Aufgabensammlung der höheren Mathematik, 15. Auflage

Vasili P. Minorski Aufgabensammlung der höheren Mathematik 15., aktualisierte Auflage Vasili P. Minorski Aufgabensa...

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Vasili P. Minorski

Aufgabensammlung der höheren Mathematik

15., aktualisierte Auflage

Vasili P. Minorski

Aufgabensammlung der höheren Mathematik Gute Studienergebnisse setzen in der Mathematik neben Kenntnissen auch Fertigkeiten voraus. Die Fertigkeiten kann man sich nur durch Üben aneignen. Mehr als 2500 Aufgaben wurden dafür in diesem Buch zusammengestellt. Ihre Lösungen, teils sogar mit Lösungsweg, sind am Ende der Sammlung zu finden. Diese moderne Aufgabensammlung, gedacht vor allem für Studenten ingenieurwissenschaftlicher Studiengänge an Hochschulen, ■ ist auf den Grundkurs Mathematik (Analysis, lineare Algebra) abgestimmt, ■ enthält viele Aufgaben mit technikorientierten Problemstellungen, ■ ermöglicht effektive Wiederholung und optimale Prüfungsvorbereitung. Aber auch Studenten der Mathematik und naturwissenschaftlicher Studiengänge können aus der Aufgabensammlung Nutzen ziehen.

www.hanser.de ISBN 978-3-446-41616-1

9

783446 416161

Vasili P. Minorski

Aufgabensammlung der höheren Mathematik Bearbeitet von Prof. Klaus Dibowski und Dr. Horst Schlegel

15., aktualisierte Auflage Mit 68 Bildern und 2670 Aufgaben mit Lösungen

Fachbuchverlag Leipzig im Carl Hanser Verlag

Aus dem Russischen übersetzt von Eberhardt Lacher, Schwarzenberg und Gerhard Liebold, Chemnitz Bearbeitung der deutschsprachigen Ausgabe von Heinz Birnbaum, Leipzig Titel der Originalausgabe: Сборник задач по высшей математике, 7. Auflage, Staatlicher Verlag für physikalisch-mathematische Literatur, Moskau 1962

Bearbeiter der 15. Auflage Prof. Dr. Klaus Dibowski Hochschule für Technik, Wirtschaft und Kultur Leipzig (FH) FB Informatik, Mathematik und Naturwissenschaften Dr. rer. nat. Horst Schlegel, Leipzig

Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. ISBN 978-3-446-41616-1

Dieses Werk ist urheberrechtlich geschützt. Alle Rechte, auch die der Übersetzung, des Nachdruckes und der Vervielfältigung des Buches, oder Teilen daraus, vorbehalten. Kein Teil des Werkes darf ohne schriftliche Genehmigung des Verlages in irgendeiner Form (Fotokopie, Mikrofilm oder ein anderes Verfahren), auch nicht für Zwecke der Unterrichtsgestaltung, reproduziert oder unter Verwendung elektronischer Systeme verarbeitet, vervielfältigt oder verbreitet werden.

Fachbuchverlag Leipzig im Carl Hanser Verlag © 2008 Carl Hanser Verlag München www.hanser.de Lektorat: Christine Fritzsch Herstellung: Renate Roßbach Satz: Klaus Dibowski, Leipzig Druck und Binden: Druckhaus „Thomas Müntzer“ GmbH, Bad Langensalza Printed in Germany

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(A B)

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M ' N ( ! N %) * {x | x ∈ R ∧ 0 ≤ x ≤ 1} $ ! +

, ! '% A = {x | x ∈ R ∧ x3 + x2 − 2x = 0}

B = {x | x ∈ R ∧ x+

4 ≤ 4 ∧ x > 0} x

! C = {x | x ∈ N ∧ x2 − 4x − 5 ≤ 0} 1 ≤ 2x < 6}

D = {x | x ∈ Z ∧ 8 , x, y . / 0! 2

A = {(x, y) | (x, y) ∈ R ∧ x + y − 3 = 0}

B = {(x, y) | (x, y) ∈ R2 ∧ 4x2 − y 2 < 0} ! C = {(x, y) | (x, y) ∈ R2 ∧ (x2 − 4)(y + 1) = 0} 2

D = {(x, y) | (x, √ y) ∈ R ∧ y > x + 1 ∧ x ≥ −1}

1 2 A ∪ B A ∩ B A \ B B \ A A × B B × A

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B = {x | x ∈ R ∧ x2 − 4x = 5}

3 A, B, C 4 05' ' 6 0 7 8 '5 ! ( 6 , A ∩ B ⊆ A A ∩ B ⊆ B

, A ⊆ A ∪ B B ⊆ A ∪ B ! (A) = A

A ⊆ B ' A ⊇ B A ∪ B = A ∩ B ' A ∩ B = A ∪ B (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C) (A \ B) ∩ (A ∪ B) = A 9 6 '5 A, B & A ∩ B = B ⇐⇒ B ⊆ A

A ∪ B = B ⇐⇒ A ⊆ B : $ ! A = {m | m ∈ N ∧ m = n2 + 1 ∧ n ∈ N}

%

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F = {(a, b), (c, b), (d, e)} P1 (x) = 0 G = {(c, f ), (c, b), (g, e)} P2 (x) = 0 ! " P3 (x) = 0 M = {a, c, h, d, g} N = {b, e, f, k} & + ' &) # $ " %& A × B x, y 2 ! ' # A = {x | x ∈ R ∧ (1 ≤ x ≤ 2 # ()& ∨ x = 3)} %&' B = {y | y ∈ R ∧ (2 ≤ y ≤ 3 * P1 (x), P2 (x), P3 (x) +', ∨ y = 4} ') - % # A = {x | x = i ∧ i ∈ {1, 2, 3, 4}} ' . &/ B = {y | y ∈ R ∧ 1 ≤ y < 3} Li = {x | Pi (x) = 0} (i = 1, 2, 3) ) 0 L1 , L2 M = {a, b, c} N = {α, β} 3 L3 1-)

" # P1 (x) · P2 (x) · P3 (x) = 0 # ,)

# ' M ! N

# ' ! N M

4 ' 5 Q

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211 90

[a, b] := {x | x ∈ R ∧ a ≤ x ≤ b}; [a, b) := {x | x ∈ R ∧ a ≤ x < b}; [a, ∞) := {x | x ∈ R ∧ a ≤ x}; (∞, b] := {x | x ∈ R ∧ x ≤ b};

(a, b] := {x | x ∈ R ∧ a < x ≤ b} (a, b) := {x | x ∈ R ∧ a < x < b} (a, ∞) := {x | x ∈ R ∧ a < x} (∞, b) := {x | x ∈ R ∧ x < b}

[a, b] ! " P0 P # ! $R1 %& ' $R2 % $R3 % ( ε $ε > 0% # ! P0 # ! P & ) | P − P0 |=| x − x0 |= (x − x0 )2 < ε $* % | P − P0 |= (x − x0 )2 + (y − y0 )2 < ε $* +,- % | P − P0 |= (x − x0 )2 + (y − y0 )2 + (z − z0 )2 < ε $* +% M . Ri (i = 1, 2, 3) / M # ! P ∈ Ri &

0 ε1( P # ! M ") - Ri

Ri

' H(M ) 2- ! M ⊆ R H(M ) • - !&

H(M ) M : sup H(M ) = lim sup M = lim M • - !&

3 H(M ) M : inf H(M ) = lim inf M = lim M

x √ 2+5 2 x = √ 2−1 √ √ 1, 47 − 2, 45 √ √ x = 5− 3 x = log9 5 · log25 27 √

√ a = 7 + 0, 2 b = 7 − 0, 2 ! a − b, a2 + b2 a3 − b3 " # # $ (a − b ∈ Q ∧ a2 + b2 ∈ Q) =⇒ a3 − b 3 ∈ Q % & # $ 0, 321 0, 132 2, 59 ' ( )

I1 = [−2, 3) I2 = [1, 5) I1 = (−5, 1) I2 = (−2, 0] I1 = [−1, 5; 3, 5)

I2 = (0, 5; 4, 5) $ * I1 ∪ I2 , I1 ∩ I2 , I1 \ I2 , I2 \ I1

+ ) , M

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lim M lim M !

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M = {(x, y, z)|x = 2 + n 4 ∧y = 4 + n 5 ∧z = 5 + ∧ n ∈ N} ⊂ R3 n

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1

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1 1% 1' 1+

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((a < b) ∧ (c < d)) =⇒ a + c 1b a · b > 0 a 0 x2 + 2x − 8 < x − 2 x3 − x2 ≤ 4x − 4 5−x <7 5+x 4x + 10 >8 4x − 3 3x + 2 ≤ −2 2x − 3 4x + 3 ≤6 & 2, 5 − x 2x2 + 12x + 8 ≤2 x2 + x − 6 √ 4x − 8 < 1 lg (2 + x) ≤ 1 !'$ ( %%

a, b |a + b| ≥ |a| − |b|

% |a + b| ≥ |a| − |b|

! "

x# ) *+ %,' $ x+3 ≥3 2x − 5 % |2x − 3| ≤ 6 |3 − 2x| > 5 7x − 3 <1 8x − 5 3x + 2 ≥1 |x + 5| 3 (2x + 1) = |x − 2| 4 |6x − 5| − |7x + 3| = 3x − 4

|2x − 1| = |x − 1| |x − 5| < |x + 2| & |x − 5| + |3 − x| ≤ 2 |x + 1| <4 x+2 |x − 3| < 10 ∧ |x + 2| ≥ 3 - ( ' ./ (x, y) ) !, 0 1 23 , + 4 (x − 3y + 1)(2x + y − 1) < 0 % x2 − 3xy + 2y 2 ≥ 0 x + |y − 2| < 5

n ∈ N \ {0} n! := 1 · 2 · 3 · . . . · n

n

0! := 1 n! = n · (n − 1)! α α k k α k α α(α − 1)(α − 2) . . . (α − k + 1) := k k!

! α 0 n := 1; := 1; = 0 n ∈ N k > n 0 0 k n n n! = 0 ≤ k ≤ n = n−k k k!(n − k)! α α α+1 + = k k+1 k+1 n α+i α+1+n = i n i=0 n−1 k + i n + k = k k+1 i=0 n α β α+β · = i n−i n i=0

" # $ % &

n n

an−k bk , n ∈ N

'

( ) * +, - ( + n ./ *

0 ( ) n, k ./

n

(a + b) =

k=0

k

(n + 3)!

n + 3!

3n + n!

3n! − 5

(2n − 3)!

n!/3

(n + 1) · (n + 2) · n! n · (n − 1) · (n − 2)!

(n + 1)! n · (n − 1)

(n + 1)! n · (n − 1)!

(n + 3)! n+3 (n − k)! (n − 2 − k)!

(2n + 1)! (2n − 1)! (2n − 2)! (2n)!

n 1 1 + n! (n + 1)! 1 1 + (2n − 1)! (2n + 1)! 1 1 + (n − 1)! n! 1 1 + (2n − 2)! (2n)! ! "# $ 7 100 3 2 96 7 1 −3 0 2 7 6 2 3 1 −n −4 −3 n 4 3 % & & % ' ( ) ( *+ , ( ' - . / #$ (3xy −2 − 7z)2 (a + 2b)3 (2x−1 − 3x2 )5 1, 13 = (1 + 0, 1)3 ) 0

/ #$ n n = 2n i i=0

n (−1) =0 i i=0 n

i

1 2 2 √ n - ( a x * √ +

! a x / # 3

4"# #( 4"#

2 $ * + 5

/ # 2 - ( √ √ 5 3 *

3 + 2 3 ## x 3 2

5 - ( * xlg x − x

/ # (−106 ) - % ( Fn+2 = Fn+1 + Fn n ≥ 0 F0 = F1 = 1. 2 5 % ( *+ , ! ( $ n n−1 Fn = , i i=0 m = 0 k > m #(

k - . / # ( - 6(! # $ (x − 1)25

n an − a−n

(a2 + b)n

7 * 2 - ! √ 13 √ 4 5 ( *

a2 x + a−1 x−2

/ # ! + % x8

%

"

!

%

#

!

!

$

!

!

!

&n '

(

Pn = n! = 1 · 2 · 3 · . . . · n

& (

&n ' ' i) ni ) * i = 1, 2, . . . , s) : (n ,n2 ,...,ns )

PWn1

=

n! ; n1 + n2 + . . . + ns = n n1 !n2 ! . . . ns !

&

(

& n k ) + , ( Vn(k) =

n! = n · (n − 1) · . . . · (n − k + 1); 1 ≤ k ≤ n (n − k)!

& -(

& n k )

+ , ' k ) ( (k)

VWn = nk

& .(

& n k + , ( n (k) Cn = ; 1≤k≤n & /( k

n k ! k " #$ n+k−1 (k) CWn = ; k≥1 %# k & ' '!( ) * )( + , )'" ' ) ' '. & !( ) ' )( / ) )( " .

# + , # , 0 + & !( ) )(

# + + + # 0 0 + , / ) / )( .

6' 0 6' ' , 2 ! 9 ' . : ; 6 " a, b, c, d, e, f, g " a, c, e " !

# 9 ' # 9 '. - * " 5 <" * ' '! " & * 2 5. * 2 " 5 " 9 ' " ' #

& 12 ) 3% -4.

0 & +0 6 ' ' a, b, c, d, e = " 2 " 6 3 caebd4.

, & & ' + & ' 5 & '

& * ! + % a, b, c, d, e, f "

# % ) 6 7 " 8 & 9 ' 5.

# ca # dab .

# d < ! + & " a, b, c, d, e, f, g, h 3 bdf 4 = 2 " .

& 12

+ ' ' ! 7 "

# .

* "2 * - " ! " $ # " $ (2 *-

% & ' ( " " " (" 9:$ ( ) * + 2 *- , " " (" " ( 9:$ ) - , 2 *- $ " ("

. / ) 9:$ k + 0(-

; ) / < " " 11 ≤ k ≤ 252 = . & / "2 ' . +- ( >" " k + ( $ , , + (2 ' . - ( " " " ( " k ( < ( 1, 2, . . . , 25 "$ "2 ? ." N2 +- ( "

3 4 +-" ." . "$ ( ( &" 5 ) "(

&" (2 ? 8"" S 5 +- ($ ' . 2 ? ." S1 " 6 ( ( $ +- (

7 . ) 8 . "$

2

= −1;

•

z =a+b a = Re z z¯ = a − b •

a, b

z

z, b = Im z

z

z = r(cos ϕ + sin ϕ) r = |z| z ; ϕ = Arc z z r ≥ 0; 0 ≤ ϕ < 2π b b |z| = r = a2 + b2 ; tan ϕ = ϕ = arctan a a z

•

z = reϕ e ϕ = cos ϕ + sin ϕ e (ϕ+k2π) = e ϕ (k ) #

z 1 = a1 + b 1

!

"

z 2 = a2 + b 2

$

z1 ± z2 = (a1 + b1 ) ± (a2 + b2 ) = (a1 ± a2 ) + (b1 ± b2 ) z1 · z2 = (a1 + b1 ) · (a2 + b2 ) = (a1 a2 − b1 b2 ) + (a1 b2 + a2 b1 ) a1 + b 1 a2 − b 2 a1 a2 + b 1 b 2 a2 b 1 − a1 b 2 z1 : z2 = · = + · 2 2 a2 + b 2 a2 − b 2 a2 + b 2 a22 + b22 ! #

!

%& %%

%'

(

"

z1 = r1 e ϕ1 = r1 (cos ϕ1 + sin ϕ1 )

z2 = r2 e ϕ2 = r2 (cos ϕ2 + sin ϕ2 )

z1 · z2 = r1 r2 e (ϕ1 +ϕ2 ) = r1 r2 cos (ϕ1 + ϕ2 ) + sin (ϕ1 + ϕ2 ) r1 (ϕ1 −ϕ2 ) r1 cos (ϕ1 − ϕ2 ) + sin (ϕ1 − ϕ2 ) z1 : z2 = e = r2 r2

$

%) %*

z = r e ϕ = r(cos ϕ + sin ϕ)

z n = (r · e ϕ )n = rn e nϕ = rn (cos nϕ + sin nϕ); n ∈ Z √ √ n zk+1 = n z = r e ϕ = ϕ 2kπ + √ √ ϕ 2kπ ϕ 2kπ n n + + = n r cos re n + sin n n n n

(k = 0, 1, 2, . . . , n − 1; n ∈ N; n > 1)

! z1 + z2 , z1 − z2 , z1 · z2 " z1 : z2 , z1 · z2 # z2 : z1 $! %& '' ! ' ! ( !' #" ) z1 = 9 − 7 , z2 = 3 + 2 4 1 4 1 % z1 = + , z2 = − 3 2 3 2 z1 = 2(cos 15◦ + sin 15◦ ), z2 = 3 e π/6 z1 = (1 + 2 )2 , z2 = (1 − )3 z1 = 2 e

5π/12

, z2 = 4 e

π/6

! ! * ! % x, y %) u, x, y, z #' (1 + 2 )x + (3 − 5 )y = 1 − 3 ⎧ ⎪ (1 + )u + (1 + 2 )x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ +(1 + 3 )y + (1 + 4 )z = % 1 + 5 ⎪ ⎪ ⎪ (3 − )u + (4 − 2 )x ⎪ ⎪ ⎪ ⎩ +(1 + )y + 4 z = 2 − ! $! %'' ! & ' ! ( !' # √ · x2 · · y 2 x, y ∈ R % (−)18 −17 √ √ 1 1 (2 3 − 3 2)2 4 + 7 i i 123 −99

+ ! ,# " ! - ! ! )!

5 17 − % 1 − 2√ 3 √ 3 3+ 2 √ √ √ 2 + 5 2 3− 2 . * % / & ' ! ' " !& !' # ( !' # # %

z = −5

% z = 9 π r = 3 , ϕ = 2 r = 8 , ϕ = −π π r = 2 , ϕ = 3 5π r = 7 , ϕ = 6 2π r = 1 , ϕ = − 3

r = 5 , ϕ = −127◦ Re z = 4 , Im z = −6 0 Re z = 0 , Im z = 2 / z = cos 60◦ + sin 30◦ Re z = −0, 5 , Im z = 8 √ Re z = 3 , Im z = −1 Re z = − Im z = −2 √ Re z = − 3 , Im z = −3 z = cos 30◦ − sin 30◦ 1 z = 9(− cos 270◦ + sin 270◦ ) ! z = e−3 π ' z = e2+3 π z = e4−11,5 π

Re z Im z 3+4 3

z = z = 2+ 1− 1+ 1− + −1 +1 z = (1 + )7 + (1 − )7 √ √ z = (1 − 3)10 − (1 + 3)10 √ (−1 + 3)15 z = (1 − )20 √ (−1 − 3)15 + (1 + )20

z =

!

e ϕ + e− ϕ

cos ϕ = 2 e ϕ − e− ϕ sin ϕ = 2 "

# $%√! z = −1 + 3 # $% z = 3 + 4

# $% √ z = 2 3 + 2 # $% z = −16 # $% & z = cos 225◦ − sin 225◦ '() * ) + ,- ) . ) /) # $% 0 '() * )1

x6 = 1

x2 = −3 + x4 = 1 −

x5 = −1 x3 = −2 − 3 x4 = −

/

2 f (x ) 34 # 0

' "

f (x) = Pn (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ; an = 0

* Pn (x) : (x − x0 ) & 1 an

x0

? bn−1

an−1

an−2

...

x0 bn−1 x0 bn−2 . . . * * bn−2 bn−3 . . .

a2

a1

a0

x0 b2

x0 b1 *

x0 b0 *

b1

b0

f (x0 )

f (x0 ) ' + x − x0 5 f (x0 ) = 0 x0 6 34 2 , # Pn (x) $ 1 Pn (x) : (x − x0 ) = bn−1 xn−1 + bn−2 xn−2 + · · · + b1 x + b0 +

Pn (x) = (x − x0 )(bn−1 xn−1 + bn−2 xn−2 + · · · + b1 x + b0 )

' 7

n Pn (x) = 0 an , an−1 , . . . , a1 , a0 C n ! "# $% &

' ! ( ) * + x1 , x2 , . . . , xk , # ( α1 , α2 , . . . , αk α1 + α2 + · · · + αk = n% - "*..$

Pn (x) = an (x − x1 )α1 (x − x2 )α2 . . . (x − xk )αk

/ Pn (x) = 0 % • 0 x1 = α + iβ x2 = x1 = α − iβ # "β = 0$* • 1 n % ! *

1 Pn (x) = 0 an = 0 ,% ) pn (x) = xn + bn−1 xn−1 + · · · + b1 x + b0 = 0 an−i = bn−i (i = 1, 2, . . . , n)* an + x1 , x2 , . . . , xn n # , pn (x) = 0 "αi # αi 2 $% & # xi bi−1 (i = 1, 2, . . . , n)

x1 + x2 + · · · + xn = x1 x2 + x1 x3 + · · · + xn−1 xn =

n

xi = −bn−1

i=1 n

xi xj = bn−2

i,j=1 (i<j)

x1 x2 x3 + x1 x2 x4 + · · · + xn−2 xn−1 xn =

n

xi xj xk = −bn−3

i,j,k=1 (i<j
** * x1 x2 . . . xn = (−1)n b0 + b0 , b1 , b2 , . . . , bn−1 ) pn (x) = 0 # % # 3 b0 * + -4 Pn (x) -4 (x − x0 ) & % f (x) = Pn (x) = a0n (x − x0 )n + a0,n−1 (x − x0 )n−1 + · · · + a01 (x − x0 ) + a00 ,

& a00 , a01 , . . . , a0n

5

,

an

? bn−1

x0

? cn−2

x0

an−1

...

a3

a2

a1

a0

x0 bn−1 *

...

x0 b3

x0 b2 *

x0 b1 *

x0 b0 *

bn−2

...

b2

b1

b0

x0 cn−2 *

...

x0 c2

x0 c1 *

x0 c0 *

cn−3

...

c1

c0

a01 =

a00 = f (x0 )

f (x0 ) 1!

2x4 + x3 + 2x2 + x = 0 ! "# "# !

$

% x2 + (5 − 2)x + 5(1 − i) = 0 % x2 + (1 − 2)x − 2i = 0

% x3 + 5x2 − 16x − 80 = 0 % x4 − 11x3 + 49x2 − 101x + 78 = 0 % x3 − 3x2 + 3x − 1 = 0 !% x4 − 2x3 − 3x2 + 4x + 4 = 0 "#

% x4 + 2x3 + 2x2 + 10x + 25 = 0 & x1 = 1 + 2i % x5 + 2x4 − 3x3 − 4x2 + 4x = 0

% 0, 5x3 − 2x2 − 1, 5x + 9 = 0 % x4 − 3x3 − 27x2 − 13x + 42 = 0 % x4 − 2x3 + 2x2 + √4x − 8 = 0 & x1 = 1 − i 3 !% x4 − 6x3 + 10x2 − 2x − 3 = 0 % x4 − 2x3 + 4x2 + 2x − 5 = 0

% x4 + 2x3 − 2x − 1 = 0 % 16x4 − 64x3 + 56x2 + 16x − 15 = 0 '% 12x6 + 16x5 + 7x4 − 17x3 + 2x = 0 √ √ 3 (% 2x − 22x+6 + 5 3 2x − 2 = 0 √ % (logx 5)3 − 44(logx 5)2 +17, 5 logx 25 − 25 = 0 ) * + & ( +

% P6 (x) = x6 − 2x5 − 2x4 + 8x2 % P5 (x) = 2x5 + 2x4 − 12x3 − 2x2 −2x + 12

% P5 (x) = x5 + 3x4 + 4x3 − 18x2 −104x − 120 % P7 (x) = (x3 − 17x2 + 63x +81)(x4 + 16) , & ( ! - +

% P3 (x) = 2x3 − 9x2 + 11x + 3 (x − 3)

% P5 (x) = −x5 + x3 − x2 + 1 (x + 2)

% P3 (x) = 2(x − 1)3 − 4(x − 1) + 6 x

R3

a

e 1 , e 2 , e3

x, y, z

a = a 1 e1 + a 2 e2 + a 3 e3

!"

! # $

• a1 , a2 , a3

• a 1 e1 , a 2 e2 , a 3 e3

a%

a!

&$ # ' $ ( ( ) * (+

T, ⎛ ⎞ a1 a = ⎝a2 ⎠ = (a1 , a2 , a3 )Ì a3 o = (0, 0, 0)Ì # $+!

y

$+!

e1 , e2 , e3 z ' !

!-

!

. " ( /

|a| = a21 + a22 + a23 −−→ r = OP = (x, y, z)Ì

.

1 + 2

a

P (x; y; z)

!0

'3( (

α, β, γ 5 + 1 r = (x, y, z)Ì x $+! y $+! z ' % +

cos α =

x , |r|

cos β =

y , |r|

cos γ =

z |r|

x

!4

(

!6

# ,

cos2 α + cos2 β + cos2 γ = 1

!7

R3

6 z

e3 6 γ Oα

*P β - e2

- y

e1

x

a λ ∈ R ⎛ ⎞ a1 λa = λ(a1 e1 + a2 e2 + a3 e3 ) = λa1 e1 + λa2 e2 + λa3 e3 = λ ⎝a2 ⎠ a3

= λ(a1 , a2 , a3 )Ì = (λa1 , λa2 , λa3 )Ì

a0 =

1 a |a|

a |a| = 0

>

! " # a + b = s

s

b 1 a

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a1 b1 a1 + b 1 s1 ⎝a2 ⎠ + ⎝b2 ⎠ = ⎝a2 + b2 ⎠ = ⎝s2 ⎠ a3 b3 a3 + b 3 s3 a+b=b+a (% ) (a + b) + c = a + (b + c) (! ) a+o=a λa = aλ

(o '

) λ∈R

(λ + μ)a = λa + μa λ(a + b) = λa + λb |a + b| ≤ |a| + |b|

λ, μ ∈ R λ∈R (Dreiecksungleichung)

$

& ( ) *

a − b = d b a 1

d

−b

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ b1 a1 − b 1 d1 a1 ⎝a2 ⎠ − ⎝b2 ⎠ = ⎝a2 − b2 ⎠ = ⎝d2 ⎠ a3 b3 a3 − b 3 d3

d = a + (−b)

!

R

" a # $ % P1 (x1 , y1 , z1 ) % P2 (x2 , y2 , z2 ) & # ' r1

O

P1

r2

⎞ x2 − x1 a = ⎝ y2 − y1 ⎠ z2 − z1 ⎛

a R - P2

()

−−−→ −−−→ " # $ # P1 P2 a = P1 P2

* a0 , b0 , a + b, b − a a − b, −2a + 3b +, a = (−3, 2, −1)Ì # b = 5e1 − 3e2 + 2e3 *& +, a + b # +, −2a + 3b # " ( * # .a / # −−→ −−→ a = AB + CD # A(0; 0; 1), B(3; 2; 1), C(4; 6; 5) # D(1; 6; 3) * # * # a = λe1 + (λ + 1)e2 + λ(λ + 1)e3 .

4 # # " " 5$ # % P # # y - 6 $ 60◦ # # z - 6 $ 45◦ 7 * * # .# # % P x-.# $ 8 % ABCD # # 9 / A(3; −4; 7), B(−5; 3; −2) # C(1; 2; −3) * # $ 9 / D # # % B ,

0 * # 1& # a = (20, 30, −60)Ì # 2 . cos2 α + cos2 β + cos2 γ = 1

# # " $ # 1&

3 " ABC A(1; 2; 3), B(3; 2; 1) # C(1; 4; 1)

" x # * √ |x| = 5 6 # # 2 #

a = (7, −4, −4)Ì b = (−2, −1, 2)Ì x.

$" v " A P −−→ −−→ AC = b AB = c % ( P x , " ( A(2; −4; 5) B(−3; 2; 7) $ ,

r A , r B , rC

! " % '$ # ABC # ABC A(1; 1; 1), B(2, 1; 0) C(1; 2; 3) ! " rS # S - ./$ a, c, b # # A(2; 3; 4), B(3; 1; 2) - 0

Ma , Mb , Mc C(4; −1; 3) $$ # %& '$ ( A(3; 3; 3) - m " A Ma

B(−1; 5; 7) ( |m|

C D AB $ ) %1 ( x, y %% * # ABC $ ( P + BC |BP | : |P C| = λ : 1 $ '

" ( A(1; −1; 5), B(3; 4; 4) C(4; 6; 1) $ , 2

Skalarprodukt a · b = |a| · |b| · cos ϕ, 0 ≤ ϕ ≤ π

3 %-

ϕ " a b $ ϕ = ∠(a, b) ⎛ ⎞ ⎛ ⎞ b1 a1 a · b = ⎝a2 ⎠ · ⎝b2 ⎠ = a1 b1 + a2 b2 + a3 b3 a3 b3 a1 b 1 + a2 b 2 + a3 b 3 a·b = 2 cos ϕ = |a| |b| a1 + a22 + a23 b21 + b22 + b23 a · b = b · a 34 "$ a · (b + c) = a · b + a · c 3# "$ -

3 3 -

o·a=0 λ(a · b) = (λa) · b = a · (λb), λ ∈ R

3 3 3 3

156-

a · a = a2 = |a|2

3 7-

a = o, b = o a · b = 0 ⇐⇒ a, b (a ⊥ b)

e21

=

e22

=

e23

= 1, e1 · e2 = e1 · e3 = e2 · e3 = 0

Vektorprodukt p=a×b 6

a × b = p ! "

0 ≤ ϕ ≤ π

p ⊥ a p ⊥ b a, b, p ! " # # |p| = |a| · |b| · sin ϕ a b

b ϕ

a × b = −(b × a) a × (b + c) = a × b + a × c λ(a × b) = (λa) × b = a × (λb), λ ∈ R o×a=o ⎛ ⎞ ⎛ ⎞ e1 b1 a1 a × b = ⎝a2 ⎠ × ⎝b2 ⎠ = a1 b1 a3 b3

e2 a2 b2

e3 a3 b3

a

$ % &

= (a2 b3 − a3 b2 )e1 − (a1 b3 − a3 b1 )e2 + (a1 b2 − a2 b1 )e3

' a = o, b = o # a × b = o ⇐⇒ a, b (

⇐⇒ b = λa Spatprodukt (a × b) · c = [a, b, c]

" #

)

[a, b, c] = [b, c, a] = [c, a, b] = −[a, c, b] = −[c, b, a] = −[b, a, c] a1 a2 a3 [a, b, c] = b1 b2 b3 = a1 (b2 c3 − b3 c2 ) − a2 (b1 c3 − b3 c1 ) + a3 (b1 c2 − b2 c1 ) c1 c2 c3

c

b a

* + , [a, b, c]

V - . ( a, b, c / ' a = o, b = o, c = o0 # [a, b, c] > 0 ⇔ a, b, c ! [a, b, c] < 0 ⇔ a, b, c 1(! [a, b, c] = 0 ⇔ a, b, c ( ⇔ c = λa + μb.

a = e1 + 2e2 + 3e3 , b = 6e1 + 4e2 − 2e3 a · b a × b

7 A(2; 2; 2) C(4; 5; 4) D(5; 5; 6)

B(4; 3; 3)

[(a − b), (b − c), (c − a)]

a = (2, 2, 1)Ì b = (6, 3, 2)Ì + 8 $ a ! a = b $ b ! a 3e1 + 4e2 + 7e3 b = (2, −5, 2)Ì ! $ "# $ λ a b 4a + 3b + 2c 7a + 6b + 5c,

a = λe1 +3e2 +4e3 |a| = 3, |b| = 2, |c| = 1, π b = 4e1 + λe2 − 7e3 % ∠(a, b) = ∠(a, c) = ∠(b, c) = 3 & (5a + 3b) · (2a − b) |a| = 3, |b| = 5 a b ' & ,, 7 $ s◦ ! a = (1, 1, 2)Ì b = (2, 1, 1)Ì ( ) "* + ,, ( !# λ Ì a = a = (−6, −3, 2) Ì (1, 1, λ)Ì , b = (1, 1, λ + 1)Ì c = b = (−3, 2, −6) (1, −1, λ)Ì , a + 5b 5a + b |a| = |b| = 3 ∠(a, b) = 30◦ % - . a, b, c $ 9* ' - "* ,, c . A B C a = e1 + e2 b = e2 + e3 A(4; 3; 2), B(2; 3; 4) C(1; 1; 1) / , :' ϕ a b

/ a = (3, 4, 5)Ì b = (−4, −5, 3)Ì

, 7 A(1; 1; −1), B(1; −1; 1), C(−1; 1; 1), D(1; 1; 1)

0 ' 0 a × (a × b) +

a = (2, −1, −1)Ì , (b × a) × a !# a = (2, 1, −3)Ì b = (−1, −3, 1)Ì c = e1 + e2 + 4e3 b = (1, −1, 1)Ì a = $ + (2, 5, 7)Ì , b = (1, 1, −1)Ì c = A(1; 2; −1), B(−1; 3; −4), C(0; 5; −7) (1, 2, 2)Ì , D(2; 4; −4) a , 1! b $ + c 7 23 4 a = λb + μc 5 , +6, ,

7 + ,,

!

"! #$ % & ' $ ( ) * ! +,-

. / !

0 1 (c2 = a2 + b2 −2ab cos γ) ,- (c2 = a2 + b2 )!

g g

r0 e3 O6 e2 e1

P0 (x0 ; y0 ; z0 ) w

v

1 P (x; y; z) r

g : r = r0 + λv

+!2

r 0 % - / P0 g v % % g λ / (−∞ < λ < +∞)

/ P1 (x1 ; y1 ; z1 ) P2 (x2 ; y2 ; z2 ) +!3

r = r1 + λ(r 2 − r1 )

r 1 = (x1 , y1 , z1 )Ì r2 = (x2 , y2 , z2 )Ì 4 5 F (xF ; yF ; zF ) 6 l % / P1 (x1 ; y1 ; z1 ) g g : r = r 0 + λv r F = (xF , yF , zF )Ì = r0 +

(r 1 − r0 ) · v ·v v2

+!

, S(xS ; yS ; zS ) -$ g1 g2 g1 : r = r 1 + λv 7 R3

g2 : r = r2 + μw

r 1 + λv = r2 + μw

( λ μ! #- +! 8 8! 69 =⇒ : ; , S !

+! 8

=⇒ S r S = (xS , yS , zS )Ì = r 1 + λv = r 2 + μw =⇒ g1 ≡ g2 ! |l| "# # $ g1 g2 g1 : r = r1 + λv g2 : r = r 2 + μw !" l : l = r 2 − r1 + μw − λv $% & l & g1 g2 $% $ "# $% λ μ' l·v =0

l·w =0

g1 g2 # $( & ) *

( * P1 P2 ( # P1 (−2; 3; −5) P2 (1; −4; −1) P1 (3; −2; 1) P2 (1; −2; 2) *%$( & * A(−5; 1; 2) B(3; −3; 1) $ r = (1, −2, 5)Ì + λ(2, −1, 1)Ì +& $& , "# # ", r = 5e1 + e2 − 2e3 + λ(4e1 − e2 −3e3 ) r = (7 − 3μ)e1 + (2μ − 2)e2 +(11 − 5μ)e3 √ 2 Ì 1 , 1) r = (1, 2, 1)Ì + λ( √ , − 2 3 r = (−4, 3, −1)Ì

√ 3 +μ( 3, − √ , 3)Ì 2

r = (2 + 4λ, −1 + 2λ, 3 − λ)Ì r = (1 + 2μ, 2 − μ, 2 + 3μ)Ì √ r = (2 + 2 3, −3, 7)Ì

√ +λ( 12, −2, 4)Ì √ r = (2 − 3, 0, 1)Ì √ +μ(− 3, 1, −2)Ì

r = (3, −1, 2)Ì + λ(2, 4, 10)Ì r = (−1, 5, 3)Ì + μ(−4, 4, 6)Ì

$ r = (3, −1, 2)Ì + λ(2, 4, 3)Ì r = (−1, 5, 10)Ì + μ(−4, 4, 6)Ì

−∞ < λ < ∞,

−∞ < μ < ∞)

+& # +

r = (−2, 5, 1)Ì + λ(−1, 2, 3)Ì r = (3, −1, 2)Ì + μ(1, −1, 1)Ì ? (−∞ < λ, μ < ∞)

. * P1 (3; −1; 2) P2 (1; 2; −1) *

g ( P1 P2 / 01 F & & 2

$ g / ! 2

, & g 3 / !

g1 g2 01 2 , 3# F 4 l - P1 (−2; 1; 1) g1 : r = (−1, 0, 1)Ì + λ(1, 1, 2)Ì P2 (0; 0; 1) P3 (1; −1; 0) g2 : r = μe1 +(3μ−1)e2 +(4μ+2)e3 4- l (−∞ < λ, μ < +∞) 4. |l| g 1

P1 (1; −2; 1) P2 (−2; 3; 5)! 00 5 g2 A(1; 2; 3) B(−2; 3; 1) Q1 (1; −5; −2) Q2 (10; −11; −5) C(2; −3; −1) " #$ " + 2 ,. g1 g2 % 4. 6 7 ABC. & ' '+ ' * P (2; −3; 4) y 89 :

( ' g : r = (1 + λ, −2 + 2λ, 5 − 2λ)Ì (−∞ < λ < +∞) P0 (1; −2; 5) g )* λ = 0+ ,* " - λ . / - g - P0 + ' '+ ' 0& ' %

" '

+ 5 6 # :# - 7 ; $ 9 < * + ' * A

E E : r = r0 + λv + μw

) +

r0 = - $ P0 (x0 ; y0 ; z0 ) 5' E v, w $ ) # + E λ, μ (−∞ < λ < ∞, −∞ < μ < ∞)

E E : n·r = d

)

+

n > - - E d = n · r 0 , ' r 0 = (x0 , y0 , z0 )Ì '$ P0 (x0 ; y0 ; z0 ) - E

⎛ ⎞ a n = ⎝b ⎠ c

z 6

n = (a, b, c)Ì E

P0 (x0 ; y0 ; z0 ) q P (x; y; z)

ax + by + cx = d

- y x

A P1 (x1 ; y1 ; z1 ) E E : n · r = d !" ax + by + cz = d

E E:

n·r−d =0 |n|

!"

ax + by + cz − d √ =0 a 2 + b 2 + c2

#

A=

|n · r 1 − d| |n|

!" A =

|ax1 + by1 + cz1 − d| √ a 2 + b 2 + c2

$

%" E & ' r = r 0 + λv + μw ( n = v × w d = n · r0 ) n · r = d %" & ' E

E : n · r = d n = (n1 , n2 , n3 )Ì v w ) n · v = 0 n · w = 0 ! * v = (0, n3 , −n2 )Ì w = (n2 , −n1 , 0)Ì + , r 0 * n · r 0 = d '+ Ì d , 0, 0 ' " ' n1 = 0 - r0 , v, w ! * r0 = n1 E : r = r0 + λv + μw ,. S(xS ; yS ; zS ) ) g E g : r = r0 + λv

E : n · r = d

r S = (xS , yS , zS )Ì = r 0 +

d − n · r0 ·v n·v

F (xF ; yF ; zF ) ! " P1 (x1 ; y1 ; z1 )Ì # $% E E : n·r =d r1 = (x1 , y1 , z1 )Ì r F = (xF , yF , zF )Ì = r1 +

d − n · r1 ·n n2

&

' g () $% E1 E2 E1 : n1 · r = d1 E2 : n2 · r = d2 ( #* g : r = r0 + λ · v % v = n1 × n2 + r 0 = (x0 , y0 , z0 )Ì () n1 · r0 = d1 n2 · r0 = d2 #* ,% x0 , y0 , z0 * (

% ,% - ) ( . () ,% %/ %

' P2 (x2 ; y2 ; z2 ) " P1 (x1 ; y1 ; z1 ) %(* $% E . % r 1 = (x1 , y1 , z1 )Ì E : n · r = d r 2 = r1 + 2 ·

d − n · r1 ·n n2

2 3 % 4

% " $% E1 ≡ E2 ( . % E1 E2 ( 5# *% ) E1 : r = (4, −2, −11)Ì +λ(−2, 5, 15)Ì + μ(2, 10, 21)Ì E2 : r = (2, 3, 4)Ì + λ(4, 5, 6)Ì +μ(0, 5, 12)Ì (−∞ < λ, μ < ∞)

6 . % " P1 ! $% E ) E : x + y − z = −1; P1 (2; 1; 1) % E : 6x − 3y + 2z = 28;

01

P1 (3; 5; −8)

3 $% E '# ) E " P (2; 3; 5) a = (4, 3, 2)Ì # E 7 % $% E : 2x + 3y + z = 6 '(( 8 ! E 2 9 : (x ≥ 0, y ≥ 0, z ≥ 0) 0 * % $% " ( 4

% E : (3, −2, 5) · r = 8

E : −y + 7z = 13

& & 0* E ) .*

E1 E2 +1 2* # P1 (1; 2; 1) . E1 : 2x + y − 2z − 4 = 0 E : x − 2y + z = 7 E2 : 3x + 6y − 2z − 12 = 0 3* ./ 4** E1 5 %

E2

+ ( 6 # P1 (2; 3; 4)

# P2 6 E1 : x − 2y + z = 1 . E E2 : (−2, 4, −2) · r = −1 E : x − 3y + 5z + 22 = 0

! " # $% + & # & g E A(−3; 2; 5), B(−2; 1; 6), C(1; 3; 2) g : x − 1 = 2y + 2 = z + 3 = λ

& −∞ < λ < +∞ g : r = (5 + 9λ)e1 − (5 + 6λ)e2 E : x+y−z+1=0 −(4 + 3λ)e3 g : r = (1, 2, 1)Ì + λ(2, −1, 2)Ì (−∞ < λ < +∞). −∞ < λ < +∞ ( E : 2x + y − z − 4 = 0 & E ' ( ) g # . E1 E2 *%

) 7 *% S & E1 : 2x − y + 3z = 1 g E E2 : (1, 1, −1) · r = 2 + & 6 E1 : 2x − 2y + 2z = 3 E : x − 2y − 3z = 0 E2 : −2x − 3y + 6z = 7 E2 # E1 : −x + 3y − 3z = 2 P1 (2; 2; −2) E2 : 3x + 2y + z = 5 + ) & +, &

# E . # A(2; 1; −2), B(0; 2; 1), C(1; 2; 0) (P1 (−1; −1; 2) . E1 E2 & E E1 : (1, −2, 1) · r = 4 A, B, C ) .* E2 : (1, 2, −2) · r = −4 & & g + ( ) S A . E E1 , E2 , E3 E1 : (2, −1, 3) · r − 9 = 0 ++ E / # E2 : (1, 2, 2) · r − 3 = 0 P (3; 2; −1) & E3 : (3, 1, −4) · r + 6 = 0 r = (−2, 0, 1)Ì + λ(−1, 3, −2)Ì (−∞ < λ < +∞).

+! &

E P1 (1; 2; 4) E1 E2 E1 : 2x − y + 3z − 6 = 0 E2 : x + 2y − z + 3 = 0

! " # E1 E2 $ % # E : 2x + 2y + z − 8 = 0 & A = 4 !

" + E g1 g2 , + P1 (4; −2; 5) & E

+ g3 E g1 P2 x = x2 = −1 # + $ S & g2 g3 !

'(! ) # E ''! - a = −e3 P1 (1; 2; 4) . M1 & # E1 E2 * / * E3 E1 : 4x − y + 3z − 6 = 0 E : x+y−z =2 E2 : x + 5y − z + 10 = 0 & , * . E3 : 2x − y + 5z − 5 = 0 M2 P2 (2; 3; −3)! '! $ + 0 A 1 / E Ì Ì g1 : r = (−2, 5, 2) + λ(1, 0, −1) + * & , * / . M2 g2 : r = (2, −4, 2)Ì + μ(1, 0, −1)Ì

+ ) 2 " & # (−∞ < λ, μ < +∞) sin α1 , n = sin α2

n ⎞ a11 a12 . . . a1n ⎜ a21 a22 . . . a2n ⎟ ⎟ A=⎜ ⎝ . . . . . . . . . . . . . . . . . . .⎠ an1 an2 . . . ann ⎛

n2 n ! " n # $ % aik i " k $ & " % % n ' Dn n a11 a12 . . . a1n a a22 . . . a2n Dn = det A = 21 . . . . . . . . . . . . . . . . . . . an1 an2 . . . ann

(

" % ) % ( ' D2 * & a D2 = 11 a21

a12 = a11 · a22 − a12 · a21 a22

+

Dn i " k $ , Aik ' n − 1 !# Dn i "Dn = (−1)i+1 ai1 · Ai1 + (−1)i+2 ai2 · Ai1 + . . . + (−1)i+n ain · Ain

!# Dn k $ Dn = (−1)1+k a1k · A1k + (−1)2+k a2k · A2k + . . . + (−1)n+k ank · Ank

/0 1

23 (−1)i+k 41

5 + − + · · · − + − · · · + − + · · · . . . . . . . . . . . . .

.

! "# a11 , a22 , . . . , ann # $ %& & & ! "

' !

λ & λ & & ! # λ() ! # * + ! # " Dn λ() ! # Dn = 0 ,

- ). / Dn = a11 · a22 · . . . · ann

0 −3 10 9 D3 = 1 −4 −2 5 −16 12 # 1 $ # 1 $ ' ! # &" () ") # 1(

$ 0 ( 3 −2 2 3 # # 6 −10 4 6 √ a −1 sin α cos α √ # # − cos α sin α a a ' 0 # 1

! # & 1(

2 3 4 5 −2 1 1 2 3 0 ) ( & 1 "( ) / a −a a # a a −a a −a −a 1 2 5 7 # 3 −4 −3 12 −15 2 x x 1 2 # y y 1 z 2 z 1 1 + cos α 1 + sin α 1 # 1 − sin α 1 + cos α 1 1 1 1 1 −2 −6 4 −3 1 2 −5 # 0 −4 3 4 6 0 1 8

−1 2

−3 1 2 −3 7 −9 1

0 2 −3 1 −3 2 2 1 −1 3 2 1

6 0 1 2 5 −1 2 −1 −1 3 −2 3 2 2 3 5 1 1 −1 −2

x

2 ! x 4 9 x 2 3 = 0 1 1 1 2 x 3 2 x −1 1 = 0 0 1 4 2 − x 1 −1 2 = 0 −2 4 − x x − 3 3 − x 3 − x

" A # $ %&' (m, n) (& # m · n )

* aik + + m * n (' , ⎞ ⎛ a11 a12 . . . a1n ⎜ a21 a22 . . . a2n ⎟ ⎟ A = A(m,n) = (aik )(m,n) = ⎜ ) - ⎝. . . . . . . . . . . . . . . . . . . . . ⎠ am1 am2 . . . amn ! aik i * k (' *. A = (aik ) B = (bik ) + . A B $ aik = bik / i k *. A = (aik ) B = (bik ) # $ .

+ ' ) , A(m,n) ± B (m,n) = S (m,n) aik ± bik = sik (i = 1, 2, . . . , m; k = 1, 2, . . . , n)

) 0

! , A + B = B + A; A + (B + C) = (A + B) + C

) 1

" A = (aik ) . λ + 2 aik # A λ ' , λA = Aλ = (λaik ) (−1) · A = −A; λ(A + B) = λA + λB (λ + μ)A = λA + μB

) ) 34 ) 33

! A · B A B + # 5 / ) 3 $5 A # %&' )m, n $5 B # %&' )n, r+

P m, r pik i ! "# $ A k % &# $ B ' A(m,n) · B (n,r) = P (m,r) ai1 · b1k + ai2 · b2k + . . . + ain · bnk = pik

#"&

(i = 1, 2, . . . , m; k = 1, 2, . . . , r)

' #"( #"

(A + B) · C = AC + BC; (AB)C = A(BC) AB = BA ) (λA)B = A(λB)

! * +, - +, . %+,

A·B a11 ## #

...

a1n ## #

ai1 ## #

...

ain ## #

am1

. . . amn

b11 ## #

...

bn1

. . . bnk

→

b1k ## #

...

'

b1r ## #

. . . bnr

↓ pik

0 / # 0 - 1 $ m, n# 2 3. -' #"

A + 0 = A; A − A = 0 ; 0 · A = 0

$ n, n ,4 a11 , a22 , . . . , ann # 6 +, ,4

#

5,

7 a = 0 .3 i > k# • 7 a = 0 .3 i < k # • 7 a = 0 .3 i = k # 8 a = 1 .3 i ,4 E , •

ik

ik

ik

. +,.'

ii

E (n,n) · A(n,n) = A(n,n) · E (n,n) = A(n,n) E (m,m) · A(m,n) = A(m,n) · E (n,n) = A(m,n)

$3 E (n,n) +, 3 - E n #

#"9 #":

A = (aik )(m,n)

AÌ = (aki )(n,m)

! "#$%" &! ' (AÌ )Ì = A; (A + B)Ì = AÌ + B Ì ; (AB)Ì = B Ì AÌ

(

) A *

+ $#

A = AÌ ! aik = aki (i, k = 1, 2, . . . , n).

, ! % 1 3 0 A= 2 5 −1 −1 4 0 B= 3 0 2 ⎛ ⎞ ⎛ ⎞ 4 3 −1 2 C = ⎝ 2 1 −2⎠ , d = ⎝−1⎠ −1 3 4 3 - $! .#" $ / ' 5A − 3B, 2B + 7C (A + B)C, 2AB, BA 2AÌ B, 3B Ì A, −BC, CB

% Ad, Bd, Cd, dA, dÌ C dÌ AÌ

AA, AÌ A, AAÌ , CC, CC Ì C Ì C Ì , dd, dÌ dÌ , ddÌ , dÌ d

-

⎛ $#

⎞ 1 −2 2 3 1 A = ⎝−2 3 ⎠ , B = 4 6 2 4 −5 AB % BA 0 - det (AAÌ ) + 3 2 1 2 A= 4 1 1 3 -

⎛ ABC 2 −1 A = ⎝−2 0 1 2

+ ⎞ 3 1 2 4 1 −3⎠ 3 1 0

⎛

3 ⎜7 ⎜ B=⎜ ⎜−1 ⎝2 −4 ⎛ 1 ⎜−3 C =⎜ ⎝2 −1

-

⎛ 1 A = ⎝0 0

⎞ −2 1 3 −4⎟ ⎟ 1 3⎟ ⎟ 0 2⎠ 2 1 ⎞ −1 1⎟ ⎟ 1⎠ 2 0 2 5 1 2

det A + ⎞ ⎛ 0 0 0 0 1 0⎠ · ⎝0 b 0 1 c ⎛ 0 0 × ⎝0 1

⎞ a 0⎠ 0 ⎞ 0 1 1 0⎠ 0 0

1 -

2 3 3 X 453 ! , ! 4 0 X · XÌ = 0 9 6 - + ' 7 A % %! )3 & 8 (n, n) ! det (λA) = λn · det A $# λ ∈ R A & 8 0 D & 8 0 + 83 # B

C AB − 2(C + D)

!( Tk * $%&' * Bi )( Bi * $%&' * Ej & $%&' * Tk (k = 1, 2, 3, 4) +, - & . / $%&' * E1 / $%&' * E2 *

T1 , T2 , T3 , T4 B1 , B2 , B3 E1 E2 ! " # $%&'( ./ ( 0 A 1 ! ) & - 2 - 2 T1 T2 T3 T4 E1 E2 - 2 S = A + AÌ 3 & B1 4 2 0 5 2 1 B2 1 2 1 3 3 2 3 4 1 1 1 B3 0

* n! 4 & n* 5'* ⎛ ⎞ a1 ⎜ a2 ⎟ ⎜ ⎟ a = ⎜ ⎟ , ai ∈ R 6/7 ⎝ ⎠ an

ai 8 9** 5'* a - ' a 6n, 17 - 2 , , 6$'* 7 & ! * 61, n7 - 2 & 64'* 7( aÌ = (a1 , a2 , . . . , an )

:% *& n* 5'*

Ì λ '" ( Ì

6.7

0 a = (a1 , a2 , . . . , an ) b = (b1 , b2 , . . . , bn ) * ( aÌ + bÌ = (a1 + b1 , a2 + b2 , . . . , an + bn ) λaÌ = (λa1 , λa2 , . . . , λan )

67 6;7

Rn & < n* 5'* 9** :% a, b, c ∈ Rn λ, μ ∈ R ( a + b = b + a; (a + b) + c = a + (b + c); (λ + μ)a = λa + μa λ(a + b) = λa + λb; λ(μa) = (λμ)a

67 6 7

- a − b = a + (−1) · b $ '* , * %&' ,%

S = {a1 , a2 , . . . , am } m ai ∈ Rn m

λi i = 1, 2, . . . , m λ1 a1 + λ2 a2 + . . . + λm am =

m

!"#

λi ai

i=1

S. $ x ∈ Rn S % & S '

x S S % # ak ∈ S ( S\{ak } '

S

S ( % m

λi (i = 1, 2, . . . , m) ) m m '* % λ2i > 0 λi ai = o o +

i=1

i=1

o = (0, 0, . . . , 0)Ì .# $ T x ∈ Rn $ B = {b1 , b2 , . . . , bm } T % , • bi ∈ T, i = 1, 2, . . . , m • B ( • -* . x ∈ T )

α1 , . . . , αm

m x= αk bk k=1

α1 , α2 , . . . , αm / x 0* B $ ' 1 Rn 2

{e1 , e2 , . . . , en } ei = (0 , . . . , 0 , 1 , 0 , . . . , 0)Ì ↑ i 3

5

a1 = (2, 4, 4)Ì , a2 = (−3, 2, −2)Ì a3 = (2, −1, 4)Ì (

b = (2, 2, 8)Ì & a1 , a2 , a3 ! S1 = {a, b, c|a, b, c ∈ R3 }

!4#

(

S2 = {a, a + b, a + b + c}

(

S = {(3, −1, 1)Ì , (−1, 3, 1)Ì , (1, 1, 1)Ì }

(

a1 = (4, 1, 3, −2)Ì a2 = (1, 2, −3, 2)Ì a3 = (16, 9, 1, −3)Ì a4 = (0, 1, 2, 3)Ì a5 = (1, −1, 15, 0)Ì 3a1 + 5a2 − a3 , 2a1 + 4a3 − 2a5 1 1 a1 + 3a3 − a4 + a5 2 2 x 2x + a1 − 2a2 − a5 = o

" x = (1, −3, 5)Ì ! S = {(1, −1, 0)Ì , (3, 5, 0)Ì } # b1 = (1, 1, 1, 1, 1)Ì b2 = (0, 1, 1, 1, 1)Ì b3 = (0, 0, 1, 1, 1)Ì b4 = (0, 0, 0, 1, 1)Ì b5 = (0, 0, 0, 0, 1)Ì

2(a1 − x) + 5(a4 + x) = o 3(a3 + 2x) − 2(a5 − x) = o

x = (19, 1, 10)Ì ! S = {(1, 1, −1)Ì , (0, −1, 1)Ì , (5, 0, 3)Ì , (−2, 1 − 3)Ì }

B = {b1 , b2 , b3 , b4 , b5 } R5 $ % & x '( B ) x = (1, 1, 0, 1, 0)Ì x = (3, 5, 4, −1, −2)Ì x = (−5, 4, −3, 2, −1)Ì

m, n A r, r r, r r A ! A " r

# $ %& $ " r +1

# A r A' ( (A) = r(A) = r;

r ≤ min (m, n)

)*+

% • , ! • • ! -. . ! ( -. % A = (aik )(m,n) r(A) = r /

⎛ d11 d12 ⎜ 0 d22 ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0

. . . d1r . . . d2r . . . drr

d1,r+1 d2,r+1

. . . d1n . . . d2n

dr,r+1

. . . drn 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

r ! "

# $ ! % & "' A ( a11 = 0 ai1 ' & ( ) − a11

i* & i = 2, 3, . . . , m ( +

⎛ ⎞ a12 ... a1n a11 ⎜ 0 b11 ... b1,n−1 ⎟ a11 a12 . . . a1n ⎜ ⎟ ,- ⎜ ⎟= o B ⎝ ⎠ 0 bm−1,1 . . . bm−1,n−1 , B A "

$ .! !

+ r(A) r /*

D (r,r) = (dik )(r,r) ( m n* ./ aÌ i = (ai1 , ai2 , . . . , ain ), i = 1, 2, . . . , m * 0 ! (m, n)* ⎛ Ì⎞ ⎛ ⎞ a1 a11 a12 . . . a1n Ì ⎜ a2 ⎟ ⎜ a21 a22 . . . a2n ⎟ ⎜ ⎟ ⎟ A=⎜ ⎟=⎜ ,$ ⎝ ⎠ ⎝. . . . . . . . . . . . . . . . . . . . .⎠ am1 am2 . . . amn aÌ m

1 r(A) = r #

• 2 r = m m ./ ai 3 • 2 r < m m ./ ai 3 4 m ./ ai ) r ./ 3 1 (dik )(rr)

⎛ ⎞ 2 1 −4 1 3 ⎜−4 7 5 −2 0 ⎟ ⎟ A=⎜ ⎝ 5 6 9 −3 −3⎠ 0 3 −1 0 2

⎛ ⎞ 2 −1 3 −2 4 A = ⎝4 −2 5 1 7⎠ 2 −1 1 8 2 ⎛ ⎞ 1 3 5 −1 ⎜2 −1 −3 4 ⎟ ⎟

B=⎜ ⎝5 1 −1 7 ⎠ 7 7 9 1 ⎛ ⎞ 4 3 −5 2 3 ⎜8 6 −7 4 2 ⎟ ⎜ ⎟ ⎟ C =⎜ ⎜4 3 −8 2 7 ⎟ ⎝4 3 1 2 −5⎠ 8 6 −1 4 −6 ⎛ ⎞ 0 2 −4 ⎜3 1 7 ⎟ ⎜ ⎟ ⎜ 5 −10⎟ D=⎜ 0 ⎟ ⎝−1 −4 5 ⎠ 2 3 0 ⎛ ⎞ 2 2 −1 1 2 ⎜4 3 −1 2 1 ⎟ ⎟ F =⎜ ⎝8 5 −3 4 −1⎠ 3 3 −2 2 1 ⎛ ⎞ 25 31 17 43 ⎜75 94 53 132⎟ ⎟ G=⎜ ⎝75 94 54 134⎠ 25 32 20 48

! " #

$ % & λ ∈ R' ⎛ ⎞ 3 1 1 4 ⎜λ 4 10 1⎟ ⎟ A=⎜ ⎝ 1 7 17 3⎠ 2 2 4 3 ⎛ ⎞ 1 λ −1 2

B = ⎝2 −1 λ 5 ⎠ 1 10 −6 λ

# ( #

)% $ $ a1 a2 a3 a4

= (1, 1, 1, 1)Ì = (1, 1, −1, −1)Ì = (1, −1, −1, 1)Ì = (1, −1, 1, −1)Ì

a1 = (4, −1, 5, 6)Ì a2 = (4, −5, 2, 6)Ì a3 = (2, −2, 1, 3)Ì a4 = (6, −3, 3, 9)Ì

* & )%# T = {a1 , a2 , a3 , a4 } + B T ( B , )% & T % # & B ( + a1 = (−3, −6, 0, 0)Ì a2 = (1, 2, 3, 4)Ì a3 = (1, 2, 0, 0)Ì

a1 = (3, 4, −1, 2)Ì a2 = (1, 1, −1, −2)Ì a3 = (4, 1, −2, 3)Ì a4 = (5, 2, −3, 1)Ì

n n a11 x1 + a12 x2 + . . . + a1n xn = a1 a21 x1 + a22 x2 + . . . + a2n xn = a2 ................................. an1 x1 + an2 x2 + . . . + ann xn = an

!

" #$% aik & ' ! ai ( i, k = 1, 2, . . . , n! " ! ' ) *

a11 a12 . . . a1n a21 a22 . . . a2n = D= 0 . . . . . . . . . . . . . . . . . . . an1 an2 . . . ann

!

" Dk (k = 1, 2, . . . , n) a11 . . . a1,k−1 a1 a1,k+1 . . . a1n a21 . . . a2,k−1 a2 a2,k+1 . . . a2n Dk = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . an1 . . . an,k−1 an an,k+1 . . . ann

!

+, ) x1 =

D1 D2 Dn , x2 = , . . . , xn = D D D

!

' ! - ' . / 0. 1 ! 2ax1 − 3bx2 = 0 3ax1 − 6bx2 = ab (ab = 0) ! 3x1 − x2 + 2x3 = 3 −x1 + 3x2 − 2x3 = −1 2x1 + 2x2 + 3x3 = 14

! x1 − x2 + x3 − x4 = −2 x1 + x2 + x3 + x4 = 0 = 5 x1 + 2x2 3x3 + 4x4 = −10 ! 3x1 + 4x2 + x3 + 2x4 = −3 3x1 + 5x2 + 3x3 + 5x4 = −6 6x1 + 8x2 + x3 + 5x4 = −8 3x1 + 5x2 + 3x3 + 7x4 = −8

P2 (x) = ax2 + bx + c P2 (1) = −1, P2 (−1) = 9 P2 (2) = −3

P3 (x) = ax3 + bx2 + cx + d P3 (1) = 2, P3 (−1) = 8 P3 (2) = −10, P3 (−2) = 26

! " ! #$ " ! % ! R · i = u i = (I1 , I2 , I3 , I4 , I5 , I6 )Ì

u = (0, 0, 0, 0, 0, E)Ì R ⎛= ⎞ 0 −1 0 −1 1 0 ⎜ 0 0 −1 1 0 1⎟ ⎜ ⎟ ⎜ −1 ⎟ 1 0 0 0 −1 ⎜ ⎟ ⎜ 0 −R2 0 R4 0 −R6 ⎟ ⎜ ⎟ ⎝−R1 0 R3 0 0 R6 ⎠ R1 R2 0 0 0 0 E = 10 &' R1 = R2 = R3 = 2 Ω R4 = R5 = R6 = 4 Ω ! ( ) " I6 !

% ( m % ! n & * a11 x1 + a12 x2 + . . . + a1n xn = b1 a21 x1 + a22 x2 + . . . + a2n xn = b2 ................................................. am1 x1 + am2 x2 + . . . + amn xn = bm

+,

- $ * ⎞ ⎛x ⎞ ⎛ b ⎞ 1 1 a11 a12 . . . a1n ⎟ ⎜ b2 ⎟ x ⎜ a21 a22 . . . a2n ⎟ ⎜ 2 ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎟=⎜ ⎟ ⎝. . . . . . . . . . . . . . . . . . . . .⎠ ⎜ ⎝ ⎠ ⎝ ⎠ am1 am2 . . . amn xn bm ⎛

+.

" $* A · x = b A = (aik )(m,n) ' x = (x1 , x2 , . . . , xn )Ì b = (b1 , b2 , . . . , bm )Ì ∈ Rn / (0 " 12$ 3 !4 B = (A|b) / ( +, !4 ' b = o ' !4 / ! ( $ 0 ' ) ! 5 x = o ∈ Rn 6)! ! ! ( m = n det A = 0 ! )"' ! ! 0 5 > 7 m = n 8 ! ! A

<

B = (A|b) ! xˆ1 d11 0

x ˆ2 d12 d21

0

0

. . . xˆr . . . d1r . . . d2r . . . drr

x ˆr+1 d1,r+1 d2,r+1 dr,r+1

0

. . . xˆn . . . d1n . . . d2n . . . drn 0

1 ˆb1 ˆb2 ˆbr

" #

%$ˆbr+1 ˆbr+s o

ˆi , ˆbk & ' det(dik )(r,r) = 0 " x A ( ) ! B = (A|b) ( % * ' " #+% , & r(A) = r(B) = r

" #%

& " #$% ˆbr+1 = ˆbr+2 = . . . = ˆbr+s = 0

. ' " #+% , / ! , 0 , " #$% & 1 'd11 xˆ1 + d12 x ˆ2 + . . . + d1r x ˆr = ˆb1 − d1,r+1 x ˆr+1 − . . . − d1n x ˆn d22 x ˆ2 + . . . + d2r x ˆr = ˆb2 − d2,r+1 x ˆr+1 − . . . − d2n x ˆn drr xˆr = ˆbr − dr,r+1 xˆr+1 − . . . − drn x ˆn

" 2%

(n − r) ) "3 %x ˆr+1 = t1 , x ˆr+2 = t2 , . . . , x ˆn = tn−r −∞ < ti < +∞ (i = 1, 2, . . . , n − r)

" 4%

4x1 + 6x2 + 17x3 + 8x4 = −20 − x3 + 2x4 = 4 2x1 + 3x2 + 7x3 + 7x4 = −4 2x1 + 3x2 + 8x3 + 5x4 = −8

4x1 + 6x2 + 17x3 + 8x4 = −20 − x3 + 2x4 = 5 2x1 + 3x2 + 7x3 + 7x4 = −4 2x1 + 3x2 + 8x3 + 5x4 = −8

x1 − x2 + x3 = 2 2x1 + x2 − x3 = 1 5x1 + x2 − x3 = 4

x1 − x2 + x3 = 2 2x1 + x2 − x3 = 1 5x1 + x2 − x3 = 0

x1 − x2 + x3 = 2 2x1 + x2 − x3 = 1 −x1 + 2x2 − x3 = 0

x1 + x2 + x3 = 3 x1 − 2x2 − x3 = −2 2x1 + x2 − 3x3 = 0

x1 + x2 + x3 = 3 x1 − 2x2 − x3 = −2 −x1 + 5x2 + 3x3 = 7

x1 + x2 + x3 + x4 = 4 x1 + x2 − x3 − x4 = 0 3x1 + 3x2 + x3 + x4 = 8 −3x1 + 3x2 − 2x3 = 0 2x1 − 2x2 + 5x3 = 0 x1 + 3x2 + 2x3 = 0

x1 − x2 + 5x3 + 8x4 = 0 4x1 + x2 − 3x3 + x4 = 0 2x1 − 4x2 + 3x3 − 4x4 = 0 3x1 + x2 − 2x3 + 2x4 = 0 ⎛ ⎞ ⎛ ⎞ x1 2 11 6 −4 −2 ⎜ ⎟ ⎜ 6 −9 −2 4 2 ⎟ ⎜x2 ⎟ ⎟⎜ ⎟ ⎜ ⎝−10 6 −2 −3 −1⎠ ⎜x3 ⎟ ⎝x4 ⎠ 1 −5 −2 2 1 x5 =o ⎛ ⎞ ⎛ ⎞ x1 12 3 4 5 6 ⎜ ⎟ ⎜2 4 3 2 1 0⎟ ⎜x2 ⎟ ⎜ ⎟ ⎜x3 ⎟ ⎟⎜ ⎟ ⎜ ⎜0 0 3 5 4 3⎟ ⎜x4 ⎟ ⎝3 6 9 11 10 9⎠ ⎜ ⎟ ⎝x5 ⎠ 12 2 2 2 2 x6 = (21, 12, 15, 48, 11)Ì ⎛ ⎞ ⎛ ⎞ 6 1 2 3 ⎛ ⎞ ⎜6⎟ ⎜3 2 1⎟ x1 ⎜ ⎟ ⎜ ⎟ ⎟⎝ ⎠ ⎜ ⎟ ⎜ ⎜1 1 1⎟ x2 = ⎜3⎟ ⎝7⎠ ⎝2 4 1⎠ x3 2 1 −1 2 ⎛ ⎞ ⎛ ⎞ x ⎛ ⎞ 1 2 3 2 1 ⎜ 1⎟ 6 x ⎜3 2 1 2 3⎟ ⎜ 2 ⎟ ⎜6⎟ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎝1 1 1 1 1⎠ ⎜x3 ⎟ = ⎝3⎠ ⎝x4 ⎠ 12 3 45 6 x5 2x1 + 3x2 + 4x3 + 3x4 = −4 x1 − 2x2 − 3x3 − 4x4 = 6 3x1 + x2 + 2x3 + 4x4 = 7 4x1 − 3x2 − 5x3 − 8x4 = 13 2x1 − 2x2 + 3x3 + 5x4 = 7 3x1 + 3x2 + 2x3 − 4x4 = 6

3x1 − 2x2 + 5x3 = 5 −2x1 + 3x2 − 6x3 = −7 5x1 − 7x2 + 4x3 = −4 ⎛ ⎞⎛ ⎞ 2 −3 −2 3 x1 ⎜ 3 5 −3 −5⎟ ⎜x2 ⎟ ⎟⎜ ⎟ ! ⎜ ⎝ 4 2 −4 −2⎠ ⎝x3 ⎠ x4 −5 4 5 −4 = (12, −1, 8, −23)Ì

⎛

2 ⎜−3 ⎜ ⎝5 3

3 2 −4 3

−2 3 2 −5

⎞⎛ ⎞ −11 x1 ⎜x2 ⎟ 10 ⎟ ⎟⎜ ⎟ −2 ⎠ ⎝x3 ⎠ x4 −19 = (−9, 7, 10, −18)Ì

λ

!" # $ %& '( ) $ *$ x+ y+ z= 3 3x + 5y + z = 9 2x + 3y + z = λ2 − 4λ + 6 5x + 6y + λz = 15 + , ( " # ' ## - ./) # - ./) ( # *$ %0 ) 1 x1 + x2 + x3 = 3 x1 + x2 − 3x3 = −1 2x1 + x2 − 2x3 = 1 x1 + 2x2 − 3x3 = 1

%

x1 − 2x2 − 3x3 = −3 x1 + 3x2 − 5x3 = 0 −x1 + 4x2 + x3 = 3 3x1 + x2 − 13x3 = −6

2x1 + x2 − x3 − x4 + x5 = 1 x1 − x2 + x3 + x4 − 2x5 = 0 3x1 + 3x2 − 3x3 − 3x4 + 4x5 = 2 4x1 + 5x2 − 5x3 − 5x4 + 7x5 = 3 2 ' ## *$ ( , ⎛ " # 1 ⎞ ⎛ ⎞ x1 −2 1 −3 3 ⎜ ⎟ ⎝ 1 −3 −1 4⎠ ⎜x2 ⎟ ⎝x3 ⎠ 2 −1 3 2 x4 = (5, 9, 10)Ì % *$ 3( )% |x1 − x2 + 3x3 − 3x4 | = 4?

4 A(n,n) = (aik )(n,n) 0 # + # det A(n,n) ) det A = 05 6 A 5 6 3 7 8 A = (aik )(n,n) A−1 = (bik )(n,n)

9

1 AA−1 = A−1 A = E n

E n : # + A−1 6 A 9 bik = (−1)i+k

Aki det A

Aki A ⎞ ⎛ A11 −A21 A31 . . . ⎟ 1 ⎜ ⎜−A12 A22 −A32 . . .⎟ A−1 = B = (bik )(n,n) = ⎝ A13 −A23 A33 . . .⎠ det A .........................

(A−1 )−1 = A; (A−1 )Ì = (AÌ )−1 (AB)−1 = B −1 · A−1 ; (λA)−1 det A−1 = (det A)−1

1 = A−1 (λ = 0) λ

! "

# $ A−1 A %& ' x = A−1 y (%& ) (* y = Ax & + ' %& (* %& &, -. -#$ / %&$ -& *%& - *%& *) $ n & #. n 0

yi =

n

aik xk = ai1 x1 + ai2 x2 + . . . + ain xn (i = 1, 2, . . . , n)

1

k=1

0 xk 0 yi %& + *) 1 %&%& S1 y1

x1 a11

yz yn

az1 an1

... ...

xs ... a1s . . . . . . azs . . . ...

ans

xn a1n

azn . . . ann

2 #' xs yz %& . , - azs = 0 , s. + z . 3 &4 *5 &4

/ %& xs yz & *%& S1 *%& S2 S2 y1

x1 b11

... ...

xs

bz1

...

yn

bn1

. . . bns

bik = aik −

yz b1s

... ...

bzs

...

xn b1n bzn

. . . bnn

+-

1 azs ais bis = - i = z azs azk 7 bzk = − - k = z azs

6 bzs =

ais · azk - i = z k = s azs

! "# !

$ %& ' ( " %& ' ( ! ''' ! )! ( ' *+ #', + # '' ! ( t p a ... p p ... a t ... b b ... t a ... p p ... a t . . . b b . . . t . ( t + #' ' & t − / ) !' 0 X

!' ' X(A + B) + C = 0 &1 $ 2

X ' !' '3 ! ) !' % ' #4 AXB + 2XB − 3D = 5C & AXB + 4AX = 8C − 3AXB

S1 y1 y2 y3 y4

a·b p

x1 x2 x3 x4 4 1 −1 −1 −9 −3 3 2 1 −1 0 −1 9 4 −2 −2

2 A−1 ! ⎛ ⎞ 2 2 3 A = ⎝ 1 −1 0⎠ −1 2 1 ' 6 7 &

''& 5 S1 ) & x3 y2 ! 5 S2

7 2 ! ⎛ ⎞ 3 −1 0 A = ⎝−2 1 1⎠ 2 −1 4 & 0 ' AA−1 = A−1 A = E

) & 5 S2 y4 ! & x4 S3

& 2 X 8 ' AÌ XA−1 + A = (A−1 )Ì

3

⎞ 2 1 1 D = ⎝3 2 1⎠ 1 2 0 ⎛ ⎞ 1 2 −1 −2 ⎜3 8 0 −4⎟ ⎟ F = ⎜ ⎝2 2 −4 −3⎠ 3 8 −1 −6 ⎛ ⎞ −2 −2 −7 G = ⎝ 1 −2 5 ⎠ −3 3 −12 ⎛ ⎞ 2 5 4 3 ⎜0 3 4 2⎟ ⎟ H = ⎜ ⎝0 0 3 1⎠ 0 0 0 4 ⎛ ⎞ 3 −2 0 −1 ⎜0 2 2 1⎟ ⎟ K = ⎜ ⎝1 −2 −3 −2⎠ 0 1 2 1 ⎛ ⎞ 8 0 0 L = ⎝0 −3 0⎠ 0 0 2 ⎛

A X 2(B+X)+XA = X(B−A)+3X

⎛ ⎞ 2 1 1 A = ⎝0 1 −2⎠ ⎛2 2 2 ⎞ 4 3 1 B = ⎝3 2 −1⎠ 8 5 3 1 2 A = 2 5 −3 4 B = −2 3 ⎛ ⎞ 1 0 0 C = ⎝3 1 0⎠ 4 −5 1

! y1 = a11 x1 + a12 x2 + . . . + a1n xn + c1 y2 = a21 x1 + a22 x2 + . . . + a2n xn + c2 ...................................................... yn = an1 x1 + an2 x2 + . . . + ann xn + cn

"#$

! ⎛

a11 ⎜ A=⎝ an1

...

⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ a1n x1 y1 c1 ⎟ , x = ⎜ ⎟ , y = ⎜ ⎟ , c = ⎜ ⎟ ⎝ ⎠ ⎝⎠ ⎝⎠ ⎠

. . . ann

xn

yn

cn

% y = Ax + c

'! Rn (

"#&

•

•

•

•

c = o

c = o

det A = 0 det A = 0

y = Ax

R ! n

• " i# $%& ei x#'( y #'( i# ')%& ai = (ai1 , ai2 , . . . , ain )Ì % A • ' ai = (ai1 , ai2 , . . . , ain )Ì i = 1, 2, . . . , n *& x#'( bi = (bi1 , bi2 , . . . , bin )Ì i = 1, 2, . . . , n y #'( + ')) & b11 b12 . . . b1n a11 a12 . . . a1n a21 a22 . . . a2n = det A · b21 b22 . . . b2n , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bn1 bn2 . . . bnn an1 an2 . . . ann

" y = Ax & -

( det A > 0 .+ det A < 0 & / " *& c = o * / -

( $ 0 1 A = A(n,n) 2

AÌ A = E n

3 A det A = ±14 / & $ 5# / 0 1 ! A−1 = AÌ " 6%& ')%& % A ) $# %& 0 + i = k ai1 ak1 + ai2 ak2 + . . . + ain akn = 1 + i = k k 0 + i = a1i a1k + a2i a2k + . . . + ani ank = 1 + i = k $ 2 0# 1 A 3 . det A = 1 & " -

(

y1 = x1 + 1, 5 y2 = x2 − 2, 5 4 3 x1 + x2 y1 = 5 5 3 4 y2 = − x1 + x2 5 5 4 3 y1 = − x1 + x2 + 1, 5 5 5 3 4 x1 + x2 − 2, 5 y2 = 5 5 1 5 4 A= 3 4 5

⎞ ⎛ 1 3 5 A = ⎝2 4 −1⎠ 1 7 3 * + ai + bi / ,$ (31 3 [b1 , b2 , b3 ] 1

4 [b1 , b2 , b3 ] = det A · [a1 , a2 , a3 ]

% & ' R3 ( {e1 , e2 , e3 } ) * + x , -* . E : x1 + x2 + x3 = 0 / 0 P 1 2

5 " ⎛ ⎞ 1 2 −2 1 2⎠ A = ⎝2 1 3 2 −2 −1 #$ A−1 " (6 3) $ + x1 = (4, 5, −2)Ì x2 = (1, −2, −3)Ì x3 = (19, −10, 13)Ì $ $ 1 y = Ax A ) (6 y 1 , y 2 , y 3 3) $1

+ 7 7$ )

& ' R3 ( {e1 , e2 , e3 } + Ì aÌ 1 = (2, 3, −1), a2 = (1, −1, −3)" Ì a3 = (1, 9, −11) ,$ (33 [a1 , a2 , a3 ] $ 1 bi = Aai

" / #$ / ! 1 1 1 A = √ 2 −1 1 ⎛ ⎞ 1 1 1 −1 1 ⎜−1 1 1 1 ⎟ ⎟ A = ⎜ 2 ⎝−1 −1 1 −1⎠ 1 −1 1 1

! det A = 1" A #$

& A = A(n,n) 8$ " det (A − λE n ) = 0

955

$ $ n1 λ ( $: A

n λ1 , λ2 , . . . , λn A xi ! " #$

(A − λi E)x = o

xi % & μxi (μ = 0) '( ) ri =

xi |xi |

" #$

*+ % ! (AÌ = A) , $ '- )$ . '- '( $ / k '- k 0 - '( $ . n '( r i ( -1 - ) , 1 + i = k 2 3( 0 ( (r i )Ì · rk = 0 + i = k 4 ) " #5$

R = (r 1 , r2 , . . . , r n )

det R = 1 6 R 1 A . , ⎛ ⎞ λ1 0 . . . 0 ⎜ ⎟ ⎜ 0 λ2 ⎟ ⎟ RÌ AR = D = ⎜ ⎜ 0 ⎟ ⎝ ⎠ 0 . . . 0 λn 2

) ! ⎛ ⎞ 6 2 2 A = ⎝2 3 −4⎠ 2 −4 3 ' $ '- A )$ '(

" #7$

8! ) $ 9 RÌ AR = D 4 : '( ) ! A 8! ) (

⎞ λ1 0 0 RÌ AR = ⎝ 0 λ2 0 ⎠ 0 0 λ3 λ1 , λ2 , λ3 A ⎛ ⎞ 15 6 0 1 ⎝ A = 6 22 6 ⎠ 11 0 6 29 ⎛ ⎞ −2 2 −4 1

A = ⎝ 2 −5 −2⎠ 3 −4 −2 −2 ⎛

⎛ ⎞ 0 1 0 0 1⎠ A = ⎝ 0 −6 −1 4 ⎛ ⎞ 2 −1 2

B = ⎝ 5 −3 3 ⎠ −1 0 −2 ⎛ ⎞ 4 −5 2 C = ⎝5 −7 3⎠ 6 −9 4

⎛

4 −5 D = ⎝ 1 −4 −4 0 ⎛ 5 1 3 ⎜2 3 1 F = ⎜ ⎝4 2 4 3 1 3 ⎛ 0 1 ⎜0 0 G = ⎜ ⎝0 0 48 −28

⎞ 7 9⎠ 5 ⎞ 2 3⎟ ⎟ 6⎠ 4 0 1 0 −8

⎞ 0 0⎟ ⎟ 1⎠ 7

! " #$ ⎛ ⎞ 11 −6 2 A = ⎝−6 10 −4⎠ 2 −4 6 ⎛ ⎞ 1 2 2

B = ⎝2 1 −2⎠ 2 −2 1

% &'! P ' ( ) ! x" *' +' xÌ Ax + aÌ x + a0 = 0

, -

$ A (AÌ = A) ." / •

,0 "

A= •

a11 a21

a12 , a= a22

a1 , x= a2

⎞" ⎛

x1 x2

⎛ ⎞ ⎛ ⎞ a1 x1 a11 a12 a13 A = ⎝a21 a22 a23 ⎠ , a = ⎝a2 ⎠ , x = ⎝x2 ⎠ a31 a32 a33 a3 x3 , 1 a12 = a21 , a13 = a31 , a23 = a32

A det A = 0

x = y − v v =

1 −1 A a 2

! 1 y Ì Ay + b0 = 0 b0 = a0 − aÌ A−1 a 4

#

"

$

y = Rz

" % z Ì Dz + b0 = 0

! #! D # & ' A( R ) & *+ ', det R = 1 - . %/ " 0 1 .2* ( / ! * λ1 > 0 3 3 ! )4, 5 *

. " 0 6 λ1 z12 + λ2 z22 + b0 = 0 λ1 >0 >0 >0

λ2 >0 <0 <0

b0 <0 = 0 =0

'4 784 ! *

%/ " 0 6 λ1 z12 + λ2 z22 + λ3 z32 + b0 = 0 λ1 >0 >0 >0 >0

λ2 >0 >0 >0 = 0

λ3 >0 <0 <0 =0

b0 <0 = 0 =0 = 0

'4 784 ! # 44, 98

A (det A = 0) ) ' : * % */ x = Ry

;

y Ì Dy + aÌ Ry + a0 = 0

λ = 0 ! " #$% % & %% % % % a % % % ' #$% %

y =z−v

% " ( λi = 0 % ' ) " ( ' #%& * '' ) & + 2 2 k k k i i λi yi2 + ki yi = λi yi + − i , % vi = − , 2λi 4λi 2λi

-' . $" % #% % ( '! ' k0 k0 kj yj + k0 = kj yj + / , % vj = − kj kj

0 # 1 2 λ1 z12 + c1 z2 + c2 = 0 λ1 >0 >0 =0

c1 = 0 =0 = 0

c2

<0

3 4 ( (

5& 1 2 λ1 z12 + λ2 z22 + c1 z3 = 0 λ1 >0 >0 >0

λ2 >0 <0 =0

c1 <0 <0 = 0

4 3 %% .6& 4 % -7

8 5 94 % '! % .$ 3x21 + 10x1 x2 + 3x22 − 2x1 − 14x2

−13 = 0

4x21 + x22 + 4x23 − 4x1 x2 − 8x1 x3 +4x2 x3 − 28x1 + 2x2 + 16x3 + 45 =0

x2 − y 2 + 2(x − y) = 1 6x1 − x22 − 4x2 − 16 = 0 x2 + y 2 − 8x + 2y + 13 = 0 16x2 + 9y 2 + 64x − 18y = 71

!" # x21 + x22 + x23 − 3x1 + 5x2 − 4x3 = 0 x2 + y 2 + z 2 = 2az (a = 0)

$ %& & 41x21 + 41x22 − 18x1 x2 + 46x1 +146x2 − 631 = 0 2 2 5x1 + 5x2 − 26x1 x2 − 36x1 +36x2 − 36 = 0

'! ( ) * + -

,

13x21 − 12x1 x2 + 4x22 + 88x1 − 48x2 +144 = 0 3x21 − 4x1 x2 − 2x1 + 4x2 − 5 = 0 x21 − 2x1 x2 + x22 − 10x1 − 6x2 +25 = 0 2 2 4x1 −4x1 x2 +x2 −6x1 +3x2 −4 = 0 * + .

/ +

0

1 * %2 , 7x21 − 4x1 x2 − 8x1 x3 + 4x22 + 4x2 x3 +7x23 + 2x1 − 4x2 + 16x3 + 18 = 0 −2x21 − 4x1 x2 − 4x1 x3 + x22 +8x2 x3 + x23 + 16x1 − 14x2 −8x3 + 6 = 0 2 x1 − 4x1 x2 + 4x1 x3 − 2x22 + 8x2 x3 −2x23 + 4x1 − 12x2 + 4x3 + 7 = 0 2x21 + 4x1 x2 − 4x1 x3 + 5x22 −8x2 x3 + 5x23 + 16x1 + 32x2 −30x3 + 54 = 0 2 2 2

2x1 − 7x2 − 4x3 + 4x1 x2 − 16x1 x3 +20x2 x3 + 60x1 − 12x2 +12x3 − 90 = 0

2x21 + 5x22 + 2x23 − 2x1 x2 − 4x1 x3 +2x2 x3 + 2x1 − 10x2 − 2x3 − 1 = 0 * %2 . . %2

3 * %2 , 5x21 + 6x22 + 7x23 −4x1 x2 + 4x2 x3 −10x1 + 8x2 + 14x3 − 6 = 0 2x21 + 5x22 + 11x23 −20x1 x2 + 4x1 x3 + 16x2 x3 −24x1 − 6x2 − 6x3 − 18 = 0 2 x1 + x22 + x23 −6x1 + 8x2 + 10x3 + 1 = 0 * %2 . % &

4 /!" + 56 76 0

a1 , a2 , a3 , . . . {an } • K > 0 n ∈ N \ {0} | an |≤ K • n ∈ N \ {0} an+1 ≥ an an+1 ≤ an • a ε > 0 n0 = n0 (ε) | an − a |< ε

• •

n > n0 .

! n→∞ lim an = a lim an = a an → a ! " lim an = ∞ lim an = −∞ K > 0 n0 = n0 (K) an > K

an < −K)

n > n0

#

" ! " •

n 1 # lim 1 + =e n n √ √ $ lim n c = 1 (c > 0) % lim n n = 1 n & lim 1 + a = ea , a ∈ R lim ln n = 0 n n ⎧ 0 |q| < 1 ⎪ ⎪ ⎪ ⎨1 q = 1 n ' lim q = ⎪ +∞ ( ") q > 1 ⎪ ⎪ ⎩ " q ≤ −1

lim 1

=0

( lim an = a lim bn = b ! )* lim(can ) = c · lim an = c · a

(c

+ )

lim(an ± bn ) = lim an ± lim bn = a ± b lim(an · bn ) = lim an · lim bn = a · b an lim an a (bn = lim = = 0 ∧ b = 0) bn lim bn b lim an bn = (lim an )lim bn = ab

(an > 0 ∧ a > 0)

lim log an = log lim an = log a (an > 0 ∧ a > 0)

{an } {sn } sn = a1 + a 2 + · · · + a n =

n

ai

i=1

s = a1 + a2 + · · · =

∞

ai

s = lim sn

n→∞

i=1

! " # $% lim sn = s & ' s ( ∞ ) ai

i=1

# an % " * $ ∞ • | ai | ! " i=1

•

+ sn = sn =

n

n

2a1 + (n − 1)d a1 + (i − 1)d = 2 i=1 n

a1 q i−1 = a1

i=1

s =

∞

a1 q i−1 =

i=1

∞ 1 i=1 ∞ i=1

i

"

qn − 1 , q = 1 q−1

a1 , | q |< 1 ! " 1−q

∞ (−1)i+1 i=1

i

1 ! " , a > 1 " , a ≤ 1 ia

! "

∞ ai =⇒ lim an = 0 n→∞

i=1

∞ an+1 n lim |an | lim ai n→∞ n→∞ an i=1 <1 >1 =1

! "

∞

(−1)i+1 ai ai > 0 ! i # {an } $

i=1

# # lim an = 0 an+1 ≤ an ! n ≥ n0 n→∞

%& ∞

ai

# %&

∞

i=1

i=1

% ∞ ai

∞

i=1

i=1

' # ( | an |≤ bn ! n ≥ n0 .

# %

' # ( an ≥ bn ! n0 ≥ n0 .

bi

bi

) " ! * + " ! ,- . ∞ ∞ / 0 ai 1 2 c = 0# cai i=1

1 2 0 3 (

i=1 ∞ i=1

c · ai = c

∞

ai

i=1

" ' ! ' # 4

3 2n − 5n2 + 8 7n3 + 2 n+7 ! 2 1− n−3 √ 2 { 4n + 3n − 2n} a √ 3n + 4 1+ n an = an = 2n + 1 n3

n0 = n0 (ε) | an − a |< ε n > n0 !" ε = 0, 001 # $ % ε & ' ( % ) a n 2 3n − 2 2 n 5n + 1 3n 2n n2 2 2 n +1 n +1 * +

an+1 = q < 1 $ lim n→∞ an lim an = 0 n→∞ # + n n lim n = 0 lim 2 = 0 n→∞ 2 n→∞ n! n! (n!)2 lim n = 0 lim =0 n→∞ n n→∞ (2n)!

, + 6n − 3 an = 6 − 5n

an =

2n(n − 1)2 (n + 2)3

(2n − 1)3 (4n − 1)2 (1 − 5n) 1 an = (−1)n · 2 n +1 1 n an = (−1) · 1 + n 10 3 an = − √ n n 3 5n − 2

an = 3n − 1 n − 10 an = 3 n−1 an = 3 8n + 10 n - an = √ 3 3 8n − n − n √ an = n 2 + n − n (−0, 3)n an = 3n − 2 n 1 an = 1 + 2n n +3 5 4 an = 1 − n √ n an = 3

an =

27log3 n 16log2 n n n+3 / an = n−5

. an =

0 . a = 7 a = 3 1 2 a = −6

Ko = −3 Ku = −12

" 3 -

s ** 0 2 = 2%% 1%+ 5% {sn } ! 1 1 1 *, > 1 2 2 + + + ... " 1·2 2·3 3·4 4 1 2 3 k(k 1+ 1) = k1 − k +1 1 + ... " + + + 2 5 8 11 1 1 1 1 2 3 + + + ... " 2 3 1 1·3 3·5 5·7 " + + 5 7 9 4 1 (2k − 1)(2k = 4 + 1) + + ... 11 1 1 1 2

2k − 1

−

2k + 1

# $ % & '

! " 0, 25 " 0, 49 " 0, 562 ( & ) * + , ) * " ) ) * " ) ) ./ */ 0 1 2 r 3 4 5 + 6 1 + 3 2 4 5 + 7 ) 8 " 1 9: " ; 9: 3 " <9: 3 " 1

103 104 10 102 + + + + ... 1! 3! 5! 7! 2 3 4 1 " − 2 + 2 − 2 ±. . . 2 2 +1 3 +1 4 +1 1 1 1 1 " 2 + 2 + 2 + 2 + . . . 2 5 8 11 *? 0 6 1 2

61 3 5 7 " + + + + . . . 2 4 6 8 1 1 1 1 " + + + + . . . 1 3 5 7 3 4 5 2 3 1 " + + 3 4 5 6 4 + + ... 6 1 1 1 + √ + ... " 1 + √ + √ 3 4 2 3 4 *. > ' 1 2 ! 2 3 4 2 3 4 1 " + + + +... 3 5 7 9 2k ∞ 2k " 3k + 1

"

k=1

k ∞ k+1 2k − 1 k=1 3k−1 ∞ 2k + 1 5k − 1 k=1 k (k2 ) ∞ 2 1 1+ 3 k

1 1 1 1 + √ + √ + √ + . . . 2 3 4 sin α sin 2α sin 3α + + ... + 1 22 32 1 1 1 + + k=1 1 + 22 1 + 24 1 + 26 1 + + ... 1 + 28 1 1 1 1 + + + + ... 2 3 4 ln 2 ln 3 ln 4 ln 5 2 2 2 2 + 15 + 15 + 15 + . . . $ 2 3 4 6 8 2 4 + + ... + + 1 1 1 3 9 27 81 1 + √ + √ + √ + . . . 4 8 16 2 3 3 5 5 7 7 + ... 1 + + + + 2! 3! 4! 5! 1 2 3 + + + ... ∞ 2k 2 1 + 14 1 + 24 1 + 34 ∞ ∞ k! sin 2k (k + 1) · 3k k=1 105k 3k 23 34 45 12 k=1 k=1 + + + + ... ∞ ∞ 2! 3! 4! 5! k5 (k + 1)! k 3 kk k=1 k=1 % &' " ( ) x * 1, 1 − 1, 01 + 1, 001 − 1, 0001 ± . . .

+ 1 1 1 1 − √ + √ − √ ± . . . ∞ ∞ xk 7k k 2 3 4 √ x kk k 1 22 33 44 k=1 k=1 − 2 + 3 − 4 ± . . . k ∞ 2 3 4 5 x−1 ; x = −1 1 1 1 1 x+1 − + − ± ... k=1 ln 2 ln 3 ln 4 ln 5 ∞ ∞ 1 1 1 1 1 xk k! ±. . . 1− + − ±· · ·+ 3 − x 2 8 3 n n+1 2k k=1 k=1 ! ∞ ∞ xk "

k · xk k2 "# k=1 k=1

x • y = f (x) • D(f ) ⊆ R x ∈ D(f ) y W (f ) y ∈ W (f ) ⊆ R !" • # y = f (x), x ∈ D(f ) • # F (x, y) = 0, x ∈ D(f ) • # x = ϕ(t), y = ψ(t), t ∈ D(ϕ) = D(ψ) f : y = f (x), x ∈ D(f ) $ f −1 f #

f −1 : y = f −1 (x), x ∈ D f −1 = W (f )

%&

' (" ! y = f (x) y = f −1 (x) ) * ( y = x y = f (x), x ∈ D(f ) + ,! I ⊆ D(f ) "- x1 , x2 ∈ I x1 < x2 " . # & f (x1 ) ≤ f (x2 ) / f (x1 ) < f (x2 ) 0 f (x1 ) ≥ f (x2 ) 1 f (x1 ) > f (x2 ) y = f (x), x ∈ D(f ) •

" E ⊆ D(f ) #

| f (x) |≤ K •

+

K > 0 $

"- x ∈ E

f (−x) = f (x)

%/

"- x ∈ D(f ) #

f (−x) = −f (x))

%0

•

p p > 0 f (x + p) = f (x)

x ∈ D(f )

!

" y = f (u), u ∈ D(f ) u = g(x), x ∈ D(g) # " W (g) ⊆ D(f ) $ x%

y = f g(x) , x ∈ D(g) &

'( )*) + , ( D(f ) ⊆ R f (x) √ f (x) = x + 1 1 ( f (x) = 4 − x2 √ f (x) = x4 − 2x2 √ f (x) = x − x3 √ 1 f (x) = −x + √ 2+x x2 − 3x + 2 f (x) = lg x+1 2x

f (x) = arccos 1+x x f (x) = arcsin lg 10 √ 1 − lg(2x − 3) f (x) = x + 3 x−2 - f (x) = 21/(1−x) f (x) = x − arctan x x−3 − lg(4 − x)

f (x) = arcsin 2 1 f (x) = | x2 − 2x − 3 |

. '( )*) + , ( D(f ) ⊆ R " ),

/ ( W (f ) 0 , ( 3 y = f (x) = √ x−5

x−3 x2 − x − 6 1 1 − y = f (x) = x+2 x−2 y = f (x) =⎧ ⎨3 − x2 | x |≤ 1 2 ⎩ | x |> 1 |x|

( y = f (x) =

1 2"" % y =

10 x2 + 1

( y =

2x − 3 3x + 2

y = | sin x| 3 y = − 25 − x2 5 π y = 5 sin 2x − 2 y = |x|

y = x|x| 1 x2 − 1 y = arcsin y = 2 x x −4 ! 2

, (" 3 4 5 f (x) = −7 sin x cos x ( f (x) = | sin x| f (x) = 6 e−2x f (x) = 3x + 7 f (x) = 5x4 − 2x2 + 3 1+x f (x) = lg 1−x

1 + x2 f (x + 1) = x2 − 3x + 2 3 x −1 1 1 ' f x + = x2 + 2 ; x = 0 x x √ 1 1 − 1 + 4x √ y = f (x) = ; = x +

f 1 + x2 ; x > 0 x " 11 + 1+ 4x 2 3 4 * D(f ) = − , +∞ 4 [−a , a] & f (x) ! 5 $ " # $ + % & $ "'$ ( 22 ' $ f .% . & $ & ' D(f ) $ ' D(f ) ⊆ R 6 7 - 89 : −1 )' * f + 5 : ; $ & '

' 4 y = f (x) = (x − 5)3 ; ' D(f ) = [5 , +∞) ' + 3 5

' y = f (x) = x2 + 1; D(f ) = (−∞ , 0] x3 − 2x2 − 9x + 18 y = f (x) = x2 − 7x + 12

y = f (x) = (x − 3)2 ; D(f ) = (−∞ , 3] x2 − 4 ' y = f (x) = 2 1−x x − 2x + 1 ; y = f (x) = 1+x 2< / =: * D(f ) = (−∞ , −1) ∪ (−1 , +∞) 7 p0 ' :$ , -. x f (x) = sin 2x + 3 sin (3x − 2) 3 − 2x y = ln arcsin −0, 5 cos (0, 8x + 1) + 2 5 √ 5x x x+1+5 − 3 sin 4x ' f (x) = sin + 2 cos ' y = √ 2 2 x+1−3 x x

f (x) = tan − 2 tan

y = 53x−1 + 7 2 3 x + 1 5 f (x) = cos2 2x y = ln x−1 2> ' $

: / f f f (x) + f (x) = √ y = f (x) = cos x; 1 D(f ) = [0 , +∞) 1−x ' y = f (x) = cos x2 ; D(f ) = R 0 )' f (x) + 1

f (x) = ln

f U (x0 ) x0

! " x0 # $ f " %& x x0 g

lim f (x) = g f (x) → g %& x → x0

#

x→x0

lim f (xn ) = g %& ' {xn } xn ∈ U (x0 ) \ {x0 } lim xn = x0 n→∞ n→∞ ()" " xn > x0 xn < x0 # ") lim f (x) = gr

lim f (x) = g # x→x0 −0

x→x0 +0

l

$ lim f (x) = g ⇐⇒ gr = gl = g x→x0

$ "* + " %& x → x0 , x → x0 + 0, x → x0 − 0, x → +∞ x → −∞

,"#

• ," +∞ −∞ •

•

,"

$- lim f1 (x) = g1 lim f2 (x) = g2 - ," %

x→x0

x→x0

lim (cf1 (x)) = c · g1 (c . )

lim f1 (x) ± f2 (x) = g1 ± g2 ; lim f1 (x) · f2 (x) = g1 · g2

x→x0 x→x0

x→x0

g1 f1 (x) = 0, g2 = 0 f2 (x) = x→x0 f2 (x) g2

f2 (x)

lim f1 (x) = g1g2 f1 (x) > 0, g1 > 0

1#

x→x0

lim

x→+∞

# 0#

lim

(," sin ax =a lim x→0 x

/#

x 1 + x1 = e

1

lim (1 + x) x = e x→0

1

lim e x = ∞; x→+0

1

lim e x = 0

x→−0

1 x

lim x = 1

lim x = 0

1 = +∞ x π 1 lim arctan = x→+0 x 2

1 = −∞ x π 1 ! lim arctan = − x→−0 x 2

x

x→+0

lim

x→+0

" # $ $ % & x2 − 2x − 15 ' lim x→5 x2 − 25 sin 3x (' lim x→0 x x + 1 x2 lim x→+∞ 3x + 2 2x − 2 lim x→1−0 |x − 1| arcsin 7x lim x→0 x

x2 − 4 x→2 x2 − 3x + 2 x3 − 3x + 2 lim 4 x→1 x − 4x + 3 1 3 lim − x→1 1 − x 1 − x3 √ 2− x−3 lim x→7 x2 − 49 sin 7x lim x→0 2x sin 5x lim x→0 sin 2x 1 − cos x lim x→0 x2 sin x − cos x lim π 1 − tan x x→ lim

4

x→+0

lim

x→−0

arcsin x x arctan 2x lim x→0 sin 3x 1 − x2 lim x→1 sin πx 2x2 − x lim x→+∞ x2 + 10 lim ( x2 + 1 − x2 − 1)

lim

x→0

x→+∞

lim (1 + tan x)cot x x→0

lim

x + 1 x

x→+∞

x−1

!

sin x " x = 0 |x| 2+x f (x) = " x = 2 4 − x2 f (x) =

1

f (x) = e 1−x3 " x = 1

x+2 x " x = −2 |x + 2| x f (x) = " x = −1 1+x x−1 f (x) = " x = 1 | tan (x − 1)| 1 f (x) = arctan " x = 1 1−x √ x + 12 − 3 f (x) = " x = −3 x2 − 9 f (x) =

y = f (x) U (x0 ) x = x0

f • x0 ⇐⇒

lim f (x) = f (x0 )

x→x0

• x0 ⇐⇒ • x0 ⇐⇒

lim

x→x0 −0

lim

f (x) = f (x0 )

x→x0 +0

f (x) = f (x0 )

• I ⇐⇒ f x ∈ I • x0 ⇐⇒ f x0 f

x0

f x0

x = x0 • f (x0 ) = g lim f (x) = g x→x0

f (x0 ) = g x0 ∈ D(f ) • lim

x→x0 +0

f (x) = gr gl = gr

•

lim

x→x0 −0

f (x) =

lim

x→x0 +0

f (x) = gl ,

lim

f (x) = +∞ ! −∞"

x→x0 +0

•

lim

x→x0 −0

lim

x→x0 −0

f (x) = +∞ ! −∞"

f (x) = −∞ ! +∞"

# $ % & ' ( $ ) f1 f −1 f * + , - & x $ . & % . / " f (x) = e1/x 1 1 + 21/(x−1) x " f (x) = 2 x −4 x−2 " f (x) = 2 x − 3x + 2

" f (x) =

x sin x |x − 2| " f (x) = x−2 sin x " f (x) = |x| 1 " f (x) = (x − 1)2

" f (x) =

0 - -

. $ $

x f (x) = √ x = 0 x+1−1 1 f (x) = x sin x = 0 x sin x

f (x) = x = 0 x x2 + 3x − 10 f (x) = 2 x = 2 x + 4x − 12 (1 + x)n − 1 f (x) = x = 0; x n ∈ N \ {0} ln (1 + x) − ln (1 − x) f (x) = x x = 0 1 f (x) = x2 sin x = 0 x !" "# $

1 x−2 x = 2 % $ f (x) x = 2 y = f (x) = arctan

& 2x x ∈ [0 , 1) y = f (x) = 3 − x x ∈ [1 , 2] " & ' [0 , 2] ( ) ! ' a * x ln (x2 ) x = 0 y = f (x) = a x = 0 " & ' (−∞ , +∞) ! f (0) % $ " 1 f (x) = 1 − x sin " & ' x I = (−∞ , +∞)

+ f : y = f (x), x ∈ D(f ) " x0 ∈ U (x0 ) ⊆ D(f )$ ) , ) - lim

x→x0

f (x) − f (x0 ) f (x0 + h) − f (x0 ) = lim h→0 x − x0 h

- ..

/ 0 ) ) 1 y f (x) = = f (x0 ) x x=x0 x x=x0

- .2

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y 6

y = f (x)

y0

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f (x) · f2 (x) − f1 (x) · f2 (x) f1 (x) • = 1 f2 (x) = 0!

2 f2 (x) f2 (x)

$ " g x0

f z0 = g(x0 ) ! " % y = f g(x) x0 ! "

f g(x) = f (z0 ) · g (x0 ) z0 = g(x0 )

x x=x0

#

y

y z = ·

x

z x

&' " y = f (x) & U x0 " f (x) = 0 ( x ∈ U ! " " f )' '! ' " *" x = f −1 (y) ( y0 = f (x0 )

−1 1 f (y0 ) = f −1 (y0 ) =

y f (x0 )

#

1

x = y

y x

g(x) y = f (x)

g(x) f (x) g(x) f (x) > 0! " )' y = f (x)

g(x) y = f (x) = e g(x)·ln f (x)

g(x) g(x) · f (x) g =⇒ y = (f ) = f (x) · g (x) ln f (x) + f (x)

! y = f (x) x = x0 " # $ ! f (x) 2 y 2 f (x) y (x0 ) = = f (x0 ) = = % x x=x0 x2 x=x0 x2 x=x0 & ' ( n) ! n y n f (x) (n) (n) y (x0 ) = f (x0 ) = = xn x=x0 xn x=x0

f (x) ! f : y = f (x), x ∈ D(f ) * x" # y = f (x) x

+

, f * x y 6 f (x + x)

6

f (x)

x

6 y

Δy

?

?

x + x

y = f (x) x Δy = f (x + x) − f (x)

-

x

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$/

x3 y = − 2x2 + 4x − 57 3 √ √ 10 3 4 y = 3 − 6 x2 + 4 x3 x √ √ 5x2 − 3 x + 3x3 4 x √ y = 3 2 x2 πx

y = 1 − x2 √ y = x 3x 2x 3x2 + 6x − 1 x3 − 2x + 7 y = ln |x| sin x + 27 tan x y = 3x2 − 2 cos x tan x y = 2 − √ x x cos x y= 1 − sin x √ x y = √ x+1 y = x ln x − lg (5x3 ) 1 + ln x ! y = x 3 x y = 2x e −3x + e5 7x2 − 6x y = ex " # ! $ % & ' & ! y = u(x) · v(x) · w(x) = uvw ! y = u(x) · v(x) · w(x) · z(x) = uvwz (

y =

) y = f (x) = −x + (x − 2)2 & * $ x ∈ (−∞ , +∞)

+ ,! & & & - ,! ( . f (x) / .0 y = f (x) + ,! & %

& - ,!

( . y = f (x) = |x|, Df = (−∞ , +∞) y = f (x) = | sin x|, Df = [0 , 2π] 1 5 √ y = (1 + x2 ) 1 + x2 √ y = x + 2 x; f (1) =? y = ln tan (3x + 2)

y = ln (x + x2 + a2 ); a = 0 1−x y = ln 1+x y = a · ebx cos cx y = ln | ln x| x y = ln tan 2 1 − sin x y = ln 1 + sin x x π + y = ln tan 2 4 4 √ x 1 2 y = (ln x) − ln x + 4 8 x 2 y = x − 25 2 25 − ln x + x2 − 25 2 √ 1 + x2 + 1 2 − x +1 ! y = ln x y = 2 ex/2 − e−x/2

y =

1 + e−3x 1 − e−3x

!

y = y(t) = sin t · cos t

# ( # ( ,( # # + ,( #

x = x(t) = cos2 t s v = v(s) = s−1 w = w(s) = s4 ln s ! x3 y = x · arctan x − ln 1 + x2

y = y(x) = x−1 1 y = arccos " x # ! y = arcsin 2x 1 − x2

y = sin x y = ln x √ y = cosh3 x2 y = x + 1 y = cos 3x y = ln cosh (ax + b), a = 0 $% ##

1 2 f (x) &'# ( x + sinh2 x y = 2 # y = x arsinh x − x2 + 1 ln (1 + x)

y = , α %

y = (arctan x)x , x > 0 (1 + x)α # &'# y = ln x2 + x + 1 y √ 2x + 1 + α y = (1 + x)−α (1 + x) + 3 arctan √ x 3 y = C1 e−x +C2 e−2x # &'( x − 1 # y + 3y + 2y = 0 + 2 arctan x - y = ln x+1 √ ) y = arctan x3 ! y = ln arctan x

y = 10 x−3x ; x > 0 . y = cosh (sin x) y = x1/ ln x ; x > 0 y = tanh3 x2 π 1/ cos x y = (tan x) ; 0<x< y = arcsin (tanh x) 2 x−1 , y = 1 + sinh2 4x y = exp √ √ x+1 / y = 4x − 1 + arccot 4x − 1 x y = x ; x > 0 y = x1/x ; x > 0

* +#!

#.. . # y (n) % + #!

1+x 1−x √ y = x

y =

y = eax

y = sin2 x 1 f (x) = 1 − x2 n! ! n = 2m (n) f (0) = 0 ! n = 2m − 1

1 1 = 1 − x2 2

1 1 + 1+x 1−x

" # $ % (x−1)(x2 +x3 +· · ·+xn ) = xn+1 −x2 " x "& ' ( x = 1 ' ## n (n + 1)n(n − 1) k(k − 1) = 3 k=1

"&" ' ## ) ( n !" " * "&" 0 n k 2 = 12 + 22 + ... + n2 = k=1

##

n(n + 1)(2n + 1) 6

+ ," f (2), f (2) f (2) ! 1 f (x) = arcsin x - ! f (x) = xn f (1) f (1) f (1) + + + ... 1! 2! f (n) (1) = 2n + n! . / 0 1 " x t x = f (t) x v = t 2 v x = 2 a = t t

x x

t3 = − 2t2 + 3t

3

! " #

$

%

"

x = A cos ωt

x

&

±A

=

x = 0

' 2 x t2

x ( x ¨ = −ω 2 x

! !) * + , * "

r

- * .

v

&

$ - ' ,

h

' $

+!

/0! +&

h V = V (h) = πh2 r − 3

t Vt = v ! %

x

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$% "#! A(−1; 1) B(3; 9) & "' ( $ )* " ! ) " " & ) " ξ + f (x) = x − x3 ; x ∈ [−2, 1] "+ f (x) = arctan x; x ∈ [0, 1] + f (x) = arcsin x; x ∈ [−1, 0] + f (x) = ln x; x ∈ [1, 2] , $ f (x) = x(x + 1)(x + 2)(x + 3) -% . f (x) = 0 /

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f (x + Δx) − f (x) = Δx · f (x + ϑ · Δx) % 0 < ϑ < 1 $ )* ! + f (x) = x2 1 "+ f (x) = x3 ) -% ϑ & x " % " ! " & x Δx "

2

3! ) 45 + ex > 1 + x )* x > 0 √ x "+ 1 + x < 1 + )* x > 0 2 x 1 >1− + √ )* x > 0 2 1+x 6 -% 5 √ 1 101 = 10 + √ 2 100 + ϑ )* 0 < ϑ < 1 " 7" √ 1 1 < 101 < 10 + 10 + 22 20 $ )* ) ! ) " ξ + f (x) = sin x; g(x) = cos x; π x ∈ [0, ] 2 √ "+ f (x) = x2 ; g(x) = x; x ∈ [1, 4] + f (x) = x2 + 2; g(x) = x3 − 1; x ∈ [1, 2]

f (x) g(x) 4" U & x0 8" x0 "+ " % ) g (x) = 0 9 ! & U lim f (x) = lim g(x) = 0,

x→x0

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lim f (x) = lim g(x) = +∞ ( ∞),

x→x0

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lim

x→x0

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lim x→0 1 − cos bx 1 − cos x lim x→0 x2 x − sin x lim x→0 x3 tan x − sin x # lim x→0 x − sin x ex lim 3 x→+∞ x ex lim 3 x→−∞ x ln x / lim x→+∞ x ln x lim x→+0 cot x

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x

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x = x0 f (x) = f (x0 ) +

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f (x0 ) f (x0 ) (x − x0 ) + (x − x0 )2 + . . . 1! 2! f (n) (x0 ) (x − x0 )n + Rn + n!

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+!

f (n+1) (x0 + ϑ(x − x0 )) (x − x0 )n+1 ; 0 < ϑ < 1 . (n + 1)!

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f (x) % . ) I • ! f (x) ≥ 0 %& x ∈ I, • ! f (x) ≤ 0 %& x ∈ I.

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x→+∞

2 ! g 2 ! f ( 3 - ( 4! * 5 (−∞, a] x → −∞6 0 * + 4 6 7 898 :1 + ! - 5 - 6 f (x) = cosh2 x + 1 2x2 − 1 +6 f (x) = x4 6 f (x) = x 1 − x2 x 6 f (x) = ln x 6 f (x) = x − 2 sin x ; - 5 + 7 < 6 f (x) = x3 ; D(f ) = (−∞, +∞) +6 f (x) = ex ; D(f ) = (−∞, +∞) x 6 f (x) = ; 1 − x2 D(f ) = (−∞, −1) ∪ (−1, 1) ∪(1, +∞) " 0 ! 7 =

√ 3 6 f (x) = 2x + 3 x2 ; D(f ) = (−∞, +∞) +6 f (x) = ln 1 + x2 + arctan x; D(f ) = (−∞, +∞)

6 f (x) = 2x ex−2 +4 ex−2 −x2 − 6x; D(f ) = (−∞, +∞) 6 f (x) = sin3 x + cos3 x; D(f ) = [0, 2π) 1 6 f (x) = (x − 4) · cosh (2x + 3) 2 1 − sinh (2x + 3); 4 D(f ) = (−∞, +∞) 1 6 f (x) = (x2 − 6x + 5) · ln (x − 1) 2 5x x2 ; − + 4 2 D(f ) = (1, +∞) 6 f (x) = x(ln x)2 − x ln x + x; D(f ) = (0, +∞) 0 89 % + 7 :1 + = 6 f (x) = x(10 − x); D(f ) = [0, 10]

f (x) = x3 − 3x + 3; 3 5 D(f ) = − , 2 2

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y = f (x) = e−x ; −∞ < x < +∞. ) * ( y +,

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x

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, -./

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lim xn = x0 .

n→∞

3 )# * 4 5 " ( "

# 0 6 / 2x − ln x − 4 = 0 1 / lg x − = 0 x / 2x − 4, 2x = 0 / xx + 2x − 6 = 0 1 / x2 + 2 − 10x = 0 x / x ln x − 14 = 0 / ex + e−3x −4 = 0 / sin x + 3x − 3 = 0 / x arctan x − 1 = 0

7/ ln x − arctan x = 0 - 8 " y = sin x, 0 ≤ x ≤ π x(5 ' (

0$ / ! $ " ## ' / $ 94 " ## ' & ! $(

94 x(5 3 : ' 0 . & ;

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x ∈ I,

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# f (x) x = F (x) + const = F (x) + C,

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f (x) x = f (x) + C # # kf (x) x = k f (x) x, (k 2 , k = 0) # # #

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1

3

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u = u(x) v = v(x) I

u v I I # # u(x) · v (x) x = u(x) · v(x) − u (x) · v(x) x

u = g(x) [a1 , b1 ]

f (u) [a2 , b2 ]

a2 ≤ g(x) ≤ b2 [a1 , b1 ] # #

f g(x) · g (x) x = f (u) u|u=g(x) ! ! ! "# $ % F (u) & u ! g(x) g (x) = 0 [a1 , b1 ] !' # #

f (x) x = f g(z) · g (z) z|z=g−1 (x)

#

f (x) x = ln |f (x)| + C; (f (x) = 0) f (x)

# f (ax + b) x =

1 F (ax + b) + C; (a = 0), F (z) = f (z) a

) ! ! *+ %+!

# 1 2 x x + 2x + x # 10x8 + 3 x x4 # x−2 x x3 # (x2 + 1)2 x x3 # √ √

x + 3 x x # 1 1 √ − √ x 4 x x3

√ ( x − 1)3 x x # x−1 √ ! x 3 x2 √ # 2x 3x √

x 4 x3 √ √ # 1 + x2 − 1 − x2 √ , x 1 − x4 # 2 3 √ % − x 1 + x2 1 − x2 # x−1 √ x x−1 √ # 3 − 1 + x2 x 2(1 + x2 ) #

(

e−x ex 1 − 2 x x # a−x ax 1 + √ x x3 # 2 x + 5x + 6 x x+3 # 2 sin x − cos x + cos2 x #

#

# # # #

7 − 2 x sin x

x sin2 x cos2 x x x 2 sin − cos x 2 2 # 4 x 2 x tan x x 1 + x2 # sin 2x x coth2 x x cos x # x x cos2 x sin2 x 2 2

!" #

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x cos x x x2 cos x x # # 2 x x e x ex sin x x # # √ (ln x)2 x x ln x x '" # ( & % # # 2x x2 x & x 2 3 x −5 7x + 8 # # √

(5x + 6)7 x 2x − π x # # x+2 cot x x x 2 + 3x2

#

2x + 3 x # 2x + 1 32x−1 x # 6 x 8 − 3x # * sin2 ϕ ϕ

#

cos #

#

x

#

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e−x x

x 5 + 2x2 # x x(1 + ln x) # t sin 2t )

− 3 x

2 3 x 2 cos (9x + 8)

tan (2x − 1) x √ 2 x − 6 x

tanh 3 # 2πt + ϕ0 t sin T # u (2u + 3)5 #

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# # # #

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sin3 x cos x x sin x x cos3 x ecos x sin x x 3

ex x2 x e− sin x cos x x √

e x √ x x # x x2 + 1 x # 2x2 x √ 3 1 + x3 # sin x x √ 1 + 2 cos x

# √ 1 + ln x x x # √ 1 + 4 sinh x cosh x x

#

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arcsin t t 1 − t2

ln2 x x x √ # 3 arctan x x 1 + x2 # cos (ln s) s s # sin3 ϕ dϕ cos5 ϕ # x√ arctan ex e x 1 + e2x # x sin (4x − 2) # x 1 + cos2 x

# arctan x x # (2x − 3) cos (5x + 1) x # 3x + 2 √ x 7x − 5 # x 1 − 1 ln 3x + x 2 5 # 3x − 1 x 2 cos (5x + 6) # 2 x − 5 e4−3x x 3 # x−5

x cos2 (3x + 1)

# arcsin x 2 x √ 2−x # x cos x x

sin3 x # √ arctan 2x − 1 x # eat cos ωt t # eat sin ωt t # x (2x − 1) arctan x 3 ! " # $ # 1 sinn x x = − cos x sinn−1 x n # n−1 · sinn−2 x x + n # 1 cosn x x = sin x cosn−1 x n # n−1 · cosn−2 x x + n %$ # # sin6 x x & cos6 x x # # 5 ' sin x x ( cos5 x x

) # # x x cos3 x sin3 x

% * # x ! sin x # x cos x + , % sin 2x sin x cos x = 2 # * ,

! $

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sin2 x cos2 x x

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sin x cos x x

1 sin α cos β = sin (α + β) 2

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− cos (α + β) : # sin 3x cos x x # cos mx cos nx x #

sin 3x sin 5x x # sin mx sin nx x # sin mx cos nx x

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(1 − x2 )3 # x3 x √ 2 − x2 # x √ x2 4 − x2 # 2 a − 2x2 √ x, a > 0 a2 − x2

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tan3 x x "# x = arctan u cot3 x x sin5 x x "# u = cos x cos7 x x

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g(x) =

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(

m < n )*" +#, -.% )*" & xn & ), *" & xn Qn (x) 1 ! f (x) m < n ! $ " f (x) /

' & 0! " + 0 1" 2

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& 3 #

x x 1 = arctan x2 + a2 a a # 1 x x x + = 2 2 arctan (x2 + a2 )2 2a (x + a2 ) 2a3 a

4 5

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x 2ax + b 2 · arctan √ =√ 2 ax2 + bx + c 4ac − b 4ac − b2 # mx + n m x = ln (ax2 + bx + c) ax2 + bx + c 2a 2ax + b 2an − bm · arctan √ + √ 2 a 4ac − b 4ac − b2 # x 1 2ax + b = · (ax2 + bx + c)2 4ac − b2 ax2 + bx + c 2ax + b 4a · arctan √ + 2 3 4ac − b2 (4ac − b ) # 1 bx + 2c x x =− · (ax2 + bx + c)2 4ac − b2 ax2 + bx + c 2b 2ax + b − · arctan √ 2 3 4ac − b2 (4ac − b )

!

" " # $ $ %& Qn (x) ' ( ) * + ", -. # x = x1 k + $/ , (x − x1 )k 0 x = α ± βi *

1 %2 $

l/ (x2 + px + q)l p = −2α q = α2 + β 2 , $+ . A1 Pm (x) A2 Ak = + + ... + (x − x1 )k (x2 + px + q)l ... x − x1 (x − x1 )2 (x − x1 )k M 2 x + N2 M 1 x + N1 M l x + Nl + 2 + 2 + ... + 2 + ... x + px + q (x + px + q)2 (x + px + q)l

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%

Qm (x) ; (x2 − 4)3 (x2 + 4)2 (x − 1)4 m < 14

& '( # 5x3 + 9x2 − 22x − 8 x x3 − 4x # x2 + x − 1 x 2x3 + 2x2 − 12x # 2x2 + 41x − 91 x x3 − 2x2 − 11x + 12 # 3x + 2 x x4 + 3x3 + 3x2 + x # x2 x x4 + 12x3 + 52x2 + 96x + 64 # 3x + 2 x x(x + 1)3

x3 − 2x + 2 x − 2x3 + 2x2 − 2x + 1 3x3 − x2 − 4x + 13 x x4 − 4x3 + 13x2 x 5 x − x2 x3 + 3 x (x + 1)(x2 + 1)2 x x 2 (x − 1) (x2 + 2x + 2) x4

! " # $ Qm (x)/ (x + 1)(x − 1) 3 (x2 + 1) ×(x2 + 2x + 3)2 ; m < 10 Qm (x)/ (3x − 15)2 (x + 2)

×(1 + x2 )2 (x2 − 2x + 5) ; m < 9

)

! ! " #$ % & ' ( " " # !

#

#

#

#

x x(x + a)# (x + a) − x 1 x = a (x # + a)x 1 1 1 ( − ) x = a x x+a x (x + a)(x + b) x 4 x − x2 − 2 x (x2 − 3)(x2 + 2) # x x x2 − 2x x4 − x2

#

x 3 x + 4x

#

#

x 2 x + 5x

x 4 x + 3x2

#

x −1

x4

! "

# ! R x, f (x), g(x) $ x, f (x), g(x)# #

#

√ n R x, ax + b x

√ n 1 n ax + b ⇒ x = (tn − b), x = tn−1 t a a % $ # n ax + b &# x R x, cx + d d · tn − b ax + b ⇒ x= t = n , cx + d a − c · tn tn−1 x = n(ad − bc) · t (a − ctn )2 # R(ex ) x

% t =

t = ex ⇒ x = ln t,

x=

t t

# R(tan x) x

t = tan x ⇒ x = arctan t,

x=

t 1 + t2

# R(sin x, cos x) x

t = tan cos x =

t = sin x

1 − t2 1 + t2

x ⇒ x = 2 arctan t, 2

x=

2 t 2t , sin x = , 1 + t2 1 + t2

t = cos x

#

R(sinh x, cosh x) x ex + e−x ex − e−x , cosh x = sinh x = 2 2 t = ex .

# x x √ 2x + 1 + 1 # x √ √ 3 x+ x # 2x e −2 ex x e2x +1 # 3x e x e2x −1 # tan4 x x # tan5 x x # x

5 + 3 cos x # x ! 3 sin x + 4 cos x # x

sinh x + 2 cosh x # x " cosh x # x3 x √ # x−1 # x 1 + sin x + cos x

#

#

#

#

$ #

% #

#

#

#

#

#

& #

' #

(

ex +1 x ex −1 x e2x + ex −2 x sin x + cos x x3 x √ 1 + 3 x4 + 1 x+1 √ x 3 3x + 1 e3x x ex +2 e2x x ex −1 x √ √ 2x − 1 − 4 2x − 1 √ x a − x x; a > 0 1+x 1 x (1 − x)(1 + x)2 1 − x x √ x + 1 + (x + 1)3 1−x 1 x 2 (1 − x) 1+x 1 3 1 − x · x 1 + x (1 + x)2

# √ 1+x x x # arctan x x 1 + x2

#

x3 #

x + ax2

x 1 + sin x

#

x x(1 − x) # x sin2 x cos2 x + a2 b2 # x cos2 x x # x

2x e + ex # 1−x x 1+x # cos2 x x sin4 x # x tan2 x x # cos2 x x sin x # sin x x b2 + cos2 x # x √ √ 3 2 x +2 x # ax − b x (ax + b)4 # x 4 x + x2 # x (sin x + cos x)2 # x √ x a + b ln x # x2 x (a − bx3 )n # 1 − 2x − x2 x # x √ (1 + x)3 # arctan x x x2 # x e −2 x e2x +4

#

#

! " #

(2x + 1)−1 (1 + cot4 x x

# √ 4 − x2 x x2 # cos x x cos 3x # sin x x sin 3x # x √ √ x+a+ x # x √ x2 + 1 − x # x4 + 1 x x3 − x2 √ # x2 + 2x x x3 # x √ x x3 − 1 # x 1 + tan x √ # arcsin x √ x x # sin 2x x cos4 x # cos 2x x sin4 x # ln (cos x) x sin2 x # x e3x − ex # sin3 x x cos5 x # ln (x + 1) x x2 # √ 1 − sin x x

√ 2x + 1)−1 x

#

x 2 # 1 + sin x x x x4 − x2 − 2 # √ e− x x √ # arctan x √ x x √ # tan x x sin 2x # ln (x2 + 1) x x3 # x a x a2x + 1 √ # 1 − sin x √ x x

# (x + 1)3 x x−1 # x arcsin x √ x 1 − x2 # x √ x2 x2 − 1 # x2 x (x + 1)4 # 4x + 1 x 3 2x + x2 − x # cos3 x + 1 x sin2 x # x 4 x +4

f (x) [a, b] ! " [a, b] # x1 , x2 , ..., xn−1 a = x0 < x1 < x2 < ... < xn−1 < xn = b n $ Ii = [xi−1 , xi ] (i = 1, 2, ..., n) % & % Δxi = xi − xi−1 ' " ξi ∈ [xi−1 , xi ] (

Sn =

n

f (ξi )Δxi

)

i=1

* Sn + % S ' , Sn → S - n → ∞ max Δi → 0

)

[a, b] ξi ∈ Ii " ' f (x) [a, b] )%, - . ,

#b f (x) x

S= a

) /

# f (x) ≥ 0 x ∈ [a, b] f (x) x b

a

y = f (x) x x = a x = b ! f (x) ≤ 0 x ∈ [a, b] " #

$ #b − f (x) x! a

f (x) [a, b] F (x) % && f (x) [a, b] #b f (x) x = F (x)|ba = F (b) − F (a)

'!()*

a

% f (x), f1 (x), f2 (x) [a, b] c ∈ (a, b) #b

#c f (x) x =

a

#b f (x) x +

a

#b

f (x) x

'!(+*

c

#a f (x) x = −

a #a

f (x) x

'!(,*

b

f (x) x = 0

'!(*

a

#b

#b (k1 f1 (x) + k2 f2 (x)) x = k1

a

#b f1 (x) x + k2

a

f2 (x) x

'!(-*

a

(k1 , k2 . )

f (t) I = [α, β] a x / I #x f (t) t = f (x) . x a

! 0 %& & Sn ' & '!((** && !

#a

#a x x

* 0

* 0

x2 x

#a

#1

ex x

ex x 1 + e2x

)

0

0 a 2

#

#a #3 (x2 − ax) x x3 x

0

#2

1

#4

x x

√ a # 3

x 2 a + x2

#4 0

#3

#2

0 √

#3

0

#1

0

x x √ 4 − x2

x √ x2 + 1

2

#3

π

x x2 x

e 3 x

+

π

#4

sin 4x x

x √

! x−1 4 √ t = x " # $ % & ' $ x 4 9 t 2 3 #1 #5 x x x √

x e +1 4x + 5 0

1

π 6

#4

π 8

a − x2 x

-

π #3 #4 x 3 tan x x x + x2

x cos2 2x

x2

, 0

#9

#

sin x cos2 x x

0 √ #a

0

0

0

x √ 1 + 2x + 1

&

a

#1 1 + x2 x

1 + tan2 x x (1 + tan x)2

π 4

#1

x x a−x

π

1

x √ 4 − x2

0

x2 x √ 4 − x2

0

*

ln (x + 1) x #3

1 x2 + 4 x x √

#1

#1

0

1

( 1

x √ (1 + x)2

1√

0π

#2

x cos x x 0 √

#3

1

# 2 2 − x2 x 1

x (1 + x2 )3

. / 0 * / 1 / " $ π π #2 #2 n − 1 sinn x x = sinn−2 x x n 0

0

π

π

#2

cosn x x = 0

n−1 n

π

#2

#2 cosn−2 x x

0

#4

0

sin4 x x

cos2 x x

0

# π

sin2 x x

π 2

π

sin6 x x

0

#2

π

#2

π

cos4 x x

0

#2

cos6 x x

0

0

! " # #∞

#b f (x) x = lim

f (x) x;

b→+∞

a

$%&'

a

#b

#b f (x) x = lim

f (x) x;

$%&

+∞ # #b f (x) x = lim f (x) x.

$%&(

a→−∞

−∞

a

a→−∞ b→+∞ a

−∞

$%&( a b −∞ " % +∞% ) * $%&( a = −b = z + #∞ −∞

#z f (x) x = lim

CH

z→+∞ −z

f (x) x

$%&&

$%&(%

f (x) [a, b] , x = c - "* ! " # . /

#b

#b f (x) x = lim

f (x) x,

c = a;

b−ε # f (x) x = lim f (x) x,

c = b;

ε→+0 a+ε

a

#b

ε→+0

a

a

#b

c−ε # 1

f (x) x = lim

f (x) x + lim

ε1 →+0

a

#b ε2 →+0 c+ε2

a

f (x) x,

c ∈ (a, b).

ε1 ε2 +0 ! ε1 = ε2 = ε " # $ % &' ( ⎞ ⎛ c−ε #b # #b CH f (x) x = lim ⎝ f (x) x + f (x) x⎠ ε→+0

a

a

c+ε

( % ) *%

#∞ f (x) x a

% # ) +,' -$ " ' ( - f (x) g(x) ) x ≥ x0 ≥ a f (x) ≤ g(x)# #∞ #∞ ." g(x) x ." f (x) x a

#∞

/"

a

#∞

f (x) x /" a

g(x) x a

A 0 " g(x) = α , (A > 0, α > 1) x

1 #∞

1

#∞

1

x x2 x √ x

#∞

1

#∞

1

#∞

e

−x

#∞

x

0

x x

#∞

1

x xn

#∞

1

2

x e−x x

0

x 1 + x2 x 2 x +x

#∞

1

#∞

2 0

x2

x √ x2 − 1 x

x2 e− 2 x

#∞

x √ x x2 − 1

2

#∞

1

x (x − 1)2

0

1

arctan x x x2 3

#2 0

3

x2 e−x x

0

#e

1 #−1

1

#0

x x2

−∞

#∞ 1 #∞

x x ln x

3

−∞

%

−x2

e 0

2m+1

x

x2

+∞ #

x + 4x + 9

x (4 − x)2

%

x (x − 1)2

+∞ # arctan x x

x √ x 1 + x2

−∞

#0,5

0

x 4 + x2

π

#2 1+x x cot x x 1 + x2 0

x x ln2 x

#2

0,5

1

1

#5

#2

0

#

x (x2 + x + 1)2

x x ln x

* + % #

," %"( -" #∞ #∞ x −x2 √ x % e 1 + x5

m! x = 2

& ' %"( ) (

−∞

−∞ +∞ #

ln x x x2

! m "" # $ #∞ e−x xm x = m! 0 #∞

+∞ #

−∞

#6 2

#2

#∞

x 2 (x + 1)2

#∞

cos x √ x x

#∞

2

3x +

1

x √ x4 − 1

x √ 5 x+7

-

[a, b] ( n . / h = - $ % #b n−1 y0 + yn + f (x) x ≈ h yi 2 i=1 a

yi = f (a + hi), i = 0, 1, 2, ..., n.

b−a " n

0&

$ ε(h) ≤

! " # $ ε(h)

(b − a)h2 |y |max 12

!"

#% % |y |max % &' ( |f (x)| )%(

[a, b]

)%%%(

[a, b] 2n ( %(

b−a * h = % 2n #b

h f (x) x ≈ 3

$ y0 + y2n + 4

n

y2i−1 + 2

i=1

a

n−1

%

+"

y2i

i=1

% yi = f (a + hi), i = 0, 1, 2, ..., 2n #% $ # ε(h) ≤

+",

(b − a)h4 (4) |y |max % |y (4) |max = max |f (4) (x)| a≤x≤b 180 #3

- ln 3 = 1

$ 10"

x x

#" % 2n = 10"

#5

% $ $. !" # " # / -

2

e−x x

0

n = 10" % $ $ !" # #1

2

ex x

! - 0

" # 2n =

- )%

" % n = 10"

#1

"

% $.

"

3

x x 1

#2

#"

x4 x

0

% $ " 0 $ # ( 1# % '2% 3. % )% 4 - % 3% ( π #1 x π=6 √ 4 − x2 0

#2 1 + x3 x (2n = 4) 0 π

#2 0

√ 3 − cos 2x x (2n = 6)

#4 0

x 1 + x4

(2n = 4)

h4 |y (4) |max ≈ |Δ4 y|max

x, y !

y = f (x)

" #$

F (x, y) = 0

y = y(x)

%

x = x(t), y = y(t)

&

%

&

%

%' )&

&

r = x(t)e1 + y(t)e2 = (x(t), y(t))Ì

r = r(ϕ)

%

%' (&

%' *&

&

%' +&

, " -

. / " 0 - 0

x

- $

x = r cos ϕ, r = x2 + y 2 ,

y = r sin ϕ

%' 1&

y ϕ = arctan x

%' 2&

x = x(t), y = y(t), z = z(t)

%

F (x, y, z) = 0, G(x, y, z) = 0

%

&

r = x(t)e1 + y(t)e2 + z(t)e3 = (x(t), y(t), z(t))Ì

( -

&

&

"

F (x, y) = 0

5

" 67

&

x = x(t) = 8t2 − 7 y = y(t) = 16t2 + 4 x = x(t) = 5t2 y = y(t) = 3t

%' 3& %' 4&

x = x(t) = r cos t y = y(t) = r sin t

x = x(t) = a cos t y = y(t) = b sin t x = x(t) = 3 cos t y = y(t) = 4 − 3 sin t x = x(t) = 2 + 3 cos t y = y(t) = −3 + 4 sin t

4 0 1

"5# # 06 x = x(t) = a(t − sin t) y = y(t) = a(1 − cos t) (a > 0; a # #

' # a−y x = x(y) = a · arccos a − y(2a − y)

"

F (x, y) = 0 ! "# # # $" % r = r(ϕ) = 4(2 cos ϕ − sin ϕ) a r = r(ϕ) = √ cos 2ϕ & ' ( x2 + y 2 − 25 = 0 9x2 − 25y 2 = 225 y − 4x2 = 0

y = x2 + 9x + 105 y = f (x)

7 '

38 r = r(ϕ) = 1 + cos ϕ

) * x2 + y 2 = 36 " # P (x, y) + #"

ω

# # , - '

.

/ " # 0 t 1 " x(0) = 6; y(0) = 0 x(0) = 0; y(0) = −6

# A(ϕ = α; r = a) # #

+ β

2 ' # # -,

b r = r(ϕ) = aϕ + b; ϕ ≥ − a 3a, b 1 a = 0 ' .

+

# , F (x, y) = 0%

-

'

9 -

' # ' 1

# * # a # # # # A(ϕ = α; r = a) # , # :

-

' # ; , M (ϕ = 0; r = a) <

a = -

1

# ( x = x(t) = 1 + 2 cos t y = y(t) = −1 + 3 sin t (0 ≤ t ≤ 2π) x = x(t) = t − sin t y = y(t) = 1 − cos t (−2π ≤ t ≤ 2π) -

1

#

1 r = r(ϕ) = ϕ (0 ≤ ϕ ≤ 3π) 2 r = r(ϕ) = e0,2ϕ (−π ≤ ϕ ≤ 2π) r = r(ϕ) = 2 + cos 3ϕ

r = 2a sin ϕ r cos ϕ = a 2 2 r sin 2ϕ = 2a √ π r sin (ϕ + ) = a 2 4 r = a(1 + cos ϕ)

rmax rmin

$ % & x = x(t) = 4 + cos t + 2 sin t y = y(t) = −4 + 2 cos t + sin t ! " " z = z(t) = 2 + 2 cos t − 2 sin t # ' # ( ) M (x, y, z) * 2 2 2 2 2 2 x − y = a x + y = a & r x cos α + y sin α − p = 0 +, y = x x, y - . + # x2 + y 2 = ax ) , z / # " (x2 + y 2 )2 = a2 (x2 − y 2 ) #0 -) 1 ! # -) ! " " 2 F (x, y) = 03 # x, y

# y = f (x) % ) P (x0 , y0 ) #0 "

y − y0 = f (x0 ) · (x − x0 ) 1 (x − x0 )

y − y0 = − f (x )

• •

0

f (x0 )

•

κ =

•

ξ = x0 −

3

(1 + (f (x0 ))2 ) 2

1 + (f (x0 ))2 1 + (f (x0 ))2 · f (x ); η = y + 0 0 f (x0 ) f (x0 )

# x = x(t), y = y(t) % ) P (x(t0 ), y(t0 )) #0 "

• • •

x(t ˙ 0 ) · (y − y(t0 )) = y(t ˙ 0 ) · x − x(t0 )

y(t ˙ 0 ) · y − y(t0 ) = −x(t ˙ 0 ) · x − x(t0 ) x¨ ˙ y − y˙ x ¨ κ = t = t0 (x˙ 2 + y˙ 2 )

•

3 2

ξ =x−

x˙ 2 + y˙ 2 x˙ 2 + y˙ 2 y; ˙ η=y+ x˙ t = t0 x¨ ˙ y − y˙ x ¨ x¨ ˙ y − y˙ x ¨

r = r(ϕ) x = x(ϕ) = r(ϕ) cos ϕ, y = y(ϕ) = r(ϕ) sin ϕ ! " # ! " $%#$ & '# ( ) % (ϕ0 , r(ϕ0 )) * #$ •

κ=

r2 + 2(r )2 − r · r 3

(r2 + (r )2 ) 2

ϕ = ϕ0 , r =

r 2 r ,r = ϕ ϕ2

F (x, y) = 0 ) % P (x0 , y0 ) * #$

Fy (x0 , y0) · (y − y0) + Fx (x0 , y0) · (x − x0 ) = 0 Fx (x0 , y0 ) · (y − y0 ) − Fy (x0 , y0 ) · (x − x0 ) = 0

• • •

κ= •

−Fy2 · Fxx + 2Fx Fy Fxy − Fx2 Fyy 3

(Fx2 + Fy2 ) 2

x = x0 , y = y0

ξ = x0 −

Fx (Fx2 + Fy2 ) Fy2 Fxx − 2Fx Fy Fxy + Fx2 Fyy

η = y0 −

Fy (Fx2 + Fy2 ) x = x0 , y = y0 Fy2 Fxx − 2Fx Fy Fxy + Fx2 Fyy

+ , % (x0 , y0 ) % ! 1 • = |κ| •

(x − ξ)2 + (y − η)2 = 2

S(x0 , y0 ) -

) % y = y1 (x) y = y2 (x). * ϑ tan ϑ =

y2 (x0 ) − y1 (x0 ) 1 + y1 (x0 ) · y2 (x0 )

&(

y˙ x = x(t), y = y(t) y = x˙ Fx F (x, y) = 0 y = − Fy 1 + (f (x))2 · f (x) f (x) 1 + (f (x))2 η = η(x) = f (x) + f (x)

ξ = ξ(x) = x −

!"

# $ y = f (x)

% & F (x, y) = 0 ' ( y ) *& F (x, y) = 0 & x + ' y x , -. " - *& / & & - y 0 1 ( y + ' *& F (x, y) = 0 '. & x . # 2 '

3

% 4 # x = x(t), y = y(t), z = z(t) '.

r = x(t)e1 + y(t)e2 + z(t)e3 = (x(t), y(t), z(t))Ì

"

r Ì = r˙ = x(t)e ˙ ˙ ˙ ˙ y(t), ˙ z(t)) ˙ 1 + y(t)e 2 + z(t)e 3 = (x(t), t

"

) t ) 4& & ) .& 5 # t" 4 " # P (x(t0 ), y(t0 ), z(t0 )) # x = x(λ) = x(t0 ) + λx(t ˙ 0 ), y = y(λ) = y(t0 ) + λy(t ˙ 0 ), z = z(λ) = z(t0 ) + λz(t ˙ 0 ); −∞ < λ < ∞

6"

'. (x, y, z)Ì = (x(t0 ), y(t0 ), z(t0 ))Ì + λ(x(t ˙ 0 ), y(t ˙ 0 ), z(t ˙ 0 ))Ì

7"

! " # $ % $ &

! ' P (x(t0 ), y(t0 ), z(t0 )) ( ) *! +,

*! -+

x(t ˙ 0 )(x − x(t0 )) + y(t ˙ 0 )(y − y(t0 )) + z(t ˙ 0 )(z − z(t0 )) = 0

t . r = (x(t), y(t), z(t))Ì / )

r /0 $

= r˙ = v t !

2

tr2 = r¨ = v˙ = a

! / 1 ) 2 , x3 + y = x = −1 3 + y 2 = x3 x1 = 0, x2 = 1 8 +y = *3 )+ 4 + x2 x=2 + y = sin x x = π + y = sinh x x = −2 + x = x(t) = a(t − sin t) y = y(t) = a(1 − cos t) π 3π *.4+ t1 = , t2 = 2 2 + x = x(t) = a cos3 t y = y(t) = a sin3 t π *5 + t = 4 + x = x(t) = a(cos t + t sin t) y = y(t) = a(sin t − t cos t) π *1 )) + t = 4 ! / 1 ) # % 2 , x−1 + y = arcsin 2

2 x5

+ y = arccos 3x 2

y 5 2

+ y = e1−x 2

# y = 1 + y 2 = 4 − x 2

y 5 1+t + x = x(t) = 3 t 3 1 y = y(t) = 2 + 2t 2t ' x = 2 + x = x(t) = t3 − 2t y = y(t) = t2 + 1 ' t = 1 + x3 + y 2 + 2x − 6 = 0 '

y=3 1 + y = x3 − 2x2 + 3x − 1 6 3

7! ' ' 4

8 x, y 24 $ ) x5 ' ! 1 ) ' , r = r(ϕ) . . ' $ 2 m = m(ϕ) 1 )

% 2y = x2 2y = 8 − x2 r (ϕ) sin ϕ + r(ϕ) cos ϕ m = m(ϕ) = % x2 + y 2 = 5 y 2 = 4x r (ϕ) cos ϕ − r(ϕ) sin ϕ 4 y = x2 x = y 2 ! " , # + /$, x2 − y 2 = 9 " x, y # * (5; 4) 5$, "$ /$, % r = r(ϕ) = 2a cos ϕ (a > 0) π 6 " ! ϕ = * (x0 ; y0 ) # 4 π ' % r = r(ϕ) = 5ϕ ϕ = 3 x2 y2 % + =1 & # a2 b2 " % y 2 = 2px " ' # % x = x(t) = 2 cos t − cos 2t, y = y(t) = −2 sin t + sin 2t * ' π t = 6 % y = x4 − 4x3 − 18x2 x = y = 0 % x = x(t) = sin 2t, % x2 + xy + y 2 = 3 x = y = 1 y = y(t) = sin2 t π x2 y2 t = % 2 + 2 = 1 " 8 a b A(a; 0) B(0; b) t % x = x(t) = arcsin √ , 1 + t2 % x = t2 , y = t3 x = y = 1 1 y = y(t) = arccos √ % r2 = 2a2 cos 2ϕ ϕ = 0 2 1 + t √ (a > 0) t = 5 ( ) * P1 (1; 0) + x y = e + # " , ' # 1 , % y = 1 + x2 - . + % x2 − y 2 = 4 /$, % y = sin x xy = a2 " 0 , # % 2y = x2 + 4x " % y = ex 1 2 2 % y = e−x , 3 % x = a(t − sin t), y = a(1 − cos t) % y = x2 y = x−1 % y = x e−x % y = sin x y = 0

! " r = r(ϕ) = a(1 − cos ϕ), a > 0 2

2

" r = a cos 2ϕ, a > 0 #" r2 =

2

a ,a>0 cos 2ϕ

x2 2 #" x = 2 cos t, y = sin t

" y = 1 −

" x = a cosh t, y = a sinh t " x = a(cos t + t sin t) y = a(sin t − t cos t) *" y 2 = 2x + 2

$ %#& y 2 = 8x #& 0, 128'

t3 3 3 &" x = a cos t, y = a sin3 t

( #&

) * * + !

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) y ,#& x3 + 1 #" y = − +#&

)

3 x,#& " y = e +#&

) y ,#& x

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. &3 & 1 #& * r = r(t) ' " r = ae1 cos t + ae2 sin t + cte3 " r = e1 cos t + e2 + e3 sin t #" r = t(1, 1, 1)Ì

" r = (1, 1, t)Ì

" r = (cosh t, 0, sinh t)Ì

4 / % #& #& x3 /#&% #& " y = x = −1 3 * 5 ) ! *" y 2 = x3 (1; 1) π π " r = (3 cos t, 3 sin t, 4t)Ì , t = " y = cos x x = 4 4 2 3 " r = te1 + t e2 + t e3 , t = 1 - . /#& 51 6 r r˙ * y = x2 − 6x + 10 P0 (3; 1)' #& & 1 %

r = (cos t, sin t, 1)Ì 0 %1 x

y = a cosh a + * #& y 2 /a /#& #& 2 #&

7 2 %#& x,#& (a > *! √ 0) π " r = (cos t, sin t, t 3)Ì , t = 2 + * 2 " r = (a sin t, b sin t cos t, c cos2 t)Ì , /#& *! π t= " y = x2 4

√ √ 2 t 2 t e sin t, 1, e cos t)Ì , r = ( 2 2 t=0 r = (t, t2 , t3 ), P (2; 4; 8) sin 2t π

r = R(cos2 t, , sin t), t = 2 4 Ì r = t(cos t, sin t, b) , P (0; 0; 0) x2 + y 2 = 10, y 2 + z 2 = 25, P (1; 3; 4)

x : y : z

x1 + 3x2 + 2x3 = 0 $

% & ' P (x, y, z)

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*

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+ , (( ( - r = r(u) = (u + 1)e1 + u2 e2 + (2u − 1)e3

4 3 2 r = r(v) = 2v 2 e1 + (3v − 2)e2 + v 2 e3 t t t r = e1 + e2 + e3 ' ( 4 3 2 ! " #

* &

. ( & / ( ( .0" ( ( 1 " / (/ ( ( 2 " / ("( " # ( 2 3 ( (

• y

y

6

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f2 (x) − f1 (x) x,

a

5 +

( f2 (x) ≥ f1 (x) 2 4 ( x ∈ [a, b].

#t2 y(t)x(t) ˙ t,

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5 6

y d6 g1 g2 -x c

#d (g2 (y) − g1 (y)) y,

A= c

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y 6 d x = x(t) y = y(t) -x c

#t2 x(t)y(t) ˙ t,

A=

t1

c = y(t1 ), d = y(t2 ) x(t) ≥ 0 t ∈ [t1 , t2 ].

• y

6

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1 A= 2

#ϕ2

1 A= 2

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t1

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r2 (ϕ) ϕ k : r = r(ϕ), ϕ ∈ [ϕ1 , ϕ2 ]

ϕ1

OP1 = r(ϕ1 ), OP2 = r(ϕ2 ).

!" " #$!" % & ' % &

( ) y = 4 − x2 , y = 0 & y = 6x − x2 , y = 0 !

2

2

x y + 2 =1 2 a b

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y = 3 − 2x − x2 , y = 0 y = x3 , y = 8, x = 0 xy = 4, x = 1, x = 4, y = 0 " y 2 = 1 − x, x = −3 y = ln x, x = e, y = 0 * y 2 = 2x + 4, x = 0 + y 2 = x3 , y = 8, x = 0 y 2 = (4 − x)3 , x = 0

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0 # 73+ 0 8 + + 9+ * + + + : 4 # +

s AB y = f (x), x1 ≤ x ≤ x2 #x2 s= 1 + (y )2 x x1

s = 1 + (y )2 x = x2 + y 2

x = x(t), y = y(t), t1 ≤ t ≤ t2 #t2 s= x˙ 2 + y˙ 2 t

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x˙ 2 + y˙ 2 t

t1

r = r(ϕ), ϕ1 ≤ ϕ ≤ ϕ2 #ϕ2 s= r2 + (r )2 ϕ

s =

r2 + (r )2 ϕ

ϕ1

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t1

r t

2

t

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+- ) +- #" $ r = r(s) = (x(s), y(s), z(s))Ì .

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/

n

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( !

,

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=

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) *" ) & + + , # !* y 2 = x3 ) ' *" 4 - x = +*"

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9 *" x = −1 * x2 + y 2 = a2

2

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(

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12 3 ≤x≤ 4 5 2 y = 2px +*"

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0

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0 x1*"

. y = ln x,

e2y tanh x = 1, 1 ≤ x ≤ 2 ) 23 x = a(t − sin t), y = a(1 − cos t) t4 t6 0 x = , y = 2 − '*" 6 4 /*"

0 , *" 1 4 x = t2 , y = t(t2 − 3) ' 3 *" /*"

0 x1*" , r = a(1 − cos ϕ)

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, N & %)- ") t, b n ' t = 0 x = 1 − sin t, y = cos t, z = t 4 ) , / )# % x = t, y = t2 , z = t3 ' t = 1 ( 5 4 ) , / / x = et , y = e−t , z = t ' t = 0 ( 6 4# r = (t cos t, t sin t, t)Ì 1 ) )/ # 4 / ) , / 3

( 7 4) # # / r = r(t) ' t % ' s 8 .( t = 0 # )/ ( r = r(s) ") t, b, n & κ w #

$/ 8 ( .(/ # 9 √ r = r(t) = (2 cos t, 2 sin t, t 5)Ì # # # r = r(t) =

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) ( √ r = r(t) = (et , e−t , t 2)Ì b ' t = 0 / )

3 T B : 4 ) , /

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κ

1 2!" !" , &r = r(t) = (a cos t, a sin t, bt)Ì ) ( 3 !" , &r = r(t) = (a cos t, −a sin t, bt)Ì ) (a > 0, b > 0)

" , !" , !" " !" 45" "

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" #

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x1

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r(ϕ)

ϕ1 #ϕ2

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2 π 3

ϕ1

r2

+

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2 sin ϕ ϕ

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' ! 3 0 h2 ) -. ! 0 a b

7' 8+ 72 3 + $1 R' 2 # /

8+ , . 9 , α

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7' y = 2

1 , x = ±1 1 + x2 2

2

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' y = a −

x2 y2 + =1 a2 b2 ' y = x3 , x = 0, y = 8

'

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. ) x = b (b > a > 0)

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y = x y = 4 ! x = 2

y = cos x, y = −1, x = ±π

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) %* x ,

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y = x3 , x = ±2 3

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y = a(1 − cos t), 0 ≤ t ≤ 2π

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. # + ) / x2 + y 2 = a2

x = b (b > a > 0)

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# 0 * - # * 1 2& * # ,

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y = M M , y = x = A A A

0

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7 3 8 % 93: 7 ; ' ' - " < R = 2

S

y

S

x

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#

x1

# y x; I = x y x 3

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100 Ω +! V0 = 0, 2 3 p0 = - 10330 ! "* &. ,%. 1 & . ' V1 = 0, 1 3 "# ( 0

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$ % .!

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i=1

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#

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lim sn (x) = s(x), x ∈ M.

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1 1 1 − 2 + 2 2 x +1 x +4 x +9 1 ± ... − 2 x + 16 / ) , n ! x# |rn (x)| < 0, 0001.

2 /

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+, ∞ n=0 ∞

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)

n=0

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r#

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4

r = 0 r = ∞ x ∈ R 0 < r < ∞ x |x − x0 | < r x |x − x0 | > r x = x0 ± r !" # $ % & !" ! " & $ ' !" ! " % (

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k=0

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% $ ·

i

ai x

i=0

∞

% k

bk x

∞

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n=0

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n

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l=0

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$

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∞

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n=0

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n=0

n=0

%

n

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n=0

#x $ ∞ 0

∞

!" !

an ·

n=0

bn+1 − an+1 , n+1

-

[a , b] ⊂ (−r , r)

% a n tn

∞

t =

∞ n=0

an ·

xn+1 , x ∈ (−r , r) n+1

.

* f (x) * ! I # x0 * /# $ $ " 0 $# 1 "$ ∞ f (k) (x0 ) k=0

k!

(x − x0 )k

2

f (x) =

f (x) x0

∞ f (k) (x0 )

k!

k=0

(x − x0 )k x ∈ I

lim Rn (x) = 0 x ∈ I,

n→∞

Rn (x) ! " # $ %& ' ( f (x) ) "! x0 = 0* ∞ f (k) (0) k=0

k!

xk

+ ( $ $ " ( , -. ( . / -. ( . ! 0 $ ! $ -. ( |x| = r x2 x4 x + 2 + 3 + ... 1 + 3·2 3 ·3 3 ·3 x3 x5 x 1 − √ + √ − √ ± . . . 5 2 52 3 53 4 4x2 2x $ 1 + 2 √ + √ 3 3 52 32 8x3 + √ + ... 72 33 2 4x 2x +√ 1 + √ 5·5 9 · 52 8x3 +√ + ... 13 · 53 x4 x2 √ + √ 1 − 3 · 2 2 32 · 3 3 x6 √ ± ... − 33 · 4 4 ∞ xn n! n=1

∞ (−x)n−1 n n=1

1

∞

3n xn (3n − 2) · 2n n=1 ∞ 10n xn √ n n=1 ∞

(−1)n−1 ·

n=1

∞

x2n−1 2n − 1

xn−1 n!

n=1

∞ n! n x nn n=1

(x + 1)3 (x + 1)2 + 2·4 3 · 42 (x + 1)4 + + ... 4 · 43 2x − 3 (2x − 3)2 − 1 3 (2x − 3)3 + ∓ ... 5

(x + 1) +

x − 1 (x − 1)2 + 1·2 3 · 22 +

(x − 1)3 + ... 5 · 23

2x + 1 (2x + 1)2 + 1 4 (2x + 1)3 + + ... 7 s(x) ! 1 + 2x + 3x2 + 4x3 + . . . ! 1 − 3x2 + 5x4 − 7x6 ± . . . x5 x7 x3 + − ± ... " x − 3 5 7 2 3 x x + + ... x + 2 3 1 + 3x + 5x2 + 7x3 + . . . # 1 − 4x + 7x2 − 10x3 ± . . .

s(x)x s(x) x s + s · x

#

s(t) t

0

s−s·x

$ #% ! & ' #( ! ! ) *!" $ + " #% n → ∞ f (x) = cosh x ! f (x) = sin2 x

#% & ' f (x) = ex/a , ' x0 = a #( ! "

- . ! #% # & ' " " /

. ! , ' x0 0 f (x) = ecos x , x0 = 0 √ ! f (x) = x3 , x0 = 1 1 " f (x) = , x0 = 2 x f (x) = ln cos x, x0 = 0 1 . ! #% # & ' / "' , ' x0 = 0

2 3/ !' 0 1 f (x) = (1 + x)3 1 ! f (x) = √ 1 + x2 1+x " f (x) = ln 1−x f (x) = ln (2 − 3x + x2 ) f (x) = ln (1 − x + x2 ) 1 # f (x) = √ 1 − x2 f (x) = e−x

2

f (x) = x e−2x 3x − 5 f (x) = 2 x − 4x + 3 x 4 f (x) = 9 + x2 ' f (x) = sin 3x + x cos 3x f (x) = cos2 x 1 f (x) = √ 3 + 2x . ! #% # & ' # , ' x0 / 2 3 / !' 0

f (x) = ln x, x0 = 1

f (x) =

x2

1 , x0 = −4 + 3x + 2

1 , x0 = −2 x x π f (x) = cos , x0 = 2 2

f (x) =

π f (x) = sin 3x, x0 = − 3 √ f (x) = 3 x, x0 = −1 f (x) = x4 − 4x2 , x0 = −2 πx , x0 = 1 f (x) = sin √ 3

f (x) = x, x0 = 4 f (x) = ex , x0 = −2 1 f (x) = 2 , x0 = −1 x 1 f (x) = 1 + x2 ! " " # $ % arctan x 1 & ! x = √ 3 $ % arctan x ' $ ! π π (

$ ) $ % '

* # ) $ + $ , # # x sin x e x x x x & #x 2 Φ(x) = e−t dt - ! 0

(x0 = 0) Φ( 13 ).

$ $ $ / . 0, 000001 ! & #x 3 Φ(x) = 1 + t2 t ' 0

Φ( 15 ) ) $ ! ' $ / . 0, 00001 ! 0 " $ #x t2 Φ(x) = cos t 4 0

Φ( 12 ) ' 0, 000001 1 2 " + x = a cos t, y = b sin t, a > b > 0

3 # I = [−l , l] f (x) ( . %$ . %$ . % x ∈ I , f (x) =

1 f (x − 0) + f (x + 0) . 2

5 ( f (x)

4 00

,

a0 nπx nπx an cos + + bn sin 2 l l n=1 ∞

f (x) =

4 01

1 an = l

#l −l

nπx f (x) cos x ; l

1 bn = l

#l f (x) sin −l

n = 0, 1, 2, . . .

nπx x l

n = 1, 2, 3, . . .

!" " "# (−∞, +∞) " !! $ #"!! % 2l "# I " &" #'!!

f (x)

%

f (−x) = f (x) (

∞

f (x) =

an =

2 l

f (x)

f (x) =

a0 nπx + an cos 2 l n=1 #l f (x) cos 0

∞ n=1

bn =

2 l

bn sin

nπx x, n = 0, 1, 2, . . . l

%

f (x) sin

+

nπx x, n = 1, 2, 3, . . . l

- ! #! % 2l = 2π

. !# "!

/ ! " ! #' 0 1 2" ! ( " f (x) = 1 #' x ∈ (0 , π) f (−x) = −f (x); π 1 1 1 1 − + − + ··· = 3 5 7 4

*

f (−x) = −f (x) (

nπx l

#l 0

)

,

f (x) = x #' x ∈ [0 , π] f (−x) = f (x); 1 1 1 π2 1 + 2 + 2 + 2 + ··· = 3 5 7 8 2 f (x) = x #' x ∈ [−π , π]; 1 1 π2 1 1 − 2 + 2 − 2 + ··· = 2 3 4 12 1 1 1 π2 1 + 2 + 2 + 2 + ··· = 2 3 4 6 π #' x ∈ (−π , 0) f (x) = π − x #' x ∈ [0 , π]

π−x x ∈ (0 , π] 2 f (−x) = f (x) f (x) = | sin x|; 1 1 1 1 + + + ··· = 1·3 3·5 5·7 2 x ∈ [0 , π/2] x f (x) = π − x x ∈ [π/2 , π]

f (x) =

f (x) =

1 x

x ∈ [−1 , 0) x ∈ (0 , 1]

l = 1 f (x) = ex x ∈ (−l , l) " f (x) # $ [0 , 2] % ! y 6

f (−x) = −f (x) 2l f (x) = 1 x ∈ (0 , l) f (−x) = −f (x) ! f (x) = 1 − x x ∈ [0 , 1], f (−x) = f (x) l = 1 0 x ∈ (−l , 0] f (x) = x x ∈ [0 , l) f (x) = x x ∈ [0 , l) f (−x) = f (x)

-x & '

! 2l = 4 !

f n x1 , x2 , . . . , xn n (x1 , x2 , . . . , xn ) ∈ D(f ) ⊆ Rn y ∈ W (f ) ⊆ R

y = f (x1 , x2 , . . . , xn ) = f (x) = f (P ), x ∈ D(f )

x = x1 e1 + x2 e2 + · · · + xn en = (x1 , x2 , . . . , xn )Ì

! "! P (x1 , x2 , . . . , xn ) #

$ % $ n = 2 z = f (x, y)& w = f (x, y, z); u = f (x, y, z) n = 3 ' d(P, Q) "! Q(q1 , q2 , . . . , qn ) d(P, Q) =

P (p1 , p2 , . . . , pn )

(p1 − q1 )2 + (p2 − q2 )2 + · · · + (pn − qn )2 = |p − q|

z = f (x, y), (x, y) ∈ D(f )

R3

P (x1 , x2 , . . . , xn ) ∈ Rn

! "

f (x1 , x2 , . . . , xn ) = c = const. # $

% f (x , x , . . . , x 1

R2

2

n ) &%

c

&% '( ) % w = f (x, y, z) R3

&% '( ) %

*%

!+ % ,

% + + .

-$ - .

+

.

z = f (x, y)

/ +

.

% &% &.

+ , 0

% 1

z = f (x, y) = −3x + 4y + 8 z = f (x, y) = 25 − x2 − y 2

z = f (x, y) = xy z = f (x, y) = x2 + y 2 4 z = f (x, y) = 2 x + y2 √ z = f (x, y) = xy ! " # " x, y "$ 1 f (x, y) = 4 − x2 − y 2 x √ f (x, y) = arcsin + xy 2

f (x, y) = ln (x + y) 1 f (x, y) = √ y− x f (x, y) = 1 − x2 + 1 − y 2 f (x, y) = x2 + y 2 − 1 + ln (x2 + y) f (x, y) = ln (1 − ex+y )

% &' " x, y "$ (c = −4, −1, 0, +1, +4 z = f (x, y) = x + y z = f (x, y) = x2 − y 2 ) *# &'+ ' w = f (x, y, z) w = x + y + z w = x2 + y 2 + z 2

w = x2 + y 2 − z 2 , f (x, y) = x4 + y 4 − 2xy - ## f (tx, ty) = t2 · f (x, y) x . , f (x, y) = - ## x−y f (a, b) + f (b, a) = 1 / *# !" # " w = f (x, y, z) √ √ √ w = x + y + z w = ln (xyz)

w = arcsin x + arcsin y + arcsin z w = 1 − x2 − y 2 − z 2

$ {xm } ' 0 Pm (x1m , x2m , . . . , xnm ), m = 1, 2, . . . - 0 A(a1 , a2 , . . . , an )- 1 lim xim = ai i = 1, 2, . . . , n

m→∞

(

.

(

/

2 # a = (a1 , a2 , . . . , an )Ì 3 lim xm = a = lim Pm = A

m→∞

m→∞

y = f (x) = f (x1 , x2 , . . . xn ) # 4 U ' x0 !" - ' 5# ' x0 $ ( , α ' f (x) x x0 - 1 {xm } xm ∈ D(f ), xm = x0 lim xm = x0 3 m→∞

lim f (xm ) = α

m→∞

(

6

lim f (x) = α

x→x0

! "! ε > 0 δ = δ(ε) > 0 !

#

! x

0 < |x − x0 | < δ $ |f (x) − α| < ε.

%

& '

$ f (x) ( ! x0 f ) U *$ x0 !+ !

lim f (x) = f (x0 )

x→x0

' z = f (x, y) $ ! , !+ Δz = f (x + Δx, y + Δy) − f (x, y) Δx z = f (x + Δx, y) − f (x, y) Δy z = f (x, y + Δy) − f (x, y) z = x2 − xy + y 2 Δz, Δx z, Δy z Δz, Δx z, Δy z ! ' ! x *$ 2 2, 1 ! y *$ 2 1, 9 .! / ! !

y lim (x;y)→(0;0) x − y #

! ! 0.1 ! 2 (x; y) ! 2 (0; 0) . ! ! y = mx ! 3. 4 1 ! ! 0. ! / 3 2 −2 1 !x2 − y 2 f (x, y) = 2 x + y2 x2 y 2 f (x, y) = 2 2 x y + (x − y)2 !

lim lim f (x, y) ! y→0 x→0

lim lim f (x, y)

x→0 y→0

/ ! #

lim

(x;y)→(0;0)

f (x, y)

5 ! $ ! #

-√ 2 − xy + 4 lim (x;y)→(0;0) xy sin (xy) lim xy (x;y)→(0;0) sin (xy) lim (x;y)→(0;0) x sin (xy) ! lim x (x;y)→(0;2) 1 lim (x2 + y 2 ) sin xy (x;y)→(0;0) x lim (x;y)→(0;0) x + y x2 + (y − 2)2 + 1 − 1 lim x→0 x2 + (y − 2)2 y→2 4−x y−x 2(2x + y) ln (x2 y 3 )

lim exp x→−0,5 y 2 − 4x2

lim

(x;y)→(4;4)

y→1

) ! $ ! '

$

z = f (x, y) (0; 0) f (x, y)⎧ ⎨ xy x2 + y 2 > 0 = x2 + y 2 ⎩0 x = y = 0

z =

x2 + y 2 > 0

f (x, y)⎧ 2 ⎨ x y 4 = x + y2 ⎩ 0

x2 + y 2 > 0

x = y = 0

x = y = 0

! """ # z = f (x, y)

1 + (y + 1)2

(x − 1 z = sin x sin y

z = ln (1 − x2 − y 2 ) z =

f (x, y)⎧ 2 ⎨ x y = x2 + y 2 ⎩ 0

1)2

x2 + y 2 (x + y)(y 2 − x)

$ % " w = f (x, y, z) "& 1 w = xyz 1 w = 2 x + y2 − z 2 1

w = 2 2 x + y − z2 + 1

' ( ) " " " ) ( * + , " " lim

h→0

f (x10 , . . . , xi−1,0 , xi0 + h, xi+1,0 , . . . , xn0 ) − f (x10 , . . . , xn0 ) h ∂f = ∂xi x=x0

+

-

. ' / f xi x0 (i = 1, 2, . . . , n) " " 0 ) 1" " 2 . ' " . ' "( " " 3 # ( 4 fx1 x2 = fx2 x1 , fx1 x1 x4 = fx1 x4 x1 = fx4 x1 x1 ") 5" y = f (x) x0 xi " ( "

∂f ∂f ∂f , ,..., ∂x1 ∂x2 ∂xn

/ f

Ì

x=x0

x0

= grad f (x)x=x0

+

6

∇=

∂ ∂ ∂ , ,..., ∂x1 ∂x2 ∂xn

Ì

grad f (x) = ∇f (x)

! "# f (x) $

x0 s = (s1 , s2 , . . . , sn )Ì %

∂f (x) s Ì = f (x ) = (grad f (x)) · s 0 ∂s x=x0 |s| x=x0

1 fxk (x0 ) · sk |s| n

=

&

k=1

' "# # (

)* f (x, y) = x3 + 3x2 y − y 3

f (x, y) = ln x2 + y 2 y f (x, y) = arctan x 1 1 √ − g(x, t) = ln √ 3 3 x t 2 2 c(a, b, γ) = a + b − 2ab cos γ z x y f (x, y, z) = + − x y z −yx f (x, y) = x e 2x − t g(x, t) = x + 2t √ α(x, t) = arcsin (t x) + f (x, y) = cos (ax − by) y f (x, y) = arcsin x x

f (x, y) = 3y − 2x h(x, t) = ln sin (x − 2t) g(x, y) = sin2 (x + y) − sin2 x − sin2 y

# f (x, y) = xy y ( f (x, y) = exp sin x

, f (x, y) = arcsin

x2 − y 2 x2 + y 2

f (x, y, z) = (xy)z f (x, y, z) = z xy f (x, y, z) = exyz cos y

f (x, y) = ln cos x - ' ! * # # (

*

z = ln (y − x2 ). % zxx , zxy , zyy u+v w = arctan . % 1 − uv w ,w ,w uu

uv

vv

z = x3 + x2 y + y 3 . % zxxx, zxxy , zxyy , zyyy / ! * # !

√ √

z = ln x + y

∂z 1 ∂z +y = ∂x ∂y 2 √ y z = x sin x ∂z ∂z z x +y = ∂x ∂y 2 x

u = exp 2 t ∂u ∂u +t =0 2x ∂x ∂t u = x2 + y 2 + z 2 2 2 2 ∂u ∂u ∂u + + =1 ∂x ∂y ∂z xy z = x−y ∂ 2z ∂2z ∂2z 2 + + 2 = 2 2 ∂x ∂x∂y ∂y x−y x

z = ex/y ∂z ∂z ∂2z = − y ∂x∂y ∂y ∂x !! y xf (x) +ϕ u= y x " # f ϕ $ xyuxy + y 2 uyy + xux + 2yuy = 0 % & ∂2z ∂2z = ∂x∂y ∂y∂x ' z = sin (ax − by) x2 z = 2 y

z = ln (x − 2y) & ! z = f (x, y) = x2 + y 2 ( grad z ) (3; 4) ) (3; 4) * +*

&

( , f - α = 30◦ $!* x#, ! ) (3; 4) . ( z = f (x, y) = x2 − y 2 & ) P (2; 1) ) P * +* & / ( , f (x, y) = x3 − 2x2 y + xy 2 + 1 ) M (1; 2) - * M N (4; 6) "! 0 ( grad u ) P (1; 2; 3) u = f (x, y, z) = xyz 1 ( 2 "! & y z = f (x, y) = ln x 1 1 ; ) A 2 4 B(1; 1) ( , # z = f (x, y) = x2 − xy − 2y 2 ) P (1; 2) - x#, ! 2 * 60◦ 3 ( , # z = f (x, y) = ln x2 + y 2 ) P (1; 1) - 2 ! 1. 4 # 5 x, y #6 7 ( , # w = f (x, y, z) = x2 − 3yz + 5 ) M (1; 2; −1) - #

! " # $% & !

w = f (x, y, z) = xy + yz + zx ' ! M (2; 1; 3) (

M N (5; 5; 15) )* # grad f (x) )* f (x1 , x2 , x3 , x4 ) = 6x1 x2 + 3x21 − cos x3 + x4 ex2 + # ," $ &- z 2 = xy (z ≥ 0) ' ! (4; 2)

y = f (x) . x0

/ h1 = Δx1 , h2 = Δx2 , . . . , hn = Δxn Δy = Δf = f (x10 + h1 , x20 + h2 , . . . , xn0 + hn ) − f (x10 , x20 , . . . , xn0 ) = f (x10 + Δx1 , x20 + Δx2 , . . . , xn0 + Δxn ) − f (x10 , x20 , . . . , xn0 )

0% 1 x0 ∈ D(f ), x0 + h ∈ D(f )

2

h = (h1 , h2 , . . . , hn )Ì = (Δx1 , Δx2 , . . . , Δxn )Ì .

3

f (x) " . x0 0

. x0 + h ∈ D(f ) ) 1 Δy = Δf = fx1 (x0 )h1 + fx2 (x0 )h2 + · · · + fxn (x0 )hn + η · ρ n = fxk (x0 ) · Δxk + η · ρ

45

k=1

(Δx1 )2 + (Δx2 )2 + · · · + (Δxn )2

4

η = η(x10 , . . . , xn0 ; Δx1 , . . . , Δxn )

44

lim η = 0.

46

ρ=

ρ→0

$

/ 45 " y & ! y = f (x) . x0 1 y =

n k=1

fxk (x0 ) · Δxk

4

Δy = y + η · ρ

! y = xi %

" # $

x1 = Δx1 , x2 = Δx2 , . . . , xn = Δxn

&

y = f (x1 , x2 , . . . , xn )% y =

∂y ∂y ∂y x1 + x2 + · · · + xn ∂x1 ∂x2 ∂xn

ρ ( ) ) y = f (x1 , x2 , . . . , xn )%

'

Δy ≈ y

f (x1 + Δx1 , . . . , xn + Δxn ) ≈ f (x1 , . . . , xn ) + f (x1 , . . . , xn )

*

z = f (x, y) + , ! % 2 z =

∂2z 2 ∂2z ∂2z 2 x + 2 x y + y ∂x2 ∂x∂y ∂y 2

-.

-

/ -. 0) 1 )+ % 2 z =

∂ ∂ x + y ∂x ∂y

2 z

2 f (x1 , x2 , . . . , xn ) # ) k $ 3 4 k %

k f (x1 , . . . , xn ) =

∂ ∂ x1 + · · · + xn ∂x1 ∂xn

2 ) $ z = f (x, y) 5 6 $ Δz $ ) 1 (x; y) ) 5$ ) 6 Δx Δy 7 ) 5 Δz z )

k f (x1 , . . . , xn )

-

z = xy 1 (5; 4) Δx = 0, 1; Δy = −0, 2 ) z = x2 y 1 (−3; 2) Δx = 0, 01; Δy = −0, 02

z = x2 − 3xy + y 2 1 (2; 1) Δx = −0, 1; Δy = 0, 2

y " (2; 1) f (x, y) = x Δx = 0, 1; Δy = 0, 2 g(x, y) = exy " (1; 2) Δx = −0, 1; Δy = 0, 1 y ϕ(x, y) = arctan " x (2; 3) Δx = 0, 1; Δy = −0, 5

xy f (x, y) = x−y g(s, t) = es/t f (x, y) = x2 + y 2 u(x, y, z) = x2 + y 2 + z 2 y f (x, y) = ln tan x z x h(x, y, z) = xy + y ϕ(x, y) = ex cos y + y sin 3x z ψ(x, y, z) = x2 + y 2 f (x1 , x2 , x3 , x4 ) = xx1 2 −x3 ln x4

!

# Δz z $ z = ln (x2 + y 2 )% &

x 2 2, 1 y 1 0, 9 ' ! 2 u y2 u(x, y) = 2 x y u(x, y) = x ln x u(x, y, z) = xy + yz + xz u(x, y) = cos (mx + ny)

( f (x1 , x2 , . . . , xn ) ) x1 , x2 , . . . , xn ' *+% ! , - fxi % . xi . . Δxi . (i = 1, 2, . . . , n)% $ Δf *+ f - ' n ∂f (x1 , . . . , xn ) · |Δxi | |Δf | ≈ |f | ≤ / ∂x i i=1 $ - ' Δf f f ≈ f

Δf f

.0 /

#

- & - ' / / # % |Δxi | %

' 1 |Δf | |f | ∂f 2 $ ∂x xi − |Δxi | ≤ xi ≤ xi + |Δxi | i (i = 1, 2, . . . , n) " " . / % ' . $ |Δf |

x z = f (x, y) = x = 2 ± 0, 1 y y = 4±0, 3 ! " f (1, 9; 3, 7), f (1, 9; 4, 3), f (2, 1; 3, 7), f (2, 1; 4, 3)# $ 1, 9 ≤ x ≤ 2, 1 3, 7 ≤ y ≤ 4, 3 %& '$ |Δz|max ( f (2; 4) ) ! " ' ( Δz * ∂z ∂z |Δz| ≤ · |Δx| + ∂x (2;4) ∂y (2;4) ×|Δy| ( |Δz|max

! " + ∂z ∂z " ∂x ∂y ( 1, 9 ≤ x ≤ 2, 1; 3, 7 ≤ y ≤ 4, 3 + |Δz| |Δz|max , $ R (

$ - R1 R2 R1 R2 R= R1 + R2 $ R1 = (550 ± 3)Ω R2 = (150 ± 1)Ω ! " ' ( ( R . //! ! " + ( / 0 $ 1 r = (5±0, 01) *% h = (12 ± 0, 04) 2 0

3 A0 ! " ' ( ( A0 ! " + ( 4 0 - $ 1 R = (400 ± 5), r = (300 ± 6) *% h = (500 ± 8) " ' ( 5 " & + a b γ 6 $ a = (92, 5 ± 0, 2)# b = (65, 6 ± 0, 1) γ = (57, 8◦ ± 0, 3◦ ) " ' ( # " 7 c + ( 8 9: # $ ; < ( ( / = *% ( ( 4 = $ > " 2 ( 6 p1 V1κ = p2 V2κ " ' ( " ( V1 # $ V2 , p1 , p2 ( #5 = $ #4 = $ # = $ (κ = 1, 4) ? 7 P '& $ Ra Ra Pa = E 2 · . (Ri + Ra )2 " ' ( " (

Pa |ΔRi | |ΔRa | = = 10 Ra Ri

Ra = 100Ri E = const.

P0 (x0 ; y0 ; z0 ) F (x, y, z) = 0

(x − x0 )Fx (x0 , y0 , z0 ) + (y − y0 )Fy (x0 , y0 , z0 ) + (z − z0 )Fz (x0 , y0 , z0 ) = 0

!

"

Ì n = Fx (x0 , y0 , z0 ), Fy (x0 , y0 , z0 ), Fz (x0 , y0 , z0 )

r = (x0 , y0 , z0 )Ì + λn, −∞ < λ < ∞

P (x0 ; y0 ; z0 ) F (x, y, z) = 0 #$ % z = f (x, y) &'( %) f (x, y) − z = 0 ' $ % F (x, y, z) = 0 * '+ Fx = Fy = Fz = 0 , #

- * ( ' z = 1 + x2 + y 2 - x=y=1 x2 + 2y 2 + 3z 2 − 21 = 0 P0 (1; 2; 2) z = ln (x2 + y 2 ) - x = 1, y = 0 z = sin x cos y - π x=y= 4 . - * ( ' xy = z 2 P0 (x0 ; y0 ; z0 )

xyz = a3

P0 (x0 ; y0 ; z0 )

x2 y2 z2 + 2 − 2 =1 2 a b c P1 (x1 ; y1 ; z1 ) P2 (a; b; c)

x2 + 4y 2 + z 2 = 36 x + y −

z=0

!"

x2 + y 2 = z 2

(3; 4; 5) #$ % & ' #

( !") * + !"

x = 0, y = 2 x2 + y 2 − xz − yz = 0 ! " xyz = a3 # $%

$ x0 = 4, z0 = 0 y0 > 0 0 1 ( # 2 ) " # 3 4 (2a2 − z 2 )x2 − a2 y 2 = 0 $ (a; a; a)

& ! '# ( ! ) " 2 2 2 2 x 3 + y 3 + z 3 = a 3 # * ! +,- a2

5 6 * $ " # z = 4 − x2 − y 2 2

. + / x2 + y 2 − (z − 5)2 = 0

+ " #

4 x, y #7 ! 4 7 2x + 2y + z = 08

6 y = f (x1 , x2 , . . . , xn ) 2 # x1 , x2 , . . . , xn ! * # t 9 x1 = ϕ1 (t), x2 = ϕ2 (t), . . . , xn = ϕn (t)

3

:.4

3

:04

( y = f (ϕ1 (t), ϕ2 (t), . . . , ϕn (t))

t ; 3 49 ∂y x1 ∂y x2 ∂y xn y = + + ···+ = · · · t ∂x1 t ∂x2 t ∂xn t ∂y ∂y ∂y x˙ 1 + x˙ 2 + · · · + x˙ n ∂x1 ∂x2 ∂xn

3

<54

6 y = f (x1 , x2 , . . . , xn ) 2 # x1 , x2 , . . . , xn * 2 m t1 , t2 , . . . , tm 9 x1 = ϕ1 (t1 , t2 , . . . , tm ), x2 = ϕ2 (t1 , t2 , . . . , tm ), . . . xn = ϕn (t1 , t2 , . . . , tm ) 3

< 4

m 2 (

y = f ϕ1 (t1 , t2 , . . . , tm ), ϕ2 (t1 , t2 , . . . , tm ), . . . , ϕn (t1 , t2 , . . . , tm ) 3

<4

m t1 , t2 , . . . , tm

∂y ∂x1 ∂y ∂y = · + ∂t1 ∂x1 ∂t1 ∂x2 ∂y ∂y ∂x1 ∂y = · + ∂t2 ∂x1 ∂t2 ∂x2 ∂y ∂y ∂x1 ∂y = · + ∂tm ∂x1 ∂tm ∂x2

∂x2 + ···+ ∂t1 ∂x2 · + ···+ ∂t2 ·

∂y ∂xn · ∂xn ∂t1 ∂y ∂xn · ∂xn ∂t2

∂x2 ∂y ∂xn + ···+ · ∂tm ∂xn ∂tm F (x1 , x2 , . . . , xn , y) = 0 y = y(x1 , x2 , . . . , xn ) ! " F # $%

#! # & ' % y (

x0 = (x10 , x20 , . . . , xn0 )Ì ∂y Fx (x10 , . . . , xn0 , y0 ) (k = 1, 2, . . . ) =− k ∂xk x0 Fy (x10 , . . . , xn0 , y0 ) ·

" ) # #% " ## Fy (x10 , . . . , xn0 , y0 ) = 0; y0 = y(x10 , . . . , xn0 ) F (x10 , . . . , xn0 , y0 ) = 0

z

t ,! " #

+

z = x2 + xy + y 2 , x = t2 , y = t z = x2 + y 2 , x = sin t, y = cos t - . /#% ) x(t) y(t) z 0 +

z t

y

z = , x = et , y = 1 − e2t x z = Ax2 + 2Bxy + Cy 2 , x = sin t, y = cos t y z = arctan , x = e2t +1, x

*

y = e2t −1 vw , u = et , v = ln t, z = u w = t2 − 1

z = uvw, u = t2 + 1, v = ln t, w = tan t z ! " # x v

z = u , u = u(x), v = v(x)

+

z = x ey , y = y(x) / f (x1 , x2 , . . . , xn ) 1 ) k ! " f (tx1 , tx2 , . . . , txn ) = tk ×f (x1 , x2 , . . . , xn ) 2$% ( #

t t = 1 ! "

# $ n ∂f xi = k · f ∂x i i=1 % # &' z ∂z ! (' ) ∂x x $ 1 & z = ln (ex + ey ), y = x3 3 y & z = x , y = ϕ(x) ' ) 2

z=

∂z ∂u

∂z ∂v

!

x , x = u − 2v, y = v + 2u' y

∂z *' z = f (x, y)' ∂x ∂z ∂z ∂z ! ∂y ∂u ∂v & u = mx + ny, v = px + qy y & u = xy, v = x √

& u = xy, v = x + y

+' u = f (x, y),

∂u x = r cos ϕ, y = r sin ϕ' ∂r ∂u ∂u ∂u ∂ϕ ∂x ∂y

$ 2 2 1 ∂u ∂u · + ∂r r ∂ϕ 2 2 ∂u ∂u = + ' ∂x ∂y

,' z = y + f (u) ! u = x2 − y 2 ' - yzx + xzy = x . f (u) '

y /' ) x

$ & x2 + y 2 − 4x + 6y = 0 & x2/3 + y 2/3 = a2/3

& x e2y −y e2x = 0 & xy + ln y + ln x = 0 & y + x = ey/x & 2 cos (x − 2y) = 2y − x ∂z ∂z ' ) ∂x ∂y $ & x2 + y 2 + z 2 − 6x = 0 & z 2 = xy

& cos (ax + by − cz) = k · (ax + by − cz) 2 2 2 & x + y + z − 2zx = a2 0' ) 1 2# 3 $ & x2 + y 2 = 10y " 4 x = 3 & x3 + y 3 − 2axy = 0 3 x=y=a

& y 2 − xy = 4 " 4 x = 3 5' 3 1 2# x2 + y 2 + 2x − 2y = 2 & 4 x 6 & 4 y 6 # ' ' - xyz = a3 $ xzx + yzy = −2z ' (' - xzx +yzy = z z y % z 7 ' & = ϕ x x

y = f (x) x0 = (x10 , x20 , . . . , xn0 )Ì U x0 (m + 1) x0 + h ∈ U ! % $m (hÌ · ∇)i f (x) f (x0 + h) = + Rm i! i=0 x=x0 " #$ (hÌ · ∇)m+1 f (x) Rm = (m + 1)! x=x0 +ϑh h = (h1 , h2 , . . . , hn )Ì , 0 < ϑ < 1 % ∇ &' ( ' n = 2 m = 3! f (x0 + h, y0 + k) = f (x0 , y0 ) + fx (x0 , y0 )h + fy (x0 , y0 )k 1 fxx (x0 , y0 )h2 + 2fxy (x0 , y0 )hk + fyy (x0 , y0 )k 2 + 2! 1 + fxxx(x0 , y0 )h3 + 3fxxy (x0 , y0 )h2 k + 3fxyy (x0 , y0 )hk 2 3! + fyyy (x0 , y0 )k 3 + R3 4 ∂ 1 ∂ +k R3 = f (x0 + ϑh, y0 + ϑk) h 4! ∂x ∂y 1 ∂ 4f ∂4f ∂4f ∂4f + 6h2 k 2 2 2 + 4hk 3 = h4 4 + 4h3 k 3 4! ∂x ∂x ∂y ∂x ∂y ∂x∂y 3 ∂ 4 f +ϑh + k 4 4 xy00 +ϑk ∂y

"

#)$

* +%,- -

f (x0 + h, y0 + k) .

% f (x, y) = x2 + xy + y 2

# +%,- f (x, y) = ln (x − y) ,0 1 x (y + 1) * 2 * ,0 3,0 4

/ +%,- f (x, y) = x3 + 2xy 2 ,0 1 (x − 1) (y − 2)

+%,- f (x, y) = sin (mx + ny) ,0 1 x y * 3 * ,0 3,0 4

x0 = 1 y0 = 2

h = x − 1 k = y − 2

2 +%,- f (x, y) = x2 y ,0 1

(x − 1) (y + 1) $ . $ 0 ,0

+%,- f (x, y, z) = x2 + y 2 + z 2 + 2xy − yz − 4x − 3y − z − 4 x = y = z = 1 .

" 1, 12,1 ) f (x, y, z) = ln (xy + z 2 ), x = y = 1, z = 0

! " # 2 # " " $ %& ' " ("" y

) f (x, y) = arctan , x = 1, x y=0 ) f (x, y) = y x , x = 2, y = 1 "!

* z(x, y) " # z 3 − 2xz + y = 0 & + !

" , z = 1 x = y = 1 ! z - . (x − 1) (y − 1) " # 2 # " " $ %& ' " (""

f (x) = f (x1 , x2 , . . . , xn ) " / x0 = x10 , x20 , . . . , xn0 )Ì & 0" f x0 1 .) %2 " , fx1 (x0 ) = 0, fx2 (x0 ) = 0, . . . , fxn (x0 ) = 0

1

3*)

1

34)

, grad f (x)|x=x0 = o

)

1

5" f " & / x0 2 " $ "6" 7 % ⎛ ⎞ fx1 x1 (x0 ) . . . fx1 xn (x0 ) H(x0 ) = ⎝. . . . . . . . . . . . . . . . . . .⎠ 1 89) fxn x1 (x0 ) . . . fxn xn (x0 )

. f x 5

"

D1 = fx1 x1 (x0 ), fx1 x1 (x0 ) D3 = fx2 x1 (x0 ) fx3 x1 (x0 )

0

fx1 x2 (x0 ) fx2 x2 (x0 ) fx1 x3 (x0 ) fx2 x3 (x0 ) , . . . , Dn = det H(x0 ) fx3 x3 (x0 )

f (x ) D2 = x1 x1 0 fx2 x1 (x0 ) fx1 x2 (x0 ) fx2 x2 (x0 ) fx3 x2 (x0 )

1

8 )

" % x0 " 2 "" , grad f (x)|x=x0 = o :

" 7 % . f x0 ,

D1 > 0, D2 > 0, D3 > 0, . . . , Dn > 0 D1 < 0, D2 > 0, D3 < 0, . . . , Dn (−1)n > 0 n = 2 z = f (x, y) ! (x0 , y0 ) " # $ $ f (x, y) (x0 , y0 ) " $ ! (x0 , y0 ) fx (x0 , y0 ) = 0, fy (x0 , y0 ) = 0 2 % D(x0 , y0 ) = fxx (x0 , y0 ) · fyy (x0 , y0 ) − fxy (x0 , y0 ) > 0

& fxx (x0 , y0 ) > 0 " fyy (x0 , y0 ) > 0 fxx (x0 , y0 ) < 0 " fyy (x0 , y0 ) < 0 ' ( )* % D(x0 , y0 ) < 0$ + , D(x0 , y0 ) = 0$ )* - )* . " $

+ )* / 0 1 z = f (x, y) z = x2 − xy + y 2 + 9x − 6y + 20 √ z = y x − y 2 − x + 6y ) z = x3 + 8y 3 − 6xy + 1 z = 2xy − 4x − 2y z = 2x3 − xy 2 + 5x2 + y 2 z = ex/2 (x + y 2 ) z = x(x2 − 3) cos y, 0 ≤ y < π * z = x4 + y 4 − 2x2 + 4xy − 2y 2 z = x2 − 6x + (3 − x)(y 3 − 3y) % / 0 1 u = f (x, y, z) u = x2 + y 2 + z 2 − 4x + 6y − 2z u = xy 2 z 3 (1 − x − 2y − 3z) (x > 0, y > 0, z > 0) z 2 y ) u = x + + + x y z

z2 2 y2 + + 4x y z (x > 0, y > 0, z > 0) u = x2 + 2y 3 + 4z 2 + 2yz + 6x −2y − 6z + 13 u = x2 + y 2 + z 2 − xy + x − 2z

u=x+

& / )* 23 4 z = f (x, y) / )* B z = x3 + y 3 − 3xy, B : 0 ≤ x ≤ 2, −1 ≤ y ≤ 2 z = x2 − y 2 , B : x2 + y 2 ≤ 1 ) z = sin x + sin y + sin (x + y), π π B:0≤x≤ , 0≤y≤ 2 2 5 / 0

)* " # / + .)* / 1 $ 6* )* V 7

8+ 1 9√ $ 6 )* 2 3

b N

2 N yi − f (xi ) −→ M in Q(a, b) = ! (xi , yi ) "#$ i=1 % # y = f (x) # % & (0; −1), (1; 2), (2; 4), (3; 6), (4; 9); ' ( ) "* ⎛ ⎞ y = ax + b 1 2 2 & (1; 5), (2; 3), (4; 1), (5; −1); A = ⎝2 1 −2⎠ b 2 −2 1 y =a+ x & + , # A

& (1; 5), (2; 18), (4; 129), (5; 250), & +*, b (8; 1025); y = ax3 + $- R. x xÌ Ax & (1; 7, 1), (2; 27, 8), (3; 62, 1), R = R(x) = Ì ; b x x (4; 110), (5; 161); y = ax xÌ = (x1 , x2 , x3 )

lg y = lg a + b lg x a

&

/$ # ( + 0

+ f, ϕ1 , . . . , ϕm # n 1 & x1 , x2 , . . . , xn (m < n) ( +* # f (x) = f (x1 , x2 , . . . , xn )

ϕ1 (x) = ϕ1 (x1 , x2 , . . . , xn ) = 0 ϕ2 (x) = ϕ2 (x1 , x2 , . . . , xn ) = 0 ϕm (x) = ϕm (x1 , x2 , . . . , xn ) = 0

1

23&

f, ϕ1 , . . . , ϕm ( G ⊆ Rn ! $ " / λ1 , λ2 , . . . , λm 1 " ! $ & ,

L = f (x) +

m

λi · ϕi (x)

1 24&

i=1

5 f 6 x0 = (x10 , x20 , . . . , xn0 )Ì ∈ G 7 1 23& +*, 5

Lxk = fxk (x0 ) +

m i=1

λi ·

∂ϕi = 0 (k = 1, 2, . . . , n) ∂xk x=x0

Lλl = ϕl (x0 ) = 0 (l = 1, 2, . . . , m)

! " # $% & (n + m) ' (n + m) () x10 , x20 , . . . , xn0 , λ1 , λ2 , . . . , λm * + $ x0 ! & , " - # .)

1 1

z = f (x, y) = + , x + y = 2 x y 1 1 1 z = f (x, y) = x+y, 2 + 2 = x y 2 z = f (x, y) = xy, x2 + y 2 = 2 u = f (x, y, z) = x − 2y + 2z, x2 + y 2 + z 2 = 9 u = f (x, y, z) = xyz, x + y − z = 3, x − y − z = 8 / # 0) 1%23 x2 − y 2 = 4, & 0) (0; 2) 4 5 # 4 6% & 78 9 3 # , 3 :! A = 6π 2 !

2 x2 + 3y 2 = 12 ) ) ;3 , # 8 4 2 2 #3 < ):!, " 8 " 7 x34 ; ;" 3 2 = + + > & 0)

(0; a) ; 0) (c; −b) , ; 7 ); 6 ; 7 3 (a > 0, b > 0, c > 0)?

α b

T = T (α, β) = v cos α + v cos β

a tan α + b tan β = c v v α β 1

1

2

2

x2 y2 + 2 = 1 2 a b @) 78 .! 3 ; # .!3

2

A # @ r 173 h B ) & 78 9 # 3 , = :! A C # $ ! a, b, c ) & 78 .! , B @ 3 r ) # n D x1 , x2 , . . . , xn , 2 3 & 6 c% ; " $ n xi = c , 0) i=1

n x1 · x2 · . . . · xn xi > 0; i = 1, 2, . . . , n!" " #$ %&$$ a, b

c $ '$( & & )$&* A $ +, &$ "

##

# f (x, y) b = B

z=f(x,y)

z

f (x, y) x y B

Z

Z B x, y

z = f (x, y)

• f (x, y) ≥ 0 :

# V =

y B

x

f (x, y) b

Z

B

# • f (x, y) ≡ 1 :

A=

#

b

B

B

αf (x, y) + βg(x, y) b = α

B

#

# g(x, y) b , α, β ∈ R

f (x, y) b + β B

B

! B = B1 ∪ B2 " B1 B2 # $ % & #

#

f (x, y) b = B

f (x, y) b + B1

B2

& ' & %% & ( #b f (x, y) b = B

y

a≤x≤b f1 (x) ≤ y ≤ f2 (x)

#

B2

f (x, y) b

B x B:

B1

#

x=a

⎡ ⎢ ⎣

f# 2 (x)

y=f1 (x)

6

a

⎤

⎥ f (x, y) y ⎦ x =

#b

B

f# 2 (x)

x=a y=f1 (x)

f2

f1 -x b

f (x, y) y x ) ! *

B B:

y

y d

c≤y≤d ϕ1 (y) ≤ x ≤ ϕ2 (y)

#

#d

ϕ#2 (y)

f (x, y) b =

ϕ1

6

ϕ2

B

c

-x

f (x, y) x y y=c x=ϕ1 (y)

B

! B x, y " #

= (x, y) $ % &

'

% (! x y = (x, y) , rA = rA (x, y) $! ## % A x, y " !) ! ! ## % A ⊥ x, y " ! ! ## #

x b B & * B

+ * # , y = x2 y = x + 2 % & - & # x

y

% , % # . ! & * + * % % / 0 1%% y = x , xy = 4 , x = 4 y 2 = 4 + x , x + 3y = 0

(x − 2)(y − 2) = 4 (x + 3)(y + 3) = 4 y = ln x , x − y = 1 , y = −1 2 1%% & ! ' ' ! 3 4 ' #2

x+2 #

I =

y x x=−1 y=x2

#a

I =

√ 2 2 a −y # x y

y=0 x=a−y

#1 2−x # 2 I = y x x=0 y=x

#a

√

2a2 −y 2

#

I =

x y √

y=0 x= ay

##

xy 2 x y B

B

y 2 = 2px ! " x = p # (p > 0) $ % &" B ' # ( #

xy b , B : (x − 2)2 + y 2 = 1 B

#

(x2 + y 2 ) b

B

B : y = x2 , y = 1

) & & ## xy − y 2 x y B

B

* + ,+-+ O(0; 0), A(10; 1) B(1; 1) & ##

ex/y x y B B

y 2 = x (y ≥ 0) ! y = 1 x = 0 " # ## xy x y B B

x".& / " + & (x − 2)2 + y 2 = 1

#

& 0 &

### f (x, y, z) b =

B

1 /2- xy = 4 ! x + y = 5 # # ! y = x ( ≡ 1) # x 3 b B & x2 + y 2 B

& & & 1

y = x2 ! 2 y = x # 4 * + ,+-+ (0; 0) (0; 1) (1; 0) & - 0 & " # & 5-+&

≡ 1 & 6 & 7 0 " & Jx , Jy & -" 0 & Jp # & 5-+& 1 & +& & ' x = 0 x = a y = 0 y = b

# ≡ 1 & 8 -+ 1" % ' " # ≡ 1(

y = 0 / - 8&+ y = sin x

y = x2 x = 4 y = 0 # xy b B B

' & .&" x = a cos3 t , y = a sin3 t , 0 ≤ t ≤ π/2 (a > 0)

#

#

f (x, y, z) x y z B

B R3 # f (x, y, z) ≡ 1 V = b B B

! " ! #$ % &' (

z

x, y

B

B : (x, y) ∈ B g1 (x, y) ≤ z ≤ g2 (x, y)

g2 B

g1

' B )' x, y "' * x ⎡ ⎤ g2#(x,y) ## # ⎢ ⎥ f (x, y, z) b = f (x, y, z) z ⎦ x y ⎣ B

B

y

B' % +(

z=g1 (x,y)

, B '! )' '- x & . B = {(x, y, z)| a ≤ x ≤ b , f1 (x) ≤ y ≤ f2 (x) , g1 (x, y) ≤ z ≤ g2 (x, y)}

/ 0 ' !* #

#b

f# 2 (x)

g2#(x,y)

f (x, y, z) b = B

f (x, y, z) z y x

% + (

x=a y=f1 (x) z=g1 (x,y)

x, z

B

B = {(x, y, z) ∈ R3 | (x, z) ∈ B , g1 (x, z) ≤ y ≤ g2 (x, z)}

' B )' x, z "' * ⎡ ⎤ g2#(x,z) ## # ⎢ ⎥ f (x, y, z) b = f (x, y, z) y ⎦ x z ⎣ B

+ B

B

y=g1 (x,z)

y, z B = {(x, y, z) ∈ R3 | (y, z) ∈ B , g1 (y, z) ≤ x ≤ g2 (y, z)}

% (

B y, z ! ##

# f (x, y, z) b =

B

B

⎡ ⎢ ⎣

g2#(y,z)

⎤ ⎥ f (x, y, z) x⎦ y z

x=g1 (y,z)

= (x, y, z) " #$ P (x; y; z) %& ' B = const. ' (

)

! " * +, '$* ) %& ' B # m = b " B

+, '$ $ )

- . A

xS =

1 m #

JA =

# x b B 2 rA

b

B

/0 ( 0 1 %& ' * ( 0 2 - , !

y = x2 , x + y + z = 2, y = 1, z = 0

z = xy, x + y = 1, x = 0, y = 0, z = 0 # (x + y + z) b* , B B

( x = 0* x = 1* y = 0* y = 1* z = 0 z = 1 - , 3 ###

x y z * (x + y + z + 1)3

B

yS =

, B ( % x+y +z = 1 - ,

1 m

# y b B

zS =

1 m

# z b B

rA = rA (x, y, z) . ( P (x; y; z) ∈ B - . A

4 1 %& ' * - , ( 2 z = x2 + y 2 * x + y = 4* x = 0* y = 0* z = 0 #

x3 y 2 z b* , B 5

B

0 ≤ x ≤ 1* 0 ≤ y ≤ x * 0 ≤ z ≤ xy 6

7 " #8 *

x + y + z = a* x = 0* y = 0* z = 0

- , * , 9 #$ z : /& #$ " %& ' * 2 x = a* 2x+z = 2a* x + z = a* y 2 = ax* y = 0 - , 0 y > 0* ,

y

!" # ≡ 1$% &' () * $ x + y + z = a% x = 0% y = 0% z = 0 (a > 0) ($ az = a2 − x2 − y 2 % z = 0 (a > 0)

+ ,' ( )- z . - !"% &' () #/ ≡ 1$* $ x = 0% y = 0% y = a% z = 0 x + z = a (a > 0) √ ($ x + y + z = a 2% x2 + y 2 = a2 % z = 0 (a > 0) $ z 2 = 2ax, z = 0, x2 + y 2 = ax (a > 0)

. - ! T : x = x(u, v, w) y = y(u, v, w)

# $

z = z(u, v, w)

/( () u, v, w ! B 0 B ∗ 0 T * B = T (B ∗ ) 1- T 1 % % # #

∂(x, y, z) ∗ b # $ f (x, y, z) b = f x(u, v, w), y(u, v, w), z(u, v, w) ∂(u, v, w) B

B∗

∂x ∂u ∂y ∂(x, y, z) = ∂(u, v, w) ∂u ∂z ∂u

∂x ∂v ∂y ∂v ∂z ∂v

∂x ∂w ∂y . . . F unktionaldeterminante ∂w ∂z ∂w

. # $ ( #, b$* ∂(x, y, z) ∗ ∂(x, y, z) b b = x y z = ∂(u, v, w) u v w ∂(u, v, w)

# ($

# $

z w ! " ∂x ∂(x, y) ∂u = ∂(u, v) ∂y ∂u

∂x ∂v ∂y ∂v

# $ % $

b

(r, ϕ) r # $ % ϕ & " $ % $ x# ' $" % $

0 ≤ r < ∞, 0 ≤ ϕ < 2π

y

6 r Y ϕ

P

-x

( " T : x = r cos ϕ y = r sin ϕ

) *+

{(r, ϕ)| r = const. , ϕ ∈ [0, 2π)} ϕ, ) -% " +

(r, ϕ) , 0 ≤ r < ∞ , 0 ≤ ϕ < 2π ( . T : x = a r cos ϕ

y = b r sin ϕ , a, b % $

) /+

ϕ, % 0 r · a r · b 1 % $

z

(r, ϕ, z) P P x, y r, ϕ P z x, y

6

P z

ϕ

−∞ < z < ∞

x

!" #

> r

T : x = r cos ϕ y = r sin ϕ

-y

P

$

%

z=z

& #'( {(r, ϕ, z)| r = const. , ϕ ∈ [0, 2π) , z ∈ (−∞, ∞)} #)*"(

z

(r, ϑ, ϕ) r " # + ϑ , + z ) " - ϕ . /* 0 ≤ r < ∞ , 0 ≤ ϑ ≤ π , 0 ≤ ϕ < 2π

!" #

x

6

P U r ϑ ϕ P

-y

T : x = r sin ϑ cos ϕ

$ %

y = r sin ϑ sin ϕ z = r cos ϑ

& #'( {(r, ϑ, ϕ)| r = const. , ϑ ∈ [0, π] , ϕ ∈ [0, 2π)} # '(

(r, ϕ)

b = r r ϕ

+ # (r, ϕ) b = rab r ϕ /* (r, ϕ, z) b = r r ϕ z # (r, ϑ, ϕ) b = r2 sin ϑ r ϑ ϕ

## 1 − x2 − y 2 x y

xy = u

B (0, 0) ! r = 1 ## (x2 + y 2 ) x y " # 2

y = vx

2

B

B

2

B $ $ x + y = 2ax (a > 0) # % &

' ( #2 #x f x2 + y 2 y x x=0 y=0

)

* ## x y " # B & 2 a − x2 − y 2 B

+ $ ! a (0; 0) " ( , ## & xy x y " # B $ B

x- . (x − 2)2 + y 2 = 1 #

/ ' ( #1 #1 f (x, y) y x x=0 y=0

0 u v )1 u = x + y" v = x − y

2 34 " 5 $ # 1 a2 & xy = , xy = 2a2 , 2 x y = , y = 2x 2

& y = ax, y 2 = 16ax, ay 2 = x3 , 16ay 2 = x3 (a > 0)

y 2 = ux, vy 2 = x3 .

& x2/3 + y 2/3 = a2/3 (a > 0)

x = r cos3 ϕ , y = r sin3 ϕ .

. 5 6 34" $ $ ' y 2 = x" y 2 = 16x" y 2 = x3 " 16y 2 = x3 # 3) 0 √ 5 1 x = uv" √ 4 y = u3 v 7 +4 34" $ $ x2 + y 2 = 1" x = 0" y = 0 # " ) y -" # 34 ≡ 1 . 5 6 5 - # 0 5 x = r cos ϕ" y = 2r sin ϕ" z = z 0 8 " $ 34 y2 − z 2 = 1 ' x2 + 4 √ z = ± 3 # 9 : 8 # $ ;< x2 + z 2 = a2 : y = 0" z = 0" y = x 0 8 ### z x y z " # B B

h2 2 (x + y 2 ) R2 : z = h (h > 0) #

$ z 2 =

### x2 + y 2 + z 2 x y z B

B x2 + y 2 + z 2 = z ### z x2 + y 2 x y z B

!" B #$ y = 0 z = 0 z = a > 0 x2 +y 2 = 2x % ### & x2 x y z B B

x2 y2 z2 ' () 2 + 2 + 2 = 1 a b c ### z x y z B & B

( z = 0 *$ () x2 y2 z2 + 2 + 2 = 1 2 a b c + , ' -) #$ . x2 z y2 & = 1 − 2 − 2 z = 0 c a b

x = ar cos ϕ y = br sin ϕ, z = z & x2/3 + y 2/3 + z 2/3 = a2/3 , a > 0 /0 x = r cos3 ϕ y = r sin3 ϕ, z = z &

& 4z = 16 − x2 − y 2 z = 0 x2 + y 2 = 4 / 1 !"& 2 & z = (x + a)2 x2 + y 2 = a2 4 & z = 2 z = 0 x2 + y 2 = 1 x + y2 x2 + y 2 = 4 & az = x2 + y 2 z = 0 x2 + y 2 ± ax = 0 (a > 0) & az = a2 − x2 − y 2 z = 0 x2 + y 2 ± ax = 0, a > 0 / !"&

2 '' * 3

x2 + y 2 + z 2 = a2 x2 + y 2 + z 2 = 4a2 '' 4 5' 6 ' ) ) ' 7 6 ' ) , 8 '' , ' -) #$ & (x2 + y 2 + z 2 )2 = a3 x & (x2 + y 2 + z 2 )2 = az(x2 + y 2 ) (a > 0)

!

! x2 + y 2 − z 2 = 0 , ' x2 + y 2 + z 2 = 2az ' , $ . (a > 0) 9 '' 3' Q #$ x2 + y 2 − z 2 = 0

z = h > 0 - ) :

(x, y, z) = z

n

F x, y, y , . . . , y (n) = 0 implizite

(n) (n−1) y = f x, y, y , . . . , y explizite

n y(x) x ∈ [a, b] [a, b]! • • •

! ! ! "# $ # % % x = a #& ' ( ) $ # # #* ! % ## % a, b, . . . +# ( ) $ # # !

y = f (x, y)

f (x, y) # , $ ## - P (x; y) ! . / - x, y0 ,

& # ! ! ! ! 1 2 # 2 ## (f (x, y) = d = const.) # + t & 3 & ! # 4 yt = y˙ ! 5! ' ) (y − y0 )2 = 2px 1 # /) y = c1 e2t +c2 e−t 2 ) y = cx ) y = ct ) y = c1 cos 2t + c2 sin 2t 2 ) y = 2cx ) x2 + y2 = c2 ) y = (c1 + c2 t) et +c3 ) y = c et ) t3 = c(t2 − y2 ) 6! + # ' %

) y2 + x1 = 2 + c e−y /2 ) 1 # x2 + y2 = 2cx x

) ln y = 1 + ay ) - y = x2 + 2cx 2

!

" xy = 2y, y = 5x2 1 " (y) ˙ 2 = t2 + y 2 , y = t c2 − x2

" (x+y) x+x y = 0, y = 2x " y¨ + y˙ = 0, y = 3 sin t − 4 cos t 2 x " 2 + ω 2 x = 0 t x = c1 cos ωt + c2 sin ωt " y¨ − 2y˙ + y = 0 y = t et y = t2 et " y − (λ1 + λ2 )y + λ1 λ2 y = 0 y = c1 eλ1 x +c2 eλ2 x

# $ y =

cx3 (c ∈ R) ! 3y − xy = 0 $ % & ' ( 1 1 " 1; " (1; 1) " 1; − 3 3

) * + ( y " y˙ = y − t " y˙ = t 2

" y˙ = y + t , y˙ = f (t, y) = t2 + y 2 * - . %

f (t, y) = d = const. / 1 d = ; 1; 2; 3 + 2 0 &1 2

3 % & 4

y˙ =

u(t) , v(y) = 0 v(y)

1 - 5 & y˙ =

v(y) y = u(t) t

- % / - ! ( # # v(y) y = u(t) t =⇒ V (y) = U (t) + C V (y) v(y) u(t)

y ( t

x2 y + y = 0 x + xy + y (y + xy) = 0 φ2 r + (r − a)φ = 0 2st2 s = (1 + t2 )t t 1 + y 2 + y 1 + t2 y˙ = 0 ty y˙ = 1 − t2 y − ty˙ = a + at2 y˙ y tan x = y y x3 = 2y ! (x2 + x)y = 2y + 1 " y a2 + x2 = y

(1 + x2 )y + 1 + y 2 = 0 # $ √ 2y x = y, y(4) = 1 π 1 = y = (2y + 1) cot x, y 4 2 x2 y + y 2 = 0, y(−1) = 1 y y t t − =0 1+y 1+t y(1) = 1 y(0) = 1 t t (1 + e )y y˙ = e , y(1) = 1 r + r tan φ φ = 0, r(π) = 2 √ y = 2 y ln x, y(e) = 1 (1 + x2 )y + y 1 + x2 = xy y(0) = 1 % $ & "' ( ) * + 100 ◦, $ +

& -* 20 ◦ , 25 ◦ , + $

10 . 60 ◦ , ) "' / 01 1$ 2) 2$ " #) "' * * -* ) 3 4 ( ( * 5 C $ 6 )

R 7 8 2 *

) 9

8*

U = const. 4 q = q(t) ( + $

q(0) = 0 : 6 8 5 " I

2 " U = 3 ;, R = 6 Ω, L = 0, 06 < & t = 0, 01 + $

8 & t = 0 $ / 1 $ & 8 5 " I $ / 2 R L

U

= 8 0 ) >" <5 '"

! 5

- 5 q ? '" 8 $ ? ( 8 + $

< ) " 8 ) @ " H

P FT

FL

y

FH

6 FT * F P H α

a

FL ?

-x

A 2$ " B 2a 2)

|v F | = v0 x x2 !! " |v F | = v0 1 − 2 # a $ % !! ! $ |v S | = vS = const. !!& ! ! '( & %! # ) * +, # % p(x) (x) ! - . x ' $/- # $ ! - - /- 1 . Δx g (x)Δx# %! - - - Δp := p(x+Δx)−p(x) ≈ −g (x)Δx# 0 ! ! 1 2! 23 p0 p = = const. % p0 := p(0)

0 # 0 := (0) # ! ' $ /- # 4 ! . +! 4 + + # 5# y(x)

6+ !! 7 ! %' x e# ! 7 6+ ' # & y !

! 6+ y ! 7 " λ y = − (λ > 0& 3# y x $! ! 4 # )

y (μ > 0& 3 x+μ ,

y = −λ

8# $ ' ! 7 * p * $- # 9 t ' ' : +! y(t)# ; : +! 9 k > 0 < ! 7 # = >! < ' - # % +' y(t) + ! 4 &

' ! 7 ?!+ # # = !! <& 7 P (−a; a) &

% A B ! <

< ! > = ' M # @# 2 > + < " 3 cy = x2 17 3 3 xy = c 1.A 3

3 x2 + 4y 2 = c2 1$ 3 3 x2 − 2y 2 = c2 1.A 3 # = !! <& +' B 7 4- 0!

< x # # = !! <& 7 P (−1; −1) +' % OT & + x < ! 7 & ! C =' #

r 2r !

"

#$ % 100 ◦ & ' " % 20 ◦ & ( #$

) * ) )

') e % + $ (r ≤ e ≤ 2r) ,- e = 1, 6 r

/ 2 1 ')) 3

,-

0 % ' , 1 $ ) 2 y = a 1 + (y ) , y(0) = b , y (0) = 0 . y

6

F -H

−F H

T / e

. / % ) ),

0 "

1 *

b

-x

4FH ,3 H := |FH |3 q a := 3 q 0 1 $ 5 H 6

v(x) := y (x)

y˙ = f (at + by + c) 0) z = at + by + c ,- ,

7 8 z˙ = a + by˙ =⇒ z˙ = a + bf (z)

y˙ = f

y t

y 0) z = ,- ,

7

t

8 y y˙ − yt ˙ −y t =⇒ z˙ = f (z) − z z˙ = = 2 t t t

y˙ =

at + by ct + dy

y y t y˙ = y =f t c+d t a+b

y˙ =

at + by + e ct + dy + h a D := c

b d

! "

• #$ D = 0 % t = tS + τ , y = yS + η & $ ! ' S(tS , yS ) ( )

g1 : at + by + e = 0 g2 : ct + dy + h = 0

* aτ + bη η = τ cτ + dη • + # D = 0 $ (' z = at + by + e $ b = 0 z = ct + dy + h $ d = 0

,& !

! - . / y " ty˙ = y 1 + ln t '" 2t2 y = (t2 + y 2 )t x

" x ln y − y x = 0 y " yy = 2y − x " x2 + y 2 − 2xyy = 0 s t s = − t t s y y " xy cos = y cos − x x x

"

" x2 y = y 2 + xy " ty˙ − y = 2t 0! . 1 ' y2 y − , y(−1) = 1 2 x x 1 y , y(1) = √ '" xy = y 1 + ln x e

" y =

! # () 2 ( 2 / & ' 3 2 ) ' 4 5 !

y M OM y ! " y

ON = OM

* "

+', % - # vF vH .

/0 1

! " a #

$ % &' $

' % ( % ! # ) # ' " vF = const.

(|v H | = vH = const.) %

2 y˙ = (8t + 2y + 1)2 '2 y = (x + y)2 2 (2x + 3y − 1)x +(4x + 6y − 5)y = 0 2 (2t − y)t + (4t − 2y + 3)y = 0 2 y = (x − y)2 + 1 2 (t + y)2 y˙ = 4 t+y−3 2 y˙ = −t + y − 1 t+y+1 2 y˙ = t−y+3

y˙ + a(t)y = r(t)

3

4 ! y(t) = z(t) + y¯(t)

!

$

r(t) ≡ 0 r(t) ≡ 0

!

5 2

z(t)

/0 0 z˙ + a(t)z = 0 y¯(t) 6

/0 5 2 +

z(t) 0 z˙ + a(t)z = 0 , 7 #

* , 0 8 , z(t) = C e−A(t)

' A(t) 9 % a(t)

5 2

y¯(t)

y¯(t) = C(t) eA(t)

# C(t)

! "

$

% & & ' " '& y˙ − e−t +y − ty˙ = ty

ue = uo sin ωt , ω = 0 ; ua (0) = 0 + ua = ua (t) " -

& y˙ − 4y = e2t 1 y˙ − y tanh t = 2 y˙ sin t cos t + y cos2 t = sin2 t x y + y = 1, |y| < 1 1 − x2 y(0) = 1 ϕ = a sin t + bϕ ' t y˙ cos t = y sin t + cos2 t

ty˙ − (t + 1)y − t2 + t3 = 0 2s z 3s2 + z = s 1 + s2 1 + s2 t−1 y ( y˙ − = t t y + t2 = 0; y(0) = 1 ) y˙ + 1+t y

y˙ + + et = 0; y(1) = 0 t * & + RC " , &

R = const. ue

C = const.

ua

τ = RC

. xy + y = ln x + 1 / 00 p(t) 1 2 ) 1 r(t) = 1/(1 + t) 1 0 , &0) #3 ! 42 5 - a ,2 ) / (a = const.) &" 1 50 & 6

y˙ + g(t)y = h(t)y r , r ∈ R \ {0; 1} 7 ' ,& v = y 1−r # v˙ + (1 − r)g(t)v = (1 − r)h(t)

8

yt ˙ + y = −ty 2 2

y˙ − ty = −y 3 e−t

y˙ + ty = ty 3 y2 y y˙ = 2 − ; y(−1) = 1 t t 3y 2 y˙ + y 3 = t + 1; y(1) = −1 (1 − t2 )y˙ − ty = ty 2 ; y(0) = 0, 5

P (t, y) + Q(t, y)y˙ = 0 P (t, y) t + Q(t, y) y = 0

! "

# $ Φ(t, y) % & % $ % $ P (t, y) Φt (t, y) = grad Φ = ' ( Q(t, y) Φy (t, y) )

Φ % $ P (t, y) Q(t, y) & *$ R = {(t, y)| a < t < b , c < y < d}

R % $"

Py (t, y) = Qt (t, y) ( )

' +

, - && $ Φ % $ $

"

'

Φ(t, y) = C Φ

• Φt = P ' ' ( . &

# Φ(t, y) = P (t, y) t = Ψ(t, y) +

g(y) -./0

/ Φy = Q Ψy + g = Q . . . 0 g(y)

) 1

g¯(y) Φ(t, y) = Ψ(t, y) + g¯(y) •

2 Φt = P $

& 2 Φy = Q 0

-

Φ(t, y) =

1

Q(t, y) y = Ψ(t, y) + g(t)

Ψt + g˙ = P Φ(t, y) = Ψ(t, y) + g¯(t) P (t, y) t + Q(t, y) y = 0

! "#$ % ¯ y) m(t, $ && m(t, ¯ y)P (t, y) t + m(t, ¯ y)Q(t, y) y = 0 $ & ' $ ( $ & δ := Qt − Py 1.

a := −

2.

b :=

3.

c :=

4. 5.

δ Q

δ P

δ P −Q δ d := − P +Q δ e := tP − yQ

a = a(t)

m ¯ = m(t) ¯

b = b(y)

m ¯ = m(y) ¯

c = c(t + y)

m ¯ = m(t ¯ + y)

x=t+y

d = d(t − y)

m ¯ = m(t ¯ − y)

x=t−y

e = e(t · y)

m ¯ = m(t ¯ · y)

x=t·y

m ˙ = a(t) m m = b(y) m m = c(x) m m = d(x) m m = e(x) m

) ( & *$ + , $ c -. c / t + y

¯ / t + y 0 )!# -$

m x = t + y & m ¯ . ' m m = c(x) 1 && & ' /

2y y2 t+ y=0 % 4 − 2 t t 3t2 ey y˙ = 1 − t3 e y e−y y˙ = −y t e −1 3x2 ey x + (x3 ey −1) y = 0

e−y x + (1 − x e−y ) y = 0 2x cos2 y x

+(2y − x2 sin 2y) y = 0

(e2t −y 2 ) t + y y = 0 2t tan y t + (t2 − 2 sin y) y = 0 (1 + 3t2 sin y) t − t cot y y = 0 (t sin y + y) t

+(t2 cos y + t ln t) y = 0

(3ty 2 + y) t + (2t2 y − t) y = 0

cos t + y 2 − sin t

x

+ cos t + y + 2y y = 0 2

(x2 − y)x + x y = 0

(x2 − 3y 2 )x + 2xy y = 0

y 2 x + (yx − 1)y = 0

(sin x + ey )x + cos x y = 0

an y (n) + · · · + a1 y˙ + a0 y = r(t) , a0 , . . . , an ∈ R

y¯ r(t) ≡ 0 z an z (n) + · · · + a1 z˙ + a0 z = 0

y = z + y¯

z !" n # $% {z1 (t), . . . , zn (t)} "& ' ( ) z = C1 z1 (t) + C2 z2 (t) + · · · + Cn zn (t) *" + y = eλt # , % & + -. - an λn + · · · + a1 λ + a0 = 0

/

0 n 1 λk ∈ C ! 2 ' , ! , % 3 - 4"

y¯ + 2 ' r(t) ( , ! 5 -. + -. y¯ 4" / 0 • p q -. 6 % $ m 3 ⎧ ⎫ m ⎪ p = p m tm + · · · + p 1 t + p 0 ⎪ ⎨ A = Am t + · · · + A1 t + A0 ⎬ =⇒ ⎪ ⎪ ⎩ ⎭ m q = qm t + · · · + q1 t + q0 B = Bm tm + · · · + B1 t + B0

5 7 6 % 1 %

! " #

$ # &

λk = a ± b

eat cos bt , eat sin bt eat cos bt , t eat cos bt , . . . , tν−1 eat cos bt eat sin bt , t eat sin bt , . . . , tν−1 eat sin bt

ν '&

( & b = 0 eat , t eat , . . . , tν−1 eat

ν '&

λk = a

%

) &* y¯ r(t)

) &* y¯

eαt (p cos ωt + q sin ωt)

eαt (A cos ωt + B sin ωt)

) + N := α + ω

( & α = 0 p cos ωt + q sin ωt

A cos ωt + B sin ωt

N := ω

( & ω = 0 p eαt

A eαt

N := α

, N ν '& # -'

. ) tν ' /

• # % ) &* y¯ 0 -

1 2 Ak Bk * 1 2 • , r(t) ( - .

y¯ /

3 1

) 4 z = C1 z1 (t) + · · · + Cn zn (t) ' &* 5 -0 y¯ = C1 (t)z1 (t) + · · · + Cn (t)zn (t)

! "

0 &* -

) & - Ck (t) '

⎞ ⎛ ⎞ 0 ⎛ ⎞ ... zn z1 ⎜ ⎟ ⎜ ⎟ C˙ 1 ⎟ ⎜ z˙1 ⎟ ⎜ ⎟ ⎜ . . . z ˙ ⎟ n ⎜ ⎟⎝ ⎠ = ⎜ ⎟ ⎜ ⎜ ⎟ 0 ⎟ ⎜ ⎝. . . . . . . . . . . . . . . . . . . ⎠ ˙ ⎝ r(t) ⎠ Cn (n−1) (n−1) . . . zn z1 an ⎛

C˙ 1 , . . . , C˙ n ! C˙ k = fk (t) " #$ y¯ % & ' ' ( " ) " ' y (4) − 2y = 0 ' y (4) + 8y + 16y = 0 y − 4y + 3y = 0 y − 4y + 4y = 0 y − 4y + 13y = 0 " y − 4y = 0

y + 4y = 0 y + 4y = 0 x 2 x − 4x = 0 +3 t 2 t 2 * 4 2 + = 0 φ 2 s s $ 2 + 2 + 2s = 0 t t t0 = 0, s0 = 1, s0 = 1 y − 5y + 8y − 4y = 0 y (4) − 16y = 0 y − 8y = 0 ! y + 3ay + 3a2 y + a3 y = 0 + y (4) + 4y = 0 , 4y (4) − 3y − y = 0

- % ". / # # ) ' 0 m1

2 & l " 3 !4 / +) !5

ϕ sin ϕ ≈ ϕ

6 0 7 ) 8$ 2 3$ 0) 8 2 b % % 01 0 ) " &" ' ' ') .$ % / )

+!

m y m ! "# & (" ' %.$)

3 1 +!+!) ! $ %)

2. #!+!! ) $! α α2 b − 4m2 g < 0 9 % & !! ' ( ". + & !! )

y (4) − y − y¨ + y˙ = t2 + t et +(t3 − 1) e−2t sin t y (4) + y − y¨ + y˙ − 2y = 2 + 4 cos t +(t2 − t + 1) e0,5t y + 2y˙ = (2t − 4) e−t sin 3t

y + 3¨ − e−2t +5 ! " y¨ + 9y = e5t y¨ − y = et (t2 − 1)

y¨ + 4y = cos 2t y¨ − 6y˙ + 8y = et + e2t y − 2y + y = e2x y − 4y = 8x3 y + 3y + 2y = sin 2x + 2 cos 2x y + y = x + 2 ex y + 3y = 9x # y + 4y + 5y = 5x2 − 32x + 5 $ y − 3y + 2y = ex 2 x + k 2 x = 2k sin kt t2 y − 2y = x e−x

y − 2y = x2 − x y + 5y + 6y = e−x + e−2x % x ¨ + 2k x˙ + 2k 2 x = 5k 2 sin kt & y + y = 6x + e−x y (4) − 81y = 27 e−3x x + x˙ = 3t2 y + 8y = e−2x x ¨ + 4x˙ + 4x = e−2t ' a3 x ¨ + ax = 1 ( ) " y + 2¨ y + 2y˙ + y = t y(0) = y(0) ˙ = y¨(0) = 0 y¨ + 2y˙ + 6y = 3t2 y(0) = 0, y(0) ˙ =3

y¨ + 4y = sin t y(0) = y(0) ˙ =1 2t y¨ − 2y˙ = e +t2 − 1 1 ˙ =1 y(0) = , y(0) 8 * 1 y + 4y = sin 2x e2x y − 4y + 5y = cos x

y + y = tan x 1 y + y = 1 + ex e−2x y + 4y + 4y = 3 x y + 4y + 4y = e−2x ln x 1 y + y = cos3 x 1 y + 4y = sin2 x ex y − 2y + y = √ 4 − x2 + ,- ' $ r . m ' $ ) / 0 !

+ l 1 0 2 ! 2

,- !

/ !

3 ! 3 !

y !

4 5' ! $ 3 l2 + %

! + $

F ! α "#$ % & 2 1 + y ≈ 1) ' () * + x, + - % $ +% κ = x . x, $/ "0 1 2 / $ 2 1 + y ≈ 1 3 4 5 $ 4 6 / x, $ + F x, $ $ 7 / . # $ $ # $

$ 8 t = 0 $ x = 0 # $ v = 0 4 +- / 9 h 4 6 :$ 2 - - # $ , $ 4 ; % 9 y +- t : $ ! < 4 = 5 m ; "; k > 01 / x, $ $ - ; ! = > - "# $ 1 7 = # $

$ 4

? x(t) :$ $ ; $ , < , 5 $ / /

8$ F = 2ωm cos ωt ex (ω > 0) 7/ @ ω 2 : = +A 7 ? :$ B 7 % t −→ ∞B C 4 =/ ?/ l $ : q(x) = kx ;% ! α / w(x) ? k w(4) = x , w(0) = w(l) = 0, α w (0) = w (l) = 0. q(x)

)

? ? ? ? ? ? ?l - x

# D 4 =/ ?/ l $ ,E F |F | = F F F -)

l

-x

! α % w(x) w(4) + λ2 w = 0

F & Fk

λ2 := α w(0) = w(l) = 0, w (0) = 0, λ w (l) = 0 λ Fk w(x) ≡ 0 ! " # !" #$ %& ' () $ "

an tn y (n) + an−1 tn−1 y (n−1) + · · · + a1 ty˙ + a0 y = r(t) a0 , . . . , an ∈ R . . . &$

• *+ t > 0 #+ , $ t = ex # $ &$- #+ * $ v(x) := y (ex ) an λ(λ − 1) . . . (λ − n + 1) + · · · + a2 λ(λ − 1) + a1 λ + a0 = 0

. / " 0+ $

y(t) = v(ln t) • *+ t < 0

0+ ) $ t −t ( / 1$ 2 ' t2 y¨ − ty˙ + 2y = t ln t ' t3 y − t2 y¨ + 2ty˙ − 2y = t3 + 3t 1

' t2 y¨ − 2y = t2 + t 2 ' t y¨ + ty˙ − y = t ' t2 y¨ + ty˙ − y = t−1 #' t2 y¨ + ty˙ + y = cos ln |t| ' x3 y − 3xy + 3y = 0

' x2 y − 2y = 0 ' x2 y + 2xy − n(n + 1)y = 0 3' x2 y + 5xy + 4y = 0 ' x2 y + xy + y = 0 ' xy + 2y = 10x ' x2 y − 6y = 12 ln x

' x2 y − 2xy + 2y = 4x $' x3 y + 3x2 y + xy = 6 ln x ' x2 y − 4xy + 6y = x5 4' x2 y + xy + y = x

y˙ 1 = a11 y1 + · · · + a1n yn + r1 (t) y˙ n = an1 y1 + · · · + ann yn + rn (t)

. . . aij yi = yi (t) . . .

! " y˙ = Ay + r

#

%$& y¯ ' () * (r(t) ≡ o) * z * () * ) z˙ = Az + *

y = z + y ¯ * () , # $% - , z . * () , z˙ = Az

# /%

* 0 z = x eλt - # /% 1. 1 * - ! 2 ' (A − λE)x = o •

λk ν 21 - ! , A - ,

z˜k := eλk t c0 + c1 t + · · · + cν−1 tν−1

# /% 1 . , *

3 c0 , . . . , cν−1 3 4 , ν 1 •

5 * λk = a ± b 6 7 ' + ν 21 - ! , A+ "

z˜k := eat c0 + c1 t + · · · + cν−1 tν−1 cos bt 5

+ d0 + d1 t + · · · + dν−1 tν−1 sin bt

# /% 1 . , * 3 ci , di (i = 0, . . . , ν − 1)+ * , 2ν 1 4 () z , # /% 4

z˜k : & z= z˜k z

1 *

k

z = C1 z 1 + · · · + Cn z n

# %

Ci z i ! " # $% & " y¯ ' ( ) ! y¯ *+

' •

•

' " # ,% ' y ¯ = C1 (t)z 1 (t)+· · ·+Cn (t)z n (t) & ' # -% + . /

0 Ck (t)1 C˙ 1 , . . . , C˙ n 2 3 4 C˙ k = fk (t) y¯

& ) 5

6 / ) 3 0 r(t) ' / y¯ 1 ) ' r(t) /

7 ! "

% z˙1 = −z1 − z2 z˙2 = − z2 % x˙ 1 = x2 x˙ 2 = x1 + et + e−t

% z˙1 = −5z1 − z2 z˙2 = 2z1 − 3z2 % z˙1 = −z1 + z2 + z3 z˙2 = z1 − z2 + z3 z˙3 = z1 + z2 − z3 % z˙1 = −5z1 − z2 z˙2 = z1 − 3z2 % z˙1 = z1 − 2z2 − z3 z˙2 = −z1 + z2 + z3 z˙3 = z1 − z3 % x˙ 1 = −x1 − x2 + t2 x˙ 2 = − x2 − x3 + 2t x˙ 3 = − x3 + t

% z˙1 = −z1 + z2 z˙2 = − z2 + 4z3 z˙3 = z1 − 4z3 % y˙ 1 = y1 + 4y2 y˙ 2 = 2y1 + 3y2 + et x + x − y = et 8% t y − x + y = et t x y −2 + 4x − y = e−t % 5 t t x + 8x − 3y = 5 e−t t % x˙ = y y˙ = x + 2 sinh t % x ¨ − 4x˙ + 4x − y = 0 y¨ + 4y˙ − 25x + 4y = 16 et 6 ! ' 9 % x˙ 1 = 3x1 − x2 + 8 x˙ 2 = −x1 + 3x2 + 8 e3t x1 (0) = 4, x2 (0) = −6 % x˙ + 3x + y = 0 y˙ − x + y = 0

x(0) = 1 y(0) = 1

y˙ = f t, y(t) , y(t0 ) = y0

y˜ y(t)

! " # $ y(t) % & " !' ( $ f " ) y˜(t) = y0 +

y(t ˙ 0) y (k) (t0 ) (t − t0 ) + · · · + (t − t0 )k 1! k!

y(t ˙ 0 ) = f (t0 , y0 ) y¨(t0 ) = ft (t0 , y0 ) + fy (t0 , y0 )y(t ˙ 0) 2 y (t0 ) = ftt (t0 , y0 ) + 2fty (t0 , y0 )y(t ˙ 0 ) + fyy (t0 , y0 ) y(t ˙ 0) + fy (t0 , y0 )¨ y (t0 )

"

! " * %

" % " !' (% %" n + (n−1) y (n) = f t, y, y, ˙ . . . , y (n−1) , y(t0 ) = y0 , y(t ˙ 0 ) = y˙ 0 , . . . , y (n−1) (t0 ) = y0 && , y(t) - % " $ %% $ " .% y¨ − (t2 + 1)y = t 1 y(1) = 1 , y(1) ˙ = 2

y¨ − y˙ + et y = 0 y(0) = 0 , y(0) ˙ = −1 " y¨ + ty˙ + et y = t2 y(0) = 1 , y(0) ˙ =0 y˙ = t2 + y 2 , y(0) = 1

y

n / " 0%% tn

h > 0 tn = t0 + nh , n = 0, 1, . . . , N

! "# $ % & '( ) $ * ! +,+"# - % #

.$ ' ( +)/ $ ! #

yn+1 = yn +

h (k1 + 2k2 + 2k3 + k4 ) 6

'$ )

h h k1 = f (tn , yn ) k2 = f tn + , yn + k1 2 2 h h k3 = f tn + , yn + k2 k4 = f (tn + h, yn + hk3 ) 2 2

'$ ) '$ )

0 1 2 % 2 '% # )3 ! # yn+1 # ' - )$ % , M > 03 ( h # n & |yn − y(tn )| ≤ M h4

'$ )

'.$ / )$ % 4 2 ( 3 ( #

5 $ ! +,+"# & # 6 # #

6 # # y˙ = f t, y(t) , y(t0 ) = y 0 ! +,+"# y n+1 = y n +

h (k1 + 2k2 + 2k3 + k4 ) 6

k1 = f (tn , y n ) h h k 3 = f tn + , y n + k 2 2 2

k2 = f

h h tn + , y n + k 1 2 2

k4 = f (tn + h, y n + hk3 )

t = τ ! ! h " # ! ! h/2 $ y˙ − y = t , y(0) = 0 τ = 0, 4 ; h = 0, 2 y−2 $ y˙ = 2 , y(1) = 1 2t + t τ = 1, 2 ; h = 0, 1 !$ y˙ 1 = −y1 − y2 y˙ 2 = − y2

y1 (0) = 0 y2 (0) = 1

τ = 0, 4 ; h = 0, 2

$ y¨ + y˙ = t − 2 y(0) = 0 , y(0) ˙ = −3 τ = 0, 2 ; h = 0, 1

y˙ = −1 + 2t +

y2 2

(2 + t2 ) τ = 1 ; h = 0, 2

, y(0) = 2

v : D ⊆ Rn −→ Rn

n

f : D ⊆ Rn −→ R

n = 2 n = 3 n = 3 ⎛ ⎞ x ⎜ ⎟ x = ⎝y ⎠ z

⎛

⎞ ⎛ ⎞ P (x) P (x, y, z) ⎜ ⎟ ⎜ ⎟ v(x) = ⎝Q(x)⎠ = ⎝Q(x, y, z)⎠ R(x) R(x, y, z)

n = 2 ! " # ! $ % D &' v(x) !( ( ) ! * "'+ , v(x) % -( ' .( "' * (+ /+ !0+ 1 2 3 ( .

* ∇ =

∂ ∂ ∂ , , ∂x ∂y ∂z

Ì

2, f * grad f = ∇f = • grad f

∂f ∂f ∂f , , ∂x ∂y ∂z

Ì

!( -( (4 f

∂Q ∂R + + ) , v = (P , Q , R)Ì : div v = ∇ · v = ∂P ∂x ∂y ∂z • div v(x0 )

v(x) x0 • / div v(x) = 0+ ) , v

e x e y ez ∂ ∂ ∂ v = (P , Q , R)Ì : rot v = ∇ × v = ∂x ∂y ∂z P Q R = (Ry − Qz , Pz − Rx , Qx − Py )Ì

• rot v(x0 ) ! x0 " | rot v(x0 )| ! # • $ % rot v(x) = o & v ' 2

2

2

∂ ∂ ∂ + 2+ 2, ( Δ = ∇ · ∇ = ∂x 2 ∂y ∂z ∇(αf1 + βf2 ) = α ∇f1 + β ∇f2 ∇ · (α v 1 + β v 2 ) = α ∇ · v 1 + β ∇ · v 2 ∇ × (α v 1 + β v 2 ) = α ∇ × v 1 + β ∇ × v 2

⎫ ⎪ ⎬ ⎪ ⎭

Δ f = fxx + fyy + fzz

α, β ∈ R

Δ v = (Δ P , Δ Q , Δ R)Ì

' (rot v ≡ o)

) (div v ≡ 0)

*

+,

- ,

!

/ 0 '

1

.

v f ! ∇f (x) = v(x) 2

ϕ(x) := −f (x) . . . 3 v 4 ! ϕ 4

v P (x, y, z)

R(x, y, z)

Q(x, y, z)

v rot v = o

5 - 3 v # ' ! ( # f (x, y, z) = P (x, y, z) x = SP (x, y, z) +

g(y, z) - ./ 0

∂SP + gy (y, z) = Q(x, y, z) =⇒ gy (y, z) ∂y # g(y, z) = gy (y, z) y = Sgy (y, z) +

h(z) -./0

f (x, y, z) = SP (x, y, z) + Sgy (y, z) + h(z) ∂ SP (x, y, z) + Sgy (y, z) + h (z) = R(x, y, z) =⇒ h(z) ∂z ϕ(x, y, z) = −f (x, y, z) 1 SP (x, y, z) P (x, y, z) x Sgy (y, z) rot v = o Qx = Py ! " " # $ • % &"'( )

• % * +,- '(

. / v(x, y) = (x , y) v(x, y) = (−y , x) v(x, y) = (−x , −y) v(x, y) = (x , −y) ⎛ ⎞ x 1 ⎜ ⎟ v(x, y, z) = ⎝y ⎠ 2 x + y2 + z 2 z . v(x, y) = (x , y) v(x, y) = (−y , x) v(x, y) = (x , −y) # div rot v = 0 rot grad f = o div grad f = Δ f grad div v = Δ v + rot rot v f v 0.

*

1! grad(f · g) = f grad g + g grad f div(f v) = f div v + v · grad f rot(f v) = f rot v + grad f × v div(v × w) = w · rot v − v · rot w f ! g v ! w 0 * $ 1! grad |x|n = n

x , n ∈ Z, |x|2−n x = o

2 3 ( f (x) = 4x + 9y · z −1 . * ) P (1; 2; 3) 3 . &"'( 3 3 ) P 4 * ! 3 5

/

f (x) = xz + yz + zx + 10 (−1; 2; −3) f (x) = xy (2; 0)

Ì v(x) = x2 , −x−1 yz 2 , xz f (x) = 4x2 + 6y 2 + z 3 ! " P (1; −2; 4) # rot v $ !%

&' ( ) % f * +, v = (y + z , x + z , x + y)Ì v = 2xyz + y 2 z + yz 2 , x2 z +2xyz + xz 2 , x2 y + xy 2 + 2xyz

Ì

v = (yz , xz , xy)Ì - .

f = f (|x|)/ " v = xf (|x|) , x = o 0

1

, . ( * 2 , ϕ f (x) := −ϕ(x) 3

" Δ f , .4 5 6 ! 1 % - 7 , % # + ! - % 84 ! 7 ! 6

/ - 84% !

1

- 1 ! x, y, z % 9

"! 7 #

+ ! , # / x, y %- - 84 ! z % ( ! :&! " -% 84 ! ( ; m/ ! P (x, y, z) + / 1

1 ! F = (0 , 0 , −mg)Ì < rot F = o #% - m = %

#

# % -

x = (x, y, z)Ì ∈ R3

>

k : x = x(t) , t ∈ [α, β] x(α) . . . (

# k x(β) . . . - # k

(α < β)

?

=⇒ k

k

, ! 1 / ! 1 ?

−k : x = x(−t) , t ∈ [−β, −α]

" ,1 " k1 : x1 = x1 (t) , t ∈ [α1 , β1 ] ;

k2 : x2 = x2 (t) , t ∈ [α2 , β2 ]

x1 (β1 ) = x2 (α2 ) k = k1 ⊕ k2 k ! t " [α, β] # k % & & ' $

k1 , . . . , kn k = k1 ⊕ k2 ⊕ · · · ⊕ kn

• k( x(t) y(t) z(t) [α, β] k

t1 , t2 ∈ [α, β] , t1 = t2 =⇒ x(t1 ) = x(t2 ) ) x(α) = x(β) k

*** # *

• + , ( x(t) y(t) z(t) + , -. ' [α, β]

•

k( k - ' ' / 0 (x(t) ˙ = o)

< · · · > ( < · · · > • ( * •

" k % - '

1 s 2 3 ' ( k : x = x(s) , s ∈ [0 , L]

+,

+, 4

k 5 ' L 6

k

7 • D ⊆ R3 • k : x = x(t) , t ∈ [α, β] % - ' • k⊂D • f : D −→ R 8 # • v : D −→ R3 # +P (x) Q(x) R(x)

,

#

#β f (x) s =

k

f x(t) |x(t)| ˙ t

+9,

α

#

#β v(x) · x =

k

α

v x(t) · x(t) ˙ t

+:,

x = (x, y, z)Ì v(x) · x = P (x) x + Q(x) y + R(x) z ! " #"

$ % & ! ' ( &

" k ) ! * +

' (

" k , x = x(s) , s ∈ [0 , L] -

" [0 , L] : 0 = s0 < s1 < · · · < sn = L (j = 0, . . . , n) xj := x(sj ) Δsi := si − si−1 , Δxi := xi − xi−1 (i = 1, . . . , n) . + -

, n

# f (x) s ;

f (xi−1 ) Δsi ≈

i=1

n

# v(x) · x

v(xi−1 ) · Δxi ≈

i=1

k

k

/ *

+) ,

• , k 0 $ (x) ! " 1

(x) s 1 " k k 1 • , F (x) + + ! " k F (x) · x * ! .

0 %

k + +

F (x0 ) x1 1 Δx1

x0

#

k

Δxn : xn Δx2 z

xn−1 x2

R F (xn−1 )

#

f (x) s = k

F (x1 ) 7

#

# f (x) s

−k

v(x) · x = −

v(x) · x

−k

k

α, β ∈ R # # # α f1 (x) + β f2 (x) s = α f1 (x) s + β f2 (x) s k

#

k

k

α v 1 (x) + β v 2 (x) · x = α

#

k

#

v 1 (x) · x + β k

v 2 (x) · x k

k = k1 ⊕ k2 ⊕ · · · ⊕ kn # # # f (x) s = f (x) s + · · · + f (x) s k

k1

#

v(x) · x = k

kn

#

#

v(x) · x + · · · + k1

v(x) · x kn

• M ⊆ R

x ∈ M ! " x n

# M $

• % M ∈ Rn

& # ' M ( "! M !

• G ∈ Rn # 1 • )* "! k v(x) · x ! + ( , - !

! "! k

"!

! " v

: G −→ Rn (n = 2 , 3) v

G ⊆#R v(x) · x G n

k

k . v ϕ !

v

(/ "! / !

A B ! ( "!0 # v(x) · x = ϕ(A) − ϕ(B) 12

AB

#$

, , 3 G ⊆ Rn - , 45 6 7- ( # * 3 R2 # R3 0 R2

R3 G G

v : G −→ Rn

G ⊆ Rn ! !

Py = Qx rot v = o

A(4; 2) B(2; 0) # [(x + y) x − x y] k

OA

OBA # !"

"# $

(y x + x y) %&

k

% $ '& ( ) $ ' ) * + A(a; 0; 0), B(a; a; 0) # C(a; a; a) $

(y x + z y + x z) k

OC

OABC - " " && F = (x − y , x)Ì '

" ' F . & -/ 0 & x = ±a y = ±a !( , ' ) "# 1 (

2 " & ' 3&"

0 4 - " " ) && F = (P , Q)Ì = (x + y , 2x)Ì ' ( " ' F & ! ( "

/ . 1 ' x = a cos t, y = a sin t ( ! , ' ) "# 1

2 ( " & &"

!"

n=2 n=3

P = x + y , Q = x %& ! *

5 - " " ) " F = (y , a)Ì && -& ! , ' ) "# (

1

& m

, '( ' 1 -/ &"

x = a cos t, y = b sin t ) 6 - 7 ) " F = (x , y , z)Ì && (

! , ' ) "# 1

2

OABCO, O(0; 0; 0), A(0; a; 0), B(a; a; 0) C(a; a; a) 8 "' 9 #

2xy x + x2 y

AB

#

(cos 2y x − 2x sin 2y y)

AB

# tan y x +

x y cos2 y

AB

'& π π A 1;

B 2; 6 4 A(0; 1), B(2; 5) C(0; 5) # [(x + y) x − 2y y] k

AB

AB y = x2 + 1 ACB

k : x= t y = t2 z = t3 , 0≤t≤1 k 0 - % & " -

A(−a; 0) B(0; a) ! ( ! " F = (y , y − x)Ì # cx # v(x)· x "' 2 v(x) = |x|3 k AB c > 0 3 ! ( k AOB Ì AB t , x = cos t , sin t , x2 2π y =a− a 0 ≤ t ≤ 2π 6 Ì x = (1 , 0 , t) , 0 ≤ t ≤ 1 $ [y x + (x + y) y] %

2

0 k - & - t % && ' %

" - ! (

) * '" + & 4 #$ ! ( x , ! ( · x ' & " % - y = x2 y = 4 |x|3 k & . && - % " v = v(x) = (y , z , x)Ì # 5 6 ! ( # ! ( v(x) · x v(x) · x ' & " % k

- ! ( k &

k : x = a cos t y = a sin t z = bt , 0 ≤ t ≤ 2π / k 0 - & - %

" -

1 " v(x) =

Ì 2 y − z 3 , 2yz , −x#2 ! (

v(x) · x k

-

! ( k %

&

k

&& " 7% - v(x) = (−y , x , z)Ì v(x) = (x , y , z)Ì v(x) = (y , x , 0)Ì v(x) = (z , x , y)Ì 8 6 , (z −y) x+(x−z) y +(y −x) z k

' 3 ABC & +* A(a; 0; 0) B(0; a; 0) C(0; 0; a) - (a > 0)

x = (x, y, z)Ì ∈ R3

Ì F : x = x(u, v) = x(u, v), y(u, v), z(u, v) , (u, v) ∈ B ⊆ R2

B !" #$ ◦

B %

$& B • •

◦

x(u, v) ' B x(u, v) ( )

xu =

∂x(u, v) ∂y(u, v) ∂z(u, v) , , ∂u ∂u ∂u

Ì

, xv =

◦

∂x(u, v) ∂y(u, v) ∂z(u, v) , , ∂v ∂v ∂v

Ì

' (u, v) ∈ B

xu × xv = o

F F &*** !' + $ " , ! $-

•

. ! / )" & . ! 0 * !

1 * 2! #0* . 3 • F : x = x(u, v) , (u, v) ∈ B . ! ◦

• x0 = x(u0 , v0 ) , (u0 , v0 ) ∈ B ! ' F

$& F

x0

n(x0 ) = xu (u0 , v0 )×xv (u0 , v0 ) & n(x0 ) = xv (u0 , v0 )×xu (u0 , v0 )

3 " 4 )" . * . 1 5! " 6 $& & 7,4 8 1 9& $!&

& ) " 74 8 )

• G ⊆ R3

• F : x = x(u, v) , (u, v) ∈ B

• F ⊂G

v : G −→ R3

• f : G −→ R •

#

f (x) σ =

F

#

f x(u, v) n x(u, v) b

B

#

v(x) · σ =

F

#

v x(u, v) · n x(u, v) b

B

Ì σ = (σ1 , σ2 , σ3 ) v(x) · σ = P (x) σ1 + Q(x) σ2 + R(x) σ3

! "# # $ % & # " σ $ % n 0 '# σ σ (" σ = σ = n σ

|n|

) *# + • + ! F , (x) & 1 Q = F (x) σ Q - F • + ! v(x) # . 1 / & U = F v(x) · σ - v F & F 0 ' / U > 0& / 1 2 - 3 " * f (x) ≡ 1 A - F + # A= F

σ

F : z = z(x, y) , (x, y) ∈ B ! " F # $ |n| = (−zx , −zy , 1)Ì = zx2 + zy2 + 1 ! Φ(x, y, z) = 0 ! F : z = z(x, y)# (x, y) ∈ B Ì Φ Φ Φ2x + Φ2y + Φ2z x y |n| = , ,1 = Φz Φz |Φz |

% ! 2

! &' 2z = x # x ($ ) y = # y = 2 √ 2x# x = 2 2 * ! + z 2 = 2xy # ($ ) x = a y = a , x ≥ 0 y ≥ 0 " * 2

2

2

+ y + z = x #

&' x2 + y 2 = a2 ! az = xy #

&' x2 + y 2 = a2 + x2 + y 2 = z 2 #

&' z 2 = 2px , ! &' x2 +y 2 = a2 #

&' x2 + y 2 = a2 ! + x2 + y 2 + z 2 = a2 #

&' " x2 + y 2 ± ax = 0 ! -$$ x2 + y 2 = 2az #

&'" x2 + y 2 = 3a2 % ., $ " ! /

)$ # 0" 0◦ β # 12" $ - α * % 3, α = 30◦ # β = 60◦ # % v(x) · σ F

Ì v = x3 , y 3 , z 3 # *

F 4" 5 -' # ) x + y + z = a# x = 0# y = 0# z = 0 (a > 0) *

# % v(x) · σ # *

F

Ì

v = (3x, 3y, 3z) F " + # 0" +$$ 6

7 % " # v(x) · σ , 8" F

cx # c > 0# *$ |x|3 2 2 y (z + 3)2 x + + = 1 F : 16 16 25 9$ ($ n(x) $ * # 5 *

$, v(x) =

x = 4r sin ϑ cos ϕ y = 4r sin ϑ cos ϕ z = −3 + 5r cos ϑ cx v(x) = (c > 0) |x|3 #

v(x) · σ ! F F

" # $ %

(z − 3)2 1 2 = 1 x + y2 + 16 25 −2 ≤ z ≤ 3 & ' ( ) n(x) * ! + ! ,& - . / ) , . F

0 - #

v(x) · σ ( v = (x, y, z)Ì

F

1 ,2 + 3 x + y + z = a ,a > 0 (

4 # -

v(x) · σ ( v = (x2 , y 2 , z 2 )Ì

F

,2+ 5 x2 + y 2 + 2az = a2 ,a > 0 ( *! , x < 0 y > 0 z > 0

3 • G ⊆ R3 / • v : G −→ R3 6 * • B ⊂ G ( ,n(x) + F

& # # v(x) · σ = div v(x) b F

,7

B

-( . v(x) , /! 8( . 2 ) ,79 F B

B

- * R3 R2 * * B ⊂ G ) ! : ) k (

k B $ % 6 # P v(x) · x = (Qx − Py ) b , v = !" Q k

6

B

Z = v(x) · x # k

% B &

v

k $ %

'( )% x0 ∈ G ' {Bn } * Bn ⊂ G + , 1 -% x0 Fn . / Bn n 4 π Vn = 0 " 3 n3 # 1 div v(x)x=x = lim v(x) · σ 0 n−→∞ Vn Fn

1 • G ⊆ R3 2 • v : G −→ R3 3 0% • F ⊂ G n(x) # " ' (% * k • k F 4% 5#

# 6

rot v(x) · σ v(x) · x = k

6"

F

6" 5 7 8$ % v k 9 : 8'# rot v F 9 : 82 % F 9

'( )% x0 ∈ G + n0 |n0 | = 1"

1 x0 n π n0 kn ! Fn An = 2 n " # kn $ n0 # % &' 6 1 rot v x=x · n0 = lim v(x) · x 0 n−→∞ An

{Fn } Fn ⊂ G

kn

#

(

v(x) · σ

) * $ v = 2 2 2 Ì x ,y ,z F + ," ./ " a 0 1 + F

( + ," 2 3% 3' ) * 3 ( 2 &% ' ) *

x + y + z = a 2 4 x2 + 2y 2 − z 2 = 1 $ 0 z = 0 z = 3 5 " + 6/ F * $ ( #

rot v(x) · σ / F

Ì x , 0 , ln (1 + z) 1 + y2 4 6*

v(x) =

7 . 6* v = u1 grad u2 %u1 u2 *$ 8 * 6 '

# # # (u1 Δ u2 B

6*

+ grad u1#· grad u2 ) x y z (u1 grad u2 ) · σ

= F

.

# # # 6* (u1 Δ u2 − u2 Δ u1 ) x y z B#

(u1 grad u2 − u2 grad u1 ) · σ ?

= F

( 9 v = (1 , x)Ì k = k1 ⊕ k2 ⊕ k3 k1 : y = x2 , x ∈ [0 , 1] 2 1 1 −k2 : x− + (y − 1)2 = , 2 4 x ∈ [0 , 1] , y ≥ 1 −k3 : x = (0 , t)Ì , t ∈ [0 , 1] .

& ) : 2ε # = |x| %ε # *" ' # = : * E x 9$ : |x| (* = ε div E

B a # Q = b B

! " # $ !% v = (2x − y, −yz 2 , −y 2 z)Ì & ' F ( x2 + y 2 + z 2 = 1

y = 0& x + y = a ( 4 % 5

6 ! ' 2( 6 2 y x + (x + y)2 y k

7 4 % ABC 2%% A(a; 0)& B(a; a) C(0; a)

) ! " # $ !% v = (4x, −2y 2 , z 2 )Ì & ' B * x2 + y 2 = 4& z = 0 z = 3 ( '

! " 2( Ì $ !% 1 1 ,− $( 7 v = y x 4 % ABC 2%% A(1; 1)& B(2; 1) C(2; 2)

+ , -&# *

8 #

U = F

x v(x) · σ # v = |x|3

. * & /% &

(, U & ' 0 x2 +y 2 +z 2 = r2 ' ' , 6-& - % Z = 1 v(x) · x # v = 2 x + y2 k

×(−y, x, 0)Ì . #& z 1 &

(, Z & ' x2 + y 2 = r2 # ' '

! " 2( $ !% v = (x + y, −2x)Ì & ' k # & # 3 x = 0&

v(x) · σ & '( v = (x, y, z)Ì &

F

$( 9( 0 x2 + y 2 + z 2 = a2 % '

: #

v(x)· σ & '( v = (x2 , y 2 , z 2 )Ì &

F

$( 5 * ( 9% ; % ' & * x2 + y 2 + 2az = a2 & x = 0& y = 0& z = 0 ( ' v = (x, y, z)Ì * $ #!< 1 V = (x σ1 + y σ2 + z σ3 ) 3

*$

F

* ! 2 x2 y2 z2 + + = 1 a2 b2 c2 - = # &

#

6 (yz x + xz y + xy z)

k

! " # $% & OAB % ' O(0; 0; 0) A(1; 1; 0) B(1; 1; 1) ( ) " * " # 3

x σ1 + y 3 σ2 + z 3 σ3 F

+ ,# x2 + y 2 + z 2 = a2 - ' ! + ,#

. ) " * "

x(z − y) x + y(x − z) y +z(y − x) z

k

- $% & % ' A(a; 0; 0) B(0; a; 0) C(0; 0; a) * ' ! / v = grad u % " * ### # ∂u Δ u x y z = σ ∂n B

F

0u *-% 1

* 2 3

4 - 5 %

u = x2 + y 2 + z 2 + ,# F : x2 + y 2 + z 2 = a2

2x2 − 5x 2

2x − 17x + 109 667 − x+6

−3x2 + 2x − 1 2x − 3 − 2 x + 2x + 8

1 1 x+ 3 2

0, 25

0, 375x + 0, 45 2x2 − 0, 1x + 3 3 ax − bx2 − c −

a = q ; p = −q 2 − 1

(x + 3)2 (2x − 5)2 (x − 1)2 (x + 1)2 (1 + a)2 (1 − a) 8x2 (x4 + 1)

144

x10 , ax = 0 2a13 5x + 2y n a + 5b

|a| · b3 · b≥0

√ a8 b 9

12

|2x − 5y| √ n xn · x2 , x ≥ 0 √ 12 a11 , a ≥ 0 x3 + 1 x > 0 − x3 + 1 x < 0

|a| = 5|b| 5|x| = 2|y| r5 u3 (s + t)5 ru = 0, s = −t

x − y , x ≥ y

|x|3 |x − y| 3 √ 5−2 2x − 1 x ≥ 1 1 x < 1 23 36

√

17 − 12 2 √ 36 + 11 10 √43 4 − 15 √ 15 a6 b 5 ab √ 28 − 4 2 √ √ 2 + 6

1 √ 5 25 −2 16 16

lg 17 = 2, 57890 . . . lg 3 √ w· u

√ 4 v3 6 a2 − b2

0, 8

4(3 − a) 3+a

3 lg a + 4 lg |b| − lg c a > 0, b = 0, c > 0

1

− lg (a2 + b2 ) 2 a2 + b2 = 0

1 lg b 2 +2 lg (a + b)

4 lg a +

7 5 − lg a − lg b 2 2 a > 0, b > 0

−7; 9 2 5

− ; 3 6

−3; 3; −5; 5 2 2 − ; ; −1; 1 3 3

x1 = x2 = 0 x3; 4 −7; −3 7; 2

λ=

1 5

λ = n2 + n n = 1, 2, 3, 4, . . . a ≥

1 16

9

−4

1 7 27

5; −

30 127

−2; 2

! "# k = 0, ±1, ±2, . . . # # 2π 2π 2π 5π 193, 22135◦+k·360◦ , +k· + k · 2π

k · 3 9 3 6 346, 77865◦+k·360◦ π 7

k · 2π, (2k + 1) · π + k · 2π k · π 6 6 k · π 11π π 3π + k · 2π + k ·2π, + k ·π

π 6 2 3 (2k + 1) · π 16 2π + k · 2π +k·π 2π 2 3 (2k + 1)π, k · ◦ ◦ 3 48, 47127 + k · 360 π + k · π ◦ ◦ 4 172, 64082 +k·360 4 π+k·π 15 π π π π π +k·π k · π, + k · , + k · 4 1, 24905 + k · π 6 5 9 3 3π π +k·π 0, 24498 + k · π k · 2π, + k · 2π 4 2 5π π π 3π +k·2π +k·2π, 5π π + k · π, +k ·π 3 3 +k·2π +k·2π, 4 3 6 6 ◦ ◦ 2π % 46, 77865 + k · 360 π π +k·π $ + k · π, + k · 2π 133, 22135◦+k·360◦ 3 2 6

B : a2 + b2 < 2ab

a, b a2 − 2ab + b2 < 0 A : (a − b)2 < 0

a, b. A

! "# B A :

(a − b)2 ≥ 0

a, b. a2 − 2ab + b2 ≥ 0 B : a2 + b2 ≥ 2ab

a, b

3

! α 0 < cos α < 1

"

× (k + 2) k+1 # i P (k + 1) :

$ % i=1 (k + 1)(k + 2) = 2 & '

* P (n) ( n ≥ 1 + , a > 0 b > 0 b a =⇒ + ≥ 2 ) P (n) : b a n n(n + 1) # i= 2 i=1 !

P (n0 ) = P (1) : 1·2 . 1= 2 P (k) : k k(k + 1)

i= 2 i=1 k ≥ 1. $ k % i + (k + 1) i=1

k(k + 1) +(k+1) = 2 k+1 (k + 1) i= 2 i=1

/ A = {−2, 0, 1} B = {2} C = {1, 2, 3, 4, 5} D = {−3, −2, −1,

0, 1, 2}

(2x − y)(2x

+y) < 0 =⇒ (y > 2x ∧ y > −2x) ∨ (y < 2x

∧ y < −2x)

0 1 2 3 1 4 5

y ¾ ½ ¹½ ¹½ ¹¾

6 -x

½

A ∪ B = {−3, −1, 5} A ∩ B = {5} A \ B = {−3} B \ A = {−1} 7 A × B = (−3, −1), 8 (−3, 5), (5, 7 −1), (5, 5) B × A = (−1, −3), 8 (−1, 5), (5, −3), (5, 5)

! !

4 ( x ∈ B ⇒ x∈ A∩B ⇒ x ∈ A ⇒ B ⊆ A; x∈B⇒x∈A⇒ (x ∈ A ∧ x ∈ B) ⇒ x∈ A∩B ⇒ (1) B ⊆ A ∩ B (2) A ∩ B ⊆ B &

((

0 &'( & ( % A∩B =B

! " X Y # 5 + ,% m ↔ n2 + 1 ! $ ! ∧n ∈ {1, 2, . . . } % &'( X ⊆ Y & ( Y ⊆ X 0

) &'($ & ( X = Y * + 6 A ∼ B $ ,% &'( ! - . 1 2n ↔ n−1 x / 10 (n = 1, 2, 3, . . . ) ,% X $ %

% ! / ,% Y 0 7 ( F 1 %$ G

% ! & ( 12 ( G−1 1 %$ ( &'( ! F −1

x∈A∪B ⇒ x ∈ A ∪ B 8 ( L1 ∪ L2 ∪ L3 ⇒ (x ∈ A ∧ x ∈ B) ( L1 ∩ (L2 ∪ L3 ) ⇒ (x ∈ A ∧ x ∈ B) ( L1 ∩ L2 ∩ L3 ⇒x∈A∩B ⇒ '' ( F1 = (1) A ∪ B ⊆ A ∩ B {(a, α), (b, α), (c, α)} 3 % F2 = (2) A ∩ B ⊆ A ∪ B

{(a, α), (b, α), (c, β)} F3 = {(a, α), (b, β), (c, α)} F4 = {(a, β), (b, α), (c, α)} F5 = {(a, α), (b, β), (c, β)} F6 = {(a, β), (b, α), (c, β)} F7 = {(a, β), (b, β), (c, α)} F8 = {(a, β), (b, β), (c, β)}

( G1 = {(α, a), (β, a)} G2 = {(α, a), (β, b)} G3 = {(α, a), (β, c)} G4 = {(α, b), (β, a)} G5 = {(α, b), (β, b)} G6 = {(α, b), (β, c)} G7 = {(α, c), (β, a)} G8 = {(α, c), (β, b)} G9 = {(α, c), (β, c)} G10 = {(α, a)} G11 = {(α, b)} G12 = {(α, c)} G13 = {(β, a)} G14 = {(β, b)} G15 = {(β, c)}

x = −0, 7

x = 0, 75 x = log9 5

1 =⇒ log25 3 = 4x

a3 − b3 = (a − b)(a2 +ab + b2 )

107 333

131 990

13 5 [−2, 5), [1, 3) [−2, 1), [3, 5) (−5, 1), (−2, 0] (−5, 2] ∪ (0, 1), ∅ [−1, 5; 3, 5), (0, 5; 3, 5) [−1, 5; 0, 5], [3, 5; 4, 5)

{1}

{−1, 1}

lim M = 1 lim M = −1

{3} − 32 , 23

2 3 2 lim M = − 3 {(2, 4, 5)} lim M =

L 12 5 −∞, 11 (−∞, 2)

[4, +∞) (−2, 1) (−3, 2) (−∞, −2] ∪ [1, 2] (−∞, −5) 3 ∪ −3 , +∞ 4 43 , 17 14 "4 3 7 , 2 (−∞; 1, 2] ∪ (2, 5; +∞)

! (−∞, −3) ∪ [−2, 2) " 2; 49 " (−2, 8] L # $

%& ' (! ## x+3 ≥3 2x − 5 |x + 3| ≥3 ⇔ |2x − 5| ⇔ (∗) |x + 3| ≥ 3|2x

5 −5| ∧ x = 2 ' x ≤ −3 (∗) ⇔ −(x + 3) ≥ −3(2x − 5) 18 ⇔x≥ 5 L1 = ∅ ' −3 < x < 52 (∗) ⇔ x + 3 ≥ −3(2x − 5) 12 ⇔x≥ 7 " 12 5 , L2 = 7 2 ' 52 < x (∗) ⇔ x + 3 ≥ 3(2x − 5)

18 ⇔x≤ 55 18 5 L3 = , 2 5

) " & "

L=L 1 ∪ L 2 ∪ L3 " 12 5 , L= 7 2 5 18 5 ∪ , 2 5 " 3 95 − 2 , 2

(−∞, −1)∪(4, +∞) −∞, 85 " 32 , +∞ 1 , − 11 2 2 38 0, 23 23 , +∞ [3, 5]

(−∞, −2) 9 ∪ − , +∞ 5

y

6

½ ½

(−7, −5] ∪ [1, 13)

-x

1 · 2 · . . . · n · (n + 1) ×(n + 2) · (n + 3) n + 1 · 2 · 3 3n + 1 · 2 · 3 · . . . · n 3 · 1 · 2 · . . . · n − 5 1 · 2 · . . . · (2n − 4) ×(2n − 3) 1 ·1·2·. . .·(n−1)·n 3

(n + 2)! n! (n − 2)!(n + 1) n + 1 (n + 2)! 2n · (2n + 1) (n − k − 1) · (n − k) 1 (2n − 1) · 2n n+2 (n + 1)! 4n2 + 2n + 1 (2n + 1)! n+1 n!

4n2 − 2n + 1 (2n)!

21 3921225 0 −36

2n = (1 + 1)n 0n = (1 − 1)n 495a4 x−2

0

60

1 8 1155 2048 14 − 81 (−1)n · (2n − 1)! n!(n − 1)!

x1 = 10−2,5 ; x2 = 10

−

9x2 y −4 − 42xy −2 z +49z 2 a3 + 6a2 b + 12ab2 +8b3 32x−5 − 240x−2 +720x − 1080x4 +810x7 − 243x10

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 25 25 25−k (−1)k x k k=0 n n 2(n−k) k b a k k=0 n k n (−1) k k=0

1 + 0, 3 + 0, 03 +0, 001 = 1, 331 !

2

×an

−2kn

P3 · P5 = 720

VW(5) = 3125 5

P4 = 24 P4 − P3 = 18 PW(2,3,1) = 60 PW(2,1,2,1) − 5 (1,2,1)

×PW6

(2) 10 · PW4

25

(k)

C25 = 225 − 1

k=1

= 33 554 431

4

6 6

VW(15) = 1 073 741 824 b a e d c 54. V5(2) = 20 V4(1) = 4

1 28.

= 120 = 120

C49(6) = 13 983 816 CW(3) = 56 VW(3) = 216 6

6

P10 = 3 628 800 V10(5) = 30 240 P5 = 120

5

(i)

C5 = 31

i=1

5

(i)

VW2 = 62

i=1

C25(k)

25 = k

C20(15) · C25(20)

= 823 727 520

VW(10) = 1024 PW(7,3) = 120 2

10

10

(i,10−i)

PW10

= 176

i=7

968 N2 = C97(3) · C3(2) S =

= 442 320

(5) C100

= 75 287 520 (5) S1 = S − C97 = 10 841 496

12 − 5 ; 6 − 9 41 − 3 ; 1 − 3 3 41 + 13+39 ; 130 130

− 2 + 14 ;

83 ; ;

73 55 48 ; + 36 73 73 55 4 − ; 1 36 3 4, 52993 + 2, 01764 − 0, 66622 −0, 98236 4, 24264 + 4, 24264 0, 64395 − 0, 17255 5, 79555 + 1, 55291 1, 06066 − 1, 06066

−5 + 2 ; −1 + 6 1 7 14 − 2 ; − − 4 4

2 14 + 25 25 3, 98174 + 3, 93185 − 2, 94646 −0, 06815 − 2, 07055 +7, 72741 0, 35355 + 0, 35355 − 0, 51764 −1, 93185

5 4 ;y= x = − 11 11

3 u = −2; x = 2 1 y = 2; z = − 2

−|xy|

−1

− 1 + √ −6 − 12 6 − e−π/2 1 + 2 − 31 − 17 3 √

15 + 2 5 6 √ 5 2 + 18

6

0 − 5 = 5 e (−π/2)

= 5 cos (−90◦ )

+ sin (−90)

9 + 0 = 9 e ·0 = 9(cos 0◦ + sin 0◦ ) π/2

0 + 3 = 3 e = 3(cos 90◦ + sin 90◦ ) −8 + 0 · = 8 e π = 8(cos 180◦ + sin 180◦ ) √ 1 + 3 = 2 e π/3 = 2(cos 60◦ + sin 60◦ ) 7√ 7 − 3+

2 2 = 7 e 5π/6 = 7(cos 150◦ + sin 150◦ ) √ 1 3 − −

2 2 (−2π/3) =e = cos (−120◦) + sin (−120◦ ) −3, 00908 −3, 99318

= 5 e (−2,21657) = 5 cos (−127◦)

+ sin (−127◦)

4 −√ 6

= 52 e√(−0,98279) = 52 cos (−56, 30993◦)+

sin (−56, 30993◦) 0 + 2 = 2 e π/2 = 2(cos 90◦ + sin 90◦ ) 0, 5 + √ · 0, 5 2 π/4 = e √2 2 (cos 45◦ = 2 + sin 45◦ ) −0, 5 + 8

=

·1,63322 64, 25 e = 64, 25( cos 93, 57633◦ + sin 93, 57633◦)

√ 3 − = 2 e (−π/6) = 2 cos (−30◦ )

+ sin (−30◦ ) −2 + 2 √ =√ 2 2 e 3π/4 = 2 2(cos 135◦ + sin 135◦) √ − 3− √ 3

=√ 2 3 e (−2π/3) = 2 3 cos (−120◦)

+ sin (−120◦) 1√ 1 3−

2 2 = e (−π/6) = cos (−30◦ ) + sin (−30◦ ) 0 − 9 = 9 e (−π/2) = 9 cos (−90◦ )

+ sin (−90◦ )

−1 + 0 = e π = cos 180◦ + sin 180◦ − e2 +0 = e2 e π = e2 (cos 180◦ + sin 180◦) π 0 + e4 · = e4 · e 2 = e4 (cos 90◦ + sin 90◦ ) Re z = 2 Im z = 1 3 Re z = − 2 3 Im z = 2

Re z = 0 Im z = −2 Re z = 16 Im z = 0

Re z = 0 √ Im z = 1024 3 Re z = −64 Im z = 0 √

1 √ 2+ 6 2 1 √ 2 z2 = − 2 √

+ 6

! z1 =

z1 = 1, 62894 +0, 52017

z2 = −1, 26495 +1, 15061

z3 = −0, 36398 −1, 67079

z1 = 1, 40211 +0, 18459

z2 = −0, 18459 +1, 40211

z3 = −1, 40211 −0, 18459

z4 = 0, 18459 −1, 40211

√ √ 2+ √ z1 = √ 2 z2 = −√2 + √2 z3 = − √ 2 −√ 2 z4 = 2 − 2 z1 = 0, 89101 +0, 45399

z2 = −0, 15643 +0, 98769

z3 = −0, 98769 +0, 15643

z4 = −0, 45399 −0, 89101

z5 = 0, 70711 −0, 70711

√

1 3 " 1; + 2 2

√3; −1 1 + 2 2 1 √3 − − 2 2 1 √3 − 2 2 −

0, 80902 + 0, 58779 − 0, 30902 +0, 95106 − 1; −0, 30902 −0, 95106

0, 80902 − 0, 58779

+0, 21275 − 0, 21275 −1, 06955 1, 06955 − 0, 21275

0, 28485 + 1, 75532 − 0, 28485 −1, 75532

0, 29863 + 1, 50405

− 1, 45186 −049340 1, 15323 − 1, 01065 0, 21275 + 1, 06955 − 1, 06955

0, 38268 + 0, 92388 − 0, 92388 +0, 38268 − 0, 38268 −0, 92388 0, 92388 − 0, 38268

x(2x3 +x2 +2x+1) = 0 x1 = 0 2x3 + x2 + 2x + 1 = 0

! " # $"" % & ' ! ( "

an−1 = a23 = 22 : n 3 3 2 x + 22 x2 + 22 · 2x +22 = 0

)

y = an x = a3 x = 2x : y 3 + y 2 + 4y + 4 = 0

$"" * " b0 = 4 = 22 "+ #

±1, ±2, ±4 , ' ,

y1 = −1& 1 1 4 4 −1 0 −4 −1 1 0 4 0 y2 + 4 = 0 y2 = 2 ; y3 = −2 xk = yk 2− 1 + k = 2, 3, 4 - " x2 , x3 , x4 : 1 x1 = 0; x2 = − ; 2 x3 = ; x4 = −

x1 x2 + x1 x3 +x2 x3 = −16 = b1 x1 x2 x3 = 80 = −b0

x1 = 2;

x2 = 3 x3,4 = 3 ± 2 x1 + x2 + x3 + x4 = 11 = −b3 x1 x2 + x1 x3 + x1 x4 +x2 x3 + x2 x4 +x3 x4 = 49 = b2 x1 x2 x3 + x1 x2 x4 +x1 x3 x4 + x2 x3 x4 = 101 = −b1 x1 x2 x3 x4 = 78 = b0

"(&

y 3 + y 2 + 4y + 4 = 0

(y 2 + 4)(y + 1) = 0

y1 = −1; y2,3 = ±2

x1 = −3 + x2 = −2 +

x1 + x2 = −5 + 2 = −b1 x1 x2 = 5 − 5 = b0 x1 = 2 ; x2 = −1 x1,2 = ±4 x3 = −5 x1 + x2 + x3 = −5 = −b2

x1 = x2 = x3 = 1 x1 = x2 = −1 x3 = x4 = 2

* x1,2 = 1 ± 2 x3,4 = −2 ± x1 = 0; x2,3 = 1; x4,5 = −2

x1 = −2; x2,3 = 3 x1 = 1; x2 = −2

x3 = −3; x4 = 7 √ x1,2 = 1 ± √ 3 x3,4 = ± 2

x1 = 1; x2√= 3 x3,4 = 1 ± 2

x1,2 = ±1 x3,4 = 1 ± 2

x1 = 1; x2 = x3 = x4 = −1 2x = y 1 1 x1 = − ; x2 = 2 2 3 5 x3 = ; x4 = 2 2 12x = y 1 x1 = 0; x2 = − 3 1 x3 = x4 = 2 x5,6 = −1 ±

y = 2x/3 x1 = 0; x2 = 3 y = log√ x5 5 x1 = 5; x2 = 5

3 2 −3 2 9 6 9 3 2 3 11 6 3 2 9

x2 (x − 2)2 (x2 + 2x +2) 2(x − 1)(x − 2)(x +3)(x2 + x + 1) (x − 3)(x + 2)2 (x2 +2x + 10) (x + √ 1)(x − 9)2 (x2 −2 2 + √ 4) · (x2 +2 2x + 4)

a1 = 0, a2 = 2, a3 = −2 λ2 + λ + 1 2 |a| = 70, cos α = 7 6 3 cos β = , cos γ = − 7 7 √ −−→ −−→ −−→ |AB | = |AC | = |BC | = 2 2 √ ! P (−4; 4; 4 2)

" D(9; −5; 6) −−→ AC = (−2, 6, −10)Ì −−→ BD = (14, −8, 8)Ì √ −−→ −−→ |AC | = 2 35, |BD | = 18

P3 (x) = 2(x − 3)3 +9(x − 3)2 +11(x − 3) + 9

−(x + 2)5 + 10(x +2)4 − 39(x + 2)3 +73(x + 2)2 − 64(x +2) + 21

2 −9 11 3 6 −9 6 3 2 −3

2x3 − 6x2 + 2x + 8 x = −1

2 9

3 2 1 Ì a0 = − √ , √ , − √ 14 14 14 5 3 2 Ì 0 b = √ , −√ , √ 38 38 38 a + b = (2, −1, 1)Ì b − a = (8, −5, 3)Ì − 2a + 3b = (21, −13, 8)Ì

3 2

x =

5 3

,−

35 10 Ì , 3 3

rA + rB + rC 3 S(3; 1; 3)

# rS =

5 11 13 1 13 17 ; ; ,D ; ; 3 3 3 3 3 3 c + λb v = 1+λ 17 P − ; 0; 0 10 √ √ √ a = 11, b = 5, c = 2 3 3 3 3 , Mb 1; ; 2 Ma ; ; 2 2 2 2 3 1 Mc ; 1; 2 2 1 1 1 Ì 1√ , , m = , |m| = 3 2 2 2 2

$ C

P (16; −5; 0)

8 (−16, 20, −8)Ì

ϕ ≈ 73, 39845◦ a · b = 0 λ = 4 15 49 108

[a, b, c] = 0

a = −b + 3c 76 0

!"

20 , 20 7 3 547 s0 = ± √1

√ 2 6

11

(1, −3, 1)Ì

r = (−2, 3, −5)Ì

+λ(3, −7, 4)Ì r = (3, −2, 1)Ì +λ(−2, 0, 1)Ì − ∞ < λ < +∞

18 23 1 ; ;− 22 √ 11 22 1826 22

F

√

33 A " &' S(4; −7; −3) (λ = −3) B ϕ ≈ 29, 62048◦ S(1; 2; 1) r = (2, −3, 4)Ì +λ(1, 0, 2)Ì !""" −∞<λ<∞

( λ = ±1, P1 (2; 0; 3) "" ) P2 (0; −4; 7) ( 11 1 λ = ±6, P3 (7; 10; −7) S 3 ; 3 ; 3 P4 (−5; −14; 17) S(−1; 3; −2), ϕ = π F (−1; 1; 2) 2

r = (3, −1, 2)Ì

+λ(−2, 3, −3)Ì − ∞ < λ < +∞

(1, 0, 1)Ì 1 4 1 − , ,− 3 3 3

Ì

43 (−36, 24, −16)Ì −→ −−→ −−→ AB , AC , AD ] [−

# [a, b, c] = 0 $ % !

ϕ ≈ 109, 87687◦ −33

Ì l = (1, √ 0, 1) |l| = 2 √ AΔ =√7 3 a = 2 14

=0 −−→ −−→ AB = −CD −−→ −−→ BC = −DA −−→ |AB | = . . . √ −−→ = |DA | = 14 −−→ −−→ AC · BD = 0

√ √ b = 42, c = 14 α = 90◦ β = 60◦ γ = 30◦

*+,

r = (1, 2, 3)Ì 5 Ì 1 +λ −2, − , − 2 2 r = (−2, 3, 1)Ì +μ(3, −1, 2)Ì r = (2, −3, −1)Ì +γ(−1, 5, 4)Ì − ∞ < λ, μ, γ < +∞

-

" ,

Ì 1 5 r = λ 2, , 2 2 3 1 Ì ,− ,1 r= 2 2 +μ(3, −1, 2)Ì 1 5 Ì r= − , ,2 2 2

+γ(1, −5, −4)Ì − ∞ < λ, μ, γ < +∞ Ì r = (1, 2, 3) √ √ +λ(−3 √3 + 1, Ì 3− 5, −2 3− 4) r = (−2, 3, 1)Ì

+μ(5, −4, 1)Ì r = (2, −3,√−1)Ì √+γ(−2√3 − 1, Ì 3 3 + 5, 3 + 4) − ∞ < λ, μ, γ < +∞ 1 2 ; ;1 S 3 3 5√ MI 3 − 3; −1 3

1√ 4√ 3; 2 − 3 3 3 MU (0; 0;0) √ ρ= 7 2− 3 √ r = 14

+

r = (1, 2, 3)Ì +λ(1, 1, −1)Ì − ∞ < λ < +∞

E1 : 15x − 24y + 10z = −2 E2 : 15x − 24y + 10z = −2

√ 3

41 7

(4, 3, 2)Ì · r = 27 Ì 8 , 0, 0 3 +λ(0, 5, 2)Ì +μ(−2, −3, 0)Ì r = (0, −13, 0)Ì +λ(0, 7, 1)Ì +μ(−1, 0, 0)Ì − ∞ < λ, μ < +∞

r =

ϕ ≈ 40, 36759◦ √ 6

A = 12 S(−7; −5; −11) S(3; 1; 3)

r = (1, 1, 0)Ì x − 2y − 3z = 4 +λ(−2, 5, 3)Ì 2x + 3y + 4z = 3 ϕ ≈ 72, 02472◦ Ì

S(1; −1; 2) 1 r = − , −2, 0 x − 8y + 9z = 21 2 +λ(3, 8, 5)Ì 2x + 2y + z = 20 ϕ ≈ 54, 73561◦ 2x + 2y + z = −4

r = (1, 1, 0)Ì +λ(9, −8, −11)Ì 7x + 14y + 24 = 0 ϕ = 90◦ 9x + 4y + 9z = 20 − ∞ < λ < +∞ √ 53 178 (−1, 1, −1)Ì · r = 1 178 r = (2, 1, −2)Ì Ì

r = (−1, 5, 1)Ì +λ(−1, 1, −1) +ν(2, −9, 2)Ì − ∞ < λ < +∞ − ∞ < ν < +∞ (2, 12, 17)Ì · r = 13 S(1; −4; 3) 5 xF = zF = , A(1; 2; 1) 2 yF = −1 r = (1, 2, 1)Ì P2 (0; 9; −6) +μ(1, 1, −4)Ì 0 ≤ μ < +∞ (2, 7, 5)Ì · r = 33 √

n = 2/2 S(−13; 7; 2)

−4 −2 (−3) −16 12 1 −2 −10 5 12 1 −4 +9 = 56 5 −16

10 9 − (+1) −16 12 −3 9 +(−4) 5 12 −3 10 −(−2) 5 −16 = 56 1 −2 −(+10) 5 12 −3 9 +(−4) 5 12 −3 9 −(−16) 1 −2 = 56 1 −4 9 5 −16 −3 10 −(−2) 5 −16

−3 10 +12 1 −4 = 56

2A + 7C, 2AB BA, CB 5A − 3B =

1 −4 −2 − 0 −2 3 0 4 22

2

3 1 −4 −2 − 0 −2 3 = 0 0 28 (−5) −1 · (−2) · 28 = 56 26 !" −38 # $ 0 −2 3

2a 1 1 −4 −2 = 0 4 22 −10 −2 3 (−1) = 56 −4a3 4 22 144

(x− y)(x− z)(y − z) 1 1 −4 −2 35 − −3 10 9 86 5 −16 12 470 3 % x1 = 2, x2 = 3

x1 = 0, x2 = −2

x1 = 1, x2 = 2 (−5) x3 = 3

& '

$

% 8 3 0 1 25 −11 (A + $B)C = % 14 7 −14 29 23 −11

2AÌ⎛ B= ⎞ 10 8 8 ⎜ ⎟ ⎝ 24 24 20 ⎠ −6 0 −4 3B Ì A =

⎛

⎞ 15 36 −9 ⎜ ⎟ ⎝12 36 0 ⎠ 12 30 −6 − BC = % $ −4 −1 7 −10 −15 −5

$ % dA −1 Ad = −4 $ % −6 Bd = 12 ⎛ ⎞ 2 ⎜ ⎟ Cd = ⎝−3⎠ 7 dÌ C = (3, 14, 12) dÌ AÌ = (Ad)Ì = (−1, −4)

AA, dd, dÌ dÌ AÌ A =

⎛

⎞ 5 13 −2 ⎜ ⎟ ⎝ 13 34 −5⎠ −2 −5$ 1 % 10 17 Ì AA = 17 30 C ·⎛ C= ⎞ 23 12 −14 ⎟ ⎜ ⎝ 12 1 −12⎠ −2 12 11 Ì= C · C ⎞ ⎛ 26 13 1 ⎟ ⎜ ⎝13 9 −7⎠ 1 −7 26 C ÌC Ì = (CC)Ì = ⎞ ⎛ 23 12 −2 ⎟ ⎜ ⎝ 12 1 12 ⎠ −14 −12 d · d⎛Ì = 4 −2 ⎜ ⎝−2 1 6 −3

11 ⎞ 6 ⎟ −3⎠ 9

dÌ d = 14 ⎛ ⎞ −6 −9 −3 ⎜ ⎟ ⎝ 8 12 4 ⎠ −12 −18 −6 $ % 0 0

0 0

45 ⎛ ⎞ −83 66 ⎜ ⎟ ⎝−46 31⎠ −34 19 abc $ 2 0 $ 2 0

% $ % 0 −2 0 , 3 0 3 % % $ −2 0 0 , 0 −3 −3

B (2, 4) C (3, 4) 3150 4200 2550 6000

! ! ! !

T1 T2 T3 T4

" # $ % & $ ' ! () * + & , 1 b = a1 + a2 + 2a3 2 - *. (3, −1, 1)Ì +(−1, 3, 1)Ì −2(1, 1, 1)Ì = o (1, 4, −7, 7)Ì (70, 40, −20, −16)Ì 1 (51, 26, 18 , 2

1 −11 )Ì 2 Ì 1

x = − , 1, 3, 3 2 8 7 x= − ,− , 3 3 11 Ì 16 − ,− 3 3 23 29 x= − ,− , 4 8 27 9 Ì , 8 8 - *. 2(1, 1, −1)Ì +3(5, 0, 3)Ì −(−2, 1, −3)Ì = x

/ !

& S % ( 0 !1 2, %%

3 α1 = 1, α2 = 0 α3 = −1, α4 = 1 α5 = −1 α1 = 3, α2 = 2 α3 = −1, α4 = −5 α5 = −1 α1 = −5, α2 = 9 α3 = −7, α4 = 5 α5 = −3

A ⎞ ⎛ 1 ⎜ ⎜−2 ⎜ ⎜ ⎝−3 0

1 7 6 3

−4 5 9 −1

2 −4 5 0

3 ← ⎟ 0⎟ ⎟ ⎟ −3⎠ 2

← 2! " 3! # # " $ ⎞ ⎛ 1 1 ⎜ ⎜0 9 ⎜ ⎜ ⎝0 9 0 3

−4

2 3

−3 0 6 −3 11 6 −1 0 2

⎟ ⎟ ⎟ ⎟ ⎠

%# # $ & B ⎞ ⎛ 1 1 ⎜ ⎜0 3 ⎜ ⎜ ⎝0 9 0 3 ⎛ 1 1 ⎜ ⎜0 3 ⎜ ⎜ ⎝0 0 0 0

−4 −1 −3 −1

−4 −1 0 0

# ' # ( B ⎞ ⎛ 1 1 −4 2 3 ⎜ ⎟ ⎜ 0 3 −1 0 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 11 0 0 ⎠ 0 0 0 0 0

r(A) = 3 )# *+## , ! "#! # " ! # # - " # . " ###! - $ ! # ! # A:

2 −4 5 0

2 3

⎟ 0 2⎟ ⎟← ⎟ 11 6 ⎠ 0 2 ⎞ 2 3 ⎟ 0 2⎟ ⎟ ⎟ 11 0 ⎠← 0 0

1 −4 3 7 5 −2 0 6 9 −3 −3 3 −1 0 2

0 9 −3 11 9 −3 0 3 ! 0 0 0

6 6 2

r(A) = 3

2 3 2 2 4 3 r(A)⎧ = ⎨2

+# λ = 0 ⎩3 +# λ = 0 r(B) = 3 +# λ∈R

# / (r = m = 4)

# /

(r = 3 < m = 4)

0

0 0

r=2<m=3 B = {a2, a3} a1 = −3a3 r=3<m=4

B = {a1 , a2 , a3 } a4 = a2 + a3

x1 = −b 2 x2 = − a 3 x1 = −1; x2 = 2 x3 = 4

x1 = 1; x2 = 2 x3 = −2; x4 = −1 x1 = 2; x2 = −2 x3 = 1; x4 = −1

x2 − 5x + 3 −2x3 + x2 − x + 4

5 A I6 = 19

! " # $ % & % ' &

( ) * % +& ! ,+ -, .

/ 0 #, 1,* +

* / " + . 0 # * x1 x2 x3 x4 1 4 6 17 8 −20 4 0 0 −1 2 3 7 7 −4 2 3 8 5 −8 0 3 −6 −12 0 2 4 0 1 −2 −4 0 0

0 0

0 0

2

' & &

( # & % 3#

& %

(#

& %

r(A) = r(B) =2=r 0 + 4 (n − r) = 2 − 2 = 2 # " + # $ x2 x4

* / . 0 5 2x1 + 7x3 = −4 − 3x2 − 7x4 − x3 = 4 − 2x4 ) $ 5 3 x1 = 12 − x2 2 21 − x4 2 x3 = −4 + 2x4 6, # " + & & *$7 x2 = 2t1 , x4 = 2t2 $ 5 x1 = 12 − 3t1 − 21t2 x2 = 2t1 x3 = −4 + 4t2 x4 = 2t2 −∞ < t1 , t2 < +∞ 5 x= (12, 0, −4, 0)Ì +

+ t1 (−3, 2, 0, 0)Ì +t2 (−21, 0, 4, 2)Ì

+ x1 x2 x3 x4 4 6 17 0 0 −1 3 7 2 3 8

1

8 −20 2 5 7 −4 5 −8

0 3 −6 −12 0 2 5 0 1 −2 −4 0 0

0 0

3 1

r(A) = 2, r(B) = 3 r(A) = r(B) 0 +

x = (1, −1, 0)Ì +t(0, 1, 1)Ì + + x = (1, 2, 3)Ì x1 = x2 = x3 = 1 Ì 4 5 , ,0

x = 3 3 Ì 2 1 +t − , − , 1 3 3 # x1 = 2 − t1 x2 = t1 x3 = 2 − t2 x4 = t2 x1 = x2 = x3 = 0 (! x = t(1, 3, 2, −1)Ì x = t1 (1, −4, 0, −13, 5)Ì +t2 (0, −2, 1, −6, 4)Ì x1 = 21 − 2t1 − 6t2

−12t3 x2 = t1 x3 = −20 + 7t2 +14t3 x4 = 15 − 5t2 − 9t3 x5 = t2 , x6 = t3

x3 x4 x5

3 x2 = − + 5t1 4 +23t2 x3 = 12t1 x4 = 12t2

x = (−1, 1, 2)Ì

x = (1, 1, 1)Ì x1 = t1 − t2 x2

−13t1 −7t2 x1 = 11 4

x=

7 5 4 , −2, − , 3 3 3

Ì

y = t, z = t

r(A) = 3, r(B) = 4 r(A) = r(B)

=r=n=3

r(A) = r(B)

x1 = 3 + t1

9 = − 2t1 + 2t2 2 = t1 3 = − − 2t2 2 = t2

=r=2 n−r =4−2=2

x2 = −2 + t2 x3 = t1 , x4 = t2

x1 = 2t

x2 = −1 + t x3 = 3 − 2t, x4 = t

λ1 = 0 :

x = (3, 2, 0, 3)Ì

x = 3, y = z = 0 λ2 = 0 : x = 3 − 2t

+t(−2, −1, 1, 0)Ì t1 = 2 : x = (−1, 0, 2, 3)Ì t2 = 6 : x = (−9, −4, 6, 3)Ì

X = −C · (A + B)−1 A, B ! " (n, n) C ! " (m, n) det(A + B) = 0

A23 = 6, A31 = 3 A32 = −3 A33 = −4 = A−1⎛ ⎞ −1 4 3 1 ⎜ ⎟ · ⎝−1 5 3 ⎠ −1 1 −6 −4 ⎞ ⎛ 1 −4 −3 ⎟ ⎜ = ⎝ 1 −5 −3⎠

S3 x1 x2 y2 y4 y1 − 12

−1 x3 0 −1 y3 − 72 −4 x4

9 2

0 1 1

3 −1

1 2

1 3 2 − 32

det A= X = (A + 2E)−1

×(5C + 3D) · B −1

X = 2A−1 C(B + E)−1

S2 x1 x2 y2 x4 y1 1

0 − 13 − 13 1 3

− 23

x3 3 1 y3 1 −1 0 −1 y4 3 2 − 23 − 23

2 2 3 1 −1 0 −1 2 1 = −1 = 0 #A $ −1 0 A11 = = −1 2 1 1 0 A12 = =1 −1 1 1 −1 A13 = =1 −1 2 A21 = −4, A22 = 5

−1 6

4

S1 x1 x2 x3 y1 2 2 y2 1 −1 y3 −1 2

3 0 1

S2 x1 y2 x3 y1 x2 y3

4 −2 1 −1 1 −2

3 0 1

S3 x1 y2 y3 y1 1 4 3 x2 1 −1 0 x3 −1 2 1

X = 2B(B − 2A +E)−1 ⎛ ⎞ −5 −1 4 ⎜ ⎟ X = ⎝ 8 2 −2⎠ −3 1

S4 y 1 y 2 y 3 x1 1 −4 −3 x2 1 −5 −3 x3 −1 6 4 S4 A−1

A−1 ⎞ ⎛ 5 4 −1 1 ⎜ ⎟ = · ⎝10 12 −3⎠ 5 0 1 1 X = ⎞ ⎛ −4 3, 8 −0, 6 ⎟ ⎜ ⎝−3 3, 84 −4, 88⎠ −3 0, 64 −2, 48

$ 5 −2 $ −3 = −2

A−1 = B −1 C

−1

4 % −2 1 % 4 3

= ⎛

⎞ 1 0 0 ⎟ ⎜ ⎝ −3 1 0⎠ −19 5 1

D−1 = ⎞ ⎛ −2 2 −1 ⎟ ⎜ ⎝ 1 −1 1 ⎠ 4 −3 1

G−1 = ⎛ ⎞ 3 −15 −8 1 ⎜ ⎟ · ⎝−1 1 1⎠ 3 −1 4 2 −1 K = ⎛ 1 1 ⎜ ⎜0 1 ⎜ ⎜ ⎝−1 −1 2 1

⎞ −2 −4 ⎟ 0 −1 ⎟ ⎟ ⎟ 3 6 ⎠ −6 −10

L−1 =⎛

⎞ 1 0 0 ⎜8 ⎟ ⎜ ⎟ ⎜0 −1 0⎟ ⎜ 3 ⎟ ⎝ 1⎠ 0 0 2

$

% 1 0 A = = AÌ 0 1 $ % 1, 5 c= −2, 5 A = E2 ! " (E Ì · E = E) det A = 1 #

! $ % ! & ! ' % $ % 1 4 3 A = 5 −3 4 $ % 1 4 −3 Ì A = 5 3 4

det A = 1 AÌ A = E 2 A ! " #

!

( & ! ' % ) A−1 = AÌ % *! 4 3 x1 = y1 − y2 5 5 3 4 x2 = y1 + y2 5 5 +,

( ! ! & ! '

) ϕ ! - ! % . ' / # ! + y1 = x1 cos ϕ +x2 sin ϕ y2 = −x1 sin ϕ +x2 cos ϕ $ % 1 −4 3 A = 5 3 4 det A = $ −1 % 1 −4 3 Ì A = 5 3 4

AÌ A = E 2

! 90

! 48 · 90 = 4320 z = Ax " ! # $ % A−1 = AÌ AÌ A = E (2,2) ⎛ ⎞ 2 −1 −1 1⎜ ⎟ & ' % y = z + c P = ⎝−1 2 −1⎠ det A = ±1 3 c = (1, 5; −2, 5)Ì AÌ A = E −1 −1 2

( ! ) *

6 − λ 2 2 2 3 − λ −4 2 −4 3 − λ =0

λ3 − 12λ2 + 21λ +98 = 0

+, - $

λ1 = λ2 = 7 λ3 = −2

! . ' , - / % λ1 = λ2 = 7 : x1 x2 x3 1 −1 2 2 0 2 −4 −4 0 2 −4 −4 0

+, %

x1 = 2t1 + 2t2 x2 = t1 , x3 = t2 −∞ < t1 , t2 < +∞

'$ 0 / - / %

x1 = (2t1 + 2t2 )e1 +t1 e2 + t2 e3

0 ' . 1 t1 = 0 t2 = 1 - / x1 = 2e1 + e3 2 $ t1 t2 $ 3 x1 x2 = (2t1 + 2t2 )e1 +t1 e2 + t2 e3

3

2(2t1 + 2t2 ) + 0 · t1 +1 · t2 = 0

3 ' .%

t1 = 5, t2 = −4 : x2 = 2e1 + 5e2 −4e3 λ3 = −2 : x1 x2 x3 1 8 2 2 0 2 5 −4 0 2 −4 5 0

+, %

x1 = t, x2 = −2t x3 = −2t − ∞ < t < +∞

0 / - / % x3 = te1 − 2te2 −2te3

0 ' . 1 t = 1 - / x3 = e1 − 2e2 − 2e3

0 %

xi |xi | (i = 1, 2, 3)

ri =

%

1 r1 = √ (2, 0, 1)Ì 5 r2 = 1 √ (2, 5, −4)Ì 3 5 1 r3 = (1, −2, −2)Ì 3

R ⎛=2

√ 5

⎜ ⎜ 0 ⎝

√1 5

2 √ 3√ 5 5 3 4 − 3√ 5

⎞

1 3 ⎟ − 23 ⎟ ⎠ 2 −3

4 det R = −1

( 0 %

˜= R ⎛

2 √ ⎜ 3√55 ⎜ ⎝ 3 4 − 3√ 5

√2 5

0 √1 5

⎞

1 3 ⎟ − 32 ⎟ ⎠ 2 −3

˜ = 1 det R λ1 = λ2 = 7 r˜1 = 1 √ (2, 5, −4)Ì 3 5 1 r˜2 = √ (2, 0, 1)Ì 5 λ3 = −2 1 r˜3 = (1, −2, −2)Ì 3

˜ Ì AR ˜=D R ⎛ ⎞ 7 0 0 ⎜ ⎟ = ⎝0 7 0 ⎠ 0 0 −2 ! λ1 = 1, λ2 = 2 λ3 = 3 R= ⎛ 9 6 1 ⎜ ⎝−6 7 11 2 −6

⎞ 2 ⎟ 6⎠ 9

! λ1 = −2, λ2 = −2 λ3 = 1 −1; 2; 3

−1; −1; −1 0; 0; 1 1; 2 ± 3i

0; 2; 2; 12 " −2; 2; 3; 4 1 # rÌ 1 = (1, 2, 2) 3 1 (2, 1, −2) rÌ = 2 3 1 rÌ 3 = (−2, 2, −1) 3 Ì r1 = 1 √ (1, −1, −1) 3 1 Ì √ (1, 1, 0) r2 = 2 1 rÌ 3 = √ (1, −1, 2) 6

% 3 5 $ A = 5 3 a = (−2, −14)Ì a0 = −13 det A = 0 $ % ! & ! x=y−v '( 2Av = a ) 1 −1 A a 2 v = (−2, 1)Ì 1 b0 = a0 − aÌ v 2 = −13 + 5 = −8 * +#, ! 3y12 + 10y1 y2 + 3y22 −8 = 0 ! y = Rz -

$

A: λ1 = 8, λ2 = −2 x1 = (1, 1)Ì x2 = (−1, 1)Ì % $ 1 1 −1 R= √ 2 1 1 det R = 1 % $ 8 0 D= 0 −2

* +#,#! 8z12 − 2z22 − 8 = 0 z2 z12 − 2 = 1 4 +./0 % "1

" ! x = Rz − v

1 x1 = √ z1 2

1 − √ z2 + 2 2

1 x2 = √ z1 2

1 + √ z2 − 1 2

z2

x2

z1

6

I ½

- x1

½

½ ¹½

⎛

⎞ 4 −2 −4 ⎜ ⎟ A = ⎝−2 1 2 ⎠ −4 2 4 det A = 0

a = (−28, 2, 16)Ì a0 = 45

x = Ry A λ1 = 9, λ2 = λ3 = 0 ! "

l1 = (−2, 1, 2)Ì l2;3 = (t1 + t2 , 2t1 , t2 )Ì

"

a ⊥ l3 # 2t1 + t2 = 0

l3 = (−t1 , 2t1 , −2t1 )Ì

" l2 ⊥ l3 t1 (t1 − t2 ) = 0 $ t1 = 0 % & l3 = o t1 = t2 = t

l2 = (2t, 2t, t)Ì l3 = (−t, 2t, −2t)Ì

'% (

t = 1 l2 = (2, 2, 1)Ì l3 = (−1, 2, −2)Ì )" ⎛ ⎞ −2 2 −1 1⎜ ⎟ R= ⎝ 1 2 2 ⎠ 3 2 1 −2

det R = 1 ⎛

⎞

9 0 0 ⎟ ⎜ D=⎝0 0 0⎠ 0 0 0 aÌ R

= (30, −12, 0)

* +,,-

9y12 + 30y1 − 12y2 +45 = 0

. y =z−v

- 3 M (−2; 1), -

a = 3, b = 4 5 3 ;− ;2 M 2 2 √ 5 r= 2 2 M (0; 0; a), r = |a|

/ % - 32z12 + 50z22 0 −800 = 0 z2 z12 + 2 =1 " " 25 16 3

1 2 5 - 18z12 − 8z22 − 72 = 0 9 y1 + z12 z2 3 − 2 =1 " 5 4 9 −12 y2 − = 0 543 3 2 . 8 - 16z12 + z22 − 16 = 0 3 5 z1 = y1 + - 4z12 − z22 − 4 = 0 3 543 5 √ z2 = y2 −

- z12 = 4 2z2 3 % 6 9z12 − 12z2 = 0 - z12 − 45 = 0 +3" 04 3 1 6 3 y4 2 , - 12z12 + 3z22 + 3z32 0 3 "

4 z 2

−1 = 0

- 6z12 − 3z22 − 3z32

0

−1 = 0

2 x

0

2

- 543 M (−1; −1), a=b=1

- 6 S(2; −2), p=3

- 7 M (4; −1), r=2

+ - 54 3""

- 2z12 + 2z22 − 7z32

−1 = 0

+ - 54 3""

- 10z12 + z22 + z32 3 "

−1 = 0

9z12 − 18z22 +36z3 = 0 18z22 − 9z12 −36z3 = 0 ! 3z12 + 6z22 = 6

"

# 3z12 + 6z22 + 9z32 −18 = 0 $ 2 2 x1 = z1 − z2 3 3

1 − z3 + 1 3 2 1 x2 = z1 + z2 3 3 2 + z3 3 1 2 x3 = − z1 − z2 3 3 2 + z3 − 1 3 O(x1 = 1; x2 = 0; x3 = −1)

2 1 z1 − z2 3 3 2 + z3 − 1 3 2 2 x3 = z1 + z2 3 3 1 − z3 + 1 3 O(x1 = 0; x2 = −1; x3 = 1)

x2 =

y12 +y22 +y32 −49 = 0 & x1 = y1 + 3 x2 = y2 − 4 x3 = y3 − 5 O(x1 = 3; x2 = −4; x3 = −5)

18z12 + 9z22 − 9z32 −18 = 0 % 1 2 x1 = − z1 + z2 3 3 2 + z3 3

2n3 − 5n2 + 8 n→∞ 7n3 + 2 2 − n5 + n83 = lim n→∞ 7 + n23 2 2−0+0 = = 7+0 7 lim 1 n→∞ 2 n+7 − = n−3

' lim

m = n

− 3 n → ∞ m → ∞

$

m −2 1+ lim m→∞ m 10 % 2 = × 1− m m −2 lim 1 + m→∞ m 10 2 × lim 1 − m→∞ m

= e−2 ·(1 − 0)10 = 4n2 + 3n n→∞

−2n = lim

1 e2

3 a = 5

a = 0 n = 1, 2, . . . , 5 n > 5

lim

n→∞

(4n2 + 3n) − 4n2 √ 4n2 + 3n + 2n 3 = lim 4 + n3 + 2 3 3 = = √ 4 4+2 3 2 1 5 − n0 (ε) = 4ε 2 a = 0 2 n0 (ε) = ε

a =

a = 0 a = 1

∞ i=1

ai

! lim an = 0 1 q = 2

q = 0

q = 1e q = 14

0 125 27 +∞ 12 1 12 0 √ e 1 e √ e 1 0 e8 an = 7nn− 2 6 − 5 2 1 − 10 0

4

!

2 an = 3n n2+ n an = −6 5 +(−1)n · √ n3 1 sn = 1 − n+1 s=1 1 1 sn = 2

1 2n + 1 1 s= 2 25 25 + + ... 100 10000 1 25 · = 1 100 1 − 100 25 = 99 1 2 557 990 a1 = 3, 5

−

"

# s40 = 2246 $ 8πr2 4πr2

6πr2 3 √ 4+ 2 8πr 21 27 πr3

√ 2 2+1

14, 2067 . . . % & '( )( * + lim an

n→∞

n 3n − 1 1 1 = lim = n→∞ 3 − 1 3 n = 0 n n an = 2n + 3 = lim

n→∞

, + lim

n→∞

n

|an |

= lim

n→∞

n 2n + 3

= lim

n→∞

⇒

1 2 + n3 1 = <1 2

) n

an = (2n10− 1)!

10n+1 (2n + 1)! - + an+1 lim n→∞ an 10 = lim 2n(2n + 1) =0<1 ⇒ )

an+1 =

) + lim an

n→∞

n n2 + 1 1 = lim n→∞ n + 1 n =0 . an+1 ≤ an * n ≥ 1/ = lim

n→∞

+

an+1 > an / n+1 (n + 1)2 + 1 n > 2 n +1

* , ( + 1 > n2 + n

* n ≥ 1 0 1 / 1 n ≥ n2 + 1 2n n ≥ 1 0

an =

*

∞ i 2 i +1 i=1 ∞ 11 ≥ 2 i=1 i

! "! n |an | = √ n n 1 √ n 2 ( n) (1 + n12 ) n ⇒ lim n |an | n→∞ 1 = 2 0 = 1# 1 ·1 # 1 an = (3n − 1)2 1 < 2 $% n ≥ 1! n

)# $% a = 2# & # !

#

# ',! # )# ) !# # ) '(! #

# )# & '-! # # )#

# '! # )# #

# # '*! # )#

#

# ) # $# '.! # −1 ≤ x < 1 )# −∞ < x < +∞ # 0 < x < +∞

#

# |x| < 1

# #

# |x| < 2

'+! #

$# −1 ≤ x < 1 # −1 < x < 1

y

y

y

2 3 2 3 32

5

x

3 2

34Π

Π4

2 Π 4

Π 2

1

x

3Π 4

2

5

1

1

2

x

y

y Π 2

1 2

x

1 2

1

1

x

Π2

[−1; +∞) (−∞; −2)∪

(−2; 2) ∪ (2; +∞) √ (−∞; − √ 2] ∪ {0} ∪[ 2; +∞)

(−∞; −1] ∪ [0; 1] (−2; 0] (−1; 1) ∪ (2; +∞) − 31 ; 1 [1; 100] (2; +∞) (−∞; 1) ∪ (1; +∞) (−∞; +∞) [1; 4) (−∞; −1)∪

(−1; 3) ∪ (3; +∞)

D(f ) =

{x | x ∈ R ∧ x > 5} W (f ) = {y | y ∈ R ∧ y > 0}

D(f ) = {x | x ∈ R

∧x = −2 9 ∧ x = 3}

W (f ) = y | y ∈ R 1: ∧y = 0 ∧ y = 5 D(f ) = {x | x ∈ R ∧x = −2 ∧ x = 2} W (f ) = {y | y ∈ R ∧y ∈ (−∞; 0) ∪[1; +∞)}

D(f ) = {x | x ∈ R}

W (f ) = {y | y ∈ R ∧y ∈ (0; 3]}

f (−x) = 7 sin x cos x = −f (x)

! !

y

y

1 2 9 5 3

1 2 4

x

2 4

x

−1

y = f (x) 2 1 1−x −1 = 4 1+x D(f −1 ) = (−1; 1] W (f −1 ) " 1 = − ; +∞ 4

x =

8 +3 y−1

1 y − 7 ln x = 3 5

2 −1

+1

x = x

ey/5 +1 ey/5 −1

%&' x = 4; y = x + 5 f (x) = x + 5 26x − 42 + 2 x − 7x + 12 ( xN1 = −2; xN2 = 2 xP = 1 )* # y x=0 = −4

%&' x = 1; y = 1 f (x) = 1 2x − 5 + 2 x − 2x + 1 (

! x2 − 5x + 6 y = f −1 (x) = 5 √ x2 − 2 +x 3 x √ −1 D(f ) = [0; +∞) 1 + 1 + x2 x y = − x3 − 1 p0 = 10π [1; +∞) 1 p0 = 4π √ " f (x) + f (−x) y = 3 − x 2 p0 = 6π 1 [0; +∞) + f (x) − f (−x) π 2 p0 = 1−x 2 ; (−∞; −1) y = 1+x xN1 = 2; xN2 = −3 + , - ∪(−1; +∞) xP = 4; xL = 3 . 2 1 3 3 − 5 sin ey x = #$ x = 0 y = 2 2

(x − 5)(x + 3) x→5 lim (x − 5)(x + 5) = lim

x→5

=

3

4 8 = 10 5 lim

x→0

x→∞ lim

x+3 x+5

sin ax =a x

1+ 3+

1 x 2 x

x2

4 12

−1 1 − 56 72

=0 2(x − 1) lim x→1−0 −(x − 1) = −2

z = arcsin 7x x→0⇒z→0 1 x = sin z 7 z lim 1 z→0 7 sin z 1 = 7 × lim sin z z→0

=7·

z

52 12

√ 2 − 2 1 2 3 2 π 2

1 =7 1

0 e e2 −1 x → −0; + 1 x → +0 −∞ x → 2 − 0 + ∞ x → 2 + 0

+∞ x → 1 − 0 0 x → 1 + 0 2 x → −2 − 0 − 2 x → −2 + 0 +∞ x → −1 − 0 −∞ x → −1+0 −1 x → 1 − 0 + 1 x → 1 + 0 π2 x → 1 − 0 π − x → 1 + 0 2 1 − 36

x → −3±0

! x0 = 0 " # #

" ! x0 = 1 " # #

"

! x01 = −2 x02 = 2 # $ " " ! x01 = 1 " " x02 = 2

% & #

" ' # ! x0 = 0 " # ( # f (2) = 1

" ! x01 = 0 # ! x0 = 1 '# ' # ( f (0) = 2 f (0) = 1 ! f (0) = 0 x0 = kπ

f (0) = 1 (k = ±1, ±2, . . . ) " f (2) = 78 " # f (0) = n1 = n ! x0 = 2 " # # f (0) = 2

f (0) = 0

lim f (x)

lim f (x) x→2−0

x→1−0

= lim f (x)

= lim f (x) = 2

a = 0 f (0) = 1

x→1+0

x→2+0

x2 − 4x + 4 = (x − 2)2 y = 2 3 10x−3 − 6x 3 + 4x 4 y = −30x−4 − 4 · 3 −1 12 − 1 x 3+ x 4 3 4 30 4 y = − 4 − √ 3 x x 3 +√ 4 x y = 1 13 5x2 − 3x 2 + 3x 4 2 2x 3 5 4 3 1 = x 3 − x− 6 2 2 3 31 + x 12 2 5 4 1 y = · x 3 23 7 1 3 − · − x− 6 2 6 3 31 19 + · x 12 2 12 10 √ 3 y = x+ 3 √ 31 1 12 √ + ·x· x7 6 4x x 8 x y =π· 1 − x2 y = π×

1(1 − x2 ) − x(−2x) (1 − x2 )2 1 + x2 y = π · (1 − x2 )2

2 cos x sin x − 2 + x3 x tan x 1 √ −√ 2x x x cos2 x 1 # 1 − sin x

1 √ $ √ √ y= x2 · 3x 2x 2 x( x + 1)2 √ √ 4 = 3 2 x6 · x y = x ln x − lg 5 √ 4 −3 lg x = 3 2 x7/8 1 1 7 4 √ y = ln x + x · y = · 3 2 x− 8 x 8 1 √ −3 lg e 4 7· 3 2 x √ y = 8· 8x 3 lg e y = 1 + ln x − 4 3 2 x − 3x + 12x + 3x

3 ln x −42x − 40 / x % − 2

2 x −2x + 7 6x2 ex +2x3 ex −3x ln 3 y⎧= ln |x| = −7x2 + 20x − 6 & ⎨ln x ex ,x>0

u vw + uv w + uvw ⎩ln (−x) , x < 0 u vwz + uv wz +uvw z + uvwz y = ⎧ ⎧ ⎨1 ⎨−2x + 2 , x ≤ 2 ,x>0 x y = ⎩ (−1) ⎩−2 1 ,x>2 ,x<0 (−x) = x ⎧ ⎨−2 !" x < 2 1 = y y = !" x = 0 ⎩0 !" x > 2 x

6x −

27 cos x + 2 cos2 x

−

!" x ∈ (−∞, +∞)

x

12 1 y = x + 2x 2

− cos1 x

− 12 1 1 y = x + 2x 2 2 1 × 1 + x− 2

! cos1 x x3 (ln x)2 √x2 − 25

∈ (−∞, 2) ∪ (2, +∞)

x = 2, y = −2 ⎧ ⎨ y = ⎩−x , x ≤ 0 x ,x>0 ⎧ ⎨−1 , x < 0 y = ⎩1 ,x>0

y =

x ∈ (−∞, +∞)

x ∈ (−∞, 0) ∪(0, +∞) y⎧= ⎪ ⎪ ⎪ ⎨sin x , x ∈ [0, π] − sin x , ⎪ ⎪ ⎪ ⎩ x ∈ (π, 2π]

y=

y = ⎧ ⎪ ⎪ ⎪ ⎨cos x , x ∈ [0, π)

1 1 y = − · 2 1−x 1 1 − · 2 1+x 1 1+x+1−x = · 2 (1 − x)(1 + x)

x ∈ [0, 2π]

− 3 y = 5 1 + x2 2 3 y = 5 − 1 2 5

− +x2 2 · 2x

y = −

15x √ 2 (1 + x )2 1 + x2

1 ln (1 − x) 2 1 − ln (1 + x) 2

− cos x , ⎪ ⎪ ⎪ ⎩ x ∈ (π, 2π]

x ∈ [0, π) ∪(π, +2π]

√ x+1 √ √ 2 x+2 x· x 2 f (1) = √ 2 1 + 2√ 3 = 3 6 sin (6x + 4) 1 √ x2 + a2

2

" − x x+ 1 e + e− −3x # −6 (1 −e e−3x )2 y˙ = cos 2t x 2

y¨ = −2 sin 2t

x˙ = − sin 2t

1 −1 a ebx (b cos cx −c sin cx) y =

x2

y⎧=

⎪ ⎪ ⎪ ⎨ln (− ln x) , 0<x<1 ⎪ ⎪ ⎪ ⎩ln (ln x) , x > 1 y =

sin1 x

1 x ln x

x 2

x ¨ = −2 cos 2t v = − 1 s (s − 1)2 2 v= 2 s2 (s − 1)3 w = s3 (4 ln s + 1) s 2 w = s2 (12 ln s s2 +7) 2 (2x − 3) x y = (x − 1)2 y =2 x(x3 − 3x + 3) × (x − 1)3

cos x, − sin x,

− cos x, sin x 1 2 1 6 , − 2, 3, − 4 x x x x 1 √ , 2 x+1 1 , − 4 (x + 1)3 3 , 8 (x + 1)5 15 − 16 (x + 1)7

−3 sin 3x, − 9 cos 3x, 27 sin 3x, 81 cos 3x

f (x + p) = f (x) p ⇒

f (x + p) = f (x)

p

y (n) =

2 · n! (1 − x)n+1

y (n) = an · eax y (n) = (−1)n−1

×

(2n − 3)! 22n−2 (n − 2)!

arctan x 1 ×x 2 −n ln y = ln 10−3x ln x 1 √ y (n) = 2n−1 |x| x2 − 1 y n−1 = 0 − 3 ln x π × sin 2x + y y = ⎧ 2 1 ⎪ √ 2 −3x · ⎪ , |x| < √12 ⎪ ⎨ 1−x2 x = −3(ln x + 1) √ 2 − , 2 f (n) (x) $ = 1−x ⎪ ⎪ ⎪ ⎩ 1 −3x (−1)n 1 √ < |x| < 1 y = −30x (ln x 2 · n! 2 (1 + x)n+1 +1) 6x cosh2 x2 sinh x2 % y 1 a · tanh (ax + b) =0 ln y = 1, + y (1 − x)n+1 1 y = 0 x + sinh 2x 1 2 ! ! x ln x = e = const.

arsinh x 1 " # (tan x) cos x x (arctan x) 1 n × 2 × ln arctan x sin x cos x 3 i(i − 1)xi−2 sin x x i=2 + 2 ln tan x + n cos x (1 + x2 ) arctan x +(x − 1) · i(i − 1) x−1 2 x+2 e x+1 i=3 2 (x + 1) x2 + x + 1 ×(i − 2)xi−3 x 2 x (1 + ln x) 4x = (n + 1)n(n − 1)xn−2 4 $% x = 1 1 1 − ln x x −1 √ x x · n 2 x 3 x 3 i(i − 1) f (−x) = f (x) 2(1 + x3 ) i=2

⇒ 1 = (n + 1)n(n − 1) 2 − f (−x) = f (x) & k − 1 = 0 % (1 + x ) arctan x ⇒ k = 1 cos x sinh (sin x) n f (−x) = −f (x) 2 2 k(k − 1) 6x tanh x

2 2 k=1 cosh x f (−x) = −f (x) (n + 1)n(n − 1) 1 = ⇒ 3 cosh x $% ' − f (−x) = −f (x) 4 sinh 4x ( ) ⇒ √ n 4x − 1 f (−x) = f (x) k(k − 1)

2x k=1

=

n

k2 −

k=1

n

n

k

k=1

√ √ k

k=1

π6 , − 63 , 7363 f (k) (x) =

n

n k!xn−k k

n = (1 + 1)n k k=0

! v = t2 − 4t + 3 a = 2t − 4

" #$ v = 0 :

!

t1 = 1, t2 = 3 x = A cos ωt x˙ = −Aω sin ωt x ¨ = −Aω 2 cos ωt = −ω 2 A cos ωt = −ω 2 x x=0 π ⇒ ωt = + kπ 2 π x˙ + kπ = ±Aω 2 π x ¨ + kπ = 0 2

!

x=A ⇒ ωt = 2kπ x(2kπ) ˙ =0 x ¨(2kπ) = −Aω 2 x = −A ⇒ ωt = (2k + 1)π x˙ (2k + 1)π = 0 x ¨ (2k + 1)π = Aω 2 (k ∈ Z) V = πr · 2h · h t t 2 h −h t = v v h = t π(2r − h)h v = π 2

! v = x˙ = Ak e−kt =

k·A e−kt = k(A−x)

! x = 10+20t− 12 ·gt2 v = 20 − gt a = −g = −9, 81 2 v=0: t=

20 ≈ 2, 04 g

% ! y = nxn−1 x ! y = √ x 2 x 1+x

! r = 4 sin2 ϕ ϕ ! (sin2 t) = sin 2t t ! cos ϕ2 ϕ 1 = − sin ϕ 2 2 ! y = 0, 04 Δy = 0, 0401

! y = 0, 05

Δy ≈ 0, 049876

& ! VV !

3x2 x x3 = 0, 6% 3b s f = 8f =

! x = 5x2 √yx <

x < 0, 005 ! VV = 3r r r 1% = 3 r r = 1 % r

2 · 0, 1 √ 5·4 4

3

' ( ) * 13 %

+ ) x1 = 1; x2 = 3

[−1, 1] - . /

+

0 AB x = 2 ∈ (1, 3)

+ , " x = 0 -

9−1 =2 3 − (−1) f (x) = 2x = 2 x=1 m=

x = 1 / $ ( − f (−2) ! f (1) 1 − (−2) 0−6 = −2 3 f (x) = 1 − 3x2 1 − 3ξ 2 = −2 =

ξ = −1 ∈ (−2, 1) 4 −1 ξ = π ∈ (0, 1) 4

ξ = − 1 − 2 π ∈ (−1, 0) 1 ξ = ln 2 = 1, 4426 . . . ξ ∈ (1, 2)

[−3, −2]

[−2, −1], [−1, 0] (x + Δx)2 − x2 = Δx · 2 · (x + ϑ · Δx) 1 ϑ= 2 ϑ !

" #

x 2x ϑ− Δx Δx 1 =0 + 3 $ ex = e0 +x · eϑx >1+x eϑx > 1 x>0 √ √ 1 + x = 1 x + √ 2 1 + ϑx x <1+ 2 x>0 1 1 =√

√ 1+x 1 x − 2 (1 + ϑx)3 x >1− 2 x>0 ϑ2 +

% & √ f (x) = x :

√ x0 + h = x0 h + √ 2 x0 + ϑh ' x0 = 100, h = 1 ( √ 10 < 100 + ϑ √ < 101 0 < ϑ < 1

)

sin π2 − sin 0 cos π2 − cos 0

cos ξ − sin ξ ⇒ tan ξ = 1 π π ⇒ ξ = ∈ 0, 4 2 2 15 3 ξ = 4 ∈ (1, 4) =

ξ =

14 ∈ (1, 2) 9

sin 5x 3x 5 cos 5x = lim x→0 3 5 5·1 = = 3 3

* lim

x→0

1 2

1 nan−1 1 a sin ax = lim x→0 b sin bx 2 a2 a cos ax lim 2 = 2 x→0 b cos bx b 1 2 1 6

= lim ex ln x

" 3 +∞ + 0 , 0

x→+0

lim x ln x

= ex→+0 = e0 = 1 1

1

0

- 1∞ .

' 3 - 0 · ∞ =.

tan x2 x→π−0 (π − x)−1 = ... = 2 lim

0 nxn−1 / lim = ... x→+∞ ex n! = lim x = 0 x→+∞ e 0 - 00 .

3 lim x ln (1+ x ) = ex→+∞ ln (3+x)−ln x

lim

= ex→+∞ = . . . = e3

x−1

−2 - ∞0 .

lim

ln x x

= ex→+∞ 1 lim = ex→+∞ x = e0 = 1

- ∞ − ∞.

$

2 x→1 x2 − 1 1 (x − 1) × x−1 % 2 x − 1 − = 2

= lim

=−

e

16

1 2

12

f (2) = 5

f (x) = e2x−x

f (2) = −1

f (2) = 6 f (2) = 12

2 2 3 = f 2!(2) 2 = f 3!(2)

f (x) = −125 + 75x 2

3

−15x + x

1 −30 300 −1000 5 −125 875 5 1 −25 175 −125 5 −100 5 1 −20 75 5 5 1 −15 1

f (x) = 72 − 102(x 2

+3) + 53(x + 3) − 12(x+ 3)3 + (x+ 3)4

f (x) = 5 − (x − 2) + 3(x− 2)2 + 2(x− 3)3

f (0) = 2 f (0) = −4 f (x) = 1 + 2x + x2 2 − x3 + R3 3 f (x) = (x−1)+(x− 1 1)2 + (x− 1)3 + R3 2 1 f (x) = x− x3 +R3 3 1 f (x) = √ 2 x−2 − √ 4 2 3 + √ (x − 2)2 32 2 5 √ (x − 2)3 − 128 2 35 √ (x − 2)4 + 2048 2 +R4

f (x) = 1 + 2x

+2x2 − 2x4 + R4

ex = 1e + x +e 1 +

(x + 1)2 1 · 2! e

(x + 1)3 1 · + 3! e 4 (x + 1) −1+ϑ(x+1) e 4! 0<ϑ<1 ln x = (x − 1) 1 − (x − 1)2 2 (x − 1)3 +

3 3 1 + ϑ(x − 1) 0<ϑ<1

+

f (0) = 1 2 f (x) = e2x−x ×(2 − 2x) f (0) = 2

2 −9 11 3 4 −10 2 2 2 −5 1 5 =f (2) 4 −2 2 2 −1 −1 = f 1!(2) 4 2 2 3 = f 2!(2)

2

| sin (x0 + h) − sin x0

−h cos x0 | h2 | sin (x0 + ϑh)| = 2 h2 ≤ 2 2 x3 x + ex = 1 + x + 2! 3! x4 x5 ϑx + + e 4! 5! 0<ϑ<1 1 1 e1 = 1 + 1 + + 2! 3! eϑ 1 + + 4! 5! ϑ 1 e e 3 1 < < = 5! 5! 5! 40 2

x a cosh xa = a + 2a

x4 ϑx cosh 3 4!a a |x| ≤ a +

=δ 4 x ϑx 0, 0008 < δ < 0, 0063 24a3 cosh a ln 1, 5 ≈ 0, 401 |x| 0, 4018 < ln 1, 5 < cosh 1 24 < 0, 4073 |x| · 1, 6 < 0, 07|x| < 24 0, 822317 x2 ln (1 + x) = x − 2 23 2 sin2 x = x2 − x4 x3 x4 2! 4! + − 25 6 27 3 4 + ... + x − x5 1 6! 8! + · +R2n 5 (1 + ϑx)5 R2n =

1 0, 55 22n+1 2n+2 · x × 5 5 (1 + ϑ · 0, 5) (2n + 2)!

π sin 2ϑx + (2n + 1) 2 1 (2n)2n+2 = · × 2 (2n + 2)! π sin 2ϑx + (2n + 1) 2 1 × 2 π sin 2ϑx + (2n + 1) 2 1 ≤ 2 (2x)2n+2 lim =0 n→∞ (2n + 2)! |x| < ∞ lim R2n = 0 n→∞

D(f ) = (−∞, +∞) xMin = 0, yMin = 2 (0, +∞) (−∞, 0) ! " D(f ) = (−∞, 0) ∪ (0, +∞) xMax1 = −1, yMax1 = 1 xMax2 = yMax2 = 1 (−∞, −1) ∪ (0, 1) ! (−1, 0) ∪ (1, +∞) ! D(f ) = [−1, √ +1] 2 1 , yMin = − xMin = − 2 2 √ 2 1 , yMax = xMax = 2√ 2 √ 2 2 ∪ , 1 −1, − 2 2 √ √ 2 2 , − 2 2 D(f ) = (0, 1) ∪ (1, +∞)

xMin = yMin = e (0, 1) ∪ (1, e) (e, +∞)

D(f ) = (−∞, +∞) π xMin = + k · 2π 3 π √ yMin = − 3 + k · 2π 3 5π + k · 2π xMax = 3 5π √ + 3 + k · 2π yMax = 3 π π (6k−1) , (6k+1) 3 3 π π (6k+1) , (6k+5) 3 3 # k ∈ Z (0, +∞) $ %$ & (−∞, 0) $ %%$ " (−∞, +∞) $ %$ &

(−∞, −1) ∪ (0, 1) (−1, 0) ∪ (1, +∞) xMax = −1, yMax = 1 xMin = yMin = 0 x = 0

xMin = −1,

xMax = −3, xMin

xW = 1, yW = −4 &' & # y = −6x + 2 lim = +∞; lim y = −∞ x→+∞

√

2−

W (f ) = (−∞; 20, 25] f (−x) = f (x) ( xN1 = 0, xN2 = −3, xN3 = 3 xMin = yMin = 0 3√ xMax1 = − 2, yMax1 = 20, 25 2 √ 3 2, yMax2 = 20, 25 xMax2 = 2 xW1,2 = ± 1, 5, yW1,2 = 11, 25 lim y = −∞ x→±∞ W1,2 : y = ±12 1, 5x − 6, 75

π 4

yMax = 9 − 2 e−5 = 2, yMin = −8

! "# x1 = 0,

y1 = 1 π x2 = , y2 = 1 2 5 1√ x3 = π, y3 = − 2 4 2 √ π ! "# x4 = 4 , y4 = 21 2 x5 = π, y5 = −1 3 x6 = π, y6 = −1 2 xMax = 4, yMax ≈ −7484, 2677 xMin = −1, 5, yMin = −2, 75

D(f ) = W (f ) = (−∞, +∞)√

xN1 = 0, xN2,3 = 1, 5 ± 1, 5 5 2 xMax = −1, yMax = 1 3 xMin = 3, yMin = −9 2 xW = 1, yW = −3 3 1 W : y = −4x + 3 lim y = +∞, lim y = −∞

xMax = 2,

yMax = 4 xMin = 3, yMin = 5, 25 − ln 4 ≈ 3, 86371

xMax = e xMin

−1

x→+∞

∪(3, +∞) 15 W (f ) = −∞, − ∪ (2, +∞) 9 f (−x) = f (x) (

f (5) = 5, f (0) = f (10) = 0 f 52 = 11 18 , f (1) = 1 xW = ±

1

x→±∞

$ D(f ) = W (f ) = (−∞, +∞) xN = 0, xN = 3 % 2

xMax = yMax = 0 xMin = 2, yMin = −8

xP2 = 3

lim y = 2

x→±∞

&'

1 9 D(f ) = (−∞, −2) ∪ (−2, +∞) W (f ) = (−∞, −11] ∪ [5, +∞) f (0) = 7* % +"' # y = x − 1 xP = −2 xMax = −6, yMax = −11 xMin = 2, yMin = 5 xMax = 0, yMax = −

yW = e−0,5 ≈ 0, 60653 lim f (x) = 0

1

% ) # xP = −3,

yMax = 1

2 2

x→−∞

D(f ) = (−∞, −3) ∪ (−3, 3)

−1

, yMax = 3 e = yMin = 1

xMax = 0,√

x→−∞

D(f ) = (−∞, +∞)

yMin = ln

" 1 W (f ) = − , +∞ ; xN = 0 e lim y = +∞, lim y = −0

D(f ) = (−∞, 3) ∪ (3, +∞) W (f ) = (−∞, 1) ∪ (1, +∞) f (0) = −1, xN = −3, xP = 3 lim f (x) = 1

x→+∞

x→±∞

D(f ) = (−∞, −1) ∪ (−1, +∞) " 1 W (f ) = − , +∞ ; xP = −1 8 xN1 = 1, xN2 = 2 lim y = 3

D(f ) = (−∞, +∞) W (f ) = (0, 2] xMax = 0, yMax = 2 lim y = 0

x→±∞

7 1 , yMin = − 5 8 2 xW = 2, 6; yW = 9 W : y ≈ 0, 39x − 0, 78

xMin =

x→±∞

xW1,2 = ±1, yW1,2 = 1, 5 3 W1,2 : y = ± x + 2, 25 4 D(f ) = (0, +∞) " √ W (f ) = 2 2, +∞ lim y = +∞, lim y = +∞ x→+∞ x→+0 √ xMin = 2, yMin = 2 2 4√ xW = 6, yW = 6 3 √ √ 6 x+ 6 W :y= 18 D(f ) = [−3, 0] ∪ [3, +∞) W (f ) = [0; +∞) lim y = +∞ x→+∞

xN1 = −3, xN2 = 0, xN 3 = 3 √ √ xMax = − 3, yMax = 6 3 xW ≈ 4, 40367, yW ≈ 6, 76493 W : y ≈ 3, 63x − 9, 24

D(f ) = (0, +∞) " 1 W (f ) = − , +∞ ; xN = 1 2e lim y = −0, lim y = +∞ x→+∞

x→+0

1 , yMin = − 2e 1, 5 xW = e−1,5 , yW = − 3 e W : y ≈ −0, 446x + 0, 025 xMin = e

−0,5

D(f ) = (−∞, +∞)

x→−∞

xMin = −1, yMin = − e−1 xW = −2, yW = −2 e−2 W : y = − e−2 x − 4 e−2

! 30 × 60 "! 5 5 #! AMax =

ah 4

a 6 $$! 4 × 4 × 2

$%!

$ ! 20 & $'! 60◦ 18 $! ≈ 2, 52 4+π $! ( $" ) * + &, ! s / *

$-! & . t = 2v s ! 2 d√ d $ !x= , y= 3 2 2 √ $"! VK ÷ VZ = 3 ÷ 1 $#! l ≈ 5, 619 0 / 1+ 1, 6 2, 4 + l = l(α) = sin α cos α

μG 3 VMax = 128π F = F (α) = % 9 cos α + μ sin α 2 tan α = μ = 0, 25 α ≈ 14, 036◦ AMaxr= r2 x= √ &' (""" " )(( 2 √ * 24 " ab +" √ 4 3 , Q · a + q · x · x2 = F · x

! FMin = 2Qaq d√ " 2 2 π ρ - αMax = − 4 2 √ √ #$ α = 2π 23 = 120◦ 6 3 Q − h ≈ 293, 94◦ x = vh 6

x1 = 0, 01903; x2 = 2, 44754 x = 2, 50618 x1 = 0, 29138; x2 = 4, 10927 x = 1, 72310

x1 = 0, 47170; x2 = 9, 99900 x = 7, 12813 x1 = −0, 40103; x2 = 1, 38233

x = 0, 76835 x1 = 1, 16234; x2 = −1, 16234 x = 3, 69259 AMax = 1, 12219 AMax = 1, 81971 h = 1, 15769R

x3 + x2 + ln |x| + C 3 1 2x5 − 3 + C x 1−x +C x2 x2 1 + 2 ln |x| − 2 + C 2 2x 2 √ 3 √ x x+ x3x+C 3 4 √ √ 4 2 x−4 x+C √ 2 √ x x − 3x + 6 x − ln |x| + C 3

√ 3 √ x3x−33x+C 4 √ 2 3x + C

arcsin x − arsinh C= x+ arcsin x − ln x + 1 + x2 + C 2 arctan x − 3 arcsin x + C 2 √ x x + x + C 3 1 3 arctan x − arsinh x + C 2 2 1 ex + + C x

2 ax − √ +C ln a x

x2

+ 2x + C 2 − cos x−sin x+2 tan x+7 cot x+C

tan x − cot x + C x + cos x + C 1 x3 − x + arctan x + C 3 tan x − x + C − coth x + x + C −2 cos x + C 1 (x − sin x) + C 2 1 (x + sin x) + C 2 x2 x2 ln x − +C 2 4 x ln x − x + C x sin x + cos x + C x2 sin x + 2x cos x − 2 sin x + C

x2 ex −2x ex +2 ex +C ex (sin x − cos x) + C 2 x[(ln x − 1)2 + 1] + C 2 √ 2 x x ln x − +C 3 3 ln x2 − 5 + C 1 ln 7x3 + 8 + C 21 1 (5x + 6)8 + C 40 √ 1 (2x − π) 2x − π + C 3

ln | sin x| + C 1 ln (2 + 3x2 ) 6 $ % 2 3 arctan x +C + 3 2

x + ln |2x + 1| + C − e−x +C 32x−1 +C 2 ln 3 $ % √ 10 2 arctan x +C 10 5

− ln (8 − 3x)2 + C ln |1 + ln x| + C ϕ sin 2ϕ +C ! − 2 4 1 ln | tan t| + C 2 x −3 +C 2 sin 2 1

tan (9x + 8) + C 3 1 − ln | cos (2x + 1)| + C 2 √ 2 3 x− 6 +C ln cosh 2 3 2πt T + ϕ0 + C − cos 2π T 1 √ − +C 3(2u + 3) 2u + 3 1 " sin4 x + C 4 1 +C 2 cos2 x − ecos x +C 1 3 ex +C 3

− e− sin x +C √

2 e x +C 1 2 (x + 1)3 + C 3 3 (1 + x3 )2 + C √ − 1 + 2 cos x + C 2 (1 + ln x)3 + C 3

16 23

(1 + 4 sinh x)3 + C

x arctan

(arcsin t)3 + C

a2 +1 ω2 (a cos ωt+ω sin ωt) eat +C

13 ln3 x + C 3

34 (arctan x)4 + C sin ln s + C 14 tan4 ϕ + C 23

(arctan ex )3 + C

√ tan x 2 arctan √ +C 2 2

1 x arctan x − ln 1 + x2 + C 2 2x − 3 sin (5x + 1) 5 2 + cos (5x + 1) + C 25 2√ 7x − 5 · (7x + 24) + C 49 2 x 61 1 −x− · ln 3x + 4 900 5 1 2 61 − x + x+C 8 60 3x − 1 tan (5x + 6) 5 3 + ln | cos (5x + 6)| + C 25 43 2 − x e4−3x +C 27 9 x−5 tan (3x + 1) 3 1 + ln | cos (3x + 1)| + C 9 √ √ x 4 2 + x − 2 2 − x · arcsin + C 2 x 1 + cot x + C − 2 sin2 x

√ 1√ 2x − 1 − 2x − 1 + C 2

a2 +1 ω2 (−ω cos ωt+a sin ωt) eat +C (x2 − x + 9) arctan x3 − 3x

3 + ln (x2 + 9) + C 2

14 ln | tan (2x − 1)| + C

$ sin5 x 5 sin3 x 5 x − cos x + 16 5 24 % 5 sin x + +C 16 $ cos5 x 5 cos3 x 5 x + sin x + 16 5 24 % 5 cos x +C + 16 cos x 4 8 4 2 − sin x + sin x + 5 3 3 +C sin x 4 8 cos4 x + cos2 x + +C 5 3 3 x cos x 1 − + ln tan + C 2 2 2 sin x 2 x π 1 1 sin x + ln tan + +C 2 cos2 x 2 2 4 x sin 4x − +C 8 32 1 1 cos3 2x − cos 2x + C 48 16 1 − (cos 4x + 2 cos 2x) + C 8 m = n : 1 sin (m + n)x sin (m − n)x + 2 m+n m−n +C m 1= n : x + sin 2mx + C 2 4m

1 1 sin 2x − sin 8x + C 4 16

m = n : sin (m − n)x sin (m + n)x − +C 2(m − n) 2(m + n) m = n : x sin 2mx − +C 2 4m

m = n : cos (m + n)x cos (m − n)x − − 2(m + n) 2(m − n) +C m = n : sin2 mx cos 2mx +C = + C1 − 4m 2m x x 2 a2 arcsin + a − x2 + C 2 a 2 x 1 1 − x2 + C arcsin x − 2 2 x √ +C 1 − x2 1

− (x2 + 4) 2 − x2 + C 3√ 4 − x2 +C

− 4x x a2 − x2 + C 1 tan2 x + ln | cos x| + C 2 1 − cot2 x − ln | sin x| + C 2 2 1 − cos x + cos3 x − cos5 x + C 3 5 3 3 5 sin x − sin x + sin x 5 1 − sin7 x + C 7 x 2 a + x2 2 a2 + ln x + a2 + x2 + C 2 x 2 x −4 2 +2 ln x + x2 − 4 + C x − 2x − x2 + C 2 arcsin 2

2 arcsin

x x − 4 − x2 2 2 x3 + 4 − x2 + C 4

x 1 √ +C 9 9 + x2

1 1 + x2 · x2 − 2 + C 3 2x + 3 +C arcsin √ 17 8x + 3 1 +C arcsin √ 2 41 1 ln x + + x2 + x + 1 + C 2 √ 5 ln 10x − 1

5 + 20(5x2 − x − 1) + C 1

ln 3x + 1 + 9x2 + 6x + 5 3 +C x + 2 2 x + 4x + 29 2 25 + ln x + 2 + x2 + 4x + 29 2 +C 3 2 2x − x 2 23 + √ ln 4x − 1 + 8(2x2 − x) 4 2 +C 1 +C ln (x + x2 + 1) + √ 2 x +1 1 2 (x + x + 1)3 3 1 3 1 2 x +x+1− − x+ 4 2 16 1 × ln x + + x2 + x + 1 + C 2 1 2 4x + 4x + 3 4 5 + ln 2x + 1 + 4x2 + 4x + 3 4 +C

ln |x − 7| + C − 41 · (x +1 2)4 + C

14 arctan x4 + C 52 ln (x2 + 8) + C − 2(x2 +3 15)2 + C 4 x arctan + C 9(x24x+ 9) + 27 3 4x − 5 2 √ arctan √ + C 31 31 3 2 ln (x2 − x + 1) 2x − 1 1 +C + √ arctan √ 3 3 − 2(x2 x++2x2 + 3) √ x+1 2 − arctan √ + C 4 2 2x − 1 2(x2 + 2x + 2) + arctan (x + 1) + C 3 − x + 17 + 2(x −9 6)2 +19 ln |x + 6| + C

In = 2(n −1 1)a2 (x2 + xa2 )n−1 +(2n − 3)In−1

D Ex + F + 2 (x − 1)3 x +1 Ix + K Gx + H + + 2 x + 2x + 3 (x2 + 2x + 3)2 B A + 3(x − 5) 9(x − 5)2 Dx + E C + 2 + x+2 x +1 Hx + I Fx + G + 2 + 2 (x + 1)2 x − 2x + 5 B A C + + 2 x − 2 (x − 2) (x − 2)3 E D F + + + 2 x + 2 (x + 2) (x + 2)3 Ix + K Gx + H L + 2 + 2 + 2 x +4 (x + 4) x−1 N P M + + + 2 3 (x − 1) (x − 1) (x − 1)4

+

n ≥ 2

1 x 1 x I2 = 2 2 + arctan +C 2a x + a2 a a " 2 2 1 x(3x + 5a ) I3 = 2 4a 2a2 (x2 + a2 )2 3 x5 +C + 2 arctan 2a a

x A+ 1 + x B− 1 + (x −C 1)2

5x + 2 ln |x| + 3 ln |x − 2|

+4 ln |x + 2| + C

1 1 12 ln |x| + ln |x − 2| 4

1 + ln |x + 3| + C 6 4 ln |x − 1| − 7 ln |x + 3| +5 ln |x − 4| + C x 4x + 3 +C + 2 ln 2(x + 1)2 x + 1 x + 4 1 4 +C − − + 2 ln x+2 x+4 x + 2 2 2 ln |x| − 2 ln |x + 1| + x+1 1 − +C 2(x + 1)2 1 1 + ln (x2 + 1) − 2(x − 1) 2 3 + arctan x + C 2 1 3 − + ln (x2 − 4x + 13) x 2

x−2 4 +C + arctan 3 3 1 1 1 + ln |x − 1|− ln (x2 + x + 1) x 3 6 √ 2x + 1 3 arctan √ + +C 3 3 1 x+2 + ln |x + 1| 2(x2 + 1) 2 1 − ln (x2 + 1) + 2 arctan x + C 4 1 (x − 1)2 1 + ln 2 − 5(x − 1) 50 x + 2x + 2 7 − arctan (x + 1) + C 25 |x − 1| x2 +x+ln √ −arctan x+C 2 x2 + 1 1 x2 + x + 1 ln 4 x2 − x + $1 1 2x + 1 + √ arctan √ 2 3 3 % 2x − 1 +C + arctan √ 3

√ x−1 3 arctan √ + C + 12 3

1 x ln +C a x + a x + b 1 +C ln a−b x + a √ √ 2 x − 2 1 √ − arctan x + C ln 12 x + 2 3 √ √ 3 x − 3 √ ln 30 x + 3 √ 2 x arctan √ + C − 10 2 1 2 ln 1 − + C 2 x 1 1 x − 1 +C + ln x 2 x + 1

|x| 1 ln √ +C 4 x2 + 4 1 x ln +C 5 x + 5 √ x 1 3 − arctan √ + C − 3x 9 3 x − 1 1 1 − arctan x + C

ln 4 x + 1 2

2 x2 + 6x + ln |x − 1| 2 3 x 10 95 − ln (x2 + 5) − √ arctan √ 3 3 5 5 +C (x + 2)2 1 ln 2 24 x − 2x + 4

2x + 1 √ (2 2x + 1 − 3) + C 12 √ √ √ 2 x−33x+66x √ −6 ln (1 + 6 x) + C 1 ln (e2x +1) − 2 arctan (ex ) + C 2 1 ex −1 +C ex + ln x 2 e +1 1 tan3 x − tan x + x + C 3

1 1 tan4 x − tan2 x − ln | cos x| + C 4 2 1 x 1 tan arctan +C 2 2 2 $ x 1 1 ln tan + 5 2 2 % x − ln tan − 2 + C 2

√

√

2 3 3 arctan 3 ex x 2 arctan (e )+C

2

+C

√ (x − 1)3 x−1 7

3(x − 1)2 +x +C + 5

ln 1 + tan x2 + C 2 ln | ex −1| − x + C − 21 x + 31 ln | ex −1|

1 + ln (ex +2) + C 6 √ √ 2 2 − 1 + tan x2 ln √ +C 2 + 1 − tan x2 2 3 3 3 3 (x4 + 1)2 − x4 + 1 8 4

3 3 ln x4 + 1 + 1 + C 4 x+2 5 3 (3x + 1)2 + C 2x e2 − 2 ex +4 ln (ex +2) + C e√x + ln | ex −1| +C √

2x − 1 + 2 4 2x −√1

2 + ln 4 2x − 1 − 1 + C √ 2 (3x2 − ax − 2a2 ) a − x + C 15 √ x 2 + C 1−x √ 2 arctan x + 1 + C x +C 11 + −x $ %4 3 3 1−x −8 +C 1+x +

2

√ 1+x √ 2 + x − 2 1 + x +C + ln x

12 (arctan x)2 + C

1 +C

a12 ln x +a a − ax x−1 +C ! sincos x √ 2 arcsin x + C "# $ x = sin2 t

ab · arctan

1 4

b tan x + C a

1 2 x + x sin 2x + cos 2x + C 2

−x − e−x + ln (ex +1) + C arcsin x + 1 − x2 + C

3

− cot3 x + C x tan x − 12 x2 + ln | cos x| + C

ln tan x2 + cos x + C

− 1b arctan cosb x + C √ √ √ 3 x − 12 x + 24 ln ( x + 2) + C b − 3ax 6a(ax +C + b)3 "# $ t = ax + b − x1 − arctan x + C 3

6

− tan x1 + 1 + C √

2b a + b ln x + C % n = 1 :

6

1 1 · +C 3b(n − 1) (a − bx3 )n−1

n = 1 : 1 ln a − bx3 + C − 3b x+1

arcsin √ 2 x + 1 + 1 − 2x − x2 + C 2 √ 2 x+1 √ − +C ( x + 1)2 x2 arctan x 1 + ln +C x 2 1 + x2

1 1 − x + ln e2x +4 2 4 ex 1 +C + arctan 2 2 √ 2x + 1 √ +C ln 1 + 2x + 1 1 − cot3 x + cot x + x + C 3 √ x 4 − x2 − arcsin + C − x 2 √ √ 3 3 + cot x ln √ +C 3 − cot x 6 √ √ 3 3 + tan x ln √ +C 3 − tan x 6 √ 2 (x + a)3 − x3 + C 3a 1" 2 x x + 1 + ln (x + x2 + 1) 2 5 +x2 + C

!

−

"

1 (x − 1)2 x2 + x + + ln +C 2 x |x| 3 x+2 1 − +C 3 x x = t−1 2 arctan x3 − 1 + C 3

#

1 1 x + ln | sin x + cos x| + C 2 2 √ √

√ 2 x arcsin x + 1 − x + C 1 + C1 tan2 x + C = cos2 x 1 cot x − cot3 x + C 3 − cot x ln (cos x) − x + C 1 ex −1 −x +C e + ln x 2 e +1 1 tan4 x + C 4 x+1 ln |x + 1| + C ln |x| − x √ 2 1 + sin x + C √ √ 2 arctan ( 2 tan x) + C 2 1 |x2 − 2| ln 2 +C 6 x +1 √ √ −2 e− x ( x + 1) + C √ 2 x arctan x − ln |1 + x| + C √ tan x + C

ln |x| −

x2 + 1 ln (x2 + 1) + C 2x2

1 arctan (ax ) + C ln a √ √ 2 x + cos x + C

2(x + 7) √ x+1 3 √ √ x + 1 − 2 √ √ +C +2 2 ln √ x+1+ 2 x − 1 − x2 arcsin x + C √ x2 − 1 +C x 3x2 + 3x + 1 − +C 3(x + 1)3

$

ln

(2x − 1)2 +C |x2 + x|

2

− 1 + cossinx +x sin

x

+

+C

2

1 x + 2x + 2 16 ln 2 x − 2x + 2

1 arctan (x + 1) 8 + arctan (x − 1) + C

1 1+ n a2 S = lim Sn = n→∞ 2 a3 1 Sn = 1+ 6 n 1 × 2+ n a3 S= 3 a(ea −1)n Sn = 1 − e−a/n 2

Sn = a2

S = ea −1

3 − a6

3(e−1) √ ln 1 + 2 12 2(1 + ln 2) e ln e2+1 17 6

√ 3−1

2 ln 1, 5 − 31

12

arctan e − π4

5 2 8 14 3 π 12a π 6 1 6

√ π 3 − 3 2 π−2 a· 4 √ √ 2 + ln (1 + 2) 2 2 ln 2 − 1 √ 2− 3 2

2 − ln 2 1 3 πa2 16 √ √ 3− 2 2 ln 1, 5 π−2 4 π −1 2 1 − ln 2 2 1 π · 2 2 1·3 π · 2·4 2 1·3·5 π · 2·4·6 2 1 π · 2 2 1·3 π · 2·4 2 1·3·5 π · 2·4·6 2

1

+∞

+∞ 1 n > 1 n−1 n ≤ 1

1 1 2 π 4 1

ln 2

16 π 6 π ln 2 + 4 2

π−2 8 √ 3 6 2

6 1 3 1 √ ln (1 + 2)

x2 = z π √ 5 4π √ 3 3 0 π

1 ! ln 2 0

1 π 4 # ∞ " # e−x xm x 0

∞ 2 = − e−x xm 0 | e−x | ≤ e−x !$ # ∞ x 1 +m e−x xm−1 x #≥ ∞ #0 ∞ e−x x " −x m−1 1 =m e x x 0 −→ m! " #

1 0< √ 1 + x5 1 ≤ 5/2 !$ x ≥ 1 x # ∞ x " 5/2 x 1

" " # √ 1 x4 − 1 !$ x > 1

1 < √ x−1

% 1 4x + 7 1 √ < 3x + 5 x + 7 !$ # ∞n ≥ 2 x 4x +7 2

& ln3 ≈1, 1016 1 ε < 0, 014 5 ln 3 ≈ 1, 0987 1 < 0, 00043 ε 5 ≈ 0, 746 1 ε < 0, 002 10 ≈ 1, 4627 1 ε < 0, 00012 10

156; ε(h) = 0;

'" ( 156 2n = 2 4 < 0, 3 |ε(h)| ≤ 15 ) π ≈ 3, 141593 !$ 2n = 20 ≈ 3, 239 *+ , - &

h4 · |y (4) |Max ≈ |Δ4 y|Max = 0, 1244 1 ε ≤ 2 2 · 0, 1244 < 0, 002 180

≈2, 701 π < 0, 0002 ε 12 ≈ 1, 06 ε(1) < 0, 02

Δ1

y

Δ2

Δ3

Δ4

1, 0000 0, 0607 1, 0607

0, 2928 0, 3535

1, 4142

0, 0312 −0, 1244

0, 3240 −0, 0932

0, 6775 2, 0917

0, 2308 0, 9083

3, 0000

Ì r = t, t2 + 9t + 105

2x − y + 18 = 0 9x − 5y2 = 0 !

x2 + y2 − r2 = 0 " 2

) x = x(t) = t

y = y(t) = f (t) r = t · e1 + f (t)e2

2

xa2 + yb2 − 1 = 0 # $ x2 + (y − 4)2 − 32 = 0 " 2

2

* x = 6 cos ωt, y = 6 sin ωt x = 6 cos ωt − π2

π (x −322) + (y +423) − 1 = 0 #% y = 6 sin ωt − 2 $ (x − 4)2 + (y + 2)2 − 20 = 0 " x = x(ϕ) = (aϕ + b) cos ϕ y = y(ϕ) = (aϕ + b) sin ϕ x2 − y2 − a2 = 0 &'$ r = (aϕ + b) cos ϕ e1 ( ) x = x(t) = 5 cos t, +(aϕ + b) sin ϕ e2 y = y(t) = 5 sin t; r = 5 cos t · e1 + 5 sin t · e2

) x = x(t) = 5 cosh t y = y(t) = 3 sinh t r = (5 cosh t, 3 sinh t)Ì

) x = x(t) = t

y = y(t) = 4t2 r = t · e1 + 4t2 e2

) x = x(t) = t y = y(t) =

t2 + 9t + 105

y x2 + y 2 = a arctan + b x & + x = at − a sin t a−y a−y , t = arccos cos t = a a sin t = 1 − cos2 t a − y 2 = 1− a

, & + - %

ϕ 0◦ 15◦ . . . 360◦ r

2 1, 97 . . .

2

x = x(ϕ) = (1 + cos ϕ) cos ϕ y = y(ϕ) = (1 + cos ϕ) sin ϕ r = r(ϕ) = (1 + cos ϕ) cos ϕ e1 +(1 + cos ϕ) sin ϕ e2 a r = r(ϕ) = cos ϕ a sin α

r = r(ϕ) = sin ϕ a sin (β − α) r = r(ϕ) = sin (β − ϕ) r = 2a cos ϕ

"& $& r = a ekϕ (k > 0) a = 1, k = 0, 2 (, - rMax = 3 ϕ = 0◦ , 120◦ , 240◦; rMin = 1 ϕ = 60◦ , 180◦ , 300◦ a2 cos 2ϕ r = a (a > 0) p r= cos (ϕ − α) π ϕ= 4 r = a cos ϕ r2 = a2 cos 2ϕ

r2 =

)

. x = a M (1; −1) a = 3

x2 + y 2 = 2ay y b = 2 x xy = a2

!" a = 1# $!%% & x + y = 2a ' () & * & (x2 + y 2 − ax)2 = a2 (x2 + y 2 )

/ M (4; −4; 2); r = 3 +& $& r = aϕ

5x2 + 5y 2 − 8xy − 72x + 72y 1 a= +279 = 0 2

y = x +

2 3

T1 : y = 0

1 T2 : y = ± (3x − 1) 2

1 y = − x + 2 2

y = −x + π

y ≈ 3, 76220x + 3, 897531 (4 − π)a 23π +2 T2 : y = −x + a 2

) T1 : y = x +

√ a 2 y = −x + 2 √ a 2 π y =x− 4 1 1 T : y = x− 2 2 N : y = −2x + 2 π T : y = −3x + 2 π 1 N : y = x+ 3 2 S1,2 (±1; 1) T1 : y = −2x + 3 1 1 N1 : y = x + 2 2

T2 : y = 2x + 3 1 1 N2 : y = − x + 2 2 1 T1 : y = − x + 2 4 N1 : y = 4x + 2 1 T2 : y = x − 2 4 N2 : y = −4x − 2 3 7 x+ T : y= 10 5 34 10 N : y =− x+ 7 7 T : y = 2x + 4 3 1 N : y =− x+ 2 2 13 5 T : y =− x+ 6 6 6 21 N : y = x+ 5 5 5 T : y = −x + 3 7 N : y =x− 3

$! % & P (x0 ; y0 ) & AB ϑ ≈ 71, 565◦ (108, 435◦) 45◦ 135◦

ϑ1 ≈ 53, 130◦; ϑ2 ≈ 126, 870◦ ϑ1 ≈ 71, 565◦; ϑ2 ≈ 108, 435◦ ' T : 4x + 4y + 1 = 0 40 ; ; S2 (−40; 40) S1 40 9 9 yy0 0 () xx + 2 =1 a2 b yy0 = p(x + x0 ) (( κ = −36 √ κ = − 62

κA = − ba2 ; √

κ = 616913

κB = −

√ 3 2 κ = 2a ( = 21 (0; 1) 10 2 = 2 (±2; 0) π = 3

= 1 √ √ N : (3 − π 3)x + (3 3 + π)y = 1 (−2; −2) = 10π 3√

= 3 2 ϕ = 135◦ = 21 ϕ ≈ 26, 565◦

ϕ = 45◦ = 4a = e (1; e−1 ) x0 = 2; y0 = e2 2√ (

= 2ar 3 2 −2 y − a2 x−1 a2 0 = −a x0 (x − x0 )

=

! 3r 3 "##

= ar 2 A(2x0 ; 0) B(0; 2y0 ) y(ϕ) ˙ m(ϕ) = x(ϕ) ˙ T : y = a; N : x = a √ √ T : (3 3 + π)x − (3 − π 3)y

b a2

P1

9

9 ; 3 ; P2 ; −3 8 8

45◦ !) * !

2 2 2 + x13 + y13 = (2a) 3 % ! %

,- $ & # $ . - "" */ 45◦ ! r1 = r1 (ϕ) = eϕ

(4; 4) (3; −2) (0; 1) (−2; 3) 0 1! . 4

0; − x2 + y 2 = a2 , z = ct 3 11 16 . x2 + z 2 = 1, y = 1 − ; 2 3 * ! 2" % π 3 √ ! !

− ;− 2 4 2 3 3 2 1 2 * ! (1; 1; 0)% " 4 (x − 3)2 + y − = 2 2 z #$!

32

x2 − z 2 = 1, y = 05 ! # 1 + sinh2 xa0 67" x, z #,

= cosh xa0 8 v = (−3 sin t, 3 cos t, 4)Ì y02 Ì 2 x0 a = = a cosh = cos t, −3 sin t, 0) (−3 a a π π v = 5; a =3 P0 (x0 ; y0 ) 4 4 √ √ n(xN ; 0) |v(1)| = 14; |a(1)| = 40 !" x#$! % & 6& . r · r˙ = 0 2 2 π √ Ì P0 N = (xN − x0 ) + y0

T : r = 0, 1, 3 2 2 √ x y 0 2 0 Ì + y02 = = y02 sinh +λ(−1, 0, 3) √ ; −∞ < λ < ∞ a a EN : 2x − 2z 3 + 3π = 0 2 1 ξ 1 3 T : r = (a, b, c)Ì + λ(a, 0, −c)Ì η = + 3 2 2 4 EN : 2ax − 2cz − a2 + c2 = 0 $ √ %Ì 27ξ 2 + 8η 2 = 0 2 2 2 2 T : r = 0, 1, (2ξ) 3 + η 3 = 3 3 2 2 2 2 ξ 3 − η 3 = (2a) 3 Ì √ +λ(1, 0, 1)

ξ 2 + η 2 = a2 EN : 2x + 2z − 2 = 0 8ξ 3 − 27η 2 = 0 T : r = (2, 4, 8)Ì + λ(1, 4, 12)Ì 2 t EN : x + 4y + 12z = 114 ξ = ξ(t) = −t2 1 + √ R 3

T : r = (1, 1 2)Ì 1 2 2 η = η(t) = 4t 1 + t √ 3 2 Ì 2 2 2 +λR −1, 0, 3 3 3 ! (ξ + η) + (ξ − η) = 2a 2 √ ' ( ! $! EN : x 2 − z = 0

1

1 1 Ì ,− , 4 3 2 EN +λ(1, −1, 1)Ì Ì Ì T : r = (1, 3, 4) + λ(12, −4, 3) 1 EN : 12x − 4y + 3z = 12 P1 (0; 0; −1); P2 23 ; − 98 ; − 27 ϕ ≈ 70, 53◦ 8 Ì T1 : r = 4, − 3 , 2 + λ(4, −2, 1)Ì S(2; 1; 1); ϕ ≈ 29, 94◦

T :

r = λ(1, 0, b)Ì : x + bz = 0

T2 : r =

4 A = 32 A= 3 3 14 A = 36 A = 3

A = πab $ A = 2 2 A = 3 2 2ph h % A = 83 2/3 A = 16 3 2 2ph ! h A = 20 56

A = 17, 5 − 6 ln 6 A = 10 23 A = πa2 A = 12 & A = 0, 8 A = 8 ln 2 8 ' A = 15 32 A = 3 ( A = a2 · 4 −2 π A = 1 ) A = 2a2 sinh 1 " A = 16 ≈ 2, 3504a2 3 A = 19, 2 A = 3πa2 A = 25, 6 A = 38 πa2 8 # A = 8 15

A = a2

A = 3πa 2

A1 = 19π 8

3π 4 2 a sinh 2π A= 2 ≈ 133, 87a2 πa2 A= 2 πa2 A= 4 πa2 A= *+ 2 7a2 A= 4π 11 2 πa A= 8 A = (10π √ a2 +27 3) 64 3a2 A= 2

A2 =

" #

s = 112 27

s = 2πa s = ln 3

s = 28 3

s = 670 27

s = 6a s = 2 ln 3 − 1

6 + ln 2 + s = 2a sinh 1

√

2

√

√ 3

s = 1, 35 + ln 2 √ √

s = p 2 + ln 1 + 2 √ s = ln 2 + 3 √ s = 4 3 1 s = ln (2 cosh 2) 2 1 s = 8a s = 4 √ 3 s = 4 3 s = 8a s = πa 1 + 4π2 a + ln (2π + 1 + 4π2 ) 2 3 s = πa 2 # b s=2 1 + (y )2 x 0 # b 1 1 + (y )2 x ≈2 2 0 f 2 y = 2 x b s = 5t0 s = 21 s = 7, 5 s = 2π 3 + ln 2

s = 2 sinh 1 s = 2 √ s = 8 2 T = (−1, 0, 1)Ì ; B = (1, 0, 1)Ì N = (0, −2, 0)Ì 1 t = √ (−1, 0, 1)Ì ; 2 1 b = √ (1, 0, 1)Ì ; n = (0, −1, 0)Ì 2 HN : r = (1, 1, 1)Ì + λ(11, 8, −9)Ì, λ ∈ R BN : r = (1, 1, 1)Ì + λ(3, −3, 1)Ì , λ ∈ R ES : 3x − 3y + z = 1 HN : r = (1, 1, 0)Ì + λ(1, 1, 0)Ì BN : r = (1, 1, 0)Ì + λ(−1, 1, 2)Ì

x2 + y 2 = z 2 . HN : r = λ(0, 1, 0)Ì (y !" # BN : r = λ(−1, 0, 1)Ì $ r = λ(1, 0, 1)Ì # t % # s = x˙ 2 + y˙ 2 + z˙ 2 dt = 3t 0

r = r(s)

√ s s 5 Ì s = 2 cos , 2 sin , 3 3√ 3 2 s 2 s 5 Ì t = − sin , cos , 3 3 3 3 3 s Ì s n = − cos , − sin , 0 3 3 √ 1 √ s s Ì b= 5 sin , − 5 cos , 2 3 3 3 √ 2 5 κ= ; w= 9 9 2 s s Ì κ·n= − cos , − sin , 0 9 3 3 t = s − w ·$ n %Ì √ 5 s s 5 1 cos , sin , 0 = 3 3 3 3 3 b = s s # t = ; r = r(s) s2 s s s Ì = − sin , 1 − cos , 4 sin 2 2 2 4 s 1 s s Ì 1 1 − cos , sin , cos t= 2 2 2 2 2 4 &' s = 2π : Ì t = (1, 0, 0)√ √ 2 2 Ì ,− n = 0, − 2 √ 2 √2 2 Ì ,− b = 0, 2 √ 2 2 ; w=0 κ= 4

( α1 = 120◦ ; α2 = 60◦ ; α3 = 45◦ t x = t cos t, y = t sin t, z = t )* HN : r = (1, 1, 1)Ì

+λ(26, 31, −22)Ì BN : r = (1, 1, 1)Ì + λ(8, −6, 1)Ì

κ = |r˙|r|×˙ 3r¨|

√ 2 9t4 + 9t2 + 1 = (1 + 4t2 + 9t4 )3

t = vt t ˙ +v a = t (vt) = vt t ˙ + v · 2s 1 v s = vt × ˙ + n t ˙ + v · nv = vt

Ì v = r˙ = (1, 1 − 2t, 0)

w = [|r,˙r˙ ×r¨,r¨r|2]

|r˙ × r¨| 2 = |r| ˙ 3 v 3 v = |r| ˙ = 2 − 4t + 4t2 4t − 2 2 at = v˙ = √ ; an = 2 v 2 − 4t + 4t t√= 0 : √ √ v = 2; at = − 2; an = 2 7 sin 2t 12 κ = 12 ; at = ; an = 3 v 2v v t = π4 : √ √ 24 2 ; at = 0, 7 2 κ= 125√ an = 2, 4 2 κ = v22 ; at = 4t; an = 2 t = 1 : κ = 29 ; at = 4; an = 2

! "!"" # ; ; ; t t2 ; ; ; 1 2t ; ; 0 2 ; ; ; 0 0 ; r˙ × r¨ ;6t2 −6t r r˙ r¨ r

t3 ; ; ; 3t2 ; ; 6t ; ; ; 6 ; ; 2 ;

$ % |r|˙ = 1 + 4t2 + 9t4 |r˙ × r¨| = 2 9t4 + 9t2 + 1 [r,˙ r¨, r ] = (r˙ × r¨) · r = 12

12 + 9t2 + 1) t = 0√: κ = 2; w = 3 2 κ= t −t )2 (e + e√ 2 w=− t (e + e−t )2√ √ 2 t = 0 : κ = 4 ; w = − 42 √ t4 + 4t2 + 1 κ= (1 + t2 + t4 )3 2 w= 4 t + 4t2 + 1√ t = 1 : κ = 32 ; w = 31 2t κ= 2 (2t + 1)2 2t w=− 2 (2t + 1)2 t = 1 : κ = 29 ; w = − 92 √ 9t4 + 4t6 + 1 κ= (t2 + 1 + t6 )3 −6t w= 4 9t + 4t6 + 1√ t = 1 : κ = 942 ; w = − 37 √ 2 1 ; w=− κ= 3 3 =

a = r¨ = (0, −2, 0)Ì κ=

4(9t4

& ' # w = a2 +b b2 > 0 ( ! # w = − a2 +b b2 < 0

ab

Q(x) = 2 x2 h VB # ab h 2 = 2 x x h 0 1 = abh 3 4π 3 R VB = 3 108 Q(x) = 2 x VB = 72

Q(x) =

A−a x a+ h B−b × b+ x h 1 VB = AB 3 Ab + aB + ab + 2

Q(x) =

ab π 2 (h − x)2 h 1 VB = πabh 3 2 VB = R3 tan α 3

Vx = 12π Vx = πa3

Vx = πph2

sinh 2

+1 √ 3 π 5π + Vx = 4 3 2 2

π2 2 (π + 2)π Vx = 4 32πa3 Vx = 105 8πa3 Vx = 3 Vx = 72π Vx = 5π2 a3 32 3 πa Vx = 105 8πa3 Vx = 15 8 2 Vy = πa b 3 2048 π Vy = 35 512π Vy = 15 512π Vy = 7 πa3 Vy = 6 64π Vy = 3 4 2 Vy = πa b 3 Vy = 19, 2π 272 π Vx = 15 11 π Vx = 4 V = 2π2 a2 b 128π V = 3

Vx =

V = 3π2 8πa3 3 32 πa3 Vp = 105 4πa3 Vp = 21 Ax = 4πr2

Vp =

Ax = πa2 (sinh 2 + 2) "√ Ax = 2π 2 √ 5 + ln 1 + 2 √ 34 17 − 2 Ax = π 9 62 π Ax = 3 64 2 πa Ax = 3 Ax = 2, 4πa2 Ax = 29, 6π Ax = 3π A = 4π2 ab 14π Ay = "3√ Ax = π 2 √ 5 + ln 1 + 2 12 2 πa 5 √ Ap = 2πa2 (2 − 2) 128 2 πa Ap = 5

Ay =

1, 413 1, 059 F = 61 g W ah2

xS = 0;

Ix = 13 ah3

1 3 a h 3 1 3 ab Ix = 12 1 3 a b Iy = 12 Iy = 6, 4

yS =

xS = 43 a;

Iy =

F = 32 g W r3 F = 2, 3544 F = 31 g W ah2 πa4 I = x 167, 424 16 h a3 a3 x = √ Mx = ; My = 2 6 6 a a ; y x = = S S 47, 088 3 3

4a 3π

yS = 0, 3b

xS = 0; yS = 85 ≈ 34, 517 ! "" $ ≈ 1, 432 ! ≈ 15, 6729 ! 2880 % ≈ 5108, 3

#

- x && " ' s(x) = 1 −1 x ()# ⎧ & ' *n ⎨1 0 0 < x ≤ 1 x s(x) = ' rn (x) = 1 − x ⎩0 0 x = 0 " 15 ⎧ ⎨(1 − x)n 0 0 < x ≤ 1 "& + ,-- 0; 2 r(x) = 1 ⎩0 |rn (x)| ≤ n−1 < 0, 001. )-" 0 x = 0 2 2 - &. & n " lg 1000 n−1> . - ) n ≥ 11 r 34. 2 34 - 0, 9. )# n lg 2 -" x < 1 − 0, 9. " "& /0 ), - " ' # + ,-- [0, 1] ), " ' # ∞ - &54 "& + # (−1)i+1 ai & ai > 0 0 -- "1 5 ,-- I = 2 , 1 ), i=1 i - - &54 . " " - 0 " n

∞ (−1)i+1 ai < an+1 i=n+1

" "& $ |rn (x)| <

1 x2 + (n + 1)2

1 1 < 2 (n + 1)2 n 0 -- x ∈ (−∞, +∞) 1 - $ n12 ≤ 10−4 0 n ≥ 100 " ≤

x∈I :

)-" n > −lglg2ε 6 7 # -- |rn (x)| < 0, 01 0 n ≥ 7 /0 " ' " - " ' - |rn (x)| < ε

1 xn+1 ≤ n+1 n+1 x ∈ [0, 1] 1 |rn (x)| < 0, 1.

|rn (x)| <

0

n ≥ 9 ⎧ ⎨1 + x3 x > 0 s(x) = ⎩0 x = 0 ⎧ 1 ⎪ ⎨ x > 0 3 )n−1 (1 + x r(x) = ⎪ ⎩0 x = 0 n rn (x) √ 0, 1 n−1 x3 < 10 − 1 x ≥ 0 ! " " # $ x ≥ 1

"# % x≥1: 1 |rn (x)| ≤ n−1 |rn (x)| < ε 2 lg ε n > 1 − & lg 2 ' ε = 0, 001 : n ≥ 11 1 1 − ( fi (x) = x+i−1 x+i 1 1 ) sn (x) = − x x+n 1 s(x) = lim sn (x) = n→∞ x 1 1 ≤ x ≥ 0 |rn (x)| = x+n n ⇒ |rn (x)| < 0, 1 n ≥ 10 * $ % x ≥ 0 + ! , "* + -

1 1 1 1 + + 2 + 3 + ... 3 3 3 . " / ! "# x ≥ 0 rn (x) " " ! - 1 |rn (x)| ≤ ) 2 · 3n−1 |rn (x)| < 0, 01 n ≥ 5

x ≥ 0

* $ x ≥ 0 + ! " " + -

1 1 1 1 + + + + ... 2 4 8 0 !

"# x ≥ 0; 1 |rn (x)| ≤ n−1 . 2 |rn (x)| < 0, 01 n ≥ 8 ∞ cos nx 1 1 1 * n−1 ≤ n−1 n−1 2 2 2 n=1 ∞ 1 1 x 1 * 2 sin ≤ 2 n n n n2 2 * x > 0 * x > 1 "* |x| > 1

" √ √ 5 * − 5, 5

√ √

√ √ % 3 3 5 5 , , "* − * − 2 2 2 2

3 * [−3, 3)

√ √

* [− 3, 3]

* (−∞, ∞)& #

n=1

* (−1, 1]

√ √ % 2 2 , * − 3 3

1 1 %* [−1, 1] * − , 10 10 *

* r = e

* [−5, 3)

* (1, 2]

* [−1, 3)

[−1, 0) s(x) = (1 −1 x)2 ,

sin2 x = |x| < 1

1 − 2x s(x) = , |x| < 1 (1 + x)2 ∞ x2n |x| < ∞ (2n)! n=0

x2n+1 sinh ϑx R2n (x) = (2n + 1)! ϑx e − e−ϑx ≤ e|x| | sinh ϑx| = 2 0 < ϑ < 1 |x|2n+1 |x| e |R2n (x)| ≤ (2n + 1)! ∞ |x|2n+1 ! (2n + 1)! n=0

" # $ % &#' (( ) 2n+1 |x| = 0 |x| < ∞ lim n→∞ (2n + 1)! (( ) n→∞ lim |R2n (x)| = 0 |x| < ∞$ ∞ x2n cosh x = |x| < ∞ (2n)! n=0

* 0 < ϑ < 1 R2n (x) = (−1)n 2n+1

x · 22n sin (2ϑx). (2n + 1)! |2x|2n+1 |R2n (x)| < (2n + 1)! lim |R2n (x)| = 0 +#( ,- ×

n→∞

# $ (

∞

(−1)n+1

n=1

|x| < ∞

2

s(x) = (11+−xx2 )2 , |x| < 1

s(x) = arctan x, |x| ≤ 1 s(x) = − ln (1 − x), x ∈ [−1, 1) s(x) = (11−+x)x 2 , |x| < 1

.

22n−1 2n x (2n)!

∞ (x − a)n e =e , |x| < ∞ n!an n=0 x (x − a)n+1 · e1+ϑ( a −1) Rn (x) = (n + 1)!an+1 lim |Rn (x)| = 0 x/a

n→∞

x4 x2 + − ... 1− 2 6 3 (x − 1)2 3 + ... 1 + (x − 1) + 2 4 2! 1 x − 2 (x − 2)2 − + − ... 2 4 8 x4 x6 x2 − − ... − − 2 12 45 ∞ −3 n f (x) = x n n=0 ∞ (n + 1)(n + 2) n x ; = (−1)n · 2 n=0 |x| < 1 ∞ 1 − 2 2n f (x) = x n n=0 ∞ (2n)! 2n = (−1)n · 2n x ; 2 (n!)2 n=0 |x| < 1 ∞ x2k−1 ; |x| < 1 f (x) = 2 2k − 1

e

/

k=1

f (x) = ln 2 + ln (1 − x)

x = ln 2 + ln 1 − 2 ∞ xn 1 + 2−n − ; |x| < 1 n n=1

f (x) = ln∞(1 + x3 ) − ln (1 + x) = −2

n=1

cos

nπ xn ; |x| < 1 3 n

∞

(2n)! 2n f (x) = x ; |x| < 1 2n (n!)2 2 n=0 ∞

x2n ; |x| < ∞ f (x) = (−1)n n! n=0

f (x) =

∞

(−1)n

n=0

2n n+1 x ; n!

|x| < ∞

2 1 1 − · 1 − x 3 1 − x3 ∞ 2 1 + n+1 xn ; |x| < 1 =− 3 n=0

f (x) = −

f (x) =

∞

(−1)n

n=0 ∞

f (x) = 2

x2n+1 ; |x| < 3 9n+1

(−1)n · 32n

n=0

(n + 2)x2n+1 ; |x| < ∞ (2n + 1)!x2n+1 ∞ (2x)2n 1 ; f (x) = 1 + (−1)n 2 n=1 (2n)! |x| < ∞ ∞ (2n)! n 1 √ f (x) = (−1)n n x ; 6 (n!)2 3 n=0 3 |x| < 2 f (x) = ln [1 + (x − 1)] = ∞ (x − 1)n ; 0<x<2 (−1)n+1 n n=1 ×

1 1 f (x) = − · 3 1 − x+4 3 ∞ 1 1 1 + · = x+4 n+1 2 1− 2 2 n=0 1 − n+1 (x + 4)n ; −6 < x < −2 3 ∞ (x + 2)n f (x) = − ; 2n+1 n=0 −4<x<0

f (x) =

n−1 ∞ x − π2

(n − 1)!2n−1 (2n − 1)π × cos |x| < ∞ 4 ∞ 32n+1

f (x) = (−1)n+1 (2n + 1)! n=0 π 2n+1 × x+ |x| < ∞ 3 x+1 + f (x) = −1 + 3 ∞ 2 · 5 · 8 · . . . · (3n − 4) (x + 1)n n!3n n=2 −2 < x < 0 n=1

f (x) = −16(x + 2) + 20(x + 2)2 −8(x + 2)3 + (x + 2)4 ∞ πn (x − 1)n

f (x) = 3n n! n=0 π π × sin +n |x| < ∞ 3 2 x−4 + f (x) = 2 + 4 ∞ (2n)!(x − 4)n (−1)n−1 · 4n−1 2 (n!)2 (2n − 1) n=2 0 < x < 8 ∞ (x + 2)n −2 f (x) = e n! n=0 |x| < ∞

f (x) =

∞

(n + 1)(x + 1)n

n=0

−2 < x < 0 f (x) =

∞

(−1)n+1 x2n−2

n=1

|x| < 1 # x t = arctan x 2 0 1+t ∞ x2n−1 = ; |x| < 1 (−1)n+1 2n − 1 n=1

1 1 1 (−1)n+1 ≈ + Φ 5 5 9 · 53 (2n − 1) · 3n−1 n=1 2 1 √ 1 1 1 < 2 · 6 < 0, 00001 π ≈ 2 3 1− + − 3 · 2! 5 1 3 · 3 5 · 9 7 · 27 1 Φ = 0, 2008 ≈ 3, 142 + 5 9 · 81 x9 x5 ∞ 2n−1 + ∓. . . Φ(x) = x− x 2! · 42 · 5 4! · 44 · 9 C + (−1)n+1 1 (2n − 1)!(2n − 1) n=1 Φ = 0, 499805 2 ∞ xn # π2 C + ln x + 1 n!n U = 4a (1 − ε2 cos2 t) 2 t n=1 √ 0 ∞ 2n+1 a2 − b 2 x ε= Φ(x) = (−1)n+2 a n!(2n + 1) n=0 1 1 1 1 " ε2 1 · 3 2 ε4 ≈ − 4+ 5 Φ U = 2πa 1 − 2 + · 3 3 3 3 · 10 2 2·4 3 1 6 2 ≈ 0, 321388 − 7 1·3·5 ε 3 · 42 − · 2 x5 12 ·· 34 ·· 56 · 7 25 ε8 5 1 x3 − 2 · Φ(x) = x + · − − ... · 3 3 3 2! 5 2·4·6·8 7 2 · 5 x7 2 · 5 · 8 x9 + 3 · − 4 · ± ... 3 · 3! 7 3 · 4! 9 ∞ √ π=2 3

l = π # bn =

2 π

π

0

1 · sin nx x

b2n = 0; b2n−1

2 1 − (−1)n = · π n 4 = π(2n − 1)

n = 1, 2, 3, . . . ∞ 4 sin (2n − 1)x f (x) = π n=1 2n − 1 ∞ π 4 sin (2n − 1) π2 f = =1 2 π n=1 2n − 1 ∞ (−1)n+1 π = ! 2n − 1 4 n=1

f (x) =

∞ 4 cos (2n − 1)x π − 2 π n=1 (2n − 1)2

f (x) =

∞ cos nx π2 +4 (−1)n 3 n2 n=1

f (x) =

∞ 2 cos (2n − 1)x 3π + 4 π n=1 (2n − 1)2

∞ (−1)n sin nx + n n=1

f (x) =

∞ 2 cos (2n − 1)x π + 4 π n=1 (2n − 1)2

" f (x) =

2 π −

f (x) =

∞ cos 2nx 4 π n=1 (2n − 1)(2n + 1)

∞ 4 (−1)n+1 π n=1 (2n − 1)2 × sin (2n − 1)x

f (x) =

∞ 4 1 π n=1 2n − 1

(2n − 1)πx l ∞ cos (2n − 1)πx 1 4 f (x) = + 2 2 π n=1 (2n − 1)2 × sin

l f (x) = − 4 ∞ 2l 1 (2n − 1)πx cos 2 2 π n=1 (2n − 1) l ∞ n+1 l nπx (−1) + sin π n=1 n l 4l l f (x) = − 2 2 π ∞ 1 (2n − 1)πx × cos 2 (2n − 1) l n=1

f (x) =

∞ 3 2 cos (2n − 1)πx − 2 4 π n=1 (2n − 1)2

∞ 1 sin nπx π n=1 n $ ∞ 1 +2 f (x) = sinh l (−1)n l n=1

−

nπx l · cos nπx l − πn sin l × l 2 + n2 π 2

%

3 4 ∞ n cos nπ 4 nπx 2 − (−1) + 2 cos 2 π n=1 n 2

f (x) =

f (x) =

∞ nπx 21 sin + π n=1 n 2

∞ 4 (−1)n+1 (2n − 1)πx sin π2 n=1 (2n − 1)2 2

D(f ) = {(x, y)| x ∈ R ∧ y ∈ R} = R2 − 3x + 4y + 8 = c

3 m = . 4 z = f (x, y) R3 8 7 D(f ) = (x, y)| x2 + y 2 ≤ 52 ! "#$ % & M (0; 0) r ≤ 5. z = f (x, y) ' " M (0; 0; 0)( r = 5 0 ≤ z ≤ 5 D(f ) = R2 % ') * + c > 0 , ,,,( + c < 0 ,, ,- . z = f (x, y) %) #%

/ ## 0 12%

D(f ) = R2 "#$ % & M (0; 0), c ≥ 0. z = f (x, y) 3# # ##

D(f ) = R2 \{(0; 0)} "#$ % & M (0; 0) c > 0. z = f (x, y) 3# #12 % ( &

4 z = 2 , y = 0 3# # x z 4% $ (z > 0) D(f ) = {(x, y)| (x ≥ 0 ∧ y ≥ 0) ∨(x ≤ 0 ∧ y ≤ 0)} ! % ') , ,,, . (c > 0) 5+ c = 0

4

4 2 0 5

5

z

0y x0 5

5

3

6

22

6

24 1

2 4

0

2 6

2

4 0

2

4

2 0

0 1

26 4 4

2 6

2 0

6 0 3 4 4 3 2 1 0 1

0

4

2

2 3

! " #

x = a, z 2 = ay y = b, z 2 = bx $% % & ' ( z = f (x, y)

7

8

( D(f ) = (x, y)| x2 + y2 < 4 ( D(f ) = {(x, y)| (−2 ≤ x ≤ 0

∧ y ≤ 0) ∨ (0 ≤ x ≤ 2 ∧ y ≥ 0)}

( D(f ) = {(x, y)| y > −x} 7 8 √ ( D(f ) = (x, y)| y > x ∧ x ≥ 0 ( D(f ) = {(x, y)| − 1 ≤ x ≤ 1

*

- * . -

( c > 0 / 0, - * z , 1 c < 0 !% / 0, - * z , 1 c = 0 2 f (tx,ty) = = t2

4 − 2tx · ty (tx)4 + (ty)

x2 + y 2 − 2xy = t2 · f (x, y)

3 a −a b + b −b a = 1

( 1. 4 5 6, ! (x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0) ( ( 1., 3., 6. 8. 4 6, ! (xyz > 0)

( 78 * ! ( D(f ) = {(x, y)| y < −x} ) x = ±1 y = ±1 ( # z = ±1 78 , ( # ( ) * * +*, ( * * / r = 1 $ n = (1, 1, 1)Ì * O(0; 0; 0) ( ! * ∧ −1 ≤ y ≤ 1} 7 D(f ) = (x, y)| x2 + y 2 ≥ 1 8 ∧ y > −x2

Δx z = (2x − y + Δx)Δx Δy z = (2y − x + Δy)Δy Δz = Δx z + Δy z − ΔxΔy

x = 2, Δx = 0, 1, y = 2, Δy = −0, 1 : Δx z = 0, 21; Δy z = −0, 19 Δz = 0, 03 m lim f (x, y) = y = mx x→0 1−m 3 2 y = x y = x 4 3 y = 2x

3 2 (−2) ! lim lim f (x, y) = −1 y→0 x→0 lim lim f (x, y) = 1 x→0 y→0

1 − m2 lim f (x, y) = x→0 1 + m2 y = mx " # $ % lim lim f (x, y) = 0 y→0 x→0 lim lim f (x, y) = 0 x→0 y→0 ⎧ ⎨1 y = x lim f (x, y) = x→0 ⎩0 y = 2x y→0

& −

1 4

1

%

0 1 2 1 4 '

2 $ % $ %

% ( x = 1, y = −1 ) * # x = mπ y = nπ (m, n ∈ Z) % ) * + x2 + y 2 = 1 ) * # x + y = 0 , - y 2 = x . ) * +,, / x = 0, y = 0, z = 0 ) * + z 2 = x2 + y 2 % ) * % 012 ,, z 2 − x2 − y 2 = 1

fx = 3x(x + 2y); fy = 3(x2 − y 2 ) 2x 2y ; fy = 2 2 +y x + y2 y x % fx = − 2 ; fy = 2 2 x +y x + y2 √ 3 t √ gx = √ 3 3x( √ x − 3 t) 3 x √ gy = √ 3 3t( t − 3 x)

fx =

x2

a − b cos γ c b − a cos γ ab sin γ ; cγ = cb = c c z y 1 1 fx = − 2 − ; fy = − 2 x z x y x 1 fz = + 2 y z

ca =

fx = e−xy (1 − xy) fy = −x2 e−xy

gx = (x +5t2t)2 ;

gt = −

5 (x + 2t)2

|x|y fx = − 2 x x2 − y 2 |x| fy = x x2 − y 2 3y fx = (3y − 2x)2 3x fy = − (3y − 2x)2 hx = cot (x − 2t) ht = −2 cot (x − 2t)

gx = 2 sin y cos (2x + y)

gy = 2 sin x cos (x + 2y)

fx = yxy−1 ; fy = xy ln x fx = − xy2 esin cos xy y x

1 sin y y e x cos x x xy 2 2x2 − 2y 2 fx = |y|(x4 − y 4 ) x2 y 2x2 − 2y 2 fy = − |y|(x4 − y 4 ) fy =

fx = yz(xy)z−1

fy = xz(xy)z−1 fz = (xy)z ln (xy)

fx = yz xy ln z; fz = xyz xy−1

fx = yz exyz ; fz = xy e

fy = xz xy ln z fy = xz exyz

xyz

fx = tan x;

fy = − tan y 2

+x ) zxx = − 2(y (y − x2 )2

2x (y − x2 )2 1 zyy = − (y − x2 )2 2u wuu = − ; wuv = 0 (1 + u2 )2 2v wvv = − (1 + v 2 )2 zxxx = 6; zxxy = 2; zxyy = 0 zyyy = 6 zxy =

αx =

t √ 2 x − x2 t2 x αt = 1 − xt2 fx = −a sin (ax − by) fy = b sin (ax − by)

zx = 2(x +1√xy) zy =

1 √ 2( xy + y)

√

zx = 2√1 x sin xy − yx2x cos xy

√ y x cos zy = x x 2

ux = t12 ex/t2 ; ut = − 2x ex/t t3 ux = ux ; uy = uy ; uz = uz 2 zxx = 2 (x −y y)3 xy zxy = −2 (x − y)3 x2 zyy = 2 (x − y)3 zx = y1 e xy ; zy = − yx2 e xy 1 x x x zxy = − 2 e y − 3 e y y y

!

y f + x · f − 2 ϕ y x xf 1 uy = − 2 + ϕ y x f + x · f 1 y uxy = − − 2 ϕ − 3 ϕ y2 x x 2xf 1 uyy = 3 + 2 ϕ y x ux =

zxy = zyx = abz 4z zxy = zyx = − xy 2 zxy = zyx = (x − 2y)2

grad z = (2x, 2y)Ì grad z|(3;4) = (6, 8)Ì (0; 0) r = 5 √ ∂z =x 3+y ∂a √ ∂z =3 3+4 ∂a (3;4)

x2 − y 2 = 3 −−−→ Ì a = M N = (3, 4) 1 3 ∂z · =1 = (−1, 2) 4 ∂a 5

(6, 3, 2)Ì 3 ! cos ϑ = √ ; ϑ ≈ 18, 43◦ 10 √ 9√ 2 − 3 2√ 2 68 3 # " − 13 3

6x1 + 6x2 , 6x1 + x4 ex2 , sin x3 , ex2

grad z = (2x, −2y)Ì

grad z|(2;1) = (4, −2)Ì

Δz = yΔx + xΔy + Δx · Δy $z = y $x + x$y Δz = −0, 62; $z = −0, 6 Δz = 2xyΔx + x2 Δy + y(Δx)2 +2xΔxΔy + (Δx)2 Δy $z = 2xy $x + x2 $y Δz = −0, 298602; $z = −0, 30 Δz = (2x − 3y)Δx + (2y − 3x)Δy +(Δx)2 − 3ΔxΔy + (Δy)2 $z = (2x − 3y)$x + (2y − 3x)$y Δz = −0, 79; $z = −0, 9 $f = $g = $f = $ $u = $f =

−y 2 $x + x2 $y (x − y)2 1 s/t s e $s − $t t t x$x + y $y x2 + y 2 x$x + y $y + z $z x2 + y 2 + z 2 2(x$y − y $x) x2 sin 2y x

grad z|(4;2)

Ì

√ 2 (1, 2)Ì = 4

1 x z−1 " y+ z $x $h = xy + y y 1 + 1 − 2 xz $y y x x 5 $z + xy + ln xy + y y $ϕ = (ex cos y + 3y cos 3x)$x +(sin 3x − ex sin y)$y (x2 + y 2 )$z − z(x$x + y $y) % $ψ = (x2 + y 2 )3

$f = (x2 − x3 )x1x2 −x3 −1 ln x4 $x1 +xx1 2 −x3 ln x1 ln x4 $x2 −xx1 2 −x3 ln x1 ln x4 $x3 $x4 +xx1 2 −x3 · x4 " 0, 075 ≈ −0, 738906 −0, 1 # Δz = ln 1, 044 ≈ 0, 04306 $z = 0, 04 2

$2 u = 4 3y 2 $x2 − 4xy $x$y x

+x2 $y 2

− xy) 2 u = − (yx xy 2

2 u = 2(xy + xz + yz) 2 u = −u(mx + ny)2

2

|Δz|Max

= f (2, 1; 3, 7) −f (2; 4) = 0, 06757

|ΔR| ≤ 0, 755Ω

ΔR R ≤ 0, 64 %

|ΔR| ≤ 0, 773Ω

|Δz| ≤ 0, 0625

ΔR R ≤ 0, 66 %

< |Δz|Max

|fx| ≤ 3,17 2, 1 3, 72 1 |Δz| ≤ = 3, 72 0, 07305 > |Δz|Max

2 |ΔA 0 |≤ 1, 363

|fy | ≤

ΔA0 A0 ≤ 0, 48 %

2 |ΔA 0 |≤ 1, 373

ΔA0 A0 ≤ 0, 49 %

≤ 4, 71 % ΔV V

ΔC ≤ 0, 63 % C ΔV ≤ 2 Δa V a Δh ≤ 10 % + h ΔV1 ≤ 0, 15 % V1

|ΔPa | ≤ 10 %

2x + 2y − z = 1 r = (1, 1, 3)Ì + λ(2, 2, −1)Ì x + 4y + 6z − 21 = 0 r = (1, 2, 2)Ì + λ(1, 4, 6)Ì

2x − z − 2 = 0 r = (1, 0, 0)Ì + λ(2, 0, −1)Ì x − y − 2z + 1 = 0; π π 1 Ì , , + λ(1, −1, −2)Ì r= 4 4 2 xy0 + yx0 = 2zz0 xy0 z0 + yx0 z0 + zx0 y0 = 3a3 xx1 yy1 zz1

2 + 2 − 2 = 1 a b c x y z + − =1 a b c

x + y − z = ±9

r = (3, 4, 5)Ì + λ(3, 4, −5)Ì

(0; 0; 0) 1 cos α = − cos β = cos γ = − √ 3 9 3 V = a = const 2 √ Sx a2/3 · 3 x0 ; 0; 0 √ Sy = 0; a2/3 · 3 y0 ; 0 √ Sz = 0; 0; a2/3 · 3 z0 r = (4, 3, 0)Ì + λ(4, 3, 5)Ì a ! √ 3 " P1 (0; 0; 4); z − 4 = 0 P2 (1; 1; 2); 2x + 2y + z = 6

z = 4t3 + 3t2 + 2t t z =0 t −2 cosh t (A − C) sin 2t + 2B cos 2t 2 e2t 4t e +1 u(w + 2vt2 ) − vw et tu2 (t2 + 1) tan t

2t · ln t · tan t + t 2 (t + 1) ln t + cos2 t z u v = vuv−1 + uv ln u x x x z y = ey +x ey x x n ∂f = k · tk−1 f xi ∂(tx ) i i=1

∂z z ex ex + ey ·x2 = x = ; ∂x e + ey x ex + ey ∂z = yxy−1 ∂x z y = xy ϕ (x) ln x + x x 2x x ∂z = 1− ∂u y y x ∂z x =− 4+ ∂v y y

∂z ∂x ∂z ∂y ∂z ∂x ∂z ∂y

∂z ∂z +p ∂u ∂v ∂z ∂z =n +q ∂u ∂v ∂z y ∂z =y − · ∂u x2 ∂v ∂z 1 ∂z =x + ∂u x ∂v =m

∂z ∂z 1 y ∂z = · + ∂x ∂u 2 x ∂v ∂z ∂z 1 y ∂z = · + ∂y ∂u 2 x ∂v

ur = ux cos ϕ + uy sin ϕ uϕ = −ux r sin ϕ + uy r cos ϕ zx = f · 2x; zy = 1 − 2f y 2−x 3+y y y = − 3 x 2x − e2y 2y e y = 2x e2y − e2x y y = − x x2 + xy + y 2

y = xy 1 y = 2 3−x y ; zy = − zx = z x y x ; zy = zx = 2z 2z a b zx = ; zy = c c y zx = 1; zy = x−z 3 3 y = ; y = − 4 4 y = −1 1 4 y = ; y = 5 5 (−1; −1) (−1; 3) (−3; 1) (1; 1)

y =

z z zx = − ; zy = − x y

zx = ϕ −

y ϕ ; zy = ϕ x

x20 + x0 y0 + y02 + (2x0 + y0 )h 2

+(x0 + 2y0 )k + h + hk + k

3

2

f (x, y) = mx + ny − (mx +3! ny)

f (x, y) = 9 + 11(x − 1) + 8(y − 2)

+3(x − 1)2 + 8(x − 1)(y − 1)+ 2(y − 2)2 + (x − 1)3 + 2(x − 1)(y − 2)2

f (x, y) = −1 − 2(x − 1) + (y + 1)

−(x − 1)2 + 2(x − 1)(y + 1) +(x − 1)2 (y + 1)

x2 y = [1 + (x − 1)]2 · [−1 + (y + 1)] 2

f (x, y) = x − (y + 1) − x2

1 +x(y + 1) − (y + 1)2 + R2 2 (x − y − 1)3 1 R2 = · 3 [ϑx + 1 − ϑ(y + 1)]3 0<ϑ<1

+R3 (mx + ny)4 sin (mϑx + nϑy) R3 = 4! f (x, y, z) = (x − 1)2 + (y − 1)2 +(z − 1)2 + 2(x − 1)(y − 1) −(x − 1)(z − 1)

f (x, y) = y − (x − 1)y + R2 f (x, y) = 1 + 2(y − 1)

+(x − 2)(y − 1) + (y − 1)2 + R2 1, 12,1 ≈ 1, 22 f (x, y) = (x − 1) + (y − 1)− 1 1 (x − 1)2 − (y − 1)2 + z 2 + R2 2 2

z(x, y) = 1 + 2(x − 1) − (y − 1)

−8(x−1)2 +10(x−1)(y−1)−3(y−1)2 +R2

xMin = −4; zMin = −1

yMin = 1

xMax = yMax = 4;

zMax = 12

xMin = 1; yMin = 12 ; zMin = 0 xMin = yMin = zMin = 0 xMin = −2; yMin = 0

zMin = −

xMin = 1;

2 e

yMin = 0 zMin = −2 xMax = −1; yMax = 0 zMax = 2 √ √ xMin = 2; yMin = − 2 zMin = 8 √ √ xMin = − 2; yMin = 2 zMin = 8

Δ

z

< 0

z1 = −9 < 0

z2 = −9 < 0

z3 = −9 >0 zM in = −10 >0 zM in = −10 √ 3 xMin = 2; yMin = −3 xMin = yMin = 2V 1√ 3 zMin = 1; uMin = −14 zMin = 2V 2 1 xMax = yMax = zMax = VMax = 8 7 1 uMax = 7 7 √ √ 4 y = 2, 4 x − 0, 8 xMin = √ 2; yMin = 2√ 4 4 6, 59 zMin = 8; uMin = 4 · 2 y = −1, 21 + x 1 xMin = ; yMin = zMin = 1 3, 13 2 y = 2, 00x3 + uMin = 4 x 1 y = 7, 164 · x1,953 xMin = −3; yMin = 3 λ1 = λ2 = 3; λ3 = −3 2 44 zMin = ; uMin = 3 27 R = x21 + x22 + x23

2 1 +4x1 x2 + 4x1 x3 − 4x2 x3 xMin = − ; yMin = −

3 3 ÷ x21 + x22 + x23 4 zMin = 1; uMin = − ! " # R $ 3 R = −3; R = 3 z = −1 x = y = 1 x1 = 3; y1 = 0 √ x2 = 3; y2 = 3 √ x3 = 3; y3 = − 3 x4 = 2; y4 = 1 x5 = 4; y5 = −1

zxx 2>0 2>0 2>0 2>0 2>0

Min

zMax = 13 x = 2, y = −1 zMax = 1; zMin = −1 3√ zMax = 3; zMin = 0 2

%

&""

' ( ! )

! ") *+*

, xMin = yMin = 1; zMin = 2 xMax = yMax = −2 zMax = −4 xMin = yMin = 2; zMin = 4 xMax = yMax = ±1; zMax = 1 xMin = −yMin = ±1 zMin = −1 xMax = 1; yMax = −2

zMax = 2; uMax = 9 xMin = −1; yMin = 2 zMin = −2; uMin = −9 11 5

xMax = ; yMax = − 4 2 11 605 zMax = − ; uMax = 4 32 √ (± 5; 1)

r = 1 ;

A √ r = √ ; h = 2A π 3 π 3 √ a = b = c = r 3 xi = nc (i = 1, 2, . . . , n) ! a = b = c = A6

h = 2

AMax = 9 sin α ÷ sin β = v1 ÷ v2 AMax = 2ab

94

=1

" 6 + 4 ln 0, 5 " 20 56

" 15 − 8 ln 4 " 12 − 1e " 92 " a2

"

"

#

1 6

a

y=0

# √2a2 −y2 x y √

#

= #

x= ay x2 a # a

+ x=a

y x

y=0 x=0 √ √ a 2# 2a2 −x2

=

π 1 − 4 2 # 2−x2

√ 8 2 5 21 p # 1 y x " 0 x=0 y=x # 1 # y 88 " = x y 105 y=0 x=0 " 6 # 2 # √2−y + x y " 1 y=1 x=0 2

y=0 2

" 43 16 ln 2 − 9 38 ln 2 ! 16

y x Jx = ab3 ; 3

a (3π − 2) 12

a3 b ; 3 ab3 + a3 b Jp = 3 π π " 2 ; 8 " (3; 4, 8) Jy =

4

a 80

" 28 15 1 " 24

32 1 5 + − 16 2

a 12

1 110

"

ln 2

4

42 23 # a# 0

0

a−x# a−x−y 0

z z y x =

a4 24

"

a a a ; ; 4 4 4 a 0; 0; 3 5

" a4

√ 32 2a5 135

πa5

√ 2

2 π 3 3 πa4 2 # π/4 #

3 2 πa 8

8π ln 2 3 πa3 16 5πa3 16

1736 15 2/ cos ϕ f (r)r r ϕ π ϕ=0 r=0 16 √ aπ 8π 3 2 1

a3 4 3

3 πh2 R2 # 1 # u

4 f (x, y) π u=0 v=−u

1 10 × v u 2 8 # 2 # 2−u

a2 9 + f (x, y) u=1

6kπa2 k ! " # πa3 3 πa3

60

4

$ ! % VB = πa3 3 1 $ !% VC = πa3 3 VB − VC & =3 VC #

Q =

(x, y, z) b = # 2π #Bh # z zr r z ϕ

v=u−2

0 1 × v u 1 2

π · abc2 1 4 x = (u + v) 2 πabc 1

y = (u − v) 2 2 3 4πa

3a2 35 ln 2 2 18π 868 2 a

2πa3 15

0

0

0

=

y − xy = 0

ty˙ − 2y = 0 y − 2xy = 0

x + yy = 0 y˙ = y 2

2

3y − t = 2ty y˙

xyy xy 2 + 1 = 1

' y = xy ln

x y

x2 + y = xy (

2xy + y = 0

( y¨ − y˙ − 2y = 0

(

# y¨ + 4y = 0 ! y − 2¨ y + y˙ = 0 2

2

y − x = 2xyy

( (

()

π h4 4

O

( ) ' ! y = t O*

t" # O

# 3 + y = t $%"

&'

y = C e1/x x + y = ln C(x

+1)(y + 1)

r = C e1/φ +a 2 s2 = t − 1t + Ct 1 + y2

=0

y2 − 1

= 2 ln et +1 −2 ln(e +1)

r = C cos φ,

+ 1 + t2 = C

t2 + y2 = ln Ct2 y = a + 1 Ct + at y = C sin x y = C e−1/x 2 2y = (1Cx −1 + x)2 2

r = −2 cos φ √ y = x ln x−x+C, √ y = x ln x − x + 1 √ C 1 + x2 √ y= , x + 1 + x2 √ 1 + x2 √ y= x + 1 + x2

ln 16 kt = k · 10 ln 2 t = 40 '

, q = q(t) = CU

1 − e− RC t

I = 0, 316 * t ≈ 7 ' ( FH = H(−1 , 0)Ì

FT = FT (cos α, sin α)Ì FL = (0 , −qx)Ì

2

FH + FT + FL = o =⇒ H = FT cos α, qx = FT sin α 5 ' # FT : qx = tan α = y H

,- . & / t 0' " )$ T * 2 y = C x qx T = −k(T − 20 ◦1) , y(0) = a y = + x2 + a2 H t 2 k " qx2 +a 6$ y= C −x 3 " 2H y = 1 + Cx + * √ 7 vS = (vS , 0)Ì ln(T − 20 ◦ 1) = −kt y = C√e x vF = +C * $ y = e x−2 %Ì x2 t = 0 T = 2 0 , v0 1 − 2 100 ◦1 y = C sin 2x − 1 a C = ln 80 kt = 1 2 y = 2 sin2 x − 80 ◦ 1 v = vF + vS 2 ln * T − 20 ◦ 1 1 1 v ' 2 T1 =

x + y = C * y = −x 8# y = y(x) ◦ ◦ 25 1 T2 = 60 1 9 2 '' y = t

! " v2 =⇒ y = tan α = 2 t3 − y3

' 2 4" v1 k 2 +3 t2 − y 2 + 5

y =

v0 vS

x2 1− 2 , a y(−a) = 0

y=

p 1 + (p − 1) e−kt

*4

y PN = cos α 2 = y 1 + tan α

! " # v0 x3 2 Y − y = y (X − x) = y 1 + y2 y= x− 2 + a $ Y = 0% &' vS 3a 3 ! ,- 4v

, y(a) = 0 a

$ () A 3 vS x2 + y 2 = c2 4

# ./( x2 − y 2 = c2 y

0 x ' X = x − A p = −gp , p(0) = 0 y 2x p0 5 y = *

+ 1 −x y XA = 2x x = − p = p0 e−g x/p y k

T

! ' 6 e = − 4πe2 % y = Cx−λ C > 0

' k + c T = y(x) −→ ∞

,' 4πe 2 x −→ 0 - xy = −a % k c & ./( + k y(x) −→ Cμ−λ + c

20 ◦ 7 = 0 x2 + 2y 2 = c2 1' x −→ 0 4π2r ( k y(t + Δt) ≈ y(t) + c 100 ◦7 = 2 y − x2 = c ./(' 4πr p − y(t) +ky(t) · Δt · 160 ◦7 r p − 60 ◦ 7 T = 4 e y = cx e = 1, 6 r T = p−y

yx2 = c y, y = k 40 ◦ 7 p y(0) = 1 2 3 -)4 1 8 y = (cosh ax − 1) + b OP = x2 + y 2 % a

0

0

y = t eCt

y = t ln t2 + Ct

− 2t e y−t x

y − x = C e y−x x2 − y 2 = Cx

s2 = 2t2 ln

y + ln x = C x x y= C − ln x

sin

C t

2x % 1 − Cx2 2x y= 1 − 3x2 y = x eCx , y = x e−x/2

0 y =

Ct = x = y eCy+1

2 9 #' Y − y = y (X − x) X = 0 % &

V0 = −ON = y − xy % ON = xy − y = OM = x2 + y 2 x2 − c2 ! y = 2c

! $( 34 4( 44

5 x, y ',44 / y ' ' % x = 0 ) %

O

x=a

vF = (0 , vF )Ì vH = vH (−x, −y)Ì x2 + y 2 =⇒ v = vF + v H vH = −x, x2 + y 2 Ì vF 2 x + y2 − y vH v

y = y(x) v2 =⇒ y = v1

y

y =

x

vF − vH

1+

y 2

y = x sinh

, x y(a) = 0

vF a ln vH x

& arctan(x + y)

& x + 2y

=x+C

+3 ln |2x + 3y − 7| =C

& 5t + 10y + C

= x(vH −vF )/vH 2vF /vH 2vF /vH

−x a × 2avF /vH vF = vH : y = a x2 a − , 2 2a 2 vF < vH :

vF > vH : y = y(x) x = 0 !"

" # $ " % & 8t + 2y + 1 = 2 tan(4t + C)

& & & &

= 3 ln |10t − 5y + 6| 1 y =x− x+C t = −y y +2 tan C + 2 (y−2)2 −2(y−2)(t− 1)− (t− 1)2 + C = 0 y−1 arctan t+2 1 − ln (t + 2)2 2 +(y − 1)2 + C = 0

& y = C e−t − e−t

× ln |t − 1|

& y = − 12 e2t +C e4t

& y = (arctan et +C) & y = −1

& & &

× cosh t

π + lntan − 4 t + + C / sin t 2 y = (arcsin x + 1) × 1 − x2 ϕ = C ebt a − (b sin t 1 + b2 + cos t) t+C 1 y = sin t + 2 2 cos t

& y = t2 + Ct et ×(sin ωt − τ ω cos ωt) 3 y = ln x + Cx & z = ss2 ++ C1 '& y = t ln |t| + 1 + Ct * p˙ = r(t) " t t p p = pr(t) t & y = 1 −1 4+−t 3 +, 1 t & y = e t − 1 p = pr(t) t − a t - ., u0 p 4

3

ua (t) = 1 + (τ ω)2

×

t τ ω e− τ

+ sin ωt

( $ ) "$ −τ ω cos ωt

ua (t) =

u0 1 + (τ ω)2

= −a 1+t p = C − a ln(1 + t) ×(1 + t) $ C = p(0) p(t) = 0 =⇒ C − a ln(1 + t) = 0 =⇒ t = ep(0)/a −1 p˙ −

y =

1 t ln Ct

1 1 + C et2 2t y = 1 − Ct2 2t y= 1 − 3t2

y 2 =

2

y 2 =

et 2t + C

y 3 = t + C e−t y 3 = t − 2 e1−t 1 y = √ 3 1 − t2 − 1

4t2 + y 2 = Ct t3 ey −y = C

t e−y +y = C x3 ey −y = C y + x e−y = C x2 cos2 y + y 2 = C

m = e−2t ; y 2 = 2(C − t) e2t m = cos y; 2t2 sin y + cos 2y

=C m = 1/ sin y; t + t3 = C sin y m = 1/t; t sin y + y ln t = C m = (t2 y 2 +2ty)−1 ; t(2 + ty)5 =C y m = et+y ;

et+y y 2 + cos t

=C 1 y ; x+ = C x2 x 1 m = ; y xy − ln y = 0 1 m = 4 ; x y 2 = Cx3 + x2

m =

m = e−y ; e−y cos x = C + x

√

√

y = C1 +C2 x+C3 ex 2 +C4 e−x 2 y = (C1 + C2 x) cos 2x +(C3 + c4 x) sin 2x x

y = C1 e +C2 e3x y = (C1 + C2 x) e2x y = e2x (A cos 3x + B sin 3x) y = C1 e2x +C2 e−2x = A cosh 2x + B sinh 2x y = A cos 2x + B sin 2x = a sin(2x + φ) −4x y = C1 + C2 e x = C1 et +C2 e−4t φ φ = A cos + B sin 2 2 s = e−t (A cos t + B sin t); s = e−t (cos t + 2 sin t)

y = C1 ex +(C2 + C3 x) e2x y = C1 cosh 2x + C2 sin 2x +C3 cos 2x + C4 sin 2x 2x y = C1e + e−x √ √ × C2 cos x 3 + C3 sin x 3 y = (C1 + C2 x + C3 x2 ) e−ax y = A sin x sinh x +B sin x cosh x + C cos x sinh x +D cos x cosh x y = A cosh x + B sinh x x x +C cos + D sin 2 2 g ϕ¨ + ϕ = 0 l

g g + C2 sin t ϕ = C1 cos t l l

l g

F = −ky ey ! m¨ y + kx = 0; y(0) = −b, y(0) ˙ =0 mg k = b "# y = −b cos t gb T = 2π gb

$ % ! m¨ y + αy˙ + ky = 0 αt y = −b e− 2m cos ωt,

! ω =

T = 2π ω & ' z = C1 + (C2 + C3 t) e

'

'

( ' '

'

t

α2 g − b 4m2

+C4 e

−t

y = t(A0 + A1 t + A2 t2 ) + t2 (B0 +B1 t) et +(D0 + D1 t +D2 t2 + D3 t3 ) e−2t cos t +(E0 +E1 t+E2 t2 +E3 t3 ) e−2t sin t z = C1 e2t +C2 e−t +C3 cos t +C4 sin t

T = 2π

y = A + (B0 + B1 t + B2 t2 ) e0,5t +tD0 cos t + tE0 sin t z = C1 + C2 et +C3 e2t y = tA + B e−2t +(D0 + D1 t) e−t sin 3t +(E0 + E1 t) e−t cos 3t 1 5t e y = C1 cos 3t + C2 sin 3t + 34 1 3 1 2 1 t − t − t y = et 6 4 4 +C1 e−t +C2 et t y = C1 cos 2t + C2 + sin 2t 4

' y = 31 et + C1 − 21 t e2t +C2 e4t ' y = (C1 + C2 x) ex + e2x ' y = C1 e2x +C2 e−2x −2x3 − 3x ' y = C1 e−x +C2 e√−2x π − 2x 4 y = C1 cos x + C2 sin x + x + ex 3 y = C1 + C2 e−3x + x2 − x 2 y = e−2x (C1 cos x + C2 sin x) +x2 − 8x + 7 y = C1 e2x +(C2 − x) ex x = A sin k(t − t0 ) − t cos kt +0, 25 2 cos

' ' )'

' ' √ √ ' y = C1 ex 2 +C2 e−x 2

−(x − 2) e−x

3

' y = C1 + C2 e2x − x6

' y = 21 e−x +x e−2x +C1 e−2x ' *' ' '

' ' +' , '

+C2 e−3x x = e−kt (C1 cos kt + C2 sin kt) + sin kt − 2 cos kt y = C1 + C2 x + (C3 + x) e−x +x3 − 3x2 x e−3x y = C1 e3x + C2 − 4 +C3 cos 3x + C4 sin 3x x = C1 +C2 cos t+C3 sin t+t3 −6t x −2x e y = C1 + 12√ √ + C2 cos x 3 + C3 sin x 3 ex t2 x = C1 + C2 t + e−2t 2 t 1 t x = A cos + B sin + a a a t −t y = e$ + e− 2 √ √ % √ 3 3 3 t+ sin t × cos 2 3 2 +t − 2

t2 t 1 − − 2 3 18 √ √ e−t 61 + cos t 5 + √ sin t 5 18 5 1 1

y = cos 2t + sin 2t + sin t 3 3 1 t t2 t3 t + y = e2t + − − 8 2 4 4 6 x cos 2x y = C1 − 2 1 + C2 + ln sin 2x sin 2x 4 y = [(C1 + ln cos x) cos x + (C2 + x) sin x] e2x

y = C1 cos x + C2 sin x x π + − cos x ln tan 2 4 −x −x y = C1 + C2 e −(1 + e ) × ln(1 + ex ) + x 1

y = e−2x C1 + C2 x + 2x y = x2 ln x 3x2 − + C1 + C2 x e−2x 2 4 1 y = C1 sin x + C2 cos x + 2 cos x y = (C1− ln | sin x|) cos 2x 1 + C2 − x − cot x sin 2x 2 2 y = C1 + 4 − x x +x arcsin + C2 x ex 2 y

!

" F = yπr2 g ey # $ % & m¨ y + yπr2 g = 0, m = πr2 l & g g t + C2 sin t y = C1 cos l l

y =

' ( & T = 2π

l g

)* + # , & F y = (l − x) α - % & y(0) = y (0) = 0 F x3 2 & y = lx − 2α 3 x3 6 ) x = a (e−t +t − 1)

)) y =

). m¨ y = −mg − αy˙ (α > 0, '(/((( ) - % & y(0) = h, y(0) ˙ =0

g g & y = h − t + 2 1 − e−βt β β β = α/m ) - % & m¨ x + kx = 0; x(0) = x0 , x(0) ˙ =0 k & x = x0 cos t m )0 $ % & m¨ x + kx = 2ωm cos ωt k 1 (# ω = m & x = C1 cos ωt +C2 sin ωt ˆ sin (ωt + ϕ) + t sin ωt = x x ˆ = C12 + (C2 + t)2 lim x ˆ=∞ 2-/

t−→∞

$ % ∞

kl4 x − 2kl2 x3 + kx5 120a )4 & w(x) = C1 + C2 x +C3 cos λx + C4 sin λx w(0) = w (0) = 0 =⇒ C1 = C3 = 0 w(l) = lC2 + C4 sin λl = 0 w (l) = C2 + C4 λ cos λl = 0 "( 5# & λl cos λl − sin λl = 0 =⇒ tan λl = λl /(6 # & λl = 4, 4934 =⇒ 20, 1907α " & Fk = l2

)3 w(x) =

y = t(C1 cos ln t + C2 sin ln t) + t ln t y = − 23 t ln2 |t| + 14 t3 +

(C1 + C2 ln |t|) · |t| + C3 t2 C1 1 2 1 2

y = t + C2 t + 3 t − t ln |t| y = 2t ln |t| + C1 t−1 + C2 t y = − 21 t−1 ln |t| + C1 t−1 + C2 t y = 21 ln |t| sin ln |t| +C1 cos ln |t| + C2 sin ln |t| y = C1 x + C2 x−1 + C3 x3 y = Cx1 + C2 x2

y = C1 xn + C2 x−(n+1) y = x−2 (C1 + C2 ln x) y = C1 cos ln x + C2 sin ln x 2

y = 5x3

+ C1 x−1 + C2

y = C1 x3 + Cx22 − 2 ln x + 31 y = C1 x + C2 x2 − 4x ln x y = C1 + C2 lnx x + ln y =

3

x

x3 + C1 x + C2 x2 6

y = x2 + C1 cos ln x + C2 sin ln x

z1 = (C1 − C2 t) e−t −t

z2 = C2 e t 1 − + C1 et x1 = 2 4 1 t + − − − C2 e−t 2 4 t 1 x2 = + + C1 et 2 4 t 1 − + C2 e−t + 2 4

z1 = e−4t (A cos t + B sin t)

z2 = e−4t [(A − B) sin t −(A + B) cos t] t −2t z1 = C1 e +C2 e z2 = C1 et +C3 e−2t z3 = C1 et −C2 e−2t −C3 e−2t z1 = (C1 t + C2 ) e−4t z2 = (−C1 t − C2 − C1 ) e−4t z1 = C2 + 3C3 e2t z2 = C1 e−t −2C3 e2t z3 = −2C1 e−t +C2 + C3 e2t x1 = (C1 t2 + C2 t + C3 ) e−t

+t2 − 3t + 3 x2 = (−2C1 t − C2 ) e−t +t x3 = 2C1 e−t +t − 1 z1 = 4C1 + C2 e−3t +C3 t e−3t z2 = 4C1 − 2C2 e−3t +C3 (−2t + 1) e−3t −3t z3 = C1 + C2 e +C3 (t − 1) e−3t 1 y1 = C1 e5t +C2 e−t − et 2 1 y2 = C1 e5t − e−t 2 x = et +C1 + C2 e−2t y = et +C1 − C2 e−2t x = 2 e−t +C1 et +C2 e−2t y = 3 e−t +3C1 et +2C2 e−2t x = C1 et +C2 e−t +t cosh t x = et +C1 e3t +C2 e−3t +C3 cos(t + φ)

x1 = −3 − 3 e2t +8 e3t +2 e4t

x2 = −1 − 3 e2t −2 e4t x = e−2t (1 − 2t)

3 1 y ≈ 1 + (x − 1) + (x − 1)2 2 2 2 5 + (x − 1)3 + (x − 1)4 3 12 x4 x5 x6 x2 + + + y ≈ −x − 2 12 15 60

x2 x3 x4 x5 − + + 2 6 6 20 3 4 5x 4x + y ≈ 1 + x + x2 + 3 6

y ≈ 1 −

h = 0, 2 : tn

yn

0 0 0, 2 0, 0214 0, 4 0, 091818 h = 0, 1 : t4 = 0, 4 , y4 = 0, 0918242

h = 0, 1 : t2 = 1, 2 , y2 = 0, 941176 h = 0, 05 : t4 = 1, 2 , y4 = 0, 941176

h = 0, 2 : tn

y1n

y2n

0 0 1 0, 2 −0, 163733 0, 818733 0, 4 −0, 268108 0, 670324

h = 0, 1 : tn

y1n

y2n

0 0 1 0, 1 −0, 0904833 0, 904838 0, 2 −0, 163745 0, 818731 0, 3 −0, 222245 0, 740818 0, 4 −0, 268127 0, 67032 y1 := y, y2 := y˙ h = 0, 1 : t2 = 0, 2 , y12 = −0, 58 , y22 = −2, 8 h = 0, 05 : t4 = 0, 2 , y14 = −0, 58 , y24 = −2, 8 h = 0, 2 : t5 = 1 , y5 = 2, 99997 h = 0, 1 : t10 = 1 , y10 = 3

v(x0 , y0 ) x2 + y 2 = x20 +y02

x2 + y 2 = r2 v r v(x0 , y0 ) x2 + y 2 = x20 +y02

x2 + y 2 = r2 v r ! " # v(x0 , y0 ) v(x0 , y0 ) $% x2 − y 2 = a2 a2 = x20 − y02 " y 2 − x2 = a2 a2 = y02 − x20 & y = x " y = −x #

' x ( O ) O#

$% y · x = c * + c = 0 x " y , *-

9y 10; z = 10 − 4x Ì 9y 9 4, ,− z z √ Ì (4, 3, −2) ; 29

5;√(−1 , −4 , 1)Ì ;

3 2 0; (0 , 2)Ì ; 2

$% &%' % % f %%

− √36

11

!" "

# div v = 3f (|x|)

+|x| f (|x|) = 0 C =⇒ f (|x|) = |x|3 (C ∈ R) rot v ≡ 0 =⇒ v

ϕ = −xy − xz − yz

Δf = 0 ϕ = −x2 yz − xy 2 z −xyz 2 Δ f = div v = 2yz + 2xz + 2xy ϕ = −xyz Δf = 0

Ep = ϕ(x) = mgz

− 21

2 − √1 3 (x y + y x) = 8 −16 ( − 52 ∂P ∂Q 3 = ∂x ∂y

−12 2 ) 1, 5a2 # 3a2 a2 a2 2 8a2 11a2

6 πa2 ∂(x + y) πmab = = ∂y 4 ∂y ∂x # 1* " + % ( Py = 5π Qx %$ 6 #

13

√ 8 2 2 a 3 2πa2

2

√ 2−1

2 2πa 3 √ 2πp2 2 8a2

−πa2 1 ) 35 √ 2−1 2 c √ √ 2−1 2

√c

+ % (

rot |x|x 3

=o

( (

( ( 3a2 4a2 (π − 2) πa2 14 3

#

A= #

σ = F

β

ϕ=0

#

90◦ 2

R sin ϑ ϑ ϕ

ϑ=90◦ −α 2 ◦

= R β cos(90 − α) (R . . . )

α = 30◦ , β = 60◦ =⇒ πR2 A= 6 3 5 a 20 96π

4πc 16 πc 5 a3 2 π a4 4 + 3 5 16

! 3a4 9 − + 2 ln 2 2 "# $ % " &# ' (& # ) ' (& * v = u1 grad u2 − u2 grad u1 6 Z = v(x) · x k# (Qx − Py ) b = (14.16) B # 2 π = 1 b = + 3 8 B # Q=

b B# b div E =ε B # · σ = a2 E = (14.15) F # 2π # π × sin ϑ ϑ ϕ ϕ=0

6

k

ϑ=0

= 4a2 π ε

v · x = π #

= rot v(x) · σ F

k: x = cos ϕ, y = sin ϕ,

z = 0; ϕ ∈ [0 , 2π] F : x = cos ϕ sin ϑ, y = sin ϕ sin ϑ, z = cos ϑ; (ϕ , ϑ) ∈ B B: 0 ≤ ϕ ≤ 2π π 0≤ϑ≤ 2 # div v b = 84 π B # v(x) · σ = F

F = F1 ∪ F2 ∪ F3 F1 : x = 2 cos ϕ, y = 2 sin ϕ, z = z; (ϕ , z) ∈ B1 B1 : 0 ≤ ϕ ≤ 2π 0≤z≤3 F2 : x = r cos ϕ, y = r sin ϕ, z = 0; (r , ϕ) ∈ B2 B2 : 0≤r≤2 0 ≤ ϕ ≤ 2π F # 3 : # F2 + z = 3 ··· = 4π #F1 ··· = 0 F2

# · · · = 36 π F3

!, -' . / !, !! div v = 0 x = o 1(14.15) =⇒ v(x) · σ = 0+ F # F -0 1 ' )0 / U = 4 π !! 2 & '$ rot v = o x ) x2 + y 2 = 0. <(14.17) =⇒ k v(x) · x = 0+ # k z 0.'

' )3 .0 4 ) 3 v ' z / / Z = 2π # ! [(x + y) x − 2x y] k # a # a−x =− 3 y x x=0

y=0

3 = − a2 2

! !

2a3 3 # 2# x 1

1

1 1 + 2 y x x2 y

=

1 2

4πa3

4

a3

4 π + 5 16

abc 4π 3

a5 π 12 5 a3 8πa3

»Absolut Spitze das Buch!« Ein begeisterter Leser

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