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TEACHER'S GUIDE
I have never been satisfied with the applications of exponential functions that are presented in precalculus courses. That coverage consists primarily of supplying the requisite formula for, say, the growth of a bacterial colony or the decay of some radioactive substance, without explaining or deriving the reasons that the exponential function appears in the formula. The students are more than willing to accept the formulas and crunch through the numbers. However, unless I present a mathematical explanation describing the reasons that the exponential function appears in the formulas governing population growth and radioactive decay, I have not fostered my students' ability to recognize other phenomena, either in mathematics or in other disciplines, that use exponential functions.
Such modeling phenomena as the unrestricted growth of a population or the decay of a radioactive substance require a function whose rate of change is proportional to the function value. This activity was designed to convince precalculus students that exponential functions satisfy such a property and are therefore valuable in applications. I realize that rate of change, that is, the derivative, is a topic traditionally reserved for calculus classes, and I am not advocating that we teach derivatives in precalculus. However, such technological tools as graphing calculators, computer algebra systems, and computer graphing programs, which greatly facilitate experimentation, allow us to discuss and compute functional rates of change without defining limits, derivatives, rules for derivatives, and so forth. Furthermore, with current educational trends toward student exploration and discovery, we can push beyond traditional course boundaries.
Audience: Precalculus or math analysis students
Objectives: To introduce the concepts of average rate of change and instantaneous rate of change of a function and to explore the relationship between the value of the function f(x) = b^sup x^ and its instantaneous rate of change
Materials: Activity sheets and a graphing calculator, such as a TI-82; a computer algebra system, such as Maple; or a computer graphing program, such as Cricket Graph III
Directions: The activity sheets should be completed sequentially. Students should complete activity sheets 1 through 3 before sheet 4 is distributed. Students should complete sheet 4 before receiving the final activity sheet. If time is short, sheet 5 makes an excellent homework assignment.
This activity may begin with...