%Exponential vs. Powers %If you modify this file, please indicate here and in the footnote %Math263A \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 Exponentials vs. Powers\footnote{Copyright \copyright 2002 Steve Chapin. All rights reserved. Please address comments to young@math.ohiou.edu.}} \end{center} \begin{enumerate} \item Enter the following sequence commands: \medskip \textbf{Important note:} Do not omit the semicolons! Also, do not omit the \verb&.& before the \verb&^& ! \begin{enumerate} \item \verb&x1 = -1.15:0.01:1.15;& \dotfill (This makes $x1$ a vector with entries from $-1.15$ to $1.15$ in $.01$ increments.) \item \verb&x2 = -1.39:0.01:1.39;& \item \verb&y1 = x1.^10;& \dotfill (This evaluates $x1^{10}$ for each entry of $x1$.) \item \verb&y2 = exp(x2);& \item \verb&plot(x1, y1, 'b', x2, y2, 'r')& \end{enumerate} These plots of $y = x^{10}$ and $y = e^x$ \emph{suggest} that the equation $x^{10} = e^x$ has two solutions --- one positive and one negative. Approximate these two solutions (to three decimal places) by ``zooming''. (To ``zoom in'' click on the button that looks like a magnifying glass with a plus sign, and then click on the graph. To ``zoom out'' select the magnifying glass with the minus sign.) \item Explain why there must be another positive solution of $x^{10} = e^x$ larger than the one that you found in \#1. By changing the beginning and ending values of $x1$ and $x2$ (you may leave the increments the same) and plotting as above, determine an interval that reveals this larger solution. (Note. You can use the up-arrow key to do this, but you must reevaluate $y1$ and/or $y2$ each time you change $x1$ and/or $x2$.) Approximate this solution (to two decimal places) by ``zooming''. \item Explain why it may be necessary to use several different domain intervals when studying computer plots. \item On a separate piece of paper, prepare a brief written report giving explanations where requested and answering all the questions. Include all of the approximate solutions. Use complete sentences and use standard mathematical notation. Do {\bf not} hand in a printout. \end{enumerate} \vfill \noindent \textsf{This assignment reinforces the fact that the exponential function, exp(x), will eventually exceed any power of x. It also illustrates the importance of scale when considering computer plots.} \end{document}