hw8solutions

# hw8solutions - Math 2250 Written HW#8 Solutions 1 The...

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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 . 10-7.5-5-2.5 2.5 5 7.5 1-5-2.5 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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hw8solutions - Math 2250 Written HW#8 Solutions 1 The...

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