Consider the function

$f(x)=xcosx$

$($a$)$ Compute the $4$th degree Taylor polynomial $P_4(x)$ about the point $x=.$

$($b$)$ Find an upper bound on the error associated with $P_4(x)$ on the interval $x+0\mathrm{.}5.$

$($c$)$ Let $I_n$ be an approximation to the integral

$I={\displaystyle \underset{}{\overset{+0\mathrm{.}5}{}}}f(x)dx$

where $f(x)$ is approximated by $P_n(x).$ Find $I_4.$

$($d$)$ Using the upper bound obtained in $($b$),$ find an upper bound on the error associated with the approximation $I_4$ in part $($c$).$

$($e$)$ Using MATLAB, compute values of the approximation $I_n$ in part $($c$)$ for increasing values of

$?n$ and value of I. Find the smallest $n$ for which $I_n$ as an estimate of I is accurate to $8$ decimal