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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