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# 152mt2_2007 - discontinuity and continuity points of the...

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Last Name Name Student No Department Section Signature : : : : : : : : : : : : : Code Acad.Year Semester Instructors Date Time Duration ATILIM UNIVERSITY Department of Mathematics Calculus II. II. Midterm Math 152 2007-2008 Fall A. ¨ O., ¨ U.A., A.D. 15.12.2007 13.00 90 minutes 7 Questions on 4 Pages Total 100 Points 1 2 3 4 5 6 7 SHOW ALL YOUR YOUR WORK IN DETAILS Question 1 (15 pts.) Find the area of the triangle with vertices A (1 , 2 , 1), B (4 , - 1 , 1) and C (3 , 4 , - 2). Question 2 (10 pts.) Find the directional derivative of the function h ( x, y ) = ln( x 2 + y 2 ) at the point P 0 (1 , - 2) in the direction of ~ u = 3 i + 4 j .

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Question 3 (6+9 pts.) Let E 1 : 3 x + 4 y + 5 z = 1 and E 2 : 8 x + 4 y + z = 1 be two planes in space. (a) Find the angle between the planes E 1 and E 2 . (b) Find an equation of the intersection line of the planes E 1 and E 2 . Question 4 (15 pts.) Find all the

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Unformatted text preview: discontinuity and continuity points of the function f ( x,y ) = x 3-y 2 x 2 + y 2 , ( x,y ) 6 = (0 , 0) , ( x,y ) = (0 , 0) . Question 5 (15 pts.) Find the equations of (a) the tangent plane and, (b) the normal line to the graph of the function z = cos( x y ) at the point ( π, 2). Question 6 (10 pts.) Find and classify all the critical point(s) of the function f ( x,y ) = x 3 + y 3-3 xy Question 7 (20 pts.) Find the maximum and minimum values of the function f ( x,y ) = x + 2 y-3 z over the ellipsoid x 2 + 4 y 2 + 9 z 2 = 108.(Use the Lagrange Multipliers method for the boundary points)...
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