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\centerline{\bf\large APPM 1360 SPRING 1997 --- EXAM 3}
\vskip 5mm
\noindent
ON YOUR EXAM BOOKLET PLEASE WRITE your name, student ID, lecture number,
time of class, instructor, and a grading table.
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\noindent
\centerline{
Lectures: 010 (8am) and 020 (11am) Lomeli, 030 (1pm) Dougherty,
040 (2pm) Halburd.
}
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\noindent
\centerline{\bf SHOW ALL WORK}
\noindent
\begin{enumerate}
\item (20 points)
\be
\item
Graph the conic 
$$
x^2-{y^2\over 2}=2
$$
and calculate its eccentricity.
Identify and carefully label on your graph
all asymptotes, foci, directrices, and points where the graph crosses
the coordinate axes.
\item
Find the equation for the ellipse with foci at $(0,\pm 2)$ and
eccentricity $1/3$.
\ee
% \item
% Find the point on the ellipse $x=2\cos  t$, $y=\sin t$, $0\le t <2\pi$ closest
% to the point $(3/4,0)$.
\item (24 points)
For what values of $x$ do each of the following series {\it i.\/}~converge
absolutely,
{\it ii.\/}~converge conditionally?
%  Find the interval and radius of convergence.
$$
\mbox{(a)}\ \ \sum_{n=0}^\infty {(-1)^n(2x-1)^n\over 3^n},
\qquad \mbox{(b)}\ \ \sum_{n=1}^\infty{(x-1)^n\over n^3},
\qquad\mbox{(c)}\ \ \sum_{n=1}^\infty{(-1)^nx^n\over 4^nn}.
$$
% \item
% \be
% \item
% Calculate the first four terms in the
% Taylor series generated by $\sin x$ at $a=\pi/4$. (This is the Taylor polynomial
% of order 3 at $a=\pi/4$.)
% \item
% The first few terms in the Maclaurin series for two functions $f$ and $g$ are
% $f(x)=1+x+3x^2-x^3+\cdots$ and $g(x)=2-2x^2+2x^4+\cdots$.  Using series
% multiplication, calculate the
% Taylor polynomial of {\underline{order $3$}} generated by $f(x)g(x)$ at $a=0$?
% \ee
\item (15 points)
\be
\item
Use the Binomial Theorem to calculate the Maclaurin series for
$(1+x^2)^{3/2}$.
\item  Write out the first four nonzero terms of this series
explicitly.
% \item Use the series from part (a) to evaluate
% $$
% \lim_{x\to 0}{1+\frac32x^2+\frac38x^4-(1+x^2)^{3/2}\over x^6}.
% $$
% N.B. Do {\underline{NOT}}
% use l'H\^opital's Rule for this question --- it would be very
% messy.
\ee
\item (21 points)
\be
\item Making  use of the  Maclaurin series for $e^x$, find the 
Maclaurin series for $e^{-x^2}$.
\item Use the series from part (a) to find a series representation for 
$\ds \int_0^{0.5} e^{-x^2} \; dx$.
\item Use series to estimate the value of the integral in part (b) 
with an error of magnitude less than $10^{-3}$.  Justify your answer.
\ee
\item (20 points)
\be
\item
Calculate the first four terms in the
Taylor series generated by $\sin x$ at $a=\pi/4$. (This is the Taylor polynomial
of order 3 generated by $\sin x$ at $a=\pi/4$.)
\item
The first few terms in the Maclaurin series for two functions $f$ and $g$ are
$$f(x)=1+x+3x^2-x^3+\cdots\quad\mbox{and}\quad g(x)=2-3x+3x^4+\cdots.$$
Using series
multiplication, calculate the
Taylor polynomial of {\underline{order $3$}} generated by $f(x)g(x)$ at $a=0$.
\ee
\item {(EXTRA CREDIT, 10 points)}
Let
$$
f(x)=x^{10}{\rm e}^{-x^3}.
$$
Without using your calculator, find $f^{(19)}(0)$.
% Suppose the series
% $$
% \sum_{n=0}^\infty a_nx^n
% $$
% has radius of convergence $R$.  What is the radius of convergence of
% the series
% $$
% \sum_{n=0}^\infty a_n^3x^n
% $$
% in terms of $R$?

\end{enumerate}
\end{document}