Taylor expansion

The differential approximates $f$ with a line; the Taylor polynomial does better, approximating it with a parabola, then a cubic, and so on. It is the generalisation of the “equivalent functions” seen with limits, with a remainder (Lagrange, Peano, Cauchy) that controls the error. The section closes with the fundamental Maclaurin expansions and the applications to computing limits.

• From the idea of the differential to Taylor
• Higher-order derivatives
• The Taylor polynomial
• The remainder — Lagrange, Peano, Cauchy
• Standard Maclaurin expansions
• Applications of Taylor — limits and infinitesimals