Removing the 'Hole' in Function Graphs

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2017-11-23 02:48

This example demonstrates how to remove the 'hole' in a function graph using rotation and intersection operations. Core mathematical concepts include standard deviation, orthocenter, and definite integrals in calculus. Key construction points involve using GeoGebra commands like Intersect, Surface, and Rotate, as summarized below. Suitable for high school and above mathematics learners to understand function continuity and calculus operations.

GeoGebra Commands & Parameters

Intersect(f, g, (1.5857469257125176, 2.9998882418684047)) → A
Intersect(f, g, (4.990379094151491, 1.0383910615646903)) → B
Surface(t, (u * g(t)) + ((1 - u) * f(t)), 0, t, x(A), x(B), u, 0, 1) → a
Surface(t, (f(t) * cos(u)), (f(t) * sin(u)), t, x(A), x(B), u, 0, b) → d
Surface(t, (g(t) * cos(u)), (g(t) * sin(u)), t, x(A), x(B), u, 0, b) → e
Rotate(a, b, xAxis) → a'
Surface(t, (u * f(t)), 0, t, x(A), x(B), u, 0, 1) → h
Surface(t, (u * g(t)), 0, t, x(A), x(B), u, 0, 1) → i
Rotate(h, b, xAxis) → h'
Surface(x(B), ((u * y(B)) * cos(t)), ((u * y(B)) * sin(t)), t, c, b, u, 0, 1) → j
Surface(x(A), ((u * y(A)) * cos(t)), ((u * y(A)) * sin(t)), t, c, b, u, 0, 1) → k
Surface(t, (g(t) * cos(u)), (g(t) * sin(u)), t, x(A), x(B), u, 0, c) → l
Rotate(i, c, xAxis) → i'

AI Drawing Prompt Reference

Draw a function graph to demonstrate how to remove the 'hole' using rotation and intersection operations. Identify two specific points on the graph, construct a surface, and rotate it around the x-axis to visualize the removal of the hole, resulting in a continuous function graph. The process involves mathematical concepts like standard deviation, orthocenter, and definite integrals, suitable for high school and above learners to understand function continuity and calculus operations.

The prompt above was compiled from the original drawing commands and is for reference only.