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Intuitive Derivative Graphing Exercise: REVAMPED!!!

Intuitive Derivative Graphing Exercise: REVAMPED!!!
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2018-06-30 09:48
FunctionCalculusDerivativeOptimizationGraphing
This example demonstrates how to graph derivative functions intuitively. It involves core math concepts like derivatives, tangent lines, function graphs, and optimization problems. Key GeoGebra commands used include FitPoly, Circle, Point, Intersect, Segment, Dilate, Vector, and Translate. Suitable for high school and above students to understand the geometric meaning of derivatives.

GeoGebra Commands & Parameters

  1. FitPoly({A, B, C, D}, 3) → f
  2. Circle(A, 1) → c
  3. Circle(B, 1) → d
  4. Circle(C, 1) → e
  5. Circle(D, 1) → p
  6. Point(f) → Q
  7. Point(a) → R
  8. Circle(Q, 1) → c_1
  9. Intersect(c_1, b) → S, T
  10. Segment(S, T) → f_1
  11. Point(f) → V
  12. Point(f) → W
  13. Point(f) → Z
  14. Point(f) → A_1
  15. Point(g_1) → B_1
  16. Point(h_1) → C_1
  17. Point(i_1) → D_1
  18. Point(j_1) → E_1
  19. Circle(V, 1) → d_1
  20. Circle(W, 1) → e_1
  21. Circle(Z, 1) → p_1
  22. Circle(A_1, 1) → q_1
  23. Intersect(p_1, m_1) → F_1, G_1
  24. Intersect(q_1, n_1) → H_1, I_1
  25. Intersect(e_1, l_1) → J_1, K_1
  26. Intersect(d_1, k_1) → L_1, M_1
  27. Segment(L_1, M_1) → r_1
  28. Segment(J_1, K_1) → s_1
  29. Segment(F_1, G_1) → t_1
  30. Segment(H_1, I_1) → a_1
  31. Dilate(U, If[0 < u ≤ 1, 1 - u, 0], N_1) → U'
  32. Segment(U, U') → b_1
  33. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], V) → U'_1
  34. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], A) → U'_2
  35. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], Q) → U'_3
  36. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], W) → U'_5
  37. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], C) → U'_6
  38. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], Z) → U'_7
  39. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], D) → U'_8
  40. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], A_1) → U'_9
  41. Vector(U, U'_1) → v
  42. Vector(U, U'_2) → w
  43. Vector(U, U'_3) → u_1
  44. Vector(U, U'_5) → w_1
  45. Vector(U, U'_6) → u_2
  46. Vector(U, U'_7) → v_2
  47. Vector(U, U'_8) → w_2
  48. Vector(U, U'_9) → a_2
  49. Translate(U, w) → U'_{10}
  50. Translate(U', w) → U''
  51. Translate(U, u_1) → U'_{11}
  52. Translate(U', u_1) → U''_1
  53. Segment(U'_{11}, U''_1) → g_2
  54. Translate(U, v) → U'_{12}
  55. Translate(U', v) → U''_2
  56. Segment(U'_{12}, U''_2) → h_2
  57. Translate(U, w_1) → U'_{13}
  58. Translate(U', w_1) → U''_3
  59. Segment(U'_{13}, U''_3) → i_2
  60. Translate(U, v_2) → U'_{14}
  61. Translate(U', v_2) → U''_4
  62. Segment(U'_{14}, U''_4) → j_2
  63. Translate(U, w_2) → U'_{15}
  64. Translate(U', w_2) → U''_5
  65. Translate(U, a_2) → U'_{16}
  66. Translate(U', a_2) → U''_6
  67. Segment(U'_{16}, U''_6) → l_2
  68. Translate(U, w_1) → U'_{17}
  69. Translate(U', w_1) → U''_7
  70. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], B) → U'_4
  71. Vector(U, U'_4) → v_1
  72. Translate(U, v_1) → U'_{18}
  73. Translate(U', v_1) → U''_8
  74. Dilate(U, If[1 < u ≤ 2, 2 - u, 0], C) → U'_{19}
  75. Vector(U, U'_6) → b_2
  76. Translate(U, b_2) → U'_{20}
  77. Translate(U', b_2) → U''_9
  78. Intersect(g, xAxis) → O_1
  79. Intersect(h, xAxis) → P_1
  80. Intersect(i, xAxis) → Q_1
  81. Intersect(xAxis, j) → R_1
  82. Intersect(g_1, xAxis) → S_1
  83. Intersect(h_1, xAxis) → T_1
  84. Intersect(i_1, xAxis) → U_1
  85. Intersect(a, xAxis) → V_1
  86. Vector(S_1, B_1) → c_2
  87. Vector(V_1, R) → e_2
  88. Vector(T_1, C_1) → r_2
  89. Vector(U_1, D_1) → t_2
  90. Dilate(S_1, If[3 < u ≤ 4, 4 - u, 0], U''_2) → W_1
  91. Dilate(O_1, If[3 < u ≤ 4, 4 - u, 0], U'') → E'
  92. Dilate(V_1, If[3 < u ≤ 4, 4 - u, 0], U''_1) → R'
  93. Dilate(P_1, If[3 < u ≤ 4, 4 - u, 0], U''_8) → Z_1
  94. Dilate(T_1, If[3 < u ≤ 4, 4 - u, 0], U''_7) → A_2
  95. Dilate(Q_1, If[3 < u ≤ 4, 4 - u, 0], U''_9) → B_2
  96. Dilate(U_1, If[3 < u ≤ 4, 4 - u, 0], U''_4) → C_2
  97. Dilate(R_1, If[3 < u ≤ 4, 4 - u, 0], U''_5) → D_2
  98. Dilate(R_1, If[3 < u ≤ 4, 4 - u, 0], U''_5) → E_2
  99. Dilate(F_2, If[3 < u ≤ 4, 4 - u, 0], U''_6) → G_2
  100. Translate(W_1, c_2) → H_2
  101. Vector(W_1, H_2) → v_3
  102. Translate(R', e_2) → R''
  103. Vector(R', R'') → a_3
  104. Translate(A_2, r_2) → J_2
  105. Vector(A_2, J_2) → c_3
  106. Translate(C_2, t_2) → L_2
  107. Vector(C_2, L_2) → e_3
  108. Vector(F_2, E_1) → g_3
  109. Translate(G_2, g_3) → N_2
  110. Vector(G_2, N_2) → h_3

AI Drawing Prompt Reference

Draw a function graph and its derivative graph to illustrate the geometric meaning of derivatives. First, fit a cubic polynomial as the original function. Then, mark several points on the function graph, draw circles centered at these points, and find their intersections with the function graph to get tangent lines. Next, draw several unit circles centered at points on the original function graph, intersect these circles with horizontal lines or the derivative function graph, and connect the intersection points to form approximate segments of the derivative graph. Finally, extend two line segments from the points on the original function graph to the left and right, representing the trend of function value changes.
The prompt above was compiled from the original drawing commands and is for reference only.