Math2015_T3_Solns3

# Math2015_T3_Solns3 - 23’ Writs Math 2015 Applied...

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Unformatted text preview: 23’ Writs -. Math 2015 Applied Multivariate and Vector Calculus: Test3 Friday October 30, 2009 10:30am to 11:20am Name: SOL-OT [Odﬁ Student Number: Instructions: Complete all 5 of the following problems in the space provided. Notes and calculators are not permitted. All cell phones and pagers are to be turned off. 1. Find all relative maxima, relative minima and saddle points for S m elf k3 . f(:l:,y)=:l:3+y2—6:c2+y—1. 4mm“; 3%(x’43\$+k2»3+133‘ 2§ Jr a? "—6) Saw, O‘v‘3\: (aux-ll lit: (”‘33: 2 13:13 bum SOR® Crnkwo, Pow/ﬁg. Viewing” :0 1 lel'bﬂrﬁ ’Lvs‘l'l =0 (1“th PQ‘MlVS “"1 (40>’H7/) Mi lLl,"l/7x> £® 7/ Swmwpuﬂ =47, Duh-In}: 9300;33- 4% : [-11)[Z)*O;Z‘l ) Smw blarilmé 0 M5 is a Skbpii Vom’ 1CD \$.31.» lag-417,.) :- (”Lily-ll 317» Dl4:-'M=L(vm-n 1m 2 + L4 5mm, DLLl.‘lI’L7>O 'lwg IS 0|. Loom, Mud. {CD 1 2. Find the maximum value of f(:c, y) = my — 3333/2 over the square 0 g :L' g 1, 0 g y g 1. am: awe mm. M ¥xﬂL (Bl-[‘33 : —b1‘§?’ g‘Ss-(Dldﬁ) ; - 11/ ‘ VHmwp > 0 =3 ‘MI’SDC‘Sﬁ :0 we) J,U*’Lw7'gg=o “(2’) Egn (D lmQQA‘es 33:0 0r Ex’3:\ (h, data, 3+0) 'TWZ, OMYA \$oDM/“(W ks . C2) 13 «2011M u,=v 30 ”mm, M olovmm 1W, (mil/3 (\$ka P9W+ (‘5 “9,03. Dirac) = Saw, “9’03 ¥5ﬁl0,o‘) -— ‘YlsLonyj : 0-“) z ‘] U310) ‘6 a SADDLE YDNT \$0 M MN Moﬁ’I U) (he, On M bowvwlma. A W“ ~ Ahowb EDWDKM‘Q 9’W0 .9, MA) E 0\ C LDL=I3 ROM”: 0 WA \$(bh03 5’0 (330’) 3: (3’0) NM? ¥‘L 1 0‘8)— \ —'L ‘3 0 :3 *3 3 L 7, ¥(\ 1%.): ‘0 is m .917‘ s m LL ¥U "3 \ Is 4 PM «by 52% (AM «MIL ’ 0L0» n 5 Mar lLé; . 3. Use the method of Lagrange multipliers to ﬁnd the maximum and minimum values of the function f(\$,y)=\$—IIJ2—y2 above the circle 51:2 + y2 = 4. In your work, show the critical points and the associated Lagrange multipliers. 361‘, amp: 11+:d'1—4‘ W1. MUS'l Sal—u. ”TN, 660.3.th V? = \V% Q” = O 1.333% load V9983) = (“1:153 “233‘ ;V%°’L«ol‘ lad/W233: g® Ego (Q adiwo )5 - 0R %:0. using )\:—l m 55h ' Feswl'ls in l = 0 which LMPUM am (HMMS‘S‘lWAHO, it X 4" is 323i 5; SOJM’l my :1) TIMI 939% (D -’)® TM we, “lake, ﬁso 0W\& \$013 M3 in (3) ho W , 80:: ”El, V00, MS hWL CHEN/‘0, Ponds (210} Mi L'Z‘O) '————-> ZS (D Z 4 2 Z, X s XL; "‘ ”ﬂail/4L V k on SW M WQAAKD/vvwe- 7, N3 K s s 1 7w 1 5L ‘3 CD (a . l‘g 5 1mm lL’f? ‘ 4. Find the volume of the solid above the rectangle R and below the surface 2 = \$211, Where R={(m,y):0§x<2,0<y<l}. V: K 2ch ~——~—_® K , H M 8N (3/5 c)"- P— J g, f‘ 'L \ s: 7. 11 d» Somll 0 web -17' “L 20 | 21—131.), Lo 0 cf? 0 Q) =3 3 5 WW [Us . 5. Compute the ﬁrst and second order Taylor polynomials of the function f(:r, y) = sin(:r + y) about the point :r = 0, y = 7r/2. ([0lean Pmlva— ﬁlhxmlwag; at ‘ DDS L1¥5§ =3 Clan. (.0, Th7.) .: (’03 k‘W/L) .>_ O l g»), 2 (JOSL'DLMAW :3 YRG (/0) “17,» _> Log CRT/Z) _: 0 O \$>DL 3 "' Shkbu Val) =3 \$504. L0,“)1) : ~ swat/139‘] \ ¥vAk3 : —\$H;1L>L*l’bt) :3 P33(0,T[/L): — SM (IT/7x)="l' ® \$30) : -—-smt><cw§ => its lent/z»: — moms—r R L105) = SL01W11>+ \$3L(0)“)27(DL~ (35 +¥3 lb,1t;2)[\3—1EZ> c \ ——@ ?.L Laws) : P, [31,63 .\. PML [,0,1(]1)[J.—03'v _l_ 7. 'l SﬂAl/Oxm7/§ L3“ ﬁlL>L l2§at5l0)“/L)(DL‘03M~D)\5 L—(l) _ Z _. Ac. («5/11 > :\~€'%WE> 'L ...
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