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Quiz-2_Fall-2009-2010_Makdisi - Math 201 — Fall 2009*10...

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Unformatted text preview: Math 201 — Fall 2009*10 Calculus and Analytic Geometry III, sections 1—8, 24—26 Quiz 2, December 2 — Duration: 1 hour GRADE& 1(/15) 2(/15) 3(/15) 4(/16) ‘ 5(/19) ‘ 6(/20) ‘ TOTAL/100 l #J—M S a\o\i ans YOUR NAME: YOUR AUB ID#: PLEASE CIRCLE YOUR SECTION: Section 1 Lecture MWF 3 Professor Makdisi Recitation F 11 Ms. Nassif Section 5 Lecture MWF 10 Professor Raji Recitation T 11 Professor Raji Section 24 Lecture MWF 2 Professor Tlas Recitation F 11 Dr. Yamani Section 2 Lecture MWF 3 Professor Makdisi Recitation F 2 Ms. Nassif Section 6 Lecture MWF 10 Professor Ra ji Recitation T 3:30 Ms. Itani Section 25 Lecture MWF 2 Professor Tlas Recitation F 12 Dr. Yamani Section 3 Lecture MWF 3 Professor Makdisi Recitation F 4 Ms. Nassif Section 7 Lecture MWF 10 Professor Raji Recitation T 8 Ms. Itani Section 26 Lecture MVVF 2 Professor Tlas Recitation F 3 Professor Tlas Section 4 Lecture MWF 3 Professor Makdisi Recitation F 9 Ms. Nassif Section 8 Lecture MWF 10 Professor Raji Recitation T 2 Ms. ltani INSTRUCTIONS: 1. Write your NAME and AUB ID number, and circle your SECTION above. 2. Solve the problems inside the booklet. Explain your steps precisely and clearly to ensure full credit. Partial solutions will receive partial credit. 3. You may use the back of each page for scratchwork OR for solutions. There are three extra blank sheets at the end, for extra scratchwork or solutions. If you need to continue a solution on another page, INDICATE CLEARLY WHERE THE GRADER SHOULD CONTINUE READING. 4. Closed book and notes. NO CALCULATORS ALLOWED. Turn OFF and put away any cell phones. GOOD LUCK! An overview of the exam problems. Take a minute to look at all the questions, THEN solve each problem on its corresponding page INSIDE the booklet. 1. Let the function f(x) be given by 0, when 0 S x < 7r f(x)= m—rr, whenng<2fi and f(z) is periodic with period 27r. a) (5 pts) Sketch the graph of f(::) for m E [—27r,47r]. b) (10 pts) The Fourier series of flat) is 2,150 an cos TLE-i-anl 6,, sin 113. Find ONLY the coefficients bu. _ _ 2. a) (6 pts) Plot the polar graph of the curve C : r = 1 + sin 6. Also draw the line L : y : 4/9 on your graph. b) (3 pts) Convert the equation of L to polar coordinates. c) (6 pts) Find the (r, 6’)-coordinates of the two points of intersection on L F) C. 3. Consider the following moving point in space: 1 P(t) = (3t,\/6et, 532‘) a) (5 pts) Find the velocity and the speed of P(t) at the instant t— — 0. b) (5 pts) What is the arclength of the curve given by P(t ) for D < t < 1n5 '? Simplify your answer. 0) (5 pts) Suppose we have a function f(z, y, z) with the property 6x-(C.|(3,\/6e,%e?) =(e2l—\/6815)' Find %f(P(t)) at the instant when the point P(t ) passes through (3, V63, —52). 32:9? does not exist. 4. a 8 ts Show that lim H p ) (x.y)—.(o.20)m4+y 2:23; b) (8 pts) Show that lim does exist (hint: the limit is eqnal to 0). (MO—(0 0) 3:2 + y2 5. Consider the function f($,y, z) = 28x33. a) (6 pts) Find the gradient of f(z,y,z) at P0(1, 1, —l). b) (7 pts) Find the equation of the tangent plane to the surface f(m, y, z) = —e at P0. c) (6 pts) Determine the direction in which f(m, y, z) increaSes most rapidly when the point (z, y, 2) moves away from P0. Your answer should be a unit vector. 6. Given a function f(m, y) satisfying f(1, 2) : 4, fif (l 2)= (3, 4). a) (6 pts) Approximately how much is f(l.03, 1.99)? ‘ b) (7 pts) Find a direction 11' in which the directional derivative Dgf : 0. Your (1.2) answer “LI should be a unit vector. 0) (7 pts) Let S be the graph off. In other words, 5': {(33, M2) 63R 1 z— — f,(z y)}. Find the equation of the tangent plane to S at the point Po(l, 2 ,4) 65' .(Be careful.) ii 1. Let the function f(:.':) be given by 0, when 0 S 33 < 7T f(:c)= m—Tr, whenfl£$<2fi and fix) is periodic with period 271'. a) (5 pts) Sketch the graph of fizz) for 33 E [—271',47r]. b) (10 pts) The Fourier series of fix) is Enzo an cos nI+En21 bu sin ms. Find ONLY the coefficients an. \ 1.1: \ 1“ i i \ T \ . \Dh: :"r? S350 51"“)(9'5! 7 CE- -DO\§RM¢J)< +SfiT‘X)$1nhXiX X: 0 XP \‘ T“? (”J aw ifiWVAK 9“ O Mask '2] Cm ‘91 URL \RSWJ Jr [0, 2‘3 J 7'1: (1-1 2;“ _ \ —- Ca! nx __ /\__ [ _ 50; “x _ r'" J“ it“ “new K X 2: 'E x: T X \ Ca ‘ZAT 17‘ —— ¥ (—1? Q h B “<O\(-M-F:I __ Cal n)‘ A)( a) (6 pts) Plot the polar graph of the curve C : ”r = 1 —|— sin 6. Also draw the line = 4/9 on your graph. 2. L:y b) (3 pts) Convert the equation of L to polar coordinates. L1 ”i 3 q/CK Mae-1x9 f‘S'mE’) ” Li/fi or c) (6 pts) Find the (r, 6)—coordinates of the two points of intersection on L m C. E‘YJ‘R V: \+sir\© 7' Lt/ 51919 s- 5?.«9 +sin1© 3 W/R )iL 9}?) G G Foot .9— L.) AH}: Li/q S'H‘N’Q. U ”Lu, «0,0 .1: U“ :1 __ 3 a: "y b / H 7, fl 7. 3 .— 3 (1,; ; cm, emu —‘-t :quemmrng sin (9 .r U'. rq/S I“; RMQOSS‘AQR Sid—‘4 Sine G CAN} SPA g : >3 Kb $35.53“— C '; \ \r 59KB 3 Lf/Ll q/ E) 9—. Sin"(%\, @3 9'- 'W' “"4033 2 1*: hrth 7* 3. Consider the following moving point in space: 1 P“) : (3t, V66, 562:). a.) (5 pts) Find the velocity and the speed of Phi) at the instant t : U. = cation, 53"”) b) (5 pts) What is the al‘clength of the curve given by P(t ) for 0 < t < in 5? Simplify your answer M5 \nc) mmigwi {g \Gwc: SQWE it: 3&3? it: {A “I? ZinS ~; §L3»e”9ci‘c [3‘ ,r cé'xg : 3mg +3., 4);?— t?“ E731: L 6m \ 3?)an+ 1'6— 1" EMS ATV—L . c) (5 pts) Suppose we have a function f(1:, y, z) with the property “' _ 2 d Vflw'v’geéez) — (C ,“N/EB, 5). Find g-tflPfiD at the instant when the point P(t ) passes through (3 x/ge Es 2). 1?, 3m chm “Mi LCLPUCD _ —~ ‘3 it " filming \WL my @“m Muf P ('5 at ‘53") ”W JC" “Ll—Mfiw—a.....\._.___ Cal: (set 5e“) flqzh'fi‘b'e / So 3»;wa w 3 6Q\ u. :7\ ".1. (th r¢£CIS)O (Egflfl, C1 4+ t“ (“s g —(gx3\— 69. +52?" H p "’9 2 4. a) (8 pts) Show that lim 3: 3; does not exist. (my) (00)m4+y2 e—w w W It : kM Q\ 2.: \‘mx 1. =— 35”, x (—2430 \Ein+o?‘) {7‘99 U—a“ {HAL ’11:; UMRKK‘ AEQDCAAS. cm A )C- neepadkgkocfl qkhg 4%, team 04“ mm mu ka wkuec re— \a» firm (m EH50 Mm,h \‘m‘\ Aces NUT Q44" \33 m Juno/QSMA k% 2 b) (8 pts) Show that lim 3: 1,: does exist (hiut: the limit is equal to 0). (x. mama) 2:2 +11 . :27'” we tuna" ¥o shod XML K M \‘21311— ' 0 #30 “V“ (wt—ate»). w “J“ 0"” “”‘5-3‘4‘15 sures: r: 53‘1““ EN“ 6%] *“QIQ \i‘i—rfihx 04 rmmfmg' \ r\u&®m®\ {‘9 P ““3 r—r’b J? S- M err-ah #56:; \ boNQ Kerrfi : ?\(A>Z){A:\ a. \;5)1€>Sj\:—Da§ ’L X 4 { 41m! m 05%; >3?" w 0 5 ”5:1 .. __,—J ‘1 x «1 So _.1\\ 4 3 (3‘;— w 3 J’M\ (are ‘1‘ gm. flwkttew) w .. ”11, o “\9 \‘M \‘1\:O =\N"\ “ML N‘W‘QL’S“ W Smclwmh fltfimh 5. Consider the function f(:c,y, z) = zexay. a) (6 pts) Find the gradient of fin: ,y,z) at P0(1,1,—1). b) (7 pts) Find the equation of the tangent plane to the surface fix, 1;, z) 2 —e at P0. 632 ip, LSJ— 40*.le G G 'H’AL in“ @154“; (53 $061 i€£\?a )Mml'pj CI! "" \{M !”rco~:¢__ C'-L\ Y‘k“; via-()9. fi\P°:o meow—‘13“ Q—EeBCx—Q *eGnA) + EC? H) ‘1 O Ga, 3 fl c) (6 pts) Determine the direction in which f(:1:, y, 2) increases most rapidly when the point (x, y, 2) moves away from P0. Your answer should be a unit vector. ’0” Leaky QC— mosi' Mela “mace ‘5 Jim? °Q GQVZGECFEI‘) 1; Maxi“; Mei; ck UAW MIX-Or I M 'hab. --a "v”; \p‘ _ C'Ee, -e.« a) :- Q—"SeJ-e, e D u '— __.——-— ' Wfl—‘n‘ \SM W «I? e .4 6. Given a function f(:r,y) satisfying f(1, 2) = 4, Vf a) (6 pts) Approximately how much is f(1.03,1.99)? 2:: (15L) 2 "P;(1.03,1.€161]J :2; 13} :(mfaflfioea—OM) 1,111,111 r11):£11>13+h§ re, £191 1- ‘64“ .A? l1112):(3’4)' Lufi 4-015 Li M {QM-L 51.; th 993%?! C(P") J‘— [[51 Jr Slpa A1 . K‘ir-M 60+ filth SAW. Mm: 052 41,1 8; redfin Lgfi‘hkw laulr l” lake; lefl'“. b) (7 pts) Find a direction a in which the directional derivative D~j [(1 2): 0. Your answer 11' should be a unit vector. we I -.b modal“ 1111 D39? 1% - 61 \1. ,0 : (21,11) 11 __ o kit-aid Se kit LAJGAL' 6.1.(31‘3‘31 a U)”’J(o(3bl\ ‘1“ 1'1th guyLlf—tfi) C7 (mat Cw) 111-2132111111) <3 3) km. (or -93) Nfi\" Q LJI'VAI’ \JQClI-ot‘ we. Gm normally; ll? \fiqwgyflf‘g __. J 30 3"" -m @1133 3 (‘h‘33 —, LL ‘uf- " is Cane glukmll MM :1) l S _ ............. _ _ .. . .3 _. — ---—»~--~--* A . __ u I Om 11w (1.1.11.1 13* l ”61%) .J c) (7 pts) Let S be the graph of f. In other words, 5 = {[zr,y,z) E R3 J z = f(a:,y)}. Find the equation of the tangent plane to S at the point P0(1, 2 ,4) E .5' (Be careful.) 1‘4 elm \ch‘L '15 J‘b M‘h MM Sal‘s l|\\a Q \Luok $631— (no‘l~¢13m?l1luj 1:151“:qu KM; anlpwx Stylmiu :_ £031], [TlNlA S 1.: “AL w Ecfi‘ilaj \ 3(x,1:%):—C3‘§J («Cy’fi—IX : (”Effi'rl) Um?)3(l,lfl] L =(3,‘1.'D‘Q“‘V¢létm);{1“)@l’lll Sr we we“ mnem— J— (EH—‘t 4) - ‘ J iiLtiE? ("”3 11116111; ’QM Lflwfibnis )‘3x 441:] :1; fl ___________________ l ”(533: PM" use—MMwae‘m n 5921/ %*3E5—($-(_1)14~1 LE ...
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