Example 7

A particle moves according to the parametric equations

x = t³ - t, y = t².

Find the slope of its path.

so

We see from Figure 2.6.3 that the path of this particle is not the graph of a function, and in fact contains a loop and crosses the point (0,1) twice, at t = — 1 and t = 1. The path is vertical at the points t = ±√⅓, where there is no slope. At the point (0,1), the two slopes of the path are dy/dx = -1 when t = -1, and dy/dx = 1 when t = 1.