Question

Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = x4 − 7x2 + 2, a = 2

Answer 1

$f(x)$ is a polynomial of degree 4, so its Taylor series will also be of degree 4. We have

$f(2)=-10$

$f'(x)=4x^3-14x\implies f'(2)=4$

$f''(x)=12x^2-14\implies f''(2)=34$

$f'''(x)=24x\implies f'''(2)=48$

$f^{(4)}(x)=24\implies f^{(4)}(2)=24$

and $f^{(n)}(2)=0$ for all $n\ge5$. So

$f(x)=f(2)+f'(2)(x-2)+\dfrac{f''(2)}{2!}(x-2)^2+\dfrac{f'''(2)}{3!}(x-2)^3+\dfrac{f^{(4)}(2)}{4!}(x-2)^4$

$\boxed{f(x)=-10+4(x-2)+17(x-2)^2+8(x-2)^3+(x-2)^4}$

(Notice that expanding all the binomials and simplifying gives the same $f(x)$ you are given to begin with.)