Algebra

\(k^x-x^k\) Slider and \(x^y=y^x\) graph

Use the slider to change the value of k, and to see dynamically how the graph of \(h(x)=k^x-x^k\) changes. Negative values have \(x^k>k^x\). Which value of k gives a graph that is never negative? Why?

Here is a very pretty graph of \(x^y=y^x\) with areas where \(x^y < y^x\) shaded in green and areas where \(x^y > y^x\) shaded in purple.

I am working on an article about some problems related to these equations, including The Biggest Product Problem. There’s a lot of interesting stuff here!

The Handshake Problem

The handshake problem is an old chestnut — if everyone in the room shook hands with everyone else, how many handshakes would there be? Then generalize. What I have to add to teaching the problem is a handout (doc version, pdf version) with different (fictional, but based on reality) students’ strategies for solving the problem. This handout is good for homework after students have worked on the problems themselves and listened to their classmates’ strategies. Read more >>