Logistics Growth In Chapter 5. the exponential grov\ th equation was deri\ed from the

Chapter 7, Problem 58

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Logistics Growth In Chapter 5. the exponential grov\ th equation was deri\ed from the assinnplion that the rate ot growthwas proportional to the e\istmg qiuintuy In practice, thereoften exists some upper limit /. past w Inch growth cannot occur.In such cases, we assume the rate of growth to be proportionalnot only to the e.xisting quantity, but also to the differencebetween the existing quantity ^ and the upper hunt L. That is,^ = ML - v).In integral form, we can express this relationship as/v v(Z. - v)k,li.(a) A slope field for the differential equation Jy/tlt = y(3 - v)is shown. Draw a possible solution to the difterential equation if v(0) = 5. and another if \(0) = -, To print an enlargedcopy of the graph, go to the website i\'i\\\:iiuirhi;niplisAiini.(b) Where y(0) is greater than 3. what is the sign of the slopeof the solution'(c) For y > 0. find lim i'(/).(d) Evaluate the two integrals above and solve for y as a function of t. w here y,, is the initial quantity.(e) Use the result in part (d) to find and graph the solutions in part (a). Use a graphing utility to graph the solutions andcompare the results with the solutions in part (a),(f) The graph of the function v is called a logistics curve. Show that the rate of growth is maximum at the point ot inflection, and that this occurs when y = 1/2,