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ASSIGNMENT 4 for SECTION 001 There are three parts to this assignment. Part A is to be completed online before 7:00 a.m. on Friday, October 15. Part B and Part C, which require full solutions, are to be handed in at the beginning of class on the same date. Part A [10 marks] This part of the assignment can be found online, labelled A4, at webwork.elearning.ubc.ca — sign in using the MATH110 001 2010W button. Part B [5 marks] This part of the assignment is drawn directly from the course texts. It focuses on mathematical exposition; you will be graded primarily on the clarity and elegance of your solutions. From the Calculus: Early Transcendentals text, complete question 54 from section 2.5 and question 86 from section 2.6. Part C [15 marks] This part of the assignment consists of more challenging questions. You are expected to provide full solutions with complete arguments and justifications. n ∈ N\{1} 1. The sigmoid function P (t) = K ert KP0 + P0 1 − 1 ert models the growth of a population P as a function of time t. P0 is a positive constant denoting the initial population. K and r are also constants, known as the population’s carrying capacity and growth rate, respectively. It is assumed that K > P0 . (a) Suppose r > 0. Determine what happens to the population over time; that is, as the variable t gets very large. (b) What if r < 0? 2. Determine where the following function is discontinuous: x if x = 1, 12 , 13 , 14 , . . . f (x) = x2 otherwise 3. Let g be a function which is continuous at a, and f be a function which is continuous at g(a). Prove that f ◦ g is continuous at a.