, where P is the number of bears at time t in years. dt (a) Given P (i) Find

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1 CALCULUS BC WORKSHEET ON LOGISTIC GROWTH Work he following on noebook paper. Do no use your calculaor. 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) Given P lim. (b) Given lim. (c) Given lim. (d) How many bears are in he park when he populaion of bears is growing he fases?. Suppose a rumor is spreading hrough a dance a a rae modeled by he logisic differenial P equaion P 3. Wha is lim? Wha does his number represen in 000 he conex of his problem? TURN->>>

2 3. (From he 1998 BC Muliple Choice) The populaion of a species saisfies he logisic differenial equaion P P, 5000 where he iniial populaion is is lim? 3000 and is he ime in years. Wha (A) 500 (B) 3000 (C) 400 (D) 5000 (E) 10, 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 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 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? 6. (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?

3 CALCULUS BC WORKSHEET ON LOGISTIC GROWTH Work he following on noebook paper. Use your calculaor on (b) and (c), 3(c) and (d), and 4(b) and (c) only. 1. 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) Given 50. lim. (vi) Use he informaion you found o skech he graph of. (b) Given 300. lim. (c) Given 500. lim. (vi) Use he informaion you found o skech he graph of.. 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) and your graphing calculaor o find P when = 3 years. (c) Use your answer o (a) and your graphing calculaor o find when P = 80 animals. TURN->>>

4 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) and your graphing calculaor o deermine how many hours have passed when half he suden body has heard he rumor. (d) Use your answer o (b) and your graphing calculaor o deermine how many sudens have heard he rumor afer hours. 4. 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) and your graphing calculaor o find he number of bears in he park when = 3 years. (c) Use your soluion o (a) and your graphing calculaor o find how many years i will ake for he bear populaion o reach 50 bears. 5. Suppose a rumor is spreading a a dance aended by 00 sudens. The rumor is spreading a a rae ha is direcly proporional o boh he number of sudens who have heard he rumor and he number of sudens who have no heard he rumor. Le P be he number of sudens who have heard he rumor, and le be he ime in minues since he rumor began o spread. (a) Wrie a differenial equaion o model his rae of change. P 0 10 and P 15 50, solve for P as a funcion of. (b) If

5 Answers o Workshee 1 on Logisic Growh 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 (b) (i) 500 (ii) [1500, 500) (iii) increasing for (1500, 500) (iv) concave down for (1500, 500) (c) (i) 500 (ii) (500, 3000] (iii) decreasing for (500, 3000) (iv) concave up for (500, 3000). 6000; here are 6000 people a he dance. 3. E 4. C 5. (a) 100 (b) Close o 0? P = 0 and P = 100 Larges? P = 50 Increasing? P Decreasing? 100 (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 = (a) skech (b) Increasing? P 0 1 Decreasing? In he long run? P 0 1 lim 1 Any inflecion poins? Yes Where? When P Wha do hey mean for he populaion? The populaion is growing he fases when P

6 Answers o Workshee on Logisic Growh 1.(a) (i) 400 (ii) [50, 400) (iii) increasing for (50, 400) (iv) concave up for (50, 00) and concave down for (00, 400) (v) yes, IP when P = 00 (b) (i) 400 (ii) [300, 400) (iii) increasing for (300, 400) (iv) concave down for (300, 400) (c) (i) 400 (ii) (400, 500] (iii) decreasing for (400, 500) (iv) concave up for (400, 500) 100e 100. (a) P or P e (b) animals (c).773 years e 3. (a) 1000 sudens 6 000e 000 (b) P or P 6 6 e e (c) hours (d) so 1995 people 4. (a) e 100 P or P e (b) or 13 bears (c) 1.97 years e kp P 5. (a) 00 (b) 00k 00e k 00k e e P or P, where k ln.0006