
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,epsfig}
\newcommand{\ds}{\displaystyle}
\newcommand{\vp}{$\vphantom{\big|}$}

\newcommand{\R}{\textbf{R}}
\newcommand{\Item}{\item{$\Box$}\quad}

\newcommand{\Answer}[1]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt \fbox{$\phantom{\Big|Answer}$}} 

\newcommand{\MyTallAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 20pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\MyAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fboxsep 5pt\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\soln}{\paragraph{Solution}}
\newcommand{\sep}{{\medskip\hrule\bigskip}}
\newcommand{\blank}{\underline{$\phantom{A blank}$}}

\setlength{\textwidth}{6.5in}
\setlength{\topmargin}{-0.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{9.0in}
\begin{document}
\begin{center}

\textbf{College of the Holy Cross} \\
\textbf{MATH 135, section 1}\\
\textbf{Exam 3 -- Friday, November 22}
\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/ 35\  \hfil \\
\hline
II  & \hfil/ 20\  \hfil \\
\hline
III  & \hfil/ 20\  \hfil \\
\hline
IV  & \hfil/ 25\  \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) $\displaystyle f(x) = e^{2x} \cos(4x)$
\MyAnswer{$f'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 1.75truein
\item[B.] (5) $\displaystyle g(x)= \frac{x^2 + 2}{\ln(x)}$ 
\MyTallAnswer{$g'(x) = $}{$\phantom{A very very long answer here}$}%
\vskip 2.5truein
\item[C.] (5) $\displaystyle h(x)=\tan^{-1}(2x + 1)$
\MyTallAnswer{$h'(x) = $}{$\phantom{A very very long answer here}$}%
\vskip 2.5truein
\vfil\eject
\item[D.] (10) $\displaystyle k(x) = x^{\sin(x)}$
\MyTallAnswer{$k'(x) = $}{$\phantom{A very very very very long answer here}$}%
\vskip 3.0truein
\item[E.] (10) Find $y'$ by implicit differentiation if $\displaystyle x^3y^2 + 2y = 3x$.  \\ 
\MyTallAnswer{$y' = $}{$\phantom{A very very very very long answer here}$}%
\vskip 10pt
\end{enumerate}
\vfil\eject

\noindent
II. (20)  A rocket is launched vertically and is tracked by a ground
station $4$ miles from the launch pad.  What is the vertical speed 
of the rocket when its height above the ground is $3$ miles and 
its distance to the ground station is increasing at $3300$ miles per
hour?   
\vskip 7.0truein
\MyAnswer{Answer: }{$\phantom{A very very very very long answer here}$}%
\vfil\eject
\noindent
\noindent
III.  All parts of this question refer to the function $\displaystyle f(x) = -2x^3 + 24x + 11$.
\begin{enumerate}
\item[A.] (10) Find the critical numbers of $f(x)$.

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

\item[B.] (10) What does the Second Derivative Test tell you about the behavior of $f$ at each of
these critical numbers?

\MyTallAnswer{Answer: }{$\phantom{A very very very very very very long answer here}$}%
\vskip 3.0truein

\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.] (5) What is the domain of $f(x)$?
\MyAnswer{Domain: }{$\phantom{A very long answer here}$}%

\item[B.] (10) 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: }{$\phantom{A very very long answer here}$}%
\vfil\eject

\item[C.] (5) What is the absolute minimum value of $f(x)$ on the interval $[0,3]$?

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

\vskip 3.0truein

\item[D.] (5) Determine the concavity of $f(x)$, and find all inflection points.

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

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

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


\end{enumerate}





\end{document}
