1. [5 marks] Let 2 HI) : bx + 613' where a,b and c are nonzero constants. Find the polynomials 3(25) and [(1) such that 5(1) best approximates f (z) for small values of I, and 1(1) best approximates [(1) for large mlues of It Student number: 2. [5 marks] Find all the values of c such that 3. [5 marks] Let f(x) = 2x2 + 3x - 1. Use a definition of the derivative to find f'(0). No credit will be given for solutions using differentiation rules, but you can use those to check x2+2 ifxc f (z) = 1 4x - 1 ifx > c your answer. is continuous. Answer: Answer: Student number: Student number: 514. The differential equation (d) [4 marks] Draw the slope eld for the differential equation on the axes below. On (1P P the slope eld, draw at least four solutions. 7 = TP 1 7 hP, dt K where 7 > h and K are positive constants, describes the growth rate of a population P P over time t, (A simplied version was examined in Week 6 and Week 8 small classes.) (a) [4 marks] Find the steady states and indicate them clearly on the phase line below, along with arrows indicating the sign of % between steady states. (e) [2 marks] The term ihP in the differential equation represents mm'able harvest, in which the population is \"harvested" at a rate proportional to the present popula- P tion, In a few sentences, describe carefully what the model predicts will happen to populations if the parameter h is larger than T. (b) [1 mark] Given an initial population P(0) = 1.5K, does the model predict that the population will increase or decrease? Answer: (c) [1 mark] Given a very small positive initial population Pm), what will the popu- lation be at the moment it is changing most rapidly? Your answer may include the terms 7\