- Purpose
- Background
- Plotting and animating curves in the plane
- Defining parametric curves and vector valued functions simply in Maple

- Exercises

where and are functions defined on some interval . From our definition of a parametric curve, it should be clear that you can always associate a parametric curve with a vector-valued function by just considering the curve traced out by the head of the vector.

>with(CalcP7);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.

>ParamPlot([t,t^2],t=-2..2);the direction of the motion on the curve can be reversed by simply changing the first component from t to -t, as shown below.

>ParamPlot([-t,t^2],t=-2..2);The ParamPlot command is nice for visualization, but its output doesn't always show up in printouts. Toproduce a printable plot, you can use the VPlot command as shown below.

>VPlot([t^2,t^3-t],t=-1.5..1.5);

>f:=t->[2*cos(t),2*sin(t)];You can evaluate this function at any value of t in the usual way.

>f(0);This is how to access a single component. You would use f(t)[2] to get the second component.

>f(t)[1]

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.

- Consider the curve
- Calculate the coordinate point on the curve at and the slope of the curve at .
- Define the vector equation of the line through the point above tangent to the curve at that point.
- Plot the graph of and this tangent line on the same graphover the interval .

- The parametric description , ,
is the ellipse

First show that the two are the same shape by plotting them parametrically and with the command`implicitplot`. Find the arclength of the ellipse. - Suppose that at time zero, flight 12 is at the point (100 mi, -101.63347 mi) at an altitude of 30,000 feet and traveling northwest at 429 mph and that flight 33 is at the same altitude, but is traveling due east at a speed of 388 mph. At time zero, flight 33 is at the point (-200 mi, 30 mi).
- Write a parametric function for each flight.
- Animate the flights on one plot.
- Do the planes crash? If so, when (convert your answer to minutes)?

2015-11-18