1. Taylor’s Theorem Section 9.3a

2. While it is beautiful that certain functions can be represented exactly by infinite Taylor series, it is the inexact Taylor series that do all the work… In practical terms, we would like to be able to use Taylor polynomials to approximate functions over the intervals of convergence of the Taylor series, and we would like to keep the error of the approximation within specified bounds… Since the error results from truncating the series down to a polynomial (that is, cutting it off after some number of terms), we call it truncation error.

3. Find a formula for the truncation error if we use to approximateover the interval. This polynomial is the fourth partial sum of the geometric series for … The truncation error is the absolute value of the part that we threw away: This is the absolute value of a geometric series with: Therefore,

4. Find a formula for the truncation error if we use to approximateover the interval. Therefore, Truncation error: Graph this error in [–1, 1] by [–5, 5] The error is very small near 0, but increases as x gets closer to 1 or –1.

5. With this last example, the series (and therefore, the error) was geometric, but Taylor’s Theorem also works for nongeometric series… Every truncation splits a Taylor series into equally significant pieces: the Taylor polynomial that gives us the approximation, and the remainder that tells us whether the approximation is any good…

6. 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, where for some c between a and x.

7. Taylor’s Theorem with Remainder The first equation in Taylor’s Theorem is Taylor’s formula. The function is the remainder of order n or the error term for the approximation of f by over I. It is also called the Lagrange form of the remainder, and bounds on found using this form are Lagrange error bounds. If as for all x in I, we say that the Taylor series generated by f at x = a converges to f on I, and we write

8. Proving Convergence of a Maclaurin Series Prove that the given series converges to sin(x) for all real x. We need to see what happens to as By Taylor’s Theorem: is the (n+1)st derivative of sin(x)… which lies between –1 and 1 inclusive… So for all x:

9. Proving Convergence of a Maclaurin Series Prove that the given series converges to sin(x) for all real x. What happens to this last term as n approaches infinity? The factorial growth in the denominator eventually outstrips the power growth in the numerator: This completes the proof!!!

10. Guided Practice Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2.

11. Guided Practice Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2.

12. Guided Practice Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2. How close is this approximation to the actual function value?

13. Guided Practice Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2. Check the table on p.477: So…

14. Guided Practice Find the Taylor polynomial of order four for the function at x = 0, and use it to approximate the value of the function at x = 0.2.

15. Guided Practice Find the Maclaurin series for the given function.

16. Guided Practice Find the Maclaurin series for the given function.