Transcription
Let us look at the graphs of these functions. Is there anything PECULIAR near the origin? Do you observe anything odd?
We see that as ‘X equals to zero’, the value of ‘Y’ changes abruptly in all three functions. In the first case, as the value of ‘X’ increases till here, the value of ‘Y’ decreases, but then suddenly, it starts to increase from this point. Similarly, this happens in the second case. And in the third one, the curve almost looks like a straight line at ‘X equal to zero’. So a question arises here. If the change in these functions is sudden at ‘X equal to zero’, then what is their RATE of change at ‘X equal to zero’? What will we get if we apply the process of differentiation here? What will be the derivative of these functions at ‘X equal to zero’?
In this video, we will understand such peculiar cases. We will see what it means, when a derivative of a function does not exist at a particular value of ‘X’. Let’s get started.
In our previous video, we asked the derivative of this function at ‘X’ equal to zero? Finding the derivative at ‘X’ equal to zero means we have to observe how this ratio behaves for values of ‘X’ near 'zero.' So if we substitute ‘X not’ equal to zero here, then the ratio will depend only on the value of ‘delta X’. It can either be greater than or less than zero. So let’s take a positive number ‘h’. In the first case, delta X is equal to ‘h’ and in the second case it is equal to negative ‘h’. Now we find the average rate of change separately for the two cases.
Consider the first case. Here according to this, the value of ‘F’ of zero plus ‘h’ will be ‘h’ and that of ‘F of zero’ will be 'zero'. So after simplification, we get the average rate of change equal to one. We see that this value is a constant; it does not depend on ‘h’. So now when ‘delta X tends to zero’, that is when ‘h tends to zero’ the average rate of change will always be 'equal to one.' Here the average rate is calculated between ‘X equal to zero’ and the value of ‘X’ greater than Zero. So we say in the limit delta X tends to zero PLUS the average rate of change approaches one. The plus symbol means we are finding the average rate between ‘zero’ and values of ‘X’ greater than zero.
Now let’s consider the second case. Here delta X is equal to negative of ‘h’, so the denominator of this ratio will be equal to ‘negative of h’. Now since ‘zero minus h’ will be less than zero, we get ‘F of zero minus h’ as negative of ‘zero minus h’, which is ‘h’. So after simplification, we get the average rate of change as ‘negative one’. So in this case we see that as 'delta X tends to zero', this ratio is always equal to ‘negative one’. Similar to this, we say that as the limit "delta X tend to zero MINUS", the average rate of change approaches negative one. The minus indicates that we are finding the average rate between ‘zero’ and values of ‘X’, less than zero. But notice as delta X tends to zero; the average rate of change of these two cases does not approach the same number. This means that the limit of this ratio as delta X tends to zero does not exist. So the derivative of this function at X equal to zero does not exist.
What does this mean? Why do we get two different values for the average rate as delta X tends to zero? To understand this, let’s look at the graph of the function in the next part.
To draw a graph of this function, let us look at its conditions. For ‘X’ greater than or equal to ‘Zero’, the value of ‘Y’ is equal to ‘X’. So for ‘X’ greater than or equal to ‘Zero’, the graph will be this straight line. Now for ‘X’ less than zero, the value of ‘Y’ is equal to ‘negative of X’. So for ‘X’ less than zero, the graph will look like this. This is the same graph that we saw in the beginning.
Now let’s understand why the derivative does not exist at ‘X equal to zero’. In our previous videos, we understood geometrically what we mean by the derivative of a function. We saw that to find the rate of change at any point on a curve, we can approximate a small region on it by a straight line. Do you remember why? This is because as we zoom around a point on a curve, it almost looks like a straight line. This straight line is actually the tangent line at that point on the curve. We obtain this tangent line as the limit of these secant lines when delta X tends to zero. The slope of the tangent line is equal to the instantaneous rate of change of ‘Y’ with respect to ‘X’, that is the derivative of the function. But wait…. is this true for any curve? The graph of this function is an exception however. If we observe the region of the graph around ‘X equal to zero’, we see that no matter how much we zoom in, it does not look like one single straight line. At ‘X equal to zero’ there will always be this sharp corner formed by the intersection of these two lines. So we see that there exists no straight line by which we can approximate this region. This is the reason why we get two different values for the average rate of change between ‘X equal to zero’ and its nearby values. As we approach from the left, that is when delta X is less than zero, the average rate is equal to the slope of the secant line between these two points. We see that the secant line and the graph of the function coincide. Similarly when we approach from the right, that is for ‘Delta X’ greater than zero, the secant line coincides with this straight line. Now as delta ‘X’ tends to Zero, these secant lines remain the same. We can think of them as two different tangent lines at ‘X equal to zero’. One with a slope equal to one and the other with a slope equal to ‘negative one’. So we see that tangent lines at ‘X equal to zero’ does not exist. That is, there is no straight line by which we can approximate this region of the graph. So there is no question of finding the slope of a tangent line. So we can say that the derivative of a function does not exist at ‘X equal to zero’.
Now here, there is no special significance to the value; X equal to zero. In a graph for any function, a corner can occur at any value of 'X'. So if we encounter corners like this, we can be sure that the derivative of a function at that corner point does not exist.
Now what about the other two functions that we mentioned in the beginning? We can see that in these two cases, if we zoom in enough, we see that the curve looks almost like a vertical line. So what is the slope of a vertical line? What do you think? Share your thoughts in the comment section below. And we shall see what it means in the next lesson. Stay updated by subscribing to our channel.