solved_examples_2 - Hisham_27511_C1 m 1 f(x x n x 0 x m m n...

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Unformatted text preview: ‫‪Hisham_27511_C1‬‬ ‫◄السؤال األول ►‬ ‫نذٌُا ‪َ m 1 ٔ f (x ) x n‬مطح يخرهفح يٍ ‪ x 0‬حرى ‪x m‬‬ ‫عُذيا ‪ٌ m n‬كٌٕ عُذَا ‪َ n 1‬مطح ٔحذٔدٌح االسرٍفاء ) ‪ f ( x‬يٍ انذسجح ‪ٔ n‬تانرانً‬ ‫] ‪ ًْ f [x 0 , x 1 ,....., x m‬أيثال انحذ ‪ x n‬أي أٌ ‪f [x 0 , x 1 ,....., x m ] 1‬‬ ‫ٔعُذيا ‪ٌ m n‬كٌٕ عذد َماط االسرٍفاء ٌفٕق دسجح حذٔدٌح االسرٍفاء تاثٍٍُ عهى األلم‬ ‫ٔتانرانً ] ‪ ًْ f [x 0 , x 1 ,....., x m‬أيثال انحذ ‪ x n 1‬ارا كاَد ‪ m n 1‬أٔ ًْ أيثال‬ ‫انحذ ‪ x n 2‬ارا كاَد ‪ْٔ.... m n 2‬كزا ٔتانرانً ‪ f [x 0 , x 1 ,....., x m ] 0‬فً جًٍع‬ ‫ْزِ انحاالخ‪.‬‬ ‫◄السؤل الثاني►‬ ‫‪S 0 (x ) 2x 3 : 0 x 1‬‬ ‫‪‬‬ ‫‪S (x ) S 1 (x ) x 3 3x 2 3x 1:1 x 2‬‬ ‫‪‬‬ ‫‪2‬‬ ‫‪S 2 (x ) 9x 15x 9 : 2 x 3‬‬ ‫َرحمك يٍ اسرًشاس ) ‪ S ( x‬عُذ َماط انعمذ ‪x 2 2 ٔ x 1 1‬‬ ‫)‪S 1 (2) 15, S 2 (2) 15 S 1 (2) S 2 (2) ٔ S 0 (1) 2, S 1 (1) 2 S 0 (1) S 1 (1‬‬ ‫فاالسرًشاس يحمك ‪.‬‬ ‫َحسة انًشرك األٔل ‪:‬‬ ‫‪S 0 (x ) 6x 2 :: 0 x 1‬‬ ‫‪‬‬ ‫‪S (x ) S 1(x ) 3x 2 6x 3::1 x 2‬‬ ‫‪S (x ) 18x 15 :: 2 x 3‬‬ ‫‪ 2‬‬ ‫َرحمك يٍ اسرًشاس ) ‪ S (x‬عُذ َماط انعمذ ‪x 2 2 ٔ x 1 1‬‬ Hisham_27511_C1 S 1(2) 21, S 2 (2) 21 S 1(2) S 2 (2) ٔ S 0 (1) 6, S 1(1) 6 S 0 (1) S 1(1) . ‫فاالسرًشاس يحمك أٌضًا‬ : ًَ‫َحسة انًشرك انثا‬ S 0(x ) 12x :: 0 x 1 S (x ) S 1(x ) 6x 6 ::1 x 2 S (x ) 18 :: 2 x 3 2 x 2 2 ٔ x 1 1 ‫ عُذ َماط انعمذ‬S ( x ) ‫َرحمك يٍ اسرًشاس‬ S 1(2) 18, S 2(2) 18 S 1(1) S 2(1) ٔ S 0(1) 12, S 1(1) 12 S 0(1) S 1(1) . ‫فاالسرًشاس أٌضاً يحمك‬ S 0(0) 0, S 2(3) 18 0 : ‫َخرثش انششط انحش‬ ‫ الٌحمك ششٌحح سٍثهٍ انركعٍثٍح انحشج‬S ( x ) ً‫فانششط انحش غٍش يحمك ٔتانران‬ ► ‫◄السؤال الثالث‬ S 0 (x ) (x 1)3 : 2 x 1 S (x ) S 1 (x ) ax 3 bx 2 cx d : 1 x 1 2 S 2 (x ) (x 1) ::1 x 2 S 0 (1) S 1 (1) a b c d 0 S 1 (1) S 2 (1) a b c d 0 (1) (2) : ‫َحسة انًشرك األٔل‬ S 0 (x ) 3(x 1) 2 :: 2 x 1 S (x ) S 1(x ) 3ax 2 2bx c :: 1 x 1 S (x ) 2(x 1) ::1 x 2 2 ‫‪Hisham_27511_C1‬‬ ‫)‪(3‬‬ ‫)‪(4‬‬ ‫‪S 0 (1) S 1 (1) 3a 2b c 0‬‬ ‫‪S 1(1) S 2 (1) 3a 2b c 0‬‬ ‫َحسة انًشرك انثاًَ ‪:‬‬ ‫‪S 0(x ) 6(x 1) :: 2 x 1‬‬ ‫‪‬‬ ‫‪S (x ) S 1(x ) 6ax 2b :: 1 x 1‬‬ ‫‪S (x ) 2 ::1 x 2‬‬ ‫‪ 2‬‬ ‫)‪(5‬‬ ‫)‪(6‬‬ ‫‪S 0(1) S 1 (1) 6a 2b 0‬‬ ‫‪S 1(1) S 2 (1) 6a 2b 2‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫تحم انًعادنرٍٍ )‪َ (6) ٔ (5‬جذ أٌ ‪ٔ a ‬‬ ‫‪6‬‬ ‫‪2‬‬ ‫‪ , b ‬عُذ انرعٌٕط تانًعادنرٍٍ )‪(4) ٔ (3‬‬ ‫َالحظ عذو ايكاٍَح اٌجاد أي لًٍح ل ‪ c‬ذحمك انًعادنرٍٍ يعًا ٔتانرانً انجًهح يسرحٍهح انحم‬ ‫ٔانراتع ) ‪ S ( x‬الٌشكم ششٌحح سٍثهٍ يًٓا ذكٍ لًٍح انثٕاتد ‪.‬‬ ‫◄السؤال الرابع►‬ ‫نذٌُا ) ‪ S ( x‬انششٌحح انركعٍثٍح انًمٍذج ‪:‬‬ ‫‪3‬‬ ‫‪S 0 (x ) 22 9(x 1) (x 1) ::1 x 2‬‬ ‫‪S (x ) ‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪S 1 (x ) A B (x 2) C (x 2) D (x 2) :: 2 x 3‬‬ ‫اٌ ) ‪ S ( x‬يسرًش عُذ َمطح انعمذج ‪ٔ x 1 2‬تانرانً ‪:‬‬ ‫‪S 0 (2) S 1 (2) A 12‬‬ ‫َحسة انًشرك األٔل ‪:‬‬ ‫‪Hisham_27511_C1‬‬ ‫‪S 0 (x ) 9 3(x 1) 2 ::1 x 2‬‬ ‫‪S (x ) ‬‬ ‫‪2‬‬ ‫‪S 1(x ) B 2C (x 2) 3D (x 2) :: 2 x 3‬‬ ‫اٌ ) ‪ S (x‬يسرًش عُذ َمطح انعمذج ‪ٔ x 1 2‬تانرانً ‪:‬‬ ‫‪S 0 (2) S 1(2) B 12‬‬ ‫َحسة انًشرك انثاًَ ‪:‬‬ ‫‪S 0(x ) 6(x 1) ::1 x 2‬‬ ‫‪S (x ) ‬‬ ‫‪S 1(x ) 2C 6D (x 2) :: 2 x 3‬‬ ‫اٌ ) ‪ S ( x‬يسرًش عُذ َمطح انعمذج ‪ٔ x 1 2‬تانرانً ‪:‬‬ ‫‪S 0(2) S 1(2) 6 2C C 3‬‬ ‫ٔنذٌُا انششط انى قٌذ يحمك أي ‪ٔ f (3) S (3) ٔ f (1) S (1) :‬نكٍ )‪f (1) f (3‬‬ ‫ٔتانرانً ‪:‬‬ ‫‪S (1) S (3) 9 12 6 3D 9 3D D 3‬‬ ‫◄السؤال الخامس ►‬ ‫‪2‬‬ ‫نذٌُا انركايم ‪dx‬‬ ‫‪2‬‬ ‫‪ sin x‬‬ ‫‪َٔ I ‬شٌذ حساب لًٍرّ تاسرخذاو طشٌمح سًٍسٌٕ انًشكثح‬ ‫‪0‬‬ ‫حٍث ‪ n 4‬أي َمطع انًجال انًطهٕب ألستع يجاالخ يرسأٌح انطٕل حٍث طٕل كم يُٓا ْٕ ‪:‬‬ ‫‪b a 2 0 ‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪4‬‬ ‫‪4‬‬ ‫‪2‬‬ ‫‪ٔ h ‬عُذَا تانرانً خًسح َماط ‪:‬‬ ‫‪x 0 a 0 f (x 0 ) f (0) 0‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ f (x 1 ) f ( ) 0.624266‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪x1 x 0 h ‬‬ Hisham_27511_C1 x 2 x 0 2h f (x 2 ) f ( ) 0.430301 x 3 x 0 3h 3 3 f (x 3 ) f ( ) 0.213798 2 2 x 4 b 2 f (x 4 ) f (2 ) 0.978340 : ٌَٕ‫َطثك انما‬ 2 I sin x 2dx 0 h f (x 0 ) 4f (x 1) 2f (x 2 ) 4f (x 3 ) f (x 4 ) 3 h 3 f (0) 4 f ( ) 2 f ( ) 4 f ( ) f (2 ) 3 2 2 0.523597 0 2.497064 0.860602 0.855192 0.978340 0.523597*1.759610 0.921326 ► ‫◄السؤال السادس‬ 3 ‫ تاسرخذاو طشٌمح شثّ انًُحشف‬n , h ٍٍ‫ َٔشٌذ اٌجاد انمًٍر‬I ln(2x 1)dx ‫نذٌُا انركايم‬ 1 E E h 2 (b a ) (b a )3 | f ( ) | | f ( ) | 12 12n 2 1 ‫انًشكثح عهًًا أٌ انخطأ‬ 1000 : ‫َطثك لإٌَ حساب انخطأ نٓزِ انطشٌمح‬ | f ( ) | max | f ( x ) | ٌٕ‫ٔاٌ أعظى لًٍح نهخطأ َحصم عهٍٓا عُذيا ٌك‬ : f (x ) ln(2x 1) ‫َحسة انًشرك انثاًَ نهراتع‬ f (x ) 2 4 f (x ) 2x 1 (2x 1)2 ‫‪Hisham_27511_C1‬‬ ‫‪4‬‬ ‫ٌٔكٌٕ‬ ‫‪(2x 1)2‬‬ ‫‪ ْٕٔ | f (x ) |‬ذاتع يرُالص ذًايًا عهى انًجال ]‪ٔ [1,3‬تانرانً ٌصم‬ ‫‪4‬‬ ‫ألكثش لًٍّ نّ عُذيا ذكٌٕ ‪ x 1‬أي‬ ‫‪9‬‬ ‫‪max | f (x ) |[1,3] f (1) ‬‬ ‫َعٕض فً انمإٌَ ‪:‬‬ ‫‪(b a )3‬‬ ‫‪8‬‬ ‫‪4‬‬ ‫‪8‬‬ ‫‪‬‬ ‫‪E‬‬ ‫‪max‬‬ ‫|‬ ‫‪f‬‬ ‫(‬ ‫‪x‬‬ ‫)‬ ‫|‬ ‫‪‬‬ ‫*‬ ‫‪‬‬ ‫‪12n 2‬‬ ‫‪12n 2 9 27n 2‬‬ ‫‪1‬‬ ‫‪8‬‬ ‫‪1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪ 27 n 2 8000 n 2 296.296 n 17.213‬‬ ‫‪2‬‬ ‫‪1000‬‬ ‫‪27 n‬‬ ‫‪1000‬‬ ‫‪b a 2 1‬‬ ‫‪ ‬‬ ‫أي ٌجة أٌ ذكٌٕ ‪ n 18‬أٔ أكثش ٔتانرانً‬ ‫‪n‬‬ ‫‪18 9‬‬ ‫‪ h ‬أٔ أصغش ‪.‬‬ ‫‪E‬‬ ...
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