GeoGebra示例：函数与积分计算

几何画板及绘图内容加载中，请稍等……

2014-07-29 22:03

函数解析几何微积分

所属专题 CCGPS AA Unit 6 Lesson 1

①本范例演示了函数图像与积分计算的过程；②涉及核心数学概念包括函数、导数、积分、解析几何；③构造中使用了LaTeX、TableText、x2、x1、dx、t2、t1、dt、r、intercept、stepMax、stepper、steps、counter、RandomizationText、Original、Randomized、t0、x0、text2、text3、text1、text4、text5、text6、text8、text14、text15、text16、text17、minusy1、f、minusmx1、text19、text20、text22、text23、mx1、text0、Problem、text12、text9等GeoGebra命令与对象；④适合高中阶段学生学习微积分与函数图像。

GeoGebra 命令及参数

LaTeX(steps) → counter
LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
LaTeX("\text{Let the independent variable be time in hours, }\textcolor{#000099}{t}.") → text2
LaTeX("\text{Let the dependent variable be the distance in miles, }\textcolor{#000099}{d}.") → text3
LaTeX("\text{Name a dependent variable and an independent variable based on the given data.}") → text1
LaTeX("\text{Write ordered pairs for the data in the problem based on the identified variables.}") → text4
LaTeX((((((((("\text{The ordered pairs will be of the form }\textcolor{#000099}{(t, d)}\text{, since the independent variable is on the}\cr\text{x-axis and the dependent variable is on the y-axis. Therefore, the ordered pairs for the}\cr\text{given data are }\textcolor{#000099}{(" * t1) * ", ") * x1) * ")}\text{ and }\textcolor{#000099}{(") * t2) * ", ") * x2) * ")}.")) → text5
LaTeX("\text{Write an equation for the speed in terms of the distance and time.}") → text6
LaTeX("rate=\frac{distance}{time}\text{ or }\textcolor{#000099}{r=\frac{d}{t}}") → text8
LaTeX((((("\text{Chloe’s speed during her " * t2) * "-hour drive was ") * r) * " miles per hour.}")) → text14
LaTeX("\text{Use the point-slope formula, }\textcolor{#000099}{y – y_1 = m(x – x_1)}\text{, to write the linear equation in one}\cr\text{variable.}") → text15
LaTeX((("\text{In the point-slope formula, }\textcolor{#000099}{y – y_1 = m(x – x_1)}\text{, }\textcolor{#000099}{m}\text{ is the slope and }\textcolor{#000099}{(x_1, y_1)}\text{ is a point}\cr\text{on the line. Use the value }\textcolor{#000099}{r = " * r) * "}\text{ from the previous step for the slope and either}\cr\text{of the given ordered pairs for }\textcolor{#000099}{(x_1, y_1)}\text{ in this formula.}")) → text16
LaTeX((((("\text{Let’s use }\textcolor{#000099}{(" * t1) * ", ") * x1) * ")}\text{. Simplify and solve for }\textcolor{#000099}{d}.")) → text17
LaTeX(Simplify[(-x1)]) → minusy1
LaTeX(((-r) * t1)) → minusmx1
LaTeX("\text{Find the value of }\textcolor{#000099}{d}\text{ when }\textcolor{#000099}{t=0}.") → text19
LaTeX("\text{The time at which Chloe left college for the drive home is represented by }\textcolor{#000099}{t=0}.") → text20
LaTeX((("\text{When }\textcolor{#000099}{t=0}\text{, }\textcolor{#000099}{d=–" * intercept) * "}.")) → text22
LaTeX("\text{Interpret the results based on the information given in the problem to determine how}\cr\text{far Chloe drove before passing the roadside attraction. }") → text23
LaTeX("\text{Move the slider to solve the problem.}") → text0
LaTeX((((((((("\text{" * t1) * " hours after leaving she has driven ") * x1) * " miles past the attraction.}\cr\text{") * t2) * " hours after leaving she has driven ") * x2) * " miles past the attraction.}")) → Problem
LaTeX((((((((("\text{Let }\textcolor{#000099}{(" * t1) * ", ") * x1) * ")}\text{ represent }\textcolor{#000099}{(t_1, d_1)}\text{ and }\textcolor{#000099}{(") * t2) * ", ") * x2) * ")}\text{ represent }\textcolor{#000099}{(t_2, d_2)}\text{. Substitute these values into}\cr\text{the formula for slope.}")) → text12
LaTeX("\text{Use the two ordered pairs and the formula for the slope of a one-variable}\cr\text{equation, }\textcolor{#000099}{m = \frac{y_2 − y_1}{x_2-x_1}}\text{, to find the rate.}") → text9
LaTeX("\text{The slope formula can be used to find the rate because the slope formula is of the same}\cr\text{form as the formula }\textcolor{#000099}{r = \frac{d}{t}}.") → text10
LaTeX("\text{Recall that the two-point slope formula is }\textcolor{#000099}{m = \frac{y_2 − y_1}{x_2-x_1}}\text{ for points }\textcolor{#000099}{(x_1, y_1)}\text{ and }\cr\textcolor{#000099}{(x_2, y_2)}\text{. We can think of this as }\textcolor{#000099}{r = \frac{d_2 − d_1}{t_2-t_1}}\text{ for points }\textcolor{#000099}{(t_1, d_1)}\text{ and }\textcolor{#000099}{(t_2, d_2)}\text{. Thus,}\cr\text{the slope }\textcolor{#000099}{m}\text{ would also be the rate }\textcolor{#000099}{r}\text{ at which Chloe drives.}") → text11
TableText({{LaTeX["\textcolor{#000099}{r=\frac{d_2-d_1}{t_2-t_1}}"], LaTeX["\text{Slope formula written in terms of the rate, }\textcolor{#000099}{r}"]}, {LaTeX[(((((((("\textcolor{#000099}{r=\frac{(" * x2) * ") - (") * x1) * ")}{(") * t2) * ") - (") * t1) * ")}}")], LaTeX[(((((((("\text{Substitute }\textcolor{#000099}{(" * t1) * ",") * x1) * ")}\text{ for }\textcolor{#000099}{(t_1,d_1)}\text{ and }\textcolor{#000099}{(") * t2) * ",") * x2) * ")}\text{ for }\textcolor{#000099}{(t_2,d_2)}.")]}, {LaTeX[(((("\textcolor{#000099}{r=\frac{" * dx) * "}{") * dt) * "}}")], LaTeX["\text{Simplify.}"]}, {LaTeX[(("\textcolor{#000099}{r=" * r) * "}")], LaTeX[""]}}) → text13
TableText({{LaTeX[(((("\textcolor{#000099}{d=" * r) * "t-") * intercept) * "}")], LaTeX["\text{Equation from the previous step}"]}, {LaTeX[(((("\textcolor{#000099}{d=" * r) * "(0)-") * intercept) * "}")], LaTeX["\text{Substitute }\textcolor{#000099}{t=0}."]}, {LaTeX[(("\textcolor{#000099}{d=-" * intercept) * "}")], LaTeX["\text{Simplify.}"]}}) → text21
LaTeX("\text{This is a distance-rate-time problem in which the rate or speed, }\textcolor{#000099}{r}\text{, is the distance}\cr\text{divided by the time.}") → text7
TableText({{LaTeX["\textcolor{#000099}{y-y_1=m(x-x_1)}"], LaTeX["\text{Point-slope formula}"]}, {LaTeX[(((((("\textcolor{#000099}{(d)-(" * x1) * ")=(") * r) * ")\left[(t)-(") * t1) * ")\right]}")], LaTeX[(((((("\text{Substitute }\textcolor{#000099}{(" * t1) * ",") * x1) * ")}\text{ for }\textcolor{#000099}{(x_1,y_1)}\text{, }\textcolor{#000099}{d}\text{ for }\textcolor{#000099}{y}\text{, }\textcolor{#000099}{r=") * r) * "}\text{ for }\textcolor{#000099}{m},")]}, {LaTeX[""], LaTeX["\text{and }\textcolor{#000099}{t}\text{ for }\textcolor{#000099}{x}."]}, {LaTeX[(((((("\textcolor{#000099}{d" * minusy1) * "=") * r) * "t") * minusmx1) * "}")], LaTeX["\text{Distribute.}"]}, {LaTeX[(((("\textcolor{#000099}{d=" * r) * "t-") * intercept) * "}")], LaTeX["\text{Simplify.}"]}}) → text18
LaTeX((((((((((((((((((((((("\text{If Chloe drove at a constant speed of " * r) * " miles per hour, the ordered pair (") * t1) * ", ") * x1) * ")}\cr\text{implies that she had traveled }") * t1) * " \cdot ") * r) * "\text{ or }") * mx1) * "\text{ miles from her starting point. The}\cr\text{distance ") * x1) * " in the ordered pair implies that Chloe was ") * mx1) * " – ") * x1) * " or ") * intercept) * " miles }\cr\text{past that starting point when she passed the roadside attraction. Therefore, }\cr\text{Chloe drove ") * intercept) * " miles before passing the roadside attraction.}")) → text24

AI 绘图提示语参考

画一个函数图像，展示变量 t（时间）和 d（距离）的关系，并用积分计算 Chloe 行驶的总距离。首先标出两个关键点 (t1, x1) 和 (t2, x2)，然后根据这两个点计算速度 r。接着，用点斜式方程 y-y1=m(x-x1) 绘制速度线，并找到与 y 轴的交点，该点表示 Chloe 出发时的位置。最后，通过计算积分展示 Chloe 行驶的总距离。

以上提示语为根据原始绘图命令整理而来，仅供参考。