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2008Final - blue MATH 32A Final Exam LAST NAME FIRST NAME...

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blue MATH 32A Final Exam March 17, 2008 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) 5 (20 pts) 2 (20 pts) 6 (20 pts) 3 (20 pts) 7 (20 pts) 4 (20 pts) TOTAL
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2 PROBLEM 1 (20 Points) Let function f ( x, y ) = x 4 - x 2 - y 2 . (A) Find all critical points of f ( x, y ) and classify each one as a maximum, minimum, or saddle point. Explain your work. Answer : (B) Find the absolute maximum and minimum of f ( x, y ) on the domain x 2 + y 2 4. Explain your work. Answer :
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3 PROBLEM 2 (20 Points) Suppose that f ( x, y ) is a function of x and y , and that x = u 2 - v 2 , y = uv . Use the table of values ∂f ∂x and ∂f ∂y to compute ∂f ∂u ( u,v )=(1 , 1) and ∂f ∂v ( u,v )=(1 , 0) x y ∂f ∂x ( x, y ) ∂f ∂y ( x, y ) 0 0 2 -3 1 0 5 1 0 1 -1 -4 1 1 3 2 Answer : ∂f ∂u ( u,v )=(1 , 1) Answer : ∂f ∂v ( u,v )=(1 , 0)
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4 PROBLEM 3 (20 Points) Let r ( t ) = h cos 3 t, 0 , sin 3 t i (A) Show that || r 0 ( t ) || 2 = 9 cos 2 t sin 2 t (B) Find a formula for the curvature κ ( t ) at time t . Answer :
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5 PROBLEM 4 (20 Points) Let P be the plane through the points P 1 = (1 , 2 , 0) , P 2 = (2 , 0 , 1) , P 3 = (0 , 0 , 2) (A) Find an equation for the plane P .
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