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Recursive Functions, Iterates, and Finite Differences

Recursive Functions, Iterates, and Finite Differences. By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org. 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 –

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Recursive Functions, Iterates, and Finite Differences

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  1. Recursive Functions,Iterates, and Finite Differences By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: May 21, 2008

  2. 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

  3. Example 1 start a1 = 5 an = an-1 + 10 definition 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

  4. Example 2 start f(1) = 3 f(n) = 5•f(n-1) + 2 definition 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

  5. Example 3 f(1) = 1 f(2) = 1 f(n) = f(n-1) + f(n-2) f(3) = f(3-1) + f(3-2) = f(2) + f(1) = 1 + 1 = 2 f(4) = f(4-1) + f(4-2) = f(3) + f(2) = 2 + 1 = 3 f(5) = f(5-1) + f(5-2) = f(4) + f(3) = 3 + 2 = 5 f(6) = f(6-1) + f(6-2) = f(5) + f(4) = 5 + 3 = 8 start definition Jeff Bivin -- LZHS

  6. Write a recursive rule for the sequence 4, 12, 36, 108, 324, . . . Is it Arithmetic or Geometric? What is the pattern? multiply by 3 What is the start? a1 = 4 an = 3·an-1 What is the definition?

  7. Write a recursive rule for the sequence 7, 12, 17, 22, 27, . . . Is it Arithmetic or Geometric? What is the pattern? add 5 What is the start? a1 = 7 an = an-1 + 5 What is the definition?

  8. Write a recursive rule for the sequence 3, 4, 7, 11, 18, 29, 47, . . . Is it Arithmetic or Geometric? neither What is the pattern? 3+4 = 7, 4 + 7 = 11, 7 + 11 = 18 What is the start? a1 = 3 a2 = 4 an = an-2 + an-1 What is the definition?

  9. 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

  10. Determine the degree of the function 4, 7, 10, 13, 16, 19, 22, 25, 28 3, 3, 3, 3, 3, 3, 3, 3 1st difference 1st Degree Jeff Bivin -- LZHS

  11. Now, write the linear model 1st Degree f(1) f(2) 4, 7, 10, 13, 16, 19, 22, 25, 28 (1, 4) (2, 7) Jeff Bivin -- LZHS

  12. Determine the degreeof the function -1, 0, 5, 14, 27, 44, 65, 90, 119 1, 5, 9, 13, 17, 21, 25, 29 1st difference 4, 4, 4, 4, 4, 4, 4 2nd difference 2nd Degree Jeff Bivin -- LZHS

  13. Now write the quadratic model 2nd Degree f(1) f(2) f(3) -1, 0, 5, 14, 27, 44, 65, 90, 119 Solve the system Jeff Bivin -- LZHS

  14. Now write the quadratic model 2nd Degree f(1) f(2) f(3) -1, 0, 5, 14, 27, 44, 65, 90, 119 a = 2 b = -5 c = 2 Jeff Bivin -- LZHS

  15. Determine the degreeof the function 1, 10, 47, 130, 277, 506, 835, 1282, 1865 9, 37, 83, 147, 229, 329, 447, 583 1st difference 28, 46, 64, 82, 100, 118, 136 2nd difference 3rd Degree 18, 18, 18, 18, 18, 18 3rd difference Jeff Bivin -- LZHS

  16. Now write the quadratic model 3rd Degree f(1) f(2) f(3) f(4) 1, 10, 47, 130, 277, 506, 835, 1282, 1865 Solve the system Jeff Bivin -- LZHS

  17. Now write the quadratic model 3rd Degree f(1) f(2) f(3) f(4) 1, 10, 47, 130, 277, 506, 835, 1282, 1865 a = 3 b = -4 c = 0 d = 2 Jeff Bivin -- LZHS

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