3.1 Vector-valued functions and space curves  (Page 4/12)

Give the component functions $x=f(t)$ and $y=g(t)$ for the vector-valued function $\text{r}(t)=3\text{sec}t\text{i}+2\text{tan}t\text{j}.$

Given $\text{r}(t)=3\text{sec}t\text{i}+2\text{tan}t\text{j},$ find the following values (if possible).

1. $\text{r}\left(\frac{\pi }{4}\right)$
2. $\text{r}\left(\pi \right)$
3. $\text{r}\left(\frac{\pi }{2}\right)$

Sketch the curve of the vector-valued function $\text{r}(t)=3\text{sec}t\text{i}+2\text{tan}t\text{j}$ and give the orientation of the curve. Sketch asymptotes as a guide to the graph.

Evaluate $\underset{t\to 0}{\text{lim}}⟨e^t\text{i}+\frac{\text{sin}t}{t}\text{j}+e^{\text{−}t}\text{k}⟩.$

Given the vector-valued function $\text{r}(t)=⟨\text{cos}t,\text{sin}t⟩,$ find the following values:

1. $\underset{t\to \frac{\pi }{4}}{\text{lim}}\text{r}(t)$
2. $\text{r}\left(\frac{\pi }{3}\right)$
3. Is $\text{r}(t)$ continuous at $t=\frac{\pi }{3}?$
4. Graph $\text{r}(t).$

Given the vector-valued function $\text{r}(t)=⟨t,t^2+1⟩,$ find the following values:

1. $\underset{t\to \mathrm{-3}}{\text{lim}}\text{r}(t)$
2. $\text{r}(\mathrm{-3})$
3. Is $\text{r}(t)$ continuous at $x=\mathrm{-3}?$
4. $\text{r}(t+2)-\text{r}(t)$

Let $\text{r}(t)=e^t\text{i}+\text{sin}t\text{j}+\text{ln}t\text{k}.$ Find the following values:

1. $\text{r}\left(\frac{\pi }{4}\right)$
2. $\underset{t\to \pi \text{/}4}{\text{lim}}\text{r}(t)$
3. Is $\text{r}(t)$ continuous at $t=t=\frac{\pi }{4}?$

$\underset{t\to 4}{\text{lim}}⟨\sqrt{t-3},\frac{\sqrt{t}-2}{t-4},\text{tan}\left(\frac{\pi }{t}\right)⟩$

$\underset{t\to \pi \text{/}2}{\text{lim}}\text{r}(t)$ for $\text{r}(t)=e^t\text{i}+\text{sin}t\text{j}+\text{ln}t\text{k}$

$\underset{t\to \infty }{\text{lim}}⟨e^{\mathrm{-2}t},\frac{2t+3}{3t-1},\text{arctan}(2t)⟩$

$\underset{t\to e^2}{\text{lim}}⟨t\text{ln}(t),\frac{\text{ln}t}{t^2},\sqrt{\text{ln}\left(t^2\right)}⟩$

$\underset{t\to \pi \text{/}6}{\text{lim}}⟨{\text{cos}}^{2}t,{\text{sin}}^{2}t,1⟩$

$\underset{t\to \infty }{\text{lim}}\text{r}(t)$ for $\text{r}(t)=2e^{\text{−}t}\text{i}+e^{\text{−}t}\text{j}+\text{ln}(t-1)\text{k}$

Describe the curve defined by the vector-valued function $\text{r}(t)=(1+t)\text{i}+(2+5t)\text{j}+(\mathrm{-1}+6t)\text{k}.$

Domain: $\text{r}(t)=⟨t^2,\text{tan}t,\text{ln}t⟩$

Domain: $\text{r}(t)=⟨t^2,\sqrt{t-3},\frac{3}{2t+1}⟩$

Domain: $\text{r}(t)=⟨\text{csc}(t),\frac{1}{\sqrt{t-3}},\text{ln}(t-2)⟩$

For what values of t is $\text{r}(t)$ continuous?

Sketch the graph of $\text{r}(t).$

Find the domain of $\text{r}(t)=2e^{\text{−}t}\text{i}+e^{\text{−}t}\text{j}+\text{ln}(t-1)\text{k}.$

For what values of t is $\text{r}(t)=2e^{\text{−}t}\text{i}+e^{\text{−}t}\text{j}+\text{ln}(t-1)\text{k}$ continuous?

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

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

$\text{r}(t)=2\left(\text{sinh}t\right)\text{i}+2\left(\text{cosh}t\right)\text{j},t>0$

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

$\text{r}(t)=⟨3\text{sin}t,3\text{cos}t⟩$

[T] $\text{r}(t)=2\text{cos}{t}^2\text{i}+(2-\sqrt{t})\text{j}$

[T] $\text{r}(t)=⟨e^{\text{cos}(3t)},e^{\text{−}\text{sin}(t)}⟩$

[T] $\text{r}(t)=⟨2-\text{sin}(2t),3+2\text{cos}t⟩$

$4x^2+9y^2=36;$ clockwise and counterclockwise

$\text{r}(t)=⟨t,t^2⟩;$ from left to right

The line through P and Q where P is $\left(1,4,\mathrm{-2}\right)$ and Q is $\left(3,9,6\right)$

What is the initial point of the path corresponding to $\text{r}(0)?$

What is $\underset{t\to \infty }{\text{lim}}\text{r}(t)?$

[T] Use technology to sketch the curve.

Eliminate the parameter t to show that $z=5-\frac{r}{10}$ where $r=x^2+y^2.$

[T] Let $r(t)=\text{cos}t\text{i}+\text{sin}t\text{j}+0.3\text{sin}(2t)\text{k}.$ Use technology to graph the curve (called the roller-coaster curve ) over the interval $\left[0,2\pi \right).$ Choose at least two views to determine the peaks and valleys.

[T] Use the result of the preceding problem to construct an equation of a roller coaster with a steep drop from the peak and steep incline from the “valley.” Then, use technology to graph the equation.

Use the results of the preceding two problems to construct an equation of a path of a roller coaster with more than two turning points (peaks and valleys).

1. Graph the curve $\text{r}(t)=\left(4+\text{cos}(18t)\right)\text{cos}(t)\text{i}+\left(4+\text{cos}(18t)\text{sin}(t)\right)\text{j}+0.3\text{sin}(18t)\text{k}$ using two viewing angles of your choice to see the overall shape of the curve.
2. Does the curve resemble a “slinky”?
3. What changes to the equation should be made to increase the number of coils of the slinky?