Creation and Manipulation of Taylor Series

1. Creation and Manipulation of Taylor Series Dan KennedyBaylor SchoolChattanooga, TN dkennedy@baylorschool.org

2. The recommended impetus for this session is free-response question 6 from the 2007 BC exam:

3. The BC students in 2007 did not exactly ace this one. The mean score was 2.28 out of 9. A mere 0.3% of the students earned 9’s. Approximately 40.6% of the students earned no points at all.

4. Bad idea: Find the first three derivatives of f and use them to build the series. Good idea: Plug into the Maclaurin series for

5. Plug in : which can be simplified to…

6. Other popular simplifications of

7. Bad idea: Forget part (a). Let’s use L’Hopital’s rule to find Dissing the AP Committee is a no-no!

8. Good idea: Do what you are told.

9. Thus: All this was worth 1 point out of 9.

10. Baaaad idea: Start by antidifferentiating Good idea: Import your series from part (a).

11. By the FTC, is the antiderivative of that equals 0 at x = 0. From part (a): So:

12. Using the first two terms of this series, we estimate: Notice that the next term of the series approximation would have been This number plays a big role in (d).

13. Quel imbecile! Baaad idea: Try arguing your case using the Lagrange error bound. Good idea: Use the bound associated with the Alternating Series Test. Be sure to justify that it applies here!

14. The series in (c) for is an alternating series of terms that decrease in absolute value with a limit of zero. Thus, the truncation error after two terms is less than the absolute value of the third term:

15. So let’s talk about series manipulation. In the old days (pre-1989), the approach to series in a calculus class was quite different from what it is today (as with many other topics). 1. Convergence tests for series of constants 2. Constructing Taylor series3. Intervals of convergence For example, consider BC-4 from 1979…

16. Notice that the series is geometric.

17. Here is the actual grading standard for part (a), which was worth 5 points out of 15. Notice that it was assumed that students would build the series using nth derivatives. Since it was 1979, they all did.

18. The grading standard for part (b) assumed the standard Ratio Test with endpoint analysis. There is no visible acknowledgment that justifies the answer in one step!

19. Today’s calculus students would (I hope) recognize as a variation of ,but the students in 1979 apparently did not. How did our students get better???? We need to reflect on these beneficial changes occasionally … if only to remind ourselves that the good old days were actually not all that great.

20. What happened was a new emphasis on series as functions … the real reason for having them in the course. In fact, many teachers are talking about series as functions long before they talk about convergence tests for series of constants. What follow are some of my favorite student explorations …

21. When they find that this function is its own derivative, most students will guess that it is . You hope someone will realize why it must be .

22. The third task is particularly rich. Students might forget the constant of integration. When reminded, they’ll cheerfully add it. Then remind them that they can use tan 0 to find it!

23. This began with a series that was valid for -1 < x < 1. An interesting postscript is that we also get convergence at -1 and 1.

24. 1 Limit = tan 1

25. Students can generally do this. They will discover that the polynomial is:

26. Now they’re ready for the nicest little exploration in the course!

27. One of the most powerful visualizations in mathematics is the spectacle of the convergence of Taylor series. Here are the Taylor polynomials for sin x about x = 0:

28. Here are their graphs, superimposed on the graph of y = sin x:

29. Coming next year from the College Board: The 2008-2009 AP Calculus Focus Materials The topic: Infinite Series!

30. dkennedy@baylorschool.org