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Unformatted text preview: 1 . a) The sun is melting a rectangular block of ice. When the block’s height x is 1 ft and the edge y of its square base is 2 ft, its height is decreasing at the 2 in./h and its base edge is decreasing at 3 in./h. What is the block’s rate of change of volume V = xy 2 at that instant? Answer. By chain rule, dV dt = ∂V ∂x dx dt + ∂V ∂y dy dt = y 2 dx dt + 2 xy dy dt = 2 2 · ( 2 12 ) + 2 · 1 · 2 · ( 3 12 ) = 5 3 ( ft 3 /h ) . b) Determine the set of points at which the function is continuous: f ( x,y ) = xy x 2 +2 y 2 , if ( x,y ) 6 = (0 , 0) , 1 3 , if ( x,y ) = (0 , 0) . Answer . The function is continuous for ( x,y ) 6 = (0 , 0), as a quotient of two continuous functions (denominator nonzero). The value of f along the line y = x is x 2 3 x 2 = 1 3 , therefore it is not continuous at (0 , 0) . The set of continuity are all ( x,y ) 6 = (0 , 0) . 2. a) Find the linearization of z = f ( x,y ) = xe 2 y at (1 , 0). Approximate f (1 . 1 , . 1). Write the equation of the tangent plane to1)....
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This note was uploaded on 12/16/2011 for the course MATH 226 at USC.
 '07
 KAMIENNY
 Math, Calculus, Rate Of Change

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