ptest3 - Practice Test 3 for 1502KB 1 Are the following...

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Practice Test 3 for 1502KB 1. Are the following function continuous or discontinuous, justify your answer ( a ) f ( x, y ) = x - y x + y for ( x, y ) = (0 , 0) 0 for ( x, y ) = (0 , 0) ( b ) f ( x, y ) = x 2 y x 2 + y 2 for ( x, y ) = (0 , 0) 0 for ( x, y ) = (0 , 0) ( c ) f ( x, y ) = x | y | x 2 + | y | for ( x, y ) = (0 , 0) 0 for ( x, y ) = (0 , 0) ( d ) f ( x, y ) = 3 x 2 y 2 x 3 + y 6 for ( x, y ) = (0 , 0) 0 for ( x, y ) = (0 , 0) ( e ) f ( x, y ) = 3 x 2 y 2 x 2 + | x | y 2 for ( x, y ) = (0 , 0) 0 for ( x, y ) = (0 , 0) 2. Calculate the partial derivatives, up to second order, of the following functions ( a ) f ( x, y ) = e - y 2 cos( x 2 + y 2 ) ( b ) f ( x, y ) = sin(3 xy ) ( c ) f ( x, y ) = x 2 + y 2 3. Function y ( x )is determined by function F ( x, y ) = 0. Find dy dx . ( a ) F ( x, y ) = ( x + y ) e - xy ( b ) F ( x, y ) = ln | x 2 + y 2 + 1 | + sin( x - y ) ( c ) F ( x, y ) = x 3 + y 3 - cos( x 2 y ) 4. Find the linear approximation of the following function at the indicated point ( x 0 , y 0 ). ( a ) f ( x, y ) = ( x - y ) e xy ( x 0 , y 0 ) = (0 , 0) ( b ) F ( x, y ) = arctan( x + y ) ( x 0 , y 0 ) = (0 , 0) ( c ) F ( x, y ) = x 2 + y 2 - 2 xy ( x 0 , y 0 ) = ( - 1 , 1) 5. Find the tangent plane or tangent line to the following surfaces or curves. ( a ) z = f ( x, y ) = x + 2 y, ( x 0 , y 0 , z 0 ) = (1 , 0 , 1) ( b ) z = f ( x, y ) = ( x 2 + 1) ln y ( x 0 , y 0 , z 0 ) = (0 , 1 , 0) ( c ) F ( x, y ) = x 2 - y 3 = 0 ( x 0 , y 0 ) = (1 , 1) ( c ) F ( x, y ) = xy = 1 ( x 0 , y 0 ) = (1 , 1) 1

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6. Find the directional derivative of the following functions along the indicated direction
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