1. AP Calculus BC Monday, 07 April 2014 OBJECTIVE TSW (1) find polynomial approximations of elementary functions and compare them with the elementary functions; (2) find Taylor and Maclaurin polynomial approximations of elementary functions; and (3) use the remainder of a Taylor polynomial. ASSIGNMENTS DUE TOMORROW (TEST DAY) –Sec. 9.4 –Sec. 9.5 –Sec. 9.6 TEST TOPICS PowerPoint is already on my website.

2. Sec. 9.7: Taylor Polynomials and Approximations

3. What does the graph of look like? Just by looking at the graph, what is an approximation for f (1.5)?

4. Sec. 9.7: Taylor Polynomials and Approximations We could get a better approximation if were represented as a polynomial P. A first degree approximation at x = 1.5 would look like this:

5. Sec. 9.7: Taylor Polynomials and Approximations Now suppose we wanted to approximate at a different x-value, say x = 0. If we use a 1 st -degree polynomial, we would want P(0) = f(0) and 1 st degree

6. Sec. 9.7: Taylor Polynomials and Approximations A 2 nd -degree polynomial approximation for would have P(0) = f(0),

7. Sec. 9.7: Taylor Polynomials and Approximations A 3 rd -degree polynomial? 4 th -degree?

8. Sec. 9.7: Taylor Polynomials and Approximations n th -degree?

9. Sec. 9.7: Taylor Polynomials and Approximations These examples are centered at c = 0. In general, c could be any value, so the polynomial would be written as

10. Sec. 9.7: Taylor Polynomials and Approximations From this, we get the following:

11. Sec. 9.7: Taylor Polynomials and Approximations Ex:Find the Taylor polynomials P 0, P 1, P 2, P 3, and P 4 for centered at c = 1.

12. Sec. 9.7: Taylor Polynomials and Approximations Ex:Find the Maclaurin polynomials P 0, P 2, P 4, and P 6 for. Use to approximate the value of. Actual value: 0.995004165278

13. Sec. 9.7: Taylor Polynomials and Approximations Ex:Find the third Taylor polynomial for centered at

14. Sec. 9.7: Taylor Polynomials and Approximations Ex:Find the third Taylor polynomial for centered at

15. Sec. 9.7: Taylor Polynomials and Approximations Ex:Use a fourth Maclaurin polynomial to approximate the value of Consider this: 1.1 is closer to 1 than 0, so an approximation using and an x-value of 0.1 would be more accurate than using and an x-value of 1.1.

16. Sec. 9.7: Taylor Polynomials and Approximations Ex:Use a fourth Maclaurin polynomial to approximate the value of

17. Sec. 9.7: Taylor Polynomials and Approximations Ex:Use a fourth Maclaurin polynomial to approximate the value of Actual value: 0.0953101798

18. Sec. 9.7: Taylor Polynomials and Approximations An approximation technique is of little value without some idea of its accuracy. To measure the accuracy of approximating a functional value by the Taylor polynomial you can use the concept of a remainder, defined as follows. Exact Value Approximate Value Remainder

19. Sec. 9.7: Taylor Polynomials and Approximations Another way to look at this is that The absolute value of is called the error associated with the approximation. In other words, This is summarized in Taylor’s Theorem, which gives the Lagrange form of the remainder.

20. Sec. 9.7: Taylor Polynomials and Approximations

21. When applying Taylor’s Theorem, You will probably not find the exact value of z!!! (If you could, an approximation would not be necessary.) You try to find bounds for from which you are able to tell how large the remainder is.

22. Sec. 9.7: Taylor Polynomials and Approximations Ex:The third Maclaurin polynomial for sin x is given by Use Taylor’s Theorem to approximate by and determine the accuracy of the approximation.

23. Sec. 9.7: Taylor Polynomials and Approximations Ex:The third Maclaurin polynomial for sin x is given by Use Taylor’s Theorem to approximate by and determine the accuracy of the approximation. Using Taylor's Theorem, where 0 < z < 0.1.

24. Sec. 9.7: Taylor Polynomials and Approximations Ex:The third Maclaurin polynomial for sin x is given by Use Taylor’s Theorem to approximate by and determine the accuracy of the approximation.

25. Sec. 9.7: Taylor Polynomials and Approximations Ex:Determine the smallest degree of the Taylor polynomial expanded about c = 1 that should be used to approximate so that the error is less than 0.001.

26. Sec. 9.7: Taylor Polynomials and Approximations Ex:Determine the smallest degree of the Taylor polynomial expanded about c = 1 that should be used to approximate so that the error is less than 0.001....

27. Sec. 9.7: Taylor Polynomials and Approximations Ex:Determine the smallest degree of the Taylor polynomial expanded about c = 1 that should be used to approximate so that the error is less than 0.001. where c < z < x, or 1 < z < 1.2

28. Sec. 9.7: Taylor Polynomials and Approximations Ex:Determine the smallest degree of the Taylor polynomial expanded about c = 1 that should be used to approximate so that the error is less than 0.001. "What is the smallest value of n that will make the inequality true?" "What value of z will give us the safest estimate?" When z is smallest (so the denominator will be smallest and the fraction will be largest.) Use a calculator to determine this.

29. Sec. 9.7: Taylor Polynomials and Approximations Ex:Determine the smallest degree of the Taylor polynomial expanded about c = 1 that should be used to approximate so that the error is less than 0.001. n = 3: You should use a 3 rd degree Taylor Polynomial