Continuous functions and their derivatives.pptx

Continuous functions and their derivatives

1. Chapter 2 Exponential and Logarithmic Functions

2. Copyright © 2016 Pearson Education, Inc. 2.4 Applications: Uninhibited and Limited Growth Models OBJECTIVE Find functions that satisfy dP / dt = kP . Solve application problems using exponential growth and limited growth models.

3. Slide 3- 3 Copyright © 2016 Pearson Education, Inc. THEOREM 9 A function y = f ( x ) satisfies the equation if and only if for some constant c . 2.4 Applications: Uninhibited and Limited Growth Models

4. Slide 3- 4 Copyright © 2016 Pearson Education, Inc. Example 1: Find the general form of the function that satisfies the equation By Theorem 9, the function must be of the form 2.4 Applications: Uninhibited and Limited Growth Models

5. Slide 3- 5 Copyright © 2016 Pearson Education, Inc. 2.4 Applications: Uninhibited and Limited Growth Models Quick Check 1 Find the general form of the function that satisfies the equation: The function is , or where is an arbitrary constant. As a check, note that

6. Slide 3- 6 Copyright © 2016 Pearson Education, Inc. Uninhibited Population Growth The equation is the basic model of uninhibited (unrestrained) population growth, whether the population is comprised of humans, bacteria in a culture, or dollars invested with interest compounded continuously. So where c is the initial population P , and t is time. 2.4 Applications: Uninhibited and Limited Growth Models

7. Slide 3- 7 Copyright © 2016 Pearson Education, Inc. Example 2: Suppose that an amount P , in dollars, is invested in a savings account where the interest is compounded continuously at 7% per year. That is, the balance P grows at the rate given by a) Find the function that satisfies the equation. Write it in terms of P and 0.07. b) Suppose that $ 10,000 is invested. What is the balance after 1 yr ? c) If $ 10,000 is invested , how fast is the balance growing at t = 1 yr ? 2.4 Applications: Uninhibited and Limited Growth Models

8. Slide 3- 8 Copyright © 2016 Pearson Education, Inc. Example 2 (concluded): a) b) c) So, At t =1, the balance is growing at the rate of about per year. 2.4 Applications: Uninhibited and Limited Growth Models

9. Slide 3- 9 Copyright © 2016 Pearson Education, Inc. Example 3: The world population was approximately 6.0400 billion at the beginning of 2000. It has been estimated that the population is growing exponentially at the rate of 0.016, or 1.6%, per year. Thus, where t is the time, in years, after 2000. 2.4 Applications: Uninhibited and Limited Growth Models

10. Slide 3- 10 Copyright © 2016 Pearson Education, Inc. Example 3 (continued): a) Find the function that satisfies the equation. Assume that P = 6.0400 and k = 0.016. b) Estimate the world population at the beginning of 2020 ( t = 20). c) Find the rate of change of the population at the beginning of 2020. 2.4 Applications: Uninhibited and Limited Growth Models

11. Slide 3- 11 Copyright © 2016 Pearson Education, Inc. Example 3 (concluded): Thus , So, the rate of growth of the population at the beginning of 2020 is roughly 133 million people per year. 2.4 Applications: Uninhibited and Limited Growth Models

12. Slide 3- 12 Copyright © 2016 Pearson Education, Inc. Models of Limited Growth The logistic equation is one model for population growth, in which there are factors preventing the population from exceeding some limiting value L , perhaps a limitation on food, living space, or other natural resources. 2.4 Applications: Uninhibited and Limited Growth Models

13. Slide 3- 13 Copyright © 2016 Pearson Education, Inc. Models of Limited Growth 2.4 Applications: Uninhibited and Limited Growth Models

14. Slide 3- 14 Copyright © 2016 Pearson Education, Inc. Spread by skin-to-skin contact or via shared towels or clothing, methicillin-resistant staphylococcus aureus (MRSA) can easily spread a staph infection throughout a university. Left unchecked, the number of cases of MRSA on a university campus t weeks after the first cases occur can be modeled by 2.4 Applications: Uninhibited and Limited Growth Models Example 4 :

15. Slide 3- 15 Copyright © 2016 Pearson Education, Inc. Example 4 (continued): a) Find the number of infected students after 3 weeks; 40 weeks; 80 weeks. b) Find the rate at which the disease is spreading after 20 weeks. c) Explain why an uninhibited growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation. 2.4 Applications: Uninhibited and Limited Growth Models

16. Slide 3- 16 Copyright © 2016 Pearson Education, Inc. Example 4 (continued): a) N (3) = 11.8. So, approximately 12 students are infected after 3 weeks. N (40) = 221.6. So, approximately 222 students are infected after 40 weeks. N (80) = 547.2. So, approximately 547 students are infected after 80 weeks. 2.4 Applications: Uninhibited and Limited Growth Models

17. Slide 3- 17 Copyright © 2016 Pearson Education, Inc. Example 4 (continued): b) Find N  ( t ) = After 20 weeks, the disease is spreading through the campus at a rate of about 4 new cases per week. 2.4 Applications: Uninhibited and Limited Growth Models

18. Slide 3- 18 Copyright © 2016 Pearson Education, Inc. Example 4 (continued): c) Unrestricted growth is inappropriate for modeling this situation because as more students become infected, fewer are left to be newly infected. The logistic equation displays the rapid spread of the disease initially, as well as the slower growth in later weeks when there are fewer students left to be newly infected. 2.4 Applications: Uninhibited and Limited Growth Models

19. Slide 3- 19 Copyright © 2016 Pearson Education, Inc. 2.4 Applications: Uninhibited and Limited Growth Models

20. Slide 3- 20 Copyright © 2016 Pearson Education, Inc. 2.4 Applications: Uninhibited and Limited Growth Models Section Summary Uninhibited growth can be modeled by a differential equation of the type , whose solutions are . Certain kinds of limited growth can be modeled by equations such as ,called the logistic function and called the limited growth function . In both cases and