Langara College - Department of Mathematics and Statistics

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Internet Resources for the Calculus Student - Topics in Calculus

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Parametric Equations

Many interesting curves are not function graphs with y=f(x), but can be
described by specifying both x and y as functions of a third variable (called
a parameter). For example the circle x^2+y^2=1 can be described "parametrically"
with x=cos(t) and y=sin(t). The path of any moving point can be described
this way with the parameter, t, being the time.
There is a fair amount of precalculus
material on parametric equations.

We can use calculus to express the slope of the curve in terms of the
derivatives of x(t) and y(t).

In fact, by the chain rule, dy/dx = (dy/dt)/(dx/dt). This can be used
to help sketch the curve and to find exact locations of 'critical points'
where the tangent is horizontal or vertical.

(See, for example, this
page from the University of Tennessee's Visual Calculus site )

We can also express the area inside a closed loop of the curve by means
of a definite integral.

(See Area
Inside a Parametric Curve(MathCad+QT@odu) )

If you have come across good web-based illustrations of these or any
other calculus applications to parametric curves, please do let
us know and we will add them here.