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

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Calculate $\underset{t\to -2}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$ for the function $\text{r}\left(t\right)=\sqrt{{t}^{2}-3t-1}\phantom{\rule{0.1em}{0ex}}\text{i}+\left(4t+3\right)\phantom{\rule{0.1em}{0ex}}\text{j}+\text{sin}\phantom{\rule{0.2em}{0ex}}\frac{\left(t+1\right)\pi }{2}\phantom{\rule{0.1em}{0ex}}\text{k}.$

$\underset{t\to -2}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)=3\phantom{\rule{0.1em}{0ex}}\text{i}-5\phantom{\rule{0.1em}{0ex}}\text{j}-\text{k}$

Now that we know how to calculate the limit of a vector-valued function, we can define continuity at a point for such a function.

## Definition

Let f, g, and h be functions of t. Then, the vector-valued function $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$ is continuous at point $t=a$ if the following three conditions hold:

1. $\text{r}\left(a\right)$ exists
2. $\underset{t\to a}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$ exists
3. $\underset{t\to a}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)=\text{r}\left(a\right)$

Similarly, the vector-valued function $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k}$ is continuous at point $t=a$ if the following three conditions hold:

1. $\text{r}\left(a\right)$ exists
2. $\underset{t\to a}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$ exists
3. $\underset{t\to a}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)=\text{r}\left(a\right)$

## Key concepts

• A vector-valued function is a function of the form $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$ or $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k},$ where the component functions f, g, and h are real-valued functions of the parameter t .
• The graph of a vector-valued function of the form $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$ is called a plane curve . The graph of a vector-valued function of the form $\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k}$ is called a space curve .
• It is possible to represent an arbitrary plane curve by a vector-valued function.
• To calculate the limit of a vector-valued function, calculate the limits of the component functions separately.

## Key equations

• Vector-valued function
$\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}\text{r}\left(t\right)=f\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+g\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+h\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{k},\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}\text{r}\left(t\right)=⟨f\left(t\right),g\left(t\right)⟩\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}\text{r}\left(t\right)=⟨f\left(t\right),g\left(t\right),h\left(t\right)⟩$
• Limit of a vector-valued function
$\underset{t\to a}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)=\left[\underset{t\to a}{\text{lim}}f\left(t\right)\right]\phantom{\rule{0.1em}{0ex}}\text{i}+\left[\underset{t\to a}{\text{lim}}g\left(t\right)\right]\phantom{\rule{0.1em}{0ex}}\text{j}\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\underset{t\to a}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)=\left[\underset{t\to a}{\text{lim}}f\left(t\right)\right]\phantom{\rule{0.1em}{0ex}}\text{i}+\left[\underset{t\to a}{\text{lim}}g\left(t\right)\right]\phantom{\rule{0.1em}{0ex}}\text{j}+\left[\underset{t\to a}{\text{lim}}h\left(t\right)\right]\phantom{\rule{0.1em}{0ex}}\text{k}$

Give the component functions $x=f\left(t\right)$ and $y=g\left(t\right)$ for the vector-valued function $\text{r}\left(t\right)=3\phantom{\rule{0.1em}{0ex}}\text{sec}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{i}+2\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{j}.$

$f\left(t\right)=3\phantom{\rule{0.1em}{0ex}}\text{sec}\phantom{\rule{0.1em}{0ex}}t,g\left(t\right)=2\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}t$

Given $\text{r}\left(t\right)=3\phantom{\rule{0.1em}{0ex}}\text{sec}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{i}+2\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{j},$ 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 $\text{r}\left(t\right)=3\phantom{\rule{0.1em}{0ex}}\text{sec}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{i}+2\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{j}$ and give the orientation of the curve. Sketch asymptotes as a guide to the graph.

Evaluate $\underset{t\to 0}{\text{lim}}⟨{e}^{t}\phantom{\rule{0.1em}{0ex}}\text{i}+\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{t}\phantom{\rule{0.1em}{0ex}}\text{j}+{e}^{\text{−}t}\phantom{\rule{0.1em}{0ex}}\text{k}⟩.$

Given the vector-valued function $\text{r}\left(t\right)=⟨\text{cos}\phantom{\rule{0.1em}{0ex}}t,\text{sin}\phantom{\rule{0.1em}{0ex}}t⟩,$ find the following values:

1. $\underset{t\to \frac{\pi }{4}}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$
2. $\text{r}\left(\frac{\pi }{3}\right)$
3. Is $\text{r}\left(t\right)$ continuous at $t=\frac{\pi }{3}?$
4. Graph $\text{r}\left(t\right).$

a. $⟨\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}⟩,$ b. $⟨\frac{1}{2},\frac{\sqrt{3}}{2}⟩,$ c. Yes, the limit as t approaches $\pi \text{/}3$ is equal to $\text{r}\left(\pi \text{/}3\right),$ d.

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

1. $\underset{t\to -3}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$
2. $\text{r}\left(-3\right)$
3. Is $\text{r}\left(t\right)$ continuous at $x=-3?$
4. $\text{r}\left(t+2\right)-\text{r}\left(t\right)$

Let $\text{r}\left(t\right)={e}^{t}\phantom{\rule{0.1em}{0ex}}\text{i}+\text{sin}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{j}+\text{ln}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{k}.$ Find the following values:

1. $\text{r}\left(\frac{\pi }{4}\right)$
2. $\underset{t\to \pi \text{/}4}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$
3. Is $\text{r}\left(t\right)$ continuous at $t=t=\frac{\pi }{4}?$

a. $⟨{e}^{\pi \text{/}4},\frac{\sqrt{2}}{2},\text{ln}\left(\frac{\pi }{4}\right)⟩;$ b. $⟨{e}^{\pi \text{/}4},\frac{\sqrt{2}}{2},\text{ln}\left(\frac{\pi }{4}\right)⟩;$ c. Yes

Find the limit of the following vector-valued functions at the indicated value of t .

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

$\underset{t\to \pi \text{/}2}{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$ for $\text{r}\left(t\right)={e}^{t}\phantom{\rule{0.1em}{0ex}}\text{i}+\text{sin}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{j}+\text{ln}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{k}$

$⟨{e}^{\pi \text{/}2},1,\text{ln}\left(\frac{\pi }{2}\right)⟩$

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

$\underset{t\to {e}^{2}}{\text{lim}}⟨t\phantom{\rule{0.1em}{0ex}}\text{ln}\left(t\right),\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}t}{{t}^{2}},\sqrt{\text{ln}\phantom{\rule{0.1em}{0ex}}\left({t}^{2}\right)}⟩$

$2{e}^{2}\phantom{\rule{0.1em}{0ex}}\text{i}+\frac{2}{{e}^{4}}\phantom{\rule{0.1em}{0ex}}\text{j}+2\phantom{\rule{0.1em}{0ex}}\text{k}$

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

$\underset{t\to \infty }{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)$ for $\text{r}\left(t\right)=2{e}^{\text{−}t}\phantom{\rule{0.1em}{0ex}}\text{i}+{e}^{\text{−}t}\phantom{\rule{0.1em}{0ex}}\text{j}+\text{ln}\left(t-1\right)\phantom{\rule{0.1em}{0ex}}\text{k}$

The limit does not exist because the limit of $\text{ln}\left(t-1\right)$ as t approaches infinity does not exist.

Describe the curve defined by the vector-valued function $\text{r}\left(t\right)=\left(1+t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left(2+5t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+\left(-1+6t\right)\phantom{\rule{0.1em}{0ex}}\text{k}.$

Find the domain of the vector-valued functions.

Domain: $\text{r}\left(t\right)=⟨{t}^{2},\text{tan}\phantom{\rule{0.1em}{0ex}}t,\text{ln}\phantom{\rule{0.1em}{0ex}}t⟩$

$t>0,t\ne \left(2k+1\right)\frac{\pi }{2},$ where k is an integer

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

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

$t>3,t\ne n\pi ,$ where n is an integer

Let $\text{r}\left(t\right)=⟨\text{cos}\phantom{\rule{0.1em}{0ex}}t,t,\text{sin}\phantom{\rule{0.1em}{0ex}}t⟩$ and use it to answer the following questions.

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

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

Find the domain of $\text{r}\left(t\right)=2{e}^{\text{−}t}\phantom{\rule{0.1em}{0ex}}\text{i}+{e}^{\text{−}t}\phantom{\rule{0.1em}{0ex}}\text{j}+\text{ln}\left(t-1\right)\phantom{\rule{0.1em}{0ex}}\text{k}.$

For what values of t is $\text{r}\left(t\right)=2{e}^{\text{−}t}\phantom{\rule{0.1em}{0ex}}\text{i}+{e}^{\text{−}t}\phantom{\rule{0.1em}{0ex}}\text{j}+\text{ln}\left(t-1\right)\phantom{\rule{0.1em}{0ex}}\text{k}$ continuous?

All t such that $t\in \left(1,\infty \right)$

Eliminate the parameter t , write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. ( Hint: Let $x=2t$ and $y={t}^{2}.$ Solve the first equation for x in terms of t and substitute this result into the second equation.)

$\text{r}\left(t\right)=2t\phantom{\rule{0.1em}{0ex}}\text{i}+{t}^{2}\phantom{\rule{0.1em}{0ex}}\text{j}$

$\text{r}\left(t\right)={t}^{3}\phantom{\rule{0.1em}{0ex}}\text{i}+2t\phantom{\rule{0.1em}{0ex}}\text{j}$

$y=2\sqrt[3]{x},$ a variation of the cube-root function

$\text{r}\left(t\right)=2\left(\text{sinh}\phantom{\rule{0.1em}{0ex}}t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+2\left(\text{cosh}\phantom{\rule{0.1em}{0ex}}t\right)\phantom{\rule{0.1em}{0ex}}\text{j},t>0$

$\text{r}\left(t\right)=3\left(\text{cos}\phantom{\rule{0.1em}{0ex}}t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+3\left(\text{sin}\phantom{\rule{0.1em}{0ex}}t\right)\phantom{\rule{0.1em}{0ex}}\text{j}$

${x}^{2}+{y}^{2}=9,$ a circle centered at $\left(0,0\right)$ with radius 3, and a counterclockwise orientation

$\text{r}\left(t\right)=⟨3\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}t,3\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t⟩$

Use a graphing utility to sketch each of the following vector-valued functions:

[T] $\text{r}\left(t\right)=2\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}{t}^{2}\phantom{\rule{0.1em}{0ex}}\text{i}+\left(2-\sqrt{t}\right)\phantom{\rule{0.1em}{0ex}}\text{j}$

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

[T] $\text{r}\left(t\right)=⟨2-\text{sin}\left(2t\right),3+2\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}t⟩$

Find a vector-valued function that traces out the given curve in the indicated direction.

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

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

For left to right, $y={x}^{2},$ where t increases

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

Consider the curve described by the vector-valued function $\text{r}\left(t\right)=\left(50{e}^{\text{−}t}\text{cos}\phantom{\rule{0.1em}{0ex}}t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left(50{e}^{\text{−}t}\text{sin}\phantom{\rule{0.1em}{0ex}}t\right)\phantom{\rule{0.1em}{0ex}}\text{j}+\left(5-5{e}^{\text{−}t}\right)\phantom{\rule{0.1em}{0ex}}\text{k}.$

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

$\left(50,0,0\right)$

What is $\underset{t\to \infty }{\text{lim}}\phantom{\rule{0.1em}{0ex}}\text{r}\left(t\right)?$

[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 $r\left(t\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{i}+\text{sin}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{j}+0.3\phantom{\rule{0.1em}{0ex}}\text{sin}\left(2t\right)\phantom{\rule{0.1em}{0ex}}\text{k}.$ 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).

One possibility is $r\left(t\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{i}+\text{sin}\phantom{\rule{0.1em}{0ex}}t\phantom{\rule{0.1em}{0ex}}\text{j}+\text{sin}\left(4t\right)\phantom{\rule{0.1em}{0ex}}\text{k}.$ By increasing the coefficient of t in the third component, the number of turning points will increase.

1. Graph the curve $\text{r}\left(t\right)=\left(4+\text{cos}\left(18t\right)\right)\text{cos}\left(t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left(4+\text{cos}\left(18t\right)\text{sin}\left(t\right)\right)\phantom{\rule{0.1em}{0ex}}\text{j}+0.3\phantom{\rule{0.1em}{0ex}}\text{sin}\left(18t\right)\phantom{\rule{0.1em}{0ex}}\text{k}$ 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?

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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