91Ó°ÊÓ

Find a tangent vector at the indicated value of \(t\). \(\mathbf{r}(t)=t \mathbf{i}+\sin (2 t) \mathbf{j}+\cos (3 t) \mathbf{k} ; t=\frac{\pi}{3}\)

Find a tangent vector at the indicated value of \(t\). \(\mathbf{r}(t)=3 t^{3} \mathbf{i}+2 t^{2} \mathbf{j}+\frac{1}{t} \mathbf{k} ; t=1\)

Find a tangent vector at the indicated value of \(t\). \(\mathbf{r}(t)=3 e^{t} \mathbf{i}+2 e^{-3 t} \mathbf{j}+4 e^{2 t} \mathbf{k} ; \quad t=\ln (2)\)

Find a tangent vector at the indicated value of \(t\). \(\mathbf{r}(t)=\cos (2 t) \mathbf{i}+2 \sin t \mathbf{j}+t^{2} \mathbf{k} ; t=\frac{\pi}{2}\)

Find the unit tangent vector for the following parameterized curves. . \(\mathbf{r}(t)=6 \mathbf{i}+\cos (3 t) \mathbf{j}+3 \sin (4 t) \mathbf{k}, \quad 0 \leq t<2 \pi\)

Find the unit tangent vector for the following parameterized curves. \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\sin t \mathbf{k}, \quad 0 \leq t<2 \pi\). Two views of this curve are presented here:

Find the unit tangent vector for the following parameterized curves. \(\mathbf{r}(t)=3 \cos (4 t) \mathbf{i}+3 \sin (4 t) \mathbf{j}+5 t \mathbf{k}, 1 \leq t \leq 2\)

Find the unit tangent vector for the following parameterized curves. \(\mathbf{r}(t)=t \mathbf{i}+3 t \mathbf{j}+t^{2} \mathbf{k}\)

Let \(\quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}-t^{4} \mathbf{k} \quad\) and \(s(t)=\sin (t) \mathbf{i}+e^{t} \mathbf{j}+\cos (t) \mathbf{k}\). Here is the graph of the function: Find the following. \(\frac{d}{d t}\left[r\left(t^{2}\right)\right]\)

Let \(\quad \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}-t^{4} \mathbf{k} \quad\) and \(s(t)=\sin (t) \mathbf{i}+e^{t} \mathbf{j}+\cos (t) \mathbf{k}\). Here is the graph of the function: Find the following. \(\frac{d}{d t}[r(t) \cdot s(t)]\)