Question

Find the Taylor polynomial T2(x) for the function F(x)=(1+x^2)^1/3 centered about the point a=0.

Answer 1

\[f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2\]in your case...\[f'(x)=\frac{2x}{3}(1+x^2)^\frac{-2}{3}\]and, \[f''(x)=\frac{2}{3}(1+x^2)^\frac{-2}{3}+\frac{-8x^2}{9}(1+x^2)^\frac{-5}{3}\]so,\[1+0*(x-0)+\frac{2}{3}(x-0)^2=\frac{2}{3}x^2+1\]