{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

finalfall2004

# finalfall2004 - Mathematics 192 Fall 2004 Final exam...

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

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document
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:»[email protected]) . / 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

{[ snackBarMessage ]}

### 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
Ask a homework question - tutors are online