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\begin{document}
\begin{center}

\textbf{College of the Holy Cross, Fall 2007} \\
\textbf{Math 131, Midterm 3} (All Sections) \\
\textbf{Wednesday, November 28, 6 PM}
\end{center}

\vfil

\noindent\textbf{Your Name}: $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\vskip 0pt plus .3fil
\noindent\textbf{Your Section}: 

\vskip 15pt
\leftline{Little (8:00am) $\underline{\qquad}$ Ballantine (9:00am) $\underline{\qquad}$} \leftline{DeStefano (10:00am) $\underline{\qquad}$ DeStefano (noon) $\underline{\qquad}$}

\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/ 15\  \hfil \\
\hline
III  & \hfil/ 20\  \hfil \\
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IV  & \hfil/ 10\  \hfil \\
\hline
V  & \hfil/ 15\  \hfil \\
\hline
VI  & \hfil/ 15\  \hfil \\
\hline\hline
Total   & \hfil/100\hfil \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\noindent
I.  For each of the following functions find the derivative and simplify. 
\begin{enumerate}
\item[A.] (5) $\displaystyle f(x) = \sin(2x)\cos(4x)$
\MyAnswer{$f'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 1.75truein
\item[B.] (5) $\displaystyle g(x)= \frac{1+\ln x}{1 - x}$ 
\MyTallAnswer{$g'(x) = $}{$\phantom{A very very long answer here}$}%
\vskip 2.5truein
\item[C.] (5) $\displaystyle h(x)=\tan^{-1}(4x^2 + x)$
\MyTallAnswer{$h'(x) = $}{$\phantom{A very very long answer here}$}%
\vskip 2.5truein
\vfil\eject
\item[D.] (5) $\displaystyle k(x) = x^{\tan x}$
\MyTallAnswer{$k'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 3.0truein
\item[E.] (5) Find the equation of the tangent line to the curve $\displaystyle x^2y^3+2y=3x$
at the point $(2,1)$.   
\MyAnswer{Tangent line: }{$\phantom{A very very very very long answer here
}$}%
\end{enumerate}
\vfil\eject

\noindent
II. (15)  A rocket is launched vertically and is tracked by a ground
station $3$ miles from the launch pad.  What is the vertical speed 
of the rocket when its height above the ground is $4$ miles and 
its distance to the ground station is increasing at $3600$ miles per
hour?   
\vskip 7.0truein
\MyAnswer{Answer: }{$\phantom{A very very very very long answer here}$}%
\vfil\eject


\noindent
III.  All parts of this question refer to the function $\displaystyle f(x) = \frac{x}{(2x + 1)^2}$.
\bigskip
\begin{enumerate}
\item[A.] (2) What is the domain of $f(x)$?
\MyAnswer{Domain: }{$\phantom{A very long answer here}$}%

\item[B.] (5) Find all critical numbers and determine where $f(x)$ is
increasing and decreasing.

\MyAnswer{Critical numbers: }{$\phantom{A very long answer here}$}%

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

\MyAnswer{Decreasing: }{$\phantom{A very very long answer here}$}%
\vskip 1.0truein
\item[C.] (2) Find all local maximum and minimum values of $f(x)$.

\MyAnswer{Local maximum values: }{$\phantom{A very very long answer here}$}%

\MyAnswer{Local minimum values: }{$\phantom{A very very long answer here}$}%

\vskip 1.0truein

\item[D.] (3) Determine the concavity of $f(x)$, and find any inflection points given
that $$\displaystyle f''(x) = \frac{8(x-1)}{(2x+1)^4}.$$

\MyAnswer{Concave up: }{$\phantom{A very very long answer here}$}%

\MyAnswer{Concave down: }{$\phantom{A very very long answer here}$}%

\MyAnswer{Inflection points: }{$\phantom{A very very long answer here}$}%
\vskip 5pt
\rightline{\it (problem continues on following page)}
\vskip 0.5truein

\vfil\eject

\bigskip

\item[E.] (2) Find all asymptotes for the graph of $f(x)$. Hint: You may use the
following facts if necessary:
$$\lim_{x\rightarrow -0.5^+} f(x)=-\infty, \text{ and } \lim_{x\rightarrow -0.5^-} f(x)=-\infty$$

\MyAnswer{Asymptotes: }{$\phantom{A very very long answer here}$}%
\vskip 1.0truein
\item[F.] (6) Sketch the graph of $f(x)$ using the information above.  Label
any maximum and minimum values, inflection points, asymptotes,
and any other points which help you to give a clear and accurate picture. 
\end{enumerate}


\vfil\eject


\noindent
IV.  All parts of this question refer to the function $\displaystyle f(x)=x^3 - 12x - 7$.
\begin{enumerate}
\item[A.] (5) Find the critical numbers of $f(x)$.

\MyAnswer{Critical numbers: }{$\phantom{A very long answer here}$}%
\vskip 3.0truein

\item[B.] (5) What does the Second Derivative Test tell you about the behavior of $f$ at 
these critical numbers?

\MyTallAnswer{Answer: }{$\phantom{A very very very very very very long answer here}$}%
\end{enumerate}
\vfil\eject
\noindent
V. (15)  A rectangular outdoor pen is to be added to a barn with a $5$ meter by $10$ meter
corner notch as shown in the diagram below.  If $85$ meters of new fencing is available,
what is the maximum area that can be enclosed? No fencing is 
needed along the walls of the barn.  Be sure to say how you know your solution 
gives the maximum area.
\begin{center}
\includegraphics[width=2.8truein,height=2.8truein]{P3F1.ps} 
\end{center}
\vskip 4.5truein
\MyAnswer{Answer: }{$\phantom{A very very very very long answer here}$}%
\vfil\eject

\noindent
VI.  Find the following limits. (Just an answer is not sufficient; you must show 
work for full credit.)
\begin{enumerate}
\item[A.] (3) $\displaystyle \lim_{x \rightarrow 0} \frac{x^3+2\sqrt{x}}{5 x^3+2}$ 
\MyAnswer{Limit: }{$\phantom{A long answer here}$}%
\vskip 1.5truein
\item[B.] (4) $\displaystyle \lim_{x \rightarrow 2} \frac{e^{x^2}-e^4}{x-2}$
\MyAnswer{Limit: }{$\phantom{A long answer here}$}%
\vskip 1.75truein
\item[C.] (4) $\displaystyle \lim_{x \rightarrow 0} \frac{e^x-x-1}{\cos x - 1}$
\MyAnswer{Limit: }{$\phantom{A long answer here}$}%
\vskip 1.75truein
\item[D.] (4) $\displaystyle \lim_{x \rightarrow \infty}\frac{\ln x}{\sqrt[4]{x}}$
\MyAnswer{Limit: }{$\phantom{A long answer here}$}%
\end{enumerate}
\end{document}
