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Second derivative test

Second derivative test
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2020-10-13 22:11
FunctionCalculusDerivative
This example demonstrates the second derivative test for quadratic functions, showing how to determine local extrema using the second derivative. Key concepts include derivatives, quadratic functions, and tangents. Construction involves GeoGebra commands such as Derivative, CurveCartesian, and Polygon. Suitable for high school or undergraduate students learning calculus.

GeoGebra Commands & Parameters

  1. Derivative(f, x) → f_{x}
  2. Derivative(f, y) → f_{y}
  3. Derivative(f_{x}, x) → f_{xx}
  4. Derivative(f_{y}, y) → f_{yy}
  5. Derivative(f_{x}, y) → f_{xy}
  6. CurveCartesian(t, t, (d * t^(2)), t, -3, 3) → mixedcurve1
  7. CurveCartesian(t, (-t), ((-d) * t^(2)), t, -3, 3) → mixedcurve2
  8. CurveCartesian(t, 0, ((1 / 2 * b) * t^(2)), t, -3, 3) → xxcurve
  9. CurveCartesian(0, t, ((1 / 2 * c) * t^(2)), t, -3, 3) → yycurve
  10. Polygon(A, B, E, D) → q1, a_1, b_1, e_1, d_1
  11. Angle(v, u) → α
  12. Polygon(A, B, E, D) → q2, a_2, b_2, e_2, d_2

AI Drawing Prompt Reference

Create a 3D surface plot illustrating the second derivative test for a quadratic function. Include two mixed partial derivative curves and two second-order derivative curves along the coordinate axes, all shaped as parabolas. Mark a quadrilateral region on the surface and display an angle within it to show the geometric meaning of the discriminant. The overall layout should clearly show the surface, auxiliary curves, and the labeled angle to aid understanding of local extremum conditions.
The prompt above was compiled from the original drawing commands and is for reference only.