asst4soln - MATH 137 Assignment 4 Due: 11 am, Friday,...

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Unformatted text preview: MATH 137 Assignment 4 Due: 11 am, Friday, October 14 Your assignment consists of two parts. Part 1 is available online at http://mapleta.uwaterloo.ca and is due on—line by 4 pm on Thursday, October 13. Part 2 consists of the problems below. Place your solutions to the problems in part 2 into the correct slot in the drop boxes outside MC 4066, corresponding to the class section in which you are registered. If your assignment is put into the wrong slot, it may be lost for a long time and may never get marked. You may copy and use the assignment templates that are available on the D2L website for MATH 137. Make sure your name and your UW user ID are clearly written at the top of the first page, and underline your last name. While it is okay to get help if you are stuck, you are required to include, at the top of your assignment, the names of students who helped you or with whom you collaborated. This acknowledgement will no effect on your assignment mark. Any outright copying of assignments, in whole or in part, is an act of cheating, and it will be reported to the associate dean. Hand in your solutions to the following 5 problems. Your solutions must have legible handwriting, and must be presented in clear, concise and logical steps that fully reveal what you are doing. 1. Let f be a function such that sin(7r:z:) 3 flat) < 1 _ for all CE in the interval (0, 1) Find 1113132 f (:11), and explain your reasoning. 02—) Decide if f continuous at a 2 1/2, and explain your reasoning. 2. Leta be any real number and let f($)= 2 1—zvzwhen$<a ac wheana V Find the values of a that make f a continuous function on all of R. Then, for each suitable value of a, sketch the resulting continuous function. 3. Evaluate the following limits if they exist. (a) lirn arctan (cc —— C(74) x—aoo . , 92:5 — :1: (b) .2le gas—+1 (c) lim az—>+oo 4. The function f 2: 1 2 has the :13—axis as a horizontal asymptote. In- :5 1 deed, f(x) = Um + x —> 0 as :1: —> 00, by inspection. So, we expect that for every 6 > 0 there is a large number K such that I f < 6 when x > K. For 6 > 0, find a suitable such K, and show that it is suitable. 5. Let f 9:3 + bx? + cm + d be a general cubic polynomial with the coefficient in front of 51:3 adjusted to be a 1. (a) Explain why f > 0 when a: > 0 and very large, and why f < 0 when :1: < 0 and very large. Hint. Rewrite f as fix) 2 5133(1 + b/a: + (2/222 + d/x3) and then in— spect the sign of the part in brackets when a: is very large. (b) Use .the above information to show that every cubic must cut the x—axis in at least one place, i.e. prove every cubic has a real root. (c) Does every degree 4 polynomial cut the :IJ—axis? Explain your answer. EXTRAS Once you have seen the concept of derivative, you should work on these extra problems, but do not hand these in. Problems such as these could be on exams. 1. Suppose f is a function with the property that If g 2:2 for every real number 3:. (a) Show that f(0) = 0. (b) Show that f’(0) = O. (C) Let flit) = { Does f’(0) exist? If so, what is f’(0)? :52 sin 1 when x 74 0 Cl: 0 whenx = O. 2. (a) Sketch the graph of the function f = (b) Show that f is differentiable for all :13. (c) Find a formula for f’ Hint. For a: > 0 and for a: < 0, you can find f’ by just looking at f (:13), and remembering a bit of high school calculus. To get f’ (0), use the definition of derivative. 3. (a) If g(;v) 2 1162/3, use the definition of the derivative to show that g’(0) does not exist. (b) Is the function 9 continuous at 0? Explain your answer. The following problems from Stewart’s book are strongly recommended for you to practice on your own. Problems such as these are eligible for the mid—term exam. Do not hand these in but make sure you know how to solve problems such as these. If you get stuck, solutions can be found in your Student Solutions Manual. 0 Section 2.5, pp. 127—130 # 41, 45, 49, 51, 53, 55, 67 0 Section 2.6, pp. 140—143 # 19, 25, 27, 35, 37, 53, 57 0 Section 2.7, pp. 150—153 # 19,21, 23, 25, 29, 53 0 Section 2.8, pp. 162—165 # 51, 53, 57 Hint for # 57. Given f(—:1:) : f(:z:), you need to prove that f’(—x) = —~f’(:z:). Well, WM) 2 km f(—rv +11) — f(—:v) h-—>0 h ton quotient, use the fact f is even, and observe that h —> 0 if and only if t——>0. . Now putt = —h into the above New— 0 Section 3.1, pp. 180-182, # 9,19, 23, 31, 33, 51, 53, 59, 61, 73, 77 MW /% / firfiglgc/VMENTZé‘LLF WWW HS (Lack SM Y are Cc'an’llZmu-L‘LL5 4Q ’Fvnchg'o so 1%)“ S/VI (77X) '- 711% jinn S/thTX) ; 5/1,» x-ai; M )""'+ 1"‘L‘53 0P Co-«+7am+y {MW Hx<"‘><’) ' ' ' : LI\( ‘5 X("J?:): (RI) l_$ nomzepv 50 WQ (,5,‘\ u>¢ m éufihe’fi+ IIMI+ -‘M, 9x000 I‘m WG-20 I 1m (Chat/EL ’7 oh; f0!“ 11% £47 UCCZ e arc how $z/JTHFTeJL 9a .p5 Cir‘c+'c~fi>( :whjwéL‘cQ-i 70"? ‘00 . 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asst4soln - MATH 137 Assignment 4 Due: 11 am, Friday,...

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