4. What was determined from
the In-Class Activity
If 1st differences are equal, you
have a linear equation.
Wednesday, March 25, 2009
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. 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
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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
33. Method of Finite
Differences
When you apply the Polynomial-Difference
Theorem.
Wednesday, March 25, 2009
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. 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. 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
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. 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