“But how do we measure the slope of a line that is curved? Straight lines were easy, but curvy lines? We could try to estimate the slope by drawing a straight line, called a tangent , which just touches that curved line in a way that tries to be at the same gradient as the curve just at the point. This is in fact how people used to do it before other ways were invented.
Let’s try this rough and ready method, just so we have some sympathy with that approach. The following diagram shows the speed graph with a tangent touching the speed curve at time 6 minutes.
To work out the slope, or gradient, we know from school maths that we need to divide the height of the incline by the extent. In the diagram this height (speed) is shown as Δs , and the extent (time) is shown as Δt . That symbol, Δ called “delta”, just means a small change. So Δt is a small change in t .
T he slope is Δs / Δt . We could chose any sized triangle for the incline, and use a ruler to measure the height and extent. With my measurements I just happen to have a triangle with Δs measured as 9.6, and Δt as 0.8. That gives the slope as follows:
We have a key result! The rate of change of speed at time 6 minutes is 12.0 mph per min.
You can see that relying on a ruler, and even trying to place a tangent by hand, isn’t going to be very accurate. So let’s get a tiny bit more sophisticated.
Calculus Not By Hand
Look at the following graph which has a new line marked on it. It isn’t the tangent because it doesn’t touch the curve only at a single point. But is does seem to be centred around time 3 minutes in some way.
In fact there is connection to time 3 minutes. What we’ve done is chosen a time above and below this point of interest at t =3. Here, w e’ve selected points 2 minutes above and below t =3 minutes. That is, t =1 and t =5 minutes.
Using our mathematical notation, we say we have a Δx of 2 minutes. And we have chosen points x-Δx and x+Δx . Remember that symbol Δ just means a “small change”, so Δx is a small change in x .
Why have we done this? It’ll become clear very soon – hang in there just a bit.
If we look at the speeds at times x-Δx and x+Δx , and draw a line between between those two points, we have something that very roughly has the same slope as a tangent at the middle point x . Have a look again at the diagram above to see that straight line. Sure, it’s not going to have exactly the same slope as a true tangent at x , but we’ll fix this.
Let’s work out the gradient of this line. We use the same approach as before where the gradient is the height of the incline divided by the extent. The following diagram makes clearer what the height and extent is here .
The height is the difference between the two speeds at x-Δx and x+Δx , that is, 1 and 5 minutes. We know the speeds are 1 2 =1 and 5 2 =25 mph at these points so the difference is 24. The extent is the very simple distance between x-Δx and x+Δx , that is, between 1 and 5, which is 4. So we have:
The gradient of the line, which is approximates the tangent at t =3 minutes, is 6 mph per min.
Let’s pause and have a think about what we’ve done. We first tried to work out the slope of a curved line by hand drawing a tangent. This approach will never be accurate, and we can’t do it many many times because, being human, we’ll get tired, bored, and make mistakes. The next approach doesn’t need us to hand draw a tangent, instead we follow a recipe to create a different line which seems to have approximately the right slope. This second approach can be automated by a computer and done many times and very quickly, as no human effort is needed.
That’s good but not good enough yet!
That second approach is only an approximation. How can we improve it so it’s not an approximation? That’s our aim after all, to be able to work out how things change, the gradient, in a mathematically precise way.”