The goal of this discussion is to have students investigate the derivative of a parametrization and to use the derivative to analyze the motion represented by the parametrization. This discussion assumes that students are familiar with the definition of the derivative of a parametrization, with its interpretation as a velocity vector for a motion, and with the symbolic calculation of the derivative. It should be preceded by an introduction to these ideas. Students should read Section 2.2: The Derivative of a Parametrization simultaneously with this discussion.
The discussion focuses on interpreting the derivative of a parametrization of the motion of an object. The examples considered are a parametrization of projectile motion and the motion along the cycloid from Discussion 8. In each case, students are asked to sketch the velocity vectors on the plot of the curve and to interpret these vectors in terms of the motion represented by the parametrization. Students should be able to complete this discussion in one class period.