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math16B_review_exer1-1

# math16B_review_exer1-1 - 5 Repeat Exercise 4 for the...

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Math 16B Spring 2011 Sarason REVIEW EXERCISES 1 The exercises here are suggestive of what might appear on Midterm Examination 1 (although some of them are more involved than would be appropriate for a 50-minute exam). 1. Find the first partial derivatives of the functions f ( x, y ) = tan( x 2 y ) , g ( x, y ) = ln(tan( x 2 y )) . 2. Evaluate the integrals I 1 = ZZ R sin y dxdy, I 2 = ZZ R cos y dxdy, where R is the triangle with vertices (0 , π 2 ) , ( π 2 , 0) , ( π 2 , π 2 ). 3. Evaluate the integral I = ZZ R ( x + y ) 3 dxdy, where R is the triangle with vertices (1 , 0)(0 , 1) , ( - 1 , 0). 4. Find the critical points of the function f ( x, y ) = x 3 + y 3 - 3 x - 3 y, and use the second-derivative test to determine the nature of each critical point.
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Unformatted text preview: 5. Repeat Exercise 4 for the function f ( x,y ) = x 3 + y 3-6 xy. 6. Let f ( x,y ) = x 2 / 5 y 3 / 5 , g ( x,y ) = 2 x + 3 y . (a) Use Lagrange’s method to ﬁnd the point ( x,y ) that majorizes f ( x,y ) under the constraint g ( x,y ) = 5 in the region x > 0, y > 0. (b) Use Lagrange’s method to ﬁnd the point ( x,y ) that minimizes g ( x,y ) under the constraint f ( x,y ) = 1 in the region x > 0, y > 0. 7. Use Lagrange’s method to ﬁnd the maximum and minimum values of the function f ( x,y ) = 3 x 3 + 4 y 3 under the constraint x 2 + y 2 = 1....
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