1CONSTRUIMOS FUTURO

TAYLOR SERIESBY:DUBAN CASTRO FLOREZ

NUMERICS METHOS IN ENGINEERINGPETROLEUM ENGINEERING

2010

CONSTRUIMOS FUTURO

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. If the series is centered at zero, the series is also called a Maclaurin series. 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.

2

TAYLOR SERIES

CONSTRUIMOS FUTURO

3

The Taylor series of a real or complex function ƒ(x) that is infinitely differentiable in a neighborhood of a real or complex number a is the power series:

which can be written in the more compact sigma notation as:

where n! denotes the factorial of n and ƒ (n)(a) denotes the nth derivative of ƒ evaluated at the point a. The zeroth derivative of ƒ is defined to be ƒ itself and (x − a)0 and 0! are both defined to be 1.

2 31

'( ) ''( ) '''( )( ) ( ) ( ) ( ) .... ( )

1 2 6 i

f a f a f af a x a x a x a f x

( )

0

( )( )

!

nn

n

f ax a

n

TAYLOR SERIES

CONSTRUIMOS FUTURO

4

)(xf

xix 1ix

order zero

order two

order one

order n

2 31

'( ) ''( ) '''( )( ) ( ) ( ) ( ) .... ( )

1 2 6 i

f a f a f af a x a x a x a f x

TAYLOR SERIES

h x a

CONSTRUIMOS FUTURO

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In the particular case where a = 0, the series is also called a Maclaurin series:

2 31

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

1 2 6 i

f f ff x x x f x

MACLAURIN SERIES

CONSTRUIMOS FUTURO

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2

1'( )

1f x

x

3

2 2

''( )

(1 )

xf x

x

2

3 52 22 2

1 3'''( )

(1 ) (1 )

xf x

x x

For f(x) = arccos (x) a) to Write the polynomial of Mclaurin P3(x) for f(x).

2

3 3 52 2 22 2 2

2 31 1 1 12

1 3

1 (1 ) (1 ) (1 )( ) arccos( ) ( ) ( ) ( )

2 61i i i i

x

x x xf x x x x x

x

31 1 1

1( ) ( ) ( )

2 6i i if x x x

EXAMPLE

CONSTRUIMOS FUTURO

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b) to Complete the following chart for P3(x) and it stops f(x) (to Use radianes).

*100true approximate

true

Value Value

Value

x -0,75 -0,5 -0,25 0 0,25 0,5 0,75

f(x) 2.4188 2.0943 1.82347 1.5707 1.3181 1.0471 0.7227

P3(x) 2.3911 2.0916 1.8234 1.5707 1.3181 1.0499 0.7505

%E 1.1471 0.1289 3.83x10-3 2.3x10-4 6.05x10-3 0.2644 3.8400

CONSTRUIMOS FUTURO

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

f(x)

P3(x)

%E

GRÁFICA SERIE DE TAYLOR

CONSTRUIMOS FUTURO

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BIBLIOGRAPHY

•http://en.wikipedia.org/wiki/Taylor_series

•CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2002