futurebird – Page 4 – mathematical sandbox

Fixed points.

"Prove that if f,g are continuous functions on [0,1] with range [0,1], then there exists a point c such the $f(c)=g(c)$"

This is a variation on a popular problem given to encourage students to use the [[wiki|intermediate value theorem|intermediate value theorem]]. The proof involves a function $h(x)=f(x)-g(x)$, this function can't be exclusively positive or negative, so we can find c so that $h(c)=0$.

Visit the official website here.

The proof is really about getting the given situation in to a form where it will "plug in" to the definition nicely. It has no deeper insight about why this happens. For that we must look at the fixed point theorems.
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An enjoyable lecture on Partitions for beginners.

I just found this on iTunes U. (Click 'New Theories Revel the Natural Numbers') It outlines the basic theory and major questions in Partition Theory.

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Enjoy it!

Rational points on the folium.

My students are studying implicit differentiation, we look at the folium of Descartes:

$x^3 + y^3- 6xy =0$

They found the equation of the tangent at $(3,3)$

For their homework I want to have them look at another point, but I wanted it to be rational, I almost gave them the origin... But that's good. The expression given by implicit differentiation is undefined there. (Though, one can find the slope of the two tangents using limits.) Also sometimes the homework has more to do with writing than with math. When that happens sometimes you may require assistance, and if that's the case you can find it online. You could hire a college assignment writing service for example.

Next I searched for rational points between 0 and 3. I'm confidant that there are only two x values that give at least one rational y value these are 36/217 and 216/217. (I used the rational root theorem to show this.)

Sadly, these numbers are too messy to use in a homework! Especially when they are mad at me for giving a very tough problem the other day.

Each of these x values has 3 corresponding y values, what are they?
Well, it's time to look for another relation... I want something that will let them find more than one tangent line at a given x value. Oh, and it must be rational... And pretty!

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