Quadrature rules with (not too many) derivatives

Abstract

Quadrature formulas for ${\int }_a^{b}f(x)dx$ where derivative terms need only be evaluated at $a$ and $b$ in the composite rule are identified. Error bounds are given when $f:[a,b]\to \mathbb{R}$ satisfies $f^{(n-1)}$ is absolutely continuous so that $f^{(n)}\in L^p([a,b])$, and when $f^{(n-1)}$ is merely continuous.