\documentclass{amsart}
\usepackage{graphicx}

\def\R{{\mathchoice{\rm I\hskip-.2em R}{\rm I\hskip-.2em R}
{\rm I\hskip-.15em R}{\rm I\hskip-.1em R}}}

\title{The Hyperbolic Functions}
\author{Put your name here}

\begin{document}
\maketitle
\section{Definitions}

In this lab, we explore the functions known as the hyperbolic functions.

\noindent {\bf Definition:}  The hyperbolic sine function is defined as:
$$
\sinh(x)=\frac{{\rm e}^x-{\rm e}^{-x}}{2}
$$
Similarly, the hyperbolic cosine is defined as:
$$
\cosh(x)=\frac{{\rm e}^x+{\rm e}^{-x}}{2}
$$
\begin{enumerate}
\item Use Maple to plot $\sinh(x)$, $\frac{1}{2}{\rm e}^{x}$,
$-\frac{1}{2}{\rm e}^{-x}$.
\item Use Maple to plot $\cosh(x)$, $\frac{1}{2}{\rm e}^{x}$,
$\frac{1}{2}{\rm e}^{-x}$. 
\item Use Maple's {\tt animate} command to determine how the parameter
$a$ changes the graph of $\sinh(ax)$ and $\cosh(ax)$.  (Look up the 
use of {\tt animate} using Maple's help)
\item Look up the definition of {\it even} and {\it odd} functions.
What do these mean graphically?
Show (graphically) that the sine and hyperbolic sine are odd.  
Show (graphically) that both the cosine
and hyperbolic cosine are even.
\item Using the definitions from trigonometry, how should the hyperbolic
tangent, hyperbolic cotangent, hyperbolic secant, and hyperbolic cosecant 
be defined?  Write them in terms of $\sinh(x)$ and $\cosh(x)$, then as
exponential functions.
\item Show that $\cosh^2(x)-\sinh^2(x)=1$ (This is a bit different than
the original trig functions).
\item Are the hyperbolic sine, cosine and tangent functions 1-1?  If not,
restrict the domain so that each is 1-1.
\item Since the hyperbolic trig functions are written in terms of 
exponentials, we can write the inverse hyperbolic trig functions
in terms of logarithms.  Compute the inverse hyperbolic sine
cosine and tangent in terms of logarithms.  Hint:  In general,
how do we find an inverse?
\end{enumerate}

\section{Calculus with the Hyperbolic Functions}

\begin{enumerate}
\item Calculate the derivatives of $\sinh(x)$, $\cosh(x)$,
and $\tanh(x)$.  Write your final answers in terms of hyperbolic functions
(rather than exponentials).
\item Have Maple calculate the derivatives of the inverse hyperbolic
sine, cosine and tangent.
\item Plot $\sinh(x)$ together with its derivative.
\item Plot $\cosh(x)$ together with its inverse.
\end{enumerate}
\section{Applying the Hyperbolic Functions}
\begin{enumerate}
\item A flexible cable strung between two telephone poles always
hangs in the shape of a catenary curve, defined by:
$$
y(x)=c+a\cosh(x/a)
$$
where $c, a$ are constants and $a>0$.  Choose a value of $c$
and determine how the value of $a$ changes the graph. (Use Maple's
{\tt animate} command).
\item In physics, it can be shown that when a cable is hung
between two poles, it takes the shape of a curve $y$ that
satisfies the differential equation:
$$
y''=\frac{pg}{T}\sqrt{1+(y')^2}
$$
where $y'$ is $\frac{dy}{dx}$, $p, g, T$ are constants.  Show
that
$$
y=\frac{T}{pg}\cosh\left( \frac{pgx}{T}\right)
$$
is a solution to the differential equation (That is, actually
substitute the formula given for $y$ into the differential
equation, and show that it is true.).
\end{enumerate}

\end{document}
