Exercise $2.$ A function and its approximations $(30$ points$)$

Consider ${\displaystyle \underset{n=0}{\overset{}{}}}\frac{x^{3n}}{(3n)!},$ that is$,$ the power series obtained by keeping every third term in the

Taylor expansion of exp$(x).$ This power series corresponds to the true function

$f_{\text{t}\text{r}\text{u}\text{e}}(x)=\frac{1}{3}(exp(x)+2$exp$(-\frac{x}{2})cos(\frac{\sqrt[\phantom{2}]{3}x}{2}))$

For any possible integer $k,$ define the $k$ term power series approximation $f_{\text{a}\text{p}\text{p}\text{r}\text{o}\text{x}}(x,k)$:$=$

${\displaystyle \underset{n=0}{\overset{k-1}{}}}\frac{x^{3n}}{(3n)!}.$

Submit a MATLAB code $(.$m file$)$ named YourlastnameYourfirstnameHW$2$p$2.$m that plots

$2D$ line plots for the functions $f_{\text{t}\text{r}\text{u}\text{e}}($in black solid line$)$ versus xin$[-5,5]($in the horizontal

axis$).$ In the same figure window, plot $f_{\text{a}\text{p}\text{p}\text{r}\text{o}\text{x}}(x,k)$ for $k=2($in red dashed line$),k=3($in

green dashed line$),k=4($in blue dashed line$).$

We shared a starter code YourlastnameYourfirstnameHW$2$p$2.$m inside the the CANVAS File

section Files$/$MATLAB Files$/$Homework $2.$ You only need to complete lines $11$ and $20$ in that

starter code, then rename the file appropriately with your first and last names.

Hint: Look up sqrt$,$ exp, cos$,$ sum, factorial and power $($ hat$(.))$ in MATLAB documen$-$

tation. Also, intuition suggests that as $k$ increases, $f_{\text{a}\text{p}\text{p}\text{r}\text{o}\text{x}}$ should get closer to $f_{\text{t}\text{r}\text{u}\text{e}}.$