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Задать вопрос экспертам

Консультация № 178515

20.05.2010, 11:26

0.00 руб.

0 1 1

Образование

найти три первых отличных от нуля члена разложения в ряд маклорена функции 1/cos x

Обсуждение

Неизвестный

20.05.2010, 15:25

общий

это ответ

Здравствуйте, Лиса.

f(x)=1/cos(x)
Ряд Маклорена:

f(x)=[$8721$]_k=0^[$8734$] f^(k)(o)*x^k/k!

k=0
f(0)=1/cos(0)=1
k=1
f'(x)=(1/cos(x))'=sin(x)/cos²(x)
f'(0)=0
k=2
f''(x)=(sin(x)/cos²(x))'=(cos(x)*cos²(x)-sin(x)*2*cos(x)*(-sin(x)))/cos⁴(x)=1/cos(x)+2*sin²(x)/cos³(x)
f''(0)=1
k=3
f⁽³⁾(x)=(1/cos(x)+2*sin²(x)/cos³(x))'=sin(x)/cos²(x)+(4*sin(x)*cos⁴(x)+6*sin³(x)*cos²(x))/cos⁶(x)=5*sin(x)/cos²(x)+6*sin³(x)/cos⁴(x)
f⁽³⁾(0)=0
k=4
f⁽⁴⁾(x)=(5*sin(x)/cos²(x)+6*sin³(x)/cos⁴(x))'=24*sin⁴(x)/cos⁵(x)+28*sin²(x)/cos³(x)+5/cos(x)
f⁽⁴⁾(0)=5

Получим
f(x)=1+x²/2!+5*x⁴/4!+... или

f(x)=1+x²/2+5*x⁴/24+...

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