Calculus

\(k^x-x^k\) 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 Biggest Product

I start this activity by handing out buckets of tiles (any small objects will do) and asking students to take 10. They can sort their ten tiles into any number of piles, and their “prize” is the product of the sizes of their piles — so if they sort them into piles of size 5 and 5; their prize is 5 x 5, and if they sort into piles of sizes 1, 2, 3, and 4, their prize is 1 x 2 x 3 x 4. The goal, of course, is to get the highest possible prize, and then to generalize for other numbers of tiles. Read more >>