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Gradients & Directional Derivatives

Gradients & Directional Derivatives
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2021-09-13 09:56
CalculusDerivativeGeometric Construction
In collections MATH1052-edX
This example demonstrates the geometric and calculus concepts of gradients and directional derivatives using GeoGebra. Key mathematical concepts include derivatives, directional derivatives, and vector analysis. Construction involves using commands like Circle, Point, Derivative to create function graphs and compute gradient vectors. Suitable for high school students or college freshmen to understand basic calculus concepts.

GeoGebra Commands & Parameters

  1. Circle(P, 1) → t
  2. Point(t) → B
  3. Derivative(f(x, y), x) → fx
  4. Derivative(f(x, y), y) → fy
  5. Vector(P, P + (fx(x(P), y(P)), fy(x(P), y(P)))) → grad
  6. ImplicitCurve(f(x, y) - f(x(P), y(P))) → c
  7. ImplicitCurve(f(x, y)) → d
  8. ImplicitCurve(f(x, y) - 1) → g
  9. ImplicitCurve(f(x, y) - 2) → h
  10. ImplicitCurve(f(x, y) - 3) → i
  11. ImplicitCurve(f(x, y) - 4) → j
  12. ImplicitCurve(f(x, y) - 5) → k
  13. ImplicitCurve(f(x, y) + 1) → l
  14. ImplicitCurve(f(x, y) + 2) → m
  15. ImplicitCurve(f(x, y) + 3) → n
  16. ImplicitCurve(f(x, y) + 4) → o
  17. ImplicitCurve(f(x, y) + 5) → p
  18. Plane(P, P', P + (fx(x(P), y(P)), fy(x(P), y(P)))) → r
  19. CurveCartesian(x(P) + (fx(x(P), y(P)) * w), y(P) + (fy(x(P), y(P)) * w), w, -100, 100) → q
  20. CurveCartesian(x(P') + (fx(x(P), y(P')) * Q), y(P') + (fy(x(P), y(P)) * Q), z(P') + (((fx(x(P), y(P)))^(2) + (fy(x(P), y(P)))^(2)) * Q), Q, -100, 100) → e
  21. Textfield(f) → InputBox1
  22. Textfield(Px) → InputBox2
  23. Textfield(Py) → InputBox3
  24. Length(grad) → normgrad
  25. Segment(P'', P') → s
  26. Function(f(x(P'), y(P')) + (fx(x(P'), y(P')) * (x - x(P'))) + (fy(x(P'), y(P')) * (y - y(P'))), x(P) - 1, x(P) + 1, y(P) - 1, y(P) + 1) → b
  27. Rotate(B, θ, P) → A
  28. Vector(P, A) → u
  29. OrthogonalLine(P, u, xOyPlane) → f_1
  30. OrthogonalPlane(P'', f_1) → v
  31. IntersectPath(v, f) → w
  32. IntersectPath(v, b) → a_1
  33. Angle(B, P, A) → β
  34. Textfield(θ) → InputBox4
  35. Point(zAxis) → C
  36. Intersect(zAxis, xAxis) → D
  37. Vector(C, D) → u_1
  38. Point(yAxis) → E
  39. Vector(E, D) → v_1
  40. Point(xAxis) → F
  41. Vector(F, D) → w_1
  42. Element(list1, faces) → Selectedvector

AI Drawing Prompt Reference

Draw a 3D coordinate system with a surface representing a binary function and its contour lines (equally spaced from function value minus 5 to plus 5). Pick a point on the surface, show its gradient vector (pointing in the steepest ascent direction), and construct a plane through that point containing the gradient vector. Intersect this plane with the surface to get a curve, and also draw the tangent plane at that point. Take another point on the surface, connect the two points with a segment, and show a direction vector at the second point representing a directional derivative. Add input boxes to modify the function, point coordinates, and rotation angle, and display the length of the gradient vector.
The prompt above was compiled from the original drawing commands and is for reference only.