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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

  • always start with it in most science and engineering

  • 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

  • about everything.

  • 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.

  • How about red?

  • 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.