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GeoGebra Sketch of y = ln(x) Showing Concavity

GeoGebra Sketch of y = ln(x) Showing Concavity
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2021-01-03 21:36
FunctionCalculusDerivativeGeometric ConstructionGraphing
This example demonstrates how to draw the curve of the natural logarithm function y = ln(x) in GeoGebra and visually highlight its concavity by using local geometric constructions. Key concepts include the function graph of ln(x), domain behavior, concavity, and the relationship between a curve and its tangent line. Construction highlights: the workflow uses Min and Max to select and bound the parameter/visual window; Vector and Intersect to define and compute key geometric points and directions; and Tangent to construct tangent lines at selected locations, making the curvature and concavity apparent from the changing slope. A set of labeled points and text objects (e.g., multiple points and text1–text16) is used to annotate the construction steps and emphasize the geometric meaning. Suitable for advanced secondary education and introductory calculus instruction, especially for teaching function properties through dynamic graphing.

GeoGebra Commands & Parameters

  1. Min(f, -4, 0) → A
  2. Max(f, 0, 6) → B
  3. Vector(H, G) → u
  4. Vector(G, H) → v
  5. Vector(I, G) → w
  6. Vector(G, I) → a
  7. Vector(I, J) → b
  8. Vector(J, I) → c
  9. Vector(J, K) → d
  10. Vector(K, J) → e
  11. Vector(G, K) → g
  12. Intersect(f', xAxis, (2.718281828459033, 0)) → L
  13. Tangent(P, f) → l
  14. Intersect(p, xAxis, (2.7182818284324286, 0)) → Q

AI Drawing Prompt Reference

Plot the curve of the natural logarithm function y = ln(x) on a coordinate plane, together with the coordinate axes to show its domain behavior. Choose a few key locations on the graph to visually reflect how the slope changes, and use the x-axis point corresponding to e ≈ 2.71828182845 as the main reference for a tangent. Construct the tangent line to the ln(x) curve at that location and mark the tangent’s intersection with the x-axis. Arrange the build roughly by first selecting bounding/visual reference points, then defining direction/connection via geometric relationships, and finally drawing the tangent and highlighting concavity.
The prompt above was compiled from the original drawing commands and is for reference only.