MF75: Cubics and the prettiest theorem in calculus

We introduce cubic polynomials, and the basic algebraic calculus for them, involving their Taylor expansions, subderivatives and tangent lines and tangent conics. The tangent conics are particularly interesting, and lead to the (arguably!) prettiest theorem in calculus.

Category:

Education

Tags:

• MathFoundations75
• MF75
• cubics
• mathematics
• foundations
• Wildberger
• calculus
• prettiest
• theorem
• polynomials
• algebraic
• tangent
• lines
• conics
• disjoint
• subderivatives
• Taylor
• education

[richiedon100]

When you have to solve the cubic equation for (r-x) couldn't that

lead to an irrational value for r?

[njwildberger]

@richiedon100 It is better to say: we cannot necessarily solve for r-x. But when we can, there is at most one solution---that is the key point. Irrational values/numbers don't really exist. A modern mirage!

[richiedon100]

@njwildberger I've heard you briefly mention sqrt2 and sqrt3. And what about

pi. Don't we need that for volume and area problems? So I'm still a bit confused

by irrationals.

[njwildberger]

@richiedon100 Don't worry, the confusion with pi and related so called irrational numbers is a widespread phenomenon. This year I will try to clarify some of the issues in this series.

[Toxie207]

Very nice - so this leads us to the idea of polarity?

[njwildberger]

@Toxie207 I think the idea of polarity is more closely hinted at by the last video on Line and parabolas II. However it is a very important notion that I will be talking about at some length in the future.

[savio2006]

There is a typo on page 5 (@13:40) at the example: it should be T(1, 1) p = 8 - 8(alpha)

[teavea10]

Does the disjoint tangent conic theorem carry up to any higher degree polynomials?

[MrElephantMemory]

You have a beautiful handwriting, which is one of the rarest things in mathematicians. Besides, you are very talented at explaining things and speak clearly. All in all, you are an excellent instructor.

[paul1964uk]

Amazing result. Very elegant.