The purpose of this lab is to give you practice with parametric curves in the plane and in visualizing parametric curves as representing motion.

cp /math/calclab/MA1023/Parametric_start_B08.mws ~/My_Documents

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Parametric_start_B08.mws`, and go through it
carefully. Then you can start working on the exercises.

where is time and is the acceleration due to gravity.

To help you to visualize parametric curves as representing motion, a
Maple routine called `ParamPlot` has been written. It uses the
Maple `animate` command to actually show the particle moving along
its trajectory. You actually used this command before for the lab
on polar coordinates. Examples are in the `Getting Started`
worksheet.

It is clear that this formula doesn't make sense if at some particular value of . If at that same value of , then it turns out the graph has a vertical tangent at that point. If both and are zero at some value of , then the curve often doesn't have a tangent line at that point. What you see instead is a sharp corner, called a cusp. An example of this appears in the second exercise.

While the concept of arc length is very useful for the theory of parametric curves, it turns out to be very difficult to compute in all but the simplest cases.

>with(plots): >with(CalcP7): >implicitplot(x^2=y,x=-2..2,y=0..4,scaling=constrained); >ParamPlot([t,t^2],t=-2..2,scaling=constrained); >ParamPlot([-t,t^2],t=-2..2,scaling=constrained);The ParamPlot command produces an animated plot. To see the animation, execute the command and then click on the plot region below to make the controls appear in the Context Bar just above the worksheet window. To enter a function parametrically

>f:=t->[t*cos(3*t),t^2]; >VPlot(f(t),t=-2*Pi..2*Pi);

- Animate the following two parametrization for
and state how the two parametrization are different.

- Given the family of parametric curves defined by and
, use a parametric plot to see how the graph changes for the following values of :
.
- What can you say about the curve for . What is different about the case ? What is the relationship between and the asymptote?
- Use the formula for the slope of a parametric curve to find .
- Evaluate the numerator and denominator of
separately for each of the following and explain the difference between the two in terms of slope of the graph.
- , .
- , .

- The parametric description , ,
is the ellipse

First show that the two are the same shape by plotting them parametrically and with the command`implicitplot`. Use the formula above to set up an integral for the arc length of the ellipse. You should find that Maple can't do the integral exactly. This is because this integral can't be done analytically. You can get a numerical approximation to the integral by putting an`evalf`command on the outside of the`int`command.

2008-11-17