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\section*{Maple and LaTeX Exercises}

The purpose of this document is twofold:  Learn to use Maple to help you answer
mathematics questions, and to write up the solutions using \LaTeX.  The \LaTeX
commands that produced this document are available on our class website to help
get you started.

When you are finished,
\begin{itemize}
\item Create a PDF file of your result (in Texmaker, go to \verb+Tools+, then \verb+PDFLatex+)
\item Go to CLEo and upload this PDF file to your dropbox (Due: Feb 09).
\end{itemize}

\subsection*{Maple Questions (In-Class)}
We'll answer the following together once you've had a chance to work with Maple a bit.
\begin{enumerate}
\item What is the difference between the following?
\begin{verbatim}
f:=x^2+2*x;
f:=x->x^2+2*x;
f=x^2+2*x;
\end{verbatim}
\item How is the percent sign \verb+%+ used?
\item How might we plot several functions on one axes?
\item How can we input our Maple code into LaTeX?
\end{enumerate}

\subsection*{Maple Homework}

These are the homework questions, whose solutions will be due on Feb 9.
Please write the solutions up using \LaTeX, and put in figures where appropriate.
Your solutions should include the Maple code that you used (discussed in the
in-class portion).

\begin{enumerate}
\item Consider the function
$$
f(x)=\sqrt{ \frac{x^4-x+1}{x^4+x+1} }
$$
\begin{enumerate}
\item Use Maple to differentiate and simplify the result. 
\item Where does the graph of $f$ have horizontal tangents?
\item Graph $f$ and $f'$ on the same axes, and provide the plot in a figure.
\end{enumerate}
\item Consider the curve with equation:
$$
2y^3+y^2-y^5=x^4-2x^3+x^2.
$$
\begin{enumerate}
\item Use the \verb+implicitplot()+ command from the \verb+plots+ package
in Maple to graph the curve.
\item At how many points does this curve have horizontal tangent lines?
Find the $x-$coordinates of these points.
\end{enumerate}
\item Prove that the equation
$$
100{\rm e}^{-x/100}=\frac{x^2}{100}
$$
has at least one solution using the Intermediate Value Theorem.  Use the 
\verb+solve()+ command in Maple to approximate the solution.
\item Biologists have observed that the chirping rate of crickets of a certain
species appears to be related to temperature.  Here is some data:
$$
\begin{array}{lccccccccc}
\mbox{Temp} & 50 & 55 & 60& 65 & 70 & 75 & 80 & 85 & 90\\
\hline
\mbox{Chirps/min} & 20 & 46 & 79 & 91 & 113 & 140 & 173 & 198 & 211\\
\end{array}
$$
\begin{enumerate}
\item Make a scatter plot of the data
\item Find and graph the regression line (also called the line of best fit).
\item Use the regression line to estimate the chirp rate at 100 degrees $F$.
\end{enumerate}
There are several commands you might use-  Here are some:  \verb+pointplot()+
and \verb+LinearFit()+
\item Use the Squeeze Theorem to show that
$$
\lim_{x\rightarrow 0}\sqrt{x^3+x^2}\,\sin\left( \frac{\pi}{x}\right)=0
$$
Illustrate the result by graph the three functions (used in the
theorem) on the same plot.
\item The sine integral function is defined in kind of a strange way:
$$
{\rm Si}(x)=\int_0^x\frac{\sin(t)}{t}\,dt
$$
and it is important in electrical engineering.  As a technical note, the formal
definition of the integrand is:
$$
f(t)=\left\{ \begin{array}{cl}
\sin(t)/t &\mbox{ if } t\neq 0\\
 1        &\mbox{ if }t=0
\end{array}\right.
$$
which makes $f$ continuous everywhere.
\begin{enumerate}
\item Use Maple to plot ${\rm Si}(x)$.
\item At what value(s) does ${\rm Si}(x)$ have local maximums?
\item Approximate the solution of ${\rm Si}(x)=1$ using the \verb+solve()+
command.
\end{enumerate}
\end{enumerate}
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