, 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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1 CALCULUS BC WORKSHEET 1 ON LOGISTIC GROWTH Work he following on noebook paper. Use your calculaor on 4(b) and 4(c) only. 1. Suppose he populaion of bears in a naional park grows according o he logisic differenial equaion 5P 0.00P, where P is he number of bears a ime in years. (a) If 0 100, lim P. Is he soluion curve increasing or decreasing? (b) If (c) If P find 1500, find lim P 3000, find lim P P.. Is he soluion curve increasing or decreasing? P.. Is he soluion curve increasing or decreasing? P. (d) How many bears are in he park when he populaion of bears is growing he fases? Jusify your answer.. Suppose a populaion of wolves grows according o he logisic differenial equaion 3P 0.01P, where P is he number of wolves a ime in years. Which of he following saemens are rue? lim P 300 I. II. The growh rae of he wolf populaion is greaes a P = 150. III. If P > 300, he populaion 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. The rae of change,, of he number of people a a dance who have heard a rumor is modeled by a logisic differenial equaion. There are 000 people a he dance. A 9PM, he number of people who have heard he rumor is 400 and is increasing a a rae of 500 people per hour. Wrie a differenial equaion o model he siuaion. 4. A populaion of animals is modeled by a funcion P ha saisfies he logisic differenial equaion 0.01P 100 P, where is measured in years. (a) If P 0 0, solve for P as a funcion of. (b) Use your answer o (a) o find P when = 3 years. (c) Use your answer o (a) o find when P = 80 animals.
2 TURN->>> 5. Suppose ha a populaion develops according o he logisic equaion 0.05P P where is measured in weeks. (a) Wha is he carrying capaciy? (b) A slope field for his equaion is shown a he righ. Where are he slopes close o 0? Where are hey larges? Which soluions are increasing? Which soluions are decreasing? (c) Use he slope field o skech soluions for iniial populaions of 0, 60, and 10. Wha do hese soluions have in common? How do hey differ? Which soluions have inflecion poins? A wha populaion level do hey occur?
3 CALCULUS BC WORKSHEET ON LOGISTIC GROWTH Work he following on noebook paper. Use your calculaor on 3(c), 5(b), and 5(c) only. 1. Suppose a rumor is spreading hrough a dance a a rae modeled by he logisic differenial P equaion P 3. Wha is lim P? Wha does his number represen in 000 he conex of his problem?. Suppose you are in charge of socking a fish pond wih fish for which he rae of populaion growh is modeled by he differenial equaion 8P 0.0P. (a) If lim P P. (b) If (c) If 50, find 300, find lim P 500, find lim P.. P.. P. (d) Which of hese graphs, a, b, or c, has an inflecion poin? Which are increasing? Which are decreasing? Jusify your answers. 3. The rae a which a rumor spreads hrough a high school of 000 sudens can be modeled by he differenial equaion 0.003P 000 P, where P is he number of sudens who have heard he rumor hours afer 9AM. (a) How many sudens have heard he rumor when i is spreading he fases? Jusify your answer. (b) If P 0 5, solve for P as a funcion of. (c) Use your answer o (b) o deermine how many hours have passed when half he suden body has heard he rumor. (d) How many sudens have heard he rumor afer hours? 4. (a) On he slope field shown on he righ for 3P 3P, skech hree soluion curves showing differen ypes of behavior for he populaion P. (b) Describe he meaning of he shape of he soluion curves for he populaion. Where is P increasing? Decreasing? Wha happens in he long run? Are here any inflecion poins? Where? Wha do hey mean for he populaion? TURN->>>
4 5. A cerain naional park is known o be capable of supporing no more han 100 grizzly bears. Ten bears are in he park a presen. The populaion growh of bears can be modeled by he logisic differenial equaion 0.1P 0.001P, where is measured in years. (a) Solve for P as a funcion of. (b) Use your soluion o (a) o find he number of bears in he park when = 3 years. (c) Use your soluion o (a) o find how many years i will ake for he bear populaion o reach 50 bears.
5 Answers o Workshee 1 on Logisic Growh 1. (a) 500; increasing (b) 500; increasing (c) 500; decreasing (d) 150. C 1 3. P 000 P e (a) P e e (b) animals (c).773 years 5. (a) 100 (b) Close o 0? P = 0 and P = 100 Larges? P = 50 Increasing? Decreasing? 100 P (c) In common? All have a limi of 100. Differ? Two are increasing; one is decreasing. Inflecion poins? The one wih iniial condiion of 0. A wha pop. level does he inflecion poin occur? When P = 50. Answers o Workshee on Logisic Growh ; he number of people a he dance.. (a) 400 (b) 400 (c) 400 (d) Only (a) has an inflecion poin. (a) and (b) are increasing; (c) is decreasing. 3. (a) 1000 sudens 6 000e 000 (b) P 6 6 e e (c) hours (d) so 1995 people P (b) Increasing? Decreasing? P 5. (a) In he long run? 0 1 lim P 1 Any inflecion poins? Yes Where? When P Wha do hey mean for he populaion? The populaion is growing he fases when P e 100 P e (b) or 13 bears (c) 1.97 years e
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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