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( ) : ( ) (CIP) /.. :,003. () ISBN 7 56 448 0.... O4 44 CIP (00) 007344 : : 7 : 7007 : (09 )8493844 : www.nwpup.com : :850 mm 68 mm / 3 :0.5 :60 :00 003 3 :0 006 000 :3: 00
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... 3 3. 4 4. 5 3 5. 5. 5. 3 8. 4 0. 5 4 3 44 3. 44 3. 44 3. 3 48 3. 4 49 3. 5 73
4 75 4. 75 4. 75 4. 3 79 4. 4 80 4. 5 8 5 30 5. 30 5. 30 5. 3 3 5. 4 33 5. 5 57 6 60 6. 60 6. 60 6. 3 6 6. 4 63 6. 5 86 7 88 7. 88 7. 88 7. 3 93 7. 4 94 7. 5 9
8 8. 8. 8. 3 3 8. 4 4 8. 5 60 9 6 9. 6 9. 6 9. 3 65 9. 4 65 9. 5 93 95 () 95 () 96 () 97 () 300 () 30 () 30 () 304 () 305 () 307 () 308
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. ) ; ) ; 3) ; 4) ; 5).. ( ) ) :.,.,, ( ). ) : ( )( ).x * x.e( x * ) = x * - x, er ( x * ) = e( x * ) x e( x * ) x *. ( x * ) r ( x * ) e( x * ) er ( x * ). 3) x *
x * = 0 m 0.x x xn xp, x 0 (.) n, e( x * ) 0m - n, x * n.n = p, x * 4).. (. ) x * n, er ( x * ) x 0 - n.. (.) x * er ( x * ) ( x + ) 0 - n, x * n. ( ) s + t + ( s, t, ), - s s -, - t. s -,,. - s,. (3 ),,,,. : e( f ( x * e( f ( x * ) ) f ( x * ) e( x * ) (.) n,, x * n ) ) i = f ( x *,, x * n ) x i e( x * i ) (.3) ( f ( x * ) ) f ( x * ) ( x * ) (.4)
3 ( f( x * n,, x n * ) ) i = f ( x *,, x * n ) x i ( x i * ) (.5) (. ) (.3 ). ( )., Condr ( f ) = (4 ) ) ; x f ( x) f( x) (.6) ), ; 3), ; 4) ; 5).. 3 ( ),.. ( ),,,.., x * = 3.4 ( 3 ), x * = 3.4.,.
4 (3 ),,...,.. 4. x =,,. x * = 3.4, x * = 3.4, x * 3 = 3.5, x * 4 = / 7,,,.,.,. x = 3. x * i m =, ( ) x - x * = - 3.4 = 0.000 50.005 = 0-3 x * 3, r ( x * ) = 3 0-3 0.00 7 ( ) x - x * = - 3.4 = 0.00 0.005 = 0-3 x * 3, r ( x * ) = 3 0-3 0.00 7
5 (3 ) x - x * 3 = - 3.5 = 0.008 0.05 = 0 - x * 3, r ( x 3 * ) = 3 0-0.07 (4 ) x - x * 4 = - / 7 = 0.00 0.005 = 0-3 x * 4 3, r ( x * 4 ) = 3 0-3 0.00 7, r ( x * ) = - x * / x * 0.000 9, r ( x * ) 0.000 5, r ( x 3 * ) 0.003, r ( x 4 * ) 0.000 4.,.,. n,., x * 5 = 3.36, x - x * 3 = 0.004 0.005 = 0-3 x * 5 3, r ( x 5 * ) = - x 5 * / x 5 * 0.00 3. 0.00 7.,,,.,.?. 0 0.%,, 0.%,..,.. 0 4. x * n,.
6 er ( x * ) 4 0 - n 4 0 - n 0.% n 3.097, 4, 0.%..,.,,, 3, 4.47, 0-4.47 / 4.47 0.048 %..3 x sin x, sin x x, x [0, ]., 0-7. [ 0, ], sin x ( Taylor), n sin x, n +. sin x = x - x 3 3! + x5 5! -, x [0, ]. x 3 / 6 0-7. sin x - x x 3 / 3!, " x [ 0, ],, x 0.008 44. = 0.008 44 ( Lagrange), sin x - sin0 = cosx,..4 x * = - 3.05, x * = 0.00, x * 3 = 0.00, y = x * + x * + x 3 *, y = x * x * x 3 * y3 = x * / x * 3 (
7 ), (.5 )., ( x * ) = ( x * ) = ( x 3 * ) = 0.000 5, r ( x * ) = 0.000 6 r ( x * ) = 0.5 r ( x 3 * ) = 0.005 (.5 ) ( y * ) ( x * ) + ( x * ) + ( x 3 * ) = 0.00 5 ( y * ) x * x * 3 ( x * ) + x * ( y 3 * ) x 3 *.. x * ( x * ) + 3 ( x * ) + x * x * ( x * ( x 3 * ) x * 3 ) ( x * r ( y * ) ( y * )/ y * 0.00 5/ 3.004 0.000 5 r ( y * ) ( y * )/ y * r ( x * ) + r ( x * ) + r ( x 3 * ) 0.505 r ( y 3 * ) ( y 3 * )/ y 3 * r ( x * ) + r ( x 3 * ) = 0.505.5 x = 0 5 %, f ( x) = 3 ) n x, x = 0 5 % x * = 5%. f ( x * ) = n ( x * ) n - = n x * nx * er ( f * ) = e( f * ) f ( x * ) f ( x * ) e( x * ) f( x * ) er ( n x * ) e( x * ) nx * = 0, ( x * ), = er ( x * ) n ( x * ) = 0.005 nx * n
8 r ( n x * ) 0.005/ n., n x, n..6 f( x) = sin( n 3 x) x * = 0.000.n = 00, f( x *, r ( f ( x *. ) ) ) = 0.%.x * er ( f ( x * ) ) = f ( x * ) - f( x) f( x) x * x = f ( ) e( x * ) f ( x) f ( x * ) e( x * ), ( x, x * ) f ( x * ), (.6), r ( f ( x * ) ) Condr ( f ( x * ) ) r ( x * ) (.7),. Condr ( f ) = n = 00 x f ( x) f ( x) = xn 3 cos( n 3 x) sin( n 3 x) = n 3 x tan( n 3 x) Cond r ( f( x * ) ) = 0 6 x * / tan( 00) 70.3 r ( f ( x * r ( x * ) ) = 0.%, (.7) ) r ( f ( x * ) )/ Condr ( f( x * ) ) 0.587 0-5 ( x * ) r ( x * ) x * 0.587 0-9 (.7).5.
9.7 4, - cos?.:.,., cos 0.999 4, - cos - 0.999 4 = 0.000 6,. - cos= sin, sin 0.07 5, - cos 0.07 5 = 0.000 6 5 3 - cos= sin + cos, 0.034 9 sin 0.034 9, cos 0.999 4, - cos + 0.999 4 0.000 609.. 3,, - cos= 0.000 609 79 80,, 3., - cos x x! - x 4,x = =, - cos0.000 609 7 978, 4! 90..8 783 7.98 x, 4. - 56 x + = 0 x, = ( 56 56-4 )/, x - bx + c = 0, x = b + sgn ( b) b - c, x = c/ x
0 sgn( b) b,, sgn( b) = -, sgn( b) =. b = 8, c =, x = b + sgn( b) b - c = 8 + 8 - = 8 + 783 55.98 x = c/ x = / 55.98 0.07 863 x = 55.98 37, x = 0.07 86 8. 4.,,. 5..9 In. = e - 0 xn e x d x In = - ni n -, I0 = - e -,, I0 ni n -,,,,.In -,,. In = e - 0 xn e x d x I n = - ni n -, In - = ( - In )/ n (.8), In n, In - n - = n n ( ). In - k n - k = n/ [ n( n - )( n - k + ) ] (.9),.
. In e - n + = e- 0 n, x n d x < In < e - 0 In e - n + + n + x n ed x = n + - e - n + = - e- ( n + ) n + (.0) (.), (.0 ), (.8), (.) (.9 )..0 x 56,.,.x 56 = ( x 8 ), x 8, x 56 ; x 8 = ( x 64 ),, x 64, x 8 ;,. x 5 6 = x 8 x 8 = x 6 4 x 6 4 x 8 = x 3 x 3 x 64 x 8 = x 6 x 6 x 3 x 64 x 8 = x 8 x 8 x 6 x 3 x 64 x 8 = x x x x 4 x 8 x 6 x 3 x 64 x 8 8... 64, 0, 5,..ts
, s - - s, - t., t = 5, s = 0.x s - = 03 = 0 03 lg 0 308 - s 0-30 8 r ( f l( x) ) - t = - 5 = 0-5 lg - 5.65 0. 3 e -. 9 : e - n = 0 : e - 9 n = 0 ( - ) n ; n! n! 9 3: e - n = 0 e x - ; (9 - n)! -. x = 0 9, - ; 9 e, e -.. 3, e,. e x e x 9 9 = n = 0 x n x n! + e 0! x0, 0 < < x = - 9,
3 9 R = e - - n = 0 ( - ) n n! = e- 0!, 0 < < (.) x = 9, R = e - - e - 9 9 n = 0 n! - = 9 e - n! n = 0 9 e n = 0 n! = e 0! 9 n! e 0! / n = 0 n!, 0 < < (.3) 0 < <,0 < - <, R R 9 n = 0 n = 0 n! >, 3. 3 e (, ),.., 3.. 5 =. e,. x * =.7, x * =.78, x * 3 =.7. x > 0, x x *,ln x *. 3. 50 cm,0 cm 0 cm. cm. 4. s = gt, g, t 0.. t, s,. 5. p( x) = x 3 + 0 x - 7 x + 8. ( ) p( x).
4 ( ) x * =.. ( 3) x *,. 6. p( x) = 0 x 7-3 x 7. 7.,. ( - ) 6, ( + ) 6, ( 3 - )3, 99-70 3 ( 3 + ) 8.,. ( ) ln( x - x - ), x ( ) ln( x + ) - ln x, x ( 3) ( 4) 3 x x + - 3 x, x - cos x, x 0, x x ( 5) sin x - tan x, x 0, x
. ) ; ) ( ) ; 3) ; 4) ; 5) ; 6) ; 7) ; 8) B.. ( ). [ a, b] f( x) { x i} n i = 0 { f ( x i ) } n i = 0.n p n ( x) pn ( xi ) = f ( xi ) i = 0,,,, n (.) pn ( x) f ( x) { x i} n i = 0, { x i} n i = 0, (.).
6.. (. ) pn ( x). pn ( x) Rn ( x) def f [ x, x0 f( x) - pn ( x) =, x,, x n ] n + ( x) (.) f ( x) [ a, b] n +, pn ( x) Rn ( x) = f ( n+ ) ( ) ( n + )! n+ ( x), ( a, b) (.3) Rn ( x) Mn+ ( n + )! n+ ( x) (.4) n n+ ( x) = ( x - x i = 0 i ) ; Mn+ = max ax b f ( n+ ) ( x). ( ) pn ( x) ) ( Lagrange) : n pn ( x) = Ln ( x) = f ( xi ) li ( x) (.5) i = 0 { li ( x) } n i = 0 n x - xk li ( x) = x i - xk k = 0 ki ) ( Newton ) : pn ( x) = = n+ ( x) ( x - xi ) n+ ( x i ) (.6) n Nn ( x) = f [ x0, x,, xi ] i ( x) (.7) i = 0 i- f[ x0 ] = f ( x0 ) ; 0 ( x) ; i ( x) = k = 0 x k ) 3) :
7 xi = a + ih ( i = 0,,, n), h = ( b - a)/ n, x = a + sh, n Nn ( x) = Nn ( a + sh) = i f ( x0 ) i = 0 0 f( x0 ) = f ( x0 ), s i = s( s - )( s - i + ) i! s i (.8) { xi } n i = 0, k f [ x0, x,, xk ] = k f ( x0 ) = f ( k) k!h k (3 ) ( Hermite) k! ( ) (.9).3 n + H n+ ( x). H n+ ( x i ) = f( xi ), H n+ ( x i ) = f ( xi ), i = 0,,, n.4 f ( x) [ a, b] n +, H n+ ( x) R n+ ( x) = f ( x) - H n+ ( x) = f ( n+ ) ( ) ( n + )! n+ ( x), ( a, b) : n H n+ ( x) = f ( x i ) i = 0 n i ( x) + i = 0 i ( x) = ( - ( x - x i ) l i ( xi ) ) l i ( x) = n - ( x - xi ) i ( x) = ( x - xi ) l i ( x) k = 0 ki xi f ( xi ) i ( x) (.0) - xk l i ( x)
8 (4 ) [ ].. 3 ( ) n + n,,,. ( ),. n, n Ln ( x) = i = 0 n f ( xi ) li ( x) = i = 0 n f ( x i ) k = 0 ki x - xk x i - xk, n( n + ) n( n + ) + n ; n Ln ( x) = f( xi ) li ( x) = i = 0 n n+ ( x) i = 0 f ( xi ) ( x - xi ) n+ ( x i ), ( n + ) ( n + ) ( n + ) ( n+ ) + n,., n( n + ) 3 n( n + ) + n, n( n + ) + n., ( ).
9,,. (3 ) f ( x) C n+ [ a, b], (.4 ). x *, R( x) Mn+ ( n + )! n+ ( x), x [ a, b],, R( x) Mn+ ( n + )! max x[ a, b]. Mn+. max x [ a, b] (4 ) n+ ( x), " x [ a, b] n+ ( x),,,,,. { f ( x i ) } i = 0 { f ( x i ) } i = 0, { f( xi )} i = 0 { f ( xi )} i = 0, { f ( x ) }, A P4 ( x) = H3 ( x) + A( x - x0 ) ( x - x ), H3 ( x), A P4 ( x ) = f( x ).,. x, x.
0. 4. (. ),.. x 0 4 f ( x) 9 3 3 ( ) l0 ( x) = ( x - ) ( x - ) ( x - 4) ( 0 - ) ( 0 - ) (0-4) l ( x) = ( x - 0 ) ( x - ) ( x - 4) ( - 0 ) ( - ) ( - 4) l ( x) = ( x - 0 ) ( x - ) ( x - 4) ( - 0 ) ( - ) ( - 4) l3 ( x) = ( x - 0 ) ( x - ) ( x - ) ( 4-0 ) ( 4 - ) (4 - ) 3 L3 ( x) = f ( xi ) li ( x) = i = 0 ( ) = - 8 x3 + 7 8 x - = 3 x3 - x + 8 3 x = - 4 x3 + 5 4 x - x = 4 x3-8 x + x l0 ( x) + 9 l ( x) + 3 l ( x) + 3 l3 ( x) = - 4 x3 + 45 4 x - x + 7 4 x +
N3 ( x) = + 8 ( x - 0 ) + 3( x - 0 ) ( x - ) - (3 ) ( x - 0 ) ( x - ) ( x - ) = 4-4 x3 + 45 4 x - x +, N3 ( x) L3 ( x).,,.,,,,,.,,.... x.0.5.0 f( x) = sin x 0.84 5 0.997 5 0.909 3 sin.8 L (.8 ),, sin.8 - L (.8)?.x [, ], L ( x) sin x. ( )
R [ f ] = x =.8 R [ f ] = f ( ) ( x -.0) ( x -.5) ( x -.0), (.0,.0) 3! ( ) f ( ) (.8 -.0 ) (.8 -.5 ) (.8 -.0) 3! 0.008 = f [.0,.5] = ( 0.997 5-0.84 5)/ (.5 -.0 ) = 0.3 f [.5,.0] = ( 0.909 3-0.997 5)/ (.0 -.5 ) = - 0.76 4 f [.0,.5,.0] = ( - 0.76 4-0.3 )/ (.0 -.0) = - 0.488 4 L (.8 ) = 0.84 5 + 0.3 (.8 -.0) - 0.488 4 (.8 -.0 ) (.8 -.5) = 0.973 884 L (.8) - sin.8 0.000 036 37 L (.8) - sin.8,. (3 ) x [, ] L ( x) - sin x = f( ) ( x -.0 ) ( x -.5) ( x -.0) 3! 6 max x 0.008 0 ( x -.0) ( x -.5 ) ( x -.0) = x [, ],..3 x0 = 0, x = y = e - x, e - 0.5 e -.5,.
3 y0 = e 0 =, y = e - 0.367 9 N ( x) = + - 0.367 9 ( x - 0) = - 0.63 x 0 - e - 0.5 N (0.5 ) = 0.683 9 e -.5 N (.5 ) = 0.05 8 e( y * (0.5 ) ) = e - 0.5 - N (0.5 ) 0.077 4 er ( y * ( 0.5) ) = 0.077 4 e - 0.5 0.7 6 e( y * (.5 ) ) = e -.5 - N (.5 ) 0.7 3 er ( y * (.5) ) = 0.7 3 e -.5 0.767 8,.5 0.5.{0,} 0.5,.5...4 p( x) = x 4 - x 3 + x - x + { ( xi, p( xi ) ) } 5 i = 0 (.3)..3 x i - - 0 3 p( x i ) 3 5 6 5 q( x) { ( x i, q( xi ) )} 5 i = 0 (.4)..4 x i - - 0 3 q( x i ) 3 5
4 q( x),. q( x) p( x) 5, r( x) = p( x) - q( x), q( x)., 5 r( x) = p( x) - q( x) {( xi, r( xi ) ) } 5 i = 0 (.5)..5 xi - - 0 3 r( x i ) 0 0 0 0 0 60 r( x) = L5 ( x) = 60 ( x + ) ( x + ) ( x - 0) ( x - ) ( x - ) (3 + ) (3 + ) ( 3-0 ) ( 3 - ) (3 - ) x5-5 x3 + x q( x) = p( x) - r( x) = - x5 + x 4 + 3 x3 + x - 3 x + r( x),..5 f ( x) { x i} n - i = { xi } n i = n - g( x) h( x) { xi } n i = n - q( x). = q( x) g( x) { xi } i n = -, q( x) = g( x) + A n - ( x), n - n - ( x) = i = ( x - xi ), x = xn q( x) Ag( x) h( x) n - { x i} n - i =, n - ( x) n -
5 ( x - x ) [ g( x) - h( x) ], q( x) = g( x) + A( x - x ) [ g( x) - h( x) ], xn A. n - q( x) n - g( x) { xi } n - i =, q( x) = g( x) + An - ( x), n - ( x) = ( x - x ) ( x - x )( x - xn - ), A. q( xn ) = A = f ( xn ), f ( xn ) = g( xn ) + A n - ( xn ) f ( xn ) - g( xn ) n - ( xn ), q( x), q( x) = g( x) + [ f ( xn ) - g( xn ) ] n - ( x) n - ( x n ) { xi } n - i = q( x) = g( x) + An - ( x) n - g( x) h( x) n -, B g( x) - h( x) = B( x - x ) ( x - x3 )( x - xn - ), B 0 q( x), n - ( x) = ( x - x ) [ g( x) - h( x) ] B q( x) = g( x) + A( x - x ) [ g( x) - h( x) ], A = A B x = x n, q( xn ) = f( xn ) = h( xn ) h( xn ) = g( xn ) + A( x n - x ) [ g( xn ) - h( xn ) ] A = - xn - x q( x) = g( x) + x - x xn - x [ h( x) - g( x) ]
6.6.6?.6 x - - 0 3 p( x) 4 6 3-4 ( ), n n ( )., n + n ( ), n.,.,,..,,,..7 f ( x).7..7 x - - 3 f ( x) - 0-5 8
7 f ( x) = 0., x = f - ( 0).,, x = f - ( y)..8..8 y - 0-5 8 x = f - ( y) - - 3 N4 ( y) = - + 0.( y + 0) + 0.0 ( y + 0) ( y + 5) - 0.00 7 ( y + 0) ( y + 5) ( y - ) + 0.000 07 ( y + 0) ( y + 5) ( y - ) ( y - ) f - ( 0) N4 ( 0) 0.709 50,..8 p ( x).9..9 x - - 0 p( x) 3 6 5
8 p ( x),.,,.,, 5 3 p [ -, -,0 ] =, p [ -,0, ] = 5, p [ 0,, ] = 4,. p ( - ) p ( ) p [ -, -, 0] p [ 0,, ], p ( - ) p ( ),. p ( - ),, p [ -, 0, ] = p [0,, ] p [ -, 0, ] =, p [ 0,, ] =,, p [ -, 0, ] = p ( - ) = 0. p ( ),, p [ -, -, 0 ] = p [ -, 0, ] p [ -, -, 0 ] =, p [ -,0, ] = 7/ 3,, p ( ). p (0 ),, p [ -, -, ] = p [ -,, ] p [ -, -, ] = 3/, p [ -,, ] = 3/ 6,, p ( 0)., p ( - ),
9 p ( - ) = 0.9 pn ( x) f ( x) { xi } n i = 0 [ a, b] n.f ( x) [ a, b], M, f ( k) ( x) M, " x [ a, b], k = 0,,,. { pn ( x)} n = f ( x). [ a, b] x [ a, b], lim n [ pn ( x) - f ( x) ] = 0,, lim Rn ( x) = lim n n f ( n+ ) ( ) ( n + )! n+ ( x) = 0. x [ a, b], f( x) { x i} n i = 0 n Rn ( x) = f ( x) - pn ( x) = f ( n+ ) ( ) ( n + )! n + ( x), ( a, b) n+ ( x) = ( x - x0 ) ( x - x )( x - xn ) Rn ( x) M ( n + )! n+ ( x) < M b - a n+ ( n + )! lim ( b - a) n + n ( n + )! = 0 lim n Rn ( x) = 0, lim f ( x) - p n ( x) = 0, " x [ a, b] n n lim pn ( x) = f( x), " x [ a, b].0 f( x) k, { x i} n i = n > k n f [ x, x,, xn ] 0, n k k - n., k k +, k. k
30 k -,. n f [ x, x,, xn ] = f ( n) ( ) n! k f ( x), n > k f [ x, x,, xn ] 0; n = k f [ x, x,, xn ] = f[ x, x,, xk ] ak k! k! = ak, ak k f( x) x k ; n = k -, f [ x, x,, xk - ] = f [ x, x,, xk ] + f [ x, x,, xk ] ( x - xk ) = f [ x, x,, xk ] + ak ( x - xk )... ( ) f [ x0, x, x k,, xk ] = i = 0 k [ f ( x i )/ j = 0 ji ( xi - xj ) ],,.,. f( x) { xi } k i = 0 k Lk ( x) = f( x i ) li ( x) = i = 0 k i = 0 ( x - x0 )( x - xi- ) ( x - xi+ )( x - xk ) ( xi - x0 )( xi - xi- ) ( xi - xi+ )( xi - xk ) f( x i ) Nk ( x) = f ( x0 ) + f [ x0, x ] ( x) + + f [ x0, x,, xk - ] k - ( x) + f [ x0, x,, xk ] k ( x)
3 i ( x) = ( x - x0 ) ( x - x )( x - xi - ) ( i k) i. Nk ( x) Lk ( x), x k.. { x i} n i = an n p ( x). n i = x k i p ( xi ) = 0, 0 k n - an -, k = n - p( x) = an n ( x), f ( x) = x k,.,. f ( x) = x k, n ( x) = ( x - x ) ( x - x )( x - xn ), { xi } n i = an n p ( x), p( x) = an n ( x). (.) n i = x k i p ( xi ) = n an i = an n i = f ( xi ) an n ( xi ) = f( x i ) ( x i - x )( x i - xi - ) ( xi - xi + )( xi - xn ) f [ x, x,, xn ] = an f ( n - ) ( ) ( n - )! = 0, 0 k n - an -, k = n -.3.0,,..0 x 0 f ( x) 0 f ( x) 0 - =
3 (.0)., x0 = 0, x =, f ( x0 ) =, f( x ) = 0, f ( x0 ) = 0, f ( x ) = -. H3 ( x) = f ( x i ) i = 0 0 ( x) - ( x) i ( x) + i = 0 f ( xi ) i ( x) = i ( x), i ( x) ( i = 0, ) 0 ( x) = ( - ( x - x0 ) l 0 ( x0 ) ) l 0 ( x) = - x 0 - x 3-3 x + x - 0 - ( x) = ( x - x ) l ( x) = ( x - ) = ( + x) ( x - ) = x - 0-0 H3 ( x) = x 3 - x + R3 ( x) = = x 3 - x f ( 4 ) ( ) ( x - 0 ) ( x - ), ( 0, ) 4!.4. 3,.. x 0 f( x) 9 f ( x) 3,,.
33 3,,.,,.. N ( x) = f ( 0) + f[ 0, ] ( x - 0 ) + f [ 0,, ] ( x - 0) ( x - ) = + ( x - 0 ) + 3( x - 0 ) ( x - ) = 3 x - x + H3 ( x) = N ( x) + k( x - 0 ) ( x - ) ( x - ) H 3 ( ) = N ( ) + k( - 0) ( - ) = 4 - k H 3 ( ) = f ( ) = 3 k =, 3 H3 ( x) = N ( x) + ( x - 0) ( x - ) ( x - ) = x 3 +, f( x). ; f( 0) - H3 ( 0) = 0, f( ) - H3 ( ) = 0 f( ) - H3 ( ) = 0, f ( ) - H 3 ( ) = 0 R( x) = f ( x) - H3 ( x) = k( x) x( x - ) ( x - ) (.) x = 0,, [0, ] k( x) x [ 0, ] 0,, t g( t) = f ( t) - H3 ( t) - k( x) t( t - ) ( t - ) (.3) g( t),
34 g(0 ) = g( ) = g( ) = g( x) = 0, g ( ) = 0 (.4) 0,, x ( Rolle),,, 3 ( 0, ) 0,, x, g ( ) = g ( ) = g ( 3 ) = 0 (.5) (.4 ), g ( t) (0, ) 4,, 3.g ( t), g ( t) (0, ) 3 ; g ( t) g( t), ( 0, ), (.3) t, (.6) (.7), g ( 4 ) ( ) = 0 (.6) g ( 4 ) ( t) = f ( 4 ) ( t) - k( x)4! (.7) k( x) = f ( 4 ) ( ) 4! (.), R( x) = f ( x) - H3 ( x) = f ( 4 ) ( ) x( x - ) ( x - ) 4! ( 0, ) x 0,,. ( ),. 3 f (). H 3 ( x) = f (0) + f[ 0, ] ( x - 0 ) + f [0,, ] ( x - 0 ) ( x - ) + f [0,,, ] ( x - 0) ( x - ) ( x - ) =
35 + ( x - 0) + ( x - 0) ( x - ) + ( x - 0 ) ( x - ) ( x - ) = x 3 + ( ). l 0 ( x), l ( x), l ( x) l ( x), l 0 (0 ) =, l 0 ( ) = 0, l 0 ( ) = 0, l 0 ( ) = 0 l (0 ) = 0, l ( ) =, l ( ) = 0, l ( ) = 0 l (0 ) = 0, l ( ) = 0, l ( ) =, l ( ) = 0 l ( 0) = 0, l ( ) = 0, l () = 0, l 0 ( ) = l0 ( x) = ( x - ) ( x - ) (0 - ) (0 - ) = - l ( x) = ( x - 0) ( x - ) ( - 0) ( - ) = ( x - ) ( x - ) x( x - ) l ( x) = - x( x - ), l ( x) = - x( x - ) ( x - ) H 3 ( x) = f (0) l 0 ( x) + f () l ( x) + f () l ( x) + f ( ) l ( x) = x 3 +,..5 n - k = 0 n - f kgk = f n g n - f0 g0 - k = 0 gk+ f k,, g, g,, gn. n - f kgk = f0 ( g - g0 ) + f ( g - g ) + f ( g3 - g ) + k = 0 + f n - ( gn - - gn - ) + f n - ( gn - gn - ) = - g0 f0 + g ( f 0 - f ) + g ( f - f ) + + gn - ( f n - - f n - ) + gn ( f n - - f n ) + gn f n = n - gn f n - g0 f0 - gk+ f k k = 0
36.6 f ( x) x0, xi = x0 + ih ( i = 0,,, n).. lim h0 f [ x0, x,, xn ] = f ( n) ( x0 ) n! n (.9), ( x0, x0 + nh ) f [ x0, x,, xn ] = f ( n) ( ) n! h 0, x0.f ( x) x0 n, lim h0 f[ x0, x,, xn ] = f ( n) (lim h0 n! ) = f ( n) ( x0 ) n!, h 0, N n ( x) f( x) x 0 n..7 f ( x) = x 7 + x 4 + 3 x +, f [ x, 0,,, 6 ] f [ x, 0,,, 7 ]. f( x) 7, 7 8, (.9 ). f ( 8 ) f ( x) x f ( 7 ) ( x) = 7!, ( x) = 0, f [ x, 0,,, 6 ] = f ( 7 ) ( ) = 7! f[ x, 0,,, 7 ] = f ( 8 ) ( ) = 0 8!.8 + + + n = n( n + ) ( n + )/ 6 g( n) = + + + n,
37 g( n) n, g( n) 3. g( n) = + + + n, n n g ( n) n, g( n) n 3 (.6 ). g( ) =, g( ) = 5, g(3 ) = 4 g( 4) = 30,, g( n) = N3 ( n) = + 4 n - + 4 n -! + 5 n - + n - 3 + 5 ( n - ) ( n - )! ( n - ) ( n - ) ( n - 3 ) 3! n 3 + 3 n + n 6 = = + n( n + ) ( n + ) 6.9 f ( x) [ a, b]. f ( x).. h : a = x0 < x < < xn = b[ a, b], h def max in hi, def hi xi - x i- ; Lh ( x) f( x) =
38 x i n i = 0. Rh ( x). ( ) : x [ a, b], i, x [ xi -, x i ], M Rh ( x) = R ( h i) ( x) = f ( ) ( x - xi- ) ( x - xi )! 0 Rh ( x) f ( ) ( x - xi - ) ( x - xi )! def M h i max ax b M h f ( x) () (.0) (.0 ), lim h0 Rh ( x) = 0,. ( ) : [ xi-, xi ], Lh ( x) = L ( i) h ( x) = f( x i- ) x - xi x i - - xi + f( x i ) x - xi - x i - x i- (.) f * ( x i- ) = f( x i- ) i-, f * ( xi ) = f( x i ) i, i-, i. L * h ( x) = L ( i) * h ( x) = f * ( xi- ) x - x i x i- - xi + f * ( xi ) x - x i- xi - xi- (.) (.), x [ x i- L * h ( x) - Lh ( x) i- xi - x x i - xi-..0, xi ], x - xi- + i xi - xi - (.) max 0kn k
39 [ -, ] e x, 0-6., h * N + ( b - a)/ h * a = -, b =. e x,. [ -, ] h Lh ( x), Rh ( x). x [ xi-, xi ], x( i- )/, Rh ( x) = R ( i) h ( x) = f( ) ( x - xi- ) ( x - xi - / ) ( x - x i ) = 3! x = x i- e h3 ( s + ) s( s - ) 6 8, ( x i-, xi ) + s h, - s Rh ( x) e 48 h3 max - s s( s - ) ( s + ) = e 3 48 h3 9 0-6, e 3 48 h3 9 0-6 h 0.08 43 d ef h * N = + - ( - ) h 43. + - ( - ) 4.8 0.08 43,.,
40 + ( b - a)/ h *, N. h = ( b - a)/ ( N - ) h *. h/. h, h,.. f( x) C 4 [ a, b], H ( h) 3 ( x) f ( x) a = x0 < x < < xn = b, h = b - a n. f( x) - H ( h) 3 ( x). [ x i, xi + ],, f ( x) - H ( h) 3 ( x) = ( x - xi ) ( x - xi+ ) f ( 4 ) ( i )/ 4!, i ( xi, x i+ ).,. f ( x) - H ( h ) 3 ( x) = max ax b max 0in - f( x) - H( h) 3 ( x) = max f ( x) - H 3 ( h) ( x) x xx i i+ x [ xi, xi+ ], f ( x) - H ( h) 3 ( x) = f ( 4 ) ( i ) ( x - x i ) ( x - x i+ ) 4! max f ( x) - H ( h) 3 ( x) = x x x i i+ f ( 4 ) ( i ) max ( x - xi ) ( x - x i+ ) x x x i i+ 4! M4 4 max ( x - xi ) ( x - xi+ ) x x x i i+ x = x i + s h (.3) M4 4 h4 max 0 s s ( s - ) = M4 384 h4 (.4) M4 def max f ( 4 ) ax b ( x)
4 (.4) (.3) f ( x) - H ( h) 3 ( x) M4 384 h4.... x 3 f( x) 4 f ( x) -, [ ]. ( ) h = h =, = = 0.5 f [ x0, x ] - m0 h f [ x0, x, x ] = - =, 0.5 0.5 ( ) M0 = 3 - M0 M M m - f [ x, x ] h = 6-54 M, M = - 7-7 M = 8, M = - 0, M0 = 8 (3 ) s( x) = s ( x), x [, ] s ( x), x [, 3] = = M, M = 8
4 8-6 x + 3 x - 3 x 3, x [, ] - 40 + 56 x - 3 x + 3 x 3, x [, 3].5..3, x = 0.35..3 x 0. 0. 0.3 0.4 y..65.8.68 :.. 00, 44 5. 3. { x i } n i = 0 { l i ( x)} n i = 0 l 0 ( x) + l ( x) + + l n ( x). :,. 4. f( x) [ a, b], : max axb f ( x) - f ( a) + f ( b) - f ( a) b - a 8 ( b - a) max f ( x) axb :. 5. k p ( x) ( x - a) n p( x j ) l j ( x) p( x), n k j = 0 6., f( x) g( x) k ( f + g ) [ x0, x,, xk ] = f [ x0, x,, xk ] + g[ x0, x,, xk ] 7. n i = i k ( i - )( i - i + ) ( i - i - ) ( i - n) = 0 ( k = 0,,, n - )
43 :,. 8. ( fk gk ) = f kgk + gk+ fk f k = g kf k - f k g k g k g k g k+ n- f( a + ih) = f ( a + nh) - f ( a) i = 0 n - f( a + ih) = f( a + nh) - f ( a) i = 0 9. p( x), p(0 ) = p (0 ) = 0, p( ) = p ( ) =, p( ) =. 0. [ -,] e x, 0-6.. f ( x) = ( x - x 0 ) ( x - x ) ( x - x n ), { x i } n i = 0, f[ x0, x,, xp ], p n +. :.. f ( x) m, f( x) = f( x + h) - f ( x), f( x) k k f ( x) (0 k m) m - k, k f ( x) 0 ( k m + ). 3. 3 + 3 + + n 3 = [ n( n + ) ]
3 3. ), ; ) ; 3) ; 4) ; 5) ; 6). 3. ( ) ) V : () g 0, " g V ; g = 0 g= 0 () rg= r g, " r R, g V () f + g f+ g, " f, g V V, V. ) f ( x) C [ a, b], n + = span{ 0,,, n } C[ a, b] * ( x), f - * = minf - ( 3.) f ( x).
3 45. g, h V ( ) ) V V (, ) () ( f, f ) 0; f = 0 ( f, f ) = 0 () ( f, g) = ( g, f ), " f, g V () ( r f + r h, g) = r ( f, g) + r ( h, g), " r, r R ; f, (, ) V, V. V ( f, f) = f. ) ( f, g) = 0, f g (, ).. () : f k+ ( x) = ( ak x + bk ) f k ( x) + ck - ) ak = A( k+ A ( ) k k + ). ck - = - ak k a k - k - = - ), bk = A( k+ A ( ) k f k - ( x), ( k =,, ) A ( k+ ) A ( ) k+ - A ( k ) A ( ) k A ( ) k+ A ( ) k - k, ( A ( k ) ) k = ( f k, f k ) 0 k - ( 3.) A ( ) m, A ( ) m f m ( x) ( m = k -, k, () [ -, ] n Pn, Pn ( x) = Pn ( x)/ A ( ) n T n ( x) = Tn ( x)/ n -,
46 - ( Pn ( x) ) d x - n - = max - x Tn ( x) max - x ( g( x) ) d x, " g( x) Pn ( 3.3) g( x), " g( x) Pn.[ ]. ( 3.4) (3 ) { i} n i = 0 n +, f ( x) {c * i } n i = 0 ( 0, 0 ) ( 0, ) ( 0, n ) (, 0 ) (, ) (, n ) ( n, 0 ) ( n, ) ( n, n ) * = c * c 0 * c * n i = 0 = c * n i i, ( f, 0 ) ( f, ) ( f, n ) ( 3.5),. Gram. * f = * - f = ( f, f) - ( f, * ) ( 3.6), { i} n i = 0, f ( x) * = ( 0, f ) ( 0, 0 ) 0 + (, f ) (, ) ( + + n, f) ( n, n ) n ( 3.7) = n * - f = ( f, f ) - i = 0 ( f, i ) ( i, i ) ( 3.8)
3 47 (4 ) { ( xi, yi )} m i = 0 n + * = c * n i = 0 i i, { c i * } n i = 0 ( 3.5), ( 3.6), { i } m i = 0 A T WA, A = m ( f, g) = i f ( xi ) g( xi ) ( 3.9) i = 0. Gram 0 ( x0 ) ( x0 ) n ( x0 ) 0 ( x ) ( x ) n ( x ) 0 ( xm ) ( xm ) n ( xm ), W = diag(,,, m ) A T W f, f = [ f ( x ) f ( x ) f( xm ) ] T., AC = n i i = 0 f A T W,. (3.9 ), (3.7 ) ( 3.8). (5 ) 3. ( ) f ( x) [ a, b],, p( x), f ( x) - p( x) <. ( Bernstein) n B f n ( x) = f ( k k = 0 n ) n k x k ( - x) n - k, n = 0,,, (3.0) lim n+ f ( x) - Bf n ( x) = 0, [ a, b] = [0, ].
48 3. ( ) n n, f( x) C[ a, b], n p * n ( x). 3.3 ( ) p * n ( x) n f ( x) C[ a, b] n [ a, b] p * n ( x) n + { x i} n+ i = : ak a x < x < < xn+ b, p * n n k = 0 n ( x) = ak x k k = 0 n ak x i - f( x i ) = ( - ) i E n ( f ), i n + (3.) k = 0 = -, En ( f) = p * n ( x) - f ( x), { x i} n+ i = p * n ( x) - f( x) [ a, b] (n +, ).,, [ ]. 3. 3 ( ),.,.,. ( ),.
3 49,. (3 ), ( ),.,.,,. (4 )., C = ( A T A) - A T f, A, f, ( A T A) - A T A, A +, C = A + f. 3. 4 3. f g Cauchy Schwarz ( f, g) ( f, f ) ( g, g) (3.) ( f + g, f + g) ( f, f ) + ( g, g) (3.3) ( 3. ), (3.3) (3.). f, g t 0 ( f + tg, f + tg ) = ( f, f ) + t( f, g) + t ( g, g)
50 f g, t,, = 4 ( f, g) (3.). (3.) - 4 ( g, g) ( f, f ) 0 ( f + g, f + g) = ( f, f) + ( f, g) + ( g, g) ( f, f) + ( f, f) ( g, g) + ( g, g) = ( f, f) + ( g, g), ( 3.3 ). ( f, f ) /, Schwarz ( 3.3),.(3. ) (3.3) Cauchy Sch warz,. 3. f C[ a, b]. = [ b f ( x) d x] a,. f Cauchy., f( x), g( x) C [ a, b] ) f = [ b f ( x) d x] a f = 0 f ( x) 0 ) r r f = [ b a r f ( x) d x] = 0 r [ b f ( x) d x] = r f a
3 5 3) ( f, g) = b f ( x) g( x) d x a C[ a, b]. ( f, f ) = f (3.3) f + g f + g Schwarz C[ a, b]. 3.3 { gi} + i = 0 g0 g gk+ = x - 0 = ( x - k ) gk k = k - = ( xg k, gk ) ( gk, gk ) - k - gk -, ( k =,, ), k = 0,,, ( gk, gk ), k =,, ( gk -, gk - ) (3.4), ( 3.),. ( 3.), { gi } i = 0 gk+ = ( x + bk ) gk + ck - gk -, k =,, (3.5) bk = A ( ) k+ - A ( ) k, ck - = - (3.5) gk bk = - bk, ck - ( gk, g k ) ( gk -, g k - ) = - k -, 0 = ( ( x + bk ) gk, gk ) ( xg k, gk ) ( gk, gk ) = - k, k =,, (3.5) (3.4)., g0 =, g = x - d,
5 ( g, g0 ) = ( xg0 - dg0, g0 ) = ( xg0, g0 ) - d( g0, g0 ) = 0 d =. ( xg0, g0 ) ( g0, g0 ) = 0. ( 3.4 ),. ( 3.4),. 3.4 [0, ] ( x) = - ln x. (3.4). ( f, g) = f g d x = 0-0 (3.4), = 0 = ( xg, g ) ( g, g ) ( xg0, g0 ) ( g0, g0 ) g0 ( x) = = - xln xd x 0 - ln xd x 0 g ( x) = x - 0 = x - = 0 = - x( x - 0 - ( x - 0 ( g, g ) ( g0, g0 ) = ln x f gd x, = / 4 4 4 ) ln xd x 4 ) ln xd x 7/ 44 = = 7 44 = 4 g ( x) = ( x - ) g ( x) - 0 g0 ( x) = ( x - 3 8 ) ( x - 4 ) - 7 44 = x - 5 7 x + 7 5 3/ 576 7/ 44 = 3 8
3 53,.,. 3.5 f ( x) = sin x, x [0, 0. ] = span{, x, x },.,,. ( h, g) = 0. h( x) g( x) d x. 0 p ( x) = c0 + c x + c x, c0, c c G = (, ) (, x) (, x ) ( x, ) ( x, x) ( x, x ) G C = F ( x, ) ( x, x) ( x, x ) F = ( sin x, ) ( sin x, x) ( sin x, x ) G C = F, = = 0 00 3 000 - cos0. + sin0. - 0.cos0. 00 3 000 40 000.99cos0. + 0.sin0. - C = ( - 0.83 440 7 0-5,.000 999, f ( x) = sin x sin x p ( x) = - 0.83 440 7 0-5 +.000 999 x - 0.04 985 x = 0. 0 300 0 40 000 500 000-0.04 985 ) T ( sin x - p ( x) ) d x = 0.989 0 -
54 = 0. 0 sin xd x - C T F = 0.338 0 - ( 3.4 ) = span{, x, x }, ( h, g) = 0. h( x) g( x) d x 0 0 = = ( xg0, g0 ) ( g0, g0 ) 0 = g ( x) = ( xg, g ) ( g, g ) g0 ( x) 0. 0 = 0. 0 xd x d x = 0.005 0. x - 0 = x - 0.05 0. 0 = 0. 0 / 40 000 / 000 = 0 ( g, g ) ( g0, g0 ) = = 0.05 x( x - 0.05 ) d x / 000 0. ( x - 0.05) d x = 00 g ( x) = ( x - ) g ( x) - 0 g0 ( x) = ( x - x - 0 ) ( x - 0 ) - 00 = 0 x + 600 f( x) = sin x ( f, g0 ) = 0. sin xd x = - cos0. 0.004 995 834 7 0 ( f, g ) = 0. sin x( x - 0.05) d x = 0 sin0. - 0.05cos0. - 0.05 0.83 083 89 0-4 =
3 55 ( f, g ) = 0. sin x( x - 0. x + / 600) d x = 0 99 99 cos0. + sin0. - 600 0 600-0.38 806 55 0-8 ( g, g ) = 0. ( x - 0. x + / 600 ) d x = 0 ( g0, g0 ) = 0., ( g, g ) = (3.7 ) 8 000 000 p ( x) = ( f, g0 ) ( f, g ) ( f, g ) g0 + g + ( g0, g0 ) ( g, g ) ( g, g ) g = 0.049 958 347 g0 + 0.998 500 595 g - 0.04 985 6 0 g = - 0.83 44 7 0-5 +.000 999 x - 0.04 985 6 x (3.8 ) = f - p = ( f, f) - 0.4 0 - ( f, g0 ) ( g0, g0 ) + ( f, g ) ( g, g ) + ( f, g ) ( g, g ) 000, 3 f ( x) = sin x, ( 0 x 0. ), x = 0.05 + 0.05 t, f( t) = sin( 0.05 + 0.05 t), ( - t ) f( t). P0 ( t) = ; P ( t) = t, P ( t) = ( 3 t - )/ ( P0, P0 ) =, ( P, P ) = 3, ( P, P ) = 5
56 ( P0, f ) = sin( 0.05 + 0.05 t) d t = 0.099 96 694 4 - ( P, f ) = tsin( 0.05 + 0.05 t) d t = 0.033 83 353 - ( P, f ) = (.5 - t - 0.5 ) sin(0.05 + 0.05 t) d t = - 0.66 567 483 5 0-4 (3.7 ) f( t) P ( t) = ( P0, f) ( P0, P0 ) ( P, f ) ( P, f ) P0 ( t) + P ( t) + P ( t) = ( P, P ) ( P, P ) 0.049 958 347 + 0.049 95 030 t - 0.46 48 7 0-4 (.5 t - 0.5 ) = 0.049 979 68 + 0.049 95 030 t - 0.64 68 063 0-4 t (3.8 ) P ( t) f ( t) = f - P = ( f, f) - ( f, P0 ) ( P0, P0 ) + ( f, P ) ( P, P ) + ( f, P ) ( P, P ) ( sin(0.05 + 0.05 t) ) dt - 0.006 653 345 989 06 = - 0.356 339 4 0 0-0 3 f ( x) = sin x p ( x) = P ( 0 x - ) = p ( x) f ( x) = d x dt - 0.83 440 7 0-5 + 0.996 00 08 x - 0.04 985 3 x = =
3 57 0.356 339 4 0 0-0 0.05 0.78 0-3, = = 0. 0 ( f ( x) - p ( x) ) d x - ( f ( t) - P ( t) ) d t (3.8),, x = 0.05 + 0.05 t = 0.05., x = a + b + b - a t = b a ( f ( x) - p( x) ) d x = b - a ( f ( t) - P ( t) ) d t = b - a - 3.6 a, b c, I( a, b, c) = [ ax + bx + c - - x ] - - x d x. f ( x) = - x [ -, ] ( x) = / - x..,. ( g, h) = g( x) h( x) ( d x, ( x) = - - x.. T0 ( x), T ( x) = x, T ( x) = x - ( T0, T0 ) =, ( T, T ) = ( T, T ) = ( f, T0 ) = - x - - x d x = ( f, T ) = - x x - - x d x = 0
58 ( f, T ) = - x x - - - x d x = - 3 (3.7 ) f ( x) p ( x) = ( f, T 0 ) ( T0, T0 ) T0 + ( f, T ) ( T, T ) T + ( f, T ) ( T, T ) T = + 0-4 3 ( x - ) = 0 3-8 3 x, ax + bx + c 0 3-8 3 x, I( a, b, c), a = - 8 0, b = 0, c = 3 3 3.7 { -, -, 0,, } { i } + i = 0 { i} 3 i = 0. (3.4),.,. { xi } 4 i = 0 -, -, 0,, x0, x,, x4 4 ( f, g) = f ( x i ) g( xi ). i = 0 (3.4) = ( x, ) (, ) = ( x, ) (, ) 0 ( x) 0 = ( x 0, 0 ) = 0 ( 0, 0 ) 5 = 0 ( x) = x - 0 = x = 0 0 = 0, 0 = (, ) ( 0, 0 ) = 0 5 = ( x) = ( x - ) - 0 0 = x - = 0 4 = 0, = (, ) (, ). = 4 0 =.4
3 59 3 ( x) = ( x - ) - = x 3-3.4 x 3.8 ( 3. ), y = c0 + c x + c x. 3. x - - 0 y 0 0 3., ; Ax = b, A T Ax = A T b;, (3.7 ). -, -,0,, x0, x,, x4, { xi } 4 i = 0 4 ( f, g) = f ( x i ) g( xi ). i = 0 y = c0 + c x + c x span{, x, x }, G = (, ) (, x) (, x ) ( x, ) ( x, x) ( x, x ) ( x, ) ( x, x) ( x, x ) = 4 i = 0 4 xi i = 0 4 x i i = 0 x i 4 i = 0 4 x i i = 0 4 x 3 i i = 0 4 x i i = 0 4 x 3 i i = 0 4 x 4 i i = 0 = 5 0 0 0 0 0 0 0 34
60 F = ( y, ) ( y, x) ( y, x ) G C = F = 4 yi i = 0 4 yi x i i = 0 4 yi x i i = 0 C = ( c0, c, c ) T = ( 58/ 35 0-3/ 7) T y = 58 35-3 7 x - ( - ) - ( - ) 0 0 c0 c c = 0 0 = 4 0 (Ac = b), A T Ac = A T b. A T A = 5 0 0 0 0 0 0 0 34 A T Ac = A T b,, A T b = c = ( c0, c, c ) T = ( 58/ 35 0-3/ 7) T, y = 58 35-3 7 x. 3 { -, -, 0,, } 4 0
3 6, 0 ( x), ( x) = x, ( x) = x -. 3.7. ( 3.7) p ( x) = ( y, 0 ) ( 0, 0 ) 0 + ( y, ) (, ) + ( y, ) (, ) 4 5 + 0 0 x + - 6 4 ( x - ) = 58 35-3 7 x, y = p ( x) = 58 35-3 7 x =. 3,,3.y = a + bx + cx 3 x, y = a + bsin x + ce x, 3,,. 3.9 Ar ( N) 4, Ar ( O) 6. 3. 6 ( ). 3. NO N O NO N O 3 N O 5 N O 4 30.006 44.03 46.006 76.0 08.00 9.0,, 6,.,. (),,
6. Ar ( N) Ar ( O), Ax = b A T = 3 5 4, xt = [ Ar ( N) Ar ( O) ] b T = [ 30.006 44.03 46.006 76.0 08.00 9.0], A T A = x = A T Ax = A T b 8 9 9 56, AT b = Ar ( N) Ar ( O) = 4.006 9 5.999 3 76.04 30.6 3.0 y = ae bx b, 3.3. 3.3 xi 3 4 y i 60 30 0 5 a,,,.. y = ae bx, ln y = ln a + bx, Y = ln y, A = ln a, Y = A + bx. A b. (, ) (, x) 4 0 G = = ( x, ) ( x, x) 0 30
3 63 F = (, Y ) ( x, Y ) = 4 Y i i = 4 xi Y i i = G[ A b] = F, = 4 ln yi i = 4 xi ln y i i = A = 4.440 98, b = - 0.456 435 a = e A = 84.85 84-0.456 4 35 x y = 84.85 84e = 3.99 34 4 30.76 37 0 3. f( x) = x 4 + x 3 + 5 x + [ -, ] p( x). f ( x) p( x), f ( x) - p( x), f( x) -. f ( x) - p( x). p( x), f ( x) - p( x) T4 ( x), [ -, ], p( x) = f( x) - T4 ( x). f( x) - p( x). (3.4 ) f ( x) - p( x) = T4 ( x) = x4 - x + 8 f( x) - p( x) = max - x f ( x) - p( x)
64, p( x) f ( x) - p( x), f ( x) p( x) = f ( x) - T4 ( x) = x 3 + 7 x + 3 4, [ -,] a n n f ( x) n - pn- ( x) = f ( x) - an T n ( x), Tn ( x) n. [ a, b], x = a + b + t b - a f ( x) t,f( t), n ;f( t) tn - P n- ( t) ; x f ( x) n - pn - ( x) = Pn - ( b - a - a + b b - a ). 3. f ( x) = + x [ 0, ] p ( x)., [ 0, ] f ( x), f ( x), f ( x) - p ( x) = 0 [0, ],,,. f ( x) p ( x) = a0 + a x. p ( x) [0, ] 3. f ( x) = x + x, f ( x) = ( + x ) 3/ f ( x) [ 0, ] f ( x), f ( x) - p ( x) = x - a = 0 (3.6) + x [0, ],, p ( x),.,
3 65 p ( 0) - f (0 ) = - ( p( ) - f ( ) ) = p ( ) - f( ) a0 - = - ( a0 + a- + ) = a0 + a - (3.7) a = - 0.44, (3.6) = - 0.455, ( 3.7 ), a0 0.955. f ( x) p ( x) = 0.955 + 0.44 x.,[ a, b] f( x) p ( x), p ( x),. 3.3 a, b, max 0x e x - ax - b. f( x) = e x,. [ 0, ] max 0x ex - ax - b, e x [0, ] ax + b, p ( x). e x, 0 p ( x), ( e x )- p ( x) = 0 e x - a = 0 (3.8) (0, ). p ( x) a = f( ) - f (0 ) - 0 = e -, (3.8) ln( e - )., p (0 ) - e 0 = - [ p (ln(e - ) ) - e ln ( e - ) ] b - = - [ aln( e - ) + b - (e - ) ]
66 b = e - aln (e - ) a b. = e - (e - )ln( e - ) 3.4, f ( x) = e x ( - x ) p6, 3 ( x), f ( x) - p6, 3 ( x). p6, 3 ( x). f ( x) 6,, p6, 3 ( x). 6, 3.. f( x) x = 0, f ( x) = p6 ( x) + e 7! x7, ( -, ) (3.9) p6 ( x) = + x +! x + + 6! x6 f( x) - p6 ( x) e 7! x 6 - T6 ( x) = x 6 - [ x 6-3 x4-3 x4 + 9 6 x - 9 6 x + 3 3 ] = p6 ( x) = + x +! x + 3! x3 + 4! x4 + 5! x5 + 6! ( 3 x4 - + 9 6 x + 3 + T6 ( x) ) = (3.0) 3 6! + x + (! - 9 6 6! ) x + 3! x3 + ( 4! + 3 6! ) x4 + 5! x5 + 6! T6 ( x) def
3 67 p6, 5 ( x) + 6! p6 ( x) - p6, 5 ( x) = 6! x 5 - T5 ( x) = x 5 - x 5 - p6, 5 ( x) = + T6 ( x) (3.) T6 ( x) 5 6! 5 4 x3 + 5 6 x = 5 4 x3-3 6! + x +! - 9 6 6! 4! + 3 6! + x 4 + 5! 3 6! + -! - 9 6 6! 4! + 3 6! p6, 4 ( x) + 5! x + 5 4 x3-6 4! x + 3! + 4 4! x 4 + 5! T5 ( x) def (3.) 5 6 x x + 3! x3 + 5 6 x + 5! x 3 + T5 ( x) = T5 ( x) (3.3) p6, 5 ( x) - p6, 4 ( x) = T5 ( x) 5! 4 5! x 4 - T4 ( x) = x 4 - [ x 4 - x + 8 ] = x - p6, 4 ( x) = + 3 6! + - 6 4!! - 9 6 6! 4! + 3 6! x + x - x + 3! + 4 4! 8 + x 3 + 4! + 3 6! 8 (3.4) T4 ( x) =
68 + 3 6! - 8 4! + 3 6! + -! - 9 6 6! + 4! + 3 6! 3! + 4 4! p6, 3 ( x) + p6, 4 ( x) - p6, 3 ( x) = x 3 + 4! + 3 6! x + 4! + 3 6! 4! + 3 6! 3 4! + 3 6! T4 ( x) 6 4! d ef x + T4 ( x) (3.5) T4 ( x) (3.5) f ( x) p6, 3 ( x) = 0.994 574 65 + 0.997 395 83 x + 0.54 968 75 x + 0.77 083 33 x 3 (3.9), ( 3. ), (3.3), ( 3.5 ) f ( x) = p6, 3 ( x) + 4! + 3 6! 5! T 5 ( x) + 6! T 6 ( x) + e 7! x7 T4 ( x) + (3.0), ( 3. ), (3.4), ( 3.6 ) f( x) - p6, 3 ( x) 3 4! + 3 6! + 4 5! + 5 6! + e 7! = 0.006 573 f( x) - p6, 3 ( x) 0.006 573 (3.6),T 6 ( x), T 5 ( x) T 4 ( x),,.
3 69 [ a, b], x = a + b f( x) f ( a + b + t b - a, t [ -,] + t b - a ), f ( t). f( t) P( t). t = x b - a - a + b P( t), b - a f( x) p( x) = P( t( x) ).. 3.5 f ( x) = e x ( - x ) T4 ( x) 3 p3 ( x), e x - p3 ( x). T4 ( x) = cos (4arccos x) cos def = 0.93 879 53 8 cos 3 def = 0.38 683 43 8 cos 5 8 = - 0.38 683 43 d ef cos 7 8 = - 0.93 879 53 d ef e x = 0.396 975 97, e x = 0.68 08 77 e x 3 =.466 3 80, e x 4 =.59 044 7, p3 ( x) = 0.396 975 97 + 0.56 708 90( x + 0.93 879 53 ) + 0.38 059 50 ( x + 0.93 879 53 ) ( x + 0.38 683 43 ) + 0.75 75 683 3 ( x + 0.93 879 53 ) ( x + 0.38 683 43 ) ( x - 0.38 683 43) = 0.994 65 3 + 0.998 933 4 x + 0.54 900 73 x + 0.75 75 68 x 3 x4 x3 x x
70 e x - p3 ( x) = e ( x - x ) ( x - x ) ( x - x3 ) ( x - x4 ) = 4! e 4 e x - p3 ( x) e 4 max - x T4 ( x), ( -, ) (3.7) T4 ( x) = e 4 3 = 0.04 58 (3.8) e x - p3 ( x) 0.04 58 (3.7 ) e x - p 3 ( x) max - x, p 3 ( x) e x e 4 T 4 ( x) e 4 max - x. T 4 ( x) (3.7) e e,. e x - p 3 ( x) = max - x ex - p 3 ( x) 0.006 66., e x, e x e x ( T aylor), ( Taylor). = + x + x x6 + +! 6! + e 7! x7 = p 6 ( x) + e 7! x7 max - x ex - p3 ( x) max - x, ( 3.8). = 3.6 f ( x) = e x p6 ( x) - p3 ( x) + max - x 7! e x 7 e x - p3 ( x) 0.006 8 ( 3.9) ( - x ) ( x) - x S3 ( x), f ( x) - S3 ( x) f( x) - S3 ( x).,
3 7. ( g, h) = ghd x, ( 3.7) f ( x) - S3 ( x) = ( f, T0 ) ( T0, T0 ) ( f, T ) ( T, T ) T0 + ( f, T ) ( T, T ) T + T + ( f, T3 ) ( T3, T3 ), T0, T, T T3, T3 (3.30) ( T0, T0 ) =, ( T, T ) = ( T, T ) = ( T3, T3 ) = (). ( f, T0 ) = 3.977 463 7, ( f, T ) =.775 499 69 ( f, T ) = 0.46 463 88, ( f, T3 ) = 0.069 644 6 (3.30) S3 ( x) =.66 065 88 +.30 38 x + (3.8 ) 0.7 495 34 ( x - ) + 0.044 336 85 (4 x 3-3 x) = 0.994 57 + 0.997 308 x + 0.54 99 x + 0.77 347 x 3 3 f ( x) - S3 ( x) = ( f, f ) - 3.4, max - x ex - S3 ( x) max 0.006 895 i = 0 - x 6 i = 0 x i i! max x 7 - x 7! e ( f, Ti ) ( Ti, Ti ) - S3 ( x) + =
7 f ( x) - S3 ( x) 0.006 443 [ -, ] ( x) = / - x. f ( x) C[ -,], Fourier, f( x) - Sn ( x) ( f, T n+ ) T n+ ( x) f( x) - S n ( x) ( f, T n+ ) T n+ ( x) S n f ( x). 3.7 f( x) [ - a, a] ( a > 0 ), p( x). : ( ) f ( x) p( x) ; ( ) f ( x) p( x). f( x),,. max - ax a f ( x) - p( x) = max - axa f( x) - p( - x) f( x) [ - a, a] p( x). t = - x p( - max - axa f ( x) - p( x) = max - ata max - ata max - ax a f ( - t) - p( - t) = f ( t) - p( - t) = f ( x) - p( - x) x) f( x),, p( - x) = p( x), x [ - a, a], p( x) [ - a, a]..
3 73. 3.8 f( x) = arcsin x [ -, ] ak = - f ( x), a l+ = - a0 = - a l f ( x) - x d x f( x) Tk ( x) d x, k =,, - x = 0, l = 0,,, arcsin xcos [ ( l + )arccos x] d x x = co s - x ( 0 - )cos( l + ) d= 4, l = 0,,, ( l + ) arcsin x [ -, ] arcsin x = 4 [ T ( x) + 9 + T3 ( x) + T5 ( x) + 5 ( l + ) T l+ ( x) + ] []. 3. 5. C [ a, b] ( ) ( f, g) ( ) ( f, g) d ef d ef b f ( x) g ( x)d x a b f ( x) g ( x)d x + f ( a) g( a) a C [ a, b].
74. f( x) = + x,f ( x) [ 0, ] p ( x),. 3. A(0,), B(,3) C(, ). : ABC AC. 4. y = a + bx ab, 3.4,. 3.4 x i 9 5 3 38 44 y i 9.0 3.3 49.0 73.3 97.8. 5. n + { f( x i )} n i = 0, f ( x) 6. [ a, b] f ( x). 7. sin x 0,. 8. a, max 0x x3 - ax. :. 9. [ 0, ] f ( x) = - x + x - x 3 + x 4, 0.008 f ( x). :, [0,] [ -, ],,,. 0. f( x) = e x, x [0,] 3 p 3 ( x), e x - p 3 ( x).. B f n ( x) f( x). m f( x) M, m B f n ( x) M.
4 4. ) ; ) ; 3) ; 4) ; 5) ; 6). 4. I = b f ( x) d x, a f ( x). ( ). [ a, b] n, h = ( b - a), n xk = a + kh, k = 0,,,, n, xk f ( xk ),
76 b a n f ( x) d x =, Ak ; C ( k n). b k = 0 A k f ( xk ) + R[ f ] = n ( b - a)c ( k n) f ( xk ) + R[ f ] k = 0 ; R[ f ] n =, f ( x) d x = b - a [ f( a) + f ( b) ] - ( b - a) 3 a f ( ), ( a, b) n =, b f ( x) d x = b - a a 6 ( b - a) 5 880 [ f( a) + 4 f( a + b ) + f( b) ] - f ( 4 ) ( ), ( a, b) n = 4, [ ] [ 5]. ( ) b n f ( x) d x a A k f ( xk ) m k = 0 f( x), m + f( x) = x m+, m. b a n f ( x) d x k = 0 Ak f ( xk ) n :.. n, n ; n, n +.
4 77 (3 ) [ a, b] n, h = ( b - a), n xk = a + kh, k = 0,,,, n. [ xk, xk+ ], ( k = 0,,,, n - ) I = b a n - f ( x) d x =. k = 0 x k+ f ( x) d x x k+ x k x k f ( x) d x,, b f ( x) d x = Tn + Rn [ f ] = a n - h [ f ( a) + f( xk ) + f ( b) ] - k = b - a h f ( ), ( a, b), b f ( x) d x = Sn + Rn [ f ] = a n - h 6 [ f ( a) + 4 f ( xk+ b - a 880 h4 f ( 4 ) k = 0 ( ), ( a, b) n - ) + f ( xk ) + f( b) ] - k =, [ 5]. (4 ) 4.,
78 [ ]. k 4. n = k T k S k - C k - R k- 3 0 0 = T = T S = 4 T 4 S C 3 3 = 8 T 8 S 4 C R 4 4 = 6 T 6 S 8 C 4 R 5 5 = 3 T 3 S 6 C 8 R 4 (5 ) [ a, b] ( x) n+ ( x) ax0 < x < xn b, n +. n + : b a n ( x) f ( x) d x = k = 0 Ak f ( xk ) + R[ f ],. R[ f ] = ) b ( n + )! f ( n+ ) (( x) n+ ( x) d x, ( a, b) a :. Ak, xk ( k = 0,,,, n) [ ].
4 79 (6 ) : ; f ( x) f ( x). [ ]. 4. 3, x m ), f ( x) =, x, x,, f( x) = x m+ (, m).,,, f ( x) =, x, x, x 3,,. ), h,./, ( ), h. 3),, M = max ax b f ( n) ( x) M, (),. M,., M,.,,. 4), [ a, b]
80 [ -, ].,.,. 4. 4 4. e x d x,. 0 ( ) R[ f ] = - e x d x 0 [e0 + e ].859 40 9 ( ) e ( - 0) 3 e 0.6 53 5 e x d x 0 6 [e0 + 4e + e ].78 86 R[ f] = - (3 ) 880 e e 0.000 943 9 880 e x d x 0 90 [7e0 + 3e 4 + e + 3e 3 4 + 7e ] R[ f] = - ( - 0 ) 945.78 8 7 ( - 0 ) 6 e e 0.000 00 4 4 945 4 6 e x d x.78 8 8, 0
4 8. 4. 0 xn d x ( n =,, 3, 4),. f( x) = x, x, x 3, x 4, a = 0, b =, I = [ f( 0) + f ( ) ] I = 6 [ f (0 ) + 4 f( ) + f( ) ] I3 = 90 [7 f (0 ) + 3 f ( 4 ) + f ( ) + 3 f ( 3 4 ) + 7 f ( ) ] 4.. 4. f( x) x x x 3 x 4 I 0.5 0.500 000 0.500 000 0.500 000 I 0.5 0.333 333 0.50 000 0.08 333 I 3 0.5 0.333 333 0.50 000 0.00 000 0.5 0.333 333 0.50 000 0.00 000 f ( x) =. 4. : f( x) = x, ; f ( x) = x, x, x 3, ; f( x) = x, x, x 3, x 4,.,. 4.3 f ( x) > 0, b f ( x) d x,. a
8,. b f ( x) d x = b - a [ f ( a) + f( b) ] - ( b - a) 3 a : f ( x) > 0, R[ f] = - ( b - a) 3 f ( ) < 0 f ( ), b - a [ f ( a) + f( b) ] > b f ( x) d x a b f ( x) d x a. :. f ( x) > 0, f ( x) ( 4.),. 4. 4.4 3 : ( )
4 83 b f ( x) d x = ( b - a) f ( a) + a ( ) b f ( x) d x = ( b - a) f ( b) - a (3 ) b a f ( ) ( b - a), ( a, b) f ( ) ( b - a), ( a, b) f ( x) d x = ( b - a) f ( a + b ) + 4 f ( ) ( b - a) 3, ( a, b) 3,. f ( x),. ( ) f ( x) a, f( x) = f ( a) + f ( ) ( x - a), ( a, x) [ a, b], b a f ( x) d x = b a f ( a) d x + b a f ( ) ( x - a) d x ( x - a) [ a, b], ( a, b), b b a ) b a f ( x) d x = ( b - a) f( a) + f ( ( x - a) d x f ( x) d x = ( b - a) f ( a) + a f ( ) ( b - a), ( a, b) ( ) f ( x) b,
84 f ( x) = [ a, b], b a f( b) + f ( ) ( x - b), ( x, b) f ( x) d x = b f ( b) d x + a b a f ( ) ( x - b) d x ( x - b) [ a, b], ( a, b), b a b ) f ( x) d x = ( b - a) f( b) + f ( ( x - b) d x a b f ( x) d x = ( b - a) f ( b) - a (3 ) f ( x) a + b, f ( x) = f( a + b [ a, b], b a ( x - f ( x) d x = b a ( a, b), b a f ( ) ( b - a), ( a, b) ) + f ( a + b ) ( x - a + b ) + f ( ) ( x - a + b ), ( a, b) f ( a + b b a ) d x + b a f ( ) ( x - a + b ) d x f ( a + b ) ( x - a + b ) d x + a + b ) [ a, b], b ) f ( x) d x = ( b - a) f( a + b ) + f ( ( x - a + b ) d x a b a f ( x) d x = ( b - a) f ( a + b ) +
4 85 4 f ( ) ( b - a) 3, ( a, b) 4.5,,. ( ) h f ( x) d x A- f ( - h) + A0 f (0 ) + A f( h) - h ( ) f( x) d x - 3 [ f ( - ) + f ( x ) + 3 f ( x ) ] (3 ) h f ( x) d x h 0 [ f (0 ) + f ( h) ] + h [ f (0 ) - f ( h) ],.,.. ( ) 3 A-, A0 f ( x) =, x, x, A- + A0 + A = h - h( A- - A ) = 0 h ( A- + A ) = h3 3 A- = A = h 3, A0 = 4 h 3 h - h x3 d x = h 3 ( - h) 3 + h 3 h3 h x 4 d x h - h 3 ( - h) 4 + h 3 h4 h f ( x) d x h - h 3 f ( - h) + 4 h 3 f (0 ) + h 3 ( )., A, f ( h) ( ) x, x, f ( x) =,
86 f( x) d x - 3 [ f ( - ) + f ( x ) + 3 f ( x ) ] x, x, x + 3 x = x = - 0.6 6 0.56 6 x + 3 x =, x = 0.689 9-0.89 9 f ( x) = x 3, f ( x) d x - 3 x 3 d x - 3 [ - + x3 + 3 x 3 ] [ f( - ) + f ( x ) + 3 f( x ) ] x = 0.689 9, x = - 0.6 6 x = - 0.89 9, x = 0.56 6,. (3 ), f ( x) =, x, h d x h 0 [ + ] + 0 h xd x h 0 [ 0 + h] + h [ - ] f ( x) = x, h 0 x d x = h [ 0 + h ] + h [ 0 - h] = h 0 x3 d x = h [0 + h3 ] + h [0-3 h ] h x 4 d x h 0 [0 + h4 ] + h [0-4 h3 ]
4 87 h f( x) d x h 0. h [ f (0 ) + f( h) ] + [ f ( 0) - f ( h) ] 4.6 h = x - x0, x ( x - x0 ) f( x) d x = h [ A f ( x0 ) + B f ( x ) ] + x 0 h 3 [ Cf ( x0 ) + D f ( x ) ] + R[ f ] A, B, C, D,,. ( x) = x - x0,.. x - x0, ( ), f ( x) =, x - x0, ( x - x0 ), ( x - x0 ) 3, A, B, C, D., n f ( x) n.. f ( x) =, x - x0, ( x - x0 ), ( x - x0 ) 3, ( x - x0 ) x x 0 = h [ A + B] ( x - x0 ) 3 3 x x 0 = h [ A0 + Bh ] + h 3 [ C + D] ( x - x0 ) 4 4 x x 0 = h [ A0 + Bh ] + h 3 [ 0 + hd] ( x - x0 ) 5 5 x x 0 = h [ A0 + Bh 3 ] + h 3 [ 0 + 3 h D]
88 A + B = B + C + D = 3 B + D = 4 B + 3 D = 5 A = 3 0, B = 7 0, C = 30, D = - 0 x ( x - x0 ) ( x - x0 ) 4 d x h [ A0 + Bh 4 ] + h 3 [ 0 + 4 h 3 D] x 0 x ( x - x0 ) f( x) d x h [ 3 x 0 0 f ( x0 ) + 7 0 f ( x ) ] +. R[ f ] : h 3 [ 30 f ( x0 ) - 0 f ( x ) ] f ( x) [ x0, x ] H( x), H( x0 ) = f( x0 ), H( x ) = f ( x ), H ( x0 ) = f ( x0 ), H ( x ) = f ( x ),, H( x), f ( x) = H( x) + f ( 4 ) ( ) ( x - x0 ) ( x - x ), ( x0, x ) 4! x ( x - x0 ) f( x) d x x 0 = x x ( x - x0 ) x 0 x 0 ( x - x0 ) H( x) d x + f ( 4 ) ( ) ( x - x0 ) ( x - x ) d x = 4!
4 89 h [ AH ( x0 ) + BH ( x ) ] + h 3 [ CH ( x0 ) + D H ( x ) ] + R[ f ] = h [ A f ( x0 ) + B f ( x ) ] + h 3 [ Cf ( x0 ) + D f ( x ) ] + R[ f ] ( x - x0 ) 3 ( x - x ) R[ f ] = x x 0 f ( 4 ) f ( 4 ) ( ) [ x0, x ], ( ) ( x - x0 ) 3 ( x - x ) d x = 4! 4! x x 0 ( x - x0 ) 3 ( x - x ) d x R[ f ] = f ( 4 ) ( ) 440 h6, ( x0, x ).,.,,.,. 4.7 P ( x) 0, h, h f ( x), P ( x) I = 3 h f( x) d x 0 I h, : I - I h = 3 8 h4 f( 0) + O( h 5 ) P ( x) = li ( x) f( x i ), xi = ih( i = 0,, i = 0 ), Ih = Ai f ( ih). I - I h i = 0, I h x = 0, Ih.I = 3 h f ( x) d x ( f ( x) 0
90 ), I - Ih. ( ) 0, h, h, f ( x) P ( x) = l0 ( x) f (0 ) + l ( x) f ( h) + l ( x) f ( h) l0 ( x) = ( x - h) ( x - h) (0 - h) (0 - h) l ( x) = ( x - 0 ) ( x - h) ( h - 0 ) ( h - h) l ( x) = = h ( x - 3 hx + h ) = - h ( x - hx ) ( x - 0) ( x - h) ( h - 0) ( h - h) = ( x - hx ) h P ( x), Ih = 3 h P ( x) d x = 3 0 4 ( ) f ( h) x = 0, h[ f( 0) + 3 f( h) ] f ( h) = f ( 0) + hf (0 ) + ( h) f (0 ) + Ih 6 ( h) 3 f ( 0) + 4 ( h) 4 f ( 4 ) ( ), (0, h) Ih = 3 hf (0 ) + 9 h f ( 0) + 9 h3 f ( 0) + 3 h 4 f(0 ) + O( h 5 ) f ( x) x = 0, f ( x) = f( 0) + f (0 ) x + f (0 ) x +, 6 f(0 ) x3 + 4 f ( 4 ) ( ) x 4, (0, x) I = 3 h f( x) d x = 3 hf ( 0) + 0 ( 3 h) f (0 ) + 6 (3 h) 3 f ( 0) + 4 ( 3 h) 4 f(0 ) + O( h 5 ) =
4 9 3 hf ( 0) + 9 h f (0 ) + 9 h3 f (0 ) + 7 8 h4 f(0 ) + O( h 5 ) I - I h = 7 8 h4 f ( 0) - 3 h 4 f( 0) + O( h 5 ) = 3 8 h4 f(0 ) + O( h 5 ) 4.8 7 ( ), I = ln xd x ( ). 4.3. 4.3 x 7/ 6 8/ 6 9/ 6 0/ 6 / 6 ln x 0 0.54 5 0.87 68 0.405 465 0.50 86 0.606 36 0.693 47 ( ) : T6 = / / 6[ 0 + ( 0.54 5 + 0.87 68 + 0.405 465 + 0.50 86 + 0.606 36) + 0.693 47] 0.38539 ( ) : S3 = / 6 / 3[ 0 + 4 ( 0.54 5 + 0.405 465 + 0.606 36) + (0.87 68 + 0.50 86) + 0.693 47] 0.386 87 I = ln xd x 0.386 94 36,.
9 4.9 f ( x) C [ a, b], b f ( x) d x, [ a, b], a ( ). f( x) C [ a, b], f ( x) [ a, b],. f( x) C [ a, b], M = max ax b Rn [ f ] = - b - a h f ( ) ( b - a) 3 f ( x), M n n ( b - a) 3 M, Rn [ f]. n, n = [ ( b - a) 3 M ] + [ ( b - a) 3 3 b - a) M ] ( M. 4.0 ln xd x, 4,? f( x) = ln x, f ( 4 ) ( ) 6, (, ), 0.5ln.5 < ln x.5 < 4, Rn [ f] = - Rn [ f ] 0-4 ln xd x < ln b - a 880 h4 f ( 4 ) ( ), < <
4 93 Rn [ f] = f ( 4 ) ( ) 6 880 n 4 880 n 4 0-4 n.54. n = 3, n + = 7., ( ) ( ),. M = max f ( 4 ) ( x) ( M = axb max f ( x) ) M, axb.,,,. 4. I = sin x 0 x d x ( ), 0-3. ( ),,? (3 ) 0-6,,? 4.0, M = f ( x) M= max f ( 4 ) ax b. f ( x) = sin x x = cos( xt) dt 0 f ( k) ( x) = d k cos ( xt) d t 0 d x = k f ( k) ( x) t k 0 max axb ( x), M M, cos( xt + k 0 tk cos ( xt + k ) dt ) d t 0 t k dt = k +
94 ( ), 0-3, Rn [ f ] = - b - a h f ( ) n 3 0-3 n 7.5.9, I T8 = 8 [ + (0.997 397 9 + 0.989 65 9 + 0.976 76 6 + 0.958 85 + 0.936 55 7 + 0.908 85 7 + 0.877 9 6) + 0.84 47 0] 0.946 ( ) 9, h = / 4,, Rn [ f] = - b - a 880 h4 f ( 4 ) ( ) 880 ( 4 )4 5 = 0.70-6 (3 ), 0-6, Rn [ f ] = - b - a 880 h4 f ( 4 ) ( ) 880 5 0-6 n 4 n.9. [ 0, ], 3 + = 7. (),, 4.. 4. I = sin x d x., 0 x 0-3 0-6. I T n + 3 ( T n - Tn )
4 95 I S n + ( S n - Sn ) 5, 3 T n - Tn < 0-3,, 5 S n - Sn < 0-6 n T n = Tn + b - a n f ( a + ( j - ) b - a n ) j =. n = k ; k = 0,,, I T n + 3 ( T n - Tn ) Sn = 4 3 T n - 3 Tn I S n + ( S n - Sn ) 5 4.4 : 4.4 k k T k 3 T k - T k - S k - 5 S k - - S k - 0 0.90 735 49 0.939 793 8 0.946 45 88 4 0.944 53 5 0.00 573 4 0.946 086 93 0.000 003 93 3 8 0.945 690 86 0.000 39 45 < 0.946 083 3 0.000 000 4 < 0.5 0-3 0.5 0-6, I T8, I S4 = 0.946 ; = 0.946 083. 4.3 I = d x 0 ( + x) x,
96 3. f ( x) =, I., I = 0 x = 0 ( + x) x d x ( + x) x = d x 0 + x = d t 0 + t, g( t) = T = - 0 [ g(0 ) + g( ) ] = [ + + 0 + t, + ] =.5 T = T + g(0.5 ) = [.5 + + 0.5 ] =.55 T4 = T + 4 ( g( 0.5 ) + g(0.75) ) = [.55 + ( + + 0.5.565 6 + 0.75 ) ] T8 = T4 + 8 ( g( 0.5) + g( 0.375) + g(0.875 ) ) = [.565 6 + 4 ( + + 0.5 + 0.65 + + 0.875 ) ].569 5 + 0.35 + T8 - T4 0 - I.57, ( ), M = max f ( x) M. axb
4 97, M M,.. 4.4 ( ) b f ( x) d x = b - a [ f( a) + f ( b) ] - a ( b - a) R[ f ] = ( b - a) 5 4!30 [ f ( b) - f ( a) ] + R[ f ] f ( 4 ) ( ), ( a, b) ( ) x N f( x) d x TN - x 0 h [ f ( x N ) - f ( x0 ) ] T N = h [ f ( x0 ) + f ( x ) + f ( x ) + + f( x N - ) + f( x N ) ] xi = x0 + ih, ( i = 0,,,, N ), Nh = xn - x0 f( x), b f ( x) d x ( ) ; a [ a, b] N [ xi-, xi ] ( i =,,, N), ( ) ( ). ( ) x0 = a, x = b 0 ( x) = ( - x - a a - b ) ( x - b a - b ) ( x) = ( - x - b b - a ) ( x - a b - a )
98 0 ( x) = ( x - a) ( x - b a - b ) ( x) = ( x - b) ( x - a b - a ) H3 ( x) = ( - x - a a - b ) ( x - b a - b ) f ( a) + ( - x - b b - a ) ( x - a b - a ) f ( b) + ( x - a) ( x - b) f ( a) + ( a - b), b a x N x 0 f ( x) d x b a H 3 ( x) d x = R[ f ] = b a 4! f ( 4 ) ( ) b - a [ f( a) + f ( b) ] - ( b - a) f ( 4 ) ( ) ( x - b) ( x - a) f ( b) ( b - a) ( ) ( x - a) ( x - b) d x = 4! b R[ f ] = ( b - a) 5 4!30 f( x) d x = x x 0 f( x) d x + x a ( x - a) ( x - b) d x f ( 4 ) ( ), ( a, b) x f( x) d x + + x N [ f ( b) - f ( a) ] x N- f ( x) d x { h [ f( x0 ) + f ( x ) ] - h [ f ( x ) - f ( x0 ) ] } + { h h [ f( x ) + f( x )] - [ f ( x ) - f ( x ) ]} + +
4 99 { h h [ f( xn - ) + f( xn )] - [ f ( xn ) - f ( xn - ) ]} = h [ f ( x0 ) + f ( x ) + f ( x ) + + f ( x N - ) + f( x N ) ] - h x N f( x) d x TN - x 0 [ f ( xn ) - f ( x0 ) ] h [ f ( x N ) - f ( x0 ) ] T N = h [ f ( x 0 ) + f ( x ) + f ( x ) + + f ( x N - ) + f ( x N ) ] 4.5 f ( x) C[ a, b], f ( a) f ( b).xi = a + ih, y i = f ( xi ), i = 0,,,, n, h = b - a n.s( x) S ( a) = f ( a), S ( b) = f ( b). : b S( x) d x = h a n - [ f ( a) + f ( b) + f ( xi ) ] + i = h [ f ( a) - f ( b) ] f ( x),. f ( x) S( x) = 6 h Mi ( xi + - x) 3 + 6 h M i + ( x - xi ) 3 + ( yi - 6 Mi h ) x i+ - x h + ( yi+ - 6 Mi+ h ) x - xi h x ( x i, xi + ), i = 0,,,, n - ( 4.)
00 yi = f ( xi ) ( i = 0,,,, n), M0, M,, Mn di = 6 d0 = 6 f [ xi, xi+ ] - f[ x i-, xi ] h f [ x0, x ] - f ( a) h M0 M M Mn - Mn - Mn = d0 d d dn - dn - dn, i =,,, n -, dn = 6 f ( b) - f [ xn -, xn ] h ( 4.) (4. ), (4. ), 3 M0 + 3 M + 3 M + + 3 Mn - + 3 Mn - + 3 Mn = 3 h [ f ( b) - f ( a) ] M0 n - + i = Mi + Mn = h [ f ( b) - f ( a) ] ( 4.3) (4. ) [ xi, xi+ ], x i+ S ( x) d x = Mi x i 6 h x i+ x i ( xi+ - x) 3 d x + Mi + 6 h x i+ x i ( x - xi ) 3 d x +
4 0 ( yi - ( yi+ - 6 ) x i+ h Mi Mi+ x i + - x d x + x i h 6 ) x i+ h x i x - xi d x = h 4 Mi h3 + 4 Mi+ h3 + ( yi - Mi 6 h ) h + ( yi+ - Mi+ 6 h ) h = ( yi + yi + ) h - i 0 n -, b a S( x) d x = (4.3 ), b a 4 ( Mi + Mi + ) h3, i = 0,,,, n - n - i = 0 x i+ x i S ( x) d x = n - h ( yi + yi + ) - i = 0 n - h( y 0 + i = n - y i + yn ) - 4 h3 ( M0 + i = n - 4 h3 ( Mi + Mi+ ) = Mi + Mn ) i = 0 S( x) d x = h n - [ f ( a) + f ( b) + h [ f ( a) - f ( b) ] i = f ( xi ) ] + S ( a) = f ( a), S ( b) = f ( b) b f ( x)d x, 4.4. a 4.6 x 4 + y =,
0 5.,,.,,.. x = cos, y = sin l = 4 0 x + y d= 4 + 3sin d= 4 I 0 < + 3sin d< =, I, 0, l = 4 I 4 R[ f] 0-5, I R[ f ] 8 0-4 ( f ( ) = 0 + 3 sin d ). 4.5 I = + 3 sin d 0. 4.5 k k T k S k - C k - R k- 3 R k - 3 - R k - 4 0.356 94.49 9.44 63 4.4 03.4 830.4 608 3 8.4.4 5.4 067.4 074 4 6.4.4.4.4 3 0.000 039 5 3.4.4.4.4 0.000 00 < 0.5 0-4
4 03 I.4, l = 4 I 9.688 4. 4.7 f( x) [ a, b]. ( ) b f ( x) d x a, n b f ( x) d x. a? ( ) { R k } k = 0 k b f ( x) d x a ( ),, ( ) ( ). ( ) : f ( x) [ a, b], xi = xi - xi -, [ a, b], n [ xi -, x i ] i f ( i )xi, maxxi 0 ( n ), n lim maxx i 0f ( i )xi i = i = = b f ( x) d x ( 4.4) a, ( 4.4) i. h = b - a, n Tn = h n [ f ( xi- ) + f ( x i ) ] = n i = i = n i = n f ( xi- ) h i = [ f ( xi- ) h + f ( x i ) h] f ( x i ) h [ a, b] n, i. lim n Tn = n lim n i = [ f ( xi - ) h + lim n n i = f ( x i ) h =
04 b [ a f ( x) d x + b a f ( x) d x] = b a f ( x) d x ( 4.5). Sn = 4 3 T n - 3 lim n Sn Tn, ( 4.5), = lim n [ 4 3 T n - 3 Tn ] = 4 3 lim n T n - 3 lim n Tn = 4 3 b a f ( x) d x - 3 b f ( x) d x = a b a f ( x) d x ( 4.6), n b f ( x) d x. a ( ) : R k = 64 63 C k+ - (4.6 ) lim k C k = 6 5 lim k S k+ - lim k R k 6 5 b a 63 C k, C k = 6 5 S k+ - f ( x) d x - b f ( x) d x a = k lim [ 64 63 C k+ - 64 63 lim k C k+ - 64 63 b, { R k } k = 0 a f ( x) d x - 5 lim k S k = 5 b f ( x) d x = a 63 C k ] = 63 lim k C k = 5 S k 63 b f ( x) d x = a b a f ( x) d x k b f ( x) d x. a
4 05 4.8, d x, ln x. ( ) ( 0-6 ) ; ( ) ; (3 ) N = 4. ( ) 4.6. 4.6 k k T k S k- C k - R k - 3 R k - 3 - R k- 4 0 0.75 0.708 333 34 0.694 444 44 4 0.697 03 8 0.693 53 97 0.693 74 60 3 8 0.694 85 0.693 54 53 0.693 47 90 0.693 47 48 4 6 0.693 39 0 0.693 47 65 0.693 47 9 0.693 47 8 0.000 000 3 < 0-6 d x 0.693 47 8, ln = 0.693 47 8. x ( ), x = ( t + 3 ), x [, ], t [ -, ]. a. x d x = - t + 3 d t 5 9 f( - 0.6 ) + 8 9 f( 0) + 5 9 f ( 0.6 ) = 5 9 3-0.6 + 8 9 3 + 5 9 3 + 0.6 0.693 69
06 f ( t) = b. x d x = - t + 3 d t t + 3 0.568 888 888 9 f ( 0) + 0.36 96 885 [ f ( - 0.906 79 845 9) + f (0.906 79 845 9) ] + 0.478 68 670 5 [ f ( - 0.538 469 30 ) + f( 0.538 469 30 ) ] 0.693 47 6 f ( t) = t + 3 (3 ) [, ] 4 : [,.5 ], [.5,.5 ], [.5,.75], [.75, ],,. x = b - a b a I t + b + a, x [ a, b], t [ -, ]. x d x = ( b - a) dt - ( b - a) t + ( b + a) f i ( t) = =.5 ( b - a) [ f ( - f ( t) = 3 ) + f( 3 ) ] ( b - a) t + ( b + a), i =,,3,4 0.5 t +.75 + 0.5 i x d x = 0.5 d t - 0.5t +.5
4 07 I I3 I4 0.5 [ f ( - 0.3 40 495 =.5.5 3 ) + f ( 3 ) ] x d x = 0.5 d t - 0.5t +.75 0.5 [ f ( - 0.8 30 44 =.75.5 3 ) + f ( 3 ) ] x d x = 0.5 d t - 0.5t + 3.5 0.5 [ f3 ( - 0.54 50 97 =.75 3 ) + f3 ( 3 ) ] x d x = 0.5 d t - 0.5t + 3.75 0.5 [ f4 ( - 0.33 53 57 3 ) + f4 ( 3 ) ] d x I + I + I3 + I4 0.693 4 9 x :.,,,. 4.9 x f ( x) d x A0 f ( x0 ) + A f( x ) 0 x0, x A0, A. x0 ( 0, ), x ( 0, )
08. A0, A x0, x, ; [ 0, ] ( x) = x ( x [ 0, ] ),,., 3. f ( x) =, x, x, x 3., A0 + A = xd x = 0 3 x0 A0 + x A = x x d x = 0 5 x 0 A0 + x A = 0 x xd x = 7 x 3 0 A0 + x 3 A = x 3 xd x = 0 9 A0 A = x0 x - 3 ( x0 x ) = 7 ( x 0 + x0 x + x ) - 3 5 5 x0 x = 7 ( x0 + x ) - 9 x0 x = u, x0 + x = v, 7 v -, A0 A = x 0 x x 3 0 x 3 9 9 v - 7 u - 3 u = 0 7 v - 5 u - 9 = 0-7 9 ( x0 + x )
4 09 u = 5 v = 0 9, ( u = 0, v = 7 9, x0 = 0, x = 7 9 ). x0 = 0.89 949, x = 0.8 6 A0 = 0.77 556, A = 0.389 x f ( x) d x 0.77 556 f (0.89 949) + 0 0.389 f (0.8 6) 0 ( x) =, x [0, ], ( x) = x ( x [0, ] ) ( x) = ( x - ) 0 ( x) ( x) = ( x - ) ( x) - 0 ( x). (, 0 ) = 0, ( x) = x - = ( x 0, 0 ) ( 0, 0 ) 3 5 0 x/ = 0 xd x x / d x (, ) = 0 (, 0 ) = 0, = ( x, ) (, ) = (, ) ( 0, 0 x/ = 0 = 0 0 ) 0 x/ = 3 5 3 x( x - 5 ) d x x / ( x - 3 5 ) d x x / ( x - 3 5 ) d x d x ( x) = ( x - 3 45 ) ( x - 3 5 ) - 75 = 3 45, = 75 x -. x + 0.38 096 66