152MT2Makeup2007_2 - most rapidly at the point(1 1 Question...

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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. Make-Up Exam for Midterm II Math 152 2007-2008 Fall A. ¨ O., ¨ U.A., A.D. 28.12.2007 18.30 90 minutes 8 Questions on 1 Pages Total 100 Points 1 2 3 4 5 6 7 8 Question 1 (15 pts.) Find an equation of the plane passing through the points A (1 , 2 , 4), B (3 , - 1 , 2) and C (2 , 1 , 1). Question 2 (10 pts.) Let f ( x,y ) = sin 2 x sin 2 y ( x 2 + y 2 ) 2 , ( x,y ) 6 = (0 , 0) 0 , ( x,y ) = (0 , 0) . Check whether or not f is continuous at (0 , 0)? Question 3 (20 pts.) Let S be a surface defined by z 2 = x 3 z +ln( x 2 + y 2 ). Find the equations of (a) the tangent plane and, (b) the normal line to the S passing through the point P (1 , 0 , 1). Question 4 (10 pts.) Find the direction in which f ( x,y ) = ( x 2 + y 2 ) / 2 increases
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Unformatted text preview: most rapidly at the point (1 , 1). Question 5 (20 pts.) Let f ( x,y ) has first and second order partial derivatives satisfying f x =-6 ,f y = 3 ,f xx = 3 ,f xy = f yx = 5 ,f yy =-4 at (0 , 0). If x = t 2 s and y = ts , find (a) ∂f ∂s , (b) ∂ 2 f ∂t∂s ± ± ± ± ( t,s )=(1 , 0) Question 6 (15 pts.) Find the local extrema and saddle points of the function f ( x,y ) = x 3 3 + 2 y 2-x 2-3 x-4 y-3 . Question 7 (20 pts.) Find the absolute maxima and minima of f ( x,y ) = x 2-4 xy + y 3 + 4 y on triangular region T bounded by the lines x = 7, y =-1 and y = x ....
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