<< Chapter < Page Chapter >> Page >

Key concepts

  • Taylor polynomials are used to approximate functions near a value x = a . Maclaurin polynomials are Taylor polynomials at x = 0 .
  • The n th degree Taylor polynomials for a function f are the partial sums of the Taylor series for f .
  • If a function f has a power series representation at x = a , then it is given by its Taylor series at x = a .
  • A Taylor series for f converges to f if and only if lim n R n ( x ) = 0 where R n ( x ) = f ( x ) p n ( x ) .
  • The Taylor series for e x , sin x , and cos x converge to the respective functions for all real x .

Key equations

  • Taylor series for the function f at the point x = a
    n = 0 f ( n ) ( a ) n ! ( x a ) n = f ( a ) + f ( a ) ( x a ) + f ( a ) 2 ! ( x a ) 2 + + f ( n ) ( a ) n ! ( x a ) n +

In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point.

f ( x ) = 1 + x + x 2 at a = 1

Got questions? Get instant answers now!

f ( x ) = 1 + x + x 2 at a = −1

f ( −1 ) = 1 ; f ( −1 ) = −1 ; f ( −1 ) = 2 ; f ( x ) = 1 ( x + 1 ) + ( x + 1 ) 2

Got questions? Get instant answers now!

f ( x ) = cos ( 2 x ) at a = π

Got questions? Get instant answers now!

f ( x ) = sin ( 2 x ) at a = π 2

f ( x ) = 2 cos ( 2 x ) ; f ( x ) = −4 sin ( 2 x ) ; p 2 ( x ) = −2 ( x π 2 )

Got questions? Get instant answers now!

f ( x ) = ln x at a = 1

f ( x ) = 1 x ; f ( x ) = 1 x 2 ; p 2 ( x ) = 0 + ( x 1 ) 1 2 ( x 1 ) 2

Got questions? Get instant answers now!

f ( x ) = e x at a = 1

p 2 ( x ) = e + e ( x 1 ) + e 2 ( x 1 ) 2

Got questions? Get instant answers now!

In the following exercises, verify that the given choice of n in the remainder estimate | R n | M ( n + 1 ) ! ( x a ) n + 1 , where M is the maximum value of | f ( n + 1 ) ( z ) | on the interval between a and the indicated point, yields | R n | 1 1000 . Find the value of the Taylor polynomial p n of f at the indicated point.

[T] ( 28 ) 1 / 3 ; a = 27 , n = 1

d 2 d x 2 x 1 / 3 = 2 9 x 5 / 3 −0.00092 when x 28 so the remainder estimate applies to the linear approximation x 1 / 3 p 1 ( 27 ) = 3 + x 27 27 , which gives ( 28 ) 1 / 3 3 + 1 27 = 3. 037 ¯ , while ( 28 ) 1 / 3 3.03658 .

Got questions? Get instant answers now!

[T] sin ( 6 ) ; a = 2 π , n = 5

Got questions? Get instant answers now!

[T] e 2 ; a = 0 , n = 9

Using the estimate 2 10 10 ! < 0.000283 we can use the Taylor expansion of order 9 to estimate e x at x = 2 . as e 2 p 9 ( 2 ) = 1 + 2 + 2 2 2 + 2 3 6 + + 2 9 9 ! = 7.3887 whereas e 2 7.3891 .

Got questions? Get instant answers now!

[T] cos ( π 5 ) ; a = 0 , n = 4

Got questions? Get instant answers now!

[T] ln ( 2 ) ; a = 1 , n = 1000

Since d n d x n ( ln x ) = ( −1 ) n 1 ( n 1 ) ! x n , R 1000 1 1001 . One has p 1000 ( 1 ) = n = 1 1000 ( −1 ) n 1 n 0.6936 whereas ln ( 2 ) 0.6931 .

Got questions? Get instant answers now!

Integrate the approximation sin t t t 3 6 + t 5 120 t 7 5040 evaluated at πt to approximate 0 1 sin π t π t d t .

Got questions? Get instant answers now!

Integrate the approximation e x 1 + x + x 2 2 + + x 6 720 evaluated at − x 2 to approximate 0 1 e x 2 d x .

0 1 ( 1 x 2 + x 4 2 x 6 6 + x 8 24 x 10 120 + x 12 720 ) d x

= 1 1 3 3 + 1 5 10 1 7 42 + 1 9 9 · 24 1 11 120 · 11 + 1 13 720 · 13 0.74683 whereas 0 1 e x 2 d x 0.74682 .

Got questions? Get instant answers now!

In the following exercises, find the smallest value of n such that the remainder estimate | R n | M ( n + 1 ) ! ( x a ) n + 1 , where M is the maximum value of | f ( n + 1 ) ( z ) | on the interval between a and the indicated point, yields | R n | 1 1000 on the indicated interval.

f ( x ) = sin x on [ π , π ] , a = 0

Got questions? Get instant answers now!

f ( x ) = cos x on [ π 2 , π 2 ] , a = 0

Since f ( n + 1 ) ( z ) is sin z or cos z , we have M = 1 . Since | x 0 | π 2 , we seek the smallest n such that π n + 1 2 n + 1 ( n + 1 ) ! 0.001 . The smallest such value is n = 7 . The remainder estimate is R 7 0.00092 .

Got questions? Get instant answers now!

f ( x ) = e −2 x on [ −1 , 1 ] , a = 0

Got questions? Get instant answers now!

f ( x ) = e x on [ −3 , 3 ] , a = 0

Since f ( n + 1 ) ( z ) = ± e z one has M = e 3 . Since | x 0 | 3 , one seeks the smallest n such that 3 n + 1 e 3 ( n + 1 ) ! 0.001 . The smallest such value is n = 14 . The remainder estimate is R 14 0.000220 .

Got questions? Get instant answers now!

In the following exercises, the maximum of the right-hand side of the remainder estimate | R 1 | max | f ( z ) | 2 R 2 on [ a R , a + R ] occurs at a or a ± R . Estimate the maximum value of R such that max | f ( z ) | 2 R 2 0.1 on [ a R , a + R ] by plotting this maximum as a function of R .

Questions & Answers

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 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
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul
Practice Key Terms 5

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 2' conversation and receive update notifications?

Ask