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Constructing Graphs of Derivative Functions (Intuitively)

Constructing Graphs of Derivative Functions (Intuitively)
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2015-09-29 17:37
FunctionCalculusDerivativeGraphing
This example demonstrates how to intuitively understand the construction of derivative function graphs by moving points. Core math concepts include derivatives, function graphs, upper and lower sums or Riemann sums, and tangent functions. Construction highlights using GeoGebra commands/objects from the summary: Random, RandomElement, Point, Circle, Intersect, Vector; objects: list1, f, b, c, d, a, g, vert_A, A, line_A, circle_A, A_2, A_3, A_1, u_A, v_A, vert_B, B, circle_B, line_B, B_1, B_2, B_3, u_B, v_B, vert_C, C, C_1, circle_C, line_C, C_2, C_3, u_C, v_C, vert_D, D, line_D, circle_D, D_1, D_2. Suitable for high school or introductory calculus students.

GeoGebra Commands & Parameters

  1. Random(1, 2) → b
  2. Random(-3, 3) → c
  3. Random(-3, 3) → d
  4. RandomElement(list1) → a
  5. Point(vert_A) → A
  6. Circle((x(A), g(x(A))), 1) → circle_A
  7. Intersect(circle_A, line_A, 1) → A_2
  8. Intersect(circle_A, line_A, 2) → A_3
  9. Intersect(g, vert_A, (0, 1)) → A_1
  10. Vector(A_1, A_3) → u_A
  11. Vector(A_1, A_2) → v_A
  12. Point(vert_B) → B
  13. Circle((x(B), g(x(B))), 1) → circle_B
  14. Intersect(g, vert_B, (1.5707963267948966, 3.5)) → B_1
  15. Intersect(circle_B, line_B, 1) → B_2
  16. Intersect(circle_B, line_B, 2) → B_3
  17. Vector(B_1, B_2) → u_B
  18. Vector(B_1, B_3) → v_B
  19. Point(vert_C) → C
  20. Intersect(g, vert_C, (3.141592653589793, 1)) → C_1
  21. Circle((x(C), g(x(C))), 1) → circle_C
  22. Intersect(circle_C, line_C, 1) → C_2
  23. Intersect(circle_C, line_C, 2) → C_3
  24. Vector(C_1, C_3) → u_C
  25. Vector(C_1, C_2) → v_C
  26. Point(vert_D) → D
  27. Circle((x(D), g(x(D))), 1) → circle_D
  28. Intersect(g, vert_D, (4.71238898038469, -1.5)) → D_1
  29. Intersect(circle_D, line_D, 1) → D_2
  30. Intersect(circle_D, line_D, 2) → D_3
  31. Vector(D_1, D_2) → u_D
  32. Vector(D_1, D_3) → v_D
  33. Point(vert_E) → E
  34. Circle((x(E), g(x(E))), 1) → circle_E
  35. Intersect(g, vert_E, (6.283185307179586, 1)) → E_1
  36. Intersect(circle_E, line_E, 1) → E_2
  37. Intersect(circle_E, line_E, 2) → E_3
  38. Vector(E_1, E_2) → u_E
  39. Vector(E_1, E_3) → v_E
  40. Point(vert_F) → F
  41. Circle((x(F), g(x(F))), 1) → circle_F
  42. Intersect(g, vert_F, (-1.5707963267948966, -1.5)) → F_1
  43. Intersect(circle_F, line_F, 1) → F_3
  44. Intersect(circle_F, line_F, 2) → F_2
  45. Vector(F_1, F_3) → u_F
  46. Vector(F_1, F_2) → v_F
  47. Point(vert_G) → G
  48. Circle((x(G), g(x(G))), 1) → circle_G
  49. Intersect(g, line_G, (-3.141592653589793, 1)) → G_1
  50. Intersect(circle_G, line_G, 1) → G_2
  51. Intersect(circle_G, line_G, 2) → G_3
  52. Vector(G_1, G_2) → u_G
  53. Vector(G_1, G_3) → v_G
  54. Point(vert_H) → H
  55. Circle((x(H), g(x(H))), 1) → circle_H
  56. Intersect(g, line_H, (-4.71238898038469, 3.5)) → H_1
  57. Intersect(circle_H, line_H, 1) → H_2
  58. Intersect(circle_H, line_H, 2) → H_3
  59. Vector(H_1, H_2) → u_H
  60. Vector(H_1, H_3) → v_H
  61. Point(vert_I) → I
  62. Circle((x(I), g(x(I))), 1) → circle_I
  63. Intersect(circle_I, line_I, 1) → I_2
  64. Intersect(circle_I, line_I, 2) → I_3
  65. Intersect(g, line_I, (-7.853981633974411, -2)) → I_1
  66. Vector(I_1, I_2) → u_I
  67. Vector(I_1, I_3) → v_I

AI Drawing Prompt Reference

Draw a function graph and its derivative graph, intuitively demonstrating the concept of derivatives by randomly selecting points and drawing circles and line segments. For each point on the function graph, draw a circle and find its intersections with the function. Connect these intersection points to form line segments that approximate tangents. Use these line segments and circles to show the local rate of change of the function, thereby intuitively understanding the geometric meaning of derivatives.
The prompt above was compiled from the original drawing commands and is for reference only.