Family of Curves, Tangents, and Intuition

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.
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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