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Lagrange Multipliers Example 1

Lagrange Multipliers Example 1
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2020-07-12 20:06
VectorCalculusOptimization
This example demonstrates how to use Lagrange multipliers to solve optimization problems with constraints. The core mathematical concepts involved include Lagrange multipliers, gradients, contour lines, and optimal solutions. The construction utilizes GeoGebra commands and objects such as LaTeX, LeftSide, Simplify, Point, UnitVector, Angle, Zip, Length, CurveCartesian, Text, Segment, RotateText, Surface, and Textfield to plot function graphs, contour lines, gradient vectors, and optimal solution points. It is suitable for university-level students in mathematics or physics to learn and understand optimization problems.

GeoGebra Commands & Parameters

  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