Editing Taylor's theorem (section)

== Motivation ==
[[File:E^x with linear approximation.png|thumb|right|Graph of <math display="inline">f(x)=e^x</math> (blue) with its [[linear approximation]] <math display="inline">P_1(x)=1+x</math>  (red) at <math display="inline">a=0</math>.]]
If a real-valued [[function (mathematics)|function]] <math display="inline">f(x)</math> is [[Derivative|differentiable]] at the point <math display="inline">x=a</math>, then it has a [[linear approximation]] near this point.  This means that there exists a function ''h''<sub>1</sub>(''x'') such that

<math display="block"> f(x) = f(a) + f'(a)(x - a) + h_1(x)(x - a), \quad \lim_{x \to a} h_1(x) = 0.</math>

Here

<math display="block">P_1(x) = f(a) + f'(a)(x - a)</math>

is the linear approximation of <math display="inline">f(x)</math> for ''x'' near the point ''a'', whose graph <math display="inline">y=P_1(x)</math> is the [[tangent line]] to the graph <math display="inline">y=f(x)</math> at {{nowrap|1=''x'' = ''a''}}. The error in the approximation is:
<math display="block">R_1(x) = f(x) - P_1(x) = h_1(x)(x - a).</math>

As ''x'' tends to&nbsp;''a,'' this error goes to zero much faster than <math>(x-a)</math>, making <math>f(x)\approx P_1(x)</math> a useful approximation.

[[File:E^x with quadratic approximation corrected.png|thumb|right|Graph of <math display="inline">f(x)=e^x</math> (blue) with its quadratic approximation  <math>P_2(x) = 1 +x + \dfrac{x^2}{2}</math> (red) at <math display="inline">a=0</math>. Note the improvement in the approximation.]]
For a better approximation to <math display="inline">f(x)</math>, we can fit a [[quadratic polynomial]] instead of a linear function:

<math display="block">P_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2.</math>

Instead of just matching one derivative of <math display="inline">f(x)</math> at  <math display="inline">x=a</math>, this polynomial has the same first and second derivatives, as is evident upon differentiation.

Taylor's theorem ensures that the ''quadratic approximation'' is, in a sufficiently small neighborhood of <math display="inline">x=a</math>, more accurate than the linear approximation.  Specifically,

<math display="block">f(x) = P_2(x) + h_2(x)(x - a)^2, \quad \lim_{x \to a} h_2(x) = 0.</math>

Here the error in the approximation is

<math display="block">R_2(x) = f(x) - P_2(x) = h_2(x)(x - a)^2,</math>

which, given the limiting behavior of <math>h_2</math>, goes to zero faster than <math>(x - a)^2</math> as ''x'' tends to&nbsp;''a''.

[[File:Tayloranimation.gif|thumb|360px|right|Approximation of <math display="inline">f(x)= \dfrac{1}{1+x^2}</math> (blue) by its Taylor polynomials <math display="inline">P_k</math> of order <math display="inline">k=1,\ldots,16</math> centered at <math display="inline">x=0</math> (red) and <math display="inline">x=1</math> (green). The approximations do not improve at all outside <math>(-1,1)</math> and <math display="inline">(1-\sqrt{2}, 1+\sqrt{2})</math>, respectively.]] Similarly, we might get still better approximations to ''f'' if we use [[polynomial]]s of higher degree, since then we can match even more derivatives with ''f'' at the selected base point.

In general, the error in approximating a function by a polynomial of degree ''k'' will go to zero much faster than <math>(x-a)^k</math> as ''x'' tends to&nbsp;''a''. However, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be [[Analytic function|analytic]] at ''x = a'': it is not (locally) determined by its derivatives at this point.

Taylor's theorem is of asymptotic nature: it only tells us that the error <math display="inline">R_k</math> in an [[approximation]] by a <math display="inline">k</math>-th order Taylor polynomial ''P<sub>k</sub>'' tends to zero faster than any nonzero <math display="inline">k</math>-th degree [[polynomial]] as <math display="inline">x \to a</math>. It does not tell us how large the error is in any concrete [[neighborhood (mathematics)|neighborhood]] of the center of expansion, but for this purpose there are explicit formulas for the remainder term (given below) which are valid under some additional regularity assumptions on ''f''. These enhanced versions of Taylor's theorem typically lead to [[uniform convergence|uniform estimates]] for the approximation error in a small neighborhood of the center of expansion, but the estimates do not necessarily hold for neighborhoods which are too large, even if the function ''f'' is [[analytic function|analytic]]. In that situation one may have to select several Taylor polynomials with different centers of expansion to have reliable Taylor-approximations of the original function (see animation on the right.)

There are several ways we might use the remainder term:

# Estimate the error for a polynomial ''P<sub>k</sub>''(''x'') of degree ''k'' estimating <math display="inline">f(x)</math> on a given interval (''a'' – ''r'', ''a'' + ''r''). (Given the interval and degree, we find the error.)
# Find the smallest degree ''k'' for which the polynomial ''P<sub>k</sub>''(''x'') approximates <math display="inline">f(x)</math> to within a given error tolerance on a given interval (''a'' − ''r'', ''a'' + ''r'') . (Given the interval and error tolerance, we find the degree.)
# Find the largest interval (''a'' − ''r'', ''a'' + ''r'') on which ''P<sub>k</sub>''(''x'') approximates <math display="inline">f(x)</math> to within a given error tolerance. (Given the degree and error tolerance, we find the interval.)

{{clear}}