91Ó°ÊÓ

Find the domain of the vector-valued function. $$ \mathbf{r}(t)=\sin t \mathbf{i}+4 \cos t \mathbf{j}+t \mathbf{k} $$

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? $$ \mathbf{r}(t)=(1+t) \mathbf{i}+t^{3} \mathbf{j}, \quad t_{0}=1 $$

Sketch the plane curve and find its length over the given interval. Function \(\quad\) Interval \(\mathbf{r}(t)=(t+1) \mathbf{i}+t^{2} \mathbf{j}\)

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? $$ \mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}, \quad t_{0}=\frac{\pi}{2} $$

Sketch the plane curve and find its length over the given interval. Function \(\quad\) Interval \(\mathbf{r}(t)=a \cos ^{3} t \mathbf{i}+a \sin ^{3} t \mathbf{j} \quad[0,2 \pi]\)

Find the domain of the vector-valued function. $$ \begin{aligned} &\mathbf{r}(t)=\mathbf{F}(t)+\mathbf{G}(t) \text { where } \\ &\mathbf{F}(t)=\cos t \mathbf{i}-\sin t \mathbf{j}+\sqrt{t} \mathbf{k}, \quad \mathbf{G}(t)=\cos t \mathbf{i}+\sin t \mathbf{j} \end{aligned} $$

Find the unit tangent vector to the curve at the specified value of the parameter. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}, \quad t=1 $$

Find the domain of the vector-valued function. $$ \begin{aligned} &\mathbf{r}(t)=\mathbf{F}(t)-\mathbf{G}(t) \text { where } \\ &\mathbf{F}(t)=\ln t \mathbf{i}+5 t \mathbf{j}-3 t^{2} \mathbf{k}, \quad \mathbf{G}(t)=\mathbf{i}+4 t \mathbf{j}-3 t^{2} \mathbf{k} \end{aligned} $$

Sketch the plane curve and find its length over the given interval. Function \(\quad\) Interval \(\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j} \quad[0,2 \pi]\)

Find the unit tangent vector to the curve at the specified value of the parameter. $$ \mathbf{r}(t)=t^{3} \mathbf{i}+2 t^{2} \mathbf{j}, \quad t=1 $$