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\centerline{\bf Math 112\hskip20mm Final Examination \hskip20mm 
December 13,2001}

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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 8 problems worth a total of 150 points.  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.} (30 points) 

a) (18 points) Evaluate the following integrals: 
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\hskip3mm i) $\dsize \int \frac{\sin(1/x)}{x^{2}} dx$ \hskip16mm 
ii) $\dsize \int \frac{\ln x}{x^{2}}dx$ \hskip16mm 
iii) $\dsize \int_{0}^{\pi}\sin^{3}\! x\, \cos x\, dx$.

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b) (12 points) Find the positive values of $p$ for 
which the integral
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\centerline{$\dsize \int_{2}^{\infty} \frac{1}{x(\ln x)^{p}}dx$}
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\noindent converges, and evaluate the integral for those values of $p$. 

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\noindent {\bf 2.} (15 points) Find the arc length of the parametric curve
$x = e^t - t$, $y = 4e^{t/2}$, $0 \leq t \leq 2$.

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\noindent {\bf 3.} (15 points) Find the volume of the solid obtained by
rotating the region bounded by the curves $y=x^2$ and $y=x$ about the
$y$-axis.

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\noindent{\bf 4.} (12 points) For each of the following sequences,
determine if it converges or diverges. 
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a) $\dsize a_n = (-1)^n  \ln{\frac{n^2+1}{n^2}},\, n=1,2,\ldots$ 
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b) $\dsize b_n =  \frac{5n^3+2}{3n^2+4n+1}, \ n=1,2,\ldots$

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\noindent{\bf 5.} (18 points) For each of the following series,
determine if it converges or diverges. 
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a) $\dsize \sum_{n=1}^{\infty} (-1)^n \frac{n^2 \, 2^n}{n!} $
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b) $\dsize \sum_{n=1}^{\infty} \frac{7^{n+2}}{3^n} $
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c) $\dsize \sum_{n=2}^{\infty} \frac{2}{n \ln{n}}$.


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\noindent{\bf 6.} (12 points) Sketch a direction field for the 
differential equation $y' = 1-y$. Then use it to sketch the solution 
curve that passes through the point $(0,1)$ and the solution curve 
that passes through the point $(0,0)$.

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\noindent{\bf 7.} (24 points) Water is draining from a hole in the 
bottom of a cylindrical tank. The water level $y$ in the tank 
(measured in feet) satisfies the differential equation 
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\centerline{$\dsize \frac{dy}{dt} = -\frac{1}{50}\sqrt{y}$,}
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\noindent where the time $t$ is measured in minutes. 
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a) Find the general solution of this differential equation.
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b)
If the water level at time $t=0$ is 25 feet, how long will it take for
the water to drain completely?

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\noindent{\bf 8.} (24 points) Two functions $f(x)$ and $g(x)$ are given
by:
$$ 
f(x) = \sum_{n=0}^\infty \frac{x^n}{n+1} = 1 + \frac{x}{2} +
\frac{x^2}{3} + \frac{x^3}{4} + \cdots, 
$$
$$
g(x) = \sum_{n=0}^\infty (-1)^n \ n \ x^n = 0 - x + 2 x^2 - 3 x^3 +
\cdots.
$$

a) What is the radius of convergence of each of the above power series?
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b) What are the power series for $f'(x)$ and $g'(x)$?
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c) A new function $h(x)$ is defined by $h(x)=f(x)g(x)$.  Calculate
$h'(0)$.
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d) Find the first three terms of the power series for $h(x)$.


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