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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 LemLew of +14? OSCuIahinj 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—
dQJAMM Ole \DOCMS; If— SluolemS . 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/kdQM‘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 \MSfg Paramde 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)]ntgw§{.+ (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$1]+['olJC‘S.LI)) =ta2'(~'+“’"“0”"”1) ‘95)“ Q3 2. On Friday October 2'“, we introduced two “opposite” methods to ﬁnd the extreme of Z = x/ 2 —— x2 — y2 subject to x + y = 1 and the only constrained extreme found is %. Now use the Lagrange Multiplier method to ﬁnd the extreme ofz = 3 — x2 — y2 subject to x + y = —2. [An appropriate sketch of the problem is required (1 pt).] 
Z Soln FYOM Lagrange MulHlolI'e/Y'
1*: 3—23—31— MU‘jN) I P6 /'
lQA Z*x=;x—)=o ——u)
*/ _
*’ +3_: —(?l ‘
ZA= “‘1 { maxmm
l? ((MHYW‘WCA) a
Pluglnj (4—)cmd(§) “4+0 ‘3)
—;\_.A +2=O ”/23?"
1 7. “,2
x=~A=3—='l (x 1+ Hawsdrew
z a. \ ‘
 —A c a 1" h+l¢m1$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 (Oi7&1”. 1‘s nonZero WQYUquere 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+ alefi'mod WM; {Ah} (‘8 , (9' (Vet/(lat [9—2, OWH/ DUOSIQ, (IV) radx'c‘n)
amwg 350° (an). Xx ...
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