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Directional Derivatives Demo by Mike Hopper

Directional Derivatives Demo by Mike Hopper
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2017-05-07 05:54
VectorFunctionCalculusDerivative
This example demonstrates the geometric interpretation of directional derivatives, visualizing the rate of change of a function at a point along any direction. Key math concepts include derivatives, vectors, gradients, and function graphs. Construction highlights the use of GeoGebra commands and objects like Circle, Point, Vector, Function, Derivative, grad to build the function graph, gradient vectors, and directional derivative lines. Suitable for university calculus courses or math enthusiasts to gain an intuitive understanding of directional derivatives.

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 function graph and show its gradient vector and directional derivatives at a point. Start by plotting the origin and drawing a circle around it, connecting points on the circle to the origin to form vectors. Then, based on the partial derivatives at the point, draw the gradient vector and implicit curves for the function's contour lines. Add multiple parallel curves to represent different heights. Finally, from the gradient vector, draw directional derivative lines along arbitrary directions to show the rate of change of the function in those directions.
The prompt above was compiled from the original drawing commands and is for reference only.