Unformatted text preview: MATH 192, FALL 2008
PRELIM 2 1. [10 points] (This 1s one of the homework problems ) Find the linearization L($, y) of the function
f (9:: y)— -- mg2 + y cos(:r —— 1) at the point Pg(1, 2) Then ﬁnd an upper bound for the magnitude E of the error in the approximation f(r,y) es L(:r,y) over the rectangle R . [:13 -—- 1| 3 0.1, [y— 2| 3 0.1. 2.. [20 points] Consider the function f(:1:, y) = 11:2 —|— kyg —— 211:3; where k is a constant. (a) For what value(s) of k: does f(:r, 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 I42 other than the value(s ) in (a). Are these points local maximum, local minimum, or saddle points? I 3. [20 points] A rectangular box without a lid—is to have 12 In2 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? _. 1 S :1: S 1 __ yz
(8) Find the area of the region. I (b) Write integral(s) in which you reverse the order of integration in (a). 5. [10 points] Consider the trapezoid with vertices (0,0), (0,111), (6,111 + 2), and (6,0). Given that the :1:— I coordinate of the center of mass is 7/2, what is 11)? Show your calculations; no credit for a guess. 6. [14 points] Consider a thin semicircular disk 332+ 3,12 g 25, y 2 0.
_ (a) The disk has uniform density. Using polar Coordinates (r, 6) write an expression, in terms of double integrals, that determines 37, the y—coordinate of the center of mass of the semiCircular disk. Calculate i}. (b) Repeat (3.) With disk density 6(1", 9) = r. 7. [14 points] A tetrahedron has vertices (0,0,0), (1,0,0), (0,2,0), and (0,0,3).
(3.) write a triple integral tocalculate the volume of the tetrahedron. (b) Find the volume by evaluating this triple integral. ...
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