Exercise · ZW-FINITE-DIFFERENCES-FOR-DERIVATIVES-008 · ★★★ Advanced

Expanding \(f(x+h)=f(x)+hf'(x)+\frac{h^2}{2}f''(x)+\cdots\), what is the leading term of the error in \(\frac{f(x+h)-f(x)}{h}-f'(x)\)?

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  • C. \(\frac{h}{2}f''(x)\)

Explanation

Dividing the expansion by \(h\) gives \(\frac{f(x+h)-f(x)}{h}=f'(x)+\frac{h}{2}f''(x)+\cdots\), so the leading error is \(\frac{h}{2}f''(x)\).

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Review the concept: Finite differences for derivatives

Finite differences are approximations of the derivatives of a function \(f\) from its values at discrete points separated by a step \(h\). They turn differential problems into algebraic systems that a computer can solve.

\[f'(x)\approx \frac{f(x+h)-f(x)}{h}\]

More practice problems: finite differences for derivatives

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