Question $2$

Let $f(x)=e^{x}cosx.$

$A.$ Compute the first four derivatives $off(x),$

showing all work.

$B.$ Verify that your derivatives satisfy the following

properties $of{f}^{(n)}(\frac{}{2})$ :

I. When $n=4k$ for some $k,$ then

$f^{(n)}(\frac{}{2})=0.$

$II.$ When $n=4k+1$ for some $k,$ then

$f^{(n)}(\frac{}{2})=(-1{)}^{k+1}{2}^{2k}{e}^{\frac{}{2}}.$

III. When $n=4k+2orn=4k+3$ for some

$k,$ then $f^{(n)}(\frac{}{2})=(-1{)}^{k+1}{2}^{k+1}{e}^{\frac{}{2}}.$

$C.$ Use the properties listed above $to$ write the Taylor

series expansion $off(x),$ centered $atx=\frac{}{2}.$