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PROFESSOR: Hi.
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Welcome once again to our
Calculus Revisited lecture,
12
00:00:37,390 --> 00:00:41,750
where today we shall discuss the
concept of infinitesimals,
13
00:00:41,750 --> 00:00:45,510
a rather elusive but very
important concept.
14
00:00:45,510 --> 00:00:49,650
And because most textbooks
illustrate this topic in terms
15
00:00:49,650 --> 00:00:53,150
of approximations, our topic
today will be called
16
00:00:53,150 --> 00:00:55,650
approximations and
infinitesimals.
17
00:00:55,650 --> 00:01:00,240
Now, how shall we introduce our
subject in terms of topics
18
00:01:00,240 --> 00:01:02,220
that you may be more
familiar with?
19
00:01:02,220 --> 00:01:05,870
Perhaps the easiest way is to
go back to an elementary
20
00:01:05,870 --> 00:01:09,200
algebra course, to distance,
rate and time problems, when
21
00:01:09,200 --> 00:01:14,100
one talked about distance
equaling rate times time.
22
00:01:14,100 --> 00:01:17,750
The question, of course, is what
rate do you use if the
23
00:01:17,750 --> 00:01:19,755
rate is not constant?
24
00:01:19,755 --> 00:01:22,430
You see, the question of
distance equals rate times
25
00:01:22,430 --> 00:01:27,430
time presupposes that you are
dealing with a constant rate.
26
00:01:27,430 --> 00:01:31,000
Now what does this mean, and how
is it directly connected
27
00:01:31,000 --> 00:01:33,210
with the development of
our calculus course?
28
00:01:33,210 --> 00:01:37,750
This shall be the subject of
our investigation today.
29
00:01:37,750 --> 00:01:39,950
See, the idea is this.
30
00:01:39,950 --> 00:01:47,010
Let's consider the curve 'y'
equals 'f of x', and let's
31
00:01:47,010 --> 00:01:48,640
suppose that the curve
is smooth.
32
00:01:48,640 --> 00:01:51,490
That is, that it possesses a
derivative, say, at the point
33
00:01:51,490 --> 00:01:53,780
'x' equals 'x1'.
34
00:01:53,780 --> 00:01:58,430
Let's draw the tangent line to
the curve at 'x' equals 'x1'.
35
00:01:58,430 --> 00:02:01,120
36
00:02:01,120 --> 00:02:02,390
Now the idea is this.
37
00:02:02,390 --> 00:02:06,200
In general what we investigate
in a calculus course is the
38
00:02:06,200 --> 00:02:09,660
concept known as 'delta y'.
39
00:02:09,660 --> 00:02:12,220
'Delta y' geometrically
is what?
40
00:02:12,220 --> 00:02:15,650
It's how much 'y' has changed
along the curve
41
00:02:15,650 --> 00:02:17,690
with respect to 'x'.
42
00:02:17,690 --> 00:02:21,260
It turns out that there is a
simpler thing that we could
43
00:02:21,260 --> 00:02:22,670
have computed.
44
00:02:22,670 --> 00:02:26,960
Notice, if we look at this
particular diagram, that since
45
00:02:26,960 --> 00:02:30,400
the tangent line never changes
its slope-- and by the way,
46
00:02:30,400 --> 00:02:33,980
when I say tangent line I
mean at the point 'x1'--
47
00:02:33,980 --> 00:02:39,830
that once we leave the point
'x1' the tangent line, in a
48
00:02:39,830 --> 00:02:42,610
way, no longer resembles
the curve.
49
00:02:42,610 --> 00:02:44,960
But the point that is important
is that I can
50
00:02:44,960 --> 00:02:47,290
compute the change in
'y' to the tangent
51
00:02:47,290 --> 00:02:49,180
line here very easily.
52
00:02:49,180 --> 00:02:52,520
And the reason for this, you
see, is rather apparent.
53
00:02:52,520 --> 00:02:55,900
Namely, the slope of the tangent
line is, on the one
54
00:02:55,900 --> 00:03:00,650
hand, 'delta y-tan' divided
by 'delta x'.
55
00:03:00,650 --> 00:03:05,210
'Delta y-tan' divided
by 'delta x'.
56
00:03:05,210 --> 00:03:09,150
On the other hand, by definition
of slope, the slope
57
00:03:09,150 --> 00:03:12,710
of the line 'L' is also
equal to what?
58
00:03:12,710 --> 00:03:13,960
It's 'dy dx'.
59
00:03:13,960 --> 00:03:18,960
60
00:03:18,960 --> 00:03:23,750
It's 'dy dx' evaluated at the
point, or the value, 'x'
61
00:03:23,750 --> 00:03:25,850
equals 'x1'.
62
00:03:25,850 --> 00:03:29,030
You see, in general the slope of
the curve varies from point
63
00:03:29,030 --> 00:03:32,220
to point, so when we talk about
the tangent line we must
64
00:03:32,220 --> 00:03:35,110
emphasize at what point
on the curve we've
65
00:03:35,110 --> 00:03:36,385
drawn the tangent line.
66
00:03:36,385 --> 00:03:40,130
At any rate, from this
particular diagram it is not
67
00:03:40,130 --> 00:03:44,150
difficult to see that to compute
the change in 'y' to
68
00:03:44,150 --> 00:03:48,860
the tangent line, that this
is nothing more than what?
69
00:03:48,860 --> 00:03:54,910
'dy dx' evaluated at 'x' equals
'x1' times 'delta x'.
70
00:03:54,910 --> 00:03:57,690
And you see this is not
an approximation.
71
00:03:57,690 --> 00:04:03,280
This is precisely the value
of 'delta y-tan'.
72
00:04:03,280 --> 00:04:11,530
The approximation seems to be
when we say let 'delta y-tan'
73
00:04:11,530 --> 00:04:13,240
represent 'delta y'.
74
00:04:13,240 --> 00:04:16,529
In other words, we get the
intuitive feeling that as
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00:04:16,529 --> 00:04:20,709
'delta x' gets small, the
difference between the true
76
00:04:20,709 --> 00:04:24,610
'delta y' and 'delta y-tan'
also gets small.
77
00:04:24,610 --> 00:04:26,760
Another way of saying
this is what?
78
00:04:26,760 --> 00:04:30,440
Our intuitive feeling is that,
in a neighborhood of the point
79
00:04:30,440 --> 00:04:35,050
of tangency, the tangent line
serves as a good approximation
80
00:04:35,050 --> 00:04:36,640
to the curve itself.
81
00:04:36,640 --> 00:04:39,360
Now let's see what this
means in terms
82
00:04:39,360 --> 00:04:41,920
of a specific example.
83
00:04:41,920 --> 00:04:44,740
I've taken the liberty
of computing the
84
00:04:44,740 --> 00:04:48,510
cube of 4.01 in advance.
85
00:04:48,510 --> 00:04:53,990
It turns out to be 64.481201.
86
00:04:53,990 --> 00:04:57,680
And it's sort of arbitrary,
like cubing this.
87
00:04:57,680 --> 00:05:00,330
If this if this doesn't look
messy enough for you we could
88
00:05:00,330 --> 00:05:02,680
have taken this to the sixth
power and then we could have
89
00:05:02,680 --> 00:05:03,310
squared this.
90
00:05:03,310 --> 00:05:04,860
But that part is irrelevant.
91
00:05:04,860 --> 00:05:07,990
A simple check shows that, more
or less, this will be a
92
00:05:07,990 --> 00:05:09,100
correct statement.
93
00:05:09,100 --> 00:05:12,280
And what I would like to do, you
see, is simply illustrate
94
00:05:12,280 --> 00:05:15,940
what our earlier comments
mean in terms of
95
00:05:15,940 --> 00:05:17,540
this specific example.
96
00:05:17,540 --> 00:05:21,840
Let's suppose I want to find
an approximation for 4.01
97
00:05:21,840 --> 00:05:23,800
fairly rapidly.
98
00:05:23,800 --> 00:05:25,460
The idea is this.
99
00:05:25,460 --> 00:05:31,240
What I do know is one number
that's very easy to cube,
100
00:05:31,240 --> 00:05:34,420
which is near 4.01,
is 4 itself.
101
00:05:34,420 --> 00:05:38,820
In other words, I know
that 4 cubed is 64.
102
00:05:38,820 --> 00:05:41,540
And by the way, I've
deliberately drawn this
103
00:05:41,540 --> 00:05:44,670
slightly distorted according
to scale so that we can see
104
00:05:44,670 --> 00:05:46,390
what's happening over here.
105
00:05:46,390 --> 00:05:55,060
What 64.481201 represents is
the actual change in height
106
00:05:55,060 --> 00:05:58,880
from here to here
along the curve.
107
00:05:58,880 --> 00:06:02,550
In other words, it would be
the length of the segment
108
00:06:02,550 --> 00:06:06,310
joining the point 'P' to
the point 'Q' here.
109
00:06:06,310 --> 00:06:11,520
What I claim is that if I
instead tried to find the
110
00:06:11,520 --> 00:06:16,190
length of 'PR', the change in
'y' not along the curve but
111
00:06:16,190 --> 00:06:21,030
along the line tangent to the
curve at the point 4.64, this
112
00:06:21,030 --> 00:06:23,300
is what I can find
fairly rapidly.
113
00:06:23,300 --> 00:06:28,350
In other words, what I am going
to do is to work this
114
00:06:28,350 --> 00:06:33,060
same idea here with
a special case.
115
00:06:33,060 --> 00:06:41,040
You see, I'm going to take 'x1'
to equal 4 and 'delta x'
116
00:06:41,040 --> 00:06:43,410
to be 0.01.
117
00:06:43,410 --> 00:06:48,260
Now you see the curve is
'y' equals 'x cubed'.
118
00:06:48,260 --> 00:06:51,935
From this I can compute 'dy
dx' rather quickly.
119
00:06:51,935 --> 00:06:54,860
120
00:06:54,860 --> 00:06:58,150
Now, I don't want 'dy dx' at
any old point, I want to
121
00:06:58,150 --> 00:07:00,830
compute it when 'x' is
4 so I can find the
122
00:07:00,830 --> 00:07:02,080
slope of the line 'L'.
123
00:07:02,080 --> 00:07:07,890
124
00:07:07,890 --> 00:07:11,060
And when 'x' is 4 this, of
course, simply is what?
125
00:07:11,060 --> 00:07:15,420
4 squared is 16,
times 3 is 48.
126
00:07:15,420 --> 00:07:17,280
So what do I have?
127
00:07:17,280 --> 00:07:21,790
I have that the slope is 48.
128
00:07:21,790 --> 00:07:24,860
I also have that 'delta
x' is 0.01.
129
00:07:24,860 --> 00:07:30,370
So according to my recipe,
'delta y-tan' is what?
130
00:07:30,370 --> 00:07:35,870
It's 'dy dx' evaluated at 'x'
equals 4, which is 48, times
131
00:07:35,870 --> 00:07:38,870
'delta x', which is 0.01.
132
00:07:38,870 --> 00:07:42,960
And that's 0.48.
133
00:07:42,960 --> 00:07:45,850
See again, let's just
juxtaposition these two.
134
00:07:45,850 --> 00:07:50,300
All I have done now is computed
this recipe in the
135
00:07:50,300 --> 00:07:56,150
particular example of trying
to find the cube of 4.01.
136
00:07:56,150 --> 00:08:01,530
And you see, now notice that
this point 0.48 is exactly the
137
00:08:01,530 --> 00:08:04,680
length of the accented
line here.
138
00:08:04,680 --> 00:08:06,670
It's the length of 'PR'.
139
00:08:06,670 --> 00:08:10,420
And what we do know is that the
height from the x-axis to
140
00:08:10,420 --> 00:08:13,610
'R' is now exactly what?
141
00:08:13,610 --> 00:08:20,210
Well, it's the 64
plus the 0.48.
142
00:08:20,210 --> 00:08:22,470
This then is our
approximation.
143
00:08:22,470 --> 00:08:24,960
And notice that this
compares with what?
144
00:08:24,960 --> 00:08:32,270
The precise answer, which
is 64.481201.
145
00:08:32,270 --> 00:08:35,159
In other words, notice what a
small error we happen to have
146
00:08:35,159 --> 00:08:37,039
in this particular case.
147
00:08:37,039 --> 00:08:39,280
And this is the way the subject
is usually brought up.
148
00:08:39,280 --> 00:08:43,559
It is not a very important thing
from my point of view.
149
00:08:43,559 --> 00:08:46,480
In other words, I think it's
rather easy to see that, first
150
00:08:46,480 --> 00:08:50,680
of all, this approximation is
rather nebulous in the sense
151
00:08:50,680 --> 00:08:55,160
that it requires a knowledge of
how fast the tangent line
152
00:08:55,160 --> 00:08:56,650
is separating from the curve.
153
00:08:56,650 --> 00:08:59,530
And this is a rather difficult
topic in its own right.
154
00:08:59,530 --> 00:09:03,040
And secondly, this was a
rather simple example.
155
00:09:03,040 --> 00:09:06,280
And we had the luxury here, you
see, of being able to find
156
00:09:06,280 --> 00:09:09,520
the exact answer so we could
compare our approximation with
157
00:09:09,520 --> 00:09:11,070
the exact answer.
158
00:09:11,070 --> 00:09:14,600
In many cases it is difficult
or impossible to find the
159
00:09:14,600 --> 00:09:16,000
exact answer.
160
00:09:16,000 --> 00:09:19,810
To emphasize this more
abstractly and more generally,
161
00:09:19,810 --> 00:09:22,080
let's consider the following.
162
00:09:22,080 --> 00:09:27,060
Instead of trying to find
the cube of 4.01, the
163
00:09:27,060 --> 00:09:28,980
generalization here is what?
164
00:09:28,980 --> 00:09:31,980
That we could take the curve
'y' equals 'x cubed'.
165
00:09:31,980 --> 00:09:34,990
The derivative of 'y' with
respect to 'x' would then be
166
00:09:34,990 --> 00:09:36,450
'3 x squared'.
167
00:09:36,450 --> 00:09:40,130
Evaluated at an arbitrary point
'x' equals 'x1', we
168
00:09:40,130 --> 00:09:42,450
would get '3 x sub 1 squared'.
169
00:09:42,450 --> 00:09:47,820
In which case 'delta y-tan'
would be '3 x1 squared'
170
00:09:47,820 --> 00:09:49,625
times 'delta x'.
171
00:09:49,625 --> 00:09:52,260
Could we have computed
the exact value of
172
00:09:52,260 --> 00:09:54,560
'delta y' had we wished?
173
00:09:54,560 --> 00:09:56,290
And the answer, of
course, is yes.
174
00:09:56,290 --> 00:09:59,690
Namely, what is the exact
value of 'delta y'?
175
00:09:59,690 --> 00:10:04,510
Well, we want to compute
this between 'x' equals
176
00:10:04,510 --> 00:10:06,290
'x1' plus 'delta x'.
177
00:10:06,290 --> 00:10:11,840
Well, what is the value of 'y'
when 'x' is 'x1 plus delta x'?
178
00:10:11,840 --> 00:10:14,270
It's ''x1 plus delta
x' cubed'.
179
00:10:14,270 --> 00:10:17,520
Then we subtract
off 'x1 cubed'.
180
00:10:17,520 --> 00:10:19,850
And if we expand this,
watch what happens.
181
00:10:19,850 --> 00:10:23,460
By the binomial theorem we get
an 'x1 cubed' term here, which
182
00:10:23,460 --> 00:10:26,370
cancels with the 'x1 cubed'
term over here.
183
00:10:26,370 --> 00:10:27,830
Then we get what?
184
00:10:27,830 --> 00:10:32,660
A '3 x1 squared delta
x', so using the
185
00:10:32,660 --> 00:10:34,040
binomial theorem here.
186
00:10:34,040 --> 00:10:35,360
And then what else do we get?
187
00:10:35,360 --> 00:10:46,940
We get plus '3x1 delta x
squared' plus 'delta x cubed'.
188
00:10:46,940 --> 00:10:51,140
And you see, what I'd like to
have us view over here is if
189
00:10:51,140 --> 00:10:59,270
we look at just this much of the
answer, this part here is
190
00:10:59,270 --> 00:11:03,780
precisely what we sought to
be 'delta y-tan' before.
191
00:11:03,780 --> 00:11:08,050
See, this is 'delta y-tan'.
192
00:11:08,050 --> 00:11:12,070
And what's left over is the
difference, of course, between
193
00:11:12,070 --> 00:11:13,990
'delta y-tan' and delta y.
194
00:11:13,990 --> 00:11:17,930
After all, delta y is just
'delta y-tan' plus this
195
00:11:17,930 --> 00:11:18,740
portion here.
196
00:11:18,740 --> 00:11:22,390
And and now if you look at this
particular portion over
197
00:11:22,390 --> 00:11:26,540
here, observe that, as we
expected, what the size of
198
00:11:26,540 --> 00:11:31,570
this thing is depends both on
where we're measuring 'x1',
199
00:11:31,570 --> 00:11:34,910
and also on how big the size
of the interval is,
200
00:11:34,910 --> 00:11:36,490
namely 'delta x'.
201
00:11:36,490 --> 00:11:38,040
And notice something
that we're going to
202
00:11:38,040 --> 00:11:39,220
come back to here.
203
00:11:39,220 --> 00:11:42,820
Notice that the 'delta x' factor
here appears at least
204
00:11:42,820 --> 00:11:44,230
to the second power.
205
00:11:44,230 --> 00:11:47,180
In other words, notice over
here that as 'delta x' get
206
00:11:47,180 --> 00:11:52,450
small, both of these terms
get small very rapidly.
207
00:11:52,450 --> 00:11:57,480
Because, you see, when 'delta x'
is close to 0 the square of
208
00:11:57,480 --> 00:12:01,370
a number close to 0 is
even closer to 0.
209
00:12:01,370 --> 00:12:03,290
But you see the important thing
for now is to notice
210
00:12:03,290 --> 00:12:06,440
that this is the error in
approximating 'delta y' by
211
00:12:06,440 --> 00:12:07,760
'delta y-tan'.
212
00:12:07,760 --> 00:12:12,800
And that this error depends both
on 'xy' and 'delta x'.
213
00:12:12,800 --> 00:12:16,010
And the question is, can we
state that in a little bit
214
00:12:16,010 --> 00:12:17,890
more of a mathematical
language?
215
00:12:17,890 --> 00:12:20,620
The answer, of course,
is that we can.
216
00:12:20,620 --> 00:12:23,800
In particular, what we're
saying is let's take the
217
00:12:23,800 --> 00:12:28,200
difference between 'dy dx' at
'x' equals 'x1', namely the
218
00:12:28,200 --> 00:12:33,170
slope of the tangent line, and
what the average change of 'y'
219
00:12:33,170 --> 00:12:36,400
with respect to 'x' is
along the curve.
220
00:12:36,400 --> 00:12:38,630
Now, this is very typical
in mathematics.
221
00:12:38,630 --> 00:12:41,220
And it's a very nice trick
to learn, and a very
222
00:12:41,220 --> 00:12:42,450
quick trick to learn.
223
00:12:42,450 --> 00:12:46,200
We know that these two things
are not equal and so all we do
224
00:12:46,200 --> 00:12:48,890
is tack on a correction
factor.
225
00:12:48,890 --> 00:12:50,230
We tack on 'k'.
226
00:12:50,230 --> 00:12:51,370
What is 'k'?
227
00:12:51,370 --> 00:12:55,130
'k' is the difference between
these two expressions.
228
00:12:55,130 --> 00:12:58,720
In other words, by definition
I add on the difference of
229
00:12:58,720 --> 00:13:01,920
these two numbers and that
makes this an equality.
230
00:13:01,920 --> 00:13:04,540
Notice, by the way, that
'k' is a variable.
231
00:13:04,540 --> 00:13:08,000
It depends on how big
'delta x' is.
232
00:13:08,000 --> 00:13:12,400
You see, notice that once
'x1' is chosen--
233
00:13:12,400 --> 00:13:14,780
and this is an important
observation.
234
00:13:14,780 --> 00:13:19,350
As messy as this thing looks,
it's a fixed number for a
235
00:13:19,350 --> 00:13:21,040
particular value of 'x1'.
236
00:13:21,040 --> 00:13:23,570
Namely, we compute the
derivative and evaluate it
237
00:13:23,570 --> 00:13:24,990
when 'x' equals 'x1'.
238
00:13:24,990 --> 00:13:26,110
This is a number.
239
00:13:26,110 --> 00:13:28,080
This, of course, clearly
depends on the
240
00:13:28,080 --> 00:13:29,600
size of 'delta x'.
241
00:13:29,600 --> 00:13:32,090
'Delta y' depends
on 'delta x'.
242
00:13:32,090 --> 00:13:34,520
At any rate, here's
what we do.
243
00:13:34,520 --> 00:13:39,330
We recognize that 'dy dx' is
the limit of 'delta y' over
244
00:13:39,330 --> 00:13:42,080
'delta x' as 'delta
x' approaches 0.
245
00:13:42,080 --> 00:13:46,420
So with this as a hint, we take
the limit of both sides
246
00:13:46,420 --> 00:13:49,210
here as 'delta x'
approaches 0.
247
00:13:49,210 --> 00:13:51,180
Notice the structure again.
248
00:13:51,180 --> 00:13:53,630
In our last series
of lectures we
249
00:13:53,630 --> 00:13:55,110
talked about limit theorems.
250
00:13:55,110 --> 00:13:57,490
We talked about the limit of
a sum being the sum of the
251
00:13:57,490 --> 00:13:59,100
limits, and things
of this type.
252
00:13:59,100 --> 00:14:02,140
And now, you see, we're going
to use these results.
253
00:14:02,140 --> 00:14:05,360
What we do is we take the limit
of both sides here as
254
00:14:05,360 --> 00:14:06,830
'delta x' approaches 0.
255
00:14:06,830 --> 00:14:14,360
256
00:14:14,360 --> 00:14:16,220
The point being what?
257
00:14:16,220 --> 00:14:22,440
That the limit of a sum is
the sum of the limits.
258
00:14:22,440 --> 00:14:26,060
So when I take the limit of this
entire side I can express
259
00:14:26,060 --> 00:14:31,880
that as the sum of the limits.
260
00:14:31,880 --> 00:14:35,000
It is the limit of this term as
'delta x' approaches 0 plus
261
00:14:35,000 --> 00:14:38,380
the limit of this term as
'delta x' approaches 0.
262
00:14:38,380 --> 00:14:41,950
Now let's stop to think what
these things mean.
263
00:14:41,950 --> 00:14:45,180
The limit of 'delta x'
approaching 0 of 'delta y'
264
00:14:45,180 --> 00:14:49,070
divided by 'delta x' is
precisely the definition of
265
00:14:49,070 --> 00:14:50,050
derivative.
266
00:14:50,050 --> 00:14:54,820
In other words, this term here
is just the 'dy dx'.
267
00:14:54,820 --> 00:14:57,150
And keep in mind we've
evaluated this
268
00:14:57,150 --> 00:14:58,610
at 'x' equals 'x1'.
269
00:14:58,610 --> 00:15:01,290
That's the point at which
we're starting.
270
00:15:01,290 --> 00:15:07,480
Now the fact that this is a
constant, and we've already
271
00:15:07,480 --> 00:15:10,600
learned that the limit of
a constant as 'delta x'
272
00:15:10,600 --> 00:15:17,290
approaches 0 is that constant,
this term here is also 'dy dx'
273
00:15:17,290 --> 00:15:20,940
evaluated at 'x' equals 'x1'.
274
00:15:20,940 --> 00:15:24,600
And of course this term here
is just the limit of 'k' as
275
00:15:24,600 --> 00:15:26,810
'delta x' approaches 0.
276
00:15:26,810 --> 00:15:32,510
And now you see if we look at
this, since all we have to do
277
00:15:32,510 --> 00:15:34,770
is observe that this term
appears on both sides of the
278
00:15:34,770 --> 00:15:37,750
equation, canceling, or
subtracting equals from
279
00:15:37,750 --> 00:15:43,570
equals, we wind up with the
result that the limit of 'k'
280
00:15:43,570 --> 00:15:46,810
as 'delta x' approaches
0 is 0.
281
00:15:46,810 --> 00:15:50,300
282
00:15:50,300 --> 00:15:53,810
And we're going to talk more
about that pictorially and
283
00:15:53,810 --> 00:15:55,100
analytically in a
little while.
284
00:15:55,100 --> 00:16:01,760
But from this result, all I'm
going to do now is go back to
285
00:16:01,760 --> 00:16:04,280
our first statement here.
286
00:16:04,280 --> 00:16:08,320
Remembering that 'delta x' is
a nonzero number, I will
287
00:16:08,320 --> 00:16:14,210
simply multiply both sides of
this equation by 'delta x'.
288
00:16:14,210 --> 00:16:17,550
And if I do that, what
do I wind up with?
289
00:16:17,550 --> 00:16:25,840
I wind up with that 'delta y'
is 'dy dx' evaluated at 'x'
290
00:16:25,840 --> 00:16:34,940
equals 'x1' times 'delta
x' plus 'k delta x'.
291
00:16:34,940 --> 00:16:40,280
Now, if I keep in mind here that
if I take the slope at a
292
00:16:40,280 --> 00:16:44,550
particular point and multiply
that by the change in 'x',
293
00:16:44,550 --> 00:16:49,330
that that's precisely what we
define to be 'delta y-tan'.
294
00:16:49,330 --> 00:16:52,240
If I put this all together
I get what?
295
00:16:52,240 --> 00:16:57,050
That 'delta y' is equal to
''delta y-tan' plus 'k delta
296
00:16:57,050 --> 00:17:02,380
x'', where the limit of 'k' as
'delta x' approaches 0 is 0.
297
00:17:02,380 --> 00:17:06,020
Now you see, when we started
this program and started to
298
00:17:06,020 --> 00:17:09,470
talk at the beginning of our
lecture about approximations,
299
00:17:09,470 --> 00:17:12,349
we did something that
was quite crude.
300
00:17:12,349 --> 00:17:14,119
The crude part was
simply this.
301
00:17:14,119 --> 00:17:18,440
What we had said was, why don't
we compute 'delta y' by
302
00:17:18,440 --> 00:17:21,640
computing 'delta
y-tan' instead.
303
00:17:21,640 --> 00:17:24,900
In other words, this is a very
easy thing to compute.
304
00:17:24,900 --> 00:17:28,270
It's just a derivative
multiplied by a change in 'x'.
305
00:17:28,270 --> 00:17:31,430
This is a very simple
thing to compute.
306
00:17:31,430 --> 00:17:34,190
And we'll use that as an
approximation for the thing
307
00:17:34,190 --> 00:17:36,650
that we really want,
which is 'delta y'.
308
00:17:36,650 --> 00:17:39,120
The question that we
deliberately overlooked at
309
00:17:39,120 --> 00:17:43,930
this point was how big was the
error when you do this?
310
00:17:43,930 --> 00:17:46,470
And what we discovered was
something very interesting
311
00:17:46,470 --> 00:17:50,060
and, by the way, may be
something which teaches us why
312
00:17:50,060 --> 00:17:53,440
the analysis is better than the
picture even though the
313
00:17:53,440 --> 00:17:55,430
picture is easier
to visualize.
314
00:17:55,430 --> 00:18:00,820
You see, we knew intuitively
that as you picked a smaller
315
00:18:00,820 --> 00:18:02,850
neighborhood at the
point of tangency.
316
00:18:02,850 --> 00:18:04,920
the tangent line became
a better and better
317
00:18:04,920 --> 00:18:07,710
approximation to the
curve itself.
318
00:18:07,710 --> 00:18:11,640
This boxed-in result tells
us much more than that.
319
00:18:11,640 --> 00:18:16,360
What this tells us is, look, if
this term by itself became
320
00:18:16,360 --> 00:18:19,720
small as 'delta x' approached
0, this would still say the
321
00:18:19,720 --> 00:18:20,780
same thing.
322
00:18:20,780 --> 00:18:22,760
This says much more than that.
323
00:18:22,760 --> 00:18:24,620
You see, what this says is--
324
00:18:24,620 --> 00:18:27,010
and watch, this is very,
very profound--
325
00:18:27,010 --> 00:18:33,500
as 'delta x' approaches 0, 'k'
is also approaching 0.
326
00:18:33,500 --> 00:18:38,230
In other words, the term 'k'
times 'delta x' seems to be
327
00:18:38,230 --> 00:18:42,820
approaching 0 much more rapidly
than 'delta x' itself.
328
00:18:42,820 --> 00:18:47,060
That, by the way, in as simple
a way as I know how, defines
329
00:18:47,060 --> 00:18:49,190
what an infinitesimal is.
330
00:18:49,190 --> 00:18:54,450
See, 'k' times 'delta x' is
called an infinitesimal
331
00:18:54,450 --> 00:18:58,790
because it goes to 0 faster
than 'delta x' itself.
332
00:18:58,790 --> 00:19:01,650
In other words, anything that
approaches 0 faster than
333
00:19:01,650 --> 00:19:05,640
'delta x' itself approaches 0
is called an infinitesimal
334
00:19:05,640 --> 00:19:08,070
with respect to 'delta x'.
335
00:19:08,070 --> 00:19:11,270
Now I'm not going to go into a
long philosophical discussion
336
00:19:11,270 --> 00:19:12,140
about that.
337
00:19:12,140 --> 00:19:15,682
Rather, I'm going to let the
actions speak louder than the
338
00:19:15,682 --> 00:19:19,560
words, and try to show why this
is such a crucial idea.
339
00:19:19,560 --> 00:19:21,220
In other words, again let
me emphasize that.
340
00:19:21,220 --> 00:19:25,820
What's crucial here is not so
much that the error goes to 0,
341
00:19:25,820 --> 00:19:29,850
it's that the error goes
to 0 much faster than
342
00:19:29,850 --> 00:19:31,490
'delta x' goes to 0.
343
00:19:31,490 --> 00:19:32,620
See, what is the error?
344
00:19:32,620 --> 00:19:34,790
The error is 'k' times
'delta x'.
345
00:19:34,790 --> 00:19:37,810
That's the difference between
these two things.
346
00:19:37,810 --> 00:19:41,080
Again, if you want to see what
this thing means pictorially,
347
00:19:41,080 --> 00:19:44,370
and I wish I knew a better way
of doing this but I don't,
348
00:19:44,370 --> 00:19:49,540
notice that 'k' times 'delta
x' was defined to be the
349
00:19:49,540 --> 00:19:53,220
difference between 'delta
y' and 'delta y-tan'.
350
00:19:53,220 --> 00:19:58,610
In other words, since this
length here is 'delta y-tan',
351
00:19:58,610 --> 00:20:01,600
and since this entire length
would be called 'delta y',
352
00:20:01,600 --> 00:20:06,890
again the accented line, this
length here, is 'k'
353
00:20:06,890 --> 00:20:09,160
times 'delta x'.
354
00:20:09,160 --> 00:20:13,490
And what does it mean to say
that 'k' approaches 0 as
355
00:20:13,490 --> 00:20:15,020
'delta x' approaches 0?
356
00:20:15,020 --> 00:20:16,640
What it means is this.
357
00:20:16,640 --> 00:20:22,400
It means that not only does this
vertical difference get
358
00:20:22,400 --> 00:20:29,490
small as 'delta x' goes to 0,
but more importantly, it means
359
00:20:29,490 --> 00:20:34,350
that this vertical distance
gets small very rapidly
360
00:20:34,350 --> 00:20:36,610
compared to how 'delta
x' get small.
361
00:20:36,610 --> 00:20:37,840
Now how can I prove that?
362
00:20:37,840 --> 00:20:40,550
Well I'm not even going
to try to prove that.
363
00:20:40,550 --> 00:20:42,690
What I'm going to just try to do
is to have you see from the
364
00:20:42,690 --> 00:20:43,870
picture what's happening here.
365
00:20:43,870 --> 00:20:47,080
See, here is a fairly
large 'delta x'.
366
00:20:47,080 --> 00:20:50,030
And the difference between
'delta y' and 'delta y-tan'
367
00:20:50,030 --> 00:20:53,670
for that large 'x' is this
length over here.
368
00:20:53,670 --> 00:20:56,150
Now suppose we take a
smaller 'delta x'.
369
00:20:56,150 --> 00:20:58,530
In other words, let's
take 'delta x' to be
370
00:20:58,530 --> 00:21:00,420
this length over here.
371
00:21:00,420 --> 00:21:03,920
Notice that the length of 'delta
x' here is still quite
372
00:21:03,920 --> 00:21:04,790
significant.
373
00:21:04,790 --> 00:21:07,110
I think if you look at this
you can see it's a fairly
374
00:21:07,110 --> 00:21:08,770
significant length.
375
00:21:08,770 --> 00:21:12,510
On the other hand, notice what
the difference between 'delta
376
00:21:12,510 --> 00:21:14,780
y' and 'delta y-tan' is now.
377
00:21:14,780 --> 00:21:17,755
It's just this little tiny
thing over here.
378
00:21:17,755 --> 00:21:22,160
In other words, notice that
as 'delta x' gets small,
379
00:21:22,160 --> 00:21:25,840
ratio-wise as 'delta x' get
small, the vertical difference
380
00:21:25,840 --> 00:21:29,600
between the tangent line and the
curve is getting small at
381
00:21:29,600 --> 00:21:31,070
a much faster rate.
382
00:21:31,070 --> 00:21:34,270
Now of course the question is
why is that so important?
383
00:21:34,270 --> 00:21:36,740
And the answer will always
come back to our
384
00:21:36,740 --> 00:21:40,440
old friend of 0/0.
385
00:21:40,440 --> 00:21:46,070
In general, if you tell a person
that the numerator of a
386
00:21:46,070 --> 00:21:51,200
fraction is small, he jumps to
the conclusion that the size
387
00:21:51,200 --> 00:21:53,380
of the fraction must be small.
388
00:21:53,380 --> 00:21:56,980
But you see, the trouble is when
the denominator is also
389
00:21:56,980 --> 00:22:00,970
small then we cannot
conclude that the
390
00:22:00,970 --> 00:22:03,120
fraction itself is small.
391
00:22:03,120 --> 00:22:06,190
And therefore in trying to
cancel out a term as being
392
00:22:06,190 --> 00:22:08,650
insignificant, when we're
dealing with something like
393
00:22:08,650 --> 00:22:13,010
this, it is no longer enough
to say but the numerator is
394
00:22:13,010 --> 00:22:14,150
going to 0.
395
00:22:14,150 --> 00:22:17,900
Let me jump ahead to the topic
that will be covered in our
396
00:22:17,900 --> 00:22:22,650
next lecture just to use this
as an insight as to what we
397
00:22:22,650 --> 00:22:27,060
have to see and what's
going on over here.
398
00:22:27,060 --> 00:22:29,910
Let me just show you what
I mean by this.
399
00:22:29,910 --> 00:22:34,220
let's go back to our recipe,
which says 'delta y' is this
400
00:22:34,220 --> 00:22:37,310
thing over here where what?
401
00:22:37,310 --> 00:22:40,560
This is important to put in
here, the limit of 'k' as
402
00:22:40,560 --> 00:22:44,410
'delta x' approaches 0 is 0.
403
00:22:44,410 --> 00:22:46,440
Now all I'm going to do
something like this.
404
00:22:46,440 --> 00:22:50,540
Let's suppose we're dealing in
a situation where 'y' and 'x'
405
00:22:50,540 --> 00:22:54,320
happen to be functions, say of,
't' as well, some third
406
00:22:54,320 --> 00:22:55,460
variable 't'.
407
00:22:55,460 --> 00:22:59,240
And what we really want
to find is 'dy dt'.
408
00:22:59,240 --> 00:23:02,380
You see, what we'd be tempted
to do is simply say what?
409
00:23:02,380 --> 00:23:06,870
Let's divide through
by 'delta t'.
410
00:23:06,870 --> 00:23:09,340
We do this, we divide through
by 'delta t'.
411
00:23:09,340 --> 00:23:12,130
Then we take the limit of
this thing as 'delta
412
00:23:12,130 --> 00:23:13,800
t' approaches 0.
413
00:23:13,800 --> 00:23:14,950
Now what happens?
414
00:23:14,950 --> 00:23:18,280
As 'delta t' approaches
0, this is a sum.
415
00:23:18,280 --> 00:23:20,840
The limit of a sum is the
sum of the limits.
416
00:23:20,840 --> 00:23:23,590
Each of the terms in the
sum is a product.
417
00:23:23,590 --> 00:23:26,590
And the limit of a product is
the product of the limits.
418
00:23:26,590 --> 00:23:30,010
So working out the regular limit
theorems, this is what?
419
00:23:30,010 --> 00:23:33,230
The limit, 'delta t' approaches
0, 'dy dx'
420
00:23:33,230 --> 00:23:36,830
evaluated at 'x' equals 'x1',
times the limit of 'delta x'
421
00:23:36,830 --> 00:23:40,250
divided by 'delta t' as 'delta
t' approaches 0, plus the
422
00:23:40,250 --> 00:23:43,730
limit of 'k' as 'delta t'
approaches 0, times the limit
423
00:23:43,730 --> 00:23:48,060
of 'delta x' divided by 'delta
t' as 'delta t' approaches 0.
424
00:23:48,060 --> 00:23:52,670
Now, if we remember what this
thing here means, this is just
425
00:23:52,670 --> 00:23:56,110
by definition 'dy dt'.
426
00:23:56,110 --> 00:23:58,940
For the sake of brevity I will
leave out subscripts now, but
427
00:23:58,940 --> 00:24:01,700
we'll just talk about this.
428
00:24:01,700 --> 00:24:06,170
This is a constant, and the
limit of constant as 'delta t'
429
00:24:06,170 --> 00:24:08,440
approaches 0 is just that
constant, which
430
00:24:08,440 --> 00:24:10,520
I'll call 'dy dx'.
431
00:24:10,520 --> 00:24:13,700
It's understood this is
evaluated at 'x' equals 'x1'.
432
00:24:13,700 --> 00:24:19,120
By definition the limit of
'delta x' divided by 'delta t'
433
00:24:19,120 --> 00:24:25,020
as 'delta t' approaches 0,
that's called 'dx dt'.
434
00:24:25,020 --> 00:24:27,020
Now we come to this
term over here.
435
00:24:27,020 --> 00:24:28,340
And here's the key point.
436
00:24:28,340 --> 00:24:31,460
Many people say, well, as
'delta t' goes to 0
437
00:24:31,460 --> 00:24:32,860
so does 'delta x'.
438
00:24:32,860 --> 00:24:36,150
That makes the numerator here
0 and that makes the whole
439
00:24:36,150 --> 00:24:37,220
fraction 0.
440
00:24:37,220 --> 00:24:39,080
But the point is that's
not true.
441
00:24:39,080 --> 00:24:42,440
What is true here is that both
the numerator and denominator
442
00:24:42,440 --> 00:24:43,760
are going to 0.
443
00:24:43,760 --> 00:24:47,680
This does not make this 0, but
rather it makes it what?
444
00:24:47,680 --> 00:24:48,530
'dx dt'.
445
00:24:48,530 --> 00:24:50,870
That's the definition
of 'dx dt'.
446
00:24:50,870 --> 00:24:52,870
The important point is what?
447
00:24:52,870 --> 00:24:56,630
That as 'delta t' approaches
0 so does 'delta x'.
448
00:24:56,630 --> 00:24:59,230
And what property
does 'k' have?
449
00:24:59,230 --> 00:25:03,110
'k' has the property that the
limit of 'k' as 'delta x'
450
00:25:03,110 --> 00:25:08,140
approaches 0 is 0 itself.
451
00:25:08,140 --> 00:25:11,900
In other words, the topics that
we'll be talking about in
452
00:25:11,900 --> 00:25:15,460
our next lecture concerns
this recipe.
453
00:25:15,460 --> 00:25:19,260
And in the development of this
recipe, the reason that the
454
00:25:19,260 --> 00:25:23,170
error term drops out is not
because the numerator of this
455
00:25:23,170 --> 00:25:28,310
term was small, it was because
the number multiplying limit
456
00:25:28,310 --> 00:25:30,610
happened to be 0.
457
00:25:30,610 --> 00:25:34,240
Now, we will talk about that
in more detail next time.
458
00:25:34,240 --> 00:25:38,690
But the message I want you to
see for right now is how we
459
00:25:38,690 --> 00:25:43,630
get hung up on this 0/0 bit,
and why we must always be
460
00:25:43,630 --> 00:25:46,700
careful in how we handle
something like this.
461
00:25:46,700 --> 00:25:49,460
Now, by the way, there is
something rather interesting
462
00:25:49,460 --> 00:25:50,820
that does happen over here.
463
00:25:50,820 --> 00:25:54,210
If you look at this thing, you
get the feeling that it's
464
00:25:54,210 --> 00:25:58,020
almost as if you could cancel a
common factor from both the
465
00:25:58,020 --> 00:26:00,240
numerator and the denominator.
466
00:26:00,240 --> 00:26:02,390
We have to be very,
very careful.
467
00:26:02,390 --> 00:26:04,560
And I would like to introduce
the language which you will be
468
00:26:04,560 --> 00:26:07,700
reading in the text in this
assignment, called
469
00:26:07,700 --> 00:26:09,250
'Differentials Now'.
470
00:26:09,250 --> 00:26:12,370
And what that means
is simply this.
471
00:26:12,370 --> 00:26:18,120
Notice that 'dy dx' is one
symbol it is not 'dy' divided
472
00:26:18,120 --> 00:26:23,090
by 'dx' because these things
have not been defined
473
00:26:23,090 --> 00:26:23,420
separately.
474
00:26:23,420 --> 00:26:27,400
In other words, you can't just
operate with symbols the way
475
00:26:27,400 --> 00:26:28,600
you might like to.
476
00:26:28,600 --> 00:26:31,800
Here's an interesting little
aside that you may find
477
00:26:31,800 --> 00:26:35,570
entertaining if it has no other
value at all: the idea
478
00:26:35,570 --> 00:26:38,870
of being told that you can
cancel common factors from
479
00:26:38,870 --> 00:26:40,710
both numerator and
denominator.
480
00:26:40,710 --> 00:26:43,660
The uninitiated may say,
you know, here's a 6
481
00:26:43,660 --> 00:26:44,700
and here's a 6.
482
00:26:44,700 --> 00:26:46,890
I'll cancel them out.
483
00:26:46,890 --> 00:26:49,070
Now this is not what
cancellation really meant,
484
00:26:49,070 --> 00:26:51,810
even though you had the same
number both upstairs and
485
00:26:51,810 --> 00:26:53,310
downstairs.
486
00:26:53,310 --> 00:26:54,925
This is not what we meant
by cancellation.
487
00:26:54,925 --> 00:26:59,730
Yet, notice that you happen to,
by accident, get the right
488
00:26:59,730 --> 00:27:01,650
answer in this particular
case.
489
00:27:01,650 --> 00:27:03,180
You see, if you do cancel
this, this does
490
00:27:03,180 --> 00:27:06,120
happen to be 1/4.
491
00:27:06,120 --> 00:27:09,650
This lecture has been going on
in kind of a dull way for you.
492
00:27:09,650 --> 00:27:13,120
Just in case you'd like some
comic relief, there are a few
493
00:27:13,120 --> 00:27:15,860
other examples that
work the same way.
494
00:27:15,860 --> 00:27:18,860
And you can amaze your friends
at the next party with them.
495
00:27:18,860 --> 00:27:23,980
I mean 19/95 happens
to be 1/5.
496
00:27:23,980 --> 00:27:30,410
26/65 happens to be 2/5.
497
00:27:30,410 --> 00:27:31,970
And I think there's one more.
498
00:27:31,970 --> 00:27:36,570
49/98 happens to be 4/8.
499
00:27:36,570 --> 00:27:39,880
But these, I believe, are
the only four fractions
500
00:27:39,880 --> 00:27:41,340
that this works for.
501
00:27:41,340 --> 00:27:43,490
In other words, the mere fact
that it looks like something
502
00:27:43,490 --> 00:27:46,700
is going to work is no guarantee
that you can do
503
00:27:46,700 --> 00:27:48,980
these things without running
into trouble.
504
00:27:48,980 --> 00:27:52,380
On the other hand, it's fair
to assume that the men who
505
00:27:52,380 --> 00:27:57,310
invented differential calculus
must've been clever enough not
506
00:27:57,310 --> 00:27:59,650
to have invented a symbolism
that would have
507
00:27:59,650 --> 00:28:01,370
gotten us into trouble.
508
00:28:01,370 --> 00:28:03,450
That, by the way, is a big
assumption to make, that they
509
00:28:03,450 --> 00:28:04,620
must have been clever enough.
510
00:28:04,620 --> 00:28:08,010
We will find, in places during
our course, that this was not
511
00:28:08,010 --> 00:28:09,400
always the case.
512
00:28:09,400 --> 00:28:10,870
The point is this though.
513
00:28:10,870 --> 00:28:16,950
That when you write 'dy dx' in
this way, if it were not going
514
00:28:16,950 --> 00:28:21,010
to be somehow identifiable with
a quotient such as 'dy'
515
00:28:21,010 --> 00:28:24,430
divided by 'dx', the chances
are we would never have
516
00:28:24,430 --> 00:28:27,100
invented this notation
in the first place.
517
00:28:27,100 --> 00:28:28,580
And so what I'd like
to just point out
518
00:28:28,580 --> 00:28:30,860
briefly now is the following.
519
00:28:30,860 --> 00:28:34,780
Can we, in fact we should,
how shall we--
520
00:28:34,780 --> 00:28:35,940
let's put it this way--
521
00:28:35,940 --> 00:28:42,160
how shall we define separate
symbols dy and dx so that when
522
00:28:42,160 --> 00:28:46,030
we write down 'dy dx' it will
make no difference whether you
523
00:28:46,030 --> 00:28:48,770
think of this as being the
derivative or whether you
524
00:28:48,770 --> 00:28:52,230
think of it as being the
quotient of two numbers?
525
00:28:52,230 --> 00:28:54,730
And by the way, you see the
answer to this question will
526
00:28:54,730 --> 00:28:57,520
not be that difficult if we
stop to think for just a
527
00:28:57,520 --> 00:28:59,610
moment over here.
528
00:28:59,610 --> 00:29:01,650
We've already answered this
question except that we
529
00:29:01,650 --> 00:29:03,950
haven't concentrated
on the fact that
530
00:29:03,950 --> 00:29:05,220
we solved the problem.
531
00:29:05,220 --> 00:29:07,350
See, let's go back to
this line 'L' again.
532
00:29:07,350 --> 00:29:10,190
What is the slope
of the line 'L'?
533
00:29:10,190 --> 00:29:13,130
On the one hand we've looked
at slope as the
534
00:29:13,130 --> 00:29:15,580
derivative 'dy dx'.
535
00:29:15,580 --> 00:29:18,560
On the other hand, just by
looking at this little
536
00:29:18,560 --> 00:29:23,140
accentuated triangle here, the
slope of the line 'L' is also
537
00:29:23,140 --> 00:29:30,710
given by 'delta y-tan'
divided by 'delta x'.
538
00:29:30,710 --> 00:29:37,690
And now one of the best ways
to define 'dy' and 'dx'
539
00:29:37,690 --> 00:29:41,690
separately, it would seem
to me, is what?
540
00:29:41,690 --> 00:29:45,880
Notice that this number divided
by this number yields
541
00:29:45,880 --> 00:29:47,800
the symbol 'dy dx'.
542
00:29:47,800 --> 00:29:56,460
Therefore why not define this
number to be 'dy' and define
543
00:29:56,460 --> 00:29:59,660
this number to be 'dx'?
544
00:29:59,660 --> 00:30:02,950
In other words the symbol 'dx'
will just be a fancy
545
00:30:02,950 --> 00:30:04,510
name for 'delta x'.
546
00:30:04,510 --> 00:30:09,110
On the other hand, the symbol
'dy' will not be a fancy name
547
00:30:09,110 --> 00:30:10,020
for 'delta y'.
548
00:30:10,020 --> 00:30:12,200
In fact, let's emphasize that.
549
00:30:12,200 --> 00:30:17,720
It's not equal to 'delta y', it
is equal to 'delta y-tan'.
550
00:30:17,720 --> 00:30:22,480
In other words, if I allow
'dy' to stand for 'delta
551
00:30:22,480 --> 00:30:27,740
y-tan' and 'dx' to stand for
'delta x', I can now treat 'dy
552
00:30:27,740 --> 00:30:30,260
dx' as if it were what?
553
00:30:30,260 --> 00:30:33,850
'dy' divided by 'dx'.
554
00:30:33,850 --> 00:30:39,160
Well, that's easy to show in
terms of an example just
555
00:30:39,160 --> 00:30:40,670
mechanically.
556
00:30:40,670 --> 00:30:43,240
Remember, given 'y' equals
'x cubed' we've
557
00:30:43,240 --> 00:30:44,730
already found what?
558
00:30:44,730 --> 00:30:47,550
That 'dy dx' is '3 x squared'.
559
00:30:47,550 --> 00:30:51,320
This thing practically begs to
be treated like a fraction.
560
00:30:51,320 --> 00:30:54,350
We would like to be able to say,
hey, if this divided by
561
00:30:54,350 --> 00:30:59,700
this is this, why isn't this
equal to this times this?
562
00:30:59,700 --> 00:31:04,420
In other words, why isn't 'dy'
equal to '3 x squared' dx'?
563
00:31:04,420 --> 00:31:08,290
And the answer is that since
we've identified or defined
564
00:31:08,290 --> 00:31:13,130
'dy' to be 'delta y-tan', and
since we've defined 'dx' to be
565
00:31:13,130 --> 00:31:17,230
'delta x', and since we already
know that this recipe
566
00:31:17,230 --> 00:31:20,530
is correct, it means, in
particular, that we can now
567
00:31:20,530 --> 00:31:24,590
write things like this without
having to worry about whether
568
00:31:24,590 --> 00:31:26,150
it's proper or not.
569
00:31:26,150 --> 00:31:29,710
Now in later lectures this
is going to play a
570
00:31:29,710 --> 00:31:31,320
very important role.
571
00:31:31,320 --> 00:31:34,890
This is what is known as the
language of differentials.
572
00:31:34,890 --> 00:31:39,390
And differentials are the
backbone off both differential
573
00:31:39,390 --> 00:31:41,390
and integral calculus.
574
00:31:41,390 --> 00:31:45,500
But the thing that I want you
to really get into our minds
575
00:31:45,500 --> 00:31:50,100
today is the basic
overall recipe.
576
00:31:50,100 --> 00:31:54,270
And I've taken the liberty of
boxing this off over here.
577
00:31:54,270 --> 00:31:56,530
I want you to practice
with approximations.
578
00:31:56,530 --> 00:32:00,403
I want you to think carefully
about how you can get quick
579
00:32:00,403 --> 00:32:01,110
approximations.
580
00:32:01,110 --> 00:32:03,580
I don't want you to come away
with the feeling that the
581
00:32:03,580 --> 00:32:05,680
approximation is what
was important.
582
00:32:05,680 --> 00:32:07,660
What was important was what?
583
00:32:07,660 --> 00:32:11,910
That the change in 'y', the true
'delta y', is ''dy dx'
584
00:32:11,910 --> 00:32:16,760
times 'delta x'' plus 'k delta
x', where the limit of 'k' as
585
00:32:16,760 --> 00:32:21,140
'delta x' approaches 0 is 0.
586
00:32:21,140 --> 00:32:24,070
And by the way, again, most
books write it this way.
587
00:32:24,070 --> 00:32:28,450
I prefer that we emphasize
that the derivative is
588
00:32:28,450 --> 00:32:32,680
evaluated or taken at a
particular value of 'x'.
589
00:32:32,680 --> 00:32:36,840
And finally what I'd like to
point out is that again, and
590
00:32:36,840 --> 00:32:39,400
we've done this many, many
times, whereas the language
591
00:32:39,400 --> 00:32:44,240
we're used to in terms of
intuition is geometry, that
592
00:32:44,240 --> 00:32:48,310
these results make perfectly
good sense without reference
593
00:32:48,310 --> 00:32:50,190
to any diagram at all.
594
00:32:50,190 --> 00:32:54,280
In other words, then, the same
result stated in analytic
595
00:32:54,280 --> 00:32:58,960
language simply says this: if
'f' is a function of 'x' and
596
00:32:58,960 --> 00:33:02,940
is differentiable when 'x'
equals 'x1', then 'f of 'x1
597
00:33:02,940 --> 00:33:05,830
plus delta x'' minus
'f of x1'--
598
00:33:05,830 --> 00:33:08,450
you see that's geometrically
what corresponds
599
00:33:08,450 --> 00:33:10,970
to your 'delta y'.
600
00:33:10,970 --> 00:33:15,860
That's ''f prime of x1' times
'delta x'' plus 'k delta x'
601
00:33:15,860 --> 00:33:21,510
where the limit of 'k' as 'delta
x' approaches 0 is 0.
602
00:33:21,510 --> 00:33:27,840
Now these two recipes summarize
precisely what it is
603
00:33:27,840 --> 00:33:30,930
that we are interested in when
we deal with the subject
604
00:33:30,930 --> 00:33:32,780
called infinitesimals.
605
00:33:32,780 --> 00:33:35,600
As I said before, and I can't
emphasize this thing strongly
606
00:33:35,600 --> 00:33:39,540
enough, when it comes time to
make approximations we will
607
00:33:39,540 --> 00:33:43,120
find better ways of getting
approximations than by the
608
00:33:43,120 --> 00:33:46,570
method known as differentials
and 'delta y-tan'.
609
00:33:46,570 --> 00:33:49,630
What is crucial is that you
use the language of
610
00:33:49,630 --> 00:33:53,780
approximations enough so as you
can see pictorially what's
611
00:33:53,780 --> 00:33:58,190
going on and then cement down
the final recipe, which I've
612
00:33:58,190 --> 00:33:59,260
boxed in here.
613
00:33:59,260 --> 00:34:03,080
This will be the building block,
as we shall see you
614
00:34:03,080 --> 00:34:06,480
next time, in the lecture which
develops the derivative
615
00:34:06,480 --> 00:34:08,389
of composite functions.
616
00:34:08,389 --> 00:34:10,860
But more about that next time.
617
00:34:10,860 --> 00:34:12,260
And until next time, goodbye.
618
00:34:12,260 --> 00:34:15,250
619
00:34:15,250 --> 00:34:18,449
Funding for the publication of
this video was provided by the
620
00:34:18,449 --> 00:34:22,500
Gabriella and Paul Rosenbaum
Foundation.
621
00:34:22,500 --> 00:34:26,679
Help OCW continue to provide
free and open access to MIT
622
00:34:26,679 --> 00:34:30,870
courses by making a donation
at ocw.mit.edu/donate.
623
00:34:30,870 --> 00:34:35,626