Copyright © Cengage Learning. All rights reserved. 7.4 SECTION Logistic Functions Copyright © Cengage Learning. All rights reserved.

Learning Objectives 1 Graph logistic functions from equations and tables 2 Use logistic models to predict and interpret unknown results

Logistic Growth

Logistic Growth Figure 7.23 represents the total number of people who have had the flu as a combination of very slow exponential growth (labeled a), rapid exponential growth (labeled b), followed by a slower increase (labeled c), and then a leveling off (labeled d). Figure 7.23

Logistic Growth At the inflection point the rate at which the flu is spreading is the greatest. The horizontal asymptote (red dashed horizontal line) represents the limiting value for the number of people who will contract the flu. The mathematical model for such behavior is called a logistic function.

Logistic Growth

Example 1 – Exploring Logistic Functions in a Real-World Context The Centers for Disease Control monitor flu infections annually, paying particular attention to flu-like symptoms in children from birth to 4 years old. The cumulative number of children 0–4 years old who visited the CDC’s sentinel providers with flu-like symptoms during the 2006–2007 flu season are displayed in Figure 7.24, together with a logistic model. Children (0–4) with Flu-Like Symptoms Figure 7.24

Example 1 – Exploring Logistic Functions in a Real-World Context cont’d a. Explain why a logistic function may better model the 2006–2007 flu infection rate than an exponential function. Figure 7.24 Children (0–4) with Flu-Like Symptoms

Example 1(a) – Solution A logistic function models the growth in the cumulative number of children with flu-like symptoms better than an exponential function because a logistic function grows slowly at first, then more quickly, and finally levels off at a limiting value. An exponential function, on the other hand, may grow slowly at first but then will increase at an ever increasing rate, ultimately exceeding the available number of children who could conceivably become infected with flu-like illnesses.

Example 1 – Exploring Logistic Functions in a Real-World Context cont’d b. Using the language of rate of change, describe the behavior of the graph and relate this to what it tells about the real-world context. Children (0–4) with Flu-Like Symptoms

Example 1(b) – Solution cont’d The graph of the function model is increasing throughout the interval; however, the rate at which the graph is increasing varies. Initially, the graph is concave up, indicating that the cumulative number of reported cases is increasing at an ever increasing rate. Around week 17, the graph changes to concave down, indicating that the cumulative number of reported cases are increasing at a lesser and lesser rate.

Example 1(b) – Solution cont’d Around week 32, the graph is increasing at such a slow rate that it appears to level off. This indicates that the number of newly reported cases in weeks 32 and beyond is so small that it has a negligible effect on the cumulative number of cases.

Example 1 – Exploring Logistic Functions in a Real-World Context cont’d c. The formula for the logistic function that models the cumulative number of reported children with flu-like symptoms is where w is the week of the flu season. What does the limiting value mean in the real-world context? How is the limiting value represented in the formula for the function H(w) and its graph?

Children (0–4) with Flu-Like Symptoms Example 1(c) – Solution cont’d The limiting value is approximately 75,700 children. It appears that no more than 75,700 children were reported to have flu-like symptoms during the season. This value is represented in the formula for H(w) by the value 75,700 found in the numerator and on the graph by the horizontal asymptote at 75,700. Children (0–4) with Flu-Like Symptoms Figure 7.24

Example 1 – Exploring Logistic Functions in a Real-World Context cont’d d. Estimate the coordinates for the point of inflection and explain what each coordinate means in the real-world context. Children (0–4) with Flu-Like Symptoms

Example 1(d) – Solution cont’d From the graph of H(w), an estimate for the coordinates of the inflection point is approximately (17, 40,000). This means at week 17 of the 2006–2007 flu season (late January), the number of children with flu-like symptoms was increasing most rapidly.

Example – Exploring Logistic Functions From a Table Between what intervals is the inflection point? What is the meaning of the inflection point within the context of the problem?

Example – Exploring Logistic Functions From a Table The percentage of households with a telephone increase throughout the table; however, the rate of increase between 10 and 15 years since 1935 begins to decrease indicating an inflection point. So between 1945 and 1950 the percentage of households with telephones increased at it’s highest rate.

Logistic Decay

Logistic Decay Many real-world data sets are modeled with decreasing rather than increasing logistic functions. As was the case with logistic growth functions, logistic decay functions have an upper limiting value, L, and a lower limiting value of y = 0.

Example 2 – Recognizing a Logistic Decay Function As shown in Table 7.19, the infant mortality rate in the United States has been falling since 1950. (An infant is a child under 1 year of age.) Table 7.19

Example 2 – Recognizing a Logistic Decay Function cont’d a. Using Table 7.19, show how the rate of change records the decline in the infant death mortality rate and discuss why this suggests that a logistic model may fit the data.

Example 2(a) – Solution If we calculate the decrease in the infant mortality rate (M) and the yearly rate of change of the infant mortality rate over each interval of time given in Table 7.20, we can determine the behavior of the function M(y). Table 7.20

Example 2(a) – Solution cont’d Note: We must exercise caution because the intervals between the years provided are not the same. From the rate of change, we can see that the decline in the infant mortality rate tends to drop slowly at first, then more dramatically, and then levels off. The change in the infant mortality rate suggests a logistic model.

Example 2 – Recognizing a Logistic Decay Function cont’d b. From the table, predict the future lower limiting value and explain what the numerical value means in the real-world context. Does this value seem reasonable?

Example 2(b) – Solution cont’d We estimate the lower limiting value to be approximately 6.0 deaths per 1000 live births because we expect the values to level off near 6.0.

Example 2 – Recognizing a Logistic Decay Function cont’d c. Create a scatter plot of the data and estimate the upper and lower limiting values.

Example 2(c) – Solution The scatter plot is shown in Figure 7.25. cont’d The scatter plot is shown in Figure 7.25. From the scatter plot, we predict the upper limiting value will be around 35 and the lower limiting value will be around 6. Figure 7.25

Logistic Decay We said earlier that all logistic functions have a lower limiting value at y = 0. We can use logistic regression to model data sets with this lower limiting value. But when a data set appears to be logistic but has a different lower limiting value, as in Example 2, we need to align the data before using logistic regression to model the function.

Example: Lower Limiting Value Alignment a.) Use logistic regression to find the logistic model for the data. (Remember to align your data for the lower limiting value.)

Example: Lower Limiting Value Alignment a.) Use logistic regression to find the logistic model for the data. (Remember to align your data for the lower limiting value.)

Example 4 – Extrapolating Exponential and Logistic Growth cont’d The sales of DVD hardware from 1997 to 2006 are given in Table 7.22. Table 7.22

Example 4 – Extrapolating Exponential and Logistic Growth cont’d a. Use logistic regression to determine the function, D(y), of the form that best models the growth in DVD sales from 1997 to 2004. Determine the value of L and explain what this number means in the real-world context. b. Based on the data from 2005 and 2006, does it appear the model can be used to accurately forecast future sales? Explain why or why not.

Example 4(a) – Solution Using the Technology Tip and logistic regression on the data from 1997 to 2004, we find million dollars is the best-fit logistic function for DVD sales (see Figure 7.29). The limiting value, L, is 40.69, which means DVD sales will level off at $40,690,000 per year. Figure 7.29

Example 4(b) – Solution cont’d For the years 2005 and 2006, we evaluate the function at t = 8 and t = 9. The model forecasts 39.3 million dollars in sales in 2005 and 40.1 million dollars in sales in 2006. In actuality, there were 36.7 and 32.7 million dollars in sales, respectively.

Example 4(b) – Solution cont’d The model predicts a leveling off of sales whereas the actual data shows a decline in sales in 2005 and 2006 Consequently, the model does not appear to accurately model future sales.