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final_98

final_98 - of decrease of f at P(2-1 2 and the direction in...

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Math 100 Final Examination Fall 1998/99 (1) (a) Let C be the upper half of the circle x 2 + y 2 = 2 ax , y 0 oriented counterclockwise. Calculate the line integral R C ( e x sin y - 2 y ) dx + ( e x cos y - 2 x ) dy . (b) Find the value of n such that the integral R C e ( x n + y n ) ( xdx + ydy ) is independent of path. For this n , ﬁnd the potential function of the vector ﬁeld h e ( x n + y n ) x, e ( x n + y n ) y i . (2) (a) Find the surface area of the portion of the cylinder x 2 + z 2 = 4 that lies inside the cylinder x 2 + y 2 = 4. (b) Evaluate the integral R 3 - 3 R 9 - y 2 - 9 - y 2 R 9 - x 2 - y 2 - 9 - x 2 - y 2 p x 2 + y 2 + z 2 dzdxdy . (3) (a) Find the shortest distance from the points (2 , 0 , 0) to the surface z = 1 xy , where x > 0 and y > 0. (b) Find the absolute maximum of the function f ( x, y ) = 4 xy - x 4 - 2 y 2 on the region R = { ( x, y ) | - 2 x 2 , - 2 y 2 } . (4) Locate all relative extrema and saddle points of f ( x, y ) = ( x + y ) e - ( x 2 + y 2 ) / 2 . (5) (a) Let f ( x, y, z ) = ( x + y ) 2 + ( y + z ) 2 + ( z + x ) 2 . Find the maximal rate
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Unformatted text preview: of decrease of f at P (2 ,-1 , 2) and the direction in which this rate of decrease occurs. (b) Find all points on the ellipsoid 2 x 2 + 3 y 2 + 4 z 2 = 9 at which the tan-gent plane is parallel to the plane x-2 y + 3 z = 5. (6) (a) Let σ denote the boundary of the solid bounded by the paraboloids z = x 2 + y 2 and z = 18-x 2-y 2 . Find the surface integral RR σ F · n dS , where F = x i-y j + z k and the surface σ is oriented outward. (b) Calculate the surface integral RR σ F · n dS , where σ is the upper half hemisphere x 2 + y 2 + z 2 = 4, z ≥ 0 oriented upward and F = x 4 yz 4 i-2 x 3 y 2 z 4 j-x 2 k . 1...
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