1. 9.7 and 9.10 Taylor Polynomials and Taylor Series

2. Suppose we wanted to find a fourth degree polynomial of the form: atthat approximates the behavior of If we make f ( 0 ) = P ( 0 ), f’(0 ) = P ’( 0 ), f’’(0 ) = P’’(0), and so on, then we would have a pretty good approximation.

4. This is called the Taylor Polynomial of degree 4.

5. Our polynomial has the form: or: If we plot both functions, we see that near zero the functions match very well!

6. Definition The series of the form Theorem If f ( x ) is represented by a power series for all x in an open interval I containing c, then is called the Taylor Series for f ( x ) at c. If c = 0, then the series is called the Maclaurin series for f.

7. Example Find the Maclaurin series for

8. Rather than start from scratch, we can use the function that we already know: Example Find the Maclaurin series for

9. Example Find the Taylor series for

10. Example Find the Maclaurin series for

11. This is a geometric series with a = 1 and r = x. Example Find the Maclaurin series for

12. Example

13. Example

14. An amazing use for infinite series: Substitute xi for x. Factor out the i terms. Euler’s Formula

15. This is the series for cosine. This is the series for sine. Let This amazing identity contains the five most famous numbers in mathematics, and shows that they are interrelated.

16. Taylor series are used to estimate the value of functions (at least theoretically - nowadays we can usually use the calculator or computer to calculate directly.) An estimate is only useful if we have an idea of how accurate the estimate is. When we use part of a Taylor series to estimate the value of a function, the end of the series that we do not use is called the remainder. If we know the size of the remainder, then we know how close our estimate is. Ex: Use to approximate over.the truncation error is, which is. When you “truncate” a number, you drop off the end.

17. Taylor’s Theorem with Remainder If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I : Lagrange Form of the Remainder Remainder after partial sum S n where c is between a and x. This is also called the remainder of order n or the error term.

18. This is called Taylor’s Inequality. Taylor’s InequalityLagrange Form of the Remainder If M is the maximum value of on the interval between a and x, then: Note that this looks just like the next term in the series, but “ a ” has been replaced by the number “ c ” in.

19. Find the Lagrange Error Bound when is used to approximate and. Remainder after 2nd order term On the interval, decreases, so its maximum value occurs at the left end-point. Lagrange Error Bound error