Question C.1.4: a. Fit the linear function y = mx to the power function y = ......
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.
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,
J=\int_0^L\left(m x-a x^n\right)^2 d xTo obtain the value of m that minimizes J , we must solve ∂ J/∂m = 0.
\frac{\partial J}{\partial m}=2 \int_0^L x\left(m x-a x^n\right) d x=0This gives
m=\frac{3 a}{n+2} L^{n-1} (1)
b. For the Aerobee drag function D = 0.00056v², a = 0.00056, n = 2, and L = 1000. Thus,
m=\frac{3(0.00056)}{2+2} 1000^{2-1}=0.42and 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.