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).

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"

Output is a linear function of input:

Recombine: changes in independent variable(input).

Provide linear changes in the dependent variable(output).

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

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

Factoring out the common exponential term :

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

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

Derivatives of exponential about origin x = 0

Zero:

First:

Second:

Third:

Fourth:

Fifth:

Sixth:

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

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

Taylor Series in Two-Dimensions

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

Linear terms:

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

Variation of function about its mean:

Variance and covariance:

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

Mean: ¼ _Y = a ¼ _{X 1} ¼ _{X 2}

Variance and covariance:

Division of Random Variables

Mean: ¼ _Y =

Variance and covariance:

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

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