%% MATH 135-01 Final Exam, Fall 2016
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\begin{document}

\begin{center}

\textbf{Holy Cross College, Fall Semester, 2016} \\
\textbf{MATH 135, Section 01, Final Exam} \\
\textbf{Friday, December~16, 8:00~AM} \\
\end{center}

\vfil

\noindent{\bf Your Name}: \Name
\vskip 0pt plus .3fil

\noindent
{\it Instructions.} Clearly mark your answers, and show work on
the test itself. Use the back of the preceding page if you need more
space for scratch work. {\it You must show all work for full
credit}, but please place answers in the boxes provided where appropriate.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Possible    \Strut \\
\hline\hline
1       & \hfil/\  25\hfil \Strut \\
\hline
2       & \hfil/\  20\hfil \Strut \\
\hline
3       & \hfil/\  25\hfil \Strut \\
\hline
4       & \hfil/\  15\hfil \Strut \\
\hline
5       & \hfil/\  25\hfil \Strut \\
\hline
6       & \hfil/\  20\hfil \Strut \\
\hline
7       & \hfil/\  30\hfil \Strut \\
\hline
8       & \hfil/\  20\hfil \Strut \\
\hline
9       & \hfil/\  20\hfil \Strut \\
\hline\hline
Exam I Subscore & \hfil/60\hfil \Strut\\
\hline\hline
Total   & \hfil/200\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\vfil

\[
\mathscr{H\!A\!P\!P\!Y\ \  H\!O\!L\!I\!D\!A\!Y\!S}!
\]

\eject


\begin{figure}
\centering
\subfloat[Plot I]{\includegraphics[width=1.2in]{135FIc.png}}
\subfloat[Plot II]{\includegraphics[width=1.2in]{135FIe.png}}\\
\subfloat[Plot III]{\includegraphics[width=1.2in]{135FIa.png}}
\subfloat[Plot IV]{\includegraphics[width=1.2in]{135FId.png}}\\
\subfloat[Plot V]{\includegraphics[width=1.2in]{135FIb.png}}
\end{figure}


\noindent
1. [5~points each]\quad Circle the number of the graph
   showing each of the following functions.  Note that there is one ``extra''
graph that does not match any of these functions.


\begin{enumerate}
  \item[(a)] $f(x) = e^{-x} + 2$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V


  \item[(b)] $f(x) = x^3 - 4x$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V


  \item[(c)] $f(x) = 2\cos(2\pi x)$\hfill 
    ~I\quad ~II\quad
    ~III\quad ~IV\quad ~V


  \item[(d)] $f(x) = \ds \frac{1}{x^2 - 9}$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

  \item[(e)] Give a formula of a function that matches the graph you 
     did not circle.

  \Answer{$f(x) = $}

\end{enumerate}

\vfil\eject
\noindent 
2. [20~points] One of the functions given in the following table is linear
and the other is exponential.  Find a formula for the linear one and place it
in the appropriate box.  In the box for the other one, write ``Exponential.''

\[
\begin{array}{|c||r|r|r|r|r|}
\hline
x          & 1 & 2 & 3& 4& 5\Strut \\
\hline
f(x)       & 1.2 & 0.6 & 0.3 & 0.15 & 0.075 \Strut \\
\hline
g(x)       & -2.3& -0.6& 1.1 & 2.8 & 4.5\Strut \\
\hline
\end{array}
\]

\vfill
\Answer{$f(x) = $}\\
\Answer{$g(x) = $}

\eject
\noindent
3.  
\begin{enumerate}
\item[(a)] [15~points]  The depth of water in a tank oscillates sinusoidally
once every 4 hours according to $d(t) = 2\cos\left(\frac{\pi t}{2}\right) + 4$.
Sketch the graph of the depth versus time.
\vskip 3.0truein


\sep

\item[(b)] [10~points]  Find the average rate of change of the depth on the interval $[1,1.1]$.
\vfill
\Answer{Average rate of change $=$}
\end{enumerate}
\eject

\noindent
4.  Compute the following limits [5~points each].  Any legal method is OK.
\begin{enumerate}
\item[(a)]  $\ds \lim_{x\to 2} \frac{x^3 + 2x}{x - 4}$
\vskip 1.7truein
\Answer{Limit $=$}
\sep
\item[(b)]  $\ds \lim_{x\to 2} \frac{x^2 - 7x + 10}{x^2 - 5x + 6}$
\vskip 1.7truein
\Answer{Limit $=$}
\sep
\item[(c)]  $\ds \lim_{x\to \infty} \frac{5x^2 - x + 21}{8x^2 - 9x + 1}$
\vskip 1.7truein
\Answer{Limit $=$}
\end{enumerate}
\eject

\noindent
5. 
 \begin{enumerate}
\item[(a)] [5~points] State the limit definition of the derivative:

\Answer{$f'(x)=$}

\sep

\item[(b)] [10~points] Use the definition to compute $f'(x)$ for
$f(x) = 3\sqrt{x + 2}$.
\vskip 3.0truein

\sep

\item[(c)] [10~points] Find the equation of the tangent line
to the graph $y = 3\sqrt{x + 2}$ at the point $(2,6)$.
\vskip 2.0truein
\Answer{Tangent line $=$}
\end{enumerate}
\eject

\noindent
6.  Compute the following derivatives using the derivative rules.
You need not simplify. [5~points each]
\begin{enumerate} 
\item[(a)]  $\ds f(t) = t^4 - \frac {1}{\sqrt[5]{t}} + e^t$.
\vskip 1.5truein
\Answer{$f'(t) = $}

\sep

\item[(b)]  $\ds g(x) = \frac{x^2 - 2}{\cos(x) + 1}$

\vskip 1.5truein
\Answer{$g'(x) = $}

\sep

\item[(c)] $\ds h(z) = \ln(4z^2 + 2 \tan^{-1}(z))$

\vskip 1.5truein
\Answer{$h'(z) = $}

\sep
\eject

\item[(d)] Find $\frac{dy}{dx}$ if $5 x^2 y^2 - 2y^5 + x = 1$.

\vskip 1.5truein
\Answer{$\frac{dy}{dx} = $}
\end{enumerate}

\sep
\noindent
7.  All parts of this question refer to the functions defined by 
$f(x) = x^4 + 2ax^2$, where $a$ is any fixed real number.  

\begin{enumerate}
\item[(a)]  [10~points]  Assuming $a < 0$, find the {\it critical points} of $f$, 
and construct a sign diagram for $f'(x)$.  Which of your 
critical points are local maxima and which are local minima?

\vskip 2.5truein

\sep 
\item[(b)]  [10~points] Repeat part a, but assume now that $a > 0$.  

\end{enumerate}
\eject

\begin{enumerate}
\item[(c)] [10~points] How many different {\it inflection points} does
the graph $y = f(x)$ have if $a < 0$?  Explain.
\end{enumerate}

\vskip 2.0truein

\sep

\noindent
8. [20~points] The radius and the height of a circular cone increase at
a rate of 2 cm/sec.  How fast is the volume of the cone increasing when $r = 10$
and $h = 20$?

\vfill
\Answer{Answer:}

\eject
\noindent
9.  [20~points] A rectangular poster is to have total area 600 square inches, including
blank 1 inch wide margins on all four sides of a central printed area.  What overall
dimensions will maximize the printed area?  
\vfill
\Answer{Dimensions:}
\eject


\end{document}
