math16B_review_exer1 - Math 16B, Section 1 Spring 2006...

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Unformatted text preview: Math 16B, Section 1 Spring 2006 Sarason REVIEW EXERCISES 1 1. In each part, find all first and second partial derivatives of the given function. (a) f ( x, y ) = ( x 2 + y 2 ) 3 / 2 (b) f ( x, y ) = e ( x 2 + y 2- z 2 ) (c) f ( x, y ) = ln( x 2 + y 2 ) 2. (Midterm, F05) In the manufacture of CuddlyWuddly dolls, the number of dolls produced with the utilization of x units of labor and y units of capital is given by the production function f ( x, y ) = p 6 x 2 + y 2 . Determine the marginal productivities of labor and capital when x = 4 and y = 2. 3. In each part, determine the critical points of the given function f ( x, y ), and classify the critical points, to the extent possible, by means of the second-derivative test. (a) f ( x, y ) = x 3 + y 3- 15 xy (b) f ( x, y ) = x 4 + y 3 + 32 x- 3 y (c) (Midterm, F05) f ( x, y ) = 16 x 5 + 5 y 2- 10 xy 4. For which values of a and b does the straight line ax + b give the best least-squares approxi- mation to the curve y = x 2 at the four points (0...
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