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PROFESSOR: Hi.
00:00:33.900 --> 00:00:37.390
Welcome once again to our
Calculus Revisited lecture,
00:00:37.390 --> 00:00:41.750
where today we shall discuss the
concept of infinitesimals,
00:00:41.750 --> 00:00:45.510
a rather elusive but very
important concept.
00:00:45.510 --> 00:00:49.650
And because most textbooks
illustrate this topic in terms
00:00:49.650 --> 00:00:53.150
of approximations, our topic
today will be called
00:00:53.150 --> 00:00:55.650
approximations and
infinitesimals.
00:00:55.650 --> 00:01:00.240
Now, how shall we introduce our
subject in terms of topics
00:01:00.240 --> 00:01:02.220
that you may be more
familiar with?
00:01:02.220 --> 00:01:05.870
Perhaps the easiest way is to
go back to an elementary
00:01:05.870 --> 00:01:09.200
algebra course, to distance,
rate and time problems, when
00:01:09.200 --> 00:01:14.100
one talked about distance
equaling rate times time.
00:01:14.100 --> 00:01:17.750
The question, of course, is what
rate do you use if the
00:01:17.750 --> 00:01:19.755
rate is not constant?
00:01:19.755 --> 00:01:22.430
You see, the question of
distance equals rate times
00:01:22.430 --> 00:01:27.430
time presupposes that you are
dealing with a constant rate.
00:01:27.430 --> 00:01:31.000
Now what does this mean, and how
is it directly connected
00:01:31.000 --> 00:01:33.210
with the development of
our calculus course?
00:01:33.210 --> 00:01:37.750
This shall be the subject of
our investigation today.
00:01:37.750 --> 00:01:39.950
See, the idea is this.
00:01:39.950 --> 00:01:47.010
Let's consider the curve 'y'
equals 'f of x', and let's
00:01:47.010 --> 00:01:48.640
suppose that the curve
is smooth.
00:01:48.640 --> 00:01:51.490
That is, that it possesses a
derivative, say, at the point
00:01:51.490 --> 00:01:53.780
'x' equals 'x1'.
00:01:53.780 --> 00:01:58.430
Let's draw the tangent line to
the curve at 'x' equals 'x1'.
00:02:01.120 --> 00:02:02.390
Now the idea is this.
00:02:02.390 --> 00:02:06.200
In general what we investigate
in a calculus course is the
00:02:06.200 --> 00:02:09.660
concept known as 'delta y'.
00:02:09.660 --> 00:02:12.220
'Delta y' geometrically
is what?
00:02:12.220 --> 00:02:15.650
It's how much 'y' has changed
along the curve
00:02:15.650 --> 00:02:17.690
with respect to 'x'.
00:02:17.690 --> 00:02:21.260
It turns out that there is a
simpler thing that we could
00:02:21.260 --> 00:02:22.670
have computed.
00:02:22.670 --> 00:02:26.960
Notice, if we look at this
particular diagram, that since
00:02:26.960 --> 00:02:30.400
the tangent line never changes
its slope-- and by the way,
00:02:30.400 --> 00:02:33.980
when I say tangent line I
mean at the point 'x1'--
00:02:33.980 --> 00:02:39.830
that once we leave the point
'x1' the tangent line, in a
00:02:39.830 --> 00:02:42.610
way, no longer resembles
the curve.
00:02:42.610 --> 00:02:44.960
But the point that is important
is that I can
00:02:44.960 --> 00:02:47.290
compute the change in
'y' to the tangent
00:02:47.290 --> 00:02:49.180
line here very easily.
00:02:49.180 --> 00:02:52.520
And the reason for this, you
see, is rather apparent.
00:02:52.520 --> 00:02:55.900
Namely, the slope of the tangent
line is, on the one
00:02:55.900 --> 00:03:00.650
hand, 'delta y-tan' divided
by 'delta x'.
00:03:00.650 --> 00:03:05.210
'Delta y-tan' divided
by 'delta x'.
00:03:05.210 --> 00:03:09.150
On the other hand, by definition
of slope, the slope
00:03:09.150 --> 00:03:12.710
of the line 'L' is also
equal to what?
00:03:12.710 --> 00:03:13.960
It's 'dy dx'.
00:03:18.960 --> 00:03:23.750
It's 'dy dx' evaluated at the
point, or the value, 'x'
00:03:23.750 --> 00:03:25.850
equals 'x1'.
00:03:25.850 --> 00:03:29.030
You see, in general the slope of
the curve varies from point
00:03:29.030 --> 00:03:32.220
to point, so when we talk about
the tangent line we must
00:03:32.220 --> 00:03:35.110
emphasize at what point
on the curve we've
00:03:35.110 --> 00:03:36.385
drawn the tangent line.
00:03:36.385 --> 00:03:40.130
At any rate, from this
particular diagram it is not
00:03:40.130 --> 00:03:44.150
difficult to see that to compute
the change in 'y' to
00:03:44.150 --> 00:03:48.860
the tangent line, that this
is nothing more than what?
00:03:48.860 --> 00:03:54.910
'dy dx' evaluated at 'x' equals
'x1' times 'delta x'.
00:03:54.910 --> 00:03:57.690
And you see this is not
an approximation.
00:03:57.690 --> 00:04:03.280
This is precisely the value
of 'delta y-tan'.
00:04:03.280 --> 00:04:11.530
The approximation seems to be
when we say let 'delta y-tan'
00:04:11.530 --> 00:04:13.240
represent 'delta y'.
00:04:13.240 --> 00:04:16.529
In other words, we get the
intuitive feeling that as
00:04:16.529 --> 00:04:20.709
'delta x' gets small, the
difference between the true
00:04:20.709 --> 00:04:24.610
'delta y' and 'delta y-tan'
also gets small.
00:04:24.610 --> 00:04:26.760
Another way of saying
this is what?
00:04:26.760 --> 00:04:30.440
Our intuitive feeling is that,
in a neighborhood of the point
00:04:30.440 --> 00:04:35.050
of tangency, the tangent line
serves as a good approximation
00:04:35.050 --> 00:04:36.640
to the curve itself.
00:04:36.640 --> 00:04:39.360
Now let's see what this
means in terms
00:04:39.360 --> 00:04:41.920
of a specific example.
00:04:41.920 --> 00:04:44.740
I've taken the liberty
of computing the
00:04:44.740 --> 00:04:48.510
cube of 4.01 in advance.
00:04:48.510 --> 00:04:53.990
It turns out to be 64.481201.
00:04:53.990 --> 00:04:57.680
And it's sort of arbitrary,
like cubing this.
00:04:57.680 --> 00:05:00.330
If this if this doesn't look
messy enough for you we could
00:05:00.330 --> 00:05:02.680
have taken this to the sixth
power and then we could have
00:05:02.680 --> 00:05:03.310
squared this.
00:05:03.310 --> 00:05:04.860
But that part is irrelevant.
00:05:04.860 --> 00:05:07.990
A simple check shows that, more
or less, this will be a
00:05:07.990 --> 00:05:09.100
correct statement.
00:05:09.100 --> 00:05:12.280
And what I would like to do, you
see, is simply illustrate
00:05:12.280 --> 00:05:15.940
what our earlier comments
mean in terms of
00:05:15.940 --> 00:05:17.540
this specific example.
00:05:17.540 --> 00:05:21.840
Let's suppose I want to find
an approximation for 4.01
00:05:21.840 --> 00:05:23.800
fairly rapidly.
00:05:23.800 --> 00:05:25.460
The idea is this.
00:05:25.460 --> 00:05:31.240
What I do know is one number
that's very easy to cube,
00:05:31.240 --> 00:05:34.420
which is near 4.01,
is 4 itself.
00:05:34.420 --> 00:05:38.820
In other words, I know
that 4 cubed is 64.
00:05:38.820 --> 00:05:41.540
And by the way, I've
deliberately drawn this
00:05:41.540 --> 00:05:44.670
slightly distorted according
to scale so that we can see
00:05:44.670 --> 00:05:46.390
what's happening over here.
00:05:46.390 --> 00:05:55.060
What 64.481201 represents is
the actual change in height
00:05:55.060 --> 00:05:58.880
from here to here
along the curve.
00:05:58.880 --> 00:06:02.550
In other words, it would be
the length of the segment
00:06:02.550 --> 00:06:06.310
joining the point 'P' to
the point 'Q' here.
00:06:06.310 --> 00:06:11.520
What I claim is that if I
instead tried to find the
00:06:11.520 --> 00:06:16.190
length of 'PR', the change in
'y' not along the curve but
00:06:16.190 --> 00:06:21.030
along the line tangent to the
curve at the point 4.64, this
00:06:21.030 --> 00:06:23.300
is what I can find
fairly rapidly.
00:06:23.300 --> 00:06:28.350
In other words, what I am going
to do is to work this
00:06:28.350 --> 00:06:33.060
same idea here with
a special case.
00:06:33.060 --> 00:06:41.040
You see, I'm going to take 'x1'
to equal 4 and 'delta x'
00:06:41.040 --> 00:06:43.410
to be 0.01.
00:06:43.410 --> 00:06:48.260
Now you see the curve is
'y' equals 'x cubed'.
00:06:48.260 --> 00:06:51.935
From this I can compute 'dy
dx' rather quickly.
00:06:54.860 --> 00:06:58.150
Now, I don't want 'dy dx' at
any old point, I want to
00:06:58.150 --> 00:07:00.830
compute it when 'x' is
4 so I can find the
00:07:00.830 --> 00:07:02.080
slope of the line 'L'.
00:07:07.890 --> 00:07:11.060
And when 'x' is 4 this, of
course, simply is what?
00:07:11.060 --> 00:07:15.420
4 squared is 16,
times 3 is 48.
00:07:15.420 --> 00:07:17.280
So what do I have?
00:07:17.280 --> 00:07:21.790
I have that the slope is 48.
00:07:21.790 --> 00:07:24.860
I also have that 'delta
x' is 0.01.
00:07:24.860 --> 00:07:30.370
So according to my recipe,
'delta y-tan' is what?
00:07:30.370 --> 00:07:35.870
It's 'dy dx' evaluated at 'x'
equals 4, which is 48, times
00:07:35.870 --> 00:07:38.870
'delta x', which is 0.01.
00:07:38.870 --> 00:07:42.960
And that's 0.48.
00:07:42.960 --> 00:07:45.850
See again, let's just
juxtaposition these two.
00:07:45.850 --> 00:07:50.300
All I have done now is computed
this recipe in the
00:07:50.300 --> 00:07:56.150
particular example of trying
to find the cube of 4.01.
00:07:56.150 --> 00:08:01.530
And you see, now notice that
this point 0.48 is exactly the
00:08:01.530 --> 00:08:04.680
length of the accented
line here.
00:08:04.680 --> 00:08:06.670
It's the length of 'PR'.
00:08:06.670 --> 00:08:10.420
And what we do know is that the
height from the x-axis to
00:08:10.420 --> 00:08:13.610
'R' is now exactly what?
00:08:13.610 --> 00:08:20.210
Well, it's the 64
plus the 0.48.
00:08:20.210 --> 00:08:22.470
This then is our
approximation.
00:08:22.470 --> 00:08:24.960
And notice that this
compares with what?
00:08:24.960 --> 00:08:32.270
The precise answer, which
is 64.481201.
00:08:32.270 --> 00:08:35.159
In other words, notice what a
small error we happen to have
00:08:35.159 --> 00:08:37.039
in this particular case.
00:08:37.039 --> 00:08:39.280
And this is the way the subject
is usually brought up.
00:08:39.280 --> 00:08:43.559
It is not a very important thing
from my point of view.
00:08:43.559 --> 00:08:46.480
In other words, I think it's
rather easy to see that, first
00:08:46.480 --> 00:08:50.680
of all, this approximation is
rather nebulous in the sense
00:08:50.680 --> 00:08:55.160
that it requires a knowledge of
how fast the tangent line
00:08:55.160 --> 00:08:56.650
is separating from the curve.
00:08:56.650 --> 00:08:59.530
And this is a rather difficult
topic in its own right.
00:08:59.530 --> 00:09:03.040
And secondly, this was a
rather simple example.
00:09:03.040 --> 00:09:06.280
And we had the luxury here, you
see, of being able to find
00:09:06.280 --> 00:09:09.520
the exact answer so we could
compare our approximation with
00:09:09.520 --> 00:09:11.070
the exact answer.
00:09:11.070 --> 00:09:14.600
In many cases it is difficult
or impossible to find the
00:09:14.600 --> 00:09:16.000
exact answer.
00:09:16.000 --> 00:09:19.810
To emphasize this more
abstractly and more generally,
00:09:19.810 --> 00:09:22.080
let's consider the following.
00:09:22.080 --> 00:09:27.060
Instead of trying to find
the cube of 4.01, the
00:09:27.060 --> 00:09:28.980
generalization here is what?
00:09:28.980 --> 00:09:31.980
That we could take the curve
'y' equals 'x cubed'.
00:09:31.980 --> 00:09:34.990
The derivative of 'y' with
respect to 'x' would then be
00:09:34.990 --> 00:09:36.450
'3 x squared'.
00:09:36.450 --> 00:09:40.130
Evaluated at an arbitrary point
'x' equals 'x1', we
00:09:40.130 --> 00:09:42.450
would get '3 x sub 1 squared'.
00:09:42.450 --> 00:09:47.820
In which case 'delta y-tan'
would be '3 x1 squared'
00:09:47.820 --> 00:09:49.625
times 'delta x'.
00:09:49.625 --> 00:09:52.260
Could we have computed
the exact value of
00:09:52.260 --> 00:09:54.560
'delta y' had we wished?
00:09:54.560 --> 00:09:56.290
And the answer, of
course, is yes.
00:09:56.290 --> 00:09:59.690
Namely, what is the exact
value of 'delta y'?
00:09:59.690 --> 00:10:04.510
Well, we want to compute
this between 'x' equals
00:10:04.510 --> 00:10:06.290
'x1' plus 'delta x'.
00:10:06.290 --> 00:10:11.840
Well, what is the value of 'y'
when 'x' is 'x1 plus delta x'?
00:10:11.840 --> 00:10:14.270
It's ''x1 plus delta
x' cubed'.
00:10:14.270 --> 00:10:17.520
Then we subtract
off 'x1 cubed'.
00:10:17.520 --> 00:10:19.850
And if we expand this,
watch what happens.
00:10:19.850 --> 00:10:23.460
By the binomial theorem we get
an 'x1 cubed' term here, which
00:10:23.460 --> 00:10:26.370
cancels with the 'x1 cubed'
term over here.
00:10:26.370 --> 00:10:27.830
Then we get what?
00:10:27.830 --> 00:10:32.660
A '3 x1 squared delta
x', so using the
00:10:32.660 --> 00:10:34.040
binomial theorem here.
00:10:34.040 --> 00:10:35.360
And then what else do we get?
00:10:35.360 --> 00:10:46.940
We get plus '3x1 delta x
squared' plus 'delta x cubed'.
00:10:46.940 --> 00:10:51.140
And you see, what I'd like to
have us view over here is if
00:10:51.140 --> 00:10:59.270
we look at just this much of the
answer, this part here is
00:10:59.270 --> 00:11:03.780
precisely what we sought to
be 'delta y-tan' before.
00:11:03.780 --> 00:11:08.050
See, this is 'delta y-tan'.
00:11:08.050 --> 00:11:12.070
And what's left over is the
difference, of course, between
00:11:12.070 --> 00:11:13.990
'delta y-tan' and delta y.
00:11:13.990 --> 00:11:17.930
After all, delta y is just
'delta y-tan' plus this
00:11:17.930 --> 00:11:18.740
portion here.
00:11:18.740 --> 00:11:22.390
And and now if you look at this
particular portion over
00:11:22.390 --> 00:11:26.540
here, observe that, as we
expected, what the size of
00:11:26.540 --> 00:11:31.570
this thing is depends both on
where we're measuring 'x1',
00:11:31.570 --> 00:11:34.910
and also on how big the size
of the interval is,
00:11:34.910 --> 00:11:36.490
namely 'delta x'.
00:11:36.490 --> 00:11:38.040
And notice something
that we're going to
00:11:38.040 --> 00:11:39.220
come back to here.
00:11:39.220 --> 00:11:42.820
Notice that the 'delta x' factor
here appears at least
00:11:42.820 --> 00:11:44.230
to the second power.
00:11:44.230 --> 00:11:47.180
In other words, notice over
here that as 'delta x' get
00:11:47.180 --> 00:11:52.450
small, both of these terms
get small very rapidly.
00:11:52.450 --> 00:11:57.480
Because, you see, when 'delta x'
is close to 0 the square of
00:11:57.480 --> 00:12:01.370
a number close to 0 is
even closer to 0.
00:12:01.370 --> 00:12:03.290
But you see the important thing
for now is to notice
00:12:03.290 --> 00:12:06.440
that this is the error in
approximating 'delta y' by
00:12:06.440 --> 00:12:07.760
'delta y-tan'.
00:12:07.760 --> 00:12:12.800
And that this error depends both
on 'xy' and 'delta x'.
00:12:12.800 --> 00:12:16.010
And the question is, can we
state that in a little bit
00:12:16.010 --> 00:12:17.890
more of a mathematical
language?
00:12:17.890 --> 00:12:20.620
The answer, of course,
is that we can.
00:12:20.620 --> 00:12:23.800
In particular, what we're
saying is let's take the
00:12:23.800 --> 00:12:28.200
difference between 'dy dx' at
'x' equals 'x1', namely the
00:12:28.200 --> 00:12:33.170
slope of the tangent line, and
what the average change of 'y'
00:12:33.170 --> 00:12:36.400
with respect to 'x' is
along the curve.
00:12:36.400 --> 00:12:38.630
Now, this is very typical
in mathematics.
00:12:38.630 --> 00:12:41.220
And it's a very nice trick
to learn, and a very
00:12:41.220 --> 00:12:42.450
quick trick to learn.
00:12:42.450 --> 00:12:46.200
We know that these two things
are not equal and so all we do
00:12:46.200 --> 00:12:48.890
is tack on a correction
factor.
00:12:48.890 --> 00:12:50.230
We tack on 'k'.
00:12:50.230 --> 00:12:51.370
What is 'k'?
00:12:51.370 --> 00:12:55.130
'k' is the difference between
these two expressions.
00:12:55.130 --> 00:12:58.720
In other words, by definition
I add on the difference of
00:12:58.720 --> 00:13:01.920
these two numbers and that
makes this an equality.
00:13:01.920 --> 00:13:04.540
Notice, by the way, that
'k' is a variable.
00:13:04.540 --> 00:13:08.000
It depends on how big
'delta x' is.
00:13:08.000 --> 00:13:12.400
You see, notice that once
'x1' is chosen--
00:13:12.400 --> 00:13:14.780
and this is an important
observation.
00:13:14.780 --> 00:13:19.350
As messy as this thing looks,
it's a fixed number for a
00:13:19.350 --> 00:13:21.040
particular value of 'x1'.
00:13:21.040 --> 00:13:23.570
Namely, we compute the
derivative and evaluate it
00:13:23.570 --> 00:13:24.990
when 'x' equals 'x1'.
00:13:24.990 --> 00:13:26.110
This is a number.
00:13:26.110 --> 00:13:28.080
This, of course, clearly
depends on the
00:13:28.080 --> 00:13:29.600
size of 'delta x'.
00:13:29.600 --> 00:13:32.090
'Delta y' depends
on 'delta x'.
00:13:32.090 --> 00:13:34.520
At any rate, here's
what we do.
00:13:34.520 --> 00:13:39.330
We recognize that 'dy dx' is
the limit of 'delta y' over
00:13:39.330 --> 00:13:42.080
'delta x' as 'delta
x' approaches 0.
00:13:42.080 --> 00:13:46.420
So with this as a hint, we take
the limit of both sides
00:13:46.420 --> 00:13:49.210
here as 'delta x'
approaches 0.
00:13:49.210 --> 00:13:51.180
Notice the structure again.
00:13:51.180 --> 00:13:53.630
In our last series
of lectures we
00:13:53.630 --> 00:13:55.110
talked about limit theorems.
00:13:55.110 --> 00:13:57.490
We talked about the limit of
a sum being the sum of the
00:13:57.490 --> 00:13:59.100
limits, and things
of this type.
00:13:59.100 --> 00:14:02.140
And now, you see, we're going
to use these results.
00:14:02.140 --> 00:14:05.360
What we do is we take the limit
of both sides here as
00:14:05.360 --> 00:14:06.830
'delta x' approaches 0.
00:14:14.360 --> 00:14:16.220
The point being what?
00:14:16.220 --> 00:14:22.440
That the limit of a sum is
the sum of the limits.
00:14:22.440 --> 00:14:26.060
So when I take the limit of this
entire side I can express
00:14:26.060 --> 00:14:31.880
that as the sum of the limits.
00:14:31.880 --> 00:14:35.000
It is the limit of this term as
'delta x' approaches 0 plus
00:14:35.000 --> 00:14:38.380
the limit of this term as
'delta x' approaches 0.
00:14:38.380 --> 00:14:41.950
Now let's stop to think what
these things mean.
00:14:41.950 --> 00:14:45.180
The limit of 'delta x'
approaching 0 of 'delta y'
00:14:45.180 --> 00:14:49.070
divided by 'delta x' is
precisely the definition of
00:14:49.070 --> 00:14:50.050
derivative.
00:14:50.050 --> 00:14:54.820
In other words, this term here
is just the 'dy dx'.
00:14:54.820 --> 00:14:57.150
And keep in mind we've
evaluated this
00:14:57.150 --> 00:14:58.610
at 'x' equals 'x1'.
00:14:58.610 --> 00:15:01.290
That's the point at which
we're starting.
00:15:01.290 --> 00:15:07.480
Now the fact that this is a
constant, and we've already
00:15:07.480 --> 00:15:10.600
learned that the limit of
a constant as 'delta x'
00:15:10.600 --> 00:15:17.290
approaches 0 is that constant,
this term here is also 'dy dx'
00:15:17.290 --> 00:15:20.940
evaluated at 'x' equals 'x1'.
00:15:20.940 --> 00:15:24.600
And of course this term here
is just the limit of 'k' as
00:15:24.600 --> 00:15:26.810
'delta x' approaches 0.
00:15:26.810 --> 00:15:32.510
And now you see if we look at
this, since all we have to do
00:15:32.510 --> 00:15:34.770
is observe that this term
appears on both sides of the
00:15:34.770 --> 00:15:37.750
equation, canceling, or
subtracting equals from
00:15:37.750 --> 00:15:43.570
equals, we wind up with the
result that the limit of 'k'
00:15:43.570 --> 00:15:46.810
as 'delta x' approaches
0 is 0.
00:15:50.300 --> 00:15:53.810
And we're going to talk more
about that pictorially and
00:15:53.810 --> 00:15:55.100
analytically in a
little while.
00:15:55.100 --> 00:16:01.760
But from this result, all I'm
going to do now is go back to
00:16:01.760 --> 00:16:04.280
our first statement here.
00:16:04.280 --> 00:16:08.320
Remembering that 'delta x' is
a nonzero number, I will
00:16:08.320 --> 00:16:14.210
simply multiply both sides of
this equation by 'delta x'.
00:16:14.210 --> 00:16:17.550
And if I do that, what
do I wind up with?
00:16:17.550 --> 00:16:25.840
I wind up with that 'delta y'
is 'dy dx' evaluated at 'x'
00:16:25.840 --> 00:16:34.940
equals 'x1' times 'delta
x' plus 'k delta x'.
00:16:34.940 --> 00:16:40.280
Now, if I keep in mind here that
if I take the slope at a
00:16:40.280 --> 00:16:44.550
particular point and multiply
that by the change in 'x',
00:16:44.550 --> 00:16:49.330
that that's precisely what we
define to be 'delta y-tan'.
00:16:49.330 --> 00:16:52.240
If I put this all together
I get what?
00:16:52.240 --> 00:16:57.050
That 'delta y' is equal to
''delta y-tan' plus 'k delta
00:16:57.050 --> 00:17:02.380
x'', where the limit of 'k' as
'delta x' approaches 0 is 0.
00:17:02.380 --> 00:17:06.020
Now you see, when we started
this program and started to
00:17:06.020 --> 00:17:09.470
talk at the beginning of our
lecture about approximations,
00:17:09.470 --> 00:17:12.349
we did something that
was quite crude.
00:17:12.349 --> 00:17:14.119
The crude part was
simply this.
00:17:14.119 --> 00:17:18.440
What we had said was, why don't
we compute 'delta y' by
00:17:18.440 --> 00:17:21.640
computing 'delta
y-tan' instead.
00:17:21.640 --> 00:17:24.900
In other words, this is a very
easy thing to compute.
00:17:24.900 --> 00:17:28.270
It's just a derivative
multiplied by a change in 'x'.
00:17:28.270 --> 00:17:31.430
This is a very simple
thing to compute.
00:17:31.430 --> 00:17:34.190
And we'll use that as an
approximation for the thing
00:17:34.190 --> 00:17:36.650
that we really want,
which is 'delta y'.
00:17:36.650 --> 00:17:39.120
The question that we
deliberately overlooked at
00:17:39.120 --> 00:17:43.930
this point was how big was the
error when you do this?
00:17:43.930 --> 00:17:46.470
And what we discovered was
something very interesting
00:17:46.470 --> 00:17:50.060
and, by the way, may be
something which teaches us why
00:17:50.060 --> 00:17:53.440
the analysis is better than the
picture even though the
00:17:53.440 --> 00:17:55.430
picture is easier
to visualize.
00:17:55.430 --> 00:18:00.820
You see, we knew intuitively
that as you picked a smaller
00:18:00.820 --> 00:18:02.850
neighborhood at the
point of tangency.
00:18:02.850 --> 00:18:04.920
the tangent line became
a better and better
00:18:04.920 --> 00:18:07.710
approximation to the
curve itself.
00:18:07.710 --> 00:18:11.640
This boxed-in result tells
us much more than that.
00:18:11.640 --> 00:18:16.360
What this tells us is, look, if
this term by itself became
00:18:16.360 --> 00:18:19.720
small as 'delta x' approached
0, this would still say the
00:18:19.720 --> 00:18:20.780
same thing.
00:18:20.780 --> 00:18:22.760
This says much more than that.
00:18:22.760 --> 00:18:24.620
You see, what this says is--
00:18:24.620 --> 00:18:27.010
and watch, this is very,
very profound--
00:18:27.010 --> 00:18:33.500
as 'delta x' approaches 0, 'k'
is also approaching 0.
00:18:33.500 --> 00:18:38.230
In other words, the term 'k'
times 'delta x' seems to be
00:18:38.230 --> 00:18:42.820
approaching 0 much more rapidly
than 'delta x' itself.
00:18:42.820 --> 00:18:47.060
That, by the way, in as simple
a way as I know how, defines
00:18:47.060 --> 00:18:49.190
what an infinitesimal is.
00:18:49.190 --> 00:18:54.450
See, 'k' times 'delta x' is
called an infinitesimal
00:18:54.450 --> 00:18:58.790
because it goes to 0 faster
than 'delta x' itself.
00:18:58.790 --> 00:19:01.650
In other words, anything that
approaches 0 faster than
00:19:01.650 --> 00:19:05.640
'delta x' itself approaches 0
is called an infinitesimal
00:19:05.640 --> 00:19:08.070
with respect to 'delta x'.
00:19:08.070 --> 00:19:11.270
Now I'm not going to go into a
long philosophical discussion
00:19:11.270 --> 00:19:12.140
about that.
00:19:12.140 --> 00:19:15.682
Rather, I'm going to let the
actions speak louder than the
00:19:15.682 --> 00:19:19.560
words, and try to show why this
is such a crucial idea.
00:19:19.560 --> 00:19:21.220
In other words, again let
me emphasize that.
00:19:21.220 --> 00:19:25.820
What's crucial here is not so
much that the error goes to 0,
00:19:25.820 --> 00:19:29.850
it's that the error goes
to 0 much faster than
00:19:29.850 --> 00:19:31.490
'delta x' goes to 0.
00:19:31.490 --> 00:19:32.620
See, what is the error?
00:19:32.620 --> 00:19:34.790
The error is 'k' times
'delta x'.
00:19:34.790 --> 00:19:37.810
That's the difference between
these two things.
00:19:37.810 --> 00:19:41.080
Again, if you want to see what
this thing means pictorially,
00:19:41.080 --> 00:19:44.370
and I wish I knew a better way
of doing this but I don't,
00:19:44.370 --> 00:19:49.540
notice that 'k' times 'delta
x' was defined to be the
00:19:49.540 --> 00:19:53.220
difference between 'delta
y' and 'delta y-tan'.
00:19:53.220 --> 00:19:58.610
In other words, since this
length here is 'delta y-tan',
00:19:58.610 --> 00:20:01.600
and since this entire length
would be called 'delta y',
00:20:01.600 --> 00:20:06.890
again the accented line, this
length here, is 'k'
00:20:06.890 --> 00:20:09.160
times 'delta x'.
00:20:09.160 --> 00:20:13.490
And what does it mean to say
that 'k' approaches 0 as
00:20:13.490 --> 00:20:15.020
'delta x' approaches 0?
00:20:15.020 --> 00:20:16.640
What it means is this.
00:20:16.640 --> 00:20:22.400
It means that not only does this
vertical difference get
00:20:22.400 --> 00:20:29.490
small as 'delta x' goes to 0,
but more importantly, it means
00:20:29.490 --> 00:20:34.350
that this vertical distance
gets small very rapidly
00:20:34.350 --> 00:20:36.610
compared to how 'delta
x' get small.
00:20:36.610 --> 00:20:37.840
Now how can I prove that?
00:20:37.840 --> 00:20:40.550
Well I'm not even going
to try to prove that.
00:20:40.550 --> 00:20:42.690
What I'm going to just try to do
is to have you see from the
00:20:42.690 --> 00:20:43.870
picture what's happening here.
00:20:43.870 --> 00:20:47.080
See, here is a fairly
large 'delta x'.
00:20:47.080 --> 00:20:50.030
And the difference between
'delta y' and 'delta y-tan'
00:20:50.030 --> 00:20:53.670
for that large 'x' is this
length over here.
00:20:53.670 --> 00:20:56.150
Now suppose we take a
smaller 'delta x'.
00:20:56.150 --> 00:20:58.530
In other words, let's
take 'delta x' to be
00:20:58.530 --> 00:21:00.420
this length over here.
00:21:00.420 --> 00:21:03.920
Notice that the length of 'delta
x' here is still quite
00:21:03.920 --> 00:21:04.790
significant.
00:21:04.790 --> 00:21:07.110
I think if you look at this
you can see it's a fairly
00:21:07.110 --> 00:21:08.770
significant length.
00:21:08.770 --> 00:21:12.510
On the other hand, notice what
the difference between 'delta
00:21:12.510 --> 00:21:14.780
y' and 'delta y-tan' is now.
00:21:14.780 --> 00:21:17.755
It's just this little tiny
thing over here.
00:21:17.755 --> 00:21:22.160
In other words, notice that
as 'delta x' gets small,
00:21:22.160 --> 00:21:25.840
ratio-wise as 'delta x' get
small, the vertical difference
00:21:25.840 --> 00:21:29.600
between the tangent line and the
curve is getting small at
00:21:29.600 --> 00:21:31.070
a much faster rate.
00:21:31.070 --> 00:21:34.270
Now of course the question is
why is that so important?
00:21:34.270 --> 00:21:36.740
And the answer will always
come back to our
00:21:36.740 --> 00:21:40.440
old friend of 0/0.
00:21:40.440 --> 00:21:46.070
In general, if you tell a person
that the numerator of a
00:21:46.070 --> 00:21:51.200
fraction is small, he jumps to
the conclusion that the size
00:21:51.200 --> 00:21:53.380
of the fraction must be small.
00:21:53.380 --> 00:21:56.980
But you see, the trouble is when
the denominator is also
00:21:56.980 --> 00:22:00.970
small then we cannot
conclude that the
00:22:00.970 --> 00:22:03.120
fraction itself is small.
00:22:03.120 --> 00:22:06.190
And therefore in trying to
cancel out a term as being
00:22:06.190 --> 00:22:08.650
insignificant, when we're
dealing with something like
00:22:08.650 --> 00:22:13.010
this, it is no longer enough
to say but the numerator is
00:22:13.010 --> 00:22:14.150
going to 0.
00:22:14.150 --> 00:22:17.900
Let me jump ahead to the topic
that will be covered in our
00:22:17.900 --> 00:22:22.650
next lecture just to use this
as an insight as to what we
00:22:22.650 --> 00:22:27.060
have to see and what's
going on over here.
00:22:27.060 --> 00:22:29.910
Let me just show you what
I mean by this.
00:22:29.910 --> 00:22:34.220
let's go back to our recipe,
which says 'delta y' is this
00:22:34.220 --> 00:22:37.310
thing over here where what?
00:22:37.310 --> 00:22:40.560
This is important to put in
here, the limit of 'k' as
00:22:40.560 --> 00:22:44.410
'delta x' approaches 0 is 0.
00:22:44.410 --> 00:22:46.440
Now all I'm going to do
something like this.
00:22:46.440 --> 00:22:50.540
Let's suppose we're dealing in
a situation where 'y' and 'x'
00:22:50.540 --> 00:22:54.320
happen to be functions, say of,
't' as well, some third
00:22:54.320 --> 00:22:55.460
variable 't'.
00:22:55.460 --> 00:22:59.240
And what we really want
to find is 'dy dt'.
00:22:59.240 --> 00:23:02.380
You see, what we'd be tempted
to do is simply say what?
00:23:02.380 --> 00:23:06.870
Let's divide through
by 'delta t'.
00:23:06.870 --> 00:23:09.340
We do this, we divide through
by 'delta t'.
00:23:09.340 --> 00:23:12.130
Then we take the limit of
this thing as 'delta
00:23:12.130 --> 00:23:13.800
t' approaches 0.
00:23:13.800 --> 00:23:14.950
Now what happens?
00:23:14.950 --> 00:23:18.280
As 'delta t' approaches
0, this is a sum.
00:23:18.280 --> 00:23:20.840
The limit of a sum is the
sum of the limits.
00:23:20.840 --> 00:23:23.590
Each of the terms in the
sum is a product.
00:23:23.590 --> 00:23:26.590
And the limit of a product is
the product of the limits.
00:23:26.590 --> 00:23:30.010
So working out the regular limit
theorems, this is what?
00:23:30.010 --> 00:23:33.230
The limit, 'delta t' approaches
0, 'dy dx'
00:23:33.230 --> 00:23:36.830
evaluated at 'x' equals 'x1',
times the limit of 'delta x'
00:23:36.830 --> 00:23:40.250
divided by 'delta t' as 'delta
t' approaches 0, plus the
00:23:40.250 --> 00:23:43.730
limit of 'k' as 'delta t'
approaches 0, times the limit
00:23:43.730 --> 00:23:48.060
of 'delta x' divided by 'delta
t' as 'delta t' approaches 0.
00:23:48.060 --> 00:23:52.670
Now, if we remember what this
thing here means, this is just
00:23:52.670 --> 00:23:56.110
by definition 'dy dt'.
00:23:56.110 --> 00:23:58.940
For the sake of brevity I will
leave out subscripts now, but
00:23:58.940 --> 00:24:01.700
we'll just talk about this.
00:24:01.700 --> 00:24:06.170
This is a constant, and the
limit of constant as 'delta t'
00:24:06.170 --> 00:24:08.440
approaches 0 is just that
constant, which
00:24:08.440 --> 00:24:10.520
I'll call 'dy dx'.
00:24:10.520 --> 00:24:13.700
It's understood this is
evaluated at 'x' equals 'x1'.
00:24:13.700 --> 00:24:19.120
By definition the limit of
'delta x' divided by 'delta t'
00:24:19.120 --> 00:24:25.020
as 'delta t' approaches 0,
that's called 'dx dt'.
00:24:25.020 --> 00:24:27.020
Now we come to this
term over here.
00:24:27.020 --> 00:24:28.340
And here's the key point.
00:24:28.340 --> 00:24:31.460
Many people say, well, as
'delta t' goes to 0
00:24:31.460 --> 00:24:32.860
so does 'delta x'.
00:24:32.860 --> 00:24:36.150
That makes the numerator here
0 and that makes the whole
00:24:36.150 --> 00:24:37.220
fraction 0.
00:24:37.220 --> 00:24:39.080
But the point is that's
not true.
00:24:39.080 --> 00:24:42.440
What is true here is that both
the numerator and denominator
00:24:42.440 --> 00:24:43.760
are going to 0.
00:24:43.760 --> 00:24:47.680
This does not make this 0, but
rather it makes it what?
00:24:47.680 --> 00:24:48.530
'dx dt'.
00:24:48.530 --> 00:24:50.870
That's the definition
of 'dx dt'.
00:24:50.870 --> 00:24:52.870
The important point is what?
00:24:52.870 --> 00:24:56.630
That as 'delta t' approaches
0 so does 'delta x'.
00:24:56.630 --> 00:24:59.230
And what property
does 'k' have?
00:24:59.230 --> 00:25:03.110
'k' has the property that the
limit of 'k' as 'delta x'
00:25:03.110 --> 00:25:08.140
approaches 0 is 0 itself.
00:25:08.140 --> 00:25:11.900
In other words, the topics that
we'll be talking about in
00:25:11.900 --> 00:25:15.460
our next lecture concerns
this recipe.
00:25:15.460 --> 00:25:19.260
And in the development of this
recipe, the reason that the
00:25:19.260 --> 00:25:23.170
error term drops out is not
because the numerator of this
00:25:23.170 --> 00:25:28.310
term was small, it was because
the number multiplying limit
00:25:28.310 --> 00:25:30.610
happened to be 0.
00:25:30.610 --> 00:25:34.240
Now, we will talk about that
in more detail next time.
00:25:34.240 --> 00:25:38.690
But the message I want you to
see for right now is how we
00:25:38.690 --> 00:25:43.630
get hung up on this 0/0 bit,
and why we must always be
00:25:43.630 --> 00:25:46.700
careful in how we handle
something like this.
00:25:46.700 --> 00:25:49.460
Now, by the way, there is
something rather interesting
00:25:49.460 --> 00:25:50.820
that does happen over here.
00:25:50.820 --> 00:25:54.210
If you look at this thing, you
get the feeling that it's
00:25:54.210 --> 00:25:58.020
almost as if you could cancel a
common factor from both the
00:25:58.020 --> 00:26:00.240
numerator and the denominator.
00:26:00.240 --> 00:26:02.390
We have to be very,
very careful.
00:26:02.390 --> 00:26:04.560
And I would like to introduce
the language which you will be
00:26:04.560 --> 00:26:07.700
reading in the text in this
assignment, called
00:26:07.700 --> 00:26:09.250
'Differentials Now'.
00:26:09.250 --> 00:26:12.370
And what that means
is simply this.
00:26:12.370 --> 00:26:18.120
Notice that 'dy dx' is one
symbol it is not 'dy' divided
00:26:18.120 --> 00:26:23.090
by 'dx' because these things
have not been defined
00:26:23.090 --> 00:26:23.420
separately.
00:26:23.420 --> 00:26:27.400
In other words, you can't just
operate with symbols the way
00:26:27.400 --> 00:26:28.600
you might like to.
00:26:28.600 --> 00:26:31.800
Here's an interesting little
aside that you may find
00:26:31.800 --> 00:26:35.570
entertaining if it has no other
value at all: the idea
00:26:35.570 --> 00:26:38.870
of being told that you can
cancel common factors from
00:26:38.870 --> 00:26:40.710
both numerator and
denominator.
00:26:40.710 --> 00:26:43.660
The uninitiated may say,
you know, here's a 6
00:26:43.660 --> 00:26:44.700
and here's a 6.
00:26:44.700 --> 00:26:46.890
I'll cancel them out.
00:26:46.890 --> 00:26:49.070
Now this is not what
cancellation really meant,
00:26:49.070 --> 00:26:51.810
even though you had the same
number both upstairs and
00:26:51.810 --> 00:26:53.310
downstairs.
00:26:53.310 --> 00:26:54.925
This is not what we meant
by cancellation.
00:26:54.925 --> 00:26:59.730
Yet, notice that you happen to,
by accident, get the right
00:26:59.730 --> 00:27:01.650
answer in this particular
case.
00:27:01.650 --> 00:27:03.180
You see, if you do cancel
this, this does
00:27:03.180 --> 00:27:06.120
happen to be 1/4.
00:27:06.120 --> 00:27:09.650
This lecture has been going on
in kind of a dull way for you.
00:27:09.650 --> 00:27:13.120
Just in case you'd like some
comic relief, there are a few
00:27:13.120 --> 00:27:15.860
other examples that
work the same way.
00:27:15.860 --> 00:27:18.860
And you can amaze your friends
at the next party with them.
00:27:18.860 --> 00:27:23.980
I mean 19/95 happens
to be 1/5.
00:27:23.980 --> 00:27:30.410
26/65 happens to be 2/5.
00:27:30.410 --> 00:27:31.970
And I think there's one more.
00:27:31.970 --> 00:27:36.570
49/98 happens to be 4/8.
00:27:36.570 --> 00:27:39.880
But these, I believe, are
the only four fractions
00:27:39.880 --> 00:27:41.340
that this works for.
00:27:41.340 --> 00:27:43.490
In other words, the mere fact
that it looks like something
00:27:43.490 --> 00:27:46.700
is going to work is no guarantee
that you can do
00:27:46.700 --> 00:27:48.980
these things without running
into trouble.
00:27:48.980 --> 00:27:52.380
On the other hand, it's fair
to assume that the men who
00:27:52.380 --> 00:27:57.310
invented differential calculus
must've been clever enough not
00:27:57.310 --> 00:27:59.650
to have invented a symbolism
that would have
00:27:59.650 --> 00:28:01.370
gotten us into trouble.
00:28:01.370 --> 00:28:03.450
That, by the way, is a big
assumption to make, that they
00:28:03.450 --> 00:28:04.620
must have been clever enough.
00:28:04.620 --> 00:28:08.010
We will find, in places during
our course, that this was not
00:28:08.010 --> 00:28:09.400
always the case.
00:28:09.400 --> 00:28:10.870
The point is this though.
00:28:10.870 --> 00:28:16.950
That when you write 'dy dx' in
this way, if it were not going
00:28:16.950 --> 00:28:21.010
to be somehow identifiable with
a quotient such as 'dy'
00:28:21.010 --> 00:28:24.430
divided by 'dx', the chances
are we would never have
00:28:24.430 --> 00:28:27.100
invented this notation
in the first place.
00:28:27.100 --> 00:28:28.580
And so what I'd like
to just point out
00:28:28.580 --> 00:28:30.860
briefly now is the following.
00:28:30.860 --> 00:28:34.780
Can we, in fact we should,
how shall we--
00:28:34.780 --> 00:28:35.940
let's put it this way--
00:28:35.940 --> 00:28:42.160
how shall we define separate
symbols dy and dx so that when
00:28:42.160 --> 00:28:46.030
we write down 'dy dx' it will
make no difference whether you
00:28:46.030 --> 00:28:48.770
think of this as being the
derivative or whether you
00:28:48.770 --> 00:28:52.230
think of it as being the
quotient of two numbers?
00:28:52.230 --> 00:28:54.730
And by the way, you see the
answer to this question will
00:28:54.730 --> 00:28:57.520
not be that difficult if we
stop to think for just a
00:28:57.520 --> 00:28:59.610
moment over here.
00:28:59.610 --> 00:29:01.650
We've already answered this
question except that we
00:29:01.650 --> 00:29:03.950
haven't concentrated
on the fact that
00:29:03.950 --> 00:29:05.220
we solved the problem.
00:29:05.220 --> 00:29:07.350
See, let's go back to
this line 'L' again.
00:29:07.350 --> 00:29:10.190
What is the slope
of the line 'L'?
00:29:10.190 --> 00:29:13.130
On the one hand we've looked
at slope as the
00:29:13.130 --> 00:29:15.580
derivative 'dy dx'.
00:29:15.580 --> 00:29:18.560
On the other hand, just by
looking at this little
00:29:18.560 --> 00:29:23.140
accentuated triangle here, the
slope of the line 'L' is also
00:29:23.140 --> 00:29:30.710
given by 'delta y-tan'
divided by 'delta x'.
00:29:30.710 --> 00:29:37.690
And now one of the best ways
to define 'dy' and 'dx'
00:29:37.690 --> 00:29:41.690
separately, it would seem
to me, is what?
00:29:41.690 --> 00:29:45.880
Notice that this number divided
by this number yields
00:29:45.880 --> 00:29:47.800
the symbol 'dy dx'.
00:29:47.800 --> 00:29:56.460
Therefore why not define this
number to be 'dy' and define
00:29:56.460 --> 00:29:59.660
this number to be 'dx'?
00:29:59.660 --> 00:30:02.950
In other words the symbol 'dx'
will just be a fancy
00:30:02.950 --> 00:30:04.510
name for 'delta x'.
00:30:04.510 --> 00:30:09.110
On the other hand, the symbol
'dy' will not be a fancy name
00:30:09.110 --> 00:30:10.020
for 'delta y'.
00:30:10.020 --> 00:30:12.200
In fact, let's emphasize that.
00:30:12.200 --> 00:30:17.720
It's not equal to 'delta y', it
is equal to 'delta y-tan'.
00:30:17.720 --> 00:30:22.480
In other words, if I allow
'dy' to stand for 'delta
00:30:22.480 --> 00:30:27.740
y-tan' and 'dx' to stand for
'delta x', I can now treat 'dy
00:30:27.740 --> 00:30:30.260
dx' as if it were what?
00:30:30.260 --> 00:30:33.850
'dy' divided by 'dx'.
00:30:33.850 --> 00:30:39.160
Well, that's easy to show in
terms of an example just
00:30:39.160 --> 00:30:40.670
mechanically.
00:30:40.670 --> 00:30:43.240
Remember, given 'y' equals
'x cubed' we've
00:30:43.240 --> 00:30:44.730
already found what?
00:30:44.730 --> 00:30:47.550
That 'dy dx' is '3 x squared'.
00:30:47.550 --> 00:30:51.320
This thing practically begs to
be treated like a fraction.
00:30:51.320 --> 00:30:54.350
We would like to be able to say,
hey, if this divided by
00:30:54.350 --> 00:30:59.700
this is this, why isn't this
equal to this times this?
00:30:59.700 --> 00:31:04.420
In other words, why isn't 'dy'
equal to '3 x squared' dx'?
00:31:04.420 --> 00:31:08.290
And the answer is that since
we've identified or defined
00:31:08.290 --> 00:31:13.130
'dy' to be 'delta y-tan', and
since we've defined 'dx' to be
00:31:13.130 --> 00:31:17.230
'delta x', and since we already
know that this recipe
00:31:17.230 --> 00:31:20.530
is correct, it means, in
particular, that we can now
00:31:20.530 --> 00:31:24.590
write things like this without
having to worry about whether
00:31:24.590 --> 00:31:26.150
it's proper or not.
00:31:26.150 --> 00:31:29.710
Now in later lectures this
is going to play a
00:31:29.710 --> 00:31:31.320
very important role.
00:31:31.320 --> 00:31:34.890
This is what is known as the
language of differentials.
00:31:34.890 --> 00:31:39.390
And differentials are the
backbone off both differential
00:31:39.390 --> 00:31:41.390
and integral calculus.
00:31:41.390 --> 00:31:45.500
But the thing that I want you
to really get into our minds
00:31:45.500 --> 00:31:50.100
today is the basic
overall recipe.
00:31:50.100 --> 00:31:54.270
And I've taken the liberty of
boxing this off over here.
00:31:54.270 --> 00:31:56.530
I want you to practice
with approximations.
00:31:56.530 --> 00:32:00.403
I want you to think carefully
about how you can get quick
00:32:00.403 --> 00:32:01.110
approximations.
00:32:01.110 --> 00:32:03.580
I don't want you to come away
with the feeling that the
00:32:03.580 --> 00:32:05.680
approximation is what
was important.
00:32:05.680 --> 00:32:07.660
What was important was what?
00:32:07.660 --> 00:32:11.910
That the change in 'y', the true
'delta y', is ''dy dx'
00:32:11.910 --> 00:32:16.760
times 'delta x'' plus 'k delta
x', where the limit of 'k' as
00:32:16.760 --> 00:32:21.140
'delta x' approaches 0 is 0.
00:32:21.140 --> 00:32:24.070
And by the way, again, most
books write it this way.
00:32:24.070 --> 00:32:28.450
I prefer that we emphasize
that the derivative is
00:32:28.450 --> 00:32:32.680
evaluated or taken at a
particular value of 'x'.
00:32:32.680 --> 00:32:36.840
And finally what I'd like to
point out is that again, and
00:32:36.840 --> 00:32:39.400
we've done this many, many
times, whereas the language
00:32:39.400 --> 00:32:44.240
we're used to in terms of
intuition is geometry, that
00:32:44.240 --> 00:32:48.310
these results make perfectly
good sense without reference
00:32:48.310 --> 00:32:50.190
to any diagram at all.
00:32:50.190 --> 00:32:54.280
In other words, then, the same
result stated in analytic
00:32:54.280 --> 00:32:58.960
language simply says this: if
'f' is a function of 'x' and
00:32:58.960 --> 00:33:02.940
is differentiable when 'x'
equals 'x1', then 'f of 'x1
00:33:02.940 --> 00:33:05.830
plus delta x'' minus
'f of x1'--
00:33:05.830 --> 00:33:08.450
you see that's geometrically
what corresponds
00:33:08.450 --> 00:33:10.970
to your 'delta y'.
00:33:10.970 --> 00:33:15.860
That's ''f prime of x1' times
'delta x'' plus 'k delta x'
00:33:15.860 --> 00:33:21.510
where the limit of 'k' as 'delta
x' approaches 0 is 0.
00:33:21.510 --> 00:33:27.840
Now these two recipes summarize
precisely what it is
00:33:27.840 --> 00:33:30.930
that we are interested in when
we deal with the subject
00:33:30.930 --> 00:33:32.780
called infinitesimals.
00:33:32.780 --> 00:33:35.600
As I said before, and I can't
emphasize this thing strongly
00:33:35.600 --> 00:33:39.540
enough, when it comes time to
make approximations we will
00:33:39.540 --> 00:33:43.120
find better ways of getting
approximations than by the
00:33:43.120 --> 00:33:46.570
method known as differentials
and 'delta y-tan'.
00:33:46.570 --> 00:33:49.630
What is crucial is that you
use the language of
00:33:49.630 --> 00:33:53.780
approximations enough so as you
can see pictorially what's
00:33:53.780 --> 00:33:58.190
going on and then cement down
the final recipe, which I've
00:33:58.190 --> 00:33:59.260
boxed in here.
00:33:59.260 --> 00:34:03.080
This will be the building block,
as we shall see you
00:34:03.080 --> 00:34:06.480
next time, in the lecture which
develops the derivative
00:34:06.480 --> 00:34:08.389
of composite functions.
00:34:08.389 --> 00:34:10.860
But more about that next time.
00:34:10.860 --> 00:34:12.260
And until next time, goodbye.
00:34:15.250 --> 00:34:18.449
Funding for the publication of
this video was provided by the
00:34:18.449 --> 00:34:22.500
Gabriella and Paul Rosenbaum
Foundation.
00:34:22.500 --> 00:34:26.679
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00:34:26.679 --> 00:34:30.870
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