Calculus II: Dr. Staples Section 9.1 Approximating Functions with Polynomials

Definition: Taylor Polynomials centered at 𝒙 = 𝒂 Let f be a function with derivatives, 𝑓!, 𝑓!! , 𝑓!!!, …𝑓! defined at a. Then the nth-order Taylor polynomial centered at 𝑥 = 𝑎 is given by 𝑝! 𝑥 = 𝑓 𝑎 + 𝑓!(𝑎)(𝑥 − 𝑎)+ !

!!(!)!!

 (𝑥 − 𝑎)! + !!!!(!)!!

(𝑥 − 𝑎)! +⋯ !! !!!

(𝑥 − 𝑎)! Note that you only need to compute the coefficients to determine the Taylor Polynomial. Observe that coefficient 𝑐! of (𝑥 − 𝑎)! follows the rule 𝑐! =

!! !!!  and  𝑝! 𝑥 =   𝑐!!

!!! (𝑥 − 𝑎)! The nth-order Taylor polynomial centered at 𝒙 = 𝒂 matches f in value, slope, and all derivatives at a. Thus it serves as a very useful approximation function. In fact, the error term or so-called remainder that arises from approximating a function with a Taylor Polynomial is well understood. Definition: Let  𝑝! 𝑥  be the Taylor polynomial of order n for f. The remainder (i.e. error) in using  𝑝! 𝑥 to approximate f at the point x is

𝑅! 𝑥 = 𝑓 𝑥 −    𝑝! 𝑥 . 

Estimating the Remainder Let n be a fixed positive integer. Suppose that there exists a number M such that the (n+1)st derivative satisfies | 𝑓!!! 𝑐 | ≤ 𝑀 for all c in [a,x]. Then the remainder in the nth-order Taylor polynomial centered at 𝑥 = 𝑎, satisfies

|𝑅! 𝑥 | = |𝑓 𝑥 −    𝑝! 𝑥 | ≤ 𝑀|𝑥 − 𝑎|!!!

𝑛 + 1 !

If you work with numerical algorithms or computer programs, it is important to be able to estimate the accuracy of your approximation.