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assignment3

# assignment3 - MATH1211/10-11(2/A3/NKT THE UNIVERSITY OF...

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MATH1211/10-11(2)/A3/NKT THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 2nd Semester: Assignment 3 Due date: 2011 February 24 (Thursday), 5:00pm. (The Exercise/problem numbers in the following refer to those in the textbook.) 1. ( § 2.4 no.22) Let f ( x, y ) = xy x 2 - y 2 x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . (a) Find f x ( x, y ) and f y ( x, y ) for ( x, y ) 6 = (0 , 0). (b) Find f x (0 , y ) and f y ( x, 0). (Note: For y 6 = 0, you may use your result in part (a) to get f x (0 , y ); but you need to compute f x (0 , 0) by other means.) (c) Find the values of f xy (0 , 0) and f yx (0 , 0). Reconcile your answer with Theorem 4.3 on p.129.) 2. ( § 2.5 no.30) Let f ( x, y ) = x 2 y x 2 + y 2 if ( x, y ) 6 = (0 , 0) 0 if ( x, y ) = (0 , 0) . (a) Use the definition of the partial derivative to find f x (0 , 0) and f y (0 , 0). (b) Let a be a nonzero constant and let x ( t ) = ( t, at ). Show that f x is differentiable, and find D ( f x )(0) directly. (c) Calculate Df (0 , 0) D x (0). How can you reconcile your answer with your answer in part (b) and the chain rule?

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assignment3 - MATH1211/10-11(2/A3/NKT THE UNIVERSITY OF...

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