A Vector’s Derivatives

A question on the AP Calculus Community bulletin board this past Sunday inspired me to write this brief outline of what the derivatives of parametric equations mean and where they come from. The Position Equation or Position Vector A parametric equation gives the coordinates of points (x, y) in the plane as functions of a…

Roulettes and Art – 1

For the last few post we have been exploring roulettes using the roulette generator (RG) for either Winplot or Geometer’s Sketchpad. These files use the equations      The derivative is given by the equations Notice that the derivative is also a form of roulette. To generate various roulettes by changing the values of R…

Roulettes and Calculus

Roulettes – 5: Calculus Considerations. In the first post of this series Roulette Generators (RG) are explained. Here are the files for Winplot or Geometer’s Sketchpad. Use them to quickly see the graphs of these curves by adjusting one or two parameters. While writing this series of posts I was intrigued by the cusps that appear in some of the curves.…

Rolling Circles

A few weeks ago I covered some trigonometry classes for another teacher. They were studying polar and parametric graphs and the common curves limaçon, rose curves, cardioids, etc. I got to thinking about these curves. the next few posts will discuss what I learned. To help me see what was happening I made a Winplot…

Parametric and Vector Equations

AP Type Questions 8 Particle moving on a plane for BC – the parametric/vector question. I have always had the impression that the AP exam assumed that parametric equations and vectors were first studied and developed in a pre-calculus course. In fact many schools do just that. It would be nice if students knew all…