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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Fall 2016} \\
\textbf{Math 135, Section 1, Midterm 1}\\
\textbf{Friday, September 23}
\end{center}

\vskip 30pt

\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.

\vskip 100pt

\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}

\vfill\eject

\vskip 10pt
\noindent
I.  Match the plots below with the following formulas. Note that there is an extra plot that does not
match any of these formulas.
\begin{itemize}
\item[(5) A)] $y = (x + 1)^2 + 2$ is Plot: $\underline{\qquad\qquad\qquad\qquad}$
\item[(5) B)] $y = 3\cos(2x)$ is Plot: $\underline{\qquad\qquad\qquad\qquad}$
\item[(5) C)] $y = 1 + e^{-x}$ is Plot :$\underline{\qquad\qquad\qquad\qquad}$
\item[(5) D)] $y = \sin(x/2)$ is Plot: $\underline{\qquad\qquad\qquad\qquad}$
\end{itemize}
\vskip 0.5in
\begin{center}
Plot 1: \includegraphics[width=2truein,height=2truein]{plot1.png} 
Plot 2: \includegraphics[width=2truein,height=2truein]{plot2.png}
\end{center}
\begin{center}
Plot 3:
\includegraphics[width=2truein,height=2truein]{plot3.png}
Plot 4:
\includegraphics[width=2truein,height=2truein]{plot4.png}
\end{center}
\begin{center}
Plot 5: \includegraphics[width=2truein,height=2truein]{plot5.png}
\end{center}
\vfill\eject

\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) $C$ & Manufacturing Cost (in dollars) $M$ \\

\hline 100 & 2400 \\
\hline 150 & 3100 \\
\hline 250 & 4500 \\
\hline 300 & 5200 \\
\hline
\end{tabular}
\end{center}
\begin{itemize}
\item[(10) A)]  Given that $M$ is a linear function of $C$, determine a formula for it, using
the correct labeling of the variables (that is, an answer expressed in terms of $x$ and $y$
will not receive full credit).
\vskip 10pt
\vskip 50pt
\vskip 10pt
\MyAnswer{Cost function:}{\qquad\qquad\qquad\qquad\qquad}%
\item[(5) B)] How much additional cost is incurred by manufacturing each additional chair?
\vskip 10pt
\vskip 50pt
\MyAnswer{Additional cost:}{\qquad\qquad\qquad\qquad\qquad}% 
\vskip 10pt
\item[(5) C)] What does the $M$-intercept represent in terms of cost?
\vskip 10pt
\vskip 40pt
\vskip 10pt 
\item[(5) D)]  Using your formula, determine how much it will cost to produce $350$ chairs per day.
\vskip 10pt
\MyAnswer{Cost:}{\qquad\qquad\qquad\qquad\qquad}%
\end{itemize}


%%%%%%%%%% Combining functions and finding domains %%%%%%%%%%%%

\vfill\eject
\noindent
III. Given $f(x)=x^2 - 6x + 1$ 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 10pt
\vskip 100pt
\vskip 10pt
\MyAnswer{Domain of $f$:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\vskip 10pt
\MyAnswer{Domain of $g$:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(5) B)] What is the domain of the function $g(x)/f(x)$?
\vskip 10pt
\vskip 100pt
\vskip 10pt
\MyAnswer{Domain of $g(x)/f(x)$:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(5) C)] Find the function $(g\circ f)(x)$.
\vskip 100pt
\MyAnswer{}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\end{itemize}

\vfill\eject
\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 
$\displaystyle \left| \tan x \right| > 1$.
\vskip 10pt
\vskip 100pt
\vskip 10pt
\MyAnswer{Values of $x$:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\vskip 10pt
\noindent
\vskip 10pt

\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 10pt
\vskip 100pt
\vskip 10pt
\MyAnswer{$\cos \theta$:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(5) C)] Express as a single logarithm: $\displaystyle \frac{1}{2} \ln 5 - 4 \ln 2 + \ln 10.$  (This
means your answer should be in the form $\ln({\rm some\ exact\ number})$, not a decimal approximation.)
\vskip 10pt
\vskip 100pt
\vskip 10pt
\MyAnswer{Single logarithm:}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%

\end{itemize}
\vfill\eject
\noindent
V.  Consider the function $\displaystyle f(x)= \frac{1}{5} e^{x+2} - 3$. 
\begin{itemize}
\item[(10) A)] Given that $f$ is one-to-one, find a formula for the inverse function of $f$. 
\vskip 10pt
\vskip 100pt
\vskip 10pt
\MyAnswer{$f^{-1}(x) = $}{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}%
\item[(10) B)] The plot above shows the graph 
of the function $y = f(x)$.  Sketch in the graph $y = f^{-1}(x)$ on the same set of axes.
Label one point on each graph with its coordinates.
\vskip 10pt
\begin{figure}
\begin{center}
\includegraphics[width=4truein,height=4truein]{FiveLess.png}
\caption{$y = f(x)$ in Question V.}
\end{center}
\end{figure}
\end{itemize}
\end{document}
