\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
{\bf Exam I}\quad{\sl February 9, 1996}\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
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 or 
program capabilities.  Doing so is a violation of the Honor Code.
\bigskip\noindent
You are required to hand in the answer sheet and the problems.  You may tear off the 
blank sheets at the back for scratch work. \bf{Please tear off these scratch sheets
gently, one page at a time.}

\def\answerskip{\vskip0.25in}

\note %1
\problem 
Which of the following differential equations is linear?
(A)  $y ^{\prime\prime} + sin y = x$ \quad \ \ \ \ (B) $ y ^{\prime\prime} + y y ^{\prime} = y - x$
(C) $ y ^{\prime} -  {1\over\sqrt2x}\ y = y - x$\quad (D) $ x\ y ^\prime - 3 y ^2 = 0$
\correct
Only C.
\wrong
Only A.
\wrong
Only B.
\wrong
Only D.
None of them
\note %2
\problem
Solve the initial value problem
$$x y ^\prime - 3y = x^5 \ \ ,\ \ y (1) = 3/2.$$
Then find $y (2).$
\correct
$24$
\wrong
${39\over2}$
\wrong
$20$
\wrong
$16$
\wrong
$32$
\note %3
\problem
	Find the largest in which the solution to the initial value problem
$$x (x^2 - 9) y \prime  +  e^x y = \sqrt{ 4-x^2}\ ,\ y(1) = 3$$
is certain to exist.
\correct
 $(0,2)$
\wrong
$(0, 3)$
\wrong
$(- 2, 2)$
\wrong
(-3, 2)$
\wrong
$
(0, \infty)$
\note %4
\problem
Classify the differential equation
$$y\prime = {sin (x) + y\ arctan (x)\over x ^2 }\ \ x > 0 .$$
Linear.
Exact.
Separable.
Homogeneous.
None of the others.
\note %5
\problem
	Classify the differential equation
$$(x y + x)\ dx  +  (x y + y)\ dy = 0$$
Separable.
Separable but not exact.
Linear.
Homogeneous.
None of the others.
\note %6
\problem
Consider the equation
$$(x^2 - 1) y \prime = \(x^2) ^{3/2}$$
For how many of the following points $(x_0\ ,\ y_0) $ does the Existence-uniqueness Theorem guarantee a unique solution with the initial condistion (x_0\ ,\ y_0) = y_0\ ?$
$ (a)\  $(0,0)$ \quad (b)\ $(1,2)$ \quad (c)  \ $(-1, -2)$ \quad (d)  \ $(2,2)$$
One.
Two
Three
Four
None
\note %7
\problem
Suppose a radioactive substance decays at a rate r proportional to the amount present.  Find the time T until only $1/5$ of the original amount is left.
${ln 5\over r}$
${ln r\over 5}$
${ln 5\over ln r}$
${ln r\over ln 5}$
\note %8
\problem
A tank contains 100 pounds of salt dissolved in 200 gallons of water.  Fresh water, i.e., water with no salt in it, enters the tank at a rate of 4 gallons per minute, with the well-stired misture leaving the tank at the same rate.  After how many minutes will the tank contain 50 pounds of salt?
$34.66$
$138.63$
$25.02$
$7.82$
$29.44$
\note %9
\problem
You invest $\$2,000$ in an account paying $\$6\%$ a year, compounded continously on the amount in excess of $\$500.$   After ten years how much is in the account?
$\$3233.18$
$\$3186.27$
$\$3072.94
$\$2927.52$
$\$2989.48$
\note %10
\problem
The population equation ${dn\over dt}$ = $4N - N^2 $ , $N (0) = 2, $ has solution $N (t)$ =
${4\over 1 + e^{-4t}$
${4\over 1 + e^{-2t}$
$2\over 1 + e^{-4t}$
${e^{2t}\over 4 + e^{2t}$
\note %11.
\problem
Consider the equation
$$N (t) = 4 N (t) - N (t)^2 -3\ ,\ t > 0,\ N (1) = 2$$
$N (t)$ is concave up and $\lim\limits_{t\to \infty}$ $N (t) = 3$
$N (t)$ is concave down and $\lim\limits_{t\to\infty}\ N(t) = 0$
$N (t)$ is concave up and $\lim\limits_{t\to\infty}\$ N(t) = \infty$
$N (t)$ is concave down and $\lim\limits_{t\to\infty}$ $N (t) = -\infty$
\note %12.
\problem
A 32 pound weight is dropping through a gas near the earth's surface.  The resistance from the gas is equal to twice the square of the velocity.  Find the magnitude of the terminal velocity of the body.
$2$
$4$
$6$
$8$
$10$
\note %13.
\problem
Consider the equation
$$(s + y^2 )\ dx + (2xy + 4y) dy = 0$$
\Fist find $f(x, y)$ so that the equation $f(x, y) = c$ define solutions implicitly.  Then compute $f(1,1) - f(0,0).$
$4$
$0$
$-5$
$11$
$-1$
\note %14.
\problem
Which of the following is an integrating factor for the equation?
$$(2xy - e^{2x})\ dx + x dy = 0$$
${e^2x}\over x}
$2xe^x
$ye^{-y}
$e^2 e^{-1/x}$
\bigskip
$x y e^{2xy}
\note %15.
\problem
Which of the following equations defines implicitly or explictly a solution of the inital value problem
$$y\prime = {x^2 + 3y^2\over 3xy}\ \ , \ y(1) = 3?$$
$y = x\  \root\of{9 + ln x}$
$y^2 = x^4 + 8$
$y = 3 e^{(x-1) ^2}
$\root\of{ x^2 + 3y^2 } =  ln x + \root 2\of {7}$
$e ^{y-3} =  2x^2 - 1$
\note %16.
\problem
Which of the following equations are NOT homogeneous?
(A)	$(x^3 y + x^2 y^2 )\ dy + xy^3 dx = 0$
(B)	$y\prime = e^{x -y} + x + y$
(C)	$2xy\  y\prime = 4x^3 + 3y^3$
(B) and (D)
(A) and (C)
(A) and (B)
all three
None

 
  









\end
