Objectives: In this tutorial, we find the derivative and second derivative of parametric equations and use these derivatives to find information about the graph of the parametric equations. After working through these materials, the student should be able

• to find the slope of the tangent line to a curve defined by parametric functions;
• to find the intervals for which the parametric functions are increasing and /or decreasing;
• to find horizontal and vertical tangent lines;
• to calculate the second derivative of a function defined implicitly by para metric functions;
• to determine the concavity of a curve defined by parametric functions.

Modules:

 Theorem. Let C be a curve with parametric equations x = x(t) and y = y(t) which are differentiable functions. If there exists a differentiable function f such that y(t) = f(x(t)) for t in some open interval, then

• Discussion [Using Flash]

• Example:

• LiveMath notebook on finding the tangent line to a parametric curve.

• Drill problems on finding the tangent line to a parametric curve.

• Using the TI-85 to find the tangent line to a parametric curve.

 Corollary. Let C be a curve with parametric equations x = x(t) and y = y(t) which are twice differentiable functions. If there exists a twice differentiable function f such that y(t) = f(x(t)) for t in some open interval, then

• Discussion [Using Flash]

• Example: