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正弦函数的导数直观发现 by Tim Brzezinski

正弦函数的导数直观发现 by Tim Brzezinski
几何画板及绘图内容加载中，请稍等……
2015-10-07 23:23
函数三角函数导数几何作图函数图像
①本范例演示学生如何直观发现正弦函数y=sin(x)的导数为余弦函数y=cos(x)；②涉及核心数学概念：导数、正弦函数、余弦函数、几何作图；③作图要点：使用了Random, RandomElement, Point, Circle, Intersect, Vector等GeoGebra命令和对象，如点、圆、交点、向量等；④适合高中数学教学，用于帮助学生理解导数的几何意义和三角函数的性质。

GeoGebra 命令及参数

  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, 0)) → 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, 1)) → 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, 1.2246467991473532E-16)) → 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)) → 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, -2.4492935982947064E-16)) → 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)) → 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, -1.2246467991473532E-16)) → 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, 1)) → 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, (-6.283185307195264, -1.5677880535186782E-11)) → I_1
  66. Vector(I_1, I_2) → u_I
  67. Vector(I_1, I_3) → v_I

AI 绘图提示语参考

画一个正弦函数 y=sin(x) 的图像，并在曲线上随机选取多个点，以每个点为圆心画半径为1的圆，圆与正弦曲线的切线相交于两点，连接切点与交点形成向量，展示正弦函数的导数是余弦函数的直观几何意义，包含多个点的向量构造过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。