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Unit 11: Parametric Differentiation

To assist us in plotting a graph of this curve we have also plotted graphs of cos t and sin t in Notes
Figure 11.1. Clearly,
when t = 0, x = cos 0 = 1; y = sin 0 = 0

when t , x cos 0; y sin 1.
2 2 2
In this way we can obtain the x and y coordinates of lots of points given by Equations (1). Some
of these are given in Table 11.1.
Figure 11.1: Graphs of sin t and cos t

Table 11.1: Values of x and y given by Equations (1)

Plotting the points given by the x and y coordinates in Table 1, and joining them with a smooth
curve we can obtain the graph. In practice you may need to plot several more points before you
can be confident of the shape of the curve. We have done this and the result is shown in
Figure 11.2.
Figure 11.2. The parametric equations define a circle centered
at the origin and having radius 1

So x = cos t, y = sin t, for t lying between 0 and 2 , are the parametric equations which describe
a circle, centre (0, 0) and radius 1.

11.2 Differentiation of A Function Defined Parametrically

It is often necessary to find the rate of change of a function defined parametrically; that is, we
dy
want to calculate . The following example will show how this is achieved.
dx

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