Suppose that f(t) and g(t) are differentiable on [a, b]. What can be said about the smoothness of the curve parameterized by x = f(t) and y = g(t) on [a, b]? (Smooth here is used in the sense of having no corners or cusps.)
A) It must be smooth B) It might or might not be smooth. C) It cannot be smooth.
How is the answer B? I thought it would always be smooth when you differentiate a function. Can someone explain?
The fact that the components depend smoothly on a parameter does not imply that they depend smoothly on each other - that would be the intuitive idea behind "no corners or cusps."
For example, if you use ParametricPlot[{t^3, t^2},{t,-1,1}] at www.wolframalpha.com , you get an idea of what can happen.