%% Math 131 Midterm 1, Fall 2004
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\markright{Math 131, Midterm 1}

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\newcommand{\Blank}{$\underline{\phantom{Please write your name in this
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\begin{document}

\begin{center}

\textbf{College of the Holy Cross, Fall Semester, 2004} \\
\textbf{Math 131, Midterm 1} (All Sections) \\
\textbf{Wednesday, September 22, 6 PM}
\end{center}

\vfil

\noindent\textbf{Name}: \Blank
\vskip 0pt plus .3fil

\paragraph{Instructions} Please write your answers in the spaces
provided, and show work on the test itself. Use the back of the
preceding page if you need more space for scratch work.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
1  & \hfil/\  15\hfil \Strut \\
\hline
2  & \hfil/\  20\hfil \Strut \\
\hline
3  & \hfil/\  20\hfil \Strut \\
\hline
4  & \hfil/\  10\hfil \Strut \\
\hline
5  & \hfil/\  20\hfil \Strut \\
\hline
6  & \hfil/\  15\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\begin{enumerate}
  
\item{} The graph $y=f(x)$ and four graphs obtained by transforming it
  are shown. Match the given formulas with the corresponding graph.
  Note that there is an extra graph.

  \begin{center}
    \input{midterm1a.eepic}
  \end{center}

  (a) $y=f(\frac{1}{2}x)$:\blank\hfil 
  (b) $y=\frac{1}{2}f(x)$:\blank\hfil 
  (c) $y=\frac{1}{2}-f(x)$:\blank

\bigskip

\sep
\vfil

\begin{center}
  \input{midterm1b.eepic}\hspace*{0.5in}\input{midterm1c.eepic}
\end{center}
\vfil

\begin{center}
  \input{midterm1d.eepic}\hspace*{0.5in}\input{midterm1e.eepic}
\end{center}
\vfil

\eject

\item{} [20~points]\quad The desert temperature~$H$ varies
  sinusoidally from a high of~$80^\circ\textrm{F}$ at 5~PM to a low
  of~$40^\circ\textrm{F}$ at 5~AM. Find a formula for~$H$ as a
  function of~$t$, with $t$~measured in hours from 5~PM. You may use
  the graph below for reference; it's a good idea to start by labeling
  the vertical and horizontal scales.

  \begin{center}
    \input{midterm1f.eepic}
  \end{center}

\vfil

\eject

\item{} [10~points each]\quad In each part, fill in the table as
  indicated. \\
  Give \textbf{exact answers}: Use fractions, square roots,
  logarithms, etc.\ as needed.

\bd

\item{(a)} Assuming~$f$ is a \textbf{linear function}. 
  \[
    \begin{array}{|c||r|r|r|r|}
      \hline
      x    & \ \ -1 &  \phantom{f()}2 & 5 & 10 \Strut \\
      \hline
      f(x) & 24 & 12 & \phantom{f(x)} & \phantom{f(x)} \Strut \\
      \hline
    \end{array}
  \]

\vfil
\sep

\item{(b)} Assuming~$f$ is an \textbf{exponential function}. 
  \[
    \begin{array}{|c||r|r|r|r|}
      \hline
      x    & \ \ -1 &  \phantom{f()}2 & 5 & 10 \Strut \\
      \hline
      f(x) & 24 & 12 & \phantom{f(x)} & \phantom{f(x)} \Strut \\
      \hline
    \end{array}
  \]

\vfil

\ed

\eject

\item{} [10~points]\quad Find the polynomial whose graph is shown;
  expand (multiply out) your answer.
\begin{center}
  \input{midterm1g.eepic}
\end{center}
\vskip 0pt plus 0.5fil
\sep

\item{} [10~points each]\quad Let $y=f(x)=e^x-1$.

  \bd

  \item{(a)} Find the inverse function; express your answer in the
    form $y=f^{-1}(x)$. 
  
    \vfil

  \item{(b)} Find the domain of~$f^{-1}$.

    \vfil

  \ed

\eject

\item{} Alice buys an automobile for \$25,000 that depreciates in
  value by 20\%~per year. Bob buys an automobile for \$18,000 that
  depreciates in value by 10\%~per year.

\bd

\item{(a)} [5~points]\quad Find formulas for $A(t)$~and $B(t)$, the
  values of the cars after $t$~years.  

\vfil

\item{(b)} [10~points]\quad After how many years are their cars worth
  the same amount?  Give an \textbf{exact answer} (using logarithms)
  and round your answer to two decimal places.

\ed

\vfil

\end{enumerate}

\end{document}
