MathPlotAI Logo MathPlotAI

Study Graphs of Trigonometric Functions

Study Graphs of Trigonometric Functions
Loading the math board and drawing, please wait…
2014-03-17 16:33
FunctionTrigonometryDerivativeGraphing
This example demonstrates how to graph and compare two trigonometric functions along with their upper and lower bounds. Key math concepts include trigonometric functions, function graphs, derivatives, and limits. It utilizes GeoGebra commands like Function, Derivative, Intersect, and Vector. Suitable for high school and above students for self-study and exploration of function properties.

GeoGebra Commands & Parameters

  1. If(angledegree, lowerlimit_d * π / 180, lowerlimit_r) → lowerlimit
  2. If(angledegree, upperlimit_d / 180, upperlimit_r) → upperlimit
  3. Function(g_{hidden}, lowerlimit * π, upperlimit * π) → g
  4. Derivative(g_{hidden}) → g'
  5. RemoveUndefined(Sequence[x(Root[g', i]), i, -31.41593, 31.41593, 0.31416]) → sincosturningptslistg
  6. Function(f_{hidden}, lowerlimit * π, upperlimit * π) → f
  7. Derivative(f_{hidden}) → f'
  8. RemoveUndefined(Sequence[x(Root[f', i]), i, -31.41593, 31.41593, 0.31416]) → sincosturningptslistf
  9. Point(g) → P_{02}
  10. Point(f) → P_{01}
  11. If(Length[sincosturningptslistg] > 0, false, true) → gchecktanfunction2
  12. Derivative(g') → g''
  13. Derivative(f') → f''
  14. If(Length[sincosturningptslistf] > 0, false, true) → fchecktanfunction1
  15. Textfield(f_{hidden}) → inputBox1
  16. Textfield(g_{hidden}) → inputBox4
  17. Sequence(Text[i + "°", (π / 180 * i - 0.05, -0.3)], i, 0, 1080, 90) → listxaxisdegreesby90
  18. If(fchecktanfunction1 ≟ false, (y(Max[f, lowerlimit * π, upperlimit * π * 10]) + y(Min[f, lowerlimit * π, upperlimit * π * 10])) / 2, meanline_{1fortan}) → meanline_1
  19. If(gchecktanfunction2 ≟ false, (y(Max[g, lowerlimit * π, upperlimit * π * 10]) + y(Min[g, lowerlimit * π, upperlimit * π * 10])) / 2, meanline_{2fortan}) → meanline_2
  20. Textfield(h_{hidden}) → inputBox5
  21. Function(h_{hidden}, lowerlimit * π, upperlimit * π) → h
  22. Function(meanline_1, lowerlimit * π, upperlimit * π) → f_{meanline1}
  23. Function(meanline_2, lowerlimit * π, upperlimit * π) → g_{meanline2}
  24. Textfield(lowerlimit_d) → inputBox6
  25. Textfield(upperlimit_d) → inputBox7
  26. Function(0, lowerlimit * π, upperlimit * π + 1) → h_{x-axis}
  27. Textfield(upperlimit_r) → inputBox3
  28. Textfield(lowerlimit_r) → inputBox2
  29. Sequence(Text[FractionText[i] + "π", (π * i - 0.15, -0.1), true, true], i, lowerlimit + xunitdistance, upperlimit, xunitdistance) → listxaxisradianlabel
  30. Textfield(xaxisnumdivquad) → inputBox8
  31. Sequence((i, 0), i, lowerlimit * π, upperlimit * π, xunitdistance * π) → listxaxisticks
  32. Sequence(abs(f(i)), i, lowerlimit * π, upperlimit * π, 0.01) → valuestanfunction
  33. Asymptote(f) → list5
  34. Asymptote(Function[g, lowerlimit * π, upperlimit * π]) → list6
  35. If(fchecktanfunction1 ≟ false, y(Max[f, lowerlimit * π, upperlimit * π * 10]), 0) → maxline_1
  36. If(fchecktanfunction1 ≟ false, y(Min[f, lowerlimit * π, upperlimit * π * 10]), 0) → minline_1
  37. Function(maxline_1, lowerlimit * π, upperlimit * π) → f_{maxline1}
  38. Function(minline_1, lowerlimit * π, upperlimit * π) → f_{minline1}
  39. Intersect(f_{maxline1}, vertline1) → A_1
  40. Intersect(f_{minline1}, vertline1) → B_1
  41. Intersect(f_{meanline1}, vertline1) → C_1
  42. Distance(A_1, C_1) → distanceA1C1
  43. Vector(C_1, A_1) → u_1
  44. Vector(C_1, B_1) → v_1
  45. Midpoint(A_1, C_1) → M_{A1C1}
  46. Midpoint(B_1, C_1) → M_{C1B1}
  47. If(gchecktanfunction2 ≟ false, y(Max[g, lowerlimit * π, upperlimit * π * 10]), 0) → maxline_2
  48. If(gchecktanfunction2 ≟ false, y(Min[g, lowerlimit * π, upperlimit * π * 10]), 0) → minline_2
  49. Function(maxline_2, lowerlimit * π, upperlimit * π) → g_{maxline1}
  50. Function(minline_2, lowerlimit * π, upperlimit * π) → g_{minline1}
  51. Intersect(g_{maxline1}, vertline2) → A_2
  52. Intersect(g_{minline1}, vertline2) → B_2
  53. Intersect(g_{meanline2}, vertline2) → C_2
  54. Midpoint(A_2, C_2) → M_{A2C2}
  55. Midpoint(B_2, C_2) → M_{B2C2}
  56. Distance(A_2, C_2) → distanceAC
  57. Vector(C_2, A_2) → u_2
  58. Vector(C_2, B_2) → v_2
  59. If(fchecktanfunction1, tancyclesf, sincoscycles1) → cycles1
  60. If(gchecktanfunction2, tancyclesg, sincoscycles2) → cycles2
  61. Intersect(g, vertline3, (0, -1.5)) → P_1
  62. Midpoint(P_1, Q_1) → M_{P1Q1}
  63. Vector(M_{P1Q1}, P_1) → u_{p1l}
  64. Vector(M_{P1Q1}, Q_1) → u_{p1r}
  65. Intersect(f, vertline4, (0.08997312945277669, 1.9515599378378266)) → P_2
  66. Midpoint(P_2, Q_2) → M_{P2Q2}
  67. Vector(M_{P2Q2}, Q_2) → u_{p2r}
  68. Vector(M_{P2Q2}, P_2) → u_{p2l}
  69. Derivative(f'') → f'''
  70. Derivative(g'') → g'''
  71. Sequence(Text[i + "°", (π / 180 * i - 0.15, -0.55)], i, lowerlimit * 180 + 180 * xunitdistance, upperlimit * 180, 180 * xunitdistance) → listxaxisdegreesby45

AI Drawing Prompt Reference

Draw two trigonometric functions along with their upper and lower bounds, showing function graphs, derivatives, and limits. First, plot the graphs of the two trigonometric functions and their upper and lower bounds. Then, calculate and plot the derivatives of these functions, marking the zeros of the derivatives. Finally, calculate and plot the mean lines of the functions, and mark the maximum and minimum values of the functions.
The prompt above was compiled from the original drawing commands and is for reference only.