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指数增长与线性增长的动态对比

指数增长与线性增长的动态对比
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2020-10-09 17:23
函数几何作图
本范例通过动态几何演示展示指数增长与线性增长的差异。核心数学概念包括指数函数、线性方程及几何变换。作图要点涉及使用GeoGebra命令如Vector、Translate、Polygon、Dilate等构建图形。适合高中生学习函数图像与几何变换的应用场景。

GeoGebra 命令及参数

  1. Vector((0, 0), (1, 0)) → u
  2. Translate(G, u) → G'
  3. Translate(B, u) → B'
  4. Translate(C, u) → C'
  5. Translate(D, u) → D'
  6. Translate(A, u) → A'
  7. Polygon(A, A', B) → poly1, b, a_1, a'
  8. Polygon(B, B', C) → poly2, c, b_1, b'
  9. Polygon(C, C', D) → poly3, d, c_1, c'
  10. Polygon(D, D', E) → poly4, e, d_1, d'
  11. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], A') → A'_1
  12. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], B') → B'_1
  13. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], C') → C'_1
  14. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], D') → D'_1
  15. Segment(A, A'_1) → p
  16. Segment(B, B'_1) → q
  17. Segment(C, C'_1) → r
  18. Segment(D, D'_1) → s
  19. Dilate(A'_1, If[1 < o ≤ 2, 2 - o, 0], B) → H
  20. Segment(A'_1, H) → t
  21. Dilate(A'_1, If[2 < o ≤ 3, 3 - o, 0], B'_1) → I
  22. Vector(A'_1, I) → v
  23. Translate(A'_1, v) → J
  24. Translate(H, v) → H'
  25. Segment(J, H') → t'
  26. Dilate(B'_1, If[3 < o ≤ 4, 4 - o, 0], C'_1) → K
  27. Vector(B'_1, K) → w
  28. Translate(J, w) → J'
  29. Translate(H', w) → H''
  30. Segment(J', H'') → t''
  31. Dilate(C'_1, If[4 < o ≤ 5, 5 - o, 0], D'_1) → M
  32. Vector(C'_1, M) → u_1
  33. Translate(J', u_1) → J''
  34. Translate(H'', u_1) → H'''
  35. Segment(J'', H''') → t'''
  36. Translate(L, u) → L'
  37. Translate(F, u) → F'
  38. Line(A, B) → f_1
  39. IntegralBetween(f, g, 0, 1) → e_1
  40. Translate(O, u) → O'
  41. IntegralBetween(f, g_1, 1, 2) → h_1
  42. IntegralBetween(f, h, 2, 3) → i_1
  43. Dilate(O, If[0 < p_1 ≤ 1, 1 - p_1, 0], O') → O'_1
  44. Dilate(L, If[0 < p_1 ≤ 1, 1 - p_1, 0], L') → L'_1
  45. Dilate(G, If[0 < p_1 ≤ 1, 1 - p_1, 0], G') → G'_1
  46. Segment(G, G'_1) → k_1
  47. Segment(O, O'_1) → l_1
  48. Segment(L, L'_1) → m_1
  49. Dilate(G'_1, If[1 < p_1 ≤ 2, 2 - p_1, 0], O) → P
  50. Segment(G'_1, P) → q_1
  51. Midpoint(O'_1, L) → Q
  52. Midpoint(L'_1, F) → R
  53. Dilate(G'_1, If[2 < p_1 ≤ 3, 3 - p_1, 0], O'_1) → S
  54. Vector(G'_1, S) → v_1
  55. Translate(G'_1, v_1) → T
  56. Translate(P, v_1) → P'
  57. Segment(T, P') → j_1
  58. Dilate(G'_1, If[3 < p_1 ≤ 4, 4 - p_1, 0], Q) → U
  59. Vector(G'_1, U) → w_1
  60. Translate(G'_1, w_1) → V
  61. Translate(P, w_1) → P'_1
  62. Segment(V, P'_1) → n_1
  63. Segment(O'_1, L) → r_1
  64. Dilate(O'_1, If[4 < p_1 ≤ 5, 5 - p_1, 0], L'_1) → W
  65. Vector(O'_1, W) → s_1
  66. Translate(O'_1, s_1) → Z
  67. Translate(L, s_1) → L'_2
  68. Segment(Z, L'_2) → t_1
  69. Dilate(O'_1, If[5 < p_1 ≤ 6, 6 - p_1, 0], R) → A_1
  70. Vector(O'_1, A_1) → u_2
  71. Translate(O'_1, u_2) → B_1
  72. Translate(L, u_2) → L'_3
  73. Segment(B_1, L'_3) → f_2
  74. Translate(G, u) → G'_2
  75. Translate(C_1, u) → F_1
  76. Translate(D_1, u) → G_1
  77. Dilate(G, If[0 < a_2 ≤ 1, 1 - a_2, 0], G'_2) → G'_3
  78. Dilate(C_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], F_1) → H_1
  79. Dilate(D_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], G_1) → I_1
  80. Segment(G, G'_3) → g_2
  81. Segment(C_1, H_1) → h_2
  82. Segment(D_1, I_1) → i_2
  83. Dilate(G'_2, If[1 < a_2 ≤ 2, 2 - a_2, 0], C_1) → J_1
  84. Segment(G'_2, J_1) → j_2
  85. IntegralBetween(f, g, 0, 1) → b_2
  86. IntegralBetween(f, g_1, 1, 2) → c_2
  87. IntegralBetween(f, h, 2, 3) → d_2
  88. Segment(D_1, F_1) → k_2
  89. Circle(F_1, e_2) → p_2
  90. Intersect(p_2, k_2) → K_1,
  91. Circle(K_1, e_2) → q_2
  92. Intersect(q_2, k_2) → L_1, M_1
  93. Dilate(G'_2, If[2 < a_2 ≤ 3, 3 - a_2, 0], M_1) → N_1
  94. Vector(G'_2, N_1) → v_2
  95. Translate(G'_2, v_2) → O_1
  96. Translate(J_1, v_2) → P_1
  97. Segment(O_1, P_1) → l_2
  98. Rotate(O_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → R_1
  99. Rotate(P_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → S_1
  100. Segment(R_1, S_1) → m_2
  101. Rotate(R_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → T_1
  102. Rotate(S_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → U_1
  103. Segment(T_1, U_1) → n_2
  104. Segment(M_1, D_1) → r_2
  105. Dilate(M_1, If[5 < a_2 ≤ 6, 6 - a_2, 0], G_1) → Q_1
  106. Vector(M_1, Q_1) → w_2
  107. Translate(M_1, w_2) → V_1
  108. Translate(D_1, w_2) → W_1
  109. Segment(V_1, W_1) → s_2
  110. Midpoint(W_1, E_1) → Z_1
  111. Rotate(V_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → A_2
  112. Rotate(W_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → B_2
  113. Segment(A_2, B_2) → t_2
  114. Rotate(A_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → C_2
  115. Rotate(B_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → D_2
  116. Segment(C_2, D_2) → f_3

AI 绘图提示语参考

绘制一个动态演示指数增长与线性增长对比的图形。画面中有五条垂直的阶梯状多边形，从左到右依次排列，每一级阶梯的高度递增，体现指数加速增长的特征；同时有另一组平行排列的阶梯状多边形，毎级高度均匀递增，体现线性匀速增长的特征。图形下方有函数曲线围成的积分区域阴影。随着画面从左到右动态展示，阶梯的每一级逐步展开，积分区域的面积随之变化，清晰对比两种增长模式的差异。整体呈现为数学图表形式，包含坐标轴、曲线和多边形填充区域。
以上提示语为根据原始绘图命令整理而来，仅供参考。