1. All the World’s a Polynomial Chris Harrow chrish@westminster.net http://casmusings.wordpress.com chrish@westminster.net http://casmusings.wordpress.com

2. All the World’s a Polynomial Historically, students struggle to understand the utility and origins of Taylor Series. This session makes use of local linearity and statistical regressions to explain tangent lines in a way that is useful to all AP Calculus students before extending the approach to create Taylor Series for AP Calculus BC. This introduction is understandable by both pre-calculus and calculus students. The session will conclude with a student project around a famous Euler problem and techniques for using series to connect circular and hyperbolic trigonometry.

3. How can you compute ? Perhaps a graph of near could help.

4. But isn’t linear... So make it linear by zooming in (LOCAL LINEARITY), and pick some ordered pairs from the resulting “line”.

5. Compute a linear “equivalent” to. But this equation is very close to. So, near,, making.

6. How close was the estimate? and So the percentage error is

7. Analyzing error Look at the residuals for.

8. That looks quadratic! Compute an equation for the linear residuals and use that to enhance your approximation. But this equation is very close to. So,, making.

9. Improving the estimate And the percentage error is

10. Analyzing error again Look at the residuals for.

11. And that looks cubic! Compute an equation for the cubic residuals and use that to enhance your approximation. And this equation is very close to. So,, making.

12. A faster way. Compute a cubic regression on the original data. This gives the same result, but faster. I prefer the build up rather than the “black box.”

13. Quartic Regressions

14. What about sine & cosine?

16. That’s suspicious These regressions suggest

17. Connections If you can evaluate, then

18. Another Strange Result

19. Euler You have one series for sine: But what if you thought of sine as a polynomial via its factors? Then,

20. Euler 2 But what if you thought of sine as a polynomial via its factors? Then,

21. Euler 3 Two polynomials representing the same curve must be equivalent, so Comparing linear terms gives.

22. Euler 4 Comparing cubic terms gives... QED