Finite Differences
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Transcript of Finite Differences
Jeff Bivin -- LZHS
Finite Differences
By: Jeffrey Bivin
Lake Zurich High School
Updated 12-5-08
Jeff Bivin -- LZHS
Determine the degree of the function given the set of finite differences.
x 1 2 3 4 5 6 7 8 9
f(x) 0 13 32 57 88 125 168 217 272
13 19 25 31 37 43 49 55
6 6 6 6 6 6 6
1st
2nd
Jeff Bivin -- LZHS
Now, identify the function of degree 2.
f(x) = ax2 + bx + c
f(2) = a(2)2 + b(2) + c = 4a + 2b + c = 13
f(1) = a(1)2 + b(1) + c = 1a + 1b + c = 0
f(3) = a(3)2 + b(3) + c = 9a + 3b + c = 6
For a 2nd degreefunction, how manypoints do we need to write the equation? 3
We will use this system to solve for
a, b, & c.
Jeff Bivin -- LZHS
Now, identify the function of degree 2.
f(x) = ax2 + bx + c
f(2) = a(2)2 + b(2) + c = 4a + 2b + c = 13
f(1) = a(1)2 + b(1) + c = 1a + 1b + c = 0
f(3) = a(3)2 + b(3) + c = 9a + 3b + c = 6
6139
13124
0111
RREF
a = 3b = 4c = -7
f(x) = ax2 + bx + c
f(x) = 3x2 + 4x - 7
Jeff Bivin -- LZHS
Determine the degree of the function given the set of finite differences.
x 1 2 3 4 5 6 7 8 9
f(x) -4 -7 38 227 704 1661 3338 6023 10052
-3 45 189 477 957 1677 2685 4029
48 144 288 480 720 1008 1344
1st
2nd
96 144 192 240 288 3363rd
48 48 48 48 484th
Jeff Bivin -- LZHS
Now, identify the function of degree 4.
f(x) = ax4 + bx3 + cx2 + dx + e
f(2) = 16a + 8b + 4c + 2d + e = -7
f(1) = 1a + 1b + 1c + 1d + e = -4
f(3) = 81a + 27b + 9c + 3d + e = 38
For a 4th degreefunction, how manypoints do we need towrite the equation? 5
We will use this system to solve for
a, b, & c.
f(4) = 256a + 64b + 16c + 4d + e = 227
f(5) = 625a + 125b + 25c + 5d + e = 704
Jeff Bivin -- LZHS
Solve the system:
7041525125625
227141664256
381392781
7124816
411111
a = 2b = -4c = -2
16a + 8b + 4c + 2d + e = -7
1a + 1b + 1c + 1d + e = -4
81a + 27b + 9c + 3d + e = 38
256a + 64b + 16c + 4d + e = 227
625a + 125b + 25c + 5d + e = 704
RREF
d = 1e = -1
f(x) = ax4 + bx3 + cx2 + dx + e
f(x) = 2x4 – 4x3 – 2x2 + x – 1
Jeff Bivin -- LZHS
Determine the degree of the function
4, 7, 10, 13, 16, 19, 22, 25, 28
3, 3, 3, 3, 3, 3, 3, 31st difference
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Now, write the linear model
4, 7, 10, 13, 16, 19, 22, 25, 28
f(1) f(2)
(1, 4)
(2, 7)
313
1247
m
)1(34 xy334 xy13 xy
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Determine the degreeof the function
-1, 0, 5, 14, 27, 44, 65, 90, 119
1, 5, 9, 13, 17, 21, 25, 291st difference
4, 4, 4, 4, 4, 4, 4 2nd difference
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Now write the quadratic model
-1, 0, 5, 14, 27, 44, 65, 90, 119
f(1) f(2) f(3)
cbnannf 2)(
1)1()1()1( 2 cbacbaf
024)2()2()2( 2 cbacbaf
539)3()3()3( 2 cbacbaf
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Now write the quadratic model
-1, 0, 5, 14, 27, 44, 65, 90, 119
f(1) f(2) f(3)
cbnannf 2)(
1)1()1()1( 2 cbacbaf
024)2()2()2( 2 cbacbaf
539)3()3()3( 2 cbacbaf
Jeff Bivin -- LZHS
5139
0124
1111
RREF
a = 2
b = -5
c = 2
252)( 2 nnnf
Jeff Bivin -- LZHS
Determine the degreeof the function
1, 10, 47, 130, 277, 506, 835, 1282, 1865
9, 37, 83, 147, 229, 329, 447, 583
28, 46, 64, 82, 100, 118, 136
18, 18, 18, 18, 18, 183rd difference
2nd difference
1st difference
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Now write the quadratic model
f(1) f(2) f(3)
dcnbnannf 23)(
1)1()1()1()1( 23 dcbadcbaf
1, 10, 47, 130, 277, 506, 835, 1282, 1865f(4)
10248)2()2()2()2( 23 dcbadcbaf
473927)3()3()3()3( 23 dcbadcbaf
13041664)4()4()4()4( 23 dcbadcbaf
Jeff Bivin -- LZHS
Jeff Bivin -- LZHS
Now write the quadratic model
f(1) f(2) f(3)
dcnbnannf 23)(
1)1()1()1()1( 23 dcbadcbaf
1, 10, 47, 130, 277, 506, 835, 1282, 1865f(4)
10248)2()2()2()2( 23 dcbadcbaf
473927)3()3()3()3( 23 dcbadcbaf
13041664)4()4()4()4( 23 dcbadcbaf
130141664
4713927
101248
11111
RREF
a = 3 b = -4 c = 0 d = 2 243)( 23 nnnf