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# AA Section 11-9

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Finite Differences

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• ### AA Section 11-9

1. 1. Section 11-9 Finite Differences Wednesday, March 25, 2009
2. 2. In-Class Activity p. 723-4 Wednesday, March 25, 2009
3. 3. What was determined from the In-Class Activity Wednesday, March 25, 2009
4. 4. What was determined from the In-Class Activity If 1st differences are equal, you have a linear equation. Wednesday, March 25, 2009
5. 5. What was determined from the In-Class Activity If 1st differences are equal, you have a linear equation. If 2nd differences are equal, you have a quadratic equation. Wednesday, March 25, 2009
6. 6. What was determined from the In-Class Activity If 1st differences are equal, you have a linear equation. If 2nd differences are equal, you have a quadratic equation. If 3rd differences are equal, you have a cubic equation. Wednesday, March 25, 2009
7. 7. Polynomial-Difference Theorem Wednesday, March 25, 2009
8. 8. Polynomial-Difference Theorem y = f(x) is a polynomial function of degree n IFF for any arithmetic sequence of independent variables, the n th difference of the dependent variables are equal and the (n-1)st differences are not equal. Wednesday, March 25, 2009
9. 9. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 Wednesday, March 25, 2009
10. 10. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 Wednesday, March 25, 2009
11. 11. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 Wednesday, March 25, 2009
12. 12. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 Wednesday, March 25, 2009
13. 13. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 Wednesday, March 25, 2009
14. 14. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 Wednesday, March 25, 2009
15. 15. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 Wednesday, March 25, 2009
16. 16. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 Wednesday, March 25, 2009
17. 17. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 Wednesday, March 25, 2009
18. 18. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 Wednesday, March 25, 2009
19. 19. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 Wednesday, March 25, 2009
20. 20. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 Wednesday, March 25, 2009
21. 21. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 Wednesday, March 25, 2009
22. 22. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 15 Wednesday, March 25, 2009
23. 23. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 15 2 Wednesday, March 25, 2009
24. 24. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 15 2 2 Wednesday, March 25, 2009
25. 25. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 15 2 2 2 Wednesday, March 25, 2009
26. 26. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 15 2 2 2 2 Wednesday, March 25, 2009
27. 27. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 15 2 2 2 2 2 Wednesday, March 25, 2009
28. 28. Example 1 Consider the data in the table. Is f(n) a polynomial function? Justify. n 1 2 3 4 5 6 7 8 f(n) 1 5 14 30 55 91 140 204 4 9 16 25 36 49 64 5 7 9 11 13 15 3rd row 2 2 2 2 2 Wednesday, March 25, 2009
29. 29. Example 1 Wednesday, March 25, 2009
30. 30. Example 1 Yes, this is a polynomial, as the 3rd differences are equal. Wednesday, March 25, 2009
31. 31. Example 1 Yes, this is a polynomial, as the 3rd differences are equal. This makes this a cubic equation. Wednesday, March 25, 2009
32. 32. Method of Finite Differences Wednesday, March 25, 2009
33. 33. Method of Finite Differences When you apply the Polynomial-Difference Theorem. Wednesday, March 25, 2009
34. 34. Method of Finite Differences When you apply the Polynomial-Difference Theorem. Examine the differences of the dependent variables to determine if a set of data represents a polynomial, where the degree n will be the row where the equal differences occur. Wednesday, March 25, 2009
35. 35. Example 2 A sequence is defined by a = 1 1  2 an = (an − 1 ) − 10an − 1 + 8, for int. n ≥ 2  Is there an explicit polynomial formula for this? Justify! Wednesday, March 25, 2009
36. 36. Example 2 A sequence is defined by a = 1 1  2 an = (an − 1 ) − 10an − 1 + 8, for int. n ≥ 2  Is there an explicit polynomial formula for this? Justify! Create a table and examine the differences. Wednesday, March 25, 2009
37. 37. Wednesday, March 25, 2009
38. 38. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 Wednesday, March 25, 2009
39. 39. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 Wednesday, March 25, 2009
40. 40. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 Wednesday, March 25, 2009
41. 41. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 Wednesday, March 25, 2009
42. 42. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 Wednesday, March 25, 2009
43. 43. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 Wednesday, March 25, 2009
44. 44. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 Wednesday, March 25, 2009
45. 45. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 Wednesday, March 25, 2009
46. 46. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 Wednesday, March 25, 2009
47. 47. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 Wednesday, March 25, 2009
48. 48. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 915244160 Wednesday, March 25, 2009
49. 49. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 915244160 118 Wednesday, March 25, 2009
50. 50. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 915244160 118 29780 Wednesday, March 25, 2009
51. 51. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 915244160 118 29780 915214240 Wednesday, March 25, 2009
52. 52. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 915244160 118 29780 915214240 29662 Wednesday, March 25, 2009
53. 53. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 915244160 118 29780 915214240 29662 915184460 Wednesday, March 25, 2009
54. 54. n 1 2 3 4 5 6 an 1 -1 19 179 30259 915304499 -2 20 160 30080 915274240 22 140 29920 915244160 118 29780 915214240 29662 915184460 There is no common difference, so there does not seem to be a polynomial formula to represent the sequence. Wednesday, March 25, 2009
55. 55. Homework p. 727 #1-23 “The only way of finding the limits of the possible is by going beyond them into the impossible.” - Arthur C. Clarke Wednesday, March 25, 2009