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\title
\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}

\note %1
\problem
Let $y(t)$ be a solution of the equation \quad $t^2y'-y=-1$\quad defined
for $t>0$.  Find
\quad $\lim\limits_{t\rightarrow 0^+}y(t).$
\correct
1
\wrong
0
\wrong
$\infty$
\wrong
$-1$
\wrong
It can be different for different solutions.



\note %2
\problem
Classify the stable equilibrium points for the equation
$${dN\over dt}=\left|\matrix{N^2&0&2N-2\cr 0&1&0\cr 2N-2&0&(N-1)^2}\right|.$$
\correct
One stable, one unstable, and one semistable.
\wrong
Two stable, one unstable, and none semistable.
\wrong
One stable, two unstable and none semistable.
\wrong
Two stable and two semistable.
\wrong
Two unstable and two semistable.


\note %3
\problem
For what value of the constant $b$ will the following differential equation
be exact?
$$(ye^{3xy}-x)dx+bxe^{3xy}dy=0$$
\correct
1
\wrong
3
\wrong
$-1$
\wrong
0
\wrong


No value of $b$.
\note %4
\problem
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.
\correct
$Q(t)={5\over \ln 2}(1-({1\over 2})^{t\over 5})$
\wrong
$Q(t)={5\over \ln 2}(({1\over 2})^{t\over 5}-1)$
\wrong
$Q(t)={5 \ln 2}(({1\over 2})^{t\over 5}-1)$
\wrong
$Q(t)={5\ln 2}(1-({1\over 2})^{t\over 5})$
\wrong
$Q(t)=(1-({1\over 2})^{t\over 5})$


\note %5
\problem
The equation \quad $y'={x^2-14xy+3y^2\over 7x^2+y^2}-1$\quad is
\correct
homogeneous.
\wrong
linear.
\wrong
separable but not exact.
\wrong
exact with a suitable choice of integrating factor.
\wrong
none of the others.





\note %6
\problem
A particular solution of \quad $y''+y=t\cos t$\quad has the form
\correct
$(At^2+Bt)\cos t+(Ct^2+Dt)\sin t.$
\wrong
$(At+B)\cos t+(Ct+D)\sin t.$
\wrong
$(At+B)\cos t+(At+B)\sin t.$
\wrong
$At^2\cos t+Bt^2\sin t.$
\wrong
$(At+B)\cos t$

\note %7
\problem
The general solution of the equation \quad $y''+y=\csc t$,\quad for
$0<t<\pi$ is
\correct
$c_1\cos t+c_2\sin t-t\cos t+(\sin t)\ln \sin t$.
\wrong
$c_1\sin t+c_2\cos t+t\sin t-(\cos t)\ln \sin t$.
\wrong
$c_1e^t\sec t+c_2e^t\csc t$.
\wrong
$c_1t\sin t+c_2(\cos t)\ln\sin t$.
\wrong
$(c_1+t)\cos t+(c_2+\ln \cos t)\sin t$.



\note %8
\problem
Find \quad $y(1)$ \quad if\quad $y''+4y'+5y=0, \quad y(0)=1,\quad y'(0)=0.$
\correct
0.301
\wrong
0.187
\wrong
0.241
\wrong
0.282
\wrong
0.666


\note %9
\problem
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
\correct
${1\over x}.$
\wrong
$x.$
\wrong
${1\over x^2}.$
\wrong
$x^2.$
\wrong
1.

\note %10
\problem
Find \quad $y(1)$\quad if $y(t)$ is the solution of the initial value problem
$$y''-y'=-2t,\quad y(0)=0,\quad y'(0)=2.$$
\correct
3
\wrong
$4+e$
\wrong
4
\wrong
$e+2$
\wrong
1



\note %11
\problem
Find the first three nonzero terms of a power series solution to the
initial value problem
$$(1-x^2)y''+3y=0,\quad y(0)=0,\quad y'(0)=1.$$
\correct
$x-{1\over2}x^3-{3\over40}x^5$
\wrong
$1+{1\over2}x^3-{3\over8}x^5$
\wrong
$x-{1\over2}x^3+{3\over20}x^5$
\wrong
$1+x+{1\over3}x^3$
\wrong
$x-{1\over6}x^3-{1\over40}x^5$


\note %12
\problem
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
\correct
$\rho\ge 5$.
\wrong
$\rho\ge 4$.
\wrong
$\rho\ge 3$.
\wrong
$\rho\ge 16$.
\wrong
$\rho =\infty$.

\note %13
\problem
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 $a_0\ne 0$?
\correct
4
\wrong
3
\wrong
2
\wrong
0
\wrong
No value of $r$.



\note %14
\problem
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.


\correct
All except D.
\wrong
All except C.
\wrong
All except B.
\wrong
All except A.
\wrong
All are true.

\note %15
\problem
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 $A^{-1}$?
\correct
Only the first row.
\wrong
Only the second row.
\wrong
Only the third row.
\wrong
All the rows.
\wrong
None of the rows.

\note %16
\problem
The vectors \quad $v_1=(1,0,1,1),\quad v_2=(2,1,1,4), \quad v_3=(0,2,3,4)$
\correct
are linearly independent and do not span ${\bf R}^4$.
\wrong
are linearly independent and span ${\bf R}^4$.
\wrong
are linearly dependent and do not span ${\bf R}^4$.
\wrong
are linearly dependent and span ${\bf R}^4$.
\wrong
are linearly independent and span a two-dimensional subspace of ${\bf R}^4$.

\note %17
\problem
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
\correct
$n=2m$.
\wrong
$n=m$.
\wrong
$n=r=2m$.
\wrong
$n+r=m$.
\wrong
$n=r=m$.



\note %18
\problem
Find \quad $|A|$\quad if $A$ is the matrix of the linear transformation $L$
defined by
 $$L\left[\matrix{x_1\cr x_2\cr x_3\cr x_4}\right]=
\left[\matrix{3x_1+5x_2-2x_3+6x_4\cr
\hphantom{3}x_1+2x_2-\hphantom{2}x_3+\hphantom{3}x_4\cr
2x_1+4x_2+\hphantom{2}x_3+5x_4\cr 3x_1+7x_2+5x_3+3x_4}\right]$$
\correct
$-18$
\wrong
$-36$
\wrong
$-9$
\wrong
$-27$
\wrong
0


\note %19
\problem
In the system below, if the coefficient matrix has determinant 25, find  $x_1$.
$$\left[\matrix{\hphantom{-}11&0&2&0&3\cr \hphantom{-1}2&4&0&5&0\cr\hphantom{1}
-1&0&7&0&8\cr \hphantom{-}14&9&0&10&0\cr -12&0&12&0&13}\right]
\left[\matrix{x_1\cr x_2\cr x_3\cr x_4\cr x_5}\right]
=\left[\matrix{1\cr 0\cr 6\cr 0\cr 11}\right]$$
\correct
0
\wrong
${4\over5}$
\wrong
$-{2\over 5}$
\wrong
${3\over5}$
\wrong
$-{1\over5}$

\note %20
\problem
Find the entry in the fourth row and second column of $A^{-1}$ if
$$A=\left[\matrix{2&0&0&0&0\cr1&2&0&0&0\cr 1&1&2&0&0\cr1&1&1&2&0\cr
1&1&1&1&2}\right].$$
\correct
$-{1\over8}$
\wrong
4
\wrong
0
\wrong
${1\over4}$
\wrong
$-{1\over 16}$



\note %21
\problem
Find the polynomial whose roots are the eigenvalues of the matrix
$A=\left[\matrix{4&0&1\cr 2&3&2\cr 1&0&4}\right]$.
\correct
$(3-\lambda)^2(5-\lambda)$
\wrong
$(3-\lambda)(5-\lambda)^2$
\wrong
$(3-\lambda)(4-\lambda)^2$
\wrong
$(3-\lambda)((4-\lambda)^2+1)$
\wrong
$15\lambda -8\lambda^2+\lambda^3$


\note %22
\problem
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?
\correct
$P=\left[\matrix{-1&-1&1\cr \hphantom{-}1&\hphantom{-}0&1\cr
\hphantom{-}0&\hphantom{-}1&1}\right]$
\wrong
$P=\left[\matrix{-2&-1&-1\cr \hphantom{-}1&\hphantom{-}1&\hphantom{-}0\cr
\hphantom{-}1&\hphantom{-}0&\hphantom{-}1}\right]$
\wrong
$P=\left[\matrix{1&1&0\cr 1&0&1\cr 0&1&1}\right]$
\wrong
$P=\left[\matrix{\hphantom{-}1&-1&\hphantom{-}1\cr -1&\hphantom{-}0&-1\cr
\hphantom{-}0&\hphantom{-}1&\hphantom{-}1}\right]$
\wrong
$P=\left[\matrix{-1&0&1\cr \hphantom{-}0&1&1\cr \hphantom{-}1&2&1}\right]$

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