Multivariable Calculus · free preview

Parametrized Curves and Motion

Curves in the plane and space described by vector-valued functions, their limits and continuity, and the derivative as the velocity of a moving point.

1. A vector-valued function traces a curve

A vector-valued function traces a curve

2. Limits and continuity pass through the components

Limits and continuity pass through the components

3. The derivative is velocity

The derivative is velocity

4. Rules of differentiation carry over

Rules of differentiation carry over

5. Reading velocity off a parametrization

A particle moves along the helix r(t)=(cos⁡t,sin⁡t,t)\mathbf{r}(t) = (\cos t, \sin t, t). What is its velocity v(t)\mathbf{v}(t)?

6. Constant length forces perpendicular velocity

A particle's position r(t)\mathbf{r}(t) always has length 33. What can be said about its velocity?

7. Limits pass through components

What is lim⁡t→π(cos⁡t,sin⁡t,t)\lim_{t \to \pi} (\cos t, \sin t, t)?

8. Speed on a circle

A particle follows the circle r(t)=(3cos⁡t,3sin⁡t)\mathbf{r}(t) = (3\cos t, 3\sin t). Give its speed, as a plain number.

9. A derivative product

For r(t)=(cos⁡t,sin⁡t,t)\mathbf{r}(t) = (\cos t, \sin t, t), compute ddt(r(t)⋅r(t))\dfrac{d}{dt}\big(\mathbf{r}(t) \cdot \mathbf{r}(t)\big) at t=0t = 0 and give the value, as a plain number.

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