积极 AI Logo 积极 AI

拉格朗日乘数法示例1

拉格朗日乘数法示例1
几何画板及绘图内容加载中，请稍等……
2020-07-12 20:06
向量微积分最值优化
这个范例演示了如何使用拉格朗日乘数法求解带约束条件的优化问题。涉及的核心数学概念包括拉格朗日乘数、梯度、等高线和最优解。作图要点使用了GeoGebra的LaTeX、LeftSide、Simplify、Point、UnitVector、Angle、Zip、Length、CurveCartesian、Text、Segment、RotateText、Surface、Textfield等命令和对象来构建函数图像、等高线、梯度向量和最优解点。适合大学数学或物理专业的学生学习和理解最值优化问题。

GeoGebra 命令及参数

  1. LaTeX(Cu_g, true) → text8b
  2. LeftSide(Cu_g) → g
  3. Simplify(Derivative[g, x, 1]) → gx'
  4. Simplify(Derivative[g, y, 1]) → gy'
  5. Simplify(Derivative[f, y, 1]) → fy'
  6. Simplify(Derivative[f, x, 1]) → fx'
  7. Point(Cu_g) → B
  8. UnitVector(Vector[B, Df]) → u_f
  9. UnitVector(Vector[B, Dg]) → u_g
  10. Angle(u_g, u_f) → α
  11. Zip((x(a), y(a), f(a)), a, Sequence[Point[Cu_g, t / 100], t, 0, 100]) → lg_1
  12. Zip((x(a), y(a), f(a)), a, Flatten[l1]) → l1f
  13. Zip((x(a), y(a), f(a)), a, Flatten[l2]) → l2f
  14. Zip(Segment[a, b], a, l1, b, l1f) → l1seg
  15. Length(l1) → lä1
  16. Length(l2) → lä2
  17. Zip(Segment[a, b], a, l2, b, l2f) → l2seg
  18. CurveCartesian(3 + (4 * cos(t)), (sqrt(8) * sin(t)), f(3 + (4 * cos(t)), (sqrt(8) * sin(t))), t, 0, (2 * π)) → q3c
  19. Text("∇f", Point[u_f, 1]) → text25f
  20. Text("∇g", Point[u_g, 1]) → text25g
  21. Segment(B, C) → f_1
  22. RotateText(text14, 90°) → text15
  23. Surface(3 + (4 * cos(t)), (sqrt(8) * sin(t)), u, t, 0, (2 * π), u, 10, 140) → g3c
  24. Textfield(txtCu01_x) → InputBox1
  25. LaTeX(Min, true, true) → text4
  26. Simplify(fx'(x, y) - (gx'(x, y) * fy'(x, y) / gy'(x, y))) → Cu_xf
  27. Simplify(fy'(x, y) - (gy'(x, y) * fx'(x, y) / gx'(x, y))) → Cu_yf
  28. Textfield(txtCu01_y) → InputBox2