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Concept
Taylor's theorem
Summary
In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.
Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory.
Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the exponential function and trigonometric functions.
It is the starting point of the study of analytic functions, and is fundamental in various areas of mathematics, as well as in numerical analysis and mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions.
If a real-valued function is differentiable at the point , then it has a linear approximation near this point. This means that there exists a function h1(x) such that
Here
is the linear approximation of for x near the point a, whose graph is the tangent line to the graph at x = a. The error in the approximation is:
As x tends to a, this error goes to zero much faster than , making a useful approximation.
For a better approximation to , we can fit a quadratic polynomial instead of a linear function:
Instead of just matching one derivative of at , this polynomial has the same first and second derivatives, as is evident upon differentiation.
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