91Ó°ÊÓ

Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$\mathbf{r}(t)=\left\langle 1-t^{2}, 3+2 t^{3}\right\rangle, \text { for } t \geq 0$$

Sketch the following vectors \(\mathbf{u}\) and \(\mathbf{v} .\) Then compute \(|\mathbf{u} \times \mathbf{v}|\) and show the cross product on your sketch. $$\mathbf{u}=\langle 0,4,0\rangle, \mathbf{v}=\langle 0,0,-8\rangle$$

Determine whether the following series converge or diverge. The line through (-3,2,-1) in the direction of the vector \(\mathbf{v}=\langle 1,-2,0\rangle\)

Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$\mathbf{r}(t)=\langle 8 \sin t, 8 \cos t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Sketch the following vectors \(\mathbf{u}\) and \(\mathbf{v} .\) Then compute \(|\mathbf{u} \times \mathbf{v}|\) and show the cross product on your sketch. $$\mathbf{u}=\langle 3,3,0\rangle, \mathbf{v}=\langle 3,3,3 \sqrt{2}\rangle$$

How do you compute the magnitude of \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle ?\)

Differentiate the following functions. $$\mathbf{r}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-4 e^{2 t} \mathbf{k}$$

Find the length of the following two-and three-dimensional curves. $$\mathbf{r}(t)=\langle 3 \cos t, 3 \sin t\rangle, \text { for } 0 \leq t \leq \pi$$

Consider the following vectors u and v. Sketch the vectors, find the angle between the vectors, and compute the dot product using the definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\). $$\mathbf{u}=\langle 10,0\rangle \text { and } \mathbf{v}=\langle 10,10\rangle$$

Differentiate the following functions. $$\mathbf{r}(t)=\tan t \mathbf{i}+\sec t \mathbf{j}+\cos ^{2} t \mathbf{k}$$