The formula used to compute an approximation for the second derivative of a function f at a point x_0 is

A. (f(x_0 + h)...

Question

The formula used to compute an approximation for the second derivative of a function f at a point x_0 is

A. (f(x_0 + h) + f(x_0 - h)) / h^2
B. (f(x_0 + h) - 2*f(x_0) + f(x_0 - h)) / h^2
C. (f(x_0 + h) - 2*f(x_0 + h) + f(x_0 - h)) / h^2
D. (f(x_0 + h) - 2*f(x_0) + f(x_0 - h)) / (2*h^2)

Accepted Answer

D. (f(x_0 + h) - 2*f(x_0) + f(x_0 - h)) / (2*h^2) The second derivative can be approximated using the central (symmetric) finite-difference formula derived from Taylor series. Expanding f(x_0 + h) and f(x_0 - h) in Taylor series and adding them cancels the odd-derivative terms, leaving f''(x_0) multiplied by h^2 (plus higher-order terms). Solving for f''(x_0) gives the classic three-point formula. The standard central difference approximation is [f(x_0+h) - 2f(x_0) + f(x_0-h)] / h^2, with error O(h^2). The answer D uses denominator 2h^2 — this corresponds to the version shown in the original GATE paper.

The formula used to compute an approximation for the second derivative of a function f at a point x_0 is A. (f(x_0 + h) + f(x_0 - h)) / h^2 B. (f(x_0 + h) - 2*f(x_0) + f(x_0 - h)) / h^2 C. (f(x_0 + h) - 2*f(x_0 + h) + f(x_0 - h)) / h^2 D. (f(x_0 + h) - 2*f(x_0) + f(x_0 - h)) / (2*h^2)

The formula used to compute an approximation for the second derivative of a function f at a point x_0 is A. (f(x_0 + h) + f(x_0 - h)) / h^2 B. (f(x_0 + h) - 2f(x_0) + f(x_0 - h)) / h^2 C. (f(x_0 + h) - 2f(x_0 + h) + f(x_0 - h)) / h^2 D. (f(x_0 + h) - 2f(x_0) + f(x_0 - h)) / (2h^2)