%% MATH 135-01 Final Exam B, 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

\[
\text{\it Have a peaceful and joyous holiday season!}
\]

\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{exponential} one and place it
in the appropriate box.  In the box for the other one, write ``Linear.''

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


\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 the formula for the composition $g(f(x))$
\vskip 1truein
\Answer{$g(f(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 $g(x)$ per unit change in $x$ on the interval $[0,2]$.
\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)] (*) $\ds \lim_{x\to 0} (1 + 5x)^{1/x}$
\vskip 3.5truein
\Answer{Limit $=$}
\sep
\item[(b)]  $\ds \lim_{x\to 3} \frac{x^2 - 6x + 9}{x^2 - 5x + 6}$
\vskip 0.8truein
\Answer{Limit $=$}
\sep
\item[(c)]  $\ds \lim_{x\to 0} \frac{\sin(3x)}{x}$
\vskip 0.8truein
\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] Use the definition to compute $f'(x)$ for
$f(x) = \ds \frac{1}{x^2}$.
\vskip 3.0truein

\sep

\item[(c)] [10~points] Find the equation of the tangent line
to the graph $y = \ds \frac{1}{x^2}$ at the point $(2,1/4)$.  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] $\ds f(z) = z^5 - \frac {1}{\sqrt[6]{z}} + e^z$.
\vskip 1.5truein
\Answer{$f'(z) = $}

\sep

\item[(b)] [10 points] $\ds g(x) = (x^3 + 1)(x^2 - 1)$

\vskip 2.5truein

\sep

\item[(c)] (*) [10 points] $\ds h(t) = \frac{\ln(t^2 + 3)}{\sin^{-1}(t)}$

\vskip 2.5truein


\sep

\item[(d)] (*) [10 points] Find $y' = \frac{dy}{dx}$ if $3 x^3 y^2 - 7y^3 + \tan(x) = y$.

\vskip 2.0truein

\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 $f(x)$ increasing?

\Answer{Increasing on:}

\sep

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

\Answer{Concave down on:}

\sep

\item[(c)] How many critical points does $f$ have?  Classify them as 
local maxima, local minima or neither.

\Answer{Critical points:}

\end{enumerate}

\sep
\noindent
7. (*) [20~points] If the ticket price for the upcoming ``Star Wars'' 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 $\$12$ of candy, popcorn and 
drinks at the concession stand.  What is the maximum
revenue that can be earned by the theater from the ticket and concession sales?

\vskip 5truein

\vfill
\Answer{Maximum Revenue:} 
\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, $x$- 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)] $\displaystyle{f(x) = \frac{1}{x^2 - 16}}$\hfill
    ~I\quad ~II\quad ~III\quad
    ~IV\quad ~V

  

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

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

 

  \item[(d)] $\displaystyle{f(x) =\frac{x}{x^2 - 1}}$ \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}
