91Ó°ÊÓ

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

Sketch the curve represented by the vectorvalued function and give the orientation of the curve. $$ \mathbf{r}(t)=\left\langle t, t^{2}, \frac{2}{3} t^{3}\right\rangle $$

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

Find \(\mathrm{T}(t), \mathrm{N}(t), a_{\mathrm{T}}\), and \(a_{\mathrm{N}}\) at the given time \(t\) for the plane curve \(\mathbf{r}(t)\). $$ \mathbf{r}(t)=\left(t-t^{3}\right) \mathbf{i}+2 t^{2} \mathbf{j}, \quad t=\mathbf{I} $$

Find the curvature \(K\) of the curve. \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+\frac{t^{2}}{2} \mathbf{k}\)

Find the open interval(s) on which the curve given by the vector-valued function is smooth. $$ \mathbf{r}(t)=\sqrt{t} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}+\frac{1}{4} t \mathbf{k} $$

Sketch the curve represented by the vectorvalued function and give the orientation of the curve. $$ \mathbf{r}(t)=\langle\cos t+t \sin t, \sin t-t \cos t, t\rangle $$

Find \(\mathrm{T}(t), \mathrm{N}(t), a_{\mathrm{T}}\), and \(a_{\mathrm{N}}\) at the given time \(t\) for the plane curve \(\mathbf{r}(t)\). $$ \mathbf{r}(t)=\left(t^{3}-4 t\right) \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}, \quad t=0 $$

Find the curvature \(K\) of the curve. \(\mathbf{r}(t)=2 t^{2} \mathbf{i}+t \mathbf{j}+\frac{1}{2} t^{2} \mathbf{k}\)

Use the properties of the derivative to find the following. (a) \(\mathbf{r}^{\prime}(t)\) (b) \(\mathbf{r}^{\prime \prime}(t)\) (c) \(D_{t}[\mathbf{r}(t) \cdot \mathbf{u}(t)]\) (d) \(D_{t}[3 \mathbf{r}(t)-\mathbf{u}(t)]\) (e) \(D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]\) (f) \(D_{t}[\|\mathbf{r}(t)\|], \quad t>0\) $$ \mathbf{r}(t)=t \mathbf{i}+3 t \mathbf{j}+t^{2} \mathbf{k}, \quad \mathbf{u}(t)=4 t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} $$