Second Derivative Approximations

Numerical Analysis · Axiom Academy

LESSON Second Derivative Approximations Numerical formulas for curvature and acceleration The second derivative tells us about the curvature of a function: f'' > 0: Curve is concave up (bowl-shaped ∪) f'' Curve is concave down (dome-shaped ∩) f'' = 0: Inflection point (curvature changes) 2 The Central Difference Formula The most common formula for approximating f''(x): This uses three points: one on each side of x, plus x itself. It has O(h²) accuracy. The coefficients [1, -2, 1]/h² can be visualized as a stencil : Interpretation: f''(x) ≈ [f(x+h) - 2f(x) + f(x-h)] / h². This measures how much the middle value differs from the average of its neighbors. For better accuracy, use a five-point formula: This achieves O(h⁴) accuracy by using five points instead of three. Second derivatives appear throughout science and engineering:

This is the written version of the interactive lesson above. See the full Numerical Analysis course.