Piecewise Functions: Continuity & Differentiability
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This example helps students explore the continuity and differentiability of functions, as well as different types of discontinuities and removable discontinuities through four problems. Key mathematical concepts include the continuity and differentiability of functions. It uses If, LaTeX, p, Problem, button1, button2, f, A, B, C, text1, text2, q, text3, text4, text5, text6, text7, text8, text9, a, b, c, d, h, text10, text11 commands and objects in GeoGebra. Suitable for high school mathematics education and learning.
GeoGebra Commands & Parameters
If(x < -2, 1 / (x + 3) - 1, If[-2 < x < 0, (x + 1)^(2 / 3) - 1, (x - 1)^(1 / 3) + 1]) → pIf(x ≤ -1, 3 * x + 1, If[x < 1, x^(2) + 2 * x, 3 * x^(3)]) → fLaTeX(f, true, true) → text1LaTeX(p, true, true) → text2If(x < 0, (x^(2) - 1) / (x + 1), If[x < 2, 2, If[x > 2, 1 / (x - 3) + 3]]) → qLaTeX(q, true, true) → text3If(x < -2, b, If[x < 2, a * x^(2), If[x > 2, c * x - d]]) → h
AI Drawing Prompt Reference
Draw a piecewise function with the following parts: for x < -2, the function is 1/(x+3)-1; for -2 ≤ x < 0, the function is (x+1)^(2/3)-1; for 0 ≤ x < 2, the function is (x-1)^(1/3)+1; for x ≥ 2, the function is 1/(x-3)+3. Also, draw a function f: for x ≤ -1, the function is 3x+1; for -1 < x < 1, the function is x^2+2x; for x ≥ 1, the function is 3x^3. Finally, draw a function h: for x < 0, the function is b; for 0 ≤ x < 2, the function is ax^2; for x ≥ 2, the function is cx-d.
The prompt above was compiled from the original drawing commands and is for reference only.