Goals

Today, we will introduce the idea of the rate of change of a function by looking at an applied example. We will also consider some of the information about the function that can be determined by looking at its rate of change. Since this is also the first of the discussion classes of the semester, a few words about this way of working are probably in order. In the discussion meetings of this class, we will be aiming for truly collaborative learning -- that is, for an integrated group effort in analyzing and attacking the discussion questions. The ideal is for everyone in each of the groups to be fully involved in the process. The idea is that, by actively participating in the class through talking about the ideas yourself in your own words, you can come to a better first understanding of what is going on than if you simply listen to someone else (even me!) talk about it.

However, it must be said that to get the most out of this kind of work, you may have to adjust some of your preconceptions. In particular:

• This is not a competition in any sense. You and your fellow group members are working as a team, and the goal is to have everyone understand what the group does fully.
• At different times, it is inevitable that different people within the group will have a more complete grasp of what you are working on and others will have a less complete grasp. Dealing with this a group setting is excellent preparation for real work in a team; it also offers opportunities for significant educational experiences:
  • If you feel totally ``clueless'' at some point, you need to feel free to ask questions and even pester your fellow group members until the point has been explained to your full satisfaction. (Don't forget, the others may be jumping to unwarranted conclusions, and your questions may save the group from pursuing an erroneous train of thought!)
  • On the other hand, when you think you do see something, you may need to explain it carefully to others. (Don't forget, the absolutely best way to make sure you really understand something is to try to explain it to someone else. If you are skipping over an important point in your thinking, it can become very apparent when you set out to convey your ideas to a team member.)

In short, everyone has something to contribute, and everyone will contribute in different ways at different times.

Background

An ecologist has been studying the population of white-tail deer in a particular herd. Over two years she has gathered the following monthly data:

Discussion Questions

1. How would you find the average rate of change of the population (per month) over a general time span covered by the table? Explain in words, and with a formula.
2. What is the average rate of change over the first year? Over the second year? Over the whole two years?
3. Suppose you wanted a reasonably accurate rate of change of the population at the start of a particular month? Explain your method in words, and with a formula or formulas? Generate a table of values for the rate of change at the start of each month.
4. At the start of which month was the herd growing the fastest? At the start of which month was the herd shrinking the fastest?
5. By hand (or otherwise, if you know a way!) sketch rough plots of the population and the rate of change of the population as functions of time over the two years. (This will involve interpolation of the data points from the tables with a smooth curve.) What happens to the rate of change at the times when the population reaches a peak or hits a minimum?

Assignment

Prepare a group write-up of your answers to the questions above. Due: In class, Monday, September 15.