## Sunday, 1 May 2011

### 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...