M138.W08.SampleTest1

M138.W08.SampleTest1 - Faculty of Mathematics University of...

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Unformatted text preview: Faculty of Mathematics University of Waterloo Math 138 Term Test 1 - Winter Term 2007 Time: 7:00 — 8:30 pm, FDR, Date: February 12. 2007. AIDS: ‘PINK TIE’, CALCULATORS ONLY Family Name: _——____ First Name: ID. Number: _._________ Signature: Check the box next to your section: [II Section 01 D. Park (MC 2035) 8:30 am. El Section 02 S. Drekic (MC 4061) r 9:30 am. {3 Section 03 L. Krivodonova (MC 4021) 12:30 pm. CI Section 04 M. Mastnnk (MC 2038) 12:30 pm. CI Section 05 C. Roberts (MC 2035) 1:30 pm. CI Section 06 C. Struthers (STJ 2009) 11:30 am. [:1 Section 07 B. .l. Mersliman (MC 406]) 11:30 am. 1:) Section 08 E. Samei (MC 4021) 2:30 pm. 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 I by P (integration by parts), PFD (partial fractions de- composition), DE (differential equations), IVP (initial value problem), and FTC2 (the fundamental theorem). 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. Any 60 marks worth of questions constitutes a complete test. MATH 138 [41 1. [6] TERM TEST #1 PAGE 2 Q n csewers-S a) Use integration by parts to find the general antiderivative / ad'sec2 ardxr V.me + kaosad +C1 1 b) Evaluate the definite integral / 1111:5112, where 0 < a < 1. 1-0.151/“(1- -] (ii) Determine whether the improper integral / z lnx dz converges or diverges 0 4-12 To ’kk d. $4+4x2 x c) Find the general antiderivative / L- if; -— ancTamOéZ» +C] lVlATH 138 TERM TEST #1 PAGE 3 3 [6] 2, a) (i) Find the ‘aree’ of the region bounded by y = f(a:) = 9:36“ , y = 0., and :c = 11 l ") *4 b £3 2 \ 1 fl. .— .. . . . °° 2 d; , 3 3 (11) Determme whether the xmproper Integral a: 6 dz: converges or dwerges. o (Cowl. ‘5 V31 00 [7] b) Show that / ip dz: converges for p > 1. l 3: (ii) Use a suitable comparison theorem for integrals to determine whether the improper 00 _ cos x , mtegral 2 dz converges or dlverges. 1 :1: MATH 138 TERM TEST #1 PAGE 4 [6] 3‘ [4] a) The shape of a. rubber pet food dish is formed by revolving about the y—axis the region bounded by y = 2:4, I = 0 , y = 0, and :1: = 1. Make a sketch of the region, including a suitable volume element AV. Then derive and evaluate a definite integral for the volume of the dish. f‘l L\l= “31 b) The temperature in °C on early spring days in a. certain‘city is given by 7rt Tt=lO '— () +Qsm12 ‘ where t is the time in hours since 9 am. (1) Find the average temperature during the period from 9 am. to 9 13.111. each day. (ii) Using symmetry, state the average temperature during the period from 9 am one day to 9 the following day. MATH 138 TERM TEST #1 PAGE 5 ,_ , . . d [o] 4. a) (1) Find the general solution of the DE E5 = 2zy2‘ and any equilibrium solutions. [ —— l a s q 3 1 ‘A y a- C. (ii) Find the particular solution satisfying y(1) = a r S E O ] \ \ : J A — 'x‘ d [5] b) Find the general solution of the DE 3—:— + i—y = c051, and show that all solutions approach a periodic function as a: —-r 00‘ Eng: Sl’M'Xl- C05”: *93 ’X 'X (0.500 'ZlDDO h‘lATH 138 TERM TEST #1 PAGE 6 17] 5. Li a) Once her parachute opens, the vertical velocity um ni/s [metres per second) of an 80 kg Sky-diver at time t Z 0 follows the IV? do A? a? — 9.8 — En, um) : 0.2 iii/'5. Where the drag coefficient k is a positive constant. (i) Solve the given IVP, and find the ‘teranal' velocity or. (Your answer will contain the constant k.) a (ii) Given that, for a safe landing, UT should not exceed 1 m/s, iind a value of k which guarantees this. Then sketch the solution you found in (i) for this value of k. b) One model for the spread of information assumes that the number of peeple y(t) who ‘know' the information increases at a rate proportional to both fit) and to the number of people who don’t know. Thus, in a city of 10,000, a suitable DE is ill. dt Jl~.y(10‘1 ~ y) , where k > Dis a constant. (i) If no one initially knows the information (is, yfil) = 0), does the DE predict what you would expect for y(t)? Explain. (ii) Sketch typical solutions of the DE for initial conditions y(0) = 1000, 5000. 7000, using a qualitative {direction field) analysis. DO NOT SOLVE THE DE. Lo (iii) According to this model, what happens eventually in this situation? MM‘H 6‘3 mamms' CommENTs on TERMTEST1 wmrER Zoo? wfiau flu 111:? mm “Nu-Am ,ou-mcLM «cm uJ-UJ. Mme. phobia—ms men/31% us note. J 4:145 ijecfixdlx 1‘ 0631 {by P , Sou. mam? 4V: AQCI’XAM (ma chlst , w=oeczoc To AJ-l whip ML“) Some did mo? wank 1Noujfi J-Cm £91m; (0-60) , bqu Aoflm - “1'0" - Q“ dufl %um¢d o , m «a: . 3*Mcbwm lu-umeg cm 4 um Luv. ya UHQ. Q" 4} maink an‘ufixmwc. WM ) “Ix a. ’ZoqmFm 5 we-XBM I some Rind 1‘53 P! wwa‘ WW wink. (Incogan ocz is 'olmosi' C135,. 3 bm33c>me non-mowc’YLb-e compam'sons (e3, \CDSXl .1 \co$x\ 3 a - KEV IDEA'. From 2W) ‘ S 3;,M who-macs 3 moxzubgfi ‘ L Sauna.“ x “'71 ‘ x‘ “‘3. a.) mxShc-nupfi'e—n M i 1"“ PAan “3% Tc Svkd. [kn- VMW % W Mmd to mar-kn.“ d-Lbk ‘, Bud mow-ta “MAW; goMAM Cctpac'u’tg 93 M Ais‘n -o. 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M138.W08.SampleTest1 - Faculty of Mathematics University of...

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