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ASSIGNMENT 4 There are three parts to this assignment. Part A is to be completed online before 10:00 a.m. on Tuesday, October 12. Part B, which should be typeset (preferably with LaTeX), is to be submitted to the assignment box by the same time. Part C is a question to be completed by your Small Group — only one submission per group, with the names of all participating members at the top. Include your full name and student ID, along with the subject and assignment number, at the top of the front page. Part A [20 marks] This part of the assignment can be found online, labelled Assignment 4, at www.mathxl.com. Questions about the MathXL program itself should be sent to Eric; in fact, questions sent through the site will be directed to him. To ask me questions, email me directly or come to my office. Part B [20 marks] 1. Let f (x) = x x2 if x = n1 , where n = 1, 2, 3, . . . otherwise (a) Find all the points at which f is continuous. (b) Find all the points at which f is differentiable. 2. Note that 1/2 1/4 1 1 = . 2 4 Prove that there are infinitely many pairs of distinct positive numbers a and b such that aa = bb . Part C [5 marks] It is sometimes recounted that Leibniz himself originally stated that the derivative of a product is the product of the derivatives; that is, that given differentiable functions f and g, 0 (f (x)g(x)) = f 0 (x)g 0 (x). This is not true in general, but there are functions f and g for which it is true. Find as many such functions as you can. Marks will be awarded according to the number of functions your group can find and justify.