A New First Order Taylor-like Theorem With An Optimized Reduced Remainder

This paper is devoted to a new first order Taylor-like formula where the corresponding remainder is strongly reduced in comparison with the usual one which which appears in the classical Taylor's formula. To derive this new formula, we introduce a linear combination of the first derivatives of the concerned function which are computed at $n+1$ equally spaced points between the two points where the function has to be evaluated. Therefore, we show that an optimal choice of the weights of the linear combination leads to minimize the corresponding remainder. Then, we analyze the Lagrange $P_1$- interpolation error estimate and also the trapezoidal quadrature error to assess the gain of accuracy we get due to this new Taylor-like formula.