Высшая математика. Том 2

' 2: $ , f ′(0)

	− cos	(	xsin 1	x)	, x ≠ 0;
1
f (x) =

0, x = 0.

2

$ , %:

(∆x sin 1∆x)

f ′

(

0

)

= lim

f (0 + ∆x)− f (0)

= lim

1 − cos

.

(

∆x→0

)

x

∆x

∆x→0

∆x

x→0

2

lim 1 − cos x =

2

, :

1

(∆x2 sin2

1

∆x

)

sin

1

∆x

sin

1

∆x 1

lim

= 1

lim

∆x

= 1

2

lim 1

∆x

= ∞ .3

∆x

1

1

2 ∆x→0

2 ∆x→0

∆x

∆x

∆x→0

3.2. " # !$ " % 3.1

/ ’ , # (

.

$ , f ′(0).

				3		2			2
1.	tg x				+ x		sin			,	x ≠ 0;
	f (x) =								x
			x = 0.
	0,
								1
3.	arctg xcos								,		x ≠ 0;
	f (x) =							5x
			x = 0.
	0,

						3
5.	sin xsin							,				x ≠ 0;
	f (x) =						x
			x = 0.
	0,
				x2 sin		5
			e				x		−1				+ x,				x ≠ 0;
7.	sin
	f (x) =
		x = 0.
	0,
					3				3				1
									2
9.	arctg				x − x						sin			3x		,	x ≠ 0;
	f (x) =
		x = 0.
	0,

arcsin x2 cos

1

+

2

x,

x ≠ 0;

2.

f (x)=

9x 3

0,

x = 0.

3

1

4.

ln

1

− sin x

sin

,

x

≠ 0;

x

f (x)=

x = 0.

0,

1

6.

1+ln 1+ x2 sin

−1,

x ≠ 0;

x

f (x)=

x = 0.

0,

2 cos

4

x

2

8.

x

+

,

x ≠ 0;

3x

f (x) =

2

x = 0.

0,

				5
	10.	sin x cos			,	x ≠ 0;
				x
		f (x) =
			x = 0.
		0,

111

																				2						6
	11.	x + arcsin x																				sin								,	x ≠ 0;
																											x
		f (x) =
							x = 0.
		0,
																		7
	13.	arctgx sin																				,				x ≠ 0;
																				x
		f (x) =
							x = 0.
		0,
				2							2		11
	15.	x				cos											,						x ≠ 0;
																x
		f (x) =
							x = 0.
		0,
		ln cos x											,					x			≠ 0;
	17.						x
		f (x) =
							x = 0.
		0,
			e	x2			− cos x
	19.																			,				x ≠ 0;
								x
		f (x) =
							x = 0.
		0,
				x	2		sin			2
	21.	f (x) = 3								x			− 1 + 2x,																	x ≠ 0;
							x = 0.
		0,
				xsin				3
	23.	f (x) = e						5x						− 1,									x ≠ 0;
							x = 0.
		0,
							3x									− x2 sin										1					x ≠ 0;
	25.	arctg																												,
		f (x)=					2																				x
						x = 0.
		0,
				1		− 2x3 sin															5			− 1 + x, x ≠ 0;
	27.	3
		f (x) =																				x
							x = 0.
		0,
		ln (1 + 2x2 + x3 )
	29.	f (x) =																										,			x ≠ 0;
													x
							x = 0.
		0,
																						1
	31.	1		− cos xsin																					,					x ≠ 0;
																							x
		f (x) =
							x = 0.
		0,

						(		)
				x2 cos 1 8x							−1+ x ,							x≠ 0;
	12.	tg 2
		f (x)=	(										)
				x=0.
		0,
														1
	14.	f (x) =	2x2		+ x2 cos												,	x ≠ 0;
														9x
					x = 0.
			0,
															1
	16.	f (x) =	2x2		+ x2 cos											, x ≠ 0;
															x
					x = 0.
			0,
									1
	18.	f (x) =	6x + xsin									,				x ≠ 0;
										x
					x = 0.
			0,
				xsin		5
		f (x) =
						x
	20.		e					− 1,					x ≠ 0;
					x = 0.
			0,

2

22.

1+ln 1+ 3x2 cos

−1,

x≠ 0;

f (x)=

x

x= 0.

0,

2tgx − 2sin x

,

x ≠ 0;

24.

x2

f (x) =

x = 0.

0,

3

2

sin x2 sin

26.

f (x)= e

x

−1

+ x2,

x ≠ 0;

x = 0.

0,

2

x

1

x

e

sin

,

x ≠ 0;

28.

x2

f (x) =

x = 0.

0,

cos x − cos3x

,

x ≠ 0;

30.

x

f (x) =

x = 0.

0,

112

3.3. & " ! ' ( )# * +

, # 3.1

, # .

# *

uα

a uα −1

sin u

cosu

cosu

− sin u

tg u

1/ cos2 u

ctg u

−1/ sin2 u

arcsin u

1

1− u2

arccosu

−

1

1

− u2

arctg u

1

1+ u2

arcctg u

−

1

1+ u2

bu

bu ln b

eu

eu

logb u

1

uln b

ln u

1/ u

x

x

x

(

)

x

1

+ ln x

! # sh u =

eu − e−u

ch u

2

! #

ch u =

eu + e−u

sh u

2

! #

th u =

eu − e−u

1

ch2 u

eu + e−u

eu + e−u

−

1

! #

cth u = eu − e−u

sh2 u

113

4 # u=u(x) v=v(x) – -

, - , x.

3.4.,& " " ! ) *-"

1.' , # (C)′ = 0 , C= const.

2.' 1, # x′ = 1

3.' # -

% , # (u + v)′ = u′ + v′ .

4.' # , - , #

( # ( -

, # (u v)′ = u′ v + u v′ .

! &(" 1. ' , #-

	u ′				1
:	(C u)′ = C u′ ;			=		u′ .
					C
	C

! &(" 2. ' # , - ,

# (, :

(uvw)' = u '·v·w + u·v '·w + u·v·w '.

5. ' , - , #

u

′

′

′

u v − uv

( v≠0):

=

.

v

2

v

′

′

! &(" 3.

C

= −C

v

.

v

2

v

!

1

& .

' 1:

y = 3

.

x

1

′

− 1

′

1

− 1

−1

3

3

2 y′ = 3

= 3

(x

2 ) = 3

−

x

2

= −

= −

.3

x3

x

2

2

2

x x

' 2: y = x3 – 3x2 + 5x + 2. 2 y ' = 3x2 – 6x+ 5.3

' 3: y = ln x · cos x.

2 y ' = (ln x) ' cos x + ln x (cos x) ' =1/x·cos x – ln x · sin x.3

114

' 4: y =		x3
			.
		cos x
2 y′ =	(x3 )′ cos x − x3 (cos x)′			=	3x2	cos x + x3 sin x
		cos2 x					.3
						cos2 x
		3.5. .' (

$ . ) - % 0 ( . . 3.2).

* ( % -
M0M. 0-
% - ,
0 ( % , -
% % . 0- #-
#
0 #- -
0 ,
0 %
0.
, ,
0 % -	*. 3.2.
0 ,

0.

* # y=f(x)

( . . 3.3).

' 0 % y0=f(x0). 5 -

x0 y0 % 0(x0; y0). x0

+ . % -

y0++y=f(x0–+x). % (x0++x; y0++y). ' 0 -

! , ". /-

( ∆y ,

∆x

- tgϕ = ∆y .

∆x

0- +x60, -

# + 60,

,

, # # %-

0. , 0

0,

!#$ +x60, 7 *. 3.3. !

115

".

tg ! # ! !68/2, !67 tg ! 6tg $ , , -

% #:

tgα = lim tgϕ = lim		∆y = f ′(x), # f '(x) = tg $ .
∆x→0	∆x→0	∆x

, , '(x0) % % -

% x0, # x,

, f(x) -

0(x; ) ".

3.6./

&, - % s = v·t, s –

(, t, v – ( .

', # ( # ,

(, , , s t, # # % .

) % -

s=s(t). $ t0. - ( ( s=s(t0). $ ( v t0.

#- ( t0++t. 9 -

% ( s=s(t0++t). +t (

( +s=s(t0++t)–s(t).

* ( ∆s = v . $ % (-

∆t .

+t. / (

( t0 ( ). , -#

( ( (, -

# ( +t.

, ( t0 (% () -

% ( t0 t0++t, +t60:

v = lim ∆s = s′(t),

∆t→0 ∆t

# % ( -

.

3.7. " ( ' ! "

* , % y=f(x). $

M(x0, y0) M, , - Oy.

* % % =kx+b.

k= f '(x0), % y = f '(x0) · x + b. '-

116

b , - M(x0, y0). ,

: y0 =f '(x0)·x0 + b.

& b=y0– f '(x0)·x0.

, , % :

y= f '(x0)·x +y0 – f '(x0)·x0

# = f '(x0)·(x – x0) + f(x0).

0- , -

(x0, y0)

(#

%), x= x0.

*

-

. &

% , -

-

.

& -

%, - % kn ' -

*. 3.4.

% -

k :
kn	= tg β = tg (900 + α )= −ctgα = −	1	= −	1	= −	1	.
		tgα				f ′(x0 )
				k

$ , - M(x0, y0), -

y= f(x) M % :

	y = −	1	(x − x0 )+ f ′(x0 ).
		f (x0 )

&, - ", # f '(x0)= 0 -

% y= y0, #

". , % x= x0.

: # 3.2. , # 3.2.

$

	0' "		" (		" ' !
	′		x = x0				y = y0
	f (x0 )= ∞
	′		′		y − y	0	= − 1	′	(x − x	0	)
	f (x0 )≠ 0		y − y0 = y0 (x − x0 )
								y0
	′		y = y0				x = x0
	f (x0 )= 0
			!
	' 1: /				y = 6 3		x − 16 4		x 3 -
# x0 = 1.			′
	2 * :
			y − y0 = y0 (x − x0 ).

117

: %: y					= 6 3		1 −			16 4 1		=		6 −		16		=		2	.
				0						3						3				3
	′			1			16				1				2					4				′		2		4		2
y		= 6				−						=					−					.			= − = .
											4 4 x3													y0
				33 x2			3								3 x2				34 x3					3 (x − 1)	1 3 3
% y − 2																						3 = 2			# y = 2					3 x − 2		3 + 2	3 .
$ : y =							2		x .3
							3
' 2: /																									y = x2 10 + 3

# x0 = 2.

= − 1 y′ (x − x0 ).

2 * : y − y0

0

2x

2 2

2

: %: y0

2

2

+ 3 = 3,4 .

y

′

′

=

=

.

=

= .

10

y0

10

10

5

% :

y − 3,4 = −

1

(x − 2) # y = −

5x

+ 5 + 3,4 . $ : y = 8,4 − 2,5x .3

2 / 5

2

3.8. " # !$ " % 3.2

/ ’ , # (

.

/ ( 1 – 15) # (

16 – 31) # x0 .

1.

y = (4x − x2 )

4, x0 = 2.

2.

y = 2x2 + 3x − 1, x0 = −2.

3.

y = x − x3, x = −1.

4.

y = x2 + 8 x − 32, x = 4.

0

0

5.

y = x + x3 ,

x

= 1.

6.

y = 3 x2 − 20,

x

= −8.

0

0

7.

y =

1 +

x

,

x

= 4.

8.

y = 84 x − 70,

x

= 16.

1 −

x

0

0

y = (x2 − 3x + 6)

9.

y = 2x2 − 3x + 1,

x0 = 1.

10.

x2 ,

x0 = 3.

11.

y =

x − 33 x,

x0 = 64.

12.

y = (x3 + 2)

(x3 − 2),

x0 = 2.

13.

y = 2x2 + 3,

x

= −1.

14.

y =

x29 + 6

,

x

= 1.

0

x4 + 1

0

15.

y = 2x +

1

,

x0 = 1.

16.

y = −2(x8 + 2) (3(x4 + 1)), x0 = 1.

x

118

17.

y =

x5 + 1

,

x

= 1.

18.

y =

x16 + 9

, x

= 1.

x4 + 1

0

1 − 5x2

0

19.

y = 3(3 x − 2 x ),

x0 = 1.

20.

y = 1 (

3x + 2),

x0 = 2.

21.

y = x

(x2 + 1),

x0 = −2.

22.

y = (x2 − 3x + 3)

3,

x0 = 3.

23.

y = 2x

(x2 + 1), x0 = 1.

24.

y = −2

(3 x + 3

x ),

x0 = 1.

25.

y =

1 + 3x2

,

x

= 1.

26.

y = 14

x − 153 x + 2,

x = 1.

3 + x2

0

y = (3x − 2x3 )

0

27.

y = 34 x −

x,

x0 = 1.

28.

3, x0 = 1.

29.

y = x2 10 + 3,

x0 = 2.

30.

y = (x2 − 2x − 3)

4,

x0 = 4.

31.

y = 63 x − 164 x 3,

x = 1.

0

3.9. ) * + " &($ )# * +. 1 " &($ )# *, 2 ) *-3($&

; y=f(x) % , -

x0, - % , # - (

∆y % .

∆x

0- % [ ; b] # ( ; b), , - [ ; b] # ( ; b).

/ , - % ' , - # .

4 ' 1. 0- y=f(x) % x0,

#.

, ,

. &-

:
# , -
% , -
(# -
).
* . 3.5 ,
b, . 4 +x60 (
∆y % ( -
∆x
+x60–0
+x60+0). 4 ' % -	*. 3.5. ,
, % -

119

% k1 k2. , -

.

4 b +x60 (	∆y	% lim	∆y	= +∞, #
	∆x	∆x→0	∆x

% . 4 % . , – « » .

4 c % -

. 4 % , - . , – « » – -.

!

#

' 1: y=|x|.

2 5 # x

= 0, lim

x

= 0 = f

(0).

x→0

', - %

. f(0++x) = f(+x) = |+x|.

, +y = f(+x) – f(0) = |+x|.

< +x< 0 (# +x

0 )

lim

∆x

= −

∆x = −1.

∆x→0−0

∆x

∆x

∆x

∆x =1.

*. 3.6. ; y=|x|

< +x > 0

lim

=

∆x→0+0

∆x

∆x

∆y

, , ( ∆x +x> 0

% ,

%, - ( %, # x= 0

y=|x| %. ! %, - x= 0 « » % -

( ).3

' 2: y = 3 x .

2 5 -

# -

. &'%, %

x= 0.

f (x + ∆x) = f (∆x) = 3 ∆x

∆y = f (∆x)− f (0) = 3 ∆x

*. 3.7. ; y = 3 x

120