91Ó°ÊÓ

Use a computer algebra system to graph the vector-valued function and identify the common curve. $$ \mathbf{r}(t)=-\frac{1}{2} t^{2} \mathbf{i}+t \mathbf{j}-\frac{\sqrt{3}}{2} t^{2} \mathbf{k} $$

Find \(\mathrm{T}(t), \mathrm{N}(t), a_{\mathrm{T}}\), and \(a_{\mathrm{N}}\) at the given time \(t\) for the plane curve \(\mathbf{r}(t)\). $$ \mathbf{r}(t)=e^{t} \mathbf{i}+e^{-2 r} \mathbf{j}, \quad t=0 $$

Find the curvature \(K\) of the curve. \(\mathbf{r}(t)=4 t \mathbf{i}+3 \cos t \mathbf{j}+3 \sin t \mathbf{k}\)

Use a graphing utility to graph the paths of a projectile for the given values of \(\theta\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and range of the projectile. (Assume that the projectile is launched from ground level.) (a) \(\theta=10^{\circ}, \quad v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (b) \(\theta=10^{\circ}, \quad v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (c) \(\theta=45^{\circ}, \quad v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (d) \(\theta=45^{\circ}, \quad v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (e) \(\theta=60^{\circ}, \quad v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (f) \(\theta=60^{\circ}, \quad v_{0}=146 \mathrm{ft} / \mathrm{sec}\)

Use the properties of the derivative to find the following. (a) \(\mathbf{r}^{\prime}(t)\) (b) \(\mathbf{r}^{\prime \prime}(t)\) (c) \(D_{t}[\mathbf{r}(t) \cdot \mathbf{u}(t)]\) (d) \(D_{t}[3 \mathbf{r}(t)-\mathbf{u}(t)]\) (e) \(D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]\) (f) \(D_{t}[\|\mathbf{r}(t)\|], \quad t>0\) $$ \begin{aligned} &\mathbf{r}(t)=t \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k} \\ &\mathbf{u}(t)=\frac{1}{t} \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k} \end{aligned} $$

Use a computer algebra system to graph the vector-valued function and identify the common curve. $$ \mathbf{r}(t)=t \mathbf{i}-\frac{\sqrt{3}}{2} t^{2} \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k} $$

Find the curvature \(K\) of the curve. \(\mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+e^{t} \mathbf{k}\)

Find the angle at which an object must be thrown to obtain (a) the maximum range and (b) the maximum height.

Find(a) \(D_{t}[\mathbf{r}(t) \cdot \mathbf{u}(t)]\) and (b) \(D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]\) by differentiating the product, then applying the properties of Theorem \(12.2\). \(\mathbf{r}(t)=t \mathbf{i}+2 t^{2} \mathbf{j}+t^{3} \mathbf{k}, \quad \mathbf{u}(t)=t^{4} \mathbf{k}\)

Determine the maximum height and range of a projectile fired at a height of \(1.5\) meters above the ground with an initial velocity of 100 meters per second and at an angle of \(30^{\circ}\) above the horizontal.