\documentclass[11pt]{article}
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\textwidth = 6.5 in
\textheight = 9 in
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\parskip = 0.2in
\parindent = 0.0in

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}

\begin{document}


% Macros

% Preface

\input epsf
\input calc
\def\problemskip{\vskip0.5in}
\def\ds{\displaystyle}

% Problems

\vbox{
%1

\noindent\llap{1.\ }Solve the equation \quad $2(y-4x^2)dx+xdy=0$.

\beginanswers
{(a)~$y=2x^2+{C\over x^2}$\answerspace}
{(b)~$y={2\over x^2}+Cx^2$\answerspace}
{(c)~$y=2x^4+Cx^2$\answerspace}
{(d)~$y={8x^3\over 3}-{x^2\over2}+C$\answerspace}
{(e)~$y=8x^2(\ln |x|+C)$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%2

\noindent\llap{2.\ }Find the solution of the initial value problem \quad
$ty'+(1-t)y=e^t,\quad y(1)=0$.

\beginanswers
{(a)~$e^t(1-{1\over t})$\answerspace}
{(b)~$e^{-t}(1-t)$\answerspace}
{(c)~$e^{t-1}-{1\over t}$\answerspace}
{(d)~${1\over t}-e^{1-t}$\answerspace}
{(e)~$e^{-t}({1\over t}-1)$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%3

\noindent\llap{3.\ }Find the largest interval in which a unique solution to
the initial value problem
$$(t^2+2t)y'+(t^2-2t)y=e^{-(t-3)},\quad y(1)=3,$$ is certain to exist.

\beginanswers
{(a)~$(0,\infty)$\answerspace}
{(b)~$(0,2)$\answerspace}
{(c)~$(-2,2)$\answerspace}
{(d)~$(-2,3)$\answerspace}
{(e)~$(-\infty,2)$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%4

\noindent\llap{4.\ }Solve the initial value problem
$$y'+2y^2=0,\quad y(0)={1\over2},$$ and then give the domain of the solution.

\beginanswers
{(a)~$(-1,\infty)$\answerspace}
{(b)~$(-\infty,1)$\answerspace}
{(c)~$(-2,\infty)$\answerspace}
{(d)~$(-\infty,2)$\answerspace}
{(e)~$(-1,1)$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%5

\noindent\llap{5.\ }Consider the equation \quad ${dy\over dt}={y+t\over
t(y-t)^2}$.
\smallskip
\noindent
For which of the following points $(t_0,y_0)$  does the
Existence-Uniqueness Theorem guarantee a unique
solution with the initial condition $y(t_0)=y_0$?
$$ (1,0)\quad\quad  (0,1)\quad\quad (1,1)\quad\quad (-1,1)$$

\beginanswers
{(a)~Only $(1,0)$ and $(-1,1)$.\answerspace}
{(b)~Only $(1,0)$ and $(0,1)$.\answerspace}
{(c)~Only $(0,1)$ and $(1,1)$.\answerspace}
{(d)~Only $(1,0)$ and $(1,1)$.\answerspace}
{(e)~Only $(0,1)$ and $(-1,1)$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%6

\noindent\llap{6.\ }A certain radioactive material has a half-life of 200
years.  What percent of a given sample will remain
after fifty years?

\beginanswers
{(a)~84\%\answerspace}
{(b)~79\%\answerspace}
{(c)~87\%\answerspace}
{(d)~92\%\answerspace}
{(e)~90\%\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%7

\noindent\llap{7.\ }The sum of \$10,000 is borrowed at the annual interest
rate of 10.8\%, which we assume to be compounded
continuously.  If the loan is paid off at the rate of \$225 per month, how
many years will it take to pay
off the loan?

\beginanswers
{(a)~4.73\answerspace}
{(b)~4.55\answerspace}
{(c)~4.13\answerspace}
{(d)~3.98\answerspace}
{(e)~4.27\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%8

\noindent\llap{8.\ }A 200 gallon tank initially has a 30\% concentration of
salt.

\noindent
Salt solution enters the tank from two different sources.

\noindent
One source provides water containing one pound of salt per gallon at a rate
of two gallons per minute.

\noindent
A second source provides three gallons per minute at a varying
concentration of $2e^{-2t}$ pounds
of salt per gallon.

\noindent
The contents are stirred and the uniform solution is drained off at a rate
of five gallons per
minute.

\noindent
Which initial value problem is satisfied by the quantity $Q(t)$ of salt in
the tank at time t?

\beginanswers
{(a)~$Q'+{1\over 40}Q=2+6e^{-2t},\quad Q(0)=60$.\answerspace}
{(b)~$Q'+{1\over 40}Q=5+10e^{-2t},\quad Q(0)=60$.\answerspace}
{(c)~$Q'=-55+10e^{-2t},\quad Q(0)=30$.\answerspace}
{(d)~$Q'={79\over 40}+6e^{-2t},\quad Q(0)=0.30$.\answerspace}
{(e)~$Q'+{1\over 40}Q=5e^{-2t},\quad Q(0)=60$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%9

\noindent\llap{9.\ }An object weighing 200 pounds falls in a medium
offering a resistive force equal to $-kv^{1.2}$, where $v$
is the velocity.  The terminal velocity is measured to be 17.5 feet per
second.  Estimate $k$.

\beginanswers
{(a)~6.45\answerspace}
{(b)~2.11\answerspace}
{(c)~10.42\answerspace}
{(d)~5.28\answerspace}
{(e)~13.64\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%10

\noindent\llap{10.\ }Classify the equilibrium solutions of the equation
\quad ${dy\over dt}=y(1-y)^2$ \quad as stable,
semistable or unstable.

\beginanswers
{(a)~$y=0$ is unstable and $y=1$ is semistable.\answerspace}
{(b)~Both $y=0$ and $y=1$ are semistable.\answerspace}
{(c)~$y=0$ is unstable and $y=1$ is stable.\answerspace}
{(d)~$y=0$ is semistable and $y=1$ is unstable.\answerspace}
{(e)~$y=0$ is stable and $y=1$ is unstable.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%11

\noindent\llap{11.\ }Solve the population equation \quad
$y'=3(y-1)(y-2),\quad y(0)=0$.

\beginanswers
{(a)~$y={2-2e^{3t}\over 1-2e^{3t}}$\answerspace}
{(b)~$y={1-e^{3t}\over 1+e^{3t}}$\answerspace}
{(c)~$y={2-2e^{3t}\over 2+e^{3t}}$\answerspace}
{(d)~$y={e^{3t}-1\over 2e^{3t}+1}$\answerspace}
{(e)~$y={2-2e^{-3t}\over 1-2e^{-3t}}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%12

\noindent\llap{12.\ }If \quad $(xy^2+y-x)dx+(x^2y+x+2)dy=0$,\quad and \quad
$y=1$ when $x=2$,\quad find $y$ when $x=0$.

\beginanswers
{(a)~2\answerspace}
{(b)~$-3$\answerspace}
{(c)~$1\over2$\answerspace}
{(d)~$-1$\answerspace}
{(e)~1\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%13

\noindent\llap{13.\ }Classify the following equation.
$$({y\over x^2+y^2}-{y^2\over x^2})dx+(2{y\over x}-{x\over x^2+y^2})dy=0$$

\beginanswers
{(a)~Exact but not separable.\answerspace}
{(b)~Separable.\answerspace}
{(c)~Linear.\answerspace}
{(d)~Homogeneous.\answerspace}
{(e)~None of the others.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%14

\noindent\llap{14.\ }Which of the following is an integrating factor for
the equation \quad ${dy\over dx}=-{4xy+3y^2-x\over
x^2+2xy}$

\beginanswers
{(a)~$x^2$\answerspace}
{(b)~$2x$\answerspace}
{(c)~$e^{2x}$\answerspace}
{(d)~${1\over x^3}$\answerspace}
{(e)~$e^{-2x}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%15

\noindent\llap{15.\ }Classify the equation \quad ${dy\over dx}=(x+y)^2$.

\beginanswers
{(a)~None of the others.\answerspace}
{(b)~Linear.\answerspace}
{(c)~Separable.\answerspace}
{(d)~Exact but not separable.\answerspace}
{(e)~Homogeneous.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%16

\noindent\llap{16.\ }Determine which of the following defines a solution to
the the equation \quad $x^2{dy\over
dx}=x^2+xy+y^2$.

\beginanswers
{(a)~$y=x\,\tan(\ln|x|+C)$\answerspace}
{(b)~$x^2+y^2=Cx^3$\answerspace}
{(c)~${x\over x+y}=-\ln|x|+C$\answerspace}
{(d)~$\arcsin(1+{y\over x})=C\,\ln|x|$\answerspace}
{(e)~$y=x+x\tan(x+C)$\answerspace}
\endanswers\problemskip}\vfil\goodbreak


% Appendix

{}

% Blank answer sheet

\vfil\eject\output={}
\vbox{{\noindent {\bf Math 226: Calculus IV}\hfill Sign your Name:\hbox to
2.5truein{\hrulefill}\medskip\noindent
%
%Put in the correct exam number and date%%%%%%%%
 {\bf Exam I}\quad{\sl February 6, 1998}\hfill Print your Name:\hbox to
2.5truein{\hrulefill}\medskip\hfill Section:\hbox to
2.5truein{\hrulefill}\medskip\hfill TA:\hbox to
2.5truein{\hrulefill}

\bigskip\noindent Do not turn this page until you are told to begin.
\bigskip\noindent {\bf Do not detach the answer sheet from the test.}
\bigskip\noindent You are required to hand in both the answer sheet and the
problems.
\bigskip\noindent Record your answers to the multiple choice problems by
placing an $\times$ through one
letter for each problem on this answer sheet. There are 16 multiple choice
questions worth 6 points each.
You start with 10 points, and the highest  possible score is 106.  Fill in
the answers as you go along.
You will not be  allowed to fill in the answers after the time is up.
\bigskip\noindent You may use a calculator, but only the standard functions
found on very inexpensive
scientific calculators.  In particular you may not use graphing,
integration, formula, matrix or  program
capabilities.
\bigskip\noindent



\def\answerskip{\vskip0.25in}


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kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{12.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}\ens
kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{13.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}\ens
kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
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kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
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kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{16.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}\ens
kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
}}}


% Marked answer sheet

\vfil\eject
\vbox{\vbox to0pt{\vskip-22.5pt\hbox to \hsize{\hfil\ansbox{70pt}{\ Answer
Key 1\ }\hfil}\vss}
\prevdepth=-1000pt
{\noindent {\bf Math 226: Calculus IV}\hfill Sign your Name:\hbox to
2.5truein{\hrulefill}\medskip\noindent
%
%Put in the correct exam number and date%%%%%%%%
 {\bf Exam I}\quad{\sl February 6, 1998}\hfill Print your Name:\hbox to
2.5truein{\hrulefill}\medskip\hfill Section:\hbox to
2.5truein{\hrulefill}\medskip\hfill TA:\hbox to
2.5truein{\hrulefill}

\bigskip\noindent Do not turn this page until you are told to begin.
\bigskip\noindent {\bf Do not detach the answer sheet from the test.}
\bigskip\noindent You are required to hand in both the answer sheet and the
problems.
\bigskip\noindent Record your answers to the multiple choice problems by
placing an $\times$ through one
letter for each problem on this answer sheet. There are 16 multiple choice
questions worth 6 points each.
You start with 10 points, and the highest  possible score is 106.  Fill in
the answers as you go along.
You will not be  allowed to fill in the answers after the time is up.
\bigskip\noindent You may use a calculator, but only the standard functions
found on very inexpensive
scientific calculators.  In particular you may not use graphing,
integration, formula, matrix or  program
capabilities.
\bigskip\noindent



\def\answerskip{\vskip0.25in}


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{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
}}}






 \end{document} 