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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.1 Exponential and Logistic Functions.

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Presentation on theme: "Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.1 Exponential and Logistic Functions."— Presentation transcript:

1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.1 Exponential and Logistic Functions

2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 2 Quick Review

3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 3 Quick Review Solutions

4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 4 What you’ll learn about Exponential Functions and Their Graphs The Natural Base e Logistic Functions and Their Graphs Population Models … and why Exponential and logistic functions model many growth patterns, including the growth of human and animal populations.

5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 5 Exponential Functions

6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6 Example Computing the Exact Value of an Exponential Function

7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7 Example Finding an Exponential Function from its Table of Values

8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8 Exponential Growth and Decay

9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 9 Example Transforming Exponential Functions

10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Exponential Functions

11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Natural Base e

12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Exponential Functions and the Base e

13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Exponential Functions and the Base e

14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Exponential Functions

15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Population Growth The 1990 population in Austin was 465,622, and in 2000 it was 656,562. Assuming exponential growth, when will it pass 800,000?

16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Logistic Growth Functions

17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Population Growth The population of New York state can be modeled by Where P is population in millions and t is the number of years since Based on the model, a) What was the New York population in 1850? b) What will the population be in 2010? c) What is New York’s maximum sustainable population?

18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Carbon Dating The amount C in grams of carbon-14 present in a certain substance after t years is given by C = 20e – t. a) What was the initial amount of carbon-14? b) How much carbon-14 is left after 10,400 years? c) When will the amount left be 10 grams?

19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Review Section 3.1 Page 286, Exercises: 1 – 69 (EOO)

20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.2 Exponential and Logistic Modeling

21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review

22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions

23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.

24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown. If r > 0, then P(t) is an exponential growth function. If r < 0, then P(t) is an exponential decay function.

25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Exponential Population Model

26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Growth and Decay Rates

27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.

28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Bacteria Growth

29 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Radioactive Decay

30 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Growth with a Logistic Model

31 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Atmospheric Pressure

32 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for 2003.

33 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population. These situations are best modeled using a logistic model.

34 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling a Rumor

35 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling a Rumor


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