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math16B_review_exer1

# math16B_review_exer1 - Math 16B Section 1 Sarason REVIEW...

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Math 16B, Section 1 Fall 2005 Sarason REVIEW EXERCISES 1 1. In each part, find the first partial derivatives of the given function f ( x, y ) (a) f ( x, y ) = x 5 - 10 x 3 y 2 + 5 xy 4 (b) f ( x, y ) = 20 x 3 / 5 y 2 / 5 (c) f ( x, y ) = e - x 2 /y 2. In each part, evaluate the integral I = R f ( x, y ) dydx of the given function f over the given region R . (a) f ( x, y ) = x 2 + y 4 ; R is the rectangle given by - 1 x 1, 0 y 2. (b) f ( x, y ) = x 2 - y 2 ; R is the triangle with vertices (0 , 0) , (1 , 0) , (1 , 2). (c) f ( x, y ) = x 2 y 3 ; R is defined by the inequalities x 2 + y 2 1, y 0. 3. For each part, determine the critical points of the function f ( x, y ), and determine the nature of each critical point, to the extent possible, by means of the second derivative test. (a) f ( x, y ) = x 3 - y 2 - 3 x + 4 y (b) f ( x, y ) = x 3 + y 3 - 9 xy (c) f ( x, y ) = 4 x 4 - 2 y 2 - xy (d) f ( x, y ) = 4 x 8 - 8 x 4 - xy 4. (a) Find the maximum and minimum values of the function f ( x, y ) = 7 x + 9 y on the curve 7 x 4 + 9 y 4 = 64. (b) Where does the function f ( x, y ) = x 2 + 3 y 2 + 10 attain its minimum value under the constraint x + y = 8?
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