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f ( x ) #H lim x→ +
f (x ) #Y lim x→ −
f ( x ) #t lim x→
nV( 5
, >
F f & > M(%
% 5(
= PU C A
f ( x ) #H lim x→ +
f ( x ) #Y lim x→ −
f ( x ) #t lim x→
%0. %$ V
= 1
x
=
~)(
I=B !
*M
: - #
~)( I=B !
( 'D > (Q ' F
Q J G8 # x
V
V= V4 '*2! ! 5V > %U '
*M Z 54 ?@A
2 '*
A2
4?F '
5M4
M2
!
? 4'
>? 4 < @ &B
*
(
(
'J5
~)(
ZQ
!
%E (
! 1 D] V4 'V*2! V @B Z %V ' i%R!
?F '
5M4
! x→x
4 '*2! (.G s]
> F ? %J (x − x ) C A ? 4 (.G t59 ! C %, #? ' ( 'P( 4 ( " V Z t59 ! C %, ? % ' ? 4 (.G ? *M %* 4 '*2! 5(~e:9 sb C VA nV( t59 ! C %, ? % ' : !Q (" >? 4 ? M= 4 5X, #>? 4 ? M= ( Y5R ' ( m ? 4 ' X*F Z 54 (%E % P( U 5M4 t59 ! C %, 4 '*2! sS 5XV, V 3V 5 V( V T 5V4 Y5R ' P( C A t! . t59 ! C %, >? 4 ' `~ t59 ! C %, 4 5XV, VF V #& V Z 4"& % C A ! *M ' & % 4 '( sl 'V*2! 4 F ' ( ? >? 4 ' X*F ? > 4' >? 4 •% ? x = 'P( . '@ : *B 4 ' % x → ?VP
(.G
(5
sin x ≈ x tan x ≈ x − cos x ≈
x
x→∞
>? 4 ' 5c B5,
Z%
x
=
! ?(5 E ' 5c
xZ%
(5*E .
#
4 '*2!
x→∞
[ ] D [%CM \34 # ! > V !Q '
F
E
-.V & V C ,%\VJ
F ! } ! T *M / +, -. U 4 > %U ' i51 ! ( 'P( a% X*VF V >?(5V E ' 5c A ! n %4 M ?(
! %F € M7• ε T / +,
[− ε ] = − ! [ε ] = V
a
5(
% 5B
>?( ! Z!5 / +, -.
J
? % ' '+ +, 5 > (5 ) 5c / +,
[a + ε ] = a + [ε ] = a [a − ε ] = a + [− ε ] = a − ? ?
' 52 a + ε ZQ ' 52 a − ε ZQ
q7 F 5*: ' 4 a q7 F 5* 4 ' 4 a
x x
Z% x → a + 4 Z% x → a − 4
! !
#1 D x + x− x −
lim x→
Z% >? 4 < @ &B Z 54
F!
lim
? 4'
~)(
n(
> F t59 ! C %, ? (x − ) > %U '
ZQ Z 54 (.G
x→
5M4
( q7 ? F '
(x + x + x− = lim (x − x − x→
)(x − ) = (x + ) = lim )(x + ) x → (x + ) x+ − x + x
x→−
C A t! . x+ − × x + x
lim x→−
>? 4 ' = lim x→−
x+
x(x +
)(
x+ +
t59 ! C %, q7 ?( x+
+
x+
+
= lim x→−
(x
+ x
)
x→−
x+
x(x +
)(
x+ +
x→
X*F tan x ≈ x
?
4'
lim x→
?
'
52 [
−ε]
5X,
F
q7 x → x→
> (Q ' Z!5 / +, -.
lim (5
+
)=
)= −
=
×
−
tan x tan x x
{ Z 4 Z%
−
[x] −
lim x
> F x − = −ε
x+
tan x tan x x× x x = lim = lim = x x x x→ x→ [x ] − lim x→ − x −
T [x]
x→
)(
(x + )
F t59 ! C %, = lim
P(
x+ −
lim >? 4 '
4 '*2!
q7 x → t59 ! C %,
lim ? 4 ' Y5R ' P(
X
−
[
−
J
−
! ?(
/ +, -.
= lim x→
/ +,
[
− ε ]− x −
−
?
'
− ε ]− + [− ε ] − − = lim = = +∞ −ε x − x − x→ −
> % > %U '
%J l 5* 4 x q7 x → ( @ ' 5 5 ε 5 ? M=
−
Z% 5 !
sin x − cos x
lim x→
− cos x ≈
x
?
4'
5X,
F
K% M4 ! K% F Z 4 Z%
= lim
x = x
x
lim x→
sin x = − cos x lim x x→
x→
>? 4 '
X*F sin x ≈ x !
lim x→
> M
(@ '
(
4?
' ' ! %U '
lim π +
(@ '
π
π
tan x
+
{
4?
π+
q7 F &2 ! <%F
tan x = −∞
'
x→
lim x→
5X,
( q7 >? F '
5M4
? 4'
~)(
x +x− x −
n(
4 '*2!
(x − ) 4 5 ‚ (x − ) 'E F ! t! . + 4 t59 >? 4 F t59 ! C %, >?( !Q F C A ( ? % ' C %, ' ! > %U ' x + x − 4 ? *M >? F ' x + x + ? M= >? 4 ? M= (x − )
lim x→
x +x− x −
= lim x→
(x − )(x + x + (x − )(x + )
)=
(x
+x+
lim (x + )
)=
=
x→
cot x lim π x→
> M
(@ '
(
4?
' ' ! %U ' cot x = −∞ lim π x→
−
(@ ' π { *4 4 ? q7 F &2 ! <%F
lim x→
>? 4 '
X*F
lim x→
?
−
4'
x sin x x = lim = − cos x x x→
5X,
F
' π−
x sin x − cos x x
Z%
lim (x + ) x→−
>?
'
52
− −ε
T
lim (−
−ε +
)
−
x →−
4'
x
=
(− ε )
=
+ε
sS
−
}
F Z%
= +∞ x +
lim (x + ) x → −∞
>? M(%
*F C %,
t59 ! C %, ! x +
lim (x + ) x → −∞
= lim x → −∞
>? 4 ' 5c `5, = = lim x → −∞
4'
(@ '
! *U
)
t59 ! C %, Z %
' (.G 'E F
lim x→
! F t! .
x − x+ x −
= lim x→
5M4
? 4'
~)(
ST
x − x+ x −
'*2!
x
(x − ) F t59 ! C %, ! & 5 . C %, > %U
+ T t59 >? 4 > F
(x − ) = (x − ) = lim (x − )(x + ) x→ (x + )
=
lim x→ −
>?
[x] +
lim x − x→
=
[
−
'
52
= lim x→
x− −x
x−
− ε ]+ + [− ε ] + − + == = = −∞ −ε − −ε − −ε
(x − )
lim x→ x −
[x] +
−ε Tx
lim x→ x − t59 ! C %,
(5*:
x = x
x→
( q7 > ? F '
x
x + x + x+
lim 4 5X,
F
! >? 4 ' Y5R t59 t! . ×
x+ x+
(
)
(x − ) x + − x = −x = lim x − ( − x) −x x→
(x − )(x + − x ) = (x − )(x + )
x− −x
t59 ! C %, >? 4 ' F
( x − )(x +
lim x → +∞ C %, 4 (
5 #5* 4 5*: Z % " >? M(% ' >? 4 ' 5c `5, = ! *U )
lim x → +∞
(x
)
− (x + x −
)=
x −
*F C %, t59 ! C %, Z % (5*E . ! >? 4 ' Y5R
x + x −x− x = lim = x − x →+∞ x
lim x →+∞
− cos x x
lim x→ >? 4 '
X*F
?
)
q7
4'
5X,
F
K% M4 Z 4 Z%
( x) − cos x = lim x x→
lim x→
= x x−
sin
limπ
x−
x→
q7 > 4 '
5X,
F
>?(5 E ' %*4 B b C %,
K% F Z 4 ! >? 4 '
sin
limπ
x− x−
x→
~)(
π
π
π
4'
x→
π
Z%
? x x t59
=
! ?(
' )
Z%
x−
+ x!
F ] t59 ! C %, Z % _ ! >? 4 ' 5c B5, x
lim x → +∞ x = x
lim x → +∞
!
x =x
(5*: (5*: = x + x x
*B5E
x+ x x = lim = x x → +∞ x
F
π
x
π
x− = lim x→
'*2!
π
4 '(
pM C %, P(
=
π
x + x
x→∞
x
x−
π
x + x+ x
lim x → +∞ 5c B5,
π
Z 4 %J TZ 4 K% F
x− = lim
π
Z% ( (
pM !
T? 4 ' > F n( x
> %U '
= ( A
lim x →∞
Z%
2 a
(5*: !
! 54 5c `5, P( 5( ] T t59 >? 4 ' 5c B5, = ! *U )
x =x
x+
lim x →∞
x+ x −
6!
4> F
= lim x →∞
U
x x+ x
!Q &
= lim x →∞
A2
x+ x+ x −
a
x x = lim = x + x x →∞ x
F
5*:
5( > *M
#
0: F
2- +! %KE 5*E . ' f (x )
n(
R
! > U 5*P %4 '
& C %, (
>
4'
n(
' g ( x ) ≤ f ( x ) ≤ h( x ) > U ! 5 h(x ) ! g (x ) T x → a 4 '*2! ! > 4 A ? ! ( .
A
4 (5 ) 5c
f (x ) = A lim x→a
55
ZQ 4 '*2! ! F qeF ! } F & * > F F !} = 5 5 ? '1F! &
(
%F %) ( 4 ?(5 ) G *
5 U
2- +! %KE 1 D Y% 1 T − sin x ≤ f (x ) ≤ − tan x ? U
l
*U
−
π π ,
x
5
5 5E ]
lim x→
X
π
f (x ) +
F
lim − sin x = x→
π
lim − tan x→
x
π
→ lim f ( x ) = =
x→
π
> F *F %J
'E 5:B D2
Y% 1
π
− cos x ≤ f ( x ) ≤ + x
π
%F ' !
f (x ) + = × + =
lim x→
f (x ) +
lim x→
X*F
?U
*U
−
π π ,
x
5
5E b
lim f ( x )
AF +
x→
lim + x = x→
lim − cos x =
→ lim f ( x ) = x→
'E 5:B D2
F X*F
x→
> %U i , (− π , π ) C %,
(
! > F I o %F G (
4> U
π
U (
l
&
T
− x ≤ f (x ) ≤
?U
+x
*U
> lim x→
lim x→
− x =
4
→ lim f ( x ) =
'*2!
x→
> F *F %J
f (x )
lim f (x ) =
→ lim
x→
AF +
lim f ( x )
x→
f (x )
'E 5:B D2
x→
+x =
4 •5B S
− ≤x≤
) Q − x ≤ f (x ) ≤ cos x ? U
*U
x
x→
5 -
X*F %F ' ! f (x )
=
5E l >
4
lim cos x = x→
lim − x =
→ lim f ( x ) =
'E 5:B D2
x→
X*F
x→
&' %$ > 54 ?F ? 2 h%
#> U > U 55 >?(
*U 5 Z!
F
% ' ( > Z % *F% 7 ' ')*F% 7 Z% Q # U f c " > U *U % ! f (c ) #] ƒ} " > U *U % ! lim f (x ) #b x →c
=
f ( x ) = f (c ) #S lim x →c
1= ZQ
&
=
F
&
= 4
%
•
} ')*F% 7 U 5 5 }
&
= 4
%
•
F ')*F% 7 U 5 5 >?(
&
b - 8- 9 - # a+b = a+ b=
> %U ' `~ !
U&
+ b
− ×( a +b = a+ b=
> %U '
−
)
> ( !Q F
>? 4 ' Y5R − b3 5 ( ? 4' &
→
b
!
=
a
!
!
− a− b=− * a + b = **
* + * * → −a = − → a = 7. b = a
!
'P(
SZ
52
a+ b= → × + b= → b=− →b=− a=
b
8- 9
#
a+ b−c = a −b+c = a+b+ c =
„ ,
S n( 5
4 'W5U
? 4
7
!
!
(` > 4 ( A %@G ! ! %@G 'P( `~ %@G F T S %c ( 5 >? 4 ' #> %U ' `~ T *M (52 c 3(5R Z% " >? 4 ' & ? 4 ' & ' %F ! 54 Y5R b ! qeF >? 4 c
a+ b−c = a −b+c = a+b+ c =
= ! 54
~)(
!
b
!
a
b
a
* ** → ***
Y % ! 54 a+b = a+ b=
= 4 F
* + ** → a + b = * + * ** → a + b =
%@G ! > (Q '
! F
c
→ b= →b=
a + b = ,b = → a = a+ b−c = → + −c = →c = −
&' %$ 1 D •5 %+ &
4
%W
b
!
X
= f (x ) = x + ax + b &
a
> ~) . (− , ) 1=
!
4 &12 sb •5
1=
f ( )= − →b = f (−
> ( !Q F (
(5
b
!
= > U
a
> U 'P( t59 ! C %,
lim x →−∞
&
x= | x − | +bx
>? 4
bx +
=
5E
a− = →a=
= →b=
4 ( !Q F
%W
b
!
=
a
x>
> U l55 F x>
b
+
x<
f (x ) =
x=
= →a=
(5*E . ( a! U b = 4 ( 5 >? 4 5X, x 3(5R b
[x ] + a
)= → −a + (a − )x + x
> U S 55 & } ! F ')*F% 7 F >? U *U F ')*F% 7 P ( 5 ( ?( !Q F F 4 ( 5 !
lim+ | x − | +bx =| − | + b = + b = → b =
x→
lim− [x ] + a = → [ − ε ] + a = → − + a = → a =
x→
[x] − > U
*U
> U 55 ? lim+ [x ] − x = − = −
x→
→
x<
F !}
&
4 ( !Q F (
U
*U
%W
=
a
x=
4 (
4
%W
5
a+ =− → a= →a= −
> U
lim x→ a x
!
x≥
f ( x ) = ax + x−
x=
ax + a+ lim− = x→ x− −
x
> U S55 }
x− a = − a
→ lim x→ a
lim x→ a x
x− a − a
=
x− a = = = (x − a )(x + a ) lim a x → a (x + a )
a
→a=
ax +
> U
|x− | x−
x>
f (x ) =
*F% 7 x =
&
x= bx + x +
> U S 55
4 ( !Q F
%W
b !a
=
x<
F !}
(
U
*U
4 (
x=
5
|x− | =a+ = →a= x→ x− lim− bx + x + = b + = → b = lim+ ax +
x→
a [x ] + − bx −
f (x ) =
x=
x< x≥
> U 5X, 5 5 (
&
F !}
4 ( !Q F
%W
> U *U 5X,
T? U
b
!
=
a
*U 5X, 5 5 x= 4 (
5
lim+ − bx − = − b − = → b = −
x→
lim− a[x ] + = a + = → a = −
x→
!
5)(
y = x + a ! y = ax + x + b
& ! 4 %W > 4 &12 s] •5 > 4„ , ! 5
= •5 %+ ( ( ,− ) 1= b
!
a
y = ax + x + b → − = + + b → b = − y= x+ a →− = + a→ a=−
> U
lim x →a >
4 AF +
( fog )(x ) − (gof )(x ) = f (g (x )) − g ( f (x )) = ( > U
*F% 7 x = −
( fog )(x ) − (gof )(x ) , x+
*F% 7 &
sin ( x − a ) = x −a
4
%W
(x − a ) = sin ( x − a ) = lim = = lim a x −a x → a ( x − a )( x + a ) x →a (x + a )
)+
−(
(
x+
b[x ] + f (x ) = x + − ax
U
lim x →a
U 5X, 5 5
F !}
U
)+ ) =
x+
x>− x=−
&
g (x ) = x +
! f (x ) =
+ −
− =
x−
4 ( !Q F
a
→a=
x+
−
%W b ! a
5E
=−
=
x<−
( q7> %U ' 5X, 5 5 s] &
=
lim+ b[x ] + = −b +
x →−
lim − − ax = − a
x →−
−b+ = →b =
,
− a= →a=
*'+ *'+ O: 9 ' Z : ∆y
>?
( )
f / x = lim ∆x→
(
AM y C 5 N … Q
) ( )
f x + ∆x − f x ∆y = lim ∆ x → ∆x ∆x
M . 5 X ' J 4 a % 5B 5)( PU x → x 'B5W q7 ∆x = x − x → x = x + ∆x >? F ' ( PU ' A2 % 5B ~)(
∆x →
( )
f / x = lim
F
x
∆x
x→ x
U *F %J 0*: H(5 > F 5* >? 4 ' X*F 5(
( )
f (x ) − f x ∆y = lim ∆x x → x x−x
X*F 1= n( 0*: = 4 '( %F ! 5*@ 54 X*F 6! ! 5 Z % ' % 5B F U *F %J 5 N … Q 4 '( %F
( )
f (x ) − f x x −x
b
x
'*2!
x
C5 N
AM
(
5* '* F
x
&R
) ( )
∆V V x + ∆x − V x V ( )−V ( = = ∆x ∆x ∆x 5F a! > U ' x = t − t + C
x
( IF%* C5 N 'Q … Q
^ > ( !Q F
t=
)= %,
t=
'A P ?G 5 N … Q > ( !Q F T 4 5 N j V ( x ) = x 3 P ?G − = −
'45+* 45 c+ ' ,B > ( !Q F
( ) ( ) → ∆x = X ( ) − X ( ) =
∆X X t − X t = ∆t t −t
# U
%J 'F 5
[9
∆t
−
h5+* t=
=
" ! ?(5 E ' 0*: ' Q … Q
#H 5 #Y
x / (t ) = t − → → x/ ( ) = × − = − t=
> ( !Q F
(
!
∆x = /
) ( )= f (
+ / )− f ( /
∆y f x + ∆x − f x = ∆x ∆x
> ( !Q F
( )
x−x
x→ x
> ( !Q F
( )
f / x = lim
x−x
> ( !Q F
(
/
∆x = /
) = ( × ( / )− )− (
x→
x→
! /
−
)=
/ − = /
& 0*M T 0*: H(5
x+ −
x−
X*F
−x− −x = lim x + = lim x→ x→ x( x + x
f (x ) = x &
0*: T0*: H(5
)
=−
X*F
x− x− x+ x− = lim × = lim x→ x − x→ ( x − ) x + x− x+
(
f (x ) = x −
x=
+ /
& 5 N IF%* … Q
/
( ) = lim x +
( ) = lim
) ( )= f (
∆y f x + ∆x − f x = ∆x ∆x
/
1=
x=
( )→ f
f (x ) − f x
x→ x
( )→ f
f (x ) − f x
f / x = lim
f (x ) =
1=
x=
f (x ) = x −
x=
) − f ( ) = ((
/
)
)(
− − /
−
)=
& 5 N IF%* … Q
)=
− /
=
= /
*'+ 1 D &
y
0*:
y′
C + R% > F 5X, T'* ^
a
y/ =
y=
5* 4 'P( Z %
nx n−
xn
y/ = × x
y=x y/ =
y= x f
( )f
n f
x +x
y=
x
/
y/ =
y =
+
sin
( x) K% M4 0*:
( )
−
cos x
x
( )
cos x
{ 0*:
( + tan ( f ))
( x)
' ! 0*:
)
K% F 0*:
− f / sin ( f )
x
(
'!
(x ) x
/
−
(x
f / cos( f )
f
−
x − x × x
x
( x)
( x)
(x)
)
( 7 0*: 5 5 Z %
t59 0*: @ t59 C %, 0*: " ! Z % t59 5 ? M=
/
cos( f )
cot
x +x
' ! 0*: !
# C %,
g/ f − f /g f
x
cot ( f )
)(
x
sin ( f )
tan
( 7 %J
y/ = x x +
x
tan ( f )
=
x
× x ×
'!
g/ f + f /g
g (x ) f (x )
sin
−
=
y/ = ×( × x +
( x )( x )
y=
( 7 0*: 5 5 Z %
= x
5* 4 'P( Z %
n−
)
g (x ) × f (x )
y=
−
×x
−
( 7 %J
x
n
(
y=
5 0*:
( + tan ( x ))
)
− f / + cot ( f )
{ *4 0*: − x
( + cot ( x ))
!
*'+ 1 D # M ' . Z 54
F" > ( !Q F
X
5( & % 0*:
f (x ) = x + x x=x
!
x
ƒ x >? M(%
5M4 C %,
P(
#H
*
•
(
f (x ) = x + x = x + x
f
n
( )f (x + x )
n f
/
5* 4 'P( Z %
n−
−
( 7 %J
( 7 %Jm (
5* 4 'P( Z %
f
/
( 7 0*: 5 5 Z %
) ƒ ( 7 0*: m ƒ5 5 Z %
x+
(x ) =
# xz x] #> M ' . Z 54
)(x
×( x +
+ x
)
−
q7
F Ca %F" 0*: C (5 F" > 4 7 5( & % 0*: x− x+
f (x ) =
!
)
] 5* 4 'P( Z % ( 7 %J ( 7 0*: 5 5 Z % % C A 0*: Z : C A ! 5 a%4Q ‡ ( 7 0*: † T 5 E ( @M @ T0*: > F U Z : C A 5( a%4Q ‡5* 4 'P( Z % ( 7 %J f (x ) =
x− x+
→ f (x ) = × /
×( x +
)−
(
x+
× (x −
)
)
×
(
f (x ) = x − x +
(
f (x ) = x − x +
)
→ f / (x ) = ×
(
x −
)× ( x
x=x
x
)
b
− x+
f ( x ) = sin x − cos f ( x ) = sin x − cos
−
x− x+
x
S
x → f / ( x ) = cos xsin x − (− sin )
> F
n
Z% '
n
f (x ) =
x x+
l
5B
P(
Z%
)
−
−
4? ('
=
>? 4 A f (x ) =
!
−
=
!
=
−
Z % ! 54 K%P
x x → f (x ) = x+ x+
(
=
(7? % '
→ f / (x ) =
F 'X Z % 4 '*2!
f (x ) =
x+ x+
+
( x)
j
− x)
z
+
x+
( x)
+ x → f / (x ) =
−
× ( ) × (x )
+
(x + )
−
f (x ) =
f ( x ) = ( − x ) → f / (x ) = × (−
)× (
− x)
−
x + cot
x
(
→ f / (x ) =
f ( x ) = sin f ( x ) = sin
−
× (x + ) − × x x × x+ (x + )
×
f (x ) = f (x ) =
'X Z %
→ f / ( x ) = × ( cos x ) × sin
−
x + cot x−
(
− x)
x
y
−
x
+ cot
f (x ) = cos x + + tan x
f ( x ) = cos x + + tan x → f / ( x ) =
(
× ( x)× x +
)
−
(x
f (x ) =
f (x ) =
(x − ) x+
( x )(x − )( → f / (x ) =
→ f / (x ) = ×
x+
)−
)
x
(x − )
+ x
v
x +
f (x ) =
x+ x+
− x+
( x+ ) f (x ) = x x
f (x ) = x x + x → f (x ) = x + x → f / (x ) =
f (x ) =
( + tan ( x ))
× sin x + +
x+ x+
×( x +
)−
×( x +
(
x+
)
)
]w ×
x+ x+
( ) ]] ) + sin x → f (x ) = x( + tan ( x + )) + cos x f ( x ) = tan ( x + )× sin x ]b f ( x ) = tan x + + sin x
(
f ( x ) = tan x +
/
'!
(
)
f ( x ) = tan x + × sin x → f / ( x ) =
' ! 0*: r ' !
( x( + tan ( x
' ! 0*:
)))sin x + cos x(tan( x
+
f (x ) = + x +
−x x
f (x ) =
]l
−
(
× ( x)× x +
/
−
)
× ( x)× x +
f (x ) =
f (x ) =
]S
(
f (x ) = + x + → f / (x ) = +
))
+
−x → f / (x ) = x
)
×x − x
−x
(x )
−
(
x × x +
)
− x
−x
x
( − x ) + tan x ]j ( + tan ( x )) × (− x cos( − x ))sin ( − x ) + x f ( x ) = sin
f ( x ) = sin
( − x ) + tan
x → f / (x ) =
πx
f (x ) = sin
f ( x ) = sin
πx
→ f / (x ) = ×
π
cos
]z
πx
f ( x ) = cos x + x f ( x ) = cos x + x → f / ( x ) =
x
+ x
→ f / (x ) = ×
−
x− x(x +
)
→ f (x ) =
]x
+ x
x
− + x
f (x ) = f (x ) =
]y
× (− sin x )(cos x )
f (x ) =
f (x ) =
πx
sin
x
x
x− x( x +
)
x− ×x + x−( x+ → f / (x ) = x +x x +x
(
)
+ x
]v
)(x − )
f (x ) = f (x ) =
− x
→ f (x ) = − x
−
− x
→ f / (x ) = × × x
f ( x ) = x( x −
) → f (x ) = (x
f (x ) = x(x −
− x
)
f
/
(x ) =
)
+
(
→ f (x ) = x − x
x
)× (x
×( x −
− x
)
)
) b]
× x− × x − x
(
(
− −
(
→ f / (x ) =
f (x ) = x − x f (x ) = x − x
bw
)
+
x
bb
)
+
)
−
+ x−
+(− ) × × x − =
)× (x
×( x −
− x
− x
sin x bS + cos x cos x( + cos x ) − (− sin x )(sin x ) f (x ) =
f (x ) =
sin x → f / (x ) = + cos x
(
+ cos x )
(5 )
f ( x ) = sin x + cos x
)
f ( x ) = tan
)
g (x ) = sin x + cos x
)
f ( x ) = sin x cos x
)
f (x ) = ( x − x +
)
f (x ) =
)
f ( x ) = sin x. cos x
)
f (x ) = x −
)
− x f (x ) = x +x+
)
f (x ) = x − x
)
f (x ) =
)
f (x ) = x + x
)
−x−x
(
(x
)
)
+
x
)
− x+
)
f ( x ) = sin x + cos x
)
f (x ) =
)
f (x ) = ( x −
−x
x + x+
f (x ) = x − x
)
f ( x ) = tan x + cos x
)
f (x ) = ( − x
)
)
x2 − x −
)
f (x ) =
x − x+
) (x
f (x ) =
x
x− x
)
*'+ -
.
W F cd 8- 9 5V &V2 ! b %VW V
f (x ) = x − x +
V1=
(
>?(
A x ,y
& ' + 5 K IJ > ( !Q F & ('+ IJ *U% 5 1= n( ! m 3 U n(
)
(
y− y =m x−x
>?( !Q '
F
)
1= C \*9 T f ( )=
>? 4 '
7 b 1=
0*:
(
)(x
(
x−
f ′( ) =
(
× −
> (Q '
F
K
y− =
)(
−x+
)
− +
IJ
52
K
)
3U
5 !
)
−
)
−
=
×
T IJ
(x − ) → y =
Z
→ f ( )= → A = ( ,
− +
= ! *B5E 0*:
f (x ) = x − x + = x − x + f ′( x ) =
x=
x−
=
=m
% 5B + =
m
!A Z
52
x+
=: F ' + 5 &2 !
x=
' + 5 &2 ! > ( !Q F
x=
%W
1=
%W
1=
p( ,
) 1=
IJ > ( !Q F y=x − x & ' + 5 K IJ 3 U > ( !Q F y=x −x & ' + 5 K IJ y = x +x+
'+ 5 K
Qe E cd 8- 9 > ( !Q F
% 5 &2 ! b %W
1=
(
>?(
)
A x ,y
1= n( !
y= m3
(
y− y =m x−x
>?( !Q ' y=
>? 4 '
7 b 1= m′ = −
y′ = ×
− → y′( x
m
)= m = −
% 5 ?L 2 IJ
U n(
F
0*:
&
IJ
1=
K
F
m′
?L 2 IJ % 5B ? m′ = −
,
(x − ) → y =
52
3U
5 !
)
= ! *B5E 0*:
=−
5
Z
x=
=− −
y− =
*U%
)
= →A=( ,
>?( !Q '
x
−
?L 2 IJ 3 U =
x−
=: F > M(% •5 %+ ZQ '2 + > ( !Q F ' + 5 &2 ! n( %W > ( !Q F ' + 5 x = − %W 'V + 5V &V2 ! b %W
1=
' + 5 ?L 2 IJ y = x + x − ' + 5 ?L 2 y = x − x ' + 5 ?L 2
y = x − x+
1= 1=
f (x ) = x − x +
& '+ 5 K > ( !Q F
IJ & (
1- 9M ZQ qeVF ! V*B5E 0*:V & V 0*:V V4 ! % ,&
83
`%a+
V* T& Z % % , ( ! ' !. ; 9: 5 % A 0*: 4 >? 4' > F ' !. & % 'X
SQF%V. V( ? 5*MV4 | = > (%E ' & ? 5*M4 F 5X, 5 5 >? 4 ' X*F F ? M4 T? 5*M4 1= f / (x ) = , f // ( x ) > ? f / (x ) = , f // ( x ) < ? M4
QF%0% R 0*: = 4 ? M4 U a ! 4 ?(5 E ' 1=
1 D Q 'V. ZQ (! 5= G* 0*: 0*:
4 'W = *M ? 4 '* U (7 ! ! 0*: ! ! 0*:
O5T 5 > G VF 'V V y %V+ > F
(7
! 5=
V@ V
'V + 5V = > V (%E 'V ' + 5 =
'+
) Q U f // (x ) < 5E !
Fa ! '+ 5= U f ! '+ '* M2 5 = T ' + % ! 1= ' + ( %W >? (%E H1 1= 1=
`5VW V 'V + 5)( M2 5 = ! a •%V V1= V( ' + 5 = 4
9
Q
//
%E
(x ) > 5E
'D > F (7 ( > %U '
%%f
4
>?(5 ) 0*: & >? 4 ' 7 F 5X, 0*: ZQ 4 'W = % V, & V VF A 0*: 4 '( >? 4 ' 0*: > F ' !. & F 'X 0*: 4 ! F #H1 1= ">? 4 ' 7 ? 4 ' ?F 5( PU C 5 N ! x
y/
x
−∞
>? 4 '
7
x
s
( )
yx
%9 Z >? 4 ' ?F
sl sj ∞
x
r
y
s] sb sS
r
( )
yx
|= C %, % ! !
sz sy
%0. Q
- F
y = x − x+
%%f
4
?(5 E ' 0*: >? 4 ' 0*: 0*: :( % 0AW q7 y/ = x −
>?( !Q '
F
x − = → x = →x = →x=±
5V2 &
@ Q T| = •5
( 5 >0B % :( ! t J ! H 9 :( ! x = ± ? 5*M4 | = %W *B ( 5 a > Q F y( ) = × − + = − → ( ,− ) y (− ) = × − + + = → (− , )
x
−∞
r
y/
y
>? 4 '
7
F H1
−
>?
'
∞
+
s
r −
1= Z
4 ZQ :( ! ?(5 E '
)=
5
y // = x x= →x= ( , ) H1 1=
>? :4 ' '* ! ! % ! K V IVJ q7 F 5X, 0*: ZQ Z% >? 4 ' ;9: ? 5*M4 | = * >? 4 ' ?F n %4 '=B IJ n( ZQ (5 F '=B 'V (− , ) 1= ! ? :4 ' A 3 U '1J F % , & − A2 Z% ! >? F ?V ( ,− ) V1= V ! VF 'V !. % ( ,− ) 1= (− , ) 1= > F %, %
<
5
Ca %F %
=: F C \*9 C \*9 > U & ?
'
4 4 %W c ! b ! a 5( = > ~) . C \*9 A ! U *U ( ,− ) ' 5*M4 y = x + ax + bx + c 4 4 %W c ! b ! a 5( = > 4 &12 S 1= •5 %+ T ' + ! U *U ( , ) (, ) & 4 %W b ! a = f (x ) = x + ax + b & y = x + ax + bx + c