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• Professor: So, again welcome to 18.01.

• We're getting started today with what we're calling

• Unit One, a highly imaginative title.

• And it's differentiation.

• So, let me first tell you, briefly, what's in store in

• the next couple of weeks.

• The main topic today is what is a derivative.

• And, we're going to look at this from several different

• points of view, and the first one is the

• geometric interpretation.

• That's what we'll spend most of today on.

• And then, we'll also talk about a physical interpretation

• of what a derivative is.

• And then there's going to be something else which I guess is

• maybe the reason why Calculus is so fundamental, and why we

• schools, which is the importance of derivatives, of

• this, to all measurements.

• So that means pretty much every place.

• That means in science, in engineering, in economics,

• in political science, etc.

• Polling, lots of commercial applications, just

• Now, that's what we'll be getting started with, and then

• there's another thing that we're gonna do in this unit,

• which is we're going to explain how to differentiate anything.

• So, how to differentiate any function you know.

• And that's kind of a tall order, but let me just

• give you an example.

• If you want to take the derivative - this we'll see

• today is the notation for the derivative of something - of

• some messy function like e ^ x arctanx.

• We'll work this out by the end of this unit.

• All right?

• Anything you can think of, anything you can write down,

• we can differentiate it.

• All right, so that's what we're gonna do, and today as I said,

• we're gonna spend most of our time on this geometric

• interpretation.

• So let's begin with that.

• So here we go with the geometric interpretation

• of derivatives.

• And, what we're going to do is just ask the geometric problem

• of finding the tangent line to some graph of some

• function at some point.

• Which is to say (x0, y0).

• So that's the problem that we're addressing here.

• Alright, so here's our problem, and now let me

• show you the solution.

• So, well, let's graph the function.

• Here's it's graph.

• Here's some point.

• All right, maybe I should draw it just a bit lower.

• So here's a point P.

• Maybe it's above the point x0. x0, by the way, this

• was supposed to be an x0.

• That was some fixed place on the x-axis.

• And now, in order to perform this mighty feat, I will use

• another color of chalk.

• OK.

• So here it is.

• There's the tangent line, Well, not quite straight.

• Close enough.

• All right?

• I did it.

• That's the geometric problem.

• I achieved what I wanted to do, and it's kind of an interesting

• question, which unfortunately I can't solve for you in

• this class, which is, how did I do that?

• That is, how physically did I manage to know what to do

• to draw this tangent line?

• But that's what geometric problems are like.

• We visualize it.

• We can figure it out somewhere in our brains.

• It happens.

• And the task that we have now is to figure out how to do it

• analytically, to do it in a way that a machine could just as

• well as I did in drawing this tangent line.

• So, what did we learn in high school about what

• a tangent line is?

• Well, a tangent line has an equation, and any line through

• a point has the equation y - y0 is equal to m the

• slope, times x - x0.

• So here's the equation for that line, and now there are two

• pieces of information that we're going to need to work

• out what the line is.

• The first one is the point.

• That's that point P there.

• And to specify P, given x, we need to know the level of y,

• which is of course just f(x0).

• That's not a calculus problem, but anyway that's a very

• important part of the process.

• So that's the first thing we need to know.

• And the second thing we need to know is the slope.

• And that's this number m.

• And in calculus we have another name for it.

• We call it f prime of x0.

• Namely, the derivative of f.

• So that's the calculus part.

• That's the tricky part, and that's the part that we

• have to discuss now.

• So just to make that explicit here, I'm going to make a

• definition, which is that f '(x0) , which is known as the

• derivative, of f, at x0, is the slope of the tangent line to y

• = f (x) at the point, let's just call it p.

• All right?

• So, that's what it is, but still I haven't made any

• progress in figuring out any better how I drew that line.

• So I have to say something that's more concrete, because

• I want to be able to cook up what these numbers are.

• I have to figure out what this number m is.

• And one way of thinking about that, let me just try this, so

• I certainly am taking for granted that in sort of

• non-calculus part that I know what a line through a point is.

• So I know this equation.

• But another possibility might be, this line here, how do I

• know - well, unfortunately, I didn't draw it quite straight,

• but there it is - how do I know that this orange line is not a

• tangent line, but this other line is a tangent line?

• Well, it's actually not so obvious, but I'm gonna

• describe it a little bit.

• It's not really the fact, this thing crosses at some other

• place, which is this point Q.

• But it's not really the fact that the thing crosses at two

• place, because the line could be wiggly, the curve could be

• wiggly, and it could cross back and forth a number of times.

• That's not what distinguishes the tangent line.

• So I'm gonna have to somehow grasp this, and I'll

• first do it in language.

• And it's the following idea: it's that if you take this

• orange line, which is called a secant line, and you think of

• the point Q as getting closer and closer to P, then the slope

• of that line will get closer and closer to the slope

• of the red line.

• And if we draw it close enough, then that's gonna

• be the correct line.

• So that's really what I did, sort of in my brain when

• I drew that first line.

• And so that's the way I'm going to articulate it first.

• Now, so the tangent line is equal to the limit of

• so called secant lines PQ, as Q tends to P.

• And here we're thinking of P as being fixed and Q as variable.

• All right?

• Again, this is still the geometric discussion, but now

• we're gonna be able to put symbols and formulas

• to this computation.

• And we'll be able to work out formulas in any example.

• So let's do that.

• So first of all, I'm gonna write out these

• points P and Q again.

• So maybe we'll put P here and Q here.

• And I'm thinking of this line through them.

• I guess it was orange, so we'll leave it as orange.

• All right.

• And now I want to compute its slope.

• So this, gradually, we'll do this in two steps.

• And these steps will introduce us to the basic notations which

• are used throughout calculus, including multi-variable

• calculus, across the board.

• So the first notation that's used is you imagine here's

• the x-axis underneath, and here's the x0, the location

• directly below the point P.

• And we're traveling here a horizontal distance which

• is denoted by delta x.

• So that's delta x, so called.