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Copyright © Cengage Learning. All rights reserved. 2 Nonlinear Functions and Models.

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1 Copyright © Cengage Learning. All rights reserved. 2 Nonlinear Functions and Models

2 Copyright © Cengage Learning. All rights reserved. 2.4 Logistic Functions and Models

3 3 Logistic Function A logistic function has the form for nonzero constants N, A, and b (A and b positive and b ≠ 1).

4 4 Logistic Functions and Models Quick Example N = 6, A = 2, b = 1.1 gives 6 / (1+2*1.1^ –x) The y-intercept is N / (1 + A). When x is large, f (x) ≈ N

5 5 Logistic Functions and Models Graph of a Logistic Function

6 6 Logistic Functions and Models Properties of the Logistic Curve The graph is an S-shaped curve sandwiched between the horizontal lines y = 0 and y = N. N is called the limiting value of the logistic curve. If b > 1 the graph rises; if b < 1, the graph falls. The y-intercept is The curve is steepest when

7 7 Logistic Functions and Models Logistic Function for Small x and the Role of b For small values of x, we have Thus, for small x, the logistic function grows approximately exponentially with base b.

8 8 Logistic Functions and Models Quick Example Let Then for small values of x. The following figure compares their graphs: The upper curve is the exponential curve. N = 50, A = 24, b = 3

9 9 Modeling with the Logistic Function

10 10 Example 1 – Epidemics A flu epidemic is spreading through the U.S. population. An estimated 150 million people are susceptible to this particular strain, and it is predicted that all susceptible people will eventually become infected. There are 10,000 people already infected, and the number is doubling every 2 weeks. Use a logistic function to model the number of people infected. Hence predict when, to the nearest week, 1 million people will be infected.

11 11 Example 1 – Solution Let t be time in weeks, and let P(t) be the total number of people infected at time t. We want to express P as a logistic function of t, so that We are told that, in the long run, 150 million people will be infected, so that N = 150,000,000. At the current time (t = 0), 10,000 people are infected, so Limiting value of P Value of P when t = 0

12 12 Example 1 – Solution Solving for A gives 10,000(1 + A) = 150,000,000 1 + A = 15,000 A = 14,999. What about b? At the beginning of the epidemic (t near 0), P is growing approximately exponentially, doubling every 2 weeks. We found that the exponential curve passing through the points (0, 10,000) and (2, 20,000) is giving us cont’d

13 13 Example 1 – Solution Now that we have the constants N, A, and b, we can write down the logistic model: The graph of this function is shown in Figure 18. cont’d Figure 18

14 14 Example 1 – Solution Now we tackle the question of prediction: When will 1 million people be infected? In other words: When is P(t) = 1,000,000? cont’d

15 15 Example 1 – Solution Thus, 1 million people will be infected by about the 13th week. cont’d Logarithmic form

16 16 Logistic Regression

17 17 Example 2 – Broadband Penetration Here are the data graphed in Figure 17: Figure 17

18 18 Example 2 – Broadband Penetration Find a logistic regression curve of the form In the long term, what percentage of broadband penetration in the United States does the model predict? cont’d

19 19 Example 2 – Solution We can use technology to obtain the following regression model: Its graph and the original data are shown in Figure 19. Figure 19 Coefficients rounded to 3 decimal places

20 20 Example 2 – Solution Because N = 29.842, this model predicts that, in the long term, the percentage of broadband penetration in the United States will be 29.842%, or about 30%. cont’d


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