WEBVTT
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PROFESSOR: Welcome
back to recitation.
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Today what I want to do is
something maybe a little bit
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more theoretical,
but the goal is
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to show that
something that you are
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going to be
repeatedly doing when
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you use quadratic approximations
is, in fact, true.
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So I'm going to
explain the situation,
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give a quick example,
and then show you
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what we're setting out to do.
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So the situation is
as follows: we're
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going to-- any time you
see a Q of f, that's
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going to represent the
quadratic approximation to f
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at x equals 0.
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So what I've done
is say, Q of f I'm
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going to define to be the thing
on the right, which is exactly
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the formula you
were given in class
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for the quadratic approximation
of a function f at x equals 0.
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So f is approximately the
thing on the right near 0.
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Our goal is to
show that if I want
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to take the quadratic
approximation
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of a product of two
functions, that I
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can do it in a different way.
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I can do it in the way written
on the right hand side, which
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actually looks more complicated
in this notation, but is,
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in fact, easier in reality.
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So let me explain
what's happening
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and then I'll give
you an example.
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If I wanted to take the
quadratic approximation
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of a product of two
functions, what I want to show
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is that instead, I could take
the quadratic approximation
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of each individual function,
multiply those together,
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and then take the quadratic
approximation of what
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I get as a result.
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So let me give you
an easy example.
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For instance, let's
let f of x equal
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e to the x and let's
let g of x equal sine x.
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Then what is Q of f?
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Q of f is the quadratic
approximation to e to the x
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at x equals 0.
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And that's going to be 1 plus
x plus x squared over 2--
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your already knew this.
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And g, the quadratic
approximation of sine x,
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is just x.
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So if I wanted to find the
quadratic approximation
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to e to the x sine
x, what I could
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do-- what this is
claiming I can do--
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is instead I can take the
quadratic approximation
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of this function
times this function.
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So instead I can take the
quadratic approximation
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of-- f was the e to the
x-- 1 plus x plus x squared
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over 2, times x.
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That's not a 4, sorry.
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That's a parentheses.
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1 plus x plus x squared
over 2, times x.
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And what is that?
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The quadratic
approximation to that
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is the quadratic
approximation to x plus x
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squared plus x cubed over 2.
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And at x equals 0, if
I have a polynomial,
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the quadratic approximation
to a polynomial at x equals 0
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is just all the terms up
to the quadratic term.
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So I drop off
higher-order terms.
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So I just get x plus x squared.
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So that's the idea.
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The idea is I have a
product of two functions,
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I know their individual
quadratic approximations,
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and so what I want
to do is I want
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to find the quadratic
approximation of this product
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by exploiting the
fact that I already
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know their individual
ones, and explain
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the fact that quadratic
approximation of polynomials
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at x equals 0 is very easy.
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So that's the example,
that's the idea.
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So now let's see
if we can do it.
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OK.
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So we have a cheat
sheet up here that I'm
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going to refer back to.
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I didn't want to use it again
and I didn't want to have
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to derive this for you, but
we have the product rule:
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f*g prime is equal to
what's on the right,
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and f*g double prime is
equal to what's on the right.
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So my goal here is
to show, remember,
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that the quadratic
approximation-- let
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me point over here again.
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The goal is to show the
quadratic approximation of f*g
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is equal to the quadratic
approximation of quadratic
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approximation of f times the
quadratic approximation of g.
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So let's do, well let's do
the right-hand side first
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because that's a little nicer.
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And then we'll show
the right-hand side
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and then we'll show
the left-hand side
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and we'll show they're equal.
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So let me start here
with the right-hand side.
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OK?
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So, let's look at what's the
quadratic approximation of f
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and what's the quadratic
approximation of g
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and then we'll take their
final quadratic approximation.
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So Q of f, we have exactly
what we need there.
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f of 0 plus f prime of 0 times x
plus f double prime of 0 over 2
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x squared.
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Q of g is equal to g of
0 plus g prime of 0 times
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x plus g double prime
at 0 over 2 x squared.
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So now what I'm going to do is
multiply those two together.
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And I'm actually going to
swing this way a tiny bit,
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if that's OK, to write
Q of f times Q of g
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because it's going
to be a little long.
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And I'm going to group
them carefully so
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that I have all the
higher-order terms at the end.
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OK?
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So I'm going to
get f of 0 g of 0
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by multiplying
these two together.
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And then I'm going to get
two terms involving an x.
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I'm going to get an f prime
times g and a g prime times f.
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Actually, if you'll
allow me, we'll
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know that anywhere
we see an f or a g
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or an f prime or a g
prime, or an f double prime
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or g double prime, they're
all evaluated at 0.
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So I'm going to drop the
0's from here on out.
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Anywhere you see those,
I'm evaluating them at 0.
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Otherwise this will
be way too long.
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So let me write
this just as f*g.
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I can even write just a
single one evaluated at 0.
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It's the product evaluated at 0.
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And then I have f prime g--
so I'll just evaluate it
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at 0 at the end of the product--
plus f g prime evaluated at 0.
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This whole thing is times x.
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I get an x here,
I get an x here.
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Now I need to figure out what
terms give me an x squared.
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OK?
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So the terms that
give me an x squared
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are f of 0 times g
double prime over 2.
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That gives me an x squared.
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f double prime times g
gives me an x squared.
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And f prime g prime
gives me an x squared.
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So let's write out those terms.
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I get f g double prime at 0 over
2-- from those two-- plus g f
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double prime at 0 over
2-- from those two-- plus
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f prime g prime at 0,
all times x squared.
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So there's an x times an x
there, gives you an x squared.
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x squared there,
x squared there.
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Now I could keep
going, and I will
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mention the higher-order
terms, but I'm not
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going to write them all
the way out because of what
00:07:09.385 --> 00:07:10.900
we're about to do.
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Let me show you
where they come from.
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You get an x cubed
term from here
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and an x cubed term from here.
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Maybe I'll write the x cubes,
but I won't write the x
00:07:21.500 --> 00:07:22.240
to the fourth.
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So the x cubed terms are f
prime g double prime 0 over 2
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plus g prime f double
prime at 0 over 2.
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And those are my x cubed terms.
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And then I got some x
to the fourth terms.
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And where do the x to the
fourth terms come from?
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They come from this product.
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Right?
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But I want to point
out something.
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What I'm going to do, I'm
going to work some magic
00:07:54.370 --> 00:07:56.450
on the board.
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This is a quadratic
approximation
00:07:57.840 --> 00:08:00.190
of f times a quadratic
approximation of g.
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Let me come over
here and remind you
00:08:02.250 --> 00:08:06.690
that I want the quadratic
approximation of that product.
00:08:06.690 --> 00:08:08.280
So what I'm going
to do is go back
00:08:08.280 --> 00:08:09.957
and look at what
I need from there,
00:08:09.957 --> 00:08:12.040
to be the quadratic
approximation of that product.
00:08:12.040 --> 00:08:13.820
So we come back over here.
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If I apply the
quadratic approximation
00:08:15.880 --> 00:08:17.930
to this thing,
which means then I'm
00:08:17.930 --> 00:08:23.910
applying it to this whole giant
thing, what do I actually get?
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This is actually a polynomial.
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I have something-- I have
a linear term, I have an x,
00:08:30.390 --> 00:08:32.015
I have an x squared,
I have an x cubed,
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I have an x to the fourth.
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So if I apply that
quadratic approximation,
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let's see what stays.
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This term stays.
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This term stays.
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This term stays.
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I'm going to erase what
disappears because I don't want
00:08:49.390 --> 00:08:51.010
us to get confused by that.
00:08:51.010 --> 00:08:52.360
So these two terms disappear.
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Why?
00:08:52.860 --> 00:08:55.630
Because again, this
is a polynomial.
00:08:55.630 --> 00:08:58.840
I have linear-- or I
have constant, I'm sorry,
00:08:58.840 --> 00:09:01.210
I think I called this maybe
linear earlier-- constant,
00:09:01.210 --> 00:09:03.500
linear, quadratic term.
00:09:03.500 --> 00:09:06.760
And what I need is just
those if I'm looking
00:09:06.760 --> 00:09:08.510
for a quadratic approximation.
00:09:08.510 --> 00:09:10.740
So constant, linear,
quadratic term.
00:09:10.740 --> 00:09:13.155
I immediately drop the
cubic and the quartic term
00:09:13.155 --> 00:09:15.030
when I'm looking at a
quadratic approximation
00:09:15.030 --> 00:09:17.340
of a polynomial at 0.
00:09:17.340 --> 00:09:19.760
So if I want the
right-hand side,
00:09:19.760 --> 00:09:23.361
I just need what's
underlined in blue.
00:09:23.361 --> 00:09:25.860
So now I'm going to put a big
box around that because that's
00:09:25.860 --> 00:09:27.270
going to be important.
00:09:27.270 --> 00:09:29.370
Whatever else happens,
we don't lose that.
00:09:29.370 --> 00:09:32.530
So now we've done
the right-hand side.
00:09:32.530 --> 00:09:35.110
And now let's write out
what is the left-hand side.
00:09:35.110 --> 00:09:37.117
And that's just going
to be plugging it
00:09:37.117 --> 00:09:38.200
straight into the formula.
00:09:40.820 --> 00:09:42.080
And using our cheat sheet.
00:09:46.170 --> 00:09:48.640
So Q of f*g, let me write
out the definition and then
00:09:48.640 --> 00:09:50.590
we'll use the cheat sheet.
00:09:50.590 --> 00:09:54.960
It's f*g at 0-- again,
this is f at 0, g at 0,
00:09:54.960 --> 00:09:57.040
that's what this
notation means--
00:09:57.040 --> 00:10:06.140
plus f*g quantity prime at
0 times x plus quantity f*g
00:10:06.140 --> 00:10:10.890
double prime at 0 over
2 times x squared.
00:10:10.890 --> 00:10:12.670
And now what we're
hoping, remember,
00:10:12.670 --> 00:10:15.960
is that what's in the box
is what shows up over here.
00:10:15.960 --> 00:10:18.160
Because this is the long
way to do the problem.
00:10:18.160 --> 00:10:20.530
This would be if I took
either the x sine x
00:10:20.530 --> 00:10:22.586
and I took all the derivatives.
00:10:22.586 --> 00:10:23.960
And this, in fact,
even though it
00:10:23.960 --> 00:10:26.980
looks more confusing
in evaluating
00:10:26.980 --> 00:10:29.890
such a quadratic
approximation, this way
00:10:29.890 --> 00:10:31.890
would be the easier way.
00:10:31.890 --> 00:10:34.480
We just want to show we
don't lose anything by doing
00:10:34.480 --> 00:10:37.280
what would be the easier way.
00:10:37.280 --> 00:10:41.484
So I get f*g at 0.
00:10:41.484 --> 00:10:43.900
And that's good-- we can see
we already have one of those,
00:10:43.900 --> 00:10:45.450
so that's nice.
00:10:45.450 --> 00:10:46.460
What do I get here?
00:10:46.460 --> 00:10:55.240
I get f prime g at 0 plus
g prime f at 0 times x.
00:10:55.240 --> 00:10:56.740
That comes-- let
me remind you, I'll
00:10:56.740 --> 00:10:59.420
walk over here-- comes
from the cheat sheet.
00:10:59.420 --> 00:11:00.510
The first thing.
00:11:00.510 --> 00:11:02.550
f*g prime is f prime
g plus g prime f.
00:11:02.550 --> 00:11:06.450
We know that one pretty
well, but just to remind you.
00:11:06.450 --> 00:11:08.570
So that's where this
term comes from.
00:11:08.570 --> 00:11:11.160
This looks promising
because if we come back
00:11:11.160 --> 00:11:13.980
to our quadratic of quadratic
of f times quadratic of g,
00:11:13.980 --> 00:11:17.130
it looks exactly like
the second term here.
00:11:17.130 --> 00:11:20.090
So now we're hoping that the x
squared term looks like this.
00:11:20.090 --> 00:11:21.840
The only thing that
might make you nervous
00:11:21.840 --> 00:11:23.904
is this doesn't have
an over 2, but if you
00:11:23.904 --> 00:11:25.570
were paying attention
to the cheat sheet
00:11:25.570 --> 00:11:27.236
you'll probably see
where that's coming.
00:11:27.236 --> 00:11:29.060
And I'll point it
out in one moment.
00:11:29.060 --> 00:11:32.100
So f*g double prime,
using the cheat sheet,
00:11:32.100 --> 00:11:39.040
is f double prime g plus g
double prime f plus 2 g prime f
00:11:39.040 --> 00:11:40.010
prime.
00:11:40.010 --> 00:11:41.450
I should have put 0's in there.
00:11:41.450 --> 00:11:44.000
Just to be consistent let
me put these 0's in there.
00:11:49.540 --> 00:11:52.890
2 f prime g prime at 0.
00:11:52.890 --> 00:11:55.540
And then I have to divide
the whole thing by 2
00:11:55.540 --> 00:11:57.800
because there's a
divided by 2 there,
00:11:57.800 --> 00:11:59.332
and then times x squared.
00:11:59.332 --> 00:12:01.730
So let me move out of
the way for a moment.
00:12:01.730 --> 00:12:05.340
So this numerator came
from the cheat sheet
00:12:05.340 --> 00:12:07.710
for the second derivative.
00:12:07.710 --> 00:12:09.280
And if you need, we can go back.
00:12:09.280 --> 00:12:11.260
Let me just remind
you, here it is.
00:12:11.260 --> 00:12:12.666
You can work it
out for yourself.
00:12:12.666 --> 00:12:15.040
You can just take the derivative
of the first derivative.
00:12:15.040 --> 00:12:16.835
But that's where
this comes from.
00:12:16.835 --> 00:12:19.210
So let me go back and we're
almost done with the problem.
00:12:21.800 --> 00:12:23.160
So what do we see?
00:12:23.160 --> 00:12:29.630
Well, we see that
we get f*g at 0.
00:12:29.630 --> 00:12:32.810
We get the second term
we want, f prime g at 0
00:12:32.810 --> 00:12:38.090
plus g prime f at 0 times x.
00:12:38.090 --> 00:12:39.590
And then the third
term is, in fact,
00:12:39.590 --> 00:12:45.080
exactly what we want because
we get f double prime g at 0
00:12:45.080 --> 00:12:52.810
over 2 plus g double prime f at
0 over 2 plus-- the 2's divide
00:12:52.810 --> 00:12:58.280
out-- and I get f prime g
prime at 0 time x squared.
00:12:58.280 --> 00:13:01.190
And if we look at
this last term and we
00:13:01.190 --> 00:13:04.130
look at the squared term
in the box we see, in fact,
00:13:04.130 --> 00:13:06.250
they are exactly the same.
00:13:06.250 --> 00:13:08.850
So let me summarize because
this was kind of a long video.
00:13:08.850 --> 00:13:10.580
So I'm going to go
back to the beginning,
00:13:10.580 --> 00:13:12.829
give you the example, and
tell you what we were really
00:13:12.829 --> 00:13:14.382
trying to do here.
00:13:14.382 --> 00:13:15.590
So let's come back over here.
00:13:15.590 --> 00:13:18.000
And let me remind
you, the goal was
00:13:18.000 --> 00:13:22.060
to show that if I wanted to
take a quadratic approximation
00:13:22.060 --> 00:13:24.780
of a product of two
functions, if I already
00:13:24.780 --> 00:13:27.890
knew their individual
quadratic approximations,
00:13:27.890 --> 00:13:31.670
you were told that you
could take those two
00:13:31.670 --> 00:13:34.020
quadratic approximations,
multiply them, and drop off
00:13:34.020 --> 00:13:36.230
the higher-order terms.
00:13:36.230 --> 00:13:38.180
The higher order than 2.
00:13:38.180 --> 00:13:39.680
So we had an example.
00:13:39.680 --> 00:13:42.210
We knew these two
quadratic approximations.
00:13:42.210 --> 00:13:44.370
And you've been told that
quadratic approximation
00:13:44.370 --> 00:13:47.300
of their product is just
the quadratic approximation
00:13:47.300 --> 00:13:50.000
of the product of their
quadratic approximations.
00:13:50.000 --> 00:13:52.220
And so our goal
today was to show
00:13:52.220 --> 00:13:54.000
that you don't drop
any of the terms
00:13:54.000 --> 00:13:58.130
that you get if you do it by
this method or by this method.
00:13:58.130 --> 00:13:59.630
And we've done that.
00:13:59.630 --> 00:14:01.208
So I think I'll stop there.