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Unformatted text preview: show that the limit doesnt exist. Problem 14.3.15: Find the partial derivatives of the function z = f ( x, y ) = xe 3 y . Find f x (2 , 1) . 3 Solution: For f x we (temporarily) hold y constant, so e 3 y constant, giving f as constant x . We then take the derivative in x , with (constant x ) = constant, or f x = e 3 y . Likewise, with x constant, f y = x y ( e 3 y ) = x d dy ( e 3 y ) = 3 xe 3 y . Finally, f x (2 , 1) = e 3 gives the rate of change of f at (2 , 1) with respect to x ; which we may constrast with f y (2 , 1) = 6 e 3 , the rate of change of f at (2 , 1) with respect to y . For example, f is growing six times more rapidly in y than in x . We also solved #49, 14.3, and in particular veried that the second partials z xy = z yx ....
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