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Задать вопрос экспертам

Консультация № 182923

23.04.2011, 21:18

132.34 руб.

24.04.2011, 23:21

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Образование

Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
найти первые и вторые частные производные функции F и записать формулу Тейлора в указаной точке х: F=((x+y+2z)^4)+ln(x+y+z)+2^(y+x-2z), x=(1,1,1)

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

23.04.2011, 22:04

общий

это ответ

Здравствуйте, Ольга Никанова!
1) Производные:
F_x=4(x+y+2z)³+1/(x+y+z)+2^y+x-2zln2
F_y=4(x+y+2z)³+1/(x+y+z)+2^y+x-2zln2
F_z=8(x+y+2z)³+1/(x+y+z)-2*2^y+x-2zln2

F_xx=12(x+y+2z)²-1/((x+y+z)²)+2^y+x-2zln²2
F_yy=12(x+y+2z)²-1/((x+y+z)²)+2^y+x-2zln²2
F_zz=48(x+y+2z)²-1/((x+y+z)²)+4*2^y+x-2zln²2
F_xy=12(x+y+2z)²-1/((x+y+z)²)+2^y+x-2zln²2
F_xz=24(x+y+2z)²-1/((x+y+z)²)-2*2^y+x-2zln²2
F_yz=24(x+y+2z)²-1/((x+y+z)²)-2*2^y+x-2zln²2

В точке (1;1;1):
F=256+ln3+1=257+ln3

F_x=256+1/3+ln2=(769/3)+ln2
F_y=256+1/3+ln2=(769/3)+ln2
F_z=512+1/3-2*ln2=(1537/3)-2*ln2

F_xx=192-1/9+ln²2=(1727/9)+ln²2
F_yy=192-1/9+ln²2=(1727/9)+ln²2
F_zz=768-1/9+4*ln²2=(6911/9)+4*ln²2
F_xy=192-1/9+ln²2=(1727/9)+ln²2
F_xz=384-1/9-2*ln²2=(3455/9)-2*ln²2
F_yz=384-1/9-2*ln²2=(3455/9)-2*ln²2

2) Формула Тэйлора:
T=F(1,1,1)+dF(1,1,1)+(1/2)d²F(1,1,1)

F(1,1,1)=257+ln3

dF(1,1,1)=F_xdx+F_ydy+F_zdz=[(769/3)+ln2](x-1)+[(769/3)+ln2](y-1)+[(1537/3)-2*ln2](z-1)

d²F(1,1,1)=F_xx(dx)²+F_yy(dx)²+F_zz(dx)²+
+2*F_xydxdy+2*F_xzdxdz+2*F_zzdydz=[(1727/9)+ln²2](x-1)²+
+[(1727/9)+ln²2](y-1)²+[(6911/9)+4*ln²2](z-1)²+
+2[(1727/9)+ln²2](x-1)(y-1)+2[(3455/9)-2*ln²2](x-1)(z-1)+2[(3455/9)-2*ln²2](y-1)(z-1)

Неизвестный

23.04.2011, 22:27

общий

Простите,я напутала в выражении со знаком:F=((x+y+2z)^4)+ln(x+y-z)+2^(y+x-2z), x=(1,1,1). Но все поняла.Спасибо.

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