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ASSIGNMENT 6 There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course webpage. The second part consists of the questions on this page. You are expected to provide full solutions with complete arguments and justifications. You will be graded on the correctness, clarity and elegance of your solutions. Your answers must be typed or very neatly written. They must be stapled, with your name and student number at the top of each page. This assignment is due at 10:00 a.m. on Friday, November 15. The written assignment is to be handed in by that time. The online assignment will also close at that time. 1. Determine the domain, intercepts, extrema, intervals of increase and decrease, inflection points and intervals of concavity for the function f (x) = x ln |x|. Then sketch the graph of the function, indicating all of this information in the sketch. 2. Let f (x) be a twice-differentiable function and F (x) be its linear approximation at x0 . In this question you will prove the following result which was stated in class, and which you will use in question 3(b): f (x) = F (x) + f 00 (c) (x − x0 )2 for some c between x and x0 . 2 (1) (a) Let g1 (x) and g2 (x) be functions which are continuous on [a, b] and differentiable on (a, b). Prove that there exists some c ∈ (a, b) with g10 (c) g1 (b) − g1 (a) . = g20 (c) g2 (b) − g2 (a) (You may assume that neither g20 (c) nor g2 (b) − g2 (a) is equal to 0.) This result, which generalizes the Mean Value Theorem, is called Cauchy’s Mean Value Theorem. Hint: apply Rolle’s Theorem to the function h(x) = g1 (x) − g1 (b) − g1 (a) g2 (x). g2 (b) − g2 (a) (b) Let E(t) be the “error term” f (t) − F (t). Prove that E(x) E 0 (d) = for some d between x0 and x 2 (x − x0 ) 2(d − x0 ) (2) Hint: apply Cauchy’s Mean Value Theorem to the functions E(t) and (t − x0 )2 on the interval [x0 , x]. (c) Prove that E 0 (d) = f 00 (c)(d − x0 ) for some c between x0 and d. Hint: show that E 0 (d) = f 0 (d) − f 0 (x0 ); that is, the error term for a constant approximation to the function f 0 (t) at x0 and evaluated at d. (d) Use your result in part (c) and the equation (2) to conclude (1). 2x = 0. x→∞ ex 3. (a) Show, using l’Hospital’s Rule, that lim (b) Show the same equality as in part (a) using an appropriate linear approximation for ex and a bound on its error term, and without using l’Hospital’s Rule.