%% MATH 135-01 Final Exam A, Fall 2019
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\begin{document}

\begin{center}

\textbf{Holy Cross College, Fall Semester, 2019} \\
\textbf{MATH 135, Section 01, Final Exam} \\
\textbf{Saturday, December~21, 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.
The questions that will be included in the Exam III subscore are marked with an asterisk ($*$).

\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/\  20\hfil \Strut \\
\hline
2       & \hfil/\  30\hfil \Strut \\
\hline
3       & \hfil/\  30\hfil \Strut \\
\hline
4       & \hfil/\  30\hfil \Strut \\
\hline
5       & \hfil/\  40\hfil \Strut \\
\hline
6       & \hfil/\  15\hfil \Strut \\
\hline
7       & \hfil/\  20\hfil \Strut \\
\hline
8       & \hfil/\  15\hfil \Strut \\
\hline\hline
Exam III Subscore & \hfil/80\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

\noindent 
1. [20~points] One of the functions given in the following table is linear
and the other is exponential.  Find a formula for the \emph{linear} one and place it
in the appropriate box.  In the box for the other one, write ``Exponential.''

\[
\begin{array}{|c||r|r|r|r|r|}
\hline
x          & 1 & 2 & 3& 4& 5\Strut \\
\hline
f(x)       & 1.2 & 2.4 & 4.8 & 9.6 & 19.2 \Strut \\
\hline
g(x)       & -3.3& -4.4& -5.5 & -6.6 & -7.7\Strut \\
\hline
\end{array}
\]

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

\eject
\noindent
2.  Let $f(x) = x^3 + 3x + 1$ and $g(x) = \sqrt{x^2 + 3}$.
\begin{enumerate}
\item[(a)] [10~points]  Find a formula for the composition $f(g(x))$
\vskip 1truein
\Answer{$f(g(x)) = $}
\sep
\item[(b)] [10~points] Write $g(x) = h(k(x))$ for two other functions $h$ and $k$, where 
$h(x) \ne x$ and $k(x) \ne x$.  
\vskip 10pt
\Answer{$h(x) = $}\\
\Answer{$k(x) = $}
\sep
\item[(c)] [10~points]  Find the average rate of change of $f(x)$ per unit change in $x$
on the interval $[1,3]$.
\vfill
\Answer{Average rate of change $=$}
\end{enumerate}
\eject

\noindent
3.  Compute the following limits [10~points each]. \emph{Any legal method is OK.}
\begin{enumerate}
\item[(a)] $\ds \lim_{x\to 0} \frac{\sin(4x)}{x}$
\vskip 1.0truein
\Answer{Limit $=$}
\sep
\item[(b)] $\ds \lim_{x\to 2} \frac{x^2 - 8x + 12}{x^2 - 2x}$
\vskip 1.5truein
\Answer{Limit $=$}
\sep
\item[(c)] (*) $\ds \lim_{x\to \infty} \left(1 + \frac{3}{x}\right)^x$
\vskip 2.5truein
\Answer{Limit $=$}
\end{enumerate}
\eject

\noindent
4. 
 \begin{enumerate}
\item[(a)] [10~points] State the limit definition of the derivative:

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

\sep

\item[(b)] [10~points] \emph{Use the definition} to compute $f'(x)$ for
$f(x) = \ds \frac{1}{x + 1}$.
\vskip 3.0truein

\sep

\item[(c)] [10~points] Find the equation of the tangent line
to the graph $y = \ds \frac{1}{x + 1}$ at the point $(2,1/3)$.  Note: you can do this one even
if you were not able to complete part b above.
\vskip 2.0truein
\Answer{Tangent line:}
\end{enumerate}
\eject

\noindent
5.  Compute the following derivatives using the derivative rules.
You need not simplify. 
\begin{enumerate} 
\item[(a)] [10 points] $\ds f(x) = x^{4/5} - \frac {1}{\sqrt[3]{x}} + e^x$.
\vskip 1.5truein

\sep

\item[(b)] [10 points]  $\ds g(x) = \frac{x^2 + 1}{x^4 + 1}$

\vskip 2.5truein

\sep

\item[(c)] (*) [10 points] $\ds h(s) = \ln(4s^2 + 2)\tan^{-1}(s)$

\vskip 2.5truein

\sep
\eject

\item[(d)] (*) [10 points] Find $y' = \frac{dy}{dx}$ if $4 x^2 y^3 - 2y^2 + \sin(x) = x^4$.

\vskip 1.5truein
\end{enumerate}

\begin{figure}
\begin{center}
\includegraphics[width = .5 \linewidth]{135F6.png}
\caption{Plot for question 6.}
\end{center}
\end{figure}
\sep
\noindent
6. (*) [5~points each] All parts of this question refer to the function $f(x)$ whose
\emph{derivative} $f'(x)$ is plotted in Figure 1 above.  Assume the domain of $f$
contains only the interval shown in the plot.

\begin{enumerate}
\item[(a)]  On which interval(s) is $f(x)$ decreasing?

\Answer{Decreasing on:}

\sep

\item[(b)] On which interval(s) is $y = f(x)$ concave up?

\Answer{Concave up on:}

\sep

\item[(c)] How many inflection points does $f$ have?  

\Answer{Inflection points:}

\end{enumerate}

\sep

\noindent
7. (*). [20~points] If the ticket price for the upcoming ``Cats'' movie is set at
$\$10$, then 1000 tickets for the first showing will be sold.  However, for each
$\$0.25$ increase in the ticket price, the number of tickets sold will go down by
10 tickets.  Each person who buys a ticket will also purchase $\$10$ of nachos and 
drinks at the concession stand.  How should the ticket price be set to maximize
the revenue earned by the theater from the ticket and concession sales?

\vskip 4truein

\vfill
\Answer{Price:}
\eject

\begin{figure}
\centering
\subfloat[Plot I]{\includegraphics[width=1.4in]{135F8I.png}}
\subfloat[Plot II]{\includegraphics[width=1.4in]{135F8II.png}}\\
\subfloat[Plot III]{\includegraphics[width=1.4in]{135F8III.png}}
\subfloat[Plot IV]{\includegraphics[width=1.4in]{135F8IV.png}}\\
\subfloat[Plot V]{\includegraphics[width=1.4in]{135F8V.png}}
\end{figure}


\noindent
8. (*) [3~points each]\quad By considering the location of vertical and 
horizontal asymptotes in the plots above, $x$- and $y$-axis intercepts, etc. determine which of 
the following functions matches each graph.  Circle the number of the graph
   showing each of the following functions. 


\begin{enumerate}
  \item[(a)] $\displaystyle{f(x) =\frac{x}{x^2 - 1}}$ \hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V


  \item[(b)] $\displaystyle{f(x) = \frac{x}{x^2 + 1}}$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V


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


  \item[(d)] $\displaystyle{f(x) = \frac{1}{x^2 - 16}}$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

  \item[(e)] $\displaystyle{f(x) = \frac{3x + 1}{x - 1}}$\hfill
     ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

\end{enumerate}


\end{document}
