%% LyX 1.1 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[11pt,oneside]{amsart}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\setlength\parskip{\medskipamount}
\setlength\parindent{0pt}
\usepackage{graphics}

\makeatletter


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
\providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
 \theoremstyle{plain}    
 \newtheorem{thm}{Theorem}[section]
 \numberwithin{equation}{section} %% Comment out for sequentially-numbered
 \numberwithin{figure}{section} %% Comment out for sequentially-numbered

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{2}
\setcounter{subsection}{3}

\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Applications of the derivative.}


\subsubsection*{Aside}

There are {}``special{}'' right triangles, all of whose sides are integer
lengths. The simplest is a {}``3-4-5{}'' triangle, whose side lengths are,
respectively, 3, 4, and, um, 5. These correspond to the coincidence that \begin{eqnarray*}
3^{2}+4^{2} & = & 9+16\\
 & = & 25\\
 & = & 5^{2}.
\end{eqnarray*}
 so these must be the sides of a right triangle. 

\vspace{0.3cm}
{\par\centering \includegraphics{345.ps} \par}
\vspace{0.3cm}

Of course, if you multiply each of these numbers by, say, 2, the relationship
between them will still hold, so you have {}``6-8-10{}'' or {}``9-12-15{}''
right triangles as well. These triangles occur a lot in related-rates problems,
and it should be a signal that you are approaching such a problem correctly
when you see one.

The other common triangles that fit this pattern are {}``5-12-13{}'' triangles.
Again, since \begin{eqnarray*}
5^{2}+12^{2} & = & 25+144\\
 & = & 169\\
 & = & 13^{2},
\end{eqnarray*}
these are the sides of a right triangle. There are actually infinitely many
distinct such ratios (\emph{not} counting a {}``10-24-26{}'' as distinct from
a {}``5-12-13{}''), but these two are the only ones you're likely to see in
these problems, or that you are likely to recognize.

As a \emph{real} aside, it is curious that this equation \[
x^{2}+y^{2}=z^{2}\]
has so many solutions, with \( x \), \( y \) and \( z \) integers. If you
ask naively whether similar equations have solutions like that, you might consider
the equation \[
x^{n}+y^{n}=z^{n},\]
and ask how many solutions it may have, with \( x, \) \( y, \) and \( z \)
all positive integers. This is one of the most famous equations in mathematics,
all the more curiously because it was the assertion that it has no solutions
(for \( n>2 \), and \( x, \) \( y, \) and \( z \) all positive integers)
that made it so famous. 

The statement that the equation \[
x^{n}+y^{n}=z^{n},\]
 had no solutions for \( x, \) \( y \), and \( z \) positive integers, \( n>2 \)
a positive integer as well, has been known for 350 years as Fermat's Last Theorem.
He claimed to have proved this fact when he wrote it down in the margin of a
text, saying regrettably that: {}``I have discovered a truly remarkable proof
which this margin is too small to contain.{}'' 

Well, that margin would have had to be pretty large indeed, it turned out. This
{}``theorem{}'' was one of the most celebrated open problems in mathematics
for hundreds of years. Many partial results (for some values of \( n \), or
some arithmetical relationships among \( x,\, y,\, z \) and \( n \)) were
shown, and vast mathematical industries sprung up around the questions surrounding
this problem. It wasn't until 1994 that Andrew Wiles, at Princeton University,
completed a chain of results linking this theorem to other, previous, theorems.
See, for example, http://www-groups.dcs.st-and.ac.uk/\textasciitilde{}history/HistTopics/Fermat's\_last\_theorem.html
for a good explanation of the result, and a brief synopsis of Fermat's life. 
\end{document}
