Quadratics Tangent to a Cubic
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This Demonstration shows a cubic polynomial, which you can modify using the locators, and a quadratic polynomial that is tangent to the cubic at the point determined by the slider. That the family of quadratics parametrized by the point of tangency do not intersect and fill the plane is the content of an amazing theorem referenced in the details below.
Contributed by: Robert L. Brown (November 2012)
Open content licensed under CC BY-NC-SA
The "choose quadratic" control determines the point at which and touch with second-order contact (or osculate, which means kiss in Latin).
Check the "family" box to show all quadratics for which the value of the tangent point is an integer.
The family of quadratics that osculate a given cubic has very interesting properties not shared in general with all osculating functions.
1. is unique for a given tangent point.
2. No two intersect.
3. The set of all fills the entire plane.
Arbitrary can be chosen by dragging the locator.
 N. J. Wildberger. Cubics and the prettiest theorem in calculus [Video]. (Nov 9, 2012) www.youtube.com/watch?v=DAHBgcDJQjw&feature=edu&list=PL5A714C94D40392AB.