Review Exercises

$\frac{d}{dt}\left[\text{u}(t)\times \text{u}(t)\right]=2{\text{u}}^{\prime }(t)\times \text{u}(t)$

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

$\text{r}(t)=\langle e^t,\frac{1}{\sqrt{4-t}},\text{sec}(t)\rangle$

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

Intersection of the cone $z=\sqrt{x^2+y^2}$ and plane $z=y-4$

$\text{u}(t)=\langle t^2,2t+6,4t^5-12\rangle$

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

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

$\text{r}(t)=\text{cos}(2t)\text{i}+8t\text{j}-\text{sin}(2t)\text{k}$

$\text{r}(t)=\sqrt{2}{e}^t\text{i}+\sqrt{2}{e}^{\text{−}t}\text{j}+2t\text{k}$

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

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?

Find the position vector $\text{r}(t)$ and the parametric representation for the position.