MATH-137-1069-Test2_exam

MATH-137-1069-Test2_exam - Faculty of Mathematics...

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Unformatted text preview: Faculty of Mathematics University of Waterloo Math 137 Term Test 2 - Fall Term 2006 Time: 7:00 — 9:00 p.m. Date: November 20, 2006. AIDS: ‘PINK TIE’ CALCULATORS ONLY I V Family Name: MM 5 ID. Number: Initials: Signature: Check the box next tolyour section: 1:! 001 B. J. Marshman (11:30 am.) [:1 008 S. Sivaloganathan (10:30 am.) - 1:! 002 O. Winkler (12:30 pm.) [:3 009 S. Gupta (2:30 pm.) E] 003 L. Marcoux (1:30 pm.) ‘I:l 010 S. Gupta (1:30 pm.) II] 004 N. Spronk (2:30 pm.) 1:] 011 C. Struthers (9:30 am.) I:] 005 B. Ingalls (1:30 pm.) El» 012 D. Harmsworth ( 8:30 am.) 1:} 006 D. Park ( 9:30 am.) 1:] 013 D. Wolczuk (8:30 am.) 1:] 007 Y—R. Liu (10:30 am.) 1 Your answers must be‘stated in a clear and logical form in order to receive full marks. Reference any theorems or rules used by their name, or the appropriate abbreviation. Useful abbreviations for this test include LSR (limit sum rule), LPR (limit product rule),.LQR, (limit quotient rule), LCR (limit composite rule for basic functions), the corresponding continuity rules (CSR, CPR, CQR, OCR), and derivative rules (DSR, DPR, DQR, DCR), as well as the Intermediate Value Theorem (IVT), and the Mean Value Theorem (MVT). Note: 1. Complete the information section above, indicating your instructor’s name by a checkmark in the appro— priate box. '/ 2. Place your initials and ID No. at the top right corner of each page. 3. Tear off the blank last page of the test to use for rough work. You may use the reverse of any page if you need extra space. [4] 1. [4] ' MATH 137 TERM TEST #2 PAGE 2 dy a) Find dag if y = (m2 +1)3/2 + 62 —— marctanx. b) Find f’(m) if f(3:) = ln(a:3€‘“’). c) Find the tangent line to the curve cos(:1:y) = —a: + 1 at the point (1,7r/2). (1) Given that the points (2:,y) on a curve satisfy the equation xy = ym for :r > 0 and y > 0, use logarithmic differentiation to find 31’ in terms of x and y, and hence Show that y’ is indeterminate at the point (e, e). The graph of P(:c) shown at left represents the profit a certain manufacturer makes by selling as units of product. Sketch a graph of P’ (the marginal profit) on the given axes, justifying the behaviour of your graph at x1, x2, and £63. TERM TEST #2 : Valuate each limit. Justify your method by stating what rules/ Theorems are used (e.g., fnit quotient rule LQR, 1’H6pital’s Rule L’HR, etc.) Ina: . 1. (1) mgi sin(7ra:) (ii) iim at 6“” CE-—>OO (iii) 1im(1 + 3m)1/x m—>0 1 [3] b) Use the Squeeze Theorem to find , Inn) 3:2 cos(;). » ‘ tana: if x 74 0 [5] 0) Consider the function f = a: ’ 1 if a: = 0 (i) Use the definition of continuity to ShOW that f is continuous at a: = 0. (ii) Use the definition of derivative to Show that f’ (0) = 0. I [4] l3] ‘ fl, , 7' ’ MATH 137 TERM TEST #2 PAGE 4 7r 3. Consider the fuction f = 2:1: - cos(—2—:c). a) Using an appropriate theorem, show that there is a c in [0, 1] such that f (c) = 0. b) What is the smallest number of steps of the Method of Bisection required to estimate 0 with an error of at most 0.01? Explain your reasoning. i 1 l l g i c) Use £1(:r), the linear approximation of f at a: 2 1, to estimate f (0.9). d) Explain why f has an inverse f '1 on R. l e) Find f’1(2) and f‘1 (2) and hence state the equation of the tangent line to y = f'1(x) at i5 " MATH 137 TERM TEST #2 PAGE 5 . j Wm— CB 4. Consider the function f = :3 [2] a) Find liIJn f(a:) and lini [2] b) Find 1118+ f and lirgr f I [3] c) Find all critical numbers (points) of f. l [2] (:1) Find the intervals on which f is increasing, and on which f is decreasing. I [2] e) Show that f”($) > 0 for an > O, and f”(;c) < 0 for a: < O. [4] f) Use the results of a) — e) to sketch the graph of y = f (:13), indicating any local extremes, points of inflection, or asymptotes. :3 ea+b > 62. [1] g) Show that, for all a > 0, b > 0, b _ a ¢ MATH 137 TERM TEST #2 PAGE 6 5. Label each statement as TRUE or FALSE in the blank provided. (Use the space between questions to justify your answer. Guesses will not be graded.) [2] a) If f(:I:) = ln(:v3e‘x), then the domain of f’(a:) is aé O}. [2] b) The function f = [1:3|is diiferentiablie at 23 = 0. [2] c) If f’(:1:) > 0 and g’(a:) > 0 on an interval I, then y = f(a;)g(x) is increasing on I. [3] d) The function f = cos(arcsin x) —- m + 2, is a constant function. [3] e) If f is differentiable on IR and has exactly two real roots, then f’ also has two real roots. If the positions of two horses are given by differentiable functions f (t) and g(t) for t 2 0, and the second horse runs faster than the first (i.e., f’(t) < g’(t) for every t Z 0), then if they start at the same point (i.e., f (0) = 9(0)), the second horse always leads the (first horse (i.e., f(t) < g(t) for every t > 0). ...
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MATH-137-1069-Test2_exam - Faculty of Mathematics...

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