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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

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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

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f ( x ) = cos ( 2 x ) at a = π

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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 )

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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

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f ( x ) = e x at a = 1

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

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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 .

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[T] sin ( 6 ) ; a = 2 π , n = 5

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[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 .

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[T] cos ( π 5 ) ; a = 0 , n = 4

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[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 .

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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 .

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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 .

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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

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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 .

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

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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 .

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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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