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Lagrange Multipliers Example 7

Lagrange Multipliers Example 7
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2020-07-16 01:39
CalculusDerivativeOptimization
This example demonstrates the application of Lagrange multipliers in solving optimization problems. Key mathematical concepts include derivatives and finding extrema under constraints. Using GeoGebra commands such as Point, Vector, and Simplify, an optimization problem is geometrically modeled. Suitable for high school calculus education and learning optimization problems.

GeoGebra Commands & Parameters

  1. Point(Cu_g) → B
  2. Simplify(Derivative[f, x, 1]) → fx'
  3. Simplify(Derivative[f, y, 1]) → fy'
  4. LeftSide(Cu_g) → g
  5. Simplify(Derivative[g, y, 1]) → gy'
  6. Simplify(Derivative[g, x, 1]) → gx'
  7. Vector(B, B + UnitVector[Vector[B, Df]] / n_v) → u_f
  8. Vector(B, B + UnitVector[Vector[B, Dg]] / n_v) → u_g
  9. Angle(u_g, u_f) → α
  10. LaTeX(f, true) → text8a
  11. LaTeX(Cu_g, true) → text8b
  12. LaTeX(Cu0_y, true) → txtCu0_y
  13. Textfield(f) → InputBox3
  14. Textfield(Cu_g) → InputBox4
  15. Surface(sqrt(cos(t)), (0.8 - cos(t)^(2))^(0.25), u, t, (-π), π, u, -3, 1) → g1
  16. Surface((-sqrt(cos(t))), (0.8 - cos(t)^(2))^(0.25), u, t, (-π), π, u, -3, 1) → g2
  17. Surface(sqrt(cos(t)), (-(0.8 - cos(t)^(2))^(0.25)), u, t, (-π), π, u, -3, 1) → g1a
  18. Surface((-sqrt(cos(t))), (-(0.8 - cos(t)^(2))^(0.25)), u, t, (-π), π, u, -3, 1) → g2a
  19. Textfield(n) → InputBox5
  20. Element(l1, Indexmax1) → Amax1
  21. Element(l1, Indexmin1) → Amin1
  22. Element(l2, Indexmin2) → Amin2
  23. Element(l2, Indexmax2) → Amax2
  24. Element(l1f, Indexmax1) → Amax1_{3D}
  25. Element(l2f, Indexmax2) → Amax2_{3D}
  26. Element(l1f, Indexmin1) → Amin1_{3D}
  27. Element(l2f, Indexmin2) → Amin2_{3D}
  28. TableText(Append["N", 1…Length[l1]], Append["\fgcolor{red}{Cp}", l1], Append["\fgcolor{blue}{f}(\fgcolor{red}{C_p })", l1fa], Append["(\dot{f_x},\dot{f_y})", l1xy'_fa], Append["(\dot{g_x},\dot{g_y})", l1xy'_ga], Append["λ_x", lλ1ax], Append["λ_y", lλ1ay], Append["\widehat{\dot{g}\dot{f}} ", lα1], "c|_") → tTab1
  29. TableText(Append["N", 1…Length[l2]], Append["\fgcolor{red}{Cp}", l2], Append["\fgcolor{blue}{f}(\fgcolor{red}{C_p })", l2fa], Append["(\dot{f_x},\dot{f_y})", l2xy'_fa], Append["(\dot{g_x},\dot{g_y})", l2xy'_ga], Append["λ_x", lλ2ax], Append["λ_y", lλ2ay], Append["\widehat{\dot{g}\dot{f}} ", lα2], "c|_") → tTab2
  30. Element(l1f, 5) → Amax15_{3D}
  31. Element(l1, 5) → Amax51
  32. Element(l1, 7) → Amin17
  33. Element(l1f, 7) → Amin17_{3D}
  34. Text("∇f", Point[u_f, 1]) → text25f
  35. Text("∇g", Point[u_g, 1]) → text25g
  36. RotateText(text14, 90°) → text15
  37. Segment(B, C) → f_1
  38. Sequence((0, 0, ((5 * 10^(-6)) * k / n_z)), k, 1, n_z) → lz_n
  39. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / n_v], a, l1xy'_g, c, l1]) → lvec1_g
  40. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / n_v], a, l1xy'_f, c, l1]) → lvec1_f
  41. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / n_v], a, l2xy'_f, c, l2]) → lvec2_f
  42. RemoveUndefined(Zip[Vector[c, c + UnitVector[a] / n_v], a, l2xy'_g, c, l2]) → lvec2_g
  43. Simplify(fx'(x, y) - (gx'(x, y) * fy'(x, y) / gy'(x, y))) → Cu_xf
  44. Simplify(fy'(x, y) - (gy'(x, y) * fx'(x, y) / gx'(x, y))) → Cu_yf
  45. Textfield(txtCu01_x) → InputBox1
  46. Textfield(txtCu01_y) → InputBox2
  47. Zip(Segment[a, b], a, l1, b, l1f) → l1seg
  48. Sequence(Text[i, l1(i) - (0.2, 0.1)], i, 1, lä1) → l1txt
  49. Sequence(Text[i, l2(i) - (0.2, 0.1)], i, 1, lä2) → l2txt
  50. Zip(Segment[a, b], a, l2, b, l2f) → l2seg
  51. Sequence(Translate[Cu_g, Vector[lz_n(k)]], k, 1, 10) → lz_c

AI Drawing Prompt Reference

Draw a 3D surface to illustrate the application of Lagrange multipliers in optimization problems. Mark several key points on the surface and draw the gradient vectors of the function and the constraint at these points, with the angle between them indicated by an arrow. Also, label the extrema of the function on the surface with different colors. Finally, list the key equations and parameters used in the calculations next to the graph in text boxes.
The prompt above was compiled from the original drawing commands and is for reference only.