The formula for a degree- Taylor polynomial of about is In the formula above, denotes the order- derivative of .
Derivation of the formula
We want the Taylor polynomial to have -th derivative at . The Power Rule for derivatives gives that the derivative of is for all positive integers , and for (because when the function is a constant ). Here the Chain Rule is used implicitly with the fact that has derivative for all .
For , the degree- term in has th derivative , because after differentiations the degree of the term will have reached and then at least one more differentiation ensures that the term is eliminated.
For , the degree- term in has th derivative at , because the differentiations leave a term with a positive power of , which is zero at .
The degree- term undergoes differentiations, leaving a constant term and accumulating all of the factors for . As such, its th derivative is times its original coefficient for all , so the coefficient of should be defined as .
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.
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.
Letting be the degree- Taylor polynomial of about , the Lagrange Error Bound states that if is defined and has absolute value at most on the entire interval if or if .
The Lagrange Error Bound bounds the true value of both above and below.
The Maclaurin series is the Taylor series chosen with . The partial sums of the Maclaurin series are the Maclaurin polynomials of of each degree.