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Lecture2 - REVIEW OF THE CHAPTER 12 Problem 1 Find the...

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REVIEW OF THE CHAPTER 12 Problem 1. Find the total differential of the function f ( x, y ) = xye x y at the point (0 , 1) . Solution. The total differential of the function z = f ( x, y ) given by formula dz = f x ( x, y ) dx + f y ( x, y ) dy. For our function we have f x ( x, y ) = ye x y + xe x y , f y ( x, y ) = xe x y - x 2 y e x y Next we need to find the values of the partial derivatives at the point (0 , 1) . f x (0 , 1) = 1 f y (0 , 1) = 0 . Hence dz = dx. Problem 2. The surface is given by equation F ( x, y, z ) = x 2 + y 2 - z 2 = 0 . Find equations of a normal line and a tangent plane to this surface at the point P = (1 , 0 , 1) . Solution. The equation of the tangent plane to the surface F ( x, y, z ) = 0 at the point P = ( x 0 , y 0 , z 0 ) is given by formula F x ( x 0 , y 0 , z 0 )( x - x 0 )+ F y ( x 0 , y 0 , z 0 )( y - y 0 )+ F z ( x 0 , y 0 , z 0 )( z - z 0 ) = 0 . (1) And the equation of the normal line to the surface F ( x, y, z ) = 0 at the point P = ( x 0 , y 0 , z 0 ) is given by formula x y z = x 0 y 0 z 0 + F x ( x 0 , y 0 , z 0 ) F y ( x 0 , y 0 , z 0 ) F z ( x 0 , y 0 , z 0 ) t. (2) Typeset by A M S -T E X 1
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Let us find the partial derivatives of the function F F x ( x, y, z ) = 2 x F
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