Высшая математика. Том 2

(y′ )′t

(1 − t2 )

(1 − t2 )

′′

x

′′

= (1 + t2 )2 t .

= (1 + t2 )2 t . 3

yxx =

xt′

$ : yxx

;,

sinx, cosx,

tg x,

ctg x, secx =

1

,

csecx =

1

-

cosx

sinx

. % # 3.1.

3.28. " # !$ " % 3.12

/ ’ , # (

.

&

x = cos 2t,

y = 2sec2 t.

x = et cost,

3.

y = et sin t.

x = t + sin t, 5. y = 2 − cost.

7.	x =	t,
		1 − t .
	y = 1

x = tgt,

9.y = 1sin 2t.

x = t,

11.y = 3 t − 1.

13.	x =	t3 − 1,
	y = ln t.
15.	x =	t − 1,
	y = 1	t .
17.	x =	t − 3,
	y = ln (t − 2).

y′′

xx

, .

2.	x =	1 − t2 ,
	y = 1 t.
			2	t,
4.	x = sh
	y = 1 ch2 t.
	x = 1 t,
6. y = 1 (1 + t2 ).

x = sin t,

8.y = sect.

10.	x =	t − 1,
	y = t	1 − t .

x = cost (1 + 2cost),

12.y = sin t(1 + 2cost).

14.x = sh t,y = th2 t.

16. x = cos2 t,y = tg2 t.

18. x = sin t,

y = ln cost.

141

19.xy

21.xy

23.x

y

25.x

y

27.x

y

x

y

31.xy

=t + sin t,

=2 + cost.

=cost,

=ln sin t.

=et ,

=arcsin t.

=ch t,

=3 sh2 t.

=2(t − sin t),

=4(2 + cost).

=1t2 ,

=1 (t2 + 1).

=ln t,

=arctgt.

20. x = t − sin t,y = 2 − cost.

22. x = cost + tsin t,y = sin t − tcost.

24. x = cost,

=sin4 (t 2).

x = arctg t,

26.y = t22.

x = sin t − tcost,

28.y = cost + tsin t.

x = cost + sin t,

30.y = sin 2t.y

3.29. 4 ' )# *, 2 ) *--($&

4 ' 4 (!!). 0- , - % y=f(x) #-

[ ; b] ( -

# % f(a)=f(b)=0, ( ; b) #

( ; b), f '(c)= 0.

5 % . 0- % #-

, - % , % " x=a x=b,

# # , < < b,

".

	*. 3.10. !				x2
		*. 3.11.	; y = 1 − 3
	*

142

&, - ( , - , - % f(a)=f(b), #'

.

1 , , - [ ; b] #

, f(x) %, -

.
, y = 1 − 3 x2 # [–1;		1] # %
( . 3.11). < f ′(x) = −	2		# %
33		x
.

- * % ". , " . *

y=f(x), '% %
[ ; b] AB. 0 , (	f (b)− f	(a)	= tgα -
	b − a

AB, f '(c) % % . , %, -

y=f(x) # ,

, - % .

4 ' 5 (8 6). 0- y=f(x) # [ ; b] -

% ( ,

[ ; b] # c, a<c<b , - f(b) – f(a)=f '(c)(b – a).

*. 3.12. ! "

4 ' 6 (9 :). 0- f(x) g(x) – , # [ ; b]

, g'(x) = 0 x ( ; b), -

[ ; b] # ( ; b), -
	f (b)− f	(a)	=	f ′(x)
					.
	b − a			g′(x)

143

3.30. " ! ( ! (( " &( + 0/0 ( ;/;

* ( - ( # , # -

0/0 ?/?. &

.

4 ' 7 (" ! ( !). ) f(x) g(x) -

, , , ,

x→a

(

x

)

x→a

(

x

)

= 0

x→a

(

x

)

x→a

(

x

)

= ∞ . ,, - %

lim f

= lim g

# lim f

= lim g

( lim

f ′(x)

, % ( -

x→a g′(x)

f (x)

f ′(x)

f(x) /g(x) x6 , lim

= lim

.

x→a g(x)

x→a g′(x)

' " : - ( #

( .

5 #" 6. $ , -

, - , - , %. : , - -

, - %, , - ,

%. , lim	x + sin x	% lim	x + sin x	= lim 1	+	sin x		= 1.
x→∞	x	x→∞	x	x→∞		x

< ( (1+cosx)/1=1+cos x x6?

.

1 ( % 0/0 # ?/?,

, # - ( .

!

& "

ln x

∞

1

1

sin x

1

' 1. 2lim

=

= lim

x

=

lim

=

.3

+ 2ln(sin x)

∞

x→0 1

x→0 2 1

cos x

2 x→0 xcos x

2

sin x

' 2.

2lim

1 − cos 4x

=

0

= lim

4sin 4x

= lim

16cos 4x

= 8.3

x→0

x2

0

x→0 2x

x→0

2

ln x

∞

1

' 3.

2lim xln x = 0 ∞ = lim

=

= lim

x

= − lim x = 0 .3

x→0

x→0

1x

∞

x→0 − 1x2

x→0

144

1

1

ln x − x +1

0

1

−1

' 4. 2lim

−

= ∞ − ∞ = lim

=

= lim

x

=

x −1

x→1

x −1 ln x

x→1

(x −1)ln x 0

x→1

ln x +

1 − x

−1

1

x

= lim

= lim

= −

.3

x→1 xln x + x − 1

x→1 ln x + 2

2

' 5: 2lim(1 + x2 )1 x = 1∞ .

x→0

1?, 10, ?0 # -

"-

.

' y = (1 + x2 )1 x .

ln (1 + x2 )

" % ln y =

1

ln (1 + x2 )=

.

ln (1 + x2 )

x

x

& limln y = lim

=

∞

= lim

2x

= lim

2

= 0 .

∞

+ x2

x→∞

x→∞

x

x→∞ x(1 + x2 )

x→∞ 1

ln y #, ln lim y = limln y = 0 .

x→∞

x→∞

, lim y = e0

# lim(1 + x2 )1 x = 1.3

x→∞

x→0

3.31. " # !$ " % 3.13

/ ’ , # (

.

# ":

1.

lim

2x2

+ 5x − 3

.

2.

lim

5x2

− 4x − 1

.

x + 3

x − 1

x → -3

x → 1

3.

lim

3x2

+ 5x − 2

.

4.

lim

4x2

− 14x + 6

.

x + 2

x − 3

x → -2

x → 3

5.

lim

6x2 + x − 1

.

6.

lim

6x2 − x − 1

.

x → -1 2 x + 1 2

x → 1 2 x − 1 2

7.

lim

9x2 − 1

.

8.

lim

3x2

− 5x − 2

.

x − 2

x → -1 3 x + 1 3

x → 2

9.

lim

3x

2 − 2x − 1

.

10.

lim

7x2 + 8x + 1

.

x + 1 3

x + 1

x → -1 3

x →

-1

145

11.	lim		x2 − 4x + 3								.																	12.	lim						2x2 + 3x − 2								.
																																							x − 1 2
	x → 3 x − 3																												x →	1 2
13.	lim					6x			2 − 5x + 1								.											14.	lim								10x2 + 9x − 7									.
									x − 1 3																														x + 7 5
	x →	1 3																											x →	-7 5
15.	lim							2x2 + 13x + 21																			.	16.	lim							2x2 − 9x + 10									.
									2x +		7																												2x − 5
	x →	-7 2																											x →	5 2
17.	lim					6x			2 + x − 1				.															18.	lim								6x2 − 75x − 39										.
																																							x + 1 2
	x →	1 3 x − 1 3																											x →	-1 2
19.	lim				2x2 − 21x − 11																		.					20.	lim		5x2								− 24x − 5				.
									x − 11																														x − 5
	x →	11																											x → 5
21.	lim			2x2 + 15x + 7																.								22.	lim				2x2 + 6x − 8								.
									x + 7																														x + 4
	x →	-7																											x →	-4			x2
23.	lim						6x2 − x − 1								.													24.	lim										+ 2x − 15		.
									3x + 1																														x + 5
	x →	-1 3																											x →	-5
25.	lim		3x2						− 40x + 128																.			26.	lim					5x					2 − 51x + 10							.
									x − 8																														x − 10
	x → 8																												x →	10
27.	lim					2x2 − 5x + 2													.									28.	lim				3x2 + 17x − 6										.
									x − 1 2																														x + 6
	x →	1 2																											x →	-6
29.	lim					3x			2 + 17x − 6													.						30.	lim							15x2 − 2x − 1									.
									x − 1 3																														x + 1 5
	x →	1 3																											x →	-1 5
31.	lim					15x2 − 2x − 1														.
									x − 1 3
	x →	1 3

3.32. '#! 4 +!

) y= f(x) (a, b) x0 (a, b). / %

: # P(x), x0 #

% f(x) . , # -

, - f(x)(P(x) # f(x) x0

( # P(x).

) ( # % n P(x)= Pn(x). @

Pn (x) = f (x0 )+	f ′(x0 )	(x − x0 )+	f ′′(x0 )	(x − x0 )2 + +	f (n) (x0 )	(x − x0 )n .
			2!
1!					n!
' Rn (x) = f (x)− Pn (x) n- (
f(x) x0. &			f (x) = Pn (x)+ Rn (x)		, , f (x) ≈ Pn (x)

146

- ( Rn (x) # .

$%, - x0 ( , b) x ( , b) % f (n+1)(x),

x ( , b) % ξ , - ( x0 x,

- (

R (x) =

f (n+1) (ξ )

(x − x )

n+1

.

(n + 1)!

n

0

5 ) * ( .

;

f ′(x0 )

f ′′(x0 )

f (x) = Pn (x)+ Rn (x) = f (x0 )+

(x − x0 )+

(x − x0 )2 +

f (n) (x )

1!

2!

n

f (n+1) (ξ )

n+1

+

0

(x − x0 )

+

(x

− x0 )

, x (x0, x)

n!

(n + 1)!

%

.

0- x0

= 0, (

f (x)

= f (0)+

f ′(0)

x + +

f (n) (0)

xn +

f (n+1) (ξ )

xn+1 , x

( x0, x).

1!

n!

(n + 1)!

5 , ).

3.33.! ) '#!- /

! ' ( )# * +

* f(x)=ex. ' : "

# (. -

( (n+1) : f (x) = ex , f (0) = 1;

f ′(x) = ex , f ′(0) = 1;

……………………..

f (n) (x) = ex , f (n) (0) = 1;

f (n+1) (x) = ex , f (n) (ξ ) = eξ .

, , %

f (n) (0)

f (n+1) (ξ )

f (x) = f (0)+

f ′(0)

x + +

xn

+

xn+1 =

1!

n!

(n + 1)!

x x2

x3

xn

xn+1

ξ

ξ ( 0, x).

= 1 +

+

+

+ +

+

e

,

1!

2!

3!

n!

(n + 1)!

$ x ,

# ex.

,

x=1, n=8,

, - %

# e : e = 1 + 1 + 1 + 1 + + 1 = 2,71828 - 2! 3! 8!

147

( R <								e		< 10−5 .
							8	9!
			$ ,							- #- x R (
			xn+1			ξ	→ 0 - n → ∞ .
R	=					e
		(		)
n
			n + 1 !

, , #- x, ( % ,

# ex #- .

& ( : " f(x)=sin x. & f(x)=sin x:

f (x) = sin x , f (0) = 0 ;

π

f

′(0) = 1;

f ′(x) = cos x = sin x +

,

2

π

2π

f

′′(x) = cos x

+

2

= sin x

+

,

f

′′(

0) = 0 ;

2

2π

3π

f

′′′(x) = cos x +

= sin x

+

,

f ′′′(0) = −1;

2

2

IV

3π

4π

IV

) (0) = 0 ;

f

(

) (x) = cos x

+

= sin x +

,

f (

2

2

………………………………………………………..

f

(

n

)

π

f (

n

)

(0) = sin

π n

;

(x) = sin x + n

,

2

2

f

(n+1) (x) = sin x +

(n + 1)π

,

f (n+1) (ξ ) = sin

ξ + (n + 1)π

.

2

2

' # : ", -

:

ξ + (n + 1)π

x

3

x

5

sin π n

sin

sin x = x −

+

− +

2

xn +

2

xn+1 .

(

)

3!

5!

n!

n + 1 !

, - ( n- ,

x

3

x

5

− + (−1)n−1

x

2n−1

sin ξ +

(n + 1)π

sin x = x −

+

+

2

xn+1 .

(

2n

)

n

)

3! 5!

(

− 1 !

+ 1 !

sin

ξ + (n + 1)π

≤ 1, ex -

2

, - lim Rn (x) = 0 x.

x→∞

148

!

' 1: # # sin 20°

: ".

2 & % n=3. : %:

x

3

x

5

− + (−1)n−1

x

2n−1

sin ξ +

(n + 1)π

sin x = x −

+

+

2

xn+1 .

(

2n − 1 !

(

3! 5!

n + 1 !

)

)

π 3

π

π

π

π

3

9

$ , - 20°=π/9: sin

9

= 9

−

= 9

−

= 0,342 .

3!

6 93

# (, ( :

sin

ξ + (n + 1)π

= π

4

sin (ξ + 2π )

π

4

1

R

=

2

xn+1

≤

≈ 0,00062 < 0,001

n

(n + 1)!

4!

9

9

4!

, , sin 20°= 0,342 0,001.3

' 2: * : " f(x) = cos x.

2 < -

:

x

2

x

4

cos π n

cos

ξ

+ (n + 1)π

cos x = 1 −

+

− +

2

xn +

2

xn+1 #

n!

(

)

4!

4!

n + 1 !

x

2

x

4

− + (−1)n−1

x

2n

cos ξ

+ (n + 1)π

cos x = 1 −

+

+

2

xn+1 .

)

(

)

2!

4!

(

!

2n

n + 1 !

, lim Rn (x) = 0 x.3

x→∞

' 3:

* : " f(x)=ln(1+x).

2 &, - # % D(y)=(–1; +?).
& : " .
f	(			x		)					(					)		, f	(		0		)			= 0 ;
									= ln 1 + x															)
f ′		(			x		)			=	(	+ x		)	−1 , f				′	(							= 1;
			(					)			1											0		(
f ′′					x					= −		(	+ x				)	−2 ,		f ′′								)		= −1;
									)			1														0			)
f ′′′				(		x						(	+ x				)	−3 ,		f ′′′					(
										= 2 1																	0			= 2 ;			)
f IV					(					)			3				(					)	−4 , f IV								(			= −2 3 ;
								x			= −2							1 + x														0

………………………………………………………..

149

f (n) (x) = (−1)n−1 (n − 1)!(1 + x)− n , f (n) (0) = (−1)(n − 1)!;

f (n+1) (x) = (−1)n n!(1 + x)−(n+1) , f (n+1) (ξ ) = (−1)n n!(1 + ξ )−(n+1) .

' : ".

ln (1 + x) = x −

x2

+

2x3

−

2 3x4

+ + (−1)n−1 (n − 1)! xn + (−1)n

n!(1 + ξ )−(n+1)

xn+1 ,

(n + 1)!

2!

3!

4!

n!

# ln (1 + x) = x −

x2

+

x3

−

x4

+ + (−1)n−1

xn

+ (−1)n

xn+1

.

n

(n + 1)(1 + ξ )(n+1)

2

3 4

: , - x (–1;1], lim Rn (x) = 0 , # -

x→∞

x ( –1;1].3

' 4: * : " f(x) = (1+x)m, m R,

m=0.

2 ' m+Z x> –1. & :- " % :

f (x) = (1 + x)m , f (0) = 1;

f ′(x) = m(1 + x)m−1 , f ′(0) = m;

f ′′(x) = m(m − 1)(1 + x)m−2 , f ′′(0) = m(m − 1);

f ′′′(x) = m(m − 1)(m − 2)(1 + x)m−3 , f ′′′(0) = m(m − 1)(m − 2);

………………………………………………………..

f (n) (x) = m(m − 1) (m − n + 1)(1 + x)m−n , f (n) (0) = m(m − 1) (m − n + 1)!; f (n+1) (x) = m(m − 1) (m − n)(1 + x)m−n−1 ,

f (n+1) (ξ ) = m(m − 1) (m − n)(1 + ξ )m−n−1 .

,

(1 + x)m = 1 + m x + m(m − 1) x2 + + m(m − 1) (m − n + 1) xn +

1! 2! n!

+ m(m − 1) ((m − n))(1 + ξ )m−n−1 xn+1 .

n + 1 !

: , - |x|<1 lim Rn (x) = 0 .3

x→∞

3.34. 7 ( &( ( #' "

&( ( #7#" )# *

' % # . ; y=f(x), [ , b], %

, - # ( x [ , b] - % # ( , # - x1 < x2, f(x1) < f(x2).

150