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

\title
\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}

\note %1
\problem Solve the equation \quad $2(y-4x^2)dx+xdy=0$.
\correct
$y=2x^2+{C\over x^2}$
\wrong
$y={2\over x^2}+Cx^2$
\wrong
$y=2x^4+Cx^2$
\wrong
$y={8x^3\over 3}-{x^2\over2}+C$
\wrong
$y=8x^2(\ln |x|+C)$



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


\note %3
\problem 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.
\correct
$(0,\infty)$
\wrong
$(0,2)$
\wrong
$(-2,2)$
\wrong
$(-2,3)$
\wrong
$(-\infty,2)$

\note %4
\problem Solve the initial value problem
$$y'+2y^2=0,\quad y(0)={1\over2},$$ and then give the domain of the solution.
\correct
$(-1,\infty)$
\wrong
$(-\infty,1)$
\wrong
$(-2,\infty)$
\wrong
$(-\infty,2)$
\wrong
$(-1,1)$



\note %5
\problem
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 $y(t_0)=y_0$?
$$ (1,0)\quad\quad  (0,1)\quad\quad (1,1)\quad\quad (-1,1)$$
\correct
Only $(1,0)$ and $(-1,1)$.
\wrong
Only $(1,0)$ and $(0,1)$.
\wrong
Only $(0,1)$ and $(1,1)$.
\wrong
Only $(1,0)$ and $(1,1)$.
\wrong
Only $(0,1)$ and $(-1,1)$.


\note %6
\problem
A certain radioactive material has a half-life of 200 years.  What percent
of a given sample will remain
after fifty years?
\correct
84\%
\wrong
79\%
\wrong
87\%
\wrong
92\%
\wrong
90\%

\note %7
\problem
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?
\correct
4.73
\wrong
4.55
\wrong
4.13
\wrong
3.98
\wrong
4.27



\note %8
\problem
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?
\correct
$Q'+{1\over 40}Q=2+6e^{-2t},\quad Q(0)=60$.
\wrong
$Q'+{1\over 40}Q=5+10e^{-2t},\quad Q(0)=60$.
\wrong
$Q'=-55+10e^{-2t},\quad Q(0)=30$.
\wrong
$Q'={79\over 40}+6e^{-2t},\quad Q(0)=0.30$.
\wrong
$Q'+{1\over 40}Q=5e^{-2t},\quad Q(0)=60$.


\note %9
\problem
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 $k$.
\correct
6.45
\wrong
2.11
\wrong
10.42
\wrong
5.28
\wrong
13.64

\note %10
\problem
Classify the equilibrium solutions of the equation \quad ${dy\over
dt}=y(1-y)^2$ \quad as stable,
semistable or unstable.
\correct
$y=0$ is unstable and $y=1$ is semistable.
\wrong
Both $y=0$ and $y=1$ are semistable.
\wrong
$y=0$ is unstable and $y=1$ is stable.
\wrong
$y=0$ is semistable and $y=1$ is unstable.
\wrong
$y=0$ is stable and $y=1$ is unstable.



\note %11
\problem
Solve the population equation \quad $y'=3(y-1)(y-2),\quad y(0)=0$.
\correct
$y={2-2e^{3t}\over 1-2e^{3t}}$
\wrong
$y={1-e^{3t}\over 1+e^{3t}}$
\wrong
$y={2-2e^{3t}\over 2+e^{3t}}$
\wrong
$y={e^{3t}-1\over 2e^{3t}+1}$
\wrong
$y={2-2e^{-3t}\over 1-2e^{-3t}}$


\note %12
\problem
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 $x=0$.
\correct
2
\wrong
$-3$
\wrong
$1\over2$
\wrong
$-1$
\wrong
1

\note %13
\problem
Classify the following equation.
$$({y\over x^2+y^2}-{y^2\over x^2})dx+(2{y\over x}-{x\over x^2+y^2})dy=0$$
\correct
Exact but not separable.
\wrong
Separable.
\wrong
Linear.
\wrong
Homogeneous.
\wrong
None of the others.



\note %14
\problem
Which of the following is an integrating factor for the equation \quad
${dy\over dx}=-{4xy+3y^2-x\over
x^2+2xy}$
\correct
$x^2$
\wrong
$2x$
\wrong
$e^{2x}$
\wrong
${1\over x^3}$
\wrong

$e^{-2x}$

\note %15
\problem
Classify the equation \quad ${dy\over dx}=(x+y)^2$.
\correct
None of the others.
\wrong
Linear.
\wrong
Separable.
\wrong
Exact but not separable.
\wrong
Homogeneous.

\note %16
\problem
Determine which of the following defines a solution to the the equation
\quad $x^2{dy\over
dx}=x^2+xy+y^2$.
\correct
$y=x\,\tan(\ln|x|+C)$
\wrong
$x^2+y^2=Cx^3$
\wrong
${x\over x+y}=-\ln|x|+C$
\wrong
$\arcsin(1+{y\over x})=C\,\ln|x|$
\wrong
$y=x+x\tan(x+C)$


\end

