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.