AA Section 11-9

Section 11-9
Finite Differences

Wednesday, March 25, 2009

In-Class Activity
p. 723-4


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 2nd differences are equal, you
have a quadratic equation.

If 3rd differences are equal, you
have a cubic equation.


Polynomial-Difference
Theorem


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.


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


Example 1

n 1 2 3 4 5 6 7 8

f(n) 1 5 14 30 55 91 140 204

4


Example 1

n 1 2 3 4 5 6 7 8

f(n) 1 5 14 30 55 91 140 204

4 9


Example 1

n 1 2 3 4 5 6 7 8

f(n) 1 5 14 30 55 91 140 204

4 9 16


Example 1

n 1 2 3 4 5 6 7 8

f(n) 1 5 14 30 55 91 140 204

4 9 16 25


Example 1

n 1 2 3 4 5 6 7 8

f(n) 1 5 14 30 55 91 140 204

4 9 16 25 36


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1

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


Example 1


Example 1

Yes, this is a polynomial, as the 3rd
differences are equal.


Example 1

Yes, this is a polynomial, as the 3rd
differences are equal.

This makes this a cubic equation.


Method of Finite
Differences


Method of Finite
Differences
When you apply the Polynomial-Difference
Theorem.


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.


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!


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.


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160 30080


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160 30080 915274240


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160 30080 915274240

22


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160 30080 915274240

22 140


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160 30080 915274240

22 140 29920


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160 30080 915274240

22 140 29920 915244160


n 1 2 3 4 5 6

an 1 -1 19 179 30259 915304499

-2 20 160 30080 915274240

22 140 29920 915244160
118


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


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


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


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


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.

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

AA Section 11-9

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