91Ó°ÊÓ

Find the derivative of the function. $$ \mathbf{F}(t)=(1+t)^{3 / 2} \mathbf{i}-(1-t)^{3 / 2} \mathbf{j}+\frac{3}{2} t \mathbf{k} $$

Compute the limit or explain why it does not exist. $$ \lim _{t \rightarrow \pi}(\tan t \mathbf{i}+3 t \mathbf{j}-4 \mathbf{k}) $$

Determine which of the parametrizations are smooth, which are piecewise smooth, and which are neither. $$ \mathbf{r}(t)=|t| \mathbf{i}+t \mathbf{j}+t \mathbf{k} $$

Determine the domain and the component functions of the given function. $$ \mathbf{F}(t)=\tanh t \mathbf{i}-\frac{1}{t^{2}-4} \mathbf{k} $$

Compute the limit or explain why it does not exist. $$ \lim _{t \rightarrow 0}\left(\frac{\sin t}{t} \mathbf{i}+e^{t} \mathbf{j}+(t+\sqrt{2}) \mathbf{k}\right) $$

Find the speed of a satellite orbiting the earth in a circular orbit with radius 5000 miles. (Hint: Take \(M\) equal to the mass \(M_{e}\) of the earth.)

First show that \(\left\|\mathbf{r}^{\prime}(t)\right\|\) is as given. Then find he tangent and the normal of the curve parametrized by \(\mathbf{r}\).IIirst show that \(\left\|\mathbf{r}^{\prime}(t)\right\|\) is as given. Then find the tangent and the normal of the curve parametrized by \(\mathbf{r}\). $$ \mathbf{r}(t)=\frac{1}{3}(1+t)^{3 / 2} \mathbf{i}+\frac{1}{3}(1-t)^{3 / 2} \mathbf{j}+\frac{\sqrt{2}}{2} t \mathbf{k} ;\left\|\mathbf{r}^{\prime}(t)\right\|=1 $$

Find the derivative of the function. $$ \mathbf{F}(t)=t^{2} \cos t \mathbf{i}+t^{3} \sin t \mathbf{j}+t^{4} \mathbf{k} $$

Determine the domain and the component functions of the given function. $$ \begin{aligned} \mathbf{F}(t)=\left[\left(t^{2}-1\right) \mathbf{i}+\ln t \mathbf{j}+\cot t \mathbf{k}\right] \\ & \times\left[\left(4-t^{2}\right) \mathbf{i}+e^{-5 t} \mathbf{j}+\frac{1}{t} \mathbf{k}\right] \end{aligned} $$

Determine which of the parametrizations are smooth, which are piecewise smooth, and which are neither. $$ \mathbf{r}(t)=t^{2 / 3} \mathbf{i}+t \mathbf{j}+t^{2} \mathbf{k} $$