Матан Лекции

0 +!+ (# $ #3 #2

& +!)#* #/ # &

b

n xk

f (x)dx =

f (x)dx =

a

k =1 x

k -1

- f (xk -1 )(x - xk )

+ f (xk )(x - xk -1 )dx + r1,n =

n

xk

k =1 x

k -1

Dk

= b - a f (xk -1 )+ f (xk ) + r1,n =

k =1

2

n

k =1

2

b - a

1

[ f (x0 ) + f (x1 ) + f (x

) + f (x

) + ... + f (x

) + f (x

)]+ r

2

k -1

k

n 2

1

1n

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

n-1

+ f (xk )

+ r1n .

2

n

k =1

-( *

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

n-1

+ f

(xk

)

n

2

k =1

b

b - a

f (a)+ f (b)

n-1

(4)

f (x)dx =

+ f (xk ) + r1n

n

2

a

k =1

2 0!+ */ % # #3 #2

& +!)#* #/ # &. 3 ( # # * + '

$' %#'.

r

£

.

1n

12n2

. .* * ) n = 1. % # & 7 * r1 = r11 $! %#' (3). .# // %

R

(x)

=

f (x)- L (x)

£

M

(x - a)(x - b),

a £ x £ b,

1

1

2

& 7 * # #3 #2 #/ $ %## f # /%#

! & ) , & 2 + ' * # L1, #0+ (

% #+ #

r

=

b ( f (x)- L (x))dx

=

b

R

(x)dx

£ b

R (x)

dx £

b

M

(x - a)(b - x)dx.

2

11

1

1

1

a

a

a

a

* ( # & * + 7# 0

' $

y = x -

a + b

2

&(

x - a = y +

b - a

, b - x =

b - a

- y,

2

2

# # & & +!)#* / */

b-a

b-a

(b

- a)2

b

- a b - a

2

b

2

(x - a)(b - x)dx =

y +

- y dy =

4

a

-

b-a

2 2

-

b-a

2

2

(b - a)3

1

(b - a)3

(b - a)3

-

+

=

.

4

3

8

8

6

. (* + // +!)#* !' # & + (!( 6 % )

r11

£

M

(b - a)3 ,

) ( 0!+ 2 ( / n = 1.

12

/ #0+ & n # # '( %

( # |Dk| 0 Dk, |Dk| = (b - a)/n. .& 7 * r1n $! %#'

0 [a, b] +!7 * !

' # %

n

M

b - a 3

M (b - a)3

M (b - a)3

r

£

= n

=

1n

k =1

12

n

12 n3

12n2

, %#/ 32

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

1.

314

2.

319

1. .

#7 ( #3 #2 & +!)#* #/ # & + # , * + !'

0 (! & ' $ %## f

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

L (x) = f (a)

+ f (c)

+ f (b)

=

(a - c)(a - b)

(c - a)(c - b)

(b - a)(b - c)

2

2

(b - a)2

& & ) + ' * # L2

- // ( 0 # & #

0 [a, b] $

%# f L2, )# #3 #2 0 ) # # &

b

b

f (x)dx = L2

(x)dx + r2 =

a

a

2

b

(b - a)2

a

+ 7# + # & 0

' y = x - c. &(

x - a = y +

b - a

# x - b = y -

b - a

,

2

2

b

[ f (a)(x - c)(x - b)- 2 f (c)(x - a)(x - b)+ f (b)(x - a)(x - c)]dx =

a

b-a

b - a

b - a

b - a

- a

2

b

f (a) y y -

-

2 f (c) y +

y

-

+ f (b) y +

y dy =

-

b-a

2

2

2

2

2

1

(b - a)3

(b - a)3

b - a

(b - a)2

(b - a)2

f (a)

+

-

-

-

3

8

8

4

4

4

1

(b - a)3

(b - a)3

(b - a)2

2 f (c)

+

-

(b - a) +

3

8

8

4

1

(b - a)3

(b - a)3

b - a

(b - a)2

(b - a)2

f (b)

+

+

-

=

3

8

8

4

4

4

f (a)

(b - a)3

+ f (c)

(b - a)3

+ f (b)

(b - a)3

.

12

3

12

b

(x)dx =

b - a

[ f (a)+ 4 f (c)+ f

(b)]+ r2 .

(1)

f

a

6

. 2( )

' # % & 7 * #, * 0( (# 3 36 $ # * . / 1 & 0 3 0 [a, b] n

0 + +

' ( # !. .* P = {x0, x1, x2, ..., xn-1, xn} - 03# #,

x = a + b - a k, k = 0,1,..., n .

k

n

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

)# ) 0 ck,

x

+ x

b - a

1

c =

k -1

k

= a +

k -

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

k

2

n

2

0 +!+ (# $ #3 #2

& +!)#* #/ # &

b

f

a

k =1 x

k -1

n

2

xk

[ f (xk -1 )(x - ck

)(x - xk )- 2 f (ck )(x - xk -1 )(x - xk )+ f (xk )(x - xk -1 )(x - ck )]dx + r2n =

D

2

k =1

k

xk -1

2

2

Dk

3

[ f (xk -1 )+ 4 f (ck )+ f

n

n

k =1

Dk

12

6n k =1

b - a

[ f (x

)+ 4 f (c )+ f

(x

)+ f (x )+ 4 f (c

)+ f (x

)+... + f

(x

)+ 4 f (c

)]+ r

2

2

n-1

n

6n

0

1

1

1

2n

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

)+ 4 f (ck )

+ r2n .

n-1

n