( ) : ( ) (CIP) /.. :,003. () ISBN O4 44 CIP (00) : : 7 : 7007 : (09 ) : : :850 mm 68 mm / 3 :0.5 :60 :00 0
|
|
|
- 带带 巴
- 5 years ago
- Views:
Transcription
1 ( )
2 ( ) : ( ) (CIP) /.. :,003. () ISBN O4 44 CIP (00) : : 7 : 7007 : (09 ) : : :850 mm 68 mm / 3 :0.5 :60 : : :3: 00
3 00, ( ),,,,,,,, 003 8
4 ( ) 00, : ( ),,,,,,,, ( ),,, (3 ), 0,, (4 ),,,,,, 00 0
5
6
7 () 95 () 96 () 97 () 300 () 30 () 30 () 304 () 305 () 307 () 308
8
9 . ) ; ) ; 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 *
10 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)
11 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.,.
12 4 (3 ),,...,.. 4. x =,,. x * = 3.4, x * = 3.4, x * 3 = 3.5, x * 4 = / 7,,,.,.,. x = 3. x * i m =, ( ) x - x * = = = 0-3 x * 3, r ( x * ) = ( ) x - x * = = = 0-3 x * 3, r ( x * ) =
13 5 (3 ) x - x * 3 = = = 0 - x * 3, r ( x 3 * ) = (4 ) x - x * 4 = - / 7 = = 0-3 x * 4 3, r ( x * 4 ) = , r ( x * ) = - x * / x * , r ( x * ) , r ( x 3 * ) 0.003, r ( x 4 * ) ,.,. n,., x * 5 = 3.36, x - x * 3 = = 0-3 x * 5 3, r ( x 5 * ) = - x 5 * / x 5 * ,,,.,.?. 0 0.%,, 0.%,.., x * n,.
14 6 er ( x * ) n n 0.% n 3.097, 4, 0.%..,.,,, 3, 4.47, / %..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 / sin x - x x 3 / 3!, " x [ 0, ],, x = ( Lagrange), sin x - sin0 = cosx,..4 x * = , x * = 0.00, x * 3 = 0.00, y = x * + x * + x 3 *, y = x * x * x 3 * y3 = x * / x * 3 (
15 7 ), (.5 )., ( x * ) = ( x * ) = ( x 3 * ) = , r ( x * ) = r ( x * ) = 0.5 r ( x 3 * ) = (.5 ) ( y * ) ( x * ) + ( x * ) + ( x 3 * ) = ( 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 * / r ( y * ) ( y * )/ y * r ( x * ) + r ( x * ) + r ( x 3 * ) r ( y 3 * ) ( y 3 * )/ y 3 * r ( x * ) + r ( x 3 * ) = 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 * ) = nx * n
16 8 r ( n x * ) 0.005/ n., n x, n..6 f( x) = sin( n 3 x) x * = 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 * ) ) ( x * ) r ( x * ) x * (.7).5.
17 9.7 4, - cos?.:.,., cos , - cos = ,. - cos= sin, sin , - cos = cos= sin + cos, sin , cos , - cos ,, - cos= ,, 3., - cos x x! - x 4,x = =, - cos , 4! x, x + = 0 x, = ( )/, x - bx + c = 0, x = b + sgn ( b) b - c, x = c/ x
18 0 sgn( b) b,, sgn( b) = -, sgn( b) =. b = 8, c =, x = b + sgn( b) b - c = = x = c/ x = / x = , x = ,, 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),.
19 . 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 , 0, 5,..ts
20 , s - - s, - t., t = 5, s = 0.x s - = 03 = 0 03 lg s r ( f l( x) ) - t = - 5 = 0-5 lg 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,
21 3 9 R = e - - n = 0 ( - ) n n! = e- 0!, 0 < < (.) x = 9, R = e - - e 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 (, ),.., =. e,. x * =.7, x * =.78, x * 3 =.7. x > 0, x x *,ln x * cm,0 cm 0 cm. cm. 4. s = gt, g, t 0.. t, s,. 5. p( x) = x x - 7 x + 8. ( ) p( x).
22 4 ( ) x * =.. ( 3) x *,. 6. p( x) = 0 x 7-3 x 7. 7.,. ( - ) 6, ( + ) 6, ( 3 - )3, ( 3 + ) 8.,. ( ) ln( x - x - ), x ( ) ln( x + ) - ln x, x ( 3) ( 4) 3 x x x, x - cos x, x 0, x x ( 5) sin x - tan x, x 0, x
23 . ) ; ) ( ) ; 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, (.).
24 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) :
25 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)
26 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., ( ).
27 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.
28 0. 4. (. ),.. x 0 4 f ( x) ( ) 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 x x - = 3 x3 - x x = - 4 x x - x = 4 x3-8 x + x l0 ( x) + 9 l ( x) + 3 l ( x) + 3 l3 ( x) = - 4 x x - x x +
29 N3 ( x) = + 8 ( x - 0 ) + 3( x - 0 ) ( x - ) - (3 ) ( x - 0 ) ( x - ) ( x - ) = 4-4 x x - x +, N3 ( x) L3 ( x).,,.,,,,,.,,.... x f( x) = sin x sin.8 L (.8 ),, sin.8 - L (.8)?.x [, ], L ( x) sin x. ( )
30 R [ f ] = x =.8 R [ f ] = f ( ) ( x -.0) ( x -.5) ( x -.0), (.0,.0) 3! ( ) f ( ) ( ) ( ) (.8 -.0) 3! = f [.0,.5] = ( )/ ( ) = 0.3 f [.5,.0] = ( )/ ( ) = f [.0,.5,.0] = ( )/ (.0 -.0) = L (.8 ) = (.8 -.0) ( ) (.8 -.5) = L (.8) - sin L (.8) - sin.8,. (3 ) x [, ] L ( x) - sin x = f( ) ( x -.0 ) ( x -.5) ( x -.0) 3! 6 max x ( x -.0) ( x -.5 ) ( x -.0) = x [, ],..3 x0 = 0, x = y = e - x, e e -.5,.
31 3 y0 = e 0 =, y = e N ( x) = ( x - 0) = x 0 - e N (0.5 ) = e -.5 N (.5 ) = e( y * (0.5 ) ) = e N (0.5 ) er ( y * ( 0.5) ) = e e( y * (.5 ) ) = e N (.5 ) er ( y * (.5) ) = e , {0,} 0.5, p( x) = x 4 - x 3 + x - x + { ( xi, p( xi ) ) } 5 i = 0 (.3)..3 x i p( x i ) q( x) { ( x i, q( xi ) )} 5 i = 0 (.4)..4 x i q( x i ) 3 5
32 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 r( x i ) 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 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 -
33 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) ]
34 6.6.6?.6 x p( x) ( ), n n ( )., n + n ( ), n.,.,,..,,,..7 f ( x).7..7 x f ( x)
35 7 f ( x) = 0., x = f - ( 0).,, x = f - ( y) y x = f - ( y) N4 ( y) = ( y + 0) ( y + 0) ( y + 5) ( y + 0) ( y + 5) ( y - ) ( y + 0) ( y + 5) ( y - ) ( y - ) f - ( 0) N4 ( 0) ,..8 p ( x).9..9 x p( x) 3 6 5
36 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 ( - ),
37 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
38 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)
39 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 - =
40 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 ( x) = ( x - x ) l ( x) = ( x - ) = ( + x) ( x - ) = x 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,,.
41 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),
42 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 - ) =
43 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
44 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 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! n = n( n + ) ( n + )/ 6 g( n) = n,
45 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 n -! + 5 n - + n ( n - ) ( n - )! ( n - ) ( n - ) ( n - 3 ) 3! n 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) =
46 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
47 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 h , e 3 48 h h d ef h * N = + - ( - ) h ( - ) ,.,
48 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)
49 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 ] = - =, ( ) M0 = 3 - M0 M M m - f [ x, x ] h = 6-54 M, M = M = 8, M = - 0, M0 = 8 (3 ) s( x) = s ( x), x [, ] s ( x), x [, 3] = = M, M = 8
50 4 8-6 x + 3 x - 3 x 3, x [, ] x - 3 x + 3 x 3, x [, 3].5..3, x = x y :.. 00, { 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 - )
51 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, 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 + ) n 3 = [ n( n + ) ]
52 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).
53 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 -,
54 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)
55 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, ].
56 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 +, ).,, [ ] ( ),.,.,. ( ),.
57 3 49,. (3 ), ( ),.,.,,. (4 )., C = ( A T A) - A T f, A, f, ( A T A) - A T A, A +, C = A + f 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)
58 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
59 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,
60 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 ( x - 0 ( g, g ) ( g0, g0 ) = ln x f gd x, = / ) ln xd x 4 ) ln xd x 7/ 44 = = 7 44 = 4 g ( x) = ( x - ) g ( x) - 0 g0 ( x) = ( x ) ( x - 4 ) = x x / 576 7/ 44 = 3 8
61 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, = = cos0. + sin cos cos sin0. - C = ( , , f ( x) = sin x sin x p ( x) = x x = ) T ( sin x - p ( x) ) d x =
62 54 = 0. 0 sin xd x - C T F = ( 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 = x - 0 = x = 0. 0 / / 000 = 0 ( g, g ) ( g0, g0 ) = = 0.05 x( x ) d x / ( x ) d x = 00 g ( x) = ( x - ) g ( x) - 0 g0 ( x) = ( x - x - 0 ) ( x - 0 ) - 00 = 0 x f( x) = sin x ( f, g0 ) = 0. sin xd x = - cos ( f, g ) = 0. sin x( x ) d x = 0 sin cos =
63 3 55 ( f, g ) = 0. sin x( x - 0. x + / 600) d x = cos0. + sin ( g, g ) = 0. ( x - 0. x + / 600 ) d x = 0 ( g0, g0 ) = 0., ( g, g ) = (3.7 ) p ( x) = ( f, g0 ) ( f, g ) ( f, g ) g0 + g + ( g0, g0 ) ( g, g ) ( g, g ) g = g g g = x x (3.8 ) = f - p = ( f, f) ( f, g0 ) ( g0, g0 ) + ( f, g ) ( g, g ) + ( f, g ) ( g, g ) 000, 3 f ( x) = sin x, ( 0 x 0. ), x = t, f( t) = sin( t), ( - t ) f( t). P0 ( t) = ; P ( t) = t, P ( t) = ( 3 t - )/ ( P0, P0 ) =, ( P, P ) = 3, ( P, P ) = 5
64 56 ( P0, f ) = sin( t) d t = ( P, f ) = tsin( t) d t = ( P, f ) = (.5 - t ) sin( t) d t = (3.7 ) f( t) P ( t) = ( P0, f) ( P0, P0 ) ( P, f ) ( P, f ) P0 ( t) + P ( t) + P ( t) = ( P, P ) ( P, P ) t (.5 t ) = t t (3.8 ) P ( t) f ( t) = f - P = ( f, f) - ( f, P0 ) ( P0, P0 ) + ( f, P ) ( P, P ) + ( f, P ) ( P, P ) ( sin( t) ) dt = f ( x) = sin x p ( x) = P ( 0 x - ) = p ( x) f ( x) = d x dt x x = =
65 , = = 0. 0 ( f ( x) - p ( x) ) d x - ( f ( t) - P ( t) ) d t (3.8),, x = 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 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
66 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 = ( x - ) = x, ax + bx + c x, I( a, b, c), a = - 8 0, b = 0, c = { -, -, 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 - ) = x - = 0 4 = 0, = (, ) (, ). = 4 0 =.4
67 ( x) = ( x - ) - = x x 3.8 ( 3. ), y = c0 + c x + c x. 3. x y , ; 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 =
68 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/ / 7) T y = x - ( - ) - ( - ) 0 0 c0 c c = 0 0 = 4 0 (Ac = b), A T Ac = A T b. A T A = A T Ac = A T b,, A T b = c = ( c0, c, c ) T = ( 58/ / 7) T, y = x. 3 { -, -, 0,, } 4 0
69 3 6, 0 ( x), ( x) = x, ( x) = x ( 3.7) p ( x) = ( y, 0 ) ( 0, 0 ) 0 + ( y, ) (, ) + ( y, ) (, ) x ( x - ) = x, y = p ( x) = x =. 3,,3.y = a + bx + cx 3 x, y = a + bsin x + ce x, 3,,. 3.9 Ar ( N) 4, Ar ( O) ( ). 3. NO N O NO N O 3 N O 5 N O ,, 6,.,. (),,
70 6. Ar ( N) Ar ( O), Ax = b A T = 3 5 4, xt = [ Ar ( N) Ar ( O) ] b T = [ ], A T A = x = A T Ax = A T b , AT b = Ar ( N) Ar ( O) = y = ae bx b, xi 3 4 y i 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
71 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 = , b = a = e A = x y = e = f( x) = x 4 + x 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)
72 64, p( x) f ( x) - p( x), f ( x) p( x) = f ( x) - T4 ( x) = x 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),.,
73 3 65 p ( 0) - f (0 ) = - ( p( ) - f ( ) ) = p ( ) - f( ) a0 - = - ( a0 + a- + ) = a0 + a - (3.7) a = , (3.6) = , ( 3.7 ), a f ( x) p ( x) = 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 - ) ]
74 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 x x x ] = p6 ( x) = + x +! x + 3! x3 + 4! x4 + 5! x5 + 6! ( 3 x x T6 ( x) ) = (3.0) 3 6! + x + (! ! ) x + 3! x3 + ( 4! + 3 6! ) x4 + 5! x5 + 6! T6 ( x) def
75 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 x x = 5 4 x3-3 6! + x +! ! 4! + 3 6! + x 4 + 5! 3 6! + -! ! 4! + 3 6! p6, 4 ( x) + 5! x x3-6 4! x + 3! + 4 4! x 4 + 5! T5 ( x) def (3.) 5 6 x x + 3! x 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! !! ! 4! + 3 6! x + x - x + 3! + 4 4! 8 + x 3 + 4! + 3 6! 8 (3.4) T4 ( x) =
76 ! - 8 4! + 3 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) = x x 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! = f( x) - p6, 3 ( x) (3.6),T 6 ( x), T 5 ( x) T 4 ( x),,.
77 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) ) f ( x) = e x ( - x ) T4 ( x) 3 p3 ( x), e x - p3 ( x). T4 ( x) = cos (4arccos x) cos def = cos 3 def = cos 5 8 = d ef cos 7 8 = d ef e x = , e x = e x 3 = , e x 4 = , p3 ( x) = ( x ) ( x ) ( x ) ( x ) ( x ) ( x ) = x x x 3 x4 x3 x x
78 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 = (3.8) e x - p3 ( x) (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) , 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) ( 3.9) ( - x ) ( x) - x S3 ( x), f ( x) - S3 ( x) f( x) - S3 ( x).,
79 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 ) = , ( f, T ) = ( f, T ) = , ( f, T3 ) = (3.30) S3 ( x) = x + (3.8 ) ( x - ) (4 x 3-3 x) = x x x 3 3 f ( x) - S3 ( x) = ( f, f ) - 3.4, max - x ex - S3 ( x) max i = 0 - x 6 i = 0 x i i! max x 7 - x 7! e ( f, Ti ) ( Ti, Ti ) - S3 ( x) + =
80 7 f ( x) - S3 ( x) [ -, ] ( 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]..
81 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) T3 ( x) + T5 ( x) + 5 ( l + ) T l+ ( x) + ] [] 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].
82 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 y i 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, 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.
83 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 ),
84 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) [ 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 +.
85 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.,
86 78 [ ]. k 4. n = k T k S k - C k - R k = 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) [ ].
87 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]
88 80 [ -, ].,., e x d x,. 0 ( ) R[ f ] = - e x d x 0 [e0 + e ] ( ) e ( - 0) 3 e e x d x 0 6 [e0 + 4e + e ] R[ f] = - (3 ) 880 e e e x d x 0 90 [7e0 + 3e 4 + e + 3e e ] R[ f] = - ( - 0 ) ( - 0 ) 6 e e e x d x , 0
89 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 ( ) ] f( x) x x x 3 x 4 I I I 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
90 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.), : ( )
91 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,
92 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 ) +
93 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) =,
94 86 f( x) d x - 3 [ f ( - ) + f ( x ) + 3 f ( x ) ] x, x, x + 3 x = x = x + 3 x =, x = 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 = , x = x = , x = ,. (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 ]
95 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 [ h D]
96 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 [ h 3 D] x 0 x ( x - x0 ) f( x) d x h [ 3 x 0 0 f ( x0 ) 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!
97 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
98 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 = ( ) 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 ) =
99 4 9 3 hf ( 0) + 9 h f (0 ) + 9 h3 f (0 ) 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 ) ( ), I = ln xd x ( ) x 7/ 6 8/ 6 9/ 6 0/ 6 / 6 ln x ( ) : T6 = / / 6[ 0 + ( ) ] ( ) : S3 = / 6 / 3[ ( ) + ( ) ] I = ln xd x ,.
100 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 ) ( ), < <
101 4 93 Rn [ f] = f ( 4 ) ( ) n n 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 +
102 94 ( ), 0-3, Rn [ f ] = - b - a h f ( ) n n 7.5.9, I T8 = 8 [ + ( ) ] ( ) 9, h = / 4,, Rn [ f] = - b - a 880 h4 f ( 4 ) ( ) 880 ( 4 )4 5 = (3 ), 0-6, Rn [ f ] = - b - a 880 h4 f ( 4 ) ( ) n 4 n.9. [ 0, ], 3 + = 7. (),, I = sin x d x., 0 x I T n + 3 ( T n - Tn )
103 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 ) : 4.4 k k T k 3 T k - T k - S k - 5 S k - - S k < < , I T8, I S4 = ; = I = d x 0 ( + x) x,
104 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( ) ] = [ t, + ] =.5 T = T + g(0.5 ) = [ ] =.55 T4 = T + 4 ( g( 0.5 ) + g(0.75) ) = [.55 + ( ) ] T8 = T4 + 8 ( g( 0.5) + g( 0.375) + g(0.875 ) ) = [ ( ) ] T8 - T4 0 - I.57, ( ), M = max f ( x) M. axb
105 4 97, M M, ( ) 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 )
106 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 ) ]} + +
107 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) 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.)
108 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 Mn Mn 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 +
109 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 =,
110 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 ] ( f ( ) = sin d ). 4.5 I = + 3 sin d k k T k S k - C k - R k- 3 R k R k <
111 4 03 I.4, l = 4 I 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 =
112 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 = 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 [ C k lim k C k 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
113 , d x, ln x. ( ) ( 0-6 ) ; ( ) ; (3 ) N = 4. ( ) k k T k S k- C k - R k - 3 R k R k < 0-6 d x , ln = x ( ), x = ( t + 3 ), x [, ], t [ -, ]. a. x d x = - t + 3 d t 5 9 f( ) f( 0) f ( 0.6 ) =
114 06 f ( t) = b. x d x = - t + 3 d t t f ( 0) [ f ( ) + f ( ) ] [ f ( ) + f( ) ] 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 i x d x = 0.5 d t - 0.5t +.5
115 4 07 I I3 I4 0.5 [ f ( = ) + f ( 3 ) ] x d x = 0.5 d t - 0.5t [ f ( = ) + f ( 3 ) ] x d x = 0.5 d t - 0.5t [ f3 ( =.75 3 ) + f3 ( 3 ) ] x d x = 0.5 d t - 0.5t [ f4 ( ) + f4 ( 3 ) ] d x I + I + I3 + I x :.,,,. 4.9 x f ( x) d x A0 f ( x0 ) + A f( x ) 0 x0, x A0, A. x0 ( 0, ), x ( 0, )
116 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 ) x0 x = 7 ( x0 + x ) - 9 x0 x = u, x0 + x = v, 7 v -, A0 A = x 0 x x 3 0 x v - 7 u - 3 u = 0 7 v - 5 u - 9 = ( x0 + x )
117 4 09 u = 5 v = 0 9, ( u = 0, v = 7 9, x0 = 0, x = 7 9 ). x0 = , x = A0 = , A = x f ( x) d x f ( ) 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 ) x/ = 0 xd x x / d x (, ) = 0 (, 0 ) = 0, = ( x, ) (, ) = (, ) ( 0, 0 x/ = 0 = 0 0 ) 0 x/ = x( x - 5 ) d x x / ( x ) d x x / ( x ) d x d x ( x) = ( x ) ( x ) - 75 = 3 45, = 75 x -. x
Ps22Pdf
) ,,, :,,,,,,, ( CIP) /. :, 2001. 9 ISBN 7-5624-2368-7.......... TU311 CIP ( 2001) 061075 ( ) : : : : * : : 174 ( A ) : 400030 : ( 023) 65102378 65105781 : ( 023) 65103686 65105565 : http: / / www. cqup.
Ps22Pdf
1, : ( ),?, :,,,, ( ), 1 180,, ( ) 1 1,, 2 180 ;,, 3 180 ;, n ( n - 2 ),, ( n - 2) 180 1 1, : ( ),.,, 2, (, ) 1 , 3 x + y = 14, 2 x - y = 6 : 1 ( ) : + 5 x = 20, x = 4 x = 4 y = 2, x = 4, y = 2 2 ( ) :
untitled
+ lim = + + lim = + lim ( ) + + + () f = lim + = + = e cos( ) = e f + = e cos = e + e + + + sin + = = = = = + = + cos d= () ( sin ) 8 cos sin cos = ( ) ( sin ) cos + d= ( + ) = cos sin cos d sin d 4 =
: : : ( CIP ) : ( ) /. :, ISBN :. G7. 4 CIP ( 00 ) 005 : : ( ) : : ( 0 : 0004) : : : / 6 : 7 ( ) : 408 () : 00
() ( ) ( : ) : : : ( CIP ) : ( ) /. :, 00. 7 ISBN 7-8008 - 958-8... :. G7. 4 CIP ( 00 ) 005 : : ( ) : : ( 0 : 0004) : : 00 7 00 7 : 78709 / 6 : 7 ( ) : 408 () : 000 : ISBN 7-8008 - 958-8/ G89 : 9 98. 00
!!!!"#$ " " %& ( " # " " " " " "$%%& " $%% " "!!
! "##$ % % % % % % % % &#!" % % #$%& (%&!! !!!!"#$ " " %& ( " # " " " " " "$%%& " $%% " "!! ! "#!"#$ $ $ $ $ %# %& $ &# ()*$ " & %!! ! " "!! !!!!" "! #$%& "# $()*$(+, $%& ())* $% (+,-.. /-.. 0&* 0%* "!!
80000 400 200 X i X1 + X 2 + X 3 + + X n i= 1 x = n n x n x 17 + 15 + 18 + 16 + 17 + 16 + 14 + 17 + 16 + 15 + 18 + 16 = 12 195 = = 1625. ( ) 12 X X n i = = 1 n i= 1 X f i f Xf = f n i= 1 X f ( Xf). i i
996,,,,,,, 997 7, 40 ; 998 4,,, 6, 8, 3, 5, ( ),, 3,,, ;, ;,,,,,,,,,
,, AB,, ( CIP) /, 000 ( /, ) ISBN 704009448 F47 CIP ( 000) 86786 55 00009 0064054588 ht tp www hep edu cn ht tp www hep com cn 006404048 787960/ 6 05 370 000 730,, 996,,,,,,, 997 7, 40 ; 998 4,,, 6, 8,
<4D F736F F D20CAFDD1A7D1A7D4BABACFB2A2B3F6C6ACCEC4BCFE2E646F63>
课 程 质 量 标 准 汇 编 ( 数 学 学 院 2010) 教 务 处 编 印 PDF 文 件 使 用 "pdffactory Pro" 试 用 版 本 创 建 www.fineprint.cn 目 录 高 等 代 数 方 法 选 讲 课 程 简 介 教 学 大 纲 考 核 大 纲 1 数 学 分 析 方 法 选 讲 课 程 简 介 教 学 大 纲 考 核 大 纲 7 数 学 分 析 课 程 简
1 V = h a + ab + b 3 = 1 = 1 + = + = BAC Quod erat demonstrandum Q E D AB p( EF) p = = AB AB CD q( EF) q p q 1 p q, EF = ED BF G G BG = FG EH a = b + c a - b = c FG = BG = HG = a EF = FG - EG = a - b
Ps22Pdf
,, (CIP) /.:, 2006 ISBN 7-5629-2480-5... -. U415.6 CIP (2006) 160794 : ( 122 :430070 ) http: ww w.t ech book.com.cn E-mail: w u tpyyk@163.com : : :7871092 1/ 16 :12.25 :302 :2006 12 1 :2006 12 1 :12000
( ) Wuhan University
Email: huangzh@whueducn, 47 Wuhan Univesity i L A TEX,, : http://affwhueducn/huangzh/ 8 4 49 7 ii : : 4 ; 8 a b c ; a b c 4 4 8 a b c b c a ; c a b x y x + y y x + y x x + y x y 4 + + 8 8 4 4 + 8 + 6 4
:,,,, ( CIP ) /,. :, ISBN CIP ( 2001) : : 127, : : : ht t p: / / www. nwpup. com : :
:,,,, ( CIP ) /,. :, 2001. 8 ISBN 7 5612 1363 8............. 0342 CIP ( 2001) 027392 : : 127, : 710072 : 029-8493844 : ht t p: / / www. nwpup. com : : 787mm1 092mm : 19. 75 : 480 : 2001 8 1 2001 8 1 :
1 2 / 3 1 A (2-1) (2-2) A4 6 A4 7 A4 8 A4 9 A ( () 4 A4, A4 7 ) 1 (2-1) (2-2) ()
(39mm E-Mail ( )( ), : : 1 1 ( ) 2 2 ( ) 29mm) WSK ( 1 2 / 3 1 A4 2 1 3 (2-1) 2-1 4 (2-2) 2-2 5 A4 6 A4 7 A4 8 A4 9 A4 10 11 ( () 4 A4, 5 6 7 8 A4 7 ) 1 (2-1) (2-2) () 1 2 (2-1) 3 (2-2) 4 5 6 7 (8 ) 9
( m+ n) a 6 4 4 4 4 7 4 4 4 48 m n m+ n a a = a 4 a 4 3 a a 4 a 4 3 a = a 4 a 4 4 a 4 == 3 = a ma na ( m+ n) a A 0 a m a n m n a m+n 0 B a m a n m n m>n a m-n C 0 (a m ) n m n a mn D (ab) n n a n b n (
4 / ( / / 5 / / ( / 6 ( / / 7 1 2 / 3 ( 4 ( 2003 8 ( 2
: / ( 6 (2003 8 : ( 1 ( ( / / (,, ( ( - ( - (39mm 29mm 2 ( 1 2 3-6 3 6-24 6-48 12-24 8-12 WSK / WSK WSK 1 4 / ( / / 5 / / ( / 6 ( / / 7 1 2 / 3 ( 4 ( 2003 8 ( 2 9 5 ( 10 3 11 / (600 4 5 AA 710 AB 720 730
( ) 001 ( CIP ) /. :,2005 ISBN CIP (2005) : : ( 147 : ) : : 850mm 1168mm : 333 :
( ) 001 ( CIP ) /. :,2005 ISBN 7-224 - 07274-5.......... 222 CIP (2005) 045547 : : ( 147 : 710003 ) : : 850mm 1168mm 32 15. 375 : 333 : 2005 7 1 2005 7 1 : 1-1000 : ISBN 7-224 - 07274-5/ I 1182 : 29. 00
幻灯片 1
Digital Signal Processing mailfzh@nwpu.edu.cn /gary/ 1. FT FT. 3. 4. DFT 5. 6. DFT 7. 1. FT FT (FS) (FT) ( ) xt () Dirichlet (, ), 1 T () = ( Ω), ( Ω ) = () T T jkωt jkωt xt X k e X k xte dt e jkω t k
,, ;,, ;,,,, ( CIP ) /,. :, ( ) ISBN T P212 CIP ( 2006) : : : : * : : 174 ( A ) : : (023 )
,, ;,, ;,,,, ( CIP ) /,. :, 2006.2 ( ) ISBN 7-5624-3583-9.............T P212 CIP ( 2006) 003761 : : : : * : : 174 ( A ) : 400030 : (023 ) 65102378 65105781 : (023 ) 65103686 65105565 : http : / /w ww.c
考 纲 解 读 14 浙 江 省 普 通 高 考 语 文 科 考 纲 研 读 吴 美 琴 今 年 的 考 试 说 明, 我 用 了 八 个 字 进 行 概 括, 那 就 是 稳 中 微 调, 关 注 生 活 稳 中 微 调 :14 年 的 语 文 考 试 说 明 是 近 几 年 来 调 整 幅 度
14 年 第 1 期 ( 总 第 87 期 ) 目 录 考 纲 解 读 语 文 吴 美 琴 (1) 数 学 王 芳 (3) 英 语 王 文 伟 (8) 物 理 季 倬 (1) 浙 江 省 义 乌 中 学 信 息 科 研 处 主 办 化 学 杨 军 (14) 生 物 吴 贵 忠 (16) 政 治 王 雪 娟 (17) 历 史 陈 旭 明 (7) 总 编 : 方 维 华 主 编 : 陈 平 执 行 主
32 91320301136426860D 江 苏 汉 邦 建 设 集 团 有 限 公 司 201601 41.50 45.00 0.00 86.50 33 05021063-9 江 苏 邦 实 建 设 工 程 有 限 公 司 201601 37.00 49.50 0.00 86.50 34 703
2016 年 徐 州 市 建 筑 业 施 工 企 业 上 半 年 综 合 信 用 评 价 得 分 表 日 期 :2016-06-14 序 号 组 织 机 构 代 码 单 位 名 称 批 次 基 本 信 用 分 综 合 考 核 得 分 日 常 扣 分 信 用 考 核 总 分 1 13641102-8 徐 州 市 政 建 设 集 团 有 限 责 任 公 司 201601 50.00 49.33 0.00
Ps22Pdf
(3 ) ,,, ;,, (CIP) /. 3. :, 003. 11 () ISBN 75610994.... TB301 CIP (000) 75084 : : 17, :71007 :09-8493844 : www.nwpup.com : : 787 mm1 09 mm 1/ 16 : 1.5 : 509 : 1997 10 1 003 11 3 5 : 15 000 : 7.00 : (,,,
a( a 0) a a( a 0) a = a ( a) = a a( a 0 ) a = a( a ) 0 a = a 4 f x 1 = x a ai a R sinx + a b ab sin x sinx = sinx sin x = 4 y = sinx + sinx - ysinx 4 = 0 sinx sinx x - 3 3= x x- 3 - x- 3 = 0
例15
cos > g g lim lim cos lim lim lim g lim ) ) lim lim g ) cos lim lim lim 3 / ) ) y, ) ) y o y y, ) y y y) y o y) ) e, ), ) y arctan y y Ce y) C y ) e y) y ) e g n www.tsinghuatutor.com [ g ] C k n n) n
Ps22Pdf
2001 ( ) 063,, ( CIP ) : : : (100054, 8 ) : : (021) 73094, ( 010 )63223094 : : : 850 1168 1/ 32 : : : : 2001 2001 : : ISBN 7-113 - 04319-4/ U 1192 : 24 00,, : ( 021 ) 73169, ( 010) 63545969 : : : : : :
监 制 制 : 中 华 人 民 共 和 国 国 务 院 侨 务 办 公 室 监 制 人 : 刘 泽 彭 顾 制 问 : ( 按 姓 氏 笔 画 排 列 ) 杨 启 光 陈 光 磊 陈 学 超 周 小 兵 赵 金 铭 班 弨 郭 熙 主 制 编 : 贾 益 民 编 制 写 : ( 按 姓 氏 笔 画 排 列 ) 干 红 梅 于 珊 王 劼 刘 潇 潇 刘 慧 许 迎 春 孙 清 忠 李 艳 吴 玉 峰
CIP / 005 ISBN X Ⅰ Ⅱ Ⅲ - - Ⅳ G CIP ISBN X/G http / /cbs pku edu cn pku edu
CIP / 005 ISBN 7-30-08496-X Ⅰ Ⅱ Ⅲ - - Ⅳ G634 603 CIP 004 353 ISBN 7-30-08496-X/G 380 0087 http / /cbs pku edu cn 67505 58874083 67656 xxjs@pup pku edu cn 675490 787 09 6 4 75 383 005 005 9 00 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
-2 4 - cr 5 - 15 3 5 ph 6.5-8.5 () 450 mg/l 0.3 mg/l 0.1 mg/l 1.0 mg/l 1.0 mg/l () 0.002 mg/l 0.3 mg/l 250 mg/l 250 mg/l 1000 mg/l 1.0 mg/l 0.05 mg/l 0.05 mg/l 0.01 mg/l 0.001 mg/l 0.01 mg/l () 0.05 mg/l
(CIP ) /. - :, ISBN TS1028 CIP (2002) ( ) mm1168 mm 1 /
2003 (CIP ) /. - :, 2002. 12 ISBN7-80624 - 659-6... - - -. TS1028 CIP (2002) 087823 ( 3 61 001 2 ) 2002 12 2002 12 850 mm1168 mm 1 /32 8. 75 200 1-3 000 13. 80 ( ) : : : : ( ) ,, :,,,, :, : :,, :,,,,,
( CIP) /, - :, ISBN H15 CIP ( 1999) ( 1 : ) : * : 6000 : 56
( ) ( CIP) /, - :, 1999. 10 ISBN 7-5045-2677-0... - - - - - -. H15 CIP ( 1999) 44365 ( 1 : 100029) : * 8501168 16 41. 5 1078 1999 9 1 1999 9 1 : 6000 : 56. 00 ( 14. 00 ) ( ),,,, :,,,,,,,,,,,,,,, 160, 28
Ps22Pdf
1 9 9 9 ( ) 063, ;,, ;,, ( CIP) / - :, 1998 10 ISBN 7 113 03130 7 T U 476 CIP ( 1998) 28879 : : : ( 100054, 8 ) : : : : 787 1092 1/ 16 : 15 : 383 : 1999 2 1 1999 2 1 : 1 : ISBN 7 113 03130 7/ T U 588 :
B3C1
- B(. AB. A( ( 3. AA PP 0 a a a 4. ( 5. Ex. ABCDEF Ans8305 Ex. ABCDE Ans00. a+ b a+ b b. a+ b = b + a a b a ( a+ b + c = a+ ( b + c a+ 0= a = 0+a a + ( a = 0 = ( a + a b a b 3. a b = a+ ( b a 4.(P AB =
365 1 4 29 499 940 91 5 16 23 11 16 29 43 81 3 3 4 365 1 4 21 1 4 10 730 = 31466. 10 232 29 773 1457 20 4 365 145 589 1825 7 3.1 47 1825 7 47 27 3303 104 5969 1488
Ps22Pdf
CIP ) / :,2006.2 ISBN 7-80702 - 113-6..........G.206 CIP (2006)080133 :8501168mm 1/ 32 : 120 :2000 2006 3 1 : 5000 ISBN 7-80702 - 113-6/ G206 : 348.00 (16 ) ,?, :,,,,,,,,,!,?,,,,,,?,, ,,,,,,,,,,,,,,,!,!,!
Ps22Pdf
()63,,, POI SSO N MARKOV BROWN, ( CIP ) /, :, 7 ISBN 7-3 - 3673 - - 6 CIP ()397 : : : (54, 8 ) : : : : : 787 9 : 3 5 : : : : I SBN 7-3 - 3673 - / O74 : 5 8,, ,,,,,,,,,,,,,, ( ),,,,,,,,,,,!, ( ) 3 3 4 6
lim f(x) lim g(x) 0, lim f(x) g(x),
2016 11 14 1 15 lim f(x) lim g(x) 0, lim f(x) g(x), 0 0. 2 15 1 f(x) g(x) (1). lim x a f(x) = lim x a g(x) = 0; (2). a g (x) f (x) (3). lim ( ). x a g (x) f(x) lim x a g(x) = lim f (x) x a g (x). 3 15
1-1 + 1 + + 2 + + 3 + 4 5 + 6 + 7 8 + 9 + 1-2 1 20000 20000 20000 20000 2 10000 30000 10000 30000 3 5000 5000 30000 4 10000 20000 10000 20000 5 3000 3000 20000 6 3000 3000 20000 7 5000 15000 8 5000 15000
<4D6963726F736F667420576F7264202D2032303135C4EAC8EBD1A74D4241C1AABFBCD7DBBACFB2CEBFBCB4F0B0B8BCB0CFEABDE22E646F6378>
05 年 入 学 MBA 联 考 综 合 试 卷 参 考 答 案 及 详 解 说 明 : 由 于 05 年 入 学 MBA 联 考 试 题 为 一 题 多 卷, 因 此 现 场 试 卷 中 的 选 择 题 顺 序 及 每 道 题 的 选 项 顺 序, 不 同 考 生 有 所 不 同 请 在 核 对 答 案 时 注 意 题 目 和 选 项 的 具 体 内 容 所 有 解 析 来 自 网 络, 仅 供
Ps22Pdf
( ) ( ) 2006 ( C I P ) /. - :, 2002. 6 ( ) ISBN7-80176 - 020-4... - -. I207. 62 CIP ( 2002 ) 035654 ( ) : : : ( 100089) 1 : : : 8501168 1 /32 : 2, 030 : 78. 125 : 2002 6 1 : 2006 5 3 : ISBN7-80176 - 020-4
校园之星
I V X V L C D M n n n X X X M M VI X X DC IV IX XL C D X V I I X V D CLXV I V X XCIX C X IC C I X X quatre vingt quatre vingt dix thousand million billion p M sinl sinl e x ii xii a ba bi MI sinl
1991 707 708 1972 36 1990 2 126 130 21 1656 1742 1705 1972 523 334 420 342 423 1433 1435 1975 205 = + + 1 2 2 2 2 2 2 2 2 1 2 2 + [ ( )] 4 2 1 2 2 2 2 2 2
1991 707 708 1972 36 1990 2 126 130 21 1656 1742 1705 1972 523 334 420 342 423 1433 1435 1975 205 = + + 1 2 2 2 2 2 2 2 2 1 2 2 + [ ( )] 4 2 1 2 2 2 2 2
WL100079ZW.PDF
ε I x = r + R + R + R g o x = R ε + v v 2 v1 a = = t t t 2 1 R x { ( 1) ( 2)" " ( 3) ( 4), ( 5)" " ( 6) ( 7) ( 8)" " ( 9) ( 10) ( 11) ( 12) ( 13) ( 14) ( 15) ( 17) {
* 1992.10 43 (91.49%) 4 9.51% 26 60.46% 13 4 30.2% 9.31 % 21 6 16 13 45 6 X1=8.16X=40.6 X2 X1 p 0.01 n =43 n =64 51 13 25 18 X1=6.635 X2=18.6 18.6 6.635 P 0.01 n =64 n =43
中华人民共和国海关进出口税则
! " " # $!"#$%&# &(%)$ *+, -.(%)$ /*)011 %1 $23 43%(536# 73("8509 %1!20+* %& ()& *+),-. +/ 01 *234+53 *+-46+7 *+-),4,+-3 8(93 (-) :&.27(4,+-3 ;(? %&4(,7&) *234+53 %24,&3 8&@,&) +- *+55,33,+-
Ps22Pdf
( ) ( ) 2006 ( C I P ) /. - :, 2002. 6 ( ) ISBN7-80176 - 020-4... - -. I207. 62 CIP ( 2002 ) 035654 ( ) : : : ( 100089) 1 : : : 8501168 1 /32 : 2, 030 : 78. 125 : 2002 6 1 : 2006 5 3 : ISBN7-80176 - 020-4
Ps22Pdf
: : ( 124 ) 7871092 16 : 3900 2003 3 1 2003 3 1 :3000 ISBN7806066187/ K29 (4 ) : 396.00 , ( ) 1959,,,,,, (1711),, ( ),,,,,,, 12, 222,,,,,,,, (, ),,,,,,,,,,,,, ,,,, 80,,,,,,,,,, 2002 11 : : : : ( 1 ) (
五花八门宝典（一）.doc
BBS...5... 11...23...26...31...46...49...54...55...57...59...62... 110... 114... 126... 132... 149 I "108" 1 2 3 4 BBS 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 M ( ) Kg S ( ) A ( ) K (
#,$./-,0+0 1&*",( &2 3&& &)# H-! " #$ %& ()*+ %& ()*+ %& ()*+!"#$%& ()*+ $*,-$* *+ %& ()*+! $/6DC $/$!DC,$$ E)$/# HC - $$$ E) 5 & $/# E<)FG
!"#$%&% ()*$+, (-.((,(/0!""# $!$ %!& %($ """"""""""""""""""""""""""""""""""""""""""""""""""""""""" "!"#$%&!!%""&) *+,-./0 1*023(,+/)& 2)$#2),)& *+,-./0 *+,-./0 *+,-./0 *+,-./0 1&)) 2 "!)"3(!4(!""# "$3%!&3"4
( CIP).:,3.7 ISBN TB CIP (3) ( ) ISBN O78 : 3.
( CIP).:,3.7 ISBN 7 568 383 3.......... TB CIP (3) 334 3 37 ( ) 64536 www.hdlgpress.com.c 7879 6 9.75 479 3 7 3 7 45 ISBN 7 568 383 3O78 : 3. 995,.,.,.,. :,,,,.. :,,,,,,.,,,,.,,. ,,.,,,.,,,.,,,,.,.,,,
: : : mm 1/ 32 :92 : : 3000 ISBN / G190 : (10 )
: : : 8501168mm 1/ 32 :92 :2000 2006 1 1 : 3000 ISBN 7-80702 - 286-8/ G190 : 228.00 (10 ) ( 1 ) ( 4 ) 5( 6 ) ( 9 ) 1. ( 9 ) 2. (10) 3. (13) 1 > 10,? (16) (18) (20) (21) 1. (21) 2. (24) 3. (25) 4.3129(26)
数 学 高 分 的 展 望 一 管 理 类 联 考 分 析 第 一 篇 大 纲 解 析 篇 编 写 : 孙 华 明 1 综 合 能 力 考 试 时 间 :014 年 1 月 4 日 上 午 8:30~11:30 分 值 分 配 : 数 学 :75 分 逻 辑 :60 分 作 文 :65 分 ; 总
目 录 数 学 高 分 的 展 望... 1 第 一 篇 大 纲 解 析 篇... 1 一 管 理 类 联 考 分 析... 1 二 最 新 大 纲 解 析... 1 三 考 前 复 习 资 料 及 方 法... 第 二 篇 总 结 篇... 4 1 应 用 题 考 点 总 结 与 技 巧 归 纳... 4 代 数 模 块 题 型 归 纳 及 考 点 总 结... 9 3 数 列 模 块 题 型 归
(CIP) /. :, ISBN T U1 4 CIP (2005) : : 127 : : : : : 787 mm mm 1/ 16 : 4
(CIP) /. :,2005.7 ISBN 7 5612 1935 0....T U1 4 CIP (2005) 040538 : : 127 :710072 : 029-88493844 88491757 : www.nwpup.com : : 787 mm 1 092 mm 1/ 16 : 4.5 : 109 : 2005 8 1 : 6.00 : : : ,,,,,, ; ;,,, 2005
#$%&% () % ()*% +,-. /01 % + (/) " " " 2- %** -340 $%&% 5!$%&% () % ()*% +,-. /01 % + (/) " " " 2- %** -340 /64 7%,(8(, *--9( ()6 /-,%/,65 :$%&
! " "!! " "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! " #$$% & ()*+,-.(*/!0%1 23)4-(4 5).67*(*8. #$$%!9 #$$% #!$1#$!1 #9 19 :9 %; :< #$$% = 0!$ ; = : : : = 1 % #!9 #$%&% () % ()*% +,-. /01 % + (/) " " " 2- %**
i n i ho n n n n n ng
A i i i i i i i i n i ho n n n n n ng o o o o o o o b B b b p b b b b b b b p b b b b b b i b i b b i b b i b i b i b bn bn bn bn bn bn bn b bn bn bn bn bn bn bng bng bng b pn bng bng bng pn bng bo bo
1991 707 708 1972 36 1990 2 126 130 21 1656 1742 1705 1972 523 334-420 342-423 1433 1435 1975 205 = + = + = 1 2 ( ) 2 2 = 2 2 2 2 2 2 2 2 = 1 4 [ + ( ) ] 2 1 2 2 2
(CIP) ().: ISBN O4-44 CIP (2004)
( ) (CIP) ().:. 2004.11 ISBN 7-04 - 015565-6... - -.O4-44 CIP (2004)092157 010-64054588 4 800-810 - 0598 100011 http:www.hep.edu.cn 010-58581000 http:www.hep.com.cn 8501168 1 32 1 8.625 220 000 12.00,.
6CO2 6H 2O C6H 2O6 6O2 = = n i= P X i n H X - P X logp X i= i n X X X P i P 0 i l n i n n P i= i H X = - p log P n i= i i i + H X - P X logp X dx - 2 2 2 2 3 2 4 2 d( Q) d( Q) > 0 = 0 di di d(
J = m i r 2 i i ϕ 1 = Rg R 2πR R T = = 2π = 84. 4 min ( 1) g 1 T = 2π m / k k T = 2π m/ k = 2π R / g = 84. 4min ( 2) T = 2π l / g ( 3) / 0 20 40 60 80 100 120 /Pa 600.66 2338.1
數學教育學習領域
高 中 数 学 课 程 补 充 资 料 013/14 学 年 就 读 中 四 学 生 适 用 013 ( 空 白 页 ) 目 录 页 数 1. 概 论 1 1.1 背 景 1 1. 关 注 事 项 及 考 虑 因 素 1 1.3 短 期 方 案 摘 要 1 1.4 评 核 设 计 概 要. 修 订 后 的 高 中 数 学 课 程 学 习 内 容 3.1 修 订 后 的 必 修 部 分 学 习 内 容
Microsoft Word - 5 框架结构设计.doc
主 要 内 容 : 结 构 布 置 框 架 结 构 的 计 算 简 图 第 5 章 竖 向 荷 载 作 用 下 框 架 结 构 内 力 的 简 化 计 算 框 架 结 构 设 计 水 平 荷 载 作 用 下 框 架 结 构 内 力 和 侧 移 的 简 化 计 算 荷 载 效 应 组 合 和 构 件 设 计 框 架 结 构 的 构 造 要 求 重 点 : 结 构 布 置 框 架 结 构 的 计 算 简
" #" #$$" "#$$% # & $%& ()*+,- #$$% " & " & ( % ( ( ( % & ( % #" #" #" #"
"#$ "##$ %& ()* "##% "##$ "##$ & () " (" (* + ( " "*, ( " - % & $ "##$ " " #" #$$" "#$$% # & $%& ()*+,- #$$% " & " & ( % ( ( ( % & ( % #" #" #" #" " # $ %&& %&&( %&& %&&) "%&&) #$%& (()*+ "* %&&) %&& %*
Ps22Pdf
/ : : : / : ISBN 7-81055 - 930-7 / F 313 : 5. 00 : 2006 4 1 CIP ( 2006) 045879 ;, ;, ( ), :,, :, ;, ; 1 , ; :,, :,,,??,? ;,?,??,? 2 ? ;,,,! ;, ;,, ;,,, : ;,, 3 ?,,,,,,, ;,, 4 , ;, ;!, ;,, 5 , :, :, - :?
校园之星
y cu jin k rn ju zo ho shu m i lng b ng png bi s bo cn zho dn ji kn hu ch x r u q xn w i kn j ju xin xin bng nin dng l ng zo j l i so ch l jin xi su tu y j ng chn ru m zh sng shn O AB AB AB O B AB
,,,, ( CIP ) /. :, ( ) ISBN K CIP ( 2005) : : 66 : : ( 0371) : : : : : : : 140mm 202mm :
,,,, ( CIP ) /. :, 2005. 8 ( ) ISBN 7-5349 - 3157-6... -. K928. 957 CIP ( 2005) 014330 : : 66 : 450002 : ( 0371) 65737028 : : : : : : : 140mm 202mm : 6. 875 : 164 : 4 : 2005 8 1 2005 8 1 : 1 4 000 : 16.
é SI 12g C = 6 12 = 1 H2( g) + O2( g) H2O( l) + 286kJ ( 1) 2 1 1 H 2( g) + O2( g) H2O( l) H = 286kJ mol ( 2) 2 1 N 2 ( g) + O2( g) NO 2 ( g) 34kJ 2 1 1 N 2 ( g) + O2( g) NO 2 ( g) H = + 34kJ mol 2 1 N
(CIP) /. :, 2005 ISBN 7-5375 - 3325-3.... R247.1 TS972.161 CIP (2005) 152967 / / 330 / 050061 / / / / / 880 1230 1/ 32 / 15 / 370 / 2006 1 1 / 2006 1
(CIP) /. :, 2005 ISBN 7-5375 - 3325-3.... R247.1 TS972.161 CIP (2005) 152967 / / 330 / 050061 / / / / / 880 1230 1/ 32 / 15 / 370 / 2006 1 1 / 2006 1 1 / 24.80 ,,,,,,,,,,,,,,,, ;,,,,, 2006 1 ( ) ( ) (
(CIP) /. :, 2004.1 ISBN7-81060 - 342-6......... - -.TS971-49 CIP (2003) 085575 818 : 200433 / : 021-65493093 : 850 1168 1/ 32 : 9.625 : 255 2004 1 1 2
(CIP) /. :, 2004.1 ISBN7-81060 - 342-6......... - -.TS971-49 CIP (2003) 085575 818 : 200433 / : 021-65493093 : 850 1168 1/ 32 : 9.625 : 255 2004 1 1 2004 1 1 : 13 000 ISBN 7-81060 - 342-6/ T012 : 19.70
,, :, ;, 20012003 10 90, 20012003 8,;, ; (CIP) /. :,2004. 1 ( ) ISBN 7 5612 1720 X.... 0441. 4 44 CIP (2003) 109559 : : 127 710072 : ( 029 ) 88493844
,, :, ;, 20012003 10 90, 20012003 8,;, ; (CIP) /. :,2004. 1 ( ) ISBN 7 5612 1720 X.... 0441. 4 44 CIP (2003) 109559 : : 127 710072 : ( 029 ) 88493844 : www.nwpup.com : : 787 mm1 092 mm 1/ 16 : 19.5 : 462
! * # + + *! # $ # #. 1 #! % &)# * 1 *! * ! % # * # * # + + +!!!! # * % # # + # + * & $ *! * # + * # % #& % &* # & # * %! + * # #!
!!#$! # $ %! ##&&!)*+, -##. -&/01& -#!!!% $! $!#!!!$23.1! 4)5 %#&&& &.1 %!!!! %! &!!!!!!!! /!!!!!!!!!!!!!!! %.&!!! #&!!!!0&&&& #!!!!!!!!!!!!!!! %.& /&&&/.! 0.!!!!!!!!!!!!!!! %1& 6 /. 1!!!!!!!!!!!!!!! %&
9,, (CIP) /. :, ISBN T U767 CI P ( 2004 ) : 122 : / mail.whut.edu.c
9,, (CIP) /. :, 2005.2 ISBN 7 5629 2097 4....T U767 CI P ( 2004 )003594 : 122 : 430070 http:/ / www.techbook.com.cn E-mail: yangxuezh@ mail.whut.edu.cn : : : 7871092 1/ 16 : 17 : 421 : 2005 2 1 : 2006
《分析化学辞典》_数据处理条目_1.DOC
3 4 5 6 7 χ χ m.303 B = f log f log C = m f = = m = f m C = + 3( m ) f = f f = m = f f = n n m B χ α χ α,( m ) H µ σ H 0 µ = µ H σ = 0 σ H µ µ H σ σ α H0 H α 0 H0 H0 H H 0 H 0 8 = σ σ σ = ( n ) σ n σ /
: ( ) ( ) : () ( ) ( )
: ( ) ( ) : () ( ) ( ) ( CIP ) /,. - :,2004. 4 ISBN 7-81100 - 114-4... - -. G861. 1 CIP (2004) 020066 100084 8501168 1/ 32 6 2004 4 1 1 6000 ISBN 7-81100 - 114-4/ G105 13. 00 ( ) , 2002 8,,,,,,,,,,,,,,,,,,,
第一章.doc
= c < < + + = S = c( ) = k =, k =,,, Λ < < + = 4 = = = = 4 k = k =,,, Λ X R X X = f () X X = f ( ) k = + k =,,, Λ = f () X X f ( ) = = = = n n = an + an +... + a + a a n =a +a +a = a + a + a a n f ( )
05. = 8 0. = 5 05. = = 0.4 = 0. = 0.75 6. 5 = 6 5 0 4 4-6 4 8. 4 5 5 + 0.9 4 = 84 5 9-6 + 0 0 4 5 4 0 = 0-5 + = + 0-4 0 = 0-4 0 = 7 0.5 [ 9 6 0.7-0.66 ] 4.9 = 9 9 7 49 [ ] 0 50 0 9 49 = [ ] 9 5 0 = 49
要 求 服 装 统 一 各 队 自 带 比 赛 球 槌 队 长 及 教 练 标 志 大 会 提 供 比 赛 用 球 和 号 码 布 ( 五 ) 比 赛 所 用 球 槌 须 为 中 国 门 球 协 会 2016 年 度 专 业 器 材 供 应 商 企 业 的 产 品, 企 业 名 称 和 品 牌 请
竞 赛 规 程 一 比 赛 时 间 和 地 点 时 间 :2016 年 8 月 7 日 至 13 日 地 点 : 湖 北 省 利 川 市 二 竞 赛 织 指 导 单 位 : 中 国 门 球 协 会 主 办 单 位 : 中 国 门 球 协 会 门 球 之 苑 编 辑 部 利 川 市 人 民 政 府 承 办 单 位 : 湖 北 省 门 球 协 会 恩 施 州 老 年 人 体 育 协 会 利 川 市 文
" "##$ """ $ $%%& %& ()(*(+,& " - - # -./ % # - - $( +?5 - %( +?5 ;&77&12 0 # -01,( 2+0 1% ( ) &) 3 $, -01,( 2+0 "##$ %4# %
""# $ "# ""# "#$%&% ()*$+, (- "*(.+/(0*+1"2 $%& $ (% % %% & %% "#$ - - $(" % 7 7 7 7+89"0%:+ ; < 8B (". $ 7?7)A@C,
,, : ;,,, (CIP) /. :, 005. ISBN TB301 CIP (005) : : 17, : : ( 09 ) : : : 787 mm1 09 mm 1/ 16 : 5.75
,, : ;,,, (CIP) /. :, 005. ISBN 7 561 1901 6.... TB301 CIP (005) 007098 : : 17, : 71007 : ( 09 )8493844 : www.nwpup.com : : 787 mm1 09 mm 1/ 16 : 5.75 : 630 : 005 1 005 1 : 8. 00 ( ) 1,,,,,,, 80100,,,,,,
图 书 在 版 编 目 (CIP) 数 据 中 文 第 九 册 教 学 参 考 / 中 国 暨 南 大 学 华 文 学 院 编. 修 订 版. 广 州 : 暨 南 大 学 出 版 社,2007.9 ISBN 978-7-81029-811-7 Ⅰ. 中 Ⅱ. 中 Ⅲ. 对 外 汉 语 教 学 Ⅳ. H195 暨 南 大 学 出 版 社 出 版 发 行 ( 中 国 广 州 ) 电 话 / 传 真 :0086-20-85221583
初 啼 八 集 2006 至 2007 年 度 出 地 版 : 伯 特 利 中 學 址 : 元 朗 錦 繡 花 園 F 段 第 四 街 11 號 電 話 :2471 2622 傳 真 :2471 5171 製 作 : 同 理 心 創 念 有 限 公 司 出 版 日 期 :2007 年 7 月 序 初 啼, 是 由 本 校 中 文 科 和 活 力 組 合 辦 的 文 集, 提 供 給 學 生 發 表
Ps22Pdf
( ) 158,,,,,, ( CIP) /. :, 1996. 12 ISBN 7 302 02353 0... :. F275 CIP ( 96) 20860 : ( :, 100084) : : : 850 1168 1/ 32 : 13. 25 : 344 : 1996 12 1 1996 12 1 : ISBN 7 302 02353 0/ F 130 : 0001 5000 : 16.
例題. y = x x = 0 y = x 0 li 0 li 0 li = y = x x = 0 = f x) x = a x = a 2
y = x x = 0 y 2 0 2 x Figure : y = x f x) x = a f x) x = a f a) dy dx x=a f a) x a f x) f a) x a f a + ) f a) f x) x = a f x) x = a y = x x = 0 例題. y = x x = 0 y = x 0 li 0 li 0 li = y = x x = 0 = f x)
<D5FDCEC42E733932>
图 书 在 版 编 目 (CIP) 数 据 与 考 场 / 张 克 敏, 兰 燕 主 编. 重 庆 : 西 南 师 范 大 学 出 版 社,2003.10 ISBN 7 唱 5621 唱 2973 唱 8 Ⅰ 畅 畅 畅 畅 Ⅱ 畅 1 张 畅 畅 畅 2 兰 畅 畅 畅 Ⅲ 畅 文 课 高 中 教 材 Ⅳ 畅 G634 畅 341 中 国 版 本 图 书 馆 CIP 数 据 核 字 (2003)
(CIP) :, 2003 ISBN 7 81010 715 1 /...........R151.2 CIP (2002) 105855 ( 530 200032) 850 mm1168 mm 1/ 32 11.25 283 1 4 100 2003 4 1 2003 4 1 ISBN 7 810
(CIP) :, 2003 ISBN 7 81010 715 1 /...........R151.2 CIP (2002) 105855 ( 530 200032) 850 mm1168 mm 1/ 32 11.25 283 1 4 100 2003 4 1 2003 4 1 ISBN 7 81010 715 1/ R680 22.00 ,,,,,,,,,,,,,,,,,,,,, 20,,,,,,,,,,,,,,,,,,,,
CIP ) / :,2006.2 ISBN 7-80702 - 188-8..........K.23 CIP (2006) 080136 :960 640mm 1/ 16 : 55 :1080 2006 3 2 1 :5000 ISBN 7-80702 - 188-8/ K 23 ( 3 ) :
CIP ) / :,2006.2 ISBN 7-80702 - 188-8..........K.23 CIP (2006) 080136 :960 640mm 1/ 16 : 55 :1080 2006 3 2 1 :5000 ISBN 7-80702 - 188-8/ K 23 ( 3 ) : 260.00 ( 1 ) ( 1 ) ( 1 ) ( 2 ) ( 3 ) ( 3 ) ( 4 ) (
Ps22Pdf
: : (CIP) / :, 2006.3 ISBN 7-5385 - 2026-0..........I222 CIP (2006) 021873 :8501168mm 1/ 32 :200 :3000 2006 1 2 : 2000 ISBN 7-5385 - 2026-0/ I393 :498.00 ( 22 ) : : : ,,,,,,,,,,, : 7 8, 8 8, 8 9, 9 10
Ps22Pdf
: : (CIP) / :, 2006.3 ISBN 7-5385 - 2026-0..........I222 CIP (2006) 021873 :8501168mm 1/ 32 :200 :3000 2006 1 2 : 2000 ISBN 7-5385 - 2026-0/ I393 :498.00 ( 22 ) : : : ,,,,,,,,,,, : 7 8, 8 8, 8 9, 9 10
Ps22Pdf
( CIP). :, 2006 ISBN 7-224-07586-8... ( ). E289. 413 - - CIP (2006) 022989 ( 147 : 710003) 787mm 1092mm 16 29 24 650 2006 3 1 2006 3 1 1 1000 ISBN 7-224-07586-8 E 57 128.00 ISBN 7-224-07586-8 E 57 : 128.00
Microsoft Word - 烘焙食品乙級第二部份 doc
烘 焙 食 品 乙 級 技 術 士 技 能 檢 定 術 科 參 考 資 料 試 題 編 號 :077-900201-3 審 定 日 期 : 年 月 日 修 訂 日 期 :96 年 7 月 1 日 97 年 1 月 30 日 97 年 10 月 27 日 98 年 6 月 20 日 98 年 12 月 17 日 99 年 08 月 20 日 烘 焙 食 品 乙 級 技 術 士 技 能 檢 定 術 科
校园之星
DI QIU ZHI L min xin i y yun TAN XUN RENNAO DE MI MI DI QIU ZHI L t jun png TAN XUN RENNAO DE MI MI DI QIU ZHI L o su TAN XUN RENNAO DE MI MI DI QIU ZHI L bo x mio c b TAN XUN RENNAO DE MI MI DI QIU ZHI
t H θ m [] Q Q q [] 9. 69kJ g q 32.0 620kJmol - 1 0. 500g mol C(s) 1 2 O (g) = CO(g) 2 SI n mol H 2H+H22H2+O2 [N2H41+O2g N2g+2H2O1] 1.2.3 qp qv [4] 1 C( ) O 2 (g) = CO(g), ( 2), 2 CO(g) CO (g),co(g) +
考 研 数 学 三 部 曲 之 大 话 高 等 数 学 0. 考 研 数 学 高 等 数 学 部 分 其 实 就 是 一 座 大 楼 房 间 80 房 间 80 第 八 层 房 间 80 房 间 804 房 间 805 房 间 70 房 间 70 房 间 70 第 七 层 房 间 704 房 间 7
第 0 章 超 级 导 读 ( 必 看 ) 本 书 共 8 章, 此 章 虽 不 讲 具 体 的 知 识 点, 但 其 地 位 是 相 当 重 要 的 因 此, 强 烈 建 议 大 家 阅 读 本 章 的 内 容 考 研 数 学 三 部 曲 之 大 话 高 等 数 学 0. 考 研 数 学 高 等 数 学 部 分 其 实 就 是 一 座 大 楼 房 间 80 房 间 80 第 八 层 房 间 80
! "! #!$$%!$$% &!!$$( # ) (
! " "!! " "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! " #$$% #$$%!!% % & %!$ ( # ) #$$% *!!% ! "! #!$$%!$$% &!!$$( # ) ( " #$ %&!#& ( )*+,* -) " " "./012 )*+ 302 4056 7+1.6 0 3*8(*/.0-96 :*+/26) -+. 80;6
3 = 90 - = 5 80 - = 57 5 3 3 3 = 90 = 67 5 3 AN DE M DM BN ABN DM BN BN OE = AD OF = AB OE= AD=AF OF= AB=AE A= 90 AE=AF 30 BF BE BF= BE= a+b =a+ b BF=BC+CF=a+CF CF= b CD=b FD= b AD= FC DFC DM=
x y 7 xy = 1 b c a b = x x = 1. 1 x + 17 + x 15 = 16 x + 17 x 15 + 17 15 x + 17 - x 15 = (x x ) ( ). x + 17 + x 15 x + y + 9 x + 4 y = 10 x + 9 y + 4 = 4xy. 9 4 ( x + ) + ( y + ) = 10 x y 9 ( x + )( ).
) E F EF F F = = FE = F = F 5 E O E F O O O O O O O O O EFO E F 6 8cm 3cm 5cm cm 3cm 5cm cm 7 5% x + 3 x =
(@ ) 005 4 36 8,,84 0 0 3 () (36 ) ( 36. 3 ) 7.3 6.5 08 08 008 008 a b ab > 0 a > b a b> 0 b 0 a x a + b> 0 3 005 3355.55 3 http://czsx.xicp.net; E-mail: shenyufu086@sina.com 8 (@ ) 3.3 0 3.4 0 3 8 3.4
Ps22Pdf
si equation equation geome triageo metria geometria geo lin xin zng chu m x y y x h b ng sh kn x y z x y z x y z x y z x y z u c xin sho jin bio A B C zh f i q an an an an n n lim n an an gng chng
Ps22Pdf
(CIP) /. :,2005 ISBN 7-5375 - 3271-0.... R16 CIP (2005) / / 330 / 050061 / / / / / 880 1230 1/ 32 / 15 / 320 / 2005 9 1 / 2005 9 1 / 24.80 ,,,,,,,,,,,,,,,,,, 1 ,, 2005 6 2 (3 ) (3 ) (4 ) (5 ) (6 ) (7 )
SM-A7000 Android SM-A7009 Android SM-A700F Android SM-A700FD Android Galaxy A7 (2015) SM-A700FQ Android SM-A70
SM-A3000 Android 5.0 2.3 SM-A3009 Android 5.0 2.3 SM-A300F Android 5.0 2.3 SM-A300FU Android 6.0 2.6 Galaxy A3 (2015) SM-A300G Android 5.0 2.3 SM-A300H Android 5.0 2.3 SM-A300HQ Android 5.0 2.3 SM-A300M
Ps22Pdf
( ) 158 10, :,,,, ( MBA),,, ( CIP) /. :, 1999 ISBN 7302037884.... 022 CIP ( 1999) 63321 : (, 100084) ht tp: / / w ww. tup. tsinghua. edu. cn : : : 7871092 1/ 16: 18. 5: 456 : 2000 1 1 2000 1 1 : ISBN 7302037884/
Microsoft Word - Z8I11A0-102.doc
自 然 考 科 學 測 試 題 關 鍵 解 析 前 言 物 理 科 今 年 自 然 科 考 題 共 68 題, 其 中 物 理 科 占 了 17 題, 今 年 的 題 目 難 度 屬 中 偏 易, 考 題 分 配 十 分 平 均 由 於 是 新 課 綱 第 一 次 學 測, 所 以 新 的 內 容 大 致 上 均 入 題, 大 都 為 物 理 科 普 知 識 就 算 是 第 二 部 分 的 考 題,