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Key concepts

  • To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function.
  • Many of the properties of differentiation from the Introduction to Derivatives also apply to vector-valued functions.
  • The derivative of a vector-valued function r ( t ) is also a tangent vector to the curve. The unit tangent vector T ( t ) is calculated by dividing the derivative of a vector-valued function by its magnitude.
  • The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function.
  • The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function.

Key equations

  • Derivative of a vector-valued function
    r ( t ) = lim Δ t 0 r ( t + Δ t ) r ( t ) Δ t
  • Principal unit tangent vector
    T ( t ) = r ( t ) r ( t )
  • Indefinite integral of a vector-valued function
    [ f ( t ) i + g ( t ) j + h ( t ) k ] d t = [ f ( t ) d t ] i + [ g ( t ) d t ] j + [ h ( t ) d t ] k
  • Definite integral of a vector-valued function
    a b [ f ( t ) i + g ( t ) j + h ( t ) k ] d t = [ a b f ( t ) d t ] i + [ a b g ( t ) d t ] j + [ a b h ( t ) d t ] k

Compute the derivatives of the vector-valued functions.

r ( t ) = t 3 i + 3 t 2 j + t 3 6 k

3 t 2 , 6 t , 1 2 t 2

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r ( t ) = sin ( t ) i + cos ( t ) j + e t k

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r ( t ) = e t i + sin ( 3 t ) j + 10 t k . A sketch of the graph is shown here. Notice the varying periodic nature of the graph.

This figure is a 3 dimensional graph. It is a curve inside of a box. The curve starts at the bottom of the box and spirals around the middle, with upward orientation.

e t , 3 cos ( 3 t ) , 5 t

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r ( t ) = e t i + 2 e t j + k

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r ( t ) = i + j + k

0 , 0 , 0

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r ( t ) = t e t i + t ln ( t ) j + sin ( 3 t ) k

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r ( t ) = 1 t + 1 i + arctan ( t ) j + ln t 3 k

−1 ( t + 1 ) 2 , 1 1 + t 2 , 3 t

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r ( t ) = tan ( 2 t ) i + sec ( 2 t ) j + sin 2 ( t ) k

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r ( t ) = 3 i + 4 sin ( 3 t ) j + t cos ( t ) k

0 , 12 cos ( 3 t ) , cos t t sin t

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r ( t ) = t 2 i + t e −2 t j 5 e −4 t k

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For the following problems, find a tangent vector at the indicated value of t .

r ( t ) = t i + sin ( 2 t ) j + cos ( 3 t ) k ; t = π 3

1 2 1 , −1 , 0

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r ( t ) = 3 t 3 i + 2 t 2 j + 1 t k ; t = 1

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r ( t ) = 3 e t i + 2 e −3 t j + 4 e 2 t k ; t = ln ( 2 )

1 1060.5625 6 , 3 4 , 32

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r ( t ) = cos ( 2 t ) i + 2 sin t j + t 2 k ; t = π 2

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Find the unit tangent vector for the following parameterized curves.

r ( t ) = 6 i + cos ( 3 t ) j + 3 sin ( 4 t ) k , 0 t < 2 π

1 9 sin 2 ( 3 t ) + 144 cos 2 ( 4 t ) 0 , −3 sin ( 3 t ) , 12 cos ( 4 t )

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r ( t ) = cos t i + sin t j + sin t k , 0 t < 2 π . Two views of this curve are presented here:

This figure has two graphs. The first graph is inside a 3 dimensional box. It has a lattice-look to the graph in the middle of the box, crossing over itself. The second graph is the same as the first, with a different position of the box for a different perspective of the lattice-looking curve.
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r ( t ) = 3 cos ( 4 t ) i + 3 sin ( 4 t ) j + 5 t k , 1 t 2

T ( t ) = −12 13 sin ( 4 t ) i + 12 13 cos ( 4 t ) j + 5 13 k

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r ( t ) = t i + 3 t j + t 2 k

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Let r ( t ) = t i + t 2 j t 4 k and s ( t ) = sin ( t ) i + e t j + cos ( t ) k . Here is the graph of the function:

This figure is a 3 dimensional graph. It is inside of a box. The box represents an octant. The curve in the graph starts at the lower left corner of the box and bends upward and out towards the other end of the box.

Find the following.

d d t [ r ( t 2 ) ]

2 t , 4 t 3 , −8 t 7

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d d t [ t 2 · s ( t ) ]

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d d t [ r ( t ) · s ( t ) ]

sin ( t ) + 2 t e t 4 t 3 cos ( t ) + t cos ( t ) + t 2 e t + t 4 sin ( t )

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Compute the first, second, and third derivatives of r ( t ) = 3 t i + 6 ln ( t ) j + 5 e −3 t k .

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Find r ( t ) · r ( t ) for r ( t ) = −3 t 5 i + 5 t j + 2 t 2 k .

900 t 7 + 16 t

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The acceleration function, initial velocity, and initial position of a particle are
a ( t ) = −5 cos t i 5 sin t j , v ( 0 ) = 9 i + 2 j , and r ( 0 ) = 5 i .
Find v ( t ) and r ( t ) .

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The position vector of a particle is r ( t ) = 5 sec ( 2 t ) i 4 tan ( t ) j + 7 t 2 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 π / 4 (if it exists).

  1. This figure is a graph of a curve in 3 dimensions. The curve has asymptotes and from the above view, the curve resembles the secant function.

  2. Undefined or infinite
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Find the velocity and the speed of a particle with the position function r ( t ) = ( 2 t 1 2 t + 1 ) i + ln ( 1 4 t 2 ) j . The speed of a particle is the magnitude of the velocity and is represented by r ' ( t ) .

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Questions & Answers

do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
Smarajit Reply
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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