\input calc
\def\problemspace{.25in}

%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Final Exam}\quad{\sl May 5, 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 25 multiple
choice questions worth 6 points each.

%% PROBLEM
Let $A = [1,2,3]$ and $B = \bmatrix{4\cr5\cr6\cr}$. Compute the {\sl matrix
product} $B\cdot A$.

%% CORRECT
$\bmatrix{4&8&12\cr
5&10&15\cr
6&12&18\cr}$
%% WRONG
$[32]$
%% WRONG
$\bmatrix{1&2&3\cr
4&5&6\cr}$
%% WRONG
$\bmatrix{4&5&6\cr
8&10&12\cr
12&15&18\cr}$
%% WRONG
{\sl not defined}


%% PROBLEM
Let $A = \bmatrix{0&-1&1\cr
1&1&-1\cr
2&1&0\cr}$.
The $(3,2)$-entry of the inverse matrix $A^{-1}$ is:
%% CORRECT
-2
%% WRONG
2
%% WRONG
1
%% WRONG
-1
%% WRONG
0


%% PROBLEM
Calculate the determinant of the matrix
$\bmatrix{1&3&-1&4\cr
2&1&0&1\cr
0&0&2&-1\cr
0&0&0&1\cr}$.
%% CORRECT
-10
%% WRONG
2
%% WRONG
-4
%% WRONG
-6
%% WRONG
8


%% PROBLEM
Find the general solution to the following system of linear equations:
$$\eqalign{
x_1 + x_3 &= 1\cr
2x_1 + x_2 + 4x_3 - x_4 &= 5\cr
3x_1 + x_2 + 6x_3 &= 10\cr
4x_1 + x_2 + 6x_3 - x_4 &= 7\cr}$$
%% CORRECT
\vtop{
\hbox{$x_1 = -3 + x_4$}
\hbox{$x_2 = -5 + 3x_4$}
\hbox{$x_3 = 4 - x_4$}}
%% WRONG
\vtop{
\hbox{$x_1 = 1 - x_3$}
\hbox{$x_2 = 5 - 2x_3$}
\hbox{$x_4 = -7$}}
%% WRONG
\vtop{
\hbox{$x_1 = -3$}
\hbox{$x_2 = -5$}
\hbox{$x_3 = 4$}
\hbox{$x_4 = 0$}}
%% WRONG
\vtop{
\hbox{$x_1 = -{1\over2} + {1\over2}x_2 - {1\over2}x_4$}
\hbox{$x_2 = {3\over2} - {1\over2}x_2 - {1\over2}x_4$}}
%% WRONG
{\sl the system is inconsistent}


%% PROBLEM
If the Gram-Schmidt process is applied to the vectors
$$X_1 = [1,1,0,1], \qquad X_2 = [0,0,1,0], \qquad X_3 = [1,0,1,1]$$
to obtain the vectors $Q_1$, $Q_2$, $Q_3$, then:
%% CORRECT
$Q_3 = {1\over\sqrt{6}}[1,-2,0,1]$
%% WRONG
$Q_3 = {1\over\sqrt{2}}[1,-1,0,0]$
%% WRONG
$Q_3 = {1\over\sqrt{2}}[0,1,0,-1]$
%% WRONG
$Q_3 = {1\over\sqrt{14}}[2,-3,0,1]$
%% WRONG
$Q_3 = {1\over\sqrt{6}}[1,1,0,-2]$



%% PROBLEM
Suppoose $y$ satisfies
$x y' - 3 y = 4x$ and $y(1) = 0$.
Calculate $y(2)$.
%% CORRECT
12
%% WRONG
4
%% WRONG
8
%% WRONG
6
%% WRONG
10


%% PROBLEM
Determine which of the following equations defines the integral curves of the
differential equation $$x^2 y' + y^2 = 1$$
%% CORRECT
$y = {c - e^{2/x}\over c + e^{2/x}}$
%% WRONG
$y = {c - e^{2x}\over c + e^{2x}}$
%% WRONG
$y = {e^{x/2} - c\over e^{x/2} + c}$
%% WRONG
$y = {e^{x} - c\over e^{x} + c}$
%% WRONG
$y = {e^{-x} - c\over ce^{x} + 1}$


%% PROBLEM
A 50~gal drum is filled with water having a 10\% concentration of cleaning
fluid. Fresh water is added to the drum at a rate of 2~gal/min and the
well-stirred mixture is siphoned out of the drum at the same rate.
Calculate how long it will take the concentration of cleaning fluid to fall
below 1\%.
%% CORRECT
57.6 min
%% WRONG
45.3 min
%% WRONG
37.5 min
%% WRONG
69.2 min
%% WRONG
28.4 min


%% PROBLEM
Classify the equilibrium solutions of the logistic equation
$${dN\over dt} = N (1-N^2)$$
%% CORRECT
$N = 1$ stable, $N = 0$ unstable, $N = -1$ stable
%% WRONG
$N = 1$ unstable, $N = 0$ stable, $N = -1$ stable
%% WRONG
$N = 1$ stable, $N = 0$ unstable, $N = -1$ unstable
%% WRONG
$N = 1$ unstable, $N = 0$ stable, $N = -1$ unstable
%% WRONG
$N = 1$ stable, $N = 0$ stable, $N = -1$ unstable


%% PROBLEM
Air resistance on a 160~lb skydiver with an unopened parachute is proportional to the square of the velocity, $kv^2$. After falling 20 seconds, the skydiver is observed to have a steady rate of descent of 200~ft/sec. What value of $k$ yields this terminal velocity? (Use the given units of lbs, ft, sec.)
%% CORRECT
$0.004$
%% WRONG
$0.16$
%% WRONG
$0.02$
%% WRONG
$0.8$
%% WRONG
$5.0$


%% PROBLEM
Find the equation of the integral curve for the differential equation
$$(e^x \cos(y) + 2x - y^2)\,dx - (e^x\sin(y) + 2xy - 1)\, dy = 0$$
that passes through the point $(0,0)$.

%% CORRECT
$e^x\cos(y) + x(x-y^2) + y = 1$
%% WRONG
$e^x\sin(y) + xy^2 - y = 0$
%% WRONG
$e^x\cos(y) + x^2 - xy^2 = 1$
%% WRONG
$e^x\sin(y) + xy^2 - y = 0$
%% WRONG
$e^x(\cos(y) - \sin(y)) - x^2y + y = 1$

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

%% CORRECT
$\mu = e^{-y^2}$
%% WRONG
$\mu = e^{-x}$
%% WRONG
$\mu = e^{y^2}$
%% WRONG
$\mu = e^x$
%% WRONG
$\mu = {1\over2}y$

%% PROBLEM
Solve the differential equation
$${dy\over dx} = {x\tan(y/x) + y\over x}$$
%% CORRECT
$\sin(y/x) = cx$
%% WRONG
$\ln|\sin(y/x)| = {1\over2}x^2 + c$
%% WRONG
$\sec^2(y/x) = {1\over2}x^2 + c$
%% WRONG
$\cos(y/x) = cy$
%% WRONG
$\ln|\cos(y/x)| = y + c$


%% PROBLEM
Solve the initial value problem
$$y^2y'' - (y')^3 = 0, \quad y(0) = 1, \quad y'(0) = 2$$
%% CORRECT
$x = {1\over2}(1 - y) + \ln|y|$
%% WRONG
$y = e^{2x}$
%% WRONG
$y = \sqrt{4x + 1}$
%% WRONG
$y = -2 + \sqrt{4x + 9}$
%% WRONG
$x = {1\over2}(1-y)^2$


%% PROBLEM
Given that $y_1 =  e^x$ is a solution to the differential equation
$$(x-1)y'' - xy' + y = 0$$
A second linearly independent solution can be found of the form $y_2 = v
y_1$ where $v$ is a non-constant function of $x$. Determine the differential
equation $v$ must satisfy.
%% CORRECT
$(x-1)v'' + (x-2) v' = 0$
%% WRONG
$(x-1)v'' - x v' + 2 v = 0$
%% WRONG
$(x-1)v'' + (x+1) v' = 0$
%% WRONG
$e^x(x-1)v'' - x v' = 0$
%% WRONG
$(x-1)v'' - x v' + 2e^x v = 0$


%% PROBLEM
Determine a suitable form for a particluar solution $y_p$ of the
differential equation
$$y'' - 4 y' + 4y = 4xe^{2x} + x \sin(2x)$$
%% CORRECT
$y_p = x^2(A_1x+A_0)e^{2x}+(B_1x+B_0)\sin(2x)+(C_1x+C_0)\cos(2x)$
%% WRONG
$y_p = x(A_1x+A_0)e^{2x}+x(B_1x+B_0)\sin(2x)+x(C_1x+C_0)\cos(2x)$
%% WRONG
$y_p = (A_1x+A_0)e^{2x}+(B_1x+B_0)\sin(2x)+(C_1x+C_0)\cos(2x)$
%% WRONG
$y_p = Axe^{2x} + Bx\sin(2x) + Cx\cos(2x)$
%% WRONG
$y_p = Ax^2 e^{2x}+(B_1x+B_0)\sin(2x)+(C_1x+C_0)\cos(2x)$


%% PROBLEM
Let $L[y] = y'' + p(x) y' + q(x)$. Suppose $x$ and  $x^{-1}$ are solutions to
$L[y] = 0$. Calculate a particular solution $y_p = u_1(x) x + u_2(x)
x^{-1}$ to the differential equation $L[y] = x^3$ using the method of
variation of parameters.
%% CORRECT
$y_p = {1\over24}x^5$
%% WRONG
$y_p = {1\over8}x^4 - {1\over12}x^6$
%% WRONG
$y_p = {1\over6}x^6$
%% WRONG
$y_p = {1\over2}(x^4 - x)$
%% WRONG
$y_p = {1\over20}x^5 - {1\over6}x^3$


%% PROBLEM
Suppose a man suspended from a bungee cord stretches the cord 16~feet
longer than its natural length of 34~ft. The man jumps from a bridge while
attached to this cord and arrives at a distance of 50~ft below the bridge
with a downward velocity of 75~ft/sec. At this point the cord starts acting
like a spring (i.e., it obeys Hooke's Law.) Assuming there are no
frictional forces, how long will it take for the man to return to the point
50~ft below the bridge? 
%% CORRECT
$\pi\sqrt{2}/2$
%% WRONG
$\pi\sqrt{6}/2$
%% WRONG
$\pi/2$
%% WRONG
$\pi\sqrt{3}/3$
%% WRONG
$\pi\sqrt{6}/3$


%% PROBLEM
Find the recurrence relation for the coefficients of a power series
solution to the differential equation
$$(1+x^2)y'' - x y' + y = 0$$
%% CORRECT
$a_n = {(n-3)^2\over n(n-1)} a_{n-2}$
%% WRONG
$a_{n+2} = {n\over (n+2)(n+1)} a_{n}$
%% WRONG
$a_{n+2} = -{1\over (n+2)(n+1)} a_{n}$
%% WRONG
$a_n = {n-1\over n} a_{n-2}$
%% WRONG
$a_n = {(n-1)^2\over (n+2)(n+1)} a_{n-2}$


%% PROBLEM
Find the first three non-zero terms of a power series solution to the
initial value problem
$$(4 - x^2)y'' + 2y = 0,\quad y(0) = 0, \quad y'(0) = 1$$
%% CORRECT
$y = x - {1\over12}x^3 - {1\over240}x^5 - \cdots$
%% WRONG
$y = x - {1\over24}x^3 + {1\over20}x^5 - \cdots$
%% WRONG
$y = x - {1\over24}x^3 - {1\over480}x^5 - \cdots$
%% WRONG
$y = x - {1\over12}x^3 + {1\over120}x^5 - \cdots$
%% WRONG
$y = x - {1\over6}x^3 - {1\over80}x^5 - \cdots$


%% PROBLEM
Classify the singular points of the differential equation
$$x^2(x-1)^2 y'' + 2xy' + (x^2-1)y = 0$$
%% CORRECT
$x=0$ regular, $x=1$ irregular
%% WRONG
$x=0$ regular, $x=1$ regular, $x=-1$ irregular
%% WRONG
$x=0$ regular, $x=1$ regular
%% WRONG
$x=0$ irregular, $x=1$ irregular
%% WRONG
$x=0$ irregular, $x=1$ regular, $x=-1$ regular


%% PROBLEM
Determine the general solution of the differential equation
$$x^2 y'' - 5 x y' + 10y = 0$$
%% CORRECT
$y = x^3(c_1\cos(\ln|x|) + c_2\sin(\ln|x|))$
%% WRONG
$y=x^{5/2}(c_1\cos({\sqrt{15}\over2}\ln|x|)+c_2\sin({\sqrt{15}\over2}\ln|x|))$
%% WRONG
$y=e^{5/2}(c_1\cos({\sqrt{15}\over2}x)+c_2\sin({\sqrt{15}\over2}x))$
%% WRONG
$y = e^{3x}(c_1\cos(x) + c_2\sin(x))$
%% WRONG
$y = c_1 x^{-3} + c_2 x^{-5}$


%% PROBLEM
The equation
$$x^2 y'' + (x + \sin(x))y' - y = 0$$
has a regular singular point at $x = 0$. Determine the exponents of the
this singularity.
%% CORRECT
$r_1 = {-1+\sqrt{5}\over2},\quad r_2 = {-1-\sqrt{5}\over2}$
%% WRONG
$r_1 = -{1\over2}+i{\sqrt{3}\over2},\quad r_2 = -{1\over2}-i{\sqrt{3}\over2}$
%% WRONG
$r_1 = 1, \quad r_2 = -1$
%% WRONG
$r_1 = 1,\quad r_2 = 0$
%% WRONG
$r_1 = r_2 = 1$


%% PROBLEM
The differential equation
$$ 2x^2 y'' + 3x y' + (2x^2 - 1)y = 0$$
has a regular singular point at $x = 0$. Find a series solution
corresponding to the larger of the exponents of the singularity.
%% CORRECT
$y = x^{1/2}\left(1 - {1\over7}x^2 + {1\over154}x^4 - \cdots\right)$
%% WRONG
$y = x^{1/2}\left(1 - {1\over4}x^2 + {1\over64}x^4 - \cdots\right)$
%% WRONG
$y = x\left(1 - x^2 + {1\over10}x^4 - \cdots\right)$
%% WRONG
$y = x^{-1}\left(1 - {1\over14}x^2 + {1\over182}x^4 - \cdots\right)$
%% WRONG
$y = x\left(1 - {1\over10}x^2 + {1\over440}x^4 - \cdots\right)$


%% PROBLEM
Solve the initial value problem
$$x^2 y'' + 7x y' + 9y = 0,\qquad y(1) = 0, \qquad y'(1) = 2$$
%% CORRECT
$y = 2\ln|x|/x^3$
%% WRONG
$y = (x^3 - x^{-3})/3$
%% WRONG
$y = -{2\over3}x^{-3}$
%% WRONG
$y = {2\over7}(x^{7/2} - x^{-7/2})$
%% WRONG
$y = {1\over\sqrt{13}}x^{7/2}\sin({\sqrt{13}\over2}\ln|x|)$




%% END
\bye
