91Ó°ÊÓ

Sketch the circle \(\mathbf{r}(t)=\cos t \mathbf{i}-\sin t \mathbf{j},\) and in each part draw the vector with its correct length. \(\begin{array}{llll}{\text { (a) } \mathbf{r}^{\prime}(\pi / 4)} & {\text { (b) } \mathbf{r}^{\prime \prime}(\pi)} & {\text { (c) } \mathbf{r}(2 \pi)-\mathbf{r}(3 \pi / 2)} & { \text { ( }3 \pi / 2)}\end{array}\)

Find the arc length of the parametric curve. $$ x=\frac{1}{2} t, y=\frac{1}{3}(1-t)^{3 / 2}, z=\frac{1}{3}(1+t)^{3 / 2} ;-1 \leq t \leq 1 $$

Find the parametric equations that correspond to the given vector equation. $$ \mathbf{r}=(2 t-1) \mathbf{i}-3 \sqrt{t} \mathbf{j}+\sin 3 t \mathbf{k} $$

Find \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) at the given point. $$ \mathbf{r}(t)=\ln t \mathbf{i}+t \mathbf{j} ; \quad t=e $$

Find the velocity, speed, and acceleration at the given time t of a particle moving along the given curve. $$ \mathbf{r}(t)=e^{t} \sin t \mathbf{i}+e^{t} \cos t \mathbf{j}+t \mathbf{k} ; t=\pi / 2 $$

Find the arc length of the graph of \(\mathbf{r}(t)\) $$ \mathbf{r}(t)=t^{3} \mathbf{i}+t \mathbf{j}+\frac{1}{2} \sqrt{6} t^{2} \mathbf{k} ; \quad 1 \leq t \leq 3 $$

Find \(\mathbf{r}^{\prime}(t)\) $$ \mathbf{r}(t)=4 \mathbf{i}-\cos t \mathbf{j} $$

Describe the graph of the equation. $$ \mathbf{r}=(3-2 t) \mathbf{i}+5 t \mathbf{j} $$

Find \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) at the given point. $$ \mathbf{r}(t)=4 \cos t \mathbf{i}+4 \sin t \mathbf{j}+t \mathbf{k} ; t=\pi / 2 $$

Suppose that a particle vibrates in such a way that its position function is \(\mathbf{r}(t)=16 \sin \pi t \mathbf{i}+4 \cos 2 \pi t \mathbf{j}\), where distance is in millimeters and \(t\) is in seconds. (a) Find the velocity and acceleration at time \(t=1\) s. (b) Show that the particle moves along a parabolic curve. (c) Show that the particle moves back and forth along the curve.