% Final exam for projects 112, Fall 2001

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\noindent{\bf Projects Math 112\hfill Final Exam\hfill December 13, 2001}

\vspace{4mm}

\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 250 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:}

\begin{enumerate}
%--------------------------------------------------------------
% 5.5-5.8:

\item (40 points) Evaluate the following integrals:
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a) $\displaystyle\int \frac{e^{x^{-1}}}{x^2} \ dx$
%\hfill
%or $\displaystyle\int \cos x \ e^{\sin x} \ dx$
\hskip40mm
b) $\displaystyle\int_{-1}^{1} \frac{x^2 \sin(x^3)}{\cos x} \ dx$
%\hfill
%or $\displaystyle\int_{0}^{\pi} \sin^3 x \cos x \ dx$
\vskip3mm
%c) $\displaystyle\int \frac{\ln \sqrt{x}}{x} \ dx$
%vskip3mm
c) 
%$\displaystyle\int_{1}^{e} x \ln x \ dx$
%\hfill or 
$\displaystyle\int \frac{\ln x}{x^2} \ dx$
\hskip42mm
d) $\displaystyle\int \frac{x}{(x+5)^3}  \ dx$

\vskip4mm

%--------------------------------------------------------------
% 5.10 cntd

\item (25  points) For what positive values of $p$ will the
following integral be finite?  Justify your answer.
\[
\int_2^\infty \frac{1}{x (\ln{x})^p} \ dx
\]

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%--------------------------------------------------------------
% 6.3
%
%\noindent {\bf 3.} (?? points) Find the arc length of the parametric curve
%given by: $x(t) = e^t - t$ and $y(t) = 4 e^{t/2}$ for $0 \leq t \leq 2$.
%
%\vskip4mm
%
%--------------------------------------------------------------
% 6.2

\item (20 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.

\vskip 4mm

%------------------------------------------------------------
% 6.7
%
%\item (8 points) Let $f$ be a continuous function such that
%$ \int_0^1 f(x) \ dx = 2. $
%Explain why there exists a value $c$ in the interval $[0, 1]$ such that
%$f(c) = 2$.
%
%\vskip4mm
%

%--------------------------------------------------------------
% 8.1

\item (35 points) For each of the following sequences,
determine if it converges or diverges. 
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a) $a_n = (-1)^n \displaystyle \ln{\frac{n^2+1}{n^2}}, \ n=1,2,\ldots. $
\hfill
b) $b_n = \displaystyle \frac{5n^3+2}{3n^2+4n+1}, \ n=1,2,\ldots. $
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c) $c_1 = 2; \ c_{n+1} = \frac{1}{3} c_n +2, \ n=1, 2, \ldots.$


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%--------------------------------------------------------------
% 8.2-8.4

\item (30 points) For each of the following series,
determine if it converges or diverges. 
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a) $\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac{n^2 \ 2^n}{n!} $
\hfill
b) $\displaystyle \sum_{n=1}^{\infty} \frac{7^{n+2}}{3^n} $
\hfill
%c) $\displaystyle \sum_{n=2}^{\infty} \frac{2}{n \ln{n}} $
%\hskip45mm
c) $\displaystyle \sum_{n=1}^{\infty} \frac{\sin(2\pi n)}{2\pi n} $


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%-------------------------------------------------------------
% 8.5-8.7

\item (40 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 series?\\
b) What are series for $f'(x)$ and $g'(x)$?\\
c) A new function $h(x)$ is defined by $h(x)=f(x)g(x)$.  Calculate
$h'(0)$.\\
d) Find the first 3 non-zero terms of the series for $h(x)$.


%-------------------------------------------------------------
%--------------------------------------------------------------
% 7.1-7.2

\item (30 points) Suppose we know that the graph below is
the graph of the right hand side $f(y)$ of the differential equation 
$dy/dt = f(y)$.          

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\scalebox{.75}{\includegraphics{finalfig2}}
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\noindent a) Sketch the direction field that corresponds to the
differential equation.  (Be sure to include enough data points in your
sketch so that all the major features of the differential equation
appear.)        
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\noindent b) Over your direction field, sketch the solutions to the
differential equation with initial conditions $y(0)=0, \
y(0)=1,$ and $y(0)=2.$

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%--------------------------------------------------------------
% 7.3

%\noindent{\bf 7.} (?? points) A cylindrical bucket with a hole in the
%bottom is being filled with water at a rate of 5 $cm^3/min$.  It is 20 
%$cm$ tall and the radius of the bottom is 10 $cm$.  When the height of
%the water is $h \ cm$, water leaks from the bucket at a rate of $\sqrt{h}
%\ cm^3/min$.  Assuming that the bucket starts out empty, how long does it
%take to fill completely?
%
%\vskip2mm

\item (30 points) A cylindrical bucket full of water is
draining from a hole in the bottom. The bucket is 20 $cm$ tall and the
radius of the bottom is 10 $cm$.  When the height of the water is $h
\ cm$, water leaks from the bucket at a rate of $\sqrt{h} \ cm^3/min$.  
Assuming that the bucket starts out full, how long does it take to
become empty?

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\end{enumerate}

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