Family of Curves, Tangents, and Intuition

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Visualizing a family of curves can give an intuitive feeling for their properties, but sometimes that intuition is wrong. So it is helpful to be able to compute, check, and display results.

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Each curve has a vertical tangent at , which comes from taking the limit. So there is no derivative at .

At , the curves with odd never have the same tangent and are never horizontal.

At , the curves with even alternate between having a derivative of and .

At , the curves with odd always have the same tangent. At (, 0), the curves with even always have the same tangent (different from the odd case).

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Contributed by: Jacqueline Zizi (March 2011)
Additional contributions by: Michael Schreiber
Open content licensed under CC BY-NC-SA


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