Let

$f(x)=\sqrt[3]{7+5x^2}$

$($a$)$ Write the first five terms in the Taylor expansion for $f(x)$ centered at $x=2.(30$ points$)$

$($b$)$ Using the terms you calculated above, calculate an approximation to $f(3).$ Show the calculations and compare it to the exact value found by a calculator. $(10$ points$)$

$($c$)$ Let

$g(x)={\displaystyle \underset{2}{\overset{x}{}}}f(t)dt$

Using the Taylor expansion you calculated for $f(x),$ write the first five terms in the Taylor expansion for $g(x)$ and approximate $g(3)$ with it$.$ If you can, compare it to the exact value as determined by a calculator. $(15$ points$)$