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Professor: So, again
welcome to 18.01.
00:00:24.180 --> 00:00:27.200
We're getting started
today with what
00:00:27.200 --> 00:00:36.000
we're calling Unit One, a
highly imaginative title.
00:00:36.000 --> 00:00:43.050
And it's differentiation.
00:00:43.050 --> 00:00:45.960
So, let me first
tell you, briefly,
00:00:45.960 --> 00:00:49.320
what's in store in the
next couple of weeks.
00:00:49.320 --> 00:01:02.170
The main topic today is
what is a derivative.
00:01:02.170 --> 00:01:09.840
And, we're going to look at this
from several different points
00:01:09.840 --> 00:01:18.820
of view, and the first one is
the geometric interpretation.
00:01:18.820 --> 00:01:21.710
That's what we'll
spend most of today on.
00:01:21.710 --> 00:01:32.330
And then, we'll also talk
about a physical interpretation
00:01:32.330 --> 00:01:37.240
of what a derivative is.
00:01:37.240 --> 00:01:44.910
And then there's going to
be something else which
00:01:44.910 --> 00:01:48.030
I guess is maybe the reason
why Calculus is so fundamental,
00:01:48.030 --> 00:01:53.240
and why we always start with it
in most science and engineering
00:01:53.240 --> 00:02:01.420
schools, which is the importance
of derivatives, of this,
00:02:01.420 --> 00:02:09.290
to all measurements.
00:02:09.290 --> 00:02:11.390
So that means pretty
much every place.
00:02:11.390 --> 00:02:15.140
That means in science,
in engineering,
00:02:15.140 --> 00:02:23.780
in economics, in
political science, etc.
00:02:23.780 --> 00:02:27.730
Polling, lots of
commercial applications,
00:02:27.730 --> 00:02:29.610
just about everything.
00:02:29.610 --> 00:02:33.040
Now, that's what we'll
be getting started with,
00:02:33.040 --> 00:02:35.480
and then there's
another thing that we're
00:02:35.480 --> 00:02:41.500
gonna do in this unit, which
is we're going to explain
00:02:41.500 --> 00:02:49.740
how to differentiate anything.
00:02:49.740 --> 00:03:01.490
So, how to differentiate
any function you know.
00:03:01.490 --> 00:03:04.260
And that's kind of a
tall order, but let
00:03:04.260 --> 00:03:05.656
me just give you an example.
00:03:05.656 --> 00:03:07.072
If you want to
take the derivative
00:03:07.072 --> 00:03:09.640
- this we'll see
today is the notation
00:03:09.640 --> 00:03:13.750
for the derivative of something
- of some messy function like e
00:03:13.750 --> 00:03:15.420
^ x arctan x.
00:03:19.730 --> 00:03:25.730
We'll work this out by
the end of this unit.
00:03:25.730 --> 00:03:26.390
All right?
00:03:26.390 --> 00:03:29.090
Anything you can think of,
anything you can write down,
00:03:29.090 --> 00:03:32.140
we can differentiate it.
00:03:32.140 --> 00:03:37.820
All right, so that's what we're
gonna do, and today, as I said,
00:03:37.820 --> 00:03:39.680
we're gonna spend
most of our time
00:03:39.680 --> 00:03:44.690
on this geometric
interpretation.
00:03:44.690 --> 00:03:50.080
So let's begin with that.
00:03:50.080 --> 00:04:01.570
So here we go with the geometric
interpretation of derivatives.
00:04:01.570 --> 00:04:11.160
And, what we're going to do is
just ask the geometric problem
00:04:11.160 --> 00:04:25.490
of finding the tangent
line to some graph
00:04:25.490 --> 00:04:31.580
of some function at some point.
00:04:31.580 --> 00:04:33.250
Which is to say (x_0, y_0).
00:04:33.250 --> 00:04:42.460
So that's the problem that
we're addressing here.
00:04:42.460 --> 00:04:46.480
Alright, so here's
our problem, and now
00:04:46.480 --> 00:04:49.440
let me show you the solution.
00:04:49.440 --> 00:04:58.530
So, well, let's
graph the function.
00:04:58.530 --> 00:05:00.510
Here's its graph.
00:05:00.510 --> 00:05:02.590
Here's some point.
00:05:02.590 --> 00:05:09.620
All right, maybe I should
draw it just a bit lower.
00:05:09.620 --> 00:05:13.950
So here's a point P.
Maybe it's above the point
00:05:13.950 --> 00:05:19.080
x_0. x_0, by the way, this
was supposed to be an x_0.
00:05:19.080 --> 00:05:26.870
That was some fixed
place on the x-axis.
00:05:26.870 --> 00:05:32.240
And now, in order to
perform this mighty feat,
00:05:32.240 --> 00:05:36.700
I will use another
color of chalk.
00:05:36.700 --> 00:05:37.710
How about red?
00:05:37.710 --> 00:05:38.900
OK.
00:05:38.900 --> 00:05:42.080
So here it is.
00:05:42.080 --> 00:05:44.440
There's the tangent line,
well, not quite straight.
00:05:44.440 --> 00:05:45.860
Close enough.
00:05:45.860 --> 00:05:46.820
All right?
00:05:46.820 --> 00:05:49.040
I did it.
00:05:49.040 --> 00:05:50.700
That's the geometric problem.
00:05:50.700 --> 00:05:56.490
I achieved what I
wanted to do, and it's
00:05:56.490 --> 00:05:58.810
kind of an interesting
question, which unfortunately I
00:05:58.810 --> 00:06:01.520
can't solve for you
in this class, which
00:06:01.520 --> 00:06:03.240
is, how did I do that?
00:06:03.240 --> 00:06:04.870
That is, how
physically did I manage
00:06:04.870 --> 00:06:07.890
to know what to do to
draw this tangent line?
00:06:07.890 --> 00:06:10.610
But that's what geometric
problems are like.
00:06:10.610 --> 00:06:12.000
We visualize it.
00:06:12.000 --> 00:06:14.050
We can figure it out
somewhere in our brains.
00:06:14.050 --> 00:06:15.420
It happens.
00:06:15.420 --> 00:06:18.850
And the task that we
have now is to figure out
00:06:18.850 --> 00:06:23.670
how to do it analytically,
to do it in a way
00:06:23.670 --> 00:06:28.636
that a machine could
just as well as I did
00:06:28.636 --> 00:06:32.230
in drawing this tangent line.
00:06:32.230 --> 00:06:39.620
So, what did we learn in high
school about what a tangent
00:06:39.620 --> 00:06:40.770
line is?
00:06:40.770 --> 00:06:42.740
Well, a tangent line
has an equation,
00:06:42.740 --> 00:06:45.700
and any line through a
point has the equation y
00:06:45.700 --> 00:06:52.190
- y_0 is equal to m, the
slope, times x - x_0.
00:06:52.190 --> 00:06:58.880
So here's the equation
for that line,
00:06:58.880 --> 00:07:02.200
and now there are two
pieces of information
00:07:02.200 --> 00:07:07.310
that we're going to need to
work out what the line is.
00:07:07.310 --> 00:07:10.860
The first one is the point.
00:07:10.860 --> 00:07:13.230
That's that point P there.
00:07:13.230 --> 00:07:16.670
And to specify P,
given x, we need
00:07:16.670 --> 00:07:23.020
to know the level of y, which
is of course just f(x_0).
00:07:23.020 --> 00:07:25.080
That's not a calculus
problem, but anyway that's
00:07:25.080 --> 00:07:28.350
a very important
part of the process.
00:07:28.350 --> 00:07:31.830
So that's the first
thing we need to know.
00:07:31.830 --> 00:07:39.490
And the second thing we
need to know is the slope.
00:07:39.490 --> 00:07:42.140
And that's this number m.
00:07:42.140 --> 00:07:45.340
And in calculus we have
another name for it.
00:07:45.340 --> 00:07:48.170
We call it f prime of x_0.
00:07:48.170 --> 00:07:51.520
Namely, the derivative of f.
00:07:51.520 --> 00:07:53.079
So that's the calculus part.
00:07:53.079 --> 00:07:54.870
That's the tricky part,
and that's the part
00:07:54.870 --> 00:07:57.760
that we have to discuss now.
00:07:57.760 --> 00:08:00.910
So just to make
that explicit here,
00:08:00.910 --> 00:08:05.940
I'm going to make a definition,
which is that f '(x_0) ,
00:08:05.940 --> 00:08:19.110
which is known as the
derivative, of f, at x_0,
00:08:19.110 --> 00:08:40.940
is the slope of the tangent
line to y = f(x) at the point,
00:08:40.940 --> 00:08:47.860
let's just call it P.
00:08:47.860 --> 00:08:50.000
All right?
00:08:50.000 --> 00:08:55.270
So, that's what
it is, but still I
00:08:55.270 --> 00:08:59.190
haven't made any progress in
figuring out any better how
00:08:59.190 --> 00:09:01.120
I drew that line.
00:09:01.120 --> 00:09:03.700
So I have to say
something that's
00:09:03.700 --> 00:09:06.210
more concrete, because I
want to be able to cook up
00:09:06.210 --> 00:09:07.410
what these numbers are.
00:09:07.410 --> 00:09:11.430
I have to figure out
what this number m is.
00:09:11.430 --> 00:09:16.870
And one way of thinking about
that, let me just try this,
00:09:16.870 --> 00:09:19.094
so I certainly am
taking for granted that
00:09:19.094 --> 00:09:20.760
in sort of non-calculus
part that I know
00:09:20.760 --> 00:09:22.820
what a line through a point is.
00:09:22.820 --> 00:09:24.440
So I know this equation.
00:09:24.440 --> 00:09:31.967
But another possibility
might be, this line here,
00:09:31.967 --> 00:09:34.050
how do I know - well,
unfortunately, I didn't draw
00:09:34.050 --> 00:09:34.170
it quite straight,
but there it is -
00:09:34.170 --> 00:09:37.770
how do I know that this orange
line is not a tangent line,
00:09:37.770 --> 00:09:45.070
but this other line
is a tangent line?
00:09:45.070 --> 00:09:53.010
Well, it's actually
not so obvious,
00:09:53.010 --> 00:09:56.200
but I'm gonna describe
it a little bit.
00:09:56.200 --> 00:09:58.540
It's not really the
fact-- this thing
00:09:58.540 --> 00:10:01.050
crosses at some
other place, which
00:10:01.050 --> 00:10:04.490
is this point Q. But
it's not really the fact
00:10:04.490 --> 00:10:06.510
that the thing
crosses at two place,
00:10:06.510 --> 00:10:07.885
because the line
could be wiggly,
00:10:07.885 --> 00:10:10.570
the curve could be wiggly,
and it could cross back
00:10:10.570 --> 00:10:11.990
and forth a number of times.
00:10:11.990 --> 00:10:17.120
That's not what distinguishes
the tangent line.
00:10:17.120 --> 00:10:19.830
So I'm gonna have to
somehow grasp this,
00:10:19.830 --> 00:10:23.560
and I'll first do
it in language.
00:10:23.560 --> 00:10:27.213
And it's the
following idea: it's
00:10:27.213 --> 00:10:31.050
that if you take this
orange line, which
00:10:31.050 --> 00:10:37.990
is called a secant line,
and you think of the point Q
00:10:37.990 --> 00:10:42.510
as getting closer and closer to
P, then the slope of that line
00:10:42.510 --> 00:10:47.860
will get closer and closer
to the slope of the red line.
00:10:47.860 --> 00:10:53.247
And if we draw it close
enough, then that's
00:10:53.247 --> 00:10:54.330
gonna be the correct line.
00:10:54.330 --> 00:10:57.030
So that's really what I did,
sort of in my brain when
00:10:57.030 --> 00:10:58.400
I drew that first line.
00:10:58.400 --> 00:11:01.010
And so that's the way I'm
going to articulate it first.
00:11:01.010 --> 00:11:13.890
Now, so the tangent line is
equal to the limit of so called
00:11:13.890 --> 00:11:24.040
secant lines PQ,
as Q tends to P.
00:11:24.040 --> 00:11:31.550
And here we're thinking of P as
being fixed and Q as variable.
00:11:31.550 --> 00:11:35.420
All right?
00:11:35.420 --> 00:11:38.320
Again, this is still the
geometric discussion,
00:11:38.320 --> 00:11:42.090
but now we're gonna be able
to put symbols and formulas
00:11:42.090 --> 00:11:43.570
to this computation.
00:11:43.570 --> 00:11:56.230
And we'll be able to work
out formulas in any example.
00:11:56.230 --> 00:11:58.700
So let's do that.
00:11:58.700 --> 00:12:05.420
So first of all, I'm gonna write
out these points P and Q again.
00:12:05.420 --> 00:12:10.979
So maybe we'll put
P here and Q here.
00:12:10.979 --> 00:12:12.770
And I'm thinking of
this line through them.
00:12:12.770 --> 00:12:16.190
I guess it was orange, so
we'll leave it as orange.
00:12:16.190 --> 00:12:19.570
All right.
00:12:19.570 --> 00:12:24.010
And now I want to
compute its slope.
00:12:24.010 --> 00:12:27.080
So this, gradually, we'll
do this in two steps.
00:12:27.080 --> 00:12:28.970
And these steps
will introduce us
00:12:28.970 --> 00:12:31.960
to the basic notations which
are used throughout calculus,
00:12:31.960 --> 00:12:35.350
including multi-variable
calculus, across the board.
00:12:35.350 --> 00:12:37.960
So the first
notation that's used
00:12:37.960 --> 00:12:42.440
is you imagine here's
the x-axis underneath,
00:12:42.440 --> 00:12:47.490
and here's the x_0, the location
directly below the point P.
00:12:47.490 --> 00:12:51.500
And we're traveling here a
horizontal distance which
00:12:51.500 --> 00:12:53.650
is denoted by delta x.
00:12:53.650 --> 00:12:58.980
So that's delta x, so called.
00:12:58.980 --> 00:13:06.960
And we could also call
it the change in x.
00:13:06.960 --> 00:13:09.920
So that's one thing we want
to measure in order to get
00:13:09.920 --> 00:13:12.290
the slope of this line PQ.
00:13:12.290 --> 00:13:14.530
And the other thing
is this height.
00:13:14.530 --> 00:13:18.050
So that's this distance here,
which we denote delta f,
00:13:18.050 --> 00:13:21.980
which is the change in f.
00:13:21.980 --> 00:13:29.310
And then, the slope is just
the ratio, delta f / delta x.
00:13:29.310 --> 00:13:39.780
So this is the
slope of the secant.
00:13:39.780 --> 00:13:44.280
And the process I just described
over here with this limit
00:13:44.280 --> 00:13:46.380
applies not just to
the whole line itself,
00:13:46.380 --> 00:13:48.920
but also in particular
to its slope.
00:13:48.920 --> 00:13:53.990
And the way we write that is
the limit as delta x goes to 0.
00:13:53.990 --> 00:13:56.970
And that's going
to be our slope.
00:13:56.970 --> 00:14:10.850
So this is the slope
of the tangent line.
00:14:10.850 --> 00:14:11.860
OK.
00:14:11.860 --> 00:14:19.460
Now, This is still
a little general,
00:14:19.460 --> 00:14:26.460
and I want to work out
a more usable form here,
00:14:26.460 --> 00:14:28.500
a better formula for this.
00:14:28.500 --> 00:14:30.710
And in order to
do that, I'm gonna
00:14:30.710 --> 00:14:36.360
write delta f, the numerator
more explicitly here.
00:14:36.360 --> 00:14:41.130
The change in f, so
remember that the point P
00:14:41.130 --> 00:14:43.450
is the point (x_0, f(x_0)).
00:14:43.450 --> 00:14:51.180
All right, that's what we got
for the formula for the point.
00:14:51.180 --> 00:14:54.955
And in order to
compute these distances
00:14:54.955 --> 00:14:57.480
and in particular the
vertical distance here,
00:14:57.480 --> 00:15:00.820
I'm gonna have to get a
formula for Q as well.
00:15:00.820 --> 00:15:05.300
So if this horizontal
distance is delta x,
00:15:05.300 --> 00:15:11.020
then this location
is x_0 + delta x.
00:15:11.020 --> 00:15:13.810
And so the point
above that point
00:15:13.810 --> 00:15:20.870
has a formula, which
is x_0 plus delta
00:15:20.870 --> 00:15:31.680
x, f of - and this is a
mouthful - x_0 plus delta x.
00:15:31.680 --> 00:15:33.950
All right, so there's the
formula for the point Q.
00:15:33.950 --> 00:15:36.416
Here's the formula
for the point P.
00:15:36.416 --> 00:15:47.560
And now I can write a different
formula for the derivative,
00:15:47.560 --> 00:15:52.200
which is the following:
so this f'(x_0) ,
00:15:52.200 --> 00:15:58.380
which is the same as m, is going
to be the limit as delta x goes
00:15:58.380 --> 00:16:05.400
to 0 of the change in f, well
the change in f is the value
00:16:05.400 --> 00:16:12.580
of f at the upper point
here, which is x_0 + delta x,
00:16:12.580 --> 00:16:19.880
and minus its value at the
lower point P, which is f(x_0),
00:16:19.880 --> 00:16:23.250
divided by delta x.
00:16:23.250 --> 00:16:24.670
All right, so this
is the formula.
00:16:24.670 --> 00:16:28.300
I'm going to put
this in a little box,
00:16:28.300 --> 00:16:32.830
because this is by far the
most important formula today,
00:16:32.830 --> 00:16:35.410
which we use to derive
pretty much everything else.
00:16:35.410 --> 00:16:37.160
And this is the way
that we're going to be
00:16:37.160 --> 00:16:46.260
able to compute these numbers.
00:16:46.260 --> 00:17:06.280
So let's do an example.
00:17:06.280 --> 00:17:13.360
This example, we'll
call this example one.
00:17:13.360 --> 00:17:19.790
We'll take the function
f(x) , which is 1/x .
00:17:19.790 --> 00:17:23.590
That's sufficiently complicated
to have an interesting answer,
00:17:23.590 --> 00:17:27.660
and sufficiently straightforward
that we can compute
00:17:27.660 --> 00:17:32.670
the derivative fairly quickly.
00:17:32.670 --> 00:17:36.130
So what is it that
we're gonna do here?
00:17:36.130 --> 00:17:42.550
All we're going to do is we're
going to plug in this formula
00:17:42.550 --> 00:17:44.570
here for that function.
00:17:44.570 --> 00:17:47.580
That's all we're going
to do, and visually
00:17:47.580 --> 00:17:52.010
what we're accomplishing is
somehow to take the hyperbola,
00:17:52.010 --> 00:17:55.050
and take a point
on the hyperbola,
00:17:55.050 --> 00:18:00.890
and figure out
some tangent line.
00:18:00.890 --> 00:18:03.132
That's what we're
accomplishing when we do that.
00:18:03.132 --> 00:18:04.840
So we're accomplishing
this geometrically
00:18:04.840 --> 00:18:07.060
but we'll be doing
it algebraically.
00:18:07.060 --> 00:18:14.860
So first, we consider this
difference delta f / delta x
00:18:14.860 --> 00:18:16.570
and write out its formula.
00:18:16.570 --> 00:18:18.190
So I have to have a place.
00:18:18.190 --> 00:18:21.670
So I'm gonna make it again
above this point x_0, which
00:18:21.670 --> 00:18:22.550
is the general point.
00:18:22.550 --> 00:18:25.940
We'll make the
general calculation.
00:18:25.940 --> 00:18:30.310
So the value of f at the top,
when we move to the right
00:18:30.310 --> 00:18:35.920
by f(x), so I just read off
from this, read off from here.
00:18:35.920 --> 00:18:40.900
The formula, the first
thing I get here is 1 /
00:18:40.900 --> 00:18:43.530
(x_0 + delta x).
00:18:43.530 --> 00:18:46.560
That's the left hand term.
00:18:46.560 --> 00:18:50.252
Minus 1 / x_0, that's
the right hand term.
00:18:50.252 --> 00:18:54.180
And then I have to
divide that by delta x.
00:18:54.180 --> 00:18:57.920
OK, so here's our expression.
00:18:57.920 --> 00:19:00.460
And by the way this has a name.
00:19:00.460 --> 00:19:10.240
This thing is called
a difference quotient.
00:19:10.240 --> 00:19:12.330
It's pretty complicated,
because there's always
00:19:12.330 --> 00:19:13.579
a difference in the numerator.
00:19:13.579 --> 00:19:16.100
And in disguise, the
denominator is a difference,
00:19:16.100 --> 00:19:18.400
because it's the difference
between the value
00:19:18.400 --> 00:19:26.310
on the right side and the
value on the left side here.
00:19:26.310 --> 00:19:34.740
OK, so now we're going to
simplify it by some algebra.
00:19:34.740 --> 00:19:35.860
So let's just take a look.
00:19:35.860 --> 00:19:40.260
So this is equal to, let's
continue on the next level
00:19:40.260 --> 00:19:41.170
here.
00:19:41.170 --> 00:19:45.100
This is equal to 1
/ delta x times...
00:19:45.100 --> 00:19:49.220
All I'm going to do is put
it over a common denominator.
00:19:49.220 --> 00:19:56.420
So the common denominator
is (x_0 + delta x) * x_0.
00:19:56.420 --> 00:20:00.720
And so in the numerator for the
first expressions I have x_0,
00:20:00.720 --> 00:20:05.290
and for the second expression
I have x_0 + delta x.
00:20:05.290 --> 00:20:08.790
So this is the same thing as
I had in the numerator before,
00:20:08.790 --> 00:20:11.450
factoring out this denominator.
00:20:11.450 --> 00:20:17.040
And here I put that numerator
into this more amenable form.
00:20:17.040 --> 00:20:20.360
And now there are two
basic cancellations.
00:20:20.360 --> 00:20:33.160
The first one is that x_0 and
x_0 cancel, so we have this.
00:20:33.160 --> 00:20:38.894
And then the second step is that
these two expressions cancel,
00:20:38.894 --> 00:20:40.310
the numerator and
the denominator.
00:20:40.310 --> 00:20:44.090
Now we have a cancellation
that we can make use of.
00:20:44.090 --> 00:20:48.680
So we'll write that under here.
00:20:48.680 --> 00:20:57.690
And this is equals -1 over
x_0 plus delta x times x_0.
00:20:57.690 --> 00:21:03.480
And then the very last step
is to take the limit as delta
00:21:03.480 --> 00:21:09.550
x tends to 0, and
now we can do it.
00:21:09.550 --> 00:21:10.690
Before we couldn't do it.
00:21:10.690 --> 00:21:11.560
Why?
00:21:11.560 --> 00:21:15.220
Because the numerator and the
denominator gave us 0 / 0.
00:21:15.220 --> 00:21:17.810
But now that I've made
this cancellation,
00:21:17.810 --> 00:21:19.430
I can pass to the limit.
00:21:19.430 --> 00:21:22.100
And all that happens is
I set this delta x to 0,
00:21:22.100 --> 00:21:22.910
and I get -1/x_0^2.
00:21:25.950 --> 00:21:31.728
So that's the answer.
00:21:31.728 --> 00:21:33.644
All right, so in other
words what I've shown -
00:21:33.644 --> 00:21:36.010
let me put it up here - is
that f'(x_0) = -1/x_0^2.
00:21:52.700 --> 00:21:55.910
Now, let's look at the
graph just a little
00:21:55.910 --> 00:22:01.610
bit to check this for
plausibility, all right?
00:22:01.610 --> 00:22:04.940
What's happening here is,
first of all it's negative.
00:22:04.940 --> 00:22:08.320
It's less than 0,
which is a good thing.
00:22:08.320 --> 00:22:16.510
You see that slope
there is negative.
00:22:16.510 --> 00:22:20.860
That's the simplest check
that you could make.
00:22:20.860 --> 00:22:24.530
And the second thing that I
would just like to point out
00:22:24.530 --> 00:22:29.380
is that as x goes to infinity,
that as we go farther
00:22:29.380 --> 00:22:32.710
to the right, it gets
less and less steep.
00:22:32.710 --> 00:22:46.050
So as x_0 goes to infinity,
less and less steep.
00:22:46.050 --> 00:22:48.660
So that's also
consistent here, when
00:22:48.660 --> 00:22:51.460
x_0 is very large, this is
a smaller and smaller number
00:22:51.460 --> 00:22:54.270
in magnitude, although
it's always negative.
00:22:54.270 --> 00:23:00.750
It's always sloping down.
00:23:00.750 --> 00:23:03.860
All right, so I've managed
to fill the boards.
00:23:03.860 --> 00:23:06.010
So maybe I should stop
for a question or two.
00:23:06.010 --> 00:23:06.510
Yes?
00:23:06.510 --> 00:23:11.430
Student: [INAUDIBLE]
00:23:11.430 --> 00:23:18.640
Professor: So the question
is to explain again
00:23:18.640 --> 00:23:22.320
this limiting process.
00:23:22.320 --> 00:23:26.710
So the formula here is we
have basically two numbers.
00:23:26.710 --> 00:23:29.030
So in other words, why is
it that this expression,
00:23:29.030 --> 00:23:33.920
when delta x tends to 0,
is equal to -1 / x_0^2 ?
00:23:33.920 --> 00:23:37.890
Let me illustrate it by
sticking in a number for x_0
00:23:37.890 --> 00:23:39.740
to make it more explicit.
00:23:39.740 --> 00:23:42.770
All right, so for
instance, let me stick
00:23:42.770 --> 00:23:46.110
in here for x_0 the number 3.
00:23:46.110 --> 00:23:52.450
Then it's -1 over 3
plus delta x times 3.
00:23:52.450 --> 00:23:54.420
That's the situation
that we've got.
00:23:54.420 --> 00:23:56.030
And now the question
is what happens
00:23:56.030 --> 00:23:58.680
as this number gets smaller
and smaller and smaller,
00:23:58.680 --> 00:24:01.690
and gets to be practically 0?
00:24:01.690 --> 00:24:04.535
Well, literally what we can
do is just plug in 0 there,
00:24:04.535 --> 00:24:07.410
and you get 3 plus 0 times
3 in the denominator.
00:24:07.410 --> 00:24:08.840
-1 in the numerator.
00:24:08.840 --> 00:24:16.030
So this tends to
-1/9 (over 3^2).
00:24:16.030 --> 00:24:20.871
And that's what I'm saying in
general with this extra number
00:24:20.871 --> 00:24:21.370
here.
00:24:21.370 --> 00:24:25.360
Other questions?
00:24:25.360 --> 00:24:25.860
Yes.
00:24:25.860 --> 00:24:34.680
Student: [INAUDIBLE]
00:24:34.680 --> 00:24:39.360
Professor: So the
question is what
00:24:39.360 --> 00:24:43.650
happened between this
step and this step, right?
00:24:43.650 --> 00:24:45.750
Explain this step here.
00:24:45.750 --> 00:24:48.150
Alright, so there were
two parts to that.
00:24:48.150 --> 00:24:53.440
The first is this delta x which
is sitting in the denominator,
00:24:53.440 --> 00:24:56.010
I factored all
the way out front.
00:24:56.010 --> 00:24:57.970
And so what's in
the parentheses is
00:24:57.970 --> 00:25:00.600
supposed to be
the same as what's
00:25:00.600 --> 00:25:03.120
in the numerator of
this other expression.
00:25:03.120 --> 00:25:05.570
And then, at the
same time as doing
00:25:05.570 --> 00:25:08.080
that, I put that
expression, which
00:25:08.080 --> 00:25:10.332
is the difference
of two fractions,
00:25:10.332 --> 00:25:12.040
I expressed it with
a common denominator.
00:25:12.040 --> 00:25:13.498
So in the denominator
here, you see
00:25:13.498 --> 00:25:17.040
the product of the denominators
of the two fractions.
00:25:17.040 --> 00:25:20.230
And then I just figured out what
the numerator had to be without
00:25:20.230 --> 00:25:22.790
really...
00:25:22.790 --> 00:25:27.660
Other questions?
00:25:27.660 --> 00:25:32.840
OK.
00:25:32.840 --> 00:25:39.670
So I claim that on
the whole, calculus
00:25:39.670 --> 00:25:43.100
gets a bad rap,
that it's actually
00:25:43.100 --> 00:25:47.220
easier than most things.
00:25:47.220 --> 00:25:52.040
But there's a perception
that it's harder.
00:25:52.040 --> 00:25:56.840
And so I really have a duty
to give you the calculus made
00:25:56.840 --> 00:25:59.210
harder story here.
00:25:59.210 --> 00:26:03.630
So we have to make things
harder, because that's our job.
00:26:03.630 --> 00:26:06.280
And this is actually what
most people do in calculus,
00:26:06.280 --> 00:26:09.560
and it's the reason why
calculus has a bad reputation.
00:26:09.560 --> 00:26:15.020
So the secret is
that when people
00:26:15.020 --> 00:26:19.360
ask problems in calculus, they
generally ask them in context.
00:26:19.360 --> 00:26:22.700
And there are many, many
other things going on.
00:26:22.700 --> 00:26:25.420
And so the little piece of
the problem which is calculus
00:26:25.420 --> 00:26:28.490
is actually fairly routine and
has to be isolated and gotten
00:26:28.490 --> 00:26:28.990
through.
00:26:28.990 --> 00:26:31.130
But all the rest of it,
relies on everything else
00:26:31.130 --> 00:26:35.987
you learned in mathematics up
to this stage, from grade school
00:26:35.987 --> 00:26:36.820
through high school.
00:26:36.820 --> 00:26:39.889
So that's the complication.
00:26:39.889 --> 00:26:41.930
So now we're going to do
a little bit of calculus
00:26:41.930 --> 00:26:49.080
made hard.
00:26:49.080 --> 00:26:53.940
By talking about a word problem.
00:26:53.940 --> 00:26:57.890
We only have one sort of word
problem that we can pose,
00:26:57.890 --> 00:27:03.090
because all we've talked about
is this geometry point of view.
00:27:03.090 --> 00:27:06.080
So far those are the only kinds
of word problems we can pose.
00:27:06.080 --> 00:27:08.870
So what we're gonna do is
just pose such a problem.
00:27:08.870 --> 00:27:23.660
So find the areas of
triangles, enclosed
00:27:23.660 --> 00:27:39.770
by the axes and the
tangent to y = 1/x.
00:27:39.770 --> 00:27:43.570
OK, so that's a
geometry problem.
00:27:43.570 --> 00:27:46.600
And let me draw a picture of it.
00:27:46.600 --> 00:27:52.590
It's practically the same as
the picture for example one.
00:27:52.590 --> 00:27:54.910
We only consider
the first quadrant.
00:27:54.910 --> 00:27:55.820
Here's our shape.
00:27:55.820 --> 00:28:00.020
All right, it's the hyperbola.
00:28:00.020 --> 00:28:02.640
And here's maybe one
of our tangent lines,
00:28:02.640 --> 00:28:05.090
which is coming in like this.
00:28:05.090 --> 00:28:12.470
And then we're trying
to find this area here.
00:28:12.470 --> 00:28:14.300
Right, so there's our problem.
00:28:14.300 --> 00:28:16.070
So why does it have
to do with calculus?
00:28:16.070 --> 00:28:17.819
It has to do with
calculus because there's
00:28:17.819 --> 00:28:19.790
a tangent line in
it, so we're gonna
00:28:19.790 --> 00:28:24.200
need to do some calculus
to answer this question.
00:28:24.200 --> 00:28:30.500
But as you'll see, the
calculus is the easy part.
00:28:30.500 --> 00:28:34.060
So let's get started
with this problem.
00:28:34.060 --> 00:28:37.150
First of all, I'm gonna
label a few things.
00:28:37.150 --> 00:28:39.770
And one important thing
to remember of course,
00:28:39.770 --> 00:28:42.770
is that the curve is y = 1/x.
00:28:42.770 --> 00:28:44.830
That's perfectly
reasonable to do.
00:28:44.830 --> 00:28:48.850
And also, we're gonna calculate
the areas of the triangles,
00:28:48.850 --> 00:28:51.530
and you could ask
yourself, in terms of what?
00:28:51.530 --> 00:28:54.296
Well, we're gonna have to pick
a point and give it a name.
00:28:54.296 --> 00:28:55.670
And since we need
a number, we're
00:28:55.670 --> 00:28:57.169
gonna have to do
more than geometry.
00:28:57.169 --> 00:28:59.290
We're gonna have to do
some of this analysis
00:28:59.290 --> 00:29:01.010
just as we've done before.
00:29:01.010 --> 00:29:04.130
So I'm gonna pick a point and,
consistent with the labeling
00:29:04.130 --> 00:29:08.320
we've done before, I'm
gonna to call it (x_0, y_0).
00:29:08.320 --> 00:29:13.370
So that's almost half the
battle, having notations, x
00:29:13.370 --> 00:29:16.220
and y for the variables,
and x_0 and y_0,
00:29:16.220 --> 00:29:18.960
for the specific point.
00:29:18.960 --> 00:29:24.310
Now, once you see that
you have these labelings,
00:29:24.310 --> 00:29:28.070
I hope it's reasonable
to do the following.
00:29:28.070 --> 00:29:31.380
So first of all, this
is the point x_0,
00:29:31.380 --> 00:29:33.630
and over here is the point y_0.
00:29:33.630 --> 00:29:37.730
That's something that
we're used to in graphs.
00:29:37.730 --> 00:29:40.200
And in order to figure out
the area of this triangle,
00:29:40.200 --> 00:29:41.950
it's pretty clear
that we should find
00:29:41.950 --> 00:29:45.740
the base, which is that we
should find this location here.
00:29:45.740 --> 00:29:47.950
And we should find
the height, so we
00:29:47.950 --> 00:29:55.590
need to find that value there.
00:29:55.590 --> 00:29:58.810
Let's go ahead and do it.
00:29:58.810 --> 00:30:02.390
So how are we going to do this?
00:30:02.390 --> 00:30:14.960
Well, so let's just take a look.
00:30:14.960 --> 00:30:17.150
So what is it that
we need to do?
00:30:17.150 --> 00:30:21.073
I claim that there's
only one calculus step,
00:30:21.073 --> 00:30:25.630
and I'm gonna put a star
here for this tangent line.
00:30:25.630 --> 00:30:28.472
I have to understand
what the tangent line is.
00:30:28.472 --> 00:30:30.430
Once I've figured out
what the tangent line is,
00:30:30.430 --> 00:30:33.230
the rest of the problem
is no longer calculus.
00:30:33.230 --> 00:30:35.890
It's just that
slope that we need.
00:30:35.890 --> 00:30:38.410
So what's the formula
for the tangent line?
00:30:38.410 --> 00:30:45.830
Put that over here. it's going
to be y - y_0 is equal to,
00:30:45.830 --> 00:30:48.180
and here's the magic number,
we already calculated it.
00:30:48.180 --> 00:30:50.970
It's in the box over there.
00:30:50.970 --> 00:30:58.270
It's -1/x_0^2 ( x - x_0).
00:30:58.270 --> 00:31:12.670
So this is the only bit of
calculus in this problem.
00:31:12.670 --> 00:31:15.500
But now we're not done.
00:31:15.500 --> 00:31:16.750
We have to finish it.
00:31:16.750 --> 00:31:19.170
We have to figure out all
the rest of these quantities
00:31:19.170 --> 00:31:27.300
so we can figure out the area.
00:31:27.300 --> 00:31:31.160
All right.
00:31:31.160 --> 00:31:40.920
So how do we do that?
00:31:40.920 --> 00:31:44.680
Well, to find this
point, this has a name.
00:31:44.680 --> 00:31:52.730
We're gonna find the
so called x-intercept.
00:31:52.730 --> 00:31:54.630
That's the first thing
we're going to do.
00:31:54.630 --> 00:31:57.800
So to do that,
what we need to do
00:31:57.800 --> 00:32:02.450
is to find where this horizontal
line meets that diagonal line.
00:32:02.450 --> 00:32:10.910
And the equation for the
x-intercept is y = 0.
00:32:10.910 --> 00:32:13.315
So we plug in y = 0, that's
this horizontal line,
00:32:13.315 --> 00:32:15.240
and we find this point.
00:32:15.240 --> 00:32:18.440
So let's do that into star.
00:32:18.440 --> 00:32:22.830
We get 0 minus, oh one
other thing we need to know.
00:32:22.830 --> 00:32:28.770
We know that y0 is f(x_0)
, and f(x) is 1/x ,
00:32:28.770 --> 00:32:31.060
so this thing is 1/x_0.
00:32:33.780 --> 00:32:38.600
And that's equal to -1/x_0^2.
00:32:38.600 --> 00:32:41.920
And here's x, and here's x_0.
00:32:41.920 --> 00:32:46.500
All right, so in order
to find this x value,
00:32:46.500 --> 00:32:53.800
I have to plug in one
equation into the other.
00:32:53.800 --> 00:32:59.170
So this simplifies a bit.
00:32:59.170 --> 00:33:03.250
This is -x/x_0^2.
00:33:03.250 --> 00:33:09.420
And this is plus 1/x_0 because
the x_0 and x0^2 cancel
00:33:09.420 --> 00:33:10.480
somewhat.
00:33:10.480 --> 00:33:12.830
And so if I put this
on the other side,
00:33:12.830 --> 00:33:20.810
I get x / x_0^2 is
equal to 2 / x_0.
00:33:20.810 --> 00:33:27.878
And if I then multiply through
- so that's what this implies -
00:33:27.878 --> 00:33:39.930
and if I multiply through
by x_0^2 I get x = 2x_0.
00:33:39.930 --> 00:33:42.270
OK, so I claim that this
point we've just calculated,
00:33:42.270 --> 00:33:51.840
it's 2x_0.
00:33:51.840 --> 00:33:57.320
Now, I'm almost done.
00:33:57.320 --> 00:34:00.210
I need to get the other one.
00:34:00.210 --> 00:34:03.280
I need to get this one up here.
00:34:03.280 --> 00:34:06.600
Now I'm gonna use a very
big shortcut to do that.
00:34:06.600 --> 00:34:27.490
So the shortcut to the
y-intercept is to use symmetry.
00:34:27.490 --> 00:34:30.900
All right, I claim I can stare
at this and I can look at that,
00:34:30.900 --> 00:34:33.540
and I know the formula
for the y-intercept.
00:34:33.540 --> 00:34:39.891
It's equal to 2y_0.
00:34:39.891 --> 00:34:40.390
All right.
00:34:40.390 --> 00:34:42.120
That's what that one is.
00:34:42.120 --> 00:34:44.320
So this one is 2y_0.
00:34:44.320 --> 00:34:48.060
And the reason I know this
is the following: so here's
00:34:48.060 --> 00:34:52.690
the symmetry of the situation,
which is not completely direct.
00:34:52.690 --> 00:34:56.350
It's a kind of mirror
symmetry around the diagonal.
00:34:56.350 --> 00:35:05.380
It involves the exchange
of (x, y) with (y, x);
00:35:05.380 --> 00:35:06.850
so trading the roles of x and y.
00:35:06.850 --> 00:35:08.720
So the symmetry
that I'm using is
00:35:08.720 --> 00:35:11.980
that any formula I get that
involves x's and y's, if I
00:35:11.980 --> 00:35:14.520
trade all the x's and
replace them by y's and trade
00:35:14.520 --> 00:35:16.645
all the y's and replace
them by x's, then
00:35:16.645 --> 00:35:18.720
I'll have a correct
formula on the other way.
00:35:18.720 --> 00:35:20.910
So if everywhere I see
a y I make it an x,
00:35:20.910 --> 00:35:22.876
and everywhere I see
an x I make it a y,
00:35:22.876 --> 00:35:24.000
the switch will take place.
00:35:24.000 --> 00:35:27.230
So why is that?
00:35:27.230 --> 00:35:30.450
That's just an accident
of this equation.
00:35:30.450 --> 00:35:46.070
That's because, so the
symmetry explained...
00:35:46.070 --> 00:35:48.160
is that the equation is y = 1/x.
00:35:48.160 --> 00:35:52.690
But that's the same
thing as xy = 1,
00:35:52.690 --> 00:35:54.710
if I multiply
through by x, which
00:35:54.710 --> 00:35:58.740
is the same thing as x = 1/y.
00:35:58.740 --> 00:36:05.450
So here's where the x
and the y get reversed.
00:36:05.450 --> 00:36:08.470
OK now if you don't
trust this explanation,
00:36:08.470 --> 00:36:23.190
you can also get the
y-intercept by plugging x = 0
00:36:23.190 --> 00:36:28.720
into the equation star.
00:36:28.720 --> 00:36:29.300
OK?
00:36:29.300 --> 00:36:34.060
We plugged y = 0 in
and we got the x-value.
00:36:34.060 --> 00:36:43.080
And you can do the same thing
analogously the other way.
00:36:43.080 --> 00:36:47.160
All right so I'm almost done
with the geometry problem,
00:36:47.160 --> 00:36:58.280
and let's finish it off now.
00:36:58.280 --> 00:37:00.930
Well, let me hold off for one
second before I finish it off.
00:37:00.930 --> 00:37:05.030
What I'd like to say is just
make one more tiny remark.
00:37:05.030 --> 00:37:09.200
And this is the hardest part
of calculus in my opinion.
00:37:09.200 --> 00:37:11.890
So the hardest
part of calculus is
00:37:11.890 --> 00:37:17.560
that we call it one
variable calculus,
00:37:17.560 --> 00:37:20.080
but we're perfectly
happy to deal
00:37:20.080 --> 00:37:25.500
with four variables at a
time or five, or any number.
00:37:25.500 --> 00:37:29.800
In this problem, I had an
x, a y, an x_0 and a y_0.
00:37:29.800 --> 00:37:32.080
That's already four
different things
00:37:32.080 --> 00:37:35.464
that have various
relationships between them.
00:37:35.464 --> 00:37:37.880
Of course the manipulations
we do with them are algebraic,
00:37:37.880 --> 00:37:39.820
and when we're doing
the derivatives
00:37:39.820 --> 00:37:43.216
we just consider what's known
as one variable calculus.
00:37:43.216 --> 00:37:45.590
But really there are millions
of variable floating around
00:37:45.590 --> 00:37:46.930
potentially.
00:37:46.930 --> 00:37:49.280
So that's what makes
things complicated,
00:37:49.280 --> 00:37:51.380
and that's something that
you have to get used to.
00:37:51.380 --> 00:37:53.580
Now there's something
else which is more subtle,
00:37:53.580 --> 00:37:57.360
and that I think many
people who teach the subject
00:37:57.360 --> 00:38:00.380
or use the subject aren't
aware, because they've already
00:38:00.380 --> 00:38:03.960
entered into the language and
they're so comfortable with it
00:38:03.960 --> 00:38:06.820
that they don't even
notice this confusion.
00:38:06.820 --> 00:38:10.180
There's something deliberately
sloppy about the way
00:38:10.180 --> 00:38:12.700
we deal with these variables.
00:38:12.700 --> 00:38:14.770
The reason is very simple.
00:38:14.770 --> 00:38:16.750
There are already
four variables here.
00:38:16.750 --> 00:38:20.680
I don't wanna create six names
for variables or eight names
00:38:20.680 --> 00:38:23.620
for variables.
00:38:23.620 --> 00:38:26.710
But really in this problem
there were about eight.
00:38:26.710 --> 00:38:29.280
I just slipped them by you.
00:38:29.280 --> 00:38:30.750
So why is that?
00:38:30.750 --> 00:38:35.910
Well notice that the first time
that I got a formula for y_0
00:38:35.910 --> 00:38:39.580
here, it was this point.
00:38:39.580 --> 00:38:44.680
And so the formula for y_0,
which I plugged in right here,
00:38:44.680 --> 00:38:50.280
was from the equation of
the curve. y_0 = 1 / x_0.
00:38:50.280 --> 00:38:55.320
The second time I did it,
I did not use y = 1/x.
00:38:55.320 --> 00:39:01.640
I used this equation here,
so this is not y = 1/x.
00:39:01.640 --> 00:39:03.480
That's the wrong thing to do.
00:39:03.480 --> 00:39:05.960
It's an easy mistake to
make if the formulas are
00:39:05.960 --> 00:39:08.810
all a blur to you and
you're not paying attention
00:39:08.810 --> 00:39:11.140
to where they are
on the diagram.
00:39:11.140 --> 00:39:16.740
You see that x-intercept
calculation there involved
00:39:16.740 --> 00:39:21.420
where this horizontal line met
this diagonal line, and y = 0
00:39:21.420 --> 00:39:25.890
represented this line here.
00:39:25.890 --> 00:39:31.520
So the sloppiness is that y
means two different things.
00:39:31.520 --> 00:39:34.450
And we do this constantly
because it's way, way more
00:39:34.450 --> 00:39:37.640
complicated not to do it.
00:39:37.640 --> 00:39:40.060
It's much more convenient
for us to allow ourselves
00:39:40.060 --> 00:39:42.660
the flexibility
to change the role
00:39:42.660 --> 00:39:47.730
that this letter plays in
the middle of a computation.
00:39:47.730 --> 00:39:50.110
And similarly, later
on, if I had done this
00:39:50.110 --> 00:39:54.110
by this more straightforward
method, for the y-intercept,
00:39:54.110 --> 00:39:55.360
I would have set x equal to 0.
00:39:55.360 --> 00:39:59.990
That would have been this
vertical line, which is x = 0.
00:39:59.990 --> 00:40:03.520
But I didn't change the letter
x when I did that, because that
00:40:03.520 --> 00:40:06.180
would be a waste for us.
00:40:06.180 --> 00:40:09.340
So this is one of the main
confusions that happens.
00:40:09.340 --> 00:40:12.460
If you can keep
yourself straight,
00:40:12.460 --> 00:40:15.310
you're a lot better
off, and as I
00:40:15.310 --> 00:40:21.720
say this is one of
the complexities.
00:40:21.720 --> 00:40:24.910
All right, so now let's
finish off the problem.
00:40:24.910 --> 00:40:30.880
Let me finally get
this area here.
00:40:30.880 --> 00:40:33.730
So, actually I'll just
finish it off right here.
00:40:33.730 --> 00:40:41.240
So the area of the
triangle is, well
00:40:41.240 --> 00:40:42.880
it's the base times the height.
00:40:42.880 --> 00:40:46.550
The base is 2x_0, the height
is 2y_0, and a half of that.
00:40:46.550 --> 00:40:54.720
So it's 1/2 (2x_0) * (2y_0) ,
which is 2x_0 y_0, which is,
00:40:54.720 --> 00:40:57.780
lo and behold, 2.
00:40:57.780 --> 00:40:59.370
So the amusing
thing in this case
00:40:59.370 --> 00:41:02.010
is that it actually didn't
matter what x_0 and y_0 are.
00:41:02.010 --> 00:41:05.870
We get the same
answer every time.
00:41:05.870 --> 00:41:10.220
That's just an accident
of the function 1 / x.
00:41:10.220 --> 00:41:19.740
It happens to be the
function with that property.
00:41:19.740 --> 00:41:23.790
All right, so we have
some more business today,
00:41:23.790 --> 00:41:24.960
some serious business.
00:41:24.960 --> 00:41:30.980
So let me continue.
00:41:30.980 --> 00:41:41.270
So, first of all, I want to
give you a few more notations.
00:41:41.270 --> 00:41:49.420
And these are just
other notations
00:41:49.420 --> 00:41:51.790
that people use to
refer to derivatives.
00:41:51.790 --> 00:41:53.920
And the first one
is the following:
00:41:53.920 --> 00:41:56.850
we already wrote y = f(x).
00:41:56.850 --> 00:41:59.630
And so when we write
delta y, that means
00:41:59.630 --> 00:42:01.960
the same thing as delta f.
00:42:01.960 --> 00:42:04.350
That's a typical notation.
00:42:04.350 --> 00:42:13.670
And previously we wrote f
prime for the derivative,
00:42:13.670 --> 00:42:20.530
so this is Newton's
notation for the derivative.
00:42:20.530 --> 00:42:22.520
But there are other notations.
00:42:22.520 --> 00:42:27.830
And one of them is df/dx,
and another one is dy/dx,
00:42:27.830 --> 00:42:29.840
meaning exactly the same thing.
00:42:29.840 --> 00:42:32.870
And sometimes we
let the function
00:42:32.870 --> 00:42:40.520
slip down below so that becomes
d/dx of f and d/dx of y.
00:42:40.520 --> 00:42:44.340
So these are all notations that
are used for the derivative,
00:42:44.340 --> 00:42:49.150
and these were
initiated by Leibniz.
00:42:49.150 --> 00:42:55.120
And these notations are used
interchangeably, sometimes
00:42:55.120 --> 00:42:56.740
practically together.
00:42:56.740 --> 00:42:59.750
They both turn out to
be extremely useful.
00:42:59.750 --> 00:43:03.314
This one omits - notice
that this thing omits
00:43:03.314 --> 00:43:07.140
- the underlying
base point, x_0.
00:43:07.140 --> 00:43:09.100
That's one of the nuisances.
00:43:09.100 --> 00:43:11.440
It doesn't give you
all the information.
00:43:11.440 --> 00:43:18.070
But there are lots of situations
like that where people leave
00:43:18.070 --> 00:43:20.090
out some of the
important information,
00:43:20.090 --> 00:43:23.150
and you have to fill
it in from context.
00:43:23.150 --> 00:43:28.360
So that's another
couple of notations.
00:43:28.360 --> 00:43:33.330
So now I have one more
calculation for you today.
00:43:33.330 --> 00:43:35.530
I carried out this
calculation of the derivative
00:43:35.530 --> 00:43:45.780
of the function 1 / x.
00:43:45.780 --> 00:43:48.640
I wanna take care of
some other powers.
00:43:48.640 --> 00:43:59.150
So let's do that.
00:43:59.150 --> 00:44:08.730
So Example 2 is going to
be the function f(x) = x^n.
00:44:08.730 --> 00:44:14.160
n = 1, 2, 3; one of these guys.
00:44:14.160 --> 00:44:18.270
And now what we're trying to
figure out is the derivative
00:44:18.270 --> 00:44:21.470
with respect to x of
x^n in our new notation,
00:44:21.470 --> 00:44:27.070
what this is equal to.
00:44:27.070 --> 00:44:33.450
So again, we're going to form
this expression, delta f /
00:44:33.450 --> 00:44:35.370
delta x.
00:44:35.370 --> 00:44:38.830
And we're going to make some
algebraic simplification.
00:44:38.830 --> 00:44:44.950
So what we plug in for
delta f is ((x delta x)^n -
00:44:44.950 --> 00:44:48.240
x^n)/delta x.
00:44:48.240 --> 00:44:50.460
Now before, let
me just stick this
00:44:50.460 --> 00:44:52.350
in then I'm gonna erase it.
00:44:52.350 --> 00:44:56.490
Before, I wrote x_0
here and x_0 there.
00:44:56.490 --> 00:44:59.390
But now I'm going
to get rid of it,
00:44:59.390 --> 00:45:01.980
because in this particular
calculation, it's a nuisance.
00:45:01.980 --> 00:45:03.540
I don't have an x
floating around,
00:45:03.540 --> 00:45:06.310
which means something
different from the x_0.
00:45:06.310 --> 00:45:08.110
And I just don't
wanna have to keep
00:45:08.110 --> 00:45:10.250
on writing all those symbols.
00:45:10.250 --> 00:45:13.850
It's a waste of
blackboard energy.
00:45:13.850 --> 00:45:15.500
There's a total
amount of energy,
00:45:15.500 --> 00:45:18.264
and I've already filled
up so many blackboards
00:45:18.264 --> 00:45:21.710
that, there's just
a limited amount.
00:45:21.710 --> 00:45:23.650
Plus, I'm trying
to conserve chalk.
00:45:23.650 --> 00:45:25.880
Anyway, no 0's.
00:45:25.880 --> 00:45:28.840
So think of x as fixed.
00:45:28.840 --> 00:45:40.030
In this case, delta x moves and
x is fixed in this calculation.
00:45:40.030 --> 00:45:42.380
All right now, in order
to simplify this, in order
00:45:42.380 --> 00:45:44.810
to understand algebraically
what's going on,
00:45:44.810 --> 00:45:48.430
I need to understand what
the nth power of a sum is.
00:45:48.430 --> 00:45:50.100
And that's a famous formula.
00:45:50.100 --> 00:45:52.550
We only need a little
tiny bit of it,
00:45:52.550 --> 00:45:56.040
called the binomial theorem.
00:45:56.040 --> 00:46:06.350
So, the binomial theorem
which is in your text
00:46:06.350 --> 00:46:12.820
and explained in
an appendix, says
00:46:12.820 --> 00:46:15.880
that if you take
the sum of two guys
00:46:15.880 --> 00:46:18.190
and you take them to the
nth power, that of course
00:46:18.190 --> 00:46:24.750
is (x + delta x) multiplied
by itself n times.
00:46:24.750 --> 00:46:29.900
And so the first term is
x^n, that's when all of the n
00:46:29.900 --> 00:46:31.690
factors come in.
00:46:31.690 --> 00:46:35.620
And then, you could have this
factor of delta x and all
00:46:35.620 --> 00:46:36.786
the rest x's.
00:46:36.786 --> 00:46:39.160
So at least one term of the
form (x^(n-1)) times delta x.
00:46:41.820 --> 00:46:43.650
And how many times
does that happen?
00:46:43.650 --> 00:46:45.930
Well, it happens when
there's a factor from here,
00:46:45.930 --> 00:46:48.230
from the next factor, and
so on, and so on, and so on.
00:46:48.230 --> 00:46:54.330
There's a total of n possible
times that that happens.
00:46:54.330 --> 00:46:59.110
And now the great thing
is that, with this alone,
00:46:59.110 --> 00:47:05.120
all the rest of the
terms are junk that we
00:47:05.120 --> 00:47:06.640
won't have to worry about.
00:47:06.640 --> 00:47:11.650
So to be more specific,
there's a very careful notation
00:47:11.650 --> 00:47:12.460
for the junk.
00:47:12.460 --> 00:47:14.840
The junk is what's called
big O of (delta x)^2.
00:47:18.070 --> 00:47:25.740
What that means is that
these are terms of order,
00:47:25.740 --> 00:47:33.676
so with (delta x)^2,
(delta x)^3 or higher.
00:47:33.676 --> 00:47:38.780
All right, that's how.
00:47:38.780 --> 00:47:42.430
Very exciting,
higher order terms.
00:47:42.430 --> 00:47:47.540
OK, so this is the only
algebra that we need to do,
00:47:47.540 --> 00:47:50.890
and now we just need to combine
it together to get our result.
00:47:50.890 --> 00:47:54.750
So, now I'm going to just
carry out the cancellations
00:47:54.750 --> 00:48:02.480
that we need.
00:48:02.480 --> 00:48:03.790
So here we go.
00:48:03.790 --> 00:48:11.990
We have delta f / delta x, which
remember was 1 / delta x times
00:48:11.990 --> 00:48:25.420
this, which is this times, now
this is x^n plus nx^(n-1) delta
00:48:25.420 --> 00:48:35.320
x plus this junk
term, minus x^n.
00:48:35.320 --> 00:48:38.380
So that's what we
have so far based
00:48:38.380 --> 00:48:41.670
on our previous calculations.
00:48:41.670 --> 00:48:48.110
Now, I'm going to do the main
cancellation, which is this.
00:48:48.110 --> 00:48:49.420
All right.
00:48:49.420 --> 00:48:56.730
So, that's 1/delta x times
nx^(n-1) delta x plus this term
00:48:56.730 --> 00:49:01.220
here.
00:49:01.220 --> 00:49:05.020
And now I can divide
in by delta x.
00:49:05.020 --> 00:49:10.530
So I get nx^(n-1) plus,
now it's O(delta x).
00:49:10.530 --> 00:49:12.230
There's at least
one factor of delta
00:49:12.230 --> 00:49:14.390
x not two factors of
delta x, because I
00:49:14.390 --> 00:49:17.610
have to cancel one of them.
00:49:17.610 --> 00:49:19.860
And now I can just
take the limit.
00:49:19.860 --> 00:49:22.410
In the limit this
term is gonna be 0.
00:49:22.410 --> 00:49:25.850
That's why I called
it junk originally,
00:49:25.850 --> 00:49:26.960
because it disappears.
00:49:26.960 --> 00:49:31.010
And in math, junk is
something that goes away.
00:49:31.010 --> 00:49:37.350
So this tends to, as delta
x goes to 0, nx^(n-1).
00:49:37.350 --> 00:49:43.260
And so what I've shown you is
that d/dx of x to the n minus--
00:49:43.260 --> 00:49:47.790
sorry, n, is equal to nx^(n-1).
00:49:51.180 --> 00:49:53.710
So now this is gonna be
super important to you
00:49:53.710 --> 00:49:56.520
right on your problem set
in every possible way,
00:49:56.520 --> 00:49:59.040
and I want to tell you one
thing, one way in which it's
00:49:59.040 --> 00:50:00.200
very important.
00:50:00.200 --> 00:50:02.240
One way that extends
it immediately.
00:50:02.240 --> 00:50:10.950
So this thing extends
to polynomials.
00:50:10.950 --> 00:50:13.970
We get quite a lot out
of this one calculation.
00:50:13.970 --> 00:50:21.960
Namely, if I take d/dx of
something like (x^3 + 5x^10)
00:50:21.960 --> 00:50:25.900
that's gonna be equal to 3x^2,
that's applying this rule
00:50:25.900 --> 00:50:27.240
to x^3.
00:50:27.240 --> 00:50:35.110
And then here, I'll
get 5*10 so 50x^9.
00:50:35.110 --> 00:50:37.760
So this is the type of
thing that we get out of it,
00:50:37.760 --> 00:50:49.727
and we're gonna make more
hay with that next time.
00:50:49.727 --> 00:50:50.227
Question.
00:50:50.227 --> 00:50:50.727
Yes.
00:50:50.727 --> 00:50:51.690
I turned myself off.
00:50:51.690 --> 00:50:52.190
Yes?
00:50:52.190 --> 00:50:56.030
Student: [INAUDIBLE]
00:50:56.030 --> 00:51:01.030
Professor: The question
was the binomial theorem
00:51:01.030 --> 00:51:04.370
only works when
delta x goes to 0.
00:51:04.370 --> 00:51:06.930
No, the binomial theorem
is a general formula
00:51:06.930 --> 00:51:10.030
which also specifies
exactly what the junk is.
00:51:10.030 --> 00:51:11.950
It's very much more detailed.
00:51:11.950 --> 00:51:13.730
But we only needed this part.
00:51:13.730 --> 00:51:18.550
We didn't care what all
these crazy terms were.
00:51:18.550 --> 00:51:23.430
It's junk for our
purposes now, because we
00:51:23.430 --> 00:51:27.650
don't happen to need any more
than those first two terms.
00:51:27.650 --> 00:51:29.910
Yes, because delta x goes to 0.
00:51:29.910 --> 00:51:32.380
OK, see you next time.