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Unformatted text preview: Section 12.2 12.2.1: Because f ( x, y ) = 4 3 x 2 y is defined for all x and y , the domain of f is the entire twodimensional plane. 12.2.2: Because x 2 + 2 y 2 = 0 for all x and y , the domain of f ( x, y ) = p x 2 + 2 y 2 is the entire two dimensional plane. 12.2.3: If either x or y is nonzero, then x 2 + y 2 &gt; 0, and so f ( x, y ) is definedbut not if x = y = 0. Hence the domain of f consists of all points ( x, y ) in the plane other than the origin. 12.2.4: If x 6 = y then the denominator in f ( x, y ) is nonzero, and thus f ( x, y ) is definedbut not if x = y . So the domain of f consists of all those points ( x, y ) in the plane for which y 6 = x . 12.2.5: The real number z has a unique cube root z 1 / 3 regardless of the value of z . Hence the domain of f ( x, y ) = ( y x 2 ) 1 / 3 consists of all points in the xyplane. 12.2.6: The real number z has a unique cube root z 1 / 3 regardless of the value of z . But 2 x is real if and only if x = 0. Therefore the domain of f ( x, y ) = (2 x ) 1 / 2 + (3 y ) 1 / 3 consists of all those points ( x, y ) for which x = 0. 12.2.7: Because arcsin z is a real number if and only if 1 5 z 5 1, the domain of the given function f ( x, y ) = sin 1 ( x 2 + y 2 ) consists of those points ( x, y ) in the xyplane for which x 2 + y 2 5 1; that is, the set of all points on and within the unit circle. 12.2.8: Because arctan z is defined for every real number z , the only obstruction to the computation of f ( x ) = arctan( y/x ) is the possibility that x = 0. This obstruction is insurmountable, and therefore the domain of f consists of all those points ( x, y ) in the xyplane for which x 6 = 0; that is, all points other than those on the yaxis. 12.2.9: For every real number z , exp( z ) is defined and unique. Therefore the domain of the given function f ( x, y ) = exp( x 2 y 2 ) consists of all points ( x, y ) in the entire xyplane. 12.2.10: Because ln z is a unique real number if and only if z &gt; 0, the domain of f ( x, y ) = ln( x 2 y 2 1) consists of those points ( x, y ) for which x 2 y 2 1 &gt; 0; that is, for which y 2 &lt; x 2 1. This is the region bounded by the hyperbola with equation x 2 y 2 = 1, shown shaded in the following figure; the bounding hyperbola itself is not part of the domain of f . 1441 x y 12.2.11: Because ln z is a unique real number if and only if z &gt; 0, then domain of f ( x, y ) = ln( y x ) consists of those points ( x, y ) for which y &gt; x . This is the region above the graph of the straight line with equation y = x (the line itself is not part of the domain of f ). 12.2.12: Because z is a unique real number if and only if z = 0, the domain of the given function f ( x, y ) = p 4 x 2 y 2 consists of those points ( x, y ) for which x 2 + y 2 5 4. That is, the domain consists of all those points ( x, y ) on and within the circle with center (0 , 0) and radius 2....
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This document was uploaded on 11/28/2010.
 Spring '09

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