Sergei Yakovenko's blog: on Math and Teaching

Saturday, December 19, 2015

Lecture 8, Dec 15

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.

Advertisements

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: