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Матан Лекции

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#*. 7. * %#/ ( 0 * + ! 3.

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j ¢¢¢(x7) = j ¢¢¢(x8) = 0.

	%, /, # ' $						%## j ¢¢¢ 0 [x7, x8], '( */ ) x Î (x7, x8) /, )
							j (4)(x) = 0.
(L (x) + K(x - x )(x - x )(x - x						))(4) 2( * +	3 6 */ +	+* ' )#* + ' *#, ) L (x) +
2			0	1	2			2
K(x - x0)(x - x1)(x - x2) /+ / / & ) ' #. # # +!)#* #/ # " (#, )
[(x - x )(x - x )2(x - x				)](4) = 4!. ( + , * +# j (4)(x) = 0 +				*# + #
0		1	2

0 = j (4)(x) = f (4)(x) - 4!C,

#0 & +#(#, )

C = f (4 )(x ) .

" (# % ( / R2(x¢) - K(x¢ - x0)(x¢ - x1)(x¢ - x2)

|R2(x¢) - K(x¢ - x0)(x¢ - x1)(x¢ - x2)| = |g(x¢)| =

|j (x¢) + C(x¢ - x0)(x¢ - x1)2(x¢ - x2)| = |C|(x¢ - x0)(x¢ - x1)2(x2 - x¢) =

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f	(4 ) (x )	(x¢ - x0 )(x¢ - x1 )2	(x2	- x¢) £	M	(x¢ - x0 )(x¢ - x1 )2 (x2 - x¢)


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& ) + ' * # L0, , # +#(
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#*. 1. * %#/ # /%# & & ) + ' * # L0, # # 6 & (# + * $ %# ' f 0 ) # + ) c

- // ( 0 # & # 0 [a, b] $ %# f L0, )# #3 #2 0 ) # # &

b	b	b
f (x)dx = L0		(x)dx + r0 = f (c)dx + r0 = f (c)(b - a)+ r0 .
a	a	a
-( * 0 ) # f(c)(b - a) /+ / */ #3 #2	! 0 ) # # & $ %## f 0 [a, b], r0 +! 2
& 7 * #3 #2 #/.
		b
(1)		f (x)dx = f (c)(b - a)+ r0

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, %#/ 31

1. / & # + #3 #2 & +!)#* #/ # & +

0!+ */ % #3 #2

& +!)#* #/ # &.

) # 0 #/ 0 + / + 2 0 #3 #2

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+ 2 % #+ & 7 * #3 #2

& 0 ) #/.

. 2( ) ' # % & 7 * #, * 0( (# 3 36 $ / & # +, 3 6 6 0 )#

+!*# ) * #3 #2 #/. / 1 & 0 3 0 [a, b]

n 0 + + ' ( # !. .* P = {x0, x1, x2, ..., xn-1,

xn} - 03# #,

= a +

b - a

k = 0,1,..., n.

(# 2( & #0 0 + Dk = [xk-1, xk], k = 1, 2, ..., n, 3 0 )# ) 0 ck,

c =

xk -1

+ xk

= a +

b - a

k -

, k =1,2,..., n,

# 2( #0 0 + Dk # # $ (1).

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& & )

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+ 2( ' ) ck.

0 +!+ (# $ #3 #2

& +!)#* #/ # &

n xk

b - a

f (x)dx = f (x)dx = f (ck )dx + r0n = f (ck )

+ r0n

f (ck ) + r0n .

k =1 x

k -1

k =1 x

k -1

k =1

-( *

b - a

=1 f (ck ) /+ / */ #3 #2

! 0 ) # # & $ %## f

0 [a, b], r0n +! 2

& 7 * #3 #2 #/.

b - a

(2)

f (x)dx =

f (ck )+ r0n ,

k =1

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1 ! % # f %% # [a, b] | f ¢¢(x)| £ M. " # ( r0n %

M (b - a)3 r0n £ 24n2 .

. .* * ) n = 1. % # & 7 * r0 = r01 $! / & # + (1). . $

' * * , & 2, # ' $ %## f * ) ' x0 = c = (a + b)/2, ( / 3 ' ) # x Î [a, b], x ¹ c, '( */ ) x, " (/6 /*/ 2( x # c, /, ) * + ( #+ $

f (x) = f (c)+ f ¢(c)(x - c)+ f ¢¢(x )(x - c)2

, 0 /* 1 # 0 2 # # $' (1), #0+ ( %

r01 = ( f (x)- L0 (x))dx =

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(c)(x - c)+

f ¢¢(x )

f (c)+ f

(x - c)

- f (c) dx

f ¢(c) ((b - c)2 - (a - c)2 )+ f ¢¢(x )(x - c)2 dx

f ¢¢(x )(x - c)2 dx

f ¢¢(x )

(x - c)2 dx £

M (x - c)2 dx =

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(b - a) 3

(b - a)3

M (b - a)3

) ( 0!+ 1 ( / n = 1.

/ #0+ & n # # '( %

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0 [a, b] +!7 * !

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' # %

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3. %#' #3 #2 & +!)#* #/ # & +

3. .

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b]. /%# !' & )

+ ' * # L1, # # 6#' (# +! * $ %# ' f 0 ) # + ) " a # b, #

+#(

L1 (x) = f (a)

x - b

+ f (b)

x - a

- f (a)(x - b)+ f (b)(x - a)

a - b

b - a

#*. 3. * %#/ # /%# & & ) + ' * # L1, # # 6 & (# + * $ %# ' f 0 ) # + ) " a # b.

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= - f (a)(x - b)+ f (b)(x - a)dx + r1 =

b - a

f (a)(a - b)2 + f (b)(b - a)2

+ r =

f (a)+ f (b)

(b - a)+ r .

2(b - a)

-( * 0 ) # (f(a) + f(b))/2(b - a) /+ / */ #3 #2

! 0 ) # # & $ %## f 0 [a, b], r1

+! 2 & 7 * #3 #2 #/.

f (a)+ f (b)

(3)

f (x)dx =

(b - a)+ r1

0!+ */ % # #3 #2 & +!)#* #/ # &.

. 2( )

' # % & 7 * # #3 #2 #/, * 0( (# 3 36 $ %#'. / 1 & 0 3

0 [a, b]

n 0 + + ' ( #

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= a +

b - a

k, k = 0,1,..., n.

2( #0 0 + Dk = [xk-1, xk], k = 1, 2, ..., n, # # $ (3).

309

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