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\begin{center}

{\bf APPM 1360 \hspace{1.0in}  EXAM \#3  \hspace{1.0in} FALL 2000}
% November 29, 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 100 points and has 5 questions.   A list of 
formulas is given on the back of this exam.  
{\bf Show all work!}   Answers with no justification will
receive no points. 

\be

\item (15 points)  Consider the ellipse $16x^2 +25y^2 = 400$.
\be
\item Put the equation into standard form.
\item Sketch a graph of the ellipse.  Your graph should include
the coordinates of the vertices, all $x$ and $y$ intercepts and the foci.
\ee 

%\item (10 points) Discuss the convergence/divergence of the
%series
% $\ds \sum_{k=1}^{\infty} (-1)^{k+1} \frac{\sqrt{k} + 1}{k+1}$


\item (25 points) Consider the power series given by
$ \ds \sum_{n=0}^{\infty} \frac{(-1)^n (2x-3)^n}{n+2} $
\be
\item For what values of $x$ does the power series converge
absolutely?
\item For what values of $x$ does the power series converge
conditionally (but not absolutely)?
\item For what values of $x$ does the power series diverge?
\item What is the radius and interval of convergence for this
power series?
\ee

\item  (20 points) Let $\ds f(x) = \sin (3x)$.
\be
\item Find the Maclaurin series for $\ds f(x) = \sin (3x)$.
\item Use the Taylor polynomial of order 3, $P_3(x)$, to estimate
$\ds f(x)  = \sin (3x) $ at $x=1/2$.  You should leave your answer
in fractional form. 
\item Estimate the error being made in using $P_3(1/2) $ to estimate
$\ds f(x) = \sin (3x) $ at $x=1/2$.
\item Does $\ds P_3(1/2)$ overestimate or underestimate 
$\ds f(1/2) = \sin(1.5)$?  Explain. 
\ee


\item  (15 points)
\be
\item Now find the Maclaurin series for $\ds \frac{\sin (3x)}{x^3}$.
(You may use your work from 3(a).)
\item Find constants $r$ and $s$ for which
\begin{eqnarray*} 
\lim_{x \rightarrow 0} \left( \frac{ \sin (3x)}{x^3} + \frac{r}{x^2}
+s \right) = 0.
\end{eqnarray*} 
\ee

\item (25 points) Let $\ds f(x) = \frac{1}{x^2}$.
\be
\item Find the Taylor series for $f(x)=1/x^2$ at $a=1$.
\item What is the interval of convergence for the series you found in
part (a)?
\item Integrate the series in part (a) term by term. What new series do
you
obtain? What is the sum of this new series and what is its interval of
convergence?
\ee

%\item Recall that $\ds \cosh x = \frac{e^x + e^{-x}}{2}$
%\be
%\item Find the Maclaurin series for $\cosh x$ and its interval of
%convergence.  (Hint: Use the series for $e^x$ given on the back of this exam.)
%\item Use the Taylor polynomial of order 2, $P_2(x)$, to estimate $\cosh
%(1/2)$.
%\item Use the Taylor series remainder to estimate the error being made in
%using $P_2(1/2)$ to estimate $\cosh (1/2)$. (Note: You may use the fact
%that $\cosh (1/2) \leq 1.5$.)
%\ee

\ee

\noindent Extra Credit (5 points): Find the sum of the series in problem 2.
 
\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 APPM 1360 \hspace{1.0in}  Exam \#3 Formula Sheet } 
% November 27, 2000, ~~~  7:00 - 8:30 p.m.
\end{center}

\be

\item Some useful limits.
\be
\item $\ds \lim_{n \rightarrow \infty} \frac{\ln n}{n} = 0$.
\item $\ds \lim_{n \rightarrow \infty} \sqrt[n]{n} = 1$.
\item $\ds \lim_{n \rightarrow \infty} x^{1/n} = 1$ for $x > 0$.
\item $\ds \lim_{n \rightarrow \infty} x^{n} = 0$ for $|x| < 1$.
\item $\ds \lim_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^{n} 
= e^x$ for any $x$. 
\item $\ds \lim_{n \rightarrow \infty} \frac{x^n}{n!} = 0$ for any $x$.
\ee
Note: In limits (c) through (f), $x$ remains fixed as $n \rightarrow \infty$. 

\item Frequently used Maclaurin series
\be
\item $ \ds \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$, valid for $|x| < 1$
\item $ \ds e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$, valid for $|x| < \infty$  
\item $ \ds \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, 
valid for $|x| < \infty$  
\item $ \ds \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$, valid
for $|x| < \infty$  
\item $ \ds \ln (1+ x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n}$, valid 
for $-1< x \leq 1$  
\item $ \ds \tan^{-1} x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)}$, 
valid for $|x| \leq 1$  
\item  binomial series: $\ds (1+x)^m = 1 + \sum_{k=1}^{\infty} \left( 
\begin{array}{c} m \\ k \end{array} \right) x^k $ valid for $|x| < 1$
\ee

\ee


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