Tangent at self-intersection point

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Homework Help Overview

The problem involves finding the slopes of the tangent lines at the self-intersection point (0,0) of the curve defined by the equation x^3 - y^2 + x^2 = 0. The context is centered around implicit differentiation and the challenges posed by undefined derivatives at specific points.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss implicit differentiation and the implications of an undefined derivative at the intersection point. There is a suggestion to solve for y explicitly and differentiate, but questions arise regarding the feasibility of this approach for more complex equations.

Discussion Status

The discussion is ongoing, with participants exploring different methods of differentiation. Some guidance has been offered regarding explicit differentiation, but there is no consensus on the best approach, particularly for cases where explicit formulas are not easily obtainable.

Contextual Notes

Participants are considering the limitations of implicit differentiation in scenarios where derivatives may be undefined, and the implications for finding tangent lines at self-intersection points.

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Homework Statement



Find the slopes of the two tangent lines of x^3-y^2+x^2=0 at 0,0.

Homework Equations


Differentiating implicitly we get (dy(x))/(dx) = (x (2+3 x))/(2 y).


The Attempt at a Solution


I'm not sure how to deal with the derivative being undefined at 0,0 when there are clearly two tangent lines.
 
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Try solving the equation for y and differentiate each branch explicitly.
 
I'm aware it can be done that way. However, is there a way to do it without separating the curve into branches, for equations where that is not so easy (or impossible)?
 
Good question. In the situation where you can't write an explicit formula for the function I don't think there is any guarantee you can do it. I could be wrong, but I don't think so.
 

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