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.[more]
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).[less]