Editing Differential calculus (section)

=== Taylor polynomials and Taylor series ===
{{Main|Taylor polynomial|Taylor series}}

The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued function {{math|''f''(''x'')}} at the point {{math|''x''<sub>0</sub>}} is a linear [[polynomial]] {{math|''a'' + ''b''(''x'' − ''x''<sub>0</sub>)}}, and it may be possible to get a better approximation by considering a quadratic polynomial {{math|''a'' + ''b''(''x'' − ''x''<sub>0</sub>) + ''c''(''x'' − ''x''<sub>0</sub>)<sup>2</sup>}}. Still better might be a cubic polynomial {{math|''a'' + ''b''(''x'' − ''x''<sub>0</sub>) + ''c''(''x'' − ''x''<sub>0</sub>)<sup>2</sup> + ''d''(''x'' − ''x''<sub>0</sub>)<sup>3</sup>}}, and this idea can be extended to arbitrarily high degree polynomials. For each one of these polynomials, there should be a best possible choice of coefficients {{math|''a''}}, {{math|''b''}}, {{math|''c''}}, and {{math|''d''}} that makes the approximation as good as possible.

In the [[Neighbourhood (mathematics)|neighbourhood]] of {{math|''x''<sub>0</sub>}}, for {{math|''a''}} the best possible choice is always {{math|''f''(''x''<sub>0</sub>)}}, and for {{math|''b''}} the best possible choice is always {{math|''f<nowiki>'</nowiki>''(''x''<sub>0</sub>)}}. For {{math|''c''}}, {{math|''d''}}, and higher-degree coefficients, these coefficients are determined by higher derivatives of {{math|''f''}}. {{math|''c''}} should always be {{math|{{sfrac|''f<nowiki>''</nowiki>''(''x''<sub>0</sub>)|2}}}}, and {{math|''d''}} should always be {{math|{{sfrac|''f<nowiki>'''</nowiki>''(''x''<sub>0</sub>)|3!}}}}. Using these coefficients gives the '''Taylor polynomial''' of {{math|''f''}}. The Taylor polynomial of degree {{math|''d''}} is the polynomial of degree {{math|''d''}} which best approximates {{math|''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 {{math|''f''}} is a polynomial of degree less than or equal to {{math|''d''}}, then the Taylor polynomial of degree {{math|''d''}} equals {{math|''f''}}.

The limit of the Taylor polynomials is an infinite series called the '''Taylor series'''. The Taylor series is frequently a very good approximation to the original function. Functions which are equal to their Taylor series are called [[analytic function]]s. It is impossible for functions with discontinuities or sharp corners to be analytic; moreover, there exist [[smooth function]]s which are also not analytic.