recursive functions, iterates, and finite differences
DESCRIPTION
Recursive Functions, Iterates, and Finite Differences. By: Jeffrey Bivin Lake Zurich High School [email protected]. Last Updated: May 21, 2008. Recursive Function. A recursive function is a function whose domain is the set of nonnegative integers and is made up of two parts – - PowerPoint PPT PresentationTRANSCRIPT
Recursive Functions,Iterates,
and Finite DifferencesBy: Jeffrey Bivin
Lake Zurich High School
Last Updated: May 21, 2008
Recursive Function
A recursive function is a function whose domain is the set of nonnegative integers and is made up of two parts –
1. Start
2. Definition
Jeff Bivin -- LZHS
Example 1a1 = 5
an = an-1 + 10
n = 2
a2 = a(2-1) + 10
a2 = a1 + 10
a2 = 5 + 10
a2 = 15
n = 3
a3 = a (3-1) + 10
a3 = a2 + 10
a3 = 15 + 10
a3 = 25
n = 4
a4 = a(4-1) + 10
a4 = a3 + 10
a4 = 25 + 10
a4 = 35
Jeff Bivin -- LZHS
Example 2f(1) = 3
f(n) = 5•f(n-1) + 2
n = 2
f(2) = 5•f(2-1) + 2
f(2) = 5•f(1) + 2
f(2) = 5•3 + 2
f(2) = 17
n = 3
f(3) = 5•f(3-1) + 2
f(3) = 5•f(2) + 2
f(3) = 5•17 + 2
f(3) = 87
n = 4
f(4) = 5•f(4-1) + 2
f(4) = 5•f(3) + 2
f(4) = 5•87 + 2
f(4) = 437
Jeff Bivin -- LZHS
Example 3
f(1) = 1f(2) = 1f(n) = f(n-1) + f(n-2)
f(3) = f(3-1) + f(3-2) = f(2) + f(1) = 1 + 1 = 2f(4) = f(4-1) + f(4-2) = f(3) + f(2) = 2 + 1 = 3f(5) = f(5-1) + f(5-2) = f(4) + f(3) = 3 + 2 = 5f(6) = f(6-1) + f(6-2) = f(5) + f(4) = 5 + 3 = 8
Jeff Bivin -- LZHS
Write a recursive rule for the sequence4, 12, 36, 108, 324, . . .
Is it Arithmetic or Geometric? What is the pattern? multiply by 3
What is the start?
What is the definition?
a1 = 4
an = 3·an-1
Is it Arithmetic or Geometric?
Write a recursive rule for the sequence7, 12, 17, 22, 27, . . .
What is the pattern? add 5
What is the start?
What is the definition?
a1 = 7
an = an-1 + 5
Is it Arithmetic or Geometric?
Write a recursive rule for the sequence
3, 4, 7, 11, 18, 29, 47, . . .
What is the pattern? 3+4 = 7, 4 + 7 = 11, 7 + 11 = 18
What is the start?
What is the definition?
a1 = 3
an = an-2 + an-1
neither
a2 = 4
Find the first three iterates of the function for the given initial value.
f(x) = 5x + 3, x0 = 2
x1 = f(x0) = f(2) = 5(2) + 3 = 13
x2 = f(x1) = f(13) = 5(13) + 3 = 68
x3 = f(x2) = f(68) = 5(68) + 3 = 343
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
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
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
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
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
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
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
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
130141664
4713927
101248
11111
RREF
a = 3 b = -4 c = 0 d = 2 243)( 23 nnnf