%Exponential vs. Powers
%If you modify this file, please indicate here and in the footnote
%Math263A

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\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.}

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