Матан Лекции

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Саратовский государственный университет им. Н.Г. Чернышевского

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[НЕСОРТИРОВАННОЕ]

Файл:

C ( f (xk ) - f (xk -1 )) dx = C ( f (xk ) - f (xk -1 ))

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2 ) % # f [a, b], % # F,

F (x) = f (t)dt

%% # [a, b] x Î [a, b] %

F¢ (x) = f(x).

. !3 #0+ ) x0 Î [a, b] # ( 2, ) $ %#/ F (#$$ %# + ) x0.

!+ * $ %## f # ( + ( ## #0+ ( '

	F ' (x0 ) = lim	F (x) - F (x0 )

		x - x
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		x	x0		x
F (x) - F (x )	=	f (t)dt - f (t)dt		=	f (t)dt	=	f (c)(x - x )	= f (c) ,
0		a	a		x0		0
x - x0			x - x0		x - x0		x - x0
x - x0			x - x0		x - x0		x - x0

&( c Î [x0, x]. * # x * # */ x0, # c * # */ x0. ." (/ + * ( + * + ( # x® x0 # 0 /*

!+ * $ %## f + ) x0, )#

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, %#/ 25 4. -, '3 #%

lim	F (x) - F (x0 )		=	lim	f (c) = f (x )

x®x0 +0	x - x			c®x0 +0	0

	0
# #) ! # ** 2( #/ # * + #+, )
lim	F (x) - F (x0 )	=		lim	f (c) = f (x )

x®x0 -0	x - x			c®x0 -0	0

	0

) ( 0!+ (#$$ %# * $ %## F + ) x0 # $

F¢(x0) = f(x0).

* # ) x0 * + * ( ' #0 ' #" ) a # # b 0 [a, b], ** 2( #/ */ */ ( *

( + 1 ' ). .* x0 - #0+ / ) 0, $ %#/ F (#$$ %# + 2( ' ) 0 [a, b] # +* ( * + ( #+ '( / $ ( / #0+ ( ', ) 0 )#+ ( 0 * + ! 2.

4 & ( / +/0# 2( # & #	! + " # #		! #2 # ( # # & # + #/
b	b	x	b
G(x) = f (t)dt = f (t)dt - f (t)dt = f (t)dt - F (x),
x	a	a	a
+#(#, ) $ %#/ G 2 (#$$ %#	0 [a, b], + +* " ) " & * + ( #+ $

G¢(x) = -F¢(x) = -f(x).

4. . !"- -#

. # * (* + & +!+ (, 2 + ) & 3 # * (* +#/. , $ ( / #0+ ( ' F¢ 0 ), ) $ %#/ F /+ / */ + 3 0 ' $ %## f. ! 0# 1 $ * (* +# ! 2.

!+ / 0 [a, b] $ %#/ f # + 3 0 1 0.

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	x
	F (x) = f (t)dt .
	a
6 3 + 2 0 ) # * ( #0 ! 2, * # 2# +		' x=b #, # # / + + # #, ) F(a) =0,
#* $ + +#(
	b
	f (t)dt = F (b) = F (b) - F (a)
	a
	%#/ F 0( * /+ / / # & ! + " # ( # & # + #/ # + 2 + / + 3 0 ' ( /
$ %## f. 3! 0 , # 6 # + 3 0 ' F(b)-F(a)		0 +#*# +!3 + 3 0 '. $#
	) !' 0 ( .

3 ) % # f [a, b], F

	b
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(#$$ %# & #*)#* #/.

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+ # & # , * * + ! ( / & +!& /(# 3 0 ) #

f (x)dx,

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1.	& # + # ) * / + # & # .......................................................................................................					254
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% 3
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f (x)dx = f (j(t))j' (t)dt.

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!+	!" $ %#' f # j # #0+ ( # ( f o j)j' !+ !	[a, b]. # 3 0, 3 # & + $# +
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x Î [a, b] , %

b	x	b
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a	a	x
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0.

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/, ) * + ( #+ $

b x

( fg )(x)dx = f (a) g (x)dx.

a a

0( (# 03# # P = {x0, x1, x2, ..., xn-1, xn} 0 [a, b] * 0 # Dk = [xk-1, xk], k = 1, 2, ..., n, (# + ' ( # !, + ' (b - a)/n. 4 & ( / *+ '* + ((# #+ * # # # ' * # # & #

n xk

( fg )(x)dx = ( fg )(x)dx = ( f (x) - f (xk -1 ) + f (xk -1 ))g (x)dx =

k =1 x

k -1

k =1 x

k -1

(1)

( f (x) - f (xk -1 ))g (x)dx + f (xk -1 ) g (x)dx.

k =1 x

k -1

k =1

k -1

3 0 )#

n xk

Sn1 = ( f (x) - f (xk -1 ))g (x)dx, Sn2 = f (xk -1 ) g (x)dx.

k =1 x

k -1

k =1

k -1

% # Sn1

Sn1

( f (x) - f (xk -1 ))g (x)dx

k =1 x

k -1

k =1

k -1

f (x) - f (xk -1 )

g (x)

dx £ C

f (x) - f (xk -1 )

dx =

k =1 x

k -1

k =1 x

k -1

256

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, %#/ 26 3. / * ( 0 ) ##

n xk		n xk
C ( f (x) - f (xk -1 ))dx £ C ( f (xk ) - f (xk -1 ))dx =
k =1 x	k -1	k =1 x	k -1

Cb - a ( f (x1 ) - f (x0 ) + f (x2 ) - f (x1 ) +... + f (xn ) - f (xn-1 )) = n

C(b - a)( f (xn ) - f (x0 )) = C(b - a)( f (b) - f (a)) .

n		n
. '( + 1 + * + ( # n®¥ # +#(#, )
lim Sn		= 0.
n®¥	1

** # $ %#

G(x) = g (t)dt.

%#/ G !+ 0 [a, b], G(a) = G(x0) = 0, # + ' ' 7 ** ( * #& [a, b] *+ & # # # *#. 3 0 )#

m = min G(x) # M = max G(x).

xÎ[a ,b ] xÎ[ a,b ]

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, %#/ 26 3. / * ( 0 ) ##

. (* +# * Sn2 + +#(

n	xk	n	xk	xk -1
Sn2 = f (xk -1 ) g (t)dt = f (xk -1 ) g (t)dt - g (t)dt =
k =1	xk -1	k =1	a	a

f (xk -1 )(G(xk ) - G(xk -1 )) =

k =1

f(x0)G(x1) + f(x1)(G(x2) - G(x1)) + ... + f(xn-1)(G(xn) - G(xn-1)) =

G(x1 )( f (x0 ) - f (x1 )) - G(x2 )( f (x1 ) - f (x2 )) +... + G(xn-1 )( f (xn-2 ) - f (xn-1 )) + G(xn ) f (xn-1 ) =

n-1

G(xk )( f (xk -1 ) - f (xk )) + G(xn ) f (xn-1 ).

k =1

. 0 /* , ) f(xk-1) - f(xk) ³ 0 # f(xn-1) ³ 0 # ) m £ G(x) £ M, * ( )# % # ( / Sn2

n-1		n-1
m( f (xk -1 ) - f (xk )) + mf (xk -1 ) £ Sn	£	M ( f (xk -1 ) - f (xk )) + Mf (xk -1 )
k =1	2	k =1
k =1		k =1

# #

m(f(x0) - f(x1) + f(x1) - f(x2) + ... + f(xn-2) - f(xn-1) + f(xn-1)) £ Sn2 £

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, %#/ 26 3. / * ( 0 ) ##

M(f(x0) - f(x1) + f(x1) - f(x2) + ... + f(xn-2) -f(xn-1) + f(xn-1)),

) + *# + * +

mf(a) £ Sn2 £ Mf(a).

* # f(a) = 0, f = 0 2( * + [a, b] # *$# + + 2( # * +# */ ) +#( !. ( 2#, ) f(a) > 0. &( ( # * ( # + * + f(a) # )#

m £ Sn2 £ M . f (a)

0+ 6 /* $ (1), " (#, )

( fg )(x)dx - Sn1

m £	a			£ M .
m £		f (a)		£ M .
		f (a)
. " (/ + 1 #" + * + " ( # n®¥, )#
		b
m £		( fg )(x)dx	£ M .
		a

f (a)

3 0 )#

( fg )(x)dx

a		= m .

	f (a)

. 7# 2 ) 0 ) ## !+ / $ %#/ G # # 0 [a, b] +* 2 ) ! 0 ) #/ 2( # # ! m # *# ! M. ) * * #, '( */ ) x Î [a, b] /, )

259

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