Taylor and MacLaurin Series Lesson 8.8
Taylor & Maclaurin Polynomials Consider a function f(x) that can be differentiated n times on some interval I Our goal: find a polynomial function M(x) which approximates f at a number c in its domain Initial requirements M(c) = f(c) M '(c) = f '(c) Centered at c or expanded about c
Example Let f(x) = e x Then a function with same value M 1 (x) = 1 + x With same value and slope at x = 0 M 2 (x) = 1 + x x 2 We determined by noting Why?
Improved Approximating Now we know f "(x) = e x and f "(0) = 1 which are same values we want for M 2 (x) So M 2 (x) = 1 + x x 2
Improved Approximating We can keep going with further derivatives For f(x) = e x at x = 0 The n th -degree Maclaurin polynomial for f is
Improved Approximating We can choose some other value for x, say x = c Then for f(x) = e x the n th degree Taylor polynomial at x = c
Power Series Representation Let's not stop at the n th term This chapter is about infinite series Recall the power series Must answer two questions 1. Existance: when does f(x) have a power series representation 2. Uniqueness: is there exactly 1 such power series?
Power Series Representation Given a differentiable function f(x) represented by power series for –R < x – c < R Then there is exactly one such representation The coefficients of a k are of the form
Try It Out Find the first several terms of the Maclaurin series for the function. (Hint: use half angle identity)
Taylor Remainder Function Conditions where given function f(x) has a power series representation Remainder function We claim f(x) is represented by its Taylor series in interval I when
Taylor's Function Requirements f and all its derivatives exist in open interval I x = c is in I Then f(x) represented by And for some z n < |x – c|
Another Trial Find the first several terms of the Taylor series for
Power Series Table Note power series for elementary functions Pg 563 Partial listing Exponential Sine, Cosine Geometric Logarithmic
Assignment Lesson 8.8 Page 564 Exercises 3 – 43 EOO