Embed Size (px)
Citation preview
ESCUELA DE INGENIERÍA DE PETROLEOS
ESCUELA DE INGENIERÍA DE PETROLEOS
A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function about a point is given by
ESCUELA DE INGENIERÍA DE PETROLEOS
It is common practice to use a finite number of terms of the series to approximate a function. The Taylor series may be regarded as the limit of the Taylor polynomials.
Taylor's theorem (actually discovered first by Gregory) states that any function satisfying certain conditions can be expressed as a Taylor series.
ESCUELA DE INGENIERÍA DE PETROLEOS
The Taylor (or more general) series of a function about a point up to order may be found using Series[f, x, a, n]. The th term of a Taylor series of a function can be computed in Mathematica using SeriesCoefficient[f, x, a, n] and is given by the inverse Z-transform
ESCUELA DE INGENIERÍA DE PETROLEOS
Taylor series of some common functions include
=
=
=
=
=
=
ESCUELA DE INGENIERÍA DE PETROLEOS
To derive the Taylor series of a function , note that the integral of the st derivate of from the point to an arbitrary point is given by
ESCUELA DE INGENIERÍA DE PETROLEOS
If the expansion is known as a Maclaurin series.
Maclaurin series are a type of series expansion in which all terms are nonnegative integer powers of the variable. Other more general types of series include the Laurent series and the Puiseux series.
ESCUELA DE INGENIERÍA DE PETROLEOS
1. Determine el n-ésimo polinomio de Taylor centrado en c de:
a) n=4, c=-2 =xi1
( ) ,1
f xx
2
1'( ) ,
( 1)f x
x
,
)1(
2)(''
3x
xf ,)1(
6)('''
4
x
xf
•⟹
⟹
⟹
⟹,
)1(
24)(
5iv
x
xf
ESCUELA DE INGENIERÍA DE PETROLEOS
2 31 1
1 1
''( )( ) '''( )( )( ) ( ) '( )( ) ...
2! 3!i i i i i i
i i i i i
f x x x f x x xf x f x f x x x
4
1( )( )...
4!
IVi i if x x x
2 31 1 12 3 4
415
1 1 2 6( ) ( ) ( ) ( ) ...
1 ( 1) 2!( 1) 3!( 1)
24... ( )
4!( 1)
i i i i i
i i
f x f x x x x xx x x x
x xx
Reemplazando por xi=2
2 3 41 1 1 1 1
1 1 1 1 1( ) ( 2) ( 2) ( 2) ( 2)
3 9 9 27 243i i i i if x x x x x Rta
ESCUELA DE INGENIERÍA DE PETROLEOS
1. Para f(x) = arccos (x)
• Escribir el polinomio de Mclaurin P3(x) para f(x).
2 31 1
1 1
''( )( ) '''( )( )( ) ( ) '( )( )
2! 3!i i i i i i
i i i i i
f x x x f x x xf x f x f x x x
,1
1)('
2xxf
,
)1(
)(''
2
32x
xxf
,
)1(
3
)1(
1)('''
2
52
2
2
32 x
x
x
xf
⟹ ⟹
ESCUELA DE INGENIERÍA DE PETROLEOS
Rta
22 3
1 1 1 11 3 3 52 2 2 22 2 2 2
1 1 1 1 3( ) arccos( ) ( ) ( ) ( )
2! 3!(1 ) (1 ) (1 ) (1 )
i i i i
x xf x x x x x
x x x x
31 1 1
1( ) ( ) ( )
2 6i i if x x x
ESCUELA DE INGENIERÍA DE PETROLEOS
b) Completar la siguiente tabla para P3(x) y para f(x) (Utilizar radianes).
100*verdadero
aproximadoverdadero
Valor
ValorValor
x -0,75 -0,5 -0,25 0 0,25 0,5 0,75
f(x) 2,4189 2,0943 1,8235 1,5708 1,3182 1,0472 0,7227
P3(x) 2,3911 2,0916 1,8234 1,5708 1,3182 1,0499 0,7505
%E 1,1471 0,1319 3,9713x10-3 2,3384x10-4 6,0512x10-3 0,2644 3,8400
ESCUELA DE INGENIERÍA DE PETROLEOS
c) Dibujar sus gráficas en los mismos ejes coordenados.
ESCUELA DE INGENIERÍA DE PETROLEOS
http://mathworld.wolfram.com/MaclaurinSeries.html
Eduardo Carrillo, Class en Presentation Ppt Methods Numeric's. Universidad Industrial de Santander 2010.
Steven C. Chapra, “Methods Numeric's for Engineering” Quinta Edition. Mac Graw Hill.
Stewart, James. "Calculus, Early Transcendent." 4 ed. Tr. Andrew Sesti. Mexico, Ed Thomson, 2002. p. 1151
