Notes for Math 220, Calculus 2

Many interesting curves in the plane cannot be described as the graph of a function \(y = f(x)\text{;}\) the circle \(x^2 + y^2 = 1\) is a very familiar example.

Relatedly, many curves describe the position of a moving object as a function of time; an object moving around the above circle might have coordinates at time \(t\) given by \(x(t)=\cos t, y(t)=\sin t\text{.}\) This is an example of a parametric description of a curve, with the new, “auxilliary” variable \(t\) called a parameter.

Note that this parametric form can convey more information that the equation \(x^2 + y^2 = 1\) for the curve, because of the information about time and place — this is important for example for computing the velocity of a moving object.