test2_fall07 - xe y + ye x =-sin ( x + y ) . Find the slope...

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Math 2401 M1 M2 M3 Test 2. Total: 20 points Section: Name: Student Number: 1. Find 5 (cos 2 r ), where r = x 2 + y 2 + z 2 . (4 points). 2. Suppose u = u ( x, y ) has continuous second par- tial derivatives. Let u x (1 , 0) = 1 , u y (1 , 0) = 0 , u xx (1 , 0) = - 1 , u xy (1 , 0) = 1 and u yy (1 , 0) = 2. By using the change of variables to polar coordinates x = r cos θ and y = r sin θ , ±nd 2 u ∂r∂θ at the point ( x, y ) = (1 , 0). (6 points). 3(a).Assume y is a di²erentiable function of x that sat- is±es the equation
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Unformatted text preview: xe y + ye x =-sin ( x + y ) . Find the slope and normal vector of the curve y = y ( x ). (2 points). (b). Let f ( x, y ) = xe y + ye x . Find a unit vector in the direction in which f decreases most rapidly at the point P (0 , 1) and give the rate of change in that direction. (2 points). 4. Maximize x 2 + y 2 on the curve x 4 + y 4 = 1. (6 points). 1...
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This note was uploaded on 01/09/2009 for the course MATH 2401 taught by Professor Morley during the Fall '08 term at Georgia Tech.

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