  A straightforward 3-point quadrature formula of closed type is derived that
improves on Simpson's rule. Just using the additional information of the
integrand's derivative at the two endpoints we show the error is sixth order in
grid spacing. Various error bounds for the quadrature formula are obtained to
quantify more precisely the errors. Applications in numerical integration are
given. With these error bounds, which are generally better than the usual Peano
bounds, the composite formulas can be applied to integrands with lower order
derivatives.
