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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
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 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.
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.
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.
One way to graph the function is with the VPlot command.
Notice how the derivative command works.
The output shows the derivative of each component, therefore if you want the derivative of the parametric equation make sure you use the definition of the derivative along with the proper component syntax.
- Enter the parametric function
- Plot the parametric equation with a domain of
- Find the value of the function points at the following five t-values , ,
- Plot the function and the five points.
- By looking at the graph, which points will have a slope of zero, a positive slope, a negative slope, or an undefined slope.
- Find the derivative of the points with a positive or negative slope.
- What do the tangent lines look like at the points with undefined slopes? The equation of the derivatives will have what value in the denominator? Looking at the formula for the derivative of a parametric equation, where does the denominator come from? Evaluate the denominator at these points.
- Find the arclength of the inner curve of the function and the outer curve of the function.
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Jane E Bouchard