# 3.2 Calculus of vector-valued functions

 Page 1 / 11
• Write an expression for the derivative of a vector-valued function.
• Find the tangent vector at a point for a given position vector.
• Find the unit tangent vector at a point for a given position vector and explain its significance.
• Calculate the definite integral of a vector-valued function.

To study the calculus of vector-valued functions, we follow a similar path to the one we took in studying real-valued functions. First, we define the derivative, then we examine applications of the derivative , then we move on to defining integrals. However, we will find some interesting new ideas along the way as a result of the vector nature of these functions and the properties of space curves.

## Derivatives of vector-valued functions

Now that we have seen what a vector-valued function is and how to take its limit, the next step is to learn how to differentiate a vector-valued function. The definition of the derivative of a vector-valued function is nearly identical to the definition of a real-valued function of one variable. However, because the range of a vector-valued function consists of vectors, the same is true for the range of the derivative of a vector-valued function.

## Definition

The derivative of a vector-valued function     $\text{r}\left(t\right)$ is

${r}^{\prime }\left(t\right)=\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{r}\left(t+\text{Δ}t\right)-\text{r}\left(t\right)}{\text{Δ}t},$

provided the limit exists. If ${r}^{\prime }\left(t\right)$ exists, then r is differentiable at t. If ${r}^{\prime }\left(t\right)$ exists for all t in an open interval $\left(a,b\right),$ then r is differentiable over the interval $\left(a,b\right).$ For the function to be differentiable over the closed interval $\left[a,b\right],$ the following two limits must exist as well:

${r}^{\prime }\left(a\right)=\underset{\text{Δ}t\to {0}^{+}}{\text{lim}}\frac{\text{r}\left(a+\text{Δ}t\right)-\text{r}\left(a\right)}{\text{Δ}t}\phantom{\rule{0.4em}{0ex}}\text{and}\phantom{\rule{0.4em}{0ex}}{r}^{\prime }\left(b\right)=\underset{\text{Δ}t\to {0}^{-}}{\text{lim}}\frac{\text{r}\left(b+\text{Δ}t\right)-\text{r}\left(b\right)}{\text{Δ}t}.$

Many of the rules for calculating derivatives of real-valued functions can be applied to calculating the derivatives of vector-valued functions as well. Recall that the derivative of a real-valued function can be interpreted as the slope of a tangent line or the instantaneous rate of change of the function. The derivative of a vector-valued function can be understood to be an instantaneous rate of change as well; for example, when the function represents the position of an object at a given point in time, the derivative represents its velocity at that same point in time.

We now demonstrate taking the derivative of a vector-valued function.

## Finding the derivative of a vector-valued function

Use the definition to calculate the derivative of the function

$\text{r}\left(t\right)=\left(3t+4\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left({t}^{2}-4t+3\right)\phantom{\rule{0.1em}{0ex}}\text{j}.$

$\begin{array}{cc}\hfill {r}^{\prime }\left(t\right)& =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{r}\left(t+\text{Δ}t\right)-\text{r}\left(t\right)}{\text{Δ}t}\hfill \\ & =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\left[\left(3\left(t+\text{Δ}t\right)+4\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left({\left(t+\text{Δ}t\right)}^{2}-4\left(t+\text{Δ}t\right)+3\right)\phantom{\rule{0.1em}{0ex}}\text{j}\right]-\left[\left(3t+4\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left({t}^{2}-4t+3\right)\phantom{\rule{0.1em}{0ex}}\text{j}\right]}{\text{Δ}t}\hfill \\ & =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\left(3t+3\text{Δ}t+4\right)\phantom{\rule{0.1em}{0ex}}\text{i}-\left(3t+4\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left({t}^{2}+2t\text{Δ}t+{\left(\text{Δ}t\right)}^{2}-4t-4\text{Δ}t+3\right)\phantom{\rule{0.1em}{0ex}}\text{j}-\left({t}^{2}-4t+3\right)\phantom{\rule{0.1em}{0ex}}\text{j}}{\text{Δ}t}\hfill \\ & =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\left(3\text{Δ}t\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left(2t\text{Δ}t+{\left(\text{Δ}t\right)}^{2}-4\text{Δ}t\right)\phantom{\rule{0.1em}{0ex}}\text{j}}{\text{Δ}t}\hfill \\ & =\underset{\text{Δ}t\to 0}{\text{lim}}\left(3\phantom{\rule{0.1em}{0ex}}\text{i}+\left(2t+\text{Δ}t-4\right)\phantom{\rule{0.1em}{0ex}}\text{j}\right)\hfill \\ & =3\phantom{\rule{0.1em}{0ex}}\text{i}+\left(2t-4\right)\phantom{\rule{0.1em}{0ex}}\text{j}.\hfill \end{array}$

Use the definition to calculate the derivative of the function $\text{r}\left(t\right)=\left(2{t}^{2}+3\right)\phantom{\rule{0.1em}{0ex}}\text{i}+\left(5t-6\right)\phantom{\rule{0.1em}{0ex}}\text{j}.$

${r}^{\prime }\left(t\right)=4t\phantom{\rule{0.1em}{0ex}}\text{i}+5\phantom{\rule{0.1em}{0ex}}\text{j}$

Notice that in the calculations in [link] , we could also obtain the answer by first calculating the derivative of each component function, then putting these derivatives back into the vector-valued function. This is always true for calculating the derivative of a vector-valued function, whether it is in two or three dimensions. We state this in the following theorem. The proof of this theorem follows directly from the definitions of the limit of a vector-valued function and the derivative of a vector-valued function.

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
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!