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\centerline{MATH 135, section 1 -- Calculus 1}
\centerline{Lab Day 1 Demo -- Parametric Curves}

\vskip 10pt
\noindent
{\it Background and Goals}
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Suppose a particle is moving along a curve in the $xy$-plane.
Then, at a time $t$ the particle will be at a point on the curve with coordinates
$(x,y)$, where $x$ and $y$ are two functions of $t$, $x=f(t)$ and $y=g(t)$.
The two equations for $x$ and $y$ are called {\it parametric equations} with {\it
parameter} $t$.  Today's lab will show you several additional examples of 
this idea and give you a way to visualize how curves are traced out this way.
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\noindent
There is {\it no assignment} connected with this first lab, but you will be creating
and saving a worksheet as a record of what you did.  The goal is to
use the software to demonstrate some of the ideas we were discussing 
in class on Monday and work through some more complicated examples this way.  
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\noindent 
There is some general information about Maple available on the course homepage.  
The main features you will need will also be demonstrated in the lab class meeting.  
\noindent
{\it Lab Demonstration}
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\noindent
A.  Launch Maple and open a worksheet.  Save it under the name {\tt LabDay1}.  At the top, 
enter your name and today's date in a text region.
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\noindent
B.  Consider a moving particle whose coordinates at
time $t$ are $x=2t+1$ and $y=3t+2$, for all real values of $t$.  When $t = -3$, for instance, we have
$x = 2(-3) + 1 = -5$ and $y = 3(-3) + 2) = -7$, so the point $[-5,-7]$ is one location of 
the moving particle.
\begin{enumerate}
\item[1)]  Calculating by hand, find the
coordinates $[x,y]$ of the particle at times $t=-2,-1,0, 1, 2$.  
\item[2)]  Then plot 
the points $[x,y]$ to guess the curve, using a Maple command of the following format:
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\begin{center}
{\tt plot( -your point list- , style=point,symbol=circle);}
\end{center}
where {\tt -your point list- } is the list of $[x,y]$ coordinates you generated,
in the form {\tt [[x1,y1],[x2,y2],[x3,y3],[x4,y4],[x5,y5]]}.  (Note the outer
set of square brackets which defines the whole list of five points.) 
In a text region in your Maple worksheet, answer this question: What curve do 
you expect these points lie on?
\item[3)]  Parametric curves can also be plotted in Maple.  Type
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\begin{center}
{\tt plot([2*t+1,3*t+2,t=-2..2]);}
\end{center}
Of course, you may also change the interval for $t$.
\item[4)]  To see how the particle traverses the curve, we will use the 
animation feature of Maple. Type
\begin{center}
{\tt with(plots):}\\ \vspace{.1in}
{\tt animatecurve([2*t+1,3*t+2,t=-10..10],frames=30);}
\end{center}
After entering the command you will see a set of axes. Click on the picture. At the
top of the screen you will see a button labelled {\Large {\bf $\triangleright$}}.
Click on it. You will see the curve being traced. You can also look at the curve
being traced one piece at a time by clicking repeatedly the button to the right of
the  {\Large {\bf $\triangleright$}} button. \vspace{.1in}
\item[5)] What happens if we replace $t$ by $-t$ in the parametric equations?
\item[6)] For after the lab:  You probably recognize the curve here, but it is 
not presented in the usual form of an $x,y$ equation.  Solve for $t$ from the 
equation $x = 2t + 1$ and substitute into the equation $y = 3t+2$.  What is 
the equation you get and how does it relate to what you saw before?
\end{enumerate}
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\noindent
C.  Now we will study the motion of a particle with coordinates given by 
\[
x = 4 \cos(t) \qquad y = 3 \sin(t)
\]
\begin{enumerate}
\item[1)]  First reason without Maple's help (i.e. use pen and paper).  Combine
$x$ and $y$ to eliminate the $t$ using a trigonometric identity.
What is the shape of the curve traversed by the particle?   
For which interval of $t$ values does the particle traverse the curve once?
\item[2)]  Use this interval for $t$ 
to plot the parameterized curve. (The Maple syntax for $\pi$ is {\tt Pi}.) 
Then animate the plot. In what direction is the curve being traversed?
\item[3)]  What happens if you change $t$ to $-t$ in the parametric equations?
\item[4)]  What if you change $t$ to $2t$?
\item[5)]  What if you change $t$ to $t/3$?
\item[6)]  In each case, how long does it take the particle to go once around the curve?
\item[7)]  What if you just change the interval for $t$ 
(say, you have $0\leq t \leq 10\pi$)?  The graph produced by Maple in this case
may look ``strange!''  Can you explain what happened?  Can you fix it so the
graph is mathematically correct?  
\end{enumerate}
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\noindent 
D.  Some additional interesting examples:  For each $n=2,3,4,5$, animate the motion of an object moving
along the curve 
\[
x=\cos(t) \qquad y=\sin(nt),  \qquad 0 \leq t \leq 2\pi.
\]
Write a short paragraph describing the effect of varying $n$ on the curve above
and enter it in a text region. (The patterns described by
these curves are called {\it Lissajous figures}.)
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