Rate of Change and Limits

In the previous post, we learnt about What is Calculus and let’s now learn another important concept of Rate of Change.

We understand the change or small and continuous change but it’s very important to know at what rate this small change is being introduced.

Let’s see that using the linear equation of,

and we have also plotted it on the graph.

So now what we mean by the rate of change is, the change in “y” or also denoted by delta “y” with respect to the corresponding change in “x” or also known as delta “x”.

If you look at the graph above and if we need to find out the change in “x” and “y”, we can do that by using the coordinates of any of the two points on this graph.

So change in Y would be (13–9) and corresponding change in x will be (5–3). So the rate of change here is 2.

Now think for a moment that on the x-axis we have the time in minutes and on y-axis we have the distance traveled by a car in kilometers. In that case, the rate of change will be change in distance for a corresponding change in minutes. That will now be 2 km/minutes. This is nothing but the speed.

So as you can see, various such things can be derived using the Rate of Change. But there is much more to the rate of change than simply deriving these new metrics. It is fundamental to the concept of derivative which essentially forms the basis of Limits which then becomes the basis for Gradient Descent in Machine Learning.

Limits

So we have seen this function of

y= 1/x

and we know when we plot, it appears something like this and here X can never be zero as we can not divide anything by zero.

So lets try to see what happens when X is either too large or too small.

So as we start increasing the value of X, you will notice that the value of Y is decreases. As we increase it a bit further and you will see, that the value of Y dropped significantly. So we can safely say that as the value of X increases, Y will start approaching closer to zero or it will be almost zero when X is infinity.

Now, look at the graph and the table below.

If we decrease the value of X, you will see that there is exponential rise in the value of Y. As we decrease X further, Y becomes larger and larger. So as X approaches zero, Y will become almost infinity.

Remember, in both these examples, X has not reached those values. It is only approaching closer and closer to those values. The mathematicians love to call this as Limits. It is denoted as follows,

So we read this as, the limit of function 1/x as x approaches 0.5 is 2.

Similarly the limit of function 1/x, as x approaches infinity, is 0.

Summary

In this section, we saw an example of how a small change led us to finding the area of the circle. We also learnt, what is the rate of change as well as what do we mean by Limit of a function as the variable within that function approaches a particular value.

This article is part of my course, Complete Data Science and Machine Learning using Python where we not only go through all such mathematical concepts but also implement the Machine Learning Algorithms in depth.

Thanks for reading….