finalfall2004 - Mathematics 192 Fall 2004 Final exam...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Mathematics 192 Fall 2004 Final exam Thursday December 9, 2004. 9:00 - 11:30 AM No calculators. An 8.5 X 11 in. sheet of paper, with information on both sides, is allowed. Show all of your work. 1. Find the following limits, or Show that they don’t exist: . . tan(cc + y) a 5 omts 11m ————-———, ) ( p ) (m,y)-*(0,0) 2 ~ :2 + y (L‘ . . ’ y b 5 ornts hm , ) ( p ) (ND—+0.1) SC - y . , 6xy2 c 5 omts hm ‘ . H p ) (z,y)-+<o,o)x2+y4 2. (15 points) Find equations for the planes that contain the line L: a: = t, y = t, z = 0 and make an angle of 7r/ 3 with the plane y = 0. 3. (15 points) Find the absolute maximum and minimum values of flay) = x3 - 393?; + 3/3 on the region bounded by a: = 0, y = 0, a: + y = 4. 4. a) (10 points) Sketch the circle 7" := 2 and the cardioid 7" = 2(1 + cos 0), and find the point(s) of intersection. b) (10 points) Find the area outside the circle 7‘ = 2 and inside the cardioid r = 2(1+cos 0). 5. (15 points) Find the volume of the solid bounded below by the sphere p = 3 and above by the cone 45 = 27r/3. 6. a) (10 points) Let F = 2my3i + (3.732112 + zeyz)j + ye:ka be a vector field. Show that F is conservative. Find a potential function for F. b) (10 points) Compute the line integral fCF - dr, where F = 2:1:i + 5$yj + 2k and C: (t,t2,t3), 0 g t _<_ 2. 7. (15 points) Let F = (33:2 + x — 3my2)i + (y3 - 6$y)j. Compute the outward flux of F across the curve C, where C’ is the boundary of the region T : 0 S 3:, 0 S y, cc + 2y 3 4. Please turn over 1 2 8. (15 points) Find the outward (away from the origin) flux of the vector field F = yi—rcj+k across the surface 5' cut from the sphere x2 + y2 + Z2 = 1 by the first octant (a: Z 0, y _>_ 0, 2 Z 0). 9. a) (10 points) Show that the value of the outward flux of F = (my — 2:2)i + yzj + z(1—— y)k across any sphere equals the volume of the sphere. ( Hint: Use the Divergence Theorem.) b) (10 points) Let 5'1 be the upper hemisphere :02 + y2 + 22 = 4, 2 2 O, and let 82 be the portion of the cone 2 = —2 + V272 + 1/2 that lies beneath the my~plane. Let n1 and 112 be the unit normal vectors to S; and 5'2 respectively, directed outward (away from the origin). Let F = (3:2y3)i + j + zk be a vector field. Find the sum of the two curl integrals 112/] (VXF)-n1da and 122/ (VXF)-n2d0. 51 S2 (Hint: You can use either Stokes’s theorem or the Divergence Theorem.) TE a? 3% NJ Am YET 2x137- WM WWW m 56> We. cmx mafia i EM m: 9”“ Q‘zwmmwl . Jim ‘23 < ~ M V Qfli‘aepa) M X?»- 3 :1 C K23? 72+ X?) \ 673%?) {-3 WW i g @ Z M W ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, @9*®6) f“%%fi"*fi> AkmE X=ngufiv fl fiQm {EEJE,=Qmw§ifz3 ‘1‘)?0 1‘76 ‘H Li ’ §Cg #3’3 1 1—1 So A615 SGT QXLZT. 7 T 7 \Dlgmzs meWaénm-B/Tm hag L: x: {Bf—£53120, v a K { ‘ / fl ‘ I- _ , ‘63. “452 Sigwéa moi? M waw W“o7 *C‘i‘c‘ 31}? HQ :me Cgmiihong . m cm Ekoéflwéfii‘ T: we} is wig, 3-} “i 5 To my, . ~ «aim in a T262 wmfi ‘“ W‘W‘ W“ § ‘ ¢ § . ’4'C«Z‘::&~’*}7 nmmvl '“>- «$1007 5 K ’ “DR-MW We? 5’ J ’ ‘ :6 M43» {amid 5"" = (WHO? K flaw l "—7 w”) 7' L) w e. (\1 a {\2 v z 0.51 3:, ‘ wCLC'zs , > 0‘ m“: H‘s»? Tha‘cmz L3 .. Tf gal : L s is .. ... Coch: @“CESBEL ‘1’” C9 > *l § B K 3) 2— K 2g. £20 aWiA 936% CW {AwNy f ! ax+lay+c&:3 a» , A Tm; I 3 mg gm‘xg be True, kw ~ km, W \ =3 an; C33» I M H832 w 2‘35: £31: Wig: 2:] 2 4913:]: BWX g“) 2 GK =6 >0 Pi is a, mimm . {Dz 2 WE; “SS-ESE. SIP2:\: >43”. Zi~3+1:~\~ v/“"\ x:7‘:‘ x i W (TBAng \MQ, Canasta m EEK—{WEEK m m ‘ a We Cloflfi fin 3 ;. 23 W" “ ,2 ‘ V 3 ‘ i é 954$: eém «r» 7f: exam-2:5; £4 6 cm oskgé‘; :9, L; {:33 Sb GER M: I OéfiéL} £3 3:, E3» 3CL‘-E\E _i_ (umEXB 3:73.me 2» gm : 3&2 w.— 3Ctfl=3 + 3%; + géw-E‘V‘Cwq 319* + 13%; M ism-4% 4% 3 (PH; :1 Wig/a) 3L1+3k~12+g g :5 (1634512533; f» "1 Qiré-32maggwfl£z+ 14E 3 3943-530 1 i :> <5me dig? 19:2. i I 5 a ? i g E i ? E H . i _- L“ “a "m '3 a 566ch C3 (t)l2— 30>?) I Ewg ca. [06611 min grfitfi Weir 3(2): 8~3ga+ 23: $~12¢$z+lfl? Q L TLQdMQfiChf’oleb/ have : 3H3}: [+33 fig; 30*): ngé” $3 (39 “EC; “Wm, mm \ICEziEflQ {‘3 q mfi'Tm' max ES i’ f K _ A _ MCLX I .5 q ‘ «3 “*~\‘__ Nag; we; m 15% 69 l s.‘ ,i- 51, g Ynm mmg O may Vfiifié (El/i 1% 3 ‘37 Com migw , we sews: 1? (:11: TM mm vfima :3 (31*: (Mg m m’m wake Rs «4, % Wm M j b m min/max fmaz me (figQgegg r. F: x3” xy+ 73 : iJi> “’5 BMUQM X+yiwifzo n ‘35 3’8 “tax a ' 3’“? 3‘7 3&2—72‘)‘: 3CX~73 «=3 3Cx~7¥x+73256x~3 we, Maw :1 wa‘fiibflmeg ) ’ a M10 27> *37 “17:4 :> 7:2 (swim) z> W293”. *- ~—— -— ~ 1 ' {mi 1%. min 61>sz- 1' ii- w “A; we :9 24:... ....> 4,0 imffié‘bb Li »/ M 6?.6L tTKCg g : i; P/chg 3g“. W max Cm Bcggq‘ @ 63% CflT€f%,flQi\, WM WWAWMWWM 3 Q 1”: 2(l+a:»s@) . / iii—K32; (it; cm 2 i 1)? 3 (@221 mgzm rn 7mm :2 a, ” gmw‘mwmmmymwflwflwhMWMMWWWWWW mm W MMAWanm “ ' " “WW M E E i «’~ L. AW )3 j». g X: f‘ 8059 Fab}, y r» (S‘zflg (So Weir {AA 3. r aiming) Mums; a ‘ 3"" 9E2 9' m (Tm R KS .RQEI; age K‘s :23; a“ 2 fi/z 3mg “[2 =15 a 47/5 L rqiralg"; W X: Sélfiufi} C936} we anfgs; ysfséocb 5mg (ij $G=§2 Zs§6zzsd3 M Wm R augmw Ly m. l q.» f ‘ \ €53 : ’ “W ‘WU i. E) 33 (czfigiffiflve “W52. “x - t W ‘9‘ ” Z 23’153 (B VD} ‘g 1 a,” N f @ $2 3 o .3 5 815% I 2 ‘ § lhftrgrofg? “59"”: X 22> z: 2 ‘54» $0313) : $593+ (353%) DEM—Y‘wé R: UG‘QT. :3 : 3X232% COWQGES WEE-fix @ 3:» CEbUME) :: 3 @932 s; ' -~ ‘ 9' a 3 ww— § a _ 33‘. 16%?ng (E3 W5“ 3 =9) ficfiia 32 a &3 +44%) » a + (~29 . {3% pk:ch % 5h M, g6? 4&1 :7 $2: X2133 + 63 +\‘5(%) D€R€r€5$®j€ “’5‘: 1?: =5 $2: 532+ VCa) $t§i€€Tt * 3‘1” +CL WM” CI. (Egg?) C’ékél. + Swag}. + 2 E 036‘ “ ’ / ’ ._ x: 3: .x Ia; .. a» WWWWW , r7M7iy=EZ 2:» cl?” «a? 22:63; 2-: Flak: {axist3j2>a<x,2£,37>}ch ® Zzt3 . ; (2x+i0~,<51: +32%?)3T;(1£+10£u+3t§3$f ( Pius in 171,477.— "WTfizéchx g A2? , u gs» M ,2 S. g “‘2- =J (2£+10{+35 )chz £+Lg£+3g - C a g g D ’[4 5" 2' is + 2Q] 3 1-; 4 3? f: 2013, "MW/WWW; , M ~ Ci) 5: (5x2 +x-is 3x53) + (jgngjm) W pa) \ _ mm C; Lama Comgfiafi’mw gmin amg "my, 2% 2, v q a is Th1. ‘bcmc‘imB 19%):{29‘1 Déxj 835} )1” A w T” w an ‘ x If?“ A 33¢ (33%) U K \ 1 F7 :5 CL QAOWCRQ Cf ma T; 3 ya { QélcijC is (E; clo%€& C23%*0€. m The. 393 New: WW5: gab???“ 3’ B Cfcefifigg Wm 2L;”“\W‘ w“ ....»w .’ A {A 2, Z, 2 J) 337" 4:942 GT& Thais 2:2 W. WW WWMWWWWWWN wwwwwWWW, H mm f? a (53%.:33; 4» L323}; + 2&3»ng . amagg amt: figmr M Wiumz 39m, X?,+>tg+zzfl» X M 1&3 be m, WT ‘- - “flare; . a," I a? ‘m (@fifiz Z:-Z+gj—m 'ng’ has wwm Wx~y~§16aw a Y Lu“ if? 126: WT mea‘ “(ELSE S‘M S2 "Tu; cam/Caged), wTwam’x (awaj i M i : (X233); +3+2~£ , Fst W, m\ “Rue Carl :fiizcgfafis L : Cfik?)¢ K? A}? ...
View Full Document

This note was uploaded on 06/01/2008 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).

Page1 / 11

finalfall2004 - Mathematics 192 Fall 2004 Final exam...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online