1. 9.7 Taylor Series

2. 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.

3. Identify this graph:What if we zoom out a bit? And a bit more? What the?! The actual function is:

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

5. 0 1 0

6. 0 1 0

7. 0 1

8. 0 1

9. P(x) is the fifth degree polynomial that has the same first through fifth derivatives at x = 0 that f(x) = sin x has. We call P(x) the 5 th degree Taylor Polynomial of f(x) at x = 0.

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

13. This pattern occurs no matter what the original function was! Our polynomial: has the form: or: This is called a Taylor Polynomial. If we continue this pattern infinitely, we get a Taylor Series. When the series is centered at x = 0, it is called a Maclaurin Series.

14. 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 ) Maclaurin Series: (generated by f at )

15. Example: Generate the Maclaurin Series for The formula for Maclaurin Series is:

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

17. example:

18. When referring to Taylor polynomials, we can talk about number of terms, order or degree. This is a polynomial in 3 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.

19. A Taylor Polynomial is used to approximate a function. A Taylor Series is equal to a function.

20. In practice, we usually use a Taylor Polynomial to approximate a function so we also want to know the error in using the approximation. The error (also known as the remainder, R n (x)) is the difference between the actual value of the function, f(x), and the polynomial of degree n, P n (x).

21. Taylor’s Theorem specifies that the remainder R n (x) (also called the Lagrange Error or Lagrange form of the remainder) is given by the formula: This looks a lot like the next term in the Taylor Series except that it is not evaluated at a but at some number z which is between x and a. z z

22. Taylor’s Theorem guarantees that there exists a value of z between a and x such that: It usually isn’t possible to find the value of but you can often find an upper bound on it. This will be called max

23. Example: Therefore the Lagrange error term