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\begin{center}

\textbf{College of the Holy Cross} \\
\textbf{MATH 135, section 1}\\
\textbf{Exam 3 -- Friday, December 6}
\end{center}

\vfil

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


\vskip 30pt
\noindent\textbf{Instructions}:  For full credit, you must show 
{\it all work} on the test pages and place your final answer
in the box 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.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss \\
\hline\hline
I  & \hfil/ 25\  \hfil \\
\hline
II  & \hfil/ 30\  \hfil \\
\hline
III  & \hfil/ 20\  \hfil \\
\hline
IV  & \hfil/ 25\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
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\vfil\eject

\noindent
I.  For each of the following functions find the derivative.  You do not need to simplify. 
\begin{enumerate}
\item[A.] (5) $\displaystyle f(x) = e^{x} \sin(5x) + \sec(x)$\\
\MyAnswer{$f'(x) = $}{$\phantom{A very very very very very long answer here}$}%
\vskip 2truein
\item[B.] (5) $\displaystyle g(x)= \frac{\cos(x)}{\ln(x)}$ 
\MyTallAnswer{$g'(x) = $}{$\phantom{A very very very long answer here}$}%
\vskip 2.5truein
\item[C.] (5) $\displaystyle h(x)=\tan^{-1}(x^2 - 4)$
\MyTallAnswer{$g'(x) = $}{$\phantom{A very very very long answer here}$}%
\vfill\eject
\item[D.] (5) Find $\frac{dy}{dx}$ by implicit differentiation if $\displaystyle x^2y^3 + 2y = 3$.  \\ 
\MyTallAnswer{$\frac{dy}{dx} = $}{$\phantom{A very very very very long answer here}$}%
\vskip 3truein
\item[E.] (5) Find all the $c$ in the open interval $(0,4)$ for which the conclusion of 
the Mean Value Theorem is true for $f(x) = x^3 - 4x^2 + 3x$ on the interval $[a,b] = [0,4]$.\\
\MyAnswer{$c = $}{$\phantom{A very very very very very long answer here}$}%
\end{enumerate}
\vfil\eject

\noindent
II. A poster is to be printed on a rectangular sheet of paper with 1 inch margins along 
the sides and 1.5 inch margins at the top and bottom.  The rectangular printed region inside the margins 
must be 500 square inches in area.  Find the dimensions of the overall sheet of the smallest 
possible area meeting these requirements.  Let $x$, $y$ be the width and height \emph{of the printed area}.  
\begin{enumerate}
\item[A.] (5) Draw a picture representing the whole poster and label the sides of length $x$ and $y$ clearly.
\vskip 2truein
\item[B.] (10) Express the area of the whole poster in terms of the one variable
$x$--the width of the printed area.
\vskip 1.5truein
\item[C.] (10) Determine a critical point of your function from B.  Why is it a minimum?
\vskip 2truein
\item[D.] (5) What are the dimensions of the whole poster of minimum area?
\end{enumerate}
\vfil\eject
\noindent
\noindent
III.  All parts of this question refer to the plot in Figure 1, which shows $y = f'(x)$ for some function $f$.
\begin{figure}
\begin{center}
\includegraphics[width=.4\linewidth]{P3QualProb.png}
\caption{Figure for Problem III.}
\end{center}
\end{figure}
\begin{enumerate}
\item[A.] (10) Recall that the plot shows $y = f'(x)$(!) 
Approximately where does the graph $y = f(x)$ have inflection points?

\MyAnswer{Inflection points at $x = $ }{$\phantom{A very long answer here}$}%

\item[B.] (10) Recall that the plot shows $y = f'(x)$(!)  Where are the critical points of $f$ and 
what types are they (i.e. local maximum, local minimum, neither)?

\MyTallAnswer{Answer: }{$\phantom{A very very very very very very long answer here}$}%

\end{enumerate}
\vfil\eject
\noindent
IV.  All parts of this question refer to the function $\displaystyle f(x) = \frac{x}{(2x + 1)^2}$,
for which the first two derivatives are, in simplified form:
\begin{eqnarray*}
f'(x) &=& \frac{1 - 2x}{(2x + 1)^3}\\
f''(x) &=& \frac{8x - 8}{(2x + 1)^4}
\end{eqnarray*} 
\bigskip
\begin{enumerate}

\item[A.] (10) Does $y = f(x)$ have any horizontal asymptotes?  Use L'Hopital's Rule.
\vskip 2truein

\MyAnswer{Horizontal asymptote at $y = $}{$\phantom{A very long answer here}$}%\

\item[B.] (10) What is the maximum value of $f(x)$ on the interval $[0,3]$?

\vskip 2truein

\MyAnswer{Maximum value: }{$\phantom{A very very long answer here}$}%

\item[C.] (5)  Over which interval(s) is the graph $y = f(x)$ concave up?\\
\MyAnswer{Concave up on:}{$\phantom{A very very very very very long answer here}$}%
\end{enumerate}





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