Using a centered difference with h = 0.1, approximate f'(2) for f(x) = x^2.

Using a centered difference with h = 0.1, approximate f'(2) for f(x) = x^2.

• A. 3.95
• B. 4.05
• C. 4.1
• D. 4.0

Answer: (D)

The centered difference method approximates f'(a) by [f(a+h) − f(a−h)] / (2h). With a = 2 and h = 0.1, compute f(2.1) = 4.41 and f(1.9) = 3.61. The difference is 0.80, and 2h = 0.2, so the quotient is 0.80 / 0.2 = 4.0. For f(x) = x^2, the exact derivative is f'(x) = 2x, which at x = 2 is 4.0. The centered difference gives the exact value here because a quadratic has zero third derivative, making the centered difference error term vanish. Thus the approximation is 4.0.

The centered difference method approximates f'(a) by [f(a+h) − f(a−h)] / (2h). With a = 2 and h = 0.1, compute f(2.1) = 4.41 and f(1.9) = 3.61. The difference is 0.80, and 2h = 0.2, so the quotient is 0.80 / 0.2 = 4.0. For f(x) = x^2, the exact derivative is f'(x) = 2x, which at x = 2 is 4.0. The centered difference gives the exact value here because a quadratic has zero third derivative, making the centered difference error term vanish. Thus the approximation is 4.0.