The Tangent Line of an Implicit Relation

This Demonstration shows the tangent line to an implicitly defined curve in the Cartesian plane. Drag the point along the curve to see the tangent line, the point of tangency, and the equation of the line tangent to the curve. Since the algorithm for finding the point of tangency may not converge to the nearest point of solution to the implicit equation, the location of the point of tangency may be suprising. And it is possible that the algorithm may not converge at all to a point on the curve; in these instances, the tangent line will take on an undefined slope or a zero-valued slope, though of course, there are points of convergence that result in undefined slope or zero-valued slope for the tangent line.



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