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Applications: Uninhibited and Limited Growth Models OBJECTIVES  Find functions that satisfy dP/dt = kP.  Convert between growth rate and doubling time.  Solve application problems using exponential growth and limited growth models. 3.3

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 8 A function y = f (x) satisfies the equation if and only if for some constant c. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 1: Find the general form of the function that satisfies the equation By Theorem 8, the function must be 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Uninhibitied 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 0, and t is time. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4: Suppose that an amount P 0, 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 0 and b) Suppose that $100 is invested. What is the balance after 1 yr? c) In what period of time will an investment of $100 double itself? 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 4 (concluded): a) b) c) 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THEOREM 9 The growth rate k and the doubling time T are related by 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 5: Worldwide use of the Internet is increasing at an exponential rate, with traffic doubling every 100 days. What is the exponential growth rate? The exponential growth rate is approximately 0.69% per day. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6: The world population was approximately billion at the beginning of 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 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (continued): a) Find the function that satisfies the equation. Assume that P 0 = and k = b) Estimate the world population at the beginning of 2020 (t = 20). c) After what period of time will the population be double that in 2000? 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 6 (concluded): 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Models of Limited Growth 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8: 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 0 cases occur can be modeled by 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8 (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 unrestricted growth model is inappropriate but a logistic equation is appropriate for this situation. Then use a calculator to graph the equation. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8 (continued): a) N(3) = So, approximately 12 students are infected after 3 weeks. N(40) = So, approximately 222 students are infected after 40 weeks. N(80) = So, approximately 547 students are infected after 80 weeks. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8 (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. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 8 (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. 3.3 Applications: Uninhibited and Limited Growth Models

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.3 Applications: Uninhibited and Limited Growth Models