  For most purposes, one can replace the use of Rolle's theorem and the mean
value theorem, which are not constructively valid, by the law of bounded
change. The proof of two basic results in numerical analysis, the error term
for Lagrange interpolation and Simpson's rule, however seem to require the full
strength of the classical Rolle's Theorem. The goal of this note is to justify
these two results constructively, using ideas going back to Amp\`ere and
Genocchi.
