<< Chapter < Page Chapter >> Page >

Now for some examples using these properties.

Using the properties of derivatives of vector-valued functions

Given the vector-valued functions

r ( t ) = ( 6 t + 8 ) i + ( 4 t 2 + 2 t 3 ) j + 5 t k

and

u ( t ) = ( t 2 3 ) i + ( 2 t + 4 ) j + ( t 3 3 t ) k ,

calculate each of the following derivatives using the properties of the derivative of vector-valued functions.

  1. d d t [ r ( t ) · u ( t ) ]
  2. d d t [ u ( t ) × u ( t ) ]
  1. We have r ( t ) = 6 i + ( 8 t + 2 ) j + 5 k and u ( t ) = 2 t i + 2 j + ( 3 t 2 3 ) k . Therefore, according to property iv.:
    d d t [ r ( t ) · u ( t ) ] = r ( t ) · u ( t ) + r ( t ) · u ( t ) = ( 6 i + ( 8 t + 2 ) j + 5 k ) · ( ( t 2 3 ) i + ( 2 t + 4 ) j + ( t 3 3 t ) k ) + ( ( 6 t + 8 ) i + ( 4 t 2 + 2 t 3 ) j + 5 t k ) · ( 2 t i + 2 j + ( 3 t 2 3 ) k ) = 6 ( t 2 3 ) + ( 8 t + 2 ) ( 2 t + 4 ) + 5 ( t 3 3 t ) + 2 t ( 6 t + 8 ) + 2 ( 4 t 2 + 2 t 3 ) + 5 t ( 3 t 2 3 ) = 20 t 3 + 42 t 2 + 26 t 16 .
  2. First, we need to adapt property v. for this problem:
    d d t [ u ( t ) × u ( t ) ] = u ( t ) × u ( t ) + u ( t ) × u″ ( t ) .

    Recall that the cross product of any vector with itself is zero. Furthermore, u″ ( t ) represents the second derivative of u ( t ) :


    u″ ( t ) = d d t [ u ( t ) ] = d d t [ 2 t i + 2 j + ( 3 t 2 3 ) k ] = 2 i + 6 t k .

    Therefore,


    d d t [ u ( t ) × u ( t ) ] = 0 + ( ( t 2 3 ) i + ( 2 t + 4 ) j + ( t 3 3 t ) k ) × ( 2 i + 6 t k ) = | i j k t 2 3 2 t + 4 t 3 3 t 2 0 6 t | = 6 t ( 2 t + 4 ) i ( 6 t ( t 2 3 ) 2 ( t 3 3 t ) ) j 2 ( 2 t + 4 ) k = ( 12 t 2 + 24 t ) i + ( 12 t 4 t 3 ) j ( 4 t + 8 ) k .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Given the vector-valued functions r ( t ) = cos t i + sin t j e 2 t k and u ( t ) = t i + sin t j + cos t k , calculate d d t [ r ( t ) · r ( t ) ] and d d t [ u ( t ) × r ( t ) ] .

d d t [ r ( t ) · r ( t ) ] = 8 e 4 t

d d t [ u ( t ) × r ( t ) ] = ( e 2 t ( cos t + 2 sin t ) + cos 2 t ) i + ( e 2 t ( 2 t + 1 ) sin 2 t ) j + ( t cos t + sin t cos 2 t ) k

Got questions? Get instant answers now!

Tangent vectors and unit tangent vectors

Recall from the Introduction to Derivatives that the derivative at a point can be interpreted as the slope of the tangent line to the graph at that point. In the case of a vector-valued function, the derivative provides a tangent vector to the curve represented by the function. Consider the vector-valued function r ( t ) = cos t i + sin t j . The derivative of this function is r ( t ) = sin t i + cos t j . If we substitute the value t = π / 6 into both functions we get

r ( π 6 ) = 3 2 i + 1 2 j and r ( π 6 ) = 1 2 i + 3 2 j .

The graph of this function appears in [link] , along with the vectors r ( π 6 ) and r ( π 6 ) .

This figure is the graph of a circle represented by the vector-valued function r(t) = cost i + sint j. It is a circle centered at the origin with radius of 1, and counter-clockwise orientation. It has a vector from the origin pointing to the curve and labeled r(pi/6). At the same point on the circle there is a tangent vector labeled “r’(pi/6)”.
The tangent line at a point is calculated from the derivative of the vector-valued function r ( t ) .

Notice that the vector r ( π 6 ) is tangent to the circle at the point corresponding to t = π / 6 . This is an example of a tangent vector    to the plane curve defined by r ( t ) = cos t i + sin t j .

Definition

Let C be a curve defined by a vector-valued function r, and assume that r ( t ) exists when t = t 0 . A tangent vector v at t = t 0 is any vector such that, when the tail of the vector is placed at point r ( t 0 ) on the graph, vector v is tangent to curve C. Vector r ( t 0 ) is an example of a tangent vector at point t = t 0 . Furthermore, assume that r ( t ) 0 . The principal unit tangent vector    at t is defined to be

T ( t ) = r ( t ) r ( t ) ,

provided r ( t ) 0 .

The unit tangent vector is exactly what it sounds like: a unit vector that is tangent to the curve. To calculate a unit tangent vector, first find the derivative r ( t ) . Second, calculate the magnitude of the derivative. The third step is to divide the derivative by its magnitude.

Finding a unit tangent vector

Find the unit tangent vector for each of the following vector-valued functions:

  1. r ( t ) = cos t i + sin t j
  2. u ( t ) = ( 3 t 2 + 2 t ) i + ( 2 4 t 3 ) j + ( 6 t + 5 ) k

  1. First step: r ( t ) = −sin t i + cos t j Second step: r ( t ) = ( sin t ) 2 + ( cos t ) 2 = 1 Third step: T ( t ) = r ( t ) r ( t ) = −sin t i + cos t j 1 = −sin t i + cos t j

  2. First step: u ( t ) = ( 6 t + 2 ) i 12 t 2 j + 6 k Second step: u ( t ) = ( 6 t + 2 ) 2 + ( −12 t 2 ) 2 + 6 2 = 144 t 4 + 36 t 2 + 24 t + 40 = 2 36 t 4 + 9 t 2 + 6 t + 10 Third step: T ( t ) = u ( t ) u ( t ) = ( 6 t + 2 ) i 12 t 2 j + 6 k 2 36 t 4 + 9 t 2 + 6 t + 10 = 3 t + 1 36 t 4 + 9 t 2 + 6 t + 10 i 6 t 2 36 t 4 + 9 t 2 + 6 t + 10 j + 3 36 t 4 + 9 t 2 + 6 t + 10 k
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 5

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask