91Ó°ÊÓ

Graph the curves described by the following functions, indicating the direction of positive orientation. Try to anticipate the shape of the curve before using a graphing utility. $$\mathbf{r}(t)=e^{-t / 20} \sin t \mathbf{i}+e^{-t / 20} \cos t \mathbf{j}+t \mathbf{k}, \text { for } 0 \leq t < \infty$$

Arc length of polar curves Find the length of the following polar curves. The spiral \(r=4 \theta^{2},\) for \(0 \leq \theta \leq 6\)

Find a vector normal to the given vectors. $$\langle 1,2,3\rangle \text { and }\langle-2,4,-1\rangle$$

Let $$\mathbf{u}(t)=2 t^{3} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}-8 \mathbf{k} \text { and } \mathbf{v}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-e^{2 t} \mathbf{k}$$ Compute the derivative of the following functions. $$\mathbf{u}(t) \times \mathbf{v}(t)$$

For the given vectors \(\mathbf{u}\) and \(\mathbf{v},\) calculate proj\(_{\mathbf{v}} \mathbf{u}\) and \(\operatorname{scal}_{\mathbf{v}} \mathbf{u}\). \(\mathbf{u}=\mathbf{i}+4 \mathbf{j}+7 \mathbf{k}\) and \(\mathbf{v}=2 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k}\)

Consider the following trajectories of moving objects. Find the tangential and normal components of the acceleration. $$\mathbf{r}(t)=\langle 10 \cos t,-10 \sin t\rangle$$

Compute the following derivatives. $$\frac{d}{d t}\left[t^{2}(\mathbf{i}+2 \mathbf{j}-2 t \mathbf{k}) \cdot\left(e^{t} \mathbf{i}+2 e^{t} \mathbf{j}-3 e^{-t} \mathbf{k}\right)\right]$$

Arc length of polar curves Find the length of the following polar curves. $$\text { The spiral } r=2 e^{2 \theta}, \text { for } 0 \leq \theta \leq \ln 8$$

Calculate the work done in the following situations. A suitcase is pulled \(50 \mathrm{ft}\) along a flat sidewalk with a constant force of 30 lb at an angle of \(30^{\circ}\) above the horizontal.

Let \(\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle-1,0\rangle .\) Carry out the following computations. Find \(|2 \mathbf{u}+3 \mathbf{v}-4 \mathbf{w}|\)