### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

### Little and Large

A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?

### Approximating Pi

By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?

# How Does Your Function Grow?

##### Stage: 5 Challenge Level:

Think about the meaning of each of the functions carefully. Try writing them out in full, explicit notation and comparing them. The log function is somewhat different in definition to the other three, but you can look at the effect of log on the other functions to develop a sense for its speed of growth.