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(1 point) Consider the slope field below for a differential equation. Use the graph to find the equilibrium solutions. TITITTT 11111 1111 1 Answer (separate by commas):y =(1 point) Dead leaves accumulate on the ground in a forest at a rate of 4 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 60 percent per year. A. Write a differential equation for the total quantity Q of dead leaves (per square centimeter) at time I: dQ _ (Tr _ B. Sketch a solution to your differential equation showing that the quantity of dead leaves tends toward an equilibrium level. Assume that initially (t = 0) there are no leaves on the ground. What is the initial quantity of leaves? Q(0) = What is the equilibrium level? Qeq = Does the equilibrium value attained depend on the initial condition? A. yes B. no (1 point) A population P obeys the logistic model. It satisfies the equation dP 5 Z P(5 - P) for P > 0. dt 500 (a) The population is increasing when (1 point) According to a simple physiological model, an athletic adult male needs 20 calories per day per pound of body weight to maintain his weight. If he consumes more or fewer calories than those required to maintain his weight, his weight changes at a rate proportional to the difference between the number of calories consumed and the number needed to maintain his current weight; the constant of proportionality is 1/3500 pounds per calorie. Suppose that a particular person has a constant caloric intake of H calories per day. Let W0) be the person's weight in pounds at time I (measured in days). (a) What differential equation has solution W(I)? M: all (Your answer may involve W, H and values given in the problem.) (b) If the person starts out weighing 175 pounds and consumes 3900 calories a day. What happens to the person's weight ast > oo? W) (1 point) Recall that one model for population growth states that a population grows at a rate proportional to its size. dP 1 (a) We begin with the differential equation = P. Find an equilibrium solution: (it P: Is this equilibrium solution stable or unstable? A. stable B. unstable Describe the longterm behavior of the solution to % = %P when P(0) is positive. A. The value of P approaches a nonzero constant. B. The value of P approaches zero. C. The value of P oscillates and does not approach a limit. D. The value of P increases without bound. E. None of the above . . . . . . . (JP 1 (b) Let's now conSIder a modified differential equation given by = EP(3 P). dt Find a stable equilibrium solution: P = Find an unstable equilibrium solution: dP 1 (b) Let's now consider a modified differential equation given by I = 5P8 P). Find a stable equilibrium solution: P: Find an unstable equilibrium solution: P = If P(0) is positive, describe the longterm behavior of the solution to % = %P(3 P). A. The value of P approaches zero. B. The value of P approaches a nonzero constant. C. The value of P oscillates and does not approach a limit. D. The value of P increases without bound. E. None of the above