prelim2 - MATH 192 FALL 2006 PRELIM 2 SHOW ALL WORK NO...

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MATH 192, FALL 2006 PRELIM 2 SHOW ALL WORK. NO CALCULATORS. WRITE YOUR SECTION NUMBER AND TA’S NAME ON THE EXAM BOOKLET. 1. [10 points] (This is one of the homework problems.) Find the linearization L ( x, y ) of the function f ( x, y ) = xy 2 + y cos( x - 1) at the point P 0 (1 , 2). Then find an upper bound for the magnitude E of the error in the approximation f ( x, y ) L ( x, y ) over the rectangle R : | x - 1 | ≤ 0 . 1 , | y - 2 | ≤ 0 . 1. 2. [20 points] Consider the function f ( x, y ) = x 2 + ky 2 - 2 xy where k is a constant. (a) For what value(s) of k does f ( x, y ) have only one critical point? Is the critical point a local maximum, a local minimum, or a saddle point? (b) Identify the locations of critical points for values of k other than the value(s) in (a). Are these points local maximum, local minimum, or saddle points? 3. [20 points] A rectangular box without a lid is to have 12 m 2 of external surface area. Use the method of Lagrange Multipliers to find the maximum volume of such a box. 4. [12 points] Consider the region defined by y 2 - 1 x 1 - y 2 . (a) Find the area of the region.
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Unformatted text preview: (b) Write integral(s) in which you reverse the order of integration in (a). 5. [10 points] Consider the trapezoid with vertices (0 , 0) , (0 , w ) , (6 , w + 2), and (6 , 0). Given that the x-coordinate of the center of mass is 7 / 2, what is w ? Show your calculations; no credit for a guess. 6. [14 points] Consider a thin semicircular disk x 2 + y 2 ≤ 25 , y ≥ 0. (a) The disk has uniform density. Using polar coordinates ( r, θ ) write an expression, in terms of double integrals, that determines ¯ y , the y-coordinate of the center of mass of the semicircular disk. Calculate ¯ y . (b) Repeat (a) with disk density δ ( r, θ ) = r . 7. [14 points] A tetrahedron has vertices (0 , , 0), (1 , , 0), (0 , 2 , 0), and (0 , , 3). (a) Write a triple integral to calculate the volume of the tetrahedron. (b) Find the volume by evaluating this triple integral....
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