Averaging the forward-difference formula f ′(x) ≈ [ f (x + h) − f (x)]/h and the backward-difference formula f ′(x) ≈ [ f (x) − f (x − h)]/h, each with error term O(h), results in the central-difference formula f ′(x) ≈ [ f (x + h) − f (x − h)]/(2h) with error O(h2). Show why.
Hint: Determine at least the first term in the error series for each formula.
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