Taylor/Maclaurin series

dangerous_dave

New member
Joined
Mar 13, 2008
Messages
21
I have quite a few questions relating to these. The first is:

find the tdegree 8 taylor polynomial p(x) of f(x) = sin x about c= pi/2

Estimate the error in p(2) to approximate sin (2).

I am having trouble finding anything that helps with this. And anything I did find I couldn't understand.

Please help.
 
general form for an nth degree Taylor polynomial for f(x) centered at x = a ...

f(x)P(x)=f(a)+f(a)(xa)+f(a)(xa)22!+f(a)(xa)33!+...+fn(a)(xa)nn!\displaystyle f(x) \approx P(x) = f(a) + f'(a)(x-a) + f''(a)\frac{(x-a)^2}{2!} + f'''(a) \frac{(x-a)^3}{3!} + ... + f^n(a) \frac{(x-a)^n}{n!}

the "a" value is pi/2.

f(x)=sin(x)\displaystyle f(x) = \sin(x) ... f(π2)=1\displaystyle f\left(\frac{\pi}{2}\right) = 1

f(x)=cos(x)\displaystyle f'(x) = \cos(x) ... f(π2)=0\displaystyle f'\left(\frac{\pi}{2}\right) = 0

f(x)=sin(x)\displaystyle f''(x) = -\sin(x) ... f(π2)=1\displaystyle f''\left(\frac{\pi}{2}\right) = -1

f(x)=cos(x)\displaystyle f'''(x) = -\cos(x) ... f(π2)=0\displaystyle f'''\left(\frac{\pi}{2}\right) = 0

see a pattern here? so what would the 8th degree Taylor polynomial for f(x) = sin(x) be?

note that the terms of the Taylor polynomial are alternating in sign ... what do you know about the error bound for alternating series?
 
Top