08W term2 - Faculty of Mathematics University of Waterloo...

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Unformatted text preview: Faculty of Mathematics University of Waterloo Math 138 Term Test 2 - Winter Term 2008 Time: 7:00 ~ 8:45 pm. 8 0L “T (ens Date: March 24, 2008. AIDS: NO AIDS PERNIITTED Family Name: — First Name: ID. Number: — Signature: Check the box next to your section: I: Section 01 P. Roh '(MC 4021) 8:30 am. D Section 02 S. Wu (MC 4061) 9:30 am. II) Section 03 K. G. Hare (MC 1056) 12:30 pm. D Section 04 Y—R. Liu (MC 2038) 12:30 pm. II: Section 05 K. W'ilkie (MC 2035) 1:30 pm, D Section 06 C. Hewitt (STJ 2009) 11:30 am. D Section 07 B. Marshman (MC 2017) 11:30 am. D Section 08 \’-R.. Liu (MC 4021) 2:30 pm. II! Section 09 D. Wolczuk (MC 2038) 1:30 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 n‘h TT (nu‘ term test), GST (geometric series test), CT (comparison test). LCT (limit comparison test). IT (integral test), ACT (absolute convergence theorem), RT (ratio test). LSR, LPR. LQR. LCR (limit sum, product, quotient, and composite rules), and L‘HR (l‘Hépital‘s Rule). USeful Information: c0520 = 1 — 25in20 = 2 cos2 0 — 1 sin 20 = 2 sin 9 c059 Note: 1. Complete the information section Mark 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 TERM TEST #2 PAGE 2 [16] 1. A particle mares along the path x(t) = y(t)) = (sin to cos(2t) + 1) in the myeplane for t Z 0. a) Use a suitable trigonometric identity to show that the path lies on y = 2(1 — 1.2). Find the points where t = t = 1r, and t = and sketch the path for Z;- S t S labelling these points and the direction of motion, ‘ ‘2 Cos2~L= \—2$tm2+. =b J5= l—2Smt+l —' ’2 C I— Dc.“ )5 ("/13 = (l ,0) a; (r) = (o, 2) z 9% = t“ ‘20) b) Find the velocity vector v = x’ of :he'particle at the points where t = 0 and t = 7r. Show these vectors accurately on your sketch in a). 21' ; \L : (cost, —’2$(m2+.3 =§ 2'0»: (l, 05 t:od>?&(°)= (oz?) Sjtcwsml ct 1400: (~i,o) t=rr=> 35(0): (0,2) “Pam‘- c) Show that the particle has velocity v = 0 at. every point where cost = O (fie, , t. = 3—2:”: . . .), and indicate on your sketch the two points where this occurs Hence describe the motion of the particle for 72: S t g 3T”; 3?" g t 5 E21,? . .. . i=0 CostZO Md. ... '25th :0 ~ _-== lg. -2.25W\f CDS'lv =0 llmuo V :O at — N " l‘QMDE. n=o,i,2,... 1; L t 5 1“ flu. Pan—“€14 moo-u: mm‘hn—dockwl-SQ £(OMQ303 “(‘50) «stops (“’05 “5 Q03, + 50395 t 5\ 1. . 73515 t 4 9312-. u dockunse 8mm d) Show that. if the particle leaves its path at t = 2w , and follows the tangent line at this point. then it passes through (1, 2) at t = 217 + 1. TC»; ngzm" Man. at t = an M ‘l:= 2TH : gtofl i-CJc—arfl 237.31) = C0,2\+C’c—:m3 ( l , D) £TCQW+‘) = + .\(\)03 = Qazb , ab MOW. loot ~T =9 e) Develop a definite integral representing the distance the particle travels for 0 S t S starting with a suitable element AL. [DO NOT evaluate the integral] «Ee’r Al- luflu- Jumq‘llx mmd 8pm 3001?: @Mimoch, AL. (old ileum“ @er ‘ adfiat). AL 53‘ m3: b‘d 14,4 m 2 (we)At)‘+(‘d'0=3l>f)z wmflm mm 0W— II o s e .4 1U M4 W MW 58:22 M51“ w z 11/; 3 ‘Léuxigj‘J‘ m: :- 5 3 d1: MATH 138 TERM TEST #2 PAGE 3 [12] 2‘ Find lim :3" for each sequence or Show that it does not exist. You may use n—ac known limits, limit theorems. or the idea that lim f (n) = lim f for continuous f . 11-000 1—000 Reference any theorems used (see the cover page). Some algebraic manipulation may be required. a)zn=C°S("-7F)+‘/ir—l 8&6“ cos n“. :60“ ) km no (1') Mel Am .L (if osc.(\\o.b.n3 “403 = O n > “M X“ «ism “0* MRS"- (Zuen 1n=Zk—_$ DCQ‘D‘: 1+ .L fi\ m 61am d «Fifi. o A n=2k—\ =-t> "x = A... h—voe 'Zhd \* m “'4‘ ‘l M B b)In= e”— 2+1r" I ‘ '“ . n , \ann W“:- QVW" e/‘Tn = kw“ (e/TA n—wo n—Mo 2/“,‘H “a QATVH Sm“ 0:“:0 \a\<\ ) Jqfimx (Pi/g“ so n-aao Wimg LQR, M x“: O = 0 “am O+‘ C) $n=ciszn anu -\ LECDS n 4: \ ; __l_. C (2—13 5 L _. ht —. fl$ n1- m . 34AM 4— =0=QM- “L- O‘PQ’L" n—pco n fi—aoo “2' -::> M C55“ :0 m h—saoo “7' 1. d)xfl=(mnn)2 &Q* —- ‘ (emo‘ m = Lam .— W $43; $00 1-. 00 x 00 1+00 MATH 138 TERM TEST #2 PAGE 4 [11] 3. 3) Use the 5 — 6 definition of a limit to prove that lim I2 + I — 6 =5. 2—0 :zr—2 we. mum skew, Caioemcmua avo ,fiw {s QSPOWCAA (\Rak \ 1341-10 ‘8\ <5 “reanalnren O < \x_z\ 4 xx). . 1 -e. - “pea-'JL—lo-SXHO _ xz-‘Wfl' anu \ 1*):1 —S\ - \ _x__?_ \‘\ x-z = \C’C‘Z31\ = \DL-2\ 857 1*2) X4. 4' —l we wfkofi ) \x_2\4 a) 1‘ \xjcic26-5\<£‘ So ckoom'ncs 8: E. 3 Lug (Awe \x‘+1-b _S\ ____ kahfl <5 o<\>c-Z\<3. OC-l b) For each statement, either show it is TRUE, or give a counter-example or theorem which shows that it is FALSE. (i) If the series 2 0.,l diverges, then the sequence {an} has no limit. ' \ F R LS E 1 CL: WM Womplk M .- 0° n L . n AW W \ Cp—W‘P=() bUA‘ 3MB? (ii) If a... > 0, and 2a“? converga. then Ea"? also converges. h < ant3“ M) bid CT , icing“ conu. ) 50 aim Zanz" canv. TRuEn Fm ano, 0.“.2 —_‘ (iii) Theseriesinhil) haspartialsumsShrzl—fi. “:1 TRLLE. nCm-D 71 7‘44 N _ \ _.L. M SN — Z L;- N am \ L \ 2. J— - i x,” (33),.” + ~ —L\”2\*(133+34 NN“ = \ __ _L_.. NH MATH 138 TERM TEST #2 PAGE 5 [13] 44 a) Use an appropriate test to determine whether each series converges. converges conditionally. or diverges. Name the test you use (see cover page). . °° «112+! ‘ v (1)2 3n+1 %\V\Q J»an \‘ “71-\ "=‘ inane anlfl = \‘\+\/n1- _ l“+o “*“° ‘ LSK I 3+ /r\ 3+0 LCR w n WWW) lot! NMT'T‘TE 1/5 *0) n=2 " WR’T: LQM \Clrx+i\ “700 a“ 1: a :0 "n‘. ham n.“ < an ' Ni :1 Mam e. / “’w Q’HD‘. ‘t‘c‘ 3L7/3. M 6%? Mm» so naa- so, w Ag, m w : b) (i) Show that the series Z] in? converges ‘ 71:] on GO SLhCn a < 13-. MA "" Cbnv so 2 \+h‘ “1 ’ “:1: “1 Z :1 Cgrnu v “3| b ‘ so mo 2 an ‘5 J —— 0 lo Z H—n‘ ‘A CT- \‘l-DC“ ie; an IT. “you; Via—2 Au Conww] 1 (ii) Describe how you would find an upper bound on the error if the partial sum 5.; is used to approximate the sum S of the series in b)(i). Justify your answer. Corolluwyafi’l': sum 2&— cmwxgeo, 305 to s i i 560' mp ‘ \+n" m:is—38\ < 3‘7. 4“ : @mdmfl: 8 H—x‘ = 2(11‘1- anan a), SMW ma. .- 'L s umkmma 0. MATH 138 Tam TEST #2 PAGE 6 [ll] 5. a) A sequence {mu} is defined recursively by zit-H = V 213m (i) Use mathematical induction to Show that 1:,I < mu“ < 3 for n = 1,2,3, . . .. 11:1. "=1‘2,3,.H. ffwehnzl? 7Cl=l +xzzm ’50 l<fi<3 i'S-Utkm. MWW 80’! 'n‘. xn<an <3_ (4) PPOLRM fiel'. l gun L3=> 2xn< Q an < 5 glrtcg J“ L5 m Unng <ng , dzxn < \Jfitan =="l> xm—x < <JZ 2L 6 <3_ ma, < M Xn<xnu<3\8dfiab&n lad a;‘\&.LLCJ-LG;\ (ii) State a theorem which guarantees that this sequence has a limit, and find that limit. SHRCA G) Show €xn15 U3 McLloomqud whoa-QJ b) Portions of a square of side length 20 are shaded in the manner shown at. right only the upper right quarter is shaded at each stage). Find the total shaded area if this process continues indefinitely. (Sula c.) had) mu m u 7. %>= MP“ 1. A_ 03+ 9.», 9.4, 2;. . ‘ 4 \e ~54 ...
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08W term2 - Faculty of Mathematics University of Waterloo...

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