%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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