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\begin{document}
\begin{center}

\textbf{College of the Holy Cross} \\
\textbf{MATH 133, Calculus With Fundamentals 1}\\
\textbf{Final Examination -- Friday, December 15}
\end{center}

\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$


\vskip 30pt
\noindent\textbf{Instructions}:  For full credit, you must show 
{\it all 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.  There are
200 total points on this exam.

\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/ 25\  \hfil \\
\hline
II  & \hfil/ 25\  \hfil \\
\hline
III  & \hfil/ 40\  \hfil \\
\hline
IV  & \hfil/ 45\  \hfil \\
\hline
V  & \hfil/ 25\  \hfil \\
\hline
VI  & \hfil/ 20\  \hfil \\
\hline
VII  & \hfil/ 20\  \hfil \\
\hline\hline
Total   & \hfil/ 200\hfil \\
\hline
\end{tabular}
\end{center}

\begin{center}
$$\mathcal{HAPPY\ HOLIDAYS!!!}$$
\end{center}

\vfil\eject
\noindent
I.  The  graph $y = f(x)$ is given in light blue.  Match each equation with one of the 
numbered magenta graphs.
\vskip 10pt
\begin{center}
\includegraphics[width=3truein,height=2truein]{FI.png} 
\end{center} 

\begin{itemize}
\item[(5) A)] $y = 2 f(x + 6)$ is plot number: $\underline{\qquad\qquad}$
\item[]
\item[(5) B)] $y = \frac{1}{3} f(x) $ is plot number: $\underline{\qquad\qquad}$
\item[]
\item[(5) C)] $y = f(x) + 3$ is plot number: $\underline{\qquad\qquad}$
\item[]
\item[(5) D)] $y = f(x - 4)$ is plot number: $\underline{\qquad\qquad}$
\item[]
\item[(5) E)] $y = -f(x + 4)$ is plot number: $\underline{\qquad\qquad}$
\item[]
\end{itemize}

\vfill\eject
\noindent
II.  A cup of hot chocolate is set out on a counter at $t = 0$.  The 
temperature of the chocolate $t$ minutes later is
$C(t) = 65 + 55 e^{-t/4}$ (in degrees F).
\vskip 10pt
\begin{enumerate}
\item[A)] (5) What is the temperature of the chocolate at $t = 0$?
\vskip 1.5truein
\item[B)] (10) What is the (instantaneous) rate of change of the temperature at $t = 10$ minutes?
\vskip 2.0truein
\item[C)] (10) How long does it take for the temperature to 
reach $100^\circ F$? 
\end{enumerate}
\vfill\eject
\noindent
III.  Compute the following limits.  Any legal method is OK.  
\vskip 10pt
\begin{enumerate}
\item[(A)] (10) $\displaystyle{\lim_{x \to 3} \frac{x^2 + x - 12}{x^2 - 5x + 7}}$
\vskip 1.5truein
\item[(B)] (10) $\displaystyle{\lim_{x \to 1} \frac{x - 1}{x^2 - 1}}$
\vskip 2.0truein
\item[(C)] (10) $\displaystyle{\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x}}$
\vskip 2.5truein
\item[(D)] (10) $\displaystyle{\lim_{x \to 0} \frac{\tan(x)}{x}}$
\end{enumerate}
\vfill\eject
\noindent
IV.  
\begin{enumerate}
\item[A)] (10) {\it Using the limit definition}, and showing all necessary steps
to justify your answer, compute $f'(x)$ for $f(x) = 5x^2 + x$.
\end{enumerate}
\vfill\eject
\noindent
IV. (continued)  Using appropriate derivative rules, compute the derivatives
of the following functions.  \emph{You do not need to simplify your answers}.
\begin{enumerate}
\item[B)] (5) $\displaystyle{g(x) = 4x^3 +  \sqrt{x} + \frac{2}{\sqrt[4]{x}} + e^2}$
\vskip 2.0truein 
\item[C)] (10) $\displaystyle{h(x) = \frac{\cos(x) + x}{\sin(x)}}$
\vskip 2.5truein
\item[D)] (10) $i(x) = \ln(x^3+e^x)$
\end{enumerate}
\vfill\eject
\noindent
IV. (continued)
\begin{enumerate}
\item[E)] (10) $j(x) = \tan^{-1}(3x + 7)$
\vskip 2.0truein
\end{enumerate}
\vskip 10pt
\noindent
V.  The following graph shows the {\it derivative}
$f'(x)$ for some function $f(x)$ defined on $0\le x \le 4$.
Note:  This \emph{is not} $y = f(x)$, it is $y = f'(x)$.  
\begin{center}
\includegraphics[width=3truein,height=2truein]{FV.png} 
\end{center} 
Using the graph, {\it estimate}
\begin{enumerate}
\item[A)] (5) The interval(s) on which $f(x)$ is increasing.\\
\MyAnswer{Increasing on:}{$\phantom{A very very very long answer here}$}
\item[B)] (10) The critical numbers of $f(x)$ \emph{in the open interval} $(0,4)$.  
Say what the behavior of $f(x)$ is at each critical number (local max, local min, neither).\\
\MyTallAnswer{Critical Points and behaviors:}{$\phantom{A very very very very long answer here}$}
\item[C)] (10) The interval(s) on which $y = f(x)$ is concave down. \\
\MyAnswer{Concave down on:}{$\phantom{A very very very very long answer here}$}
\end{enumerate}
\vfill\eject
\noindent
VI. A town wants to build a pipeline from a water station on a small
island 2 miles from the shore of its water reservoir to the town. 
One possible route is shown dotted in red. The town is 6 miles along the shore
from the point nearest the island.  It costs \$3 million   
per mile to lay pipe under the water and \$2 million per mile
to lay pipe along the shoreline.
\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{FVI.png} 
\end{center} 

\begin{enumerate} 
\item[A)] (5)  Give the cost $C(x)$ of the pipeline
as a function of the location $x$ as shown. \\
\vskip 1.0in
\item[B)] (5) Differentiate your $C(x)$ to find $C'(x)$.\\
\vskip 1.0in
\item[C)] (5) Find all critical points of $C(x)$.\\
\vskip 2.0in
\item[D)] (5)  Where on the 
shoreline should the pipeline hit land to minimize the costs
of construction?  Say how you know this gives the minimum cost. \\
\end{enumerate}
\vfill\eject
\noindent
VII. (20)  A block of dry ice (solid $CO_2$) is 
evaporating and losing volume at the rate of $10$ cm${}^3$/min. 
It has the shape of a cube at all times.  How 
fast are the edges of cube shrinking when the block has volume 
$216$ cm${}^3$?  
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