CALCULUS BC., where P is the number of bears at time t in years. dt (a) Given P (i) Find lim Pt.
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1 CALCULUS BC WORKSHEET 1 ON LOGISTIC GROWTH NAME Do not use your calculator. 1. Suppose the population of bears in a national park grows according to the logistic differential equation 5P 0.00P, where P is the number of bears at time t in years. (a) Given P (i) Find lim Pt. (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. d P (iv) Find and use it to find the values of P for which the solution curve is concave up and concave down. Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of Pt. (b) Given (i) Find lim Pt. (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) For what values of P is the solution curve concave up? Concave down? Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of Pt.
2 (c) Given (i) Find lim Pt. (ii) What is the range of the solution curve? (iii) For what values of P is the solution curve increasing? Decreasing? Justify your answer. (iv) For what values of P is the solution curve concave up? Concave down? Justify your answer. (v) Does the solution curve have an inflection point? Justify your answer. (vi) Use the information you found to sketch the graph of Pt. (d) How many bears are in the park when the population of bears is growing the fastest?. Suppose a rumor is spreading through a dance at a rate modeled by the logistic differential P equation P 3. What is lim Pt? What does this number represent in 000 the context of this problem? 3. (From the 1998 BC Multiple Choice) The population Pt of a species satisfies the logistic differential equation P P, 5000 where the initial population is is lim Pt? 3000 and t is the time in years. What (A) 500 (B) 3000 (C) 400 (D) 5000 (E) 10, Suppose a population of wolves grows according to the logistic differential equation 3P 0.01P, where P is the number of wolves at time t in years. Which of the following statements are true? lim Pt 300 I. II. The growth rate of the wolf population is greatest at P = 150. III. If P > 300, the population of wolves is increasing. (A) I only (B) II only (C) I and II only (D) II and III only (E) I, II, and III
3 5. Suppose that a population develops according to the logistic equation 0.05P P where t is measured in weeks. (a) What is the carrying capacity? (b) A slope field for this equation is shown at the right. Where are the slopes close to 0? Where are they largest? Which solutions are increasing? Which solutions are decreasing? (c) Use the slope field to sketch solutions for initial populations of 0, 60, and 10. What do these solutions have in common? How do they differ? Which solutions have inflection points? At what population level do they occur? 6. (a) On the slope field shown on the right for 3P 3P, sketch three solution curves showing different types of behavior for the population P. (b) Describe the meaning of the shape of the solution curves for the population. Where is P increasing? Decreasing? What happens in the long run? Are there any inflection points? Where? What do they mean for the population?
4 CALCULUS BC WORKSHEET ON LOGISTIC GROWTH NAME Use your calculator on (c) and (d), 3(b) and (c), and 4(c) and (d) only. 1. A population of rabbits is given by the formula 1000 Pt () t 1 e where t is the number of months after a few rabbits are released. Show that P(t) is a solution of a logistic differential equation. Identify k and the carrying capacity. Estimate P(0) and explain its meaning.. The rate at which a rumor spreads through a high school of 000 students can be modeled by the differential equation 0.003P 000 P, where P is the number of students who have heard the rumor t hours after 9AM. (a) How many students have heard the rumor when it is spreading the fastest? Justify your answer. (b) If 5, solve for P as a function of t. (c) Use your answer to (b) and your graphing calculator to determine how many hours have passed when half the student body has heard the rumor. (d) Use your answer to (b) and your graphing calculator to determine how many students have heard the rumor after hours. 3. A certain national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. The population growth of bears can be modeled by the logistic differential equation 0.1P 0.001P, where t is measured in years. (a) Solve for P as a function of t. (b) Use your solution to (a) and your graphing calculator to find the number of bears in the park when t = 3 years. (c) Use your solution to (a) and your graphing calculator to find how many years it will take for the bear population to reach 50 bears.
5 4. Suppose a rumor is spreading at a dance attended by 00 students. The rumor is spreading at a rate that is directly proportional to both the number of students who have heard the rumor and the number of students who have not heard the rumor. Let P be the number of students who have heard the rumor, and let t be the time in minutes since the rumor began to spread. (a) Write a differential equation to model this rate of change. (b) If P P 0 10 and 15 50, solve for P as a function of t. (c) Use your solution to (b) to find the number of students who have heard the rumor after 1 hour. (d) Use your solution to (b) to find the time it takes for 175 students to hear the rumor.
6 Answers to Worksheet 1 on Logistic Growth 1. (a) (i) 500 (ii) [100, 500) (iii) increasing for [100, 500) (iv) concave up for (100, 150) and concave down for (150, 500) (v) yes, IP when P = 150 (vi) sketch (b) (i) 500 (ii) [1500, 500) (iii) increasing for [1500, 500) (iv) concave down for (1500, 500) (v) no (vi) sketch (c) (i) 500 (ii) (500, 3000] (iii) decreasing for (500, 3000] (iv) concave up for (500, 3000) (v) no (vi) sketch (d) ; there are 6000 people at the dance. 3. E 4. C 5. (a) 100 (b) Close to 0? P = 0 and P = 100 Largest? P = 50 Increasing? P Decreasing? 100 (c) In common? All have a limit of 100. Differ? Two are increasing; one is decreasing. Inflection points? The one with initial condition of 0. At what pop. level does the inflection point occur? When P = (a) sketch (b) Increasing? P 0 1 Decreasing? In the long run? P 0 1 lim Pt 1 t Any inflection points? Yes Where? When P What do they mean for the population? The population is growing the fastest when P
7 Answers to Worksheet on Logistic Growth 1. Mk =.7; M = 1000; k =.0007; P(0)~ 8.initially there were about 8 rabbits ( p369 #31). (a) 1000 students 6t 000e 000 (b) P or P 6t 6 e e (c) hours (d) so 1995 people t 3. (a) 0.1t 100e 100 P or P e (b) or 13 bears (c) 1.97 years 0.1t 0.1t 9 1 9e kp P 19 00kt ln 00e 00 3 (b) P or P where k 00kt 00kt 19 e 19e (a) 00
where people/square mile. In
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, where P is the number of bears at time t in years. dt (a) If 0 100, lim Pt. Is the solution curve increasing or decreasing?
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