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2019-01-14 04:55
三角函数解析几何微积分最值优化几何作图
①本范例通过动态图形演示了连续函数的介值定理；②涉及核心数学概念包括连续性、斜率、三角形性质；③作图要点：使用Point, Mirror, Semicircle, Intersect, CircleArc, Line, AngularBisector, OrthogonalLine, Circle, Polygon, Midpoint, Rotate, Angle, If等GeoGebra命令构建几何模型；④适合高中及以上学生及数学爱好者用于理解抽象概念的可视化表达。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Mirror(A, yAxis) → A'
  3. Semicircle(A', A) → c
  4. Intersect(c, yAxis, 1) → B
  5. Intersect(xAxis, yAxis) → C
  6. CircleArc(C, B, A') → d
  7. Point(d) → D
  8. Line(A', D) → f
  9. Line(D, A) → g
  10. Line(A, A') → h
  11. AngularBisector(f, h) → i, j
  12. AngularBisector(h, g) → k, l
  13. Intersect(j, k) → E
  14. Intersect(i, l) → F
  15. Intersect(i, k) → G
  16. OrthogonalLine(E, g) → m
  17. OrthogonalLine(F, f) → n
  18. OrthogonalLine(G, h) → p
  19. Intersect(h, p) → H
  20. Intersect(g, m) → I
  21. Intersect(f, n) → J
  22. Circle(F, J) → e
  23. Circle(E, I) → q
  24. Circle(G, H) → r
  25. Polygon(A', D, 4) → poly1, s, t, a, b, K, L
  26. Polygon(D, A, 4) → poly2, f_1, g_1, h_1, i_1, M, N
  27. Polygon(A, A', 4) → poly3, j_1, k_1, l_1, m_1, O, P
  28. Midpoint(L, D) → Q
  29. Midpoint(D, M) → R
  30. Mirror(J, Q) → J'
  31. Mirror(I, R) → I'
  32. Rotate(J, 90°, Q) → J'_1
  33. Mirror(J'_1, Q) → S
  34. Polygon(S, J, 4) → poly4, n_1, p_1, q_1, r_1, T, U
  35. Rotate(I, 90°, R) → II
  36. Polygon(I, II, 4) → poly5, s_1, t_1, a_1, b_1, V, W
  37. Midpoint(A', P) → Z
  38. Polygon(A', S, J) → t1, j_2, a', s_2
  39. Polygon(S, L, J') → t2, j', s_3, l_2
  40. Polygon(J', K, J'_1) → t3, j'_1, c_1, k_2
  41. Polygon(J'_1, D, J) → t4, j_3, d_1, d_2
  42. Polygon(W, I', N) → t5, n_2, w, i'
  43. Polygon(W, D, I) → t6, i_2, w_1, d_3
  44. Polygon(I, A, II) → t7, i'_1, i_3, a_2
  45. Polygon(II, I', M) → t8, m_2, e_1, f_2
  46. Polygon(II, R, I') → t9, g_2, h_2, r_2
  47. Polygon(II, R, I) → t10, i_4, p_2, r_3
  48. Polygon(I, R, W) → t11, w_2, i_5, r_4
  49. Polygon(R, W, I') → t12, q_2, r_5, w_3
  50. Angle(A, A', D) → α
  51. Angle(D, A, A') → β
  52. If(0 < o ≤ 1, Dilate[S, 1 - o, O], O) → A_1
  53. If(0 < o ≤ 1, Dilate[U, 1 - o, A'], A') → B_1
  54. If(0 < o ≤ 1, Dilate[T, 1 - o, A], A) → C_1
  55. If(0 < o ≤ 1, Dilate[J, 1 - o, P], P) → D_1
  56. If(0 < o ≤ 1, Rotate[Polygon[Translate[A', A_1 - S], Translate[S, A_1 - S], Translate[J, A_1 - S]], (o * β), A_1], Rotate[Polygon[Translate[A', A_1 - S], Translate[S, A_1 - S], Translate[J, A_1 - S]], β, A_1]) → t13
  57. If(0 < o ≤ 1, Rotate[Polygon[Translate[S, B_1 - U], Translate[L, B_1 - U], Translate[J', B_1 - U]], (o * β), B_1], Rotate[Polygon[Translate[S, B_1 - U], Translate[L, B_1 - U], Translate[J', B_1 - U]], β, A']) → t14
  58. If(0 < o ≤ 1, Rotate[Polygon[Translate[J', C_1 - T], Translate[K, C_1 - T], Translate[J'_1, C_1 - T]], (o * β), C_1], Rotate[Polygon[Translate[J', C_1 - T], Translate[K, C_1 - T], Translate[J'_1, C_1 - T]], β, A]) → t15
  59. If(0 < o ≤ 1, Rotate[Polygon[Translate[J'_1, D_1 - J], Translate[D, D_1 - J], Translate[J, D_1 - J]], (o * β), D_1], Rotate[Polygon[Translate[J'_1, D_1 - J], Translate[D, D_1 - J], Translate[J, D_1 - J]], β, P]) → t16
  60. If(2 < o ≤ 3, Dilate[I, 3 - o, O], O) → E_1
  61. If(2 < o ≤ 3, Dilate[W, 3 - o, A'], A') → F_1
  62. If(2 < o ≤ 3, Dilate[V, 3 - o, A], A) → G_1
  63. If(2 < o ≤ 3, Dilate[II, 3 - o, P], P) → H_1
  64. If(2 < o ≤ 3, Rotate[Polygon[Translate[W, E_1 - I], Translate[D, E_1 - I], Translate[I, E_1 - I]], ((o - 2) * (β - 90°)), E_1], Rotate[Polygon[Translate[W, E_1 - I], Translate[D, E_1 - I], Translate[I, E_1 - I]], β - 90°, O]) → t17
  65. If(2 < o ≤ 3, Rotate[Polygon[Translate[W, F_1 - W], Translate[I', F_1 - W], Translate[N, F_1 - W]], ((o - 2) * (β - 90°)), F_1], Rotate[Polygon[Translate[W, F_1 - W], Translate[I', F_1 - W], Translate[N, F_1 - W]], β - 90°, A']) → t18
  66. If(2 < o ≤ 3, Rotate[Polygon[Translate[II, G_1 - V], Translate[I', G_1 - V], Translate[M, G_1 - V]], ((o - 2) * (β - 90°)), G_1], Rotate[Polygon[Translate[II, G_1 - V], Translate[I', G_1 - V], Translate[M, G_1 - V]], β - 90°, A]) → t19
  67. If(2 < o ≤ 3, Rotate[Polygon[Translate[I, H_1 - II], Translate[A, H_1 - II], Translate[II, H_1 - II]], ((o - 2) * (β - 90°)), H_1], Rotate[Polygon[Translate[I, H_1 - II], Translate[A, H_1 - II], Translate[II, H_1 - II]], β - 90°, P]) → t20
  68. If(4 < o ≤ 5, Dilate[I, 5 - o, O], O) → I_1
  69. If(4 < o ≤ 5, Dilate[W, 5 - o, A'], A') → J_1
  70. If(4 < o ≤ 5, Dilate[V, 5 - o, A], A) → K_1
  71. If(4 < o ≤ 5, Dilate[II, 5 - o, P], P) → L_1
  72. If(4 < o ≤ 5, Rotate[Polygon[Translate[I, I_1 - I], Translate[R, I_1 - I], Translate[W, I_1 - I]], ((o - 4) * (β - 90°)), I_1], Rotate[Polygon[Translate[I, I_1 - I], Translate[R, I_1 - I], Translate[W, I_1 - I]], β - 90°, O]) → t21
  73. If(4 < o ≤ 5, Rotate[Polygon[Translate[R, J_1 - W], Translate[W, J_1 - W], Translate[I', J_1 - W]], ((o - 4) * (β - 90°)), J_1], Rotate[Polygon[Translate[R, J_1 - W], Translate[W, J_1 - W], Translate[I', J_1 - W]], β - 90°, A']) → t22
  74. If(4 < o ≤ 5, Rotate[Polygon[Translate[II, K_1 - V], Translate[R, K_1 - V], Translate[I', K_1 - V]], ((o - 4) * (β - 90°)), K_1], Rotate[Polygon[Translate[II, K_1 - V], Translate[R, K_1 - V], Translate[I', K_1 - V]], β - 90°, A]) → t23
  75. If(4 < o ≤ 5, Rotate[Polygon[Translate[II, L_1 - II], Translate[R, L_1 - II], Translate[I, L_1 - II]], ((o - 4) * (β - 90°)), L_1], Rotate[Polygon[Translate[II, L_1 - II], Translate[R, L_1 - II], Translate[I, L_1 - II]], β - 90°, P]) → t24
  76. If(6 < o ≤ 7, Dilate[Q, 7 - o, Z], Z) → M_1
  77. Angle(A', S, J) → δ
  78. Angle(W, I, D) → ε
  79. If(6 < o ≤ 7, Rotate[Translate[poly4, M_1 - Q], ((o - 6) * β), M_1], Rotate[Translate[poly4, M_1 - Q], β, Z]) → q1
  80. Angle(A', D, A) → ζ

AI 绘图提示语参考

画一个动态几何图形，展示连续函数的中值定理。包含一个半圆和两个相交的圆弧，通过几何构造展示连续性、斜率和三角形性质。先画一条横线和一条纵线，以纵线为轴镜像半圆，再交点构造圆弧和线段，绘制角平分线和垂线，最后形成多个多边形和圆，体现几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。