Tangent Sequences of a Cubic

Old mathematicians never die;
they just go off on a tangent.


Recently re-introduced to a pretty property of the tangents to cubics from Ross Honsberger's book, More Mathematical Morsels (Dolciani Mathematical Expositions).  I will illustrate a simple part of the property, then provide a source for a really nice extension that seems to be pretty new.

As the magician says, take a cubic, take any cubic... OK, so graph any y=f(x) where f(x) is a third degree polynomial. (you really don't have to graph it, but it might help you see a second property).  

1)  Pick any point other than the inflection point and record the x-value as A

2) Write the equation of the tangent line at that point
3)  Find where the tangent line intersects f(x) , call this B
4)  Repeat the process starting at Point B to get a point C,
5)  Do it again starting at C to get point D...
Look at the sequence A, B, C, D... What do you observe?


Ok, so let's practice a little integration. Find the area between the tangents and the curve.  Now think of the ratio of the two areas big/little...
Focus deeply, I'm reading your mind...
AHA, the ratio was 16.

"How does he do it?"

There is an additional nice extension about this idea explained by Alexander Bogomolny at his Cut-the-Knot web site.  Enjoy.