x 0, x 1,...,x n f(x) p n (x) = f[x 0, x 1,..., x n, x]w n (x),
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- Constance King
- 6 years ago
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1 ÛÜØ Þ ÜÒ Ô ÚÜ Ô Ü Ñ Ü Ô Ð Ñ Ü ÜØ º½ ÞÜ Ò f Ø ÚÜ ÚÛÔ Ø Ü Ö ºÞ ÜÒ Ô ÚÜ Ô Ð Ü Ð Þ Õ Ô ÞØÔ ÛÜØ Ü ÚÛÔ Ø Ü Ö L(f) = f(x)dx ÚÜ Ô Ü ÜØ Þ Ü Ô, b] Ö Û Þ Ü Ô Ñ ÒÖØ k Ü f Ñ Df(x) = f (x) ÐÖ D Ü Ü ÜØ Þ Ü Ô Ñ Ü ÜØ Ñ Ð Ñ Ü ÜØ ºD k (f) = D(D k f) ÐÖ k ÜÕÒ Ü Ü ÜØ ºL(αf + βg) = αl(f) + βl(g) Ñ Ò ÛÒ Ñ ÜÒ Ð Ñ Ü Ô Ð L Ü Ô Ð Ü ÜØ Ñ ºÑ Ô Ð Ø p, b] Ö Û ÞÜ Ò ÚÛÔ Ø f Þ e(x) = f(x) p(x) ÐÖ Ü Ò e ÜÝ L(f) L(p) = L(e) p ÐÖ f ÐÖ Ü Ò p(x) = n j=0 f(x j)l j (x) Ñ Ò ºp Ñ Ô Ð Ø ÐÖ f ÚÛÔ Ø Ü Û Ý j = 0,,..., n L(l j ) = w j Ñ x 0, x,...,x n Þ ÛÔ f ÐÝ ÚÐ ØÜ Ô Ñ Ô Ð Ø Ñ ºL(p) Ý ÐÖ L(f) Þ ÜÖ ÐÝ Ý E(f) = L(f) L(p) = L(e) ºL(f) L(p) Ó Û E(f) Þ ÜÒ Ô Ü º¾ Þ ÛÔ f ÐÝ ÚÐ ØÜ Ô Ñ Ô Ð Ø P n, b] Ö Û Ü f Þ x 0, x,...,x n ³ ÛÜØ µ º½ Õ Ô Ø ÐÖ º, b] Ö ÛÐ Þ Ý µ f(x) p n (x) = fx 0, x,..., x n, x]w n (x), Ó ºw n (x) = n j=0 (x x j) ÜÝ d fx fx dx 0,..., x n, x] = lim 0,...,x n,x+h] fx 0,...,x n,x] h 0 h = lim h 0 fx 0,...,x n, x, x + h] = fx 0,...,x n, x, x], ½
2 ¾ Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ Ó Ð º½µ f (x) p n (x) = fx 0,...,x n, x, x]w n (x) + fx 0, x,...,x n, x]w n (x). Ð ÛÔ ³ ÛÜØ ÐÝ º ØÝÒ ØÐ E(f) = Df Dp n = f (x) P n (x) = f(n+) (ξ) (n + )! w n(x)+ f(n+) (η) (n + )! w n (x), ξ, η (, b). º¾µ Ñ Ü Ñ Ó Ø Ð Ð ÚÐ ØÜ Ô Þ ÛÔÒ Þ ÞÜ Ô Þ Ý Ð Ñ Ú Ü Ñ ÞÜ Ô Þ Ñ Ý Ò ÛÔ Ó Ò Þ Ý ÚÐ ØÜ Ô Þ ÛÔ Þ Ð ÔÓ ÔÝ Ü Ò Þ ÛÜ Ð Ò µ º½ ÐÝ Ó Ò Ó Ð ÝÐ k Ü Ö w n (x) = w n (x k ) µ º¾ Þ Ð Ð º µ E(f) = f(n+) (η) (n + )! w n(x k ) = f(n+) (η) (n + )! n j=0, j k (x k x j ). Ó Ò ÜÛÒ ºw n(x) = 0 ÛÔ ÞÜ Ô Þ Ý Ð ÞÜ Ô Þ ÜÝØ µ º¾ Þ Ð Ð Ð ÔÓ Ó Ý Ü Ü Ò Þ ÛÜ Ð Ò µ º½ ÐÝ º µ E(f) = f(n+) (ξ) (n + )! w n(x) = f(n+) (ξ) (n + )! n (x x j ). j=0 º Ý Þ ÜÖÔ n ÐÝ Ñ ÜÖ Ò Ü Ö D(f) ¹Ð Ü Û D(p n ) Þ Ý Ô ºD(f) ¹Ð ÜÖ Ð D(p 0 ) = 0 Ó Ò Ñ Ð ÛÒ n = 0 Ü Ö p (x) = f(x 0 )+fx 0, x ](x x 0 ) ÞÒ ºD(f) ¹Ð Þ Ô Ý Þ ÜÖ ÞÝ Ð ÛÔ n = Ü Ö x 0 ÛÔ ÞÜ Ô Þ ÜÖ ºD(p ) = fx 0, x ] Ð ÛÔ º µ f (x 0 ) fx 0, x 0 + h] = f(x 0 + h) f(x 0 ). h µ º Ø ÐÖ Ý Þ ÜÖ º µ E(f) = hf (η). ¹Ü Ô Þ ÛÔ x = x + h x 0 = x h Þ ÛÔ ÞÜ ÐÖ Þ ÝÖÔ ÞÜ ÜÖ ÜÛÒ º ÚÐ Ø º µ f (x) fx h, x + h] = f(x + h) f(x h). h µ º Ø ÐÖ Ý Þ ÜÖ º µ E(f) = 6 h f (ξ).
3 Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ ÞÜ ÔÐ Ý fx 0, x ] Lgrnge ØÝÒ Ø ÐÖ Þ Ö ÞØÒ ÓÔ n = Ü Ö Þ Ú Þ Ö Û Þ ÛÔ Ð Ð Ü Û ÛÐ Ò ÝÜØ Ó Ð ºx Ó Ð x 0 Ó ÞÒ ÕÒ ÛÔ º ÝÐ ÜÖ Ñ ÔÐ Û Ó ºÓ Û x x 0 ÝÜØ Ý ÔÞ ÞÒ Þ Þ ÜÖ Þ Ð ÛÔ n = Ü Ö p (x) = f(x 0 ) + fx 0, x ](x x 0 ) + fx 0, x, x ](x x 0 )(x x ), x = x 0 +h x = x 0 +h Ü Ô Ñ ºp (x) = fx 0, x ]+fx 0, x, x ](x x 0 x ) Ð ÛÔ º µ f (x 0 ) 3f(x 0) + 4f(x 0 + h) f(x 0 + h). h º½¼µ E(f) = 3 h f (η). º½½µ f (x) f(x 0 + h) f(x 0 h). h Ð ÛÔ Ý Þ ÜÖ Ð ÛÔ x = x 0 + h x = x 0 h Ü Ô Ñ º½¾µ E(f) = 6 h f (η). ÐÖÒÒ ÚÐ ØÜ Ô Ò Ô Ð Ø ÞÜ Ö Þ ÜÒ Ô Ü Ð Þ ØÕ Ô Þ Õ Ô Ð ÛÐ ÜÝØ º½ µ p n (x) = n j=0 fx 0, x,...,x j ]w j (x), w (x), w j (x) = (x x j )w j (x), j. ÜÞ º½ µ e n (x) = fx 0, x,...,x n, x]w n (x). Ý Ü Ü Ð Ô ÐÖ ÑÝÐ ºÜÞ ÜÕÒ Þ Ü Ô Ñ Ü Û Ý Ð ÞÜÝØ Ò µ º½ Õ Ô Ô Ü ºÑ ÒÖØ ÜØÕÒ µ º½ Õ Ô Ý ÛÐ Ò ÝÜØ Þ d dx fx 0,...,x n, x] = fx 0,...,x n, x, x]. Ñ ÛÞÒ k > Ü Ö Ô º½ µ d k dx kfx 0, x,...,x n, x] = k!fx 0, x,...,x n, x,...,x]. }{{} k ºÑ ÒÖØ k Þ ØÐ Þ Ø ÚÜ Ü f ÚÛÔ Ø Ý ÔÞ
4 Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ k ÐÖ ÚÛ Ô d k dx kfx 0,...,x n, x] = d ( ) d k dx dx k fx 0,...,x n, x] = d dx ((k )!fx 0,...,x n, x,...,x]. }{{} k ÓÒÕÔ ºy () = = y (k) = x ÜÝ fx 0,...,x n, y (),...,y (k) ] ÐÖ Þ ÓÒÕÔ Ð ÜÝ Ñ ÔÞÝÒ k ÐÝ ÚÛÔ Ø ρ ρ(y (),...,y (k) ) = fx 0,..., x n, y (),...,y (k) ] Ð ÛÔ ÞÜÝÜÝ ÐÐ ÝÒÞÝÔ º(y (j) = x) x ÐÝ ÚÛÔ Ø Ñ Ò d dx ρ(y(),...,y (k) ) = k dρ dy (j) dy (j) dx = k dρ dy = kfx 0,...,x (j) n, x, x,...,x], }{{} º ÔÖ Þ Ô k º½ µ f (k) (x) n j=0 fx 0, x,..., x j ] dj dx j w j (x). Ð ÛÔ µ º½ ¹ µ º½ Þ Õ ÔÒ º½ µ E(f) = dk dx k (fx 0,..., x n, x]w n (x)). Ý f ÐÝ k ÜÕÒ ÞÜ ÔÐ Ü Ü Û º½ µ f (k) (x) k!fx 0, x,...,x k ]. Ü Ñ Ü Û ºx 0, x,..., x k Þ ÛÔ Þ Ð Ò Ð Ò Ô Ò Ö ÛÐ Ý x ÜÝ µ º½ Õ Ô ÐÝ ³ ÛÜØ º ØÝÒ ÐÝ Ú Þ µ º½ Õ Ô º Ó Û Ö Û ÐÝ ºn = k Ü Ö Þ ÜÒ Ô ÚÜ Ô º Þ ÛÔ f ÐÝ ÚÐ ØÜ Ô Ñ Ô Ð Ø p n, b] Ö Û Ø ÚÜ ÚÛÔ Ø f Þ º½ µ I(f) = ³ ÛÜØ µ º½ Õ Ô Ø ÐÖ º, b] Ö ÛÐ Þ Ý µ x 0, x,...,x n f(x) p n (x) = fx 0,...,x n, x]w n (x), f(x)dx = p n (x)dx + Ó Ð ºw n (x) = n j= (x x j) ÜÝ fx 0, x,...,x n, x]w n (x)dx
5 Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ º¾¼µ E(f) = I(f) I(p n ) = fx 0, x,...,x n, x]w n (x)dx ØÐ Ö Û ÓÒ Õ Ý w n ÚÛÔ ØÐ, b] Ö Û Ñ º Õ Ô ÝØÐ ÓÞ Ô Ñ Ü Ñ ÜÛÒ º¾½µ E(f) = fx 0, x,...,x n, η] ÐÜ Ô Ó Ý ÐÝ Ñ Ô ÜÖ ØÝÒ w n (x)dx Þ Ø ÚÜ Þ Ü Ô n+ Ý f ÚÛÔ ØÐ Ð Õ Ô Ñ ºη (, b) ÞÒ ÕÒ ÛÔ Ü Ö º¾¾µ E(f) = (n + )! f(n+) (ξ) Ü ÖÝ ξ (, b) ÛÔ ÞÒ Û w n (x)dx. µ º¾¼ Þ ÝØÐ Ð Ô w n(x)dx = 0 Ð, b] Ö Û Ö Û ÓÒ Õ Ó w n ÚÛÔ ØÐ Ñ fx 0,...,x n, x]fx 0,..., x n, x n+ ] + fx 0,...,x n, x n+, x](x x n+ ). ÓØ E(f) = fx 0, x,...,x n+ ]w n (x)dx + º¾ µ E(f) = Ð ÛÔ µ º¾¾ Õ Ô Þ ÚÔ fx 0, x,...,x n+ ](x x n+ )w n (x)dx. Ó Ð ÔÞ Ô Ø ÐÖ ÕØ Ð Ý Ó Ý Ü ÐÜ Ô fx 0, x,...,x n+, x]w n+ (x)dx., b] Ö Û Ö Û ÓÒ Õ Ý w n+ ÚÛÔ ØÐÝ Ü Ð ÓÞ Ô x n+ ÛÔ Þ Ñ º¾ µ E(f) = fx 0, x,...,x n+, η] w n+ (x)dx Ñ ÝÜÐ Ð Ô Þ Ø ÚÜ Þ Ü Ô n + Ý f ÚÛÔ ØÐ Ð Õ Ô Ñ ºη (, b) ÞÒ ÕÒ ÛÔ Ü Ö º¾ µ E(f) = (n + )! f(n+) (ξ) Ü ÖÝ ξ (, b) ÛÔ ÞÒ Û w n+ (x)dx. Þ Ý Þ ÜÖÔ Þ Ü ÛÒ ÚÜ Ô Þ Õ Ô Ð ÛÐ ÞÔÒ ÐÖ µ º½ Õ Ô ÝÒÞÝÔ ºµ º¾ ¹ µ º¾¾ Þ Õ Ô Þ ÞÜ Ö Ü Û
6 Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ ÕØ ÐÖÒÒ ÚÐ ØÜ Ô Ñ Ô Ð Ø ÞÜ Ö ÚÜ Ô Þ Õ Ô º º½ f(x) = f(x 0 ) + fx 0, x](x x 0 ), I(p 0 ) f(x 0 )dx = f(x 0 )(b ). º¾ µ º¾ µ E(f) = f (ξ) Ð ÛÔ x 0 = Ñ I(f) f()(b ). µ º¾¾ Õ Ô ÝÒÞÝÔ º, b] Ö Û w 0 (x) = x 0 (x )dx = f (ξ)(b ). ºÑ Ô ÐÒ Þ Ý Þ ÜÛÔ ÚÜ Ô Þ Ý (I) Ð ÛÔ x 0 = +b Ñ ( ) + b º¾ µ I(f) f (b ). w (x) = ( ) x +b b Ð ÛÔ x = x 0 Ü Ô º w 0(x)dx = ( ) x +b = 0 Ó µ º¾ Õ Ô ÝÒÞÝÔ º, b] Ö Û 0 (II) º¾ µ E(f) = 4 f (ξ)(b ) 3. ºÖÚÒ Þ ÛÔ Þ Ý Þ ÜÛÔ ÚÜ Ô Þ Ý Ô Ý Ü ÐÖÒÒ ÚÐ ØÜ Ô Ñ Ô Ð Ø ÞÜ Ö ÚÜ Ô Þ Õ Ô º º¾ f(x) = f(x 0 ) + fx 0, x ](x x 0 ) + fx 0, x, x]w (x), I(p 0 ) = (f(x 0) + fx 0, x ](x x 0 ))dx = f(x 0 )(b ) + fx 0, x ](b )(b + x 0 ). Ð ÛÔ x = b ¹ x 0 = Ñ (III) º ¼µ I(f) (b )(f() + f(b)). µ º¾¾ Õ Ô ÝÒÞÝÔ º, b] Ö Û w (x) = (x )(x b) 0 º ½µ E(f) = f (ξ)(b ) 3. º ØÜ ÐÐ Þ ÜÛÔ ÚÜ Ô Þ Ý
7 Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ ÔÝ ÐÖÒÒ ÚÐ ØÜ Ô Ñ Ô Ð Ø ÞÜ Ö ÚÜ Ô Þ Õ Ô º º f(x) = f(x 0 ) + fx 0, x ]w 0 (x) + fx 0, x, x ]w (x) + fx 0, x, x, x]w (x), I(p 0 ) = (f(x 0) + fx 0, x ](x x 0 ) + fx 0, x, x ](x x 0 )(x x ))dx = f(x 0 )(b ) ( + fx 0, x ](b )(b + x 0 ) ) +fx 0, x, x ] (b x0 ) (b x 6 + x 0 ) ( x 0) ( x 6 + x 0 ). Ü Ô x = x Ü Ô º w (x) = 0 Ð ÛÔ x = b ¹ x = +b º ¾µ I(f) (b ) f() + f ( +b ( ( = b 6 f() + 4f +b ) ) + f(b), º µ E(f) = w 3 (x)(x x 3 )w (x) = (x ) ( x +b ) f() + 6 ( f(b) f ( +b x 0 = Ñ ) (IV ) (x b) ) + f(b) )] fx 0, x, x, x 3, x]w 3 (x)dx = 90 f(4) (ξ)(b ) 5. º(Simpson s rule) Ó ÕØÒ Õ ÐÐ Þ ÜÛÔ Þ ÜÛÔ ÚÜ Ô Þ Ý Þ Ý ÐÝ ÐÖÒÒ ÚÐ ØÜ Ô Ñ Ô Ð Ø ÞÜ Ö ÚÜ Ô Þ Õ Ô º º f(x) = f(x 0 ) + fx 0, x ]w 0 (x) + fx 0, x, x ]w (x) + fx 0, x, x, x 3 ]w (x) +fx 0, x, x, x 3, x]w 3 (x), ÚÛÔ Ø ºw 3 (x) = (x ) (x b) Ð ÛÔ x = x 3 = b ¹ x 0 = x = Ñ (V ) Ý Ý Ð µ º¾¾ Õ Ô ÝÒÞÝÔ Ó Ð, b] Ö Û Þ º µ I(f) b (b ) (f() + f(b)) + (f () f (b)), º µ E(f) = 70 f(4) (ξ)(b ) 5. Ñ Ø Ð Ò Ý ØÜ ÐÐ Ð Ô ºÓÛ ÞÒ ØÜ ÐÐ Þ ÜÛÔ ÚÜ Ô Þ Ý ÜÝ x = b x = y = 0 Ñ ÜÝ ÐÖ Ü Ò ØÜ Ý ÐÜ Ô Þ Ü ÜÝ Ó Ý Ý Ñ Ý ÞÒ Ó (b, f(b)) (, f()) Þ ÛÔ Ü Ü Ö º ÚÛÔ Ø Þ ÝÐ Þ ÜÖ ÔÐ Û (I) (V ) Ñ ÜÛÒ Ü Ö Ô Þ ØÝ ÚÜ Ô Þ Õ Ô Ü Ö k ÜÖÒ ÜÝ (b ) k ÑÜ Þ Ñ ÐÐ Ð Ñ º Ð Ö Þ Ý ÐРе Þ ÞÒ ÖÐ ºÐ b Ö Û Ü ÜÝ Û Ð ÛÔ Ð Ó Ð º¹Ò Ð ÑÐÝ ÜØÕÒ Ñ Ý Ü k > ¹Ý Ó Ò Ó ÐÖ ÜÞ º Ô Û Ý Ñ Ó Û Ü ÐÖ Ñ Ö Û Ü Ö
8 Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ Þ Ý Ñ Õ Ñ Ý Ü Ó Û Ñ Ò Ð Ü Ý Ñ Ö Û n ¹Ð, b] Ö Û Þ ÛÐ Ô º Ó Û Ñ Ö Û n Ð Ö Û ÞÛ Ð ÐÖ (I) (V ) Ñ ÜÛÒ ÐÖ Ñ ÕÕ Ò ÚÜ Ô Þ Õ Ô ÞÖ ÞØÔ j = x j = + jh Û Ð Þ ÛÔ ºh = b Ý Ü ÐÖ Ñ Ö Û N ¹Ð, b] N º0,,...,N Ñ Ô ÐÒ Þ Ý (I) º º xj º µ µ º¾ ¹ µ º¾ Þ Õ Ô Ø ÐÖ Ñ Ð ÛÒ x j, x j ] Ö Û Ð x j f(x)dx = (x j x j )d(x j ) + f (ξ(x j x j ) = hf(x j ) + f (ξ)h. I(f) h Ñ ÝÜÐ Ð Ô N f(x j ), ÞÜ Ô Ô Ô Ý Ó Ò Ð N f(x)dx = N xj x j f(x)dx ¹Ý Ó Ò E(f) = h f (ξ)(b ). f (ξ j )h µ º¾ ¹Ò ÞÐ ÛÞÒ Ý Þ Õ Ô ºµ º Þ Ñ ÛÒ ξ (, b) Ñ Û Ø ÚÜ ÖÚÒ Þ ÛÔ Þ Ý (II) º º º µ I(f) h N ( xj + x j f ), E(f) = b 4 f (ξ)h. Ð ÛÔ µ º¾ ¹ µ º¾ Þ Õ ÔÒ Ñ ØÜ Þ Õ Ô (III) º º Ð ÛÔ µ º ½ ¹ µ º ¼ Þ Õ ÔÒ º µ I(f) h N f(x j ) + f(x j ) = h(f(x 0) + f(x N )) + h N f(x j), E(f) = b f (ξ)h. Ó ÕØÒ Õ Þ Õ Ô (IV ) º º º µ Ð ÛÔ µ º ¹ µ º ¾ Þ Õ ÔÒ I(f) h f() + f(b) + N 6 f(x j) + 4 N f ( x j +x j ) ], E(f) = b 80 f(4) (ξ) h4. 4
9 Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ ÞÔÛ ÞÒ Ñ ØÜ Þ Õ Ô (V ) º º Ð ÛÔ µ º ¹ µ º Þ Õ ÔÒ º ¼µ I(f) h(f() + f(b)) + h N f(x j) + h(f () f (b)), E(f) = b 70 f(4) (ξ)h 4. Ñ ÝÐ ÝÒ Ñ Ð Ø Ñ ÐÜ Ô Ü Û º ÐÜ Ô ÐÝ Ü Û Ò Ü Ý ÓØ Ý Ò Ý ÐÖ ÓÞ Ô Ñ ÝÐ ÝÒ Ñ Ð Ø Ñ ÐÜ Ô Ü Û Ü ÛÐ Ñ Ò Û Ñ Ø Ö Ô ÜÝ Þ Ý Þ Ð Ð Ô Ô Þ Û Ò ÜÞ Ñ ÐÒ Ò Ò Ð Ø ÐÜ Ô Ð Ü Û ÞÞÐ Ó Ô Ô Ý Þ Ö Þ ºÑ ÝÐ ÝÒ Ñ Ð Ø Ñ ÐÜ Ô R = {(x, y) x b, c y b} ÜÒ Ð Ü Ý Ò ÝÐ Ó ÐÒ R ÜÝ f(x, y)dxdy Þ Ý Ñ Ó ÕØÒ Õ Û Ü ÖÔ Ð Ø ÐÜ Ô Ü ÛÐ ºÑ Ö Û < b c < d Ü Ö Ñ ÖÚ ÐÝ Ñ Ñ ÒÐÝ m n Ü Ú Newton-cotes R º ½µ º ¾µ R { h = b n k = d c m Ü Ú Ò ÞÜÝÜÝ ÐÜ Ô Ð Ø ÐÜ Ô Þ Þ Ô f(x, y)dx = ( d f(x, y)dy c ) dx d f(x, y)dy Ý ÜÒ Ð µ º ¾ ¹ Ò ÔØ ÐÜ Ô Ý Ð Ó ÕØÒ Õ Û ÝÒÞÝÔ c j = 0,,,..., m Ð Ü Ö y j = c + jk ºÖ Û x ÜÝ d c = k 3 f(x, y)dy f(x, y 0 ) + m f(x, y j) + 4 m f(x, y j ) + f(x, y m ) ] (d c)k f(x,µ) y 4, Ó Ð (c, d) Ö Û µ ÛÔ Ý Ü Ö º µ d c = k 3 f(x, y)dydx + k 3 f(x, y 0 )dx + k 3 f(x, y m )dx m (d c)k4 80 f(x, y j )dx + 4k 3 4 f(x, µ) y 4 m f(x, y j )dx
10 ½¼ Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ x i = +ih µ º ÝÒ ÐÜ Ô Ð Ð Ó ÕØÒ Õ Û ÐÝ Ü Û Þ Ý Ð ÖØÔ Ý Ö 0 i n ÜÝ f(x, y j )dx f(x 0, y j ) + n f(x i, y j ) + 4 n = h 3 i= i= ] f(x i, y j ) + f(x n, y j ) b 80 h4 4 f(ξ,y j ) x 4 ÐÖ Ó ÞÔ Ð Ø ÐÜ Ô ÐÝ Ü Û Õ ξ (, b) ÛÔ Ý Ü Ö d º µ º µ c f(x, y)dydx f(x 0, y 0 ) + n f(x i, y 0 ) + 4 n = hk 9 i= i= +4 m f(x 0, y j ) + f(x 0, y m ) + 4 m f(x i, y 0 ) + f(x n, y 0 ) + m n i= f(x i, y j ) + 8 m i= f(x 0, y j ) n f(x i, y j ) +8 m n f(x i, y j ) + 6 m n f(x i, y j ) + m f(x n, y j ) i= i= ] +4 m f(x n, y j ) + m f(x i, y m ) + 4 n f(x i, y m ) + f(x n, y m ) + E, k(b )h f(ξ 0,y 0 ) x 4 i= + m k(b )h4 E = f(ξ j,y j ) x m (d c)(b ) º µ E = h 4 4 f 80 Þ ÜÛÐ Ñ Ô Ô ÖÒ Ô º I i=0 j=0 i= ÐÖ Ô ÞÔ E Ý ÜÝ ] 4 f(ξ j,y j ) + 4 f(ξ m,y m ) x 4 x 4 (d c)k f(x, µ) dx y 4 Ò Þ Ð ÞÔÞ Ô Ý R Ñ Þ Þ Ø ÚÜ 4 f 4 f y 4 ] m 4 f (d c)(b )k4 (η, µ) x x (η, µ) + f 4 k4 4 y 4 f (η, µ) y4 ] ( η, µ) 4 x 4 Ñ k = d c m ÚÔ ºR Ñ Þ η, µ, η, µ Þ ÛÔ Ó Ý Ü Ö ln(x + y)dydx Ð Ø ÐÜ Ô ÔÐ Ó ÞÔ º½ Ò w ij ln(x i +y j ) = k = 0.5 h = 0.5 m = n = ÔÐ Ó ÞÔ ÜÝ º
11 ½½ Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ f x 4 (x, y) = 6 3 f = f = (x+y) 4 x 3 (x+y) 3 x f = (x+y) x x+y Ó Ð f = ln(x+y) Ó ÞÔ ÞÒ ÛÒ E Ý Ó Ð f = y 4 ] E (y y 0 )(x 4 x 0 ) h 4 4 f (η, µ) 80 x + k 4 4 f ( η, µ) 4 x mx 80 (x,y) R mx (x+y) 4 (x,y) R ] mx (x+y) 4 = ] (x 0 +y ) 4 (x+y) 4 ] 96 (x+y) 4 º Þ ÜØÕ 5 ÞÐÖ Õ ÔÒ ÑÖ ØÚ ÛÔ I Ó Ð Ñ ÐÜ Ô Ü ÛÐ Þ Ð Ð Ð ÜÝØ Ñ Ð Ø Ñ ÐÜ Ô Ü ÛÐ Ô ÜÝ Ý ÐÝ Ð ÐÜ Ô ÞÜ ÐÖ ÝÖÔ ºÑ ÝÐ ÝÒ Ñ ÐÜ Ô ÜØ Ñ ÜÒ ÐÜ Ô Þ ÜÖ ÐÝÒÐ ÝÖÔÝ Ý ÐÝ Þ ÑÚÒÚÐ ºÓ ÕØÒ Õ Þ Õ Ô Ý Ò Ý Þ Ñ ÔÝ ÔØÐ ºI = ºÕ Þ ÝÒ Ò Þ ÐÖ ÜÞ Þ Ý ÝÒÞÝ Ð ÜÝØ Ð Ø ln(x + y)dydx Ð Ø ÐÜ Ô ÔÐ Ó ÞÔ º¾ Ò R = {(x, y).4 x ÚÜ Ô Ñ Þ Þ ÔÝÔ I Ð Ø ÐÜ Ô Ü ÛÐ Õ Þ Ý Þ Ý Ö Þ Ü Ô Ð ºR = {(u, v) u, v } ¹Ð {.0,.0 y.5} u = (x.4.0) Ý µ ÚÜ Ô ÔÞÝÒ Ô Ý º.0.4 v = ÐÖ ÞÜ ÒÝ Ö (y.0.5).5.0 I = ln(0.3u + 0.5v + u)dvdu. Ð ÛÔ Ó ÛÖ ÝÒÞÝ Ð I Ý Ð ÛÔ v ¹Ð Ñ u ¹Ð n = 3 Ü Ö Õ ÐÝ Þ Ô Þ ÝÒÞÝ Ð ÓÞ Ô Ý Ö ºÞ ÜØÕ 8 ÞÐÖ Õ ÔÒ ÐÝ ØÚ Þ ÛÔ I ÔÐ Ò Ý Ó Ð Ø ÐÜ Ô Þ ÜÖ Ð Ñ Ô Ô ÖÒ Ô ÐÐ ÜÞ ÜÛÒ ÓÔ ÞÔ Ý Ö º µ º µ d(x) c(x) d (y) c (y) f(x, y)dxdy f(x, y)dxdy. Þ Þ Ó ÕØÒ Õ ÐÐ ÑÖØ Ö ÝÒÞÝÔ Ô µ º ÕÒ Ñ Ð Ø Ñ ÐÜ Ô Þ ÜÖ Ð Ò x Ü Ú ÐÖ Ñ ÖÚ Ð Þ Ö ÛÔ Ü Ú ÝÖÔ ºÑ ÔÞÝÒ Þ ÞÖ Û
12 ½¾ Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ d(x) = h 3 { c(x) f(x, y)dydx k(x) 3 f(x, c(x)) + 4f(x, c(x) + k(x)) + f(x, d(x))]dx k() (f(, c()) + 4f(, c() + k()) + f(, d())) 3 h = b + 4k(+b) 3 f( + h, c( + h)) + 4f( + b, c( + h) + k( + h)) + f( + h, d( + h))] + k(b) 3 f(b, c(b)) + 4f(b, c(b) + k(b)) + f(b, d(b))] }. h = b n J = 0 J = 0 ºµ º ÕÒ Ð Ø ÐÜ Ô ÐÝ Þ ÜÖ Ð ÓÞ Ô Ò Ü Ú ºI = d(x) c(x) (Ó Ý Ü Ü ÐÝ ÑÛÒ ). ( Ü ÐÝ ÑÛÒ ). ( ¹ Ü ÐÝ ÑÛÒ ). J 3 = 0 For i = 0,,,..., n, do x = + ih Hx = d(x) c(x) m k = f(x, c(x)) + f(x, d(x)), k = 0 f(x, y)dydx Ü ÛÐ ÑÞÜ Ð º½ ÑÞ Ü Ð n, m Ñ Ñ ÒÐÝ Ñ ÜØÕÒ, b ÚÛ Þ ÛÔ ÐÛ ºI ÜÖÐ ÜÛ J ÜØÕÒ ÐØ (Ñ Þ Ò ÛÒ Þ ÛÔ ÐÝ Ñ Õ Ü Ö) (Ñ ¹ Þ Ò ÛÒ Þ ÛÔ ÐÝ Ñ Õ Ü Ö) k 3 = 0 For j =,,..., m, do y = c(x) + j Hx z = f(x, y) If j mod = 0, k + z k else k 3 + z k 3 L = (k + k + 4k 3 ) Hx 3 ( ) d(xi ) f(x i, y)dy Ý Ò L : ÜÖ c(x i ) If i = 0 or i = n, J + L J elseif i mod = 0, J + L J, else J 3 + L J 3 J = (J + J + 4J 3 ) h 3 print J. Stop.
13 ½ Þ ÜÒ Ô ÚÜ Ô Ü º ÛÜØ Ñ Þ ÐÖ f(x, y) = e y x ÚÛÔ Ø Ü Ö ÑÞ Ü Ð ÑÝ Ô Ñ º Ò Þ Ò ÐÜ Ô ÜÖ º R = {(x, y) 0. x 0.5, x 3 y x }, 0.5 x 0. x 3 e y x dydx Ð ÛÔ n = m = 5 ÜÝ º Þ ÜØÕ 7 ÞÐÖ Õ ÔÒ ÐÝ ØÚ Þ ÛÔ Ñ ÜÒ Ñ Ð Ü Ô Þ ÜÖ ÐÖ Ñ Ð ÜÞ º Ó ÞÔ ÜÝ ÛÜØ ÖÚ Ý ÑÞÜ Ð Ø Ö Ñ Ñ Ð Ø Ñ Ð Ü Ô Þ ÜÖ º½ Ð ÜÞ Ý Þ ÜÖ n = m = xy dydx º½. x.0 x (x + y )dydx º¾ º e y x dxdy º 0 0 n = Ó ÞÔ ÜÝ ÛÜØ ÖÚ Ý ÑÞÜ Ð Ø Ö Ð Ø Ð Ü Ô Þ ÜÖ º¾ Ð ÜÞ Ý Þ ÜÖ 3, m = 5 π y cos( x)dydx. 0 0 ÝÐ ÝÒ Ð Ü Ô Þ ÜÖ Ð Ó ÕÒ Õ ÐÐ ÝÒÞÝ e xyz dxdydz, º Ð ÜÞ ºÑ Ý Ñ ÛÐ Ý ÐÝÐ Ñ Ô Ý Ñ Ü Ú ÐÖ Ñ Ö Û Þ ÛÐ ÜÝ Ý Ð ÝÒ Ð Ü Ô Þ ÜÖ Ð Ð Ø Ð Ü Ô Þ ÜÖ ÐÝ ÑÞÜ Ð Ò ÑÞÜ Ð Ô º Ð ÜÞ d(x) f(x,y) c(x) e(x,y) H(x, y, z)dzdydx. Ü Ú Ò
u x + u y = x u . u(x, 0) = e x2 The characteristics satisfy dx dt = 1, dy dt = 1
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