In the first chapter we looked at two different kinds of change -- discrete change, or discrete dynamical systems -- and continuous change, or continuous dynamical systems. Discrete dynamical systems are very useful for three reasons.

• In many situations change really is discrete -- it occurs at well-defined time intervals. Examples of this kind of change include farm prices and the population growth of temperate zone insects.

• Continuous change can often be approximated very well by discrete change. Discrete change is often easier to work with and is conceptually more simple. Thus, even when a continuous model is better, a discrete model may work almost as well and may be more widely understood.

• Data is usually discrete rather than continuous. So even when the underlying model is continuous we may be forced to work with a discrete model because of the limitations of the available data.

For these reasons we begin our systematic study of models with discrete dynamical systems.

The graph at the left shows data obtained from the United States Census Bureau estimating world population in the middle of each year for the years 1950-1995. Click here to see the same data in tabular form. A great deal of data that is useful in class can be obtained at no charge over the World Wide Web from the United States Census Bureau. The links below lead to some examples. Historical World Population Data Midyear World Population Data 1950 -- 2050 Midyear Population Data by Country 1950 -- 2050 For additional information on creating extracts from the 1990 United States Census click here.

The data in the graph above is available in your CAS window. Open your CAS window now to follow along as we examine this data.

Data Snooping 101

We are primarily interested in change -- the way in which the world's population is changing from one year to the next. We begin by examining the data. The technical term for this is data snooping. The two most common ways to describe this kind of change are

• The change in the number of people from one year to the next.
• The change in population from one year to the next expressed as a percentage.

The graph at the left shows the population change expressed as a number for each year from 1950 -- 1995. Notice this graph shows an increase in the rate of growth -- from about 37,000,000 people per year in 1950 to about 80,000,000 people per year in the 1980s and 1990s. Around 1960 there is a dramatic drop in the population growth rate. Can you suggest any possible explanations? This graph is rather worrisome. We see a dramatic increase in the rate of population increase over the course of this 45 year period with the rate of increase apparently settling down at about 80,000,000 people per year.

The graph at the left shows the population change expressed as a percentage for each year from 1950 -- 1995. Population growth is often expressed as a percentage. If the population of a particular city is growing, for example, at the rate of 1,000 people per year the impact of this growth on the city could be very small (if the population of the city were 5,000,000) or it could be very big (if the population of the city were 25,000). The percentage rate of increase is a better indicator of the impact of the population growth. This graph paints a very different picture than the graph above. The percentage rate of increase of population is the same at the end of this 45 year period as at the beginning.

In this module we work with three different ways of describing the rate at which a sequence

• The difference between each term and the preceding term

  

• The ratio between each term and the preceding term

  

• The percentage difference between each term and the preceding term

  

The ratio and the percentage difference get at the same idea. They both compare the growth from one term to the next to the first of these two terms. The percentage rate of increase is commonly used in everyday language but the ratio is used more often in modeling.

The difference and the ratio are most useful in different situations. For example, if water was evaporating from an open cylindrical storage tank and the seqeunce

The simplest models are ones in which either the differences or the ratios are constant.

Theorem

If the differences

Prove the theorem above. Hint: Use mathematical induction.

Theorem

If the ratios

This kind of model is called an exponential model.

Prove the theorem above. Hint: Use mathematical induction.

The first steps in examining data are often computing the differences and the ratios described above. If the differences are nearly constant then a linear model may be in order. If the ratios are nearly constant then an exponential model may be in order.

• Make tables for the differences and the ratios for the world population data in your CAS window.

• Click here to go to a United States Census Bureau site from which you can obtain midyear population data for many different countries for the years 1950 -- 2050. The "data" for the future is not data but predictions, or projections. Use the data for 1950 -- 1995 for one country of your choice. Make tables showing the differences and ratios for the population of the country you chose.

• Look at another country. Choose one that is quite different than the first country you chose. Compare your results for the two countries.

• Based on your analysis for the world population make several different predictions, or projections, for the world population in the year 2050. One projection should be based on a low estimate for the population growth rate over the years 1995 -- 2050. Another should be based on a high estimate. A third should be based on a middle estimate.

• Make similar projections for the two countries that you studied above.

• Compare your projections with the projections at the United States Census Bureau Web site.

Data Snooping with Physical Data

Now we want to look at some physical data. This data is interesting in itself and it will also be useful for doing additional experiments later on. You will need some food dye, your TI-CBL with a light probe, and one of the TI graphing calculators. You will also need a dark room.

We will do two sets of experiments to investigate the effect that food dye has on the amount of light that passes through water. For the first experiment we work with a solution of dye of a given concentration, we vary the depth of the liquid and measure how much light passes through the solution with different depths. For the second experiment we fix the depth of the solution and vary the concentration of the food dye. Once we find a relationship between the concentration of the food dye and the amount of light that passes through we can do some additional experiments to see how a polluted lake is cleaned up by the normal flow of water.

The four icons below lead to instructions and programs for using each of the TI graphing calculators with the TI CBL to collect the data. For this experiment you should use the TI-CBL as a light meter, making but not recording light intensity readings. As you make the readings record them with paper-and-pencil.

Whichever calculator you use you will make the following measurements with the calculator connected to the CBL by the linking cable and the light probe connected to the CH1 port of the CBL. Make sure that both the CBL and the TI-92 are turned on.

• A "base-line" measurement with a black solid object between the light probe and the solution. This will give us a reference point for later measurements.

• One series of ten measurements using a solution of fixed concentration but with ten different depths in multiples k, 2k, 3k, ... 10k of the first depth.

• One series of ten measurements with the same depth but with ten different concentrations in multiples k, 2k, 3k, ... 10k of the first concentration.

After you've collected the data do some data snooping. If your measurements are

and the base-line measurement is A then work with the sequence

Does this look like an exponential model? -- or like a linear model? Can you find a formula that will enable you to determine concentration from the light probe reading? Can you find a formula that will let you determine depth from the light probe reading? These formulas can be used in other experiments.