# Julian's musings

## Increasing functions and functions increasing

Here’s the graph of $y=-\dfrac{1}{x}$ for $x\ne0$.

Where is this function increasing? Is it an increasing function?

Looking at various recent examination papers, it has become clear to me that there is significant confusion between these two questions. This post is intended to bring some clarity to the situation.

At the start of this post, I will give an example of the confusion as it appears in exam questions (and probably elsewhere), and clarify what the two different phrases mean using the above example. I will then delve more deeply into the mathematics of these two things, going beyond A-level content, and use some undergraduate analysis to find equivalent conditions for them in terms of the derivatives of the functions. It is fine to skip over the technical stuff and just look at the results (theorems)!

(Exactly the same applies to the use of the term “decreasing”, but for simplicity we will focus on increasing functions in this post.)

## A visit to Michaela

Having recently listened to about 5.5 hours of Craig Barton interviewing Dani Quinn (part 1 and part 2), the Head of Mathematics at Michaela Community School, I decided that it was worth visiting the school to see their principles in action for myself, so last week, I took to the buses to visit Wembley.

## Small angle approximations - an application

I thought a bit more about my previous post on small angle approximations, and decided it might be helpful to describe an application of the small angle approximations. While this example contains non-examinable aspects (at least in single maths A-level), the context should be fairly familiar (or can easily be demonstrated), and the mathematics is accessible to single maths students (at least as a demonstration). It also ties together ideas from mechanics and pure maths, so is helpful in this regard.

The question is: what is the period of a pendulum?