The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem (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. By hand, sketch a solution curve of the given initial-value problem.(b) Solve the initial-value problem and then use a graphing utility to verify the solution curve in part (a). How many companies are expected to adopt the new technology when t = 10?

Solution:Step-1:a)In this problem we need to find by using the phase portrait concept of section 2.1 to predict number of super markets are expected to adopt the new procedure over a long period of time.By hand ,sketch a solution curve of the given initial value problem.Step-2:Given initial value problem is We know that , at initial time .Now, we have to find the equilibrium solutions by solving ., since 2000 is the constant value.Therefore, N= 2000 is the limiting value.Step-3:If,Case(1):N<0,(1-0.0005N)>0Therefore, the common interval is Case(2):N>0,(1-0.0005N)<0Therefore, the common interval is Step-4:If,Therefore, the interval is .Step-5:The hand graph of the function is shown below: Step-6:Therefore,2000 number of super markets are expected to adopt the new procedure over a long period of time.Step-7:b)In this problem we need to solve the initial value problem and then use a graphing utility to verify the solution curve in part (a), and we need to find number of companies are expected to adopt the new technology when t = 10.Step-8:We know that , the given is logistic equation and the solution of the logistic equation is , since N(0) is the common term.Where, N(0) is the initial value.That is, N(0) = 1.k is the limiting value of N(t).That is , k=2000 and r = 1.Substitute these values in the above equation we get. Step-9:Substitute, t=10 in the above equation we get. , since Step-10:Therefore, , when t = 10, then approximately 1834 companies are expected to adopt the new technology.