math53midterm2

math53midterm2 - x 2-2 x-y 2-2 on the set D = { ( x, y ) |...

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Mock Second Midterm Exam – Multivariable Calculus Math 53M, Fall 2009. Instructor: E. Frenkel 1. Find the equations of the tangent plane and the normal line to the surface z + 1 = xy cos z at the point (1 , 0 , 0). 2. Suppose that over a certain region of space the electrical potential V is given by V ( x, y, z )= x 2 - xy - xyz. (a) Find the rate of change of the potential at the point P = (2 , 2 , 1) in the direction of the vector ± 1 , 1 , - 1 ² . (b) In which direction does V increase most rapidly at the point P ? (c) What is the maximum rate of change of V at the point P ? 3. Find the absolute maximum and minimum values of the function f ( x, y )=
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Unformatted text preview: x 2-2 x-y 2-2 on the set D = { ( x, y ) | y -x, x 2 + y 2 4 } . List all points where these values are attained. 4. Find the mass and the center of mass of the lamina that occupies the region D = { ( x, y ) | x 2 + y 2 4 , y , y 3 x } , and has mass density ( x, y ) = x 2 + y 2 . 5. Evaluate the integral 1 1-x 2 2-x 2-y 2 x 2 + y 2 xy dz dy dx. 6. Evaluate the integral R e 4 x +8 y dA, where R is the parallelogram with vertices (-1 , 3), (1 ,-3), (3 ,-1), and (1 , 5)....
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