The document discusses finite differences and polynomial functions. It is shown that if the nth differences of a function are equal, the function is a polynomial of degree n. The document provides examples of using difference tables to determine if a function is polynomial and, if so, its degree. It is concluded that applying the polynomial difference theorem involves examining differences in a table to determine if the data represents a polynomial function.

1. Section 11-9
Finite Differences
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2. In-Class Activity
p. 723-4
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3. What was determined from
the In-Class Activity
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4. What was determined from
the In-Class Activity
If 1st differences are equal, you
have a linear equation.
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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.
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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.
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7. Polynomial-Difference
Theorem
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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29. Example 1
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30. Example 1
Yes, this is a polynomial, as the 3rd
differences are equal.
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31. Example 1
Yes, this is a polynomial, as the 3rd
differences are equal.
This makes this a cubic equation.
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32. Method of Finite
Differences
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33. Method of Finite
Differences
When you apply the Polynomial-Difference
Theorem.
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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.
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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!
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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.
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37. Wednesday, March 25, 2009

38. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
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39. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
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40. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2
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41. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20
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42. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20 160
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43. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20 160 30080
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44. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20 160 30080 915274240
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45. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20 160 30080 915274240
22
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46. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20 160 30080 915274240
22 140
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47. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20 160 30080 915274240
22 140 29920
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48. n 1 2 3 4 5 6
an 1 -1 19 179 30259 915304499
-2 20 160 30080 915274240
22 140 29920 915244160
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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
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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
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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
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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
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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
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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.
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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
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