U1S1V08 The Derivative at a point.txt

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We've been thinking about average velocity
over a period of time.
We know that if we want the average velocity between eight
o'clock and some other time b, then
that's given by this formula.
We have the change in position divided by the change in time.
But if we want the instantaneous velocity at eight o'clock
exactly, then this formula is a little bit problematic.
If we try plugging in b is 8:01, then that's
giving it as an average velocity over a period of one minute.
That's not the same thing as the instantaneous velocity
at eight o'clock.
We could try b is eight o'clock and one second,
but that's still an average velocity.
Now, it's an OK approximation if we're talking
about the velocity of a car.
So the car's velocity is not going
to change very much over that one second.
But if we're talking about a dragonfly,
then its velocity fluctuates all over the place.
The issue is this other variable b.
If we want an instantaneous velocity at eight o'clock,
there's no good way to choose one specific value
for b in this formula that's going
to work in every situation and that everyone can just
agree on.
We know that the closer b is to 8, the better.
But we can't plug in b equals eight,
because we'd get 0 over 0, and that's just ridiculous.
So our solution is to take a limit
as b approaches eight o'clock.
So our formula for the instantaneous velocity
at eight o'clock is a limit as b approaches
eight of f of b minus f of 8-- that's
the change in f-- divided by b minus 8, the change in time.
In other words, we're taking the limit of average velocities
as b approaches 8, or as the time interval gets
shorter and shorter.

This is a massive concept, the idea
that you can take the limit of a bunch of average velocities
and get an instantaneous velocity.
And we want to apply this not just to instantaneous velocity,
but we want to talk about instantaneous rates of change
of any function.
So let me erase this.
If we want the instantaneous rate of change of a function
f of x, at some point, x equals a.
We're going to give this a special name.
So we're going to call this the derivative of f
of x at the point x equals a.
So this is our big idea.
The derivative at a point is measuring
the instantaneous rate of change of the function at that point.
And the formula for it is exactly the same
as what we had below.
We've got f of b minus f of a divided by b minus a.
So that's an average rate of change.
And then we take the limit as b approaches a.
And that's the derivative.

So we give this a special notation.
We're going to say that the derivative of f at a
will be denoted by this f prime of a.
So all of these things-- this notation,
this formula, this idea of an instantaneous rate of change--
all of these are wrapped up in this word "derivative."
And that's what we're going to be
learning about for the next several weeks.
So let's start getting used to it.