MATH118MW10S_0

MATH118MW10S_0 - UW Student ID Number: 1" i...

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Unformatted text preview: UW Student ID Number: 1" i " University of Waterloo Midterm Examination Math 118 (Calculus 2 for Engineering ) Instructor: See table below I Section: See table below Date: Monday, March 1, 2010 Time: 7:00 pm. Term: 1101 Duration of exam: 2 hours Number of exam pages: 11 Exam type: Closed Book (including cover page) Additional material allowed: None Circle your instructor’s name and section number Instructor Section aid . .: ems n - . :. , M J g V I R. S. Malinowski 003 ME students R. S. Malinowski 004 MGTE students Instructions FOR EXAMINERS' USE ONLY 1. Write your name and ID number at the top of this page. Please circle your instructor’s name and your section number up above. 2. Answer the questions in the spaces provided, using the backs of pages for overflow or rough work. 3. Show all your work required to obtain your answers. 4. The use of calculators or any other aids will not be allowed for the test. Formulas: o sin2a = 25inacosa o sin2a = (l — cos2a)/2 0 cos2 a = (1+ cos 200/2 Kb—a5 "SM—i965} _ 3 . lTnISM 12712 WM“ Math 118 Midterm Exam, Winter 2010 Page 2 of 11 [18] 1. Compute the following integrals. t=ar (a) / tan 2:1: sec4 2m d9: cl t“ = 2011 = (lying/16:) SCC4C’9d t télewwm+mfiu$stedt < if u (HuZ‘Mu : Jifu—Hfiogu .L . *‘ 2<’2L)u2+ étfiifl‘fic 4 {amfixfl % mfibx) m (M/e 335:6) DUES \nuc) #2 C) X 1130* “mu CU C11 filaCi) moo fc xamcfi W Jiobc MM 1% maw’ 1:4: if? Hm ‘ l X, l J— OWW” Qfinfiqi)? "2mm? I l ,1. 00 4 12: O + L N ame: u = 1am Gt) d u: 96C} Ha) CHI Math 118 Midterm Exam, Winter 2010 Page 3 of 11 Name; (c) M - W93} WWWWW M \/3——2m-—:1:2 ¢ 7 U I“ If \ 'J'CX'fDZ'F‘S“ GLU‘ rd} _ at U ‘ u * WWJZTZ: = avcsm +C «11—0 2 ‘ 1+ = avcsm ( ) +c, ch! = 1+ (1 3c :1" UV ~5Vdu ‘3 t ‘3 -— '35 L e»! 3c 5 “I; 3 $1?th) ~ Jgjmom 1": ‘ _ Eifl' HUI} "' J§[E))LS +C I S” A ._ M ’ —~ 3C MC JL 2M~«’“— ,L ~ '°’> ~ '5 2g 7:1,, ~\ L Math 118 Midterm Exam, Winter 2010 Page 5 of 11 Name: 1 m ,2 i [5} 2. Consider the definite integral /0 cos (2x) also Effz) (a) Subdividing the interval [0, 1] into n = 6 equal subintervals write explicitly the terms used in the Trapezoid rule approximation of this integral. (Do not evaluate the individual terms or determine the sum.) A1 = l:_Q : _.L (0 (0 To a A. [00$(O)+ 20051423) + 2ws(—%)+2cosm + 12 Fm MW zooswn i lens .5) + mam] (b) Use one of the error bound formulas on the front page of this exam to determine the required number of subdivisions if we wish to approximate the value of the above integral with an error _<_ 10 1000by using Simpson’s approximation. (By rounding off numbers compute a reasonable value for Wis Karma < “3'4 \ )zn2 Rik 003(21.) 7, i be > =-2sin C 2x3 ' "Wm; smut cos (7230 < K (at 13 highest vaW—e H13 =4 7 fl (See 93mph) 0" DC ’ E" ><\3 ‘ Q<%<\ ‘ I, Din?— 4‘ if“ 3 ‘4 “V4 é “1 h/I? U<J§<fij, I 1404 S n1 8 3 3 \ $3 < 2 + .1. 2 \L _, , O S {3’ ' Vi I > > 1?: «Ea reasonable Van o? n 5 (pg 3&3?“ Mm .3552 EM Math 118 Midterm Exam, Winter 2010 Page 6 of 11 Name: 2+2cost _2 + gsint Where 0 S t g 2’”. [5] 3. Consider the parametric equations { ma) W) (a) Plot the parametric curve of the equations in the plane. Indicate the initial point and the direction of the curve. costs): at » 2 amas- W Z weary «minim as C1r1)2:“i’ c 12;?“ "a MKWM _ 4. (:1 «21+ (36132:; :22” air ole t x “1%” ini’nm éw—MW. mm! 1 VOW“ 2 T . <3 —2 3% 2 ~4 2W 4 —2. (b) Find the slope of the tangent line to this curve when t 2 7T / 6. at era Q 00% “t. i , mafia “‘1 age-Mos ab W2; :2 s'mt ti 1% «grim/o ‘ ‘ E." T) ’ “E [4] 4. (a) Give the Cartesian coordinates of the point Whose polar coordinates are (4, 37r). x : 400mg“) == '—4 OWN “ 3=4smégifiw O ( 4W6) / (b) Give the polar coordinates (7“, 6), Where 7" > O and 0 S 6 < 27r, of the point Whose Cartesian coordinates are (—1, Quadran ‘I‘ I r: (124%; a w +an ( ‘d/x) WWWWW W I w “Veg—1M Alene—rm): m ‘15-.) (we) ~ < 2.. 2y «l s + 3 :21“ (c) Find a polar equation for the curve represented by the Cartesian equation $2 = 4y. (No need to simplify the polar expression.) =1. (errata - (200529 .3,» 4v 5mg ii a ire/031% "a 4stn9 / r Atari? C8318 ((1) Find a Cartesian equation representing the curve described by the polar equation 2 - 7“ = Sin 26. r2 :: 25MB 0196 V2 2' new Y4: 231 9C4+2>C2ij 2+ (fr = (12+52'Y'7: 2323c Math 118 Midterm Exam, Winter 2010 Page 7 of 11 Name: [6] 5. Consider the polar equation 7" = 1 + 200s 19. (a) Sketch the graph of the given polar equation from 6’ = O to 27r by first plotting the equation in a QT—rectangular system. Indicate the initial point and the direction of the curve. [0' l z ooSCflaf ) l —— \ ‘ “\.,\ x1) 2 1+ 200 5% "*0 ' ease “’"J/Q (b) For the above polar equation determine the following values: i. All angles 0 between 0 and 27r Where the radial coordinate 7“ is zero. r:\+ 2.00539 «a: ‘6 3' arc we {Mt/2.7) (LOst '3 "V2 fir fl at ) fl 3 3 V/ ii. All angles 6 between 0 land 27r where the curve meets the positive polar axis (other than the pole). "6' “1 ’0 («0mm awash) v,” » e =.- TV 0 a“ [4] Math 118 Midterm Exam, Winter 2010 Page 8 of 11 Name: 6. Consider the circle r1 = 3sin0 and the cardioid 7‘2 = 1 + sind whose graphs appear below. (a) Determine the polar coordinates of all points of intersection of these two polar cur . , V88 r‘ 2 o r: 3 30% BM‘KB“ 1‘5; 0 ~51? (A? if? :52 ‘0‘ “v; 2W). . r2 7: O trainer i < 3/9, “7(0) r/ “n43 (a is (A, gaitfiri‘nfif” 633* la : 1&1 2‘} (3/2, grille) ( O. ‘07 a 0g 3391) (b) Set up an integral expression whose solution will give the area of the region repre— senting the intersection of the interior of both curves. Do not solve the integral expression. 1V; A ‘ ' veaw O [BSMQDJ'le'i‘ Tr memora— / /(9 (m7 7. Determine the length of the polar curve 7" = e29 for O S 6 3 7r. ‘ 9 M-mwmwwn.tW length "if Jaye» 2+ (473%)}: 4&9 OW .,. 2629' q T? _ I W ~~~~~~~ “’i s [\j (fife)?— + < Q€215>1 0(3— 0 W W ‘ fig-l 4&9 are» r 1,. 353‘ smug {x13 always PosifiVc ,2 T a Math 118 Midterm Exam, Winter 2010 Page 10 of 11 Name: [2] 10. Consider the recursive sequence S : {$71}le whose first term x1 is positive and all other terms are defined recursively by 1 4 $n+1=§ $371+; This sequence is known to converge to a particular limit L regardless of the value of the first term IE1 provided x1 > 0. Determine the limit L of this sequence. h e L (“9921:1900 523$ 1 8 2(\ was Total = 55 pts l 2 > 9:3 decreasing c ._ I fl “3 Wm convevgc mavflé Claim: 3(6me .5 cle£;.,r~eqsiach (Hawgficsg Mi. 1‘) gorse casc- xl >13 13 714 59 no ‘7 at ' 8M 2072>W§itxgé§0“”“ ...
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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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MATH118MW10S_0 - UW Student ID Number: 1&amp;quot; i...

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