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\begin{center}

\textbf{College of the Holy Cross, Fall 2013} \\
\textbf{Math 135, Section 1, Midterm 1}\\
\textbf{Friday, September 20}
\end{center}

\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 0pt plus .3fil


\vskip 30pt
\noindent\textbf{Instructions}: Please write your answers in the boxes
provided, and show work on the test pages. Use the back of the
preceding page if you need more space for scratch work.  The numbers
next to each part of the questions are their point values.

\vfil

\noindent\textbf{Please do not write in the space below}

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/\  20\hfil \\
\hline
II  & \hfil/\  25\hfil \\
\hline
III  & \hfil/\  20\hfil \\
\hline
IV  & \hfil/\  15\hfil \\
\hline
V  & \hfil/\  20\hfil \\
\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I.  Match the plots below with the following formulas. Note that there is an extra plot.
\begin{itemize}
\item[(5) A)] $y = 3 - (x - 1)^2 $ is Plot: $\underline{\qquad\qquad}$
\item[(5) B)] $y = \sin(x/2)$ is Plot: $\underline{\qquad\qquad}$
\item[(5) C)] $y = 1 - e^{-x}$ is Plot: $\underline{\qquad\qquad}$
\item[(5) D)] $y = \sin(2x)$ is Plot: $\underline{\qquad\qquad}$
\end{itemize}
\vskip 0.5in
\begin{center}
Plot 1: \includegraphics[width=2truein,height=2truein]{realplot1.ps} 
Plot 2: \includegraphics[width=2truein,height=2truein]{realplot2.ps}
\end{center}
\begin{center}
Plot 3:
\includegraphics[width=2truein,height=2truein]{realplot3.ps}
Plot 4:
\includegraphics[width=2truein,height=2truein]{realplot4.ps}
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\begin{center}
Plot 5: \includegraphics[width=2truein,height=2truein]{realplot5.ps}
\end{center}
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\noindent
II.  The manager of a furniture factory has collected the following data for the cost of manufacturing chairs.
\bigskip
\begin{center}
\begin{tabular}{|c|c|}
\hline \# Chairs (per day) $x$ & Cost (in dollars) $y$ \\

\hline 100 & 2400 \\
\hline 150 & 3100 \\
\hline 250 & 4500 \\
\hline 300 & 5200 \\
\hline
\end{tabular}
\end{center}
\begin{itemize}
\item[(10) A)]  Given that $y$ is a linear function of $x$, determine a formula for it.  
\vskip 1.0truein
\MyAnswer{Cost function:}{$\phantom{A pretty long answer goes here}$}%
\item[(5) B)] What does the slope represent in real-world terms?
\vskip 1.5truein
\item[(5) C)] What does the $y$-intercept represent in terms of cost?
\vskip 1.5truein
\item[(5) D)]  Using your model, determine how much it will cost to produce $350$ chairs per day.
\vskip 1.0truein
\MyAnswer{Cost:}{$\phantom{A pretty long answer goes here}$}%
\end{itemize}
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%%%%%%%%%% Combining functions and finding domains %%%%%%%%%%%%


\noindent
III. Given $f(x)=4-x^2$ and $g(x)=\sqrt{3x-2}$, answer the following questions.

\begin{itemize}
\item[(10) A)] Find the domain of $f(x)$ and the domain of $g(x)$.

\vskip 2.0truein
\MyAnswer{Domain of $f$:}{$\phantom{A long answer here}$}%
\MyAnswer{Domain of $g$:}{$\phantom{A long answer here}$}%
\item[(5) B)] What is the domain of the function $g(x)/f(x)$?

\vskip 2.0truein
\MyAnswer{Domain of $g(x)/f(x)$:}{$\phantom{A long answer here}$}%
\item[(5) C)] Find the function $g\circ f$.
\vskip 2.0truein
\MyAnswer{$g\circ f$:}{$\phantom{A very very very very very long answer here}$}%
\end{itemize}

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\noindent
IV.  Answer the following questions. 

\begin{itemize}
\item[(5) A)] Find all values of $x$ in 
$\displaystyle \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ for which 
$\displaystyle \left| \tan x \right| > 1$.


\MyAnswer{Values of $x$:}{$\phantom{A very long answer here}$}%
\vskip 1.0in 


\item[(5) B)] If $\displaystyle \sin \theta =\frac{2}{3}$ and $\displaystyle \frac{\pi}{2}<\theta<\pi$, give the exact value of 
$\cos \theta$. 

\vskip 1.0in 
\MyAnswer{$\cos \theta$:}{$\phantom{A long answer here}$}%
\item[(5) C)] Express as a single logarithm: $\displaystyle \frac{1}{2} \ln 3 - 3 \ln 2 + \ln 6.$

\vskip 1.0in
\MyAnswer{Single logarithm:}{$\phantom{A very long answer here}$}%

\end{itemize}
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\noindent
V.  Consider the function $\displaystyle f(x)= \frac{1}{2} e^{x+1} + 1$. 
\begin{itemize}
\item[(15) A)] Given that $f$ is one-to-one, find a formula for the inverse function of $f$. 
\vskip 2.0in
\MyAnswer{$f^{-1}(x) = $}{$\phantom{A very very very very long answer here}$}%
\item[(10) B)] In the space below, plot the graphs 
of the functions $f$ and $f^{-1}$ on the same set of axes.
Label one point on each graph with its coordinates.
\end{itemize}
\vfil

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