\documentclass[12pt]{article} 
%  Note the package graphpap, which you can use to make graphpaper
\usepackage{color,graphicx,graphpap} 
%\pagestyle{empty}
\oddsidemargin 0pt \textwidth 6.5in 
\textheight 9.5in
\setlength{\unitlength}{.5in}  % a picture dimension
\topmargin -.5in 
\newtheorem{problem}{Problem} 
\begin{document}
\begin{center}
\Large\bf  Going from numerical to graphical and back again
\end{center}

More often than not, when we \textcolor{blue}{collect data}, it is done numerically. For example,
we might measure the
temperature in a room at various times during a given time interval.   The data here
is a set of ordered pairs of the form (time,temperature).  It is natural to \textcolor{blue}{plot the
data} on some Cartesian coordinate paper to get a feel for what happened to the temperature
in the room over the time period in question.   After we have plotted the data points,
we might \textcolor{blue}{interpolate the data}, that is, sketch in a plausible graph of the temperature
function for the room during the time interval that data was collected. 

\begin{problem}  Here are some time and temperature measurements taken in a room.

\begin{center}
\begin{tabular}[.5in]{|l|l|l|l|l|l|l|l|l|}
\hline
time t (hours)    &  0  &  1   & 2   &  3 &  4 & 5 & 6 & 7 \\ \hline
temp f(t) $C^{\circ}$  & 2.3&4.9 &4.2&1.5&1 & 3.2  & 4.1 & 1\\ \hline
\end{tabular}
\end{center}


Describe briefly what happens to the temperature in the room over the time interval.
Speculate on what kind of a room we have here.   Plot the data on the graph paper
given below and draw a plausible temperature function for the room based on the
data.  In what time period is the temperature changing most rapidly?  Increasing most 
rapidly?


\vspace{1.25in}

\begin{center} 
  \begin{picture}(7,5)\thicklines
  \multiput(0,0)(1,0){8}{\color{blue}{\line(0,1){5}}}
  \multiput(0,0)(0,1){6}{\color{blue}{\line(1,0){7}}}
  \put(3,-.5){\ttfamily\large T I M E }
  \put(-.5,2){\shortstack{\ttfamily\large T\\ \\E\\ \\M\\ \\P }}
  \end{picture}
\end{center}

\newpage

{\ttfamily  An alternative is to use the graphpaper package to produce 
graphpaper.}

\vspace{2cm}

\hspace{3in} \graphpaper[1](0,0)(7,5)  
  
\vspace{2cm}

{\ttfamily\color{red}  Or, if we wanted the table to be to the left of the graph
   we could use a 'parbox' command}
\vspace{2cm}

\parbox[b][3in][c]{3in}{
\begin{tabular}[.5in]{|r|r|} \hline
time t (hours) & temp f(t) (C$^{\circ}$  \\ \hline\hline
0 & 2.3  \\ \hline
1 & 4.9  \\ \hline
2 & 4.2  \\ \hline
3 & 1.5  \\ \hline
4 & 1   \\ \hline
5 & 3.2  \\ \hline
6 & 4.1  \\ \hline
7 & 1   \\ \hline
\end{tabular} }
  \begin{picture}(7,5)\thicklines
  \multiput(0,0)(1,0){8}{\color{blue}{\line(0,1){5}}}
  \multiput(0,0)(0,1){6}{\color{blue}{\line(1,0){7}}}
  \put(3,-.5){\ttfamily\large T I M E }
  \put(-.5,2){\shortstack{\ttfamily\large T\\ \\E\\ \\M\\ \\P }}
  \end{picture}
 
\end{problem}

\newpage


On the other hand, one is often faced with the problem of {\color{blue} 
reading data} off of a graph of some sort.  For example, we might be given
a polar coordinate plot of a curve and ask some numerical questions about the
plot.
\begin{problem}  For the polar plot $r = f(\theta)$ given below, what is the maximum value of
$r$ as $\theta$ ranges from 0 to $\pi$?  The minimum value?  The average value?
Explain your reasoning.
\end{problem}

\begin{center}
\includegraphics[height=2in,width=3in,angle=-90]{graph.eps} 
\end{center} 


{\small\ttfamily\color{red}  The eps files generated by Maple can be improved
by editing the postscript file. For example, you can comment out the
postscript commands which draw the box around the file. We can also make the lines thicker
by bumping up the values of thin, med, and thick in the file.} 

\begin{center}
\includegraphics[height=5in,width=4in,angle=-90]{graph2.eps} 
\end{center} 
\end{document}
