1 The function of has a derivative given by f'(x)=- a positive constant. x = R, x 0, x = k where k is x(k-x) (a) The expression for f'(x) can be written in the form Find a and b in terms of k. (b) Hence, find an expression for f(x). а b + where a, b = R. bЄ 3 x k-x Consider P, the population of a colony of ants, which has an initial value of 1200. dP P(k-P) dt The rate of change of the population can be modelled by the differential equation where t is the time measured in days, t≥0, and k is the upper bound for the population. (c) By solving the differential equation, show that P=- 1200k (k-1200)e 5+1200 At t = 10 the population of the colony has doubled in size from its initial value. (d) Find the value of k, giving your answer correct to four significant figures. 5k (e) Find the value of t when the rate of change of the population is at its maximum.

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