3361_soln_quiz2_F2009

3361_soln_quiz2_F2009 - ENGM 3361 Fall 2009 Engineering...

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Unformatted text preview: ENGM 3361 Fall 2009 Engineering Mathematics IV_C Quiz 2 Full Name (Print): . ID: B00 < oi manly/m Signature: Time: 3:05pm — 4:55 pm Date: Tuesday October 6th, 2009 Room: Chapter House 107 Prof: Edward Yao INSTRUCTIONS _ _._- “’1’."‘SHGW'all stepmhareva‘ni‘s‘ applicable. ”"7" 7 A i W" A 2. Use reverse sides as scrap paper. 3. All features of sketches must be clearly justified. 4. Keen y0___a- tid. 1) No questions will be answered on any aspect of the quiz. 2) If a student requests to Use a washroom, an invigilator has to accompany the student as he/ she leaves the exam room. 3) All Dal student IDs will be strictly checked. 4) Every student must sign on the front page of her/his paper. 5) A student must obtain permission to leave his/her seat. 6) No communication or exchange of information of any kind between students. 7) No scrap paper, calculators, class notes, textbooks, or assignments' solutions will be allowed. 8) No students are allowed to have cell phones or any other communication device with them, even if they don't use them, and even if they are switched off. 9) Possession of a prohibited item is considered cheating, even if use of the item cannot be proved. 10) All bags must be left at the front of the room. ll) If cheating is suspected, the student’s paper and any suspect material will be confiscated, a notation made on the paper, and a new paper provided for the remainder of the examination. No exceptions will be made to the rules for any reason. Any student who does not conform will be ejected from the exam. Questions Part I _ 2 F fimomdfl L Part II 4 A Manda Part III I 7 S :5 m T, Part IV 2 'm Total 15 A Mp,” dot Part1 [2 points] Ql Find the equation of an osculating circle to a smooth curve C (traced by a smooth function 7(5)) at time t1 . Hints: 1) Use the figure. ' the osculating circle at time [1 x E (t) 2) Let EO (Q) be the center of the osculating circle to the curve C at time [1. 3) Let E (t) be any position vector that traces out the osculating circle that lies on the osculating plane determined by the unit tangent vector T(tl) and the normal vector N ([1) at the time II. The equation of the plane is given by Hal) - (E(t) — E0(tl)) = 0. Sol.” Th1 OSCW‘AH‘V‘j Cl‘Ycle l/ms +l/H’ Same curt/alum fig!) as “H’lf’ CHYl/e C 0h" til/we ii. The YMciI‘HS V€c+or I‘m‘+—u'a«I—.'nj a+ cA ’Poi‘yw ( 5:17) and firmi'naJ—viaj 61+ ‘HM Lem-Lew of +14? OSCuIa-hinj CI'YC/e I3 fill/e by Wm?” FYOm TYl‘CAV‘file LOU/O, we have. ._.x _ ’—> ,1— _ rec.) + [Haul/VH0 E°‘*')VV0.LS So 76.9 99491519” of “UM oSCu/aH/ij Cl'rc/e is fll‘vem 105 H ECt)—— ( m.) * ma.) (+0)“ l<'(+,n) Bl?!) ' FEES ~ E,(+.))=o IJE Student—8 didn’t wilpg "tLLQ geogvml ew’am’ clot/1’1— d-QJAMM Ole \DOCMS; If— S-l-uolem-S . A30l [4498+ Povp‘. Y'I\€)l’)+ th‘ve “new 30/? “A - I Part II [4 points, V2 each.] Q2 1. In class we discussed “arc length so far” function s(t). Did you remember its properties? Please give two properties of s(t). Property 1: s’(t) = Y it) i NL‘MQ YH") 1'5 0‘ 31440014 jcw’IC/Iviom' Property 2: s(t) IS an {V1 (NO/(Aging If S‘h/kd-QM‘K WM—Q‘ V(t) and a(t)? Here V(t)— — ”9%) L 61%) POW function of t L9H Hmuew‘s k” 02$ " Su/C. ‘H‘FLM 01's. £01- 991) :3“); 2. If \“V(t)| constant, then what kind of geometric relation 1s in between r'(t), a(t)— ~ ?"(t), and?” ”(t) is continuous. 3. Suppose ?(t) is a smooth function. Give the definitions for the following vectors. Unit tangent vector: fit) = Normal vector: 1V (t) = @— ?“ 0.15 WW 4. Sketch a smooth function fit) and its smooth curve C along with PU), 1V (t) and E (t) at time t in the following Cartesian system. (sketch all necessary features) TFlt) z A ix NH) fie 0 x 5. Give the decomposition formula of acceleration €1(t) o 75 , 0.1? _ - do) = _ U T(t) + I€U____ 6. Is the curvature K:- -:—” 7;“? (Answe1 Yes or No ) 7. lf 2 = f(x, y) has continuous second partial derivatives, then II II 0‘; ny E,»— Zyx 8. A70?) X Ht) 2 §(t) is wrong somewhere. Correct it below. ._5 A A T66) X Nit): B (t) O? Kiw) xérttk - lift) Ea) N(t) U‘ “Wall No 1141‘s “Egg“,SVHM m c \MS-fg Paramd-e 015‘ oY V“) [‘5 PMPQMI‘CHlar'hJ—Ztt). 1 W, Part III [7 points, 3.5 each] ‘ Q3 1. Letw=f(x,y,z):x—y—22+x2y+ey‘,x=x(s,t)=§+st, l y = y(t) = lnt , and Z -= 2(t) :rcost (i) Find the composite function w(s, t) ofindependent variables 5 and t. WtS,’c)=y£[1(§/t), aft), 2&9] 2 (whose S i =(,§.+Sf)-]nt-gw§{.+ (15.44%)th +€ ‘7 1 W E, (ii) Draw a tree diagram of the composite function w(s, t) in (i). '3 \ 5 t OY' 9’ >Z % / X’H—fli .g W ——>>‘O——~»t fig \Z—t . (iii) Based on the tree diagram in (ii), find %. [DO NOT SIMPLIFY] V: 7:. M 3.x— + M“ (loll; + M 6%; Y 3X H’ M i' 37‘ a L z _ -3 +5) + (O—l— o+x2+ Zfly )-t+(o~o—J+o+ye )GW :3 :(VO‘OflxflmM/fi YZ: Jr Y‘ts't) L” =Q+4X1)(’/J§i+$)+ivl+xl+2‘£ )'f+(”‘l+\/e ) M 1“" +)(OS:]IJE + 7' Um [1+ a(%+se)|mt](%+s)+[—I+ ( 75545!) +£9ch 6 H .. Int) (oSt , ' t 2° P 3' , ‘ g (iv) Find the angle in radian between a tangent line with slope 2—? 0 1 ‘ ‘< s: ,t: O— and the positive direction of r—axis ' o! = tan?” 3;? §=o t=,)=176':{0 +[—l—to+(o$1]+['olJC‘S.LI)) =ta2'(~'+“’"“0”"”1) ‘95)“ Q3 2. On Friday October 2'“, we introduced two “opposite” methods to find the extreme of Z = x/ 2 —— x2 — y2 subject to x + y = 1 and the only constrained extreme found is %. Now use the Lagrange Multiplier method to find the extreme ofz = 3 — x2 — y2 subject to x + y = —2. [An appropriate sketch of the problem is required (1 pt).] - Z Soln FYOM Lagrange MulHlolI'e/Y' 1*: 3—23—31— MU‘jN) I P6- /' lQA- Z*x=-;x—)=o ——u) */ _ *’ +3_: -—(?l ‘ ZA= “‘1 {- maxmm l? ((MHYW‘WCA) a Pluglnj (4—)cmd(§) “4+0 ‘3) —;\_-.A +2=O ”/23?" 1 7. “,2 x=~A=-3—='l (x 1+ Haws-drew z a. \ ‘ - —A c- a 1" h+l¢m1$gsmkr ‘1=-“;' 1 ' 30“" 1» Ha QLM’gwe um 1 pence, (,lecpA (Jolt—wt L 4,4, 02) lg obi/3mg A total MMI‘WIH AHA also A global Wtfi/Xt‘wq. W60! 38 Part IV [2 points] Q4 (i) Find the missing info below. Implicit Function Theorem (multivariable case) Let a system (1) of two equations f (x, y, u, v) = 0 and g(x, y, u, v) = 0 be satisfied by the real number x0, yo, no, and v0. Suppose that both functions f (x, y, u, v) and g(x, y, u, U) have continuous first partial derivatives in the same neighborhood of( x0, yo, uo, 170) with «9.7g 5”? 9152i w av i 0, 3,1 13 i“ 9’“ mowioluowowaw Then the system (i) uniquely implies functions u(x, y) and v(x, y) in the same neighborhood N of ( x0, yo) such that u(x0, yo) = uo and 17(x0, yo) = 120, where u(x, y) and v(x, y) have continuous first partial derivatives in N.[1pt] (ii) Consider the familiar change of variables from the Cartesian x, y coordinates to the polar coordinates r, 9, or, equivalently, x = rcos (9, y = rsinO f(x,y,r,9) = x — rcosO = 0, g(x,y,r,9) = y — rsinO = 0. Do these relations define implicit functions r(x, y) and 9(x, y) throughout some neighborhood of any given point P on the Cartesian plane? [lpt] gels/i “‘1 0.25 /%kill air H) "LQ’SQ YS‘LLO . g; 33 = ' = (~Mwli~mswl "“ 3; ’37» , scan ~YwSL9 (-g‘wursfnw) 7— 7. = T (050 +S~LLO : r (Oi-7&1”. 1‘s nonZero WQYUqu-er-e exwpt at ’the origi‘m (r=0). Freya—tin IMJL‘LH 113mm». WW [Ad/H raluhbns Do (MAG/wt (MAN/“i— ’(Mv‘fzh‘sm rtxsq) Ami (QCXHj) “‘4 Same Mghbmlnwl of any @I'w/n perm {0 on 7A4 (LC/«lamb». plan/1e, wue/‘n F} P (I? (at Hue Driaxin. In #14 Case 0% P {s m, Hm COYiijI'ml‘tqu, Imph‘cil— Fuch—Jav. Theorem eiodls . Tht‘s is no+ SMYPh‘Si‘wg Scha a’r +l~c OHSM ((‘20 ), (9 {3 140+ al-efi'mod WM; {Ah} (‘8 , (9' (Vet/(lat [9—2, OWH/ DUOSIQ, (IV) radx'c‘n) amwg 350° (an). Xx ...
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