%Taylor Series %If you modify this file, please indicate here and in the footnote %Math263C \documentclass[12pt]{article} \usepackage{times} \pagestyle{empty} \addtolength{\textwidth}{1.2in} \addtolength{\textheight}{1.2in} \addtolength{\oddsidemargin}{-.58in} \addtolength{\evensidemargin}{-.58in} \renewcommand{\baselinestretch}{1.0} \parindent = 0cm \parskip = .1cm \begin{document} \begin{center} {\Large Taylor Series \footnote{Copyright \copyright 2002 Larry Snyder and Todd Young. All rights reserved. Please address comments to young@math.ohiou.edu.}} \end{center} \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} \vfill \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.} \end{document}