- [Instructor] So we're told that this table
gives select values of the differentiable function F.
So it gives us the value of the function
at a few values for X, in particular,
five different values for X and
it tells us what the corresponding f(x) is.
And they say, what is the best estimate
for f'(4)?
So this is the derivative of our function F
when X is equal to four.
Or another way to think about it,
what is the slope of the tangent line
when X is equal to four for f(x)?
So what is the best estimate for f'(4)
we can make based on this table?
So, let's just visualize what's going on
before we even look at the choices.
So let me draw some axes here.
And let me plot these points.
We know that these would sit on the curve of Y
is equal to f(x).
When X is zero, f(x) is 72.
So this is the point (0,72).
This is the point (3,95)
clearly, two different scales on the X and Y axes.
This is the point
(5,112).
This is (6,77).
This is (9,54), actually, let me
write out
the
this is one, two, three,
four, five, six,
seven, eight, nine,
and ten.
Now they want us to know, they want to
what is the derivative of our function
when F is equal to four?
Well, they haven't told us
even what the value of F is at four.
We don't know what that point is.
But what they're trying to do is,
well, we're trying to make a best estimate.
And using these points, we don't even know
exactly what the curve looks like.
It could look like all sorts of things.
We could try to fit a reasonably smooth curve.
The curve might look something like that.
But it might be wackier.
It might do something like this.
Let me try to do it.
It might look something
like this.
So we don't know, for sure.
All we know is that
it needs to go through those points.
'Cause they've just sampled to the function
at those points.
But let's just, for the sake of
this exercise, let's assume the simplest,
let's say it's a nice smooth
curve without too many twists and turns
that goes through these points, just like that.
So what they're asking, okay,
when X is equal to four
if this yellow curve were the actual curve
then what is the slope of the tangent line, at that point?
So we would be visualizing that.
Now to be clear,
this tangent line that I just drew
this would be for
this version of our function that I did
connecting these points.
That does not have to be
the actual function.
We know that the actual function has to go
through those points.
But I'm just doing this for visualization purposes.
One of the whole ideas here is
that all we do have is the sample
and we're trying to get a best estimate.
We don't know
if it's even gonna be a good estimate.
It's just going to be a best estimate.
So what we generally do
when we just have some data
around a point, is, let's use a data points
that are closest
to that point
and find slopes of secant lines
pretty close around that point.
And that's going to give us our best estimate
for the slope of the tangent line.
So what points do we have near
F of four, or near the point
(4,f(4))?
Well, they give us what F is equal to
when X is equal to three.
They give us this point, right over here.
Let me do this in another color.
So, (3,95)
that is that right over there.
And they also give us
(5,112).
That is that point right over there.
And so what we could do
we could say, well, what is the average
rate of change between these two points?
Another way to think about it is,
what is the slope of the secant line
between those two points.
And, that would be our best estimate
for the slope of the tangent line
at X equals four.
Do we know that it's a good estimate?
Do we know that it's even close?
No, we don't know for sure, but that would
be the best estimate.
It would be better than trying to take the
the average rate of change between
When X equals three
and X equals six.
Or between when X equals zero
and X equals nine.
These are pretty close around four.
And so, let's do that.
Let's find the average rate of change
between when X goes from three to five.
So, we can see here
our change in X.
Let me do this in a new color.
So our change in X here
is equal to
plus two.
And I can draw that out.
My change in X here
is
plus two.
And, my change in Y
is going to be
when my X increased by two, my change in Y is
plus, let's see, this is
if I add ten I get to
I get to 105.
If I add another seven
so this is plus 17.
So this is plus 17 right over here.
Plus 17.
And so my change in Y over change in X.
Change in Y
over my change in X.
For this secant line between
when X is equaling three
and X is equaling five
is going to be equal to 17 over two.
Seventeen over
two.
Which is equal to 8.5.
So the slope of this green line here is 8.5.
And that would be our best estimate
for the slope of the tangent line when X equals four
of the curve Y is equal to f(x).
And so, lucky for us
the people who wrote this question had
the exact same logic, and they did it right over there.
So you wouldn't have to graph it the way I did.
I did it just to help us visualize what's going on.
In general, when you see a question like this
they're really saying, look,
you don't have all the data you need
to figure out exactly what f'(4) is.
But if you can find points close to, or around
f'(4)
and find the secant line, the average rate
of the slope of the secant line.
Or the average rate of change between those points
that's going to be our best estimate
for the instantaneous rate of change
when X equals four.
Or the derivative when X equals four.