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\noindent{\bf Math 112\hskip38mm Prelim 3 \hskip32mm Fall 2001}\newline 
\centerline{\bf November 27, 2001}

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\noindent{\bf Instructions:} 1) Write your name, your instructor's 
name, and your section number on the front cover of the exam booklet. 
Sign the academic integrity statement.
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\noindent 2) Show all your work on all questions and explain your 
answers carefully.\newline 
A correct answer without justification gets no credit.
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\noindent 3) Do not simplify your answers. 
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\noindent 4) There are 7 problems worth a total of 100 points.  
(Problems 6 and 7 are on the back of this sheet.) Do them in whatever 
order you wish, but label each solution clearly.
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\noindent 5) No calculators are allowed on this exam. 

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\noindent{\bf Problems:} 

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\noindent{\bf 1.} (20 points) Find the radius and interval of 
convergence of the following power series. 
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\centerline{ a) $\displaystyle \sum_{n=0}^{\infty}\frac{x^{3n}}{n!}$
\hskip25mm 
 b) $\displaystyle \sum_{n=1}^{\infty} \frac{(x-1)^{n}}{n}$.}

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\noindent{\bf 2.} (20 points) a) Find a power series representation 
for the function\newline 
\centerline{$f(x) = \displaystyle\frac{x^{2}}{(1-x)^{2}}$.}
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\noindent (Start by differentiating a suitable geometric series.)
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b) Determine the radius of convergence of the series you found in part 
a).
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c) Is the series you found in part a) the Maclaurin series of $f(x)$?
Explain your answer.

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\noindent{\bf 3.} (10 points) Using the definition of Taylor's series 
find the first three terms of the Taylor series for $\sin x$ centered 
at $a = \pi/4$. (The constant term counts as the first term. You may 
find it useful to remember that $\sin(\pi/4) = \cos(\pi/4) = 1/\sqrt{2}$.)

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\noindent{\bf 4.} (8  points) Use Euler's method with step size $1/2$ 
to estimate $y(2)$, where $y(x)$ is the solution of the initial-value 
problem $y' = 3x-2y+1$, $y(1)=2$.  Remember to show your work.

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\noindent{\bf 5.} (12  points) Solve the initial-value problem \ \ 
$\displaystyle\frac{dy}{dx} = 2x+2xy^{2}$, $y(0) = 1$. 

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\noindent{\bf 6.} (15 points) A tank contains 500 liters of pure 
water.  Brine that contains 0.1 kilograms of salt per liter of water 
enters the tank at a rate of 5 liters per minute.  The solution is 
kept thoroughly mixed and drains from the tank at the same rate.  How 
much salt is in the tank after $t$ minutes?

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\noindent{\bf 7.} (15 points) Match each differential equation with 
its direction field (labeled I--IV). Give reasons for your answers. 
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\centerline{ (a) $y'=x-y$, \hskip4mm (b) $y' = 2x-1$, \hskip4mm (c) 
$y' = x^{2} + y - 1$, \hskip4mm (d) $y'=y$.}

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