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Саратовский государственный университет им. Н.Г. Чернышевского

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[НЕСОРТИРОВАННОЕ]

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Матан Лекции

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# 14
1.	.* $ %#/ f	!+ R ,
		lim f (x) = 0,	lim f (x) =1.
		x®-¥	x®¥
	. # # # $ %#/ 2 ) ! 0 ) #/ 2( 0 # 1?
2.	# #0 * ( 6#" $ %#' # 3 ! $ %##		0 [0,1]
	a) f (x) = x, X = [0,1];
	b) f (x) =\| x \|,	X = [-1,1];
	c) f (x) = x2 ,	X = [-1,1];
	d) f (x) = log x,	X = [1, e];
	e) f (x) = sin x,	X = [0,p ];

3.# #0 * ( 6#" $ %#' (#$$ %# ! + ) x0 = 0

a)f (x) =| x |;

b)f (x) =| sin x |;

c)f (x) =| x3 |;

d)f (x) = | x |;

4.	.* $ %##	f	# g (#$$ %# ! + ) x0 ,		f (x0 ) = g (x0 )	#	f (x) < g (x)	( / x > x0 . #, ) 3/0
	* + ( #+ +			* +
				f ¢(x0 ) < g¢(x0 );
5.	.* $ %##	f	# g (#$$ %# ! + ) x0 ,		f (x0 ) = g (x0 )	#	f (x) < g (x)	( / x > x0 . #, ) 3/0
	* + ( #+ +			* +
				f ¢(x0 ) £ g¢(x0 );
6. .* $ %#/		f		!+ 0 [—1,1] # (#$$ %# + ) 0. .2#

g (x) = f (x) - f (0) ; x

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#, ) $ %#/ g 3/0 & #) [-1,0) È (0,1] ?

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# 15

.* $ %#/

(#$$ %#

)

x0 ,

f (x) ³ 0 .

)

$ %#/

y =

3/0

f (x)

(#$$ %# + ) x0 ?

.* $ %#/

(#$$ %#

)

x0 ,

f (x) > 0 .

)

$ %#/

y =

3/0

f (x)

(#$$ %# + ) x0 ?

.* y = f (x) = x3 . ;+ / */ # 3 / $ %#/

f -1 (#$$ %# ' + +* " ) " x Î R ?

.* $ %#/ y = f (x) (#$$ %# + ) x0 , $ %#/ z = g ( y)

!+ ,

(#$$ %# + )

y0 ,

y0 = f (x0 ) . #, ) 0#%#/ g o f 2 0 */ (#$$ %# ' + ) x0 ?

.* $ %#/ y = f (x)

!+ ,

(#$$ %# + ) x0 , $ %#/ z = g ( y) (#$$ %# + )

y0 ,

y0 = f (x0 ) . #, ) 0#%#/ g o f 2 0 */ (#$$ %# ' + ) x0 ?

.* $ %#/

y = f (x)

!+ ,

(#$$ %# + )

x0 , $ %#/

z = g ( y)

!+ ,

(#$$ %# + ) y0 , y0 = f (x0 ) . #, ) 0#%#/ g o f

2 0 */ (#$$ %# ' + )

x0 ?

.* $ %#/ f

# !' *# + ) x0

# * 6 * + ( *

#' (

f ¢(x

- 0) =

lim

f (x) - f (x0 )

;

x®x0 -0

x - x

#, ) f ¢(x0 - 0) ³ 0 ?

.* $ %#/ f

# !' *# + ) x0

# * 6 * + ( *

#' (

f ¢(x

+ 0) =

lim

f (x) - f (x0 )

;

x - x

x®x0 +0

#, ) f ¢(x0 + 0) £ 0 ?

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1.	.* $ %## f # g ( + + / * +#/: !+ !	0 [a,b];(#$$ %# ! # +
	(a,b) g (a) ¹ g (b) # f ' (x) # g ' (x) 3 6 */ + 0 ( +	[a,b].
	#, ) 3/0 '( */ ) Î(a,b) /, )

f ' (c) = f (b) - f (a) . g ' (c) g (b) - g (a)

2.#0 * ( 6#" $ %#' g * ( +!3 + 7# * ( 0 ) ##, ) 3! + # * +

, & 2 * ( 0 ) ##

a)g(x)=x; b)g(x)=|x|; c)g(x)=f(x);

3.	.* $ %## f # g ( + + / * +#/:
	(#$$ %# ! +		' d -* * # ) # x0 , 0 #* ) # * ' ) # x0;
	limx®x	f (x) = 0 # limx®x	g (x) = ¥ ;
		0	0
	* 6 * + (# #0 ( +
( )			lim	f ' (x)g 2 (x)	= l.

				g ' (x)
			x®x0	g ' (x)
(3)			lim	g ' (x) f 2 (x)	= l.

				f ' (x)
			x®x0	f ' (x)
	#0 * ) + (
			lim f (x)g (x)
			x®x0
	* 6 * + # + –l.
4.	.* $ %## f # g ( + + / * +#/:
	(#$$ %# ! +		' d -* * # ) # 0, 0 #* ) # * ' ) # 0;

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, %#/ 16

lim x®x f (x) =1# limx®x		g (x) = ¥ ;
0		0
* 6 * + (
			lim	f ' (x)g 2	(x)	= l.


			x®x0 g ' (x) f (x)
#, ) (
				lim f (x)g ( x )
				x®x0
* 6 * + # +	e-l.
5. .* $ %## f # g ( + + / * +#/ :
(#$$ %# ! +		' d -* * # ) # 0, 0 #* ) # * ' ) # 0;
f (x) > 0, lim x®x f (x) = 0 # limx®x			g (x) = 0;
0			0
* 6 * + (
			lim	f ' (x)g 2	(x)	= l.


			x®x0 g ' (x) f (x)
#, ) (
				lim f (x)g ( x )
* 6 * + # +				x®x0
* 6 * + # +	e-l

6. .* $ %## f # g ( + + / * +#/:

(#$$ %# ! + ' d -* * # ) # 0, 0 #* ) # * ' ) # 0;

lim f (x) = ¥ # lim g (x) = 0;

x®x0 x®x0

* 6 * + (

lim f ' (x)g 2 (x) = l. x®x0 g ' (x) f (x)

#, ) (

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, %#/ 16

lim f (x)g ( x )

x®x0

	* 6 * + # +		e-l.
7.	.* $ %## f # g ( + + / * +#/:
	(#$$ %# ! + ' d -* * # ) # 0, 0 #* ) # * ' ) # 0;
	lim f (x) = ¥ # lim g (x) = ¥;
	x®x0	x®x0
	* 6 * + (
					lim	e f ( x ) f '(x)	= l > 0.


					x®x0 e g ( x ) g '(x)
	#, ) (
					lim ( f (x) - g (x))
					x®x0
	* 6 * + # +		log l.
	.* f (x) = x # g (x) =				. #0 * ( 6#" ( + # # +# , # / 1$$#+ #+ (#
8.	.* f (x) = x # g (x) =			x2 +1	. #0 * ( 6#" ( + # # +# , # / 1$$#+ #+ (#
	+

f(x)

a)limx®¥ g (x) ;

g(x)

b)limx®¥ f (x) ;

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# 17.

1.	.* $ %#/ f (#$$ %#	# + (a,b) # f			¢
					(x) > 0 +* (, 0 #* ) # ) & )#* ).
	#, ) $ %#/ f 3/0 /+ / / & + 0 * 6 ' (a,b)?
2.	.* $ %#/ f (#$$ %#	# + (a,b) # f			¢
					(x) > 0 +* (, 0 #* ) # ) & )# ).
	#, ) $ %#/ f 3/0 /+ / / & + 0 * 6 ' (a,b)?
3.	.* $ %#/ a 3 * ) # x ® x0 # (#$$ %# + d -* * # ) # x0 , 0 #* ) # ) # x0 .
	#, ) $ %#/		f (x) = a(x)(x - x0 )2

	3/0 (+ 2(! (#$$ %# + d -* * # ) # x0 ?
4.	.* $ %#/ f (#$$ %# (n+1) 0 + d -* * # ) # x0 # * , & 2 + $ ' +
			f (n+1) (c)		n+1
				(x	- x )	.
			(n +1) !
					0

	#, ) f (n+1)(c)(x - x )* # */ 0 # x ® x ?
	0	0
5.	.( 2#, ) $ %#/ f #2(! (#$$ %# + ) x0 # ( + + / * +#/:

f¢(x0 ) = f "(x0 ) = 0 ;

f"'(x0 ) > 0 .

#0 * ( 6#" + 2( #' 2 +!* 0 $ %## f

a)# !' *# + ) x0 ;

b)# !' # # + ) x0 ;

c)# & 1 * + ) x0 ;

d)0/ ( ( * + & +.

6. .( 2#, ) $ %#/ f ) ! 2(! (#$$ %# + ) x0 # ( + + / * +#/:

346

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f '(x0 ) = f "(x0 ) = f "'(x0 ) = 0 ;

f (4 ) > 0 .

#0 * ( 6#" + 2( #' 2 +!* 0 $ %## f

a)# !' *# + ) x0 ;

b)# !' # # + ) x0 ;

c)# & 1 * + ) x0 ;

d)0/ ( ( * + & +.

7. .( 2#, ) $ %#/ f 3 * ) )#* 0 (#$$ %# + ) x0 # +* #0+ ( ! 3 &

/( + ) x0 3 6 */ + 0.

#0 * ( 6#" + 2( #' 2 +!* 0 $ %## f

a)# !' *# + ) x0 ;

b)# !' # # + ) x0 ;

c)# & 1 * + ) x0 ;

d)0/ ( ( * + & +.

347

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# 18.

1. .* A # B - +! ! 2 * +. #0 * ( 6#" 2 * + 3/0 +!

a)AÈB;

b)AÇB;

c)A\B.

2. #0 * ( 6#" 2 * + /+ / */ +! ! (a > 0, b > 0, p > 0) a)

	x2	+		y 2	£1;
	a2			b2

b)
	x2	-	y 2		£1;
	a2
				b2

y 2 £ 2 px.

3. # #0 * ( 6#" $ %#' +! ! *+ #" 2 * + " ( #/

a) y=sinx,

0 £ x £ 2p;

b)y=sinx, p/2 £ x £ 5p/2;

c)y=x, -¥ £ x £ ¥;

d)y=x2, -¥ £ x £ ¥;

e)y=x3, -¥ £ x £ ¥;

348

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				! ( / # + #/ # /
				, %#/ 18.
	f) y=\|x\|,		-¥ £ x £ ¥;
	g) y=\|\|x\|-1\|,		-¥ £ x £ ¥;
	h) y=\|\|x\|-1\|,		0 £ x £ ¥.
4.	.# # # # #' +! * # $ %## y=\|x\|?
5.	.* $ %#/ y=f(x) (#$$ %# # +! 0 [a, b]. 4 ( # $ %#/ y=f¢(x) 3/0 +! '
	[a,b]?
6.	.#+ (# # $ %##, / +! + 6 * +			' # + * *+ # # #0+ ( ! # 3 & /(.
7.	/ #"	!" n $ %#/ y=xn /+ / */ +! '		+ 6 * + ' *#?
8.	/ #"	!" n $ %#/ y=xn # ) x=0?
9.	.* $ %## f(x) # g(x) +! ! 0 [a,b]. / #0 * ( 6#" $ %#' 3/0 +! [a,b]

a)f(x) + g(x);

b)f(x) - g(x);

c)f(x)g(x).

10. .* $ %#/ y=f(x) +! 0 [a,b]. 4 ( # $ %#/ y=-f(x) 1 0

a)+! ';

b)+ & ';

c)# ) # .

349

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