  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 appears in the classical Taylor's formula. To derive this new formula, we
introduce a linear combination of the first derivative of the concerned
function, which is computed at n+1 equally-spaced points between the two points
where the function has to be evaluated. We show that an optimal choice of the
weights in the linear combination leads to minimizing the corresponding
remainder. Then, we analyze the Lagrange P1- interpolation error estimate and
also the trapezoidal quadrature error, in order to assess the gain of accuracy
we obtain using this new Taylor-like formula.
