Objective: Provide an early Introduction to related rates of change using escalator motion and average rates of change.
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.
Related rate problems cause anxiety in many calculus students for a variety of reasons. (Click here to go to a demo on related rates that use instantaneous rates of change.) This demo can be used before a formal development of the variety of differentiation techniques studied in first term calculus. We use the familiar setting of escalator motion and average rates of change to relate vertical, horizontal, and "hypotenuse" rates. We give a brief review of average rates of change and then introduce the escalator.
The average rate of change of a function f(x) is used as an introductory step to viewing a derivative as an instantaneous rate of change. A common definition is stated as
This is then illustrated geometrically as shown in Figure 1 and we say
that the quotient Dy / Dx is the slope of the secant line from (a, f(a)) to (b, f(b)). The concept of average rate of change is then 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.
To lay a foundation for related rates which come after various differentiation techniques like power rule, chain rule, and implicit differentiation are developed, we can use the motion of an escalator. One approach is the following.
An escalator employs a right triangle so that people can move from a lower floor to an upper floor (or vice versa) as shown in Figure 2. We have labeled the base of the triangle horizontal (denoted H), the altitude of the triangle vertical (denoted V), and the hypotenuse of the triangle people (denoted P, for the distance people travel between floors).
The following animation provides a reasonable visualization of the motion of an escalator.
The gear mechanisms that are used to move the treads of the escalator are set a certain average speed, denote this as s inches per second. Hence the people move between floors along the hypotenuse of the triangle at the average rate DP = s . Natural questions that arise involve the related rates of change both horizontally and vertically. For instance,
Our approach to answer these questions depends upon the information we have available concerning the escalator's "dimensions", that is the lengths of the sides of the right triangle and the value of s. We give several scenarios next.
Scenario #1. We are given s = 10"/sec, H = 400", and V = 300". Find DH and DV.
Scenario #2. We given H and V and it has been measured by experiment that the average travel time between floors is t seconds. Find DH, DV, and DP.
Scenario #3. This is an outline of a project for groups in your class. Discuss the average rate of change of a function in the usual way. Talk briefly about an escalator and show the animation used above. Each group is to find an escalator and estimate the average rates of change DH, DV, and DP as shown in Figure 2. They are to develop a write up for the project including all pertinent data, calculations, and conclusions. They will need a tape measure and a watch that displays seconds. Emphasize that they may need to do several timings to get an average time to move from floor to floor. If someone has a digital camera it would be nice to include a picture of the "their" escalator. A class discussion of "differences" in escalators is a nice follow up.
A very nice resource for a project
of this type is available at the web site of William Thayer (his email is below)
He has a detailed outline for students to follow which incorporates work sheets
and lots of guidance. The outline uses the steps Speed, Function Notation, Y Rate,
X Rate, Slope, Linear Functions, *Related Rate Equations, Linear Motion, and
History of d = r t. In addition there are links to associated pages developed by
his students, aptly called "math artists".
William V. Thayer
We appreciate his cooperation in the development of this demo. The features of this demo and the link to his work sheets are used with his permission. This demo was constructed by
and is included in Demos with Positive Impact with his permission.
DRH 2/11/2002 Last updated 5/4/2004