taylor series and maclaurin with exercices

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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

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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.

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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

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Taylor series of some common functions include

=

=

=

=

=

=

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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

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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.

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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

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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

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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

⟹ ⟹

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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

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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

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c) Dibujar sus gráficas en los mismos ejes coordenados.

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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