Math Help

Parametrics – Curves, Calculus, Polar Coordinates

In a Nut Shell: Sometimes it is convenient to express a function, y = f(x), in

terms of a third variable t. Here t is called a parameter. Then the

“parametric representation” of the function, y = f(x), is as follows:

x = x(t) y = y(t)

Example of the parametric representation of a curve.

Let x = cos t and y = sin t be the parametric representation.

What does this curve look like in the x-y plane?

Note that x² + y² = 1. So the function is a circle of radius one centered at the origin.

Further note that as t increases one “marches” counterclockwise around the circle

starting at x = 1 and y = 0 at t = 0.

Parametric representation is useful for multi-valued functions as shown below.

Here the arrows indicate the direction traveled along the curve as the

parameter t increases.

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc