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We're first exposed to the idea of a slope of a line early on

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in our algebra careers, but I figure it never hurts

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to review it a bit.

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So let me draw some axes.

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That is my y-axis.

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

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y is equal to f of x.

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

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

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

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find the slope of that line?

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

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

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

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And then what would this be?

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

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going to be some line.

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

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

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a little bit of review.

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So this is a.

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

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

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And then we could take another point on this line.

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

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And then this coordinate up here is going to be

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the point b, f of b.

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Right?

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Because this is just the point when you evaluate

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the function at b.

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

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So let me just draw a little line right there.

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So that is f of b, right there.

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Actually, let me make it clear that this coordinate right

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is the point a, f of a.

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

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generally, of this entire line?

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Because whole the slope is consistent the

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whole way through it.

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

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actually going to be the value of this m.

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

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Well, a couple of ways to think about it.

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Slope is equal to rise over run.

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You might have seen that when you first learned algebra.

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

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y over change in x.

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

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in x is for this particular case.

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So the change in y is equal to what?

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

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

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

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let's start with him.

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

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distance, right here.

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So let me draw a little triangle.

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That distance right there is a change in y.

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

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This is the change in y.

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That is your change in y, that distance.

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So what is that distance?

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It's f of b minus f of a.

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So it equals f of b minus f of a.

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That is your change in y.

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Now what is your change in x The slope is change

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in y over change in x.

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So what our change in x?

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What's this distance?

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

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

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

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point x minus this point x.

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So this point's x-coordinate is b.

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So it's going to be b minus a.

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

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

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

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That straightforward.

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And that comes straight out of your Algebra 1 class.

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

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

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

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

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

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do that over 5 minus 2.

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

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would be your change in x.

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5 minus 2.

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

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slope would be 4/3.

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Now let's see if we can generalize this.

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

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to be learning as we delve into calculus.

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

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So let's say I have a curve.

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We have to have a curve before we can generalize

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it to a curve.

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Let me scroll down a little.

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

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you the similarity.

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

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Let's say I have a curve.

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I'll make it a familiar-looking curve.

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

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looks something like that.

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And I want to find the slope.

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

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

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

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

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But on a curve your slope is changing.

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

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the slope over here?

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

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The line just barely touches it.

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That's the slope over there.

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It's a negative slope.

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

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little bit less negative.

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It goes like that.

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

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Let me do it in a different color.

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Let me do it in purple.

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So over here, your slope is slightly less negative.

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It's a slightly less downward-sloping line.

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

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

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equals 0, is tangent to this curve.

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

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slope starts increasing.

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I'm trying to draw a tangent line.

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

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

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of the big change that happens when you go from a

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line to a curve.

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

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

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

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the entire line.

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

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bit more nuanced when we do it for a curve.

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Because it depends what point we're talking about.

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

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The slope is different at every point along the curve.

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It changes.

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

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It's going to look something like that.

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So let's try a bit of an experiment.

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

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be too much of a risk.

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Let me draw better than that.

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

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

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this the f of x axis.

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Either way.

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And let me draw my curve again.

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

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That's my curve.

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

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What can I do?

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

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to find a slope, right?

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

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So let's just call this point right here,

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that's going to be x.

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We're going to be general.

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This is going to be our point x.

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But to find our slope, according to our traditional

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

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So let's get another point in here.

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

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

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

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

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

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

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let me just say h bigger.

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So this is x plus h.

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That's what that point is right there.

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So what going to be their corresponding y-coordinates

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on the curve?

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

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

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particular x right here.

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

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I'll do a little 0 here.

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

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This is f of x naught.

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

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here, that point up here?

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

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which I shifted over a little bit.

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

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f of x naught plus h.

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That's its y-coordinate.

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

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are relatively close to each other?

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Remember, this isn't going to be the slope

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just at this point.

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

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

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secant line between, to the curve.

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

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once at this point.

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You can't see it.

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

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something like this.

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

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

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

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would be f of x naught plus h.

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

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x-coordinate That's all it is.

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So these are the 2 points.

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

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slope of this secant line?

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

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

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Let me draw it here.

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

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then your change in x would be that right there.

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

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

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

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

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

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I just evaluated this guy up here.

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

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The slightly larger x evaluate its y-coordinate.

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Where the curve is at that value of x.

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

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a f of x naught plus h.

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That's just the y-coordinate up here.

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Minus this y-coordinate over here.

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So minus f of x naught.

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So that equals our change in y.

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

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So what is this?

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This is the larger x-value.

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We started with this coordinate, so we start

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with its x-coordinate.

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So it's x naught plus h, x naught plus h.

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Minus this x-coordinate.

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Well, we just picked a general number.

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It's x naught.

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So that is over your change in x.

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Just like that.

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So this is the slope of the secant line.

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We still haven't answered what the slope is right at that

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point, but maybe this will help us get there.

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If we simplify this, so let me write it down like this.

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The slope of the secant, let me write that properly.

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The slope of the secant line is equal to the value of the

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function at this point, f of x naught plus h, minus the value

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of the function here, mine f of x naught.

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So that just tells us the change in y.

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It's the exact same definition of slope we've always used.

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Over the change in x.

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And we can simplify this.

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We have x naught plus h minus x naught.

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So x naught minus x naught cancel out, so you

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have that over h.

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So this is equal to our change in y over change in x.

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Fair enough.

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But I started off saying, I want to find the slope of

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the line at that point, at this point, right here.

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This is the zoomed-out version of it.

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So what can I do?

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Well, I defined second point here as just the first

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point plus some h.

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And we have something in our toolkit called a limit.

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This h is just a general number.

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It could be 10, it could be 2, it could be 0.02, it could be 1

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times 10 to the negative 100.

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It could be an arbitrarily small number.

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So what happens, what would happen, at least theoretically,

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if I were take the limit as h approaches 0?

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So, you know, first, maybe h is this fairly large number over

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here, and then if I take h a little bit smaller, then I'd be

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finding the slope of this secant line.

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If I took h to be even a little bit smaller, I'd be finding the

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slope of that secant line.

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If h is a little bit smaller, I'd be finding

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the slope of that line.

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So as h approaches 0, I'll be getting closer and closer to

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finding the slope of the line right at my point in question.

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Obviously, if h is a large number, my secant line is going

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to be way off from the slope at exactly that point right there.

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But if h is 0.0000001, if it's an infinitesimally small

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number, then I'm going to get pretty close.

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So what happens if I take the limit as h

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approaches zero of this?

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So the limit as h approaches 0 of my secant slope.

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Of, let me switch to green.

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f of x naught plus h minus f of x naught, that was my change in

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y, over h, which is my change in x.

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And now just to clarify something, and sometimes you'll

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see it in different calculus books, sometimes instead of an

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h, they'll write a delta x here.

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Where this second point would have been defined as x naught

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plus delta x, and then, this would have simplified to just

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delta x over there, and we'd be taking the limit as

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delta x approaches 0.

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The exact same thing.

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h, delta x, doesn't matter.

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We're taking h as the difference between one x point

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and then the higher x point, and then we're just going to

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take the limit as that approaches zero.

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We could have called that delta x just as easily.

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But I'm going to call this thing, which equals the slope

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of the tangent line, and it does equal the slope of the

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tangent line, I'm going to call this the derivative of f.

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Let me write that down.

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And I'm going to say that this is equal to f prime of x.

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And this is going to be another function.

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Because remember, the slope changes at every x-value.

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No matter what x-value you pick, the slope is

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going to be different.

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Doesn't have to be, but the way I drew this curve,

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it is different.

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It can be different.

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So now, you give me an x-value in here, I'll apply this

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formula over here, and then I can tell you the

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slope at that point.

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And it all seems very confusing and maybe

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abstract at this point.

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In the next video, I'll actually do an example of

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calculating a slope, and it'll make it everything a

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little bit more concrete.

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