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Unformatted text preview: Homework Week 6 Solutions 14.1.3 Substituting ( x,y,z ) = (3 , 8 , 2) and ( x,y,z ) = (3 , 2 , 6) in the function, we obtain h (3 , 8 , 2) = 3 8 2 2 = 3 8 1 4 = 6 h (3 , 2 , 6) = 3 ( 2) ( 6) 2 = 6 1 36 = 1 6 14.1.7 The function is defined if y 2 x > 0 or y > 2 x . This is the region in the xyplane that is above the line y = 2 x . D = { ( x,y ) : y > 2 x } x y y = 2 x y > 2 x 14.1.11 The function is defined for all x 6 = 0. The domain is the xyplane with the yaxis excluded. D = { ( x,y ) : x 6 = 0 } x y x 6 = 0 14.1.20 (a)  x  +  y  . The level curves are  x  +  y  = c , y = c  x  , or y = c +  x  . The graph (D) corresponds to the function with these level curves. (b) cos( x y ). The vertical trace in the plane x = c is the curve z = cos( c y ) in the plane x = c . These traces correspond to the graph (C). (c) 1 1 + 9 x 2 + y 2 (e) 1 1 + 9 x 2 + 9 y 2 . The level curves of the two functions are: 1 1+9 x 2 + y 2 = c 1 1+9 x 2 +9 y 2 = c 1 + 9 x 2 + y 2 = 1 c 1 + 9 x 2 + 9 y 2 = 1 c 9 x 2 + y 2 = 1 1 c 9 x 2 + 9 y 2 = 1 1 c x 2 + y 2 = 1+ c 9 c For suitable values of c , the level curves of the function in (c) are ellipses as in (E), and the level curves of the function (e) are circles as in (A). (d) cos( x 2 ) e 1 / ( x 2 + y 2 ) (f) cos( x 2 + y 2 ) e 1 / ( x 2 + y 2 ) . The value of  z  is decreasing to zero as x or y are decreasing, hence the possible graphs are (B) and (F). In (f), z is constant whenever x 2 + y 2 is constant, that is, z is constant whenever ( x,y ) varies on a circle. Hence (f) corresponds to the graph (F) and (d) corresponds to (B). To summarize, we have the following matching: (a) (D) (b) (C) (c) (E) (d) (B) (e) (A) (f) (F) 14.1.22 The graph is shown in the figure: 1 2 1 0.5 1 0.5 1 0.5 0.5 1 1 y x z The horizontal trace at height c is x 2 + y = c or y = x 2 + c . This is a parabola in the plane z = c ....
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This note was uploaded on 10/26/2011 for the course MATH 32A taught by Professor Gangliu during the Spring '08 term at UCLA.
 Spring '08
 GANGliu

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