1. Do Now: Find both the local linear and the local quadratic approximations of f(x) = e x at x = 0 Aim: How do we make polynomial approximations for a given function ?

2. we want to find a second degree polynomial of the form: atthat approximates the behavior of If we make, and the first, and the second, derivatives the same, then we would have a pretty good approximation.

5. Suppose we wanted to find a fourth degree polynomial of the form: atthat approximates the behavior of If we make, and the first, second, third and fourth derivatives the same, then we would have a pretty good approximation.

8. If we plot both functions, we see that near zero the functions match very well!

9. This pattern occurs no matter what the original function was! Our polynomial: has the form: or:

10. Maclaurin Series: (generated by f at ) If we want to center the series (and it’s graph) at some point other than zero, we get the Taylor Series: Taylor Series: (generated by f at )

11. Brook Taylor 1685 - 1731 Taylor Series Brook Taylor was an accomplished musician and painter. He did research in a variety of areas, but is most famous for his development of ideas regarding infinite series. Greg Kelly, Hanford High School, Richland, Washington

12. Write the Taylor Series for f(x) = cos x centered at x = 0. What are the Taylor Series for some common functions ?

14. The more terms we add, the better our approximation.

15. For use the Ratio Test to determine the interval of convergence.

16. If the limit of the ratio between consecutive terms is less than one, then the series will converge. The interval of convergence is

17. When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial with 3 positive terms. It is a 4th order Taylor polynomial, because it was found using the 4th derivative. It is also a 4th degree polynomial, because x is raised to the 4th power. The 3rd order polynomial for is, but it is degree 2. The x 3 term drops out when using the third derivative. This is also the 2nd order polynomial. A recent AP exam required the student to know the difference between order and degree.

18. Both sides are even functions. Cos (0) = 1 for both sides.

19. Both sides are odd functions. Sin (0) = 0 for both sides.

20. What is the interval of convergence for the sine series?

21. Find the Taylor Series for centered at a =1 And determine the interval of convergence

23. If the limit of the ratio between consecutive terms is less than one, then the series will converge.

24. The interval of convergence is (0,2]. The radius of convergence is 1. If the limit of the ratio between consecutive terms is less than one, then the series will converge.

25. Power series for elementary functions Interval of Convergence