1. Remainder Theorem

2. The n-th Talor polynomial The polynomial is called the n-th Taylor polynomial for f about c

3. The n-th Maclaurin polynomial The polynomial is called the n-th Maclaurin polynomial for f

4. Taylor formula for f with the Remainder The difference is called the n-th remainder for the Taylor series of f about c.

5. Thr Remainder Theorem Assume that all the first n+1 derivatives exist on an interval l containing c and

6. Example (1) Approximate e to five decimal places

9. Why we chose n=9 ?

10. Example (2) Approximate sin3 ○ to five decimal places

13. Approximating the sum of a convergent alternating series

14. Theorem Let T be the sum of an alternating series satisfying the main convergence test for an alternating series. Then: 1. T lies between any two successive partial sums of the series 2. If T is approximated by a partial sum T n, then: a. |T - T n | ≤ a n+1 b. The sign of the error is the same as that of the coefficient of a n+1

15. Using power series to approximate definite integrals

16. Example Give an estimation of the integral on the right accurate to 3 decimal places

17. A power series for the given integral

18. Approximating the integral

19. Homework