Math $231$

Summer $2025$

Written Homework $2$

Due Friday, June $13$

Please carefully read and follow the accompanying instructions for each problem in Canvas.

Evaluate the following using derivative rules.

$($a$)\frac{d}{dx}\sqrt[\phantom{2}]{(x^{2}7x-\frac{3}{x^2})(5x^3-2x3)}$

$($b$)\frac{d}{dx}[\frac{(4x3{)}^5}{(3x^{2}2x1)\sqrt[\phantom{2}]{x\frac{1}{x}}}]$

Draw a graph for a continuous function $f(x)$ which satisfies all of the following criteria:

$\underset{x-}{lim}f(x)=5$ and $\underset{x}{lim}f(x)=-$

$f^{\text{'}}(x)<mn>0</mn>$ on the intervals $(-,-2)$ and $(3,),f^{\text{'}}(x)>0$ on the interval $(-2,3)$ and $f^{\text{'}}(x)=0$ at $x=-2,3.$

$f^{\text{'}\text{'}}(x)<mn>0</mn>$ on the intervals $(-,-4)$ and $(0,5),f^{\text{'}\text{'}}(x)>0$ on the intervals $(-4,0)$ and $(5,),$ and $f^{\text{'}\text{'}}(x)=0$ at $x=-4,0,5.$

A country's gross domestic production is modeled by the equation $P=2L^{\frac{3}{5}}{K}^{\frac{2}{5}}$ where $L$ is the amount the country devotes to labor and $K$ is the amount spent on capital. Currently the country spends $5$ trillion dollars on labor and $4$ trillion on capital. If the country is currently increasing their labor expenditures at a rate of $0\mathrm{.}1$ trillion dollars$/$year and their capital expenditures at $0\mathrm{.}15$ trillion dollars$/$year$,$ at what rate is the gross domestic production of the country increasing?