Final_Fall-2008-2009_Kobeissi

Final_Fall-2008-2009_Kobeissi - American University of...

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American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2008-2009 Final Exam Exercise 1 a. (5 points) If f ( u,v,w ) is a differentiable function and if u = x - y , v = y - z , and w = z - x , show that ∂f ∂x + ∂f ∂y + ∂f ∂z = 0 b. (10 points) Use the method of Lagrange multipliers to find the maximum and minimum values of f ( x,y ) = 3 x - y + 6 on the circle x 2 + y 2 = 4 Exercise 2 (10 points) Convert to polar coordinates, then evaluate the following integral Z 2 0 Z 0 - 1 - ( y - 1) 2 xy 2 dxdy Exercise 3 (12 points) Here is the region of integration of the integral = j x 6 z = y 2 y z (0,-1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1,-1,1) (1,-1,0) Z 1 0 Z 0 - 1 Z y 2 0 dz dy dx Rewrite the integral as an equivalent iterated integral in the other 5 orders, then evaluate one of them Exercise 4 Let V be the volume of the region D that is bounded by the paraboloid z = x 2 + y 2 , and the plane z = 2 y . a)
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This note was uploaded on 03/01/2010 for the course MATH 201 taught by Professor Variousteachers during the Spring '10 term at American University of Beirut.

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Final_Fall-2008-2009_Kobeissi - American University of...

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