jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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5DUQ3-Y_gX4	used a lot in computer science, less so outside of that,
5DUQ3-Y_gX4	but it is used in calculus as well, maybe not in 18.01,
5DUQ3-Y_gX4	is this asymptotic notation.
5DUQ3-Y_gX4	In computer science, probably many of you
5DUQ3-Y_gX4	will have seen big O notation.
5DUQ3-Y_gX4	Here, I'm going to use a variant called little o notation.
5DUQ3-Y_gX4	So this is not a capital O. This is a lowercase o.
5DUQ3-Y_gX4	So these terms we call little o of delta x.
5DUQ3-Y_gX4	It's a little o of delta x.
5DUQ3-Y_gX4	It denotes any function that goes to 0 faster than linear,
5DUQ3-Y_gX4	faster than delta x.
5DUQ3-Y_gX4	So delta x squared goes to 0 faster than delta
5DUQ3-Y_gX4	x as delta x goes to 0.
5DUQ3-Y_gX4	I should say as delta x goes to 0.
5DUQ3-Y_gX4	And so you look at this thing and this probably
5DUQ3-Y_gX4	looks a lot like a Taylor series.
5DUQ3-Y_gX4	You do a Taylor expansion of f of x
5DUQ3-Y_gX4	plus delta x around f of x.
5DUQ3-Y_gX4	This is the first term.
5DUQ3-Y_gX4	This is the second term.
5DUQ3-Y_gX4	The third term, remember, is 1/2 f double prime delta x squared
5DUQ3-Y_gX4	or something like that.
5DUQ3-Y_gX4	But that's the wrong way to look at this.
5DUQ3-Y_gX4	A Taylor series is a much more advanced concept.
5DUQ3-Y_gX4	It's something you can do much later in calculus
5DUQ3-Y_gX4	and for good reason.
5DUQ3-Y_gX4	Because not every function even has a Taylor series
5DUQ3-Y_gX4	that converges.
5DUQ3-Y_gX4	This is more basic.
5DUQ3-Y_gX4	This is really the definition of a derivative.
5DUQ3-Y_gX4	The derivative is the linearization.
5DUQ3-Y_gX4	If you make a small change in delta x, the change in f
5DUQ3-Y_gX4	is a linear term plus smaller stuff.
5DUQ3-Y_gX4	And that smaller stuff only gives you a Taylor series
5DUQ3-Y_gX4	if it's basically a polynomial.
5DUQ3-Y_gX4	It might be smaller stuff like delta x to the 1.1
5DUQ3-Y_gX4	in which case it doesn't have a Taylor series.
5DUQ3-Y_gX4	But this is always true.
5DUQ3-Y_gX4	This is just this is what it means
5DUQ3-Y_gX4	to be the slope of a tangent.
5DUQ3-Y_gX4	And so the nice thing about this is
5DUQ3-Y_gX4	Alan says this notion, if we keep the delta x on the right,
5DUQ3-Y_gX4	this is going to be much easier to generalize
5DUQ3-Y_gX4	to other kinds of x's that are vectors or matrices or even
5DUQ3-Y_gX4	other functions, other kinds of things.
5DUQ3-Y_gX4	So this is the linearization.
5DUQ3-Y_gX4	Whoops.
5DUQ3-Y_gX4	So we're going to have a delta f, which I'm
5DUQ3-Y_gX4	going to have to define as--
5DUQ3-Y_gX4	whoops, let me put it in black--
5DUQ3-Y_gX4	f of x plus delta x.
5DUQ3-Y_gX4	And again, the delta, the Greek letter delta,
5DUQ3-Y_gX4	is not infinitesimal.
5DUQ3-Y_gX4	It's just a small number.
5DUQ3-Y_gX4	It's just a number.
5DUQ3-Y_gX4	This is that thing.
5DUQ3-Y_gX4	But I'm going to drop terms, higher order terms.
5DUQ3-Y_gX4	So there'll be an error there.
5DUQ3-Y_gX4	Well, yeah, so actually let me put this another way.
5DUQ3-Y_gX4	So this is going to be approximately f prime of x
5DUQ3-Y_gX4	delta x plus higher order terms.
5DUQ3-Y_gX4	This is the higher order.
5DUQ3-Y_gX4	So this is the small change in the input of the function.
5DUQ3-Y_gX4	And this is the resulting small change in the output.
5DUQ3-Y_gX4	And this is going to be the definition of the derivative.
5DUQ3-Y_gX4	The derivative is whatever you do to delta x to first order
5DUQ3-Y_gX4	to give you the linear change in the output for a small change
5DUQ3-Y_gX4	in the input.
5DUQ3-Y_gX4	It's a little annoying, though, to keep these o's around.
5DUQ3-Y_gX4	So we keep always having to-- whenever
5DUQ3-Y_gX4	we have a finite change in the input
5DUQ3-Y_gX4	and a finite change in the output, this is never exact.
5DUQ3-Y_gX4	This is an approximate relationship.
5DUQ3-Y_gX4	And we have to keep saying plus higher order terms,
5DUQ3-Y_gX4	plus higher order terms, plus little o delta x.
5DUQ3-Y_gX4	And it's annoying.
5DUQ3-Y_gX4	So it's easier to use to switch to differential notation.
5DUQ3-Y_gX4	So I'm just going to change my delta to d.
5DUQ3-Y_gX4	So df is going to be f of x plus dx minus f of x.
5DUQ3-Y_gX4	And this is going to be f prime of x dx where this is--
5DUQ3-Y_gX4	I'm just going to right equal.
5DUQ3-Y_gX4	So we can think of this differential in dx
5DUQ3-Y_gX4	as being arbitrarily small.
5DUQ3-Y_gX4	So it's really a limit, some kind of limit, of course.
5DUQ3-Y_gX4	And so you can also think of it as really it's just this.
5DUQ3-Y_gX4	But I'm just implicitly going to drop any higher order terms.
5DUQ3-Y_gX4	ALAN EDELMAN: That's how I like to think of it.
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Right?
5DUQ3-Y_gX4	So we don't have to get too fancy
5DUQ3-Y_gX4	with defining differentials.
5DUQ3-Y_gX4	I mean, this is a definition, right?
5DUQ3-Y_gX4	This is just shorthand for this where
5DUQ3-Y_gX4	I don't have to write plus a little o over delta x
5DUQ3-Y_gX4	all the time.
5DUQ3-Y_gX4	Let's see.
5DUQ3-Y_gX4	And so it's important to keep in-- so this f prime to delta
5DUQ3-Y_gX4	x, this is the derivative.
5DUQ3-Y_gX4	Here df is the differential.
5DUQ3-Y_gX4	So if I ask you for the derivative,
5DUQ3-Y_gX4	I'm asking for f prime.

5DUQ3-Y_gX4

used a lot in computer science, less so outside of that,

5DUQ3-Y_gX4

but it is used in calculus as well, maybe not in 18.01,

5DUQ3-Y_gX4

is this asymptotic notation.

5DUQ3-Y_gX4

In computer science, probably many of you

5DUQ3-Y_gX4

will have seen big O notation.

5DUQ3-Y_gX4

Here, I'm going to use a variant called little o notation.

5DUQ3-Y_gX4

So this is not a capital O. This is a lowercase o.

5DUQ3-Y_gX4

So these terms we call little o of delta x.

5DUQ3-Y_gX4

It's a little o of delta x.

5DUQ3-Y_gX4

It denotes any function that goes to 0 faster than linear,

5DUQ3-Y_gX4

faster than delta x.

5DUQ3-Y_gX4

So delta x squared goes to 0 faster than delta

5DUQ3-Y_gX4

x as delta x goes to 0.

5DUQ3-Y_gX4

I should say as delta x goes to 0.

5DUQ3-Y_gX4

And so you look at this thing and this probably

5DUQ3-Y_gX4

looks a lot like a Taylor series.

5DUQ3-Y_gX4

You do a Taylor expansion of f of x

5DUQ3-Y_gX4

plus delta x around f of x.

5DUQ3-Y_gX4

This is the first term.

5DUQ3-Y_gX4

This is the second term.

5DUQ3-Y_gX4

The third term, remember, is 1/2 f double prime delta x squared

5DUQ3-Y_gX4

or something like that.

5DUQ3-Y_gX4

But that's the wrong way to look at this.

5DUQ3-Y_gX4

A Taylor series is a much more advanced concept.

5DUQ3-Y_gX4

It's something you can do much later in calculus

5DUQ3-Y_gX4

and for good reason.

5DUQ3-Y_gX4

Because not every function even has a Taylor series

5DUQ3-Y_gX4

that converges.

5DUQ3-Y_gX4

This is more basic.

5DUQ3-Y_gX4

This is really the definition of a derivative.

5DUQ3-Y_gX4

The derivative is the linearization.

5DUQ3-Y_gX4

If you make a small change in delta x, the change in f

5DUQ3-Y_gX4

is a linear term plus smaller stuff.

5DUQ3-Y_gX4

And that smaller stuff only gives you a Taylor series

5DUQ3-Y_gX4

if it's basically a polynomial.

5DUQ3-Y_gX4

It might be smaller stuff like delta x to the 1.1

5DUQ3-Y_gX4

in which case it doesn't have a Taylor series.

5DUQ3-Y_gX4

But this is always true.

5DUQ3-Y_gX4

This is just this is what it means

5DUQ3-Y_gX4

to be the slope of a tangent.

5DUQ3-Y_gX4

And so the nice thing about this is

5DUQ3-Y_gX4

Alan says this notion, if we keep the delta x on the right,

5DUQ3-Y_gX4

this is going to be much easier to generalize

5DUQ3-Y_gX4

to other kinds of x's that are vectors or matrices or even

5DUQ3-Y_gX4

other functions, other kinds of things.

5DUQ3-Y_gX4

So this is the linearization.

5DUQ3-Y_gX4

Whoops.

5DUQ3-Y_gX4

So we're going to have a delta f, which I'm

5DUQ3-Y_gX4

going to have to define as--

5DUQ3-Y_gX4

whoops, let me put it in black--

5DUQ3-Y_gX4

f of x plus delta x.

5DUQ3-Y_gX4

And again, the delta, the Greek letter delta,

5DUQ3-Y_gX4

is not infinitesimal.

5DUQ3-Y_gX4

It's just a small number.

5DUQ3-Y_gX4

It's just a number.

5DUQ3-Y_gX4

This is that thing.

5DUQ3-Y_gX4

But I'm going to drop terms, higher order terms.

5DUQ3-Y_gX4

So there'll be an error there.

5DUQ3-Y_gX4

Well, yeah, so actually let me put this another way.

5DUQ3-Y_gX4

So this is going to be approximately f prime of x

5DUQ3-Y_gX4

delta x plus higher order terms.

5DUQ3-Y_gX4

This is the higher order.

5DUQ3-Y_gX4

So this is the small change in the input of the function.

5DUQ3-Y_gX4

And this is the resulting small change in the output.

5DUQ3-Y_gX4

And this is going to be the definition of the derivative.

5DUQ3-Y_gX4

The derivative is whatever you do to delta x to first order

5DUQ3-Y_gX4

to give you the linear change in the output for a small change

5DUQ3-Y_gX4

in the input.

5DUQ3-Y_gX4

It's a little annoying, though, to keep these o's around.

5DUQ3-Y_gX4

So we keep always having to-- whenever

5DUQ3-Y_gX4

we have a finite change in the input

5DUQ3-Y_gX4

and a finite change in the output, this is never exact.

5DUQ3-Y_gX4

This is an approximate relationship.

5DUQ3-Y_gX4

And we have to keep saying plus higher order terms,

5DUQ3-Y_gX4

plus higher order terms, plus little o delta x.

5DUQ3-Y_gX4

And it's annoying.

5DUQ3-Y_gX4

So it's easier to use to switch to differential notation.

5DUQ3-Y_gX4

So I'm just going to change my delta to d.

5DUQ3-Y_gX4

So df is going to be f of x plus dx minus f of x.

5DUQ3-Y_gX4

And this is going to be f prime of x dx where this is--

5DUQ3-Y_gX4

I'm just going to right equal.

5DUQ3-Y_gX4

So we can think of this differential in dx

5DUQ3-Y_gX4

as being arbitrarily small.

5DUQ3-Y_gX4

So it's really a limit, some kind of limit, of course.

5DUQ3-Y_gX4

And so you can also think of it as really it's just this.

5DUQ3-Y_gX4

But I'm just implicitly going to drop any higher order terms.

5DUQ3-Y_gX4

ALAN EDELMAN: That's how I like to think of it.

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Right?

5DUQ3-Y_gX4

So we don't have to get too fancy

5DUQ3-Y_gX4

with defining differentials.

5DUQ3-Y_gX4

I mean, this is a definition, right?

5DUQ3-Y_gX4

This is just shorthand for this where

5DUQ3-Y_gX4

I don't have to write plus a little o over delta x

5DUQ3-Y_gX4

all the time.

5DUQ3-Y_gX4

Let's see.

5DUQ3-Y_gX4

And so it's important to keep in-- so this f prime to delta

5DUQ3-Y_gX4

x, this is the derivative.

5DUQ3-Y_gX4

Here df is the differential.

5DUQ3-Y_gX4

So if I ask you for the derivative,

5DUQ3-Y_gX4

I'm asking for f prime.