# How do you find the first three terms of the Taylor series for f(x)=cos(5x) for x=0?

Mar 8, 2015

$f \left(x\right) \approx 1 + 0 \left(x - 0\right) + \frac{- 25}{2 \cdot 1} {\left(x - 0\right)}^{2} = 1 - \frac{25}{2} {x}^{2}$.

Method:
The Taylor series centered at $a$ for $f \left(x\right)$ is:

f(a)+f'(a)(x-a)+ (f''(a))/(2!)(x-a)^2+(f'''(a))/(3!)(x-a)^3+ * * * +(f^(n+1)(a))/((n+1)!)(x-a)^(n+1)+ * * *

The first three terms will involve $f \left(x\right) = \cos 5 x$, $f ' \left(x\right) = - 5 \sin 5 x$, and $f ' ' \left(x\right) - 25 \cos 5 x$, each evaluated at $a = 0$

We find: $f \left(0\right) = 1$, $f ' \left(0\right) = 0$, and $f ' ' \left(0\right) = - 25$.

Substitute into the series and simplify is necessary.

$f \left(x\right) \approx 1 + 0 \left(x - 0\right) + \frac{- 25}{2 \cdot 1} {\left(x - 0\right)}^{2} = 1 - \frac{25}{2} {x}^{2}$.