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152sumFinal2003

152sumFinal2003 - x 2 y 2 = 1 and bounded by the planes y =...

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Last Name Name Student No Department Section Signature : : : : : : : : : : : : : Code Acad. Year Semester Instructors Date Time Duration M E T U Department of Mathematics Calculus-II Final Exam Math 152 2004 Summer A. ¨ O., M.K., A. ¨ O. 13.08.2004 19.00 90 minutes 4 Questions on 4 Pages Total 80 Points 1 2 3 4 Question 1 (4+20 pts.) (a) Find and classify the critical point(s) of the function f ( x, y ) = 16 - x 2 - 4 y 2 . (b) Find maximum and minimum values of the function f ( x, y ) in (a) over the region x 4 + 2 y 4 1.

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Question 2 (16 pts.) Evaluate Z 12 3 Z 2 y/ 3 cos( x 3 ) dxdy + Z 3 0 Z 2 1 cos( x 3 ) dxdy.
Question 3 (12+12 pts.) Consider the solid in the first quadrant inside the paraboloid z = 4 - x 2 - y 2 , outside the cylinder

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Unformatted text preview: x 2 + y 2 = 1 and bounded by the planes y = 0 and x √ 3-y = 0. (a) Sketch the solid and write down the volume of the solid in rectangular (cartesian) coordinates. Do NOT Evaluate! (b) Write down the volume of the solid in polar coordinates and evaluate it. Question 4 (7+9 pts.) Let f ( x,y ) = 2 x + x 2 y + y sin y and let ~u = a i + b j be a unit vector. (a) Express D ~u f (1 , 0) in terms of a and b . (b) Find the values of a and b for which D ~u f (1 , 0) is maximum....
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