2014_FALL_MATH2011.yflo.Homework-3.pdf - Yik Fung LO...

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Yik Fung LO MATH 2011 (Fall 2014-2015) WeBWork Homework-3 due 10/23/2014 at 11:55pm HKT You may need to give 4 or 5 significant digits for some (floating point) numerical answers in order to have them accepted by the computer. 1. (2 pts) Consider the function f ( x , y ) = ( 2 xy ( x 2 + y 2 ) 2 , ( x , y ) 6 = ( 0 , 0 ) 0 , ( x , y ) = ( 0 , 0 ) (a) Use a computer to draw a contour diagram for f . Which of the following is the contour diagram? ? The function f is differentiable at all points ( x , y ) 6 = ( 0 , 0 ) , as it is a rational fraction with denominator ( x 2 + y 2 ) 2 = 0 only when ( x , y ) = ( 0 , 0 ) . (c) The partial derivatives of f at points ( x , y ) 6 = ( 0 , 0 ) are given by f x ( x , y ) = 2 y ( x 2 + y 2 ) 2 - 8 x 2 y ( x 2 + y 2 ) (b) Is f differentiable at all points ( x , y ) 6 = ( 0 , 0 ) ? (c) Calculate the partial derivatives of f for ( x , y ) 6 = ( 0 , 0 ) : (d) A first test for whether f is differentiable at ( 0 , 0 ) is to see if it is continuous there. Calculate each of the following limits to determine if f is continuous at ( 0 , 0 ) : lim h 0 f ( 0 , h ) = lim h 0 f ( h , 0 ) = lim h 0 f ( h , h ) = (In each case, enter DNE if the limit does not exist.) Is f continuous at ( 0 , 0 ) ? ? Is f differentiable at ( 0 , 0 ) ? ? (e) Find the partial derivative f y at ( 0 , 0 ) by calculating it di- rectly with a limit: f y = lim h 0 1 h ( f ( , ) - f ( , )) = Do the partial derivatives f x and f y exist and are they contin- uous at ( 0 , 0 ) ? (Hint: to test continuity, you may want to use a similar calculation as you used to test the continuity of f) ? (b) The function f is differentiable at all points ( x , y ) 6 = ( 0 , 0 ) , as it is a rational fraction with denominator ( x 2 + y 2 ) 2 = 0 only when ( x , y ) = ( 0 , 0 ) . (c) The partial derivatives of f at points ( x , y ) 6 = ( 0 , 0 ) are given by f x ( x , y ) = 2 y ( x 2 + y 2 ) 2 - 8 x 2 y ( x 2 + y 2 )
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