91Ó°ÊÓ

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 \(\mathbf{r}\). $$ \mathbf{r}(t)=\left(t^{2}+4\right) \mathbf{i}+2 t \mathbf{j} ;\left\|\mathbf{r}^{\prime}(t)\right\|=2 \sqrt{t^{2}+1} $$

Find a piece wise smooth (smooth if possible) parametrization with the given orientation for the curve. The circle in the plane \(x=-2\) centered at the point \((-2,-2,-1)\), with radius 3 and with a clockwise orientation as viewed from the \(y z\) plane.

Find the derivative of the function. $$ \mathbf{F}(t)=\mathbf{i}+t \mathbf{j}+t^{5} \mathbf{k} $$

Determine the domain and the component functions of the given function. $$ \mathbf{F}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} $$

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

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

Compute the limit or explain why it does not exist. $$ \lim _{t \rightarrow-1}\left(3 \mathbf{i}+t \mathbf{j}+t^{5} \mathbf{k}\right) $$

Determine the domain and the component functions of the given function. $$ \mathbf{F}(t)=\sqrt{t+1} \mathbf{i}+\sqrt{1-t} \mathbf{j}+\mathbf{k} $$

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

Determine which of the parametrizations are smooth, which are piecewise smooth, and which are neither. $$ \mathbf{r}(t)=(t-1) \mathbf{i}+(t-1) \mathbf{j}+(t-1) \mathbf{k} $$