From the idea of the differential to Taylor

The differential $df = f'(x_0)\,dx$ provides a linear approximation of $f$ near $x_0$: the tangent line. Can we do better? Yes: the idea is to approximate $f$ with a parabola, then with a cubic, and so on, obtaining an approximation that is better and better the higher the degree we go. The result is the Taylor polynomial.

Remark — Taylor as a "generalisation of equivalent functions"

In the chapter on limits we already used substitutions of the type ”$\sin x \sim x$ as $x\to 0$” or ”$1-\cos x \sim x^2/2$ as $x\to 0$”: these are the equivalent functions. The Taylor expansion is the generalisation of this idea: instead of a single equivalence to first or second order, we write a whole polynomial expression approximating $f$ to an arbitrary order, with a controlled remainder measuring the error made. Knowing Taylor is equivalent to having to hand all the equivalent functions in the book, and infinitely many more besides.

Links

Topics: Derivatives
Concepts: Differential · Equivalent functions · Taylor polynomial
People: Brook Taylor