ESCUELA DE INGENIERÍA DE PETROLEOS

RUBEN DARIO ARISMENDI RUEDA

ESCUELA DE INGENIERÍA DE PETROLEOS

CHAPTER 3: ‘TAYLOR’S APPROXIMATION’

ESCUELA DE INGENIERÍA DE PETROLEOS

Taylor's Series. Is a theorem that let us to obtain polynomics approximations of a function in an specific point where the function is diferenciable. As well, with this theorem we can delimit the range of error in the estimation.This is a finitive serie, and the Residual term is include to considerate all the terms from (n+1) to infinitive.

Taylor's Serie

Residual term

1

1

1

!1

nii

n

n xxnf

R

'' 2

' ...1 1 1 12! !

nf x f x ni if x f x f x x x x x x x Ri i i i ii i i in

ESCUELA DE INGENIERÍA DE PETROLEOS

McLaurin Serie.

2 31 1 1

1 1

(0)( ) (0)( ) (0)( )( ) ( ) (0) (0)( ) ..

2 ! 3 ! !

( 0)

n ni i i

i i

i

f x f x f xf x f x f f x

n

Serie de de Mclaurin x

ESCUELA DE INGENIERÍA DE PETROLEOS

How is Taylor’s Serie Used and Why is it important?

Taylor’s serie is used with a finitive number of terms that will provide us an approximation really close to the real solution of the function.

ESCUELA DE INGENIERÍA DE PETROLEOS

1 2 3 4When the number of derivates (number of terms) in the Serie increase, the result is goning to be closer to the real value of the function.

'' 2

' ...1 1 1 12! !

nf x f x ni if x f x f x x x x x x x Ri i i i ii i i in

ESCUELA DE INGENIERÍA DE PETROLEOS

NUMERICAL DIFFERENTIATION .

From the Taylor’s serie of first order.

We reflect the First derivate:

ii

iii xx

xfxfxf

1

1' ii xx 1 ; = h

PROGRESSIVE DIFFERENTIATION

iiiii xxxfxfxf 11 '

ESCUELA DE INGENIERÍA DE PETROLEOS

From the Taylor’s serie of first order.

iiiii xxxfxfxf 11 '

We reflect the First derivate:

ii

iii xx

xfxfxf

1

1' ; =h ii xx 1

REGRESSIVE DIFFERENTIATION

ESCUELA DE INGENIERÍA DE PETROLEOS

From the Taylor’s serie of first order (Progressive and Regressive)

iiiii xxxfxfxf 11 '

iiiii xxxfxfxf 11 '-

h

xfxfxf iii 2

' 11

We reflect the First derivate:

CENTRATE DIFFERENTIATION

ESCUELA DE INGENIERÍA DE PETROLEOS

EXAMPLE.

Determine the Taylor’s Polynom

,1

)(x

xf n = 4 , c = 1 = xi

11

)( i

i xxf

11

)('2

i

ix

xf

22

)(''3

i

ix

xf

66

)('''4

i

ix

xf

5

24( ) 24IVi

i

f xx

DEVELOPMENT.1.Find all the derivates that is needed.

ESCUELA DE INGENIERÍA DE PETROLEOS

2. Replace the values of the derivates in the Taylor’s Serie To find the Polynom.

2 3 42 ( 6) 241 ( 1) 1 1 1 11 1 1 1 12! 3! 4!

2 3 41 1 1 1 11 1 1 1 1

12 3 2 4 3 22 2 1 3 3 1 4 6 4 1

4 3 25 10 10 5

f x x x x xi i i i i

f x x x x xi i i i ix xi

f x x x x x x x x x x x

f x x x x x

At the end, We will have the polynom to get the approximate value of the function