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# notes001 (82) - @[email protected] 2 [email protected]@y and 2 [email protected]@x Do not just...

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MATH 2203 °Quiz 4 (Version 1) Solutions October 4, 2010 NAME____________________________ 1) By considering di/erent paths of approach, show that lim ( x;y ) ! (0 ; 0) x + y x ° y does not exist. Solution: First we try approaching (0 ; 0) along the y axis (where x = 0 ). This gives us lim ( x;y ) ! (0 ; 0) with x =0 x + y x ° y = lim y ! 0 0 + y 0 ° y = ° 1 . Now let us try approaching (0 ; 0) along the x axis (where y = 0 ). This gives us lim ( x;y ) ! (0 ; 0) with y =0 x + y x ° y = lim x ! 0 x + 0 x ° 0 = 1 . Since two di/erent results were obtained for di/erent paths of approach, we conclude that lim ( x;y ) ! (0 ; 0) x + y x ° y does not exist. 2) For the function f ( x; y ) = sin (2 x ° 3 y ) , ±nd ,

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Unformatted text preview: @[email protected] , @ 2 [email protected]@y , and @ 2 [email protected]@x . Do not just write down an-swers. You must include intermediate steps in your calculations. Solution: @f @x = cos (2 x & 3 y ) ± (2) = 2 cos (2 x & 3 y ) . @f @y = cos (2 x & 3 y ) ± ( & 3) = & 3 cos (2 x & 3 y ) . 1 @ 2 f @[email protected] = @ @x ( & 3 cos (2 x & 3 y )) = & 3 ( & sin (2 x & 3 y )) (2) = 6 sin (2 x & 3 y ) . @ 2 f @[email protected] = @ @y (2 cos (2 x & 3 y )) = 2 ( & sin (2 x & 3 y )) ( & 3) = 6 sin (2 x & 3 y ) . 2...
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notes001 (82) - @[email protected] 2 [email protected]@y and 2 [email protected]@x Do not just...

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