%% Math 131-01 Midterm 3, Fall 2003

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

\usepackage{math131}

\pagestyle{myheadings}
\markright{Math 131, Early 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{$\vphantom{\Big|}$}

\begin{document}

\begin{center}

\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Midterm 3} \\
\textbf{Friday, November 21, 2 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/\  10\hfil \Strut \\
\hline
2       & \hfil/\  10\hfil \Strut \\
\hline
3       & \hfil/\  20\hfil \Strut \\
\hline
4       & \hfil/\  20\hfil \Strut \\
\hline
5       & \hfil/\  20\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

\be

\item{} [5 points each]\quad Evaluate the following limits:

\bd

\item{(a)} $\ds\lim_{x\to0}\frac{1-\cos x}{x^2}$

\vfil

\item{(b)} $\ds\lim_{x\to\infty}\frac{\ln(e^x+1)}{x}$

\vfil

\ed

\hrulefill

\item{} [10 points]\quad A small portion of the graph $y=f(x)$ (with
  horizontal and vertical scales as shown) is given below. Use linear
  approximation to determine which of the following functions might be
  depicted. (Show work, and circle your answer(s).)
  \[
  f(x)=e^x\qquad f(x)=\sin x\qquad f(x)=\cos x\qquad f(x)=\sqrt{1+2x}
  \]
\begin{comment}
  \begin{align*}
  f(x)&=e^x    & f(x)&=\sin x    & f(x)&=\sqrt{1+2x} \\
  f(x)&=\cos x & f(x)&=\ln (1+x) & f(x)&=\frac{1}{1-x}
  \end{align*}
\end{comment}

\vfil\vfil

  \begin{center}
    \input{midterm3-01a.eepic}
  \end{center}

\eject

\item{} [10 points each]\quad Both parts of this question refer to the
  function $f(x)=4x^3-x^4$.

  \bd

  \item{(a)} Find and classify the critical point(s) of~$f$.

    \vfil

    \hrulefill

  \item{(b)} Find the absolute maximum and minimum values of~$f$ on
  the interval~$[1,4]$.

\vfil
  \ShortAnswer{Maximum value:}
  \ShortAnswer{Minimum value:}

  \ed

\eject

\item{} [20~points]\quad A spherical balloon is being inflated at
  $20~\textrm{in}^3/\textrm{sec}$. When the radius is 6~inches, at
  what rate is the radius increasing? The volume of a sphere of
  radius~$r$ is $V=\frac{4}{3}\pi r^3$.

\vfil

\hrulefill

\item{} [20~points]\quad Let $a>0$. A rectangle is inscribed under the
  curve $y=\sqrt{x}$:
  \begin{center}
    \input{midterm3-01b.eepic}
  \end{center}
  Find the value of~$x$, with $0<x<a$, for which the area of the
  rectangle is largest. Show all work for full credit.

\vfil

\ShortAnswer{$x=$}
\eject

\item{} [10~points]\quad Find the inflection point(s) of the function
  $f(x)=x^{2/3}-\frac{1}{5}x^{5/3}$. 

\vfil

\hrulefill

\item{} [10~points]\quad As a car brakes to a stop, the speed
  (measured in feet per second) after $t$~seconds is given by the
  table below. 
  \[
    \begin{array}{|c||c|c|c|c|}
      \hline
      \vp t & 0  &  1 &  2 & 3 \\
      \hline
      \vp v & 66 & 40 & 18 & 0 \\
      \hline
    \end{array}
  \]
  On the basis of this information, give the most accurate estimate of
  the distance required for the car to come to a stop.

  \vfil

\eject

\ee

\end{document}

