Basic Objects:

For the following vectors, find $\mathbf{r}'(t), \mathbf{T}(1), \mathbf{r}''$, and $\mathbf{r}'(t) \times \mathbf{r}''(t)$.

• $$\mathbf{r}(t) = \langle t+1, t^2 + 2, t^3 + 3\rangle$$
• $$\mathbf{r}(t) = \langle e^{t}, t e^{t}, t^2 e^{t} \rangle$$

Tangent Lines:

Find parametric equations for the tangent line to the curve at the given point:

• $$\mathbf{r}(t) = \langle \cos t, \sin t, 1 \rangle; \hspace{5mm} (1,0,1)$$
• $$\mathbf{r}(t) = \langle e^t, t e^t, te^{t^2} \rangle; \hspace{5mm} (1,0,0)$$

More Interesting Problem

Find the points on the curve $\mathbf{r}(t)$ where $\mathbf{r}(t)$ and $\mathbf{r}'(t)$ have the opposite direction. $$\mathbf{r}(t) = \langle 5t, 3+t^2, 0 \rangle.$$