%Taylor Series
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{\Large
Taylor Series
\footnote{Copyright \copyright 2002 Larry Snyder and Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
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\textsc{MatLab} has an interactive Taylor series calculator
called \verb&taylortool&. It plots \verb$f$ and the \verb$N$-th degree
Taylor polynomial on an interval. After \verb&taylortool& is
started, we can change \verb$f$, \verb$N$, the interval,
or the point \verb$a$.
\begin{enumerate}
\item
\begin{enumerate}
\item Enter the command: \verb& taylortool('sin(x)')&
\item In the taylortool window, change \verb$N$ to be 3.
You can change the degree \verb$N$ using the
buttons \verb&>>& or \verb&<<&. Also you can
just enter the value for \verb$N$
in the box for \verb$N$.
\item For what domain does the Taylor polynomial appear
to be a good approximation of the function?
\item Now use the button \verb&>>& to increase \verb$N$ until
the approximation appears to be accurate on the
whole interval.
\item For the degree \verb$N$ above, use Taylor's Formula (by hand) to
find an upper bound on the error of the approximation.
\end{enumerate}
\item In the \verb&taylortool& window, change the function to
$f(x) = e^x$
(use \verb& exp(x)&), the interval to $[-3, 3]$
and $N$ to 3. Repeat the process above.
\item Repeat the above process for $\sin(e^x)$
on the interval $[0,3]$.
What problems do you encounter. What do you think causes this?
Does $\sin(e^x)$ equal its Taylor series?
For roughly what range of $x$ and $N$ would $T_N(x)$ be a
practical approximation tool? What might be a more reasonable
strategy for approximating $\sin(e^x)$?
\item Prepare a brief (\verb$< $1 page) written report describing
what happened and answering the questions.
Use complete sentences and standard mathematical notation. Do {\bf not} get a printout.
\end{enumerate}
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\noindent
\textsf{The taylortool can help us gain some appreciation for the loss of accuracy of the
Taylor approximation as $x$ varies farther from the approximation point $a$.
We also encounter the difficulty of approximating a function that oscillates.
Although a Taylor Series does actually equal a certain function,
computers can only do polynomial operations. So for instance,
the sine function on calculators or computers {\bf must} be approximated using
polynomial computations and knowing the accuracy is important.}
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