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Final_Fall-2005-2006 - American University of Beirut MATH...

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Unformatted text preview: American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2005-2006 Final Exam Name: ............................... ID #: ................................. good luck Exercise 1 (10 points) Given the surface z = x 2- 4 xy + y 3 + 4 y- 2 containing the point P (1 ,- 1 ,- 2) a. Find an equation of the tangent plane to the surface at P . b. Find an equation of the normal line to the surface at P . Exercise 2 (15 points) a. Which of the following series converges and which diverges? justify. i. + ∞ X n =0 e n 1 + e 2 n ii. + ∞ X n =1 1- cos n n ln n iii. + ∞ X n =1 n 10 + n 2 b. Find the radius of convergence of the series + ∞ X n =0 x 2 n 2 n , then find its sum. Exercise 3 (10 points) a. If w = f ( x,y ) is differentiable and x = r + s,y = r- s , show that ∂w ∂r × ∂w ∂s = ∂f ∂x ¶ 2- ∂f ∂y ¶ 2 b. Prove or disprove: The function f ( x,y ) = x 2 y 2 x 4 + 3 y 2 can be extended by continuity at (0 , 0)....
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Final_Fall-2005-2006 - American University of Beirut MATH...

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