91Ó°ÊÓ

Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle t \sin t, t^{2}, t \cos 2 t\right\rangle $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle t, \cos 2 t, \sin 2 t\rangle $$

\(9-14\) Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\left\langle t^{2}+1, t^{3}, t^{2}-1\right\rangle$$

\(9-14\) Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\langle 2 \cos t, 3 t, 2 \sin t\rangle$$

Graph the curve with parametric equations \(x=\sin t\) \(y=\sin 2 t, z=\sin 3 t\) . Find the total length of this curve correct to four decimal places.

Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle\tan t, \sec t, 1 / t^{2}\right\rangle $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle 1+t, 3 t,-t\rangle $$

\(7-14\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle 1, \cos t, 2 \sin t\rangle $$

Find the derivative of the vector function. $$ \mathbf{r}(t)=\mathbf{i}-\mathbf{j}+e^{4 t} \mathbf{k} $$

Let \(C\) be the curve of intersection of the parabolic cylinder \(x^{2}=2 y\) and the surface \(3 z=x y .\) Find the exact length of \(C\) from the origin to the point \((6,18,36) .\)