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Unformatted text preview: Math 2250 Written HW #8 Solutions 1. The folium of Descartes, pictured below, is determined by the equation x 3 + y 3 9 xy = 0 . 107.552.5 2.5 5 7.5 152.5 2.5 5 (a) Determine the slopes of the tangent lines to the folium at the points (4 , 2) and (2 , 4). Answer: We intend to determine y at these points using implicit differentiation, so differentiate both sides of x 3 + y 3 9 xy = 0 with respect to x : d dx ( x 3 + y 3 9 xy ) = d dx (0) 3 x 2 + 3 y 2 y 9(1 y + xy ) = 0 3 x 2 + 3 y 2 y 9 y 9 xy = 0 3 x 2 9 y + y (3 y 2 9 x ) = 0 . Therefore, we have y (3 y 2 9 x ) = 9 y 3 x 2 or, equivalently, y = 9 y 3 x 2 3 y 2 9 x . Hence, the slope of the tangent line at (4 , 2) is y = 9(2) 3(4) 2 3(2) 2 9(4) = 18 48 12 36 = 30 24 = 5 4 . 1 Likewise, the slope of the tangent line at (2 , 4) is y = 9(4) 3(2) 2 3(4) 2 9(2) = 36 12 48 18 = 24 36 = 4 5 . (b) At what point other than the origin does the folium have a horizontal tangent line? Answer: The folium will have a horizontal tangent line when y = 0. Since we know y = 9 y 3 x 2 3 y 2 9 x , this will occur exactly when 9 y 3 x 2 = 0 ....
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This note was uploaded on 01/18/2012 for the course MATH 2250 taught by Professor Chestkofsky during the Fall '08 term at University of Georgia Athens.
 Fall '08
 CHESTKOFSKY
 Calculus, Slope

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