Math 255 2nd Midterm

# Math 255 2nd Midterm - f is continuous, show that Z a Z y f...

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Math 255, Winter 2010 Review excercises for Exam 2 1. Let f ( x,y ) = ( ( x 2 + y 2 ) log( x 2 + y 2 ) for ( x,y ) 6 = (0 , 0) 0 ( x,y ) = (0 , 0) . (a) Is f ( x,y ) continuous at (0 , 0)? (b) Is f ( x,y ) diﬀerentiable at (0 , 0)? 2. 2 sin( xyz ) = x 2 + y 2 + xz deﬁnes z = f ( x,y ) implicitly. a) f (1 , 0) =? b) Compute ∂z ∂x in terms of x , y and z . c) ∂z ∂x (1 , 0) =? 3. Find the equation of the tangent plane for the surface pre- scribed by x 2 + y 2 + 2 z 2 = 4 at the point (1 , 1 , 1). 4. Find the absolute maximum and minimum for f ( x,y,z ) = - x 2 + 2 y 2 - z 2 in the region D = { ( x,y,z ) | x 2 + y 2 + 2 z 2 6 4 } . 5. Find the point(s) on the intersection curve of surfaces x = 1 and x 2 + 2 y 2 + 4 z 2 = 7 with the longest distance to the origin a) without using Lagrange multiplier. b) using Lagrange multiplier. 6. Compute R R R cos x · sin x · e y · sin x dA , R = [0 ] × [0 , 1]. 7. A lamina lies in both circles x 2 + y 2 = 1 and x 2 + y 2 = 2 y . Suppose the density is inverse proportional to the distance to the origin. Compute its center of mass. 8. If

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Unformatted text preview: f is continuous, show that Z a Z y f ( t )d t d y = Z a ( a-t ) f ( t )d t 1 Math 255, Winter 2010 9. Evaluate the integral Z Z D 1 ( x 2 + y 2 ) 2 d A over the area bounded by one loop of the lemniscate D = { ( r, ) : r 2 = cos 2 } 10. Evaluate the integral Z Z D 1 p x 2 + y 2 d A where D = [0 , 1] [0 , 1]. 11. Does the the surfaces x 2-2 yz + y 3 = 4 and x 2 + 1 = z 2-2 y 2 intersect orthogonally at the point (1 ,-1 , 2)? 12. Find the point in the paraboloid z = x 2 + y 2 that is closest to the point (3 ,-6 , 4). 13. Find the equation of the tangent plane (in cartesian coordi-nates) of the surface given in cylindrical coordinates z = r at the point ( r,z ) = (2 , 4 , 1). 2...
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## This note was uploaded on 12/31/2011 for the course MATH 255 taught by Professor Jackwaddell during the Fall '08 term at University of Michigan.

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Math 255 2nd Midterm - f is continuous, show that Z a Z y f...

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