1. Consider the following: Now, use the reciprocal function and tangent line to get an approximation. Lecture 31 – Approximating Functions 1 12 3

2. 2 2 2.01

3. First derivative gave us more information about the function (in particular, the direction). For values of x near a the linear approximation given by the tangent line should be better than the constant approximation. 3 Second derivative will give us more information (curvature). For values of x near a the quadratic approximation should be better than the linear approximation.

4. What quadratic is used as the approximation? 4 Key idea: Need to have quadratic match up with the function and its first and second derivatives at x = a.

5. 5 2 2.01 Use p 2 (x) to get a better approximation.

6. Graphical Example at x = 0 6 12 3

7. What higher degree polynomial is appropriate? 7 Key idea: Need to have n th degree polynomial match up with the function and all of its derivatives at x = a.

8. The coefficients, c k, for the n th degree Taylor polynomial approximating the function f(x) at x = a have the form: 8

9. Def: The Taylor polynomial of order n for function f at x = a: 9 The remainder term for using this polynomial: Lecture 32 – Taylor Polynomials for some c between x and a. where M provides a bound on how big the n+1 st derivative could possibly be.

10. 10 Estimate the maximum error in approximating the reciprocal function at x = 2 with an 8 th order Taylor polynomial on the interval [2, 3].

11. 11

12. 12 What is the actual maximum error in approximating the reciprocal function at x = 2 with an 8 th order Taylor polynomial on the interval [2, 3]?

13. What n th degree polynomial would you need in order to keep the error below.0001? 13

14. 14 To keep error below.0001, need to keep R n below.0001.

15. The Taylor series centered at x = a: 15 Lecture 33 – Taylor Series is a power series with The Taylor series centered at x = 0 is called a Maclaurin series:

16. Find the Maclaurin series for f (x) = sin x. 16 Example 1

17. Find the Maclaurin series for f (x) = e x. 17 Example 2

18. 18 For what values of x will the last two series converge? Ratio Test: Series converges for

19. 19 Consider the graphs:

20. 20 Example 3 Find the Maclaurin series for f (x) = ln(1 + x).

21. 21 For what values of x will the series converge?

22. 22 Creating new series for: Example 4

23. Create and use other Taylor series like was done with power series. 23 Lecture 34 – More Taylor Series

24. 24 Example 1

25. 25 Example 2

26. 26 Example 3

27. 27 Example 4

28. 28 Example 5