% %&latex
\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\newenvironment{changemargin}[2]{\begin{list}{}{
\setlength{\topsep}{0pt}\setlength{\leftmargin}{0pt}
\setlength{\rightmargin}{0pt}
\setlength{\listparindent}{\parindent}
\setlength{\itemindent}{\parindent}
\setlength{\parsep}{0pt plus 1pt}
\addtolength{\leftmargin}{#1}\addtolength{\rightmargin}{#2}
}\item }{\end{list}}
%This environment takes two arguments, and will indent the left
%and right margins by their values, respectively. Negative values
%will cause the margins to be widened, so
%\begin{changemargin}{-1cm}{-1cm} widens the left and right margins
%by 1cm. add ``\end{changemargin}'' to turn it off.
\oddsidemargin=0in
\evensidemargin=0in
\topmargin=0in
\headsep=0in
\headheight=0in
\pagestyle{empty}
\textheight=23cm
\textwidth=17cm
% \vspace*{-2.0cm}
% Uniform abbreviations for Greek letters
\def\af{{\alpha}}
\def\bt{{\beta}}
\def\gm{{\gamma}}
\def\dt{{\delta}}
\def\ep{{\varepsilon}}
\def\zt{{\zeta}}
\def\et{{\eta}}
\def\ch{{\chi}}
\def\io{{\iota}}
\def\te{{\theta}}
\def\ld{{\lambda}}
\def\sm{{\sigma}}
\def\kp{{\kappa}}
\def\ph{{\varphi}}
\def\ps{{\psi}}
\def\rh{{\rho}}
\def\om{{\omega}}
\def\ta{{\tau}}
\def\Gm{{\Gamma}}
\def\Dt{{\Delta}}
\def\Et{{\Eta}}
\def\Th{{\Theta}}
\def\Ld{{\Lambda}}
\def\Sm{{\Sigma}}
\def\Ph{{\Phi}}
\def\Ps{{\Psi}}
\def\Om{{\Omega}}
\newcommand{\andeqn}{\,\,\,\,\,\, {\rm and} \,\,\,\,\,\,}
\newcommand{\rsz}[1]{\raisebox{0ex}[0.8ex][0.8ex]{$#1$}}
\newcommand{\ts}{\textstyle}
\begin{document}
\noindent
Math 251 \hspace{0.65in}
Midterm 2\hspace{\fill}NAME:\underline{\hspace*{2.0in}}\\
W~18 May 2016
\hspace{\fill}Student ID:\underline{\hspace*{2.0in}}\\
\underline{GENERAL INSTRUCTIONS}\\
\begin{enumerate}
\item
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
\item
The exam pages are {\textbf{two sided}}.
\item
Closed book, except for a $3 \times 5$ file card.
\item
The following are all prohibited:
Calculators (of any kind), cell phones, laptops, iPods,
% electronic dictionaries,
and any other electronic devices or communication devices.
All electronic or communication devices you have with you
must be turned completely off and put inside something
(pack, purse, etc.)\ and out of sight.
\item
The point values are as indicated in each problem; total 100 points.
\item
Write all answers on the test paper.
% Use the space below the extra credit problems
% Use the extra sheet at the back
% Use the back of the page with the extra credit problems
Use the back of the last page
for long answers or scratch work.
(If you do write an answer there,
indicate on the page containing the problem
where your answer is.)
\item
Show enough of your work that your method is obvious.
Be sure that every statement you write is correct.
Cross out any material you do not wish to have considered.
Correct answers with insufficient justification
or accompanied by additional incorrect statements
will not receive full credit.
Correct guesses to problems requiring significant work,
and correct answers obtained after a sequence of mostly incorrect steps,
will receive no credit.
\item
Be sure you say what you mean, and use correct notation.
Credit will be based on what you say, not what you mean.
\item
When exact values are specified, give answers such as $\frac{1}{7}$,
$\sqrt{2}$, $\ln (2)$, or $\frac{2\pi}{9}$.
Decimal approximations will not be accepted.
\item
Final answers must always be simplified unless otherwise specified.
\item
Grading complaints must be submitted in writing at the beginning of
the class period after the one in which the exam is returned
(usually by the next class period after the exam).
\item
Time: 50 minutes.
% (unless extended, with class agreement, by an early start).
\end{enumerate}
\vfill
\noindent
\hrulefill
\vfill
% \hspace{-1cm}
\centerline{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\makebox[0.55in]{1} & \makebox[0.55in]{2} &
\makebox[0.55in]{3} & \makebox[0.55in]{4} & \makebox[0.55in]{5} &
\makebox[0.55in]{6} &
\makebox[0.55in]{7} &
\makebox[0.55in]{8} & \makebox[0.55in]{TOTAL} &
\makebox[0.55in]{EC} \\
\hline
1 & 16 & 8 & 25 & 5 & 8 & 24 & 13 & 100 & \\
\hline & & & & & & & & & \\
& & & & & & & & & \\
\end{tabular}
}
\pagebreak
\noindent
1. (1 point)
Are you awake?
\vspace{4ex}
\noindent
2. (16 points)
Helium is leaking from a spherical balloon.
At a certain time, helium is being lost at the rate of $2$
cubic centimeters per minute,
and the radius of the balloon is $10$ centimeters.
At what rate is the radius decreasing at that time?
(Be sure to include the correct units in your answer.)
\vfill
\vfill
\vfill
\vfill
\noindent
3. (8 points)
Let
$q (x) = \pi^4 - \sin (x^2 \ln (x))$.
Find $q' (x)$.
\vfill
\mbox{}
\pagebreak
\noindent
4. (25 points)
A xenobiologist wants to wall off a rectangular enclosure
for a small herd of fire breathing monsters from the planet Yuggxth.
One side of the enclosure will be a long straight river
which flows from east to west;
no wall is needed here.
A second side will be along an already existing wall which is
perpendicular to the river, and which can be used as it is.
His research grant contains enough money to build 6 kilometers of wall
for the remaining two sides.
What is the largest area he can enclose?
Include units, and be sure to verify that your maximum or minimum
really is what you claim it is.
(Show a full mathematical solution.
A correct guess with no valid work will receive no credit,
and a correct number supported only by heuristic reasoning
will receive very little credit.)
\vfill
\mbox{}
\pagebreak
\noindent
5. (5 points)
The derivative of the function
$f (x) = ( x - 3 ) e^{- x}$
is given by
$f' (x) = - (x - 4) e^{- x}$,
and the second derivative
is given by $f'' (x) = (x - 5) e^{- x}$.
Determine whether $f$ has a local minimum,
a local maximum, or neither at $x = 4$.
\vfill
\vfill
\noindent
6.
A pendulum swings back and forth on the surface of the
earth.
The relation between the period $T$ and the length $l$ of the
pendulum can be modelled by
the equation $T = \sqrt{k l}$, where $T$ is measured in seconds,
$l$ is measured in meters, and $k = 5 \, {\mbox{s}^2}/{\mbox{m}}$.
Answer the following questions, being careful to include units when
appropriate.
\bigskip
(a) {(1 point)}
When $l$ is $20$ meters, compute the period of the pendulum.
\vfill
(b) {(7 points)}
Using part (a) as your base value, give a linear approximation
for the period of the pendulum when the length $l$ is changed to $18$
meters.
\vfill
\vfill
\mbox{}
\pagebreak
\vspace*{-14ex}
\noindent
7. (24 points)
Graph the function $f (x) = x^3 - 3 x^2$ using the
methods of calculus.
In particular, determine exactly the $x$-intercept(s),
$y$-intercept(s), asymptotes, intervals of increase and decrease,
critical numbers,
local maximums, local minimums,
intervals of concavity, and inflection points.
Be sure to label your axes, and include the scales.
{\textbf{Be sure to organize your work so that it is easy to follow,
and explain what you are doing.
A list of equations, with no explanation of how they relate to
each other or to the
problem, will receive little credit.}}
Fill in the table (writing ``none'' if appropriate),
and show full work and the graph elsewhere on the page.
\vspace{0.5ex}
\noindent
\begin{tabular}{|r|c}
\cline{1-2}
Domain.
& \makebox[10em][c]{} \\ \cline{1-2}
$x$-intercepts.
& \\ \cline{1-2}
$y$-intercepts.
& \\ \cline{1-2}
Horizontal asymptotes.
& \\ \cline{1-2}
Vertical asymptotes.
& \\ \cline{1-2}
Critical numbers.
& \\ \cline{1-2}
Local minimums.
& \\ \cline{1-2}
Local maximums.
& \\ \cline{1-2}
Intervals of strict increase.
& \\ \cline{1-2}
Intervals of strict decrease.
& \\ \cline{1-2}
Intervals of upwards concavity.
& \\ \cline{1-2}
Intervals of downwards concavity.
& \\ \cline{1-2}
Inflection points.
& \\ \cline{1-2}
\end{tabular}
\vfill
\pagebreak
\noindent
8. (13 points)
Use the methods of calculus to
find the exact values of $x$ at which the function
$f (x) = {\displaystyle{\frac{x}{x^2 + 1} }}$
takes its absolute minimum and maximum on the interval $[0, 7]$.
(No credit will be given for correct guesses without
supporting work that is valid for general functions
of the sort considered in this course.)
\vfill
\pagebreak
\noindent
Extra credit.
(Do not attempt these problems until you have done and
checked your answer to all the ordinary problems on this exam.
They will only be counted if you get $75$ or more points
on the main part of this exam.)
\bigskip
\noindent
EC.
Use the following steps to prove
that if $f$ is a function defined for all real numbers,
and there is a constant $k$ such that $f' (x) = k f (x)$
for all real $x$, then there is a constant $C$ such that
$f (x) = C e^{k x}$ for all real $x$.
(That is, prove that the exponential functions are the
{\emph{only}} functions for which the growth rate is proportional
to the function value.)
\bigskip
(a) (5 extra credit points)
Suppose $f' (x) = k f (x)$
for all real $x$.
Define
\[
g (x) = \frac{f (x)}{e^{k x}}.
\]
Show that $g' (x) = 0$ for all $x$.
\vfill
(b) (10 extra credit points)
Use the Mean Value Theorem
to conclude that $g$ is a constant function.
(Show that for any two real numbers $a < b$, we have $g (a) = g (b)$.)
\vfill
(c) (5 extra credit points)
Explain why parts (a) and (b) show that $f (x)$ has the required form.
\vfill
\end{document}