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3rdsol - Calculus 119 3rd Week 1 1 Let’s first obtain the...

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Unformatted text preview: Calculus 119 3rd Week 1 1. Let’s first obtain the formula. x 4 + y 4 = x 2 y 2 ⇒ 4 x 3 + 4 y 3 y = 2 xy 2 + x 2 2 yy ⇒ y = 2 x 3- xy 2 x 2 y- 2 y 3 Why is this result meaningless? This result meaningless, because the solution set of the equation x 4 + y 4 = x 2 y 2 consists of a single point as seem below. x 4 + y 4 = x 2 y 2 ⇒ x 4- 2 x 2 y 2 + y 4 + x 2 y 2 = 0 ⇒ ( x 2- y 2 ) 2 + x 2 y 2 = 0 ⇒ ( x 2- y 2 ) 2 = 0 and x 2 y 2 = 0 since both of ( x 2- y 2 ) 2 and x 2 y 2 are positive. ⇒ x = 0 and y = 0 2. By using implicit differentiation, we obtain the following equality cos ( x + 2 y )(1 + 2 y ) = 2 cos ( y ) + 2 x (- sin ( y ) y ) The slope of the tangent line at point (0,0) (i.e origin) is ∂y ∂x | (0 , 0) . It can be computed by replacing (0,0) with (x,y) ∂y ∂x | (0 , 0) = y (0) = 1 2 Consequently, The equation is y = 1 2 x 1 Please email all corrections and suggestions to these solutions to [email protected]
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3rdsol - Calculus 119 3rd Week 1 1 Let’s first obtain the...

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