1. Section 9.3 Taylor’s Theorem

2. When you learn new things, it is a healthy to ask yourself “Why are we learning this? What makes it interesting? What makes it relevant to the corpus of knowledge the human race has acquired?” This is not a copout or an excuse to justify not wanting to learn new things, but rather a way to put what your learning into perspective and make interesting connections Hence, we ask ourselves… what makes Taylor series interesting or useful? Why do we use them? Why Taylor Series?

3. Which is faster for a computer to find for a value of x between -1 and 1? Imagine you were doing millions of these calculations per second. The time savings of the polynomial would be immense! Of course, the latter polynomial only gives you an approximation of arctan(x). In what scenario might someone care how accurate your approximation is? Taylor Series Are a Tool for Approximation

4. Imagine there’s an asteroid barreling towards Earth. If it hits, it will destroy the planet. Example 1: Taylor Series as a Tool for Approximation

5. The world’s best scientists have calculated that a rocket launched at an angle of arctan(0.8) radians will hit the asteroid dead-center.

6. The problem is that the rocket needs to be launched pronto or else the asteroid is going to hit us. He doesn’t have time to wait around for his computer to calculate the billions of digits of arctan(0.8) for the near-perfect hit… 0.67474094222355266330565209736

7. But we don’t need to hit this particular asteroid dead center to destroy it. Rather, we just need to make contact to destroy it.

8. But we don’t need to hit this particular asteroid dead center to destroy it. Rather, we just need to make contact to destroy it.

9. Our scientists tell us we really only need to be within 5% of the actual value of arctan(0.8) radians to make contact. Anything beyond this will miss…

10. As long as we can calculate arctan(0.8) within this ±5% range, we’re saved!

12. We’re saved!! We needed to be within 5% and we were actually within 2.983%!!!!

13. Example 1

15. Example 2

16. All of our examples so far have problem with them: In the Asteroid Problem (and Example 1), we computed the (Actual – Estimate)/Actual to find our error of our approximation. Why did we find an ESTIMATE if checking if it was accurate enough to use required us to compute the ACTUAL value?? And Example 2 didn’t need a Taylor series expansion in the first place… It was just 1/(1+x 2 )

17. Example 3: An Error ESTIMATE for arctan(0.8)

19. We needed to be within 5% and we have an estimate that says our error is NO WORSE than 4.646%!

20. Alternating Series Remainder Estimate

21. Example 4: What If We Don’t Alternate?

22. Remainder Estimate Theorem

23. Example 4

25. Example 5

28. MEMORIZE!!