In the last tutorial, I told you that calculus is just a set of tools for describing the relationship between a function and the change in its variables.

How Calculus is useful 

And so, in this tutorial, we're going to explore what this means and how it might be useful. Let's start by having a look at a classic example, a speed versus time graph for a car. The most obvious thing this graph tells us is that the car's speed is not constant because a constant speed would just be a flat horizontal line.

Furthermore, starting from zero speed at zero time, this car's speed is initially increasing with time, which is another way of saying that it is accelerating. Towards the end of the time period, the car's speed is shown to be rapidly decreasing, meaning that it is decelerating. There's a lot of information in this graph, and calculus will allow us to extract much more than just the speed. As we've already said, a horizontal line implies a constant speed. And then, the more the line is sloping up, the greater the acceleration. In fact, acceleration can be defined as the local gradient of a speed-time graph. And clearly, acceleration itself is also a function of time in our example.

We refer to the gradient at a single point as the local gradient.  And we can illustrate this concept by drawing a tangent line, which is a straight line that touches the curve at a point and is also the same gradient of the curve at that point. After the initial acceleration, the car's speed reaches a peak and then begins to decelerate again.

Deceleration has a negative slope. By recording the slope of these tangent lines at every point, we could plot an entirely new graph that would show us acceleration against time rather than speed against time.

Before we plot this, for the complex case, let's think about what this acceleration time graph would look like for a car traveling at constant speed. Constant speed, we've got a flat horizontal line on our speed-time graph. Then, we can say that its gradient is, of course, zero. So the acceleration time graph would also just be a horizontal line. But in this case, our acceleration equals zero. Going back to a more complex case, let's just talk through what we should expect before we have a go plotting it.

So, initially, the gradient is positive and fairly constant, before it drops to zero at the peak. It then becomes negative for a period before returning to zero.

Let's now take a look at the graph for acceleration versus time overlaid onto the speed-time graph. Don't forget the vertical axis for the blue line is speed and will have units of distance per time, whereas the vertical axis for the orange line is acceleration and will have units of distance per time squared. Because they have got different units, we can scale either of these two lines vertically in this block, and the meaning would still be identical. However, these have been scaled just to make the most use of this plot area available.

You can see the points at which the acceleration function is zero, i.e. where it crosses the horizontal axis, coincide with where the speed-time graph is flat and has zero gradients as we would expect. Although we will be discussing the formal definition of a derivative in a later tutorial, what we've just done by eye is the essence of calculus, where we took a continuous function and described its slope at every point by constructing a new function, which is its derivative. We can in principle plot the derivative of the acceleration function following the same procedure, where we simply take the slope of the acceleration function at every point.

This is the rate of change of acceleration which we can also think of as being the second derivative of the speed, and it's actually referred to as the jerk of the car. Think of the jerky motion of a car as it stops and starts. Now, you may have never heard of this concept before, but hopefully, just by telling you that it's the derivative of the acceleration curve, this should be all you need to know to approximately sketch the jerk. Also very interesting is the idea of taking our baseline speed function and trying to imagine what function this would have been the gradient of as in applying the inverse procedure to the one that we have just discussed. We can refer to this as the anti-derivative, which for those who have done calculus before, you may remember that it's closely related to something called the integral. For the example we are discussing here, it would represent the distance of the car from its starting position. This should make more sense when you consider the change in the distance with respect to time, i.e. the slope of the distance time graph, i.e. how much distance you are covering per unit time, is just the speed. This analysis of slopes is all we are going to discuss in the video. And even though we haven't laid out yet the formal definition of calculus, you should already be able to answer lots of gradient type questions in the following exercise, which will put you in a very strong position to start thinking about differentiation more formally.