Taylor polynomial
The degree- Taylor polynomial of a function about is the unique polynomial of degree whose value and first derivatives match the value and first derivatives of at .
The formula for a degree- Taylor polynomial of about is In the formula above, denotes the order- derivative of .
Taylor polynomials are often used to approximate non-polynomial functions that cannot be calculated exactly, such as trigonometric functions, exponential functions, and logarithms.
Contents
Derivation of the formula
Special cases
Maclaurin polynomial
A Maclaurin polynomial is a Taylor series with . Setting simplifies the appearance of the polynomial somewhat, since every instance of in the formula is replaced with .
For some functions, like and , Maclaurin polynomials are generally effective across the domain (although using a different -value might allow greater accuracy for the same choice of degree). However, for functions like , Maclaurin polynomials cannot be defined because the function and its derivatives are undefined at . For other functions, Maclaurin polynomials can be defined, but do not in general approximate the function well (see Taylor series), so a value of closer to the -value of the desired approximation must be chosen.
Tangent-line approximation
A tangent-line approximation is a first-degree Taylor polynomial, given by . The name "tangent-line approximation" comes from the fact that the graph is a line tangent to the graph of at . Tangent-line approximations are used in Euler's method and Newton's method.
Error bound
Taylor series
The Taylor series of an infinitely differentiable function is the infinite series The partial sums of the Taylor series are the Taylor polynomials of about of each degree.
The Taylor series is the Maclaurin series is the Taylor series chosen with . The partial sums of the Maclaurin series are the Maclaurin polynomials of of each degree.