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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. |
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