# Rate of change

Here, we discuss rate of change ...

## Introduction - Average rates/speeds

Q: How fast was he going?

This is quite an ordinary question, but it could be referring to a number of things. The obvious interpretation is "how far did he travel per unit of time", but it could also refer to the number of burgers eaten in an hour during a burger-eating contest! Whatever it was he was doing, the question addresses how much of it was done in a period of time, maybe a second, minute, hour, or even a year.

So imagine two scales that start together and running alongside one another. Take the burger-eating example. The first scale is a count of the number of burgers eaten (as they happen) and the second is time. If 18 burgers are eaten by a contestant during the entire contest which lasts for an hour, then his average speed of eating is 18 burgers/hr. However, with all those burgers, if we looked at how many burgers were eaten during each of the 60 minutes making up the hour, chances are that he will have eaten more in some minutes than during others.

The diagram shows how many burgers are eaten during the first few minutes. The two horizontal lines are the two scales: the top line is the 'burgers' scale and the bottom one is the time-scale. On the top scale, each burger-icon represents a burger eaten. You can see that the contestant started off well ... eating three in the first three minutes, but then slowed down and only managed a further three in the following six minutes.

So, his average speed was 1 burger/minute for the first three minutes and half a burger per minute in the next six minutes. This is quite easy to do in our heads, but how do we work this out mathematically?

### Step 1 - Choose intervals

Well, first choose a time interval. In the diagram, this is done by choosing two points which will represent a start and a finish time respectively. The time interval is the time between these two time points. At each of the two time points, draw a vertical line which meets the burger scale. These are then burger line points.

### Step 2 - Count them up

Count the number of burgers eaten between the burger line points and establish the length of time for the time interval between the time points.

### Step 3 - Find the ratio

Divide the 'burger count' by the length of time. Be careful here to specify the time units carefully.

### Example

Consider minute number 3 to minute number 8. The time interval is 5 minutes, during which he ate only 2 burgers. So he ate

2/5 = 0.4 burgers per minute during that time interval.

### Main points of this section

The main points to observe from this section are:

• That the speed of eating is changing all the time
• When quoting an average speed in this way, you must state a time interval that goes with it
• That the speed in this example really refers to the rate of change of burgers eaten with time

But is it possible to quote a speed which is true at an instant? It would be nice to be able to say how many burgers per minute the contestant was eating at (say) the 4-minute mark. Well, this is also possible and is explained in the next section!