TAYLOR SERIES EXPANSION


Determines the value of a function f(x) at any x from the value of the function and all its derivatives at a given location x o (provided no discontinuities occur).

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Taylor series expansion ” evolves into a power series

  1. Series about x

  2. Series about origin x = 0

Observations:

  1. An arbitrary function f(x) can be expressed as a power series: a n =

  2. Coefficients of power series are related to the derivative of the function evaluated at origin.

  3. A linear function consists of only the first two terms: f = a + a 1 x

To establish linear relationship about ambient state:

  • Stress Strain constitutive relation in elasticity

  • Pressure Density equation of state

  • Voltage Current about quiescent point

  • Input Output

Linear implies: "input" disturbance (x - x ) small enough that "output"

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Output is a linear function of input:

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Recombine: changes in independent variable(input).

Provide linear changes in the dependent variable(output).

Slope m serves to adjust units and is called "sensitivity."

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Exponential function e ax ” Taylor series about x = x in interval - ˆ < x < ˆ

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Factoring out the common exponential term :

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MacLaurin series about x = 0 in interval - ˆ < x < ˆ

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Normal density-like function: ”Taylor series about x = 0

Standard Normal Distribution:

 

Exact

vs. Two

Three

Four Terms

z = ±0.5

0.3521

0.3490

0.3522

0.3519

z =±0.675 (Q 1,3 )

0.3177

0.3080

0.3184

0.3178

z = ±1.0

0.2420

0.1995

0.2494

-0.2427

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Derivatives of exponential about origin x = 0

Zero:

First:

Second:

Third:

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Fourth:

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Fifth:

Sixth:

Sine function sin x ” Taylor series about x = x in interval - ˆ < x < ˆ

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MacLaurin series about x = 0 in interval - ˆ < x < ˆ

Partial Derivatives

Dependent variable has two or more independent variables

f(x, y)

Differentiate wrt to only one independent variable while holding the other variable constant e.g., y = y o

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Taylor Series in Two-Dimensions

Taylor series of f(x, y) about point (x o , y o ):

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Linear terms:

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Taylor Series of Random Variable (RV) Functions

Arbitrary function of two random variables X 1 and X 2

Y (X 1 , X 2 )

Mean:

Variance and Covariance

Consider only linear terms of the Taylor series expansion about the mean of each random variable, ¼ Y = Y

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Variation of function about its mean:

Variance and covariance:

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Note 

If X 1 and X 2 are independent RV, covariance ƒ X 1 X 2 = 0.

Functions of Random Variables

Sum or difference: Y = a 1 X 1 ± a 2 X 2

Mean: ¼ Y = a 1 ¼ X 1 ± a 2 ¼ X 2

Variance and covariance:

Again, if X 1 and X 2 are independent RVs then the covariance is zero.

Product: Y = a o X 1 X 2

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Mean: ¼ Y = a ¼ X 1 ¼ X 2

Variance and covariance:

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Division of Random Variables

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Mean: ¼ Y =

Variance and covariance:

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or normalizing by the square of the mean of the quotient ¼ Y .

Again, if X 1 and X 2 are independent RVs, then the covariance is zero.

Powers of a Random Variable

Single RV: X 1

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Mean: ¼ Y = a ¼ X 1 ¼ X 2

Variance:

or normalizing by the square of the means

Exponential of a Random Variable

Single RV X 1 :

where units of the RV X 1 are those of 1/b, and units of the RV Y are the same as those of a o .

Mean: ¼ Y = ±a o

Variance:

or normalizing by the square of the means

Consider a constant raised to RV power:

then

Variance:

Constant Raised to RV Power

Single RV X 1 :

where units of the RV X 1 are those of 1/b and units of the RV Y are the same as those of c

then

Mean:

Variance:

Logarithm of Random Variable

Single RV X 1 : Y = a o In ( bX 1 )

where units of the RV X 1 are those of 1/b and units of the RV Y are the same as those of a o

then

Mean: ¼ Y = a o ln

Variance:

Example: Horizontal Beam Deflection

Deflection of the center of the beam of length L [m] under uniform loading W [N/m] is deterministically given by:

Y = = a o WL 3

where E = elastic modulus of the beam material [N/m] and I = moment of inertia of beam cross section about its center of area [m 4 ].

Load and length can be considered r.v. with mean and ± one standard deviation is given as:

W = ¼ W ± 1 ƒ W = 4000 N ± 40 N

L = ¼ L ± 1 ƒ L = 20 m ± 0.2 m

Find: The fractional standard deviation of the deflection Y

Mean deflection:

¼ Y = ±a o ¼ W ¼ L 3

Variance of deflection:

Fractional variance of deflection of beam: divide by

For the case given the fractional standard deviations of the two variables are equal:

Numerical value for the fractional variance of the deflection:

Numerical value for the fractional standard deviation:

= 0.032

Observations:

  1. Although W and L have the same fractional standard deviation (0.01), the length ” because it is a third power term in the deflection ” is seen to have more significance on the standard deviation of the deflection.

  2. The fractional standard deviation of the deflection Y is considerably larger than those of either the weight W or length L.

Example: Difference between Two Means

Examples:

  1. Clearance

  2. Before and after comparison (e.g., treated vs. untreated)

  3. Comparison of two suppliers

Mean: ¼ Y = ¼ X 1 - ¼ X 2 or

Variance (assume independent so covariance is zero):

Standardized form of sample difference

t-distribution form of sample difference:

Introduce "effective sample variance"

then




Six Sigma and Beyond. Design for Six Sigma (Vol. 6)
Six Sigma and Beyond: Design for Six Sigma, Volume VI
ISBN: 1574443151
EAN: 2147483647
Year: 2003
Pages: 235

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