in our algebra careers, but I figure it never hurts

to review it a bit.

So let me draw some axes.

That is my y-axis.

Maybe I should call it my f of x-axis.

y is equal to f of x.

Let me draw my x-axis, just like that, that is my x-axis.

And let me draw a line, let me draw a line like this.

And what we want to do is remind ourselves, how do we

find the slope of that line?

And what we do is, we take two points on the line,

so let's say we take this point, right here.

Let's say that that is the point x is equal to a.

And then what would this be?

This would be the point f of a, where the function is

going to be some line.

We could write f of x is going to be equal to mx plus b.

We don't know what m and b are, but this is all

a little bit of review.

So this is a.

And then the y-value is what happens to the function when

you evaluated it at a, so that's that point right there.

And then we could take another point on this line.

Let's say we take point b, right there.

And then this coordinate up here is going to be

the point b, f of b.

Right?

Because this is just the point when you evaluate

the function at b.

You put b in here, you're going to get that point right there.

So let me just draw a little line right there.

So that is f of b, right there.

Actually, let me make it clear that this coordinate right

is the point a, f of a.

So how do we find the slope between these 2 points, or more

generally, of this entire line?

Because whole the slope is consistent the

whole way through it.

And we know that once we find the slope, that's

actually going to be the value of this m.

That's all a review of your algebra, but how do we do it?

Well, a couple of ways to think about it.

Slope is equal to rise over run.

You might have seen that when you first learned algebra.

Or another way of writing it, it's change in

y over change in x.

So let's figure out what the change in why over the change

in x is for this particular case.

So the change in y is equal to what?

Well, let's just take, you can take this guy as being the

first point, or that guy as being the first point.

But since this guy has a larger x and a larger y,

let's start with him.

The change in y between that guy and that guy is this

distance, right here.

So let me draw a little triangle.

That distance right there is a change in y.

Or I could just transfer it to the y-axis.

This is the change in y.

That is your change in y, that distance.

So what is that distance?

It's f of b minus f of a.

So it equals f of b minus f of a.

That is your change in y.

Now what is your change in x The slope is change

in y over change in x.

So what our change in x?

What's this distance?

Remember, we're taking this to be the first point,

so we took its y minus the other point's y.

So to be consistent, we're going to have to take this

point x minus this point x.

So this point's x-coordinate is b.

So it's going to be b minus a.

And just like that, if you knew the equation of this line, or

if you had the coordinates of these 2 points, you would just

plug them in right here and you would get your slope.

That straightforward.

And that comes straight out of your Algebra 1 class.

And let me just, just to make sure it's concrete for you, if

this was the point 2, 3, and let's say that this, up here,

was the point 5, 7, then if we wanted to find the slope of

this line, we would do 7 minus 3, that would be our change in

y, this would be 7 and this would be 3, and then we

do that over 5 minus 2.

Because this would be a 5, and this would be a 2, and so this

would be your change in x.

5 minus 2.

So 7 minus 3 is 4, and 5 minus 2 is 3. so your

slope would be 4/3.

Now let's see if we can generalize this.

And this is what the new concept that we're going

to be learning as we delve into calculus.

Let's see if we can generalize this somehow to a curve.

So let's say I have a curve.

We have to have a curve before we can generalize

it to a curve.

Let me scroll down a little.

Well, actually, I want to leave this up here, show

you the similarity.

Let's say I have, I'll keep it pretty general right now.

Let's say I have a curve.

I'll make it a familiar-looking curve.

Let's say it's the curve y is equal to x squared, which

looks something like that.

And I want to find the slope.

Let's say I want to find the slope at some point.

And actually, before even talking about it, let's even

think about what it means to find the slope of a curve.

Here, the slope was the same the whole time, right?

But on a curve your slope is changing.

And just to get an intuition for that means, is, what's

the slope over here?

Your slope over here is the slope of the tangent line.

The line just barely touches it.

That's the slope over there.

It's a negative slope.

Then over here, your slope is still negative, but it's a

little bit less negative.

It goes like that.

I don't know if I did that, drew that.

Let me do it in a different color.

Let me do it in purple.

So over here, your slope is slightly less negative.

It's a slightly less downward-sloping line.

And then when you go over here, at the 0 point, right here,

your slope is pretty much flat, because the horizontal line, y

equals 0, is tangent to this curve.

And then as you go to more positive x's, then your

slope starts increasing.

I'm trying to draw a tangent line.

And here it's increasing even more, it's increased even more.

So your slope is changing the entire time, and this is kind

of the big change that happens when you go from a

line to a curve.

A line, your slope is the same the entire time.

You could take any two points of a line, take the change in y

over the change in x, and you get the slope for

the entire line.

But as you can see already, it's going to be a little

bit more nuanced when we do it for a curve.

Because it depends what point we're talking about.

We can't just say, what is the slope for this curve?

The slope is different at every point along the curve.

It changes.

If we go up here, it's going to be even steeper.

It's going to look something like that.

So let's try a bit of an experiment.

And I know how this experiment turns out, so it won't

be too much of a risk.

Let me draw better than that.

So that is my y-axis, and that's my x-axis.

Let's call this, we can call this y, or we can call

this the f of x axis.

Either way.

And let me draw my curve again.

And I'll just draw it in the positive coordinate, like that.

That's my curve.

And what if I want to find the slope right there?

What can I do?

Well, based on our definition of a slope, we need 2 points

to find a slope, right?

Here, I don't know how to find the slope with 1 point.

So let's just call this point right here,

that's going to be x.

We're going to be general.

This is going to be our point x.

But to find our slope, according to our traditional

algebra 1 definition of a slope, we need 2 points.

So let's get another point in here.

Let's just take a slightly larger version of this x.

So let's say, we want to take, actually, let's do it even

further out, just because it's going to get messy otherwise.

So let's say we have this point right here.

And the difference, it's just h bigger than x.

Or actually, instead of saying h bigger, let's just, well

let me just say h bigger.

So this is x plus h.

That's what that point is right there.

So what going to be their corresponding y-coordinates

on the curve?

Well, this is the curve of y is equal to f of x.

So this point right here is going to be f of our

particular x right here.

And maybe to show you that I'm taking a particular x, maybe

I'll do a little 0 here.

This is x naught, this is x naught plus h.

This is f of x naught.

And then what is this going to be up here, this point up

here, that point up here?

Its y-coordinate is going to be f of f of this x-coordinate,

which I shifted over a little bit.

It's right there. f of this x-coordinate, which is

f of x naught plus h.

That's its y-coordinate.

So what is a slope going to be between these two points that

are relatively close to each other?

Remember, this isn't going to be the slope

just at this point.

This is the slope of the line between these two points.

And if I were to actually draw it out, it would actually be a

secant line between, to the curve.

So it would intersect the curve twice, once at this point,

once at this point.

You can't see it.

If I blew it up a little bit, it would look

something like this.

This is our coordinate x naught f of x naught, and up here

is our coordinate for this point, which would be, the

x-coordinate would be x naught plus h, and the y-coordinate

would be f of x naught plus h.

Just whatever this function is, we're evaluating it at this

x-coordinate That's all it is.

So these are the 2 points.

So maybe a good start is to just say, hey, what is the

slope of this secant line?

And just like we did in the previous example, you find the

change in y, and you divide that by your change in x.

Let me draw it here.

Your change in y would be that right here, change in y, and

then your change in x would be that right there.

So what is the slope going to be of the secant line?

The slope is going to be equal to, let's start with this

point up here, just because it seems to be larger.

So we want a change in y. so this value right here, this

y-value, is f of x naught plus h.

I just evaluated this guy up here.

Looks like a fancy term, but all it means is, look.

The slightly larger x evaluate its y-coordinate.

Where the curve is at that value of x.

So that is going to be, so the change in y is going to be

a f of x naught plus h.

That's just the y-coordinate up here.

Minus this y-coordinate over here.

So minus f of x naught.

So that equals our change in y.

And you want to divide that by your change in x.

So what is this?

This is the larger x-value.

We started with this coordinate, so we start