In calculus, Taylor's theorem gives an approximation of a ${\textstyle k}$-times differentiable function around a given point by a polynomial of degree ${\textstyle k}$, called the ${\textstyle k}$-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ${\textstyle k}$ 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 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. It provided the mathematical basis for some landmark early computing machines: Charles Babbage's difference engine calculated sines, cosines, logarithms, and other transcendental functions by numerically integrating the first 7 terms of their Taylor series.

If a real-valued function ${\textstyle f(x)}$ is differentiable at the point ${\textstyle x=a}$, then it has a linear approximation near this point. This means that there exists a function h₁(x) such that

$${\displaystyle f(x)=f(a)+f'(a)(x-a)+h_{1}(x)(x-a),\quad \lim _{x\to a}h_{1}(x)=0.}$$

Here

$${\displaystyle P_{1}(x)=f(a)+f'(a)(x-a)}$$

is the linear approximation of ${\textstyle f(x)}$ for x near the point a, whose graph ${\textstyle y=P_{1}(x)}$ is the tangent line to the graph ${\textstyle y=f(x)}$ at x = a. The error in the approximation is: $${\displaystyle R_{1}(x)=f(x)-P_{1}(x)=h_{1}(x)(x-a).}$$