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The purpose of this lab is to introduce you to curve computations
using Maple for parametric curves and vector-valued functions in the
To assist you, there is a worksheet associated with this lab that
contains examples and even solutions to some of the exercises. You can
copy that worksheet to your home directory by going to your computer's Start menu and choose run. In the run field type:
when you hit enter, you can then choose MA1024 and then choose the worksheet
Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting the exercises.
By parametric curve in the plane, we mean a pair of equations
and for in some interval . A vector-valued function in
the plane is a function that associates a vector in
the plane with
each value of in its domain. Such a vector valued function can
written in component form as follows,
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.
The ParamPlot command is in the CalcP package so you have to load it first. If you get an error from this command, ask for help right away.
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.
the direction of the motion on the curve can be reversed by simply changing the first component from t to -t, as shown below.
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.
The easiest way to define a vector function or a parametric curve is to use the Maple list notaion with square brackets. Strictly speaking, this does not define something that Maple recognizes as a vector, but it will work with all of the commands you need for this lab.
You can evaluate this function at any value of t in the usual way.
This is how to access a single component. You would use f(t) to get the second component.
The graph of a parametric curve may not have a slope at every point on
the curve. When the slope exists, it must be given by the formula
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
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.
As mentioned above, parametric curves often represent the motion of a
particle or mechanical system. As we will see in class, when we think
of a parametric curve as representing motion, we need a way to measure
the distance traveled by the particle. This distance is given by the
arc length, , of a curve. For a parametric curve ,
, the arc length of the curve for
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 simple function . Animate the following two parameterizations for
and state how the two parameterizations of the same function are different.
- Consider the curve
. Plot the graph of and calculate the points at
and . Then plot the points on the graph. Calculate the slope of the curve at each of the given points.
- 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? When(convert your answer to minutes)?
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Dina J. Solitro-Rassias