Copyright © Cengage Learning. All rights reserved. 3 Exponential and Logarithmic Functions

Copyright © Cengage Learning. All rights reserved. 3.6 Nonlinear Models

3 What You Should Learn Classify scatter plots. Use scatter plots and a graphing utility to find models for data and choose the model that best fits a set of data. Use a graphing utility to find exponential and logistic models for data.

4 Classifying Scatter Plots

5 A scatter plot can be used to give you an idea of which type of model will best fit a set of data.

6 Example 1 – Classifying Scatter Plots Decide whether each set of data could best be modeled by a linear model, y = ax + b, an exponential model, y = ab x,or a logarithmic model, y = a + b ln x. a. (2, 1), (2.5, 1.2), (3, 1.3), (3.5, 1.5), (4, 1.8), (4.5, 2), (5, 2.4), (5.5, 2.5), (6, 3.1), (6.5, 3.8), (7, 4.5), (7.5, 5), (8, 6.5), (8.5, 7.8), (9, 9), (9.5, 10) b. (2, 2), (2.5, 3.1), (3, 3.8), (3.5, 4.3), (4, 4.6), (4.5, 5.3), (5, 5.6), (5.5, 5.9), (6, 6.2), (6.5, 6.4), (7, 6.9), (7.5, 7.2), (8, 7.6), (8.5, 7.9), (9, 8), (9.5, 8.2)

7 Example 1 – Classifying Scatter Plots c. (2, 1.9), (2.5, 2.5), (3, 3.2), (3.5, 3.6), (4, 4.3), (4.5, 4.7), (5, 5.2), (5.5, 5.7), (6, 6.4), (6.5, 6.8), (7, 7.2), (7.5, 7.9), (8, 8.6), (8.5, 8.9), (9, 9.5), (9.5, 9.9) Solution: a. From Figure 3.41, it appears that the data can best be modeled by an exponential function. Figure 3.41

8 Example 1 – Solution b. From Figure 3.42, it appears that the data can best be modeled by a logarithmic function. c. From Figure 3.43, it appears that the data can best be modeled by a linear function. cont’d Figure 3.42 Figure 3.43

9 Fitting Nonlinear Models to Data

10 Fitting Nonlinear Models to Data Once you have used a scatter plot to determine the type of model that would best fit a set of data, there are several ways that you can actually find the model. Each method is best used with a computer or calculator, rather than with hand calculations.

11 Example 2 – Fitting a Model to a Data Fit the following data from Example 1(a) to an exponential model and a power model. Identify the coefficient of determination and determine which model fits the data better. (2, 1), (2.5, 1.2), (3, 1.3), (3.5, 1.5), (4, 1.8), (4.5, 2), (5, 2.4), (5.5, 2.5), (6, 3.1), (6.5, 3.8), (7, 4.5), (7.5, 5), (8, 6.5), (8.5, 7.8), (9, 9), (9.5, 10)

12 Example 2 – Solution Begin by entering the data into a graphing utility. Then use the regression feature of the graphing utility to find exponential and power models for the data, as shown in Figure Figure 3.44 Power Model Exponential Model

13 Example 2 – Solution So, an exponential model for the data is y = 0.507(1.368) x, and a power model for the data is y = 0.249x Plot the data and each model in the same viewing window, as shown in Figure cont’d Figure 3.45 Power Model Exponential Model

14 Example 2 – Solution To determine which model fits the data better, compare the coefficients of determination for each model. The model whose r 2 -value is closest to 1 is the model that better fits the data. In this case, the better-fitting model is the exponential model. cont’d

15 Modeling with Exponential and Logistic Functions

16 Example 4 – Fitting an Exponential Model to Data The table below shows the amounts of revenue R (in billions of dollars) collected by the Internal Revenue Service (IRS) for selected years from 1963 through Use a graphing utility to find a model for the data. Then use the model to estimate the revenue collected in (Source: IRS Data Book)

17 Example 4 – Solution Let x represent the year, with x = 3 corresponding to Begin by entering the data into a graphing utility and displaying the scatter plot, as shown in Figure Figure 3.48

18 Example 4 – Solution From the scatter plot, it appears that an exponential model is a good fit. Use the regression feature of the graphing utility to find the exponential model, as shown in Figure Change the model to a natural exponential model, as follows. R = 96.56(1.076) x = (1.076)x  x cont’d Write original model. b = e ln b Figure 3.49 Simplify.

19 Example 4 – Solution Graph the data and the natural exponential model R = 96.56e 0.073x in the same viewing window, as shown in Figure From the model, you can see that the revenue collected by the IRS from 1963 through 2008 had an average annual increase of about 7%. From this model, you can estimate the 2013 revenue to be R = 96.56e 0.073x cont’d Figure 3.50 Write natural exponential model.

20 Example 4 – Solution = 96.56e 0.073(53)  $ billion which is more than twice the amount collected in You can also use the value feature of the graphing utility to approximate the revenue in 2013 to be $ billion, as shown in Figure cont’d Substitute 53 for x. Use a calculator.