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Partial Derivatives and Tangent Planes

Partial Derivatives and Tangent Planes
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2016-10-12 21:10
VectorCoordinate GeometrySolid GeometryCalculusDerivative
This example demonstrates the relationship between partial derivatives and tangent planes of a function in three-dimensional space. Key math concepts include partial derivatives, tangent planes, vectors, and 3D coordinate geometry. It is constructed using GeoGebra commands/objects like Textfield, Plane, CurveCartesian, Derivative, Vector, Line. Suitable for university calculus courses or introductory mathematical modeling.

GeoGebra Commands & Parameters

  1. Textfield(f) → InputBox1
  2. Plane((a, 0, 1), L_x) → P_x
  3. Plane((0, b, 1), L_y) → P_y
  4. CurveCartesian(t, b, f(t, b), t, -100, 100) → f_{resy}
  5. Derivative(f, x) → f_x
  6. Derivative(f, y) → f_y
  7. Vector((1, 0, f_{xat(a,b)})) → v_x
  8. Vector((0, 1, f_{yat(a,b)})) → v_y
  9. Line((a, b, f(a, b)), v_y) → T_y
  10. Line((a, b, f(a, b)), v_x) → T_x
  11. Plane(T_x, T_y) → T_f
  12. CurveCartesian(a, t, f(a, t), t, -100, 100) → f_{resx}

AI Drawing Prompt Reference

Visualize a function in 3D space, showing the relationship between its partial derivatives and tangent planes. Start by drawing two coordinate planes and the function's cross-sections along the x and y axes. Then compute and plot the partial derivative vectors at point (a, b), use them to draw tangent lines and the tangent plane. Finally, combine the tangent lines and plane to display the complete tangent plane effect.
The prompt above was compiled from the original drawing commands and is for reference only.