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Саратовский государственный университет им. Н.Г. Чернышевского

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

f ' (x ) = lim

f (x) - f (x0 )

f (x) - f (x0 )

.PDF

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Добавлен:

09.06.2015

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com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 16 5. .+# , # / ( / 7 #/ 3 * ) 3 7#"

* & * 7#, # ' $ %#/ f # g 0 [x, x1], '( */ ) c Î (x, x1) /, )

f ' (c) = f (x1 ) - f (x) . g ' (c) g (x1 ) - g (x)

. 3 0 1 $ +#(

f (x)

f (x1 )

-1

f ' (c)

f (x)

g ' (c)

g (x)

g (x )

-1

g (x)

0 7# 1 + * + *# f(x)/g(x)

f ' (c)

g (x1 )

-1

f (x)

g (x)

g ' (c)

f (x )

-1

f (x)

!) l #0 3 #" ) * ' + * + # + ( 2( * + ! 3 0 + #/

f (x) - l = g (x)


	g (x1 )	-1

f ' (c)	g (x)

	f (x1 )
g ' (c)			-1

	f (x)


	f ' (c)
-1 +	f ' (c)	- l.
-1 +		- l.
	g ' (c)

* ( )# % # ( /

(3)	f (x)	- l	£	f ' (c)
				g ' (c)
	g (x)

g (x1 )	-1
g (x)			-1	+	f ' (c)	- l	.
f (x )			-1	+		- l	.
f (x )		-1			g ' (c)
1		-1
f (x)

- $# *# x1 # * # x x0. &( 7 #/ (f(x1))/f(x) # (g(x1))/g(x) 3 * ) ! # 1 0 ) # $ %##

160

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, %#/ 16 5. .+# , # / ( / 7 #/ 3 * ) 3 7#"

g (x1 )

-1

g (x)

-1

f (x1 )

-1

f (x)

* # */ 0 # x® x0. - )#, * 6 * + d2 > 0 , ) ( / +* " x, x0 < x < x0+d2, * + ( #+

+ * +

g (x1 )

-1

(4)

g (x)

-1

< e.

f (x1 )

-1

f (x)

3# + * + * + (1), (2) # (4) # # / #" +

+ * + (3) #, & / d* = min{d, d , d }, )#

f (x)

"e > 0$d* > 0"x, x < x < x

+d*

- l

< e (e +

+1),

g (x)

) +! 2 */ * +#

lim

f (x)

= l.

x®x0 +0 g (x)

# #) ! # ** 2( #/ # 2, )

lim f (x) = l.

x®x0 -0 g (x)

0 (+ " * ( #" + * + * (, )

lim f (x) = l,

x®x0 g (x)

) 0 )#+ ( 0 * + ! 5.

161

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, %#/ 16 5. .+# , # / ( / 7 #/ 3 * ) 3 7#"

3 ! 4 # 5 * + ( #+! # + * ), &( x0 = -¥ # # x0 = ¥. 0 * + * & ( + / */ + * + * +## * /+#+7# #*/ #0 #/ # + + ## ) # x0 3 * ) ( ' ) #.

. +# , # / 0+ / + * ) * 7 & # #/ 0 # #* ( 7 #/ 3 * ) !" # # 3 * ) 3 7#" #* ( 7 #/ #" #0+ ( !".

162

x - x0

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		, %#/ 17
	1. # #'	* # $ %##
	17
1.	# #' * # $ %##.................................................................................................................................	163
2.	.#0+ ( ! +!*7#" /( +....................................................................................................................................	164
3.	' * * . .............................................................................................................................	165
4.	' * * , & 2.......................................................................................................................	167
5.	* ) * +# 1 * $ %## .................................................................................................................	170

1. / +

. 2, # * (* + ( * + / (#$$ %# #*)#* # ( / * + #/ * # $ %##

# +.

1. ! % # f %% # (a, b). / # f (a, b) ,

f¢ (x) ³ 0 (a, b). / # f (a, b) , f¢ (x) £ 0 (a, b).

. .+ ( ( 0 * + ( / 3!+ 6 ' $ %## f, ( / + 0 * 6 ' $ %## ** 2( #/ *# #) !.

) * 3" (# * # * +#/. .* $ %#/ f 3!+ (a, b). !3 #0+ ) x0 Î (a, b) # '(

#0+ (

f ' (x0 ) = lim f (x) - f (x0 ) .

x®x0

. * x > x0. &( x - x0 > 0 # f(x) - f(x0) ³ 0 3 & ( / , ) $ %#/ f 3!+. ( + ,

f(x) - f (x0 ) ³ 0 x - x0

# 1

163

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,%#/ 17 2. .#0+( ! +!*7#" /(+

. * x0 - #0+ / ) #0 (a, b), f¢ (x) ³ 0 ( / +* " x Î (a, b).

( 2 ( * ) * * +#/. .* f¢ (x) ³ 0 (a, b). !3 #0+ ) # x1 Î (a, b) # x2 Î (a, b), x1 < x2.

%#/ f (#$$ %# [x1, x2]. .# # , & 2 f [x1, x2], * & * ' '( */ ) c Î (x1, x2) /, )

f(x2) - f(x1) = f¢ (c)(x2 - x1).

f¢ (c) ³ 0 # x2 - x1 > 0, f(x2) - f(x1) ³ 0, ) 0 ) 3!+ # $ %## f (a, b) # 0 )#+ ( 0 * +

! 1.

2. ( '

(# # ( #+ ( # #0+ ( !" + & # #0+ & /( +.

( # 1. ! d- x0 % # f f¢, , ,

% #, %% # x0. " % # f %% # x0

(f¢ )¢ (x0)

% # f x0

f¢¢ (x0).

!, d- x0 % # f f(k-1), , ,

% #, %% # x0. " % # f k %% # x0

(f(k-1))¢ (x0)

k % # f x0

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, %#/ 17 3. ' * * .

f(k)(x0).

# 3 0, f(k)(x0) = (f(k-1))¢ (x0).

3. . % (

& * ( # (#$$ %# ' + ) x0 $ %## f 2 #3 #0# # ' ' $ %# ' * & 7 * 3 +!* & /( * # # x® x0. . / # #0+ ( !" +!*7#" /( + 3 36 1 ( #, 0 #+ # ' $ %# & ) * #, + ' /( (#$$ %# * #. 6 # & (" ( * ( 2# */ + * ( 6 ' .

2. ! % # f %% # n x0. "

x® x0 % # a

, %

f (x) = f (x ) + f ' (x )(x - x ) +

f ' ' (x )

(x - x )2 +...

(n ) (x )

(x - x

)n +a(x)(x - x )n .

( n

f ' ' (x0 )

( n)

(x0 )

(k )

(x0 )

Tn (x) = f (x0 ) + f ' (x0 )(x - x0 ) +

(x - x0 ) 2 + ... +

(x - x0 ) n

(x - x0 ) k .

k =0

", (

a(x)(x - x0)n

% # f " !. 2 (* + / $ ' * * . .

. .+ ( ( 0 * + # ( %##. / n = 1 + 2( # ! 2 # $ '

+ 6 */ + ( # (#$$ %# ' + ) x0 $ %## # 1 * + ( #+!.

. ( 2#, ) + 2( # ! 2 # $ ' * + ( #+! ( / k = 1, 2, ..., n-1 # ( 2 #" ( / k = n.

165

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, %#/ 17 3. ' * * .

3 0 )#

Rn(x) = f(x) - Tn(x).

!* ! 2 * * # + , ) Rn(x) = a(x)(x - x0)n. 4 ( ( 0!+ 1 * 7 #.

. 2( +* & 0 #, ) +*/ #' & ) /+ / */ * & ( 0 (#$$ %# ' $ %# ' + 3 ' ).

. 1 $ %#/ Rn (#$$ %# n 0 + ) x0. '( #0+ ( Rn¢ + #0+ ' ) (#$$ %# * # Rn

								n	f	(k )	(x0 )
(1)	R' (x) = f ' (x) - T ' (x) = f ' (x) -								f		(x0 )		(x - x			) k -1 =
(1)	R' (x) = f ' (x) - T ' (x) = f ' (x) -								(k -1)!				(x - x		0	) k -1 =
			n						(k -1)!						0
								k =0
	f ' (x) - f ' (x		) - f ' ' (x		)(x - x		) - ... -			f (n ) (x		0	)	(x - x				) n-1
												0
		0		0		0											0
		0		0		0				(n -1)!							0
										(n -1)!

( & ' * !, * $ %#/ f (#$$ %# n 0 + ) x0, #0+ ( / f¢ (#$$ %# (n - 1) 0 + ) x0. & * 0 # ( %## ( / f¢ * + ( #+ $ ' * & ) ' /( (n - 1).

- )#, * 6 * + 3 *	) / # x® x0						$ %#/ b /, ) * + ( #+ $
(2)	f ' (x) = f ' (x			) + ( f ' )'(x					)(x - x				) + ... +		( f ' )( n-1)			(x	0	)	(x - x				)n-1 + b (x)(x - x				) n-1	=
(2)	f ' (x) = f ' (x		0	) + ( f ' )'(x				0	)(x - x			0	) + ... +		(n -1)!						(x - x			0	)n-1 + b (x)(x - x			0	) n-1	=
			0					0				0												0				0

	f ' (x		) + f ' ' (x			)(x - x					) + ... +			( f )(n ) (x		0	)	(x - x					) n-1	+ b (x)(x - x			) n-1 .
																0
		0			0					0				(n -1)!								0				0
		0			0					0												0				0

+ #+ / $! (1) # (2), #" (# +!+ (, )
(3)									R			¢(x) = b(x)(x - x )n-1.
											n							0

166

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 17 4. ' * * , & 2

3 # + # #, ) Rn(x0) = f(x0) - Tn(x0) = f(x0) - f(x0) = 0. !3 #0+ ) x #0 d-* * # ) # x0, &( Rn (#$$ %#, # # # Rn , & 2 0 * % +! # ) # x # x0, * & * ' 2( x

# x0 '( */ ) c /, ) * + ( #+ $

Rn(x) = Rn(x) - Rn(x0) = Rn¢ (c)(x - x0).

8 $ + * * (3) ( + * +

Rn(x) = b(c)(c - x0)n-1(x - x0).

) +#( , )

c - x

x - x0

#* # x * # */ x0, # 2 ) / ) c * # */ x0. .10

lim	R (x)	= lim	b (c)(c - x )n-1	(x - x )	= lim b (c)	(c - x )n-1	= 0.
	n		0	0		0
			(x - x )n			(x - x )n-1
x®x0 (x - x )n		x®x0			x®x0
0			0			0

# 3 0, ! ( 0 #, ) $ %#/

	a(x) =	Rn (x)
	a(x) =	(x - x )n
		(x - x )n
		0

/+ / */ 3 * ) ' # x® x0, ) 0 )#+ ( 0 * + ! 2.

4. . % #

* # $ %# f 2# * +#/ 3 2 * #, ) + 2, # # $%#/ & 7 * # #3 #2 #/ $ %## & ) ' 3 ( 3 & ).

3. ! % # f %% # (n + 1) d- x0. " x d-

x0 c, x x0, , %

167

(n +1)!

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 17 4. ' * * , & 2

f (x) = f (x

) + f ' (x

)(x - x

) +

f ' ' (x

)

(x - x

+ ... +

f ( n) (x

)

(x - x

) n +

f (n+1) (c)

(x - x

) n+1 .

(n +1)!

. , Tn % 3 ", (

f ( n+1) (c) (x - x0 ) n+1

% # f " 1. 3 (* + / $ ' * * , & 2.

. 3 0 )#

Rn(x) = f(x) - Tn(x).

!* ! 3 * * # + , ) Rn(x) +	* , & 2. -#, ) $ %#/ Rn (#$$ %# (n + 1) 0 +
d-* * # ) # x0, Rn(x0) = 0. / ( 0 * + +!3 #0+				) x #0 d-*				* # ) # x0 # 0 *
% +! # ) # x # x # # 7# $ %#' F		1	= R	n	# G	, &( G	(x) = (x - x )n+1. 3 $ %##
0					1	1		0

(#$ %# ! ** #+ 0, G1(x0) = 0, G1(x) ¹ 0 # G1¢(x) ¹ 0, * # x ¹ x0. & * 7# 2( x

# x0 '( */ ) c1 /, ) * + ( #+ $


Rn (x)	=	F1 (x)	=	F1 (x) - F1 (x0 )			=	F1 ' (c1 )		=	f ' (c1 ) -Tn ' (c1 )		.
(x - x )n+1				G (x) - G (x )
		G (x)						G ' (c )			(n +1)(c - x )n
0	1		1		1	0	1		1	1		0

+ 0 * % +! # ) # c1 # x0 # # 7# $ %#' F2 = F1¢ # G2 = G1¢. 3 $ %##

(#$$ %# ! ** #+ 0, F2(x0) = f¢ (x0) - Tn¢(x0) = 0, G2(x0) = 0, G2(x) ¹ 0 # G2¢(x) ¹ 0, * # x ¹ x0. & * 7# 2( c1 # x0 '( */ ) c2 /, ) * + ( #+ $

F2 (c1 ) = F2 (c1 ) - F2 (x0 ) = F2 ' (c2 ) = f ' ' (c2 ) -Tn ' ' (c2 ) . G2 (c1 ) G2 (c1 ) - G2 (x0 ) G2 ' (c2 ) (n +1)n(c2 - x0 )n-1

168

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, %#/ 17 4. ' * * , & 2

#*. 1. * %#/ ( 0 * + ! $ ' * * , & 2

(. n- 7 & 0 * % +! # ) # c

# x

# # 7# $ %#' F

= F

(n-1)

# G

n-1

= G

(n-1). 3 $ %## (#$$ %# !

** #+ 0, F (x ) = f(n-1)(x ) - T

(n-1)(x ) = 0, G

(x ) = 0, G

(x) ¹ 0 #

Gn¢(x) ¹ 0, * # x ¹ x0. & * 7# 2( cn-1 # x0

'( */ ) cn /, ) * + ( #+ $

F (c

n-1

)

F (c

n-1

) - F (x )

F ' (c

)

f ( n) (c

) -T (n) (c

)

Gn (cn-1 ) Gn (cn-1 ) - Gn (x0 )

Gn ' (cn )

(n +1)n...2(cn - x0 )

* ( 7 &

0 * % +! # ) # c # x # # , & 2 $ %## F

n+1

= F

(n). 8 $ %#/

) - T

(#$$ %#

** #+ 0, F

n+1

(x ) = f(n)(x

(n)(x ) = 0.

& * , & 2 2( c

# x

'( */ ) cn+1 /, ) * + ( #+ $

)

) - F

(x )

' (c

n+1

)

(n+1) (c

n+1

)

n+1

Gn+1 (cn )

(n +1)!(cn - x0 )

(n +1)!

+ ' ! ) #, ) (n + 1)-/ #0+ ( / Tn(n+1)(x) & ) * #

+!7 n 2( * +

3 6 */ +

0+ 6 /* ) % ) # + * + #

) ! 3 0 ) #/, 3 2( */, )

R (x)

f (n+1) (c)

(x - x )n+1

(n +1)!

&( c = cn+1, ) * + * + *$# +

+#( & 7 * # #3 #2 #/ $ %## f & ) ' #

0 )#+ ( 0 * + ! 3.

169

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