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

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

• A. 4.0
• B. 4.2
• C. 3.9
• D. 4.1

Answer: (D)

Forward difference estimates the derivative at a point by the slope of the secant over a small step: f'(x) ≈ [f(x+h) − f(x)]/h. For f(x) = x^2 at x = 2 and h = 0.1, we have f(2) = 4 and f(2.1) = 4.41. The slope is (4.41 − 4)/0.1 = 0.41/0.1 = 4.1. This is the forward-difference approximation of f′(2). The exact derivative is f′(x) = 2x, which gives 4 at x = 2, so 4.1 is a close approximation with small error due to using a finite h.

Forward difference estimates the derivative at a point by the slope of the secant over a small step: f'(x) ≈ [f(x+h) − f(x)]/h. For f(x) = x^2 at x = 2 and h = 0.1, we have f(2) = 4 and f(2.1) = 4.41. The slope is (4.41 − 4)/0.1 = 0.41/0.1 = 4.1. This is the forward-difference approximation of f′(2). The exact derivative is f′(x) = 2x, which gives 4 at x = 2, so 4.1 is a close approximation with small error due to using a finite h.