Objective: The goal of this demo is to provide students with a concrete understanding of the average rate of change for physical situations and for functions described in tabular or graphic form.
Platform: No particular software package is required. Support for a viewer of gif or mov files is required. Viewers within a browser, Windows media player, QuickTime, or a commercial program can be used. It is recommended that a viewer that contains a stop/start feature be used when incorporating the animation in a lecture format or when students view the animation on an individual basis. A set of interactive Excel demos that use graphs of functions is included.
Instructor's Notes: In mathematics average rate of change is a stepping stone to instantaneous rates of change and the fundamental concept of a limit. Thus it is important to provide a variety of learning experiences about average rates of change so that students understand this fundamental notion of change. This demo provides visual experiences which connect to the algebraic expressions for average rates of change.
Average Rates for Objects in Motion: To provide a focus, we start with two common visualizations; a falling ball and a moving vehicle. Figure 1 shows an animation of a ball falling from rest under the influence of gravity, while Figure 2 displays a car traveling along a straight track with constant acceleration. (Click here to download a zipped file containing the animations in Figures 1 and 2 in both gif and QuickTime formats.)
Average rate of change is often introduced by saying it is the change in distance over the change in time:
Let s denote distance and t denote time, then we use the symbols for change in distance and for change in time. Thus we have
In situations involving the motion of an object like in Figures 1 and 2 we use the terminology average velocity in place of average rate of change. In such cases we denote average velocity by and we have
In Figure 1 we can compute the average velocity of the falling ball between two marks on the ruler displayed beside the path of the ball. For instance, we can measure the time it takes the ball to drop from the top (0 meter mark) to the 3 meter mark. In this case we have
and so the average velocity of the ball from s = 0 to s = 3 is
(The symbol means approximately equal to, since your calculator will display more than two decimal digits when you compute the ratio of 3 to 0.72. Displaying several decimal places is sufficient for our work.)
Many students don't recognize that things like falling bodies, moving vehicles, rising populations, and decaying radioactive materials do not change at constant rates. For instance, the falling ball has different average velocities as it passes various meter marks. Figure 3 shows a sample of the average velocities of the ball.
A display of the average velocities as the ball passes meter marks is available as an animation. By clicking here you can download a zipped file containing both an animated gif and a QuickTime file that illustrates the different average velocities of the falling ball. We recommend that when you show it to a class that you use the QuickTime file. This will let you start and stop the animation so that you can discuss portions with your students. Figure 4 contains a segment of the animation. The full animation starts tracking the average velocities as the ball starts at the top.
A similar display of the average velocities as the car passes successive 10 meter marks is available as an animation. By clicking here you can download a zipped file containing both an animated gif and a QuickTime file that illustrates the different average velocities of the car. A preview of this animation appears in Figure 5.
A good classroom activity for students is to have them construct a table of distances covered, elapsed times, and average velocities of the falling ball animation and/or the moving car animation. One suggestion is first show the animation discussing it as it progresses. Using the QuickTime file lets you start and stop the animation. After the initial discussion show the animation a second time during which students construct a table as shown in Figure 6. A brief discussion of these tables reinforces the idea that the average velocity of the objects in these demos changes.
To provide computational experience for calculating average velocities, imagine that the ball was falling on the moon or on mars. Basically the animation remains the same, but the times to fall from the top to a meter mark will vary. Tables for each of these scenarios appear in Figure 7. A brief discussion of why the times and hence average velocities change is a natural link to basic physics ideas.
If you prefer to use the moving car animation we can alter its behavior by giving the car an initial velocity or changing the value of the constant acceleration. Two such cases appear in the tables shown in Figure 8. Again a brief discussion of the changes in time and average velocity lead to an easy link to familiar physical concepts.
The tables in Figures 7 and 8 can be downloaded as a pdf file by clicking here. Each table appears on a separate page for ease of duplication for class handouts.
Average Rate of Change of a Function: Each of the tables in Figures 6 - 8 is a discrete sample of a function. Next we connect average rates of change to the slope of a line segment between two points on a curve.
Figure 9a displays a plot of the time vs. distance data for the falling ball. (We have included the point (0, 0) since at time = 0, the distance traveled is s = 0.) The points shown in Figure 9a are a sample of the points along the curve shown in Figure 9b which is plot of time vs. distance for all distances along the ruler shown in Figure 1. The data points in Figure 9a are from the falling ball data in Figure 6.
In Figure 10 we display an animation that draws a line segment from the origin to each data point in Figure 9a and displays the slope of that segment. Comparing the slopes of the line segments with the average velocities for the falling ball that are displayed in Figure 6 we see that they are the same.
Click here to download a zipped file containing the animation of Figure 10 in both gif and QuickTime formats. (For class discussion we recommend using the QuickTime file so that you can start and stop the animation as you discuss the ideas.)
An animation similar to that in Figure 10 for the moving car is available. Click here to download a zipped file containing the moving car animation in both gif and QuickTime formats. (For class discussion we recommend using the QuickTime file so that you can start and stop the animation as you discuss the ideas.)
Average Rate of Change of a Function over an Interval
The average rate of change of a function y = f(x) over an interval [a, b] in its domain is defined as follows:
This is illustrated geometrically as shown in Figure 11 and we say
that the quotient Dy / Dx is the slope of the secant line from point (a, f(a)) to point (b, f(b)). (Note: a secant line is any line connecting two points on the same curve.) If function y = f(x) measures the distance covered as time varies from x = a to x = b, then the slope of the secant line from (a, f(a)) to (b, f(b)) is interpreted as an average velocity. This situation was illustrated for the falling ball by the animation in Figure 10.
The falling ball and moving car examples discussed above are familiar situations to most students. The data shown in Figure 6 and used in the animations for these examples is limited to intervals starting at 0 and ending at a certain meter mark. We have constructed an interactive Excel demo for each of these examples that permits computation of the average rate of change over many intervals. Figure 12 shows the screen for the falling ball. The idea is to move the sliders to calculate the average rate of change between the two points on the curve. (The curve shown was generated by creating an interpolant to the discrete data in Figure 6 which is displayed in Figure 9a.) This Excel file for the falling ball can be executed or downloaded by clicking here. For a corresponding Excel file for the moving car click here.
For a set of interactive Excel files for computing average rates of change along a curve see the auxiliary resources below.
1. We have constructed at set of five interactive Excel demos involving average rates of change. You can execute or download this collection by clicking here. These demos could be used in class by the instructor, used with groups in lab setting, or assigned as out-of-class investigations. A set of suggestions for questions that could be assigned as part of the student investigations is available by clicking here.
2. Radar Guns: At the January 2005 Joint Mathematics Meeting in Atlanta, Ga., Melvin Royer of Indiana Wesleyan University gave a talk entitled Calculus Demonstrations: Economical Radar Guns in the contributed paper session MY FAVORITE DEMO—Innovative Strategies for Mathematics Instructors organized by David R. Hill and Lila F. Roberts. in his talk he discussed a classroom demonstration that demonstrates how to use economical radar guns to measure average velocity. The economical radar gun consists of a meter stick and a stop watch. The set up is to have a ball which is rolling down an inclined plane that has a meter stick fixed along its edge and use the stop watch to get time it takes cover a particular distance. By having multiple stop watches so that students can work in teams data like that given in Figure 6 can easily be recorded. Students can then determine the average rate of change over different time intervals and get first hand experience with a situation in which the moving object does not have constant velocity. Professor Royer kindly gave us permission to include his abstract in this demo abstract; click here for a pdf file of the abstract. In his abstract he doesn't stop with average velocity and we will refer to it again in a demo on instantaneous velocity.
A similar class demo with a complete lesson plan can be found at http://www.tvgreen.com/Spectrum08/document/MotionLab.htm. The materials needed are easily obtained.
For some basic information on how actual radar guns work go to http://electronics.howstuffworks.com/radar-detector1.htm. Portions of the information give a nice description that provides a connection between math and physics.
3. The demo Escalator Motion and Average Rates of Change provides an early introduction to related rates of change using escalator motion and average rates of change.
4. Connections to other topics. The concept of average rate of change is often illustrated through examples and exercises that use applications like average velocity, average acceleration, average weight gain, average cost, and so on. These are followed by tying the average rate of change to the instantaneous rate of change by a limit process.
5. The animations for the falling ball and the moving car
were made using applet at the following URLs respectively.
and is included in Demos with Positive Impact with his permission.