参数曲线与运动

A vector-valued function traces a curve

Limits and continuity pass through the components

The derivative is velocity

Rules of differentiation carry over

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

A particle's position $\mathbf{r}(t)$ always has length $3$. What can be said about its velocity?

What is $\lim_{t \to \pi} (\cos t, \sin t, t)$?

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

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