It isn’t just in science that we find ‘change’ – humans are fascinated with change and instinctively sense its importance. But it is hard to find an area of science that is not heavily dependent on ‘rates of change’, or in other words ‘how quickly things are changing’. ‘How quickly’ doesn’t have to mean ‘with time’ (e.g. ‘per second’), as you can see in the diagram below.
Each of the examples above is a rate of change. Notice how their unit includes the word ‘per’. This is typical for a quantity that is a rate of change. In fact, the third quantity has the word ‘per’ in it twice. See if you can work out the difference between ‘litres per second’ (which is itself a rate of change!) and ‘litres per second per degree Celsius’…
In this site, we’ll use this blue indented text for asides, and things that don’t really fit in the flow of the main narrative. You can skip them without breaking the flow of the main text. So, for our first one, just to stretch your mind a bit, try to imagine what it would mean to talk about the rate of change of the quantities in the diagram.
– For example, can you visualise what is meant by the ‘rate of change of global population rise’? This isn’t the rate of the rise – it’s how quickly that rate is changing.
– We can also imagine different rates of change for the temperature gradient across the wall: how quickly it changes ‘in time’ (in ‘degrees Celsius per centimetre per second’) or how quickly it changes ‘in space’ across the wall (in ‘degrees Celsius per centimetre per centimetre’)
What we have just described are ‘rates of change of rates of change’. Don’t worry if it was confusing at this stage – we’ll do more of it later on.
Rates of change with time
In this example, we are going to look at rates of change with time. We are doing this first as they are probably the easiest rates to think about. And we are going to follow the tried and tested method of looking at motion to exemplify the ideas.
We tried to find another example, just to be different. We looked at plant growth, bacteria populations, all sorts, but there was always a reason why it wasn’t as good an example for explanatory purposes. Quite often, that was because it was hard to visualise the ‘rate of change of the rate of change’. With motion, as you will see in a minute, you can have a quantity, the rate of change, and the ‘rate of change of the rate of change’ all meaningful in terms of everyday experience. We couldn’t get bacteria to do that for us.
So let’s have a look at a simple example of motion – an acceleration from rest, followed by a period of time at constant velocity. WARNING: learning from these videos may require effort. Pause any of our videos if they are going too fast, and think things through to get back on track. Understanding science is tricky if you ‘get behind the curve’, so when concepts build upon each other, try to understand each step in the sequence.
So to summarise the video:
- A rate of change of a quantity is the same thing as the gradient of the graph of the quantity
- The inverse operation (which would find the quantity from the rate of change) is the area under the line of the rate of change
There’s one important caveat when talking about areas under curves being the inverse of finding the gradient, and we’ll deal with this in Section 3…
But this example isn’t very realistic, because all the lines on our graph are perfectly straight – in real life, things don’t move as perfectly as that. We have done this in Section 1 to ‘ease you in’ – straight lines make it easy to work out the gradients of the lines, and the areas under them.
The dreaded calculus
Those of you who have studied calculus will probably know where we are heading here. Those of you who haven’t may well be terrified of the very word, because it seems to be the done thing to be scared of ‘calculus’ until you meet it. Well, we’ll put that right in Sections 2 and 3. Calculus is just ‘what you do when the lines aren’t straight’ (which is science is most of the time!). Admittedly, it can take some effort to get used to (it did for us), but when you do, it’s just part of your toolkit, like multiplying and dividing. If you do find it tricky, remember – you probably found adding and multiplying hard when you were younger, but you persevered and now you don’t!
By the way, if you think it’s a cop-out to oversimplify our first page by assuming straight line graphs, then we beg to differ! When science gets hard, the way to proceed is often to make simplifications, solve the simplified problem, and then see if you can make things more realistic. You could also have a reasonable philosophical argument with someone by claiming that, to some degree, ALL science works like this… We’ll discuss questions like this in our accompanying blog, and in Section 10 – Approximations.
OK, let’s carry on. Some sensible places to go next are listed below. Of course, you could go back to the home page and from there to somewhere entirely different…
Link to Section 2 – Differentiation
Makes sense to go here, because differentiation is how to calculate rates of change when the lines aren’t straight
Link to Section 10 – Approximations (Coming Soon)
If you were intrigued by the final ‘blue section’, you could go here first… (when we have written it…)
Link to Home Page
And if you don’t like our suggestions, just go back here…