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Fall 2006 - Hall's Class - Practice Exam 2

# Fall 2006 - Hall's Class - Practice Exam 2 - 1 Let D be the...

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Unformatted text preview: 1. Let D be the region deﬁned by D={(z,y):231+2ys4.0§\$7y53). (b) (8 points) Find a rectangular region D' = [(2,2)] x [c, d] and a linear transformation T such than D = T(D‘)r v u:¥*zj a u+lv=3X 3 vzx—‘j X: “1,3,, 3 62 [2,11% Y0151 (c) (8 points) Compute ffD (Ii—ghdzdy by making a change of varir ables. .‘ .7: ‘ _’I 30‘) = )— “(iii— 3 =» 1m 9 } '7 I4 ‘1 2 v _ or“: _| L2: 3 5130 V ‘50“, § 1; (ZuL)D&V _! ‘1: 3 = %§:ﬁ1d“:%(u)r g Pngc 2 2. Let T : 1W — 1R2 he the transformation T(u,v) = (u+2v,2u~uz). (a) (10 points) Draw and label the image of the rectangle D‘ = [0, 2] X [0,1] under T. V ("9 L.) \ @@ @t L“ (9.- (+,o),oe~€£—z ran-ILHIOeteL @ ( (Mu) : 2,4. , ostH - 1 - ) —-) (2am t 3,021“ «“(a‘m7wﬂ:z> @3 (in) ,obléL -—-; (++Z,2£—I)Iol;téL ‘5 “POLE-‘3 (ha) | u ®= (o,t),osket » {ﬁfth/0261:: z (“Ml W” ) 3:41,? (1)) (10 points) Is T one~to~one on D"? Explain. sane» TOM)": :— 1(uL,Vpl 9° ‘MZV: “Aviva SHE—w. Lulu-m 2u\'V.L =Zu2—VL" . 7 3206 “Ms “4‘7 VF": - ' a. So v\=v1_ laud ul— Y{\$. "PK {\MLl'u» +fﬂl: ’X‘q vc surku hank f- \s uol Vb" 0" R I bul- l = , we: unf. ulaz//‘ Hf (S ‘_%V‘ m loin ' — ML LA“ 3° Page 5 3. Let 51(t) and am be the paths CW) = (cos(t).sin(t)).03tsvr, 52“) = (cos(t),7s'm(t)),0 51,5 7r. (:1) (5 points) Draw graphs of (710) and 52(t). Be sure to label Lhe (b) (5 points) Show than the vccmr ﬁeld F(w,y) = (33%;, ﬂ) is a gradient vector ﬁeld - # may 9x )3 ’ (guy-)L Q'uqz)‘ f 1: XI' 1 L _ X 1" j l '— o “my” , 04 *3le ("‘*7‘)1 “fl ‘ 3:: E 0 4 mold-d Vaclav (c) (5 points) Compute #13115 and F— I ds‘ directly (by using the line integral formula). _ "(T/t) = (-Sln+,(0'7“‘) a7“ - (’SW‘I'C‘S‘t) .. _. 1r S‘Z' E ml; = §W(\$M§, dost) ’ (-943, (east )Jf ga’F '0‘ 5 = 306947455" )' (‘5‘;‘3 ‘ a 05.45 I- wk) - (—Suk’ #951 = \$5)“ :Eﬂ 5:0 Wu»: ' EVA (d) (5 points) Why doesn't your answer to part (c) contradict the damental Thevrem of Line '.7 Tu wax luau. lam = w' ()5) :3 ml A»);ch ’QD/ 3’30, whbk ‘5 a} “La Meolu") 06¢“me , §D 4H1. MW dos «cl aRI-7 . {I} \.\.plwl, “cum/k) 4. Let S be the part of the cone 2 = 3 I2 + if that lies below the plane 2 : 6. (a) (7 points) Find a parametrization of S. (b) (7 points) Draw a picture of S, showing the grid lines for your paramctr ization. i=9'7 ,Rx r ad. data} (M “44 W i‘)‘ 9 %' ink:- —?.=§r \vl an”) 6 points) Sigposc S is oriented so that the “inside” of the cone is the part that would hold water. Is you parametrization orientation: preserving or orientation reversing? Explain. i, ’Ymmb u‘, 7T; 70:49 r’g‘iftwloolmhvsk 90 L7 “all” M (Via. [2&7} KT“; VD“? ‘ib'umls Wukdz aha)“ - g OWiR’in-s P014193 O .. .. g 14(40ka7, a»qu TrxTD : Q (twelﬁq'e/g) x (-m.tiér(osa,a) .: (-3m591‘5r5m9’ f) Lot/WA (mm) 1'» (M U?) 5. Let S be the surface of the unit cube [0,1] x [0,1] >< [0,1]. Let F(r,y.z) = (I, ZyVO)‘ (a) (4 points) Explain Why the surface integrals of F over the bottom (2 = (J) and top (2 : 1) of the cube are both zero. :‘vl, L‘A- F TLLL “M3,. 4m\ vulvrs m “ l- \«us We v~ coapm - 90 (5.: :o m bah £0549; WA SfE-R‘JSW) Slagz C;‘ (5“ LbHovk, g7, "s (b) (11 points) Compute the surface integrals of I? over the left side (y = 0) and right side (y = l) of the cube (2+ 9} \,4 M bﬁ— doll-{QM 9‘4 4“ . _ % OEXe', 092’ 9; . @- (X/e) — (X/o/ )1 (DI-U0) v0!” ‘ 1 ﬁle = u,o,o)xzom = P \$83.41 = S“ (x,o,o"(°/-'I°)""‘°‘* 1E S; o o ruﬁhfS-Vb - L L el/o,%/\ : ,2 : Cxl'I‘l’) ’0éx g“ ) LAM :l—yX-Fa = (0"llol 'POMK ’ SS” Edi; ;'S‘0‘(o\(¥,2,o)-(0,-I/o)clxol% 5% : _§\§0\(;1)AYA£ :E] D my.» a (#5 continued) (0) (4 points) Compute the surface imegrals of 15 over the back (I = 0) and fron (\$21) of he Cube. _ m %; ML mic/M Sb l“ l“ l“ ££b\ \$51 Q—(yptl : (0,7,%),Déj£\, 0 “A N g - (o V o) x(a D \) : (“0,0) (30ml) muo — I, / / A l \ :- S‘S‘E.Dl§=—Sa§b(0,27,0)-(|/O,0)A7e 101 s; 09 5‘, obZél Saga/7,2): (we), 3 J ,n = 0,00) 179' T” \e ’ .(l/D’o)d7JQ:§D I \ J gm SC §;§;(\,2’Io) go"? —2 (e) (/1 points) Suppose F is the velomty ﬁeld of a liquid, What does the sign of your answer imply about the flow of the liquid relative to ihe cube? Page 7 ...
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