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    Taylor Series

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    The Taylor series expansion approximates analytical functions using polynomials that are tangent to the function at a selected point. It expresses a function as an infinite sum of terms involving its derivatives at that point. The Taylor series can be truncated to a finite number of terms, yielding a polynomial approximation with a truncation error dependent on omitted higher-order terms. For example, truncating the Taylor series of sin(t) about π/2 to the second and fifth orders provides increasingly accurate approximations with quantifiable errors.

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    Taylor Series

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    The Taylor series expansion approximates analytical functions using polynomials that are tangent to the function at a selected point. It expresses a function as an infinite sum of terms involving its derivatives at that point. The Taylor series can be truncated to a finite number of terms, yielding a polynomial approximation with a truncation error dependent on omitted higher-order terms. For example, truncating the Taylor series of sin(t) about π/2 to the second and fifth orders provides increasingly accurate approximations with quantifiable errors.

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    Mathcad for Taylor series

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    The Taylor series expansion approximates analytical functions using polynomials that are tangent to the function at a selected point. It expresses a function as an infinite sum of terms involving its derivatives at that point. The Taylor series can be truncated to a finite number of terms, yielding a polynomial approximation with a truncation error dependent on omitted higher-order terms. For example, truncating the Taylor series of sin(t) about π/2 to the second and fifth orders provides increasingly accurate approximations with quantifiable errors.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    169 views3 pages

    Taylor Series

    Uploaded by

    books_uk
    The Taylor series expansion approximates analytical functions using polynomials that are tangent to the function at a selected point. It expresses a function as an infinite sum of terms involving its derivatives at that point. The Taylor series can be truncated to a finite number of terms, yielding a polynomial approximation with a truncation error dependent on omitted higher-order terms. For example, truncating the Taylor series of sin(t) about π/2 to the second and fifth orders provides increasingly accurate approximations with quantifiable errors.

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    Download as PDF, TXT or read online from Scribd
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    The main use of Taylor series in engineering is to approximate analytical functions with polynomials that are tangent to the function at a chosen point. A Taylor series expansion expresses a function as an infinite sum of terms involving derivatives of the function evaluated at a nominal point.

    A Taylor series expansion is used to approximate an analytically 'complicated' function by a polynomial that is tangent to the function at a selected point of choice.

    A Taylor series expansion states that a function f(x) that is analytical within a region of interest can be expressed as the infinite sum from n=0 to infinity of (f^n(x0)/(n!))(x-x0)^n, where f^n(x0) is the nth derivative of f(x) evaluated at the nominal point x0.

    TAYLOR SERIES EXPANSION

    The main use of Taylor Series in engineering is to approximate an analytically "complicated" function by a polynomial that is tangent to the function in question at a selected point of choice.
    The Taylor Series Expansion states that a function that is analytical within a region of interest can be expressed as follows:

    n (x xo )n f ( x ) = f ( xo ) + f ( x) n n! n =1 x x = xo

    (1)

    This expansion is often called the " infinite " Taylor Series Expansion because of the summation over infinite terms. A function is analytical at a point xo when all its derivatives can be calculated at that point. The Taylor Series can be truncated by using a limited number of derivatives. The result will be an approximation of the original function f(x). For example the following truncated Taylor Series is of the fourth order because it uses only 4 derivative terms:

    n ( x x o )n f ( x) f ( xo ) + f ( x) n! n =1 x n x=x
    4
    o

    (2)

    The error by approximating (1) by (2) is called a TRUNCATION ERROR because (2) is obtained by truncating the infinite series from (1). In this case the truncation error is of order five because all the derivatives above and including the 5th order are ommited in the approximation. This truncation error is often expressed as follows:

    n ( x x o )n O (5) = f ( x) n! n =5 x n x=x

    (3)

    Where the nomenclature "O(5)" implies that the fifth and higher derivative terms are ommited. In general, if an approximation of f(x) uses only N-1 derivate terms from the original Taylor series then the truncation error will be

    n ( x x o )n O( N ) = f ( x) n! n= N x n x= x

    (4)

    Example : Consider the following function.

    f ( t) := sin( t)

    This function is clearly analytical for all "t" because all the derivatives of the sine wave can be readily calculated. The 2nd order (truncated) Taylor Series approximation about the nominal point can be obtained from the following mathcad comand: to = 2

    series , t = f2 ( t) := f ( t) float , 3

    ,3 1. .500 ( t 1.57)

    2.

    Note how the series command works: series , t =

    part set the nominal 2 2 point of the expansion and the 3 that follows sets the order of the truncation error. The float,3 command limits to three the number of decimal places in the result.

    , 3 the t =

    Similarly, the 5th order approximation about the nominal point to = is: 2

    series , t = f5 ( t) := f ( t) float , 3

    ,6 1. .500 ( t 1.57)

    2.

    + 4.17 10 ( t 1.57)

    -2

    4.

    The plots below show how good the approximations are. Remember that the nominal point for the expansion is t = = 1.571 so the functions match at this point. 2

    f ( t) f2( t) f5( t) f

    1 0 1 2 3 t, t, t, 2 4 5 6

    The truncation error of f5 ( t) at time t=5 is equal to the difference between the f(t) (red curve) and f5 ( t) (green curve) at t=5. This error is shown in black in the plot below.

    1 f ( t) f5( t)

    f ( 5) f ( 5) 5

    1 0 1 2 3 t, t, 4 5 6

    5 5

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