152supExam2007 - f ( x,y ) = 4 xy-2 x 2-y 4 . Q4 (10...

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Last Name Name Student No Department Section Signature : : : : : : : : : : : : : Code Acad.Year Semester Instructor Date Time Duration ATILIM UNIVERSITY Department of Mathematics Calculus II Supplementary Exam Math 152 2007-2008 Fall 04.02.2008 90 minutes 6 Questions on 1 Pages Total 100 Points 1 2 3 4 5 6 Q1 (15 points) Test the following series for convergence. (a) X n =0 ( n !) 2 (2 n )! (b) X n =0 1 2 + 3 n (c) X n =1 158 n + cos( ) 2005 n Q2 (20 points) Let a plane π : 2 x + y - z = 4 and a point P 0 = ( - 1 , 1 , 2) be given. (a) Is P 0 on the plane π ? (b) Find the equation of the line l perpendicular to π through the point P 0 . (c) Find the intersection point P 1 , of l and π . (d) Find the distance between P 0 and P 1 . Q3 (15 points) Find and classify the critical points of the function
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Unformatted text preview: f ( x,y ) = 4 xy-2 x 2-y 4 . Q4 (10 points) Evaluate the following limit or explain why it does not exist: lim ( x,y ) (1 , 0) ( x-1) y 2( x-1) 2 + y 4 Q5 (15 points) Write and evaluate the volume of the solid lying between the paraboloids z = x 2 + y 2 and 3 z = 4-x 2-y 2 in cylindrical coordinates. Q6 (25 points) Evaluate the following double integrals (a) Z / 2 Z 5 y sin x x dxdy (b) Z Z D e-( x 2 + y 2 ) dxdy given that D is the quarter disc bounded by x = 0, y = 0 and x 2 + y 2 = ln5....
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