Logistic Growth: The growth rate of many different populations depends not only on the number of individuals (leading to exponential growth) but also on a "carrying capacity" of the environment. If x is the population at time t and the growth rate of x is proportional to the product of the population and the carrying capacity M minus the population, then the growth rate is described by the differential equation dx/dt = kx(M - x) Where k and M are constants for a given species in a given environment. 15. Let k = 1 and M = 100, and assume the initial population is (0) = 5 . a. Solve the differential equation dx/dt = *[100 - x] for x . b. Graph the population *[t] for 0 S t$ 20. c. When will the population be 20? 50? 90? 100? d. What is the population after a "long" time? (Find the limit, as t becomes arbitrarily large, of x) 16. Write 0.39999.... using sigma notation. 17. The P-Test to determine whether the given series converges, and then [b) use the Integral Test to verify your convergence conclusion of part (=]. 1 1/9