%Exponential vs. Powers
%If you modify this file, please indicate here and in the footnote
%Math263A
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{\Large
Exponentials vs. Powers\footnote{Copyright \copyright 2002 Steve Chapin.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
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\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.}
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