1. Taylor and MacLaurin Series Lesson 8.8

2. 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

3. 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 + 0.5 x 2 We determined by noting Why?

4. 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 + 0.5 x 2

5. 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

6. 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

7. 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?

8. 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

9. Try It Out Find the first several terms of the Maclaurin series for the function. (Hint: use half angle identity)

10. 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

11. 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|

12. Another Trial Find the first several terms of the Taylor series for

13. Power Series Table Note power series for elementary functions  Pg 563 Partial listing  Exponential  Sine, Cosine  Geometric  Logarithmic

14. Assignment Lesson 8.8 Page 564 Exercises 3 – 43 EOO