Question 30 A function f has a Maclaurin series given by 2 + 3x + x2 + 2x3 + ..., and the Maclaurin series converges to f() for all real numbers a. If g is the function defined by g (x) = ef(), what is the coefficient of a2 in the Maclaurin series for g ? A B C e2 D He2Question 37 Let h be a continuous function of I. Which of the following could be a slope field for a differential equation of the form - = h(x) ? A B C DQuestion 43 Let f be the function defined by f (x) = 14 - 2x3 + 2x2 - x. For how many values of a in the open interval (0, 1.565) is the instantaneous rate of change of f equal to the average rate of change of f on the closed interval [0, 1.565] ? A Zero B One (C Three D FourQuestion 44 The population P of rabbits on a small island grows at a rate that is jointly proportional to the size of the rabbit population and the difference between the rabbit population and the carrying capacity of the population. If the carrying capacity of the population is 2400 rabbits, which of the following differential equations best models the growth rate of the rabbit population with respect to time t, where k is a constant? A = 2400 - KP B " = k (2400 - P) C ap = k- (2400 - P) D dp = kP (2400 - P)Question 45 A region is bounded by two concentric circles, as shown by the shaded region in the figure above. The radius of the outer circle, R, is increasing at a constant rate of 2 inches per second. The radius of the inner circle, r, is decreasing at a constant rate of 1 inch per second. What is the rate of change, in square inches per second, of the area of the region at the instant when R is 4 inches and r is 3 inches? A) B 6TT C 107 D 227