函数平均变化率的计算示例

GeoGebra 命令及参数

1. LaTeX(steps) → counter
2. LaTeX("\text{Move the slider to solve the problem.}") → text0
3. Corner(1) → A
4. Corner(3) → B
5. Polynomial((((x - a) * (x - b)) * (x - c))) → f
6. Zip(If[a ≟ 1, b, 100], a, map, b, ends) → pickEnds
7. Sort(pickEnds \ {100}) → usedEnds
8. Sequence(LaTeX[(((("[" * Element[usedEnds, i]) * ",") * Element[usedEnds, i + 1]) * "]")], i, 1, Length[usedEnds] - 1) → intervals
9. TableText({intervals}) → intervalDisplay
10. LaTeX((((("\textcolor{#000099}{f(x)=" * f) * "}\cr\textcolor{#000099}{") * intervalDisplay) * "}")) → Problem
11. Element(intervals, chooseInterval) → chosenInterval
12. LaTeX((("\text{Calculate the average rate of change of }\textcolor{#000099}{f(x)}\text{ over the}\cr\text{interval }\textcolor{#000099}{" * chosenInterval) * "}.")) → text1
13. LaTeX((((("\text{First find the values of }\textcolor{#000099}{f(" * Element[usedEnds, chooseInterval]) * ")}\text{ and }\textcolor{#000099}{f(") * Element[usedEnds, chooseInterval + 1]) * ")}.")) → text2
14. Coefficients(f) → coeffs
15. LaTeX((("\text{Calculate }\textcolor{#000099}{f(" * Element[usedEnds, chooseInterval]) * ")}.")) → text3
16. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → text5
17. LaTeX((((("\textcolor{#000099}{f(" * Element[usedEnds, chooseInterval]) * ")=") * f(Element[usedEnds, chooseInterval])) * "}")) → text7
18. LaTeX((("\text{Calculate }\textcolor{#000099}{f(" * Element[usedEnds, chooseInterval + 1]) * ")}.")) → text8
19. LaTeX((("\textcolor{#000099}{f(x)=" * f) * "}")) → text10
20. LaTeX((((("\textcolor{#000099}{f(" * Element[usedEnds, chooseInterval + 1]) * ")=") * f(Element[usedEnds, chooseInterval + 1])) * "}")) → text12
21. Element(usedEnds, chooseInterval) → Lend
22. Element(usedEnds, chooseInterval + 1) → Rend
23. LaTeX("\text{Simplify.}") → text18
24. LaTeX((("\textcolor{#000099}{r=" * r) * "}")) → text19
25. Zip((a, f(a)), a, usedEnds) → showEnds
26. Sequence((f(Element[usedEnds, i]) - f(Element[usedEnds, i + 1])) / (Element[usedEnds, i] - Element[usedEnds, i + 1]), i, 1, Length[usedEnds] - 1) → rateList
27. Zip(If[a < 0, "negative", If[a > 0, "positive", If[a ≟ 0, "zero", "undefined"]]], a, rateList) → rateWord
28. TableText({intervals, rateList, rateWord}, "v") → text24
29. If(Defined[IndexOf["positive", rateWord]], LaTeX["\text{When the average rate of change is positive on}\cr\text{an interval, the function rises more than it falls}\cr\text{on that interval.}\cr"], "") → wordpos
30. If(Defined[IndexOf["zero", rateWord]], LaTeX["\text{When the average rate of change is zero on an}\cr\text{interval, the function either is constant}\cr\text{throughout the interval (a horizontal line) or has}\cr\text{the same value at the ends of the interval and}\cr\text{moves away from that value and then back. The}\cr\text{second possibility results in at least one}\cr\text{maximum or minimum on the interval.}\cr"], "") → wordzero
31. If(Defined[IndexOf["negative", rateWord]], LaTeX["\text{When the average rate of change is negative on}\cr\text{an interval, the function falls more than it rises}\cr\text{on that interval.}\cr"], "") → wordneg
32. If(Defined[IndexOf["undefined", rateWord]], LaTeX["\text{When the average rate of change is undefined on}\cr\text{an interval, either the interval has zero width or}\cr\text{at least one of the endpoints is undefined.}\cr"], "") → wordunde
33. Sequence(Sequence[If[Element[rateList, i] ≟ Element[rateList, j], 1, 0], i, j + 1, Length[rateList]], j, 1, Length[rateList]) → match
34. Flatten(match) → flatmatch
35. Sum(flatmatch) → summatch
36. If(summatch > 0, LaTeX["\text{}\cr\text{Note that just because the average rate of}\cr\text{change in two intervals is the same does not}\cr\text{mean that the function behaves identically in}\cr\text{the two intervals.}\cr"], "") → matched
37. LaTeX(((((wordpos * wordneg) * wordzero) * wordunde) * matched)) → text25
38. LaTeX("") → text22
39. LaTeX("\text{Original function}") → text4
40. LaTeX("\text{Original function}") → text9
41. LaTeX("\text{Rate of change formula}") → text14
42. LaTeX((((("\text{The rate of change of }\textcolor{#000099}{f(x)}\text{ over the interval }\textcolor{#000099}{" * chosenInterval) * "}\text{ is }\textcolor{#000099}{") * r) * "}.")) → text20
43. LaTeX((((("\text{Substitute the values for }\textcolor{#000099}{f(" * Lend) * ")}\text{ and }\textcolor{#000099}{f(") * Rend) * ")}\text{ into the rate of change formula and solve.}")) → text13
44. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
45. LaTeX((("\text{Substitute }\textcolor{#000099}{" * Element[usedEnds, chooseInterval]) * "}\text{ for }\textcolor{#000099}{x}\text{ and simplify.}")) → text6
46. LaTeX((("\text{Substitute }\textcolor{#000099}{" * Element[usedEnds, chooseInterval + 1]) * "}\text{ for }\textcolor{#000099}{x}\text{ and simplify.}")) → text11
47. LaTeX("\textcolor{#000099}{r=\frac{f_1(x) - f_2(x)}{x_1-x_2}}") → text15
48. LaTeX((((((((("\text{Substitute }\textcolor{#000099}{" * fleft) * "}\text{ for }\textcolor{#000099}{f_1(x)}\text{, }\textcolor{#000099}{") * fright) * "}\text{ for }\textcolor{#000099}{f_2(x)}\text{, }\cr\textcolor{#000099}{") * Lend) * "}\text{ for }\textcolor{#000099}{x_1}\text{, and }\textcolor{#000099}{") * Rend) * "}\text{ for }\textcolor{#000099}{x_2}.")) → text16
49. LaTeX((((((((("\textcolor{#000099}{r=\frac{(" * f(Lend)) * ")-(") * f(Rend)) * ")}{(") * Lend) * ")-(") * Rend) * ")}}")) → text17
50. LaTeX("\text{Graph the function and the endpoints of the intervals.}") → text21
51. LaTeX("\text{Determine if the rate of change over each interval is positive, negative, 0, or undefined, }\cr\text{and whether the rates of change for different intervals are the same.}") → text23