Activity 10.2.3

Vector-valued functions can be used to generate many interesting curves. Graph each of the following using an appropriate technological tool, and then write one sentence for each function to describe the behavior of the resulting curve.

1. $\vr(t) = \langle t\cos(t), t\sin(t) \rangle$

2. $\vr(t) = \langle \sin(t)\cos(t), t\sin(t) \rangle$

3. $\vr(t) = \langle \sin(5t), \sin(4t) \rangle$

4. $\vr(t) = \langle t^2\sin(t)\cos(t), 0.9t\cos(t^2), \sin(t) \rangle$. (Note that this defines a curve in three-dimensional space.)