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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 7, 2015}
\end{center}
\vskip 10pt

\noindent
{\it General Information}
\bigskip
\begin{itemize}
\item  The final examination will be
given 11:30am to 2:00pm on Tuesday, December 15, in Swords 321 
(where our class has met on Mondays and Fridays).
\item The final will be similar in format to the 
midterm exams but perhaps 1.5 times as long.  If you are well prepared 
and you work steadily, then you should be able to finish the exam in about 
1.75 hours.  However, you will have the full 2.5 hour period to work on 
the exam if you need that much time.
\item The final will be a {\it comprehensive exam} -- 
it will cover all the topics we have studied this semester,
with roughly equal weight given to the four sections of the 
course corresponding to the four midterm exams.  
\item The review sheets for the four midterms are 
reposted on the course homepage if you need another copy of any of them.
\item It may not be necessary to say this, but
here goes anyway:  {\it You should take this exam seriously} -- 
it is worth $20\%$ of your course average and it can 
pull your course grade up {\it or down} depending on how you do.  
\item Get started reviewing over the weekend.  Don't try to ``cram'' 
the night before.
\item Review the videos, your class notes, and the 
text, especially for topics where you lost points on the midterms.  
There are a lot of worked-out examples and discussions of all of 
the topics we have covered there.
\item Be sure you actually do 
enough practice problems so that you have the 
facility to solve exam-type questions in a limited
amount of time.   {\it Even if you 
have saved solutions for practice problems from the midterms},
it is going to be much more beneficial to do practice problems
starting ``from scratch'' rather than just reading old
solutions.
\item You may use a calculator, but
no graphing features.  No cell-phones, computers, or other electronic
devices beyond a basic calculator may be used during the exam.  Please do not bring them
with you; they will be subject to confiscation for the period
of the exam if you use them. 
\item Lauren is arranging
a review session before the exam.  We can discuss this in 
class on Friday, December 11.
\item I will also be available 8 - 10am and 3 - 5pm on Monday, December 14
for ``last minute'' questions. 
\item The following is (a slightly edited version of) the final I gave the last time I taught
a Calculus 1 course, in fall 2013.  It's a good guide for what our exam will look like.
\end{itemize}
\vfill\eject

\noindent
I.  The  graph $y = f(x)$ is given in blue (more like cyan).  Match each equation with one of the 
numbered pink (actually, magenta) graphs.
\vskip 10pt
\begin{figure}
\begin{center}
\includegraphics[width=3truein,height=2truein]{FI.png} 
\caption{Figure for problem I}
\end{center} 
\end{figure}

\begin{itemize}
\item[A)] $y = f(x - 4)$ is plot number: $\underline{\qquad\qquad}$
\item[B)] $y = f(x) + 3$ is plot number: $\underline{\qquad\qquad}$
\item[C)] $y = \frac{1}{3} f(x) $ is plot number: $\underline{\qquad\qquad}$
\item[D)] $y = -f(x + 4)$ is plot number: $\underline{\qquad\qquad}$
\item[E)] $y = 2 f(x + 6)$ is plot number: $\underline{\qquad\qquad}$
\end{itemize}
\vskip 10pt
\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
$C(t) = 70 + 80e^{-t/3}$ (in degrees F).
\vskip 10pt
\begin{enumerate}
\item[A)] What is the temperature of the chocolate at $t = 0$?
\item[B)] What is the rate of change of the temperature at $t = 10$ minutes?
\item[C)] How long does it take for the temperature to 
reach $100^\circ F$?  
\end{enumerate}
\noindent
III.  Compute the following limits.  Any legal method is OK.  
\vskip 10pt
\begin{enumerate}
\item[(A)]  $\displaystyle{\lim_{x \to 3} \frac{x^2 + x - 12}{x^2 - 5x + 6}}$.
\item[(B)]  $\displaystyle{\lim_{x \to 1^-} \frac{|x - 1|}{x^2 - 1}}$.
\item[(C)]  $\displaystyle{\lim_{x \to 0} \frac{\tan(x)}{x}}$
\end{enumerate}
\vskip 10pt
\noindent
IV.  
\begin{enumerate}
\item[A)]  {\it Using the limit definition}, and showing all necessary steps
to justify your answer, compute $f'(x)$ for $f(x) = 5x^2 - x + 3$.
\item[] Using appropriate derivative rules, compute the derivatives
of the following functions.  You do not need to simplify your answers.
\item[B)]  $\displaystyle{g(x) = 4x^3 +  \sqrt{x} + \frac{2}{\sqrt[4]{x}} + e^2}$
\item[C)]  $\displaystyle{h(x) = \frac{\sin(x) + x}{\sec(x)}}$
\item[D)]  $i(x) = \ln(x^3+3)$
\item[E)]  $j(x) = \tan^{-1}(12x + 2)$\
\end{enumerate}
\vskip 10pt
\noindent
V.  The graph in Figure 2 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{figure} 
\begin{center}
\includegraphics[width=3truein,height=2truein]{FV.png} 
\caption{Figure for problem V.}
\end{center} 
\end{figure}
Using the graph, {\it estimate}
\begin{enumerate}
\item[A)] The interval(s) on which $f(x)$ is increasing.
\item[B)] The critical points 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).
\item[C)] The interval(s) on which $y = f(x)$ is concave down. 
\end{enumerate}
\vskip10pt
\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{figure}
\begin{center}
\includegraphics[width=2.5truein,height=2.5truein]{FVI.png} 
\end{center} 
\caption{Figure for problem VI.}
\end{figure}

\begin{enumerate} 
\item[A)] Give the cost $C(x)$ of the pipeline
as a function of the location $x$ as shown. 
\item[B)] Where along the 
shoreline should the pipeline hit land to minimize the costs
of construction?  Say how you know this gives the minimum cost.
\end{enumerate}
\vskip 10pt
\noindent
VII.  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${}^3$? 
\vskip 10pt
\noindent
VIII.  True or false:  The graph
obtained by stretching $y = e^{-x}$ vertically by a factor of 2 can
{\it also} be obtained from $y = e^{-x}$ by a horizontal shift.  Explain 
your answer.
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