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二元函数极值演示

二元函数极值演示
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2017-04-07 21:37
函数微积分最值优化
所属专题 ₁ΛM 0 -Critical points and their nature using hessian matrix
①本范例演示二元函数的极值点及其几何意义；②涉及核心数学概念包括偏导数、海森矩阵、函数极值判定；③构造要点：使用Derivative计算偏导数，IntersectPath求交点，Vector绘制向量，OrthogonalPlane创建法平面，Length测量距离等；④适合大学数学、微积分课程教学与学习。

GeoGebra 命令及参数

  1. Derivative(f, x) → f_x
  2. Derivative(f_x, y) → f_{xy}
  3. Derivative(f_x, x) → f_{xx}
  4. Derivative(f, y) → f_y
  5. Derivative(f_y, y) → f_{yy}
  6. IntersectPath(g, f_x) → ndx
  7. IntersectPath(g, f_y) → ndy
  8. Derivative(f_y, x) → f_{yx}
  9. Textfield(f) → CasillaDeEntrada1
  10. Vector(((-f_x(P_1)), (-f_y(P_1)), 1) / sqrt(1 + f_x(P_1)^(2) + f_y(P_1)^(2))) → u
  11. Vector(P_1, P_1 + u) → u'
  12. OrthogonalPlane(P_1, u') → a
  13. IntersectPath(a, f) → b
  14. Textfield(P) → CasillaDeEntrada2
  15. Length(lPc) → n
  16. Join({{{"(x, y)", "f(x,y)", "f_x(x,y)", "f_y(x,y)", "f_{xx}(x,y)", "H(x,y)", "Tipo"}}, Sequence[{lPc(k), f(lPc(k)), f_x(lPc(k)), f_y(lPc(k)), f_{xx}(lPc(k)), H(lPc(k)), If[f_x(lPc(k)) ≠ 0 ∨ f_y(lPc(k)) ≠ 0, "No crítico", f_{xx}(lPc(k)) > 0 ∧ H(lPc(k)) > 0, "Mínimo relativo", f_{xx}(lPc(k)) < 0 ∧ H(lPc(k)) > 0, "Máximo relativo", H(lPc(k)) ≠ 0, "Punto de silla", "??"]}, k, 1, Length[lPc]]}) → tPc
  17. TableText(tPc, "|_") → texto2
  18. Sequence((x(lPc(k)), y(lPc(k)), f(lPc(k))), k, 1, n) → lPcz
  19. Sequence({lPc(k), f_x(lPc(k)), f_y(lPc(k)), f_{xx}(lPc(k)), H(lPc(k)), If[f_x(lPc(k)) ≠ 0 ∨ f_y(lPc(k)) ≠ 0, "No crítico", f_{xx}(lPc(k)) > 0 ∧ H(lPc(k)) > 0, "Mínimo relativo", f_{xx}(lPc(k)) < 0 ∧ H(lPc(k)) > 0, "Máximo relativo", H(lPc(k)) ≠ 0, "Punto de silla", "??"]}, k, 1, Length[lPc]) → lPc0