\def\problemspace{.25in}
\input calc
\let\ds=\displaystyle

%% TITLE
{\bf Math 226: Calculus IV}\hfill Name:\hbox to 2.5truein{\hrulefill}\medskip
{\bf Final Exam}\quad{\sl May 6, 1991}\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
Compute the matrix product
$$\bmatrix{0&-1&2&3\cr4&1&0&-2\cr0&6&1&0\cr}
\bmatrix{1&1\cr2&-2\cr-3&3\cr4&4\cr}$$
%% CORRECT
{$\bmatrix{4&20\cr-2&-6\cr9&-9\cr}$}
%% WRONG
{$\bmatrix{0&-2&-6&12\cr4&-2&0&-8\cr0&12&-3&0\cr}$}
%% WRONG
{$\bmatrix{0&-1\cr8&-2\cr0&18\cr}$}
%% WRONG
{$\bmatrix{0&-1&2&3\cr8&-2&0&4\cr0&18&-3&0\cr}$}
%% WRONG
{{\sl not defined}} 


%% PROBLEM
Solve the following system of linear equations.
$$\matrix{
x_1&+&2x_2&+&x_3&+&2x_4&+&x_5 &=& -4\cr
2x_1&+&x_2&+&2x_3&+&x_4&+&2x_5 &=& 1\cr
3x_1&+&2x_2&+&3x_3&+&2x_4&+&3x_5 &=& 0\cr
4x_1&+&5x_2&+&4x_3&+&5x_4&+&4x_5 &=& -7\cr
}$$
%% CORRECT
{\vtop{\halign{$#$&$#$\hfill\cr
x_1&=2 - x_3-x_5\cr
x_2&=-3-x_4\cr
x_3,&x_4, x_5{\rm\ free}\cr
}}}
%% WRONG
{\vtop{\halign{$#$&$#$\hfill\cr
x_1&=2\cr
x_2&=-3\cr
x_3&=x_4= x_5 = 0\cr
}}}
%% WRONG
{\vtop{\halign{$#$&$#$\hfill\cr
x_1&=-4 - x_3\cr
x_2&=-3-x_4\cr
x_5&= 0\cr
x_3,&x_4{\rm\ free}\cr
}}}
%% WRONG
{\vtop{\halign{$#$&$#$\hfill\cr
x_1&=0\cr
x_2&=-4\cr
x_3&=x_4=1\cr
x_5 &= 0\cr
}}}
%% WRONG
{{\sl no solution}}


%% PROBLEM
Compute the inverse of the matrix
$$\bmatrix{
0&0&1\cr1&1&1\cr1&0&0\cr
}$$
The (3,2)-entry of the inverse is:
%% CORRECT
{$0$}
%% WRONG
{$-1$}
%% WRONG
{$1$}
%% WRONG
{$1/2$}
%% WRONG
{$-1/2$}


%% PROBLEM
Compute the determinant of
$$\bmatrix{
1&0&2&0\cr
0&3&0&4\cr
5&0&6&0\cr
0&7&0&8\cr
}$$
%% CORRECT
{$16$}
%% WRONG
{$-20$}
%% WRONG
{$24$}
%% WRONG
{$-12$}
%% WRONG
{$0$}


%% PROBLEM
Find a maximal linearly independent subset of the vectors
$$\eqalign{
X_1 &= (1,0,1,1,0,0)\cr
X_2 &= (1,1,0,0,1,1)\cr
X_3 &= (1,1,1,1,1,1)\cr
X_4 &= (0,0,1,1,0,0)\cr
X_5 &= (0,1,0,0,1,1)\cr
}$$
%% CORRECT
{$X_1$, $X_2$, $X_3$}
%% WRONG
{$X_1$, $X_2$, $X_3$, $X_4$}
%% WRONG
{$X_2$, $X_3$, $X_4$}
%% WRONG
{$X_1$, $X_3$, $X_4$, $X_5$}
%% WRONG
{$X_1$, $X_3$, $X_5$}




%% PROBLEM
Find the general solution to
$$y' + y = e^x$$
%% CORRECT
{$y = {1\over\ds2}e^x + c e^{-x}$}
%% WRONG
{$y = {1\over\ds2} + c e^{-x}$}
%% WRONG
{$y = 2e^x$}
%% WRONG
{$y = e^{2x} + c e^x$}
%% WRONG
{$y = {1\over\ds2}e^{2x} + c e^{-x}$}



%% PROBLEM
Solve the initial value problem
$${dy\over\ds dx} ={xy \over\ds 1+x^2},\qquad y(0) = 1$$
%% CORRECT
{$y = \sqrt{1 + x^2}$}
%% WRONG
{$y = 1+\ln(\sqrt{1 + x^2})$}
%% WRONG
{$y = e^{\tan^{-1}(x)}$}
%% WRONG
{$y = 1+\ln(1 + \tan^{-1}(x))$}
%% WRONG
{$y = 0$}



%% PROBLEM
Ground water seeps into a cylindrical retention pond at a rate of $100~{\rm ft}^3$
per minute. There is storm drain above the pond and when the water reaches this
level it drains out at a rate proportional to the height~$h$ of the pond {\sl
above} the storm drain. A county engineer measures the flow in the storm drain
to be $50~{\rm ft}^3$ per minute when~$h$ is 3~in. Find the limiting value
for~$h$.
%% CORRECT
{6 in}
%% WRONG
{1 ft}
%% WRONG
{4 in}
%% WRONG
{2 ft}
%% WRONG
{8 in}



%% PROBLEM
Suppose the population~$N$ of spotted owls in a certain national park satisfies a
logistic equation of the form
$${dN\over\ds dt} = \left({500\over\ds N}-1\right)\left(1-{1000\over\ds N}\right)N^3$$
If the current population of spotted owls in the park is~100, what is the long
term population predicted by this model.
%% CORRECT
{$0$}
%% WRONG
{$100$}
%% WRONG
{$500$}
%% WRONG
{$1000$}
%% WRONG
{{\sl no limit}}



%% PROBLEM
A body of mass $m$ is projected upward with initial veliocity~$v_0$ in a medium
offering resistance equal to the magnitude of the velocity. Find an
expression for the time~$t$ when the body reaches its maximum height.
%% CORRECT
{$t = m \ln({\ds v_0\over\ds mg} + 1)$}
%% WRONG
{$t = m \ln({\ds v_0\over\ds mg})$}
%% WRONG
{$t = {\ds v_0\over\ds g}$}
%% WRONG
{$t = {\ds v_0\over\ds mg}$}
%% WRONG
{$t = \sqrt{{\ds m v_0 \over\ds g}}$}



%% PROBLEM
Find the unique solution to
$$(3x^2y + 2y)\,dx + (x^3 - 2y + 2x)\, dy = 0$$
satisfying $y(1) = 1$.
%% CORRECT
{$x^3y - y^2 + 2xy = 2$}
%% WRONG
{$x^3y + y^2 = 2$}
%% WRONG
{$x^3 y + 2xy = 3$}
%% WRONG
{$3x^2y + 2xy + y^2 = 6$}
%% WRONG
{$x^3 - y^2 = 0$}



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



%% PROBLEM
Solve the equation
$${dy\over\ds dx} = {xy + y^2 \over\ds x^2}$$
by making a substitution $v = {y\over\ds x}$.
%% CORRECT
{$y = {x\over\ds c - \ln(x)}$}
%% WRONG
{$y = x (\ln(x) + c)$}
%% WRONG
{$y = {c\over\ds x^2}$}
%% WRONG
{$y = -{1\over\ds \ln(x)} + c$}
%% WRONG
{$y = {x\over\ds c - x^2}$}



%% PROBLEM
Compute the Wronskian of the functions
$$y_1  = x\cos(x)\,\qquad y_2 = x\sin(x)$$
%% CORRECT
{$x^2$}
%% WRONG
{$x^2(1 - 2\sin^2(x))$}
%% WRONG
{$x^2(1 - 2\cos^2(x))$}
%% WRONG
{$2x\sin(x)\cos(x)$}
%% WRONG
{$0$}



%% PROBLEM
Solve the initial value problem
$$\displaylines{
y'' + y(y')^3 = 0\cr
y(0) = 1,\quad y'(0) = 2\cr
}$$
%% CORRECT
{$y = (6x + 1)^{1/3}$}
%% WRONG
{$y = (4x + 1)^{1/2}$}
%% WRONG
{$y = y = e^{2x}$}
%% WRONG
{$y = (1 - 4x)^{-1/2}$}
%% WRONG
{$y = (1 - 6x)^{-1/3}$}




%% PROBLEM
Find the general solution of
$$6y'' - 7y' + 2y = 0$$
%% CORRECT
{$y = c_1 e^{x/2} + c_2 e^{2x/3}$}
%% WRONG
{$y = c_1 e^{3x/2} + c_2 e^{x/3}$}
%% WRONG
{$y = c_1 e^{-x} + c_2 e^{2x/3}$}
%% WRONG
{$y = c_1 e^{2x} + c_2 e^{-2x/3}$}
%% WRONG
{$y = c_1 e^{-2x/3} + c_2 e^{x/3}$}




%% PROBLEM
Find the solution of
$$y'' - 4y' + 4y = 0$$
satisfying
$y(0) = 0$ and $y'(0) = 2$.
%% CORRECT
{$y = 2x e^{2x}$}
%% WRONG
{$y = {1\over\ds 2}(e^{2x} - e^{-2x})$}
%% WRONG
{$y = \sin(2x)$}
%% WRONG
{$ y = \cos(2x)$}
%% WRONG
{$y = x(e^{2x} + e^{-2x})$}



%% PROBLEM
A mass weighing 2~lbs stretches a spring 6~in. The mass is pushed upward,
contracting the spring a distance of 1~in, and then released. Suppose air
resistance exerts a damping force given by ~$0.5 u'(t)$ where $u(t)$ is the
displacement (in ft) from equilibrium with the positive direction down. Determine
the subsequent motion. ($g = 32 {\rm\ ft/sec}^2$)
%% CORRECT
{$u = -{1\over\ds 12}e^{-4t}(\cos(4\sqrt{3} t)
+ {1\over\ds\sqrt{3}}\sin(4\sqrt{3}t))$}
%% WRONG
{$u = -{1\over\ds 12}e^{-t/8}(\cos({\sqrt{107}\over\ds8} t) +
{1\over\ds\sqrt{107}}\sin({\sqrt{107}\over\ds8}t))$}
%% WRONG
{$u = -{1\over\ds 12}e^{-2t}(\cos({\sqrt{15}\over\ds2} t) +
{4\over\ds\sqrt{15}}\sin({\sqrt{15}\over\ds2}t))$}
%% WRONG
{$u = -{1\over\ds 12}e^{-8t}(\cos({\sqrt{63}\over\ds4}t) +
{32\over\ds\sqrt{63}}\sin({\sqrt{63}\over\ds4}t))$}
%% WRONG
{$u = -{1\over\ds 12}e^{-t/4}(\cos({\sqrt{83}\over\ds6} t) +
{3\over\ds2\sqrt{83}}\sin({\sqrt{83}\over\ds6}t))$}




%% PROBLEM
Find the form of a particular solution of
$$y'' - 4y' + 3y = e^x + \cos(x)$$
%% CORRECT
{$y_p = A x e^x + B \cos(x) + C\sin(x)$}
%% WRONG
{$y_p = A e^x + B \cos(x) + C\sin(x)$}
%% WRONG
{$y_p = A x^2 e^x + x(B \cos(x) + C\sin(x))$}
%% WRONG
{$y_p = (A x^2 + Bx + C) e^x + x(D \cos(x) + E\sin(x))$}
%% WRONG
{$y_p = A e^x + x^2( B \cos(x) + C\sin(x))$}




%% PROBLEM
Given that $y_1 = x$ and $y_2 = x^{-1}$ are solutions to a second order
homogeneous linear equation
$$L[y] = 0,\qquad (x > 0)$$
find a particular solution~$y_p$ to the equation
$$L[y] = x^3,\qquad (x > 0)$$
%% CORRECT
{$ y_p = {1\over\ds 24}x^5$}
%% WRONG
{$ y_p = {1\over\ds 4}x^3 - x^2 + 1$}
%% WRONG
{$ y_p = {1\over\ds 12}x^5 - {1\over\ds 8} x^3$}
%% WRONG
{$ y_p = {1\over\ds 8}x^3$}
%% WRONG
{$ y_p = {1\over\ds 3}x^{-3} - {1\over\ds2}x^3$}





%% PROBLEM
Classify the singular points of the equation
$$x(x^2 - 1)y'' + x(1-x)y' + y = 0$$
%% CORRECT
{$x = 0$, $x = \pm1$ regular}
%% WRONG
{$x = 0$ regular, $x = \pm1$ irregular}
%% WRONG
{$x = 0$, $x = 1$ regular; $x = -1$ irregular}
%% WRONG
{$x = 0$ irregular, $x = \pm1$ regular}
%% WRONG
{$x = 0$, $x = -1$ regular, $x = 1$ irregular}




%% PROBLEM
Compute the power series expansion of $f(x) = {1\over\ds 1 - x}$ centered at
$x_0 = -1$.
%% CORRECT
{$\sum_{n=0}^\infty {1\over\ds 2^{n+1}} (x+1)^n$}
%% WRONG
{$\sum_{n=0}^\infty {1\over\ds 2^n} (x-1)^n$}
%% WRONG
{$\sum_{n=0}^\infty {1\over\ds n!} (x+1)^n$}
%% WRONG
{$\sum_{n=0}^\infty x^n$}
%% WRONG
{$\sum_{n=0}^\infty {1\over\ds n} (x-1)^n$}





%% PROBLEM
Determine the recurrence relation for the coefficients of a power series solution
centered at $x_0 = 0$ of the equation
$$y'' + x^2 y = 0$$
%% CORRECT
{$a_n = -{1\over\ds n(n-1)} a_{n-4}$}
%% WRONG
{$a_{n+1} = -{1\over\ds (n+1)n} a_n$}
%% WRONG
{$a_n = -{1\over\ds n(n+1)} a_{n-2}$}
%% WRONG
{$a_{n+1} = -{1\over\ds (n+1)} a_{n-2} + {1\over\ds n(n-1) }a_{n-4}$}
%% WRONG
{$a_{n+2} = -{1\over\ds (n+1)} a_{n-2}$}




%% PROBLEM
Find the general solution to
$$x^2 y'' - 5 x y' + 13 y = 0,\qquad (x > 0)$$
%% CORRECT
{$y = x^3(c_1\cos(2\ln(x)) + c_2 \sin(2\ln(x)))$}
%% WRONG
{$y = x^{-3}(c_1\cos(\ln(x)) + c_2 \sin(\ln(x)))$}
%% WRONG
{$y = c_1 x^{-3} + c_2 x^2$}
%% WRONG
{$y = e^{-5x/2}(c_1\cos(\sqrt{17}x) + c_2 \sin(\sqrt{17}x))$}
%% WRONG
{$y = c_1x^{5x/2} + c_2 x^{5x/2}\ln(x)$}





%% PROBLEM
The equation
$$x^2 y'' + x(1+x)y' - {1\over\ds 4}y = 0$$
has a regular singular point at $x_0 = 0$. Find a series solution corresponding
to the larger of the exponents of the singularity.
%% CORRECT
{$y = x^{1/2}(1 - {1\over\ds 4}x + {1\over\ds 16}x^2 - {5\over\ds 384}x^3 +
\ldots)$}
%% WRONG
{$y = x^{-1/2}(1 - x + {1\over\ds 4}x^2 - {1\over\ds 120}x^3 + \ldots)$}
%% WRONG
{$y = x^{-(1+\sqrt{2})/2}(1 - {1\over2}x + {1\over\ds 4}x^2 - {3\over\ds 196}x^3 + \ldots)$}
%% WRONG
{$y = x^{(1+\sqrt{2})/2}(1 - x + {1\over\ds 12}x^2 - {1\over\ds 360}x^3 + \ldots)$}
%% WRONG
{$y = x^{3/2}(1 - x + {1\over\ds 15}x^2 - {13\over\ds 240}x^3 + \ldots)$}





%% END
\end
