Approximating the value of an $n$th degree taylor polynomial

Question

First I need to derive the Taylor polynomial for degree n for the two functions below:

f(x) = $\sqrt(1+x)$ and f(x) = cos(x)

Afterwards I need to find the approximate value of both functions at x = $\pi$/4 by hand calculator (up to two decimal places).

I understand how to derive the taylor polynomial, but what I don't understand is how to approximate a value for degree n.

Answer 1

Hints:

1) Taylor series at a point $x=a$ is given by

$$ f(x) = \sum_{k=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n $$

2) try to find few derivatives to see a general formula for the $n$th derivative

$$\sqrt{1+x}= \sum_{k=0}^{\infty} {1/2\choose k} x^k .$$

3) use the identity

$$\cos(x) = \frac{e^{ix}+e^{-ix}}{2}.$$