MATH118MW03S

MATH118MW03S - Time: 2 hours Instructors: [J Chem / Enviro...

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Unformatted text preview: Time: 2 hours Instructors: [J Chem / Enviro Chem R. Malinowski E] Civil R. Malinowski ../ [3/ Mechanical M. Schpigel No Additional Materials Allowed Instructions: 1. Indicate your lecture section by checking the appropriate box above. 2. Fill in the following: 3. A complete paper has 9. pages, including this page. Check that you have a complete test paper. s should be written in the spaces provided. If more space is required for a 4. Answer the exam or the solution or for rough work, use either the blank page at the end of back of the previous page. 5. Marks allotted to each question are indicated in the left margin. 6. Your grade will be influenced by how clearly you express your ideas and how you organize your solutions. Untidy and/or illegible solutions will be penalized. .__—u_—..—u.._——_——————- ___...__——u————_.——r——_.v ll numufifll 1 l 0 [-2 i i mm ./ \ ‘ I . I v I 6* I Mark Obtained Math 118 Midterm Winter 2003 Integration Formulas: use as required I 1 dx = -1—Tan"(ZC—) + C x2+a2 a J-tan(x)dx = lnlsec(x)| + C Marks 1. Evaluate each of the following integrals: [3] a) I sin5(x)cos3(x)dx : Sfiaqééb C0445 (@334ch } :3 Maer (mm : §(6ms@.5:n7(«33 Cara Clix u a: 6; mm A UN mséo A ¢< :S<e\5— dlcoséax 6‘“ id: M a mag C0 96 J :Jfi; _ {,3 e g + C/ Math 118 Midterm Name: Winter 2003 ID: x [4] b) °< 0% ~ 1-x: Jame Ax:-fi5€g9;;g % Sl%t<‘“l)0zc 74 H $ny 9 x kw fig 1 :1 S Wf’J/Z/séédfi) 9% \ ‘ 3’27”? m3 Q - $2: MAGXYZ Mew W ‘ ‘ j E éfztjim : “ :Elmne) 3 figkLmé “366:6 A9 a 2. ' y» ml 4? sag , SsaCQAe 3‘2“ x + " ; 3 Se #2. m '55 ' / 7» “aaww d“ o vysxdx : 5W» Aw: 4/ “(’34 *3» IX 541+?) :Y; , «A oLx AIfiSA $064er «ax-r é />< W3 ' A32 : _ (M Rf. ZN“, SW5 Zuaé +6¢+Crx=o<io<+é : ZM’A ’4 Swing” ‘ xii” BOEA'C'X "4-" 4X: K 4+5 RSV! >C:-l j; Z‘nl’fi‘ "$11.4 (—3? +C_ - 0W Name: IDI ._————— Math 118 Midterm Winter 2003 2. For the parametric curve with equations x = —t2 + 4t +1, y = sin(—2- ) on the interval 0 .<_ t S 4: [4] a) find a general expression for —y— in terms of t; “mu—MW 3”“ m '3sz 3‘sz WE . d . When IS 1 = 0? When does 3% not ex1st? What can you say about the tangent lines at these points? Indicate these properties on your sketch in part (b). 1t 9&0 M n thaalalamlncaséléiged) aw ) War“ “3 "’ "" C,o>5(1%,£\50 631,533) 2 {brigame “we «32% Come it: «(heal mack Am. / {+743 0 t: Z [3] b) plot the curve on the xy axes; (It is not necessary to eliminate I; you may plot some specific points and use the information from part a) set up, but do not evaluate, an integral expression for the arc [3] 0) length of this curve. uwv'vrn‘wmm-wm ~ l m 534:“, Math 118 Midterm Name: Winter 2003 4. Consider the sequence a1=2 a+1=1_ 1 for n=1,2,3,... " 2+an [2] a) Write out the first four terms of the sequence, c11,czz,a3,a4 \ a; 2 at: 12.1.” as: 12:; at = i - “M 7' q — ~— it 2 \*—‘L . : é ’ H \g 2“ LI v.1 " i I H ’ E Does the sequence appear to be increasing or decreasing? JANE," sequence, appmrs +5 be decraaéiflw” [4] b) Show by induction that the sequence either increases or decreases, based on your statement in part a). b IR?- i‘m mme Mm)? @w‘iqw “Z QK: \ jrixere is edema (M sock X T: Gt :4? [3] b) Given that the sequence is bounded above by 2 and below by 0, determine whether or not the sequence converges. If it converges, find the limit. i? 04m; "I: Z 0m ‘X l f :, b6} seqoema '15 bounded 0m \1’0 2 Old Q} :00 Ova Niki Gm: m \.. 7‘”! Comrargg. oi 1.x ‘ w 5i w w bo" lg: an“ a. "[355"? \ “Ewan Q‘p:fl%~ OW; wfichZFMMA Winter 2003 ID: M 5. Do the following series converge or diverge? For the ones that converge, determine the sum. r . Facs’t ) check {he n“ Mm «314/?» cc n2 1 n7. [2] a) 2524 hm fl ,lglowa ,\‘mwlw;i n=1 n + new Snlzwq fl*%& m 5 “in km O 5,, We series, Q\\\ Amecge new iggl‘imeck if?“ Term g °° 1 [31 b) 2 f “m \ l' ‘ "=2n Inn $309 m3 1—, “W2 T ,0 :3on mnveffie ham check HM injfafiL raw/(M7- iw Tu dx lei- 0% Ma- Z, [X \MA m: a“ flaw \ 644.; : e} m) j: géfl Auk 1"; no :33" 03 a?" - . g: Jame series of.“ Jive e :: C} if“ (in . r3 '1 a“ [3] C) :3”? bkm 233:, M: m 14332:) n: 6" new n 1 6 when Mn‘é W668 *Mew “do Atheong Storm 0 :- 00} oxwa geriet Amaze?» 3r: 3+2 , .52, haw: 13,. +6: W? A '/""‘”’exz'fi"” '54 53’ " 10% *0 [3. N61“) (he‘d; 36 Q!» flaw “81f? Math 118 Midterm Name: Winter 2003 ' H): . . °° n . L [6] 6. leen the series 21n£n+1). 2 g 71:1 a) write out the partial sum s”, which is the sum of the first n terms; n k / 3‘? Z \n ? , Z Kt} v b) find a closed form expression for the nth partial sum s"; (hint: a property of logarithms is necessary) \. (2/7 0) determine whether the series converges or diverges. If the series [0 converges, find its sum. If the series diverges, explain why. Z Awkéflbk j CMCEK lifle @«d..d\‘l_t3 MQWQMQA 1 com er‘Vim \c: 300 who) sacks wit“ diverse. in: 1n<afiantégwgln<a “margin (Sm gear} “Hoe $ma¥tm bang \n’adx (2‘? 3w will "7 us\\ 95* \oflkfi‘ and \oxgex ) bo‘f new; CK\'\V\\OW§¥Y\C Stern‘s £61305 QAAQA KS.“ 32* fillhr‘glfllf\1\3 close. (A Va lamb big“ “mac (hemix‘ 8; 9&3. ots Sock “Were no COWVQ’CCIBQY‘AC; Q .. Total: [54] ...
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This note was uploaded on 05/18/2011 for the course MATH 118 taught by Professor Zhou during the Spring '08 term at Waterloo.

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MATH118MW03S - Time: 2 hours Instructors: [J Chem / Enviro...

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