91Ó°ÊÓ

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\frac{2 t}{8+t^{3}} \mathbf{i}+\frac{2 t^{2}}{8+t^{3}} \mathbf{j} $$

Find \(v(t), a(t), T(t)\), and \(N(t)\) (if it exists) for an object moving along the path given by the vector-valued function \(r(t)\). Use the results to determine the form of the path. Is the speed of the object constant or changing? $$ \mathbf{r}(t)=t^{2} \mathbf{j}+\mathbf{k} $$

Find the curvature \(K\) of the curve. \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\)

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=(t-1) \mathbf{i}+\frac{1}{t} \mathbf{j}-t^{2} \mathbf{k} \quad \text { 36. } \mathbf{r}(t)=e^{t} \mathbf{i}-e^{-t} \mathbf{j}+3 t \mathbf{k} $$

Sketch the curve represented by the vectorvalued function and give the orientation of the curve. $$ \mathbf{r}(t)=2 \sin t \mathbf{i}+2 \cos t \mathbf{j}+e^{-t} \mathbf{k} $$

Find the curvature \(K\) of the curve. \(\mathbf{r}(t)=\langle a(\omega t-\sin \omega t), a(1-\cos \omega t)\rangle\)

Find \(\mathrm{T}(t), \mathrm{N}(t), a_{\mathrm{T}}\), and \(a_{\mathrm{N}}\) at the given time \(t\) for the plane curve \(\mathbf{r}(t)\). $$ \mathbf{r}(t)=t \mathbf{i}+\frac{1}{t} \mathbf{j}, \quad t=\mathbf{1} $$

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \begin{aligned} &\mathbf{r}(t)=(t-1) \mathbf{i}+\frac{1}{t} \mathbf{j}-t^{2} \mathbf{k}\\\ &\text { 36. } \mathbf{r}(t)=e^{t} \mathbf{i}-e^{-t} \mathbf{j}+3 t \mathbf{k} \end{aligned} $$

Sketch the curve represented by the vectorvalued function and give the orientation of the curve. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}+\frac{3}{2} t \mathbf{k} $$

Find \(\mathrm{T}(t), \mathrm{N}(t), a_{\mathrm{T}}\), and \(a_{\mathrm{N}}\) at the given time \(t\) for the plane curve \(\mathbf{r}(t)\). $$ \mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}, \quad t=1 $$