Lesson 9 - Taylor's Theorem v3 (Notes Template).pdf -...

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1 | P a g e Taylor’s Theorem While it is truly elegant that Taylor series exactly converge to functions (on their interval of convergence), it is the inexact Taylor polynomials that essentially do all of the work. Let’s ponder this for a minute… While sin x can be found exactly by summing an infinite Taylor series, if we want to use that information to find sin 3 , we will have to evaluate Taylor polynomials until we arrive at an approximation with which we are satisfied. Example 1: Approximating a Function to Specifications Find a Taylor polynomial that will serve as a substitute for sin x on the interval ,   such that the error between the Taylor polynomial approximation and sin x is less than 0.0001 for all x in ,   .
2 | P a g e Goal: Use Taylor polynomials to approximate functions over the interval of convergence of the Taylor series while keeping the error of the approximation within specified bounds. Let’s examine where this error comes from… Example 2: Truncation Error for a Geometric Series Find a formula for the truncation error if we use 2 4 6 1 x x x to approximate 2 1 1 x over the interval 1,1 .
3 | P a g e The Remainder Every truncation splits a Taylor series into two equally significant pieces: The Taylor polynomial n P x that gives us the approximation, and The remainder n R x that tells us whether the approximation is any good Taylor’s Theorem is about both pieces. Taylor’s Theorem with Remainder If f has derivatives of all orders in an open interval I containing a , then for each positive integer n and for each x in I ,           2 ... 2! ! n n n f a f a f x f a f a x a x a x a R x n  where   1 1 1 ! n n n f c R x x a n