Question

How do you find the Taylor polynomial of degree n=4 for x near the point a=pi for the function cosx?

Answer 1

`-1+(x-pi)^2/2-(x-pi)^4/24`

Explanation:

`cos(pi) = -1`
1st Derivative: `-sin(x) " then " f'(pi)=0`
2nd Derivative: `-cos(x) " then "f''(pi)=-1`
3rd Derivative: `sin(x) " then " "f'''(pi)=0`
4th Derivative: `cos(x) " then " f''''(pi)=-1`

Odd terms except the first is zero then use only the even terms.
Putting it all together:

`-1+(x-pi)^2/(2!)-(x-pi)^4/(4!)...`