
%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Exam II}\quad{\sl March 19, 1992}\hfill Score:\hbox to 2.5truein{\hrulefill}

\bigskip
Record your answers to the multiple choice problems by placing an $\times$
through one letter for each problem on this answer sheet. There are 15 multiple
choice questions worth 6 points each. You start with 10~points.

\def\problemspace{.25in}


%% PROBLEM
Find the general solution of the differential equation
\quad $y' + 2x y = x$.
%% CORRECT
$y = {1\over2}(c e^{-x^2} + 1)$
%% WRONG
$y = {1\over2}(c + e^{x^2})$
%% WRONG
$y = c + x^2$
%% WRONG
$y = {1\over2}(c + e^{2x})$
%% WRONG
$y =  c e^{-2x} + 1$



%% PROBLEM
Solve the initial value problem
\quad $y' - x y^2 = 0, \quad y(0) = 3$.
%% CORRECT
$y = {6\over 2 - 3x^2}$
%% WRONG
$y = (3\ln(x))^{1/3} + 3$
%% WRONG
$y = {3\over x^2 + 1}$
%% WRONG
$y = {3\over\displaystyle\ln(x+1)}$
%% WRONG
$y = {1\over2}x^2 + 3$


%% PROBLEM
Solve the differential equation
$(3x^2y - y^2 - 2x)\,dx + (x^3 - 2xy + 1)\,dy = 0$.
%% CORRECT
$xy^2 - (x^3 + 1)y + x^2 = c$
%% WRONG
$x^3y^2 - {1\over3}y^3 - x^2 +{4\over3} = c$
%% WRONG
$f(x,y) = x^3y - xy^2 - x^2$
%% WRONG
$f(x,y) = x^3y - xy^2 + y$
%% WRONG
$y = {1\over2}[(x^3 + 1) + \sqrt{(x^3+1)^2 - x^2 + c}]$


%% PROBLEM
Find the integral curve of the differential equation
$${dy\over dx} = {x (y^2 + 1)\over 1 - x^2y}$$
that passes through the point (1,0).
%% CORRECT
$x^2y^2 - 2y + x^2 = 1$
%% WRONG
$x^2y^2 + x^2 = 1$
%% WRONG
${1\over2} x^2y^2 - y = 0$
%% WRONG
$y = {\sqrt{2 - 2x^2}\over x}$
%% WRONG
$y = {3x^2(y^2+1)\over 6 - 2x^3y} - {1\over2}$


%% PROBLEM
Solve the differential equation
$${dy\over dx} = {y(x^2 - y^2)\over x^3}$$
subject to the initial condition $y(1) = -{1\over2}$.
%% CORRECT
$y = -{x\over\sqrt{2\ln(x) + 4}}$
%% WRONG
${2\over3}x^3 y-{1\over3}y^3 = -{1\over2}$
%% WRONG
$y = {4y(x^3 - 3xy^2)\over x^4}$
%% WRONG
${1\over3}x^3 y - {1\over 4}y^4$ 
%% WRONG
$y = {x\over 2\ln(x) + 2}$


%% PROBLEM
Two populations of bacteria, type~A and type~B, are increasing in a lab
experiment at rates proportional to their current sizes. The type~A bacteria
double their numbers every week and the type~B bacteria triple their numbers
every week. If there are twice as many type~A bacteria as type~B initially,
determine how long it will take for the number of type~B bacteria to equal the
number of type~A bacteria.
%% CORRECT
11.97 days
%% WRONG
13.51 days
%% WRONG
9.33 days
%% WRONG
10.15 days
%% WRONG
8.67 days


%% PROBLEM
Suppose a population $N$ grows according to the law
$${dN\over dt} = \alpha (N^2 - 10N + 24)(N^2 - 5N + 6)$$
where $\alpha$ is a positive constant. If $N(0) = 5$, determine the limiting value
for $N(t)$ as $t\to\infty$.
%% CORRECT
4
%% WRONG
2
%% WRONG
3
%% WRONG
6
%% WRONG
5


%% PROBLEM
A solution of orange dye is being mixed in a 100 liter tank. The tank starts with
100~$\ell$ of pure water to which two streams of dye solution are being added.
The red stream contains 1~$g/\ell$ of dye and the yellow stream contains
2~$g/\ell$ of dye. Both streams are flowing in at a rate of 1~$\ell/{\rm sec}$. The
mixture is well-stirred and siphoned off so that the tank remains at 100~$\ell$.
Find an expression for the concentration $c(t)$ of dye in the orange solution
leaving the tank.
%% CORRECT
$c(t) = 1.5(1 - e^{-t/50})$
%% WRONG
$c(t) = 0.5(1 + e^{-t/100})$
%% WRONG
$c(t) = 100(1 - e^{-2t})$
%% WRONG
$c(t) = 50(1 + e^{-2t})$
%% WRONG
$c(t) = {2\over3}(1 - {1\over2}e^{-t/50})$


%% PROBLEM
An object weighing 1~lb falls from rest in a medium offering resistance
numerically equal to the square of the velocity.  Find the velocity of the object
as a function of~$t$. (Assume $g = 32 {\rm\ ft}/{\rm sec}^2$.)
%% CORRECT
$v = {1-e^{-64t}\over1+e^{-64t}}$ ft/sec
%% WRONG
$v = (1-e^{-32t})$ ft/sec
%% WRONG
$v = -16t^2$ ft/sec
%% WRONG
$v = {1+e^{-32t}\over1-e^{-32t}}$ ft/sec
%% WRONG
$v = (1+e^{-64t})$ ft/sec



%% PROBLEM
Suppose a glass of ice-cold lemonade ($32^\circ F$) is placed outside on a hot
day ($90^\circ F$) and warms up at a rate proportional to the difference
between its temperature and the surrounding air. Ten minutes later the lemonade
is~$50^\circ F$. How long will it take before the lemonade reaches~$70^\circ F$.
%% CORRECT
28.65 mins
%% WRONG
25.35 mins
%% WRONG
23.55 mins
%% WRONG
30.45 mins
%% WRONG
20.75 mins


%% PROBLEM
A bank loans money at an annual rate of $r$ compounded continuously. Suppose $P$ dollars are borrowed and paid back continuously over $N$ years. Find a formula for the amount $k$ that must be paid back annually.
%% CORRECT
$k = {rP\over 1 - e^{-rN}}$
%% WRONG
$k = {Pe^{rN}\over N}$
%% WRONG
$k = {rP\over N}$
%% WRONG
$k = {rP\over e^{rN} - 1}$
%% WRONG
$k = {P(1+e^{rN})\over N}$


%% PROBLEM
Find an integrating factor for the differential equation
$$2x\,dx + (x^2 + \cos(y))\,dy = 0$$
%% CORRECT
$e^y$
%% WRONG
$e^{-x}$
%% WRONG
$\sin(y)$
%% WRONG
${1\over x}$
%% WRONG
$e^x$


%% PROBLEM
Convert the second order equation
$$y y'' - (y')^3 = 0$$
into a first order equation by making the substitution $v = y'$.
%% CORRECT
${dv\over dy} = {v^2\over y}$
%% WRONG
$y v' - v^3 = 0$
%% WRONG
$v' = {v\over y}$
%% WRONG
${dv\over dx} = {v^3\over x}$
%% WRONG
${dv\over dy} - {1\over y} v = 0$


%% PROBLEM
Solve the initial value problem
$$x^3 y'' = (y')^2,\quad y(1) = 1,\quad y'(1) = 2$$
by making an appropriate substitution.
%% CORRECT
$y = {1\over3}(2x^3 + 1)$
%% WRONG
$y = {1\over 2x^2 -1}$
%% WRONG
$y = {x^2\over 2x^2 -1}$
%% WRONG
$y = x^2$
%% WRONG
$y= x + {1\over2}\ln(2x-1)$


%% PROBLEM
Determine the interval on which one can expect the solution to the following
initial value problem to be valid.
$$y'' = {e^x - xy\over x^2 - 1},\quad y(2) = 0, \quad y'(2) = 1$$
%% CORRECT
$(1,\infty)$
%% WRONG
$(-1,1)$
%% WRONG
$(-\infty,-1)$
%% WRONG
$(-\infty,\infty)$
%% WRONG
{{\sl cannot be determined}}

%% END
\bye
