OPTIMIZED FIRST-ORDER TAYLOR-LIKE FORMULAS AND GAUSS QUADRATURE ERRORS

In this article, we derive an optimal first-order Taylor-like formula. In a seminal paper [15], we introduced a new first-order Taylor-like formula that yields a reduced remainder compared to the classical Taylor’s formula. In this work, we relax the assumption of equally spaced points in our formula. Instead, we consider a sequence of unknown points and a sequence of unknown weights. We then solve an optimization problem to determine the optimal distribution of points and weights that minimizes the corresponding remainder. Numerical results are provided to illustrate our findings.