\input calc
\input epsf
\def\problemspace{0.2in}

%% TITLE
\noindent
{\bf Math 226: Calculus IV}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam II}\quad{\sl Mar. 18, 1993}\hfill
Sextion:\hbox to 2.5truein{\hrulefill}

\bigskip\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 15 multiple
choice questions worth 6 points each. You start with 10 points.

\bigskip\bigskip
\def\answerspace{0.5in}


%% PROBLEM
Solve the differential equation $y' + 2y = 3e^x$ subject to the initial
condition $y(1) = 0$.

%% CORRECT
$y = e^{x} - e^{3-2x}$
%% WRONG
$y = e^{3x} - e^{3}$
%% WRONG
$y = e^{2x} - e^{3-x}$
%% WRONG
$y = e^{x} - 1$
%% WRONG
$y = e^{x}$


%% PROBLEM
On what interval is the solution to the initial value problem
$$\cos(x) y' = y + e^{1/x},\qquad y(\pi) = 0$$ guaranteed to exist? 

%% CORRECT
$(\pi/2, 3\pi/2)$
%% WRONG
$(0,\infty)$
%% WRONG
$(0, 3\pi/2)$
%% WRONG
$(-\pi/2, \pi/2)$
%% WRONG
$(\pi, \infty)$


%% PROBLEM
Determine which of the direction fields below is associated to the
differential equation $y' + x/y = 0$.

%% CORRECT
\epsfxsize=1.5truein\epsfbox{exam2.1.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.2.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.3.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.4.eps}
%% WRONG
\epsfxsize=1.5truein\epsfbox{exam2.5.eps}


%% PROBLEM
Gases from the ground below a house seep into its 3000 ${\rm ft}^3$
basement at the rate of 2 ${\rm ft}^3$ per day. Air leaves the
basement at the same rate.  The incoming gases contain 5\% radon.
Assuming the basement is initially clear of radon gas,
calculate how many days it will take for the basement to reach the
dangerous level of 1\% radon gas.

%% CORRECT
$334.7$
%% WRONG
$300.0$
%% WRONG
$351.8$
%% WRONG
$387.0$
%% WRONG
$405.6$

%% PROBLEM
Let $\phi(x)$ be the  solution of the differential equation
$y' = {xy^4 - y\over x^2y^3 - x}$ that satisfies $\phi(1) = 2$. Find
$\phi(3)$.

%% CORRECT
$6$
%% WRONG
$-8$
%% WRONG
$5/2$
%% WRONG
$\sqrt{21}$
%% WRONG
$5^{1/3}$


%% PROBLEM
Suppose a certain colony of protozoa grows at a rate $r$ proportional
to its total population. Find the time $T$ required for the colony
of protozoa to reach ten times its original number.

%% CORRECT
$T = {\ln(10)\over r}$
%% WRONG
$T = {\ln(r)\over 10}$
%% WRONG
$T = {\ln(r)\over \ln(10)}$
%% WRONG
$T = {\ln(10)\over \ln(r)}$
%% WRONG
$T = {r\over \ln(10)}$


%% PROBLEM
Find the {\sl stable} equilibrium solutions of the differential equation
${dx\over dt} = x^2 - 6x + 8$

%% CORRECT
$x = 2$
%% WRONG
$x = 4$
%% WRONG
$x = 2$ and $x = 4$
%% WRONG
$x = 0$
%% WRONG
{\sl there are no stable equilibria}

%% PROBLEM
Suppose the water in St.~Mary's lake is at $40^\circ F$ and cools
at a rate proportional to the difference between its temperature
and the ambient air temperature of $10^\circ F$. If the temperature
of the water drops 5 degrees in 2 hours, how many hours will it take
all together for the lake to begin to freeze ($32^\circ F$).

%% CORRECT
3.4
%% WRONG
3.8
%% WRONG
4.0
%% WRONG
3.2
%% WRONG
3.6

%% PROBLEM
A projectile is launched vertically upward from sea level with an
initial velocity of $v_0$. Neglecting air resistance, but considering
the change of gravitational attraction with altitude, $x$, find
the maximum altitude attained by the projectile. The appropriate
differential equation is $$v{dv\over dx} = -{gR^2\over(R+x)^2}$$
where $v$ is the velocity of the projectile, $R$ is the radius of
the earth, and $g$ is the gravitational constant $g = 32 {\rm ft}/{\rm sec}^2$.

%% CORRECT
${v_0^2R\over 2gR - v_0^2}$
%% WRONG
${gR^2\over gR + v_0^2}$
%% WRONG
$\sqrt{2gR\over v_0}$
%% WRONG
$\sqrt{v_0\over gR}$
%% WRONG
${2gR^2\over 2gR - v_0^2}$


%% PROBLEM
Find the integral curve for the differential equation
$$(3x^2y^2 - 2x - y)\,dx + (2x^3y + 2y - x)\,dy = 0$$
that passes through the point $(0,2)$.

%% CORRECT
$x^3y^2 - x^2 + y^2 - xy = 4$
%% WRONG
$f(x,y) = x^3y^2 - x^2 - xy$
%% WRONG
$x^3y^2 - x^2 + - xy = 0$
%% WRONG
$f(x,y) = x^2y^3 - 2xy - {1\over2}y^2$
%% WRONG
$x^2y^3 - 2xy - {1\over2}y^2 = -2$


%% PROBLEM
Find an integrating factor $\mu$ for the differential equation
$$y\,\tan(x)\, dx + (y\, \sec(x) - 1 )\,dy = 0$$

%% CORRECT
$\mu = \cos(x)$
%% WRONG
$\mu = \cot(x)$
%% WRONG
$\mu = \ln|\sin(x)|$
%% WRONG
$\mu = \csc(x)$
%% WRONG
$\mu = e^{-x}$


%% PROBLEM
Suppose a certain population of red squirrels satisfies
the logistic equation $${dN\over dt} = -0.05\left(1 - {N\over
700}\right)\left(1 - {N\over 2000}\right)N$$ If the current population
is 600, determine the long-term population of red squirrels predicted
by this model.

%% CORRECT
$0$
%% WRONG
$700$
%% WRONG
$2000$
%% WRONG
$\ln(700)/0.05$
%% WRONG
$\ln(2000)/0.05$


%% PROBLEM
Solve the initial value problem
$$y' = {x + y\over x}, \quad y(1) = 1$$

%% CORRECT
$y = x\ln(x) + x$
%% WRONG
$y = {x\over 1-\ln(x)}$
%% WRONG
$y = {3 - x^2\over 2x}$
%% WRONG
$y = {x\over 2x - 1}$
%% WRONG
$y = {2x\over x + 1}$


%% PROBLEM
Reduce the second order differential equation
$$(1+y')y'' - y y' = 0,\quad y(0) = 1, \quad y'(0) = 0$$
to a first order equation in $y$.

%% CORRECT
$y' = y - 1, \quad y(0) = 1$
%% WRONG
$(1 + y) y' - x y^2 = 0, \quad  y(1) = 1$
%% WRONG
$y' = {y\over y + 1}, \quad y(0) = 1$
%% WRONG
$y' = -1 + \sqrt{4 + y^2}, \quad y(0) = 1$
%% WRONG
$y' = {x y\over y - 1}, \quad y(1) = 0$


%% PROBLEM
Classify the following differential equation
$$y' = {\cos(x) + y\sin(x)\over \ln(x)},\quad x > 0$$

%% CORRECT
{\sl linear}
%% WRONG
{\sl separable}
%% WRONG
{\sl exact}
%% WRONG
{\sl homogeneous}
%% WRONG
{\sl none of the above}

%% END
\end
