### Review Exercises

*True or False*? Justify your answer with a proof or a counterexample.

A parametric equation that passes through points P and Q can be given by $\text{r}(t)=\langle {t}^{2},3t+1,t-2\rangle ,$ where $P(1,4,\mathrm{-1})$ and $Q(16,11,2).$

$\frac{d}{dt}\left[\text{u}(t)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}(t)\right]=2{\text{u}}^{\prime}(t)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}(t)$

The curvature of a circle of radius $r$ is constant everywhere. Furthermore, the curvature is equal to $1\text{/}r.$

The speed of a particle with a position function $\text{r}(t)$ is $\left({\text{r}}^{\prime}(t)\right)\text{/}\left(\left|{\text{r}}^{\prime}(t)\right|\right).$

Find the domains of the vector-valued functions.

$\text{r}(t)=\langle \text{sin}(t),\text{ln}(t),\sqrt{t}\rangle $

Sketch the curves for the following vector equations. Use a calculator if needed.

**[T]** $\text{r}(t)=\langle {t}^{2},{t}^{3}\rangle $

**[T]** $\text{r}(t)=\langle \text{sin}\left(20t\right){e}^{\text{\u2212}t},\text{cos}\left(20t\right){e}^{\text{\u2212}t},{e}^{\text{\u2212}t}\rangle $

Find a vector function that describes the following curves.

Intersection of the cylinder ${x}^{2}+{y}^{2}=4$ with the plane $x+z=6$

Find the derivatives of $\text{u}(t),$ ${\text{u}}^{\prime}(t),$ ${\text{u}}^{\prime}(t)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}(t),$ $\text{u}(t)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{\text{u}}^{\prime}(t),$ and $\text{u}(t)\xb7{\text{u}}^{\prime}(t).$ Find the unit tangent vector.

$\text{u}(t)=\langle {e}^{t},{e}^{\text{\u2212}t}\rangle $

Evaluate the following integrals.

$\int \left(\text{tan}(t)\text{sec}(t)\text{i}-t{e}^{3t}\text{j}\right)dt$

$\underset{1}{\overset{4}{\int}}\text{u}(t)dt},$ with $\text{u}(t)=\langle \frac{\text{ln}(t)}{t},\frac{1}{\sqrt{t}},\text{sin}\left(\frac{t\pi}{4}\right)\rangle $

Find the length for the following curves.

$\text{r}(t)=\langle 3t,4\phantom{\rule{0.1em}{0ex}}\text{cos}(t),4\phantom{\rule{0.1em}{0ex}}\text{sin}\left(t\right)\rangle $ for $1\le t\le 4$

$\text{r}(t)=2\text{i}+\text{t}\phantom{\rule{0.2em}{0ex}}\text{j}+3{t}^{2}\text{k}$ for $0\le t\le 1$

Reparameterize the following functions with respect to their arc length measured from $t=0$ in direction of increasing $t.$

$\text{r}(t)=2t\phantom{\rule{0.1em}{0ex}}\text{i}+(4t-5)\text{j}+(1-3t)\text{k}$

$\text{r}(t)=\text{cos}(2t)\text{i}+8t\phantom{\rule{0.1em}{0ex}}\text{j}-\text{sin}(2t)\text{k}$

Find the curvature for the following vector functions.

$\text{r}(t)=(2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t)\text{i}-4t\phantom{\rule{0.1em}{0ex}}\text{j}+(2\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t)\text{k}$

$\text{r}(t)=\sqrt{2}{e}^{t}\text{i}+\sqrt{2}{e}^{\text{\u2212}t}\text{j}+2t\phantom{\rule{0.1em}{0ex}}\text{k}$

Find the unit tangent vector, the unit normal vector, and the binormal vector for $\text{r}(t)=2\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{i}+3t\phantom{\rule{0.1em}{0ex}}\text{j}+2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{k}.$

Find the tangential and normal acceleration components with the position vector $\text{r}(t)=\langle \text{cos}\phantom{\rule{0.1em}{0ex}}t,\text{sin}\phantom{\rule{0.1em}{0ex}}t,{e}^{t}\rangle .$

A Ferris wheel car is moving at a constant speed $v$ and has a constant radius $r.$ Find the tangential and normal acceleration of the Ferris wheel car.

The position of a particle is given by $\text{r}(t)=\langle {t}^{2},\text{ln}\left(t\right),\text{sin}\left(\pi t\right)\rangle ,$ where $t$ is measured in seconds and $\text{r}$ is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1 sec?

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle $\theta $ and initial velocity ${\mathbf{\text{v}}}_{0}.$ The only force acting on the cannonball is gravity, so we begin with a constant acceleration $\text{a}(t)=\text{\u2212}g\phantom{\rule{0.1em}{0ex}}\text{j}.$

Find the velocity vector function $\text{v}(t).$

At what angle do you need to fire the cannonball for the horizontal distance to be greatest? What is the total distance it would travel?