1. Home
  2. Calculus volume 3
  3. Vector-valued functions
  4. Calculus of vector-valued

3.2 Calculus of vector-valued functions

<< Chapter < Page
  Calculus volume 3     Page 1 / 11
Chapter >> Page >
  • 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     r ( t ) is

r ′ ( t ) = lim Δ t → 0 r ( t + Δ t ) − r ( t ) Δ t ,

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

r ′ ( a ) = lim Δ t → 0 + r ( a + Δ t ) − r ( a ) Δ t and r ′ ( b ) = lim Δ t → 0 − r ( b + Δ t ) − r ( b ) 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

r ( t ) = ( 3 t + 4 ) i + ( t 2 − 4 t + 3 ) j .

Let’s use [link] :

r ′ ( t ) = lim Δ t → 0 r ( t + Δ t ) − r ( t ) Δ t = lim Δ t → 0 [ ( 3 ( t + Δ t ) + 4 ) i + ( ( t + Δ t ) 2 − 4 ( t + Δ t ) + 3 ) j ] − [ ( 3 t + 4 ) i + ( t 2 − 4 t + 3 ) j ] Δ t = lim Δ t → 0 ( 3 t + 3 Δ t + 4 ) i − ( 3 t + 4 ) i + ( t 2 + 2 t Δ t + ( Δ t ) 2 − 4 t − 4 Δ t + 3 ) j − ( t 2 − 4 t + 3 ) j Δ t = lim Δ t → 0 ( 3 Δ t ) i + ( 2 t Δ t + ( Δ t ) 2 − 4 Δ t ) j Δ t = lim Δ t → 0 ( 3 i + ( 2 t + Δ t − 4 ) j ) = 3 i + ( 2 t − 4 ) j .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the definition to calculate the derivative of the function r ( t ) = ( 2 t 2 + 3 ) i + ( 5 t − 6 ) j .

r ′ ( t ) = 4 t i + 5 j

Got questions? Get instant answers now!

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.

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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