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\def\pt#1{\hbox{#1) }}
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\def\R{{\Bbb R}}
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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 17\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 17} 
\bigskip


\itemitem{1.\hskip12pt a)}   Give the conditions necessary for a 
function $f\colon\R\to\R$ to have a Taylor series expansion about a
point $c\in\R$. 

\medskip
\itemitem{b)}  Graph the function $f(x)=\ln x$.  What is the domain of
$f$?  

\medskip
\itemitem{c)}  For what values of $c\in\R$ does $f$ has a Taylor
series centered at $c$?  Compute the Taylor series for two of these
values $c_1$ and $c_2$.

\medskip
\itemitem{d)}  What is the interval of convergence for these two
Taylor series?

\medskip
\itemitem{e)}  Write out $P_5(x;c_1)$ and $P_5(x;c_2)$, the $5^{\rm
th}$-order Taylor polynomials for $f(x)=\ln x$ centered about $c_1$
and $c_2$ respectively.  Choose some real number $x_0$ that is inside
the interval of convergence for both Taylor series, and compute
$P_5(x_0;c_1)$ and $P_5(x_0;c_2)$.

\medskip
\itemitem{f)}  From Lagrange's Theorem, what can be said about the
error involved in using these two Taylor polynomials to estimate $\ln
x_0$. 
\itemitem{}  (Hint: For each expansion, what is the maximum value of
$f^{(6)}(x)$ on the interval from $x_0$ to the center of the
expansion?) 

\medskip
\itemitem{g)}  Use your calculator to find $\ln x_0$.  How does your
calculator come up with this number?

\medskip 
\item{2.}  Let $f\colon\R\to\R$ be a smooth function.  (What does that
mean?)  Only some of the following statements are true.  For those
that are true, indicate why.  For those that are false, provide a
counterexample.

\medskip
\itemitem{a)}  If a Taylor series expansion for $f$ converges at
$x\in\R$, then it converges to $f(x)$.

\medskip
\itemitem{b)}  If the remainder $R_n(x;c)$ tends to zero as
$n\to\infty$, then the Taylor series converges at $x$.

\medskip
\itemitem{c)}  If the remainder $R_n(x;c)$ tends to zero as
$n\to\infty$, then the Taylor series converges at $x$, and it
converges to $f(x)$.

\medskip
\itemitem{d)}  If a Taylor series converges to $f(x)$ for some values
of $x$, then it converges to $f(x)$ for all values of $x$ in the
interval of convergence.

\medskip
\itemitem{e)}  If all the derivatives of $f$ are bounded on some
interval containing the center of expansion, then the Taylor series
converges to $f$ on this interval.

\eject
\item{3.}  Taylor series may be found for each of the following
functions by a long method or a short method.  Describe both methods,
then use the short one.  Determine the interval of convergence in each
case.
$$
\eqalign{
\pt a f(x)&=x^3e^x\cr
\pt d f(x)&=\ln(x^2+1)\cr}\qquad
\eqalign{
\pt b f(x)&={1\over1-x^2}\cr
\pt e f(x)&=\arctan x\cr}\qquad
\eqalign{
\pt c f(x)&={1\over1+x^2}\cr
\cr}\hskip1in
$$

\medskip
\item{4.}  A flock (or is it gaggle?) of geese in flight will always
travel in a ``V-shaped'' formation.  One side of this ``V'', however,
is almost always longer than the other.  Using rigorous mathematics,
explain why this happens.





\bye