% 226S98Final Version 1, Mon May  4 13:43:11 1998


% Page Setup
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\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}{Definition}


\begin{document}






% Problems

\vbox{
%1

\noindent\llap{1.\ }Let $y(t)$ be a solution of the equation \quad
$t^2y'-y=-1$\quad defined for $t>0$.  Find
\quad $\lim\limits_{t\rightarrow 0^+}y(t).$

\beginanswers
{(a)~1\answerspace}
{(b)~0\answerspace}
{(c)~$\infty$\answerspace}
{(d)~$-1$\answerspace}
{(e)~It can be different for different solutions.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%2

\noindent\llap{2.\ }Classify the stable equilibrium points for the equation
$${dN\over dt}=\left|\matrix{N^2&0&2N-2\cr 0&1&0\cr 2N-2&0&(N-1)^2}\right|.$$

\beginanswers
{(a)~One stable, one unstable, and one semistable.\answerspace}
{(b)~Two stable, one unstable, and none semistable.\answerspace}
{(c)~One stable, two unstable and none semistable.\answerspace}
{(d)~Two stable and two semistable.\answerspace}
{(e)~Two unstable and two semistable.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%3

\noindent\llap{3.\ }For what value of the constant $b$ will the following
differential equation be exact?
$$(ye^{3xy}-x)dx+bxe^{3xy}dy=0$$

\beginanswers
{(a)~1\answerspace}
{(b)~3\answerspace}
{(c)~$-1$\answerspace}
{(d)~0\answerspace}
{(e)~No value of $b$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%4

\noindent\llap{4.\ }The water bath surrounding a certain nuclear reactor
contains radioactive material that
decays with a half-life of 5 days.  New radioactive material is added to
the bath at the
constant rate of 1 gram per day.  Assume the bath contains no radioactive
material
initially.  Find an expression for the amount $Q(t)$ of grams of
radioactive material
after $t$ days.

\beginanswers
{(a)~$Q(t)={5\over \ln 2}(1-({1\over 2})^{t\over 5})$\answerspace}
{(b)~$Q(t)={5\over \ln 2}(({1\over 2})^{t\over 5}-1)$\answerspace}
{(c)~$Q(t)={5 \ln 2}(({1\over 2})^{t\over 5}-1)$\answerspace}
{(d)~$Q(t)={5\ln 2}(1-({1\over 2})^{t\over 5})$\answerspace}
{(e)~$Q(t)=(1-({1\over 2})^{t\over 5})$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%5

\noindent\llap{5.\ }The equation \quad $y'={x^2-14xy+3y^2\over
7x^2+y^2}-1$\quad is

\beginanswers
{(a)~homogeneous.\answerspace}
{(b)~linear.\answerspace}
{(c)~separable but not exact.\answerspace}
{(d)~exact with a suitable choice of integrating factor.\answerspace}
{(e)~none of the others.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%6

\noindent\llap{6.\ }A particular solution of \quad $y''+y=t\cos t$\quad has
the form

\beginanswers
{(a)~$(At^2+Bt)\cos t+(Ct^2+Dt)\sin t.$\answerspace}
{(b)~$(At+B)\cos t+(Ct+D)\sin t.$\answerspace}
{(c)~$(At+B)\cos t+(At+B)\sin t.$\answerspace}
{(d)~$At^2\cos t+Bt^2\sin t.$\answerspace}
{(e)~$(At+B)\cos t$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%7

\noindent\llap{7.\ }The general solution of the equation \quad $y''+y=\csc
t$,\quad for $0<t<\pi$ is

\beginanswers
{(a)~$c_1\cos t+c_2\sin t-t\cos t+(\sin t)\ln \sin t$.\answerspace}
{(b)~$c_1\sin t+c_2\cos t+t\sin t-(\cos t)\ln \sin t$.\answerspace}
{(c)~$c_1e^t\sec t+c_2e^t\csc t$.\answerspace}
{(d)~$c_1t\sin t+c_2(\cos t)\ln\sin t$.\answerspace}
{(e)~$(c_1+t)\cos t+(c_2+\ln \cos t)\sin t$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%8

\noindent\llap{8.\ }Find \quad $y(1)$ \quad if\quad $y''+4y'+5y=0, \quad
y(0)=1,\quad y'(0)=0.$

\beginanswers
{(a)~0.301\answerspace}
{(b)~0.187\answerspace}
{(c)~0.241\answerspace}
{(d)~0.282\answerspace}
{(e)~0.666\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%9

\noindent\llap{9.\ }For $x>0$ the Wronskian of two independent solutions of
the Euler equation
\quad$x^2y''+xy'-y=0$\quad is a constant multiple of

\beginanswers
{(a)~${1\over x}.$\answerspace}
{(b)~$x.$\answerspace}
{(c)~${1\over x^2}.$\answerspace}
{(d)~$x^2.$\answerspace}
{(e)~1.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%10

\noindent\llap{10.\ }Find \quad $y(1)$\quad if $y(t)$ is the solution of
the initial value problem
$$y''-y'=-2t,\quad y(0)=0,\quad y'(0)=2.$$

\beginanswers
{(a)~3\answerspace}
{(b)~$4+e$\answerspace}
{(c)~4\answerspace}
{(d)~$e+2$\answerspace}
{(e)~1\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%11

\noindent\llap{11.\ }Find the first three nonzero terms of a power series
solution to the initial value problem
$$(1-x^2)y''+3y=0,\quad y(0)=0,\quad y'(0)=1.$$

\beginanswers
{(a)~$x-{1\over2}x^3-{3\over40}x^5$\answerspace}
{(b)~$1+{1\over2}x^3-{3\over8}x^5$\answerspace}
{(c)~$x-{1\over2}x^3+{3\over20}x^5$\answerspace}
{(d)~$1+x+{1\over3}x^3$\answerspace}
{(e)~$x-{1\over6}x^3-{1\over40}x^5$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%12

\noindent\llap{12.\ }The equation \quad $(x^2+16)y''+3y'-4x^2y=0$\quad has
a series solution in powers of
$x-3$.  From the general theory, its radius of convergence $\rho$ must satisfy

\beginanswers
{(a)~$\rho\ge 5$.\answerspace}
{(b)~$\rho\ge 4$.\answerspace}
{(c)~$\rho\ge 3$.\answerspace}
{(d)~$\rho\ge 16$.\answerspace}
{(e)~$\rho =\infty$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%13

\noindent\llap{13.\ }Consider the equation \quad
$x(1-x)y''+(2+x)y'+y=0$.\quad  For which values of $r$ is
there sure to be a solution of the form\quad $|x-1|^r\sum\limits_{n=0}^\infty
a_n(x-1)^n$,\quad with \quad $a_0\ne 0$?

\beginanswers
{(a)~4\answerspace}
{(b)~3\answerspace}
{(c)~2\answerspace}
{(d)~0\answerspace}
{(e)~No value of $r$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%14

\noindent\llap{14.\ }Which of the following statements about the matrix $A$
are true?
$$A=\left[\matrix{2&4&2&-1&0&2&-2\cr 4&8&4&-2&1&7&-4\cr
0&0&0&\hphantom{-}2&0&4&\hphantom{-}4}\right]$$

A.  The first row of the reduced echelon form of $A$ is
$\left[\matrix{1&2&1&0&0&2&0}\right]$.

B.  The system $Ax=b$, where $x$ is $7\times 1$ and $b$ is $3\times 1$, has
a solution for
any choice of
$b$.

C.  The dimension of the column space of $A$ is 3.

D.  The dimension of the solution space of $A$ is 3.

\beginanswers
{(a)~All except D.\answerspace}
{(b)~All except C.\answerspace}
{(c)~All except B.\answerspace}
{(d)~All except A.\answerspace}
{(e)~All are true.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%15

\noindent\llap{15.\ }If\quad
$A=\left[\matrix{1&-1&0\cr 0&\hphantom{-}2&1\cr
1&\hphantom{-}1&0}\right]$,\quad which
row of $A^{-1}$ is orthogonal to the other two rows of $A^{-1}$?

\beginanswers
{(a)~Only the first row.\answerspace}
{(b)~Only the second row.\answerspace}
{(c)~Only the third row.\answerspace}
{(d)~All the rows.\answerspace}
{(e)~None of the rows.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%16

\noindent\llap{16.\ }The vectors \quad $v_1=(1,0,1,1),\quad v_2=(2,1,1,4),
\quad v_3=(0,2,3,4)$

\beginanswers
{(a)~are linearly independent and do not span ${\bf R}^4$.\answerspace}
{(b)~are linearly independent and span ${\bf R}^4$.\answerspace}
{(c)~are linearly dependent and do not span ${\bf R}^4$.\answerspace}
{(d)~are linearly dependent and span ${\bf R}^4$.\answerspace}
{(e)~are linearly independent and span a two-dimensional subspace of ${\bf
R}^4$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%17

\noindent\llap{17.\ }Let $A$ be an $m\times n$ matrix of rank $r$.  Suppose
the row space and the solution
space of $A$ have the same dimension, and that $Ax=b$ always has a
solution.  Then

\beginanswers
{(a)~$n=2m$.\answerspace}
{(b)~$n=m$.\answerspace}
{(c)~$n=r=2m$.\answerspace}
{(d)~$n+r=m$.\answerspace}
{(e)~$n=r=m$.\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%18

\noindent\llap{18.\ }Find \quad $|A|$\quad if $A$ is the matrix of the
linear transformation $L$ defined by
 $$L\left[\matrix{x_1\cr x_2\cr x_3\cr x_4}\right]=
\left[\matrix{3x_1+5x_2-2x_3+6x_4\cr
\hphantom{3}x_1+2x_2-\hphantom{2}x_3+\hphantom{3}x_4\cr
2x_1+4x_2+\hphantom{2}x_3+5x_4\cr 3x_1+7x_2+5x_3+3x_4}\right]$$

\beginanswers
{(a)~$-18$\answerspace}
{(b)~$-36$\answerspace}
{(c)~$-9$\answerspace}
{(d)~$-27$\answerspace}
{(e)~0\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%19

\noindent\llap{19.\ }In the system below, if the coefficient matrix has
determinant 25, find  $x_1$.
$$\left[\matrix{\hphantom{-}11&0&2&0&3\cr \hphantom{-1}2&4&0&5&0\cr\hphantom{1}
-1&0&7&0&8\cr \hphantom{-}14&9&0&10&0\cr -12&0&12&0&13}\right]
\left[\matrix{x_1\cr x_2\cr x_3\cr x_4\cr x_5}\right]
=\left[\matrix{1\cr 0\cr 6\cr 0\cr 11}\right]$$

\beginanswers
{(a)~0\answerspace}
{(b)~${4\over5}$\answerspace}
{(c)~$-{2\over 5}$\answerspace}
{(d)~${3\over5}$\answerspace}
{(e)~$-{1\over5}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%20

\noindent\llap{20.\ }Find the entry in the fourth row and second column of
$A^{-1}$ if
$$A=\left[\matrix{2&0&0&0&0\cr1&2&0&0&0\cr 1&1&2&0&0\cr1&1&1&2&0\cr
1&1&1&1&2}\right].$$

\beginanswers
{(a)~$-{1\over8}$\answerspace}
{(b)~4\answerspace}
{(c)~0\answerspace}
{(d)~${1\over4}$\answerspace}
{(e)~$-{1\over 16}$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%21

\noindent\llap{21.\ }Find the polynomial whose roots are the eigenvalues of
the matrix
$A=\left[\matrix{4&0&1\cr 2&3&2\cr 1&0&4}\right]$.

\beginanswers
{(a)~$(3-\lambda)^2(5-\lambda)$\answerspace}
{(b)~$(3-\lambda)(5-\lambda)^2$\answerspace}
{(c)~$(3-\lambda)(4-\lambda)^2$\answerspace}
{(d)~$(3-\lambda)((4-\lambda)^2+1)$\answerspace}
{(e)~$15\lambda -8\lambda^2+\lambda^3$\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%22

\noindent\llap{22.\ }Let \quad$A=\left[\matrix{0&1&1\cr 1&0&1\cr
1&1&0}\right]$.\quad  Then $A$ has 2 as an
eigenvalue, and $-1$ as an eigenvalue of multiplicity two.  Use ${\bf
reduced}$ echelon
form to find the eigenvectors of $A$.  For which of the following matrices
$P$ will
$P^{-1}AP$ be a diagonal matrix?

\beginanswers
{(a)~$P=\left[\matrix{-1&-1&1\cr \hphantom{-}1&\hphantom{-}0&1\cr
\hphantom{-}0&\hphantom{-}1&1}\right]$\answerspace}
{(b)~$P=\left[\matrix{-2&-1&-1\cr \hphantom{-}1&\hphantom{-}1&\hphantom{-}0\cr
\hphantom{-}1&\hphantom{-}0&\hphantom{-}1}\right]$\answerspace}
{(c)~$P=\left[\matrix{1&1&0\cr 1&0&1\cr 0&1&1}\right]$\answerspace}
{(d)~$P=\left[\matrix{\hphantom{-}1&-1&\hphantom{-}1\cr -1&\hphantom{-}0&-1\cr
\hphantom{-}0&\hphantom{-}1&\hphantom{-}1}\right]$\answerspace}
{(e)~$P=\left[\matrix{-1&0&1\cr \hphantom{-}0&1&1\cr
\hphantom{-}1&2&1}\right]$\answerspace}
\endanswers\problemskip}\vfil\goodbreak


% Appendix

{}

% Blank answer sheet

\vfil\eject\output={}
\vbox{{\noindent
{\bf Math 226: Calculus IV}\hfill
Sign your Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl May 7, 1998}\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
{\bf When you are told to begin, tear off the answer sheet (gently) and
keep it under
your test while you are working.
\bigskip\noindent
 At the end of the exam please turn in only the answer sheet.}
\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 22 multiple
choice questions worth 6 points each. You start with 24 points, and the
highest
possible score is 156.  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.
\bigskip\noindent


\def\answerskip{\vskip0.25in}


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kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
}\hfil\vtop{
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kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
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kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{22.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}\ens
kip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
}}}


% Marked answer sheet

\vfil\eject
\vbox{\vbox to0pt{\vskip-22.5pt\hbox to \hsize{\hfil\ansbox{70pt}{\ Answer
Key 1\ }\hfil}\vss}
\prevdepth=-1000pt
{\noindent
{\bf Math 226: Calculus IV}\hfill
Sign your Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Final Exam}\quad{\sl May 7, 1998}\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
{\bf When you are told to begin, tear off the answer sheet (gently) and
keep it under
your test while you are working.
\bigskip\noindent
 At the end of the exam please turn in only the answer sheet.}
\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 22 multiple
choice questions worth 6 points each. You start with 24 points, and the
highest
possible score is 156.  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.
\bigskip\noindent


\def\answerskip{\vskip0.25in}


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}}}

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