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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.
Rxxn
xfxx
xfxxxfxfxf n
iii
n
iii
iiiii 12
111 !...
!2
'''
Taylor's Serie
Residual term
1
1
1
!1
nii
n
n xxnf
R
ESCUELA DE INGENIERÍA DE PETROLEOS
ni
n
iii xn
fx
fxffxf 1
2111 !
0...
!2
0''0'0
McLaurin Serie.
0ix
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 4
Rxxn
xfxx
xfxxxfxfxf n
iii
n
iii
iiiii 12
111 !...
!2
'''
When 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.
ESCUELA DE INGENIERÍA DE PETROLEOS
NUMERICAL DIFFERENTIATION .
From the Taylor’s serie of first order.
iiiii xxxfxfxf 11 '
We reflect the First derivate:
ii
iii xx
xfxfxf
1
1' ii xx 1 ; = h
PROGRESSIVE DIFFERENTIATION
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
