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Math Help - Regarding implicit differentiation

  1. #1
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    Regarding implicit differentiation

    How do I show that the curve x^3+y^3 = 3xy is symmetrical about the line y=x??
    The curve is not one-to-one so I cant show that it is a self inverse. All I can think of is to show that the expression is the same when x and y are swapped. Is that enough??

    Suppose that |x| and |y| (in the above expression) are both very large. Explain why x+y~k, where k is a constant?? Because the two expressions are the same when the variables are swapped??

    Thanks
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  2. #2
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    Quote Originally Posted by woollybull View Post
    All I can think of is to show that the expression is the same when x and y are swapped. Is that enough??

    Yes, that is enough (because reflection in the line y=x is what happens when you interchange x and y).

    Quote Originally Posted by woollybull View Post
    Suppose that |x| and |y| (in the above expression) are both very large. Explain why x+y~k, where k is a constant??

    Let m=y/x. Then the equation x^3+y^3=3xy can be written \frac{1+m^3}m = \frac3x. The right-hand side of that goes to 0 as |x|\to\infty, from which you can see that m\to-1
    as |x|\to\infty.

    Now factorise
    x^3+y^3 as (x+y)(x^2-xy+y^2) to see that x+y = \frac{3m}{m^2-m+1}\to-1 as m\to-1.

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    Thanks a lot!!
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