Math_138__Term_Test_2_Solutions

Math_138__Term_Test_2_Solutions - Faculty of Mathematics...

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Unformatted text preview: Faculty of Mathematics University of Waterloo Math 138 Term Test 2 - Winter Term 2006 Time: 7:00 - 9:00 pm. Date: March 13, 2006. 50 L off 10 M 5 AIDS: ‘PINK TIE’ CALCULATORS ONLY Flemin Name; Initials: _ I.D. Number: ._.._— Signature: Check the box next to your section: El Section 01 J. Emerson 8:30 an. El Section 02 S. A. Campbell 8:30 mm. :1 Section 03 J. Emerson 12:30 pm. El Section 04 A-M. Allison 12:30 pm. El Section 05 C. Small 1:30 pm. El Section 06 C. Struthers 11:30 mm. C! Section 07 B. J. Marshmsn 11:30 am. El Section 08 B. Ferguson 2:30 p.m. 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 PFD (partial fractions decomposition), nch TT (nth term test), GS'I‘ (geometric series test), CT (comparison test), LCT (limit comparison test), IT (integrals test), ACT (absolute convergence theorem), LSR, LPR., LQR, LCR (limit sum, product, quotient, and composite rules). Note: 1. Complete the information section above, indicating your instructor’s name by a checlanark in the appro— priate box. 2. Place your initials and ID No. at the top right corner of each page. 3. Tear ofi 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 Team Tee-r #2 PAGE 2 1. A moving particle in the z—y plane follows the path 3: a F(t) = (cos t,.sir.12 t) , for t 2 0. a) Find an equation relating x and y, and sketch the path of the particle in the 2-1; plane for D s t 5 1r, labelling the points where t = 0, E and 7r, and showing the direction of motion. X=¢O5'\', 3:5m‘t=\-5°5‘t gt.x_a "9 n5: \-:ur.'t , 5:11: niacith palmth mapmabela. Eb}: Up) , EL 17%) = (0.0, Eér} = (4,03 s) Marie-n right 13 hat. b) Find the velocity of the particle at t = 0, g, and 1r, and show on your sketch. gm- E‘m a(- saver, 250;: mm =9 ico\=(o,o) , 3%: been , 1(a): (go) (2) How does the motion of the particle for 1:: s t 5 2w difier from U s t 5 7r? What happens for t a 21? Fe: 1':- ts a1; , x‘mx —sont >_.o , sompafliéfli WWW’ We“ ,M‘Daachq Wu. omnipoimt‘e .MOUfng. re':the._,fl¢fih+_-...T F01 h‘a 31!, it Mm‘ls Than to» Motion: , writ. PMLDG‘ a'n'rJ d) Develop a definite integral representing the distance the particle travels for 0 5 t 5 7r, start— ing with a suitable element AL. [DO NOT evaluate the integral] item 0. PmTCtté-n .1 tom] irate whom At gunmdtmflne 3&- w m- e H mm“ EH 3 rT—— ' 1‘ M i (Ewan of“ I “a: boTh M e be -= ‘ILXUIWCQQL at - t o Summ'ch AL's Sch. 09-h 611' mime] The. ’Q‘I‘M'd "D A a s n' . - it ~ oawwfiw Math L=A3€To ‘2' AL :1 I slht++4 thtt aha-Inuit Ill-'6de 3” ° ° 5mm Sultafi e) Suppose the particle flies off its path at to = —2- , and foflows the tangent line I at this point. a“ to I “,3 ' Find the vector equation for l, and the position of the particle at t = 2r. met sum 35, = E (to + E‘ (m C-E - t.) = (on) - (-an 12E,25M_§.:c°s;zt3(t-W2\ = CO. I) t- Ct- 3%) (go) ZirCZfl = ‘9.” + ‘2.ij =LJ§,|) . f) 'If another particle leaves (gull) at t = 33", moving on the path it = (-311.0— what value of the constant u will guarantee that it collides with the first particle as it travels along I? The-second. pantie}... "Vinnie m1!» QGM tun?“ ‘5= “(it-35;) . Sthfl'fiu Sow? PM'h'JJ Mack”:- 4): = E at 1:: 911' (gram PM E) , we meat lnqu-t s5=i get tgug;‘ 85.11... moo-nah whiten-QM. “Wu-J: Mat-i =1 , cg. ' 'Thmucm 4 ‘2“ n u «r (Thst ,uumot mu.) MATH 138 TERM TEST #2 PAGE 3 2. Find the limit of each sequence {In}, or show that it does not exist. You my use known limits, limit theorems, er the idea that. nlim flu) = bm' fix) for continuous f. Reference any theorems “W 1—” used by the appropriate abbreviations. Some algebraic manipulation may be required. 1 wa=em+g um L,=o ,bqu Mm co“ ONE- hqm h “an ' (11“- OM W5 amend" -\ MJ. “u, wanna“: Woo-din +\ § M m J.le ONE. b} xn=‘/n!+n—n I J {NH-n -h- = Jht-H" ‘“\L4“1*“H° = 4-“- n. M mm W“ * I. x“ = 1“ “3000 141-909 “14" am a J.‘ __\_————- =|/ L. RLSR,LCR .D§fu.4\t-"ajxfi‘ “V2...” f. ' ' C) xfl=nae'“ S a '1 a 'x. £2): S—Cfl ace .6} m “g. on w —-m.««= awn - m9$*" e (as) Elfin—[m 23‘ L: — ’X-oa "a?! _.. - 74 m m n‘e'“=o — it: :1 =° \1\ 3 \CDS < _L_ .r—q o n3 “1’ '° W M 0: :0 (us had-:0 '4' “n'w‘ n-un " .L. e) sn=nsin(%). £91. = 7" sm 1'5: a at“ . - . ‘k wa .K = 1mg“): Qum- nwnn “ “+00 rx—eoe V1 11+ M.# 5 1% my ‘X-vMo-PQ-PO" u—ho‘ u = ‘\ MATH 138 TERM TEST #2 PAGE 4 -————_....._.—.__—.._____ DO 3. a} (i) Use a partial Erections decomposition to show that the series 2 1 . SmBSN=1“m- J...— =- L _ J—u-d I N “(n+0 '1'! NH Us.” Zea—‘31,; =(4-§)i(%-%\,(si_§\+..,.x__;m “3| Sake '11».- ..lufi- mm in In.ch Hen-nu.“ me. tin {ubi- my. I'M Thnud' CL, an. Mai- punuiueSudu. 1 «.4 “I‘m, “3- 3.1;“ {5'11 (ii) Define what it means to my 2 an converges, and hence determine whether our not the run-1 Mn + 1) has partial as: w series in a) (i) converges. a,“ dancing!» '1“ Wrap SN utters . Hm,f-nG), 'kmasfi = mono- gm): K-O bat-SR. b) For each statement, either show it is TRUE, or give a. counter-example or theorem which shows that it. is FALSE. ‘ I I _ | Hm |a.,,| = 1, The series Ea... my converge, or it may diverge. r ‘ u m I e» i. L.” _ .. r .. u & - v M m . 9.... _ .. r q I .'. FALSE (ii) :(IMP has sum 1 ‘_ 1m "8y Ger , ELM)“ Cum?» T° La ‘19: “MM, for all values ofthe constant]: suchthat k aé m . l-\<1r __-_ PAUSE i- _ t < 6.11- <1. so no _ _ . . (iii) Z2"+ zfiisewnvergsnt series. SLfl-M % Z“ is a. gt.th nun, n-l n=51 n;- _ Ma, 2.3“ is, a. coho-MW“ P—mlm: 1k». m urnuw, The bottom half of a circle of radius 1 is shaded. Then a circle of radius 1/2 is drawn in the upper half of the first circle, and its bottom is shaded. This process 7 is repeated indefinitely, with the radius halved at each stage. Show that the total shaded area is 31. Ma. 0% 8mm: Mme. (s LIV-i; u I. u n n 3'- “ 1. 3- 13m. Mi GIN-0- L5 Iifii1+Gfll+ (ff-+(é—‘Yhnls .L— I _. _ .__._ --—z'K§|+i-+ .li:+‘-L+...1._.\i.1f “.mrl 1/3.; h MATH 138 TERM T251- #2 PAGE 5 4. 3) Use an appropriate test to determine whether each series converges, converges absolutely, or _ diverges. a)” “+1 J; a __._M.w~ 3+4... _ a,“ My" 33‘“ “*“W w- mm = 4 in, LQR,L$R, LCR / ‘Qn\=' ‘msnfi‘ J— L .5— _.—————-‘= (fl) icofimr) H n-lzfi‘tblu 2n+; ant-J 2“ 9mm as n. aean mn'ua 5.3 651' Quirk Maid) a C-Dsut C . ‘fi 50 We: gnmw 83-d- T . M “nu-yew absotunlnd . an] znlkl (iii) 2 . “=3 7‘ Vstnu an > ‘1 Sm n >e (is, n13) , ,... . -- “aflwvm. ..' . -- ~ 7. ,...F_- _-—-—“ >lT—‘->o ‘01 nee. a Swan 2% «saw Pm“: 9% 9.9.50 Mew but CT “:3 (w)§:;% W 21': z Aim Cmfl s 5”“ (EMA we» n-eoh a. “4"” Ln“)! = Linn "run"- 5 _. = Q' 1 “*‘F (T ' 11H {FL—\- , . ..mWC‘—MW (LPR,L_QR\--— \-O=O<l on b) (1) Use the Integral Test to show that Z 1 in: converges. all Q00: ‘54:; t‘5» ED‘In'l'LJe ffiumdng, '- d's. in“ wlepufihim Sap—L..— clg = Mm EMTMJJB a 1”“ MeWb —- m‘mnt l-V-x“ baa- ‘ b-oee ‘ = 1%.. I: e I. 4 4 .l'. 16.; W emu- (ii) Use the Integral Test Corollary (ie, the Remainder Estimate) to find an upper bound on the error if the partial sum 53 is used to approximate the series in b)(i), 3,9“ m m 5'% t3 WWl'ma-t! TL; bud-6,1Lw w . J... E 9. ‘0 _ (3-3% 4 “)0de 5T; widen each“? u. 1;; 4.44m e: mqu MATH 138 TERM Tear #2 PAGE 6 5. a.) State the precise definition of what it means tn say “13:: 2,. a: p for a sequence {:3} of real numbers. Gwen OM41 €>o , 3 K70 nut-cir- l‘kaui' \rx“_9\ <3 Liam “’H" b) Use your definition in a) to prove that lim 1 = 0. n—um g \zé_“-o\<g 1g Ence ,ngi",:i‘_.=>e"> =1?“ Choose K-enfig- We“ \"_-.‘C>\=ae""<2e'K in “>K c) State the Monotonic Sequence Theorem.‘ Eu-uu-k baumdsd. manoTe-ne sequin“ gate 4. MR. d) Use the theorem in c) to show that. the sequence defined recursively by :1 = 1 and In.” = u/1+‘2.I‘zfl for n=1.2,3,--- converges. Then find its limit. . {HINTz First use mathematical induction to show that z“ < In+1 < 3 for n. = 1. 2, 3, - - Chechgp‘“:\i x\=‘: Iz=m=fi<3. -.- H“- MQW’I .bwloLo go: ‘nal. Maw-w. thud. 801m 1c“ < 'xnh <3 on Prom M 8m n+I: 61):» 2.94“ < 23M” < ‘3 =13 H-‘2’Kn < H- ZMM‘ < ‘4 6 L- ' =9 . <3. Le, Kn“ < 1:,“1 < 3 MN Mme; [n CALGAL‘MC‘ and. loamy-um, Suwu is xn¢1emi < 3 gmmn=\,2,... i-Ltma. 1* Leno-theses b3 Th. Mmifianle 3 ' 'Xo-e? "‘ “an a? ‘9 'P= “*2? a 5'1 91—29-1 =0, _.p 9: QiJz-Z-DA gang; ..p PaHFL' Mia P>i, ...
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This note was uploaded on 05/23/2010 for the course MATH 138 taught by Professor Anoymous during the Fall '07 term at Waterloo.

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Math_138__Term_Test_2_Solutions - Faculty of Mathematics...

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