The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem dN dt (a) Use the phase portrait concept of Section 2.1 to predict how many supermarkets are expected to adopt the new procedure over a long period of time. supermarkets By hand, sketch a solution curve of the given initial-value problem. N 1200 1000 800 600 400 200 N 1200 1000 800 600 400 = N(1 0.0004N), N(0) = 1. 200 N(t) = 5 5 10 10 15 15 20 20 t t N 2500 2000 1500 1000 DO 500 N 2500 2000 SU 1500 1000 500 t 20 @O 5 10 15 (b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a). How many supermarkets are expected to adopt the new technology when t = 10? (Round your answer to the nearest integer.) supermarkets 5 10 15 20 t

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During the period from 1790 to 1930, a country's population P(t) (t in years) grew from 3.7 million to 118.6 million. Throughout this period, P(t) remained close to the solution of the initial value dP problem = 0.03145P-0.0001486P², P(0) = 3.7. dt (a) What 1930 population does this logistic equation predict? (b) What limiting population does it predict? (c) The country's population in 2000 was 350 million. Has this logistic equation continued since 1930 to accurately model the country's population?

1. log,x = -2 3. logzx + log;9 = 1 4. logzx + logz(x -6) = 4 2. 4 +3 log x = 10 %3D