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Mean Value Theorem GeoGebra Demo

Mean Value Theorem GeoGebra Demo
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2018-02-20 06:00
FunctionCalculusDerivativeIntegralGeometric Construction
This example demonstrates the geometric interpretation of the Mean Value Theorem for continuous functions on a closed interval, showing the relationship between the derivative, integral, and parallelism of tangent and secant lines. Core math concepts include derivatives, definite integrals, and similarity transformations. Constructed using commands like Point, Segment, Vector, Derivative, Intersect, If, Textfield, Line, and objects such as E, f, A, J, PTA, B, H, PTB, etc. Suitable for high school students and above to learn and understand key calculus concepts, or for teacher demonstrations.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Point(yAxis) → B
  3. Segment(PTA, PTB) → q
  4. Vector((0, 0), (0, m_{sec})) → v_{sec}
  5. Derivative(g) → g'
  6. Intersect(g', xAxis) → C, D
  7. Derivative(f) → f'
  8. If(y(A) < MVPT1 < y(B), MVPT1, If[y(A) < MVPT2 < y(B), MVPT2, MVPT3]) → MVPT
  9. Segment(B, A) → m
  10. Segment(H, J) → p
  11. Textfield(tanL) → inputBox1
  12. Line(c_{MVT}, f_c) → k
  13. Vector(c_{MVT}, f_c) → u
  14. Textfield(f) → inputBox2
  15. Vector(A, J) → j
  16. Vector(B, H) → n
  17. Line(E, q) → DynSecL
  18. Intersect(f, DynSecL) → K, L, M, N
  19. Point(O, v_{sec}) → P
  20. Segment(O, P) → r
  21. Vector(O, P) → s
  22. Point(m) → F
  23. Vector(F, G) → v
  24. Vector(I, Q) → w
  25. Vector(f_c, (2, y(f_c) + m_{sec})) → b
  26. Point(m) → T
  27. Vector(U, (2, y(U) + f'(y(T)))) → c
  28. Vector(T, U) → d
  29. Line(V, Vector[(1, f'(x(V)))]) → h

AI Drawing Prompt Reference

Draw the graph of a continuous function on a closed interval, showing the parallelism between the tangent and secant lines, and labeling the derivative and integral. Start by plotting two points on the x and y axes and connecting them with a line segment. Then, find the intersection points of the derivative with the x-axis and draw a vector parallel to the secant line. Next, determine a point within the interval based on the function's value and draw the corresponding tangent line. Finally, illustrate the parallelism between the tangent and secant lines and label the derivative, integral, and tangent equation.
The prompt above was compiled from the original drawing commands and is for reference only.