Applications Uninhibited and Limited Growth Models 3 3

Applications: Uninhibited and Limited Growth Models 3. 3 OBJECTIVE • Find functions that satisfy d. P/dt = k. P. • Convert between growth rate and doubling time. • Solve application problems using exponential growth and limited growth models. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 1

3. 3 Applications: Uninhibited and Limited Growth Models Quick Check 1 Differentiate . Then

3. 3 Applications: Uninhibited and Limited Growth Models THEOREM 8 A function y = f (x) satisfies the equation if and only if for some constant c. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 3

3. 3 Applications: Uninhibited and Limited Growth Models Example 1: Find the general form of the function that satisfies the equation By Theorem 8, the function must be Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 4

3. 3 Applications: Uninhibited and Limited Growth Models Quick Check 2 Find the general form of the function that satisfies the equation: The function is constant. , or where is an arbitrary As a check, note that Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 5

3. 3 Applications: Uninhibited and Limited Growth Models 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. In the absence of inhibiting or stimulation factors, a population normally reproduces at a rate proportional to its size, and this is exactly what d. P/dt = k. P says. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 6

3. 3 Applications: Uninhibited and Limited Growth Models Uninhibited Population Growth The only function that satisfies this differential equation is given by where t is time and k is the rate expressed in decimal notation. Note that so c represents the initial population, which we denoted P 0: Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 7

3. 3 Applications: Uninhibited and Limited Growth Models Example 2: 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 0. 07. 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 ? Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 8

S l 2 i 0 1 d 2 e 3 P. e a 3 -r s 9 o n E d u c a t i o n , I n c. 3. 3 Applications: Uninhibited and Limited Growth Models Example 2 (concluded): a) b) c) A l l r i g Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 9

3. 3 Applications: Uninhibited and Limited Growth Models THEOREM 9 The exponential growth rate

3. 3 Applications: Uninhibited and Limited Growth Models Quick Check 3 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. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 11

3. 3 Applications: Uninhibited and Limited Growth Models 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. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 12

3. 3 Applications: Uninhibited and Limited Growth Models Example 3 (continued): a) Find the function that satisfies the equation. Assume that P 0 = 6. 0400 and k = 0. 016. 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? Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 13

3. 3 Applications: Uninhibited and Limited Growth Models Example 3 (concluded): Copyright © 2016,

3. 3 Applications: Uninhibited and Limited Growth Models of Limited Growth The logistic equation, or logistic function 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. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 15

3. 3 Applications: Uninhibited and Limited Growth Models of Limited Growth Copyright © 2016,

3. 3 Applications: Uninhibited and Limited Growth Models Example 4: 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 Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 17

3. 3 Applications: Uninhibited and Limited Growth Models 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. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 18

3. 3 Applications: Uninhibited and Limited Growth Models 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. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 19

3. 3 Applications: Uninhibited and Limited Growth Models Example 4 (continued): b) Find N (t) = After 20 weeks, the disease is spreading about 4 new cases per week. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 20

3. 3 Applications: Uninhibited and Limited Growth Models 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. Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 21

3. 3 Applications: Uninhibited and Limited Growth Models Copyright © 2016, 2012 Pearson Education,

3. 3 Applications: Uninhibited and Limited Growth Models of Limited Growth Another model of

3. 3 Applications: Uninhibited and Limited Growth Models Section Summary • Uninhibited growth can be modeled by a differential equation of the form , whose solutions are . • The exponential growth rate k and the doubling time T are related by the equation , or • Certain kinds of limited growth can be modeled by equations such as and Copyright © 2016, 2012 Pearson Education, Inc. 3. 3 - 24