19-----bluefinalsol[1]

# 19-----bluefinalsol[1] - blue MATH 32A FINAL EXAM December...

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Sheet1 Page 1 blue MATH 32A FINAL EXAM December 13, 2007 LASTNAME FIRST NAME IDNO. YourTA: To receive credit, you must write your answer in the space provided. DO NOT WRITE BELOW THIS LINE 1 (20 pts) 4 (20 pts) 2 (20 pts) 5 (20 pts) 3 (20 pts) 6 (20 pts) TOTAL FOR WRITTEN PROBLEMS

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Sheet1 Page 2 1 PROBLEM 1 (20 Points) Thetemperature atposition(x,y,z)in a room is oC f(x,y,z)=18+2z + xy A woman walksdown a spiral staircasein the middle of the room. She holds a thermometer whose position at time t (seconds)is tt t c(t)= hcos ,sin ,8- 333 (t in seconds). How fast is temperature reading on the thermometer changing at t =6 s. Solution: We have .f = hy,x,2i 1 t 1 t 1 c (t)= h- sin , cos ,- 3 33 33 At t = 6, c(6)= (x, y, z)= h1,0,8-2i .f = h0,1,2i c
Sheet1 Page 3 (6)= h0, 11 ,- 33 By the ChainRule forpaths, the rate of change of tempera ture is d (t) dt The temperature reading at t =6 is changing at the rate 1112 1 ,- - Err:520 oC/s 3333 3 3 PROBLEM 2 (20 Points) Findthe maximum value of f(x,y,z)= xyz, subject to the constraint 222 g(x,y,z)=4x + y + z =48 Solution: Use the method of Lagrange multipliers: .f = hyz,xz,xyi = .g = h8x, 2y,2zi Case 1: x,y,z are all non-zero. Then yz xz xy Err:520 8x 2y 2z f(c(t))= .f h c .f h c ' (6)= h0,1,2ihh0,

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Sheet1 Page 4 This yields 2 22 2 y =4x, z =4x Plug into the constraint: 222 22 4x + y + z =4x 2 +4x 2 +4x =48 . x =4 f(2,4,4)=2(4)(4)=32 Case2: Atleastone of x,y,z is zero. Inthis casef(x,y,z)= xyz =0. We conclude that the maximum value of f = xyz subject to the constraint is 32. t PROBLEM 3 (20 Points) Let 2 f(x,y)=2xy - 1 x 3 -y 6 (A)Find the criticalpoints of f(x,y). (B)Determinethe nature ofthe criticalpoints(min, max or saddle). Solution: (A)To find thecriticalpoints,wesolve We obtain x = h2, y = h2x = h4, z = h2x = h4, and More generally, f(h2,h4,h4)= h32.
Sheet1 Page 5 1 2 fx =2y - x =0 2 fy =2x -2y =0 11 We obtain x = y and y = x2 . Hence x =

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## This test prep was uploaded on 04/21/2008 for the course MATH 31A taught by Professor Jonathanrogawski during the Fall '07 term at UCLA.

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19-----bluefinalsol[1] - blue MATH 32A FINAL EXAM December...

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