152mt2_2006 - passing through the point P (1 , , 1)....

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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 2005-2006 Summer U. Y., E. S. 24.07.2006 10.00 120 minutes 8 Questions on 6 Pages Total 115 Points 1 2 3 4 5 6 7 8 SHOW ALL YOUR YOUR WORK IN DETAILS 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).
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Question 2 (10 pts.) Find an equation of the line parallel to the line ~ r ( t ) = (2 t + 1 ,t - 1 , 2 - t ) passing through the point P 0 ( - 1 , 1 , 2). Question 3 (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)?
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Question 4 (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
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Unformatted text preview: passing through the point P (1 , , 1). Question 5 (10 pts.) Find the direction in which f ( x,y ) = ( x 2 + y 2 ) / 2 increases most rapidly at the point (1 , 1). Question 6 (15 pts.) Find Maclaurin series of f ( x ) = x (2-x ) 2 . Surname, Name: Question 7 (20 pts.) Let f ( x,y ) has rst 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 , nd (a) f s (b) 2 f ts ( t,s )=(1 , 0) Question 8 (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 ....
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152mt2_2006 - passing through the point P (1 , , 1)....

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