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# MATHT2sols - 71 1(20 points 1(6 State with hypotheses the...

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Unformatted text preview: 71% 1 (20 points) 1. (6) State, with hypotheses, the second derivative test. Tl 5;an YEW? we M “’l 105;) Mi: {m 0x wa d'f‘f; 0,”) (QIL) jf 5. {MIL fa ,a M H: p” €ij Tl 1+>0 a“, (Into m f £3 JFJFX>0 mum at] l", H CO 9.:ng “:0 ,choeL'f. 2. (7) For f(x,y) = 631(3)2 — 3:2), ﬁnd fx,fy,fm,f1y,fyy and all critical points . ‘lxi {[4 ’2x) 1% —_ 2‘151' +7323“ ‘3 z y (3941 '15:} lC‘H ~.—_ gag/'1 +253 4 21.1824 3;, 1C 2er _ @WJW )8 ya '1 , _ ~ 'L a? so ’2) w «we (a W7) WW OW+ Pkg (0/ 0) 7—. 0, ’2, 2 (20 points) 1. (8) Draw level curves for ﬂat, y) = 163:2 — y2 for k = —1,0, 1, carefully labelling each branch and intercept. v\ 2. (12) Use Lagrange multipliers to ﬁnd the maximal value of f(m,y) subject to the constraint y — I2 = ~2. Show work. J ,_ —_ 0( ﬁl, \Vj 73/ 31K: ‘7X)‘l ») X ’ g "L —J—"“"'""_4 )C'lla “u Yet—ﬂ?)— —— —- M \ ’23 2 >‘ \ 91: (1611100 amt. “F“! q , 3 VJ— &Q\ .25) : 'XC—lla L1 ’ \$11: —2. ”X473 M406 . \X 1 ﬂ lb(\1+2)‘l4 ‘ x l 1 lbﬂfgl‘ﬁ/L‘Rj) - — ”f 150/75,?) Ho ’23; ; 37% / :11 M44 CIA“ 1°”: ’1 A. @W \/ 3 (“Oath/NC ”fl" 0L2~l.]-max. 3 (20 points) 1. (6) For the region 22 S x2 + y2 + 22 g 42, 2 S y S 3, sketch a slice for .. .. W some y between 2 and 3. - WHOM SW] EM If] 2. (8) Set up agﬁntegra] to ﬁnd the volume ef'the region I“? I. [6 i5), / 3M (2% -/6+3 3. (6) Compute the volume. _ 6 b 5'41 ”L T A W : ”Zia .77r 34F H 4 (20 points) 1. (6) For a region R 1n three dimensions write down formulas for center of mass ﬁrst. 111 terms of m, My: 1W“,M1y, and then in terms of integrals. m = «gov 5” = M734“: “(*ng 2 = mfg/w " QQP 230W 2. (8) For the region in Problem 3 Fwith 6(33, y,z z)-— — 3/, write down appro— priate limits and integrals to compute center of mass. (if q '_ I“ i V Mm ” ('15 V d 5 (20 points) ) Draw the region R in the positive orthant (a: 2 07 y 2 0, z 2 0) /0 here :1: + y S 1 and 51:2 + y2 + 22 S 1. Hint: you’re chopping a slice off a. portion of a sphere. H,,....—___ @ xx 2. @Determine appropriate limits to compute (order cannot be changed) VaouM f f [R dzdydz 1-K ——""-’ mm" Ae- dg Ax W '0 II, m ix 3. E Compute the volume of the region. . \S. 737‘: bﬁm US , .ﬂx ...
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