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yellowfinalsol

# yellowfinalsol - yellow MATH 32A FINAL EXAM LAST NAME FIRST...

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Unformatted text preview: yellow MATH 32A FINAL EXAM December 13, 2007 LAST NAME FIRST NAME ID NO. Your TA: 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 1 2 PROBLEM 1 (20 Points) The plane x 2 + y + z 3 = 1 intersects the x , y , and z axes in points P , Q , and R . (A) Determine P , Q , and R . (B) Find the area of the triangle △ PQR . Solution: (A) The points are P = (2 , , 0) , Q = (0 , 1 , 0) , R = (0 , , 3) (B) The area of △ PQR is one-half the length of the cross product: → PQ × → PR = ( j − 2 i ) × (3 k − 2 i ) = 3 i + 2 k + 6 j = ( 3 , 6 , 2 ) Thus, the area is: 1 2 || → PQ × → PR || = 1 2 radicalbig 3 2 + 6 2 + 2 2 = 7 2 3 PROBLEM 2 (20 Points) The temperature at position ( x, y, z ) in a room is f ( x, y, z ) = xy + 2 z + 17 o C A woman walks down a spiral staircase in the middle of the room. She holds a thermometer whose position at time t (seconds) is c ( t ) = ( cos t 3 , sin t 3 , 8 − t 3 ) ( t in seconds). How fast is temperature reading on the thermometer changing at t = 6 π s. Solution: We have ∇ f = ( y, x, 2 ) c ′ ( t ) = (− 1 3 sin t 3 , 1 3 cos t 3 , − 1 3 ) At t = 6 π , c (6 π ) = ( x, y, z ) = ( 1 , , 8 − 2 π ) ∇ f = ( , 1 , 2 ) c ′ (6 π ) = ( , 1 3 , − 1 3 ) By the Chain Rule for paths, the rate of change of temper- ature is d dt f ( c ( t )) = ∇ f · c ′ ( t ) The temperature reading at t = 6 π is changing at the rate ∇ f · c ′ (6 π ) = ( , 1 , 2 ) · ( , 1 3 , − 1 3 ) = − 1 3 o C/s 4 PROBLEM 3 (20 Points) Find the maximum value of f ( x, y, z ) = xyz , subject to the constraint g ( x, y, z ) = 9 x 2 + y 2 + z 2 = 243 Solution: Use the method of Lagrange multipliers: ∇ f = ( yz, xz, xy ) = λ ∇ g = λ ( 18 x, 2 y, 2 z ) Case 1: x, y, z are all non-zero. Then yz 18 x = xz 2 y = xy 2 z This yields y 2 = 9 x 2 , z 2 = 9 x 2 Plug into the constraint: 9 x 2 + y 2 + z 2 = 9 x 2 + 9 x 2 + 9 x 2 = 243 ⇒ x 2 = 243 / 27 = 9 We obtain x = ± 3,...
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yellowfinalsol - yellow MATH 32A FINAL EXAM LAST NAME FIRST...

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