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Unformatted text preview: Math 23 B. Dodson Week 6 Homework: 14.3 partial derivatives, 2nd order deriv 14.4 tangent plane, differentials 14.5 chain rule Week 6 Homework: Problem 14.3.15: Find the partial derivatives of the function z = f ( x, y ) = xe 3 y . Find f x (2 , 1) . 2 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 verified that the second partials z xy = z yx . Week 6 Homework: 14.4 tangent plane, differentials Problem 14.4.3: Find the tangent plane to the surface the z = f ( x, y ) = p 4- x 2- 2 y 2 at (1 ,- 1 , 1) . 3...
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