Question

Undetermined Coefficients

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Answer 1

\[\large \text{Problem:}\] Derive the two point formula \[f'(x)\approx \frac{f(x+h)-f(x-h)}{2h}\] using the method of UC.

\[\large \text{Solution: }\]
\[f'(0)=\frac{1}{2h}[f(h)-f(-h)]\]
Assume we want a solution of the form \[f'(0)\approx Af(h)+Bf(-h)+Cf(0)\]
Choose base functions {1,x,x^2}
Then follows:\[f(x)=1: \text{ }0=A+B+C \\f(x)=x: \text{ }1=(h)A+(-h)B+(0)C \\f(x)=x^2: \text{ }0=(h)^2A+(-h)^2B+(0)C\]
Solving simultaneously, A=h/2, B=-h/2, C=0, giving the required equation.




\[\large \large \text{Here comes my questions:}\]
1. Why do we choose x=0 at the beginning?
2. Where does the 1 come frome in the second substitution?

Answer 2

@.Sam. maybe you can shed some light on this?

Answer 3

1.) Because for most whatever f is, the compare/contrast of f(h) with f(-h) will be illustrative.

2.) because if f(x) = x then f'(x) = 1 (no matter what x is. In part 3, f'(x) = 2x, but f'(0) is 2*0 = 0

Answer 4

And shouldn't it be A=1/2h, B = -1/2h, C=0?