LOGISTIC GROWTH WS 2 1. Suppose you are in charge of stocking a fish pond with fish for which the rate of population growth is modeled by the differential equation dP dt =8P-0.02P. (a) Given P(0)=50. (i) Find lim P(t). 818 (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) Find d P dt 2 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 P(t). (b) Given P(0)=300. (i) Find lim P(t). 198 (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 P(t). (c) Given P(0)=500. (i) Find lim P(t). 18 (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 P(t). 2. A population of animals is modeled by a function P that satisfies the logistic differential equation dP = = 0.01P (100 P), where t is measured in years. dt (a) If P(0)=20, solve for P as a function of t. (b) Use your answer to (a) and your graphing calculator to find P when t = 3 years. (c) Use your answer to (a) and your graphing calculator to find t when P = 80 animals.