%% Math 131-01 Midterm 2, Fall 2003

\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,epic,eepic}

\pagestyle{myheadings}
\markright{Math 131-01, Midterm}

%% \setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in}

\def\Answer#1{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|Put answer here}$}}
\def\ShortAnswer#1{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|answer}$}}

\def\Blank{$\underline{\phantom{Please write your name in this
        space}}$} 
\def\Strut{$\vphantom{\Big|}$}

\def\bd{\begin{description}}
\def\ed{\end{description}}
\def\ds{\displaystyle}
\def\vp{\vphantom{\big|}}

\begin{document}

\begin{center}

\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Midterm 2, Early Version} \\
\textbf{Thursday, October 30, 1 PM} \\
Professor Hwang
\end{center}

\vfil

\noindent{\bf Name}: \Blank
\vskip 0pt plus .3fil

\paragraph{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. \textbf{You must show all work for full
  credit}.

\vfil

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

\begin{center}
\begin{tabular}{|l||r|}
\hline
Problem & Points/Poss    \Strut \\
\hline\hline
1       & \hfil/\  30\hfil \Strut \\
\hline
2       & \hfil/\  10\hfil \Strut \\
\hline
3       & \hfil/\  20\hfil \Strut \\
\hline
4       & \hfil/\  10\hfil \Strut \\
\hline
5       & \hfil/\  10\hfil \Strut \\
\hline
6       & \hfil/\  10\hfil \Strut \\
\hline
7       & \hfil/\  10\hfil \Strut \\
\hline\hline
Total   & \hfil/100\hfil \Strut \\
\hline
\end{tabular}
\end{center}

\vfil\eject

\begin{enumerate}

\item{} [5 points each]\quad Compute the derivatives of the given
  functions. 

\bd

\item{(a)} $f(t)=(t^2+1)^4$

\vfil

\hrulefill

\item{(b)} $g(x)=x^2\ln x$

\vfil

\hrulefill

\item{(c)} $z(u)=\ds\frac{u}{e^u+1}$

\vfil

\hrulefill

\item{(d)} $s(t) = te^{-t^2}$

\vfil

\hrulefill

\item{(e)} $b(x) = \ln(e^x+e^{-x})$

\vfil

\hrulefill

\item{(f)} $k(s)=\ds\frac{s^3+\ln s}{s^2}$

\vfil\eject

\item{} [10 points]\quad Find the tangent line to the curve
  $xy^2=xy+2$ at the point~$(1,2)$.

\vfil

\hrulefill

\item{} [10 points]\quad Suppose
  $x'(t)=\sqrt{e^{2x}+1}$. Find~$x''(t)$ as a function of~$x$.

\vfil

\eject

\item{} In both parts of this question, let $f(x)=x^2$.

\bd

\item{(a)} [10~points]\quad Compute the derivative~$f'(x)$ \emph{from
    the definition}.

\vfil\vfil

\item{(b)} [10~points]\quad Find the equation of the line obtained by
  zooming in on the graph at the point~$(-1,1)$, and sketch the line
  in the grid provided.

\begin{center}
  \input{midterm2a-01.eepic}
\end{center}

\vfil

\ed

\eject

\item{} [5~points each]\quad The position of a car (measured in feet)
  after $t$~seconds is:
  \[
    \begin{array}{|c||c|c|c|}
      \hline
      \vp t    & 0 & 1 & 1.5 \\
      \hline
      \vp s(t) & 0 & 80 & 110 \\
      \hline
    \end{array}
  \]

  \bd

  \item{(a)} Estimate the speed of the car when $t=0.5$ and when
  $t=1.25$. Give units.

  \vfil

  \ShortAnswer{Speed at $t=0.5$}
  \ShortAnswer{Speed at $t=1.25$}

  \item{(b)} When $t=1$, does the car appear to be accelerating or
    decelerating? Write a sentence or two in support of your answer.

  \vfil

  \ed

\eject

\item{} [10~points]\quad Let $f(x) = e^{-x^2/2}$. Find the interval(s)
  on which $f$~is both increasing and concave down.

\eject

\item{} [10 points]\quad Define a function by
  \[
  f(x) = 
  \begin{cases}
    e^x & \text{ if $x<0$}, \\
    \cos x & \text{ if $x\geq0$}.
  \end{cases}
  \]
  Is~$f$ differentiable at $x=0$? Justify your answer for full credit.

\ed

\end{enumerate}

\end{document}

