Simpson's theorem

Simpson's theorem states that if a function is continuous and twice differentiable on an interval, then the change in the average rate of change over the interval relates directly to the function's second derivative. In simple terms, it connects the overall change in a function's slope to how the curvature (or concavity) of the function behaves. This theorem underpins methods in numerical analysis and approximation, such as Simpson's rule for estimating integrals, showing how the way a function bends influences the accuracy of approximations of its total change.