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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Fall 2007} \\
\textbf{Math 131, Midterm 1} (All Sections) \\
\textbf{Wednesday, September 19, 6 PM}
\end{center}

\vfil

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

\vskip 15pt
\leftline{Little (8:00am) $\underline{\qquad}$ Ballantine (9:00am) $\underline{\qquad}$} \leftline{DeStefano (10:00am) $\underline{\qquad}$ DeStefano (noon) $\underline{\qquad}$}

\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/\  12\hfil \\
\hline
II  & \hfil/\  15\hfil \\
\hline
III  & \hfil/\  18\hfil \\
\hline
IV  & \hfil/\  20\hfil \\
\hline
V  & \hfil/\  20\hfil \\
\hline
VI  & \hfil/\  15\hfil \\
\hline\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 graph.
\begin{itemize}
\item[(4) A)] $y = 3 - (x - 1)^2$ is Plot: $\underline{\qquad\qquad}$
\item[(4) B)] $y = \sin(x/2)$ is Plot: $\underline{\qquad\qquad}$
\item[(4) C)] $y = 1 - e^{-x}$ is Plot: $\underline{\qquad\qquad}$
\end{itemize}
\vskip 0.5in
\begin{center}
Plot 1: \includegraphics[width=2truein,height=2truein]{plot1.ps} 
Plot 2: \includegraphics[width=2truein,height=2truein]{plot2.ps}
\end{center}
\begin{center}
Plot 3:
\includegraphics[width=2truein,height=2truein]{plot3.ps}
Plot 4:
\includegraphics[width=2truein,height=2truein]{plot4.ps}
\end{center}

\vfil\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) & Cost (in dollars) \\

\hline 100 & 2400 \\
\hline 150 & 3100 \\
\hline 250 & 4500 \\
\hline 300 & 5200 \\
\hline
\end{tabular}
\end{center}
\begin{itemize}
\item[(6) A)]  Express the cost ($y$) as a function of the number of chairs produced 
per day ($x$), assuming that it is linear.
\vskip 1.0truein
\MyAnswer{Cost function:}{$\phantom{A pretty long answer goes here}$}%
\item[(3) B)] What is the slope, and what does it represent in real-world terms?
\vskip 1.5truein
\item[(3) C)] What is the $y$-intercept, and what does it represent in terms of cost?
\vskip 1.5truein
\item[(3) 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}
\vfil\eject

%%%%%%%%%% 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[(6) 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[(6) B)] Find the function $f/g$ and its domain.

\vskip 2.0truein
\MyAnswer{The function $f/g$:}{$\phantom{A long answer here}$}% 
\MyAnswer{Domain of $f/g$:}{$\phantom{A long answer here}$}%
\item[(6) C)] Find the function $f\circ g$ and its domain.
\vskip 2.0truein
\MyAnswer{$f\circ g$:}{$\phantom{A very long answer here}$}%
\MyAnswer{Domain of $f\circ g$:}{$\phantom{A long answer here}$}%
\end{itemize}

\vfil\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 
$\displaystyle \left| \tan x \right| > 1$.


\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:}{$\phantom{A very long answer here}$}%



\item[(5) E)] Complete the following table with the values for the functions $f$, $g$ and $h$ given that  $f$ is even,  $g$ is odd, and $h =f \circ g$. 

\begin{center}
\begin{tabular}{|c|c|c|c|} \hline $x$ & $f(x)$ & $g(x)$ & $h(x)$\\  \hline
$ -3$ & $-1$ & $0$ & $1$ \\ \hline $-2$ & $3$ & $1$ & $-2$ \\ \hline $ -1$ & $-2$ & $-3$ & $-1$ \\ \hline $0$ & $1$ & $2$ & $3$ \\ \hline $1$ & & & \\ \hline $2$ & & & \\ \hline $3$ & & & $1$\\ \hline 
\end{tabular}
\end{center}
\end{itemize}
\vfil\eject

\noindent
V.  Consider the function $\displaystyle f(x)=2 e^{x+3}+1$. 
\begin{itemize}
\item[(5) A)] What are the domain and range of $f$? 

\vskip 1.5in
\MyAnswer{Domain of $f$:}{$\phantom{A long answer here}$}%
\MyAnswer{Range of $f$:}{$\phantom{A long answer here}$}%
\item[(5) B)] 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 long answer here}$}%
\item[(5) C)] What are the domain and range of $f^{-1}$? 
\vskip 1.5in
\MyAnswer{Domain of $f^{-1}$:}{$\phantom{A long answer here}$}%
\MyAnswer{Range of $f^{-1}$:}{$\phantom{A long answer here}$}%
\item[(5) D)] On the next page, plot the graph 
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\eject
\noindent
Place graph for V D on this page.
\vfil\eject
\noindent
VI.  The half-life of radioactive polonium-210 is about 140 days.
\bigskip
\begin{itemize}
\item[(5) A)] If a sample contains 200g of polonium-210 originally, find the
amount remaining after 560 days.
\vskip 2.0in
\MyAnswer{}{$\phantom{A long long answer here}$}%
\item[(5) B)] Find the amount remaining after $t$ days.
\vskip 2.0in 
\MyAnswer{}{$\phantom{A long long answer here}$}%
\item[(5) C)] How long will it take the amount in the sample to decay to 10g?
(Give your answer in exact form.)
\vskip 2.0in
\MyAnswer{}{$\phantom{A long long answer here}$}%
\end{itemize}

\end{document}
