value at x = 1. Series (b) converges for -1 … x … 1 and diverges elsewhere. –1

0

1

x

un + 1 ƒxƒ x n + 1 # n! (c) ` un ` = ` : 0 for every x. n` = n + 1 sn + 1d! x

n! 1#2#3Án = # # Á # sn + 1d! 1 2 3 n sn + 1d

The series converges absolutely for all x. 0

x

sn + 1d!x n + 1 un + 1 ` = sn + 1d ƒ x ƒ : q unless x = 0. (d) ` un ` = ` n!x n The series diverges for all values of x except x = 0. 0

x

The previous example illustrated how a power series might converge. The next result shows that if a power series converges at more than one value, then it converges over an entire interval of values. The interval might be finite or infinite and contain one, both, or none of its endpoints. We will see that each endpoint of a finite interval must be tested independently for convergence or divergence.

532

Chapter 9: Infinite Sequences and Series

THEOREM 18—The Convergence Theorem for Power Series q

If the power series

n 2 Á converges at x = c Z 0, then it converges a an x = a0 + a1 x + a2 x +

n=0

absolutely for all x with ƒ x ƒ 6 ƒ c ƒ . If the series diverges at x = d, then it diverges for all x with ƒ x ƒ 7 ƒ d ƒ .

Proof The proof uses the Comparison Test, with the given series compared to a converging geometric series. q Suppose the series g n = 0 an c n converges. Then limn: q an c n = 0 by the nth-Term Test. Hence, there is an integer N such that ƒ an c n ƒ 6 1 for all n 7 N, so that ƒ an ƒ 6 series diverges

series converges

d R c

0

c

d

for n 7 N.

(5)

Now take any x such that ƒ x ƒ 6 ƒ c ƒ , so that ƒ x ƒ > ƒ c ƒ 6 1. Multiplying both sides of Equation (5) by ƒ x ƒ n gives

series diverges

R

1 n ƒcƒ

x

FIGURE 9.16 Convergence of gan xn at x = c implies absolute convergence on the interval - ƒ c ƒ 6 x 6 ƒ c ƒ ; divergence at x = d implies divergence for ƒ x ƒ 7 ƒ d ƒ . The corollary to Theorem 18 asserts the existence of a radius of convergence R Ú 0. For ƒ x ƒ 6 R the series converges absolutely and for ƒ x ƒ 7 R it diverges.

ƒ an ƒ ƒ x ƒ n 6

ƒ x ƒn ƒ c ƒn

for n 7 N. q

Since ƒ x>c ƒ 6 1, it follows that the geometric series g n = 0 ƒ x>c ƒ n converges. By the Comq parison Test (Theorem 10), the series g n = 0 ƒ an ƒ ƒ x n ƒ converges, so the original power series q g n = 0 an x n converges absolutely for - ƒ c ƒ 6 x 6 ƒ c ƒ as claimed by the theorem. (See Figure 9.16.) q Now suppose that the series g n = 0 an x n diverges at x = d. If x is a number with ƒ x ƒ 7 ƒ d ƒ and the series converges at x, then the first half of the theorem shows that the series also converges at d, contrary to our assumption. So the series diverges for all x with ƒ x ƒ 7 ƒ d ƒ. To simplify the notation, Theorem 18 deals with the convergence of series of the form gan x n . For series of the form gansx - adn we can replace x - a by x¿ and apply the results to the series gansx¿dn .

The Radius of Convergence of a Power Series The theorem we have just proved and the examples we have studied lead to the conclusion that a power series gcnsx - adn behaves in one of three possible ways. It might converge only at x = a, or converge everywhere, or converge on some interval of radius R centered at x = a. We prove this as a Corollary to Theorem 18.

COROLLARY TO THEOREM 18 The convergence of the series gcnsx - adn is described by one of the following three cases: 1. There is a positive number R such that the series diverges for x with ƒ x - a ƒ 7 R but converges absolutely for x with ƒ x - a ƒ 6 R. The series may or may not converge at either of the endpoints x = a - R and x = a + R. 2. The series converges absolutely for every x sR = q d. 3. The series converges at x = a and diverges elsewhere sR = 0d.

9.7

533

Power Series q

Proof We first consider the case where a = 0, so that we have a power series g n = 0 cnx n centered at 0. If the series converges everywhere we are in Case 2. If it converges only at q x = 0 then we are in Case 3. Otherwise there is a nonzero number d such that g n = 0 cnd n q diverges. Let S be the set of values of x for which g n = 0 cnx n converges. The set S does not include any x with ƒ x ƒ 7 ƒ d ƒ , since Theorem 18 implies the series diverges at all such values. So the set S is bounded. By the Completeness Property of the Real Numbers (Appendix 7) S has a least upper bound R. (This is the smallest number with the property that all elements of S are less than or equal to R.) Since we are not in Case 3, the series converges at some number b Z 0 and, by Theorem 18, also on the open interval s - ƒ b ƒ , ƒ b ƒ d. Therefore R 7 0. If ƒ x ƒ 6 R then there is a number c in S with ƒ x ƒ 6 c 6 R, since otherwise R would not be the least upper bound for S. The series converges at c since c H S, so by Theorem 18 the series converges absolutely at x. Now suppose ƒ x ƒ 7 R. If the series converges at x, then Theorem 18 implies it converges absolutely on the open interval s - ƒ x ƒ , ƒ x ƒ d, so that S contains this interval. Since R is an upper bound for S, it follows that ƒ x ƒ … R, which is a contradiction. So if ƒ x ƒ 7 R then the series diverges. This proves the theorem for power series centered at a = 0. For a power series centered at an arbitrary point x = a, set x¿ = x - a and repeat the argument above, replacing x with x¿ . Since x¿ = 0 when x = a, convergence of the series q g n = 0 ƒ cnsx¿d ƒ n on a radius R open interval centered at x¿ = 0 corresponds to convergence q of the series g n = 0 ƒ cnsx - ad ƒ n on a radius R open interval centered at x = a. R is called the radius of convergence of the power series, and the interval of radius R centered at x = a is called the interval of convergence. The interval of convergence may be open, closed, or half-open, depending on the particular series. At points x with ƒ x - a ƒ 6 R, the series converges absolutely. If the series converges for all values of x, we say its radius of convergence is infinite. If it converges only at x = a, we say its radius of convergence is zero.

How to Test a Power Series for Convergence 1. Use the Ratio Test (or Root Test) to find the interval where the series converges absolutely. Ordinarily, this is an open interval ƒx - aƒ 6 R

or

a - R 6 x 6 a + R.

2. If the interval of absolute convergence is finite, test for convergence or divergence at each endpoint, as in Examples 3a and b. Use a Comparison Test, the Integral Test, or the Alternating Series Test. 3. If the interval of absolute convergence is a - R 6 x 6 a + R, the series diverges for ƒ x - a ƒ 7 R (it does not even converge conditionally) because the nth term does not approach zero for those values of x.

Operations on Power Series On the intersection of their intervals of convergence, two power series can be added and subtracted term by term just like series of constants (Theorem 8). They can be multiplied just as we multiply polynomials, but we often limit the computation of the product to the first few terms, which are the most important. The following result gives a formula for the coefficients in the product, but we omit the proof. (Power series can also be divided in a way similar to division of polynomials, but we do not give a formula for the general coefficient here.)

534

Chapter 9: Infinite Sequences and Series

THEOREM 19—The Series Multiplication Theorem for Power Series If q q Asxd = g n = 0 an x n and Bsxd = g n = 0 bn x n converge absolutely for ƒ x ƒ 6 R, and n

cn = a0 bn + a1 bn - 1 + a2 bn - 2 + Á + an - 1b1 + an b0 = a ak bn - k , k=0

then

q g n = 0 cn x n

converges absolutely to A(x)B(x) for ƒ x ƒ 6 R: q

q

q

n=0

n=0

n=0

a a an x n b # a a bn x n b = a cn x n .

Finding the general coefficient cn in the product of two power series can be very tedious and the term may be unwieldy. The following computation provides an illustration of a product where we find the first few terms by multiplying the terms of the second series by each term of the first series: q

q

xn+1 a a x n b # a a s -1dn b n + 1 n=0 n=0 x2 x3 = s1 + x + x 2 + Á d ax + - Áb 2 3 = ax -

x2 x3 x4 x3 x4 x5 + - Á b + ax 2 + - Á b + ax 3 + - Áb + Á 2 3 2 3 2 3

('''')''''* by 1

= x +

Multiply second series . . .

('''')''''* by x

5x 3 x4 Á x2 + . 2 6 6

('''')''''* by x 2

and gather the first four powers.

We can also substitute a function ƒsxd for x in a convergent power series. q

THEOREM 20 If g n = 0 an x n converges absolutely for ƒ x ƒ 6 R, then q g n = 0 an sƒsxddn converges absolutely for any continuous function ƒ on ƒ ƒsxd ƒ 6 R. q

Since 1>s1 - xd = g n = 0 x n converges absolutely for ƒ x ƒ 6 1, it follows from Theoq rem 20 that 1>s1 - 4x 2 d = g n = 0 s4x 2 dn converges absolutely for ƒ 4x 2 ƒ 6 1 or ƒ x ƒ 6 1>2. A theorem from advanced calculus says that a power series can be differentiated term by term at each interior point of its interval of convergence.

THEOREM 21—The Term-by-Term Differentiation Theorem radius of convergence R 7 0, it defines a function

If gcnsx - adn has

q

ƒsxd = a cnsx - adn n=0

on the interval

a - R 6 x 6 a + R.

This function ƒ has derivatives of all orders inside the interval, and we obtain the derivatives by differentiating the original series term by term: q

ƒ¿sxd = a ncnsx - adn - 1, n=1 q

ƒ–sxd = a nsn - 1dcnsx - adn - 2 , n=2

and so on. Each of these derived series converges at every point of the interval a - R 6 x 6 a + R.

9.7

EXAMPLE 4

Power Series

535

Find series for ƒ¿sxd and ƒ–sxd if ƒsxd =

1 = 1 + x + x2 + x3 + x4 + Á + xn + Á 1 - x q

= a xn,

-1 6 x 6 1.

n=0

Solution

We differentiate the power series on the right term by term: ƒ¿sxd =

1 = 1 + 2x + 3x 2 + 4x 3 + Á + nx n - 1 + Á s1 - xd2 q

= a nx n - 1, n=1

ƒ–sxd =

-1 6 x 6 1;

2 = 2 + 6x + 12x 2 + Á + nsn - 1dx n - 2 + Á s1 - xd3 q

= a nsn - 1dxn - 2,

-1 6 x 6 1.

n=2

Caution Term-by-term differentiation might not work for other kinds of series. For example, the trigonometric series q sin sn!xd a n2 n=1 converges for all x. But if we differentiate term by term we get the series q n!cos sn!xd , a n2 n=1 which diverges for all x. This is not a power series since it is not a sum of positive integer powers of x. It is also true that a power series can be integrated term by term throughout its interval of convergence. This result is proved in a more advanced course.

THEOREM 22—The Term-by-Term Integration Theorem

Suppose that

q

ƒsxd = a cnsx - adn n=0

converges for a - R 6 x 6 a + R sR 7 0d. Then q

a cn

n=0

(x - a) n+1 n + 1

converges for a - R 6 x 6 a + R and q

L

ƒsxd dx = a cn n=0

sx - adn + 1 + C n + 1

for a - R 6 x 6 a + R.

EXAMPLE 5

Identify the function q

ƒsxd = a

n=0

s -1dn x 2n + 1 x3 x5 = x + - Á, 5 2n + 1 3

-1 … x … 1.

536

Chapter 9: Infinite Sequences and Series Solution

We differentiate the original series term by term and get ƒ¿sxd = 1 - x 2 + x 4 - x 6 + Á , -1 6 x 6 1.

Theorem 21

This is a geometric series with first term 1 and ratio -x , so 2

1 1 = . 2 1 - s -x d 1 + x2

ƒ¿sxd =

We can now integrate ƒ¿sxd = 1>s1 + x 2 d to get L

dx = tan-1 x + C. 2 L1 + x

ƒ¿sxd dx =

The series for ƒ(x) is zero when x = 0, so C = 0. Hence ƒsxd = x -

The Number P as a Series q s -1dn p = tan-1 1 = a 4 n = 0 2n + 1

x3 x5 x7 + + Á = tan-1 x, 5 7 3

-1 6 x 6 1.

(6)

It can be shown that the series also converges to tan-1 x at the endpoints x = ;1, but we omit the proof. Notice that the original series in Example 5 converges at both endpoints of the original interval of convergence, but Theorem 22 can guarantee the convergence of the differentiated series only inside the interval.

EXAMPLE 6

The series 1 = 1 - t + t2 - t3 + Á 1 + t

converges on the open interval -1 6 t 6 1. Therefore, x

x

t2 t3 t4 1 dt = t + + Ád 2 3 4 0 L0 1 + t 2 3 4 x x x = x + + Á 2 3 4

ln s1 + xd =

or

q

Alternating Harmonic Series Sum

ln s1 + xd = a

q

n=1

s -1dn - 1 ln 2 = a n

s -1dn - 1 x n , n

Theorem 22

-1 6 x 6 1.

It can also be shown that the series converges at x = 1 to the number ln 2, but that was not guaranteed by the theorem.

n=1

Exercises 9.7 Intervals of Convergence In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally? q

1. a x

q

2. a sx + 5d

n

n

n=0

n=0

q

q

3. a s -1dns4x + 1dn n=0 q

sx - 2d 10 n n=0

5. a

n

4. a

n=1

s3x - 2dn n

q

6. a s2xdn n=0

q

nx n 7. a n=0 n + 2 q

9. a

x

n

n2n 3n s -1dnx n 11. a n! n=0 n=1 q

q

13. a

n=1 q

15. a

n=0

4nx 2n n xn 2n 2 + 3

q

8. a

n=1 q

10. a

s -1dnsx + 2dn n sx - 1dn

2n 3nx n 12. a n = 0 n! n=1 q

q

14. a

n=1 q

16. a

n=0

sx - 1dn n 3 3n s -1dnx n + 1 2n + 3

9.7 q

q

nsx + 3dn 5n n=0

q

47. a a

nx n 18. a n 2 n = 0 4 sn + 1d

17. a q

q

2nx n 19. a 3n n=0

n

n=0

20. a 2ns2x + 5d

n

n=1

q

21. a s2 + s -1dn d # sx + 1dn - 1 s -1dn 32nsx - 2dn 3n n=1

q

n

1 23. a a1 + n b x n

24. a sln ndx n

25. a n nx n

26. a n!sx - 4dn

s -1dn + 1sx + 2dn 27. a n2n n=1

28. a s -2dnsn + 1dsx - 1dn

n=1 q

Exercise 54. 2n + 1

s4x - 5d

n=1 q

n=0

Get the information you need about a 1>(n ln n) from Section 9.3,

q

31. a

n=0 q

2 a 1>(n(ln n) ) from Section 9.3, Exercise 55.

xn 30. a n = 2 n ln n q

n=1 q

Get the information you need about

q

xn 29. a 2 n = 2 nsln nd

n 3>2

48. a a n=0

x2 - 1 b 2

n

n

22. a

n=1 q

q

n

537

Theory and Examples 49. For what values of x does the series 1 -

n=1 q

q

x2 + 1 b 3

Power Series

q

s3x + 1dn + 1 32. a 2n + 2 n=1

1 xn 33. a # # Á 2 4 8 s2nd n=1 q 3 # 5 # 7 Á s2n + 1d n + 1 x 34. a n 2 # 2n

1 1 1 sx - 3d + sx - 3d2 + Á + a- b sx - 3dn + Á 2 4 2

converge? What is its sum? What series do you get if you differentiate the given series term by term? For what values of x does the new series converge? What is its sum? 50. If you integrate the series in Exercise 49 term by term, what new series do you get? For what values of x does the new series converge, and what is another name for its sum? 51. The series sin x = x -

x3 x5 x7 x9 x 11 + + + Á 3! 5! 7! 9! 11!

converges to sin x for all x. a. Find the first six terms of a series for cos x. For what values of x should the series converge? b. By replacing x by 2x in the series for sin x, find a series that converges to sin 2x for all x. c. Using the result in part (a) and series multiplication, calculate the first six terms of a series for 2 sin x cos x. Compare your answer with the answer in part (b). 52. The series ex = 1 + x +

x2 x3 x4 x5 + + + + Á 2! 3! 4! 5!

converges to e x for all x.

n=1 q

a. Find a series for sd>dxde x . Do you get the series for e x ? Explain your answer.

1 + 2 + 3 + Á + n xn 35. a 2 2 2 2 n=1 1 + 2 + 3 + Á + n 36. a A 2n + 1 - 2n B sx - 3d

b. Find a series for 1 e x dx . Do you get the series for e x ? Explain your answer.

In Exercises 37–40, find the series’ radius of convergence.

c. Replace x by -x in the series for e x to find a series that converges to e -x for all x. Then multiply the series for e x and e -x to find the first six terms of a series for e -x # e x .

q

n

n=1

q

n! xn 37. a # # Á 3n n=1 3 6 9 q 2 2 # 4 # 6 Á s2nd b xn 38. a a # # Á s3n - 1d n=1 2 5 8

53. The series tan x = x +

q sn!d2 n x 39. a n n = 1 2 s2nd! q

converges to tan x for -p>2 6 x 6 p>2 . a. Find the first five terms of the series for ln ƒ sec x ƒ . For what values of x should the series converge?

n2

n b xn 40. a a n=1 n + 1

b. Find the first five terms of the series for sec2 x . For what values of x should this series converge?

(Hint: Apply the Root Test.) In Exercises 41–48, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x. q

42. a se x - 4dn

sx - 1d2n 43. a 4n n=0

44. a

n 2x - 1b 45. a a 2 n=0

46. a sln xdn

q

c. Check your result in part (b) by squaring the series given for sec x in Exercise 54. 54. The series

q

41. a 3nx n n=0 q

17x 7 62x 9 x3 2x 5 + + + + Á 3 15 315 2835

n=0 q

sx + 1d2n 9n n=0 q

n=0

sec x = 1 +

61 6 277 8 Á 5 4 x2 + x + x + x + 2 24 720 8064

converges to sec x for -p>2 6 x 6 p>2 . a. Find the first five terms of a power series for the function ln ƒ sec x + tan x ƒ . For what values of x should the series converge?

538

Chapter 9: Infinite Sequences and Series q

b. Find the first four terms of a series for sec x tan x. For what values of x should the series converge?

b. Show that if g n = 0 an x n = 0 for all x in an open interval s -c, cd, then an = 0 for every n.

56. The sum of the series g n = 0 sn2>2n d To find the sum of this series, express 1>s1 - xd as a geometric series, differentiate both sides of the resulting equation with respect to x, multiply both sides of the result by x, differentiate again, multiply by x again, and set x equal to 1> 2. What do you get? q

c. Check your result in part (b) by multiplying the series for sec x by the series given for tan x in Exercise 53. 55. Uniqueness of convergent power series q

q

a. Show that if two power series g n = 0 an x n and g n = 0 bn x n are convergent and equal for all values of x in an open interval s -c, cd, then an = bn for every n. (Hint: Let q q ƒsxd = g n = 0 an x n = g n = 0 bn x n . Differentiate term by term to show that an and bn both equal ƒ snds0d>sn!d .)

9.8

Taylor and Maclaurin Series This section shows how functions that are infinitely differentiable generate power series called Taylor series. In many cases, these series can provide useful polynomial approximations of the generating functions. Because they are used routinely by mathematicians and scientists, Taylor series are considered one of the most important topics of this chapter.

Series Representations We know from Theorem 21 that within its interval of convergence the sum of a power series is a continuous function with derivatives of all orders. But what about the other way around? If a function ƒ(x) has derivatives of all orders on an interval I, can it be expressed as a power series on I? And if it can, what will its coefficients be? We can answer the last question readily if we assume that ƒ(x) is the sum of a power series q

ƒsxd = a ansx - adn n=0

= a0 + a1sx - ad + a2sx - ad2 + Á + ansx - adn + Á with a positive radius of convergence. By repeated term-by-term differentiation within the interval of convergence I, we obtain ƒ¿sxd = a1 + 2a2sx - ad + 3a3sx - ad2 + Á + nansx - adn - 1 + Á ,

ƒ–sxd = 1 # 2a2 + 2 # 3a3sx - ad + 3 # 4a4sx - ad2 + Á ,

ƒ‡sxd = 1 # 2 # 3a3 + 2 # 3 # 4a4sx - ad + 3 # 4 # 5a5sx - ad2 + Á , with the nth derivative, for all n, being ƒsndsxd = n!an + a sum of terms with sx - ad as a factor. Since these equations all hold at x = a, we have ƒ¿sad = a1,

ƒ–sad = 1 # 2a2,

ƒ‡sad = 1 # 2 # 3a3,

and, in general, ƒsndsad = n!an . q

These formulas reveal a pattern in the coefficients of any power series g n = 0 ansx - adn that converges to the values of ƒ on I (“represents ƒ on I”). If there is such a series (still an open question), then there is only one such series, and its nth coefficient is an =

ƒsndsad . n!

9.8

Taylor and Maclaurin Series

539

If ƒ has a series representation, then the series must be ƒsxd = ƒsad + ƒ¿sadsx - ad + + Á +

ƒ–sad sx - ad2 2!

ƒsndsad sx - adn + Á . n!

(1)

But if we start with an arbitrary function ƒ that is infinitely differentiable on an interval I centered at x = a and use it to generate the series in Equation (1), will the series then converge to ƒ(x) at each x in the interior of I? The answer is maybe—for some functions it will but for other functions it will not, as we will see.

Taylor and Maclaurin Series The series on the right-hand side of Equation (1) is the most important and useful series we will study in this chapter.

HISTORICAL BIOGRAPHIES Brook Taylor (1685–1731) Colin Maclaurin (1698–1746)

DEFINITIONS Let ƒ be a function with derivatives of all orders throughout some interval containing a as an interior point. Then the Taylor series generated by ƒ at x = a is q

a

k=0

ƒ–sad ƒ skdsad sx - ad2 sx - adk = ƒsad + ƒ¿sadsx - ad + 2! k! + Á +

ƒ sndsad sx - adn + Á . n!

The Maclaurin series generated by ƒ is q

ƒ–s0d 2 ƒsnds0d n ƒ skds0d k Á + = ƒs0d + ƒ¿s0dx + + x x x + Á, a k! 2! n! k=0 the Taylor series generated by ƒ at x = 0.

The Maclaurin series generated by ƒ is often just called the Taylor series of ƒ. Find the Taylor series generated by ƒsxd = 1>x at a = 2. Where, if anywhere, does the series converge to 1> x?

EXAMPLE 1 Solution

We need to find ƒs2d, ƒ¿s2d, ƒ–s2d, Á . Taking derivatives we get

ƒsxd = x -1,

ƒ¿sxd = -x -2,

ƒ–sxd = 2!x -3,

Á,

ƒsndsxd = s -1dnn!x -sn + 1d,

so that ƒs2d = 2-1 =

1 , 2

ƒ¿s2d = -

1 , 22

ƒ–s2d 1 = 2-3 = 3 , 2! 2

Á,

ƒsnds2d s -1dn = n+1 . n! 2

The Taylor series is ƒs2d + ƒ¿s2dsx - 2d + =

ƒ–s2d ƒsnds2d sx - 2d2 + Á + sx - 2dn + Á 2! n!

n sx - 2d2 sx - 2d 1 Á + s -1dn sx - 2d + Á . + 2 22 23 2n + 1

540

Chapter 9: Infinite Sequences and Series

This is a geometric series with first term 1> 2 and ratio r = -sx - 2d>2. It converges absolutely for ƒ x - 2 ƒ 6 2 and its sum is 1>2 1 1 = = x. 1 + sx - 2d>2 2 + sx - 2d In this example the Taylor series generated by ƒsxd = 1>x at a = 2 converges to 1> x for ƒ x - 2 ƒ 6 2 or 0 6 x 6 4.

Taylor Polynomials The linearization of a differentiable function ƒ at a point a is the polynomial of degree one given by P1sxd = ƒsad + ƒ¿sadsx - ad. In Section 3.11 we used this linearization to approximate ƒ(x) at values of x near a. If ƒ has derivatives of higher order at a, then it has higher-order polynomial approximations as well, one for each available derivative. These polynomials are called the Taylor polynomials of ƒ.

DEFINITION Let ƒ be a function with derivatives of order k for k = 1, 2, Á , N in some interval containing a as an interior point. Then for any integer n from 0 through N, the Taylor polynomial of order n generated by ƒ at x = a is the polynomial Pnsxd = ƒsad + ƒ¿sadsx - ad + y

+

3.0

ƒskdsad ƒsndsad sx - adk + Á + sx - adn . n! k!

y e x y P3(x)

1.5

We speak of a Taylor polynomial of order n rather than degree n because ƒsndsad may be zero. The first two Taylor polynomials of ƒsxd = cos x at x = 0, for example, are P0sxd = 1 and P1sxd = 1. The first-order Taylor polynomial has degree zero, not one. Just as the linearization of ƒ at x = a provides the best linear approximation of ƒ in the neighborhood of a, the higher-order Taylor polynomials provide the “best” polynomial approximations of their respective degrees. (See Exercise 40.)

1.0

EXAMPLE 2

y P2(x)

2.5

2.0

y P1(x)

at x = 0. 0.5

–0.5

ƒ–sad sx - ad2 + Á 2!

Find the Taylor series and the Taylor polynomials generated by ƒsxd = e x

Since ƒsndsxd = e x and ƒsnds0d = 1 for every n = 0, 1, 2, Á , the Taylor series generated by ƒ at x = 0 (see Figure 9.17) is

Solution

0

0.5

1.0

x

FIGURE 9.17 The graph of ƒsxd = e x and its Taylor polynomials P1sxd = 1 + x P2sxd = 1 + x + sx 2>2!d P3sxd = 1 + x + sx 2>2!d + sx 3>3!d. Notice the very close agreement near the center x = 0 (Example 2).

ƒs0d + ƒ¿s0dx +

ƒsnds0d n ƒ–s0d 2 x + Á + x + Á 2! n! = 1 + x +

x2 xn + Á + + Á 2 n!

q

xk = a . k = 0 k! This is also the Maclaurin series for e x . In the next section we will see that the series converges to e x at every x.

9.8

Taylor and Maclaurin Series

541

The Taylor polynomial of order n at x = 0 is Pnsxd = 1 + x +

EXAMPLE 3 at x = 0. Solution

x2 xn + Á + . 2 n!

Find the Taylor series and Taylor polynomials generated by ƒsxd = cos x

The cosine and its derivatives are cos x,

ƒ¿sxd =

-sin x,

ƒ–sxd = -cos x, o ƒs2ndsxd = s -1dn cos x,

ƒ sxd =

sin x,

ƒsxd =

s3d

o ƒs2n + 1dsxd = s -1dn + 1 sin x.

At x = 0, the cosines are 1 and the sines are 0, so ƒs2n + 1ds0d = 0.

ƒs2nds0d = s -1dn, The Taylor series generated by ƒ at 0 is

ƒ–s0d 2 ƒ‡s0d 3 ƒsnds0d n x + x + Á + x + Á 2! 3! n! x2 x4 x 2n + 0 # x3 + + Á + s -1dn = 1 + 0#x + Á 2! 4! s2nd!

ƒs0d + ƒ¿s0dx +

q

= a

k=0

s -1dkx 2k . s2kd!

This is also the Maclaurin series for cos x. Notice that only even powers of x occur in the Taylor series generated by the cosine function, which is consistent with the fact that it is an even function. In Section 9.9, we will see that the series converges to cos x at every x. Because ƒs2n + 1ds0d = 0, the Taylor polynomials of orders 2n and 2n + 1 are identical: P2nsxd = P2n + 1sxd = 1 -

x4 x2 x 2n + - Á + s -1dn . 2! 4! s2nd!

Figure 9.18 shows how well these polynomials approximate ƒsxd = cos x near x = 0. Only the right-hand portions of the graphs are given because the graphs are symmetric about the y-axis. y 2 1

P4

P0

P8

P12

P16 y cos x

0

1

2

3

4

5

6

7

8

9

–1 P2

P6

P10

P14

P18

–2

FIGURE 9.18

The polynomials n

P2nsxd = a

k=0

s -1dkx 2k s2kd!

converge to cos x as n : q . We can deduce the behavior of cos x arbitrarily far away solely from knowing the values of the cosine and its derivatives at x = 0 (Example 3).

x

542

Chapter 9: Infinite Sequences and Series

EXAMPLE 4 y

It can be shown (though not easily) that

x0

0,

ƒsxd = e

2

e–1/x , x 0

1

x = 0 x Z 0

0, 2 e -1>x ,

(Figure 9.19) has derivatives of all orders at x = 0 and that ƒsnds0d = 0 for all n. This means that the Taylor series generated by ƒ at x = 0 is –2

–1

0

1

x

2

FIGURE 9.19 The graph of the 2 continuous extension of y = e -1>x is so flat at the origin that all of its derivatives there are zero (Example 4). Therefore its Taylor series is not the function itself.

ƒs0d + ƒ¿s0dx +

ƒ–s0d 2 ƒsnds0d n x + Á + x + Á 2! n!

= 0 + 0 # x + 0 # x2 + Á + 0 # xn + Á = 0 + 0 + Á + 0 + Á.

The series converges for every x (its sum is 0) but converges to ƒ(x) only at x = 0. That is, the Taylor series generated by ƒsxd in this example is not equal to the function ƒsxd itself. Two questions still remain. 1. 2.

For what values of x can we normally expect a Taylor series to converge to its generating function? How accurately do a function’s Taylor polynomials approximate the function on a given interval?

The answers are provided by a theorem of Taylor in the next section.

Exercises 9.8 Finding Taylor Polynomials In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by ƒ at a. 1. ƒsxd = e 2x,

a = 0

2. ƒsxd = sin x,

3. ƒsxd = ln x,

a = 1

4. ƒsxd = ln s1 + xd,

a = 0

5. ƒsxd = 1>x,

a = 2

6. ƒsxd = 1>sx + 2d,

a = 0

7. ƒsxd = sin x,

a = p>4

9. ƒsxd = 2x,

a = 4

8. ƒsxd = tan x,

a = 0

a = p>4

10. ƒsxd = 21 - x,

a = 0

Finding Taylor Series at x 0 (Maclaurin Series) Find the Maclaurin series for the functions in Exercises 11–22. 11. e -x 13.

12. xe x 14.

15. sin 3x 17. 7 cos s -xd

18. 5 cos px

e x + e -x 19. cosh x = 2

20. sinh x =

21. x 4 - 2x 3 - 5x + 4

22.

24. ƒsxd = 2x 3 + x 2 + 3x - 8,

a = 1

a = 1

28. ƒsxd = 1>s1 - xd3, 29. ƒsxd = e ,

a = 2

30. ƒsxd = 2x,

a = 1

x

a = -1

a = 0

31. ƒsxd = cos s2x + (p>2)d, 32. ƒsxd = 2x + 1,

a = p>4

a = 0

In Exercises 33–36, find the first three nonzero terms of the Maclaurin series for each function and the values of x for which the series converges absolutely.

35. ƒsxd = (sin x) ln (1 + x) 36. ƒsxd = x sin2 x e x - e -x 2

x2 x + 1

a = 2

27. ƒsxd = 1>x , 2

34. ƒsxd = s1 - x + x 2 d e x

Finding Taylor and Maclaurin Series In Exercises 23–32, find the Taylor series generated by ƒ at x = a . 23. ƒsxd = x 3 - 2x + 4,

a = -2

26. ƒsxd = 3x 5 - x 4 + 2x 3 + x 2 - 2,

33. ƒsxd = cos x - s2>s1 - xdd

2 + x 1 - x x 16. sin 2

1 1 + x

25. ƒsxd = x 4 + x 2 + 1,

Theory and Examples 37. Use the Taylor series generated by e x at x = a to show that e x = e a c1 + sx - ad +

sx - ad2 + Á d. 2!

38. (Continuation of Exercise 37.) Find the Taylor series generated by e x at x = 1 . Compare your answer with the formula in Exercise 37. 39. Let ƒ(x) have derivatives through order n at x = a . Show that the Taylor polynomial of order n and its first n derivatives have the same values that ƒ and its first n derivatives have at x = a .

9.9 40. Approximation properties of Taylor polynomials Suppose that ƒ(x) is differentiable on an interval centered at x = a and that gsxd = b0 + b1sx - ad + Á + bnsx - adn is a polynomial of degree n with constant coefficients b0, Á , bn . Let Esxd = ƒsxd - gsxd . Show that if we impose on g the conditions i) Esad = 0 ii) lim

x: a

The approximation error is zero at x = a .

Esxd = 0, sx - adn

The error is negligible when compared to sx - adn .

gsxd = ƒsad + ƒ¿sadsx - ad +

+

9.9

Quadratic Approximations The Taylor polynomial of order 2 generated by a twice-differentiable function ƒ(x) at x = a is called the quadratic approximation of ƒ at x = a . In Exercises 41–46, find the (a) linearization (Taylor polynomial of order 1) and (b) quadratic approximation of ƒ at x = 0 . 43. ƒsxd = 1> 21 - x

ƒ–sad sx - ad2 + Á 2!

543

Thus, the Taylor polynomial Pnsxd is the only polynomial of degree less than or equal to n whose error is both zero at x = a and negligible when compared with sx - adn .

41. ƒsxd = ln scos xd

then

Convergence of Taylor Series

42. ƒsxd = e sin x 2

45. ƒsxd = sin x

44. ƒsxd = cosh x 46. ƒsxd = tan x

snd

ƒ sad sx - adn . n!

Convergence of Taylor Series In the last section we asked when a Taylor series for a function can be expected to converge to that (generating) function. We answer the question in this section with the following theorem.

THEOREM 23—Taylor’s Theorem If ƒ and its first n derivatives ƒ¿, ƒ–, Á , ƒsnd are continuous on the closed interval between a and b, and ƒsnd is differentiable on the open interval between a and b, then there exists a number c between a and b such that ƒsbd = ƒsad + ƒ¿sadsb - ad + +

ƒ–sad sb - ad2 + Á 2!

ƒsndsad ƒsn + 1dscd sb - adn + sb - adn + 1 . n! sn + 1d!

Taylor’s Theorem is a generalization of the Mean Value Theorem (Exercise 45). There is a proof of Taylor’s Theorem at the end of this section. When we apply Taylor’s Theorem, we usually want to hold a fixed and treat b as an independent variable. Taylor’s formula is easier to use in circumstances like these if we change b to x. Here is a version of the theorem with this change.

Taylor’s Formula If ƒ has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I, ƒsxd = ƒsad + ƒ¿sadsx - ad + +

ƒ–sad sx - ad2 + Á 2!

ƒsndsad sx - adn + Rnsxd, n!

(1)

where Rnsxd =

f sn + 1dscd sx - adn + 1 sn + 1d!

for some c between a and x.

(2)

544

Chapter 9: Infinite Sequences and Series

When we state Taylor’s theorem this way, it says that for each x H I, ƒsxd = Pnsxd + Rnsxd. The function Rnsxd is determined by the value of the sn + 1dst derivative ƒsn + 1d at a point c that depends on both a and x, and that lies somewhere between them. For any value of n we want, the equation gives both a polynomial approximation of ƒ of that order and a formula for the error involved in using that approximation over the interval I. Equation (1) is called Taylor’s formula. The function Rnsxd is called the remainder of order n or the error term for the approximation of ƒ by Pnsxd over I. If Rnsxd : 0 as n : q for all x H I, we say that the Taylor series generated by ƒ at x = a converges to ƒ on I, and we write q

ƒsxd = a

k=0

ƒskdsad sx - adk . k!

Often we can estimate Rn without knowing the value of c, as the following example illustrates. Show that the Taylor series generated by ƒsxd = e x at x = 0 converges to ƒ(x) for every real value of x.

EXAMPLE 1

The function has derivatives of all orders throughout the interval I = s - q , q d. Equations (1) and (2) with ƒsxd = e x and a = 0 give Solution

ex = 1 + x +

xn x2 + Á + + Rnsxd 2! n!

Polynomial from Section 9.8, Example 2

and Rnsxd =

ec xn+1 sn + 1d!

for some c between 0 and x.

Since e x is an increasing function of x, e c lies between e 0 = 1 and e x . When x is negative, so is c, and e c 6 1. When x is zero, e x = 1 and Rnsxd = 0. When x is positive, so is c, and e c 6 e x . Thus, for Rn(x) given as above, ƒ Rnsxd ƒ …

ƒ x ƒn+1 sn + 1d!

when x … 0,

ec 6 1

and ƒ Rnsxd ƒ 6 e x

xn+1 sn + 1d!

when x 7 0.

ec 6 ex

Finally, because xn+1 = 0 n: q sn + 1d! lim

for every x,

Section 9.1, Theorem 5

lim Rnsxd = 0, and the series converges to e x for every x. Thus,

n: q

q

xk xk x2 ex = a + Á + = 1 + x + + Á. 2! k! k = 0 k! The Number e as a Series q

1 e = a n! n=0

We can use the result of Example 1 with x = 1 to write e = 1 + 1 +

1 1 + Á + + Rns1d, 2! n!

(3)

9.9

Convergence of Taylor Series

545

where for some c between 0 and 1, Rns1d = e c

3 1 6 . sn + 1d! sn + 1d!

ec 6 e1 6 3

Estimating the Remainder It is often possible to estimate Rnsxd as we did in Example 1. This method of estimation is so convenient that we state it as a theorem for future reference.

THEOREM 24—The Remainder Estimation Theorem If there is a positive constant M such that ƒ ƒsn + 1dstd ƒ … M for all t between x and a, inclusive, then the remainder term Rnsxd in Taylor’s Theorem satisfies the inequality ƒ Rnsxd ƒ … M

ƒ x - a ƒ n+1 . sn + 1d!

If this inequality holds for every n and the other conditions of Taylor’s Theorem are satisfied by ƒ, then the series converges to ƒ(x).

The next two examples use Theorem 24 to show that the Taylor series generated by the sine and cosine functions do in fact converge to the functions themselves.

EXAMPLE 2 Solution

Show that the Taylor series for sin x at x = 0 converges for all x.

The function and its derivatives are ƒsxd =

sin x,

ƒ¿sxd =

cos x,

ƒ–sxd =

- sin x,

ƒ‡sxd =

- cos x,

o ƒ

o (2k + 1)

sxd = s -1d sin x,

(2k)

k

ƒ

sxd = s -1dk cos x,

so f s2kds0d = 0

and

f s2k + 1ds0d = s -1dk .

The series has only odd-powered terms and, for n = 2k + 1, Taylor’s Theorem gives sin x = x -

s -1dkx 2k + 1 x3 x5 + R2k + 1sxd. + - Á + 3! 5! s2k + 1d!

All the derivatives of sin x have absolute values less than or equal to 1, so we can apply the Remainder Estimation Theorem with M = 1 to obtain ƒ R2k + 1sxd ƒ … 1 #

ƒ x ƒ 2k + 2 . s2k + 2d!

From Theorem 5, Rule 6, we have s ƒ x ƒ 2k + 2>s2k + 2d!d : 0 as k : q , whatever the value of x, so R2k + 1sxd : 0 and the Maclaurin series for sin x converges to sin x for every x. Thus, q s -1dkx 2k + 1 x3 x5 x7 sin x = a = x + + Á. 3! 5! 7! k = 0 s2k + 1d!

(4)

546

Chapter 9: Infinite Sequences and Series

Show that the Taylor series for cos x at x = 0 converges to cos x for every

EXAMPLE 3 value of x.

We add the remainder term to the Taylor polynomial for cos x (Section 9.8, Example 3) to obtain Taylor’s formula for cos x with n = 2k:

Solution

cos x = 1 -

x2 x 2k x4 + R2ksxd. + - Á + s -1dk 2! 4! s2kd!

Because the derivatives of the cosine have absolute value less than or equal to 1, the Remainder Estimation Theorem with M = 1 gives ƒ R2ksxd ƒ … 1 #

ƒ x ƒ 2k + 1 . s2k + 1d!

For every value of x, R2k (x) : 0 as k : q . Therefore, the series converges to cos x for every value of x. Thus, q

cos x = a

k=0

s -1dkx 2k x4 x6 x2 + + Á. = 1 2! 4! 6! s2kd!

(5)

Using Taylor Series Since every Taylor series is a power series, the operations of adding, subtracting, and multiplying Taylor series are all valid on the intersection of their intervals of convergence.

EXAMPLE 4

Using known series, find the first few terms of the Taylor series for the given function using power series operations. (a)

1 (2x + x cos x) 3

(b) e x cos x

Solution

(a)

x 2k x2 x4 1 2 1 (2x + x cos x) = x + x a1 + Áb + - Á + (-1) k 3 3 3 2! 4! (2k)! =

x3 x5 x5 x3 2 1 x + x + - Á + # - Á = x 3 3 6 72 3! 3 4!

(b) e x cos x = a1 + x + = a1 + x +

x2 x3 x4 x2 x4 + + + Á b # a1 + - Áb 2! 3! 4! 2! 4!

x2 x3 x4 x2 x3 x4 x5 + + + Áb - a + + + + Áb 2! 3! 4! 2! 2! 2!2! 2!3! + a

= 1 + x -

Multiply the first series by each term of the second series.

x4 x5 x6 + + + Áb + Á 4! 4! 2!4!

x3 x4 + Á 3 6

By Theorem 20, we can use the Taylor series of the function ƒ to find the Taylor series of ƒ(u(x)) where u(x) is any continuous function. The Taylor series resulting from this substitution will converge for all x such that u(x) lies within the interval of convergence of the Taylor

9.9

Convergence of Taylor Series

547

series of ƒ. For instance, we can find the Taylor series for cos 2x by substituting 2x for x in the Taylor series for cos x: q

s -1dks2xd2k s2xd4 s2xd6 s2xd2 + + Á = 1 2! 4! 6! s2kd!

cos 2x = a

k=0

= 1 -

Eq. (5) with 2x for x

22x 2 24x 4 26x 6 + + Á 2! 4! 6!

q

22kx 2k . = a s -1dk s2kd! k=0 For what values of x can we replace sin x by x - sx 3>3!d with an error of magnitude no greater than 3 * 10 -4 ?

EXAMPLE 5

Here we can take advantage of the fact that the Taylor series for sin x is an alternating series for every nonzero value of x. According to the Alternating Series Estimation Theorem (Section 9.6), the error in truncating Solution

x3 x5 + - Á 3! 5! ------

sin x = x after sx 3>3!d is no greater than

`

ƒ x ƒ5 x5 ` = . 120 5!

Therefore the error will be less than or equal to 3 * 10 -4 if 5

ƒxƒ 6 3 * 10 -4 120

Rounded down, to be safe

5 ƒ x ƒ 6 2360 * 10 -4 L 0.514.

or

The Alternating Series Estimation Theorem tells us something that the Remainder Estimation Theorem does not: namely, that the estimate x - sx 3>3!d for sin x is an underestimate when x is positive, because then x 5>120 is positive. Figure 9.20 shows the graph of sin x, along with the graphs of a number of its approximating Taylor polynomials. The graph of P3sxd = x - sx 3>3!d is almost indistinguishable from the sine curve when 0 … x … 1. y 2 P1

P5

P9

P13

P17

5

6

7

1

0

1

2

3

4

y sin x

8

9

–1 P3

P7

P11

P15

P19

–2

FIGURE 9.20 The polynomials n s -1dkx 2k + 1 P2n + 1sxd = a k = 0 s2k + 1d!

converge to sin x as n : q . Notice how closely P3sxd approximates the sine curve for x … 1 (Example 5).

x

548

Chapter 9: Infinite Sequences and Series

A Proof of Taylor’s Theorem We prove Taylor’s theorem assuming a 6 b. The proof for a 7 b is nearly the same. The Taylor polynomial Pnsxd = ƒsad + ƒ¿sadsx - ad +

ƒ–sad f sndsad sx - ad2 + Á + sx - adn 2! n!

and its first n derivatives match the function ƒ and its first n derivatives at x = a. We do not disturb that matching if we add another term of the form Ksx - adn + 1 , where K is any constant, because such a term and its first n derivatives are all equal to zero at x = a. The new function fnsxd = Pnsxd + Ksx - adn + 1 and its first n derivatives still agree with ƒ and its first n derivatives at x = a. We now choose the particular value of K that makes the curve y = fnsxd agree with the original curve y = ƒsxd at x = b. In symbols, ƒsbd = Pnsbd + Ksb - adn + 1,

or

K =

ƒsbd - Pnsbd sb - adn + 1

.

(6)

With K defined by Equation (6), the function Fsxd = ƒsxd - fnsxd measures the difference between the original function ƒ and the approximating function fn for each x in [a, b]. We now use Rolle’s Theorem (Section 4.2). First, because Fsad = Fsbd = 0 and both F and F¿ are continuous on [a, b], we know that F¿sc1 d = 0

for some c1 in sa, bd.

Next, because F¿sad = F¿sc1 d = 0 and both F¿ and F– are continuous on [a, c1], we know that F–sc2 d = 0

for some c2 in sa, c1 d.

Rolle’s Theorem, applied successively to F–, F‡, Á , F sn - 1d implies the existence of c3

in sa, c2 d

such that F‡sc3 d = 0,

c4

in sa, c3 d

such that F s4dsc4 d = 0,

o cn

in sa, cn - 1 d

such that F sndscn d = 0.

Finally, because F snd is continuous on [a, cn] and differentiable on sa, cn d, and F sndsad = F sndscn d = 0, Rolle’s Theorem implies that there is a number cn + 1 in sa, cn d such that F sn + 1dscn + 1 d = 0.

(7)

If we differentiate Fsxd = ƒsxd - Pnsxd - Ksx - adn + 1 a total of n + 1 times, we get F sn + 1dsxd = ƒsn + 1dsxd - 0 - sn + 1d!K.

(8)

Equations (7) and (8) together give K =

ƒsn + 1dscd sn + 1d!

for some number c = cn + 1 in sa, bd.

(9)

9.9

Convergence of Taylor Series

549

Equations (6) and (9) give ƒsbd = Pnsbd +

ƒsn + 1dscd sb - adn + 1 . sn + 1d!

This concludes the proof.

Exercises 9.9 Finding Taylor Series Use substitution (as in Example 4) to find the Taylor series at x = 0 of the functions in Exercises 1–10. 1. e -5x 4. sin a

px b 2

7. ln s1 + x d 2

2. e -x>2

3. 5 sin s -xd

5. cos 5x 2

6. cos A x 2>3> 22 B

8. tan

-1

4

s3x d

1 10. 2 - x

1 9. 1 + 34 x 3

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 11–28. 11. xe x 14. sin x - x +

x3 3!

x2 - 1 + cos x 2

12. x 2 sin x

13.

15. x cos px

16. x 2 cos sx 2 d

17. cos2 x (Hint: cos2 x = s1 + cos 2xd>2 .) 18. sin2 x 21.

1 s1 - xd2

24. sin x # cos x 27.

x ln (1 + x 2) 3

39. How close is the approximation sin x = x when ƒ x ƒ 6 10 -3 ? For which of these values of x is x 6 sin x ? 40. The estimate 21 + x = 1 + sx>2d is used when x is small. Estimate the error when ƒ x ƒ 6 0.01.

41. The approximation e x = 1 + x + sx 2>2d is used when x is small. Use the Remainder Estimation Theorem to estimate the error when ƒ x ƒ 6 0.1 .

42. (Continuation of Exercise 41.) When x 6 0 , the series for e x is an alternating series. Use the Alternating Series Estimation Theorem to estimate the error that results from replacing e x by 1 + x + sx 2>2d when -0.1 6 x 6 0 . Compare your estimate with the one you obtained in Exercise 41. Theory and Examples 43. Use the identity sin2 x = s1 - cos 2xd>2 to obtain the Maclaurin series for sin2 x . Then differentiate this series to obtain the Maclaurin series for 2 sin x cos x. Check that this is the series for sin 2x.

19.

x2 1 - 2x

20. x ln s1 + 2xd

22.

2 s1 - xd3

23. x tan-1 x 2

44. (Continuation of Exercise 43. ) Use the identity cos2 x = cos 2x + sin2 x to obtain a power series for cos2 x .

26. cos x - sin x

45. Taylor’s Theorem and the Mean Value Theorem Explain how the Mean Value Theorem (Section 4.2, Theorem 4) is a special case of Taylor’s Theorem.

25. e x +

1 1 + x

28. ln (1 + x) - ln (1 - x)

Find the first four nonzero terms in the Maclaurin series for the functions in Exercises 29–34. ln (1 + x) 29. e x sin x 30. 31. (tan-1 x) 2 1 - x 32. cos2 x # sin x

38. If cos x is replaced by 1 - sx 2>2d and ƒ x ƒ 6 0.5 , what estimate can be made of the error? Does 1 - sx 2>2d tend to be too large, or too small? Give reasons for your answer.

33. e sin x

46. Linearizations at inflection points Show that if the graph of a twice-differentiable function ƒ(x) has an inflection point at x = a , then the linearization of ƒ at x = a is also the quadratic approximation of ƒ at x = a . This explains why tangent lines fit so well at inflection points. 47. The (second) second derivative test Use the equation

34. sin stan-1 xd ƒsxd = ƒsad + ƒ¿sadsx - ad +

Error Estimates 35. Estimate the error if P3(x) = x - sx 3>6d is used to estimate the value of sin x at x = 0.1. 36. Estimate the error if P4(x) = 1 + x + (x 2>2) + (x 3>6) + (x 4>24) is used to estimate the value of e x at x = 1>2.

37. For approximately what values of x can you replace sin x by x - sx 3>6d with an error of magnitude no greater than 5 * 10 -4 ? Give reasons for your answer.

ƒ–sc2 d sx - ad2 2

to establish the following test. Let ƒ have continuous first and second derivatives and suppose that ƒ¿sad = 0 . Then a. ƒ has a local maximum at a if ƒ– … 0 throughout an interval whose interior contains a; b. ƒ has a local minimum at a if ƒ– Ú 0 throughout an interval whose interior contains a.

550

Chapter 9: Infinite Sequences and Series

48. A cubic approximation Use Taylor’s formula with a = 0 and n = 3 to find the standard cubic approximation of ƒsxd = 1>s1 - xd at x = 0 . Give an upper bound for the magnitude of the error in the approximation when ƒ x ƒ … 0.1. 49. a. Use Taylor’s formula with n = 2 to find the quadratic approximation of ƒsxd = s1 + xdk at x = 0 (k a constant). b. If k = 3 , for approximately what values of x in the interval [0, 1] will the error in the quadratic approximation be less than 1> 100? 50. Improving approximations of P

T b. Try it with a calculator. ˆ

51. The Taylor series generated by ƒsxd = g n = 0 an x n is ˆ q g n = 0 an x n A function defined by a power series g n = 0 an x n with a radius of convergence R 7 0 has a Taylor series that converges to the function at every point of s -R, Rd. Show this by q showing that the Taylor series generated by ƒsxd = g n = 0 an x n is q the series g n = 0 an x n itself. An immediate consequence of this is that series like x4 x6 x8 + + Á 3! 5! 7!

and x4 x5 + + Á, 2! 3! obtained by multiplying Taylor series by powers of x, as well as series obtained by integration and differentiation of convergent power series, are themselves the Taylor series generated by the functions they represent. x 2e x = x 2 + x 3 +

52. Taylor series for even functions and odd functions (Continuq ation of Section 9.7, Exercise 55.) Suppose that ƒsxd = g n = 0 an x n converges for all x in an open interval s -R, Rd . Show that a. If ƒ is even, then a1 = a3 = a5 = Á = 0 , i.e., the Taylor series for ƒ at x = 0 contains only even powers of x. b. If ƒ is odd, then a0 = a2 = a4 = Á = 0 , i.e., the Taylor series for ƒ at x = 0 contains only odd powers of x. COMPUTER EXPLORATIONS Taylor’s formula with n = 1 and a = 0 gives the linearization of a function at x = 0 . With n = 2 and n = 3 we obtain the standard quadratic

9.10

a. For what values of x can the function be replaced by each approximation with an error less than 10 -2 ? b. What is the maximum error we could expect if we replace the function by each approximation over the specified interval? Using a CAS, perform the following steps to aid in answering questions (a) and (b) for the functions and intervals in Exercises 53–58. Step 1: Plot the function over the specified interval.

a. Let P be an approximation of p accurate to n decimals. Show that P + sin P gives an approximation correct to 3n decimals. (Hint: Let P = p + x .)

x sin x = x 2 -

and cubic approximations. In these exercises we explore the errors associated with these approximations. We seek answers to two questions:

Step 2: Find the Taylor polynomials P1sxd, P2sxd, and P3sxd at x = 0. Step 3: Calculate the sn + 1dst derivative ƒsn + 1dscd associated with the remainder term for each Taylor polynomial. Plot the derivative as a function of c over the specified interval and estimate its maximum absolute value, M. Step 4: Calculate the remainder Rnsxd for each polynomial. Using the estimate M from Step 3 in place of ƒsn + 1dscd, plot Rnsxd over the specified interval. Then estimate the values of x that answer question (a). Step 5: Compare your estimated error with the actual error Ensxd = ƒ ƒsxd - Pnsxd ƒ by plotting Ensxd over the specified interval. This will help answer question (b). Step 6: Graph the function and its three Taylor approximations together. Discuss the graphs in relation to the information discovered in Steps 4 and 5. 53. ƒsxd =

1 21 + x

,

54. ƒsxd = s1 + xd3>2,

3 ƒxƒ … 4 -

1 … x … 2 2

x , ƒxƒ … 2 x + 1 56. ƒsxd = scos xdssin 2xd, ƒ x ƒ … 2 57. ƒsxd = e -x cos 2x, ƒ x ƒ … 1 55. ƒsxd =

2

58. ƒsxd = e x>3 sin 2x,

ƒxƒ … 2

The Binomial Series and Applications of Taylor Series We can use Taylor series to solve problems that would otherwise be intractable. For example, many functions have antiderivatives that cannot be expressed using familiar functions. In this section we show how to evaluate integrals of such functions by giving them as Taylor series. We also show how to use Taylor series to evaluate limits that lead to indeterminate forms and how Taylor series can be used to extend the exponential function from real to complex numbers. We begin with a discussion of the binomial series, which comes from the Taylor series of the function f (x) = (1 + x)m, and conclude the section with a table of commonly used series.

9.10

The Binomial Series and Applications of Taylor Series

551

The Binomial Series for Powers and Roots The Taylor series generated by ƒsxd = s1 + xdm , when m is constant, is 1 + mx +

msm - 1d 2 msm - 1dsm - 2d 3 Á x + x + 2! 3! msm - 1dsm - 2d Á sm - k + 1d k Á + x + . k!

(1)

This series, called the binomial series, converges absolutely for ƒ x ƒ 6 1. To derive the series, we first list the function and its derivatives: ƒsxd = s1 + xdm ƒ¿sxd = ms1 + xdm - 1 ƒ–sxd = msm - 1ds1 + xdm - 2 ƒ‡sxd = msm - 1dsm - 2ds1 + xdm - 3 o ƒ sxd = msm - 1dsm - 2d Á sm - k + 1ds1 + xdm - k . skd

We then evaluate these at x = 0 and substitute into the Taylor series formula to obtain Series (1). If m is an integer greater than or equal to zero, the series stops after sm + 1d terms because the coefficients from k = m + 1 on are zero. If m is not a positive integer or zero, the series is infinite and converges for ƒ x ƒ 6 1. To see why, let uk be the term involving x k . Then apply the Ratio Test for absolute convergence to see that uk + 1 m - k ` uk ` = ` x` :ƒxƒ k + 1

as k : q .

Our derivation of the binomial series shows only that it is generated by s1 + xdm and converges for ƒ x ƒ 6 1. The derivation does not show that the series converges to s1 + xdm . It does, but we leave the proof to Exercise 64.

The Binomial Series For -1 6 x 6 1, q m s1 + xdm = 1 + a a b x k , k k=1

where we define m a b = m, 1

msm - 1d m a b = , 2! 2

and msm - 1dsm - 2d Á sm - k + 1d m a b = k! k

EXAMPLE 1

for k Ú 3.

If m = -1, a

-1 b = -1, 1

a

-1s -2d -1 = 1, b = 2! 2

552

Chapter 9: Infinite Sequences and Series

and a

-1s -2ds -3d Á s -1 - k + 1d -1 k! = s -1dk a b = s -1dk . b = k! k! k

With these coefficient values and with x replaced by -x, the binomial series formula gives the familiar geometric series q

s1 + xd-1 = 1 + a s -1dkx k = 1 - x + x 2 - x 3 + Á + s -1dkx k + Á . k=1

We know from Section 3.11, Example 1, that 21 + x L 1 + sx>2d for x small. With m = 1>2, the binomial series gives quadratic and higher-order approximaƒ ƒ tions as well, along with error estimates that come from the Alternating Series Estimation Theorem:

EXAMPLE 2

s1 + xd1>2

x = 1 + + 2

3 1 1 1 1 a b a- b a b a- b a- b 2 2 2 2 2 x2 + x3 2! 3! 3 5 1 1 a b a- b a- b a- b 2 2 2 2 + x4 + Á 4!

= 1 +

5x 4 x x2 x3 + + Á. 2 8 16 128

Substitution for x gives still other approximations. For example, 21 - x 2 L 1 -

A

x2 x4 2 8

1 1 1 1 - x L 1 2x 8x 2

for ƒ x 2 ƒ small

1 for ` x ` small, that is, ƒ x ƒ large.

Evaluating Nonelementary Integrals Sometimes we can use a familiar Taylor series to find the sum of a given power series in terms of a known function. For example, x2 -

(x 2) 3 (x 2) 5 (x 2) 7 x6 x 10 x 14 + + Á = (x 2) + + Á = sin x 2 . 3! 5! 7! 3! 5! 7!

Additional examples are provided in Exercises 59–62. Taylor series can be used to express nonelementary integrals in terms of series. Integrals like 1 sin x 2 dx arise in the study of the diffraction of light.

EXAMPLE 3

Express 1 sin x 2 dx as a power series.

From the series for sin x we substitute x 2 for x to obtain

Solution

sin x 2 = x 2 -

x6 x 10 x 14 x 18 + + - Á. 3! 5! 7! 9!

Therefore, L

sin x 2 dx = C +

x3 x 11 x 15 x 10 x7 + + - Á. - # # # 3 7 3! 11 5! 15 7! 19 # 9!

9.10

The Binomial Series and Applications of Taylor Series

553

1

Estimate 10 sin x 2 dx with an error of less than 0.001.

EXAMPLE 4

From the indefinite integral in Example 3, we easily find that

Solution

1

L0

sin x 2 dx =

1 1 1 1 1 - # + + - Á. 3 7 3! 11 # 5! 15 # 7! 19 # 9!

The series on the right-hand side alternates, and we find by numerical evaluations that 1 L 0.00076 11 # 5! is the first term to be numerically less than 0.001. The sum of the preceding two terms gives 1

L0

sin x 2 dx L

1 1 L 0.310. 3 42

With two more terms we could estimate 1

L0

sin x 2 dx L 0.310268

with an error of less than 10 -6 . With only one term beyond that we have 1

L0

sin x 2 dx L

1 1 1 1 1 + + L 0.310268303, 3 42 1320 75600 6894720

with an error of about 1.08 * 10 -9 . To guarantee this accuracy with the error formula for the Trapezoidal Rule would require using about 8000 subintervals.

Arctangents In Section 9.7, Example 5, we found a series for tan-1 x by differentiating to get d 1 = 1 - x2 + x4 - x6 + Á tan-1 x = dx 1 + x2 and then integrating to get tan-1 x = x -

x3 x5 x7 + + Á. 5 7 3

However, we did not prove the term-by-term integration theorem on which this conclusion depended. We now derive the series again by integrating both sides of the finite formula n + 1 2n + 2 1 2 4 6 Á + s -1dnt 2n + s -1d t = 1 t + t t + , 1 + t2 1 + t2

(2)

in which the last term comes from adding the remaining terms as a geometric series with first term a = s -1dn + 1t 2n + 2 and ratio r = -t 2 . Integrating both sides of Equation (2) from t = 0 to t = x gives tan-1 x = x -

x3 x5 x7 x 2n + 1 + + Á + s -1dn + Rnsxd, 5 7 3 2n + 1

where s -1dn + 1t 2n + 2 dt. 1 + t2 L0 x

Rnsxd =

The denominator of the integrand is greater than or equal to 1; hence ƒ Rnsxd ƒ …

ƒxƒ

L0

t 2n + 2 dt =

ƒ x ƒ 2n + 3 . 2n + 3

554

Chapter 9: Infinite Sequences and Series

If ƒ x ƒ … 1, the right side of this inequality approaches zero as n : q . Therefore limn: q Rnsxd = 0 if ƒ x ƒ … 1 and q

s -1dnx 2n + 1 , 2n + 1 n=0

tan-1 x = a

tan-1 x = x -

ƒ x ƒ … 1. (3)

x3 x5 x7 + + Á, 5 7 3

ƒ x ƒ … 1.

We take this route instead of finding the Taylor series directly because the formulas for the higher-order derivatives of tan-1 x are unmanageable. When we put x = 1 in Equation (3), we get Leibniz’s formula: s -1dn p 1 1 1 1 = 1 - + - + - Á + + Á. 5 7 4 3 9 2n + 1 Because this series converges very slowly, it is not used in approximating p to many decimal places. The series for tan-1 x converges most rapidly when x is near zero. For that reason, people who use the series for tan-1 x to compute p use various trigonometric identities. For example, if a = tan-1

1 2

and

1 b = tan-1 , 3

then tan sa + bd =

1 tan a + tan b 2 + = 1 - tan a tan b 1 -

1 3 1 6

= 1 = tan

p 4

and p 1 1 = a + b = tan-1 + tan-1 . 4 2 3 Now Equation (3) may be used with x = 1>2 to evaluate tan-1 (1>2) and with x = 1>3 to give tan-1 (1>3). The sum of these results, multiplied by 4, gives p.

Evaluating Indeterminate Forms We can sometimes evaluate indeterminate forms by expressing the functions involved as Taylor series.

EXAMPLE 5

Evaluate lim

x:1

ln x . x - 1

We represent ln x as a Taylor series in powers of x - 1. This can be accomplished by calculating the Taylor series generated by ln x at x = 1 directly or by replacing x by x - 1 in the series for ln (1 + x) in Section 9.7, Example 6. Either way, we obtain Solution

ln x = sx - 1d -

1 sx - 1d2 + Á , 2

from which we find that lim

x:1

EXAMPLE 6

ln x 1 = lim a1 - sx - 1d + Á b = 1. x - 1 2 x:1

Evaluate lim

x:0

sin x - tan x . x3

9.10

Solution

The Binomial Series and Applications of Taylor Series

555

The Taylor series for sin x and tan x, to terms in x 5 , are x5 x3 + - Á, 3! 5!

sin x = x -

tan x = x +

x3 2x 5 + + Á. 3 15

Subtracting the series term by term, it follows that sin x - tan x = -

x3 x5 x2 1 - Á = x3 a- - Á b. 2 8 2 8

Division of both sides by x3 and taking limits then gives sin x - tan x x2 1 = lim a- - Áb 3 2 8 x:0 x:0 x lim

1 = - . 2 If we apply series to calculate limx:0 ss1>sin xd - s1/xdd, we not only find the limit successfully but also discover an approximation formula for csc x.

EXAMPLE 7 Solution

Find lim a x:0

1 1 - xb. sin x

Using algebra and the Taylor series for sin x, we have x - sin x 1 1 - x = = sin x x sin x x3 a

x - ax x # ax -

x5 x3 + - Áb 3! 5! x3 x5 + - Áb 3! 5!

x2 1 + Áb 3! 5!

x2 1 + Á 3! 5! = . = x x2 x2 1 + Á x2 a1 + Áb 3! 3! Therefore, x2 1 + Á 3! 5! 1 1 lim a - x b = lim § x ¥ = 0. x:0 sin x x:0 x2 Á 1 + 3! From the quotient on the right, we can see that if ƒ x ƒ is small, then x 1 1 1 - x L x# = 6 3! sin x

or

x 1 csc x L x + . 6

Euler’s Identity A complex number is a number of the form a + bi, where a and b are real numbers and i = 2-1 (see Appendix 8). If we substitute x = iu (u real) in the Taylor series for e x and use the relations i 2 = -1,

i 3 = i 2i = -i,

i 4 = i 2i 2 = 1,

i 5 = i 4i = i,

556

Chapter 9: Infinite Sequences and Series

and so on, to simplify the result, we obtain e iu = 1 + = a1 -

i 2u2 i 3u3 i 4u4 i 5u5 i 6u6 iu + + + + + + Á 1! 2! 3! 4! 5! 6!

u4 u6 u3 u5 u2 + + Á b + i au + - Á b = cos u + i sin u. 2! 4! 6! 3! 5!

This does not prove that e iu = cos u + i sin u because we have not yet defined what it means to raise e to an imaginary power. Rather, it says how to define e iu to be consistent with other things we know about the exponential function for real numbers.

DEFINITION For any real number u, e iu = cos u + i sin u.

(4)

Equation (4), called Euler’s identity, enables us to define e a + bi to be e a # e bi for any complex number a + bi. One consequence of the identity is the equation e ip = -1. When written in the form e ip + 1 = 0, this equation combines five of the most important constants in mathematics. TABLE 9.1 Frequently used Taylor series q

1 = 1 + x + x 2 + Á + x n + Á = a x n, 1 - x n=0

ƒxƒ 6 1

q

1 = 1 - x + x 2 - Á + s -xdn + Á = a s -1dnx n, 1 + x n=0 q

ex = 1 + x + sin x = x cos x = 1 -

x2 xn xn + Á + + Á = a , 2! n! n = 0 n!

ƒxƒ 6 1

ƒxƒ 6 q

q s -1dnx 2n + 1 x3 x5 x 2n + 1 + - Á + s -1dn + Á = a , 3! 5! s2n + 1d! n = 0 s2n + 1d! q s -1dnx 2n x2 x4 x 2n + - Á + s -1dn + Á = a , 2! 4! s2nd! s2nd! n=0

ln s1 + xd = x tan-1 x = x -

ƒxƒ 6 q

ƒxƒ 6 q

q s -1dn - 1x n x2 x3 xn + - Á + s -1dn - 1 n + Á = a , n 2 3 n=1

q s -1dnx 2n + 1 x3 x5 x 2n + 1 + - Á + s -1dn + Á = a , 5 3 2n + 1 2n + 1 n=0

-1 6 x … 1 ƒxƒ … 1

Exercises 9.10 Binomial Series Find the first four terms of the binomial series for the functions in Exercises 1–10. 1. s1 + xd1>2

2. s1 + xd1>3

3. s1 - xd-1>2

4. s1 - 2xd1>2 7. s1 + x 3 d-1>2

5. a1 +

x b 2

-2

6. a1 -

8. s1 + x 2 d-1>3

x b 3

4

9.10 1 9. a1 + x b

1>2

10.

sin 3x 2 x:0 1 - cos 2x

x

39. lim

3 2 1 + x

Find the binomial series for the functions in Exercises 11–14. 11. s1 + xd4

12. s1 + x 2 d3

13. s1 - 2xd3

14. a1 -

x b 2

The Binomial Series and Applications of Taylor Series

4

0.2

0.2

sin x 2 dx

L0 0.1

17.

L0

16.

L0

e -x - 1 dx x

21 + x 4

dx

18.

3 2 1 + x 2 dx

L0

T Use series to approximate the values of the integrals in Exercises 19–22 with an error of magnitude less than 10 -8 . 0.1

19.

L0

0.1

sin x x dx

20.

21 + x 4 dx

22.

L0

0.1

21.

L0

e -x dx

1

L0

2

1

integral 10 cos t 2 dt .

in the integral

t4 t8 + in the 2 4! t t2 t3 + 2 4! 6!

2t dt .

In Exercises 25–28, find a polynomial that will approximate F(x) throughout the given interval with an error of magnitude less than 10 -3 . x

25. Fsxd =

sin t 2 dt,

L0 x

26. Fsxd =

t 2e -t dt,

L0

x

27. Fsxd =

L0

2

[0, 1] [0, 1]

ln s1 + td dt, (a) [0, 0.5] t L0

(b) [0, 1]

Indeterminate Forms Use series to evaluate the limits in Exercises 29–40. e x - s1 + xd e x - e -x 29. lim 30. lim x 2 x: 0 x: 0 x 31. lim

1 - cos t - st 2>2d t4

t:0

33. lim

y - tan

y: 0

-1

y

y3

34. lim

tan

y: 0

2

x: q

ln s1 + x 2 d x: 0 1 - cos x

32 34 36 + 4 - 6 + Á # # 4 2! 4 4! 4 # 6! 2

44.

1 1 1 1 + + Á 2 2 # 22 3 # 23 4 # 24

45.

p p7 p5 p3 - 7 + Á - 3 + 5 3 3 # 5! 3 # 7! 3 # 3!

25 2 27 23 + 5 - 7 + Á - 3 # # 3 3 5 3 #7 3 3 47. x 3 + x 4 + x 5 + x 6 + Á 46.

22x 4 23x 5 2 4x 6 + - Á 2! 3! 4! 51. -1 + 2x - 3x 2 + 4x 3 - 5x 4 + Á 50. x 2 - 2x 3 +

52. 1 +

x x2 x3 x4 + + + + Á 5 2 3 4

Theory and Examples 53. Replace x by -x in the Taylor series for ln s1 + xd to obtain a series for ln s1 - xd . Then subtract this from the Taylor series for ln s1 + xd to show that for ƒ x ƒ 6 1 ,

-1

u5 y - sin y

y 3 cos y

36. lim sx + 1d sin x: q

x2 - 4 x: 2 ln sx - 1d

38. lim

x3 x5 1 + x = 2 ax + + + Á b. 5 1 - x 3

55. According to the Alternating Series Estimation Theorem, how many terms of the Taylor series for tan-1 1 would you have to add to be sure of finding p>4 with an error of magnitude less than 10 -3 ? Give reasons for your answer. 56. Show that the Taylor series for ƒsxd = tan-1 x diverges for ƒ x ƒ 7 1.

sin u - u + su3>6d

u: 0

35. lim x 2se -1>x - 1d 37. lim

32. lim

6

54. How many terms of the Taylor series for ln s1 + xd should you add to be sure of calculating ln (1.1) with an error of magnitude less than 10 -8 ? Give reasons for your answer.

(b) [0, 1]

x

28. Fsxd =

43. 1 -

ln

tan-1 t dt, (a) [0, 0.5]

5

32x 2 34x 4 3 6x 6 + + Á 2! 4! 6! 49. x 3 - x 5 + x 7 - x 9 + x 11 - Á

24. Estimate the error if cos 2t is approximated by 1 1 10 cos

4

48. 1 -

1 - cos x dx x2

23. Estimate the error if cos t 2 is approximated by 1 -

x # sin x 2

1 1 1 1 42. a b + a b + a b + a b + Á 4 4 4 4

0.25

1

x:0

1 1 1 + + + Á 2! 3! 4!

3

15.

ln s1 + x 3 d

Using Table 9.1 In Exercises 41–52, use Table 9.1 to find the sum of each series. 41. 1 + 1 +

Approximations and Nonelementary Integrals T In Exercises 15–18, use series to estimate the integrals’ values with an error of magnitude less than 10 -3 . (The answer section gives the integrals’ values rounded to five decimal places.)

40. lim

557

1 x + 1

T 57. Estimating Pi About how many terms of the Taylor series for tan-1 x would you have to use to evaluate each term on the righthand side of the equation p = 48 tan-1

1 1 1 + 32 tan-1 - 20 tan-1 57 18 239

with an error of magnitude less than 10 -6 ? In contrast, the conq vergence of g n = 1s1>n 2 d to p2>6 is so slow that even 50 terms will not yield two-place accuracy.

558

Chapter 9: Infinite Sequences and Series

58. Integrate the first three nonzero terms of the Taylor series for tan t from 0 to x to obtain the first three nonzero terms of the Taylor series for ln sec x. 59. a. Use the binomial series and the fact that d sin-1 x = s1 - x 2 d-1>2 dx to generate the first four nonzero terms of the Taylor series for sin-1 x . What is the radius of convergence? b. Series for cos-1 x Use your result in part (a) to find the first five nonzero terms of the Taylor series for cos-1 x . 60. a. Series for sinh-1 x Taylor series for

Find the first four nonzero terms of the

sinh-1 x =

by integrating the series 1 1 = 2 1 + t2 t

Euler’s Identity 67. Use Equation (4) to write the following powers of e in the form a + bi . a. e -ip

61. Obtain the Taylor series for 1>s1 + xd from the series for -1>s1 + xd . 2

62. Use the Taylor series for 1>s1 - x 2 d to obtain a series for 2x>s1 - x 2 d2 . T 63. Estimating Pi The English mathematician Wallis discovered the formula 2#4#4#6#6#8# Á p = # # # # # # Á . 4 3 3 5 5 7 7

to show that mƒ(x) ƒ¿(x) = , 1 + x

-1 6 x 6 1.

b. Define g(x) = (1 + x) - m ƒ(x) and show that g¿(x) = 0. c. From part (b), show that ƒ(x) = (1 + x) m. 65. Series for sin-1 x Integrate the binomial series for s1 - x 2 d-1>2 to show that for ƒ x ƒ 6 1 , q 1 # 3 # 5 # Á # s2n - 1d x 2n + 1 sin-1 x = x + a . 2 # 4 # 6 # Á # s2nd 2n + 1 n=1 66. Series for tan

-1

x for x>1 Derive the series

p 1 1 1 + Á, x 7 1 - x + 2 5x 5 3x 3 p 1 1 1 + Á, x 6 -1 , tan-1 x = - - x + 2 5x 5 3x 3 tan-1 x =

and

sin u =

e iu - e -iu . 2i

70. Show that a. cosh iu = cos u ,

b. sinh iu = i sin u .

71. By multiplying the Taylor series for e x and sin x, find the terms through x 5 of the Taylor series for e x sin x . This series is the imaginary part of the series for e x # e ix = e s1 + idx . Use this fact to check your answer. For what values of x should the series for e x sin x converge? 72. When a and b are real, we define e sa + ibdx with the equation e sa + ibdx = e ax # e ibx = e axscos bx + i sin bxd . Differentiate the right-hand side of this equation to show that d sa + ibdx = sa + ibde sa + ibdx . e dx

a. Differentiate the series q m ƒ(x) = 1 + a a b x k k=1 k

e iu + e -iu 2

69. Establish the equations in Exercise 68 by combining the formal Taylor series for e iu and e -iu .

Find p to two decimal places with this formula. 64. Use the following steps to prove that the binomial series in Equation (1) converges to (1 + x)m.

c. e -ip>2

b. e ip>4

68. Use Equation (4) to show that

x

T b. Use the first three terms of the series in part (a) to estimate sinh-1 0.25 . Give an upper bound for the magnitude of the estimation error.

1 1 1 1 1 = 2 - 4 + 6 - 8 + Á 1 + s1>t 2 d t t t t

in the first case from x to q and in the second case from - q to x.

cos u =

dt . L0 21 + t 2

#

Thus the familiar rule sd>dxde kx = ke kx holds for k complex as well as real. 73. Use the definition of e iu to show that for any real numbers u, u1 , and u2 , a. e iu1e iu2 = e isu1 + u2d,

b. e -iu = 1>e iu .

74. Two complex numbers a + ib and c + id are equal if and only if a = c and b = d . Use this fact to evaluate

L

e ax cos bx dx

and

L

e ax sin bx dx

from L

e sa + ibdx dx =

a - ib sa + ibdx e + C, a2 + b2

where C = C1 + iC2 is a complex constant of integration.

Chapter 9

Chapter 9

Practice Exercises

559

Questions to Guide Your Review

1. What is an infinite sequence? What does it mean for such a sequence to converge? To diverge? Give examples. 2. What is a monotonic sequence? Under what circumstances does such a sequence have a limit? Give examples. 3. What theorems are available for calculating limits of sequences? Give examples. 4. What theorem sometimes enables us to use l’Hôpital’s Rule to calculate the limit of a sequence? Give an example. 5. What are the six commonly occurring limits in Theorem 5 that arise frequently when you work with sequences and series? 6. What is an infinite series? What does it mean for such a series to converge? To diverge? Give examples. 7. What is a geometric series? When does such a series converge? Diverge? When it does converge, what is its sum? Give examples. 8. Besides geometric series, what other convergent and divergent series do you know? 9. What is the nth-Term Test for Divergence? What is the idea behind the test?

18. What is an alternating series? What theorem is available for determining the convergence of such a series? 19. How can you estimate the error involved in approximating the sum of an alternating series with one of the series’ partial sums? What is the reasoning behind the estimate? 20. What is absolute convergence? Conditional convergence? How are the two related? 21. What do you know about rearranging the terms of an absolutely convergent series? Of a conditionally convergent series? 22. What is a power series? How do you test a power series for convergence? What are the possible outcomes? 23. What are the basic facts about a. sums, differences, and products of power series? b. substitution of a function for x in a power series? c. term-by-term differentiation of power series? d. term-by-term integration of power series? Give examples.

10. What can be said about term-by-term sums and differences of convergent series? About constant multiples of convergent and divergent series?

24. What is the Taylor series generated by a function ƒ(x) at a point x = a ? What information do you need about ƒ to construct the series? Give an example.

11. What happens if you add a finite number of terms to a convergent series? A divergent series? What happens if you delete a finite number of terms from a convergent series? A divergent series?

25. What is a Maclaurin series?

12. How do you reindex a series? Why might you want to do this?

27. What are Taylor polynomials? Of what use are they?

13. Under what circumstances will an infinite series of nonnegative terms converge? Diverge? Why study series of nonnegative terms? 14. What is the Integral Test? What is the reasoning behind it? Give an example of its use.

28. What is Taylor’s formula? What does it say about the errors involved in using Taylor polynomials to approximate functions? In particular, what does Taylor’s formula say about the error in a linearization? A quadratic approximation?

15. When do p-series converge? Diverge? How do you know? Give examples of convergent and divergent p-series.

29. What is the binomial series? On what interval does it converge? How is it used?

16. What are the Direct Comparison Test and the Limit Comparison Test? What is the reasoning behind these tests? Give examples of their use.

30. How can you sometimes use power series to estimate the values of nonelementary definite integrals? To find limits?

17. What are the Ratio and Root Tests? Do they always give you the information you need to determine convergence or divergence? Give examples.

Chapter 9

s -1dn n

1 - 2n 3. an = 2n

31. What are the Taylor series for 1>s1 - xd, 1>s1 + xd, e x, sin x, cos x, ln s1 + xd, and tan-1 x ? How do you estimate the errors involved in replacing these series with their partial sums?

Practice Exercises

Determining Convergence of Sequences Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence. 1. an = 1 +

26. Does a Taylor series always converge to its generating function? Explain.

2. an =

1 - s -1dn 2n

4. an = 1 + s0.9dn

5. an = sin

np 2

7. an =

ln sn 2 d n

9. an =

n + ln n n

11. an = a

n - 5 n b

6. an = sin np

n

8. an =

ln s2n + 1d n

10. an =

ln s2n 3 + 1d n

1 12. an = a1 + n b

-n

560

Chapter 9: Infinite Sequences and Series

13. an =

3 14. an = a n b

3n An n

n

15. an = ns21>n - 1d 17. an =

Maclaurin Series Each of the series in Exercises 51–56 is the value of the Taylor series at x = 0 of a function ƒ(x) at a particular point. What function and what point? What is the sum of the series?

1>n

16. an = 22n + 1

sn + 1d! n!

s -4dn n!

18. an =

51. 1 52.

Convergent Series Find the sums of the series in Exercises 19–24. q

q

-2 20. a n = 2 nsn + 1d

1 19. a n = 3 s2n - 3ds2n - 1d q

q

-8 22. a n = 3 s4n - 3ds4n + 1d

9 21. a n = 1 s3n - 1ds3n + 2d q

n=0

n=1

q

2n

s -1d 29. a ln sn + 1d n=1

s -1dn n 2n 2 + 1

n + 1 n! n=1 q

1 30. a 2 n = 2 n sln nd

q s -1dn 3n 2 34. a 3 n=1 n + 1 q s -1dnsn 2 + 1d 36. a 2 n = 1 2n + n - 1 q

s -3dn 37. a n! n=1

2n 3n 38. a n n=1 n

q n=1

2n

q

ln n 32. a n = 3 ln sln nd

35. a

39. a

n

s -1dn

q

ln n 31. a 3 n=1 n

q

n=1

q

q

n=1

27. a

n=1

1 28. a 3 n = 1 2n

q

q

-5 26. a n

1

q

33. a

q n=1 q

1 2nsn + 1dsn + 2d

s -1dn - 1s3x - 1dn n

2

xn 45. a n n=1 n q

sn + 1dx 2n - 1 47. a 3n n=0 q

49. a scsch ndx n n=1

p3 p2n + 1 p5 + - Á + s - 1dn + Á 3! 5! s2n + 1d!

54. 1 -

p2n p2 p4 + Á + - Á + s -1dn 2n 9 # 2! 81 # 4! 3 s2nd!

55. 1 + ln 2 +

q

40. a

n=2

1

-

23

sln 2d2 sln 2dn + Á + + Á 2! n!

1 9 23

+

+ s -1dn - 1

1

1 n 2n 2 - 1

- Á

4523

1

+ Á

s2n - 1d A 23 B 2n - 1

Find Taylor series at x = 0 for the functions in Exercises 57–64. 57.

1 1 - 2x

1 1 + x3 2x 60. sin 3 x3 62. cos 25 2 64. e -x 58.

59. sin px 61. cos sx 5>3 d 63. e spx>2d

Taylor Series In Exercises 65–68, find the first four nonzero terms of the Taylor series generated by ƒ at x = a . 65. ƒsxd = 23 + x 2

at

x = -1

66. ƒsxd = 1>s1 - xd

at

x = 2

at

x = 3

67. ƒsxd = 1>sx + 1d

Power Series In Exercises 41–50, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally. q q sx + 4dn sx - 1d2n - 2 41. a 42. a n n3 n=1 n = 1 s2n - 1d! 43. a

53. p -

56.

n=1

Determining Convergence of Series Which of the series in Exercises 25–40 converge absolutely, which converge conditionally, and which diverge? Give reasons for your answers. 25. a

8 2 2n 4 + - Á + s -1dn - 1 n + Á 3 18 81 n3

q

3 24. a s -1dn n 4

23. a e -n

q

1 1 1 + - Á + s -1dn n + Á 4 16 4

68. ƒsxd = 1>x

at

x = a 7 0

Nonelementary Integrals Use series to approximate the values of the integrals in Exercises 69–72 with an error of magnitude less than 10 -8 . (The answer section gives the integrals’ values rounded to 10 decimal places.) 1>2

69.

L0 1>2

71.

L0

1

e -x dx

70.

tan-1 x x dx

72.

3

L0

x sin sx 3 d dx 1>64

tan-1 x

L0

2x

dx

q

sn + 1ds2x + 1dn s2n + 1d2n n=0

44. a q

46. a

xn

2n s -1dnsx - 1d2n + 1 48. a 2n + 1 n=0 n=1 q

q

50. a scoth ndx n n=1

Using Series to Find Limits In Exercises 73–78: a. Use power series to evaluate the limit. T b. Then use a grapher to support your calculation. 73. lim

x:0

7 sin x e 2x - 1

75. lim a t: 0

1 1 - 2b 2 - 2 cos t t

74. lim

u :0

76. lim

h:0

e u - e - u - 2u u - sin u ssin hd>h - cos h h2

Chapter 9

77. lim

z:0

1 - cos2 z ln s1 - zd + sin z

y2 y: 0 cos y - cosh y

78. lim

Theory and Examples 79. Use a series representation of sin 3x to find values of r and s for which sin 3x r lim a 3 + 2 + sb = 0 . x:0 x x T 80. Compare the accuracies of the approximations sin x L x and sin x L 6x>s6 + x 2 d by comparing the graphs of ƒsxd = sin x - x and gsxd = sin x - s6x>s6 + x 2 dd . Describe what you find. 81. Find the radius of convergence of the series 2 # 5 # 8 # Á # s3n - 1d n a 2 # 4 # 6 # Á # s2nd x . n=1 q

82. Find the radius of convergence of the series q 3 # 5 # 7 # Á # s2n + 1d n a 4 # 9 # 14 # Á # s5n - 1d sx - 1d . n=1

83. Find a closed-form formula for the nth partial sum of the series q g n = 2 ln s1 - s1>n 2 dd and use it to determine the convergence or divergence of the series.

Additional and Advanced Exercises

b. Show that the function defined by the series satisfies a differential equation of the form d 2y dx 2

86. a. Find the Maclaurin series for the function x 2>s1 + xd . b. Does the series converge at x = 1 ? Explain. q

q

87. If g n = 1 an and g n = 1 bn are convergent series of nonnegative q numbers, can anything be said about g n = 1 an bn ? Give reasons for your answer. q

q

88. If g n = 1 an and g n = 1 bn are divergent series of nonnegative numq bers, can anything be said about g n = 1 an bn ? Give reasons for your answer. 89. Prove that the sequence 5xn6 and the series g k = 1 sxk + 1 - xk d both converge or both diverge. q

90. Prove that g n = 1 san>s1 + an dd converges if an 7 0 for all n and q g n = 1 an converges. q

91. Suppose that a1, a2 , a3 , Á , an are positive numbers satisfying the following conditions: i) a1 Ú a2 Ú a3 Ú Á ; ii) the series a2 + a4 + a8 + a16 + Á diverges. Show that the series

84. Evaluate - 1dd by finding the limits as n : q of the series’ nth partial sum. 1 3 1 6 Á x + x + 6 180 # # # Á # 1 4 7 s3n - 2d 3n + x + Á. s3nd!

y = 1 +

Chapter 9

q

1 1. a n + s1>2d n = 1 s3n - 2d q

3. a s -1dn tanh n n=1

q

diverges. 92. Use the result in Exercise 91 to show that q

1 1 + a n = 2 n ln n

diverges.

Choosing Centers for Taylor Series Taylor’s formula ƒsxd = ƒsad + ƒ¿sadsx - ad +

stan-1 nd2

2. a 2 n=1 n + 1 q logn sn!d 4. a n3

+

n=2

q

Which of the series g n = 1 an defined by the formulas in Exercises 5–8 converge, and which diverge? Give reasons for your answers. an + 1 =

nsn + 1d an sn + 2dsn + 3d

(Hint: Write out several terms, see which factors cancel, and then generalize.) 6. a1 = a2 = 7,

an + 1 =

n an sn - 1dsn + 1d

7. a1 = a2 = 1,

an + 1 =

1 1 + an

8. an = 1>3n

a3 a1 a2 + + + Á 1 2 3

Additional and Advanced Exercises

Determining Convergence of Series q Which of the series g n = 1 an defined by the formulas in Exercises 1–4 converge, and which diverge? Give reasons for your answers.

5. a1 = 1,

= x ay + b

and find the values of the constants a and b.

q g k = 2 s1>sk 2

85. a. Find the interval of convergence of the series

561

if n is odd,

if n Ú 2

if n Ú 2

an = n>3n

if n is even

ƒ–sad sx - ad2 + Á 2!

ƒsn + 1dscd ƒsndsad sx - adn + sx - adn + 1 n! sn + 1d!

expresses the value of ƒ at x in terms of the values of ƒ and its derivatives at x = a . In numerical computations, we therefore need a to be a point where we know the values of ƒ and its derivatives. We also need a to be close enough to the values of ƒ we are interested in to make sx - adn + 1 so small we can neglect the remainder. In Exercises 9–14, what Taylor series would you choose to represent the function near the given value of x? (There may be more than one good answer.) Write out the first four nonzero terms of the series you choose. 9. cos x 11. e x

near

near

13. cos x

x = 1

x = 0.4

near

x = 69

10. sin x 12. ln x

x = 6.3

near near

14. tan-1 x

x = 1.3

near

x = 2

562

Chapter 9: Infinite Sequences and Series

Theory and Examples 15. Let a and b be constants with 0 6 a 6 b . Does the sequence 5sa n + b n d1>n6 converge? If it does converge, what is the limit?

24. Find values of a and b for which lim

7 3 7 3 2 2 2 + + + + + + 10 10 5 10 7 10 2 10 3 10 4 10 6 7 3 + + Á. + 10 8 10 9

1 +

n+1

Ln

n=0

1 dx . 1 + x2

18. Find all values of x for which q

nx n a sn + 1ds2x + 1dn n=1 converges absolutely.

ƒsnd un C un + 1 = 1 + n + n 2 , q

where ƒ ƒsnd ƒ 6 K for n Ú N , then g n = 1 un converges if C 7 1 and diverges if C … 1 . Show that the results of Raabe’s Test agree with what you q q know about the series g n = 1 s1>n 2 d and g n = 1 s1>nd . q

26. (Continuation of Exercise 25. ) Suppose that the terms of g n = 1 un are defined recursively by the formulas u1 = 1,

T 19. a. Does the value of cos sa>nd n b , n

lim a1 -

n: q

a constant ,

b. Does the value of a and b constant, b Z 0 ,

appear to depend on the value of b? If so, how? c. Use calculus to confirm your findings in parts (a) and (b). q

20. Show that if g n = 1 an converges, then a a

n=1

1 + sin san d b 2

q

q

a. Show that g n = 1 a n2 converges. q g n = 1 an >s1

21. Find a value for the constant b that will make the radius of convergence of the power series n n

b x a ln n

n=2

- an d converge? Explain. q

28. (Continuation of Exercise 27. ) If g n = 1 an converges, and if q 1 7 an 7 0 for all n, show that g n = 1 ln s1 - an d converges.

(Hint: First show that ƒ ln s1 - an d ƒ … an>s1 - an d . ) 29. Nicole Oresme’s Theorem Prove Nicole Oresme’s Theorem that 1 +

n

converges.

q

1 2

#

2 +

1 4

#

3 + Á +

22. How do you know that the functions sin x, ln x, and e x are not polynomials? Give reasons for your answer. 23. Find the value of a for which the limit lim

sin saxd - sin x - x

is finite and evaluate the limit.

x3

n + Á = 4. 2n - 1

(Hint: Differentiate both sides of the equation 1>s1 - xd = q 1 + g n = 1 x n .) 30. a. Show that q

nsn + 1d 2x 2 = n x sx - 1d3 n=1 a

for ƒ x ƒ 7 1 by differentiating the identity q

equal to 5.

x: 0

s2n - 1d2 un . s2nds2n + 1d

Apply Raabe’s Test to determine whether the series converges.

b. Does

q

un + 1 =

27. If g n = 1 an converges, and if an Z 1 and an 7 0 for all n,

appear to depend on the value of a? If so, how? cos sa>nd n lim a1 b , bn n: q

= -1 .

25. Raabe’s (or Gauss’s) Test The following test, which we state without proof, is an extension of the Ratio Test. q Raabe’s Test: If g n = 1 un is a series of positive constants and there exist constants C, K, and N such that

17. Evaluate a

2x 2

x:0

16. Find the sum of the infinite series

q

cos saxd - b

ax

n+1

n=1

=

x2 1 - x

twice, multiplying the result by x, and then replacing x by 1> x. b. Use part (a) to find the real solution greater than 1 of the equation q

x = a

n=1

nsn + 1d . xn

10 PARAMETRIC EQUATIONS AND POLAR COORDINATES OVERVIEW In this chapter we study new ways to define curves in the plane. Instead of thinking of a curve as the graph of a function or equation, we consider a more general way of thinking of a curve as the path of a moving particle whose position is changing over time. Then each of the x- and y-coordinates of the particle’s position becomes a function of a third variable t. We can also change the way in which points in the plane themselves are described by using polar coordinates rather than the rectangular or Cartesian system. Both of these new tools are useful for describing motion, like that of planets and satellites, or projectiles moving in the plane or space. Parabolas, ellipses, and hyperbolas (called conic sections, or conics, and reviewed in Appendix 4) model the paths traveled by projectiles, planets, or any other object moving under the sole influence of a gravitational or electromagnetic force.

10.1

Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. We are now going to introduce another way to describe a curve by expressing both coordinates as functions of a third variable t.

Parametric Equations

Position of particle at time t

Figure 10.1 shows the path of a moving particle in the xy-plane. Notice that the path fails the vertical line test, so it cannot be described as the graph of a function of the variable x. However, we can sometimes describe the path by a pair of equations, x = ƒstd and y = gstd, where ƒ and g are continuous functions. When studying motion, t usually denotes time. Equations like these describe more general curves than those like y = ƒsxd and provide not only the graph of the path traced out but also the location of the particle sx, yd = sƒstd, gstdd at any time t. ( f (t), g(t))

DEFINITION

If x and y are given as functions x = ƒstd,

FIGURE 10.1 The curve or path traced by a particle moving in the xy-plane is not always the graph of a function or single equation.

y = gstd

over an interval I of t-values, then the set of points sx, yd = sƒstd, gstdd defined by these equations is a parametric curve. The equations are parametric equations for the curve.

The variable t is a parameter for the curve, and its domain I is the parameter interval. If I is a closed interval, a … t … b, the point (ƒ(a), g(a)) is the initial point of the curve and (ƒ(b), g(b)) is the terminal point. When we give parametric equations and a parameter

563

564

Chapter 10: Parametric Equations and Polar Coordinates

interval for a curve, we say that we have parametrized the curve. The equations and interval together constitute a parametrization of the curve. A given curve can be represented by different sets of parametric equations. (See Exercises 19 and 20.)

EXAMPLE 1

Sketch the curve defined by the parametric equations x = t 2,

y = t + 1,

- q 6 t 6 q.

Solution We make a brief table of values (Table 10.1), plot the points (x, y), and draw a smooth curve through them (Figure 10.2). Each value of t gives a point (x, y) on the curve, such as t = 1 giving the point (1, 2) recorded in Table 10.1. If we think of the curve as the path of a moving particle, then the particle moves along the curve in the direction of the arrows shown in Figure 10.2. Although the time intervals in the table are equal, the consecutive points plotted along the curve are not at equal arc length distances. The reason for this is that the particle slows down at it gets nearer to the y-axis along the lower branch of the curve as t increases, and then speeds up after reaching the y-axis at (0, 1) and moving along the upper branch. Since the interval of values for t is all real numbers, there is no initial point and no terminal point for the curve. y

TABLE 10.1 Values of

t53

x = t2 and y = t + 1 for selected values of t. t

x

y

-3 -2 -1 0 1 2 3

9 4 1 0 1 4 9

-2 -1 0 1 2 3 4

t52 (4, 3)

t51 (1, 2) t 5 0 (0, 1) (1, 0) (4, –1) t 5 –1 t 5 –2

(9, 4)

x (9, –2) t 5 –3

FIGURE 10.2 The curve given by the parametric equations x = t 2 and y = t + 1 (Example 1).

EXAMPLE 2

Identify geometrically the curve in Example 1 (Figure 10.2) by eliminating the parameter t and obtaining an algebraic equation in x and y.

We solve the equation y = t + 1 for the parameter t and substitute the result into the parametric equation for x. This procedure gives t = y - 1 and Solution

x = t 2 = ( y - 1) 2 = y 2 - 2y + 1.

y t5 p 2

x2 1 y2 5 1

The equation x = y 2 - 2y + 1 represents a parabola, as displayed in Figure 10.2. It is sometimes quite difficult, or even impossible, to eliminate the parameter from a pair of parametric equations, as we did here.

P(cos t, sin t) t

t5p 0

t50 (1, 0)

x

t 5 3p 2

FIGURE 10.3 The equations x = cos t and y = sin t describe motion on the circle x 2 + y 2 = 1 . The arrow shows the direction of increasing t (Example 3).

EXAMPLE 3 (a) x = cos t, (b) x = a cos t,

Graph the parametric curves y = sin t, y = a sin t,

0 … t … 2p. 0 … t … 2p.

Solution

(a) Since x 2 + y 2 = cos2 t + sin2 t = 1, the parametric curve lies along the unit circle x 2 + y 2 = 1. As t increases from 0 to 2p, the point sx, yd = scos t, sin td starts at (1, 0) and traces the entire circle once counterclockwise (Figure 10.3).

10.1

Parametrizations of Plane Curves

565

(b) For x = a cos t, y = a sin t, 0 … t … 2p, we have x 2 + y 2 = a 2 cos2 t + a 2 sin2 t = a 2 . The parametrization describes a motion that begins at the point (a, 0) and traverses the circle x 2 + y 2 = a 2 once counterclockwise, returning to (a, 0) at t = 2p. The graph is a circle centered at the origin with radius r = a and coordinate points (a cos t, a sin t). y

EXAMPLE 4

The position P(x, y) of a particle moving in the xy-plane is given by the equations and parameter interval

y 5 x 2, x 0 t 5 4 (2, 4)

x = 1t,

We try to identify the path by eliminating t between the equations x = 1t and y = t. With any luck, this will produce a recognizable algebraic relation between x and y. We find that Solution

(1, 1) x

0 Starts at t50

FIGURE 10.4 The equations x = 1t and y = t and the interval t Ú 0 describe the path of a particle that traces the righthand half of the parabola y = x 2 (Example 4).

y

y x2

(–2, 4)

y = t =

A 1t B 2 = x 2 .

Thus, the particle’s position coordinates satisfy the equation y = x 2 , so the particle moves along the parabola y = x 2 . It would be a mistake, however, to conclude that the particle’s path is the entire parabola y = x 2 ; it is only half the parabola. The particle’s x-coordinate is never negative. The particle starts at (0, 0) when t = 0 and rises into the first quadrant as t increases (Figure 10.4). The parameter interval is [0, q d and there is no terminal point. The graph of any function y = ƒ(x) can always be given a natural parametrization x = t and y = ƒ(t). The domain of the parameter in this case is the same as the domain of the function ƒ.

(2, 4) t2

t –2

t Ú 0.

Identify the path traced by the particle and describe the motion.

P(t, t)

t51

y = t,

P(t, t 2 )

EXAMPLE 5

t1

A parametrization of the graph of the function ƒ(x) = x 2 is given by

(1, 1) 0

x = t,

x

FIGURE 10.5 The path defined by x = t, y = t 2, - q 6 t 6 q is the entire parabola y = x 2 (Example 5).

y = ƒ(t) = t 2,

- q 6 t 6 q.

When t Ú 0, this parametrization gives the same path in the xy-plane as we had in Example 4. However, since the parameter t here can now also be negative, we obtain the lefthand part of the parabola as well; that is, we have the entire parabolic curve. For this parametrization, there is no starting point and no terminal point (Figure 10.5). Notice that a parametrization also specifies when (the value of the parameter) a particle moving along the curve is located at a specific point along the curve. In Example 4, the point (2, 4) is reached when t = 4; in Example 5, it is reached “earlier” when t = 2. You can see the implications of this aspect of parametrizations when considering the possibility of two objects coming into collision: They have to be at the exact same location point P(x, y) for some (possibly different) values of their respective parameters. We will say more about this aspect of parametrizations when we study motion in Chapter 12.

EXAMPLE 6

Find a parametrization for the line through the point (a, b) having slope m.

Solution A Cartesian equation of the line is y - b = m(x - a). If we set the parameter t = x - a, we find that x = a + t and y - b = mt. That is,

x = a + t,

y = b + mt,

-q 6 t 6 q

parametrizes the line. This parametrization differs from the one we would obtain by the technique used in Example 5 when t = x . However, both parametrizations give the same line.

566

Chapter 10: Parametric Equations and Polar Coordinates

TABLE 10.2 Values of x = t + s1>td and y = t - s1>td for selected values of t.

t

1> t

x

y

0.1 0.2 0.4 1.0 2.0 5.0 10.0

10.0 5.0 2.5 1.0 0.5 0.2 0.1

10.1 5.2 2.9 2.0 2.5 5.2 10.1

9.9 4.8 2.1 0.0 1.5 4.8 9.9

EXAMPLE 7

Sketch and identify the path traced by the point P(x, y) if 1 x = t + t,

1 y = t - t,

t 7 0.

We make a brief table of values in Table 10.2, plot the points, and draw a smooth curve through them, as we did in Example 1. Next we eliminate the parameter t from the equations. The procedure is more complicated than in Example 2. Taking the difference between x and y as given by the parametric equations, we find that Solution

1 1 2 x - y = at + t b - at - t b = t . If we add the two parametric equations, we get 1 1 x + y = at + t b + at - t b = 2t. We can then eliminate the parameter t by multiplying these last equations together: 2 sx - ydsx + yd = a t bs2td = 4,

y t 10

10

(10.1, 9.9) 5

t5

or, multiplying together the terms on the left-hand side, we obtain a standard equation for a hyperbola (reviewed in Appendix 4):

(5.2, 4.8)

t2 t 1 (2.5, 1.5) 5 0 (2, 0) (2.9, –2.1) t 0.4 (5.2, –4.8) –5 t 0.2

x 2 - y 2 = 4.

(1)

Thus the coordinates of all the points P(x, y) described by the parametric equations satisfy Equation (1). However, Equation (1) does not require that the x-coordinate be positive. So there are points (x, y) on the hyperbola that do not satisfy the parametric equation x = t + (1>t), t 7 0, for which x is always positive. That is, the parametric equations do not yield any points on the left branch of the hyperbola given by Equation (1), points where (10.1, –9.9) the x-coordinate would be negative. For small positive values of t, the path lies in the –10 fourth quadrant and rises into the first quadrant as t increases, crossing the x-axis when t 0.1 t = 1 (see Figure 10.6). The parameter domain is (0, q ), and there is no starting point and FIGURE 10.6 The curve for x = t + s1>td, no terminal point for the path. 10

y = t - s1>td, t 7 0 in Example 7. (The part shown is for 0.1 … t … 10.)

x

Examples 4, 5, and 6 illustrate that a given curve, or portion of it, can be represented by different parametrizations. In the case of Example 7, we can also represent the right-hand branch of the hyperbola by the parametrization x = 24 + t 2,

y = t,

- q 6 t 6 q,

which is obtained by solving Equation (1) for x Ú 0 and letting y be the parameter. Still another parametrization for the right-hand branch of the hyperbola given by Equation (1) is x = 2 sec t,

y = 2 tan t,

-

p p 6 t 6 . 2 2

This parametrization follows from the trigonometric identity sec2 t - tan2 t = 1, so x 2 - y 2 = 4 sec2 t - 4 tan2 t = 4(sec2 t - tan2 t) = 4. HISTORICAL BIOGRAPHY Christian Huygens (1629–1695)

As t runs between -p>2 and p>2, x = sec t remains positive and y = tan t runs between - q and q , so P traverses the hyperbola’s right-hand branch. It comes in along the branch’s lower half as t : 0 - , reaches (2, 0) at t = 0, and moves out into the first quadrant as t increases steadily toward p>2. This is the same hyperbola branch for which a portion is shown in Figure 10.6.

10.1

Parametrizations of Plane Curves

567

Cycloids Guard cycloid

Guard cycloid

The problem with a pendulum clock whose bob swings in a circular arc is that the frequency of the swing depends on the amplitude of the swing. The wider the swing, the longer it takes the bob to return to center (its lowest position). This does not happen if the bob can be made to swing in a cycloid. In 1673, Christian Huygens designed a pendulum clock whose bob would swing in a cycloid, a curve we define in Example 8. He hung the bob from a fine wire constrained by guards that caused it to draw up as it swung away from center (Figure 10.7).

Cycloid

FIGURE 10.7 In Huygens’ pendulum clock, the bob swings in a cycloid, so the frequency is independent of the amplitude. y

EXAMPLE 8 A wheel of radius a rolls along a horizontal straight line. Find parametric equations for the path traced by a point P on the wheel’s circumference. The path is called a cycloid.

P(x, y) (at a cos , a a sin )

We take the line to be the x-axis, mark a point P on the wheel, start the wheel with P at the origin, and roll the wheel to the right. As parameter, we use the angle t through which the wheel turns, measured in radians. Figure 10.8 shows the wheel a short while later when its base lies at units from the origin. The wheel’s center C lies at (at, a) and the coordinates of P are Solution

t

M

at

0

a C(at, a)

x

x = at + a cos u,

FIGURE 10.8 The position of P(x, y) on the rolling wheel at angle t (Example 8).

y = a + a sin u.

To express u in terms of t, we observe that t + u = 3p>2 in the figure, so that

y

u = (x, y)

3p - t. 2

This makes

t a O

2a

x

FIGURE 10.9 The cycloid curve for t Ú 0 given by Eqs. (2) derived in Example 8.

O

a

2a

a

2a

3p - tb = -sin t, 2

sin u = sin a

3p - tb = -cos t. 2

The equations we seek are x = at - a sin t,

y = a - a cos t.

x

P(at a sin t, a a cos t)

a

cos u = cos a

These are usually written with the a factored out: x = ast - sin td,

2a

y = as1 - cos td.

(2)

B(a, 2a)

Figure 10.9 shows the first arch of the cycloid and part of the next.

y

FIGURE 10.10 To study motion along an upside-down cycloid under the influence of gravity, we turn Figure 10.9 upside down. This points the y-axis in the direction of the gravitational force and makes the downward y-coordinates positive. The equations and parameter interval for the cycloid are still x = ast - sin td, y = as1 - cos td,

t Ú 0.

The arrow shows the direction of increasing t.

Brachistochrones and Tautochrones If we turn Figure 10.9 upside down, Equations (2) still apply and the resulting curve (Figure 10.10) has two interesting physical properties. The first relates to the origin O and the point B at the bottom of the first arch. Among all smooth curves joining these points, the cycloid is the curve along which a frictionless bead, subject only to the force of gravity, will slide from O to B the fastest. This makes the cycloid a brachistochrone (“brah-kiss-toe-krone”), or shortest-time curve for these points. The second property is that even if you start the bead partway down the curve toward B, it will still take the bead the same amount of time to reach B. This makes the cycloid a tautochrone (“taw-toekrone”), or same-time curve for O and B. It can be shown that the cycloid from O to B is the one and only brachistochrone for O and B. We omit the argument here.

568

Chapter 10: Parametric Equations and Polar Coordinates

Exercises 10.1 Finding Cartesian from Parametric Equations Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. 1. x = 3t, y = 9t 2, - q 6 t 6 q 2. x = - 1t,

y = t,

t Ú 0

3. x = 2t - 5,

y = 4t - 7,

4. x = 3 - 3t,

y = 2t,

5. x = cos 2t,

y = sin 2t,

6. x = cos sp - td,

-q 6 t 6 q

10. x = 11. x = 12. x = 13. x =

y = sin sp - td,

y = 1t,

15. x = sec t - 1, 2

16. x = -sec t,

28. Find parametric equations and a parameter interval for the motion of a particle that moves along the graph of y = x 2 in the following way: beginning at (0, 0) it moves to (3, 9), and then travels back and forth from (3, 9) to (-3, 9) infinitely many times. x 2 + y 2 = a 2,

0 … t … p 0 … t … p

0 … t … 2p 0 … t … 2p p p sin t, y = cos 2t, - … t … 2 2 1 + sin t, y = cos t - 2, 0 … t … p t 2, y = t 6 - 2t 4, - q 6 t 6 q t - 2 t , y = , -1 6 t 6 1 t - 1 t + 1 t, y = 21 - t 2, -1 … t … 0

14. x = 2t + 1,

27. Find parametric equations and a parameter interval for the motion of a particle starting at the point (2, 0) and tracing the top half of the circle x 2 + y 2 = 4 four times.

29. Find parametric equations for the semicircle

0 … t … 1

7. x = 4 cos t, y = 2 sin t, 8. x = 4 sin t, y = 5 cos t, 9. x =

26. the ray (half line) with initial point s -1, 2d that passes through the point (0, 0)

y 7 0,

using as parameter the slope t = dy>dx of the tangent to the curve at (x, y). 30. Find parametric equations for the circle x 2 + y 2 = a 2, using as parameter the arc length s measured counterclockwise from the point (a, 0) to the point (x, y). 31. Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle u in the accompanying figure as the parameter. y

t Ú 0

y = tan t,

-p>2 6 t 6 p>2

-p>2 6 t 6 p>2 17. x = -cosh t, y = sinh t, - q 6 t 6 q 18. x = 2 sinh t, y = 2 cosh t, - q 6 t 6 q

2 (x, y)

y = tan t,

Finding Parametric Equations 19. Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x 2 + y 2 = a 2 a. once clockwise.

b. once counterclockwise.

c. twice clockwise.

d. twice counterclockwise.

u 4

0

32. Find a parametrization for the curve y = 2x with terminal point (0, 0) using the angle u in the accompanying figure as the parameter. y y 5 x (x, y)

(There are many ways to do these, so your answers may not be the same as the ones in the back of the book.) 20. Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the ellipse sx 2>a 2 d + sy 2>b 2 d = 1 a. once clockwise.

b. once counterclockwise.

c. twice clockwise.

d. twice counterclockwise.

u

21. the line segment with endpoints s - 1, -3d and (4, 1)

x

0

33. Find a parametrization for the circle sx - 2d2 + y 2 = 1 starting at (1, 0) and moving clockwise once around the circle, using the central angle u in the accompanying figure as the parameter.

(As in Exercise 19, there are many correct answers.) In Exercises 21–26, find a parametrization for the curve.

y (x, y)

1 1

22. the line segment with endpoints s - 1, 3d and s3, -2d 23. the lower half of the parabola x - 1 = y 2 24. the left half of the parabola y = x 2 + 2x 25. the ray (half line) with initial point (2, 3) that passes through the point s -1, -1d

x

u 0

1

2

3

x

10.1 34. Find a parametrization for the circle x 2 + y 2 = 1 starting at (1, 0) and moving counterclockwise to the terminal point (0, 1), using the angle u in the accompanying figure as the parameter.

Parametrizations of Plane Curves

569

37. As the point N moves along the line y = a in the accompanying figure, P moves in such a way that OP = MN . Find parametric equations for the coordinates of P as functions of the angle t that the line ON makes with the positive y-axis.

y y

(0, 1)

(x, y)

A(0, a)

1

N

u (1, 0)

–2

M

x t P

x

O

35. The witch of Maria Agnesi The bell-shaped witch of Maria Agnesi can be constructed in the following way. Start with a circle of radius 1, centered at the point (0, 1), as shown in the accompanying figure. Choose a point A on the line y = 2 and connect it to the origin with a line segment. Call the point where the segment crosses the circle B. Let P be the point where the vertical line through A crosses the horizontal line through B. The witch is the curve traced by P as A moves along the line y = 2. Find parametric equations and a parameter interval for the witch by expressing the coordinates of P in terms of t, the radian measure of the angle that segment O A makes with the positive x-axis. The following equalities (which you may assume) will help. a. x = AQ

b. y = 2 - AB sin t

c. AB # OA = (AQ) 2

38. Trochoids A wheel of radius a rolls along a horizontal straight line without slipping. Find parametric equations for the curve traced out by a point P on a spoke of the wheel b units from its center. As parameter, use the angle u through which the wheel turns. The curve is called a trochoid, which is a cycloid when b = a . Distance Using Parametric Equations 39. Find the point on the parabola x = t, y = t 2, - q 6 t 6 q , closest to the point (2, 1> 2). (Hint: Minimize the square of the distance as a function of t.) 40. Find the point on the ellipse x = 2 cos t, y = sin t, 0 … t … 2p closest to the point (3> 4, 0). (Hint: Minimize the square of the distance as a function of t.)

y y52

Q

(0, 1)

T GRAPHER EXPLORATIONS If you have a parametric equation grapher, graph the equations over the given intervals in Exercises 41–48.

A P(x, y)

B

41. Ellipse

y = 2 sin t,

a. 0 … t … 2p

t x

O

36. Hypocycloid When a circle rolls on the inside of a fixed circle, any point P on the circumference of the rolling circle describes a hypocycloid. Let the fixed circle be x 2 + y 2 = a 2, let the radius of the rolling circle be b, and let the initial position of the tracing point P be A (a, 0). Find parametric equations for the hypocycloid, using as the parameter the angle u from the positive x-axis to the line joining the circles’ centers. In particular, if b = a>4, as in the accompanying figure, show that the hypocycloid is the astroid x = a cos3 u,

x = 4 cos t,

y = a sin3 u.

b. 0 … t … p

c. -p>2 … t … p>2 .

42. Hyperbola branch x = sec t (enter as 1> cos (t)), y = tan t (enter as sin (t)> cos (t)), over a. -1.5 … t … 1.5

b. -0.5 … t … 0.5

c. -0.1 … t … 0.1 . 43. Parabola

x = 2t + 3,

44. Cycloid

x = t - sin t,

a. 0 … t … 2p

y = t 2 - 1,

-2 … t … 2

y = 1 - cos t,

over

b. 0 … t … 4p

c. p … t … 3p . 45. Deltoid x = 2 cos t + cos 2t,

y

over

y = 2 sin t - sin 2t;

0 … t … 2p

What happens if you replace 2 with -2 in the equations for x and y? Graph the new equations and find out. 46. A nice curve

u O

C b P

x = 3 cos t + cos 3t, A(a, 0) x

y = 3 sin t - sin 3t;

0 … t … 2p

What happens if you replace 3 with -3 in the equations for x and y? Graph the new equations and find out.

570

Chapter 10: Parametric Equations and Polar Coordinates 48. a. x = 6 cos t + 5 cos 3t, 0 … t … 2p

47. a. Epicycloid x = 9 cos t - cos 9t,

y = 9 sin t - sin 9t;

0 … t … 2p

b. x = 6 cos 2t + 5 cos 6t, 0 … t … p

b. Hypocycloid x = 8 cos t + 2 cos 4t,

y = 8 sin t - 2 sin 4t;

c. x = 6 cos t + 5 cos 3t, 0 … t … 2p

0 … t … 2p

d. x = 6 cos 2t + 5 cos 6t, 0 … t … p

c. Hypotrochoid x = cos t + 5 cos 3t,

10.2

y = 6 cos t - 5 sin 3t;

0 … t … 2p

y = 6 sin t - 5 sin 3t; y = 6 sin 2t - 5 sin 6t; y = 6 sin 2t - 5 sin 3t; y = 6 sin 4t - 5 sin 6t;

Calculus with Parametric Curves In this section we apply calculus to parametric curves. Specifically, we find slopes, lengths, and areas associated with parametrized curves.

Tangents and Areas A parametrized curve x = ƒstd and y = gstd is differentiable at t if ƒ and g are differentiable at t. At a point on a differentiable parametrized curve where y is also a differentiable function of x, the derivatives dy>dt, dx>dt, and dy>dx are related by the Chain Rule: dy dy dx # . = dt dx dt If dx>dt Z 0, we may divide both sides of this equation by dx>dt to solve for dy>dx. Parametric Formula for dy/dx If all three derivatives exist and dx>dt Z 0, dy>dt dy = . dx dx>dt

(1)

If parametric equations define y as a twice-differentiable function of x, we can apply Equation (1) to the function dy>dx = y¿ to calculate d 2y>dx 2 as a function of t: d 2y dx 2

=

dy¿>dt d s y¿d = . dx dx>dt

Eq. (1) with y¿ in place of y

Parametric Formula for d 2y/dx 2 If the equations x = ƒstd, y = gstd define y as a twice-differentiable function of x, then at any point where dx>dt Z 0 and y¿ = dy>dx, d 2y dx

2

=

dy¿>dt . dx>dt

(2)

10.2

EXAMPLE 1

y

Find the tangent to the curve

2

1

0

x = sec t, t5 p 4 (2, 1)

y = tan t,

-

p p 6 t 6 , 2 2

at the point (12, 1), where t = p>4 (Figure 10.11).

x 2 x 5 sec t, y 5 tan t, –p ,t, p 2 2

1

Calculus with Parametric Curves

Solution

FIGURE 10.11 The curve in Example 1 is the right-hand branch of the hyperbola x 2 - y 2 = 1.

The slope of the curve at t is dy>dt dy sec2 t sec t = = sec t tan t = tan t . dx dx>dt

Eq. (1)

Setting t equal to p>4 gives sec (p>4) dy ` t = p>4 = dx tan (p>4) =

12 = 12. 1

The tangent line is y - 1 = 12 sx - 12d y = 12 x - 2 + 1 y = 12 x - 1. Find d 2y>dx 2 as a function of t if x = t - t 2, y = t - t 3 .

EXAMPLE 2 Solution Finding d2y>dx 2 in Terms of t 1. Express y¿ = dy>dx in terms of t. 2. Find dy¿>dt. 3. Divide dy¿>dt by dx>dt.

1.

Express y¿ = dy>dx in terms of t. y¿ =

2.

Differentiate y¿ with respect to t. dy¿ 2 - 6t + 6t 2 d 1 - 3t 2 = a b = dt dt 1 - 2t s1 - 2td2

y 1 x cos3 t y sin3 t 0 t 2

–1

0

dy>dt dy 1 - 3t 2 = = 1 - 2t dx dx>dt

3.

1

x

–1

FIGURE 10.12 The astroid in Example 3.

Derivative Quotient Rule

Divide dy¿>dt by dx>dt. d 2y dx 2

EXAMPLE 3

=

dy¿>dt s2 - 6t + 6t 2 d>s1 - 2td2 2 - 6t + 6t 2 = = 1 - 2t dx>dt s1 - 2td3

Find the area enclosed by the astroid (Figure 10.12) x = cos3 t,

y = sin3 t,

0 … t … 2p.

Eq. (2)

571

572

Chapter 10: Parametric Equations and Polar Coordinates

By symmetry, the enclosed area is 4 times the area beneath the curve in the first quadrant where 0 … t … p>2. We can apply the definite integral formula for area studied in Chapter 5, using substitution to express the curve and differential dx in terms of the parameter t. So, Solution

1

A = 4

y dx

L0

p>2

= 4

L0

sin3 t # 3 cos2 t sin t dt

p>2

= 12

L0

a

Substitution for y and dx

2

1 - cos 2t 1 + cos 2t b a b dt 2 2

sin4 t = a

1 - cos 2t 2 b 2

3 2 L0

p>2

=

3 2 L0

p>2

= =

3 c 2 L0

=

3 1 1 1 1 1 c at - sin 2tb - at + sin 2tb + asin 2t - sin3 2tb d 2 2 2 4 2 3 0

=

3 p 1 p 1 c a - 0 - 0 - 0b - a + 0 - 0 - 0b + (0 - 0 - 0 + 0) d 2 2 2 2 2

=

3p . 8

(1 - 2 cos 2t + cos2 2t)(1 + cos 2t) dt

Expand square term.

(1 - cos 2t - cos2 2t + cos3 2t) dt

Multiply terms.

p>2

p>2

(1 - cos 2t) dt -

L0

p>2

cos2 2t dt +

L0

cos3 2t dt d p>2

Section 8.2, Example 3

Evaluate.

Length of a Parametrically Defined Curve Let C be a curve given parametrically by the equations y

x = ƒstd

Pk Pk –1

B Pn

P2 P1

0

y = gstd,

a … t … b.

We assume the functions ƒ and g are continuously differentiable (meaning they have continuous first derivatives) on the interval [a, b]. We also assume that the derivatives ƒ ¿(t) and g ¿(t) are not simultaneously zero, which prevents the curve C from having any corners or cusps. Such a curve is called a smooth curve. We subdivide the path (or arc) AB into n pieces at points A = P0, P1, P2, Á , Pn = B (Figure 10.13). These points correspond to a partition of the interval [a, b] by a = t0 6 t1 6 t2 6 Á 6 tn = b, where Pk = sƒstk d, gstk dd. Join successive points of this subdivision by straight line segments (Figure 10.13). A representative line segment has length

C

A P0

and

x

Lk = 2s¢xk d2 + s¢yk d2 = 2[ƒstk d - ƒstk - 1 d]2 + [gstk d - gstk - 1 d]2

FIGURE 10.13 The smooth curve C defined parametrically by the equations x = ƒstd and (see Figure 10.14). If ¢tk is small, the length Lk is approximately the length of arc Pk - 1Pk . y = gstd, a … t … b . The length of the By the Mean Value Theorem there are numbers tk* and tk** in [tk - 1, tk] such that curve from A to B is approximated by the sum of the lengths of the polygonal path ¢xk = ƒstk d - ƒstk - 1 d = ƒ¿stk* d ¢tk , (straight line segments) starting at A = P0 , ¢yk = gstk d - gstk - 1 d = g¿stk**d ¢tk . then to P1 , and so on, ending at B = Pn .

10.2 y Pk ( f (tk), g(tk))

Lk

573

Assuming the path from A to B is traversed exactly once as t increases from t = a to t = b, with no doubling back or retracing, an approximation to the (yet to be defined) “length” of the curve AB is the sum of all the lengths Lk :

Δyk

n

n

2 2 a Lk = a 2s¢xk d + s¢yk d

Δxk Pk –1 ( f (tk –1 ), g(tk –1 )) 0

Calculus with Parametric Curves

k=1

k=1 n

= a 2[ƒ¿stk* d]2 + [g¿stk** d]2 ¢tk .

x

FIGURE 10.14 The arc Pk - 1 Pk is approximated by the straight line segment shown here, which has length Lk = 2s¢xk d2 + s¢yk d2 .

k=1

Although this last sum on the right is not exactly a Riemann sum (because ƒ¿ and g¿ are evaluated at different points), it can be shown that its limit, as the norm of the partition tends to zero and the number of segments n : q , is the definite integral n

b

lim a 2[ƒ ¿ (t k*)]2 + [g¿(t k**)]2 ¢tk = ƒ ƒ P ƒ ƒ :0 k=1

La

2[ƒ¿std]2 + [g¿std]2 dt.

Therefore, it is reasonable to define the length of the curve from A to B as this integral.

DEFINITION If a curve C is defined parametrically by x = ƒstd and y = gstd, a … t … b, where ƒ¿ and g¿ are continuous and not simultaneously zero on [a, b], and C is traversed exactly once as t increases from t = a to t = b, then the length of C is the definite integral b

L =

2[ƒ¿std]2 + [g¿std]2 dt.

La

A smooth curve C does not double back or reverse the direction of motion over the time interval [a, b] since sƒ¿d2 + sg¿d2 7 0 throughout the interval. At a point where a curve does start to double back on itself, either the curve fails to be differentiable or both derivatives must simultaneously equal zero. We will examine this phenomenon in Chapter 12, where we study tangent vectors to curves. If x = ƒstd and y = gstd, then using the Leibniz notation we have the following result for arc length: b

L =

2 dy 2 dx b + a b dt. dt La B dt

a

(3)

What if there are two different parametrizations for a curve C whose length we want to find; does it matter which one we use? The answer is no, as long as the parametrization we choose meets the conditions stated in the definition of the length of C (see Exercise 41 for an example).

EXAMPLE 4

Using the definition, find the length of the circle of radius r defined para-

metrically by x = r cos t

and

y = r sin t,

0 … t … 2p.

574

Chapter 10: Parametric Equations and Polar Coordinates Solution

As t varies from 0 to 2p, the circle is traversed exactly once, so the circumfer-

ence is 2p

L =

L0

2 dy 2 dx b + a b dt. dt B dt

a

We find dy = r cos t dt

dx = -r sin t, dt and a

2 dy 2 dx b + a b = r 2ssin2 t + cos2 td = r 2. dt dt

So 2p

L =

EXAMPLE 5

L0

2r 2 dt = r C t D 0 = 2pr. 2p

Find the length of the astroid (Figure 10.12) y = sin3 t,

x = cos3 t,

0 … t … 2p.

Because of the curve’s symmetry with respect to the coordinate axes, its length is four times the length of the first-quadrant portion. We have

Solution

x = cos3 t,

y = sin3 t

2

a

dx b = [3 cos2 ts -sin td]2 = 9 cos4 t sin2 t dt

a

dy 2 b = [3 sin2 tscos td]2 = 9 sin4 t cos2 t dt

2 dy 2 dx 2 b + a b = 29 cos2 t sin2 tscos t + sin2 td ('')''* dt B dt

a

1

= 29 cos2 t sin2 t = 3 ƒ cos t sin t ƒ

cos t sin t Ú 0 for 0 … t … p>2

= 3 cos t sin t. Therefore, p>2

Length of first-quadrant portion = =

L0 3 2 L0

3 cos t sin t dt sin 2t dt p>2

= -

cos t sin t = s1>2d sin 2t

p>2

3 cos 2t d 4 0

=

The length of the astroid is four times this: 4s3>2d = 6.

3 . 2

10.2

Calculus with Parametric Curves

575

HISTORICAL BIOGRAPHY

Length of a Curve y = ƒsxd

Gregory St. Vincent (1584–1667)

The length formula in Section 6.3 is a special case of Equation (3). Given a continuously differentiable function y = ƒsxd, a … x … b, we can assign x = t as a parameter. The graph of the function ƒ is then the curve C defined parametrically by x = t

y = ƒstd,

and

a … t … b,

a special case of what we considered before. Then, dx = 1 dt

dy = ƒ¿std. dt

and

From Equation (1), we have dy>dt dy = = ƒ¿std, dx dx>dt giving a

2 dy 2 dx b + a b = 1 + [ƒ¿std]2 dt dt

= 1 + [ƒ¿sxd]2 .

t = x

Substitution into Equation (3) gives the arc length formula for the graph of y = ƒsxd, consistent with Equation (3) in Section 6.3.

The Arc Length Differential Consistent with our discussion in Section 6.3, we can define the arc length function for a parametrically defined curve x = ƒ(t) and y = g(t), a … t … b, by t

s(t) =

La

2[ƒ¿(z)]2 + [g¿(z)]2 dz.

Then, by the Fundamental Theorem of Calculus, 2 dy 2 dx ds a b + a b . = 2[ƒ¿(t)]2 + [g¿(t)]2 = dt dt B dt

The differential of arc length is ds =

2 dy 2 dx b + a b dt. dt B dt

a

(4)

Equation (4) is often abbreviated to ds = 2dx 2 + dy 2. Just as in Section 6.3, we can integrate the differential ds between appropriate limits to find the total length of a curve. Here’s an example where we use the arc length formula to find the centroid of an arc.

EXAMPLE 6

Find the centroid of the first-quadrant arc of the astroid in Example 5.

We take the curve’s density to be d = 1 and calculate the curve’s mass and moments about the coordinate axes as we did in Section 6.6.

Solution

576

Chapter 10: Parametric Equations and Polar Coordinates

The distribution of mass is symmetric about the line y = x, so x = y. A typical segment of the curve (Figure 10.15) has mass

y B(0, 1) ~~ (x, y) (cos 3 t, sin3 t)

dm = 1 # ds =

x~ ds ~y

2 dy 2 dx b + a b dt = 3 cos t sin t dt. dt B dt

a

From Example 5

c.m.

The curve’s mass is 0

A(1, 0)

x p>2

M =

FIGURE 10.15 The centroid (c.m.) of the astroid arc in Example 6.

L0

p>2

dm =

3 cos t sin t dt =

L0

3 . 2

Again from Example 5

The curve’s moment about the x-axis is p>2

Mx =

L

y˜ dm =

L0

sin3 t # 3 cos t sin t dt

p>2

= 3

L0

sin4 t cos t dt = 3 #

p>2

sin5t d 5 0

=

3 . 5

It follows that y =

3>5 Mx 2 = = . M 5 3>2

The centroid is the point (2>5, 2>5).

Areas of Surfaces of Revolution In Section 6.4 we found integral formulas for the area of a surface when a curve is revolved about a coordinate axis. Specifically, we found that the surface area is S = 1 2py ds for revolution about the x-axis, and S = 1 2px ds for revolution about the y-axis. If the curve is parametrized by the equations x = ƒ(t) and y = g(t), a … t … b, where ƒ and g are continuously differentiable and (ƒ¿) 2 + (g¿) 2 7 0 on [a, b], then the arc length differential ds is given by Equation (4). This observation leads to the following formulas for area of surfaces of revolution for smooth parametrized curves. Area of Surface of Revolution for Parametrized Curves If a smooth curve x = ƒstd, y = gstd, a … t … b, is traversed exactly once as t increases from a to b, then the areas of the surfaces generated by revolving the curve about the coordinate axes are as follows. 1. Revolution about the x-axis ( y » 0): b

S =

2py

La

2 dy 2 dx b + a b dt dt B dt

a

(5)

2. Revolution about the y-axis (x » 0): b

S =

La

2px

2 dy 2 dx b + a b dt dt B dt

a

(6)

As with length, we can calculate surface area from any convenient parametrization that meets the stated criteria.

10.2

577

EXAMPLE 7 The standard parametrization of the circle of radius 1 centered at the point (0, 1) in the xy-plane is

y

Circle x 5 cos t y 5 1 1 sin t 0 t 2p

Calculus with Parametric Curves

x = cos t,

y = 1 + sin t,

0 … t … 2p.

Use this parametrization to find the area of the surface swept out by revolving the circle about the x-axis (Figure 10.16).

(0, 1)

Solution

We evaluate the formula b

S =

x

La

2py

Eq. (5) for revolution about the x-axis; y = 1 + sin t Ú 0

2 dy 2 dx b + a b dt dt B dt

a

2p

=

L0

2ps1 + sin td2s(''')'''* -sin td2 + scos td2 dt 1 2p

= 2p s1 + sin td dt L0

= 2p C t - cos t D 0 = 4p2.

FIGURE 10.16 In Example 7 we calculate the area of the surface of revolution swept out by this parametrized curve.

2p

Exercises 10.2 Tangents to Parametrized Curves In Exercises 1–14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d 2y>dx 2 at this point.

18. x sin t + 2x = t, 19. x = t + t, 3

t sin t - 2t = y,

y + 2t = 2x + t , 3

20. t = ln (x - t),

2

y = te t,

t = p t = 1

t = 0

1. x = 2 cos t, y = 2 sin t, t = p>4 2. x = sin 2pt, y = cos 2pt, t = -1>6 3. x = 4 sin t, y = 2 cos t, t = p>4

Area 21. Find the area under one arch of the cycloid

4. x = cos t,

22. Find the area enclosed by the y-axis and the curve

5. x = t,

y = 23 cos t,

y = 2t,

6. x = sec2 t - 1, 7. x = sec t,

8. x = - 2t + 1, 9. x = 2t + 3, 10. x = 1>t,

y = t , 4

y = 1 + sin t,

1 , 13. x = t + 1

t y = , t - 1 y = 1 - e t,

a. x = t 2,

t = 1 t = p>3

16. x = 25 - 1t, 17. x + 2x

3>2

t = 2

26. x = t 3,

27. x = t >2,

y = t9

t = 2 t = 4 t = 0

0 … t … 13

y = (2t + 1) 3>2>3,

28. x = (2t + 3) 3>2>3,

y2t + 1 + 2t2y = 4,

b. x = t 3,

y = 3t 2>2,

2

t = 0

y(t - 1) = 2t,

= t + t, 2

y = t6

Lengths of Curves Find the lengths of the curves in Exercises 25–30. 25. x = cos t, y = t + sin t, 0 … t … p

t = p>2

2y 3 - 3t 2 = 4,

0 … t … 2p .

24. Find the area under y = x over [0, 1] using the following parametrizations.

Implicitly Defined Parametrizations Assuming that the equations in Exercises 15–20 define x and y implicitly as differentiable functions x = ƒ(t), y = g(t), find the slope of the curve x = ƒ(t), y = g(t) at the given value of t. 15. x 3 + 2t 2 = 9,

y = b sin t, 3

t = 3

y = 1 - cos t,

12. x = cos t,

14. x = t + e t,

x = a cos t,

t = -1

y = -2 + ln t,

11. x = t - sin t,

23. Find the area enclosed by the ellipse

t = -p>4

t = p>6

y = 23t,

y = 1 + e -t .

x = t - t 2,

t = 1>4

y = tan t,

2

t = 2p>3

y = tan t,

y = a(1 - cos t).

x = a(t - sin t),

0 … t … 4

y = t + t 2>2,

29. x = 8 cos t + 8t sin t y = 8 sin t - 8t cos t, 0 … t … p>2

0 … t … 3

30. x = ln (sec t + tan t) - sin t y = cos t, 0 … t … p>3

Surface Area Find the areas of the surfaces generated by revolving the curves in Exercises 31–34 about the indicated axes. 31. x = cos t,

y = 2 + sin t,

0 … t … 2p;

x-axis

578

Chapter 10: Parametric Equations and Polar Coordinates

32. x = s2>3dt 3>2,

y = 2 2t,

0 … t … 23;

y-axis

y

33. x = t + 22, y = st >2d + 22t, - 22 … t … 22; y-axis 34. x = ln ssec t + tan td - sin t, y = cos t, 0 … t … p>3; x-axis 2

3

35. A cone frustum The line segment joining the points (0, 1) and (2, 2) is revolved about the x-axis to generate a frustum of a cone. Find the surface area of the frustum using the parametrization x = 2t, y = t + 1, 0 … t … 1 . Check your result with the geometry formula: Area = psr1 + r2 dsslant heightd. 36. A cone The line segment joining the origin to the point (h, r) is revolved about the x-axis to generate a cone of height h and base radius r. Find the cone’s surface area with the parametric equations x = ht, y = rt, 0 … t … 1 . Check your result with the geometry formula: Area = prsslant heightd.

1

44. The curve with parametric equations x = t,

y = sin t - t cos t,

0 … t … p>2.

y = e t sin t,

0 … t … 2p

y

38. Find the coordinates of the centroid of the curve x = e t cos t,

y = 1 - cos t,

is called a sinusoid and is shown in the accompanying figure. Find the point (x, y) where the slope of the tangent line is a. largest b. smallest.

Centroids 37. Find the coordinates of the centroid of the curve x = cos t + t sin t,

x

1

–1

0 … t … p.

2

39. Find the coordinates of the centroid of the curve x = cos t,

y = t + sin t,

0 … t … p.

T 40. Most centroid calculations for curves are done with a calculator or computer that has an integral evaluation program. As a case in point, find, to the nearest hundredth, the coordinates of the centroid of the curve x = t 3,

y = 3t 2>2,

0 … t … 13.

Theory and Examples 41. Length is independent of parametrization To illustrate the fact that the numbers we get for length do not depend on the way we parametrize our curves (except for the mild restrictions preventing doubling back mentioned earlier), calculate the length of the semicircle y = 21 - x 2 with these two different parametrizations: a. x = cos 2t,

y = sin 2t,

b. x = sin pt,

y = cos pt,

0 … t … p>2.

x

2p

0

T The curves in Exercises 45 and 46 are called Bowditch curves or Lissajous figures. In each case, find the point in the interior of the first quadrant where the tangent to the curve is horizontal, and find the equations of the two tangents at the origin. 45.

46. y

–1

y

x sin t y sin 2t

1

-1>2 … t … 1>2.

1

x

–1

x sin 2t y sin 3t

1

x

–1

42. a. Show that the Cartesian formula d

L =

1 + a

2

dx b dy dy

Lc B for the length of the curve x = g(y), c … y … d (Section 6.3, Equation 4), is a special case of the parametric length formula b

2 dy 2 dx b + a b dt. dt La B dt Use this result to find the length of each curve.

L =

a

b. x = y 3>2, 0 … y … 4>3 3 c. x = y 2>3, 0 … y … 1 2

47. Cycloid a. Find the length of one arch of the cycloid x = a(t - sin t),

y = a(1 - cos t).

b. Find the area of the surface generated by revolving one arch of the cycloid in part (a) about the x-axis for a = 1. 48. Volume Find the volume swept out by revolving the region bounded by the x-axis and one arch of the cycloid x = t - sin t,

y = 1 - cos t

about the x-axis.

43. The curve with parametric equations x = (1 + 2 sin u) cos u,

y = (1 + 2 sin u) sin u

is called a limaçon and is shown in the accompanying figure. Find the points (x, y) and the slopes of the tangent lines at these points for a. u = 0.

b. u = p>2 .

c. u = 4p>3 .

COMPUTER EXPLORATIONS In Exercises 49–52, use a CAS to perform the following steps for the given curve over the closed interval. a. Plot the curve together with the polygonal path approximations for n = 2, 4, 8 partition points over the interval. (See Figure 10.13.)

10.3 1 3 t , 3

579

1 2 t , 0 … t … 1 2

b. Find the corresponding approximation to the length of the curve by summing the lengths of the line segments.

49. x =

c. Evaluate the length of the curve using an integral. Compare your approximations for n = 2, 4, 8 with the actual length given by the integral. How does the actual length compare with the approximations as n increases? Explain your answer.

50. x = 2t 3 - 16t 2 + 25t + 5,

y =

Polar Coordinates

y = t 2 + t - 3,

0 … t … 6 51. x = t - cos t, 52. x = e cos t, t

y = 1 + sin t, y = e sin t, t

-p … t … p

0 … t … p

Polar Coordinates

10.3

In this section we study polar coordinates and their relation to Cartesian coordinates. You will see that polar coordinates are very useful for calculating many multiple integrals studied in Chapter 14.

P(r, ) r Origin (pole)

O

Initial ray

x

Definition of Polar Coordinates

FIGURE 10.17 To define polar coordinates for the plane, we start with an origin, called the pole, and an initial ray.

To define polar coordinates, we first fix an origin O (called the pole) and an initial ray from O (Figure 10.17). Usually the positive x-axis is chosen as the initial ray. Then each point P can be located by assigning to it a polar coordinate pair sr, ud in which r gives the directed distance from O to P and u gives the directed angle from the initial ray to ray OP. So we label the point P as Psr, ud

P ⎛⎝2, ⎛⎝ P ⎛⎝2, – 11 ⎛⎝ 6 6

Directed distance from O to P

– 11 6 6 x

O

Initial ray 0

FIGURE 10.18 Polar coordinates are not unique.

7p6

u 5 p6

O

u50

P ⎛⎝2, 7p⎛⎝ 5 P ⎛⎝–2, p ⎛⎝ 6 6

FIGURE 10.19 Polar coordinates can have negative r-values.

As in trigonometry, u is positive when measured counterclockwise and negative when measured clockwise. The angle associated with a given point is not unique. While a point in the plane has just one pair of Cartesian coordinates, it has infinitely many pairs of polar coordinates. For instance, the point 2 units from the origin along the ray u = p>6 has polar coordinates r = 2, u = p>6. It also has coordinates r = 2, u = -11p>6 (Figure 10.18). In some situations we allow r to be negative. That is why we use directed distance in defining Psr, ud. The point Ps2, 7p>6d can be reached by turning 7p>6 radians counterclockwise from the initial ray and going forward 2 units (Figure 10.19). It can also be reached by turning p>6 radians counterclockwise from the initial ray and going backward 2 units. So the point also has polar coordinates r = -2, u = p>6.

EXAMPLE 1

p6 x

Directed angle from initial ray to OP

Find all the polar coordinates of the point Ps2, p>6d.

We sketch the initial ray of the coordinate system, draw the ray from the origin that makes an angle of p>6 radians with the initial ray, and mark the point s2, p>6d (Figure 10.20). We then find the angles for the other coordinate pairs of P in which r = 2 and r = -2. For r = 2, the complete list of angles is Solution

p , 6

p ; 2p, 6

p ; 4p, 6

p ; 6p, Á . 6

580

Chapter 10: Parametric Equations and Polar Coordinates

For r = -2, the angles are

⎛2, p⎛ 5 ⎛–2, – 5p ⎛ ⎝ 6⎝ ⎝ 6 ⎝

7p ⎛ 5 ⎛⎝–2, 6 ⎝ etc.

7p6

-

p 6

5p , 6

-

5p ; 2p, 6

-

5p ; 4p, 6

-

5p ; 6p, Á . 6

The corresponding coordinate pairs of P are Initial ray

O

x

–5p6

a2,

p + 2npb, 6

n = 0, ;1, ;2, Á

and

FIGURE 10.20 The point Ps2, p>6d has infinitely many polar coordinate pairs (Example 1).

a-2, -

5p + 2npb, 6

n = 0, ;1, ;2, Á .

When n = 0, the formulas give s2, p>6d and s -2, -5p>6d. When n = 1, they give s2, 13p>6d and s -2, 7p>6d, and so on.

ra a

Polar Equations and Graphs

x

O

FIGURE 10.21 The polar equation for a circle is r = a .

y

(a)

p 1 r 2, 0 u 2

0

1

2

x

2 0

EXAMPLE 2

A circle or line can have more than one polar equation.

(a) r = 1 and r = -1 are equations for the circle of radius 1 centered at O. (b) u = p>6, u = 7p>6, and u = -5p>6 are equations for the line in Figure 10.20. Equations of the form r = a and u = u0 can be combined to define regions, segments, and rays.

y

(b)

If we hold r fixed at a constant value r = a Z 0, the point Psr, ud will lie ƒ a ƒ units from the origin O. As u varies over any interval of length 2p, P then traces a circle of radius ƒ a ƒ centered at O (Figure 10.21). If we hold u fixed at a constant value u = u0 and let r vary between - q and q , the point Psr, ud traces the line through O that makes an angle of measure u0 with the initial ray. (See Figure 10.19 for an example.)

u 5 p, 4 p –3 r 2 4 x

3

EXAMPLE 3

Graph the sets of points whose polar coordinates satisfy the following

conditions. (a) 1 … r … 2 (b) -3 … r … 2

2p 3

(c)

y

(c)

5p 6

2p 5p u 3 6 0

x

5p 2p … u … 3 6

Solution

0 … u …

and and

u =

p 2

p 4

sno restriction on rd

The graphs are shown in Figure 10.22.

Relating Polar and Cartesian Coordinates FIGURE 10.22 The graphs of typical inequalities in r and u (Example 3).

When we use both polar and Cartesian coordinates in a plane, we place the two origins together and take the initial polar ray as the positive x-axis. The ray u = p>2, r 7 0,

10.3 y

P(x, y) P(r, ) r

0

581

becomes the positive y-axis (Figure 10.23). The two coordinate systems are then related by the following equations.

Ray 2

Common origin

Polar Coordinates

Equations Relating Polar and Cartesian Coordinates

y

x

x = r cos u,

0, r 0 x Initial ray

FIGURE 10.23 The usual way to relate polar and Cartesian coordinates.

y = r sin u,

r 2 = x 2 + y 2,

y tan u = x

The first two of these equations uniquely determine the Cartesian coordinates x and y given the polar coordinates r and u. On the other hand, if x and y are given, the third equation gives two possible choices for r (a positive and a negative value). For each (x, y) Z (0, 0), there is a unique u H [0, 2pd satisfying the first two equations, each then giving a polar coordinate representation of the Cartesian point (x, y). The other polar coordinate representations for the point can be determined from these two, as in Example 1.

EXAMPLE 4 Here are some plane curves expressed in terms of both polar coordinate and Cartesian coordinate equations. Polar equation

Cartesian equivalent

r cos u = 2 r cos u sin u = 4 2 r cos2 u - r 2 sin2 u = 1 r = 1 + 2r cos u r = 1 - cos u

x = 2 xy = 4 2 x - y2 = 1 y 2 - 3x 2 - 4x - 1 = 0 4 4 x + y + 2x 2y 2 + 2x 3 + 2xy 2 - y 2 = 0

2

Some curves are more simply expressed with polar coordinates; others are not. y x 2 ( y 3) 2 9 or r 6 sin

Solution

We apply the equations relating polar and Cartesian coordinates: x 2 + ( y - 3) 2 x 2 + y 2 - 6y + 9 x 2 + y 2 - 6y r 2 - 6r sin u r = 0 or r - 6 sin u r

(0, 3)

0

Find a polar equation for the circle x 2 + s y - 3d2 = 9 (Figure 10.24).

EXAMPLE 5

x

FIGURE 10.24 The circle in Example 5.

= = = = = =

9 9 0 0 0 6 sin u

Expand s y - 3d2. Cancellation x2 + y2 = r2, y = r sin u Includes both possibilities

EXAMPLE 6 Replace the following polar equations by equivalent Cartesian equations and identify their graphs. (a) r cos u = -4 (b) r 2 = 4r cos u 4 (c) r = 2 cos u - sin u Solution

We use the substitutions r cos u = x, r sin u = y, and r2 = x2 + y2 .

(a) r cos u = -4 The Cartesian equation: The graph:

r cos u = -4 x = -4

Substitution

Vertical line through x = -4 on the x-axis

582

Chapter 10: Parametric Equations and Polar Coordinates

(b) r 2 = 4r cos u The Cartesian equation:

r 2 = 4r cos u x 2 + y 2 = 4x

Substitution

x 2 - 4x + y 2 = 0 x 2 - 4x + 4 + y 2 = 4

Completing the square

sx - 2d + y = 4

Factoring

2

The graph: (c) r =

2

Circle, radius 2, center sh, kd = s2, 0d

4 2 cos u - sin u

The Cartesian equation:

The graph:

rs2 cos u - sin ud = 4 2r cos u - r sin u = 4

Multiplying by r

2x - y = 4

Substitution

y = 2x - 4

Solve for y.

Line, slope m = 2, y-intercept b = -4

Exercises 10.3 Polar Coordinates 1. Which polar coordinate pairs label the same point? a. (3, 0)

b. s -3, 0d

c. s2, 2p>3d

d. s2, 7p>3d

e. s -3, pd

f. s2, p>3d

g. s -3, 2pd

h. s -2, -p>3d

2. Which polar coordinate pairs label the same point? a. s -2, p>3d

b. s2, -p>3d

c. sr, ud

d. sr, u + pd

e. s -r, ud

f. s2, -2p>3d

g. s -r, u + pd

h. s -2, 2p>3d

3. Plot the following points (given in polar coordinates). Then find all the polar coordinates of each point. a. s2, p>2d

b. (2, 0)

c. s -2, p>2d

d. s -2, 0d

4. Plot the following points (given in polar coordinates). Then find all the polar coordinates of each point. a. s3, p>4d

b. s -3, p>4d

c. s3, -p>4d

d. s -3, - p>4d

Polar to Cartesian Coordinates 5. Find the Cartesian coordinates of the points in Exercise 1. 6. Find the Cartesian coordinates of the following points (given in polar coordinates). a.

A 22, p>4 B

c. s0, p>2d

b. (1, 0) d.

A - 22, p>4 B

e. s -3, 5p>6d g. s -1, 7pd

f. s5, tan-1 s4>3dd h. A 223, 2p>3 B

Cartesian to Polar Coordinates 7. Find the polar coordinates, 0 … u 6 2p and r Ú 0, of the following points given in Cartesian coordinates. a. (1, 1)

b. (-3, 0d

c. (23, -1)

d. (-3, 4)

8. Find the polar coordinates, -p … u 6 p and r Ú 0, of the following points given in Cartesian coordinates. a. (-2, -2)

b. (0, 3)

c. (- 23, 1)

d. (5, -12)

9. Find the polar coordinates, 0 … u 6 2p and r … 0, of the following points given in Cartesian coordinates. a. (3, 3)

b. (-1, 0d

c. (-1, 23)

d. (4, -3)

10. Find the polar coordinates, -p … u 6 2p and r … 0, of the following points given in Cartesian coordinates. a. (-2, 0)

b. (1, 0)

c. (0, -3)

d. a

13 1 , b 2 2

Graphing in Polar Coordinates Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises 11–26. 11. r = 2 13. r Ú 1

12. 0 … r … 2 14. 1 … r … 2

10.4 15. 0 … u … p>6,

r Ú 0

16. u = 2p>3,

17. u = p>3,

-1 … r … 3

18. u = 11p>4,

19. u = p>2,

r Ú 0

20. u = p>2,

21. 0 … u … p,

r = 1

23. p>4 … u … 3p>4,

r Ú -1 r … 0

22. 0 … u … p,

r = -1

0 … r … 1

24. -p>4 … u … p>4,

-1 … r … 1

25. -p>2 … u … p>2,

1 … r … 2

26. 0 … u … p>2,

r … -2

Graphing in Polar Coordinates

43. r 2 + 2r 2 cos u sin u = 1

44. cos2 u = sin2 u

45. r 2 = -4r cos u

46. r 2 = -6r sin u

47. r = 8 sin u

48. r = 3 cos u

49. r = 2 cos u + 2 sin u

50. r = 2 cos u - sin u

51. r sin au +

52. r sin a

p b = 2 6

583

2p - ub = 5 3

Cartesian to Polar Equations Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

1 … ƒrƒ … 2

53. x = 7

54. y = 1

Polar to Cartesian Equations Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.

56. x - y = 3

57. x + y = 4

27. r cos u = 2

28. r sin u = -1

59.

29. r sin u = 0

30. r cos u = 0

61. y 2 = 4x

62. x 2 + xy + y 2 = 1

31. r = 4 csc u

32. r = -3 sec u

63. x 2 + s y - 2d2 = 4

64. sx - 5d2 + y 2 = 25

33. r cos u + r sin u = 1

34. r sin u = r cos u

65. sx - 3d2 + s y + 1d2 = 4

66. sx + 2d2 + s y - 5d2 = 16

35. r 2 = 1

36. r 2 = 4r sin u

67. Find all polar coordinates of the origin.

37. r =

5 sin u - 2 cos u

38. r 2 sin 2u = 2

39. r = cot u csc u

40. r = 4 tan u sec u

41. r = csc u e r cos u

42. r sin u = ln r + ln cos u

10.4

Graphing in Polar Coordinates

y2 x2 + = 1 9 4

2

55. x = y 2

58. x 2 - y 2 = 1

60. xy = 2

68. Vertical and horizontal lines a. Show that every vertical line in the xy-plane has a polar equation of the form r = a sec u . b. Find the analogous polar equation for horizontal lines in the xy-plane.

It is often helpful to have the graph of an equation in polar coordinates. This section describes some techniques for graphing these equations using symmetries and tangents to the graph.

Symmetry Figure 10.25 illustrates the standard polar coordinate tests for symmetry. The following summary says how the symmetric points are related.

Symmetry Tests for Polar Graphs 1. Symmetry about the x-axis: If the point sr, ud lies on the graph, then the point sr, -ud or s -r, p - ud lies on the graph (Figure 10.25a). 2. Symmetry about the y-axis: If the point sr, ud lies on the graph, then the point sr, p - ud or s -r, -ud lies on the graph (Figure 10.25b). 3. Symmetry about the origin: If the point sr, ud lies on the graph, then the point s -r, ud or sr, u + pd lies on the graph (Figure 10.25c).

584

Chapter 10: Parametric Equations and Polar Coordinates

Slope

y

The slope of a polar curve r = ƒsud in the xy-plane is still given by dy> dx, which is not r¿ = dƒ>du. To see why, think of the graph of ƒ as the graph of the parametric equations

(r, ) x

0

x = r cos u = ƒsud cos u,

(r, –) or (–r, )

If ƒ is a differentiable function of u, then so are x and y and, when dx>du Z 0, we can calculate dy> dx from the parametric formula

(a) About the x-axis

dy>du dy = dx dx>du

y

(r, ) or (–r, –)

(r, )

Section 10.2, Eq. (1) with t = u

d sƒsud # sin ud du = d sƒsud # cos ud du

x

0

y = r sin u = ƒsud sin u.

(–

df sin u + ƒsud cos u du = df cos u - ƒsud sin u du

(b) About the y-axis y (r, )

Product Rule for derivatives

Therefore we see that dy>dx is not the same as dƒ>du. x

0

(–r, ) or (r, )

x

Slope of the Curve r = ƒ(u)

(c) About the origin

FIGURE 10.25 Three tests for symmetry in polar coordinates.

dy ƒ¿sud sin u + ƒsud cos u ` = dx sr, ud ƒ¿sud cos u - ƒsud sin u provided dx>du Z 0 at sr, ud.

If the curve r = ƒsud passes through the origin at u = u0 , then ƒsu0 d = 0, and the slope equation gives dy ƒ¿su0 d sin u0 ` = = tan u0 . dx s0, u0d ƒ¿su0 d cos u0 If the graph of r = ƒsud passes through the origin at the value u = u0 , the slope of the curve there is tan u0 . The reason we say “slope at s0, u0 d” and not just “slope at the origin” is that a polar curve may pass through the origin (or any point) more than once, with different slopes at different u-values. This is not the case in our first example, however.

EXAMPLE 1 Solution

Graph the curve r = 1 - cos u.

The curve is symmetric about the x-axis because sr, ud on the graph Q r = 1 - cos u Q r = 1 - cos s -ud Q sr, -ud on the graph.

cos u = cos s -ud

10.4

U

r 1 cos U

0 p 3 p 2 2p 3 p

0 1 2

3 2 2 (a) ⎛ 3 , 2 ⎛ y ⎝2 3 ⎝ ⎛1, ⎛ ⎝ 2⎝

(, 2)

1

0

2

⎛ 1, ⎛ ⎝2 3⎝ x

Graph the curve r 2 = 4 cos u.

EXAMPLE 2

The equation r 2 = 4 cos u requires cos u Ú 0, so we get the entire graph by running u from -p>2 to p>2. The curve is symmetric about the x-axis because

Solution

sr, ud on the graph Q r 2 = 4 cos u

(b) ⎛ 3 , 2⎛ y ⎝2 3 ⎝

r 1 cos

1 (, 2)

2

⎛ 3 , 4 ⎛ ⎝2 3 ⎝

0

585

As u increases from 0 to p, cos u decreases from 1 to -1, and r = 1 - cos u increases from a minimum value of 0 to a maximum value of 2. As u continues on from p to 2p, cos u increases from -1 back to 1 and r decreases from 2 back to 0. The curve starts to repeat when u = 2p because the cosine has period 2p. The curve leaves the origin with slope tan s0d = 0 and returns to the origin with slope tan s2pd = 0. We make a table of values from u = 0 to u = p, plot the points, draw a smooth curve through them with a horizontal tangent at the origin, and reflect the curve across the x-axis to complete the graph (Figure 10.26). The curve is called a cardioid because of its heart shape.

1

3 2

Graphing in Polar Coordinates

Q r 2 = 4 cos s -ud ⎛1, ⎛ ⎝ 2⎝

Q sr, -ud on the graph. The curve is also symmetric about the origin because sr, ud on the graph Q r 2 = 4 cos u

⎛ 1 , ⎛ ⎝2 3⎝ x

Q s -rd2 = 4 cos u

⎛ 1 , – ⎛ ⎝2 3⎝ ⎛ 1, 3 ⎛ ⎝ 2 ⎝

(c)

FIGURE 10.26 The steps in graphing the cardioid r = 1 - cos u (Example 1). The arrow shows the direction of increasing u .

cos u = cos (-u)

Q s -r, ud on the graph. Together, these two symmetries imply symmetry about the y-axis. The curve passes through the origin when u = -p>2 and u = p>2. It has a vertical tangent both times because tan u is infinite. For each value of u in the interval between -p>2 and p>2, the formula r 2 = 4 cos u gives two values of r: r = ;22cos u. We make a short table of values, plot the corresponding points, and use information about symmetry and tangents to guide us in connecting the points with a smooth curve (Figure 10.27).

y r 2 4 cos U

cos U

0 p ; 6 p ; 4

1 13 2 1 12 1 2

p 3 p ; 2 ;

r — 21cos U ;2 L ;1.9

2

2 0

x

L ;1.7 L ;1.4

0

0 (a)

Loop for r –2cos , Loop for r 2cos , – – 2 2 2 2 (b)

FIGURE 10.27 The graph of r 2 = 4 cos u . The arrows show the direction of increasing u . The values of r in the table are rounded (Example 2).

586

Chapter 10: Parametric Equations and Polar Coordinates

(a) r 2 1

Converting a Graph from the rU- to xy-Plane r2

2 0

One way to graph a polar equation r = ƒsud in the xy-plane is to make a table of (r, u)-values, plot the corresponding points there and connect them in order of increasing u. This can work well if enough points have been plotted to reveal all the loops and dimples in the graph. Another method of graphing is to

sin 2 3 2

4

2

1. 2.

–1 No square roots of negative numbers

This method is sometimes better than simple point plotting because the first Cartesian graph, even when hastily drawn, shows at a glance where r is positive, negative, and nonexistent, as well as where r is increasing and decreasing. Here’s an example.

(b) r 1

0

r sin 2

2

first graph the function r = ƒsud in the Cartesian ru-plane, then use that Cartesian graph as a “table” and guide to sketch the polar coordinate graph in the xy-plane

3 2

parts from square roots

EXAMPLE 3

Graph the lemniscate curve r 2 = sin 2u.

–1 r sin 2

Here we begin by plotting r 2 (not r) as a function of u in the Cartesian r u-plane. See Figure 10.28a. We pass from there to the graph of r = ; 2sin 2u in the ru-plane (Figure 10.28b), and then draw the polar graph (Figure 10.28c). The graph in Figure 10.28b “covers” the final polar graph in Figure 10.28c twice. We could have managed with either loop alone, with the two upper halves, or with the two lower halves. The double covering does no harm, however, and we actually learn a little more about the behavior of the function this way. Solution

(c)

2

y r 2 sin 2 0

x

USING TECHNOLOGY FIGURE 10.28 To plot r = ƒsud in the Cartesian ru-plane in (b), we first plot r 2 = sin 2u in the r 2u-plane in (a) and then ignore the values of u for which sin 2u is negative. The radii from the sketch in (b) cover the polar graph of the lemniscate in (c) twice (Example 3).

Graphing Polar Curves Parametrically

For complicated polar curves we may need to use a graphing calculator or computer to graph the curve. If the device does not plot polar graphs directly, we can convert r = ƒsud into parametric form using the equations x = r cos u = ƒsud cos u,

y = r sin u = ƒsud sin u.

Then we use the device to draw a parametrized curve in the Cartesian xy-plane. It may be necessary to use the parameter t rather than u for the graphing device.

Exercises 10.4 Symmetries and Polar Graphs Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves.

Graph the lemniscates in Exercises 13–16. What symmetries do these curves have? 13. r 2 = 4 cos 2u

14. r 2 = 4 sin 2u

15. r = -sin 2u

16. r 2 = -cos 2u

1. r = 1 + cos u

2. r = 2 - 2 cos u

3. r = 1 - sin u

4. r = 1 + sin u

5. r = 2 + sin u

6. r = 1 + 2 sin u

Slopes of Polar Curves Find the slopes of the curves in Exercises 17–20 at the given points. Sketch the curves along with their tangents at these points.

7. r = sin su>2d

8. r = cos su>2d

17. Cardioid r = -1 + cos u;

u = ;p>2

18. Cardioid r = -1 + sin u;

u = 0, p

9. r 2 = cos u 11. r 2 = -sin u

10. r 2 = sin u 12. r 2 = -cos u

2

19. Four-leaved rose r = sin 2u;

u = ;p>4, ;3p>4

20. Four-leaved rose r = cos 2u;

u = 0, ;p>2, p

10.5

Graphing Limaçons Graph the limaçons in Exercises 21–24. Limaçon (“lee-ma-sahn”) is Old French for “snail.” You will understand the name when you graph the limaçons in Exercise 21. Equations for limaçons have the form r = a ; b cos u or r = a ; b sin u . There are four basic shapes. 21. Limaçons with an inner loop 1 1 a. r = + cos u b. r = + sin u 2 2 22. Cardioids a. r = 1 - cos u

b. r = -1 + sin u

23. Dimpled limaçons

Areas and Lengths in Polar Coordinates

T 29. Which of the following has the same graph as r = 1 - cos u ? a. r = -1 - cos u b. r = 1 + cos u Confirm your answer with algebra. T 30. Which of the following has the same graph as r = cos 2u ? a. r = -sin s2u + p>2d b. r = -cos su>2d Confirm your answer with algebra. T 31. A rose within a rose

Graph the equation r = 1 - 2 sin 3u .

T 32. The nephroid of Freeth Graph the nephroid of Freeth: 3 b. r = - sin u 2

3 a. r = + cos u 2 24. Oval limaçons

u r = 1 + 2 sin . 2 T 33. Roses

a. r = 2 + cos u

b. r = -2 + sin u

Graphing Polar Regions and Curves 25. Sketch the region defined by the inequalities -1 … r … 2 and -p>2 … u … p>2 .

Graph the roses r = cos mu for m = 1>3, 2, 3 , and 7.

T 34. Spirals Polar coordinates are just the thing for defining spirals. Graph the following spirals. a. r = u b. r = -u c. A logarithmic spiral: r = e u>10

26. Sketch the region defined by the inequalities 0 … r … 2 sec u and -p>4 … u … p>4 .

d. A hyperbolic spiral: r = 8>u

In Exercises 27 and 28, sketch the region defined by the inequality.

e. An equilateral hyperbola: r = ;10> 2u

28. 0 … r … cos u

27. 0 … r … 2 - 2 cos u

10.5

2

This section shows how to calculate areas of plane regions and lengths of curves in polar coordinates. The defining ideas are the same as before, but the formulas are different in polar versus Cartesian coordinates.

k rn

Area in the Plane

( f (k ), k) rk

The region OTS in Figure 10.29 is bounded by the rays u = a and u = b and the curve r = ƒsud. We approximate the region with n nonoverlapping fan-shaped circular sectors based on a partition P of angle TOS. The typical sector has radius rk = ƒsuk d and central angle of radian measure ¢uk . Its area is ¢uk>2p times the area of a circle of radius rk , or

r f () r2 r1 k O

(Use different colors for the two branches.)

Areas and Lengths in Polar Coordinates

y

S

587

T

x

Ak =

2 1 2 1 r ¢uk = A ƒsuk d B ¢uk . 2 k 2

The area of region OTS is approximately FIGURE 10.29 To derive a formula for the area of region OTS, we approximate the region with fan-shaped circular sectors.

2 1 a Ak = a 2 A ƒsuk d B ¢uk . k=1 k=1 n

n

If ƒ is continuous, we expect the approximations to improve as the norm of the partition P goes to zero, where the norm of P is the largest value of ¢uk . We are then led to the following formula defining the region’s area:

588

Chapter 10: Parametric Equations and Polar Coordinates 2 1 A = lim a A ƒsuk d B ¢uk 2 ‘ P ‘:0 k = 1 n

y dA 1 r 2 d 2

=

r

d

x

O

Area of the Fan-Shaped Region Between the Origin and the Curve r = ƒsUd, A … U … B

FIGURE 10.30 The area differential dA for the curve r = ƒ(u). y

b

1 2 r du. La 2 This is the integral of the area differential (Figure 10.30) A =

r 5 2(1 1 cos u) P(r, u)

2

r

2 1 2 1 r du = A ƒ(u) B du. 2 2

dA =

u 5 0, 2p

0

2 1 A ƒsud B du. 2 La b

P(r, )

EXAMPLE 1

Find the area of the region in the plane enclosed by the cardioid r = 2s1 + cos ud.

x

4

Solution We graph the cardioid (Figure 10.31) and determine that the radius OP sweeps out the region exactly once as u runs from 0 to 2p. The area is therefore

–2

FIGURE 10.31 The cardioid in Example 1.

u = 2p

Lu = 0

y

1 2 r du = 2 L0

2p

1# 4s1 + cos ud2 du 2

2p

u5b

= r2

L0

2s1 + 2 cos u + cos2 ud du 2p

=

r1

=

x

y

L0

1 + cos 2u b du 2

s3 + 4 cos u + cos 2ud du

= c3u + 4 sin u +

FIGURE 10.32 The area of the shaded region is calculated by subtracting the area of the region between r1 and the origin from the area of the region between r2 and the origin.

r1 5 1 2 cos u

a2 + 4 cos u + 2

2p

u5a

0

L0

Upper limit u 5 p2

2p

sin 2u d = 6p - 0 = 6p. 2 0

To find the area of a region like the one in Figure 10.32, which lies between two polar curves r1 = r1sud and r2 = r2sud from u = a to u = b , we subtract the integral of s1>2dr 12 du from the integral of s1>2dr 22 du. This leads to the following formula.

Area of the Region 0 … r1sUd … r … r2sUd, A … U … B

r2 5 1

1 2 1 2 1 r 2 du r 1 du = A r 22 - r 12 B du La 2 La 2 La 2 b

A =

b

b

(1)

u x

0

Find the area of the region that lies inside the circle r = 1 and outside the cardioid r = 1 - cos u.

EXAMPLE 2 Lower limit u 5 –p2

FIGURE 10.33 The region and limits of integration in Example 2.

Solution We sketch the region to determine its boundaries and find the limits of integration (Figure 10.33). The outer curve is r2 = 1, the inner curve is r1 = 1 - cos u, and u runs from -p>2 to p>2. The area, from Equation (1), is

10.5

Areas and Lengths in Polar Coordinates

589

1 A r 22 - r 12 B du L-p>2 2 p>2

A =

p>2

= 2

L0

1 A r 2 - r 12 B du 2 2

Symmetry

p>2

=

s1 - s1 - 2 cos u + cos2 udd du

L0 p>2

=

L0

p>2

s2 cos u - cos2 ud du =

= c2 sin u -

L0

p>2

sin 2u u d 2 4 0

Square r1.

a2 cos u -

1 + cos 2u b du 2

p . 4

= 2 -

The fact that we can represent a point in different ways in polar coordinates requires extra care in deciding when a point lies on the graph of a polar equation and in determining the points in which polar graphs intersect. (We needed intersection points in Example 2.) In Cartesian coordinates, we can always find the points where two curves cross by solving their equations simultaneously. In polar coordinates, the story is different. Simultaneous solution may reveal some intersection points without revealing others, so it is sometimes difficult to find all points of intersection of two polar curves. One way to identify all the points of intersection is to graph the equations.

Length of a Polar Curve We can obtain a polar coordinate formula for the length of a curve r = ƒsud, a … u … b , by parametrizing the curve as x = r cos u = ƒsud cos u,

y = r sin u = ƒsud sin u,

a … u … b.

(2)

The parametric length formula, Equation (3) from Section 10.2, then gives the length as b

L =

2 dy 2 dx b + a b du. du La B du

a

This equation becomes b

L =

r2 + a

La B

2

dr b du du

when Equations (2) are substituted for x and y (Exercise 29).

Length of a Polar Curve If r = ƒsud has a continuous first derivative for a … u … b and if the point Psr, ud traces the curve r = ƒsud exactly once as u runs from a to b , then the length of the curve is

y r 1 cos P(r, ) r

b

1

L =

2

0

x

EXAMPLE 3

FIGURE 10.34 Calculating the length of a cardioid (Example 3).

r2 + a

La B

2

dr b du. du

(3)

Find the length of the cardioid r = 1 - cos u.

Solution We sketch the cardioid to determine the limits of integration (Figure 10.34). The point Psr, ud traces the curve once, counterclockwise as u runs from 0 to 2p, so these are the values we take for a and b .

590

Chapter 10: Parametric Equations and Polar Coordinates

With r = 1 - cos u,

dr = sin u, du

we have r2 + a

2

dr b = s1 - cos ud2 + ssin ud2 du 2 u + sin2 u = 2 - 2 cos u = 1 - 2 cos u + cos ('')''* 1

and b

L =

r2 + a

La B 2p

=

L0 A

4 sin 2

2p

=

L0

2 ` sin

2p

=

L0

2 sin

2

dr b du = du L0 u du 2

2p

22 - 2 cos u du

1 - cos u = 2 sin2 (u>2)

u ` du 2 u du 2

sin (u>2) Ú 0

for

0 … u … 2p

2p

u = c-4 cos d = 4 + 4 = 8. 2 0

EXERCISES 10.5 Finding Polar Areas Find the areas of the regions in Exercises 1–8. 1. Bounded by the spiral r = u for 0 … u … p y

4. Inside the cardioid r = as1 + cos ud,

a 7 0

5. Inside one leaf of the four-leaved rose r = cos 2u 6. Inside one leaf of the three-leaved rose r = cos 3u y

⎛p , p⎛ ⎝ 2 2⎝

r5u

r 5 cos 3u (p, p)

x

0

1

x

2. Bounded by the circle r = 2 sin u for p>4 … u … p>2 y ⎛2 , p⎛ ⎝ 2⎝

7. Inside one loop of the lemniscate r 2 = 4 sin 2u 8. Inside the six-leaved rose r 2 = 2 sin 3u

r 5 2 sin u

Find the areas of the regions in Exercises 9–16. u5 p 4 0

9. Shared by the circles r = 2 cos u and r = 2 sin u x

10. Shared by the circles r = 1 and r = 2 sin u 11. Shared by the circle r = 2 and the cardioid r = 2s1 - cos ud 12. Shared by the cardioids r = 2s1 + cos ud and r = 2s1 - cos ud

3. Inside the oval limaçon r = 4 + 2 cos u

13. Inside the lemniscate r 2 = 6 cos 2u and outside the circle r = 23

10.6 14. Inside the circle r = 3a cos u and outside the cardioid r = as1 + cos ud, a 7 0 15. Inside the circle r = -2 cos u and outside the circle r = 1 16. Inside the circle r = 6 above the line r = 3 csc u 17. Inside the circle r = 4 cos u and to the right of the vertical line r = sec u 18. Inside the circle r = 4 sin u and below the horizontal line r = 3 csc u

27. The curve r = cos3 su>3d,

r tan – 2 2

x = ƒsud cos u,

b

L =

2 dy 2 dx b + a b du du La B du

a

into L =

r (22) csc

x

1

b. It looks as if the graph of r = tan u, -p>2 6 u 6 p>2 , could be asymptotic to the lines x = 1 and x = -1 . Is it? Give reasons for your answer. 20. The area of the region that lies inside the cardioid curve r = cos u + 1 and outside the circle r = cos u is not 2p

1 [scos u + 1d2 - cos2 u] du = p . 2 L0

a. r = a

b. r = a cos u

Theory and Examples 31. Average value If ƒ is continuous, the average value of the polar coordinate r over the curve r = ƒsud, a … u … b , with respect to u is given by the formula b

24. The curve r = a sin su>2d,

b. The circle r = a

0 … u … p,

a 7 0

25. The parabolic segment r = 6>s1 + cos ud,

0 … u … p>2

26. The parabolic segment r = 2>s1 - cos ud,

p>2 … u … p

10.6

1 ƒsud du . b - a La

a. The cardioid r = as1 - cos ud

0 … u … p

23. The cardioid r = 1 + cos u 2

c. r = a sin u

Use this formula to find the average value of r with respect to u over the following curves sa 7 0d .

0 … u … 25

22. The spiral r = e u> 22,

2

dr b du . du

30. Circumferences of circles As usual, when faced with a new formula, it is a good idea to try it on familiar objects to be sure it gives results consistent with past experience. Use the length formula in Equation (3) to calculate the circumferences of the following circles sa 7 0d.

rav = Finding Lengths of Polar Curves Find the lengths of the curves in Exercises 21–28.

r2 + a

La B

Why not? What is the area? Give reasons for your answers.

21. The spiral r = u2,

y = ƒsud sin u

(Equations 2 in the text) transform

b

0

0 … u … p 22

29. The length of the curve r ƒ(U), A … U … B Assuming that the necessary derivatives are continuous, show how the substitutions

(1, 4)

–1

591

0 … u … p>4

28. The curve r = 21 + sin 2u,

19. a. Find the area of the shaded region in the accompanying figure. y

Conics in Polar Coordinates

c. The circle r = a cos u,

-p>2 … u … p>2

32. r ƒ(U) vs. r 2ƒ(U) Can anything be said about the relative lengths of the curves r = ƒsud, a … u … b , and r = 2ƒsud, a … u … b ? Give reasons for your answer.

Conics in Polar Coordinates Polar coordinates are especially important in astronomy and astronautical engineering because satellites, moons, planets, and comets all move approximately along ellipses, parabolas, and hyperbolas that can be described with a single relatively simple polar coordinate equation. (The standard Cartesian equations for the conics are reviewed in Appendix 4.) We develop the polar coordinate equation here after first introducing the idea of a conic section’s eccentricity. The eccentricity reveals the conic section’s type (circle, ellipse, parabola, or hyperbola) and the degree to which it is “squashed” or flattened.

592

Chapter 10: Parametric Equations and Polar Coordinates

Eccentricity

y Directrix x –c D

Although the center-to-focus distance c (see Appendix 4) does not appear in the standard Cartesian equation

P(x, y)

y2 x2 + = 1, a2 b2

x

0 F(c, 0)

for an ellipse, we can still determine c from the equation c = 2a 2 - b 2 . If we fix a and vary c over the interval 0 … c … a, the resulting ellipses will vary in shape. They are circles if c = 0 (so that a = b) and flatten as c increases. If c = a, the foci and vertices overlap and the ellipse degenerates into a line segment. Thus we are led to consider the ratio e = c>a. We use this ratio for hyperbolas as well—only in this case c equals 2a 2 + b 2

FIGURE 10.35 The distance from the focus F to any point P on a parabola equals the distance from P to the nearest point D on the directrix, so PF = PD.

Directrix 2 x ae

b

F1(–c, 0)

F2(c, 0)

x D2

P(x, y) –b c ae a

D1

F1(–c, 0)

y

a e

0 a e

Whereas a parabola has one focus and one directrix, each ellipse has two foci and two directrices. These are the lines perpendicular to the major axis at distances ;a>e from the center. From Figure 10.35 we see that a parabola has the property PF = 1 # PD

(1)

for any point P on it, where F is the focus and D is the point nearest P on the directrix. For an ellipse, it can be shown that the equations that replace Equation (1) are

Directrix 2 x ae D2

2a 2 + b 2 c . e = a = a The eccentricity of a parabola is e = 1.

FIGURE 10.36 The foci and directrices of the ellipse sx 2>a 2 d + s y 2>b 2 d = 1 . Directrix 1 corresponds to focus F1 and directrix 2 to focus F2 .

Directrix 1 x – ae

DEFINITION The eccentricity of the ellipse sx 2>a 2 d + s y 2>b 2 d = 1 sa 7 bd is

The eccentricity of the hyperbola sx 2>a 2 d - s y 2>b 2 d = 1 is

0 D1

instead of 2a 2 - b 2—and define these ratios with the term eccentricity.

2a 2 - b 2 c e = a = . a

y Directrix 1 x – ae

sa 7 bd

PF1 = e # PD1,

PF2 = e # PD2 .

(2)

P(x, y)

x F2(c, 0)

a c ae

FIGURE 10.37 The foci and directrices of the hyperbola sx 2>a 2 d - s y 2>b 2 d = 1 . No matter where P lies on the hyperbola, PF1 = e # PD1 and PF2 = e # PD2 .

Here, e is the eccentricity, P is any point on the ellipse, F1 and F2 are the foci, and D1 and D2 are the points on the directrices nearest P (Figure 10.36). In both Equations (2) the directrix and focus must correspond; that is, if we use the distance from P to F1 , we must also use the distance from P to the directrix at the same end of the ellipse. The directrix x = -a>e corresponds to F1s -c, 0d, and the directrix x = a>e corresponds to F2sc, 0d. As with the ellipse, it can be shown that the lines x = ;a>e act as directrices for the hyperbola and that PF1 = e # PD1

and

PF2 = e # PD2 .

(3)

Here P is any point on the hyperbola, F1 and F2 are the foci, and D1 and D2 are the points nearest P on the directrices (Figure 10.37).

10.6

Conics in Polar Coordinates

593

In both the ellipse and the hyperbola, the eccentricity is the ratio of the distance between the foci to the distance between the vertices (because c>a = 2c>2a).

Eccentricity =

distance between foci distance between vertices

In an ellipse, the foci are closer together than the vertices and the ratio is less than 1. In a hyperbola, the foci are farther apart than the vertices and the ratio is greater than 1. The “focus–directrix” equation PF = e # PD unites the parabola, ellipse, and hyperbola in the following way. Suppose that the distance PF of a point P from a fixed point F (the focus) is a constant multiple of its distance from a fixed line (the directrix). That is, suppose PF = e # PD,

(4)

where e is the constant of proportionality. Then the path traced by P is (a) a parabola if e = 1, (b) an ellipse of eccentricity e if e 6 1, and (c) a hyperbola of eccentricity e if e 7 1. There are no coordinates in Equation (4), and when we try to translate it into coordinate form, it translates in different ways depending on the size of e. At least, that is what happens in Cartesian coordinates. However, as we are about to see, in polar coordinates the equation PF = e # PD translates into a single equation regardless of the value of e. Given the focus and corresponding directrix of a hyperbola centered at the origin and with foci on the x-axis, we can use the dimensions shown in Figure 10.37 to find e. Knowing e, we can derive a Cartesian equation for the hyperbola from the equation PF = e # PD, as in the next example. We can find equations for ellipses centered at the origin and with foci on the x-axis in a similar way, using the dimensions shown in Figure 10.36.

EXAMPLE 1

Find a Cartesian equation for the hyperbola centered at the origin that has a focus at (3, 0) and the line x = 1 as the corresponding directrix. y x1 D(1, y)

y2 x2 1 3 6

P(x, y)

0 1

x

We first use the dimensions shown in Figure 10.37 to find the hyperbola’s eccentricity. The focus is (see Figure 10.38)

Solution

sc, 0d = s3, 0d,

so

c = 3.

Again from Figure 10.37, the directrix is the line

F(3, 0)

a x = e = 1,

so

a = e.

When combined with the equation e = c>a that defines eccentricity, these results give FIGURE 10.38 The hyperbola and directrix in Example 1.

3 c e = a = e,

so

e2 = 3

and

e = 23.

594

Chapter 10: Parametric Equations and Polar Coordinates

Knowing e, we can now derive the equation we want from the equation PF = e # PD. In the coordinates of Figure 10.38, we have PF = e # PD 2sx - 3d2 + s y - 0d2 = 23 ƒ x - 1 ƒ

Eq. (4) e = 23

x 2 - 6x + 9 + y 2 = 3sx 2 - 2x + 1d 2x 2 - y 2 = 6 y2 x2 = 1. 3 6

Polar Equations To find a polar equation for an ellipse, parabola, or hyperbola, we place one focus at the origin and the corresponding directrix to the right of the origin along the vertical line x = k (Figure 10.39). In polar coordinates, this makes

Directrix

PF = r

P Focus at origin

D

and

r

F B r cos

ion

ect

cs oni

k

x

PD = k - FB = k - r cos u. The conic’s focus–directrix equation PF = e # PD then becomes r = esk - r cos ud,

xk

C

which can be solved for r to obtain the following expression. FIGURE 10.39 If a conic section is put in the position with its focus placed at the origin and a directrix perpendicular to the initial ray and right of the origin, we can find its polar equation from the conic’s focus–directrix equation.

Polar Equation for a Conic with Eccentricity e r =

ke , 1 + e cos u

(5)

where x = k 7 0 is the vertical directrix.

EXAMPLE 2 Here are polar equations for three conics. The eccentricity values identifying the conic are the same for both polar and Cartesian coordinates. ellipse

r =

k 2 + cos u

e = 1:

parabola

r =

k 1 + cos u

e = 2:

hyperbola

r =

2k 1 + 2 cos u

e =

1 : 2

You may see variations of Equation (5), depending on the location of the directrix. If the directrix is the line x = -k to the left of the origin (the origin is still a focus), we replace Equation (5) with r =

ke . 1 - e cos u

10.6

Conics in Polar Coordinates

595

The denominator now has a s - d instead of a s + d. If the directrix is either of the lines y = k or y = -k, the equations have sines in them instead of cosines, as shown in Figure 10.40. r

ke 1 e cos

r

Focus at origin

r

Focus at origin x

x

Directrix x k (a)

ke 1 e cos

Directrix x –k (b)

ke 1 e sin y

r

Directrix y k

ke 1 e sin y Focus at origin

Focus at origin Directrix y –k (d)

(c)

FIGURE 10.40 Equations for conic sections with eccentricity e 7 0 but different locations of the directrix. The graphs here show a parabola, so e = 1.

EXAMPLE 3 x = 2. Solution

Find an equation for the hyperbola with eccentricity 3> 2 and directrix

We use Equation (5) with k = 2 and e = 3>2: r =

EXAMPLE 4

2s3>2d 1 + s3>2d cos u

Solution

Focus at Center origin

r =

6 . 2 + 3 cos u

Find the directrix of the parabola r =

Directrix xk

or

25 . 10 + 10 cos u

We divide the numerator and denominator by 10 to put the equation in standard

polar form: r =

5>2 . 1 + cos u

r =

ke 1 + e cos u

x

This is the equation ea a a e

FIGURE 10.41 In an ellipse with semimajor axis a, the focus– directrix distance is k = sa>ed - ea , so ke = as1 - e 2 d .

with k = 5>2 and e = 1. The equation of the directrix is x = 5>2. From the ellipse diagram in Figure 10.41, we see that k is related to the eccentricity e and the semimajor axis a by the equation a k = e - ea.

596

Chapter 10: Parametric Equations and Polar Coordinates

From this, we find that ke = as1 - e 2 d. Replacing ke in Equation (5) by as1 - e 2 d gives the standard polar equation for an ellipse.

Polar Equation for the Ellipse with Eccentricity e and Semimajor Axis a r =

as1 - e 2 d 1 + e cos u

(6)

Notice that when e = 0, Equation (6) becomes r = a, which represents a circle.

Lines Suppose the perpendicular from the origin to line L meets L at the point P0sr0, u0 d, with r0 Ú 0 (Figure 10.42). Then, if Psr, ud is any other point on L, the points P, P0 , and O are the vertices of a right triangle, from which we can read the relation

y

P(r, )

r0 = r cos su - u0 d.

r P0(r0 , 0 )

r0

The Standard Polar Equation for Lines

L

0

If the point P0sr0, u0 d is the foot of the perpendicular from the origin to the line L, and r0 Ú 0, then an equation for L is

x

O

FIGURE 10.42 We can obtain a polar equation for line L by reading the relation r0 = r cos su - u0 d from the right triangle OP0 P .

r cos su - u0 d = r0 .

(7)

For example, if u0 = p>3 and r0 = 2, we find that r cos au r acos u cos

p b = 2 3

p p + sin u sin b = 2 3 3

23 1 r cos u + r sin u = 2, 2 2

or

x + 23 y = 4.

Circles To find a polar equation for the circle of radius a centered at P0sr0, u0 d, we let Psr, ud be a point on the circle and apply the Law of Cosines to triangle OP0 P (Figure 10.43). This gives

y P(r, )

a 2 = r 02 + r 2 - 2r0 r cos su - u0 d.

a r 0 O

r0

P0(r0 , 0 )

If the circle passes through the origin, then r0 = a and this equation simplifies to x

FIGURE 10.43 We can get a polar equation for this circle by applying the Law of Cosines to triangle OP0 P .

a 2 = a 2 + r 2 - 2ar cos su - u0 d r 2 = 2ar cos su - u0 d r = 2a cos su - u0 d. If the circle’s center lies on the positive x-axis, u0 = 0 and we get the further simplification r = 2a cos u.

(8)

10.6

Conics in Polar Coordinates

597

If the center lies on the positive y-axis, u = p>2, cos su - p>2d = sin u, and the equation r = 2a cos su - u0 d becomes r = 2a sin u.

(9)

Equations for circles through the origin centered on the negative x- and y-axes can be obtained by replacing r with -r in the above equations.

EXAMPLE 5

Here are several polar equations given by Equations (8) and (9) for circles through the origin and having centers that lie on the x- or y-axis.

Radius

Center (polar coordinates)

3 2 1> 2

(3, 0) s2, p>2d s -1>2, 0d s -1, p>2d

1

Polar equation r r r r

= = = =

6 cos u 4 sin u -cos u -2 sin u

Exercises 10.6 Ellipses and Eccentricity In Exercises 1–8, find the eccentricity of the ellipse. Then find and graph the ellipse’s foci and directrices. 1. 16x 2 + 25y 2 = 400

2. 7x 2 + 16y 2 = 112

3. 2x + y = 2

4. 2x + y = 4

5. 3x + 2y = 6

6. 9x 2 + 10y 2 = 90

7. 6x + 9y = 54

8. 169x 2 + 25y 2 = 4225

2

2

2

2

2

2

2

2

Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates. 9. Foci: s0, ;3d Eccentricity: 0.5

10. Foci: s ;8, 0d Eccentricity: 0.2

11. Vertices: s0, ;70d Eccentricity: 0.1

12. Vertices: s ;10, 0d Eccentricity: 0.24

21. 8x 2 - 2y 2 = 16

22. y 2 - 3x 2 = 3

23. 8y 2 - 2x 2 = 16

24. 64x 2 - 36y 2 = 2304

Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates. 25. Eccentricity: 3 Vertices: s0, ;1d

26. Eccentricity: 2 Vertices: s ;2, 0d

27. Eccentricity: 3 Foci: s ;3, 0d

28. Eccentricity: 1.25 Foci: s0, ;5d

Eccentricities and Directrices Exercises 29–36 give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. 29. e = 1,

x = 2

30. e = 1,

y = 2

Exercises 13–16 give foci and corresponding directrices of ellipses centered at the origin of the xy-plane. In each case, use the dimensions in Figure 10.36 to find the eccentricity of the ellipse. Then find the ellipse’s standard-form equation in Cartesian coordinates.

31. e = 5,

y = -6

32. e = 2,

x = 4

13. Focus:

Parabolas and Ellipses Sketch the parabolas and ellipses in Exercises 37–44. Include the directrix that corresponds to the focus at the origin. Label the vertices with appropriate polar coordinates. Label the centers of the ellipses as well.

A 25, 0 B

Directrix:

x =

14. Focus: 9

25

15. Focus: s -4, 0d Directrix: x = -16

(4, 0)

Directrix:

x =

16 3

16. Focus: A - 22, 0 B Directrix: x = -222

Hyperbolas and Eccentricity In Exercises 17–24, find the eccentricity of the hyperbola. Then find and graph the hyperbola’s foci and directrices. 17. x 2 - y 2 = 1

18. 9x 2 - 16y 2 = 144

19. y - x = 8

20. y 2 - x 2 = 4

2

2

33. e = 1>2,

x = 1

34. e = 1>4,

x = -2

35. e = 1>5,

y = -10

36. e = 1>3,

y = 6

1 1 + cos u 25 39. r = 10 - 5 cos u 400 41. r = 16 + 8 sin u 8 43. r = 2 - 2 sin u 37. r =

6 2 + cos u 4 40. r = 2 - 2 cos u 12 42. r = 3 + 3 sin u 4 44. r = 2 - sin u 38. r =

598

Chapter 10: Parametric Equations and Polar Coordinates

Lines Sketch the lines in Exercises 45– 48 and find Cartesian equations for them.

73. r = 1>s1 + 2 sin ud

74. r = 1>s1 + 2 cos ud

75. Perihelion and aphelion A planet travels about its sun in an ellipse whose semimajor axis has length a. (See accompanying figure.)

45. r cos au -

p b = 22 4

46. r cos au +

3p b = 1 4

a. Show that r = as1 - ed when the planet is closest to the sun and that r = as1 + ed when the planet is farthest from the sun.

47. r cos au -

2p b = 3 3

48. r cos au +

p b = 2 3

b. Use the data in the table in Exercise 76 to find how close each planet in our solar system comes to the sun and how far away each planet gets from the sun.

Find a polar equation in the form r cos su - u0 d = r0 for each of the lines in Exercises 49–52. 49. 22 x + 22 y = 6 51. y = -5

Aphelion (farthest from sun)

50. 23 x - y = 1 52. x = -4

Planet

Circles Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii. 53. r = 4 cos u 54. r = 6 sin u 55. r = -2 cos u 56. r = -8 sin u Find polar equations for the circles in Exercises 57– 64. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. 57. sx - 6d2 + y 2 = 36 58. sx + 2d2 + y 2 = 4

Perihelion (closest to sun)

a

Sun

76. Planetary orbits Use the data in the table below and Equation (6) to find polar equations for the orbits of the planets. Semimajor axis (astronomical units)

Eccentricity

59. x 2 + s y - 5d2 = 25

60. x 2 + s y + 7d2 = 49

Planet

61. x 2 + 2x + y 2 = 0

62. x 2 - 16x + y 2 = 0 4 64. x 2 + y 2 - y = 0 3

Mercury

0.3871

0.2056

Venus

0.7233

0.0068

63. x + y + y = 0 2

2

Graphs of Polar Equations, and Examples T Graph the lines and conic sections in Exercises 65–74. 65. r = 3 sec su - p>3d 66. r = 4 sec su + p>6d 67. r = 4 sin u 68. r = -2 cos u 69. r = 8>s4 + cos ud 70. r = 8>s4 + sin ud 71. r = 1>s1 - sin ud

Chapter 10

72. r = 1>s1 + cos ud

Earth

1.000

0.0167

Mars

1.524

0.0934

Jupiter

5.203

0.0484

Saturn

9.539

0.0543

Uranus

19.18

0.0460

Neptune

30.06

0.0082

Questions to Guide Your Review

1. What is a parametrization of a curve in the xy-plane? Does a function y = ƒ(x) always have a parametrization? Are parametrizations of a curve unique? Give examples. 2. Give some typical parametrizations for lines, circles, parabolas, ellipses, and hyperbolas. How might the parametrized curve differ from the graph of its Cartesian equation? 3. What is a cycloid? What are typical parametric equations for cycloids? What physical properties account for the importance of cycloids? 4. What is the formula for the slope dy>dx of a parametrized curve x = ƒ(t), y = g(t)? When does the formula apply? When can you expect to be able to find d 2y>dx 2 as well? Give examples. 5. How can you sometimes find the area bounded by a parametrized curve and one of the coordinate axes? 6. How do you find the length of a smooth parametrized curve x = ƒ(t), y = g(t), a … t … b? What does smoothness have to do with length? What else do you need to know about the parametrization in order to find the curve’s length? Give examples.

7. What is the arc length function for a smooth parametrized curve? What is its arc length differential? 8. Under what conditions can you find the area of the surface generated by revolving a curve x = ƒ(t), y = g(t), a … t … b, about the x-axis? the y-axis? Give examples. 9. How do you find the centroid of a smooth parametrized curve x = ƒ(t), y = g(t), a … t … b? Give an example. 10. What are polar coordinates? What equations relate polar coordinates to Cartesian coordinates? Why might you want to change from one coordinate system to the other? 11. What consequence does the lack of uniqueness of polar coordinates have for graphing? Give an example. 12. How do you graph equations in polar coordinates? Include in your discussion symmetry, slope, behavior at the origin, and the use of Cartesian graphs. Give examples. 13. How do you find the area of a region 0 … r1sud … r … r2sud, a … u … b , in the polar coordinate plane? Give examples.

Chapter 10 14. Under what conditions can you find the length of a curve r = ƒsud, a … u … b , in the polar coordinate plane? Give an example of a typical calculation.

Practice Exercises

599

16. Explain the equation PF = e # PD . 17. What are the standard equations for lines and conic sections in polar coordinates? Give examples.

15. What is the eccentricity of a conic section? How can you classify conic sections by eccentricity? How are an ellipse’s shape and eccentricity related?

Chapter 10

Practice Exercises

Identifying Parametric Equations in the Plane Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle. 1. x = t>2, y = t + 1; - q 6 t 6 q 2. x = 2t,

y = 1 - 2t;

3. x = (1>2) tan t, 4. x = - 2 cos t,

t Ú 0

y = (1>2) sec t; y = 2 sin t;

18. x = t - 6t , 3

19. x = 3 cos u,

3

0 … t … p

6. x = 4 cos t,

y = 9 sin t;

0 … t … 2p

2

y = 3 sin u,

shown here. The loop starts at t = - 23 and ends at t = 23 . y

Finding Parametric Equations and Tangent Lines 7. Find parametric equations and a parameter interval for the motion of a particle in the xy-plane that traces the ellipse 16x 2 + 9y 2 = 144 once counterclockwise. (There are many ways to do this.)

t 0

t0

t 3 0

1

2

8. Find parametric equations and a parameter interval for the motion of a particle that starts at the point (-2, 0) in the xy-plane and traces the circle x 2 + y 2 = 4 three times clockwise. (There are many ways to do this.)

9. x = (1>2) tan t, 10. x = 1 + 1>t 2,

y = (1>2) sec t; y = 1 - 3>t;

t = p>3

t = 2

11. Eliminate the parameter to express the curve in the form y = ƒ(x). a. x = 4t 2,

y = t3 - 1

b. x = cos t,

12. Find parametric equations for the given curve. a. Line through (1, -2) with slope 3 b. (x - 1) 2 + ( y + 2) 2 = 9 c. y = 4x 2 - x 2

Surface Areas Find the areas of the surfaces generated by revolving the curves in Exercises 21 and 22 about the indicated axes. 21. x = t 2>2,

y = 2t,

22. x = t + 1>(2t), 2

13. y = x 1>2 - s1>3dx 3>2, 14. x = y 2>3,

1 … x … 4

1 … y … 8

15. y = s5>12dx 6>5 - s5>8dx 4>5, 16. x = sy >12d + s1>yd, 3

1 … x … 32

1 … y … 2

0 … t … 15;

y = 41t,

x-axis

1> 12 … t … 1;

y-axis

Polar to Cartesian Equations Sketch the lines in Exercises 23–28. Also, find a Cartesian equation for each line. p b = 223 3

25. r = 2 sec u Lengths of Curves Find the lengths of the curves in Exercises 13–19.

x

4

t 0

–1

23. r cos au +

d. 9x + 4y = 36 2

y = tan t

0 … t … 1 3p 2

0 … u …

1

In Exercises 9 and 10, find an equation for the line in the xy-plane that is tangent to the curve at the point corresponding to the given value of t. Also, find the value of d 2y>dx 2 at this point.

0 … t … p>2

20. Find the length of the enclosed loop x = t 2, y = st 3>3d - t

0 … t … p

y = cos t;

y = 5 sin t - sin 5t,

y = t + 6t ,

2

-p>2 6 t 6 p>2

5. x = -cos t,

2

17. x = 5 cos t - cos 5t,

27. r = -s3>2d csc u

24. r cos au -

22 3p b = 4 2

26. r = - 22 sec u

28. r = A 323 B csc u

Find Cartesian equations for the circles in Exercises 29–32. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. 29. r = -4 sin u

30. r = 323 sin u

31. r = 222 cos u

32. r = -6 cos u

600

Chapter 10: Parametric Equations and Polar Coordinates

Cartesian to Polar Equations Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. 33. x 2 + y 2 + 5y = 0

34. x 2 + y 2 - 2y = 0

35. x + y - 3x = 0

36. x 2 + y 2 + 4x = 0

2

2

38. -4 sin u … r … 0

Match each graph in Exercises 39–46 with the appropriate equation (a)– (1). There are more equations than graphs, so some equations will not be matched. 6 a. r = cos 2u b. r cos u = 1 c. r = 1 - 2 cos u d. r = sin 2u e. r = u f. r 2 = cos 2u 2 g. r = 1 + cos u h. r = 1 - sin u i. r = 1 - cos u j. r 2 = sin 2u k. r = -sin u l. r = 2 cos u + 1 39. Four-leaved rose

40. Spiral

y

Area in Polar Coordinates Find the areas of the regions in the polar coordinate plane described in Exercises 47–50. 47. Enclosed by the limaçon r = 2 - cos u 48. Enclosed by one leaf of the three-leaved rose r = sin 3u 49. Inside the “figure eight” r = 1 + cos 2u and outside the circle r = 1 50. Inside the cardioid r = 2s1 + sin ud and outside the circle r = 2 sin u

51. r = -1 + cos u

x x

52. r = 2 sin u + 2 cos u, 53. r = 8 sin su>3d, 3

41. Limaçon y

y x x

44. Cardioid y

y

x

x

Exercises 59–62 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. 61. e = 1>2,

r cos u = 2 r sin u = 2

60. e = 1, 62. e = 1>3,

r cos u = -4 r sin u = -6

Additional and Advanced Exercises

Polar Coordinates 1. a. Find an equation in polar coordinates for the curve x = e 2t cos t,

-p>2 … u … p>2

Conics in Polar Coordinates Sketch the conic sections whose polar coordinate equations are given in Exercises 55–58. Give polar coordinates for the vertices and, in the case of ellipses, for the centers as well. 8 2 55. r = 56. r = 1 + cos u 2 + cos u 6 12 57. r = 58. r = 1 - 2 cos u 3 + sin u

59. e = 2,

Chapter 10

0 … u … p>2

0 … u … p>4

54. r = 21 + cos 2u,

42. Lemniscate

43. Circle

x

Length in Polar Coordinates Find the lengths of the curves given by the polar coordinate equations in Exercises 51–54.

y

y

46. Lemniscate y

x

Graphs in Polar Coordinates Sketch the regions defined by the polar coordinate inequalities in Exercises 37 and 38. 37. 0 … r … 6 cos u

45. Parabola

y = e 2t sin t;

-q 6 t 6 q.

b. Find the length of the curve from t = 0 to t = 2p . 2. Find the length of the curve r = 2 sin3 su>3d, 0 … u … 3p , in the polar coordinate plane.

Exercises 3–6 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section. 3. e = 2, 5. e = 1>2,

r cos u = 2 r sin u = 2

4. e = 1, 6. e = 1>3,

r cos u = -4 r sin u = -6

Chapter 10 Theory and Examples 7. Epicycloids When a circle rolls externally along the circumference of a second, fixed circle, any point P on the circumference of the rolling circle describes an epicycloid, as shown here. Let the fixed circle have its center at the origin O and have radius a. y

x = r cos u

dy dr = r cos u + sin u . du du

(2)

(3)

Since c = f - u from (1), x A(a, 0)

tan c = tan sf - ud =

Let the radius of the rolling circle be b and let the initial position of the tracing point P be A(a, 0). Find parametric equations for the epicycloid, using as the parameter the angle u from the positive x-axis to the line through the circles’ centers.

y = as1 - cos td;

0 … t … 2p .

The Angle Between the Radius Vector and the Tangent Line to a Polar Coordinate Curve In Cartesian coordinates, when we want to discuss the direction of a curve at a point, we use the angle f measured counterclockwise from the positive x-axis to the tangent line. In polar coordinates, it is more convenient to calculate the angle c from the radius vector to the tangent line (see the accompanying figure). The angle f can then be calculated from the relation f = u + c, (1) which comes from applying the Exterior Angle Theorem to the triangle in the accompanying figure. y r ⫽ f ()

P(r, )

tan f =

dy>du dy = dx dx>du

because tan f is the slope of the curve at P. Also, y tan u = x . Hence dy>du y dy dx - x x - y dx>du du du tan c = . = dy y dy>du dx x + y 1 + x du du dx>du 9. From Equations (2), (3), and (4), show that r tan c = . dr>du

(4)

(5)

This is the equation we use for finding c as a function of u .

r

tan f - tan u . 1 + tan f tan u

Furthermore,

8. Find the centroid of the region enclosed by the x-axis and the cycloid arch

y = r sin u

are differentiable functions of u with

C

O

0

and

dx dr = -r sin u + cos u , du du

␪

x = ast - sin td,

601

Suppose the equation of the curve is given in the form r = ƒsud , where ƒsud is a differentiable function of u . Then

P b

Additional and Advanced Exercises

10. Find the value of tan c for the curve r = sin4 su>4d . 11. Find the angle between the radius vector to the curve r = 2a sin 3u and its tangent when u = p>6 . T 12. a. Graph the hyperbolic spiral ru = 1 . What appears to happen to c as the spiral winds in around the origin?

x

b. Confirm your finding in part (a) analytically.

11 VECTORS AND THE GEOMETRY OF SPACE OVERVIEW To apply calculus in many real-world situations and in higher mathematics, we need a mathematical description of three-dimensional space. In this chapter we introduce three-dimensional coordinate systems and vectors. Building on what we already know about coordinates in the xy-plane, we establish coordinates in space by adding a third axis that measures distance above and below the xy-plane. Vectors are used to study the analytic geometry of space, where they give simple ways to describe lines, planes, surfaces, and curves in space. We use these geometric ideas later in the book to study motion in space and the calculus of functions of several variables, with their many important applications in science, engineering, economics, and higher mathematics.

Three-Dimensional Coordinate Systems

11.1

z

z = constant (0, 0, z) (0, y, z) (x, 0, z) 0

P(x, y, z) (0, y, 0) y

(x, 0, 0) y = constant x

x = constant

(x, y, 0)

FIGURE 11.1 The Cartesian coordinate system is right-handed.

602

To locate a point in space, we use three mutually perpendicular coordinate axes, arranged as in Figure 11.1. The axes shown there make a right-handed coordinate frame. When you hold your right hand so that the fingers curl from the positive x-axis toward the positive y-axis, your thumb points along the positive z-axis. So when you look down on the xy-plane from the positive direction of the z-axis, positive angles in the plane are measured counterclockwise from the positive x-axis and around the positive z-axis. (In a left-handed coordinate frame, the z-axis would point downward in Figure 11.1 and angles in the plane would be positive when measured clockwise from the positive x-axis. Right-handed and left-handed coordinate frames are not equivalent.) The Cartesian coordinates (x, y, z) of a point P in space are the values at which the planes through P perpendicular to the axes cut the axes. Cartesian coordinates for space are also called rectangular coordinates because the axes that define them meet at right angles. Points on the x-axis have y- and z-coordinates equal to zero. That is, they have coordinates of the form (x, 0, 0). Similarly, points on the y-axis have coordinates of the form (0, y, 0), and points on the z-axis have coordinates of the form (0, 0, z). The planes determined by the coordinates axes are the xy-plane, whose standard equation is z = 0; the yz-plane, whose standard equation is x = 0; and the xz-plane, whose standard equation is y = 0. They meet at the origin (0, 0, 0) (Figure 11.2). The origin is also identified by simply 0 or sometimes the letter O. The three coordinate planes x = 0, y = 0, and z = 0 divide space into eight cells called octants. The octant in which the point coordinates are all positive is called the first octant; there is no convention for numbering the other seven octants. The points in a plane perpendicular to the x-axis all have the same x-coordinate, this being the number at which that plane cuts the x-axis. The y- and z-coordinates can be any numbers. Similarly, the points in a plane perpendicular to the y-axis have a common y-coordinate and the points in a plane perpendicular to the z-axis have a common z-coordinate. To write equations for these planes, we name the common coordinate’s value. The plane x = 2 is the plane perpendicular to the x-axis at x = 2. The plane y = 3 is the plane perpendicular to the y-axis

11.1

Three-Dimensional Coordinate Systems

603

z z

(0, 0, 5)

(2, 3, 5)

xz-plane: y 0

Line y 3, z 5 Plane z 5 Line x 2, z 5

yz-plane: x 0 Plane x 2

x y-plane: z 0

Origin

Plane y 3 0

(0, 3, 0)

(2, 0, 0) (0, 0, 0)

y

y x

x

Line x 2, y 3

FIGURE 11.2 The planes x = 0, y = 0 , and z = 0 divide space into eight octants.

FIGURE 11.3 The planes x = 2, y = 3 , and z = 5 determine three lines through the point (2, 3, 5).

at y = 3. The plane z = 5 is the plane perpendicular to the z-axis at z = 5. Figure 11.3 shows the planes x = 2, y = 3, and z = 5, together with their intersection point (2, 3, 5). The planes x = 2 and y = 3 in Figure 11.3 intersect in a line parallel to the z-axis. This line is described by the pair of equations x = 2, y = 3. A point (x, y, z) lies on the line if and only if x = 2 and y = 3. Similarly, the line of intersection of the planes y = 3 and z = 5 is described by the equation pair y = 3, z = 5. This line runs parallel to the x-axis. The line of intersection of the planes x = 2 and z = 5, parallel to the y-axis, is described by the equation pair x = 2, z = 5. In the following examples, we match coordinate equations and inequalities with the sets of points they define in space.

EXAMPLE 1

We interpret these equations and inequalities geometrically.

(a) z Ú 0 (b) x = -3 (c) z = 0, x … 0, y Ú 0 (d) x Ú 0, y Ú 0, z Ú 0 (e) -1 … y … 1

z The circle x 2 1 y 2 5 4, z 5 3

(f) y = -2, z = 2 (0, 2, 3)

The half-space consisting of the points on and above the xy-plane. The plane perpendicular to the x-axis at x = -3. This plane lies parallel to the yz-plane and 3 units behind it. The second quadrant of the xy-plane. The first octant. The slab between the planes y = -1 and y = 1 (planes included). The line in which the planes y = -2 and z = 2 intersect. Alternatively, the line through the point s0, -2, 2d parallel to the x-axis.

(2, 0, 3) The plane z53 (0, 2, 0) (2, 0, 0) x

EXAMPLE 2

x2 + y2 = 4

y x2

1

y2

5 4, z 5 0

FIGURE 11.4 The circle x 2 + y 2 = 4 in the plane z = 3 (Example 2).

What points P(x, y, z) satisfy the equations and

z = 3?

The points lie in the horizontal plane z = 3 and, in this plane, make up the circle x 2 + y 2 = 4. We call this set of points “the circle x 2 + y 2 = 4 in the plane z = 3” or, more simply, “the circle x 2 + y 2 = 4, z = 3” (Figure 11.4).

Solution

604

Chapter 11: Vectors and the Geometry of Space z

Distance and Spheres in Space P1( 1, y , z 1)

P2(x 2, y 2, z 2 )

The formula for the distance between two points in the xy-plane extends to points in space.

The Distance Between P1sx1 , y1 , z1 d and P2sx2 , y2 , z2 d 0

ƒ P1 P2 ƒ = 2sx2 - x1 d2 + s y2 - y1 d2 + sz2 - z1 d2

B(x , y 2, z 1) y

A(x , y1 z 1) x

FIGURE 11.5 We find the distance between P1 and P2 by applying the Pythagorean theorem to the right triangles P1 AB and P1 BP2 .

Proof We construct a rectangular box with faces parallel to the coordinate planes and the points P1 and P2 at opposite corners of the box (Figure 11.5). If Asx2 , y1 , z1 d and Bsx2 , y2 , z1 d are the vertices of the box indicated in the figure, then the three box edges P1 A, AB, and BP2 have lengths ƒ P1 A ƒ = ƒ x2 - x1 ƒ ,

ƒ AB ƒ = ƒ y2 - y1 ƒ ,

ƒ BP2 ƒ = ƒ z2 - z1 ƒ .

Because triangles P1 BP2 and P1 AB are both right-angled, two applications of the Pythagorean theorem give 2 2 2 ƒ P1 P2 ƒ = ƒ P1 B ƒ + ƒ BP2 ƒ

and

2 2 2 ƒ P1 B ƒ = ƒ P1 A ƒ + ƒ AB ƒ

(see Figure 11.5). So ƒ P1 P2 ƒ 2 = ƒ P1 B ƒ 2 + ƒ BP2 ƒ 2 = ƒ P1 A ƒ 2 + ƒ AB ƒ 2 + ƒ BP2 ƒ 2

Substitute ƒ P1 B ƒ 2 = ƒ P1 A ƒ 2 + ƒ AB ƒ 2 .

= ƒ x2 - x1 ƒ 2 + ƒ y2 - y1 ƒ 2 + ƒ z2 - z1 ƒ 2 = sx2 - x1 d2 + sy2 - y1 d2 + sz2 - z1 d2. Therefore ƒ P1 P2 ƒ = 2sx2 - x1 d2 + s y2 - y1 d2 + sz2 - z1 d2.

EXAMPLE 3

The distance between P1s2, 1, 5d and P2s -2, 3, 0d is ƒ P1 P2 ƒ = 2s -2 - 2d2 + s3 - 1d2 + s0 - 5d2 = 216 + 4 + 25 = 245 L 6.708.

z P0(x 0 , y0 , z 0 )

P(x, y, z)

We can use the distance formula to write equations for spheres in space (Figure 11.6). A point P(x, y, z) lies on the sphere of radius a centered at P0sx0 , y0 , z0 d precisely when ƒ P0 P ƒ = a or sx - x0 d2 + sy - y0 d2 + sz - z0 d2 = a 2 .

a

The Standard Equation for the Sphere of Radius a and Center sx0 , y0 , z0 d sx - x0 d2 + sy - y0 d2 + sz - z0 d2 = a 2

0 y x

EXAMPLE 4

Find the center and radius of the sphere x 2 + y 2 + z 2 + 3x - 4z + 1 = 0.

FIGURE 11.6 The sphere of radius a centered at the point sx0 , y0 , z0 d.

We find the center and radius of a sphere the way we find the center and radius of a circle: Complete the squares on the x-, y-, and z-terms as necessary and write each

Solution

11.1

Three-Dimensional Coordinate Systems

605

quadratic as a squared linear expression. Then, from the equation in standard form, read off the center and radius. For the sphere here, we have x 2 + y 2 + z 2 + 3x - 4z + 1 = 0 sx 2 + 3xd + y 2 + sz 2 - 4zd = -1 2

2

2

3 3 -4 -4 ax 2 + 3x + a b b + y 2 + az 2 - 4z + a b b = -1 + a b + a b 2 2 2 2 ax +

2

2

3 9 21 b + y 2 + sz - 2d2 = -1 + + 4 = . 2 4 4

From this standard form, we read that x0 = -3>2, y0 = 0, z0 = 2, and a = 221>2. The center is s -3>2, 0, 2d. The radius is 221>2.

EXAMPLE 5

Here are some geometric interpretations of inequalities and equations involving spheres. (a) x 2 + y 2 + z 2 6 4 (b) x 2 + y 2 + z 2 … 4

The interior of the sphere x 2 + y 2 + z 2 = 4. The solid ball bounded by the sphere x 2 + y 2 + z 2 = 4. Alternatively, the sphere x 2 + y 2 + z 2 = 4 together with its interior. The exterior of the sphere x 2 + y 2 + z 2 = 4. The lower hemisphere cut from the sphere x 2 + y 2 + z 2 = 4 by the xy-plane (the plane z = 0).

(c) x 2 + y 2 + z 2 7 4 (d) x 2 + y 2 + z 2 = 4, z … 0

Just as polar coordinates give another way to locate points in the xy-plane (Section 10.3), alternative coordinate systems, different from the Cartesian coordinate system developed here, exist for three-dimensional space. We examine two of these coordinate systems in Section 14.7.

Exercises 11.1 Geometric Interpretations of Equations In Exercises 1–16, give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. 1. x = 2,

y = 3

2. x = -1,

3. y = 0,

z = 0

4. x = 1,

z = 0 y = 0

5. x 2 + y 2 = 4,

z = 0

6. x 2 + y 2 = 4,

z = -2

7. x 2 + z 2 = 4,

y = 0

8. y 2 + z 2 = 1,

x = 0

9. x + y + z = 1, 2

2

x = 0

2

10. x 2 + y 2 + z 2 = 25,

y = -4

11. x 2 + y 2 + sz + 3d2 = 25, 12. x + s y - 1d + z = 4, 2

2

13. x 2 + y 2 = 4,

2

z = y

14. x 2 + y 2 + z 2 = 4,

y = x

z = 0 y = 0

Geometric Interpretations of Inequalities and Equations In Exercises 17–24, describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. 17. a. x Ú 0,

y Ú 0,

z = 0

b. x Ú 0,

y … 0,

b. 0 … x … 1,

18. a. 0 … x … 1 c. 0 … x … 1,

0 … y … 1,

z = 0

0 … y … 1

0 … z … 1

19. a. x 2 + y 2 + z 2 … 1

b. x 2 + y 2 + z 2 7 1

20. a. x 2 + y 2 … 1,

b. x 2 + y 2 … 1,

z = 0

c. x + y … 1, 2

2

z = 3

no restriction on z

21. a. 1 … x 2 + y 2 + z 2 … 4 b. x 2 + y 2 + z 2 … 1, 22. a. x = y, 23. a. y Ú x , 2

z = 0 z Ú 0

15. y = x 2,

z = 0

24. a. z = 1 - y,

16. z = y 2,

x = 1

b. z = y 3,

z Ú 0 b. x = y, b. x … y ,

no restriction on x

x = 2

2

no restriction on z 0 … z … 2

606

Chapter 11: Vectors and the Geometry of Space

In Exercises 25–34, describe the given set with a single equation or with a pair of equations.

44. P1s3, 4, 5d, 45. P1s0, 0, 0d,

P2s2, -2, -2d

25. The plane perpendicular to the

46. P1s5, 3, -2d,

P2s0, 0, 0d

a. x-axis at (3, 0, 0)

b. y-axis at s0, - 1, 0d

c. z-axis at s0, 0, -2d 26. The plane through the point s3, -1, 2d perpendicular to the a. x-axis

b. y-axis

c. z-axis

27. The plane through the point s3, -1, 1d parallel to the a. xy-plane

b. yz-plane

c. xz-plane

28. The circle of radius 2 centered at (0, 0, 0) and lying in the a. xy-plane

b. yz-plane

c. xz-plane

29. The circle of radius 2 centered at (0, 2, 0) and lying in the a. xy-plane

b. yz-plane

b. yz-plane

Spheres Find the centers and radii of the spheres in Exercises 47–50. 47. sx + 2d2 + y 2 + sz - 2d2 = 8 48. sx - 1d2 + ay +

2

1 b + sz + 3d2 = 25 2

49. A x - 22 B + A y - 22 B + A z + 22 B = 2 2

50. x 2 + ay +

2

2

b. y-axis

2

16 1 1 b + az - b = 3 3 9

c. xz-plane

Find equations for the spheres whose centers and radii are given in Exercises 51–54. Center

Radius

31. The line through the point s1, 3, -1d parallel to the a. x-axis

2

c. plane y = 2

30. The circle of radius 1 centered at s -3, 4, 1d and lying in a plane parallel to the a. xy-plane

P2s2, 3, 4d

c. z-axis

32. The set of points in space equidistant from the origin and the point (0, 2, 0) 33. The circle in which the plane through the point (1, 1, 3) perpendicular to the z-axis meets the sphere of radius 5 centered at the origin 34. The set of points in space that lie 2 units from the point (0, 0, 1) and, at the same time, 2 units from the point s0, 0, - 1d Inequalities to Describe Sets of Points Write inequalities to describe the sets in Exercises 35–40.

51. (1, 2, 3)

214

52. s0, -1, 5d

2

1 2 53. a -1, , - b 2 3

4 9

54. s0, -7, 0d

7

Find the centers and radii of the spheres in Exercises 55–58. 55. x 2 + y 2 + z 2 + 4x - 4z = 0 56. x 2 + y 2 + z 2 - 6y + 8z = 0 57. 2x 2 + 2y 2 + 2z 2 + x + y + z = 9

35. The slab bounded by the planes z = 0 and z = 1 (planes included)

58. 3x 2 + 3y 2 + 3z 2 + 2y - 2z = 9

36. The solid cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2 , and z = 2

Theory and Examples

37. The half-space consisting of the points on and below the xy-plane 38. The upper hemisphere of the sphere of radius 1 centered at the origin 39. The (a) interior and (b) exterior of the sphere of radius 1 centered at the point (1, 1, 1) 40. The closed region bounded by the spheres of radius 1 and radius 2 centered at the origin. (Closed means the spheres are to be included. Had we wanted the spheres left out, we would have asked for the open region bounded by the spheres. This is analogous to the way we use closed and open to describe intervals: closed means endpoints included, open means endpoints left out. Closed sets include boundaries; open sets leave them out.) Distance In Exercises 41–46, find the distance between points P1 and P2. 41. P1s1, 1, 1d,

P2s3, 3, 0d

42. P1s -1, 1, 5d,

P2s2, 5, 0d

43. P1s1, 4, 5d,

P2s4, -2, 7d

59. Find a formula for the distance from the point P(x, y, z) to the a. x-axis.

b. y-axis.

c. z-axis.

60. Find a formula for the distance from the point P(x, y, z) to the a. xy-plane.

b. yz-plane.

c. xz-plane.

61. Find the perimeter of the triangle with vertices As -1, 2, 1d, Bs1, -1, 3d, and C(3, 4, 5). 62. Show that the point P(3, 1, 2) is equidistant from the points As2, -1, 3d and B(4, 3, 1). 63. Find an equation for the set of all points equidistant from the planes y = 3 and y = -1. 64. Find an equation for the set of all points equidistant from the point (0, 0, 2) and the xy-plane. 65. Find the point on the sphere x 2 + sy - 3d2 + sz + 5d2 = 4 nearest

a. the xy-plane.

b. the point s0, 7, -5d.

66. Find the point equidistant from the points (0, 0, 0), (0, 4, 0), (3, 0, 0), and (2, 2, -3).

11.2 Vectors

607

Vectors

11.2

Some of the things we measure are determined simply by their magnitudes. To record mass, length, or time, for example, we need only write down a number and name an appropriate unit of measure. We need more information to describe a force, displacement, or velocity. To describe a force, we need to record the direction in which it acts as well as how large it is. To describe a body’s displacement, we have to say in what direction it moved as well as how far. To describe a body’s velocity, we have to know where the body is headed as well as how fast it is going. In this section we show how to represent things that have both magnitude and direction in the plane or in space.

Component Form

Terminal point B

AB

Initial point A

FIGURE 11.7 The directed line segment 1 AB is called a vector.

A quantity such as force, displacement, or velocity is called a vector and is represented by a directed line segment (Figure 11.7). The arrow points in the direction of the action and its length gives the magnitude of the action in terms of a suitably chosen unit. For example, a force vector points in the direction in which the force acts and its length is a measure of the force’s strength; a velocity vector points in the direction of motion and its length is the speed of the moving object. Figure 11.8 displays the velocity vector v at a specific location for a particle moving along a path in the plane or in space. (This application of vectors is studied in Chapter 12.) y

y

z v

v

B A

D C P O

x F

(a) two dimensions

E

FIGURE 11.9 The four arrows in the plane (directed line segments) shown here have the same length and direction. They therefore represent the same vector, 1 1 1 1 and we write AB = CD = OP = EF . z P(x 1, y1, z 1)

0

Q(x 2 , y 2 , z 2 )

Position vector of PQ v v1, v 2, v3 v2

x

0

x

0

x

y

(b) three dimensions

FIGURE 11.8 The velocity vector of a particle moving along a path (a) in the plane (b) in space. The arrowhead on the path indicates the direction of motion of the particle.

1 DEFINITIONS The vector represented by the directed line segment AB has 1 initial point A and terminal point B and its length is denoted by ƒ AB ƒ . Two vectors are equal if they have the same length and direction.

(v1, v 2 , v3)

v3 v1

y

1 FIGURE 11.10 A vector PQ in standard position has its initial point at the origin. 1 The directed line segments PQ and v are parallel and have the same length.

The arrows we use when we draw vectors are understood to represent the same vector if they have the same length, are parallel, and point in the same direction (Figure 11.9) regardless of the initial point. In textbooks, vectors are usually written in lowercase, boldface letters, for example u, v, and w. Sometimes we use uppercase boldface letters, such as F, to denote a force vector. u, In handwritten form, it is customary to draw small arrows above the letters, for example s s. y, w s s , and F We need a way to represent vectors algebraically so that we can be more precise about 1 1 the direction of a vector. Let v = PQ . There is one directed line segment equal to PQ whose initial point is the origin (Figure 11.10). It is the representative of v in standard position and is the vector we normally use to represent v. We can specify v by writing the

608

Chapter 11: Vectors and the Geometry of Space

coordinates of its terminal point sv1, v2 , v3 d when v is in standard position. If v is a vector in the plane its terminal point sv1, v2 d has two coordinates.

DEFINITION If v is a two-dimensional vector in the plane equal to the vector with initial point at the origin and terminal point sv1, v2 d, then the component form of v is v = 8v1, v29.

If v is a three-dimensional vector equal to the vector with initial point at the origin and terminal point sv1, v2, v3 d, then the component form of v is v = 8v1, v2 , v39.

HISTORICAL BIOGRAPHY Carl Friedrich Gauss (1777–1855)

So a two-dimensional vector is an ordered pair v = 8v1, v29 of real numbers, and a three-dimensional vector is an ordered triple v = 8v1, v2 , v39 of real numbers. The numbers v1, v2 , and v3 are the components of v. 1 If v = 8v1, v2 , v39 is represented by the directed line segment PQ , where the initial point is Psx1, y1, z1 d and the terminal point is Qsx2 , y2 , z2 d, then x1 + v1 = x2, y1 + v2 = y2 , and z1 + v3 = z2 (see Figure 11.10). Thus, v1 = x2 - x1, v2 = y2 - y1 , 1 and v3 = z2 - z1 are the components of PQ . In summary, given the points Psx1, y1, z1 d and Qsx2 , y2 , z2 d, the standard position vec1 tor v = 8v1, v2 , v39 equal to PQ is v = 8x2 - x1, y2 - y1, z2 - z19.

If v is two-dimensional with Psx1, y1 d and Qsx2 , y2 d as points in the plane, then v = 8x2 - x1, y2 - y19. There is no third component for planar vectors. With this understanding, we will develop the algebra of three-dimensional vectors and simply drop the third component when the vector is two-dimensional (a planar vector). Two vectors are equal if and only if their standard position vectors are identical. Thus 8u1, u2 , u39 and 8v1, v2 , v39 are equal if and only if u1 = v1, u2 = v2 , and u3 = v3 . 1 The magnitude or length of the vector PQ is the length of any of its equivalent directed line segment representations. In particular, if v = 8x2 - x1, y2 - y1, z2 - z19 is the 1 standard position vector for PQ , then the distance formula gives the magnitude or length of v, denoted by the symbol ƒ v ƒ or ƒ ƒ v ƒ ƒ . 1 The magnitude or length of the vector v = PQ is the nonnegative number 2 2 2 ƒ v ƒ = 2v12 + v22 + v32 = 2sx2 - x1 d + s y2 - y1 d + sz2 - z1 d

(see Figure 11.10). The only vector with length 0 is the zero vector 0 = 80, 09 or 0 = 80, 0, 09. This vector is also the only vector with no specific direction.

EXAMPLE 1

Find the (a) component form and (b) length of the vector with initial point Ps -3, 4, 1d and terminal point Qs -5, 2, 2d.

Solution

1 (a) The standard position vector v representing PQ has components v1 = x2 - x1 = -5 - s -3d = -2,

v2 = y2 - y1 = 2 - 4 = -2,

11.2 Vectors

609

and v3 = z2 - z1 = 2 - 1 = 1. 1 The component form of PQ is v = 8-2, -2, 19. 1 (b) The length or magnitude of v = PQ is

y

ƒ v ƒ = 2s -2d2 + s -2d2 + s1d2 = 29 = 3.

F = a, b 45°

EXAMPLE 2 x

FIGURE 11.11 The force pulling the cart forward is represented by the vector F whose horizontal component is the effective force (Example 2).

A small cart is being pulled along a smooth horizontal floor with a 20-lb force F making a 45° angle to the floor (Figure 11.11). What is the effective force moving the cart forward? Solution

The effective force is the horizontal component of F = 8a, b9, given by 22 b L 14.14 lb. a = ƒ F ƒ cos 45° = s20d a 2

Notice that F is a two-dimensional vector.

Vector Algebra Operations Two principal operations involving vectors are vector addition and scalar multiplication. A scalar is simply a real number, and is called such when we want to draw attention to its differences from vectors. Scalars can be positive, negative, or zero and are used to “scale” a vector by multiplication. Let u = 8u1, u2 , u39 and v = 8v1, v2 , v39 be vectors with k a

DEFINITIONS scalar.

Addition: u + v = 8u1 + v1, u2 + v2 , u3 + v39 Scalar multiplication: ku = 8ku1, ku2 , ku39 We add vectors by adding the corresponding components of the vectors. We multiply a vector by a scalar by multiplying each component by the scalar. The definitions apply to planar vectors except there are only two components, 8u1, u29 and 8v1, v29. The definition of vector addition is illustrated geometrically for planar vectors in Figure 11.12a, where the initial point of one vector is placed at the terminal point of the other. Another interpretation is shown in Figure 11.12b (called the parallelogram law of y

y u1 v1, u 2 v 2

v2

v

u+v

u+v v

v1

u

u

u2 0

x

u1 (a)

x

0 (b)

FIGURE 11.12 (a) Geometric interpretation of the vector sum. (b) The parallelogram law of vector addition.

610

Chapter 11: Vectors and the Geometry of Space

1.5u

2u

u

–2u

addition), where the sum, called the resultant vector, is the diagonal of the parallelogram. In physics, forces add vectorially as do velocities, accelerations, and so on. So the force acting on a particle subject to two gravitational forces, for example, is obtained by adding the two force vectors. Figure 11.13 displays a geometric interpretation of the product ku of the scalar k and vector u. If k 7 0, then ku has the same direction as u; if k 6 0, then the direction of ku is opposite to that of u. Comparing the lengths of u and ku, we see that 2 2 2 2 2 2 2 ƒ ku ƒ = 2sku1 d + sku2 d + sku3 d = 2k su 1 + u 2 + u 3 d

FIGURE 11.13 Scalar multiples of u.

= 2k 2 2u 12 + u 22 + u 32 = ƒ k ƒ ƒ u ƒ . The length of ku is the absolute value of the scalar k times the length of u. The vector s -1du = -u has the same length as u but points in the opposite direction. The difference u - v of two vectors is defined by

uv v

u - v = u + s -vd.

If u = 8u1, u2 , u39 and v = 8v1, v2 , v39, then

u

u - v = 8u1 - v1, u2 - v2, u3 - v39.

(a)

Note that su - vd + v = u, so adding the vector su - vd to v gives u (Figure 11.14a). Figure 11.14b shows the difference u - v as the sum u + s -vd.

EXAMPLE 3

v

(a) 2u + 3v

u

Let u = 8-1, 3, 19 and v = 84, 7, 09. Find the components of (b) u - v

–v u (–v) (b)

FIGURE 11.14 (a) The vector u - v, when added to v, gives u. (b) u - v = u + s -vd .

(c) `

1 u` . 2

Solution

(a) 2u + 3v = 28-1, 3, 19 + 384, 7, 09 = 8-2, 6, 29 + 812, 21, 09 = 810, 27, 29 (b) u - v = 8-1, 3, 19 - 84, 7, 09 = 8-1 - 4, 3 - 7, 1 - 09 = 8-5, -4, 19 (c) `

2

2

2

3 1 1 3 1 1 1 1 u ` = ` h- , , i ` = a- b + a b + a b = 211. 2 2 2 2 2 2 2 2 C

Vector operations have many of the properties of ordinary arithmetic.

Properties of Vector Operations Let u, v, w be vectors and a, b be scalars. 1. u + v = v + u

3. 5. 7. 9.

u + 0 = u 0u = 0 asbud = sabdu sa + bdu = au + bu

2. 4. 6. 8.

su + vd + w = u + sv + wd u + s -ud = 0 1u = u asu + vd = au + av

These properties are readily verified using the definitions of vector addition and multiplication by a scalar. For instance, to establish Property 1, we have u + v = 8u1, u2, u39 + 8v1, v2, v39 = 8u1 + v1, u2 + v2, u3 + v39 = 8v1 + u1, v2 + u2, v3 + u39 = 8v1, v2, v39 + 8u1, u2, u39 = v + u.

11.2 Vectors

611

When three or more space vectors lie in the same plane, we say they are coplanar vectors. For example, the vectors u, v, and u + v are always coplanar.

Unit Vectors A vector v of length 1 is called a unit vector. The standard unit vectors are i = 81, 0, 09,

j = 80, 1, 09,

and

k = 80, 0, 19.

Any vector v = 8v1, v2 , v39 can be written as a linear combination of the standard unit vectors as follows: v = 8v1, v2 , v39 = 8v1, 0, 09 + 80, v2 , 09 + 80, 0, v39 z

= v181, 0, 09 + v280, 1, 09 + v380, 0, 19

OP2 x 2 i y2 j z 2 k

= v1 i + v2 j + v3 k.

P2(x2, y 2 , z 2 )

We call the scalar (or number) v1 the i-component of the vector v, v2 the j-component, and v3 the k-component. In component form, the vector from P1sx1, y1, z1 d to P2sx2 , y2 , z2 d is

k O

1 P1P2 = sx2 - x1 di + s y2 - y1 dj + sz2 - z1 dk

P1P2

j

i y P1(x 1, y 1 , z 1 )

x

(Figure 11.15). Whenever v Z 0, its length ƒ v ƒ is not zero and

OP1 x 1i y 1 j z 1k

FIGURE 11.15 The vector from P1 to P2 1 is P1P2 = sx2 - x1 di + s y2 - y1 dj + sz2 - z1 dk.

`

1 1 v` = ƒ v ƒ = 1. ƒvƒ ƒvƒ

That is, v> ƒ v ƒ is a unit vector in the direction of v, called the direction of the nonzero vector v.

EXAMPLE 4

Find a unit vector u in the direction of the vector from P1s1, 0, 1d to

P2s3, 2, 0d. Solution

1 We divide P1P2 by its length: 1 P1P2 = s3 - 1di + s2 - 0dj + s0 - 1dk = 2i + 2j - k 1 2 2 2 ƒ P1P2 ƒ = 2s2d + s2d + s -1d = 24 + 4 + 1 = 29 = 3 u =

1 2i + 2j - k P1P2 2 2 1 = i + j - k. 1 = 3 3 3 3 ƒ P1P2 ƒ

1 The unit vector u is the direction of P1P2 . If v = 3i - 4j is a velocity vector, express v as a product of its speed times a unit vector in the direction of motion.

EXAMPLE 5 Solution

Speed is the magnitude (length) of v: 2 2 ƒ v ƒ = 2s3d + s -4d = 29 + 16 = 5.

HISTORICAL BIOGRAPHY Hermann Grassmann (1809–1877)

The unit vector v> ƒ v ƒ has the same direction as v: 3i - 4j v 3 4 = = i - j. 5 5 5 v ƒ ƒ

612

Chapter 11: Vectors and the Geometry of Space

So 3 4 v = 3i - 4j = 5 a i - jb . 5 5 (')'* Length (speed)

Direction of motion

In summary, we can express any nonzero vector v in terms of its two important features, v . length and direction, by writing v = ƒ v ƒ ƒvƒ If v Z 0, then v 1. is a unit vector in the direction of v; ƒvƒ v 2. the equation v = ƒ v ƒ expresses v as its length times its direction. v ƒ ƒ

EXAMPLE 6 A force of 6 newtons is applied in the direction of the vector v = 2i + 2j - k. Express the force F as a product of its magnitude and direction. Solution

The force vector has magnitude 6 and direction F = 6

v , so ƒvƒ

2i + 2j - k 2i + 2j - k v = 6 = 6 3 ƒvƒ 222 + 22 + s -1d2

2 2 1 = 6 a i + j - kb . 3 3 3

Midpoint of a Line Segment Vectors are often useful in geometry. For example, the coordinates of the midpoint of a line segment are found by averaging. The midpoint M of the line segment joining points P1sx1, y1, z1 d and P2sx2 , y2 , z2 d is the point

P1(x 1, y1, z 1)

a

⎛ x x 2 y1 y 2 z1 z 2⎛ , , M⎝ 1 2 2 2 ⎝

x1 + x2 y1 + y2 z1 + z2 , , b. 2 2 2

To see why, observe (Figure 11.16) that P2(x 2, y 2 , z 2 )

1 1 1 1 1 1 1 1 OM = OP1 + sP1P2 d = OP1 + sOP2 - OP1 d 2 2 1 1 1 = sOP1 + OP2 d 2 y1 + y2 x1 + x2 z1 + z2 = i + j + k. 2 2 2

O

FIGURE 11.16 The coordinates of the midpoint are the averages of the coordinates of P1 and P2.

EXAMPLE 7

The midpoint of the segment joining P1s3, -2, 0d and P2s7, 4, 4d is a

3 + 7 -2 + 4 0 + 4 , , b = s5, 1, 2d. 2 2 2

11.2 Vectors

613

Applications An important application of vectors occurs in navigation.

EXAMPLE 8 A jet airliner, flying due east at 500 mph in still air, encounters a 70-mph tailwind blowing in the direction 60° north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they?

N

v 30˚ 70

uv 500

u

E

NOT TO SCALE

FIGURE 11.17 Vectors representing the velocities of the airplane u and tailwind v in Example 8.

If u = the velocity of the airplane alone and v = the velocity of the tailwind, then ƒ u ƒ = 500 and ƒ v ƒ = 70 (Figure 11.17). The velocity of the airplane with respect to the ground is given by the magnitude and direction of the resultant vector u + v. If we let the positive x-axis represent east and the positive y-axis represent north, then the component forms of u and v are

Solution

u = 8500, 09

v = 870 cos 60°, 70 sin 60°9 = 835, 35 239.

and

Therefore, u + v = 8535, 35239 = 535i + 3523 j ƒ u + v ƒ = 25352 + s3513d2 L 538.4 and u = tan-1

3523 L 6.5°. 535

Figure 11.17

The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5° north of east. Another important application occurs in physics and engineering when several forces are acting on a single object.

40°

55° F1 F2 55°

The force vectors F1 and F2 have magnitudes ƒ F1 ƒ and ƒ F2 ƒ and components that are measured in newtons. The resultant force is the sum F1 + F2 and must be equal in magnitude and acting in the opposite (or upward) direction to the weight vector w (see Figure 11.18b). It follows from the figure that

Solution

40° 75 (a) F F1 F2 0, 75

F1

EXAMPLE 9 A 75-N weight is suspended by two wires, as shown in Figure 11.18a. Find the forces F1 and F2 acting in both wires.

F1 = 8- ƒ F1 ƒ cos 55°, ƒ F1 ƒ sin 55°9 and F2 = 8ƒ F2 ƒ cos 40°, ƒ F2 ƒ sin 40°9.

Since F1 + F2 = 80, 759, the resultant vector leads to the system of equations - ƒ F1 ƒ cos 55° + ƒ F2 ƒ cos 40° = 0 ƒ F1 ƒ sin 55° + ƒ F2 ƒ sin 40° = 75.

F1 F2 55°

F2

40°

w 0, 75 (b)

FIGURE 11.18 The suspended weight in Example 9.

Solving for ƒ F2 ƒ in the first equation and substituting the result into the second equation, we get ƒ F2 ƒ =

ƒ F1 ƒ cos 55° cos 40°

and

ƒ F1 ƒ sin 55° +

ƒ F1 ƒ cos 55° sin 40° = 75. cos 40°

It follows that 75 ƒ F1 ƒ = sin 55° + cos 55° tan 40° L 57.67 N,

614

Chapter 11: Vectors and the Geometry of Space

and 75 cos 55° ƒ F2 ƒ = sin 55° cos 40° + cos 55° sin 40° =

75 cos 55° L 43.18 N. sins55° + 40°d

The force vectors are then F1 = 8-33.08, 47.249 and F2 = 833.08, 27.769.

Exercises 11.2 Vectors in the Plane In Exercises 1–8, let u = 83, -29 and v = 8-2, 59 . Find the (a) component form and (b) magnitude (length) of the vector. 1. 3u

2. -2v

3. u + v

4. u - v

5. 2u - 3v

6. -2u + 5v

7.

3 4 u + v 5 5

8. -

Geometric Representations In Exercises 23 and 24, copy vectors u, v, and w head to tail as needed to sketch the indicated vector. 23. w u

5 12 u + v 13 13

In Exercises 9–16, find the component form of the vector. 1 9. The vector PQ , where P = s1, 3d and Q = s2, -1d 1 10. The vector OP where O is the origin and P is the midpoint of segment RS, where R = s2, -1d and S = s -4, 3d

v

a. u + v

b. u + v + w

c. u - v

d. u - w

24.

11. The vector from the point A = s2, 3d to the origin 1 1 12. The sum of AB and CD , where A = s1, -1d, B = s2, 0d, C = s -1, 3d , and D = s - 2, 2d

v

13. The unit vector that makes an angle u = 2p>3 with the positive x-axis

u

14. The unit vector that makes an angle u = -3p>4 with the positive x-axis

w

15. The unit vector obtained by rotating the vector 80, 19 120° counterclockwise about the origin 16. The unit vector obtained by rotating the vector 81, 09 135° counterclockwise about the origin

a. u - v

b. u - v + w

c. 2u - v

d. u + v + w

Vectors in Space In Exercises 17–22, express each vector in the form v = v1 i + v2 j + v3 k . 1 17. P1P2 if P1 is the point s5, 7, -1d and P2 is the point s2, 9, -2d 1 18. P1P2 if P1 is the point (1, 2, 0) and P2 is the point s -3, 0, 5d 1 19. AB if A is the point s -7, -8, 1d and B is the point s -10, 8, 1d 1 20. AB if A is the point (1, 0, 3) and B is the point s -1, 4, 5d

Length and Direction In Exercises 25–30, express each vector as a product of its length and direction.

22. -2u + 3v if u = 8-1, 0, 29 and v = 81, 1, 19

29.

21. 5u - v if u = 81, 1, -19 and v = 82, 0, 39

25. 2i + j - 2k

26. 9i - 2j + 6k

27. 5k

28.

1 26

i -

1 26

j -

1 26

k

30.

3 4 i + k 5 5 i 23

+

j 23

+

k 23

11.2 Vectors 31. Find the vectors whose lengths and directions are given. Try to do the calculations without writing. Length

Direction

a. 2

i

b. 23 1 c. 2

-k 3 j + 5 6 i 7

d. 7

30˚

4 k 5 3 2 j + k 7 7

Direction

a. 7

-j 3 4 - i - k 5 5 3 4 12 i j k 13 13 13 1 1 1 i + j k 22 23 26

13 12

d. a 7 0

F2

100

Length

c.

45˚

F1

46. Consider a 50-N weight suspended by two wires as shown in the accompanying figure. If the magnitude of vector F1 is 35 N, find angle a and the magnitude of vector F2.

32. Find the vectors whose lengths and directions are given. Try to do the calculations without writing.

b. 22

615

33. Find a vector of magnitude 7 in the direction of v = 12i - 5k . 34. Find a vector of magnitude 3 in the direction opposite to the direction of v = s1>2di - s1>2dj - s1>2dk .

a

60˚

F1 F2

50

47. Consider a w-N weight suspended by two wires as shown in the accompanying figure. If the magnitude of vector F2 is 100 N, find w and the magnitude of vector F1. 40˚

35˚

F1

Direction and Midpoints In Exercises 35–38, find 1 a. the direction of P1P2 and b. the midpoint of line segment P1 P2 . 35. P1s -1, 1, 5d

P2s2, 5, 0d

36. P1s1, 4, 5d

P2s4, -2, 7d

37. P1s3, 4, 5d

P2s2, 3, 4d

P2s2, -2, -2d 38. P1s0, 0, 0d 1 39. If AB = i + 4j - 2k and B is the point (5, 1, 3), find A. 1 40. If AB = -7i + 3j + 8k and A is the point s -2, -3, 6d , find B. Theory and Applications 41. Linear combination Let u = 2i + j, v = i + j , and w = i - j . Find scalars a and b such that u = av + bw . 42. Linear combination Let u = i - 2j , v = 2i + 3j , and w = i + j . Write u = u1 + u2 , where u1 is parallel to v and u2 is parallel to w. (See Exercise 41.) 43. Velocity An airplane is flying in the direction 25° west of north at 800 km> h. Find the component form of the velocity of the airplane, assuming that the positive x-axis represents due east and the positive y-axis represents due north. 44. (Continuation of Example 8.) What speed and direction should the jetliner in Example 8 have in order for the resultant vector to be 500 mph due east? 45. Consider a 100-N weight suspended by two wires as shown in the accompanying figure. Find the magnitudes and components of the force vectors F1 and F2.

F2

w

48. Consider a 25-N weight suspended by two wires as shown in the accompanying figure. If the magnitudes of vectors F1 and F2 are both 75 N, then angles a and b are equal. Find a. a

b

F1

F2 25

49. Location A bird flies from its nest 5 km in the direction 60° north of east, where it stops to rest on a tree. It then flies 10 km in the direction due southeast and lands atop a telephone pole. Place an xy-coordinate system so that the origin is the bird’s nest, the x-axis points east, and the y-axis points north. a. At what point is the tree located? b. At what point is the telephone pole? 50. Use similar triangles to find the coordinates of the point Q that divides the segment from P1sx1, y1, z1 d to P2sx2 , y2 , z2 d into two lengths whose ratio is p>q = r . 51. Medians of a triangle Suppose that A, B, and C are the corner points of the thin triangular plate of constant density shown here. a. Find the vector from C to the midpoint M of side AB. b. Find the vector from C to the point that lies two-thirds of the way from C to M on the median CM.

616

Chapter 11: Vectors and the Geometry of Space

c. Find the coordinates of the point in which the medians of ¢ABC intersect. According to Exercise 19, Section 6.6, this point is the plate’s center of mass. z

54. Vectors are drawn from the center of a regular n-sided polygon in the plane to the vertices of the polygon. Show that the sum of the vectors is zero. (Hint: What happens to the sum if you rotate the polygon about its center?)

C(1, 1, 3)

c.m.

y B(1, 3, 0)

x

53. Let ABCD be a general, not necessarily planar, quadrilateral in space. Show that the two segments joining the midpoints of opposite sides of ABCD bisect each other. (Hint: Show that the segments have the same midpoint.)

M A(4, 2, 0)

55. Suppose that A, B, and C are vertices of a triangle and that a, b, and c are, respectively, the midpoints of the opposite sides. Show 1 1 1 that Aa + Bb + Cc = 0 . 56. Unit vectors in the plane Show that a unit vector in the plane can be expressed as u = scos udi + ssin udj , obtained by rotating i through an angle u in the counterclockwise direction. Explain why this form gives every unit vector in the plane.

52. Find the vector from the origin to the point of intersection of the medians of the triangle whose vertices are As1, -1, 2d,

11.3

Bs2, 1, 3d,

and

The Dot Product

F

Cs -1, 2, -1d .

v

Length ⎥ F⎥ cos

FIGURE 11.19 The magnitude of the force F in the direction of vector v is the length ƒ F ƒ cos u of the projection of F onto v.

If a force F is applied to a particle moving along a path, we often need to know the magnitude of the force in the direction of motion. If v is parallel to the tangent line to the path at the point where F is applied, then we want the magnitude of F in the direction of v. Figure 11.19 shows that the scalar quantity we seek is the length ƒ F ƒ cos u, where u is the angle between the two vectors F and v. In this section we show how to calculate easily the angle between two vectors directly from their components. A key part of the calculation is an expression called the dot product. Dot products are also called inner or scalar products because the product results in a scalar, not a vector. After investigating the dot product, we apply it to finding the projection of one vector onto another (as displayed in Figure 11.19) and to finding the work done by a constant force acting through a displacement.

Angle Between Vectors When two nonzero vectors u and v are placed so their initial points coincide, they form an angle u of measure 0 … u … p (Figure 11.20). If the vectors do not lie along the same line, the angle u is measured in the plane containing both of them. If they do lie along the same line, the angle between them is 0 if they point in the same direction and p if they point in opposite directions. The angle u is the angle between u and v. Theorem 1 gives a formula to determine this angle.

THEOREM 1—Angle Between Two Vectors The angle u between two nonzero vectors u = 8u1, u2 , u39 and v = 8v1, v2 , v39 is given by

v u

u = cos-1 a

u1 v1 + u2 v2 + u3 v3 b. ƒuƒ ƒvƒ

FIGURE 11.20 The angle between u and v.

Before proving Theorem 1, we focus attention on the expression u1 v1 + u2 v2 + u3 v3 in the calculation for u. This expression is the sum of the products of the corresponding components for the vectors u and v.

11.3

The Dot Product

617

DEFINITION The dot product u # v s“u dot v”d of vectors u = 8u1, u2 , u39 and v = 8v1, v2 , v39 is the scalar u # v = u1 v1 + u2 v2 + u3 v3 .

EXAMPLE 1

(a) 81, -2, -19 # 8-6, 2, -39 = s1ds -6d + s -2ds2d + s -1ds -3d = -6 - 4 + 3 = -7 1 1 (b) a i + 3j + kb # s4i - j + 2kd = a bs4d + s3ds -1d + s1ds2d = 1 2 2 The dot product of a pair of two-dimensional vectors is defined in a similar fashion: 8u1, u29 # 8v1, v29 = u1 v1 + u2 v2 .

We will see throughout the remainder of the book that the dot product is a key tool for many important geometric and physical calculations in space (and the plane), not just for finding the angle between two vectors. Proof of Theorem 1 Applying the law of cosines (Equation (8), Section 1.3) to the triangle in Figure 11.21, we find that w

2 2 2 ƒ w ƒ = ƒ u ƒ + ƒ v ƒ - 2 ƒ u ƒ ƒ v ƒ cos u

u

Law of cosines

2 ƒ u ƒ ƒ v ƒ cos u = ƒ u ƒ + ƒ v ƒ - ƒ w ƒ . 2

v

2

2

Because w = u - v, the component form of w is 8u1 - v1, u2 - v2 , u3 - v39. So

A 2u 12 + u 22 + u 32 B = u 12 + u 22 + u 32 2 2 ƒ v ƒ = A 2v12 + v22 + v32 B = v12 + v22 + v32 2 2 2 2 2 ƒ w ƒ = A 2su1 - v1 d + su2 - v2 d + su3 - v3 d B 2

2 ƒuƒ =

FIGURE 11.21 The parallelogram law of addition of vectors gives w = u - v.

= su1 - v1 d2 + su2 - v2 d2 + su3 - v3 d2 = u 12 - 2u1v1 + v12 + u 22 - 2u2v2 + v22 + u 32 - 2u3v3 + v32 and ƒ u ƒ 2 + ƒ v ƒ 2 - ƒ w ƒ 2 = 2su1 v1 + u2v2 + u3 v3). Therefore, 2 ƒ u ƒ ƒ v ƒ cos u = ƒ u ƒ 2 + ƒ v ƒ 2 - ƒ w ƒ 2 = 2su1 v1 + u2 v2 + u3 v3 d ƒ u ƒ ƒ v ƒ cos u = u1 v1 + u2 v2 + u3 v3 u1 v1 + u2 v2 + u3 v3 cos u = . ƒuƒ ƒvƒ Since 0 … u 6 p, we have u = cos-1 a

u1 v1 + u2 v2 + u3 v3 b. ƒuƒ ƒvƒ

The Angle Between Two Nonzero Vectors u and v u = cos-1 a

u#v b u ƒ ƒ ƒvƒ

618

Chapter 11: Vectors and the Geometry of Space

EXAMPLE 2 Solution

Find the angle between u = i - 2j - 2k and v = 6i + 3j + 2k.

We use the formula above: u # v = s1ds6d + s -2ds3d + s -2ds2d = 6 - 6 - 4 = -4 2 2 2 ƒ u ƒ = 2s1d + s -2d + s -2d = 29 = 3

ƒ v ƒ = 2s6d2 + s3d2 + s2d2 = 249 = 7 u = cos-1 a

u#v -4 b L 1.76 radians. b = cos-1 a s3ds7d ƒuƒ ƒvƒ

The angle formula applies to two-dimensional vectors as well. y

EXAMPLE 3

Find the angle u in the triangle ABC determined by the vertices A = s0, 0d, B = s3, 5d, and C = s5, 2d (Figure 11.22).

B(3, 5)

C(5, 2)

1 A

1

FIGURE 11.22 The triangle in Example 3.

1 1 The angle u is the angle between the vectors CA and CB . The component forms of these two vectors are 1 1 CA = 8-5, -29 and CB = 8-2, 39. Solution

x

First we calculate the dot product and magnitudes of these two vectors. 1 1 CA # CB = s -5ds -2d + s -2ds3d = 4 1 ƒ CA ƒ = 2s -5d2 + s -2d2 = 229 1 ƒ CB ƒ = 2s -2d2 + s3d2 = 213 Then applying the angle formula, we have 1 1 CA # CB u = cos-1 ¢ 1 1 ≤ ƒ CA ƒ ƒ CB ƒ = cos-1 ¢ L 78.1°

4

A 229 B A 213 B or

≤

1.36 radians.

Orthogonal Vectors Two nonzero vectors u and v are perpendicular if the angle between them is p>2. For such vectors, we have u # v = 0 because cos sp>2d = 0. The converse is also true. If u and v are nonzero vectors with u # v = ƒ u ƒ ƒ v ƒ cos u = 0, then cos u = 0 and u = cos-1 0 = p>2. The following definition also allows for one or both of the vectors to be the zero vector.

DEFINITION

EXAMPLE 4

Vectors u and v are orthogonal if u # v = 0.

To determine if two vectors are orthogonal, calculate their dot product.

(a) u = 83, -29 and v = 84, 69 are orthogonal because u # v = s3ds4d + s -2ds6d = 0. (b) u = 3i - 2j + k and v = 2j + 4k are orthogonal because u # v = s3ds0d + s -2ds2d + s1ds4d = 0.

11.3

The Dot Product

619

(c) 0 is orthogonal to every vector u since

0 # u = 80, 0, 09 # 8u1, u2, u39 = s0dsu1 d + s0dsu2 d + s0dsu3 d = 0.

Dot Product Properties and Vector Projections The dot product obeys many of the laws that hold for ordinary products of real numbers (scalars).

Properties of the Dot Product If u, v, and w are any vectors and c is a scalar, then 1. u # v = v # u 2. scud # v = u # scvd = csu # vd # # # 3. u sv + wd = u v + u w 4. u # u = ƒ u ƒ 2 5. 0 # u = 0. Proofs of Properties 1 and 3 The properties are easy to prove using the definition. For instance, here are the proofs of Properties 1 and 3. 1.

u # v = u1 v1 + u2 v2 + u3 v3 = v1 u1 + v2 u2 + v3 u3 = v # u

3.

u # sv + wd = 8u1, u2 , u39 # 8v1 + w1, v2 + w2 , v3 + w39 = u1sv1 + w1 d + u2sv2 + w2 d + u3sv3 + w3 d

Q

= u1 v1 + u1 w1 + u2 v2 + u2 w2 + u3 v3 + u3 w3

u

= su1 v1 + u2 v2 + u3 v3 d + su1 w1 + u2 w2 + u3 w3 d v P

R

S

Q

We now return to the problem of projecting one vector onto another, posed in the 1 1 opening to this section. The vector projection of u = PQ onto a nonzero vector v = PS 1 (Figure 11.23) is the vector PR determined by dropping a perpendicular from Q to the line PS. The notation for this vector is

u v R

P

S

FIGURE 11.23 The vector projection of u onto v.

Force 5 u v

= u#v + u#w

projv u

s“the vector projection of u onto v”d.

If u represents a force, then projv u represents the effective force in the direction of v (Figure 11.24). If the angle u between u and v is acute, projv u has length ƒ u ƒ cos u and direction v> ƒ v ƒ (Figure 11.25). If u is obtuse, cos u 6 0 and projv u has length - ƒ u ƒ cos u and direction -v> ƒ v ƒ . In both cases, projv u = s ƒ u ƒ cos ud

projv u

FIGURE 11.24 If we pull on the box with force u, the effective force moving the box forward in the direction v is the projection of u onto v.

= a

u#v v b ƒvƒ ƒvƒ

= a

u#v bv. ƒvƒ2

v ƒvƒ ƒ u ƒ cos u =

ƒ u ƒ ƒ v ƒ cos u u#v = v ƒ ƒ ƒvƒ

620

Chapter 11: Vectors and the Geometry of Space

u

u

proj v u

proj v u

v

Length u cos (a)

v

Length –u cos (b)

FIGURE 11.25 The length of projv u is (a) ƒ u ƒ cos u if cos u Ú 0 and (b) - ƒ u ƒ cos u if cos u 6 0 .

The number ƒ u ƒ cos u is called the scalar component of u in the direction of v (or of u onto v). To summarize,

The vector projection of u onto v is the vector projv u = a

u#v bv. ƒvƒ2

(1)

The scalar component of u in the direction of v is the scalar ƒ u ƒ cos u =

u#v v = u# . ƒvƒ ƒvƒ

(2)

Note that both the vector projection of u onto v and the scalar component of u onto v depend only on the direction of the vector v and not its length (because we dot u with v> ƒ v ƒ , which is the direction of v).

EXAMPLE 5 Find the vector projection of u = 6i + 3j + 2k onto v = i - 2j - 2k and the scalar component of u in the direction of v. Solution

We find projv u from Equation (1):

u#v 6 - 6 - 4 projv u = v # v v = si - 2j - 2kd 1 + 4 + 4 = -

8 8 4 4 si - 2j - 2kd = - i + j + k. 9 9 9 9

We find the scalar component of u in the direction of v from Equation (2): ƒ u ƒ cos u = u #

v 1 2 2 = s6i + 3j + 2kd # a i - j - kb 3 3 3 ƒvƒ

= 2 - 2 -

4 4 = - . 3 3

Equations (1) and (2) also apply to two-dimensional vectors. We demonstrate this in the next example. Find the vector projection of a force F = 5i + 2j onto v = i - 3j and the scalar component of F in the direction of v.

EXAMPLE 6

11.3 Solution

The Dot Product

621

The vector projection is projv F = ¢

F#v ≤v ƒvƒ2

5 - 6 1 si - 3jd = si - 3jd 1 + 9 10 3 1 = i + j. 10 10 =

The scalar component of F in the direction of v is ƒ F ƒ cos u =

5 - 6 F#v 1 = = . v ƒ ƒ 21 + 9 210

A routine calculation (see Exercise 29) verifies that the vector u - projv u is orthogonal to the projection vector projv u (which has the same direction as v). So the equation u#v u#v v + ¢u - ¢ 2 ≤v≤ 2 ≤ ƒvƒ ƒvƒ (')'* ('')''*

u = projv u + su - projv ud = ¢

Parallel to v

Orthogonal to v

expresses u as a sum of orthogonal vectors.

Work

F P

D

Q

u _F_ cos u

In Chapter 6, we calculated the work done by a constant force of magnitude F in moving an object through a distance d as W = Fd. That formula holds only if the force is directed 1 along the line of motion. If a force F moving an object through a displacement D = PQ has some other direction, the work is performed by the component of F in the direction of D. If u is the angle between F and D (Figure 11.26), then scalar component of F Work = ain the direction of D bslength of Dd

FIGURE 11.26 The work done by a constant force F during a displacement D is s ƒ F ƒ cos ud ƒ D ƒ , which is the dot product F # D.

= s ƒ F ƒ cos ud ƒ D ƒ = F # D.

DEFINITION

1 ment D = PQ is

The work done by a constant force F acting through a displaceW = F # D.

If ƒ F ƒ = 40 N (newtons), ƒ D ƒ = 3 m, and u = 60°, the work done by F in acting from P to Q is

EXAMPLE 7

Work = F # D

Definition

= ƒ F ƒ ƒ D ƒ cos u = s40ds3d cos 60°

Given values

= s120ds1>2d = 60 J s joulesd. We encounter more challenging work problems in Chapter 15 when we learn to find the work done by a variable force along a path in space.

622

Chapter 11: Vectors and the Geometry of Space

Exercises 11.3 a. Show that

Dot Product and Projections In Exercises 1–8, find

cos a =

v # u,

ƒvƒ, ƒuƒ b. the cosine of the angle between v and u a.

c. the scalar component of u in the direction of v u = -2i + 4j - 25k

2. v = s3>5di + s4>5dk,

u = 5i + 12j

3. v = 10i + 11j - 2k,

u = 3j + 4k

4. v = 2i + 10j - 11k,

u = 2i + 2j + k

6. v = -i + j,

u = 22i + 23j + 2k

7. v = 5i + j,

u = 2i + 217j

8. v = h

1

, i, 22 23 1

u = h

1 22

11. u = 23i - 7j,

,-

23

i

East

rth

No

v = 3i + 4k v = 23i + j - 2k

12. u = i + 22j - 22k,

c , ƒvƒ

1

v = i + 2j - k

10. u = 2i - 2j + k,

cos g =

16. Water main construction A water main is to be constructed with a 20% grade in the north direction and a 10% grade in the east direction. Determine the angle u required in the water main for the turn from north to east.

Angle Between Vectors T Find the angles between the vectors in Exercises 9–12 to the nearest hundredth of a radian. 9. u = 2i + j,

b , ƒvƒ

b. Unit vectors are built from direction cosines Show that if v = ai + bj + ck is a unit vector, then a, b, and c are the direction cosines of v.

u = i + j + k

5. v = 5j - 3k,

cos b =

and cos2 a + cos2 b + cos2 g = 1 . These cosines are called the direction cosines of v.

d. the vector projv u . 1. v = 2i - 4j + 25k,

a , ƒvƒ

v = -i + j + k

Theory and Examples 17. Sums and differences In the accompanying figure, it looks as if v1 + v2 and v1 - v2 are orthogonal. Is this mere coincidence, or are there circumstances under which we may expect the sum of two vectors to be orthogonal to their difference? Give reasons for your answer.

13. Triangle Find the measures of the angles of the triangle whose vertices are A = s -1, 0d, B = s2, 1d , and C = s1, -2d . 14. Rectangle Find the measures of the angles between the diagonals of the rectangle whose vertices are A = s1, 0d, B = s0, 3d, C = s3, 4d , and D = s4, 1d .

v2 v1 v 2

15. Direction angles and direction cosines The direction angles a, b, and g of a vector v = ai + bj + ck are defined as follows:

v1

a is the angle between v and the positive x-axis s0 … a … pd b is the angle between v and the positive y-axis s0 … b … pd

–v 2

v1 v 2

g is the angle between v and the positive z-axis s0 … g … pd .

18. Orthogonality on a circle Suppose that AB is the diameter of a circle with center O and that C is a point on one of the two arcs 1 1 joining A and B. Show that CA and CB are orthogonal.

z

C v 0

v

A

–u

O

u

B

y

x

19. Diagonals of a rhombus Show that the diagonals of a rhombus (parallelogram with sides of equal length) are perpendicular.

11.3 20. Perpendicular diagonals Show that squares are the only rectangles with perpendicular diagonals. 21. When parallelograms are rectangles Prove that a parallelogram is a rectangle if and only if its diagonals are equal in length. (This fact is often exploited by carpenters.) 22. Diagonal of parallelogram Show that the indicated diagonal of the parallelogram determined by vectors u and v bisects the angle between u and v if ƒ u ƒ = ƒ v ƒ .

623

32. Line parallel to a vector Show that the vector v = ai + bj is parallel to the line bx - ay = c by establishing that the slope of the line segment representing v is the same as the slope of the given line. In Exercises 33–36, use the result of Exercise 31 to find an equation for the line through P perpendicular to v. Then sketch the line. Include v in your sketch as a vector starting at the origin. 33. Ps2, 1d,

v = i + 2j

35. Ps -2, -7d,

u

The Dot Product

v = -2i + j

34. Ps -1, 2d,

v = -2i - j

36. Ps11, 10d,

v = 2i - 3j

In Exercises 37–40, use the result of Exercise 32 to find an equation for the line through P parallel to v. Then sketch the line. Include v in your sketch as a vector starting at the origin.

v

23. Projectile motion A gun with muzzle velocity of 1200 ft> sec is fired at an angle of 8° above the horizontal. Find the horizontal and vertical components of the velocity. 24. Inclined plane Suppose that a box is being towed up an inclined plane as shown in the figure. Find the force w needed to make the component of the force parallel to the inclined plane equal to 2.5 lb. w

33˚ 15˚

25. a. Cauchy-Schwartz inequality Since u # v = ƒ u ƒ ƒ v ƒ cos u, show that the inequality ƒ u # v ƒ … ƒ u ƒ ƒ v ƒ holds for any vectors u and v. b. Under what circumstances, if any, does ƒ u # v ƒ equal ƒ u ƒ ƒ v ƒ ? Give reasons for your answer.

37. Ps -2, 1d, 39. Ps1, 2d,

v = i - j

38. Ps0, -2d,

v = -i - 2j

40. Ps1, 3d,

42. Locomotive The Union Pacific’s Big Boy locomotive could pull 6000-ton trains with a tractive effort (pull) of 602,148 N (135,375 lb). At this level of effort, about how much work did Big Boy do on the (approximately straight) 605-km journey from San Francisco to Los Angeles? 43. Inclined plane How much work does it take to slide a crate 20 m along a loading dock by pulling on it with a 200-N force at an angle of 30° from the horizontal? 44. Sailboat The wind passing over a boat’s sail exerted a 1000-lb magnitude force F as shown here. How much work did the wind perform in moving the boat forward 1 mi? Answer in foot-pounds.

60°

28. Cancellation in dot products In real-number multiplication, if uv1 = uv2 and u Z 0 , we can cancel the u and conclude that v1 = v2 . Does the same rule hold for the dot product? That is, if u # v1 = u # v2 and u Z 0 , can you conclude that v1 = v2? Give reasons for your answer.

30. A force F = 2i + j - 3k is applied to a spacecraft with velocity vector v = 3i - j. Express F as a sum of a vector parallel to v and a vector orthogonal to v.

1000 lb magnitude force

F

Angles Between Lines in the Plane The acute angle between intersecting lines that do not cross at right angles is the same as the angle determined by vectors normal to the lines or by the vectors parallel to the lines. n1

n2

Equations for Lines in the Plane 31. Line perpendicular to a vector Show that v = ai + bj is perpendicular to the line ax + by = c by establishing that the slope of the vector v is the negative reciprocal of the slope of the given line.

v = 3i - 2j

Work 41. Work along a line Find the work done by a force F = 5i (magnitude 5 N) in moving an object along the line from the origin to the point (1, 1) (distance in meters).

26. Dot multiplication is positive definite Show that dot multiplication of vectors is positive definite; that is, show that u # u Ú 0 for every vector u and that u # u = 0 if and only if u = 0. 27. Orthogonal unit vectors If u1 and u2 are orthogonal unit vectors and v = au1 + bu2 , find v # u1 .

29. Using the definition of the projection of u onto v, show by direct calculation that su - projv ud # projv u = 0.

v = 2i + 3j

L2 L2

v2

v1 L1

L1

624

Chapter 11: Vectors and the Geometry of Space

Use this fact and the results of Exercise 31 or 32 to find the acute angles between the lines in Exercises 45–50. 45. 3x + y = 5,

2x - y = 4

46. y = 23x - 1,

47. 23x - y = - 2, 48. x + 23y = 1, 49. 3x - 4y = 3,

y = - 23x + 2

50. 12x + 5y = 1,

x - 23y = 1

A 1 - 23 B x + A 1 + 23 B y = 8

x - y = 7 2x - 2y = 3

The Cross Product

11.4

In studying lines in the plane, when we needed to describe how a line was tilting, we used the notions of slope and angle of inclination. In space, we want a way to describe how a plane is tilting. We accomplish this by multiplying two vectors in the plane together to get a third vector perpendicular to the plane. The direction of this third vector tells us the “inclination” of the plane. The product we use to multiply the vectors together is the vector or cross product, the second of the two vector multiplication methods. We study the cross product in this section.

The Cross Product of Two Vectors in Space

uv

v

n

We start with two nonzero vectors u and v in space. If u and v are not parallel, they determine a plane. We select a unit vector n perpendicular to the plane by the right-hand rule. This means that we choose n to be the unit (normal) vector that points the way your right thumb points when your fingers curl through the angle u from u to v (Figure 11.27). Then we define a new vector as follows.

u

DEFINITION

The cross product u * v (“u cross v”) is the vector u * v = s ƒ u ƒ ƒ v ƒ sin ud n.

FIGURE 11.27 The construction of u * v.

Unlike the dot product, the cross product is a vector. For this reason it’s also called the vector product of u and v, and applies only to vectors in space. The vector u * v is orthogonal to both u and v because it is a scalar multiple of n. There is a straightforward way to calculate the cross product of two vectors from their components. The method does not require that we know the angle between them (as suggested by the definition), but we postpone that calculation momentarily so we can focus first on the properties of the cross product. Since the sines of 0 and p are both zero, it makes sense to define the cross product of two parallel nonzero vectors to be 0. If one or both of u and v are zero, we also define u * v to be zero. This way, the cross product of two vectors u and v is zero if and only if u and v are parallel or one or both of them are zero. Parallel Vectors Nonzero vectors u and v are parallel if and only if u * v = 0 . The cross product obeys the following laws. Properties of the Cross Product If u, v, and w are any vectors and r, s are scalars, then 1. srud * ssvd = srsdsu * vd 2. u * sv + wd = u * v + u * w 3. v * u = -su * vd 4. sv + wd * u = v * u + w * u 5. 0 * u = 0 6. u * sv * wd = (u # w)v - (u # v)w

11.4

v

–n u

vu

FIGURE 11.28 The construction of v * u.

The Cross Product

625

To visualize Property 3, for example, notice that when the fingers of your right hand curl through the angle u from v to u, your thumb points the opposite way; the unit vector we choose in forming v * u is the negative of the one we choose in forming u * v (Figure 11.28). Property 1 can be verified by applying the definition of cross product to both sides of the equation and comparing the results. Property 2 is proved in Appendix 8. Property 4 follows by multiplying both sides of the equation in Property 2 by -1 and reversing the order of the products using Property 3. Property 5 is a definition. As a rule, cross product multiplication is not associative so su * vd * w does not generally equal u * sv * wd. (See Additional Exercise 17.) When we apply the definition and Property 3 to calculate the pairwise cross products of i, j, and k, we find (Figure 11.29) i * j = -sj * id = k

z kij

k

j

j * k = -sk * jd = i k * i = -si * kd = j

jki

and y

i Diagram for recalling cross products

i * i = j * j = k * k = 0.

ƒ u * v ƒ Is the Area of a Parallelogram

x ijk

Because n is a unit vector, the magnitude of u * v is

FIGURE 11.29 The pairwise cross products of i, j, and k.

Area base ⋅ height u ⋅ vsin u × v

ƒ u * v ƒ = ƒ u ƒ ƒ v ƒ ƒ sin u ƒ ƒ n ƒ = ƒ u ƒ ƒ v ƒ sin u. This is the area of the parallelogram determined by u and v (Figure 11.30), ƒ u ƒ being the base of the parallelogram and ƒ v ƒ ƒ sin u ƒ the height.

Determinant Formula for u * v

v h v sin u

FIGURE 11.30 The parallelogram determined by u and v.

Our next objective is to calculate u * v from the components of u and v relative to a Cartesian coordinate system. Suppose that u = u1 i + u2 j + u3 k

and

v = v1 i + v2 j + v3 k.

Then the distributive laws and the rules for multiplying i, j, and k tell us that u * v = su1 i + u2 j + u3 kd * sv1 i + v2 j + v3 kd = u1 v1 i * i + u1 v2 i * j + u1 v3 i * k + u2 v1 j * i + u2 v2 j * j + u2 v3 j * k + u3 v1 k * i + u3 v2 k * j + u3 v3 k * k = su2 v3 - u3 v2 di - su1 v3 - u3 v1 dj + su1 v2 - u2 v1 dk. The component terms in the last line are hard to remember, but they are the same as the terms in the expansion of the symbolic determinant i 3 u1 v1

j u2 v2

k u3 3 . v3

626

Chapter 11: Vectors and the Geometry of Space

So we restate the calculation in this easy-to-remember form.

Determinants 2 * 2 and 3 * 3 determinants are evaluated as follows: a ` c a1 3 b1 c1

a2 b2 c2 - a2 `

a3 b2 b3 3 = a1 ` c2 c3 b1 c1

Calculating the Cross Product as a Determinant If u = u1 i + u2 j + u3 k and v = v1 i + v2 j + v3 k, then

b ` = ad - bc d

i 3 u * v = u1 v1

b3 ` c3

b3 b1 ` + a3 ` c3 c1

b2 ` c2

EXAMPLE 1 Solution

j u2 v2

k u3 3 . v3

Find u * v and v * u if u = 2i + j + k and v = -4i + 3j + k.

We expand the symbolic determinant: i u * v = 3 2 -4

j 1 3

k 1 13 = ` 3 1

1 2 `i - ` 1 -4

1 2 `j + ` 1 -4

1 `k 3

= -2i - 6j + 10k v * u = -su * vd = 2i + 6j - 10k.

Property 3

Find a vector perpendicular to the plane of Ps1, -1, 0d, Qs2, 1, -1d, and Rs -1, 1, 2d (Figure 11.31).

EXAMPLE 2

1 1 The vector PQ * PR is perpendicular to the plane because it is perpendicular to both vectors. In terms of components, 1 PQ = s2 - 1di + s1 + 1dj + s -1 - 0dk = i + 2j - k 1 PR = s -1 - 1di + s1 + 1dj + s2 - 0dk = -2i + 2j + 2k

Solution

i 1 1 3 PQ * PR = 1 -2

z R(–1, 1, 2)

j 2 2

k 2 -1 3 = ` 2 2

-1 1 `i - ` 2 -2

-1 1 `j + ` 2 -2

2 `k 2

= 6i + 6k. Find the area of the triangle with vertices Ps1, -1, 0d, Qs2, 1, -1d, and Rs -1, 1, 2d (Figure 11.31).

EXAMPLE 3 0 P(1, –1, 0)

x

y

Solution

The area of the parallelogram determined by P, Q, and R is 1 1 Values from Example 2 ƒ PQ * PR ƒ = ƒ 6i + 6k ƒ = 2s6d2 + s6d2 = 22 # 36 = 622.

Q(2, 1, –1)

1 1 FIGURE 11.31 The vector PQ * PR is perpendicular to the plane of triangle PQR (Example 2). The area of triangle PQR is 1 1 half of ƒ PQ * PR ƒ (Example 3).

The triangle’s area is half of this, or 322 .

EXAMPLE 4 and Rs -1, 1, 2d.

Find a unit vector perpendicular to the plane of Ps1, -1, 0d, Qs2, 1, -1d,

1 1 Since PQ * PR is perpendicular to the plane, its direction n is a unit vector perpendicular to the plane. Taking values from Examples 2 and 3, we have 1 1 PQ * PR 6i + 6k 1 1 n = 1 = i + k. 1 = 6 22 22 22 PQ * PR ƒ ƒ

Solution

11.4

The Cross Product

627

For ease in calculating the cross product using determinants, we usually write vectors in the form v = v1 i + v2 j + v3 k rather than as ordered triples v = 8v1, v2 , v39.

Torque When we turn a bolt by applying a force F to a wrench (Figure 11.32), we produce a torque that causes the bolt to rotate. The torque vector points in the direction of the axis of the bolt according to the right-hand rule (so the rotation is counterclockwise when viewed from the tip of the vector). The magnitude of the torque depends on how far out on the wrench the force is applied and on how much of the force is perpendicular to the wrench at the point of application. The number we use to measure the torque’s magnitude is the product of the length of the lever arm r and the scalar component of F perpendicular to r. In the notation of Figure 11.32,

n

Torque

r

Magnitude of torque vector = ƒ r ƒ ƒ F ƒ sin u,

Component of F perpendicular to r. Its length is F sin .

F

or ƒ r * F ƒ . If we let n be a unit vector along the axis of the bolt in the direction of the torque, then a complete description of the torque vector is r * F, or Torque vector = s ƒ r ƒ ƒ F ƒ sin ud n.

FIGURE 11.32 The torque vector describes the tendency of the force F to drive the bolt forward.

Recall that we defined u * v to be 0 when u and v are parallel. This is consistent with the torque interpretation as well. If the force F in Figure 11.32 is parallel to the wrench, meaning that we are trying to turn the bolt by pushing or pulling along the line of the wrench’s handle, the torque produced is zero.

EXAMPLE 5 3 ft bar P

Q 70° 20 lb magnitude force

The magnitude of the torque generated by force F at the pivot point P in

Figure 11.33 is 1 1 ƒ PQ * F ƒ = ƒ PQ ƒ ƒ F ƒ sin 70° L s3ds20ds0.94d F

FIGURE 11.33 The magnitude of the torque exerted by F at P is about 56.4 ft-lb (Example 5). The bar rotates counterclockwise around P.

L 56.4 ft-lb. In this example the torque vector is pointing out of the page toward you.

Triple Scalar or Box Product

The product su * vd # w is called the triple scalar product of u, v, and w (in that order). As you can see from the formula ƒ su * vd # w ƒ = ƒ u * v ƒ ƒ w ƒ ƒ cos u ƒ ,

the absolute value of this product is the volume of the parallelepiped (parallelogram-sided box) determined by u, v, and w (Figure 11.34). The number ƒ u * v ƒ is the area of the base uv w Height w cos

v

Area of base u v

u Volume area of base · height u v w cos (u v) · w

FIGURE 11.34 The number ƒ su * vd # w ƒ is the volume of a parallelepiped.

628

Chapter 11: Vectors and the Geometry of Space

parallelogram. The number ƒ w ƒ ƒ cos u ƒ is the parallelepiped’s height. Because of this geometry, su * vd # w is also called the box product of u, v, and w. By treating the planes of v and w and of w and u as the base planes of the parallelepiped determined by u, v, and w, we see that su * vd # w = sv * wd # u = sw * ud # v.

Since the dot product is commutative, we also have The dot and cross may be interchanged in a triple scalar product without altering its value.

su * vd # w = u # sv * wd.

The triple scalar product can be evaluated as a determinant: su * vd # w = c `

u3 u1 `i - ` v3 v1

u2 v2

= w1 `

u3 u1 ` - w2 ` v3 v1

u2 v2

u1 3 = v1 w1

u3 u1 `j + ` v3 v1

u2 v2 w2

u2 ` kd # w v2

u3 u1 ` + w3 ` v3 v1

u2 ` v2

u3 v3 3 . w3

Calculating the Triple Scalar Product as a Determinant u1 3 v1 w1

su * vd # w =

u2 v2 w2

u3 v3 3 w3

EXAMPLE 6 Find the volume of the box (parallelepiped) determined by u = i + 2j - k, v = -2i + 3k, and w = 7j - 4k. Solution

Using the rule for calculating a 3 * 3 determinant, we find

1 2 -1 0 3 -2 3 -2 0 3 3 = (1) ` ` - (2) ` ` + (-1) ` ` = -23. su * vd # w = 3 -2 0 7 -4 0 -4 0 7 0 7 -4 The volume is ƒ su * vd # w ƒ = 23 units cubed.

Exercises 11.4 Cross Product Calculations In Exercises 1–8, find the length and direction (when defined) of u * v and v * u. 1. u = 2i - 2j - k, 2. u = 2i + 3j,

v = i - k

v = -i + j

3. u = 2i - 2j + 4k, 4. u = i + j - k, 5. u = 2i,

v = -i + j - 2k

v = 0

v = -3j

6. u = i * j,

v = j * k

7. u = -8i - 2j - 4k, 8. u =

3 1 i - j + k, 2 2

v = 2i + 2j + k v = i + j + 2k

In Exercises 9–14, sketch the coordinate axes and then include the vectors u, v, and u * v as vectors starting at the origin. 10. u = i - k,

v = j

11. u = i - k,

v = j + k

12. u = 2i - j,

v = i + 2j

13. u = i + j,

v = i - j

14. u = j + 2k,

v = i

9. u = i,

v = j

11.4

d. scud # v = u # scvd = csu # vd

a. Find the area of the triangle determined by the points P, Q, and R. b. Find a unit vector perpendicular to plane PQR. 16. Ps1, 1, 1d,

629

c. s -ud * v = -su * vd

Triangles in Space In Exercises 15–18,

15. Ps1, -1, 2d,

The Cross Product

Qs2, 0, -1d, Qs2, 1, 3d,

Rs3, - 1, 1d

Qs3, -1, 2d,

Rs3, - 1, 1d

18. Ps -2, 2, 0d,

Qs0, 1, -1d,

Rs - 1, 2, -2d

e. csu * vd = scud * v = u * scvd

sany number cd

f. u # u = ƒ u ƒ 2 g. su * ud # u = 0

h. su * vd # u = v # su * vd

Rs0, 2, 1d

17. Ps2, -2, 1d,

sany number cd

29. Given nonzero vectors u, v, and w, use dot product and cross product notation, as appropriate, to describe the following. a. The vector projection of u onto v b. A vector orthogonal to u and v

Triple Scalar Products In Exercises 19–22, verify that su * vd # w = sv * wd # u = sw * ud # v and find the volume of the parallelepiped (box) determined by u, v, and w. u

v

w

19. 2i

2j

2k

20. i - j + k

2i + j - 2k

-i + 2j - k

21. 2i + j

2i - j + k

i + 2k

22. i + j - 2k

-i - k

2i + 4j - 2k

a. su * vd # w

In Exercises 25 and 26, find the magnitude of the torque exerted by F 1 on the bolt at P if ƒ PQ ƒ = 8 in . and ƒ F ƒ = 30 lb . Answer in footpounds.

b. u * sv # wd c. u * sv * wd

d. u # sv # wd

32. Cross products of three vectors Show that except in degenerate cases, su * vd * w lies in the plane of u and v, whereas u * sv * wd lies in the plane of v and w. What are the degenerate cases? 33. Cancellation in cross products If u * v = u * w and u Z 0, then does v = w? Give reasons for your answer. 34. Double cancellation If u Z 0 and if u * v = u * w and u # v = u # w, then does v = w? Give reasons for your answer. Area of a Parallelogram Find the areas of the parallelograms whose vertices are given in Exercises 35–40.

26. F

F

135° Q

Q P

P

e. A vector orthogonal to u * v and u * w

31. Let u, v, and w be vectors. Which of the following make sense, and which do not? Give reasons for your answers.

24. Parallel and perpendicular vectors Let u = i + 2j - k, v = -i + j + k, w = i + k, r = - sp>2di - pj + sp>2dk. Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

60°

d. The volume of the parallelepiped determined by u, v, and w f. A vector of length ƒ u ƒ in the direction of v 30. Compute si * jd * j and i * sj * jd. What can you conclude about the associativity of the cross product?

Theory and Examples 23. Parallel and perpendicular vectors Let u = 5i - j + k, v = j - 5k, w = -15i + 3j - 3k. Which vectors, if any, are (a) perpendicular? (b) Parallel? Give reasons for your answers.

25.

c. A vector orthogonal to u * v and w

27. Which of the following are always true, and which are not always true? Give reasons for your answers. 2u # u

35. As1, 0d,

Bs0, 1d,

Cs -1, 0d,

36. As0, 0d,

Bs7, 3d,

Cs9, 8d,

37. As -1, 2d,

Bs2, 0d,

38. As -6, 0d,

Bs1, -4d,

39. As0, 0, 0d,

Bs3, 2, 4d,

40. As1, 0, -1d,

a. ƒ u ƒ = b. u # u = ƒ u ƒ c. u * 0 = 0 * u = 0

Ds0, -1d Ds2, 5d

Cs7, 1d,

Ds4, 3d

Cs3, 1d,

Ds -4, 5d

Cs5, 1, 4d,

Ds2, -1, 0d

Cs2, 4, -1d,

Bs1, 7, 2d,

Ds0, 3, 2d

d. u * s -ud = 0

Area of a Triangle Find the areas of the triangles whose vertices are given in Exercises 41–47.

e. u * v = v * u

41. As0, 0d,

Bs -2, 3d,

Cs3, 1d

f. u * sv + wd = u * v + u * w

42. As -1, -1d,

h. su * vd # w = u # sv * wd

43. As -5, 3d,

Bs1, -2d,

44. As -6, 0d,

Bs10, -5d,

Cs -2, 4d

45. As1, 0, 0d,

Bs0, 2, 0d,

Cs0, 0, -1d

46. As0, 0, 0d,

Bs -1, 1, -1d,

g. su * vd # v = 0

28. Which of the following are always true, and which are not always true? Give reasons for your answers. a.

u#v

= v#u

b. u * v = -sv * ud

47. As1, -1, 1d,

Bs3, 3d,

Bs0, 1, 1d,

Cs2, 1d Cs6, -2d

Cs3, 0, 3d Cs1, 0, -1d

630

Chapter 11: Vectors and the Geometry of Space 50. Triangle area Find a concise 3 3 determinant formula that gives the area of a triangle in the xy-plane having vertices sa1, a2 d, sb1, b2 d , and sc1, c2 d .

48. Find the volume of a parallelepiped if four of its eight vertices are As0, 0, 0d, Bs1, 2, 0d, Cs0, -3, 2d, and Ds3, -4, 5d. 49. Triangle area Find a 2 2 determinant formula for the area of the triangle in the xy-plane with vertices at s0, 0d, sa1, a2 d , and sb1, b2 d . Explain your work.

Lines and Planes in Space

11.5

This section shows how to use scalar and vector products to write equations for lines, line segments, and planes in space. We will use these representations throughout the rest of the book.

Lines and Line Segments in Space

z P0(x 0 , y0 , z 0 ) P(x, y, z)

L v 0

y x

FIGURE 11.35 A point P lies on L through P0 parallel to v if and only if 1 P0 P is a scalar multiple of v.

In the plane, a line is determined by a point and a number giving the slope of the line. In space a line is determined by a point and a vector giving the direction of the line. Suppose that L is a line in space passing through a point P0sx0 , y0 , z0 d parallel to a 1 vector v = v1 i + v2 j + v3 k. Then L is the set of all points P(x, y, z) for which P0 P is 1 parallel to v (Figure 11.35). Thus, P0 P = tv for some scalar parameter t. The value of t depends on the location of the point P along the line, and the domain of t is s - q , q d. 1 The expanded form of the equation P0 P = tv is sx - x0 di + s y - y0 dj + sz - z0 dk = tsv1 i + v2 j + v3 kd, which can be rewritten as xi + yj + zk = x0 i + y0 j + z0 k + tsv1 i + v2 j + v3 kd.

(1)

If r(t) is the position vector of a point P(x, y, z) on the line and r0 is the position vector of the point P0sx0, y0, z0 d, then Equation (1) gives the following vector form for the equation of a line in space. Vector Equation for a Line A vector equation for the line L through P0sx0 , y0 , z0 d parallel to v is rstd = r0 + tv,

-q 6 t 6 q,

(2)

where r is the position vector of a point P(x, y, z) on L and r0 is the position vector of P0sx0 , y0 , z0 d. Equating the corresponding components of the two sides of Equation (1) gives three scalar equations involving the parameter t: x = x0 + tv1,

y = y0 + tv2,

z = z0 + tv3 .

These equations give us the standard parametrization of the line for the parameter interval -q 6 t 6 q. Parametric Equations for a Line The standard parametrization of the line through P0sx0 , y0 , z0 d parallel to v = v1 i + v2 j + v3 k is x = x0 + tv1,

y = y0 + tv2,

z = z0 + tv3,

-q 6 t 6 q

(3)

11.5

631

Find parametric equations for the line through s -2, 0, 4d parallel to v = 2i + 4j - 2k (Figure 11.36).

z

EXAMPLE 1 P0(–2, 0, 4) t0

4

With P0sx0 , y0 , z0 d equal to s -2, 0, 4d and v1 i + v2 j + v3 k equal to 2i + 4j - 2k, Equations (3) become Solution

x = -2 + 2t,

P1(0, 4, 2)

2

t1

0

EXAMPLE 2

4

2

8

4 t2

x

Lines and Planes in Space

v 2i 4j 2k

y P2(2, 8, 0)

FIGURE 11.36 Selected points and parameter values on the line in Example 1. The arrows show the direction of increasing t.

Qs1, -1, 4d. Solution

y = 4t,

z = 4 - 2t.

Find parametric equations for the line through Ps -3, 2, -3d and

The vector 1 PQ = s1 - s -3ddi + s -1 - 2dj + s4 - s -3ddk = 4i - 3j + 7k

is parallel to the line, and Equations (3) with sx0 , y0 , z0 d = s -3, 2, -3d give x = -3 + 4t,

y = 2 - 3t,

z = -3 + 7t.

We could have chosen Qs1, -1, 4d as the “base point” and written x = 1 + 4t,

y = -1 - 3t,

z = 4 + 7t.

These equations serve as well as the first; they simply place you at a different point on the line for a given value of t. Notice that parametrizations are not unique. Not only can the “base point” change, but so can the parameter. The equations x = -3 + 4t 3, y = 2 - 3t 3 , and z = -3 + 7t 3 also parametrize the line in Example 2. To parametrize a line segment joining two points, we first parametrize the line through the points. We then find the t-values for the endpoints and restrict t to lie in the closed interval bounded by these values. The line equations together with this added restriction parametrize the segment. Q(1, –1, 4)

Parametrize the line segment joining the points Ps -3, 2, -3d and Qs1, -1, 4d (Figure 11.37).

EXAMPLE 3

z

t1

We begin with equations for the line through P and Q, taking them, in this case, from Example 2:

Solution

–3 –1 1

x = -3 + 4t,

0 2

z = -3 + 7t.

We observe that the point sx, y, zd = s -3 + 4t, 2 - 3t, -3 + 7td

y x

y = 2 - 3t,

t0 P(–3, 2, –3)

FIGURE 11.37 Example 3 derives a parametrization of line segment PQ. The arrow shows the direction of increasing t.

on the line passes through Ps -3, 2, -3d at t = 0 and Qs1, -1, 4d at t = 1. We add the restriction 0 … t … 1 to parametrize the segment: x = -3 + 4t,

y = 2 - 3t,

z = -3 + 7t,

0 … t … 1.

The vector form (Equation (2)) for a line in space is more revealing if we think of a line as the path of a particle starting at position P0sx0 , y0 , z0 d and moving in the direction of vector v. Rewriting Equation (2), we have rstd = r0 + tv = r0 + t ƒ v ƒ æ

æ

Initial position

Time

v . ƒvƒ

æ æ Speed Direction

(4)

632

Chapter 11: Vectors and the Geometry of Space

In other words, the position of the particle at time t is its initial position plus its distance moved sspeed * timed in the direction v> ƒ v ƒ of its straight-line motion.

EXAMPLE 4 A helicopter is to fly directly from a helipad at the origin in the direction of the point (1, 1, 1) at a speed of 60 ft> sec. What is the position of the helicopter after 10 sec? Solution

We place the origin at the starting position (helipad) of the helicopter. Then the

unit vector u =

1 1 1 i + j + k 23 23 23

gives the flight direction of the helicopter. From Equation (4), the position of the helicopter at any time t is rstd = r0 + tsspeeddu = 0 + ts60d ¢

1 1 1 i + j + k≤ 23 23 23

= 2023tsi + j + kd. When t = 10 sec, rs10d = 20023 si + j + kd = h 200 23, 20023, 20023 i . After 10 sec of flight from the origin toward (1, 1, 1), the helicopter is located at the point s200 23, 20023, 20023d in space. It has traveled a distance of s60 ft>secds10 secd = 600 ft, which is the length of the vector r(10).

The Distance from a Point to a Line in Space S

PS sin

P

To find the distance from a point S to a line that passes through a point P parallel to a vec1 tor v, we find the absolute value of the scalar component of PS in the direction of a vector normal to the line (Figure 11.38). In the notation of the figure, the absolute value of the 1 ƒ PS * v ƒ 1 . scalar component is ƒ PS ƒ sin u, which is ƒvƒ

v

FIGURE 11.38 The distance from S to the line through P parallel to v is 1 ƒ PS ƒ sin u , where u is the angle between 1 PS and v.

Distance from a Point S to a Line Through P Parallel to v 1 ƒ PS * v ƒ d = ƒvƒ

EXAMPLE 5

(5)

Find the distance from the point S(1, 1, 5) to the line L:

x = 1 + t,

y = 3 - t,

z = 2t .

We see from the equations for L that L passes through P(1, 3, 0) parallel to v = i - j + 2k. With

Solution

1 PS = s1 - 1di + s1 - 3dj + s5 - 0dk = -2j + 5k

11.5

Lines and Planes in Space

633

and i 1 3 PS * v = 0 1

j -2 -1

k 5 3 = i + 5j + 2k, 2

Equation (5) gives d =

n

1 ƒ PS * v ƒ 21 + 25 + 4 230 = = = 25. ƒvƒ 21 + 1 + 4 26

An Equation for a Plane in Space Plane M P(x, y, z)

P0(x 0 , y0 , z 0 )

FIGURE 11.39 The standard equation for a plane in space is defined in terms of a vector normal to the plane: A point P lies in the plane through P0 normal to n if and 1 only if n # P0 P = 0.

A plane in space is determined by knowing a point on the plane and its “tilt” or orientation. This “tilt” is defined by specifying a vector that is perpendicular or normal to the plane. Suppose that plane M passes through a point P0sx0 , y0 , z0 d and is normal to the nonzero vector n = Ai + Bj + Ck. Then M is the set of all points P(x, y, z) for which 1 1 P0 P is orthogonal to n (Figure 11.39). Thus, the dot product n # P0 P = 0. This equation is equivalent to sAi + Bj + Ckd # [sx - x0 di + s y - y0 dj + sz - z0 dk] = 0 or Asx - x0 d + Bs y - y0 d + Csz - z0 d = 0.

Equation for a Plane The plane through P0sx0 , y0 , z0 d normal to n = Ai + Bj + Ck has Vector equation: Component equation: Component equation simplified:

1 n # P0 P = 0 Asx - x0 d + Bs y - y0 d + Csz - z0 d = 0 where Ax + By + Cz = D, D = Ax0 + By0 + Cz0

Find an equation for the plane through P0s -3, 0, 7d perpendicular to n = 5i + 2j - k.

EXAMPLE 6 Solution

The component equation is 5sx - s -3dd + 2s y - 0d + s -1dsz - 7d = 0.

Simplifying, we obtain 5x + 15 + 2y - z + 7 = 0 5x + 2y - z = -22. Notice in Example 6 how the components of n = 5i + 2j - k became the coefficients of x, y, and z in the equation 5x + 2y - z = -22. The vector n = Ai + Bj + Ck is normal to the plane Ax + By + Cz = D.

634

Chapter 11: Vectors and the Geometry of Space

EXAMPLE 7

Find an equation for the plane through A(0, 0, 1), B(2, 0, 0), and C(0, 3, 0).

We find a vector normal to the plane and use it with one of the points (it does not matter which) to write an equation for the plane. The cross product

Solution

i 1 1 AB * AC = 3 2 0

j 0 3

k -1 3 = 3i + 2j + 6k -1

is normal to the plane. We substitute the components of this vector and the coordinates of A(0, 0, 1) into the component form of the equation to obtain 3sx - 0d + 2s y - 0d + 6sz - 1d = 0 3x + 2y + 6z = 6.

Lines of Intersection Just as lines are parallel if and only if they have the same direction, two planes are parallel if and only if their normals are parallel, or n1 = kn2 for some scalar k. Two planes that are not parallel intersect in a line.

EXAMPLE 8

Find a vector parallel to the line of intersection of the planes 3x - 6y - 2z = 15 and 2x + y - 2z = 5. The line of intersection of two planes is perpendicular to both planes’ normal vectors n1 and n2 (Figure 11.40) and therefore parallel to n1 * n2 . Turning this around, n1 * n2 is a vector parallel to the planes’ line of intersection. In our case, Solution

i n1 * n2 = 3 3 2

E N A PL

n1

2

n2

k -2 3 = 14i + 2j + 15k. -2

Any nonzero scalar multiple of n1 * n2 will do as well. E1

AN

PL

n1 n 2

j -6 1

FIGURE 11.40 How the line of intersection of two planes is related to the planes’ normal vectors (Example 8).

EXAMPLE 9 Find parametric equations for the line in which the planes 3x - 6y - 2z = 15 and 2x + y - 2z = 5 intersect. We find a vector parallel to the line and a point on the line and use Equations (3). Example 8 identifies v = 14i + 2j + 15k as a vector parallel to the line. To find a point on the line, we can take any point common to the two planes. Substituting z = 0 in the plane equations and solving for x and y simultaneously identifies one of these points as s3, -1, 0d. The line is Solution

x = 3 + 14t,

y = -1 + 2t,

z = 15t.

The choice z = 0 is arbitrary and we could have chosen z = 1 or z = -1 just as well. Or we could have let x = 0 and solved for y and z. The different choices would simply give different parametrizations of the same line. Sometimes we want to know where a line and a plane intersect. For example, if we are looking at a flat plate and a line segment passes through it, we may be interested in knowing what portion of the line segment is hidden from our view by the plate. This application is used in computer graphics (Exercise 74).

11.5

EXAMPLE 10

Lines and Planes in Space

635

Find the point where the line x =

8 + 2t, 3

y = -2t,

z = 1 + t

intersects the plane 3x + 2y + 6z = 6. Solution

The point a

8 + 2t, -2t, 1 + tb 3

lies in the plane if its coordinates satisfy the equation of the plane, that is, if 3a

8 + 2tb + 2s -2td + 6s1 + td = 6 3 8 + 6t - 4t + 6 + 6t = 6 8t = -8 t = -1.

The point of intersection is sx, y, zd ƒ t = -1 = a

8 2 - 2, 2, 1 - 1b = a , 2, 0b . 3 3

The Distance from a Point to a Plane If P is a point on a plane with normal n, then the distance from any point S to the plane is 1 the length of the vector projection of PS onto n. That is, the distance from S to the plane is 1 n d = ` PS # ` ƒnƒ where n = Ai + Bj + Ck is normal to the plane.

EXAMPLE 11

(6)

Find the distance from S(1, 1, 3) to the plane 3x + 2y + 6z = 6.

We find a point P in the plane and calculate the length of the vector projection 1 of PS onto a vector n normal to the plane (Figure 11.41). The coefficients in the equation 3x + 2y + 6z = 6 give Solution

n = 3i + 2j + 6k. z n 3i 2j 6k S(1, 1, 3)

3x 2y 6z 6

(0, 0, 1) Distance from S to the plane

0

(2, 0, 0)

P(0, 3, 0)

y

x

FIGURE 11.41 The distance from S to the plane is the 1 length of the vector projection of PS onto n (Example 11).

636

Chapter 11: Vectors and the Geometry of Space

The points on the plane easiest to find from the plane’s equation are the intercepts. If we take P to be the y-intercept (0, 3, 0), then 1 PS = s1 - 0di + s1 - 3dj + s3 - 0dk = i - 2j + 3k, ƒ n ƒ = 2s3d2 + s2d2 + s6d2 = 249 = 7. The distance from S to the plane is 1 n d = ` PS # ` ƒnƒ

1 length of projn PS

6 3 2 = ` si - 2j + 3kd # a i + j + kb ` 7 7 7 = `

Angles Between Planes

n2

3 18 17 4 - + . ` = 7 7 7 7

The angle between two intersecting planes is defined to be the acute angle between their normal vectors (Figure 11.42).

n1

EXAMPLE 12 2x + y - 2z = 5. Solution

Find the angle between the planes 3x - 6y - 2z = 15 and

The vectors n1 = 3i - 6j - 2k,

n2 = 2i + j - 2k

are normals to the planes. The angle between them is u = cos-1 a

FIGURE 11.42 The angle between two planes is obtained from the angle between their normals.

= cos-1 a

n1 # n2 b ƒ n1 ƒ ƒ n2 ƒ

4 b 21

L 1.38 radians.

About 79 deg

Exercises 11.5 9. The line through s0, -7, 0d perpendicular to the plane x + 2y + 2z = 13

Lines and Line Segments Find parametric equations for the lines in Exercises 1–12. 1. The line through the point Ps3, -4, -1d parallel to the vector i + j + k

10. The line through (2, 3, 0) perpendicular to the vectors u = i + 2j + 3k and v = 3i + 4j + 5k

2. The line through Ps1, 2, -1d and Qs -1, 0, 1d

11. The x-axis

12. The z-axis

3. The line through Ps -2, 0, 3d and Qs3, 5, -2d 4. The line through P(1, 2, 0) and Qs1, 1, -1d 5. The line through the origin parallel to the vector 2j + k 6. The line through the point s3, -2, 1d parallel to the line x = 1 + 2t, y = 2 - t, z = 3t 7. The line through (1, 1, 1) parallel to the z-axis 8. The line through s2, 4, 5d 3x + 7y - 5z = 21

perpendicular

to

the

plane

Find parametrizations for the line segments joining the points in Exercises 13–20. Draw coordinate axes and sketch each segment, indicating the direction of increasing t for your parametrization. 13. (0, 0, 0),

(1, 1, 3> 2)

14. (0, 0, 0),

(1, 0, 0)

15. (1, 0, 0),

(1, 1, 0)

16. (1, 1, 0),

(1, 1, 1)

17. s0, 1, 1d, s0, -1, 1d

18. (0, 2, 0),

(3, 0, 0)

19. (2, 0, 2),

20. s1, 0, -1d,

s0, 3, 0d

(0, 2, 0)

11.5

Lines and Planes in Space

Planes Find equations for the planes in Exercises 21–26.

Angles Find the angles between the planes in Exercises 47 and 48.

21. The plane through P0s0, 2, -1d normal to n = 3i - 2j - k

47. x + y = 1,

22. The plane through s1, -1, 3d parallel to the plane

48. 5x + y - z = 10,

3x + y + z = 7 23. The plane through s1, 1, -1d, s2, 0, 2d , and s0, -2, 1d 24. The plane through (2, 4, 5), (1, 5, 7), and s -1, 6, 8d 25. The plane through P0s2, 4, 5d perpendicular to the line y = 1 + 3t,

x = 5 + t,

z = 4t

26. The plane through As1, -2, 1d perpendicular to the vector from the origin to A 27. Find the point of intersection of the lines x = 2t + 1, y = 3t + 2, z = 4t + 3 , and x = s + 2, y = 2s + 4, z = -4s - 1 , and then find the plane determined by these lines. 28. Find the point of intersection of the lines x = t, y = -t + 2, z = t + 1 , and x = 2s + 2, y = s + 3, z = 5s + 6 , and then find the plane determined by these lines. In Exercises 29 and 30, find the plane containing the intersecting lines. 29. L1: x = -1 + t, y = 2 + t, z = 1 - t; - q 6 t 6 q z = 2 - 2s; - q 6 s 6 q 30. L1: x = t, y = 3 - 3t, z = -2 - t; - q 6 t 6 q L2: x = 1 + s, y = 4 + s, z = -1 + s; - q 6 s 6 q L2: x = 1 - 4s, y = 1 + 2s,

31. Find a plane through P0s2, 1, -1d and perpendicular to the line of intersection of the planes 2x + y - z = 3, x + 2y + z = 2 . 32. Find a plane through the points P1s1, 2, 3d, P2s3, 2, 1d and perpendicular to the plane 4x - y + 2z = 7 . Distances In Exercises 33–38, find the distance from the point to the line. 33. s0, 0, 12d;

x = 4t,

y = -2t,

34. s0, 0, 0d;

x = 5 + 3t,

35. s2, 1, 3d;

x = 2 + 2t,

z = 2t

y = 5 + 4t,

z = -3 - 5t

y = 1 + 6t,

36. s2, 1, -1d;

x = 2t,

y = 1 + 2t,

37. s3, -1, 4d;

x = 4 - t,

38. s -1, 4, 3d;

x = 10 + 4t,

z = 3 z = 2t

y = 3 + 2t, y = -3,

z = -5 + 3t z = 4t

In Exercises 39–44, find the distance from the point to the plane. 39. s2, -3, 4d,

x + 2y + 2z = 13

40. s0, 0, 0d,

3x + 2y + 6z = 6

41. s0, 1, 1d,

4y + 3z = -12

42. s2, 2, 3d,

2x + y + 2z = 4

43. s0, -1, 0d,

2x + y + 2z = 4

44. s1, 0, -1d,

-4x + y + z = 4

637

2x + y - 2z = 2 x - 2y + 3z = -1

T Use a calculator to find the acute angles between the planes in Exercises 49–52 to the nearest hundredth of a radian. 49. 2x + 2y + 2z = 3,

2x - 2y - z = 5

z = 0

50. x + y + z = 1,

sthe xy-planed

x + 2y + z = 2

51. 2x + 2y - z = 3, 52. 4y + 3z = - 12,

3x + 2y + 6z = 6

Intersecting Lines and Planes In Exercises 53–56, find the point in which the line meets the plane. 53. x = 1 - t, 54. x = 2,

y = 3t,

y = 3 + 2t,

55. x = 1 + 2t,

z = 1 + t;

z = -2 - 2t; 6x + 3y - 4z = -12

y = 1 + 5t,

56. x = -1 + 3t,

2x - y + 3z = 6

y = -2,

z = 3t; z = 5t;

x + y + z = 2 2x - 3z = 7

Find parametrizations for the lines in which the planes in Exercises 57–60 intersect. 57. x + y + z = 1,

x + y = 2

58. 3x - 6y - 2z = 3, 59. x - 2y + 4z = 2, 60. 5x - 2y = 11,

2x + y - 2z = 2 x + y - 2z = 5

4y - 5z = - 17

Given two lines in space, either they are parallel, or they intersect, or they are skew (imagine, for example, the flight paths of two planes in the sky). Exercises 61 and 62 each give three lines. In each exercise, determine whether the lines, taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection. 61. L1: x = 3 + 2t, y = -1 + 4t, z = 2 - t; - q 6 t 6 q -q 6 s 6 q -q 6 r 6 q 62. L1: x = 1 + 2t, y = -1 - t, z = 3t; - q 6 t 6 q L2: x = 2 - s, y = 3s, z = 1 + s; - q 6 s 6 q L3: x = 5 + 2r, y = 1 - r, z = 8 + 3r; - q 6 r 6 q L2: x = 1 + 4s, y = 1 + 2s, z = -3 + 4s;

L3: x = 3 + 2r, y = 2 + r, z = -2 + 2r;

Theory and Examples 63. Use Equations (3) to generate a parametrization of the line through Ps2, -4, 7d parallel to v1 = 2i - j + 3k . Then generate another parametrization of the line using the point P2s -2, -2, 1d and the vector v2 = -i + s1>2dj - s3>2dk . 64. Use the component form to generate an equation for the plane through P1s4, 1, 5d normal to n1 = i - 2j + k . Then generate another equation for the same plane using the point P2s3, -2, 0d and the normal vector n2 = - 22i + 2 22j - 22k . 65. Find the points in which the line x = 1 + 2t, y = -1 - t, z = 3t meets the coordinate planes. Describe the reasoning behind your answer.

45. Find the distance from the plane x + 2y + 6z = 1 to the plane x + 2y + 6z = 10 .

66. Find equations for the line in the plane z = 3 that makes an angle of p>6 rad with i and an angle of p>3 rad with j. Describe the reasoning behind your answer.

46. Find the distance from the line x = 2 + t, y = 1 + t, z = -s1>2d - s1>2dt to the plane x + 2y + 6z = 10 .

67. Is the line x = 1 - 2t, y = 2 + 5t, z = -3t parallel to the plane 2x + y - z = 8 ? Give reasons for your answer.

638

Chapter 11: Vectors and the Geometry of Space

68. How can you tell when two planes A1 x + B1 y + C1 z = D1 and A2 x + B2 y + C2 z = D2 are parallel? Perpendicular? Give reasons for your answer.

b. Test the formulas obtained for y and z in part (a) by investigating their behavior at x1 = 0 and x1 = x0 and by seeing what happens as x0 : q . What do you find?

69. Find two different planes whose intersection is the line x = 1 + t, y = 2 - t, z = 3 + 2t . Write equations for each plane in the form Ax + By + Cz = D .

z

70. Find a plane through the origin that is perpendicular to the plane M: 2x + 3y + z = 12 in a right angle. How do you know that your plane is perpendicular to M?

P(0, y, z) P1(x1, y1, z1)

71. The graph of sx>ad + s y>bd + sz>cd = 1 is a plane for any nonzero numbers a, b, and c. Which planes have an equation of this form?

0 (x1, y1, 0)

72. Suppose L1 and L2 are disjoint (nonintersecting) nonparallel lines. Is it possible for a nonzero vector to be perpendicular to both L1 and L2 ? Give reasons for your answer. 73. Perspective in computer graphics In computer graphics and perspective drawing, we need to represent objects seen by the eye in space as images on a two-dimensional plane. Suppose that the eye is at Esx0, 0, 0d as shown here and that we want to represent a point P1sx1, y1, z1 d as a point on the yz-plane. We do this by projecting P1 onto the plane with a ray from E. The point P1 will be portrayed as the point P(0, y, z). The problem for us as graphics designers is to find y and z given E and P1 . 1

1

a. Write a vector equation that holds between EP and EP1 . Use the equation to express y and z in terms of x0 , x1, y1 , and z1 .

11.6

y

E(x 0, 0, 0) x

74. Hidden lines in computer graphics Here is another typical problem in computer graphics. Your eye is at (4, 0, 0). You are looking at a triangular plate whose vertices are at (1, 0, 1), (1, 1, 0), and s -2, 2, 2d . The line segment from (1, 0, 0) to (0, 2, 2) passes through the plate. What portion of the line segment is hidden from your view by the plate? (This is an exercise in finding intersections of lines and planes.)

Cylinders and Quadric Surfaces Up to now, we have studied two special types of surfaces: spheres and planes. In this section, we extend our inventory to include a variety of cylinders and quadric surfaces. Quadric surfaces are surfaces defined by second-degree equations in x, y, and z. Spheres are quadric surfaces, but there are others of equal interest which will be needed in Chapters 13–15.

Cylinders z

Generating curve (in the yz-plane)

A cylinder is a surface that is generated by moving a straight line along a given planar curve while holding the line parallel to a given fixed line. The curve is called a generating curve for the cylinder (Figure 11.43). In solid geometry, where cylinder means circular cylinder, the generating curves are circles, but now we allow generating curves of any kind. The cylinder in the following example is generated by a parabola.

EXAMPLE 1 Find an equation for the cylinder made by the lines parallel to the z-axis that pass through the parabola y = x 2, z = 0 (Figure 11.44). y

x

Lines through generating curve parallel to x-axis

FIGURE 11.43 A cylinder and generating curve.

The point P0sx0 , x 02, 0d lies on the parabola y = x 2 in the xy-plane. Then, for any value of z, the point Qsx0 , x 02, zd lies on the cylinder because it lies on the line x = x0 , y = x 02 through P0 parallel to the z-axis. Conversely, any point Qsx0 , x 02, zd whose y-coordinate is the square of its x-coordinate lies on the cylinder because it lies on the line x = x0 , y = x 02 through P0 parallel to the z-axis (Figure 11.44). Regardless of the value of z, therefore, the points on the surface are the points whose coordinates satisfy the equation y = x 2 . This makes y = x 2 an equation for the cylinder. Because of this, we call the cylinder “the cylinder y = x 2.” Solution

11.6 z Q 0(x 0, x 02, z)

P0(x 0, x 02, 0) 0 x

Cylinders and Quadric Surfaces

639

As Example 1 suggests, any curve ƒsx, yd = c in the xy-plane defines a cylinder parallel to the z-axis whose equation is also ƒsx, yd = c. For instance, the equation x 2 + y 2 = 1 defines the circular cylinder made by the lines parallel to the z-axis that pass through the circle x 2 + y 2 = 1 in the xy-plane. In a similar way, any curve gsx, zd = c in the xz-plane defines a cylinder parallel to the y-axis whose space equation is also gsx, zd = c. Any curve hs y, zd = c defines a cylinder parallel to the x-axis whose space equation is also hs y, zd = c. The axis of a cylinder need not be parallel to a coordinate axis, however.

y PA

Quadric Surfaces

BO RA

LA

y

x2

A quadric surface is the graph in space of a second-degree equation in x, y, and z. In this section we study quadric surfaces given by the equation Ax 2 + By 2 + Cz 2 + Dz = E,

EXAMPLE 2

The ellipsoid y2 x2 z2 + + 2 = 1 2 2 a b c

(Figure 11.45) cuts the coordinate axes at s ; a, 0, 0d, s0, ; b, 0d, and s0, 0, ; cd. It lies within the rectangular box defined by the inequalities ƒ x ƒ … a, ƒ y ƒ … b, and ƒ z ƒ … c. The surface is symmetric with respect to each of the coordinate planes because each variable in the defining equation is squared. z Elliptical cross-section in the plane z z 0

z c y2 x2 2 1 2 a b in the xy-plane The ellipse

z0

a

EL LIPSE

y

b

y

SE

ELLIPSE

x

LIP

x

EL

FIGURE 11.44 Every point of the cylinder in Example 1 has coordinates of the form sx0 , x 02, zd . We call it “the cylinder y = x 2 .”

where A, B, C, D, and E are constants. The basic quadric surfaces are ellipsoids, paraboloids, elliptical cones, and hyperboloids. Spheres are special cases of ellipsoids. We present a few examples illustrating how to sketch a quadric surface, and then give a summary table of graphs of the basic types.

The ellipse x2 z2 2 1 2 a c in the xz-plane

The ellipse

y2 b2

z2 1 c2

in the yz-plane

FIGURE 11.45 The ellipsoid y2 x2 z2 + 2 + 2 = 1 2 a b c in Example 2 has elliptical cross-sections in each of the three coordinate planes.

The curves in which the three coordinate planes cut the surface are ellipses. For example, y2 x2 + = 1 a2 b2

when

z = 0.

Chapter 11: Vectors and the Geometry of Space

The curve cut from the surface by the plane z = z0 , ƒ z0 ƒ 6 c, is the ellipse y2 x2 + = 1. a 2s1 - sz0>cd2 d b 2s1 - sz0>cd2 d If any two of the semiaxes a, b, and c are equal, the surface is an ellipsoid of revolution. If all three are equal, the surface is a sphere.

EXAMPLE 3

The hyperbolic paraboloid y2 b

-

2

x2 z = c, a2

c 7 0

has symmetry with respect to the planes x = 0 and y = 0 (Figure 11.46). The crosssections in these planes are c 2 y . b2

x = 0:

the parabola z =

y = 0:

the parabola z = -

(1)

c 2 x . a2

(2)

In the plane x = 0, the parabola opens upward from the origin. The parabola in the plane y = 0 opens downward. If we cut the surface by a plane z = z0 7 0, the cross-section is a hyperbola, y2 b

2

-

z0 x2 = c, 2 a

with its focal axis parallel to the y-axis and its vertices on the parabola in Equation (1). If z0 is negative, the focal axis is parallel to the x-axis and the vertices lie on the parabola in Equation (2). The parabola z c2 y 2 in the yz-plane b z Part of the hyperbola HY

PER

in the plane z c

y2 b2

z

x2 1 a2

BOLA

PA

BO

RA

x

RA

LA

Saddle point PA

640

The parabola z – c2 x2 a in the xz-plane

B OLA

y 2 y2 Part of the hyperbola x 2 2 1 a b in the plane z –c

y

x

FIGURE 11.46 The hyperbolic paraboloid s y 2>b 2 d - sx 2>a 2 d = z>c, c 7 0 . The cross-sections in planes perpendicular to the z-axis above and below the xy-plane are hyperbolas. The cross-sections in planes perpendicular to the other axes are parabolas.

Near the origin, the surface is shaped like a saddle or mountain pass. To a person traveling along the surface in the yz-plane the origin looks like a minimum. To a person traveling the xz-plane the origin looks like a maximum. Such a point is called a saddle point of a surface. We will say more about saddle points in Section 13.7. Table 11.1 shows graphs of the six basic types of quadric surfaces. Each surface shown is symmetric with respect to the z-axis, but other coordinate axes can serve as well (with appropriate changes to the equation).

11.6

Graphs of Quadric Surfaces

TABLE 11.1

z

Elliptical cross-section in the plane z z0

z c

z

2

y x2 2 1 a2 b in the xy-plane

The parabola z

The ellipse z0

c 2 x a2

in the xz-plane

ELLIPSE

zc b

a y

SE

y

OL

x

A

LIP

ELLIPSE

AB

b

ELLIPSE

x

The parabola z

EL

x2 z2 2 1 a2 c in the xz-plane

The ellipse

y

2

b2 in the yz-plane

x2 y2 1 a2 b2 in the plane z c

c z The line z – y b in the yz-plane z c a

y

y x

Part of the hyperbola PSE

x y2 z2 2 1 2 b c

x ELLIPSE

in the yz-plane

y z2 x2 + 2 = 2 2 a b c

ELLIPTICAL CONE

HYPERBOLOID OF ONE SHEET

The parabola z c2 y 2 in the yz-plane b z HY

PER

a

RB

PE

The hyperbola z2 x2 1 c2 a2 in the xz-plane

H

0

BO RA

2 y2 Part of the hyperbola x 2 2 1 a b in the plane z –c

RB

LA

(0, 0, –c) Vertex

x2 1 a2

x2 1 a2

x

z

y x

OLA

PE

HY

O

The hyperbola y2 (0, 0, c) z2 1 Vertex c2 b2 in the yz-plane y

y

HYPERB

x

Y

y

The parabola z – c2 x2 a in the xz-plane

OL

ELLIPSE

y2 b2

B OLA

LA

RA

PA

x

A

b

Part of the hyperbola in the plane z c

Saddle point

z

y2 z2 x2 + = 1 a2 b2 c2

B OL A PA

2

x2 y 1 a2 b2 in the plane z c2 The ellipse

y

LIPSE

2

z

z

HY PERBOLA

SE

x

ELLI

y2 z x2 2 + 2 = c a b

2 2 Part of the hyperbola x 2 z 2 1 in the xz-plane a c z 2 x2 y zc The ellipse 2 2 2 a b in the plane z c b2 a2 2 ELLIPSE x2 y The ellipse 2 2 1 a b in the x y-plane b y EL a

b

ELLIP

x

ELLIPTICAL PARABOLOID

z

The ellipse

y

y x

y2 z2 x2 2 + 2 + 2 = 1 a b c

ELLIPSOID

c 2 y b2

in the yz-plane

z2 1 c2

HYPERBOLA

The ellipse

The line z ac x in the xz-plane

z

y2 x2 2 1 a2 b in the plane z c The ellipse

PAR

a

641

Cylinders and Quadric Surfaces

x

ELLIPSE

HYPERBOLOID OF TWO SHEETS

y2 z2 x2 - 2 - 2 = 1 2 c a b

HYPERBOLIC PARABOLOID

y2 2

b

-

z x2 = c, a2

c 7 0

642

Chapter 11: Vectors and the Geometry of Space

Exercises 11.6 Matching Equations with Surfaces In Exercises 1–12, match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a)–(1). 1. x 2 + y 2 + 4z 2 = 10

2. z 2 + 4y 2 - 4x 2 = 4

3. 9y 2 + z 2 = 16

4. y 2 + z 2 = x 2

5. x = y - z

6. x = -y - z

2

2

2

7. x 2 + 2z 2 = 8

x

y

Drawing Sketch the surfaces in Exercises 13–44.

12. 9x 2 + 4y 2 + 2z 2 = 36 z

a.

CYLINDERS

z

b.

y

x

10. z = -4x 2 - y 2

11. x 2 + 4z 2 = y 2

z

l.

2

8. z 2 + x 2 - y 2 = 1

9. x = z 2 - y 2

z

k.

13. x 2 + y 2 = 4

14. z = y 2 - 1

15. x 2 + 4z 2 = 16

16. 4x 2 + y 2 = 36

ELLIPSOIDS y

x

x y

17. 9x 2 + y 2 + z 2 = 9

18. 4x 2 + 4y 2 + z 2 = 16

19. 4x 2 + 9y 2 + 4z 2 = 36

20. 9x 2 + 4y 2 + 36z 2 = 36

PARABOLOIDS AND CONES 21. z = x 2 + 4y 2

c.

23. x = 4 - 4y - z

z

d.

z

22. z = 8 - x 2 - y 2

2

24. y = 1 - x 2 - z 2

2

25. x 2 + y 2 = z 2

26. 4x 2 + 9z 2 = 9y 2

HYPERBOLOIDS 27. x 2 + y 2 - z 2 = 1 x

y

y

x

28. y 2 + z 2 - x 2 = 1

30. s y 2>4d - sx 2>4d - z 2 = 1

29. z - x - y = 1 2

2

2

HYPERBOLIC PARABOLOIDS 31. y 2 - x 2 = z z

e.

z

f.

ASSORTED 33. z = 1 + y 2 - x 2

34. 4x 2 + 4y 2 = z 2

35. y = -sx + z d

36. 16x 2 + 4y 2 = 1

37. x 2 + y 2 - z 2 = 4

38. x 2 + z 2 = y

39. x + z = 1

40. 16y 2 + 9z 2 = 4x 2

2

x

y

2

x

y

h.

z

2

2

41. z = -sx 2 + y 2 d

42. y 2 - x 2 - z 2 = 1

43. 4y + z - 4x = 4

44. x 2 + y 2 = z

2

g.

32. x 2 - y 2 = z

2

2

z

Theory and Examples 45. a. Express the area A of the cross-section cut from the ellipsoid x2 + y

x

x

i.

by the plane z = c as a function of c. (The area of an ellipse with semiaxes a and b is pab .) b. Use slices perpendicular to the z-axis to find the volume of the ellipsoid in part (a).

z

j.

z

y

y2 z2 + = 1 4 9

c. Now find the volume of the ellipsoid

x

y

x

y

y2 x2 z2 + 2 + 2 = 1. 2 a b c Does your formula give the volume of a sphere of radius a if a = b = c?

Chapter 11 46. The barrel shown here is shaped like an ellipsoid with equal pieces cut from the ends by planes perpendicular to the z-axis. The crosssections perpendicular to the z-axis are circular. The barrel is 2h units high, its midsection radius is R, and its end radii are both r. Find a formula for the barrel’s volume. Then check two things. First, suppose the sides of the barrel are straightened to turn the barrel into a cylinder of radius R and height 2h. Does your formula give the cylinder’s volume? Second, suppose r = 0 and h = R so the barrel is a sphere. Does your formula give the sphere’s volume? z

b. Express your answer in part (a) in terms of h and the areas A0 and Ah of the regions cut by the hyperboloid from the planes z = 0 and z = h . c. Show that the volume in part (a) is also given by the formula h V = sA0 + 4Am + Ah d , 6 where Am is the area of the region cut by the hyperboloid from the plane z = h>2 .

49. z = y 2,

-2 … x … 2,

50. z = 1 - y 2, y

-3 … x … 3, over

47. Show that the volume of the segment cut from the paraboloid 2

y x2 z + 2 = c 2 a b by the plane z = h equals half the segment’s base times its altitude. 48. a. Find the volume of the solid bounded by the hyperboloid y2 x2 z2 + 2 - 2 = 1 2 a b c and the planes z = 0 and z = h, h 7 0 .

Chapter 11

-2 … y … 2

52. z = x 2 + 2y 2

r

–h

-0.5 … y … 2

-2 … x … 2,

51. z = x + y , 2

R

x

643

Viewing Surfaces T Plot the surfaces in Exercises 49–52 over the indicated domains. If you can, rotate the surface into different viewing positions. r

h

Questions to Guide Your Review

2

-3 … y … 3

a. -3 … x … 3,

-3 … y … 3

b. -1 … x … 1,

-2 … y … 3

c. -2 … x … 2,

-2 … y … 2

d. -2 … x … 2,

-1 … y … 1

COMPUTER EXPLORATIONS Use a CAS to plot the surfaces in Exercises 53–58. Identify the type of quadric surface from your graph. 53.

y2 x2 z2 + = 1 9 36 25

55. 5x 2 = z 2 - 3y 2 57.

y2 x2 z2 - 1 = + 9 16 2

y2 x2 z2 = 1 9 9 16 y2 x2 = 1 + z 56. 16 9 54.

58. y - 24 - z 2 = 0

Questions to Guide Your Review

1. When do directed line segments in the plane represent the same vector? 2. How are vectors added and subtracted geometrically? Algebraically? 3. How do you find a vector’s magnitude and direction? 4. If a vector is multiplied by a positive scalar, how is the result related to the original vector? What if the scalar is zero? Negative? 5. Define the dot product (scalar product) of two vectors. Which algebraic laws are satisfied by dot products? Give examples. When is the dot product of two vectors equal to zero? 6. What geometric interpretation does the dot product have? Give examples. 7. What is the vector projection of a vector u onto a vector v? Give an example of a useful application of a vector projection. 8. Define the cross product (vector product) of two vectors. Which algebraic laws are satisfied by cross products, and which are not? Give examples. When is the cross product of two vectors equal to zero? 9. What geometric or physical interpretations do cross products have? Give examples.

10. What is the determinant formula for calculating the cross product of two vectors relative to the Cartesian i, j, k-coordinate system? Use it in an example. 11. How do you find equations for lines, line segments, and planes in space? Give examples. Can you express a line in space by a single equation? A plane? 12. How do you find the distance from a point to a line in space? From a point to a plane? Give examples. 13. What are box products? What significance do they have? How are they evaluated? Give an example. 14. How do you find equations for spheres in space? Give examples. 15. How do you find the intersection of two lines in space? A line and a plane? Two planes? Give examples. 16. What is a cylinder? Give examples of equations that define cylinders in Cartesian coordinates. 17. What are quadric surfaces? Give examples of different kinds of ellipsoids, paraboloids, cones, and hyperboloids (equations and sketches).

644

Chapter 11: Vectors and the Geometry of Space

Chapter 11

Practice Exercises

Vector Calculations in Two Dimensions In Exercises 1–4, let u = 8-3, 49 and v = 82, -59 . Find (a) the component form of the vector and (b) its magnitude.

In Exercises 25 and 26, find (a) the area of the parallelogram determined by vectors u and v and (b) the volume of the parallelepiped determined by the vectors u, v, and w.

1. 3u - 4v

2. u + v

25. u = i + j - k,

3. -2u

4. 5v

26. u = i + j,

In Exercises 5–8, find the component form of the vector.

5. The vector obtained by rotating 80, 19 through an angle of 2p>3 radians 6. The unit vector that makes an angle of p>6 radian with the positive x-axis 7. The vector 2 units long in the direction 4i - j 8. The vector 5 units long in the direction opposite to the direction of s3>5di + s4>5dj

Express the vectors in Exercises 9–12 in terms of their lengths and directions. 9. 22i + 22j

10. - i - j

11. Velocity vector v = s -2 sin tdi + s2 cos tdj when t = p>2 . 12. Velocity vector v = se t cos t - e t sin tdi + se t sin t + e t cos tdj when t = ln 2. Vector Calculations in Three Dimensions Express the vectors in Exercises 13 and 14 in terms of their lengths and directions. 13. 2i - 3j + 6k

14. i + 2j - k

15. Find a vector 2 units long in the direction of v = 4i - j + 4k. 16. Find a vector 5 units long in the direction opposite to the direction of v = s3>5d i + s4>5d k. In Exercises 17 and 18, find ƒ v ƒ , ƒ u ƒ , v # u, u # v, v * u, u * v, ƒ v * u ƒ , the angle between v and u, the scalar component of u in the direction of v, and the vector projection of u onto v. 17. v = i + j

18. v = i + j + 2k

u = 2i + j - 2k

u = -i - k

u = i + j - 5k

v = j,

w = -i - 2j + 3k

w = i + j + k

Lines, Planes, and Distances 27. Suppose that n is normal to a plane and that v is parallel to the plane. Describe how you would find a vector n that is both perpendicular to v and parallel to the plane. 28. Find a vector in the plane parallel to the line ax + by = c . In Exercises 29 and 30, find the distance from the point to the line. 29. (2, 2, 0); x = -t,

y = t,

30. (0, 4, 1); x = 2 + t,

z = -1 + t

y = 2 + t,

z = t

31. Parametrize the line that passes through the point (1, 2, 3) parallel to the vector v = -3i + 7k. 32. Parametrize the line segment joining the points P(1, 2, 0) and Qs1, 3, -1d . In Exercises 33 and 34, find the distance from the point to the plane. 33. s6, 0, -6d,

x - y = 4

34. s3, 0, 10d,

2x + 3y + z = 2

35. Find an equation for the plane that passes through the point s3, -2, 1d normal to the vector n = 2i + j + k. 36. Find an equation for the plane that passes through the point s -1, 6, 0d perpendicular to the line x = -1 + t, y = 6 - 2t, z = 3t . In Exercises 37 and 38, find an equation for the plane through points P, Q, and R. 37. Ps1, -1, 2d, 38. Ps1, 0, 0d,

Rs -1, 2, -1d

Qs2, 1, 3d, Qs0, 1, 0d,

Rs0, 0, 1d

39. Find the points in which the line x = 1 + 2t, y = -1 - t, z = 3t meets the three coordinate planes. 40. Find the point in which the line through the origin perpendicular to the plane 2x - y - z = 4 meets the plane 3x - 5y + 2z = 6 . 41. Find the acute angle between the planes x = 7 and x + y + 22z = -3 .

In Exercises 19 and 20, find projv u. 19. v = 2i + j - k

v = 2i + j + k,

20. u = i - 2j v = i + j + k

42. Find the acute angle between the planes x + y = 1 and y + z = 1.

In Exercises 21 and 22, draw coordinate axes and then sketch u, v, and u * v as vectors at the origin.

43. Find parametric equations for the line in which the planes x + 2y + z = 1 and x - y + 2z = -8 intersect.

21. u = i,

44. Show that the line in which the planes

v = i + j

22. u = i - j,

v = i + j

23. If ƒ v ƒ = 2, ƒ w ƒ = 3 , and the angle between v and w is p>3 , find ƒ v - 2w ƒ . 24. For what value or values of a will the vectors u = 2i + 4j - 5k and v = -4i - 8j + ak be parallel?

x + 2y - 2z = 5

and

5x - 2y - z = 0

intersect is parallel to the line x = -3 + 2t,

y = 3t,

z = 1 + 4t .

Chapter 11 45. The planes 3x + 6z = 1 and 2x + 2y - z = 3 intersect in a line. b. Find equations for the line of intersection. 46. Find an equation for the plane that passes through the point (1, 2, 3) parallel to u = 2i + 3j + k and v = i - j + 2k. 47. Is v = 2i - 4j + k related in any special way to the plane 2x + y = 5 ? Give reasons for your answer. 1 48. The equation n # P0 P = 0 represents the plane through P0 normal 1 to n. What set does the inequality n # P0 P 7 0 represent?

645

a. s2i - 3j + 3kd # ssx + 2di + s y - 1dj + zkd = 0 b. x = 3 - t,

a. Show that the planes are orthogonal.

Practice Exercises

y = -11t,

z = 2 - 3t

c. sx + 2d + 11s y - 1d = 3z d. s2i - 3j + 3kd * ssx + 2di + s y - 1dj + zkd = 0

e. s2i - j + 3kd * s -3i + kd # ssx + 2di + s y - 1dj + zkd =0

62. The parallelogram shown here has vertices at As2, -1, 4d, Bs1, 0, -1d, Cs1, 2, 3d , and D. Find

49. Find the distance from the point P(1, 4, 0) to the plane through A(0, 0, 0), Bs2, 0, -1d, and Cs2, -1, 0d .

z D

50. Find the distance from the point (2, 2, 3) to the plane 2x + 3y + 5z = 0 . 51. Find a vector parallel to the plane 2x - y - z = 4 and orthogonal to i + j + k. A(2, –1, 4)

52. Find a unit vector orthogonal to A in the plane of B and C if A = 2i - j + k, B = i + 2j + k, and C = i + j - 2k.

C(1, 2, 3)

53. Find a vector of magnitude 2 parallel to the line of intersection of the planes x + 2y + z - 1 = 0 and x - y + 2z + 7 = 0 . 54. Find the point in which the line through the origin perpendicular to the plane 2x - y - z = 4 meets the plane 3x - 5y + 2z = 6 .

y

55. Find the point in which the line through P(3, 2, 1) normal to the plane 2x - y + 2z = -2 meets the plane.

x

B(1, 0, –1)

a. the coordinates of D,

56. What angle does the line of intersection of the planes 2x + y - z = 0 and x + y + 2z = 0 make with the positive x-axis?

b. the cosine of the interior angle at B, 1 1 c. the vector projection of BA onto BC ,

57. The line

d. the area of the parallelogram, L:

x = 3 + 2t,

y = 2t,

z = t

intersects the plane x + 3y - z = - 4 in a point P. Find the coordinates of P and find equations for the line in the plane through P perpendicular to L. 58. Show that for every real number k the plane x - 2y + z + 3 + k s2x - y - z + 1d = 0 contains the line of intersection of the planes x - 2y + z + 3 = 0

and

e. an equation for the plane of the parallelogram, f. the areas of the orthogonal projections of the parallelogram on the three coordinate planes. 63. Distance between lines Find the distance between the line L1 through the points As1, 0, -1d and Bs -1, 1, 0d and the line L2 through the points Cs3, 1, -1d and Ds4, 5, -2d . The distance is to be measured along the line perpendicular to the two lines. First find 1 a vector n perpendicular to both lines. Then project AC onto n. 64. (Continuation of Exercise 63.) Find the distance between the line through A(4, 0, 2) and B(2, 4, 1) and the line through C(1, 3, 2) and D(2, 2, 4).

2x - y - z + 1 = 0 .

59. Find an equation for the plane through As -2, 0, -3d and Bs1, -2, 1d that lies parallel to the line through Cs -2, -13>5, 26>5d and Ds16>5, -13>5, 0d . 60. Is the line x = 1 + 2t, y = - 2 + 3t, z = -5t related in any way to the plane -4x - 6y + 10z = 9 ? Give reasons for your answer. 61. Which of the following are equations for the plane through the points Ps1, 1, -1d , Q(3, 0, 2), and Rs -2, 1, 0d ?

Quadric Surfaces Identify and sketch the surfaces in Exercises 65–76. 65. x 2 + y 2 + z 2 = 4

66. x 2 + s y - 1d2 + z 2 = 1

67. 4x 2 + 4y 2 + z 2 = 4

68. 36x 2 + 9y 2 + 4z 2 = 36

69. z = -sx + y d

70. y = -sx 2 + z 2 d

71. x 2 + y 2 = z 2

72. x 2 + z 2 = y 2

73. x + y - z = 4

74. 4y 2 + z 2 - 4x 2 = 4

75. y 2 - x 2 - z 2 = 1

76. z 2 - x 2 - y 2 = 1

2

2

2

2

2

646

Chapter 11: Vectors and the Geometry of Space

Chapter 11

Additional and Advanced Exercises

1. Submarine hunting Two surface ships on maneuvers are trying to determine a submarine’s course and speed to prepare for an aircraft intercept. As shown here, ship A is located at (4, 0, 0), whereas ship B is located at (0, 5, 0). All coordinates are given in thousands of feet. Ship A locates the submarine in the direction of the vector 2i + 3j - s1>3dk, and ship B locates it in the direction of the vector 18i - 6j - k. Four minutes ago, the submarine was located at s2, -1, -1>3d . The aircraft is due in 20 min. Assuming that the submarine moves in a straight line at a constant speed, to what position should the surface ships direct the aircraft?

clockwise when we look toward the origin from A. Find the velocity v of the point of the body that is at the position B(1, 3, 2). z

B(1, 3, 2)

A(1, 1, 1) O

1

1

z

v

x

Ship B

Ship A (4, 0, 0)

(0, 5, 0)

x

y

5 ft a 3 ft

Submarine

H1:

x = 6 + 40t,

y = -3 + 10t,

z = -3 + 2t

H2:

x = 6 + 110t,

y = -3 + 4t,

z = -3 + t .

Time t is measured in hours, and all coordinates are measured in miles. Due to system malfunctions, H2 stops its flight at (446, 13, 1) and, in a negligible amount of time, lands at (446, 13, 0). Two hours later, H1 is advised of this fact and heads toward H2 at 150 mph. How long will it take H1 to reach H2 ? 3. Torque The operator’s manual for the Toro® 21-in. lawnmower says “tighten the spark plug to 15 ft-lb s20.4 N # md .” If you are installing the plug with a 10.5-in. socket wrench that places the center of your hand 9 in. from the axis of the spark plug, about how hard should you pull? Answer in pounds.

b

F2

F1

2. A helicopter rescue Two helicopters, H1 and H2 , are traveling together. At time t = 0 , they separate and follow different straight-line paths given by

y

5. Consider the weight suspended by two wires in each diagram. Find the magnitudes and components of vectors F1 and F2, and angles a and b. a.

NOT TO SCALE

3

4 ft 100 lbs

b.

13 ft b

a 5 ft

F1

F2

12 ft

200 lbs

(Hint: This triangle is a right triangle.) 6. Consider a weight of w N suspended by two wires in the diagram, where T1 and T2 are force vectors directed along the wires. a

b T2

T1

9 in.

b

a w

a. Find the vectors T1 and T2 and show that their magnitudes are w cos b ƒ T1 ƒ = sin sa + bd 4. Rotating body The line through the origin and the point A(1, 1, 1) is the axis of rotation of a right body rotating with a constant angular speed of 3> 2 rad> sec. The rotation appears to be

and ƒ T2 ƒ =

w cos a . sin sa + bd

Chapter 11 b. For a fixed b determine the value of a which minimizes the magnitude ƒ T1 ƒ .

Additional and Advanced Exercises

11. Use vectors to show that the distance from P1sx1, y1 d to the line ax + by = c is

c. For a fixed a determine the value of b which minimizes the magnitude ƒ T2 ƒ . 7. Determinants and planes y1 - y y2 - y y3 - y

z1 - z z2 - z 3 = 0 z3 - z

b. What set of points in space is described by the equation y y1 y2 y3

z z1 z2 z3

1 14 = 0? 1 1

y = a2 s + b2 ,

z = a3 s + b3 ,

-q 6 s 6 q

y = c2 t + d2 ,

z = c3 t + d3 ,

-q 6 t 6 q,

intersect or are parallel if and only if a1 3 a2 a3

ƒ D1 - D2 ƒ . ƒ Ai + Bj + C k ƒ

c. Find an equation for the plane parallel to the plane 2x - y + 2z = -4 if the point s3, 2, -1d is equidistant from the two planes. d. Write equations for the planes that lie parallel to and 5 units away from the plane x - 2y + z = 3 . 14. Prove that four points A, B, C, and D are coplanar (lie in a com1 1 1 mon plane) if and only if AD # sAB * BC d = 0 . 15. The projection of a vector on a plane Let P be a plane in space and let v be a vector. The vector projection of v onto the plane P, projP v, can be defined informally as follows. Suppose the sun is shining so that its rays are normal to the plane P. Then projP v is the “shadow” of v onto P. If P is the plane x + 2y + 6z = 6 and v = i + j + k, find projP v.

b1 - d1 b2 - d2 3 = 0 . b3 - d3

c1 c2 c3

13. a. Show that the distance between the parallel planes Ax + By + Cz = D1 and Ax + By + Cz = D2 is

b. Find the distance between the planes 2x + 3y - z = 6 and 2x + 3y - z = 12 .

and x = c1 t + d1,

9. Consider a regular tetrahedron of side length 2. a. Use vectors to find the angle u formed by the base of the tetrahedron and any one of its other edges.

16. The accompanying figure shows nonzero vectors v, w, and z, with z orthogonal to the line L, and v and w making equal angles b with L. Assuming ƒ v ƒ = ƒ w ƒ , find w in terms of v and z. z

D 2

2 u

C

A

v

1

w

P

2

L

1 B

b. Use vectors to find the angle u formed by any two adjacent faces of the tetrahedron. This angle is commonly referred to as a dihedral angle. 10. In the figure here, D is the midpoint of side AB of triangle ABC, and E is one-third of the way between C and B. Use vectors to prove that F is the midpoint of line segment CD. C

F D

17. Triple vector products The triple vector products su * vd * w and u * sv * wd are usually not equal, although the formulas for evaluating them from components are similar: su * vd * w = su # wdv - sv # wdu. u * sv * wd = su # wdv - su # vdw.

Verify each formula for the following vectors by evaluating its two sides and comparing the results. u v w a. 2i

E

A

.

. 2A2 + B 2 + C 2 b. Find an equation for the sphere that is tangent to the planes x + y + z = 3 and x + y + z = 9 if the planes 2x - y = 0 and 3x - z = 0 pass through the center of the sphere.

d =

8. Determinants and lines Show that the lines x = a1 s + b1,

2a 2 + b 2

ƒ Ax1 + By1 + Cz1 - D ƒ

d =

is an equation for the plane through the three noncollinear points P1sx1, y1, z1 d, P2sx2 , y2 , z2 d , and P3sx3 , y3 , z3 d .

x x 4 1 x2 x3

ƒ ax1 + by1 - c ƒ

d =

12. a. Use vectors to show that the distance from P1sx1, y1, z1 d to the plane Ax + By + Cz = D is

a. Show that x1 - x 3 x2 - x x3 - x

647

B

2j

2k

b. i - j + k

2i + j - 2k

-i + 2j - k

c. 2i + j

2i - j + k

i + 2k

d. i + j - 2k

-i - k

2i + 4j - 2k

648

Chapter 11: Vectors and the Geometry of Space

18. Cross and dot products vectors, then

Show that if u, v, w, and r are any

a. u * sv * wd + v * sw * ud + w * su * vd = 0

b. u * v = su # v * idi + su # v * jdj + su # v * kdk c. su * vd # sw * rd = ` 19. Cross and dot products

u#w u#r

v#w `. v#r

Prove or disprove the formula

u * su * su * vdd # w = - ƒ u ƒ 2 u # v * w.

20. By forming the cross product of two appropriate vectors, derive the trigonometric identity sin sA - Bd = sin A cos B - cos A sin B . 21. Use vectors to prove that sa 2 + b 2 dsc 2 + d 2 d Ú sac + bdd2 for any four numbers a, b, c, and d. (Hint: Let u = ai + bj and v = ci + dj.) 22. Show that w = ƒ v ƒ u + ƒ u ƒ v bisects the angle between u and v. 23. Show that ƒ u + v ƒ … ƒ u ƒ + ƒ v ƒ for any vectors u and v. 24. Show that ƒ v ƒ u + ƒ u ƒ v and ƒ v ƒ u - ƒ u ƒ v are orthogonal.

12 VECTOR-VALUED FUNCTIONS AND MOTION IN SPACE OVERVIEW Now that we have learned about vectors and the geometry of space, we can combine these ideas with our earlier study of functions. In this chapter we introduce the calculus of vector-valued functions. The domains of these functions are real numbers, as before, but their ranges are vectors, not scalars. We use this calculus to describe the paths and motions of objects moving in a plane or in space, and we will see that the velocities and accelerations of these objects along their paths are vectors. We will also introduce new quantities that describe how an object’s path can turn and twist in space.

Curves in Space and Their Tangents

12.1

When a particle moves through space during a time interval I, we think of the particle’s coordinates as functions defined on I: x = ƒstd,

z

r

P( f (t), g(t), h(t))

O y x

FIGURE 12.1 The position vector 1 r = OP of a particle moving through space is a function of time.

y = g std,

z = hstd,

t H I.

(1)

The points sx, y, zd = sƒstd, gstd, hstdd, t H I, make up the curve in space that we call the particle’s path. The equations and interval in Equation (1) parametrize the curve. A curve in space can also be represented in vector form. The vector 1 (2) rstd = OP = ƒstdi + gstdj + hstdk from the origin to the particle’s position P(ƒ(t), g(t), h(t)) at time t is the particle’s position vector (Figure 12.1). The functions ƒ, g, and h are the component functions (components) of the position vector. We think of the particle’s path as the curve traced by r during the time interval I. Figure 12.2 displays several space curves generated by a computer graphing program. It would not be easy to plot these curves by hand. z

z

z

y

x

x

y

x

y r(t) 5 (sin 3t)(cos t)i 1 (sin 3t)(sin t)j 1 tk (a)

r(t) 5 (cos t)i 1 (sin t)j 1 (sin 2t)k

(b)

r(t) 5 (4 1 sin20t)(cos t)i 1 (4 1 sin20t)(sint)j 1 (cos20t)k (c)

FIGURE 12.2 Space curves are defined by the position vectors r(t).

649

650

Chapter 12: Vector-Valued Functions and Motion in Space

Equation (2) defines r as a vector function of the real variable t on the interval I. More generally, a vector-valued function or vector function on a domain set D is a rule that assigns a vector in space to each element in D. For now, the domains will be intervals of real numbers resulting in a space curve. Later, in Chapter 15, the domains will be regions in the plane. Vector functions will then represent surfaces in space. Vector functions on a domain in the plane or space also give rise to “vector fields,” which are important to the study of the flow of a fluid, gravitational fields, and electromagnetic phenomena. We investigate vector fields and their applications in Chapter 15. Real-valued functions are called scalar functions to distinguish them from vector functions. The components of r in Equation (2) are scalar functions of t. The domain of a vector-valued function is the common domain of its components. z

EXAMPLE 1

Graph the vector function rstd = scos tdi + ssin tdj + tk.

Solution

The vector function

2

is defined for all real values of t. The curve traced by r winds around the circular cylinder x 2 + y 2 = 1 (Figure 12.3). The curve lies on the cylinder because the i- and j-components of r, being the x- and y-coordinates of the tip of r, satisfy the cylinder’s equation:

t t0

x 2 + y 2 = scos td2 + ssin td2 = 1.

t 2

r

0 (1, 0, 0)

rstd = scos tdi + ssin tdj + tk

t

t 2

The curve rises as the k-component z = t increases. Each time t increases by 2p, the curve completes one turn around the cylinder. The curve is called a helix (from an old Greek word for “spiral”). The equations

P x2 y2 1

y

x = cos t,

x

y = sin t,

parametrize the helix, the interval - q 6 t 6 q being understood. Figure 12.4 shows more helices. Note how constant multiples of the parameter t can change the number of turns per unit of time.

FIGURE 12.3 The upper half of the helix rstd = scos tdi + ssin tdj + t k (Example 1). z

z

x

z

x y r(t) (cos t)i (sin t)j tk

FIGURE 12.4

z = t

x y

r(t) (cos t)i (sin t)j 0.3tk

y r(t) (cos 5t)i (sin 5t)j tk

Helices spiral upward around a cylinder, like coiled springs.

Limits and Continuity The way we define limits of vector-valued functions is similar to the way we define limits of real-valued functions.

12.1

Curves in Space and Their Tangents

651

DEFINITION Let rstd = ƒstdi + gstdj + hstdk be a vector function with domain D, and L a vector. We say that r has limit L as t approaches t0 and write lim rstd = L

t:t0

if, for every number P 7 0, there exists a corresponding number d 7 0 such that for all t H D ƒ rstd - L ƒ 6 P

whenever

0 6 ƒ t - t0 ƒ 6 d.

If L = L1i + L2 j + L3k, then it can be shown that limt:t0 rstd = L precisely when lim ƒstd = L1,

lim gstd = L2,

t:t0

t:t0

and

lim hstd = L3 .

t:t0

We omit the proof. The equation lim rstd = a lim ƒstdbi + a lim gstdbj + a lim hstdbk

t:t0

t:t0

t:t0

t:t0

(3)

provides a practical way to calculate limits of vector functions.

EXAMPLE 2

If rstd = scos tdi + ssin tdj + tk, then lim rstd = a lim cos tbi + a lim sin tbj + a lim tbk

t:p>4

t:p>4

=

t:p>4

t:p>4

22 22 p i + j + k. 2 2 4

We define continuity for vector functions the same way we define continuity for scalar functions.

DEFINITION A vector function r(t) is continuous at a point t = t0 in its domain if limt:t0 rstd = rst0 d. The function is continuous if it is continuous at every point in its domain. From Equation (3), we see that r(t) is continuous at t = t0 if and only if each component function is continuous there (Exercise 31).

EXAMPLE 3 (a) All the space curves shown in Figures 12.2 and 12.4 are continuous because their component functions are continuous at every value of t in s - q , q d. (b) The function gstd = scos tdi + ssin tdj + :t;k

is discontinuous at every integer, where the greatest integer function :t; is discontinuous.

Derivatives and Motion Suppose that rstd = ƒstdi + gstdj + hstdk is the position vector of a particle moving along a curve in space and that ƒ, g, and h are differentiable functions of t. Then the difference between the particle’s positions at time t and time t + ¢t is ¢r = rst + ¢td - rstd

652

Chapter 12: Vector-Valued Functions and Motion in Space z

(Figure 12.5a). In terms of components,

P

¢r = rst + ¢td - rstd

r(t t) r(t) Q

= [ƒst + ¢tdi + gst + ¢tdj + hst + ¢tdk] - [ƒstdi + gstdj + hstdk]

r(t) r(t t)

C

= [ƒst + ¢td - ƒstd]i + [gst + ¢td - gstd]j + [hst + ¢td - hstd]k.

O

y (a)

x

As ¢t approaches zero, three things seem to happen simultaneously. First, Q approaches P along the curve. Second, the secant line PQ seems to approach a limiting position tangent to the curve at P. Third, the quotient ¢r>¢t (Figure 12.5b) approaches the limit lim

z

r'(t)

¢t:0

r(t t) r(t) t Q

ƒst + ¢td - ƒstd gst + ¢td - gstd ¢r = c lim di + c lim dj ¢t ¢t ¢t ¢t:0 ¢t:0 + c lim

¢t:0

P

= c

r(t) r(t t) C

hst + ¢td - hstd dk ¢t

dƒ dg dh di + c dj + c dk. dt dt dt

We are therefore led to the following definition. O

y (b)

x

FIGURE 12.5 As ¢t : 0 , the point Q approaches the point P along the curve C. 1 In the limit, the vector PQ> ¢t becomes the tangent vector r¿std .

C1 C2

C3

C4 C5

FIGURE 12.6 A piecewise smooth curve made up of five smooth curves connected end to end in a continuous fashion. The curve here is not smooth at the points joining the five smooth curves.

DEFINITION The vector function rstd = ƒstdi + gstdj + hstdk has a derivative (is differentiable) at t if ƒ, g, and h have derivatives at t. The derivative is the vector function r¿std =

dƒ dg rst + ¢td - rstd dh dr = lim = i + j + k. dt dt dt dt ¢t ¢t:0

A vector function r is differentiable if it is differentiable at every point of its domain. The curve traced by r is smooth if dr> dt is continuous and never 0, that is, if ƒ, g, and h have continuous first derivatives that are not simultaneously 0. The geometric significance of the definition of derivative is shown in Figure 12.5. 1 The points P and Q have position vectors r(t) and rst + ¢td, and the vector PQ is represented by rst + ¢td - rstd. For ¢t 7 0, the scalar multiple s1>¢tdsrst + ¢td - rstdd 1 points in the same direction as the vector PQ . As ¢t : 0, this vector approaches a vector that is tangent to the curve at P (Figure 12.5b). The vector r¿std, when different from 0, is defined to be the vector tangent to the curve at P. The tangent line to the curve at a point sƒst0 d, gst0 d, hst0 dd is defined to be the line through the point parallel to r¿st0 d. We require dr>dt Z 0 for a smooth curve to make sure the curve has a continuously turning tangent at each point. On a smooth curve, there are no sharp corners or cusps. A curve that is made up of a finite number of smooth curves pieced together in a continuous fashion is called piecewise smooth (Figure 12.6). Look once again at Figure 12.5. We drew the figure for ¢t positive, so ¢r points forward, in the direction of the motion. The vector ¢r>¢t, having the same direction as ¢r, points forward too. Had ¢t been negative, ¢r would have pointed backward, against the direction of motion. The quotient ¢r>¢t, however, being a negative scalar multiple of ¢r, would once again have pointed forward. No matter how ¢r points, ¢r>¢t points forward and we expect the vector dr>dt = lim¢t:0 ¢r>¢t, when different from 0, to do the same. This means that the derivative dr> dt, which is the rate of change of position with respect to time, always points in the direction of motion. For a smooth curve, dr> dt is never zero; the particle does not stop or reverse direction.

12.1

Curves in Space and Their Tangents

653

DEFINITIONS If r is the position vector of a particle moving along a smooth curve in space, then vstd =

dr dt

is the particle’s velocity vector, tangent to the curve. At any time t, the direction of v is the direction of motion, the magnitude of v is the particle’s speed, and the derivative a = dv>dt, when it exists, is the particle’s acceleration vector. In summary, dr . dt 2. Speed is the magnitude of velocity: Speed = ƒ v ƒ . dv d 2r = 2. 3. Acceleration is the derivative of velocity: a = dt dt 4. The unit vector v> ƒ v ƒ is the direction of motion at time t. 1. Velocity is the derivative of position:

v =

EXAMPLE 4

Find the velocity, speed, and acceleration of a particle whose motion in space is given by the position vector rstd = 2 cos t i + 2 sin t j + 5 cos2 t k. Sketch the velocity vector vs7p>4d. Solution

The velocity and acceleration vectors at time t are vstd = r œ std = -2 sin t i + 2 cos t j - 10 cos t sin t k

z

r′ ⎛⎝

= -2 sin t i + 2 cos t j - 5 sin 2t k,

7p ⎛ 4 ⎝

astd = r ﬂ std = -2 cos t i - 2 sin t j - 10 cos 2t k, and the speed is

y

ƒ vstd ƒ = 2s -2 sin td2 + s2 cos td2 + s -5 sin 2td2 = 24 + 25 sin2 2t.

t 7p 4

When t = 7p>4, we have x

FIGURE 12.7 The curve and the velocity vector when t = 7p>4 for the motion given in Example 4.

va

7p b = 22 i + 22 j + 5 k, 4

aa

7p b = - 22 i + 22 j, 4

`va

7p b ` = 229. 4

A sketch of the curve of motion, and the velocity vector when t = 7p>4, can be seen in Figure 12.7. We can express the velocity of a moving particle as the product of its speed and direction: Velocity = ƒ v ƒ a

v b = sspeeddsdirectiond. v ƒ ƒ

Differentiation Rules Because the derivatives of vector functions may be computed component by component, the rules for differentiating vector functions have the same form as the rules for differentiating scalar functions.

654

Chapter 12: Vector-Valued Functions and Motion in Space

Differentiation Rules for Vector Functions Let u and v be differentiable vector functions of t, C a constant vector, c any scalar, and ƒ any differentiable scalar function. 1. Constant Function Rule:

d C = 0 dt

2. Scalar Multiple Rules:

d [custd] = cu¿std dt d [ƒstdustd] = ƒ¿stdustd + ƒstdu¿std dt

When you use the Cross Product Rule, remember to preserve the order of the factors. If u comes first on the left side of the equation, it must also come first on the right or the signs will be wrong.

3. Sum Rule:

d [ustd + vstd] = u¿std + v¿std dt

4. Difference Rule:

d [ustd - vstd] = u¿std - v¿std dt

5. Dot Product Rule:

d [ustd # vstd] = u¿std # vstd + ustd # v¿std dt

6. Cross Product Rule:

d [ustd * vstd] = u¿std * vstd + ustd * v¿std dt

7. Chain Rule:

d [usƒstdd] = ƒ¿stdu¿sƒstdd dt

We will prove the product rules and Chain Rule but leave the rules for constants, scalar multiples, sums, and differences as exercises. Proof of the Dot Product Rule Suppose that u = u1stdi + u2stdj + u3stdk and v = y1stdi + y2stdj + y3stdk. Then d d su # vd = su y + u2 y2 + u3 y3 d dt dt 1 1 = u 1œ y1 + u 2œ y2 + u 3œ y3 + u1y 1œ + u2y 2œ + u3y 3œ . (''')''''* ('''')'''* u¿ # v u # v¿ Proof of the Cross Product Rule We model the proof after the proof of the Product Rule for scalar functions. According to the definition of derivative, ust + hd * vst + hd - ustd * vstd d su * vd = lim . dt h h:0 To change this fraction into an equivalent one that contains the difference quotients for the derivatives of u and v, we subtract and add ustd * vst + hd in the numerator. Then d su * vd dt ust + hd * vst + hd - ustd * vst + hd + ustd * vst + hd - ustd * vstd h h:0

= lim

ust + hd - ustd vst + hd - vstd * vst + hd + ustd * d h h h:0 ust + hd - ustd vst + hd - vstd = lim * lim vst + hd + lim ustd * lim . h h h:0 h:0 h:0 h:0 = lim c

12.1

655

Curves in Space and Their Tangents

The last of these equalities holds because the limit of the cross product of two vector functions is the cross product of their limits if the latter exist (Exercise 32). As h approaches zero, vst + hd approaches v(t) because v, being differentiable at t, is continuous at t (Exercise 33). The two fractions approach the values of du> dt and dv> dt at t. In short, dv du d * v + u * . su * vd = dt dt dt As an algebraic convenience, we sometimes write the product of a scalar c and a vector v as vc instead of cv. This permits us, for instance, to write the Chain Rule in a familiar form:

Proof of the Chain Rule Suppose that ussd = assdi + bssdj + cssdk is a differentiable vector function of s and that s = ƒstd is a differentiable scalar function of t. Then a, b, and c are differentiable functions of t, and the Chain Rule for differentiable real-valued functions gives da db dc d [ussd] = i + j + k dt dt dt dt

du d u ds = , dt ds dt

=

da ds db ds dc ds i + j + k ds dt ds dt ds dt

=

ds da db dc a i + j + kb dt ds ds ds

=

ds du dt ds

where s = ƒstd .

= ƒ¿stdu¿sƒstdd.

s = ƒstd

Vector Functions of Constant Length When we track a particle moving on a sphere centered at the origin (Figure 12.8), the position vector has a constant length equal to the radius of the sphere. The velocity vector dr> dt, tangent to the path of motion, is tangent to the sphere and hence perpendicular to r. This is always the case for a differentiable vector function of constant length: The vector and its first derivative are orthogonal. By direct calculation,

z

dr dt

rstd # rstd = c 2

P

d [rstd # rstd] = 0 dt

r(t)

y

r¿std # rstd + rstd # r¿std = 0

ƒ rstd ƒ = c is constant. Differentiate both sides.

2r¿std # rstd = 0.

Rule 5 with rstd = ustd = vstd

The vectors r œ std and r(t) are orthogonal because their dot product is 0. In summary, x

If r is a differentiable vector function of t of constant length, then FIGURE 12.8 If a particle moves on a sphere in such a way that its position r is a differentiable function of time, then r # sd r>dtd = 0 .

r#

dr = 0. dt

(4)

We will use this observation repeatedly in Section 12.4. The converse is also true (see Exercise 27).

Exercises 12.1 Motion in the Plane In Exercises 1–4, r(t) is the position of a particle in the xy-plane at time t. Find an equation in x and y whose graph is the path of the particle. Then find the particle’s velocity and acceleration vectors at the given value of t. 1. rstd = st + 1di + st 2 - 1dj,

t = 1

2. rstd =

t 1 i + t j, t + 1

3. rstd = e t i +

2 2t e j, 9

t = -1>2 t = ln 3

4. rstd = scos 2tdi + s3 sin 2tdj,

t = 0

656

Chapter 12: Vector-Valued Functions and Motion in Space

Exercises 5–8 give the position vectors of particles moving along various curves in the xy-plane. In each case, find the particle’s velocity and acceleration vectors at the stated times and sketch them as vectors on the curve. 5. Motion on the circle x 2 + y 2 = 1 rstd = ssin tdi + scos tdj;

t = p>4 and p>2

i) Does the particle have constant speed? If so, what is its constant speed?

6. Motion on the circle x + y = 16 2

2

t t rstd = a4 cos bi + a4 sin bj; 2 2 7. Motion on the cycloid x = t - sin t, rstd = st - sin tdi + s1 - cos tdj;

t = p and 3p>2

ii) Is the particle’s acceleration vector always orthogonal to its velocity vector?

y = 1 - cos t

iii) Does the particle move clockwise or counterclockwise around the circle?

t = p and 3p>2

t = -1, 0, and 1

Motion in Space In Exercises 9–14, r(t) is the position of a particle in space at time t. Find the particle’s velocity and acceleration vectors. Then find the particle’s speed and direction of motion at the given value of t. Write the particle’s velocity at that time as the product of its speed and direction. 9. rstd = st + 1di + st 2 - 1dj + 2t k, 10. rstd = s1 + tdi +

t2

j +

t3 k, 3

4 12. rstd = ssec tdi + stan tdj + t k, 3 13. rstd = s2 ln st + 1ddi + t 2 j +

t = 1

t = p>2

t = p>6

2

t k, 2

t = 1

14. rstd = se -t di + s2 cos 3tdj + s2 sin 3tdk,

t = 0

In Exercises 15–18, r(t) is the position of a particle in space at time t. Find the angle between the velocity and acceleration vectors at time t = 0. 15. rstd = s3t + 1di + 23t j + t 2k 16. rstd = a

22 22 tbi + a t - 16t 2 b j 2 2

17. rstd = sln st 2 + 1ddi + stan-1 tdj + 2t 2 + 1 k 18. rstd =

4 1 4 s1 + td3>2 i + s1 - td3>2 j + t k 9 9 3

t0 = 0

20. rstd = t 2 i + s2t - 1dj + t 3 k, t0 = 2 t - 1 j + t ln t k, t0 = 1 21. rstd = ln t i + t + 2 22. rstd = scos tdi + ssin tdj + ssin 2tdk,

t0 =

t Ú 0 t Ú 0

c. rstd = cos st - p>2di + sin st - p>2dj, d. rstd = scos tdi - ssin tdj, e. rstd = cos st di + sin st dj, 2

2

24. Motion along a circle

t Ú 0

t Ú 0 t Ú 0

Show that the vector-valued function

rstd = s2i + 2j + kd 1

1

j ≤ + sin t ¢

1

1

1

k≤ 22 22 23 23 23 describes the motion of a particle moving in the circle of radius 1 centered at the point s2, 2, 1d and lying in the plane x + y - 2z = 2 . i -

i +

j +

25. Motion along a parabola A particle moves along the top of the parabola y 2 = 2x from left to right at a constant speed of 5 units per second. Find the velocity of the particle as it moves through the point (2, 2). 26. Motion along a cycloid A particle moves in the xy-plane in such a way that its position at time t is rstd = st - sin tdi + s1 - cos tdj. T a. Graph r(t). The resulting curve is a cycloid. b. Find the maximum and minimum values of ƒ v ƒ and ƒ a ƒ . (Hint: Find the extreme values of ƒ v ƒ 2 and ƒ a ƒ 2 first and take square roots later.) 27. Let r be a differentiable vector function of t. Show that if r # sdr>dtd = 0 for all t, then ƒ r ƒ is constant.

Tangents to Curves As mentioned in the text, the tangent line to a smooth curve rstd = ƒstdi + gstdj + hstdk at t = t0 is the line that passes through the point sƒst0 d, gst0 d, hst0 dd parallel to vst0 d , the curve’s velocity vector at t0 . In Exercises 19–22, find parametric equations for the line that is tangent to the given curve at the given parameter value t = t0 . 19. rstd = ssin tdi + st 2 - cos tdj + e t k,

b. rstd = cos s2tdi + sin s2tdj,

+ cos t ¢

t = 1

22 11. rstd = s2 cos tdi + s3 sin tdj + 4t k,

iv) Does the particle begin at the point (1, 0)? a. rstd = scos tdi + ssin tdj,

8. Motion on the parabola y = x 2 + 1 rstd = t i + st 2 + 1dj;

Theory and Examples 23. Motion along a circle Each of the following equations in parts (a)–(e) describes the motion of a particle having the same path, namely the unit circle x 2 + y 2 = 1 . Although the path of each particle in parts (a)–(e) is the same, the behavior, or “dynamics,” of each particle is different. For each particle, answer the following questions.

p 2

28. Derivatives of triple scalar products a. Show that if u, v, and w are differentiable vector functions of t, then dv du # dw d v * w + u# su # v * wd = * w + u#v * . dt dt dt dt b. Show that dr dr d d 2r d 3r ar # * 2b = r# a * 3b. dt dt dt dt dt (Hint: Differentiate on the left and look for vectors whose products are zero.)

12.2 29. Prove the two Scalar Multiple Rules for vector functions. 30. Prove the Sum and Difference Rules for vector functions. 31. Component Test for Continuity at a Point Show that the vector function r defined by rstd = ƒstdi + g stdj + hstdk is continuous at t = t0 if and only if ƒ, g, and h are continuous at t0 . 32. Limits of cross products of vector functions Suppose that r1std = ƒ1stdi + ƒ2stdj + ƒ3stdk, r2std = g1stdi + g2stdj + g3stdk, lim t:t0 r1std = A, and lim t:t0 r2std = B. Use the determinant formula for cross products and the Limit Product Rule for scalar functions to show that

Integrals of Vector Functions; Projectile Motion

657

35. rstd = ssin t - t cos tdi + scos t + t sin tdj + t 2k, 0 … t … 6p, t0 = 3p>2 36. rstd = 22t i + e t j + e -t k,

-2 … t … 3,

37. rstd = ssin 2tdi + sln s1 + tddj + t k, t0 = p>4

t0 = 1

0 … t … 4p,

38. rstd = sln st 2 + 2ddi + stan-1 3tdj + 2t 2 + 1 k, -3 … t … 5, t0 = 3 In Exercises 39 and 40, you will explore graphically the behavior of the helix

lim sr1std * r2stdd = A * B.

rstd = scos atdi + ssin atdj + bt k

t:t0

33. Differentiable vector functions are continuous Show that if rstd = ƒstdi + gstdj + hstdk is differentiable at t = t0 , then it is continuous at t0 as well. 34. Constant Function Rule Prove that if u is the vector function with the constant value C, then du>dt = 0. COMPUTER EXPLORATIONS Use a CAS to perform the following steps in Exercises 35–38. a. Plot the space curve traced out by the position vector r.

b. Find the components of the velocity vector dr> dt.

c. Evaluate dr> dt at the given point t0 and determine the equation of the tangent line to the curve at rst0 d .

as you change the values of the constants a and b. Use a CAS to perform the steps in each exercise. 39. Set b = 1 . Plot the helix r(t) together with the tangent line to the curve at t = 3p>2 for a = 1, 2, 4, and 6 over the interval 0 … t … 4p . Describe in your own words what happens to the graph of the helix and the position of the tangent line as a increases through these positive values. 40. Set a = 1 . Plot the helix r(t) together with the tangent line to the curve at t = 3p>2 for b = 1>4, 1>2, 2 , and 4 over the interval 0 … t … 4p . Describe in your own words what happens to the graph of the helix and the position of the tangent line as b increases through these positive values.

d. Plot the tangent line together with the curve over the given interval.

12.2

Integrals of Vector Functions; Projectile Motion In this section we investigate integrals of vector functions and their application to motion along a path in space or in the plane.

Integrals of Vector Functions A differentiable vector function R(t) is an antiderivative of a vector function r(t) on an interval I if dR>dt = r at each point of I. If R is an antiderivative of r on I, it can be shown, working one component at a time, that every antiderivative of r on I has the form R + C for some constant vector C (Exercise 35). The set of all antiderivatives of r on I is the indefinite integral of r on I.

DEFINITION The indefinite integral of r with respect to t is the set of all antiderivatives of r, denoted by 1 rstd dt. If R is any antiderivative of r, then L

rstd dt = Rstd + C.

The usual arithmetic rules for indefinite integrals apply.

658

Chapter 12: Vector-Valued Functions and Motion in Space

EXAMPLE 1 L

To integrate a vector function, we integrate each of its components.

sscos tdi + j - 2tkd dt = a

cos t dtbi + a dtbj - a 2t dtbk L L L = ssin t + C1 di + st + C2 dj - st 2 + C3 dk = ssin tdi + tj - t k + C 2

(1) (2)

C = C1i + C2 j - C3k

As in the integration of scalar functions, we recommend that you skip the steps in Equations (1) and (2) and go directly to the final form. Find an antiderivative for each component and add a constant vector at the end. Definite integrals of vector functions are best defined in terms of components. The definition is consistent with how we compute limits and derivatives of vector functions.

DEFINITION If the components of rstd = ƒstdi + gstdj + hstdk are integrable over [a, b], then so is r, and the definite integral of r from a to b is b

La

EXAMPLE 2

rstd dt = a

b

La

ƒstd dtbi + a

La

gstd dtbj + a

b

La

hstd dtbk.

As in Example 1, we integrate each component.

p

L0

b

sscos tdi + j - 2tkd dt = a

p

L0

cos t dtbi + a

p

L0

dtbj - a

= C sin t D 0 i + C t D 0 j - C t 2 D k p

p

p

L0

2t dtbk

p 0

= [0 - 0]i + [p - 0]j - [p 2 - 0 2]k = pj - p 2k The Fundamental Theorem of Calculus for continuous vector functions says that rstd dt = Rstd D a = Rsbd - Rsad

b

La

b

where R is any antiderivative of r, so that R¿std = rstd (Exercise 36).

EXAMPLE 3 Suppose we do not know the path of a hang glider, but only its acceleration vector astd = -s3 cos tdi - s3 sin tdj + 2k. We also know that initially (at time t = 0) the glider departed from the point (4, 0, 0) with velocity vs0d = 3j. Find the glider’s position as a function of t. Solution

Our goal is to find r(t) knowing

The differential equation: The initial conditions:

d 2r = -s3 cos tdi - s3 sin tdj + 2k dt 2 vs0d = 3j and rs0d = 4i + 0j + 0k. a =

Integrating both sides of the differential equation with respect to t gives vstd = -s3 sin tdi + s3 cos tdj + 2tk + C1. We use vs0d = 3j to find C1 : 3j = -s3 sin 0di + s3 cos 0dj + s0dk + C1 3j = 3j + C1 C1 = 0.

12.2

Integrals of Vector Functions; Projectile Motion

659

The glider’s velocity as a function of time is dr = vstd = -s3 sin tdi + s3 cos tdj + 2tk. dt

z

Integrating both sides of this last differential equation gives rstd = s3 cos tdi + s3 sin tdj + t 2k + C2. We then use the initial condition rs0d = 4i to find C2 : 4i = s3 cos 0di + s3 sin 0dj + s02 dk + C2 4i = 3i + s0dj + s0dk + C2

(4, 0, 0) x

C2 = i.

y

FIGURE 12.9 The path of the hang glider in Example 3. Although the path spirals around the z-axis, it is not a helix.

The glider’s position as a function of t is rstd = s1 + 3 cos tdi + s3 sin tdj + t 2 k. This is the path of the glider shown in Figure 12.9. Although the path resembles that of a helix due to its spiraling nature around the z-axis, it is not a helix because of the way it is rising. (We say more about this in Section 12.5.)

The Vector and Parametric Equations for Ideal Projectile Motion

y

v0

A classic example of integrating vector functions is the derivation of the equations for the motion of a projectile. In physics, projectile motion describes how an object fired at some angle from an initial position, and acted upon by only the force of gravity, moves in a vertical coordinate plane. In the classic example, we ignore the effects of any frictional drag on the object, which may vary with its speed and altitude, and also the fact that the force of gravity changes slightly with the projectile’s changing height. In addition, we ignore the long-distance effects of Earth turning beneath the projectile, such as in a rocket launch or the firing of a projectile from a cannon. Ignoring these effects gives us a reasonable approximation of the motion in most cases. To derive equations for projectile motion, we assume that the projectile behaves like a particle moving in a vertical coordinate plane and that the only force acting on the projectile during its flight is the constant force of gravity, which always points straight down. We assume that the projectile is launched from the origin at time t = 0 into the first quadrant with an initial velocity v0 (Figure 12.10). If v0 makes an angle a with the horizontal, then

_v0 _ sin a j

a x

_v0 _ cos a i r 5 0 at time t 5 0 a 5 –gj (a)

y (x, y)

v0 = s ƒ v0 ƒ cos adi + s ƒ v0 ƒ sin adj.

v

If we use the simpler notation y0 for the initial speed ƒ v0 ƒ , then

a 5 –gj

v0 = sy0 cos adi + sy0 sin adj.

r 5 x i 1 yj 0

x

The projectile’s initial position is r0 = 0i + 0j = 0.

R Horizontal range (b)

FIGURE 12.10 (a) Position, velocity, acceleration, and launch angle at t = 0 . (b) Position, velocity, and acceleration at a later time t.

(3)

(4)

Newton’s second law of motion says that the force acting on the projectile is equal to the projectile’s mass m times its acceleration, or msd 2r>dt 2 d if r is the projectile’s position vector and t is time. If the force is solely the gravitational force -mgj, then m

d 2r = -mg j dt 2

and

d 2r = -gj dt 2

660

Chapter 12: Vector-Valued Functions and Motion in Space

where g is the acceleration due to gravity. We find r as a function of t by solving the following initial value problem. d 2r = -gj dt 2

Differential equation:

r = r0

Initial conditions:

dr = v0 dt

and

when t = 0

The first integration gives dr = -sgtdj + v0 . dt A second integration gives r = -

1 2 gt j + v0 t + r0 . 2

Substituting the values of v0 and r0 from Equations (3) and (4) gives r = -

1 2 gt j + sy0 cos adti + sy0 sin adtj + 0. 2 ('''''')''''''* v0t

Collecting terms, we have

Ideal Projectile Motion Equation r = sy0 cos adti + asy0 sin adt -

1 2 gt bj. 2

(5)

Equation (5) is the vector equation for ideal projectile motion. The angle a is the projectile’s launch angle (firing angle, angle of elevation), and y0 , as we said before, is the projectile’s initial speed. The components of r give the parametric equations x = sy0 cos adt

and

y = sy0 sin adt -

1 2 gt , 2

(6)

where x is the distance downrange and y is the height of the projectile at time t Ú 0.

EXAMPLE 4 A projectile is fired from the origin over horizontal ground at an initial speed of 500 m> sec and a launch angle of 60°. Where will the projectile be 10 sec later? We use Equation (5) with y0 = 500, a = 60°, g = 9.8, and t = 10 to find the projectile’s components 10 sec after firing. Solution

r = sy0 cos adti + asy0 sin adt -

1 2 gt bj 2

23 1 1 = s500d a bs10di + as500d a b10 - a bs9.8ds100dbj 2 2 2 L 2500i + 3840j Ten seconds after firing, the projectile is about 3840 m above ground and 2500 m downrange from the origin.

12.2

Integrals of Vector Functions; Projectile Motion

661

Ideal projectiles move along parabolas, as we now deduce from Equations (6). If we substitute t = x>sy0 cos ad from the first equation into the second, we obtain the Cartesian coordinate equation y = -a

g 2y 02

cos2 a

b x 2 + stan adx.

This equation has the form y = ax 2 + bx, so its graph is a parabola. A projectile reaches its highest point when its vertical velocity component is zero. When fired over horizontal ground, the projectile lands when its vertical component equals zero in Equation (5), and the range R is the distance from the origin to the point of impact. We summarize the results here, which you are asked to verify in Exercise 27.

Height, Flight Time, and Range for Ideal Projectile Motion For ideal projectile motion when an object is launched from the origin over a horizontal surface with initial speed y0 and launch angle a:

y

Maximum height:

ymax =

v0

t =

Flight time:

a (x 0 , y0 )

sy0 sin ad2 2g 2y0 sin a g

y 02 R = g sin 2a.

Range: x

0

FIGURE 12.11 The path of a projectile fired from sx0 , y0 d with an initial velocity v0 at an angle of a degrees with the horizontal.

If we fire our ideal projectile from the point sx0, y0 d instead of the origin (Figure 12.11), the position vector for the path of motion is r = sx0 + sy0 cos adtdi + ay0 + sy0 sin adt as you are asked to show in Exercise 29.

Exercises 12.2 Integrating Vector-Valued Functions Evaluate the integrals in Exercises 1–10.

1

6.

1

1.

[t 3i + 7j + st + 1dk] dt

L0 2

2.

L1

4 cs6 - 6tdi + 3 2t j + a 2 b k d dt t

L-p>4

4

2

[te t i + e t j + ln t k] dt p>2

9.

1 1 1 c i + j + k d dt 5. 5 - t 2t L1 t

23 k d dt 1 + t2

[te t i + e -t j + k] dt

L1

[ssin tdi + s1 + cos tdj + ssec2 tdk] dt [ssec t tan tdi + stan tdj + s2 sin t cos tdk] dt

L0

L0

i +

ln 3

8.

p>3

4.

2 21 - t 2

1

7.

p>4

3.

c

L0

[cos t i - sin 2t j + sin2 t k] dt

L0 p/4

10.

L0

[sec t i + tan2 t j - t sin t k] dt

1 2 gt bj, 2

(7)

662

Chapter 12: Vector-Valued Functions and Motion in Space

Initial Value Problems Solve the initial value problems in Exercises 11–16 for r as a vector function of t. 11. Differential equation: Initial condition: 12. Differential equation: Initial condition: 13. Differential equation: Initial condition: 14. Differential equation: Initial condition: 15. Differential equation: Initial conditions:

dr = -t i - t j - t k dt rs0d = i + 2j + 3k dr = s180tdi + s180t - 16t 2 dj dt rs0d = 100j

Initial conditions:

23. Firing golf balls A spring gun at ground level fires a golf ball at an angle of 45°. The ball lands 10 m away. a. What was the ball’s initial speed? b. For the same initial speed, find the two firing angles that make the range 6 m.

3 dr 1 k = st + 1d1>2 i + e -t j + 2 t + 1 dt rs0d = k

24. Beaming electrons An electron in a TV tube is beamed horizontally at a speed of 5 * 10 6 m>sec toward the face of the tube 40 cm away. About how far will the electron drop before it hits?

dr = st 3 + 4tdi + t j + 2t 2 k dt rs0d = i + j

25. Equal-range firing angles What two angles of elevation will enable a projectile to reach a target 16 km downrange on the same level as the gun if the projectile’s initial speed is 400 m> sec?

d 2r = -32k dt 2 rs0d = 100k and

26. Finding muzzle speed Find the muzzle speed of a gun whose maximum range is 24.5 km.

dr ` = 8i + 8j dt t = 0 16. Differential equation:

22. Throwing a baseball A baseball is thrown from the stands 32 ft above the field at an angle of 30° up from the horizontal. When and how far away will the ball strike the ground if its initial speed is 32 ft> sec?

d 2r = -si + j + kd dt 2 rs0d = 10i + 10j + 10k and dr ` = 0 dt t = 0

Motion Along a Straight Line 17. At time t = 0 , a particle is located at the point (1, 2, 3). It travels in a straight line to the point (4, 1, 4), has speed 2 at (1, 2, 3) and constant acceleration 3i - j + k. Find an equation for the position vector r(t) of the particle at time t. 18. A particle traveling in a straight line is located at the point s1, -1, 2d and has speed 2 at time t = 0 . The particle moves toward the point (3, 0, 3) with constant acceleration 2i + j + k. Find its position vector r(t) at time t. Projectile Motion Projectile flights in the following exercises are to be treated as ideal unless stated otherwise. All launch angles are assumed to be measured from the horizontal. All projectiles are assumed to be launched from the origin over a horizontal surface unless stated otherwise. 19. Travel time A projectile is fired at a speed of 840 m> sec at an angle of 60°. How long will it take to get 21 km downrange?

27. Verify the results given in the text (following Example 4) for the maximum height, flight time, and range for ideal projectile motion. 28. Colliding marbles The accompanying figure shows an experiment with two marbles. Marble A was launched toward marble B with launch angle a and initial speed y0 . At the same instant, marble B was released to fall from rest at R tan a units directly above a spot R units downrange from A. The marbles were found to collide regardless of the value of y0 . Was this mere coincidence, or must this happen? Give reasons for your answer. B

1 2 gt 2

R tan

v0 A

R

29. Firing from sx0 , y0 d Derive the equations x = x0 + sy0 cos adt, y = y0 + sy0 sin adt -

20. Range and height versus speed a. Show that doubling a projectile’s initial speed at a given launch angle multiplies its range by 4. b. By about what percentage should you increase the initial speed to double the height and range? 21. Flight time and height A projectile is fired with an initial speed of 500 m> sec at an angle of elevation of 45°. a. When and how far away will the projectile strike? b. How high overhead will the projectile be when it is 5 km downrange? c. What is the greatest height reached by the projectile?

1 2 gt 2

(see Equation (7) in the text) by solving the following initial value problem for a vector r in the plane. Differential equation:

d 2r = -g j dt 2

Initial conditions:

rs0d = x0 i + y0 j dr s0d = sy0 cos adi + sy0 sin adj dt

12.2 30. Where trajectories crest For a projectile fired from the ground at launch angle a with initial speed y0 , consider a as a variable and y0 as a fixed constant. For each a, 0 6 a 6 p>2 , we obtain a parabolic trajectory as shown in the accompanying figure. Show that the points in the plane that give the maximum heights of these parabolic trajectories all lie on the ellipse x 2 + 4 ay -

Integrals of Vector Functions; Projectile Motion

The Rule for Negatives, b

b

s -rstdd dt = -

La La is obtained by taking k = -1 .

rstd dt ,

b. The Sum and Difference Rules: b

y 02 2 y 04 , b = 4g 4g 2

663

La

b

sr1std ; r2stdd dt =

La

b

r1std dt ;

La

r2std dt

c. The Constant Vector Multiple Rules:

where x Ú 0 .

C # rstd dt = C #

b

y

La

b

La

rstd dt

sany constant vector Cd

and

Ellipse

b

⎛ 1 R, y ⎛ max ⎝ ⎝2

La

Parabolic trajectory

La

rstd dt

sany constant vector Cd

34. Products of scalar and vector functions Suppose that the scalar function u(t) and the vector function r(t) are both defined for a … t … b . x

0

b

C * rstd dt = C *

a. Show that ur is continuous on [a, b] if u and r are continuous on [a, b]. b. If u and r are both differentiable on [a, b], show that ur is differentiable on [a, b] and that

31. Volleyball A volleyball is hit when it is 4 ft above the ground and 12 ft from a 6-ft-high net. It leaves the point of impact with an initial velocity of 35 ft> sec at an angle of 27° and slips by the opposing team untouched. a. Find a vector equation for the path of the volleyball. b. How high does the volleyball go, and when does it reach maximum height? c. Find its range and flight time. d. When is the volleyball 7 ft above the ground? How far (ground distance) is the volleyball from where it will land? e. Suppose that the net is raised to 8 ft. Does this change things? Explain. 32. Shot put In Moscow in 1987, Natalya Lisouskaya set a women’s world record by putting an 8 lb 13 oz shot 73 ft 10 in. Assuming that she launched the shot at a 40° angle to the horizontal from 6.5 ft above the ground, what was the shot’s initial speed? Theory and Examples 33. Establish the following properties of integrable vector functions. a. The Constant Scalar Multiple Rule: b

d dr du surd = u + r . dt dt dt 35. Antiderivatives of vector functions a. Use Corollary 2 of the Mean Value Theorem for scalar functions to show that if two vector functions R1std and R2std have identical derivatives on an interval I, then the functions differ by a constant vector value throughout I. b. Use the result in part (a) to show that if R(t) is any antiderivative of r(t) on I, then any other antiderivative of r on I equals Rstd + C for some constant vector C. 36. The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus for scalar functions of a real variable holds for vector functions of a real variable as well. Prove this by using the theorem for scalar functions to show first that if a vector function r(t) is continuous for a … t … b , then t

d rstd dt = rstd dt La at every point t of (a, b). Then use the conclusion in part (b) of Exercise 35 to show that if R is any antiderivative of r on [a, b] then

b

k rstd dt = k r std dt La La

sany scalar kd

b

La

rstd dt = Rsbd - Rsad .

664

Chapter 12: Vector-Valued Functions and Motion in Space

Arc Length in Space

12.3

In this and the next two sections, we study the mathematical features of a curve’s shape that describe the sharpness of its turning and its twisting. Base point 3

0

Arc Length Along a Space Curve

2

1

–2

–1

4 s

FIGURE 12.12 Smooth curves can be scaled like number lines, the coordinate of each point being its directed distance along the curve from a preselected base point.

One of the features of smooth space and plane curves is that they have a measurable length. This enables us to locate points along these curves by giving their directed distance s along the curve from some base point, the way we locate points on coordinate axes by giving their directed distance from the origin (Figure 12.12). This is what we did for plane curves in Section 10.2. To measure distance along a smooth curve in space, we add a z-term to the formula we use for curves in the plane.

DEFINITION The length of a smooth curve rstd = xstdi + ystdj + zstdk, a … t … b, that is traced exactly once as t increases from t = a to t = b, is b

L =

2 2 dy 2 dx dz b + a b + a b dt. dt dt La C dt

a

(1)

Just as for plane curves, we can calculate the length of a curve in space from any convenient parametrization that meets the stated conditions. We omit the proof. The square root in Equation (1) is ƒ v ƒ , the length of a velocity vector dr> dt. This enables us to write the formula for length a shorter way.

Arc Length Formula b

L =

La

ƒ v ƒ dt

(2)

z

A glider is soaring upward along the helix rstd = scos tdi + ssin tdj + tk. How long is the glider’s path from t = 0 to t = 2p?

EXAMPLE 1 2 t

t 2

0 (1, 0, 0)

The path segment during this time corresponds to one full turn of the helix (Figure 12.13). The length of this portion of the curve is Solution

t0

r

b

t 2 P

L = y

x

FIGURE 12.13 The helix in Example 1, rstd = scos tdi + ssin tdj + tk.

La

2p

ƒ v ƒ dt =

L0

2s -sin td2 + scos td2 + s1d2 dt

2p

=

L0

22 dt = 2p22 units of length.

This is 22 times the circumference of the circle in the xy-plane over which the helix stands.

12.3 z

Arc Length in Space

665

If we choose a base point Pst0 d on a smooth curve C parametrized by t, each value of t determines a point Pstd = sxstd, ystd, zstdd on C and a “directed distance” t

sstd = r

P(t)

0 Base point s(t) P(t0) x

FIGURE 12.14 The directed distance along the curve from Pst0 d to any point P(t) is t

sstd =

Lt0

ƒ vstd ƒ dt.

y

Lt0

ƒ vstd ƒ dt,

measured along C from the base point (Figure 12.14). This is the arc length function we defined in Section 10.2 for plane curves that have no z-component. If t 7 t0 , s(t) is the distance along the curve from Pst0 d to P(t). If t 6 t0 , s(t) is the negative of the distance. Each value of s determines a point on C, and this parametrizes C with respect to s. We call s an arc length parameter for the curve. The parameter’s value increases in the direction of increasing t. We will see that the arc length parameter is particularly effective for investigating the turning and twisting nature of a space curve.

Arc Length Parameter with Base Point Pst0 d t

sstd =

Lt0

t

2[x¿std]2 + [y¿std]2 + [z¿std]2 dt =

Lt0

ƒ vstd ƒ dt

(3)

We use the Greek letter t (“tau”) as the variable of integration in Equation (3) because the letter t is already in use as the upper limit. If a curve r(t) is already given in terms of some parameter t and s(t) is the arc length function given by Equation (3), then we may be able to solve for t as a function of s: t = tssd. Then the curve can be reparametrized in terms of s by substituting for t: r = rstssdd. The new parametrization identifies a point on the curve with its directed distance along the curve from the base point.

EXAMPLE 2

This is an example for which we can actually find the arc length parametrization of a curve. If t0 = 0, the arc length parameter along the helix rstd = scos tdi + ssin tdj + tk from t0 to t is t

sstd =

Lt0

ƒ vstd ƒ dt

Eq. (3)

22 dt

Value from Example 1

t

=

L0

= 22 t. Solving this equation for t gives t = s> 22. Substituting into the position vector r gives the following arc length parametrization for the helix: rstssdd = ¢ cos

s 22

≤ i + ¢ sin

s 22

≤j +

s 22

k.

Unlike Example 2, the arc length parametrization is generally difficult to find analytically for a curve already given in terms of some other parameter t. Fortunately, however, we rarely need an exact formula for s(t) or its inverse t(s).

666

Chapter 12: Vector-Valued Functions and Motion in Space

HISTORICAL BIOGRAPHY

Speed on a Smooth Curve

Josiah Willard Gibbs (1839–1903)

Since the derivatives beneath the radical in Equation (3) are continuous (the curve is smooth), the Fundamental Theorem of Calculus tells us that s is a differentiable function of t with derivative ds = ƒ vstd ƒ . dt

(4)

Equation (4) says that the speed with which a particle moves along its path is the magnitude of v, consistent with what we know. Although the base point Pst0 d plays a role in defining s in Equation (3), it plays no role in Equation (4). The rate at which a moving particle covers distance along its path is independent of how far away it is from the base point. Notice that ds>dt 7 0 since, by definition, ƒ v ƒ is never zero for a smooth curve. We see once again that s is an increasing function of t.

Unit Tangent Vector

z

We already know the velocity vector v = dr>dt is tangent to the curve rstd and that the vector

v

T = T5 v _v_

v v ƒ ƒ

P(t 0 )

is therefore a unit vector tangent to the (smooth) curve, called the unit tangent vector (Figure 12.15). The unit tangent vector T is a differentiable function of t whenever v is a differentiable function of t. As we will see in Section 12.5, T is one of three unit vectors in a traveling reference frame that is used to describe the motion of objects traveling in three dimensions.

x

EXAMPLE 3

r

0

y

s

FIGURE 12.15 We find the unit tangent vector T by dividing v by ƒ v ƒ .

Find the unit tangent vector of the curve rstd = s1 + 3 cos tdi + s3 sin tdj + t2k

representing the path of the glider in Example 3, Section 12.2. Solution

In that example, we found v =

dr = -s3 sin tdi + s3 cos tdj + 2tk dt

and ƒ v ƒ = 29 + 4t 2 . y

Thus, T5v

x2 1 y2 5 1

T =

P(x, y) r

For the counterclockwise motion

t 0

v 3 sin t 3 cos t 2t = i + j + k. ƒvƒ 29 + 4t 2 29 + 4t 2 29 + 4t 2

(1, 0)

x

rstd = scos tdi + ssin tdj around the unit circle, we see that

FIGURE 12.16 Counterclockwise motion around the unit circle.

v = s -sin tdi + scos tdj is already a unit vector, so T = v and T is orthogonal to r (Figure 12.16). The velocity vector is the change in the position vector r with respect to time t, but how does the position vector change with respect to arc length? More precisely, what is the derivative dr>ds? Since ds>dt 7 0 for the curves we are considering, s is one-to-one and

12.3

Arc Length in Space

667

has an inverse that gives t as a differentiable function of s (Section 3.8). The derivative of the inverse is dt 1 1 = = . ds ds>dt ƒvƒ This makes r a differentiable function of s whose derivative can be calculated with the Chain Rule to be v dr dt dr 1 = = v = = T. ds dt ds ƒvƒ ƒvƒ

(5)

This equation says that dr> ds is the unit tangent vector in the direction of the velocity vector v (Figure 12.15).

Exercises 12.3 Finding Tangent Vectors and Lengths In Exercises 1–8, find the curve’s unit tangent vector. Also, find the length of the indicated portion of the curve. 1. rstd = s2 cos tdi + s2 sin tdj + 25t k,

0 … t … p

2. rstd = s6 sin 2tdi + s6 cos 2tdj + 5t k,

0 … t … p

3. rstd = t i + s2>3dt 3>2 k,

0 … t … 8

4. rstd = s2 + tdi - st + 1dj + t k,

0 … t … 3

5. rstd = scos3 t dj + ssin3 t dk,

0 … t … p>2

6. rstd = 6t 3 i - 2t 3 j - 3t 3 k,

1 … t … 2

7. rstd = st cos tdi + st sin tdj + A 2 22>3 B t 3>2 k, 8. rstd = st sin t + cos tdi + st cos t - sin tdj,

0 … t … p 22 … t … 2

rstd = s5 sin tdi + s5 cos tdj + 12t k at a distance 26p units along the curve from the point s0, 5, 0d in the direction of increasing arc length. 10. Find the point on the curve rstd = s12 sin tdi - s12 cos tdj + 5t k at a distance 13p units along the curve from the point s0, -12, 0d in the direction opposite to the direction of increasing arc length. Arc Length Parameter In Exercises 11–14, find the arc length parameter along the curve from the point where t = 0 by evaluating the integral t

L0

rstd =

A 22t B i + A 22t B j + s1 - t 2 dk

from (0, 0, 1) to A 22, 22, 0 B . 16. Length of helix The length 2p22 of the turn of the helix in Example 1 is also the length of the diagonal of a square 2p units on a side. Show how to obtain this square by cutting away and flattening a portion of the cylinder around which the helix winds. 17. Ellipse

9. Find the point on the curve

s =

Theory and Examples 15. Arc length Find the length of the curve

ƒ vstd ƒ dt

from Equation (3). Then find the length of the indicated portion of the curve. 11. rstd = s4 cos tdi + s4 sin tdj + 3t k,

0 … t … p>2

12. rstd = scos t + t sin tdi + ssin t - t cos tdj, 13. rstd = se t cos tdi + se t sin tdj + e t k,

p>2 … t … p

-ln 4 … t … 0

14. rstd = s1 + 2tdi + s1 + 3tdj + s6 - 6tdk,

-1 … t … 0

a. Show that the curve rstd = scos tdi + ssin tdj + s1 - cos tdk, 0 … t … 2p , is an ellipse by showing that it is the intersection of a right circular cylinder and a plane. Find equations for the cylinder and plane. b. Sketch the ellipse on the cylinder. Add to your sketch the unit tangent vectors at t = 0, p>2, p , and 3p>2 . c. Show that the acceleration vector always lies parallel to the plane (orthogonal to a vector normal to the plane). Thus, if you draw the acceleration as a vector attached to the ellipse, it will lie in the plane of the ellipse. Add the acceleration vectors for t = 0, p>2, p , and 3p>2 to your sketch. d. Write an integral for the length of the ellipse. Do not try to evaluate the integral; it is nonelementary. T e. Numerical integrator Estimate the length of the ellipse to two decimal places. 18. Length is independent of parametrization To illustrate that the length of a smooth space curve does not depend on the parametrization you use to compute it, calculate the length of one turn of the helix in Example 1 with the following parametrizations. a. rstd = scos 4tdi + ssin 4tdj + 4t k,

0 … t … p>2

b. rstd = [cos st>2d]i + [sin st>2d] j + st>2dk, c. rstd = scos tdi - ssin td j - t k,

0 … t … 4p

-2p … t … 0

668

Chapter 12: Vector-Valued Functions and Motion in Space

19. The involute of a circle If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end P traces an involute of the circle. In the accompanying figure, the circle in question is the circle x 2 + y 2 = 1 and the tracing point starts at (1, 0). The unwound portion of the string is tangent to the circle at Q, and t is the radian measure of the angle from the positive x-axis to segment OQ. Derive the parametric equations x = cos t + t sin t,

y = sin t - t cos t,

t 7 0

20. (Continuation of Exercise 19.) Find the unit tangent vector to the involute of the circle at the point P(x, y). 21. Distance along a line Show that if u is a unit vector, then the arc length parameter along the line rstd = P0 + t u from the point P0sx0, y0, z0 d where t = 0, is t itself. 22. Use Simpson’s Rule with n = 10 to approximate the length of arc of r(t) = t i + t 2 j + t 3k from the origin to the point (2, 4, 8).

of the point P(x, y) for the involute.

y String

Q

P(x, y) t O

1

(1, 0)

x

Curvature and Normal Vectors of a Curve

12.4

In this section we study how a curve turns or bends. We look first at curves in the coordinate plane, and then at curves in space.

y

Curvature of a Plane Curve As a particle moves along a smooth curve in the plane, T = dr>ds turns as the curve bends. Since T is a unit vector, its length remains constant and only its direction changes as the particle moves along the curve. The rate at which T turns per unit of length along the curve is called the curvature (Figure 12.17). The traditional symbol for the curvature function is the Greek letter k (“kappa”).

T

P

T

s P0 0

T x

FIGURE 12.17 As P moves along the curve in the direction of increasing arc length, the unit tangent vector turns. The value of ƒ d T>ds ƒ at P is called the curvature of the curve at P.

DEFINITION of the curve is

If T is the unit vector of a smooth curve, the curvature function k = `

dT `. ds

If ƒ dT>ds ƒ is large, T turns sharply as the particle passes through P, and the curvature at P is large. If ƒ dT>ds ƒ is close to zero, T turns more slowly and the curvature at P is smaller.

12.4

Curvature and Normal Vectors of a Curve

669

If a smooth curve r(t) is already given in terms of some parameter t other than the arc length parameter s, we can calculate the curvature as k = `

dT dT dt ` = ` ` ds dt ds

=

dT 1 ` ` ƒ ds>dt ƒ dt

=

1 dT ` `. ƒ v ƒ dt

Chain Rule

ds = ƒvƒ dt

Formula for Calculating Curvature If r(t) is a smooth curve, then the curvature is k =

1 dT ` `, ƒ v ƒ dt

(1)

where T = v> ƒ v ƒ is the unit tangent vector.

Testing the definition, we see in Examples 1 and 2 below that the curvature is constant for straight lines and circles.

EXAMPLE 1 A straight line is parametrized by rstd = C + tv for constant vectors C and v. Thus, r œ std = v, and the unit tangent vector T = v> ƒ v ƒ is a constant vector that always points in the same direction and has derivative 0 (Figure 12.18). It follows that, for any value of the parameter t, the curvature of the straight line is zero: k =

1 dT 1 ` ` = ƒ 0 ƒ = 0. ƒ v ƒ dt ƒvƒ

T

EXAMPLE 2

Here we find the curvature of a circle. We begin with the parametrization rstd = sa cos tdi + sa sin tdj

FIGURE 12.18 Along a straight line, T always points in the same direction. The curvature, ƒ d T>ds ƒ , is zero (Example 1).

of a circle of radius a. Then, v =

dr = -sa sin tdi + sa cos tdj dt

ƒ v ƒ = 2s -a sin td2 + sa cos td2 = 2a 2 = ƒ a ƒ = a.

Since a 7 0, ƒ a ƒ = a.

From this we find T =

v = -ssin tdi + scos tdj ƒvƒ

dT = -scos tdi - ssin tdj dt

`

dT ` = 2cos2 t + sin2 t = 1. dt

Hence, for any value of the parameter t, the curvature of the circle is k =

1 dT 1 1 1 ` ` = a s1d = a = . dt radius v ƒ ƒ

Although the formula for calculating k in Equation (1) is also valid for space curves, in the next section we find a computational formula that is usually more convenient to apply.

670

Chapter 12: Vector-Valued Functions and Motion in Space

N 5 1 dT κ ds

T

T P2

P1

Among the vectors orthogonal to the unit tangent vector T is one of particular significance because it points in the direction in which the curve is turning. Since T has constant length (namely, 1), the derivative dT> ds is orthogonal to T (Equation 4, Section 12.1). Therefore, if we divide dT> ds by its length k, we obtain a unit vector N orthogonal to T (Figure 12.19).

s P0

N 5 1 dT κ ds

FIGURE 12.19 The vector d T> ds, normal to the curve, always points in the direction in which T is turning. The unit normal vector N is the direction of d T> ds.

DEFINITION At a point where k Z 0, the principal unit normal vector for a smooth curve in the plane is 1 dT N = k . ds The vector dT> ds points in the direction in which T turns as the curve bends. Therefore, if we face in the direction of increasing arc length, the vector dT> ds points toward the right if T turns clockwise and toward the left if T turns counterclockwise. In other words, the principal normal vector N will point toward the concave side of the curve (Figure 12.19). If a smooth curve r(t) is already given in terms of some parameter t other than the arc length parameter s, we can use the Chain Rule to calculate N directly: N =

dT>ds ƒ dT>ds ƒ

sdT>dtdsdt>dsd ƒ dT>dt ƒ ƒ dt>ds ƒ dT>dt . = ƒ dT>dt ƒ =

dt 1 = 7 0 cancels. ds ds>dt

This formula enables us to find N without having to find k and s first.

Formula for Calculating N If r(t) is a smooth curve, then the principal unit normal is N =

d T>dt , ƒ dT>dt ƒ

where T = v> ƒ v ƒ is the unit tangent vector.

EXAMPLE 3

Find T and N for the circular motion rstd = scos 2tdi + ssin 2tdj.

Solution

We first find T: v = -s2 sin 2tdi + s2 cos 2tdj ƒ v ƒ = 24 sin2 2t + 4 cos2 2t = 2 T =

v = -ssin 2tdi + scos 2tdj. ƒvƒ

From this we find dT = -s2 cos 2tdi - s2 sin 2tdj dt

`

dT ` = 24 cos2 2t + 4 sin2 2t = 2 dt

(2)

12.4

Curvature and Normal Vectors of a Curve

671

and N =

dT>dt ƒ dT>dt ƒ

= -scos 2tdi - ssin 2tdj.

Eq. (2)

Notice that T # N = 0, verifying that N is orthogonal to T. Notice too, that for the circular motion here, N points from r(t) toward the circle’s center at the origin.

Circle of Curvature for Plane Curves Circle of curvature Center of curvature

Curve Radius of curvature N

T P(x, y)

The circle of curvature or osculating circle at a point P on a plane curve where k Z 0 is the circle in the plane of the curve that 1. 2. 3.

is tangent to the curve at P (has the same tangent line the curve has) has the same curvature the curve has at P has center that lies toward the concave or inner side of the curve (as in Figure 12.20).

The radius of curvature of the curve at P is the radius of the circle of curvature, which, according to Example 2, is 1 Radius of curvature = r = k .

FIGURE 12.20 The center of the osculating circle at P(x, y) lies toward the inner side of the curve.

To find r, we find k and take the reciprocal. The center of curvature of the curve at P is the center of the circle of curvature.

EXAMPLE 4

Find and graph the osculating circle of the parabola y = x 2 at the origin.

We parametrize the parabola using the parameter t = x (Section 10.1, Example 5)

Solution

rstd = ti + t 2j. First we find the curvature of the parabola at the origin, using Equation (1): v =

dr = i + 2tj dt

ƒ v ƒ = 21 + 4t 2 so that T =

v = s1 + 4t 2 d-1>2 i + 2ts1 + 4t 2 d-1>2 j. ƒvƒ

From this we find dT = -4ts1 + 4t 2 d-3>2 i + [2s1 + 4t 2 d-1>2 - 8t 2s1 + 4t 2 d-3>2] j. dt At the origin, t = 0, so the curvature is ks0d = =

dT 1 ` s0d ` vs0d ƒ ƒ dt 1 ƒ 0i + 2j ƒ 21

= s1d20 2 + 22 = 2.

Eq. (1)

672

Chapter 12: Vector-Valued Functions and Motion in Space

Therefore, the radius of curvature is 1>k = 1>2. At the origin we have t = 0 and T = i, so N = j. Thus the center of the circle is (0, 1>2). The equation of the osculating circle is therefore

y y x2 Osculating circle

sx - 0d2 + ay -

2

2

1 1 b = a b . 2 2

1 2

0

x

1

FIGURE 12.21 The osculating circle for the parabola y = x 2 at the origin (Example 4).

You can see from Figure 12.21 that the osculating circle is a better approximation to the parabola at the origin than is the tangent line approximation y = 0.

Curvature and Normal Vectors for Space Curves If a smooth curve in space is specified by the position vector r(t) as a function of some parameter t, and if s is the arc length parameter of the curve, then the unit tangent vector T is dr>ds = v> ƒ v ƒ . The curvature in space is then defined to be k = `

dT 1 dT ` = ` ` ds ƒ v ƒ dt

(3)

just as for plane curves. The vector dT> ds is orthogonal to T, and we define the principal unit normal to be dT>dt 1 dT N = k . = ds dT>dt ƒ ƒ z

EXAMPLE 5

Find the curvature for the helix (Figure 12.22)

rstd = sa cos tdi + sa sin tdj + btk, 2b

Solution

t

t 2

(4)

a, b Ú 0,

a 2 + b 2 Z 0.

We calculate T from the velocity vector v: v = -sa sin tdi + sa cos tdj + bk ƒ v ƒ = 2a 2 sin2 t + a 2 cos2 t + b 2 = 2a 2 + b 2

0 (a, 0, 0)

t0

r

P

t 2

x 2 y 2 a2

T = y

Then using Equation (3),

x

FIGURE 12.22 The helix rstd = sa cos tdi + sa sin tdj + bt k, drawn with a and b positive and t Ú 0 (Example 5).

v 1 = [-sa sin tdi + sa cos tdj + bk]. 2 ƒvƒ 2a + b 2

k =

1 dT ` ` ƒ v ƒ dt

[-sa cos tdi - sa sin tdj] ` 2a + b 2a + b 2 a = 2 ƒ -scos tdi - ssin tdj ƒ a + b2 1

=

=

2

2

`

1

2

a a 2scos td2 + ssin td2 = 2 . 2 a + b a + b2 2

From this equation, we see that increasing b for a fixed a decreases the curvature. Decreasing a for a fixed b eventually decreases the curvature as well. If b = 0, the helix reduces to a circle of radius a and its curvature reduces to 1> a, as it should. If a = 0, the helix becomes the z-axis, and its curvature reduces to 0, again as it should.

12.4

EXAMPLE 6 Solution

Curvature and Normal Vectors of a Curve

673

Find N for the helix in Example 5 and describe how the vector is pointing.

We have dT 1 = [sa cos tdi + sa sin tdj] 2 dt 2a + b 2 a dT 1 ` ` = 2a 2 cos2 t + a 2 sin2 t = 2 2 2 dt 2a + b 2a + b 2 dT>dt N = ƒ dT>dt ƒ = -

2a 2 + b 2 a

#

1

2a + b 2 = -scos tdi - ssin tdj. 2

Example 5

Eq. (4)

[sa cos tdi + sa sin tdj]

Thus, N is parallel to the xy-plane and always points toward the z-axis.

Exercises 12.4 Plane Curves Find T, N, and k for the plane curves in Exercises 1– 4. 1. rstd = t i + sln cos tdj,

-p>2 6 t 6 p>2

2. rstd = sln sec tdi + t j,

-p>2 6 t 6 p>2

7. Normals to plane curves a. Show that nstd = -g¿stdi + ƒ¿stdj and -nstd = g¿stdi ƒ¿stdj are both normal to the curve rstd = ƒstdi + g stdj at the point (ƒ(t), g(t)). To obtain N for a particular plane curve, we can choose the one of n or -n from part (a) that points toward the concave side of the curve, and make it into a unit vector. (See Figure 12.19.) Apply this method to find N for the following curves.

3. rstd = s2t + 3di + s5 - t 2 dj t 7 0

4. rstd = scos t + t sin tdi + ssin t - t cos tdj,

5. A formula for the curvature of the graph of a function in the xy-plane a. The graph y = ƒsxd in the xy-plane automatically has the parametrization x = x, y = ƒsxd , and the vector formula rsxd = x i + ƒsxdj. Use this formula to show that if ƒ is a twice-differentiable function of x, then ksxd =

ƒ ƒ–sxd ƒ

C 1 + sƒ¿sxdd2 D 3>2

c. Show that the curvature is zero at a point of inflection. 6. A formula for the curvature of a parametrized plane curve a. Show that the curvature of a smooth curve rstd = ƒstdi + g stdj defined by twice-differentiable functions x = ƒstd and y = g std is given by the formula #$ #$ ƒx y - yxƒ . # # sx 2 + y 2 d3>2

The dots in the formula denote differentiation with respect to t, one derivative for each dot. Apply the formula to find the curvatures of the following curves. b. rstd = t i + sln sin tdj, c. rstd = [tan

-1

0 6 t 6 p

ssinh td]i + sln cosh tdj

c. rstd = 24 - t 2 i + t j,

-2 … t … 2

8. (Continuation of Exercise 7.) a. Use the method of Exercise 7 to find N for the curve rstd = ti + s1>3dt 3 j when t 6 0 ; when t 7 0 . b. Calculate N for t Z 0 directly from T using Equation (4) for the curve in part (a). Does N exist at t = 0 ? Graph the curve and explain what is happening to N as t passes from negative to positive values.

.

b. Use the formula for k in part (a) to find the curvature of y = ln scos xd, -p>2 6 x 6 p>2 . Compare your answer with the answer in Exercise 1.

k =

b. rstd = t i + e 2t j

Space Curves Find T, N, and k for the space curves in Exercises 9–16. 9. 10. 11. 12. 13. 14. 15.

rstd rstd rstd rstd rstd rstd rstd

= = = = = = =

s3 sin tdi + s3 cos tdj + 4t k scos t + t sin tdi + ssin t - t cos tdj + 3k se t cos tdi + se t sin tdj + 2k s6 sin 2tdi + s6 cos 2tdj + 5t k st 3>3di + st 2>2dj, t 7 0 scos3 tdi + ssin3 tdj, 0 6 t 6 p>2 t i + sa cosh st>addj, a 7 0

16. rstd = scosh tdi - ssinh tdj + t k More on Curvature 17. Show that the parabola y = ax 2, a Z 0 , has its largest curvature at its vertex and has no minimum curvature. (Note: Since the curvature of a curve remains the same if the curve is translated or rotated, this result is true for any parabola.)

674

Chapter 12: Vector-Valued Functions and Motion in Space

18. Show that the ellipse x = a cos t, y = b sin t, a 7 b 7 0 , has its largest curvature on its major axis and its smallest curvature on its minor axis. (As in Exercise 17, the same is true for any ellipse.) 19. Maximizing the curvature of a helix In Example 5, we found the curvature of the helix rstd = sa cos tdi + sa sin tdj + bt k sa, b Ú 0d to be k = a>sa 2 + b 2 d . What is the largest value k can have for a given value of b? Give reasons for your answer. 20. Total curvature We find the total curvature of the portion of a smooth curve that runs from s = s0 to s = s1 7 s0 by integrating k from s0 to s1 . If the curve has some other parameter, say t, then the total curvature is s1

t1

t

1 ds K = k ds = k dt = k ƒ v ƒ dt , dt Ls0 Lt0 Lt0

where t0 and t1 correspond to s0 and s1 . Find the total curvatures of a. The portion of the helix rstd = s3 cos tdi + s3 sin tdj + t k, 0 … t … 4p . b. The parabola y = x 2, - q 6 x 6 q . 21. Find an equation for the circle of curvature of the curve rstd = t i + ssin tdj at the point sp>2, 1d . (The curve parametrizes the graph of y = sin x in the xy-plane.) 22. Find an equation for the circle of curvature of the curve rstd = s2 ln tdi - [t + s1>td] j, e -2 … t … e 2 , at the point s0, -2d , where t = 1 . T The formula ksxd =

ƒ ƒ–sxd ƒ

C 1 + s ƒ¿sxdd D

2 3>2

25. y = sin x,

0 … x … 2p

b. Calculate the curvature k of the curve at the given value t0 using the appropriate formula from Exercise 5 or 6. Use the parametrization x = t and y = ƒstd if the curve is given as a function y = ƒsxd . c. Find the unit normal vector N at t0 . Notice that the signs of the components of N depend on whether the unit tangent vector T is turning clockwise or counterclockwise at t = t0 . (See Exercise 7.) d. If C = a i + b j is the vector from the origin to the center (a, b) of the osculating circle, find the center C from the vector equation C = rst0 d +

e. Plot implicitly the equation sx - ad2 + s y - bd2 = 1>k2 of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure it is square. 27. rstd = s3 cos tdi + s5 sin tdj,

26. y = e x,

3

29. rstd = t 2i + st 3 - 3tdj,

-2 … x … 2 -1 … x … 2

1 Nst0 d . kst0 d

The point Psx0 , y0 d on the curve is given by the position vector rst0 d .

28. rstd = scos tdi + ssin tdj,

,

24. y = x 4>4,

-2 … x … 2

a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like.

3

derived in Exercise 5, expresses the curvature ksxd of a twicedifferentiable plane curve y = ƒsxd as a function of x. Find the curvature function of each of the curves in Exercises 23–26. Then graph ƒ(x) together with ksxd over the given interval. You will find some surprises. 23. y = x 2,

COMPUTER EXPLORATIONS In Exercises 27–34 you will use a CAS to explore the osculating circle at a point P on a plane curve where k Z 0 . Use a CAS to perform the following steps:

0 … t … 2p, 0 … t … 2p,

-4 … t … 4,

30. rstd = st 3 - 2t 2 - tdi +

t0 = p>4 t0 = p>4

t0 = 3>5

3t

j, -2 … t … 5, 21 + t 2 31. rstd = s2t - sin tdi + s2 - 2 cos tdj, 0 … t … 3p, t0 = 3p>2 32. rstd = se -t cos tdi + se -t sin tdj, 33. y = x 2 - x,

-2 … x … 5,

34. y = xs1 - xd2>5,

0 … t … 6p,

t0 = 1

t0 = p>4

x0 = 1

-1 … x … 2,

x0 = 1>2

Tangential and Normal Components of Acceleration

12.5 z

B5T×N N 5 1 dT κ ds r s

P

T 5 dr ds

If you are traveling along a space curve, the Cartesian i, j, and k coordinate system for representing the vectors describing your motion is not truly relevant to you. What is meaningful instead are the vectors representative of your forward direction (the unit tangent vector T), the direction in which your path is turning (the unit normal vector N), and the tendency of your motion to “twist” out of the plane created by these vectors in the direction perpendicular to this plane (defined by the unit binormal vector B = T * N). Expressing the acceleration vector along the curve as a linear combination of this TNB frame of mutually orthogonal unit vectors traveling with the motion (Figure 12.23) is particularly revealing of the nature of the path and motion along it.

y

P0 x

FIGURE 12.23 The TNB frame of mutually orthogonal unit vectors traveling along a curve in space.

The TNB Frame The binormal vector of a curve in space is B = T * N, a unit vector orthogonal to both T and N (Figure 12.24). Together T, N, and B define a moving right-handed vector frame that plays a significant role in calculating the paths of particles moving through space. It is called the Frenet (“fre-nay”) frame (after Jean-Frédéric Frenet, 1816–1900), or the TNB frame.

12.5

Tangential and Normal Components of Acceleration

675

Tangential and Normal Components of Acceleration When an object is accelerated by gravity, brakes, or a combination of rocket motors, we usually want to know how much of the acceleration acts in the direction of motion, in the tangential direction T. We can calculate this using the Chain Rule to rewrite v as

B

P

v =

N T

dr ds dr ds = = T . dt ds dt dt

Then we differentiate both ends of this string of equalities to get a =

FIGURE 12.24 The vectors T, N, and B (in that order) make a right-handed frame of mutually orthogonal unit vectors in space.

dv ds d d 2s ds dT = aT b = 2 T + dt dt dt dt dt dt

=

d 2s ds dT ds d 2s ds ds T + a b = T + akN b 2 2 dt ds dt dt dt dt dt

=

ds d 2s T + k a b N. 2 dt dt

dT = kN ds

2

DEFINITION

If the acceleration vector is written as a = aTT + aNN,

a

(1)

then

N

T

ds 2 a N 5 κ ⎛⎝ ⎛⎝ dt

2 a T 5 d 2s dt

aT =

d 2s d = v dt ƒ ƒ dt 2

and

aN = k a

2

ds b = kƒ v ƒ2 dt

(2)

are the tangential and normal scalar components of acceleration.

s P0

FIGURE 12.25 The tangential and normal components of acceleration. The acceleration a always lies in the plane of T and N, orthogonal to B.

d 2s T dt 2 a

P _v_ 2 k_v_2 N 5 r N

C

FIGURE 12.26 The tangential and normal components of the acceleration of an object that is speeding up as it moves counterclockwise around a circle of radius r .

Notice that the binormal vector B does not appear in Equation (1). No matter how the path of the moving object we are watching may appear to twist and turn in space, the acceleration a always lies in the plane oƒ T and N orthogonal to B. The equation also tells us exactly how much of the acceleration takes place tangent to the motion sd 2s>dt 2 d and how much takes place normal to the motion [ksds>dtd2] (Figure 12.25). What information can we discover from Equations (2)? By definition, acceleration a is the rate of change of velocity v, and in general, both the length and direction of v change as an object moves along its path. The tangential component of acceleration aT measures the rate of change of the length of v (that is, the change in the speed). The normal component of acceleration a N measures the rate of change of the direction of v. Notice that the normal scalar component of the acceleration is the curvature times the square of the speed. This explains why you have to hold on when your car makes a sharp (large k), high-speed (large ƒ v ƒ ) turn. If you double the speed of your car, you will experience four times the normal component of acceleration for the same curvature. If an object moves in a circle at a constant speed, d 2s>dt 2 is zero and all the acceleration points along N toward the circle’s center. If the object is speeding up or slowing down, a has a nonzero tangential component (Figure 12.26). To calculate a N , we usually use the formula a N = 2ƒ a ƒ 2 - a T2 , which comes from solving the equation ƒ a ƒ 2 = a # a = a T2 + a N2 for a N . With this formula, we can find aN without having to calculate k first.

Formula for Calculating the Normal Component of Acceleration a N = 2ƒ a ƒ 2 - a T2

(3)

676

Chapter 12: Vector-Valued Functions and Motion in Space

EXAMPLE 1

Without finding T and N, write the acceleration of the motion rstd = scos t + t sin tdi + ssin t - t cos tdj,

T a P(x, y)

tN

y

ing

in the form a = a TT + aNN. (The path of the motion is the involute of the circle in Figure 12.27. See also Section 12.3, Exercise 19.) Solution

Str

r

Q

t 7 0

We use the first of Equations (2) to find a T : dr = s -sin t + sin t + t cos tdi + scos t - cos t + t sin tdj dt = st cos tdi + st sin tdj

v =

t O

(1, 0)

x

ƒ v ƒ = 2t 2 cos2 t + t 2 sin2 t = 2t 2 = ƒ t ƒ = t aT =

x2 1 y2 5 1

d d v = std = 1. dt ƒ ƒ dt

t 7 0 Eq. (2)

Knowing a T , we use Equation (3) to find a N : FIGURE 12.27 The tangential and normal components of the acceleration of the motion rstd = scos t + t sin tdi + ssin t - t cos tdj, for t 7 0 . If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end P traces an involute of the circle (Example 1).

a = scos t - t sin tdi + ssin t + t cos tdj 2 2 ƒaƒ = t + 1

After some algebra

a N = 2ƒ a ƒ 2 - a T2 = 2st 2 + 1d - s1d = 2t 2 = t. We then use Equation (1) to find a: a = aTT + aNN = s1dT + stdN = T + tN.

Torsion How does dB> ds behave in relation to T, N, and B? From the rule for differentiating a cross product, we have d(T * N) dN dB dT = = * N + T * . ds ds ds ds Since N is the direction of dT> ds, sdT>dsd * N = 0 and dN dN dB = 0 + T * = T * . ds ds ds From this we see that dB> ds is orthogonal to T since a cross product is orthogonal to its factors. Since dB> ds is also orthogonal to B (the latter has constant length), it follows that dB> ds is orthogonal to the plane of B and T. In other words, dB> ds is parallel to N, so dB> ds is a scalar multiple of N. In symbols, dB = -tN. ds The negative sign in this equation is traditional. The scalar t is called the torsion along the curve. Notice that dB # N = -tN # N = -ts1d = -t. ds We use this equation for our next definition.

12.5

Binormal Rectifying plane

DEFINITION Normal plane

Tangential and Normal Components of Acceleration

677

Let B = T * N. The torsion function of a smooth curve is t = -

dB # N. ds

(4)

B P T

Unit tangent

N

Principal normal

Osculating plane

FIGURE 12.28 The names of the three planes determined by T, N, and B.

Unlike the curvature k, which is never negative, the torsion t may be positive, negative, or zero. The three planes determined by T, N, and B are named and shown in Figure 12.28. The curvature k = ƒ dT>ds ƒ can be thought of as the rate at which the normal plane turns as the point P moves along its path. Similarly, the torsion t = -sdB>dsd # N is the rate at which the osculating plane turns about T as P moves along the curve. Torsion measures how the curve twists. Look at Figure 12.29. If P is a train climbing up a curved track, the rate at which the headlight turns from side to side per unit distance is the curvature of the track. The rate at which the engine tends to twist out of the plane formed by T and N is the torsion. It can be shown that a space curve is a helix if and only if it has constant nonzero curvature and constant nonzero torsion.

dB ds

The torsion at P is –(d B/ds)⋅ N.

B The curvature at P is (dT/ds). s increases

N T P

s0

FIGURE 12.29 Every moving body travels with a TNB frame that characterizes the geometry of its path of motion.

Computational Formulas The most widely used formula for torsion, derived in more advanced texts, is

t =

# x 3 x$ % x

# y $ y % y

# z $3 z % z

ƒ v * a ƒ2

sif v * a Z 0d.

(5)

The dots in Equation (5) denote differentiation with respect to t, one derivative for # $ % each dot. Thus, x (“x dot”) means dx> dt, x (“x double dot”) means d 2x>dt 2 , and x (“x triple # 3 3 dot”) means d x>dt . Similarly, y = dy>dt, and so on. There is also an easy-to-use formula for curvature, as given in the following summary table (see Exercise 21).

678

Chapter 12: Vector-Valued Functions and Motion in Space

Computation Formulas for Curves in Space Unit tangent vector:

T =

v ƒvƒ

Principal unit normal vector:

N =

dT>dt ƒ dT>dt ƒ

Binormal vector:

B = T * N

Curvature:

k = `

ƒv * aƒ dT ` = ds ƒ v ƒ3 # x 3 x$ % x dB # N = t = ds ƒv

Torsion: Tangential and normal scalar components of acceleration:

# y $ y % y

# z $3 z % z

* a ƒ2

a = a TT + aNN aT =

d v dt ƒ ƒ

a N = k ƒ v ƒ 2 = 2ƒ a ƒ 2 - a T2

Exercises 12.5 Finding Tangential and Normal Components In Exercises 1 and 2, write a in the form a = a TT + aNN without finding T and N. 1. rstd = sa cos tdi + sa sin tdj + bt k 2. rstd = s1 + 3tdi + st - 2dj - 3t k In Exercises 3–6, write a in the form a = a TT + aNN at the given value of t without finding T and N. 3. rstd = st + 1di + 2tj + t 2k,

t = 1

4. rstd = st cos tdi + st sin tdj + t 2k,

t = 0

5. rstd = t i + st + s1>3dt dj + st - s1>3dt 3 dk, 2

3

6. rstd = se cos tdi + se sin tdj + 22e k, t

t

t

t = 0

t = 0

Finding the TNB Frame In Exercises 7 and 8, find r, T, N, and B at the given value of t. Then find equations for the osculating, normal, and rectifying planes at that value of t. 7. rstd = scos tdi + ssin tdj - k,

t = p>4

8. rstd = scos tdi + ssin tdj + t k,

t = 0

In Exercises 9–16 of Section 12.4, you found T, N, and k. Now, in the following Exercises 9–16, find B and t for these space curves. 9. rstd = s3 sin tdi + s3 cos tdj + 4t k 10. rstd = scos t + t sin tdi + ssin t - t cos tdj + 3k 11. rstd = se t cos tdi + se t sin tdj + 2k

12. rstd = s6 sin 2tdi + s6 cos 2tdj + 5t k 13. rstd = st 3>3di + st 2>2dj,

t 7 0

14. rstd = scos tdi + ssin tdj,

0 6 t 6 p>2

15. rstd = ti + sa cosh st>addj,

a 7 0

3

3

16. rstd = scosh tdi - ssinh tdj + t k Physical Applications 17. The speedometer on your car reads a steady 35 mph. Could you be accelerating? Explain. 18. Can anything be said about the acceleration of a particle that is moving at a constant speed? Give reasons for your answer. 19. Can anything be said about the speed of a particle whose acceleration is always orthogonal to its velocity? Give reasons for your answer. 20. An object of mass m travels along the parabola y = x 2 with a constant speed of 10 units> sec. What is the force on the object due to its acceleration at (0, 0)? at s21>2, 2d ? Write your answers in terms of i and j. (Remember Newton’s law, F = ma.) Theory and Examples 21. Vector formula for curvature For a smooth curve, use Equation (1) to derive the curvature formula

k =

ƒv * aƒ . ƒ v ƒ3

12.6 22. Show that a moving particle will move in a straight line if the normal component of its acceleration is zero. 23. A sometime shortcut to curvature If you already know ƒ aN ƒ and ƒ v ƒ , then the formula aN = k ƒ v ƒ 2 gives a convenient way to find the curvature. Use it to find the curvature and radius of curvature of the curve rstd = scos t + t sin tdi + ssin t - t cos tdj,

Velocity and Acceleration in Polar Coordinates

v # k = 0 for all t in [a, b]. Show that hstd = 0 for all t in [a, b]. (Hint: Start with a = d 2r>dt 2 and apply the initial conditions in reverse order.). 28. A formula that calculates T from B and v If we start with the definition t = -sd B>dsd # N and apply the Chain Rule to rewrite dB> ds as dB d B dt dB 1 , = = ds dt ds dt ƒ v ƒ

t 7 0.

(Take aN and ƒ v ƒ from Example 1.)

we arrive at the formula

24. Show that k and t are both zero for the line

t = -

rstd = sx0 + Atdi + s y0 + Btdj + sz0 + Ctdk. 25. What can be said about the torsion of a smooth plane curve rstd = ƒstdi + gstdj? Give reasons for your answer. 26. The torsion of a helix

Show that the torsion of the helix

rstd = sa cos tdi + sa sin tdj + bt k,

679

a, b Ú 0

is t = b>sa 2 + b 2 d . What is the largest value t can have for a given value of a? Give reasons for your answer. 27. Differentiable curves with zero torsion lie in planes That a sufficiently differentiable curve with zero torsion lies in a plane is a special case of the fact that a particle whose velocity remains perpendicular to a fixed vector C moves in a plane perpendicular to C. This, in turn, can be viewed as the following result. Suppose rstd = ƒstdi + gstdj + hstdk is twice differentiable for all t in an interval [a, b], that r = 0 when t = a , and that

1 dB # Nb . a ƒ v ƒ dt

The advantage of this formula over Equation (5) is that it is easier to derive and state. The disadvantage is that it can take a lot of work to evaluate without a computer. Use the new formula to find the torsion of the helix in Exercise 26. COMPUTER EXPLORATIONS Rounding the answers to four decimal places, use a CAS to find v, a, speed, T, N, B, k, t , and the tangential and normal components of acceleration for the curves in Exercises 29–32 at the given values of t. 29. rstd = st cos tdi + st sin tdj + t k,

t = 23

30. rstd = se cos tdi + se sin tdj + e k, t

t

t

t = ln 2

31. rstd = st - sin tdi + s1 - cos tdj + 2-t k, t = -3p 32. rstd = s3t - t 2 di + s3t 2 dj + s3t + t 3 dk,

t = 1

Velocity and Acceleration in Polar Coordinates

12.6

In this section we derive equations for velocity and acceleration in polar coordinates. These equations are useful for calculating the paths of planets and satellites in space, and we use them to examine Kepler’s three laws of planetary motion.

Motion in Polar and Cylindrical Coordinates

y

When a particle at P(r, u) moves along a curve in the polar coordinate plane, we express its position, velocity, and acceleration in terms of the moving unit vectors

u

r

O

P(r, )

ur = scos udi + ssin udj,

ur

x

FIGURE 12.30 The length of r is the positive polar coordinate r of the point P. Thus, ur , which is r> ƒ r ƒ , is also r> r. Equations (1) express ur and uu in terms of i and j.

uu = -ssin udi + scos udj, (1) 1 shown in Figure 12.30. The vector ur points along the position vector OP , so r = rur . The vector uu , orthogonal to ur , points in the direction of increasing u. We find from Equations (1) that dur = -ssin udi + scos udj = uu du duu = -scos udi - ssin udj = -ur . du When we differentiate ur and uu with respect to t to find how they change with time, the Chain Rule gives # dur # # ur = u = u uu, du

# du u # # uu = u = -u ur . du

(2)

680

Chapter 12: Vector-Valued Functions and Motion in Space

y

Hence, we can express the velocity vector in terms of ur and uu as v

# d # # # # arur b = r ur + rur = r ur + ru uu . v = r = dt

. ru . r ur

P(r, )

r

x

O

FIGURE 12.31 In polar coordinates, the velocity vector is # # v = r ur + ru uu.

To extend these equations of motion to space, we add zk to the right-hand side of the equation r = rur . Then, in these cylindrical coordinates, we have

z

k

r = rur + zk # # # v = r ur + ru uu + z k # $ $ ## $ a = sr - ru2 dur + sru + 2r u duu + z k.

u

r r ur z k

zk

See Figure 12.31. As in the previous section, we use Newton’s dot notation for time deriv# # atives to keep the formulas as simple as we can: ur means dur>dt, u means du>dt, and so on. The acceleration is $ # # # $ # # ## a = v = sr ur + r ur d + sruuu + ru uu + ruuu d. # # When Equations (2) are used to evaluate ur and uu and the components are separated, the equation for acceleration in terms of ur and uu becomes # $ $ ## a = sr - ru2 dur + sru + 2ru duu .

ur

The vectors ur , uu , and k make a right-handed frame (Figure 12.32) in which ur * uu = k,

y

r ur

(3)

uu * k = ur ,

k * ur = uu .

x

FIGURE 12.32 Position vector and basic unit vectors in cylindrical coordinates. Notice that ƒ r ƒ Z r if z Z 0 .

Planets Move in Planes Newton’s law of gravitation says that if r is the radius vector from the center of a sun of mass M to the center of a planet of mass m, then the force F of the gravitational attraction between the planet and sun is F = -

GmM r ƒ r ƒ2 ƒ r ƒ

(Figure 12.33). The number G is the universal gravitational constant. If we measure mass in kilograms, force in newtons, and distance in meters, G is about 6.6726 * 10 -11 Nm2 kg-2 . $ Combining the gravitation law with Newton’s second law, F = mr, for the force acting on the planet gives GmM r $ mr = , 2 ƒrƒ ƒrƒ GM r $ r = . ƒ r ƒ2 ƒ r ƒ M r r

m

r

r F – GmM r2 r

FIGURE 12.33 The force of gravity is directed along the line joining the centers of mass.

The planet is accelerated toward the sun’s center of mass at all times. $ Since r is a scalar multiple of r, we have $ r * r = 0. From this last equation, d # # # $ $ sr * r d = r(')'* * r + r * r = r * r = 0. dt 0

It follows that # r * r = C for some constant vector C.

(4)

12.6

681

# Equation (4) tells us that r and r always lie in a plane perpendicular to C. Hence, the planet moves in a fixed plane through the center of its sun (Figure 12.34). We next see how Kepler’s laws describe the motion in a precise way.

. Cr×r Sun

r Planet

Velocity and Acceleration in Polar Coordinates

Kepler’s First Law (Ellipse Law)

. r

Kepler’s first law says that a planet’s path is an ellipse with the sun at one focus. The eccentricity of the ellipse is

FIGURE 12.34 A planet that obeys Newton’s laws of gravitation and motion travels in the plane through the sun’s center # of mass perpendicular to C = r * r.

e =

r0y 02 - 1 GM

(5)

and the polar equation (see Section 10.7, Equation (5)) is r =

s1 + edr0 . 1 + e cos u

(6)

Here y0 is the speed when the planet is positioned at its minimum distance r0 from the sun. We omit the lengthy proof. The sun’s mass M is 1.99 * 10 30 kg.

Kepler’s Second Law (Equal Area Law) Planet r Sun

FIGURE 12.35 The line joining a planet to its sun sweeps over equal areas in equal times.

Kepler’s second law says that the radius vector from the sun to a planet (the vector r in our model) sweeps out equal areas in equal times (Figure 12.35). To derive the law, we use # Equation (3) to evaluate the cross product C = r * r from Equation (4): # C = r * r = r * v # # # Eq. (3), z = 0 = rur * sru r + ru u u d # # = rrsur * ur d + rsru dsur * uu d ('')''*

('')''*

0

k

# = rsru dk.

(7)

Setting t equal to zero shows that

# C = [rsru d]t = 0 k = r0 y0 k.

Substituting this value for C in Equation (7) gives # r0 y0 k = r 2u k, or

# r 2u = r0 y0 .

This is where the area comes in. The area differential in polar coordinates is dA =

1 2 r du 2

(Section 10.5). Accordingly, dA> dt has the constant value dA 1 # 1 = r 2u = r0 y0. 2 2 dt

(8)

So dA>dt is constant, giving Kepler’s second law. HISTORICAL BIOGRAPHY

Kepler’s Third Law (Time–Distance Law)

Johannes Kepler (1571–1630)

The time T it takes a planet to go around its sun once is the planet’s orbital period. Kepler’s third law says that T and the orbit’s semimajor axis a are related by the equation 4p2 T2 = . 3 GM a Since the right-hand side of this equation is constant within a given solar system, the ratio of T 2 to a 3 is the same ƒor every planet in the system.

682

Chapter 12: Vector-Valued Functions and Motion in Space

Here is a partial derivation of Kepler’s third law. The area enclosed by the planet’s elliptical orbit is calculated as follows: T

Area =

dA

L0 T

=

1 r0 y0 dt L0 2

=

1 Tr y . 2 0 0

Eq. (8)

If b is the semiminor axis, the area of the ellipse is pab, so 2pab 2pa 2 T = r0 y0 = r0 y0 21 - e 2 .

Planet r rmax

Sun

r0

(9)

It remains only to express a and e in terms of r0, y0, G, and M. Equation (5) does this for e. For a, we observe that setting u equal to p in Equation (6) gives rmax = r0

FIGURE 12.36 The length of the major axis of the ellipse is 2a = r0 + rmax.

For any ellipse, b = a 21 - e 2

1 + e . 1 - e

Hence, from Figure 12.36, 2a = r0 + rmax =

2r0 2r0GM . = 1 - e 2GM - r0y 02

(10)

Squaring both sides of Equation (9) and substituting the results of Equations (5) and (10) produces Kepler’s third law (Exercise 9).

Exercises 12.6 In Exercises 1–5, find the velocity and acceleration vectors in terms of ur and uu. du 1. r = a(1 - cos u) and = 3 dt du 2. r = a sin 2u and = 2t dt du 3. r = e au and = 2 dt 4. r = a(1 + sin t) and u = 1 - e -t 5. r = 2 cos 4t

and u = 2t

6. Type of orbit For what values of y0 in Equation (5) is the orbit in Equation (6) a circle? An ellipse? A parabola? A hyperbola? 7. Circular orbits Show that a planet in a circular orbit moves with a constant speed. (Hint: This is a consequence of one of Kepler’s laws.)

Chapter 12

8. Suppose that r is the position vector of a particle moving along a plane curve and dA> dt is the rate at which the vector sweeps out area. Without introducing coordinates, and assuming the necessary derivatives exist, give a geometric argument based on increments and limits for the validity of the equation dA 1 # = ƒr * rƒ . 2 dt 9. Kepler’s third law Complete the derivation of Kepler’s third law (the part following Equation (10)). 10. Find the length of the major axis of Earth’s orbit using Kepler’s third law and the fact that Earth’s orbital period is 365.256 days.

Questions to Guide Your Review

1. State the rules for differentiating and integrating vector functions. Give examples.

3. What is special about the derivatives of vector functions of constant length? Give an example.

2. How do you define and calculate the velocity, speed, direction of motion, and acceleration of a body moving along a sufficiently differentiable space curve? Give an example.

4. What are the vector and parametric equations for ideal projectile motion? How do you find a projectile’s maximum height, flight time, and range? Give examples.

Chapter 12 5. How do you define and calculate the length of a segment of a smooth space curve? Give an example. What mathematical assumptions are involved in the definition? 6. How do you measure distance along a smooth curve in space from a preselected base point? Give an example. 7. What is a differentiable curve’s unit tangent vector? Give an example. 8. Define curvature, circle of curvature (osculating circle), center of curvature, and radius of curvature for twice-differentiable curves in the plane. Give examples. What curves have zero curvature? Constant curvature?

Practice Exercises

683

10. How do you define N and k for curves in space? How are these quantities related? Give examples. 11. What is a curve’s binormal vector? Give an example. How is this vector related to the curve’s torsion? Give an example. 12. What formulas are available for writing a moving body’s acceleration as a sum of its tangential and normal components? Give an example. Why might one want to write the acceleration this way? What if the body moves at a constant speed? At a constant speed around a circle? 13. State Kepler’s laws.

9. What is a plane curve’s principal normal vector? When is it defined? Which way does it point? Give an example.

Chapter 12

Practice Exercises

Motion in the Plane In Exercises 1 and 2, graph the curves and sketch their velocity and acceleration vectors at the given values of t. Then write a in the form a = a TT + aNN without finding T and N, and find the value of k at the given values of t.

b. Find v and a at t = 0, 1, 2 , and 3 and add these vectors to your sketch. c. At any given time, what is the forward speed of the topmost point of the wheel? Of C?

A 22 sin t B j, t = 0 and p>4 2. rstd = A 23 sec t B i + A 23 tan t B j, t = 0

y

1. rstd = s4 cos tdi +

3. The position of a particle in the plane at time t is r =

1 21 + t

2

i +

t 21 + t 2

j.

Find the particle’s highest speed. 4. Suppose rstd = se t cos tdi + se t sin tdj. Show that the angle between r and a never changes. What is the angle? 5. Finding curvature At point P, the velocity and acceleration of a particle moving in the plane are v = 3i + 4j and a = 5i + 15j . Find the curvature of the particle’s path at P. 6. Find the point on the curve y = e x where the curvature is greatest. 7. A particle moves around the unit circle in the xy-plane. Its position at time t is r = xi + yj, where x and y are differentiable functions of t. Find dy> dt if v # i = y . Is the motion clockwise or counterclockwise? 8. You send a message through a pneumatic tube that follows the curve 9y = x 3 (distance in meters). At the point (3, 3), v # i = 4 and a # i = -2 . Find the values of v # j and a # j at (3, 3). 9. Characterizing circular motion A particle moves in the plane so that its velocity and position vectors are always orthogonal. Show that the particle moves in a circle centered at the origin. 10. Speed along a cycloid A circular wheel with radius 1 ft and center C rolls to the right along the x-axis at a half-turn per second. (See the accompanying figure.) At time t seconds, the position vector of the point P on the wheel’s circumference is r = spt - sin ptdi + s1 - cos ptdj. a. Sketch the curve traced by P during the interval 0 … t … 3 .

C t

r

1

P

x

0

Projectile Motion 11. Shot put A shot leaves the thrower’s hand 6.5 ft above the ground at a 45° angle at 44 ft> sec. Where is it 3 sec later? 12. Javelin A javelin leaves the thrower’s hand 7 ft above the ground at a 45° angle at 80 ft> sec. How high does it go? 13. A golf ball is hit with an initial speed y0 at an angle a to the horizontal from a point that lies at the foot of a straight-sided hill that is inclined at an angle f to the horizontal, where 0 6 f 6 a 6

p . 2

Show that the ball lands at a distance 2y 02 cos a g cos2 f

sin sa - fd ,

measured up the face of the hill. Hence, show that the greatest range that can be achieved for a given y0 occurs when a = sf>2d + sp>4d , i.e., when the initial velocity vector bisects the angle between the vertical and the hill.

684

Chapter 12: Vector-Valued Functions and Motion in Space

T 14. Javelin In Potsdam in 1988, Petra Felke of (then) East Germany set a women’s world record by throwing a javelin 262 ft 5 in. a. Assuming that Felke launched the javelin at a 40° angle to the horizontal 6.5 ft above the ground, what was the javelin’s initial speed? b. How high did the javelin go? Motion in Space Find the lengths of the curves in Exercises 15 and 16. 15. rstd = s2 cos tdi + s2 sin tdj + t 2k,

0 … t … p>4

16. rstd = s3 cos tdi + s3 sin tdj + 2t 3>2k,

0 … t … 3

30. Radius of curvature Show that the radius of curvature of a twice-differentiable plane curve rstd = ƒstdi + g stdj is given by the formula # # x2 + y2 d $ #2 #2 r = $ $ $ , where s = dt 2x + y . 2x 2 + y 2 - s 2 31. An alternative definition of curvature in the plane An alternative definition gives the curvature of a sufficiently differentiable plane curve to be ƒ df>ds ƒ , where f is the angle between T and i (Figure 12.37a). Figure 12.37b shows the distance s measured counterclockwise around the circle x 2 + y 2 = a 2 from the point (a, 0) to a point P, along with the angle f at P. Calculate the circle’s curvature using the alternative definition. (Hint: f = u + p>2 .)

In Exercises 17–20, find T, N, B, k , and t at the given value of t.

y

T

4 4 1 17. rstd = s1 + td3>2 i + s1 - td3>2j + t k, t = 0 9 9 3 18. rstd = se t sin 2tdi + se t cos 2tdj + 2e t k, t = 0 1 19. rstd = ti + e 2tj, 2

t = ln 2

i x

0

20. rstd = s3 cosh 2tdi + s3 sinh 2tdj + 6t k,

t = ln 2

(a) y

In Exercises 21 and 22, write a in the form a = a TT + aNN at t = 0 without finding T and N.

T

x 2 y 2 a2

P

21. rstd = s2 + 3t + 3t 2 di + s4t + 4t 2 dj - s6 cos tdk

a

22. rstd = s2 + tdi + st + 2t 2 dj + s1 + t 2 dk

s

O

23. Find T, N, B, k , and t as functions of t if rstd = ssin tdi +

x s 0 at (a, 0)

A 22 cos t B j + ssin tdk.

24. At what times in the interval 0 … t … p are the velocity and acceleration vectors of the motion rstd = i + s5 cos tdj +s3 sin tdk orthogonal? 25. The position of a particle moving in space at time t Ú 0 is t t rstd = 2i + a4 sin b j + a3 - p bk. 2 Find the first time r is orthogonal to the vector i - j. 26. Find equations for the osculating, normal, and rectifying planes of the curve rstd = t i + t 2j + t 3k at the point (1, 1, 1). 27. Find parametric equations for the line that is tangent to the curve rstd = e t i + ssin tdj + ln s1 - tdk at t = 0 . 28. Find parametric equations for the line tangent to the helix r(t) =

A 22 cos t B i + A 22 sin t B j + tk at the point where t = p>4 .

(b)

FIGURE 12.37 Figures for Exercise 31. 32. The view from Skylab 4 What percentage of Earth’s surface area could the astronauts see when Skylab 4 was at its apogee height, 437 km above the surface? To find out, model the visible surface as the surface generated by revolving the circular arc GT, shown here, about the y-axis. Then carry out these steps: 1. Use similar triangles in the figure to show that y0>6380 = 6380>s6380 + 437d . Solve for y0 . 2. To four significant digits, calculate the visible area as 6380

VA =

2px

Ly0

C

1 + a

2

dx b dy . dy

3. Express the result as a percentage of Earth’s surface area. y

Theory and Examples 29. Synchronous curves equations

By eliminating a from the ideal projectile

x = sy0 cos adt,

y = sy0 sin adt -

1 2 gt , 2

show that x + s y + gt >2d = This shows that projectiles launched simultaneously from the origin at the same initial speed will, at any given instant, all lie on the circle of radius y0 t centered at s0, -gt 2>2d , regardless of their launch angle. These circles are the synchronous curves of the launching. 2

2

2

y 02 t 2 .

S (Skylab) ⎧ 437 ⎨ G ⎩ x (6380) 2 y 2 T y0 6380 0

x

Chapter 12

Chapter 12

Additional and Advanced Exercises

685

Additional and Advanced Exercises

Applications 1. A frictionless particle P, starting from rest at time t = 0 at the point (a, 0, 0), slides down the helix rsud = sa cos udi + sa sin udj + buk

sa, b 7 0d

under the influence of gravity, as in the accompanying figure. The u in this equation is the cylindrical coordinate u and the helix is the curve r = a, z = bu, u Ú 0 , in cylindrical coordinates. We assume u to be a differentiable function of t for the motion. The law of conservation of energy tells us that the particle’s speed after it has fallen straight down a distance z is 22gz , where g is the constant acceleration of gravity. a. Find the angular velocity du>dt when u = 2p. b. Express the particle’s u- and z-coordinates as functions of t. c. Express the tangential and normal components of the velocity dr> dt and acceleration d 2r>dt 2 as functions of t. Does the acceleration have any nonzero component in the direction of the binormal vector B?

a r

x The helix r a, z b

P

y

Positive z-axis points down.

Motion in Polar and Cylindrical Coordinates 3. Deduce from the orbit equation s1 + edr0 1 + e cos u that a planet is closest to its sun when u = 0 and show that r = r0 at that time. r =

T 4. A Kepler equation The problem of locating a planet in its orbit at a given time and date eventually leads to solving “Kepler” equations of the form 1 sin x = 0 . 2 a. Show that this particular equation has a solution between x = 0 and x = 2 . ƒsxd = x - 1 -

b. With your computer or calculator in radian mode, use Newton’s method to find the solution to as many places as you can. 5. In Section 12.6, we found the velocity of a particle moving in the plane to be # # # # v = x i + y j = r ur + ru uu . # # # # a. Express x and y in terms of r and ru by evaluating the dot products v # i and v # j. # # # # b. Express r and r u in terms of x and y by evaluating the dot products v # ur and v # uu . 6. Express the curvature of a twice-differentiable curve r = ƒsud in the polar coordinate plane in terms of ƒ and its derivatives. 7. A slender rod through the origin of the polar coordinate plane rotates (in the plane) about the origin at the rate of 3 rad> min. A beetle starting from the point (2, 0) crawls along the rod toward the origin at the rate of 1 in.> min. a. Find the beetle’s acceleration and velocity in polar form when it is halfway to (1 in. from) the origin.

z

2. Suppose the curve in Exercise 1 is replaced by the conical helix r = au, z = bu shown in the accompanying figure. a. Express the angular velocity du>dt as a function of u . b. Express the distance the particle travels along the helix as a function of u .

T b. To the nearest tenth of an inch, what will be the length of the path the beetle has traveled by the time it reaches the origin? 8. Conservation of angular momentum Let r(t) denote the position in space of a moving object at time t. Suppose the force acting on the object at time t is c rstd , rstd ƒ ƒ3 where c is a constant. In physics the angular momentum of an object at time t is defined to be Lstd = rstd * mvstd , where m is the mass of the object and v(t) is the velocity. Prove that angular momentum is a conserved quantity; i.e., prove that L(t) is a constant vector, independent of time. Remember Newton’s law F = ma. (This is a calculus problem, not a physics problem.) Fstd = -

x y Conical helix r a, z b P Cone z ba r Positive z-axis points down. z

13 PARTIAL DERIVATIVES OVERVIEW Many functions depend on more than one independent variable. For instance, the volume of a right circular cylinder is a function V = pr 2h of its radius and its height, so it is a function V(r, h) of two variables r and h. In this chapter we extend the basic ideas of single-variable calculus to functions of several variables. Their derivatives are more varied and interesting because of the different ways the variables can interact. The applications of these derivatives are also more varied than for single-variable calculus, and in the next chapter we will see that the same is true for integrals involving several variables.

13.1

Functions of Several Variables In this section we define functions of more than one independent variable and discuss ways to graph them. Real-valued functions of several independent real variables are defined similarly to functions in the single-variable case. Points in the domain are ordered pairs (triples, quadruples, n-tuples) of real numbers, and values in the range are real numbers as we have worked with all along.

DEFINITIONS Suppose D is a set of n-tuples of real numbers sx1, x2 , Á , xn d. A real-valued function ƒ on D is a rule that assigns a unique (single) real number w = ƒsx1, x2 , Á , xn d to each element in D. The set D is the function’s domain. The set of w-values taken on by ƒ is the function’s range. The symbol w is the dependent variable of ƒ, and ƒ is said to be a function of the n independent variables x1 to xn. We also call the xj’s the function’s input variables and call w the function’s output variable.

If ƒ is a function of two independent variables, we usually call the independent variables x and y and the dependent variable z, and we picture the domain of ƒ as a region in the xy-plane (Figure 13.1). If ƒ is a function of three independent variables, we call the independent variables x, y, and z and the dependent variable w, and we picture the domain as a region in space. In applications, we tend to use letters that remind us of what the variables stand for. To say that the volume of a right circular cylinder is a function of its radius and height, we might write V = ƒsr, hd. To be more specific, we might replace the notation ƒ(r, h) by the formula that calculates the value of V from the values of r and h, and write V = pr 2h. In either case, r and h would be the independent variables and V the dependent variable of the function.

686

13.1

Functions of Several Variables

687

y f

f (a, b)

(x, y) 0

D

x

0

(a, b)

f (x, y)

z

FIGURE 13.1 An arrow diagram for the function z = ƒsx, yd.

As usual, we evaluate functions defined by formulas by substituting the values of the independent variables in the formula and calculating the corresponding value of the dependent variable. For example, the value of ƒsx, y, zd = 2x 2 + y 2 + z 2 at the point (3, 0, 4) is ƒs3, 0, 4d = 2s3d2 + s0d2 + s4d2 = 225 = 5.

Domains and Ranges In defining a function of more than one variable, we follow the usual practice of excluding inputs that lead to complex numbers or division by zero. If ƒsx, yd = 2y - x 2, y cannot be less than x 2. If ƒsx, yd = 1>sxyd, xy cannot be zero. The domain of a function is assumed to be the largest set for which the defining rule generates real numbers, unless the domain is otherwise specified explicitly. The range consists of the set of output values for the dependent variable.

EXAMPLE 1 (a) These are functions of two variables. Note the restrictions that may apply to their domains in order to obtain a real value for the dependent variable z. Function

Domain

Range

z = 2y - x 2

y Ú x2

[0, q d

1 z = xy

xy Z 0

s - q , 0d ´ s0, q d

z = sin xy

Entire plane

[-1, 1]

(b) These are functions of three variables with restrictions on some of their domains. Function

Domain

Range

w = 2x 2 + y 2 + z 2

Entire space

[0, q d

1 x2 + y2 + z2 w = xy ln z

sx, y, zd Z s0, 0, 0d

s0, q d

Half-space z 7 0

s - q, q d

w =

Functions of Two Variables Regions in the plane can have interior points and boundary points just like intervals on the real line. Closed intervals [a, b] include their boundary points, open intervals (a, b) don’t include their boundary points, and intervals such as [a, b) are neither open nor closed.

688

Chapter 13: Partial Derivatives

DEFINITIONS A point sx0 , y0 d in a region (set) R in the xy-plane is an interior point of R if it is the center of a disk of positive radius that lies entirely in R (Figure 13.2). A point sx0 , y0 d is a boundary point of R if every disk centered at sx0 , y0 d contains points that lie outside of R as well as points that lie in R. (The boundary point itself need not belong to R.) The interior points of a region, as a set, make up the interior of the region. The region’s boundary points make up its boundary. A region is open if it consists entirely of interior points. A region is closed if it contains all its boundary points (Figure 13.3).

(x0 , y0)

R

(a) Interior point

y

R

y

(x0 , y0)

x

0

0

y

x

0

x

(b) Boundary point

FIGURE 13.2 Interior points and boundary points of a plane region R. An interior point is necessarily a point of R. A boundary point of R need not belong to R.

{(x, y) x 2 y 2 1} Open unit disk. Every point an interior point.

FIGURE 13.3

{(x, y) x 2 y 2 1} Boundary of unit disk. (The unit circle.)

{(x, y) x 2 y 2 1} Closed unit disk. Contains all boundary points.

Interior points and boundary points of the unit disk in the plane.

As with a half-open interval of real numbers [a, b), some regions in the plane are neither open nor closed. If you start with the open disk in Figure 13.3 and add to it some of but not all its boundary points, the resulting set is neither open nor closed. The boundary points that are there keep the set from being open. The absence of the remaining boundary points keeps the set from being closed.

DEFINITIONS A region in the plane is bounded if it lies inside a disk of fixed radius. A region is unbounded if it is not bounded.

Examples of bounded sets in the plane include line segments, triangles, interiors of triangles, rectangles, circles, and disks. Examples of unbounded sets in the plane include lines, coordinate axes, the graphs of functions defined on infinite intervals, quadrants, half-planes, and the plane itself.

y Interior points, where y x 2 0

EXAMPLE 2

Describe the domain of the function ƒsx, yd = 2y - x 2.

Since ƒ is defined only where y - x 2 Ú 0, the domain is the closed, unbounded region shown in Figure 13.4. The parabola y = x 2 is the boundary of the domain. The points above the parabola make up the domain’s interior. Solution

Outside, y x2 0

The parabola y x2 0 is the boundary.

1

–1

0

1

x

FIGURE 13.4 The domain of ƒsx, yd in Example 2 consists of the shaded region and its bounding parabola.

Graphs, Level Curves, and Contours of Functions of Two Variables There are two standard ways to picture the values of a function ƒ(x, y). One is to draw and label curves in the domain on which ƒ has a constant value. The other is to sketch the surface z = ƒsx, yd in space.

13.1

Functions of Several Variables

689

DEFINITIONS The set of points in the plane where a function ƒ(x, y) has a constant value ƒsx, yd = c is called a level curve of ƒ. The set of all points (x, y, ƒ(x, y)) in space, for (x, y) in the domain of ƒ, is called the graph of ƒ. The graph of ƒ is also called the surface z f sx, yd.

z 100 f (x, y) 5 75

Graph ƒsx, yd = 100 - x 2 - y 2 and plot the level curves ƒsx, yd = 0, ƒsx, yd = 51, and ƒsx, yd = 75 in the domain of ƒ in the plane.

EXAMPLE 3

The surface z 5 f (x, y) 5 100 2 x 2 2 y 2 is the graph of f.

f (x, y) 5 51 (a typical level curve in the function’s domain) 10 10 x

y

f (x, y) 5 0

FIGURE 13.5 The graph and selected level curves of the function ƒsx, yd in Example 3.

Plane z 75

ƒsx, yd = 100 - x 2 - y 2 = 0,

or

x 2 + y 2 = 100,

which is the circle of radius 10 centered at the origin. Similarly, the level curves ƒsx, yd = 51 and ƒsx, yd = 75 (Figure 13.5) are the circles ƒsx, yd = 100 - x 2 - y 2 = 51,

or

x 2 + y 2 = 49

ƒsx, yd = 100 - x 2 - y 2 = 75,

or

x 2 + y 2 = 25.

The level curve ƒsx, yd = 100 consists of the origin alone. (It is still a level curve.) If x 2 + y 2 7 100, then the values of ƒsx, yd are negative. For example, the circle 2 x + y 2 = 144, which is the circle centered at the origin with radius 12, gives the constant value ƒsx, yd = -44 and is a level curve of ƒ.

The contour curve f (x, y) 100 x 2 y 2 75 is the circle x 2 y 2 25 in the plane z 75. z

The domain of ƒ is the entire xy-plane, and the range of ƒ is the set of real numbers less than or equal to 100. The graph is the paraboloid z = 100 - x 2 - y 2, the positive portion of which is shown in Figure 13.5. The level curve ƒsx, yd = 0 is the set of points in the xy-plane at which

Solution

z 100 x 2 y 2

100 75

The curve in space in which the plane z = c cuts a surface z = ƒsx, yd is made up of the points that represent the function value ƒsx, yd = c. It is called the contour curve ƒsx, yd = c to distinguish it from the level curve ƒsx, yd = c in the domain of ƒ. Figure 13.6 shows the contour curve ƒsx, yd = 75 on the surface z = 100 - x 2 - y 2 defined by the function ƒsx, yd = 100 - x 2 - y 2. The contour curve lies directly above the circle x 2 + y 2 = 25, which is the level curve ƒsx, yd = 75 in the function’s domain. Not everyone makes this distinction, however, and you may wish to call both kinds of curves by a single name and rely on context to convey which one you have in mind. On most maps, for example, the curves that represent constant elevation (height above sea level) are called contours, not level curves (Figure 13.7).

Functions of Three Variables 0 y x The level curve f (x, y) 100 x 2 y 2 75 is the circle x 2 y 2 25 in the xy-plane.

FIGURE 13.6 A plane z = c parallel to the xy-plane intersecting a surface z = ƒsx, yd produces a contour curve.

In the plane, the points where a function of two independent variables has a constant value ƒsx, yd = c make a curve in the function’s domain. In space, the points where a function of three independent variables has a constant value ƒsx, y, zd = c make a surface in the function’s domain.

DEFINITION The set of points (x, y, z) in space where a function of three independent variables has a constant value ƒsx, y, zd = c is called a level surface of ƒ. Since the graphs of functions of three variables consist of points (x, y, z, ƒ(x, y, z)) lying in a four-dimensional space, we cannot sketch them effectively in our three-dimensional frame of reference. We can see how the function behaves, however, by looking at its threedimensional level surfaces.

EXAMPLE 4

Describe the level surfaces of the function ƒsx, y, zd = 2x 2 + y 2 + z 2 .

690

Chapter 13: Partial Derivatives

x 2 y 2 z 2 1 z

x 2 y 2 z 2 2 x 2 y 2 z 2 3

1 y

2 3

FIGURE 13.7 Contours on Mt. Washington in New Hampshire. (Reproduced by permission from the Appalachian Mountain Club.)

x

FIGURE 13.8 The level surfaces of ƒsx, y, zd = 2x 2 + y 2 + z 2 are concentric spheres (Example 4).

z

(x 0 , y0 , z 0 )

y x

The value of ƒ is the distance from the origin to the point (x, y, z). Each level surface 2x 2 + y 2 + z 2 = c, c 7 0, is a sphere of radius c centered at the origin. Figure 13.8 shows a cutaway view of three of these spheres. The level surface 2x 2 + y 2 + z 2 = 0 consists of the origin alone. We are not graphing the function here; we are looking at level surfaces in the function’s domain. The level surfaces show how the function’s values change as we move through its domain. If we remain on a sphere of radius c centered at the origin, the function maintains a constant value, namely c. If we move from a point on one sphere to a point on another, the function’s value changes. It increases if we move away from the origin and decreases if we move toward the origin. The way the values change depends on the direction we take. The dependence of change on direction is important. We return to it in Section 13.5.

Solution

The definitions of interior, boundary, open, closed, bounded, and unbounded for regions in space are similar to those for regions in the plane. To accommodate the extra dimension, we use solid balls of positive radius instead of disks.

(a) Interior point (x 0 , y0 , z 0 )

z

y x

(b) Boundary point

FIGURE 13.9 Interior points and boundary points of a region in space. As with regions in the plane, a boundary point need not belong to the space region R.

DEFINITIONS A point sx0 , y0 , z0 d in a region R in space is an interior point of R if it is the center of a solid ball that lies entirely in R (Figure 13.9a). A point sx0 , y0 , z0 d is a boundary point of R if every solid ball centered at sx0 , y0 , z0 d contains points that lie outside of R as well as points that lie inside R (Figure 13.9b). The interior of R is the set of interior points of R. The boundary of R is the set of boundary points of R. A region is open if it consists entirely of interior points. A region is closed if it contains its entire boundary.

Examples of open sets in space include the interior of a sphere, the open half-space z 7 0, the first octant (where x, y, and z are all positive), and space itself. Examples of closed sets in space include lines, planes, and the closed half-space z Ú 0. A solid sphere

13.1

691

Functions of Several Variables

with part of its boundary removed or a solid cube with a missing face, edge, or corner point is neither open nor closed. Functions of more than three independent variables are also important. For example, the temperature on a surface in space may depend not only on the location of the point P(x, y, z) on the surface but also on the time t when it is visited, so we would write T = ƒsx, y, z, td.

Computer Graphing Three-dimensional graphing programs for computers and calculators make it possible to graph functions of two variables with only a few keystrokes. We can often get information more quickly from a graph than from a formula. w

x

25

EXAMPLE 5

The temperature w beneath the Earth’s surface is a function of the depth x beneath the surface and the time t of the year. If we measure x in feet and t as the number of days elapsed from the expected date of the yearly highest surface temperature, we can model the variation in temperature with the function

15

w = cos s1.7 * 10 -2t - 0.2xde -0.2x. t

FIGURE 13.10 This graph shows the seasonal variation of the temperature below ground as a fraction of surface temperature (Example 5).

(The temperature at 0 ft is scaled to vary from +1 to -1, so that the variation at x feet can be interpreted as a fraction of the variation at the surface.) Figure 13.10 shows a graph of the function. At a depth of 15 ft, the variation (change in vertical amplitude in the figure) is about 5% of the surface variation. At 25 ft, there is almost no variation during the year. The graph also shows that the temperature 15 ft below the surface is about half a year out of phase with the surface temperature. When the temperature is lowest on the surface (late January, say), it is at its highest 15 ft below. Fifteen feet below the ground, the seasons are reversed. Figure 13.11 shows computer-generated graphs of a number of functions of two variables together with their level curves.

z z

z

y x

y

x

y

x

y y y x

x

x

(a) z = sin x + 2 sin y

FIGURE 13.11

(b) z = (4x2 + y2)e-x

2

- y2

Computer-generated graphs and level curves of typical functions of two variables.

(c) z = xye-y

2

692

Chapter 13: Partial Derivatives

Exercises 13.1 21. ƒsx, yd = xy

Domain, Range, and Level Curves In Exercises 1–4, find the specific function values. a. ƒs0, 0d

b. ƒs - 1, 1d

c. ƒs2, 3d

d. ƒs -3, -2d

2. ƒsx, yd = sin sxyd a. ƒ a2,

b. ƒ a -3,

p b 6

p b 12

216 - x 2 - y 2

24. ƒsx, yd = 29 - x 2 - y 2

25. ƒsx, yd = ln sx 2 + y 2 d

26. ƒsx, yd = e -sx

27. ƒsx, yd = sin-1 s y - xd

y 28. ƒsx, yd = tan-1 a x b

2

+ y 2d

29. ƒsx, yd = ln sx 2 + y 2 - 1d 30. ƒsx, yd = ln s9 - x 2 - y 2 d

p d. ƒ a - , -7 b 2

1 c. ƒ ap, b 4 3. ƒsx, y, zd =

1

23. ƒsx, yd =

1. ƒsx, yd = x 2 + xy 3

22. ƒsx, yd = y>x 2

x - y

Matching Surfaces with Level Curves Exercises 31–36 show level curves for the functions graphed in (a)–(f ) on the following page. Match each set of curves with the appropriate function.

y2 + z2

a. ƒs3, -1, 2d

1 1 b. ƒ a1, , - b 2 4

1 c. ƒ a0, - , 0b 3

d. ƒs2, 2, 100d

31.

32. y

y

4. ƒsx, y, zd = 249 - x 2 - y 2 - z 2 a. ƒs0, 0, 0d

b. ƒs2, -3, 6d

c. ƒs -1, 2, 3d

d. ƒ a

4

,

5

,

6

22 22 22

b x

x

In Exercises 5–12, find and sketch the domain for each function. 5. ƒsx, yd = 2y - x - 2 6. ƒsx, yd = ln sx 2 + y 2 - 4d sx - 1dsy + 2d 7. ƒsx, yd = sy - xdsy - x 3 d sin sxyd 8. ƒsx, yd = 2 x + y 2 - 25 9. ƒsx, yd = cos-1 sy - x 2 d

33.

34. y

y

10. ƒsx, yd = ln sxy + x - y - 1d 11. ƒsx, yd = 2sx 2 - 4dsy 2 - 9d 12. ƒsx, yd =

x

1 ln s4 - x 2 - y 2 d

x

In Exercises 13–16, find and sketch the level curves ƒsx, yd = c on the same set of coordinate axes for the given values of c. We refer to these level curves as a contour map. 13. ƒsx, yd = x + y - 1, 14. ƒsx, yd = x 2 + y 2, 15. ƒsx, yd = xy,

c = -3, -2, -1, 0, 1, 2, 3

c = 0, 1, 4, 9, 16, 25

c = -9, -4, -1, 0, 1, 4, 9

16. ƒsx, yd = 225 - x 2 - y 2 ,

19. ƒsx, yd = 4x + 9y 2

18. ƒsx, yd = 2y - x 2

36. y

y

c = 0, 1, 2, 3, 4

In Exercises 17–30, (a) find the function’s domain, (b) find the function’s range, (c) describe the function’s level curves, (d) find the boundary of the function’s domain, (e) determine if the domain is an open region, a closed region, or neither, and (f) decide if the domain is bounded or unbounded. 17. ƒsx, yd = y - x

35.

20. ƒsx, yd = x 2 - y 2

x

x

13.1 a.

f.

z

Functions of Several Variables

693

z

y x

b. z–

y

x

z (cos x)(cos y) e –x 2 y 2 /4 z

xy 2 x2 y2

z y2 y4 x2 x

y

c.

Functions of Two Variables Display the values of the functions in Exercises 37–48 in two ways: (a) by sketching the surface z = ƒsx, yd and (b) by drawing an assortment of level curves in the function’s domain. Label each level curve with its function value.

z

37. ƒsx, yd = y 2

38. ƒsx, yd = 2x

39. ƒsx, yd = x + y 2

40. ƒsx, yd = 2x 2 + y 2

2

41. ƒsx, yd = x - y

42. ƒsx, yd = 4 - x 2 - y 2

43. ƒsx, yd = 4x 2 + y 2

44. ƒsx, yd = 6 - 2x - 3y

2

45. ƒsx, yd = 1 - ƒ y ƒ x

y z

47. ƒsx, yd = 2x + y + 4

1 4x 2 y 2

48. ƒsx, yd = 2x 2 + y 2 - 4

2

Finding Level Curves In Exercises 49–52, find an equation for and sketch the graph of the level curve of the function ƒ(x, y) that passes through the given point.

z

d.

46. ƒsx, yd = 1 - ƒ x ƒ - ƒ y ƒ

2

49. ƒsx, yd = 16 - x 2 - y 2, 50. ƒsx, yd = 2x - 1, 2

A 222, 22 B

s1, 0d

51. ƒsx, yd = 2x + y - 3 , s3, -1d 2y - x , s -1, 1d 52. ƒsx, yd = x + y + 1 2

y x

z

e–y

Sketching Level Surfaces In Exercises 53–60, sketch a typical level surface for the function.

cos x

z

e.

53. ƒsx, y, zd = x 2 + y 2 + z 2

54. ƒsx, y, zd = ln sx 2 + y 2 + z 2 d

55. ƒsx, y, zd = x + z

56. ƒsx, y, zd = z

57. ƒsx, y, zd = x + y 2

58. ƒsx, y, zd = y 2 + z 2

2

59. ƒsx, y, zd = z - x 2 - y 2

60. ƒsx, y, zd = sx 2>25d + s y 2>16d + sz 2>9d y x

z

xy(x 2 y 2 ) x2 y2

Finding Level Surfaces In Exercises 61–64, find an equation for the level surface of the function through the given point. 61. ƒsx, y, zd = 2x - y - ln z, 62. ƒsx, y, zd = ln sx + y + z d, 2

2

s3, -1, 1d s -1, 2, 1d

694

Chapter 13: Partial Derivatives

63. gsx, y, zd = 2x 2 + y 2 + z 2 , A 1, -1, 22 B x - y + z , s1, 0, -2d 64. gsx, y, zd = 2x + y - z In Exercises 65–68, find and sketch the domain of ƒ. Then find an equation for the level curve or surface of the function passing through the given point. q

n=0 q

s1, 2d

sx + yd , n!z n n=0 y

67. ƒsx, yd =

0 … x … 2p,

Use a CAS to plot the implicitly defined level surfaces in Exercises 73–76. 74. x 2 + z 2 = 1

x 76. sin a b - scos yd2x 2 + z 2 = 2 2

sln 4, ln 9, 2d

du

, s0, 1d Lx 21 - u2 y z du dt + , 68. gsx, y, zd = 2 Lx 1 + t L0 24 - u2

A 0, 1, 23 B

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for each of the functions in Exercises 69–72. a. Plot the surface over the given rectangle. b. Plot several level curves in the rectangle. c. Plot the level curve of ƒ through the given point. y 69. ƒsx, yd = x sin + y sin 2x, 0 … x … 5p, 0 … y … 5p, 2 Ps3p, 3pd 70. ƒsx, yd = ssin xdscos yde 2x 0 … y … 5p, Ps4p, 4pd

2

13.2

0.1

75. x + y 2 - 3z 2 = 1

n

66. gsx, y, zd = a

72. ƒsx, yd = e sx - yd sin sx 2 + y 2 d, -2p … y … p, Psp, -pd

73. 4 ln sx 2 + y 2 + z 2 d = 1

n

x 65. ƒsx, yd = a a y b ,

71. ƒsx, yd = sin sx + 2 cos yd, -2p … x … 2p, -2p … y … 2p, Psp, pd

+ y2>8

,

0 … x … 5p,

Parametrized Surfaces Just as you describe curves in the plane parametrically with a pair of equations x = ƒstd, y = gstd defined on some parameter interval I, you can sometimes describe surfaces in space with a triple of equations x = ƒsu, yd, y = gsu, yd, z = hsu, yd defined on some parameter rectangle a … u … b, c … y … d. Many computer algebra systems permit you to plot such surfaces in parametric mode. (Parametrized surfaces are discussed in detail in Section 15.5.) Use a CAS to plot the surfaces in Exercises 77–80. Also plot several level curves in the xy-plane. 77. x = u cos y, y = u sin y, 0 … y … 2p

z = u,

0 … u … 2,

78. x = u cos y, y = u sin y, 0 … y … 2p

z = y,

0 … u … 2,

79. x = s2 + cos ud cos y, y = s2 + cos ud sin y, 0 … u … 2p, 0 … y … 2p 80. x = 2 cos u cos y, y = 2 cos u sin y, 0 … u … 2p, 0 … y … p

z = sin u,

z = 2 sin u,

Limits and Continuity in Higher Dimensions This section treats limits and continuity for multivariable functions. These ideas are analogous to limits and continuity for single-variable functions, but including more independent variables leads to additional complexity and important differences requiring some new ideas.

Limits for Functions of Two Variables If the values of ƒ(x, y) lie arbitrarily close to a fixed real number L for all points (x, y) sufficiently close to a point sx0 , y0 d, we say that ƒ approaches the limit L as (x, y) approaches sx0 , y0 d. This is similar to the informal definition for the limit of a function of a single variable. Notice, however, that if sx0 , y0 d lies in the interior of ƒ’s domain, (x, y) can approach sx0 , y0 d from any direction. For the limit to exist, the same limiting value must be obtained whatever direction of approach is taken. We illustrate this issue in several examples following the definition.

13.2

Limits and Continuity in Higher Dimensions

695

DEFINITION We say that a function ƒ(x, y) approaches the limit L as (x, y) approaches sx0 , y0 d, and write lim

sx, yd:sx0, y0d

ƒsx, yd = L

if, for every number P 7 0, there exists a corresponding number d 7 0 such that for all (x, y) in the domain of ƒ, ƒ ƒsx, yd - L ƒ 6 P

0 6 2sx - x0 d2 + s y - y0 d2 6 d.

whenever

The definition of limit says that the distance between ƒ(x, y) and L becomes arbitrarily small whenever the distance from (x, y) to sx0 , y0 d is made sufficiently small (but not 0). The definition applies to interior points sx0 , y0 d as well as boundary points of the domain of ƒ, although a boundary point need not lie within the domain. The points (x, y) that approach sx0 , y0 d are always taken to be in the domain of ƒ. See Figure 13.12. y f (x, y)

D

δ (x 0 , y0 ) 0

x

0

(

Le

)

L

Le

z

FIGURE 13.12 In the limit definition, d is the radius of a disk centered at (x0, y0). For all points sx, yd within this disk, the function values ƒsx, yd lie inside the corresponding interval sL - P, L + Pd.

As for functions of a single variable, it can be shown that lim

x = x0

lim

y = y0

lim

k = k

sx, yd:sx0, y0d sx, yd:sx0, y0d sx, yd:sx0, y0d

sany number kd.

For example, in the first limit statement above, ƒsx, yd = x and L = x0. Using the definition of limit, suppose that P 7 0 is chosen. If we let d equal this P, we see that 0 6 2sx - x0 d2 + sy - y0 d2 6 d = P implies 2sx - x0 d2 6 P

sx - x0 d2 … sx - x0 d2 + sy - y0 d2

ƒ x - x0 ƒ 6 P

2a 2 = ƒ a ƒ

ƒ ƒsx, yd - x0 ƒ 6 P

x = ƒsx, yd

That is, ƒ ƒsx, yd - x0 ƒ 6 P

whenever

0 6 2sx - x0 d2 + s y - y0 d2 6 d.

696

Chapter 13: Partial Derivatives

So a d has been found satisfying the requirement of the definition, and lim ƒsx, yd = lim x = x0. sx, yd:sx0 , y0d

sx, yd:sx0 , y0d

As with single-variable functions, the limit of the sum of two functions is the sum of their limits (when they both exist), with similar results for the limits of the differences, constant multiples, products, quotients, powers, and roots.

THEOREM 1—Properties of Limits of Functions of Two Variables lowing rules hold if L, M, and k are real numbers and lim

sx, yd:sx0, y0d

ƒsx, yd = L

1. Sum Rule:

and

lim

sx, yd:sx0 , y0d

The fol-

gsx, yd = M.

lim

(ƒsx, yd + gsx, ydd = L + M

lim

(ƒsx, yd - gsx, ydd = L - M

lim

kƒsx, yd = kL

lim

sƒsx, yd # gsx, ydd = L # M

sx, yd:sx0 , y0d

2. Difference Rule:

sx, yd:sx0 , y0d

3. Constant Multiple Rule:

sx, yd:sx0 , y0d

4. Product Rule:

sx, yd:sx0, y0d

lim

5. Quotient Rule:

sx, yd:sx0 , y0d

6. Power Rule:

ƒsx, yd L , = M gsx, yd

sany number kd

M Z 0

lim

[ƒsx, yd] n = L n, n a positive integer

lim

n n 2ƒsx, yd = 2L = L1>n ,

sx, yd:sx0 , y0d

7. Root Rule:

sx, yd:sx0 , y0d

n a positive integer, and if n is even, we assume that L 7 0.

While we won’t prove Theorem 1 here, we give an informal discussion of why it’s true. If (x, y) is sufficiently close to sx0 , y0 d, then ƒ(x, y) is close to L and g(x, y) is close to M (from the informal interpretation of limits). It is then reasonable that ƒsx, yd + gsx, yd is close to L + M; ƒsx, yd - gsx, yd is close to L - M; kƒ(x, y) is close to kL; ƒsx, ydgsx, yd is close to LM; and ƒ(x, y)> g(x, y) is close to L> M if M Z 0. When we apply Theorem 1 to polynomials and rational functions, we obtain the useful result that the limits of these functions as sx, yd : sx0 , y0 d can be calculated by evaluating the functions at sx0 , y0 d. The only requirement is that the rational functions be defined at sx0 , y0 d.

EXAMPLE 1

In this example, we can combine the three simple results following the limit definition with the results in Theorem 1 to calculate the limits. We simply substitute the x- and y-values of the point being approached into the functional expression to find the limiting value. (a) (b)

lim

sx, yd:s0,1d

lim

x - xy + 3 x y + 5xy - y

sx, yd:s3, -4d

EXAMPLE 2

2

3

=

0 - s0ds1d + 3 = -3 s0d s1d + 5s0ds1d - s1d3 2

2x 2 + y 2 = 2s3d2 + s -4d2 = 225 = 5

Find lim

sx, yd:s0,0d

x 2 - xy 2x - 2y

.

13.2

Limits and Continuity in Higher Dimensions

697

Since the denominator 2x - 2y approaches 0 as sx, yd : s0, 0d, we cannot use the Quotient Rule from Theorem 1. If we multiply numerator and denominator by 2x + 2y, however, we produce an equivalent fraction whose limit we can find:

Solution

x 2 - xy

lim

sx, yd:s0,0d

2x - 2y

=

= =

A x 2 - xy B A 2x + 2y B

lim

sx, yd:s0,0d

A 2x - 2y B A 2x + 2y B

x A x - y B A 2x + 2y B x - y sx, yd:s0,0d lim

lim

sx, yd:s0,0d

Algebra

x A 2x + 2y B

Cancel the nonzero factor sx - yd.

= 0 A 20 + 20 B = 0

Known limit values

We can cancel the factor sx - yd because the path y = x (along which x - y = 0) is not in the domain of the function x 2 - xy 2x - 2y

EXAMPLE 3

Find

4xy 2

lim

sx, yd:s0,0d

x2 + y2

.

if it exists.

We first observe that along the line x = 0, the function always has value 0 when y Z 0. Likewise, along the line y = 0, the function has value 0 provided x Z 0. So if the limit does exist as (x, y) approaches (0, 0), the value of the limit must be 0. To see if this is true, we apply the definition of limit. Let P 7 0 be given, but arbitrary. We want to find a d 7 0 such that Solution

`

4xy 2 x2 + y2

- 0` 6 P

0 6 2x 2 + y 2 6 d

whenever

or 4ƒ x ƒ y 2 x2 + y2

6 P

whenever

0 6 2x 2 + y 2 6 d.

Since y 2 … x 2 + y 2 we have that 4ƒ x ƒ y 2 x + y 2

2

… 4 ƒ x ƒ = 42x 2 … 42x 2 + y 2 .

y2 x + y2 2

… 1

So if we choose d = P>4 and let 0 6 2x 2 + y 2 6 d, we get

`

4xy 2 x2 + y2

P - 0 ` … 42x 2 + y 2 6 4d = 4 a b = P. 4

It follows from the definition that lim

sx, yd:s0,0d

4xy 2 x + y2 2

= 0.

698

Chapter 13: Partial Derivatives

EXAMPLE 4

y If ƒsx, yd = x ,

does

lim

(x, y):(0, 0)

ƒsx, yd exist?

Solution The domain of ƒ does not include the y-axis, so we do not consider any points sx, yd where x = 0 in the approach toward the origin s0, 0d. Along the x-axis, the value of the function is ƒsx, 0d = 0 for all x Z 0. So if the limit does exist as sx, yd : s0, 0d, the value of the limit must be L = 0. On the other hand, along the line y = x, the value of the function is ƒsx, xd = x>x = 1 for all x Z 0. That is, the function ƒ approaches the value 1 along the line y = x. This means that for every disk of radius d centered at s0, 0d, the disk will contain points sx, 0d on the x-axis where the value of the function is 0, and also points sx, xd along the line y = x where the value of the function is 1. So no matter how small we choose d as the radius of the disk in Figure 13.12, there will be points within the disk for which the function values differ by 1. Therefore, the limit cannot exist because we can take P to be any number less than 1 in the limit definition and deny that L = 0 or 1, or any other real number. The limit does not exist because we have different limiting values along different paths approaching the point s0, 0d.

Continuity As with functions of a single variable, continuity is defined in terms of limits. z

DEFINITION

A function ƒ(x, y) is continuous at the point (x 0, y0) if

1. ƒ is defined at sx0 , y0 d, 2. lim ƒsx, yd exists, sx, yd:sx0, y0d

x –y

3.

lim

sx, yd:sx0, y0d

ƒsx, yd = ƒsx0 , y0 d.

A function is continuous if it is continuous at every point of its domain. (a)

y 0

–0.8 –1

0.8 1 0.8

–0.8

0 0.8

x

–0.8 –1

1 0.8 0

–0.8

As with the definition of limit, the definition of continuity applies at boundary points as well as interior points of the domain of ƒ. The only requirement is that each point (x, y) near sx0, y0 d be in the domain of ƒ. A consequence of Theorem 1 is that algebraic combinations of continuous functions are continuous at every point at which all the functions involved are defined. This means that sums, differences, constant multiples, products, quotients, and powers of continuous functions are continuous where defined. In particular, polynomials and rational functions of two variables are continuous at every point at which they are defined.

EXAMPLE 5

Show that

(b)

2xy

FIGURE 13.13 (a) The graph of 2xy ƒsx, yd =

x + y2 L 0, 2

,

ƒsx, yd =

sx, yd Z s0, 0d sx, yd = s0, 0).

The function is continuous at every point except the origin. (b) The values of ƒ are different constants along each line y = mx, x Z 0 (Example 5).

x + y2 L 0, 2

,

sx, yd Z s0, 0d sx, yd = s0, 0d

is continuous at every point except the origin (Figure 13.13). The function ƒ is continuous at any point sx, yd Z s0, 0d because its values are then given by a rational function of x and y and the limiting value is obtained by substituting the values of x and y into the functional expression. Solution

13.2

Limits and Continuity in Higher Dimensions

699

At (0, 0), the value of ƒ is defined, but ƒ, we claim, has no limit as sx, yd : s0, 0d. The reason is that different paths of approach to the origin can lead to different results, as we now see. For every value of m, the function ƒ has a constant value on the “punctured” line y = mx, x Z 0, because ƒsx, yd `

y = mx

=

2xy x + y 2

2

`

y = mx

=

2xsmxd 2mx 2 2m = 2 = . 2 x + smxd x + m 2x 2 1 + m2 2

Therefore, ƒ has this number as its limit as (x, y) approaches (0, 0) along the line: lim

sx, yd:s0,0d along y = mx

ƒsx, yd =

lim

sx, yd:s0,0d

cƒsx, yd `

y = mx

d =

2m . 1 + m2

This limit changes with each value of the slope m. There is therefore no single number we may call the limit of ƒ as (x, y) approaches the origin. The limit fails to exist, and the function is not continuous. Examples 4 and 5 illustrate an important point about limits of functions of two or more variables. For a limit to exist at a point, the limit must be the same along every approach path. This result is analogous to the single-variable case where both the left- and right-sided limits had to have the same value. For functions of two or more variables, if we ever find paths with different limits, we know the function has no limit at the point they approach. z

Two-Path Test for Nonexistence of a Limit If a function ƒ(x, y) has different limits along two different paths in the domain of ƒ as (x, y) approaches sx0 , y0 d, then limsx, yd:sx0, y0d ƒsx, yd does not exist.

EXAMPLE 6

Show that the function ƒsx, yd =

y

x

2x 2 y x4 + y 2

(Figure 13.14) has no limit as (x, y) approaches (0, 0). The limit cannot be found by direct substitution, which gives the indeterminate form 0> 0. We examine the values of ƒ along curves that end at (0, 0). Along the curve y = kx 2, x Z 0, the function has the constant value Solution

(a) y 0 1

ƒsx, yd ` 1

y = kx 2

=

2x 2y x4 + y 2

`

y = kx 2

=

2x 2skx 2 d 2kx 4 2k = 4 = . 4 2 2 x + skx d x + k 2x 4 1 + k2

Therefore, 0

0

x

lim

sx, yd:s0,0d along y = kx2

ƒsx, yd =

lim

sx, yd:s0,0d

cƒsx, yd `

y = k x2

d =

2k . 1 + k2

0

This limit varies with the path of approach. If (x, y) approaches (0, 0) along the parabola y = x 2, for instance, k = 1 and the limit is 1. If (x, y) approaches (0, 0) along the x-axis, k = 0 and the limit is 0. By the two-path test, ƒ has no limit as (x, y) approaches (0, 0).

(b)

It can be shown that the function in Example 6 has limit 0 along every path y = mx (Exercise 53). We conclude that

–1

–1

FIGURE 13.14 (a) The graph of ƒsx, yd = 2x 2y>sx 4 + y 2 d. (b) Along each path y = kx 2 the value of ƒ is constant, but varies with k (Example 6).

Having the same limit along all straight lines approaching (x0, y0) does not imply a limit exists at (x0, y0).

700

Chapter 13: Partial Derivatives

Whenever it is correctly defined, the composite of continuous functions is also continuous. The only requirement is that each function be continuous where it is applied. The proof, omitted here, is similar to that for functions of a single variable (Theorem 9 in Section 2.5).

Continuity of Composites If ƒ is continuous at sx0 , y0 d and g is a single-variable function continuous at ƒsx0 , y0 d, then the composite function h = g ⴰ f defined by hsx, yd = gsƒsx, ydd is continuous at sx0 , y0 d.

For example, the composite functions e x - y,

cos

xy x2 + 1

ln s1 + x 2y 2 d

,

are continuous at every point (x, y).

Functions of More Than Two Variables The definitions of limit and continuity for functions of two variables and the conclusions about limits and continuity for sums, products, quotients, powers, and composites all extend to functions of three or more variables. Functions like ln sx + y + zd

y sin z x - 1

and

are continuous throughout their domains, and limits like e x+z

lim

P:s1,0,-1d

z 2 + cos 2xy

=

e1-1 1 = , 2 s -1d + cos 0 2

where P denotes the point (x, y, z), may be found by direct substitution.

Extreme Values of Continuous Functions on Closed, Bounded Sets The Extreme Value Theorem (Theorem 1, Section 4.1) states that a function of a single variable that is continuous throughout a closed, bounded interval [a, b] takes on an absolute maximum value and an absolute minimum value at least once in [a, b]. The same holds true of a function z = ƒsx, yd that is continuous on a closed, bounded set R in the plane (like a line segment, a disk, or a filled-in triangle). The function takes on an absolute maximum value at some point in R and an absolute minimum value at some point in R. Similar results hold for functions of three or more variables. A continuous function w = ƒsx, y, zd, for example, must take on absolute maximum and minimum values on any closed, bounded set (solid ball or cube, spherical shell, rectangular solid) on which it is defined. We will learn how to find these extreme values in Section 13.7.

Exercises 13.2 Limits with Two Variables Find the limits in Exercises 1–12. 3x 2 - y 2 + 5 1. 2. lim sx, yd:s0,0d x 2 + y 2 + 2

3. lim

sx, yd: s0,4d

lim

sx, yd: s3,4d

2x 2 + y 2 - 1

4.

sec x tan y

6.

lim

sx, yd: s2, -3d

1 1 ax + y b

x 2y

5.

lim

sx, yd: s0,p>4d

lim

sx, yd: s0,0d

cos

2

x2 + y3 x + y + 1

13.2

7.

lim

sx, yd:s0,ln 2d

e x-y

8.

lim

sx, yd: s1,1d

ln ƒ 1 + x 2 y 2 ƒ

y

9. 11.

lim

sx, yd:s0,0d

e sin x x

10.

x sin y

lim

12.

x2 + 1

sx, yd:s1, p>6d

lim

sx, yd: s1>27, p3d

lim

sx, yd: sp>2,0d

3 cos2 xy

cos y + 1 y - sin x

Limits of Quotients Find the limits in Exercises 13–24 by rewriting the fractions first. x 2 - 2xy + y 2 x2 - y2 lim lim 13. 14. x - y sx, yd:s1,1d sx, yd: s1,1d x - y xZy

xZy

xy - y - 2x + 2 lim 15. x - 1 sx, yd:s1,1d xZ1

16.

y Z -4, x Z x

17.

18.

20.

21.

23.

y + 4

lim

sx, yd:s2, -4d 2

x - y + 2 2x - 2 2y

lim

2x - 2y x + y - 4

lim

sx, yd:s2,2d x+yZ4

lim

sx, yd:s4,3d xZy+1

2x + y - 2

lim

lim

sx, yd: s2,0d 2x - y Z 4

22x - y - 2 2x - y - 4

x2 + y2

x3 + y3 sx, yd:s1,-1d x + y lim

Continuity in Space At what points (x, y, z) in space are the functions in Exercises 35–40 continuous? 35. a. ƒsx, y, zd = x 2 + y 2 - 2z 2 b. ƒsx, y, zd = 2x 2 + y 2 - 1 36. a. ƒsx, y, zd = ln xyz

b. ƒsx, y, zd = e x + y cos z

1 37. a. hsx, y, zd = xy sin z

b. hsx, y, zd =

1 1 b. hsx, y, zd = ƒyƒ + ƒzƒ ƒ xy ƒ + ƒ z ƒ 39. a. hsx, y, zd = ln sz - x 2 - y 2 - 1d 1 b. hsx, y, zd = z - 2x 2 + y 2

No Limit at a Point By considering different paths of approach, show that the functions in Exercises 41–48 have no limit as sx, yd : s0, 0d.

lim

x 2x + y 2

2

42. ƒsx, yd =

z

x - y

sx, yd: s2,2d

x

x4 - y4

x

y

Limits with Three Variables Find the limits in Exercises 25–30. 25. 27. 28. 30.

lim

1 1 1 ax + y + z b

lim

ssin2 x + cos2 y + sec2 zd

P:s1,3,4d

P:sp,p,0d

lim

P:s-1>4,p>2,2d

lim

P:s2, -3,6d

tan-1 xyz

26.

29.

2xy + yz

lim

P: s1,-1,-1d

lim

P: sp,0,3d

x + z 2

2

ze -2y cos 2x

ln 2x 2 + y 2 + z 2

Continuity in the Plane At what points (x, y) in the plane are the functions in Exercises 31–34 continuous? 31. a. ƒsx, yd = sin sx + yd

b. ƒsx, yd = ln sx 2 + y 2 d

x + y 32. a. ƒsx, yd = x - y

b. ƒsx, yd =

1 33. a. g sx, yd = sin xy

b. g sx, yd =

x2 + y2

x4 x + y2 4

z

1 - cos sxyd lim 22. xy sx, yd: s0,0d 24.

1 x2 + z2 - 1

38. a. hsx, y, zd =

41. ƒsx, yd = -

2x - 2y + 1 x - y - 1 sin sx 2 + y 2 d

sx, yd:s0,0d

19.

701

40. a. hsx, y, zd = 24 - x 2 - y 2 - z 2 1 b. hsx, y, zd = 2 4 - 2x + y 2 + z 2 - 9

x y - xy + 4x 2 - 4x 2

sx, yd:s0,0d xZy

Limits and Continuity in Higher Dimensions

y x + 1 2

x + y 2 + cos x

1 34. a. g sx, yd = 2 b. g sx, yd = 2 x - 3x + 2 x - y

43. ƒsx, yd =

x4 - y 2

x + y x - y 45. gsx, yd = x + y 47. hsx, yd =

4

2

x2 + y y

44. ƒsx, yd =

y

xy xy ƒ ƒ

x2 - y 46. g sx, yd = x - y 48. hsx, yd =

x 2y x + y2 4

Theory and Examples In Exercises 49 and 50, show that the limits do not exist. xy 2 - 1 xy + 1 lim lim 49. 50. (x, y):(1,1) y - 1 (x, y):(1, -1) x 2 - y 2 1, 51. Let ƒsx, yd = • 1, 0,

y Ú x4 y … 0 otherwise.

Find each of the following limits, or explain that the limit does not exist. a. b. c.

lim

ƒsx, yd

lim

ƒsx, yd

lim

ƒsx, yd

(x, y):(0,1) (x, y):(2,3) (x, y):(0,0)

702

Chapter 13: Partial Derivatives

52. Let ƒsx, yd = e

x Ú 0 . x 6 0

x 2, x 3,

59. (Continuation of Example 5.) a. Reread Example 5. Then substitute m = tan u into the formula

Find the following limits. a. b. c.

lim

ƒsx, yd

lim

ƒsx, yd

(x, y): (3, -2) (x, y): (-2, 1)

lim

(x, y): (0, 0)

ƒsx, yd `

=

2m 1 + m2

and simplify the result to show how the value of ƒ varies with the line’s angle of inclination.

ƒsx, yd

53. Show that the function in Example 6 has limit 0 along every straight line approaching (0, 0). 54. If ƒsx0 , y0 d = 3, what can you say about lim

sx, yd: sx0 , y0d

ƒsx, yd

if ƒ is continuous at sx0 , y0 d? If ƒ is not continuous at sx0 , y0 d? Give reasons for your answers. The Sandwich Theorem for functions of two variables states that if gsx, yd … ƒsx, yd … hsx, yd for all sx, yd Z sx0 , y0 d in a disk centered at sx0 , y0 d and if g and h have the same finite limit L as sx, yd : sx0 , y0 d, then lim

sx, yd:sx0 , y0d

ƒsx, yd = L.

Use this result to support your answers to the questions in Exercises 55–58.

b. Use the formula you obtained in part (a) to show that the limit of ƒ as sx, yd : s0, 0d along the line y = mx varies from -1 to 1 depending on the angle of approach. 60. Continuous extension Define ƒ(0, 0) in a way that extends x2 - y2 ƒsx, yd = xy 2 x + y2 to be continuous at the origin. Changing to Polar Coordinates If you cannot make any headway with limsx, yd:s0,0d ƒsx, yd in rectangular coordinates, try changing to polar coordinates. Substitute x = r cos u, y = r sin u, and investigate the limit of the resulting expression as r : 0. In other words, try to decide whether there exists a number L satisfying the following criterion: Given P 7 0, there exists a d 7 0 such that for all r and u, ƒrƒ 6 d

x 2y 2 tan-1 xy 1 6 1 6 xy 3

lim

sx, yd: s0,0d

lim

tan-1 xy xy ? sx, yd: s0,0d Give reasons for your answer.

sx, yd: s0,0d

ƒsx, yd = lim ƒsr cos u, r sin ud = L.

tell you anything about 4 - 4 cos 2ƒ xy ƒ lim ? sx, yd:s0,0d ƒ xy ƒ Give reasons for your answer. 57. Does knowing that ƒ sin s1>xd ƒ … 1 tell you anything about lim

sx, yd: s0,0d

1 y sin x ?

Give reasons for your answer. 58. Does knowing that ƒ cos s1>yd ƒ … 1 tell you anything about lim

sx, yd:s0,0d

Give reasons for your answer.

1 x cos y ?

r: 0

r 3 cos3 u x3 = lim = lim r cos3 u = 0. 2 r: 0 r: 0 x + y r2 2

ƒrƒ 6 d

x y 6 4 - 4 cos 2ƒ xy ƒ 6 2 ƒ xy ƒ 6

(1)

To verify the last of these equalities, we need to show that Equation (1) is satisfied with ƒsr, ud = r cos3 u and L = 0. That is, we need to show that given any P 7 0, there exists a d 7 0 such that for all r and u,

56. Does knowing that 2 2

ƒ ƒsr, ud - L ƒ 6 P.

For instance,

tell you anything about lim

Q

If such an L exists, then

55. Does knowing that

2 ƒ xy ƒ -

y = mx

Since

Q

ƒ r cos3 u - 0 ƒ 6 P.

ƒ r cos3 u ƒ = ƒ r ƒ ƒ cos3 u ƒ … ƒ r ƒ # 1 = ƒ r ƒ ,

the implication holds for all r and u if we take d = P. In contrast, r 2 cos2 u x2 = = cos2 u 2 2 x + y r2 takes on all values from 0 to 1 regardless of how small ƒ r ƒ is, so that limsx, yd:s0,0d x 2>sx 2 + y 2 d does not exist. In each of these instances, the existence or nonexistence of the limit as r : 0 is fairly clear. Shifting to polar coordinates does not always help, however, and may even tempt us to false conclusions. For example, the limit may exist along every straight line (or ray) u = constant and yet fail to exist in the broader sense. Example 5 illustrates this point. In polar coordinates, ƒsx, yd = s2x 2yd>sx 4 + y 2 d becomes ƒsr cos u, r sin ud =

r cos u sin 2u r 2 cos4 u + sin2 u

13.3 for r Z 0. If we hold u constant and let r : 0, the limit is 0. On the path y = x 2, however, we have r sin u = r 2 cos2 u and ƒsr cos u, r sin ud = =

r cos u sin 2u r 2 cos4 u + sr cos2 ud2 r sin u 2r cos2 u sin u = 2 = 1. 2r 2 cos4 u r cos2 u

2x 2 + y 2 6 d 69. ƒsx, yd = x 2 + y 2,

68. ƒsx, yd =

3x 2y

ƒ ƒsx, yd - ƒs0, 0d ƒ 6 P.

Q

P = 0.01

70. ƒsx, yd = y>sx 2 + 1d,

P = 0.05

71. ƒsx, yd = sx + yd>sx + 1d,

In Exercises 67 and 68, define ƒ(0, 0) in a way that extends ƒ to be continuous at the origin. 3x 2 - x 2y 2 + 3y 2 b 67. ƒsx, yd = ln a x2 + y2

703

Using the Limit Definition Each of Exercises 69–74 gives a function ƒ(x, y) and a positive number P. In each exercise, show that there exists a d 7 0 such that for all (x, y),

2

In Exercises 61–66, find the limit of ƒ as sx, yd : s0, 0d or show that the limit does not exist. x 3 - xy 2 x3 - y3 61. ƒsx, yd = 2 62. ƒsx, yd = cos a b x + y2 x2 + y2 y2 2x 63. ƒsx, yd = 2 64. ƒsx, yd = 2 x + y2 x + x + y2 ƒxƒ + ƒyƒ b 65. ƒsx, yd = tan-1 a 2 x + y2 x2 - y2 66. ƒsx, yd = 2 x + y2

Partial Derivatives

P = 0.01

72. ƒsx, yd = sx + yd>s2 + cos xd, 73. ƒsx, yd = 74. ƒsx, yd =

xy

P = 0.02

2

x + y2 2

x3 + y4 x2 + y2

and f (0, 0) = 0,

P = 0.04

and f (0, 0) = 0,

P = 0.02

Each of Exercises 75–78 gives a function ƒ(x, y, z) and a positive number P. In each exercise, show that there exists a d 7 0 such that for all (x, y, z), 2x 2 + y 2 + z 2 6 d

Q

75. ƒsx, y, zd = x 2 + y 2 + z 2, 76. ƒsx, y, zd = xyz, 77. ƒsx, y, zd =

ƒ ƒsx, y, zd - ƒs0, 0, 0d ƒ 6 P. P = 0.015

P = 0.008 x + y + z

, P = 0.015 x2 + y2 + z2 + 1 78. ƒsx, y, zd = tan2 x + tan2 y + tan2 z, P = 0.03 79. Show that ƒsx, y, zd = x + y - z is continuous at every point sx0 , y0 , z0 d.

x2 + y2

80. Show that ƒsx, y, zd = x 2 + y 2 + z 2 is continuous at the origin.

13.3

Partial Derivatives The calculus of several variables is similar to single-variable calculus applied to several variables one at a time. When we hold all but one of the independent variables of a function constant and differentiate with respect to that one variable, we get a “partial” derivative. This section shows how partial derivatives are defined and interpreted geometrically, and how to calculate them by applying the rules for differentiating functions of a single variable. The idea of differentiability for functions of several variables requires more than the existence of the partial derivatives, but we will see that differentiable functions of several variables behave in the same way as differentiable single-variable functions.

Partial Derivatives of a Function of Two Variables If sx0 , y0 d is a point in the domain of a function ƒ(x, y), the vertical plane y = y0 will cut the surface z = ƒsx, yd in the curve z = ƒsx, y0 d (Figure 13.15). This curve is the graph of the function z = ƒsx, y0 d in the plane y = y0. The horizontal coordinate in this plane is x; the vertical coordinate is z. The y-value is held constant at y0, so y is not a variable. We define the partial derivative of ƒ with respect to x at the point sx0 , y0 d as the ordinary derivative of ƒsx, y0 d with respect to x at the point x = x0. To distinguish partial derivatives from ordinary derivatives we use the symbol 0 rather than the d previously used. In the definition, h represents a real number, positive or negative.

704

Chapter 13: Partial Derivatives z Vertical axis in the plane y y 0 P(x 0 , y 0 , f (x 0, y 0)) z f (x, y)

The curve z f (x, y 0 ) in the plane y y 0 Tangent line 0

x0

y0 x (x 0 h, y 0 )

(x 0, y 0 )

y

Horizontal axis in the plane y y 0

FIGURE 13.15 The intersection of the plane y = y0 with the surface z = ƒsx, yd, viewed from above the first quadrant of the xy-plane.

DEFINITION sx0 , y0 d is

The partial derivative of ƒ(x, y) with respect to x at the point 0ƒ ƒsx0 + h, y0 d - ƒsx0 , y0 d ` = lim , 0x sx0, y0d h:0 h

provided the limit exists.

An equivalent expression for the partial derivative is d ƒ(x, y0) ` . dx x = x0 The slope of the curve z = ƒsx, y0 d at the point Psx0 , y0, ƒsx0 , y0 dd in the plane y = y0 is the value of the partial derivative of ƒ with respect to x at sx0 , y0 d. (In Figure 13.15 this slope is negative.) The tangent line to the curve at P is the line in the plane y = y0 that passes through P with this slope. The partial derivative 0ƒ>0x at sx0 , y0 d gives the rate of change of ƒ with respect to x when y is held fixed at the value y0 . We use several notations for the partial derivative: 0ƒ sx , y d or ƒxsx0 , y0 d, 0x 0 0

0z ` , 0x sx0, y0d

and

ƒx,

0ƒ 0z , z , or . 0x x 0x

The definition of the partial derivative of ƒ(x, y) with respect to y at a point sx0 , y0 d is similar to the definition of the partial derivative of ƒ with respect to x. We hold x fixed at the value x0 and take the ordinary derivative of ƒsx0 , yd with respect to y at y0 .

DEFINITION sx0 , y0 d is

The partial derivative of ƒ(x, y) with respect to y at the point

0ƒ ƒsx0 , y0 + hd - ƒsx0 , y0 d d ` = ƒsx0 , yd ` = lim , 0y sx0, y0d dy h h:0 y = y0 provided the limit exists.

13.3

Vertical axis in the plane x x0

P(x 0 , y 0 , f (x 0, y 0 ))

0ƒ sx , y d, 0y 0 0

z f (x, y) 0 y0

x (x 0 , y 0 )

705

The slope of the curve z = ƒsx0 , yd at the point Psx0 , y0 , ƒsx0 , y0 dd in the vertical plane x = x0 (Figure 13.16) is the partial derivative of ƒ with respect to y at sx0 , y0 d. The tangent line to the curve at P is the line in the plane x = x0 that passes through P with this slope. The partial derivative gives the rate of change of ƒ with respect to y at sx0 , y0 d when x is held fixed at the value x0. The partial derivative with respect to y is denoted the same way as the partial derivative with respect to x:

z

Tangent line

x0

Partial Derivatives

y

ƒysx0 , y0 d,

0ƒ , 0y

ƒy .

Notice that we now have two tangent lines associated with the surface z = ƒsx, yd at the point Psx0 , y0 , ƒsx0 , y0 dd (Figure 13.17). Is the plane they determine tangent to the surface at P? We will see that it is for the differentiable functions defined at the end of this section, and we will learn how to find the tangent plane in Section 13.6. First we have to learn more about partial derivatives themselves.

(x 0 , y 0 k) The curve z f (x 0 , y) in the plane x x0

z

Horizontal axis in the plane x x 0 This tangent line P(x 0 , y 0, f (x 0 , y 0 )) has slope fy(x 0 , y 0 ).

FIGURE 13.16 The intersection of the plane x = x0 with the surface z = ƒsx, yd, viewed from above the first quadrant of the xy-plane.

The curve z f (x 0, y) in the plane x x 0

This tangent line has slope fx (x 0 , y 0 ). The curve z f (x, y0) in the plane y y0 z f (x, y)

x y y0

(x 0 , y0 )

x x0

y

FIGURE 13.17 Figures 13.15 and 13.16 combined. The tangent lines at the point sx0 , y0 , ƒsx0 , y0 dd determine a plane that, in this picture at least, appears to be tangent to the surface.

Calculations The definitions of 0ƒ>0x and 0ƒ>0y give us two different ways of differentiating ƒ at a point: with respect to x in the usual way while treating y as a constant and with respect to y in the usual way while treating x as a constant. As the following examples show, the values of these partial derivatives are usually different at a given point sx0 , y0 d.

EXAMPLE 1

Find the values of 0ƒ>0x and 0ƒ>0y at the point s4, -5d if ƒsx, yd = x 2 + 3xy + y - 1.

Solution

To find 0ƒ>0x, we treat y as a constant and differentiate with respect to x: 0ƒ 0 2 = sx + 3xy + y - 1d = 2x + 3 # 1 # y + 0 - 0 = 2x + 3y. 0x 0x

The value of 0ƒ>0x at s4, -5d is 2s4d + 3s -5d = -7.

706

Chapter 13: Partial Derivatives

To find 0ƒ>0y, we treat x as a constant and differentiate with respect to y: 0ƒ 0 2 = sx + 3xy + y - 1d = 0 + 3 # x # 1 + 1 - 0 = 3x + 1. 0y 0y The value of 0ƒ>0y at s4, -5d is 3s4d + 1 = 13. Find 0ƒ>0y as a function if ƒsx, yd = y sin xy.

EXAMPLE 2 Solution

We treat x as a constant and ƒ as a product of y and sin xy: 0ƒ 0 0 0 = s y sin xyd = y sin xy + ssin xyd s yd 0y 0y 0y 0y = s y cos xyd

EXAMPLE 3

0 sxyd + sin xy = xy cos xy + sin xy. 0y

Find ƒx and ƒy as functions if ƒsx, yd =

Solution

2y . y + cos x

We treat ƒ as a quotient. With y held constant, we get 2y 0 a ƒx = b = 0x y + cos x =

s y + cos xd

s y + cos xds0d - 2ys -sin xd s y + cos xd2

0 0 s2yd - 2y s y + cos xd 0x 0x s y + cos xd2 2y sin x

=

s y + cos xd2

.

With x held constant, we get 2y 0 a ƒy = b = 0y y + cos x =

s y + cos xd

s y + cos xds2d - 2ys1d s y + cos xd

2

0 0 s2yd - 2y s y + cos xd 0y dy s y + cos xd2

=

2 cos x . s y + cos xd2

Implicit differentiation works for partial derivatives the way it works for ordinary derivatives, as the next example illustrates.

EXAMPLE 4

Find 0z>0x if the equation yz - ln z = x + y

defines z as a function of the two independent variables x and y and the partial derivative exists. Solution We differentiate both sides of the equation with respect to x, holding y constant and treating z as a differentiable function of x:

0y 0x 0 0 s yzd ln z = + 0x 0x 0x 0x 0z 1 0z y - z = 1 + 0 0x 0x 1 0z ay - z b = 1 0x 0z z = . 0x yz - 1

With y constant, 0 0z s yzd = y . 0x 0x

13.3

Partial Derivatives

707

The plane x = 1 intersects the paraboloid z = x 2 + y 2 in a parabola. Find the slope of the tangent to the parabola at (1, 2, 5) (Figure 13.18).

EXAMPLE 5

z

Solution

The slope is the value of the partial derivative 0z>0y at (1, 2): 0z 0 2 sx + y 2 d ` ` = = 2y ` = 2s2d = 4. 0y s1,2d 0y s1,2d s1,2d

Surface z x2 y2 Plane x1

Tangent line

(1, 2, 5)

As a check, we can treat the parabola as the graph of the single-variable function z = s1d2 + y 2 = 1 + y 2 in the plane x = 1 and ask for the slope at y = 2. The slope, calculated now as an ordinary derivative, is d dz ` = s1 + y 2 d ` = 2y ` = 4. dy y = 2 dy y=2 y=2

2

1

y x1

x

FIGURE 13.18 The tangent to the curve of intersection of the plane x = 1 and surface z = x 2 + y 2 at the point (1, 2, 5) (Example 5).

Functions of More Than Two Variables The definitions of the partial derivatives of functions of more than two independent variables are like the definitions for functions of two variables. They are ordinary derivatives with respect to one variable, taken while the other independent variables are held constant.

EXAMPLE 6

If x, y, and z are independent variables and ƒsx, y, zd = x sin s y + 3zd,

then 0ƒ 0 0 = [x sin s y + 3zd] = x sin s y + 3zd 0z 0z 0z 0 = x cos s y + 3zd s y + 3zd = 3x cos s y + 3zd. 0z

R1

EXAMPLE 7 If resistors of R1, R2 , and R3 ohms are connected in parallel to make an R-ohm resistor, the value of R can be found from the equation

R2

1 1 1 1 = + + R R1 R2 R3

R3

(Figure 13.19). Find the value of 0R>0R2 when R1 = 30, R2 = 45, and R3 = 90 ohms.

FIGURE 13.19 Resistors arranged this way are said to be connected in parallel (Example 7). Each resistor lets a portion of the current through. Their equivalent resistance R is calculated with the formula 1 1 1 1 + + . = R R1 R2 R3

Solution To find 0R>0R2, we treat R1 and R3 as constants and, using implicit differentiation, differentiate both sides of the equation with respect to R2 :

0 0 1 1 1 1 a b = a + + b 0R2 R 0R2 R1 R2 R3 -

1 1 0R = 0 - 2 + 0 R 2 0R2 R2 2

0R R2 R = 2 = a b . 0R2 R2 R2 When R1 = 30, R2 = 45, and R3 = 90, 3 + 2 + 1 6 1 1 1 1 1 = + + = = = , R 30 45 90 90 90 15

708

Chapter 13: Partial Derivatives

so R = 15 and 2

2

15 0R 1 1 = a b = a b = . 0R2 45 3 9 Thus at the given values, a small change in the resistance R2 leads to a change in R about 1>9th as large.

Partial Derivatives and Continuity A function ƒ(x, y) can have partial derivatives with respect to both x and y at a point without the function being continuous there. This is different from functions of a single variable, where the existence of a derivative implies continuity. If the partial derivatives of ƒ(x, y) exist and are continuous throughout a disk centered at sx0 , y0 d, however, then ƒ is continuous at sx0, y0 d, as we see at the end of this section. z

0, xy 0 1, xy 0

EXAMPLE 8

Let

z

ƒsx, yd = e

L1 1

xy Z 0 xy = 0

0, 1,

(Figure 13.20). 0 L2 x

y

(a) Find the limit of ƒ as (x, y) approaches (0, 0) along the line y = x. (b) Prove that ƒ is not continuous at the origin. (c) Show that both partial derivatives 0ƒ>0x and 0ƒ>0y exist at the origin. Solution

FIGURE 13.20 The graph of 0, ƒsx, yd = e 1,

xy Z 0 xy = 0

consists of the lines L1 and L2 and the four open quadrants of the xy-plane. The function has partial derivatives at the origin but is not continuous there (Example 8).

(a) Since ƒ(x, y) is constantly zero along the line y = x (except at the origin), we have lim

sx, yd:s0,0d

ƒsx, yd `

y=x

=

lim

0 = 0.

sx, yd:s0,0d

(b) Since ƒs0, 0d = 1, the limit in part (a) proves that ƒ is not continuous at (0, 0). (c) To find 0ƒ>0x at (0, 0), we hold y fixed at y = 0. Then ƒsx, yd = 1 for all x, and the graph of ƒ is the line L1 in Figure 13.20. The slope of this line at any x is 0ƒ>0x = 0. In particular, 0ƒ>0x = 0 at (0, 0). Similarly, 0ƒ>0y is the slope of line L2 at any y, so 0ƒ>0y = 0 at (0, 0). Example 8 notwithstanding, it is still true in higher dimensions that differentiability at a point implies continuity. What Example 8 suggests is that we need a stronger requirement for differentiability in higher dimensions than the mere existence of the partial derivatives. We define differentiability for functions of two variables (which is slightly more complicated than for single-variable functions) at the end of this section and then revisit the connection to continuity.

Second-Order Partial Derivatives When we differentiate a function ƒ(x, y) twice, we produce its second-order derivatives. These derivatives are usually denoted by 0 2ƒ 0x 2

0 2ƒ

or ƒxx ,

0 2ƒ or ƒyx , 0x0y

0y 2 and

or ƒyy , 0 2ƒ or ƒxy . 0y0x

13.3

Partial Derivatives

709

The defining equations are 0 2ƒ 0x 2

=

0 0ƒ a b, 0x 0x

0 2ƒ 0 0ƒ = a b, 0x0y 0x 0y

and so on. Notice the order in which the mixed partial derivatives are taken:

HISTORICAL BIOGRAPHY

0 2ƒ 0x0y

Differentiate first with respect to y, then with respect to x.

ƒyx = sƒy dx

Means the same thing.

EXAMPLE 9

If ƒsx, yd = x cos y + ye x, find the second-order derivatives 0 2ƒ

Pierre-Simon Laplace (1749–1827)

0x Solution

, 2

0 2ƒ , 0y0x

0 2ƒ 0y

, 2

and

0 2ƒ . 0x0y

The first step is to calculate both first partial derivatives.

0ƒ 0 = sx cos y + ye x d 0x 0x

0ƒ 0 = sx cos y + ye x d 0y 0y

= cos y + ye x = -x sin y + e x Now we find both partial derivatives of each first partial: 0 2ƒ 0 0ƒ = a b = -sin y + e x 0y0x 0y 0x 0 2ƒ 0x

2

=

0 2ƒ 0 0ƒ = a b = -sin y + e x 0x0y 0x 0y 0 2ƒ

0 0ƒ a b = ye x. 0x 0x

0y

2

0 0ƒ a b = -x cos y. 0y 0y

=

The Mixed Derivative Theorem You may have noticed that the “mixed” second-order partial derivatives 0 2ƒ 0y0x

0 2ƒ 0x0y

and

in Example 9 are equal. This is not a coincidence. They must be equal whenever ƒ, ƒx , ƒy , ƒxy , and ƒyx are continuous, as stated in the following theorem.

THEOREM 2—The Mixed Derivative Theorem If ƒsx, yd and its partial derivatives ƒx , ƒy , ƒxy , and ƒyx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then ƒxysa, bd = ƒyxsa, bd.

HISTORICAL BIOGRAPHY Alexis Clairaut (1713–1765)

Theorem 2 is also known as Clairaut’s Theorem, named after the French mathematician Alexis Clairaut who discovered it. A proof is given in Appendix 10. Theorem 2 says that to calculate a mixed second-order derivative, we may differentiate in either order, provided the continuity conditions are satisfied. This ability to proceed in different order sometimes simplifies our calculations.

EXAMPLE 10

Find 0 2w>0x0y if w = xy +

ey . y + 1 2

710

Chapter 13: Partial Derivatives

The symbol 0 2w>0x0y tells us to differentiate first with respect to y and then with respect to x. However, if we interchange the order of differentiation and differentiate first with respect to x we get the answer more quickly. In two steps, Solution

0w = y 0x

and

0 2w = 1. 0y0x

If we differentiate first with respect to y, we obtain 0 2w>0x0y = 1 as well. We can differentiate in either order because the conditions of Theorem 2 hold for w at all points ( x0, y0).

Partial Derivatives of Still Higher Order Although we will deal mostly with first- and second-order partial derivatives, because these appear the most frequently in applications, there is no theoretical limit to how many times we can differentiate a function as long as the derivatives involved exist. Thus, we get third- and fourth-order derivatives denoted by symbols like 0 3ƒ 0x0y 2

= ƒyyx,

0 4ƒ

= ƒyyxx , 0x 20y 2 and so on. As with second-order derivatives, the order of differentiation is immaterial as long as all the derivatives through the order in question are continuous.

EXAMPLE 11

Find ƒyxyz if ƒsx, y, zd = 1 - 2xy 2z + x 2y.

Solution We first differentiate with respect to the variable y, then x, then y again, and finally with respect to z: ƒy = -4xyz + x 2 ƒyx = -4yz + 2x ƒyxy = -4z ƒyxyz = -4

Differentiability The starting point for differentiability is not the difference quotient we saw in studying singlevariable functions, but rather the idea of increment. Recall from our work with functions of a single variable in Section 3.11 that if y = ƒsxd is differentiable at x = x0 , then the change in the value of ƒ that results from changing x from x0 to x0 + ¢x is given by an equation of the form ¢y = ƒ¿sx0 d¢x + P¢x in which P : 0 as ¢x : 0. For functions of two variables, the analogous property becomes the definition of differentiability. The Increment Theorem (proved in Appendix 10) tells us when to expect the property to hold.

THEOREM 3—The Increment Theorem for Functions of Two Variables Suppose that the first partial derivatives of ƒ(x, y) are defined throughout an open region R containing the point sx0 , y0 d and that ƒx and ƒy are continuous at sx0 , y0 d. Then the change ¢z = ƒsx0 + ¢x, y0 + ¢yd - ƒsx0 , y0 d in the value of ƒ that results from moving from sx0 , y0 d to another point (x0 + ¢x, y0 + ¢yd in R satisfies an equation of the form ¢z = ƒxsx0 , y0 d¢x + ƒysx0 , y0 d¢y + P1 ¢x + P2 ¢y in which each of P1, P2 : 0 as both ¢x, ¢y : 0.

13.3

Partial Derivatives

711

You can see where the epsilons come from in the proof given in Appendix 10. Similar results hold for functions of more than two independent variables.

DEFINITION A function z = ƒsx, yd is differentiable at sx0 , y0 d if ƒxsx0 , y0 d and ƒysx0 , y0 d exist and ¢z satisfies an equation of the form ¢z = ƒxsx0 , y0 d¢x + ƒysx0 , y0 d¢y + P1 ¢x + P2 ¢y in which each of P1, P2 : 0 as both ¢x, ¢y : 0. We call ƒ differentiable if it is differentiable at every point in its domain, and say that its graph is a smooth surface.

Because of this definition, an immediate corollary of Theorem 3 is that a function is differentiable at sx0, y0 d if its first partial derivatives are continuous there.

COROLLARY OF THEOREM 3 If the partial derivatives ƒx and ƒy of a function ƒ(x, y) are continuous throughout an open region R, then ƒ is differentiable at every point of R. If z = ƒsx, yd is differentiable, then the definition of differentiability assures that ¢z = ƒsx0 + ¢x, y0 + ¢yd - ƒsx0 , y0 d approaches 0 as ¢x and ¢y approach 0. This tells us that a function of two variables is continuous at every point where it is differentiable.

THEOREM 4—Differentiability Implies Continuity If a function ƒsx, yd is differentiable at sx0 , y0 d, then ƒ is continuous at sx0 , y0 d. As we can see from Corollary 3 and Theorem 4, a function ƒ(x, y) must be continuous at a point sx0 , y0 d if ƒx and ƒy are continuous throughout an open region containing sx0 , y0 d. Remember, however, that it is still possible for a function of two variables to be discontinuous at a point where its first partial derivatives exist, as we saw in Example 8. Existence alone of the partial derivatives at that point is not enough, but continuity of the partial derivatives guarantees differentiability.

Exercises 13.3 Calculating First-Order Partial Derivatives In Exercises 1–22, find 0ƒ>0x and 0ƒ>0y. 1. ƒsx, yd = 2x 2 - 3y - 4

2. ƒsx, yd = x 2 - xy + y 2

y

21. ƒsx, yd =

Lx

g std dt

sg continuous for all td

q

3. ƒsx, yd = sx 2 - 1ds y + 2d 4. ƒsx, yd = 5xy - 7x 2 - y 2 + 3x - 6y + 2 5. ƒsx, yd = sxy - 1d2

6. ƒsx, yd = s2x - 3yd3

7. ƒsx, yd = 2x 2 + y 2

8. ƒsx, yd = sx 3 + s y>2dd2>3

9. ƒsx, yd = 1>sx + yd

20. ƒsx, yd = logy x

19. ƒsx, yd = x y

10. ƒsx, yd = x>sx 2 + y 2 d

11. ƒsx, yd = sx + yd>sxy - 1d 12. ƒsx, yd = tan-1 s y>xd

22. ƒsx, yd = a sxydn n=0

s ƒ xy ƒ 6 1d

In Exercises 23–34, find ƒx , ƒy , and ƒz . 23. ƒsx, y, zd = 1 + xy 2 - 2z 2 24. ƒsx, y, zd = xy + yz + xz 25. ƒsx, y, zd = x - 2y 2 + z 2 26. ƒsx, y, zd = sx 2 + y 2 + z 2 d-1>2

13. ƒsx, yd = e sx + y + 1d

14. ƒsx, yd = e -x sin sx + yd

15. ƒsx, yd = ln sx + yd

16. ƒsx, yd = e xy ln y

27. ƒsx, y, zd = sin-1 sxyzd

17. ƒsx, yd = sin2 sx - 3yd

18. ƒsx, yd = cos2 s3x - y 2 d

29. ƒsx, y, zd = ln sx + 2y + 3zd

28. ƒsx, y, zd = sec-1 sx + yzd

712

Chapter 13: Partial Derivatives 31. ƒsx, y, zd = e -sx

30. ƒsx, y, zd = yz ln sxyd 32. ƒsx, y, zd = e

2

+ y2 + z2d

-xyz

33. ƒsx, y, zd = tanh sx + 2y + 3zd

Using the Partial Derivative Definition In Exercises 57–60, use the limit definition of partial derivative to compute the partial derivatives of the functions at the specified points. 0ƒ 0x

34. ƒsx, y, zd = sinh sxy - z 2 d

57. ƒsx, yd = 1 - x + y - 3x 2y,

In Exercises 35–40, find the partial derivative of the function with respect to each variable.

58. ƒsx, yd = 4 + 2x - 3y - xy 2,

35. ƒst, ad = cos s2pt - ad

59. ƒsx, yd = 22x + 3y - 1,

36. g su, yd = y 2e s2u>yd

37. hsr, f, ud = r sin f cos u 39. Work done by the heart

38. g sr, u, zd = r s1 - cos ud - z (Section 3.11, Exercise 61)

WsP, V, d, y, gd = PV +

Vdy 2g

2

40. Wilson lot size formula (Section 4.6, Exercise 53) hq km Asc, h, k, m, qd = q + cm + 2 Calculating Second-Order Partial Derivatives Find all the second-order partial derivatives of the functions in Exercises 41–50. 41. ƒsx, yd = x + y + xy

42. ƒsx, yd = sin xy

43. g sx, yd = x 2y + cos y + y sin x 44. hsx, yd = xe y + y + 1 46. ssx, yd = tan 2

48. w = ye x 50. w =

-1

45. r sx, yd = ln sx + yd 47. w = x 2 tan sx yd

s y>xd

-y

49. w = x sin sx 2yd

x - y x2 + y

60. ƒsx, yd = •

sin sx 3 + y 4 d x2 + y2

and

0ƒ 0y

0ƒ 0x

0ƒ 0x

and 0ƒ 0y

and

0ƒ 0y 0ƒ 0y

at s1, 2d at s -2, 1d

at s -2, 3d

(x, y) Z s0, 0d sx, yd = s0, 0d,

0, 0ƒ 0x

,

and

at

s0, 0d

61. Let ƒsx, yd = 2x + 3y - 4. Find the slope of the line tangent to this surface at the point s2, -1d and lying in the a. plane x = 2 b. plane y = -1. 62. Let ƒsx, yd = x 2 + y 3. Find the slope of the line tangent to this surface at the point s -1, 1d and lying in the a. plane x = - 1 b. plane y = 1. 63. Three variables Let w = ƒsx, y, zd be a function of three independent variables and write the formal definition of the partial derivative 0ƒ>0z at sx0 , y0 , z0 d. Use this definition to find 0ƒ>0z at (1, 2, 3) for ƒsx, y, zd = x 2yz 2. 64. Three variables Let w = ƒsx, y, zd be a function of three independent variables and write the formal definition of the partial derivative 0ƒ>0y at sx0 , y0 , z0 d. Use this definition to find 0ƒ>0y at s -1, 0, 3d for ƒsx, y, zd = -2xy 2 + yz 2. Differentiating Implicitly 65. Find the value of 0z>0x at the point (1, 1, 1) if the equation xy + z 3x - 2yz = 0

Mixed Partial Derivatives In Exercises 51–54, verify that wxy = wyx. 51. w = ln s2x + 3yd

52. w = e + x ln y + y ln x

53. w = xy 2 + x 2y 3 + x 3y 4

54. w = x sin y + y sin x + xy

x

55. Which order of differentiation will calculate fxy faster: x first or y first? Try to answer without writing anything down. a. ƒsx, yd = x sin y + e y b. ƒsx, yd = 1>x

66. Find the value of 0x>0z at the point s1, -1, -3d if the equation xz + y ln x - x 2 + 4 = 0 defines x as a function of the two independent variables y and z and the partial derivative exists. Exercises 67 and 68 are about the triangle shown here.

c. ƒsx, yd = y + sx>yd

B

d. ƒsx, yd = y + x y + 4y - ln s y + 1d 2

3

2

e. ƒsx, yd = x 2 + 5xy + sin x + 7e x 56. The fifth-order partial derivative 0 5ƒ>0x 20y 3 is zero for each of the following functions. To show this as quickly as possible, which variable would you differentiate with respect to first: x or y? Try to answer without writing anything down. a. ƒsx, yd = y 2x 4e x + 2 b. ƒsx, yd = y + yssin x - x d 2

4

c. ƒsx, yd = x 2 + 5xy + sin x + 7e x >2

2

c a C

f. ƒsx, yd = x ln xy

d. ƒsx, yd = xe y

defines z as a function of the two independent variables x and y and the partial derivative exists.

b

A

67. Express A implicitly as a function of a, b, and c and calculate 0A>0a and 0A>0b. 68. Express a implicitly as a function of A, b, and B and calculate 0a>0A and 0a>0B. 69. Two dependent variables Express yx in terms of u and y if the equations x = y ln u and y = u ln y define u and y as functions of the independent variables x and y, and if yx exists. (Hint: Differentiate both equations with respect to x and solve for yx by eliminating ux .)

13.3 70. Two dependent variables Find 0x>0u and 0y>0u if the equations u = x 2 - y 2 and y = x 2 - y define x and y as functions of the independent variables u and y, and the partial derivatives exist. (See the hint in Exercise 69.) Then let s = x 2 + y 2 and find 0s>0u. y 3, 71. Let ƒsx, yd = b 2 -y ,

y Ú 0 y 6 0.

2x, x 2,

0x 2

0 2ƒ

+

0y 2

0 2ƒ 0z 2

= 0

is satisfied by steady-state temperature distributions T = ƒsx, y, zd in space, by gravitational potentials, and by electrostatic potentials. The two-dimensional Laplace equation 0 2ƒ 0x 2

+

73. ƒsx, y, zd = x 2 + y 2 - 2z 2 74. ƒsx, y, zd = 2z 3 - 3sx 2 + y 2 dz 75. ƒsx, yd = e -2y cos 2x 77. ƒsx, yd = 3x + 2y - 4 x 78. ƒsx, yd = tan-1 y 80. ƒsx, y, zd = e 3x + 4y cos 5z

Theory and Examples The three-dimensional Laplace equation +

Show that each function in Exercises 73–80 satisfies a Laplace equation.

79. ƒsx, y, zd = sx 2 + y 2 + z 2 d-1>2

x Ú 0 x 6 0.

Find ƒx, ƒy, ƒxy, and ƒyx, and state the domain for each partial derivative.

0 2ƒ

713

76. ƒsx, yd = ln2x 2 + y 2

Find ƒx, ƒy, ƒxy, and ƒyx, and state the domain for each partial derivative. 72. Let ƒsx, yd = b

Partial Derivatives

0 2ƒ 0y 2

The Wave Equation If we stand on an ocean shore and take a snapshot of the waves, the picture shows a regular pattern of peaks and valleys in an instant of time. We see periodic vertical motion in space, with respect to distance. If we stand in the water, we can feel the rise and fall of the water as the waves go by. We see periodic vertical motion in time. In physics, this beautiful symmetry is expressed by the one-dimensional wave equation 0 2w 0 2w = c2 2 , 2 0t 0x where w is the wave height, x is the distance variable, t is the time variable, and c is the velocity with which the waves are propagated.

= 0,

w

obtained by dropping the 0 2ƒ>0z 2 term from the previous equation, describes potentials and steady-state temperature distributions in a plane (see the accompanying figure). The plane (a) may be treated as a thin slice of the solid (b) perpendicular to the z-axis. x

∂ 2f ∂x 2

∂ 2f ∂y2

0

x

In our example, x is the distance across the ocean’s surface, but in other applications, x might be the distance along a vibrating string, distance through air (sound waves), or distance through space (light waves). The number c varies with the medium and type of wave. Show that the functions in Exercises 81–87 are all solutions of the wave equation.

(a)

81. w = sin sx + ctd 82. w = cos s2x + 2ctd ∂ 2f ∂ 2f ∂ 2f 2 2 0 ∂x 2 ∂y ∂z

(b)

83. w = sin sx + ctd + cos s2x + 2ctd 84. w = ln s2x + 2ctd 86. w = 5 cos s3x + 3ctd + e

85. w = tan s2x - 2ctd x + ct

87. w = ƒsud, where ƒ is a differentiable function of u, and u = asx + ctd, where a is a constant 88. Does a function ƒ(x, y) with continuous first partial derivatives throughout an open region R have to be continuous on R? Give reasons for your answer.

Boundary temperatures controlled

89. If a function ƒsx, yd has continuous second partial derivatives throughout an open region R, must the first-order partial derivatives of ƒ be continuous on R? Give reasons for your answer.

714

Chapter 13: Partial Derivatives Show that ƒxs0, 0d and ƒys0, 0d exist, but ƒ is not differentiable at s0, 0d. (Hint: Use Theorem 4 and show that ƒ is not continuous at s0, 0d.)

90. The heat equation An important partial differential equation that describes the distribution of heat in a region at time t can be represented by the one-dimensional heat equation 0ƒ 0 2ƒ = 2. 0t 0x

92. Let ƒsx, yd = b

Show that usx, td = sin saxd # e -bt satisfies the heat equation for constants a and b . What is the relationship between a and b for this function to be a solution? xy 2

91. Let ƒsx, yd = • x 2 + y 4 0,

13.4

,

0, 1,

x 2 6 y 6 2x 2 otherwise.

Show that ƒxs0, 0d and ƒys0, 0d exist, but ƒ is not differentiable at s0, 0d.

(x, y) Z s0, 0d sx, yd = s0, 0d.

The Chain Rule The Chain Rule for functions of a single variable studied in Section 3.6 says that when w = ƒsxd is a differentiable function of x and x = gstd is a differentiable function of t, w is a differentiable function of t and dw> dt can be calculated by the formula dw dw dx = . dt dx dt For functions of two or more variables the Chain Rule has several forms. The form depends on how many variables are involved, but once this is taken into account, it works like the Chain Rule in Section 3.6.

Functions of Two Variables The Chain Rule formula for a differentiable function w = ƒsx, yd when x = xstd and y = ystd are both differentiable functions of t is given in the following theorem.

0ƒ 0w , , ƒ indicates the partial 0x 0x x derivative of ƒ with respect to x. Each of

THEOREM 5—Chain Rule for Functions of One Independent Variable and Two Intermediate Variables If w = ƒsx, yd is differentiable and if x = xstd, y = ystd are differentiable functions of t, then the composite w = ƒsxstd, ystdd is a differentiable function of t and dw = ƒxsxstd, ystdd # x¿std + ƒysxstd, ystdd # y¿std, dt or

0ƒ dx 0ƒ dy dw = + . 0x dt 0y dt dt

Proof The proof consists of showing that if x and y are differentiable at t = t0 , then w is differentiable at t0 and a

dy dw 0w dx 0w b = a b a b + a b a b , 0x P0 dt t0 0y P0 dt t0 dt t0

where P0 = sxst0 d, yst0 dd. The subscripts indicate where each of the derivatives is to be evaluated.

13.4

The Chain Rule

715

Let ¢x, ¢y, and ¢w be the increments that result from changing t from t0 to t0 + ¢t. Since ƒ is differentiable (see the definition in Section 13.3), ¢w = a

0w 0w b ¢x + a b ¢y + P1 ¢x + P2 ¢y, 0x P0 0y P0

where P1, P2 : 0 as ¢x, ¢y : 0. To find dw> dt, we divide this equation through by ¢t and let ¢t approach zero. The division gives ¢y ¢y ¢x ¢x ¢w 0w 0w = a b + a b + P1 + P2 . 0x P0 ¢t 0y P0 ¢t ¢t ¢t ¢t Letting ¢t approach zero gives a

dw ¢w b = lim dt t0 ¢t:0 ¢t = a

dy dy 0w dx 0w dx b a b + a b a b + 0# a b + 0# a b . 0x P0 dt t0 0y P0 dt t0 dt t0 dt t0

Often we write 0w>0x for the partial derivative 0ƒ>0x, so we can rewrite the Chain Rule in Theorem 5 in the form

To remember the Chain Rule, picture the diagram below. To find dw> dt, start at w and read down each route to t, multiplying derivatives along the way. Then add the products. Chain Rule w f (x, y)

0w 0x

Dependent variable

0w 0y y Intermediate variables

x dx dt

dy dt

t dw 0 w dx 0 w dy dt 0 x dt 0 y dt

Independent variable

dw 0w dx 0w dy = + . 0x dt 0y dt dt However, the meaning of the dependent variable w is different on each side of the preceding equation. On the left-hand side, it refers to the composite function w = ƒsxstd, ystdd as a function of the single variable t. On the right-hand side, it refers to the function w = ƒsx, yd as a function of the two variables x and y. Moreover, the single derivatives dw>dt, dx>dt, and dy>dt are being evaluated at a point t0, whereas the partial derivatives 0w>0x and 0w>0y are being evaluated at the point sx0, y0 d, with x0 = xst0 d and y0 = yst0 d. With that understanding, we will use both of these forms interchangeably throughout the text whenever no confusion will arise. The branch diagram in the margin provides a convenient way to remember the Chain Rule. The “true” independent variable in the composite function is t, whereas x and y are intermediate variables (controlled by t) and w is the dependent variable. A more precise notation for the Chain Rule shows where the various derivatives in Theorem 5 are evaluated: 0ƒ dw st d = sx , y ) 0x 0 0 dt 0

EXAMPLE 1

#

0ƒ dx st d + sx , y d 0y 0 0 dt 0

#

dy st d. dt 0

Use the Chain Rule to find the derivative of w = xy

with respect to t along the path x = cos t, y = sin t. What is the derivative’s value at t = p>2? Solution

We apply the Chain Rule to find dw> dt as follows: dw 0w dx 0w dy = + 0x dt 0y dt dt 0sxyd d # scos td + 0sxyd = 0x 0y dt

#

d ssin td dt

= s yds -sin td + sxdscos td = ssin tds -sin td + scos tdscos td = -sin2 t + cos2 t = cos 2t.

716

Chapter 13: Partial Derivatives

In this example, we can check the result with a more direct calculation. As a function of t, 1 w = xy = cos t sin t = sin 2t, 2 so d 1 dw 1 = a sin 2tb = 2 dt dt 2

#

2 cos 2t = cos 2t .

In either case, at the given value of t, a

dw p b = cos a2 # b = cos p = -1 . 2 dt t = p>2

Functions of Three Variables You can probably predict the Chain Rule for functions of three intermediate variables, as it only involves adding the expected third term to the two-variable formula.

THEOREM 6—Chain Rule for Functions of One Independent Variable and Three Intermediate Variables If w = ƒsx, y, zd is differentiable and x, y, and z are differentiable functions of t, then w is a differentiable function of t and dw 0w dx 0w dy 0w dz = + + . 0x dt 0y dt 0z dt dt

Here we have three routes from w to t instead of two, but finding dw> dt is still the same. Read down each route, multiplying derivatives along the way; then add.

The proof is identical with the proof of Theorem 5 except that there are now three intermediate variables instead of two. The branch diagram we use for remembering the new equation is similar as well, with three routes from w to t.

EXAMPLE 2

w = xy + z,

Chain Rule w f (x, y, z)

0w 0x

0w 0z

0w 0y

x

y dx dt

dy dt

Dependent variable

y = sin t,

z = t.

Using the Chain Rule for three intermediate variables, we have dw 0w dx 0w dy 0w dz = + + 0x dt 0y dt 0z dt dt

Intermediate variables

= s yds -sin td + sxdscos td + s1ds1d

dz dt

Independent t variable dw 0 w dx 0 w dy 0 w dz 0 x dt 0 y dt 0 z dt dt

x = cos t,

In this example the values of w(t) are changing along the path of a helix (Section 12.1) as t changes. What is the derivative’s value at t = 0? Solution

z

Find dw> dt if

= ssin tds -sin td + scos tdscos td + 1 = -sin2 t + cos2 t + 1 = 1 + cos 2t,

Substitute for the intermediate variables.

so a

dw b = 1 + cos s0d = 2. dt t = 0

For a physical interpretation of change along a curve, think of an object whose position is changing with time t. If w = Tsx, y, zd is the temperature at each point (x, y, z) along a curve C with parametric equations x = xstd, y = ystd, and z = zstd, then the composite function w = Tsxstd, ystd, zstdd represents the temperature relative to t along the curve. The derivative dw> dt is then the instantaneous rate of change of temperature due to the motion along the curve, as calculated in Theorem 6.

Functions Defined on Surfaces If we are interested in the temperature w = ƒsx, y, zd at points (x, y, z) on the earth’s surface, we might prefer to think of x, y, and z as functions of the variables r and s that give

13.4

717

The Chain Rule

the points’ longitudes and latitudes. If x = gsr, sd, y = hsr, sd, and z = ksr, sd, we could then express the temperature as a function of r and s with the composite function w = ƒsgsr, sd, hsr, sd, ksr, sdd. Under the conditions stated below, w has partial derivatives with respect to both r and s that can be calculated in the following way.

THEOREM 7—Chain Rule for Two Independent Variables and Three Intermediate Suppose that w = ƒsx, y, zd, x = gsr, sd, y = hsr, sd, and z = ksr, sd. If all four functions are differentiable, then w has partial derivatives with respect to r and s, given by the formulas

Variables

0w 0x 0w 0w = + 0r 0x 0r 0y 0w 0w 0x 0w = + 0s 0x 0s 0y

0y 0w 0z + 0r 0z 0r 0y 0w 0z + . 0s 0z 0s

The first of these equations can be derived from the Chain Rule in Theorem 6 by holding s fixed and treating r as t. The second can be derived in the same way, holding r fixed and treating s as t. The branch diagrams for both equations are shown in Figure 13.21. w f (x, y, z)

w f (x, y, z)

Dependent variable

w

0w 0x

f Intermediate variables

x

y g

Independent variables

h

x

z k

r, s w f ( g(r, s), h (r, s), k (r, s))

0y 0r

z

0w 0y

x

0w 0z y

0y 0s

0x 0s

0z 0r

r 0w 0w 0x 0w 0y 0w 0z 0r 0x 0r 0y 0r 0z 0r

z

0z 0s

s 0w 0 w 0x 0w 0y 0w 0z 0s 0x 0s 0y 0s 0z 0s (c)

(b)

(a)

FIGURE 13.21

y

0x 0r

0w 0x

0w 0z

0w 0y

Composite function and branch diagrams for Theorem 7.

EXAMPLE 3

Solution

Express 0w>0r and 0w>0s in terms of r and s if r w = x + 2y + z 2, x = s, y = r 2 + ln s,

z = 2r.

Using the formulas in Theorem 7, we find 0w 0w 0x 0w 0y 0w 0z = + + 0r 0x 0r 0y 0r 0z 0r 1 = s1d a s b + s2ds2rd + s2zds2d 1 1 = s + 4r + s4rds2d = s + 12r 0w 0w 0x 0w 0y 0w 0z = + + 0s 0x 0s 0y 0s 0z 0s = s1d a-

Substitute for intermediate variable z.

r 1 2 r b + s2d a s b + s2zds0d = s - 2 2 s s

718

Chapter 13: Partial Derivatives Chain Rule

If ƒ is a function of two intermediate variables instead of three, each equation in Theorem 7 becomes correspondingly one term shorter.

w f (x, y)

0w 0x

0w 0y

If w = ƒsx, yd, x = gsr, sd, and y = hsr, sd, then

x

y

0x 0r

0w 0x 0w 0y 0w = + 0r 0x 0r 0y 0r

0y 0r

and

0w 0x 0w 0y 0w = + . 0s 0x 0s 0y 0s

r

0w 0w 0x 0w 0y 0r 0 x 0r 0y 0r

FIGURE 13.22 Branch diagram for the equation 0w 0w 0x 0w 0y = + . 0r 0x 0r 0y 0r

Figure 13.22 shows the branch diagram for the first of these equations. The diagram for the second equation is similar; just replace r with s.

EXAMPLE 4

Express 0w>0r and 0w>0s in terms of r and s if w = x 2 + y 2,

Solution

x = r - s,

y = r + s.

The preceding discussion gives the following. 0w 0w 0x 0w 0y = + 0r 0x 0r 0y 0r

0w 0x 0w 0y 0w = + 0s 0x 0s 0y 0s

= s2xds1d + s2yds1d

= s2xds -1d + s2yds1d

= 2sr - sd + 2sr + sd

= -2sr - sd + 2sr + sd

= 4r

= 4s

Substitute for the intermediate variables.

If ƒ is a function of a single intermediate variable x, our equations are even simpler.

If w = ƒsxd and x = gsr, sd, then dw 0x 0w = 0r dx 0r

and

0w dw 0x = . 0s dx 0s

In this case, we use the ordinary (single-variable) derivative, dw> dx. The branch diagram is shown in Figure 13.23.

Chain Rule w f (x) dw dx

Implicit Differentiation Revisited The two-variable Chain Rule in Theorem 5 leads to a formula that takes some of the algebra out of implicit differentiation. Suppose that

x

0x 0r

0x 0s

r

1. 2. s

0w dw 0 x dx 0r 0r 0w dw 0 x dx 0s 0s

FIGURE 13.23 Branch diagram for differentiating ƒ as a composite function of r and s with one intermediate variable.

The function F(x, y) is differentiable and The equation Fsx, yd = 0 defines y implicitly as a differentiable function of x, say y = hsxd.

Since w = Fsx, yd = 0, the derivative dw> dx must be zero. Computing the derivative from the Chain Rule (branch diagram in Figure 13.24), we find 0 =

dy dw dx = Fx + Fy dx dx dx

= Fx # 1 + Fy #

dy . dx

Theorem 5 with t = x and ƒ = F

13.4

Fy 0 w 0y y h(x)

x dx 1 dx

719

If Fy = 0w>0y Z 0, we can solve this equation for dy> dx to get

w F(x, y)

0w F x 0x

The Chain Rule

dy Fx = - . Fy dx We state this result formally.

dy h'(x) dx

THEOREM 8—A Formula for Implicit Differentiation Suppose that F(x, y) is differentiable and that the equation Fsx, yd = 0 defines y as a differentiable function of x. Then at any point where Fy Z 0,

x dw dy Fx • 1 Fy • dx dx

FIGURE 13.24 Branch diagram for differentiating w = Fsx, yd with respect to x. Setting dw>dx = 0 leads to a simple computational formula for implicit differentiation (Theorem 8).

dy Fx = - . Fy dx

EXAMPLE 5 Solution

(1)

Use Theorem 8 to find dy> dx if y 2 - x 2 - sin xy = 0.

Take Fsx, yd = y 2 - x 2 - sin xy. Then -2x - y cos xy dy Fx = = Fy 2y - x cos xy dx =

2x + y cos xy . 2y - x cos xy

This calculation is significantly shorter than a single-variable calculation using implicit differentiation. The result in Theorem 8 is easily extended to three variables. Suppose that the equation Fsx, y, zd = 0 defines the variable z implicitly as a function z = ƒsx, yd. Then for all sx, yd in the domain of ƒ, we have Fsx, y, ƒsx, ydd = 0. Assuming that F and ƒ are differentiable functions, we can use the Chain Rule to differentiate the equation Fsx, y, zd = 0 with respect to the independent variable x: 0 =

0F 0x 0F 0y 0F 0z + + 0x 0x 0y 0x 0z 0x

= Fx # 1 + Fy # 0 + Fz #

0z , 0x

y is constant when differentiating with respect to x.

so Fx + Fz

0z = 0. 0x

A similar calculation for differentiating with respect to the independent variable y gives Fy + Fz

0z = 0. 0y

Whenever Fz Z 0, we can solve these last two equations for the partial derivatives of z = ƒsx, yd to obtain

Fx 0z = 0x Fz

and

Fy 0z = - . 0y Fz

(2)

720

Chapter 13: Partial Derivatives

An important result from advanced calculus, called the Implicit Function Theorem, states the conditions for which our results in Equations (2) are valid. If the partial derivatives Fx, Fy, and Fz are continuous throughout an open region R in space containing the point sx0, y0, z0 d, and if for some constant c, Fsx0, y0, z0 d = c and Fzsx0, y0, z0 d Z 0, then the equation Fsx, y, zd = c defines z implicitly as a differentiable function of x and y near sx0, y0, z0 d, and the partial derivatives of z are given by Equations (2).

EXAMPLE 6 Solution

0z 0z and at s0, 0, 0d if x 3 + z 2 + ye xz + z cos y = 0. 0x 0y

Find

Let Fsx, y, zd = x 3 + z 2 + ye xz + z cos y. Then

Fx = 3x 2 + zye xz,

Fy = e xz - z sin y,

and

Fz = 2z + xye xz + cos y.

Since Fs0, 0, 0d = 0, Fzs0, 0, 0d = 1 Z 0, and all first partial derivatives are continuous, the Implicit Function Theorem says that Fsx, y, zd = 0 defines z as a differentiable function of x and y near the point s0, 0, 0d. From Equations (2), 3x 2 + zye xz Fx 0z = = 0x Fz 2z + xye xz + cos y

and

Fy e xz - z sin y 0z . = = 0y Fz 2z + xye xz + cos y

At s0, 0, 0d we find 0 0z = - = 0 0x 1

and

0z 1 = - = -1. 0y 1

Functions of Many Variables We have seen several different forms of the Chain Rule in this section, but each one is just a special case of one general formula. When solving particular problems, it may help to draw the appropriate branch diagram by placing the dependent variable on top, the intermediate variables in the middle, and the selected independent variable at the bottom. To find the derivative of the dependent variable with respect to the selected independent variable, start at the dependent variable and read down each route of the branch diagram to the independent variable, calculating and multiplying the derivatives along each route. Then add the products found for the different routes. In general, suppose that w = ƒsx, y, Á , yd is a differentiable function of the intermediate variables x, y, Á , y (a finite set) and the x, y, Á , y are differentiable functions of the independent variables p, q, Á , t (another finite set). Then w is a differentiable function of the variables p through t, and the partial derivatives of w with respect to these variables are given by equations of the form 0w 0x 0w 0y Á 0w 0y 0w = + + + . 0p 0x 0p 0y 0p 0y 0p The other equations are obtained by replacing p by q, Á , t, one at a time. One way to remember this equation is to think of the right-hand side as the dot product of two vectors with components a

0w 0w 0w , ,Á, b 0x 0y 0y

and

a

0x 0y 0y , , Á , b. 0p 0p 0p

('''')''''*

('''')''''*

Derivatives of w with respect to the intermediate variables

Derivatives of the intermediate variables with respect to the selected independent variable

13.4

721

The Chain Rule

Exercises 13.4 Chain Rule: One Independent Variable In Exercises 1–6, (a) express dw> dt as a function of t, both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t. Then (b) evaluate dw> dt at the given value of t. 1. w = x 2 + y 2,

x = cos t,

y = sin t;

5. w = 2ye - ln z, t = 1 x

6. w = z - sin xy,

x = cos t,

y = sin t,

x = ln st + 1d,

-1

2

x = t,

y = ln t,

y = tan

z = e t - 1;

z = 42t ;

8. z = tan-1 sx>yd, x = u cos y, su, yd = s1.3, p>6d

z = e;

t,

t = 1

y = u sin y; y = u sin y;

In Exercises 9 and 10, (a) express 0w>0u and 0w>0y as functions of u and y both by using the Chain Rule and by expressing w directly in terms of u and y before differentiating. Then (b) evaluate 0w>0u and 0w>0y at the given point (u, y). 9. w = xy + yz + xz, su, yd = s1>2, 1d

x = u + y,

y = u - y, y

10. w = ln sx + y + z d, x = ue sin u, z = ue y; su, yd = s -2, 0d 2

2

2

z = uy ; y

y = ue cos u,

In Exercises 11 and 12, (a) express 0u>0x, 0u>0y, and 0u>0z as functions of x, y, and z both by using the Chain Rule and by expressing u directly in terms of x, y, and z before differentiating. Then (b) evaluate 0u>0x, 0u>0y, and 0u>0z at the given point (x, y, z). p - q 11. u = q - r , p = x + y + z, q = x - y + z, r = x + y - z;

sx, y, zd =

A 23, 2, 1 B

12. u = e qr sin-1 p, p = sin x, q = z 2 ln y, sx, y, zd = sp>4, 1>2, -1>2d

r = 1>z;

Using a Branch Diagram In Exercises 13–24, draw a branch diagram and write a Chain Rule formula for each derivative. dz for z = ƒsx, yd, x = gstd, y = hstd dt dz for z = ƒsu, y, wd, u = gstd, y = hstd, w = kstd 14. dt 0w 0w and for w = hsx, y, zd, x = ƒsu, yd, y = gsu, yd, 15. 0u 0y z = ksu, yd 13.

r = gsx, yd,

s = hsx, yd,

0w 0w and for w = gsx, yd, x = hsu, yd, y = ksu, yd 0u 0y 0w 0w 18. and for w = gsu, yd, u = hsx, yd, y = ksx, yd 0x 0y 19.

0z 0z and for z = ƒsx, yd, 0t 0s

x = gst, sd,

y = hst, sd

0y for y = ƒsud, u = gsr, sd 0r 0w 0w 21. and for w = gsud, u = hss, td 0s 0t 0w 22. for w = ƒsx, y, z, yd, x = gs p, qd, 0p z = js p, qd, y = ks p, qd 20.

t

Chain Rule: Two and Three Independent Variables In Exercises 7 and 8, (a) express 0z>0u and 0z>0y as functions of u and y both by using the Chain Rule and by expressing z directly in terms of u and y before differentiating. Then (b) evaluate 0z>0u and 0z>0y at the given point su, yd. 7. z = 4e x ln y, x = ln su cos yd, su, yd = s2, p>4d

0w 0w and for w = ƒsr, s, td, 0x 0y t = ksx, yd

17.

t = p

2. w = x 2 + y 2, x = cos t + sin t, y = cos t - sin t; t = 0 y x 3. w = z + z , x = cos2 t, y = sin2 t, z = 1>t; t = 3 4. w = ln sx 2 + y 2 + z 2 d, t = 3

16.

y = hs p, qd,

0w 0w and for w = ƒsx, yd, x = gsrd, y = hssd 0r 0s 0w 24. for w = gsx, yd, x = hsr, s, td, y = ksr, s, td 0s 23.

Implicit Differentiation Assuming that the equations in Exercises 25–28 define y as a differentiable function of x, use Theorem 8 to find the value of dy> dx at the given point. 25. x 3 - 2y 2 + xy = 0,

s1, 1d

26. xy + y 2 - 3x - 3 = 0,

s -1, 1d

27. x + xy + y - 7 = 0,

s1, 2d

2

2

28. xe y + sin xy + y - ln 2 = 0,

s0, ln 2d

Find the values of 0z>0x and 0z>0y at the points in Exercises 29–32. 29. z 3 - xy + yz + y 3 - 2 = 0, 1 1 1 30. x + y + z - 1 = 0,

s1, 1, 1d

s2, 3, 6d

31. sin sx + yd + sin s y + zd + sin sx + zd = 0, 32. xe + ye + 2 ln x - 2 - 3 ln 2 = 0, y

z

sp, p, pd

s1, ln 2, ln 3d

Finding Partial Derivatives at Specified Points 33. Find 0w>0r when r = 1, s = -1 if w = sx + y + zd2, x = r - s, y = cos sr + sd, z = sin sr + sd. 34. Find 0w>0y when u = -1, y = 2 x = y 2>u, y = u + y, z = cos u. 35. Find 0w>0y when u = 0, y = 0 x = u - 2y + 1, y = 2u + y - 2. 36. Find 0z>0u when u = 0, y = 1 x = u 2 + y 2, y = uy. 37. Find 0z>0u and 0z>0y when 5 tan-1 x and x = e u + ln y.

if if

if

w = xy + ln z, w = x 2 + s y>xd,

z = sin xy + x sin y,

u = ln 2, y = 1

if

z=

38. Find 0z>0u and 0z>0y when u = 1, y = -2 if z = ln q and q = 1y + 3 tan-1 u.

722

Chapter 13: Partial Derivatives

Theory and Examples 0w 0w 39. Assume that w = ƒss 3 + t 2 d and ƒ¿sxd = e x. Find and . 0t 0s 0f 0f x2 2 s 40. Assume that w = ƒ Qts , t R , sx, yd = xy, and sx, yd = . 0x 0y 2 0w 0w Find and . 0t 0s 41. Changing voltage in a circuit The voltage V in a circuit that satisfies the law V = IR is slowly dropping as the battery wears out. At the same time, the resistance R is increasing as the resistor heats up. Use the equation

47. Extreme values on a helix Suppose that the partial derivatives of a function ƒ(x, y, z) at points on the helix x = cos t, y = sin t, z = t are

0V dI 0V dR dV = + 0I dt 0R dt dt

49. Temperature on a circle Let T = ƒsx, yd be the temperature at the point (x, y) on the circle x = cos t, y = sin t, 0 … t … 2p and suppose that

to find how the current is changing at the instant when R = 600 ohms, I = 0.04 amp, dR>dt = 0.5 ohm>sec, and dV>dt = -0.01 volt>sec. V

ƒx = cos t,

ƒy = sin t,

ƒz = t 2 + t - 2.

At what points on the curve, if any, can ƒ take on extreme values? 48. A space curve Let w = x 2e 2y cos 3z . Find the value of dw> dt at the point s1, ln 2, 0d on the curve x = cos t, y = ln st + 2d, z = t.

0T = 8x - 4y, 0x

0T = 8y - 4x. 0y

a. Find where the maximum and minimum temperatures on the circle occur by examining the derivatives dT> dt and d 2T>dt 2.

Battery

b. Suppose that T = 4x 2 - 4xy + 4y 2. Find the maximum and minimum values of T on the circle.

I

50. Temperature on an ellipse Let T = g sx, yd be the temperature at the point (x, y) on the ellipse

R

42. Changing dimensions in a box The lengths a, b, and c of the edges of a rectangular box are changing with time. At the instant in question, a = 1 m, b = 2 m, c = 3 m, da>dt = db>dt = 1 m>sec, and dc>dt = -3 m>sec. At what rates are the box’s volume V and surface area S changing at that instant? Are the box’s interior diagonals increasing in length or decreasing? 43. If ƒ(u, y, w) is differentiable and u = x - y, y = y - z, and w = z - x, show that 0ƒ 0ƒ 0ƒ + + = 0. 0x 0y 0z 44. Polar coordinates Suppose that we substitute polar coordinates x = r cos u and y = r sin u in a differentiable function w = ƒsx, yd.

x = 222 cos t,

y = 22 sin t,

0 … t … 2p,

and suppose that 0T = y, 0x

0T = x. 0y

a. Locate the maximum and minimum temperatures on the ellipse by examining dT> dt and d 2T>dt 2. b. Suppose that T = xy - 2. Find the maximum and minimum values of T on the ellipse. Differentiating Integrals Under mild continuity restrictions, it is true that if

a. Show that

b

Fsxd =

0w = ƒx cos u + ƒy sin u 0r

La

g st, xd dt,

b

and

then F¿sxd =

g x st, xd dt. Using this fact and the Chain Rule, we La can find the derivative of

1 0w r 0u = -ƒx sin u + ƒy cos u. b. Solve the equations in part (a) to express ƒx and ƒy in terms of 0w>0r and 0w>0u.

ƒsxd

Fsxd =

c. Show that sƒx d2 + sƒy d2 = a

2

g st, xd dt

La

2

0w 1 0w b + 2 a b. 0r 0u r

by letting u

45. Laplace equations Show that if w = ƒsu, yd satisfies the Laplace equation ƒuu + ƒyy = 0 and if u = sx 2 - y 2 d>2 and y = xy, then w satisfies the Laplace equation wxx + wyy = 0. 46. Laplace equations Let w = ƒsud + g syd, where u = x + iy, y = x - iy, and i = 2 -1. Show that w satisfies the Laplace equation wxx + wyy = 0 if all the necessary functions are differentiable.

Gsu, xd =

La

g st, xd dt,

where u = ƒsxd. Find the derivatives of the functions in Exercises 51 and 52. x2

51. Fsxd =

L0

1

2t 4 + x 3 dt

52. Fsxd =

Lx

2

2t 3 + x 2 dt

13.5

Directional Derivatives and Gradient Vectors

723

Directional Derivatives and Gradient Vectors

13.5

If you look at the map (Figure 13.25) showing contours within Yosemite National Park in California, you will notice that the streams flow perpendicular to the contours. The streams are following paths of steepest descent so the waters reach lower elevations as quickly as possible. Therefore, the fastest instantaneous rate of change in a stream’s elevation above sea level has a particular direction. In this section, you will see why this direction, called the “downhill” direction, is perpendicular to the contours.

FIGURE 13.25 Contours within Yosemite National Park in California show streams, which follow paths of steepest descent, running perpendicular to the contours. (Source: http://www.usgs.gov)

Directional Derivatives in the Plane We know from Section 13.4 that if ƒ(x, y) is differentiable, then the rate at which ƒ changes with respect to t along a differentiable curve x = gstd, y = hstd is

y Line x x 0 su 1, y y 0 su 2

dƒ 0ƒ dx 0ƒ dy = + . 0x dt 0y dt dt

u u1i u 2 j Direction of increasing s R P0(x 0, y0 ) 0

x

FIGURE 13.26 The rate of change of ƒ in the direction of u at a point P0 is the rate at which ƒ changes along this line at P0 .

At any point P0sx0, y0 d = P0sgst0 d, hst0 dd, this equation gives the rate of change of ƒ with respect to increasing t and therefore depends, among other things, on the direction of motion along the curve. If the curve is a straight line and t is the arc length parameter along the line measured from P0 in the direction of a given unit vector u, then dƒ> dt is the rate of change of ƒ with respect to distance in its domain in the direction of u. By varying u, we find the rates at which ƒ changes with respect to distance as we move through P0 in different directions. We now define this idea more precisely. Suppose that the function ƒ(x, y) is defined throughout a region R in the xy-plane, that P0sx0 , y0 d is a point in R, and that u = u1 i + u2 j is a unit vector. Then the equations x = x0 + su1,

y = y0 + su2

parametrize the line through P0 parallel to u. If the parameter s measures arc length from P0 in the direction of u, we find the rate of change of ƒ at P0 in the direction of u by calculating dƒ> ds at P0 (Figure 13.26).

724

Chapter 13: Partial Derivatives

DEFINITION The derivative of ƒ at P0(x 0 , y0) in the direction of the unit vector u u 1i u 2 j is the number a

dƒ ƒsx0 + su1, y0 + su2 d - ƒsx0 , y0 d , b = lim s ds u,P0 s:0

(1)

provided the limit exists.

The directional derivative defined by Equation (1) is also denoted by “The derivative of ƒ at P0 in the direction of u”

sDu ƒdP0.

The partial derivatives ƒxsx0, y0 d and ƒysx0, y0 d are the directional derivatives of ƒ at P0 in the i and j directions. This observation can be seen by comparing Equation (1) to the definitions of the two partial derivatives given in Section 13.3.

EXAMPLE 1

Using the definition, find the derivative of ƒsx, yd = x 2 + xy

at P0s1, 2d in the direction of the unit vector u = A 1> 22 B i + A 1> 22 B j. Solution

¢

Applying the definition in Equation (1), we obtain dƒ ƒsx0 + su1, y0 + su2 d - ƒsx0 , y0 d = lim ≤ s ds u,P0 s:0 ƒ¢1 + s # = lim

s:0

¢1 + = lim

s 22

1 1 ,2 + s# ≤ - ƒs1, 2d 22 22 s 2

≤ + ¢1 +

s 22

s:0

¢1 + z Surface S: z f (x, y)

= lim f (x 0 su1, y0 su 2 ) f (x 0 , y0 )

= lim

⎫ ⎬ ⎭

P(x 0 , y0 , z 0 )

s:0

22

≤ - s12 + 1 # 2d

3s s2 s2 + ¢2 + + ≤ - 3 ≤ 2 2 22 s

+ s2 s

s

s

= lim ¢ s:0

5 22

+ s≤ =

5 22

.

y

C

P0(x 0 , y0 )

22

+

≤ ¢2 +

The rate of change of ƒsx, yd = x 2 + xy at P0s1, 2d in the direction u is 5> 12.

s

x

22

s:0

5s Tangent line Q

2s

Eq. (1)

Interpretation of the Directional Derivative (x 0 su1, y0 su 2 ) u u 1i u 2 j

FIGURE 13.27 The slope of curve C at P0 is lim slope (PQ); this is the Q:P

directional derivative dƒ a b = sDu ƒdP0. ds u, P0

The equation z = ƒsx, yd represents a surface S in space. If z0 = ƒsx0 , y0 d, then the point Psx0 , y0 , z0 d lies on S. The vertical plane that passes through P and P0sx0 , y0 d parallel to u intersects S in a curve C (Figure 13.27). The rate of change of ƒ in the direction of u is the slope of the tangent to C at P in the right-handed system formed by the vectors u and k. When u = i, the directional derivative at P0 is 0ƒ>0x evaluated at sx0 , y0 d. When u = j, the directional derivative at P0 is 0ƒ>0y evaluated at sx0 , y0 d. The directional derivative generalizes the two partial derivatives. We can now ask for the rate of change of ƒ in any direction u, not just the directions i and j.

13.5

725

Directional Derivatives and Gradient Vectors

For a physical interpretation of the directional derivative, suppose that T = ƒsx, yd is the temperature at each point (x, y) over a region in the plane. Then ƒsx0 , y0 d is the temperature at the point P0sx0 , y0 d and sDu ƒdP0 is the instantaneous rate of change of the temperature at P0 stepping off in the direction u.

Calculation and Gradients We now develop an efficient formula to calculate the directional derivative for a differentiable function ƒ. We begin with the line x = x0 + su1,

y = y0 + su2 ,

(2)

through P0sx0 , y0 d, parametrized with the arc length parameter s increasing in the direction of the unit vector u = u1 i + u2 j. Then by the Chain Rule we find a

dƒ 0ƒ 0ƒ dy dx = a b b + a b 0x P0 ds 0y P0 ds ds u,P0 = a

Chain Rule for differentiable ƒ

0ƒ 0ƒ b u + a b u2 0x P0 1 0y P0

= ca

From Eqs. (2), dx>ds = u1 and dy>ds = u2

0ƒ 0ƒ b i + a b j d # cu1 i + u2 j d. 0x P0 0y P0

144442444443 Gradient of ƒ at P0

(3)

14243 Direction u

Equation (3) says that the derivative of a differentiable function ƒ in the direction of u at P0 is the dot product of u with the special vector called the gradient of ƒ at P0 .

DEFINITION

The gradient vector (gradient) of ƒ(x, y) at a point P0sx0 , y0 d

is the vector §ƒ =

0ƒ 0ƒ i + j 0x 0y

obtained by evaluating the partial derivatives of ƒ at P0. The notation §ƒ is read “grad ƒ” as well as “gradient of ƒ” and “del ƒ.” The symbol § by itself is read “del.” Another notation for the gradient is grad ƒ.

THEOREM 9—The Directional Derivative Is a Dot Product differentiable in an open region containing P0sx0 , y0 d, then a

dƒ b = s§ƒdP0 # u, ds u,P0

If ƒsx, yd is

(4)

the dot product of the gradient §ƒ at P0 and u. Find the derivative of ƒsx, yd = xe y + cos sxyd at the point (2, 0) in the direction of v = 3i - 4j.

EXAMPLE 2 Solution

The direction of v is the unit vector obtained by dividing v by its length: u =

v v 3 4 = = i - j. 5 5 5 v ƒ ƒ

726

Chapter 13: Partial Derivatives

y

The partial derivatives of ƒ are everywhere continuous and at (2, 0) are given by fxs2, 0d = se y - y sin sxydds2,0d = e 0 - 0 = 1

∇f i 2j

2

fys2, 0d = sxe y - x sin sxydds2,0d = 2e 0 - 2 # 0 = 2.

The gradient of ƒ at (2, 0) is

1

§ƒ ƒ s2,0d = ƒxs2, 0di + ƒys2, 0dj = i + 2j 0 –1

1 P0 (2, 0)

3

4

x

(Figure 13.28). The derivative of ƒ at (2, 0) in the direction of v is therefore sDuƒd ƒ s2,0d = §ƒ ƒ s2,0d # u

u 3i 4j 5 5

FIGURE 13.28 Picture §ƒ as a vector in the domain of ƒ. The figure shows a number of level curves of ƒ. The rate at which ƒ changes at (2, 0) in the direction u = s3>5di - s4>5dj is §ƒ # u = -1 (Example 2).

Eq. (4)

8 3 3 4 = si + 2jd # a i - jb = - = -1. 5 5 5 5 Evaluating the dot product in the formula

Du ƒ = §ƒ # u = ƒ §ƒ ƒ ƒ u ƒ cos u = ƒ §ƒ ƒ cos u,

where u is the angle between the vectors u and §ƒ, reveals the following properties. Properties of the Directional Derivative Duƒ = §ƒ # u = ƒ §ƒ ƒ cos u 1. The function ƒ increases most rapidly when cos u = 1 or when u = 0 and u is the direction of §ƒ. That is, at each point P in its domain, ƒ increases most rapidly in the direction of the gradient vector §ƒ at P. The derivative in this direction is Duƒ = ƒ §ƒ ƒ cos s0d = ƒ §ƒ ƒ . 2. Similarly, ƒ decreases most rapidly in the direction of - §ƒ. The derivative in this direction is Duƒ = ƒ §ƒ ƒ cos spd = - ƒ §ƒ ƒ . 3. Any direction u orthogonal to a gradient §f Z 0 is a direction of zero change in ƒ because u then equals p>2 and Duƒ = ƒ §ƒ ƒ cos sp>2d = ƒ §ƒ ƒ # 0 = 0. As we discuss later, these properties hold in three dimensions as well as two.

EXAMPLE 3

Find the directions in which ƒsx, yd = sx 2>2d + s y 2>2d

(a) increases most rapidly at the point (1, 1). (b) decreases most rapidly at (1, 1). (c) What are the directions of zero change in ƒ at (1, 1)? Solution

(a) The function increases most rapidly in the direction of §ƒ at (1, 1). The gradient there is s§ƒds1,1d = sxi + yjds1,1d = i + j. Its direction is u =

i + j i + j 1 1 = = i + j. 2 2 ƒi + jƒ 2s1d + s1d 22 22

(b) The function decreases most rapidly in the direction of - §ƒ at (1, 1), which is -u = -

1 1 i j. 22 22

13.5

727

(c) The directions of zero change at (1, 1) are the directions orthogonal to §ƒ:

z f (x, y) x2 y2 2 2

z

Directional Derivatives and Gradient Vectors

n = -

1 1 i + j 22 22

and

-n =

1 1 i j. 22 22

See Figure 13.29.

(1, 1, 1)

Gradients and Tangents to Level Curves 1 1

x

–∇f

y

(1, 1)

Zero change in f

Most rapid decrease in f

If a differentiable function ƒsx, yd has a constant value c along a smooth curve r = gstdi + hstdj (making the curve a level curve of ƒ), then ƒsgstd, hstdd = c. Differentiating both sides of this equation with respect to t leads to the equations d d ƒsgstd, hstdd = scd dt dt 0ƒ dg 0ƒ dh + = 0 0x dt 0y dt 0ƒ 0ƒ dg dh a i + jb # a i + jb = 0. 0x 0y dt dt

∇f i j

Most rapid increase in f

FIGURE 13.29 The direction in which ƒsx, yd increases most rapidly at (1, 1) is the direction of §ƒ ƒ s1,1d = i + j. It corresponds to the direction of steepest ascent on the surface at (1, 1, 1) (Example 3).

('')''* §ƒ

Chain Rule

(5)

('')''* dr dt

Equation (5) says that §ƒ is normal to the tangent vector dr> dt, so it is normal to the curve. The level curve f (x, y) f (x 0 , y 0 )

At every point sx0 , y0 d in the domain of a differentiable function ƒ(x, y), the gradient of ƒ is normal to the level curve through sx0 , y0 d (Figure 13.30).

(x 0 , y 0 ) ∇f (x 0 , y 0 )

FIGURE 13.30 The gradient of a differentiable function of two variables at a point is always normal to the function’s level curve through that point.

Equation (5) validates our observation that streams flow perpendicular to the contours in topographical maps (see Figure 13.25). Since the downflowing stream will reach its destination in the fastest way, it must flow in the direction of the negative gradient vectors from Property 2 for the directional derivative. Equation (5) tells us these directions are perpendicular to the level curves. This observation also enables us to find equations for tangent lines to level curves. They are the lines normal to the gradients. The line through a point P0sx0 , y0 d normal to a vector N = Ai + Bj has the equation Asx - x0 d + Bs y - y0 d = 0 (Exercise 39). If N is the gradient s§ƒdsx0, y0d = ƒxsx0 , y0 di + ƒysx0 , y0 dj, the equation is the tangent line given by ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 dsy - y0 d = 0.

y x 2y –4

∇f (–2, 1) – i 2j

–2

–1

1 0

Find an equation for the tangent to the ellipse x2 + y2 = 2 4

x2 y2 2 4

2 (–2, 1)

EXAMPLE 4

1

2

x 22

(Figure 13.31) at the point s -2, 1d. Solution

The ellipse is a level curve of the function ƒsx, yd =

FIGURE 13.31 We can find the tangent to the ellipse sx 2>4d + y 2 = 2 by treating the ellipse as a level curve of the function ƒsx, yd = sx 2>4d + y 2 (Example 4).

x2 + y 2. 4

The gradient of ƒ at s -2, 1d is x §ƒ ƒ s-2,1d = a i + 2yjb = -i + 2j. 2 s-2,1d

(6)

728

Chapter 13: Partial Derivatives

The tangent is the line s -1dsx + 2d + s2ds y - 1d = 0

Eq. (6)

x - 2y = -4. If we know the gradients of two functions ƒ and g, we automatically know the gradients of their sum, difference, constant multiples, product, and quotient. You are asked to establish the following rules in Exercise 40. Notice that these rules have the same form as the corresponding rules for derivatives of single-variable functions.

Algebra Rules for Gradients 1. 2. 3. 4.

Sum Rule: Difference Rule: Constant Multiple Rule: Product Rule:

5. Quotient Rule:

EXAMPLE 5

§sƒ + gd = §ƒ + §g §sƒ - gd = §ƒ - §g §skƒd = k§ƒ sany number kd §sƒgd = ƒ§g + g§ƒ ƒ g§ƒ - ƒ§g § ag b = g2

We illustrate two of the rules with ƒsx, yd = x - y §ƒ = i - j

gsx, yd = 3y §g = 3j.

We have 1. §sƒ - gd = §sx - 4yd = i - 4j = §ƒ - §g 2.

Rule 2

§sƒgd = §s3xy - 3y 2 d = 3yi + s3x - 6ydj = 3ysi - jd + 3yj + s3x - 6ydj = 3ysi - jd + s3x - 3ydj = 3ysi - jd + sx - yd3j = g§ƒ + ƒ§g

Rule 4

Functions of Three Variables For a differentiable function ƒ(x, y, z) and a unit vector u = u1 i + u2 j + u3 k in space, we have §ƒ =

0ƒ 0ƒ 0ƒ i + j + k 0x 0y 0z

and Duƒ = §ƒ # u =

0ƒ 0ƒ 0ƒ u1 + u2 + u. 0x 0y 0z 3

The directional derivative can once again be written in the form Du ƒ = §ƒ # u = ƒ §ƒ ƒ ƒ u ƒ cos u = ƒ §ƒ ƒ cos u, so the properties listed earlier for functions of two variables extend to three variables. At any given point, ƒ increases most rapidly in the direction of §ƒ and decreases most rapidly in the direction of - §ƒ. In any direction orthogonal to §ƒ, the derivative is zero.

13.5

Directional Derivatives and Gradient Vectors

729

EXAMPLE 6 (a) Find the derivative of ƒsx, y, zd = x 3 - xy 2 - z at P0s1, 1, 0d in the direction of v = 2i - 3j + 6k. (b) In what directions does ƒ change most rapidly at P0 , and what are the rates of change in these directions? Solution

(a) The direction of v is obtained by dividing v by its length: ƒ v ƒ = 2s2d2 + s -3d2 + s6d2 = 249 = 7 u =

3 6 v 2 = i - j + k. 7 7 7 ƒvƒ

The partial derivatives of ƒ at P0 are ƒx = s3x 2 - y 2 ds1,1,0d = 2,

ƒy = -2xy ƒ s1,1,0d = -2,

ƒz = -1 ƒ s1,1,0d = -1.

The gradient of ƒ at P0 is §ƒ ƒ s1,1,0d = 2i - 2j - k. The derivative of ƒ at P0 in the direction of v is therefore 3 6 2 sDuƒds1,1,0d = §ƒ ƒ s1,1,0d # u = s2i - 2j - kd # a i - j + kb 7 7 7 =

6 6 4 4 + - = . 7 7 7 7

(b) The function increases most rapidly in the direction of §ƒ = 2i - 2j - k and decreases most rapidly in the direction of - §ƒ. The rates of change in the directions are, respectively, ƒ §ƒ ƒ = 2s2d2 + s -2d2 + s -1d2 = 29 = 3

- ƒ §ƒ ƒ = -3 .

and

Exercises 13.5 Calculating Gradients In Exercises 1– 6, find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point. 1. ƒsx, yd = y - x, 3. gsx, yd = xy ,

s2, 1d s2, -1d

2

5. ƒsx, yd = 22x + 3y, 6. ƒsx, yd = tan

-1

2x y ,

2. ƒsx, yd = ln sx 2 + y 2 d, y2 x2 4. gsx, yd = - , 2 2

A 22, 1 B

s -1, 2d s4, -2d

In Exercises 7–10, find §f at the given point. 7. ƒsx, y, zd = x 2 + y 2 - 2z 2 + z ln x,

9. ƒsx, y, zd = sx 2 + y 2 + z 2 d-1>2 + ln sxyzd, 10. ƒsx, y, zd = e

x+y

cos z + s y + 1d sin

-1

x,

11. ƒsx, yd = 2xy - 3y 2,

P0s5, 5d,

u = 4i + 3j

12. ƒsx, yd = 2x + y ,

P0s -1, 1d,

u = 3i - 4j

2

13. gsx, yd =

2

x - y , xy + 2

s1, 1, 1d s -1, 2, -2d s0, 0, p>6d

P0s1, -1d,

u = 12i + 5j

14. hsx, yd = tan-1 sy>xd + 23 sin-1 sxy>2d, u = 3i - 2j

P0s1, 1d,

15. ƒsx, y, zd = xy + yz + zx,

u = 3i + 6j - 2k

P0s1, -1, 2d,

16. ƒsx, y, zd = x + 2y - 3z , 2

s1, 1, 1d

8. ƒsx, y, zd = 2z 3 - 3sx 2 + y 2 dz + tan-1 xz,

s1, 1d

Finding Directional Derivatives In Exercises 11–18, find the derivative of the function at P0 in the direction of u.

2

2

17. gsx, y, zd = 3e cos yz, x

18. hsx, y, zd = cos xy + e u = i + 2j + 2k

P0s1, 1, 1d,

P0s0, 0, 0d, yz

+ ln zx,

u = i + j + k

u = 2i + j - 2k

P0s1, 0, 1>2d,

730

Chapter 13: Partial Derivatives

In Exercises 19–24, find the directions in which the functions increase and decrease most rapidly at P0. Then find the derivatives of the functions in these directions. 19. ƒsx, yd = x 2 + xy + y 2,

P0s -1, 1d

20. ƒsx, yd = x 2y + e xy sin y,

P0s1, 0d

21. ƒsx, y, zd = sx>yd - yz, 22. gsx, y, zd = xe y + z 2,

P0s1, ln 2, 1>2d P0s1, 1, 1d

24. hsx, y, zd = ln sx + y - 1d + y + 6z, 2

2

P0s1, 1, 0d

Tangent Lines to Level Curves In Exercises 25–28, sketch the curve ƒsx, yd = c together with §ƒ and the tangent line at the given point. Then write an equation for the tangent line. 25. x 2 + y 2 = 4, 26. x - y = 1, 2

27. xy = -4,

A 22, 22 B

34. Changing temperature along a circle Is there a direction u in which the rate of change of the temperature function T sx, y, zd = 2xy - yz (temperature in degrees Celsius, distance in feet) at Ps1, -1, 1d is -3°C>ft ? Give reasons for your answer. 35. The derivative of ƒ(x, y) at P0s1, 2d in the direction of i + j is 212 and in the direction of -2j is -3 . What is the derivative of ƒ in the direction of -i - 2j ? Give reasons for your answer. 36. The derivative of ƒ(x, y, z) at a point P is greatest in the direction of v = i + j - k. In this direction, the value of the derivative is 2 13.

A 22, 1 B

s2, -2d

28. x 2 - xy + y 2 = 7,

32. Zero directional derivative In what directions is the derivative of ƒsx, yd = sx 2 - y 2 d>sx 2 + y 2 d at P(1, 1) equal to zero? 33. Is there a direction u in which the rate of change of ƒsx, yd = x 2 - 3xy + 4y 2 at P(1, 2) equals 14? Give reasons for your answer.

P0s4, 1, 1d

23. ƒsx, y, zd = ln xy + ln yz + ln xz,

31. Zero directional derivative In what direction is the derivative of ƒsx, yd = xy + y 2 at P(3, 2) equal to zero?

a. What is §ƒ at P? Give reasons for your answer.

s -1, 2d

b. What is the derivative of ƒ at P in the direction of i + j ?

Theory and Examples 29. Let ƒsx, yd = x 2 - xy + y 2 - y. Find the directions u and the values of Du ƒs1, -1d for which a. Du ƒs1, -1d is largest

b. Du ƒs1, -1d is smallest

c. Du ƒs1, -1d = 0

d. Du ƒs1, -1d = 4

e. Du ƒs1, -1d = -3 (x - y) . Find the directions u and the values of 30. Let ƒsx, yd = (x + y) 1 3 Du ƒ a- , b for which 2 2 1 3 a. Du ƒ a- , b is largest 2 2

1 3 b. Du ƒ a- , b is smallest 2 2

1 3 c. Du ƒ a- , b = 0 2 2

1 3 d. Du ƒ a- , b = -2 2 2

37. Directional derivatives and scalar components How is the derivative of a differentiable function ƒ(x, y, z) at a point P0 in the direction of a unit vector u related to the scalar component of s§ƒdP0 in the direction of u? Give reasons for your answer. 38. Directional derivatives and partial derivatives Assuming that the necessary derivatives of ƒ(x, y, z) are defined, how are D i ƒ, D j ƒ, and D k ƒ related to ƒx , ƒy , and ƒz? Give reasons for your answer. 39. Lines in the xy-plane Show that Asx - x0 d + Bsy - y0 d = 0 is an equation for the line in the xy-plane through the point sx0 , y0 d normal to the vector N = Ai + Bj. 40. The algebra rules for gradients Given a constant k and the gradients

1 3 e. Du ƒ a- , b = 1 2 2

13.6

§ƒ =

0ƒ 0ƒ 0ƒ i + j + k, 0x 0y 0z

§g =

0g 0g 0g i + j + k, 0x 0y 0z

establish the algebra rules for gradients.

Tangent Planes and Differentials In this section we define the tangent plane at a point on a smooth surface in space. Then we show how to calculate an equation of the tangent plane from the partial derivatives of the function defining the surface. This idea is similar to the definition of the tangent line at a point on a curve in the coordinate plane for single-variable functions (Section 3.1). We then study the total differential and linearization of functions of several variables.

Tangent Planes and Normal Lines If r = gstdi + hstdj + kstdk is a smooth curve on the level surface ƒsx, y, zd = c of a differentiable function ƒ, then ƒsgstd, hstd, kstdd = c. Differentiating both sides of this

13.6

Tangent Planes and Differentials

731

equation with respect to t leads to ∇f

d d ƒsgstd, hstd, kstdd = scd dt dt 0ƒ dh 0ƒ dk 0ƒ dg + + = 0 0x dt 0y dt 0z dt 0ƒ 0ƒ 0ƒ dg dh dk a i + j + kb # a i + j + kb = 0. 0x 0y 0z dt dt dt

v2 P0

v1 f (x, y, z) c

FIGURE 13.32 The gradient §ƒ is orthogonal to the velocity vector of every smooth curve in the surface through P0. The velocity vectors at P0 therefore lie in a common plane, which we call the tangent plane at P0.

('''')''''* §ƒ

Chain Rule

(1)

('''')''''* dr>dt

At every point along the curve, §ƒ is orthogonal to the curve’s velocity vector. Now let us restrict our attention to the curves that pass through P0 (Figure 13.32). All the velocity vectors at P0 are orthogonal to §ƒ at P0, so the curves’ tangent lines all lie in the plane through P0 normal to §ƒ. We now define this plane.

DEFINITIONS The tangent plane at the point P0sx0 , y0 , z0 d on the level surface ƒsx, y, zd = c of a differentiable function ƒ is the plane through P0 normal to §ƒ ƒ P0. The normal line of the surface at P0 is the line through P0 parallel to §ƒ ƒ P0. From Section 11.5, the tangent plane and normal line have the following equations:

Tangent Plane to ƒsx, y, zd = c at P0sx0 , y0 , z0 d ƒxsP0 dsx - x0 d + ƒy sP0 dsy - y0 d + ƒzsP0 dsz - z0 d = 0

(2)

Normal Line to ƒsx, y, zd = c at P0sx0 , y0 , z0 d x = x0 + ƒxsP0 dt, z P0(1, 2, 4)

The surface x 2 1 y2 1 z 2 9 5 0

EXAMPLE 1

y = y0 + ƒy sP0 dt,

z = z0 + ƒzsP0 dt

(3)

Find the tangent plane and normal line of the surface ƒsx, y, zd = x 2 + y 2 + z - 9 = 0

A circular paraboloid

at the point P0s1, 2, 4d. Normal line

Tangent plane

The surface is shown in Figure 13.33. The tangent plane is the plane through P0 perpendicular to the gradient of ƒ at P0 . The gradient is Solution

§ƒ ƒ P0 = s2xi + 2yj + kds1,2,4d = 2i + 4j + k. The tangent plane is therefore the plane

y x

FIGURE 13.33 The tangent plane and normal line to this surface at P0 (Example 1).

2sx - 1d + 4s y - 2d + sz - 4d = 0,

or

2x + 4y + z = 14.

The line normal to the surface at P0 is x = 1 + 2t,

y = 2 + 4t,

z = 4 + t.

To find an equation for the plane tangent to a smooth surface z = ƒsx, yd at a point P0sx0 , y0 , z0 d where z0 = ƒsx0 , y0 d, we first observe that the equation z = ƒsx, yd is

732

Chapter 13: Partial Derivatives

equivalent to ƒsx, yd - z = 0. The surface z = ƒsx, yd is therefore the zero level surface of the function Fsx, y, zd = ƒsx, yd - z. The partial derivatives of F are 0 sƒsx, yd - zd = fx - 0 = fx 0x 0 Fy = sƒsx, yd - zd = fy - 0 = fy 0y 0 Fz = sƒsx, yd - zd = 0 - 1 = -1. 0z

Fx =

The formula FxsP0 dsx - x0 d + FysP0 dsy - y0 d + FzsP0 dsz - z0 d = 0 for the plane tangent to the level surface at P0 therefore reduces to ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 dsy - y0 d - sz - z0 d = 0.

Plane Tangent to a Surface z = ƒxx, yc at xx0 , y0 , ƒxx0 , y0cc The plane tangent to the surface z = ƒsx, yd of a differentiable function ƒ at the point P0sx0 , y0 , z0 d = sx0 , y0 , ƒsx0 , y0 dd is ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 dsy - y0 d - sz - z0 d = 0.

EXAMPLE 2

(4)

Find the plane tangent to the surface z = x cos y - ye x at (0, 0, 0).

We calculate the partial derivatives of ƒsx, yd = x cos y - ye x and use Equation (4): Solution

ƒxs0, 0d = scos y - ye x ds0,0d = 1 - 0 # 1 = 1

z

The plane x z 4 0 g(x, y, z)

ƒys0, 0d = s -x sin y - e x ds0,0d = 0 - 1 = -1. The tangent plane is therefore

1 # sx - 0d - 1 # s y - 0d - sz - 0d = 0,

Eq. (4)

or x - y - z = 0.

EXAMPLE 3

The surfaces ƒsx, y, zd = x 2 + y 2 - 2 = 0

∇g

A cylinder

and gsx, y, zd = x + z - 4 = 0

The ellipse E ∇f ∇g

∇f

(1, 1, 3)

A plane

meet in an ellipse E (Figure 13.34). Find parametric equations for the line tangent to E at the point P0s1, 1, 3d . The tangent line is orthogonal to both §ƒ and §g at P0, and therefore parallel to v = §ƒ * §g. The components of v and the coordinates of P0 give us equations for the line. We have §ƒ ƒ s1,1,3d = s2xi + 2yjds1,1,3d = 2i + 2j Solution

y x

The cylinder x2 y2 2 0 f (x, y, z)

FIGURE 13.34 This cylinder and plane intersect in an ellipse E (Example 3).

§g ƒ s1,1,3d = si + kds1,1,3d = i + k i 3 v = s2i + 2jd * si + kd = 2 1

j 2 0

k 0 3 = 2i - 2j - 2k. 1

13.6

Tangent Planes and Differentials

733

The tangent line is x = 1 + 2t,

y = 1 - 2t,

z = 3 - 2t.

Estimating Change in a Specific Direction The directional derivative plays the role of an ordinary derivative when we want to estimate how much the value of a function ƒ changes if we move a small distance ds from a point P0 to another point nearby. If ƒ were a function of a single variable, we would have dƒ = ƒ¿sP0 d ds.

Ordinary derivative * increment

For a function of two or more variables, we use the formula dƒ = s§ƒ ƒ P0 # ud ds,

Directional derivative * increment

where u is the direction of the motion away from P0. Estimating the Change in ƒ in a Direction u To estimate the change in the value of a differentiable function ƒ when we move a small distance ds from a point P0 in a particular direction u, use the formula dƒ = s§ƒ ƒ P0 # ud ds }

14243 Directional Distance derivative increment

EXAMPLE 4

Estimate how much the value of ƒsx, y, zd = y sin x + 2yz

will change if the point Psx, y, zd moves 0.1 unit from P0s0, 1, 0d straight toward P1s2, 2, -2d. 1 We first find the derivative of ƒ at P0 in the direction of the vector P0 P1 = 2i + j - 2k. The direction of this vector is 1 1 P0 P1 P0 P1 2 1 2 = i + j - k. u = 1 = 3 3 3 3 ƒ P0 P1 ƒ Solution

The gradient of ƒ at P0 is §ƒ ƒ s0,1,0d = ss y cos xdi + ssin x + 2zdj + 2ykds0,1,0d = i + 2k. Therefore, 2 1 2 2 4 2 §ƒ ƒ P0 # u = si + 2kd # a i + j - kb = - = - . 3 3 3 3 3 3 The change dƒ in ƒ that results from moving ds = 0.1 unit away from P0 in the direction of u is approximately 2 dƒ = s§ƒ ƒ P0 # udsdsd = a- bs0.1d L -0.067 unit. 3

How to Linearize a Function of Two Variables Functions of two variables can be complicated, and we sometimes need to approximate them with simpler ones that give the accuracy required for specific applications without being so difficult to work with. We do this in a way that is similar to the way we find linear replacements for functions of a single variable (Section 3.11).

734

Chapter 13: Partial Derivatives

A point near (x 0 , y 0 )

(x, y)

Suppose the function we wish to approximate is z = ƒsx, yd near a point sx0 , y0 d at which we know the values of ƒ, ƒx , and ƒy and at which ƒ is differentiable. If we move from sx0 , y0 d to any nearby point (x, y) by increments ¢x = x - x0 and ¢y = y - y0 (see Figure 13.35), then the definition of differentiability from Section 13.3 gives the change ƒsx, yd - ƒsx0 , y0 d = fxsx0 , y0 d¢x + ƒysx0 , y0 d¢y + P1 ¢x + P2 ¢y,

Dy 5 y 2 y0 A point where f is differentiable

where P1, P2 : 0 as ¢x, ¢y : 0. If the increments ¢x and ¢y are small, the products P1 ¢x and P2 ¢y will eventually be smaller still and we have the approximation ƒsx, yd L ƒsx0 , y0 d + ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 ds y - y0 d.

(x 0 , y0 ) D x x x 5 2 0

FIGURE 13.35 If ƒ is differentiable at sx0 , y0 d, then the value of ƒ at any point (x, y) nearby is approximately ƒsx0 , y0 d + ƒxsx0 , y0 d¢x + ƒysx0 , y0 d¢y.

(''''''''''''')'''''''''''''* Lsx, yd

In other words, as long as ¢x and ¢y are small, ƒ will have approximately the same value as the linear function L.

DEFINITIONS The linearization of a function ƒ(x, y) at a point sx0 , y0 d where ƒ is differentiable is the function Lsx, yd = ƒsx0 , y0 d + ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 ds y - y0 d.

(5)

The approximation ƒsx, yd L Lsx, yd is the standard linear approximation of ƒ at sx0 , y0 d.

From Equation (4), we find that the plane z = Lsx, yd is tangent to the surface z = ƒsx, yd at the point sx0 , y0 d. Thus, the linearization of a function of two variables is a tangent-plane approximation in the same way that the linearization of a function of a single variable is a tangent-line approximation. (See Exercise 55.)

EXAMPLE 5

Find the linearization of ƒsx, yd = x 2 - xy +

1 2 y + 3 2

at the point (3, 2). Solution

We first evaluate ƒ, ƒx , and ƒy at the point sx0 , y0 d = s3, 2d: ƒs3, 2d = ax 2 - xy +

1 2 = 8 y + 3b 2 s3,2d

ƒxs3, 2d =

0 1 ax 2 - xy + y 2 + 3b = s2x - yds3,2d = 4 0x 2 s3,2d

ƒys3, 2d =

0 1 ax 2 - xy + y 2 + 3b = s -x + yds3,2d = -1, 0y 2 s3,2d

giving Lsx, yd = ƒsx0 , y0 d + ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 ds y - y0 d = 8 + s4dsx - 3d + s -1ds y - 2d = 4x - y - 2. The linearization of ƒ at (3, 2) is Lsx, yd = 4x - y - 2. When approximating a differentiable function ƒ(x, y) by its linearization L(x, y) at sx0 , y0 d, an important question is how accurate the approximation might be.

13.6 y

735

If we can find a common upper bound M for ƒ ƒxx ƒ , ƒ ƒyy ƒ , and ƒ ƒxy ƒ on a rectangle R centered at sx0 , y0 d (Figure 13.36), then we can bound the error E throughout R by using a simple formula (derived in Section 13.9). The error is defined by Esx, yd = ƒsx, yd - Lsx, yd.

k h (x 0 , y0 ) R

0

Tangent Planes and Differentials

x

FIGURE 13.36 The rectangular region R: ƒ x - x0 ƒ … h, ƒ y - y0 ƒ … k in the xy-plane.

The Error in the Standard Linear Approximation If ƒ has continuous first and second partial derivatives throughout an open set containing a rectangle R centered at sx0 , y0 d and if M is any upper bound for the values of ƒ ƒxx ƒ , ƒ ƒyy ƒ , and ƒ ƒxy ƒ on R, then the error E(x, y) incurred in replacing ƒ(x, y) on R by its linearization Lsx, yd = ƒsx0 , y0 d + ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 dsy - y0 d satisfies the inequality 1 ƒ Esx, yd ƒ … 2 Ms ƒ x - x0 ƒ + ƒ y - y0 ƒ d2. To make ƒ Esx, yd ƒ small for a given M, we just make ƒ x - x0 ƒ and ƒ y - y0 ƒ small.

Differentials Recall from Section 3.11 that for a function of a single variable, y = ƒsxd, we defined the change in ƒ as x changes from a to a + ¢x by ¢ƒ = ƒsa + ¢xd - ƒsad and the differential of ƒ as dƒ = ƒ¿sad¢x. We now consider the differential of a function of two variables. Suppose a differentiable function ƒ(x, y) and its partial derivatives exist at a point sx0 , y0 d. If we move to a nearby point sx0 + ¢x, y0 + ¢yd, the change in ƒ is ¢ƒ = ƒsx0 + ¢x, y0 + ¢yd - ƒsx0 , y0 d. A straightforward calculation from the definition of L(x, y), using the notation x - x0 = ¢x and y - y0 = ¢y, shows that the corresponding change in L is ¢L = Lsx0 + ¢x, y0 + ¢yd - Lsx0 , y0 d = ƒxsx0 , y0 d¢x + ƒysx0 , y0 d¢y. The differentials dx and dy are independent variables, so they can be assigned any values. Often we take dx = ¢x = x - x0 , and dy = ¢y = y - y0 . We then have the following definition of the differential or total differential of ƒ.

DEFINITION If we move from sx0 , y0 d to a point sx0 + dx, y0 + dyd nearby, the resulting change dƒ = ƒxsx0 , y0 d dx + ƒysx0 , y0 d dy in the linearization of ƒ is called the total differential of ƒ.

EXAMPLE 6

Suppose that a cylindrical can is designed to have a radius of 1 in. and a height of 5 in., but that the radius and height are off by the amounts dr = +0.03 and dh = -0.1. Estimate the resulting absolute change in the volume of the can. Solution

To estimate the absolute change in V = pr 2h, we use ¢V L dV = Vr sr0 , h0 d dr + Vhsr0 , h0 d dh.

736

Chapter 13: Partial Derivatives

With Vr = 2prh and Vh = pr 2, we get dV = 2pr0 h0 dr + pr 02 dh = 2ps1ds5ds0.03d + ps1d2s -0.1d = 0.3p - 0.1p = 0.2p L 0.63 in3

EXAMPLE 7 Your company manufactures right circular cylindrical molasses storage tanks that are 25 ft high with a radius of 5 ft. How sensitive are the tanks’ volumes to small variations in height and radius? With V = pr 2h, the total differential gives the approximation for the change in

Solution

volume as

r5 r 25

h 25

h5 (a)

(b)

FIGURE 13.37 The volume of cylinder (a) is more sensitive to a small change in r than it is to an equally small change in h. The volume of cylinder (b) is more sensitive to small changes in h than it is to small changes in r (Example 7).

dV = Vr s5, 25d dr + Vhs5, 25d dh = s2prhds5,25d dr + spr 2 ds5,25d dh = 250p dr + 25p dh. Thus, a 1-unit change in r will change V by about 250p units. A 1-unit change in h will change V by about 25p units. The tank’s volume is 10 times more sensitive to a small change in r than it is to a small change of equal size in h. As a quality control engineer concerned with being sure the tanks have the correct volume, you would want to pay special attention to their radii. In contrast, if the values of r and h are reversed to make r = 25 and h = 5, then the total differential in V becomes dV = s2prhds25,5d dr + spr 2 ds25,5d dh = 250p dr + 625p dh. Now the volume is more sensitive to changes in h than to changes in r (Figure 13.37). The general rule is that functions are most sensitive to small changes in the variables that generate the largest partial derivatives.

Functions of More Than Two Variables Analogous results hold for differentiable functions of more than two variables. 1.

The linearization of ƒ(x, y, z) at a point P0sx0 , y0 , z0 d is Lsx, y, zd = ƒsP0 d + ƒxsP0 dsx - x0 d + ƒy sP0 ds y - y0 d + ƒzsP0 dsz - z0 d.

2.

3.

Suppose that R is a closed rectangular solid centered at P0 and lying in an open region on which the second partial derivatives of ƒ are continuous. Suppose also that ƒ ƒxx ƒ , ƒ ƒyy ƒ , ƒ ƒzz ƒ , ƒ ƒxy ƒ , ƒ ƒxz ƒ , and ƒ ƒyz ƒ are all less than or equal to M throughout R. Then the error Esx, y, zd = ƒsx, y, zd - Lsx, y, zd in the approximation of ƒ by L is bounded throughout R by the inequality 1 ƒ E ƒ … 2 Ms ƒ x - x0 ƒ + ƒ y - y0 ƒ + ƒ z - z0 ƒ d2. If the second partial derivatives of ƒ are continuous and if x, y, and z change from x0 , y0 , and z0 by small amounts dx, dy, and dz, the total differential dƒ = ƒxsP0 d dx + ƒysP0 d dy + ƒzsP0 d dz gives a good approximation of the resulting change in ƒ.

EXAMPLE 8

Find the linearization L(x, y, z) of ƒsx, y, zd = x 2 - xy + 3 sin z

at the point sx0 , y0 , z0 d = s2, 1, 0d. Find an upper bound for the error incurred in replacing ƒ by L on the rectangle R: ƒ x - 2 ƒ … 0.01, ƒ y - 1 ƒ … 0.02, ƒ z ƒ … 0.01. Solution

Routine calculations give

ƒs2, 1, 0d = 2,

ƒxs2, 1, 0d = 3,

ƒys2, 1, 0d = -2,

ƒzs2, 1, 0d = 3.

13.6

737

Tangent Planes and Differentials

Thus, Lsx, y, zd = 2 + 3sx - 2d + s -2ds y - 1d + 3sz - 0d = 3x - 2y + 3z - 2. Since ƒxx = 2,

ƒyy = 0,

ƒzz = -3 sin z,

ƒxy = -1,

ƒxz = 0,

ƒyz = 0,

and ƒ -3 sin z ƒ … 3 sin 0.01 L 0.03, we may take M = 2 as a bound on the second partials. Hence, the error incurred by replacing ƒ by L on R satisfies 1 ƒ E ƒ … 2 s2ds0.01 + 0.02 + 0.01d2 = 0.0016.

Exercises 13.6 Tangent Planes and Normal Lines to Surfaces In Exercises 1–8, find equations for the

Estimating Change 19. By about how much will ƒsx, y, zd = ln 2x 2 + y 2 + z 2

(a) tangent plane and (b) normal line at the point P0 on the given surface. 1. x 2 + y 2 + z 2 = 3, 2. x 2 + y 2 - z 2 = 18, 3. 2z - x = 0, 2

P0s3, 5, -4d

20. By about how much will ƒsx, y, zd = e x cos yz

P0s2, 0, 2d

4. x 2 + 2xy - y 2 + z 2 = 7, 5. cos px - x y + e 2

xz

2

7. x + y + z = 1,

P0s1, -1, 3d

+ yz = 4,

6. x - xy - y - z = 0, 2

change if the point P(x, y, z) moves from P0s3, 4, 12d a distance of ds = 0.1 unit in the direction of 3i + 6j - 2k?

P0s1, 1, 1d

change as the point P(x, y, z) moves from the origin a distance of ds = 0.1 unit in the direction of 2i + 2j - 2k?

P0s0, 1, 2d

21. By about how much will

P0s1, 1, -1d

gsx, y, zd = x + x cos z - y sin z + y

P0s0, 1, 0d

8. x 2 + y 2 - 2xy - x + 3y - z = -4,

change if the point P(x, y, z) moves from P0s2, -1, 0d a distance of ds = 0.2 unit toward the point P1s0, 1, 2)?

P0s2, -3, 18d

In Exercises 9–12, find an equation for the plane that is tangent to the given surface at the given point. 9. z = ln sx 2 + y 2 d, 11. z = 2y - x,

s1, 0, 0d

10. z = e -sx

2

+ y 2d

,

12. z = 4x 2 + y 2,

s1, 2, 1d

s0, 0, 1d s1, 1, 5d

Tangent Lines to Space Curves In Exercises 13–18, find parametric equations for the line tangent to the curve of intersection of the surfaces at the given point. 13. Surfaces: x + y 2 + 2z = 4, Point:

(1, 1, 1)

14. Surfaces: xyz = 1, Point:

x 2 + 2y 2 + 3z 2 = 6

(1, 1, 1)

15. Surfaces: x 2 + 2y + 2z = 4, Point:

(1, 1, 1> 2)

16. Surfaces: x + y 2 + z = 2, Point:

x = 1

(1> 2, 1, 1> 2)

y = 1 y = 1

17. Surfaces: x 3 + 3x 2y 2 + y 3 + 4xy - z 2 = 0, x 2 + y 2 + z 2 = 11 Point:

(1, 1, 3)

18. Surfaces: x 2 + y 2 = 4, Point:

A 22, 22, 4 B

x2 + y2 - z = 0

22. By about how much will hsx, y, zd = cos spxyd + xz 2 change if the point P(x, y, z) moves from P0s -1, -1, -1d a distance of ds = 0.1 unit toward the origin? 23. Temperature change along a circle Suppose that the Celsius temperature at the point (x, y) in the xy-plane is T sx, yd = x sin 2y and that distance in the xy-plane is measured in meters. A particle is moving clockwise around the circle of radius 1 m centered at the origin at the constant rate of 2 m> sec. a. How fast is the temperature experienced by the particle changing in degrees Celsius per meter at the point P A 1>2, 23>2 B ? b. How fast is the temperature experienced by the particle changing in degrees Celsius per second at P? 24. Changing temperature along a space curve The Celsius temperature in a region in space is given by T sx, y, zd = 2x 2 - xyz. A particle is moving in this region and its position at time t is given by x = 2t 2, y = 3t, z = -t 2, where time is measured in seconds and distance in meters. a. How fast is the temperature experienced by the particle changing in degrees Celsius per meter when the particle is at the point Ps8, 6, -4d? b. How fast is the temperature experienced by the particle changing in degrees Celsius per second at P?

738

Chapter 13: Partial Derivatives

Finding Linearizations In Exercises 25–30, find the linearization L(x, y) of the function at each point. 25. ƒsx, yd = x 2 + y 2 + 1 at

a. (0, 0),

b. (1, 1)

26. ƒsx, yd = sx + y + 2d2 at

a. (0, 0),

b. (1, 2)

27. ƒsx, yd = 3x - 4y + 5 at

a. (0, 0),

b. (1, 1)

28. ƒsx, yd = x y at

a. (1, 1),

b. (0, 0)

29. ƒsx, yd = e cos y at

a. (0, 0),

b. s0, p>2d

3 4 x

30. ƒsx, yd = e

2y - x

at

a. (0, 0),

34. ƒsx, yd = s1>2dx 2 + xy + s1>4dy 2 + 3x - 3y + 4 at P0s2, 2d, R:

ƒ x - 2 ƒ … 0.1,

ƒ y - 2 ƒ … 0.1

35. ƒsx, yd = 1 + y + x cos y at P0s0, 0d, R:

ƒ x ƒ … 0.2,

ƒ y ƒ … 0.2

sUse ƒ cos y ƒ … 1 and ƒ sin y ƒ … 1 in estimating E.d 36. ƒsx, yd = xy 2 + y cos sx - 1d at P0s1, 2d, R:

ƒ x - 1 ƒ … 0.1,

ƒ y - 2 ƒ … 0.1

37. ƒsx, yd = e x cos y at P0s0, 0d,

b. (1, 2)

31. Wind chill factor Wind chill, a measure of the apparent temperature felt on exposed skin, is a function of air temperature and wind speed. The precise formula, updated by the National Weather Service in 2001 and based on modern heat transfer theory, a human face model, and skin tissue resistance, is

R:

ƒ x ƒ … 0.1,

ƒ y ƒ … 0.1

sUse e … 1.11 and ƒ cos y ƒ … 1 in estimating E.d x

38. ƒsx, yd = ln x + ln y at P0s1, 1d, R:

ƒ x - 1 ƒ … 0.2,

ƒ y - 1 ƒ … 0.2

W = Wsy, Td = 35.74 + 0.6215 T - 35.75 y0.16 + 0.4275 T # y0.16,

where T is air temperature in °F and y is wind speed in mph. A partial wind chill chart is given. T (°F) 5 10

15

10

5

0

5 25 19 13

7

1

-5

-11

-16

-22

10 21 15

9

3

-4

-10

-16

-22

-28

15 19 13

6

0

-7

-13

-19

-26

-32

4

-2

-9

-15

-22

-29

-35

25 16

9

3

-4

-11

-17

-24

-31

-37

30 15

8

1

-5

-12

-19

-26

-33

-39

35 14

7

0

-7

-14

-21

-27

-34

-41

a. Use the table to find Ws20, 25d, Ws30, -10d, and Ws15, 15d. b. Use the formula to find Ws10, -40d, Ws50, -40d, and Ws60, 30d. c. Find the linearization Lsy, Td of the function Wsy, Td at the point s25, 5d. d. Use Lsy, Td in part (c) to estimate the following wind chill values. i) Ws24, 6d

ii) Ws27, 2d

iii) Ws5, -10d (Explain why this value is much different from the value found in the table.) 32. Find the linearization Lsy, Td of the function Wsy, Td in Exercise 31 at the point s50, -20d. Use it to estimate the following wind chill values. a. Ws49, -22d

b. Ws53, -19d

c. Ws60, -30d

Bounding the Error in Linear Approximations In Exercises 33–38, find the linearization L(x, y) of the function ƒ(x, y) at P0. Then find an upper bound for the magnitude ƒ E ƒ of the error in the approximation ƒsx, yd L Lsx, yd over the rectangle R. 33. ƒsx, yd = x 2 - 3xy + 5 at P0s2, 1d, R:

ƒ x - 2 ƒ … 0.1,

ƒ y - 1 ƒ … 0.1

39. ƒsx, y, zd = xy + yz + xz at a. (1, 1, 1)

30 25 20

Y (mph) 20 17 11

Linearizations for Three Variables Find the linearizations L(x, y, z) of the functions in Exercises 39–44 at the given points. b. (1, 0, 0)

c. (0, 0, 0)

40. ƒsx, y, zd = x + y + z at 2

2

a. (1, 1, 1)

2

b. (0, 1, 0)

c. (1, 0, 0)

41. ƒsx, y, zd = 2x + y + z at 2

2

a. (1, 0, 0)

2

b. (1, 1, 0)

c. (1, 2, 2)

42. ƒsx, y, zd = ssin xyd>z at a. sp>2, 1, 1d

b. (2, 0, 1)

43. ƒsx, y, zd = e x + cos s y + zd at b. a0,

a. (0, 0, 0)

c. a0,

p , 0b 2

p p , b 4 4

44. ƒsx, y, zd = tan-1 sxyzd at a. (1, 0, 0)

b. (1, 1, 0)

c. (1, 1, 1)

In Exercises 45–48, find the linearization L(x, y, z) of the function ƒ(x, y, z) at P0 . Then find an upper bound for the magnitude of the error E in the approximation ƒsx, y, zd L Lsx, y, zd over the region R. 45. ƒsx, y, zd = xz - 3yz + 2 R:

ƒ x - 1 ƒ … 0.01,

P0s1, 1, 2d,

at

ƒ y - 1 ƒ … 0.01,

46. ƒsx, y, zd = x 2 + xy + yz + s1>4dz 2 R:

ƒ x - 1 ƒ … 0.01,

ƒ y - 1 ƒ … 0.01,

47. ƒsx, y, zd = xy + 2yz - 3xz R:

ƒ x - 1 ƒ … 0.01,

at

R:

ƒ x ƒ … 0.01,

ƒ y ƒ … 0.01,

P0s1, 1, 2d, ƒ z - 2 ƒ … 0.08

P0s1, 1, 0d,

ƒ y - 1 ƒ … 0.01,

48. ƒsx, y, zd = 22 cos x sin s y + zd

ƒ z - 2 ƒ … 0.02 at

at

ƒ z ƒ … 0.01 P0s0, 0, p>4d,

ƒ z - p>4 ƒ … 0.01

Estimating Error; Sensitivity to Change 49. Estimating maximum error Suppose that T is to be found from the formula T = x se y + e -y d, where x and y are found to be 2 and ln 2 with maximum possible errors of ƒ dx ƒ = 0.1 and ƒ dy ƒ = 0.02. Estimate the maximum possible error in the computed value of T.

13.6 50. Variation in electrical resistance The resistance R produced by wiring resistors of R1 and R2 ohms in parallel (see accompanying figure) can be calculated from the formula 1 1 1 + . = R R1 R2

Tangent Planes and Differentials

739

Theory and Examples 55. The linearization of ƒ(x, y) is a tangent-plane approximation Show that the tangent plane at the point P0sx0 , y0, ƒsx0 , y0 dd on the surface z = ƒsx, yd defined by a differentiable function ƒ is the plane ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 ds y - y0 d - sz - ƒsx0 , y0 dd = 0

a. Show that dR = a

2

2

R R b dR1 + a b dR2 . R1 R2

b. You have designed a two-resistor circuit, like the one shown, to have resistances of R1 = 100 ohms and R2 = 400 ohms, but there is always some variation in manufacturing and the resistors received by your firm will probably not have these exact values. Will the value of R be more sensitive to variation in R1 or to variation in R2? Give reasons for your answer.

or z = ƒsx0 , y0 d + ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 ds y - y0 d. Thus, the tangent plane at P0 is the graph of the linearization of ƒ at P0 (see accompanying figure).

z f (x, y) V

z R1

(x 0,

y0 , f (x 0 , y0 ))

R2

z L(x, y)

y

c. In another circuit like the one shown, you plan to change R1 from 20 to 20.1 ohms and R2 from 25 to 24.9 ohms. By about what percentage will this change R? 51. You plan to calculate the area of a long, thin rectangle from measurements of its length and width. Which dimension should you measure more carefully? Give reasons for your answer. 52. a. Around the point (1, 0), is ƒsx, yd = x 2s y + 1d more sensitive to changes in x or to changes in y? Give reasons for your answer. b. What ratio of dx to dy will make dƒ equal zero at (1, 0)? 53. Value of a 2 : 2 determinant If ƒ a ƒ is much greater than ƒ b ƒ , ƒ c ƒ , and ƒ d ƒ , to which of a, b, c, and d is the value of the determinant ƒsa, b, c, dd = `

a c

b ` d

most sensitive? Give reasons for your answer. 54. The Wilson lot size formula The Wilson lot size formula in economics says that the most economical quantity Q of goods (radios, shoes, brooms, whatever) for a store to order is given by the formula Q = 22KM>h , where K is the cost of placing the order, M is the number of items sold per week, and h is the weekly holding cost for each item (cost of space, utilities, security, and so on). To which of the variables K, M, and h is Q most sensitive near the point sK0 , M0 , h0 d = s2, 20, 0.05d? Give reasons for your answer.

(x 0 , y0 ) x

56. Change along the involute of a circle Find the derivative of ƒsx, yd = x 2 + y 2 in the direction of the unit tangent vector of the curve rstd = scos t + t sin tdi + ssin t - t cos tdj,

t 7 0.

57. Tangent curves A smooth curve is tangent to the surface at a point of intersection if its velocity vector is orthogonal to §f there. Show that the curve rstd = 2t i + 2t j + s2t - 1dk is tangent to the surface x 2 + y 2 - z = 1 when t = 1. 58. Normal curves A smooth curve is normal to a surface ƒsx, y, zd = c at a point of intersection if the curve’s velocity vector is a nonzero scalar multiple of §ƒ at the point. Show that the curve rstd = 2t i + 2t j -

1 st + 3dk 4

is normal to the surface x 2 + y 2 - z = 3 when t = 1.

740

Chapter 13: Partial Derivatives

Extreme Values and Saddle Points

13.7

Continuous functions of two variables assume extreme values on closed, bounded domains (see Figures 13.38 and 13.39). We see in this section that we can narrow the search for these extreme values by examining the functions’ first partial derivatives. A function of two variables can assume extreme values only at domain boundary points or at interior domain points where both first partial derivatives are zero or where one or both of the first partial derivatives fail to exist. However, the vanishing of derivatives at an interior point (a, b) does not always signal the presence of an extreme value. The surface that is the graph of the function might be shaped like a saddle right above (a, b) and cross its tangent plane there.

HISTORICAL BIOGRAPHY Siméon-Denis Poisson (1781–1840)

Derivative Tests for Local Extreme Values To find the local extreme values of a function of a single variable, we look for points where the graph has a horizontal tangent line. At such points, we then look for local maxima, local minima, and points of inflection. For a function ƒ(x, y) of two variables, we look for points where the surface z = ƒsx, yd has a horizontal tangent plane. At such points, we then look for local maxima, local minima, and saddle points. We begin by defining maxima and minima.

x y

FIGURE 13.38 The function z = scos xdscos yde -2x

2

+ y2

has a maximum value of 1 and a minimum value of about -0.067 on the square region ƒ x ƒ … 3p>2, ƒ y ƒ … 3p>2 .

DEFINITIONS (a, b). Then

Let ƒsx, yd be defined on a region R containing the point

1. ƒ(a, b) is a local maximum value of ƒ if ƒsa, bd Ú ƒsx, yd for all domain

points (x, y) in an open disk centered at (a, b). 2. ƒ(a, b) is a local minimum value of ƒ if ƒsa, bd … ƒsx, yd for all domain

points (x, y) in an open disk centered at (a, b). Local maxima correspond to mountain peaks on the surface z = ƒsx, yd and local minima correspond to valley bottoms (Figure 13.40). At such points the tangent planes, when they exist, are horizontal. Local extrema are also called relative extrema. As with functions of a single variable, the key to identifying the local extrema is a first derivative test.

z

Local maxima (no greater value of f nearby) x y

FIGURE 13.39 The “roof surface” 1 z = A ƒ ƒ x ƒ - ƒ yƒ ƒ - ƒ x ƒ - ƒ y ƒB 2 has a maximum value of 0 and a minimum value of -a on the square region ƒ x ƒ … a, ƒ y ƒ … a.

Local minimum (no smaller value of f nearby)

FIGURE 13.40 A local maximum occurs at a mountain peak and a local minimum occurs at a valley low point.

THEOREM 10—First Derivative Test for Local Extreme Values If ƒ(x, y) has a local maximum or minimum value at an interior point (a, b) of its domain and if the first partial derivatives exist there, then ƒxsa, bd = 0 and ƒysa, bd = 0.

13.7 z

0f 0 0x z f (x, y) 0f 0 0y

0 a

g(x) f (x, b) b

741

Proof If ƒ has a local extremum at (a, b), then the function gsxd = ƒsx, bd has a local extremum at x = a (Figure 13.41). Therefore, g¿sad = 0 (Chapter 4, Theorem 2). Now g¿sad = ƒxsa, bd, so ƒxsa, bd = 0. A similar argument with the function hsyd = ƒsa, yd shows that ƒysa, bd = 0. If we substitute the values ƒxsa, bd = 0 and ƒysa, bd = 0 into the equation

h( y) f (a, y) y

(a, b, 0)

x

Extreme Values and Saddle Points

FIGURE 13.41 If a local maximum of ƒ occurs at x = a, y = b , then the first partial derivatives ƒxsa, bd and ƒysa, bd are both zero.

ƒxsa, bdsx - ad + ƒysa, bds y - bd - sz - ƒsa, bdd = 0 for the tangent plane to the surface z = ƒsx, yd at (a, b), the equation reduces to 0 # sx - ad + 0 # s y - bd - z + ƒsa, bd = 0 or z = ƒsa, bd. Thus, Theorem 10 says that the surface does indeed have a horizontal tangent plane at a local extremum, provided there is a tangent plane there.

z

DEFINITION An interior point of the domain of a function ƒ(x, y) where both ƒx and ƒy are zero or where one or both of ƒx and ƒy do not exist is a critical point of ƒ. y

Theorem 10 says that the only points where a function ƒ(x, y) can assume extreme values are critical points and boundary points. As with differentiable functions of a single variable, not every critical point gives rise to a local extremum. A differentiable function of a single variable might have a point of inflection. A differentiable function of two variables might have a saddle point.

x

z

xy(x 2 y 2 ) x2 y2 z

DEFINITION A differentiable function ƒ(x, y) has a saddle point at a critical point (a, b) if in every open disk centered at (a, b) there are domain points (x, y) where ƒsx, yd 7 ƒsa, bd and domain points sx, yd where ƒsx, yd 6 ƒsa, bd. The corresponding point (a, b, ƒ(a, b)) on the surface z = ƒsx, yd is called a saddle point of the surface (Figure 13.42).

y

x

EXAMPLE 1

Find the local extreme values of ƒsx, yd = x 2 + y 2 - 4y + 9.

The domain of ƒ is the entire plane (so there are no boundary points) and the partial derivatives ƒx = 2x and ƒy = 2y - 4 exist everywhere. Therefore, local extreme values can occur only where ƒx = 2x = 0 and ƒy = 2y - 4 = 0. The only possibility is the point (0, 2), where the value of ƒ is 5. Since ƒsx, yd = x 2 + s y - 2d2 + 5 is never less than 5, we see that the critical point (0, 2) gives a local minimum (Figure 13.43).

Solution z

y2

y4

x2

FIGURE 13.42 Saddle points at the origin. z

EXAMPLE 2 15 5 x

2

1

The domain of ƒ is the entire plane (so there are no boundary points) and the partial derivatives ƒx = -2x and ƒy = 2y exist everywhere. Therefore, local extrema can occur only at the origin s0, 0d where ƒx = 0 and ƒy = 0. Along the positive x-axis, however, ƒ has the value ƒsx, 0d = -x 2 6 0; along the positive y-axis, ƒ has the value ƒs0, yd = y 2 7 0. Therefore, every open disk in the xy-plane centered at (0, 0) contains points where the function is positive and points where it is negative. The function has a saddle point at the origin and no local extreme values (Figure 13.44a). Figure 13.44b displays the level curves (they are hyperbolas) of ƒ, and shows the function decreasing and increasing in an alternating fashion among the four groupings of hyperbolas. Solution

10

1

2

3

4

y

FIGURE 13.43 The graph of the function ƒsx, yd = x 2 + y 2 - 4y + 9 is a paraboloid which has a local minimum value of 5 at the point (0, 2) (Example 1).

Find the local extreme values (if any) of ƒsx, yd = y 2 - x 2.

742

Chapter 13: Partial Derivatives

That ƒx = ƒy = 0 at an interior point (a, b) of R does not guarantee ƒ has a local extreme value there. If ƒ and its first and second partial derivatives are continuous on R, however, we may be able to learn more from the following theorem, proved in Section 13.9.

z z

y2

x2

THEOREM 11—Second Derivative Test for Local Extreme Values Suppose that ƒ(x, y) and its first and second partial derivatives are continuous throughout a disk centered at (a, b) and that ƒxsa, bd = ƒysa, bd = 0 . Then

y x

i) ii) iii) iv)

(a) y

f inc

3

–3 f dec

Saddle point –1

f dec

1 –1

–3

x

ƒ has a local maximum at (a, b) if ƒxx 6 0 and ƒxx ƒyy - ƒxy2 7 0 at (a, b). ƒ has a local minimum at (a, b) if ƒxx 7 0 and ƒxx ƒyy - ƒxy2 7 0 at (a, b). ƒ has a saddle point at (a, b) if ƒxx ƒyy - ƒxy2 6 0 at (a, b). the test is inconclusive at (a, b) if ƒxx ƒyy - ƒxy2 = 0 at (a, b). In this case, we must find some other way to determine the behavior of ƒ at (a, b).

The expression ƒxx ƒyy - ƒxy2 is called the discriminant or Hessian of ƒ. It is sometimes easier to remember it in determinant form, ƒxx ƒyy - ƒxy2 = `

1 3

f inc (b)

FIGURE 13.44 (a) The origin is a saddle point of the function ƒsx, yd = y 2 - x 2. There are no local extreme values (Example 2). (b) Level curves for the function ƒ in Example 2.

ƒxx ƒxy

ƒxy `. ƒyy

Theorem 11 says that if the discriminant is positive at the point (a, b), then the surface curves the same way in all directions: downward if ƒxx 6 0, giving rise to a local maximum, and upward if ƒxx 7 0, giving a local minimum. On the other hand, if the discriminant is negative at (a, b), then the surface curves up in some directions and down in others, so we have a saddle point.

EXAMPLE 3

Find the local extreme values of the function ƒsx, yd = xy - x 2 - y 2 - 2x - 2y + 4.

The function is defined and differentiable for all x and y, and its domain has no boundary points. The function therefore has extreme values only at the points where ƒx and ƒy are simultaneously zero. This leads to Solution

ƒx = y - 2x - 2 = 0,

ƒy = x - 2y - 2 = 0,

or x = y = -2. Therefore, the point s -2, -2d is the only point where ƒ may take on an extreme value. To see if it does so, we calculate ƒxx = -2,

ƒyy = -2,

ƒxy = 1.

The discriminant of ƒ at sa, bd = s -2, -2d is ƒxx ƒyy - ƒxy2 = s -2ds -2d - s1d2 = 4 - 1 = 3. The combination ƒxx 6 0

and

ƒxx ƒyy - ƒxy2 7 0

tells us that ƒ has a local maximum at s -2, -2d. The value of ƒ at this point is ƒs -2, -2d = 8.

EXAMPLE 4 Solution

Find the local extreme values of ƒsx, yd = 3y 2 - 2y 3 - 3x 2 + 6xy.

Since ƒ is differentiable everywhere, it can assume extreme values only where ƒx = 6y - 6x = 0

and

ƒy = 6y - 6y 2 + 6x = 0.

13.7

743

From the first of these equations we find x = y, and substitution for y into the second equation then gives 6x - 6x 2 + 6x = 0 or 6x s2 - xd = 0. The two critical points are therefore (0, 0) and (2, 2). To classify the critical points, we calculate the second derivatives:

z

10

5

ƒxx = -6, 1

Extreme Values and Saddle Points

2

3

ƒyy = 6 - 12y,

ƒxy = 6.

y

The discriminant is given by

2

ƒxxƒyy - ƒxy 2 = s -36 + 72yd - 36 = 72sy - 1d.

3 x

FIGURE 13.45 The surface z = 3y 2 - 2y 3 - 3x 2 + 6xy has a saddle point at the origin and a local maximum at the point (2, 2) (Example 4).

At the critical point (0, 0) we see that the value of the discriminant is the negative number -72, so the function has a saddle point at the origin. At the critical point (2, 2) we see that the discriminant has the positive value 72. Combining this result with the negative value of the second partial ƒxx = -6, Theorem 11 says that the critical point (2, 2) gives a local maximum value of ƒs2, 2d = 12 - 16 - 12 + 24 = 8. A graph of the surface is shown in Figure 13.45.

Absolute Maxima and Minima on Closed Bounded Regions We organize the search for the absolute extrema of a continuous function ƒ(x, y) on a closed and bounded region R into three steps. 1. 2. 3.

List the interior points of R where ƒ may have local maxima and minima and evaluate ƒ at these points. These are the critical points of ƒ. List the boundary points of R where ƒ has local maxima and minima and evaluate ƒ at these points. We show how to do this shortly. Look through the lists for the maximum and minimum values of ƒ. These will be the absolute maximum and minimum values of ƒ on R. Since absolute maxima and minima are also local maxima and minima, the absolute maximum and minimum values of ƒ appear somewhere in the lists made in Steps 1 and 2.

EXAMPLE 5

Find the absolute maximum and minimum values of ƒsx, yd = 2 + 2x + 4y - x2 - y2

on the triangular region in the first quadrant bounded by the lines x = 0, y = 0, y = 9 - x . Solution Since ƒ is differentiable, the only places where ƒ can assume these values are points inside the triangle (Figure 13.46a) where ƒx = ƒy = 0 and points on the boundary.

(a) Interior points. For these we have fx = 2 - 2x = 0,

fy = 4 - 2y = 0,

yielding the single point sx, yd = s1, 2d. The value of ƒ there is ƒs1, 2d = 7. (b) Boundary points. We take the triangle one side at a time: i) On the segment OA, y = 0. The function ƒsx, yd = ƒsx, 0d = 2 + 2x - x 2

744

Chapter 13: Partial Derivatives

may now be regarded as a function of x defined on the closed interval 0 … x … 9. Its extreme values (we know from Chapter 4) may occur at the endpoints

y B(0, 9) y9x

x = 0

where

ƒs0, 0d = 2

x = 9

where

ƒs9, 0d = 2 + 18 - 81 = -61

(4, 5)

x0

and at the interior points where ƒ¿sx, 0d = 2 - 2x = 0. The only interior point where ƒ¿sx, 0d = 0 is x = 1, where

(1, 2)

ƒsx, 0d = ƒs1, 0d = 3.

x

y0

O

A(9, 0)

ii) On the segment OB, x = 0 and

(a)

ƒsx, yd = ƒs0, yd = 2 + 4y - y2.

z y 9

(1, 2, 7) 6 3

9

6

3

x

As in part i), we consider ƒ(0, y) as a function of y defined on the closed interval [0, 9]. Its extreme values can occur at the endpoints or at interior points where f ¿(0, y) = 0. Since f ¿(0, y) = 4 - 2y, the only interior point where f ¿(0, y) = 0 occurs at (0, 2), with f (0, 2) = 6. So the candidates for this segment are ƒs0, 0d = 2,

ƒs0, 9d = -43,

ƒs0, 2d = 6.

iii) We have already accounted for the values of ƒ at the endpoints of AB, so we need only look at the interior points of AB. With y = 9 - x, we have

220

ƒsx, yd = 2 + 2x + 4s9 - xd - x2 - s9 - xd2 = -43 + 16x - 2x2. Setting ƒ¿sx, 9 - xd = 16 - 4x = 0 gives

240

x = 4. At this value of x, (9, 0, 261)

260

y = 9 - 4 = 5

and

ƒsx, yd = ƒ(4, 5) = -11.

(b)

FIGURE 13.46 (a) This triangular region is the domain of the function in Example 5. (b) The graph of the function in Example 5. The blue points are the candidates for maxima or minima.

Summary We list all the candidates: 7, 2, -61, 3, -43, 6, -11. The maximum is 7, which ƒ assumes at (1, 2). The minimum is -61, which ƒ assumes at (9, 0). See Figure 13.46b. Solving extreme value problems with algebraic constraints on the variables usually requires the method of Lagrange multipliers introduced in the next section. But sometimes we can solve such problems directly, as in the next example.

EXAMPLE 6

A delivery company accepts only rectangular boxes the sum of whose length and girth (perimeter of a cross-section) does not exceed 108 in. Find the dimensions of an acceptable box of largest volume.

Girth distance around here

Solution Let x, y, and z represent the length, width, and height of the rectangular box, respectively. Then the girth is 2y + 2z. We want to maximize the volume V = xyz of the box (Figure 13.47) satisfying x + 2y + 2z = 108 (the largest box accepted by the delivery company). Thus, we can write the volume of the box as a function of two variables:

Vs y, zd = s108 - 2y - 2zdyz . = 108yz - 2y 2z - 2yz 2.

V = xyz and x = 108 - 2y - 2z

z

Setting the first partial derivatives equal to zero, x

y

FIGURE 13.47 The box in Example 6.

Vys y, zd = 108z - 4yz - 2z 2 = s108 - 4y - 2zdz = 0 Vzs y, zd = 108y - 2y 2 - 4yz = s108 - 2y - 4zdy = 0,

13.7

Extreme Values and Saddle Points

745

gives the critical points (0, 0), (0, 54), (54, 0), and (18, 18). The volume is zero at (0, 0), (0, 54), (54, 0), which are not maximum values. At the point (18, 18), we apply the Second Derivative Test (Theorem 11): Vyy = -4z,

Vzz = -4y,

Vyz = 108 - 4y - 4z.

Then Vyy Vzz - V yz2 = 16yz - 16s27 - y - zd2. Thus, Vyys18, 18d = -4s18d 6 0 and

C Vyy Vzz - V yz2 D s18,18d = 16s18ds18d - 16s -9d2 7 0 imply that s18, 18d gives a maximum volume. The dimensions of the package are x = 108 - 2s18d - 2s18d = 36 in., y = 18 in., and z = 18 in. The maximum volume is V = s36ds18ds18d = 11,664 in3, or 6.75 ft3. Despite the power of Theorem 11, we urge you to remember its limitations. It does not apply to boundary points of a function’s domain, where it is possible for a function to have extreme values along with nonzero derivatives. Also, it does not apply to points where either ƒx or ƒy fails to exist.

Summary of Max-Min Tests The extreme values of ƒ(x, y) can occur only at i) boundary points of the domain of ƒ ii) critical points (interior points where ƒx = ƒy = 0 or points where ƒx or ƒy fails to exist). If the first- and second-order partial derivatives of ƒ are continuous throughout a disk centered at a point (a, b) and ƒxsa, bd = ƒysa, bd = 0, the nature of ƒ(a, b) can be tested with the Second Derivative Test: i) ii) iii) iv)

ƒxx 6 ƒxx 7 ƒxx ƒyy ƒxx ƒyy

0 and ƒxx ƒyy - ƒxy2 7 0 at sa, bd Q local maximum 0 and ƒxx ƒyy - ƒxy2 7 0 at sa, bd Q local minimum - ƒxy2 6 0 at sa, bd Q saddle point - ƒxy2 = 0 at sa, bd Q test is inconclusive

Exercises 13.7 Finding Local Extrema Find all the local maxima, local minima, and saddle points of the functions in Exercises 1–30. 1. ƒsx, yd = x 2 + xy + y 2 + 3x - 3y + 4

5. ƒsx, yd = 2xy - x 2 - 2y 2 + 3x + 4 6. ƒsx, yd = x 2 - 4xy + y 2 + 6y + 2 7. ƒsx, yd = 2x 2 + 3xy + 4y 2 - 5x + 2y

2. ƒsx, yd = 2xy - 5x 2 - 2y 2 + 4x + 4y - 4

8. ƒsx, yd = x 2 - 2xy + 2y 2 - 2x + 2y + 1

3. ƒsx, yd = x 2 + xy + 3x + 2y + 5

9. ƒsx, yd = x 2 - y 2 - 2x + 4y + 6

4. ƒsx, yd = 5xy - 7x 2 + 3x - 6y + 2

10. ƒsx, yd = x 2 + 2xy

746

Chapter 13: Partial Derivatives 39. Find two numbers a and b with a … b such that

11. ƒsx, yd = 256x 2 - 8y 2 - 16x - 31 + 1 - 8x

b

12. ƒsx, yd = 1 - 2x 2 + y 2 3

La

13. ƒsx, yd = x 3 - y 3 - 2xy + 6 14. ƒsx, yd = x 3 + 3xy + y 3

has its largest value. 40. Find two numbers a and b with a … b such that

15. ƒsx, yd = 6x - 2x + 3y + 6xy 2

3

2

16. ƒsx, yd = x + y + 3x - 3y - 8 3

3

2

2

b

17. ƒsx, yd = x + 3xy - 15x + y - 15y 3

2

3

La

18. ƒsx, yd = 2x 3 + 2y 3 - 9x 2 + 3y 2 - 12y

41. Temperatures A flat circular plate has the shape of the region x 2 + y 2 … 1. The plate, including the boundary where x 2 + y 2 = 1, is heated so that the temperature at the point (x, y) is

20. ƒsx, yd = x 4 + y 4 + 4xy 1 x2 + y2 - 1 23. ƒsx, yd = y sin x 2

25. ƒsx, yd = e x

+ y2 - 4x

27. ƒsx, yd = e -ysx 2 + y 2 d

s24 - 2x - x 2 d1>3 dx

has its largest value.

19. ƒsx, yd = 4xy - x 4 - y 4

21. ƒsx, yd =

s6 - x - x 2 d dx

1 1 22. ƒsx, yd = x + xy + y

Tsx, yd = x 2 + 2y 2 - x.

24. ƒsx, yd = e cos y 2x

Find the temperatures at the hottest and coldest points on the plate.

26. ƒsx, yd = e y - ye x 28. ƒsx, yd = e xsx 2 - y 2 d

42. Find the critical point of

29. ƒsx, yd = 2 ln x + ln y - 4x - y

ƒsx, yd = xy + 2x - ln x 2y

30. ƒsx, yd = ln sx + yd + x 2 - y Finding Absolute Extrema In Exercises 31–38, find the absolute maxima and minima of the functions on the given domains. 31. ƒsx, yd = 2x 2 - 4x + y 2 - 4y + 1 on the closed triangular plate bounded by the lines x = 0, y = 2, y = 2x in the first quadrant 32. Dsx, yd = x 2 - xy + y 2 + 1 on the closed triangular plate in the first quadrant bounded by the lines x = 0, y = 4, y = x 33. ƒsx, yd = x 2 + y 2 on the closed triangular plate bounded by the lines x = 0, y = 0, y + 2x = 2 in the first quadrant 34. Tsx, yd = x 2 + xy + y 2 - 6x 0 … x … 5, -3 … y … 3

on

the

rectangular

plate

35. Tsx, yd = x 2 + xy + y 2 - 6x + 2 on the rectangular plate 0 … x … 5, -3 … y … 0 36. ƒsx, yd = 48xy - 32x 3 - 24y 2 0 … x … 1, 0 … y … 1

on

the

rectangular

plate

37. ƒsx, yd = s4x - x 2 d cos y on the rectangular plate 1 … x … 3, -p>4 … y … p>4 (see accompanying figure) z

in the open first quadrant sx 7 0, y 7 0d and show that ƒ takes on a minimum there. Theory and Examples 43. Find the maxima, minima, and saddle points of ƒ(x, y), if any, given that a. ƒx = 2x - 4y b. ƒx = 2x - 2 c. ƒx = 9x 2 - 9

and and and

ƒy = 2y - 4x ƒy = 2y - 4 ƒy = 2y + 4

Describe your reasoning in each case. 44. The discriminant ƒxx ƒyy - ƒxy 2 is zero at the origin for each of the following functions, so the Second Derivative Test fails there. Determine whether the function has a maximum, a minimum, or neither at the origin by imagining what the surface z = ƒsx, yd looks like. Describe your reasoning in each case. a. ƒsx, yd = x 2y 2

b. ƒsx, yd = 1 - x 2y 2

c. ƒsx, yd = xy

d. ƒsx, yd = x 3y 2

2

e. ƒsx, yd = x 3y 3

f. ƒsx, yd = x 4y 4

45. Show that (0, 0) is a critical point of ƒsx, yd = x 2 + kxy + y 2 no matter what value the constant k has. (Hint: Consider two cases: k = 0 and k Z 0.) 46. For what values of the constant k does the Second Derivative Test guarantee that ƒsx, yd = x 2 + kxy + y 2 will have a saddle point at (0, 0)? A local minimum at (0, 0)? For what values of k is the Second Derivative Test inconclusive? Give reasons for your answers.

z 5 (4x 2 x 2 ) cos y

47. If ƒxsa, bd = ƒysa, bd = 0, must ƒ have a local maximum or minimum value at (a, b)? Give reasons for your answer. y x

38. ƒsx, yd = 4x - 8xy + 2y + 1 on the triangular plate bounded by the lines x = 0, y = 0, x + y = 1 in the first quadrant

48. Can you conclude anything about ƒ(a, b) if ƒ and its first and second partial derivatives are continuous throughout a disk centered at the critical point (a, b) and ƒxxsa, bd and ƒyysa, bd differ in sign? Give reasons for your answer. 49. Among all the points on the graph of z = 10 - x 2 - y 2 that lie above the plane x + 2y + 3z = 0, find the point farthest from the plane.

13.7 50. Find the point on the graph of z = x 2 + y 2 + 10 nearest the plane x + 2y - z = 0. 51. Find the point on the plane 3x + 2y + z = 6 that is nearest the origin. 52. Find the minimum distance from the point s2, -1, 1d to the plane x + y - z = 2. 53. Find three numbers whose sum is 9 and whose sum of squares is a minimum. 54. Find three positive numbers whose sum is 3 and whose product is a maximum. 55. Find the maximum x + y + z = 6.

value

of

s = xy + yz + xz

where 2

57. Find the dimensions of the rectangular box of maximum volume that can be inscribed inside the sphere x 2 + y 2 + z 2 = 4. 58. Among all closed rectangular boxes of volume 27 cm3, what is the smallest surface area? 59. You are to construct an open rectangular box from 12 ft2 of material. What dimensions will result in a box of maximum volume? 60. Consider the function ƒsx, yd = x 2 + y 2 + 2xy - x - y + 1 over the square 0 … x … 1 and 0 … y … 1.

Curves: i) The line x = 2t,

y = t + 1

ii) The line segment x = 2t,

y = t + 1,

-1 … t … 0

iii) The line segment x = 2t,

y = t + 1,

0 … t … 1

64. Functions: a. ƒsx, yd = x 2 + y 2 b. gsx, yd = 1>sx 2 + y 2 d Curves: i) The line x = t,

y = 2 - 2t y = 2 - 2t,

w = smx1 + b - y1 d2 + Á + smxn + b - yn d2 .

a a xk b a a yk b - n a xk yk

b. Find the absolute maximum value of ƒ over the square.

1 b = n a a yk - m a xk b,

a. critical points (points where dƒ> dt is zero or fails to exist), and

(1)

Show that the values of m and b that do this are

m =

b. endpoints of the parameter domain.

0 … t … 1

65. Least squares and regression lines When we try to fit a line y = mx + b to a set of numerical data points sx1, y1 d, sx2 , y2 d, Á , sxn , yn d (Figure 13.48), we usually choose the line that minimizes the sum of the squares of the vertical distances from the points to the line. In theory, this means finding the values of m and b that minimize the value of the function

a. Show that ƒ has an absolute minimum along the line segment 2x + 2y = 1 in this square. What is the absolute minimum value?

Extreme Values on Parametrized Curves To find the extreme values of a function ƒ(x, y) on a curve x = xstd, y = ystd, we treat ƒ as a function of the single variable t and use the Chain Rule to find where dƒ> dt is zero. As in any other single-variable case, the extreme values of ƒ are then found among the values at the

747

63. Function: ƒsx, yd = xy

ii) The line segment x = t,

56. Find the minimum distance from the cone z = 2x + y to the point s -6, 4, 0d. 2

Extreme Values and Saddle Points

2

a a xk b - n a x k2

(2)

,

(3)

with all sums running from k = 1 to k = n . Many scientific calculators have these formulas built in, enabling you to find m and b with only a few keystrokes after you have entered the data. The line y = mx + b determined by these values of m and b is called the least squares line, regression line, or trend line for the data under study. Finding a least squares line lets you 1. summarize data with a simple expression,

Find the absolute maximum and minimum values of the following functions on the given curves.

2. predict values of y for other, experimentally untried values of x, 3. handle data analytically.

61. Functions: a. ƒsx, yd = x + y

b. gsx, yd = xy

c. hsx, yd = 2x 2 + y 2

y Pn(xn , yn )

Curves: i) The semicircle x + y = 4, 2

2

y Ú 0

ii) The quarter circle x + y = 4, 2

2

x Ú 0,

y mx b

P1(x1, y1 )

y Ú 0

Use the parametric equations x = 2 cos t, y = 2 sin t .

P2(x 2 , y2 )

62. Functions: a. ƒsx, yd = 2x + 3y

b. gsx, yd = xy

0

c. hsx, yd = x 2 + 3y 2 Curves:

i) The semiellipse sx 2>9d + s y 2>4d = 1,

y Ú 0

ii) The quarter ellipse sx 2>9d + s y 2>4d = 1,

x Ú 0,

Use the parametric equations x = 3 cos t, y = 2 sin t .

y Ú 0

x

FIGURE 13.48 To fit a line to noncollinear points, we choose the line that minimizes the sum of the squares of the deviations.

748

Chapter 13: Partial Derivatives

In Exercises 66–68, use Equations (2) and (3) to find the least squares line for each set of data points. Then use the linear equation you obtain to predict the value of y that would correspond to x = 4. 66. s -2, 0d, 68. (0, 0),

s0, 2d, (1, 2),

s2, 3d

67. s -1, 2d,

s0, 1d,

d. Calculate the function’s second partial derivatives and find the discriminant ƒxx ƒyy - ƒxy 2. e. Using the max-min tests, classify the critical points found in part (c). Are your findings consistent with your discussion in part (c)?

s3, -4d

69. ƒsx, yd = x 2 + y 3 - 3xy,

(2, 3)

COMPUTER EXPLORATIONS In Exercises 69–74, you will explore functions to identify their local extrema. Use a CAS to perform the following steps: a. Plot the function over the given rectangle. b. Plot some level curves in the rectangle. c. Calculate the function’s first partial derivatives and use the CAS equation solver to find the critical points. How do the critical points relate to the level curves plotted in part (b)? Which critical points, if any, appear to give a saddle point? Give reasons for your answer.

13.8

2

2

-2 … x … 2,

71. ƒsx, yd = x 4 + y 2 - 8x 2 - 6y + 16, -6 … y … 6 72. ƒsx, yd = 2x 4 + y 4 - 2x 2 - 2y 2 + 3, - 3>2 … y … 3>2

-5 … y … 5 -2 … y … 2

-3 … x … 3, -3>2 … x … 3>2,

73. ƒsx, yd = 5x 6 + 18x 5 - 30x 4 + 30xy 2 - 120x 3, -4 … x … 3, -2 … y … 2 x 5 ln sx 2 + y 2 d, 0, -2 … x … 2, -2 … y … 2

74. ƒsx, yd = e

sx, yd Z s0, 0d , sx, yd = s0, 0d

Lagrange Multipliers

HISTORICAL BIOGRAPHY Joseph Louis Lagrange (1736–1813)

-5 … x … 5,

70. ƒsx, yd = x - 3xy + y , 3

Sometimes we need to find the extreme values of a function whose domain is constrained to lie within some particular subset of the plane—a disk, for example, a closed triangular region, or along a curve. We saw an instance of this situation in Example 5 of the previous section. In this section, we explore a powerful method for finding extreme values of constrained functions: the method of Lagrange multipliers.

Constrained Maxima and Minima We first consider a problem where a constrained minimum can be found by eliminating a variable.

EXAMPLE 1

Find the point Psx, y, zd on the plane 2x + y - z - 5 = 0 that is closest

to the origin. Solution

The problem asks us to find the minimum value of the function 1 ƒ OP ƒ = 2sx - 0d2 + s y - 0d2 + sz - 0d2 = 2x 2 + y 2 + z 2

subject to the constraint that 2x + y - z - 5 = 0. 1 Since ƒ OP ƒ has a minimum value wherever the function ƒsx, y, zd = x 2 + y 2 + z 2 has a minimum value, we may solve the problem by finding the minimum value of ƒ(x, y, z) subject to the constraint 2x + y - z - 5 = 0 (thus avoiding square roots). If we regard x and y as the independent variables in this equation and write z as z = 2x + y - 5, our problem reduces to one of finding the points (x, y) at which the function hsx, yd = ƒsx, y, 2x + y - 5d = x 2 + y 2 + s2x + y - 5d2

13.8

Lagrange Multipliers

749

has its minimum value or values. Since the domain of h is the entire xy-plane, the First Derivative Test of Section 13.7 tells us that any minima that h might have must occur at points where hx = 2x + 2s2x + y - 5ds2d = 0,

hy = 2y + 2s2x + y - 5d = 0.

This leads to 10x + 4y = 20,

4x + 4y = 10,

and the solution x =

5 , 3

y =

5 . 6

We may apply a geometric argument together with the Second Derivative Test to show that these values minimize h. The z-coordinate of the corresponding point on the plane z = 2x + y - 5 is 5 5 5 z = 2a b + - 5 = - . 3 6 6 Therefore, the point we seek is 5 5 5 P a , , - b. 3 6 6

Closest point:

The distance from P to the origin is 5> 26 L 2.04. Attempts to solve a constrained maximum or minimum problem by substitution, as we might call the method of Example 1, do not always go smoothly. This is one of the reasons for learning the new method of this section.

EXAMPLE 2

z

Find the points on the hyperbolic cylinder x 2 - z 2 - 1 = 0 that are clos-

est to the origin. The cylinder is shown in Figure 13.49. We seek the points on the cylinder closest to the origin. These are the points whose coordinates minimize the value of the function Solution 1

ƒsx, y, zd = x 2 + y 2 + z 2

subject to the constraint that x 2 - z 2 - 1 = 0 . If we regard x and y as independent variables in the constraint equation, then

(–1, 0, 0) (1, 0, 0) y x x 2 z2 1

Square of the distance

z2 = x2 - 1 and the values of ƒsx, y, zd = x 2 + y 2 + z 2 on the cylinder are given by the function hsx, yd = x 2 + y 2 + sx 2 - 1d = 2x 2 + y 2 - 1. To find the points on the cylinder whose coordinates minimize ƒ, we look for the points in the xy-plane whose coordinates minimize h. The only extreme value of h occurs where

FIGURE 13.49 The hyperbolic cylinder x 2 - z 2 - 1 = 0 in Example 2.

hx = 4x = 0

and

hy = 2y = 0,

that is, at the point (0, 0). But there are no points on the cylinder where both x and y are zero. What went wrong? What happened was that the First Derivative Test found (as it should have) the point in the domain of h where h has a minimum value. We, on the other hand, want the points on the cylinder where h has a minimum value. Although the domain of h is the entire

750

Chapter 13: Partial Derivatives The hyperbolic cylinder x 2 z 2 1 On this part,

On this part,

z

x z 2 1

x –z 2 1

xy-plane, the domain from which we can select the first two coordinates of the points (x, y, z) on the cylinder is restricted to the “shadow” of the cylinder on the xy-plane; it does not include the band between the lines x = -1 and x = 1 (Figure 13.50). We can avoid this problem if we treat y and z as independent variables (instead of x and y) and express x in terms of y and z as x 2 = z 2 + 1.

x

1

With this substitution, ƒsx, y, zd = x 2 + y 2 + z 2 becomes

–1

x1

x –1

y

FIGURE 13.50 The region in the xy-plane from which the first two coordinates of the points (x, y, z) on the hyperbolic cylinder x 2 - z 2 = 1 are selected excludes the band -1 6 x 6 1 in the xy-plane (Example 2).

ks y, zd = sz 2 + 1d + y 2 + z 2 = 1 + y 2 + 2z 2 and we look for the points where k takes on its smallest value. The domain of k in the yz-plane now matches the domain from which we select the y- and z-coordinates of the points (x, y, z) on the cylinder. Hence, the points that minimize k in the plane will have corresponding points on the cylinder. The smallest values of k occur where ky = 2y = 0

kz = 4z = 0,

and

or where y = z = 0. This leads to x 2 = z 2 + 1 = 1,

x = ;1.

The corresponding points on the cylinder are s ;1, 0, 0d. We can see from the inequality ks y, zd = 1 + y 2 + 2z 2 Ú 1 that the points s ;1, 0, 0d give a minimum value for k. We can also see that the minimum distance from the origin to a point on the cylinder is 1 unit. x2 z2 1 0

Another way to find the points on the cylinder closest to the origin is to imagine a small sphere centered at the origin expanding like a soap bubble until it just touches the cylinder (Figure 13.51). At each point of contact, the cylinder and sphere have the same tangent plane and normal line. Therefore, if the sphere and cylinder are represented as the level surfaces obtained by setting Solution 2

z

x2

y2

z2

a2

0

ƒsx, y, zd = x 2 + y 2 + z 2 - a 2 y

and

gsx, y, zd = x 2 - z 2 - 1

equal to 0, then the gradients §ƒ and §g will be parallel where the surfaces touch. At any point of contact, we should therefore be able to find a scalar l (“lambda”) such that §ƒ = l§g,

x

FIGURE 13.51 A sphere expanding like a soap bubble centered at the origin until it just touches the hyperbolic cylinder x 2 - z 2 - 1 = 0 (Example 2).

or 2xi + 2yj + 2zk = ls2xi - 2zkd. Thus, the coordinates x, y, and z of any point of tangency will have to satisfy the three scalar equations 2x = 2lx,

2y = 0,

2z = -2lz.

For what values of l will a point (x, y, z) whose coordinates satisfy these scalar equations also lie on the surface x 2 - z 2 - 1 = 0? To answer this question, we use our knowledge that no point on the surface has a zero x-coordinate to conclude that x Z 0. Hence, 2x = 2lx only if 2 = 2l,

or

l = 1.

For l = 1, the equation 2z = -2lz becomes 2z = -2z. If this equation is to be satisfied as well, z must be zero. Since y = 0 also (from the equation 2y = 0), we conclude that the points we seek all have coordinates of the form sx, 0, 0d.

13.8

Lagrange Multipliers

751

What points on the surface x 2 - z 2 = 1 have coordinates of this form? The answer is the points (x, 0, 0) for which x 2 - s0d2 = 1,

x 2 = 1,

or

x = ;1.

The points on the cylinder closest to the origin are the points s ;1, 0, 0d.

The Method of Lagrange Multipliers In Solution 2 of Example 2, we used the method of Lagrange multipliers. The method says that the extreme values of a function ƒ(x, y, z) whose variables are subject to a constraint gsx, y, zd = 0 are to be found on the surface g = 0 among the points where §ƒ = l§g for some scalar l (called a Lagrange multiplier). To explore the method further and see why it works, we first make the following observation, which we state as a theorem.

THEOREM 12—The Orthogonal Gradient Theorem Suppose that ƒ(x, y, z) is differentiable in a region whose interior contains a smooth curve C:

rstd = gstdi + hstdj + kstdk.

If P0 is a point on C where ƒ has a local maximum or minimum relative to its values on C, then §ƒ is orthogonal to C at P0 .

Proof We show that §ƒ is orthogonal to the curve’s velocity vector at P0 . The values of ƒ on C are given by the composite ƒ(g(t), h(t), k(t)), whose derivative with respect to t is dƒ 0ƒ dg 0ƒ dh 0ƒ dk = + + = §ƒ # v. 0x dt 0y dt 0z dt dt At any point P0 where ƒ has a local maximum or minimum relative to its values on the curve, dƒ>dt = 0, so §ƒ # v = 0.

By dropping the z-terms in Theorem 12, we obtain a similar result for functions of two variables.

COROLLARY OF THEOREM 12 At the points on a smooth curve rstd = gstdi + hstdj where a differentiable function ƒ(x, y) takes on its local maxima and minima relative to its values on the curve, §ƒ # v = 0, where v = dr>dt.

Theorem 12 is the key to the method of Lagrange multipliers. Suppose that ƒ(x, y, z) and g(x, y, z) are differentiable and that P0 is a point on the surface gsx, y, zd = 0 where ƒ has a local maximum or minimum value relative to its other values on the surface. We assume also that §g Z 0 at points on the surface gsx, y, zd = 0. Then ƒ takes on a local maximum or minimum at P0 relative to its values on every differentiable curve through P0 on the surface gsx, y, zd = 0. Therefore, §ƒ is orthogonal to the velocity vector of every such differentiable curve through P0 . So is §g , moreover (because §g is orthogonal to the level surface g = 0, as we saw in Section 13.5). Therefore, at P0, §ƒ is some scalar multiple l of §g.

752

Chapter 13: Partial Derivatives

The Method of Lagrange Multipliers Suppose that ƒsx, y, zd and gsx, y, zd are differentiable and §g Z 0 when gsx, y, zd = 0. To find the local maximum and minimum values of ƒ subject to the constraint gsx, y, zd = 0 (if these exist), find the values of x, y, z, and l that simultaneously satisfy the equations §ƒ = l§g

and

gsx, y, zd = 0.

(1)

For functions of two independent variables, the condition is similar, but without the variable z.

Some care must be used in applying this method. An extreme value may not actually exist (Exercise 41). y

EXAMPLE 3

Find the greatest and smallest values that the function

x2 y2 1 8 2

2

ƒsx, yd = xy takes on the ellipse (Figure 13.52) x

0

y2 x2 + = 1. 8 2

22

Solution

We want to find the extreme values of ƒsx, yd = xy subject to the constraint

FIGURE 13.52 Example 3 shows how to find the largest and smallest values of the product xy on this ellipse.

gsx, yd =

y2 x2 + - 1 = 0. 8 2

To do so, we first find the values of x, y, and l for which §ƒ = l§g

and

gsx, yd = 0.

The gradient equation in Equations (1) gives yi + xj =

l xi + lyj, 4

from which we find xy –2

y

xy 2

∇g 1 i j 2

1 0

∇f i 2j

1

x

y =

l x, 4

x = ly,

xy –2

FIGURE 13.53 When subjected to the constraint gsx, yd = x 2>8 + y 2>2 - 1 = 0, the function ƒsx, yd = xy takes on extreme values at the four points s ;2, ;1d. These are the points on the ellipse when §ƒ (red) is a scalar multiple of §g (blue) (Example 3).

y =

l l2 slyd = y, 4 4

so that y = 0 or l = ;2. We now consider these two cases. Case 1: If y = 0 , then x = y = 0. But (0, 0) is not on the ellipse. Hence, y Z 0. Case 2: If y Z 0, then l = ;2 and x = ;2y. Substituting this in the equation gsx, yd = 0 gives s ;2yd2 y2 + = 1, 8 2

xy 2

and

4y 2 + 4y 2 = 8

and

y = ;1.

The function ƒsx, yd = xy therefore takes on its extreme values on the ellipse at the four points s ;2, 1d, s ;2, -1d. The extreme values are xy = 2 and xy = -2. The Geometry of the Solution The level curves of the function ƒsx, yd = xy are the hyperbolas xy = c (Figure 13.53). The farther the hyperbolas lie from the origin, the larger the absolute value of ƒ. We want to find the extreme values of ƒ(x, y), given that the point (x, y) also lies on the ellipse x 2 + 4y 2 = 8. Which hyperbolas intersecting the ellipse lie farthest from the origin? The hyperbolas that just graze the ellipse, the ones that are tangent to it, are

13.8

Lagrange Multipliers

753

farthest. At these points, any vector normal to the hyperbola is normal to the ellipse, so §ƒ = yi + xj is a multiple sl = ;2d of §g = sx>4di + yj. At the point (2, 1), for example, §g =

1 i + j, 2

and

§ƒ = 2§g.

§g = -

1 i + j, 2

and

§ƒ = -2§g.

§ƒ = i + 2j, At the point s -2, 1d, §ƒ = i - 2j,

Find the maximum and minimum values of the function ƒsx, yd = 3x + 4y on the circle x 2 + y 2 = 1.

EXAMPLE 4 Solution

We model this as a Lagrange multiplier problem with gsx, yd = x 2 + y 2 - 1

ƒsx, yd = 3x + 4y,

and look for the values of x, y, and l that satisfy the equations §ƒ = l§g:

3i + 4j = 2xli + 2ylj

gsx, yd = 0:

x 2 + y 2 - 1 = 0.

The gradient equation in Equations (1) implies that l Z 0 and gives 3 , 2l

x =

y =

2 . l

These equations tell us, among other things, that x and y have the same sign. With these values for x and y, the equation gsx, yd = 0 gives a

2

2

3 2 b + a b - 1 = 0, l 2l

so

∇f 3i 4j

9 4 + 2 = 1, 4l2 l

5 ∇g 2

4l2 = 25,

and

5 l = ; . 2

Thus,

y ∇g

x2 y2 1

9 + 16 = 4l2,

6i 8j 5 5

⎛ 3 , 4⎛ ⎝ 5 5⎝ x 3x 4y 5 3x 4y –5

FIGURE 13.54 The function ƒsx, yd = 3x + 4y takes on its largest value on the unit circle gsx, yd = x 2 + y 2 - 1 = 0 at the point (3> 5, 4> 5) and its smallest value at the point s -3>5, -4>5d (Example 4). At each of these points, §ƒ is a scalar multiple of §g . The figure shows the gradients at the first point but not the second.

x =

3 3 = ; , 5 2l

y =

2 4 = ; , 5 l

and ƒsx, yd = 3x + 4y has extreme values at sx, yd = ;s3>5, 4>5d. By calculating the value of 3x + 4y at the points ;s3>5, 4>5d, we see that its maximum and minimum values on the circle x 2 + y 2 = 1 are 3 25 4 3a b + 4a b = = 5 5 5 5

and

3 25 4 3 a- b + 4 a- b = = -5. 5 5 5

The level curves of ƒsx, yd = 3x + 4y are the lines 3x + 4y = c (Figure 13.54). The farther the lines lie from the origin, the larger the absolute value of ƒ. We want to find the extreme values of ƒ(x, y) given that the point (x, y) also lies on the circle x 2 + y 2 = 1 . Which lines intersecting the circle lie farthest from the origin? The lines tangent to the circle are farthest. At the points of tangency, any vector normal to the line is normal to the circle, so the gradient §ƒ = 3i + 4j is a multiple sl = ;5>2d of the gradient §g = 2xi + 2yj. At the point (3> 5, 4> 5), for example, The Geometry of the Solution

§ƒ = 3i + 4j,

§g =

6 8 i + j, 5 5

and

§ƒ =

5 §g. 2

754

Chapter 13: Partial Derivatives

Lagrange Multipliers with Two Constraints

g2 0

Many problems require us to find the extreme values of a differentiable function ƒ(x, y, z) whose variables are subject to two constraints. If the constraints are g1sx, y, zd = 0

∇g1

and

g2sx, y, zd = 0

and g1 and g2 are differentiable, with §g1 not parallel to §g2 , we find the constrained local maxima and minima of ƒ by introducing two Lagrange multipliers l and m (mu, pronounced “mew”). That is, we locate the points P(x, y, z) where ƒ takes on its constrained extreme values by finding the values of x, y, z, l, and m that simultaneously satisfy the three equations

∇f ∇g2

g1 0

C

§ƒ = l§g1 + m§g2 ,

FIGURE 13.55 The vectors §g1 and §g2 lie in a plane perpendicular to the curve C because §g1 is normal to the surface g1 = 0 and §g2 is normal to the surface g2 = 0 .

g1sx, y, zd = 0,

g2sx, y, zd = 0

(2)

Equations (2) have a nice geometric interpretation. The surfaces g1 = 0 and g2 = 0 (usually) intersect in a smooth curve, say C (Figure 13.55). Along this curve we seek the points where ƒ has local maximum and minimum values relative to its other values on the curve. These are the points where §ƒ is normal to C, as we saw in Theorem 12. But §g1 and §g2 are also normal to C at these points because C lies in the surfaces g1 = 0 and g2 = 0. Therefore, §ƒ lies in the plane determined by §g1 and §g2 , which means that §ƒ = l§g1 + m§g2 for some l and m . Since the points we seek also lie in both surfaces, their coordinates must satisfy the equations g1sx, y, zd = 0 and g2sx, y, zd = 0, which are the remaining requirements in Equations (2). The plane x + y + z = 1 cuts the cylinder x 2 + y 2 = 1 in an ellipse (Figure 13.56). Find the points on the ellipse that lie closest to and farthest from the origin.

EXAMPLE 5

z Cylinder x 2 y 2 1

Solution

We find the extreme values of ƒsx, y, zd = x 2 + y 2 + z 2

P2

(the square of the distance from (x, y, z) to the origin) subject to the constraints (0, 1, 0)

(1, 0, 0) x

y

g1sx, y, zd = x 2 + y 2 - 1 = 0

(3)

g2sx, y, zd = x + y + z - 1 = 0.

(4)

The gradient equation in Equations (2) then gives

P1

§ƒ = l§g1 + m§g2 2xi + 2yj + 2zk = ls2xi + 2yjd + msi + j + kd 2xi + 2yj + 2zk = s2lx + mdi + s2ly + mdj + mk

Plane xy z 1

or 2x = 2lx + m, FIGURE 13.56 On the ellipse where the plane and cylinder meet, we find the points closest to and farthest from the origin. (Example 5).

2y = 2ly + m,

2z = m.

(5)

The scalar equations in Equations (5) yield 2x = 2lx + 2z Q s1 - ldx = z, 2y = 2ly + 2z Q s1 - ldy = z.

(6)

Equations (6) are satisfied simultaneously if either l = 1 and z = 0 or l Z 1 and x = y = z>s1 - ld. If z = 0, then solving Equations (3) and (4) simultaneously to find the corresponding points on the ellipse gives the two points (1, 0, 0) and (0, 1, 0). This makes sense when you look at Figure 13.56.

13.8

Lagrange Multipliers

755

If x = y, then Equations (3) and (4) give x2 + x2 - 1 = 0 2x 2 = 1 x = ;

x + x + z - 1 = 0 z = 1 - 2x 22 2

z = 1 < 22.

The corresponding points on the ellipse are P1 = a

22 22 , , 1 - 22b 2 2

P2 = a-

and

22 22 ,, 1 + 22b. 2 2

Here we need to be careful, however. Although P1 and P2 both give local maxima of ƒ on the ellipse, P2 is farther from the origin than P1. The points on the ellipse closest to the origin are (1, 0, 0) and (0, 1, 0). The point on the ellipse farthest from the origin is P2.

Exercises 13.8 Two Independent Variables with One Constraint 1. Extrema on an ellipse Find the points on the ellipse x 2 + 2y 2 = 1 where ƒsx, yd = xy has its extreme values. 2. Extrema on a circle Find the extreme values of ƒsx, yd = xy subject to the constraint gsx, yd = x 2 + y 2 - 10 = 0. 3. Maximum on a line Find the maximum value of ƒsx, yd = 49 - x 2 - y 2 on the line x + 3y = 10. 4. Extrema on a line Find the local extreme values of ƒsx, yd = x 2y on the line x + y = 3. 5. Constrained minimum Find the points on the curve xy 2 = 54 nearest the origin. 6. Constrained minimum Find the points on the curve x 2y = 2 nearest the origin. 7. Use the method of Lagrange multipliers to find a. Minimum on a hyperbola The minimum value of x + y, subject to the constraints xy = 16, x 7 0, y 7 0 b. Maximum on a line The maximum value of xy, subject to the constraint x + y = 16. Comment on the geometry of each solution. 8. Extrema on a curve Find the points on the curve x 2 + xy + y 2 = 1 in the xy-plane that are nearest to and farthest from the origin. 9. Minimum surface area with fixed volume Find the dimensions of the closed right circular cylindrical can of smallest surface area whose volume is 16p cm3. 10. Cylinder in a sphere Find the radius and height of the open right circular cylinder of largest surface area that can be inscribed in a sphere of radius a. What is the largest surface area? 11. Rectangle of greatest area in an ellipse Use the method of Lagrange multipliers to find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse x 2>16 + y 2>9 = 1 with sides parallel to the coordinate axes. 12. Rectangle of longest perimeter in an ellipse Find the dimensions of the rectangle of largest perimeter that can be inscribed in

the ellipse x 2>a 2 + y 2>b 2 = 1 with sides parallel to the coordinate axes. What is the largest perimeter? 13. Extrema on a circle Find the maximum and minimum values of x 2 + y 2 subject to the constraint x 2 - 2x + y 2 - 4y = 0. 14. Extrema on a circle Find the maximum and minimum values of 3x - y + 6 subject to the constraint x 2 + y 2 = 4. 15. Ant on a metal plate The temperature at a point (x, y) on a metal plate is Tsx, yd = 4x 2 - 4xy + y 2. An ant on the plate walks around the circle of radius 5 centered at the origin. What are the highest and lowest temperatures encountered by the ant? 16. Cheapest storage tank Your firm has been asked to design a storage tank for liquid petroleum gas. The customer’s specifications call for a cylindrical tank with hemispherical ends, and the tank is to hold 8000 m3 of gas. The customer also wants to use the smallest amount of material possible in building the tank. What radius and height do you recommend for the cylindrical portion of the tank? Three Independent Variables with One Constraint 17. Minimum distance to a point Find the point on the plane x + 2y + 3z = 13 closest to the point (1, 1, 1). 18. Maximum distance to a point Find the point on the sphere x 2 + y 2 + z 2 = 4 farthest from the point s1, -1, 1d. 19. Minimum distance to the origin Find the minimum distance from the surface x 2 - y 2 - z 2 = 1 to the origin. 20. Minimum distance to the origin Find the point on the surface z = xy + 1 nearest the origin. 21. Minimum distance to the origin Find the points on the surface z 2 = xy + 4 closest to the origin. 22. Minimum distance to the origin Find the point(s) on the surface xyz = 1 closest to the origin. 23. Extrema on a sphere Find the maximum and minimum values of ƒsx, y, zd = x - 2y + 5z on the sphere x + y 2 + z 2 = 30. 2

756

Chapter 13: Partial Derivatives

24. Extrema on a sphere Find the points on the sphere x 2 + y 2 + z 2 = 25 where ƒsx, y, zd = x + 2y + 3z has its maximum and minimum values. 25. Minimizing a sum of squares Find three real numbers whose sum is 9 and the sum of whose squares is as small as possible. 26. Maximizing a product Find the largest product the positive numbers x, y, and z can have if x + y + z 2 = 16. 27. Rectangular box of largest volume in a sphere Find the dimensions of the closed rectangular box with maximum volume that can be inscribed in the unit sphere.

34. Minimize the function ƒsx, y, zd = x 2 + y 2 + z 2 subject to the constraints x + 2y + 3z = 6 and x + 3y + 9z = 9. 35. Minimum distance to the origin Find the point closest to the origin on the line of intersection of the planes y + 2z = 12 and x + y = 6. 36. Maximum value on line of intersection Find the maximum value that ƒsx, y, zd = x 2 + 2y - z 2 can have on the line of intersection of the planes 2x - y = 0 and y + z = 0. 37. Extrema on a curve of intersection Find the extreme values of ƒsx, y, zd = x 2yz + 1 on the intersection of the plane z = 1 with the sphere x 2 + y 2 + z 2 = 10.

28. Box with vertex on a plane Find the volume of the largest closed rectangular box in the first octant having three faces in the coordinate planes and a vertex on the plane x>a + y>b + z>c = 1, where a 7 0, b 7 0, and c 7 0.

38. a. Maximum on line of intersection Find the maximum value of w = xyz on the line of intersection of the two planes x + y + z = 40 and x + y - z = 0.

29. Hottest point on a space probe A space probe in the shape of the ellipsoid

b. Give a geometric argument to support your claim that you have found a maximum, and not a minimum, value of w.

4x 2 + y 2 + 4z 2 = 16

39. Extrema on a circle of intersection Find the extreme values of the function ƒsx, y, zd = xy + z 2 on the circle in which the plane y - x = 0 intersects the sphere x 2 + y 2 + z 2 = 4.

enters Earth’s atmosphere and its surface begins to heat. After 1 hour, the temperature at the point (x, y, z) on the probe’s surface is Tsx, y, zd = 8x 2 + 4yz - 16z + 600 . Find the hottest point on the probe’s surface. 30. Extreme temperatures on a sphere Suppose that the Celsius temperature at the point (x, y, z) on the sphere x 2 + y 2 + z 2 = 1 is T = 400xyz 2 . Locate the highest and lowest temperatures on the sphere. 31. Maximizing a utility function: an example from economics In economics, the usefulness or utility of amounts x and y of two capital goods G1 and G2 is sometimes measured by a function U(x, y). For example, G1 and G2 might be two chemicals a pharmaceutical company needs to have on hand and U(x, y) the gain from manufacturing a product whose synthesis requires different amounts of the chemicals depending on the process used. If G1 costs a dollars per kilogram, G2 costs b dollars per kilogram, and the total amount allocated for the purchase of G1 and G2 together is c dollars, then the company’s managers want to maximize U(x, y) given that ax + by = c . Thus, they need to solve a typical Lagrange multiplier problem. Suppose that Usx, yd = xy + 2x and that the equation ax + by = c simplifies to 2x + y = 30. Find the maximum value of U and the corresponding values of x and y subject to this latter constraint. 32. Locating a radio telescope You are in charge of erecting a radio telescope on a newly discovered planet. To minimize interference, you want to place it where the magnetic field of the planet is weakest. The planet is spherical, with a radius of 6 units. Based on a coordinate system whose origin is at the center of the planet, the strength of the magnetic field is given by Msx, y, zd = 6x - y 2 + xz + 60. Where should you locate the radio telescope? Extreme Values Subject to Two Constraints 33. Maximize the function ƒsx, y, zd = x 2 + 2y - z 2 subject to the constraints 2x - y = 0 and y + z = 0.

40. Minimum distance to the origin Find the point closest to the origin on the curve of intersection of the plane 2y + 4z = 5 and the cone z 2 = 4x 2 + 4y 2. Theory and Examples 41. The condition §ƒ l§g is not sufficient Although §ƒ = l§g is a necessary condition for the occurrence of an extreme value of ƒ(x, y) subject to the conditions gsx, yd = 0 and §g Z 0, it does not in itself guarantee that one exists. As a case in point, try using the method of Lagrange multipliers to find a maximum value of ƒsx, yd = x + y subject to the constraint that xy = 16. The method will identify the two points (4, 4) and s -4, -4d as candidates for the location of extreme values. Yet the sum sx + yd has no maximum value on the hyperbola xy = 16. The farther you go from the origin on this hyperbola in the first quadrant, the larger the sum ƒsx, yd = x + y becomes. 42. A least squares plane The plane z = Ax + By + C is to be “fitted” to the following points sxk , yk , zk d: s0, 0, 0d,

s0, 1, 1d,

s1, 1, 1d,

s1, 0, -1d.

Find the values of A, B, and C that minimize 4 2 a sAxk + Byk + C - zk d ,

k=1

the sum of the squares of the deviations. 43. a. Maximum on a sphere Show that the maximum value of a 2b 2c 2 on a sphere of radius r centered at the origin of a Cartesian abc-coordinate system is sr 2>3d3. b. Geometric and arithmetic means Using part (a), show that for nonnegative numbers a, b, and c, sabcd1>3 …

a + b + c ; 3

that is, the geometric mean of three nonnegative numbers is less than or equal to their arithmetic mean. 44. Sum of products Let a1, a2 , Á , an be n positive numbers. Find the maximum of © ni = 1 ai xi subject to the constraint © ni = 1 x i 2 = 1.

Chapter 13 COMPUTER EXPLORATIONS In Exercises 45–50, use a CAS to perform the following steps implementing the method of Lagrange multipliers for finding constrained extrema: a. Form the function h = ƒ - l1 g1 - l2 g2 , where ƒ is the function to optimize subject to the constraints g1 = 0 and g2 = 0. b. Determine all the first partial derivatives of h, including the partials with respect to l1 and l2 , and set them equal to 0. c. Solve the system of equations found in part (b) for all the unknowns, including l1 and l2. d. Evaluate ƒ at each of the solution points found in part (c) and select the extreme value subject to the constraints asked for in the exercise.

Questions to Guide Your Review

46. Minimize ƒsx, y, zd = xyz subject x 2 + y 2 - 1 = 0 and x - z = 0.

to

the

757 constraints

47. Maximize ƒsx, y, zd = x 2 + y 2 + z 2 subject to the constraints 2y + 4z - 5 = 0 and 4x 2 + 4y 2 - z 2 = 0. 48. Minimize ƒsx, y, zd = x 2 + y 2 + z 2 subject to the constraints x 2 - xy + y 2 - z 2 - 1 = 0 and x 2 + y 2 - 1 = 0. 49. Minimize ƒsx, y, z, wd = x 2 + y 2 + z 2 + w 2 subject to the constraints 2x - y + z - w - 1 = 0 and x + y - z + w - 1 = 0. 50. Determine the distance from the line y = x + 1 to the parabola y 2 = x. (Hint: Let (x, y) be a point on the line and (w, z) a point on the parabola. You want to minimize sx - wd2 + sy - zd2.)

45. Minimize ƒsx, y, zd = xy + yz subject to the constraints x 2 + y 2 - 2 = 0 and x 2 + z 2 - 2 = 0.

Chapter 13

Questions to Guide Your Review

1. What is a real-valued function of two independent variables? Three independent variables? Give examples. 2. What does it mean for sets in the plane or in space to be open? Closed? Give examples. Give examples of sets that are neither open nor closed.

13. What is the general Chain Rule? What form does it take for functions of two independent variables? Three independent variables? Functions defined on surfaces? How do you diagram these different forms? Give examples. What pattern enables one to remember all the different forms?

3. How can you display the values of a function ƒ(x, y) of two independent variables graphically? How do you do the same for a function ƒ(x, y, z) of three independent variables?

14. What is the derivative of a function ƒ(x, y) at a point P0 in the direction of a unit vector u? What rate does it describe? What geometric interpretation does it have? Give examples.

4. What does it mean for a function ƒ(x, y) to have limit L as sx, yd : sx0 , y0 d? What are the basic properties of limits of functions of two independent variables?

15. What is the gradient vector of a differentiable function ƒ(x, y)? How is it related to the function’s directional derivatives? State the analogous results for functions of three independent variables.

5. When is a function of two (three) independent variables continuous at a point in its domain? Give examples of functions that are continuous at some points but not others.

16. How do you find the tangent line at a point on a level curve of a differentiable function ƒ(x, y)? How do you find the tangent plane and normal line at a point on a level surface of a differentiable function ƒ(x, y, z)? Give examples.

6. What can be said about algebraic combinations and composites of continuous functions? 7. Explain the two-path test for nonexistence of limits. 8. How are the partial derivatives 0ƒ>0x and 0ƒ>0y of a function ƒ(x, y) defined? How are they interpreted and calculated? 9. How does the relation between first partial derivatives and continuity of functions of two independent variables differ from the relation between first derivatives and continuity for real-valued functions of a single independent variable? Give an example. 10. What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples. 11. What does it mean for a function ƒ(x, y) to be differentiable? What does the Increment Theorem say about differentiability? 12. How can you sometimes decide from examining ƒx and ƒy that a function ƒ(x, y) is differentiable? What is the relation between the differentiability of ƒ and the continuity of ƒ at a point?

17. How can you use directional derivatives to estimate change? 18. How do you linearize a function ƒ(x, y) of two independent variables at a point sx0 , y0 d? Why might you want to do this? How do you linearize a function of three independent variables? 19. What can you say about the accuracy of linear approximations of functions of two (three) independent variables? 20. If (x, y) moves from sx0 , y0 d to a point sx0 + dx, y0 + dyd nearby, how can you estimate the resulting change in the value of a differentiable function ƒ(x, y)? Give an example. 21. How do you define local maxima, local minima, and saddle points for a differentiable function ƒ(x, y)? Give examples. 22. What derivative tests are available for determining the local extreme values of a function ƒ(x, y)? How do they enable you to narrow your search for these values? Give examples. 23. How do you find the extrema of a continuous function ƒ(x, y) on a closed bounded region of the xy-plane? Give an example. 24. Describe the method of Lagrange multipliers and give examples.

758

Chapter 13: Partial Derivatives

Chapter 13

Practice Exercises

Domain, Range, and Level Curves In Exercises 1–4, find the domain and range of the given function and identify its level curves. Sketch a typical level curve. 1. ƒsx, yd = 9x 2 + y 2

2. ƒsx, yd = e x + y

3. gsx, yd = 1>xy

4. gsx, yd = 2x 2 - y

In Exercises 5–8, find the domain and range of the given function and identify its level surfaces. Sketch a typical level surface. 5. ƒsx, y, zd = x 2 + y 2 - z

6. gsx, y, zd = x 2 + 4y 2 + 9z 2

1 7. hsx, y, zd = 2 x + y2 + z2 8. ksx, y, zd =

2

Evaluating Limits Find the limits in Exercises 9–14. 9. 11. 13.

lim

sx,yd:sp, ln 2d

lim

x2 - y2

P: s1, -1, ed

ln ƒ x + y + z ƒ

2 + y x + cos y x 3y 3 - 1 lim 12. sx,yd: s1,1d xy - 1 14.

lim

sx,yd: s0,0d

lim

P: s1,-1,-1d

tan-1 sx + y + zd

By considering different paths of approach, show that the limits in Exercises 15 and 16 do not exist. y x2 + y2 lim 15. sx,ydlim 16. xy 2 : s0,0d sx,yd : s0,0d x - y y Z x2

xy Z 0

17. Continuous extension Let ƒsx, yd = sx 2 - y 2 d>sx 2 + y 2 d for sx, yd Z s0, 0d . Is it possible to define ƒ(0, 0) in a way that makes ƒ continuous at the origin? Why? 18. Continuous extension ƒsx, yd =

26. gsx, yd = e x + y sin x

27. ƒsx, yd = x + xy - 5x 3 + ln sx 2 + 1d 28. ƒsx, yd = y 2 - 3xy + cos y + 7e y Chain Rule Calculations 29. Find dw> dt at t = 0 if w = sin sxy + pd, x = e t, and y = ln st + 1d.

31. Find 0w>0r and 0w>0s when r = p and s = 0 if w = sin s2x - yd, x = r + sin s, y = rs. 10.

x - y

sx,yd:s1,1d

lim

e y cos x

x 25. gsx, yd = y + y

30. Find dw> dt at t = 1 if w = xe y + y sin z - cos z, x = 22t, y = t - 1 + ln t , and z = pt.

1 x + y + z2 + 1 2

Second-Order Partials Find the second-order partial derivatives of the functions in Exercises 25–28.

Let

sin sx - yd , ƒxƒ + ƒyƒ L 0,

ƒxƒ + ƒyƒ Z 0 sx, yd = s0, 0d.

Is ƒ continuous at the origin? Why?

32. Find

0w>0u

19. gsr, ud = r cos u + r sin u y 1 20. ƒsx, yd = ln sx 2 + y 2 d + tan-1 x 2 21. ƒsR1, R2 , R3 d =

1 1 1 + + R1 R2 R3

24. ƒsr, l, T, wd =

nRT (the ideal gas law) V

T 1 2rl A pw

if

w =

34. Show that if w = ƒssd is any differentiable function of s and if s = y + 5x , then 0w 0w - 5 = 0. 0x 0y Implicit Differentiation Assuming that the equations in Exercises 35 and 36 define y as a differentiable function of x, find the value of dy> dx at point P. 35. 1 - x - y 2 - sin xy = 0, 36. 2xy + e

x+y

- 2 = 0,

Ps0, 1d

Ps0, ln 2d

Directional Derivatives In Exercises 37–40, find the directions in which ƒ increases and decreases most rapidly at P0 and find the derivative of ƒ in each direction. Also, find the derivative of ƒ at P0 in the direction of the vector v. 2 -2y

38. ƒsx, yd = x e

,

P0sp>4, p>4d,

P0s1, 0d,

v = 3i + 4j

v = i + j

39. ƒsx, y, zd = ln s2x + 3y + 6zd,

P0s -1, -1, 1d,

v = 2i + 3j + 6k 40. ƒsx, y, zd = x 2 + 3xy - z 2 + 2y + z + 4,

P0s0, 0, 0d,

v = i + j + k 41. Derivative in velocity direction Find the derivative of ƒsx, y, zd = xyz in the direction of the velocity vector of the helix rstd = scos 3tdi + ssin 3tdj + 3t k

22. hsx, y, zd = sin s2px + y - 3zd 23. Psn, R, T, Vd =

u = y = 0

when

33. Find the value of the derivative of ƒsx, y, zd = xy + yz + xz with respect to t on the curve x = cos t, y = sin t, z = cos 2t at t = 1 .

37. ƒsx, yd = cos x cos y, Partial Derivatives In Exercises 19–24, find the partial derivative of the function with respect to each variable.

0w>0y

and

ln 21 + x 2 - tan-1 x and x = 2e u cos y.

at t = p>3. 42. Maximum directional derivative What is the largest value that the directional derivative of ƒsx, y, zd = xyz can have at the point (1, 1, 1)?

Chapter 13 43. Directional derivatives with given values At the point (1, 2), the function ƒ(x, y) has a derivative of 2 in the direction toward (2, 2) and a derivative of -2 in the direction toward (1, 1). a. Find ƒxs1, 2d and ƒys1, 2d . b. Find the derivative of ƒ at (1, 2) in the direction toward the point (4, 6). 44. Which of the following statements are true if ƒ(x, y) is differentiable at sx0 , y0 d ? Give reasons for your answers. a. If u is a unit vector, the derivative of ƒ at sx0 , y0 d in the direction of u is sƒxsx0 , y0 di + ƒysx0 , y0 djd # u. b. The derivative of ƒ at sx0 , y0 d in the direction of u is a vector. c. The directional derivative of ƒ at sx0 , y0 d has its greatest value in the direction of §ƒ . d. At sx0 , y0 d , vector §ƒ is normal to the curve ƒsx, yd = ƒsx0 , y0 d.

46. y 2 + z 2 = 4;

s0, -1, ;1d,

s2, ;2, 0d,

s0, 0, 0d

s2, 0, ;2d

In Exercises 47 and 48, find an equation for the plane tangent to the level surface ƒsx, y, zd = c at the point P0 . Also, find parametric equations for the line that is normal to the surface at P0 . 47. x 2 - y - 5z = 0,

P0s2, -1, 1d

48. x 2 + y 2 + z = 4,

P0s1, 1, 2d

In Exercises 49 and 50, find an equation for the plane tangent to the surface z = ƒsx, yd at the given point. 49. z = ln sx + y d,

s0, 1, 0d

50. z = 1>sx 2 + y 2 d,

s1, 1, 1>2d

2

2

P0sp, 1d

52.

y2 3 x2 = , 2 2 2

P0s1, 2d

Tangent Lines to Curves In Exercises 53 and 54, find parametric equations for the line that is tangent to the curve of intersection of the surfaces at the given point. 53. Surfaces: x 2 + 2y + 2z = 4, Point:

(1, 1, 1> 2)

54. Surfaces: x + y + z = 2, 2

Point:

(1> 2, 1, 1> 2)

y = 1 y = 1

Linearizations In Exercises 55 and 56, find the linearization L(x, y) of the function ƒ(x, y) at the point P0 . Then find an upper bound for the magnitude of the error E in the approximation ƒsx, yd L Lsx, yd over the rectangle R. 55. ƒsx, yd = sin x cos y, R:

`x -

p ` … 0.1, 4

P0sp>4, p>4d

`y -

ƒ x - 1 ƒ … 0.1,

P0s1, 1d

ƒ y - 1 ƒ … 0.2

Find the linearizations of the functions in Exercises 57 and 58 at the given points. 57. ƒsx, y, zd = xy + 2yz - 3xz at (1, 0, 0) and (1, 1, 0) 58. ƒsx, y, zd = 22 cos x sin s y + zd at s0, 0, p>4d and sp>4, p>4, 0d Estimates and Sensitivity to Change 59. Measuring the volume of a pipeline You plan to calculate the volume inside a stretch of pipeline that is about 36 in. in diameter and 1 mile long. With which measurement should you be more careful, the length or the diameter? Why?

61. Change in an electrical circuit Suppose that the current I (amperes) in an electrical circuit is related to the voltage V (volts) and the resistance R (ohms) by the equation I = V>R . If the voltage drops from 24 to 23 volts and the resistance drops from 100 to 80 ohms, will I increase or decrease? By about how much? Is the change in I more sensitive to change in the voltage or to change in the resistance? How do you know? 62. Maximum error in estimating the area of an ellipse If a = 10 cm and b = 16 cm to the nearest millimeter, what should you expect the maximum percentage error to be in the calculated area A = pab of the ellipse x 2>a 2 + y 2>b 2 = 1 ? 63. Error in estimating a product Let y = uy and z = u + y, where u and y are positive independent variables. a. If u is measured with an error of 2% and y with an error of 3%, about what is the percentage error in the calculated value of y ? b. Show that the percentage error in the calculated value of z is less than the percentage error in the value of y.

In Exercises 51 and 52, find equations for the lines that are tangent and normal to the level curve ƒsx, yd = c at the point P0 . Then sketch the lines and level curve together with §ƒ at P0 . 51. y - sin x = 1,

R:

759

60. Sensitivity to change Is ƒsx, yd = x 2 - xy + y 2 - 3 more sensitive to changes in x or to changes in y when it is near the point (1, 2)? How do you know?

Gradients, Tangent Planes, and Normal Lines In Exercises 45 and 46, sketch the surface ƒsx, y, zd = c together with §ƒ at the given points. 45. x 2 + y + z 2 = 0;

56. ƒsx, yd = xy - 3y 2 + 2,

Practice Exercises

p ` … 0.1 4

64. Cardiac index To make different people comparable in studies of cardiac output, researchers divide the measured cardiac output by the body surface area to find the cardiac index C: C =

cardiac output . body surface area

The body surface area B of a person with weight w and height h is approximated by the formula B = 71.84w 0.425h 0.725 , which gives B in square centimeters when w is measured in kilograms and h in centimeters. You are about to calculate the cardiac index of a person 180 cm tall, weighing 70 kg, with cardiac output of 7 L>min. Which will have a greater effect on the calculation, a 1-kg error in measuring the weight or a 1-cm error in measuring the height? Local Extrema Test the functions in Exercises 65–70 for local maxima and minima and saddle points. Find each function’s value at these points. 65. ƒsx, yd = x 2 - xy + y 2 + 2x + 2y - 4 66. ƒsx, yd = 5x 2 + 4xy - 2y 2 + 4x - 4y

760

Chapter 13: Partial Derivatives

67. ƒsx, yd = 2x 3 + 3xy + 2y 3 68. ƒsx, yd = x + y - 3xy + 15 3

3

69. ƒsx, yd = x 3 + y 3 + 3x 2 - 3y 2 70. ƒsx, yd = x 4 - 8x 2 + 3y 2 - 6y Absolute Extrema In Exercises 71–78, find the absolute maximum and minimum values of ƒ on the region R. 71. ƒsx, yd = x 2 + xy + y 2 - 3x + 3y R: The triangular region cut from the first quadrant by the line x + y = 4 72. ƒsx, yd = x 2 - y 2 - 2x + 4y + 1 R: The rectangular region in the first quadrant bounded by the coordinate axes and the lines x = 4 and y = 2

87. Extrema on curve of intersecting surfaces Find the extreme values of ƒsx, y, zd = xs y + zd on the curve of intersection of the right circular cylinder x 2 + y 2 = 1 and the hyperbolic cylinder xz = 1. 88. Minimum distance to origin on curve of intersecting plane and cone Find the point closest to the origin on the curve of intersection of the plane x + y + z = 1 and the cone z 2 = 2x 2 + 2y 2. Theory and Examples 89. Let w = ƒsr, ud, r = 2x 2 + y 2 , and u = tan-1 sy>x). Find 0w>0x and 0w>0y and express your answers in terms of r and u. 90. Let z = ƒsu, yd, u = ax + by, and y = ax - by. Express zx and zy in terms of fu , fy , and the constants a and b. 91. If a and b are constants, w = u 3 + tanh u + cos u, and u = ax + by, show that

73. ƒsx, yd = y 2 - xy - 3y + 2x a

R: The square region enclosed by the lines x = ;2 and y = ;2 74. ƒsx, yd = 2x + 2y - x 2 - y 2 R: The square region bounded by the coordinate axes and the lines x = 2, y = 2 in the first quadrant 75. ƒsx, yd = x 2 - y 2 - 2x + 4y R: The triangular region bounded below by the x-axis, above by the line y = x + 2, and on the right by the line x = 2 76. ƒsx, yd = 4xy - x 4 - y 4 + 16

92. Using the Chain Rule If w = ln sx 2 + y 2 + 2zd, x = r + s, y = r - s, and z = 2rs, find wr and ws by the Chain Rule. Then check your answer another way. 93. Angle between vectors The equations e u cos y - x = 0 and e u sin y - y = 0 define u and y as differentiable functions of x and y. Show that the angle between the vectors 0u 0u i + j 0x 0y

R: The triangular region bounded below by the line y = -2, above by the line y = x, and on the right by the line x = 2 77. ƒsx, yd = x 3 + y 3 + 3x 2 - 3y 2 R: The square region enclosed by the lines x = ;1 and y = ;1 78. ƒsx, yd = x 3 + 3xy + y 3 + 1 R: The square region enclosed by the lines x = ;1 and y = ;1 Lagrange Multipliers 79. Extrema on a circle Find the extreme values of ƒsx, yd = x 3 + y 2 on the circle x 2 + y 2 = 1. 80. Extrema on a circle Find the extreme values of ƒsx, yd = xy on the circle x 2 + y 2 = 1. 81. Extrema in a disk Find the extreme values of ƒsx, yd = x 2 + 3y 2 + 2y on the unit disk x 2 + y 2 … 1. 82. Extrema in a disk Find the extreme values of ƒsx, yd = x 2 + y 2 - 3x - xy on the disk x 2 + y 2 … 9. 83. Extrema on a sphere Find the extreme values of ƒsx, y, zd = x - y + z on the unit sphere x 2 + y 2 + z 2 = 1. 84. Minimum distance to origin Find the points on the surface x 2 - zy = 4 closest to the origin. 85. Minimizing cost of a box A closed rectangular box is to have volume V cm3. The cost of the material used in the box is a cents>cm2 for top and bottom, b cents>cm2 for front and back, and c cents>cm2 for the remaining sides. What dimensions minimize the total cost of materials? 86. Least volume Find the plane x>a + y>b + z>c = 1 that passes through the point (2, 1, 2) and cuts off the least volume from the first octant.

0w 0w = b . 0y 0x

and

0y 0y i + j 0x 0y

is constant. 94. Polar coordinates and second derivatives Introducing polar coordinates x = r cos u and y = r sin u changes ƒ(x, y) to gsr, ud. Find the value of 0 2g>0u2 at the point sr, ud = s2, p>2d, given that 0 2ƒ 0ƒ 0ƒ 0 2ƒ = = 2 = 2 = 1 0x 0y 0x 0y at that point. 95. Normal line parallel to a plane

Find the points on the surface

s y + zd + sz - xd2 = 16 2

where the normal line is parallel to the yz-plane. 96. Tangent plane parallel to xy-plane Find the points on the surface xy + yz + zx - x - z 2 = 0 where the tangent plane is parallel to the xy-plane. 97. When gradient is parallel to position vector Suppose that §ƒsx, y, zd is always parallel to the position vector xi + yj + zk. Show that ƒs0, 0, ad = ƒs0, 0, -ad for any a. 98. One-sided directional derivative in all directions, but no gradient The one-sided directional derivative of ƒ at P (x0, y0, z0) in the direction u = u1i + u2 j + u3k is the number lim

s: 0 +

f (x0 + su1, y0 + su2, z0 + su3) - f (x0, y0, z0) . s

Chapter 13 Show that the one-sided directional derivative of ƒsx, y, zd = 2x + y + z 2

2

Additional and Advanced Exercises

761

100. Tangent plane and normal line a. Sketch the surface x 2 - y 2 + z 2 = 4 .

2

at the origin equals 1 in any direction but that ƒ has no gradient vector at the origin.

b. Find a vector normal to the surface at s2, -3, 3d. Add the vector to your sketch.

99. Normal line through origin Show that the line normal to the surface xy + z = 2 at the point (1, 1, 1) passes through the origin.

c. Find equations for the tangent plane and normal line at s2, -3, 3d.

Chapter 13

Additional and Advanced Exercises

Partial Derivatives 1. Function with saddle at the origin If you did Exercise 60 in Section 13.2, you know that the function xy

ƒsx, yd =

L

x2 - y2 x2 + y2

,

sx, yd Z s0, 0d sx, yd = s0, 0d

0,

(see the accompanying figure) is continuous at (0, 0). Find ƒxys0, 0d and ƒyxs0, 0d.

5. Homogeneous functions A function ƒ(x, y) is homogeneous of degree n (n a nonnegative integer) if ƒstx, tyd = t nƒsx, yd for all t, x, and y. For such a function (sufficiently differentiable), prove that 0ƒ 0ƒ a. x + y = nƒsx, yd 0x 0y 0 2ƒ 0 2ƒ b + y 2 a 2 b = nsn - 1dƒ. 0x0y 0x 0y 6. Surface in polar coordinates Let b. x 2 a

0 2ƒ

2

b + 2xy a

z

ƒsr, ud =

L

sin 6r , 6r

r Z 0

1,

r = 0,

where r and u are polar coordinates. Find a. lim ƒsr, ud y

r: 0

b. ƒrs0, 0d

c. ƒusr, ud,

r Z 0.

z f (r, )

x

2. Finding a function from second partials Find a function w = ƒsx, yd whose first partial derivatives are 0w>0x = 1 + e x cos y and 0w>0y = 2y - e x sin y and whose value at the point (ln 2, 0) is ln 2. 3. A proof of Leibniz’s Rule Leibniz’s Rule says that if ƒ is continuous on [a, b] and if u(x) and y(x) are differentiable functions of x whose values lie in [a, b], then ysxd

d dy du ƒstd dt = ƒsysxdd - ƒsusxdd . dx Lusxd dx dx

gsu, yd =

Lu

ƒstd dt,

a. Show that §r = r>r. b. Show that §sr n d = nr n - 2r.

Prove the rule by setting y

Gradients and Tangents 7. Properties of position vectors Let r = xi + yj + zk and let r = ƒ rƒ.

u = usxd,

y = ysxd

and calculating dg> dx with the Chain Rule. 4. Finding a function with constrained second partials Suppose that ƒ is a twice-differentiable function of r, that r = 2x 2 + y 2 + z 2 , and that ƒxx + ƒyy + ƒzz = 0. Show that for some constants a and b, a ƒsrd = r + b.

c. Find a function whose gradient equals r.

d. Show that r # dr = r dr.

e. Show that §sA # rd = A for any constant vector A.

8. Gradient orthogonal to tangent Suppose that a differentiable function ƒ(x, y) has the constant value c along the differentiable curve x = gstd, y = hstd; that is, ƒsgstd, hstdd = c for all values of t. Differentiate both sides of this equation with respect to t to show that §ƒ is orthogonal to the curve’s tangent vector at every point on the curve.

762

Chapter 13: Partial Derivatives

9. Curve tangent to a surface Show that the curve rstd = sln tdi + st ln tdj + t k is tangent to the surface

at (0, 0, 1). 10. Curve tangent to a surface

Show that the curve

t3 4 - 2 bi + a t - 3 bj + cos st - 2dk 4

is tangent to the surface x 3 + y 3 + z 3 - xyz = 0 at s0, -1, 1d. Extreme Values 11. Extrema on a surface Show that the only possible maxima and minima of z on the surface z = x 3 + y 3 - 9xy + 27 occur at (0, 0) and (3, 3). Show that neither a maximum nor a minimum occurs at (0, 0). Determine whether z has a maximum or a minimum at (3, 3). 12. Maximum in closed first quadrant Find the maximum value of ƒsx, yd = 6xye -s2x + 3yd in the closed first quadrant (includes the nonnegative axes). 13. Minimum volume cut from first octant Find the minimum volume for a region bounded by the planes x = 0, y = 0, z = 0 and a plane tangent to the ellipsoid y2 x2 z2 + + 2 = 1 2 2 a b c at a point in the first octant. 14. Minimum distance from a line to a parabola in xy-plane By minimizing the function ƒsx, y, u, yd = sx - ud2 + sy - yd2 subject to the constraints y = x + 1 and u = y 2, find the minimum distance in the xy-plane from the line y = x + 1 to the parabola y 2 = x. Theory and Examples 15. Boundedness of first partials implies continuity Prove the following theorem: If ƒ(x, y) is defined in an open region R of the xy-plane and if ƒx and ƒy are bounded on R, then ƒ(x, y) is continuous on R. (The assumption of boundedness is essential.) 16. Suppose that rstd = gstdi + hstdj + kstdk is a smooth curve in the domain of a differentiable function ƒ(x, y, z). Describe the relation between dƒ> dt, §ƒ , and v = dr>dt. What can be said about §ƒ and v at interior points of the curve where ƒ has extreme values relative to its other values on the curve? Give reasons for your answer. 17. Finding functions from partial derivatives Suppose that ƒ and g are functions of x and y such that 0ƒ 0g = 0y 0x

0ƒ = 0, 0x

ƒs1, 2d = gs1, 2d = 5

and

ƒs0, 0d = 4 .

Find ƒ(x, y) and g(x, y).

xz 2 - yz + cos xy = 1

rstd = a

and suppose that

and

0ƒ 0g = , 0x 0y

18. Rate of change of the rate of change We know that if ƒ(x, y) is a function of two variables and if u = ai + bj is a unit vector, then Du ƒsx, yd = ƒxsx, yda + ƒysx, ydb is the rate of change of ƒ(x, y) at (x, y) in the direction of u. Give a similar formula for the rate of change of the rate of change of ƒ(x, y) at (x, y) in the direction u. 19. Path of a heat-seeking particle A heat-seeking particle has the property that at any point (x, y) in the plane it moves in the direction of maximum temperature increase. If the temperature at (x, y) is Tsx, yd = -e -2y cos x, find an equation y = ƒsxd for the path of a heat-seeking particle at the point sp>4, 0d. 20. Velocity after a ricochet A particle traveling in a straight line with constant velocity i + j - 5k passes through the point (0, 0, 30) and hits the surface z = 2x 2 + 3y 2. The particle ricochets off the surface, the angle of reflection being equal to the angle of incidence. Assuming no loss of speed, what is the velocity of the particle after the ricochet? Simplify your answer. 21. Directional derivatives tangent to a surface Let S be the surface that is the graph of ƒsx, yd = 10 - x 2 - y 2 . Suppose that the temperature in space at each point (x, y, z) is Tsx, y, zd = x 2y + y 2z + 4x + 14y + z. a. Among all the possible directions tangential to the surface S at the point (0, 0, 10), which direction will make the rate of change of temperature at (0, 0, 10) a maximum? b. Which direction tangential to S at the point (1, 1, 8) will make the rate of change of temperature a maximum? 22. Drilling another borehole On a flat surface of land, geologists drilled a borehole straight down and hit a mineral deposit at 1000 ft. They drilled a second borehole 100 ft to the north of the first and hit the mineral deposit at 950 ft. A third borehole 100 ft east of the first borehole struck the mineral deposit at 1025 ft. The geologists have reasons to believe that the mineral deposit is in the shape of a dome, and for the sake of economy, they would like to find where the deposit is closest to the surface. Assuming the surface to be the xy-plane, in what direction from the first borehole would you suggest the geologists drill their fourth borehole? The one-dimensional heat equation If w(x, t) represents the temperature at position x at time t in a uniform wire with perfectly insulated sides, then the partial derivatives wxx and wt satisfy a differential equation of the form wxx =

1 wt . c2

This equation is called the one-dimensional heat equation. The value of the positive constant c 2 is determined by the material from which the wire is made. 23. Find all solutions of the one-dimensional heat equation of the form w = e rt sin px, where r is a constant. 24. Find all solutions of the one-dimensional heat equation that have the form w = e rt sin k x and satisfy the conditions that ws0, td = 0 and wsL, td = 0. What happens to these solutions as t : q ?

14 MULTIPLE INTEGRALS OVERVIEW In this chapter we consider the integral of a function of two variables ƒ(x, y) over a region in the plane and the integral of a function of three variables ƒ(x, y, z) over a region in space. These multiple integrals are defined to be the limit of approximating Riemann sums, much like the single-variable integrals presented in Chapter 5. We illustrate several applications of multiple integrals, including calculations of volumes, areas in the plane, moments, and centers of mass.

Double and Iterated Integrals over Rectangles

14.1

In Chapter 5 we defined the definite integral of a continuous function ƒ(x) over an interval [a, b] as a limit of Riemann sums. In this section we extend this idea to define the double integral of a continuous function of two variables ƒ(x, y) over a bounded rectangle R in the plane. In both cases the integrals are limits of approximating Riemann sums. The Riemann sums for the integral of a single-variable function ƒ(x) are obtained by partitioning a finite interval into thin subintervals, multiplying the width of each subinterval by the value of ƒ at a point ck inside that subinterval, and then adding together all the products. A similar method of partitioning, multiplying, and summing is used to construct double integrals.

Double Integrals We begin our investigation of double integrals by considering the simplest type of planar region, a rectangle. We consider a function ƒ(x, y) defined on a rectangular region R, R:

d

Ak

yk

c … y … d.

We subdivide R into small rectangles using a network of lines parallel to the x- and y-axes (Figure 14.1). The lines divide R into n rectangular pieces, where the number of such pieces n gets large as the width and height of each piece gets small. These rectangles form a partition of R. A small rectangular piece of width ¢x and height ¢y has area ¢A = ¢x¢y. If we number the small pieces partitioning R in some order, then their areas are given by numbers ¢A1, ¢A2 , Á , ¢An , where ¢Ak is the area of the kth small rectangle. To form a Riemann sum over R, we choose a point sxk , yk d in the kth small rectangle, multiply the value of ƒ at that point by the area ¢Ak, and add together the products:

y R

a … x … b,

(xk , yk )

xk

c 0

a

b

x

n

Sn = a ƒsxk , yk d ¢Ak . k=1

FIGURE 14.1 Rectangular grid partitioning the region R into small rectangles of area ¢Ak = ¢xk ¢yk.

Depending on how we pick sxk , yk d in the kth small rectangle, we may get different values for Sn.

763

764

Chapter 14: Multiple Integrals

We are interested in what happens to these Riemann sums as the widths and heights of all the small rectangles in the partition of R approach zero. The norm of a partition P, written 7P7, is the largest width or height of any rectangle in the partition. If 7P7 = 0.1 then all the rectangles in the partition of R have width at most 0.1 and height at most 0.1. Sometimes the Riemann sums converge as the norm of P goes to zero, written 7P7 : 0. The resulting limit is then written as n

lim a ƒsxk , yk d ¢Ak . ƒ ƒ P ƒ ƒ :0 k=1

As 7P7 : 0 and the rectangles get narrow and short, their number n increases, so we can also write this limit as n

lim a ƒsxk , yk d ¢Ak , n: q k=1

with the understanding that 7P7 : 0, and hence ¢Ak : 0, as n : q . There are many choices involved in a limit of this kind. The collection of small rectangles is determined by the grid of vertical and horizontal lines that determine a rectangular partition of R. In each of the resulting small rectangles there is a choice of an arbitrary point sxk , yk d at which ƒ is evaluated. These choices together determine a single Riemann sum. To form a limit, we repeat the whole process again and again, choosing partitions whose rectangle widths and heights both go to zero and whose number goes to infinity. When a limit of the sums Sn exists, giving the same limiting value no matter what choices are made, then the function ƒ is said to be integrable and the limit is called the double integral of ƒ over R, written as

6

ƒsx, yd dA

or

R

6

ƒsx, yd dx dy.

R

It can be shown that if ƒ(x, y) is a continuous function throughout R, then ƒ is integrable, as in the single-variable case discussed in Chapter 5. Many discontinuous functions are also integrable, including functions that are discontinuous only on a finite number of points or smooth curves. We leave the proof of these facts to a more advanced text.

Double Integrals as Volumes

z z 5 f(x, y)

f(xk, yk) d y

When ƒ(x, y) is a positive function over a rectangular region R in the xy-plane, we may interpret the double integral of ƒ over R as the volume of the 3-dimensional solid region over the xy-plane bounded below by R and above by the surface z = ƒsx, yd (Figure 14.2). Each term ƒsxk , yk d¢Ak in the sum Sn = g ƒsxk , yk d¢Ak is the volume of a vertical rectangular box that approximates the volume of the portion of the solid that stands directly above the base ¢Ak. The sum Sn thus approximates what we want to call the total volume of the solid. We define this volume to be

b x

R (xk, yk)

Δ Ak

Volume = lim Sn = n: q

6

ƒsx, yd dA,

R

FIGURE 14.2 Approximating solids with rectangular boxes leads us to define the volumes of more general solids as double integrals. The volume of the solid shown here is the double integral of ƒ(x, y) over the base region R.

where ¢Ak : 0 as n : q . As you might expect, this more general method of calculating volume agrees with the methods in Chapter 6, but we do not prove this here. Figure 14.3 shows Riemann sum approximations to the volume becoming more accurate as the number n of boxes increases.

14.1

Double and Iterated Integrals over Rectangles

(a) n 16

(b) n 64

765

(c) n 256

FIGURE 14.3 As n increases, the Riemann sum approximations approach the total volume of the solid shown in Figure 14.2.

Fubini’s Theorem for Calculating Double Integrals

z

Suppose that we wish to calculate the volume under the plane z = 4 - x - y over the rectangular region R: 0 … x … 2, 0 … y … 1 in the xy-plane. If we apply the method of slicing from Section 6.1, with slices perpendicular to the x-axis (Figure 14.4), then the volume is

4 z4xy

x=2

Asxd dx,

Lx = 0

(1)

where A(x) is the cross-sectional area at x. For each value of x, we may calculate A(x) as the integral Asxd =

1 x

y

2 x

A(x) ⌠ ⌡

y1

y0

(4 x y) dy

FIGURE 14.4 To obtain the crosssectional area A(x), we hold x fixed and integrate with respect to y.

y=1

Ly = 0

s4 - x - yd dy,

(2)

which is the area under the curve z = 4 - x - y in the plane of the cross-section at x. In calculating A(x), x is held fixed and the integration takes place with respect to y. Combining Equations (1) and (2), we see that the volume of the entire solid is Volume = =

x=2

Lx = 0 x=2

Lx = 0

Asxd dx =

x=2

Lx = 0

c4y - xy -

a

y=1

Ly = 0

s4 - x - yd dyb dx

x=2 y 2 y=1 7 d dx = a - xb dx 2 y=0 2 Lx = 0

2

7 x2 = c x d = 5. 2 2 0

(3)

If we just wanted to write a formula for the volume, without carrying out any of the integrations, we could write 2

Volume =

L0 L0

1

s4 - x - yd dy dx.

The expression on the right, called an iterated or repeated integral, says that the volume is obtained by integrating 4 - x - y with respect to y from y = 0 to y = 1, holding x fixed, and then integrating the resulting expression in x with respect to x from x = 0 to x = 2. The limits of integration 0 and 1 are associated with y, so they are placed on the integral closest to dy. The other limits of integration, 0 and 2, are associated with the variable x, so they are placed on the outside integral symbol that is paired with dx.

766

Chapter 14: Multiple Integrals z

What would have happened if we had calculated the volume by slicing with planes perpendicular to the y-axis (Figure 14.5)? As a function of y, the typical cross-sectional area is

4 z4xy

As yd =

x=2

Lx = 0

s4 - x - yd dx = c4x -

x=2

x2 - xy d = 6 - 2y. 2 x=0

(4)

The volume of the entire solid is therefore Volume = y

1 y

y=1

Ly = 0

As yd dy =

y=1

Ly = 0

s6 - 2yd dy = C 6y - y 2 D 0 = 5, 1

in agreement with our earlier calculation. Again, we may give a formula for the volume as an iterated integral by writing

2 1

x2

x

A(y) ⌠ (4 x y) dx ⌡x 0

FIGURE 14.5 To obtain the cross-sectional area A( y), we hold y fixed and integrate with respect to x.

Volume =

L0 L0

2

s4 - x - yd dx dy.

The expression on the right says we can find the volume by integrating 4 - x - y with respect to x from x = 0 to x = 2 as in Equation (4) and integrating the result with respect to y from y = 0 to y = 1. In this iterated integral, the order of integration is first x and then y, the reverse of the order in Equation (3). What do these two volume calculations with iterated integrals have to do with the double integral s4 - x - yd dA 6 R

HISTORICAL BIOGRAPHY Guido Fubini (1879–1943)

over the rectangle R: 0 … x … 2, 0 … y … 1? The answer is that both iterated integrals give the value of the double integral. This is what we would reasonably expect, since the double integral measures the volume of the same region as the two iterated integrals. A theorem published in 1907 by Guido Fubini says that the double integral of any continuous function over a rectangle can be calculated as an iterated integral in either order of integration. (Fubini proved his theorem in greater generality, but this is what it says in our setting.)

THEOREM 1—Fubini’s Theorem (First Form) If ƒ(x, y) is continuous throughout the rectangular region R: a … x … b, c … y … d, then d

6 R

ƒsx, yd dA =

Lc La

b

b

ƒsx, yd dx dy =

La Lc

d

ƒsx, yd dy dx.

Fubini’s Theorem says that double integrals over rectangles can be calculated as iterated integrals. Thus, we can evaluate a double integral by integrating with respect to one variable at a time. Fubini’s Theorem also says that we may calculate the double integral by integrating in either order, a genuine convenience. When we calculate a volume by slicing, we may use either planes perpendicular to the x-axis or planes perpendicular to the y-axis.

EXAMPLE 1

Calculate 4R ƒsx, yd dA for

ƒsx, yd = 100 - 6x 2y

and

R:

0 … x … 2,

-1 … y … 1.

14.1

1

100

6

ƒsx, yd dA =

R

1 x

=

L-1L0

1

s100 - 6x 2yd dx dy =

L-1

L-1

C 100x - 2x 3y D xx == 20 dy

s200 - 16yd dy = C 200y - 8y 2 D -1 = 400. 1

Reversing the order of integration gives the same answer: 1

R

2

2

1

50

–1

767

Figure 14.6 displays the volume beneath the surface. By Fubini’s Theorem,

Solution

z

z 5 100 2 6x 2y

Double and Iterated Integrals over Rectangles

y

2

1

L0 L-1

2

s100 - 6x 2yd dy dx =

L0

y=1 C 100y - 3x 2y 2 D y = -1 dx

2

FIGURE 14.6 The double integral 4R ƒ(x, y) dA gives the volume under this surface over the rectangular region R (Example 1).

=

[s100 - 3x 2 d - s -100 - 3x 2 d] dx

L0 2

=

L0

200 dx = 400.

z z 5 10 1 x 2 1 3y 2

EXAMPLE 2

Find the volume of the region bounded above by the ellipitical paraboloid z = 10 + x 2 + 3y 2 and below by the rectangle R: 0 … x … 1, 0 … y … 2.

10

R

1

2

y

Solution The surface and volume are shown in Figure 14.7. The volume is given by the double integral

x

1

FIGURE 14.7 The double integral 4R ƒ(x, y) dA gives the volume under this surface over the rectangular region R (Example 2).

V =

6

s10 + x 2 + 3y 2 d dA =

R

= =

1

y=2 C 10y + x 2y + y 3 D y = 0 dx

1

(20 + 2x 2 + 8) dx = c20x +

L0 L0

L0 L0

2

s10 + x 2 + 3y 2 d dy dx

1

86 2 3 x + 8x d = . 3 3 0

Exercises 14.1 Evaluating Iterated Integrals In Exercises 1–12, evaluate the iterated integral. 2

1.

L1 L0 0

3.

2.

(x + y + 1) dx dy

4.

L0 L0

6.

1

y dx dy L0 L0 1 + xy L0 L1

ln 5

e 2x + y dy dx

a1 -

x2 + y2 b dx dy 2

4

(x y - 2xy) dy dx

L1 L0 L0 L1

a

x + 2yb dx dy 2

L-1L0

6

s6y 2 - 2xd dA,

12.

Lp L0

(sin x + cos y) dx dy

R: 0 … x … 1,

0 … y … 2

R

14.

6 R

2x b dA, y2

R:

0 … x … 4,

xy cos y dA,

R:

-1 … x … 1,

a

2

xye x dy dx

p

2p

y sin x dx dy

Evaluating Double Integrals over Rectangles In Exercises 13–20, evaluate the double integral over the given region R. 13.

0

L0 L-2

1

10.

(x - y) dy dx

2

4

8.

1

L0 L0 3

(4 - y ) dy dx

1

L0 L-1 1

2

ln 2

9.

2xy dy dx

2

1

7.

2

1

L-1L-1 3

5.

4

p/2

2

11.

15.

6 R

1 … y … 2

0 … y … p

768 16.

6

Chapter 14: Multiple Integrals

y sin (x + y) dA,

R:

-p … x … 0,

0 … y … p

R

17.

6

e x - y dA,

0 … x … ln 2,

R:

0 … y … ln 2

R

18.

2

6

xye xy dA,

R:

0 … x … 2,

0 … y … 1

R

19.

xy 3 2 6x + 1

dA,

0 … x … 1,

R:

0 … y … 2

R

20.

y 2 2 6x y + 1

dA,

R:

0 … x … 1,

0 … y … 1

R

In Exercises 21 and 22, integrate ƒ over the given region. 21. Square ƒ(x, y) = 1>(xy) over the square 1 … x … 2, 1 … y … 2 22. Rectangle ƒ(x, y) = y cos xy over the rectangle 0 … x … p, 0 … y … 1

Volume Beneath a Surface z = ƒ(x, y) 23. Find the volume of the region bounded above by the paraboloid z = x 2 + y 2 and below by the square R: -1 … x … 1, -1 … y … 1. 24. Find the volume of the region bounded above by the ellipitical paraboloid z = 16 - x 2 - y 2 and below by the square R: 0 … x … 2, 0 … y … 2. 25. Find the volume of the region bounded above by the plane z = 2 - x - y and below by the square R: 0 … x … 1, 0 … y … 1. 26. Find the volume of the region bounded above by the plane z = y>2 and below by the rectangle R: 0 … x … 4, 0 … y … 2. 27. Find the volume of the region bounded above by the surface z = 2 sin x cos y and below by the rectangle R: 0 … x … p>2, 0 … y … p>4. 28. Find the volume of the region bounded above by the surface z = 4 - y 2 and below by the rectangle R: 0 … x … 1, 0 … y … 2.

Double Integrals over General Regions

14.2

In this section we define and evaluate double integrals over bounded regions in the plane which are more general than rectangles. These double integrals are also evaluated as iterated integrals, with the main practical problem being that of determining the limits of integration. Since the region of integration may have boundaries other than line segments parallel to the coordinate axes, the limits of integration often involve variables, not just constants.

Double Integrals over Bounded, Nonrectangular Regions

Ak

R

yk

(xk , yk )

xk

FIGURE 14.8 A rectangular grid partitioning a bounded nonrectangular region into rectangular cells.

To define the double integral of a function ƒ(x, y) over a bounded, nonrectangular region R, such as the one in Figure 14.8, we again begin by covering R with a grid of small rectangular cells whose union contains all points of R. This time, however, we cannot exactly fill R with a finite number of rectangles lying inside R, since its boundary is curved, and some of the small rectangles in the grid lie partly outside R. A partition of R is formed by taking the rectangles that lie completely inside it, not using any that are either partly or completely outside. For commonly arising regions, more and more of R is included as the norm of a partition (the largest width or height of any rectangle used) approaches zero. Once we have a partition of R, we number the rectangles in some order from 1 to n and let ¢Ak be the area of the kth rectangle. We then choose a point sxk , yk d in the kth rectangle and form the Riemann sum n

Sn = a ƒsxk , yk d ¢Ak . k=1

As the norm of the partition forming Sn goes to zero, 7P7 : 0, the width and height of each enclosed rectangle goes to zero and their number goes to infinity. If ƒ(x, y) is a continuous function, then these Riemann sums converge to a limiting value, not dependent on any of the choices we made. This limit is called the double integral of ƒ(x, y) over R: n

lim a ƒsxk , yk d ¢Ak = ƒ ƒ P ƒ ƒ :0 k=1

6 R

ƒsx, yd dA.

14.2

769

Double Integrals over General Regions

The nature of the boundary of R introduces issues not found in integrals over an interval. When R has a curved boundary, the n rectangles of a partition lie inside R but do not cover all of R. In order for a partition to approximate R well, the parts of R covered by small rectangles lying partly outside R must become negligible as the norm of the partition approaches zero. This property of being nearly filled in by a partition of small norm is satisfied by all the regions that we will encounter. There is no problem with boundaries made from polygons, circles, ellipses, and from continuous graphs over an interval, joined end to end. A curve with a “fractal” type of shape would be problematic, but such curves arise rarely in most applications. A careful discussion of which type of regions R can be used for computing double integrals is left to a more advanced text.

Volumes If ƒ(x, y) is positive and continuous over R, we define the volume of the solid region between R and the surface z = ƒsx, yd to be 4R ƒsx, yd dA, as before (Figure 14.9). If R is a region like the one shown in the xy-plane in Figure 14.10, bounded “above” and “below” by the curves y = g2sxd and y = g1sxd and on the sides by the lines x = a, x = b, we may again calculate the volume by the method of slicing. We first calculate the cross-sectional area Asxd =

y = g2sxd

Ly = g1sxd

ƒsx, yd dy

and then integrate A(x) from x = a to x = b to get the volume as an iterated integral: b

V =

La

b

Asxd dx =

g2sxd

La Lg1sxd

ƒsx, yd dy dx.

(1)

z

z 5 f(x, y)

z

Height 5 f(xk, yk)

z f (x, y)

0 x

0

a

b

x

y

y g1(x)

x y

A(x) R R (xk, yk)

y g2(x) D Ak

Volume 5 lim S f(xk, yk) D Ak 5

f (x, y) dA R

FIGURE 14.9 We define the volumes of solids with curved bases as a limit of approximating rectangular boxes.

FIGURE 14.10 The area of the vertical slice shown here is A(x). To calculate the volume of the solid, we integrate this area from x = a to x = b: b

La

b

Asxd dx =

g2sxd

La Lg1sxd

ƒsx, yd dy dx.

770

Chapter 14: Multiple Integrals

Similarly, if R is a region like the one shown in Figure 14.11, bounded by the curves x = h2s yd and x = h1s yd and the lines y = c and y = d, then the volume calculated by slicing is given by the iterated integral

z z f (x, y)

A( y) c

y

d

d

Volume =

y

x x h1( y)

h2s yd

Lc Lh1s yd

ƒsx, yd dx dy.

(2)

That the iterated integrals in Equations (1) and (2) both give the volume that we defined to be the double integral of ƒ over R is a consequence of the following stronger form of Fubini’s Theorem.

x h 2( y)

FIGURE 14.11 The volume of the solid shown here is d

Lc

d

As yd dy =

h2syd

Lc Lh1syd

ƒsx, yd dx dy.

For a given solid, Theorem 2 says we can calculate the volume as in Figure 14.10, or in the way shown here. Both calculations have the same result.

THEOREM 2—Fubini’s Theorem (Stronger Form) region R.

Let ƒ(x, y) be continuous on a

1. If R is defined by a … x … b, g1sxd … y … g2sxd, with g1 and g2 continuous on [a, b], then b

6

ƒsx, yd dA =

R

g2sxd

ƒsx, yd dy dx.

La Lg1sxd

2. If R is defined by c … y … d, h1syd … x … h2s yd, with h1 and h2 continuous on [c, d], then d

6

ƒsx, yd dA =

R

h2s yd

ƒsx, yd dx dy.

Lc Lh1s yd

EXAMPLE 1 Find the volume of the prism whose base is the triangle in the xy-plane bounded by the x-axis and the lines y = x and x = 1 and whose top lies in the plane z = ƒsx, yd = 3 - x - y. See Figure 14.12. For any x between 0 and 1, y may vary from y = 0 to y = x (Figure 14.12b). Hence,

Solution

1

V =

L0

1

s3 - x - yd dy dx =

L0 L0 1

=

x

a3x -

L0

c3y - xy -

y 2 y=x dx d 2 y=0

x=1

3x 2 3x 2 x3 b dx = c d = 1. 2 2 2 x=0

When the order of integration is reversed (Figure 14.12c), the integral for the volume is 1

V = = =

1

L0 Ly 1

a3 -

1

a

L0

L0

1

s3 - x - yd dx dy =

L0

c3x -

x=1

x2 - xy d dy 2 x=y

y2 1 - y - 3y + + y 2 b dy 2 2

y3 y=1 5 3 5 - 4y + y 2 b dy = c y - 2y 2 + d = 1. 2 2 2 2 y=0

The two integrals are equal, as they should be.

14.2

771

Double Integrals over General Regions

z y

x1 yx

(3, 0, 0) yx

z 5 f(x, y) 5 3 2 x 2 y

R x

y0 1

0

(1, 0, 2)

(b) y

x1 yx

(1, 1, 1)

xy

y

x1

(1, 1, 0)

(1, 0, 0)

R

x51

R x

0

y5x

1

x

(c)

(a)

FIGURE 14.12 (a) Prism with a triangular base in the xy-plane. The volume of this prism is defined as a double integral over R. To evaluate it as an iterated integral, we may integrate first with respect to y and then with respect to x, or the other way around (Example 1). (b) Integration limits of x=1

y=x

ƒsx, yd dy dx.

Lx = 0 Ly = 0

If we integrate first with respect to y, we integrate along a vertical line through R and then integrate from left to right to include all the vertical lines in R. (c) Integration limits of y=1

x=1

Ly = 0 Lx = y

ƒsx, yd dx dy.

If we integrate first with respect to x, we integrate along a horizontal line through R and then integrate from bottom to top to include all the horizontal lines in R.

Although Fubini’s Theorem assures us that a double integral may be calculated as an iterated integral in either order of integration, the value of one integral may be easier to find than the value of the other. The next example shows how this can happen.

EXAMPLE 2

Calculate 6

sin x x dA,

R

where R is the triangle in the xy-plane bounded by the x-axis, the line y = x, and the line x = 1.

772

Chapter 14: Multiple Integrals

y

Solution The region of integration is shown in Figure 14.13. If we integrate first with respect to y and then with respect to x, we find

x1 yx

1

1

L0

a

x

x

1

y=x

b dx =

1

sin x dx

If we reverse the order of integration and attempt to calculate 1

FIGURE 14.13 The region of integration in Example 2.

L0

1

Ly

sin x x dx dy,

we run into a problem because 1 sssin xd>xd dx cannot be expressed in terms of elementary functions (there is no simple antiderivative). There is no general rule for predicting which order of integration will be the good one in circumstances like these. If the order you first choose doesn’t work, try the other. Sometimes neither order will work, and then we need to use numerical approximations.

y x2 y 2 1

1

sin x ay x d

y=0 L0 L0 = -cos s1d + 1 L 0.46.

L0

R 0

1

sin x x dyb dx =

R

Finding Limits of Integration We now give a procedure for finding limits of integration that applies for many regions in the plane. Regions that are more complicated, and for which this procedure fails, can often be split up into pieces on which the procedure works.

xy1 0

1

x

Using Vertical Cross-sections When faced with evaluating 4R ƒsx, yddA, integrating first with respect to y and then with respect to x, do the following three steps:

(a) y Leaves at y 1 x 2

1 R

1.

Sketch. Sketch the region of integration and label the bounding curves (Figure 14.14a).

2.

Find the y-limits of integration. Imagine a vertical line L cutting through R in the direction of increasing y. Mark the y-values where L enters and leaves. These are the y-limits of integration and are usually functions of x (instead of constants) (Figure 14.14b). Find the x-limits of integration. Choose x-limits that include all the vertical lines through R. The integral shown here (see Figure 14.14c) is

Enters at y1x L

0

x

1

x

3.

(b)

6

ƒsx, yd dA =

R

y

Leaves at y 1 x 2

1 R

Enters at y1x

0

x

1

ƒsx, yd dy dx.

Lx = 0 Ly = 1 - x

1

ƒsx, yd dA =

R

Smallest x is x 0

y = 21 - x2

Using Horizontal Cross-sections To evaluate the same double integral as an iterated integral with the order of integration reversed, use horizontal lines instead of vertical lines in Steps 2 and 3 (see Figure 14.15). The integral is 6

L

x=1

21 - y 2

ƒsx, yd dx dy.

L0 L1 - y

x

Largest x is x 1

Largest y y is y 1 1

Enters at x1y R

(c) y

FIGURE 14.14 Finding the limits of integration when integrating first with respect to y and then with respect to x.

Smallest y is y 0 0

1

Leaves at x 1 y2 x

FIGURE 14.15 Finding the limits of integration when integrating first with respect to x and then with respect to y.

14.2

EXAMPLE 3

y 4

Double Integrals over General Regions

773

Sketch the region of integration for the integral 2

(2, 4)

2x

s4x + 2d dy dx

L0 Lx2

y 5 2x

and write an equivalent integral with the order of integration reversed.

y 5 x2

The region of integration is given by the inequalities x 2 … y … 2x and 0 … x … 2. It is therefore the region bounded by the curves y = x 2 and y = 2x between x = 0 and x = 2 (Figure 14.16a). To find limits for integrating in the reverse order, we imagine a horizontal line passing from left to right through the region. It enters at x = y>2 and leaves at x = 2y. To include all such lines, we let y run from y = 0 to y = 4 (Figure 14.16b). The integral is Solution

x

2

0

(a) y

2y

4

4

s4x + 2d dx dy.

(2, 4)

L0 Ly>2

The common value of these integrals is 8. y x5 2

x 5 y

Properties of Double Integrals

2 (b)

0

Like single integrals, double integrals of continuous functions have algebraic properties that are useful in computations and applications.

x

If ƒ(x, y) and g(x, y) are continuous on the bounded region R, then the following properties hold.

FIGURE 14.16 Region of integration for Example 3.

1. Constant Multiple:

6 R

cƒsx, yd dA = c ƒ (x, yd dA 6

sany number cd

R

2. Sum and Difference: sƒsx, yd ; gsx, ydd dA = ƒsx, yd dA ; gsx, yd dA 6 6 6 R

R

R

3. Domination: y

(a)

6

ƒsx, yd dA Ú 0

ƒsx, yd Ú 0 on R

if

R

R1

(b)

6

ƒsx, yd dA Ú

R

R2

R R1 ∪ R2

0 ⌠⌠ f(x, y) dA ⌠⌠ f(x, y) dA ⌠⌠ f (x, y) dA ⌡⌡ ⌡⌡ ⌡⌡ R1 R R2

FIGURE 14.17 The Additivity Property for rectangular regions holds for regions bounded by smooth curves.

4. Additivity: x

6

gsx, yd dA

if

ƒsx, yd Ú gsx, yd on R

R

6 R

ƒsx, yd dA =

6 R1

ƒsx, yd dA +

6

ƒsx, yd dA

R2

if R is the union of two nonoverlapping regions R1 and R2

Property 4 assumes that the region of integration R is decomposed into nonoverlapping regions R1 and R2 with boundaries consisting of a finite number of line segments or smooth curves. Figure 14.17 illustrates an example of this property.

774

Chapter 14: Multiple Integrals

The idea behind these properties is that integrals behave like sums. If the function ƒ(x, y) is replaced by its constant multiple cƒ(x, y), then a Riemann sum for ƒ n

Sn = a ƒsxk , yk d ¢Ak

z

k=1

is replaced by a Riemann sum for cƒ

16 z 5 16 2 x 2 2 y 2

n

n

a cƒsxk , yk d ¢Ak = c a ƒsxk , yk d ¢Ak = cSn .

k=1

2 1

y 5 4x 2 2

x

y

y 5 2x

(a) y

y 5 4x 2 2

2 x5

y2 4

y 5 2x

x5

R 0

(1, 2) y1 2 4

0.5

1

x

(b)

FIGURE 14.18 (a) The solid “wedgelike” region whose volume is found in Example 4. (b) The region of integration R showing the order dx dy.

k=1

Taking limits as n : q shows that c limn: q Sn = c 4R ƒ dA and limn: q cSn = 4R cƒ dA are equal. It follows that the Constant Multiple Property carries over from sums to double integrals. The other properties are also easy to verify for Riemann sums, and carry over to double integrals for the same reason. While this discussion gives the idea, an actual proof that these properties hold requires a more careful analysis of how Riemann sums converge. Find the volume of the wedgelike solid that lies beneath the surface z = 16 - x 2 - y 2 and above the region R bounded by the curve y = 22x, the line y = 4x - 2, and the x-axis.

EXAMPLE 4

Solution Figure 14.18a shows the surface and the “wedgelike” solid whose volume we want to calculate. Figure 14.18b shows the region of integration in the xy-plane. If we integrate in the order dy dx (first with respect to y and then with respect to x), two integrations will be required because y varies from y = 0 to y = 2 1x for 0 … x … 0.5, and then varies from y = 4x - 2 to y = 21x for 0.5 … x … 1. So we choose to integrate in the order dx dy, which requires only one double integral whose limits of integration are indicated in Figure 14.18b. The volume is then calculated as the iterated integral:

6

(16 - x 2 - y 2) dA

R

2

=

L0 Ly >4 2

2

=

L0 2

=

( y + 2)>4

L0

= c

c16x -

(16 - x 2 - y 2) dx dy x = ( y + 2)>4

x3 - xy 2 d 3 x = y2>4

c4( y + 2) -

dx

( y + 2) 3 ( y + 2)y 2 y6 y4 - 4y 2 + # + d dy # 3 64 4 3 64 4

191y 63y 2 145y 3 49y 4 y5 y7 2 20803 + + + d = L 12.4. 24 32 96 768 20 1344 0 1680

Exercises 14.2 Sketching Regions of Integration In Exercises 1–8, sketch the described regions of integration. 1. 0 … x … 3, 0 … y … 2x 2. -1 … x … 2,

x - 1 … y … x

3. -2 … y … 2,

y2 … x … 4

4. 0 … y … 1,

y … x … 2y

2

5. 0 … x … 1,

ex … y … e

6. 1 … x … e 2,

0 … y … ln x

7. 0 … y … 1,

0 … x … sin-1 y

8. 0 … y … 8,

1 y … x … y 1>3 4

14.2 Finding Limits of Integration In Exercises 9–18, write an iterated integral for 4R dA over the described region R using (a) vertical cross-sections, (b) horizontal crosssections. 9.

10. y

31. y 5 2x

y58

x53 x

x

12. 35.

y 5 ex

y 5 3x

x52

39.

15. Bounded by y = e -x, y = 1, and x = ln 3 17. Bounded by y = 3 - 2x, y = x, and x = 0

34.

0

L0 Ly - 2

36.

ln 2

38.

L0

9 - 4x 2

40.

3y dx dy

42.

y dx dy 24 - x 2

L0 L-24 - x 2 p>6

ln x

44.

(x + y) dx dy

L0

46.

6x dy dx

1>2

Lsin x

xy 2 dy dx

tan-1 y

13

ey

L0 L1

4 - y2

2

xy dy dx

dx dy

L0 L0

21 - y 2

L1 L0

2

Le y 2

16x dy dx

dy dx

L0 L1 - x

ex

dy dx

dx dy

1 - x2

1

dx dy

L0 L-21 - y 2

3

45.

2

L0 L0

e

43.

16. Bounded by y = 0, x = 0, y = 1, and y = ln x

sthe uy-planed

2y

L0 L1

1

41.

13. Bounded by y = 1x, y = 0, and x = 9 14. Bounded by y = tan x, x = 0, and y = 1

4 - 2u dy du y2

dy dx

3>2

x

sthe tu-planed

4 - 2x

L0 Ly

y51 x

4 - 2u

L0 L2

1

37. y 5 x2

3 cos t du dt

L0 L1

1

y

sec t

L-p>3 L0

1

y

sthe st-planed

Reversing the Order of Integration In Exercises 33–46, sketch the region of integration and write an equivalent double integral with the order of integration reversed. 33.

11.

8t dt ds

L0 L0

3>2

32.

sthe py-planed

21 - s2

p>3

y

y 5 x3

2 dp dy

L-2 Ly 1

30.

775

-y

0

29.

Double Integrals over General Regions

2xy dx dy

L0 L0

18. Bounded by y = x 2 and y = x + 2

In Exercises 47–56, sketch the region of integration, reverse the order of integration, and evaluate the integral.

Finding Regions of Integration and Double Integrals In Exercises 19–24, sketch the region of integration and evaluate the integral.

47.

p

19.

p

x

x sin y dy dx

L0 L0 ln 8

21.

L1

e

L0 1

23.

ln y

L0 L0

x+y

20.

dx dy

22.

y2

L1 Ly 4

3y 3e xy dx dy

24.

y dy dx

L1 L0

p

48.

x 2e xy dx dy

50.

1

L0 Ly

2ln 3

dx dy

51.

L0

Ly>2 1>16

2x

3 y>2x dy dx e 2

In Exercises 25–28, integrate ƒ over the given region. 25. Quadrilateral ƒsx, yd = x>y over the region in the first quadrant bounded by the lines y = x, y = 2x, x = 1, and x = 2 26. Triangle ƒsx, yd = x 2 + y 2 over the triangular region with vertices (0, 0), (1, 0), and (0, 1) 27. Triangle ƒsu, yd = y - 2u over the triangular region cut from the first quadrant of the uy-plane by the line u + y = 1 28. Curved region ƒss, td = e ln t over the region in the first quadrant of the st-plane that lies above the curve s = ln t from t = 1 to t = 2 s

Each of Exercises 29–32 gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral.

53.

8

54.

4 - x2

L0 L0 3

52.

2y 2 sin xy dy dx

1

L0 L2x>3

xe 2y dy dx 4 - y 3

e y dy dx

1>2

Ly1>4

L0

2

e x dx dy

2

L0 Lx 2

22ln 3

y2

2

sin y y dy dx

L0 Lx 1

49.

sin x

L0 L0 2

p

2

cos s16px 5 d dx dy dy dx

4 L0 L2 x y + 1 3

55. Square region 4R s y - 2x 2 d dA where R is the region bounded by the square ƒ x ƒ + ƒ y ƒ = 1 56. Triangular region 4R xy dA where R is the region bounded by the lines y = x, y = 2x, and x + y = 2 Volume Beneath a Surface z = ƒ(x, y) 57. Find the volume of the region bounded above by the paraboloid z = x 2 + y 2 and below by the triangle enclosed by the lines y = x, x = 0, and x + y = 2 in the xy-plane. 58. Find the volume of the solid that is bounded above by the cylinder z = x 2 and below by the region enclosed by the parabola y = 2 - x 2 and the line y = x in the xy-plane.

776

Chapter 14: Multiple Integrals

59. Find the volume of the solid whose base is the region in the xy-plane that is bounded by the parabola y = 4 - x 2 and the line y = 3x, while the top of the solid is bounded by the plane z = x + 4. 60. Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder x 2 + y 2 = 4, and the plane z + y = 3. 61. Find the volume of the solid in the first octant bounded by the coordinate planes, the plane x = 3, and the parabolic cylinder z = 4 - y 2. 62. Find the volume of the solid cut from the first octant by the surface z = 4 - x 2 - y. 63. Find the volume of the wedge cut from the first octant by the cylinder z = 12 - 3y 2 and the plane x + y = 2. 64. Find the volume of the solid cut from the square column ƒ x ƒ + ƒ y ƒ … 1 by the planes z = 0 and 3x + z = 3. 65. Find the volume of the solid that is bounded on the front and back by the planes x = 2 and x = 1, on the sides by the cylinders y = ;1>x, and above and below by the planes z = x + 1 and z = 0. 66. Find the volume of the solid bounded on the front and back by the planes x = ;p>3, on the sides by the cylinders y = ;sec x, above by the cylinder z = 1 + y 2, and below by the xy-plane. In Exercises 67 and 68, sketch the region of integration and the solid whose volume is given by the double integral. 3

67.

a1 -

L0 L0 4

68.

2 - 2x>3

216 - y 2

L0 L-216 - y

2

q

76. Unbounded region Integrate ƒsx, yd = 1>[sx 2 - xds y - 1d2>3] over the infinite rectangle 2 … x 6 q , 0 … y … 2. 77. Noncircular cylinder A solid right (noncircular) cylinder has its base R in the xy-plane and is bounded above by the paraboloid z = x 2 + y 2. The cylinder’s volume is 1

V =

y

L0 L0

225 - x 2 - y 2 dx dy

2-y

2

sx 2 + y 2 d dx dy +

sx 2 + y 2 d dx dy.

L1 L0

Sketch the base region R and express the cylinder’s volume as a single iterated integral with the order of integration reversed. Then evaluate the integral to find the volume. Evaluate the integral

78. Converting to a double integral 2

stan-1px - tan-1 xd dx.

L0

(Hint: Write the integrand as an integral.) 79. Maximizing a double integral maximizes the value of

What region R in the xy-plane

s4 - x 2 - 2y 2 d dA? 6 R

Give reasons for your answer. 80. Minimizing a double integral What region R in the xy-plane minimizes the value of

1 dy dx 3 L1 Le x y 1> 21 - x 2

L-1 L-1> 21 - x 2 q

L- q L- q q

72.

75. Circular sector Integrate ƒsx, yd = 24 - x 2 over the smaller sector cut from the disk x 2 + y 2 … 4 by the rays u = p>6 and u = p>2.

sx 2 + y 2 - 9d dA? 6

-x

q

71.

Theory and Examples

1

1

70.

74. ƒsx, yd = x + 2y over the region R inside the circle sx - 2d2 + s y - 3d2 = 1 using the partition x = 1, 3> 2, 2, 5> 2, 3 and y = 2, 5> 2, 3, 7> 2, 4 with sxk , yk d the center (centroid) in the kth subrectangle (provided the subrectangle lies within R)

1 1 x - yb dy dx 3 2

Integrals over Unbounded Regions Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section 8.7. Evaluate the improper integrals in Exercises 69–72 as iterated integrals. 69.

73. ƒsx, yd = x + y over the region R bounded above by the semicircle y = 11 - x 2 and below by the x-axis, using the partition x = -1, -1>2 , 0, 1> 4, 1> 2, 1 and y = 0, 1> 2, 1 with sxk , yk d the lower left corner in the kth subrectangle (provided the subrectangle lies within R)

L0 L0

q

R

Give reasons for your answer.

s2y + 1d dy dx

1 dx dy sx 2 + 1ds y 2 + 1d

xe -sx + 2yd dx dy

Approximating Integrals with Finite Sums In Exercises 73 and 74, approximate the double integral of ƒ(x, y) over the region R partitioned by the given vertical lines x = a and horizontal lines y = c. In each subrectangle, use sxk , yk d as indicated for your approximation. n

6 R

ƒsx, yd dA L a ƒsxk , yk d ¢Ak k=1

81. Is it possible to evaluate the integral of a continuous function ƒ(x, y) over a rectangular region in the xy-plane and get different answers depending on the order of integration? Give reasons for your answer. 82 How would you evaluate the double integral of a continuous function ƒ(x, y) over the region R in the xy-plane enclosed by the triangle with vertices (0, 1), (2, 0), and (1, 2)? Give reasons for your answer. 83. Unbounded region q

q

L- qL- q

e -x

2

- y2

Prove that b

b

b: qL -b

L-b

dx dy = lim = 4a

q

L0

e -x

2

- y2

2

e -x dxb . 2

dx dy

14.3

L0

L0

3

2

x dy dx. s y - 1d2>3

1

89. COMPUTER EXPLORATIONS Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises 85–88. 3

85.

L1 L1 1

87.

x

L0 L0

1

1

1 xy dy dx

86.

tan-1 xy dy dx

14.3

1

L0 L0 1

88.

L-1 L0

e -sx

2

21 - x

+ y2d

777

Use a CAS double-integral evaluator to find the integrals in Exercises 89–94. Then reverse the order of integration and evaluate, again with a CAS.

84. Improper double integral Evaluate the improper integral 1

Area by Double Integration

L0 L2y 2

91.

dy dx

2

3 21 - x 2 - y 2 dy dx

3

2

e x dx dy

90.

422y

9

92.

x2

4 - y2

L0 L0 2

1 dy dx L1 L0 x + y

94.

x cos s y 2 d dy dx

L0 Lx2 2

sx 2y - xy 2 d dx dy

L0 Ly 3 2

93.

4

e xy dx dy

8

1 dx dy L1 Ly 3 2x 2 + y 2

Area by Double Integration In this section we show how to use double integrals to calculate the areas of bounded regions in the plane, and to find the average value of a function of two variables.

Areas of Bounded Regions in the Plane If we take ƒsx, yd = 1 in the definition of the double integral over a region R in the preceding section, the Riemann sums reduce to n

n

Sn = a ƒsxk , yk d ¢Ak = a ¢Ak . k=1

k=1

(1)

This is simply the sum of the areas of the small rectangles in the partition of R, and approximates what we would like to call the area of R. As the norm of a partition of R approaches zero, the height and width of all rectangles in the partition approach zero, and the coverage of R becomes increasingly complete (Figure 14.8). We define the area of R to be the limit n

lim a ¢Ak = ƒ ƒ P ƒ ƒ :0 k=1

DEFINITION

6

dA.

(2)

R

The area of a closed, bounded plane region R is A =

6

dA.

R

As with the other definitions in this chapter, the definition here applies to a greater variety of regions than does the earlier single-variable definition of area, but it agrees with the earlier definition on regions to which they both apply. To evaluate the integral in the definition of area, we integrate the constant function ƒsx, yd = 1 over R.

EXAMPLE 1 quadrant.

Find the area of the region R bounded by y = x and y = x 2 in the first

778

Chapter 14: Multiple Integrals

y

Solution We sketch the region (Figure 14.19), noting where the two curves intersect at the origin and (1, 1), and calculate the area as

1

(1, 1)

y5x

1

A =

y 5 x2

y5x

x

1

L0 Lx2

dy dx =

L0

1

= y 5 x2 0

sx - x 2 d dx = c

x2

1

x3 x2 1 d = . 2 3 0 6

1

x

1

L0

x

cy d dx

FIGURE 14.19 The region in Example 1.

Notice that the single-variable integral 10 sx - x 2 d dx, obtained from evaluating the inside iterated integral, is the integral for the area between these two curves using the method of Section 5.6. Find the area of the region R enclosed by the parabola y = x 2 and the line

EXAMPLE 2 y = x + 2.

If we divide R into the regions R1 and R2 shown in Figure 14.20a, we may calculate the area as Solution

y

y 5 x2

A =

4

⌠ ⌠ y dx dy ⌡1 ⌡y – 2

6

dA +

R1

6

dA =

R2

2 1

0

A =

y

⌠ ⌠ dx dy ⌡0 ⌡–y x

R1

(a)

A =

y 5 x2 (2, 4)

y5x12

2

(–1, 1)

0

L1 Ly - 2

dx dy.

x+2

L-1 Lx2

dy dx.

L-1

cy d

x+2 x2

2

dx =

L-1

sx + 2 - x 2 d dx = c

2

9 x2 x3 + 2x d = . 2 3 -1 2

Average Value

x12

⌠ ⌠ ⌡–1⌡x2

2y

This second result, which requires only one integral, is simpler and is the only one we would bother to write down in practice. The area is 2

y

L0 L-2y

4

dx dy +

On the other hand, reversing the order of integration (Figure 14.20b) gives

R2 (–1, 1)

2y

1

(2, 4)

y5x12

dy dx

x

(b)

FIGURE 14.20 Calculating this area takes (a) two double integrals if the first integration is with respect to x, but (b) only one if the first integration is with respect to y (Example 2).

The average value of an integrable function of one variable on a closed interval is the integral of the function over the interval divided by the length of the interval. For an integrable function of two variables defined on a bounded region in the plane, the average value is the integral over the region divided by the area of the region. This can be visualized by thinking of the function as giving the height at one instant of some water sloshing around in a tank whose vertical walls lie over the boundary of the region. The average height of the water in the tank can be found by letting the water settle down to a constant height. The height is then equal to the volume of water in the tank divided by the area of R. We are led to define the average value of an integrable function ƒ over a region R as follows:

Average value of ƒ over R =

1 ƒ dA. area of R 6

(3)

R

If ƒ is the temperature of a thin plate covering R, then the double integral of ƒ over R divided by the area of R is the plate’s average temperature. If ƒ(x, y) is the distance from the point (x, y) to a fixed point P, then the average value of ƒ over R is the average distance of points in R from P.

14.3

Area by Double Integration

779

Find the average value of ƒsx, yd = x cos xy over the rectangle R: 0 … x … p, 0 … y … 1.

EXAMPLE 3

The value of the integral of ƒ over R is

Solution

p

p

1

L0 L0

x cos xy dy dx =

L0

L0

y=1

dx

L

x cos xy dy = sin xy + C

y=0 p

=

csin xy d

ssin x - 0d dx = -cos x d

p

= 1 + 1 = 2. 0

The area of R is p. The average value of ƒ over R is 2>p.

Exercises 14.3 Area by Double Integrals In Exercises 1–12, sketch the region bounded by the given lines and curves. Then express the region’s area as an iterated double integral and evaluate the integral. 1. The coordinate axes and the line x + y = 2

Finding Average Values 19. Find the average value of ƒsx, yd = sin sx + yd over a. the rectangle 0 … x … p,

0 … y … p.

b. the rectangle 0 … x … p,

0 … y … p>2.

20. Which do you think will be larger, the average value of ƒsx, yd = xy over the square 0 … x … 1, 0 … y … 1, or the average value of ƒ over the quarter circle x 2 + y 2 … 1 in the first quadrant? Calculate them to find out.

2. The lines x = 0, y = 2x, and y = 4 3. The parabola x = -y 2 and the line y = x + 2 4. The parabola x = y - y 2 and the line y = -x 5. The curve y = e x and the lines y = 0, x = 0, and x = ln 2

21. Find the average height of the paraboloid z = x 2 + y 2 over the square 0 … x … 2, 0 … y … 2.

6. The curves y = ln x and y = 2 ln x and the line x = e, in the first quadrant

22. Find the average value of ƒsx, yd = 1>sxyd over the square ln 2 … x … 2 ln 2, ln 2 … y … 2 ln 2.

7. The parabolas x = y 2 and x = 2y - y 2 8. The parabolas x = y 2 - 1 and x = 2y 2 - 2

Theory and Examples

9. The lines y = x, y = x>3, and y = 2 10. The lines y = 1 - x and y = 2 and the curve y = e

x

11. The lines y = 2x, y = x>2, and y = 3 - x 12. The lines y = x - 2 and y = -x and the curve y = 1x Identifying the Region of Integration The integrals and sums of integrals in Exercises 13–18 give the areas of regions in the xy-plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. 6

13.

2y

L0 Ly 2>3 p>4

15.

L0

1-x

L-1L-2x 2

18.

14.

cos x

Lsin x 0

17.

3

dx dy

L0 Lx - 4

16. 2

dy dx +

0 2

L0 L-x 2

dy dx

L0 L-x>2 4

dy dx +

1-x

L0 L0

dy dx

2x

dy dx

xs2 - xd

dy dx

y+2

L-1Ly2

dx dy

23. Bacterium population If ƒsx, yd = s10,000e y d>s1 + ƒ x ƒ >2d represents the “population density” of a certain bacterium on the xy-plane, where x and y are measured in centimeters, find the total population of bacteria within the rectangle -5 … x … 5 and -2 … y … 0. 24. Regional population If ƒsx, yd = 100 s y + 1d represents the population density of a planar region on Earth, where x and y are measured in miles, find the number of people in the region bounded by the curves x = y 2 and x = 2y - y 2. 25. Average temperature in Texas According to the Texas Almanac, Texas has 254 counties and a National Weather Service station in each county. Assume that at time t0, each of the 254 weather stations recorded the local temperature. Find a formula that would give a reasonable approximation of the average temperature in Texas at time t0. Your answer should involve information that you would expect to be readily available in the Texas Almanac. 26. If y = ƒ(x) is a nonnegative continuous function over the closed interval a … x … b, show that the double integral definition of area for the closed plane region bounded by the graph of ƒ, the vertical lines x = a and x = b, and the x-axis agrees with the definition for area beneath the curve in Section 5.3.

780

Chapter 14: Multiple Integrals

14.4

Double Integrals in Polar Form Integrals are sometimes easier to evaluate if we change to polar coordinates. This section shows how to accomplish the change and how to evaluate integrals over regions whose boundaries are given by polar equations.

Integrals in Polar Coordinates When we defined the double integral of a function over a region R in the xy-plane, we began by cutting R into rectangles whose sides were parallel to the coordinate axes. These were the natural shapes to use because their sides have either constant x-values or constant y-values. In polar coordinates, the natural shape is a “polar rectangle” whose sides have constant r- and u-values. Suppose that a function ƒsr, ud is defined over a region R that is bounded by the rays u = a and u = b and by the continuous curves r = g1sud and r = g2sud. Suppose also that 0 … g1sud … g2sud … a for every value of u between a and b. Then R lies in a fanshaped region Q defined by the inequalities 0 … r … a and a … u … b. See Figure 14.21.

a 1 2Du DAk u5b

(rk , uk)

R

a 1 Du Dr Q

Du

r 5 g2(u)

3Dr 2Dr

r5a

r 5 g1(u)

Dr

u5p

u5a

u50

0

FIGURE 14.21 The region R: g1sud … r … g2sud, a … u … b, is contained in the fanshaped region Q: 0 … r … a, a … u … b. The partition of Q by circular arcs and rays induces a partition of R.

We cover Q by a grid of circular arcs and rays. The arcs are cut from circles centered at the origin, with radii ¢r, 2¢r, Á , m¢r, where ¢r = a>m. The rays are given by u = a,

u = a + ¢u,

u = a + 2¢u,

Á,

u = a + m¿¢u = b,

where ¢u = sb - ad>m¿. The arcs and rays partition Q into small patches called “polar rectangles.” We number the polar rectangles that lie inside R (the order does not matter), calling their areas ¢A1, ¢A2, Á , ¢An. We let srk , uk d be any point in the polar rectangle whose area is ¢Ak . We then form the sum n

Sn = a ƒsrk, uk d ¢Ak. k=1

If ƒ is continuous throughout R, this sum will approach a limit as we refine the grid to make ¢r and ¢u go to zero. The limit is called the double integral of ƒ over R. In symbols, lim Sn =

n: q

6 R

ƒsr, ud dA.

14.4

⎛r 1 Dr ⎛ ⎝k 2 ⎝ rk

⎛r 2 Dr ⎛ ⎝k 2 ⎝

DAk

Small sector

A =

Large sector 0

FIGURE 14.22

781

To evaluate this limit, we first have to write the sum Sn in a way that expresses ¢Ak in terms of ¢r and ¢u . For convenience we choose rk to be the average of the radii of the inner and outer arcs bounding the kth polar rectangle ¢Ak . The radius of the inner arc bounding ¢Ak is then rk - s¢r>2d (Figure 14.22). The radius of the outer arc is rk + s¢r>2d. The area of a wedge-shaped sector of a circle having radius r and angle u is

Du Dr

Double Integrals in Polar Form

The observation that

1 # 2 u r , 2

as can be seen by multiplying pr 2, the area of the circle, by u>2p, the fraction of the circle’s area contained in the wedge. So the areas of the circular sectors subtended by these arcs at the origin are 2

area of area of b - a b ¢Ak = a large sector small sector

Inner radius:

1 ¢r ar b ¢u 2 k 2

leads to the formula ¢Ak = rk ¢r ¢u.

Outer radius:

¢r 1 ar + b ¢u. 2 k 2

2

Therefore, ¢Ak = area of large sector - area of small sector 2

=

y

Combining this result with the sum defining Sn gives

x 2 1 y2 5 4

2 R

2

y 5 2

2

¢u ¢u ¢r ¢r c ark + b - ark b d = s2rk ¢rd = rk ¢r ¢u. 2 2 2 2

n

⎛ ⎛ ⎝2, 2⎝

Sn = a ƒsrk , uk drk ¢r ¢u. k=1

x

0

As n : q and the values of ¢r and ¢u approach zero, these sums converge to the double integral lim Sn =

(a)

n: q

Leaves at r 2 L

A version of Fubini’s Theorem says that the limit approached by these sums can be evaluated by repeated single integrations with respect to r and u as

2 R

x

Largest is . 2 L

2

0

yx

R

ƒsr, ud dA =

R

u=b

r = g2sud

Lu = a Lr = g1sud

ƒsr, ud r dr du.

Finding Limits of Integration The procedure for finding limits of integration in rectangular coordinates also works for polar coordinates. To evaluate 4R ƒsr, ud dA over a region R in polar coordinates, integrating first with respect to r and then with respect to u, take the following steps.

(b)

2

6

Enters at r 2 csc

0

y

ƒsr, ud r dr du.

R

y

r sin y 2 or r 2 csc

6

Smallest is . 4 x

(c) FIGURE 14.23 Finding the limits of integration in polar coordinates.

1. 2.

3.

Sketch. Sketch the region and label the bounding curves (Figure 14.23a). Find the r-limits of integration. Imagine a ray L from the origin cutting through R in the direction of increasing r. Mark the r-values where L enters and leaves R. These are the r-limits of integration. They usually depend on the angle u that L makes with the positive x-axis (Figure 14.23b). Find the u-limits of integration. Find the smallest and largest u-values that bound R. These are the u-limits of integration (Figure 14.23c). The polar iterated integral is

6 R

ƒsr, ud dA =

u = p>2

r=2

Lu = p>4 Lr = 22 csc u

ƒsr, ud r dr du.

782

Chapter 14: Multiple Integrals

EXAMPLE 1 Find the limits of integration for integrating ƒsr, ud over the region R that lies inside the cardioid r = 1 + cos u and outside the circle r = 1.

y

2

r 1 cos

Solution

– 2

1

Enters at r1

2

1. 2.

x

3.

L Leaves at r 1 cos

We first sketch the region and label the bounding curves (Figure 14.24). Next we find the r-limits of integration. A typical ray from the origin enters R where r = 1 and leaves where r = 1 + cos u. Finally we find the u-limits of integration. The rays from the origin that intersect R run from u = -p>2 to u = p>2. The integral is 1 + cos u

p>2

ƒsr, ud r dr du.

L-p>2 L1

FIGURE 14.24 Finding the limits of integration in polar coordinates for the region in Example 1.

If ƒsr, ud is the constant function whose value is 1, then the integral of ƒ over R is the area of R.

Area in Polar Coordinates The area of a closed and bounded region R in the polar coordinate plane is Area Differential in Polar Coordinates dA = r dr du

A =

6

r dr du.

R

This formula for area is consistent with all earlier formulas, although we do not prove this fact. y 4

Leaves at r 4 cos 2

x

EXAMPLE 2

Find the area enclosed by the lemniscate r 2 = 4 cos 2u.

Solution We graph the lemniscate to determine the limits of integration (Figure 14.25) and see from the symmetry of the region that the total area is 4 times the first-quadrant portion. p>4

Enters at r0

– 4

r 2 4 cos 2

A = 4

L0

L0

24 cos 2u

p>4

r dr du = 4

L0

p>4

FIGURE 14.25 To integrate over the shaded region, we run r from 0 to 24 cos 2u and u from 0 to p>4 (Example 2).

= 4

L0

2 cos 2u du = 4 sin 2u d

c

r = 24 cos 2u

r2 d 2 r=0

du

p>4

= 4. 0

Changing Cartesian Integrals into Polar Integrals The procedure for changing a Cartesian integral 4R ƒsx, yd dx dy into a polar integral has two steps. First substitute x = r cos u and y = r sin u, and replace dx dy by r dr du in the Cartesian integral. Then supply polar limits of integration for the boundary of R. The Cartesian integral then becomes 6 R

ƒsx, yd dx dy =

6

ƒsr cos u, r sin ud r dr du,

G

where G denotes the same region of integration now described in polar coordinates. This is like the substitution method in Chapter 5 except that there are now two variables to substitute for instead of one. Notice that the area differential dx dy is not replaced by dr du but by r dr du. A more general discussion of changes of variables (substitutions) in multiple integrals is given in Section 14.8.

14.4

EXAMPLE 3

Double Integrals in Polar Form

783

Evaluate 6

ex

2

+ y2

dy dx,

R

where R is the semicircular region bounded by the x-axis and the curve y = 21 - x 2 (Figure 14.26).

y 1 x2 r1

1

–1

In Cartesian coordinates, the integral in question is a nonelementary integral 2 2 and there is no direct way to integrate e x + y with respect to either x or y. Yet this integral and others like it are important in mathematics—in statistics, for example—and we need to find a way to evaluate it. Polar coordinates save the day. Substituting x = r cos u, y = r sin u and replacing dy dx by r dr du enables us to evaluate the integral as Solution

y

0 0

1

6

x

ex

2

+ y2

p

dy dx =

R

1

p

2

e r r dr du =

L0 L0

L0

1

1 2 c e r d du 2 0

p

=

FIGURE 14.26 The semicircular region in Example 3 is the region 0 … r … 1,

0 … u … p.

p 1 se - 1d du = se - 1d. 2 2 L0 2

The r in the r dr du was just what we needed to integrate e r . Without it, we would have been unable to find an antiderivative for the first (innermost) iterated integral.

EXAMPLE 4

Evaluate the integral 1

21 - x2

L0 L0 Solution

(x 2 + y 2) dy dx.

Integration with respect to y gives 1

L0

(1 - x 2) 3>2 b dx, 3

ax 2 21 - x 2 +

an integral difficult to evaluate without tables. Things go better if we change the original integral to polar coordinates. The region of integration in Cartesian coordinates is given by the inequalites 0 … y … 21 - x 2 and 0 … x … 1, which correspond to the interior of the unit quarter circle x 2 + y 2 = 1 in the first quadrant. (See Figure 14.26, first quadrant.) Substituting the polar coordinates x = r cos u, y = r sin u, 0 … u … p>2 and 0 … r … 1, and replacing dx dy by r dr du in the double integral, we get

z z 5 9 2 x2 2 y2

21 - x2

1

9

L0

L0

p>2

(x 2 + y 2) dy dx =

L0

L0 p>2

=

L0

1

c

(r 2) r dr du r=1

r4 d du = 4 r=0 L0

p>2

p 1 du = . 4 8

Why is the polar coordinate transformation so effective here? One reason is that x 2 + y 2 simplifies to r 2 . Another is that the limits of integration become constants.

–2

EXAMPLE 5 2

x2 1 y2 5 1

2

y

x

FIGURE 14.27 The solid region in Example 5.

Find the volume of the solid region bounded above by the paraboloid z = 9 - x 2 - y 2 and below by the unit circle in the xy-plane.

The region of integration R is the unit circle x 2 + y 2 = 1, which is described in polar coordinates by r = 1, 0 … u … 2p. The solid region is shown in Figure 14.27. The volume is given by the double integral Solution

784

Chapter 14: Multiple Integrals 2p

6

s9 - x 2 - y 2 d dA =

R

1

2p

=

=

1

L0 L0 2p

=

s9 - r 2 d r dr du

L0 L0

L0

s9r - r 3 d dr du r=1

9 1 c r2 - r4 d du 2 4 r=0

17 4 L0

2p

du =

17p . 2

EXAMPLE 6

Using polar integration, find the area of the region R in the xy-plane enclosed by the circle x 2 + y 2 + 4, above the line y = 1, and below the line y = 13x.

A sketch of the region R is shown in Figure 14.28. First we note that the line y = 13x has slope 13 = tan u, so u = p>3. Next we observe that the line y = 1 intersects the circle x 2 + y 2 = 4 when x 2 + 1 = 4, or x = 13. Moreover, the radial line from the origin through the point (13, 1) has slope 1> 13 = tan u, giving its angle of inclination as u = p>6. This information is shown in Figure 14.28. Now, for the region R, as u varies from p>6 to p>3, the polar coordinate r varies from the horizontal line y = 1 to the circle x 2 + y 2 = 4. Substituting r sin u for y in the equation for the horizontal line, we have r sin u = 1, or r = csc u, which is the polar equation of the line. The polar equation for the circle is r = 2. So in polar coordinates, for p>6 … u … p>3, r varies from r = csc u to r = 2. It follows that the iterated integral for the area then gives Solution

y

y 5 3x

2 (1, 3)

y 5 1, or r 5 csc u

R 1

0

(3, 1) p 2 p 3 x 1 y2 5 4 6 1 2

x

p>3

6

FIGURE 14.28 The region R in Example 6.

dA =

R

Lp>6 Lcsc u p>3

=

2

Lp>6 p>3

=

r dr du r=2

1 c r2 d du 2 r = csc u 1 C4 - csc2 uD du 2

Lp>6 p>3 1 = C4u + cot uD p>6 2 =

p - 13 1 4p 1 1 4p a + b - a + 13b = . 2 3 2 6 3 13

Exercises 14.4 Regions in Polar Coordinates In Exercises 1–8, describe the given region in polar coordinates. 1.

3.

4. 1

2. y

3

y

9

4 –1 1

0

y

y

9

x

0

4

x

0

1

x 0

1

x

14.4 5.

y 2

28. Cardioid overlapping a circle Find the area of the region that lies inside the cardioid r = 1 + cos u and outside the circle r = 1.

2 1

0

0

1

2

x

x

1

7. The region enclosed by the circle x 2 + y 2 = 2x. 8. The region enclosed by the semicircle x + y = 2y, y Ú 0. 2

2

Evaluating Polar Integrals In Exercises 9–22, change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. 21 - x

1

6

13.

L0 L0 L1

2

21 - x 2

L-1 L-21 - x 2 s1 + x + y d 2sln 2d2 - y 2

L0

L0

21 - y 2

L-1 L-21 - y 2

e 2x

2 2

2

+ y2

x

y dy dx

22x - x 2

L1 L0

dx dy

23.

L0 p>2

24.

L0

38. Converting to a polar integral Integrate ƒsx, yd = [ln sx 2 + y 2 d]>sx 2 + y 2 d over the region 1 … x 2 + y 2 … e 2. 39. Volume of noncircular right cylinder The region that lies inside the cardioid r = 1 + cos u and outside the circle r = 1 is the base of a solid right cylinder. The top of the cylinder lies in the plane z = x. Find the cylinder’s volume.

1 dy dx (x 2 + y 2) 2

L0

L0

I2 = a

r 5 sin2 u dr du 3 sec u

p>2

r 7 dr du +

whose top is bounded by the sphere z = 22 - r 2. Find the cylinder’s volume. a. The usual way to evaluate the improper integral q 2 I = 10 e -x dx is first to calculate its square:

2 sec u

tan-1 43

26.

40. Volume of noncircular right cylinder The region enclosed by the lemniscate r 2 = 2 cos 2u is the base of a solid right cylinder

41. Converting to polar integrals

r 2 cos u dr du

L0

cone z = 2x 2 + y 2 above the disk x 2 + y 2 … a 2 in the xy-plane.

Theory and Examples 37. Converting to a polar integral Integrate ƒsx, yd = [ln sx 2 + y 2 d]> 2x 2 + y 2 over the region 1 … x 2 + y 2 … e.

csc u

p>4

25.

the hemispherical surface z = 2a 2 - x 2 - y 2 above the disk x 2 + y 2 … a 2 in the xy-plane.

dx dy

r 3 sin u cos u dr du

Lp>6 L1

R

36. Average distance squared from a point in a disk to a point in its boundary Find the average value of the square of the distance from the point P(x, y) in the disk x 2 + y 2 … 1 to the boundary point A(1, 0).

dy dx

1

L0

Average values In polar coordinates, the average value of a function over a region R (Section 14.3) is given by 1 ƒ(r, u) r dr du. Area (R) 6

35. Average distance from interior of disk to center Find the average distance from a point P(x, y) in the disk x 2 + y 2 … a 2 to the origin.

dy dx

In Exercises 23–26, sketch the region of integration and convert each polar integral or sum of integrals to a Cartesian integral or sum of integrals. Do not evaluate the integrals. p>2

32. Overlapping cardioids Find the area of the region common to the interiors of the cardioids r = 1 + cos u and r = 1 - cos u.

34. Average height of a cone Find the average height of the (single)

y

L22 L24 - y 2

31. Cardioid in the first quadrant Find the area of the region cut from the first quadrant by the cardioid r = 1 + sin u.

33. Average height of a hemisphere Find the average height of

sx + 2yd dy dx

L0 Lx

Find the area enclosed by one leaf of the rose

dy dx

ln sx 2 + y 2 + 1d dx dy

22 - x 2

2

22.

2 2

1

21.

2

L-1 L-21 - x 2 1 + 2x 2 + y 2

1

20.

16.

0

ln 2

19.

L0 L0 2

L1

sx 2 + y 2 d dx dy

L-a L-2a2 - x 2

14.

dy dx

2

2a2 - x 2

a

x

1

18.

12.

x dx dy

0

17.

sx 2 + y 2 d dx dy

L0 L0

y

23

15.

10.

24 - y 2

L0 L0

21 - y

1

dy dx

L-1 L0 2

11.

2

29. One leaf of a rose r = 12 cos 3u.

30. Snail shell Find the area of the region enclosed by the positive x-axis and spiral r = 4u>3, 0 … u … 2p. The region looks like a snail shell.

23 2

9.

785

Area in Polar Coordinates 27. Find the area of the region cut from the first quadrant by the curve r = 2s2 - sin 2ud1>2.

6. y

Double Integrals in Polar Form

Ltan

-1 4

3

4 csc u

L0

r 7 dr du

q

L0

e -x dxb a

q

2

L0

e -y dyb = 2

q

L0 L0

q

e -sx

2

+ y 2d

dx dy.

Evaluate the last integral using polar coordinates and solve the resulting equation for I.

786

Chapter 14: Multiple Integrals 46. Area Suppose that the area of a region in the polar coordinate plane is

b. Evaluate x

lim erf sxd = lim

x: q

2e -t

x: qL 0

42. Converting to a polar integral q

q

L0 L0

2

2p

A =

Evaluate the integral

1 dx dy. s1 + x 2 + y 2 d2

43. Existence Integrate the function ƒsx, yd = 1>s1 - x 2 - y 2 d over the disk x 2 + y 2 … 3>4 . Does the integral of ƒ(x, y) over the disk x 2 + y 2 … 1 exist? Give reasons for your answer.

2 sin u

3p>4

dt.

r dr du.

Lp>4 Lcsc u

Sketch the region and find its area. COMPUTER EXPLORATIONS In Exercises 47–50, use a CAS to change the Cartesian integrals into an equivalent polar integral and evaluate the polar integral. Perform the following steps in each exercise. a. Plot the Cartesian region of integration in the xy-plane. b. Change each boundary curve of the Cartesian region in part (a) to its polar representation by solving its Cartesian equation for r and u.

44. Area formula in polar coordinates Use the double integral in polar coordinates to derive the formula b

1 2 r du La 2

c. Using the results in part (b), plot the polar region of integration in the ru-plane.

for the area of the fan-shaped region between the origin and polar curve r = ƒsud, a … u … b.

d. Change the integrand from Cartesian to polar coordinates. Determine the limits of integration from your plot in part (c) and evaluate the polar integral using the CAS integration utility.

A =

45. Average distance to a given point inside a disk Let P0 be a point inside a circle of radius a and let h denote the distance from P0 to the center of the circle. Let d denote the distance from an arbitrary point P to P0. Find the average value of d 2 over the region enclosed by the circle. (Hint: Simplify your work by placing the center of the circle at the origin and P0 on the x-axis.)

14.5

1

47.

L0

Lx 1

49.

1

1

y x + y 2

y>3

2

dy dx

y

L0 L-y>3 2x 2 + y 2

48.

L0

L0 1

dx dy

50.

L0 Ly

x>2

x dy dx x2 + y2

2-y

2x + y dx dy

Triple Integrals in Rectangular Coordinates Just as double integrals allow us to deal with more general situations than could be handled by single integrals, triple integrals enable us to solve still more general problems. We use triple integrals to calculate the volumes of three-dimensional shapes and the average value of a function over a three-dimensional region. Triple integrals also arise in the study of vector fields and fluid flow in three dimensions, as we will see in Chapter 15.

Triple Integrals

z

If F(x, y, z) is a function defined on a closed, bounded region D in space, such as the region occupied by a solid ball or a lump of clay, then the integral of F over D may be defined in the following way. We partition a rectangular boxlike region containing D into rectangular cells by planes parallel to the coordinate axes (Figure 14.29). We number the cells that lie completely inside D from 1 to n in some order, the kth cell having dimensions ¢xk by ¢yk by ¢zk and volume ¢Vk = ¢xk ¢yk ¢zk. We choose a point sxk , yk , zk d in each cell and form the sum

(x k , yk , zk )

zk

D

yk

xk

n

x y

Sn = a Fsxk , yk , zk d ¢Vk. k=1

FIGURE 14.29 Partitioning a solid with rectangular cells of volume ¢Vk .

(1)

We are interested in what happens as D is partitioned by smaller and smaller cells, so that ¢xk , ¢yk , ¢zk and the norm of the partition 7P7, the largest value among ¢xk , ¢yk , ¢zk , all approach zero. When a single limiting value is attained, no matter how the partitions and points sxk , yk , zk d are chosen, we say that F is integrable over D. As before, it can be

14.5

Triple Integrals in Rectangular Coordinates

787

shown that when F is continuous and the bounding surface of D is formed from finitely many smooth surfaces joined together along finitely many smooth curves, then F is integrable. As 7P7 : 0 and the number of cells n goes to q , the sums Sn approach a limit. We call this limit the triple integral of F over D and write lim Sn =

n: q

Fsx, y, zd dV 9

lim Sn =

or

ƒ ƒ P ƒ ƒ :0

D

9

Fsx, y, zd dx dy dz.

D

The regions D over which continuous functions are integrable are those having “reasonably smooth” boundaries.

Volume of a Region in Space If F is the constant function whose value is 1, then the sums in Equation (1) reduce to Sn = a Fsxk , yk , zk d ¢Vk = a 1 # ¢Vk = a ¢Vk . As ¢xk , ¢yk , and ¢zk approach zero, the cells ¢Vk become smaller and more numerous and fill up more and more of D. We therefore define the volume of D to be the triple integral n

lim a ¢Vk = n: q k=1

DEFINITION

9

dV.

D

The volume of a closed, bounded region D in space is V =

9

dV.

D

This definition is in agreement with our previous definitions of volume, although we omit the verification of this fact. As we see in a moment, this integral enables us to calculate the volumes of solids enclosed by curved surfaces.

Finding Limits of Integration in the Order dz dy dx We evaluate a triple integral by applying a three-dimensional version of Fubini’s Theorem (Section 14.2) to evaluate it by three repeated single integrations. As with double integrals, there is a geometric procedure for finding the limits of integration for these single integrals. To evaluate 9

Fsx, y, zd dV

D

over a region D, integrate first with respect to z, then with respect to y, and finally with respect to x. (You might choose a different order of integration, but the procedure is similar, as we illustrate in Example 2.) 1.

Sketch. Sketch the region D along with its “shadow” R (vertical projection) in the xy-plane. Label the upper and lower bounding surfaces of D and the upper and lower bounding curves of R.

788

Chapter 14: Multiple Integrals z z 5 f2(x, y)

D z 5 f1(x, y)

y y 5 g1(x)

a b

R

x

2.

y 5 g2(x)

Find the z-limits of integration. Draw a line M passing through a typical point (x, y) in R parallel to the z-axis. As z increases, M enters D at z = ƒ1sx, yd and leaves at z = ƒ2sx, yd. These are the z-limits of integration. z M Leaves at z 5 f2(x, y) D

Enters at z 5 f1(x, y) y a y 5 g1(x) R

b x

(x, y)

3.

y 5 g2(x)

Find the y-limits of integration. Draw a line L through (x, y) parallel to the y-axis. As y increases, L enters R at y = g1sxd and leaves at y = g2sxd. These are the y-limits of integration. z M

D

Enters at y 5 g1(x) a

y

x b x

R (x, y)

L Leaves at y 5 g2(x)

14.5

4.

Triple Integrals in Rectangular Coordinates

789

Find the x-limits of integration. Choose x-limits that include all lines through R parallel to the y-axis ( x = a and x = b in the preceding figure). These are the x-limits of integration. The integral is x=b

y = g2sxd

z = ƒ2sx, yd

Fsx, y, zd dz dy dx.

Lx = a Ly = g1sxd Lz = ƒ1sx, yd

Follow similar procedures if you change the order of integration. The “shadow” of region D lies in the plane of the last two variables with respect to which the iterated integration takes place. The preceding procedure applies whenever a solid region D is bounded above and below by a surface, and when the “shadow” region R is bounded by a lower and upper curve. It does not apply to regions with complicated holes through them, although sometimes such regions can be subdivided into simpler regions for which the procedure does apply. Find the volume of the region D enclosed by the surfaces z = x 2 + 3y 2 and z = 8 - x - y 2.

EXAMPLE 1

2

Solution

The volume is V =

9

dz dy dx,

D

the integral of Fsx, y, zd = 1 over D. To find the limits of integration for evaluating the integral, we first sketch the region. The surfaces (Figure 14.30) intersect on the elliptical cylinder x 2 + 3y 2 = 8 - x 2 - y 2 or x 2 + 2y 2 = 4, z 7 0. The boundary of the region R, the projection of D onto the xy-plane, is an ellipse with the same equation: x 2 + 2y 2 = 4. The “upper” boundary of R is the curve y = 1s4 - x 2 d>2. The lower boundary is the curve y = - 1s4 - x 2 d>2. Now we find the z-limits of integration. The line M passing through a typical point (x, y) in R parallel to the z-axis enters D at z = x 2 + 3y 2 and leaves at z = 8 - x 2 - y 2. M

z

Leaves at z 5 8 2 x2 2 y2

z 5 8 2 x2 2 y2 The curve of intersection

D (–2, 0, 4) z 5 x 2 1 3y 2

(2, 0, 4) Enters at z 5 x 2 1 3y 2 Enters at y 5 –(4 2 x 2 )/ 2 (2, 0, 0) x

(–2, 0, 0)

x

(x, y)

x 2 1 2y 2 5 4

R Leaves at y 5 (4 2 x 2 )/ 2

L

y

FIGURE 14.30 The volume of the region enclosed by two paraboloids, calculated in Example 1.

790

Chapter 14: Multiple Integrals

Next we find the y-limits of integration. The line L through (x, y) parallel to the y-axis enters R at y = - 2s4 - x 2 d>2 and leaves at y = 2s4 - x 2 d>2. Finally we find the x-limits of integration. As L sweeps across R, the value of x varies from x = -2 at s -2, 0, 0d to x = 2 at (2, 0, 0). The volume of D is V =

9

dz dy dx

D

2

=

8 - x2 - y2

L-2 L-2s4 - x 2d>2 Lx 2 + 3y 2 2

=

2s4 - x 2d>2

2s4 - x 2d>2

L-2 L-2s4 - x 2d>2

dz dy dx

s8 - 2x 2 - 4y 2 d dy dx y = 2s4 - x 2d>2

2

4 = cs8 - 2x dy - y 3 d dx 3 y = -2s4 - x 2d>2 L-2 2

2

=

L-2 2

=

L-2

a2s8 - 2x 2 d

8 4 - x2 4 - x2 - a b 3 2 B 2

c8 a

= 8p22.

4 - x2 b 2

3>2

-

8 4 - x2 a b 3 2

3>2

3>2

d dx =

b dx 422 2 s4 - x 2 d3>2 dx 3 L-2

After integration with the substitution x = 2 sin u

In the next example, we project D onto the xz-plane instead of the xy-plane, to show how to use a different order of integration.

EXAMPLE 2 Set up the limits of integration for evaluating the triple integral of a function F(x, y, z) over the tetrahedron D with vertices (0, 0, 0), (1, 1, 0), (0, 1, 0), and (0, 1, 1). Use the order of integration dy dz dx.

z

(0, 1, 1)

1 y5x1z

L Line x1z51

D R

y51 (0, 1, 0)

y

M (x, z)

1

x Enters at y5x1z

Leaves at y51 (1, 1, 0)

x

FIGURE 14.31 Finding the limits of integration for evaluating the triple integral of a function defined over the tetrahedron D (Examples 2 and 3).

Solution We sketch D along with its “shadow” R in the xz-plane (Figure 14.31). The upper (right-hand) bounding surface of D lies in the plane y = 1. The lower (left-hand) bounding surface lies in the plane y = x + z. The upper boundary of R is the line z = 1 - x. The lower boundary is the line z = 0. First we find the y-limits of integration. The line through a typical point (x, z) in R parallel to the y-axis enters D at y = x + z and leaves at y = 1. Next we find the z-limits of integration. The line L through (x, z) parallel to the z-axis enters R at z = 0 and leaves at z = 1 - x. Finally we find the x-limits of integration. As L sweeps across R, the value of x varies from x = 0 to x = 1. The integral is 1

L0 L0

1-x

1

Lx + z

Fsx, y, zd dy dz dx.

EXAMPLE 3 Integrate F(x, y, z) = 1 over the tetrahedron D in Example 2 in the order dz dy dx, and then integrate in the order dy dz dx. Solution First we find the z-limits of integration. A line M parallel to the z-axis through a typical point (x, y) in the xy-plane “shadow” enters the tetrahedron at z = 0 and exits through the upper plane where z = y - x (Figure 14.32). Next we find the y-limits of integration. On the xy-plane, where z = 0, the sloped side of the tetrahedron crosses the plane along the line y = x. A line L through (x, y) parallel to the y-axis enters the shadow in the xy-plane at y = x and exits at y = 1 (Figure 14.32).

14.5

z5y2x (0, 1, 1)

1

y-x

1

Fsx, y, zd dz dy dx.

L0 Lx L0 D

For example, if Fsx, y, zd = 1, we would find the volume of the tetrahedron to be (0, 1, 0)

0 x

y

L

(x, y)

y51

1

V =

y5x R 1

791

Finally we find the x-limits of integration. As the line L parallel to the y-axis in the previous step sweeps out the shadow, the value of x varies from x = 0 to x = 1 at the point (1, 1, 0) (see Figure 14.32). The integral is

z

M

Triple Integrals in Rectangular Coordinates

(1, 1, 0)

x

FIGURE 14.32 The tetrahedron in Example 3 showing how the limits of integration are found for the order dz dy dx.

dz dy dx

L0 Lx L0 1

=

y-x

1

1

L0 Lx

s y - xd dy dx y=1

1

=

1 c y 2 - xy d dx 2 y=x L0 1

=

L0

a

1 1 - x + x 2 b dx 2 2 1

1 1 1 = c x - x2 + x3d 2 2 6 0 =

1 . 6

We get the same result by integrating with the order dy dz dx. From Example 2, 1-x

1

V =

L0 L0 L0 L0 1

=

L0 1

=

L0

Lx + z 1-x

1

=

1

dy dz dx

(1 - x - z) dz dx z=1-x

c(1 - x)z -

1 2 z d 2 z=0

c(1 - x) 2 -

1 (1 - x) 2 d dx 2

dx

1

=

1 (1 - x) 2 dx 2 L0 1

1 1 = - (1 - x) 3 d = . 6 6 0

Average Value of a Function in Space The average value of a function F over a region D in space is defined by the formula Average value of F over D =

1 F dV. volume of D 9

(2)

D

For example, if Fsx, y, zd = 2x 2 + y 2 + z 2, then the average value of F over D is the average distance of points in D from the origin. If F(x, y, z) is the temperature at (x, y, z) on a solid that occupies a region D in space, then the average value of F over D is the average temperature of the solid.

792

Chapter 14: Multiple Integrals

Find the average value of Fsx, y, zd = xyz throughout the cubical region D bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2 in the first octant.

EXAMPLE 4

z 2

We sketch the cube with enough detail to show the limits of integration (Figure 14.33). We then use Equation (2) to calculate the average value of F over the cube. The volume of the region D is s2ds2ds2d = 8. The value of the integral of F over the cube is Solution

D 2 y 2

2

x

2

L0 L0 L0

2

2

xyz dx dy dz =

FIGURE 14.33 The region of integration in Example 4.

2

L0 L0 2

=

c

cy 2z d

L0

x=2

2

2

x2 yz d dy dz = 2yz dy dz 2 x=0 L0 L0 y=2 y=0

2

dz =

L0

2

4z dz = c2z 2 d = 8. 0

With these values, Equation (2) gives 1 1 Average value of = xyz dV = a bs8d = 1. 8 volume xyz over the cube 9 cube

In evaluating the integral, we chose the order dx dy dz, but any of the other five possible orders would have done as well.

Properties of Triple Integrals Triple integrals have the same algebraic properties as double and single integrals. Simply replace the double integrals in the four properties given in Section 14.2, page 773, with triple integrals.

Exercises 14.5 Triple Integrals in Different Iteration Orders 1. Evaluate the integral in Example 2 taking Fsx, y, zd = 1 to find the volume of the tetrahedron in the order dz dx dy. 2. Volume of rectangular solid Write six different iterated triple integrals for the volume of the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 1, y = 2, and z = 3. Evaluate one of the integrals. 3. Volume of tetrahedron Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane 6x + 3y + 2z = 6. Evaluate one of the integrals. 4. Volume of solid Write six different iterated triple integrals for the volume of the region in the first octant enclosed by the cylinder x 2 + z 2 = 4 and the plane y = 3. Evaluate one of the integrals. 5. Volume enclosed by paraboloids Let D be the region bounded by the paraboloids z = 8 - x 2 - y 2 and z = x 2 + y 2. Write six different triple iterated integrals for the volume of D. Evaluate one of the integrals. 6. Volume inside paraboloid beneath a plane Let D be the region bounded by the paraboloid z = x 2 + y 2 and the plane z = 2y. Write triple iterated integrals in the order dz dx dy and dz dy dx that give the volume of D. Do not evaluate either integral.

Evaluating Triple Iterated Integrals Evaluate the integrals in Exercises 7–20. 1

7.

1

L0 L1

dz dy dx

11.

e3

L1 L1 L1 p>6

1

1 xyz dx dy dz 3

L0 L-2

L0

y sin z dx dy dz

sx + y + zd dy dx dz

29 - x 2

29 - x 2

2-x

p

14.

2-x-y

16.

2x + y

L0 L-24 - y 2 L0 1 - x2

1

dz dy dx

L0

24 - y 2

2

dz dy dx

L0

L0 L0

2e

L1

cos su + y + wd du dy dw e

se s ln r

(ln t) 2 t dt dr ds

L3

suyw-spaced

srst-spaced

dz dx dy

4 - x2 - y

p

L0 L0 L0 1

18.

9.

2

L0 L0 p

17.

1

L0 L0 1

15.

L0

e2

e

dz dx dy

3 - 3x - y

L-1 L0 L0 3

13.

3 - 3x

L0 L0 1

12.

8 - x2 - y2

3y

L0 Lx 2 + 3y 2

L0 1

10.

sx 2 + y 2 + z 2 d dz dy dx

L0 L0 L0 22

8.

1

x dz dy dx

14.5 ln sec y

p>4

19.

L0

L0 7

20.

2

24 - q 2

L0 L0 L0

e x dx dt dy

793

24. The region in the first octant bounded by the coordinate planes and the planes x + z = 1, y + 2z = 2

2t

L- q

Triple Integrals in Rectangular Coordinates

styx-spaced

z

q dp dq dr r + 1

spqr-spaced

Finding Equivalent Iterated Integrals 21. Here is the region of integration of the integral 1

y

1-y

1

x

dz dy dx.

L-1 Lx 2 L0

25. The region in the first octant bounded by the coordinate planes, the plane y + z = 2, and the cylinder x = 4 - y 2

z Top: y z 1 Side: y x2

z

1 –1 (–1, 1, 0) y

1

1 x

y

(1, 1, 0)

Rewrite the integral as an equivalent iterated integral in the order a. dy dz dx

b. dy dx dz

c. dx dy dz

d. dx dz dy

x

26. The wedge cut from the cylinder x 2 + y 2 = 1 by the planes z = -y and z = 0

e. dz dx dy.

z

22. Here is the region of integration of the integral 1

y2

0

L0 L-1 L0

dz dy dx. y

z

(0, –1, 1)

x

1

(1, –1, 1)

zy

27. The tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1, 0, 0), (0, 2, 0), and (0, 0, 3)

2

(1, –1, 0)

z

y

0

(0, 0, 3)

1 x

Rewrite the integral as an equivalent iterated integral in the order a. dy dz dx

b. dy dx dz

c. dx dy dz

d. dx dz dy

(0, 2, 0) y

(1, 0, 0)

e. dz dx dy.

x

Finding Volumes Using Triple Integrals Find the volumes of the regions in Exercises 23–36. 23. The region between the cylinder z = y 2 and the xy-plane that is bounded by the planes x = 0, x = 1, y = -1, y = 1

28. The region in the first octant bounded by the coordinate planes, the plane y = 1 - x, and the surface z = cos spx>2d, 0 … x … 1 z

z

y y x

x

794

Chapter 14: Multiple Integrals

29. The region common to the interiors of the cylinders x 2 + y 2 = 1 and x 2 + z 2 = 1, one-eighth of which is shown in the accompanying figure z

33. The region between the planes x + y + 2z = 2 and 2x + 2y + z = 4 in the first octant 34. The finite region bounded by the planes z = x, x + z = 8, z = y, y = 8, and z = 0 35. The region cut from the solid elliptical cylinder x 2 + 4y 2 … 4 by the xy-plane and the plane z = x + 2 36. The region bounded in back by the plane x = 0, on the front and sides by the parabolic cylinder x = 1 - y 2, on the top by the paraboloid z = x 2 + y 2, and on the bottom by the xy-plane

x2 1 y2 5 1

Average Values In Exercises 37–40, find the average value of F(x, y, z) over the given region.

x2 1 z2 5 1

37. Fsx, y, zd = x 2 + 9 over the cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2 y

x

30. The region in the first octant bounded by the coordinate planes and the surface z = 4 - x 2 - y z

38. Fsx, y, zd = x + y - z over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 1, y = 1, and z = 2 39. Fsx, y, zd = x 2 + y 2 + z 2 over the cube in the first octant bounded by the coordinate planes and the planes x = 1, y = 1, and z = 1 40. Fsx, y, zd = xyz over the cube in the first octant bounded by the coordinate planes and the planes x = 2, y = 2, and z = 2 Changing the Order of Integration Evaluate the integrals in Exercises 41–44 by changing the order of integration in an appropriate way. 4

41. y

4 cos sx 2 d

2

L0 L0 L2y 1

42.

x

1

1

dx dy dz

22z

1

2

12xze zy dy dx dz

L0 L0 Lx

2

31. The region in the first octant bounded by the coordinate planes, the plane x + y = 4, and the cylinder y 2 + 4z 2 = 16

1

43. 44.

ln 3

pe 2x sin py 2 y2

3 L0 L2z L0

2

z

1

L0

L0

4 - x2

x

L0

dx dy dz

sin 2z dy dz dx 4 - z

Theory and Examples 45. Finding an upper limit of an iterated integral Solve for a: y

4 - a - x2

1

L0 L0 x

32. The region cut from the cylinder x + y = 4 by the plane z = 0 and the plane x + z = 3 2

2

z

4 - x2 - y

La

dz dy dx =

4 . 15

46. Ellipsoid For what value of c is the volume of the ellipsoid x 2 + s y>2d2 + sz>cd2 = 1 equal to 8p? 47. Minimizing a triple integral What domain D in space minimizes the value of the integral s4x 2 + 4y 2 + z 2 - 4d dV ? 9 D

Give reasons for your answer. 48. Maximizing a triple integral What domain D in space maximizes the value of the integral

y x

s1 - x 2 - y 2 - z 2 d dV ? 9 D

Give reasons for your answer.

14.6 COMPUTER EXPLORATIONS In Exercises 49–52, use a CAS integration utility to evaluate the triple integral of the given function over the specified solid region. 49. Fsx, y, zd = x y z over the solid cylinder x 2 + y 2 = 1 and the planes z = 0 and z = 1 2 2

bounded

51. Fsx, y, zd =

Moments and Centers of Mass

795

z over the solid bounded below by sx 2 + y 2 + z 2 d3>2

the cone z = 2x 2 + y 2 and above by the plane z = 1

by

52. Fsx, y, zd = x 4 + y 2 + z 2 over the solid sphere x 2 + y 2 + z2 … 1

50. Fsx, y, zd = ƒ xyz ƒ over the solid bounded below by the paraboloid z = x 2 + y 2 and above by the plane z = 1

Moments and Centers of Mass

14.6

This section shows how to calculate the masses and moments of two- and threedimensional objects in Cartesian coordinates. Section 14.7 gives the calculations for cylindrical and spherical coordinates. The definitions and ideas are similar to the single-variable case we studied in Section 6.6, but now we can consider more realistic situations.

z

mk (xk , yk , zk ) Vk

Masses and First Moments If dsx, y, zd is the density (mass per unit volume) of an object occupying a region D in space, the integral of d over D gives the mass of the object. To see why, imagine partitioning the object into n mass elements like the one in Figure 14.34. The object’s mass is the limit

D (xk , yk , zk )

n

n

M = lim a ¢mk = lim a dsxk, yk, zk d ¢Vk = n: q n: q k=1

y

x

FIGURE 14.34 To define an object’s mass, we first imagine it to be partitioned into a finite number of mass elements ¢mk.

k=1

dsx, y, zd dV. 9 D

The first moment of a solid region D about a coordinate plane is defined as the triple integral over D of the distance from a point (x, y, z) in D to the plane multiplied by the density of the solid at that point. For instance, the first moment about the yz-plane is the integral Myz =

9

xdsx, y, zd dV.

D

The center of mass is found from the first moments. For instance, the x-coordinate of the center of mass is x = Myz>M . For a two-dimensional object, such as a thin, flat plate, we calculate first moments about the coordinate axes by simply dropping the z-coordinate. So the first moment about the y-axis is the double integral over the region R forming the plate of the distance from the axis multiplied by the density, or

z

z 4 x2 y2

My = c.m.

6

xd(x, y) dA.

R

R 0 y x

Table 14.1 summarizes the formulas.

x2 y2 4

FIGURE 14.35 Finding the center of mass of a solid (Example 1).

EXAMPLE 1

Find the center of mass of a solid of constant density d bounded below by the disk R: x 2 + y 2 … 4 in the plane z = 0 and above by the paraboloid z = 4 - x 2 - y 2 (Figure 14.35).

796

Chapter 14: Multiple Integrals

Mass and first moment formulas

TABLE 14.1

THREE-DIMENSIONAL SOLID Mass: M =

d = dsx, y, zd is the density at (x, y, z).

d dV 9 D

First moments about the coordinate planes: Myz =

9

Mxz =

x d dV,

D

9

Mxy =

y d dV,

D

9

z d dV

D

Center of mass: x =

Myz , M

Mxz , M

y =

z =

Mxy M

TWO-DIMENSIONAL PLATE Mass:

M =

d dA 6

d = dsx, yd is the density at (x, y).

R

First moments:

My =

Center of mass:

My , x = M

6

Mx =

x d dA,

R

Solution

6

y d dA

R

Mx y = M

By symmetry x = y = 0. To find z, we first calculate Mxy =

z = 4 - x2 - y2

6Lz = 0 R

=

z = 4 - x2 - y2

z2 z d dz dy dx = c d 6 2 z=0

d dy dx

R

d s4 - x 2 - y 2 d2 dy dx 26 R

2p

2

=

d s4 - r 2 d2 r dr du 2L0 L0

=

d 2L0

2p

c-

r=2

Polar coordinates simplify the integration.

16d 1 s4 - r 2 d3 d du = 6 3 L0 r=0

2p

du =

32pd . 3

A similar calculation gives the mass M =

4 - x2 - y2

6L0

d dz dy dx = 8pd.

R

Therefore z = sMxy>Md = 4>3 and the center of mass is sx, y, zd = s0, 0, 4>3d. When the density of a solid object or plate is constant (as in Example 1), the center of mass is called the centroid of the object. To find a centroid, we set d equal to 1 and proceed to find x, y, and z as before, by dividing first moments by masses. These calculations are also valid for two-dimensional objects.

EXAMPLE 2 Find the centroid of the region in the first quadrant that is bounded above by the line y = x and below by the parabola y = x 2.

14.6

Moments and Centers of Mass

797

We sketch the region and include enough detail to determine the limits of integration (Figure 14.36). We then set d equal to 1 and evaluate the appropriate formulas from Table 14.1: Solution

y 1

(1, 1) y5x

1

M =

y 5 x2 0

L0 Lx 2

x

1

x

1

Mx =

FIGURE 14.36 The centroid of this region is found in Example 2.

=

L0 1

My =

L0

x

L0 Lx 2 1

1

1 dy dx =

a

y=x

1

dx =

L0

y = x2 1

y dy dx =

cy d

L0

c

sx - x 2 d dx = c

1

x2 x3 1 d = 2 3 0 6

y 2 y=x d dx 2 y = x2 1

x x x3 x5 1 b dx = c d = 2 2 6 10 0 15 2

4

x

L0 Lx 2

1

x dy dx =

L0

cxy d

y=x

1

dx =

L0

y = x2

sx 2 - x 3 d dx = c

1

x3 x4 1 d = . 3 4 0 12

From these values of M, Mx , and My , we find x =

My 1>12 1 = = M 2 1>6

and

y =

1>15 Mx 2 = = . M 5 1>6

The centroid is the point (1> 2, 2>5).

Moments of Inertia

Δm k yk rk

An object’s first moments (Table 14.1) tell us about balance and about the torque the object experiences about different axes in a gravitational field. If the object is a rotating shaft, however, we are more likely to be interested in how much energy is stored in the shaft or about how much energy is generated by a shaft rotating at a particular angular velocity. This is where the second moment or moment of inertia comes in. Think of partitioning the shaft into small blocks of mass ¢mk and let rk denote the distance from the kth block’s center of mass to the axis of rotation (Figure 14.37). If the shaft rotates at a constant angular velocity of v = du>dt radians per second, the block’s center of mass will trace its orbit at a linear speed of

rk

Axi

s of

rota

tion

yk = L

du d sr ud = rk = rk v. dt k dt

The block’s kinetic energy will be approximately FIGURE 14.37 To find an integral for the amount of energy stored in a rotating shaft, we first imagine the shaft to be partitioned into small blocks. Each block has its own kinetic energy. We add the contributions of the individual blocks to find the kinetic energy of the shaft.

1 1 1 ¢m y 2 = ¢mksrk vd2 = v2rk 2 ¢mk . 2 k k 2 2 The kinetic energy of the shaft will be approximately 1 2 2 a 2 v rk ¢mk . The integral approached by these sums as the shaft is partitioned into smaller and smaller blocks gives the shaft’s kinetic energy: KEshaft =

1 2 2 1 v r dm = v2 r 2 dm. 2 L L2

(1)

The factor I =

L

r 2 dm

is the moment of inertia of the shaft about its axis of rotation, and we see from Equation (1) that the shaft’s kinetic energy is 1 KEshaft = Iv2. 2

798

Chapter 14: Multiple Integrals z

x

x 2 1 y 2 y

dV

x 2 1 z 2

0 x

y 2 1 z 2

y

z x

y

x

FIGURE 14.38 Distances from dV to the coordinate planes and axes.

The moment of inertia of a shaft resembles in some ways the inertial mass of a locomotive. To start a locomotive with mass m moving at a linear velocity y, we need to provide a kinetic energy of KE = s1>2dmy 2. To stop the locomotive we have to remove this amount of energy. To start a shaft with moment of inertia I rotating at an angular velocity v, we need to provide a kinetic energy of KE = s1>2dIv2. To stop the shaft we have to take this amount of energy back out. The shaft’s moment of inertia is analogous to the locomotive’s mass. What makes the locomotive hard to start or stop is its mass. What makes the shaft hard to start or stop is its moment of inertia. The moment of inertia depends not only on the mass of the shaft but also on its distribution. Mass that is farther away from the axis of rotation contributes more to the moment of inertia. We now derive a formula for the moment of inertia for a solid in space. If r(x, y, z) is the distance from the point (x, y, z) in D to a line L, then the moment of inertia of the mass ¢mk = dsxk, yk, zk d¢Vk about the line L (as in Figure 14.37) is approximately ¢Ik = r 2sxk, yk, zk d¢mk. The moment of inertia about L of the entire object is n

n

IL = lim a ¢Ik = lim a r 2sxk , yk , zk d dsxk , yk , zk d ¢Vk = n: q n: q k=1

k=1

9

r 2d dV.

D

If L is the x-axis, then r = y + z (Figure 14.38) and 2

2

2

Ix =

s y 2 + z 2 d d(x, y, z) dV. 9 D

Similarly, if L is the y-axis or z-axis we have Iy =

sx 2 + z 2 d d(x, y, z) dV 9

and

D

Iz =

sx 2 + y 2 d d(x, y, z) dV. 9 D

Table 14.2 summarizes the formulas for these moments of inertia (second moments because they invoke the squares of the distances). It shows the definition of the polar moment about the origin as well.

EXAMPLE 3

z

Find Ix, Iy, Iz for the rectangular solid of constant density d shown in

Figure 14.39. Solution

The formula for Ix gives c>2

c

Ix = y

a x

Center of block

b

a>2

L-c>2 L-b>2 L-a>2

s y 2 + z 2 d d dx dy dz.

We can avoid some of the work of integration by observing that s y 2 + z 2 dd is an even function of x, y, and z since d is constant. The rectangular solid consists of eight symmetric pieces, one in each octant. We can evaluate the integral on one of these pieces and then multiply by 8 to get the total value. c>2

FIGURE 14.39 Finding Ix, Iy, and Iz for the block shown here. The origin lies at the center of the block (Example 3).

b>2

b>2

Ix = 8

L0 L0 L0 c

c>2

a

L0

= 4ad

L0

= 4ad a

c>2

s y 2 + z 2 d d dx dy dz = 4ad

L0 L0

b>2

s y 2 + z 2 d dy dz

y = b>2 y3 dz + z 2y d 3 y=0

c>2

= 4ad

a>2

b3 z 2b + b dz 24 2

c 3b abcd 2 b 3c M 2 + b = sb + c 2 d = sb + c 2 d. 48 48 12 12

M = abcd

14.6

Moments and Centers of Mass

799

TABLE 14.2 Moments of inertia (second moments) formulas

THREE-DIMENSIONAL SOLID About the x-axis:

Ix =

s y 2 + z 2 d d dV 9

About the y-axis:

Iy =

sx 2 + z 2 d d dV 9

About the z-axis:

Iz =

sx 2 + y 2 d d dV 9

About a line L:

IL =

r 2 d dV 9

d = d(x, y, z)

r(x, y, z) distance from the point (x, y, z) to line L

TWO-DIMENSIONAL PLATE About the x-axis:

Ix =

About the y-axis:

Iy =

About a line L:

IL =

About the origin (polar moment):

I0 =

6 6 6

d = d(x, y)

y 2 d dA x 2 d dA

r(x, y) distance from (x, y) to L

r 2sx, yd d dA

sx 2 + y 2 d d dA = Ix + Iy 6

Similarly, Iy =

M 2 sa + c 2 d 12

Iz =

and

M 2 sa + b 2 d. 12

EXAMPLE 4

A thin plate covers the triangular region bounded by the x-axis and the lines x = 1 and y = 2x in the first quadrant. The plate’s density at the point (x, y) is dsx, yd = 6x + 6y + 6. Find the plate’s moments of inertia about the coordinate axes and the origin. y

(1, 2)

2

Solution We sketch the plate and put in enough detail to determine the limits of integration for the integrals we have to evaluate (Figure 14.40). The moment of inertia about the x-axis is

y 5 2x x51

1

Ix =

L0 L0 1

0

1

x

FIGURE 14.40 The triangular region covered by the plate in Example 4.

=

2x

L0

1

y 2dsx, yd dy dx =

L0 L0 y = 2x

c2xy 3 +

2x

s6xy 2 + 6y 3 + 6y 2 d dy dx 1

3 4 y + 2y 3 d dx = (40x 4 + 16x 3) dx 2 y=0 L0

= C 8x 5 + 4x 4 D 0 = 12. 1

800

Chapter 14: Multiple Integrals

Similarly, the moment of inertia about the y-axis is

Beam A

1

Iy =

Axis

L0 L0

2x

x 2dsx, yd dy dx =

39 . 5

Notice that we integrate y 2 times density in calculating Ix and x 2 times density to find Iy. Since we know Ix and Iy, we do not need to evaluate an integral to find I0; we can use the equation I0 = Ix + Iy from Table 14.2 instead: Beam B

I0 = 12 +

39 60 + 39 99 = = . 5 5 5

Axis

The moment of inertia also plays a role in determining how much a horizontal metal beam will bend under a load. The stiffness of the beam is a constant times I, the moment of inertia of a typical cross-section of the beam about the beam’s longitudinal axis. The greater the value of I, the stiffer the beam and the less it will bend under a given load. That FIGURE 14.41 The greater the polar is why we use I-beams instead of beams whose cross-sections are square. The flanges at moment of inertia of the cross-section of a beam about the beam’s longitudinal axis, the the top and bottom of the beam hold most of the beam’s mass away from the longitudinal axis to increase the value of I (Figure 14.41). stiffer the beam. Beams A and B have the same cross-sectional area, but A is stiffer.

Exercises 14.6 Plates of Constant Density 1. Finding a center of mass Find the center of mass of a thin plate of density d = 3 bounded by the lines x = 0, y = x, and the parabola y = 2 - x 2 in the first quadrant. 2. Finding moments of inertia Find the moments of inertia about the coordinate axes of a thin rectangular plate of constant density d bounded by the lines x = 3 and y = 3 in the first quadrant. 3. Finding a centroid Find the centroid of the region in the first quadrant bounded by the x-axis, the parabola y 2 = 2x, and the line x + y = 4. 4. Finding a centroid Find the centroid of the triangular region cut from the first quadrant by the line x + y = 3. 5. Finding a centroid Find the centroid of the region cut from the first quadrant by the circle x 2 + y 2 = a 2. 6. Finding a centroid Find the centroid of the region between the x-axis and the arch y = sin x, 0 … x … p. 7. Finding moments of inertia Find the moment of inertia about the x-axis of a thin plate of density d = 1 bounded by the circle x 2 + y 2 = 4. Then use your result to find Iy and I0 for the plate. 8. Finding a moment of inertia Find the moment of inertia with respect to the y-axis of a thin sheet of constant density d = 1 bounded by the curve y = ssin2 xd>x 2 and the interval p … x … 2p of the x-axis. 9. The centroid of an infinite region Find the centroid of the infinite region in the second quadrant enclosed by the coordinate axes and the curve y = e x. (Use improper integrals in the massmoment formulas.)

10. The first moment of an infinite plate Find the first moment about the y-axis of a thin plate of density dsx, yd = 1 covering the 2 infinite region under the curve y = e -x >2 in the first quadrant. Plates with Varying Density 11. Finding a moment of inertia Find the moment of inertia about the x-axis of a thin plate bounded by the parabola x = y - y 2 and the line x + y = 0 if dsx, yd = x + y. 12. Finding mass Find the mass of a thin plate occupying the smaller region cut from the ellipse x 2 + 4y 2 = 12 by the parabola x = 4y 2 if dsx, yd = 5x. 13. Finding a center of mass Find the center of mass of a thin triangular plate bounded by the y-axis and the lines y = x and y = 2 - x if dsx, yd = 6x + 3y + 3. 14. Finding a center of mass and moment of inertia Find the center of mass and moment of inertia about the x-axis of a thin plate bounded by the curves x = y 2 and x = 2y - y 2 if the density at the point (x, y) is dsx, yd = y + 1. 15. Center of mass, moment of inertia Find the center of mass and the moment of inertia about the y-axis of a thin rectangular plate cut from the first quadrant by the lines x = 6 and y = 1 if dsx, yd = x + y + 1. 16. Center of mass, moment of inertia Find the center of mass and the moment of inertia about the y-axis of a thin plate bounded by the line y = 1 and the parabola y = x 2 if the density is dsx, yd = y + 1. 17. Center of mass, moment of inertia Find the center of mass and the moment of inertia about the y-axis of a thin plate bounded by the x-axis, the lines x = ;1, and the parabola y = x 2 if dsx, yd = 7y + 1.

14.6 18. Center of mass, moment of inertia Find the center of mass and the moment of inertia about the x-axis of a thin rectangular plate bounded by the lines x = 0, x = 20, y = - 1, and y = 1 if dsx, yd = 1 + sx>20d. 19. Center of mass, moments of inertia Find the center of mass, the moment of inertia about the coordinate axes, and the polar moment of inertia of a thin triangular plate bounded by the lines y = x, y = -x, and y = 1 if dsx, yd = y + 1.

Moments and Centers of Mass

z

z2x 2 x –2

20. Center of mass, moments of inertia Repeat Exercise 19 for dsx, yd = 3x 2 + 1. Solids with Constant Density 21. Moments of inertia Find the moments of inertia of the rectangular solid shown here with respect to its edges by calculating Ix , Iy , and Iz . z c

b y a x

22. Moments of inertia The coordinate axes in the figure run through the centroid of a solid wedge parallel to the labeled edges. Find Ix , Iy , and Iz if a = b = 6 and c = 4. z

Centroid at (0, 0, 0)

a 2 b 3

c

y

c 3

a b

x

23. Center of mass and moments of inertia A solid “trough” of constant density is bounded below by the surface z = 4y 2, above by the plane z = 4, and on the ends by the planes x = 1 and x = -1. Find the center of mass and the moments of inertia with respect to the three axes. 24. Center of mass A solid of constant density is bounded below by the plane z = 0, on the sides by the elliptical cylinder x 2 + 4y 2 = 4, and above by the plane z = 2 - x (see the accompanying figure).

1

Mxy =

y

x 2 4y 2 4

2 x

25. a. Center of mass Find the center of mass of a solid of constant density bounded below by the paraboloid z = x 2 + y 2 and above by the plane z = 4. b. Find the plane z = c that divides the solid into two parts of equal volume. This plane does not pass through the center of mass. 26. Moments A solid cube, 2 units on a side, is bounded by the planes x = ;1, z = ; 1, y = 3, and y = 5. Find the center of mass and the moments of inertia about the coordinate axes. 27. Moment of inertia about a line A wedge like the one in Exercise 22 has a = 4, b = 6, and c = 3. Make a quick sketch to check for yourself that the square of the distance from a typical point (x, y, z) of the wedge to the line L: z = 0, y = 6 is r 2 = s y - 6d2 + z 2. Then calculate the moment of inertia of the wedge about L. 28. Moment of inertia about a line A wedge like the one in Exercise 22 has a = 4, b = 6, and c = 3. Make a quick sketch to check for yourself that the square of the distance from a typical point (x, y, z) of the wedge to the line L: x = 4, y = 0 is r 2 = sx - 4d2 + y 2. Then calculate the moment of inertia of the wedge about L. Solids with Varying Density In Exercises 29 and 30, find a. the mass of the solid. b. the center of mass. 29. A solid region in the first octant is bounded by the coordinate planes and the plane x + y + z = 2. The density of the solid is dsx, y, zd = 2x. 30. A solid in the first octant is bounded by the planes y = 0 and z = 0 and by the surfaces z = 4 - x 2 and x = y 2 (see the accompanying figure). Its density function is dsx, y, zd = k xy, k a constant. z 4

z 5 4 2 x2

a. Find x and y. b. Evaluate the integral 2

x 5 y2 s1>2d24 - x 2

L-2 L-s1>2d24 - x 2L0

2-x

z dz dy dx

y 2

using integral tables to carry out the final integration with respect to x. Then divide Mxy by M to verify that z = 5>4.

801

x

(2, 2, 0)

802

Chapter 14: Multiple Integrals

In Exercises 31 and 32, find a. the mass of the solid.

z L c.m.

b. the center of mass.

c. the moments of inertia about the coordinate axes. 31. A solid cube in the first octant is bounded by the coordinate planes and by the planes x = 1, y = 1, and z = 1. The density of the cube is dsx, y, zd = x + y + z + 1.

L

32. A wedge like the one in Exercise 22 has dimensions a = 2, b = 6, and c = 3. The density is dsx, y, zd = x + 1. Notice that if the density is constant, the center of mass will be (0, 0, 0). 33. Mass Find the mass of the solid bounded by the planes x + z = 1, x - z = -1, y = 0 and the surface y = 2z. The density of the solid is dsx, y, zd = 2y + 5. 34. Mass Find the mass of the solid region bounded by the parabolic surfaces z = 16 - 2x 2 - 2y 2 and z = 2x 2 + 2y 2 if the density of the solid is dsx, y, zd = 2x 2 + y 2 . Theory and Examples The Parallel Axis Theorem Let Lc.m. be a line through the center of mass of a body of mass m and let L be a parallel line h units away from Lc.m.. The Parallel Axis Theorem says that the moments of inertia Ic.m. and IL of the body about Lc.m. and L satisfy the equation IL = Ic.m. + mh 2.

(2)

As in the two-dimensional case, the theorem gives a quick way to calculate one moment when the other moment and the mass are known. 35. Proof of the Parallel Axis Theorem a. Show that the first moment of a body in space about any plane through the body’s center of mass is zero. (Hint: Place the body’s center of mass at the origin and let the plane be the yz-plane. What does the formula x = Myz >M then tell you?)

P(r, u, z)

D x

(h, 0, 0)

b. To prove the Parallel Axis Theorem, place the body with its center of mass at the origin, with the line Lc.m. along the z-axis and the line L perpendicular to the xy-plane at the point (h, 0, 0). Let D be the region of space occupied by the body. Then, in the notation of the figure, IL =

9

ƒ v - hi ƒ 2 dm.

D

Expand the integrand in this integral and complete the proof. 36. The moment of inertia about a diameter of a solid sphere of constant density and radius a is s2>5dma 2, where m is the mass of the sphere. Find the moment of inertia about a line tangent to the sphere. 37. The moment of inertia of the solid in Exercise 21 about the z-axis is Iz = abcsa 2 + b 2 d>3. a. Use Equation (2) to find the moment of inertia of the solid about the line parallel to the z-axis through the solid’s center of mass. b. Use Equation (2) and the result in part (a) to find the moment of inertia of the solid about the line x = 0, y = 2b. 38. If a = b = 6 and c = 4, the moment of inertia of the solid wedge in Exercise 22 about the x-axis is Ix = 208. Find the moment of inertia of the wedge about the line y = 4, z = -4>3 (the edge of the wedge’s narrow end).

Integration in Cylindrical Coordinates We obtain cylindrical coordinates for space by combining polar coordinates in the xy-plane with the usual z-axis. This assigns to every point in space one or more coordinate triples of the form sr, u, zd, as shown in Figure 14.42.

z

0 r

y x

y

c.m.

When a calculation in physics, engineering, or geometry involves a cylinder, cone, or sphere, we can often simplify our work by using cylindrical or spherical coordinates, which are introduced in this section. The procedure for transforming to these coordinates and evaluating the resulting triple integrals is similar to the transformation to polar coordinates in the plane studied in Section 14.4.

z

u

v

(x, y, z)

2 hi

Triple Integrals in Cylindrical and Spherical Coordinates

14.7

x

v 5 xi 1 yj hi

y

FIGURE 14.42 The cylindrical coordinates of a point in space are r, u, and z.

DEFINITION Cylindrical coordinates represent a point P in space by ordered triples sr, u, zd in which 1. r and u are polar coordinates for the vertical projection of P on the xy-plane 2. z is the rectangular vertical coordinate.

14.7

Triple Integrals in Cylindrical and Spherical Coordinates

803

The values of x, y, r, and u in rectangular and cylindrical coordinates are related by the usual equations.

Equations Relating Rectangular (x, y, z) and Cylindrical sr, U, zd Coordinates x = r cos u,

y = r sin u,

r = x + y , 2

z u 5 u0 , r and z vary z 5 z 0, r and u vary z0

2

u0

r 5 a, u and z vary

FIGURE 14.43 Constant-coordinate equations in cylindrical coordinates yield cylinders and planes.

tan u = y>x

r = 4 p u = 3 z = 2.

y

x

z = z,

In cylindrical coordinates, the equation r = a describes not just a circle in the xy-plane but an entire cylinder about the z-axis (Figure 14.43). The z-axis is given by r = 0. The equation u = u0 describes the plane that contains the z-axis and makes an angle u0 with the positive x-axis. And, just as in rectangular coordinates, the equation z = z0 describes a plane perpendicular to the z-axis. Cylindrical coordinates are good for describing cylinders whose axes run along the z-axis and planes that either contain the z-axis or lie perpendicular to the z-axis. Surfaces like these have equations of constant coordinate value:

0 a

2

Cylinder, radius 4, axis the z-axis Plane containing the z-axis Plane perpendicular to the z-axis

When computing triple integrals over a region D in cylindrical coordinates, we partition the region into n small cylindrical wedges, rather than into rectangular boxes. In the kth cylindrical wedge, r, u and z change by ¢rk , ¢uk , and ¢zk , and the largest of these numbers among all the cylindrical wedges is called the norm of the partition. We define the triple integral as a limit of Riemann sums using these wedges. The volume of such a cylindrical wedge ¢Vk is obtained by taking the area ¢Ak of its base in the ru-plane and multiplying by the height ¢z (Figure 14.44). For a point srk , uk , zk d in the center of the kth wedge, we calculated in polar coordinates that ¢Ak = rk ¢rk ¢uk . So ¢Vk = ¢zk rk ¢rk ¢uk and a Riemann sum for ƒ over D has the form n

Sn = a ƒsrk , uk , zk d ¢zk rk ¢rk ¢uk . k=1

The triple integral of a function ƒ over D is obtained by taking a limit of such Riemann sums with partitions whose norms approach zero: Volume Differential in Cylindrical Coordinates dV = dz r dr du

lim Sn =

n: q

9 D

ƒ dV =

9

ƒ dz r dr du.

D

Triple integrals in cylindrical coordinates are then evaluated as iterated integrals, as in the following example.

z r Δr Δu

r Δu

Δz Δu r

Δr

FIGURE 14.44 In cylindrical coordinates the volume of the wedge is approximated by the product ¢V = ¢z r ¢r ¢u.

EXAMPLE 1

Find the limits of integration in cylindrical coordinates for integrating a function ƒsr, u, zd over the region D bounded below by the plane z = 0, laterally by the circular cylinder x 2 + s y - 1d2 = 1, and above by the paraboloid z = x 2 + y 2. Solution The base of D is also the region’s projection R on the xy-plane. The boundary of R is the circle x 2 + s y - 1d2 = 1. Its polar coordinate equation is

x 2 + s y - 1d2 = 1 x + y 2 - 2y + 1 = 1 r 2 - 2r sin u = 0 r = 2 sin u. 2

804

Chapter 14: Multiple Integrals

The region is sketched in Figure 14.45. We find the limits of integration, starting with the z-limits. A line M through a typical point sr, ud in R parallel to the z-axis enters D at z = 0 and leaves at z = x 2 + y 2 = r 2. Next we find the r-limits of integration. A ray L through sr, ud from the origin enters R at r = 0 and leaves at r = 2 sin u. Finally we find the u-limits of integration. As L sweeps across R, the angle u it makes with the positive x-axis runs from u = 0 to u = p. The integral is

z

Top Cartesian: z 5 x2 1 y2 Cylindrical: z 5 r2

M

D

p

9

ƒsr, u, zd dV =

D

y

2 u

x

R

(r, u)

L

Cartesian: x2 1 (y 2 1)2 5 1 Polar: r 5 2 sin u

r2

2 sin u

L0 L0

L0

ƒsr, u, zd dz r dr du.

Example 1 illustrates a good procedure for finding limits of integration in cylindrical coordinates. The procedure is summarized as follows.

How to Integrate in Cylindrical Coordinates To evaluate

FIGURE 14.45 Finding the limits of integration for evaluating an integral in cylindrical coordinates (Example 1).

9

ƒsr, u, zd dV

D

over a region D in space in cylindrical coordinates, integrating first with respect to z, then with respect to r, and finally with respect to u, take the following steps. 1.

Sketch. Sketch the region D along with its projection R on the xy-plane. Label the surfaces and curves that bound D and R. z z 5 g2(r, u)

D r 5 h1(u)

z 5 g1(r, u) y

x

R r 5 h2(u)

2.

Find the z-limits of integration. Draw a line M through a typical point sr, ud of R parallel to the z-axis. As z increases, M enters D at z = g1sr, ud and leaves at z = g2sr, ud. These are the z-limits of integration. z M

z 5 g2(r, u)

D r 5 h1(u)

z 5 g1(r, u)

y x

R (r, u) r 5 h2(u)

14.7

3.

Triple Integrals in Cylindrical and Spherical Coordinates

805

Find the r-limits of integration. Draw a ray L through sr, ud from the origin. The ray enters R at r = h1sud and leaves at r = h2sud. These are the r-limits of integration. z M z 5 g2(r, u)

D z 5 g1(r, u)

a u

x r 5 h1(u)

b y R (r, u) u5b

u5a L

4.

Find the u-limits of integration. As L sweeps across R, the angle u it makes with the positive x-axis runs from u = a to u = b. These are the u-limits of integration. The integral is

9

ƒsr, u, zd dV =

D

z

4

z 5 x 2 1 y2 5 r2

M

u=b

r = h2sud

z = g2sr, ud

Lu = a Lr = h1sud Lz = g1sr, ud

ƒsr, u, zd dz r dr du.

Find the centroid sd = 1d of the solid enclosed by the cylinder x 2 + y 2 = 4, bounded above by the paraboloid z = x 2 + y 2, and bounded below by the xy-plane.

EXAMPLE 2

We sketch the solid, bounded above by the paraboloid z = r 2 and below by the plane z = 0 (Figure 14.46). Its base R is the disk 0 … r … 2 in the xy-plane. The solid’s centroid sx, y, zd lies on its axis of symmetry, here the z-axis. This makes x = y = 0 . To find z , we divide the first moment Mxy by the mass M. To find the limits of integration for the mass and moment integrals, we continue with the four basic steps. We completed our initial sketch. The remaining steps give the limits of integration. The z-limits. A line M through a typical point sr, ud in the base parallel to the z-axis enters the solid at z = 0 and leaves at z = r 2. The r-limits. A ray L through sr, ud from the origin enters R at r = 0 and leaves at r = 2. The u-limits. As L sweeps over the base like a clock hand, the angle u it makes with the positive x-axis runs from u = 0 to u = 2p. The value of Mxy is Solution

x2 1 y2 5 4 r52 x

r 5 h2(u)

y

(r, u) L

FIGURE 14.46 Example 2 shows how to find the centroid of this solid.

Mxy

2p

2

2p

2

r2

2p

r2

2

z2 = z dz r dr du = c d r dr du L0 L0 L0 L0 L0 2 0 =

r5 dr du = L0 L0 2 L0

2p

c

2

r6 d du = 12 0 L0

2p

16 32p du = . 3 3

The value of M is 2p

M =

L0 L0 L0 2p

=

r2

2

L0 L0

2p

dz r dr du =

2

2p

r 3 dr du =

L0

L0 L0 c

4 2

2

r2

cz d r dr du 0 2p

r d du = 4 du = 8p. 4 0 L0

806

Chapter 14: Multiple Integrals

Therefore,

z

z =

P(r, f, u) f

and the centroid is (0, 0, 4 >3). Notice that the centroid lies outside the solid.

r z 5 r cos f

0 x

u

Mxy 32p 1 4 = = , M 3 8p 3

Spherical Coordinates and Integration

r

y

y x

FIGURE 14.47 The spherical coordinates r, f, and u and their relation to x, y, z, and r.

Spherical coordinates locate points in space with two angles and one distance, as shown 1 in Figure 14.47. The first coordinate, r = ƒ OP ƒ , is the point’s distance from the origin. 1 Unlike r, the variable r is never negative. The second coordinate, f, is the angle OP makes with the positive z-axis. It is required to lie in the interval [0, p]. The third coordinate is the angle u as measured in cylindrical coordinates.

DEFINITION Spherical coordinates represent a point P in space by ordered triples sr, f, ud in which 1. r is the distance from P to the origin. 1 2. f is the angle OP makes with the positive z-axis s0 … f … pd. 3. u is the angle from cylindrical coordinates s0 … u … 2pd. z f 5 f0, r and u vary

f0

u0

P(a, f0, u0)

y

On maps of the Earth, u is related to the meridian of a point on the Earth and f to its latitude, while r is related to elevation above the Earth’s surface. The equation r = a describes the sphere of radius a centered at the origin (Figure 14.48). The equation f = f0 describes a single cone whose vertex lies at the origin and whose axis lies along the z-axis. (We broaden our interpretation to include the xy-plane as the cone f = p>2.) If f0 is greater than p>2, the cone f = f0 opens downward. The equation u = u0 describes the half-plane that contains the z-axis and makes an angle u0 with the positive x-axis.

Equations Relating Spherical Coordinates to Cartesian and Cylindrical Coordinates

x r 5 a, f and u vary

x = r cos u = r sin f cos u,

z = r cos f,

y = r sin u = r sin f sin u,

(1)

r = 2x 2 + y 2 + z 2 = 2r 2 + z 2.

u 5 u0, r and f vary

FIGURE 14.48 Constant-coordinate equations in spherical coordinates yield spheres, single cones, and half-planes.

r = r sin f,

EXAMPLE 3 Solution

Find a spherical coordinate equation for the sphere x 2 + y 2 + sz - 1d2 = 1.

We use Equations (1) to substitute for x, y, and z: x 2 + y 2 + sz - 1d2 = 1 r2 sin2 f cos2 u + r2 sin2 f sin2 u + sr cos f - 1d2 = 1

Eqs. (1)

r2 sin2 fscos2 u + sin2 ud + r2 cos2 f - 2r cos f + 1 = 1 (''')'''* 1

r2ssin2 f + cos2 fd = 2r cos f (''')'''* 1

r2 = 2r cos f r = 2 cos f .

r 7 0

14.7

1

2

y2

1)2

1 (z 2 51 r 5 2 cos f

EXAMPLE 4 1

807

The angle f varies from 0 at the north pole of the sphere to p>2 at the south pole; the angle u does not appear in the expression for r, reflecting the symmetry about the z-axis (see Figure 14.49).

z x2

Triple Integrals in Cylindrical and Spherical Coordinates

Find a spherical coordinate equation for the cone z = 2x 2 + y 2.

Use geometry. The cone is symmetric with respect to the z-axis and cuts the first quadrant of the yz-plane along the line z = y. The angle between the cone and the positive z-axis is therefore p>4 radians. The cone consists of the points whose spherical coordinates have f equal to p>4, so its equation is f = p>4. (See Figure 14.50.) Solution 1

f

r

Use algebra. If we use Equations (1) to substitute for x, y, and z we obtain the

Solution 2

y

same result:

x

z = 2x 2 + y 2

FIGURE 14.49 The sphere in Example 3.

z

r cos f = 2r2 sin2 f

Example 3

r cos f = r sin f

r 7 0, sin f Ú 0

cos f = sin f p f = . 4

4

x 2

z 4

y2

Spherical coordinates are useful for describing spheres centered at the origin, half-planes hinged along the z-axis, and cones whose vertices lie at the origin and whose axes lie along the z-axis. Surfaces like these have equations of constant coordinate value: r = 4

Sphere, radius 4, center at origin

p 3 p u = . 3

Cone opening up from the origin, making an angle of p>3 radians with the positive z-axis

f = y

x

FIGURE 14.50 The cone in Example 4.

Half-plane, hinged along the z-axis, making an angle of p>3 radians with the positive x-axis

When computing triple integrals over a region D in spherical coordinates, we partition the region into n spherical wedges. The size of the kth spherical wedge, which contains a point srk , fk , uk d, is given by the changes ¢rk , ¢fk , and ¢uk in r, f, and u. Such a spherical wedge has one edge a circular arc of length rk ¢fk , another edge a circular arc of length rk sin fk ¢uk , and thickness ¢rk. The spherical wedge closely approximates a cube of these dimensions when ¢rk , ¢fk , and ¢uk are all small (Figure 14.51). It can be shown that the volume of this spherical wedge ¢Vk is ¢Vk = rk2 sin fk ¢rk ¢fk ¢uk for srk, fk, uk d a point chosen inside the wedge. The corresponding Riemann sum for a function ƒsr, f, ud is

Volume Differential in Spherical Coordinates dV = r2 sin f dr df du

z r sin f r sin f Δu

rΔf

0 … f … p

n

Sn = a ƒsrk, fk, uk d rk2 sin fk ¢rk ¢fk ¢uk . k=1

As the norm of a partition approaches zero, and the spherical wedges get smaller, the Riemann sums have a limit when ƒ is continuous: Of r Δr u x

u 1 Δu

FIGURE 14.51 In spherical coordinates dV = dr # r df # r sin f du = r2 sin f dr df du.

lim Sn =

n: q

y

9

ƒsr, f, ud dV =

D

9

ƒsr, f, ud r2 sin f dr df du.

D

In spherical coordinates, we have dV = r2 sin f dr df du. To evaluate integrals in spherical coordinates, we usually integrate first with respect to r. The procedure for finding the limits of integration is as follows. We restrict our attention to integrating over domains that are solids of revolution about the z-axis (or portions thereof ) and for which the limits for u and f are constant.

808

Chapter 14: Multiple Integrals

How to Integrate in Spherical Coordinates To evaluate 9

ƒsr, f, ud dV

D

over a region D in space in spherical coordinates, integrating first with respect to r, then with respect to f, and finally with respect to u, take the following steps. 1.

Sketch. Sketch the region D along with its projection R on the xy-plane. Label the surfaces that bound D. z

r 5 g2(f, u)

D r 5 g1(f, u)

R

y

x

2.

Find the r-limits of integration. Draw a ray M from the origin through D making an angle f with the positive z-axis. Also draw the projection of M on the xy-plane (call the projection L). The ray L makes an angle u with the positive x-axis. As r increases, M enters D at r = g1sf, ud and leaves at r = g2sf, ud. These are the r-limits of integration. z

fmax

fmin

f

M r 5 g2(f, u)

D r 5 g1(f, u)

u5a x

3.

u5b y

R θ L

Find the f-limits of integration. For any given u, the angle f that M makes with the z-axis runs from f = fmin to f = fmax. These are the f -limits of integration.

14.7

4.

Triple Integrals in Cylindrical and Spherical Coordinates

809

Find the u-limits of integration. The ray L sweeps over R as u runs from a to b. These are the u-limits of integration. The integral is

9

ƒsr, f, ud dV =

D

u=b

f = fmax

r = g2sf, ud

Lu = a Lf = fmin Lr = g1sf, ud

ƒsr, f, ud r2 sin f dr df du.

EXAMPLE 5

Find the volume of the “ice cream cone” D cut from the solid sphere r … 1 by the cone f = p>3.

z M Sphere r 5 1

D

The volume is V = 7D r2 sin f dr df du, the integral of ƒsr, f, ud = 1

Solution

Cone f 5 p 3 R u x

FIGURE 14.52 Example 5.

L

y

The ice cream cone in

over D. To find the limits of integration for evaluating the integral, we begin by sketching D and its projection R on the xy-plane (Figure 14.52). The r-limits of integration. We draw a ray M from the origin through D making an angle f with the positive z-axis. We also draw L, the projection of M on the xy-plane, along with the angle u that L makes with the positive x-axis. Ray M enters D at r = 0 and leaves at r = 1. The f-limits of integration. The cone f = p>3 makes an angle of p>3 with the positive z-axis. For any given u, the angle f can run from f = 0 to f = p>3. The u-limits of integration. The ray L sweeps over R as u runs from 0 to 2p. The volume is p>3

2p

V =

r2 sin f dr df du = L0 L0 9 p>3

2p

L0 L0 2p

=

L0

c-

c

r2 sin f dr df du

L0

D

=

1

2p p>3 r3 1 1 d sin f df du = sin f df du 3 0 3 L0 L0 p>3

1 cos f d 3 0

2p

du =

L0

a-

p 1 1 1 + b du = s2pd = . 6 3 6 3

A solid of constant density d = 1 occupies the region D in Example 5. Find the solid’s moment of inertia about the z-axis.

EXAMPLE 6

In rectangular coordinates, the moment is

Solution

Iz =

sx 2 + y 2 d dV. 9

In spherical coordinates, x 2 + y 2 = sr sin f cos ud2 + sr sin f sin ud2 = r2 sin2 f. Hence, Iz =

sr2 sin2 fd r2 sin f dr df du = r4 sin3 f dr df du. 9 9

For the region in Example 5, this becomes 2p

Iz =

p>3

L0 L0 2p

L0

=

1 5L0 L0

=

1 5L0

2p

1

L0 L0

p>3

a-

p>3

2p

r4 sin3 f dr df du =

s1 - cos2 fd sin f df du =

1 5L0

1 1 1 1 + 1 + - b du = 5L0 2 24 3

2p

2p

c

r5 1 3 d sin f df du 5 0 c-cos f +

cos3 f p>3 d du 3 0

5 p 1 du = s2pd = . 24 24 12

810

Chapter 14: Multiple Integrals

Coordinate Conversion Formulas CYLINDRICAL TO RECTANGULAR

SPHERICAL TO RECTANGULAR

SPHERICAL TO CYLINDRICAL

x = r cos u y = r sin u z = z

x = r sin f cos u y = r sin f sin u z = r cos f

r = r sin f z = r cos f u = u

Corresponding formulas for dV in triple integrals: dV = dx dy dz = dz r dr du = r2 sin f dr df du In the next section we offer a more general procedure for determining dV in cylindrical and spherical coordinates. The results, of course, will be the same.

Exercises 14.7 Evaluating Integrals in Cylindrical Coordinates Evaluate the cylindrical coordinate integrals in Exercises 1–6. 2p

1.

2p

5.

1>22 - r

1

218 - r 2

3

dz r dr du

L0 L0 Lr >3 p

dz r dr du 4.

u>p

L0 L0

324 - r 2

L-24 - r 2

z dz r dr du

7.

r dr dz du

L0 L0 L0 1

9.

1 3

L0 L0 2

2z

2p

L-1L0 L0

1 + cos u

4r dr du dz

2p

sr cos u + z d r du dr dz 2

L0 24 - r

8.

2

2

2

2p

sr sin u + 1d r du dz dr L0 Lr - 2 L0 11. Let D be the region bounded below by the plane z = 0, above by the sphere x 2 + y 2 + z 2 = 4, and on the sides by the cylinder x 2 + y 2 = 1. Set up the triple integrals in cylindrical coordinates that give the volume of D using the following orders of integration. 10.

a. dz dr du

b. dr dz du

ƒsr, u, zd dz r dr du

as an iterated integral over the region that is bounded below by the plane z = 0, on the side by the cylinder r = cos u, and on top by the paraboloid z = 3r 2.

sr 2 sin2 u + z 2 d dz r dr du

z>3

c. du dz dr

Finding Iterated Integrals in Cylindrical Coordinates 13. Give the limits of integration for evaluating the integral 9

Changing the Order of Integration in Cylindrical Coordinates The integrals we have seen so far suggest that there are preferred orders of integration for cylindrical coordinates, but other orders usually work well and are occasionally easier to evaluate. Evaluate the integrals in Exercises 7–10. 3

b. dr dz du

2

1>2

L0 L0 L-1>2

2p

a. dz dr du

3 dz r dr du

L0 L0 Lr 2p

6.

3 + 24r 2

L0 1

2.

2

u>2p

L0 L0

2p

dz r dr du

L0 L0 Lr 2p

3.

22 - r 2

1

integrals in cylindrical coordinates that give the volume of D using the following orders of integration.

14. Convert the integral 1

21 - y2

L-1L0

L0

sx 2 + y 2 d dz dx dy

to an equivalent integral in cylindrical coordinates and evaluate the result. In Exercises 15–20, set up the iterated integral for evaluating 7D ƒsr, u, zd dz r dr du over the given region D. 15. D is the right circular cylinder whose base is the circle r = 2 sin u in the xy-plane and whose top lies in the plane z = 4 - y. z z4y

c. du dz dr

12. Let D be the region bounded below by the cone z = 2x 2 + y 2 and above by the paraboloid z = 2 - x 2 - y 2. Set up the triple

x

y x

r 2 sin

14.7

Triple Integrals in Cylindrical and Spherical Coordinates

16. D is the right circular cylinder whose base is the circle r = 3 cos u and whose top lies in the plane z = 5 - x.

811

z z2x

z

2 z552x

y 1 r 5 3 cos u

y

x

x

yx

17. D is the solid right cylinder whose base is the region in the xy-plane that lies inside the cardioid r = 1 + cos u and outside the circle r = 1 and whose top lies in the plane z = 4. z

Evaluating Integrals in Spherical Coordinates Evaluate the spherical coordinate integrals in Exercises 21–26. p

4

21.

p

y

p>4

L0 L0

r1 x

r 1 cos

18. D is the solid right cylinder whose base is the region between the circles r = cos u and r = 2 cos u and whose top lies in the plane z = 3 - y. z

26.

y

r cos r 2 cos

p

1

p>3

5r3 sin3 f dr df du 2

3r2 sin f dr df du Lsec f

L0 L0

p>4

L0 L0

2

sec f

sr cos fd r2 sin f dr df du

L0

z2y

r3 sin 2f df du dr

L0 L-p Lp>4 2 csc f

2p

Lp>6 Lcsc f L0 1

29.

p>2

0

p>3

28.

z 2

r2 sin f dr df du

Changing the Order of Integration in Spherical Coordinates The previous integrals suggest there are preferred orders of integration for spherical coordinates, but other orders give the same value and are occasionally easier to evaluate. Evaluate the integrals in Exercises 27–30. 27.

19. D is the prism whose base is the triangle in the xy-plane bounded by the x-axis and the lines y = x and x = 1 and whose top lies in the plane z = 2 - y.

s1 - cos fd>2

L0 L0

2p

z3y

x

L0 2p

25.

sr cos fd r2 sin f dr df du

L0 L0 L0 3p>2

24.

2

L0 p

2p

23.

r2 sin f dr df du

L0 L0 L0 2p

22.

2 sin f

p

p>4

L0 L0 L0 p>2

r2 sin f du dr df

12r sin3 f df du dr

p/2

2

5r4 sin3 f dr du df Lp>6 L-p/2 Lcsc f 31. Let D be the region in Exercise 11. Set up the triple integrals in spherical coordinates that give the volume of D using the following orders of integration. 30.

y

a. dr df du 1 x

yx

20. D is the prism whose base is the triangle in the xy-plane bounded by the y-axis and the lines y = x and y = 1 and whose top lies in the plane z = 2 - x.

b. df dr du

32. Let D be the region bounded below by the cone z = 2x 2 + y 2 and above by the plane z = 1. Set up the triple integrals in spherical coordinates that give the volume of D using the following orders of integration. a. dr df du

b. df dr du

812

Chapter 14: Multiple Integrals

Finding Iterated Integrals in Spherical Coordinates In Exercises 33–38, (a) find the spherical coordinate limits for the integral that calculates the volume of the given solid and then (b) evaluate the integral. 33. The solid between the sphere r = cos f and the hemisphere r = 2, z Ú 0 z r 5 cos f

r52

2

volume of D as an iterated triple integral in (a) cylindrical and (b) spherical coordinates. Then (c) find V. 41. Let D be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center of the sphere. Express the volume of D as an iterated triple integral in (a) spherical, (b) cylindrical, and (c) rectangular coordinates. Then (d) find the volume by evaluating one of the three triple integrals. 42. Express the moment of inertia Iz of the solid hemisphere x 2 + y 2 + z 2 … 1, z Ú 0, as an iterated integral in (a) cylindrical and (b) spherical coordinates. Then (c) find Iz. Volumes Find the volumes of the solids in Exercises 43–48. 43.

x

2

2

44. z

z

y

z 4 4 (x 2 y 2)

34. The solid bounded below by the hemisphere r = 1, z Ú 0, and above by the cardioid of revolution r = 1 + cos f

z512r

21

21

z r 5 1 1 cos f

r51

1

1

x

y

y

x

z 5 2 1 2 r 2

z (x 2 y 2 ) 2 1

45.

z –y

46.

z

z

z x 2 y 2

y

x

35. The solid enclosed by the cardioid of revolution r = 1 - cos f 36. The upper portion cut from the solid in Exercise 35 by the xy-plane

r 3 cos

37. The solid bounded below by the sphere r = 2 cos f and above by the cone z = 2x 2 + y 2

x

z

z 5 x2 1 y2

y r –3 cos y

x

47.

48. z

z

z 1 x 2 y 2

z 5 31 2 x2 2 y2

r 5 2 cos f x

y x

38. The solid bounded below by the xy-plane, on the sides by the sphere r = 2, and above by the cone f = p>3

y x

y

r 5 cos u

z f5

p 3

49. Sphere and cones Find the volume of the portion of the solid sphere r … a that lies between the cones f = p>3 and f = 2p>3. r52

x

r sin

y

50. Sphere and half-planes Find the volume of the region cut from the solid sphere r … a by the half-planes u = 0 and u = p>6 in the first octant. 51. Sphere and plane Find the volume of the smaller region cut from the solid sphere r … 2 by the plane z = 1.

Finding Triple Integrals 39. Set up triple integrals for the volume of the sphere r = 2 in (a) spherical, (b) cylindrical, and (c) rectangular coordinates. 40. Let D be the region in the first octant that is bounded below by the cone f = p>4 and above by the sphere r = 3. Express the

52. Cone and planes Find the volume of the solid enclosed by the cone z = 2x 2 + y 2 between the planes z = 1 and z = 2. 53. Cylinder and paraboloid Find the volume of the region bounded below by the plane z = 0, laterally by the cylinder x 2 + y 2 = 1, and above by the paraboloid z = x 2 + y 2.

14.7

Triple Integrals in Cylindrical and Spherical Coordinates

813

54. Cylinder and paraboloids Find the volume of the region bounded below by the paraboloid z = x 2 + y 2, laterally by the cylinder x 2 + y 2 = 1, and above by the paraboloid z = x 2 + y 2 + 1.

73. Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius 1 and height 1 about an axis through the vertex parallel to the base. (Take d = 1.)

55. Cylinder and cones Find the volume of the solid cut from the thick-walled cylinder 1 … x 2 + y 2 … 2 by the cones z = ; 2x 2 + y 2.

74. Moment of inertia of solid sphere Find the moment of inertia of a solid sphere of radius a about a diameter. (Take d = 1.)

56. Sphere and cylinder Find the volume of the region that lies inside the sphere x 2 + y 2 + z 2 = 2 and outside the cylinder x 2 + y 2 = 1. 57. Cylinder and planes Find the volume of the region enclosed by the cylinder x 2 + y 2 = 4 and the planes z = 0 and y + z = 4. 58. Cylinder and planes Find the volume of the region enclosed by the cylinder x 2 + y 2 = 4 and the planes z = 0 and x + y + z = 4. 59. Region trapped by paraboloids Find the volume of the region bounded above by the paraboloid z = 5 - x 2 - y 2 and below by the paraboloid z = 4x 2 + 4y 2. 60. Paraboloid and cylinder Find the volume of the region bounded above by the paraboloid z = 9 - x 2 - y 2, below by the xy-plane, and lying outside the cylinder x 2 + y 2 = 1. 61. Cylinder and sphere Find the volume of the region cut from the solid cylinder x 2 + y 2 … 1 by the sphere x 2 + y 2 + z 2 = 4. 62. Sphere and paraboloid Find the volume of the region bounded above by the sphere x 2 + y 2 + z 2 = 2 and below by the paraboloid z = x 2 + y 2. Average Values 63. Find the average value of the function ƒsr, u, zd = r over the region bounded by the cylinder r = 1 between the planes z = -1 and z = 1. 64. Find the average value of the function ƒsr, u, zd = r over the solid ball bounded by the sphere r 2 + z 2 = 1. (This is the sphere x 2 + y 2 + z 2 = 1.) 65. Find the average value of the function ƒsr, f, ud = r over the solid ball r … 1. 66. Find the average value of the function ƒsr, f, ud = r cos f over the solid upper ball r … 1, 0 … f … p>2. Masses, Moments, and Centroids 67. Center of mass A solid of constant density is bounded below by the plane z = 0, above by the cone z = r, r Ú 0, and on the sides by the cylinder r = 1. Find the center of mass. 68. Centroid Find the centroid of the region in the first octant that is bounded above by the cone z = 2x 2 + y 2, below by the plane z = 0, and on the sides by the cylinder x 2 + y 2 = 4 and the planes x = 0 and y = 0. 69. Centroid Find the centroid of the solid in Exercise 38.

75. Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius a and height h about its axis. (Hint: Place the cone with its vertex at the origin and its axis along the z-axis.) 76. Variable density A solid is bounded on the top by the paraboloid z = r 2, on the bottom by the plane z = 0, and on the sides by the cylinder r = 1. Find the center of mass and the moment of inertia about the z-axis if the density is a. dsr, u, zd = z

b. dsr, u, zd = r.

77. Variable density A solid is bounded below by the cone z = 2x 2 + y 2 and above by the plane z = 1. Find the center of mass and the moment of inertia about the z-axis if the density is a. dsr, u, zd = z

b. dsr, u, zd = z 2.

78. Variable density A solid ball is bounded by the sphere r = a. Find the moment of inertia about the z-axis if the density is a. dsr, f, ud = r2

b. dsr, f, ud = r = r sin f.

79. Centroid of solid semiellipsoid Show that the centroid of the solid semiellipsoid of revolution sr 2>a 2 d + sz 2>h 2 d … 1, z Ú 0, lies on the z-axis three-eighths of the way from the base to the top. The special case h = a gives a solid hemisphere. Thus, the centroid of a solid hemisphere lies on the axis of symmetry threeeighths of the way from the base to the top. 80. Centroid of solid cone Show that the centroid of a solid right circular cone is one-fourth of the way from the base to the vertex. (In general, the centroid of a solid cone or pyramid is one-fourth of the way from the centroid of the base to the vertex.) 81. Density of center of a planet A planet is in the shape of a sphere of radius R and total mass M with spherically symmetric density distribution that increases linearly as one approaches its center. What is the density at the center of this planet if the density at its edge (surface) is taken to be zero? 82. Mass of planet’s atmosphere A spherical planet of radius R has an atmosphere whose density is m = m0 e -ch, where h is the altitude above the surface of the planet, m0 is the density at sea level, and c is a positive constant. Find the mass of the planet’s atmosphere. Theory and Examples 83. Vertical planes in cylindrical coordinates a. Show that planes perpendicular to the x-axis have equations of the form r = a sec u in cylindrical coordinates. b. Show that planes perpendicular to the y-axis have equations of the form r = b csc u.

70. Centroid Find the centroid of the solid bounded above by the sphere r = a and below by the cone f = p>4.

84. (Continuation of Exercise 83. ) Find an equation of the form r = ƒsud in cylindrical coordinates for the plane ax + by = c, c Z 0.

71. Centroid Find the centroid of the region that is bounded above by the surface z = 2r, on the sides by the cylinder r = 4, and below by the xy-plane.

85. Symmetry What symmetry will you find in a surface that has an equation of the form r = ƒszd in cylindrical coordinates? Give reasons for your answer.

72. Centroid Find the centroid of the region cut from the solid ball r 2 + z 2 … 1 by the half-planes u = -p>3, r Ú 0, and u = p>3, r Ú 0.

86. Symmetry What symmetry will you find in a surface that has an equation of the form r = ƒsfd in spherical coordinates? Give reasons for your answer.

814

Chapter 14: Multiple Integrals

Substitutions in Multiple Integrals

14.8

The goal of this section is to introduce you to the ideas involved in coordinate transformations. You will see how to evaluate multiple integrals by substitution in order to replace complicated integrals by ones that are easier to evaluate. Substitutions accomplish this by simplifying the integrand, the limits of integration, or both. A thorough discussion of multivariable transformations and substitutions, and the Jacobian, is best left to a more advanced course following a study of linear algebra.

v

(u, v) G

Substitutions in Double Integrals u

0 Cartesian uv-plane x g(u, v) y h(u, v)

The polar coordinate substitution of Section 14.4 is a special case of a more general substitution method for double integrals, a method that pictures changes in variables as transformations of regions. Suppose that a region G in the uy-plane is transformed one-to-one into the region R in the xy-plane by equations of the form x = gsu, yd,

y

(x, y)

R 0

x

as suggested in Figure 14.53. We call R the image of G under the transformation, and G the preimage of R. Any function ƒ(x, y) defined on R can be thought of as a function ƒ(g(u, y), h(u, y)) defined on G as well. How is the integral of ƒ(x, y) over R related to the integral of ƒ(g(u, y), h(u, y)) over G? The answer is: If g, h, and ƒ have continuous partial derivatives and J(u, y) (to be discussed in a moment) is zero only at isolated points, if at all, then

Cartesian xy-plane

6 FIGURE 14.53 The equations x = gsu, yd and y = hsu, yd allow us to change an integral over a region R in the xy-plane into an integral over a region G in the uy-plane by using Equation (1).

y = hsu, yd,

ƒsx, yd dx dy =

R

6

ƒsgsu, yd, hsu, ydd ƒ Jsu, yd ƒ du dy.

(1)

G

The factor J(u, y), whose absolute value appears in Equation (1), is the Jacobian of the coordinate transformation, named after German mathematician Carl Jacobi. It measures how much the transformation is expanding or contracting the area around a point in G as G is transformed into R.

DEFINITION The Jacobian determinant or Jacobian of the coordinate transformation x = gsu, yd, y = hsu, yd is Jsu, yd = 4

0x 0u

0x 0y

0y 0u

0y 0y

0y 0y 0x 4 = 0x . 0u 0y 0u 0y

(2)

The Jacobian can also be denoted by Jsu, yd = HISTORICAL BIOGRAPHY Carl Gustav Jacob Jacobi (1804–1851)

0sx, yd 0su, yd

to help us remember how the determinant in Equation (2) is constructed from the partial derivatives of x and y. The derivation of Equation (1) is intricate and properly belongs to a course in advanced calculus. We do not give the derivation here. Find the Jacobian for the polar coordinate transformation x = r cos u, y = r sin u, and use Equation (1) to write the Cartesian integral 4R ƒ(x, y) dx dy as a polar integral.

EXAMPLE 1

14.8

2 G

Jsr, ud = 4 r

1

0x 0r

0x 0u 4 = ` cos u 0y sin u 0u

0y 0r

x r cos y r sin

6

ƒsx, yd dx dy =

R

y

6

ƒsr cos u, r sin ud r dr du.

(3)

G

This is the same formula we derived independently using a geometric argument for polar area in Section 14.4. Notice that the integral on the right-hand side of Equation (3) is not the integral of ƒsr cos u, r sin ud over a region in the polar coordinate plane. It is the integral of the product of ƒsr cos u, r sin ud and r over a region G in the Cartesian ru-plane.

2

1 R

0 0

-r sin u ` = rscos2 u + sin2 ud = r. r cos u

Since we assume r Ú 0 when integrating in polar coordinates, ƒ J(r, u) ƒ = ƒ r ƒ = r, so that Equation (1) gives

Cartesian r -plane

815

Solution Figure 14.54 shows how the equations x = r cos u, y = r sin u transform the rectangle G: 0 … r … 1, 0 … u … p>2, into the quarter circle R bounded by x 2 + y 2 = 1 in the first quadrant of the xy-plane. For polar coordinates, we have r and u in place of u and y. With x = r cos u and y = r sin u, the Jacobian is

0

Substitutions in Multiple Integrals

1

x

Here is an example of a substitution in which the image of a rectangle under the coordinate transformation is a trapezoid. Transformations like this one are called linear transformations.

Cartesian xy-plane

EXAMPLE 2

Evaluate

FIGURE 14.54 The equations x = r cos u, y = r sin u transform G into R.

4

x = sy>2d + 1

L0 Lx = y>2

2x - y dx dy 2

by applying the transformation u =

2x - y , 2

y =

y 2

(4)

and integrating over an appropriate region in the uy-plane. Solution We sketch the region R of integration in the xy-plane and identify its boundaries (Figure 14.55).

y

v 2

u0

0

v2

1

y 2x

xuv y 2v

u1

G

v0

y4

4

u

R

0

1 y0

y 2x 2

x

FIGURE 14.55 The equations x = u + y and y = 2y transform G into R. Reversing the transformation by the equations u = s2x - yd>2 and y = y>2 transforms R into G (Example 2).

816

Chapter 14: Multiple Integrals

To apply Equation (1), we need to find the corresponding uy-region G and the Jacobian of the transformation. To find them, we first solve Equations (4) for x and y in terms of u and y. From those equations it is easy to see that x = u + y,

y = 2y.

(5)

We then find the boundaries of G by substituting these expressions into the equations for the boundaries of R (Figure 14.55). xy-equations for the boundary of R x x y y

= = = =

Corresponding uY-equations for the boundary of G

Simplified uY-equations

u + y = 2y>2 = y u + y = s2y>2d + 1 = y + 1 2y = 0 2y = 4

y>2 s y>2d + 1 0 4

u u y y

= = = =

0 1 0 2

The Jacobian of the transformation (again from Equations (5)) is

Jsu, yd = 4

0x 0u

0x 0y

0y 0u

0y 0y

4 = 4

0 su + yd 0u

0 su + yd 0y

0 s2yd 0u

0 s2yd 0y

4 = `1 0

1 ` = 2. 2

We now have everything we need to apply Equation (1): 4

x = sy>2d + 1

L0 Lx = y>2 v

1

2

vu

=

G 1 u1

0

y=2 u=1 2x - y dx dy = u ƒ Jsu, yd ƒ du dy 2 Ly = 0 Lu = 0

EXAMPLE 3

1

1-x

L0 L0

u v x 3 3 2u v y 3 3

0

L0

dy = 2.

2x + y s y - 2xd2 dy dx.

Solution We sketch the region R of integration in the xy-plane and identify its boundaries (Figure 14.56). The integrand suggests the transformation u = x + y and y = y - 2x. Routine algebra produces x and y as functions of u and y:

y

x =

1

R 1

u y - , 3 3

y =

2u y + . 3 3

(6)

From Equations (6), we can find the boundaries of the uy-region G (Figure 14.56).

xy1

y0

L0

2

1

cu 2 d dy =

Evaluate

–2

0

2

suds2d du dy =

u

v –2u

x0

L0 L0

1

x

FIGURE 14.56 The equations x = su>3d - sy>3d and y = s2u>3d + sy>3d transform G into R. Reversing the transformation by the equations u = x + y and y = y - 2x transforms R into G (Example 3).

xy-equations for the boundary of R x + y = 1 x = 0 y = 0

Corresponding uY-equations for the boundary of G a

2u u y y - b + a + b = 1 3 3 3 3 u y = 0 3 3 2u y + = 0 3 3

Simplified uY-equations u = 1 y = u y = -2u

14.8

Substitutions in Multiple Integrals

817

The Jacobian of the transformation in Equations (6) is

Jsu, yd = 4

0x 0u

0x 0y

0y 0u

0y 0y

4 = 4

1 3

-

2 3

1 3

4 = 1. 3 1 3

Applying Equation (1), we evaluate the integral: 1-x

1

L0 L0

2x + y s y - 2xd2 dy dx = 1

u=1

y=u

Lu = 0 Ly = -2u

u

u 1>2 y 2 ƒ Jsu, yd ƒ dy du y=u

1

=

1 1 1 u 1>2 y 2 a b dy du = u 1>2 c y 3 d du 3 3L0 3 y = -2u L0 L-2u

=

1 2 2 u 1>2su 3 + 8u 3 d du = u 7>2 du = u 9/2 d = . 9L0 9 9 0 0 L

1

1

1

In the next example we illustrate a nonlinear transformation of coordinates resulting from simplifying the form of the integrand. Like the polar coordinates’ transformation, nonlinear transformations can map a straight line boundary of a region into a curved boundary (or vice versa with the inverse transformation). In general, nonlinear transformations are more complex to analyze than linear ones, and a complete treatment is left to a more advanced course. y

y5x

2

EXAMPLE 4

y52

R

2

The square root terms in the integrand suggest that we might simplify the integration by substituting u = 2xy and y = 2y>x. Squaring these equations, we readily have u 2 = xy and y2 = y>x, which imply that u 2y2 = y 2 and u 2>y2 = x 2. So we obtain the transformation (in the same ordering of the variables as discussed before) Solution

xy 5 1

1

x

2

y

y 1xy dx dy. xe A L1 L1>y

1

0

Evaluate the integral

u x = y

FIGURE 14.57 The region of integration R in Example 4.

and

y = uy.

Let’s first see what happens to the integrand itself under this transformation. The Jacobian of the transformation is y

Jsu, yd = 4

uy 2 ⇔ y 2

2 u 1 ⇔ xy 1

0x 0u

0x 0y

0y 0u

0y 0y

4 = †

1 y y

-u y2 † 2u = y. u

G 1

If G is the region of integration in the uy-plane, then by Equation (1) the transformed double integral under the substitution is

y1⇔yx

0

1

2

u

FIGURE 14.58 The boundaries of the region G correspond to those of region R in Figure 14.57. Notice as we move counterclockwise around the region R, we also move counterclockwise around the region G. The inverse transformation equations u = 1xy, y = 1y>x produce the region G from the region R.

y 1xy 2u dx dy = ye u y du dy = 2ue u du dy. xe A 6 6 6 R

G

G

The transformed integrand function is easier to integrate than the original one, so we proceed to determine the limits of integration for the transformed integral. The region of integration R of the original integral in the xy-plane is shown in Figure 14.57. From the substitution equations u = 2xy and y = 2y>x, we see that the image of the left-hand boundary xy = 1 for R is the vertical line segment u = 1, 2 Ú y Ú 1, in G (see Figure 14.58). Likewise, the right-hand boundary y = x of R maps to the horizontal line segment y = 1, 1 … u … 2, in G. Finally, the horizontal top boundary y = 2 of R

818

Chapter 14: Multiple Integrals

maps to uy = 2, 1 … y … 2, in G. As we move counterclockwise around the boundary of the region R, we also move counterclockwise around the boundary of G, as shown in Figure 14.58. Knowing the region of integration G in the uy-plane, we can now write equivalent iterated integrals: 2

y

L1 L1>y

y 1xy e dx dy = Ax

2

2>u

L1 L1

2ue u dy du.

Note the order of integration.

We now evaluate the transformed integral on the right-hand side, 2

L1

2>u

L1

2

2ue u dy du = 2

L1

yue u D y = 1 du y = 2>u

2

= 2

s2e u - ue u d du

L1 2

= 2

L1

s2 - ude u du

= 2 C (2 - u)e u + e u D u = 1 u=2

Integrate by parts.

= 2se - (e + e)d = 2e(e - 2). 2

Substitutions in Triple Integrals The cylindrical and spherical coordinate substitutions in Section 14.7 are special cases of a substitution method that pictures changes of variables in triple integrals as transformations of three-dimensional regions. The method is like the method for double integrals except that now we work in three dimensions instead of two. Suppose that a region G in uyw-space is transformed one-to-one into the region D in xyz-space by differentiable equations of the form x = gsu, y, wd,

y = hsu, y, wd,

z = ksu, y, wd,

as suggested in Figure 14.59. Then any function F(x, y, z) defined on D can be thought of as a function Fsgsu, y, wd, hsu, y, wd, ksu, y, wdd = Hsu, y, wd defined on G. If g, h, and k have continuous first partial derivatives, then the integral of F(x, y, z) over D is related to the integral of H(u, y, w) over G by the equation 9

Fsx, y, zd dx dy dz =

D

9

Hsu, y, wd ƒ Jsu, y, wd ƒ du dy dw.

G

w

z x 5 g(u, y, w) y 5 h(u, y, w) z 5 k(u, y, w)

G

D y

u

Cartesian uyw-space

y x

Cartesian xyz-space

FIGURE 14.59 The equations x = gsu, y, wd, y = hsu, y, wd, and z = ksu, y, wd allow us to change an integral over a region D in Cartesian xyz-space into an integral over a region G in Cartesian uyw-space using Equation (7).

(7)

14.8

z

819

The factor Jsu, y, wd, whose absolute value appears in this equation, is the Jacobian determinant

Cube with sides parallel to the coordinate axes

Jsu, y, wd = 6 G u r

Substitutions in Multiple Integrals

Cartesian ruz-space

0x 0u

0x 0y

0x 0w

0y 0u

0y 0y

0y 0sx, y, zd 6 = . 0w 0su, y, wd

0z 0u

0z 0y

0z 0w

This determinant measures how much the volume near a point in G is being expanded or contracted by the transformation from (u, y, w) to (x, y, z) coordinates. As in the twodimensional case, the derivation of the change-of-variable formula in Equation (7) is omitted. For cylindrical coordinates, r, u , and z take the place of u, y, and w. The transformation from Cartesian ruz-space to Cartesian xyz-space is given by the equations

x 5 r cos u y 5 r sin u z5z z

x = r cos u,

z 5 constant

y = r sin u,

z = z

(Figure 14.60). The Jacobian of the transformation is D

u 5 constant x

Jsr, u, zd = 6

Cartesian xyz-space

FIGURE 14.60 The equations x = r cos u, y = r sin u, and z = z transform the cube G into a cylindrical wedge D.

0y 0r

0x 0u 0y 0u

0z 0r

0z 0u

0x 0r

r 5 constant y

0x 0z

cos u 0y 6 = 3 sin u 0z 0 0z 0z

-r sin u r cos u 0

0 03 1

= r cos2 u + r sin2 u = r. The corresponding version of Equation (7) is

9

Fsx, y, zd dx dy dz =

9

D

Hsr, u, zd ƒ r ƒ dr du dz.

G

We can drop the absolute value signs whenever r Ú 0. For spherical coordinates, r, f, and u take the place of u, y, and w. The transformation from Cartesian rfu-space to Cartesian xyz-space is given by x = r sin f cos u,

y = r sin f sin u,

z = r cos f

(Figure 14.61). The Jacobian of the transformation (see Exercise 19) is 0x 0r Jsr, f, ud = 6

0y 0r

0x 0f 0y 0f

0x 0u 0y 6 = r2 sin f. 0u

0z 0r

0z 0f

0z 0u

The corresponding version of Equation (7) is

9 D

Fsx, y, zd dx dy dz =

9 G

Hsr, f, ud ƒ r2 sin f ƒ dr df du.

820

Chapter 14: Multiple Integrals

u

Cube with sides parallel to the coordinate axes

z

D

(x, y, z) f

x 5 r sin f cos u y 5 r sin f sin u z 5 r cos f

f 5 constant

r

G f r

r 5 constant

u 5 constant

y

u

Cartesian rfu-space

x

Cartesian xyz-space

FIGURE 14.61 The equations x = r sin f cos u, y = r sin f sin u, and z = r cos f transform the cube G into the spherical wedge D.

We can drop the absolute value signs because sin f is never negative for 0 … f … p. Note that this is the same result we obtained in Section 14.7. Here is an example of another substitution. Although we could evaluate the integral in this example directly, we have chosen it to illustrate the substitution method in a simple (and fairly intuitive) setting.

w 1

EXAMPLE 5

Evaluate

G 3

2

1

x = sy>2d + 1

L0 L0 Lx = y>2

y

u

4

a

2x - y z + b dx dy dz 2 3

by applying the transformation

x5u1y y 5 2y z 5 3w

u = s2x - yd>2,

y = y>2,

w = z>3

(8)

and integrating over an appropriate region in uyw-space.

z Rear plane: y x 5 , or y 5 2x 2

3

Solution We sketch the region D of integration in xyz-space and identify its boundaries (Figure 14.62). In this case, the bounding surfaces are planes. To apply Equation (7), we need to find the corresponding uyw-region G and the Jacobian of the transformation. To find them, we first solve Equations (8) for x, y, and z in terms of u, y, and w. Routine algebra gives

D

x = u + y,

1 4

y = 2y,

z = 3w.

(9)

y

x Front plane: y x 5 1 1, or y 5 2x 2 2 2

FIGURE 14.62 The equations x = u + y, y = 2y, and z = 3w transform G into D. Reversing the transformation by the equations u = s2x - yd>2, y = y>2, and w = z>3 transforms D into G (Example 5).

We then find the boundaries of G by substituting these expressions into the equations for the boundaries of D: xyz-equations for the boundary of D x x y y z z

= = = = = =

y>2 s y>2d + 1 0 4 0 3

Corresponding uYw-equations for the boundary of G u + y = 2y>2 = y u + y = s2y>2d + 1 = y + 1 2y = 0 2y = 4 3w = 0 3w = 3

Simplified uYw-equations u u y y w w

= = = = = =

0 1 0 2 0 1

14.8

Substitutions in Multiple Integrals

821

The Jacobian of the transformation, again from Equations (9), is

Jsu, y, wd = 6

0x 0u

0x 0y

0y 0u

0y 0y

0z 0u

0z 0y

0x 0w

1 0y 6 = 30 0w 0 0z 0w

1 2 0

0 0 3 = 6. 3

We now have everything we need to apply Equation (7): 3

x = sy>2d + 1

4

L0 L0 Lx = y>2 1

=

2

2

su + wd ƒ Jsu, y, wd ƒ du dy dw 1

2

= 6

L0 L0

1

2

su + wds6d du dy dw = 6

L0 L0 L0 1

2x - y z + b dx dy dz 2 3

1

L0 L0 L0 1

=

a

L0 L0

a

c

1

u2 + uw d dy dw 2 0 2

1

1

y 1 + wb dy dw = 6 c + yw d dw = 6 s1 + 2wd dw 2 0 L0 2 L0

= 6 C w + w 2 D 0 = 6s2d = 12. 1

Exercises 14.8 Jacobians and Transformed Regions in the Plane 1. a. Solve the system u = x - y,

y = 2x + y

for x and y in terms of u and y. Then find the value of the Jacobian 0sx, yd>0su, yd. b. Find the image under the transformation u = x - y, y = 2x + y of the triangular region with vertices (0, 0), (1, 1), and s1, -2d in the xy-plane. Sketch the transformed region in the uy-plane. 2. a. Solve the system u = x + 2y,

y = x - y

for x and y in terms of u and y. Then find the value of the Jacobian 0sx, yd>0su, yd. b. Find the image under the transformation u = x + 2y, y = x - y of the triangular region in the xy-plane bounded by the lines y = 0, y = x , and x + 2y = 2. Sketch the transformed region in the uy-plane. 3. a. Solve the system u = 3x + 2y,

by the x-axis, the y-axis, and the line x + y = 1. Sketch the transformed region in the uy-plane. 4. a. Solve the system u = 2x - 3y,

y = -x + y

for x and y in terms of u and y. Then find the value of the Jacobian 0sx, yd>0su, yd. b. Find the image under the transformation u = 2x - 3y, y = -x + y of the parallelogram R in the xy-plane with boundaries x = -3, x = 0, y = x, and y = x + 1. Sketch the transformed region in the uy-plane. Substitutions in Double Integrals 5. Evaluate the integral 4

x = s y>2d + 1

L0 Lx = y>2

2x - y dx dy 2

from Example 1 directly by integration with respect to x and y to confirm that its value is 2. 6. Use the transformation in Exercise 1 to evaluate the integral

y = x + 4y

for x and y in terms of u and y. Then find the value of the Jacobian 0sx, yd>0su, yd. b. Find the image under the transformation u = 3x + 2y, y = x + 4y of the triangular region in the xy-plane bounded

s2x 2 - xy - y 2 d dx dy 6 R

for the region R in the first quadrant bounded by the lines y = -2x + 4, y = -2x + 7, y = x - 2, and y = x + 1.

822

Chapter 14: Multiple Integrals

7. Use the transformation in Exercise 3 to evaluate the integral

16. Use the transformation x = u 2 - y2, y = 2uy to evaluate the integral

s3x 2 + 14xy + 8y 2 d dx dy 6

L0 L0

for the region R in the first quadrant bounded by the lines y = -s3>2dx + 1, y = -s3>2dx + 3, y = -s1>4dx , and y = -s1>4dx + 1. 8. Use the transformation and parallelogram R in Exercise 4 to evaluate the integral 2sx - yd dx dy. 6 R

9. Let R be the region in the first quadrant of the xy-plane bounded by the hyperbolas xy = 1, xy = 9 and the lines y = x, y = 4x. Use the transformation x = u>y, y = uy with u 7 0 and y 7 0 to rewrite y a x + 2xyb dx dy 6 A R

as an integral over an appropriate region G in the uy-plane. Then evaluate the uy-integral over G. 10. a. Find the Jacobian of the transformation x = u, y = uy and sketch the region G: 1 … u … 2, 1 … uy … 2, in the uy-plane. b. Then use Equation (1) to transform the integral 2

2

y x dy dx

L1 L1 into an integral over G, and evaluate both integrals. 11. Polar moment of inertia of an elliptical plate A thin plate of constant density covers the region bounded by the ellipse x 2>a 2 + y 2>b 2 = 1, a 7 0, b 7 0, in the xy-plane. Find the first moment of the plate about the origin. (Hint: Use the transformation x = ar cos u, y = br sin u.) 12. The area of an ellipse The area pab of the ellipse x 2>a 2 + y 2>b 2 = 1 can be found by integrating the function ƒsx, yd = 1 over the region bounded by the ellipse in the xy-plane. Evaluating the integral directly requires a trigonometric substitution. An easier way to evaluate the integral is to use the transformation x = au, y = by and evaluate the transformed integral over the disk G: u 2 + y 2 … 1 in the uy-plane. Find the area this way. 2>3

Finding Jacobians 17. Find the Jacobian 0sx, yd>0su, yd of the transformation a. x = u cos y,

y = u sin y

b. x = u sin y,

y = u cos y.

18. Find the Jacobian 0sx, y, zd>0su, y, wd of the transformation a. x = u cos y,

y = u sin y,

b. x = 2u - 1,

y = 3y - 4,

sx + 2yde sy - xd dx dy

by first writing it as an integral over a region G in the uy-plane. 14. Use the transformation x = u + s1>2dy, y = y to evaluate the integral sy + 4d>2

L0 Ly>2

y 3s2x - yde s2x - yd dx dy

15. Use the transformation x = u>y, y = uy to evaluate the integral sum

L1

y

L1>y

4

(x 2 + y 2) dx dy +

z = s1>2dsw - 4d.

20. Substitutions in single integrals How can substitutions in single definite integrals be viewed as transformations of regions? What is the Jacobian in such a case? Illustrate with an example. Substitutions in Triple Integrals 21. Evaluate the integral in Example 5 by integrating with respect to x, y, and z. 22. Volume of an ellipsoid

Find the volume of the ellipsoid

y2 x2 z2 + + = 1. a2 b2 c2 (Hint: Let x = au, y = by, and z = cw. Then find the volume of an appropriate region in uyw-space.) 23. Evaluate 9

ƒ xyz ƒ dx dy dz

over the solid ellipsoid y2 x2 z2 + 2 + 2 … 1. 2 a b c (Hint: Let x = au, y = by, and z = cw . Then integrate over an appropriate region in uyw-space.) 24. Let D be the region in xyz-space defined by the inequalities 1 … x … 2,

0 … xy … 2,

0 … z … 1.

2

by first writing it as an integral over a region G in the uy-plane.

2

z = w

19. Evaluate the appropriate determinant to show that the Jacobian of the transformation from Cartesian rfu-space to Cartesian xyz-space is r2 sin f .

2 - 2y

L0 Ly

2x 2 + y 2 dy dx.

(Hint: Show that the image of the triangular region G with vertices (0, 0), (1, 0), (1, 1) in the uy-plane is the region of integration R in the xy-plane defined by the limits of integration.)

13. Use the transformation in Exercise 2 to evaluate the integral

2

211 - x

1

R

L2

4>y

Ly>4

(x 2 + y 2) dx dy.

Evaluate sx 2y + 3xyzd dx dy dz 9 D

by applying the transformation u = x,

y = xy,

w = 3z

and integrating over an appropriate region G in uyw-space.

Chapter 14 25. Centroid of a solid semiellipsoid Assuming the result that the centroid of a solid hemisphere lies on the axis of symmetry threeeighths of the way from the base toward the top, show, by transforming the appropriate integrals, that the center of mass of a solid semiellipsoid sx 2>a 2 d + s y 2>b 2 d + sz 2>c 2 d … 1, z Ú 0, lies on the z-axis three-eighths of the way from the base toward the top. (You can do this without evaluating any of the integrals.)

Chapter 14

Practice Exercises

823

26. Cylindrical shells In Section 6.2, we learned how to find the volume of a solid of revolution using the shell method; namely, if the region between the curve y = ƒsxd and the x-axis from a to b s0 6 a 6 bd is revolved about the y-axis, the volume of the b resulting solid is 1a 2pxƒsxd dx . Prove that finding volumes by using triple integrals gives the same result. (Hint: Use cylindrical coordinates with the roles of y and z changed.)

Questions to Guide Your Review

1. Define the double integral of a function of two variables over a bounded region in the coordinate plane. 2. How are double integrals evaluated as iterated integrals? Does the order of integration matter? How are the limits of integration determined? Give examples. 3. How are double integrals used to calculate areas and average values. Give examples.

7. How are double and triple integrals in rectangular coordinates used to calculate volumes, average values, masses, moments, and centers of mass? Give examples. 8. How are triple integrals defined in cylindrical and spherical coordinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates? 9. How are triple integrals in cylindrical and spherical coordinates evaluated? How are the limits of integration found? Give examples.

4. How can you change a double integral in rectangular coordinates into a double integral in polar coordinates? Why might it be worthwhile to do so? Give an example.

10. How are substitutions in double integrals pictured as transformations of two-dimensional regions? Give a sample calculation.

5. Define the triple integral of a function ƒ(x, y, z) over a bounded region in space.

11. How are substitutions in triple integrals pictured as transformations of three-dimensional regions? Give a sample calculation.

6. How are triple integrals in rectangular coordinates evaluated? How are the limits of integration determined? Give an example.

Chapter 14

Practice Exercises

Evaluating Double Iterated Integrals In Exercises 1–4, sketch the region of integration and evaluate the double integral. 10

1.

1>y

ye dx dy

L1 L0

3.

2.

L0 L-29 - 4t 2

2 - 2y

1

t ds dt

4.

e y>x dy dx

L0 L0

29 - 4t 2

3>2

x3

1 xy

xy dx dy

L0 L2y

In Exercises 5–8, sketch the region of integration and write an equivalent integral with the order of integration reversed. Then evaluate both integrals. 4

5.

sy - 4d>2

L0 L-24 - y

6.

L0 L-29 - 4y2

x

L0 Lx

2

29 - 4y2

3>2

7.

1

dx dy

2

y dx dy

8.

2x dy dx

4 - x2

2x dy dx

L0 L0

Evaluate the integrals in Exercises 9–12. 1

9.

L0 L2y 8

11.

2

2

4 cos sx 2 d dx dy 2

dy dx

4 3 x y + 1 L0 L2

10.

L0 Ly>2 1

12.

1

1

3 y L0 L2

2

e x dx dy 2p sin px 2 dx dy x2

Areas and Volumes Using Double Integrals 13. Area between line and parabola Find the area of the region enclosed by the line y = 2x + 4 and the parabola y = 4 - x 2 in the xy-plane. 14. Area bounded by lines and parabola Find the area of the “triangular” region in the xy-plane that is bounded on the right by the parabola y = x 2, on the left by the line x + y = 2, and above by the line y = 4. 15. Volume of the region under a paraboloid Find the volume under the paraboloid z = x 2 + y 2 above the triangle enclosed by the lines y = x, x = 0, and x + y = 2 in the xy-plane. 16. Volume of the region under parabolic cylinder Find the volume under the parabolic cylinder z = x 2 above the region enclosed by the parabola y = 6 - x 2 and the line y = x in the xy-plane. Average Values Find the average value of ƒsx, yd = xy over the regions in Exercises 17 and 18. 17. The square bounded by the lines x = 1, y = 1 in the first quadrant 18. The quarter circle x 2 + y 2 … 1 in the first quadrant

824

Chapter 14: Multiple Integrals

Polar Coordinates Evaluate the integrals in Exercises 19 and 20 by changing to polar coordinates. 1 21 - x 2 2 dy dx 19. 2 2 2 L-1L-21 - x 2 s1 + x + y d 21 - y 2

1

ln sx 2 + y 2 + 1d dx dy L-1L-21 - y 2 21. Integrating over a lemniscate Integrate the function ƒsx, yd = 1>s1 + x 2 + y 2 d2 over the region enclosed by one loop of the lemniscate sx 2 + y 2 d2 - sx 2 - y 2 d = 0. 20.

22. Integrate ƒsx, yd = 1>s1 + x 2 + y 2 d2 over a. Triangular region and A 1, 23 B .

The triangle with vertices (0, 0), (1, 0),

b. First quadrant The first quadrant of the xy-plane.

29. Average value Find the average value of ƒsx, y, zd = 30xz 2x 2 + y over the rectangular solid in the first octant bounded by the coordinate planes and the planes x = 1, y = 3, z = 1. 30. Average value Find the average value of r over the solid sphere r … a (spherical coordinates). Cylindrical and Spherical Coordinates 31. Cylindrical to rectangular coordinates Convert

p

p

32. Rectangular to cylindrical coordinates (a) Convert to cylindrical coordinates. Then (b) evaluate the new integral.

ln 7

24. 25.

Lln 6 1

Lln 4 x+y

x

z

2y

3 L1 L1 L0 z

21xy 2 dz dy dx

33. Rectangular to spherical coordinates (a) Convert to spherical coordinates. Then (b) evaluate the new integral.

e sx + y + zd dz dy dx

21 - x 2

1

1

L-1 L-21 - x L2x 2 + y 2

dz dy dx

2

s2x - y - zd dz dy dx

L0 L0 L0 e

26.

L0 x2

ln 5

sx 2 + y 2d

L0 L-21 - x 2 L-sx 2 + y 2d

cos sx + y + zd dx dy dz

ln 2

21 - x 2

1

p

L0 L0 L0

24 - r 2

3 dz r dr du, r Ú 0 L0 L0 Lr to (a) rectangular coordinates with the order of integration dz dx dy and (b) spherical coordinates. Then (c) evaluate one of the integrals.

Evaluating Triple Iterated Integrals Evaluate the integrals in Exercises 23–26. 23.

22

2p

dy dz dx

Volumes and Average Values Using Triple Integrals 27. Volume Find the volume of the wedge-shaped region enclosed on the side by the cylinder x = - cos y, -p>2 … y … p>2, on the top by the plane z = -2x, and below by the xy-plane. z

z 5 –2x

34. Rectangular, cylindrical, and spherical coordinates Write an iterated triple integral for the integral of ƒsx, y, zd = 6 + 4y over the region in the first octant bounded by the cone z = 2x 2 + y 2 , the cylinder x 2 + y 2 = 1, and the coordinate planes in (a) rectangular coordinates, (b) cylindrical coordinates, and (c) spherical coordinates. Then (d) find the integral of ƒ by evaluating one of the triple integrals. 35. Cylindrical to rectangular coordinates Set up an integral in rectangular coordinates equivalent to the integral 23

p>2

L0 x 5 –cos y

L1

24 - r 2

L1

r 3ssin u cos udz 2 dz dr du .

Arrange the order of integration to be z first, then y, then x. 36. Rectangular to cylindrical coordinates The volume of a solid is 2

–p 2

p 2

x

L0 L0 y

28. Volume Find the volume of the solid that is bounded above by the cylinder z = 4 - x 2, on the sides by the cylinder x 2 + y 2 = 4, and below by the xy-plane. z z 5 4 2 x2

y x2 1 y2 5 4 x

22x - x 2

24 - x2 - y 2

L-24 - x 2 - y 2

dz dy dx .

a. Describe the solid by giving equations for the surfaces that form its boundary. b. Convert the integral to cylindrical coordinates but do not evaluate the integral. 37. Spherical versus cylindrical coordinates Triple integrals involving spherical shapes do not always require spherical coordinates for convenient evaluation. Some calculations may be accomplished more easily with cylindrical coordinates. As a case in point, find the volume of the region bounded above by the sphere x 2 + y 2 + z 2 = 8 and below by the plane z = 2 by using (a) cylindrical coordinates and (b) spherical coordinates. Masses and Moments 38. Finding Iz in spherical coordinates Find the moment of inertia about the z-axis of a solid of constant density d = 1 that is bounded above by the sphere r = 2 and below by the cone f = p>3 (spherical coordinates).

Chapter 14 39. Moment of inertia of a “thick” sphere Find the moment of inertia of a solid of constant density d bounded by two concentric spheres of radii a and b sa 6 bd about a diameter. 40. Moment of inertia of an apple Find the moment of inertia about the z-axis of a solid of density d = 1 enclosed by the spherical coordinate surface r = 1 - cos f. The solid is the red curve rotated about the z-axis in the accompanying figure. z

r = 1 2 cos f

y

Additional and Advanced Exercises

825

bounded by the line y = x and the parabola y = x 2 in the xy-plane if the density is dsx, yd = x + 1 . 47. Plate with variable density Find the mass and first moments about the coordinate axes of a thin square plate bounded by the lines x = ;1, y = ;1 in the xy-plane if the density is dsx, yd = x 2 + y 2 + 1>3. 48. Triangles with same inertial moment Find the moment of inertia about the x-axis of a thin triangular plate of constant density d whose base lies along the interval [0, b] on the x-axis and whose vertex lies on the line y = h above the x-axis. As you will see, it does not matter where on the line this vertex lies. All such triangles have the same moment of inertia about the x-axis. 49. Centroid Find the centroid of the region in the polar coordinate plane defined by the inequalities 0 … r … 3, -p>3 … u … p>3. 50. Centroid Find the centroid of the region in the first quadrant bounded by the rays u = 0 and u = p>2 and the circles r = 1 and r = 3.

x

51. a. Centroid Find the centroid of the region in the polar coordinate plane that lies inside the cardioid r = 1 + cos u and outside the circle r = 1. b. Sketch the region and show the centroid in your sketch. 41. Centroid Find the centroid of the “triangular” region bounded by the lines x = 2, y = 2 and the hyperbola xy = 2 in the xy-plane. 42. Centroid Find the centroid of the region between the parabola x + y 2 - 2y = 0 and the line x + 2y = 0 in the xy-plane. 43. Polar moment Find the polar moment of inertia about the origin of a thin triangular plate of constant density d = 3 bounded by the y-axis and the lines y = 2x and y = 4 in the xy-plane. 44. Polar moment Find the polar moment of inertia about the center of a thin rectangular sheet of constant density d = 1 bounded by the lines a. x = ;2,

y = ;1 in the xy-plane

b. x = ;a,

y = ;b in the xy-plane.

(Hint: Find Ix. Then use the formula for Ix to find Iy and add the two to find I0.)

52. a. Centroid Find the centroid of the plane region defined by the polar coordinate inequalities 0 … r … a, -a … u … a s0 6 a … pd. How does the centroid move as a : p- ? b. Sketch the region for a = 5p>6 and show the centroid in your sketch. Substitutions 53. Show that if u = x - y and y = y, then q

L0 L0

Chapter 14

e -sx ƒsx - y, yd dy dx =

q

L0 L0

q

e -ssu + yd ƒsu, yd du dy.

54. What relationship must hold between the constants a, b, and c to make q

45. Inertial moment Find the moment of inertia about the x-axis of a thin plate of constant density d covering the triangle with vertices (0, 0), (3, 0), and (3, 2) in the xy-plane. 46. Plate with variable density Find the center of mass and the moments of inertia about the coordinate axes of a thin plate

x

q

L- qL- q

e -sax

2

+ 2bxy + cy2d

dx dy = 1?

(Hint: Let s = ax + by and t = gx + dy, where sad - bgd2 = ac - b 2. Then ax 2 + 2bxy + cy 2 = s 2 + t 2.)

Additional and Advanced Exercises 3. Solid cylindrical region between two planes Find the volume of the portion of the solid cylinder x 2 + y 2 … 1 that lies between the planes z = 0 and x + y + z = 2.

Volumes 1. Sand pile: double and triple integrals The base of a sand pile covers the region in the xy-plane that is bounded by the parabola x 2 + y = 6 and the line y = x . The height of the sand above the point (x, y) is x 2. Express the volume of sand as (a) a double integral, (b) a triple integral. Then (c) find the volume.

4. Sphere and paraboloid Find the volume of the region bounded above by the sphere x 2 + y 2 + z 2 = 2 and below by the paraboloid z = x 2 + y 2.

2. Water in a hemispherical bowl A hemispherical bowl of radius 5 cm is filled with water to within 3 cm of the top. Find the volume of water in the bowl.

5. Two paraboloids Find the volume of the region bounded above by the paraboloid z = 3 - x 2 - y 2 and below by the paraboloid z = 2x 2 + 2y 2.

826

Chapter 14: Multiple Integrals

6. Spherical coordinates Find the volume of the region enclosed by the spherical coordinate surface r = 2 sin f (see accompanying figure).

Similarly, it can be shown that x

y

u

L0 L0 L0

z r 5 2 sin f

e msx - td ƒstd dt du dy =

14. Transforming a double integral to obtain constant limits Sometimes a multiple integral with variable limits can be changed into one with constant limits. By changing the order of integration, show that 1

y

L0

ƒsxd a

x

L0

gsx - ydƒs yd dyb dx 1

=

x

L0

ƒs yd a 1

=

7. Hole in sphere A circular cylindrical hole is bored through a solid sphere, the axis of the hole being a diameter of the sphere. The volume of the remaining solid is 23

2p

V = 2

L0 L0 L1 a. Find the radius of the hole and the radius of the sphere. b. Evaluate the integral. 8. Sphere and cylinder Find the volume of material cut from the solid sphere r 2 + z 2 … 9 by the cylinder r = 3 sin u. 9. Two paraboloids Find the volume of the region enclosed by the surfaces z = x 2 + y 2 and z = sx 2 + y 2 + 1d>2. 10. Cylinder and surface z xy Find the volume of the region in the first octant that lies between the cylinders r = 1 and r = 2 and that is bounded below by the xy-plane and above by the surface z = xy.

e -ax - e -bx dx. x

b

La

a sin b

2a2 - y2

Ly cot b

L0

1 b, 2

b. Rewrite the Cartesian integral with the order of integration reversed. 13. Reducing a double to a single integral By changing the order of integration, show that the following double integral can be reduced to a single integral: L0 L0

u

e msx - td ƒstd dt du =

x

L0

sx - tde msx - td ƒstd dt .

b

L0 L0

where a 7 0 and 0 6 b 6 p>2.

x

17. Mass and polar inertia of a counterweight The counterweight of a flywheel of constant density 1 has the form of the smaller segment cut from a circle of radius a by a chord at a distance b from the center sb 6 ad. Find the mass of the counterweight and its polar moment of inertia about the center of the wheel.

a

Show, by changing to polar coordinates, ln sx 2 + y 2 d dx dy = a 2 b aln a -

1

1 gs ƒ x - y ƒ dƒsxdƒsyd dx dy. 2L0 L0

16. Polar inertia of triangular plate Find the polar moment of inertia about the origin of a thin triangular plate of constant density d = 3 bounded by the y-axis and the lines y = 2x and y = 4 in the xy-plane.

e -xy dy

to form a double integral and evaluate the integral by changing the order of integration.) 12. a. Polar coordinates that

gsx - ydƒsxd dxb dy

Theory and Examples 19. Evaluate

(Hint: Use the relation e -ax - e -bx = x

Ly

18. Centroid of boomerang Find the centroid of the boomerangshaped region between the parabolas y 2 = -4sx - 1d and y 2 = -2sx - 2d in the xy-plane.

Changing the Order of Integration 11. Evaluate the integral L0

1

Masses and Moments 15. Minimizing polar inertia A thin plate of constant density is to occupy the triangular region in the first quadrant of the xy-plane having vertices (0, 0), (a, 0), and (a, 1> a). What value of a will minimize the plate’s polar moment of inertia about the origin?

24 - z 2

r dr dz du.

q

sx - td2 msx - td ƒstd dt. e 2 L0 x

2 2

e max sb x

, a2y 2d

dy dx,

where a and b are positive numbers and max sb 2x 2, a 2y 2 d = e

b 2x 2 if b 2x 2 Ú a 2y 2 a 2y 2 if b 2x 2 6 a 2y 2.

20. Show that 0 2Fsx, yd dx dy 6 0x 0y over the rectangle x0 … x … x1, y0 … y … y1, is Fsx1, y1 d - Fsx0, y1 d - Fsx1, y0 d + Fsx0, y0 d. 21. Suppose that ƒsx, yd can be written as a product ƒsx, yd = FsxdGsyd of a function of x and a function of y. Then

Chapter 14 the integral of ƒ over the rectangle R: a … x … b, c … y … d can be evaluated as a product as well, by the formula

6

ƒsx, yd dA = a

R

b

La

Fsxd dxb a

Gs yd dyb .

Lc

(1)

6

ƒsx, yd dA =

R

Lc

FsxdGs yd dxb dy

La

d

=

Lc

= a

(i)

b

aGs yd Fsxd dxb dy Lc La d

=

Fsxd dxbGs yd dy

La

b

La

(ii)

b

a

(iii)

d

Fsxd dxb

Lc

Gs yd dy.

(iv)

a. Give reasons for steps (i) through (iv). When it applies, Equation (1) can be a time-saver. Use it to evaluate the following integrals. p>2

ln 2

b.

L0

L0

2

e x cos y dy dx

c.

q

I =

L0

e -y dy = 2

2p . 2

b. Substitute y = 1t in Equation (2) to show that ≠s1>2d = 2I = 1p.

b

a

827

a. If you have not yet done Exercise 41 in Section 14.4, do it now to show that

d

The argument is that d

Additional and Advanced Exercises

1

x dx dy 2 L1 L-1 y

22. Let Du ƒ denote the derivative of ƒsx, yd = sx 2 + y 2 d>2 in the direction of the unit vector u = u1 i + u2 j. a. Finding average value Find the average value of Du ƒ over the triangular region cut from the first quadrant by the line x + y = 1. b. Average value and centroid Show in general that the average value of Du ƒ over a region in the xy-plane is the value of Du ƒ at the centroid of the region. 23. The value of Ωs1/2d The gamma function,

24. Total electrical charge over circular plate The electrical charge distribution on a circular plate of radius R meters is ssr, ud = krs1 - sin ud coulomb>m2 (k a constant). Integrate s over the plate to find the total charge Q. 25. A parabolic rain gauge A bowl is in the shape of the graph of z = x 2 + y 2 from z = 0 to z = 10 in. You plan to calibrate the bowl to make it into a rain gauge. What height in the bowl would correspond to 1 in. of rain? 3 in. of rain? 26. Water in a satellite dish A parabolic satellite dish is 2 m wide and 1> 2 m deep. Its axis of symmetry is tilted 30 degrees from the vertical. a. Set up, but do not evaluate, a triple integral in rectangular coordinates that gives the amount of water the satellite dish will hold. (Hint: Put your coordinate system so that the satellite dish is in “standard position” and the plane of the water level is slanted.) (Caution: The limits of integration are not “nice.”) b. What would be the smallest tilt of the satellite dish so that it holds no water? 27. An infinite half-cylinder Let D be the interior of the infinite right circular half-cylinder of radius 1 with its single-end face suspended 1 unit above the origin and its axis the ray from (0, 0, 1) to q. Use cylindrical coordinates to evaluate

9

zsr 2 + z 2 d-5>2 dV.

D

q

≠sxd =

L0

t

x-1

b

-t

e dt,

extends the factorial function from the nonnegative integers to other real values. Of particular interest in the theory of differential equations is the number q

1 ≠a b = t s1>2d - 1 e -t dt = 2 L0 L0

q

e -t 2t

dt.

(2)

28. Hypervolume We have learned that 1a 1 dx is the length of the interval [a, b] on the number line (one-dimensional space), 4R 1 dA is the area of region R in the xy-plane (two-dimensional space), and 7D 1 dV is the volume of the region D in threedimensional space (xyz-space). We could continue: If Q is a region in 4-space (xyzw-space), then |Q 1 dV is the “hypervolume” of Q. Use your generalizing abilities and a Cartesian coordinate system of 4-space to find the hypervolume inside the unit 3-dimensional sphere x 2 + y 2 + z 2 + w 2 = 1.

15 INTEGRATION IN VECTOR FIELDS OVERVIEW In this chapter we extend the theory of integration to curves and surfaces in space. The resulting theory of line and surface integrals gives powerful mathematical tools for science and engineering. Line integrals are used to find the work done by a force in moving an object along a path, and to find the mass of a curved wire with variable density. Surface integrals are used to find the rate of flow of a fluid across a surface. We present the fundamental theorems of vector integral calculus, and discuss their mathematical consequences and physical applications. In the final analysis, the key theorems are shown as generalized interpretations of the Fundamental Theorem of Calculus.

Line Integrals

15.1

To calculate the total mass of a wire lying along a curve in space, or to find the work done by a variable force acting along such a curve, we need a more general notion of integral than was defined in Chapter 5. We need to integrate over a curve C rather than over an interval [a, b]. These more general integrals are called line integrals (although path integrals might be more descriptive). We make our definitions for space curves, with curves in the xy-plane being the special case with z-coordinate identically zero. Suppose that ƒ(x, y, z) is a real-valued function we wish to integrate over the curve C lying within the domain of ƒ and parametrized by rstd = gstdi + hstdj + kstdk, a … t … b. The values of ƒ along the curve are given by the composite function ƒ(g(t), h(t), k(t)). We are going to integrate this composite with respect to arc length from t = a to t = b. To begin, we first partition the curve C into a finite number n of subarcs (Figure 15.1). The typical subarc has length ¢sk. In each subarc we choose a point sxk, yk, zk d and form the sum

z t5b

r(t)

Ds k

x

t5a

(x k , yk , z k )

n

Sn = a ƒsxk , yk , zk d ¢sk ,

y

FIGURE 15.1 The curve r(t) partitioned into small arcs from t = a to t = b. The length of a typical subarc is ¢sk.

k=1

which is similar to a Riemann sum. Depending on how we partition the curve C and pick sxk, yk, zk d in the kth subarc, we may get different values for Sn. If ƒ is continuous and the functions g, h, and k have continuous first derivatives, then these sums approach a limit as n increases and the lengths ¢sk approach zero. This limit gives the following definition, similar to that for a single integral. In the definition, we assume that the partition satisfies ¢sk : 0 as n : q .

DEFINITION If ƒ is defined on a curve C given parametrically by r(t) = g(t)i + h(t)j + k(t)k, a … t … b, then the line integral of ƒ over C is n

LC

ƒsx, y, zd ds = lim a ƒsxk, yk, zk d ¢sk , n: q

provided this limit exists.

828

k=1

(1)

15.1

Line Integrals

829

If the curve C is smooth for a … t … b (so v = dr>dt is continuous and never 0) and the function f is continuous on C, then the limit in Equation (1) can be shown to exist. We can then apply the Fundamental Theorem of Calculus to differentiate the arc length equation, t

sstd =

La

ƒ vstd ƒ dt,

Eq. (3) of Section 13.3 with t0 = a

to express ds in Equation (1) as ds = ƒ vstd ƒ dt and evaluate the integral of ƒ over C as b 2 2 dy 2 ds dx dz a b + a b + a b = ƒvƒ = dt dt dt dt A

LC

ƒsx, y, zd ds =

La

ƒsgstd, hstd, kstdd ƒ vstd ƒ dt.

(2)

Notice that the integral on the right side of Equation (2) is just an ordinary (single) definite integral, as defined in Chapter 5, where we are integrating with respect to the parameter t. The formula evaluates the line integral on the left side correctly no matter what parametrization is used, as long as the parametrization is smooth. Note that the parameter t defines a direction along the path. The starting point on C is the position rsad and movement along the path is in the direction of increasing t (see Figure 15.1).

How to Evaluate a Line Integral To integrate a continuous function ƒ(x, y, z) over a curve C: 1. Find a smooth parametrization of C,

rstd = gstdi + hstdj + kstdk,

a … t … b.

2. Evaluate the integral as b

LC

ƒsx, y, zd ds =

La

ƒsgstd, hstd, kstdd ƒ vstd ƒ dt.

If ƒ has the constant value 1, then the integral of ƒ over C gives the length of C from t = a to t = b in Figure 15.1. Integrate ƒsx, y, zd = x - 3y 2 + z over the line segment C joining the origin to the point (1, 1, 1) (Figure 15.2).

EXAMPLE 1

z (1, 1, 1)

Solution

C

We choose the simplest parametrization we can think of: rstd = ti + tj + tk,

y

x (1, 1, 0)

0 … t … 1.

The components have continuous first derivatives and ƒ vstd ƒ = ƒ i + j + k ƒ = 212 + 12 + 12 = 23 is never 0, so the parametrization is smooth. The integral of ƒ over C is

FIGURE 15.2 The integration path in Example 1.

1

LC

ƒsx, y, zd ds =

L0

ƒst, t, td A 23 B dt

Eq. (2), ds = |v(t)| dt = 13 dt

1

=

L0

st - 3t 2 + td23 dt 1

= 23

L0

s2t - 3t 2 d dt = 23 C t 2 - t 3 D 0 = 0. 1

830

Chapter 15: Integration in Vector Fields

Additivity Line integrals have the useful property that if a piecewise smooth curve C is made by joining a finite number of smooth curves C1, C2 , Á , Cn end to end (Section 12.1), then the integral of a function over C is the sum of the integrals over the curves that make it up: LC z

C2

ƒ ds +

LC2

ƒ ds + Á +

LCn

ƒ ds.

(3)

Figure 15.3 shows another path from the origin to (1, 1, 1), the union of line segments C1 and C2. Integrate ƒsx, y, zd = x - 3y 2 + z over C1 ´ C2.

We choose the simplest parametrizations for C1 and C2 we can find, calculating the lengths of the velocity vectors as we go along: Solution

y

C1 x

LC1

EXAMPLE 2

(1, 1, 1)

(0, 0, 0)

ƒ ds =

(1, 1, 0)

FIGURE 15.3 The path of integration in Example 2.

C1:

rstd = ti + tj,

0 … t … 1;

C2:

rstd = i + j + tk,

ƒ v ƒ = 212 + 12 = 22

0 … t … 1;

ƒ v ƒ = 20 2 + 0 2 + 12 = 1.

With these parametrizations we find that

LC1 ´C2

ƒsx, y, zd ds =

LC1

ƒsx, y, zd ds +

LC2

ƒsx, y, zd ds

1

=

1

ƒst, t, 0d22 dt +

L0

L0

ƒs1, 1, tds1d dt

1

=

L0

Eq. (3)

Eq. (2)

1

st - 3t 2 + 0d22 dt +

= 22 c

L0

s1 - 3 + tds1d dt

1 1 22 3 t2 t2 - t 3 d + c - 2t d = - . 2 2 2 2 0 0

Notice three things about the integrations in Examples 1 and 2. First, as soon as the components of the appropriate curve were substituted into the formula for ƒ, the integration became a standard integration with respect to t. Second, the integral of ƒ over C1 ´ C2 was obtained by integrating ƒ over each section of the path and adding the results. Third, the integrals of ƒ over C and C1 ´ C2 had different values.

The value of the line integral along a path joining two points can change if you change the path between them.

We investigate this third observation in Section 15.3.

Mass and Moment Calculations We treat coil springs and wires as masses distributed along smooth curves in space. The distribution is described by a continuous density function dsx, y, zd representing mass per unit length. When a curve C is parametrized by rstd = xstdi + ystdj + zstdk, a … t … b, then x, y, and z are functions of the parameter t, the density is the function dsxstd, ystd, zstdd, and the arc length differential is given by ds =

2 2 dy 2 dx dz b + a b + a b dt. dt dt A dt

a

15.1

Line Integrals

831

(See Section 12.3.) The spring’s or wire’s mass, center of mass, and moments are then calculated with the formulas in Table 15.1, with the integrations in terms of the parameter t over the interval [a, b]. For example, the formula for mass becomes b

M =

La

dsxstd, ystd, zstdd

2 2 dy 2 dx dz b + a b + a b dt. dt dt A dt

a

These formulas also apply to thin rods, and their derivations are similar to those in Section 6.6. Notice how alike the formulas are to those in Tables 14.1 and 14.2 for double and triple integrals. The double integrals for planar regions, and the triple integrals for solids, become line integrals for coil springs, wires, and thin rods.

TABLE 15.1 Mass and moment formulas for coil springs, wires, and thin rods lying

along a smooth curve C in space M =

d ds d = d(x, y, z) is the density at (x, y, z) LC First moments about the coordinate planes: Mass:

Myz =

LC

Mxz =

x d ds,

LC

Mxy =

y d ds,

LC

z d ds

Coordinates of the center of mass: x = Myz >M,

y = Mxz >M,

z = Mxy >M

Moments of inertia about axes and other lines: Ix = IL =

LC

s y 2 + z 2 d d ds,

LC

r 2 d ds

Iy =

LC

sx 2 + z 2 d d ds,

Iz =

LC

sx 2 + y 2 d d ds,

rsx, y, zd = distance from the point sx, y, zd to line L

Notice that the element of mass dm is equal to d ds in the table rather than d dV as in Table 14.1, and that the integrals are taken over the curve C.

EXAMPLE 3

A slender metal arch, denser at the bottom than top, lies along the semicircle y 2 + z 2 = 1, z Ú 0, in the yz-plane (Figure 15.4). Find the center of the arch’s mass if the density at the point (x, y, z) on the arch is dsx, y, zd = 2 - z.

z 1 c.m.

We know that x = 0 and y = 0 because the arch lies in the yz-plane with its mass distributed symmetrically about the z-axis. To find z , we parametrize the circle as

Solution

–1

x

1 y y 2 z 2 1, z 0

FIGURE 15.4 Example 3 shows how to find the center of mass of a circular arch of variable density.

rstd = scos tdj + ssin tdk,

0 … t … p.

For this parametrization, ƒ vstd ƒ =

2 2 dy 2 dx dz b + a b + a b = 2s0d2 + s -sin td2 + scos td2 = 1, dt dt B dt

a

so ds = ƒ v ƒ dt = dt.

832

Chapter 15: Integration in Vector Fields

The formulas in Table 15.1 then give p

M =

LC

d ds =

LC

s2 - zd ds =

L0

s2 - sin td dt = 2p - 2 p

Mxy =

LC

zd ds =

LC

zs2 - zd ds =

p

= z =

L0

s2 sin t - sin2 td dt =

ssin tds2 - sin td dt

L0

8 - p 2

Mxy 8 - p 8 - p# 1 = L 0.57. = M 2 2p - 2 4p - 4

With z to the nearest hundredth, the center of mass is (0, 0, 0.57).

Line Integrals in the Plane z

There is an interesting geometric interpretation for line integrals in the plane. If C is a smooth curve in the xy-plane parametrized by rstd = xstdi + ystdj, a … t … b, we generate a cylindrical surface by moving a straight line along C orthogonal to the plane, holding the line parallel to the z-axis, as in Section 11.6. If z = ƒsx, yd is a nonnegative continuous function over a region in the plane containing the curve C, then the graph of ƒ is a surface that lies above the plane. The cylinder cuts through this surface, forming a curve on it that lies above the curve C and follows its winding nature. The part of the cylindrical surface that lies beneath the surface curve and above the xy-plane is like a “winding wall” or “fence” standing on the curve C and orthogonal to the plane. At any point (x, y) along the curve, the height of the wall is ƒsx, yd. We show the wall in Figure 15.5, where the “top” of the wall is the curve lying on the surface z = ƒsx, yd. (We do not display the surface formed by the graph of ƒ in the figure, only the curve on it that is cut out by the cylinder.) From the definition

height f (x, y)

y ta (x, y)

Δsk

x

Plane curve C

tb

FIGURE 15.5 The line integral 1C ƒ ds gives the area of the portion of the cylindrical surface or “wall” beneath z = ƒsx, yd Ú 0.

n

LC

ƒ ds = lim a ƒsxk, yk d ¢sk, n: q k=1

where ¢sk : 0 as n : q , we see that the line integral 1C ƒ ds is the area of the wall shown in the figure.

Exercises 15.1 Graphs of Vector Equations Match the vector equations in Exercises 1–8 with the graphs (a)–(h) given here. a.

c.

b. z

d. z

z

z

(2, 2, 2)

2

–1 1

1

y x

x 1 x

y

1

2 y

2 x

y

15.1 e.

833

Line Integrals

z

f. z

z

z

2 (0, 0, 1) (0, 0, 0)

(1, 1, 1) 1

C3

(0, 0, 0)

1

y

y

(1, 1, 1)

y C2

x

x

–2

(1, 1, –1)

x

y

C1

x

(1, 1, 0)

(b)

(a)

g.

h.

The paths of integration for Exercises 15 and 16. z

z

16. Integrate ƒsx, y, zd = x + 1y - z 2 over the path from (0, 0, 0) to (1, 1, 1) (see accompanying figure) given by

2 –2

2 y

2

2 x

y

2. rstd = i + j + t k, 4. rstd = ti,

0 … t … 2

7. rstd = st - 1dj + 2tk, 2

0 … t … 2p

-1 … t … 1

6. rstd = t j + s2 - 2tdk,

C2:

rstd = tj + k,

C 3:

rstd = ti + j + k,

0 … t … 1 0 … t … 1

0 … t … 2p.

Line Integrals over Plane Curves 19. Evaluate 1C x ds, where C is a. the straight-line segment x = t, y = t>2, from (0, 0) to (4, 2). b. the parabolic curve x = t, y = t 2, from (0, 0) to (2, 4).

0 … t … 1 -1 … t … 1

8. rstd = s2 cos tdi + s2 sin tdk,

0 … t … 1

rstd = sa cos tdj + sa sin tdk,

-1 … t … 1

5. rstd = ti + tj + tk,

rstd = tk,

18. Integrate ƒsx, y, zd = - 2x 2 + z 2 over the circle

0 … t … 1

3. rstd = s2 cos tdi + s2 sin tdj,

C1:

17. Integrate ƒsx, y, zd = sx + y + zd>sx 2 + y 2 + z 2 d over the path rstd = ti + tj + tk, 0 6 a … t … b.

x

1. rstd = ti + s1 - tdj,

(0, 1, 1)

C1

(1, 1, 1)

–1

C2

0 … t … p

20. Evaluate 1C 2x + 2y ds, where C is a. the straight-line segment x = t, y = 4t, from (0, 0) to (1, 4). b. C1 ´ C2; C1 is the line segment from (0, 0) to (1, 0) and C2 is the line segment from (1, 0) to (1, 2). 2

Evaluating Line Integrals over Space Curves 9. Evaluate 1C sx + yd ds where C is the straight-line segment x = t, y = s1 - td, z = 0, from (0, 1, 0) to (1, 0, 0). 10. Evaluate 1C sx - y + z - 2d ds where C is the straight-line segment x = t, y = s1 - td, z = 1, from (0, 1, 1) to (1, 0, 1). 11. Evaluate 1C sxy + y + zd ds along the curve rstd = 2ti + t j + s2 - 2tdk, 0 … t … 1. 12. Evaluate 1C 2x 2 + y 2 ds along the curve rstd = s4 cos tdi + s4 sin tdj + 3tk, - 2p … t … 2p.

21. Find the line integral of ƒsx, yd = ye x rstd = 4ti - 3tj, -1 … t … 2.

22. Find the line integral of ƒsx, yd = x - y + 3 along the curve rstd = (cos t)i + (sin t)j, 0 … t … 2p. x2 ds, where C is the curve x = t 2, y = t 3, for LC y 4>3 1 … t … 2.

23. Evaluate

24. Find the line integral of ƒsx, yd = 2y>x along the curve rstd = t 3i + t 4j, 1>2 … t … 1. 25. Evaluate 1C A x + 2y B ds where C is given in the accompanying figure. y

13. Find the line integral of ƒsx, y, zd = x + y + z over the straightline segment from (1, 2, 3) to s0, -1, 1d. 14. Find the line integral of ƒsx, y, zd = 23>sx 2 + y 2 + z 2 d over the curve rstd = ti + tj + tk, 1 … t … q .

C

15. Integrate ƒsx, y, zd = x + 1y - z over the path from (0, 0, 0) to (1, 1, 1) (see accompanying figure) given by rstd = ti + t 2 j,

C2:

rstd = i + j + tk,

0 … t … 1 0 … t … 1

(1, 1)

y5x

2

C1:

along the curve

y 5 x2 (0, 0)

x

834

Chapter 15: Integration in Vector Fields

26. Evaluate figure.

1 ds where C is given in the accompanying 2 2 x + y + 1 LC

yz-plane. Find the moments of inertia of the rod about the three coordinate axes. A spring of constant density d

39. Two springs of constant density lies along the helix

y

rstd = scos tdi + ssin tdj + tk, (0, 1)

(1, 1)

(0, 0)

(1, 0)

a. Find Iz.

x

In Exercises 27–30, integrate ƒ over the given curve. 27. ƒsx, yd = x >y, 3

C:

y = x >2, 2

28. ƒsx, yd = sx + y 2 d> 21 + x 2, (0, 0) 29. ƒsx, yd = x + y, (2, 0) to (0, 2)

C:

30. ƒsx, yd = x 2 - y, C: (0, 2) to s 12, 12d

0 … x … 2 C:

0 … t … 2p.

y = x 2>2 from (1, 1> 2) to

x 2 + y 2 = 4 in the first quadrant from x 2 + y 2 = 4 in the first quadrant from

b. Suppose that you have another spring of constant density d that is twice as long as the spring in part (a) and lies along the helix for 0 … t … 4p. Do you expect Iz for the longer spring to be the same as that for the shorter one, or should it be different? Check your prediction by calculating Iz for the longer spring. 40. Wire of constant density along the curve

rstd = st cos tdi + st sin tdj + A 222>3 B t 3>2k,

Masses and Moments 33. Mass of a wire Find the mass of a wire that lies along the curve rstd = st 2 - 1dj + 2tk, 0 … t … 1, if the density is d = s3>2dt. 34. Center of mass of a curved wire A wire of density dsx, y, zd = 15 2y + 2 lies along the curve rstd = st 2 - 1dj + 2tk, -1 … t … 1. Find its center of mass. Then sketch the curve and center of mass together.

0 … t … 1.

Find z and Iz. 41. The arch in Example 3

Find Ix for the arch in Example 3.

42. Center of mass and moments of inertia for wire with variable density Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lying along the curve

31. Find the area of one side of the “winding wall” standing orthogonally on the curve y = x 2, 0 … x … 2, and beneath the curve on the surface ƒsx, yd = x + 2y. 32. Find the area of one side of the “wall” standing orthogonally on the curve 2x + 3y = 6, 0 … x … 6, and beneath the curve on the surface ƒsx, yd = 4 + 3x + 2y.

A wire of constant density d = 1 lies

rstd = ti +

222 3>2 t2 t j + k, 3 2

0 … t … 2,

if the density is d = 1>st + 1d. COMPUTER EXPLORATIONS In Exercises 43–46, use a CAS to perform the following steps to evaluate the line integrals. a. Find ds = ƒ vstd ƒ dt for the path rstd = gstdi + hstdj + kstdk. b. Express the integrand ƒsgstd, hstd, kstdd ƒ vstd ƒ as a function of the parameter t. c. Evaluate 1C ƒ ds using Equation (2) in the text.

35. Mass of wire with variable density Find the mass of a thin wire lying along the curve rstd = 22ti + 22tj + s4 - t 2 dk, 0 … t … 1, if the density is (a) d = 3t and (b) d = 1.

43. ƒsx, y, zd = 21 + 30x 2 + 10y ; 0 … t … 2

36. Center of mass of wire with variable density Find the center of mass of a thin wire lying along the curve rstd = ti + 2tj + s2>3dt 3>2k, 0 … t … 2, if the density is d = 315 + t.

44. ƒsx, y, zd = 21 + x 3 + 5y 3 ;

37. Moment of inertia of wire hoop A circular wire hoop of constant density d lies along the circle x 2 + y 2 = a 2 in the xy-plane. Find the hoop’s moment of inertia about the z-axis.

45. ƒsx, y, zd = x1y - 3z 2 ; rstd = scos 2tdi + ssin 2tdj + 5tk, 0 … t … 2p

38. Inertia of a slender rod A slender rod of constant density lies along the line segment rstd = tj + s2 - 2tdk, 0 … t … 1, in the

15.2

rstd = ti + t 2j + 3t 2k,

rstd = ti +

0 … t … 2

46. ƒsx, y, zd = a1 + t 5>2k,

9 1>3 z b 4 0 … t … 2p

1 2 t j + 1tk, 3

1>4

;

rstd = scos 2tdi + ssin 2tdj +

Vector Fields and Line Integrals: Work, Circulation, and Flux Gravitational and electric forces have both a direction and a magnitude. They are represented by a vector at each point in their domain, producing a vector field. In this section we show how to compute the work done in moving an object through such a field by using

15.2

Vector Fields and Line Integrals: Work, Circulation, and Flux

835

a line integral involving the vector field. We also discuss velocity fields, such as the vector field representing the velocity of a flowing fluid in its domain. A line integral can be used to find the rate at which the fluid flows along or across a curve within the domain.

Vector Fields

FIGURE 15.6 Velocity vectors of a flow around an airfoil in a wind tunnel.

Suppose a region in the plane or in space is occupied by a moving fluid, such as air or water. The fluid is made up of a large number of particles, and at any instant of time, a particle has a velocity v. At different points of the region at a given (same) time, these velocities can vary. We can think of a velocity vector being attached to each point of the fluid representing the velocity of a particle at that point. Such a fluid flow is an example of a vector field. Figure 15.6 shows a velocity vector field obtained from air flowing around an airfoil in a wind tunnel. Figure 15.7 shows a vector field of velocity vectors along the streamlines of water moving through a contracting channel. Vector fields are also associated with forces such as gravitational attraction (Figure 15.8), and to magnetic fields, electric fields, and also purely mathematical fields. Generally, a vector field is a function that assigns a vector to each point in its domain. A vector field on a three-dimensional domain in space might have a formula like Fsx, y, zd = Msx, y, zdi + Nsx, y, zdj + Psx, y, zdk.

FIGURE 15.7 Streamlines in a contracting channel. The water speeds up as the channel narrows and the velocity vectors increase in length.

The field is continuous if the component functions M, N, and P are continuous; it is differentiable if each of the component functions is differentiable. The formula for a field of two-dimensional vectors could look like Fsx, yd = Msx, ydi + Nsx, ydj. We encountered another type of vector field in Chapter 12. The tangent vectors T and normal vectors N for a curve in space both form vector fields along the curve. Along a curve r(t) they might have a component formula similar to the velocity field expression v(t) = ƒ(t)i + g(t)j + h(t)k.

z

y

If we attach the gradient vector ¥ƒ of a scalar function ƒ(x, y, z) to each point of a level surface of the function, we obtain a three-dimensional field on the surface. If we attach the velocity vector to each point of a flowing fluid, we have a three-dimensional field defined on a region in space. These and other fields are illustrated in Figures 15.9–15.15. To sketch the fields, we picked a representative selection of domain points and drew the

x z

FIGURE 15.8 Vectors in a gravitational field point toward the center of mass that gives the source of the field. y x

FIGURE 15.9 A surface, like a mesh net or parachute, in a vector field representing water or wind flow velocity vectors. The arrows show the direction and their lengths indicate speed.

836

Chapter 15: Integration in Vector Fields

y

y

x

f (x, y, z) 5 c

FIGURE 15.10 The field of gradient vectors §ƒ on a surface ƒsx, y, zd = c. z

FIGURE 15.11 The radial field F = xi + yj of position vectors of points in the plane. Notice the convention that an arrow is drawn with its tail, not its head, at the point where F is evaluated.

x

FIGURE 15.12 A “spin” field of rotating unit vectors F = s -yi + xjd>sx 2 + y 2 d1>2 in the plane. The field is not defined at the origin.

vectors attached to them. The arrows are drawn with their tails, not their heads, attached to the points where the vector functions are evaluated. x 2 y 2 a2

Gradient Fields

z a2 r 2

The gradient vector of a differentiable scalar-valued function at a point gives the direction of greatest increase of the function. An important type of vector field is formed by all the

0

x

y

FIGURE 15.13 The flow of fluid in a long cylindrical pipe. The vectors v = sa 2 - r 2 dk inside the cylinder that have their bases in the xy-plane have their tips on the paraboloid z = a 2 - r 2. y

0

x

WIND SPEED, M/S FIGURE 15.14 The velocity vectors v(t) of a projectile’s motion make a vector field along the trajectory.

0

2

4 6 8 10 12 14 16+

FIGURE 15.15 NASA’s Seasat used radar to take 350,000 wind measurements over the world’s oceans. The arrows show wind direction; their length and the color contouring indicate speed. Notice the heavy storm south of Greenland.

15.2

Vector Fields and Line Integrals: Work, Circulation, and Flux

837

gradient vectors of the function (see Section 13.5). We define the gradient field of a differentiable function ƒ(x, y, z) to be the field of gradient vectors §ƒ =

0ƒ 0ƒ 0ƒ i + j + k. 0x 0y 0z

At each point sx, y, zd, the gradient field gives a vector pointing in the direction of greatest increase of ƒ, with magnitude being the value of the directional derivative in that direction. The gradient field is not always a force field or a velocity field.

EXAMPLE 1

Suppose that the temperature T at each point (x, y, z) in a region of space

is given by T = 100 - x 2 - y 2 - z 2, and that F(x, y, z) is defined to be the gradient of T. Find the vector field F. The gradient field F is the field F = ¥T = -2xi - 2yj - 2zk. At each point in space, the vector field F gives the direction for which the increase in temperature is greatest. Solution

Line Integrals of Vector Fields In Section 15.1 we defined the line integral of a scalar function ƒsx, y, zd over a path C. We turn our attention now to the idea of a line integral of a vector field F along the curve C. Such line integrals have important applications in studying fluid flows, and electrical or gravitational fields. Assume that the vector field F = Msx, y, zdi + Nsx, y, zdj + Psx, y, zdk has continuous components, and that the curve C has a smooth parametrization rstd = gstdi + hstdj + kstdk, a … t … b. As discussed in Section 15.1, the parametrization rstd defines a direction (or orientation) along C which we call the forward direction. At each point along the path C, the tangent vector T = dr>ds = v> ƒ v ƒ is a unit vector tangent to the path and pointing in this forward direction. (The vector v = dr>dt is the velocity vector tangent to C at the point, as discussed in Sections 12.1 and 12.3.) Intuitively, the line integral of the vector field is the line integral of the scalar tangential component of F along C. This tangential component is given by the dot product F#T = F#

dr , ds

so we have the following formal definition, where ƒ = F # T in Equation (1) of Section 15.1.

DEFINITION Let F be a vector field with continuous components defined along a smooth curve C parametrized by rstd, a … t … b. Then the line integral of F along C is LC

F # T ds =

LC

aF #

dr b ds = F # dr. ds LC

We evaluate line integrals of vector fields in a way similar to how we evaluate line integrals of scalar functions (Section 15.1).

838

Chapter 15: Integration in Vector Fields

Evaluating the Line Integral of F Mi Nj Pk along C: r(t) g(t)i h(t)j k(t)k 1. Express the vector field F in terms of the parametrized curve C as Fsrstdd by substituting the components x = gstd, y = hstd, z = kstd of r into the scalar components Msx, y, zd, Nsx, y, zd, Psx, y, zd of F. 2. Find the derivative (velocity) vector dr>dt . 3. Evaluate the line integral with respect to the parameter t, a … t … b, to obtain

LC

EXAMPLE 2

F # dr =

b

La

Fsrstdd #

dr dt . dt

Evaluate 1C F # dr, where Fsx, y, zd = zi + xyj - y2k along the curve C

given by rstd = t 2i + tj + 2t k, 0 … t … 1. Solution

We have Fsrstdd = 2t i + t 3j - t 2k

z = 1t, xy = t 3, -y 2 = -t 2

and dr 1 = 2t i + j + k. dt 22t Thus, LC

F # dr = =

1

Fsrstdd #

1

a2t 3>2 + t 3 -

L0 L0

dr dt dt 1 3>2 t b dt 2 1

3 2 17 1 = c a b a t 5>2 b + t 4 d = . 2 5 4 20 0

Line Integrals with Respect to dx, dy, or dz When analyzing forces or flows, it is often useful to consider each component direction separately. In such situations we want a line integral of a scalar function with respect to one of the coordinates, such as 1C M dx. This integral is not the same as the arc length line integral 1C M ds we defined in Section 15.1. To define the integral 1C M dx for the scalar function Msx, y, zd, we specify a vector field F = Msx, y, zdi having a component only in the x-direction, and none in the y- or z-direction. Then, over the curve C parametrized by rstd = gstdi + hstdj + kstdk for a … t … b, we have x = gstd, dx = g¿std dt, and F # dr = F #

dr dt = Msx, y, zdg¿std dt = Msx, y, zd dx. dt From the definition of the line integral of F along C, we define LC

Msx, y, zd dx =

LC

F # dr,

where

F = Msx, y, zd i.

In the same way, by defining F = Nsx, y, zd j with a component only in the y-direction, or as F = Psx, y, zdk with a component only in the z-direction, we can obtain the line integrals 1C N dy and 1C P dz. Expressing everything in terms of the parameter t along the curve C, we have the following formulas for these three integrals:

15.2

Vector Fields and Line Integrals: Work, Circulation, and Flux

839

b

Msx, y, zd dx =

LC

La

Msgstd, hstd, kstdd g¿std dt

(1)

Nsgstd, hstd, kstdd h¿std dt

(2)

Psgstd, hstd, kstdd k¿std dt

(3)

b

LC

Nsx, y, zd dy =

La b

LC

Psx, y, zd dz =

La

It often happens that these line integrals occur in combination, and we abbreviate the notation by writing LC

Msx, y, zd dx +

LC

Nsx, y, zd dy +

LC

Psx, y, zd dz =

LC

M dx + N dy + P dz.

Evaluate the line integral 1C -y dx + z dy + 2x dz, where C is the helix rstd = (cos t)i + (sin t)j + t k, 0 … t … 2p.

EXAMPLE 3

We express everything in terms of the parameter t, so x = cos t, y = sin t, z = t, and dx = -sin t dt, dy = cos t dt, dz = dt. Then,

Solution

2p

LC

-y dx + z dy + 2x dz =

[s -sin tds -sin td + t cos t + 2 cos t] dt

L0 2p

=

L0

[2 cos t + t cos t + sin2 t] dt

= c2 sin t + st sin t + cos td + a

2p

sin 2t t bd 2 4 0

= [0 + s0 + 1d + sp - 0d] - [0 + s0 + 1d + s0 - 0d] = p.

Work Done by a Force over a Curve in Space Suppose that the vector field F = Msx, y, zdi + Nsx, y, zdj + Psx, y, zdk represents a force throughout a region in space (it might be the force of gravity or an electromagnetic force of some kind) and that rstd = gstdi + hstdj + kstdk, Fk

Pk

(xk , yk , zk ) Pk21

Tk Fk . Tk

FIGURE 15.16 The work done along the subarc shown here is approximately Fk # Tk ¢sk, where Fk = Fsxk, yk, zk d and Tk = Tsxk, yk, zk d.

a … t … b,

is a smooth curve in the region. The formula for the work done by the force in moving an object along the curve is motivated by the same kind of reasoning we used in Chapter 6 to b derive the formula W = 1a F(x) dx for the work done by a continuous force of magnitude F(x) directed along an interval of the x-axis. For a curve C in space, we define the work done by a continuous force field F to move an object along C from a point A to another point B as follows. We divide C into n subarcs Pk - 1Pk with lengths ¢sk, starting at A and ending at B. We choose any point sxk, yk, zk d in the subarc Pk - 1Pk and let Tsxk, yk, zk d be the unit tangent vector at the chosen point. The work Wk done to move the object along the subarc Pk - 1Pk is approximated by the tangential component of the force Fsxk, yk, zk d times the arclength ¢sk approximating the distance the object moves along the subarc (see Figure 15.16).

840

Chapter 15: Integration in Vector Fields

The total work done in moving the object from point A to point B is then approximated by summing the work done along each of the subarcs, so W L a Wk L a Fsxk, yk, zk d # Tsxk, yk, zk d ¢sk. n

n

k=1

k=1

For any subdivision of C into n subarcs, and for any choice of the points sxk, yk, zk d within each subarc, as n : q and ¢sk : 0, these sums approach the line integral F # T ds. LC This is just the line integral of F along C, which is defined to be the total work done.

B tb

DEFINITION Let C be a smooth curve parametrized by rstd, a … t … b, and F be a continuous force field over a region containing C. Then the work done in moving an object from the point A = rsad to the point B = rsbd along C is

T

W =

F

A ta

FIGURE 15.17 The work done by a force F is the line integral of the scalar component F # T over the smooth curve from A to B.

LC

F # T ds =

b

La

Fsrstdd #

dr dt. dt

(4)

The sign of the number we calculate with this integral depends on the direction in which the curve is traversed. If we reverse the direction of motion, then we reverse the direction of T in Figure 15.17 and change the sign of F # T and its integral. Using the notations we have presented, we can express the work integral in a variety of ways, depending upon what seems most suitable or convenient for a particular discussion. Table 15.2 shows five ways we can write the work integral in Equation (4). In the table, the field components M, N, and P are functions of the intermediate variables x, y, and z, which in turn are functions of the independent variable t along the curve C in the vector field. So along the curve, x = g (t), y = h (t), and z = k (t) with dx = g¿(t) dt, dy = h¿(t) dt, and dz = k¿(t) dt. TABLE 15.2 Different ways to write the work integral for F = Mi + Nj + Pk over

the curve C : r(t) = g(t)i + h(t)j + k(t)k, a … t … b W = =

LC LC

F # T ds

The definition

F # dr

Vector differential form

b

=

La b

=

z

= (0, 0, 0)

(1, 1, 1)

La

F#

dr dt dt

Parametric vector evaluation

A Mg¿(t) + Nh¿(t) + Pk¿(t) B dt Parametric scalar evaluation

M dx + N dy + P dz

LC

Scalar differential form

Find the work done by the force field F = s y - x 2 di + sz - y 2 dj + sx - z dk along the curve rstd = ti + t 2j + t 3k, 0 … t … 1, from (0, 0, 0) to (1, 1, 1) (Figure 15.18).

EXAMPLE 4 2

y

Solution x r(t) ti t 2j t 3k

First we evaluate F on the curve rstd: F = s y - x 2 di + sz - y 2 dj + sx - z 2 dk

(1, 1, 0)

FIGURE 15.18 The curve in Example 4.

= st 2 - t 2 di + st 3 - t 4 dj + st - t 6 dk . (')'* 0

Substitute x = t, y = t 2, z = t 3.

15.2

Vector Fields and Line Integrals: Work, Circulation, and Flux

841

Then we find dr> dt, dr d = sti + t 2j + t 3kd = i + 2tj + 3t 2k. dt dt Finally, we find F # dr>dt and integrate from t = 0 to t = 1: F#

dr = [st 3 - t 4 dj + st - t 6 dk] # si + 2tj + 3t 2kd dt = st 3 - t 4 ds2td + st - t 6 ds3t 2 d = 2t 4 - 2t 5 + 3t 3 - 3t 8

so, 1

Work =

L0

s2t 4 - 2t 5 + 3t 3 - 3t 8 d dt 1

3 3 29 2 2 = c t5 - t6 + t4 - t9d = . 5 6 4 9 60 0

EXAMPLE 5 Find the work done by the force field F = xi + yj + zk in moving an object along the curve C parametrized by r(t) = cos (pt) i + t 2j + sin (pt) k, 0 … t … 1. Solution

We begin by writing F along C as a function of t,

Next we compute dr> dt,

Fsrstdd = cos (pt) i + t 2j + sin (pt) k.

dr = -p sin (pt) i + 2tj + p cos (pt) k. dt We then calculate the dot product, F(r(t)) #

dr = -p sin (pt) cos (pt) + 2t 3 + p sin (pt) cos(pt) = 2t 3. dt

The work done is the line integral b

La

F(r(t)) #

1

1

dr t4 1 2t 3 dt = d = . dt = 2 0 2 dt L0

Flow Integrals and Circulation for Velocity Fields Suppose that F represents the velocity field of a fluid flowing through a region in space (a tidal basin or the turbine chamber of a hydroelectric generator, for example). Under these circumstances, the integral of F # T along a curve in the region gives the fluid’s flow along, or circulation around, the curve. For instance, the vector field in Figure 15.11 gives zero circulation around the unit circle in the plane. By contrast, the vector field in Figure 15.12 gives a nonzero circulation around the unit circle.

DEFINITIONS If r(t) parametrizes a smooth curve C in the domain of a continuous velocity field F, the flow along the curve from A = r(a) to B = r(b) is F # T ds. (5) LC The integral is called a flow integral. If the curve starts and ends at the same point, so that A = B, the flow is called the circulation around the curve. Flow =

842

Chapter 15: Integration in Vector Fields

The direction we travel along C matters. If we reverse the direction, then T is replaced by -T and the sign of the integral changes. We evaluate flow integrals the same way we evaluate work integrals.

EXAMPLE 6 A fluid’s velocity field is F = xi + zj + yk. Find the flow along the helix rstd = scos tdi + ssin tdj + tk, 0 … t … p>2. Solution

We evaluate F on the curve,

F = xi + zj + yk = scos tdi + tj + ssin tdk

Substitute x = cos t, z = t, y = sin t.

and then find dr> dt:

dr = s -sin tdi + scos tdj + k. dt Then we integrate F # sdr>dtd from t = 0 to t = F#

p : 2

dr = scos tds -sin td + stdscos td + ssin tds1d dt = -sin t cos t + t cos t + sin t

so, Flow =

y

t=b

Lt = a

= c x

F#

dr dt = dt L0 p>2

cos2 t + t sin t d 2 0

p>2

s -sin t cos t + t cos t + sin td dt = a0 +

p p 1 1 b - a + 0b = - . 2 2 2 2

EXAMPLE 7 Find the circulation of the field F = sx - ydi + xj around the circle rstd = scos tdi + ssin tdj, 0 … t … 2p (Figure 15.19). Solution

On the circle, F = sx - ydi + xj = scos t - sin tdi + scos tdj, and dr = s -sin tdi + scos tdj. dt

FIGURE 15.19 The vector field F and curve r(t) in Example 7.

Then F#

dr = -sin t cos t + (''')'''* sin2 t + cos2 t dt 1

gives 2p

Circulation = Simple, not closed

Simple, closed

L0

= ct -

F#

dr dt = dt L0

2p

s1 - sin t cos td dt

2p

sin2 t d = 2p. 2 0

As Figure 15.19 suggests, a fluid with this velocity field is circulating counterclockwise around the circle, so the circulation is positive. Not simple, not closed

Not simple, closed

FIGURE 15.20 Distinguishing curves that are simple or closed. Closed curves are also called loops.

Flux Across a Simple Plane Curve A curve in the xy-plane is simple if it does not cross itself (Figure 15.20). When a curve starts and ends at the same point, it is a closed curve or loop. To find the rate at which a fluid is entering or leaving a region enclosed by a smooth simple closed curve C in the xy-plane,

15.2

Vector Fields and Line Integrals: Work, Circulation, and Flux

843

we calculate the line integral over C of F # n, the scalar component of the fluid’s velocity field in the direction of the curve’s outward-pointing normal vector. We use only the normal component of F, while ignoring the tangential component, because the normal component leads to the flow across C. The value of this integral is the flux of F across C. Flux is Latin for flow, but many flux calculations involve no motion at all. If F were an electric field or a magnetic field, for instance, the integral of F # n is still called the flux of the field across C.

DEFINITION If C is a smooth simple closed curve in the domain of a continuous vector field F = Msx, ydi + Nsx, ydj in the plane, and if n is the outwardpointing unit normal vector on C, the flux of F across C is Flux of F across C =

z

For clockwise motion, k T points outward. y C

T

x = gstd,

kT z For counterclockwise motion, T k points outward. y k x

C

Tk

(6)

y = hstd,

a … t … b,

that traces the curve C exactly once as t increases from a to b. We can find the outward unit normal vector n by crossing the curve’s unit tangent vector T with the vector k. But which order do we choose, T * k or k * T? Which one points outward? It depends on which way C is traversed as t increases. If the motion is clockwise, k * T points outward; if the motion is counterclockwise, T * k points outward (Figure 15.21). The usual choice is n = T * k, the choice that assumes counterclockwise motion. Thus, although the value of the integral in Equation (6) does not depend on which way C is traversed, the formulas we are about to derive for computing n and evaluating the integral assume counterclockwise motion. In terms of components,

T

FIGURE 15.21 To find an outward unit normal vector for a smooth simple curve C in the xy-plane that is traversed counterclockwise as t increases, we take n = T * k. For clockwise motion, we take n = k * T.

F # n ds.

Notice the difference between flux and circulation. The flux of F across C is the line integral with respect to arc length of F # n, the scalar component of F in the direction of the outward normal. The circulation of F around C is the line integral with respect to arc length of F # T, the scalar component of F in the direction of the unit tangent vector. Flux is the integral of the normal component of F; circulation is the integral of the tangential component of F. To evaluate the integral for flux in Equation (6), we begin with a smooth parametrization

n = T * k = a

dy dy dx dx i + jb * k = i j. ds ds ds ds

If F = Msx, ydi + Nsx, ydj, then F # n = Msx, yd

dy dx - Nsx, yd . ds ds

Hence, LC

F # n ds = 哷

k x

LC

LC

aM

dy dx - N b ds = M dy - N dx. ds ds F C

We put a directed circle ~ on the last integral as a reminder that the integration around the closed curve C is to be in the counterclockwise direction. To evaluate this integral, we express M, dy, N, and dx in terms of the parameter t and integrate from t = a to t = b. We do not need to know n or ds explicitly to find the flux.

844

Chapter 15: Integration in Vector Fields

Calculating Flux Across a Smooth Closed Plane Curve sFlux of F = Mi + Nj across Cd =

F

M dy - N dx

(7)

C

The integral can be evaluated from any smooth parametrization x = gstd, y = hstd, a … t … b, that traces C counterclockwise exactly once.

Find the flux of F = sx - ydi + xj across the circle x 2 + y 2 = 1 in the xy-plane. (The vector field and curve were shown previously in Figure 15.19.)

EXAMPLE 8

The parametrization rstd = scos tdi + ssin tdj, 0 … t … 2p, traces the circle counterclockwise exactly once. We can therefore use this parametrization in Equation (7). With dy = dssin td = cos t dt M = x - y = cos t - sin t,

Solution

N = x = cos t,

dx = dscos td = -sin t dt,

we find 2p

Flux =

LC

M dy - N dx = 2p

=

L0

L0 2p

cos2 t dt =

L0

scos2 t - sin t cos t + cos t sin td dt

Eq. (7)

2p

1 + cos 2t sin 2t t d = p. dt = c + 2 2 4 0

The flux of F across the circle is p. Since the answer is positive, the net flow across the curve is outward. A net inward flow would have given a negative flux.

Exercises 15.2 Vector Fields Find the gradient fields of the functions in Exercises 1–4. 1. ƒsx, y, zd = sx 2 + y 2 + z 2 d-1>2 2. ƒsx, y, zd = ln 2x 2 + y 2 + z 2 3. gsx, y, zd = e z - ln sx 2 + y 2 d 4. gsx, y, zd = xy + yz + xz 5. Give a formula F = Msx, ydi + Nsx, ydj for the vector field in the plane that has the property that F points toward the origin with magnitude inversely proportional to the square of the distance from (x, y) to the origin. (The field is not defined at (0, 0).)

a. The straight-line path C1: rstd = ti + tj + tk, b. The curved path C2: rstd = ti + t 2j + t 4 k,

c. The path C3 ´ C4 consisting of the line segment from (0, 0, 0) to (1, 1, 0) followed by the segment from (1, 1, 0) to (1, 1, 1) 8. F = [1>sx 2 + 1d]j

7. F = 3yi + 2xj + 4zk 9. F = 1zi - 2xj + 1yk

10. F = xyi + yzj + xzk

11. F = s3x 2 - 3xdi + 3zj + k 12. F = s y + zdi + sz + xdj + sx + ydk z

6. Give a formula F = Msx, ydi + Nsx, ydj for the vector field in the plane that has the properties that F = 0 at (0, 0) and that at any other point (a, b), F is tangent to the circle x 2 + y 2 = a 2 + b 2 and points in the clockwise direction with magnitude 2 2 ƒ F ƒ = 2a + b . Line Integrals of Vector Fields In Exercises 7–12, find the line integrals of F from (0, 0, 0) to (1, 1, 1) over each of the following paths in the accompanying figure.

0 … t … 1 0 … t … 1

(0, 0, 0)

C1

C2

(1, 1, 1)

C4 C3

x (1, 1, 0)

y

15.2 Line Integrals with Respect to x, y, and z In Exercises 13–16, find the line integrals along the given path C. 13. 14. 15.

LC LC LC

(x - y) dx, where C: x = t, y = 2t + 1, for 0 … t … 3 x 2 y dy, where C: x = t, y = t , for 1 … t … 2 (x 2 + y 2) dy, where C is given in the accompanying figure. y (3, 3) C (0, 0)

16.

LC

(3, 0)

x

2x + y dx, where C is given in the accompanying figure.

Vector Fields and Line Integrals: Work, Circulation, and Flux Line Integrals in the Plane

23. Evaluate 1C xy dx + sx + yd dy along the curve y = x 2 from s -1, 1d to (2, 4). 24. Evaluate 1C sx - yd dx + sx + yd dy counterclockwise around the triangle with vertices (0, 0), (1, 0), and (0, 1).

25. Evaluate 1C F # T ds for the vector field F = x 2i - yj along the curve x = y 2 from (4, 2) to s1, -1d .

26. Evaluate 1C F # dr for the vector field F = yi - xj counterclockwise along the unit circle x 2 + y 2 = 1 from (1, 0) to (0, 1). Work, Circulation, and Flux in the Plane 27. Work Find the work done by the force F = xyi + sy - xdj over the straight line from (1, 1) to (2, 3). 28. Work Find the work done by the gradient of ƒsx, yd = sx + yd2 counterclockwise around the circle x 2 + y 2 = 4 from (2, 0) to itself. 29. Circulation and flux Find the circulation and flux of the fields F1 = xi + yj

y

F2 = -yi + xj

and

around and across each of the following curves.

C

a. The circle rstd = scos tdi + ssin tdj,

(1, 3)

(0, 3)

0 … t … 2p 0 … t … 2p

b. The ellipse rstd = scos tdi + s4 sin tdj, y 5 3x

30. Flux across a circle

b.

LC

(x + y - z) dx (x + y - z) dy

(x + y - z) dz LC 18. Along the curve r(t) = (cos t)i + (sin t)j - (cos t)k, 0 … t … p, evaluate each of the following integrals. c.

a.

LC

xz dx

b.

LC

xz dy

c.

LC

xyz dz

Work In Exercises 19–22, find the work done by F over the curve in the direction of increasing t. 19. F = xyi + yj - yzk rstd = ti + t 2j + tk,

F2 = 2xi + sx - ydj

and

across the circle

17. Along the curve r(t) = ti - j + t 2k, 0 … t … 1, evaluate each of the following integrals. LC

Find the flux of the fields

F1 = 2xi - 3yj

x

(0, 0)

a.

845

21. F = zi + xj + yk rstd = ssin tdi + scos tdj + tk,

0 … t … 2p.

In Exercises 31–34, find the circulation and flux of the field F around and across the closed semicircular path that consists of the semicircular arch r1std = sa cos tdi + sa sin tdj, 0 … t … p, followed by the line segment r2std = ti, -a … t … a. 31. F = xi + yj

32. F = x 2 i + y 2 j

33. F = -yi + xj

34. F = -y 2 i + x 2 j

35. Flow integrals Find the flow of the velocity field F = sx + ydi - sx 2 + y 2 dj along each of the following paths from (1, 0) to s -1, 0d in the xy-plane. a. The upper half of the circle x 2 + y 2 = 1 b. The line segment from (1, 0) to s -1, 0d c. The line segment from (1, 0) to s0, -1d followed by the line segment from s0, -1d to s -1, 0d 36. Flux across a triangle Find the flux of the field F in Exercise 35 outward across the triangle with vertices (1, 0), (0, 1), s -1, 0d. 37. Find the flow of the velocity field F = y 2i + 2xyj along each of the following paths from (0, 0) to (2, 4). a.

0 … t … 1

20. F = 2yi + 3xj + sx + ydk rstd = scos tdi + ssin tdj + st>6dk,

rstd = sa cos tdi + sa sin tdj,

b.

y

y

(2, 4)

0 … t … 2p

(2, 4)

y 5 2x

y 5 x2

0 … t … 2p

22. F = 6zi + y j + 12xk rstd = ssin tdi + scos tdj + st>6dk, 2

0 … t … 2p

(0, 0)

2

x

(0, 0)

2

x

c. Use any path from (0, 0) to (2, 4) different from parts (a) and (b).

846

Chapter 15: Integration in Vector Fields

38. Find the circulation of the field F = yi + (x + 2y)j around each of the following closed paths. a.

y (–1, 1)

(1, 1)

45. Work and area Suppose that ƒ(t) is differentiable and positive for a … t … b. Let C be the path rstd = ti + ƒstdj, a … t … b, and F = yi. Is there any relation between the value of the work integral

x (–1, –1)

b.

(1, –1)

y

44. Two “central” fields Find a field F = Msx, ydi + Nsx, ydj in the xy-plane with the property that at each point sx, yd Z s0, 0d, F points toward the origin and ƒ F ƒ is (a) the distance from (x, y) to the origin, (b) inversely proportional to the distance from (x, y) to the origin. (The field is undefined at (0, 0).)

LC

x 2 y2 4

F # dr

and the area of the region bounded by the t-axis, the graph of ƒ, and the lines t = a and t = b? Give reasons for your answer. x

c. Use any closed path different from parts (a) and (b). Vector Fields in the Plane 39. Spin field Draw the spin field F = -

y 2x + y 2 2

i +

46. Work done by a radial force with constant magnitude A particle moves along the smooth curve y = ƒsxd from (a, ƒ(a)) to (b, ƒ(b)). The force moving the particle has constant magnitude k and always points away from the origin. Show that the work done by the force is

LC x 2x 2 + y 2

j

(see Figure 15.12) along with its horizontal and vertical components at a representative assortment of points on the circle x 2 + y 2 = 4. 40. Radial field Draw the radial field F = xi + yj (see Figure 15.11) along with its horizontal and vertical components at a representative assortment of points on the circle x 2 + y 2 = 1. 41. A field of tangent vectors a. Find a field G = Psx, ydi + Qsx, ydj in the xy-plane with the property that at any point sa, bd Z s0, 0d, G is a vector of magnitude 2a 2 + b 2 tangent to the circle x 2 + y 2 = a 2 + b 2 and pointing in the counterclockwise direction. (The field is undefined at (0, 0).) b. How is G related to the spin field F in Figure 15.12?

F # T ds = k C sb 2 + sƒsbdd2 d1>2 - sa 2 + sƒsadd2 d1>2 D .

Flow Integrals in Space In Exercises 47–50, F is the velocity field of a fluid flowing through a region in space. Find the flow along the given curve in the direction of increasing t. 47. F = - 4xyi + 8yj + 2k rstd = ti + t 2j + k, 0 … t … 2 48. F = x 2i + yzj + y 2k rstd = 3tj + 4tk, 0 … t … 1 49. F = sx - zdi + xk rstd = scos tdi + ssin tdk,

0 … t … 2p

51. Circulation Find the circulation of F = 2xi + 2zj + 2yk around the closed path consisting of the following three curves traversed in the direction of increasing t. C1:

rstd = scos tdi + ssin tdj + tk,

C2:

rstd = j + sp>2ds1 - tdk,

C3:

rstd = ti + s1 - tdj,

42. A field of tangent vectors

z

a. Find a field G = Psx, ydi + Qsx, ydj in the xy-plane with the property that at any point sa, bd Z s0, 0d, G is a unit vector tangent to the circle x 2 + y 2 = a 2 + b 2 and pointing in the clockwise direction. (1, 0, 0)

0 … t … 1

0 … t … 1

C2 (0, 1, 0)

C3 x

0 … t … p>2

⎛0, 1, ⎛ ⎝ 2⎝ C1

b. How is G related to the spin field F in Figure 15.12? 43. Unit vectors pointing toward the origin Find a field F = Msx, ydi + Nsx, ydj in the xy-plane with the property that at each point sx, yd Z s0, 0d, F is a unit vector pointing toward the origin. (The field is undefined at (0, 0).)

0 … t … p

50. F = -yi + xj + 2k rstd = s -2 cos tdi + s2 sin tdj + 2tk,

y

15.3

Path Independence, Conservative Fields, and Potential Functions

52. Zero circulation Let C be the ellipse in which the plane 2x + 3y - z = 0 meets the cylinder x 2 + y 2 = 12. Show, without evaluating either line integral directly, that the circulation of the field F = xi + yj + zk around C in either direction is zero. 53. Flow along a curve The field F = xyi + yj - yzk is the velocity field of a flow in space. Find the flow from (0, 0, 0) to (1, 1, 1) along the curve of intersection of the cylinder y = x 2 and the plane z = x . (Hint: Use t = x as the parameter.) z

COMPUTER EXPLORATIONS In Exercises 55–60, use a CAS to perform the following steps for finding the work done by force F over the given path: a. Find dr for the path rstd = gstdi + hstdj + kstdk. b. Evaluate the force F along the path. F # dr. LC 55. F = xy 6 i + 3xsxy 5 + 2dj; 0 … t … 2p c. Evaluate

3 2 i + j; 1 + x2 1 + y2 0 … t … p

56. F = zx

(1, 1, 1) y

54. Flow of a gradient field Find the flow of the field F = §sxy 2z 3 d: a. Once around the curve C in Exercise 52, clockwise as viewed from above b. Along the line segment from (1, 1, 1) to s2, 1, -1d.

15.3

rstd = s2 cos tdi + ssin tdj, rstd = scos tdi + ssin tdj,

57. F = s y + yz cos xyzdi + sx 2 + xz cos xyzdj + sz + xy cos xyzdk; rstd = (2 cos t)i + (3 sin t)j + k, 0 … t … 2p 58. F = 2xyi - y 2j + ze x k; 1 … t … 4

y x2 x

847

rstd = -ti + 1tj + 3tk,

59. F = s2y + sin xdi + sz 2 + s1>3dcos ydj + x 4 k; rstd = ssin tdi + scos tdj + ssin 2tdk, -p>2 … t … p>2 1 60. F = sx 2ydi + x 3j + xyk; rstd = scos tdi + ssin tdj + 3 s2 sin2 t - 1dk, 0 … t … 2p

Path Independence, Conservative Fields, and Potential Functions A gravitational field G is a vector field that represents the effect of gravity at a point in space due to the presence of a massive object. The gravitational force on a body of mass m placed in the field is given by F = mG. Similarly, an electric field E is a vector field in space that represents the effect of electric forces on a charged particle placed within it. The force on a body of charge q placed in the field is given by F = qE. In gravitational and electric fields, the amount of work it takes to move a mass or charge from one point to another depends on the initial and final positions of the object—not on which path is taken between these positions. In this section we study vector fields with this property and the calculation of work integrals associated with them.

Path Independence If A and B are two points in an open region D in space, the line integral of F along C from A to B for a field F defined on D usually depends on the path C taken, as we saw in Section 15.1. For some special fields, however, the integral’s value is the same for all paths from A to B.

DEFINITIONS Let F be a vector field defined on an open region D in space, and suppose that for any two points A and B in D the line integral 1C F # dr along a path C from A to B in D is the same over all paths from A to B. Then the integral # 1C F dr is path independent in D and the field F is conservative on D. The word conservative comes from physics, where it refers to fields in which the principle of conservation of energy holds. When a line integral is independent of the path C from

848

Chapter 15: Integration in Vector Fields B

point A to point B, we sometimes represent the integral by the symbol 1A rather than the usual line integral symbol 1C . This substitution helps us remember the path-independence property. Under differentiability conditions normally met in practice, we will show that a field F is conservative if and only if it is the gradient field of a scalar function ƒ—that is, if and only if F = §ƒ for some ƒ. The function ƒ then has a special name.

DEFINITION If F is a vector field defined on D and F = §ƒ for some scalar function ƒ on D, then ƒ is called a potential function for F.

A gravitational potential is a scalar function whose gradient field is a gravitational field, an electric potential is a scalar function whose gradient field is an electric field, and so on. As we will see, once we have found a potential function ƒ for a field F, we can evaluate all the line integrals in the domain of F over any path between A and B by B

LA

F # dr =

B

LA

§ƒ # dr = ƒsBd - ƒsAd.

(1)

If you think of §ƒ for functions of several variables as being something like the derivative ƒ¿ for functions of a single variable, then you see that Equation (1) is the vector calculus analogue of the Fundamental Theorem of Calculus formula b

La

ƒ¿sxd dx = ƒsbd - ƒsad.

Conservative fields have other important properties. For example, saying that F is conservative on D is equivalent to saying that the integral of F around every closed path in D is zero. Certain conditions on the curves, fields, and domains must be satisfied for Equation (1) to be valid. We discuss these conditions next.

Assumptions on Curves, Vector Fields, and Domains In order for the computations and results we derive below to be valid, we must assume certain properties for the curves, surfaces, domains, and vector fields we consider. We give these assumptions in the statements of theorems, and they also apply to the examples and exercises unless otherwise stated. The curves we consider are piecewise smooth. Such curves are made up of finitely many smooth pieces connected end to end, as discussed in Section 12.1. We will treat vector fields F whose components have continuous first partial derivatives. The domains D we consider are open regions in space, so every point in D is the center of an open ball that lies entirely in D (see Section 12.1). We also assume D to be connected. For an open region, this means that any two points in D can be joined by a smooth curve that lies in the region. Finally, we assume D is simply connected, which means that every loop in D can be contracted to a point in D without ever leaving D. The plane with a disk removed is a two-dimensional region that is not simply connected; a loop in the plane that goes around the disk cannot be contracted to a point without going into the “hole” left by the removed disk (see Figure 15.22c). Similarly, if we remove a line from space, the remaining region D is not simply connected. A curve encircling the line cannot be shrunk to a point while remaining inside D.

15.3

Path Independence, Conservative Fields, and Potential Functions

849

Connectivity and simple connectivity are not the same, and neither property implies the other. Think of connected regions as being in “one piece” and simply connected regions as not having any “loop-catching holes.” All of space itself is both connected and simply connected. Figure 15.22 illustrates some of these properties.

y

Simply connected

Caution Some of the results in this chapter can fail to hold if applied to situations where the conditions we’ve imposed do not hold. In particular, the component test for conservative fields, given later in this section, is not valid on domains that are not simply connected (see Example 5).

x

(a)

Line Integrals in Conservative Fields

z

Gradient fields F are obtained by differentiating a scalar function ƒ. A theorem analogous to the Fundamental Theorem of Calculus gives a way to evaluate the line integrals of gradient fields.

Simply connected

THEOREM 1—Fundamental Theorem of Line Integrals Let C be a smooth curve joining the point A to the point B in the plane or in space and parametrized by r(t). Let ƒ be a differentiable function with a continuous gradient vector F = ¥ƒ on a domain D containing C. Then

y

x (b)

LC

y

F # dr = ƒ(B) - ƒ(A).

Like the Fundamental Theorem, Theorem 1 gives a way to evaluate line integrals without having to take limits of Riemann sums or finding the line integral by the procedure used in Section 15.2. Before proving Theorem 1, we give an example.

C1 Not simply connected x

EXAMPLE 1

Suppose the force field F = ¥ƒ is the gradient of the function

(c)

ƒ(x, y, z) = -

z

1 . x2 + y2 + z2

Find the work done by F in moving an object along a smooth curve C joining (1, 0, 0) to (0, 0, 2) that does not pass through the origin. An application of Theorem 1 shows that the work done by F along any smooth curve C joining the two points and not passing through the origin is

Solution C2 Not simply connected

y

x

LC (d)

FIGURE 15.22 Four connected regions. In (a) and (b), the regions are simply connected. In (c) and (d), the regions are not simply connected because the curves C1 and C2 cannot be contracted to a point inside the regions containing them.

F # dr = ƒ(0, 0, 2) - ƒ(1, 0, 0) = -

3 1 - (-1) = . 4 4

The gravitational force due to a planet, and the electric force associated with a charged particle, can both be modeled by the field F given in Example 1 up to a constant that depends on the units of measurement. Proof of Theorem 1 Suppose that A and B are two points in region D and that C: rstd = gstdi + hstdj + kstdk, a … t … b, is a smooth curve in D joining A to B.

850

Chapter 15: Integration in Vector Fields

We use the abbreviated form r(t) = xi + yj + zk for the parametrization of the curve. Along the curve, ƒ is a differentiable function of t and dƒ 0ƒ dx 0ƒ dy 0ƒ dz = + + 0x dt 0y dt 0z dt dt = §ƒ # a

Chain Rule in Section 13.4 with x = gstd, y = hstd, z = kstd

dy dz dr dr dx i + j + kb = §ƒ # = F# . dt dt dt dt dt

Because F = §ƒ

Therefore,

LC

F # dr =

t=b

Lt = a

F#

b dƒ dr dt = dt dt La dt

r(a) = A, r(b) = B

b

= ƒsgstd, hstd, kstdd d = ƒsBd - ƒsAd. a

So we see from Theorem 1 that the line integral of a gradient field F = ¥ƒ is straightforward to compute once we know the function ƒ. Many important vector fields arising in applications are indeed gradient fields. The next result, which follows from Theorem 1, shows that any conservative field is of this type.

THEOREM 2—Conservative Fields are Gradient Fields Let F = Mi + Nj + Pk be a vector field whose components are continuous throughout an open connected region D in space. Then F is conservative if and only if F is a gradient field ¥ƒ for a differentiable function ƒ.

Theorem 2 says that F = ¥ƒ if and only if for any two points A and B in the region D, the value of line integral 1C F # dr is independent of the path C joining A to B in D. z D

C0 A (x0, y, z) L B0 B (x, y, z) x0

y

x x

FIGURE 15.23 The function ƒ(x, y, z) in the proof of Theorem 2 is computed by a line integral 1C0 F # dr = ƒ(B0) from A to B0, plus a line integral 1L F # dr along a line segment L parallel to the x-axis and joining B0 to B located at (x, y, z). The value of ƒ at A is ƒ(A) = 0.

Proof of Theorem 2 If F is a gradient field, then F = ¥ƒ for a differentiable function ƒ, and Theorem 1 shows that 1C F # dr = ƒ(B) - ƒ(A). The value of the line integral does not depend on C, but only on its endpoints A and B. So the line integral is path independent and F satisfies the definition of a conservative field. On the other hand, suppose that F is a conservative vector field. We want to find a function ƒ on D satisfying §ƒ = F. First, pick a point A in D and set ƒ(A) = 0. For any other point B in D define ƒ(B) to equal 1C F # dr, where C is any smooth path in D from A to B. The value of ƒ(B) does not depend on the choice of C, since F is conservative. To show that §ƒ = F we need to demonstrate that 0ƒ>0x = M, 0ƒ>0y = N , and 0ƒ>0z = P. Suppose that B has coordinates (x, y, z). By definition, the value of the function ƒ at a nearby point B0 located at (x0, y, z) is 1C0 F # dr, where C0 is any path from A to B0. We take a path C = C0 h L from A to B formed by first traveling along C0 to arrive at B0 and then traveling along the line segment L from B0 to B (Figure 15.23). When B0 is close to B, the segment L lies in D and, since the value ƒ(B) is independent of the path from A to B, ƒ(x, y, z) =

LC0

F # dr +

LL

F # dr.

Differentiating, we have 0 0 ƒ(x, y, z) = a F # dr + F # drb. 0x 0x LC0 LL

15.3

Path Independence, Conservative Fields, and Potential Functions

851

Only the last term on the right depends on x, so 0 0 ƒ(x, y, z) = F # dr. 0x 0x LL Now parametrize L as r(t) = ti + yj + zk, x0 … t … x. Then dr>dt = i, F # dr>dt = M , x and 1L F # dr = 1x0 M(t, y, z) dt. Differentiating then gives x

0 0 M(t, y, z) dt = M(x, y, z) ƒ(x, y, z) = 0x 0x L x0 by the Fundamental Theorem of Calculus. The partial derivatives 0ƒ>0y = N and 0ƒ>0z = P follow similarly, showing that F = §ƒ.

EXAMPLE 2

Find the work done by the conservative field F = yzi + xzj + xyk = §ƒ,

where

ƒ(x, y, z) = xyz,

along any smooth curve C joining the point As -1, 3, 9d to Bs1, 6, -4d. With ƒsx, y, zd = xyz, we have

Solution

LC

F # dr =

B

LA

§ƒ # dr

F = §ƒ and path independence

= ƒsBd - ƒsAd

Theorem 1

= xyz ƒ s1,6, -4d - xyz ƒ s-1,3,9d = s1ds6ds -4d - s -1ds3ds9d = -24 + 27 = 3. A very useful property of line integrals in conservative fields comes into play when the path of integration is a closed curve, or loop. We often use the notation D for integration around a closed path (discussed with more detail in the next section). C

THEOREM 3—Loop Property of Conservative Fields are equivalent.

The following statements

#

1. D F dr = 0 around every loop (that is, closed curve C) in D. C

2. The field F is conservative on D. B C2

C1

C1

A

B –C 2

A

FIGURE 15.24 If we have two paths from A to B, one of them can be reversed to make a loop.

Proof that Part 1 Q Part 2 We want to show that for any two points A and B in D, the integral of F # dr has the same value over any two paths C1 and C2 from A to B. We reverse the direction on C2 to make a path -C2 from B to A (Figure 15.24). Together, C1 and -C2 make a closed loop C, and by assumption,

LC1

F # dr -

LC2

F # dr =

LC1

F # dr +

L-C2

F # dr =

LC

F # dr = 0.

Thus, the integrals over C1 and C2 give the same value. Note that the definition of F # dr shows that changing the direction along a curve reverses the sign of the line integral.

852

Chapter 15: Integration in Vector Fields

B

C2

B

Proof that Part 2 Q Part 1 We want to show that the integral of F # dr is zero over any closed loop C. We pick two points A and B on C and use them to break C into two pieces: C1 from A to B followed by C2 from B back to A (Figure 15.25). Then

–C 2 C1

F

C1

F # dr =

C

LC1

F # dr +

LC2

F # dr =

B

LA

F # dr -

B

LA

F # dr = 0.

The following diagram summarizes the results of Theorems 2 and 3. A

A Theorem 2

FIGURE 15.25 If A and B lie on a loop, we can reverse part of the loop to make two paths from A to B.

F = §ƒ on D

3

Theorem 3

3

F conservative on D

F # dr = 0 F C over any loop in D

Two questions arise: 1. 2.

How do we know whether a given vector field F is conservative? If F is in fact conservative, how do we find a potential function ƒ (so that F = §ƒ)?

Finding Potentials for Conservative Fields The test for a vector field being conservative involves the equivalence of certain partial derivatives of the field components.

Component Test for Conservative Fields Let F = Msx, y, zdi + Nsx, y, zdj + Psx, y, zdk be a field on a connected and simply connected domain whose component functions have continuous first partial derivatives. Then, F is conservative if and only if 0N 0P = , 0y 0z

0P 0M = , 0z 0x

and

0N 0M = . 0x 0y

(2)

Proof that Equations (2) hold if F is conservative There is a potential function ƒ such that F = Mi + Nj + Pk =

0ƒ 0ƒ 0ƒ i + j + k. 0x 0y 0z

Hence, 0 2ƒ 0P 0 0ƒ a b = = 0y 0y 0z 0y 0z =

0 2ƒ 0z 0y

=

0N 0 0ƒ a b = . 0z 0y 0z

Mixed Derivative Theorem, Section 13.3

The others in Equations (2) are proved similarly. The second half of the proof, that Equations (2) imply that F is conservative, is a consequence of Stokes’ Theorem, taken up in Section 15.7, and requires our assumption that the domain of F be simply connected.

15.3

Path Independence, Conservative Fields, and Potential Functions

853

Once we know that F is conservative, we usually want to find a potential function for F. This requires solving the equation §ƒ = F or 0ƒ 0ƒ 0ƒ i + j + k = Mi + Nj + Pk 0x 0y 0z for ƒ. We accomplish this by integrating the three equations 0ƒ = M, 0x

0ƒ = N, 0y

0ƒ = P, 0z

as illustrated in the next example. Show that F = se x cos y + yzdi + sxz - e x sin ydj + sxy + zdk is conservative over its natural domain and find a potential function for it.

EXAMPLE 3

The natural domain of F is all of space, which is connected and simply connected. We apply the test in Equations (2) to

Solution

M = e x cos y + yz,

N = xz - e x sin y,

P = xy + z

and calculate 0N 0P = x = , 0y 0z

0M 0P = y = , 0z 0x

0N 0M = -e x sin y + z = . 0x 0y

The partial derivatives are continuous, so these equalities tell us that F is conservative, so there is a function ƒ with §ƒ = F (Theorem 2). We find ƒ by integrating the equations 0ƒ = e x cos y + yz, 0x

0ƒ = xz - e x sin y, 0y

0ƒ = xy + z. 0z

(3)

We integrate the first equation with respect to x, holding y and z fixed, to get ƒsx, y, zd = e x cos y + xyz + gsy, zd. We write the constant of integration as a function of y and z because its value may depend on y and z, though not on x. We then calculate 0ƒ>0y from this equation and match it with the expression for 0ƒ>0y in Equations (3). This gives -e x sin y + xz +

0g = xz - e x sin y, 0y

so 0g>0y = 0. Therefore, g is a function of z alone, and ƒsx, y, zd = e x cos y + xyz + hszd. We now calculate 0ƒ>0z from this equation and match it to the formula for 0ƒ>0z in Equations (3). This gives xy +

dh = xy + z, dz

or

so hszd =

z2 + C. 2

dh = z, dz

854

Chapter 15: Integration in Vector Fields

Hence, ƒsx, y, zd = e x cos y + xyz +

z2 + C. 2

We have infinitely many potential functions of F, one for each value of C. Show that F = s2x - 3di - zj + scos zdk is not conservative.

EXAMPLE 4

We apply the Component Test in Equations (2) and find immediately that

Solution

0P 0 = scos zd = 0, 0y 0y

0N 0 = s -zd = -1. 0z 0z

The two are unequal, so F is not conservative. No further testing is required.

EXAMPLE 5

Show that the vector field F =

-y x + y 2

2

i +

x j + 0k x + y2 2

satisfies the equations in the Component Test, but is not conservative over its natural domain. Explain why this is possible. We have M = -y>(x 2 + y 2), N = x>(x 2 + y 2), and P = 0. If we apply the Component Test, we find Solution

0N 0P = 0 = , 0y 0z

0P 0M = 0 = , 0x 0z

and

y2 - x2 0N 0M = = 2 . 0y 0x (x + y 2) 2

So it may appear that the field F passes the Component Test. However, the test assumes that the domain of F is simply connected, which is not the case. Since x 2 + y 2 cannot equal zero, the natural domain is the complement of the z-axis and contains loops that cannot be contracted to a point. One such loop is the unit circle C in the xy-plane. The circle is parametrized by r(t) = (cost)i + (sin t)j, 0 … t … 2p. This loop wraps around the z-axis and cannot be contracted to a point while staying within the complement of the z-axis. To show that F is not conservative, we compute the line integral D F # dr around the C loop C. First we write the field in terms of the parameter t: F =

-y -sin t cost x i + 2 j = i + j = (-sint)i + (cost)j. x + y2 x + y2 sin2 t + cos2 t sin2 t + cos2 t 2

Next we find dr>dt = (-sin t)i + (cost)j, and then calculate the line integral as

F

C

F # dr =

F

C

F#

dr dt = dt L0

2p

A sin2 t + cos2 t B dt = 2p.

Since the line integral of F around the loop C is not zero, the field F is not conservative, by Theorem 3. Example 5 shows that the Component Test does not apply when the domain of the field is not simply connected. However, if we change the domain in the example so that it is restricted to the ball of radius 1 centered at the point (2, 2, 2), or to any similar ball-shaped region which does not contain a piece of the z-axis, then this new domain D is simply connected. Now the partial derivative Equations (2), as well as all the assumptions of the Component Test, are satisfied. In this new situation, the field F in Example 5 is conservative on D.

15.3

Path Independence, Conservative Fields, and Potential Functions

855

Just as we must be careful with a function when determining if it satisfies a property throughout its domain (like continuity or the Intermediate Value Property), so must we also be careful with a vector field in determining the properties it may or may not have over its assigned domain.

Exact Differential Forms It is often convenient to express work and circulation integrals in the differential form

LC

M dx + N dy + P dz

discussed in Section 15.2. Such line integrals are relatively easy to evaluate if M dx + N dy + P dz is the total differential of a function ƒ and C is any path joining the two points from A to B. For then

LC

M dx + N dy + P dz =

0ƒ 0ƒ 0ƒ dx + dy + dz 0x 0y 0z LC B

=

LA

§ƒ # dr

= ƒsBd - ƒsAd.

§ƒ is conservative. Theorem 1

Thus, B

LA

df = ƒsBd - ƒsAd,

just as with differentiable functions of a single variable.

DEFINITIONS Any expression Msx, y, zd dx + Nsx, y, zd dy + Psx, y, zd dz is a differential form. A differential form is exact on a domain D in space if M dx + N dy + P dz =

0f 0ƒ 0ƒ dx + dy + dz = dƒ 0x 0y 0z

for some scalar function ƒ throughout D.

Notice that if M dx + N dy + P dz = dƒ on D, then F = Mi + Nj + Pk is the gradient field of ƒ on D. Conversely, if F = §ƒ, then the form M dx + N dy + P dz is exact. The test for the form’s being exact is therefore the same as the test for F being conservative.

Component Test for Exactness of M dx + N dy + P dz The differential form M dx + N dy + P dz is exact on a connected and simply connected domain if and only if 0N 0P = , 0y 0z

0M 0P = , 0z 0x

and

0N 0M = . 0x 0y

This is equivalent to saying that the field F = Mi + Nj + Pk is conservative.

856

Chapter 15: Integration in Vector Fields

EXAMPLE 6

Show that y dx + x dy + 4 dz is exact and evaluate the integral s2,3, -1d

y dx + x dy + 4 dz

Ls1,1,1d

over any path from (1, 1, 1) to s2, 3, -1d. Solution

We let M = y, N = x, P = 4 and apply the Test for Exactness: 0N 0P = 0 = , 0y 0z

0N 0M = 1 = . 0x 0y

0M 0P = 0 = , 0z 0x

These equalities tell us that y dx + x dy + 4 dz is exact, so y dx + x dy + 4 dz = dƒ for some function ƒ, and the integral’s value is ƒs2, 3, -1d - ƒs1, 1, 1d. We find ƒ up to a constant by integrating the equations 0ƒ 0ƒ 0ƒ = y, = x, = 4. 0x 0y 0z

(4)

From the first equation we get ƒsx, y, zd = xy + gsy, zd. The second equation tells us that 0ƒ 0g = x + = x, 0y 0y

0g = 0. 0y

or

Hence, g is a function of z alone, and ƒsx, y, zd = xy + hszd. The third of Equations (4) tells us that 0ƒ dh = 0 + = 4, 0z dz

or

hszd = 4z + C.

Therefore, ƒsx, y, zd = xy + 4z + C. The value of the line integral is independent of the path taken from (1, 1, 1) to (2, 3, -1), and equals ƒs2, 3, -1d - ƒs1, 1, 1d = 2 + C - s5 + Cd = -3.

Exercises 15.3 Testing for Conservative Fields Which fields in Exercises 1–6 are conservative, and which are not? 1. F = yzi + xzj + xyk 2. F = s y sin zdi + sx sin zdj + sxy cos zdk 3. F = yi + sx + zdj - yk

8. F = s y + zdi + sx + zdj + sx + ydk 9. F = e y + 2zsi + xj + 2xkd 10. F = s y sin zdi + sx sin zdj + sxy cos zdk 11. F = sln x + sec2sx + yddi + asec2sx + yd +

4. F = -yi + xj 5. F = sz + ydi + zj + s y + xdk 6. F = se x cos ydi - se x sin ydj + zk Finding Potential Functions In Exercises 7–12, find a potential function ƒ for the field F. 7. F = 2xi + 3yj + 4zk

12. F =

y 1 + x2 y2

i + a

y y2 + z2

bj +

z k y2 + z2

x z + bj + 1 + x2 y2 21 - y 2 z 2 a

y 21 - y 2 z 2

1 + z bk

15.3

Exact Differential Forms In Exercises 13–17, show that the differential forms in the integrals are exact. Then evaluate the integrals. s2,3, -6d

13.

Ls0,0,0d s3,5,0d

14.

Ls1,1,2d s1,2,3d

15.

Ls0,0,0d s3,3,1d

16.

Ls0,0,0d s0,1,1d

17.

Ls1,0,0d

2x dx + 2y dy + 2z dz

Ls1,1,1d s2,1,1d

20.

Ls1,2,1d

z 1

4 2x dx - y dy dz 1 + z2 2

(1, 0, 1) z 5 x2

sin y cos x dx + cos y sin x dy + dz

Ls0,2,1d s1,2,3d

a. The line segment x = 1, y = 0, 0 … z … 1 b. The helix rstd = scos tdi + ssin tdj + st>2pdk, 0 … t … 2p

2xy dx + sx 2 - z 2 d dy - 2yz dz

s1,p>2,2d

19.

29. Work along different paths Find the work done by F = sx 2 + ydi + s y 2 + xdj + ze z k over the following paths from (1, 0, 0) to (1, 0, 1).

c. The x-axis from (1, 0, 0) to (0, 0, 0) followed by the parabola z = x 2, y = 0 from (0, 0, 0) to (1, 0, 1)

yz dx + xz dy + xy dz

Finding Potential Functions to Evaluate Line Integrals Although they are not defined on all of space R 3, the fields associated with Exercises 18–22 are simply connected and the Component Test can be used to show they are conservative. Find a potential function for each field and evaluate the integrals as in Example 6. 18.

857

Path Independence, Conservative Fields, and Potential Functions

1 1 2 cos y dx + a y - 2x sin yb dy + z dz

(0, 0, 0)

x

(1, 0, 0)

30. Work along different paths Find the work done by F = e yz i + sxze yz + z cos ydj + sxye yz + sin ydk over the following paths from (1, 0, 1) to s1, p>2, 0d. a. The line segment x = 1, y = pt>2, z = 1 - t, 0 … t … 1 z

z2 3x 2 dx + y dy + 2z ln y dz x s2x ln y - yzd dx + a y - xzb dy - xy dz 2

y

1 (1, 0, 1) p 2

s2,2,2d

y x 1 1 21. y dx + a z - y 2 b dy - z 2 dz Ls1,1,1d s2,2,2d 2x dx + 2y dy + 2z dz 22. x2 + y2 + z2 Ls-1, -1, -1d Applications and Examples 23. Revisiting Example 6 Evaluate the integral s2,3, -1d

Ls1,1,1d

y

x

⎛1, p , 0⎛ ⎝ 2 ⎝

1

b. The line segment from (1, 0, 1) to the origin followed by the line segment from the origin to s1, p>2, 0d z

y dx + x dy + 4 dz

1

from Example 6 by finding parametric equations for the line segment from (1, 1, 1) to s2, 3, -1d and evaluating the line integral of F = yi + xj + 4k along the segment. Since F is conservative, the integral is independent of the path.

(1, 0, 1) p 2

(0, 0, 0)

y

24. Evaluate x 2 dx + yz dy + s y 2>2d dz LC along the line segment C joining (0, 0, 0) to (0, 3, 4). Independence of path Show that the values of the integrals in Exercises 25 and 26 do not depend on the path taken from A to B.

x

c. The line segment from (1, 0, 1) to (1, 0, 0), followed by the x-axis from (1, 0, 0) to the origin, followed by the parabola y = px 2>2, z = 0 from there to s1, p>2, 0d

B

25.

B

26.

LA

z

z 2 dx + 2y dy + 2xz dz

LA

1

x dx + y dy + z dz

(1, 0, 1)

2x 2 + y 2 + z 2

In Exercises 27 and 28, find a potential function for F. 2x 1 - x2 b j, 27. F = y i + a y2

⎛1, p , 0⎛ ⎝ 2 ⎝

1

(0, 0, 0)

ex 28. F = se x ln ydi + a y + sin zbj + s y cos zdk

y

y 5 p x2 2

{(x, y): y 7 0} x (1, 0, 0)

⎛1, p , 0⎛ ⎝ 2 ⎝

858

Chapter 15: Integration in Vector Fields

31. Evaluating a work integral two ways Let F = §sx 3y 2 d and let C be the path in the xy-plane from s -1, 1d to (1, 1) that consists of the line segment from s -1, 1d to (0, 0) followed by the line segment from (0, 0) to (1, 1). Evaluate 1C F # dr in two ways. a. Find parametrizations for the segments that make up C and evaluate the integral. b. Use ƒsx, yd = x 3y 2 as a potential function for F. 32. Integral along different paths Evaluate the line integral 2 1C 2x cos y dx - x sin y dy along the following paths C in the xy-plane. a. The parabola y = sx - 1d2 from (1, 0) to (0, 1) b. The line segment from s -1, pd to (1, 0) c. The x-axis from s -1, 0d to (1, 0) d. The astroid rstd = scos3 tdi + ssin3 tdj, 0 … t … 2p, counterclockwise from (1, 0) back to (1, 0) y

34. Gradient of a line integral Suppose that F = §ƒ is a conservative vector field and gsx, y, zd =

sx,y,zd

Ls0,0,0d

F # dr.

Show that §g = F. 35. Path of least work You have been asked to find the path along which a force field F will perform the least work in moving a particle between two locations. A quick calculation on your part shows F to be conservative. How should you respond? Give reasons for your answer. 36. A revealing experiment By experiment, you find that a force field F performs only half as much work in moving an object along path C1 from A to B as it does in moving the object along path C2 from A to B. What can you conclude about F? Give reasons for your answer. 37. Work by a constant force Show that the work done by a constant force field F = ai + bj + ck in moving a particle along 1 any path from A to B is W = F # AB.

(0, 1)

38. Gravitational field (–1, 0)

(1, 0)

x

a. Find a potential function for the gravitational field F = -GmM

(0, –1)

xi + yj + zk sx + y 2 + z 2 d3>2 2

(G, m, and M are constantsd. 33. a. Exact differential form How are the constants a, b, and c related if the following differential form is exact? say 2 + 2czxd dx + ysbx + czd dy + say 2 + cx 2 d dz b. Gradient field

For what values of b and c will

F = s y 2 + 2czxdi + ysbx + czdj + s y 2 + cx 2 dk

b. Let P1 and P2 be points at distance s1 and s2 from the origin. Show that the work done by the gravitational field in part (a) in moving a particle from P1 to P2 is 1 1 GmM a s2 - s1 b.

be a gradient field?

15.4

Green’s Theorem in the Plane If F is a conservative field, then we know F = ¥ƒ for a differentiable function ƒ, and we can calculate the line integral of F over any path C joining point A to B as # 1C F dr = ƒ(B) - ƒ(A). In this section we derive a method for computing a work or flux integral over a closed curve C in the plane when the field F is not conservative. This method, known as Green’s Theorem, allows us to convert the line integral into a double integral over the region enclosed by C. The discussion is given in terms of velocity fields of fluid flows (a fluid is a liquid or a gas) because they are easy to visualize. However, Green’s Theorem applies to any vector field, independent of any particular interpretation of the field, provided the assumptions of the theorem are satisfied. We introduce two new ideas for Green’s Theorem: divergence and circulation density around an axis perpendicular to the plane.

Divergence Suppose that Fsx, yd = Msx, ydi + Nsx, ydj is the velocity field of a fluid flowing in the plane and that the first partial derivatives of M and N are continuous at each point of a region R. Let (x, y) be a point in R and let A be a small rectangle with one corner at (x, y) that, along with its interior, lies entirely in R. The sides of the rectangle, parallel to the coordinate axes, have lengths of ¢x and ¢y. Assume that the components M and N do not

15.4

Green’s Theorem in the Plane

(x, y Δy)

859

(x Δ x, y Δy) F·j0

F · (–i) 0 A

Δy F·i0

F(x, y) F · (–j) 0

(x Δ x, y)

(x, y) Δx

FIGURE 15.26 The rate at which the fluid leaves the rectangular region A across the bottom edge in the direction of the outward normal -j is approximately F(x, y) # (-j) ¢x, which is negative for the vector field F shown here. To approximate the flow rate at the point (x, y), we calculate the (approximate) flow rates across each edge in the directions of the red arrows, sum these rates, and then divide the sum by the area of A. Taking the limit as ¢x : 0 and ¢y : 0 gives the flow rate per unit area.

change sign throughout a small region containing the rectangle A. The rate at which fluid leaves the rectangle across the bottom edge is approximately (Figure 15.26) Fsx, yd # s -jd ¢x = -Nsx, yd¢x.

This is the scalar component of the velocity at (x, y) in the direction of the outward normal times the length of the segment. If the velocity is in meters per second, for example, the flow rate will be in meters per second times meters or square meters per second. The rates at which the fluid crosses the other three sides in the directions of their outward normals can be estimated in a similar way. The flow rates may be positive or negative depending on the signs of the components of F. We approximate the net flow rate across the rectangular boundary of A by summing the flow rates across the four edges as defined by the following dot products. Fluid Flow Rates:

Top: Bottom: Right: Left:

Fsx, y + ¢yd # j ¢x Fsx, yd # s -jd ¢x = Fsx + ¢x, yd # i ¢y Fsx, yd # s -id ¢y =

= Nsx, y + ¢yd¢x -Nsx, yd¢x = Msx + ¢x, yd¢y -Msx, yd¢y.

Summing opposite pairs gives Top and bottom:

sNsx, y + ¢yd - Nsx, ydd¢x L a

Right and left:

sMsx + ¢x, yd - Msx, ydd¢y L a

0N ¢yb ¢x 0y 0M ¢xb ¢y. 0x

Adding these last two equations gives the net effect of the flow rates, or the Flux across rectangle boundary L a

0N 0M + b ¢x¢y. 0x 0y

We now divide by ¢x¢y to estimate the total flux per unit area or flux density for the rectangle: Flux across rectangle boundary 0N 0M L a + b. 0x 0y rectangle area

860

Chapter 15: Integration in Vector Fields

Source: div F (x 0 , y0 ) . 0 A gas expanding at the point (x 0 , y0 ).

Finally, we let ¢x and ¢y approach zero to define the flux density of F at the point (x, y). In mathematics, we call the flux density the divergence of F. The symbol for it is div F, pronounced “divergence of F” or “div F.”

DEFINITION

The divergence (flux density) of a vector field F = Mi + Nj at

the point (x, y) is div F =

Sink: div F (x 0 , y0 ) , 0

0N 0M + . 0x 0y

(1)

A gas compressing at the point (x 0 , y0 ).

FIGURE 15.27 If a gas is expanding at a point sx0, y0 d, the lines of flow have positive divergence; if the gas is compressing, the divergence is negative.

A gas is compressible, unlike a liquid, and the divergence of its velocity field measures to what extent it is expanding or compressing at each point. Intuitively, if a gas is expanding at the point sx0 , y0), the lines of flow would diverge there (hence the name) and, since the gas would be flowing out of a small rectangle about sx0, y0 d, the divergence of F at sx0, y0 d would be positive. If the gas were compressing instead of expanding, the divergence would be negative (Figure 15.27).

EXAMPLE 1

The following vector fields represent the velocity of a gas flowing in the xy-plane. Find the divergence of each vector field and interpret its physical meaning. Figure 15.28 displays the vector fields. y

y

x

x

(a)

(b)

y

y

x

x

(c)

(d)

FIGURE 15.28 Velocity fields of a gas flowing in the plane (Example 1).

(a) Uniform expansion or compression: F(x, y) = cxi + cyj (b) Uniform rotation: F(x, y) = -cyi + cxj (c) Shearing flow: F(x, y) = yi -y x (d) Whirlpool effect: F(x, y) = 2 i + 2 j x + y2 x + y2

15.4

Green’s Theorem in the Plane

861

Solution

0 0 (a) div F = 0x (cx) + 0y (cy) = 2c: If c 7 0, the gas is undergoing uniform expansion; if c 6 0, it is undergoing uniform compression. 0 0 (b) div F = 0x (-cy) + 0y (cx) = 0: The gas is neither expanding nor compressing. 0 (c) div F = 0x ( y) = 0: The gas is neither expanding nor compressing. 2xy 2xy -y x 0 0 b + b = 2 - 2 = 0: Again, a 2 (d) div F = 0x a 2 2 2 2 2 0y x + y x + y (x + y ) (x + y 2) 2 the divergence is zero at all points in the domain of the velocity field.

Cases (b), (c), and (d) of Figure 15.28 are plausible models for the two-dimensional flow of a liquid. In fluid dynamics, when the velocity field of a flowing liquid always has divergence equal to zero, as in those cases, the liquid is said to be incompressible.

Spin Around an Axis: The k-Component of Curl The second idea we need for Green’s Theorem has to do with measuring how a floating paddle wheel, with axis perpendicular to the plane, spins at a point in a fluid flowing in a plane region. This idea gives some sense of how the fluid is circulating around axes located at different points and perpendicular to the region. Physicists sometimes refer to this as the circulation density of a vector field F at a point. To obtain it, we return to the velocity field Fsx, yd = Msx, ydi + Nsx, ydj and consider the rectangle A in Figure 15.29 (where we assume both components of F are positive). (x, y Δy)

(x Δ x, y Δy) F · (–i) 0

F · (– j) 0

Δy

A

F(x, y)

F·j0 F·i0 (x Δ x, y)

(x, y) Δx

FIGURE 15.29 The rate at which a fluid flows along the bottom edge of a rectangular region A in the direction i is approximately F(x, y) # i ¢x, which is positive for the vector field F shown here. To approximate the rate of circulation at the point (x, y), we calculate the (approximate) flow rates along each edge in the directions of the red arrows, sum these rates, and then divide the sum by the area of A. Taking the limit as ¢x : 0 and ¢y : 0 gives the rate of the circulation per unit area.

The circulation rate of F around the boundary of A is the sum of flow rates along the sides in the tangential direction. For the bottom edge, the flow rate is approximately Fsx, yd # i ¢x = Msx, yd¢x.

862

Chapter 15: Integration in Vector Fields

Vertical axis k

This is the scalar component of the velocity F(x, y) in the tangent direction i times the length of the segment. The flow rates may be positive or negative depending on the components of F. We approximate the net circulation rate around the rectangular boundary of A by summing the flow rates along the four edges as defined by the following dot products. Top: Bottom: Right: Left:

(x 0 , y 0 )

Curl F (x 0 , y 0 ) . k 0 Counterclockwise circulation

Fsx, y + ¢yd # s -id ¢x = -Msx, y + ¢yd¢x Fsx, yd # i ¢x = Msx, yd¢x Fsx + ¢x, yd # j ¢y = Nsx + ¢x, yd¢y Fsx, yd # s -jd ¢y = -Nsx, yd¢y.

We sum opposite pairs to get Top and bottom:

Vertical axis k

Right and left:

-sMsx, y + ¢yd - Msx, ydd¢x L - a sNsx + ¢x, yd - Nsx, ydd¢y L a

0M ¢yb ¢x 0y

0N ¢xb ¢y. 0x

(x 0 , y 0 )

Adding these last two equations gives the net circulation relative to the counterclockwise orientation, and dividing by ¢x¢y gives an estimate of the circulation density for the rectangle:

Curl F (x 0 , y 0 ) . k 0 Clockwise circulation

Circulation around rectangle 0N 0M L . 0x 0y rectangle area

FIGURE 15.30 In the flow of an incompressible fluid over a plane region, the k-component of the curl measures the rate of the fluid’s rotation at a point. The k-component of the curl is positive at points where the rotation is counterclockwise and negative where the rotation is clockwise.

We let ¢x and ¢y approach zero to define the circulation density of F at the point (x, y). If we see a counterclockwise rotation looking downward onto the xy-plane from the tip of the unit k vector, then the circulation density is positive (Figure 15.30). The value of the circulation density is the k-component of a more general circulation vector field we define in Section 15.7, called the curl of the vector field F. For Green’s Theorem, we need only this k-component, obtained by taking the dot product of curl F with k.

DEFINITION The circulation density of a vector field F = Mi + Nj at the point (x, y) is the scalar expression 0N 0M . 0x 0y

(2)

This expression is also called the k-component of the curl, denoted by (curl F) # k. If water is moving about a region in the xy-plane in a thin layer, then the k-component of the curl at a point sx0, y0 d gives a way to measure how fast and in what direction a small paddle wheel spins if it is put into the water at sx0, y0 d with its axis perpendicular to the plane, parallel to k (Figure 15.30). Looking downward onto the xy-plane, it spins counterclockwise when (curl F) # k is positive and clockwise when the k-component is negative.

EXAMPLE 2 Find the circulation density, and interpret what it means, for each vector field in Example 1. Solution

(a) Uniform expansion: (curl F) # k = at very small scales.

0 0 (cy) (cx) = 0. The gas is not circulating 0x 0y

15.4

Green’s Theorem in the Plane

863

(b) Rotation: (curl F) # k =

0 0 (cx) (-cy) = 2c. The constant circulation density 0x 0y indicates rotation at every point. If c 7 0, the rotation is counterclockwise; if c 6 0, the rotation is clockwise.

y

x

(c) Shear: (curl F) # k = -

0 (y) = -1. The circulation density is constant and negative, 0y so a paddle wheel floating in water undergoing such a shearing flow spins clockwise. The rate of rotation is the same at each point. The average effect of the fluid flow is to push fluid clockwise around each of the small circles shown in Figure 15.31.

(d) Whirlpool: FIGURE 15.31 A shearing flow pushes the fluid clockwise around each point (Example 2c).

scurl Fd # k =

-y y2 - x2 y2 - x2 x 0 0 a 2 b a 2 b = 2 - 2 = 0. 2 2 2 2 0x x + y 0y x + y (x + y ) (x + y 2) 2

The circulation density is 0 at every point away from the origin (where the vector field is undefined and the whirlpool effect is taking place), and the gas is not circulating at any point for which the vector field is defined.

Two Forms for Green’s Theorem In one form, Green’s Theorem says that under suitable conditions the outward flux of a vector field across a simple closed curve in the plane equals the double integral of the divergence of the field over the region enclosed by the curve. Recall the formulas for flux in Equations (3) and (4) in Section 15.2 and that a curve is simple if it does not cross itself.

THEOREM 4—Green’s Theorem (Flux-Divergence or Normal Form) Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane. Let F = Mi + N j be a vector field with M and N having continuous first partial derivatives in an open region containing R. Then the outward flux of F across C equals the double integral of div F over the region R enclosed by C. F

F # n ds =

C

F

M dy - N dx =

C

a

0N 0M + b dx dy 0x 0y 6

(3)

R

Outward flux

Divergence integral

We introduced the notation DC in Section 15.3 for integration around a closed curve. We elaborate further on the notation here. A simple closed curve C can be traversed in two possible directions. The curve is traversed counterclockwise, and said to be positively oriented, if the region it encloses is always to the left of an object as it moves along the path. Otherwise it is traversed clockwise and negatively oriented. The line integral of a vector field F along C reverses sign if we change the orientation. We use the notation

F

F(x, y) # dr

C

for the line integral when the simple closed curve C is traversed counterclockwise, with its positive orientation. A second form of Green’s Theorem says that the counterclockwise circulation of a vector field around a simple closed curve is the double integral of the k-component of the curl of the field over the region enclosed by the curve. Recall the defining Equation (2) for circulation in Section 15.2.

864

Chapter 15: Integration in Vector Fields

THEOREM 5—Green’s Theorem (Circulation-Curl or Tangential Form) Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane. Let F = Mi + N j be a vector field with M and N having continuous first partial derivatives in an open region containing R. Then the counterclockwise circulation of F around C equals the double integral of scurl Fd # k over R. F

F # T ds =

C

F

M dx + N dy =

C

0N 0M b dx dy 0x 0y 6 a

(4)

R

Counterclockwise circulation

Curl integral

The two forms of Green’s Theorem are equivalent. Applying Equation (3) to the field G1 = Ni - Mj gives Equation (4), and applying Equation (4) to G2 = -Ni + Mj gives Equation (3). Both forms of Green’s Theorem can be viewed as two-dimensional generalizations of the Net Change Theorem in Section 5.4. The outward flux of F across C, defined by the line integral on the left-hand side of Equation (3), is the integral of its rate of change (flux density) over the region R enclosed by C, which is the double integral on the right-hand side of Equation (3). Likewise, the counterclockwise circulation of F around C, defined by the line integral on the left-hand side of Equation (4), is the integral of its rate of change (circulation density) over the region R enclosed by C, which is the double integral on the right-hand side of Equation (4).

EXAMPLE 3

Verify both forms of Green’s Theorem for the vector field Fsx, yd = sx - ydi + xj

and the region R bounded by the unit circle C:

Solution

rstd = scos tdi + ssin tdj,

0 … t … 2p.

Evaluating F(r(t)) and differentiating components, we have M = cos t - sin t,

dx = dscos td = -sin t dt,

N = cos t,

dy = dssin td = cos t dt,

0M = 1, 0x

0M = -1, 0y

0N = 1, 0x

0N = 0. 0y

The two sides of Equation (3) are

F

M dy - N dx =

C

t = 2p

Lt = 0

scos t - sin tdscos t dtd - scos tds -sin t dtd

2p

=

6 R

a

L0

cos2 t dt = p

0N 0M + b dx dy = s1 + 0d dx dy 0x 0y 6 R

=

6 R

dx dy = area inside the unit circle = p.

15.4

Green’s Theorem in the Plane

865

The two sides of Equation (4) are

y

F

T

M dx + N dy =

C

t = 2p

Lt = 0

x

2p

= T

6 R

FIGURE 15.32 The vector field in Example 3 has a counterclockwise circulation of 2p around the unit circle.

scos t - sin tds -sin t dtd + scos tdscos t dtd

a

s -sin t cos t + 1d dt = 2p

L0

0N 0M b dx dy = s1 - s -1dd dx dy = 2 dx dy = 2p. 0x dy 6 6 R

R

Figure 15.32 displays the vector field and circulation around C.

Using Green’s Theorem to Evaluate Line Integrals If we construct a closed curve C by piecing together a number of different curves end to end, the process of evaluating a line integral over C can be lengthy because there are so many different integrals to evaluate. If C bounds a region R to which Green’s Theorem applies, however, we can use Green’s Theorem to change the line integral around C into one double integral over R.

EXAMPLE 4

Evaluate the line integral

F

xy dy - y 2 dx,

C

where C is the square cut from the first quadrant by the lines x = 1 and y = 1. Solution We can use either form of Green’s Theorem to change the line integral into a double integral over the square. 1.

With the Normal Form Equation (3): Taking M = xy, N = y 2, and C and R as the square’s boundary and interior gives 1

F

xy dy - y 2 dx =

C

6 R

1

=

2.

sy + 2yd dx dy =

L0

c3xy d

x=1 x=0

L0 L0

1

3y dx dy 1

1

dy =

L0

3y dy =

3 2 3 y d = . 2 2 0

With the Tangential Form Equation (4): Taking M = -y 2 and N = xy gives the same result:

F

C

-y 2 dx + xy dy =

6

s y - s -2ydd dx dy =

3 . 2

R

Calculate the outward flux of the vector field Fsx, yd = xi + y 2j across the square bounded by the lines x = ;1 and y = ;1.

EXAMPLE 5

866

Chapter 15: Integration in Vector Fields

Calculating the flux with a line integral would take four integrations, one for each side of the square. With Green’s Theorem, we can change the line integral to one double integral. With M = x, N = y 2, C the square, and R the square’s interior, we have Solution

Flux =

F

F # n ds =

F

C

=

M dy - N dx

C

a

0N 0M + b dx dy 0x 0y 6

Green’s Theorem

R

1

=

L-1L-1 1

= y P2 (x, f2 (x))

C 2: y 5 f2 (x)

L-1

cx + 2xy d

s2 + 4yd dy = c2y + 2y 2 d

x=1 x = -1

dy

1 -1

= 4.

Proof of Green’s Theorem for Special Regions

P1(x, f1(x))

C1: y 5 f1(x) a

L-1

1

s1 + 2yd dx dy =

Let C be a smooth simple closed curve in the xy-plane with the property that lines parallel to the axes cut it at no more than two points. Let R be the region enclosed by C and suppose that M, N, and their first partial derivatives are continuous at every point of some open region containing C and R. We want to prove the circulation-curl form of Green’s Theorem,

R

0

1

x

F

x

b

a

0N 0M b dx dy. 0x 0y 6

M dx + N dy =

C

FIGURE 15.33 The boundary curve C is made up of C1, the graph of y = ƒ1sxd, and C2 , the graph of y = ƒ2sxd.

(5)

R

Figure 15.33 shows C made up of two directed parts: y = ƒ1sxd,

C1:

a … x … b,

C2:

y = ƒ2sxd,

b Ú x Ú a.

For any x between a and b, we can integrate 0M>0y with respect to y from y = ƒ1sxd to y = ƒ2sxd and obtain ƒ2sxd

Lƒ1sxd

y = ƒ sxd

2 0M = Msx, ƒ2sxdd - Msx, ƒ1sxdd. dy = Msx, yd d 0y y = ƒ1sxd

We can then integrate this with respect to x from a to b: b

ƒ2sxd

La Lƒ1sxd

b

0M dy dx = [Msx, ƒ2sxdd - Msx, ƒ1sxdd] dx 0y La a

= y d

C 1' : x 5 g1( y)

y

= = -

M dx -

LC1

Msx, ƒ1sxdd dx

M dx

F

M dx.

C

R

0

LC2

La

Q 2(g2( y), y) Q1(g1( y), y)

c

Lb

b

Msx, ƒ2sxdd dx -

C '2 : x 5 g2( y)

Therefore

x

FIGURE 15.34 The boundary curve C is made up of C 1œ , the graph of x = g1syd, and C 2œ , the graph of x = g2syd.

F

C

M dx =

6

a-

0M b dx dy. 0y

(6)

R

Equation (6) is half the result we need for Equation (5). We derive the other half by integrating 0N>0x first with respect to x and then with respect to y, as suggested by Figure 15.34.

15.4 y

867

Green’s Theorem in the Plane

This shows the curve C of Figure 15.33 decomposed into the two directed parts C 1œ : x = g1s yd, d Ú y Ú c and C 2œ : x = g2s yd, c … y … d. The result of this double integration is 0N (7) N dy = dx dy. 6 0x F

C R

C

R

Summing Equations (6) and (7) gives Equation (5). This concludes the proof. x

0 (a) y b

a

C R

Green’s Theorem also holds for more general regions, such as those shown in Figures 15.35 and 15.36, but we will not prove this result here. Notice that the region in Figure 15.36 is not simply connected. The curves C1 and Ch on its boundary are oriented so that the region R is always on the left-hand side as the curves are traversed in the directions shown. With this convention, Green’s Theorem is valid for regions that are not simply connected. While we stated the theorem in the xy-plane, Green’s Theorem applies to any region R contained in a plane bounded by a curve C in space. We will see how to express the double integral over R for this more general form of Green’s Theorem in Section 15.7. y

0

a

x

b

C1

(b)

Ch

FIGURE 15.35 Other regions to which Green’s Theorem applies.

R 0

h

1

x

FIGURE 15.36 Green’s Theorem may be applied to the annular region R by summing the line integrals along the boundaries C1 and Ch in the directions shown.

Exercises 15.4 Verifying Green’s Theorem In Exercises 1–4, verify the conclusion of Green’s Theorem by evaluating both sides of Equations (3) and (4) for the field F = Mi + Nj. Take the domains of integration in each case to be the disk R: x 2 + y 2 … a 2 and its bounding circle C: r = sa cos tdi + sa sin tdj, 0 … t … 2p. 1. F = -yi + xj

2. F = yi

3. F = 2xi - 3yj

4. F = -x 2yi + xy 2j

9. F = (xy + y 2)i + (x - y)j

y

y (1, 1)

x 5 y2

5. F = sx - ydi + sy - xdj C: The square bounded by x = 0, x = 1, y = 0, y = 1

y 5 x2

–2

2

1 4 x yj 2

12. F =

y

2

C C

1

x 2 1 y2 5 1

(2, 2)

y5x

2

8. F = sx + ydi - sx 2 + y 2 dj C: The triangle bounded by y = 0, x = 1 , and y = x

x i + A tan-1 y B j 1 + y2 y

2

2

–1

y 5 x2 2 x (0, 0)

x

–1

11. F = x 3y 2 i +

7. F = s y - x di + sx + y dj C: The triangle bounded by y = 0, x = 3 , and y = x 2

x 2 1 2y 2 5 2

x

6. F = sx + 4ydi + sx + y dj C: The square bounded by x = 0, x = 1, y = 0, y = 1 2

C 1

C

(0, 0)

Circulation and Flux In Exercises 5–14, use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F and curve C.

10. F = (x + 3y)i + (2x - y)j

x

1

–1

x

868

Chapter 15: Integration in Vector Fields

13. F = sx + e x sin ydi + sx + e x cos ydj C: The right-hand loop of the lemniscate r 2 = cos 2u y 14. F = atan-1 x bi + ln sx 2 + y 2 dj

Green’s Theorem Area Formula Area of R =

1 x dy - y dx 2F C

C: The boundary of the region defined by the polar coordinate inequalities 1 … r … 2, 0 … u … p 15. Find the counterclockwise circulation and outward flux of the field F = xyi + y 2j around and over the boundary of the region enclosed by the curves y = x 2 and y = x in the first quadrant.

The reason is that by Equation (3), run backward, Area of R =

16. Find the counterclockwise circulation and the outward flux of the field F = s -sin ydi + sx cos ydj around and over the square cut from the first quadrant by the lines x = p>2 and y = p>2.

R

x bi + se x + tan-1 ydj 1 + y2

across the cardioid r = as1 + cos ud, a 7 0. 18. Find the counterclockwise circulation of F = s y + e x ln ydi + se x>ydj around the boundary of the region that is bounded above by the curve y = 3 - x 2 and below by the curve y = x 4 + 1.

25. The circle rstd = sa cos tdi + sa sin tdj, 27. The astroid rstd = scos3 tdi + ssin3 tdj,

21.

F

0 … t … 2p 0 … t … 2p

y = 1 - cos t

28. One arch of the cycloid x = t - sin t,

29. Let C be the boundary of a region on which Green’s Theorem holds. Use Green’s Theorem to calculate

b.

Using Green’s Theorem Apply Green’s Theorem to evaluate the integrals in Exercises 21–24.

0 … t … 2p

26. The ellipse rstd = sa cos tdi + sb sin tdj,

19. F = 2xy 3i + 4x 2y 2j

C: The circle sx - 2d2 + s y - 2d2 = 4

R

C

a.

20. F = s4x - 2ydi + s2x - 4ydj

1 1 + b dy dx 2 2 6

Use the Green’s Theorem area formula given above to find the areas of the regions enclosed by the curves in Exercises 25–28.

Work In Exercises 19 and 20, find the work done by F in moving a particle once counterclockwise around the given curve. C: The boundary of the “triangular” region in the first quadrant enclosed by the x-axis, the line x = 1, and the curve y = x 3

a

1 1 = x dy - y dx. 2 2 F

17. Find the outward flux of the field F = a3xy -

6

dy dx =

F

ƒsxd dx + gsyd dy

C

F

ky dx + hx dy

sk and h constantsd.

C

30. Integral dependent only on area Show that the value of F

xy 2 dx + sx 2y + 2xd dy

C

around any square depends only on the area of the square and not on its location in the plane. 31. Evaluate the integral

s y 2 dx + x 2 dyd

F

4x3y dx + x4 dy

C

C

C: The triangle bounded by x = 0, x + y = 1, y = 0

for any closed path C. 32. Evaluate the integral

22.

F

s3y dx + 2x dyd F

C

C: The boundary of 0 … x … p, 0 … y … sin x

23.

F

s6y + xd dx + s y + 2xd dy

- y3 dy + x3 dx

C

for any closed path C. 33. Area as a line integral Show that if R is a region in the plane bounded by a piecewise smooth, simple closed curve C, then

C

C: The circle sx - 2d2 + sy - 3d2 = 4

24.

F

s2x + y 2 d dx + s2xy + 3yd dy

C

C: Any simple closed curve in the plane for which Green’s Theorem holds Calculating Area with Green’s Theorem If a simple closed curve C in the plane and the region R it encloses satisfy the hypotheses of Green’s Theorem, the area of R is given by

Area of R =

F

x dy = -

C

F

y dx.

C

34. Definite integral as a line integral Suppose that a nonnegative function y = ƒsxd has a continuous first derivative on [a, b]. Let C be the boundary of the region in the xy-plane that is bounded below by the x-axis, above by the graph of ƒ, and on the sides by the lines x = a and x = b. Show that b

La

ƒsxd dx = -

F

C

y dx.

15.4 35. Area and the centroid Let A be the area and x the x-coordinate of the centroid of a region R that is bounded by a piecewise smooth, simple closed curve C in the xy-plane. Show that 1 1 x 2 dy = xy dx = x 2 dy - xy dx = Ax. 2 F 3 F F C

C

Green’s Theorem in the Plane

a. Let ƒsx, yd = ln sx 2 + y 2 d and let C be the circle x 2 + y 2 = a 2. Evaluate the flux integral

F

§ƒ # n ds.

C

C

36. Moment of inertia Let Iy be the moment of inertia about the y-axis of the region in Exercise 35. Show that

b. Let K be an arbitrary smooth, simple closed curve in the plane that does not pass through (0, 0). Use Green’s Theorem to show that

1 1 x 3 dy = x 2y dx = x 3 dy - x 2y dx = Iy . 3 F 4 F F C

869

C

F

§ƒ # n ds

K

C

37. Green’s Theorem and Laplace’s equation Assuming that all the necessary derivatives exist and are continuous, show that if ƒ(x, y) satisfies the Laplace equation 0 2ƒ 0x

2

+

0 2ƒ 0y 2

= 0,

then 0ƒ 0ƒ dx dy = 0 0x F 0y

C

for all closed curves C to which Green’s Theorem applies. (The converse is also true: If the line integral is always zero, then ƒ satisfies the Laplace equation.) 38. Maximizing work Among all smooth, simple closed curves in the plane, oriented counterclockwise, find the one along which the work done by 1 1 F = a x 2y + y 3 bi + xj 4 3 is greatest. (Hint: Where is scurl Fd # k positive?) 39. Regions with many holes Green’s Theorem holds for a region R with any finite number of holes as long as the bounding curves are smooth, simple, and closed and we integrate over each component of the boundary in the direction that keeps R on our immediate left as we go along (see accompanying figure).

has two possible values, depending on whether (0, 0) lies inside K or outside K. 40. Bendixson’s criterion The streamlines of a planar fluid flow are the smooth curves traced by the fluid’s individual particles. The vectors F = Msx, ydi + Nsx, ydj of the flow’s velocity field are the tangent vectors of the streamlines. Show that if the flow takes place over a simply connected region R (no holes or missing points) and that if Mx + Ny Z 0 throughout R, then none of the streamlines in R is closed. In other words, no particle of fluid ever has a closed trajectory in R. The criterion Mx + Ny Z 0 is called Bendixson’s criterion for the nonexistence of closed trajectories. 41. Establish Equation (7) to finish the proof of the special case of Green’s Theorem. 42. Curl component of conservative fields Can anything be said about the curl component of a conservative two-dimensional vector field? Give reasons for your answer.

COMPUTER EXPLORATIONS In Exercises 43–46, use a CAS and Green’s Theorem to find the counterclockwise circulation of the field F around the simple closed curve C. Perform the following CAS steps. a. Plot C in the xy-plane. b. Determine the integrand s0N>0xd - s0M>0yd for the curl form of Green’s Theorem. c. Determine the (double integral) limits of integration from your plot in part (a) and evaluate the curl integral for the circulation. 43. F = s2x - ydi + sx + 3ydj,

C: The ellipse x 2 + 4y 2 = 4

44. F = s2x 3 - y 3 di + sx 3 + y 3 dj,

C: The ellipse

y2 x2 + = 1 4 9

45. F = x -1e y i + se y ln x + 2xdj, C: The boundary of the region defined by y = 1 + x 4 (below) and y = 2 (above) 46. F = xe y i + (4x 2 ln y)j, C: The triangle with vertices (0, 0), (2, 0), and (0, 4)

870

Chapter 15: Integration in Vector Fields

Surfaces and Area

15.5

We have defined curves in the plane in three different ways: u constant

y

y constant (u, y)

y = ƒsxd

Implicit form:

Fsx, yd = 0

Parametric vector form:

rstd = ƒstdi + gstdj,

u

0

Parametrization

z Curve y constant

S

Explicit form:

z = ƒsx, yd

Implicit form:

Fsx, y, zd = 0.

There is also a parametric form for surfaces that gives the position of a point on the surface as a vector function of two variables. We discuss this new form in this section and apply the form to obtain the area of a surface as a double integral. Double integral formulas for areas of surfaces given in implicit and explicit forms are then obtained as special cases of the more general parametric formula.

P Curve u constant

r(u, y) f (u, y)i g(u, y)j h(u, y)k, Position vector to surface point y

FIGURE 15.37 A parametrized surface S expressed as a vector function of two variables defined on a region R.

Parametrizations of Surfaces Suppose rsu, yd = ƒsu, ydi + gsu, ydj + hsu, ydk

EXAMPLE 1 Cone: z x 2 y 2 r

(1)

is a continuous vector function that is defined on a region R in the uy-plane and one-toone on the interior of R (Figure 15.37). We call the range of r the surface S defined or traced by r. Equation (1) together with the domain R constitute a parametrization of the surface. The variables u and y are the parameters, and R is the parameter domain. To simplify our discussion, we take R to be a rectangle defined by inequalities of the form a … u … b, c … y … d. The requirement that r be one-to-one on the interior of R ensures that S does not cross itself. Notice that Equation (1) is the vector equivalent of three parametric equations: x = ƒsu, yd,

z

y = gsu, yd,

z = hsu, yd.

Find a parametrization of the cone z = 2x 2 + y 2,

0 … z … 1.

Here, cylindrical coordinates provide a parametrization. A typical point (x, y, z) on the cone (Figure 15.38) has x = r cos u, y = r sin u, and z = 2x 2 + y 2 = r, with 0 … r … 1 and 0 … u … 2p. Taking u = r and y = u in Equation (1) gives the parametrization Solution

1

rsr, ud = sr cos udi + sr sin udj + rk,

r(r, ) (r cos )i (r sin )j rk

(x, y, z) (r cos , r sin , r) r

x

a … t … b.

We have analogous definitions of surfaces in space:

R

x

Explicit form:

0 … r … 1,

0 … u … 2p.

The parametrization is one-to-one on the interior of the domain R, though not on the boundary tip of its cone where r = 0.

y

EXAMPLE 2 FIGURE 15.38 The cone in Example 1 can be parametrized using cylindrical coordinates.

Find a parametrization of the sphere x 2 + y 2 + z 2 = a 2.

Solution Spherical coordinates provide what we need. A typical point (x, y, z) on the sphere (Figure 15.39) has x = a sin f cos u, y = a sin f sin u, and z = a cos f,

15.5

(x, y, z) 5 (a sin f cos u, a sin f sin u, a cos f)

rsf, ud = sa sin f cos udi + sa sin f sin udj + sa cos fdk, 0 … f … p, 0 … u … 2p.

a

Again, the parametrization is one-to-one on the interior of the domain R, though not on its boundary “poles” where f = 0 or f = p.

f

a

r(f, u)

EXAMPLE 3

a

Find a parametrization of the cylinder x 2 + sy - 3d2 = 9,

y

x

0 … z … 5.

In cylindrical coordinates, a point (x, y, z) has x = r cos u, y = r sin u, and z = z. For points on the cylinder x 2 + s y - 3d2 = 9 (Figure 15.40), the equation is the same as the polar equation for the cylinder’s base in the xy-plane:

Solution

FIGURE 15.39 The sphere in Example 2 can be parametrized using spherical coordinates.

x 2 + s y 2 - 6y + 9d = 9 r 2 - 6r sin u = 0

z

Cylinder: x 2 ( y 3)2 9 or r 6 sin

or r = 6 sin u,

r 6 sin

0 … u … p.

x = r cos u = 6 sin u cos u = 3 sin 2u y = r sin u = 6 sin2 u z = z.

r(, z)

x 2 + y 2 = r 2, y = r sin u

A typical point on the cylinder therefore has

z

x

871

0 … f … p, 0 … u … 2p. Taking u = f and y = u in Equation (1) gives the parametrization

z

u

Surfaces and Area

(x, y, z) (3 sin 2, 6 sin 2 , z) y

FIGURE 15.40 The cylinder in Example 3 can be parametrized using cylindrical coordinates.

Taking u = u and y = z in Equation (1) gives the one-to-one parametrization rsu, zd = s3 sin 2udi + s6 sin2 udj + zk,

0 … u … p,

0 … z … 5.

Surface Area Our goal is to find a double integral for calculating the area of a curved surface S based on the parametrization rsu, yd = ƒsu, ydi + gsu, ydj + hsu, ydk,

a … u … b,

c … y … d.

We need S to be smooth for the construction we are about to carry out. The definition of smoothness involves the partial derivatives of r with respect to u and y: 0ƒ 0g 0h 0r = i + j + k 0u 0u 0u 0u 0ƒ 0g 0h 0r = i + j + k. ry = 0y 0y 0y 0y ru =

DEFINITION A parametrized surface rsu, yd = ƒsu, ydi + gsu, ydj + hsu, ydk is smooth if ru and ry are continuous and ru * ry is never zero on the interior of the parameter domain. The condition that ru * ry is never the zero vector in the definition of smoothness means that the two vectors ru and ry are nonzero and never lie along the same line, so they always determine a plane tangent to the surface. We relax this condition on the boundary of the domain, but this does not affect the area computations.

872

Chapter 15: Integration in Vector Fields

Now consider a small rectangle ¢Auy in R with sides on the lines u = u0 , u = u0 + ¢u, y = y0, and y = y0 + ¢y (Figure 15.41). Each side of ¢Auy maps to a curve on the surface S, and together these four curves bound a “curved patch element” ¢suy. In the notation of the figure, the side y = y0 maps to curve C1, the side u = u0 maps to C2 , and their common vertex su0 , y0 d maps to P0. z C1: y y0

P0

C 2: u u 0

Parametrization y S

d

Δ uy

y0 Δy

Δ A uy

y0 R

c 0

x a

u0

u0 Δu

b

u y

ru 3 ry P0

ry C2: u 5 u0

z

ru Δsuy C1: y 5 y0 y

x

FIGURE 15.42 A magnified view of a surface patch element ¢suy.

Δyry

P0 z

C2

Δuru C1

x

y y0 Δy

u u 0 Δu

FIGURE 15.41 A rectangular area element ¢Auy in the uy-plane maps onto a curved patch element ¢suy on S.

Figure 15.42 shows an enlarged view of ¢suy. The partial derivative vector rusu0, y0 d is tangent to C1 at P0. Likewise, rysu0, y0 d is tangent to C2 at P0. The cross product ru * ry is normal to the surface at P0. (Here is where we begin to use the assumption that S is smooth. We want to be sure that ru * ry Z 0.) We next approximate the surface patch element ¢suy by the parallelogram on the tangent plane whose sides are determined by the vectors ¢uru and ¢yry (Figure 15.43). The area of this parallelogram is ƒ ¢uru * ¢yry ƒ = ƒ ru * ry ƒ ¢u ¢y.

A partition of the region R in the uy-plane by rectangular regions ¢Auy induces a partition of the surface S into surface patch elements ¢suy. We approximate the area of each surface patch element ¢suy by the parallelogram area in Equation (2) and sum these areas together to obtain an approximation of the surface area of S: a ƒ ru * ry ƒ ¢u ¢y.

Δsuy

y

FIGURE 15.43 The area of the parallelogram determined by the vectors ¢uru and ¢yry approximates the area of the surface patch element ¢suy.

(2)

(3)

n

As ¢u and ¢y approach zero independently, the number of area elements n tends to q and the continuity of ru and ry guarantees that the sum in Equation (3) approaches the d b double integral 1c 1a ƒ ru * ry ƒ du dy. This double integral over the region R defines the area of the surface S.

DEFINITION

The area of the smooth surface

rsu, yd = ƒsu, ydi + gsu, ydj + hsu, ydk,

a … u … b,

c … y … d

is d

A =

6 R

ƒ ru * ry ƒ dA =

Lc La

b

ƒ ru * ry ƒ du dy.

(4)

15.5

873

Surfaces and Area

We can abbreviate the integral in Equation (4) by writing ds for ƒ ru * ry ƒ du dy. The surface area differential ds is analogous to the arc length differential ds in Section 12.3.

Surface Area Differential for a Parametrized Surface ds = ƒ ru * ry ƒ du dy

6

ds

(5)

S

Surface area differential

EXAMPLE 4

Differential formula for surface area

Find the surface area of the cone in Example 1 (Figure 15.38).

In Example 1, we found the parametrization

Solution

rsr, ud = sr cos udi + sr sin udj + rk,

0 … r … 1,

0 … u … 2p.

To apply Equation (4), we first find rr * ru : i rr * ru = 3 cos u -r sin u

j sin u r cos u

k 13 0

= -sr cos udi - sr sin udj + sr cos2 u + r sin2 udk.

(''')'''* r

Thus, ƒ rr * ru ƒ = 2r 2 cos2 u + r 2 sin2 u + r 2 = 22r 2 = 22r. The area of the cone is 2p

A =

1

2p

=

1

L0 L0

EXAMPLE 5 Solution

ƒ rr * ru ƒ dr du

L0 L0

Eq. (4) with u = r, y = u 2p

22 r dr du =

L0

22 22 du = s2pd = p22 units squared. 2 2

Find the surface area of a sphere of radius a.

We use the parametrization from Example 2: rsf, ud = sa sin f cos udi + sa sin f sin udj + sa cos fdk, 0 … f … p, 0 … u … 2p.

For rf * ru, we get i rf * ru = 3 a cos f cos u -a sin f sin u

j a cos f sin u a sin f cos u

k -a sin f 3 0

= sa 2 sin2 f cos udi + sa 2 sin2 f sin udj + sa 2 sin f cos fdk. Thus, ƒ rf * ru ƒ = 2a 4 sin4 f cos2 u + a 4 sin4 f sin2 u + a 4 sin2 f cos2 f = 2a 4 sin4 f + a 4 sin2 f cos2 f = 2a 4 sin2 f ssin2 f + cos2 fd = a 2 2sin2 f = a 2 sin f,

874

Chapter 15: Integration in Vector Fields

since sin f Ú 0 for 0 … f … p. Therefore, the area of the sphere is p

2p

A =

2p

=

a 2 sin f df du

L0 L0 L0

p

c-a 2 cos f d du = 0

2p

L0

2a 2 du = 4pa 2

units squared.

This agrees with the well-known formula for the surface area of a sphere.

z p 2

Let S be the “football” surface formed by rotating the curve x = cos z, y = 0, -p>2 … z … p>2 around the z-axis (see Figure 15.44). Find a parametrization for S and compute its surface area.

EXAMPLE 6

x 5 cos z, y 5 0

r 5 cos z is the radius of a circle at height z Solution

u

(x, y, z) 1

1 y

x

Example 2 suggests finding a parametrization of S based on its rotation around the z-axis. If we rotate a point (x, 0, z) on the curve x = cos z, y = 0 about the z-axis, we obtain a circle at height z above the xy-plane that is centered on the z-axis and has radius r = cos z (see Figure 15.44). The point sweeps out the circle through an angle of rotation u, 0 … u … 2p. We let (x, y, z) be an arbitrary point on this circle, and define the parameters u = z and y = u. Then we have x = r cos u = cos u cos y, y = r sin u = cos u sin y, and z = u giving a parametrization for S as r(u, y) = cos u cos y i + cos u sin y j + u k,

–p 2

-

p p … u … , 2 2

0 … y … 2p.

Next we use Equation (5) to find the surface area of S. Differentiation of the parametrization gives FIGURE 15.44 The “football” surface in Example 6 obtained by rotating the curve x = cos z about the z-axis.

ru = -sin u cos y i - sin u sin y j + k and ry = -cos u sin y i + cos u cos y j. Computing the cross product we have i ru * ry = 3 -sin u cos y -cos u sin y

j -sin u sin y cos u cos y

k 13 0

= -cos u cos y i - cos u sin y j - (sin u cos u cos2 y + cos u sin u sin2 y)k. Taking the magnitude of the cross product gives ƒ ru * ry ƒ = 2cos2 u (cos2 y + sin2 y) + sin2 u cos2 u = 2cos2 u (1 + sin2 u) = cos u 21 + sin2 u.

cos u Ú 0 for -

From Equation (4) the surface area is given by the integral 2p

A =

L0

p>2

L-p>2

cos u 21 + sin2 u du dy.

p p … u … 2 2

15.5

Surfaces and Area

875

To evaluate the integral, we substitute w = sin u and dw = cos u du, -1 … w … 1. Since the surface S is symmetric across the xy-plane, we need only integrate with respect to w from 0 to 1, and multiply the result by 2. In summary, we have 2p

A = 2

L0

L0 2p

= 2

L0 2p

=

L0

1

c

21 + w 2 dw dy 1

w 1 21 + w 2 + ln aw + 21 + w 2 b d dy 2 2 0

Integral Table Formula 35

1 1 2 c 22 + ln A 1 + 22 B d dy. 2 2

= 2p C 22 + ln A 1 + 22 B D .

Implicit Surfaces Surfaces are often presented as level sets of a function, described by an equation such as F(x, y, z) = c,

Surface F(x, y, z) c

S p

R

The vertical projection or “shadow” of S on a coordinate plane

FIGURE 15.45 As we soon see, the area of a surface S in space can be calculated by evaluating a related double integral over the vertical projection or “shadow” of S on a coordinate plane. The unit vector p is normal to the plane.

for some constant c. Such a level surface does not come with an explicit parametrization, and is called an implicitly defined surface. Implicit surfaces arise, for example, as equipotential surfaces in electric or gravitational fields. Figure 15.45 shows a piece of such a surface. It may be difficult to find explicit formulas for the functions ƒ, g, and h that describe the surface in the form r(u, y) = ƒ(u, y)i + g(u, y)j + h(u, y)k. We now show how to compute the surface area differential ds for implicit surfaces. Figure 15.45 shows a piece of an implicit surface S that lies above its “shadow” region R in the plane beneath it. The surface is defined by the equation F(x, y, z) = c and p is a unit vector normal to the plane region R. We assume that the surface is smooth (F is differentiable and ¥F is nonzero and continuous on S) and that ¥F # p Z 0, so the surface never folds back over itself. Assume that the normal vector p is the unit vector k, so the region R in Figure 15.45 lies in the xy-plane. By assumption, we then have ¥F # p = ¥F # k = Fz Z 0 on S. An advanced calculus theorem called the Implicit Function Theorem implies that S is then the graph of a differentiable function z h(x, y), although the function h(x, y) is not explicitly known. Define the parameters u and y by u = x and y = y. Then z = h(u, y) and r(u, y) = ui + yj + h(u, y)k

(6)

gives a parametrization of the surface S. We use Equation (4) to find the area of S. Calculating the partial derivatives of r, we find ru = i +

0h k 0u

and

ry = j +

0h k. 0y

Applying the Chain Rule for implicit differentiation (see Equation (2) in Section 13.4) to F(x, y, z) = c, where x = u, y = y, and z = h(u, y), we obtain the partial derivatives Fx 0h = 0u Fz

and

Fy 0h = - . 0y Fz

Substitution of these derivatives into the derivatives of r gives ru = i -

Fx k Fz

and

ry = j -

Fy k. Fz

876

Chapter 15: Integration in Vector Fields

From a routine calculation of the cross product we find ru * ry =

Fy Fx i + j + k Fz Fz

=

1 (F i + Fy j + Fz k) Fz x

=

¥F ¥F = Fz ¥F # k

=

¥F . ¥F # p

Fz Z 0

p = k

Therefore, the surface area differential is given by ds = ƒ ru * ry ƒ du dy =

ƒ ¥F ƒ dx dy. ƒ ¥F # p ƒ

u = x and y = y

We obtain similar calculations if instead the vector p = j is normal to the xz-plane when Fy Z 0 on S, or if p = i is normal to the yz-plane when Fx Z 0 on S. Combining these results with Equation (4) then gives the following general formula.

Formula for the Surface Area of an Implicit Surface The area of the surface Fsx, y, zd = c over a closed and bounded plane region R is Surface area =

ƒ §F ƒ # dA, 6 ƒ §F p ƒ

(7)

R

where p = i, j, or k is normal to R and § F # p Z 0. Thus, the area is the double integral over R of the magnitude of §F divided by the magnitude of the scalar component of §F normal to R. We reached Equation (7) under the assumption that §F # p Z 0 throughout R and that §F is continuous. Whenever the integral exists, however, we define its value to be the area of the portion of the surface Fsx, y, zd = c that lies over R. (Recall that the projection is assumed to be one-to-one.) z

EXAMPLE 7

Find the area of the surface cut from the bottom of the paraboloid x 2 + y 2 - z = 0 by the plane z = 4.

4

z x2 y2 R 0 x2 y2 4

We sketch the surface S and the region R below it in the xy-plane (Figure 15.46). The surface S is part of the level surface Fsx, y, zd = x 2 + y 2 - z = 0, and R is the disk x 2 + y 2 … 4 in the xy-plane. To get a unit vector normal to the plane of R, we can take p = k. At any point (x, y, z) on the surface, we have Solution

S

y

x

FIGURE 15.46 The area of this parabolic surface is calculated in Example 7.

Fsx, y, zd = x 2 + y 2 - z §F = 2xi + 2yj - k ƒ §F ƒ = 2s2xd2 + s2yd2 + s -1d2 = 24x 2 + 4y 2 + 1

ƒ §F # p ƒ = ƒ §F # k ƒ = ƒ -1 ƒ = 1.

15.5

Surfaces and Area

877

In the region R, dA = dx dy. Therefore, Surface area =

ƒ §F ƒ # dA 6 ƒ §F p ƒ

Eq. (7)

R

=

24x 2 + 4y 2 + 1 dx dy

6 x2 + y2 … 4 2p

= = =

2

L0 L0

Polar coordinates

2

2p

c

2p

p 1 s173>2 - 1d du = A 17217 - 1 B . 12 6

L0 L0

24r 2 + 1 r dr du

1 s4r 2 + 1d3>2 d du 12 0

Example 7 illustrates how to find the surface area for a function z = ƒ(x, y) over a region R in the xy-plane. Actually, the surface area differential can be obtained in two ways, and we show this in the next example. Derive the surface area differential ds of the surface z = ƒ(x, y) over a region R in the xy-plane (a) parametrically using Equation (5), and (b) implicitly, as in Equation (7).

EXAMPLE 8

Solution

(a) We parametrize the surface by taking x = u, y = y, and z = ƒ(x, y) over R. This gives the parametrization r(u, y) = ui + yj + ƒ(u, y)k. Computing the partial derivatives gives ru + i + ƒu k, ry = j + fy k and ru * ry = -ƒu i - ƒyj + k.

i 31 0

j 0 1

k ƒu 3 ƒy

Then ƒ ru * ry ƒ du dy = 2ƒu 2 + ƒy 2 + 1 du dy. Substituting for u and y then gives the surface area differential ds = 2ƒx 2 + ƒy 2 + 1 dx dy. (b) We define the implicit function F(x, y, z) = ƒ(x, y) - z. Since (x, y) belongs to the region R, the unit normal to the plane of R is p = k. Then ¥F = ƒx i + ƒy j - k so that 2 2 ƒ ¥F # p ƒ = ƒ -1 ƒ = 1, ƒ ¥F ƒ = 2ƒx + ƒy + 1, and ƒ ¥F ƒ > ƒ ¥F # p ƒ = ƒ ¥F ƒ. The surface area differential is again given by ds = 2ƒx 2 + ƒy 2 + 1 dx dy. The surface area differential derived in Example 8 gives the following formula for calculating the surface area of the graph of a function defined explicitly as z = ƒ(x, y).

Formula for the Surface Area of a Graph z = ƒ(x, y) For a graph z = ƒ(x, y) over a region R in the xy-plane, the surface area formula is A =

6 R

2ƒx 2 + ƒy 2 + 1 dx dy.

(8)

878

Chapter 15: Integration in Vector Fields

Exercises 15.5 Finding Parametrizations In Exercises 1–16, find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) 1. The paraboloid z = x 2 + y 2, z … 4 2. The paraboloid z = 9 - x 2 - y 2, z Ú 0 3. Cone frustum The first-octant portion of the cone z = 2x 2 + y 2>2 between the planes z = 0 and z = 3 4. Cone frustum The portion of the cone z = 2 2x 2 + y 2 between the planes z = 2 and z = 4 5. Spherical cap The cap cut from the sphere x 2 + y 2 + z 2 = 9 by the cone z = 2x 2 + y 2 6. Spherical cap The portion of the sphere x 2 + y 2 + z 2 = 4 in the first octant between the xy-plane and the cone z = 2x 2 + y 2 7. Spherical band The portion of the sphere x 2 + y 2 + z 2 = 3 between the planes z = 23>2 and z = - 23>2 8. Spherical cap The upper portion cut from the sphere x 2 + y 2 + z 2 = 8 by the plane z = -2 9. Parabolic cylinder between planes The surface cut from the parabolic cylinder z = 4 - y 2 by the planes x = 0, x = 2, and z = 0 10. Parabolic cylinder between planes The surface cut from the parabolic cylinder y = x 2 by the planes z = 0, z = 3, and y = 2 11. Circular cylinder band The portion of the cylinder y 2 + z 2 = 9 between the planes x = 0 and x = 3 12. Circular cylinder band The portion of the cylinder x 2 + z 2 = 4 above the xy-plane between the planes y = -2 and y = 2 13. Tilted plane inside cylinder The portion of the plane x + y + z = 1 a. Inside the cylinder x 2 + y 2 = 9 b. Inside the cylinder y 2 + z 2 = 9 14. Tilted plane inside cylinder The portion of the plane x - y + 2z = 2 a. Inside the cylinder x + z = 3 2

2

b. Inside the cylinder y 2 + z 2 = 2 15. Circular cylinder band The portion of the cylinder sx - 2d2 + z 2 = 4 between the planes y = 0 and y = 3 16. Circular cylinder band The portion of the cylinder y 2 + sz - 5d2 = 25 between the planes x = 0 and x = 10

Surface Area of Parametrized Surfaces In Exercises 17–26, use a parametrization to express the area of the surface as a double integral. Then evaluate the integral. (There are many correct ways to set up the integrals, so your integrals may not be the same as those in the back of the book. They should have the same values, however.) 17. Tilted plane inside cylinder The portion of the plane y + 2z = 2 inside the cylinder x 2 + y 2 = 1

18. Plane inside cylinder The portion of the plane z = -x inside the cylinder x 2 + y 2 = 4 19. Cone frustum The portion of the cone z = 2 2x 2 + y 2 between the planes z = 2 and z = 6 20. Cone frustum The portion of the cone z = 2x 2 + y 2>3 between the planes z = 1 and z = 4>3 21. Circular cylinder band The portion of the cylinder x 2 + y 2 = 1 between the planes z = 1 and z = 4 22. Circular cylinder band The portion of the cylinder x 2 + z 2 = 10 between the planes y = - 1 and y = 1 23. Parabolic cap The cap cut from the paraboloid z = 2 - x 2 - y 2 by the cone z = 2x 2 + y 2 24. Parabolic band The portion of the paraboloid z = x 2 + y 2 between the planes z = 1 and z = 4 25. Sawed-off sphere The lower portion cut from the sphere x 2 + y 2 + z 2 = 2 by the cone z = 2x 2 + y 2 26. Spherical band The portion of the sphere x 2 + y 2 + z 2 = 4 between the planes z = -1 and z = 23 Planes Tangent to Parametrized Surfaces The tangent plane at a point P0sƒsu0 , y0 d, gsu0 , y0 d, hsu0 , y0 dd on a parametrized surface rsu, yd = ƒsu, ydi + gsu, ydj + hsu, ydk is the plane through P0 normal to the vector rusu0 , y0 d * rysu0 , y0 d, the cross product of the tangent vectors rusu0 , y0 d and rysu0 , y0 d at P0. In Exercises 27–30, find an equation for the plane tangent to the surface at P0. Then find a Cartesian equation for the surface and sketch the surface and tangent plane together. 27. Cone The cone rsr, ud = sr cos udi + sr sin udj + rk, r Ú 0, 0 … u … 2p at the point P0 A 22, 22, 2 B corresponding to sr, ud = s2, p>4d 28. Hemisphere The hemisphere surface rsf, ud = s4 sin f cos udi + s4 sin f sin udj + s4 cos fdk, 0 … f … p>2, 0 … u … 2p, at the point P0 A 22, 22, 223 B corresponding to sf, ud = sp>6, p>4d 29. Circular cylinder The circular cylinder rsu, zd = s3 sin 2udi + s6 sin2 udj + zk, 0 … u … p, at the point P0 A 323>2, 9>2, 0 B corresponding to su, zd = sp>3, 0d (See Example 3.) 30. Parabolic cylinder The parabolic cylinder surface rsx, yd = xi + yj - x 2k, - q 6 x 6 q , - q 6 y 6 q , at the point P0s1, 2, -1d corresponding to sx, yd = s1, 2d More Parametrizations of Surfaces 31. a. A torus of revolution (doughnut) is obtained by rotating a circle C in the xz-plane about the z-axis in space. (See the accompanying figure.) If C has radius r 7 0 and center (R, 0, 0), show that a parametrization of the torus is rsu, yd = ssR + r cos udcos ydi + ssR + r cos udsin ydj + sr sin udk, where 0 … u … 2p and 0 … y … 2p are the angles in the figure.

15.5 b. Show that the surface area of the torus is A = 4p2Rr.

C 0

879

34. Hyperboloid of one sheet a. Find a parametrization for the hyperboloid of one sheet x 2 + y 2 - z 2 = 1 in terms of the angle u associated with the circle x 2 + y 2 = r 2 and the hyperbolic parameter u associated with the hyperbolic function r 2 - z 2 = 1. (Hint: cosh 2 u - sinh 2 u = 1.)

z

r u

Surfaces and Area

x

b. Generalize the result in part (a) to the hyperboloid sx 2>a 2 d + s y 2>b 2 d - sz 2>c 2 d = 1.

R

35. (Continuation of Exercise 34.) Find a Cartesian equation for the plane tangent to the hyperboloid x 2 + y 2 - z 2 = 25 at the point sx0 , y0 , 0d, where x 02 + y 02 = 25.

z

36. Hyperboloid of two sheets Find a parametrization of the hyperboloid of two sheets sz 2>c 2 d - sx 2>a 2 d - s y 2>b 2 d = 1. Surface Area for Implicit and Explicit Forms 37. Find the area of the surface cut from the paraboloid x 2 + y 2 - z = 0 by the plane z = 2. x

y

u

32. Parametrization of a surface of revolution Suppose that the parametrized curve C: (ƒ(u), g(u)) is revolved about the x-axis, where gsud 7 0 for a … u … b. a. Show that rsu, yd = ƒsudi + sgsudcos ydj + sgsudsin ydk is a parametrization of the resulting surface of revolution, where 0 … y … 2p is the angle from the xy-plane to the point r(u, y) on the surface. (See the accompanying figure.) Notice that ƒ(u) measures distance along the axis of revolution and g(u) measures distance from the axis of revolution. y

( f (u), g(u), 0) r(u, y) C y g(u) z

f (u) x

b. Find a parametrization for the surface obtained by revolving the curve x = y 2, y Ú 0, about the x-axis. 33. a. Parametrization of an ellipsoid The parametrization x = a cos u, y = b sin u, 0 … u … 2p gives the ellipse sx 2>a 2 d + s y 2>b 2 d = 1. Using the angles u and f in spherical coordinates, show that rsu, fd = sa cos u cos fdi + sb sin u cos fdj + sc sin fdk

is a parametrization of the ellipsoid sx 2>a 2 d + s y 2>b 2 d + sz 2>c 2 d = 1.

b. Write an integral for the surface area of the ellipsoid, but do not evaluate the integral.

38. Find the area of the band cut from the paraboloid x 2 + y 2 - z = 0 by the planes z = 2 and z = 6. 39. Find the area of the region cut from the plane x + 2y + 2z = 5 by the cylinder whose walls are x = y 2 and x = 2 - y 2. 40. Find the area of the portion of the surface x 2 - 2z = 0 that lies above the triangle bounded by the lines x = 23, y = 0, and y = x in the xy-plane. 41. Find the area of the surface x 2 - 2y - 2z = 0 that lies above the triangle bounded by the lines x = 2, y = 0, and y = 3x in the xy-plane. 42. Find the area of the cap cut from the sphere x 2 + y 2 + z 2 = 2 by the cone z = 2x 2 + y 2. 43. Find the area of the ellipse cut from the plane z = cx (c a constant) by the cylinder x 2 + y 2 = 1. 44. Find the area of the upper portion of the cylinder x 2 + z 2 = 1 that lies between the planes x = ;1>2 and y = ;1>2. 45. Find the area of the portion of the paraboloid x = 4 - y 2 - z 2 that lies above the ring 1 … y 2 + z 2 … 4 in the yz-plane. 46. Find the area of the surface cut from the paraboloid x 2 + y + z 2 = 2 by the plane y = 0. 47. Find the area of the surface x 2 - 2 ln x + 215y - z = 0 above the square R: 1 … x … 2, 0 … y … 1, in the xy-plane. 48. Find the area of the surface 2x 3>2 + 2y 3>2 - 3z = 0 above the square R: 0 … x … 1, 0 … y … 1, in the xy-plane. Find the area of the surfaces in Exercises 49–54. 49. The surface cut from the bottom of the paraboloid z = x 2 + y 2 by the plane z = 3 50. The surface cut from the “nose” of the paraboloid x = 1 y 2 - z 2 by the yz-plane 51. The portion of the cone z = 2x 2 + y 2 that lies over the region between the circle x 2 + y 2 = 1 and the ellipse 9x 2 + 4y 2 = 36 in the xy-plane. (Hint: Use formulas from geometry to find the area of the region.) 52. The triangle cut from the plane 2x + 6y + 3z = 6 by the bounding planes of the first octant. Calculate the area three ways, using different explicit forms. 53. The surface in the first octant cut from the cylinder y = s2>3dz 3>2 by the planes x = 1 and y = 16>3

880

Chapter 15: Integration in Vector Fields

54. The portion of the plane y + z = 4 that lies above the region cut from the first quadrant of the xz-plane by the parabola x = 4 - z2

z

55. Use the parametrization

0

r(x, z) = xi + ƒ(x, z)j + zk

y

(x, y, z)

and Equation (5) to derive a formula for ds associated with the explicit form y = ƒ(x, z).

u

z

56. Let S be the surface obtained by rotating the smooth curve y = ƒ(x), a … x … b, about the x-axis, where ƒ(x) Ú 0.

f (x)

a. Show that the vector function x

r(x, u) = x i + ƒ(x) cos u j + ƒ(x) sin u k

b. Use Equation (4) to show that the surface area of this surface of revolution is given by

is a parametrization of S, where u is the angle of rotation around the x-axis (see the accompanying figure).

b

A =

La

2pƒ(x)21 + [ƒ¿(x)]2 dx.

Surface Integrals

15.6

To compute quantities such as the flow of liquid across a curved membrane or the upward force on a falling parachute, we need to integrate a function over a curved surface in space. This concept of a surface integral is an extension of the idea of a line integral for integrating over a curve.

Surface Integrals

Pk

Δyry

z Δuru Δsk 5 Δsuy x

(xk , yk , zk )

y

FIGURE 15.47 The area of the patch ¢sk is the area of the tangent parallelogram determined by the vectors ¢u ru and ¢y ry. The point (xk, yk, zk) lies on the surface patch, beneath the parallelogram shown here.

Suppose that we have an electrical charge distributed over a surface S, and that the function G(x, y, z) gives the charge density (charge per unit area) at each point on S. Then we can calculate the total charge on S as an integral in the following way. Assume, as in Section 15.5, that the surface S is defined parametrically on a region R in the uy-plane, r(u, y) = ƒ(u, y)i + g(u, y)j + h(u, y)k,

(u, y) H R.

In Figure 15.47, we see how a subdivision of R (considered as a rectangle for simplicity) divides the surface S into corresponding curved surface elements, or patches, of area ¢suy L ƒ ru * ry ƒ du dy. As we did for the subdivisions when defining double integrals in Section 14.2, we number the surface element patches in some order with their areas given by ¢s1, ¢s2, Á , ¢sn. To form a Riemann sum over S, we choose a point (xk, yk, zk) in the kth patch, multiply the value of the function G at that point by the area ¢sk , and add together the products: n

a G(xk, yk, zk) ¢sk.

k=1

Depending on how we pick (xk, yk, zk) in the kth patch, we may get different values for this Riemann sum. Then we take the limit as the number of surface patches increases, their areas shrink to zero, and both ¢u : 0 and ¢y : 0. This limit, whenever it exists independent of all choices made, defines the surface integral of G over the surface S as n

6 S

G(x, y, z) ds = lim a G(xk, yk, zk) ¢sk. n: q k=1

(1)

15.6

Surface Integrals

881

Notice the analogy with the definition of the double integral (Section 14.2) and with the line integral (Section 15.1). If S is a piecewise smooth surface, and G is continuous over S, then the surface integral defined by Equation (1) can be shown to exist. The formula for evaluating the surface integral depends on the manner in which S is described, parametrically, implicitly or explicitly, as discussed in Section 15.5.

Formulas for a Surface Integral 1. For a smooth surface S defined parametrically as r(u, y) = ƒ(u, y)i +

g(u, y)j + h(u, y)k, (u, y) H R, and a continuous function G(x, y, z) defined on S, the surface integral of G over S is given by the double integral over R,

6

G(x, y, z) ds =

G(ƒ(u, y), g(u, y), h(u, y)) ƒ ru * ry ƒ du dy.

6

S

(2)

R

2. For a surface S given implicitly by F(x, y, z) c, where F is a continuously

differentiable function, with S lying above its closed and bounded shadow region R in the coordinate plane beneath it, the surface integral of the continuous function G over S is given by the double integral over R,

6

G(x, y, z) ds =

S

6

G(x, y, z)

R

ƒ ¥F ƒ dA, ƒ ¥F # p ƒ

(3)

where p is a unit vector normal to R and ¥F # p Z 0. 3. For a surface S given explicitly as the graph of z = ƒ(x, y), where ƒ is a continuously differentiable function over a region R in the xy-plane, the surface integral of the continuous function G over S is given by the double integral over R,

6

G(x, y, z) ds =

6

S

G(x, y, ƒ(x, y)) 2ƒx 2 + ƒy 2 + 1 dx dy.

(4)

R

The surface integral in Equation (1) takes on different meanings in different applications. If G has the constant value 1, the integral gives the area of S. If G gives the mass density of a thin shell of material modeled by S, the integral gives the mass of the shell. If G gives the charge density of a thin shell, then the integral gives the total charge.

EXAMPLE 1

Integrate Gsx, y, zd = x 2 over the cone z = 2x 2 + y 2, 0 … z … 1.

Using Equation (2) and the calculations from Example 4 in Section 15.5, we have ƒ rr * ru ƒ = 22r and

Solution

2p

6

x 2 ds =

S

L0

1

L0

A r 2 cos2 u B A 22r B dr du

2p

x = r cos u

1

= 22 r 3 cos2 u dr du L0 L0 =

22 4 L0

2p

2p

cos2 u du =

22 u p22 1 c + sin 2u d = . 4 2 4 4 0

882

Chapter 15: Integration in Vector Fields

Surface integrals behave like other double integrals, the integral of the sum of two functions being the sum of their integrals and so on. The domain Additivity Property takes the form 6

G ds =

S

6

G ds +

S1

6

G ds + Á +

S2

6

G ds.

Sn

When S is partitioned by smooth curves into a finite number of smooth patches with nonoverlapping interiors (i.e., if S is piecewise smooth), then the integral over S is the sum of the integrals over the patches. Thus, the integral of a function over the surface of a cube is the sum of the integrals over the faces of the cube. We integrate over a turtle shell of welded plates by integrating over one plate at a time and adding the results. Integrate Gsx, y, zd = xyz over the surface of the cube cut from the first octant by the planes x = 1, y = 1, and z = 1 (Figure 15.48).

EXAMPLE 2

z

Side A

1

0

We integrate xyz over each of the six sides and add the results. Since xyz = 0 on the sides that lie in the coordinate planes, the integral over the surface of the cube reduces to Solution

6 1 y

1 Side C x

Side B

xyz ds =

Cube surface

6

xyz ds +

Side A

6

xyz ds +

Side B

6

xyz ds.

Side C

Side A is the surface ƒsx, y, zd = z = 1 over the square region Rxy: 0 … x … 1, 0 … y … 1, in the xy-plane. For this surface and region, p = k,

FIGURE 15.48 The cube in Example 2.

§ƒ = k,

ƒ §ƒ # p ƒ = ƒ k # k ƒ = 1

ƒ §ƒ ƒ = 1,

ƒ §ƒ ƒ 1 dA = dx dy = dx dy # 1 §ƒ p ƒ ƒ

ds =

xyz = xys1d = xy and 1

6

xyz ds =

Side A

6

xy dx dy =

Rxy

1

L0 L0

1

xy dx dy =

y 1 dy = . 2 4 0 L

Symmetry tells us that the integrals of xyz over sides B and C are also 1> 4. Hence, 6

xyz ds =

3 1 1 1 + + = . 4 4 4 4

Cube surface

Integrate G sx, y, zd = 21 - x 2 - y 2 over the “football” surface S formed by rotating the curve x = cos z, y = 0, -p>2 … z … p>2, around the z-axis.

EXAMPLE 3

The surface is displayed in Figure 15.44, and in Example 6 of Section 15.5 we found the parametrization Solution

p p … u … and 0 … y … 2p, 2 2 where y represents the angle of rotation from the xz-plane about the z-axis. Substituting this parametrization into the expression for G gives x = cos u cos y,

y = cos u sin y,

z = u,

-

21 - x 2 - y 2 = 21 - (cos2 u)(cos2 y + sin2 y) = 21 - cos2 u = ƒ sin u ƒ. The surface area differential for the parametrization was found to be (Example 6, Section 15.5) ds = cos u 21 + sin2 u du dy.

15.6

Surface Integrals

883

These calculations give the surface integral p>2

2p

6

21 - x 2 - y 2 ds =

S

L0

L-p>2

p>2

2p

= 2

L0

n

2

L1

L0

= 2p #

Positive direction

sin u cos u 21 + sin2 u du dy

L0

2p

=

ƒ sin u ƒ cos u 21 + sin2 u du dy

2w dw dy

w = 1 + sin2 u, dw = 2 sin u cos u du When u = 0, w = 1. When u = p>2, w = 2.

4p 2 3>2 w d = A 222 - 1 B . 3 3 1 2

Orientation

FIGURE 15.49 Smooth closed surfaces in space are orientable. The outward unit normal vector defines the positive direction at each point. d

c

a

b Start Finish db

We call a smooth surface S orientable or two-sided if it is possible to define a field n of unit normal vectors on S that varies continuously with position. Any patch or subportion of an orientable surface is orientable. Spheres and other smooth closed surfaces in space (smooth surfaces that enclose solids) are orientable. By convention, we choose n on a closed surface to point outward. Once n has been chosen, we say that we have oriented the surface, and we call the surface together with its normal field an oriented surface. The vector n at any point is called the positive direction at that point (Figure 15.49). The Möbius band in Figure 15.50 is not orientable. No matter where you start to construct a continuous unit normal field (shown as the shaft of a thumbtack in the figure), moving the vector continuously around the surface in the manner shown will return it to the starting point with a direction opposite to the one it had when it started out. The vector at that point cannot point both ways and yet it must if the field is to be continuous. We conclude that no such field exists.

Surface Integral for Flux

ac

FIGURE 15.50 To make a Möbius band, take a rectangular strip of paper abcd, give the end bc a single twist, and paste the ends of the strip together to match a with c and b with d. The Möbius band is a nonorientable or one-sided surface.

Suppose that F is a continuous vector field defined over an oriented surface S and that n is the chosen unit normal field on the surface. We call the integral of F # n over S the flux of F across S in the positive direction. Thus, the flux is the integral over S of the scalar component of F in the direction of n.

DEFINITION The flux of a three-dimensional vector field F across an oriented surface S in the direction of n is

z

Flux =

4 (1, 0, 4)

6

F # n ds.

(5)

S

y x2

The definition is analogous to the flux of a two-dimensional field F across a plane curve C. In the plane (Section 15.2), the flux is

n

LC 1

y

1 x

FIGURE 15.51 Finding the flux through the surface of a parabolic cylinder (Example 4).

F # n ds,

the integral of the scalar component of F normal to the curve. If F is the velocity field of a three-dimensional fluid flow, the flux of F across S is the net rate at which fluid is crossing S in the chosen positive direction. We discuss such flows in more detail in Section 15.7. Find the flux of F = yzi + xj - z 2 k through the parabolic cylinder y = x , 0 … x … 1, 0 … z … 4, in the direction n indicated in Figure 15.51.

EXAMPLE 4 2

884

Chapter 15: Integration in Vector Fields

On the surface we have x = x, y = x 2, and z = z, so we automatically have the parametrization rsx, zd = xi + x 2 j + zk, 0 … x … 1, 0 … z … 4. The cross product of tangent vectors is i j k rx * rz = 3 1 2x 0 3 = 2xi - j. 0 0 1 The unit normal vectors pointing outward from the surface as indicated in Figure 15.51 are Solution

2xi - j rx * rz . = ƒ rx * rz ƒ 24x 2 + 1 On the surface, y = x 2, so the vector field there is n =

F = yzi + xj - z 2 k = x 2zi + xj - z 2 k. Thus, F#n =

1 ssx 2zds2xd + sxds -1d + s -z 2 ds0dd 24x 2 + 1 2x 3z - x = . 24x 2 + 1

The flux of F outward through the surface is 6

F # n ds =

S

4

2x 3z - x

L0 L0 24x 2 + 1 4

=

1

1

2x 3z - x

L0 L0 24x 2 + 1 4

=

ƒ rx * rz ƒ dx dz 24x 2 + 1 dx dz

1

x=1

4

s2x 3z - xd dx dz =

1 1 c x 4z - x 2 d dz 2 2 x=0 0 L

L0 L0 4 4 1 1 = sz - 1d dz = sz - 1d2 d 4 0 L0 2 1 1 = s9d - s1d = 2. 4 4

There is a simple formula for the flux of F across a parametrized surface r(u, y). Since ds = ƒ ru * ry ƒ du dy and n = Flux Across a Parametrized Surface Flux =

F # (ru * ry) du dy 6 R

ru * ry ƒ ru * ry ƒ

it follows that ru * ry F # n ds = F# F # (ru * ry) du dy. ƒ r * ry ƒ du dy = ru * ry ƒ u ƒ 6 6 6 S

R

R

This integral for flux simplifies the computation in Example 4. Since

F # (rx * rz) = (x2z)(2x) + (x)(-1) = 2x3z - x,

we obtain directly F # n ds = (2x3z - x) dx dz = 2 6 L0 L0 4

Flux =

1

S

in Example 4. If S is part of a level surface gsx, y, zd = c, then n may be taken to be one of the two fields n = ;

§g ƒ §g ƒ

,

(6)

15.6

Surface Integrals

885

depending on which one gives the preferred direction. The corresponding flux is Flux =

6

F # n ds

S

=

aF #

6 R

=

6

F#

R

; §g ƒ §g ƒ b # p ƒ dA §g §g ƒ ƒ ƒ

Eqs. (6) and (3)

; §g dA. ƒ §g # p ƒ

(7)

Find the flux of F = yzj + z 2k outward through the surface S cut from the cylinder y + z 2 = 1, z Ú 0, by the planes x = 0 and x = 1.

EXAMPLE 5

2

z y2

2

1z 51

Solution The outward normal field on S (Figure 15.52) may be calculated from the gradient of gsx, y, zd = y 2 + z 2 to be §g 2yj + 2zk 2yj + 2zk n = + = = = yj + zk. 2 2 ƒ §g ƒ 24y + 4z 221

n

S 1

R xy

y

(1, –1, 0) (1, 1, 0) x

FIGURE 15.52 Calculating the flux of a vector field outward through the surface S. The area of the shadow region Rxy is 2 (Example 5).

With p = k, we also have

ƒ §g ƒ 2 1 dA = dA = z dA. # k §g 2z ƒ ƒ ƒ ƒ We can drop the absolute value bars because z Ú 0 on S. The value of F # n on the surface is F # n = syzj + z 2 kd # s yj + zkd = y 2z + z 3 = zs y 2 + z 2 d y 2 + z 2 = 1 on S = z. The surface projects onto the shadow region Rxy, which is the rectangle in the xy-plane shown in Figure 15.52. Therefore, the flux of F outward through S is ds =

6

F # n ds =

S

6

1 szd a z dAb =

Rxy

6

dA = areasRxy d = 2.

Rxy

Moments and Masses of Thin Shells Thin shells of material like bowls, metal drums, and domes are modeled with surfaces. Their moments and masses are calculated with the formulas in Table 15.3. The derivations TABLE 15.3 Mass and moment formulas for very thin shells

M =

Mass:

6

d ds

d = dsx, y, zd = density at sx, y, zd as mass per unit area

S

First moments about the coordinate planes: Myz =

6

Mxz =

x d ds,

S

6

y d ds,

Mxy =

S

Coordinates of center of mass: x = Myz >M,

y = Mxz >M,

6

z d ds

S

z = Mxy >M

Moments of inertia about coordinate axes: Ix =

6

s y 2 + z 2 d d ds,

S

IL =

6 S

Iy =

6

sx 2 + z 2 d d ds, Iz =

S

r 2d ds

rsx, y, zd = distance from point sx, y, zd to line L

6 S

sx 2 + y 2 d d ds,

886

Chapter 15: Integration in Vector Fields

are similar to those in Section 6.6. The formulas are like those for line integrals in Table 15.1, Section 15.1. z ⎛0, 0, a ⎛ ⎝ 2⎝

EXAMPLE 6

x2 1 y2 1 z2 5 a2

Find the center of mass of a thin hemispherical shell of radius a and con-

stant density d. We model the shell with the hemisphere

Solution

ƒsx, y, zd = x 2 + y 2 + z 2 = a 2,

S

a

R

y

a x2 1 y2 5 a2

x

FIGURE 15.53 The center of mass of a thin hemispherical shell of constant density lies on the axis of symmetry halfway from the base to the top (Example 6).

z Ú 0

(Figure 15.53). The symmetry of the surface about the z-axis tells us that x = y = 0. It remains only to find z from the formula z = Mxy >M. The mass of the shell is M =

6

d ds = d

6

S

ds = sddsarea of Sd = 2pa 2d.

d = constant

S

To evaluate the integral for Mxy, we take p = k and calculate ƒ §ƒ ƒ = ƒ 2xi + 2yj + 2zk ƒ = 22x 2 + y 2 + z 2 = 2a ƒ §ƒ # p ƒ = ƒ §ƒ # k ƒ = ƒ 2z ƒ = 2z ds =

ƒ §ƒ ƒ a dA = z dA. ƒ §ƒ # p ƒ

Then Mxy =

6

zd ds = d

6

S

z =

a z z dA = da

6

R

dA = daspa 2 d = dpa 3

R

Mxy pa 3d a = = . M 2 2pa 2d

The shell’s center of mass is the point (0, 0, a> 2). Find the center of mass of a thin shell of density d = 1>z 2 cut from the cone z = 2x + y 2 by the planes z = 1 and z = 2 (Figure 15.54).

EXAMPLE 7

z

2

The symmetry of the surface about the z-axis tells us that x = y = 0 . We find z = Mxy >M. Working as in Example 4 of Section 15.5, we have

Solution

2 z

x 2

y2

1

rsr, ud = (r cos u)i + (r sin u)j + rk,

1 … r … 2,

0 … u … 2p,

and x

ƒ rr * ru ƒ = 22r.

y

Therefore, FIGURE 15.54 The cone frustum formed when the cone z = 2x 2 + y 2 is cut by the planes z = 1 and z = 2 (Example 7).

2p

M =

6

d ds =

S

2p

= 22 L0

2

1 22r dr du 2 L0 L1 r

2 C ln r D 1 du = 22

= 2p22 ln 2,

2p

L0

ln 2 du

15.6 2p

Mxy =

6

dz ds =

S

2p

887

Surface Integrals

2

1 r22r dr du 2 L0 L1 r

2

= 22

dr du

L0 L1 2p

= 22

L0

z =

du = 2p22,

Mxy 2p22 1 . = = M ln 2 2p22 ln 2

The shell’s center of mass is the point (0, 0, 1> ln 2).

Exercises 15.6 Surface Integrals In Exercises 1–8, integrate the given function over the given surface.

z

1. Parabolic cylinder Gsx, y, zd = x, over the parabolic cylinder y = x 2, 0 … x … 2, 0 … z … 3

(1, 1, 2) 1

2. Circular cylinder Gsx, y, zd = z, over the cylindrical surface y 2 + z 2 = 4, z Ú 0, 1 … x … 4

z x y2

(0, 1, 1)

3. Sphere Gsx, y, zd = x 2, over the unit sphere x 2 + y 2 + z 2 = 1 4. Hemisphere Gsx, y, zd = z 2, over the hemisphere x 2 + y 2 + z 2 = a 2, z Ú 0 5. Portion of plane Fsx, y, zd = z, over the portion of the plane x + y + z = 4 that lies above the square 0 … x … 1, 0 … y … 1, in the xy-plane

(0, 0, 0)

1

6. Cone Fsx, y, zd = z - x, over the cone z = 2x 2 + y 2, 0 … z … 1

1

7. Parabolic dome Hsx, y, zd = x 2 25 - 4z, over the parabolic dome z = 1 - x 2 - y 2, z Ú 0

x

8. Spherical cap Hsx, y, zd = yz, over the part of the sphere x 2 + y 2 + z 2 = 4 that lies above the cone z = 2x 2 + y 2 9. Integrate Gsx, y, zd = x + y + z over the surface of the cube cut from the first octant by the planes x = a, y = a, z = a. 10. Integrate Gsx, y, zd = y + z over the surface of the wedge in the first octant bounded by the coordinate planes and the planes x = 2 and y + z = 1.

(0, 1, 0) y

(1, 1, 0)

16. Integrate G(x, y, z) = x over the surface given by z = x 2 + y for 0 … x … 1,

-1 … y … 1.

17. Integrate G(x, y, z) = xyz over the triangular surface with vertices (1, 0, 0), (0, 2, 0), and (0, 1, 1).

11. Integrate Gsx, y, zd = xyz over the surface of the rectangular solid cut from the first octant by the planes x = a, y = b, and z = c.

z 1

(0, 1, 1)

12. Integrate Gsx, y, zd = xyz over the surface of the rectangular solid bounded by the planes x = ;a, y = ;b, and z = ;c. 13. Integrate Gsx, y, zd = x + y + z over the portion of the plane 2x + 2y + z = 2 that lies in the first octant. 14. Integrate Gsx, y, zd = x2y 2 + 4 over the surface cut from the parabolic cylinder y 2 + 4z = 16 by the planes x = 0, x = 1, and z = 0. 15. Integrate G(x, y, z) = z - x over the portion of the graph of z = x + y 2 above the triangle in the xy-plane having vertices (0, 0, 0), (1, 1, 0), and (0, 1, 0). (See accompanying figure.)

y (0, 2, 0) x

(1, 0, 0)

18. Integrate G(x, y, z) = x - y - z over the portion of the plane x + y = 1 in the first octant between z = 0 and z = 1 (see the accompanying figure on the next page).

888

Chapter 15: Integration in Vector Fields 33. Fsx, y, zd = yi - xj + k

z 1

34. Fsx, y, zd = zxi + zyj + z 2k

(0, 1, 1)

35. Fsx, y, zd = xi + yj + zk xi + yj + zk 36. Fsx, y, zd = 2x 2 + y 2 + z 2

(1, 0, 1) y

1 x

37. Find the flux of the field Fsx, y, zd = z 2i + xj - 3zk outward through the surface cut from the parabolic cylinder z = 4 - y 2 by the planes x = 0, x = 1, and z = 0.

1

Finding Flux Across a Surface In Exercises 19–28, use a parametrization to find the flux 4S F # n ds across the surface in the given direction. 19. Parabolic cylinder F = z 2i + xj - 3zk outward (normal away from the x-axis) through the surface cut from the parabolic cylinder z = 4 - y 2 by the planes x = 0, x = 1 , and z = 0 20. Parabolic cylinder F = x 2j - xzk outward (normal away from the yz-plane) through the surface cut from the parabolic cylinder y = x 2, -1 … x … 1 , by the planes z = 0 and z = 2

38. Find the flux of the field Fsx, y, zd = 4xi + 4yj + 2k outward (away from the z-axis) through the surface cut from the bottom of the paraboloid z = x 2 + y 2 by the plane z = 1. 39. Let S be the portion of the cylinder y = e x in the first octant that projects parallel to the x-axis onto the rectangle Ryz: 1 … y … 2, 0 … z … 1 in the yz-plane (see the accompanying figure). Let n be the unit vector normal to S that points away from the yz-plane. Find the flux of the field Fsx, y, zd = -2i + 2yj + zk across S in the direction of n. z

21. Sphere F = zk across the portion of the sphere x 2 + y 2 + z 2 = a 2 in the first octant in the direction away from the origin 22. Sphere F = xi + yj + zk across the sphere x + y + z = a in the direction away from the origin 2

2

2

23. Plane F = 2xyi + 2yzj + 2xzk upward across the portion of the plane x + y + z = 2a that lies above the square 0 … x … a, 0 … y … a , in the xy-plane 24. Cylinder F = xi + yj + zk outward through the portion of the cylinder x 2 + y 2 = 1 cut by the planes z = 0 and z = a 25. Cone F = xyi - zk outward (normal away from the z-axis) through the cone z = 2x 2 + y 2, 0 … z … 1 26. Cone F = y i + xzj - k outward (normal away from the z-axis) through the cone z = 2 2x 2 + y 2, 0 … z … 2 2

27. Cone frustum F = -xi - yj + z 2k outward (normal away from the z-axis) through the portion of the cone z = 2x 2 + y 2 between the planes z = 1 and z = 2 28. Paraboloid F = 4xi + 4yj + 2k outward (normal away from the z-axis) through the surface cut from the bottom of the paraboloid z = x 2 + y 2 by the plane z = 1 In Exercises 29 and 30, find the flux of the field F across the portion of the given surface in the specified direction. 29. Fsx, y, zd = -i + 2j + 3k S: rectangular surface z = 0, direction k

0 … x … 2,

0 … y … 3,

30. Fsx, y, zd = yx i - 2j + xzk 2

S: rectangular surface y = 0, direction -j

-1 … x … 2,

2 … z … 7,

In Exercises 31–36, find the flux of the field F across the portion of the sphere x 2 + y 2 + z 2 = a 2 in the first octant in the direction away from the origin. 31. Fsx, y, zd = zk 32. Fsx, y, zd = -yi + xj

1

Ry z

2

1 x y ex

S

2

y

40. Let S be the portion of the cylinder y = ln x in the first octant whose projection parallel to the y-axis onto the xz-plane is the rectangle Rxz: 1 … x … e, 0 … z … 1. Let n be the unit vector normal to S that points away from the xz-plane. Find the flux of F = 2yj + zk through S in the direction of n. 41. Find the outward flux of the field F = 2xyi + 2yzj + 2xzk across the surface of the cube cut from the first octant by the planes x = a, y = a, z = a. 42. Find the outward flux of the field F = xzi + yzj + k across the surface of the upper cap cut from the solid sphere x 2 + y 2 + z 2 … 25 by the plane z = 3. Moments and Masses 43. Centroid Find the centroid of the portion of the sphere x 2 + y 2 + z 2 = a 2 that lies in the first octant. 44. Centroid Find the centroid of the surface cut from the cylinder y 2 + z 2 = 9, z Ú 0, by the planes x = 0 and x = 3 (resembles the surface in Example 5). 45. Thin shell of constant density Find the center of mass and the moment of inertia about the z-axis of a thin shell of constant density d cut from the cone x 2 + y 2 - z 2 = 0 by the planes z = 1 and z = 2. 46. Conical surface of constant density Find the moment of inertia about the z-axis of a thin shell of constant density d cut from the cone 4x 2 + 4y 2 - z 2 = 0, z Ú 0, by the circular cylinder x 2 + y 2 = 2x (see the accompanying figure).

15.7

Stokes’ Theorem

889

47. Spherical shells

z

a. Find the moment of inertia about a diameter of a thin spherical shell of radius a and constant density d. (Work with a hemispherical shell and double the result.) b. Use the Parallel Axis Theorem (Exercises 14.6) and the result in part (a) to find the moment of inertia about a line tangent to the shell. 48. Conical Surface Find the centroid of the lateral surface of a solid cone of base radius a and height h (cone surface minus the base).

4x 2 4y2 z 2 0 z0 y x

15.7

2

x 2 y2 2x or r 2 cos

Stokes’ Theorem

Curl F (x, y, z)

FIGURE 15.55 The circulation vector at a point (x, y, z) in a plane in a threedimensional fluid flow. Notice its right-hand relation to the rotating particles in the fluid.

As we saw in Section 15.4, the circulation density or curl component of a two-dimensional field F = Mi + Nj at a point (x, y) is described by the scalar quantity s0N>0x - 0M>0yd. In three dimensions, circulation is described with a vector. Suppose that F is the velocity field of a fluid flowing in space. Particles near the point (x, y, z) in the fluid tend to rotate around an axis through (x, y, z) that is parallel to a certain vector we are about to define. This vector points in the direction for which the rotation is counterclockwise when viewed looking down onto the plane of the circulation from the tip of the arrow representing the vector. This is the direction your right-hand thumb points when your fingers curl around the axis of rotation in the way consistent with the rotating motion of the particles in the fluid (see Figure 15.55). The length of the vector measures the rate of rotation. The vector is called the curl vector, and for the vector field F = Mi + Nj + Pk it is defined to be curl F = a

0N 0N 0P 0M 0P 0M bi + a bj + a bk. 0y 0z 0z 0x 0x 0y

(1)

This information is a consequence of Stokes’ Theorem, the generalization to space of the circulation-curl form of Green’s Theorem and the subject of this section. Notice that scurl Fd # k = s0N>0x - 0M>0yd is consistent with our definition in Section 15.4 when F = Msx, ydi + Nsx, ydj. The formula for curl F in Equation (1) is often written using the symbolic operator 0 0 0 § = i + j + k . (2) 0x 0y 0z (The symbol § is pronounced “del.”) The curl of F is § * F : i

j

k

0 § * F = 4 0x

0 0y

0 4 0z

M

N

P

= a

0N 0N 0P 0M 0P 0M bi + a bj + a bk 0y 0z 0z 0x 0x 0y

= curl F.

890

Chapter 15: Integration in Vector Fields

curl F = § * F

EXAMPLE 1 Solution

(3)

Find the curl of F = sx 2 - zdi + xe zj + xyk.

We use Equation (3) and the determinant form, so curl F = § * F i

j

k

0 0x

0 0y

0 4 0z

x2 - z

xe z

xy

= 4

= a

0 0 0 2 0 sxyd sxe z dbi - a sxyd sx - zdbj 0y 0z 0x 0z

+ a

0 0 2 sxe z d sx - zdbk 0x 0y

= sx - xe z di - s y + 1dj + se z - 0dk = x(1 - e z)i - ( y + 1)j + e zk As we will see, the operator § has a number of other applications. For instance, when applied to a scalar function ƒ(x, y, z), it gives the gradient of ƒ: §ƒ =

0ƒ 0ƒ 0ƒ i + j + k. 0x 0y 0z

It is sometimes read as “del ƒ” as well as “grad ƒ.”

Stokes’ Theorem S

n

C

FIGURE 15.56 The orientation of the bounding curve C gives it a right-handed relation to the normal field n. If the thumb of a right hand points along n, the fingers curl in the direction of C.

Stokes’ Theorem generalizes Green’s Theorem to three dimensions. The circulation-curl form of Green’s Theorem relates the counterclockwise circulation of a vector field around a simple closed curve C in the xy-plane to a double integral over the plane region R enclosed by C. Stokes’ Theorem relates the circulation of a vector field around the boundary C of an oriented surface S in space (Figure 15.56) to a surface integral over the surface S. We require that the surface be piecewise smooth, which means that it is a finite union of smooth surfaces joining along smooth curves.

THEOREM 6—Stokes’ Theorem Let S be a piecewise smooth oriented surface having a piecewise smooth boundary curve C. Let F = Mi + Nj + Pk be a vector field whose components have continuous first partial derivatives on an open region containing S. Then the circulation of F around C in the direction counterclockwise with respect to the surface’s unit normal vector n equals the integral of § * F # n over S: F

F # dr =

C

Counterclockwise circulation

6

§ * F # n ds

S

Curl integral

(4)

15.7

Stokes’ Theorem

891

Notice from Equation (4) that if two different oriented surfaces S1 and S2 have the same boundary C, their curl integrals are equal:

6

§ * F # n1 ds =

S1

6

§ * F # n2 ds.

S2

Both curl integrals equal the counterclockwise circulation integral on the left side of Equation (4) as long as the unit normal vectors n1 and n2 correctly orient the surfaces. If C is a curve in the xy-plane, oriented counterclockwise, and R is the region in the xy-plane bounded by C, then ds = dx dy and s§ * Fd # n = s§ * Fd # k = a

0N 0M b. 0x 0y

Under these conditions, Stokes’ equation becomes F # dr =

F

C

a

0N 0M b dx dy, 0x 0y 6 R

which is the circulation-curl form of the equation in Green’s Theorem. Conversely, by reversing these steps we can rewrite the circulation-curl form of Green’s Theorem for two-dimensional fields in del notation as

Green: k R

Curl

F

Circulation

F # dr =

C

6

§ * F # k dA.

(5)

R

See Figure 15.57.

Stokes:

Evaluate Equation (4) for the hemisphere S: x 2 + y 2 + z 2 = 9, z Ú 0, its bounding circle C: x 2 + y 2 = 9, z = 0, and the field F = yi - xj.

EXAMPLE 2

n Cur

S

l

a C ir c ul

The hemisphere looks much like the surface in Figure 15.56 with the bounding circle C in the xy-plane (see Figure 15.58). We calculate the counterclockwise circulation around C (as viewed from above) using the parametrization rsud = s3 cos udi + s3 sin udj, 0 … u … 2p: Solution

tion

FIGURE 15.57 Comparison of Green’s Theorem and Stokes’ Theorem.

dr = s -3 sin u dudi + s3 cos u dudj F = yi - xj = s3 sin udi - s3 cos udj

F # dr = -9 sin2 u du - 9 cos2 u du = -9 du F

F # dr =

C

z x2

1

y2

1

z2

59

2p

L0

-9 du = -18p.

For the curl integral of F, we have § * F = a

n

k y C: x 2 1 y 2 5 9 x

FIGURE 15.58 A hemisphere and a disk, each with boundary C (Examples 2 and 3).

0N 0N 0P 0M 0P 0M bi + a bj + a bk 0y 0z 0z 0x 0x 0y

= s0 - 0di + s0 - 0dj + s -1 - 1dk = -2k xi + yj + zk xi + yj + zk Outer unit normal n = = 2 2 2 3 2x + y + z Section 15.6, Example 6, 3 ds = z dA with a = 3

§ * F # n ds = -

2z 3 dA = -2 dA 3 z

892

Chapter 15: Integration in Vector Fields

and 6

§ * F # n ds =

6 2

-2 dA = -18p.

x +y …9

S

2

The circulation around the circle equals the integral of the curl over the hemisphere, as it should. The surface integral in Stokes’ Theorem can be computed using any surface having boundary curve C, provided the surface is properly oriented and lies within the domain of the field F. The next example illustrates this fact for the circulation around the curve C in Example 2.

EXAMPLE 3

Calculate the circulation around the bounding circle C in Example 2 using the disk of radius 3 centered at the origin in the xy-plane as the surface S (instead of the hemisphere). See Figure 15.58. As in Example 2, § * F = -2k. For the surface being the described disk in the xy-plane, we have the normal vector n = k so that Solution

§ * F # n ds = -2k # k dA = -2 dA

and 6

§ * F # n ds =

6 2

-2 dA = -18p,

x2 + y … 9

S

a simpler calculation than before. z

Find the circulation of the field F = sx 2 - ydi + 4zj + x 2k around the curve C in which the plane z = 2 meets the cone z = 2x 2 + y 2 , counterclockwise as viewed from above (Figure 15.59).

EXAMPLE 4 C: x 2 y 2 4, z 2

n

Stokes’ Theorem enables us to find the circulation by integrating over the surface of the cone. Traversing C in the counterclockwise direction viewed from above corresponds to taking the inner normal n to the cone, the normal with a positive k-component. We parametrize the cone as

Solution

rsr, ud = sr cos udi + sr sin udj + rk, S: r(t) (r cos )i (r sin ) j rk x

0 … r … 2,

0 … u … 2p.

We then have

y

n =

FIGURE 15.59 The curve C and cone S in Example 4.

=

-sr cos udi - sr sin udj + rk rr * ru = ƒ rr * ru ƒ r22

Section 15.5, Example 4

1 Q -scos udi - ssin udj + kb 22

ds = r22 dr du § * F = -4i - 2xj + k = -4i - 2r cos uj + k.

Section 15.5, Example 4 Routine calculation x = r cos u

Accordingly, § * F#n = =

1 a4 cos u + 2r cos u sin u + 1b 22 1 a4 cos u + r sin 2u + 1b 22

15.7

Stokes’ Theorem

893

and the circulation is

F

F # dr =

C

6

§ * F # n ds

Stokes’ Theorem, Eq. (4)

S

2p

=

L0

2

L0

1 a4 cos u + r sin 2u + 1b A r22 dr du B = 4p. 22

EXAMPLE 5

The cone used in Example 4 is not the easiest surface to use for calculating the circulation around the bounding circle C lying in the plane z = 3. If instead we use the flat disk of radius 3 centered on the z-axis and lying in the plane z = 3, then the normal vector to the surface S is n = k. Just as in the computation for Example 4, we still have § * F = -4i - 2xj + k. However, now we get § * F # n = 1, so that

6

§ * F # n ds =

6 2

1 dA = 4p.

The shadow is the disk of radius 2 in the xy-plane.

x +y …4

S

2

This result agrees with the circulation value found in Example 4.

EXAMPLE 6 Find a parametrization for the surface S formed by the part of the hyperbolic paraboloid z = y2 - x2 lying inside the cylinder of radius one around the z-axis and for the boundary curve C of S. (See Figure 15.60.) Then verify Stokes’ Theorem for S using the normal having positive k-component and the vector field F = yi - xj + k. As the unit circle is traversed counterclockwise in the xy-plane, the z-coordinate of the surface bounding the curve C is given by y2 - x2. A parametrization of C is given by

Solution

r(t) = (cos t)i + (sin t)j + (sin2 t - cos2 t)k,

0 … t … 2p

with dr = (-sin t)i + (cos t)j + (4 sin t cos t)k, 0 … t … 2p. dt z

Along the curve r(t) the formula for the vector field F is F = (sin t)i - (cos t)j + k.

1

The counterclockwise circulation along C is the value of the line integral

S

2p

21 1

1

x

L0

y

dr dt = dt L0 =

C

a-sin2 t - cos2 t + 4 sin t cos tb dt

2p

a4 sin t cos t - 1b dt =

2p

L0

a2 sin 2t - 1b dt

2p

= -2p. 0

We now compute the same quantity by integrating § * F # n over the surface S. We use polar coordinates and parametrize S by noting that above the point (r, u) in the plane, the z-coordinate of S is y2 - x2 = r 2 sin2 u - r 2 cos2 u. A parametrization of S is

1 S

1

L0

2p

= c-cos 2t - t d

z

x

F#

1

y

C

r(r, u) = (r cos u)i + (r sin u)j + r 2(sin2 u - cos2 u)k,

We next compute § * F # n ds. We have i

FIGURE 15.60 The surface and vector field for Example 6.

and

0 … r … 1,

j

k

0 § * F = 4 0x

0 0y

0 4 = -2k 0z

y

-x

1

0 … u … 2p.

894

Chapter 1: Functions

rr = (cos u)i + (sin u)j + 2r(sin2 u - cos2 u)k ru = (-r sin u)i + (r cos u)j + 4r 2(sin u cos u)k i rr * ru = † cos u -r sin u

j sin u r cos u

k 2r(sin2 u - cos2 u) † 4r 2(sin u cos u)

= 2r 2(2 sin2 u cos u - sin2 u cos u + cos3 u)i - 2r 2(2 sin u cos2 u + sin3 u + sin u cos2 u)j + rk. We now obtain LLS

§ * F # n ds =

2p

L0

1

§ * F # (rr * ru) dr du

L0 L0 2p

=

§ * F#

L0 2p

=

1

rr * ru ƒ rr * ru ƒ dr du ƒ rr + ru ƒ

1

-2r dr du = -2p.

L0 L0

So the integral of § * F # n over S equals the counterclockwise circulation of F along C, as stated by Stokes’ Theorem.

Paddle Wheel Interpretation of § * F Suppose that F is the velocity field of a fluid moving in a region R in space containing the closed curve C. Then F # dr

F

C

is the circulation of the fluid around C. By Stokes’ Theorem, the circulation is equal to the flux of § * F through any suitably oriented surface S with boundary C: F

C

F # dr =

6

§ * F # n ds.

S

Suppose we fix a point Q in the region R and a direction u at Q. Take C to be a circle of radius r, with center at Q, whose plane is normal to u. If § * F is continuous at Q, the average value of the u-component of § * F over the circular disk S bounded by C approaches the u-component of § * F at Q as the radius r : 0: s§ * F # udQ = lim

r:0

1 § * F # u ds. pr2 6 S

If we apply Stokes’Theorem and replace the surface integral by a line integral over C, we get Curl F

s§ * F # udQ = lim

r:0

1 F # dr. pr2 F

(6)

C

Q

The left-hand side of Equation (6) has its maximum value when u is the direction of § * F. When r is small, the limit on the right-hand side of Equation (6) is approximately 1 F # dr, pr2 F C

FIGURE 15.61 The paddle wheel interpretation of curl F.

which is the circulation around C divided by the area of the disk (circulation density). Suppose that a small paddle wheel of radius r is introduced into the fluid at Q, with its axle directed along u (Figure 15.61). The circulation of the fluid around C affects the rate

15.7

Stokes’ Theorem

895

of spin of the paddle wheel. The wheel spins fastest when the circulation integral is maximized; therefore it spins fastest when the axle of the paddle wheel points in the direction of § * F.

EXAMPLE 7 A fluid of constant density rotates around the z-axis with velocity F = vs -yi + xjd, where v is a positive constant called the angular velocity of the rotation (Figure 15.62). Find § * F and relate it to the circulation density.

z v

Solution

With F = -vyi + vxj, we find the curl § * F = a

P(x, y, z)

0N 0N 0P 0M 0P 0M bi + a bj + a bk 0y 0z 0z 0x 0x 0y

= s0 - 0di + s0 - 0dj + sv - s -vddk = 2vk. By Stokes’ Theorem, the circulation of F around a circle C of radius r bounding a disk S in a plane normal to § * F , say the xy-plane, is F 5 v(–yi 1 xj)

0

F

F # dr =

C

r y

P(x, y, 0)

6

§ * F # n ds =

S

6 S

Thus solving this last equation for 2v, we have s§ * Fd # k = 2v =

x

FIGURE 15.62 A steady rotational flow parallel to the xy-plane, with constant angular velocity v in the positive (counterclockwise) direction (Example 7).

2vk # k dx dy = s2vdspr2 d.

1 F # dr, pr2 F C

consistent with Equation (6) when u = k. Use Stokes’ Theorem to evaluate 1C F # dr, if F = xzi + xyj + 3xzk and C is the boundary of the portion of the plane 2x + y + z = 2 in the first octant, traversed counterclockwise as viewed from above (Figure 15.63).

EXAMPLE 8

The plane is the level surface ƒsx, y, zd = 2 of the function ƒsx, y, zd = 2x + y + z. The unit normal vector

Solution z

n = (0, 0, 2)

§ƒ s2i + j + kd 1 = = a2i + j + kb ƒ §ƒ ƒ ƒ 2i + j + k ƒ 26

is consistent with the counterclockwise motion around C. To apply Stokes’Theorem, we find n C 2x 1 y 1 z 5 2 (1, 0, 0)

j

k

0 curl F = § * F = 4 0x

0 0y

0 4 = sx - 3zdj + yk. 0z

xz

xy

3xz

R (0, 2, 0)

x

i

y 5 2 2 2x

FIGURE 15.63 The planar surface in Example 8.

On the plane, z equals 2 - 2x - y, so y

§ * F = sx - 3s2 - 2x - yddj + yk = s7x + 3y - 6dj + yk and § * F#n =

1 1 a7x + 3y - 6 + yb = a7x + 4y - 6b. 26 26 The surface area element is ds =

ƒ §ƒ ƒ 26 dA = dx dy. 1 ƒ §ƒ # k ƒ

896

Chapter 15: Integration in Vector Fields

The circulation is F

F # dr =

6

C

§ * F # n ds

S

2 - 2x

1

=

L0 L0 2 - 2x

1

= z

Stokes’ Theorem, Eq. (4)

L0 L0

1 a7x + 4y - 6b 26 dy dx 26 s7x + 4y - 6d dy dx = -1.

Let the surface S be the ellipitical paraboloid z = x 2 + 4y 2 lying beneath the plane z = 1 (Figure 15.64). We define the orientation of S by taking the inner normal vector n to the surface, which is the normal having a positive k-component. Find the flux of § * F across S in the direction n for the vector field F = yi - xzj + xz 2k.

EXAMPLE 9

C: x2 1 4y2 5 1

n

We use Stokes’ Theorem to calculate the curl integral by finding the equivalent counterclockwise circulation of F around the curve of intersection C of the paraboloid z = x 2 + 4y 2 and the plane z = 1, as shown in Figure 15.64. Note that the orientation of S is consistent with traversing C in a counterclockwise direction around the z-axis. The curve C is the ellipse x 2 + 4y 2 = 1 in the plane z = 1. We can parametrize the ellipse by x = cos t, y = 12 sin t, z = 1 for 0 … t … 2p, so C is given by Solution

z 5 x2 1 4y2

x y

FIGURE 15.64 The portion of the ellipitical paraboloid in Example 9, showing its curve of intersection C with the plane z = 1 and its inner normal orientation by n.

r(t) = (cos t)i +

1 (sin t)j + k, 2

0 … t … 2p.

To compute the circulation integral D F # dr, we evaluate F along C and find the velocity C vector dr>dt: F(r(t)) =

1 (sin t)i - (cos t)j + (cos t)k 2

and dr 1 = -(sin t)i + (cos t)j. 2 dt

E

Then,

A

F

D

F # dr =

C

= B

C

F(r(t)) #

2p

1 1 a- sin2 t - cos2 tb dt 2 2

L0 L0

= -

(a)

2p

1 2 L0

dr dt dt

2p

dt = -p.

Therefore the flux of the curl across S in the direction n for the field F is 6

§ * F # n ds = -p.

S

Proof of Stokes’ Theorem for Polyhedral Surfaces

(b)

FIGURE 15.65 (a) Part of a polyhedral surface. (b) Other polyhedral surfaces.

Let S be a polyhedral surface consisting of a finite number of plane regions or faces. (See Figure 15.65 for examples.) We apply Green’s Theorem to each separate face of S. There are two types of faces: 1. 2.

Those that are surrounded on all sides by other faces. Those that have one or more edges that are not adjacent to other faces.

15.7

Stokes’ Theorem

897

The boundary ¢ of S consists of those edges of the type 2 faces that are not adjacent to other faces. In Figure 15.65a, the triangles EAB, BCE, and CDE represent a part of S, with ABCD part of the boundary ¢. We apply a generalized tangential form of Green’s Theorem to the three triangles of Figure 15.65a in turn and add the results to get £

F

EAB

+

F

BCE

+

≥F # dr = £

F

CDE

+

6

EAB

6

BCE

+

6

≥ § * F # n ds.

(7)

CDE

In the generalized form, the line integral of F around the curve enclosing the plane region R normal to n equals the double integral of (curl F) . n over R. The three line integrals on the left-hand side of Equation (7) combine into a single line integral taken around the periphery ABCDE because the integrals along interior segments cancel in pairs. For example, the integral along segment BE in triangle ABE is opposite in sign to the integral along the same segment in triangle EBC. The same holds for segment CE. Hence, Equation (7) reduces to F

F # dr =

ABCDE

6

§ * F # n ds.

ABCDE

When we apply the generalized form of Green’s Theorem to all the faces and add the results, we get F

F # dr =

¢

6

§ * F # n ds.

S

This is Stokes’ Theorem for the polyhedral surface S in Figure 15.65a. More general polyhedral surfaces are shown in Figure 15.65b and the proof can be extended to them. General smooth surfaces can be obtained as limits of polyhedral surfaces.

Stokes’ Theorem for Surfaces with Holes

n

Stokes’ Theorem holds for an oriented surface S that has one or more holes (Figure 15.66). The surface integral over S of the normal component of § * F equals the sum of the line integrals around all the boundary curves of the tangential component of F, where the curves are to be traced in the direction induced by the orientation of S. For such surfaces the theorem is unchanged, but C is considered as a union of simple closed curves.

An Important Identity The following identity arises frequently in mathematics and the physical sciences. S

curl grad ƒ = 0

FIGURE 15.66 Stokes’ Theorem also holds for oriented surfaces with holes.

or

§ * §f = 0

(8)

Forces arising in the study of electromagnetism and gravity are often associated with a potential function ƒ. The identity (8) says that these forces have curl equal to zero. The identity (8) holds for any function ƒ(x, y, z) whose second partial derivatives are continuous. The proof goes like this: i

j

k

0 § * §ƒ = 5 0x

0 0y

0 = sƒzy - ƒyz di - sƒzx - ƒxz dj + sƒyx - ƒxy dk. 0z 5

0ƒ 0x

0ƒ 0y

0ƒ 0z

If the second partial derivatives are continuous, the mixed second derivatives in parentheses are equal (Theorem 2, Section 13.3) and the vector is zero.

898

Chapter 15: Integration in Vector Fields

C S

Conservative Fields and Stokes’ Theorem In Section 15.3, we found that a field F being conservative in an open region D in space is equivalent to the integral of F around every closed loop in D being zero. This, in turn, is equivalent in simply connected open regions to saying that § * F = 0 (which gives a test for determining if F is conservative for such regions).

(a)

THEOREM 7—Curl F = 0 Related to the Closed-Loop Property If § * F = 0 at every point of a simply connected open region D in space, then on any piecewisesmooth closed path C in D, F

F # dr = 0.

C

Sketch of a Proof Theorem 7 can be proved in two steps. The first step is for simple closed curves (loops that do not cross themselves), like the one in Figure 15.67a. A theorem from topology, a branch of advanced mathematics, states that every smooth simple closed curve C in a simply connected open region D is the boundary of a smooth two-sided surface S that also lies in D. Hence, by Stokes’ Theorem, (b)

FIGURE 15.67 (a) In a simply connected open region in space, a simple closed curve C is the boundary of a smooth surface S. (b) Smooth curves that cross themselves can be divided into loops to which Stokes’ Theorem applies.

F

F # dr =

C

6

§ * F # n ds = 0.

S

The second step is for curves that cross themselves, like the one in Figure 15.67b. The idea is to break these into simple loops spanned by orientable surfaces, apply Stokes’ Theorem one loop at a time, and add the results. The following diagram summarizes the results for conservative fields defined on connected, simply connected open regions. Theorem 2, Section 15.3

F conservative on D

F ∇f on D Vector identity (Eq. 8) (continuous second partial derivatives)

Theorem 3, Section 15.3

F • dr 0 EC over any closed path in D

∇ F 0 throughout D Theorem 7 Domain's simple connectivity and Stokes' Theorem

Exercises 15.7 Using Stokes’ Theorem to Find Line Integrals In Exercises 1–6, use the surface integral in Stokes’ Theorem to calculate the circulation of the field F around the curve C in the indicated direction.

2. F = 2yi + 3xj - z 2k C: The circle x 2 + y 2 = 9 in the xy-plane, counterclockwise when viewed from above

1. F = x 2i + 2xj + z 2k C: The ellipse 4x 2 + y 2 = 4 in the xy-plane, counterclockwise when viewed from above

3. F = yi + xzj + x 2k C: The boundary of the triangle cut from the plane x + y + z = 1 by the first octant, counterclockwise when viewed from above

15.7 4. F = sy 2 + z 2 di + sx 2 + z 2 dj + sx 2 + y 2 dk C: The boundary of the triangle cut from the plane x + y + z = 1 by the first octant, counterclockwise when viewed from above 5. F = s y 2 + z 2 di + sx 2 + y 2 dj + sx 2 + y 2 dk C: The square bounded by the lines x = ;1 and y = ;1 in the xy-plane, counterclockwise when viewed from above 6. F = x 2y 3i + j + zk C: The intersection of the cylinder x 2 + y 2 = 4 and the hemisphere x 2 + y 2 + z 2 = 16, z Ú 0, counterclockwise when viewed from above Flux of the Curl 7. Let n be the outer unit normal of the elliptical shell S:

4x 2 + 9y 2 + 36z 2 = 36,

z Ú 0,

and let F = yi + x j + sx + y d 2

2

4 3>2

sin e

2xyz

k.

Find the value of 6

8. Let n be the outer unit normal (normal away from the origin) of the parabolic shell 4x 2 + y + z 2 = 4,

y Ú 0,

and let F = a-z +

1 1 b i + stan-1 ydj + ax + bk. 2 + x 4 + z

Find the value of 6

§ * F # n ds.

S

9. Let S be the cylinder x 2 + y 2 = a 2, 0 … z … h, together with its top, x 2 + y 2 … a 2, z = h. Let F = -yi + xj + x 2k. Use Stokes’ Theorem to find the flux of § * F outward through S. 10. Evaluate 6

rsf, ud =

§ * s yid # n ds,

s4 - r 2 dk,

s9 - r 2 dk,

rk,

s5 - rdk,

A 23 sin f cos u B i + A 23 sin f sin u B j + 0 … f … p>2,

0 … u … 2p

18. F = y i + z j + xk S: rsf, ud = s2 sin f cos udi + s2 sin f sin udj + s2 cos fdk, 0 … f … p>2, 0 … u … 2p 2

(Hint: One parametrization of the ellipse at the base of the shell is x = 3 cos t, y = 2 sin t, 0 … t … 2p.)

S:

13. F = 2zi + 3xj + 5yk S: rsr, ud = sr cos udi + sr sin udj + 0 … r … 2, 0 … u … 2p 14. F = sy - zdi + sz - xdj + sx + zdk S: rsr, ud = sr cos udi + sr sin udj + 0 … r … 3, 0 … u … 2p 15. F = x 2yi + 2y 3zj + 3zk S: rsr, ud = sr cos udi + sr sin udj + 0 … r … 1, 0 … u … 2p 16. F = sx - ydi + s y - zdj + sz - xdk S: rsr, ud = sr cos udi + sr sin udj + 0 … r … 5, 0 … u … 2p 17. F = 3yi + s5 - 2xdj + sz 2 - 2dk

A 23 cos f B k,

S

899

Stokes’ Theorem for Parametrized Surfaces In Exercises 13–18, use the surface integral in Stokes’ Theorem to calculate the flux of the curl of the field F across the surface S in the direction of the outward unit normal n.

S:

§ * F # n ds.

Stokes’ Theorem

2

Theory and Examples 19. Zero circulation Use the identity § * §ƒ = 0 (Equation (8) in the text) and Stokes’ Theorem to show that the circulations of the following fields around the boundary of any smooth orientable surface in space are zero. a. F = 2xi + 2yj + 2zk b. F = §sxy 2z 3 d c. F = § * sxi + yj + zkd d. F = §ƒ 20. Zero circulation Let ƒsx, y, zd = sx 2 + y 2 + z 2 d-1>2. Show that the clockwise circulation of the field F = §ƒ around the circle x 2 + y 2 = a 2 in the xy-plane is zero a. by taking r = sa cos tdi + sa sin tdj, 0 … t … 2p, and integrating F # dr over the circle. b. by applying Stokes’ Theorem. 21. Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2 , oriented as shown here. Show that

S

where S is the hemisphere x 2 + y 2 + z 2 = 1, z Ú 0. 11. Flux of curl F Show that 6

F

2y dx + 3z dy - x dz

C

§ * F # n ds

z

S

has the same value for all oriented surfaces S that span C and that induce the same positive direction on C. 12. Let F be a differentiable vector field defined on a region containing a smooth closed oriented surface S and its interior. Let n be the unit normal vector field on S. Suppose that S is the union of two surfaces S1 and S2 joined along a smooth simple closed curve C. Can anything be said about 6

§ * F # n ds?

S

Give reasons for your answer.

2x 1 2y 1 z 5 2

2 C

O

a1

y

1 x

depends only on the area of the region enclosed by C and not on the position or shape of C. 22. Show that if F = xi + yj + zk, then § * F = 0.

900

Chapter 15: Integration in Vector Fields

23. Find a vector field with twice-differentiable components whose curl is xi + yj + zk or prove that no such field exists. 24. Does Stokes’ Theorem say anything special about circulation in a field whose curl is zero? Give reasons for your answer. 25. Let R be a region in the xy-plane that is bounded by a piecewise smooth simple closed curve C and suppose that the moments of inertia of R about the x- and y-axes are known to be Ix and Iy. Evaluate the integral F

§sr

4

F =

-y x2 + y2

i +

x j + zk x2 + y2

is zero but that F

F # dr

C

d # n ds,

C

where r = 2x 2 + y 2, in terms of Ix and Iy.

15.8

26. Zero curl, yet field not conservative Show that the curl of

is not zero if C is the circle x 2 + y 2 = 1 in the xy-plane. (Theorem 7 does not apply here because the domain of F is not simply connected. The field F is not defined along the z-axis so there is no way to contract C to a point without leaving the domain of F.)

The Divergence Theorem and a Unified Theory The divergence form of Green’s Theorem in the plane states that the net outward flux of a vector field across a simple closed curve can be calculated by integrating the divergence of the field over the region enclosed by the curve. The corresponding theorem in three dimensions, called the Divergence Theorem, states that the net outward flux of a vector field across a closed surface in space can be calculated by integrating the divergence of the field over the region enclosed by the surface. In this section we prove the Divergence Theorem and show how it simplifies the calculation of flux. We also derive Gauss’s law for flux in an electric field and the continuity equation of hydrodynamics. Finally, we unify the chapter’s vector integral theorems into a single fundamental theorem.

Divergence in Three Dimensions The divergence of a vector field F = Msx, y, zdi + Nsx, y, zdj + Psx, y, zdk is the scalar function div F = § # F =

0N 0M 0P + + . 0x 0y 0z

(1)

The symbol “div F” is read as “divergence of F” or “div F.” The notation § # F is read “del dot F.” Div F has the same physical interpretation in three dimensions that it does in two. If F is the velocity field of a flowing gas, the value of div F at a point (x, y, z) is the rate at which the gas is compressing or expanding at (x, y, z). The divergence is the flux per unit volume or flux density at the point.

EXAMPLE 1 The following vector fields represent the velocity of a gas flowing in space. Find the divergence of each vector field and interpret its physical meaning. Figure 15.68 displays the vector fields. (a) Expansion: F(x, y, z) = xi + yj + zk (b) Compression: F(x, y, z) = -xi - yj - zk

15.8

The Divergence Theorem and a Unified Theory

901

(c) Rotation about z-axis: F(x, y, z) = -yi + xj (d) Shearing along horizontal planes: F(x, y, z) = zj

z

z

y

y x

x

(b)

(a) z

z

y y x x

(c)

FIGURE 15.68

(d)

Velocity fields of a gas flowing in space (Example 1).

Solution

(a) div F =

0 0 0 (x) + (y) + (z) = 3: The gas is undergoing uniform expansion at all 0x 0y 0z

points. 0 0 0 (-x) + (-y) + (-z) = -3: The gas is undergoing uniform com0x 0y 0z pression at all points.

(b) div F =

(c) div F =

0 0 (-y) + (x) = 0: The gas is neither expanding nor compressing at any 0x 0y

point. 0 (z) = 0: Again, the divergence is zero at all points in the domain of the ve0y locity field, so the gas is neither expanding nor compressing at any point.

(d) div F =

Divergence Theorem The Divergence Theorem says that under suitable conditions, the outward flux of a vector field across a closed surface equals the triple integral of the divergence of the field over the region enclosed by the surface.

902

Chapter 15: Integration in Vector Fields

THEOREM 8—Divergence Theorem Let F be a vector field whose components have continuous first partial derivatives, and let S be a piecewise smooth oriented closed surface. The flux of F across S in the direction of the surface’s outward unit normal field n equals the integral of ¥ # F over the region D enclosed by the surface: 6

F # n ds =

S

9

§ # F dV.

(2)

D

Outward flux

Divergence integral

EXAMPLE 2

z

Evaluate both sides of Equation (2) for the expanding vector field F = xi + yj + zk over the sphere x 2 + y 2 + z 2 = a 2 (Figure 15.69). The outer unit normal to S, calculated from the gradient of ƒsx, y, zd = x 2 + y + z - a 2, is

Solution 2

2

n = y

2sxi + yj + zkd 24sx + y + z d 2

2

2

=

xi + yj + zk . a

x 2 + y 2 + z 2 = a 2 on S

Hence, x2 + y2 + z2 a2 ds = a a ds = a ds.

F # n ds =

x

Therefore, FIGURE 15.69 A uniformly expanding vector field and a sphere (Example 2).

6

F # n ds =

S

6

a ds = a

6

S

ds = as4pa 2 d = 4pa 3.

S

Area of S is 4pa 2.

The divergence of F is §#F =

0 0 0 sxd + s yd + szd = 3, 0x 0y 0z

so 9

§ # F dV =

4 3 dV = 3 a pa 3 b = 4pa 3. 3 9

D

D

Find the flux of F = xyi + yzj + xzk outward through the surface of the cube cut from the first octant by the planes x = 1, y = 1, and z = 1.

EXAMPLE 3

Instead of calculating the flux as a sum of six separate integrals, one for each face of the cube, we can calculate the flux by integrating the divergence Solution

§#F =

0 0 0 sxyd + s yzd + sxzd = y + z + x 0x 0y 0z

over the cube’s interior: Flux =

6

F # n ds =

Cube surface

1

=

9

§ # F dV

The Divergence Theorem

Cube interior 1

L0 L0 L0

1

sx + y + zd dx dy dz =

3 . 2

Routine integration

15.8 z

The Divergence Theorem and a Unified Theory

903

EXAMPLE 4

(a) Calculate the flux of the vector field F = x2i + 4xyzj + zex k out of the box-shaped region D: 0 … x … 3, 0 … y … 2, 0 … z … 1. (See Figure 15.70.)

1

2

y

3 x

FIGURE 15.70 The integral of div F over this region equals the total flux across the six sides (Example 4).

(b) Integrate div F over this region and slow that the result is the same value as in part (a), as implied by the Divergence Theorem. Solution

(a) The region D has six sides. We calculate the flux across each side in turn. Consider the top side in the plane z = 1, having outward normal k. The flux across this side is given by F # n = zex. Since z = 1 on this side, the flux at a point (x, y, z) on the top is ex. The total outward flux across this side is given by the integral 2

3

ex dx dy = 2e3 - 2. Routine integration L0 L0 The outward flux across the other sides is computed similarly, and the results are summarized in the following table. Side x x y y z z

= = = = = =

F#n

Unit normal n -i i -j j -k k

0 3 0 2 0 1

-x2 x2 -4xyz 4xyz -zex zex

= = = = = =

Flux across side 0 9 0 8xz 0 ex

0 18 0 18 0 2e3 - 2

The total outward flux is obtained by adding the terms for each of the six sides, giving 18 + 18 + 2e3 - 2 = 34 + 2e3. (b) We first compute the divergence of F, obtaining div F = § # F = 2x + 4xz + ex. The integral of the divergence of F over D is 1

2

3

div F dV = (2x + 4xz + ex) dx dy dz L0 L0 L0 9 D

1

=

L0 L0

2

(8 + 18z + e3) dy dz

1

=

(16 + 36z + 2e3) dz L0 = 34 + 2e3.

As implied by the Divergence Theorem, the integral of the divergence over D equals the outward flux across the boundary surface of D.

Proof of the Divergence Theorem for Special Regions To prove the Divergence Theorem, we take the components of F to have continuous first partial derivatives. We first assume that D is a convex region with no holes or bubbles, such as a solid ball, cube, or ellipsoid, and that S is a piecewise smooth surface. In addition, we assume that any line perpendicular to the xy-plane at an interior point of the region Rxy that is the projection of D on the xy-plane intersects the surface S in exactly two points, producing surfaces S1: z = ƒ1sx, yd, sx, yd in Rxy S2: z = ƒ2sx, yd, sx, yd in Rxy,

904

Chapter 15: Integration in Vector Fields

z

R yz

D

Rx z

S2 S1 y x

Rx y

FIGURE 15.71 We prove the Divergence Theorem for the kind of three-dimensional region shown here.

with ƒ1 … ƒ2. We make similar assumptions about the projection of D onto the other coordinate planes. See Figure 15.71. The components of the unit normal vector n = n1i + n2j + n3k are the cosines of the angles a, b, and g that n makes with i, j, and k (Figure 15.72). This is true because all the vectors involved are unit vectors. We have n1 = n # i = ƒ n ƒ ƒ i ƒ cos a = cos a n2 = n # j = ƒ n ƒ ƒ j ƒ cos b = cos b n3 = n # k = ƒ n ƒ ƒ k ƒ cos g = cos g. Thus, n = scos adi + scos bdj + scos gdk and F # n = M cos a + N cos b + P cos g. In component form, the Divergence Theorem states that 0N 0P 0M + + b dx dy dz. sM cos a + N cos b + P cos gd ds = a 0y 0z 6 9 0x S D ('''''')''''''* (''')'''* # F n div F

z

We prove the theorem by proving the three following equalities: n3 k

6

(n1, n 2, n 3)

i

n1

(3)

0N dx dy dz 9 0y

(4)

0P dx dy dz 9 0z

(5)

S

n

6

D

N cos b ds =

S

j

n2

0M dx dy dz 9 0x

M cos a ds =

y

6

x

D

P cos g ds =

S

FIGURE 15.72 The components of n are the cosines of the angles a, b, and g that it makes with i, j, and k.

D

Proof of Equation (5) We prove Equation (5) by converting the surface integral on the left to a double integral over the projection Rxy of D on the xy-plane (Figure 15.73). The surface S consists of an upper part S2 whose equation is z = ƒ2sx, yd and a lower part S1 whose equation is z = ƒ1sx, yd. On S2, the outer normal n has a positive k-component and cos g ds = dx dy

ds =

because

dx dy dA = cos g . ƒ cos g ƒ

z n D

z f2(x, y)

cos g ds = -dx dy.

S2

d

Therefore,

S1 z f1(x, y)

O

d

n y

x

See Figure 15.74. On S1, the outer normal n has a negative k-component and

6

P cos g ds =

S

P cos g ds + P cos g ds 6 6 S2

=

6

S1

Psx, y, ƒ2sx, ydd dx dy -

Rxy

Rxy dA dx dy

FIGURE 15.73 The region D enclosed by the surfaces S1 and S2 projects vertically onto Rxy in the xy-plane.

=

6

6

Psx, y, ƒ1sx, ydd dx dy

Rxy

[Psx, y, ƒ2sx, ydd - Psx, y, ƒ1sx, ydd] dx dy

Rxy

=

c

ƒ2sx,yd

6 Lƒ1sx,yd Rxy

0P 0P dz d dx dy = dz dx dy. 0z 9 0z D

15.8

k g

The Divergence Theorem and a Unified Theory

905

This proves Equation (5). The proofs for Equations (3) and (4) follow the same pattern; or just permute x, y, z; M, N, P; a, b, g, in order, and get those results from Equation (5). This proves the Divergence Theorem for these special regions.

n

Divergence Theorem for Other Regions Here g is acute, so ds 5 dx dy/cos g.

The Divergence Theorem can be extended to regions that can be partitioned into a finite number of simple regions of the type just discussed and to regions that can be defined as limits of simpler regions in certain ways. For an example of one step in such a splitting process, suppose that D is the region between two concentric spheres and that F has continuously differentiable components throughout D and on the bounding surfaces. Split D by an equatorial plane and apply the Divergence Theorem to each half separately. The bottom half, D1, is shown in Figure 15.75. The surface S1 that bounds D1 consists of an outer hemisphere, a plane washer-shaped base, and an inner hemisphere. The Divergence Theorem says that

k

g

Here g is obtuse, so ds 5 –dx dy/cos g. n

dy

F # n1 ds1 =

6

dx

S1

FIGURE 15.74 An enlarged view of the area patches in Figure 15.73. The relations ds = ;dx dy>cos g come from Eq. (7) in Section 15.5.

9

§ # F dV1 .

(6)

D1

The unit normal n1 that points outward from D1 points away from the origin along the outer surface, equals k along the flat base, and points toward the origin along the inner surface. Next apply the Divergence Theorem to D2, and its surface S2 (Figure 15.76): 6

F # n2 ds2 =

S2

9

§ # F dV2 .

(7)

D2

As we follow n2 over S2 , pointing outward from D2 , we see that n2 equals -k along the washer-shaped base in the xy-plane, points away from the origin on the outer sphere, and points toward the origin on the inner sphere. When we add Equations (6) and (7), the integrals over the flat base cancel because of the opposite signs of n1 and n2. We thus arrive at the result

z

k

6

F # n ds =

S

O y D1

n1

9

§ # F dV,

D

with D the region between the spheres, S the boundary of D consisting of two spheres, and n the unit normal to S directed outward from D.

x

EXAMPLE 5 FIGURE 15.75 The lower half of the solid region between two concentric spheres.

Find the net outward flux of the field xi + yj + zk

F =

r3

r = 2x 2 + y 2 + z 2

,

across the boundary of the region D: 0 6 b2 … x2 + y2 + z2 … a2 (Figure 15.77). Solution

The flux can be calculated by integrating § # F over D. We have 0r x 1 = sx 2 + y 2 + z 2 d-1>2s2xd = r 0x 2

z

and D2 n2

Similarly,

y

3y 2 0N 1 = 3 - 5 0y r r

and

3z 2 0P 1 = 3 - 5. 0z r r

Hence,

–k

div F =

x

FIGURE 15.76 The upper half of the solid region between two concentric spheres.

0r 3x 2 0 0M 1 = sxr-3 d = r-3 - 3xr-4 = 3 - 5. 0x 0x 0x r r

3r2 3 3 2 3 2 2 sx + y + z d = = 0 r5 r5 r3 r3

and 9 D

§ # F dV = 0.

§ # F div F

906

Chapter 15: Integration in Vector Fields

So the integral of § # F over D is zero and the net outward flux across the boundary of D is zero. There is more to learn from this example, though. The flux leaving D across the inner sphere Sb is the negative of the flux leaving D across the outer sphere Sa (because the sum of these fluxes is zero). Hence, the flux of F across Sb in the direction away from the origin equals the flux of F across Sa in the direction away from the origin. Thus, the flux of F across a sphere centered at the origin is independent of the radius of the sphere. What is this flux? To find it, we evaluate the flux integral directly for an arbitrary sphere Sa. The outward unit normal on the sphere of radius a is

z

Sa y Sb x

n = FIGURE 15.77 Two concentric spheres in an expanding vector field. The outer sphere is Sa and surrounds the inner sphere Sb.

xi + yj + zk 2x + y + z 2

2

2

=

xi + yj + zk . a

Hence, on the sphere, F#n =

x2 + y2 + z2 xi + yj + zk xi + yj + zk a2 1 # = = 4 = 2 a 3 4 a a a a

and 6

F # n ds =

Sa

1 1 ds = 2 s4pa 2 d = 4p. a 26 a Sa

The outward flux of F across any sphere centered at the origin is 4p.

Gauss’s Law: One of the Four Great Laws of Electromagnetic Theory

z

There is still more to be learned from Example 4. In electromagnetic theory, the electric field created by a point charge q located at the origin is

S

Esx, y, zd =

y x

Sphere Sa

FIGURE 15.78 A sphere Sa surrounding another surface S. The tops of the surfaces are removed for visualization.

q q q xi + yj + zk 1 r r b = a = , 2 3 4pP0 ƒ r ƒ 4pP0 ƒ r ƒ 4pP0 ƒrƒ r3

where P0 is a physical constant, r is the position vector of the point (x, y, z), and r = ƒ r ƒ = 2x 2 + y 2 + z 2. In the notation of Example 4, q E = F. 4pP0 The calculations in Example 4 show that the outward flux of E across any sphere centered at the origin is q>P0, but this result is not confined to spheres. The outward flux of E across any closed surface S that encloses the origin (and to which the Divergence Theorem applies) is also q>P0. To see why, we have only to imagine a large sphere Sa centered at the origin and enclosing the surface S (see Figure 15.78). Since §#E = §#

q q F = §#F = 0 4pP0 4pP0

when r 7 0, the integral of § # E over the region D between S and Sa is zero. Hence, by the Divergence Theorem, 6

E # n ds = 0,

Boundary of D

and the flux of E across S in the direction away from the origin must be the same as the flux of E across Sa in the direction away from the origin, which is q>P0. This statement, called Gauss’s Law, also applies to charge distributions that are more general than the one assumed here, as you will see in nearly any physics text. Gauss’s Law:

6 S

q E # n ds = P0

15.8

The Divergence Theorem and a Unified Theory

907

Continuity Equation of Hydrodynamics Let D be a region in space bounded by a closed oriented surface S. If v(x, y, z) is the velocity field of a fluid flowing smoothly through D, d = dst, x, y, zd is the fluid’s density at (x, y, z) at time t, and F = dv, then the continuity equation of hydrodynamics states that §#F +

n

v Δt

h (v Δ t) . n

0d = 0. 0t If the functions involved have continuous first partial derivatives, the equation evolves naturally from the Divergence Theorem, as we now see. First, the integral

Δ

6

F # n ds

S

S

FIGURE 15.79 The fluid that flows upward through the patch ¢s in a short time ¢t fills a “cylinder” whose volume is approximately base * height = v # n ¢s ¢t.

is the rate at which mass leaves D across S (leaves because n is the outer normal). To see why, consider a patch of area ¢s on the surface (Figure 15.79). In a short time interval ¢t, the volume ¢V of fluid that flows across the patch is approximately equal to the volume of a cylinder with base area ¢s and height sv¢td # n, where v is a velocity vector rooted at a point of the patch: ¢V L v # n ¢s ¢t. The mass of this volume of fluid is about

¢m L dv # n ¢s ¢t,

so the rate at which mass is flowing out of D across the patch is about ¢m L dv # n ¢s. ¢t This leads to the approximation a ¢m L a dv # n ¢s ¢t as an estimate of the average rate at which mass flows across S. Finally, letting ¢s : 0 and ¢t : 0 gives the instantaneous rate at which mass leaves D across S as dm = dv # n ds, dt 6 S

which for our particular flow is dm = F # n ds. dt 6 S

Now let B be a solid sphere centered at a point Q in the flow. The average value of § # F over B is 1 § # F dV. volume of B 9 B

It is a consequence of the continuity of the divergence that § # F actually takes on this value at some point P in B. Thus, s§ # FdP =

6

F # n ds

S 1 § # F dV = volume of B 9 volume of B B

rate at which mass leaves B across its surface S = . volume of B The last term of the equation describes decrease in mass per unit volume.

(8)

908

Chapter 15: Integration in Vector Fields

Now let the radius of B approach zero while the center Q stays fixed. The left side of Equation (8) converges to s§ # FdQ, the right side to s -0d>0tdQ. The equality of these two limits is the continuity equation 0d §#F = - . 0t # The continuity equation “explains” § F: The divergence of F at a point is the rate at which the density of the fluid is decreasing there. The Divergence Theorem 6

F # n ds =

S

9

§ # F dV

D

now says that the net decrease in density of the fluid in region D is accounted for by the mass transported across the surface S. So, the theorem is a statement about conservation of mass (Exercise 31).

Unifying the Integral Theorems If we think of a two-dimensional field F = Msx, ydi + Nsx, ydj as a three-dimensional field whose k-component is zero, then § # F = s0M>0xd + s0N>0yd and the normal form of Green’s Theorem can be written as F

F # n ds =

C

0N 0M § # F dA. + b dx dy = 0x 0y 6 6 a

R

R

Similarly, § * F # k = s0N>0xd - s0M>0yd, so the tangential form of Green’s Theorem can be written as F

F # dr =

C

6 R

a

0N 0M § * F # k dA. b dx dy = 0x 0y 6 R

With the equations of Green’s Theorem now in del notation, we can see their relationships to the equations in Stokes’ Theorem and the Divergence Theorem. Green’s Theorem and Its Generalization to Three Dimensions Normal form of Green’s Theorem:

F

F # n ds =

C

Divergence Theorem:

6

6 R

F # n ds =

S

Tangential form of Green’s Theorem:

F F

C

9

§ # F dV

D

F # dr =

C

Stokes’ Theorem:

§ # F dA

6

§ * F # k dA

R

F # dr =

6

§ * F # n ds

S

Notice how Stokes’ Theorem generalizes the tangential (curl) form of Green’s Theorem from a flat surface in the plane to a surface in three-dimensional space. In each case, the integral of the normal component of curl F over the interior of the surface equals the circulation of F around the boundary. Likewise, the Divergence Theorem generalizes the normal (flux) form of Green’s Theorem from a two-dimensional region in the plane to a three-dimensional region in space. In each case, the integral of § # F over the interior of the region equals the total flux of the field across the boundary. There is still more to be learned here. All these results can be thought of as forms of a single fundamental theorem. Think back to the Fundamental Theorem of Calculus in Section 5.4. It says that if ƒ(x) is differentiable on (a, b) and continuous on [a, b], then b dƒ dx = ƒsbd - ƒsad. La dx

15.8 n –i

ni a

b

x

The Divergence Theorem and a Unified Theory

909

If we let F = ƒsxdi throughout [a, b], then sdƒ>dxd = § # F. If we define the unit vector field n normal to the boundary of [a, b] to be i at b and -i at a (Figure 15.80), then ƒsbd - ƒsad = ƒsbdi # sid + ƒsadi # s -id = Fsbd # n + Fsad # n = total outward flux of F across the boundary of [a, b].

FIGURE 15.80 The outward unit normals at the boundary of [a, b] in onedimensional space.

The Fundamental Theorem now says that Fsbd # n + Fsad # n =

3

§ # F dx.

[a,b]

The Fundamental Theorem of Calculus, the normal form of Green’s Theorem, and the Divergence Theorem all say that the integral of the differential operator § # operating on a field F over a region equals the sum of the normal field components over the boundary of the region. (Here we are interpreting the line integral in Green’s Theorem and the surface integral in the Divergence Theorem as “sums” over the boundary.) Stokes’ Theorem and the tangential form of Green’s Theorem say that, when things are properly oriented, the integral of the normal component of the curl operating on a field equals the sum of the tangential field components on the boundary of the surface. The beauty of these interpretations is the observance of a single unifying principle, which we might state as follows. A Unifying Fundamental Theorem of Vector Integral Calculus The integral of a differential operator acting on a field over a region equals the sum of the field components appropriate to the operator over the boundary of the region.

Exercises 15.8 Calculating Divergence In Exercises 1–4, find the divergence of the field. 1. The spin field in Figure 15.12 2. The radial field in Figure 15.11 3. The gravitational field in Figure 15.8 and Exercise 38a in Section 15.3 4. The velocity field in Figure 15.13 Calculating Flux Using the Divergence Theorem In Exercises 5–16, use the Divergence Theorem to find the outward flux of F across the boundary of the region D. 5. Cube

F = s y - xdi + sz - ydj + s y - xdk

D: The cube bounded by the planes x = ;1, y = ;1, and z = ;1 6. F = x 2i + y 2j + z 2k a. Cube D: The cube cut from the first octant by the planes x = 1, y = 1 , and z = 1 b. Cube D: The cube bounded by the planes x = ;1, y = ;1, and z = ;1 c. Cylindrical can D: The region cut from the solid cylinder x 2 + y 2 … 4 by the planes z = 0 and z = 1 7. Cylinder and paraboloid F = yi + xyj - zk D: The region inside the solid cylinder x 2 + y 2 … 4 between the plane z = 0 and the paraboloid z = x 2 + y 2 8. Sphere F = x 2i + xzj + 3zk D: The solid sphere x 2 + y 2 + z 2 … 4

9. Portion of sphere F = x 2i - 2xyj + 3xzk D: The region cut from the first octant by the sphere x 2 + y 2 + z2 = 4 10. Cylindrical can F = s6x 2 + 2xydi + s2y + x 2zdj + 4x 2y 3k D: The region cut from the first octant by the cylinder x 2 + y 2 = 4 and the plane z = 3 11. Wedge F = 2xzi - xyj - z 2k D: The wedge cut from the first octant by the plane y + z = 4 and the elliptical cylinder 4x 2 + y 2 = 16 12. Sphere F = x 3i + y 3j + z 3k D: The solid sphere x 2 + y 2 + z 2 … a 2 13. Thick sphere F = 2x 2 + y 2 + z 2 sxi + yj + zkd D: The region 1 … x 2 + y 2 + z 2 … 2 14. Thick sphere F = sxi + yj + zkd> 2x 2 + y 2 + z 2 D: The region 1 … x 2 + y 2 + z 2 … 4 15. Thick sphere F = s5x 3 + 12xy 2 di + s y 3 + e y sin zdj + s5z 3 + e y cos zdk D: The solid region between the spheres x 2 + y 2 + z 2 = 1 and x2 + y2 + z2 = 2 y 2z 16. Thick cylinder F = ln sx 2 + y 2 di - a x tan-1 x bj + z2x 2 + y 2 k D: The thick-walled cylinder 1 … x 2 + y 2 … 2,

-1 … z … 2

910

Chapter 15: Integration in Vector Fields

Properties of Curl and Divergence 17. div (curl G) is zero a. Show that if the necessary partial derivatives of the components of the field G = Mi + Nj + Pk are continuous, then § # § * G = 0. b. What, if anything, can you conclude about the flux of the field § * G across a closed surface? Give reasons for your answer. 18. Let F1 and F2 be differentiable vector fields and let a and b be arbitrary real constants. Verify the following identities. a. § # saF1 + bF2 d = a§ # F1 + b§ # F2 b. § * saF1 + bF2 d = a§ * F1 + b§ * F2 c. § # sF1 * F2 d = F2 # § * F1 - F1 # § * F2 19. Let F be a differentiable vector field and let g(x, y, z) be a differentiable scalar function. Verify the following identities. a. § # sgFd = g§ # F + §g # F b. § * sgFd = g§ * F + §g * F 20. If F = Mi + Nj + Pk is a differentiable vector field, we define the notation F # § to mean 0 0 0 M + N + P . 0x 0y 0z

23. a. Show that the outward flux of the position vector field F = xi + yj + zk through a smooth closed surface S is three times the volume of the region enclosed by the surface. b. Let n be the outward unit normal vector field on S. Show that it is not possible for F to be orthogonal to n at every point of S. 24. Maximum flux Among all rectangular solids defined by the inequalities 0 … x … a, 0 … y … b, 0 … z … 1, find the one for which the total flux of F = s -x 2 - 4xydi - 6yzj + 12zk outward through the six sides is greatest. What is the greatest flux? 25. Volume of a solid region Let F = xi + yj + zk and suppose that the surface S and region D satisfy the hypotheses of the Divergence Theorem. Show that the volume of D is given by the formula

S

26. Outward flux of a constant field Show that the outward flux of a constant vector field F = C across any closed surface to which the Divergence Theorem applies is zero. 27. Harmonic functions A function ƒ(x, y, z) is said to be harmonic in a region D in space if it satisfies the Laplace equation

b. §sF1 # F2 d = sF1 # §dF2 + sF2 # §dF1 + F1 * s§ * F2 d + F2 * s§ * F1 d

Theory and Examples 21. Let F be a field whose components have continuous first partial derivatives throughout a portion of space containing a region D bounded by a smooth closed surface S. If ƒ F ƒ … 1, can any bound be placed on the size of 9

§ # F dV ?

0 2ƒ

§2ƒ = § # §ƒ =

For differentiable vector fields F1 and F2, verify the following identities. a. § * sF1 * F2 d = sF2 # §dF1 - sF1 # §dF2 + s§ # F2 dF1 s§ # F1 dF2

1 F # n ds. 36

Volume of D =

0x 2

+

0 2ƒ 0y 2

+

0 2ƒ 0z 2

= 0

throughout D. a. Suppose that ƒ is harmonic throughout a bounded region D enclosed by a smooth surface S and that n is the chosen unit normal vector on S. Show that the integral over S of §ƒ # n, the derivative of ƒ in the direction of n, is zero. b. Show that if ƒ is harmonic on D, then 6

ƒ §ƒ # n ds =

S

9

ƒ §ƒ ƒ 2 dV.

D

28. Outward flux of a gradient field Let S be the surface of the portion of the solid sphere x 2 + y 2 + z 2 … a 2 that lies in the first octant and let ƒsx, y, zd = ln 2x 2 + y 2 + z 2. Calculate

D

Give reasons for your answer. 22. The base of the closed cubelike surface shown here is the unit square in the xy-plane. The four sides lie in the planes x = 0, x = 1, y = 0, and y = 1. The top is an arbitrary smooth surface whose identity is unknown. Let F = xi - 2yj + sz + 3dk and suppose the outward flux of F through Side A is 1 and through Side B is -3. Can you conclude anything about the outward flux through the top? Give reasons for your answer.

6

§ƒ # n ds.

( §ƒ # n is the derivative of ƒ in the direction of outward normal n.) S

29. Green’s first formula Suppose that ƒ and g are scalar functions with continuous first- and second-order partial derivatives throughout a region D that is bounded by a closed piecewise smooth surface S. Show that

z

6

Top

S

ƒ §g # n ds =

9

sƒ § 2g + §ƒ # §gd dV.

(9)

D

Equation (9) is Green’s first formula. (Hint: Apply the Divergence Theorem to the field F = ƒ §g.) 30. Green’s second formula (Continuation of Exercise 29.) Interchange ƒ and g in Equation (9) to obtain a similar formula. Then subtract this formula from Equation (9) to show that 1 y

1 Side B x

Side A

(1, 1, 0)

6 S

sƒ §g - g§ƒd # n ds =

9

sƒ § 2g - g§ 2ƒd dV.

D

This equation is Green’s second formula.

(10)

Chapter 15 31. Conservation of mass Let v(t, x, y, z) be a continuously differentiable vector field over the region D in space and let p(t, x, y, z) be a continuously differentiable scalar function. The variable t represents the time domain. The Law of Conservation of Mass asserts that d pst, x, y, zd dV = pv # n ds, dt 9 6 D

S

where S is the surface enclosing D.

32. The heat diffusion equation Let T(t, x, y, z) be a function with continuous second derivatives giving the temperature at time t at the point (x, y, z) of a solid occupying a region D in space. If the solid’s heat capacity and mass density are denoted by the constants c and r, respectively, the quantity crT is called the solid’s heat energy per unit volume. a. Explain why - §T points in the direction of heat flow. b. Let -k§T denote the energy flux vector. (Here the constant k is called the conductivity.) Assuming the Law of Conservation of Mass with -k§T = v and crT = p in Exercise 31, derive the diffusion (heat) equation

b. Use the Divergence Theorem and Leibniz’s Rule, 0p d pst, x, y, zd dV = dV, dt 9 9 0t

0T = K §2T, 0t where K = k>scrd 7 0 is the diffusivity constant. (Notice that if T(t, x) represents the temperature at time t at position x in a uniform conducting rod with perfectly insulated sides, then §2T = 0 2T>0x 2 and the diffusion equation reduces to the onedimensional heat equation in Chapter 13’s Additional Exercises.)

D

to show that the Law of Conservation of Mass is equivalent to the continuity equation, § # pv +

Chapter 15

911

(In the first term § # pv, the variable t is held fixed, and in the second term 0p>0t , it is assumed that the point (x, y, z) in D is held fixed.)

a. Give a physical interpretation of the conservation of mass law if v is a velocity flow field and p represents the density of the fluid at point (x, y, z) at time t.

D

Practice Exercises

0p = 0. 0t

Questions to Guide Your Review

1. What are line integrals? How are they evaluated? Give examples. 2. How can you use line integrals to find the centers of mass of springs? Explain. 3. What is a vector field? A gradient field? Give examples. 4. How do you calculate the work done by a force in moving a particle along a curve? Give an example.

13. How do you calculate the area of a parametrized surface in space? Of an implicitly defined surface F(x, y, z) = 0? Of the surface which is the graph of z = ƒ(x, y)? Give examples. 14. How do you integrate a function over a parametrized surface in space? Of surfaces that are defined implicitly or in explicit form? What can you calculate with surface integrals? Give examples.

6. What is special about path independent fields?

15. What is an oriented surface? How do you calculate the flux of a three-dimensional vector field across an oriented surface? Give an example.

7. How can you tell when a field is conservative?

16. What is Stokes’ Theorem? How can you interpret it?

8. What is a potential function? Show by example how to find a potential function for a conservative field.

17. Summarize the chapter’s results on conservative fields.

9. What is a differential form? What does it mean for such a form to be exact? How do you test for exactness? Give examples.

19. How does the Divergence Theorem generalize Green’s Theorem?

5. What are flow, circulation, and flux?

10. What is the divergence of a vector field? How can you interpret it? 11. What is the curl of a vector field? How can you interpret it?

18. What is the Divergence Theorem? How can you interpret it? 20. How does Stokes’ Theorem generalize Green’s Theorem? 21. How can Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem be thought of as forms of a single fundamental theorem?

12. What is Green’s Theorem? How can you interpret it?

Chapter 15

Practice Exercises

Evaluating Line Integrals 1. The accompanying figure shows two polygonal paths in space joining the origin to the point (1, 1, 1). Integrate ƒsx, y, zd = 2x 3y 2 - 2z + 3 over each path.

z (0, 0, 0)

z (0, 0, 0)

(1, 1, 1)

(1, 1, 1)

y x

(1, 1, 0) Path 1

y x

(1, 1, 0) Path 2

912

Chapter 15: Integration in Vector Fields

2. The accompanying figure shows three polygonal paths joining the origin to the point (1, 1, 1). Integrate ƒsx, y, zd = x 2 + y - z over each path. z

b. Find the area of the portion of the cylinder that lies inside the hemisphere. (Hint: Project onto the xz-plane. Or evaluate the integral 1 h ds, where h is the altitude of the cylinder and ds is the element of arc length on the circle x 2 + y 2 = 2x in the xy-plane.)

z

(0, 0, 0)

(0, 0, 0)

(1, 1, 1)

14. a. Hemisphere cut by cylinder Find the area of the surface cut from the hemisphere x 2 + y 2 + z 2 = 4, z Ú 0, by the cylinder x 2 + y 2 = 2x.

(1, 1, 1)

(1, 0, 0) C1 x

C3

y

C4

(1, 1, 0)

C2

C3

z

y

z 5 4 2 r2

(1, 1, 0)

x

Hemisphere

z (0, 0, 1) (0, 0, 0)

C6 C5

(0, 1, 1)

C7 (1, 1, 1) y x

x

3. Integrate ƒsx, y, zd = 2x 2 + z 2 over the circle rstd = sa cos tdj + sa sin tdk,

4. Integrate ƒsx, y, zd = 2x 2 + y 2 over the involute curve rstd = scos t + t sin tdi + ssin t - t cos tdj,

0 … t … 23.

Evaluate the integrals in Exercises 5 and 6. s4,-3,0d dx + dy + dz 5. Ls-1,1,1d 2x + y + z s10,3,3d

6.

dx -

y z dy - z dz Ay A

Ls1,1,1d 7. Integrate F = -s y sin zdi + sx sin zdj + sxy cos zdk around the circle cut from the sphere x 2 + y 2 + z 2 = 5 by the plane z = -1, clockwise as viewed from above. 8. Integrate F = 3x 2yi + sx 3 + 1dj + 9z 2k around the circle cut from the sphere x 2 + y 2 + z 2 = 9 by the plane x = 2. Evaluate the integrals in Exercises 9 and 10. 9.

10.

Cylinder r 5 2 cos u

0 … t … 2p.

y

15. Area of a triangle Find the area of the triangle in which the plane sx>ad + s y>bd + sz>cd = 1 sa, b, c 7 0d intersects the first octant. Check your answer with an appropriate vector calculation. 16. Parabolic cylinder cut by planes Integrate yz a. gsx, y, zd = 24y 2 + 1 z b. gsx, y, zd = 24y 2 + 1 over the surface cut from the parabolic cylinder y 2 - z = 1 by the planes x = 0, x = 3 , and z = 0. 17. Circular cylinder cut by planes Integrate gsx, y, zd = x 4ysy 2 + z 2 d over the portion of the cylinder y 2 + z 2 = 25 that lies in the first octant between the planes x = 0 and x = 1 and above the plane z = 3.

8x sin y dx - 8y cos x dy

18. Area of Wyoming The state of Wyoming is bounded by the meridians 111°3¿ and 104°3¿ west longitude and by the circles 41° and 45° north latitude. Assuming that Earth is a sphere of radius R = 3959 mi, find the area of Wyoming.

y 2 dx + x 2 dy LC C is the circle x 2 + y 2 = 4.

Parametrized Surfaces Find parametrizations for the surfaces in Exercises 19–24. (There are many ways to do these, so your answers may not be the same as those in the back of the book.)

LC C is the square cut from the first quadrant by the lines x = p>2 and y = p>2 .

Finding and Evaluating Surface Integrals 11. Area of an elliptical region Find the area of the elliptical region cut from the plane x + y + z = 1 by the cylinder x 2 + y 2 = 1. 12. Area of a parabolic cap Find the area of the cap cut from the paraboloid y 2 + z 2 = 3x by the plane x = 1. 13. Area of a spherical cap Find the area of the cap cut from the top of the sphere x 2 + y 2 + z 2 = 1 by the plane z = 22>2.

19. Spherical band The portion of the sphere x 2 + y 2 + z 2 = 36 between the planes z = -3 and z = 323 20. Parabolic cap The portion of the paraboloid z = -sx 2 + y 2 d>2 above the plane z = -2 21. Cone

The cone z = 1 + 2x 2 + y 2, z … 3

22. Plane above square The portion of the plane 4x + 2y + 4z = 12 that lies above the square 0 … x … 2, 0 … y … 2 in the first quadrant

Chapter 15 23. Portion of paraboloid The portion of the paraboloid y = 2sx 2 + z 2 d, y … 2, that lies above the xy-plane 24. Portion of hemisphere The portion of the hemisphere x 2 + y 2 + z 2 = 10, y Ú 0, in the first octant 25. Surface area

Find the area of the surface

rsu, yd = su + ydi + su - ydj + yk, 0 … u … 1, 0 … y … 1. 26. Surface integral Integrate ƒsx, y, zd = xy - z 2 over the surface in Exercise 25. Find the surface area of the helicoid

27. Area of a helicoid

rsr, ud = (r cos u)i + (r sin u)j + uk, 0 … u … 2p, 0 … r … 1, in the accompanying figure.

Practice Exercises

913

38. Flow along different paths Find the flow of the field F = §sx 2ze y d a. once around the ellipse C in which the plane x + y + z = 1 intersects the cylinder x 2 + z 2 = 25, clockwise as viewed from the positive y-axis. b. along the curved boundary of the helicoid in Exercise 27 from (1, 0, 0) to s1, 0, 2pd. In Exercises 39 and 40, use the surface integral in Stokes’ Theorem to find the circulation of the field F around the curve C in the indicated direction. 39. Circulation around an ellipse

F = y 2i - yj + 3z 2k

C: The ellipse in which the plane 2x + 6y - 3z = 6 meets the cylinder x 2 + y 2 = 1, counterclockwise as viewed from above 40. Circulation around a circle F = sx 2 + ydi + sx + ydj + s4y 2 - zdk

z

C: The circle in which the plane z = -y meets the sphere x 2 + y 2 + z 2 = 4, counterclockwise as viewed from above

2p (1, 0, 2p)

Masses and Moments 41. Wire with different densities Find the mass of a thin wire lying along the curve rstd = 22ti + 22tj + s4 - t 2 dk, 0 … t … 1, if the density at t is (a) d = 3t and (b) d = 1.

(1, 0, 0)

42. Wire with variable density Find the center of mass of a thin wire lying along the curve rstd = ti + 2tj + s2>3dt 3>2k, 0 … t … 2, if the density at t is d = 325 + t.

y

x

28. Surface integral Evaluate the integral 4S 2x 2 + y 2 + 1 ds, where S is the helicoid in Exercise 27. Conservative Fields Which of the fields in Exercises 29–32 are conservative, and which are not?

43. Wire with variable density Find the center of mass and the moments of inertia about the coordinate axes of a thin wire lying along the curve rstd = ti +

2 22 3>2 t2 t j + k, 3 2

0 … t … 2,

if the density at t is d = 1>st + 1d .

29. F = xi + yj + zk 30. F = sxi + yj + zkd>sx 2 + y 2 + z 2 d3>2

44. Center of mass of an arch A slender metal arch lies along the semicircle y = 2a 2 - x 2 in the xy-plane. The density at the point (x, y) on the arch is dsx, yd = 2a - y. Find the center of mass.

31. F = xe yi + ye zj + ze xk 32. F = si + zj + ykd>sx + yzd Find potential functions for the fields in Exercises 33 and 34. 33. F = 2i + s2y + zdj + s y + 1dk 34. F = sz cos xzdi + e yj + sx cos xzdk Work and Circulation In Exercises 35 and 36, find the work done by each field along the paths from (0, 0, 0) to (1, 1, 1) in Exercise 1. 35. F = 2xyi + j + x k

36. F = 2xyi + x j + k

2

2

37. Finding work in two ways Find the work done by F =

xi + yj sx 2 + y 2 d3>2

over the plane curve rstd = se t cos tdi + se t sin tdj from the point (1, 0) to the point se 2p, 0d in two ways: a. By using the parametrization of the curve to evaluate the work integral. b. By evaluating a potential function for F.

45. Wire with constant density A wire of constant density d = 1 lies along the curve rstd = se t cos tdi + se t sin tdj + e t k, 0 … t … ln 2. Find z and Iz. 46. Helical wire with constant density Find the mass and center of mass of a wire of constant density d that lies along the helix rstd = s2 sin tdi + s2 cos tdj + 3tk, 0 … t … 2p. 47. Inertia and center of mass of a shell Find Iz and the center of mass of a thin shell of density dsx, y, zd = z cut from the upper portion of the sphere x 2 + y 2 + z 2 = 25 by the plane z = 3. 48. Moment of inertia of a cube Find the moment of inertia about the z-axis of the surface of the cube cut from the first octant by the planes x = 1, y = 1, and z = 1 if the density is d = 1. Flux Across a Plane Curve or Surface Use Green’s Theorem to find the counterclockwise circulation and outward flux for the fields and curves in Exercises 49 and 50. 49. Square

F = s2xy + xdi + sxy - ydj

C: The square bounded by x = 0, x = 1, y = 0, y = 1

914

Chapter 15: Integration in Vector Fields

50. Triangle

55. Spherical cap F = -2xi - 3yj + zk

F = s y - 6x 2 di + sx + y 2 dj

C: The triangle made by the lines y = 0, y = x , and x = 1 51. Zero line integral Show that F

ln x sin y dy -

cos y x dx = 0

C

for any closed curve C to which Green’s Theorem applies. 52. a. Outward flux and area Show that the outward flux of the position vector field F = xi + yj across any closed curve to which Green’s Theorem applies is twice the area of the region enclosed by the curve. b. Let n be the outward unit normal vector to a closed curve to which Green’s Theorem applies. Show that it is not possible for F = xi + yj to be orthogonal to n at every point of C. In Exercises 53–56, find the outward flux of F across the boundary of D. 53. Cube

F = 2xyi + 2yzj + 2xzk

D: The cube cut from the first octant by the planes x = 1, y = 1, z = 1 54. Spherical cap F = xzi + yzj + k D: The entire surface of the upper cap cut from the solid sphere x 2 + y 2 + z 2 … 25 by the plane z = 3

Chapter 15

D: The upper region cut from the solid sphere x 2 + y 2 + z 2 … 2 by the paraboloid z = x 2 + y 2 56. Cone and cylinder

F = s6x + ydi - sx + zdj + 4yzk

D: The region in the first octant bounded by the cone z = 2x 2 + y 2, the cylinder x 2 + y 2 = 1, and the coordinate planes 57. Hemisphere, cylinder, and plane Let S be the surface that is bounded on the left by the hemisphere x 2 + y 2 + z 2 = a 2, y … 0, in the middle by the cylinder x 2 + z 2 = a 2, 0 … y … a, and on the right by the plane y = a. Find the flux of F = yi + zj + xk outward across S. 58. Cylinder and planes Find the outward flux of the field F = 3xz 2i + yj - z 3k across the surface of the solid in the first octant that is bounded by the cylinder x 2 + 4y 2 = 16 and the planes y = 2z, x = 0, and z = 0. 59. Cylindrical can Use the Divergence Theorem to find the flux of F = xy 2i + x 2yj + yk outward through the surface of the region enclosed by the cylinder x 2 + y 2 = 1 and the planes z = 1 and z = -1. 60. Hemisphere Find the flux of F = s3z + 1dk upward across the hemisphere x 2 + y 2 + z 2 = a 2, z Ú 0 (a) with the Divergence Theorem and (b) by evaluating the flux integral directly.

Additional and Advanced Exercises

Finding Areas with Green’s Theorem Use the Green’s Theorem area formula in Exercises 15.4 to find the areas of the regions enclosed by the curves in Exercises 1–4.

3. The eight curve x = s1>2d sin 2t, y = sin t, 0 … t … p (one loop) y 1

1. The limaçon x = 2 cos t - cos 2t, y = 2 sin t - sin 2t, 0 … t … 2p y

x

0

x

1

21

4. The teardrop x = 2a cos t - a sin 2t, y = b sin t, 0 … t … 2p 2. The deltoid x = 2 cos t + cos 2t, y = 2 sin t - sin 2t, y 0 … t … 2p

0

3

x

y b

0

2a

x

Chapter 15 Theory and Applications 5. a. Give an example of a vector field F (x, y, z) that has value 0 at only one point and such that curl F is nonzero everywhere. Be sure to identify the point and compute the curl. b. Give an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that curl F is nonzero everywhere. Be sure to identify the line and compute the curl. c. Give an example of a vector field F (x, y, z) that has value 0 on a surface and such that curl F is nonzero everywhere. Be sure to identify the surface and compute the curl. 6. Find all points (a, b, c) on the sphere x 2 + y 2 + z 2 = R 2 where the vector field F = yz 2i + xz 2j + 2xyzk is normal to the surface and Fsa, b, cd Z 0. 7. Find the mass of a spherical shell of radius R such that at each point (x, y, z) on the surface the mass density dsx, y, zd is its distance to some fixed point (a, b, c) of the surface.

a. Show that the surface integral giving the magnitude of the total force on the ball due to the fluid’s pressure is n

Force = lim a ws4 - zk d ¢sk = n: q k=1

2 2x 2 + y 2. See Practice Exercise 27 for a figure. 9. Among all rectangular regions 0 … x … a, 0 … y … b, find the one for which the total outward flux of F = sx 2 + 4xydi - 6yj across the four sides is least. What is the least flux? 10. Find an equation for the plane through the origin such that the circulation of the flow field F = zi + xj + yk around the circle of intersection of the plane with the sphere x 2 + y 2 + z 2 = 4 is a maximum. 11. A string lies along the circle x 2 + y 2 = 4 from (2, 0) to (0, 2) in the first quadrant. The density of the string is r sx, yd = xy. a. Partition the string into a finite number of subarcs to show that the work done by gravity to move the string straight down to the x-axis is given by n

S

Buoyant force =

6

wsz - 4dk # n ds,

where n is the outer unit normal at (x, y, z). This illustrates Archimedes’ principle that the magnitude of the buoyant force on a submerged solid equals the weight of the displaced fluid.

LC

g xy 2 ds,

where g is the gravitational constant.

c. Use the Divergence Theorem to find the magnitude of the buoyant force in part (b). 14. Fluid force on a curved surface A cone in the shape of the surface z = 2x 2 + y 2, 0 … z … 2 is filled with a liquid of constant weight density w. Assuming the xy-plane is “ground level,” show that the total force on the portion of the cone from z = 1 to z = 2 due to liquid pressure is the surface integral F =

c. Show that the total work done equals the work required to move the string’s center of mass sx, yd straight down to the x-axis. 12. A thin sheet lies along the portion of the plane x + y + z = 1 in the first octant. The density of the sheet is d sx, y, zd = xy. a. Partition the sheet into a finite number of subpieces to show that the work done by gravity to move the sheet straight down to the xy-plane is given by n

6

g xyz ds,

S

where g is the gravitational constant. b. Find the total work done by evaluating the surface integral in part (a). c. Show that the total work done equals the work required to move the sheet’s center of mass sx, y, zd straight down to the xy-plane.

6

ws2 - zd ds.

S

Evaluate the integral. 15. Faraday’s Law If E(t, x, y, z) and B(t, x, y, z) represent the electric and magnetic fields at point (x, y, z) at time t, a basic principle of electromagnetic theory says that § * E = -0B>0t. In this expression § * E is computed with t held fixed and 0B>0t is calculated with (x, y, z) fixed. Use Stokes’ Theorem to derive Faraday’s Law,

b. Find the total work done by evaluating the line integral in part (a).

k=1

ws4 - zd ds.

S

0 … r … 1, 0 … u … 2p, if the density function is dsx, y, zd =

Work = lim a g xk yk zk ¢sk = n: q

6

b. Since the ball is not moving, it is being held up by the buoyant force of the liquid. Show that the magnitude of the buoyant force on the sphere is

rsr, ud = sr cos udi + sr sin udj + uk,

k=1

915

13. Archimedes’ principle If an object such as a ball is placed in a liquid, it will either sink to the bottom, float, or sink a certain distance and remain suspended in the liquid. Suppose a fluid has constant weight density w and that the fluid’s surface coincides with the plane z = 4. A spherical ball remains suspended in the fluid and occupies the region x 2 + y 2 + sz - 2d2 … 1.

8. Find the mass of a helicoid

Work = lim a g xk y k2 ¢sk = n: q

Additional and Advanced Exercises

F

E # dr = -

C

0 B # n ds, 0t 6 S

where C represents a wire loop through which current flows counterclockwise with respect to the surface’s unit normal n, giving rise to the voltage F

E # dr

C

around C. The surface integral on the right side of the equation is called the magnetic flux, and S is any oriented surface with boundary C. 16. Let F = -

GmM r ƒrƒ3

be the gravitational force field defined for r Z 0. Use Gauss’s Law in Section 15.8 to show that there is no continuously differentiable vector field H satisfying F = § * H.

916

Chapter 15: Integration in Vector Fields

17. If ƒ(x, y, z) and g(x, y, z) are continuously differentiable scalar functions defined over the oriented surface S with boundary curve C, prove that 6 S

18.

s§ƒ * §gd # n ds =

Suppose that § # F1

F

ƒ §g # dr.

C

§ # F2 and § * F1 =

= § * F2 over a region D enclosed by the oriented surface S with outward unit normal n and that F1 # n = F2 # n on S. Prove that F1 = F2 throughout D.

19. Prove or disprove that if § # F = 0 and § * F = 0, then F = 0.

20. Let S be an oriented surface parametrized by r(u, y). Define the notation dS = ru du * ry dy so that dS is a vector normal to the

surface. Also, the magnitude ds = ƒ dS ƒ is the element of surface area (by Equation 5 in Section 15.5). Derive the identity ds = sEG - F 2 d1>2 du dy where E = ƒ ru ƒ 2,

F = ru # ry ,

and

G = ƒ ry ƒ 2.

21. Show that the volume V of a region D in space enclosed by the oriented surface S with outward normal n satisfies the identity V =

1 r # n ds, 36 S

where r is the position vector of the point (x, y, z) in D.

APPENDICES

A.1

Real Numbers and the Real Line This section reviews real numbers, inequalities, intervals, and absolute values.

Real Numbers Much of calculus is based on properties of the real number system. Real numbers are numbers that can be expressed as decimals, such as -

3 = -0.75000 Á 4 1 = 0.33333 Á 3

22 = 1.4142 Á The dots Á in each case indicate that the sequence of decimal digits goes on forever. Every conceivable decimal expansion represents a real number, although some numbers have two representations. For instance, the infinite decimals .999 Á and 1.000 Á represent the same real number 1. A similar statement holds for any number with an infinite tail of 9’s. The real numbers can be represented geometrically as points on a number line called the real line. Rules for inequalities If a, b, and c are real numbers, then: 1. a 6 b Q a + c 6 b + c 2. a 6 b Q a - c 6 b - c 3. a 6 b and c 7 0 Q ac 6 bc 4. a 6 b and c 6 0 Q bc 6 ac Special case: a 6 b Q -b 6 -a 1 5. a 7 0 Q a 7 0 6. If a and b are both positive or both 1 1 negative, then a 6 b Q 6 a b

–2

–1 – 3 4

0

1 3

1 2

2

3

4

The symbol ⺢ denotes either the real number system or, equivalently, the real line. The properties of the real number system fall into three categories: algebraic properties, order properties, and completeness. The algebraic properties say that the real numbers can be added, subtracted, multiplied, and divided (except by 0) to produce more real numbers under the usual rules of arithmetic. You can never divide by 0. The order properties of real numbers are given in Appendix 7. The useful rules at the left can be derived from them, where the symbol Q means “implies.” Notice the rules for multiplying an inequality by a number. Multiplying by a positive number preserves the inequality; multiplying by a negative number reverses the inequality. Also, reciprocation reverses the inequality for numbers of the same sign. For example, 2 6 5 but -2 7 -5 and 1>2 7 1>5. The completeness property of the real number system is deeper and harder to define precisely. However, the property is essential to the idea of a limit (Chapter 2). Roughly speaking, it says that there are enough real numbers to “complete” the real number line, in the sense that there are no “holes” or “gaps” in it. Many theorems of calculus would fail if the real number system were not complete. The topic is best saved for a more advanced course, but Appendix 7 hints about what is involved and how the real numbers are constructed.

AP-1

AP-2

Appendices

We distinguish three special subsets of real numbers. 1. 2. 3.

The natural numbers, namely 1, 2, 3, 4, Á The integers, namely 0, ;1, ;2, ;3, Á The rational numbers, namely the numbers that can be expressed in the form of a fraction m>n, where m and n are integers and n Z 0. Examples are 1 , 3

-

4 -4 4 = = , 9 9 -9

200 , 13

57 =

and

57 . 1

The rational numbers are precisely the real numbers with decimal expansions that are either (a) terminating (ending in an infinite string of zeros), for example, 3 = 0.75000 Á = 0.75 4

or

(b) eventually repeating (ending with a block of digits that repeats over and over), for example, 23 = 2.090909 Á = 2.09 11

The bar indicates the block of repeating digits.

A terminating decimal expansion is a special type of repeating decimal, since the ending zeros repeat. The set of rational numbers has all the algebraic and order properties of the real numbers but lacks the completeness property. For example, there is no rational number whose square is 2; there is a “hole” in the rational line where 22 should be. Real numbers that are not rational are called irrational numbers. They are characterized by having nonterminating and nonrepeating decimal expansions. Examples are 3 p, 22, 2 5, and log10 3. Since every decimal expansion represents a real number, it should be clear that there are infinitely many irrational numbers. Both rational and irrational numbers are found arbitrarily close to any point on the real line. Set notation is very useful for specifying a particular subset of real numbers. A set is a collection of objects, and these objects are the elements of the set. If S is a set, the notation a H S means that a is an element of S, and a x S means that a is not an element of S. If S and T are sets, then S ´ T is their union and consists of all elements belonging either to S or T (or to both S and T ). The intersection S ¨ T consists of all elements belonging to both S and T. The empty set ¤ is the set that contains no elements. For example, the intersection of the rational numbers and the irrational numbers is the empty set. Some sets can be described by listing their elements in braces. For instance, the set A consisting of the natural numbers (or positive integers) less than 6 can be expressed as A = 51, 2, 3, 4, 56. The entire set of integers is written as 50, ;1, ;2, ;3, Á 6. Another way to describe a set is to enclose in braces a rule that generates all the elements of the set. For instance, the set A = 5x ƒ x is an integer and 0 6 x 6 66 is the set of positive integers less than 6.

Appendix 1

Real Numbers and the Real Line

AP-3

Intervals A subset of the real line is called an interval if it contains at least two numbers and contains all the real numbers lying between any two of its elements. For example, the set of all real numbers x such that x 7 6 is an interval, as is the set of all x such that -2 … x … 5. The set of all nonzero real numbers is not an interval; since 0 is absent, the set fails to contain every real number between -1 and 1 (for example). Geometrically, intervals correspond to rays and line segments on the real line, along with the real line itself. Intervals of numbers corresponding to line segments are finite intervals; intervals corresponding to rays and the real line are infinite intervals. A finite interval is said to be closed if it contains both of its endpoints, half-open if it contains one endpoint but not the other, and open if it contains neither endpoint. The endpoints are also called boundary points; they make up the interval’s boundary. The remaining points of the interval are interior points and together comprise the interval’s interior. Infinite intervals are closed if they contain a finite endpoint, and open otherwise. The entire real line ⺢ is an infinite interval that is both open and closed. Table A.1 summarizes the various types of intervals.

TABLE A.1 Types of intervals

Notation

Set description

Type

Picture

(a, b)

5x ƒ a 6 x 6 b6

Open

[a, b]

5x ƒ a … x … b6

Closed

[a, b)

5x ƒ a … x 6 b6

Half-open

(a, b]

5x ƒ a 6 x … b6

sa, q d

a

b

a

b

a

b

Half-open

a

b

5x ƒ x 7 a6

Open

a

[a, q d

5x ƒ x Ú a6

Closed

a

s - q , bd

5x ƒ x 6 b6

Open

b

s - q , b]

5x ƒ x … b6

Closed

b

⺢ (set of all real numbers)

Both open and closed

s - q, q d

Solving Inequalities The process of finding the interval or intervals of numbers that satisfy an inequality in x is called solving the inequality.

EXAMPLE 1

Solve the following inequalities and show their solution sets on the real line.

(a) 2x - 1 6 x + 3

(b) -

x 6 2x + 1 3

(c)

6 Ú 5 x - 1

AP-4

Appendices

0

1

x

4

0

x

1

x

11 5

1

Add 1 to both sides. Subtract x from both sides.

The solution set is the open interval s - q , 4d (Figure A.1a).

(b) 0

2x - 1 6 x + 3 2x 6 x + 4 x 6 4

(a)

(a) –3 7

Solution

-

(b)

x 6 2x + 1 3

-x 6 6x + 3 0 6 7x + 3 -3 6 7x

(c)

FIGURE A.1 Solution sets for the inequalities in Example 1.

-

3 6 x 7

Multiply both sides by 3. Add x to both sides. Subtract 3 from both sides. Divide by 7.

The solution set is the open interval s -3>7, q d (Figure A.1b). (c) The inequality 6>sx - 1d Ú 5 can hold only if x 7 1, because otherwise 6>sx - 1d is undefined or negative. Therefore, sx - 1d is positive and the inequality will be preserved if we multiply both sides by sx - 1d, and we have 6 Ú 5 x - 1 6 Ú 5x - 5 11 Ú 5x

Multiply both sides by sx - 1d . Add 5 to both sides.

11 Ú x. 5

Or x …

11 . 5

The solution set is the half-open interval (1, 11>5 ] (Figure A.1c).

Absolute Value The absolute value of a number x, denoted by ƒ x ƒ , is defined by the formula ƒxƒ = e

EXAMPLE 2

ƒ 3 ƒ = 3,

ƒ 0 ƒ = 0,

x, -x,

x Ú 0 x 6 0.

ƒ -5 ƒ = -s -5d = 5,

ƒ - ƒaƒƒ = ƒaƒ

Geometrically, the absolute value of x is the distance from x to 0 on the real number line. Since distances are always positive or 0, we see that ƒ x ƒ Ú 0 for every real number x, and ƒ x ƒ = 0 if and only if x = 0. Also, ƒ x - y ƒ = the distance between x and y – 5 5 –5

3 0

3

4 1 1 4 3 1

4

FIGURE A.2 Absolute values give distances between points on the number line.

on the real line (Figure A.2). Since the symbol 2a always denotes the nonnegative square root of a, an alternate definition of ƒ x ƒ is ƒ x ƒ = 2x 2 . It is important to remember that 2a 2 = ƒ a ƒ . Do not write 2a 2 = a unless you already know that a Ú 0. The absolute value has the following properties. (You are asked to prove these properties in the exercises.)

Appendix 1

Real Numbers and the Real Line

AP-5

Absolute Value Properties

a –a

a

x

0

a

x

FIGURE A.3 ƒ x ƒ 6 a means x lies between -a and a.

1. ƒ -a ƒ = ƒ a ƒ

A number and its additive inverse or negative have the same absolute value.

2. ƒ ab ƒ = ƒ a ƒ ƒ b ƒ

The absolute value of a product is the product of the absolute values.

ƒaƒ a 3. ` ` = b ƒbƒ 4. ƒ a + b ƒ … ƒ a ƒ + ƒ b ƒ

The absolute value of a quotient is the quotient of the absolute values.

Note that ƒ -a ƒ Z - ƒ a ƒ . For example, ƒ -3 ƒ = 3, whereas - ƒ 3 ƒ = -3. If a and b differ in sign, then ƒ a + b ƒ is less than ƒ a ƒ + ƒ b ƒ . In all other cases, ƒ a + b ƒ equals ƒ a ƒ + ƒ b ƒ . Absolute value bars in expressions like ƒ -3 + 5 ƒ work like parentheses: We do the arithmetic inside before taking the absolute value.

EXAMPLE 3 ƒ -3 + 5 ƒ = ƒ 2 ƒ = 2 6 ƒ -3 ƒ + ƒ 5 ƒ = 8 ƒ3 + 5ƒ = ƒ8ƒ = ƒ3ƒ + ƒ5ƒ ƒ -3 - 5 ƒ = ƒ -8 ƒ = 8 = ƒ -3 ƒ + ƒ -5 ƒ

Absolute values and intervals If a is any positive number, then 5. ƒ x ƒ = a 3 x = ;a 6. ƒ x ƒ 6 a 3 -a 6 x 6 a 7. ƒ x ƒ 7 a 3 x 7 a or x 6 -a 8. ƒ x ƒ … a 3 -a … x … a 9. ƒ x ƒ Ú a 3 x Ú a or x … -a

The triangle inequality. The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values.

The inequality ƒ x ƒ 6 a says that the distance from x to 0 is less than the positive number a. This means that x must lie between -a and a, as we can see from Figure A.3. The statements in the table are all consequences of the definition of absolute value and are often helpful when solving equations or inequalities involving absolute values. The symbol 3 is often used by mathematicians to denote the “if and only if ” logical relationship. It also means “implies and is implied by.”

EXAMPLE 4 Solution

Solve the equation ƒ 2x - 3 ƒ = 7.

By Property 5, 2x - 3 = ;7, so there are two possibilities: 2x - 3 = 7 2x = 10 x = 5

2x - 3 = -7 2x = -4 x = -2

Equivalent equations without absolute values Solve as usual.

The solutions of ƒ 2x - 3 ƒ = 7 are x = 5 and x = -2.

EXAMPLE 5 Solution

2 Solve the inequality ` 5 - x ` 6 1.

We have 2 2 ` 5 - x ` 6 1 3 -1 6 5 - x 6 1

Property 6

2 3 -6 6 - x 6 -4

Subtract 5.

1 33 7 x 7 2

1 Multiply by - . 2

3

1 1 6 x 6 . 3 2

Take reciprocals.

AP-6

Appendices

Notice how the various rules for inequalities were used here. Multiplying by a negative number reverses the inequality. So does taking reciprocals in an inequality in which both sides are positive. The original inequality holds if and only if s1>3d 6 x 6 s1>2d. The solution set is the open interval ( 1>3 , 1>2 ).

Exercises A.1 1. Express 1>9 as a repeating decimal, using a bar to indicate the repeating digits. What are the decimal representations of 2>9? 3>9? 8>9? 9>9?

Solve the inequalities in Exercises 18–21. Express the solution sets as intervals or unions of intervals and show them on the real line. Use the result 2a 2 = ƒ a ƒ as appropriate.

2. If 2 6 x 6 6 , which of the following statements about x are necessarily true, and which are not necessarily true?

18. x 2 6 2

19. 4 6 x 2 6 9

20. sx - 1d2 6 4

21. x 2 - x 6 0

a. 0 6 x 6 4 c. 1 6

b. 0 6 x - 2 6 4

x 6 3 2

d.

1 1 1 6 x 6 6 2

6 e. 1 6 x 6 3

f. ƒ x - 4 ƒ 6 2

g. -6 6 -x 6 2

h. -6 6 - x 6 -2

In Exercises 3–6, solve the inequalities and show the solution sets on the real line. 3. -2x 7 4 5. 2x -

4. 5x - 3 … 7 - 3x

7 1 Ú 7x + 2 6

6.

4 1 sx - 2d 6 sx - 6d 5 3

Solve the equations in Exercises 7–9. 7. ƒ y ƒ = 3

8. ƒ 2t + 5 ƒ = 4

9 9. ƒ 8 - 3s ƒ = 2

Solve the inequalities in Exercises 10–17, expressing the solution sets as intervals or unions of intervals. Also, show each solution set on the real line. 10. ƒ x ƒ 6 2

11. ƒ t - 1 ƒ … 3

12. ƒ 3y - 7 ƒ 6 4

13. `

1 1 14. ` 3 - x ` 6 2

15. ƒ 2s ƒ Ú 4

z - 1` … 1 5

16. ƒ 1 - x ƒ 7 1

A.2

17. `

r + 1 ` Ú 1 2

22. Do not fall into the trap of thinking ƒ -a ƒ = a . For what real numbers a is this equation true? For what real numbers is it false? 23. Solve the equation ƒ x - 1 ƒ = 1 - x . 24. A proof of the triangle inequality Give the reason justifying each of the numbered steps in the following proof of the triangle inequality. ƒ a + b ƒ 2 = sa + bd2 = a 2 + 2ab + b 2 … a2 + 2 ƒ a ƒ ƒ b ƒ + b2 = ƒaƒ2 + 2ƒaƒ ƒbƒ + ƒbƒ2 = s ƒ a ƒ + ƒ b ƒ d2 ƒa + bƒ … ƒaƒ + ƒbƒ

(1) (2) (3) (4)

25. Prove that ƒ ab ƒ = ƒ a ƒ ƒ b ƒ for any numbers a and b. 26. If ƒ x ƒ … 3 and x 7 -1>2 , what can you say about x? 27. Graph the inequality ƒ x ƒ + ƒ y ƒ … 1 . 28. For any number a, prove that ƒ -a ƒ = ƒ a ƒ . 29. Let a be any positive number. Prove that ƒ x ƒ 7 a if and only if x 7 a or x 6 -a . 30. a. If b is any nonzero real number, prove that ƒ 1>b ƒ = 1> ƒ b ƒ . ƒaƒ a for any numbers a and b Z 0 . b. Prove that ` ` = b ƒbƒ

Mathematical Induction Many formulas, like nsn + 1d 1 + 2 + Á + n = , 2 can be shown to hold for every positive integer n by applying an axiom called the mathematical induction principle. A proof that uses this axiom is called a proof by mathematical induction or a proof by induction. The steps in proving a formula by induction are the following: 1. 2.

Check that the formula holds for n = 1. Prove that if the formula holds for any positive integer n = k, then it also holds for the next integer, n = k + 1.

Appendix 2

Mathematical Induction

AP-7

The induction axiom says that once these steps are completed, the formula holds for all positive integers n. By Step 1 it holds for n = 1. By Step 2 it holds for n = 2, and therefore by Step 2 also for n = 3, and by Step 2 again for n = 4, and so on. If the first domino falls, and the kth domino always knocks over the sk + 1dst when it falls, all the dominoes fall. From another point of view, suppose we have a sequence of statements S1, S2, Á , Sn, Á , one for each positive integer. Suppose we can show that assuming any one of the statements to be true implies that the next statement in line is true. Suppose that we can also show that S1 is true. Then we may conclude that the statements are true from S1 on.

EXAMPLE 1

Use mathematical induction to prove that for every positive integer n, nsn + 1d . 1 + 2 + Á + n = 2

Solution

1.

We accomplish the proof by carrying out the two steps above.

The formula holds for n = 1 because 1 =

2.

1s1 + 1d . 2

If the formula holds for n = k, does it also hold for n = k + 1? The answer is yes, as we now show. If ksk + 1d 1 + 2 + Á + k = , 2 then ksk + 1d k 2 + k + 2k + 2 1 + 2 + Á + k + sk + 1d = + sk + 1d = 2 2 =

sk + 1dssk + 1d + 1d sk + 1dsk + 2d = . 2 2

The last expression in this string of equalities is the expression nsn + 1d>2 for n = sk + 1d. The mathematical induction principle now guarantees the original formula for all positive integers n. In Example 4 of Section 5.2 we gave another proof for the formula giving the sum of the first n integers. However, proof by mathematical induction is more general. It can be used to find the sums of the squares and cubes of the first n integers (Exercises 9 and 10). Here is another example.

EXAMPLE 2

Show by mathematical induction that for all positive integers n, 1 1 1 1 + 2 + Á + n = 1 - n. 1 2 2 2 2

Solution

1.

We accomplish the proof by carrying out the two steps of mathematical induction.

The formula holds for n = 1 because 1 1 = 1 - 1. 1 2 2

AP-8

Appendices

2.

If 1 1 1 1 + 2 + Á + k = 1 - k, 1 2 2 2 2 then 1 1 1 1 1 1 1 1#2 + k+1 + 2 + Á + k + k+1 = 1 - k + k+1 = 1 - k 1 # 2 2 2 2 2 2 2 2 2 2 1 1 = 1 - k+1 + k+1 = 1 - k+1 . 2 2 2 Thus, the original formula holds for n = sk + 1d whenever it holds for n = k.

With these steps verified, the mathematical induction principle now guarantees the formula for every positive integer n.

Other Starting Integers Instead of starting at n = 1 some induction arguments start at another integer. The steps for such an argument are as follows. 1. 2.

Check that the formula holds for n = n1 (the first appropriate integer). Prove that if the formula holds for any integer n = k Ú n1 , then it also holds for n = sk + 1d.

Once these steps are completed, the mathematical induction principle guarantees the formula for all n Ú n1 . Show that n! 7 3n if n is large enough.

EXAMPLE 3 Solution

How large is large enough? We experiment: n n! 3n

1 1 3

2 2 9

3 6 27

4 24 81

5 120 243

6 720 729

7 5040 2187

It looks as if n! 7 3n for n Ú 7. To be sure, we apply mathematical induction. We take n1 = 7 in Step 1 and complete Step 2. Suppose k! 7 3k for some k Ú 7. Then sk + 1d! = sk + 1dsk!d 7 sk + 1d3k 7 7 # 3k 7 3k + 1 . Thus, for k Ú 7, k! 7 3k

implies

sk + 1d! 7 3k + 1 .

The mathematical induction principle now guarantees n! Ú 3n for all n Ú 7.

Proof of the Derivative Sum Rule for Sums of Finitely Many Functions We prove the statement dun du1 du2 d su + u2 + Á + un d = + + Á + dx 1 dx dx dx

Appendix 2

Mathematical Induction

AP-9

by mathematical induction. The statement is true for n = 2, as was proved in Section 3.3. This is Step 1 of the induction proof. Step 2 is to show that if the statement is true for any positive integer n = k, where k Ú n0 = 2, then it is also true for n = k + 1. So suppose that duk du1 du2 d su + u2 + Á + uk d = + + Á + . dx 1 dx dx dx

(1)

Then d (u1 + u2 + Á + uk + uk + 1) dx (++++)++++* ()* Call the function defined by this sum u.

Call this function y.

=

duk + 1 d su1 + u2 + Á + uk d + dx dx

Sum Rule for

=

duk duk + 1 du1 du2 + + Á + + . dx dx dx dx

Eq. (1)

d su + yd dx

With these steps verified, the mathematical induction principle now guarantees the Sum Rule for every integer n Ú 2.

Exercises A.2 1. Assuming that the triangle inequality ƒ a + b ƒ … ƒ a ƒ + ƒ b ƒ holds for any two numbers a and b, show that ƒ x1 + x2 + Á + xn ƒ … ƒ x1 ƒ + ƒ x2 ƒ + Á + ƒ xn ƒ for any n numbers. 2. Show that if r Z 1 , then

6. Show that n! 7 n 3 if n is large enough. 7. Show that 2n 7 n 2 if n is large enough. 8. Show that 2n Ú 1>8 for n Ú -3 . 9. Sums of squares Show that the sum of the squares of the first n positive integers is n an +

1 - rn+1 1 + r + r + Á + rn = 1 - r 2

for every positive integer n. dy du d suyd = u + y , and the fact that 3. Use the Product Rule, dx dx dx d d n sxd = 1 to show that sx d = nx n - 1 for every positive intedx dx ger n. 4. Suppose that a function ƒ(x) has the property that ƒsx1 x2 d = ƒsx1 d + ƒsx2 d for any two positive numbers x1 and x2 . Show that ƒsx1 x2 Á xn d = ƒsx1 d + ƒsx2 d + Á + ƒsxn d for the product of any n positive numbers x1, x2, Á , xn . 5. Show that

1 bsn + 1d 2 . 3

10. Sums of cubes Show that the sum of the cubes of the first n positive integers is snsn + 1d>2d2 . 11. Rules for finite sums Show that the following finite sum rules hold for every positive integer n. (See Section 5.2.) n

n

n

a. a sak + bk d = a ak + a bk k=1

k=1

k=1

n

n

n

b. a sak - bk d = a ak - a bk k=1

k=1

c. a cak = c # a ak n

n

k=1

k=1

d. a ak = n # c

k=1

(any number c)

n

2 2 1 2 + 2 + Á + n = 1 - n 3 3 31 3 for all positive integers n.

k=1

(if ak has the constant value c)

12. Show that ƒ x n ƒ = ƒ x ƒ n for every positive integer n and every real number x.

AP-10

Appendices

Lines, Circles, and Parabolas

A.3

This section reviews coordinates, lines, distance, circles, and parabolas in the plane. The notion of increment is also discussed. y P(a, b)

b Positive y-axis

3 2

Negative x-axis –3

–2

–1

1

Origin

0 –1

Negative y-axis

1

2

a3

x

Positive x-axis

–2 –3

FIGURE A.4 Cartesian coordinates in the plane are based on two perpendicular axes intersecting at the origin.

HISTORICAL BIOGRAPHY René Descartes (1596–1650)

y (1, 3) 3 Second quadrant (, )

First quadrant (, )

2

1 (–2, 1)

(2, 1)

(0, 0) (1, 0)

–2

–1

0

1

2

x

(–2, –1) Third quadrant (, )

–1

Fourth quadrant (, )

–2 (1, –2)

FIGURE A.5 Points labeled in the xy-coordinate or Cartesian plane. The points on the axes all have coordinate pairs but are usually labeled with single real numbers, (so (1, 0) on the x-axis is labeled as 1). Notice the coordinate sign patterns of the quadrants.

Cartesian Coordinates in the Plane In Appendix 1 we identified the points on the line with real numbers by assigning them coordinates. Points in the plane can be identified with ordered pairs of real numbers. To begin, we draw two perpendicular coordinate lines that intersect at the 0-point of each line. These lines are called coordinate axes in the plane. On the horizontal x-axis, numbers are denoted by x and increase to the right. On the vertical y-axis, numbers are denoted by y and increase upward (Figure A.4). Thus “upward” and “to the right” are positive directions, whereas “downward” and “to the left” are considered as negative. The origin O, also labeled 0, of the coordinate system is the point in the plane where x and y are both zero. If P is any point in the plane, it can be located by exactly one ordered pair of real numbers in the following way. Draw lines through P perpendicular to the two coordinate axes. These lines intersect the axes at points with coordinates a and b (Figure A.4). The ordered pair (a, b) is assigned to the point P and is called its coordinate pair. The first number a is the x-coordinate (or abscissa) of P; the second number b is the y-coordinate (or ordinate) of P. The x-coordinate of every point on the y-axis is 0. The y-coordinate of every point on the x-axis is 0. The origin is the point (0, 0). Starting with an ordered pair (a, b), we can reverse the process and arrive at a corresponding point P in the plane. Often we identify P with the ordered pair and write P(a, b). We sometimes also refer to “the point (a, b)” and it will be clear from the context when (a, b) refers to a point in the plane and not to an open interval on the real line. Several points labeled by their coordinates are shown in Figure A.5. This coordinate system is called the rectangular coordinate system or Cartesian coordinate system (after the sixteenth-century French mathematician René Descartes). The coordinate axes of this coordinate or Cartesian plane divide the plane into four regions called quadrants, numbered counterclockwise as shown in Figure A.5. The graph of an equation or inequality in the variables x and y is the set of all points P(x, y) in the plane whose coordinates satisfy the equation or inequality. When we plot data in the coordinate plane or graph formulas whose variables have different units of measure, we do not need to use the same scale on the two axes. If we plot time vs. thrust for a rocket motor, for example, there is no reason to place the mark that shows 1 sec on the time axis the same distance from the origin as the mark that shows 1 lb on the thrust axis. Usually when we graph functions whose variables do not represent physical measurements and when we draw figures in the coordinate plane to study their geometry and trigonometry, we try to make the scales on the axes identical. A vertical unit of distance then looks the same as a horizontal unit. As on a surveyor’s map or a scale drawing, line segments that are supposed to have the same length will look as if they do and angles that are supposed to be congruent will look congruent. Computer displays and calculator displays are another matter. The vertical and horizontal scales on machine-generated graphs usually differ, and there are corresponding distortions in distances, slopes, and angles. Circles may look like ellipses, rectangles may look like squares, right angles may appear to be acute or obtuse, and so on. We discuss these displays and distortions in greater detail in Section 1.4.

Increments and Straight Lines When a particle moves from one point in the plane to another, the net changes in its coordinates are called increments. They are calculated by subtracting the coordinates of the

Appendix 3

C(5, 6) 6 B(2, 5)

4

In going from the point As4, -3d to the point B(2, 5) the increments in the x- and y-coordinates are

EXAMPLE 1

y –5, x 0

3

¢x = 2 - 4 = -2,

2 y 8

1

1

2

3

4

x

5

¢y = 5 - s -3d = 8.

From C(5, 6) to D(5, 1) the coordinate increments are

D(5, 1)

0

AP-11

starting point from the coordinates of the ending point. If x changes from x1 to x2 , the increment in x is ¢x = x2 - x1 .

y

5

Lines, Circles, and Parabolas

¢x = 5 - 5 = 0,

–1

¢y = 1 - 6 = -5.

See Figure A.6.

–2 –3

(2, –3)

A(4, –3) x –2

FIGURE A.6 Coordinate increments may be positive, negative, or zero (Example 1).

Given two points P1sx1, y1 d and P2sx2, y2 d in the plane, we call the increments ¢x = x2 - x1 and ¢y = y2 - y1 the run and the rise, respectively, between P1 and P2 . Two such points always determine a unique straight line (usually called simply a line) passing through them both. We call the line P1 P2 . Any nonvertical line in the plane has the property that the ratio ¢y y2 - y1 rise m = run = = x2 - x1 ¢x

y P2

L

has the same value for every choice of the two points P1sx1, y1 d and P2sx2, y2 d on the line (Figure A.7). This is because the ratios of corresponding sides for similar triangles are equal.

P2 (x2, y2) y (rise)

y

DEFINITION

¢y y2 - y1 rise m = run = = x2 - x1 ¢x

P1 (x1, y1) x (run) P1 x 0

The constant ratio

Q(x2, y1)

is the slope of the nonvertical line P1 P2 .

Q x

FIGURE A.7 Triangles P1 QP2 and P1 ¿Q¿P2 ¿ are similar, so the ratio of their sides has the same value for any two points on the line. This common value is the line’s slope.

The slope tells us the direction (uphill, downhill) and steepness of a line. A line with positive slope rises uphill to the right; one with negative slope falls downhill to the right (Figure A.8). The greater the absolute value of the slope, the more rapid the rise or fall. The slope of a vertical line is undefined. Since the run ¢x is zero for a vertical line, we cannot form the slope ratio m. The direction and steepness of a line can also be measured with an angle. The angle of inclination of a line that crosses the x-axis is the smallest counterclockwise angle from the x-axis to the line (Figure A.9). The inclination of a horizontal line is 0°. The inclination of a vertical line is 90°. If f (the Greek letter phi) is the inclination of a line, then 0 … f 6 180°. The relationship between the slope m of a nonvertical line and the line’s angle of inclination f is shown in Figure A.10: m = tan f. Straight lines have relatively simple equations. All points on the vertical line through the point a on the x-axis have x-coordinates equal to a. Thus, x = a is an equation for the vertical line. Similarly, y = b is an equation for the horizontal line meeting the y-axis at b. (See Figure A.11.) We can write an equation for a nonvertical straight line L if we know its slope m and the coordinates of one point P1sx1, y1 d on it. If P(x, y) is any other point on L, then we can

AP-12

Appendices

use the two points P1 and P to compute the slope, y - y1 m = x - x1 so that

y L1 6 L2

P4(3, 6)

P1(0, 5)

y - y1 = msx - x1 d,

4 3

P2(4, 2)

2

The equation

1 0 –1

y = y1 + msx - x1 d.

or

1

2

3

4

5

6

x

y = y1 + msx - x1 d is the point-slope equation of the line that passes through the point sx1, y1 d and has slope m.

P3(0, –2)

FIGURE A.8 The slope of L1 is ¢y 6 - s -2d 8 m = = = . 3 - 0 3 ¢x That is, y increases 8 units every time x increases 3 units. The slope of L2 is ¢y -3 2 - 5 m = = . = 4 - 0 4 ¢x That is, y decreases 3 units every time x increases 4 units.

y P2

L y

P1

this

m

this x

x

not this

Solution

y tan x x

not this

FIGURE A.9 Angles of inclination are measured counterclockwise from the x-axis.

EXAMPLE 2

x

FIGURE A.10 The slope of a nonvertical line is the tangent of its angle of inclination.

Write an equation for the line through the point (2, 3) with slope -3>2.

We substitute x1 = 2, y1 = 3, and m = -3>2 into the point-slope equation

and obtain y = 3 -

3 (x - 2), 2

or

y = -

3 x + 6. 2

When x = 0, y = 6 so the line intersects the y-axis at y = 6.

EXAMPLE 3

y Along this line, x2

6

Solution

Write an equation for the line through s -2, -1d and (3, 4).

The line’s slope is m =

5 Along this line, y3

4 3

We can use this slope with either of the two given points in the point-slope equation:

(2, 3)

2 1 0

-5 -1 - 4 = = 1. -5 -2 - 3

1

2

3

4

x

FIGURE A.11 The standard equations for the vertical and horizontal lines through (2, 3) are x = 2 and y = 3 .

With xx1 , y1c x2, 1c

With xx1 , y1c x3, 4c

y = -1 + x + 2

y = 4 + x - 3

y = -1 + 1 # sx - s -2dd

y = 4 + 1 # sx - 3d

y = x + 1

y = x + 1 Same result

Either way, y = x + 1 is an equation for the line (Figure A.12).

Appendix 3

(3, 4) yx1

y = b + msx - 0d, –2

0 –1 (–2, –1)

1

2

3

AP-13

The y-coordinate of the point where a nonvertical line intersects the y-axis is called the y-intercept of the line. Similarly, the x-intercept of a nonhorizontal line is the x-coordinate of the point where it crosses the x-axis (Figure A.13). A line with slope m and y-intercept b passes through the point (0, b), so it has equation

y 4

Lines, Circles, and Parabolas

y = mx + b.

or, more simply,

x

The equation y = mx + b

FIGURE A.12 The line in Example 3.

is called the slope-intercept equation of the line with slope m and y-intercept b.

Lines with equations of the form y = mx have y-intercept 0 and so pass through the origin. Equations of lines are called linear equations. The equation

y

Ax + By = C

b

sA and B not both 0d

is called the general linear equation in x and y because its graph always represents a line and every line has an equation in this form (including lines with undefined slope).

L

0

a

Parallel and Perpendicular Lines

x

Lines that are parallel have equal angles of inclination, so they have the same slope (if they are not vertical). Conversely, lines with equal slopes have equal angles of inclination and so are parallel. If two nonvertical lines L1 and L2 are perpendicular, their slopes m1 and m2 satisfy m1 m2 = -1, so each slope is the negative reciprocal of the other:

FIGURE A.13 Line L has x-intercept a and y-intercept b.

1 m1 = - m2 , y

To see this, notice by inspecting similar triangles in Figure A.14 that m1 = a>h, and m2 = -h>a. Hence, m1 m2 = sa>hds -h>ad = -1.

L1

L2

1 m2 = - m1 .

C

0

A

D

2 a

B

FIGURE A.14 ¢ADC is similar to ¢CDB . Hence f1 is also the upper angle in ¢CDB . From the sides of ¢CDB , we read tan f1 = a>h .

x

The distance between points in the plane is calculated with a formula that comes from the Pythagorean theorem (Figure A.15). y

This distance is

x2 – x12 y2 – y12 (x2 – x1)2 (y2 – y1)2

d y2

y1

Q(x2 , y2)

P(x1, y1) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

1

Distance and Circles in the Plane

Slope m 2

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1 h

Slope m1

y2 – y1

C(x2 , y1)

x2 – x1 0

x1

x2

x

FIGURE A.15 To calculate the distance between Psx1 , y1 d and Qsx2 , y2 d , apply the Pythagorean theorem to triangle PCQ.

AP-14

Appendices

Distance Formula for Points in the Plane The distance between Psx1 , y1 d and Qsx2 , y2 d is d = 2s¢xd2 + s¢yd2 = 2sx2 - x1 d2 + s y2 - y1 d2 .

EXAMPLE 4

y

(a) The distance between Ps -1, 2d and Q(3, 4) is P(x, y)

2s3 - s -1dd2 + s4 - 2d2 = 2s4d2 + s2d2 = 220 = 24 # 5 = 225 .

a

(b) The distance from the origin to P(x, y) is

C(h, k)

2sx - 0d2 + s y - 0d2 = 2x 2 + y 2 . (x h) 2 (y k) 2 a 2 0

x

By definition, a circle of radius a is the set of all points P(x, y) whose distance from some center C(h, k) equals a (Figure A.16). From the distance formula, P lies on the circle if and only if

FIGURE A.16 A circle of radius a in the xy-plane, with center at (h, k).

2sx - hd2 + s y - kd2 = a, so

(x - h) 2 + (y - k) 2 = a 2.

(1)

Equation (1) is the standard equation of a circle with center (h, k) and radius a. The circle of radius a = 1 and centered at the origin is the unit circle with equation x 2 + y 2 = 1.

EXAMPLE 5 (a) The standard equation for the circle of radius 2 centered at (3, 4) is sx - 3d2 + s y - 4d2 = 22 = 4 . (b) The circle sx - 1d2 + s y + 5d2 = 3 has h = 1, k = -5, and a = 23. The center is the point sh, kd = s1, -5d and the radius is a = 23. If an equation for a circle is not in standard form, we can find the circle’s center and radius by first converting the equation to standard form. The algebraic technique for doing so is completing the square.

EXAMPLE 6

Find the center and radius of the circle x 2 + y 2 + 4x - 6y - 3 = 0.

Appendix 3 Solution

Lines, Circles, and Parabolas

We convert the equation to standard form by completing the squares in x and y:

x + y 2 + 4x - 6y - 3 = 0 2

Start with the given equation. Gather terms. Move the constant to the right-hand side.

sx 2 + 4xd + s y 2 - 6yd = 3 2

2

-6 4 ax 2 + 4x + a b b + ay 2 - 6y + a b b = 2 2

y

AP-15

2

-6 4 3 + a b + a b 2 2

Exterior: (x h) 2 (y k) 2 a 2 On: (x h)2 (y k)2 a2

2

Add the square of half the coefficient of x to each side of the equation. Do the same for y. The parenthetical expressions on the left-hand side are now perfect squares.

sx 2 + 4x + 4d + s y 2 - 6y + 9d = 3 + 4 + 9 a

k

Write each quadratic as a squared linear expression.

sx + 2d2 + s y - 3d2 = 16

(h, k)

The center is s -2, 3d and the radius is a = 4. The points (x, y) satisfying the inequality Interior: (x h) 2 ( y k) 2 a 2 x

h

0

sx - hd2 + s y - kd2 6 a 2 make up the interior region of the circle with center (h, k) and radius a (Figure A.17). The circle’s exterior consists of the points (x, y) satisfying

FIGURE A.17 The interior and exterior of the circle sx - hd2 + s y - kd2 = a 2 . y (–2, 4)

Parabolas y x2 (2, 4)

4

The geometric definition and properties of general parabolas are reviewed in Appendix 4. Here we look at parabolas arising as the graphs of equations of the form y = ax 2 + bx + c. Consider the equation y = x 2 . Some points whose coordinates satisfy this 3 9 equation are s0, 0d, s1, 1d, a , b, s -1, 1d, s2, 4d, and s -2, 4d. These points (and all oth2 4 ers satisfying the equation) make up a smooth curve called a parabola (Figure A.18).

EXAMPLE 7

⎛3 , 9⎛ ⎝2 4⎝ (–1, 1)

1

–2

0

–1

sx - hd2 + s y - kd2 7 a 2 .

(1, 1)

1

2

x

FIGURE A.18 The parabola y = x 2 (Example 7).

The graph of an equation of the form y = ax 2 is a parabola whose axis (axis of symmetry) is the y-axis. The parabola’s vertex (point where the parabola and axis cross) lies at the origin. The parabola opens upward if a 7 0 and downward if a 6 0. The larger the value of ƒ a ƒ , the narrower the parabola (Figure A.19). Generally, the graph of y = ax 2 + bx + c is a shifted and scaled version of the parabola y = x 2 . We discuss shifting and scaling of graphs in more detail in Section 1.2. The Graph of y ax 2 bx c, a 0 The graph of the equation y = ax 2 + bx + c, a Z 0, is a parabola. The parabola opens upward if a 7 0 and downward if a 6 0. The axis is the line x = -

b . 2a

(2)

The vertex of the parabola is the point where the axis and parabola intersect. Its x-coordinate is x = -b>2a; its y-coordinate is found by substituting x = -b>2a in the parabola’s equation.

AP-16

Appendices

Notice that if a = 0, then we have y = bx + c, which is an equation for a line. The axis, given by Equation (2), can be found by completing the square.

y

symmetry

y 2x 2

y

x2 2

EXAMPLE 8

x2 10

Solution

y

Graph the equation y = -

1 2 x - x + 4. 2

Comparing the equation with y = ax 2 + bx + c we see that

1 –4 –3 –2

2

3

1 a = - , 2

x

4

Axis of

–1

Vertex at origin

x = -

FIGURE A.19 Besides determining the direction in which the parabola y = ax 2 opens, the number a is a scaling factor. The parabola widens as a approaches zero and narrows as ƒ a ƒ becomes large.

When x = -1, we have y = -

9 1 s -1d2 - s -1d + 4 = . 2 2

The vertex is s -1, 9>2d. The x-intercepts are where y = 0: -

Intercept at y 4

(–2, 4) 3

1 2 x - x + 4 = 0 2

x 2 + 2x - 8 = 0 sx - 2dsx + 4d = 0 x = 2, x = -4

(0, 4) Axis: x –1

s -1d b = = -1. 2a 2s -1>2d

y

Point symmetric with y-intercept

–3 –2

c = 4.

Since a 6 0, the parabola opens downward. From Equation (2) the axis is the vertical line

2

y –x 6 y –x 2

Vertex is ⎛–1, 9⎛ ⎝ 2⎝

b = -1,

1 y – x2 x 4 2

2 1 0

1

x

We plot some points, sketch the axis, and use the direction of opening to complete the graph in Figure A.20.

Intercepts at x –4 and x 2

FIGURE A.20 The parabola in Example 8.

Exercises A.3 Distance, Slopes, and Lines In Exercises 1 and 2, a particle moves from A to B in the coordinate plane. Find the increments ¢x and ¢y in the particle’s coordinates. Also find the distance from A to B. 1. As -3, 2d,

Bs -1, - 2d

2. As -3.2, - 2d,

Bs -8.1, -2d

Describe the graphs of the equations in Exercises 3 and 4. 3. x 2 + y 2 = 1

4. x 2 + y 2 … 3

Plot the points in Exercises 5 and 6 and find the slope (if any) of the line they determine. Also find the common slope (if any) of the lines perpendicular to line AB. 5. As -1, 2d,

Bs -2, -1d

6. As2, 3d,

Bs -1, 3d

In Exercises 7 and 8, find an equation for (a) the vertical line and (b) the horizontal line through the given point. 7. s -1, 4>3d

8. A 0, - 22 B

Appendix 3 In Exercises 9–15, write an equation for each line described. 9. Passes through s -1, 1d with slope -1 10. Passes through (3, 4) and s -2, 5d

x2 + y2 = 1

39. y = - x , 2

12. Passes through s -12, -9d and has slope 0 14. Passes through s5, -1d and is parallel to the line 2x + 5y = 15 15. Passes through (4, 10) and is perpendicular to the line 6x - 3y = 5

38. y - x = 1,

y = x2

y = 2x - 1 2

40. x + y = 1, 2

13. Has y-intercept 4 and x-intercept -1

AP-17

Theory and Examples In Exercises 37–40, graph the two equations and find the points at which the graphs intersect. 37. y = 2x,

11. Has slope -5>4 and y-intercept 6

Lines, Circles, and Parabolas

sx - 1d2 + y 2 = 1

2

41. Insulation By measuring slopes in the figure, estimate the temperature change in degrees per inch for (a) the gypsum wallboard; (b) the fiberglass insulation; (c) the wood sheathing. 80°

In Exercises 16 and 17, find the line’s x- and y-intercepts and use this information to graph the line. 17. 22x - 23y = 26

18. Is there anything special about the relationship between the lines Ax + By = C1 and Bx - Ay = C2 sA Z 0, B Z 0d ? Give reasons for your answer. 19. A particle starts at As -2, 3d and its coordinates change by increments ¢x = 5, ¢y = -6 . Find its new position. 20. The coordinates of a particle change by ¢x = 5 and ¢y = 6 as it moves from A(x, y) to Bs3, -3d . Find x and y. Circles In Exercises 21–23, find an equation for the circle with the given center C(h, k) and radius a. Then sketch the circle in the xy-plane. Include the circle’s center in your sketch. Also, label the circle’s x- and y-intercepts, if any, with their coordinate pairs. 21. Cs0, 2d,

a = 2

23. C A - 23, -2 B , a = 2

22. Cs -1, 5d,

a = 210

Fiberglass between studs

24. x 2 + y 2 + 4x - 4y + 4 = 0 26. x 2 + y 2 - 4x + 4y = 0

Parabolas Graph the parabolas in Exercises 27–30. Label the vertex, axis, and intercepts in each case. 27. y = x 2 - 2x - 3

28. y = -x 2 + 4x

29. y = -x 2 - 6x - 5

30. y =

50° Air

inside room 40° at 72°F

Air outside at 0°F

30° 20° 10° 0°

0

33. x + y 7 1, 2

2

1 2 x + x + 4 2

32. sx - 1d2 + y 2 … 4 x + y 6 4 2

34. x 2 + y 2 + 6y 6 0,

1

2 3 4 5 Distance through wall (inches)

6

7

42. Insulation According to the figure in Exercise 41, which of the materials is the best insulator? The poorest? Explain. 43. Pressure under water The pressure p experienced by a diver under water is related to the diver’s depth d by an equation of the form p = kd + 1 (k a constant). At the surface, the pressure is 1 atmosphere. The pressure at 100 meters is about 10.94 atmospheres. Find the pressure at 50 meters. 44. Reflected light A ray of light comes in along the line x + y = 1 from the second quadrant and reflects off the x-axis (see the accompanying figure). The angle of incidence is equal to the angle of reflection. Write an equation for the line along which the departing light travels.

Inequalities Describe the regions defined by the inequalities and pairs of inequalities in Exercises 31–34. 31. x 2 + y 2 7 7

Siding

The temperature changes in the wall in Exercises 41 and 42.

Graph the circles whose equations are given in Exercises 24–26. Label each circle’s center and intercepts (if any) with their coordinate pairs. 25. x 2 + y 2 - 3y - 4 = 0

60° Temperature (°F)

16. 3x + 4y = 12

Sheathing Gypsum wallboard

70°

2

y xy1 1

Angle of Angle of incidence reflection

0

1

y 7 -3

35. Write an inequality that describes the points that lie inside the circle with center s -2, 1d and radius 26 . 36. Write a pair of inequalities that describe the points that lie inside or on the circle with center (0, 0) and radius 22 , and on or to the right of the vertical line through (1, 0).

x

The path of the light ray in Exercise 44. Angles of incidence and reflection are measured from the perpendicular.

AP-18

Appendices

45. Fahrenheit vs. Celsius In the FC-plane, sketch the graph of the equation C =

48. Show that the triangle with vertices A(0, 0), B A 1, 23 B , and C(2, 0) is equilateral.

5 sF - 32d 9

linking Fahrenheit and Celsius temperatures. On the same graph sketch the line C = F . Is there a temperature at which a Celsius thermometer gives the same numerical reading as a Fahrenheit thermometer? If so, find it. 46. The Mt. Washington Cog Railway Civil engineers calculate the slope of roadbed as the ratio of the distance it rises or falls to the distance it runs horizontally. They call this ratio the grade of the roadbed, usually written as a percentage. Along the coast, commercial railroad grades are usually less than 2%. In the mountains, they may go as high as 4%. Highway grades are usually less than 5%. The steepest part of the Mt. Washington Cog Railway in New Hampshire has an exceptional 37.1% grade. Along this part of the track, the seats in the front of the car are 14 ft above those in the rear. About how far apart are the front and rear rows of seats?

A.4

47. By calculating the lengths of its sides, show that the triangle with vertices at the points A(1, 2), B(5, 5), and Cs4, -2d is isosceles but not equilateral.

49. Show that the points As2, -1d , B(1, 3), and Cs -3, 2d are vertices of a square, and find the fourth vertex. 50. Three different parallelograms have vertices at s -1, 1d , (2, 0), and (2, 3). Sketch them and find the coordinates of the fourth vertex of each. 51. For what value of k is the line 2x + ky = 3 perpendicular to the line 4x + y = 1 ? For what value of k are the lines parallel? 52. Midpoint of a line segment Show that the point with coordinates a

x1 + x2 y1 + y2 , b 2 2

is the midpoint of the line segment joining Psx1 , y1 d to Qsx2 , y2 d .

Conic Sections In this appendix we define and review parabolas, ellipses, and hyperbolas geometrically and derive their standard Cartesian equations. These curves are called conic sections or conics because they are formed by cutting a double cone with a plane (Figure A.21). This geometry method was the only way they could be described by Greek mathematicians who did not have our tools of Cartesian or polar coordinates.

Parabolas DEFINITIONS A set that consists of all the points in a plane equidistant from a given fixed point and a given fixed line in the plane is a parabola. The fixed point is the focus of the parabola. The fixed line is the directrix.

If the focus F lies on the directrix L, the parabola is the line through F perpendicular to L. We consider this to be a degenerate case and assume henceforth that F does not lie on L. A parabola has its simplest equation when its focus and directrix straddle one of the coordinate axes. For example, suppose that the focus lies at the point F(0, p) on the positive y-axis and that the directrix is the line y = -p (Figure A.22). In the notation of the figure, a point P(x, y) lies on the parabola if and only if PF = PQ. From the distance formula, PF = 2sx - 0d2 + s y - pd2 = 2x 2 + s y - pd2 PQ = 2sx - xd2 + ( y - s -pdd2 = 2s y + pd2 .

Appendix 4

Ellipse: plane oblique to cone axis

Circle: plane perpendicular to cone axis

Conic Sections

Parabola: plane parallel to side of cone

AP-19

Hyperbola: plane cuts both halves of cone

(a)

Point: plane through cone vertex only

Pair of intersecting lines

Single line: plane tangent to cone (b)

FIGURE A.21 The standard conic sections (a) are the curves in which a plane cuts a double cone. Hyperbolas come in two parts, called branches. The point and lines obtained by passing the plane through the cone’s vertex (b) are degenerate conic sections.

When we equate these expressions, square, and simplify, we get

y

y =

x 2 4py Focus p

The vertex lies halfway between p directrix and focus. Directrix: y –p

x2 4p

or

x 2 = 4py.

Standard form

(1)

F(0, p) P(x, y) x Q(x, –p)

L

FIGURE A.22 The standard form of the parabola x 2 = 4py, p 7 0 .

These equations reveal the parabola’s symmetry about the y-axis. We call the y-axis the axis of the parabola (short for “axis of symmetry”). The point where a parabola crosses its axis is the vertex. The vertex of the parabola x 2 = 4py lies at the origin (Figure A.22). The positive number p is the parabola’s focal length. If the parabola opens downward, with its focus at s0, -pd and its directrix the line y = p , then Equations (1) become y = -

x2 4p

and

x 2 = -4py.

By interchanging the variables x and y, we obtain similar equations for parabolas opening to the right or to the left (Figure A.23).

AP-20

Appendices y

y Directrix x –p

Directrix xp

y2 –4px

y2 4px

Vertex

Vertex Focus

Focus

x

0 F( p, 0)

F(–p, 0) 0

x

(b)

(a)

FIGURE A.23 (a) The parabola y 2 = 4px . (b) The parabola y 2 = -4px .

EXAMPLE 1 Solution

Find the focus and directrix of the parabola y 2 = 10x.

We find the value of p in the standard equation y 2 = 4px: 4p = 10,

p =

so

5 10 = . 4 2

Then we find the focus and directrix for this value of p: Focus: Directrix: Vertex

Focus Center

Focus

5 s p, 0d = a , 0b 2 x = -p

or

x = -

5 . 2

Vertex

Ellipses Focal axis

FIGURE A.24 Points on the focal axis of an ellipse. y

DEFINITIONS An ellipse is the set of points in a plane whose distances from two fixed points in the plane have a constant sum. The two fixed points are the foci of the ellipse. The line through the foci of an ellipse is the ellipse’s focal axis. The point on the axis halfway between the foci is the center. The points where the focal axis and ellipse cross are the ellipse’s vertices (Figure A.24).

b P(x, y) Focus F1(–c, 0)

Focus 0 Center F2(c, 0)

a

x

FIGURE A.25 The ellipse defined by the equation PF1 + PF2 = 2a is the graph of the equation sx 2>a 2 d + s y 2>b 2 d = 1, where b 2 = a 2 - c 2.

If the foci are F1s -c, 0d and F2sc, 0d (Figure A.25), and PF1 + PF2 is denoted by 2a, then the coordinates of a point P on the ellipse satisfy the equation 2sx + cd2 + y 2 + 2sx - cd2 + y 2 = 2a . To simplify this equation, we move the second radical to the right-hand side, square, isolate the remaining radical, and square again, obtaining 2

y x2 + 2 = 1. 2 a a - c2

(2)

Appendix 4

Conic Sections

AP-21

Since PF1 + PF2 is greater than the length F1 F2 (by the triangle inequality for triangle PF1 F2), the number 2a is greater than 2c. Accordingly, a 7 c and the number a 2 - c 2 in Equation (2) is positive. The algebraic steps leading to Equation (2) can be reversed to show that every point P whose coordinates satisfy an equation of this form with 0 6 c 6 a also satisfies the equation PF1 + PF2 = 2a. A point therefore lies on the ellipse if and only if its coordinates satisfy Equation (2). If b = 2a 2 - c 2 ,

(3)

then a 2 - c 2 = b 2 and Equation (2) takes the form y2 x2 + = 1. a2 b2

(4)

Equation (4) reveals that this ellipse is symmetric with respect to the origin and both coordinate axes. It lies inside the rectangle bounded by the lines x = ;a and y = ;b. It crosses the axes at the points s ;a, 0d and s0, ;bd. The tangents at these points are perpendicular to the axes because dy b 2x = - 2 dx a y

is zero if x = 0 and infinite if y = 0. The major axis of the ellipse in Equation (4) is the line segment of length 2a joining the points s ;a, 0d. The minor axis is the line segment of length 2b joining the points s0, ;bd. The number a itself is the semimajor axis, the number b the semiminor axis. The number c, found from Equation (3) as

y y2 x2 1 (0, 3) 16 9 Vertex (–4, 0) Focus (–7, 0)

c = 2a 2 - b 2 ,

Vertex (4, 0) Focus

0

Obtained from Eq. (4) by implicit differentiation

is the center-to-focus distance of the ellipse. If a = b, the ellipse is a circle. x

(7, 0) Center

EXAMPLE 2

The ellipse y2 x2 + = 1 16 9

(0, –3)

FIGURE A.26 An ellipse with its major axis horizontal (Example 2).

(5)

(Figure A.26) has Semimajor axis:

a = 216 = 4,

Center-to-focus distance: Foci:

s ;c, 0d =

Semiminor axis:

b = 29 = 3

c = 216 - 9 = 27

A ; 27, 0 B

Vertices: s ;a, 0d = s ;4, 0d Center: s0, 0d. If we interchange x and y in Equation (5), we have the equation y2 x2 + = 1. 9 16

(6)

The major axis of this ellipse is now vertical instead of horizontal, with the foci and vertices on the y-axis. There is no confusion in analyzing Equations (5) and (6). If we find the intercepts on the coordinate axes, we will know which way the major axis runs because it is the longer of the two axes.

AP-22

Appendices

Standard-Form Equations for Ellipses Centered at the Origin Foci on the x-axis:

y2 x2 + = 1 a2 b2

sa 7 bd

Center-to-focus distance: Foci: s ;c, 0d Vertices: s ;a, 0d Foci on the y-axis:

y2 x2 + = 1 b2 a2

c = 2a 2 - b 2

sa 7 bd

Center-to-focus distance: Foci: s0, ;cd Vertices: s0, ;ad

c = 2a 2 - b 2

In each case, a is the semimajor axis and b is the semiminor axis.

Hyperbolas

Vertices Focus

Focus Center Focal axis

FIGURE A.27 Points on the focal axis of a hyperbola.

DEFINITIONS A hyperbola is the set of points in a plane whose distances from two fixed points in the plane have a constant difference. The two fixed points are the foci of the hyperbola. The line through the foci of a hyperbola is the focal axis. The point on the axis halfway between the foci is the hyperbola’s center. The points where the focal axis and hyperbola cross are the vertices (Figure A.27).

If the foci are F1s -c, 0d and F2sc, 0d (Figure A.28) and the constant difference is 2a, then a point (x, y) lies on the hyperbola if and only if 2sx + cd2 + y 2 - 2sx - cd2 + y 2 = ;2a .

y x –a

To simplify this equation, we move the second radical to the right-hand side, square, isolate the remaining radical, and square again, obtaining

xa P(x, y)

F1(–c, 0)

0

(7)

x F2(c, 0)

FIGURE A.28 Hyperbolas have two branches. For points on the right-hand branch of the hyperbola shown here, PF1 - PF2 = 2a . For points on the lefthand branch, PF2 - PF1 = 2a . We then let b = 2c 2 - a 2.

2

y x2 + 2 = 1. 2 a a - c2

(8)

So far, this looks just like the equation for an ellipse. But now a 2 - c 2 is negative because 2a, being the difference of two sides of triangle PF1 F2 , is less than 2c, the third side. The algebraic steps leading to Equation (8) can be reversed to show that every point P whose coordinates satisfy an equation of this form with 0 6 a 6 c also satisfies Equation (7). A point therefore lies on the hyperbola if and only if its coordinates satisfy Equation (8). If we let b denote the positive square root of c 2 - a 2 , b = 2c 2 - a 2 ,

(9)

then a 2 - c 2 = -b 2 and Equation (8) takes the more compact form y2 x2 = 1. a2 b2

(10)

Appendix 4

Conic Sections

AP-23

The differences between Equation (10) and the equation for an ellipse (Equation 4) are the minus sign and the new relation c2 = a2 + b2.

From Eq. (9)

Like the ellipse, the hyperbola is symmetric with respect to the origin and coordinate axes. It crosses the x-axis at the points s ;a, 0d. The tangents at these points are vertical because dy b 2x = 2 dx a y

Obtained from Eq. (10) by implicit differentiation

is infinite when y = 0. The hyperbola has no y-intercepts; in fact, no part of the curve lies between the lines x = -a and x = a. The lines b y = ;a x are the two asymptotes of the hyperbola defined by Equation (10). The fastest way to find the equations of the asymptotes is to replace the 1 in Equation (10) by 0 and solve the new equation for y: y2 y2 x2 x2 b = 1 : = 0 : y = ; a x. 2 2 2 2 a b a b ('')''* ('')''* (')'* hyperbola

EXAMPLE 3

y y – 5 x 2

F(3, 0)

y2 x2 = 1 5 4

(11)

is Equation (10) with a 2 = 4 and b 2 = 5 (Figure A.29). We have Center-to-focus distance: c = 2a 2 + b 2 = 24 + 5 = 3 Foci: s ;c, 0d = s ;3, 0d, Vertices: s ;a, 0d = s ;2, 0d Center: s0, 0d

x

2

asymptotes

The equation

y 5 x 2 y2 x2 1 4 5

F(–3, 0) –2

0 for 1

Asymptotes: FIGURE A.29 The hyperbola and its asymptotes in Example 3.

y2 x2 = 0 5 4

y = ;

or

25 x. 2

If we interchange x and y in Equation (11), the foci and vertices of the resulting hyperbola will lie along the y-axis. We still find the asymptotes in the same way as before, but now their equations will be y = ;2x> 25.

Standard-Form Equations for Hyperbolas Centered at the Origin Foci on the x-axis:

y2 x2 = 1 a2 b2

Center-to-focus distance: Foci:

s ;c, 0d

Vertices:

c = 2a 2 + b 2

Foci on the y-axis:

x2 = 1 b2 c = 2a 2 + b 2

s0, ;cd

Vertices: 2

a

-

2

Center-to-focus distance: Foci:

s ;a, 0d

y2

s0, ;ad

y b x2 Asymptotes: Asymptotes: - 2 = 0 or y = ; a x 2 a b Notice the difference in the asymptote equations (b> a in the first, a> b in the second).

y2 a

2

-

x2 = 0 b2

or

a y = ; x b

AP-24

Appendices

We shift conics using the principles reviewed in Section 1.2, replacing x by x + h and y by y + k . Show that the equation x 2 - 4y 2 + 2x + 8y - 7 = 0 represents a hyperbola. Find its center, asymptotes, and foci.

EXAMPLE 4

Solution

We reduce the equation to standard form by completing the square in x and y as

follows: (x 2 + 2x) - 4( y 2 - 2y) = 7 (x 2 + 2x + 1) - 4( y 2 - 2y + 1) = 7 + 1 - 4 (x + 1) 2 - ( y - 1) 2 = 1. 4 This is the standard form Equation (10) of a hyperbola with x replaced by x + 1 and y replaced by y - 1. The hyperbola is shifted one unit to the left and one unit upward, and it has center x + 1 = 0 and y - 1 = 0, or x = -1 and y = 1. Moreover, a 2 = 4,

b 2 = 1,

c 2 = a 2 + b 2 = 5,

so the asymptotes are the two lines x + 1 - ( y - 1) = 0 2

x + 1 + ( y - 1) = 0. 2

and

The shifted foci have coordinates A -1 ; 25, 1 B .

Exercises A.4 Identifying Graphs Match the parabolas in Exercises 1– 4 with the following equations: x = 2y,

x = -6y,

2

2

y = 8x, 2

y = -4x . 2

Then find each parabola’s focus and directrix. 1.

y

2.

y

x

3.

x

4.

y

Match each conic section in Exercises 5–8 with one of these equations: y2 x2 + = 1, 4 9

x2 + y 2 = 1, 2

y2 - x 2 = 1, 4

y2 x2 = 1. 4 9

Then find the conic section’s foci and vertices. If the conic section is a hyperbola, find its asymptotes as well. y y 5. 6.

y x

x

x

x

Appendix 4 7.

AP-25

Shifting Conic Sections You may wish to review Section 1.2 before solving Exercises 39–56.

8. y

y

Conic Sections

39. The parabola y 2 = 8x is shifted down 2 units and right 1 unit to generate the parabola s y + 2d2 = 8sx - 1d . a. Find the new parabola’s vertex, focus, and directrix.

x

x

b. Plot the new vertex, focus, and directrix, and sketch in the parabola. 40. The parabola x 2 = -4y is shifted left 1 unit and up 3 units to generate the parabola sx + 1d2 = -4s y - 3d .

Parabolas Exercises 9–16 give equations of parabolas. Find each parabola’s focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. 9. y 2 = 12x

10. x 2 = 6y

11. x 2 = -8y

12. y 2 = -2x

13. y = 4x 2

14. y = -8x 2

15. x = -3y 2

16. x = 2y 2

a. Find the new parabola’s vertex, focus, and directrix. b. Plot the new vertex, focus, and directrix, and sketch in the parabola.

41. The ellipse sx 2>16d + s y 2>9d = 1 is shifted 4 units to the right and 3 units up to generate the ellipse s y - 3d2 sx - 4d2 + = 1. 16 9 a. Find the foci, vertices, and center of the new ellipse.

Ellipses Exercises 17–24 give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. 17. 16x 2 + 25y 2 = 400

18. 7x 2 + 16y 2 = 112

19. 2x 2 + y 2 = 2

20. 2x 2 + y 2 = 4

21. 3x + 2y = 6

22. 9x + 10y = 90

23. 6x 2 + 9y 2 = 54

24. 169x 2 + 25y 2 = 4225

2

2

2

42. The ellipse sx 2>9d + s y 2>25d = 1 is shifted 3 units to the left and 2 units down to generate the ellipse s y + 2d2 sx + 3d2 + = 1. 9 25

2

a. Find the foci, vertices, and center of the new ellipse.

Exercises 25 and 26 give information about the foci and vertices of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation from the given information. 25. Foci: A ; 22, 0 B

b. Plot the new foci, vertices, and center, and sketch in the new ellipse.

Vertices: s ;2, 0d

b. Plot the new foci, vertices, and center, and sketch in the new ellipse.

43. The hyperbola sx 2>16d - s y 2>9d = 1 is shifted 2 units to the right to generate the hyperbola y2 sx - 2d2 = 1. 16 9

26. Foci: s0, ;4d Vertices: s0, ;5d Hyperbolas Exercises 27–34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola’s asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch. 27. x 2 - y 2 = 1

28. 9x 2 - 16y 2 = 144

29. y 2 - x 2 = 8

30. y 2 - x 2 = 4

31. 8x 2 - 2y 2 = 16

32. y 2 - 3x 2 = 3

33. 8y 2 - 2x 2 = 16

34. 64x 2 - 36y 2 = 2304

A 0, ; 22 B

Asymptotes:

y = ;x

37. Vertices: s ;3, 0d Asymptotes:

4 y = ; x 3

36. Foci:

s ;2, 0d

Asymptotes:

y = ;

1 23

38. Vertices: s0, ;2d Asymptotes:

b. Plot the new center, foci, vertices, and asymptotes, and sketch in the hyperbola.

44. The hyperbola s y 2>4d - sx 2>5d = 1 is shifted 2 units down to generate the hyperbola s y + 2d2 x2 = 1. 5 4

Exercises 35–38 give information about the foci, vertices, and asymptotes of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation from the information given. 35. Foci:

a. Find the center, foci, vertices, and asymptotes of the new hyperbola.

1 y = ; x 2

x

a. Find the center, foci, vertices, and asymptotes of the new hyperbola. b. Plot the new center, foci, vertices, and asymptotes, and sketch in the hyperbola. Exercises 45–48 give equations for parabolas and tell how many units up or down and to the right or left each parabola is to be shifted. Find an equation for the new parabola, and find the new vertex, focus, and directrix. 45. y 2 = 4x, 47. x = 8y, 2

left 2 , down 3

46. y 2 = -12x,

right 1 , down 7

48. x = 6y, 2

right 4 , up 3

left 3 , down 2

AP-26

Appendices

Exercises 49–52 give equations for ellipses and tell how many units up or down and to the right or left each ellipse is to be shifted. Find an equation for the new ellipse, and find the new foci, vertices, and center. y2 x2 49. + = 1, 6 9

56.

x2 + y 2 = 1, 2

right 3, up 4

51.

y2 x2 + = 1, 3 2

right 2, up 3

52.

y x + = 1, 16 25

y2 x2 = 1, left 2, down 1 16 9

55. y 2 - x 2 = 1,

left 2, down 1

50.

2

54.

y2 - x 2 = 1, 3

left 1, down 1 right 1, up 3

Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57–68. 57. x 2 + 4x + y 2 = 12

2

58. 2x 2 + 2y 2 - 28x + 12y + 114 = 0

left 4, down 5

Exercises 53–56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes. y2 x2 53. = 1, right 2, up 2 5 4

A.5

59. x 2 + 2x + 4y - 3 = 0

60. y 2 - 4y - 8x - 12 = 0

61. x 2 + 5y 2 + 4x = 1

62. 9x 2 + 6y 2 + 36y = 0

63. x 2 + 2y 2 - 2x - 4y = -1 64. 4x 2 + y 2 + 8x - 2y = -1 65. x 2 - y 2 - 2x + 4y = 4

66. x 2 - y 2 + 4x - 6y = 6

67. 2x 2 - y 2 + 6y = 3

68. y 2 - 4x 2 + 16x = 24

Proofs of Limit Theorems This appendix proves Theorem 1, Parts 2–5, and Theorem 4 from Section 2.2.

THEOREM 1—Limit Laws

If L, M, c, and k are real numbers and

lim ƒsxd = L and

x:c

1. Sum Rule: 2. Difference Rule: 3. Constant Multiple Rule: 4. Product Rule: 5. Quotient Rule: 6. Power Rule: 7. Root Rule:

lim gsxd = M,

x:c

then

lim sƒsxd + gsxdd = L + M

x:c

lim sƒsxd - gsxdd = L - M

x:c

lim sk # ƒsxdd = k # L

x:c

lim sƒsxd # gsxdd = L # M

x:c

ƒsxd L = , M x:c gsxd lim

M Z 0

lim [ƒ(x)]n = L n, n a positive integer

x:c

n

n

lim 2ƒ(x) = 2L = L 1/n, n a positive integer

x:c

(If n is even, we assume that lim ƒ(x) = L 7 0.) x:c

We proved the Sum Rule in Section 2.3 and the Power and Root Rules are proved in more advanced texts. We obtain the Difference Rule by replacing gsxd by -gsxd and M by -M in the Sum Rule. The Constant Multiple Rule is the special case gsxd = k of the Product Rule. This leaves only the Product and Quotient Rules. Proof of the Limit Product Rule We show that for any P 7 0 there exists a d 7 0 such that for all x in the intersection D of the domains of ƒ and g, 0 6 ƒ x - c ƒ 6 d Q ƒ ƒsxdgsxd - LM ƒ 6 P.

Appendix 5

Proofs of Limit Theorems

AP-27

Suppose then that P is a positive number, and write ƒ(x) and g(x) as ƒsxd = L + sƒsxd - Ld,

gsxd = M + sgsxd - Md.

Multiply these expressions together and subtract LM: ƒsxd # gsxd - LM = sL + sƒsxd - LddsM + sgsxd - Mdd - LM = LM + Lsgsxd - Md + Msƒsxd - Ld + sƒsxd - Ldsgsxd - Md - LM = Lsgsxd - Md + Msƒsxd - Ld + sƒsxd - Ldsgsxd - Md. (1) Since ƒ and g have limits L and M as x : c, there exist positive numbers d1, d2, d3 , and d4 such that for all x in D 0 0 0 0

6 6 6 6

ƒx ƒx ƒx ƒx

-

cƒ cƒ cƒ cƒ

6 6 6 6

d1 d2 d3 d4

Q Q Q Q

ƒ ƒsxd ƒ gsxd ƒ ƒsxd ƒ gsxd

-

L ƒ 6 2P>3 M ƒ 6 2P>3 L ƒ 6 P>s3s1 + ƒ M ƒ dd M ƒ 6 P>s3s1 + ƒ L ƒ dd.

(2)

If we take d to be the smallest numbers d1 through d4 , the inequalities on the right-hand side of the Implications (2) will hold simultaneously for 0 6 ƒ x - c ƒ 6 d . Therefore, for all x in D, 0 6 ƒ x - c ƒ 6 d implies ƒ ƒsxd # gsxd - LM ƒ to Eq. (1) … ƒ L ƒ ƒ gsxd - M ƒ + ƒ M ƒ ƒ ƒsxd - L ƒ + ƒ ƒsxd - L ƒ ƒ gsxd - M ƒ … s1 + ƒ L ƒ d ƒ gsxd - M ƒ + s1 + ƒ M ƒ d ƒ ƒsxd - L ƒ + ƒ ƒsxd - L ƒ ƒ gsxd - M ƒ Triangle inequality applied

6

P P P P = P. + + 3 3 3 A A3

Values from (2)

This completes the proof of the Limit Product Rule. Proof of the Limit Quotient Rule We show that limx:cs1>gsxdd = 1>M. We can then conclude that ƒsxd 1 1 1 L = lim aƒsxd # b = lim ƒsxd # lim = L# = M M g(x) gsxd gsxd x:c x:c x:c x:c lim

by the Limit Product Rule. Let P 7 0 be given. To show that limx:cs1>gsxdd = 1>M, we need to show that there exists a d 7 0 such that for all x 0 6 ƒx - cƒ 6 d

Q

`

1 1 ` 6 P. M gsxd

Since ƒ M ƒ 7 0, there exists a positive number d1 such that for all x 0 6 ƒ x - c ƒ 6 d1

Q

M ƒ gsxd - M ƒ 6 2 .

(3)

For any numbers A and B it can be shown that ƒ A ƒ - ƒ B ƒ … ƒ A - B ƒ and ƒ B ƒ - ƒ A ƒ ƒ A - B ƒ , from which it follows that ƒ ƒ A ƒ - ƒ B ƒ ƒ … ƒ A - B ƒ . With A = gsxd and B = M, this becomes ƒ ƒ gsxd ƒ - ƒ M ƒ ƒ … ƒ gsxd - M ƒ ,

AP-28

Appendices

which can be combined with the inequality on the right in Implication (3) to get, in turn, ƒMƒ 2

ƒ ƒ gsxd ƒ - ƒ M ƒ ƒ 6 -

ƒMƒ ƒMƒ 6 ƒ gsxd ƒ - ƒ M ƒ 6 2 2 3ƒ M ƒ ƒMƒ 6 ƒ gsxd ƒ 6 2 2 ƒ M ƒ 6 2 ƒ gsxd ƒ 6 3 ƒ M ƒ

3 2 1 6 6 . ƒMƒ ƒ gsxd ƒ ƒ gsxd ƒ

(4)

Therefore, 0 6 ƒ x - c ƒ 6 d1 implies that

`

M - gsxd 1 1 1 # 1 # ` = ` ` … ƒ M - gsxd ƒ M gsxd Mgsxd ƒ M ƒ ƒ gsxd ƒ 6

1 # 2 # ƒ M - gsxd ƒ . ƒMƒ ƒMƒ

Inequality (4)

(5)

Since s1>2d ƒ M ƒ 2P 7 0, there exists a number d2 7 0 such that for all x 0 6 ƒ x - c ƒ 6 d2

Q

P ƒ M - gsxd ƒ 6 2 ƒ M ƒ 2 .

(6)

If we take d to be the smaller of d1 and d2 , the conclusions in (5) and (6) both hold for all x such that 0 6 ƒ x - c ƒ 6 d. Combining these conclusions gives 0 6 ƒx - cƒ 6 d

Q

`

1 1 ` 6 P. M gsxd

This concludes the proof of the Limit Quotient Rule.

THEOREM 4—The Sandwich Theorem Suppose that gsxd … ƒsxd … hsxd for all x in some open interval I containing c, except possibly at x = c itself. Suppose also that limx:c gsxd = limx:c hsxd = L. Then limx:c ƒsxd = L.

Proof for Right-Hand Limits Suppose limx:c+ gsxd = limx:c+ hsxd = L. Then for any P 7 0 there exists a d 7 0 such that for all x the interval c 6 x 6 c + d is contained in I and the inequality implies L - P 6 gsxd 6 L + P

and

L - P 6 hsxd 6 L + P.

These inequalities combine with the inequality gsxd … ƒsxd … hsxd to give L - P 6 gsxd … ƒsxd … hsxd 6 L + P, L - P 6 ƒsxd 6 L + P, - P 6 ƒsxd - L 6 P. Therefore, for all x, the inequality c 6 x 6 c + d implies ƒ ƒsxd - L ƒ 6 P.

Appendix 6

Commonly Occurring Limits

AP-29

Proof for Left-Hand Limits Suppose limx:c- gsxd = limx:c- hsxd = L. Then for any P 7 0 there exists a d 7 0 such that for all x the interval c - d 6 x 6 c is contained in I and the inequality implies L - P 6 gsxd 6 L + P

and

L - P 6 hsxd 6 L + P.

We conclude as before that for all x, c - d 6 x 6 c implies ƒ ƒsxd - L ƒ 6 P. Proof for Two-Sided Limits If limx:c gsxd = limx:c hsxd = L, then g(x) and h(x) both approach L as x : c + and as x : c - ; so limx:c+ ƒsxd = L and limx:c- ƒsxd = L . Hence limx:c ƒsxd exists and equals L.

Exercises A.5 1. Suppose that functions ƒ1sxd, ƒ2sxd, and ƒ3sxd have limits L1, L2 , and L3 , respectively, as x : c . Show that their sum has limit L1 + L2 + L3 . Use mathematical induction (Appendix 2) to generalize this result to the sum of any finite number of functions.

5. Limits of rational functions Use Theorem 1 and the result of Exercise 4 to show that if ƒ(x) and g(x) are polynomial functions and gscd Z 0 , then

2. Use mathematical induction and the Limit Product Rule in Theorem 1 to show that if functions ƒ1sxd, ƒ2sxd, Á , ƒnsxd have limits L1, L2, Á , Ln as x : c , then

ƒsxd ƒscd = . gscd x:c gsxd lim

lim ƒ1sxd # ƒ2sxd # Á # ƒnsxd = L1 # L2 # Á # Ln .

6. Composites of continuous functions Figure A.30 gives the diagram for a proof that the composite of two continuous functions is continuous. Reconstruct the proof from the diagram. The statement to be proved is this: If ƒ is continuous at x = c and g is continuous at ƒ(c), then g ⴰ ƒ is continuous at c. Assume that c is an interior point of the domain of ƒ and that ƒ(c) is an interior point of the domain of g. This will make the limits involved two-sided. (The arguments for the cases that involve one-sided limits are similar.)

x:c

3. Use the fact that limx:c x = c and the result of Exercise 2 to show that limx:c x n = c n for any integer n 7 1 . 4. Limits of polynomials Use the fact that limx:cskd = k for any number k together with the results of Exercises 1 and 3 to show that limx:c ƒsxd = ƒscd for any polynomial function ƒsxd = an x n + an - 1 x n - 1 + Á + a1 x + a0 .

g f f f

f

g g

c

g

f(c)

g( f(c))

FIGURE A.30 The diagram for a proof that the composite of two continuous functions is continuous.

A.6

Commonly Occurring Limits This appendix verifies limits (4)–(6) in Theorem 5 of Section 9.1. Limit 4: If ƒ x ƒ 6 1, lim x n = 0 We need to show that to each P 7 0 there corresponds n: ˆ

an integer N so large that ƒ x n ƒ 6 P for all n greater than N. Since P1>n : 1, while ƒ x ƒ 6 1, there exists an integer N for which P1>N 7 ƒ x ƒ . In other words, ƒ x N ƒ = ƒ x ƒ N 6 P.

(1)

AP-30

Appendices

This is the integer we seek because, if ƒ x ƒ 6 1, then n N ƒ x ƒ 6 ƒ x ƒ for all n 7 N.

(2)

Combining (1) and (2) produces ƒ x n ƒ 6 P for all n 7 N, concluding the proof. n

x Limit 5: For any number x, lim a1 n b e x n: ˆ

Let

n

x an = a1 + n b . Then n

x x ln an = ln a1 + n b = n ln a1 + n b : x, as we can see by the following application of l’Hôpital’s Rule, in which we differentiate with respect to n: lns1 + x>nd x lim n ln a1 + n b = lim 1/n n: q

n: q

a = lim

n: q

x 1 b # a- 2 b 1 + x>n n x = lim = x. n: q 1 + x/n -1/n 2

Apply Theorem 3, Section 9.1, with ƒsxd = e x to conclude that n

x a1 + n b = an = e ln an : e x .

Limit 6: For any number x, lim

n: ˆ

xn 0 Since n! -

ƒ x ƒn ƒ x ƒn xn … … , n! n! n!

all we need to show is that ƒ x ƒ n>n! : 0. We can then apply the Sandwich Theorem for Sequences (Section 9.1, Theorem 2) to conclude that x n>n! : 0. The first step in showing that ƒ x ƒ n>n! : 0 is to choose an integer M 7 ƒ x ƒ , so that s ƒ x ƒ >Md 6 1. By Limit 4, just proved, we then have s ƒ x ƒ >Mdn : 0. We then restrict our attention to values of n 7 M. For these values of n, we can write ƒ x ƒn ƒ x ƒn = # # Á # # n! 1 2 M sM + 1d # sM + 2d # Á # n ('''''')''''''* sn - Md factors n

…

n

M

n ƒxƒ ƒxƒ M MM ƒ x ƒ = = a b . M! M M!M n M!M n - M

Thus, 0 …

n ƒ x ƒn MM ƒ x ƒ … a b . n! M! M

Now, the constant M M>M! does not change as n increases. Thus the Sandwich Theorem tells us that ƒ x ƒ n>n! : 0 because s ƒ x ƒ >Mdn : 0.

Appendix 7

A.7

Theory of the Real Numbers

AP-31

Theory of the Real Numbers A rigorous development of calculus is based on properties of the real numbers. Many results about functions, derivatives, and integrals would be false if stated for functions defined only on the rational numbers. In this appendix we briefly examine some basic concepts of the theory of the reals that hint at what might be learned in a deeper, more theoretical study of calculus. Three types of properties make the real numbers what they are. These are the algebraic, order, and completeness properties. The algebraic properties involve addition and multiplication, subtraction and division. They apply to rational or complex numbers as well as to the reals. The structure of numbers is built around a set with addition and multiplication operations. The following properties are required of addition and multiplication. A1 A2 A3 A4 M1 M2 M3 M4 D

a + sb + cd = sa + bd + c for all a, b, c. a + b = b + a for all a, b. There is a number called “0” such that a + 0 = a for all a. For each number a, there is a b such that a + b = 0. asbcd = sabdc for all a, b, c. ab = ba for all a, b. There is a number called “1” such that a # 1 = a for all a. For each nonzero a, there is a b such that ab = 1. asb + cd = ab + bc for all a, b, c.

A1 and M1 are associative laws, A2 and M2 are commutativity laws, A3 and M3 are identity laws, and D is the distributive law. Sets that have these algebraic properties are examples of fields, and are studied in depth in the area of theoretical mathematics called abstract algebra. The order properties allow us to compare the size of any two numbers. The order properties are O1 O2 O3 O4 O5

For any a and b, either a … b or b … a or both. If a … b and b … a then a = b. If a … b and b … c then a … c. If a … b then a + c … b + c. If a … b and 0 … c then ac … bc.

O3 is the transitivity law, and O4 and O5 relate ordering to addition and multiplication. We can order the reals, the integers, and the rational numbers, but we cannot order the complex numbers. There is no reasonable way to decide whether a number like i = 2-1 is bigger or smaller than zero. A field in which the size of any two elements can be compared as above is called an ordered field. Both the rational numbers and the real numbers are ordered fields, and there are many others. We can think of real numbers geometrically, lining them up as points on a line. The completeness property says that the real numbers correspond to all points on the line, with no “holes” or “gaps.” The rationals, in contrast, omit points such as 22 and p, and the integers even leave out fractions like 1> 2. The reals, having the completeness property, omit no points. What exactly do we mean by this vague idea of missing holes? To answer this we must give a more precise description of completeness. A number M is an upper bound for a set of numbers if all numbers in the set are smaller than or equal to M. M is a least upper bound if it is the smallest upper bound. For example, M = 2 is an upper bound for the

AP-32

Appendices

y 0.5 y x x3 0.3

0.1 0.1

0.3

0.5

0.7

0.9 1

1/3

FIGURE A.31 The maximum value of y = x - x 3 on [0, 1] occurs at the irrational number x = 21>3 .

x

negative numbers. So is M = 1, showing that 2 is not a least upper bound. The least upper bound for the set of negative numbers is M = 0. We define a complete ordered field to be one in which every nonempty set bounded above has a least upper bound. If we work with just the rational numbers, the set of numbers less than 22 is bounded, but it does not have a rational least upper bound, since any rational upper bound M can be replaced by a slightly smaller rational number that is still larger than 22. So the rationals are not complete. In the real numbers, a set that is bounded above always has a least upper bound. The reals are a complete ordered field. The completeness property is at the heart of many results in calculus. One example occurs when searching for a maximum value for a function on a closed interval [a, b], as in Section 4.1. The function y = x - x 3 has a maximum value on [0, 1] at the point x satisfying 1 - 3x 2 = 0, or x = 11>3. If we limited our consideration to functions defined only on rational numbers, we would have to conclude that the function has no maximum, since 11>3 is irrational (Figure A.31). The Extreme Value Theorem (Section 4.1), which implies that continuous functions on closed intervals [a, b] have a maximum value, is not true for functions defined only on the rationals. The Intermediate Value Theorem implies that a continuous function ƒ on an interval [a, b] with ƒsad 6 0 and ƒsbd 7 0 must be zero somewhere in [a, b]. The function values cannot jump from negative to positive without there being some point x in [a, b] where ƒsxd = 0. The Intermediate Value Theorem also relies on the completeness of the real numbers and is false for continuous functions defined only on the rationals. The function ƒsxd = 3x 2 - 1 has ƒs0d = -1 and ƒs1d = 2, but if we consider ƒ only on the rational numbers, it never equals zero. The only value of x for which ƒsxd = 0 is x = 21>3, an irrational number. We have captured the desired properties of the reals by saying that the real numbers are a complete ordered field. But we’re not quite finished. Greek mathematicians in the school of Pythagoras tried to impose another property on the numbers of the real line, the condition that all numbers are ratios of integers. They learned that their effort was doomed when they discovered irrational numbers such as 12. How do we know that our efforts to specify the real numbers are not also flawed, for some unseen reason? The artist Escher drew optical illusions of spiral staircases that went up and up until they rejoined themselves at the bottom. An engineer trying to build such a staircase would find that no structure realized the plans the architect had drawn. Could it be that our design for the reals contains some subtle contradiction, and that no construction of such a number system can be made? We resolve this issue by giving a specific description of the real numbers and verifying that the algebraic, order, and completeness properties are satisfied in this model. This is called a construction of the reals, and just as stairs can be built with wood, stone, or steel, there are several approaches to constructing the reals. One construction treats the reals as all the infinite decimals, a.d1d2d3d4 Á In this approach a real number is an integer a followed by a sequence of decimal digits d1, d2, d3, Á , each between 0 and 9. This sequence may stop, or repeat in a periodic pattern, or keep going forever with no pattern. In this form, 2.00, 0.3333333 Á and 3.1415926535898 Á represent three familiar real numbers. The real meaning of the dots “ Á ” following these digits requires development of the theory of sequences and series, as in Chapter 9. Each real number is constructed as the limit of a sequence of rational numbers given by its finite decimal approximations. An infinite decimal is then the same as a series a +

d2 d1 + + Á. 10 100

This decimal construction of the real numbers is not entirely straightforward. It’s easy enough to check that it gives numbers that satisfy the completeness and order properties,

Appendix 8

Complex Numbers

AP-33

but verifying the algebraic properties is rather involved. Even adding or multiplying two numbers requires an infinite number of operations. Making sense of division requires a careful argument involving limits of rational approximations to infinite decimals. A different approach was taken by Richard Dedekind (1831–1916), a German mathematician, who gave the first rigorous construction of the real numbers in 1872. Given any real number x, we can divide the rational numbers into two sets: those less than or equal to x and those greater. Dedekind cleverly reversed this reasoning and defined a real number to be a division of the rational numbers into two such sets. This seems like a strange approach, but such indirect methods of constructing new structures from old are common in theoretical mathematics. These and other approaches can be used to construct a system of numbers having the desired algebraic, order, and completeness properties. A final issue that arises is whether all the constructions give the same thing. Is it possible that different constructions result in different number systems satisfying all the required properties? If yes, which of these is the real numbers? Fortunately, the answer turns out to be no. The reals are the only number system satisfying the algebraic, order, and completeness properties. Confusion about the nature of the numbers and about limits caused considerable controversy in the early development of calculus. Calculus pioneers such as Newton, Leibniz, and their successors, when looking at what happens to the difference quotient ¢y ƒsx + ¢xd - ƒsxd = ¢x ¢x as each of ¢y and ¢x approach zero, talked about the resulting derivative being a quotient of two infinitely small quantities. These “infinitesimals,” written dx and dy, were thought to be some new kind of number, smaller than any fixed number but not zero. Similarly, a definite integral was thought of as a sum of an infinite number of infinitesimals ƒsxd # dx as x varied over a closed interval. While the approximating difference quotients ¢y>¢x were understood much as today, it was the quotient of infinitesimal quantities, rather than a limit, that was thought to encapsulate the meaning of the derivative. This way of thinking led to logical difficulties, as attempted definitions and manipulations of infinitesimals ran into contradictions and inconsistencies. The more concrete and computable difference quotients did not cause such trouble, but they were thought of merely as useful calculation tools. Difference quotients were used to work out the numerical value of the derivative and to derive general formulas for calculation, but were not considered to be at the heart of the question of what the derivative actually was. Today we realize that the logical problems associated with infinitesimals can be avoided by defining the derivative to be the limit of its approximating difference quotients. The ambiguities of the old approach are no longer present, and in the standard theory of calculus, infinitesimals are neither needed nor used.

A.8

Complex Numbers Complex numbers are expressions of the form a + ib, where a and b are real numbers and i is a symbol for 1-1 . Unfortunately, the words “real” and “imaginary” have connotations that somehow place 1-1 in a less favorable position in our minds than 12. As a matter of fact, a good deal of imagination, in the sense of inventiveness, has been required to construct the real number system, which forms the basis of the calculus (see Appendix 7). In this appendix we review the various stages of this invention. The further invention of a complex number system is then presented.

AP-34

Appendices

The Development of the Real Numbers The earliest stage of number development was the recognition of the counting numbers 1, 2, 3, Á , which we now call the natural numbers or the positive integers. Certain simple arithmetical operations can be performed with these numbers without getting outside the system. That is, the system of positive integers is closed under the operations of addition and multiplication. By this we mean that if m and n are any positive integers, then m + n = p

and

mn = q

(1)

are also positive integers. Given the two positive integers on the left side of either equation in (1), we can find the corresponding positive integer on the right side. More than this, we can sometimes specify the positive integers m and p and find a positive integer n such that m + n = p. For instance, 3 + n = 7 can be solved when the only numbers we know are the positive integers. But the equation 7 + n = 3 cannot be solved unless the number system is enlarged. The number zero and the negative integers were invented to solve equations like 7 + n = 3. In a civilization that recognizes all the integers Á , -3, -2, -1, 0, 1, 2, 3, Á ,

(2)

an educated person can always find the missing integer that solves the equation m + n = p when given the other two integers in the equation. Suppose our educated people also know how to multiply any two of the integers in the list (2). If, in Equations (1), they are given m and q, they discover that sometimes they can find n and sometimes they cannot. Using their imagination, they may be inspired to invent still more numbers and introduce fractions, which are just ordered pairs m> n of integers m and n. The number zero has special properties that may bother them for a while, but they ultimately discover that it is handy to have all ratios of integers m> n, excluding only those having zero in the denominator. This system, called the set of rational numbers, is now rich enough for them to perform the rational operations of arithmetic: 2

1

1

FIGURE A.32 With a straightedge and compass, it is possible to construct a segment of irrational length.

1. (a) addition (b) subtraction

2. (a) multiplication (b) division

on any two numbers in the system, except that they cannot divide by zero because it is meaningless. The geometry of the unit square (Figure A.32) and the Pythagorean theorem showed that they could construct a geometric line segment that, in terms of some basic unit of length, has length equal to 22. Thus they could solve the equation x2 = 2 by a geometric construction. But then they discovered that the line segment representing 22 is an incommensurable quantity. This means that 22 cannot be expressed as the ratio of two integer multiples of some unit of length. That is, our educated people could not find a rational number solution of the equation x 2 = 2. There is no rational number whose square is 2. To see why, suppose that there were such a rational number. Then we could find integers p and q with no common factor other than 1, and such that p 2 = 2q 2 .

(3)

Since p and q are integers, p must be even; otherwise its product with itself would be odd. In symbols, p = 2p1 , where p1 is an integer. This leads to 2p 12 = q 2 which says q must be even, say q = 2q1 , where q1 is an integer. This makes 2 a factor of both p and q, contrary to our choice of p and q as integers with no common factor other than 1. Hence there is no rational number whose square is 2.

Appendix 8

Complex Numbers

AP-35

Although our educated people could not find a rational solution of the equation x 2 = 2, they could get a sequence of rational numbers 7 , 5

1 , 1

41 , 29

239 , 169

Á,

(4)

whose squares form a sequence 1 , 1

49 , 25

1681 , 841

57,121 , 28,561

Á,

(5)

that converges to 2 as its limit. This time their imagination suggested that they needed the concept of a limit of a sequence of rational numbers. If we accept the fact that an increasing sequence that is bounded from above always approaches a limit (Theorem 6, Section 9.1) and observe that the sequence in (4) has these properties, then we want it to have a limit L. This would also mean, from (5), that L 2 = 2, and hence L is not one of our rational numbers. If to the rational numbers we further add the limits of all bounded increasing sequences of rational numbers, we arrive at the system of all “real” numbers. The word real is placed in quotes because there is nothing that is either “more real” or “less real” about this system than there is about any other mathematical system.

The Complex Numbers Imagination was called upon at many stages during the development of the real number system. In fact, the art of invention was needed at least three times in constructing the systems we have discussed so far: 1. 2. 3.

The first invented system: the set of all integers as constructed from the counting numbers. The second invented system: the set of rational numbers m> n as constructed from the integers. The third invented system: the set of all real numbers x as constructed from the rational numbers.

These invented systems form a hierarchy in which each system contains the previous system. Each system is also richer than its predecessor in that it permits additional operations to be performed without going outside the system: 1.

In the system of all integers, we can solve all equations of the form x + a = 0,

2.

(6)

where a can be any integer. In the system of all rational numbers, we can solve all equations of the form ax + b = 0,

3.

(7)

provided a and b are rational numbers and a Z 0. In the system of all real numbers, we can solve all of Equations (6) and (7) and, in addition, all quadratic equations ax 2 + bx + c = 0

having

a Z 0

and

b 2 - 4ac Ú 0.

(8)

You are probably familiar with the formula that gives the solutions of Equation (8), namely, x =

-b ; 2b2 - 4ac , 2a

(9)

AP-36

Appendices

and are familiar with the further fact that when the discriminant, b 2 - 4ac, is negative, the solutions in Equation (9) do not belong to any of the systems discussed above. In fact, the very simple quadratic equation x2 + 1 = 0 is impossible to solve if the only number systems that can be used are the three invented systems mentioned so far. Thus we come to the fourth invented system, the set of all complex numbers a + ib. We could dispense entirely with the symbol i and use the ordered pair notation (a, b). Since, under algebraic operations, the numbers a and b are treated somewhat differently, it is essential to keep the order straight. We therefore might say that the complex number system consists of the set of all ordered pairs of real numbers (a, b), together with the rules by which they are to be equated, added, multiplied, and so on, listed below. We will use both the (a, b) notation and the notation a + ib in the discussion that follows. We call a the real part and b the imaginary part of the complex number (a, b). We make the following definitions. Equality a + ib = c + id if and only if a = c and b = d.

Two complex numbers (a, b) and (c, d) are equal if and only if a = c and b = d.

Addition sa + ibd + sc + idd = sa + cd + isb + dd

The sum of the two complex numbers (a, b) and (c, d) is the complex number sa + c, b + dd.

Multiplication sa + ibdsc + idd = sac - bdd + isad + bcd csa + ibd = ac + isbcd

The product of two complex numbers (a, b) and (c, d) is the complex number sac - bd, ad + bcd. The product of a real number c and the complex number (a, b) is the complex number (ac, bc).

The set of all complex numbers (a, b) in which the second number b is zero has all the properties of the set of real numbers a. For example, addition and multiplication of (a, 0) and (c, 0) give sa, 0d + sc, 0d = sa + c, 0d, sa, 0d # sc, 0d = sac, 0d,

which are numbers of the same type with imaginary part equal to zero. Also, if we multiply a “real number” (a, 0) and the complex number (c, d), we get sa, 0d # sc, dd = sac, add = asc, dd. In particular, the complex number (0, 0) plays the role of zero in the complex number system, and the complex number (1, 0) plays the role of unity or one. The number pair (0, 1), which has real part equal to zero and imaginary part equal to one, has the property that its square, s0, 1ds0, 1d = s -1, 0d, has real part equal to minus one and imaginary part equal to zero. Therefore, in the system of complex numbers (a, b) there is a number x = s0, 1d whose square can be added to unity = s1, 0d to produce zero = s0, 0d, that is, s0, 1d2 + s1, 0d = s0, 0d.

Appendix 8

Complex Numbers

AP-37

The equation x2 + 1 = 0 therefore has a solution x = s0, 1d in this new number system. You are probably more familiar with the a + ib notation than you are with the notation (a, b). And since the laws of algebra for the ordered pairs enable us to write sa, bd = sa, 0d + s0, bd = as1, 0d + bs0, 1d, while (1, 0) behaves like unity and (0, 1) behaves like a square root of minus one, we need not hesitate to write a + ib in place of (a, b). The i associated with b is like a tracer element that tags the imaginary part of a + ib. We can pass at will from the realm of ordered pairs (a, b) to the realm of expressions a + ib, and conversely. But there is nothing less “real” about the symbol s0, 1d = i than there is about the symbol s1, 0d = 1, once we have learned the laws of algebra in the complex number system of ordered pairs (a, b). To reduce any rational combination of complex numbers to a single complex number, we apply the laws of elementary algebra, replacing i 2 wherever it appears by -1. Of course, we cannot divide by the complex number s0, 0d = 0 + i0. But if a + ib Z 0, then we may carry out a division as follows: sc + iddsa - ibd sac + bdd + isad - bcd c + id = = . a + ib sa + ibdsa - ibd a2 + b2 The result is a complex number x + iy with x =

ac + bd , a2 + b2

y =

ad - bc , a2 + b2

and a 2 + b 2 Z 0, since a + ib = sa, bd Z s0, 0d. The number a - ib that is used as multiplier to clear the i from the denominator is called the complex conjugate of a + ib. It is customary to use z (read “z bar”) to denote the complex conjugate of z; thus z = a + ib,

z = a - ib.

Multiplying the numerator and denominator of the fraction sc + idd>sa + ibd by the complex conjugate of the denominator will always replace the denominator by a real number.

EXAMPLE 1

We give some illustrations of the arithmetic operations with complex

numbers. (a) s2 + 3id + s6 - 2id = s2 + 6d + s3 - 2di = 8 + i (b) s2 + 3id - s6 - 2id = s2 - 6d + s3 - s -2ddi = -4 + 5i (c) s2 + 3ids6 - 2id = s2ds6d + s2ds -2id + s3ids6d + s3ids -2id = 12 - 4i + 18i - 6i 2 = 12 + 14i + 6 = 18 + 14i (d)

2 + 3i 2 + 3i 6 + 2i = 6 - 2i 6 - 2i 6 + 2i 12 + 4i + 18i + 6i2 36 + 12i - 12i - 4i2 6 + 22i 3 11 = = + i 40 20 20 =

Argand Diagrams There are two geometric representations of the complex number z = x + iy: 1. 2.

as the point P(x, y) in the xy-plane § as the vector OP from the origin to P.

AP-38

Appendices

In each representation, the x-axis is called the real axis and the y-axis is the imaginary axis. Both representations are Argand diagrams for x + iy (Figure A.33). In terms of the polar coordinates of x and y, we have

y P(x, y)

x = r cos u,

r

y

and

O

x

y = r sin u,

z = x + iy = rscos u + i sin ud.

x

FIGURE A.33 This Argand diagram represents z = x + iy both as a point § P(x, y) and as a vector OP .

(10)

We define the absolute value of a complex number x + iy to be the length r of a vector § OP from the origin to P(x, y). We denote the absolute value by vertical bars; thus, 2 2 ƒ x + iy ƒ = 2x + y .

If we always choose the polar coordinates r and u so that r is nonnegative, then r = ƒ x + iy ƒ . The polar angle u is called the argument of z and is written u = arg z. Of course, any integer multiple of 2p may be added to u to produce another appropriate angle. The following equation gives a useful formula connecting a complex number z, its conjugate z, and its absolute value ƒ z ƒ , namely, z # z = ƒ z ƒ2.

Euler’s Formula The identity e iu = cos u + i sin u, The notation exp (A) also stands for e A .

called Euler’s formula, enables us to rewrite Equation (10) as z = re iu . This formula, in turn, leads to the following rules for calculating products, quotients, powers, and roots of complex numbers. It also leads to Argand diagrams for e iu . Since cos u + i sin u is what we get from Equation (10) by taking r = 1, we can say that e iu is represented by a unit vector that makes an angle u with the positive x-axis, as shown in Figure A.34. y

y

ei cos i sin

r1 arg z O

ei cos i sin (cos , sin )

x

(a)

O

x

(b)

FIGURE A.34 Argand diagrams for e iu = cos u + i sin u (a) as a vector and (b) as a point.

Products To multiply two complex numbers, we multiply their absolute values and add their angles. Let z1 = r1 eiu1,

z2 = r2 eiu2 ,

(11)

Appendix 8

AP-39

so that

y z 1z 2

Complex Numbers

ƒ z1 ƒ = r1,

1

Then r2

z1

r1

2

ƒ z2 ƒ = r2,

arg z2 = u2 .

z1 z2 = r1 e iu1 # r2 e iu2 = r1 r2 e isu1 + u2d

z2

r1r 2

arg z1 = u1;

and hence

1

ƒ z1 z2 ƒ = r1 r2 = ƒ z1 ƒ # ƒ z2 ƒ

x

O

arg sz1 z2 d = u1 + u2 = arg z1 + arg z2 . FIGURE A.35 When z1 and z2 are multiplied, ƒ z1 z2 ƒ = r1 # r2 and arg sz1 z2 d = u1 + u2 .

Thus, the product of two complex numbers is represented by a vector whose length is the product of the lengths of the two factors and whose argument is the sum of their arguments (Figure A.35). In particular, from Equation (12) a vector may be rotated counterclockwise through an angle u by multiplying it by e iu . Multiplication by i rotates 90°, by -1 rotates 180°, by -i rotates 270°, and so on.

y

2

0

EXAMPLE 2 Let z1 = 1 + i, z2 = 23 - i. We plot these complex numbers in an Argand diagram (Figure A.36) from which we read off the polar representations

z1 1 i

1

z 1z 2 22

4

3 1

12 – 6

1 3

z1 = 22e ip>4,

z2 = 2e -ip>6 .

x

Then

1 2 –1

(12)

z1 z2 = 222 exp a

z 2 3 i

= 222 acos

FIGURE A.36 To multiply two complex numbers, multiply their absolute values and add their arguments.

ip ip ip b = 222 exp a b 4 6 12 p p + i sin b L 2.73 + 0.73i. 12 12

Quotients Suppose r2 Z 0 in Equation (11). Then z1 r1 e iu1 r1 = = r2 e isu1 - u2d . z2 iu2 r2 e Hence z1 r1 ƒ z1 ƒ ` z2 ` = r2 = ƒ z2 ƒ

and

z1 arg a z2 b = u1 - u2 = arg z1 - arg z2.

That is, we divide lengths and subtract angles for the quotient of complex numbers.

EXAMPLE 3

Let z1 = 1 + i and z2 = 23 - i, as in Example 2. Then

1 + i 23 - i

=

22e ip>4 22 5pi>12 5p 5p = L 0.707 acos e + i sin b 2 12 12 2e -ip>6

L 0.183 + 0.683i.

AP-40

Appendices

Powers If n is a positive integer, we may apply the product formulas in Equation (12) to find z n = z # z # Á # z.

n factors

With z = re iu , we obtain z n = sre iu dn = r ne isu + u +

Á + ud

n summands

= r ne inu .

(13)

The length r = ƒ z ƒ is raised to the nth power and the angle u = arg z is multiplied by n. If we take r = 1 in Equation (13), we obtain De Moivre’s Theorem.

De Moivre’s Theorem scos u + i sin udn = cos nu + i sin nu.

(14)

If we expand the left side of De Moivre’s equation above by the Binomial Theorem and reduce it to the form a + ib, we obtain formulas for cos nu and sin nu as polynomials of degree n in cos u and sin u.

EXAMPLE 4

If n = 3 in Equation (14), we have scos u + i sin ud3 = cos 3u + i sin 3u.

The left side of this equation expands to cos3 u + 3i cos2 u sin u - 3 cos u sin2 u - i sin3 u. The real part of this must equal cos 3u and the imaginary part must equal sin 3u. Therefore, cos 3u = cos3 u - 3 cos u sin2 u, sin 3u = 3 cos2 u sin u - sin3 u.

Roots If z = re iu is a complex number different from zero and n is a positive integer, then there are precisely n different complex numbers w0 , w1 , Á , wn - 1 , that are nth roots of z. To see why, let w = re ia be an nth root of z = re iu , so that wn = z or rne ina = re iu . Then n

r = 2r is the real, positive nth root of r. For the argument, although we cannot say that na and u must be equal, we can say that they may differ only by an integer multiple of 2p. That is, na = u + 2kp,

k = 0, ;1, ;2, Á .

Therefore, u 2p a = n + k n .

Appendix 8

Complex Numbers

AP-41

Hence, all the nth roots of z = re iu are given by

u 2p n n 2re iu = 2r exp i a n + k n b,

y z rei

k = 0, ;1, ;2, Á .

(15)

2 3

w1

r

There might appear to be infinitely many different answers corresponding to the infinitely many possible values of k, but k = n + m gives the same answer as k = m in Equation (15). Thus, we need only take n consecutive values for k to obtain all the different nth roots of z. For convenience, we take

w0

r 1/3

x

3

O 2 3

k = 0, 1, 2, Á , n - 1.

2 3

iu

All the nth roots of re lie on a circle centered at the origin and having radius equal to the real, positive nth root of r. One of them has argument a = u>n. The others are uniformly spaced around the circle, each being separated from its neighbors by an angle equal to 2p>n . Figure A.37 illustrates the placement of the three cube roots, w0, w1, w2 , of the complex number z = re iu .

w2

FIGURE A.37 The three cube roots of z = re iu .

EXAMPLE 5

Find the four fourth roots of -16.

As our first step, we plot the number -16 in an Argand diagram (Figure A.38) and determine its polar representation re iu . Here, z = -16, r = +16, and u = p. One of the fourth roots of 16e ip is 2e ip>4 . We obtain others by successive additions of 2p>4 = p>2 to the argument of this first one. Hence,

Solution

y 2

w1 2

2

–16

w0 4

and the four roots are x 2

w2

2

p 3p 5p 7p 4 2 16 exp ip = 2 exp i a , , , b, 4 4 4 4

w3

FIGURE A.38 The four fourth roots of -16 .

w0 = 2 ccos

p p + i sin d = 22s1 + id 4 4

w1 = 2 ccos

3p 3p + i sin d = 22s -1 + id 4 4

w2 = 2 ccos

5p 5p + i sin d = 22s -1 - id 4 4

w3 = 2 ccos

7p 7p + i sin d = 22s1 - id. 4 4

The Fundamental Theorem of Algebra One might say that the invention of 2 -1 is all well and good and leads to a number system that is richer than the real number system alone; but where will this process end? Are 4 6 we also going to invent still more systems so as to obtain 1-1, 1-1, and so on? But it turns out this is not necessary. These numbers are already expressible in terms of the complex number system a + ib. In fact, the Fundamental Theorem of Algebra says that with the introduction of the complex numbers we now have enough numbers to factor every polynomial into a product of linear factors and so enough numbers to solve every possible polynomial equation.

AP-42

Appendices

The Fundamental Theorem of Algebra Every polynomial equation of the form an z n + an - 1 z n - 1 + Á + a1 z + a0 = 0, in which the coefficients a0, a1, Á , an are any complex numbers, whose degree n is greater than or equal to one, and whose leading coefficient an is not zero, has exactly n roots in the complex number system, provided each multiple root of multiplicity m is counted as m roots.

A proof of this theorem can be found in almost any text on the theory of functions of a complex variable.

Exercises A.8 Operations with Complex Numbers 1. How computers multiply complex numbers Find sa, bd # sc, dd = sac - bd, ad + bcd . a. s2, 3d # s4, -2d

c. s -1, -2d # s2, 1d

b. s2, -1d # s -2, 3d

15. cos 4u

16. sin 4u

17. Find the three cube roots of 1.

(This is how complex numbers are multiplied by computers.) 2. Solve the following equations for the real numbers, x and y. a. s3 + 4id2 - 2sx - iyd = x + iy b. a

Powers and Roots Use De Moivre’s Theorem to express the trigonometric functions in Exercises 15 and 16 in terms of cos u and sin u .

2

1 + i 1 = 1 + i b + x + iy 1 - i

c. s3 - 2idsx + iyd = 2sx - 2iyd + 2i - 1

18. Find the two square roots of i. 19. Find the three cube roots of -8i . 20. Find the six sixth roots of 64. 21. Find the four solutions of the equation z 4 - 2z 2 + 4 = 0 . 22. Find the six solutions of the equation z 6 + 2z 3 + 2 = 0 . 23. Find all solutions of the equation x 4 + 4x 2 + 16 = 0 . 24. Solve the equation x 4 + 1 = 0 .

Graphing and Geometry 3. How may the following complex numbers be obtained from z = x + iy geometrically? Sketch. a. z c. -z

b. s -zd

d. 1> z

4. Show that the distance between the two points z1 and z2 in an Argand diagram is ƒ z1 - z2 ƒ . In Exercises 5–10, graph the points z = x + iy that satisfy the given conditions. 5. a. ƒ z ƒ = 2

b. ƒ z ƒ 6 2

c. ƒ z ƒ 7 2

6. ƒ z - 1 ƒ = 2

7. ƒ z + 1 ƒ = 1

8. ƒ z + 1 ƒ = ƒ z - 1 ƒ

9. ƒ z + i ƒ = ƒ z - 1 ƒ

10. ƒ z + 1 ƒ Ú ƒ z ƒ

1 + i23 1 - i23

26. Complex arithmetic with conjugates Show that the conjugate of the sum (product, or quotient) of two complex numbers, z1 and z2, is the same as the sum (product, or quotient) of their conjugates. 27. Complex roots of polynomials with real coefficients come in complex-conjugate pairs a. Extend the results of Exercise 26 to show that ƒszd = ƒszd if ƒszd = an z n + an - 1 z n - 1 + Á + a1 z + a0 is a polynomial with real coefficients a0, Á , an .

Express the complex numbers in Exercises 11–14 in the form re iu , with r Ú 0 and -p 6 u … p . Draw an Argand diagram for each calculation. 1 + i 11. A 1 + 2-3 B 2 12. 1 - i 13.

Theory and Examples 25. Complex numbers and vectors in the plane Show with an Argand diagram that the law for adding complex numbers is the same as the parallelogram law for adding vectors.

14. s2 + 3ids1 - 2id

b. If z is a root of the equation ƒszd = 0 , where ƒ(z) is a polynomial with real coefficients as in part (a), show that the conjugate z is also a root of the equation. (Hint: Let ƒszd = u + iy = 0 ; then both u and y are zero. Use the fact that ƒszd = ƒszd = u - iy .) 28. Absolute value of a conjugate Show that ƒ z ƒ = ƒ z ƒ . 29. When z = z If z and z are equal, what can you say about the location of the point z in the complex plane?

Appendix 9

The Distributive Law for Vector Cross Products

AP-43

c. ƒ Reszd ƒ … ƒ z ƒ d. ƒ z1 + z2 ƒ 2 = ƒ z1 ƒ 2 + ƒ z2 ƒ 2 + 2Resz1z2 d e. ƒ z1 + z2 ƒ … ƒ z1 ƒ + ƒ z2 ƒ

30. Real and imaginary parts Let Re(z) denote the real part of z and Im(z) the imaginary part. Show that the following relations hold for any complex numbers z, z1 , and z2 . a. z + z = 2Reszd b. z - z = 2iImszd

A.9

The Distributive Law for Vector Cross Products In this appendix we prove the Distributive Law u * sv + wd = u * v + u * w, which is Property 2 in Section 11.4. Proof To derive the Distributive Law, we construct u * v a new way. We draw u and v from the common point O and construct a plane M perpendicular to u at O (Figure A.39). We then project v orthogonally onto M, yielding a vector v¿ with length ƒ v ƒ sin u. We rotate v¿ 90° about u in the positive sense to produce a vector v– . Finally, we multiply v– by the length of u. The resulting vector ƒ u ƒ v– is equal to u * v since v– has the same direction as u * v by its construction (Figure A.39) and ƒ u ƒ ƒ v– ƒ = ƒ u ƒ ƒ v¿ ƒ = ƒ u ƒ ƒ v ƒ sin u = ƒ u * v ƒ .

M'

v u O

M v'

v''

uv

90˚

FIGURE A.39 As explained in the text, u * v = ƒ u ƒ v– . (The primes used here are purely notational and do not denote derivatives.)

Now each of these three operations, namely, 1. 2. 3.

projection onto M rotation about u through 90° multiplication by the scalar ƒ u ƒ

when applied to a triangle whose plane is not parallel to u, will produce another triangle. If we start with the triangle whose sides are v, w, and v + w (Figure A.40) and apply these three steps, we successively obtain the following: 1. 2.

A triangle whose sides are v¿, w¿ , and sv + wd¿ satisfying the vector equation v¿ + w¿ = sv + wd¿ A triangle whose sides are v–, w– , and sv + wd– satisfying the vector equation v– + w– = sv + wd–

AP-44

Appendices

(the double prime on each vector has the same meaning as in Figure A.39)

w

u

vw w' M

v v' (v w)'

FIGURE A.40 The vectors, v, w, v + w , and their projections onto a plane perpendicular to u.

3.

A triangle whose sides are ƒ u ƒ v–, ƒ u ƒ w– , and ƒ u ƒ sv + wd– satisfying the vector equation ƒ u ƒ v– + ƒ u ƒ w– = ƒ u ƒ sv + wd– .

Substituting ƒ u ƒ v– = u * v, ƒ u ƒ w– = u * w, and ƒ u ƒ sv + wd– = u * sv + wd from our discussion above into this last equation gives u * v + u * w = u * sv + wd, which is the law we wanted to establish.

A.10

The Mixed Derivative Theorem and the Increment Theorem This appendix derives the Mixed Derivative Theorem (Theorem 2, Section 13.3) and the Increment Theorem for Functions of Two Variables (Theorem 3, Section 13.3). Euler first published the Mixed Derivative Theorem in 1734, in a series of papers he wrote on hydrodynamics.

THEOREM 2—The Mixed Derivative Theorem If ƒ(x, y) and its partial derivatives ƒx, ƒy, ƒxy , and ƒyx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b), then ƒxysa, bd = ƒyxsa, bd.

Proof The equality of ƒxysa, bd and ƒyxsa, bd can be established by four applications of the Mean Value Theorem (Theorem 4, Section 4.2). By hypothesis, the point (a, b) lies in the interior of a rectangle R in the xy-plane on which ƒ, ƒx, ƒy, ƒxy , and ƒyx are all defined. We let h and k be the numbers such that the point sa + h, b + kd also lies in R, and we consider the difference ¢ = Fsa + hd - Fsad,

(1)

Fsxd = ƒsx, b + kd - ƒsx, bd.

(2)

where

Appendix 10

The Mixed Derivative Theorem and the Increment Theorem

AP-45

We apply the Mean Value Theorem to F, which is continuous because it is differentiable. Then Equation (1) becomes ¢ = hF¿sc1 d,

(3)

where c1 lies between a and a + h. From Equation (2), F¿sxd = ƒxsx, b + kd - ƒxsx, bd, so Equation (3) becomes ¢ = h[ƒxsc1, b + kd - ƒxsc1, bd].

(4)

Now we apply the Mean Value Theorem to the function gs yd = fxsc1, yd and have gsb + kd - gsbd = kg¿sd1 d, or ƒxsc1, b + kd - ƒxsc1, bd = kƒxysc1, d1 d for some d1 between b and b + k. By substituting this into Equation (4), we get ¢ = hkƒxysc1, d1 d

(5)

for some point sc1, d1 d in the rectangle R¿ whose vertices are the four points (a, b), sa + h, bd, sa + h, b + kd, and sa, b + kd. (See Figure A.41.) By substituting from Equation (2) into Equation (1), we may also write

y

R k (a, b)

R' h

¢ = ƒsa + h, b + kd - ƒsa + h, bd - ƒsa, b + kd + ƒsa, bd = [ƒsa + h, b + kd - ƒsa, b + kd] - [ƒsa + h, bd - ƒsa, bd] = fsb + kd - fsbd,

(6)

fs yd = ƒsa + h, yd - ƒsa, yd.

(7)

where 0

x

FIGURE A.41 The key to proving fxysa, bd = fyxsa, bd is that no matter how small R¿ is, fxy and fyx take on equal values somewhere inside R¿ (although not necessarily at the same point).

The Mean Value Theorem applied to Equation (6) now gives ¢ = kf¿sd2 d

(8)

for some d2 between b and b + k. By Equation (7), f¿s yd = ƒysa + h, yd - ƒysa, yd.

(9)

Substituting from Equation (9) into Equation (8) gives ¢ = k[ƒysa + h, d2 d - ƒysa, d2 d]. Finally, we apply the Mean Value Theorem to the expression in brackets and get ¢ = khƒyxsc2, d2 d

(10)

for some c2 between a and a + h. Together, Equations (5) and (10) show that ƒxysc1, d1 d = ƒyxsc2, d2 d,

(11)

where sc1, d1 d and sc2, d2 d both lie in the rectangle R¿ (Figure A.41). Equation (11) is not quite the result we want, since it says only that ƒxy has the same value at sc1, d1 d that ƒyx has at sc2, d2 d. The numbers h and k in our discussion, however, may be made as small as we wish. The hypothesis that ƒxy and ƒyx are both continuous at (a, b) means that ƒxysc1, d1 d = ƒxysa, bd + P1 and ƒyxsc2, d2 d = ƒyxsa, bd + P2 , where each of P1, P2 : 0 as both h, k : 0. Hence, if we let h and k : 0, we have ƒxysa, bd = ƒyxsa, bd. The equality of ƒxysa, bd and ƒyxsa, bd can be proved with hypotheses weaker than the ones we assumed. For example, it is enough for ƒ, ƒx , and ƒy to exist in R and for ƒxy to be continuous at (a, b). Then ƒyx will exist at (a, b) and equal ƒxy at that point.

AP-46

Appendices

THEOREM 3—The Increment Theorem for Functions of Two Variables Suppose that the first partial derivatives of ƒsx, yd are defined throughout an open region R containing the point sx0, y0 d and that fx and ƒy are continuous at sx0, y0 d. Then the change ¢z = ƒsx0 + ¢x, y0 + ¢yd - ƒsx0, y0 d in the value of ƒ that results from moving from sx0, y0 d to another point sx0 + ¢x, y0 + ¢yd in R satisfies an equation of the form ¢z = ƒxsx0, y0 d¢x + ƒysx0, y0 d¢y + P1 ¢x + P2 ¢y in which each of P1, P2 : 0 as both ¢x, ¢y : 0. C(x0 x, y0 y)

A(x0, y0 ) B(x0 x, y0 )

Proof We work within a rectangle T centered at Asx0, y0 d and lying within R, and we assume that ¢x and ¢y are already so small that the line segment joining A to Bsx0 + ¢x, y0 d and the line segment joining B to Csx0 + ¢x, y0 + ¢yd lie in the interior of T (Figure A.42). We may think of ¢z as the sum ¢z = ¢z1 + ¢z2 of two increments, where ¢z1 = ƒsx0 + ¢x, y0 d - ƒsx0, y0 d is the change in the value of ƒ from A to B and

T

FIGURE A.42 The rectangular region T in the proof of the Increment Theorem. The figure is drawn for ¢x and ¢y positive, but either increment might be zero or negative.

¢z2 = ƒsx0 + ¢x, y0 + ¢yd - ƒsx0 + ¢x, y0 d is the change in the value of ƒ from B to C (Figure A.43). On the closed interval of x-values joining x0 to x0 + ¢x, the function Fsxd = ƒsx, y0 d is a differentiable (and hence continuous) function of x, with derivative F¿sxd = fxsx, y0 d. By the Mean Value Theorem (Theorem 4, Section 4.2), there is an x-value c between x0 and x0 + ¢x at which Fsx0 + ¢xd - Fsx0 d = F¿scd ¢x or ƒsx0 + ¢x, y0 d - ƒsx0, y0 d = fxsc, y0 d ¢x or ¢z1 = ƒxsc, y0 d ¢x.

(12)

Similarly, Gs yd = ƒsx0 + ¢x, yd is a differentiable (and hence continuous) function of y on the closed y-interval joining y0 and y0 + ¢y, with derivative G¿s yd = ƒysx0 + ¢x, yd. Hence, there is a y-value d between y0 and y0 + ¢y at which Gs y0 + ¢yd - Gs y0 d = G¿sdd ¢y or ƒsx0 + ¢x, y0 + ¢yd - ƒsx0 + ¢x, yd = ƒysx0 + ¢x, dd ¢y or ¢z2 = ƒysx0 + ¢x, dd ¢y.

(13)

Appendix 10

AP-47

The Mixed Derivative Theorem and the Increment Theorem z S

Q z f (x, y)

P0

Q'

z 2

P''

z P'

0

z 1

A(x0 , y 0 )

y0

y

B

x

y 0 y C(x0 x, y 0 y)

(x0 x, y 0 )

FIGURE A.43 Part of the surface z = ƒsx, yd near P0sx0, y0, ƒsx0, y0 dd . The points P0, P¿, and P– have the same height z0 = ƒsx0, y0 d above the xy-plane. The change in z is ¢z = P¿S . The change ¢z1 = ƒsx0 + ¢x, y0 d - ƒsx0, y0 d, shown as P–Q = P¿Q¿ , is caused by changing x from x0 to x0 + ¢x while holding y equal to y0 . Then, with x held equal to x0 + ¢x , ¢z2 = ƒsx0 + ¢x, y0 + ¢yd - ƒsx0 + ¢x, y0 d is the change in z caused by changing y0 from y0 + ¢y , which is represented by Q¿S. The total change in z is the sum of ¢z1 and ¢z2 .

Now, as both ¢x and ¢y : 0, we know that c : x0 and d : y0 . Therefore, since fx and fy are continuous at sx0, y0 d, the quantities P1 = ƒxsc, y0 d - ƒxsx0, y0 d, P2 = ƒysx0 + ¢x, dd - ƒysx0, y0 d

(14)

both approach zero as both ¢x and ¢y : 0. Finally, ¢z = ¢z1 + ¢z2 = ƒxsc, y0 d¢x + ƒysx0 + ¢x, dd¢y = [ƒxsx0, y0 d + P1]¢x + [ƒysx0, y0 d + P2]¢y

From Eqs. (12) and (13) From Eq. (14)

= ƒxsx0, y0 d¢x + ƒysx0, y0 d¢y + P1 ¢x + P2 ¢y, where both P1 and P2 : 0 as both ¢x and ¢y : 0, which is what we set out to prove. Analogous results hold for functions of any finite number of independent variables. Suppose that the first partial derivatives of w = ƒsx, y, zd are defined throughout an open region containing the point sx0, y0, z0 d and that ƒx, ƒy , and ƒz are continuous at sx0, y0, z0 d. Then ¢w = ƒsx0 + ¢x, y0 + ¢y, z0 + ¢zd - ƒsx0, y0, z0 d = ƒx ¢x + ƒy ¢y + ƒz ¢z + P1 ¢x + P2 ¢y + P3 ¢z,

(15)

AP-48

Appendices

where P1, P2, P3 : 0 as ¢x, ¢y, and ¢z : 0. The partial derivatives ƒx, ƒy, ƒz in Equation (15) are to be evaluated at the point sx0, y0, z0 d. Equation (15) can be proved by treating ¢w as the sum of three increments, ¢w1 = ƒsx0 + ¢x, y0, z0 d - ƒsx0, y0, z0 d ¢w2 = ƒsx0 + ¢x, y0 + ¢y, z0 d - ƒsx0 + ¢x, y0, z0 d ¢w3 = ƒsx0 + ¢x, y0 + ¢y, z0 + ¢zd - ƒsx0 + ¢x, y0 + ¢y, z0 d,

(16) (17) (18)

and applying the Mean Value Theorem to each of these separately. Two coordinates remain constant and only one varies in each of these partial increments ¢w1, ¢w2, ¢w3 . In Equation (17), for example, only y varies, since x is held equal to x0 + ¢x and z is held equal to z0 . Since ƒsx0 + ¢x, y, z0 d is a continuous function of y with a derivative ƒy , it is subject to the Mean Value Theorem, and we have ¢w2 = ƒysx0 + ¢x, y1, z0 d ¢y for some y1 between y0 and y0 + ¢y.

A.11

Taylor’s Formula for Two Variables In this section we use Taylor’s formula to derive the Second Derivative Test for local extreme values (Section 13.7) and the error formula for linearizations of functions of two independent variables (Section 13.6). The use of Taylor’s formula in these derivations leads to an extension of the formula that provides polynomial approximations of all orders for functions of two independent variables.

Derivation of the Second Derivative Test t1

S(a h, b k)

Parametrized segment in R (a th, b tk), a typical point on the segment

Let ƒ(x, y) have continuous partial derivatives in an open region R containing a point P(a, b) where ƒx = ƒy = 0 (Figure A.44). Let h and k be increments small enough to put the point Ssa + h, b + kd and the line segment joining it to P inside R. We parametrize the segment PS as x = a + th,

y = b + tk,

0 … t … 1.

If Fstd = ƒsa + th, b + tkd , the Chain Rule gives t0

F¿std = ƒx

P(a, b)

dy dx + ƒy = hƒx + kƒy . dt dt

Part of open region R

FIGURE A.44 We begin the derivation of the Second Derivative Test at P(a, b) by parametrizing a typical line segment from P to a point S nearby.

Since ƒx and ƒy are differentiable (they have continuous partial derivatives), F¿ is a differentiable function of t and F– =

0F¿ dx 0F¿ dy 0 0 + = shƒx + kƒy d # h + shƒx + kƒy d # k 0x dt 0y dt 0x 0y

= h 2ƒxx + 2hkƒxy + k 2ƒyy .

ƒxy = ƒyx

Since F and F¿ are continuous on [0, 1] and F¿ is differentiable on (0, 1), we can apply Taylor’s formula with n = 2 and a = 0 to obtain Fs1d = Fs0d + F¿s0ds1 - 0d + F–scd Fs1d = Fs0d + F¿s0d +

1 F–scd 2

s1 - 0d2 2 (1)

Appendix 11

Taylor’s Formula for Two Variables

AP-49

for some c between 0 and 1. Writing Equation (1) in terms of ƒ gives ƒsa + h, b + kd = ƒsa, bd + hƒxsa, bd + kƒysa, bd +

1 2 . A h ƒxx + 2hkƒxy + k 2ƒyy B ` 2 sa + ch, b + ckd

(2)

Since ƒxsa, bd = ƒysa, bd = 0, this reduces to ƒsa + h, b + kd - ƒsa, bd =

1 2 . A h ƒxx + 2hkƒxy + k 2ƒyy B ` 2 sa + ch, b + ckd

(3)

The presence of an extremum of ƒ at sa, bd is determined by the sign of ƒsa + h, b + kd - ƒsa, bd. By Equation (3), this is the same as the sign of Qscd = sh 2ƒxx + 2hkƒxy + k 2ƒyy d ƒ sa + ch, b + ckd . Now, if Qs0d Z 0, the sign of Q(c) will be the same as the sign of Q(0) for sufficiently small values of h and k. We can predict the sign of Qs0d = h 2ƒxxsa, bd + 2hkƒxysa, bd + k 2ƒyysa, bd

(4)

from the signs of ƒxx and ƒxx ƒyy - ƒxy2 at (a, b). Multiply both sides of Equation (4) by ƒxx and rearrange the right-hand side to get ƒxx Qs0d = shƒxx + kƒxy d2 + sƒxx ƒyy - ƒxy2 dk 2.

(5)

From Equation (5) we see that 1. 2. 3.

4.

If ƒxx 6 0 and ƒxx ƒyy - ƒxy2 7 0 at (a, b), then Qs0d 6 0 for all sufficiently small nonzero values of h and k, and ƒ has a local maximum value at (a, b). If ƒxx 7 0 and ƒxx ƒyy - ƒxy2 7 0 at (a, b), then Qs0d 7 0 for all sufficiently small nonzero values of h and k, and ƒ has a local minimum value at (a, b). If ƒxx ƒyy - ƒxy2 6 0 at (a, b), there are combinations of arbitrarily small nonzero values of h and k for which Qs0d 7 0, and other values for which Qs0d 6 0. Arbitrarily close to the point P0sa, b, ƒsa, bdd on the surface z = ƒsx, yd there are points above P0 and points below P0 , so ƒ has a saddle point at (a, b). If ƒxx ƒyy - ƒxy2 = 0, another test is needed. The possibility that Q(0) equals zero prevents us from drawing conclusions about the sign of Q(c).

The Error Formula for Linear Approximations We want to show that the difference E(x, y), between the values of a function ƒ(x, y), and its linearization L(x, y) at sx0 , y0 d satisfies the inequality 1 ƒ Esx, yd ƒ … 2 Ms ƒ x - x0 ƒ + ƒ y - y0 ƒ d2. The function ƒ is assumed to have continuous second partial derivatives throughout an open set containing a closed rectangular region R centered at sx0 , y0 d. The number M is an upper bound for ƒ ƒxx ƒ , ƒ ƒyy ƒ , and ƒ ƒxy ƒ on R. The inequality we want comes from Equation (2). We substitute x0 and y0 for a and b, and x - x0 and y - y0 for h and k, respectively, and rearrange the result as ƒsx, yd = ƒsx0 , y0 d + ƒxsx0 , y0 dsx - x0 d + ƒysx0 , y0 ds y - y0 d ('''''''''')'''''''''''* linearization L(x, y)

+ 1 A sx - x0 d2ƒxx + 2sx - x0 ds y - y0 dƒxy + s y - y0 d2ƒyy B ` sx0 + csx - x0d, y0 + cs y - y0dd. 2 ('''''''''''''''')'''''''''''''''''* error E(x, y)

AP-50

Appendices

This equation reveals that 1 ƒ E ƒ … 2 A ƒ x - x0 ƒ 2 ƒ ƒxx ƒ + 2 ƒ x - x0 ƒ ƒ y - y0 ƒ ƒ ƒxy ƒ + ƒ y - y0 ƒ 2 ƒ ƒyy ƒ B . Hence, if M is an upper bound for the values of ƒ ƒxx ƒ , ƒ ƒxy ƒ , and ƒ ƒyy ƒ on R, 1 ƒ E ƒ … 2 A ƒ x - x0 ƒ 2 M + 2 ƒ x - x0 ƒ ƒ y - y0 ƒ M + ƒ y - y0 ƒ 2M B =

1 Ms ƒ x - x0 ƒ + ƒ y - y0 ƒ d2. 2

Taylor’s Formula for Functions of Two Variables The formulas derived earlier for F¿ and F– can be obtained by applying to ƒ(x, y) the operators ah

0 0 + k b 0x 0y

ah

and

2

0 0 02 02 02 + k b = h 2 2 + 2hk + k2 2 . 0x 0y 0x 0y 0x 0y

These are the first two instances of a more general formula, n

F sndstd =

dn 0 0 Fstd = ah + k b ƒsx, yd, 0x 0y dt n

(6)

which says that applying d n>dt n to Fstd gives the same result as applying the operator ah

0 0 + k b 0x 0y

n

to ƒ(x, y) after expanding it by the Binomial Theorem. If partial derivatives of ƒ through order n + 1 are continuous throughout a rectangular region centered at (a, b), we may extend the Taylor formula for F(t) to Fstd = Fs0d + F¿s0dt +

F–s0d 2 F snds0d snd t + Á + t + remainder, 2! n!

and take t = 1 to obtain Fs1d = Fs0d + F¿s0d +

F snds0d F–s0d + Á + + remainder. 2! n!

When we replace the first n derivatives on the right of this last series by their equivalent expressions from Equation (6) evaluated at t = 0 and add the appropriate remainder term, we arrive at the following formula.

Taylor’s Formula for ƒ(x, y) at the Point (a, b) Suppose ƒ(x, y) and its partial derivatives through order n + 1 are continuous throughout an open rectangular region R centered at a point (a, b). Then, throughout R, 1 2 ƒsa + h, b + kd = ƒsa, bd + shƒx + kƒy d ƒ sa, bd + sh ƒxx + 2hkƒxy + k 2ƒyy d ƒ sa, bd 2! n 0 0 1 3 1 + sh ƒxxx + 3h 2kƒxxy + 3hk 2ƒxyy + k 3ƒyyy d ƒ sa, bd + Á + ah + k b ƒ` 0x 0y 3! n! sa, bd +

0 0 1 ah + k b 0x 0y sn + 1d!

n+1

ƒ`

sa + ch, b + ckd

.

(7)

Appendix 11

Taylor’s Formula for Two Variables

AP-51

The first n derivative terms are evaluated at (a, b). The last term is evaluated at some point sa + ch, b + ckd on the line segment joining (a, b) and sa + h, b + kd. If sa, bd = s0, 0d and we treat h and k as independent variables (denoting them now by x and y), then Equation (7) assumes the following form.

Taylor’s Formula for ƒ(x, y) at the Origin ƒsx, yd = ƒs0, 0d + xƒx + yƒy +

1 2 sx ƒxx + 2xyƒxy + y 2ƒyy d 2!

+

n 0n f 0n f 1 3 1 n0 f n-1 n Á ax sx ƒxxx + 3x 2yƒxxy + 3xy 2ƒxyy + y 3ƒyyy d + Á + + nx y b + + y 3! n! 0x n 0y n 0x n - 10y

+

0n + 1 f 0n + 1 f 0n + 1 f 1 ax n + 1 n + 1 + (n + 1)x ny n + Á + y n + 1 n + 1 b ` 0x 0y sn + 1d! 0x 0y scx, cyd

(8)

The first n derivative terms are evaluated at (0, 0). The last term is evaluated at a point on the line segment joining the origin and (x, y). Taylor’s formula provides polynomial approximations of two-variable functions. The first n derivative terms give the polynomial; the last term gives the approximation error. The first three terms of Taylor’s formula give the function’s linearization. To improve on the linearization, we add higher-power terms.

EXAMPLE 1 Find a quadratic approximation to ƒsx, yd = sin x sin y near the origin. How accurate is the approximation if ƒ x ƒ … 0.1 and ƒ y ƒ … 0.1? Solution

We take n = 2 in Equation (8): ƒsx, yd = ƒs0, 0d + sxƒx + yƒy d + +

1 2 sx ƒxx + 2xyƒxy + y 2ƒyy d 2

1 3 sx ƒxxx + 3x 2yƒxxy + 3xy 2ƒxyy + y 3ƒyyy dscx, cyd. 6

Calculating the values of the partial derivatives, ƒs0, 0d = sin x sin y ƒ s0,0d = 0,

ƒxxs0, 0d = -sin x sin y ƒ s0,0d = 0,

ƒxs0, 0d = cos x sin y ƒ s0,0d = 0,

ƒxys0, 0d = cos x cos y ƒ s0,0d = 1,

ƒys0, 0d = sin x cos y ƒ s0,0d = 0,

ƒyys0, 0d = -sin x sin y ƒ s0,0d = 0,

we have the result 1 2 sx s0d + 2xys1d + y 2s0dd, 2 The error in the approximation is

sin x sin y L 0 + 0 + 0 +

Esx, yd =

or

sin x sin y L xy.

1 3 sx ƒxxx + 3x 2yƒxxy + 3xy 2ƒxyy + y 3ƒyyy d ƒ scx, cyd . 6

The third derivatives never exceed 1 in absolute value because they are products of sines and cosines. Also, ƒ x ƒ … 0.1 and ƒ y ƒ … 0.1. Hence 8 1 3 3 3 3 3 ƒ Esx, yd ƒ … 6 ss0.1d + 3s0.1d + 3s0.1d + s0.1d d = 6 s0.1d … 0.00134 (rounded up). The error will not exceed 0.00134 if ƒ x ƒ … 0.1 and ƒ y ƒ … 0.1.

This page intentionally left blank

ANSWERS TO ODD-NUMBERED EXERCISES 29. (a) ƒsxd = e

CHAPTER 1 Section 1.1, pp. 11–13 1. D: s - q , q d, R: [1, q d 3. D: [-2, q d, R: [0, q d 5. D: (- q , 3) ´ (3, q ), R: (- q , 0) ´ (0, q ) 7. (a) Not a function of x because some values of x have two values of y (b) A function of x because for every x there is only one possible y 23 2 x , 4

9. A =

d

p = 3x 11. x =

220x - 20x + 25 13. L = 4 15. s - q , q d

23

A = 2d 2,

,

V =

d3

2, 0, (b) ƒsxd = d 2, 0,

323

f(x) = 5 – 2x

4

–4

–2

2

-2 … x … 0 0 6 x … 1 1 6 x … 3

x, (b) ƒsxd = • -2x + 2, -1,

y

2 1

y = –x 3

2

x

1 2 3 4 5

–2

2

F(t) =

Dec. - q 6 x 6 q 41. Symmetric about the y-axis

t | t|

1

2

3

x

43. Symmetric about the origin

y

1 –4 –3 –2 –1

1 2

Inc. - q 6 x 6 0 and 0 6 x 6 q

–2 –4

y

y = – 1x

– 2 –1 –1 –2

x

2

–2

19. s - q , 0d ´ s0, q d

1 2 3 4

-1 … x 6 0 0 6 x … 1 1 6 x 6 3

1 2

x

4

6 6 6 …

g(x) = √ |x|

–5 –4 –3 –2 –1 –1 –2

2

x x x x

y

4 3 2 1

6

… … … …

33. (a) 0 … x 6 1 (b) -1 6 x … 0 35. Yes 37. Symmetric about the origin 39. Symmetric about the origin

y

y

0 1 2 3

-x, 31. (a) ƒsxd = • 1, - 12 x + 32,

2

17. s - q , q d

0 … x … 1 1 6 x … 2

x, -x + 2,

y

t

4

1 4

–2

y = √⏐x⏐

21. (- q , -5) ´ (-5, -3] ´ [3, 5) ´ (5, q ) 23. (a) For each positive value of (b) For each value of x Z 0 , x, there are two values of y. there are two values of y. y

y2 = x2

⏐y⏐ = x

–4

–2

0

–2 2

Dec. - q 6 x … 0 ; Inc. 0 … x 6 q

y

4

2 4

x

1/8 –1 –1/8

x3 y = –– 8 x 1

2

–1

Inc. - q 6 x 6 q

1

45. No symmetry

2

y 0

2

4

x

6

–1

x

1

–2

0 –1

–1 –4

x 1

2

3

y = – x 3/2

–2 –3

25. y

27.

–4 y

⎧x 0≤x≤1 f(x) = ⎨ ⎩ 2 – x, 1 < x ≤ 2

4

1

0

1

2

–5 2

y = x + 2x

x –2

1

x

Dec. 0 … x 6 q 47. Even 49. Even 51. Odd 53. Even 55. Neither 57. Neither 59. t = 180 61. s = 2.4 63. V = x(14 - 2x)(22 - 2x) 65. (a) h (b) ƒ (c) g 67. (a) s -2, 0d ´ s4, q d 71. C = 5s2 + 12 ) h

y = 4 – x2

A-1

A-2

Chapter 1: Answers to Odd-Numbered Exercises

Section 1.2, pp. 18–21 1. Dƒ : - q 6 x 6 q , Dg : x Ú 1, Rƒ : - q Rg : y Ú 0, Dƒ + g = Dƒ # g = Dg, Rƒ + g : y Ú 3. Dƒ : - q 6 x 6 q , Dg : - q 6 x 6 q , Rg : y Ú 1, Dƒ>g : - q 6 x 6 q , Rƒ>g : 0 Dg>ƒ : - q 6 x 6 q , Rg>ƒ : y Ú 1/2 5. (a) 2

(c) x 2 + 2

(b) 22

(f) -2

(g) x + 10

11. (a) ƒ( g (x))

25.

1 x–1

(e) 5

y= 1 x

1 1

0 –1

–1

(c) g ( g (x))

37.

(d) j ( j (x))

2

x2 x (d) x - 1 1 (e) x - 1 1 (f) x

2x - 5 x x - 1 1 1 + x 1 x

2x 2 - 5

–2

39.

2

0

–3

(e) -1

0

x

6

1

43.

–2

–1

y = (x + 1)

x

y

2/3

1

1

2 x

0 –1

x

1

0

5

2

y = 1 – x 2/3

(f) 0 45.

1

y

47.

y

–1

1

y = √x – 1 – 1

0

1

x + y = 49

y=x

2

x

3

1 y = ––––– x–2

1 x

0

–2

1

2

3

4

–1 –2

49.

51.

y

y 4

3

3

2

x

1

–3 –2 –1

1 y = ––––––– 2 (x – 1)

3 2

1 y=x +2

1 0

2

(1, –1)

–1

1 –1

y

3

y + 1 = (x + 1) 3

2

–2

y = 1 + √x – 1 (1, 1)

–2 –1

y

(–2, –3) 0

3

y

x

x

4

1

2x + 1 (b) Dƒ ⴰ g = (- q , -1] ´ (0, q ), Dg ⴰ ƒ = (-1, q ) (c) Rƒ ⴰ g = [0, 1) ´ (1, q ), Rg ⴰ ƒ = (0, q ) 2x g (x) = x - 1 (a) y = -sx + 7d2 (b) y = -sx - 4d2 (a) Position 4 (b) Position 1 (c) Position 2 (d) Position 3 27. y + 1 = sx + 1d3 sx + 2d2 + s y + 3d2 = 49 2

y

1

x

1 + 1, g ( ƒ (x)) = Ax

x

0

2

41.

x

(d) 0

y = √x + 4

x

2

y = ⏐x – 2⏐ 4

2x - 7 3x + 6

(c) -2

–4

y

( ƒ ⴰ g) (x)

2x 3x

(b) 2

2

y= 1 x

(a) x - 7 (b) x + 2 (c)

21. 23.

2 y–1=

2

ƒ(x)

17. (a) ƒ (g(x)) =

y y–1= 1 x–1

(f) h ( j (ƒ(x)))

g (x)

15. (a) 1

35.

y

(h) x - 6x + 6

(b) j ( g (x))

(e) g (h ( ƒ(x)))

19.

(d) x 2 + 10x + 22 4

1 x - 1

33. y - 1 =

5x + 1 9. C 4x + 1

7. 13 - 3x

13.

6 y 6 q, 1, Rƒ # g : y Ú 0 Rƒ : y = 2, 6 y … 2,

0

1

2

y=

1 +1 x2

1 x

3

–1 0

1

2

x

3

–1 –2 2

53.

y

2

(x + 2) + (y + 3) = 49

29. y = 2x + 0.81

31. y = 2x

y

2 0.9 –0.81

y y = √ x + 0.81

y = 2x

7

–2 –1 0

y = √x

1

1

5 4 3 2 1 1

x

2

y = 2x – 7

4

7/2

x

x

55. (a) D : [0, 2],

R : [2, 3]

(b) D : [0, 2],

R : [-1, 0]

y

y –7

1

3

y = f(x) + 2 y = f (x) – 1

2 0

1

0

1

2

3

4

x –1

1

2

x

A-3

Chapter 1: Answers to Odd-Numbered Exercises (c) D : [0, 2],

R : [0, 2]

(d) D : [0, 2],

y

R : [-1, 0]

75.

y

y 3 1 2

2

y = 2 f(x)

y = – f(x)

1

1

0

y = |x 2 – 1|

x

2

–2 0

1

3

2

(e) D : [-2, 0],

R : [0, 1]

(f) D : [1, 3],

77. (a) Odd (f) Even

2 y = f (x – 1) 1

x

0

(g) D : [-2, 0],

1. (a) 8p m

0

R : [0, 1]

2

1

(h) D : [-1, 1],

1

x

0

57. y = 3x 2 - 3

B

sin u

0

cos u

-1

tan u

0

cot u

UND

23 2 1 2

-

2

1

–1

67.

-p

5. u

55p m 3. 8.4 in. 9 0 -2p>3

23 1

y = – f(x + 1) + 1

y = f(–x)

–1

1 1 59. y = + 2 2x 2

x2 4

4 -

(b)

y

2

63. y =

x

3

R : [0, 1]

y

–2

(b) Odd (c) Odd (d) Even (g) Even (h) Even (i) Odd

x

1

0

61. y = 24x + 1

69. 5 4

1 1

2

3

4

x

2

y = (x –

1

–1 –2

–3 –2 –1

1

2

1)3

3

4

5

x

π

–1

73.

2

–1

3

4

x

–4 –3 –2 –1

–1

–2

-1

1

UND

- 22

UND

1

22

1

y = cos πx

0

1

x

2

–4

y = cos ⎞⎠x – π ⎞⎠ 2

1

1 1

2

3 3

y = – √x

4

x 0 –1

3

6

x 0

–3 –4

0

y y = – sin π x 3

2 1

2

UND

22 -1

19. Period 2p

y

3

1 –1 2x

1 –4 –3 –2 –1

UND

–1

17. Period 6

4 y=

0

22 1

-

y

4 3

0

y = sin 2x

π 2

y

1

x

y = – √2x + 1

71.

1

y

1

+2

–3 –4

1

0

y

3

–2 –1

3p>4

28 , tan x = - 28 3 1 2 , cos x = 11. sin x = 25 25 13. Period p 15. Period 2

y

2

23 sec u -1 -2 2 csc u UND 23 7. cos x = -4>5, tan x = -3>4

p>2

9. sin x = -

65. y = 1 - 27x 3

y

(e) Even

Section 1.3, pp. 27–29

y = f(x + 2) 1

x

2

R : [0, 1]

y

2

–1

1 –1

–1

y

–2

–1

x

–1

π 2

π

2π

x

A-4

Chapter 1: Answers to Odd-Numbered Exercises 23. Period p>2 , symmetric about the origin

21. Period 2p y

9. [-5, 5] by [-6, 6]

s y = sin ⎞⎠x – π ⎞⎠ + 1 4

2

2

s = cot 2t

–5 –4 –π 0 4

π 4

3π 4

x

7π 4

–π

–π 2

π 2

0

y f(x) = x √9 – x 2

5 4 3 2 1

1

1

11. [-2, 6] by [-5, 4]

y 4 2 1

–2 –1

1 2 3 4 5

x

–2 –1

t

π

–1

y = 2x – 3x 2/3

3

1

–1

2

4

5

x

6

–4 –5

–2

13. [-2, 8] by [-5, 10] 25. Period 4, symmetric about the y-axis

29. D : s - q , q d, R : y = - 1, 0, 1

s

y = ⎣sin x⎦ –2π

1 –3

–2

–1

–π

8

2

π

1

t

3

10 9 8 7 6 5 4 3 2

y = 5x 2/5 – 2x

y = sin x 2

2π x

1

y

10

y πt s = sec — 2

15. [-3, 3] by [0, 10]

y

–2

2

4

6

8

x

–2

–1

–4

–1

–3

17. [-10, 10] by [-10, 10]

–2

–1

1

47. 53.

41. - cos x

2 + 22 4

49.

43.

2 - 23 4

26 + 22 4 51.

p 2p 4p 5p , , , 3 3 3 3

59. 27 L 2.65

63. a = 1.464

2 67. A = - p , B = 4, 1 C = 0, D = p

C = -p, D = -1 y

y y = 2sin (x + π) – 1

3 π

1

π 2

–1

3π 2

–10 –8 –6 –4 1

3

5

t

– π1

0.5

y

6 4

x–1 x2 – x – 6

2 4 6 8 10

x

p p , d by 125 125 [-1.25, 1.25]

1. d 3. d 5. [-3, 5] by [-15, 40]

–5

–2 –4 –6

f (x) =

5

4

x

6x 2 – 15x + 6 4x 2 – 10x x

10

[-1.25, 1.25] y

7. [-3, 6] by [-250, 50]

f(x) = x 4 – 4x 3 + 15

3

27. [-100p, 100p] by

y 1.0

1.0

y

2

y f (x) =

–2 –4 –6

1

23. [-6, 10] by [-6, 6]

25. c-

Section 1.4, p. 33

x2 + 2 x2 + 1

1.0

6 4 2

–1 –3

2 4 6 8 10

y

1 π

f (x) =

x

21. [-10, 10] by [-6, 6]

x

5π 2

2.5

–4 –3 –2 –1

πt y = – π2 sin ⎞⎠ ⎞⎠ + π1 2

x

3.0

x+3 x+2

2.0

–2 –4 –6 –8

3

y

y=

8 6 4

–10 –8 –6 –4

p p 5p 3p , , , 6 2 6 2

65. A = 2, B = 2p,

–π 2

22 + 26 4

45.

2

19. [-4, 4] by [0, 3]

y

39. -cos x

y = |x 2 – 1|

x y = cos 50

( )

y = sin 250x

f(x) = x 5 – 5x 4 + 10 0.5

50

40 30

–2

1

20

–100 10 –2 –1 –10

2

3

4

5

6

x –0.02

0.02

x

–300

300

–50

–150 1

3

4

x

–200 –250

–0.5 –1.0

x

A-5

Chapter 1: Answers to Odd-Numbered Exercises p p , d by 15 15 [-0.25, 0.25]

29. c-

y

5.

31.

y=x+

1 10

7.

(x + 1)2 + (y – 2)2 = 9 1 –4

–0.1

0.1

–2

–1

y 2x 1

5

y ex

4

4

2

0.1

y 2 –x 1

5

1 y x e

y

sin 30x

0.2

–0.2

y

y

3

3

2

2

1

x

2

1

–4 –3 –2 –1

1

2

3

4

x

x

0.2

–4 –3 –2 –1

0

1

9.

2

3

x

4

–2

y

–0.2

3 2

33.

35. y

–2

y

f(x) = –tan 2x

4

2.0

3

1.5

2

1.0

1

0.5

–1

1

2

x

3

–6

–4 –3 –2 –1 f(x) = sin 2x + cos 3x

2

4

x

6

–2.0

39. y 8

3

7

2

6

1 2

3

4

5

6

Section 1.6, pp. 49–51

4

–1

1. One-to-one 3. Not one-to-one 5. One-to-one 7. Not one-to-one 9. One-to-one 11. D: (0, 1] R: [0, q d 13. D: [-1, 1] R: [-p>2, p>2]

3

–2 y=

–3

2

1 x–3

1

–4

161>4 = 2 13. 41>2 = 2 15. 5 17. 1423 19. 4 D: - q 6 x 6 q ; R: 0 6 y 6 1>2 D: - q 6 t 6 q ; R: 1 6 y 6 q x L 2.3219 27. x L -0.6309 29. After 19 years t>14 1 31. (a) A(t) = 6.6 a b (b) About 38 days later 2 33. L 11.433 years, or when interest is paid 35. 248 L 2.815 * 10 14

y = x ⎣x⎦

5 x

y 1 ex

–3

11. 21. 23. 25.

–3

4

x

4

–5

–2

y

3

–4

–2

37.

2

–2 y 1 e –x

–1

–4

1

–1

–5 –4 –3 –2 –1

1

2

3

4

y

y

x

5

y=f

–1

–1

2 1

(x) y=x

–2 1

1.

3.

y

3–x

y 4x

6 5

15. D: [0, 6] R: [0, 3]

5 4 3 2 1

y 2– t

y 2x

4 –4 –3 –2 –1 2 1 0

y = f (x) x

1

2

3

4

x

–1 –2 –3 –4 –5

17. (a) Symmetric about the line y = x y

y y=x

3

–4 –3 –2 –1

y=x

y

y y (15)x

(x)

1 – 2

x

1

–1

1 2

– –1 2

y = f(x)

Section 1.5, pp. 38–39

y=f

1

2

3

4

t

1

6

y = √1 – x 2 0≤x≤1

y = f(x) y –2 t

3 0 y=f

3

–1

(x)

6

x

1

x

A-6

Chapter 2: Answers to Odd-Numbered Exercises

19. ƒ -1sxd = 2x - 1

77. (a) y = ln x - 3

3 21. ƒ -1sxd = 2 x + 1

23. ƒ sxd = 2x - 1 -1

(b) y = ln (x - 1)

(c) y = 3 + ln (x + 1)

-1

(e) y = ln (-x) 79. -0.7667

25. ƒ sxd = 2x ; D: - q 6 x 6 q ; R: - q 6 y 6 q 5

3 x - 1 ; D: - q 6 x 6 q ; 27. ƒ -1sxd = 2 q 6 y 6 q R: 1 ; D: x 7 0 ; R: y 7 0 29. ƒ -1sxd = 2x 2x + 3 ; D: - q 6 x 6 q , x Z 1; 31. ƒ -1(x) = x - 1 R: - q 6 y 6 q , y Z 2

(d) y = ln (x - 2) - 4

(f) y = e x

1 81. (a) Amount = 8 a b 2 83. L 43.592 years

t>12

(b) 36 hours

CHAPTER 2

33. ƒ -1(x) = 1 - 2x + 1; D: -1 … x 6 q ; R: - q 6 y … 1

Section 2.1, pp. 57–59

2x + b ; 35. ƒ - 1sxd = x - 1 D: - q 6 x 6 q , x Z 1, 1 37. (a) ƒ -1sxd = m x

323 4 3. (a) - p (b) - p 5. 1 7. (a) 4 (b) y = 4x - 7 9. (a) 2 (b) y = 2x - 7 11. (a) 12 (b) y = 12x - 16 13. (a) -9 (b) y = -9x - 2 15. Your estimates may not completely agree with these. (a) PQ1 PQ2 PQ3 PQ4 The appropriate units are m> sec. 43 46 49 50

1. (a) 19

R: - q 6 y 6 q , y Z 2

(b) The graph of ƒ -1 is the line through the origin with slope 1> m. 39. (a) ƒ -1sxd = x - 1 y=x+1 y=x

2 1 –2

–1

(b) L 50 m>sec or 180 km> h

y=x–1 1

x

2

17. (a)

–1 –2

(b) ƒ -1sxd = x - b . The graph of ƒ -1 is a line parallel to the graph of ƒ. The graphs of ƒ and ƒ -1 lie on opposite sides of the line y = x and are equidistant from that line. (c) Their graphs will be parallel to one another and lie on opposite sides of the line y = x equidistant from that line. 41. (a) ln 3 - 2 ln 2 2 (d) ln 3 3 43. (a) ln 5

49. e 2t + 4

(c) -ln 2

1 (f) s3 ln 3 - ln 2d 2 2t 2 (c) ln a b b

1 (e) ln 3 + ln 2 2 (b) ln sx - 3d

45. (a) 7.2 (b) 47. (a) 1

(b) 2 sln 2 - ln 3d

x (c) y

1 x2

(b) 1

53. y = 2xe x + 1

(b) k = s1>10d ln 2 (c) k = 1000 ln a ln .4 ln 2 57. (a) t = -10 ln 3 (b) t = (c) t = k ln .2 59. 4 sln xd2 61. (a) 7 (b) 22 (c) 75 (d) 2 (e) 0.5 (f) -1 3 ln 63. (a) 2x (b) x 2 (c) sin x 65. (a) (b) 3 (c) 2 ln 2 67. (a) -p>6 (b) p>4 (c) -p>3 (b) p>2

71. Yes, g(x) is also one-to-one. 73. Yes, ƒ ⴰ g is also one-to-one. 75. (a) ƒ -1(x) = log2 a

x b 100 - x

100 0 2000

(b) ƒ -1(x) = log1.1 a

01

02

03

x

04

Year

(b) L $56,000/year (c) L $42,000/year 19. (a) 0.414213, 0.449489, s 21 + h - 1d>h

(b) gsxd = 1x

1 + h

1.1

21 + h

1.04880 1.004987 1.0004998 1.0000499

s 21 + h - 1d>h 0.4880

1.01

1.001

0.4987

0.4998

1.0001 0.499

1.000001

1.000005 1.0000005

55. (a) k = ln 2

69. (a) p

200

1.00001

(c) -x 2 - y 2

51. e5t + b

y Profit (1000s)

y

(b) 1

x b 50 - x

0.5

0.5 (c) 0.5

(d) 0.5

21. (a) 15 mph, 3.3 mph, 10 mph (c) 20 mph when t = 3.5 hr

(b) 10 mph, 0 mph, 4 mph

Section 2.2, pp. 67–70 1. (a) Does not exist. As x approaches 1 from the right, g(x) approaches 0. As x approaches 1 from the left, g(x) approaches 1. There is no single number L that all the values g(x) get arbitrarily close to as x : 1. (b) 1 (c) 0 (d) 1> 2 3. (a) True (b) True (c) False (d) False (e) False (f) True (g) True

Chapter 2: Answers to Odd-Numbered Exercises 5. As x approaches 0 from the left, x> ƒ x ƒ approaches -1. As x approaches 0 from the right, x> ƒ x ƒ approaches 1. There is no single number L that the function values all get arbitrarily close to as x : 0. 7. Nothing can be said. 9. No; no; no 11. -9 13. -8 15. 5> 8 17. 27 19. 16 21. 3> 2 23. 1> 10 25. -7 27. 3> 2 29. -1>2 31. -1 33. 4> 3 35. 1> 6 37. 4 39. 1> 2 41. 3> 2 43. -1 45. 1 47. 1>3 49. 24 - p 51. (a) Quotient Rule (b) Difference and Power Rules (c) Sum and Constant Multiple Rules 53. (a) -10 (b) -20 (c) -1 (d) 5> 7 55. (a) 4 (b) -21 (c) -12 (d) -7>3 57. 2 59. 3 61. 1>s2 27d 63. 25 65. (a) The limit is 1. 67. (a) ƒsxd = sx 2 - 9d>sx + 3d x ƒ(x)

- 3.1 -6.1

-3.01 -6.01

-3.001 -6.001

-3.0001 -6.0001

-3.00001 -6.00001

- 3.000001 -6.000001

x ƒ(x)

-2.9 -5.9

-2.99 -5.99

-2.999 -5.999

-2.9999 -5.9999

-2.99999 -5.99999

-2.999999 -5.999999

lim ƒsxd = -6

(c)

x: -3

69. (a) Gsxd = sx + 6d>sx 2 + 4x - 12d x -5.9 G(x) -.126582 -5.99999 -.1250001

-5.999 -.1250156

-5.9999 -.1250015

-6.01 -.124843

Section 2.3, pp. 76–79 1. d = 2

(

3. d = 1>2

5

(

5. d = 1>18

x

7

(

– 7/2 – 3

x

–1/2

(

(

4/9

1/2

x

4/7

7. d = 0.1 9. d = 7>16 11. d = 25 - 2 15. s3.99, 4.01d, d = 0.01 17. s -0.19, 0.21d, 19. s3, 15d, d = 5 21. s10>3, 5d, d = 2>3 23. s - 24.5, - 23.5d, 25. s 215, 217d,

13. d = 0.36 d = 0.19

d = 24.5 - 2 L 0.12

d = 217 - 4 L 0.12

0.03 0.03 27. a2 - m , 2 + m b,

d =

0.03 m

29. a

c c c 1 1 31. L = -3, d = 0.01 - m , m + b, d = m 2 2 33. L = 4, d = 0.05 35. L = 4, d = 0.75 55. [3.384, 3.387]. To be safe, the left endpoint was rounded up and the right endpoint rounded down. 59. The limit does not exist as x approaches 3.

-6.001 -.124984

-6.0001 -.124998

-6.000001 -.124999

True True True 2, 1

(c) 3, 3 5. (a) No 7. (a)

(b) True (c) False (d) True (g) False (h) False (i) False (l) False (b) No, lim + ƒsxd Z lim - ƒsxd x:2

(d) Yes, 3 (b) Yes, 0

71. (a) ƒsxd = sx 2 - 1d>s ƒ x ƒ - 1d -1.1 2.1

-1.01 2.01

- 1.001 2.001

-1.0001 2.0001

-1.00001 2.00001

-1.000001 2.000001

x ƒ(x)

-.9 1.9

-.99 1.99

- .999 1.999

-.9999 1.9999

-.99999 1.99999

-.999999 1.999999

lim ƒsxd = 2

(c) No (b) 1, 1

(c) Yes, 1

⎧ x 3, x ≠ 1 y=⎨ ⎩0, x = 1

1

x ƒ(x)

(e) True (j) False

x:2

y

x: -6

(c)

(

1

1. (a) (f) (k) 3. (a)

lim Gsxd = -1>8 = -0.125

(c)

77. c = 0, 1, -1; the limit is 0 at c = 0, and 1 at c = 1, -1. 79. 7 81. (a) 5 (b) 5 83. (a) 0 (b) 0

Section 2.4, pp. 84–86

-5.999999 -.1250000

x -6.1 G(x) -.123456 -6.00001 -.124999

-5.99 -.1251564

x 1

–1 –1

9. (a) D : 0 … x … 2, R : 0 6 y … 1 and y = 2 (b) s0, 1d ´ s1, 2d (c) x = 2 (d) x = 0

x: -1

73. (a) gsud = ssin ud>u u .1 .01 .001 .0001 .00001 gsud .998334 .999983 .999999 .999999 .999999

.000001 .999999

u -.1 -.01 -.001 - .0001 -.00001 gsud .998334 .999983 .999999 .999999 .999999

-.000001 .999999

y 2

⎧√1 – x 2 , 0 ≤ x < 1 ⎪ y = ⎨ 1, 1≤x<2 ⎪ x=2 ⎩ 2,

1

lim gsud = 1

u :0

75. (a) ƒsxd = x 1>s1 - xd x .9 .99 .999 .9999 .99999 ƒ(x) .348678 .366032 .367695 .367861 .367877

x 0

.999999 .367879

x 1.1 1.01 1.001 1.0001 1.00001 1.000001 ƒ(x) .385543 .369711 .368063 .367897 .367881 .367878 lim ƒsxd L 0.36788

x: 1

A-7

1

2

11. 23 13. 1 15. 2> 25 17. (a) 1 (b) -1 19. (a) 1 (b) 2> 3 21. 1 23. 3> 4 25. 2 27. 1> 2 29. 2 31. 0 33. 1 35. 1> 2 37. 0 39. 3> 8 41. 3

47. d = P2, lim + 2x - 5 = 0

51. (a) 400

x:5

(b) 399

(c) The limit does not exist.

A-8

Chapter 2: Answers to Odd-Numbered Exercises

Section 2.5, pp. 95–97 1. 3. 7. 13. 17. 21. 23. 25. 31. 35. 39. 47. 73.

71. Here is one possibility.

No; discontinuous at x = 2 ; not defined at x = 2 Continuous 5. (a) Yes (b) Yes (c) Yes (d) Yes (a) No (b) No 9. 0 11. 1, nonremovable; 0, removable All x except x = 2 15. All x except x = 3, x = 1 All x 19. All x except x = 0 All x except np>2, n any integer All x except np>2, n an odd integer All x Ú -3>2 27. All x 29. All x 0; continuous at x = p 33. 1; continuous at y = 1 22>2; continuous at t = 0 37. 1; continuous at x = 0 gs3d = 6 41. ƒs1d = 3>2 43. a = 4>3 45. a = -2, 3 a = 5>2, b = -1>2 71. x L 1.8794, -1.5321, -0.3473 x L 1.7549 75. x L 3.5156 77. x L 0.7391

y y = f (x)

1

x

0

y

y

5

5

–2

1

–1

2

3

4

–3

–2

x=1

–5

–1

75. Here is one possibility.

2

x

81. 0 83. -3>4 85. 5>2 93. (a) For every positive real number B there exists a corresponding number d 7 0 such that for all x c - d 6 x 6 c Q ƒsxd 7 B . (b) For every negative real number -B there exists a corresponding number d 7 0 such that for all x c 6 x 6 c + d Q ƒsxd 6 -B . (c) For every negative real number -B there exists a corresponding number d 7 0 such that for all x c - d 6 x 6 c Q ƒsxd 6 -B . 101. y 6

6 (2, 4)

x

–3

2 3 4 y= x –4 =x+1– x–1 x–1

2

y=

0

1 2

x2 1 =x+1+ x–1 x–1 3

4

2 1

x

5

–3

1.5 y=1

y= x+3 x+2

(0, 0)

3 2 1

–4 –3 –2 –1 –3 y= 1 x+2

–2

0

103.

x

(–1, –2) –2 –3

4

x

5

105. y

y x = –2

(1, 2)

2

y=x 1 2 3 4

3

–2

y

x = –2

1

0

–2

y

y=x+1

5

3 y=x+1

69. Here is one possibility.

x

–1

5 1

5

h(x) = x , x ≠ 0 | x| 1

y= 1 2x + 4

0

4

79. At most one

y

–10

67.

2

y

99.

–5

–10

1

0

4 –4

3

1

x = –2 x

1 (x – 2)2

2

10 1 x–1

f (x) =

3

0

1. (a) 0 (b) -2 (c) 2 (d) Does not exist (e) -1 (f) q (g) Does not exist (h) 1 (i) 0 3. (a) -3 (b) -3 5. (a) 1> 2 (b) 1> 2 7. (a) -5>3 (b) -5>3 9. 0 11. -1 13. (a) 2> 5 (b) 2> 5 15. (a) 0 (b) 0 17. (a) 7 (b) 7 19. (a) 0 (b) 0 21. (a) -2>3 (b) -2>3 23. 2 25. q 27. 0 29. 1 31. q 33. 1 35. 1>2 37. q 39. - q 41. - q 43. q 45. (a) q (b) - q 47. q 49. q 51. - q 53. (a) q (b) - q (c) - q (d) q 55. (a) - q (b) q (c) 0 (d) 3> 2 57. (a) - q (b) 1> 4 (c) 1> 4 (d) 1> 4 (e) It will be - q . 59. (a) - q (b) q 61. (a) q (b) q (c) q (d) q 63. 65.

y=

5 4

–1

Section 2.6, pp. 108–110

10

73. Here is one possibility.

y

x

y = – 1x

1

–1

y= 1

–1

x2

y=

–1 x

x

√4 – x 2

1

x –2

1

–1

2

–1 –2 x=2

x

A-9

Chapter 3: Answers to Odd-Numbered Exercises 107.

y

CHAPTER 3

3

Section 3.1, pp. 119–120

2 y = x 2/3 + 1 x 1/3

1

1. P1: m1 = 1, P2: m2 = 5 x

1

–3 –2 –1 –1

2

3. P1: m1 = 5>2, P2: m2 = -1>2

5. y = 2x + 5

3

7. y = x + 1 y

y

–2 –3

5

109. At q : q , at - q : 0

y=x+1 4

y = 2x + 5

y = 2 √x

4 3 (–1, 3)

Practice Exercises, pp. 111–113 1. At x = -1:

lim - ƒsxd =

x: -1

3

2

lim + ƒsxd = 1 , so

1

lim ƒsxd = 1 = ƒs -1d ; continuous at x = -1

1

lim ƒsxd = lim + ƒsxd = 0 , so lim ƒsxd = 0 .

x:0 -

x: 0

x: 0

However, ƒs0d Z 0 , so ƒ is discontinuous at x = 0 . The discontinuity can be removed by redefining ƒ(0) to be 0. lim - ƒsxd = -1 and lim + ƒsxd = 1 , so lim ƒsxd

At x = 1:

x:1

x: 1

(1, 2)

2

x: -1

x: -1

At x = 0:

y=4–x

2

–3

–2

0

–1

0

x

2

1

1

2

3

4

9. y = 12x + 16 y

x:1

does not exist. The function is discontinuous at x = 1 , and the discontinuity is not removable.

y = x3

y = 12x + 16

x

–2

y y = f(x) 1

–1

0

x

1

(– 2, –8)

–8

–1

3. (a) -21

(b) 49 (c) 0 (d) 1 (e) 1 (f) 7 1 (g) -7 (h) 5. 4 7 7. (a) s - q , + q d (b) [0, q d (c) s - q , 0d and s0, q d (d) s0, q d 9. (a) Does not exist (b) 0 1 1 11. 13. 2x 15. 17. 2>3 19. 2>p 2 4 23. 4 25. - q 27. 0 29. 2 31. 0

21. 1

35. No in both cases, because lim ƒsxd does not exist, and x: 1

11. m = 4, y - 5 = 4sx - 2d 13. m = -2, y - 3 = -2sx - 3d 15. m = 12, y - 8 = 12st - 2d 1 1 17. m = , y - 2 = sx - 4d 4 4 19. m = -10 21. m = -1>4 23. s -2, -5d 25. y = -sx + 1d, y = -sx - 3d 27. 19.6 m> sec 29. 6p 33. Yes 35. Yes 37. (a) Nowhere 39. (a) At x = 0 41. (a) Nowhere 43. (a) At x = 1 45. (a) At x = 0

lim ƒsxd does not exist.

x: -1

37. Yes, ƒ does have a continuous extension, to a = 1 with ƒ(1) = 4>3. 39. No 41. 2>5 43. 0 45. - q 47. 0 49. 1 51. 1 53. -p>2 55. (a) x = 3 (b) x = 1 (c) x = -4

Additional and Advanced Exercises, pp. 113–115 3. 0; the left-hand limit was needed because the function is undefined for y 7 c . 5. 65 6 t 6 75 ; within 5°F 13. (a) B (b) A (c) A (d) A 21. (a) lim r+sad = 0.5, lim +r+sad = 1 a : -1

a: 0

(b) lim r-sad does not exist, lim + r-sad = 1 a : -1

a: 0

25. 0

27. 1

29. 4

31. y = 2x

33. y = x, y = -x

Section 3.2, pp. 125–129 1. -2x, 6, 0, -2 5. 11. 17.

3

3. -

2 1 2 , 2, - , 3 4 t 3 23

3

3 1 , , 2 223u 223 222 ,

-1 2sq + 1d2q + 1 -4

,

sx - 2d2x - 2 -1 21. 1>8 23. sx + 2d2

7. 6x 2

13. 1 -

9 ,0 x2

y - 4 = 25.

9.

1 s2t + 1d2 15. 3t 2 - 2t, 5

1 sx - 6d 2

-1 sx - 1d2

19. 6

27. (b)

29. (d)

x

A-10

Chapter 3: Answers to Odd-Numbered Exercises

31. (a) x = 0, 1, 4 (b) y' 4 3 2 1 –8 –6 –4 –2 0

33.

15. y¿ = 3x 2 + 10x + 2 -

y' 3.2

f ' on (– 4, 6)

2 4 6 8

19. g¿sxd =

1 0.6 –0.7

x

84 85 86 87 88

x

23. ƒ¿ssd =

–3.3

27. y¿ = i) 1.5 °F>hr ii) 2.9 °F>hr iii) 0 °F>hr iv) -3.7 °F>hr (b) 7.3 °F>hr at 12 P.M., -11 °F>hr at 6 P.M.

35. (a)

(c)

Slope dT y = –– dt

9 (ºF/hr) 6

2 4

6

8 10 12

t (hrs)

–9 –12

f (0 + h) - f (0) 37. Since lim+ = 1 h h: 0 f (0 + h) - f (0) while lim= 0, h h:0 f (0 + h) - f (0) does not exist and ƒ(x) is not difƒ¿(0) = lim h h:0 ferentiable at x = 0. f (1 + h) - f (1) 39. Since lim+ = 2 while h h: 0 f (1 + h) - f (1) f (1 + h) - f (1) 1 = , ƒ¿(1) = lim lim 2 h h h: 0 h :0

43. 45. 47.

does not exist and ƒ(x) is not differentiable at x = 1. Since ƒ(x) is not continuous at x = 0, ƒ(x) is not differentiable at x = 0. (a) -3 … x … 2 (b) None (c) None (a) -3 … x 6 0, 0 6 x … 3 (b) None (c) x = 0 (a) -1 … x 6 0, 0 6 x … 2 (b) x = 0 (c) None

Section 3.3, pp. 137–139 dy d 2y = -2x, 2 = -2 dx dx ds d 2s 2 3. = 15t - 15t 4, 2 = 30t - 60t 3 dt dt 1.

dy d 2y = 4x 2 - 1 + 2e x, 2 = 8x + 2e x dx dx 18 6 dw 1 d 2w 2 7. = - 3 + 2, 2 = 4 - 3 dz z z dz z z dy d 2y 9. = 12x - 10 + 10x -3, 2 = 12 - 30x -4 dx dx 5.

11.

1 2ss 2s + 1d

2

-19 s3x - 2d2

dy t 2 - 2t - 1 = dt s1 + t 2 d2 25. y¿ = -

-4x 3 - 3x 2 + 1 sx - 1d2sx 2 + x + 1d2 2

1 + 2x -3>2 x2

29. y¿ = -2e -x + 3e 3x

9 5>4 x - 2e -2x 4 dr ds se s - e s 2 = 3t 1>2 37. y¿ = = - ex e - 1 39. 35. 5>7 dt ds s2 7x 3 2 s4d 41. y¿ = 2x - 3x - 1, y– = 6x - 3, y‡ = 12x, y = 12, 31. y¿ = 3x 2e x + x 3e x

33. y¿ =

43. y¿ = 3x 2 + 4x - 8, y– = 6x + 4, y‡ = 6, y (n) = 0 for n Ú 4

–6

41.

21.

17. y¿ =

y snd = 0 for n Ú 5

3 0 –3

x2 + x + 4 sx + 0.5d2

1 x2

5 d 2r 5 dr 2 -2 = 3 + 2, 2 = 4 - 3 ds 3s 2s ds s s

13. y¿ = -5x 4 + 12x 2 - 2x - 3

45. y¿ = 2x - 7x -2, y– = 2 + 14x -3 dr d 2r = 3u-4, 2 = -12u-5 du du d 2w dw = -z -2 - 1, 2 = 2z -3 49. dz dz 47.

51.

dw d 2w = 6ze 2z(1 + z), 2 = 6e 2z(1 + 4z + 2z 2) dz dz

(b) -7 (c) 7>25 (d) 20 5 x 55. (a) y = - + (b) m = - 4 at (0, 1) 8 4 (c) y = 8x - 15, y = 8x + 17

53. (a) 13

57. y = 4x, y = 2 61. (2, 4) 67. 50

59. a = 1, b = 1, c = 0

63. (0, 0), (4, 2)

65. (a) y = 2x + 2

(c) (2, 6)

69. a = -3

71. P¿sxd = nan x n - 1 + sn - 1dan - 1x n - 2 + Á + 2a2 x + a1 73. The Product Rule is then the Constant Multiple Rule, so the latter is a special case of the Product Rule. d suywd = uyw¿ + uy¿w + u¿yw 75. (a) dx d su u u u d = u1 u2 u3 u4 ¿ + u1 u2 u3 ¿u4 + u1 u2 ¿u3 u4 + (b) dx 1 2 3 4 u1 ¿u2 u3 u4 d su Á un d = u1 u2 Á un - 1un ¿ + u1 u2 Á un - 2un - 1 ¿un + (c) dx 1 Á + u1 ¿u2 Á un 77.

2an 2 nRT dP + = 2 dV sV - nbd V3

Section 3.4, pp. 146–149 1. (a) (b) (c) 3. (a) (b) (c)

- 2 m, -1 m> sec 3 m> sec, 1 m> sec; 2 m>sec2, 2 m>sec2 Changes direction at t = 3>2 sec -9 m, -3 m>sec 3 m> sec, 12 m> sec; 6 m>sec2, -12 m>sec2 No change in direction

Chapter 3: Answers to Odd-Numbered Exercises

(b) y 7 0 when 0 … t 6 6.25 Q the object moves up; y 6 0 when 6.25 6 t … 12.5 Q the object moves down. (c) The object changes direction at t = 6.25 sec . (d) The object speeds up on (6.25, 12.5] and slows down on [0, 6.25). (e) The object is moving fastest at the endpoints t = 0 and t = 12.5 when it is traveling 200 ft> sec. It’s moving slowest at t = 6.25 when the speed is 0. (f) When t = 6.25 the object is s = 625 m from the origin and farthest away.

5. (a) -20 m, -5 m>sec (b) 45 m> sec, (1>5) m> sec; 140 m>sec2, s4>25d m>sec2 (c) No change in direction 7. (a) as1d = -6 m>sec2, as3d = 6 m>sec2 (b) ys2d = 3 m>sec (c) 6 m 9. Mars: L 7.5 sec, Jupiter: L 1.2 sec 11. gs = 0.75 m>sec2 13. (a) y = -32t, ƒ y ƒ = 32t ft>sec, a = -32 ft>sec2 (b) t L 3.3 sec (c) y L -107.0 ft>sec 15. (a) t = 2, t = 7 (b) 3 … t … 6 (c) (d) ⎪y⎪ (m/sec) 4 3 2 1

Speed

0

2

4

6

8

0 –1 –2 –3 –4

t (sec)

10

21. 23. 25.

27.

dy a = –– dt

6

4 y=61– t 2 12

2

1 2 3 4 5 6 7 8 9 10

t

t

12

–1

D =

6250 m 9

4

t

–5 s = t 3 – 6t 2 + 7t

–10

6 ; 215 sec 3 6 - 215 6 + 215 (b) y 6 0 when 6 t 6 Q the object 3 3 6 - 215 moves left; y 7 0 when 0 … t 6 or 3 6 + 215 6 t … 4 Q the object moves right. 3 (a) y = 0 when t =

(c) The object changes direction at t = (d) The object speeds up on a and slows down on c0,

6 ; 215 sec . 3

6 - 215 6 + 215 , 2b ´ a , 4d 3 3

6 + 215 6 - 215 b ´ a2, b. 3 3

6 + 215 the object is at position s L -6.303 3 units and farthest from the origin.

Section 3.5, pp. 153–156 1. -10 - 3 sin x

3. 2x cos x - x 2 sin x 2 5. -csc x cot x 7. sin x sec2 x + sin x 9. 0 2x -csc2 x 11. 13. 4 tan x sec x - csc2 x 15. x 2 cos x s1 + cot xd2

s s = 200t – 16t2

600 400 ds = 200 – 32t dt

12 –200

5

dy t = –1 dt 12

29. t = 25 sec

200

ds = 3t 2 – 12t + 7 dt

(f) When t =

1

31.

d2s = 6t – 12 dt2

(e) The object is moving fastest at t = 0 and t = 4 when it is 6 ; 215 moving 7 units> sec and slowest at t = sec . 3

5

3

s

10

190 ft> sec (b) 2 sec (c) 8 sec, 0 ft> sec 10.8 sec, 90 ft> sec (e) 2.8 sec Greatest acceleration happens 2 sec after launch Constant acceleration between 2 and 10.8 sec, -32 ft>sec2 4 (a) sec , 280 cm> sec (b) 560 cm> sec, 980 cm>sec2 7 (c) 29.75 flashes> sec C = position, A = velocity, B = acceleration (a) $110> machine (b) $80 (c) $79.90 (a) b¿s0d = 10 4 bacteria>h (b) b¿s5d = 0 bacteria>h (c) b¿s10d = - 10 4 bacteria>h dy t (a) = - 1 12 dt dy (b) The largest value of is 0 m> h when t = 12 and dt dy the smallest value of is -1 m>h when t = 0 . dt y (c)

17. (a) (d) (f) (g) 19.

33.

a

3

A-11

t

d2s = –32 dt2

(a) y = 0 when t = 6.25 sec

17. 3x 2 sin x cos x + x 3 cos2 x - x 3 sin2 x -2 csc t cot t 19. sec2 t + e -t 21. 23. -u su cos u + 2 sin ud s1 - csc td2 25. sec u csc u stan u - cot ud = sec2 u - csc2 u 27. sec2 q

29. sec2 q

A-12 31.

Chapter 3: Answers to Odd-Numbered Exercises

q 3 cos q - q 2 sin q - q cos q - sin q

(q - 1) 33. (a) 2 csc3 x - csc x (b) 2 sec3 x - sec x 2

35.

2

13. With u = ssx 2>8d + x - s1>xdd, y = u 4 : 4u 3 # a

y

15. y=x

y = –x –

y = sin x

1 –3/2 – –/2

–1

37.

/2

3/2

y = –1

(3/2, –1)

2

x

17.

19.

y

y = sec x

21. (–/3, 2)

2

23.

√2

/4, √2

1

–/2 –/3

/4

/2

1

47. a

(/4, 1) y = 2x – + 1 2 /2

x

y = 2x + – 1 2

sx + 1d4

49. 2ue -u sin A e -u

t + 2 t b cos a b 2st + 1d3>2 2t + 1

2

51. 2p sin spt - 2d cos spt - 2d

53.

2

B

8 sin s2td s1 + cos 2td5

55. 10t 10 tan9 t sec2 t + 10t 9 tan10 t

–1 (–/4, –1)

45. (a) y = -x + p>2 + 2 (b) y = 4 - 23 47. 0 49. 13>2 51. -1 53. 0 55. - 22 m>sec, 22 m>sec, 22 m>sec2, 22 m>sec3 57. c = 9 59. sin x 61. (a) i) 10 cm ii) 5 cm iii) -5 12 L -7.1 cm (b) i) 0 cm> sec ii) -5 13 L -8.7 cm>sec iii) -512 L -7.1 cm>sec

dy cos2 (pt - 1) 57. dt = -2p sin (pt - 1) # cos (pt - 1) # e 59.

-3t 6 (t 2 + 4) (t 3 - 4t) 4

63. a1 + tan4 a 65. -

61. -2 cos scos s2t - 5dd ssin s2t - 5dd 2

t t t b b atan3 a bsec2 a b b 12 12 12

t sin st 2 d 21 + cos st 2 d

67. 6 tan (sin3 t) sec2 (sin3 t) sin2 t cos t

69. 3(2t 2 - 5) 3 (18t 2 - 5)

Section 3.6, pp. 161–164 1. 12x 3 3. 3 cos s3x + 1d 7. 10 sec2 s10x - 5d

s4x + 3d3s4x + 7d

5 37. a x 2 - 3x + 3b e 5x>2 2 x sec x tan x + sec x 39. 2x sec2 s22xd + tan s2 2xd 41. 217 + x sec x 2 sin u 2 43. 45. -2 sin su d sin 2u + 2u cos s2ud cos su2 d (1 + cos u) 2

y = tan x

/4

1 b 2x 2

33.

2

35. (1 - x)e -x + 3e 3x

y

–/4

x 3 a4 -

x

39. Yes, at x = p 41. No p p 43. a- , -1 b ; a , 1 b 4 4

–/2

1

31. s3x - 2d6 -

4

0

3

x x2 x 1 1 1 + 1 + 2b = 4 a + x - xb a + 1 + 2b 4 8 4 x x dy dy du With u = tan x, y = sec u : = = dx du dx 2 ssec u tan udssec xd = sec stan xd tan stan xd sec2 x dy dy du With u = sin x, y = u 3 : = = 3u 2 cos x = dx du dx 3 sin2 x scos xd dy y = e u, u = -5x, = -5e -5x dx dy y = e u, u = 5 - 7x, = -7e (5 - 7x) dx csc u 4 1 25. p scos 3t - sin 5td 27. cot u + csc u 223 - t

29. 2x sin4 x + 4x 2 sin3 x cos x + cos-2 x + 2x cos-3 x sin x

√2 + √2 y = √2 x –

2√3 +2 3

y = –2√3x –

dy dy du = = dx du dx

5. -sin ssin xd cos x

dy dy du 9. With u = s2x + 1d, y = u 5 : = = 5u 4 # 2 = dx du dx 10s2x + 1d4 dy dy du 11. With u = s1 - sx>7dd, y = u -7 : = = dx du dx -8 x 1 -7u -8 # a- b = a1 - b 7 7

71.

73. 2 csc2 s3x - 1d cot s3x - 1d x2

6 1 2 a1 + x b a1 + x b x3 75. 16(2x + 1) 2 (5x + 1)

77. 2(2x + 1) e 79. 5>2 81. -p>4 87. (a) 2>3 (b) 2p + 5 (c) 15 - 8p 2

83. 0 85. -5 (d) 37>6 (e) -1

(f) 22>24 (g) 5>32 (h) -5>s3 217d 89. 5 91. (a) 1 (b) 1 93. y = 1 - 4x 95. (a) y = px + 2 - p (b) p>2 97. It multiplies the velocity, acceleration, and jerk by 2, 4, and 8, respectively. 4 2 m>sec2 99. ys6d = m>sec, as6d = 5 125

Chapter 3: Answers to Odd-Numbered Exercises

Section 3.7, pp. 168–169 1.

-2xy - y 2

-2x + 3x y - xy + x 3

5. 11.

15.

2

2

x 2y - x 3 + y -cos2 sxyd - y x

13.

2e 2x - cos (x + 3y) 3 cos (x + 3y)

x 21. y¿ = - y , y– =

7.

1 y sx + 1d2

9. cos2 y

–2

1 1 y sin a y b - cos a y b + xy

–2

19.

2u

25. y¿ = 27. -2 31. (a) y 33. (a) y 35. (a) y 37. (a) y 39. (a) y

2y + 1

-y 2 - x 2

7. 1> 9

y3

, y– =

2s 2y + 1d3

x + 3xy 2

2

dx , = - 3 dy y + 2xy

45. 2u + 3 ssin ud a

y=f –3/2

0 –3/2

(c) 2, 1> 2

–1

(x) =

x 3 – 2 2 3

x

u + 5 1 1 c - + tan u d u cos u u + 5 u

51.

x2x 2 + 1 1 x 2 c + 2 d 3sx + 1d x + 1 sx + 1d2>3 x

69. a

5 x + 4 4

77.

y=f

–1

x 5 (x) = – + 4 4

5 4

(c) -4, -1>4

5

ln 5 2 2s

b52s

2sln rd rsln 2dsln 4d

x

-

55. -2 tan u

61. e cos ts1 - t sin td y 2 - xy ln y x 2 - xy ln x

71. px sp - 1d 79.

2x b x2 + 1

73.

1 u ln 2

67. 2x ln x 75.

3 x ln 4

-2 sx + 1dsx - 1d

1 1 cos slog7 ud 83. ln 7 ln 5 x 1 1 + lnsx + 1db 85. t slog2 3d3log 2 t 87. t 89. sx + 1dx a x + 1 ln t 1 + b 91. s 2tdt a 93. ssin xdxsln sin x + x cot xd 2 2 81. sin slog7 ud +

y = f(x) = –4x + 5

5 4 0

49.

1 1 3 xsx - 2d 1 a + 3 B x2 + 1 x x - 2 1 - t 57. 59. 1>s1 + e u d t ye y cos x dy = 63. 65. dx 1 - ye y sin x

y 5

1 + cot u b 2su + 3d

53.

(b) y = f(x) = 2x + 3

23. x 3 ln x

17.

1 1 1 + d = 3t 2 + 6t + 2 47. tst + 1dst + 2d c t + t + 1 t + 2

dx 1 = dy dy>dx

3. (a) ƒ -1sxd = -

y 3

21. 2 sln td + sln td2

1 u + 1 1 - ln t 25. t2

15. -1>x

t 1 1 1 1 a b = 43. a b 2 At + 1 t t + 1 22t st + 1d3>2

Section 3.8, pp. 178–179

(b)

13. 2> t

2x + 1 1 1 1 41. a b 2xsx + 1d a x + b = 2 x + 1 22xsx + 1d

23 23 23 1 , b , m = 23 at a , b 4 2 4 2 27 27 27 45. s -3, 2d : m = - ; s -3, -2d : m = ; s3, 2d : m = ; 8 8 8 27 s3, -2d : m = 8 47. s3, -1d

3 x 2 2

11. 1> x

1 1 29. 31. 2 cos sln ud x ln x xs1 + ln xd2 tan sln ud 3x + 2 2 33. 35. 37. 2 u 2xsx + 1d t s1 - ln td 10x 1 39. 2 + 2s1 - xd x + 1

43. m = -1 at a

1. (a) ƒ -1sxd =

9. 3

27.

1

41. Points: s - 27, 0d and s 27, 0d , Slope: -2

dy y + 2xy 53. , = - 2 dx x + 3xy2

3 y = 2 x at x = 0.

19. 3> x

2

29. s -2, 1d : m = -1, s -2, -1d : m = 1 29 7 1 4 (b) y = - x + = x 7 7 4 2 8 1 = 3x + 6 (b) y = - x + 3 3 6 7 6 7 (b) y = - x = x + 7 7 6 6 p p 2 2 = - x + p (b) y = p x - p + 2 2 x 1 = 2px - 2p (b) y = + 2p 2p

3

x

2

(c) Slope of ƒ at (1, 1) : 3; slope of g at (1, 1): 1> 3; slope of ƒ at s -1, -1d : 3; slope of g at s -1, -1d : 1> 3 (d) y = 0 is tangent to y = x 3 at x = 0; x = 0 is tangent to

-r u

2 2 dy (2x 2y 2 + y 2 - 2x)e x - x 2e 2x - 1 xe x + 1 d y = , 2 = y dx dx y3

2y

1 –1

2r

y = x 1/3

–1

-y 2

17. -

y = x3

1

2

23.

y 2

1 - 2y 2x + 2y - 1

3.

x 2 + 2xy

5. (b)

A-13

ln x 2 95. sx ln x d a x b

A-14

Chapter 3: Answers to Odd-Numbered Exercises

Section 3.9, pp. 185–186 1. 3. 5. 7.

(a) (a) (a) (a)

p>4 (b) -p>3 -p>6 (b) p>4 p>3 (b) 3p>4 3p>4 (b) p>6

9. 1> 22 19. 0 25.

21.

11. -1> 23 -2x 21 - x 4 1

13. p>2 23.

27.

15. p>2

17. p>2

22

35.

-e t ƒ e t ƒ 2se t d2 - 1

=

2

-1

43.

2e 2t - 1

4

37.

2

-2s 2 21 - s 2

39. 0

41. sin-1 x 47. (a) Defined; there is an angle whose tangent is 2. (b) Not defined; there is no angle whose cosine is 2. 49. (a) Not defined; no angle has secant 0. (b) Not defined; no angle has sine 22 . 59. (a) Domain: all real numbers except those having the form p + kp where k is an integer; range: -p>2 6 y 6 p>2 2 (b) Domain: - q 6 x 6 q ; range: - q 6 y 6 q 61. (a) Domain: - q 6 x 6 q ; range: 0 … y … p (b) Domain: -1 … x … 1 ; range: -1 … y … 1 63. The graphs are identical.

Section 3.10, pp. 191–195 1. 9. 13.

15.

17.

dA dr 3. 10 5. -6 7. -3>2 = 2pr dt dt 2 31>13 11. (a) -180 m >min (b) -135 m3>min dV dV dh dr (a) (b) = pr 2 = 2phr dt dt dt dt dV dr 2 dh (c) = pr + 2phr dt dt dt 1 (a) 1 volt> sec (b) - amp> sec 3 V dI dR 1 dV (c) = a b I dt I dt dt (d) 3>2 ohms> sec, R is increasing. dx ds x (a) = 2 2 dt dt 2x + y

dy y y dy ds dx dx x (b) (c) = + = -x dt dt dt 2x 2 + y 2 dt 2x 2 + y 2 dt du dA 1 19. (a) = ab cos u 2 dt dt du dA da 1 1 (b) = ab cos u + b sin u 2 2 dt dt dt du dA da db 1 1 1 = ab cos u + b sin u + a sin u (c) 2 2 2 dt dt dt dt 21. (a) 14 cm2>sec , increasing (b) 0 cm> sec, constant (c) -14>13 cm>sec , decreasing

31. 35. 37. 41.

21 - 2t 2 -2x

sx + 1d2x + 2x ƒ 2s + 1 ƒ 2s + s -1 -1 1 29. 31. 33. -1 2 stan x)(1 + x 2d 21 - t 2 2t s1 + td 2

-1 m>min (b) r = 226y - y 2 m 24p 5 dr (c) m>min = 288p dt 2 1 ft> min, 40p ft >min 33. 11 ft> sec Increasing at 466> 1681 L> min2 -5 m>sec 39. -1500 ft>sec 5 10 2 in.>min, in >min 72p 3 (a) -32> 213 L -8.875 ft>sec

29. (a)

(c) p>6 (c) -p>3 (c) p>6 (c) 2p>3

23. (a) -12 ft>sec (b) -59.5 ft2>sec (c) -1 rad>sec 25. 20 ft> sec dh dr = 11.19 cm>min (b) = 14.92 cm>min 27. (a) dt dt

(b) du1>dt = 8>65 rad>sec, du2>dt = -8>65 rad>sec (c) du1>dt = 1>6 rad>sec, du2>dt = -1>6 rad>sec

Section 3.11, pp. 203–205 1. Lsxd = 10x - 13

3. Lsxd = 2

5. L(x) = x - p

1 4 13. 1 - x x + 12 3 k-1 15. ƒs0d = 1. Also, ƒ¿sxd = k s1 + xd , so ƒ¿s0d = k . This means the linearization at x = 0 is Lsxd = 1 + kx . 17. (a) 1.01 (b) 1.003 7. 2x

9. -x - 5

19. a3x 2 23.

3 22x

1 - y 32y + x

dx

27. (4x 2) sec2 a 29. 31. 37. 41. 43. 47. 51. 55. 61.

65. (b)

3 2x 1

11.

b dx

21. 5

25.

22x

2 - 2x 2 dx s1 + x 2 d2 cos s5 2xd dx

x3 b dx 3

scsc s1 - 2 2xd cot s1 - 22xdd dx

# e 2x dx

33.

2 2x -1

2x dx 1 + x2

2

35.

2xe x

2

1 + e 2x

dx

(b) .4 (c) .01 dx 39. (a) .41 2e -2x - 1 (a) .231 (b) .2 (c) .031 (a) -1>3 (b) -2>5 (c) 1>15 45. dV = 4pr 02 dr dS = 12x0 dx 49. dV = 2pr0 h dr (a) 0.08p m2 (b) 2% 53. dV L 565.5 in3 1 (a) 2% (b) 4% 57. % 59. 3% 3 The ratio equals 37.87, so a change in the acceleration of gravity on the moon has about 38 times the effect that a change of the same magnitude has on Earth. (a) L(x) = x ln 2 + 1 L 0.69x + 1 y

y

y = 2x 1.4

3 2

y = 2x

y = (ln 2)x + 1 1

–3

–2

–1

0

1 0.8

y = (ln 2)x + 1

0.4 1

2

3

x –1

Practice Exercises, pp. 206–211 1. 5x 4 - 0.25x + 0.25 3. 3xsx - 2d 5. 2sx + 1ds2x 2 + 4x + 1d 7. 3su2 + sec u + 1d2 s2u + sec u tan ud

–0.5

0

0.5

1

x

Chapter 3: Answers to Odd-Numbered Exercises

9.

1

3 1 5 9 99. a , b and a , - b 2 4 2 4

11. 2 sec2 x tan x

2 2t A 1 + 2t B 2

13. 8 cos3 s1 - 2td sin s1 - 2td

103. (a) s -2, 16d , (3, 11)

15. 5ssec td ssec t + tan td5

105.

17.

u cos u + sin u

19.

22u sin u

cos 22u

A-15

101. s -1, 27d and (2, 0) (b) (0, 20), (1, 7)

y y = tan x y = – –1 x + + 1 2 8

22u

(/4, 1)

1

2 2 2 21. x csc a x b + csc a x b cot a x b 23.

1 1>2 x sec s2xd2 C 16 tan s2xd2 - x -2 D 2

–/2

25. -10x csc2 sx 2 d 27. 8x 3 sin s2x 2 d cos s2x 2 d + 2x sin2 s2x 2 d -st + 1d 1 - x -1 29. 31. 33. 1>2 8t 3 sx + 1d3 1 2x 2 a1 + x b 5x + cos 2x 37. 3 22x + 1 39. -9 c 2 d s5x + sin 2xd5>2 2 sin u cos u 43. xe 4x 45. = 2 cot u sin2 u

-2 sin u 35. scos u - 1d2 41. -2e 47.

-x>5

2 sln 2dx

49. -8-tsln 8d

51. 18x 2.6

53. sx + 2dx + 2slnsx + 2d + 1d 57. 61.

67.

-1 21 - x cos 2

1 - z 2z 2 - 1

-1

+ sec-1 z

4x - 4y 1>3

1 21 - u 2

t 1 2t 1 + t2

59. tan-1std +

x

-3x 2 - 4y + 2

55. -

65. -

63. -1

75. y>x

77. -

89.

(b)

(c) 5>12

3 22e 23>2 cos A e 23>2 B 4

–1

dp 6q - 4p = dq 3p 2 + 4q

79.

d 2y dx 2

-2xy 2 - 1

=

y = f(x) (6, 1)

(e) 12

91. -

1 2

93.

(f) 9>2 -2 s2t + 1d2

97. (a)

4

x

6

1

x

0

2x + tan 2x d c 2 2cos 2x x + 1 st + 1dst - 1d 5 1 1 1 1 d c + d 137. 5 c t + 1 t - 1 t - 2 t + 3 st - 2dst + 3d 139.

1 2u

x, 0≤x≤1 2 – x, 1 < x ≤ 2

y=

1

2sx 2 + 1d

141. (a)

y

1

1

123. (a) 0, 0 (b) 1700 rabbits, L1400 rabbits 125. -1 127. 1>2 129. 4 131. 1

135.

x 4y 3

(d) 1>4

y

0

(4, 3)

2

133. To make g continuous at the origin, define g(0) = 1.

95. (a)

–1

y 3

1 2y sx + 1d2

71.

2e -tan x 1 + x2

dr = s2r - 1dstan 2sd ds d 2y -2xy 3 - 2x 4 83. (a) = y5 dx 2

87. 0

9 1 x + , normal: y = 4x - 2 4 4 7 1 113. Tangent: y = 2x - 4 , normal: y = - x + 2 2 5 11 4 115. Tangent: y = - x + 6 , normal: y = x 5 5 4 1 117. s1, 1d: m = - ; s1, -1d: m not defined 2 111. Tangent: y = -

121.

81.

85. (a) 7 (b) -2 (g) 3>4

109. 4

(–1, 2)

y 69. - x

x

119. B = graph of ƒ, A = graph of ƒ¿

y + 2 x + 3

-1

73. -1>2

1 4

/2

1x – –1 y=–– 2 8

–1

(–/4, –1)

107.

/4

–/4

1

(b) Yes

2

(c) No

ssin ud2u a

ln sin u + u cot ub 2

dS dr = s4pr + 2phd dt dt

(b)

dS dh = 2pr dt dt

(c)

dS dr dh = s4pr + 2phd + 2pr dt dt dt

(d)

dr dh r = dt 2r + h dt

x

143. -40 m2>sec

145. 0.02 ohm> sec

147. 2 m> sec

–1

f (x) =

x 2, –1 ≤ x < 0 –x 2, 0 ≤ x < 1

149. (a) r = 151. (a)

(b) Yes

(c) Yes

2 h 5

(b) -

125 ft>min 144p

3 km>sec or 600 m> sec 5

18 (b) p rpm

A-16

Chapter 4: Answers to Odd-Numbered Exercises p - 2 2

153. (a) Lsxd = 2x +

CHAPTER 4 Section 4.1, pp. 219–222

y

1. 3. 5. 7. 9. 15.

y = tan x

1 y = 2x + ( – 2)/2

(–/4, –1)

x

/4

–/4

–1

Absolute minimum at x = c2 ; absolute maximum at x = b Absolute maximum at x = c ; no absolute minimum Absolute minimum at x = a ; absolute maximum at x = c No absolute minimum; no absolute maximum Absolute maximum at (0, 5) 11. (c) 13. (d) Absolute minimum at 17. Absolute maximum at x = 0 ; no absolute x = 2 ; no absolute maximum minimum y

22s4 - pd 4

(b) Lsxd = - 22x +

y

f(x) = ⎪x⎪

y = g(x)

2

y

1

1 1 1

–1

x

2

x

2

–1

19. Absolute maximum at x = p>2; absolute minimum at x = 3p>2

√2 ⎛–/4, √ 2⎞ ⎝ ⎠

y = sec x

y –/2

–/4

/2

0

x 3

y = –√2x + √2 ⎛⎝4 – ⎞⎠ /4

155. Lsxd = 1.5x + 0.5

157. dS =

159. (a) 4%

(c) 12%

(b) 8%

prh0 2r 2 + h 02

/2

3/2

2

x

dh

Additional and Advanced Exercises, pp. 211–213 1. (a) sin 2u = 2 sin u cos u; 2 cos 2u = 2 sin u s -sin ud + cos u s2 cos ud; 2 cos 2u = -2 sin2 u + 2 cos2 u; cos 2u = cos2 u - sin2 u (b) cos 2u = cos2 u - sin2 u; -2 sin 2u = 2 cos u s -sin ud - 2 sin u scos ud; sin 2u = cos u sin u + sin u cos u; sin 2u = 2 sin u cos u 1 3. (a) a = 1, b = 0, c = (b) b = cos a, c = sin a 2 525 9 5. h = -4, k = , a = 2 2 7. (a) 0.09y (b) Increasing at 1% per year 9. Answers will vary. Here is one possibility. y

–3

21. Absolute maximum: -3 ; absolute minimum: -19>3

23. Absolute maximum: 3; absolute minimum: -1

y

–2 –1 0 –1

y

1

–2 –3 –4 –5 (–2, –19/3) Abs min

–6

2

3

x

0

(3, –3) 1

Abs max y = 23 x – 5

–1

11. (a) 2 sec, 64 ft> sec (b) 12.31 sec, 393.85 ft b 15. (a) m = - p (b) m = -1, b = p 3 9 17. (a) a = , b = 19. ƒ odd Q ƒ¿ is even 4 4 23. h¿ is defined but not continuous at x = 0; k¿ is defined and continuous at x = 0 . 27. (a) 0.8156 ft (b) 0.00613 sec (c) It will lose about 8.83 min> day.

–1 –2 –3 –4

1 (0, –1) Abs min

–7

27. Absolute maximum: 2; absolute minimum: -1 y

1

x

2

(2, – 0.25) Abs max

y = – 1 , 0.5 ≤ x ≤ 2 x2 (0.5, –4) Abs min

x

2

–2 ≤ x ≤ 3

25. Absolute maximum: -0.25 ; absolute minimum: -4

0

2 y=x –1 –1 ≤ x ≤ 2

2

y t

(2, 3) Abs max

3

3

y = √x –1 ≤ x ≤ 8

1 –1 –1 (–1, –1) Abs min

1

2

3

4

5

6

7

8

(8, 2) Abs max x

A-17

Chapter 4: Answers to Odd-Numbered Exercises 29. Absolute maximum: 2; absolute minimum: 0

31. Absolute maximum: 1; absolute minimum: -1

y y = √ 4 – x2 –2 ≤ x ≤ 1

x = -

(/2, 1) Abs max

(0, 2) Abs max 1

(–2, 0) –1 Abs min

0

1

–1

/2

5/6

–1 (–/2, –1) Abs min

73.

–1 /3

0

/2

2/3

37. Absolute maximum is 1>e at x = 1; absolute minimum is -e at x = -1. y

1

–2 (–1, –e) –3 Absolute – 4 minimum

1

2

3

x

x

Value

Undefined 0 0 Undefined

Local max Minimum Maximum Local min

0 -2 2 0

Critical point or endpoint

Derivative

Extremum

Value

x = 1

Undefined

Minimum

2

2

75.

t

3

Critical point or endpoint

Derivative

Extremum

Value

x = -1 x = 1 x = 3

0 Undefined 0

Maximum Local min Maximum

5 1 5

Abs min (3, –1)

–1

39. Absolute maximum value is (1>4) + ln 4 at x = 4; absolute minimum value is 1 at x = 1; local maximum at (1>2, 2 - ln 2). y

Absolute maximum ⎛ 1⎛ 1, ⎝ e⎝

2

1

0

⎛

Abs max at ⎝4,

⎛ 1 + ln 4 ⎝ 4

1.5 1.25 f(x) = x1 + ln x 1 Abs min at (1, 1) 0.75 0.5 0.25 1

2

3

4

5

x

41. Increasing on (0, 8), decreasing on s -1, 0d ; absolute maximum: 16 at x = 8 ; absolute minimum: 0 at x = 0 43. Increasing on s -32, 1d ; absolute maximum: 1 at u = 1; absolute minimum: - 8 at u = -32 45. x = 3 47. x = 1, x = 4 49. x = 1 51. x = 0 and x = 4 53. Minimum value is 1 at x = 2 . 4 41 55. Local maximum at s -2, 17d ; local minimum at a , - b 3 27 57. Minimum value is 0 at x = -1 and x = 1 . 59. There is a local minimum at (0, 1). 1 1 61. Maximum value is at x = 1; minimum value is - at x = -1 . 2 2 63. The minimum value is 2 at x = 0. 1 1 65. The minimum value is - e at x = e . p 67. The maximum value is at x = 0; an absolute minimum value 2 is 0 at x = 1 and x = -1.

Undefined Local min

12 1>3 10 L 1.034 25 0

Extremum

y = 2 – ⎪t⎪ –1 ≤ t ≤ 3

1

Local max

Value

Derivative

x x x x

(0, 2) Abs max

y = csc x (/2, 1) /3 ≤ x ≤ 2/3 Abs min

0

Critical point or endpoint

35. Absolute maximum: 2; absolute minimum: -1

Abs max ⎞2/3, 2/ √3⎞ ⎠ ⎠

Abs max ⎞/3, 2/ √3⎞ ⎠ ⎠

71.

y = sin , –/ 2 ≤ ≤ 5/6

y

y

4 5

x = 0

–/2 x

33. Absolute maximum: 2> 23 ; absolute minimum: 1

–3 –2 –1 –1

Critical point Derivative Extremum or endpoint

y

1

1.2 1.0 0.8 0.6 0.4 0.2

69.

= = = =

-2 - 12 12 2

77. (a) No (b) The derivative is defined and nonzero for x Z 2 . Also, ƒs2d = 0 and ƒsxd 7 0 for all x Z 2 . (c) No, because s - q , q d is not a closed interval. (d) The answers are the same as parts (a) and (b) with 2 replaced by a. 79. Yes 81. g assumes a local maximum at -c . 83. (a) Maximum value is 144 at x = 2 . (b) The largest volume of the box is 144 cubic units, and it occurs when x = 2 . y0 2 85. + s0 2g 87. Maximum value is 11 at x = 5 ; minimum value is 5 on the interval [-3, 2] ; local maximum at s -5, 9d . 89. Maximum value is 5 on the interval [3, q d ; minimum value is -5 on the interval s - q , -2] .

Section 4.2, pp. 228–230 4 1 - 2 L ;0.771 A p 1 1 A 1 + 27 B L 1.22, 3 A 1 - 27 B L -0.549 3 Does not; ƒ is not differentiable at the interior domain point x = 0. Does 13. Does not; ƒ is not differentiable at x = -1. (a) i)

1. 1> 2 7. 9. 11. 17.

5. ;

3. 1

–2

ii)

0

x

2

x –5

–4

–3

iii)

x –1

0

2

iv)

x 0

4

9

18

24

A-18 29. Yes 33. 35. 37.

39. 43.

Chapter 4: Answers to Odd-Numbered Exercises 31. (a) 4

(b) 3

(c) 3

x2 + C 2 1 x + C 1 - cos 2t 2 1 - cos 2t 2

x3 x4 (a) (b) (c) + C + C 3 4 1 1 (a) (b) x + x + C (c) 5x - x + C t (a) (b) 2 sin + C + C 2 t (c) + 2 sin + C 2 e 2x ƒsxd = x 2 - x 41. ƒ(x) = 1 + 2 1 - cos sptd 2 s = 4.9t + 5t + 10 45. s = p

47. s = e t + 19t + 4

49. s = sin s2td - 3

51. If T(t) is the temperature of the thermometer at time t, then Ts0d = -19 °C and Ts14d = 100 °C . From the Mean Value Ts14d - Ts0d Theorem, there exists a 0 6 t0 6 14 such that = 14 - 0 8.5 °C>sec = T ¿st0 d , the rate at which the temperature was changing at t = t0 as measured by the rising mercury on the thermometer. 53. Because its average speed was approximately 7.667 knots, and by the Mean Value Theorem, it must have been going that speed at least once during the trip. 57. The conclusion of the Mean Value Theorem yields 1 1 - a b a - b 1 = - 2 Q c2 a b = a - b Q c = 1ab . b - a ab c 61. ƒ(x) must be zero at least once between a and b by the Intermediate Value Theorem. Now suppose that ƒ(x) is zero twice between a and b. Then, by the Mean Value Theorem, ƒ¿sxd would have to be zero at least once between the two zeros of ƒ(x), but this can’t be true since we are given that ƒ¿sxd Z 0 on this interval. Therefore, ƒ(x) is zero once and only once between a and b. 71. 1.09999 … ƒs0.1d … 1.1

Section 4.3, pp. 233–235 1. (a) (b) (c) 3. (a) (b) (c) 5. (a) (b) (c) 7. (a) (b) (c) 9. (a) (b) (c) 11. (a) (b) (c)

0, 1 Increasing on s - q , 0d and s1, q d; decreasing on (0, 1) Local maximum at x = 0; local minimum at x = 1 -2, 1 Increasing on s -2, 1d and s1, q d; decreasing on s - q , -2d No local maximum; local minimum at x = -2 Critical point at x = 1 Decreasing on (- q , 1), increasing on (1, q ) Local (and absolute) minimum at x = 1 0, 1 Increasing on s - q , -2d and s1, q d; decreasing on s -2, 0d and (0, 1) Local minimum at x = 1 -2, 2 Increasing on (- q , -2) and (2, q ); decreasing on s -2, 0d and (0, 2) Local maximum at x = -2; local minimum at x = 2 -2, 0 Increasing on s - q , -2d and s0, q d; decreasing on s -2, 0d Local maximum at x = -2; local minimum at x = 0

13. (a)

p 2p 4p , , 2 3 3

(b) Increasing on a and a

p 2p 2p 4p p , b; decreasing on a0, b, a , b, 3 3 2 2 3

4p , 2pb 3

(c) Local maximum at x = 0 and x =

4p ; local minimum at 3

2p and x = 2p 3 Increasing on s -2, 0d and (2, 4); decreasing on (-4, -2) and (0, 2) Absolute maximum at (-4, 2); local maximum at (0, 1) and (4, -1); absolute minimum at (2, -3); local minimum at s -2, 0d Increasing on (-4, -1), (1>2, 2), and (2, 4); decreasing on (- 1, 1>2) Absolute maximum at (4, 3); local maximum at (-1, 2) and (2, 1); no absolute minimum; local minimum at (-4, -1) and (1>2, -1) Increasing on s - q , -1.5d; decreasing on s -1.5, q d Local maximum: 5.25 at t = -1.5; absolute maximum: 5.25 at t = -1.5 Decreasing on s - q , 0d; increasing on (0, 4>3); decreasing on s4>3, q d Local minimum at x = 0 s0, 0d; local maximum at x = 4>3 s4>3, 32>27d; no absolute extrema Decreasing on s - q , 0d; increasing on (0, 1>2); decreasing on s1>2, q d Local minimum at u = 0 s0, 0d; local maximum at u = 1>2 s1>2, 1>4d; no absolute extrema Increasing on s - q , q d; never decreasing No local extrema; no absolute extrema Increasing on s -2, 0d and s2, q d; decreasing on s - q , -2d and (0, 2) Local maximum: 16 at x = 0; local minimum: 0 at x = ;2; no absolute maximum; absolute minimum: 0 at x = ;2 Increasing on s - q , -1d; decreasing on s -1, 0d; increasing on (0, 1); decreasing on s1, q d Local maximum: 0.5 at x = ;1; local minimum: 0 at x = 0; absolute maximum: 1>2 at x = ;1; no absolute minimum Increasing on (10, q); decreasing on (1, 10) Local maximum: 1 at x = 1; local minimum: - 8 at x = 10; absolute minimum: -8 at x = 10 x =

15. (a) (b)

17. (a) (b)

19. (a) (b) 21. (a) (b) 23. (a) (b) 25. (a) (b) 27. (a) (b) 29. (a) (b) 31. (a) (b)

33. (a) Decreasing on s -222, -2d; increasing on s -2, 2d; decreasing on s2, 222d (b) Local minima: gs -2d = -4, gs222d = 0 ; local maxima: gs -222d = 0, gs2d = 4; absolute maximum: 4 at x = 2 ; absolute minimum: -4 at x = -2 35. (a) Increasing on s - q , 1d; decreasing when 1 6 x 6 2 , decreasing when 2 6 x 6 3; discontinuous at x = 2; increasing on s3, q d (b) Local minimum at x = 3 s3, 6d; local maximum at x = 1 s1, 2d; no absolute extrema 37. (a) Increasing on s -2, 0d and s0, q d; decreasing on s - q , -2d 3 (b) Local minimum: -62 2 at x = -2; no absolute maximum; 3 absolute minimum: -6 2 2 at x = -2

A-19

Chapter 4: Answers to Odd-Numbered Exercises 39. (a) Increasing on (- q , -2> 27) and (2> 27, q ); decreasing

69. (a)

(b)

y

on (-2> 27, 0) and (0, 2> 27)

y

(b) Local maximum: 2422>77>6 L 3.12 at x = -2> 27 ; local 3

2

3 2>77>6 L -3.12 at x = 2> 27; no minimum: -24 2

43. (a) Increasing on (e -1, q ), decreasing on (0, e -1) (b) A local minimum is -e -1 at x = e -1, no local maximum; absolute minimum is -e -1 at x = e -1, no absolute maximum 45. (a) Local maximum: 1 at x = 1 ; local minimum: 0 at x = 2 (b) Absolute maximum: 1 at x = 1 ; no absolute minimum 47. (a) Local maximum: 1 at x = 1 ; local minimum: 0 at x = 2 (b) No absolute maximum; absolute minimum: 0 at x = 2 49. (a) Local maxima: -9 at t = -3 and 16 at t = 2 ; local minimum: -16 at t = -2 (b) Absolute maximum: 16 at t = 2 ; no absolute minimum 51. (a) Local minimum: 0 at x = 0 (b) No absolute maximum; absolute minimum: 0 at x = 0 53. (a) Local maximum: 5 at x = 0; local minimum: 0 at x = -5 and x = 5 (b) Absolute maximum: 5 at x = 0; absolute minimum: 0 at x = -5 and x = 5 55. (a) Local maximum: 2 at x = 0; 23 local minimum: at x = 2 - 23 4 23 - 6 (b) No absolute maximum; an absolute minimum at x = 2 - 23 57. (a) Local maximum: 1 at x = p>4;

61. (a) Local minimum: sp>3d - 23 at x = 2p> 3 ; local maximum: 0 at x = 0 ; local maximum: p at x = 2p 63. (a) Local minimum: 0 at x = p> 4 65. Local maximum: 3 at u = 0 ; local minimum: -3 at u = 2p 67.

3

0

1

(a)

x 0

1

(b)

x

0

1

(c)

–4 –3 –2 –1 0 –1

1

y = f(x)

y = f(x)

x 0

– 4x + 3

1

(d)

x

5

2

4

1

y = f (x) 1

1

x2

y = x 3 – 3x + 3

Loc max (–1, 5)

4

y=

1

x

73. a = -2, b = 4 75. (a) Absolute minimum occurs at x = p>3 with ƒ(p>3) = -ln 2, and the absolute maximum occurs at x = 0 with ƒ(0) = 0. (b) Absolute minimum occurs at x = 1>2 and x = 2 with ƒ(1>2) = ƒ(2) = cos (ln 2), and the absolute maximum occurs at x = 1 with ƒ(1) = 1. 77. Minimum of 2 - 2 ln 2 L 0.613706 at x = ln 2; maximum of 1 at x = 0 79. Absolute maximum value of 1>2e assumed at x = 1> 2e dƒ-1 1 83. Increasing; = x -2>3 9 dx df -1 1 85. Decreasing; = - x -2>3 3 dx

y

y = f(x)

x

2

1. Local maximum: 3> 2 at x = -1; local minimum: -3 at x = 2; point of inflection at s1>2, -3>4); rising on s - q , -1d and s2, q d; falling on s - 1, 2d; concave up on s1>2, q d; concave down on s - q , 1> 2d 3. Local maximum: 3> 4 at x = 0; local minimum: 0 at x = ;1; 3 3 4 4 32 32 points of inflection at a- 23, b and a23, b; 4 4 rising on s -1, 0d and s1, q d; falling on s - q , -1d and (0, 1); concave up on s - q , - 23d and s 23, q d; concave down on s - 23, 23d 23 23 -2p p 5. Local maxima: at x = -2p>3, + at + 3 2 3 2 23 23 p 2p at x = -p>3, x = p>3; local minima: - 3 2 3 2 at x = 2p> 3; points of inflection at s -p> 2, -p> 2d , (0, 0), and sp>2, p>2d, rising on s -p>3, p>3d; falling on s -2p> 3, -p> 3d and sp>3, 2p>3d, concave up on s -p> 2, 0d and sp>2, 2p>3d; concave down on s -2p> 3, -p> 2d and s0, p> 2d 7. Local maxima: 1 at x = -p> 2 and x = p>2, 0 at x = -2p and x = 2p; local minima: -1 at x = -3p> 2 and x = 3p> 2, 0 at x = 0; points of inflection at s -p, 0d and sp, 0d; rising on s -3p>2, -p>2d, s0, p>2d, and s3p>2, 2pd; falling on s -2p, -3p> 2d, s -p> 2, 0d, and sp>2, 3p>2d; concave up on s -2p, -pd and sp, 2pd; concave down on s -p, 0d and s0, pd y 9. 11. y

local minimum: -1 at x = 3p>4 59. Local maximum: 2 at x = p>6; local maximum: 23 at x = 2p; local minimum: -2 at x = 7p>6; local minimum: 23 at x = 0

y

0

Section 4.4, pp. 243–246

local maximum: 0 at x = p; local minimum: 0 at x = 0;

y

y = g(x)

2

absolute extrema 41. (a) Increasing on ((1>3) ln (1>2), q ), decreasing on (- q , (1>3) ln (1>2)) 3 (b) Local minimum is 2>3 at x = (1>3) ln (1>2); no local 2 3 maximum; absolute minimum is 2>3 at x = (1>3) ln (1>2); 2 no absolute maximum

y

y = g(x)

–2

Inf l 1

2

3

4

x 2

(2, –1) Abs min

1 –1

(1, 1) Loc min 1

x

A-20

Chapter 4: Answers to Odd-Numbered Exercises

13.

15.

y

33.

y

35.

y

(2, 5) Loc max

y = 2x – 3 x 2/3

3 2 Infl (1, 1)

2

–3

–1

x

2

1

–2

–1

4

Cusp, Loc max (0, 0)

Infl (2, 1)

–1

1

1

y

4

1

x

5

–1 2

1

0

3

4

2

Inf l

(1, –1) Loc min

x

y = x 2/3 ⎞⎠5–2 – x⎞⎠

3

(1, 3/2) Loc max

3 ⎞ ⎞ ⎠–1/2, 3/ √4 ⎠

–1 (0, –3) Loc min

3

y = –2x +

17.

6x 2

–2

–3

19.

y

–2

1

2

⎞–1/ √3 , –5/9⎞ ⎠ ⎠

⎞1/ √3, –5/9⎞ ⎠ ⎠

Infl

Infl

–2

0

2

2

3

–2

Abs max (2, 2)

–1

x (0, 0) Inf l 1

41.

43.

6 y= x –3 x–2

Abs min

2

x

y=

(3, 6) Loc min

2

(2, 2) Abs max

2√3, √3 Infl

2 4

6

8

x x

–4

–1 –2 –2√3, –√3 (–2, –2) Infl Abs min

1 2 (0, 0) Infl

–6 Loc max

2, 2√3 – 2

4/3, 4√3/3 + 1

10 8 6 4

Infl

2

–8

Abs max

5/3, 5√3/3 – 1 Infl 3/2, 3√3/2 Loc min

45.

y = x2 – 1

x

2

y = √3x – 2 cos x

1

(0, 0) 1 2 Cusp Abs min

–4 –3 –2 –1

2 Loc max (0, 1)

y

y = √⏐x⏐

2

3

3/2

y

47.

y

/2, √3/2

/2

0 (0, –2) Abs min

27.

2 1

(1, 2) Loc max

–8 –6 –4

y

8x x2 + 4

4

–2

25.

y

8

Infl

0

x

(4, 0) Abs min

y = x √8 – x2

y

2

(4, –256) Loc min

2 ⎞ ⎞ ⎠2 √ 2 , 0⎠ Loc min

(–4, 0) Abs min

–4

(–2, –4) Abs min

(, )

(3, –162) Infl

–300

4

1

⎞ ⎞ ⎠–2 √ 2 , 0⎠

x

y = x + sin x

–100 –200

y = √16 – x2

2

x

5

(0, 4) Abs max

Loc max

2

4

3

3

4

y = sin x cos x 1

Abs max (/4, 1/2)

/4

(0, 0) Loc min

Infl (/2, 0) /2

Loc max (, 0) x

3/4

–2

(–1, 0) Abs min

(1, 0) Abs min

y

49.

(3/4, –1/2) Abs min

–1

4

2

x

51. y

y

y ln (3 x 2)

xe1x

3

29.

31.

y

–3

–2

y = x 1/5

2

y 2

2 Vert tan at x = 0

x

3

y

3

y

y = x 5 – 5x 4

39.

Abs max (2, 4)

–3

23. 1

y = 4x 3 – x 4

9 Infl (0, 0) 3 1

y

Loc max (0, 0)

y

Infl

Abs min (1, –1)

Abs min (–1, –1)

21.

(2, 16)

15

–1

2

4

21 1

37.

(3, 27) Abs max

27

x

(0, 0) 1 Cusp Loc min

–1

–5

y

y = x 4 – 2x2

Loc max (0, 0)

–2

y = (x – 2) 3 + 1

y=

1

1

x

√x2

+1

1

–1 –1 –2

–3 –2 –1 –1 x

1 2 (0, 0) Infl

3

x

–4 –3 –2 –1 –1 –2

1 2 (0, 0) Infl

3

–√3

(1, e) Loc min

4

–2

2 1

–3 –2 –1 1 2 3 4 5 6

x

Loc max (0, ln 3) √3 1

–1 –2 –3

2

3

x

x

A-21

Chapter 4: Answers to Odd-Numbered Exercises 53.

55.

y y e x 2e –x 3x

1.

y Loc Loc 3 max max (–4, 0) (–2, 0) 2 1 –4 –2

0.5 –1.5 –1 –0.5 –0.5 (0, –1) Loc max –1

x

0.5 1 1.5 2

Loc max (2, 0) 2

Loc max (4, 0) 4 x

1 3 y ⎞⎠⎞⎠ ⎞⎠ ln 2, ⎞⎠– ⎞⎠ ln 2⎞⎠ 2 2 Inflection

–2

1 -2>3 2 x + x -5>3 3 3

x = –2 Inf l

x = –1 Inf l Vert tan

Inf l vert tan x=0

y

-2, 2,

79. y– = e

59. y– = 1 - 2x Loc max

1 1 e –x 1

Infl

(0, 0.5) Inflection x 1 2 3

–0.5 –1

x=1 Abs min

y ln (cos x)

57.

–3 –2

77. y– =

(ln 2, 1 3 ln 2) Loc min

–1.5

y

2 75. y– = - sx + 1d-5>3 3

Loc max (0, 0)

y

81.

Loc max

y''

y' y

Inf l

x=2

P

x = 12

Loc min

Inf l

Loc min

x

x=0

x = –1

61. y– = 3sx - 3dsx - 1d

x 6 0 x 7 0

63. y– = 3sx - 2dsx + 2d Loc max

Infl x=3

Infl Loc min

Abs min

x=1

x=0 Infl x = –2

x = –2√3

x=0

Inf l x = 2 Abs min x = 2√3

83.

y P

y

65. y– = 4(4 - x)(5x 2 - 16x + 8)

Inf l

Inf l

Loc max x = 8/5

y' Loc min

Infl

x = 8 + 2 √6 5

Loc min x=0

Infl x = 8 – 2√6 5

x

y''

Infl x=4

85.

87.

u 1 69. y– = - csc2 , 2 2 0 6 u 6 2p

67. y– = 2 sec2 x tan x

5

2 y = 2x + x – 1 x2 – 1

4

=

y = x2

3

Abs max

Infl

y

y

y=2

2

x=0

y=

1 –1 0 –1

1

–2

2

y = 12 x

1 x–1

3

x

4 y= x +1 x2

2 1

–1

x

1

x=1

p p 71. y– = 2 tan u sec2 u, - 6 u 6 2 2 = – Loc max 4

89.

91.

y

y y= 1 x2 – 1

=0 Inf1 = 4 Loc min

1 –1

73. y– = -sin t, 0 … t … 2p t= 2

t=0 Loc min

t= Infl

x

1

–1

Loc max t = 2

Abs max

2 y=– x –2 x2 – 1

y= 1 x2 – 1

t = 3 2 Abs min

x= –1

–√2

–1

y = –1 x= 1

1

√2

–1 –2

x = –1

x=1

x

A-22

Chapter 4: Answers to Odd-Numbered Exercises

93.

125. The zeros of y¿ = 0 and y– = 0 are extrema and points of 3 2; local maximum inflection, respectively. Inflection at x = - 2 at x = -2; local minimum at x = 0 .

95.

y x = –1

y=

y

x2 x+1 3

y=

2 y= x –x+1 x–1

y=x–1

1 x–1

y

y' = 4x(x 3 + 8) 100

x –4 –3 –2

1

2

3

1

x

2

50

–1 –3 y=x

– 50

–4

x=1 y" = 16(x 3 + 2)

97.

99.

y

y=

y=

–4

–2

9 x+2

0 –4

1

Section 4.5, pp. 253–254

1 –1

0

1. -1>4

x

1

–12 –16

y = 8/(x2 + 4)

0

103.

Point

y¿

105.

y 7

+ 0 -

–200 –400

y' =

5x 3 (x –

y = x 5 – 5x 4 – 240

61. e 3 81. (b)

-1 2

x

0.4

(4, 4)

4

0.2

(2, 1) 1 0

y" = 20x 2 (x – 3)

3

59. 1

y = (sin x)

0.6

y

0

57. e 1>2

55. 1>e

0.8

2

4

6

x

(d) Positive: 5 … t … 7, 13 … t … 15; negative: 0 … t … 5, 7 … t … 13 L 60 thousand units Local minimum at x = 2; inflection points at x = 1 and x = 5> 3 b = -3 -1, 2 a = -1, b = 3, c = 9 The zeros of y¿ = 0 and y– = 0 are extrema and points of inflection, respectively. Inflection at x = 3, local maximum at x = 0, local minimum at x = 4. 200

53. 1

1 (6, 7)

107. (a) Toward origin: 0 … t 6 2 and 6 … t … 10; away from origin: 2 … t … 6 and 10 … t … 15 (b) t = 2, t = 6, t = 10 (c) t = 5, t = 7, t = 13

109. 111. 115. 119. 121. 123.

51. 1>e

y

y–

+ + 0 -

P Q R S T

13. 0

89. (a) We should assign the value 1 to ƒ(x) = (sin x) x to make it continuous at x = 0.

x

1

11. 5>7

39. - q

63. 0 65. 1 67. 3 69. 1 71. 0 73. q 27 75. (b) is correct. 77. (d) is correct. 79. c = 10 3 83. -1 87. (a) y = 1 (b) y = 0, y = 2

y 2

9. -23>7

5. 1>2 7. 1>4

17. -2 19. 1>4 21. 2 23. 3 25. -1 1 29. 31. ln 2 33. 1 35. 1>2 37. ln 2 ln 2 41. -1>2 43. -1 45. 1 47. 0 49. 2

27. ln 3

x=1

–8

101.

3. 5>7

15. -16

–1

x

4 y=x–4

–100

x x2 – 1

x = –2

–6

x

y x = –1

(x – 1)3 9 y = x2 + x – 2 2

3 2 y = 45 x5 + 16x2 – 25

4 4)

5

x

0.5

1

1.5

2

2.5

3

x

(c) The maximum value of ƒ(x) is close to 1 near the point x L 1.55 (see the graph in part (a)).

Section 4.6, pp. 260–266 1. 16 in., 4 in. by 4 in. 3. (a) sx, 1 - xd (b) Asxd = 2xs1 - xd 1 1 (c) square units, 1 by 2 2 2450 3 35 5 14 5. 7. 80,000 m2 ; 400 m by 200 m in * * in., 3 3 3 27 9. (a) The optimum dimensions of the tank are 10 ft on the base edges and 5 ft deep. (b) Minimizing the surface area of the tank minimizes its weight for a given wall thickness. The thickness of the steel walls would likely be determined by other considerations such as structural requirements. p 11. 9 * 18 in. 13. 15. h : r = 8 : p 2

Chapter 4: Answers to Odd-Numbered Exercises 17. (a) Vsxd = 2xs24 - 2xds18 - 2xd

(b) Domain: (0, 9)

V Maximum x = 3.3944487 V = 1309.9547

1600 1200

A-23

(b) The minimum distance is from the point (3> 2, 0) to the point (1, 1) on the graph of y = 1x , and this occurs at the value x = 1 , where D(x), the distance squared, has its minimum value. y, D(x)

D(x) = x2 – 2x + 9– 4

800 2.5 400

2

4

6

2

x

8

(c) Maximum volume L 1309.95 in3 when x L 3.39 in. (d) V¿sxd = 24x 2 - 336x + 864 , so the critical point is at x = 7 - 213 , which confirms the result in part (c). (e) x = 2 in. or x = 5 in. 19. L 2418.40 cm3 21. (a) h = 24, w = 18 (24, 10368) V (b) Abs max 10000 6000 V = 54h2 – 3 h3 2

2000 0

5

10

15 20

25

1 Dmin= √5 2

0.5

0.5

51 8

30

31.

9b

x

m, triangle;

b23p 9 + 23p

2p 3 m 3 m, circle

3 (b) -1 * 2 35. (a) 16 2 37. (a) ys0d = 96 ft>sec (b) 256 ft at t = 3 sec (c) Velocity when s = 0 is ys7d = -128 ft>sec. 41. (a) 6 * 6 23 in.

43. (a) 4 23 * 4 26 in. 45. (a) 10p L 31.42 cm> sec ; when t = 0.5 sec , 1.5 sec, 2.5 sec, 3.5 sec; s = 0 , acceleration is 0. (b) 10 cm from rest position; speed is 0. s = ss12 - 12td2 + 64t 2 d1> 2 -12 knots , 8 knots No 4 213 . This limit is the square root of the sums of the squares of the individual speeds. c a ka 2 49. x = , y = 51. + 50 2 4 2 2km 2km 53. (a) (b) A h A h C 57. 4 * 4 * 3 ft, $288 59. M = 65. (a) y = -1 2

47. (a) (b) (c) (e)

67. (a) The minimum distance is

h

⎧ √x , x ≥ 0 y=⎨ ⎩ √–x, x < 0

33.

39. L 46.87 ft

2.5

x

–h

(c) L L 11 in.

9 + 23p

2

h

35

27. Radius = 22 m, height = 1 m, volume = 29. 1

1.5

Section 4.7, pp. 269–271

23. If r is the radius of the hemisphere, h the height of the cylinder, 1>3 1>3 3V 3V and V the volume, then r = a b and h = a p b . 8p 25. (b) x =

1

5 13 51 5763 2387 1. x2 = - , 3. x2 = - , 5. x2 = 3 21 31 4945 2000 7. x1, and all later approximations will equal x0 . y 9.

8000

4000

y = √x

1.5

25 . 2

11. The points of intersection of y = x 3 and y = 3x + 1 or y = x 3 - 3x and y = 1 have the same x-values as the roots of part (i) or the solutions of part (iv). 13. 1.165561185 15. (a) Two (b) 0.35003501505249 and -1.0261731615301 17. ;1.3065629648764, ;0.5411961001462 19. x L 0.45 21. 0.8192 23. 0, 0.53485 25. The root is 1.17951. 27. (a) For x0 = -2 or x0 = -0.8, xi : -1 as i gets large. (b) For x0 = -0.5 or x0 = 0.25, xi : 0 as i gets large. (c) For x0 = 0.8 or x0 = 2, xi : 1 as i gets large. (d) For x0 = - 221>7 or x0 = 221>7 , Newton’s method does not converge. The values of xi alternate between - 221>7 and 221>7 as i increases. 29. Answers will vary with machine speed.

Section 4.8, pp. 277–281 1. (a) x 2 3. (a) x -3 1 5. (a) - x

x3 - x2 + x 3 1 1 (b) - x -3 (c) - x -3 + x 2 + 3x 3 3 5 5 (b) - x (c) 2x + x

(b)

x3 3

(c)

22x 3 + 2 1x 3 9. (a) x 2>3 (b) x 1>3 (c) x -1>3 11. (a) ln x (b) 7 ln x (c) x - 5 ln x 1 13. (a) cos spxd (b) -3 cos x (c) - p cos spxd + cos s3xd 7. (a) 2x 3

(b) 1x

(c)

A-24

Chapter 4: Answers to Odd-Numbered Exercises

15. (a) tan x

x (b) 2 tan a b 3

17. (a) -csc x

(b)

1 csc s5xd 5

(c) -

123. (a) y = 10t 3>2 - 6t 1>2 (b) s = 4t 5>2 - 4t 3>2 127. (a) - 1x + C (b) x + C (c) 1x + C (d) -x + C (e) x - 1x + C (f) -x - 1x + C

3x 2 tan a b 3 2

(c) 2 csc a

px b 2

1 3x e (b) -e -x (c) 2e x>2 3 x 5 1 x -1 -x 1 3 2 a b (a) (b) (c) ln 3 ln 2 ln (5>3) 3 1 1 -1 -1 (a) 2 sin x (b) tan x (c) tan-1 2x 2 2 5x 2 x4 t2 x2 + x + C + C + 7x + C 27. t 3 + 29. 2 4 2 2 3 3 x x 1 - + C -x 33. x 2>3 + C 3 3 2 3 4>3 8 2 3>2 x + x + C 37. 4y 2 - y 3>4 + C 3 4 3 2 2 x2 + x + C + C 41. 2 2t 43. -2 sin t + C 2t u 1 -21 cos + C 47. 3 cot x + C 49. - csc u + C 3 2 4x 1 3x e - 5e -x + C 53. -e -x + + C 3 ln 4 1 4 sec x - 2 tan x + C 57. - cos 2x + cot x + C 2 sin 4t t + + C 61. ln ƒ x ƒ - 5 tan-1 x + C 2 8

19. (a) 21. 23. 25. 31. 35. 39. 45. 51. 55. 59.

1. No 3. No minimum; absolute maximum: ƒs1d = 16; critical points: x = 1 and 11>3 5. Absolute minimum: g(0) = 1; no absolute maximum; critical point: x = 0 7. Absolute minimum: 2 - 2 ln 2 at x = 2; absolute maximum 1 at x = 1 9. Yes, except at x = 0 11. No 15. (b) one 1 17. (b) 0.8555 99677 2 23. Global minimum value of at x = 2 2 25. (a) t = 0 , 6, 12 (b) t = 3 , 9 (c) 6 6 t 6 12 (d) 0 6 t 6 6, 12 6 t 6 14 27.

29. y

y

15 3

y = –x 3 + 6x2 – 9x + 3 3

8 3

3 y = x2 – x 6

1 –2 –1 0

3x s23 + 1d

+ C 65. tan u + C 67. -cot x - x + C 23 + 1 69. -cos u + u + C 2x x2 d x2 a sin x + C b = sin x + cos x = 83. (a) Wrong: 2 2 2 dx x2 cos x x sin x + 2 d s -x cos x + Cd = - cos x + x sin x (b) Wrong: dx d s -x cos x + sin x + Cd = -cos x + x sin x + (c) Right: dx cos x = x sin x 3 3s2x + 1d2s2d d s2x + 1d 85. (a) Wrong: a + Cb = = 3 3 dx 2 2s2x + 1d d ss2x + 1d3 + Cd = 3s2x + 1d2s2d = (b) Wrong: dx 6s2x + 1d2 d (c) Right: ss2x + 1d3 + Cd = 6s2x + 1d2 dx 87. Right 89. (b) 91. y = x 2 - 7x + 10 63.

Practice Exercises, pp. 281–285

1

113. 117. 119. 121.

2

1

x

6

2

3

x

4

–1

31.

33. y

y y = x – 3x 2/3

500

(6, 432) y = x 3(8 – x)

400

–3 –4

9

18

27

(4, 256)

200 100 –2 –1 0 –100

2

4

6

x

8

35.

37. y

y 11 ⎞1 √2, ⎞6 4√2⎞ e1√2⎞ 9 ⎠ ⎠ ⎠ ⎠

2

x 1

–1

2

3

y (x 3)2e x

–4 –3 –2 –1 0 1 2 3 4

x

⎞1 √2, ⎞6 4√2⎞ e1√2⎞ ⎠ ⎠ ⎠ ⎠

y = x √3 – x

–1

(1, 4e)

5 3 1

1

39.

41. y

y

3

x

(8, –4)

300

–2

111. y = -sin t + cos t + t - 1 1 115. y = x - x 4>3 + y = 2x 3>2 - 50 2 y = -sin x - cos x - 2 (a) (i) 33.2 units, (ii) 33.2 units, (iii) 33.2 units (b) True t = 88>k, k = 16

109. y = x - 4x + 5 3

4

–2

2

x 1 1 93. y = - x + 95. y = 9x 1>3 + 4 2 2 97. s = t + sin t + 4 99. r = cos sp ud - 1 1 1 101. y = sec t + 103. y = 3 sec-1 t - p 2 2 1 105. y = x 2 - x 3 + 4x + 1 107. r = t + 2t - 2

2

1

ln 3

–5 –3

5 4 3 2 1 –1 –2 –3 –4

2

y ln( x 2 4 x 3)

–3 –2 –1 2

5

7

x

2

y sin–1(1x)

1 1

–1 –2

– 2

2

3

x

Chapter 4: Answers to Odd-Numbered Exercises 43. (a) Local maximum at x = 4 , local minimum at x = -4 , inflection point at x = 0 (b)

91. x = 5 - 25 hundred L 276 tires,

x=4 Loc max x=0 Infl Loc min x = –4

45. (a) Local maximum at x = 0 , local minima at x = -1 and x = 2 , inflection points at x = (1 ; 27)>3 (b)

Loc max Infl

x=0

x = 1 + √7 3 Loc min Infl x=2

x = 1 – √7 3

Loc min x = –1

47. (a) Local maximum at x = - 22 , local minimum at x = 22 , inflection points at x = ;1 and 0 (b)

Loc max x = –√2

Infl x = –1 Infl x=0 x = √2

Infl x=1

Loc min

53.

55. y

y 5

x2

+1 x = x + 1x

y=

y= x+1 =1+ 4 x–3 x–3 5

4 3

(1, 2)

2 y=x

1

2 1 2 3 4

–1

x

6

–4 –3 –2 –1 –1

1

2

3

4

Additional and Advanced Exercises, pp. 285–288

–3

(–1, –2)

–4 –5

57.

59. y

y

4 3

2 y= x 2

2

3 x = – √3

x = √3 y=1

2

y= 1 x

1 0

2 y= x –4 x2 – 3

4

1 –1

2

3

x

3 2 y= x +2 = x + 1 2 2x x

–4 –3

–1 0 –1

1

2 3

4

–2 –3

–3

61. 73. 85. 89.

y = 2s5 - 25d hundred L 553 tires 93. Dimensions: base is 6 in. by 12 in., height = 2 in.; maximum volume = 144 in.3 5 x4 + x 2 - 7x + C 95. x5 = 2.1958 23345 97. 4 2 4 1 99. 2t 3>2 - t + C 101. + C 103. (u 2 + 1d3>2 + C r + 5 s 1 105. s1 + x 4 d3>4 + C 107. 10 tan + C 3 10 x 1 1 109. 111. x - sin + C csc 22 u + C 2 2 22 x2 1 113. 3 ln x 115. e t + e -t + C + C 2 2 u2 - p 3 117. 119. sec-1 ƒ x ƒ + C + C 2 - p 2 1 121. y = x - x - 1 123. r = 4t 5>2 + 4t 3>2 - 8t 125. Yes, sin-1sxd and -cos-1sxd differ by the constant p>2 . 127. 1> 22 units long by 1> 2e units high, A = 1> 22e L 0.43 units2 129. Absolute maximum = 0 at x = e>2 , absolute minimum = -0.5 at x = 0.5 131. x = ;1 are the critical points; y = 1 is a horizontal asymptote in both directions; absolute minimum value of the function is e -22>2 at x = -1, and absolute maximum value is e 22>2 at x = 1. 133. (a) Absolute maximum of 2> e at x = e 2 , inflection point se 8>3, s8>3de -4>3 d , concave up on se 8>3, q d , concave down on s0, e 8>3 d (b) Absolute maximum of 1 at x = 0 , inflection points s ;1> 22, 1> 2ed , concave up on s - q , -1> 22d ´ s1> 22, q d , concave down on s -1> 22, 1> 22d (c) Absolute maximum of 1 at x = 0 , inflection point (1, 2> e), concave up on s1, q d , concave down on s - q , 1d

x

–2 –3

A-25

5 63. 0 65. 1 67. 3>7 69. 0 71. 1 ln 10 75. ln 2 77. 5 79. - q 81. 1 83. e bk (a) 0, 36 (b) 18, 18 87. 54 square units height = 2, radius = 22

x

1. The function is constant on the interval. 3. The extreme points will not be at the end of an open interval. 5. (a) A local minimum at x = -1 , points of inflection at x = 0 and x = 2 (b) A local maximum at x = 0 and local minima 1 ; 27 at x = -1 and x = 2 , points of inflection at x = 3 9. No 11. a = 1, b = 0, c = 1 13. Yes 15. Drill the hole at y = h>2 . RH 17. r = for H 7 2R, r = R if H … 2R 2sH - Rd 10 5 1 1 1 19. (a) (b) (c) (d) 0 (e) (f) 1 (g) 3 3 2 2 2 (h) 3 c - b c + b b 2 - 2bc + c 2 + 4ae 21. (a) (b) (c) 2e 2 4e c + b + t (d) 2 1 1 23. m0 = 1 - q , m1 = q

A-26

Chapter 5: Answers to Odd-Numbered Exercises

25. s = ce kt

(c)

y

s

(2, 3)

3

1000

s = soe

f (x) = x 2 – 1, 0≤x≤2 Midpoint

kt

800

2

600 s = 16t 2

400

1 200 0

1

2

3

4

5

6

7

27. (a) k = -38.72

(b) 25 ft

29. Yes, y = x + C

31. y0 =

t c1

0

2 22 3>4 b 3

c2

x

c4

c3

–1

35. (a)

(b)

CHAPTER 5

y

Section 5.1, pp. 296–298 (a) 0.125 (b) 0.21875 (c) 0.625 (d) 0.46875 (a) 1.066667 (b) 1.283333 (c) 2.666667 (d) 2.083333 0.3125, 0.328125 7. 1.5, 1.574603 (a) 87 in. (b) 87 in. 11. (a) 3490 ft (b) 3840 ft (a) 74.65 ft> sec (b) 45.28 ft> sec (c) 146.59 ft 31 15. 17. 1 16 19. (a) Upper = 758 gal, lower = 543 gal (b) Upper = 2363 gal, lower = 1693 gal (c) L 31.4 h, L 32.4 h 1. 3. 5. 9. 13.

(b) 222 L 2.828 p (c) 8 sin a b L 3.061 8 (d) Each area is less than the area of the circle, p . As n increases, the polygon area approaches p .

f (x) = sin x, – ≤ x ≤ Left-hand

c1 = –

y f (x) = sin x, – ≤ x ≤ Right-hand

1

c2

c3 = 0

c4

x

–

1

c1

c2 = 0

–1

c3

c4 =

–1

(c)

y f (x) = sin x, – ≤ x ≤ Midpoint 1 c1 –/2 c2

21. (a) 2

c3 /2 c4

–

x

–1

2 1 1 , 3 2n 6n 2 5 6n + 1 5 1 43. 45. , + + 6 6 2 6n 2 37. 1.2

Section 5.2, pp. 304–305 6s1d 6s2d + = 7 1 + 1 2 + 1 3. coss1dp + coss2dp + coss3dp + coss4dp = 0

39.

2 3

41. 12 +

1 1 n + 2n 2 ,

27n + 9 , 2n 2

12

1 2

1.

23 - 2 p p + sin = 7. All of them 2 3 2 6 4 5 1 1 13. a k 15. a s -1dk + 1 ak k k=1 k=1 2 k=1 (a) -15 (b) 1 (c) 1 (d) -11 (e) 16 (a) 55 (b) 385 (c) 3025 -56 23. -73 25. 240 27. 3376 (a) 21 (b) 3500 (c) 2620 (a) 4n (b) cn (c) (n 2 - n)>2 (a) (b)

5. sin p - sin 11. 17. 19. 21. 29. 31. 33.

y

Section 5.3, pp. 313–317 2

9. b

3

9. 11. 13. 17. 21. 27. 35. 45. 51.

(2, 3) 3

2

x2

f (x) = – 1, 0≤x≤2 Right-hand

f (x) = x – 1, 0≤x≤2 Left-hand 2

5

x 2 dx

L0

3.

L-7

3

sx 2 - 3xd dx

2

c1 = 0 c2 c3 = 1 c4

b

–1

1

2

x

0

–1

1 dx L2 1 - x

sec x dx L-p>4 (a) 0 (b) -8 (c) -12 (d) 10 (e) -2 (f) 16 (a) 5 (b) 5 23 (c) -5 (d) -5 (a) 4 (b) -4 15. Area = 21 square units Area = 9p>2 square units 19. Area = 2.5 square units Area = 3 square units 23. b 2>4 25. b 2 - a 2 (a) 2p (b) p 29. 1>2 31. 3p2>2 33. 7>3 2 1>24 37. 3a >2 39. b>3 41. -14 43. -2 49. 0 -7>4 47. 7 Using n subintervals of length ¢x = b>n and right-endpoint values: Area =

1

5.

0

7.

y (2, 3)

1.

c1 c2 = 1 c3 c4 = 2

x

L0

3x 2 dx = b 3

53. Using n subintervals of length ¢x = b>n and right-endpoint values: b

Area =

L0

2x dx = b 2

x

Chapter 5: Answers to Odd-Numbered Exercises 55. 61. 63. 71. 73.

1

La

1

sin sx 2 d dx …

L0

b

77.

(f) Toward the origin between t = 6 and t = 9 since the velocity is negative on this interval. Away from the origin between t = 0 and t = 6 since the velocity is positive there. (g) Right or positive side, because the integral of ƒ from 0 to 9 is positive, there being more area above the x-axis than below.

avsƒd = 0 57. avsƒd = -2 59. avsƒd = 1 (a) avsgd = -1>2 (b) avsgd = 1 (c) avsgd = 1>4 c(b - a) 65. b 3>3 - a 3>3 67. 9 69. b 4>4 - a 4>4 a = 0 and b = 1 maximize the integral. Upper bound = 1, lower bound = 1>2

75. For example,

dx = 1

L0

Section 5.5, pp. 333–335

b

ƒsxd dx Ú

La

0 dx = 0

79. Upper bound = 1>2

3. -10>3

5. 8 7. 1 9. 2 23 11. 0 2 12 p 13. -p>4 15. 1 17. 19. -8>3 4 4 4 21. -3>4 23. 22 - 28 + 1 25. -1 27. 16 1 1 29. 7>3 31. 2p>3 33. p (4p - 2p) 35. (e - 1) 2 1 41. 4t 5 b 2 1x 3 1 43. 3x 2e -x 45. 21 + x 2 47. - x -1>2 sin x 49. 0 2 2 51. 1 53. 2xe (1>2)x 55. 1 57. 28>3 59. 1>2 61. p p 22p 1 1 63. 65. d, since y¿ = x and yspd = dt - 3 = -3 2 Lp t 37. 226 - 25

1. 5.

Section 5.4, pp. 325–328 1. 6

39. scos1xd a

9. 13. 15. 17. 21. 25. 29.

0

67. b, since y¿ = sec x and ys0d =

L0

sec t dt + 4 = 4

33.

x

2 71. bh 73. $9.00 sec t dt + 3 3 L2 a. T(0) = 70°F, T(16) = 76°F T(25) = 85°F b. av(T) = 75°F 2x - 2 79. -3x + 5 (a) True. Since ƒ is continuous, g is differentiable by Part 1 of the Fundamental Theorem of Calculus. (b) True: g is continuous because it is differentiable. (c) True, since g¿s1d = ƒs1d = 0 . (d) False, since g–s1d = ƒ¿s1d 7 0 . (e) True, since g¿s1d = 0 and g–s1d = ƒ¿s1d 7 0 . (f) False: g–sxd = ƒ¿sxd 7 0 , so g– never changes sign. (g) True, since g¿s1d = ƒs1d = 0 and g¿sxd = ƒsxd is an increasing function of x (because ƒ¿sxd 7 0). t d ds (a) y = ƒsxd dx = ƒstd Q ys5d = ƒs5d = 2 m>sec = dt dt L0 (b) a = df>dt is negative since the slope of the tangent line at t = 5 is negative. 3 9 1 (c) s = ƒsxd dx = s3ds3d = m since the integral is the 2 2 L0 area of the triangle formed by y = ƒsxd , the x-axis, and x = 3. (d) t = 6 since after t = 6 to t = 9 , the region lies below the x-axis. (e) At t = 4 and t = 7 , since there are horizontal tangents there.

69. y = 75.

77. 81.

83.

A-27

37.

1 1 3. - (x 2 + 5) -3 + C (2x + 4) 6 + C 6 3 1 1 7. - cos 3x + C (3x 2 + 4x) 5 + C 10 3 1 11. -6s1 - r 3 d1>2 + C sec 2t + C 2 1 3>2 1 - 1d - sin s2x 3>2 - 2d + C sx 3 6 1 1 (a) - scot2 2ud + C (b) - scsc2 2ud + C 4 4 1 2 - s3 - 2sd3>2 + C 19. - s1 - u2 d5>4 + C 5 3 1 23. tan s3x + 2d + C s -2>s1 + 1xdd + C 3 6 1 6 x r3 27. a sin a b + C - 1b + C 2 3 18 1 2 31. - cos sx 3>2 + 1d + C + C 3 2 cos s2t + 1d sin2 s1>ud 1 35. -sin a t - 1 b + C + C 2 3>2 2 1 1 39. + C s1 + t 4 d4 + C a2 - x b 16 3 3>2

3 2 + C a1 - 3 b 27 x 1 1 43. (x - 1) 12 + (x - 1) 11 + C 12 11 4 2 1 45. - (1 - x) 8 + (1 - x) 7 - (1 - x) 6 + C 7 8 3 -1 1 1 47. sx 2 + 1d5>2 - sx 2 + 1d3>2 + C 49. + C 5 3 4 (x 2 - 4) 2 41.

51. e sin x + C

53. 2 tan A e2x + 1 B + C

57. z - ln (1 + ez ) + C 61. e sin

-1

x

+ C

63.

55. ln ƒ ln x ƒ + C

5 2r 59. tan-1 a b + C 6 3

1 ssin-1 xd3 + C 3

65. ln ƒ tan-1 y ƒ + C

6 6 (b) + C + C 2 + tan3 x 2 + tan3 x 6 (c) + C 2 + tan3 x 1 1 69. sin 23s2r - 1d2 + 6 + C 71. s = s3t 2 - 1d4 - 5 2 6 p 73. s = 4t - 2 sin a2t + b + 9 6 67. (a) -

75. s = sin a2t -

p b + 100t + 1 2

77. 6 m

A-28

Chapter 6: Answers to Odd-Numbered Exercises

Section 5.6, pp. 341–344 1. (a) 14>3 (b) 2>3 3. (a) 1>2 (b) -1>2 5. (a) 15>16 (b) 0 7. (a) 0 (b) 1>8 9. (a) 4 (b) 0 11. (a) 1>6 (b) 1>2 13. (a) 0 (b) 0 15. 2 23 17. 3>4 19. 35>2 - 1 21. 3 23. p>3 25. e 1 27. ln 3 29. sln 2d2 31. 33. ln 2 35. ln 27 37. p ln 4 39. p>12 41. 2p>3 43. 23 - 1 45. -p>12 47. 16>3 49. 25>2 51. p>2 53. 128>15 55. 4>3 57. 5>6 59. 38>3 61. 49>6 63. 32>3 65. 48>5 67. 8>3 69. 8 71. 5>3 (There are three intersection points.) 73. 18 75. 243>8 77. 8>3 79. 2 81. 104>15 4 4 83. 56>15 85. 4 87. 89. p>2 91. 2 93. 1>2 - p 3 95. 1 97. ln 16 99. 2 101. 2 ln 5 103. (a) (; 2c, c) (b) c = 42>3 (c) c = 42>3 105. 11>3 107. 3>4 109. Neither 111. Fs6d - Fs2d 113. (a) -3 (b) 3 115. I = a>2

9. 11. 23. 29. 37. 41. 45. 49. 55. 61. 65.

75. 2 85

-1

2x

+ C

77. 1 23

9. y = x 3 + 2x - 4

2x 2 + 1

11. 36>5

13. y

y = x2 ⁄ 3

–4

0

1 2 - p 2 y

4

1 yt

x

3 y = –4

0

–4

1

2

y sin t

t

–1

15. 13>3 2

4

6

y

t (sec)

8

2

(d) 0 x -1>2 s2x - 1d dx = 2 7. cos dx = 2 2 L1 L-p (a) 4 (b) 2 (c) -2 (d) -2p (e) 8>5 8>3 13. 62 15. 1 17. 1>6 19. 18 21. 9>8 22 8 22 - 7 p2 27. + - 1 25. 4 32 2 6 Min: -4 , max: 0, area: 27>4 31. 6>5 33. 1 x sin t -1 y = a t b dt - 3 39. y = sin x L5 2p y = sec-1 x + , x 7 1 43. -4scos xd1>2 + C 3 t3 4 47. u2 + u + sin s2u + 1d + C + t + C 3 1 51. tan se x - 7d + C 53. e tan x + C - cos s2t 3>2 d + C 3 -ln 7 1 57. ln (9> 25) 59. - sln xd-2 + C 3 2 3 -1 1 x2 A 3 B + C 63. 2 sin 2sr - 1d + C 2 ln 3 22 -1 x - 1 2x - 1 1 67. sec-1 ` tan a b + C ` + C 2 4 2 22

69. e sin

3 (b) 2 12

5. (a) 1>4

y=2

(c) 13

5

5.

7. ƒsxd =

(b) No x

(b) h (feet)

0

(b) 31

1. (a) Yes

–8

700 600 500 400 300 200 100

3. (a) -1>2

Additional and Advanced Exercises, pp. 349–352

2

Practice Exercises, pp. 345–348 1. (a) About 680 ft

d 1 sx ln x - x + Cd = x # x + ln x - 1 + 0 = ln x dx 1 (b) e - 1 -6 119. 25°F 121. 22 + cos3 x 123. 3 + x4 dy dy -2 1 125. = x e cos (2 ln x) 127. = 2 dx dx 21 - x 21 - 2 (sin-1 x) 2 129. Yes 131. - 21 + x 2 133. Cost L $10,899 using a lower sum estimate 117. (a)

0

71. 2 2tan - 1 y + C 79. 8

81. 27 23>160

87. 6 23 - 2p

89. -1

73. 16 83. p>2 91. 2

93. 1

95. 15>16 + ln 2 97. e - 1 99. 1> 6 101. 9> 14 9 ln 2 103. 105. p 107. p> 23 109. sec-1 ƒ 2y ƒ + C 4 111. p>12 113. (a) b (b) b

y = 1 – x2

y=1

–2

–1

1

x

2

1

19. p>2

17. 1>2

21. ln 2

23. 1>6

25.

L0

ƒsxd dx

2

27. (b) pr 29. (a) 0 (b) -1 (c) -p (d) x = 1 (e) y = 2x + 2 - p (f) x = -1, x = 2 (g) [-2p, 0] sin 4y sin y ƒxƒ 31. 2>x 33. 35. 2x ln ƒ x ƒ - x ln 1y 21y 22 37. ssin xd>x

39. x = 1

41.

1 1 , , 2:1 ln 2 2 ln 2

43. 2> 17

CHAPTER 6 Section 6.1, pp. 361–364 1. 16

3. 16>3

5. (a) 223

9. 8p

11. 10

13. (a) s 2h

17. 4 - p

19.

32p 5

29. p a

21. 36p

(b) 8 (b) s 2h 23. p

7. (a) 60

(b) 36

2p 3 p 1 25. a1 - 2 b 2 e

15.

p 11 31. 2p 33. 2p + 2 22 b 2 3 117p 2p 35. 4p ln 4 37. p2 - 2p 39. 41. 5 3 7p 4p 43. psp - 2d 45. 47. 8p 49. 3 6 27.

p ln 4 2

Chapter 6: Answers to Odd-Numbered Exercises 32p 8p 224p (c) (d) 5 3 15 16p 56p 64p 53. (a) (b) (c) 55. V = 2a 2bp2 15 15 15 ph 2(3a - h) 1 57. (a) V = (b) m>sec 3 120p 4 - b + a 61. V = 3308 cm3 63. 2 51. (a) 8p

A-29

Section 6.4, pp. 381–383

(b)

p>4

(tan x) 21 + sec4 x dx

1. (a) 2p L0 (b)

(c) S L 3.84

y 1 0.8

Section 6.2, pp. 369–371

0.6

1. 6p 7p 11. 15

0.4

3. 2p

5. 14p>3

13. (b) 4p

7. 8p 9. 5p>6 16p 15. (322 + 5) 15

y = tan x

0.2

8p 16p 4p 19. 21. 3 3 3 23. (a) 16p (b) 32p (c) 28p (d) 24p (e) 60p (f) 48p 27p 27p 72p 108p 25. (a) (b) (c) (d) 5 5 2 2 6p 4p 27. (a) (b) (c) 2p (d) 2p 5 5 2p 29. (a) About the x-axis: V = ; about the y-axis: V = 15 2p (b) About the x-axis: V = ; about the y-axis: V = 15 5p 2p 4p 31. (a) (b) (c) 2p (d) 3 3 3 7p 4p 33. (a) (b) 15 30 48p 24p 35. (a) (b) 5 5 9p 9p 37. (a) (b) 16 16 39. Disk: 2 integrals; washer: 2 integrals; shell: 1 integral 256p 244p 1 41. (a) (b) 47. p a1 - e b 3 3

0.2

0

17.

2

3. (a) 2p (b)

L1

1. 12

3 9. ln 2 + 8

17. (a)

21 + 4x 2 dx

L-1 p

1.6

p 6 p 6

xy = 1

1.4 1.2 1 0.5

0.6

53 11. 6

L-1

0.8

0.9

(c) S L 63.37

4

3 x1/2 + y1/2 = 3 2

13. 2

1

2

7. (a) 2p

L0

(b)

a

3

4

x

y

L0

(c) S L 2.08

tan t dtb sec y dy

y 1

(c) L 3.82

21 + s y + 1d2 dy

(c) L 9.29

p>6

sec x dx (c) L 0.55 L0 23. (a) y = 1x from (1, 1) to (4, 2) (b) Only one. We know the derivative of the function and the value of the function at one value of x. 25. 1 27. Yes, ƒ(x) = ; x + C where C is any real number.

0.8 0.6 y

0.4

x=

L0

tan t dt

0.2

21. (a)

35.

x

1

4

3

19. (a)

0.7

5. (a) 2p s3 - x 1>2 d2 21 + s1 - 3x -1>2 d2 dx L1 (b) y

(c) L 6.13

21 + cos2 y dy

L0

(c) S L 5.02

1.8

2

15. (a)

1 -4 y 21 + y dy

p>3

99 7. 8

x

0.8

y

Section 6.3, pp. 376–377 123 5. 32

0.6

2

1

53 3. 6

0.4

2 (10 3>2 - 1) 27

0

9. 4p25

0.1 0.2 0.3 0.4 0.5 0.6 0.7

11. 3p25

17. p(28 - 1)>9 23. 253p>20

x

13. 98p>81

19. 35p 25>3

15. 2p

21. p a

15 + ln 2b 16

27. Order 226.2 liters of each color.

A-30

Chapter 7: Answers to Odd-Numbered Exercises

Section 6.5, pp. 386–389 1. 5. 7. 13.

15. 19. 25. 31.

400 N>m 3. 4 cm, 0.08 J (a) 7238 lb> in. (b) 905 in.-lb, 2714 in.-lb 780 J 9. 72,900 ft-lb 11. 160 ft-lb (a) 1,497,600 ft-lb (b) 1 hr, 40 min (d) At 62.26 lb>ft3: a) 1,494,240 ft-lb b) 1 hr, 40 min At 62.59 lb>ft3: a) 1,502,160 ft-lb b) 1 hr, 40.1 min 37,306 ft-lb 17. 7,238,299.47 ft-lb 2446.25 ft-lb 21. 15,073,099.75 J 85.1 ft-lb 27. 98.35 ft-lb 29. 91.32 in.-oz 5.144 * 10 10 J

1. x = 0, y = 12>5 3. x = 1, y = -3>5 5. x = 16>105, y = 8>15 7. x = 0, y = p>8 9. x L 1.44, y L 0.36 ln 16 ln 4 11. x = p , y = 0 13. x = 7, y = 12 15. x = 3>2, y = 1>2 224p 17. (a) (b) x = 2, y = 0 3 (c) y y=

4

4

72p 9p 3. p2 5. 280 35 7. (a) 2p (b) p (c) 12p>5 (d) 26p>5 9. (a) 8p (b) 1088p>15 (c) 512p>15 1.

11. p(323 - p)>3 13. p(e - 1) 28p 3 10 1 15. ft 17. 19. 3 + ln 2 3 3 8 28p 22>3 23. 4p 25. 4640 J 10 ft-lb, 30 ft-lb 29. 418,208.81 ft-lb 22,500p ft-lb, 257 sec 33. x = 0, y = 8>5 x = 3>2, y = 12>5 37. x = 9>5, y = 11>10

Additional and Advanced Exercises, pp. 399–400

p 5. 30 12

A

2x - a p 7. 28> 3

7. ln s1 + 2xd + C

9. 1

5. ln ƒ 6 + 3 tan t ƒ + C

11. 2(ln 2) 4

-t 2

13. 2

+ C 15. 2e 17. -e + C 19. - e + C 1 sec pt 1 + C 21. p e 23. 1 25. ln s1 + e r d + C 27. 2 ln 2 6 1 29. 31. 33. 32760 35. 322 + 1 ln 2 ln 7 sln xd2 3 ln 2 1 a b + C 37. 39. 2sln 2d2 41. 43. ln 10 2 2 ln 10 1>x

47. y = 1 - cos se t - 2d

2 3>2 y - x 1>2 = C 11. e y - e x = C 3 13. -x + 2 tan 2y = C 15. e -y + 2e 2x = C 1 17. y = sin sx 2 + Cd 19. ln ƒ y 3 - 2 ƒ = x 3 + C 3 2 21. 4 ln (1y + 2) = e x + C 9.

Practice Exercises, pp. 397–399

1. ƒsxd =

3. ln ƒ y 2 - 25 ƒ + C

Section 7.2, pp. 418–420 x

21. x = y = 1>3 23. x = a>3, y = b>3 25. 13d>6 ap 27. x = 0, y = 4 29. x = 1>2, y = 4 31. x = 6>5, y = 8>7

21. 27. 31. 35.

2 1. ln a b 3

49. y = 2se + xd - 1 51. y = x + ln ƒ x ƒ + 2 53. p ln 16 55. 6 + ln 2 57. (b) 0.00469 69. (a) 1.89279 (b) -0.35621 (c) 0.94575 (d) -2.80735 (e) 5.29595 (f) 0.97041 (g) -1.03972 (h) -1.61181

y=– 4 √x

–4

Section 7.1, pp. 409–411

-x

√x

1

CHAPTER 7

45. sln 10d ln ƒ ln x ƒ + C

(2, 0) 0

(b) s2a>p, 2a>pd

2r

Section 6.6, pp. 396–397

4

n , s0, 1>2d 2n + 1 15. (a) x = y = 4sa 2 + ab + b 2 d>s3psa + bdd 11. x = 0, y =

3. ƒsxd = 2C 2 - 1 x + a, where C Ú 1 4h 23mh 9. 3

23. 25. 31. 33. 39. 41.

(a) -0.00001 (b) 10,536 years (c) 82% 54.88 g 27. 59.8 ft 29. 2.8147498 * 10 14 (a) 8 years (b) 32.02 years 15.28 years 35. 56,562 years (a) 17.5 min (b) 13.26 min 45. 54.62% -3°C 43. About 6693 years

Section 7.3, pp. 425–428 1. cosh x = 5>4, tanh x = -3>5, coth x = -5>3, sech x = 4>5, csch x = -4>3 3. sinh x = 8>15, tanh x = 8>17, coth x = 17>8, sech x = 15>17, csch x = 15>8 x 1 5. x + x 7. e 5x 9. e 4x 13. 2 cosh 3 tanh 2t 15. sech2 2t + 17. coth z 2t 19. sln sech udssech u tanh ud 21. tanh3 y 1 1 25. 27. - tanh-1 u 1 + u 22xs1 + xd 29.

1 22t

- coth-1 2t

31. -sech-1 x

23. 2

33.

ln 2 1 1 + a b 2 B

2u

Chapter 8: Answers to Odd-Numbered Exercises

35. ƒ sec x ƒ 43. 12 sinh a

x - ln 3 b + C 2

47. tanh ax 53.

cosh 2x + C 2

41.

1 b + C 2

3 + ln 2 32

CHAPTER 8 45. 7 ln ƒ e x>7 + e -x>7 ƒ + C 51. ln

57. 3> 4

3 + ln22 8

59.

-ln 3 61. ln (2> 3) 63. 65. ln 3 2 67. (a) sinh-1s 23d (b) ln s 23 + 2d 69. (a) coth-1s2d - coth-1s5>4d

(b) -ln a

12 b + sech-1 13

Section 8.1, pp. 436–438 1. -2x cos sx>2d + 4 sin sx>2d + C

5 2

49. -2 sech 2t + C

55. e - e -1

71. (a) -sech-1 a

7. xe x - e x + C 11. 13. 15. 17. 19.

1 1 (b) a b ln a b 2 3 4 a b 5

21.

3 = -ln a b + ln s2d = ln s4>3d 2 73. (a) 0

79. 2p

81.

6 5

1 (ln (x - 5)) 2 + C 2 ln 2 9. 3 ln 7 11. 2(22 - 1) 13. y = ln s3>2d 1 15. y = ln x - ln 3 17. y = 1 - ex 19. 1> 3 21. 1> e m> sec 23. ln 5x - ln 3x = ln s5>3d 25. 1> 2

27. y = atan-1 a

5. 2 ln 2

x + C bb 2

2 -1 29. y = sin s2 tan x + Cd 31. y = -2 + ln s2 - e -x d 35. 19,035 years

7.

2

33. y = 4x - 42x + 1

Additional and Advanced Exercises, pp. 429–430 (b) p>2 (c) p p 1 3. tan x + tan-1 A x B is a constant and the constant is for 2 p x 7 0; it is - for x 6 0. 2 1. (a) 1 -1

25.

29. 31. 33.

Practice Exercises, pp. 428–429 3. ln 8

23.

27.

(b) 0 mg 77. (b) (c) 80 25 L 178.89 ft>sec A k

35. 39. 41.

–2 –2

ln 4 7. x = p , y = 0

45. 2 2x sin 2x + 2 cos 2x + C

5p - 323 p - 4 49. 8 9 51. (a) p (b) 3p (c) 5p (d) s2n + 1dp 53. 2ps1 - ln 2d 55. (a) psp - 2d (b) 2p p 2 (e + 9) 57. (a) 1 (b) (e - 2)p (c) 2 1 1 (d) x = (e 2 + 1), y = (e - 2) 4 2 1 s1 - e -2p d 61. u = x n, dy = cos x dx 59. 2p 63. u = x n, dy = e ax dx 67. x sin-1 x + cos ssin-1 xd + C 2

47.

69. x sec-1 x - ln ƒ x + 2x 2 - 1 ƒ + C 73. (a) x sinh

-1

x - cosh ssinh

-1

71. Yes

xd + C

y= 2 y = tan –1 x + tan –1 ⎛ x1 ⎛ ⎝ ⎝

2 –1

y= – 2

y tan s yd - ln21 + y 2 + C x tan x + ln ƒ cos x ƒ + C sx 3 - 3x 2 + 6x - 6de x + C sx 2 - 7x + 7de x + C sx 5 - 5x 4 + 20x 3 - 60x 2 + 120x - 120de x + C 1 s -e u cos u + e u sin ud + C 2 e 2x s3 sin 3x + 2 cos 3xd + C 13 2 A 23s + 9 e 23s + 9 - e 23s + 9 B + C 3 p23 p2 - ln s2d 3 18 1 [-x cos sln xd + x sin sln xd] + C 2 1 ln ƒ sec x 2 + tan x 2 ƒ + C 2 1 1 1 2 x (ln x) 2 - x 2 ln x + x 2 + C 2 2 4 1 4 1 1 37. e x + C - x ln x - x + C 4 2 2 1 2 2 x (x + 1) 3>2 (x + 1) 5>2 + C 3 15 3 2 - sin 3x sin 2x - cos 3x cos 2x + C 5 5

(b) x sinh-1 x - s1 + x 2 d1>2 + C

2

–4

9. -sx2 + 2x + 2d e-x + C

3 4

-1

43. -cos e x + C

y

1

5. ln 4 -

3. t 2 sin t + 2t cos t - 2 sin t + C

1 + 21 - s4>5d2 1 + 21 - s12>13d2 b + ln a b s12>13d s4>5d

1. -cos e x + C

A-31

4

x

Section 8.2, pp. 443–444 1 1 sin 2x + C 3. - cos4 x + C 2 4 2 1 7. - cos x + cos3 x - cos5 x + C 5 3 1.

11.

1 1 4 sin x - sin6 x + C 4 6

13.

1 cos3 x - cos x + C 3 1 9. sin x - sin3 x + C 3

5.

1 1 x + sin 2x + C 2 4

A-32

Chapter 8: Answers to Odd-Numbered Exercises

15. 16> 35

17. 3p

49. y = 2 B

19. -4 sin x cos3 x + 2 cos x sin x + 2x + C 21. -cos4 2u + C 27.

3 2 3 B2

33.

1 tan2 x + C 2

23. 4

25. 2

4 3 a b 5 2

29.

35.

5>2

-

18 2 3 - a b 7 2 35

1 sec3 x + C 3

39. 2 23 + ln s2 + 23 d

41.

37.

7>2

31. 22

1 tan3 x + C 3

2 1 tan u + sec2 u tan u + C 3 3

43. 4> 3 45. 2 tan2 x - 2 ln s1 + tan2 xd + C 1 1 47. tan4 x - tan2 x + ln ƒ sec x ƒ + C 4 2 1 4 1 49. 53. p - ln23 51. cos 5x - cos x + C 3 10 2 1 1 55. sin x + sin 7x + C 2 14 1 1 1 57. sin 3u - sin u sin 5u + C 6 4 20 2 1 1 59. - cos5 u + C 61. cos u cos 5u + C 5 4 20

63. sec x - ln ƒ csc x + cot x ƒ + C 65. cos x + sec x + C 1 2 1 1 67. x - x sin 2x - cos 2x + C 69. ln s1 + 22d 4 4 8 2 8p + 3 4p 71. p2>2 73. x = ,y = 3 12p

51. y =

3 3p x tan-1 a b 2 2 8

55. (a)

1 (p + 6 23 - 12) 12

(b) x =

(b) (c)

7.

t225 - t 2 25 -1 t sin a b + + C 5 2 2

9.

24x 2 - 49 2x 1 ln ` + ` + C 7 7 2

11. 7 B

2y 2 - 49 y - sec-1 a b R + C 7 7

15. - 29 - x 2 + C

2x 2 - 1 + C x

1 17. sx 2 + 4d3>2 - 42x 2 + 4 + C 3

31.

5 1 21 - x 2 b + C a x 5

35. ln 9 - ln s1 + 210d 41. 2x 2 - 1 + C 45. 4 sin-1 47.

29. 2 tan-1 2x +

1 2 1 x + ln ƒ x 2 - 1 ƒ + C 2 2

3 3 1 2 3. + + x - 3 x - 2 x + 1 sx + 1d2 17 -2 2 -1 -12 5. z + 2 + 7. 1 + + z - 1 t - 3 t - 2 z 1 9. [ln ƒ 1 + x ƒ - ln ƒ 1 - x ƒ ] + C 2

43.

37. p>6

1 ln sx + 6d2sx - 1d5 ƒ + C 7 ƒ

13. sln 15d>2

1 1 1 ln t + ln ƒ t + 2 ƒ + ln ƒ t - 1 ƒ + C 2 ƒ ƒ 6 3

4x + C s4x + 1d

39. sec-1 ƒ x ƒ + C

1 ln 21 + x 4 + x 2 ƒ + C 2 ƒ

1x + 2x 24 - x + C 2

1 -1 1 sin 2x - 2x 21 - x (1 - 2x) + C 4 4

21. sp + 2 ln 2d>8

2 1 2 -1 2x + 1 b + C 27. 3 ln ƒ x - 1 ƒ + 6 ln ƒ x + x + 1 ƒ - 23 tan a 23 x - 1 1 1 -1 29. 4 ln ` x + 1 ` + 2 tan x + C -1 + ln (u2 + 2u + 2) - tan-1 su + 1d + C u2 + 2u + 2

33. x 2 + ln `

2

y 1 b + C a 3 21 - y 2

17. 3 ln 2 - 2

25. -ss - 1d-2 + ss - 1d-1 + tan-1 s + C

31.

3

33.

1 1 (1 - x 2) 5>2 - (1 - x 2) 3>2 + C 5 3

x + 1 x 1 ln ` ` + C 4 x - 1 2sx 2 - 1d 1 23. tan-1 y - 2 + C y + 1

-2 24 - w 2 5x 10 19. 21. + C tan-1 + C w 3 6 x 4p 23. 423 25. + C 3 2x 2 - 1 27. -

1 1 (1 - x 2) 3>2 + (1 - x 2) 5>2 + C 5 3

19.

13.

p2 + 1223p - 72

1.

15. -

5. p>6

, y =

Section 8.4, pp. 454–455

Section 8.3, pp. 447–448 3. p>4

3 23 - p

53. 3p>4

4(p + 6 23 - 12) 12(p + 6 23 - 12) 2 1 57. (a) - x 2 (1 - x 2) 3>2 (1 - x 2) 5>2 + C 3 15

11. 1. ln ƒ 29 + x 2 + x ƒ + C

2x 2 - 4 x - sec-1 a b R 2 2

x - 1 x ` + C

1 35. 9x + 2 ln ƒ x ƒ + x + 7 ln ƒ x - 1 ƒ + C y2 1 - ln ƒ y ƒ + ln s1 + y 2 d + C 2 2 sin y - 2 1 41. ln ` ` + C 5 sin y + 3 37.

43.

39. ln a

et + 1 b + C et + 2

stan-1 2xd2 6 - 3 ln ƒ x - 2 ƒ + + C 4 x - 2

45. ln `

1x - 1 ` + C 1x + 1

47. 221 + x + ln `

1x + 1 - 1 ` + C 1x + 1 + 1

A-33

Chapter 8: Answers to Odd-Numbered Exercises x4 1 ` + C ln ` 4 4 x + 1 51. x = ln ƒ t - 2 ƒ - ln ƒ t - 1 ƒ + ln 2

51.

49.

55. 3p ln 25

59. (a) x =

57. 1.10

53. x = 1000e 4t 499 + e 4t

6t - 1 t + 2 (b) 1.55 days

x - 3 b + C A 3 23 2sx - 2d + 4b + C 3. 2x - 2 a 3 5.

atan-1

2

s2x - 3d3>2sx + 1d + C 5

29 - 4x - 3 - 29 - 4x 2 ` + C - ln ` x 3 29 - 4x + 3 sx + 2ds2x - 6d24x - x 2 x - 2 + 4 sin-1 a b + C 9. 6 2 7.

11. -

1 27

ln `

27 + 27 + x 2 ` + C x

13. 24 - x 2 - 2 ln ` 15. 17. 19. 21. 25. 27.

2 + 24 - x 2 ` + C x

e 2t s2 cos 3t + 3 sin 3td + C 13 2 x 1 1 cos-1 x + sin-1 x - x21 - x 2 + C 2 4 4 x2 x3 1 tan-1 x + ln s1 + x 2 d + C 3 6 6 sin s7t>2d sin s9t>2d cos 5x cos x + C d + C 23. 8 c 7 10 2 9 6 6 sin su>12d + sin s7u>12d + C 7 x 1 1 ln (x 2 + 1) + + tan-1 x + C 2 2 2s1 + x 2 d

29. ax -

1 1 b sin-1 2x + 2x - x 2 + C 2 2

31. sin-1 2x - 2x - x 2 + C 33. 21 - sin2 t - ln `

1 + 21 - sin2 t ` + C sin t

35. ln ƒ ln y + 23 + sln yd

2

ƒ

53. 22 + ln A 22 + 1 B

+ C

37. ln ƒ x + 1 + 2x 2 + 2x + 5 ƒ + C 9 x + 2 x + 2 25 - 4x - x 2 + sin-1 a b + C 2 2 3 2 sin2 2x cos 2x 4 cos 2x sin4 2x cos 2x + C 41. 10 15 15 sin3 2u cos2 2u sin3 2u + + C 43. 10 15 45. tan2 2x - 2 ln ƒ sec 2x ƒ + C 39.

ssec pxdstan pxd 1 + p ln ƒ sec px + tan px ƒ + C p 3 csc x cot x 3 -csc3 x cot x - ln ƒ csc x + cot x ƒ + C 49. 4 8 8

55. p>3

57. 2p23 + p22 ln s 22 + 23d 59. x = 4>3, y = ln22

Section 8.5, pp. 460–461 1.

1 [sec se t - 1d tan se t - 1d + 2 ln ƒ sec se t - 1d + tan se t - 1d ƒ ] + C

63. p>8

61. 7.62

67. p>4

Section 8.6, pp. 468–470 1. I: (a) 1.5, 0 (b) 1.5, 0 (c) 0% II: (a) 1.5, 0 (b) 1.5, 0 (c) 0% 3. I: (a) 2.75, 0.08 (b) 2.67, 0.08 (c) 0.0312 L 3% II: (a) 2.67, 0 (b) 2.67, 0 (c) 0% 5. I: (a) 6.25, 0.5 (b) 6, 0.25 (c) 0.0417 L 4% II: (a) 6, 0 (b) 6, 0 (c) 0% 7. I: (a) 0.509, 0.03125 (b) 0.5, 0.009 (c) 0.018 L 2% II: (a) 0.5, 0.002604 (b) 0.5, 0.4794 (c) 0% 9. I: (a) 1.8961, 0.161 (b) 2, 0.1039 (c) 0.052 L 5% II: (a) 2.0045, 0.0066 (b) 2, 0.00454 (c) 0.2% 11. (a) 1 (b) 2 13. (a) 116 (b) 2 15. (a) 283 (b) 2 17. (a) 71 (b) 10 19. (a) 76 (b) 12 21. (a) 82 (b) 8 23. 15,990 ft3 25. L10.63 ft 27. (a) L0.00021 (b) L1.37079 (c) L0.015% 31. (a) L5.870 (b) ƒ ET ƒ … 0.0032 33. 21.07 in. 35. 14.4

Section 8.7, pp. 479–481 1. p>2 15. 25. 35. 43. 51. 59. 65. 67. 75.

7. p>2 9. ln 3 11. ln 4 13. 0 p 23 17. p 19. ln a1 + b 21. -1 23. 1 2 33. ln 2 -1/4 27. p>2 29. p>3 31. 6 Diverges 37. Converges 39. Converges 41. Converges Diverges 45. Converges 47. Converges 49. Diverges Converges 53. Converges 55. Diverges 57. Converges Diverges 61. Converges 63. Converges (a) Converges when p 6 1 (b) Converges when p 7 1 1 69. 2p 71. ln 2 73. (b) L0.88621 (a) 3. 2

5. 6

y

y 1

1.8 1.6

0.8

1.4

x

sin t dt Si(x) = t L0

1.2 1

0.6

0.8

0.2

0.6 0.4

0

0.2

–0.2

0

y = sin t t

0.4

5

10

15

20

(b) p>2 77. (a)

25

x

y 0.4 0.3 0.2

47.

0.1 –3

–2

–1

0

1

2

3

(b) L0.683, L0.954, L0.997

x

5

10

15

20

25

t

A-34

Chapter 9: Answers to Odd-Numbered Exercises et + 1 87. - 29 - 4t 2 + C 89. ln a t 4 e + 2 3>2 1 93. x 95. - tan-1 (cos 5t) + C 5 3

Practice Exercises, pp. 481–483 1. sx + 1dsln sx + 1dd - sx + 1d + C 1 ln s1 + 9x 2 d + C 6 5. sx + 1d2e x - 2sx + 1de x + 2e x + C 2e x sin 2x e x cos 2x 7. + + C 5 5 9. 2 ln ƒ x - 2 ƒ - ln ƒ x - 1 ƒ + C 3. x tan-1 s3xd -

cos u - 1 1 ` + C ln ` 3 cos u + 2 1 15. 4 ln ƒ x ƒ - ln sx 2 + 1d + 4 tan-1 x + C 2 13. -

109.

23 t 1 + C tan-1 t tan-1 2 6 23 x2 2 4 21. + ln ƒ x + 2 ƒ + ln ƒ x - 1 ƒ + C 2 3 3

29. - 216 - y 2 + C 33. ln 37. 41. 45. 51. 53. 63. 69.

1 29 - x 2

+ C

1 ln 2 x 1 35. ln ` 6 x

1. xssin-1 xd2 + 2ssin-1 xd21 - x 2 - 2x + C -s

ƒ + C

ƒ 4 - x2 ƒ + C + 3 ` + C - 3

tan5 x cos5 x cos7 x 39. + + C + C 5 7 5 cos u cos 11u 43. 4 21 - cos st>2d + C + C 2 22 At least 16 47. T = p, S = p 49. 25°F (a) L2.42 gal (b) L24.83 mi>gal 57. ln 3 59. 2 61. p>6 p>2 55. 6 Diverges 65. Diverges 67. Converges 2x 3>2 - x + 2 2x - 2 ln A 2x + 1 B + C 3

71. ln `

2x 2x 2 + 1

` -

73. -2 cot x - ln ƒ csc x + cot x ƒ + csc x + C

77.

u sin (2u + 1) cos (2u + 1) + + C 2 4

81. 2 £

85.

A 22 - x B 3 3

- 2 22 - x≥ + C

79.

7. 0 9. ln s4d - 1 11. 1 17. (a) p (b) ps2e - 5d 19. (b) p a

8sln 2d2 16sln 2d 16 + b 3 9 27

23. 21 + e 2 - ln a 25.

12p 5

13. 32p>35

27. a =

21. a

15. 2p e2 + 1 e - 2 , b 4 2

21 + e 2 1 + e b - 22 + ln A 1 + 22 B e 1 ln 2 ,2 4

29.

1 6 p … 1 2

Section 9.1, pp. 495–498

2

3 + y y 1 1 ln ` ` + tan-1 + C 12 3 - y 6 3

x21 - x 2 - sin-1 x x 2 sin-1 x + + C 2 4 1 5. aln A t - 21 - t 2 B - sin-1 tb + C 2 3.

CHAPTER 9

x 1 b + C a 2 2x 2 + 1

75.

p 4

111.

Additional and Advanced Exercises, pp. 483–485

27. ln ƒ 1 - e

31. -

1 - 21 - x 4 1 ln ` ` + C 2 x2 1 115. x tan-1 A 22 tan x B + C 12

1 ln x x + C 2

113. (b)

3 9 x2 - ln ƒ x + 3 ƒ + ln ƒ x + 1 ƒ + C 2 2 2

2x + 1 - 1 1 25. ln ` ` + C 3 2x + 1 + 1

+ C

107. ln x - ln ƒ 1 + ln x ƒ + C

sy - 2d5sy + 2d 1 ` + C ln ` 16 y6

19.

23.

91. 1>4

97. 22r - 2 ln s1 + 2rd + C 1 1 99. x 2 - ln (x 2 + 1) + C 2 2 2 1 101. ln ƒ x + 1 ƒ + ln ƒ x 2 - x + 1 ƒ + 3 6 2x - 1 1 b + C tan-1 a 13 13 8 4 4 103. A 1 + 2x B 7>2 - 5 A 1 + 2x B 5>2 + 3 A 1 + 2x B 3>2 + C 7 105. 2 ln ƒ 2x + 21 + x ƒ + C

1 11. ln ƒ x ƒ - ln ƒ x + 1 ƒ + + C x + 1

17.

1 b + C 2

1 sec2 u + C 4

83. tan-1 ( y - 1) + C

1 1 1 z 1 1 ln ƒ z ƒ - c ln (z 2 + 4) + tan-1 Q R d + C 4 4z 4 2 2 2

1. a1 = 0, a2 = -1>4, a3 = -2>9, a4 = -3>16 3. a1 = 1, a2 = -1>3, a3 = 1>5, a4 = -1>7 5. a1 = 1>2, a2 = 1>2, a3 = 1>2, a4 = 1>2 3 7 15 31 63 127 255 511 1023 7. 1, , , , , , , , , 2 4 8 16 32 64 128 256 512 1 1 1 1 1 1 1 1 9. 2, 1, - , - , , , - , - , , 2 4 8 16 32 64 128 256 11. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 13. an = s -1dn + 1, n Ú 1 2n - 1 ,n Ú 1 3(n + 2) 19. an = n 2 - 1, n Ú 1 21. an = 4n - 3, n Ú 1 1 + s -1dn + 1 3n + 2 ,n Ú 1 23. an = 25. an = ,n Ú 1 2 n! 27. Converges, 2 29. Converges, -1 31. Converges, -5 15. an = s -1dn + 1snd2, n Ú 1

17. an =

A-35

Chapter 9: Answers to Odd-Numbered Exercises 33. 39. 45. 51. 57. 63. 69. 73. 79. 85.

Diverges 35. Diverges 37. Converges, 1> 2 Converges, 0 41. Converges, 22 43. Converges, 1 Converges, 0 47. Converges, 0 49. Converges, 0 Converges, 1 53. Converges, e 7 55. Converges, 1 Converges, 1 59. Diverges 61. Converges, 4 Converges, 0 65. Diverges 67. Converges, e -1 Converges, e 2>3 71. Converges, x sx 7 0d Converges, 0 75. Converges, 1 77. Converges, 1> 2 Converges, 1 81. Converges, p>2 83. Converges, 0 Converges, 0 87. Converges, 1> 2 89. Converges, 0

91. 8

93. 4

95. 5

97. 1 + 22

99. xn = 2n - 2

101. (a) ƒsxd = x 2 - 2, 1.414213562 L 22 (b) ƒsxd = tan sxd - 1, 0.7853981635 L p>4 (c) ƒsxd = e x , diverges 103. (b) 1 111. Nondecreasing, bounded 113. Not nondecreasing, bounded 115. Converges, nondecreasing sequence theorem 117. Converges, nondecreasing sequence theorem 119. Diverges, definition of divergence 121. Converges 123. Converges 133. (b) 23

Section 9.2, pp. 505–506 2s1 - s1>3dn d 1 - s - 1>2dn , 3 3. sn = , 2>3 1 - s1>3d 1 - s -1>2d 1 1 1 1 1 4 1 , + Á, 5. sn = 7. 1 - + 5 2 n + 2 2 4 16 64 3 9 57 249 9. - + + + + Á, diverges. 4 16 64 256 5 5 5 23 1 1 1 11. s5 + 1d + a + b + a + b + a + b + Á, 2 3 4 9 8 27 2 1. sn =

13. s1 + 1d + a

17 1 1 1 1 1 1 - b + a + b + a b + Á, 5 2 4 25 8 125 6

15. Converges, 5>3 17. Converges, 1>7 19. 23>99 21. 7>9 23. 1>15 25. 41333>33300 27. Diverges 29. Inconclusive 31. Diverges 33. Diverges 1 35. sn = 1 ; converges, 1 37. sn = ln 2n + 1; diverges n + 1 p p 1 - cos-1 a b; converges, 3 n + 2 6 1 1 43. 5 45. 1 47. 49. Converges, 2 + 22 ln 2 e2 Converges, 1 53. Diverges 55. Converges, 2 e - 1 Converges, 2> 9 59. Converges, 3> 2 61. Diverges p Converges, 4 65. Diverges 67. Converges, p - e

39. sn = 41. 51. 57. 63.

69. a = 1, r = -x ; converges to 1>s1 + xd for ƒ x ƒ 6 1 71. a = 3, r = sx - 1d>2 ; converges to 6>s3 - xd for x in s -1, 3d 1 1 1 73. ƒ x ƒ 6 , 75. -2 6 x 6 0, 2 1 - 2x 2 + x p 1 77. x Z s2k + 1d , k an integer; 2 1 - sin x q q 1 1 79. (a) a (b) a n = -2 sn + 4dsn + 5d n = 0 sn + 2dsn + 3d q

1 (c) a n = 5 sn - 3dsn - 2d

89. (a) r = 3>5 (b) r = -3>10 1 + 2r 91. ƒ r ƒ 6 1, 93. 8 m2 1 - r2

Section 9.3, pp. 511–512 1. Converges 9. 13. 17. 19. 21. 23. 27. 29. 31. 35. 39. 43.

3. Converges

5. Converges

7. Diverges 1 6 1 Converges 11. Converges; geometric series, r = 10 n = 1 Z 0 15. Diverges; p-series, p 6 1 Diverges; lim n: q n + 1 1 Converges; geometric series, r = 6 1 8 Diverges; Integral Test Converges; geometric series, r = 2>3 6 1 2n Z 0 Diverges; Integral Test 25. Diverges; lim n: q n + 1 Diverges; limn: q A 2n>ln n B Z 0 1 7 1 Diverges; geometric series, r = ln 2 Converges; Integral Test 33. Diverges; nth-Term Test Converges; Integral Test 37. Converges; Integral Test Converges; Integral Test 41. a = 1 (a) y 1 y= 1 x

1 1/2 0

1

2

1/n

3

n

n+1

x

⌠n+1 1 dx < 1 + 1 + … + 1 ⎮ x 2 n ⌡1 y 1 y= 1 x

1

1/n

1/2 0

1

2

3

n–1

x

n

⌠n 1 1+ 1 +…+ 1 <1+ ⎮ dx 2 n ⌡1 x

(b) L 41.55 45. True 47. (b) n Ú 251,415 8

1 49. s8 = a 3 L 1.195 51. 10 60 n=1 n 59. (a) 1.20166 … S … 1.20253 (b) S L 1.2021, error 6 0.0005

Section 9.4, pp. 516–517 1. Converges; compare with g (1>n 2)

3. Diverges; compare with g A 1> 2n B 5. Converges; compare with g (1>n 3>2) 7. Converges; compare with g 9. Converges

n + 4n 1 = 25 g 3>2 B n4 + 0 n

11. Diverges; limit comparison with g (1>n)

13. Diverges; limit comparison with g A 1> 2n B 17. Diverges; limit comparison with g A 1> 2n B

15. Diverges

A-36

Chapter 9: Answers to Odd-Numbered Exercises

19. Converges; compare with gs1>2n d

q

17. Converges conditionally; 1> 2n : 0 but g n = 1

27. 29. 31. 33. 35. 39.

n

n

n

n n 1 b 6 a b = a b 3n + 1 3n 3 Diverges; direct comparison with gs1>nd Diverges; limit comparison with gs1>nd Diverges; limit comparison with gs1>nd Converges; compare with gs1>n 3>2 d 1 1 1 1 Converges; n … n 37. Converges; n - 1 6 n-1 n2 2 3 + 1 3 Converges; comparison with g (1>5n 2)

25. Converges; a

41. Diverges; comparison with g (1>n) 43. Converges; comparison with g

diverges (compare with g n = 1s1>nd). 23. Diverges;

3 + n :1 5 + n

25. Converges conditionally; a 27. 29. 33.

1 1 + n b : 0 but s1 + nd>n 2 7 1>n n2 Converges absolutely; Ratio Test > 0 Converges absolutely by Integral Test 31. Diverges; an : Converges absolutely by Ratio Test s -1dn + 1 cos np 1 Converges absolutely since ` ` = ` ` = 3>2 3>2 n n n2n (convergent p-series)

2e n 2 6 -n = 2n e + e e + 1 2e n 2 = n , a term from a convergent geometric series. e e 2n 1 Converges conditionally; g (-1) converges by Alter2(n + 1) 1 nating Series Test; g diverges by limit comparison 2(n + 1) with g (1>n). 51. ƒ Error ƒ 6 2 * 10 -11 ƒ Error ƒ 6 0.2 55. n Ú 4 57. 0.54030 n Ú 31 (a) an Ú an + 1 (b) -1>2

45. Converges absolutely; sech n =

47.

Section 9.5, pp. 521–522 Converges 3. Diverges 5. Converges 7. Converges Converges 11. Diverges 13. Converges 15. Converges Converges; Ratio Test 19. Diverges; Ratio Test Converges; Ratio Test Converges; compare with gs3>s1.25dn d n 3 25. Diverges; lim a1 - n b = e -3 Z 0 n: q 1. 9. 17. 21. 23.

Converges; compare with gs1>n 2 d Diverges; compare with gs1>s2ndd Diverges; compare with gs1>nd 33. Converges; Ratio Test Converges; Ratio Test 37. Converges; Ratio Test Converges; Root Test 41. Converges; compare with gs1>n 2 d Converges; Ratio Test 45. Converges; Ratio Test Diverges; Ratio Test 49. Converges; Ratio Test s1>n!d

1 53. Diverges; an = a b 3 57. Diverges; Root Test 61. Converges; Ratio Test

:1

65. Yes

Section 9.6, pp. 527–528 1. 3. 5. 7. 9. 11. 13. 15.

1 n + 3

diverges (compare with gs1> 2nd d . 43. Diverges, an : 1>2 Z 0

53. Converges; limit comparison with gs1>n d 63. Converges 65. Converges 67. Converges

55. Converges; Ratio Test 59. Converges; Root Test

q

diverges.

37. Converges absolutely by Root Test 39. Diverges; an : q 41. Converges conditionally; 2n + 1 - 2n = 1>s 2n + 2n + 1d : 0 , but series of absolute values

2

51. Converges; Ratio Test

q

21. Converges conditionally; 1>sn + 3d : 0 but g n = 1

35.

1 n(n - 1)

or limit comparison with g (1>n 2) 45. Diverges; limit comparison with gs1>nd p>2 tan-1 n 47. Converges; 6 1.1 n 1.1 n 49. Converges; compare with gs1>n 2 d 51. Diverges; limit comparison with gs1>nd

27. 29. 31. 35. 39. 43. 47.

1

2n q 19. Converges absolutely; compare with g n = 1s1>n 2 d .

21. Diverges; nth-Term Test 23. Converges; compare with g (1>n 2)

Converges by Theorem 16 Converges; Alternating Series Test Converges; Alternating Series Test > 0 Diverges; an : > 0 Diverges; an : Converges; Alternating Series Test Converges by Theorem 16 Converges absolutely. Series of absolute values is a convergent geometric series.

49. 53. 59.

n

Section 9.7, pp. 536–538 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. 43. 45.

(a) 1, -1 6 x 6 1 (b) -1 6 x 6 1 (c) none (a) 1>4, -1>2 6 x 6 0 (b) -1>2 6 x 6 0 (c) none (a) 10, -8 6 x 6 12 (b) -8 6 x 6 12 (c) none (a) 1, -1 6 x 6 1 (b) -1 6 x 6 1 (c) none (a) 3, -3 … x … 3 (b) -3 … x … 3 (c) none (a) q , for all x (b) for all x (c) none (a) 1>2, -1>2 … x 6 1>2 (b) -1>2 6 x 6 1>2 (c) -1>2 (a) 1, -1 … x 6 1 (b) -1 6 x 6 1 (c) x = -1 (a) 5, -8 6 x 6 2 (b) -8 6 x 6 2 (c) none (a) 3, -3 6 x 6 3 (b) -3 6 x 6 3 (c) none (a) 1, -2 6 x 6 0 (b) -2 6 x 6 0 (c) none (a) 1, -1 6 x 6 1 (b) -1 6 x 6 1 (c) none (a) 0, x = 0 (b) x = 0 (c) none (a) 2, -4 6 x … 0 (b) -4 6 x 6 0 (c) x = 0 (a) 1, -1 … x … 1 (b) -1 … x … 1 (c) none (a) 1>4, 1 … x … 3>2 (b) 1 … x … 3>2 (c) none (a) q , for all x (b) for all x (c) none (a) 1, -1 … x 6 1 (b) - 1 6 x 6 1 (c) -1 3 39. 8 41. -1>3 6 x 6 1>3, 1>(1 - 3x) -1 6 x 6 3, 4>s3 + 2x - x 2 d 0 6 x 6 16, 2> A 4 - 2x B

Chapter 9: Answers to Odd-Numbered Exercises 47. - 22 6 x 6 22, 3>s2 - x 2 d 49. 1 6 x 6 5, 2>sx - 1d, 1 6 x 6 5, -2>sx - 1d2 x8 x2 x4 x6 x 10 51. (a) cos x = 1 + + + Á; 2! 4! 6! 8! 10! converges for all x (b) Same answer as part (c) 23x 3 2 5x 5 27x 7 29x 9 211x 11 (c) 2x + + + Á 3! 5! 7! 9! 11! 17x 8 31x 10 x2 x4 x6 p p 53. (a) + + + + , - 6 x 6 2 12 45 2520 14175 2 2 6 8 4 17x 62x 2x p p (b) 1 + x 2 + + + + Á, - 6 x 6 3 45 315 2 2

Section 9.8, pp. 542–543 1. P0sxd = 1, P1sxd = 1 + 2x, P2sxd = 1 + 2x + 2x 2, 4 P3sxd = 1 + 2x + 2x 2 + x 3 3 1 3. P0sxd = 0, P1sxd = x - 1, P2sxd = sx - 1d - sx - 1d2 , 2 1 1 P3sxd = sx - 1d - sx - 1d2 + sx - 1d3 2 3 1 1 1 5. P0sxd = , P1sxd = - sx - 2d , 2 2 4 1 1 1 P2sxd = - sx - 2d + sx - 2d2 , 2 4 8 1 1 1 1 sx - 2d3 P3sxd = - sx - 2d + sx - 2d2 2 4 8 16 22 22 22 p , P1 sxd = + ax - b , 7. P0sxd = 2 2 2 4 2 22 22 22 p p P2sxd = + ax - b ax - b , 2 2 4 4 4 2 22 22 22 p p P3sxd = + ax - b ax - b 2 2 4 4 4 3 22 p ax - b 12 4 1 9. P0sxd = 2, P1sxd = 2 + sx - 4d , 4 1 1 P2sxd = 2 + sx - 4d sx - 4d2 , 4 64 1 1 1 sx - 4d2 + sx - 4d3 P3sxd = 2 + sx - 4d 4 64 512 q s -xdn x2 x3 x4 = 1 - x + + - Á 11. a n! 2! 3! 4! n=0 q

13. a s -1dnx n = 1 - x + x 2 - x 3 + Á n=0 q

q q s -1dn32n + 1x 2n + 1 s -1dnx 2n x 2n 17. 7 a 19. a s2n + 1d! s2nd! n=0 n=0 n = 0 s2nd! x 4 - 2x 3 - 5x + 4 8 + 10sx - 2d + 6sx - 2d2 + sx - 2d3 21 - 36sx + 2d + 25sx + 2d2 - 8sx + 2d3 + sx + 2d4 q q e2 n n 29. a sx - 2dn a s -1d sn + 1dsx - 1d n=0 n = 0 n!

5 33. -1 - 2x - x 2 - Á , -1 6 x 6 1 2 1 1 35. x 2 - x 3 + x 4 + Á , -1 6 x 6 1 2 6 41. Lsxd = 0, Qsxd = -x 2>2 43. Lsxd = 1, Qsxd = 1 + x 2>2 45. Lsxd = x, Qsxd = x

Section 9.9, pp. 549–550 q

s -5xdn 52x 2 5 3x 3 = 1 - 5x + + Á n! 2! 3! n=0

1. a q

q 5s -1dns -xd2n + 1 5s -1dn + 1x 2n + 1 = a s2n + 1d! s2n + 1d! n=0 n=0 5x 3 5x 5 5x 7 = -5x + + + Á 3! 5! 7! q s -1dn(5x 2) 2n 25x 4 625x 8 5. a = 1 + - Á 2! 4! s2nd! n=0

3. a

q

x 2n x4 x6 x8 7. a s -1dn + 1 n = x 2 + + Á 2 3 4 n=1 q

27.

q

p 22n 31. a s -1dn + 1 ax - b 4 (2n)! n=0

2n

n

3 3 32 33 9. a s -1dn a b x 3n = 1 - x 3 + 2 x 6 - 3 x 9 + Á 4 4 4 4 n=0 q n+1 3 4 5 x x x x 11. a = x + x2 + + + + Á 2! 3! 4! n = 0 n! q s -1dnx 2n x4 x6 x8 x 10 13. a = + + Á 4! 6! 8! 10! s2nd! n=2 q s -1dnp2nx 2n + 1 p 4x 5 p6x 7 p2x 3 15. x + + Á = a 2! 4! 6! s2nd! n=0 q s -1dns2xd2n 17. 1 + a = 2 # s2nd! n=1 2 4 s2xd6 s2xd s2xd s2xd8 - # 1 - # + # + # - Á 2 2! 2 4! 2 6! 2 8! q

19. x 2 a s2xdn = x 2 + 2x 3 + 4x 4 + Á n=0 q

21. a nx n - 1 = 1 + 2x + 3x 2 + 4x 3 + Á n=1 q

x 4n - 1 x7 x 11 x 15 23. a s -1dn + 1 = x3 + + Á 5 7 2n 1 3 n=1 q

3 5 25 4 Á 1 25. a a + s -1dn b x n = 2 + x 2 - x 3 + x 2 6 24 n = 0 n! q

(-1) n - 1x 2n + 1 x3 x5 x7 = + - Á 3n 3 6 9 n=1

27. a

x3 x5 + Á 3 30 23 6 2 44 8 Á x2 - x4 + x x + 3 45 105 1 1 1 + x + x2 - x4 + Á 2 8 1 6 4.2 * 10 -6 ƒ Error ƒ … 10 4 # 4! 1>5 ƒ x ƒ 6 s0.06d 6 0.56968 ƒ Error ƒ 6 s10 -3 d3>6 6 1.67 * 10 -10, -10 -3 6 x 6 0 ƒ Error ƒ 6 s30.1 ds0.1d3>6 6 1.87 * 10 -4 k(k - 1) 2 (a) Q(x) = 1 + kx + (b) 0 … x 6 100 -1>3 x 2

29. x + x 2 + 31.

15. a 21. 23. 25.

A-37

33. 35. 37. 39. 41. 49.

A-38

Chapter 10: Answers to Odd-Numbered Exercises 77. - 2

Section 9.10, pp. 556–558 1. 5. 9. 11. 13. 15. 23. 27.

29. 39. 49. 59.

5 3 Á 3 x x2 x3 1 1 + 3. 1 + x + x 2 + + x + 2 8 16 2 8 16 2 6 9 3 3 3x 3x 5x x x 1 - x + + 7. 1 4 2 2 8 16 1 1 1 1 + + 2x 8x 2 16x 3 s1 + xd4 = 1 + 4x + 6x 2 + 4x 3 + x 4 s1 - 2xd3 = 1 - 6x + 12x 2 - 8x 3 0.00267 17. 0.10000 19. 0.09994 21. 0.10000 x3 x7 x 11 1 25. L 0.00011 - # + 3 13 # 6! 7 3! 11 # 5! 2 4 x x (a) 2 12 x 32 x4 x6 x8 x2 (b) - # + # - # + Á + s -1d15 2 3 4 5 6 7 8 31 # 32 31. -1>24 33. 1>3 35. -1 37. 2 1>2 3 x3 41. e 43. cos 45. 23>2 47. 3>2 4 1 - x x3 -1 51. 55. 500 terms 57. 4 terms 1 + x2 (1 + x) 2 3x 5 5x 7 x3 (a) x + , radius of convergence = 1 + + 6 40 112

81. 2> 3

79. r = -3, s = 9>2

n + 1 1 b ; the series converges to ln a b . 83. ln a 2n 2 85. (a) q

(b) a = 1, b = 0

87. It converges.

Additional and Advanced Exercises, pp. 561–562 1. Converges; Comparison Test 5. Converges; Comparison Test 9. With a = p>3, cos x = +

3. Diverges; nth-Term Test 7. Diverges; nth-Term Test

23 1 1 sx - p>3d - sx - p>3d2 2 2 4

23 sx - p>3d3 + Á 12

x2 x3 + + Á 2! 3! 1 1 13. With a = 22p, cos x = 1 - sx - 22pd2 + sx - 22pd4 2 4! 1 sx - 22pd6 + Á 6! 1 15. Converges, limit = b 17. p>2 21. b = ; 5 23. a = 2, L = - 7>6 27. (b) Yes

11. With a = 0, e x = 1 + x +

3x 5 5x 7 x3 p - x 2 6 40 112 61. 1 - 2x + 3x 2 - 4x 3 + Á (b)

(b) A 1> 22 B (1 + i) (c) - i 1 1 5 Á 71. x + x 2 + x 3 x + , for all x 3 30 67. (a) -1

CHAPTER 10 Section 10.1, pp. 568–570 1.

3.

Practice Exercises, pp. 559–561

y

Converges to 1 3. Converges to -1 5. Diverges Converges to 0 9. Converges to 1 11. Converges to e -5 Converges to 3 15. Converges to ln 2 17. Diverges 1> 6 21. 3> 2 23. e>se - 1d 25. Diverges Converges conditionally 29. Converges conditionally Converges absolutely 33. Converges absolutely Converges absolutely 37. Converges absolutely Converges absolutely (a) 3, -7 … x 6 -1 (b) -7 6 x 6 -1 (c) x = -7 (a) 1>3, 0 … x … 2>3 (b) 0 … x … 2>3 (c) None (a) q , for all x (b) For all x (c) None (a) 23, - 23 6 x 6 23 (b) - 23 6 x 6 23 (c) None 49. (a) e, -e 6 x 6 e (b) -e 6 x 6 e (c) Empty set q 1 1 4 51. 53. sin x, p, 0 55. e x , ln 2, 2 57. a 2nx n , , 1 + x 4 5 1. 7. 13. 19. 27. 31. 35. 39. 41. 43. 45. 47.

n=0

q

59. 65. 67. 69.

q q sspxd>2dn s -1dnp2n + 1x 2n + 1 s -1dnx 10n>3 61. 63. a a a n! s2n + 1d! s2nd! n=0 n=0 n=0 2 3 9sx + 1d sx + 1d 3sx + 1d + + Á 2 + 2 # 1! 25 # 3! 23 # 2! 1 1 1 1 - 2 sx - 3d + 3 sx - 3d2 - 4 sx - 3d3 4 4 4 4 0.4849171431 71. 0.4872223583 73. 7> 2 75. 1> 12

y y=x

t<0

2

y=x

t>0

t<0

1

t>0

1 x

0

x

0

1

5.

1

7. y

y 2

x 2 + y2 = 1 2

1 t= 2 –2 –1

2

t=0 0

–1 –2

1 t=

2

x

x2 + y2 = 1 16 4

t = 0, 2 0

4

x

A-39

Chapter 10: Answers to Odd-Numbered Exercises 9.

Section 10.2, pp. 577–579

11. y

y

= - 12 dx 2 d 2y 12 1 = 3. y = - x + 212, 2 4 dx 2 2 1 d y = -2 7. y = 2x - 13, 5. y = x + , 4 dx 2 d 2y 1 = 9. y = x - 4, 2 dx 2

y = 1 – 2x

1

2

t<0

–1

changes direction at t = 0

x

1

(0, 0)

x

2 t= – 2

t= 2

–1

15. y 3

0≤t≤ 2

2 t=0

y=

√1 – x

2

1 –2

0

1

x=y

–1

19. 1

21. 3a 2p

29. p2

31. 8p2

1

2

3

4

x

–1

x

–2

– ≤t<0 2

–3

37. (x, y) = 39. (x, y) = 43. (a) x =

17.

(c) x =

y 2

2

x –y =1

t=0 –1

dx

dx 2

dx 2

= -313

= -4

= 108

15. -

23. abp

25. 4

2

d 2y

3 16

17. -6 27. 12

2

t=0 t = –1 –1

d 2y

13. y = 9x - 1,

y

d 2y

p13 + 2, 3

11. y = 13x 13.

d2 y

1. y = -x + 2 12,

2 y = x (x – 2)

45. a

12 , 1 b, 2

47. (a) 8a

x

0

52p 35. 3p15 3 12 24 24 a p - 2 , 2 - 2b p p 1 4 a ,p - b 41. (a) p (b) p 3 3 dy 1 1, y = 0, = (b) x = 0, y = 3, 2 dx 3 - 13 dy 13 - 1 213 - 1 = , y = , 2 2 dx 13 - 2 33.

y = 2x at t = 0,

(b)

dy = 0 dx

y = -2x at t = p

64p 3

Section 10.3, pp. 582–583 1. a, e; b, g; c, h; d, f y = -a sin t, 0 … t … 2p y = a sin t, 0 … t … 2p y = -a sin t, 0 … t … 4p y = a sin t, 0 … t … 4p x = -1 + 5t, y = -3 + 4t, 0 … t … 1 x = t 2 + 1, y = t, t … 0 x = 2 - 3t, y = 3 - 4t, t Ú 0 x = 2 cos t, y = 2 ƒ sin t ƒ , 0 … t … 4p a -at 29. Possible answer: x = , y = , 2 21 + t 21 + t 2 -q 6 t 6 q 19. (a) x = a cos t, (b) x = a cos t, (c) x = a cos t, (d) x = a cos t, 21. Possible answer: 23. Possible answer: 25. Possible answer: 27. Possible answer:

31. Possible answer: x =

4 , 1 + 2 tan u

y =

4 tan u , 1 + 2 tan u

0 … u 6 p>2 and x = 0, y = 2 if u = p>2 33. Possible answer: x = 2 - cos t, 35. x = 2 cot t,

y = 2 sin2 t,

37. x = a sin t tan t, 2

y = sin t,

y 2,

2 x

(–2, 0)

(2, 0) –2,

2

p p + 2npb and a -2, + (2n + 1)pb, n an integer 2 2 (b) (2, 2np) and (-2, (2n + 1)p), n an integer

(a) a2,

3p 3p + 2npb and a-2, + (2n + 1)pb , 2 2 n an integer (d) (2, (2n + 1)p) and (- 2, 2np), n an integer

(c) a2,

5. (a) (3, 0)

(b) (-3, 0)

(c)

A -1, 23 B

(f ) A 1, 23 B (g) (-3, 0) p 7. (a) a 12, b (b) (3, p) 4 (e) (3, 0)

0 … t … 2p

0 6 t 6 p

y = a sin2 t,

3.

0 … t 6 p>2

39. (1, 1)

(c) a2,

11p b 6

4 (d) a5, p - tan-1 b 3

(d) A 1, 23 B

(h)

A -1, 23 B

A-40

Chapter 10: Answers to Odd-Numbered Exercises

9. (a) a -312, (c) a -2, 11.

5p b 4

Section 10.4, pp. 586–587

(b) (-1, 0)

1. x-axis

3 (d) a -5, p - tan-1 b 4

5p b 3

13.

y

3. y-axis

y

y

r = 1 + cos

y

r = 1 – sin

1 2

r=2

0

r1

x

2

0

–1

x

1

2

17.

y

–1

y

y

0

r = 2 + sin 1

x

2

–1

–1 –2

–1

0

1

21.

y

y

x

2

√2 2

–1

9. x-axis, y-axis, origin

2 r0 =

r=1 0

11. y-axis, x-axis, origin y

y 1 r 2 = cos

0

x

O y 1

3 4 4 0r1

25.

y 2

–

x

x

1

2 2 1r2 –1

1

0

x

0 –1

r 2 = – sin

x

1

–1

23.

x

1

–

19.

√2 2

r = sin (/2)

3

2

3

x

0

7. x-axis, y-axis, origin

= 3 –1 r 3

3 0 6 r0

x

–2

5. y-axis

y

1

x

–1

15.

0

1

2

x

13. x-axis, y-axis, origin 15. Origin 17. The slope at (-1, p>2) is -1 , at (-1, -p>2) is 1. y

–2

27. 31. 33. 35. 37. 39. 41. 43. 45. 47. 49.

x = 2 , vertical line through (2, 0) 29. y = 0 , the x-axis y = 4 , horizontal line through (0, 4) x + y = 1 , line, m = -1, b = 1 x 2 + y 2 = 1 , circle, C(0, 0), radius 1 y - 2x = 5 , line, m = 2, b = 5 y 2 = x , parabola, vertex (0, 0), opens right y = e x , graph of natural exponential function x + y = ;1 , two straight lines of slope -1, y-intercepts b = ;1 (x + 2) 2 + y 2 = 4 , circle, C(-2, 0) , radius 2 x 2 + ( y - 4) 2 = 16 , circle, C(0, 4), radius 4 (x - 1) 2 + ( y - 1) 2 = 2 , circle, C(1, 1), radius 22

51. 57. 61. 65.

53. r cos u = 7 55. u = p>4 23y + x = 4 59. 4r 2 cos2 u + 9r 2 sin2 u = 36 r = 2 or r = -2 63. r = 4 sin u r sin2 u = 4 cos u 2 r = 6r cos u - 2r sin u - 6 67. (0, u) , where u is any angle

⎞–1, – ⎞ ⎠ 2⎠

r = –1 + cos

2

x

⎞–1, ⎞ ⎠ 2⎠

19. The slope at (1, p>4) is -1 , at (-1, -p>4) is 1, at (-1, 3p>4) is 1, at (1, -3p>4) is -1 . y ⎞–1, – ⎞ ⎠ 4⎠

⎞1, ⎞ ⎠ 4⎠

r = sin 2 x

⎞1, – 3 ⎞ ⎠ 4⎠

⎞–1, 3 ⎞ ⎠ 4⎠

A-41

Chapter 10: Answers to Odd-Numbered Exercises 21. (a)

(b)

y

r=

1 + cos 2

r=

1 2

1 2

3 2

x

1 + sin 2

–

–

1 2

3 + cos 2 5 2

0

x

25.

–

27.

y

x

y2 x2 + = 1 27 36 y2 x2 + = 1 13. 9 4

5 2 y

2

directrices are x = ;

19. e = 22;

1 . 12

y

x 3

y 2 y2 – x =1 8 8

x 2 – y2 = 1

2 √8

1

29. (a)

–√2

√2

–2 F1 –1

1 F2 2

Section 10.5, pp. 590–591 3. 18p

13. 3 23 - p 19. (a)

p 5. 8

7. 2

23 p 15. + 3 2 21. 19> 3

3 p 2 4

25. 3 A 22 + ln A 1 + 22 B B

p 9. - 1 2

11. 5p - 8

x

–2 –2 –√8

–2

–4 F2

–3

–6

F A 0, ; 210 B ; 2 . directrices are y = ; 110

3 , 5

1 ; F(0, ;1); 12 directrices are y = ;2.

3. e =

F(;3, 0);

25 directrices are x = ; . 3 y 4

–5

F1

F2

–3

3

5

x2 +

1

y2 =1 2

1 F2

–4

–2

2

4

–2

–1

y2 x2 2 29. r = = 1 27. x 2 = 1 8 8 1 + cos u 10 30 1 31. r = 33. r = 35. r = 2 + cos u 1 - 5 sin u 5 - sin u 25. y 2 -

x

–√2

1 √2 x

F2 –√10

–1 –4

2 √10 4

–1 –√2 –10

y

x2 + y2 = 1 25 16 √2

F1

F1 √2 –5

F1 √10 2

5

1. e =

x

x

y

10

–4 –√10 –2 –√2

4

23. e = 25;

2 y2 – x =1 2 8

F2

Section 10.6, pp. 597–598

2

y

x2 – y2 = 1 2 8

3 p + 8 8

–4

–1

F A ; 210, 0 B ; 2 directrices are x = ; . 110

23. 8 27.

3

21. e = 25;

8p 17. + 23 3

6 4 F1

2

–3

F(0, ;4);

directrices are y = ; 2.

–2

1 1. p3 6

x

3

y2 x2 + = 1 4851 4900 y2 x2 + = 1 15. 64 48

r = –1 0

√3

11.

F A ; 22, 0 B ;

17. e = 22;

0 r 2 – 2 cos

–4

–3 –√3

x

3 – sin 2

r=2

x

F2

–√6

9.

–3 2

F1

–√3

3 2 r=

=1

–1

F2

3 2

3

x2 + y2 = 1 9 6

√6

√2

–√2

y

+

2

y2

1

F1

1 2

3 2 r=

y

x2

x

1 2

(b)

y

7. e =

y

√3 1 – 2

23. (a)

13 ; F A ; 23, 0 B ; 3 directrices are x = ;3 23.

1 ; F(0, ;1); 13 directrices are y = ;3.

5. e =

3 2

1 2

–

y

x

A-42

Chapter 10: Answers to Odd-Numbered Exercises

37.

61. r = -2 cos u

39. y

63. r = -sin u y

y

y x=1

2 r= ⎛1 ⎛ ,0 ⎝2 ⎝

–2

⎛5 ⎛ ,0 ⎝3 ⎝

1 –5

–1

x

1

0

25 10 – 5 cos

r=

x = –5

1 1 + cos

⎛5 , ⎝3

(5, 0)

0

⎛ ⎝

2 2 x + ⎛y + 1 ⎛ = 1 ⎝ 2⎝ 4 r = – sin

(x + 1)2 + y2 = 1 r = –2 cos x

x

x (–1, 0)

–1 ⎛ 0, ⎝

–2

41.

65.

43.

r = 3 sec ⎛ –

⎛ 2⎝

r=

400 r= 16 + 8 sin

–2

⎛50 , ⎝3

⎝

8 2 – 2 sin

6

⎛ 3 ⎛ 2, ⎝ 2 ⎝

x x

3 ⎛ 2 ⎝

69.

45. y = 2 - x

47. y =

71. y

23 x + 2 23 3

y

y

r=

2 –6

49. r cos au -

p b = 3 4

1

51. r cos au +

p b = 5 2

55.

–1

–1

x

–2

Radius = 2

59. r = 10 sin u y

r = 10 sin x2 + (y – 5)2 = 25

2

(x – 6) + y = 36 r = 12 cos (0, 5)

x x (6, 0)

x

1 1 + 2 sin

75. (b)

r = –2 cos

Radius = 1

2

1

y

(1, )

(2, 0)

y

1

1

1 3

r=

57. r = 12 cos u

1 1 – sin

y

r = 4 cos

x

r=

8 4 + cos

73.

y

x

x

x

2

–1 –1

y = √3 x + 2√3 3

4 x +y=2

y

1 –2

53.

⎛ 3⎝

y = –4

3 ⎛ 2 ⎝

2

r = 4 sin

4

2√3 x

2

0

x

0

⎛ 50, ⎝

y

y y = 50

⎛50 , ⎝3

67. y

y

– 1⎛ 2⎝

Planet

Perihelion

Aphelion

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

0.3075 AU 0.7184 AU 0.9833 AU 1.3817 AU 4.9512 AU 9.0210 AU 18.2977 AU 29.8135 AU

0.4667 AU 0.7282 AU 1.0167 AU 1.6663 AU 5.4548 AU 10.0570 AU 20.0623 AU 30.3065 AU

x

A-43

Chapter 10: Answers to Odd-Numbered Exercises

Practice Exercises, pp. 599–600 1.

31.

A x - 22 B 2 + y 2 = 2

3. y

y

y 4y2 – 4x2 = 1

y = 2x + 1 1

r = –5 sin

r = 2√2 cos

1

x2 + ⎛y + 5 ⎛ 2 = 25 ⎝ 2⎝ 4

t=0 ⎛ ⎝√2

1 t=0 2 0

x

0

–1 2

33. r = -5 sin u

y

1

⎛

, 0⎝

x x

x ⎛ ⎝x

⎛2

⎛ 0, ⎝

2

– √2⎝ + y = 2

35. r = 3 cos u

37.

y

5.

y

y y = x2

t=0

⎛x ⎝

t=

1

2 2 – 3⎛ + y = 9 2⎝ 4

–1

7. x = 3 cos t,

y = 4 sin t,

13 1 1 9. y = x + , 2 4 4 ; ƒ x ƒ 3>2 11. (a) y = - 1 8 10 3

15.

285 8

0 … t … 2p

(b) y =

17. 10

13 x - 4 3

23. y =

x

19.

39. d

; 21 - x x

9p 2

21.

41. l

43. k

45. i

51. 8 53. p - 3 55. V(1, 0)

2

47.

9 p 2

49. 2 +

y

76p 3

3 2 0

y x=2

r=

2 1 + cos

r=

(2, )

6 1 – 2 cos

x

(1, 0)

(6, )

0

–2

x – √3 y = 4√3

–3 x

2

4√3

x

59. r =

–4

4 1 + 2 cos u

61. r =

2 2 + sin u

Additional and Advanced Exercises, pp. 600–601

3 2

29. x 2 + ( y + 2) 2 = 4

25 4p (e - 1) 2 4 2 3. r = 5. r = 1 + 2 cos u 2 + sin u a + b 7. x = (a + b) cos u - b cos a ub, b 1. (a) r = e 2u

y

y

x

r = –4 sin

2 2 x + (y + 2) = 4

x

–3 2 y=– 3 2

p 4

57. V(2, p) and V(6, p)

y

25. x = 2

y

27. y = -

x

6

0

x

1

0

0 ≤ r ≤ 6 cos

r = 3 cos ⎛3 ⎛ ,0 ⎝2 ⎝

13.

– 5⎛ 2⎝

(b)

y = (a + b) sin u - b sin a

(0, –2)

11.

p 2

a + b ub b

x

A-44

Chapter 11: Answers to Odd-Numbered Exercises 23. The vector v is horizontal and 1 in. long. The vectors u and w 11 are in . long. w is vertical and u makes a 45° angle with the 16 horizontal. All vectors must be drawn to scale.

CHAPTER 11 Section 11.1, pp. 605–606 1. 3. 7. 9. 11. 13. 15. 17. 19. 21.

23.

25. 27. 29. 31.

The line through the point (2, 3, 0) parallel to the z-axis The x-axis 5. The circle x 2 + y 2 = 4 in the xy-plane 2 The circle x + z 2 = 4 in the xz-plane The circle y 2 + z 2 = 1 in the yz-plane The circle x 2 + y 2 = 16 in the xy-plane The ellipse formed by the intersection of the cylinder x 2 + y 2 = 4 and the plane z = y The parabola y = x 2 in the xy-plane (a) The first quadrant of the xy-plane (b) The fourth quadrant of the xy-plane (a) The ball of radius 1 centered at the origin (b) All points more than 1 unit from the origin (a) The ball of radius 2 centered at the origin with the interior of the ball of radius 1 centered at the origin removed (b) The solid upper hemisphere of radius 1 centered at the origin (a) The region on or inside the parabola y = x 2 in the xy-plane and all points above this region (b) The region on or to the left of the parabola x = y 2 in the xyplane and all points above it that are 2 units or less away from the xy-plane (a) x = 3 (b) y = -1 (c) z = -2 (a) z = 1 (b) x = 3 (c) y = -1 (a) x 2 + s y - 2d2 = 4, z = 0 (b) s y - 2d2 + z 2 = 4, x = 0 (c) x 2 + z 2 = 4, y = 2 (a) y = 3, z = -1 (b) x = 1, z = -1 (c) x = 1, y = 3

33. x 2 + y 2 + z 2 = 25, z = 3

35. 0 … z … 1

37. z … 0

39. (a) sx - 1d2 + s y - 1d2 + sz - 1d2 6 1

51. sx - 1d2 + s y - 2d2 + sz - 3d2 = 14

59. (a) 2y 2 + z 2

2

16 1 2 b + az + b = 2 3 81

55. Cs -2, 0, 2d, a = 28

523 1 1 1 57. C a- , - , - b , a = 4 4 4 4

(b) 2x 2 + z 2

(c) 2x 2 + y 2

1. (a) 89, -69

5. (a) 812, -199 9. 81, -49

(b) 2505

11. 8-2, -39

23 1 ,- i 2 2 21. 3i + 5j - 8k 15. h -

(b) 3 213

v

u

(c)

(d)

–v

u u

–w

u–v u–w

1 2 2 25. 3 a i + j - kb 3 3 3 29. 31. 33. 35. 37. 39. 43.

27. 5(k)

1 1 1 1 a i j kb A 2 23 23 23 3 2 (a) 2i (b) - 23k (c) (d) 6i - 2j + 3k j + k 5 10 7 s12i - 5kd 13 3 4 1 (a) (b) (1> 2, 3, 5> 2) i + j k 5 22 5 22 22 5 7 9 1 1 1 (a) (b) a , , b i j k 2 2 2 23 23 23 3 1 As4, -3, 5d 41. a = , b = 2 2 L8-338.095, 725.0469 100 cos 45° L 73.205 N, ƒ F1 ƒ = sin 75°

100 cos 30° L 89.658 N, sin 75° F1 = 8- ƒ F1 ƒ cos 30°, ƒ F1 ƒ sin 30°9 L 8-63.397, 36.6039, F2 = 8 ƒ F2 ƒ cos 45°, ƒ F2 ƒ sin 45°9 L 863.397, 63.3979 100 sin 75° 47. w = L 126.093 N, cos 40° w cos 35° L 106.933 N ƒ F1 ƒ = sin 75° 5 5 23 49. (a) s5 cos 60°, 5 sin 60°d = a , b 2 2 a 51. (a)

3. (a) 81, 39

5 + 1022 5 23 - 1022 , b 2 2

3 3 i + j - 3k 2 2

(b) i + j - 2k

(c) (2, 2, 1)

(b) 210

1 14 7. (a) h , i 5 5

(b)

1 23 13. h - , i 2 2

17. -3i + 2j - k

w

(b) s5 cos 60° + 10 cos 315°, 5 sin 60° + 10 sin 315°d =

61. 217 + 233 + 6 63. y = 1 65. (a) (0, 3, -3) (b) (0, 5, -5)

Section 11.2, pp. 614–616

u+v+w u+v

ƒ F2 ƒ =

41. 3 43. 7 45. 2 23 47. Cs -2, 0, 2d, a = 222 49. C A 22, 22, - 22 B , a = 22 2

(b)

v

u

45.

(b) sx - 1d2 + s y - 1d2 + sz - 1d2 7 1

53. sx + 1d2 + ay -

(a)

19. -3i + 16j

2197 5

Section 11.3, pp. 622–624 (b) -1 (c) -5 (d) -2i + 4j - 25k 5 1 1 3. (a) 25, 15, 5 (b) (c) (d) s10i + 11j - 2kd 3 3 9 2 2 5. (a) 2, 234, 23 (b) (c) 23 234 234 1 (d) s5j - 3kd 17 1. (a) -25, 5, 5

Chapter 11: Answers to Odd-Numbered Exercises

7. (a) 10 + 217, 226, 221 10 + 217

(c)

(d)

9.

10 + 217

(b)

1

2546

10 + 217 s5i + jd 26

y

i x

–2i + j

0

3 2

x

17.

29. x

31. 33.

–3

37. x + y = - 1

39. 2x - y = 0 y

y P(1, 2)

P(–2, 1)

35.

1 2x – y = 0 x

–2

1

x

49. –i – 2j

x + y = –1 –1

41. 5 J

45.

43. 3464 J

p 4

47.

p 6

3. 5. 7.

1 2 2 ƒ u * v ƒ = 3 , direction is 3 i + 3 j + 3 k; ƒ v * u ƒ = 3 , 1 2 2 direction is - i - j - k 3 3 3 ƒ u * v ƒ = 0 , no direction; ƒ v * u ƒ = 0 , no direction ƒ u * v ƒ = 6 , direction is -k; ƒ v * u ƒ = 6 , direction is k 1 2 i k; ƒ v * u ƒ = 625 , ƒ u * v ƒ = 6 25 , direction is 25 25 1 2 direction is i + k 25 25

j a2 b2

k a1 03 = ` b1 0

a2 `k b2

and the triangle’s area is

49. 0.14

1 1 a1 `A * B` = ; ` 2 2 b1

a2 `. b2

The applicable sign is s + d if the acute angle from A to B runs counterclockwise in the xy-plane, and s - d if it runs clockwise.

Section 11.4, pp. 628–630 1.

(b) ;

i A * B = 3 a1 b1

i–j

45.

–2k

1 s2i + j + kd 26 22 1 (a) (b) ; si - jd 2 22 8 21. 7 23. (a) None (b) u and w 25. 1023 ft-lb (a) True (b) Not always true (c) True (d) True (e) Not always true (f) True (g) True (h) True u#v (a) projv u = v # v v (b) ; u * v (c) ; su * vd * w v (d) ƒ su * vd # w ƒ (e) su * vd * su * wd (f) ƒ u ƒ ƒvƒ (a) Yes (b) No (c) Yes (d) No No, v need not equal w. For example, i + j Z -i + j , but i * si + jd = i * i + i * j = 0 + k = k and i * s -i + jd = - i * i + i * j = 0 + k = k . 25 11 2 37. 13 39. 2129 41. 43. 2 2 221 3 47. 2 2 If A = a1 i + a2 j and B = b1 i + b2 j , then

15. (a) 226

–2x + y = –3 4

y i+j

x

x + 2y = 4

1

z

i–j

19. 27.

1

–2

i–k

x

y

i + 2j

y

j

13.

y

i–j+k

ij=k

b L 63.435 degrees .

2

z j+k

25 23. Horizontal component: L 1188 ft>sec , vertical component: L 167 ft>sec 25. (a) Since ƒ cos u ƒ … 1, we have ƒ u # v ƒ = ƒ u ƒ ƒ v ƒ ƒ cos u ƒ … ƒ u ƒ ƒ v ƒ s1d = ƒ u ƒ ƒ v ƒ . (b) We have equality precisely when ƒ cos u ƒ = 1 or when one or both of u and v are 0. In the case of nonzero vectors, we have equality when u = 0 or p , that is, when the vectors are parallel. 27. a 33. x + 2y = 4 35. -2x + y = -3

0

11. z

226 9. 0.75 rad 11. 1.77 rad 1 b L 63.435 degrees , angle at 13. Angle at A = cos-1 a 25 3 B = cos-1 a b L 53.130 degrees , angle at 5 C = cos-1 a

A-45

Section 11.5, pp. 636–638 1. 3. 5. 7. 9. 11.

x x x x x x

= = = = = =

3 + t, y = -4 + - 2 + 5t, y = 5t, 0, y = 2t, z = t 1, y = 1, z = 1 t, y = -7 + 2t, t, y = 0, z = 0

t,

z = -1 + t z = 3 - 5t

+ t z = 2t

A-46

Chapter 11: Answers to Odd-Numbered Exercises

13. x = t,

y = t,

z =

0 … t … 1

3 t, 2

z = 0,

19.

4x2 + 9y2 + 4z2 = 36 z

9x2 + y2 + z2 = 9

3

-1 … t … 0

3

z

z

z

17.

y = 1 + t,

15. x = 1,

–3

–1

–3 –2

1, 1,

3 2

y

3

1 x

y

2 3

–3

(1, 1, 0)

x

(1, 0, 0)

(0, 0, 0)

x

y

–3

x

21.

z

23.

z = x2 + 4y2

z 2 x = 4 – 4y2 – z2

4

x

17. x = 0, y = 1 - 2t, z = 1, 0 … t … 1

1

19. x = 2 - 2t, y = 2t, z = 2 - 2t, 0 … t … 1

z

2 1

y

z

(0, –1, 1)

4

y

(0, 1, 1) x

x

(2, 0, 2) (0, 2, 0)

y

23. 7x - 5y - 4z = 6 27. s1, 2, 3d, -20x + 12y + z = 7

29. y + z = 3

31. x - y + z = 0

9 242 7

41. 19> 5

55. 59. 61. 63.

27.

x 2 + y2 – z2 = 1 z

x

21. 3x - 2y - z = -3 25. x + 3y + 4z = 34

47.

z

25. x 2 + y 2 = z2

x

37.

y

39. 3

33. 2 230

43. 5> 3

–1 1

1

45. 9> 241

y

x

3 3 1 p>4 49. 1.38 rad 51. 0.82 rad 53. a , - , b 2 2 2 (1, 1, 0) 57. x = 1 - t, y = 1 + t, z = -1 x = 4, y = 3 + 6t, z = 1 + 3t L1 intersects L2; L2 is parallel to L3; L1 and L3 are skew. x = 2 + 2t, y = -4 - t, z = 7 + 3t; x = -2 - t, y = -2 + s1>2dt, z = 1 - s3>2dt

x

29.

Section 11.6, pp. 642–643

2

z

2

z –x –y =1 y2 – x 2 = z 2

y

1 √3

33.

3. (a), cylinder 5. (l), hyperbolic paraboloid 9. (k), hyperbolic paraboloid 11. (h), cone 15. z

31.

z 2

1 3 65. a0, - , - b, s -1, 0, - 3d, s1, -1, 0d 2 2 69. Many possible answers. One possibility: x + y = 3 and 2y + z = 7. 71. sx>ad + sy>bd + sz>cd = 1 describes all planes except those through the origin or parallel to a coordinate axis.

1. (d), ellipsoid 7. (b), cylinder z 13.

–1

y

35. 0

y

x

x

√3

35.

z2 = 1 + y 2 – x 2 z

z y = – (x 2 + z 2)

x2 + y2 = 4 2

y

x2 + 4z2 = 16

y x

–2 4 2 x

y

x

y

x

A-47

Chapter 11: Answers to Odd-Numbered Exercises 37.

39.

x 2 + y2 – z2 = 4 z

z

23. 31. 35. 37.

x2 + z2 = 1 1

1 3 39. a0, - , - b, s -1, 0, - 3d, s1, -1, 0d 2 2

1 y

x

y

x

41.

43.

z

z

z = –(x 2 + y 2)

4y 2 + z2 – 4x 2 = 4 y

41. p>3

x = - 5 + 5t, y = 3 - t, z = -3t (b) x = -12t, y = 19>12 + 15t, z = 1>6 + 6t Yes; v is parallel to the plane. 3 51. -3j + 3k 2 53. a5i - j - 3kb 235

43. 45. 47. 49.

55. a

x y

x

2ps9 - c 2 d 45. (a) 9

227 25. (a) 214 (b) 1 29. 278>3 33. 22 x = 1 - 3t, y = 2, z = 3 + 7t 2x + y + z = 5 -9x + y + 7z = 4

65.

4pabc (c) 3

(b) 8p

57. 59. 61. 63.

11 26 7 , ,- b 9 9 9 s1, -2, -1d; x = 1 - 5t, y = -2 + 3t, 2x + 7y + 2z + 10 = 0 (a) No (b) No (c) No (d) No 11> 2107 67.

x 2 + y2 + z2 = 4 z

z = -1 + 4t (e) Yes

4x 2 + 4y2 + z2 = 4 z 2

2 –2

Practice Exercises, pp. 644–645 1. (a) 8-17, 329 3. (a) 86, -89

–1

(b) 21313 (b) 10

x

23 1 5. h , - i [assuming counterclockwise] 2 2 7. h

8 217

,-

2 217

11. v (p>2) = 2(-i)

8 233

i -

2 233

j +

22

i +

1 22

69.

3

z x 2 + y2 = z2

y

y

3 6 2 i - j + k. 7 7 7

8 233

x

k

u * v = 2i - 2j + k, ƒ v * u ƒ = 3, u = cos-1 a 22

71.

z = – (x 2 + y 2) z

j.

17. ƒ v ƒ = 22, ƒ u ƒ = 3, v # u = u # v = 3, v * u = - 2i + 2j - k ,

ƒ u ƒ cos u =

x

x

13. Length = 7, direction is 15.

1

–2

y

1

1

–2

i

9. Length = 2, direction is

y

2

2

, projv u =

1 22

b =

73.

75.

x 2 + y2 – z2 = 4 z

p , 4

y2 – x2 – z2 = 1 z 3

√5

3 si + jd 2

3 3

–2 –1

–2

4 s2i + j - kd 3 21. u * v = k

3

2

19.

2

y

√10

x –3 z

x

i (i + j) = k

Additional and Advanced Exercises, pp. 646–648 y

i i+j x

1. s26, 23, -1>3d

3

3. ƒ F ƒ = 20 lb

5. (a) ƒ F1 ƒ = 80 lb, ƒ F2 ƒ = 60 lb, F1 = 8-48, 649, 3 4 F2 = 848, 369, a = tan-1 , b = tan-1 3 4

y

A-48

Chapter 12: Answers to Odd-Numbered Exercises

2400 1000 (b) ƒ F1 ƒ = L 184.615 lb, ƒ F2 ƒ = L 76.923 lb, 13 13 -12,000 28,800 , i L 8-71.006, 170.4149, F1 = h 169 169 F2 = h

-2 2 bi + 2tj + t k; a = a bi + 2j + k; t + 1 st + 1d2 1 2 1 speed: 26; direction: i + j + k; 26 26 26

13. v = a

12,000 5000 , i L 871.006, 29.5869, 169 169 12 , 5

a = tan-1

b = tan-1

vs1d = 26 ¢

5 12

15. p>2

9. (a) u = tan-1 22 L 54.74° (b) u = tan-1 2 22 L 70.53° 6 13. (b) (c) 2x - y + 2x = 8 214 (d) x - 2y + z = 3 + 5 26 and x - 2y + z = 3 - 526 23 13 32 i + j k 15. 41 41 41 17. (a) 0, 0 (b) -10i - 2j + 6k, -9i - 2j + 7k (c) -4i - 6j + 2k, i - 2j - 4k (d) -10i - 10k, -12i - 4j - 8k 19. The formula is always true.

19. x = t,

1

26 17. p>2

i +

y = -1,

2 26

j +

z = 1 + t

1 26

k≤

21. x = t,

y =

1 t, 3

z = t

23. (a) (i): It has constant speed 1 (ii): Yes (iii): Counterclockwise (iv): Yes (b) (i): It has constant speed 2 (ii): Yes (iii): Counterclockwise (iv): Yes (c) (i): It has constant speed 1 (ii): Yes (iii): Counterclockwise (iv): No. It starts at s0, -1d instead of (1, 0) (d) (i): It has constant speed 1 (ii): Yes (iii): Clockwise (iv): Yes (e) (i): It has variable speed (ii): No (iii): Counterclockwise (iv): Yes 25. v = 2 25i + 25j

CHAPTER 12 Section 12.1, pp. 655–657 1. y = x 2 - 2x, v = i + 2j, a = 2j 2 3. y = x 2, v = 3i + 4j, a = 3i + 8j 9 22 22 - 22 22 p 5. t = : v = i j, a = i j; 4 2 2 2 2 t = p>2: v = -j, a = -i y 1 a ⎞⎠ ⎞⎠ 4 0

1. s1>4di + 7j + s3>2dk

3. ¢

p + 2 22 ≤ j + 2k 2

5. sln 4di + sln 4dj + sln 2dk e - 1 e - 1 7. i + e j + k 2 p 9. i - j + k 4 11. rstd = a

v ⎞⎠ ⎞⎠ 4

-t 2 -t 2 -t 2 + 1 bi + a + 2 bj + a + 3b k 2 2 2

13. rstd = sst + 1d3>2 - 1di + s -e -t + 1dj + slnst + 1d + 1dk 15. rstd = 8t i + 8t j + s -16t 2 + 100dk

x

a ⎞⎠ ⎞⎠ 2

Section 12.2, pp. 661–663

v ⎞⎠ ⎞⎠ 2

3 6 1 2 t + 1≤i - ¢ t2 + t - 2≤j 17. rstd = ¢ t 2 + 2 2 211 211 7. t = p: v = 2i, y

a = -j;

t=

v()

2 a() 1

r = (t – sin t)i + (1 – cos t)j 0

t =

3p : v = i - j, 2

a = i

t = 3 2 v ⎞⎠3⎞⎠ 2 a ⎞⎠3⎞⎠ 2 x 2

9. v = i + 2tj + 2k; a = 2j; speed: 3; direction: 2 2 2 1 k; vs1d = 3 a i + j + kb 3 3 3 3 11. v = s -2 sin tdi + s3 cos tdj + 4k; a = s -2 cos tdi - s3 sin tdj; speed: 225 ; direction: A -1> 25 B i + A 2> 25 B k;

vsp>2d = 225 C A -1> 25 B i + A 2> 25 B k D

1 2 i + j + 3 3

2t 1 1 2 + ¢ t2 + t + 3≤k = ¢ t 2 + ≤ s3i - j + kd 2 2 211 211 +si + 2j + 3kd 19. 21. 23. 25. 31.

50 sec (a) 72.2 sec; 25,510 m (b) 4020 m (c) 6378 m (a) y0 L 9.9 m>sec (b) a L 18.4° or 71.6° 39.3° or 50.7° (a) (Assuming that “x” is zero at the point of impact) rstd = sxstddi + s ystddj, where xstd = s35 cos 27°dt and ystd = 4 + s35 sin 27°dt - 16t 2 . (b) At t L 0.497 sec , it reaches its maximum height of about 7.945 ft. (c) Range L 37.45 ft; flight time L 1.201 sec (d) At t L 0.254 and t L 0.740 sec, when it is L 29.555 and L 14.396 ft from where it will land (e) Yes. It changes things because the ball won’t clear the net.

Chapter 12: Answers to Odd-Numbered Exercises

Section 12.3, pp. 667–668 25 2 2 1. T = a- sin tbi + a cos tb j + k, 3p 3 3 3 1

3. T =

2t

i +

21 + t

k,

52 3

21 + t 3 5. T = -cos t j + sin t k, 2 cos t - t sin t sin t + t cos t 7. T = a bi + a bj t + 1 t + 1 +a

22t p b k, + p t + 1 2 1>2

2

5p 2

L =

(0, –1, 1)

(–1, 0, 2)

(1, 0, 0)

y

2p

(d) L =

L0

21 + sin2 t dt

(e) L L 7.64

Section 12.4, pp. 673–674 1. T = scos tdi - ssin tdj, N = s -sin tdi - scos tdj, k = cos t -t t 1 3. T = i j, N = i2 2 21 + t 21 + t 21 + t 2 1 21 + t 5. (b) cos x

2

j, k =

7. (b) N =

1

2 A 21 + t 2 B 3

-2e 2t

i +

1

j 21 + 4e 21 + 4e 4t 1 (c) N = - A 24 - t 2 i + t j B 2 3 cos t 3 sin t 4 9. T = i j + k, N = s -sin tdi - scos tdj, 5 5 5 3 k = 25 cos t + sin t cos t - sin t 11. T = a ≤i + ¢ ≤j, 22 22 N = ¢ k =

4t

-cos t - sin t 22 1

e t 22

≤i + ¢

2t + 1 2t + 1 1 k = tst 2 + 1d3>2 t t 15. T = asech a bi + atanh a bj ,

-sin t + cos t 22

i 2t + 1 2

-

tj 2t + 1 2

t t N = a -tanh a bi + asech a bj , t 1 k = a sech2 a 21. ax -

2

p b + y2 = 1 2

25. ksxd = ƒ sin x ƒ >s1 + cos2 xd3>2

Section 12.5, pp. 678–679 225 4 T + N 5. as0d = 2N 3 3 22 22 22 22 p p 7. r a b = i + j - k, T a b = i + j, 4 2 2 4 2 2 22 22 p p Na b = i j, B a b = k; osculating plane: 4 2 2 4 z = -1 ; normal plane: -x + y = 0 ; rectifying plane: 1. a = ƒ a ƒ N

3. as1d =

x + y = 22

(0, 1, 1)

x

j, N =

2

23. ksxd = 2>s1 + 4x 2 d3>2

3 23 15. 22 + ln A 1 + 22 B 4 2 2 17. (a) Cylinder is x + y = 1 , plane is x + z = 1 . z (b) and (c) 13. sstd = 23e t - 23,

1

i +

2

19. 1> (2b) L =

11. sstd = 5t,

9. s0, 5, 24pd

t

13. T =

A-49

≤j,

3 4 4 4 9. B = a cos tbi - a sin tbj - k, t = 5 5 5 25 11. B = k, t = 0 13. B = -k, t = 0 15. B = k, t = 0 17. Yes. If the car is moving on a curved path sk Z 0d , then aN = k ƒ v ƒ 2 Z 0 and a Z 0. 1 23. k = t , r = t 29. Components of v: -1.8701 , 0.7089, 1.0000 Components of a: -1.6960, -2.0307 , 0 Speed: 2.2361; Components of T: -0.8364 , 0.3170, 0.4472 Components of N: -0.4143, -0.8998, -0.1369 Components of B: 0.3590, -0.2998 , 0.8839; Curvature: 0.5060 Torsion: 0.2813; Tangential component of acceleration: 0.7746 Normal component of acceleration: 2.5298 31. Components of v: 2.0000, 0, - 0.1629 Components of a: 0, -1.0000 , - 0.0086; Speed: 2.0066 Components of T: 0.9967, 0, - 0.0812 Components of N: -0.0007, -1.0000 , - 0.0086 Components of B: - 0.0812, 0.0086, 0.9967 ; Curvature: 0.2484 Torsion: 0.0411 ; Tangential component of acceleration: 0.0007 Normal component of acceleration: 1.0000

Section 12.6, p. 682 1. v a 3. v a 5. v a

= = = = = =

(3a sin u)ur + 3a(1 - cos u)uu 9a(2 cos u - 1)ur + (18a sin u)uu 2ae auur + 2e auuu 4e au(a 2 - 1)ur + 8ae auuu (-8 sin 4t)ur + (4 cos 4t)uu (- 40 cos 4t)ur - (32 sin 4t)uu

,

A-50

Chapter 13: Answers to Odd-Numbered Exercises

Practice Exercises, pp. 683–684 1.

CHAPTER 13

y2 x2 + = 1 16 2

Section 13.1, pp. 692–694 y v ⎞⎠ ⎞⎠ 4

2 √2 1

(2√2, 1)

a ⎞⎠ ⎞⎠ 4

0

–4

1. (a) 0 (b) 0 (c) 58 (d) 33 3. (a) 4>5 (b) 8>5 (c) 3 (d) 0 5. Domain: all points (x, y) on 7. Domain: all points (x, y) or above line y = x + 2 not lying on the graph of y = x or y = x 3

v(0) x

4

a(0)

–1

y

y

At t = 0: a T = 0, aN = 4, k = 2 ; 4 22 4 22 7 p At t = : a T = , aN = ,k = 4 3 3 27 3. ƒ v ƒ max = 1 5. k = 1>5 7. dy>dt = -x ; clockwise 11. Shot put is on the ground, about 66 ft 3 in. from the stopboard. 2

Bs0d = 19. Tsln 2d =

1 3 22 1

i + i +

j +

4 3 22

k;

k =

Nsln 2d = -

j;

217 8 Bsln 2d = k; k = ;t = 0 17 217 21. as0d = 10T + 6N 23. T = ¢

1 22

N = ¢1

B =

217

1 322 4

cos t ≤ i - ssin tdj + ¢ 1

22

x (–1, –1)

9. Domain: all points (x, y) satisfying x 2 - 1 … y … x 2 + 1 y

i +

1 217

1 22

x –1

22

bg

x=2

3

sin t ≤ k; –2

x

2

1

y = –3 –3

31. k = 1>a 13.

15. y

y 4

gb 2t 2

bgt

x+y–1=c

2 –2

2 –2

2

d r = T + aa 2 b N dt 2 a + b2 2a 2 + b 2 2

y

x = –2

y=3

1

, z = 2sa 2 + b 2 d 2sa 2 + b 2 d gbt (c) vstd = T; 2a 2 + b 2 (b) u =

y = x2 – 1

j;

pgb du ` = 2 dt u = 2p A a2 + b2 gbt 2

y = x2 + 1

1

cos t ≤ k;

Additional and Advanced Exercises, p. 685

There is no component in the direction of B. dy # # dx # # 5. (a) = r cos u - r u sin u, = r sin u + r u cos u dt dt du dr # # # # = x cos u + y sin u, r = -x sin u + y cos u dt dt 7. (a) as1d = -9u r - 6u u, vs1d = - ur + 3uu (b) 6.5 in. (b)

x

11. Domain: all points (x, y) for which (x - 2)(x + 2)(y - 3)(y + 3) Ú 0

sin t ≤ i - scos tdj - ¢ 1

217

i k; k = ;t = 0 22 22 22 25. p>3 27. x = 1 + t, y = t, z = -t

1. (a)

y = x3

22 1 ;t = 3 6 4

(1, 1)

y=x

2

p p p p 1 + + ln ¢ + 1 + ≤ 4B 16 4 16 B 1 1 2 2 1 17. Ts0d = i - j + k; Ns0d = i + j; 3 3 3 22 22 15. Length =

y=x+2

4

x c: 3 2 1 0 –1 –2 –3

c = –9 c = –4 c = –1 1=c 4=c 9=c

xy = c

c=9 c=4 c=1 0=c –1 = c

x

–4 = c –9 = c

All points in the xy-plane (b) All reals The lines y - x = c (d) No boundary points Both open and closed (f) Unbounded All points in the xy-plane (b) z Ú 0 For ƒsx, yd = 0, the origin; for ƒsx, yd Z 0, ellipses with the center (0, 0), and major and minor axes along the x- and y-axes respectively (d) No boundary points (e) Both open and closed (f) Unbounded

17. (a) (c) (e) 19. (a) (c)

A-51

Chapter 13: Answers to Odd-Numbered Exercises 21. (a) All points in the xy-plane (b) All reals (c) For ƒsx, yd = 0, the x- and y-axes; for ƒsx, yd Z 0, hyperbolas with the x- and y-axes as asymptotes (d) No boundary points (e) Both open and closed (f) Unbounded 23. (a) All (x, y) satisfying x 2 + y 2 6 16 (b) z Ú 1>4 (c) Circles centered at the origin with radii r 6 4 (d) Boundary is the circle x 2 + y 2 = 16 (e) Open (f) Bounded 25. (a) sx, yd Z s0, 0d (b) All reals (c) The circles with center (0, 0) and radii r 7 0 (d) Boundary is the single point (0, 0) (e) Open (f) Unbounded 27. (a) All (x, y) satisfying -1 … y - x … 1 (b) -p>2 … z … p>2 (c) Straight lines of the form y - x = c where -1 … c … 1 (d) Boundary is two straight lines y = 1 + x and y = - 1 + x (e) Closed (f) Unbounded 29. (a) Domain: all points (x, y) outside the circle x 2 + y 2 = 1 (b) Range: all reals (c) Circles centered at the origin with radii r 7 1 (d) Boundary: x 2 + y 2 = 1 (e) Open (f ) Unbounded 31. (f) 33. (a) 35. (d) 37. (a) (b) z y z=y

43. (a)

z

(b)

z = 4x2 + y2

y 4

16

z = 16

4 2

2

z=4

z=0 4

2

1 0

y

x

45. (a)

(b) z

y

z = 1 – ⏐y⏐

(0, 0, 1)

z = –1 2

z=0

1

z=1

0 y

1

x

z=0

–1

x

z = –1

–2

47. (a)

(b) y

z z = √x2 + y2 + 4

2

z=4 z=1 x z=1 z=4

z=0

x

1 2

4 3

z = √20 z = √13

2 z = √8 1 z = √5

2

z=2

y

y

–4 –3 –2 –1

1

2

3

x

4

–1

x

–2

x

39. (a)

z

(b)

–3

y

z = x2 + y2

–4 z=4

49. x 2 + y 2 = 10

z=1 z=0 –2

–1

1

2

51. x + y 2 = 4 y

y

x

√10

2

– √10

y

√10

4

x

x

41. (a)

–2

(b) z = –3

z y

z = –1 z=0

3

z=1

2

z=2

1 y 0 –1 –2 x

– √10

z = –2

z = x2 – y

53.

55.

z

z

1

1

z=3 x 1

1

x

1

y x

–3 f(x, y, z) = x 2 + y2 + z2 = 1

f(x, y, z) = x + z = 1

y

x

A-52

Chapter 13: Answers to Odd-Numbered Exercises

57.

59.

z

z

9.

f (x, y, z) = x2 + y2 = 1 5 2 2

1

1

1 y

11. 13. 1 1

x

f(x, y, z) = z – x 2 – y 2 = 1 or z = x 2 + y2 + 1

17.

y

x

61. 2x - y - ln z = 2 63. x 2 + y 2 + z 2 = 4 65. Domain: all points (x, y) 67. Domain: all points (x, y) satisfying ƒ x ƒ 6 ƒ y ƒ satisfying -1 … x … 1 and -1 … y … 1 y

y

y=x

19. 23. 25. 27.

1

29. x –1

1

x

y = –x

–1

level curve: y = 2x

level curve: p sin-1 y - sin-1 x = 2

Section 13.2, pp. 700–703

5> 2 3. 226 5. 1 7. 1> 2 9. 1 11. 1>4 13. 0 19. 1> 4 21. 1 23. 3 25. 19>12 -1 17. 2 2 29. 3 31. (a) All (x, y) (b) All (x, y) except (0, 0) (a) All (x, y) except where x = 0 or y = 0 (b) All (x, y) (a) All (x, y, z) (b) All (x, y, z) except the interior of the cylinder x 2 + y 2 = 1 37. (a) All (x, y, z) with z Z 0 (b) All (x, y, z) with x 2 + z 2 Z 1 39. (a) All points (x, y, z) satisfying z 7 x 2 + y 2 + 1 1. 15. 27. 33. 35.

41. 43. 45. 47. 49. 51. 55. 59. 63. 69. 75.

(b) All points (x, y, z) satisfying z Z 2x2 + y2 Consider paths along y = x, x 7 0, and along y = x, x 6 0. Consider the paths y = k x 2, k a constant. Consider the paths y = mx, m a constant, m Z - 1. Consider the paths y = kx 2, k a constant, k Z 0. Consider the paths x = 1 and y = x. (a) 1 (b) 0 (c) Does not exist The limit is 1. 57. The limit is 0. (a) ƒsx, yd ƒ y = mx = sin 2u where tan u = m 61. 0 Does not exist 65. p>2 67. ƒs0, 0d = ln 3 d = 0.1 71. d = 0.005 73. d = 0.04 d = 20.015 77. d = 0.005

Section 13.3, pp. 711–714 0ƒ 0ƒ 0ƒ 0ƒ = 4x, = -3 3. = 2xsy + 2d, = x2 - 1 0x 0y 0x 0y 0ƒ 0ƒ 5. = 2ysxy - 1d, = 2xsxy - 1d 0x 0y 0ƒ 0ƒ y x 7. , = = 0x 2 2 0y 2 2x + y 2x + y 2 1.

31.

0ƒ 0ƒ -1 -1 = = , 0x sx + yd2 0y sx + yd2 0ƒ -y 2 - 1 0ƒ -x 2 - 1 , = = 2 0y 0x sxy - 1d sxy - 1d2 0ƒ 0ƒ 0ƒ 0ƒ 1 1 15. , = e x + y + 1, = ex+y+1 = = x + y 0y x + y 0x 0y 0x 0ƒ = 2 sin sx - 3yd cos sx - 3yd, 0x 0ƒ = -6 sin sx - 3yd cos sx - 3yd 0y 0ƒ 0ƒ 0ƒ 0ƒ = yx y - 1, = x y ln x 21. = -gsxd, = gs yd 0x 0y 0x 0y ƒx = y 2, ƒy = 2xy, ƒz = -4z ƒx = 1, ƒy = -ysy 2 + z 2 d-1>2, ƒz = -zsy 2 + z 2 d-1>2 yz xy xz ƒx = , ƒy = , ƒz = 2 2 2 2 2 2 21 - x y z 21 - x y z 21 - x 2y 2z 2 3 1 2 ƒx = ,ƒ = ,ƒ = x + 2y + 3z y x + 2y + 3z z x + 2y + 3z 2 2 2 2 2 2 2 2 2 ƒx = -2xe -sx + y + z d, ƒy = -2ye -sx + y + z d, ƒz = -2ze -sx + y + z d

33. ƒx = sech2sx + 2y + 3zd, ƒy = 2 sech2sx + 2y + 3zd, ƒz = 3 sech2sx + 2y + 3zd 0ƒ 0ƒ 35. = -2p sin s2pt - ad, = sin s2pt - ad 0t 0a 0h 0h 0h 37. = sin f cos u, = -r sin f sin u = r cos f cos u, 0r 0u 0f dy 2 39. WPsP, V, d, y, gd = V, WV sP, V, d, y, gd = P + , 2g Vy 2 Vdy WdsP, V, d, y, gd = , WysP, V, d, y, gd = g , 2g Vdy 2 2g 2 0ƒ 0 2ƒ 0ƒ 0 2ƒ 41. = 1 + y, = 1 + x, 2 = 0, 2 = 0, 0x 0y 0x 0y 0 2ƒ 0 2ƒ = = 1 0y 0x 0x 0y 0g 0g 43. = 2xy + y cos x, = x 2 - sin y + sin x, 0x 0y 0 2g 0 2g = 2y - y sin x, = -cos y, 2 0x 0y 2 2 2 0g 0g = = 2x + cos x 0y 0x 0x 0y WgsP, V, d, y, gd = -

0r 0 2r 0r 0 2r -1 -1 1 1 = , 2 = , = , = , 2 x + y 0y x + y 0x 2 0x sx + yd 0y sx + yd2 0 2r 0 2r -1 = = 0y 0x 0x 0y sx + yd2 0w 0w 47. = x 2y sec2 (xy) + 2x tan (xy), = x 3 sec2 (xy), 0x 0y 0 2w 0 2w = = 2x 3y sec2 (xy) tan (xy) + 3x 2 sec2 (xy) 0y 0x 0x 0y 0 2w = 4xy sec2 (xy) + 2x 2y 2 sec2 (xy) tan (xy) + 2 tan (xy) 0x 2 0 2w = 2x 4 sec2 (xy) tan (xy) 0y 2 45.

Chapter 13: Answers to Odd-Numbered Exercises 0w 0w = x 3 cos (x 2y), = sin (x 2y) + 2x 2y cos (x 2y), 0x 0y 0 2w 0 2w = = 3x 2 cos (x 2y) - 2x 4y sin (x 2y) 0y 0x 0x 0y 0 2w = 6xy cos (x 2y) - 4x 3y 2 sin (x 2y) 0x 2 0 2w = -x 5 sin (x 2y) 0y 2 3 -6 0w 0w 0 2w 0 2w 2 , , 51. = = = = 0x 0x 0y 2x + 3y 0y 2x + 3y 0y 0x s2x + 3yd2 0w 0w = y 2 + 2xy 3 + 3x 2y 4, = 2xy + 3x 2y 2 + 4x 3y 3, 53. 0x 0y 49.

55. 57. 59. 63. 69. 71.

89.

0 2w 0 2w = = 2y + 6xy 2 + 12x 2y 3 0y 0x 0x 0y (a) x first (b) y first (c) x first (d) x first (e) y first (f) y first ƒxs1, 2d = -13, ƒys1, 2d = - 2 (b) 2 ƒx (-2, 3) = 1>2, ƒy (-2, 3) = 3>4 61. (a) 3 c cos A - b a 0A 0A 12 65. -2 67. = , = 0a bc sin A 0b bc sin A ln y yx = sln udsln yd - 1 ƒx (x, y) = 0 for all points (x, y), 3y 2, y Ú 0 , ƒy (x, y) = e -2y, y 6 0 ƒxy (x, y) = ƒyx (x, y) = 0 for all points (x, y) Yes

15.

0w 0w 0x 0w 0y 0w 0z = + + , 0u 0x 0u 0y 0u 0z 0u 0y 0w 0w 0x 0w 0w 0z = + + 0y 0x 0y 0y 0y 0z 0y w

w

w

z

w

x w

y x y

y

u z

u

x

u

17.

3.

w

5. 7.

9.

11.

13.

x dy dt t

y

y

y

x

y y

0z 0z 0x 0z 0y 0z 0z 0x 0z 0y = + , = + 0t 0x 0t 0y 0t 0s 0x 0s 0y 0s z

z

z

x

z

y

z

x y

y

t

x

t

z

y

x

y

y

s

x

s

t

s

0w dw 0u 0w dw 0u = , = 0s du 0s 0t du 0t w

w dw du

dw du u

u

u

t

u

s s

t

0w dx 0w dy 0w 0w = + = 0r 0x dr 0y dr 0x dy 0w 0w dx 0w 0w = + = 0s 0x ds 0y ds 0y w

w

y

x

r

y =0

r

y

25. 4> 3 31.

dy dx since = 0, dr dr dy dx since = 0 ds ds w

w

x x

y dx dt

x

y

u

r

z

y

w

y

u

z

z

x

w

x y

x

u

23.

z

w

w

y

x

21.

y

y z

y

y

w

x

19.

w

z

0w 0w 0x 0w 0y 0w 0w 0x 0w 0y . = + , = + 0u 0x 0u 0y 0u 0y 0x 0y 0y 0y

Section 13.4, pp. 721–722 1.

x

y

u

x

dw dw (a) = 0, (b) spd = 0 dt dt dw dw (a) = 1, (b) s3d = 1 dt dt dw dw = 4t tan-1 t + 1, (b) s1d = p + 1 (a) dt dt 0z = 4 cos y ln su sin yd + 4 cos y, (a) 0u 4u cos2 y 0z = -4u sin y ln su sin yd + 0y sin y 0z 0z = 22 sln 2 + 2d, = -2 22 sln 2 - 2d (b) 0u 0y 0w 0w = 2u + 4uy, = -2y + 2u 2 (a) 0u 0y 3 0w 0w = 3, = (b) 0u 0y 2 -y 0u 0u 0u z , = 0, = = (a) 0x 0y sz - yd2 0z sz - yd2 0u 0u 0u = 0, = 1, = -2 (b) 0x 0y 0z dz 0z dx 0z dy = + 0x dt 0y dt dt

z

w

x w

y x y

27. -4>5

0z 0z = -1, = -1 0x 0y

w

x

w

y

x = 0

s

y

s

x

y

s

3 0z 1 0z 29. = , = 0x 4 0y 4 33. 12

35. -7

A-53

A-54

Chapter 13: Answers to Odd-Numbered Exercises

0z 0z = 2, = 1 0u 0y 3 2 0w 3 2 0w 39. = 2t e s + t , = 3s 2 e s + t 0t 0s 41. -0.00005 amps>sec

3 4 i - j, Du ƒ(1, -1) = 5 5 5 3 4 (b) u = - i + j, Du ƒ(1, -1) = -5 5 5

29. (a) u =

37.

49. (a) Maximum at ¢ at ¢

22 22 22 22 , ,≤ and ¢ ≤ ; minimum 2 2 2 2

22 22 22 22 , ,≤ and ¢ ≤ 2 2 2 2

(b) Max = 6, min = 2 x2

51. 2x2x + x + 8

3x 2

3

L0

2 2t + x 4

3

33. No, the maximum rate of change is 2185 6 14 . 35. -7> 25

dt

Section 13.6, pp. 737–739

Section 13.5, pp. 729–730 1.

3.

y

y

Δ

f = –i + j

2

y – x = –1

1

x= 2 y2

(2, 1) 1

0 –1

x

2

x

(2, –1)

Δ

f = i – 4j

5.

y f= 1i+ 3j 2 4

Δ

4 = 2x + 3y x 2

7. §ƒ = 3i + 2j - 4k

9. §ƒ = -

26 23 23 i + j k 27 54 54

13. 21> 13 15. 3 17. 2 1 1 1 1 19. u = i + j, sDuƒdP0 = 22; -u = i j, 22 22 22 22 11. -4

sDuƒdP0 = - 22 5 1 1 21. u = i j k, sDuƒdP0 = 323; 3 23 3 23 3 23 5 1 1 i + j + k, sD-uƒdP0 = -3 23 -u = 3 23 3 23 3 23 1 si + j + kd, sDuƒdP0 = 223; 23. u = 23 1 -u = si + j + kd, sD-uƒdP0 = -2 23 23 y 25. 27. y f 2√2i 2√2j

xy = –4

Δ

2

y=x–4

(√2, √2)

y = –x + 2√2

x

Δ

2 x2 + y2 = 4

1. (a) x + y + z = 3 (b) x = 1 + 2t, y = 1 + 2t, z = 1 + 2t 3. (a) 2x - z - 2 = 0 (b) x = 2 - 4t, y = 0, z = 2 + 2t 5. (a) 2x + 2y + z - 4 = 0 (b) x = 2t, y = 1 + 2t, z = 2 + t 7. (a) x + y + z - 1 = 0 (b) x = t, y = 1 + t, z = t 9. 2x - z - 2 = 0 11. x - y + 2z - 1 = 0 13. x = 1, y = 1 + 2t, z = 1 - 2t 1 15. x = 1 - 2t, y = 1, z = + 2t 2 17. x = 1 + 90t, y = 1 - 90t, z = 3 9 19. dƒ = 21. dg = 0 L 0.0008 11,830 23 1 sin 23 - cos 23 L 0.935°C>ft 2 2 (b) 23 sin 23 - cos 23 L 1.87°C>sec 25. (a) Lsx, yd = 1 (b) Lsx, yd = 2x + 2y - 1 27. (a) Lsx, yd = 3x - 4y + 5 (b) Lsx, yd = 3x - 4y + 5 p 29. (a) Lsx, yd = 1 + x (b) Lsx, yd = -y + 2 31. (a) W(20, 25) = 11°F, W(30, -10) = -39°F, W(15, 15) = 0°F (b) W(10, -40) L -65.5°F, W(50, -40) L -88°F, W(60, 30) L 10.2°F (c) L(y, T) L - 0.36 (y - 25) + 1.337(T - 5) - 17.4088 (d) i) L(24, 6) L -15.7°F ii) L(27, 2) L -22.1°F iii) L(5, -10) L -30.2°F 33. Lsx, yd = 7 + x - 6y; 0.06 35. Lsx, yd = x + y + 1; 0.08 37. Lsx, yd = 1 + x; 0.0222 39. (a) Lsx, y, zd = 2x + 2y + 2z - 3 (b) Lsx, y, zd = y + z (c) Lsx, y, zd = 0 1 1 41. (a) Lsx, y, zd = x (b) Lsx, y, zd = x + y 22 22 1 2 2 (c) Lsx, y, zd = x + y + z 3 3 3 43. (a) Lsx, y, zd = 2 + x p (b) Lsx, y, zd = x - y - z + + 1 2 p (c) Lsx, y, zd = x - y - z + + 1 2 23. (a)

4 3

(–1, 2)

3 3 4 4 i + j, u = - i - j 5 5 5 5 7 24 (d) u = -j, u = i j 25 25 7 24 (e) u = -i, u = i + j 25 25 7 7 2 2 31. u = i j, -u = i + j 253 253 253 253 (c) u =

47. scos 1, sin 1, 1d and scoss -2d, sins -2d, -2d

f = –2i + 2j –2

2 (2, –2)

x

Chapter 13: Answers to Odd-Numbered Exercises Lsx, y, zd = 2x - 6y - 2z + 6, 0.0024 Lsx, y, zd = x + y - z - 1, 0.00135 Maximum error (estimate) … 0.31 in magnitude Pay more attention to the smaller of the two dimensions. It will generate the larger partial derivative. 53. ƒ is most sensitive to a change in d. 45. 47. 49. 51.

Section 13.7, pp. 745–748 1. ƒ(-3, 3) = -5, local minimum 3 17 5. ƒ a3, b = , local maximum 2 2 7. ƒ(2, -1) = -6, local minimum

3. ƒ(-2, 1), saddle point

9. ƒ(1, 2), saddle point

16 16 11. ƒ a , 0 b = - , local maximum 7 7 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37.

170 2 2 ƒ(0, 0), saddle point; ƒ a - , b = , local maximum 3 3 27 ƒ(0, 0) = 0, local minimum; ƒ(1, -1), saddle point ƒ(0, ; 25), saddle points; ƒ(-2, -1) = 30, local maximum; ƒ(2, 1) = -30, local minimum ƒ(0, 0), saddle point; ƒ(1, 1) = 2, ƒ(- 1, -1) = 2, local maxima ƒ(0, 0) = -1, local maximum ƒ(np, 0), saddle points, for every integer n ƒ(2, 0) = e -4, local minimum ƒ(0, 0) = 0, local minimum; ƒ(0, 2), saddle point 1 1 ƒ a , 1b = ln a b - 3, local maximum 2 4 Absolute maximum: 1 at (0, 0); absolute minimum: -5 at (1, 2) Absolute maximum: 4 at (0, 2); absolute minimum: 0 at (0, 0) Absolute maximum: 11 at s0, -3d; absolute minimum: -10 at s4, -2d 3 22 Absolute maximum: 4 at (2, 0); absolute minimum: at 2 p p p p a3, - b, a3, b , a1, - b, and a1, b 4 4 4 4

min ƒ = -1>2 at t = -1>2; no max ii) max ƒ = 0 at t = -1, 0; min ƒ = -1>2 at t = -1>2 iii) max ƒ = 4 at t = 1; min ƒ = 0 at t = 0 20 71 9 67. y = = x + , y 13 13 ƒ x = 4 13

63. i)

Section 13.8, pp. 755–757 1. ¢ ; 7. 9. 13. 15. 17. 23. 25. 29. 33. 37.

1° 1 23 1 23 at a- , b and a- , b ; coldest is 4 2 2 2 2

1 1 - ° at a , 0b. 4 2 43. (a) ƒ(0, 0), saddle point (b) ƒ(1, 2), local minimum (c) ƒs1, -2d, local minimum; ƒs -1, -2d, saddle point 9 6 3 1 1 355 49. a , , 51. a , , b 53. 3, 3, 3 55. 12 b 7 7 7 6 3 36 4 4 4 57. 59. 2 ft * 2 ft * 1 ft * * 23 23 23 61. (a) On the semicircle, max ƒ = 2 22 at t = p>4, min ƒ = -2 at t = p . On the quarter circle, max ƒ = 222 at t = p>4, min ƒ = 2 at t = 0, p>2. (b) On the semicircle, max g = 2 at t = p>4, min g = -2 at t = 3p>4. On the quarter circle, max g = 2 at t = p>4, min g = 0 at t = 0, p>2. (c) On the semicircle, max h = 8 at t = 0, p; min h = 4 at t = p>2. On the quarter circle, max h = 8 at t = 0, min h = 4 at t = p>2.

1

1 1 1 , ≤, ¢ ; ,- ≤ 2 2 22 22

3. 39

5. A 3, ;322 B

(a) 8 (b) 64 r = 2 cm, h = 4 cm 11. Length = 4 22, width = 322 ƒs0, 0d = 0 is minimum; ƒs2, 4d = 20 is maximum. Lowest = 0°, highest = 125° 5 3 19. 1 21. (0, 0, 2), s0, 0, -2d a , 2, b 2 2 ƒs1, -2, 5d = 30 is maximum; ƒs -1, 2, -5d = -30 is minimum. 2 2 2 3, 3, 3 27. by by units 23 23 23 s ;4>3, -4>3, -4>3d 31. Us8, 14d = $128 4 35. (2, 4, 4) ƒs2>3, 4>3, -4>3d = 3 Maximum is 1 + 6 23 at A ; 26, 23, 1 B ; minimum is 1 - 623 at A ; 26, - 23, 1 B .

39. Maximum is 4 at s0, 0, ;2d; minimum is 2 at A ; 22, ; 22, 0 B .

Practice Exercises, pp. 758–761 1. Domain: all points in the xy-plane; range: z Ú 0. Level curves are ellipses with major axis along the y-axis and minor axis along the x-axis. y 3 z=9

39. a = -3, b = 2 41. Hottest is 2

A-55

–1

1

x

–3

3. Domain: all (x, y) such that x Z 0 and y Z 0; range: z Z 0. Level curves are hyperbolas with the x- and y-axes as asymptotes. y

z=1 x

5. Domain: all points in xyz-space; range: all real numbers. Level surfaces are paraboloids of revolution with the z-axis as axis. z f(x, y, z) = x 2 + y2 – z = –1 or z = x2 + y2 + 1

1 y x

A-56

Chapter 13: Answers to Odd-Numbered Exercises

7. Domain: all (x, y, z) such that sx, y, zd Z s0, 0, 0d; range: positive real numbers. Level surfaces are spheres with center (0, 0, 0) and radius r 7 0. z

h(x, y, z) = 2 12 =1 x + y + z2 or 2 2 2 x +y +z =1

1

47. Tangent: 4x - y - 5z = 4; normal line: x = 2 + 4t, y = -1 -t, z = 1 - 5t 49. 2y - z - 2 = 0 51. Tangent: x + y = p + 1; normal line: y = x - p + 1 y

y = –x + + 1 y=x–+1

2 1

x

0

1

1

y

9. -2 11. 1> 2 13. 1 15. Let y = k x 2, k Z 1 17. No; limsx,yd:s0,0d ƒsx, yd does not exist. 0g 0g 19. = -r sin u + r cos u = cos u + sin u, 0r 0u 0ƒ 0ƒ 1 1 0ƒ 1 21. = - 2, = - 2, = - 2 0R1 0R 2 R1 R 2 0R3 R3 nT 0P nR 0P nRT 0P RT 0P 23. = , = , = , = - 2 0n V 0R V 0T V 0V V 2 2 g g 0 2g 0 0 2g 0 2x 1 25. = 0, 2 = 3 , = = - 2 0x 0y 0x 2 0y y 0y 0x y 2 0 2ƒ 0 2ƒ 0 2ƒ 2 - 2x 2 0 ƒ 27. = -30x + 2 , 2 = 0, = = 1 2 2 0y 0x 0x 0y 0x sx + 1d 0y 29.

dw ` = -1 dt t = 0

0w 0w ` = 2, ` = 2 - p 0r sr, sd = sp, 0d 0s sr, sd = sp, 0d dƒ 33. ` = -ssin 1 + cos 2dssin 1d + scos 1 + cos 2dscos 1d dt t = 1 - 2ssin 1 + cos 1dssin 2d 31.

35.

dy ` = -1 dx sx, yd = s0,1d

y = 1 + sin x 1

x

2

53. x = 1 - 2t, y = 1, z = 1>2 + 2t 55. Answers will depend on the upper bound used for ƒ ƒxx ƒ , ƒ ƒxy ƒ , ƒ ƒyy ƒ. With M = 22>2, ƒ E ƒ … 0.0142. With M = 1, ƒ E ƒ … 0.02. 57. Lsx, y, zd = y - 3z, Lsx, y, zd = x + y - z - 1 59. Be more careful with the diameter. 61. dI = 0.038, % change in I = 15.83%, more sensitive to voltage change 63. (a) 5% 65. Local minimum of -8 at s -2, -2d 67. Saddle point at s0, 0d, ƒs0, 0d = 0; local maximum of 1> 4 at s -1>2, -1>2d 69. Saddle point at s0, 0d, ƒs0, 0d = 0; local minimum of -4 at (0, 2); local maximum of 4 at s -2, 0d; saddle point at s -2, 2d, ƒs -2, 2d = 0 71. Absolute maximum: 28 at (0, 4); absolute minimum: -9>4 at (3> 2, 0) 73. Absolute maximum: 18 at s2, -2d; absolute minimum: -17>4 at s -2, 1>2d 75. Absolute maximum: 8 at s -2, 0d; absolute minimum: -1 at (1, 0) 77. Absolute maximum: 4 at (1, 0); absolute minimum: -4 at s0, -1d 79. Absolute maximum: 1 at s0, ;1d and (1, 0); absolute minimum: - 1 at s -1, 0d 81. Maximum: 5 at (0, 1); minimum: - 1>3 at s0, -1>3d 83. Maximum: 23 at ¢

22 22 i j; 2 2 22 22 decreases most rapidly in the direction -u = i + j; 2 2 22 22 7 v where u1 = ; D-uƒ = ; Du1 ƒ = Duƒ = 2 2 10 ƒvƒ 3 6 2 39. Increases most rapidly in the direction u = i + j + k; 7 7 7 3 6 2 decreases most rapidly in the direction -u = - i - j - k; 7 7 7 v Du ƒ = 7; D-u ƒ = -7; Du1 ƒ = 7 where u1 = ƒvƒ 41. p> 22 43. (a) ƒxs1, 2d = ƒys1, 2d = 2 (b) 14> 5 45. z 2 2 37. Increases most rapidly in the direction u = -

x +y+z =0

∇ f⏐(0, –1, 1) = j + 2k

1 ∇ f⏐(0, 0, 0) = j y –1 x ∇ f⏐(0, –1, –1) = j – 2k

¢-

1

,

1

23 23

85. Width = ¢

,-

c 2V ≤ ab

1 23

1 23 1>3

,-

1

,

1

23 23

≤ ; minimum: - 23 at

≤

b 2V , depth = ¢ ac ≤

1>3

, height = ¢

a 2V ≤ bc

1>3

3 1 1 1 1 87. Maximum: at ¢ , , 22 ≤ and ¢ ,, - 22 ≤ ; 2 22 22 22 22 1 1 1 1 1 at ¢ , , - 22 ≤ and ¢ ,, 22 ≤ 2 22 22 22 22 sin u 0w 0w cos u 0w 0w 0w 0w 89. = cos u - r , = sin u + r 0x 0r 0r 0u 0y 0u 95. st, -t ; 4, td, t a real number minimum:

Additional and Advanced Exercises, pp. 761–762 1. ƒxys0, 0d = -1, ƒyxs0, 0d = 1 23abc r2 1 7. (c) 13. V = = sx 2 + y 2 + z 2 d 2 2 2 y 9 x 17. ƒsx, yd = + 4, gsx, yd = + 2 2 2 19. y = 2 ln ƒ sin x ƒ + ln 2 -1 1 21. (a) s2i + 7jd (b) s98i - 127j + 58kd 253 229,097 2 2 23. w = e -c p t sin px

A-57

Chapter 14: Answers to Odd-Numbered Exercises 3 ln 2 2 29. 8

Section 14.1, pp. 767–768 1. 24 13. 14 25. 1

27. -1>10

25.

CHAPTER 14 3. 1 5. 16 7. 2 ln 2 1 9. (3>2) (5 - e) 11. 3>2 15. 0 17. 1>2 19. 2 ln 2 21. (ln 2) 2 23. 8>3 27. 22

Section 14.2, pp. 774–777 1.

31. 2p y

–2

y

–2

33.

2

1

– 3

(2, –2)

1

dx dy

L2 L0

35.

x

3

dy dx

y (1, 1)

1

4 y = 4 – 2x

x

x

L0 Lx2

y

4

t

3

s4 - yd>2

4

x = y2

u = sec t (/3, 2)

2

y = –p

y y = 2x

6

p

2 y=p

(–2, –2)

3.

u (–/3, 2)

2

y=x

(1, 2)

y = x2

–2 0

5.

x

1

7.

y

e

y

37.

y = ex

e

1

L1 Lln y

1

39.

x = sin 1

(1, e)

9

y = ex –1y

y = 9 – 4x 2 (1, 1)

1

x

2

9. (a) 0 … x … 2, x … y … 8 (b) 0 … y … 8, 0 … x … y 1>3 11. (a) 0 … x … 3, x 2 … y … 3x y (b) 0 … y … 9, … x … 2y 3 13. (a) 0 … x … 9, 0 … y … 2x (b) 0 … y … 3, y 2 … x … 9 15. (a) 0 … x … ln 3, e -x … y … 1 1 (b) … y … 1, - ln y … x … ln 3 3 17. (a) 0 … x … 1, x … y … 3 - 2x

x 0

x

1

0

1

41.

–1

1

23. e - 2

3 - y 2

e3

45.

x 2 + y2 = 1

0

1

(x + y) dy dx

y x = ey

3

(ln ln 8, ln 8)

(, )

y=x

x = ln y

ln ln 8

x

e

47. 2

y

x

1

e3

e - 2 2

(1, 1)

x

x

51. 2

y

y (1, 1)

1 x = y2

0

x

3

L1 Lln x

49.

y

xy dx dy

0 0

1

1

x

L0 Ley

y

x=1 ln 8

e

1

y (, )

43.

y = ln x

21. 8 ln 8 - 16 + e

y

0

1

3y dy dx y

(b) 0 … y … 1, 0 … x … y ´ 1 … y … 3, 0 … x …

21 - x2

L-1 L0

x

3 2

3

p2 19. + 2 2

16x dx dy

L0 L0

y

y e y=e

A 29 - y B>2

9

dx dy

x

1

0

2√ln 3 y = 2x

(√ln 3, 2√ln 3)

x=y

1

x 0

1

x 0

√ln 3

x

A-58

Chapter 14: Answers to Odd-Numbered Exercises 55. -2>3

53. 1>s80pd y

1

(0.5, 0.0625)

0.0625 y = x4

ex

ln 2

5.

y

L0

L0

7.

L0 Ly 2

(ln 2, 2)

y = ex

x+y=1

1

(1, 1) x = y2

1 x

0.5

0

–1

x = 2y – y2

x

1

0

2

9.

–1

59. 625> 12

61. 16

x

ln 2

0

3y

1 dx dy = 4

L0 Ly

65. 2s1 + ln 2d

63. 20

2

x

or 6

L0 Lx>3

1 dy dx +

2

L2 Lx>3

1 dy dx = 4

z

y

y

y = 1x 3

y=x

1

2

z = 1 – 1 x – 1y 3 2

2

y=2

y 2

2

3

1 x

71. p2

69. 1

73. -

3 32

2-x

1

sx 2 + y 2 d dy dx =

L0 Lx

11.

75.

20 23 9

2x

1

4 3

L1 Lx>2

2y

L0 Ly>2

3-x

2

1 dy dx +

L0 Lx>2

3-y

2

1 dx dy +

L1 Ly>2

1 dy dx =

3 2

1 dx dy =

3 2

or

y

y

y = 2x or x = 1 y 2

3 y=

2

x

6

x

3

77.

x

1

x–y=1

–x – y = 1

57. 4> 3 67.

1 3

dx dy =

y

y

–x + y = 1

2y - y 2

1

dy dx = 1

2 –

2

x

y = 1 x or x = 2y 2 x

1 y=

1 1

x

79. R is the set of points (x, y) such that x 2 + 2y 2 6 4. 81. No, by Fubini’s Theorem, the two orders of integration must give the same result. 85. 0.603 87. 0.233

1

x

2 3 y = 3 – x or x = 3 – y

15. 22 - 1

13. 12

y

y

y = cos x

Section 14.3, p. 779 2-x

2

1.

L0 L0 2-y

2

L0 L0

6

1

dy dx = 2

or

3.

-y 2

L-2 Ly - 2

dx dy =

y= x 2

9 2

0

y

dx dy = 2

(–1, 1)

1

17.

y –4 y=x+2

2

–2

(–4, –2)

–2

2

x

0

y 2

y = –2x (0, 0)

0

y = sin x

3 2

x

(/4, √2/2)

x

12 NOT TO SCALE

(–1, 2) x = –y2

y=2–x

0

√2 2

(12, 6)

y 2 = 3x

y=1–x 2 (2, –1) y=–x 2

x

4

x

A-59

Chapter 14: Answers to Odd-Numbered Exercises 19. (a) 0 (b) 4>p2 21. 8> 3 23. 40,000s1 - e -2 dln s7>2d L 43,329

24 - x 2

2

5.

8-x2-y2

1dz dx dy,

L-2 L-24 - x 2 Lx 2 + y 2

23.

8-y2

2

3p p p … u … 2p, 0 … r … 9 3. … u … , 0 … r … csc u 2 4 4 p 5. 0 … u … , 1 … r … 223 sec u; 6 p p … u … , 1 … r … 2 csc u 6 2 p p p 7. - … u … , 0 … r … 2 cos u 9. 2 2 2 11. 2p 13. 36 15. 2 - 23 17. (1 - ln 2) p 1.

21.

1 dz dx dy,

L-28 - z - y 2 28 - z

8

2

1 dx dz dy +

8-x2

2

L-2 L4

2 (1 + 22) 3

8

L-2 Ly 2 L-2z - y 2

28 - z - y 2

28 - z - x 2

L- 28 - z - x 2 28 - z

1z

4

1 dx dy dz +

L4 L-28 - z L-28 - z - y 2

1 dy dz dx +

1 dx dz dy,

2z - y 2

1 dx dy dz,

L0 L-1z L-2z - y 2 2

2z - x 2

4

L-2 L -2z - x 2 x2 L

28 - z - x 2

L4 L-28 - z L-28 - z - x 2

2z - y 2

4

1z

4

1 dy dx dz +

1 dy dz dx,

2z - x 2

L0 L-1z L- 2z - x 2

1 dy dx dz.

The value of all six integrals is 16p. y = √1 – x2 or x = √1 – y2

1

7. 1

9. 6

15. 7> 6

11.

17. 0 1 - x2

1

21. (a) x

1

1

21 - x2

y

L0 L0

L0 L0

23. 2> 3

x

x

2

y 2 (x 2 + y 2) dy dx or

2

L0 Ly

y 2 sx 2 + y 2 d dx dy

Section 14.5, pp. 792–795 1. 1> 6

L0 L0 1

L0 L0 2

2 - 2x

L0 3 - 3x

2

3 - 3y>2

3

dy dz dx,

2 - 2z>3

L0 L0

1 - y>2 - z>3

dx dz dy,

L0

1 - y>2

L0 L0

2 - 2x - 2z>3

L0

L0 L0 3

3 - 3x - 3y>2

dz dy dx,

13. 18

1-z

1-z

2y

(b)

2y

L-2y

21 - z

1

dy dz dx

(d)

dy dx dz

L0 L-21 - z Lx2 1

dx dy dz

1-z

1-y

L0 L0

2y

L-2y

dx dz dy

1-y

L0 L-2y L0 25. 20> 3

dz dx dy 27. 1

29. 16> 3

31. 8p -

33. 2 35. 4p 37. 31> 3 39. 1 41. 2 sin 4 45. a = 3 or a = 13>3 47. The domain is the set of all points (x, y, z) such that 4x 2 + 4y 2 + z 2 … 4.

2a 2a 27. 2sp - 1d 29. 12p 31. s3p>8d + 1 33. 35. 3 3 2p 5p 4 + 37. 2p A 2 - 2e B 39. 41. (a) (b) 1 3 8 2 1 43. p ln 4, no 45. sa 2 + 2h 2 d 2

1

5 (2 - 23) 4 p 1 19. 2 8

Lx 2

L0 L0 1

(e) y=x

2

2

xy dx dy

x=2

2

L-1 L0 1

(c)

21 - y2

1

xy dy dx or

L0 L0

3.

28 - z - y 2

L-2 L4

y

25.

8-x2-y2

L-2 L-24 - y 2 Lx 2 + y 2

Section 14.4, pp. 784–786

19. (2 ln 2 - 1) (p>2)

24 - y 2

2

1 - y>2 - z>3

dx dy dz. L0 L0 L0 The value of all six integrals is 1.

L0

Section 14.6, pp. 800–802 5>14, y = 38>35 3. x = 64>35, y = 5>7 y = 4a>s3pd 7. Ix = Iy = 4p, I0 = 8p -1, y = 1>4 11. Ix = 64>105 3>8, y = 17>16 11>3, y = 14>27, Iy = 432 0, y = 13>31, Iy = 7>5 0, y = 7>10; Ix = 9>10, Iy = 3>10, I0 = 6>5 M 2 M 2 M 2 21. Ix = sb + c 2 d, Iy = sa + c 2 d, Iz = sa + b 2 d 3 3 3 23. x = y = 0, z = 12>5, Ix = 7904>105 L 75.28, 1. 5. 9. 13. 15. 17. 19.

x x x x x x x

= = = = = = =

Iy = 4832>63 L 76.70, Iz = 256>45 L 5.69

3 - 3x - 3y>2

dz dx dy,

L0 1 - z>3

32 3 43. 4

2 - 2x - 2z>3

dy dx dz,

25. (a) x = y = 0, z = 8>3 27. IL = 1386 29. (a) 4> 3 31. (a) 5> 2

(b) c = 2 22

(b) x = 4>5, y = z = 2>5 (b) x = y = z = 8>15 (c) Ix = Iy = Iz = 11>6

33. 3 37. (a) Ic.m. = (b) IL =

abcsa 2 + b 2 d a2 + b2 , Rc.m. = 12 B 12

abcsa 2 + 7b 2 d a 2 + 7b 2 , RL = 3 3 B

A-60

Chapter 14: Answers to Odd-Numbered Exercises

Section 14.7, pp. 810–813 1.

4p A 22 - 1 B

17p 5

3.

3

43. 8p>3

5. p A 6 22 - 8 B

7.

3p 10

9. p>3 2p

11. (a)

23

2p

L0 L0

p>2

L-p>2 L0

ƒsr, u, zd dz r dr du 4

ƒsr, u, zd dz r dr du

L0

2 - r sin u

sec u

ƒsr, u, zd dz r dr du

L0

29. a

27. 2p p>6

2p

31. (a)

ƒsr, u, zd dz r dr du

1 + cos u

L0

25. 5p

r du dz dr

L0

L-p>2 L1 L0

21. p2

L0 L0

8 - 5 22 bp 2

z ( f(z), , z) z

L0

csc f

r2 sin f dr df du r2 sin f df dr du +

L0 L1 Lp>6

p>2

1

Section 14.8, pp. 821–823

r2 sin f df dr du

L0 L0 Lp>6 p>2

u + y y - 2u 1 ,y = ; 3 3 3 (b) Triangular region with boundaries u = 0, y = 0, and u + y = 3 1 1 1 3. (a) x = s2u - yd, y = s3y - ud; 5 10 10 (b) Triangular region with boundaries 3y = u, y = 2u, and 3u + y = 10 1. (a) x =

2

31p r sin f dr df du = 33. 6 L0 L0 Lcos f 2p p 1 - cos f 8p r2 sin f dr df du = 35. 3 L0 L0 L0 2p p>2 2 cos f p r2 sin f dr df du = 37. 3 L0 Lp>4 L0 2

p>2

39. (a) 8

L0

p>2

L0 p>2

(b) 8

L0

dz dy dx

17. (a) `

cos y sin y

-u sin y ` = u cos2 y + u sin2 y = u u cos y

r2 sin f dr df du

(b) `

sin y cos y

u cos y ` = -u sin2 y - u cos2 y = -u -u sin y

r dz dr du

24 - x 2

2p

24 - x 2 - y 2

L0 p>3

L0 L0

52 2u su + yd y du dy = 8 + ln 2 3 L1 L1 pabsa 2 + b 2 d 225 3 1 a1 + 2 b L 0.4687 15. 11. 13. 4 3 16 e

24 - r 2

L0 L0

2p

2

Lsec f 23

24 - r 2

L0 L0

L1

23

23 - x 2

3

9.

r2 sin f dr df du

L0

(c) 8 L0 L0

2

7. 64> 5

2

2

2

y

r2 sin f df dr du +

L0 L0 L0

2p

+ x

p>6

2

2p

sin-1s1>rd

2

2p

(c)

( f(z), + , z) f(z)

f(z)

r sin f dr df du +

p>2

2p

(b)

5 p (b) sx, y, zd = a0, 0, b, Iz = 6 14 3M 81. pR 3 85. The surface’s equation r = ƒszd tells us that the point sr, u, zd = sƒszd, u, zd will lie on the surface for all u. In particular, sƒszd, u + p, zd lies on the surface whenever sƒszd, u, zd lies on the surface, so the surface is symmetric with respect to the zaxis.

2

L0 Lp>6 L0

41. (a)

23. p>3

4p A 8 - 323 B

2

2p

(b)

r dr dz du

4 - r sin u

2 sin u

p>4

19.

L0 L 0 23 L

24 - z

3r 2

L0

L0 L0

2

2p

L0

cos u

p>2

17.

L0

L0 L0

p

2p

r dr dz du +

2

3p - 4 2pa 3 49. 51. 5p>3 18 3 4 A 222 - 1 B p 55. 57. 16p 59. 5p>2 3 47.

63. 2> 3 65. 3> 4 3 67. x = y = 0, z = 3>8 69. sx, y, zd = s0, 0, 3>8d a 4 hp 71. x = y = 0, z = 5>6 73. Ix = p>4 75. 10 p 4 77. (a) sx, y, zd = a0, 0, b, Iz = 5 12 61.

1

24 - r 2

1

(c)

15.

r dz dr du

L0 L0 L0

(b)

13.

24 - r 2

1

53. p>2

45. 9> 4

L-23 L-23 - x 2 L1

21. 12

r dz dr du 24 - x 2 - y 2

dz dy dx

(d) 5p>3

23.

a 2b 2c 2 6

Chapter 15: Answers to Odd-Numbered Exercises

Practice Exercises, pp. 823–825

Additional and Advanced Exercises, pp. 825–827

3. 9> 2

1. 9e - 9

6 - x2

2

y

1. (a)

t

10 (1/10, 10)

L-3 Lx (c) 125> 4

3 2

3. 2p

y = 1x

A-61

–3

3

2

x 2 dy dx

6 - x2

L-3 Lx

x2

L0

dz dy dx

7. (a) Hole radius = 1, sphere radius = 2

5. 3p>2

s

(b) 423p 2

(b)

9. p>4

b 11. ln a a b

4 15. 1> 23

2

s + 4t = 9

0

1

x NOT TO SCALE

4 - x2

0

5.

b 17. Mass = a 2 cos-1 a a b - b2a 2 - b 2,

(1, 1)

1

L-2 L2x + 4

s1>2d29 - x2

3

4 dy dx = 3

7.

L-3 L0

y = 2x + 4

4

3 2

x = –√4 – y

–3

9. sin 4

ln 17 4

11.

p - 2 4

23. 0 22

31. (a)

15. 4> 3

17. 1> 4

25. 8> 35

27. p>2

29.

22 - y 2

p>4

2p

L0 L0

21. (b) 1

(c) 0 27. 2p c

1 1 22 - a b d 3 3 2

x

L0 L0

L0

37. (a)

L1

23 - x 2

L0

8p A 4 22 - 5 B

24 - x 2 - y 2

L1

8p A 4 22 - 5 B

15p + 32 ,y = 0 6p + 48 y r = 1 + cos

r=1

p 3

z 2 xy dz dy dx

47. M = 4, Mx = 0, My = 0 49. x =

(b) 1

c.m. 1 ≈ 1.18

–1

1. Graph (c)

2

x

3. Graph (g) 5. Graph (d) 7. Graph (f) 13 1 11. 13. 3214 15. A 525 + 9 B 2 6

b 17. 23 ln a a b

19. (a) 4 25

21.

15 16 se - e 64 d 32

25.

1 3>2 s5 + 722 - 1d 6

31.

1 s173>2 - 1) 6

2

(b) 3 3 5 5 8pdsb - a d 39. Iz = 15 1 41. x = y = 43. I0 = 104 2 - ln 4

51. (a) x =

Section 15.1, pp. 832–834

z xy dz dy dx

L0 L21 - x 2 L1 +

CHAPTER 15

9. 22 (c) 2p A 8 - 422 B

3 r2 sin f dr df du

24 - x 2 - y 2

23

2s31 - 35>2 d 3

3 dz dx dy

r2 sin f dr df du =

23 - x 2

1

19. p

24 - x 2 - y 2

2

L0 sec f

p>4

2p

35.

13. 4> 3

L-22 L-22 - y 2 L2x 2 + y 2

(b) 33.

3

1 a2b2 - 1d se ab

25. h = 220 in., h = 260 in.

x 2 + 4y2 = 9

0

a4 b b3 2 b3 cos-1 a a b 2a 2 - b 2 sa - b 2 d3>2 2 2 6

x

–2

21.

19.

y

y

I0 =

9 y dy dx = 2

23.

(b)

1 (173>2 - 1) 12

1 s40 3>2 - 133>2 d 27 27.

1025 - 2 3

29. 8

33. 222 - 1

35. (a) 422 - 2 39. (a) Iz = 2p22d

(b) 22 + ln A 1 + 22 B (b) Iz = 4p22d

37. Iz = 2pda 3

41. Ix = 2p - 2

Section 15.2, pp. 844–847 45. Ix = 2d 3 23 p ,y = 0

1. §ƒ = -sxi + yj + zkdsx 2 + y 2 + z 2 d-3>2 2y 2x 3. §g = - a 2 bi - a 2 bj + e zk x + y2 x + y2 ky kx 5. F = - 2 i - 2 j, any k 7 0 sx + y 2 d3>2 sx + y 2 d3>2 7. (a) 9> 2 (b) 13> 3 (c) 9> 2 9. (a) 1> 3 (b) -1>5 (c) 0 11. (a) 2 (b) 3> 2 (c) 1> 2 13. -15>2 15. 36 17. (a) -5>6 (b) 0 (c) -7>12 19. 1> 2 21. -p 23. 69> 4 25. -39>2 27. 25> 6 29. (a) Circ1 = 0, circ2 = 2p, flux1 = 2p, flux2 = 0 (b) Circ1 = 0, circ2 = 8p, flux1 = 8p, flux2 = 0 31. Circ = 0, flux = a 2p 33. Circ = a 2p, flux = 0 p 35. (a) (b) 0 (c) 1 37. (a) 32 (b) 32 (c) 32 2

A-62

Chapter 15: Answers to Odd-Numbered Exercises

39.

13. (a) rsr, ud = sr cos udi + sr sin udj + s1 - r cos u - r sin udk, 0 … r … 3, 0 … u … 2p (b) rsu, yd = s1 - u cos y - u sin ydi + su cos ydj + su sin ydk, 0 … u … 3, 0 … y … 2p 15. rsu, yd = s4 cos2 ydi + uj + s4 cos y sin ydk, 0 … u … 3, -sp>2d … y … sp>2d; another way: rsu, yd = s2 + 2 cos ydi + uj + s2 sin ydk, 0 … u … 3, 0 … y … 2p

y 2 x2 + y2 = 4 0

2

x

2p

17.

L0 L0 2p

41. (a) G = -yi + xj xi + yj 43. F = 2x 2 + y 2

19.

(b) G = 2x 2 + y 2 F 47. 48

49. p

51. 0

53.

1 2

25 p25 r dr du = 2 2

3

2p

r25 dr du = 8p25

21.

4

L0 L1 2p 1 525 - 1 B A p u24u 2 + 1 du dy = 23. 6 L0 L0 L0 L1

1 du dy = 6p

2 sin f df du = A 4 + 222 B p L0 Lp>4 27. 29. 2p

Section 15.3, pp. 856–858

1

p

25.

1. Conservative

3. Not conservative 3y 2 2 7. ƒsx, y, zd = x + + 2z 2 + C 2 9. ƒsx, y, zd = xe y + 2z + C

5. Not conservative

z

z z = √x 2 + y2

11. ƒsx, y, zd = x ln x - x + tan sx + yd + 15. -16 17. 1 19. 9 ln 2 x2 - 1 27. F = § a y b 29. (a) 1 (b) 1

13. 49

1 ln s y 2 + z 2 d + C 2 21. 0 23. -3

x 2 + (y – 3)2 = 9 √3x + y = 9 x + y – √2z = 0

(√2, √2, 2)

(c) 1

31. (a) 2 (b) 2 33. (a) c = b = 2a (b) c = b = 2 35. It does not matter what path you use. The work will be the same on any path because the field is conservative. 37. The force F is conservative because all partial derivatives of M, N, and P are zero. ƒsx, y, zd = ax + by + cz + C; A = sxa, ya, zad and B = sxb, yb, zb). Therefore, 1 F # dr = ƒsBd - ƒsAd = asxb - xad + bs yb - yad + cszb - zad = 1 F # AB.

x x

6

y

y 3√3 , 9/2, 0 2

p

2p

33. (b) A =

L0 L0

[a 2b 2 sin2 f cos2 f + b 2c 2 cos4 f cos2 u +

a 2c 2 cos4 f sin2 u]1>2 df du 35. x0 x + y0 y = 25

Section 15.4, pp. 867–869 1. 5. 9. 13. 17. 29.

Flux = 0, circ = 2pa 2 3. Flux = -pa 2, circ = 0 Flux = 2, circ = 0 7. Flux = -9, circ = 9 Flux = -11>60, circ = -7>60 11. Flux = 64>9, circ = 0 Flux = 1>2, circ = 1>2 15. Flux = 1>5, circ = -1>12 0 19. 2> 33 21. 0 23. -16p 25. pa 2 27. 3p>8 (a) 0 if C is traversed counterclockwise (b) sh - kdsarea of the regiond 39. (a) 0

= … = …

Section 15.6, pp. 887–889 3

1.

6

x ds =

2

u24u 2 + 1 du dy =

L0 L0

S

Section 15.5, pp. 878–880 1. rsr, ud 0 … u 3. rsr, ud 0 … u

37. 13p>3 39. 4 41. 626 - 222 p 43. p2c 2 + 1 45. A 17217 - 525 B 47. 3 + 2 ln 2 6 p 2 49. A 13213 - 1 B 51. 5p22 53. 3 A 525 - 1 B 6

p

2p

sr cos udi + sr sin udj + r k, 0 … r … 2, 2p sr cos udi + sr sin udj + sr>2dk, 0 … r … 6, p>2 2

5. rsr, ud = sr cos udi + sr sin udj + 29 - r 2 k, 0 … r … 3 22>2, 0 … u … 2p; Also: rsf, ud = s3 sin f cos udi + s3 sin f sin udj + s3 cos fdk, 0 … f … p>4, 0 … u … 2p 7. rsf, ud = A 23 sin f cos u B i + A 23 sin f sin u B j +

A 23 cos f B k, p>3 … f … 2p>3, 0 … u … 2p

9. rsx, yd = xi + yj + s4 - y dk, 0 … x … 2, -2 … y … 2 11. rsu, yd = ui + s3 cos ydj + s3 sin ydk, 0 … u … 3, 0 … y … 2p 2

3.

6

x 2 ds =

S

L0 L0 1

5.

6 S

z ds =

17 217 - 1 4

sin3 f cos2 u df du =

4p 3

1

L0 L0

s4 - u - yd23 dy du = 323

(for x = u, y = y) 1

7.

6

x 2 25 - 4z ds =

S

L0 L0 1

u24u 2 + 1 dy du = 9. 9a

3

2p

u 2 cos2 y # 24u 2 + 1 #

2p

u 3s4u 2 + 1d cos2 y dy du =

L0 L0 abc 11. sab + ac + bcd 4

1 15. A 22 + 626 B 30

17. 26>30

13. 2

11p 12

A-63

Appendices: Answers to Odd-Numbered Exercises

19. -32

21.

27. -73p>6

pa 3 6

23. 13a 4>6

29. 18

pa 31. 6

3

25. 2p>3

APPENDICES

pa 2 33. 4

Appendix 1, p. AP-6

pa 3 35. 37. -32 39. -4 41. 3a 4 2 15p22 a a a 14 d 43. a , , b 45. sx, y, zd = a0, 0, b , Iz = 2 2 2 9 2 8p 4 20p 4 a d (b) a d 47. (a) 3 3

1. 0.1, 0.2, 0.3, 0.8, 0.9 or 1 3. x 6 -2

9. 7>6, 25>6 13. 0 … z … 10

–2

1. 4p 3. -5>6 5. 0 7. -6p 9. 2pa 15. -p>4 17. -15p 25. 16Iy + 16Ix

2

13. 12p

1. 0 3. 0 5. -16 7. -8p 9. 3p 11. -40>3 13. 12p 15. 12p A 4 22 - 1 B 21. The integral’s value never exceeds the surface area of S.

t

4

0

15. s - q , -2] ´ [2, q d –2

Section 15.8, pp. 909–911

x

–1/3

11. -2 … t … 4

Section 15.7, pp. 898–900

1 3

x

–2

7. 3, -3

5. x … -

17. s - q , -3] ´ [1, q d s

2

z

10

–3

r

1

19. s -3, -2d ´ s2, 3d 21. (0, 1) 23. (- q , 1] 27. The graph of ƒ x ƒ + ƒ y ƒ … 1 is the interior and boundary of the “diamond-shaped” region. y 1 x y 1

Practice Exercises, pp. 911–914 1. Path 1: 2 23 ; path 2: 1 + 3 22 7. 8p sin s1d 15. 19.

21. 23. 25. 31.

9. 0

11. p 23

3. 4a

2

5. 0

13. 2p a1 -

1 22

b

abc 1 1 1 17. 50 + 2 + 2 2 A a2 b c rsf, ud = s6 sin f cos udi + s6 sin f sin udj + s6 cos fdk, p 2p … f … , 0 … u … 2p 6 3 rsr, ud = sr cos udi + sr sin udj + s1 + rdk, 0 … r … 2, 0 … u … 2p rsu, yd = su cos ydi + 2u 2j + su sin ydk, 0 … u … 1, 0 … y … p 27. p C 22 + ln A 1 + 22 B D 29. Conservative 26 Not conservative 33. ƒsx, y, zd = y 2 + yz + 2x + z

35. Path 1: 2; path 2: 8> 3 39. 0

37. (a) 1 - e - 2p

41. (a) 4 22 - 2

(b) 22 + ln A 1 + 22 B

723 3 ,I = 2 z 3

49. Flux: 3> 2; circ: -1>2 57. 0

47. sx, y, zd = s0, 0, 49>12d, Iz = 640p 53. 3

55.

–1

Appendix 3, pp. AP-16–AP-18 1. 2, -4; 2 25 1 5. m⊥ = 3

3. Unit circle

y 2

A(–1, 2) y = 3x + 5

1

Slope = 3

x –2

–1

B(– 2, –1)

0 –1

7. (a) x = -1 (b) y = 4>3 9. y = -x 5 x 11. y = - x + 6 13. y = 4x + 4 15. y = - + 12 4 2 17. x-intercept = 23, y-intercept = - 22 y

2p A 7 - 8 22 B 3

0

59. p

1

Additional and Advanced Exercises, pp. 914–916

x

2

√2 x – √3 y = √6

–1

1. 6p 3. 2> 3 5. (a) Fsx, y, zd = zi + xj + yk (b) Fsx, y, zd = zi + yk (c) Fsx, y, zd = zi 16pR 3 7. 9. a = 2, b = 1. The minimum flux is -4. 3 16 11. (b) g 3 16 (c) Work = a gxy dsb y = g xy 2 ds = g 3 LC LC 4 13. (c) 19. False if F = yi + xj pw 3

x

1

(b) 1 - e - 2p

16 2 232 64 56 43. sx, y, zd = a1, , b ; Ix = ,I = ,I = 15 3 45 y 15 z 9 45. z =

–1

–2

19. s3, -3d 21. x 2 + s y - 2d2 = 4

23. A x + 23 B 2 + s y + 2d2 = 4

y

y ⎛ – √3, 0⎞ ⎝ ⎠

(0 , 4)

x

–4

(0, –1)

C(0 , 2) C ⎛⎝–√3, – 2⎞⎠

(0, 0) –2

–1

1

2

x

(0, –3) –4

A-64

Appendices: Answers to Odd-Numbered Exercises

25. x 2 + s y - 3>2d2 = 25>4

27. y

y (0, 4)

y = x 2 – 2x – 3

(–1, 0)

4 3

0 C(0, 3/2)

1

(2, 0)

(– 2, 0) 1 2 –1 (0, –1) –2 2 2 x + (y – 3/2) = 25/4 –2 –1

3

x

4

1

2

(3, 0)

x

x2 + y 2 = 1, 2

9.

V A ; 22, 0 B

Fs ;1, 0d,

11.

y

x = –3

y=2

2

3

2

(0, –3)

–3

0

x

F(3, 0)

x

0 F(0, –2)

V(1, –4)

29.

y

y 2 = 12x

Axis: x = 1

2

7.

x 2 = – 8y

y V(–3, 4)

2

4

y = –x – 6x – 5

13.

15.

y

y

y = 4x 2

–3 (–1, 0)

1 4

x

0

Axis: x = –3

–6

1 6

x = –3y 2

(–5, 0)

x=

⎛ 1⎛ F 0, ⎝ 16 ⎝ 0

⎛ 1 ⎛ F – ,0 ⎝ 12 ⎝ 0

(0, –5)

(–6, –5)

17.

–

1 16

1 6

19.

y

y

2 x2 y + =1 25 16

4

√2

x2 +

y2 =1 2

F1

1

F2

F1 –5

x

1 12

x

1/4 directrix y = –

31. Exterior points of a circle of radius 27 , centered at the origin 33. The washer between the circles x 2 + y 2 = 1 and x 2 + y 2 = 4 (points with distance from the origin between 1 and 2) 35. sx + 2d2 + s y - 1d2 6 6 2 1 2 1 37. a , b , a,b 25 25 25 25 1 1 1 1 39. a, - b, a ,- b 3 3 23 23 41. (a) L -2.5 degrees> inch (b) L -16.1 degrees> inch (c) L -8.3 degrees> inch 43. 5.97 atm 45. Yes: C = F = -40°

1 12

–3

0

3

x

5

0

x

1

–1

F2

–4

C C=F

– 40

32 C=

F

21.

51. k = -8,

–40

x2 + =1 2 3

–1

√2 F2

Appendix 4, pp. AP-24–AP-26 1. y 2 = 8x,

Fs2, 0d , directrix: x = -2

3. x 2 = -6y, Fs0, - 3>2d , directrix: y = 3>2 y2 x2 = 1, F A ; 213, 0 B , Vs ;2, 0d , 5. 4 9 3 asymptotes: y = ; x 2

25.

√6

2 x2 y + =1 9 6

F1

0

k = 1>2

y

y2

√3

5 – (F 32) 9

1 (– 40, – 40)

23.

y

y2 x2 + = 1 4 2

x

F2

F1 –√3

0

√3

3

x

Appendices: Answers to Odd-Numbered Exercises 27. Asymptotes: y = ;x

29. Asymptotes: y = ;x

y

y

x2 – y2 = 1 F1

F2

–√2

√2

y2 x2 – =1 8 8

2

2√2

x

43. (a) Center: (2, 0); foci: (7, 0) and (-3, 0) ; vertices: (6, 0) and 3 (-2, 0) ; asymptotes: y = ; (x - 2) 4 (b) y

F1

4

(x – 2) 16

3 y = – (x – 2) 4

x

y=

2

–

3 (x – 2) 4

(–3, 0)

(7, 0) 0

2

33. Asymptotes: y = ;x>2

y

y y2 – =1 2 8

√10

x2

x

√10

–√10

x

–√10

35. y 2 - x 2 = 1

37.

F1

y2 x2 – =1 2 8

√2

F2

√2

F2

y2 x2 = 1 9 16

39. (a) Vertex: (1, -2) ; focus: (3, -2) ; directrix: x = -1 (b) y ( y + 2) 2 = 8(x – 1)

2 0

x

1 2 3

–2

F(3, –2)

45. ( y + 3) 2 = 4(x + 2), V(-2, -3), F(-1, -3) , directrix: x = -3 47. (x - 1) 2 = 8( y + 7), V(1, -7), F(1, -5), directrix: y = -9 (x + 2) 2 ( y + 1) 2 49. + = 1, F A -2, ; 23 - 1 B , 6 9 V(-2, ;3 - 1), C(-2, -1) 51.

53.

(x - 2) 2 ( y - 3) 2 + = 1, F(3, 3) and F(1, 3), 3 2 V A ; 23 + 2, 3 B , C(2, 3) (x - 2) 2 ( y - 2) 2 = 1, 5 4

F(5, 2) and F(-1, 2),

25 (x - 2) 2 55. ( y + 1) 2 - (x + 1) 2 = 1, C(-1, -1), F A -1, 22 - 1 B and F A -1, - 22 - 1 B , V(-1, 0) and V(-1, -2); asymptotes ( y + 1) = ;(x + 1) V(4, 2) and V(0, 2); asymptotes: ( y - 2) = ;

a = 4

57. C(-2, 0),

59. V(-1, 1),

F(-1, 0)

61. Ellipse:

41. (a) Foci: A 4 ; 27, 3 B ; vertices: (8, 3) and (0, 3); center: (4, 3) (b) y

(x - 1) 2 + ( y - 1) 2 = 1, C(1, 1), F(2, 1) and 2 F(0, 1), V A 22 + 1, 1 B and V A - 22 + 1, 1 B

63. Ellipse:

65. Hyperbola: (x - 1) 2 - ( y - 2) 2 = 1,

C(4, 3) (0, 3)

(8, 3) F2(4 + √7, 3) 4

C(1, 2),

F A 1 + 22, 2 B and F A 1 - 22, 2 B , V(2, 2) and V(0, 2); asymptotes: (y - 2) = ;(x - 1)

(x – 4) 2 ( y – 3) 2 + =1 16 9 F1(4 – √7, 3)

0

C(2, 2),

(x + 2) 2 + y 2 = 1, C(-2, 0), F(0, 0) and 5 F(-4, 0), V A 25 - 2, 0 B and V A - 25 - 2, 0 B

V(1, –2) –4

6

x

(6, 0)

(–2, 0)

F1

y =1 9

F2

–4

31. Asymptotes: y = ;2x

A-65

F(0, 0), 8

x

( y - 3) 2 x2 = 1, C(0, 3), F(0, 6) and 6 3 V A 0, 26 + 3 B and V A 0, - 26 + 3 B ;

67. Hyperbola:

asymptotes: y = 22x + 3 or y = - 22x + 3

A-66

Appendices: Answers to Odd-Numbered Exercises

Appendix 8, pp. AP-42–AP-43 1. (a) (14, 8) (b) s -1, 8d (c) s0, -5d 3. (a) By reflecting z across the real axis (b) By reflecting z across the imaginary axis (c) By reflecting z in the real axis and then multiplying the length of the vector by 1> ƒ z ƒ 2 5. (a) Points on the circle x 2 + y 2 = 4 (b) Points inside the circle x 2 + y 2 = 4 (c) Points outside the circle x 2 + y 2 = 4

7. Points on a circle of radius 1, center s -1, 0d 9. Points on the line y = -x 11. 4e 2pi>3 13. 1e 2pi>3 15. cos4 u - 6 cos2 u sin2 u + sin4 u 19. 2i, - 23 - i, 23 - i 23. 1 ; 23i, -1 ; 23i

21.

17. 1, -

23 1 ; i 2 2

26 22 26 22 ; ; i, i 2 2 2 2

INDEX

a, logarithms with base, 408–409 Abscissa, AP-10 Absolute change, 203 Absolute convergence, 524–525 Absolute Convergence Test, 525 Absolute extrema, finding, 218–219 Absolute (global) maximum, 214–216, 743–745 Absolute (global) minimum, 214–216, 743–745 Absolute value definition of, AP-4–AP-6, AP-38 properties of, AP-5 Absolute value function derivative of, 160 as piecewise-defined function, 5 Acceleration definition of, 141 derivative of (jerk), 141, 142 as derivative of velocity, 141, 142–143 in free fall, 143 free fall and, 142 normal component of, 674–678 in polar coordinates, 679–682 in space, 653 tangential component of, 674–678 velocity and position from, 226 Addition of functions, 14–15 of vectors, 609–610 Addition formulas, trigonometric, 25 Additivity double integrals and, 773 line integrals and, 830 Additivity Rule, for definite integrals, 309 Albert of Saxony, 513 Algebra, Fundamental Theorem of, AP-41–AP-42 Algebra operations, vector, 609–611 Algebra rules for finite sums, 300–301 for gradients, 728 for natural logarithm, 43 Algebra systems, computer. See Computer algebra systems (CAS) Algebraic functions, 10 Alternating series definition of, 522–524 harmonic, 522–523

Alternating Series Estimation Theorem, 524, 547 Alternating Series Test, 522 Angle convention, 22 Angle of elevation, 660 Angle of inclination, AP-11 Angles direction, 622 between planes, 636 in standard position, 22 in trigonometric functions, 21–22 between vectors, 616–618 Angular velocity of rotation, 895 Antiderivative linearity rules, 273 Antiderivatives definition of, 271 difference rule, 273 finding, 271–274 and indefinite integrals, 276–277 motion and, 274–275 of vector function, 657 Antidifferentiation, 271 Applied optimization of area of rectangle, 257 examples from economics, 259–260 examples from mathematics and physics, 257–259 solving problems, 255 using least material, 256–257 volume of can, 256 Approximations center of, 196 differential, error in, 201–202 by differentials, 196 error analysis of, 465–468 linear, error formula for, 735, AP-49–AP-50 Newton’s Method for roots, 266–268 for roots and powers, 198 by Simpson’s Rule, 465–468, 467–468 standard linear, 196, 734 tangent line, 196, 734 by Taylor polynomials, 540 trapezoidal, 461–463 by Trapezoidal Rule, 467–468 using parabolas, 463–465 Arbitrary constant, 272

Arc length along curve in space, 664–665 differential formula for, 375–376 discontinuities in dy/dx, 374–375 of a function, 375, 666 length of a curve y = f (x), 372–374 and line integrals, 828–829 Arc length differential, 575–576 Arc length formula, 374–375, 573, 664 Arc length parameter, 665 Arccosine function defining, 46–48 identities involving, 48–49 Arcsecant, 180 Arcsine function defining, 46–48 identities involving, 48–49 Arctangent, 180–181, 553–554 Area of bounded regions in plane, 777–778 cross-sectional, 353, 354 under curve or graph, 311 between curves, 338–341 as definite integral, 289 definition of, 312 by double integration, 777–779 enclosed by astroid, 571–572 and estimating with finite sums, 289–296 finite approximations for, 291 under graph of nonnegative function, 296 by Green’s Theorem, 868 infinite, 472 of parallelogram, 625, 872 in polar coordinates, 782 of smooth surface, 872 surfaces and, 379, 576–577, 870–877 of surfaces of revolution, 378–381, 576–577 total, 290, 323–325 Area differential, 588 Argand diagrams, AP-37–AP-38 Argument, AP-38 Arrow diagram for a function, 2 Associative laws, AP-31 Astroid area enclosed by, 571–572 length of, 574

I-1

I-2

Index

Asymptotes of graphs, 97–108 in graphs of rational functions, 9 horizontal, 97, 99–102, 106 of hyperbolas, AP-23–AP-24 oblique or slant, 102 vertical, 97, 105–106 vertical, integrands with, 474–475 Average rates of change, 54, 56 Average speed definition of, 52 moving bodies and, 52–54 over short time intervals, 53 Average value of continuous functions, 312–313 of multivariable functions, 778–779, 791–792 of nonnegative continuous functions, 295–296 Average velocity, 140 ax definition of, 35, 407 derivative of, 132, 173–175, 409 integral of, 409 inverse equations for, 408 laws of exponents, 407 Axis(es) coordinate, AP-10 of ellipse, AP-20 moments of inertia about, 831, 885 of parabola, AP-15, AP-19 slicing and rotation about, volumes by, 353–360 spin around, 861–863 Base, a of cylinder, 353 of exponential function, 34, 133, 407 logarithmic functions with, 42–43 logarithms with, 174–175, 408–409 Bernoulli, Daniel, 215 Bernoulli, Johann, 159, 246 Binomial series, 550–552 Binormal vector, 678 Birkhoff, George David, 331 Bolzano, Bernard, 141 Boundary points finding absolute maximum and minimum values, 743 for regions in plane, 688 for regions in space, 690 Bounded intervals, 6 Bounded regions absolute maxima and minima on, 743–745 areas of, in plane, 777–779 definition of, 688 Bounded sequences, 493–495 Box product, 627–628 Brachistochrones, 567 Branch diagram(s), for multivariable Chain rules, 715, 716, 717, 718, 719 Cable, hanging, 11, 427 Calculators to estimate limits, 64–66 graphing with, 29–33 Carbon-14 decay, 416

Cardioid definition of, 585 graphing of, 588 length of, 589–590 in polar coordinates, area enclosed by, 588 Cartesian coordinate systems, 602–605 Cartesian coordinates conversion to/from polar coordinates, 782–784 in plane, AP-10 related to cylindrical and spherical coordinates, 806 related to cylindrical coordinates, 803 related to polar coordinates, 579–582 three-dimensional. See Three-dimensional coordinate systems triple integrals in, 786–792 Cartesian integrals, changing into polar integrals, 782–784 CAS. See Computer algebra systems CAST rule, 23 Catenary, 11, 427 Cauchy, Augustin-Louis, 251 Cauchy’s Mean Value Theorem, 251–252 Cavalieri, Bonaventura, 355 Cavalieri’s principle, 355 Center of curvature, for plane curves, 671 Center of linear approximation, 196 Center of mass centroid, 395–396 coordinates of, 391, 831, 885 definition of, 390 moments and, 389–396, 795–800 of solid, 796 of thin flat plate, 391–394 of thin shell, 886–887 of wire or spring, 831 Centroids, 395–396, 796–797 Chain Rule and curvature function, 670 derivation of Second Derivative Test and, AP-48 derivatives of composite function, 156–161 derivatives of exponential functions, 174, 175, 406, 408 derivatives of inverse functions, 172 derivatives of inverse trigonometric functions, 182, 183 for differentiable parametrized curves, 570 and directional derivatives, 725 for functions of three variables, 717–718 for functions of two variables, 714–716 for implicit differentiation, 719, 875 for inverse hyperbolic functions, 424 and motion in polar coordinates, 679 “outside-inside” rule and, 158–159 for partial derivatives, 714–720 with powers of function, 159–162 proof of, 157–158, 202 related rates equations, 187, 188 repeated use of, 159 Substitution Rule and, 328–333 for two independent variables and three intermediate variables, 717–718 for vector functions, 654–655

Change of base in a logarithm, 174, 408 estimating, in special direction, 733 exponential, 411–412 rates of, 52–57, 117–118, 139–145 sensitivity to, 145, 203 Charge, electrical, 906 Circle of curvature, for plane curves, 671–672 Circles length of, 573 osculating, 671 in plane, AP-13–AP-15 polar equation for, 596–597 standard Cartesian equation for, AP-14 Circulation, flux versus, 843 Circulation density, 861–863 Circulation for velocity fields, 841–842 Clairaut, Alexis, 709 Clairaut’s Theorem, 709 Closed curve, 842 Closed region, 688, 690 Coefficients binomial, 551 determination for partial fractions, 453–454 of polynomial, 8–9 of power series, 529 undetermined, 449 Combining functions, 14–21 Combining series, 503–504 Common functions, 7–11 Common logarithm function, 43 Commutativity laws, AP-31 Comparison tests for convergence of improper integrals, 477–478 for convergence of series, 512–515 Complete ordered field, AP-31–AP-32 Completeness property of real numbers, 35, AP-31 Completing the square, AP-14–AP-15 Complex conjugate, AP-37 Complex numbers definition of, AP-33 development of, AP-34–AP-38 division of, AP-39 Euler’s formula and, AP-38 Fundamental Theorem of Algebra and, AP-41–AP-42 imaginary part of, AP-34 multiplication of, AP-38–AP-39 operations on, AP-38–AP-39 powers of, AP-40 real part of, AP-34 roots of, AP-40–AP-41 Component equation, for plane, 633 Component form of vectors, 607–609 Component functions, 649, 835 Component (scalar) of u in direction of v, 620 Component test for conservative fields, 852, 854 for exact differential form, 855 Composite functions continuity of, 90–92, 700 definition of, 15 derivative of, 156–161, 654 Compressing a graph, 16 Compression of a gas, uniform, 860–861

Index Computational formulas, for torsion, 676 Computer algebra systems (CAS) in evaluation of improper integrals, 475–476 integral tables and, 456–457 integrate command, 458 integration with, 458–459 Computer graphing of functions, 29–33 of functions of two variables, 691 Computers, to estimate limits, 64–66 Concave down graph, 235 Concave up graph, 235, 236 Concavity curve sketching and, 235–243 second derivative test for, 236 Conditional convergence, 524–525 Cones elliptical, 639, 641 parametrization of, 870 surface area of, 873 Conics defined, AP-18, AP-20, AP-22 eccentricity of, 591–594 in polar coordinates, 591–597, AP-18 polar equations of, 594–596 Connected region, 848 Connectedness, 92 Conservative fields component test for, 852, 854 finding potentials for, 852–855 as gradient fields, 850 line integrals in, 854 loop property of, 851 and Stokes’ theorem, 898 Constant arbitrary, 272 nonzero, 273 rate, 412 spring, 384 Constant force, work done by, 383–384, 621 Constant Function Rule, 654 Constant functions definition of, 7, 61 derivative of, 129 Constant Multiple Rules for antiderivatives, 273, 274, 276 for combining series, 503 for derivatives, 130–131 for finite sums, 300 for gradients, 728 for integrals, 309, 773 for limits, 62 for limits of functions of two variables, 696 for limits of sequences, 489 Constant Value Rule for finite sums, 300 Constrained maximum, 748–751 Constrained minimum, 748–751 Construction of reals, AP-32–AP-33 Continuity. See also Discontinuity of composites, 700 differentiability and, 125, 711 of function at a point, 87–89 at interior point, 87 on an interval, 89 of inverse functions, 90 at left endpoint, 87

limits and, 52–115 for multivariable functions, 698–700 partial derivatives and, 708 of vector functions, 650–651 Continuity equation of hydrodynamics, 907–908 Continuity Test, 88 Continuous extension, 94–95 Continuous function theorem for sequences, 490 Continuous functions absolute extrema of, 218–219, 700 average value of, 312–313, 778, 791 composite of, 90–92 definition of, 89–90, 651 differentiability and, 125 extreme values of, on closed bounded sets, 215, 700 integrability of, 307 Intermediate Value Theorem for, 92–93, 258–259 limits of, 86, 91 nonnegative, average value of, 295–296 at point, 698 properties of, 89 Continuous vector field, 835 Contour curve, 689 Convergence absolute, 524–525 conditional, 524–525 definition of, 307 of improper integrals, 471, 473–474 interval of, 533 of power series, 529–533 radius of, 532–533 of Riemann sums, 307 of sequence, 487–489 of series geometric, 500 Integral Test, 507–510 power, 529–533 of Taylor Series, 543–549 tests for, 476–478, 526–527 Convergence Theorem for Power Series, 532 Coordinate axes definition of, AP-10 moments of inertia about, 831, 885 Coordinate conversion formulas, 810 Coordinate frame left-handed, 602 right-handed, 602 Coordinate pair, AP-10 Coordinate planes definition of, 602 first moments about, 831, 885 Coordinate systems, three-dimensional. See Three-dimensional coordinate systems Coordinates of center of mass, 391, 831, 885 polar, integrals in, 780–781 xyz, line integrals and, 838–839 Coplanar vectors, 611 Corner, 124 Cosecant, 22 Cosecant function extended definition of, 22 integral of, 405 inverse of, 47, 180, 184

I-3

Cosine(s) extended definition of, 22 integrals of products of, 442–443 integrals of products of powers of, 439–441 law of, 25–26, 617 values of, 23 Cosine function derivative of, 149–151 graph of, 10 integral of, 439 inverse of, 47, 184 Costs fixed, 144 marginal, 144, 259 variable, 144 Cot x derivatives of, 152 integrals of, 404–405 inverses of, 180–181 Cotangent function extended definition of, 22 integral of, 405 inverse of, 47, 180, 184 Courant, Richard, 130 Critical point, 218, 240, 741, 745 Cross product with determinants, 625–627 proof of distributive law for, AP-43–AP-44 properties of, 624–625 right-hand rule for, 624 of two vectors in space, 624–625 Cross Product Rule for derivatives of vector functions, 654–655 Cross-sections horizontal, limits of integration and, 772–773 vertical, limits of integration and, 772 volumes using, 353–360 Csc x derivatives of, 152 integrals of, 404–405 inverses of, 180–181 Cube, integral over surface of, 882 Cube root function, 8 Cubic functions, 9 Curl, k-component of, 861–863 Curl vector, 889–890 Curvature calculation of, 669, 678 center of, 671 of plane curves, 668–671 radius of, 671 in space, 672 Curved patch element, 872 Curves area between, 335–341 area under, 311, 472 assumptions for vector integral calculus, 848–849 closed, 842 contour, 689 generating for cylinder surface, 638 graphing of, 338–340 initial point of, 563 level, 727–728 negatively oriented, 863 parametric, 563–564

I-4

Index

Curves (continued ) parametrically defined, length of, 572–575 parametrized, 564, 576–577 piecewise smooth, 652 plane curvature of, 668–671 flux across, 842–844 lengths of, 372–376, 572–575 parametizations of, 563–567, 649 plates bounded by two, 394–395 points of inflection of, 236–238, 241 polar graphing of, 576–577 length of, 589–590 positively oriented, 863 secant to, 54 sketching, 235–243 slope of definition of, 54–56, 116 finding, 55, 116, 584 smooth, 3–4, 372–373, 572–573 curvature of, 668–671 length of, 664 speed on, 666 torsion of, 677 in space, 649–655 arc length along, 664–665 binormals to, 674 formulas for, 678 normals to, 670 parametric equations for, 649 vector equations for. See Vector functions tangent line to, 116 tangents to, 52–57, 666, 727–728 terminal point of, 563 work done by force over, 839–841 y = f (x), length of, 373 Cusp, 124 Cycloids, 567 Cylinder(s) base of, 353 parabolic, flux through, 883 parametrization of, 871 quadric surfaces and, 638–641 slicing with, 364–366 volume of, 353 Cylindrical coordinates definition of, 802 integration with, 804–806 motion in, 679–680 parametrization by, 871 to rectangular coordinates, 803, 810 from spherical coordinates, 810 triple integrals in, 802–810 volume differential in, 803 Cylindrical shells, volumes using, 364–369 Cylindrical solid, volume of, 353–354 Cylindrical surface, 378 De Moivre’s Theorem, AP-40 Decay, exponential, 37–38, 412 Decay rate, radioactive, 37–38, 412 Decreasing function, 6, 230–231 Dedekind, Richard, 339, 504, AP-33

Definite integrals and antiderivatives, 276–277 applications of, 353–400 average value of continuous functions, 312–313 definition of, 289, 305–307, 328 evaluation of, by parts, 436 existence of, 305–307 Mean Value Theorem for, 317–320 nonnegative functions and, 311–312 notation for, 306 properties of, 308–310 substitution in, 335–337 of symmetric functions, 337–338 of vector function, 658 Definite integration by parts, 436 Definite integration by substitution, 335 Degree, of polynomial, 9 “Del f,” 725, 894–896 Density circulation, 861–863 as continuous function, 392 flux, 860 Dependent variable of function, 1, 686 Derivative product rule, 134–135, 233 Derivative quotient rule, 135–136, 152–153, 523 Derivative rule for inverses, 171 Derivative sum rule, 131–132, AP-8–AP-9 Derivative tests, for local extreme values, 216–217, 587–589, 740–743 Derivatives of absolute value function, 160 alternate formula for, 121 applications of, 214–288 calculation from definition, 121 of composite function, 156–161, 654 of constant function, 129 constant multiple rule for, 130–131 of cosine function, 149–151 Cross Product Rule, 654 definition of, 120 difference rule for, 131–132 directional. See Directional derivatives Dot Product Rule, 654 in economics, 144–145 of exponential functions, 132–133, 406–407 as function, 116, 120–125 functions from, graphical behavior of, 242–243 General Power Rule for, 130, 176–177, 407 graphing of, 122–123 higher-order, 136–137, 166 of hyperbolic functions, 421–422 of integral, 323 of inverse functions, 170–178 of inverse hyperbolic functions, 423–425 of inverse trigonometric functions, 45–46, 184 involving loga x, 175, 409 left-handed, 123–124 Leibniz’s Rule, 350 of logarithms, 170–178 notations for, 122 nth, 137 one-sided, 123–124 partial. See Partial derivatives

at point, 116–118, 124–125 of power series, 534 as rate of change, 139–145 of reciprocal function, 121 right-handed, 123–124 second-order, 136–137 of sine function, 149–150 of square root function, 122 symbols for, 137 of tangent vector, 669 third, 137 of trigonometric functions, 149–153 of vector function, 651–653 as velocity, 140, 653 Descartes, René, AP-10 Determinant(s) calculating the cross product, 628 Jacobian, 814, 816, 817, 819 Difference quotient definition of, 117 forms for, 121 limit of, 117 Difference Rules for antiderivatives, 273, 276 for combining series, 503 for derivatives, 131–132 for derivatives of constant functions, 129 for exponential functions, 132–133 for finite sums, 300 of geometric series, 503 for gradient, 728 for higher-order derivatives, 136–137 for integrals, 309 for limits, 62 for limits of functions with two variables, 696 for limits of sequences, 489 for positive integers, 129–130 for products and quotients, 134–136 for vector functions, 654 Differentiability, 123–125, 703, 708, 710–711 Differentiable functions constant multiple rule of, 130 continuity and, 711 continuous, 125, 572 definition of, 121 graph of, 199 on interval, 123–124 parametric curves and, 570 partial derivatives, 703–705 rules for, 130–137, 157–159, 654 Taylor’s formula for, 540 Differential approximation, error in, 201–202 Differential equations initial value problems and, 274 particular solution, 274 separable, 413–414 Differential forms, 855–856 Differential formula, short form of arc length, 375–376 Differentials definition of, 198 estimating with, 199–201 linearization and, 195–203 surface area, for parametrized surface, 873 tangent planes and, 735–737 total, 736

Index Differentiation Chain Rule and, 156–161 derivative as a function, 120–125 derivative as a rate of change, 139–145 derivatives of trigonometric functions, 149–153 implicit, 164–167, 718–720 and integration, as inverse processes, 323 inverse trigonometric functions and, 180–184 linearization and, 195–203 related rates, 186–191 tangents and derivative at a point, 116–118 term-by-term for power series, 534 of vector functions, rules for, 653–655 Differentiation rules, 129–137 Direct Comparison Test, 476, 477 Directed line segments, 607 Direction along a path, 563–564, 828–829 estimating change in, 733 of vectors, 611 Direction cosines, 622 Directional derivatives calculation of, 725–727 definition of, 724 as dot product, 725 estimating change with, 733 gradient vectors and, 723–729 and gradients, 725 interpretation of, 724–725 in plane, 723–724 properties of, 726 Directrix (directrices) of ellipse, 592 of hyperbola, 592 of parabola, 593, 595 Dirichlet, Lejeune, 472 Discontinuity in dy/dx, 374–375 infinite, 88 jump, 88 oscillating, 88 point of, 87 removable, 88 Discriminant (Hessian) of function, 742 Disk method, 356–359 Displacement definition of, 140, 294 versus distance traveled, 294–295, 322 Display window, 29–32 Distance in plane, AP-13–AP-15 and spheres in space, 604–605 in three-dimensional Cartesian coordinates point to line, 632–633 point to plane, 633–634, 635–636 point to point, 604 Distance formula, 604, AP-14 Distance traveled calculating, 291–293 versus displacement, 294–295, 322 total, 294, 322

Distributive Law definition of, AP-31 proof of, AP-43–AP-44 for vector cross products, 625 Divergence of improper integrals, 471 limits and, 471 nth-term test for, 502 of sequence, 487–489 to infinity, 489 to negative infinity, 489 of series, 499 tests for, 476–478, 526–527 of vector field, 858–861, 900–901 Divergence Theorem statement of, 901–902 for other regions, 905–906 for special regions, 903–905 Domain connected, 848 of function, 1–3, 686, 687 natural, 2 of vector field, 835, 848 simply connected, 848 Dominant terms, 106–107 Domination, double integrals and, 773 Domination Rule for definite integrals, 309 Dot product angle between vectors, 616–618 definition of, 617 directional derivative as, 725 orthogonal vectors and, 618–619 properties of, 619–621 Dot Product Rule for vector functions, 654 Double-angle formulas, trigonometric, 25 Double integrals Fubini’s theorem for calculating, 765–767 over bounded nonrectangular regions, 768–769 over rectangles, 763–767 in polar form, 780–784 properties of, 773–774 substitutions in, 814–818 as volumes, 764–765 Double integration, area by, 777–779 Dummy variable in integrals, 307 e definition of number, 133, 402 as limit, 177–178 natural exponential and, 36, 133, 405–406 as series, 544–545 Eccentricity of ellipse, 592–593 of hyperbola, 592–593 of parabola, 592–593 in polar coordinates, 591–594 polar equation for conic with, 594 Economics derivatives in, 144–145 examples of applied optimization from, 259–260 Electric field, 847 Electromagnetic theory (Gauss’ Law), 906 Elements of set, AP-2 Ellipse Law (Kepler’s First Law), 681

I-5

Ellipses center of, AP-20 center-to-focus distance of, AP-21 eccentricity of, 592–593 focal axis of, AP-20 major axis of, AP-21 minor axis of, AP-21 polar equations of, 594–596 vertices of, AP-20 Ellipsoids definition of, 639 graphs of, 641 of revolution, 640 Elliptical cones, 639, 641 Elliptical paraboloids, 641 Empty set, AP-2 Endpoint extreme values, 216 Endpoint values of function, 87, 218 Equal Area Law (Kepler’s Second Law), 681 Equations differential. See Differential equations for ellipses, 596, AP-22 Euler’s identity, 556–557 focus-directrix, 593 for hyperbolas, AP-22–AP-24 ideal projectile motion and, 660 inverse, 408 linear, AP-13 parametric. See Parametric equations for plane in space, 633–634 point-slope, AP-12 polar for circles, 596 polar for lines, 596 related rates, 187–191 relating polar and Cartesian coordinates, 581 relating rectangular and cylindrical coordinates, 803 relating spherical coordinates to Cartesian and cylindrical coordinates, 806 Error analysis for linear approximation, 735 for numerical integration, 465–468 in standard linear approximation, 201–202, 735 Error estimation, for integral test, 509–510 Error formula, for linear approximations, 201–202, 735, AP-49–AP-50 Error term, in Taylor’s formula, 544 Euler, Leonhard, AP-44 Euler’s formula, AP-38 Euler’s identity, 556–557 Evaluation Theorem (Fundamental Theorem, Part 2), 320–322 Even functions, 6–7 ex derivative of, 133, 406–407 integral of, 406–407 inverse equation for, 405–406 laws of exponents for, 227, 407 Exact differential forms, 855–856 Expansion, uniform, for a gas, 860–861 Exponential change (growth or decay), 411–412 Exponential functions with base a, 34, 44, 407 behavior of, 34–36 derivatives of, 132–133, 173–175, 406–407

I-6

Index

Exponential functions (continued ) description of, 10, 33–34 general, 35–36, 44, 407–408 growth and decay, 37–38 integral of, 406–407 natural, 36, 406 Exponential growth, 37–38, 412, 414–415 Exponents irrational, 176–177 Laws of, 227, 407 rules for, 36 Extrema finding of, 217–219 global (absolute), 214–216, 218 local (relative), 216–217, 231–233, 238–242, 740 Extreme Value Theorem, 215–216, 700, AP-32 Extreme values constrained, and Lagrange multipliers, 751 at endpoints, 216 of functions, 214–219, 740–743 local (relative) derivative tests, 217, 231–233 for several variables, 740, 742 for single variable functions, 216–217 Factorial notation, 493 Fan-shaped region in polar coordinates, area of, 588 Fermat, Pierre de, 55 Fermat’s principle in optics, 258 Fibonacci numbers, 493 Fields conservative, 847, 849–855, 898 electric, 847 gradient, 850 gravitational, 847 number, AP-31 ordered, AP-31 vector, 837–838 Finite (bounded) intervals, 6, AP-3 Finite limits, 97–108 Finite sums algebra rules for, 300–301 estimating with, 289–296 limits of, 301–302 and sigma notation, 299–301 Firing angle, 660 First Derivative Test, 230–233, 740, 749 First derivative theorem for local extreme values, 217–218, 231–233 First moments about coordinate axes, 796 about coordinate planes, 796, 885 masses and, 795–797 First-order differential equation topics covered online applications of first-order differential equations autonomous differential equations Bernoulli differential equation carrying capacity competitive-hunter model curve, sigmoid shape

equilibria equilibrium values Euler’s method exponential population growth model falling body, encountering resistance first-order linear equations graphical solutions of autonomous differential equations initial value problems integrating factor Law of Exponential Change limit cycle limiting population logistic population growth mixture problems motion with resistance proportional to velocity Newton’s law of cooling Newton’s second law of motion numerical method and solution orthogonal trajectories phase lines and phase planes resistance proportional to velocity rest points RL circuits slope fields solution curves solution of first-order equations standard form of linear equations steady-state value systems of differential equations terminal velocity Flat plate, center of mass of, 391–394, 796 Flight time, 661 Flow integrals, 841–842 Fluid flow rates, 859 Fluid forces, work and, 841 Flux across plane curve, 842–844 across rectangle boundary, 859–860 calculation of, 863, 883–885 versus circulation, 843 definition of, 843, 883 surface integral for, 883–885 Flux density (divergence), of vector field, 860, 900–901 Foci, AP-18–AP-21 Forces addition of, 609–610 constant, 383 field of, 839–840 variable along line, 383–384 work done by over curve in space, 839–841 through displacement, 621 Free fall, Galileo’s law for, 52, 141–142 Frenet, Jean-Frédéric, 674 Frenet frame computational formulas, 677 definition of, 674 torsion in, 676–677 Fubini, Guido, 766 Fubini’s theorem for double integrals, 765–767, 770–772, 781, 787 Functions absolute value, 5 addition of, 14–15

algebraic, 10 arcsine and arccosine, 46–47 arrow diagram of, 2, 687 combining of, 14–21 common, 7–11 component, 649 composite. See Composite functions constant, 7, 61, 129, 295–296 continuity of, 87, 651, 698 continuous. See Continuous functions continuous at endpoint, 87 continuous at point, 94–95, 651, 698 continuous extension of, 94–95 continuous on interval, 89 continuously differentiable, 372, 380, 572–573 cosine, 22, 149–151 critical point of, 218, 741 cube root, 8 cubic, 9 decreasing, 6, 230–231 defined by formulas, 14 defined on surfaces, 716–718 definition of, 1 dependent variable of, 1, 686 derivative as, 116, 120–125 derivative of, 117, 121, 124–125, 652 from derivatives, graphical behavior of, 242–243 differentiable. See Differentiable functions discontinuity of, 87–88, 698 domain of, 1–3, 14, 686, 687 even, 6–7 exponential. See Exponential functions extreme values of, 214–219, 740–745, 751, 754 gradient of, 725 graphing with calculators and computers, 29–33 graphs of, 1–11, 14–21, 689 greatest integer, 5 Hessian of function of two variables, 742 hyperbolic. See Hyperbolic functions identity, 24–25, 61, 897 implicitly defined, 164–166, 718 increasing, 6, 230–231 independent variable of, 1, 686 input variable of, 1, 686 integer ceiling, 5 integer floor, 5 integrable, 307–310, 658, 764, 786 inverse. See Inverse functions least integer, 5 left-continuous, 87 limit of, 59–67, 695 linear, 7 linearization of, 195–199, 733–735 logarithmic. See Logarithmic functions machine diagram of, 2 of many variables, 720 marginal cost, 144 maximum and minimum values of, 214, 216–217, 233, 743–745 monotonic, 230–233 of more than two variables, 700, 707–708, 736–737

Index multiplication of, 14 natural exponential, definition of, 36, 406 natural logarithm, 43, 172–175, 401–402 nondifferentiable, 124 nonintegrable, 307 nonnegative area under graph of, 311–312 continuous, 295–296 numerical representation of, 4 odd , 6 one-to-one, 39–40 output variable of, 1, 686 piecewise-continuous, 307 piecewise-defined, 5 piecewise-smooth, 848 polynomial, 8 position, 5 positive, area under graph of, 296 potential, 848 power, 7–8, 176–177 quadratic, 9 range of, 1–3, 686, 687 rational. See Rational functions real-valued, 2, 650, 686 reciprocal, derivative of, 121 representation as power series, 538–539 right-continuous, 87 scalar, 650 scaling of, 16–18 scatterplot of, 4 of several variables, 686–691 shift formulas for, 16 sine, 22, 149–151 in space, average value of, 791–792 square root, 8 derivative of, 122 symmetric, 6–7, 337–338, 583 of three variables, 689–691, 716, 728–729 total area under graph of, 324 total cost, 144 transcendental, 11, 406 trigonometric. See Trigonometric functions of two variables, 687–688, 691, 710 Chain Rule(s) for, 714–716 Increment Theorem of, 710 limits for, 694–698 linearization of, 733–735 partial derivatives of, 686–691, 703–705 unit step, 62 value of, 2 vector. See Vector functions velocity, 294, 653 vertical line test for, 4–5 Fundamental Theorem of Algebra, AP-41–AP-42 Fundamental Theorem of Calculus arc length differential and, 575 continuous functions and, 401 description of, 317–325, 908–909 evaluating definite integrals, 431 for line integrals, 829 Part 1 (derivative of integral), 319–320, 403 proof of, 320

Part 2 (Evaluation Theorem), 320–322 Net Change Theorem, 322 proof of, 320–322 path independence and, 848 Fundamental Theorem of Line Integrals, 849 Galileo Galilei free-fall formula, 52, 141–142 law of, 52 Gauss, Carl Friedrich, 301, 608 Gauss’s Law, 906 General linear equation, AP-13 General Power Rule for derivatives, 130, 176–177, 407 General sine function, 26 General solution of differential equation, 274, 413 Genetic data, and sensitivity to change, 145 Geometric series convergence of, 500 definition of, 500–501 Geometry in space, 602–648 Gibbs, Josiah Willard, 666 Global (absolute) maximum, 214, 743–745 Global (absolute) minimum, 214, 743–745 Gradient Theorem, Orthogonal, for constrained extrema, 751 Gradient vector fields conservative fields as, 850 definition of, 836–837 Gradient vectors algebra rules for, 728 curl of, 897 definition of, 725 directional derivatives and, 723–729 to level curves, 727–728 Graphing, with calculators and computers, 29–33 Graphing windows, 29–32 Graphs asymptotes of, 97–108 of common functions, 7–11 connectedness and, 92 of derivatives, 122–123 of equation, AP-10 of functions, 3–4, 14–21, 689 of functions with several variables, 686–691 of functions with three variables, 689–691 of functions with two variables, 688–689 of parametric equations, 586 in polar coordinates, 580, 583–586 of polar curves, 586 of sequence, 487 surface area of, 877 symmetric about origin, 6, 583 symmetric about x-axis, 6, 583 symmetric about y-axis, 6, 583 symmetry tests for, 583 technique for, 586 trigonometric, transformations of, 26 of trigonometric functions, 24, 31–33 of y ƒ(x), strategy for, 240–242 Grassmann, Hermann, 611 Gravitation, Newton’s Law of, 680 Gravitational constant, 680 Gravitational field

I-7

definition of, 847 vectors in, 835 Greatest integer function definition of, 5 as piecewise-defined function, 5 Green’s Theorem area by, 868 circulation curl or tangential form, 864, 866, 891, 908 comparison with Divergence Theorem, 900, 908 comparison with Stokes’ Theorem, 890, 908 divergence or normal form of, 863, 900, 908 to evaluate line integrals, 865–866 forms for, 863–865 generalization in three dimensions, 908 and the Net Change Theorem, 864 in plane, 858–867 proof of, for special regions, 866–867 Growth exponential, 412 Growth rates of functions, 412 Half-angle formulas, trigonometric, 25 Half-life, 38, 45, 416 Halley, Edmund, 231 Harmonic motion, simple, 151–152 Harmonic series alternating, 522–523 definition of, 507 Heat equation, 714 Heat transfer, 416–417 Height, maximum in projectile motion, 661 Helix, 650 Hessian of function, 742 Higher-order derivatives, 136–137, 166, 710 Hooke’s law of springs, 384–385 Horizontal asymptotes, 97, 99–102, 106 Horizontal scaling and reflecting formulas, 17 Horizontal shift of function, 16 Horizontal strips, 392–393 Huygens, Christian, 566, 567 Hydrodynamics, continuity equation of, 907–908 Hyperbolas branches of, AP-19 center of, AP-22 definition of, AP-22 directrices, 592 eccentricity of, 592–593 equation of, in Cartesian coordinates, 593 focal axis of, AP-22 foci of, AP-22 polar equation of, 594 standard-form equations for, AP-22–AP-24 vertices of, AP-22 Hyperbolic functions definitions of, 420–421 derivatives of, 421–422, 423–425 graphs of, 423 identities for, 420–421, 423 integrals of, 421–422 inverse, 422–423 six basic, 420 Hyperbolic paraboloid, 640, 641 Hyperboloids, 639, 641

I-8

Index

i-component of vector, 611 Identity function, 7, 61, 897 Image, 814 Implicit differentiation Chain Rule and, 718–720 formula for, 719 technique for, 164–167 Implicit Function Theorem, 720, 875 Implicit surfaces, 875–877 Implicitly defined functions, 164–166 Improper integrals approximations to, 478 calculating as limits, 471–478 with a CAS, 475–476 convergence of, 471, 473–474 of Type I, 471 of Type II, 474 Increasing function, 6, 230–231 Increment Theorem for Functions of Two Variables, 710, AP-46–AP-48 Increments, AP-10–AP-13 Indefinite integrals. See also Antiderivatives definition of, 276–277, 328 evaluation with substitution rule, 328–333 Independent variable of function, 1, 686 Indeterminate form 0/0, 246–250 Indeterminate forms of limits, 246–252, 554–555 Indeterminate powers, 250–251 Index of sequence, 486 Index of summation, 299 Induction, mathematical, AP-5 Inequalities rules for, AP-1 solving of, AP-3–AP-4 Inertia, moments of, 797–800 Infinite discontinuities, 88 Infinite (unbounded) intervals, 6, AP-3 Infinite limits definition of, precise, 104–105 description and examples, 102–104 of integration, 471–473 Infinite sequence, 486–495. See also Sequences Infinite series, 498–504 Infinitesimals, AP-33 Infinity divergence of sequence to, 489 limits at, 97–108 and rational functions, 99 Inflection, point of, 218, 236–238, 241 Initial point of curve, 563 of vector, 607 Initial ray in polar coordinates, 579 Initial speed in projectile motion, 660 Initial value problems definition of, 274 and differential equations, 274 separable differential equations and, 414 Inner products. See Dot product Input variable of function, 1, 686 Instantaneous rates of change derivative as, 139–140 tangent lines and, 56–57 Instantaneous speed, 52–54 Instantaneous velocity, 140–141

Integer ceiling function (Least integer function), 5 Integer floor function (Greatest integer function), 5 Integers description of, AP-34 positive, power rule for, 129–130 starting, AP-8 Integrable functions, 307–310, 764, 786 Integral form, product rule in, 432–435 Integral sign, 276 Integral tables, 456–457 Integral test for convergence of series, 507–510 error estimation, 509–510 remainder in, 509–510 Integral theorems, for vector fields, 908–909 Integrals approximation of by lower sums, 291 by midpoint rule, 291 by Riemann sum, 302–304 by Simpson’s Rule, 463–465 by Trapezoidal Rule, 461–463 by upper sums, 290 Brief Table of, 431 definite. See Definite integrals double. See Double integrals exponential change and, 411–417 of hyperbolic functions, 420–425 improper, 471–478 approximations to, 478 of Type I, 471 of Type II, 474 indefinite, 276–277, 328–333 involving log, 409 iterated, 765 line. See Line integrals logarithm defined as, 401–409 multiple, 763–827 nonelementary, 459, 552–553 polar, changing Cartesian integrals into, 782–784 in polar coordinates, 780–781 of powers of tan x and sec x, 441–442 of rate, 322–323 repeated, 765 substitution in, 329–333, 335, 814–821 surface, 880–887, 892 table of, 456–457 of tan x, cot x, sec x, and csc x, 404–405 trigonometric, 439–443 triple. See Triple integrals of vector fields, 837–838 of vector functions, 657–659 work, 383–384, 839–841 Integrands definition of, 276 with vertical asymptotes, 474–475 Integrate command (CAS), 458 Integration basic formulas, 431 with CAS, 458–459 in cylindrical coordinates, 802–810 and differentiation, relationship between, 323 limits of. See Limits, of integration

numerical, 461–468 by parts, 432–436 by parts formula, 432–433 of rational functions by partial fractions, 448–454 with respect to y, area between curves, 340–341 in spherical coordinates, 808–810 with substitution, 329, 335 techniques of, 431–485 term-by-term for power series, 535–536 by trigonometric substitution, 444–447 variable of, 276, 306 in vector fields, 828–916 of vector function, 658–659 Interest, compounded continuously, 37 Interior point continuity at, 87 finding absolute maximum and minimum values, 743 for regions in plane, 688 for regions in space, 690 Intermediate Value Property, 92 Intermediate Value Theorem continuous functions and, 92–93, 258–259, 402, AP-32 monotonic functions and, 230 Intermediate variable, 715–716 Intersection, lines of, 634–635 Intersection of sets, AP-2 Interval of convergence, 533 Intervals definition of, AP-3 differentiable on, 123–124 parameter, 563–564 types of, AP-3 Inverse equations, 408 Inverse function-inverse cofunction identities, 184 Inverse functions definition of, 40–41 derivative rule for, 171 and derivatives, 170–178 of exponential functions, 11, 39 finding, 171–172 hyperbolic, 422–423 and logarithms, 39–49 trigonometric. See Inverse trigonometric functions Inverse trigonometric functions cofunction identities, 184 definition of, 46, 180, 184 derivatives of, 45–46, 184 study of, 180–184 Inverses finding of, 41–42 integration and differentiation operations, 323 of ln x and number e, 405–406 for one-to-one functions, 40 of tan x, cot x, sec x, and csc x, 180–184 Irrational numbers definition of, AP-2 as exponents, 35, 176–177 Irreducible quadratic polynomial, 449–450 Iterated integral, 765

Index j-component of vector, 611 Jacobi, Carl Gustav Jacob, 814 Jacobian determinant, 814, 816, 817, 819 Jerk, 141, 142 Joule, James Prescott, 383 Joules, 383 Jump discontinuity, 88 k-component of curl, 861–863 k-component of vector, 611 Kepler, Johannes, 681 Kepler’s First Law (Ellipse Law), 681 Kepler’s Second Law (Equal Area Law), 681 Kepler’s Third Law (Time-Distance Law), 681–682 Kovalevsky, Sonya, 424 kth subinterval of partition, 302, 303, 306 Lagrange, Joseph-Louis, 223, 748 Lagrange multipliers method of, 751–753 partial derivatives and, 748–755 solving extreme value problems, 744 with two constraints, 754–755 Laplace, Pierre-Simon, 709 Laplace equation, 713 Launch angle, 660 Law of cooling, Newton’s, 416–417 Law of cosines, 25–26, 617 Law of refraction, 259 Laws of exponents, 227, 407 Laws of logarithms proofs of, 226–227 properties summarized, 43 Least integer function, 5 Least upper bound, 493, AP-31–AP-32 Left-continuous functions, 87 Left-hand derivatives, 123–124 Left-hand limits definition of, 80–81 informal, 79–81 precise, 70–76, 81 Left-handed coordinate frame, 602 Leibniz, Gottfried, 329, AP-33 Leibniz’s formula, 554 Leibniz’s notation, 122, 158, 195, 198, 306, 573 Leibniz’s Rule, 350 Length along curve in space, 664–665 constant, vector functions of, 655 of curves, 372–373, 572–575 of parametrically defined curve, 572–575 of polar coordinate curve, 589–590 of vector (magnitude), 608, 609–610 Lenses, light entering, 167 Level curves, of functions of two variables, 689 Level surface, of functions of three variables, 689 L’Hôpital, Guillaume de, 246 L’Hôpital’s Rule finding limits of sequences by, 247–248 indeterminate forms and, 246–252 proof of, 251–252 Limit Comparison Test, 476, 477, 514–515

Limit Laws for functions with two variables, 696 limit of a function and, 59–67 theorem, 62, 98 Limit Power Rule, 62 Limit Product Rule, proof of, AP-26–AP-27 Limit Quotient Rule, proof of, AP-27–AP-28 Limit Root Rule, 62 Limit theorems, proofs of, AP-26–AP-29 Limits of (sin u)/u, 82–84 commonly occurring, 492–493, AP-29– AP-30 continuity and, 52–115 of continuous functions, 86, 91 for cylindrical coordinates, 804–806 definition of informal, 59–60 precise, 70–76 proving theorems with, 75–76 testing of, 71–73 deltas, finding algebraically, 73–75 of difference quotient, 117 e (the number) as, 177–178 estimation of, calculators and computers for, 64–66 finite, 97–108 of finite sums, 301–302 of function values, 59–67 for functions of two variables, 694–698 indeterminate forms of, 246–252 infinite, 102–104 precise definitions of, 104–105 at infinity, 97–108 of integration for cylindrical coordinates, 804–806 for definite integrals, 335–336 finding of, for multiple integrals, 772–773, 781–782, 787–791, 804–806, 808–809 infinite, 471–473 for polar coordinates, 781–782 for rectangular coordinates, 786, 787–792 for spherical coordinates, 808–809 left-hand. See Left-hand limits nonexistence of, two-path test for functions of two variables, 699 one-sided. See One-sided limits of polynomials, 63 power rule for, 62 of rational functions, 63, 99 of Riemann sums, 305–307 right-hand. See Right-hand limits root rule for, 62 Sandwich Theorem, 66–67 of sequences, 488, 490 two-sided, 79 of vector-valued functions, 650–651 Line integrals additivity and, 830 definition of, 828 evaluation of, 829, 838 by Green’s Theorem, 865–866 fundamental theorem of, 849 integration in vector fields, 828–832 interpretation of, 832

I-9

mass and moment calculations and, 830–832 in plane, 832 vector fields and, 834–844 xyz coordinates and, 838–839 Line segments directed, 607 midpoint of, finding with vectors, 612 in space, 630–636 Linear approximations error formula for, 735, AP-49–AP-50 standard, 196, 734–735 Linear equations general, AP-13 Linear functions, 7 Linear transformations, 815–816 Linearization definition of, 196, 734 differentials and, 195–203 of functions of two variables, 733–735, 736 Lines of intersection, for planes, 634–635 masses along, 389–390 motion along, 140–144 normal, 167 normal, tangent planes and, 730–733 parallel, AP-13 parametric equations for, 630–631 perpendicular, AP-13 and planes, in space, 630–636 polar equation for, 596 secant, 54 straight equation lines, 630 tangent, 56–57, 116 vector equations for, 630–632 vertical, shell formula for revolution about, 367 work done by variable force along, 383–384 Liquids incompressible, 861 pumping from containers, 385–386 ln bx, 43, 226–227 ln x and change of base, 44 derivative of, 172–173, 403 graph and range of, 43, 403–404 integral of, 433–434 inverse equation for, 43, 406 inverse of, 405–406 and number e, 43, 405–406 properties of, 43, 403 ln xr, 43, 227 Local extrema first derivative test for, 231–233 first derivative theorem for, 217 second derivative test for, 238–242 Local extreme values definition of, 216–217, 740 derivative tests for, 217–218, 740–743, 742 first derivative theorem for, 217–218, 740 Local (relative) maximum, 216–217, 740, 745 Local (relative) minimum, 216–217, 740, 745 loga u , derivative of, 173–175, 409 loga x derivatives and integrals involving, 409 inverse equations for, 44, 408 Logarithmic differentiation, 175–178

I-10

Index

Logarithmic functions with base a, 42–43, 408–409 change of base formula and, 44 common, 43 description of, 11 natural, 43, 401–402 Logarithms algebraic properties of, 43, 409 applications of, 44–45 with base a, 42–43, 408–409 defined as integral, 401–409 derivatives of, 170–178 integral of, 433 inverse functions and, 39–49, 405 inverse properties of, 44, 408 laws of, proofs of, 43, 226–227 natural, 43, 401–402 properties of, 43–44 Loop, 842 Lower bound, 310 Lower sums, 291 Machine diagram of function, 2 Maclaurin, Colin, 539 Maclaurin series, 539–540, 541 Magnitude (length) of vector, 608, 609–610 Marginal cost, 144, 259 Marginal profit, 259 Marginal revenue, 259 Marginals, 144 Mass. See also Center of mass along line, 389–390 distributed over plane region, 390–391 formulas for, 391, 394, 796, 831 by line integral, 831 and moment calculations line integrals and, 830–832 multiple integrals and, 796, 799 moments of, 391 of thin shells, 885–887 of wire or thin rod, 830–831 Mathematical induction, AP-5 Max-Min Inequality Rule for definite integrals, 309, 317–318 Max-Min Tests, 231–232, 238, 740, 742, 745 Maximum absolute (global), 214, 743–745 constrained, 748–751 local (relative), 216–217, 740, 745 Mean value. See Average value Mean Value Theorems arbitrary constants, 272 Cauchy’s, 251–252 corollary 1, 225 corollary 2, 225–226, 403 corollary 3, 230, 233 for definite integrals, 317–320 for derivatives, 223 interpretation of, 224, 317 laws of logarithms, proofs of 226 mathematical consequences of, 225–226 for parametrically defined curves, 572 Mendel, Gregor Johann, 145 Mesh size, 462 Midpoint of line segment in space, finding with vectors, 612

Midpoint rule, 291, 292 Minimum absolute (global), 214, 743–745 constrained, 748–751 local (relative), 216–217, 740, 745 Mixed Derivative Theorem, 709, AP-44 Möbius band, 883 Moments and centers of mass, 389–396, 795–800, 831, 885 first, 795–797, 885 of inertia, 797–800, 885 and mass calculations, line integrals and, 830–832 of solids and plates, 799 of system about origin, 390 of thin shells, 885–887 of wires or thin rods, 830–831 Monotonic functions, 230–233 Monotonic Sequence Theorem, 494–495, 507 Monotonic sequences, 493–495 Motion along curve in space, 651–653, 674 along line, 140–144 antiderivatives and, 274–275 direction of, 653 in polar and cylindrical coordinates, 679–680 simple harmonic, 151–152 vector functions and, 649, 651–653 Multiple integrals. See Double integrals; Triple integrals Multiplication of complex numbers, AP-38–AP-39 of functions, 14 of power series, 534 scalar, of vectors, 609–610 Multiplier (Lagrange), 744, 748–755 Napier, John, 43 Natural domain of function, 2 Natural exponential function definition of, 36, 406 derivative of, 133, 406–407 graph of, 36, 133, 405 power series for, 544 Natural logarithm function algebraic properties of, 43, 403 definition of, 43, 401–402 derivative of, 172–175, 403 power series for, 536 Natural logarithms, 43, 401–402 Natural numbers, AP-2 Negative rule, for antiderivatives, 273 Net Change Theorem statement of, 322 and Green’s Theorem, 864 Newton, Sir Isaac, 317, AP-33 Newton-Raphson method, 266–269 Newton’s law of cooling, 416–417 Newton’s law of gravitation, 680 Newton’s method applying, 267–268 convergence of approximations, 269 procedure for, 266–267 Nondecreasing partial sums, 499

Nondecreasing sequences, 494 Nondifferentiable function, 124 Nonelementary integrals, 459, 552–553 Nonintegrable functions, 307–310 Norm of partition, 304, 764, 803 Normal component of acceleration, 674–678 Normal line, 167, 731 Normal plane, 677 Normal vector, 672–673 Notations, for derivative, 122, 704–705 nth partial sum, 498–499 nth-term test for divergence, 502 Numerical integration, 461–468 Numerical representation of functions, 4 Oblique (slant) asymptote, 102 Octants, 602 Odd functions, 6–7 One-sided derivatives, 123–124 One-sided limits. See also Left-hand limits; Right-hand limits definition of informal, 79–81 precise, 81 derivatives at endpoints, 123 involving (sin u/u), 82–84 One-to-one functions, 39–40 Open region, 688, 690 Optics Fermat’s principle in, 258 Snell’s Law of, 259 Optimization, applied. See Applied optimization Orbital period, 681 Order of Integration Rule, 309 Ordered field, AP-31–AP-32 Oresme, Nicole, 488 Orientable surface, 883 Origin of coordinate system, AP-10 moment of system about, 390 in polar coordinates, 579 Orthogonal gradient theorem, 751 Orthogonal vectors, 618–619 Oscillating discontinuities, 88 Osculating circle, 671 Osculating plane, 677 Output variable of function, 686 Outside-Inside interpretation of chain rule, 158–159 r-limits of integration, finding of, 808 p-series, 509 Paddle wheel, 894–896 Parabola(s) approximations by, 463–465 axis of, AP-15, AP-19 definition of, AP-18 directrix of, 595, AP-18, AP-20 eccentricity of, 592–593 focal length of, AP-19 focus of, AP-18, AP-20 as graphs of equations, AP-15–AP-16 parametrization of, 564–565 semicubical, 169 vertex of, AP-15, AP-19

Index Paraboloids definition of, 639 elliptical, 641 hyperbolic, 640, 641 volume of region enclosed by, 789–790 Parallel lines, 634, AP-13 Parallel planes lines of intersection, 634 slicing by, 354–355 Parallel vectors, cross product of, 624 Parallelogram area of, 625 law of addition, 609–610, 617 Parameter domain, 563, 870 Parameter interval, 563–564 Parameters, 563, 870 Parametric curve arc length of, 572–575, 664–665 calculus with, 570–577 definition of, 563 differentiable, 570 graphing, 564–565, 586 Parametric equations of circle, 564, 573–574 for curves in space, 649 of cycloid, 567 definition of, 563–564 graphing, 564–566 of hyperbola, 566, 571 of lines, 631–632 for projectile motion, 659–661 Parametric formulas, for derivatives, 570 Parametrization of cone, 870 of curves, 563–567, 649 of cylinder, 871 of line, 630–631 of sphere, 870–871 and surface area, 871–875 of surfaces, 870–875 Partial derivatives calculations of, 705–707 Chain Rule for, 714–720 and continuity, 694–700 continuous, identity for function with, 897 definitions of, 704 equivalent notations for, 704 extreme values and saddle points, 740–745 of function of several variables, 686–691 of function of two variables, 703–705 functions of several variables, 686–691 gradient vectors and, 723–729 higher-order, 710 Lagrange multipliers, 748–755 second-order, 708–709 tangent planes and, 730–737 Partial fractions definition of, 449 integration of rational functions by, 448–454 method of, 449–453 Partial sums nondecreasing, 507 nth of series, 498–499 sequence of, 499 Particular solution, of differential equation, 274

Partitions definition of, 763 kth subinterval of, 303 norm of, 304, 764 for Riemann sums, 304 Parts, integration by, 432–436 Pascal, Blaise, 499 Path independence, 847–848 Path integrals. See Line integrals Path of particle, 649 Pendulum clock, 567 Percentage change, 203 Periodicity, of trigonometric functions, 24 Perpendicular lines, AP-13 Perpendicular (orthogonal) vectors, 618–619 Physics, examples of applied optimization from, 257–259 Piecewise-continuous functions, 307, 349–350 Piecewise-defined functions, 5 Piecewise-smooth curves, 652, 848 Piecewise-smooth surface, 881, 890 Pinching Theorem. See Sandwich Theorem Plane areas for polar coordinates, 587–589 Plane curves circle of curvature for, 671–672 lengths of, 372–376 parametrizations of, 563–567 Plane regions interior point, 688 masses distributed over, 390–391 Plane tangent to surface, 731, 732 Planes angles between, 636 Cartesian coordinates in, AP-10 directional derivatives in, 723–724 distance and circles in, AP-13–AP-15 equation for, 633 Green’s Theorem in, 858–867 horizontal tangent to surface, 740 line integrals in, 832 lines of intersection for, 634–635 motion of planets in, 680–681 normal, 677 osculating, 677 parallel, 634 rectifying, 677 in space, 630–636 Planetary motion Kepler’s First Law (Ellipse Law) of, 681 Kepler’s Second Law (Equal Area Law) of, 681 Kepler’s Third Law (Time-Distance Law) of, 681–682 as planar, 680–681 Plate(s) bounded by two curves, 394–395 thin flat, center of mass of, 391–394 two-dimensional, 796, 799 Point-slope equation, AP-12 Points boundary, 690 of discontinuity, definition of, 87 of inflection, 218, 236–238 interior, 690 in three-dimensional Cartesian coordinate system, distance to plane, 635–636 Poisson, Siméon-Denis, 740

I-11

Polar coordinate pair, 579 Polar coordinates area in, 782 area of polar region, 588 Cartesian coordinates related to, 579–582 conics in, 591–597, AP-18 definition of, 579 graphing in, 580, 583–586 symmetry tests for, 583 initial ray of, 579 integrals in, 780–781 length of polar curve, 589 motion in, 679–680 pole in, 579 slope of polar curve, 584–585 velocity and acceleration in, 679–682 Polar equations of circles, 596–597 of conic sections, 594–596 graphing of, 580 of lines, 596 Polyhedral surfaces, 896–897 Polynomial functions, definition of, 8 Polynomials coefficients of, 8–9 degree of, 9 derivative of, 132 limits of, 63 quadratic irreducible, 449–450 Taylor, 540–542, 547, 548 Population growth unlimited, 414–415 Position, of particle in space over time, 649 Position function, acceleration and, 226 Position vector, 607 Positive integers definition of, AP-34 derivatives power rule for, 129–130 Potential function, 848 Potentials, for conservative fields, 852–855 Power Chain Rule, 159–162, 165 Power functions, 7–8, 176–177 Power Rule for derivatives, general version of, 130, 176–177, 407 for limits, 62 for limits of functions of two variables, 696 natural logarithms, 43, 409 for positive integers, 129–130 proof of, 176 Power series convergence of, 529–532 radius of, 532 testing of, 532 multiplication of, 533–534 operations on, 533–536 reciprocal, 529–530 term-by-term differentiation of, 534 term-by-term integration of, 535–536 Powers binomial series for, 551–552 of complex numbers, AP-40 indeterminate, 250–251 of sines and cosines, products of, 439–441 Preimage, 814 Principal unit normal vector, 672, 678

I-12

Index

Product Rule for derivatives, 134–135, 233 for gradient, 728 in integral form, integration by parts, 432–435 for limits, 62 of functions with two variables, 696 proof of, 134 for natural logarithms, 43, 409 for power series, 534 for sequences, 489 Products of complex numbers, AP-38–AP-39 of powers of sines and cosines, 439–441 and quotients, derivatives of, 134–136 of sines and cosines, 442–443 Profit, marginal, 259 Projectile motion, vector and parametric equations for, 659–661 Projection, of vectors, 619–621 Proportionality relationship, 7 Pumping liquids from containers, 385–386 Pyramid, volume of, 354–355 Pythagorean theorem, 24, 26, 27, AP-13, AP-34 Quadrants, of coordinate system, AP-10 Quadratic approximations, 543 Quadratic polynomial, irreducible, 449–450 Quadric surfaces, 639–641 Quotient Rule for derivatives, 135–136, 152–153, 523 for gradient, 728 for limits, 62 of functions with two variables, 696 proof of, AP-27–AP-28 for natural logarithms, 43, 409 for sequences, 489 Quotients for complex numbers, AP-39 products and, derivatives of, 134–136 r-limits of integration, 781, 805 f-limits of integration, finding of, 808–809 Radian measure and derivatives, 161 Radians, 21–22, 23 Radioactive decay, 37–38, 415–416 Radioactive elements, half-life of, 45, 416 Radioactivity, 415–416 Radius of circle, AP-14 of convergence, 533 of convergence of power series, 532–533 of curvature, for plane curves, 671–672 Radius units, 21 Range of function, 1–3, 686, 687 in projectile motion, 661 Rate(s) average, 54 of change, 52–57 instantaneous, derivative as, 56–57 integral of, 322–323 Rate constant, exponential change, 412 Ratio, in geometric series, 500 Ratio Test, 517–519, 530–531, 533, 551

Rational exponents, 35 Rational functions definition of, 9 domain of, 9 integration of, by partial fractions, 448–454 limits of, 63 at infinity, 99 Rational numbers, AP-2, AP-34 Real numbers construction of reals and, AP-32–AP-33 development of, AP-34–AP-35 properties of algebraic, AP-1, AP-31 completeness, 35, AP-1, AP-31 order, AP-1, AP-31 and real line, AP-1 theory of, AP-31–AP-33 Real-valued functions, 2, 650, 686 Reals, construction of, AP-32–AP-33 Rearrangement theorem, for absolutely convergent series, 526 Reciprocal function, derivative of, 121 Reciprocal Rule for natural logarithms, 43, 409 Rectangles approximating area of, 289–291 defining Riemann sums, 303–304 double integrals over, 763–767 optimizing area of, inside circle, 257–258 Rectangular coordinates. See Cartesian coordinates Rectifying plane, 677 Recursion formula, 493 Recursive definitions, 493 Reduction formula, 435, 457 Reflection of graph, 16–18 Refraction, Law of, 259 Regions bounded, 688 closed, 688, 690 connected, 848 general, double integrals over, 768–774 open, 688, 690, 898 plane interior point, 688 masses distributed over, 389 simply connected, 848 solid, volume of, 769–772 in space interior point, 690 volume of, 787 special divergence theorem for, 903–905 Green’s Theorem for, 866–867 unbounded, 688 Reindexing infinite series, 504 Related rates, 186–191 Relative change, 203 Relative (local) extrema, 216–217, 740 Remainder estimating of, in Taylor’s Theorem, 544, 545–546 in integral test, 509–510 of order n, definition for Taylor’s formula, 544 Remainder Estimation Theorem, 545, 547 Removable discontinuities, 88

Representation of function, power series, 528–539 Resultant vector, 609–610 Revenue, marginal, 259 Revolution about y-axis, 380–381 areas of surfaces of, 378–381, 576–577 ellipsoid of, 640 Shell formula for, 367 solids of disk method, 356–359 washer method, 359–360 surface of, 378 Riemann, Georg Friedrich Bernhard, 302 Riemann sums concept overview, 302–304 convergence of, 307 forming, 306 for integrals, 391 limits of, 305–308 line integrals and, 828 slicing with cylinders, 365 for surface integrals, 880 total area of rectangle, 323 for triple integrals, 803, 807 volumes using cross-sections, 354 volumes using cylindrical shells, 367 work and, 384 Right-continuous functions, 87 Right-hand limits definition of, 80–81 proof of, AP-28 Right-handed coordinate frame, 602 Right-handed derivatives, 123–124 Rise, AP-11 Rolle, Michel, 222 Rolle’s Theorem, 222–223 Root finding, 93 Root rule for limits, 62 for limits of functions of two variables, 696 Root Test, 519–520, 533 Roots binomial series for, 551–552 of complex numbers, AP-40–AP-41 finding by Newton’s Method, 267–268 and Intermediate Value Theorem, 92–93 Rotation disk method, 356–359 uniform, 860 Run, AP-11 Saddle points, 640, 741, 742, 745 Sandwich Theorem statement of, 66–67 limits at infinity, 101 limits involving (sin u)/u, 82 proof of, AP-28 for sequences, 490 Savings account growth, 34 Scalar functions, 650 Scalar Multiple Rules for vector functions, 654 Scalar multiplication of vectors, 609–610 Scalar products. See Dot product

Index Scalars, definition of, 609 Scaling, of function graph, 16–18 Scatterplot, 4 Sec x derivatives of, 152 integrals of, 404–405, 441–442 inverse of, 180–181 Secant, trigonometric function, 22 Secant function extended definition of, 22 integral of, 405 inverse of, 47, 180, 184 Secant lines, 54 Secant slope, 55 Second derivative test for concavity, 236 derivation of, two-variable function, AP-48–AP-49 for local extrema, 238–242 summary of, 745 Second moments, 797–800, 885 Second-order differential equation topics covered online applications of second-order equations auxiliary equation boundary value problems complementary equation damped vibrations critical damping overdamping underdamping electric circuits Euler equation Euler’s method existence of second-order solutions form of second-order solutions forced vibrations general solution for linear equations homogeneous equations linear combination linearly independent solutions linearity method of undetermined coefficients nonhomogeneous equations power-series solutions uniqueness of second-order solutions second-order differential equations second-order initial value problems second-order linear equations second-order series solutions simple harmonic motion solution of constant-coefficient second-order linear equations superposition principle theorem on general solution form variation of parameters Second-order partial derivatives, 708–709 Separable differential equations, 413–414 Sequences bounded, 493–495 calculation of, 489–491 convergence of, 487–489 divergence of, 487–489 index of, 486 infinite, 486–495 to infinity, 489

limits of, 488, 490 by Continuous Function Theorem, 490 by l’Hôpital’s Rule, 491–492 by Sandwich Theorem, 490 monotonic, 494 to negative infinity, 489 of partial sums, 499 nondecreasing, 494 recursively defined, 493 Series absolutely convergent, 524–525 adding or deleting terms, 504 alternating, 522–524 harmonic, 522–523 binomial, 550–552 combining, 503–504 conditionally convergent, 524 convergence of, comparison tests for, 512–514 convergent, 499 divergent, 499, 502 error estimation, 509–510 geometric, 500–501 harmonic, 507, 522–523 infinite, 498–504 integral test, 507–510 Maclaurin, 539–540 p-, 509 partial sum of, 498–499 power, 529–536 rearrangement of, 526 reindexing, 504 representations, of functions of power, 528–539 sum of, 498–499 Taylor, 539–540, 543–549, 546–547 tests for absolute convergence, 525 alternating, 522–524 comparison, 513 convergence, 512–514 integral, 507–510 limit comparison, 514–515 ratio, 517–519 root, 519–520 summary of, 526–527 Set, AP-2 Shearing flow, 860 Shell formula for revolution, 367 Shell method, 366–369 Shells, thin, masses and moments of, 885–887 Shift formulas for functions, 16 Shifting, of function graph, 16 Short differential formula, arc length, 375–376 SI units, 383 Sigma notation, 299–304 Simple harmonic motion, 151–152 Simply connected region, 848 Simpson, Thomas, 464 Simpson’s Rule approximations by, 463–465, 467–468 error analysis and, 465–468 Sine(s) extended definition of, 22 integrals of products of, 442–443

I-13

integrals of products of powers of, 439–441 values of, 23 Sine function derivative of, 149–151 graph of, 10 integral of, 439 inverse of, 47, 182 Sinusoid formula, 26 Slant (oblique) asymptote, 102 Slicing with cylinders, 364–366 by parallel planes, 354–355 volume by, 354–355 Slope of curve, 54–56 of nonvertical line, AP-11 of parametrized curves, 565–566 of polar coordinate curve, 584–585 tangent line and, 116–117 Smooth curves, 3–4, 372–373, 652 Smooth surface, 871–872, 875 Snell van Royen, Willebrord, 258 Snell’s Law, 259 Solids Cavalieri’s principle of, 355 cross-section of, 353 three-dimensional, masses and moments, 796, 799 volume calculation of, 354 by disk method, 356–359 by double integrals, 764, 765, 769–772 by method of slicing, 353–360 by triple integrals, 786–787 by washer method, 359–360 Solids of revolution by disk method, 356–359 by washer method, 359–360 Solution of differential equation, 413 particular, 274 Speed along smooth curve, 666 average, 52–54 definition of, 141 instantaneous, 52–54 over short time intervals, 53 of particle in space, 653 related rates equations, 189 Spheres concentric, in vector field, 905–906 parametrization of, 870–871 in space, distance and, 604–605 standard equation for, 604 surface area of, 873–874 Spherical coordinates definition of, 806 triple integrals in, 806–810 Spin around axis, 861–863 Spring constant, 384, 385 Springs Hooke’s law for, 384–385 mass of, 830–831 work to stretch, 385

I-14

Index

Square root function definition of, 8 derivative of, 122 Square roots, elimination of, in integrals, 441 Squeeze Theorem. See Sandwich Theorem St. Vincent, Gregory, 575 Standard linear approximation, 196, 734 Standard unit vectors, 611 Step size, 462 Stokes’ Theorem comparison with Green’s Theorem, 889, 890, 891, 908, 909 conservative fields and, 852, 898 integration in vector fields, 889–898 for polyhedral surfaces, 896–897 surface integral in, 892 for surfaces with holes, 897 Stretching a graph, 17 Substitution and area between curves, 335–341 in double integrals, 814–818 rectangular to polar coordinates, 781 indefinite integrals and, 328–333 in multiple integrals, 814–821 trigonometric, 444–447 in triple integrals, 818–821 rectangular to cylindrical coordinates, 803 rectangular to spherical coordinates, 807 Substitution formula for definite integrals, 335–337 Substitution Rule in definite integrals, 335 definition of, 329 evaluation of indefinite integrals with, 328–333 Subtraction, of vectors, 610 Sum Rule for antiderivatives, 273, 276 for combining series, 503 for definite integrals, 309 derivative, 131–132 for finite sums, 300 of functions of two variables, 696 of geometric series, 503 for gradients, 728 for limits, 62, 75 of sequences, 489 for vector functions, 654 Sums and difference, of double integrals, 773 finite, 526 finite, estimation with, 289–296 limits of, 301–302 lower, 291 partial, sequence of, 499 Riemann. See Riemann sums upper, 290 Surface area defining of, 378–380, 871–875 differential for parametrized surface, 873 of explicit surface, 881 of graph, 877 of implicit surface, 876–877, 881

parametrization of, 871–875 for revolution about y-axis, 380–381 for sphere, 873–874 Surface integrals computation of, 881–883 for flux, 883–885 formulas for, 881 integration in vector fields, 880–887 in Stoke’s Theorem, 892 Surface of revolution, 378 Surfaces and area, 576–577, 870–877 functions defined on, 716–718 with holes, 897 implicit, 875–877 implicitly defined, 875 level, 689 orientable, 883 parametrization of, 870–875 piecewise smooth, 881, 890 plane tangent to, 731–733 quadric, 639–641 smooth, 871–872, 875 two-sided, 883 of two-variable functions, 689 Symmetric functions definite integrals of, 337 graphs of, 6–7 properties of, 6 Symmetry tests, for graphs in polar coordinates, 583 u-limits of integration, finding of, 781–782, 809 System torque, systems of masses, 389–390 Table of integrals, 431, 456–457 Tan x derivatives of, 152 integrals of, 404–405, 441–442 inverses of, 180–181 Tangent(s) to curves, 52–57, 666, 727–728 of curves in space, 649–655 extended definition of, 22 to graph of function, 116–117 to level curves, 727–728 and normals, 167 at point, 116–118 slope of, 54 values of, 23 vertical, 124 Tangent function extended definition of, 22 integral of, 405 inverse of, 47, 180, 184 Tangent line approximation, 196, 734 Tangent lines to curve, 116 instantaneous rates of change and, 56–57 Tangent plane approximation, 734 Tangent planes horizontal, 740–741 and normal lines, 730–733 to a parametric surface, 871 Tangent vector, 652 Tangential component of acceleration, 674–678 Tautochrones, 567

Taylor, Brook, 539 Taylor polynomials, 540–542, 547, 548 Taylor series applying of, 546–547, 552–556 convergence of, 543–549 definition of, 539–540 frequently used, 557 Taylor’s Formula definition of, 543, 544 for functions of two variables, AP-48–AP-51 Taylor’s Theorem definition of, 543 proof of, 548–549 Term-by-term differentiation, 534 Term-by-term integration, 535–536 Term of a sequence, 486 Term of a series, 499 Terminal point of curve, 563 of vector, 607 Theorem(s) Absolute Convergence Test, 525 Algebraic Properties of Natural Logarithm, 43, 403 Alternating Series Estimation, 524, 547 angle between two vectors, 616 Cauchy’s Mean Value, 251–252 Chain Rule, 158 for functions of three variables, 716 for functions of two variables, 714 for two independent variables and three intermediate variables, 717 Comparison Test, 513 conservative fields are gradient fields, 850 Continuous function for sequences, 490 Convergence, for Power Series, 532 curl F 0 related to loop property, 898 De Moivre’s, AP-40 Derivative Rule for Inverses, 171 Differentiability implies continuity, 125, 711 Direct Comparison Test, 514 Divergence, 901–902 Evaluation, 320–322 Exactness of differential forms, 855 Extreme Value, 215–216, AP-32 First derivative test for local extreme values, 217–218, 740 Formula for Implicit Differentiation, 719 Fubini’s, 766, 770–772 Fundamental, 317–325 of Algebra, AP-41–AP-42 of Calculus part 1, 319–320 part 2, 320–322 of Line Integrals, 849 Green’s, 864 Implicit Function, 720, 875 Increment, for Functions of Two Variables, 710, AP-46–AP-48 integrability of continuous functions, 307 Integral Test, 507 Intermediate value, 92–93 Laws of Exponents for e x, 407 l’Hôpital’s Rule, 246–252, 491–492 Limit, proofs of, AP-26–AP-29 Limit Comparison Test, 514–515

Index Limit Laws, 62, 98 Loop property of conservative fields, 851 Mean Value, 222–227, 230, 272, 372–373, 572, AP-44–AP-48 corollary 1, 225 corollary 2, 225–226 for definite integrals, 317–320 Mixed Derivatives, 709, AP-44 Monotonic Sequence, 494–495, 507 Multiplication of power series, 534 Net Change, 322 Nondecreasing Sequence, 494 number e as limit, 177 Orthogonal gradient, 751 Properties of continuous functions, 89 Properties of limits of functions of two variables, 696 Ratio Test, 517 Rearrangement, for Absolutely Convergent Series, 526 Remainder Estimation, 545, 547 Rolle’s, 222–223 Root Test, 519 Sandwich, 66–67, 82, 101, 490, AP-28 Second derivative test for local extrema, 238, 742 Stokes’, 890 Substitution in definite integrals, 335 Substitution Rule, 330 Taylor’s, 543, 548–549 Term-by-Term Differentiation, 534 Term-by-Term Integration, 535–536 Thickness variable, 367 Thin shells, moments and masses of, 885–887 Three-dimensional coordinate systems Cartesian, 602–605 coordinate planes, 602 cylindrical, 804–806 right- and left-handed, 602 spherical, 808–809 Three-dimensional solid, 796, 799 Three-dimensional vectors, component form of, 608 Time-Distance Law (Kepler’s Third Law), 681–682 TNB frame, 674 Torque, 389–390, 627 Torsion, 676–677, 678 Torus, 878 Total differential, 736 Transcendental functions, 11, 406 Transcendental numbers, 406 Transformations Jacobian of, 816, 817 linear, 815–816 of trigonometric graphs, 26 Transitivity law for real numbers, AP-31 Trapezoid, area of, 312 Trapezoidal Rule approximations by, 461–463, 467–468 error analysis and, 465–468 Triangle inequality, AP-5 Trigonometric functions angles, 21–22 derivatives of, 149–153

graphs of, 10, 24, 31–33 transformations of, 26, 47 integrals of, 439–443 inverse, 45–46, 180–184 periodicity of, 24 six basic, 22–23 Trigonometric identities, 24–25 Trigonometric substitutions, 444–447 Triple integrals in cylindrical coordinates, 802–810 properties of, 786–787, 792 in rectangular coordinates, 786–792 in spherical coordinates, 806–810 substitutions in, 818–821 Triple scalar product (box product), 627–628 Tuning fork data, 4 Two-dimensional vectors, component form of, 608 Two-path test for nonexistence of limit, 699 Two-sided limits definition of, 79 proof of, AP-29 Two-sided surface, 883 Unbounded intervals, 6 Unbounded region, 688 Unbounded sequence, 493 Undetermined coefficients, 449 Unified theory, 908–909 Union of set, AP-2 Unit binormal vector, 674 Unit circle, AP-14 Unit normal vector, 670 Unit step functions, limits and, 62 Unit tangent vector, 666–667, 678 Unit vectors definition of, 611 writing vectors in terms of, 611–612 Universal gravitational constant, 680 Upper bound, 310, AP-31 Upper sums, 290 Value(s) absolute, AP-4–AP-6, AP-38 average, 312–313 extreme, 214–219, 740–745 of function, 2–3, 791–792 of improper integral, 471, 474 local maximum, 216–217, 740 local minimum, 216–217, 740 Variable force along curve, 839–841 along line, 383–384 Variable of integration, 276, 307 Variables dependent, 1 dummy, 307 functions of several, 686–691, 700, 707–708, 720 independent, 1, 714, 716 input, 686 intermediate, 715–716 output, 686 proportional, 7 thickness, 367–368

I-15

three, functions of, 689–691, 728–729 Chain Rule for, 716 two, functions of, 687–688, 691, 710 Chain Rule for, 714–716 independent, and three intermediate, 717 limits for, 694–698 linearization of, 733–735, 736 partial derivatives of, 703–705 Taylor’s formula for, AP-48–AP-51 Vector equations for curves in space, 649 for lines, 630, 631–632 of plane, 633 for projectile motion, 659–661 Vector fields conservative, 847, 850–852 continuous, 835 curl of, 889–890 definition of, 835–836 differentiable, 835 divergence of, 860–861 electric, 847 flux density of, 860 gradient, 836–837, 849–850 gravitational, 847 integration in, 828–916 and line integrals, 834–844 line integrals of, definition of, 837–838 potential function for, 848 Vector functions antiderivatives of, 657 of constant length, 655 continuity of, 650–651 curves in space and, 649–655 definite integral of, 658–659 derivatives of, definition of, 652 differentiable, 652 differentiation rules for, 653–655 indefinite integral of, 657–659 integrals of, 657–661 limits of, 650–651 Vector product. See Cross product Vector-valued functions. See Vector functions Vectors acceleration, 653, 674 addition of, 609–610, 617 algebra operations with, 609–611 angle between, 616–618 applications of, 613–614 binormal, of curve, 674 component form of, 607–609 coplanar, 611 cross product as area of parallelogram, 625 in component form, 625–627 definition of, 624 as determinant, 625–627 right-hand rule for, 624 of two vectors in space, 624–625 curl, 889–890 definition of, 607 direction of, 611 dot product, definition of, 616 equality of, 607

I-16

Index

Vectors (continued ) and geometry in space, 602–648 gradient, 725 in gravitational field, 835 i-component of, 611 initial point of, 607 j-component of, 611 k-component of, 611 length (magnitude) of, 608, 609–610 midpoint of line segments, 612–614 in navigation, 613 normal, of curve, 672–673 notation for, 607 parallel, 624 perpendicular (orthogonal), 618–619 in physics and engineering, 613–614 position, standard, 607–608 principal unit normal, 670, 678 projection of, 619–621 resultant, 609–610 scalar multiplication of, 609–610 standard position, for a point, 607–608 standard unit, 611 subtraction (difference) of, 610 tangent, of curve, 652 terminal point of, 607 three-dimensional, 608 torque, 627 Triple scalar product of, 627–628 two-dimensional, 608, 618 unit definition of, 611–612 derivative in direction of, 724 writing vectors in terms of, 611–612 unit binormal, 674 unit normal, 672 unit tangent, 666–667

velocity, 607, 653 zero vector, 608 Velocity along space curve, 653 angular, of rotation, 895 average, 140 definition of, 140 free fall and, 142 instantaneous, 140–141 in polar coordinates, 679–682 and position, from acceleration, 226 Velocity fields circulation for, 841–842 flow integral, 841–842 Velocity function acceleration and, 226, 653 speed and, 294 Vertical asymptotes. See also Asymptotes definition of, 105–106 limits and, 97 Vertical line test, 4–5 Vertical scaling and reflecting formulas, 17 Vertical shift of function, 16 Vertical strip, 364, 392 Vertical tangents, 124 Viewing windows, 29–32 Volume of cylinder, 353 differential in cylindrical coordinates, 803 in spherical coordinates, 807 by disks for rotation about axis, 356 double integrals as, 764–765 by iterated integrals, 769–772 of pyramid, 354–355 of region in space, 787 by slicing, 354–355 of solid region, 769–772 of solid with known cross-section, 354 triple integrals as, 787 using cross-sections, 353–360

using cylindrical shells, 364–369 by washers for rotation about axis, 359 von Koch, Helga, 506 Washer method, 359–360, 369 Wave equation, 713 Weierstrass, Karl, 477 Whirlpool effect, 860 Windows, graphing, 29–32 Work by constant force, 383 by force over curve in space, 839–841 by force through displacement, 621 Hooke’s Law for springs, 384–385 and kinetic energy, 388 pumping liquids from containers, 385–386 by variable force along curve, 840 by variable force along line, 383–384 x-coordinate, AP-10 x-intercept, AP-13 x-limits of integration, 789, 791 xy-plane definition of, 602 xz-plane, 602 y, integration with respect to, 340–341 y = ƒ(x) graphing of, 240–242 length of, 372–374, 575 y-axis, revolution about, 380–381 y-coordinate, AP-10 y-intercept, AP-13 y-limits of integration, 788, 790 yz-plane, 602 z-limits of integration, 788, 789, 790, 804 Zero denominators, algebraic elimination of, 64 Zero vector, 608 Zero Width Interval Rule, 309, 402

CREDITS Page ix, Copyright © 2009 Josef Hoflehner; Page 1, Chapter 1 opener photo, Gordon Wiltsie/National Geographic/Getty Images; Page 52, Chapter 2 opener photo, Light Thru My Lens/Getty Images; Page 116, Chapter 3 opener photo, Datacraft/Getty Images; Page 147, Section 3.4, photo for Exercise 19, PSSC Physics, 2nd ed., DC Heath & Co. with Education Development Center, Inc.; Page 209, Practice Exercises 123 and 124, NCPMF “Differentiation” by W.U. Walton et al., Project CALC, Education Development Center, Inc.; Page 214, Chapter 4 opener photo, Comstock/Getty Images; Page 289, Chapter 5 opener photo, Victoriano Izquierdo/Getty Images; Page 353, Chapter 6 opener photo, Livio Sinibaldi/Getty Images; Page 390, Figure 6.40, PSSC Physics, 2nd ed., DC Heath & Co. with Education Development Center, Inc.; Page 401, Chapter 7 opener photo, Datacraft/Getty Images; Page 431, Chapter 8 opener photo, Chris Leschinsky/Getty Images; Page 486, Chapter 9 opener photo, DAJ/Getty Images; Page 501, Figure 9.9, PSSC Physics, 2nd ed., DC Heath & Co. with Education Development Center, Inc.; Page 563, Chapter 10 opener photo, Yoshio Sawaragi/ImagwerksRF/Getty Images; Page 602, Chapter 11 opener photo, Ellen Isaacs/Alamy; Page 649, Chapter 12 opener photo, Datacraft/Getty Images; Page 686, Chapter 13 opener photo, Dagny Willis/Getty Images; Page 763, Chapter 14 opener photo, Lanz von Horsten/Getty Images; Page 828, Chapter 15 opener photo, Karl Weatherly/Getty Images; Page 836, Figure 15.15 NASA; Online Chapter 16 opener photo, Yoshio Sawaragi/ImagwerksRF/Getty Images.

C-1

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A BRIEF TABLE OF INTEGRALS Basic Forms 1. 3.

L L

k dx = kx + C

sany number kd

dx x = ln ƒ x ƒ + C

ax a x dx = + C 5. ln a L 7. 9. 11. 13. 15.

L L L L L

2. 4.

sa 7 0, a Z 1d

6.

cos x dx = sin x + C

8.

csc2 x dx = -cot x + C

10.

csc x cot x dx = -csc x + C

12.

cot x dx = ln ƒ sin x ƒ + C

14.

cosh x dx = sinh x + C

16.

x dx 1 = a tan-1 a + C 2 2 La + x x dx = sinh-1 a + C 19. 2 2 L 2a + x

L L L L L L L

x n dx =

xn+1 + C n + 1

sn Z -1d

e x dx = e x + C sin x dx = -cos x + C sec2 x dx = tan x + C sec x tan x dx = sec x + C tan x dx = ln ƒ sec x ƒ + C sinh x dx = cosh x + C

x dx = sin-1 a + C L 2a 2 - x 2 x dx 1 18. = a sec-1 ` a ` + C L x2x 2 - a 2 dx x = cosh-1 a + C sx 7 a 7 0d 20. 2 2 L 2x - a

17.

sa 7 0d

Forms Involving ax b 21. 22. 23. 25.

27.

L L L L L

sax + bdn dx =

sax + bdn + 1 + C, asn + 1d

xsax + bdn dx =

n Z -1

sax + bdn + 1 ax + b b c d + C, n + 2 n + 1 a2

1 sax + bd-1 dx = a ln ƒ ax + b ƒ + C xsax + bd-2 dx =

A 2ax + b B

n

b 1 c ln ƒ ax + b ƒ + d + C ax + b a2

2 A 2ax + b B dx = a n + 2

n Z -1, -2 24.

x b xsax + bd-1 dx = a - 2 ln ƒ ax + b ƒ + C a L

26.

dx x 1 ` + C = ln ` b ax + b L xsax + bd

28.

2ax + b dx = 22ax + b + b x

n+2

+ C,

n Z -2

L

dx L x2ax + b

T-1

T-2

A Brief Table of Integrals

29. (a) 30.

L

dx L x2ax + b

2ax + b - 2b 1 ln ` ` + C 2b 2ax + b + 2b

=

2ax + b 2ax + b a dx + dx = + C x 2 L x2ax + b x2

(b) 31.

dx L x2ax - b

=

ax - b 2 tan-1 + C b A 2b

2ax + b dx a dx = + C bx 2b L x2ax + b L x 2 2ax + b

Forms Involving a2 x2 x dx x x dx 1 1 + = a tan-1 a + C 33. = tan-1 a + C 2 2 2 2 2 2 3 2 2 a + x sa + x d 2a sa + x d 2a L L dx -1 x 2 2 = sinh a + C = ln A x + 2a + x B + C 34. L 2a 2 + x 2 a2 x 35. 2a 2 + x 2 dx = 2a 2 + x 2 + ln A x + 2a 2 + x 2 B + C 2 2 L x a4 ln A x + 2a 2 + x 2 B + C 36. x 2 2a 2 + x 2 dx = sa 2 + 2x 2 d2a 2 + x 2 8 8 L 32.

37.

38.

39.

40.

L

2a 2 + x 2 a + 2a 2 + x 2 dx = 2a 2 + x 2 - a ln ` ` +C x x

2a 2 + x 2 2a 2 + x 2 2 2 dx = ln x + 2a + x + C A B x x2 L x2 L 2a 2 + x 2

dx = -

x2a 2 + x 2 a2 ln A x + 2a 2 + x 2 B + + C 2 2

a + 2a 2 + x 2 1 = - a ln ` ` + C x

dx L x2a 2 + x 2

41.

dx L x 2 2a 2 + x 2

= -

2a 2 + x 2 + C a 2x

Forms Involving a2 x2 42. 44. 46.

47.

49.

51.

dx x + a 1 = ln ` x - a ` + C 2 2 2a La - x dx L 2a 2 - x 2 L L

x = sin-1 a + C

x 2 2a 2 - x 2 dx =

43. 45.

x2 L 2a - x

2

dx =

dx L x 2 2a 2 - x 2

50.

2a 2 - x 2 + C a 2x

Forms Involving x2 a2 52. 53.

dx L 2x - a 2 2

L

= ln ƒ x + 2x 2 - a 2 ƒ + C

2x 2 - a 2 dx =

x a 2 -1 x 2a 2 - x 2 + sin a + C 2 2

a x 1 sin-1 a - x2a 2 - x 2 sa 2 - 2x 2 d + C 8 8

a 2 -1 x 1 sin a - x2a 2 - x 2 + C 2 2

= -

L

2a 2 - x 2 dx =

4

2a 2 - x 2 a + 2a 2 - x 2 dx = 2a 2 - x 2 - a ln ` ` + C 48. x x

2

dx x x + a 1 = ln ` x - a ` + C + 2 2 2 2a 2sa 2 - x 2 d 4a 3 L sa - x d

x a2 2x 2 - a 2 ln ƒ x + 2x 2 - a 2 ƒ + C 2 2

2a 2 - x 2 2a 2 - x 2 x dx = -sin-1 a + C x 2 x L dx L x2a - x 2

2

a + 2a 2 - x 2 1 = - a ln ` ` + C x

A Brief Table of Integrals

54.

55.

56.

57.

58.

A 2x - a B dx =

L

dx

L A 2x - a 2

L

B

2 n

x A 2x - a 2

L

x A 2x 2 - a 2 B

2 n

2

=

n + 1

x A 2x 2 - a 2 B s2 - nda

B dx =

-

n-2 na 2 2x 2 - a 2 B dx, A n + 1L

-

n - 3 dx , 2 2 sn - 2da L A 2x - a 2 B n - 2

2-n

2

A 2x 2 - a 2 B

2 n

x 2 2x 2 - a 2 dx =

n

n Z -1 n Z 2

n+2

n + 2

+ C,

n Z -2

x a4 s2x 2 - a 2 d2x 2 - a 2 ln ƒ x + 2x 2 - a 2 ƒ + C 8 8

2x 2 - a 2 x dx = 2x 2 - a 2 - a sec-1 ` a ` + C x

L

2x 2 - a 2 2x 2 - a 2 2 2 dx = ln x + 2x a + C ƒ ƒ x x2 L x2 a2 x dx = 60. ln ƒ x + 2x 2 - a 2 ƒ + 2x 2 - a 2 + C 2 2 2 2 L 2x - a 59.

61.

dx L x2x 2 - a 2

x a 1 1 = a sec-1 ` a ` + C = a cos-1 ` x ` + C

62.

dx L x 2 2x 2 - a 2

=

2x 2 - a 2 + C a 2x

Trigonometric Forms 63. 65. 67.

L

1 sin ax dx = - a cos ax + C sin2 ax dx =

L

sin 2ax x + C 2 4a

sinn ax dx = -

L

64.

sinn - 1 ax cos ax n - 1 + n na

66.

L

cosn ax dx =

L

(b) (c) 70. 72. 74. 75.

76.

L

L L

cos2 ax dx =

sin 2ax x + + C 2 4a

cosn - 1 ax sin ax n - 1 + n na

a2 Z b2

sinsa - bdx sinsa + bdx + C, 2sa - bd 2sa + bd

a2 Z b2

cos ax cos bx dx =

sinsa - bdx sinsa + bdx + + C, 2sa - bd 2sa + bd

a2 Z b2

sin ax cos ax dx = -

cos 2ax + C 4a

cos ax 1 dx = a ln ƒ sin ax ƒ + C sin ax L L

L

sin ax sin bx dx =

L L

1 cos ax dx = a sin ax + C

sinn - 2 ax dx

cosn - 2 ax dx L cossa + bdx cossa - bdx 69. (a) sin ax cos bx dx = + C, 2sa + bd 2sa - bd L 68.

L

71. 73.

L L

sinn ax cos ax dx =

sinn + 1 ax + C, sn + 1da

cosn ax sin ax dx = -

n Z -1

cosn + 1 ax + C, sn + 1da

sin ax 1 cos ax dx = - a ln ƒ cos ax ƒ + C sinn ax cosm ax dx = sinn ax cosm ax dx =

sinn - 1 ax cosm + 1 ax n - 1 + sinn - 2 ax cosm ax dx, m + nL asm + nd

sinn + 1 ax cosm - 1 ax m - 1 sinn ax cosm - 2 ax dx, + m + nL asm + nd

n Z -m m Z -n

sreduces sinn axd sreduces cosm axd

n Z -1

T-3

T-4

A Brief Table of Integrals

77.

dx ax b - c p -2 b d + C, = tan-1 c tan a 4 2 Ab + c L b + c sin ax a2b 2 - c 2

78.

c + b sin ax + 2c 2 - b 2 cos ax dx -1 ln ` ` + C, = 2 2 b + c sin ax b + c sin ax L a2c - b

79.

dx ax p 1 = - a tan a b + C 4 2 L 1 + sin ax

81.

dx b - c ax 2 = tan-1 c tan d + C, 2 Ab + c L b + c cos ax a2b 2 - c 2

82.

c + b cos ax + 2c 2 - b 2 sin ax dx 1 = ln ` ` + C, 2 2 b + c cos ax b + c cos ax L a2c - b

83.

ax dx 1 = a tan + C 1 + cos ax 2 L

85.

L

87. 89. 91. 93.

97.

84.

x 1 sin ax - a cos ax + C a2

86.

L L

1 tan ax dx = a ln ƒ sec ax ƒ + C

90.

L

1 tan2 ax dx = a tan ax - x + C

92.

tann ax dx =

L

x n - 1 cos ax dx

n-1

ax tan tann - 2 ax dx, asn - 1d L

88.

n Z 1

94.

L

1 sec ax dx = a ln ƒ sec ax + tan ax ƒ + C

96.

L

1 sec2 ax dx = a tan ax + C

98.

secn - 2 ax tan ax n - 2 secn - 2 ax dx, secn ax dx = + 99. n - 1L asn 1d L 100.

L

101.

L

cscn ax dx = -

secn ax na + C,

n Z 0

ax dx p 1 = a tan a + b + C 4 2 L 1 - sin ax

b2 6 c2

dx ax 1 = - a cot + C 1 cos ax 2 L L

x cos ax dx =

x 1 cos ax + a sin ax + C a2

L

n xn x n cos ax dx = a sin ax - a

L

1 cot ax dx = a ln ƒ sin ax ƒ + C

L

1 cot2 ax dx = - a cot ax - x + C

L

cotn ax dx = -

L

x n - 1 sin ax dx

cotn - 1 ax cotn - 2 ax dx, asn - 1d L

L

1 csc ax dx = - a ln ƒ csc ax + cot ax ƒ + C

L

1 csc2 ax dx = - a cot ax + C

n Z 1

n Z 1

cscn - 2 ax cot ax n - 2 + cscn - 2 ax dx, n - 1L asn - 1d

secn ax tan ax dx =

b2 6 c2

b2 7 c2

n xn x n sin ax dx = - a cos ax + a

L

95.

x sin ax dx =

80.

b2 7 c2

n Z 1 102.

L

cscn ax cot ax dx = -

cscn ax na + C,

n Z 0

Inverse Trigonometric Forms 103. 105.

L L

1 sin-1 ax dx = x sin-1 ax + a 21 - a 2x 2 + C tan-1 ax dx = x tan-1 ax -

104.

L

1 cos-1 ax dx = x cos-1 ax - a 21 - a 2x 2 + C

1 ln s1 + a 2x 2 d + C 2a

xn+1 a x n + 1 dx sin-1 ax , n Z -1 n + 1 n + 1 L 21 - a 2x 2 L xn+1 a x n + 1 dx 107. cos-1 ax + , n Z -1 x n cos-1 ax dx = n + 1 n + 1 L 21 - a 2x 2 L 106.

108.

L

x n sin-1 ax dx =

x n tan-1 ax dx =

x n + 1 dx xn+1 a tan-1 ax , n + 1 n + 1 L 1 + a 2x 2

n Z -1

A Brief Table of Integrals

Exponential and Logarithmic Forms 109.

111. 113. 114. 115.

L L L L L

1 e ax dx = a e ax + C xe ax dx =

110.

e ax sax - 1d + C a2

x nb ax dx =

112.

x nb ax n x n - 1b ax dx, a ln b a ln b L

1 b ax b ax dx = a + C, ln b L L

n 1 x ne ax dx = a x ne ax - a

b 7 0, b Z 1

L

x n - 1e ax dx

b 7 0, b Z 1

e ax sin bx dx =

e ax sa sin bx - b cos bxd + C a + b2

e ax cos bx dx =

e ax sa cos bx + b sin bxd + C a + b2

2

2

116.

L

ln ax dx = x ln ax - x + C

x n + 1sln axdm m x nsln axdm - 1 dx, n Z -1 n + 1 n + 1L L sln axdm + 1 dx + C, m Z -1 x -1sln axdm dx = = ln ƒ ln ax ƒ + C 118. 119. m + 1 x ln ax L L x nsln axdm dx =

117.

Forms Involving 22ax x2, a>0 120. 121.

122.

123.

124. 125.

dx x - a = sin-1 a a b + C L 22ax - x 2 L

x - a a 2 -1 x - a 22ax - x 2 + sin a a b + C 2 2

22ax - x 2 dx =

A 22ax - x B dx = 2 n

L

dx

L A 22ax - x L L

B

2 n

sx - ad A 22ax - x 2 B n + 1

sx - ad A 22ax - x 2 B

=

sn - 2da

x22ax - x 2 dx =

n

+

n-2 na 2 22ax - x 2 B dx A n + 1L

+

n - 3 dx 2 sn - 2da L A 22ax - x 2 B n - 2

2-n

2

sx + ads2x - 3ad22ax - x 2 a 3 -1 x - a + sin a a b + C 6 2

22ax - x 2 x - a dx = 22ax - x 2 + a sin-1 a a b + C x

126.

22ax - x 2 2a - x x - a dx = -2 - sin-1 a a b + C A x x2 L

127.

x dx x - a = a sin-1 a a b - 22ax - x 2 + C L 22ax - x 2

128.

dx 1 2a - x = -a + C A x L x22ax - x 2

Hyperbolic Forms 129. 131. 133.

L L L

1 sinh ax dx = a cosh ax + C sinh2 ax dx = sinhn ax dx =

130.

sinh 2ax x - + C 4a 2 sinh

n-1

ax cosh ax n - 1 - n na

132.

L

sinhn - 2 ax dx,

L L

1 cosh ax dx = a sinh ax + C cosh2 ax dx =

n Z 0

sinh 2ax x + + C 4a 2

T-5

T-6 134. 135. 137. 139. 141. 143. 144. 145. 147. 149. 150. 151. 153. 154.

A Brief Table of Integrals

L

coshn ax dx =

coshn - 1 ax sinh ax n - 1 + n na

L

coshn - 2 ax dx,

x 1 x sinh ax dx = a cosh ax - 2 sinh ax + C a L xn n x n sinh ax dx = a cosh ax - a

L

L 1 tanh ax dx = a ln scosh axd + C

L

1 tanh2 ax dx = x - a tanh ax + C

L

L L

n Z 0

136.

x n - 1 cosh ax dx

138. 140. 142.

tanhn ax dx = -

tanhn - 1 ax + tanhn - 2 ax dx, sn - 1da L

n Z 1

cothn ax dx = -

cothn - 1 ax + cothn - 2 ax dx, sn - 1da L

n Z 1

L

1 sech ax dx = a sin-1 stanh axd + C

146.

L

1 sech2 ax dx = a tanh ax + C

148.

L L L L L

sechn ax dx =

sechn - 2 ax tanh ax n - 2 + sechn - 2 ax dx, n - 1L sn - 1da

cschn ax dx = -

sechn ax + C, na

n Z 0

152.

e ax sinh bx dx =

e ax e bx e -bx c d + C, 2 a + b a - b

a2 Z b2

e ax cosh bx dx =

e ax e bx e -bx c + d + C, 2 a + b a - b

a2 Z b2

xn n x n cosh ax dx = a sinh ax - a

L

L 1 coth ax dx = a ln ƒ sinh ax ƒ + C

L

1 coth2 ax dx = x - a coth ax + C

L

ax 1 csch ax dx = a ln ` tanh ` + C 2

L

1 csch2 ax dx = - a coth ax + C

L

n Z 1

L

cschn ax coth ax dx = -

cschn ax + C, na

Some Definite Integrals q

155.

L0

x n - 1e -x dx = ≠snd = sn - 1d!,

p>2

157.

p>2

sin x dx = n

L0

L0

q

n 7 0

156.

1 # 3 # 5 # Á # sn - 1d # p , 2#4#6# Á #n 2 n cos x dx = d # # # Á # 2 4 6 sn - 1d , 3#5#7# Á #n

x n - 1 sinh ax dx

n Z 1

cschn - 2 ax coth ax n - 2 cschn - 2 ax dx, n - 1L sn - 1da

sechn ax tanh ax dx = -

x 1 x cosh ax dx = a sinh ax - 2 cosh ax + C a L

L0

e -ax dx = 2

1 p , 2A a

a 7 0

if n is an even integer Ú2 if n is an odd integer Ú3

n Z 0

BASIC ALGEBRA FORMULAS Arithmetic Operations asb + cd = ab + ac,

a#c ac = b d bd

c ad + bc a + = , b d bd

a>b a d = #c b c>d

Laws of Signs -a a a = - = b b -b

-s -ad = a, Zero Division by zero is not defined. 0 If a Z 0: a = 0,

a 0 = 1,

0a = 0

For any number a: a # 0 = 0 # a = 0

Laws of Exponents a ma n = a m + n,

sabdm = a mb m,

sa m dn = a mn,

n

a m>n = 2a m =

n aBm A2

If a Z 0, am = a m - n, an

a 0 = 1,

a -m =

1 . am

The Binomial Theorem For any positive integer n, sa + bdn = a n + na n - 1b + +

nsn - 1d n - 2 2 a b 1#2

nsn - 1dsn - 2d n - 3 3 a b + Á + nab n - 1 + b n . 1#2#3

For instance, sa + bd2 = a 2 + 2ab + b 2,

sa - bd2 = a 2 - 2ab + b 2

sa + bd3 = a 3 + 3a 2b + 3ab 2 + b 3,

sa - bd3 = a 3 - 3a 2b + 3ab 2 - b 3.

Factoring the Difference of Like Integer Powers, n>1 a n - b n = sa - bdsa n - 1 + a n - 2b + a n - 3b 2 + Á + ab n - 2 + b n - 1 d For instance, a 2 - b 2 = sa - bdsa + bd, a 3 - b 3 = sa - bdsa 2 + ab + b 2 d, a 4 - b 4 = sa - bdsa 3 + a 2b + ab 2 + b 3 d. Completing the Square If a Z 0, ax 2 + bx + c = au 2 + C

au = x + sb>2ad, C = c -

The Quadratic Formula If a Z 0 and ax 2 + bx + c = 0, then x =

-b ; 2b 2 - 4ac . 2a

b2 b 4a

GEOMETRY FORMULAS A = area, B = area of base, C = circumference, S = lateral area or surface area, V = volume Triangle

Similar Triangles c'

c h

Pythagorean Theorem

a'

a

c

b

b'

b

b

a

a' b' c' a5b5c

A 5 1 bh 2

Parallelogram

a2 1

Trapezoid

2

5 c2

Circle a

h h

r

b

A 5 pr 2, C 5 2pr

b

A 5 bh

A 5 1 ( 1 )h 2

Any Cylinder or Prism with Parallel Bases

Right Circular Cylinder r

h

h

h

V 5 Bh

B

B V 5 pr2h S 5 2prh 5 Area of side

Any Cone or Pyramid

Right Circular Cone

h

h

B

Sphere

V5

1 Bh 3

B

V 5 1 pr2h 3 S 5 prs 5 Area of side

V 5 43 pr3, S 5 4pr2

LIMITS General Laws

Specific Formulas

If L, M, c, and k are real numbers and

If Psxd = an x n + an - 1 x n - 1 + Á + a0 , then

lim ƒsxd = L

and

x:c

lim gsxd = M,

x:c

then

lim Psxd = Pscd = an c n + an - 1 c n - 1 + Á + a0 .

x:c

lim sƒsxd + gsxdd = L + M

Sum Rule:

x:c

lim sƒsxd - gsxdd = L - M

Difference Rule:

x:c

If P(x) and Q(x) are polynomials and Qscd Z 0, then

lim sƒsxd # gsxdd = L # M

Product Rule:

Psxd Pscd = . Qscd x:c Qsxd lim

x:c

Constant Multiple Rule:

lim sk # ƒsxdd = k # L

x:c

ƒsxd L = , M x:c gsxd

Quotient Rule:

lim

M Z 0 If ƒ(x) is continuous at x = c, then

The Sandwich Theorem

lim ƒsxd = ƒscd.

If gsxd … ƒsxd … hsxd in an open interval containing c, except possibly at x = c, and if

x:c

lim gsxd = lim hsxd = L,

x:c

x:c

then limx:c ƒsxd = L.

lim

x:0

sin x x = 1

and

lim

x:0

1 - cos x = 0 x

Inequalities

L’Hôpital’s Rule

If ƒsxd … gsxd in an open interval containing c, except possibly at x = c, and both limits exist, then

If ƒsad = gsad = 0, both ƒ¿ and g¿ exist in an open interval I containing a, and g¿sxd Z 0 on I if x Z a, then

lim ƒsxd … lim gsxd.

x:c

x:c

lim

x:a

Continuity

ƒsxd ƒ¿sxd = lim , gsxd x:a g¿sxd

assuming the limit on the right side exists.

If g is continuous at L and limx:c ƒsxd = L, then lim g(ƒsxdd = gsLd.

x:c

DIFFERENTIATION RULES General Formulas

Inverse Trigonometric Functions

Assume u and y are differentiable functions of x. d Constant: scd = 0 dx du d dy su + yd = Sum: + dx dx dx d du dy su - yd = Difference: dx dx dx d du scud = c Constant Multiple: dx dx d dy du suyd = u Product: + y dx dx dx

d 1 ssin-1 xd = dx 21 - x 2

d 1 scos-1 xd = dx 21 - x 2

d 1 stan-1 xd = dx 1 + x2

d 1 ssec-1 xd = 2 dx - 1 x 2x ƒ ƒ

d 1 scot-1 xd = dx 1 + x2

d 1 scsc-1 xd = 2 dx x 2x - 1 ƒ ƒ

du dy y - u d u dx dx a b = dx y y2 d n x = nx n - 1 dx d sƒsgsxdd = ƒ¿sgsxdd # g¿sxd dx

Quotient: Power: Chain Rule:

Hyperbolic Functions d ssinh xd = cosh x dx d stanh xd = sech2 x dx d scoth xd = -csch2 x dx

d scosh xd = sinh x dx d ssech xd = -sech x tanh x dx d scsch xd = -csch x coth x dx

Inverse Hyperbolic Functions Trigonometric Functions d ssin xd = cos x dx d stan xd = sec2 x dx d scot xd = -csc2 x dx

d scos xd = -sin x dx d ssec xd = sec x tan x dx d scsc xd = -csc x cot x dx

Exponential and Logarithmic Functions d x e = ex dx d x a = a x ln a dx

d 1 ln x = x dx d 1 sloga xd = dx x ln a

d 1 ssinh-1 xd = dx 21 + x 2

d 1 scosh-1 xd = 2 dx 2x - 1

d 1 stanh-1 xd = dx 1 - x2

d 1 ssech-1 xd = dx x21 - x 2

d 1 scoth-1 xd = dx 1 - x2

d 1 scsch-1 xd = dx ƒ x ƒ 21 + x 2

Parametric Equations If x = ƒstd and y = gstd are differentiable, then y¿ =

dy>dt dy = dx dx>dt

and

d 2y dx

2

=

dy¿>dt . dx>dt

INTEGRATION RULES General Formulas a

Zero:

ƒsxd dx = 0

La a

Order of Integration:

b

ƒsxd dx = -

Lb

ƒsxd dx

La

b

b

kƒsxd dx = k ƒsxd dx La La

Constant Multiples:

b

b

-ƒsxd dx = -

La

La

b

sƒsxd ; gsxdd dx =

La b

La

b

ƒsxd dx ;

c

ƒsxd dx +

Additivity:

sk = -1d

ƒsxd dx

b

Sums and Differences:

sAny number kd

La

gsxd dx

c

ƒsxd dx =

ƒsxd dx La Lb La Max-Min Inequality: If max ƒ and min ƒ are the maximum and minimum values of ƒ on [a, b], then min ƒ # sb - ad …

ƒsxd dx … max ƒ # sb - ad.

b

La b

ƒsxd Ú gsxd

Domination:

on

[a, b]

implies

La

b

ƒsxd dx Ú

La

gsxd dx

b

ƒsxd Ú 0

on

[a, b]

implies

La

ƒsxd dx Ú 0

The Fundamental Theorem of Calculus x

Part 1 If ƒ is continuous on [a, b], then Fsxd = 1a ƒstd dt is continuous on [a, b] and differentiable on (a, b) and its derivative is ƒ(x); x

F¿(x) =

d ƒstd dt = ƒsxd. dx La

Part 2 If ƒ is continuous at every point of [a, b] and F is any antiderivative of ƒ on [a, b], then b

La

ƒsxd dx = Fsbd - Fsad.

Integration by Parts

Substitution in Definite Integrals ƒsgsxdd # g¿sxd dx =

b

La

gsbd

Lgsad

ƒsud du

ƒsxdg¿sxd dx = ƒsxdgsxd D a -

b

La

b

b

La

ƒ¿sxdgsxd dx

tan A + tan B 1 - tan A tan B tan A - tan B tan sA - Bd = 1 + tan A tan B

Trigonometry Formulas

tan sA + Bd = y

1. Definitions and Fundamental Identities y 1 Sine: sin u = r = csc u Cosine:

x 1 cos u = r = sec u

Tangent:

y 1 tan u = x = cot u

P(x, y) r 0

y

x

x

cos s -ud = cos u

sin2 u + cos2 u = 1,

sec2 u = 1 + tan2 u,

sin 2u = 2 sin u cos u, cos2 u =

1 + cos 2u , 2

p b = -cos A, 2

sin aA +

p b = cos A, 2

sin A sin B =

2. Identities sin s -ud = -sin u,

sin aA -

csc2 u = 1 + cot2 u

cos 2u = cos2 u - sin2 u sin2 u =

1 - cos 2u 2

1 1 cos sA - Bd + cos sA + Bd 2 2

sin A cos B =

1 1 sin sA - Bd + sin sA + Bd 2 2

sin A + sin B = 2 sin

1 1 sA + Bd cos sA - Bd 2 2

sin A - sin B = 2 cos

1 1 sA + Bd sin sA - Bd 2 2 1 1 sA + Bd cos sA - Bd 2 2

cos A - cos B = -2 sin

cos sA - Bd = cos A cos B + sin A sin B

1 1 sA + Bd sin sA - Bd 2 2

y

y y sin x

y cos x

Trigonometric Functions Radian Measure

Degrees

1

2 45

C ir

2

90

3 2 2

x

– – 2

2

0

3 2 2

x

Domain: (– , ) Range: [–1, 1]

2

y

1

y

y tan x

y sec x

e

it cir cl

cle of

1

4

1

r

Un

1

θ

2

0

y sinx Domain: (– , ) Range: [–1, 1]

4

45 s

– – 2

Radians

p b = -sin A 2

cos A cos B =

cos A + cos B = 2 cos

cos sA + Bd = cos A cos B - sin A sin B

cos aA +

p b = sin A 2

1 1 cos sA - Bd - cos sA + Bd 2 2

sin sA + Bd = sin A cos B + cos A sin B sin sA - Bd = sin A cos B - cos A sin B

cos aA -

s ra diu

1

r 6

30 3

2

60

u s s or u = r , r = 1 = u 180° = p radians .

90 1

– 3 – – 2 2 3

2 3

2

0 3 2 2

x

Domain: All real numbers except odd integer multiples of /2 Range: (– , )

3 2 2

– 3 – – 0 2 2

x

Domain: All real numbers except odd integer multiples of /2 Range: (– , –1] h [1, )

1 y

y

y csc x

The angles of two common triangles, in degrees and radians. 1 – – 0 2

y cot x

1 2

3 2 2

Domain: x 0, , 2, . . . Range: (– , –1] h [1, )

x

– – 0 2

2

3 2 2

Domain: x 0, , 2, . . . Range: (– , )

x

SERIES Tests for Convergence of Infinite Series 1. The nth-Term Test: Unless an : 0, the series diverges. 2. Geometric series: gar n converges if ƒ r ƒ 6 1; otherwise it diverges. 3. p-series: g1>n p converges if p 7 1; otherwise it diverges. 4. Series with nonnegative terms: Try the Integral Test, Ratio Test, or Root Test. Try comparing to a known series with the Comparison Test or the Limit Comparison Test.

5. Series with some negative terms: Does g ƒ an ƒ converge? If yes, so does gan since absolute convergence implies convergence. 6. Alternating series: gan converges if the series satisfies the conditions of the Alternating Series Test.

Taylor Series q

1 = 1 + x + x 2 + Á + x n + Á = a x n, 1 - x n=0

ƒxƒ 6 1

q

1 = 1 - x + x 2 - Á + s -xdn + Á = a s -1dnx n, 1 + x n=0 q

ex = 1 + x + sin x = x cos x = 1 -

xn x2 xn + Á + + Á = a , 2! n! n = 0 n!

ƒxƒ 6 1

ƒxƒ 6 q

q s -1dnx 2n + 1 x5 x3 x 2n + 1 + - Á + s -1dn + Á = a , 3! 5! s2n + 1d! n = 0 s2n + 1d! q s -1dnx 2n x4 x2 x 2n + - Á + s -1dn + Á = a , 2! 4! s2nd! s2nd! n=0

ln s1 + xd = x -

ƒxƒ 6 q

ƒxƒ 6 q

q s -1dn - 1x n x2 x3 xn + - Á + s -1dn - 1 n + Á = a , n 2 3 n=1

-1 6 x … 1

q

ln

1 + x x5 x3 x 2n + 1 x 2n + 1 = 2 tanh-1 x = 2 ax + + + Á b = 2a , + Á + 5 1 - x 3 2n + 1 2n + 1 n=0

tan-1 x = x -

q s -1dnx 2n + 1 x5 x3 x 2n + 1 + + Á = a , - Á + s -1dn 5 3 2n + 1 2n + 1 n=0

ƒxƒ 6 1

ƒxƒ … 1

Binomial Series s1 + xdm = 1 + mx +

msm - 1dx 2 msm - 1dsm - 2dx 3 msm - 1dsm - 2d Á sm - k + 1dx k + + Á + + Á 2! 3! k!

q m = 1 + a a bx k, k=1 k

ƒ x ƒ 6 1,

where m a b = m, 1

msm - 1d m , a b = 2! 2

msm - 1d Á sm - k + 1d m a b = k! k

for k Ú 3.

VECTOR OPERATOR FORMULAS (CARTESIAN FORM) Formulas for Grad, Div, Curl, and the Laplacian Cartesian (x, y, z) i, j, and k are unit vectors in the directions of increasing x, y, and z. M, N, and P are the scalar components of F(x, y, z) in these directions. Gradient

Divergence

§ƒ =

0ƒ 0ƒ 0ƒ i + j + k 0x 0y 0z

§#F =

The Fundamental Theorem of Line Integrals 1.

Let F = Mi + Nj + Pk be a vector field whose components are continuous throughout an open connected region D in space. Then there exists a differentiable function ƒ such that 0ƒ 0ƒ 0ƒ F = §ƒ = i + j + k 0x 0y 0z

if and only if for all points A and B in D the value of 1A F # dr is independent of the path joining A to B in D. If the integral is independent of the path from A to B, its value is B

2.

F # dr = ƒsBd - ƒsAd.

B

LA 0N 0P 0M + + 0x 0y 0z

Green’s Theorem and Its Generalization to Three Dimensions Curl

i

j

k

0 § * F = 4 0x

0 0y

0 4 0z

M

N

P

0 ƒ 2

Laplacian

§2ƒ =

0x

2

0ƒ 2

+

0y

2

Normal form of Green’s Theorem:

C

Divergence Theorem:

0ƒ 2

+

0z

2

F # n ds = § # F dA 6 F F # n ds = § # F dV 6 9 S

Tangential form of Green’s Theorem:

u * sv * wd = su # wdv - su # vdw

Stokes’ Theorem:

D

F # dr = § * F # k dA F 6

C

Vector Triple Products su * vd # w = sv * wd # u = sw * ud # v

R

R

F # dr = § * F # n ds 6 F

C

S

Vector Identities In the identities here, ƒ and g are differentiable scalar functions, F, F1 , and F2 are differentiable vector fields, and a and b are real constants. § * s§ƒd = 0 §sƒgd = ƒ§g + g§ƒ § # sgFd = g§ # F + §g # F § * sgFd = g§ * F + §g * F § # saF1 + bF2 d = a§ # F1 + b§ # F2 § * saF1 + bF2 d = a§ * F1 + b§ * F2 §sF1 # F2 d = sF1 # §dF2 + sF2 # §dF1 + F1 * s§ * F2 d + F2 * s§ * F1 d

§ # sF1 * F2 d = F2 # § * F1 - F1 # § * F2

§ * sF1 * F2 d = sF2 # §dF1 - sF1 # §dF2 + s§ # F2 dF1 - s§ # F1 dF2 § * s§ * Fd = §s§ # Fd - s§ # §dF = §s§ # Fd - §2F 1 s§ * Fd * F = sF # §dF - §sF # Fd 2