09mock2 - − y 2 − 2 on the set D = x y | y ≥ − x x...

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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 a 1 , 1 , 1 A . (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 ) = x 2 2 x
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Unformatted text preview: − 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 ρ ( x, y ) = r x 2 + y 2 . 5. Evaluate the integral i 1 i √ 1-x 2 i √ 2-x 2-y 2 √ x 2 + y 2 xy dz dy dx. 6. Evaluate the integral i i 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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This note was uploaded on 04/23/2011 for the course JAPAN 7b taught by Professor Wallace during the Spring '11 term at Berkeley.

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