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ASSIGNMENT 16 for SECTION 001 There are three parts to this assignment. Part A is to be completed online before 7:00 a.m. on Friday, March 18. 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 focuses on fundamental skills and computations. It can be found online, labelled A16, 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 questions 66 and 86 from section 4.2. 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. 1. Write down and sketch the graph of a function defined everywhere, increasing everywhere, concave down everywhere, and without any horizontal asymptotes. 2. Functions like f (x) = x3 0 and g(x) = 1−x5 have extrema that are undetectable via the Second Derivative Test. Propose a “Generalized Derivative Test” that applies when f 0 (x) = f 00 (x) = · · · = f (n−2) (x) = f (n−1) (x) = 0, and f (n) (x) 6= 0. 3. Prove that the graph of a function crosses over its tangent line at an inflection point. (Hint: let g(x) be the vertical distance between the function and the tangent line. Sketch the graphs of g(x) and g 0 (x).)