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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:
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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 .
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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(
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\/ 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
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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 '_
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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 1K ——""’ mm" Ae dg Ax
W '0 II, m ix 3. E Compute the volume of the region. .
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 Fall '08
 Kim
 Math, Calculus

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