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2007Final - blue MATH 32A FINAL EXAM LAST NAME FIRST NAME...

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blue 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
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2 PROBLEM 1 (20 Points) The temperature at position ( x, y, z ) in a room is f ( x, y, z ) = 18 + 2 z + xy 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.
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3 PROBLEM 2 (20 Points) Find the maximum value of f ( x, y, z ) = xyz , subject to the constraint g ( x, y, z ) = 4 x 2 + y 2 + z 2 = 48
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4 PROBLEM 3 (20 Points) Let f ( x, y ) = 2 xy 1 6 x 3 y 2 (A) Find the critical points of f ( x, y ). (B) Determine the nature of the critical points (min, max or saddle).
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5 PROBLEM 4 (20 Points) The plane x + y 2 + z 3 = 1 intersects the x , y , and z axes in points P , Q , and R .
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