Higher derivatives and better approximation
We discussed a few issues:
- Lagrange interpolation formula: how to estimate the difference through the derivative ?
- 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 -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.