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\problemgrouptitle{Derivative Applications Review}

\begin{problemblock}{Find the critical numbers of the given functions.} 
\item $\displaystyle f(x)=\frac{x+2}{x^2-3}$
\answer{$x=-1$, $x=-3$}

\item $\displaystyle g(t)=\sqrt{t^2-3t}$
\answer{$t=0$, $t=3$}

\item $\displaystyle F(\theta)=\cos (2\theta)+2\sin\theta$
\answer{$\displaystyle \theta\ \mbox{in}\ \left\{\frac{\pi}{6}+2n\pi\right\}\cup\left\{\frac{5\pi}{6}+2n\pi\right\}\cup\left\{\frac{\pi}{2}+2n\pi\right\}\cup\left\{\frac{3\pi}{2}+2n\pi\right\}$ ($n$ an integer)}
\end{problemblock}

\begin{problemblock}{Find the \absolute maximum and \absolute minimum values and their locations for the given function on the closed interval.}

\item $\displaystyle f(x)=\frac{x}{x^2+16}$ on the closed interval $[-1,\ 1]$.
\answer{\Absolute maximum: $\displaystyle f(1)=\frac{1}{17}$, \Absolute mininimum: $\displaystyle f(-1)=-\frac{1}{17}$}

\item $\displaystyle g(t)=t\sqrt{8-t^2}$ on the closed interval $[0,1]$.
\answer{\Absolute maximum: $\displaystyle g(1)=\sqrt{7}$, \Absolute minimum: $g(0)=0$}
\end{problemblock}

\lab{\addtoanswerkey{\clearpage}}
\begin{problemblock}{Find the domain, intercepts, asymptotes, \relative maxima and minima, intervals of increase and decrease, intervals of concave upward and downward, and inflections points. Then sketch the graph of the given functions.}

\item $\displaystyle f(x)=8x^{\frac{1}{3}}+x^{\frac{4}{3}}$
\answer{$D=\mathset{R}=(-\infty,\infty)$; $x$-int=0, -8; $y$-int=0; No asymptotes; Increasing on: $\left(-2,\infty\right)$;  Decreasing on: $\left(-\infty,-2\right)$; No \relative maxima; \Relative mininimum: $f(-2)=-2^{\frac{10}{3}}+2^{\frac{4}{3}}$; Concave upward on $\left(-\infty,0\right)\cup\left(4,\infty\right)$; Concave downward on $\left(0,4\right)$; Inflection points: $(0,0)$, $\left(4,3(4)^{\frac{4}{3}}\right)$; Graph:\begin{center}
\resizebox{5cm}{!}{\asycode{%
import graph;
size(250,250);
real f(real x) {return 8*cbrt(x)+(cbrt(x))^4;}
pair F(real x) {return (x,f(x));}
draw(graph(f,-10,6,operator ..),red+2);
xaxis("$x$",Arrow(HookHead,size=4),above=true);
yaxis("$y$",Arrow(HookHead,size=4),above=true);
dot(F(-2),4+black);
label("$(-2,-2^{\frac{10}{3}}+2^{\frac{4}{3}})$",F(-2),SW);
dot((0,0),4+black);
dot(F(4),4+black);
label("$\left(4,3(4)^{\frac{4}{3}}\right)$",F(4),E);
xtick("$-8$",-8);
label("$f(x)=8x^{\frac{1}{3}}+x^{\frac{4}{3}}$",F(1),SE);
}}
\end{center}}

\item $\displaystyle g(x)=\frac{x^2}{x-2}$
\answer{$D=\mathset{R}-\{2\}=(-\infty,2)\cup(2,\infty)$; $x$-int=0; $y$-int=0; Vertical asymptote: $x=2$; Increasing on: $\left(-\infty,0\right)\cup\left(4,\infty\right)$; Decreasing on $\left(0,2\right)\cup(2,\infty)$; \Relative maximum: $f(0)=0$; \Relative minimum: $f(4)=8$, , Concave upward on $\left(2,\infty\right)$; Concave downward on $\left(-\infty, 2\right)$; No inflection points; Graph:
\begin{center}
\resizebox{5cm}{!}{\asycode{%
import graph;
size(200,200,IgnoreAspect);
real f(real x) {return x^2/(x-2);}
pair F(real x) {return (x,f(x));}
draw(graph(f,-6,1.83,operator ..),red+2);
draw(graph(f,2.25,6,operator ..),red+2);
xaxis("$x$",Arrow(HookHead,size=4),above=true);
yaxis("$y$",Arrow(HookHead,size=4),above=true);
dot((0,0),4+black);
dot(F(4),4+black);
label("$(4,8)$",F(4),S);
xequals(2,ymin=f(1.83),ymax=f(2.25),dashed);
label("$\displaystyle f(x)=\frac{x^2}{(x-2)}$",F(0),NW);
label("$x=2$",F(1.83),SSE);
}}
\end{center}}
\end{problemblock}

\begin{problemblock}{Evaluate the given limits.}

\addtoanswerkey{\begin{multicols}{4}}

\item $\displaystyle \lim_{x\to \infty}\frac{3x^2-4x-5}{2x^5+3}$
\answer{$0$}

\item $\displaystyle \lim_{x\to -\infty}\frac{5x^4-3x+1}{x^4+7}$
\answer{$5$}

\item $\displaystyle \lim_{x\to -\infty }\frac{\sqrt{5x^2-4}}{2x+1}$
\answer{$\displaystyle -\frac{\sqrt{5}}{2}$}

\item $\displaystyle \lim_{x\to -\infty }\left(2x+\sqrt{x^2+4x+2}\right)$
\answer{$-\infty$}

\addtoanswerkey{\end{multicols}}

\end{problemblock}

\begin{problemblock}{Find the horizontal and vertical asymptotes of the given functions.}

\item $\displaystyle y=\frac{2x^2+7x+3}{x^2+x-6}$
\answer{Vertical asymptote: $x=2$, Horizontal asymptote: $y=2$}

\item $\displaystyle g(t)=\frac{\sqrt{9+4t^2}}{2t+3}$
\answer{Vertical asymptote: $\displaystyle t=-\frac{3}{2}$, Horizontal asymptotes: $\displaystyle y=-1$, $\displaystyle y=1$}
\end{problemblock}

\textbook{\clearpage}
\item A store with a rectangular floorplan is to sit in the middle of one side of a larger rectangular lot with parking on three sides as shown.
\begin{center}
\resizebox{4cm}{!}{\includegraphics{DERIVATIVE_APPLICATIONS/lot}}
\end{center}
If the narrow strips of parking on the side are to be $10$~m wide while the parking in the front is to be $18$~m wide, find the optimal dimensions of the store that will minimize the total lot area if the store itself must have an area of $1000$~m$^2$.  What is the total lot area in this case?
\answer{Vertical length in diagram is $30$~m and horizontal length is $\displaystyle \frac{100}{3}$~m; lot area=$2560$~m$^2$}

\item A farmer wants to enclose a rectangular garden on one side by a
brick wall costing \$$20$/m and on the other three sides by a metal fence
costing \$$5$/m. Find the dimensions of the garden that minimize the cost if
the area of the garden is $250$~m$^2$.
\answer{Brick side is $10$~m and exclusively fence side is $25$~m}

\item A window shaped like a Roman arch consists of a rectangle
  surmounted by a semicircle. Find the dimensions of the window that
  will allow the maximum amount of light if the perimeter of the
  window is $10$~m.
\answer{Maximal area occurs when Width=Circle diameter=$\displaystyle\frac{20}{4+\pi}$~m and Height=$\displaystyle\frac{10}{4+\pi}$~m (i.e. half the width).}

