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mthsc206-fall-2010-notes-15.1

mthsc206-fall-2010-notes-15.1 - 15 15.1 Partial Derivatives...

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15 Partial Derivatives 15.1 Functions of Several Variables Definition 15.1.1 A function of 2 variables is a rule that assigns a unique real number f ( x, y ) to each ordered pair ( x, y ) in the domain D . The domain of f is the set of ordered pairs ( x, y ) for which f is defined. The range of f is the set { f ( x, y ) | ( x, y ) Dom ( f ) } . The graph of f is the set of points { ( x, y, f ( x, y )) | ( x, y ) Dom ( f ) } . NOTE: In a real-valued function of 2 variables x and y are independent variables, and z = f ( x, y ) is the dependent variable. Example 15.1.1 Define f ( x, y ) = xy - 5 2 y - x 2 . Find the domain as a set and sketch it. Example 15.1.2 Let f ( x, y ) = 1 - x 2 - y 2 . Find the domain as a set and sketch it. Find the range of f . What does the graph of z = f ( x, y ) look like? Definition 15.1.2 The level curves of a function f of 2 variables are the (plane) curves with equations k = f ( x, y ) , where k is a constant in the range of
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