Question

How do you determine a taylor series approximation for `f(x)=cos(x)` where `n=4` and `a=3`?

Answer 1

Hello,

The formula is

`f(x) approx sum_{k=0}^n (f^((k))(a))/(k!) \cdot (x-a)^k`

Apply these formula with `f = cos`, `n=4` and `a=3`. You get

`cos(x) approx cos(3) + cos'(3)(x-3) + cos''(3)(x-3)^2/2 + cos'''(3)(x-3)^3/6 + cos''''(3)(x-3)^4/24`.

Because you know the derivatives of `cos`, you can write

`cos(x) approx cos(3) - sin(3)(x-3) - cos(3)(x-3)^2/2 + sin(3)(x-3)^3/6 + cos(3)(x-3)^4/24`.