a. Fit the linear function y = mx to the power function $y = ax^n$ over the range 0 ≤ x ≤ L. The values of a and n are given.

b. Apply the result to the Aerobee drag function D=0.00056v² over the range 0≤v≤1000, discussed in Example 1.3.4.

Question

a. Fit the linear function y = mx to the power function [latex]y = ax^n[/latex] over the range 0 ≤ x ≤ L. The values of a and n are given.

b. Apply the result to the Aerobee drag function D=0.00056v² over the range 0≤v≤1000, discussed in Example 1.3.4.

Accepted Answer

a. The appropriate least-squares criterion is the integral of the square of the difference between the linear model and the power function over the stated range. Thus,

[latex]J=\int_0^L\left(m x-a x^n\right)^2 d x[/latex]

To obtain the value of m that minimizes J, we must solve 𝜕J∕𝜕m = 0.

[latex]\frac{\partial J}{\partial m}=2 \int_0^L x\left(m x-a x^n\right) d x=0[/latex]

This gives

[latex]m=\frac{3 a}{n+2}L^{n-1}[/latex] (1)

b. For the Aerobee drag function D = 0.00056v², a = 0.00056, n = 2, and L = 1000. Thus,

[latex]m=\frac{3(0.00056)}{2+2}1000^{2-1}=0.42[/latex]

and the linear description is D = 0.42v, where D is in pounds and v is in ft /sec. This is the linear model that minimizes the integral of the squared error over 0 ≤ v ≤ 1000 ft /sec.

Question C.2.4: a. Fit the linear function y = mx to the power function y = ...

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