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\begin{document}






\title{{\large\bf Math 325, Test 3}}
\comment{ \large 
\begin{minipage}[c]{6.2in}
1. Please cross \fbox{$\times $} the correct answers. \\
2.  This test will be exactly 50 minutes in length.  
When you are told to begin, but not before, glance 
through the entire test and put your name on each
page.  It is YOUR RESPONSIBILITY to make sure your 
test consists of 5 PAGES with 10 PROBLEMS.  The
point value for each problem is 10 points with a total of 100 points.  
The blank sheet and the backs
of the test pages are for scratch work.\\
3.  On all problems, show your work, indicating clearly 
in the space provided how you arrived at your answer. 
 The points you receive for problems 6 to 10 will depend on the
extent to which the work you show convinces the grader 
(at the time of grading) that you did all or
part of the problem correctly,
 whether or not you write down the correct answer.
\end{minipage}} 
\answersheetfootline{\large\bf 
\fbox{\rule[-2mm]{0mm}{8mm}Sign your name:\hspace{6cm}}
\hspace{6cm}}
\twocolumn{12} 
\date{{\large December 4, 1991}}
 \noprofessor \nosection \vertmultchoice=2.5in
\multiplechoiceskip=0.8cm \let\ansborder=\boxborder \boxwidth=1cm
\boxdepth=0.4cm

\pagenumberstyle=1
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\problem
Suppose $f(x)$ satisfies the initial value problem $y^{\prime } =y+4x;\;
y(0)=3$.
 Use Euler's method with stepsize $h=1$ to estimate $f(2)$.(2 iterations).

\correct $16$
\wrong $0$
\wrong $-7$
\wrong $2$
\wrong $4$





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\problem
Consider the following nonlinear system: 
$$\begin{array}{clcr} x' & = & x^2 + 5x + 6 \\ 
y' & = & x - y - 4 \end{array}$$

Which of the following points is a critical point?

\correct $(-3, -7)$
\wrong $(-3, -2)$
\wrong $(-2, 0)$
\wrong $(-2, -5)$
\wrong $(-3, 5)$		


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\problem

Find the fundamental period of $f(x) = \tan \frac{1}{3}x.$

\correct $3 \pi$
\wrong $ \pi$
\wrong $2$
\wrong $\frac {\pi}{2}$
\wrong $\frac{1}{3}\pi$

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\problem

Find a solution of the heat conduction problem:
$$u_{xx} = u_t  \ \ \ 0 < x < 1, \ t > 0$$
$$u(0, t) = 0, \ \ \ u(1, t) = 0 {\rm\; for\; all\;}  t > 0,$$
and $u(x,0) = 3 \sin 2 \pi x + 4 \sin 6 \pi x.$

\correct $3e^{-4 \pi ^2 t} \sin 2 \pi x + 4 e^{-36 \pi^2t} \sin 6 \pi x$
\wrong $e^{-36 \pi^2 t} \sin 2 \pi x + 3e^{-4 \pi^2 t} \sin 6 \pi x$
\wrong $3e^t \sin 2 \pi x + 4e^t \sin 6 \pi x$
\wrong $3e^{-4 \pi^2t} \sin 2 \pi x + 4 e ^{-6 \pi^2 t} \sin 6 \pi x$
\wrong $3e^{-2 \pi t} \sin 2 \pi x + 4 e^{-6 \pi t} \sin 6 \pi x$

\vspace*{2cm}
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\problem

Find the coefficient of $\cos (2\pi x)$ in the Fourier Series of the function
$f(x) = -2\sin (2\pi x)+\cos (3\pi x)$ defined on $[- 1,1 ].$
\correct $0$
\wrong $\pi$
\wrong $\frac{\pi}{2}$
\wrong $5$
\wrong $-1$

\vspace*{7cm}
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\problem
Suppose $f(x)$ is a function whose graph passes through the point
$P=(0,1)$ in the 
$xy\! \! $--plane and
satisfies the differential equation $y^{\prime } =(1+y^{3})\cos x$.
 Which of the following lines is the tangent line of the graph of $f$ at
$P$?

\correct $y=2x+1$
\wrong $x+1$
\wrong $2x-1$
\wrong $x-1$
\wrong $3x+1$

\vspace*{6cm}

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\problem
Use the improved Euler method with stepsize $h = 0.1$ to estimate $y(1.1)$
for the initial
value problem $y' = \ ( \sqrt{x + y} \ ), \; y(1) = 3.$
(You may approximate $\sqrt{4.3}$ by $2.08$.)


\vspace*{6cm}

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\problem 
If $u(x, t) = $F(x)G(t) satisfies the partial differential equation
$$tu_{xx} - xu_t = 0$$
Then use the method of separation of variables to replace this partial
differential equation by an
ordinary differential equation for $F$ and an ordinary differential
equation for $G$.

\vspace*{6cm}

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\problem
Consider the non-linear system:
$$\begin{array}{clcr}
x' & = & x^2 - y^2 \\
y' & = & 1 - x  \end{array}$$ 

a.  Show that the point $(1,1)$ is a critical point for this system.

b.  Determine the type and stability for this equilibrium point.

c.  Find all other critical points for this system.

\vspace*{5cm}

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\problem
Let $f(x) = x$ for $-1 \leq x < 1$ and $f(x + 2) = f(x),$ so that $f$ is
periodic with period 2.
If we represent $f$ by the Fourier series
$$\frac{a_0}{2} + \sum_{m=1}^{\infty} \left(a_m \cos (m \pi x) + b_m \sin
(m \pi x)\right)$$
determine the values of $a_0, \;  a_1,$ and  $b_1.$  



\vspace*{5cm}
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