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辛普森法则与梯形法则的视觉比较

辛普森法则与梯形法则的视觉比较
几何画板及绘图内容加载中，请稍等……
2018-06-22 17:34
函数微积分积分滑块动画函数图像
①本范例通过图形演示了辛普森法则与梯形法则在数值积分中的近似效果差异；②涉及的核心数学概念包括数值积分、误差分析、函数逼近；③作图要点提炼自构造摘要，使用了Sequence（序列）、Element（元素）、FitPoly（拟合多项式）、Integral（积分）、Sum（求和）、LaTeX（公式显示）、TrapezoidalSum（梯形求和）、TableText（表格文本）、Textfield（文本字段）等GeoGebra命令/对象；④适合高中及以上数学学习者，用于理解不同积分近似方法的精度与适用场景。

GeoGebra 命令及参数

  1. Sequence(p, p, a, b, (b - a) / ((2 * n))) → L_1
  2. Sequence((Element[L_1, i], f(Element[L_1, i])), i, 1, (2 * n) + 1) → L_2
  3. Sequence(FitPoly[Take[L_2, i, i + 2], 2], i, 1, (2 * n), 2) → L_3
  4. Element(L_1, A1) → B1
  5. Element(L_1, A2) → B2
  6. Element(L_1, A3) → B3
  7. Element(L_1, A4) → B4
  8. Element(L_1, A5) → B5
  9. Element(L_1, A6) → B6
  10. Element(L_1, A7) → B7
  11. Element(L_1, A8) → B8
  12. Element(L_1, A9) → B9
  13. Element(L_1, A10) → B10
  14. Element(L_1, A11) → B11
  15. Element(L_1, A12) → B12
  16. Element(L_1, A13) → B13
  17. Element(L_1, A14) → B14
  18. Element(L_1, A15) → B15
  19. Element(L_1, A16) → B16
  20. Element(L_1, A17) → B17
  21. Element(L_1, A18) → B18
  22. Element(L_1, A19) → B19
  23. Element(L_1, A20) → B20
  24. Element(L_1, A21) → B21
  25. Element(L_1, A22) → B22
  26. Element(L_1, A23) → B23
  27. Element(L_2, A1) → C1
  28. Element(L_2, A2) → C2
  29. Element(L_2, A3) → C3
  30. Element(L_2, A4) → C4
  31. Element(L_2, A5) → C5
  32. Element(L_2, A6) → C6
  33. Element(L_2, A7) → C7
  34. Element(L_2, A8) → C8
  35. Element(L_2, A9) → C9
  36. Element(L_2, A10) → C10
  37. Element(L_2, A11) → C11
  38. Element(L_2, A12) → C12
  39. Element(L_2, A13) → C13
  40. Element(L_2, A14) → C14
  41. Element(L_2, A15) → C15
  42. Element(L_2, A16) → C16
  43. Element(L_2, A17) → C17
  44. Element(L_2, A18) → C18
  45. Element(L_2, A19) → C19
  46. Element(L_2, A20) → C20
  47. Element(L_2, A21) → C21
  48. Element(L_2, A22) → C22
  49. Element(L_2, A23) → C23
  50. FitPoly({C1, C2, C3}, 2) → D1
  51. FitPoly({C3, C4, C5}, 2) → D3
  52. FitPoly({C5, C6, C7}, 2) → D5
  53. FitPoly({C7, C8, C9}, 2) → D7
  54. FitPoly({C9, C10, C11}, 2) → D9
  55. FitPoly({C11, C12, C13}, 2) → D11
  56. FitPoly({C13, C14, C15}, 2) → D13
  57. FitPoly({C15, C16, C17}, 2) → D15
  58. FitPoly({C17, C18, C19}, 2) → D17
  59. FitPoly({C19, C20, C21}, 2) → D19
  60. FitPoly({C21, C22, C23}, 2) → D21
  61. Integral(D1, x(C1), x(C3)) → F1
  62. Integral(D3, x(C3), x(C5)) → F3
  63. Integral(D5, x(C5), x(C7)) → F5
  64. Integral(D7, x(C7), x(C9)) → F7
  65. Integral(D9, x(C9), x(C11)) → F9
  66. Integral(D11, x(C11), x(C13)) → F11
  67. Integral(D13, x(C13), x(C15)) → F13
  68. Integral(D15, x(C15), x(C17)) → F15
  69. Integral(D17, x(C17), x(C19)) → F17
  70. Integral(D19, x(C19), x(C21)) → F19
  71. Integral(D21, x(C21), x(C23)) → F21
  72. Element(L_1, A24) → B24
  73. Element(L_2, A24) → C24
  74. Element(L_1, A25) → B25
  75. Element(L_2, A25) → C25
  76. Element(L_1, A26) → B26
  77. Element(L_2, A26) → C26
  78. Element(L_1, A27) → B27
  79. Element(L_2, A27) → C27
  80. Element(L_1, A28) → B28
  81. Element(L_2, A28) → C28
  82. Element(L_1, A29) → B29
  83. Element(L_2, A29) → C29
  84. Element(L_1, A30) → B30
  85. Element(L_2, A30) → C30
  86. Element(L_1, A31) → B31
  87. Element(L_2, A31) → C31
  88. Element(L_1, A32) → B32
  89. Element(L_2, A32) → C32
  90. Element(L_1, A33) → B33
  91. Element(L_2, A33) → C33
  92. Element(L_1, A34) → B34
  93. Element(L_2, A34) → C34
  94. Element(L_1, A35) → B35
  95. Element(L_2, A35) → C35
  96. Element(L_1, A36) → B36
  97. Element(L_2, A36) → C36
  98. Element(L_1, A37) → B37
  99. Element(L_2, A37) → C37
  100. Element(L_1, A38) → B38
  101. Element(L_2, A38) → C38
  102. Element(L_1, A39) → B39
  103. Element(L_2, A39) → C39
  104. Element(L_1, A40) → B40
  105. Element(L_2, A40) → C40
  106. Element(L_1, A41) → B41
  107. Element(L_2, A41) → C41
  108. FitPoly({C23, C24, C25}, 2) → D23
  109. FitPoly({C25, C26, C27}, 2) → D25
  110. FitPoly({C27, C28, C29}, 2) → D27
  111. FitPoly({C29, C30, C31}, 2) → D29
  112. FitPoly({C31, C32, C33}, 2) → D31
  113. FitPoly({C33, C34, C35}, 2) → D33
  114. FitPoly({C35, C36, C37}, 2) → D35
  115. FitPoly({C37, C38, C39}, 2) → D37
  116. FitPoly({C39, C40, C41}, 2) → D39
  117. Integral(D23, x(C23), x(C25)) → F23
  118. Integral(D25, x(C25), x(C27)) → F25
  119. Integral(D27, x(C27), x(C29)) → F27
  120. Integral(D29, x(C29), x(C31)) → F29
  121. Integral(D31, x(C31), x(C33)) → F31
  122. Integral(D33, x(C33), x(C35)) → F33
  123. Integral(D35, x(C35), x(C37)) → F35
  124. Integral(D37, x(C37), x(C39)) → F37
  125. Integral(D39, x(C39), x(C41)) → F39
  126. Sum(Sequence[Element[L_4, (2 * i) - 1], i, 1, n]) → simpson
  127. LaTeX($1) → exact
  128. LaTeX($2) → approx
  129. LaTeX("\text{Exact value = }" + exact + " ≈ " + approx) → text2
  130. LaTeX("\text{GeoGebra's approximation} = " + (LaTeX[$3])) → text4
  131. TrapezoidalSum(f, a, b, n) → trapezium
  132. LaTeX("\text{Calculation from Trapezium Rule} = " + trapezium) → text5
  133. Sequence(p, p, a, b, (b - a) / n) → L_6
  134. Sequence(f(Element[L_6, i]), i, 1, n + 1) → L_7
  135. TableText({L_6, L_7}, "c_|h") → text6
  136. TableText({L_1, Append["", First[L_4, (2 * n)]]}, "c_|h") → text3
  137. Textfield(a) → InputBox1
  138. Textfield(b) → InputBox2
  139. Textfield(f) → InputBox3