$2\mathrm{.}15$ a$.$ Suppose the run$-$time of a serial program is given by $T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}}=n^2,$ where the units of the run$-$time are in microseconds. Suppose that a parallelization of this program has run$-$time $T_{\text{p}\text{a}\text{r}\text{a}\text{l}\text{l}\text{e}\text{l}}=\frac{n^2}{p}+lo{g}_2(p).$ Write a program that finds the speedups and efficiencies of this program for var$-$ious values of $n$ and $p.$ Run your program with $n=10,20,40,...,320,$ and $p=1,2,4,...,128.$ What happens to the speedups and efficiencies as $p$ is increased and $n$ is held fixed? What happens when $p$ is fixed and $n$ is increased?

b$.($Needs calculus.$)$ Suppose that for a given program $T_{\text{p}\text{a}\text{r}\text{a}\text{l}\text{l}\text{e}\text{l}}=\frac{T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}}}{p}+T_{\text{o}\text{v}\text{e}\text{r}\text{h}\text{e}\text{a}\text{d}}$ and consider the case when $p$ is held fixed. In this setting suppose $T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}}^{\text{'}}$ and $T_{\text{o}\text{v}\text{e}\text{r}\text{h}\text{e}\text{a}\text{d}}^{\text{'}}$ denote the derivatives of $T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}}$ and $T_{\text{o}\text{v}\text{e}\text{r}\text{h}\text{e}\text{a}\text{d}}$ with respect to $n$ for the fixed value of $p.$ Also suppose that for the fixed value of $p$ and for all $n,$ the functions $T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}},T_{\text{o}\text{v}\text{e}\text{r}\text{h}\text{e}\text{a}\text{d}},T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}}^{\text{'}},$ and $T_{\text{o}\text{v}\text{e}\text{r}\text{h}\text{e}\text{a}\text{d}}^{\text{'}}$ are positive. Show that if $\frac{T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}}^{\text{'}}}{T_{\text{s}\text{e}\text{r}\text{i}\text{a}\text{l}}^{\text{'}}}>\frac{T_{\text{o}\text{v}\text{e}\text{r}\text{h}\text{e}\text{a}\text{d}}^{\text{'}}}{T_{\text{o}\text{v}\text{e}\text{r}\text{h}\text{e}\text{a}\text{d}}}$