math16B_review_exer3

math16B_review_exer3 - Math 16B Section 1 Sarason REVIEW...

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Math 16B, Section 1 Spring 2006 Sarason REVIEW EXERCISES 3 1. In each part, find and classify (by means of the second-derivative test) the critical points of the function f . (a) f ( x, y ) = 8 x 3 + y 3 - xy (b) f ( x, y ) = 5 x 2 + y 2 - x - y - 2 xy 2. Which pair of values a, b minimizes the function E ( a, b ) = 1 0 ( ax + b - x 2 ) 2 dx ? 3. Among all rectangular boxes of side lengths x, y, z inches and diagonal measuring 28 inches, which values of x, y, z maximize 2 x + 2 y + z ? 4. (Final Exam, F04) The Mexican firm Novedades Terminador SSA manufactures Arnold masks. The number of masks it can produce in a week with the utilization of x units of labor and y units of capital is given by the production function f ( x, y ) = 500 x 3 / 5 y 2 / 5 . The expense for each unit of labor is 100 pesos per week and for each unit of capital it is 200 pesos per week. Total expenses are limited to 1000 pesos per week. How many units of labor and capital should the firm utilize so as to maximize production? 5. Evaluate the integrals. (a) π/ 2 0 sin 3 x cos 3 x dx (b) 0 x 3 e - x 2 dx (c) 1 0 ( x + 1) ln( x + 1) dx (d) R ln( x + y + 1) dxdy , where R is the triangle with vertices (0 , 0), (1 , 0), (0 , 1).
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