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\begin{center}

{\bf APPM 1360 \hspace{1.0in}  FINAL EXAM  \hspace{1.0in} FALL 2000}
% December 20, 2000, ~~~  7:00 - 8:30 p.m.
\end{center}


\noindent {\bf On the front of your bluebook, please write:
a grading key, 
your name, student ID, and
instructor (Dougherty or Biswas). }  This 
exam is worth 200 points and has 9 questions.   A list of 
formulas is given on the last page. 
{\bf Show all work!}   Answers with no justification will
receive no points. 

\be



\item (15 points) Evaluate 
$ \ds \int \frac{2x^3 + 2x + 1}{x^2(x^2+1)} \; dx  $ 

\item (20 points) Find the volume of the solids generated by revolving the
region between the two curves $y=x^2+1$ and $y=-x+3$ about
the
\be
\item $x$-axis
\item the line $x =2$.
\ee

%\item Find the center of mass of the region in the upper half plane
%that is  bounded below by the $x$-axis and above by
%$\ds \frac{x^2}{9} + \frac{y^2}{16} = 1$.
%Assume constant density.


\item (20 points) In your bluebook, either write the entire word TRUE or FALSE
for each of the following statements. For this question only, you
do not need to give a reason or show any work.
\be
\item $\ds x \int \frac{1}{x} \; dx = x \ln |x| + Cx$ where $C$ is 
some constant.
\item If $\ds \lim_{n \rightarrow \infty} a_n = L $, where $L$ is
any finite value, the sequence $a_1, a_2, \ldots$ converges.
%\item  $\ds \sum_{n=1}^{\infty} n (1+n^2)^p$ converges if $p < -1$.
\item $\ds \sum_{n=1}^{\infty} (e^x -3)^n$ is a power series.
\item $\ds \sum_{n=1}^{\infty} (e^x -3)^n$ is a geometric series.
\ee

\item (30 points)  Let $a_n = \int_0^{\infty} t^{n-1}e^{-t} \; dt $
for $n= 1, 2, 3, \ldots$.
(Note: Even if you get stuck on one of the pieces, you should try to
do the others.)
\be
\item Find $a_1$.
\item Find $a_2$.
\item Use integration by parts to show $a_{n+1} = n a_n$.
\item Use parts (a) and (c) to find a simple formula 
for $a_n$.
\item Does the sequence $a_1, a_2, a_3, \ldots$ converge?
\ee


\item (20 points)
\be
\item  For what values of $p$ does the series
$\ds \sum_{n=0}^{\infty} n (1+n^2)^p$ converge?
\item Evaluate $\ds \sum_{n=0}^{\infty} \left( 
\int_{n}^{n+1} \frac{1}{x^2+1} \; dx \right)$.
\ee


%\item Consider the function $\ds f(x) = \sqrt{1+x^4}$.
%\be
%\item Find the Maclaurin series for $f(x)$.
%\item Use the first two terms of the series
%in part (a) to estimate the value of
%$\ds \int_0^{0.1} \sqrt{1+x^4} \; dx $
%\item Estimate the error in your approximation
%in part (b).
%\ee


\item (20 points) For what values of $x$ do the following series
converge absolutely, converge conditionally or diverge? 
Be sure to name any test you use.
\begin{eqnarray*}
\begin{array}{lllll}
(a)\;\;\;  \ds \sum_{n=3}^{\infty} \left( 1- \frac{x}{3n} \right)^n & & & & 
(b) \;\;\; \ds \sum_{n=3}^{\infty} (\ln x )^n
\end{array}
\end{eqnarray*}

\newpage


\item (25 points)
\be
\item Use the Binomial Theorem to calculate the first 4 terms of the
Maclaurin series for $\ds f(x)= \frac{1}{\sqrt{1-x^3}}$.
\item Using the Maclaurin series for $\sin x$ and your answer to
part (a), find the Taylor polynomial of order four at $x=0$
for the function
$\ds g(x) = \frac{\sin x}{\sqrt{1-x^3}}$.
\ee


\item (20 points) Consider $\ds x^2 - y^2 =1$.
\be
\item Graph the equation $\ds x^2 - y^2 = 1$.  State the vertices, foci
and asymptotes and label them on the graph.
\item Find the area in the right half-plane (i.e.\ $x \geq 0$) bounded
by $\ds x^2 - y^2 =1$ and the line $x = \sqrt{2}$.
For this problem, you only need to set up the integral, you do not
need to evaluate it.
\ee


\item (30 points) Let $r_1 = 2(1-\cos \theta)$ and $r_2 = 1$.
\be
\item Carefully graph $r_1$.
\item Find the area inside $r_1$ and outside $r_2$.
\item Find the length of the $r_1$ curve.
\ee

%\item (?? points) Solve the initial value problem $\ds y^{\prime} - y =
%x$
%with $y(0) = 0$.
%(Note: You may use either of the two techniques that we covered in class 
%for solving this problem.) 

%\item DON'T FORGET TO ADD THE MATCHING PROBLEM!!!
\ee
\bigskip

{\it Verify that the following information is clearly written on the front of your bluebook:
your name and student ID number, your instructor's name (Dougherty or Biswas),
and  a grading key.}


\newpage

 
\begin{center}

{\bf Final Exam  Formula Sheet } 
% December 20, 2000, ~~~  4:30 - 7:00 p.m.
\end{center}

\be

\item A short table of integrals.  In the following, $a \neq 0$.
\be
\item $\ds \int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}(u/a) + C $ for
$u^2 < a^2$
\item $\ds \int \frac{du}{a^2+u^2} = (1/a) \tan^{-1}(u/a) + C$
\item $\ds \int \frac{du}{u \sqrt{u^2-a^2}} = (1/a) \sec^{-1} |u/a| + C$
for $u^2 > a^2 +C$
\item $\ds \int \frac{du}{\sqrt{a^2 + u^2}} = \sinh^{-1} (u/a) + C$ for
$a>0$
\item $\ds \int \frac{du}{\sqrt{u^2 - a^2}} = \cosh^{-1}(u/a) + C $ for
$u> a > 0$
\item $\ds \int \frac{du}{a^2 - u^2} = \left\{ \begin{array}{ll} (1/a)
\tanh^{-1}(u/a) + C  & \mbox{ if }
u^2 < a^2 \\ (1/a) \coth^{-1}(u/a) + C & \mbox{ if } u^2 > a^2
\end{array} \right. $
\item $\ds \int \frac{du}{u \sqrt{a^2 - u^2}} = -(1/a) \, \mbox{sech}^{-1}(u/a) +
C$ for $0 < u < a$
\item $\ds \int \frac{du}{u \sqrt{a^2 + u^2}} = -(1/a) \, \mbox{csch}^{-1} |u/a| +
C $ for $u \neq 0$
\ee

\item Some  trig identities.
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\;\; \ds \sin^2x + cos^2x=1 & & & &
(d) \;\;\;\; \ds \sin^2x= (1- \cos(2x))/2 \\

(b) \;\;\;\; \ds \cos^2x = (1 + \cos (2x))/2 & & & & 
(e) \;\;\;\; \ds \cosh^2x - \sinh^2x=1 \\
(c) \;\;\;\; \ds \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B & & & &
(f) \;\;\;\; \ds \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B
\end{array}
\end{eqnarray*}


\item Some useful trig substitutions.

\be
\item $\ds x = a \tan \theta$ replaces $a^2 + x^2$  by $a^2 \sec^2 \theta$
\item $\ds x = a \sin \theta$ replaces $a^2 - x^2$  by $a^2 \cos^2 \theta$
\item $\ds x = a \sec \theta$ replaces $x^2 - a^2$  by $a^2 \tan^2 \theta$
\ee


\item Some useful limits. 
\begin{eqnarray*}
\begin{array}{lllll}
(a) \;\;\;\; \ds \lim_{n \rightarrow \infty} \frac{\ln n}{n} = 0 & & & & 
(d) \;\;\;\; \ds \lim_{n \rightarrow \infty} \sqrt[n]{n} = 1 \\ 
(b) \;\;\;\; \ds \lim_{n \rightarrow \infty} x^{1/n} = 1 \mbox{ for $x > 0$} & & & &
(e) \;\;\;\; \ds \lim_{n \rightarrow \infty} x^{n} = 0 \mbox{ for $|x| < 1$}  \\ 
(c) \;\;\;\; \ds \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} 
= e^x \mbox{ for any $x$} & & & &
(f) \;\;\;\;  \ds \lim_{n \rightarrow \infty} \frac{x^n}{n!} = 0 \mbox{ for any $x$}.
\end{array}
\end{eqnarray*}
Note: In limits (c) through (f), $x$ remains fixed as $n \rightarrow \infty$. 

\item Frequently used Maclaurin series
\begin{eqnarray*}
\begin{array}{llll}
(a) \;\;\; \ds \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n  \mbox{ for $|x| < 1$}
& & &
(e) \;\;\;  \ds e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \mbox{ for $|x| < \infty$  } \\
(b) \;\;\; \ds \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} 
\mbox{ for $|x| < \infty$ }  & & &  
(f) \;\;\; \ds \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}
\mbox{ for $|x| < \infty$  } \\
(c) \;\;\; \ds \ln (1+ x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n}, 
\mbox{ for $-1< x \leq 1$ }   & & &
(g) \;\;\; \ds \tan^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)}, 
\mbox{ for $|x| \leq 1$  } \\
(d) \;\;\;  \ds (1+x)^m = 1 + \sum_{k=1}^{\infty} \left( 
\begin{array}{c} m \\ k \end{array} \right) x^k \mbox{ for $|x| < 1$} & & & 
\end{array}
\end{eqnarray*}

\ee


\end{document}