# Higher derivatives and better approximation

We discussed a few issues:

• Lagrange interpolation formula: how to estimate the difference $f(b)-f(a)$ through the derivative $f'$?
• Consequence: vanishing of several derivatives at a point means that a function has a “root of high order” at this point (with explanation, what does that mean).
• Taylor formula for polynomials: if you know all derivatives of a polynomial at some point, then you know it everywhere.
• Peano formula for $C^n$-smooth functions: approximation by the Taylor polynomial with asymptotic bound for the error.
• Lagrange formula: explicit estimate for the error.

The notes (updated) are available here.