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Try to Graph the Derivative Function by Marc Renault

Try to Graph the Derivative Function by Marc Renault
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2016-04-03 01:18
FunctionCoordinate GeometryCalculusDerivativeGraphing
This example demonstrates how to infer and graph the derivative function by analyzing the original function's graph. Key math concepts include the geometric meaning of derivatives, function graph characteristics, and linear fitting. GeoGebra commands/objects used in construction are Sequence, Corner, FitPoly, Function, Segment, and Point. Suitable for high school students learning introductory calculus to understand the relationship between derivatives and original function graphs.

GeoGebra Commands & Parameters

  1. Sequence((i, Random[-2, 3] + (0 * dummy)), i, -12, 12, 3) → fPointList
  2. Corner(1) → panelPt1
  3. Corner(4) → panelPt4
  4. FitPoly(fPointList, Length[fPointList] - 1) → f
  5. Function(f, -6, 6) → fRes
  6. Function(g, -6, 6) → gRes
  7. Segment((-6, -3.5), (-6, 4.5)) → segA
  8. Segment((-5, -3.5), (-5, 4.5)) → segB
  9. Segment((-4, -3.5), (-4, 4.5)) → segC
  10. Segment((-3, -3.5), (-3, 4.5)) → segD
  11. Segment((-2, -3.5), (-2, 4.5)) → segE
  12. Segment((-1, -3.5), (-1, 4.5)) → segF
  13. Segment((0, -3.5), (0, 4.5)) → segG
  14. Segment((1, -3.5), (1, 4.5)) → segH
  15. Segment((2, -3.5), (2, 4.5)) → segI
  16. Segment((3, -3.5), (3, 4.5)) → segJ
  17. Segment((4, -3.5), (4, 4.5)) → segK
  18. Segment((5, -3.5), (5, 4.5)) → segL
  19. Segment((6, -3.5), (6, 4.5)) → segM
  20. Point(segA) → A
  21. Point(segB) → B
  22. Point(segC) → C
  23. Point(segD) → D
  24. Point(segE) → E
  25. Point(segF) → F
  26. Point(segG) → G
  27. Point(segH) → H
  28. Point(segI) → I
  29. Point(segJ) → J
  30. Point(segK) → K
  31. Point(segL) → L
  32. Point(segM) → M
  33. FitPoly({A, B, C}, 2) → parABC
  34. FitPoly({C, D, E}, 2) → parCDE
  35. FitPoly({E, F, G}, 2) → parEFG
  36. FitPoly({G, H, I}, 2) → parGHI
  37. FitPoly({I, J, K}, 2) → parIJK
  38. FitPoly({K, L, M}, 2) → parKLM
  39. FitPoly({B, C, D}, 2) → parBCD
  40. FitPoly({D, E, F}, 2) → parDEF
  41. FitPoly({F, G, H}, 2) → parFGH
  42. FitPoly({H, I, J}, 2) → parHIJ
  43. FitPoly({J, K, L}, 2) → parJKL
  44. Function(parABC, x(A), x(B)) → conAB
  45. Function((parABC(x) * (1 - (x + 5))) + (parBCD(x) * (x + 5)), -5, -4) → conBC
  46. Function((parBCD(x) * (1 - (x + 4))) + (parCDE(x) * (x + 4)), -4, -3) → conCD
  47. Function((parCDE(x) * (1 - (x + 3))) + (parDEF(x) * (x + 3)), -3, -2) → conDE
  48. Function((parDEF(x) * (1 - (x + 2))) + (parEFG(x) * (x + 2)), -2, -1) → conEF
  49. Function((parEFG(x) * (1 - (x + 1))) + (parFGH(x) * (x + 1)), -1, 0) → conFG
  50. Function((parFGH(x) * (1 - (x + 0))) + (parGHI(x) * (x + 0)), 0, 1) → conGH
  51. Function((parGHI(x) * (1 - (x - 1))) + (parHIJ(x) * (x - 1)), 1, 2) → conHI
  52. Function((parHIJ(x) * (1 - (x - 2))) + (parIJK(x) * (x - 2)), 2, 3) → conIJ
  53. Function((parIJK(x) * (1 - (x - 3))) + (parJKL(x) * (x - 3)), 3, 4) → conJK
  54. Function((parJKL(x) * (1 - (x - 4))) + (parKLM(x) * (x - 4)), 4, 5) → conKL
  55. Function(parKLM, 5, 6) → conLM

AI Drawing Prompt Reference

Draw the graph of a function and its derivative. First, randomly generate the graph of a function, then fit a polynomial to the derivative based on the function's characteristics. Draw a rectangular area on the y-axis, and use the fitted polynomials to connect adjacent inflection points, constructing the derivative function's graph. Finally, use these polynomial functions to draw the continuous derivative function.
The prompt above was compiled from the original drawing commands and is for reference only.