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 functions 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, 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 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 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 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 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 functions 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 the 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 functions 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
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