3.2 Calculus of vector-valued functions  (Page 5/11)

$\text{r}(t)=t^3\text{i}+3t^2\text{j}+\frac{t^3}{6}\text{k}$

$\text{r}(t)=\text{sin}(t)\text{i}+\text{cos}(t)\text{j}+e^t\text{k}$

$\text{r}(t)=e^{\text{−}t}\text{i}+\text{sin}(3t)\text{}\text{j}+10\sqrt{t}\text{k}.$ A sketch of the graph is shown here. Notice the varying periodic nature of the graph.

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

$\text{r}(t)=\text{i}+\text{j}+\text{k}$

$\text{r}(t)=t{e}^t\text{i}+t\text{ln}(t)\text{j}+\text{sin}(3t)\text{k}$

$\text{r}(t)=\frac{1}{t+1}\text{i}+\text{arctan}(t)\text{j}+\text{ln}{t}^3\text{k}$

$\text{r}(t)=\text{tan}(2t)\text{i}+\text{sec}(2t)\text{j}+{\text{sin}}^2(t)\text{k}$

$\text{r}(t)=3\text{i}+4\text{sin}(3t)\text{j}+t\text{cos}(t)\text{k}$

$\text{r}(t)=t^2\text{i}+t{e}^{\mathrm{-2}t}\text{j}-5e^{\mathrm{-4}t}\text{k}$

$\text{r}(t)=t\text{i}+\text{sin}(2t)\text{j}+\text{cos}(3t)\text{k};t=\frac{\pi }{3}$

$\text{r}(t)=3t^3\text{i}+2t^2\text{j}+\frac{1}{t}\text{k};t=1$

$\text{r}(t)=3e^t\text{i}+2e^{\mathrm{-3}t}\text{j}+4e^{2t}\text{k};$ $t=\text{ln}(2)$

$\text{r}(t)=\text{cos}(2t)\text{i}+2\text{sin}t\text{j}+t^2\text{k};t=\frac{\pi }{2}$

$\text{r}(t)=6\text{i}+\text{cos}(3t)\text{j}+3\text{sin}(4t)\text{k},$ $0\le t<2\pi$

$\text{r}(t)=\text{cos}t\text{i}+\text{sin}t\text{j}+\text{sin}t\text{k},$ $0\le t<2\pi .$ Two views of this curve are presented here:

$\text{r}(t)=3\text{cos}(4t)\text{i}+3\text{sin}(4t)\text{j}+5t\text{k},1\le t\le 2$

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

$\frac{d}{dt}\left[r\left(t^2\right)\right]$

$\frac{d}{dt}\left[t^2·s(t)\right]$

$\frac{d}{dt}\left[r(t)·s(t)\right]$

Compute the first, second, and third derivatives of $\text{r}(t)=3t\text{i}+6\text{ln}(t)\text{j}+5e^{\mathrm{-3}t}\text{k}.$

Find $\text{r}\prime (t)·\text{r}\text{″}(t)\text{for}\text{r}(t)=\mathrm{-3}{t}^5\text{i}+5t\text{j}+2t^2\text{k}.$

The acceleration function, initial velocity, and initial position of a particle are
$\text{a}(t)=\mathrm{-5}\text{cos}t\text{i}-5\text{sin}t\text{j},\text{v}(0)=9\text{i}+2\text{j},\text{and}\text{r}(0)=5\text{i}.$
Find $\text{v}(t)\text{and}\text{r}(t).$

The position vector of a particle is $\text{r}(t)=5\text{sec}(2t)\text{i}-4\text{tan}(t)\text{j}+7t^2\text{k}.$

1. Graph the position function and display a view of the graph that illustrates the asymptotic behavior of the function.
2. Find the velocity as t approaches but is not equal to $\pi \text{/}4$ (if it exists).

Find the velocity and the speed of a particle with the position function $\text{r}(t)=\left(\frac{2t-1}{2t+1}\right)\text{i}+\text{ln}(1-4t^2)\text{j}.$ The speed of a particle is the magnitude of the velocity and is represented by $\text{‖}{r}^{\text{'}}(t)\text{‖}.$