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指数增长与衰减的动态演示

指数增长与衰减的动态演示
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2020-03-28 00:54
函数变换
这个GeoGebra范例动态展示了指数增长与衰减的过程。它涉及的核心数学概念是指数函数。通过使用Vector, Translate, Polygon, Dilate, Segment, Line, IntegralBetween, Circle, Intersect, Rotate, Midpoint, Distance, Textfield等命令，用户可以直观地理解指数函数的性质。适合高中数学教学和学习。

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(O, O'_1) → l_1
  47. Segment(L, L'_1) → m_1
  48. Dilate(G'_1, If[1 < p_1 ≤ 2, 2 - p_1, 0], O) → P
  49. Segment(G'_1, P) → q_1
  50. Dilate(G'_1, If[2 < p_1 ≤ 3, 3 - p_1, 0], O'_1) → S
  51. Vector(G'_1, S) → v_1
  52. Translate(G'_1, v_1) → T
  53. Translate(P, v_1) → P'
  54. Segment(O'_1, L) → r_1
  55. Dilate(O'_1, If[4 < p_1 ≤ 5, 5 - p_1, 0], L'_1) → W
  56. Vector(O'_1, W) → s_1
  57. Translate(O'_1, s_1) → Z
  58. Translate(L, s_1) → L'_2
  59. Translate(G, u) → G'_2
  60. Translate(C_1, u) → F_1
  61. Translate(D_1, u) → G_1
  62. Dilate(G, If[0 < a_2 ≤ 1, 1 - a_2, 0], G'_2) → G'_3
  63. Dilate(C_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], F_1) → H_1
  64. Dilate(D_1, If[0 < a_2 ≤ 1, 1 - a_2, 0], G_1) → I_1
  65. Segment(G, G'_3) → g_2
  66. Segment(C_1, H_1) → h_2
  67. Segment(D_1, I_1) → i_2
  68. Dilate(G'_2, If[1 < a_2 ≤ 2, 2 - a_2, 0], C_1) → J_1
  69. Segment(G'_2, J_1) → j_2
  70. IntegralBetween(f, g, 0, 1) → b_2
  71. IntegralBetween(f, g_1, 1, 2) → c_2
  72. IntegralBetween(f, h, 2, 3) → d_2
  73. Segment(D_1, F_1) → k_2
  74. Circle(F_1, e_2) → p_2
  75. Intersect(p_2, k_2) → K_1,
  76. Circle(K_1, e_2) → q_2
  77. Intersect(q_2, k_2) → L_1, M_1
  78. Dilate(G'_2, If[2 < a_2 ≤ 3, 3 - a_2, 0], M_1) → N_1
  79. Vector(G'_2, N_1) → v_2
  80. Translate(G'_2, v_2) → O_1
  81. Translate(J_1, v_2) → P_1
  82. Segment(O_1, P_1) → l_2
  83. Rotate(O_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → R_1
  84. Rotate(P_1, If[3 < a_2 ≤ 4, ((a_2 - 3) * π), π], K_1) → S_1
  85. Segment(R_1, S_1) → m_2
  86. Rotate(R_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → T_1
  87. Rotate(S_1, If[4 < a_2 ≤ 5, ((a_2 - 4) * π), π], L_1) → U_1
  88. Segment(T_1, U_1) → n_2
  89. Segment(M_1, D_1) → r_2
  90. Dilate(M_1, If[5 < a_2 ≤ 6, 6 - a_2, 0], G_1) → Q_1
  91. Vector(M_1, Q_1) → w_2
  92. Translate(M_1, w_2) → V_1
  93. Translate(D_1, w_2) → W_1
  94. Segment(V_1, W_1) → s_2
  95. Midpoint(W_1, E_1) → Z_1
  96. Rotate(V_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → A_2
  97. Rotate(W_1, If[6 < a_2 ≤ 7, ((a_2 - 6) * π), π], W_1) → B_2
  98. Segment(A_2, B_2) → t_2
  99. Rotate(A_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → C_2
  100. Rotate(B_2, If[7 < a_2 ≤ 8, ((a_2 - 7) * π), π], Z_1) → D_2
  101. Segment(C_2, D_2) → f_3
  102. Translate(F, u) → F'_1
  103. Intersect(f, n, (4, 12.713506250000002)) → E_2
  104. Dilate(F, If[0 < p_1 ≤ 1, 1 - p_1, 0], F'_1) → F'_2
  105. Segment(F, F'_2) → g_3
  106. Dilate(O'_1, If[2 < p_1 ≤ 3, 3 - p_1, 0], L) → Q
  107. Segment(O'_1, Q) → j_1
  108. Distance(O'_1, L) → distanceO'1L
  109. Dilate(L'_1, If[3 < p_1 ≤ 4, 4 - p_1, 0], F) → R
  110. Segment(L'_1, R) → t_1
  111. Distance(L'_1, F) → distanceL'1F
  112. Dilate(F'_1, If[4 < p_1 ≤ 5, 5 - p_1, 0], N) → U
  113. Segment(F'_1, U) → f_2
  114. IntegralBetween(f, k_1, x(F), x(F'_1)) → w_1
  115. Distance(F'_1, N) → distanceF'1N
  116. Textfield(base) → InputBox1
  117. Segment(G, G'_1) → i_3

AI 绘图提示语参考

绘制一个动态展示指数增长和衰减的图形。首先画出x轴和y轴，然后通过平移和缩放，创建一系列与x轴平行的线段，展示指数函数的性质。接着画出指数函数的图像，并用阴影表示积分区域。最后通过平移和缩放，动态展示指数增长和衰减的过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。