Transcription
What is the slope of a vertical line? Intuitively, the question doesn't make any sense, right? We know that the slope of a straight line measures its steepness. So, how steep is a vertical line?
We encounter situations like this in these types of functions. Observe the graph of the functions near X equal to zero. If we want to draw a tangent line here, we would immediately draw a vertical line like this. But previously, we've seen that the slope of a tangent line tells us the rate of change of a function—that is, its derivative. So, what's the derivative at X equal to zero here? Or indirectly, what's the slope of a vertical line? We can now understand these cases in this video.
Let's start with this function. In the previous video, we saw how we can intuitively understand whether the derivative exists or not at a point just by looking at a function's graph. If, after zooming around that point, the graph doesn't look almost like a straight line, then we know that the derivative does not exist at that point. So, what does this tell us about this function at X equal to zero? If we zoom in here, we can see that it looks like a vertical line. So, does this mean the derivative exists at this point? Let's see.
By now, we are familiar with how to find the derivative of a function. In order to find this limit of the average rate as Delta X tends to zero, we consider two cases. First, when Delta X is greater than zero. And second, when Delta X is less than zero. So, can you find the average rate of change in these two cases? Let's look at the calculations quickly. In the first case, let Delta X be equal to h, and in the second case, the negative of h. In both the cases, 'h' is a positive number.
Now, if we calculate the average rate in the first case, we will get that to be equal to one over h raised to the power 1 over 3. We proceed in a similar way for this second case. Here, it will be equal to the negative of 1 over h raised to the power 1 over 3. Now, can you tell me what's the next step? We know we need to find whether these average rates approach the same number as Delta X, that is, h tends to 0. For this, let's represent these average rates on a number line. In the first case, the average rate will be a positive number. And in the second case, it will be a negative number.
Now, as h tends to 0, in the first case, the value of the average rate will move up on the number line. That is, it will approach a larger positive number. And in the second case, its value will move down on the number line. That is, it will approach a smaller negative number. So, we see that as Delta X tends to 0 in both the cases, the average rate does not approach any particular number. In the first case, we say that as Delta X tends to 0 plus, the average rate tends to positive infinity. And in the second case, as Delta X tends to 0 minus, the average rate tends to negative infinity. It means that in both the cases, the magnitude of the average rate keeps on increasing. So, from this, we can conclude that this limit of the average rate does not exist as Delta X tends to 0. So, this means that the derivative of this function at X equal to 0 does not exist. Even though the very small region of the graph near X equal to 0 looks like a straight line, the derivative doesn't exist at this point.
So, what does all this mean geometrically? Let's continue this in the next part. Let's divide the graph of the function into two parts corresponding to these two cases. These average rates will be equal to the slope of the corresponding secant lines on the graph. For the first case, the slope of the secant line will be positive and equal to one over h raised to the power 1 over 3. In the second case, it will be equal to the negative of 1 over h raised to the power 1 over 3. Now, we can see that as Delta X tends to 0, these secant lines will approach these vertical straight lines. But now the derivative of this function does not exist because we cannot define the slope of a vertical line.
Why? For example: consider the straight line graph of a function. Let's say the slope of the straight line is 'M'. So, it means for any two points on this line, the ratio of change in Y over change in X is equal to M. So, the slope tells us how fast or slow the value of y changes with respect to X. But now observe these vertical lines. For any two points on it, the change in X will always be zero. So, we cannot define this ratio here. Hence, the slope of a vertical straight line is undefined.
Now, one subtle thing to observe is that these two vertical lines are not the same. According to this, we can say that in the first case, the secant line approaches a tangent line with infinite slope. But the slope is always positive. While in the second case, we can say that the secant line approaches a tangent line with infinite slope, but the slope is always negative. So, in the first case, the tangent is directed in this direction. And in the second case, in this direction. So, we see that we get two different tangent lines at the same point. Therefore, to start with, in this case, a tangent line at X equal to zero does not exist. And so the derivative doesn't exist. In a graph of a function, this type of pattern is called a cusp. Here, X equal to zero does not hold any special significance. A cusp can occur at any point in the graph of the function. It's formed at the point on the graph when the average rate of change in these two cases approaches infinite slope of opposite signs. So, if we find a cusp at any point on a function's graph, we can directly say that the derivative of the function does not exist at that point.
Until now, we have seen two different types of sharp turns when the function changes abruptly. One is a corner point like this, and the other is a cusp. We saw that the derivative of these types of points does not exist. In the case of a corner, we get two different tangents with a finite slope. And in the case of a cusp, we get two different tangents with infinite slopes. In short, the derivative of these points doesn't exist because we cannot find a straight line by which we can approximate the graph of the function around that point. Now, there might occur sharp turns which are combinations of these two types. As Delta X tends to 0 from one side, we get a tangent line with a finite slope. And from the other side, we get a tangent line with infinite slope. In such cases also, we can now directly see that the derivative at such points does not exist.
Now, what about this function we mentioned in the beginning? What will be the derivative at X equal to 0? Intuitively, we can see that the derivative will not exist at X equal to 0. This is because we will get that the tangent line at X equal to 0 will be a vertical straight line. How? If we find the average rate in two cases as earlier, we will get this. Here, we see that the average rate in both the cases is the same. The secant lines corresponding to these averages will look like this. Now, we can see that as Delta X tends to zero, these secant lines will approach this same vertical straight line. So, we get this vertical tangent line at X equal to zero. But as we saw earlier, the slope of this vertical tangent line is undefined. We say that it's infinite. So, the derivative does not exist here. Such vertical tangent lines can occur at any point on the graph of a function. So, in such cases, we can directly say that the derivative does not exist at that point.
So, we saw different types of situations where the derivative of a function does not exist. At these points, the function changes suddenly or rapidly. So, to find the rate of change, we cannot approximate the graph around such points by a straight line.
Now let me ask you a question. So far, we've learnt that for a function, we can find its derivative at a particular value of X. It is equal to the slope of the tangent line at that point on the function's graph. Now, the important question is, Is the derivative of a function also a function? What do you think? Share your thoughts in the comment section below and subscribe to our channel to keep learning!