%% Math 131 Midterm 1, Fall 2004
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\markright{MATH 133, section 1, Midterm 1}

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\begin{document}

\begin{center}

\textbf{College of the Holy Cross, Fall Semester, 2015} \\
\textbf{MATH 133, section 1, Midterm 1} \\
\textbf{Thursday, September 24}
\end{center}

\vfil

\noindent\textbf{Your 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.  There are
100 possible points distributed as below.

\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/\  15\hfil \Strut \\
\hline
3  & \hfil/\  20\hfil \Strut \\
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4  & \hfil/\  20\hfil \Strut \\
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5  & \hfil/\  15\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{} 
\begin{enumerate}
\item[(a)] (7.5) Express the set of all $x$ satisfying $|4x - 12| \le 8$ as an interval or union of intervals.

\vskip 3.0truein

\item[(b)] (7.5) What is the domain of the function $f(x) = \frac{1}{x \sqrt{4 - x}}$?  Any correct form is OK.
\end{enumerate}
\vfil
\eject 
\item{} (15) 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 that does not match any of the formulas.

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

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

\bigskip

\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{}
\begin{enumerate}
\item[(a)] (10) Complete the square:  $q(x) = 3x^2 + 18x + 36$.
\vskip 2.0truein
\item[(b)] (5) What is the minimum value of $q(x)$?
\vskip 2.0truein
\item[(c)] (5) Using your answers from parts (a) and (b), determine the \emph{range} of the function 
$r(x) = \sqrt{q(x)} = \sqrt{3x^2 + 18x + 36}$.  
\end{enumerate}

\vfil\eject

\item{} The temperature~$H$ in a desert varies
  sinusoidally from a high of~$80^\circ\textrm{F}$ at 5~PM to a low
  of~$40^\circ\textrm{F}$ at 5~AM.  (See graph below)
  

  \begin{center}
    \input{midterm1f.eepic}
  \end{center}
\vglue 10pt
\begin{enumerate}
\item[(a)] (5)  What is the period of this sinusoidal oscillation?  $\underline{\qquad\qquad}$ \hfil
\item[(b)] (5)  What is the amplitude? $\underline{\qquad\qquad}$ \hfil
\item[(c)] (10)  If $t = 0$ is the first 5PM time shown, give a possible formula for $H$ as a function
of $t$.
\end{enumerate}
\vfil \eject
\item{}
\begin{enumerate}
\item[(a)] (7.5)  Solve for $x$:  $4^{x + 1} = 8^{x - 3}$.
\vskip 3.0truein
\item[(b)] (7.5) A sample of a radioactive element is decaying over time.  The mass present
at time $t$ is given by 
according to $M(t) = 139 e^{-0.003 t}\ {\rm grams}$, where $t$ is in months.  When will the mass 
present reach 50 grams?
\end{enumerate}
\vfil\eject
\item{}  You are traveling by donkey along a straight line road starting from $x = 0$ (miles) 
at time $t = 0$ (hours).  For the first two hours, you move in the positive $x$-direction at 5 miles per hour.  
At $t = 2$, you realize you have dropped an important item from your saddle bag.  So you turn around and 
retrace your steps at 5 miles per hour.  You find the item at $t = 3$.  Then you turn back around
and continue at 5 miles per hour for an additional 2 hours.  
\vskip 10pt
\begin{enumerate}
\item[(a)] (7.5) Sketch the graph of your position $x$ as a function of time $t$ for $0 \le t \le 5$.  
\vglue 3.0truein
\item[(b)]  (7.5) Give your position $x$ as a piecewise-defined function of $t$.
\end{enumerate}
\vfil\eject
\end{enumerate}


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