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Newton's Method for Root Approximation

Newton's Method for Root Approximation
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2015-03-14 08:27
FunctionCalculusDerivativeOptimizationGeometric Construction
This example demonstrates the iterative process and convergence of Newton's method in finding approximate roots of a function. Core mathematical concepts include derivatives, iterative formulas, function graphs, and root approximation. GeoGebra commands/objects used in construction summary: objects f (function), A2 to A25, and B1 to B13 (representing iterative points, function values, tangents, etc.). Suitable for high school and above mathematics learners to understand classic algorithms in numerical analysis.

GeoGebra Commands & Parameters

  1. f(x) = x^(2) - 2
  2. Numeric A2(1.0)
  3. A2 + 1
  4. A3 + 1
  5. A4 + 1
  6. A5 + 1
  7. A6 + 1
  8. A7 + 1
  9. A8 + 1
  10. A9 + 1
  11. A10 + 1
  12. A11 + 1
  13. A12 + 1
  14. A13 + 1
  15. A14 + 1
  16. A15 + 1
  17. A16 + 1
  18. A17 + 1
  19. A18 + 1
  20. A19 + 1
  21. A20 + 1
  22. A21 + 1
  23. A22 + 1
  24. A23 + 1
  25. A24 + 1
  26. A25 + 1
  27. Numeric B2(3.0)
  28. B2 - f(B2) / f'(B2)
  29. B3 - f(B3) / f'(B3)
  30. B4 - f(B4) / f'(B4)
  31. B5 - f(B5) / f'(B5)
  32. B6 - f(B6) / f'(B6)
  33. B7 - f(B7) / f'(B7)
  34. B8 - f(B8) / f'(B8)
  35. B9 - f(B9) / f'(B9)
  36. B10 - f(B10) / f'(B10)
  37. B11 - f(B11) / f'(B11)
  38. B12 - f(B12) / f'(B12)
  39. B13 - f(B13) / f'(B13)
  40. B14 - f(B14) / f'(B14)
  41. B15 - f(B15) / f'(B15)
  42. B16 - f(B16) / f'(B16)
  43. B17 - f(B17) / f'(B17)
  44. B18 - f(B18) / f'(B18)
  45. B19 - f(B19) / f'(B19)
  46. B20 - f(B20) / f'(B20)
  47. B21 - f(B21) / f'(B21)
  48. B22 - f(B22) / f'(B22)
  49. B23 - f(B23) / f'(B23)
  50. B24 - f(B24) / f'(B24)
  51. B25 - f(B25) / f'(B25)
  52. g(x) = f(B2) + (f'(B2) * (x - B2))
  53. (B2, f(B2))
  54. (B3, 0)
  55. (B3, f(B3))
  56. h(x) = f(B3) + (f'(B3) * (x - B3))
  57. (B4, 0)
  58. (B2, 0)
  59. (B4, f(B4))
  60. p(x) = f(B4) + (f'(B4) * (x - B4))

AI Drawing Prompt Reference

Draw the graph of the function y=x^2-2 and use Newton's method to find its roots. Start from x=1 and iteratively calculate the new x value until convergence. Show the initial point, iteration points, tangents, and the function curve in the graph.
The prompt above was compiled from the original drawing commands and is for reference only.