91Ó°ÊÓ

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter \(t .\) $$ \mathbf{r}(t)=\left\langle 3 \cos t, 3 \sin t, t^{2}\right\rangle $$

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter \(t .\) $$ \mathbf{r}(t)=e^{-t} \mathbf{i}+t^{2} \mathbf{j}+\tan t \mathbf{k} $$

Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter \(t .\) $$\mathbf{r}(t)=2 \cos t \mathbf{j}+3 \sin t \mathbf{k} .$$ The graph is shown here:

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

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

Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\langle\sin t, t, \cos t\rangle .$$The graph is shown here:

The position function of an object is given by \(\mathbf{r}(t)=\left\langle t^{2}, 5 t, t^{2}-16 t\right\rangle .\) At what time is the speed a minimum?

Find the equations for the velocity, acceleration, and speed of the particle at any time. A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector \(\mathbf{r}(t)=(3 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+t^{2} \mathbf{k} .\) The path is similar to that of a helix, although it is not a helix. The graph is shown here:

Given that \(\mathbf{r}(t)=\left\langle e^{-5 t} \sin t, e^{-5 t} \cos t, 4 e^{-5 t}\right\rangle\) is the position vector of a moving particle, find the following quantities: The velocity of the particle

Given that \(\mathbf{r}(t)=\left\langle e^{-5 t} \sin t, e^{-5 t} \cos t, 4 e^{-5 t}\right\rangle\) is the position vector of a moving particle, find the following quantities: The speed of the particle