\documentclass{article}
\usepackage{amssymb,amsmath,color}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyfoot[C]{\bf \copyright\  2012 C. J. Schultz}
\providecommand{\abs}[1]{\lvert#1\rvert}
\providecommand{\norm}[1]{\lVert#1\rVert}

\begin{document}

\section{Examples:}

\subsection{Proof of AM-GM}

Now, if we can prove
\[ \Big(2\sqrt{\frac{a}{b}}\Big)^m + \Big(2\sqrt{\frac{b}{a}}\Big)^m
\ge 2^{m+1} \]
then we will be done. Dividing both sides of this inequality by $2^m$ yields
\[ \Big(\frac{a}{b}\Big)^m + \Big(\frac{b}{a}\Big)^m \ge 2, \]
which we can verify easily by AM-GM on the reciprocals that make up the LHS,
thus we have our desired
\[ \Big(1 + \frac{a}{b}\Big)^m + \Big(1 + \frac{b}{a}\Big)^m \ge
\Big(2\sqrt{\frac{a}{b}}\Big)^m + \Big(2\sqrt{\frac{b}{a}}\Big)^m \ge 2^{m+1}. \]

\subsection{Math Etc.}

This will be and example equation:
\[\lim_{x \to  \inf}{e^x} \neq \int_{x^2}^{x^3}{\abs{\rho}}\; d\phi \]
Now I would like to show some sub points of this equation:
\subsubsection{Form 1:}
\[ \frac{high}{low} \rightsquigarrow \frac{high\;upper}{low\;lower}\]

\subsubsection{Form 2:}
\begin{equation}\label{eq1}
a+x=b 
\end{equation}
\begin{equation}
  \label{eq2}
x=b-a
\end{equation}
\section{Next Days Notes:}

\subsection{Boxes}
I can create basic boxes for text \makebox[2in]{like this}. Notice
that there's a 2in wide space with `like this' in the middle of it.

If I want to put a box around the text, I can use a frame box. The
result looks \framebox[2in]{like this}.

I can also justify the text to the right within a box
\makebox[1.5in][r]{like so} or \framebox[2.5in][l]{like so}.

We can also use quick versions of these. We can just \mbox{do this}
or \fbox{this} to create a quick box that's exactly the size of what we put in it.

\subsection{Color}
Color can be used to make a document much more attractive:
\textcolor{red}{This could be very important text}
\colorbox{cyan}{This could be something you need to remember}

Color names are:
Apricot	Aquamarine	Bittersweet	Black
Blue	BlueGreen	BlueViolet	BrickRed
Brown	BurntOrange	CadetBlue	CarnationPink
Cerulean	CornflowerBlue	Cyan	Dandelion
DarkOrchid	Emerald	ForestGreen	Fuchsia
Goldenrod	Gray	Green	GreenYellow
JungleGreen	Lavender	LimeGreen	Magenta
Mahogany	Maroon	Melon	MidnightBlue
Mulberry	NavyBlue	OliveGreen	Orange
OrangeRed	Orchid	Peach	Periwinkle
PineGreen	Plum	ProcessBlue	Purple
RawSienna	Red	RedOrange	RedViolet
Rhodamine	RoyalBlue	RoyalPurple	RubineRed
Salmon	SeaGreen	Sepia	SkyBlue
SpringGreen	Tan	TealBlue	Thistle
Turquoise	Violet	VioletRed	White
WildStrawberry	Yellow	YellowGreen	YellowOrange

Math and color?
\textcolor{red}{
\[\frac{x}{y}\]
}
\[\mathbf{\textcolor{green}{\int_{\textcolor{blue}{x}}^{\textcolor{red}{e^x}}{y}}}\]
\end{document}
