CA I.7 Exponential Functions - Exploration of Rate of Change Relationships
Purpose: To identify and utilize rate of change relationships exhibited in exponential functions.
Classroom Procedure: After an initial introduction to this activity students will work on the first three problems in small groups. There may be points in this work that you would like to call the class together to share the progress that they have made. This activity generally takes two days in class to complete and Q5 is a good breaking point as after an initial discussion of this problem students can complete this problem at home and return ready to discuss their findings. This is followed by a class discussion where we establish that exponential functions will always have a rate of change that is proportional to the dependent variable. For Q7, students should work individually, then in small groups and then be ready to participate in whole class discussion. There is a lot in this question and it is best if you can give it a good amount of time in class so that students are comfortable with these ideas before working on the synthesis questions. The Synthesis Questions should be completed individually as homework and students should be prepared to share their work on these problems.
Ideas this Activity Builds On: This activity is really quite new for the students. Although it builds on an understanding of characterizing exponential functions as having constant growth factor or constant percent change it quickly moves away from this idea to characterize them in an entirely new way. We expect that students are comfortable with calculating rates of change and recognizing and writing down proportional relationships.
Introduction/Motivation of the Activity: To motivate this activity we talk a little about the fact that some growth and decay can be characterized using the rate of change and that we will begin investigating this idea using exponential functions first.
Need to Establish by the End of Activity/Wrap-Up: Students should leave this activity understanding that if a rate of change is proportional to the dependent variable then the original relationship was an exponential one. In addition they should be able to recognize this information in a variety of representations.
Additional Notes: The synthesis questions for this activity push students to interpret and understand rate of change equations that are not proportional relationships. These questions are not unduly difficult if students are focused on meaning in the activity however they are very difficult for students who have been trying to find patterns to work from. It is a good place to try to focus students in on the meaning of the rate of change equations. It is generally a good idea to leave some time where students could ask you questions in office hours or after class for these synthesis questions.
Note on Rate of Change Equations:
We have introduced rate of change equations in this activity as a precursor to Differential equations later in the course. In a rate of change equation we always associate the rate of change with the beginning of the interval. Another way to say this is when we have (Delta y)/(Delta t) = F(y) what that means is that (y_(n+1)-y_n)/(Delta t)=F(y_n). When the students will have to do something numerical with the rate of change equation, we provide the Delta t used to obtain the rate of change equation. This means that this is the Delta t that it is appropriate to make numerical estimations with. In general, this should provide no issues for the students as each numerical estimation that is asked for is appropriate given the Delta t that was used to create the rate of change equation. If there is no numerical work required then we do not provide the Delta t and the assumption should be that we have used a small enough Delta t to be appropriate for that question. Using a small enough Delta t makes the rate of change equation very close to the corresponding differential equation.