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\begin{center}
  \textbf{Comments on Maple Problem Set C}
\end{center}

\emph{Plotting}
\begin{itemize}
\item To plot an \emph{expression}, e.g., 
  \begin{math}
    x^2
  \end{math},
the syntax is:
\begin{center}
  plot(x$\hat{\ }$2,x=-1..1).
\end{center}
\item To plot a \emph{function}, e.g., 
  \begin{math}
    f
  \end{math},
where 
\begin{math}
  f
\end{math}
has been defined by the Maple command
\begin{center}
  f := x $\rightarrow$ x$\hat{\ }$2,
\end{center}
the syntax is:
\begin{center}
  plot(f,-1..1).
\end{center}
Note that the syntax is different for functions and expressions.
\item If the graph doesn't go far enough, you might need to increase
  the value of \textbf{maxfun.}
\item
To display several plots together, assign a name to each plot.  (End
each plot command which does that with : instead of ; unless you want
to see all the points Maple computed on the plot.)  Then give the
command:
\begin{center}
  with(plots):
\end{center}
 Finally, if for example, you want to display two plots named plot1
 and plot2 together, give the command:
\begin{center}
  display(\{plot1,plot2\})
\end{center}
\end{itemize}

\emph{Problem 2}
\begin{itemize}
\item Be sure to get explicit formulas for 
  \begin{math}
    \phi _0
  \end{math}
   and 
   \begin{math}
    \phi _1
   \end{math}
  to find out where they ``blow up.''
\item Get as accurate an estimate as you can for 
  \begin{math}
    x_*
  \end{math}
by adjusting the interval on which you plot the solution.
\end{itemize}



\emph{Problem 10}


\begin{itemize}
\item On each part, make
sure you plot on an appropriate interval to get a good idea of the
behavior of the solution as 
\begin{math}
  x
\end{math}
increases.
\item Find the limiting behavior of 
  \begin{math}
    y
  \end{math}
if there is one.
\item Estimate the blow-up time if it is finite.
\item You might need several plots.  You might need to increase
  \textbf{numpoints} if the plot looks jagged.
\end{itemize}

\emph{Problem 14}
\begin{itemize}
\item
(a)
Be sure to answer all the questions.  This includes:
\begin{itemize}
\item What appears to be happening as 
  \begin{math}
    x
  \end{math}
increases?
\end{itemize}
\item
(b)
A    good way to compare the solutions for different step sizes is
  to plot them together.
\item (d) Be sure to discuss (with illustration) the dependence of the
  solution on the initial value.
\item (e) The Maple demonstation I gave in class for the Runge-Kutta
  method
does not produce output in a very convenient form for plotting.  I
suggest
 that instead of trying to use the procedure in the demonstration, 
you make appropriate modifications to the procedure for Euler's method 
given in \emph{Differential Equations with Maple}.
\end{itemize}

\emph{Problem 16}
\begin{itemize}
\item Think about values for digits.  Explain why the number you've
  chosen should give the correct accuracy.
  \begin{itemize}
  \item You should be able to get agreement of the numerical
    approximation and 
    \begin{math}
      e^x
    \end{math}
to 15 digits.
  \end{itemize}
\item Remember to use \textbf{evalf}.
\end{itemize}

\end{document}
