91Ó°ÊÓ

Sketch the plane curve represented by the vector-valued function, and sketch the vectors \(\mathbf{r}\left(t_{0}\right)\) and \(\mathbf{r}^{\prime}\left(t_{0}\right)\) for the given value of \(t_{0}\). Position the vectors such that the initial point of \(\mathbf{r}\left(t_{0}\right)\) is at the origin and the initial point of \(\mathbf{r}^{\prime}\left(t_{0}\right)\) is at the terminal point of \(\mathbf{r}\left(t_{0}\right) .\) What is the relationship between \(\mathbf{r}^{\prime}\left(t_{0}\right)\) and the curve? $$ \mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}, \quad t_{0}=\frac{\pi}{2} $$

A baseball is hit 3 feet above the ground at 100 feet per second and at an angle of \(45^{\circ}\) with respect to the ground. (a) Find the vector-valued function for the path of the baseball. (b) Find the maximum height. (c) Find the range. (d) Find the arc length of the trajectory.

Find the unit tangent vector \(T(t)\) and find a set of parametric equations for the line tangent to the space curve at point \(P .\) $$ \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t \mathbf{k}, \quad P(0,0,0) $$

Sketch the space curve and find its length over the given interval. Function \(\quad\) Interval \(\mathbf{r}(t)=\left\langle\cos t+t \sin t, \sin t-t \cos t, t^{2}\right\rangle \quad\left[0, \frac{\pi}{2}\right]\)

Find the unit tangent vector \(T(t)\) and find a set of parametric equations for the line tangent to the space curve at point \(P .\) $$ \mathbf{r}(t)=\langle 2 \cos t, 2 \sin t, 4\rangle, \quad P(\sqrt{2}, \sqrt{2}, 4) $$

The position vector \(\mathbf{r}\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, e^{t}\right\rangle $$

Find the unit tangent vector \(T(t)\) and find a set of parametric equations for the line tangent to the space curve at point \(P .\) $$ \mathbf{r}(t)=\left\langle 2 \sin t, 2 \cos t, 4 \sin ^{2} t\right\rangle, \quad P(1, \sqrt{3}, 1) $$

Find the curvature \(K\) of the curve, where \(s\) is the arc length parameter. $$ \mathbf{r}(s)=\left(1+\frac{\sqrt{2}}{2} s\right) \mathbf{i}+\left(1-\frac{\sqrt{2}}{2} s\right) \mathbf{j} $$

Verify that the space curves intersect at the given values of the parameters. Find the angle between the tangent vectors to the curves at the point of intersection. $$ \begin{aligned} &\mathbf{r}(t)=\langle t, \cos t, \sin t\rangle, \quad t=0 \\ &\mathbf{u}(s)=\left\langle-\frac{1}{2} \sin ^{2} s-\sin s, 1-\frac{1}{2} \sin ^{2} s-\sin s,\right. \\ &\left.\frac{1}{2} \sin s \cos s+\frac{1}{2} s\right\rangle, \quad s=0 \end{aligned} $$

What is known about the speed of an object if the angle between the velocity and acceleration vectors is (a) acute and (b) obtuse?