Page "Differential calculus" Paragraph 46

In the neighbourhood of x < sub > 0 </ sub >, for a the best possible choice is always f ( x < sub > 0 </ sub >), and for b the best possible choice is always f < nowiki >'</ nowiki >( x < sub > 0 </ sub >).

For c, d, and higher-degree coefficients, these coefficients are determined by higher derivatives of f. c should always be f < nowiki ></ nowiki >( x < sub > 0 </ sub >)/ 2, and d should always be f < nowiki ></ nowiki >( x < sub > 0 </ sub >)/ 3 !.

Using these coefficients gives the Taylor polynomial of f. The Taylor polynomial of degree d is the polynomial of degree d which best approximates f, and its coefficients can be found by a generalization of the above formulas.

Taylor's theorem gives a precise bound on how good the approximation is.

If f is a polynomial of degree less than or equal to d, then the Taylor polynomial of degree d equals f.