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\begin{document}
\begin{center}

\textbf{College of the Holy Cross} \\
\textbf{MATH 133, section 1 -- Calculus with Fundamentals}\\
\textbf{Final Exam -- December 15, 2015}
\end{center}
\vskip 10pt
\vfil

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


\vskip 30pt
\noindent\textbf{Instructions}:  Except on problems I and V, you must
{\it  show all work necessary to justify your answers} on the test pages.
Place your final answer in the space provided for the problem.  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.

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 30\  \hfil \\
\hline
II  & \hfil/ 20\  \hfil \\
\hline
III  & \hfil/ 30\  \hfil \\
\hline
IV  & \hfil/ 50\  \hfil \\
\hline
V  & \hfil/ 30\  \hfil \\
\hline
VI  & \hfil/ 20\  \hfil \\
\hline
VII  & \hfil/ 15\  \hfil \\
\hline
VIII  & \hfil/ 5\  \hfil \\
\hline\hline
Total   & \hfil/ 200\hfil \\
\hline
\end{tabular}
\end{center}

\begin{center}
$$\mathcal{HAVE\ \ A\ \ PEACEFUL\ \ AND\ \ ENJOYABLE\ \ HOLIDAY\ \ SEASON}!$$
\end{center}

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\noindent
I.  A portion of the graph $y = f(x)$ is given in black.  Match each equation 
with one of the red graphs identified with lower-case letters.
\vskip 10pt
\begin{figure}
\begin{center}
\includegraphics[width=3truein]{133FI.png} 
\caption{Figure for problem I}
\end{center} 
\end{figure}

\begin{itemize}
\item[A)] (5) $y = -f(x + 5)$ is letter: $\underline{\qquad\qquad}$
\item[]
\item[B)] (5) $y = f(x + 4) - 3.2$ is letter: $\underline{\qquad\qquad}$
\item[]
\item[C)] (5) $y = 3 f(x)$ is letter: $\underline{\qquad\qquad}$
\item[]
\item[D)] (5) $y = f(x) + 2 $ is letter: $\underline{\qquad\qquad}$
\item[]
\item[E)] (5) $y =  f(x - 4)$ is letter: $\underline{\qquad\qquad}$
\item[]
\item[F)] (5) Is $f(x) = xe^{-x}$ $\underline{\qquad\qquad}$
or $f(x) = \cos(x) - 1$ $\underline{\qquad\qquad}$?  (Mark the correct formula).
\end{itemize}
\vfil\eject
\noindent
II.  The power delivered by a battery to an apparatus of resistance $R$ (in ohms)
is 
$$P(R) = \frac{5 R}{(R + 0.5)^2}$$
(in watts).  
\vskip 10pt
\begin{enumerate}
\item[A)] (5) If $R = 10$ ohms, what is the power delivered by the battery?
\vskip 2.0truein
\item[B)] (5) A power of 2 watts can be obtained with two different values of the 
resistance $R$.  What are they? 
\vskip 2.0truein 
\item[C)] (10) What is the \emph{rate of change} of the power at $R = 10$ ohms?
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\end{enumerate}
\noindent
III.  Compute the following limits.  Any legal method is OK.  
\vskip 10pt
\begin{enumerate}
\item[(A)] (10) $\displaystyle{\lim_{x \to 3} \frac{x^2 - 6x + 9}{x^2 - 5x + 6}}$.
\vskip 2.0truein
\item[(B)] (10) $\displaystyle{\lim_{x \to 2} \frac{\sqrt{x + 7} - 3}{x - 2}}$.
\vskip 3.0truein
\item[(C)] (10) $\displaystyle{\lim_{x \to 0} x\cot(x)}$
\end{enumerate}
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\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 $f(x) = \displaystyle{\frac{1}{x - 3}}$.
\vskip 3.0truein
\item[] Using appropriate derivative rules, compute the derivatives
of the following functions.  You do \emph{not} need to simplify your answers.
\item[B)] (10) $\displaystyle{g(x) = \pi x^e + \frac{3}{x^7} + \sqrt[3]{x} + 5}$
\vskip 2.5truein
\item[C)] (10) $\displaystyle{h(x) = \frac{x^2 - 4x + 1}{x^3 + 4}}$
\vskip 2.5truein
\item[D)] (10) $i(x) = x \ln(\cos(x)) + e^{\cos(x)}$
\vskip 3.0truein
\item[E)] (10) $j(x) = \sin^{-1}(3x - 4)$\
\end{enumerate}
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\noindent
V.  The graph in Figure 2 shows the {\it derivative}
$f'(x)$ for some function $f(x)$ defined on $-1.5\le x \le 2.5$.
In particular you should assume $f(1), f(2)$ are defined and finite.
Note:  This graph \emph{is not $y = f(x)$, it is $y = f'(x)$}. 
\begin{figure} 
\begin{center}
\includegraphics[width=3truein]{133FV.png} 
\caption{$y = f'(x)$ for problem V.}
\end{center} 
\end{figure}
Using the graph, answer these questions:
\begin{enumerate}
\item[A)] (5) What are the critical points of $f$ in the interval $[-1.5,2.5]$? 
\item[] Answer: $x = \underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item[B)] (5)  Classify each of the points you found in part A) as a local 
maximum, a local minimum, or neither.
\item[] Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item[C)] (5) Explain briefly how you know your answer in B) is correct.
\vskip 25pt
\item[D)] (5)  Is $x = 0$ a point of inflection of $f$?  Why or why not?
\item[] Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item[E)] (5)  Does it appear that $f''(2)$ exists? (Yes/No):  
\item[] Answer: $\underline{\qquad\qquad\qquad}$
\item[F)] (5)  Over which interval(s) contained in $[-1.5,2.5]$ is the graph $y = f(x)$
concave down?
\item[] Answer: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\end{enumerate}
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\noindent
VI.  A Spanish factory can produce $P = 2 L K^2$ million lightbulbs
if labor costing $L$ million euros is hired and equipment costing $K$ million euros 
is obtained.  If an order for $1.7$ million lightbulbs is received, which 
combination of $L$ and $K$ will \emph{minimize} the total cost of labor plus equipment
to fill the order?
\begin{enumerate} 
\item[A)] (5) Express the total cost of producing $1.7$ million lightbulbs
as a function of one of the two variables $K,L$.  (Your choice, but choose wisely!)
\vskip 2.0truein 
\item[B)] (10) Find a critical point of your function from part A which is 
a realistic solution of this problem; solve for the other variable.
\vskip 3.0truein
\item[C)] (5) How do you know that your solution \emph{minimizes} the total cost?
\end{enumerate}
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\noindent
VII. (15) A moth ball is 
evaporating and losing volume at the rate of $.1$ cm${}^3$/week. 
It has the shape of a sphere at all times.  How 
fast is the surface area of the moth ball shrinking when the
radius of the ball is $1$cm?  (Note: A sphere of radius $r$
has volume $V = \frac{4\pi r^3}{3}$ and surface area $A = 4\pi r^2$.) 
\vfil\eject
\noindent
VIII. (5) True/False:  The graph
obtained by shifting $y = \ln(x)$ vertically by 2 units can
{\it also} be obtained from $y = \ln(x)$ by a horizontal compression.  Explain 
your answer.
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