Do you know which is the hardest mountain
to climb? You would probably say Mount Everest,
since it is the highest mountain. But it is
incorrect. The hardest one is the second highest
mountain ‘K 2’. Due to the difficulty
of its ascent, it is called the SAVAGE mountain.
It is harder to climb K2 because we know that
along with height there are many other factors
to consider. Steepness is one of them.
For example let’s consider these two mountains.
The right one clearly has more altitude. But
the left one will be harder to climb as it is steeper.
Now let me ask you a question. What do we
mean by steepness?
For example consider these two straight lines.
Which one is steeper? Here the difference
is subtle, but we can see that it is the second one.
But can you think of a way to concretely prove
that the second one is steeper?
One simple way is to draw the horizontal and
vertical lines like this, measure their lengths,
and find the ratio of the vertical length
TO the horizontal length. On substituting
the values, we get one and one point five
The second ratio is clearly greater. What
does this tell us?
If we walk along these lines, this ratio tells
us how much VERTICAL distance we cover RELATIVE
to the horizontal distance! So if the ratio
is higher, it means that the slant line is STEEPER.
We can also see this as the distance covered
while going up per unit of the distance while
going across! There’s another way to know
which line is steeper.
And that is to find this angle. Greater the
angle, Steeper will be the line.
But notice that these two lines are connected
to the angle by the trigonometric TANGENT
function. This ratio is equal to tan theta
one and this is equal to tan theta two. Such
a ratio for a straight line is called its
slope. It is the measure of the STEEPNESS
of the line
Now look at this curve. What is the slope
of this curve? Does it even make sense to
ask this? What do you think?
In this video, we will understand what this
means. Here, the process of differentiation
helps us. Also we know that such a curve can
be represented algebraically by a function
between two variables. We will see that this
slope is connected to the idea of the DERIVATIVE
of a function.
So let’s continue with our question. One
thing we can clearly see here is that different
portions of the curve will have different
steepness. We can intuitively see that as
we move from the left towards the right along
the curve, it gets steeper.
So the slope of the curve in general doesn’t
make sense. So let’s focus our attention
around this point ‘P’. What will be the
slope at this point?
For this let me ask you one question. Is the
earth flat or round?
You would probably be laughing because we
all know that the earth is almost round in
shape. But notice the surroundings around
you… it seems that the earth is flat. Why
is it so?
Consider this circle as the representation
of the earth. And let’s say you are currently
situated here. If we zoom in around this portion
of curved line, we see that it becomes less
and less curved. It looks almost like a straight line.
Let’s say we draw a straight line only intersecting
the curve at this point. We can see that the
curved line is almost the same as this straight
line. We know that this line which passes
through only one point of the curve is called
the tangent line at that point.
Take a moment and just observe what we found.
This is the most important idea on which Calculus
is built. A curved line in a very small region
around a point on it, can be approximated
by a straight line .This straight line is
the TANGENT line at this point.
So this solves our problem. The slope of the
curve at a point will be equal to the slope
of the tangent line at that point. And we’re
familiar with how to find the slope of a straight
line.
So how do we find the slope of the tangent
line at this point?
But before answering this question, let’s
look at WHY we are so interested in finding this slope…
We will continue this in the next part.
Previously, we saw what we meant by a function
between two variables, ‘X’ and ‘Y’.
A function tells us how the value of one variable
DEPENDS on another variable. So if the value
of ‘X’ changes from ‘X not’ to ‘X
not plus delta X’ then the value of ‘Y’
will also change from ‘Y not’ to ‘Y
not plus delta Y’.
Now look at this ratio ‘delta Y over delta
X’. It tells us the rate at which the value
of ‘Y’ changes in proportion to the change
in ‘X’. For example, consider these two
ratios. As the ratio in the second case is
greater, the value of ‘Y’ will change
relatively faster here.
This rate of change for a function is actually
related to the slope of a curve. Let us see
how.
Let’s say the graph of this function is
a straight line like this. 'X not Y not’
is this point and when the value of ‘X’
changes by ‘delta X’, we reach this point.
Now consider the ratio ‘delta Y’ over ‘delta X’.
We saw earlier that for a straight line, this
ratio is the measure of its steepness. So,
the rate of change in ‘Y’ with respect
to ‘X’ is equal to the slope of this line.
Now we can see that for a straight line, its
slope is constant throughout. So its rate
of change will also be constant. But what
if instead of this, the graph of the function
is given by a curved line.
The slope at a point tells us the rate of
change at that point. We saw earlier that
in a very small region around a point, the
curve can be thought of as a straight line.
This is the tangent line at this point. So
we see that the slope of this line is equal
to the rate of change here.
But now, we can see that depending on the
point on the curve, the slope will change.
So this means, depending on the value of the
‘X’ variable, the rate of change will also vary.
Let us look at an example to understand this.
Let’s say in this function, ‘Y’ is the
distance travelled by an object in motion.
And ‘X’ is the time taken for the distance
travelled. Then the rate of change in ‘Y’
with respect to ‘X’ will be equal to the
speed of the object.
In the first case, we know that the rate of
change is constant. So the speed of the object
will be constant throughout its motion. While
in the second case, the rate of change is
not constant. So here, this rate will tell
us the instantaneous speed of the object.
For this reason the rate of change for a function
at a particular value of ‘X’; X not is
called as the instantaneous rate of change at X not.
So to conclude, a very small portion of a
curve around a point can be approximated by
the tangent line at that point. The slope
of this tangent line tells us the instantaneous
rate of change of the function at that point.
But how do we find this slope or instantaneous
rate of change?
Do you remember that in one of our previous
videos, we found the instantaneous speed of
an object?
To find the instantaneous speed, we have to
use the process of differentiation.
Let’s say we want to find the instantaneous
speed at this point ‘P’, that is at the
time instant ‘X not’.
For this, we first find the average speed
for some time interval ‘delta X’. At ‘X
not’ the distance covered will be this.
And at ‘X not plus delta X’ it will be this.
So the average speed in the time interval
‘delta X’ will be equal to this. For a
function, this ratio is called as the average
rate of change.
Now let’s say we draw a straight line between
these two points. It is called the secant
line. We see that the slope of this secant
line will be equal to the average speed in
the time interval ‘delta X’.
So we see that the average rate of change
between two values of ‘X’ is equal to
the slope of the secant line between the corresponding
points on the curve.
Now what in the next step? How do we find
the instantaneous speed from this average speed?
For this we find the average speed in shorter
time intervals, that is, as ‘delta X’
tends to zero.
We say that when the limit ‘delta X’ tends
to zero, the average speed approaches the
instantaneous speed. .
This instantaneous speed is the instantaneous
rate of change at ‘X not’. Now what happens
to this secant line as ‘delta X’ tends to zero?
Let’s say we have to find the average rate
of change between points ‘P’ and ‘Q
one’, ‘Q two’ and so on. We can see
that as ‘delta X’ tends to zero, these
points comes closer and closer to the point
'P'. So as the limit ‘delta X’ tends to
zero, the secant line will pass through only
one point – ‘P’ on the curve. That is
the secant line approaches the tangent line.
This is how using the process of differentiation
we find the instantaneous rate of change of
a function or the slope of a tangent line.
Now here instead of writing the instantaneous
rate of change in ‘Y’ with respect to
‘X’ at ‘X not’, we say that the derivative
of the function at ‘X not’ is this.
The derivative is usually denoted by the symbol
‘D Y by D X’.
So we saw here that the derivative of the
function at a particular value of ‘X’
is its instantaneous rate of change at that point.
In the next lesson we will further explore
the concept of derivatives. We will see how
to exactly find the derivative for a particular
function. Also we will understand what this
symbol means intuitively.
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