91Ó°ÊÓ

Given \(\quad \mathbf{r}(t)=t \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k}\) and \(\mathbf{u}(t)=\frac{1}{t} \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k},\) find the following: \(\mathbf{r}(t) \times \mathbf{u}(t)\)

Given \(\quad \mathbf{r}(t)=t \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k}\) and \(\mathbf{u}(t)=\frac{1}{t} \mathbf{i}+2 \sin t \mathbf{j}+2 \cos t \mathbf{k},\) find the following: \(\frac{d}{d t}(\mathbf{r}(t) \times \mathbf{u}(t))\)

Find the unit tangent vector \(\mathbf{T}(\mathrm{t})\) for the following vectorvalued functions. \(\mathbf{r}(t)=\left\langle t, \frac{1}{t}\right\rangle .\) The graph is shown here:

Find the unit tangent vector \(\mathbf{T}(\mathrm{t})\) for the following vectorvalued functions. \(\mathbf{r}(t)=\langle t \cos t, t \sin t\rangle\)

Find the unit tangent vector \(\mathbf{T}(\mathrm{t})\) for the following vectorvalued functions. \(\mathbf{r}(t)=\langle t+1,2 t+1,2 t+2\rangle\)

Evaluate the following integrals: \(\int\left(e^{t} \mathbf{i}+\sin t \mathbf{j}+\frac{1}{2 t-1} \mathbf{k}\right) d t\)

Evaluate the following integrals: \(\int_{0}^{1} \mathbf{r}(t) d t,\) where \(\mathbf{r}(t)=\left\langle\sqrt[3]{t}, \frac{1}{t+1}, e^{-t}\right\rangle\)

Find the arc length of the curve on the given interval. \(\mathbf{r}(t)=t^{2} \mathbf{i}+14 t \mathbf{j}, 0 \leq t \leq 7\). This portion of the graph is shown here:

Find the arc length of the curve on the given interval. \(\mathbf{r}(t)=t^{2} \mathbf{i}+\left(2 t^{2}+1\right) \mathbf{j}, 1 \leq t \leq 3\)

Find the arc length of the curve on the given interval. \(\quad \mathbf{r}(t)=\langle 2 \sin t, 5 t, 2 \cos t\rangle, 0 \leq t \leq \pi\). This portion of the graph is shown here: