91Ó°ÊÓ

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$r(t)=\left\langle\cos t^{2}, \sin t^{2}\right\rangle$$

Graph the curves described by the following functions, indicating the positive orientation. $$\mathbf{r}(t)=\langle 2 \cos t, 2 \sin t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$\mathbf{r}(t)=\langle\sqrt{3} \sin t, \sin t, 2 \cos t\rangle$$

Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$\mathbf{r}(t)=\left\langle t^{2}+3, t^{2}+10, \frac{1}{2} t^{2}\right\rangle, \text { for } t \geq 0$$

Find the length of the following twoand three-dimensional curves. \(\mathbf{r}(t)=\langle 2+3 t, 1-4 t,-4+3 t\rangle,\) for \(1 \leq t \leq 6\)

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$r(t)=\langle t, \ln \cos t\rangle$$

Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$r(t)=\left\langle 2 e^{2 t}+1, e^{2 t}-1,2 e^{2 t}-10\right\rangle, \text { for } t \geq 0$$

Differentiate the following functions. $$\mathbf{r}(t)=\left\langle(t+1)^{-1}, \tan ^{-1} t, \ln (t+1)\right\rangle$$

Find the length of the following twoand three-dimensional curves. \(\mathbf{r}(t)=\langle 4 \cos t, 4 \sin t, 3 t\rangle,\) for \(0 \leq t \leq 6 \pi\)

Velocity and acceleration from position Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$r(t)=\langle 3+t, 2-4 t, 1+6 t\rangle, \text { for } t \geq 0$$