When it comes to the instantaneous change of a function, our goal is to describe how
a function is changing. The difference between instantaneous change
and average change is that we want to describe the change using only a single point, rather
than between two points. This can be used for things like the instantaneous
speed of a rocket a few minutes after launch, or possibly the instantaneous rate of change
of a population.
One problem we run into is that when we normally describe something like the change of a function
we usually need two points to do it. This is so we can record both the change in
the y-direction and x-direction.
If we are at a single point, then it doesn't seem like we are changing in either direction.
To see this you might attempt to use our change formula for a single point.
This unfortunately would give you 0 divided by 0, which isn't really useful for describing change.
Fortunately there is a way we can find the change at a single point, and the secret
is in using the average rate but in a very clever way!
Let's see how this is done. To approximate the change at a single point
we begin by chosing a point, along with one other point.
This gives us two points. Using these two points we find the value of the
average rate of change between them.
Now rather than stopping there, we instead start moving the second point closer
and closer. Each time we do this we can record the new
value for the average rate between them. Our hope is that as we find these new values,
they will begin to get closer and possibly settle on some specific value.
This is what we will call our limiting value, and it works great as an approximation to
the instantaneous change of a function.
Let's take a look at a simple example to see this process in action.
Here we have a function, and I want to know the instantaneous rate
of change at this point here. I'll choose it, along with one other point.
Now I can measure the average rate of change between these two points, using my average
rate formula. And when I do I get a value of 1.
Now let's move that second point a bit closer. If I measure the average rate now, I get a
value of 1.5 Let's move it closer again. This time the
value is 1.8
As we continue to move this second point, we are getting different values for the rate
of change. Even though they are changing, they seem to
be getting close to a limiting value. In fact it appears they are getting close
to the number 2
Let's move them closer still to see if this is the case!
After moving them really close together, our value for the average rate is now 1.99, and
that's really close to 2, and even more evidence that our limiting value is indeed 2.
Since we are looking at the average rate, and moving the point closer,
2 can be considered an approximation for the instantaneous rate of change of our function.
It otherwords, it gives us a great way to describe how the function is changing at a
single point, which is exactly what we want!
Unfortunately, we can only call it an approximation. This is because we don't have any way of precisely
saying how close to the value of 2 we are getting.
Maybe the true value its approaching is 2.015, or it might even be approaching 1.993.
We can move these points closer together to gather more evidence,
but it would be much better if we had an an exact way of knowing what value this rate
was approaching, a method for finding the limiting value, exactly.
Fortunately Calculus has an answer for this issue, and its one of the first major tools
we'll need for many future problems. This tool is known as the limit, and it's
job is to help us describe what value a function is approaching, exactly.
We'll save this for the next lecture. For now its good to recognize that even without
the limit, we can still get a very good idea of how the
function is changing at a single point, we simply use our average rate of change and
then move our points closer and closer together so we can find that limiting value.
Thanks for watching.
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If you want to know more about approximating the instantaneious rate of change, you can
watch a few of my examples. You can also move onto my next lecture where
I talk about one of the most powerful tools of Calucus, the limit!
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