\documentclass[12pt]{report} 

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\newcommand{\pp}{\par \noindent}
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\begin{document}
\noindent
\vfil \noindent

{\LARGE{
\hfil Math 120 A - Autumn 2016 \hfil \pp
\hfil Midterm Exam Number Two\hfil  \pp
\hfil November 17th, 2016 \hfil \pp
}}

\normalsize
\vfil
\medskip
\hfil Name: \hrulefill \hrulefill \hspace{0.5in} Student ID no. : \hrulefill

\vfil

\hfil Signature: \hrulefill \hrulefill \hrulefill \hspace{0.5in} Section: \hrulefill 
 
\vfil
\begin{center}
{\Huge
\begin{tabular}{||c|c|r||} \hline 
 1 & 14 &\hspace{10mm} \hfil\\ \hline 
 2 & 15 &       \\ \hline
 3 & 15 &       \\ \hline
 4 & 16 &      \\ \hline
Total & 60 & \\ \hline 
\end{tabular}
}
\end{center}
\vfil
\begin{itemize}
\item This exam consists of FOUR problems on FIVE pages, including this cover sheet.
\item Show all work for full credit. 
\item You may use a TI-30X IIS calculator during this
exam.  Other calculators and electronic device are not permitted.
\item You do not need to simplify your answers.
\item If you use a trial-and-error or guess-and-check method
when a more rigorous method is available, you will not receive full credit.
\item If you write on the back of the page, please indicate that you have done so!
\item \textbox{Draw a box} around your final answer to each problem.
\item You may use one hand-written double-sided 8.5" by 11" page of notes.
\item You have 50 minutes to complete the exam.
\end{itemize}

\vfil
{\small --}

\newpage

\begin{enumerate}

\item Happy Thursday! I bought you this graph.

\vspace{-.2 in}

\begin{center}
\begin{tikzpicture}[auto,scale=1, line width=2pt]

\foreach \x in {-3,-2,...,5} \draw [printersafe, line width=1pt,densely dotted] (\x,-2.2)--(\x,5.2);
\foreach \x in {-3,-2,...,5} \draw (\x+.2,.19) node {\scriptsize \x};
\foreach \y in {-2,-1,...,5} \draw [printersafe, line width=1pt,densely dotted] (-3.2,\y)--(5.2,\y);
\foreach \y in {-2,-1,...,5} \draw (.2,\y+.19) node {\scriptsize \y};

\draw [line width = 1.5 pt, <->] (-3.6,0) -- (5.5,0);
\draw [line width = 1.5 pt, <->] (0,-2.6) -- (0,5.6);

\draw (5.7,0) node {$x$};
\draw (0,5.9) node {$f(x)$};

\draw (-3,2) -- (-1,3);
\draw (3,1) arc (90:0:1 and 2) -- (5,-2);
\draw plot[smooth,tension=1] coordinates{(-1,3) (-.5,4) (.5,4.5) (1,5)};

\fill[black] (5,-2) circle (.15);
\fill[black] (3,1) circle (.15);
\draw [very thick, fill=white] (-3,2) circle (.15);
\draw [very thick, fill=white] (1,5) circle (.15);
\end{tikzpicture}
\end{center}

\begin{enumerate}
\item {\bf [4 points]} Compute $f(f(f(4)))$.
\vskip .8 in
\item {\bf [5 points]} Find the domain and range of $f^{-1}(x)$.
\vskip 1.5 in
\item {\bf [5 points]} Let $g(x) = f(2x+1) + 1$. Find the domain and range of $g(x)$.
\end{enumerate}

\newpage

\item {\bf [15 points]} Gomba is on a diet. His weight is a linear-to-linear rational function of time.\\[.1 in]
Right now, Gomba weighs 21 pounds.\\[.1 in]
In 1 month, he will weigh 20.5 pounds.\\[.1 in]
In 7 months, he will weigh 18.75 pounds.\\[.1 in]
In the long run, what will Gomba's weight approach?\\[.1 in]
{\small (Assume Gomba will live forever.)}

\newpage

\item {\bf [5 points per part]} The rent for a one-bedroom apartment in Beattle is growing exponentially. (Even though the city is filled with bees.)
\begin{enumerate}
\item In the year 2000, the rent in Beattle was $\$1020$, and it increases by $2.3\%$ per year.\\[.1 in]
Write a function $f(t)$ for the rent in Beattle $t$ years after 2000.

\vskip 1.5 in
\item The average monthly rent in Tickoma is also growing exponentially.\\[.1 in]
In the year 2007, the rent in Tickoma was $\$500$ less than the rent in Beattle.\\[.1 in]
In the year 2016, the rent in Tickoma is $\$1000$.\\[.1 in]
Write a function $g(t)$ for the rent in Tickoma $t$ years after 2000.

\vskip 2.5 in

\item When will the rents in Beattle and Tickoma be equal?\\[.1 in] {\small (Round your answer to the nearest year.)}

\end{enumerate}

\newpage

\item A {\em polar rectangle} is the region bounded by two concentric circular arcs and two rays through the center of those arcs. Okay, fine, here's a picture:
\vspace{-.3 in}
\begin{flushright}
\begin{tikzpicture}[scale = .6]
\draw [very thick,pattern = north west lines] (10:5) arc (10:78.7549:5) -- (78.7549:2) arc (78.7549:10:2) -- cycle;
\draw [printersafe, dash pattern = on 2 pt off 2 pt, thick] (10:2) -- (0,0) -- (78.7549:2);
\draw [very thick, printersafe, densely dotted,<->,shift={(168.7549:.3)}] (0,0) -- node[left] {$y$}(78.7549:2);
\draw [very thick, printersafe, densely dotted,<->,shift={(-80:.3)}] (0,0) -- node[below] {$x$}(10:5);
\node at (50:.7) {$\theta$};
\end{tikzpicture}
\end{flushright}

\vspace{-1.4 in}

\begin{enumerate}
\item {\bf [4 points]} Write a formula for the {\em area} of this polar rectangle.

{\small (Your answer will involve $x$, $y$, and $\theta$. Let $\theta$ be measured in radians.)}

\vskip 1.2 in

\item {\bf [4 points]} Write a formula for the {\em perimeter} of this polar rectangle.

\vskip 1.5 in

\item {\bf [8 points]} Suppose you have 24 meters of fencing, and you want to construct a fence in the shape of a polar rectangle with central angle $\theta = 1.2$ radians. What is the maximum possible area of your fence?

\end{enumerate}

\end{enumerate}

\end{document}