91Ó°ÊÓ

Find the arc length of the curve on the given interval.Find the length of one turn of the helix given by \(\mathbf{r}(t)=\frac{1}{2} \cos t \mathbf{i}+\frac{1}{2} \sin t \mathbf{j}+\sqrt{\frac{3}{4}} t \mathbf{k}\).

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

Find the domain of the vector-valued functions. $$ \text { Domain: } \mathbf{r}(t)=\left\langle t^{2}, \sqrt{t-3}, \frac{3}{2 t+1}\right\rangle $$

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

Given that \(\mathbf{r}(t)=\left\langle e^{-5 t} \sin t, e^{-5 t} \cos t, 4 e^{-5 t}\right\rangle\) is the position vector of a moving particle, find the following quantities: The velocity of the particle

Find the domain of the vector-valued functions. $$ \text { Domain: } \mathbf{r}(t)=\left\langle\csc (t), \frac{1}{\sqrt{t-3}}, \ln (t-2)\right\rangle $$

Find the arc length of the curve on the given interval.Find the arc length of the vector-valued function \(\mathbf{r}(t)=-t \mathbf{i}+4 t \mathbf{j}+3 t \mathbf{k}\) over \([0,1]\).

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

Given that \(\mathbf{r}(t)=\left\langle e^{-5 t} \sin t, e^{-5 t} \cos t, 4 e^{-5 t}\right\rangle\) is the position vector of a moving particle, find the following quantities: The speed of the particle

Let \(\mathbf{r}(t)=\langle\cos t, t, \sin t\rangle\) and use it to answer the following questions. For what values of \(t\) is \(\mathbf{r}(t)\) continuous?