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Complex Roots of Derivatives

Complex Roots of Derivatives
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2026-04-21 04:03
DerivativeComplex NumberGeometric Construction
In collections complex-numbers
This example demonstrates the geometric construction of complex roots of derivatives. Key mathematical concepts include complex numbers, derivatives, and geometric transformations. The construction involves GeoGebra commands such as Sequence, Sum, Derivative, ComplexRoot, and ConvexHull. Suitable for high school students or college undergraduates learning complex analysis and geometric constructions.

GeoGebra Commands & Parameters

  1. Sequence(Random[-20, 20] / 10, k, 1, 10) → k, {1.9, -0.4, -0.2, 1, -1.1, -0.3, 0.1, -0.6, -0.8, 0.5}
  2. Sum((k(i + 1) * x^(i)), i, 0, 9) → f
  3. Derivative(f) → f1
  4. Derivative(f1) → f2
  5. Derivative(f2) → f3
  6. Derivative(f3) → f4
  7. Derivative(f4) → f5
  8. Derivative(f5) → f6
  9. Derivative(f6) → f7
  10. Derivative(f7) → f8
  11. Derivative(f8) → f9
  12. ComplexRoot(f) → N0_{1}, N0_{2}, N0_{3}, N0_{4}, N0_{5}, N0_{6}, N0_{7}, N0_{8}, N0_{9}
  13. ComplexRoot(f1) → N1, O1, P1, Q1, R1, S1, T1, U1
  14. ComplexRoot(f2) → N2, O2, P2, Q2, R2, S2, T2
  15. ComplexRoot(f3) → N3, O3, P3, Q3, R3, S3
  16. ComplexRoot(f4) → N4, O4, P4, Q4, R4
  17. ComplexRoot(f5) → N5, O5, P5, Q5
  18. ComplexRoot(f6) → N6, O6, P6
  19. ComplexRoot(f7) → N7, O7
  20. ConvexHull(Pu0) → Hu0
  21. ConvexHull(Pu1) → Hu1
  22. ConvexHull(Pu2) → Hu2
  23. ConvexHull(Pu3) → Hu3
  24. ConvexHull(Pu4) → Hu4
  25. ConvexHull(Pu5) → Hu5
  26. ConvexHull(Pu6) → Hu6
  27. ConvexHull(Pu7) → Hu7

AI Drawing Prompt Reference

Create a geometric construction showing the complex roots of polynomial derivatives. Start by generating 10 random complex numbers to form a polynomial function and its first five derivatives. Identify and plot the complex roots for each derivative with distinct colors. Finally, draw convex hulls around the sets of roots for each derivative to visualize their geometric arrangement.
The prompt above was compiled from the original drawing commands and is for reference only.