Skip to main content

Chapter 5 Functions and Their Graphs

earth population

World3 is a computer model developed by a team of researchers at MIT. The model tracks population growth, use of resources, land development, industrial investment, pollution, and many other variables that describe human impact on the planet.

The figure below is taken from Limits to Growth: The 30-Year Update. The graphs represent four possible answers to World3's core question: How may the global population and economy interact with and adapt to Earth's limited carrying capacity (the maximum it can sustain) over the coming decades?

models of world population limits

Each of the graphs represents a nonlinear function. A function is a special relationship between two variables, and we have already encountered linear and quadratic funtions. In this chapter we examine the properties and features of some basic nonlinear functions, and how they may be used as mathematial models.

Investigation 5.0.1. Epidemics.

A contagious disease whose spread is unchecked can devastate a confined population. For example, in the early 16th-century, Spanish troops introduced smallpox into the Aztec population in Central America, and the resulting epidemic contributed significantly to the fall of Montezuma’s empire.

Suppose that an outbreak of cholera follows severe flooding in an isolated town of 5000 people. Initially (Day 0), 40 people are infected. Every day after that, 25% of those still healthy fall ill.

  1. At the beginning of the first day (Day 1), how many people are still healthy? How many will fall ill during the first day? What is the total number of people infected after the first day?

  2. Check your results against the table below. Subtract the total number of infected residents from 5000 to find the number of healthy residents at the beginning of the second day. Then fill in the rest of the table for 10 days. (Round off decimal results to the nearest whole number.)

    Day Number
    healthy
    New
    patients
    Total
    infected
    \(0\) \(5000\) \(40\) \(40\)
    \(1\) \(4960\) \(1240\) \(1280\)
    \(2\) \(\) \(\) \(\)
    \(3\) \(\) \(\) \(\)
    \(4\) \(\) \(\) \(\)
    \(5\) \(\) \(\) \(\)
    \(6\) \(\) \(\) \(\)
    \(7\) \(\) \(\) \(\)
    \(8\) \(\) \(\) \(\)
    \(9\) \(\) \(\) \(\)
    \(10\) \(\) \(\) \(\)
    grid
  3. Use the last column of the table to plot the total number of infected residents, \(I\text{,}\) against time, \(t\text{,}\) on the grid. Connect your data points with a smooth curve.

  4. Do the values of \(I\) approach some largest value? Draw a dotted horizontal line at that value of \(I\text{.}\) Will the values of \(I\) ever exceed that value?

  5. What is the first day on which at least 95% of the population is infected?

  6. Look back at the table. What is happening to the number of new patients each day as time goes on? How is this phenomenon reflected in the graph? How would your graph look if the number of new patients every day were a constant?

  7. Summarize your work: In your own words, describe how the number of residents infected with cholera changes with time. Include a description of your graph.