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HERBERT GROSS: Hi, our lecture
today is entitled
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differentiation of inverse
functions.
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00:00:38,770 --> 00:00:40,800
And it pulls together
two previous
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00:00:40,800 --> 00:00:42,720
topics that we've discussed.
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00:00:42,720 --> 00:00:46,510
Namely, inverse functions
themselves, and secondly, the
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chain rule that we've
discussed just
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00:00:48,400 --> 00:00:49,800
a short time ago.
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00:00:49,800 --> 00:00:52,720
And perhaps the best way to
introduce the power of
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00:00:52,720 --> 00:00:56,250
differentiation of inverse
functions is to start out with
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00:00:56,250 --> 00:00:57,450
such a problem.
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00:00:57,450 --> 00:01:00,720
Let's actually try to
differentiate a particular
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00:01:00,720 --> 00:01:03,870
function, which at least up
until now, we have not been
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00:01:03,870 --> 00:01:05,430
able to differentiate.
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The function happens to be y
equals the cube root of 'x'.
24
00:01:09,440 --> 00:01:12,200
In other words, 'y' equals
'x to the 1/3'.
25
00:01:12,200 --> 00:01:16,210
Let's find 'dy dx' if 'y'
equals 'x to the 1/3'.
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00:01:16,210 --> 00:01:19,380
Now, the whole idea of inverse
functions is what?
27
00:01:19,380 --> 00:01:22,240
That it gives us a chance
to paraphrase.
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00:01:22,240 --> 00:01:25,310
That we can interchange the role
of the dependent and the
29
00:01:25,310 --> 00:01:26,810
independent variables.
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00:01:26,810 --> 00:01:29,740
And like any other form of
paraphrase, even though two
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00:01:29,740 --> 00:01:34,250
things may be synonymous,
psychologically one of the two
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00:01:34,250 --> 00:01:37,920
may be easier for us to
visualize than the other.
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00:01:37,920 --> 00:01:40,970
In particular, in this
particular case, if 'y' equals
34
00:01:40,970 --> 00:01:44,130
'x to the 1/3', another way
of writing the same
35
00:01:44,130 --> 00:01:46,280
thing is to say what?
36
00:01:46,280 --> 00:01:50,780
'x' equals 'y cubed'.
37
00:01:50,780 --> 00:01:54,590
And now, given that 'x' equals
'y cubed', in other words,
38
00:01:54,590 --> 00:01:58,280
with treating 'x' as a function
of 'y', we certainly
39
00:01:58,280 --> 00:02:01,430
know how to differentiate 'y
cubed' with respect to 'y'.
40
00:02:01,430 --> 00:02:04,470
Namely, we know that for
a positive exponent to
41
00:02:04,470 --> 00:02:07,900
differentiate, all we have to do
is bring the exponent down
42
00:02:07,900 --> 00:02:09,780
and replace it by one less.
43
00:02:09,780 --> 00:02:13,000
In other words, right away what
we can say is that 'dx
44
00:02:13,000 --> 00:02:17,280
dy' is '3y squared'.
45
00:02:17,280 --> 00:02:21,290
Now if we use the result of last
time that we talked about
46
00:02:21,290 --> 00:02:23,590
when we were discussing the
chain rule, and I'll review
47
00:02:23,590 --> 00:02:25,540
that result in just
a few moments.
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00:02:25,540 --> 00:02:29,150
But for the time being, let's
assume that we have the result
49
00:02:29,150 --> 00:02:34,190
that 'dx dy' is the reciprocal
of 'dy dx'.
50
00:02:34,190 --> 00:02:41,950
In other words, if 'dy dx' is
'1 over 'dx dy'', that tells
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00:02:41,950 --> 00:02:47,450
us that 'dy dx' is '1
over 3y squared'.
52
00:02:47,450 --> 00:02:50,770
And I guess to write that in a
more convenient form, that's
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00:02:50,770 --> 00:02:53,670
'1/3 y to the minus
2' if you're used
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00:02:53,670 --> 00:02:55,400
to exponential notation.
55
00:02:55,400 --> 00:03:00,450
If we now recall from above that
'y' is equal to 'x to the
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00:03:00,450 --> 00:03:07,500
1/3', this can now be written
as ''1/3 x to the 1/3'
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00:03:07,500 --> 00:03:09,360
to the minus 2'.
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00:03:09,360 --> 00:03:15,740
And we now arrive at an answer
'dy dx' is '1/3 x
59
00:03:15,740 --> 00:03:17,440
to the minus 2/3'.
60
00:03:17,440 --> 00:03:20,020
By the way, let me point out
there certainly was nothing
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00:03:20,020 --> 00:03:23,320
wrong with leaving our answer
in this particular form.
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00:03:23,320 --> 00:03:26,980
It's just conventional that
wherever possible, if the
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00:03:26,980 --> 00:03:30,040
original problem was given as
a function of 'x', we would
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00:03:30,040 --> 00:03:33,130
like our answer to also
be a function of 'x'.
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00:03:33,130 --> 00:03:35,110
I mean, here for example, we
could have said that the
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00:03:35,110 --> 00:03:39,020
answer is '1/3 y to the
minus 2', where 'y'
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00:03:39,020 --> 00:03:40,760
equals 'x to the 1/3'.
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00:03:40,760 --> 00:03:45,270
All we've done here is to fill
this thing in explicitly.
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00:03:45,270 --> 00:03:47,370
By the way, there may have
been a tendency--
70
00:03:47,370 --> 00:03:49,620
and here's an example of
circular reasoning.
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00:03:49,620 --> 00:03:52,110
There may have been a tendency
to say, didn't we already
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00:03:52,110 --> 00:03:55,500
learn that to differentiate a
power, we just bring the power
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00:03:55,500 --> 00:03:58,240
down and replace
it by one less?
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00:03:58,240 --> 00:04:01,490
In other words, if we did that
bringing down the 1/3 would
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00:04:01,490 --> 00:04:04,000
give us a factor of
1/3 in front.
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00:04:04,000 --> 00:04:09,030
Replacing 1/3 by one less,
1/3 minus 1 is minus 2/3.
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00:04:09,030 --> 00:04:14,100
We would then see that 'dy
dx' should be '1/3 x
78
00:04:14,100 --> 00:04:15,640
to the minus 2/3'.
79
00:04:15,640 --> 00:04:18,640
Which is exactly what
we got this way.
80
00:04:18,640 --> 00:04:20,550
The point that we should mention
at this particular
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00:04:20,550 --> 00:04:24,920
stage is that the derivative of
'x' to the 'n' being 'nx to
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00:04:24,920 --> 00:04:29,550
the 'n - 1'' was proven only
for the case that 'n' is an
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00:04:29,550 --> 00:04:31,690
integer, either positive
or negative.
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00:04:31,690 --> 00:04:35,020
One uses the binomial theorem
to prove the result for a
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00:04:35,020 --> 00:04:36,170
positive integer.
86
00:04:36,170 --> 00:04:39,810
One uses the quotient rule
to prove the result for a
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00:04:39,810 --> 00:04:40,930
negative integer.
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00:04:40,930 --> 00:04:43,700
And now, even though I didn't
do this thing in general, I
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00:04:43,700 --> 00:04:45,710
think you can see how this
will generalize.
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00:04:45,710 --> 00:04:49,470
For fractional exponents,
one uses the
91
00:04:49,470 --> 00:04:51,080
inverse function idea.
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00:04:51,080 --> 00:04:55,760
Namely, we did have to use the
fact that if 'y' is equal to
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00:04:55,760 --> 00:04:59,750
'x to the 1/3', 'x' equals
'y cubed' was
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00:04:59,750 --> 00:05:01,910
an equivalent equation.
95
00:05:01,910 --> 00:05:03,910
And we could differentiate
that.
96
00:05:03,910 --> 00:05:08,910
Now, what is the hang up here,
in so far as how certain are
97
00:05:08,910 --> 00:05:13,190
we that 'dy dx' and 'dx
dy' are reciprocals?
98
00:05:13,190 --> 00:05:16,710
You may recall that when we used
the chain rule, we showed
99
00:05:16,710 --> 00:05:19,650
that if 'y' is a differentiable
function of
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00:05:19,650 --> 00:05:22,830
'x', and if 'x' is a
differentiable function of
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00:05:22,830 --> 00:05:24,720
'u', then 'y' is a
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00:05:24,720 --> 00:05:26,520
differentiable function of 'u'.
103
00:05:26,520 --> 00:05:31,580
And in particular, 'dy du' is
''dy dx' times 'dx du''.
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00:05:31,580 --> 00:05:33,750
And if we now take the
particular case where the
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00:05:33,750 --> 00:05:36,000
first variable equals
the third, way 'u'
106
00:05:36,000 --> 00:05:38,000
equals 'y', we get what?
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00:05:38,000 --> 00:05:49,340
'dy dy', which is 1, equals
''dy dx' times 'dx dy''.
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00:05:49,340 --> 00:05:52,140
And at first glance, it might
seem that we've proven
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00:05:52,140 --> 00:05:56,290
rigorously now the result that
'dx dy' and 'dy dx' are
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00:05:56,290 --> 00:05:58,300
reciprocals of each other.
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00:05:58,300 --> 00:06:00,140
Product is 1.
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00:06:00,140 --> 00:06:02,990
The one logical hang up
that we have right
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00:06:02,990 --> 00:06:04,680
now is simply this.
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00:06:04,680 --> 00:06:09,230
In the statement of the chain
rule, it did not say that 'y'
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00:06:09,230 --> 00:06:11,440
had to be a function of
'x' and 'x' had to be
116
00:06:11,440 --> 00:06:12,880
a function of 'u'.
117
00:06:12,880 --> 00:06:14,040
It said 'y' had to be a
118
00:06:14,040 --> 00:06:15,480
differentiable function of 'x'.
119
00:06:15,480 --> 00:06:18,230
That was the first variable
had to be a differentiable
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00:06:18,230 --> 00:06:19,460
function of the second.
121
00:06:19,460 --> 00:06:21,400
And the second had to
be a differentiable
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00:06:21,400 --> 00:06:22,770
function of the third.
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00:06:22,770 --> 00:06:25,980
In other words, coming down to
here, if we know that 'y' is a
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00:06:25,980 --> 00:06:29,470
differentiable function of 'x'
and 'y' has an inverse
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00:06:29,470 --> 00:06:34,160
function, and if we also knew
that the inverse function was
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00:06:34,160 --> 00:06:35,220
differentiable.
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00:06:35,220 --> 00:06:37,740
See, in other words, this must
be a differentiable function
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00:06:37,740 --> 00:06:39,290
of this and this must be a
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00:06:39,290 --> 00:06:40,760
differentiable function of this.
130
00:06:40,760 --> 00:06:43,915
In other words, the one point
that was missing was is that
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00:06:43,915 --> 00:06:49,490
if we knew that if a function
is differentiable, then its
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00:06:49,490 --> 00:06:52,460
inverse if it exists, is
also differentiable.
133
00:06:52,460 --> 00:06:56,420
The chain rule would have given
us a rigorous proof.
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00:06:56,420 --> 00:06:59,090
The point that we're missing
though is we do not as yet
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00:06:59,090 --> 00:07:04,680
know that the inverse of a
differentiable function is
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00:07:04,680 --> 00:07:06,760
also a differentiable
function.
137
00:07:06,760 --> 00:07:10,460
Now if you recall on our earlier
lecture on inverse
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00:07:10,460 --> 00:07:14,060
functions, we pointed out that
there was a rather interesting
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00:07:14,060 --> 00:07:17,660
graphical interpretation between
'y' equals 'f of x'
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00:07:17,660 --> 00:07:19,600
and 'y' equals 'f
inverse of x'.
141
00:07:19,600 --> 00:07:22,290
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00:07:22,290 --> 00:07:25,310
By the way whenever I say, you
may recall, that's just my
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00:07:25,310 --> 00:07:27,680
polite way of saying perhaps
you don't, but you'd better
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00:07:27,680 --> 00:07:29,970
look it up because we had it.
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00:07:29,970 --> 00:07:35,370
So recall that the two curves
are symmetric with respect to
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00:07:35,370 --> 00:07:37,660
the line 'y' equals 'x'.
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00:07:37,660 --> 00:07:40,700
By the way again, just
a brief aside here.
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00:07:40,700 --> 00:07:42,840
Notice that either one of these
functions could have
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00:07:42,840 --> 00:07:45,470
been called 'f of x' and the
other one could have been
150
00:07:45,470 --> 00:07:47,280
called 'f inverse'.
151
00:07:47,280 --> 00:07:51,920
In other words, just another
piece of brief knowledge here
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00:07:51,920 --> 00:07:54,870
that the inverse of the inverse
is the original
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00:07:54,870 --> 00:07:56,090
function again.
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00:07:56,090 --> 00:07:58,350
In other words, thinking in
terms of our function machine,
155
00:07:58,350 --> 00:08:01,650
if you interchanged the input
and the output, and then of
156
00:08:01,650 --> 00:08:05,390
the resulting machine again,
interchange the input and the
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00:08:05,390 --> 00:08:08,660
output, you're back to the
original machine again.
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00:08:08,660 --> 00:08:11,650
So in this particular diagram, I
certainly could have labeled
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00:08:11,650 --> 00:08:14,730
this curve 'y' equals 'f of x'
or 'g of x', then this curve
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00:08:14,730 --> 00:08:18,020
here would have been 'y' equals
'g inverse of x'.
161
00:08:18,020 --> 00:08:19,610
But the important
point is what?
162
00:08:19,610 --> 00:08:22,890
That 'y' equals 'f of x' and
'y' equals 'f inverse of x'
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00:08:22,890 --> 00:08:26,050
are symmetric with respect
to the line 'y equals x'.
164
00:08:26,050 --> 00:08:30,450
Now you see we have a
particularly simple geometric
165
00:08:30,450 --> 00:08:34,830
argument as to why an inverse
function should be
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00:08:34,830 --> 00:08:38,039
differentiable if the original
function is differentiable.
167
00:08:38,039 --> 00:08:42,120
Namely, pictorially, what does
it mean to say that a function
168
00:08:42,120 --> 00:08:43,380
is differentiable?
169
00:08:43,380 --> 00:08:45,450
It means that when you
plot its graph,
170
00:08:45,450 --> 00:08:47,820
the graph is smooth.
171
00:08:47,820 --> 00:08:50,940
In other words, if 'f' is a
differentiable function, the
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00:08:50,940 --> 00:08:54,120
curve 'y' equals 'f of x'
will be a smooth curve.
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00:08:54,120 --> 00:08:56,620
Now simply ask yourself the
following question.
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00:08:56,620 --> 00:09:00,360
If you take a smooth curve and
take its mirror image with
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00:09:00,360 --> 00:09:03,490
respect to the line 'y' equals
'x', or for that matter with
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00:09:03,490 --> 00:09:05,550
respect to any line,
do you expect the
177
00:09:05,550 --> 00:09:08,780
curve to become un-smooth?
178
00:09:08,780 --> 00:09:12,250
You see, in other words, the
mirror image of a smooth curve
179
00:09:12,250 --> 00:09:13,700
will again, be smooth.
180
00:09:13,700 --> 00:09:16,650
And that's perhaps the most
intuitive way of picking off
181
00:09:16,650 --> 00:09:20,780
in your mind why if a function
is differentiable, its inverse
182
00:09:20,780 --> 00:09:23,240
function will also be
differentiable.
183
00:09:23,240 --> 00:09:26,240
Of course, as we've seen many
times already in this course,
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00:09:26,240 --> 00:09:29,800
we must distinguish between
geometric intuition and
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00:09:29,800 --> 00:09:31,270
mathematical analysis.
186
00:09:31,270 --> 00:09:33,880
That on more occasions than
one, what seemed to be
187
00:09:33,880 --> 00:09:37,220
happening geometrically was
complicated by something
188
00:09:37,220 --> 00:09:40,800
unforeseen when we tried to get
the results in terms of
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00:09:40,800 --> 00:09:42,360
analytic methods.
190
00:09:42,360 --> 00:09:44,250
Let me give you an illustration
of this.
191
00:09:44,250 --> 00:09:47,910
You see, what we're really
saying is, granted that a
192
00:09:47,910 --> 00:09:51,270
picture can be a good visual
aid, let's suppose we're given
193
00:09:51,270 --> 00:09:53,295
the curve 'y' equals
'f inverse of x'.
194
00:09:53,295 --> 00:09:56,240
195
00:09:56,240 --> 00:09:58,200
Well, first of all, what is
that the same as saying?
196
00:09:58,200 --> 00:10:00,920
It's the same as saying that
'x' equals 'f of y'.
197
00:10:00,920 --> 00:10:02,740
At any rate, the question
is this.
198
00:10:02,740 --> 00:10:06,780
We're assuming that 'f' is a
differentiable function.
199
00:10:06,780 --> 00:10:09,980
What we would like to do is to
prove that 'f inverse' is also
200
00:10:09,980 --> 00:10:11,580
differentiable.
201
00:10:11,580 --> 00:10:14,100
Now you see the whole thing
again is that whenever you
202
00:10:14,100 --> 00:10:18,570
want to prove anything, what
do you mean if you take 'f
203
00:10:18,570 --> 00:10:20,540
inverse' and differentiate it?
204
00:10:20,540 --> 00:10:24,500
And by the way, that may look
like a funny notation.
205
00:10:24,500 --> 00:10:26,690
Think of 'f inverse' as
being one symbol.
206
00:10:26,690 --> 00:10:27,360
Call it 'g'.
207
00:10:27,360 --> 00:10:28,430
Call it whatever you want.
208
00:10:28,430 --> 00:10:32,060
All we're saying is, how would
we, by definition, find the
209
00:10:32,060 --> 00:10:34,780
derivative of the inverse
function say,
210
00:10:34,780 --> 00:10:37,110
at 'x' equals 'x1'?
211
00:10:37,110 --> 00:10:40,120
And the answer is that by
definition, it's just what?
212
00:10:40,120 --> 00:10:44,700
It's the limit as 'delta
x' approaches 0.
213
00:10:44,700 --> 00:10:46,620
Same definition as before.
214
00:10:46,620 --> 00:10:51,060
''f of 'x1 plus 'delta
x' minus 'f of
215
00:10:51,060 --> 00:10:56,000
x1'' over 'delta x'.
216
00:10:56,000 --> 00:10:58,120
And in fact, if you want to
write that a little bit more
217
00:10:58,120 --> 00:11:01,160
explicitly, what is another
way of writing 'delta x'?
218
00:11:01,160 --> 00:11:06,790
'Delta x', of course, is ''x1
plus 'delta x' minus x1'.
219
00:11:06,790 --> 00:11:08,490
Now the idea is--
220
00:11:08,490 --> 00:11:10,600
this is 'f inverse'.
221
00:11:10,600 --> 00:11:13,580
See we want to find the
derivative of 'f inverse'.
222
00:11:13,580 --> 00:11:14,960
Who cares what the name
of the function is?
223
00:11:14,960 --> 00:11:16,920
Whatever the name of the
function is, what do you do?
224
00:11:16,920 --> 00:11:21,250
You compute the function at 'x1
plus 'delta x', subtract
225
00:11:21,250 --> 00:11:24,830
off its value at 'x1', and
divide by 'delta x'.
226
00:11:24,830 --> 00:11:27,810
So by the same definition that
we had the first time we
227
00:11:27,810 --> 00:11:31,230
defined derivative, this is
the basic definition for
228
00:11:31,230 --> 00:11:33,870
finding the derivative
of 'f inverse'.
229
00:11:33,870 --> 00:11:36,840
Now, how do we use the
fact that we already
230
00:11:36,840 --> 00:11:38,480
know what 'f' is like?
231
00:11:38,480 --> 00:11:41,170
Remember, we mentioned when we
talked about inverse functions
232
00:11:41,170 --> 00:11:43,640
before is at the
time you use--
233
00:11:43,640 --> 00:11:47,120
the way you really effectively
handle inverse functions is
234
00:11:47,120 --> 00:11:50,340
when you know properties of
the original function.
235
00:11:50,340 --> 00:11:53,220
We're not just working blindly
with 'f inverse' here, we're
236
00:11:53,220 --> 00:11:58,040
working with the case that 'f
inverse' is the inverse of the
237
00:11:58,040 --> 00:12:01,350
function 'f', and that we know
that 'f' is differentiable.
238
00:12:01,350 --> 00:12:04,000
And now we want to see if
knowing that 'f' is
239
00:12:04,000 --> 00:12:06,980
differentiable, can we prove
that 'f inverse' is
240
00:12:06,980 --> 00:12:08,340
differentiable?
241
00:12:08,340 --> 00:12:10,610
And you see, the idea is not
really that difficult.
242
00:12:10,610 --> 00:12:13,780
We can work this thing out step
by step from out little
243
00:12:13,780 --> 00:12:15,160
diagram over here.
244
00:12:15,160 --> 00:12:19,180
You see, notice that another
name for 'f inverse of 'x1
245
00:12:19,180 --> 00:12:23,310
plus 'delta x'' is 'y1
plus delta y'.
246
00:12:23,310 --> 00:12:25,130
Let me just work with what's
in the bracketed
247
00:12:25,130 --> 00:12:27,950
expression over here.
248
00:12:27,950 --> 00:12:31,330
See, the numerator is
'y1 plus 'delta y'.
249
00:12:31,330 --> 00:12:32,490
That's this term.
250
00:12:32,490 --> 00:12:34,900
Now, what's 'f inverse of x1'?
251
00:12:34,900 --> 00:12:37,480
As we come up over here,
remember the curve is 'y'
252
00:12:37,480 --> 00:12:38,760
equals 'f inverse'.
253
00:12:38,760 --> 00:12:41,230
So 'f inverse of x1'
is just 'y1'.
254
00:12:41,230 --> 00:12:45,770
255
00:12:45,770 --> 00:12:47,480
Now what's our denominator?
256
00:12:47,480 --> 00:12:52,260
'x1 plus 'delta x' maps into
'y1 plus 'delta y'.
257
00:12:52,260 --> 00:12:54,250
Well, the idea is this.
258
00:12:54,250 --> 00:13:00,810
In terms of inverse functions,
'x1 plus 'delta x' is just the
259
00:13:00,810 --> 00:13:04,150
back map of 'y1 plus
'delta y'.
260
00:13:04,150 --> 00:13:07,600
In other words, since
'f inverse'--
261
00:13:07,600 --> 00:13:08,570
let's write that down.
262
00:13:08,570 --> 00:13:14,360
Since 'f inverse of 'x1 plus
'delta x' is equal to 'y1 plus
263
00:13:14,360 --> 00:13:17,030
'delta y', that's another
way of saying that
264
00:13:17,030 --> 00:13:19,925
'x1 plus 'delta x'--
265
00:13:19,925 --> 00:13:22,160
we might as well write this
because this is what we're
266
00:13:22,160 --> 00:13:23,410
emphasizing.
267
00:13:23,410 --> 00:13:25,940
268
00:13:25,940 --> 00:13:29,340
In other words, this
becomes what?
269
00:13:29,340 --> 00:13:35,680
'x1 plus 'delta x' is just 'f
of 'y1 plus 'delta y''.
270
00:13:35,680 --> 00:13:40,130
And 'x1' is just 'f of y1'.
271
00:13:40,130 --> 00:13:45,680
272
00:13:45,680 --> 00:13:49,220
See again, I used the picture
as a visual aid.
273
00:13:49,220 --> 00:13:52,200
But notice that everything I've
written down here follows
274
00:13:52,200 --> 00:13:55,090
analytically by my basic
definitions.
275
00:13:55,090 --> 00:13:58,770
I don't want to overwhelm you
with formal proofs here.
276
00:13:58,770 --> 00:14:01,210
These are all done
in the text.
277
00:14:01,210 --> 00:14:03,700
And I think that again, for
those of you who are
278
00:14:03,700 --> 00:14:06,437
proof-oriented, the proofs are
done excellently enough so
279
00:14:06,437 --> 00:14:08,170
that you'll get them
from that.
280
00:14:08,170 --> 00:14:11,410
And for those of you who are not
proof-oriented, an extra
281
00:14:11,410 --> 00:14:14,490
few minutes here will not make
that much of a difference.
282
00:14:14,490 --> 00:14:17,570
But what I want you to see over
here is how this thing
283
00:14:17,570 --> 00:14:19,220
starts to set up now.
284
00:14:19,220 --> 00:14:22,230
In other words, notice that this
starts to look like what?
285
00:14:22,230 --> 00:14:24,820
Let me just come over here where
we have some more space.
286
00:14:24,820 --> 00:14:25,380
This is what?
287
00:14:25,380 --> 00:14:28,705
The limit as 'delta
x' approaches 0.
288
00:14:28,705 --> 00:14:32,960
289
00:14:32,960 --> 00:14:36,400
'f 'delta y' over ''f
of 'y1 plus delta
290
00:14:36,400 --> 00:14:39,470
y' minus 'f of y1''.
291
00:14:39,470 --> 00:14:42,390
And see, if you look at this
thing, remember 'f' is a
292
00:14:42,390 --> 00:14:46,060
differentiable function
of 'y'.
293
00:14:46,060 --> 00:14:52,520
If this had been a 'delta y'
approaching 0, this would have
294
00:14:52,520 --> 00:14:54,250
just been what?
295
00:14:54,250 --> 00:14:57,050
If we could assume that as
'delta x' approaches 0, 'delta
296
00:14:57,050 --> 00:15:00,130
y' approaches 0, this just
would have been what?
297
00:15:00,130 --> 00:15:04,440
This is the reciprocal of the
derivative of 'f of y' with
298
00:15:04,440 --> 00:15:05,300
respect to 'y'.
299
00:15:05,300 --> 00:15:09,650
In other words, this would be
what one would call 'dx dy'
300
00:15:09,650 --> 00:15:12,210
evaluated at 'y' equals 'y1'.
301
00:15:12,210 --> 00:15:15,940
302
00:15:15,940 --> 00:15:17,020
This is what?
303
00:15:17,020 --> 00:15:23,410
''f of 'y1 plus delta y' minus
'f of y1'' over 'delta y' as
304
00:15:23,410 --> 00:15:26,580
'delta y' approaches 0
is that derivative.
305
00:15:26,580 --> 00:15:27,390
And we have what?
306
00:15:27,390 --> 00:15:29,550
The reciprocal of this thing.
307
00:15:29,550 --> 00:15:34,000
And in other words, what we will
have proven is that 'dy
308
00:15:34,000 --> 00:15:38,340
dx' evaluated at 'x' equals
'x1' is the same as the
309
00:15:38,340 --> 00:15:44,590
reciprocal of 'dx dy' evaluated
at 'y' equals 'y1'.
310
00:15:44,590 --> 00:15:47,000
The only thing we have to be
sure of in terms of the formal
311
00:15:47,000 --> 00:15:50,880
proof is to make sure that as
'delta x' approaches 0, 'delta
312
00:15:50,880 --> 00:15:52,310
y' approaches 0.
313
00:15:52,310 --> 00:15:55,170
And that is not too difficult
a thing to do.
314
00:15:55,170 --> 00:15:57,400
As I say, the proof is
done in the book.
315
00:15:57,400 --> 00:15:59,990
We could do it here, but I think
that that would take
316
00:15:59,990 --> 00:16:04,550
away from the flavor of what
we're trying to show.
317
00:16:04,550 --> 00:16:08,840
The idea is it's fine to think
in terms of intuitive ideas.
318
00:16:08,840 --> 00:16:12,640
In fact, to level with you as
much as I can, of all of my
319
00:16:12,640 --> 00:16:16,470
mathematician friends who are
outstanding in various aspects
320
00:16:16,470 --> 00:16:20,030
of mathematics, to my knowledge
not one of them
321
00:16:20,030 --> 00:16:23,020
works without some sort of
mental picture as to
322
00:16:23,020 --> 00:16:24,390
what's going on.
323
00:16:24,390 --> 00:16:26,040
In other words, you can take
something that's very, very
324
00:16:26,040 --> 00:16:29,870
abstract and somehow or other,
you associate in your mind
325
00:16:29,870 --> 00:16:32,720
some kind of a visual aid
that gives you a hint as
326
00:16:32,720 --> 00:16:34,100
to what to do next.
327
00:16:34,100 --> 00:16:38,360
But once you know what to do
next, you always formulate the
328
00:16:38,360 --> 00:16:41,700
thing in terms of mathematical
precision.
329
00:16:41,700 --> 00:16:44,290
In other words, another way of
looking at this-- let's give
330
00:16:44,290 --> 00:16:45,500
this a broad title.
331
00:16:45,500 --> 00:16:49,950
Let's call this 'Proof
versus Intuition'.
332
00:16:49,950 --> 00:16:52,770
And this is a topic that comes
up very, very early in
333
00:16:52,770 --> 00:16:53,790
mathematics.
334
00:16:53,790 --> 00:16:57,350
Perhaps the first place that
it's extremely noticeable is
335
00:16:57,350 --> 00:16:59,690
in the subject called
plane geometry.
336
00:16:59,690 --> 00:17:01,330
Let me give you a
for instance.
337
00:17:01,330 --> 00:17:04,060
Let's take a typical
traditional
338
00:17:04,060 --> 00:17:05,890
plane geometry problem.
339
00:17:05,890 --> 00:17:09,099
We'l take an isosceles
triangle ABC,
340
00:17:09,099 --> 00:17:12,020
with AB equal to AC.
341
00:17:12,020 --> 00:17:15,030
And what we would like to prove
is that the base angles
342
00:17:15,030 --> 00:17:16,650
of this triangle are equal.
343
00:17:16,650 --> 00:17:19,400
We'd like to prove that angle
B equals angle C.
344
00:17:19,400 --> 00:17:22,329
Now you remember how you tackled
this problem in high
345
00:17:22,329 --> 00:17:24,300
school geometry.
346
00:17:24,300 --> 00:17:25,810
You said something like this.
347
00:17:25,810 --> 00:17:29,850
Well, let me draw the
angle bisector here.
348
00:17:29,850 --> 00:17:36,350
That meets BC at D. AD equals
itself by identity.
349
00:17:36,350 --> 00:17:40,350
These two angles are equal by
definition of angle bisector.
350
00:17:40,350 --> 00:17:45,000
Therefore, triangle ABD is
congruent to triangle ACD.
351
00:17:45,000 --> 00:17:48,880
And corresponding pots of
congruent triangles are equal.
352
00:17:48,880 --> 00:17:51,230
And you then proved that the
base angles of an isosceles
353
00:17:51,230 --> 00:17:53,440
triangle were equal.
354
00:17:53,440 --> 00:17:57,260
Now, at this stage of the game
if you were anything like me,
355
00:17:57,260 --> 00:17:58,850
what you would have done
has said, this is
356
00:17:58,850 --> 00:17:59,630
the end of the problem.
357
00:17:59,630 --> 00:18:01,190
Let's go onto the next one.
358
00:18:01,190 --> 00:18:04,910
But if you had passed this in
this particular way, you would
359
00:18:04,910 --> 00:18:06,150
have got a 0.
360
00:18:06,150 --> 00:18:07,360
And why'd you get a 0?
361
00:18:07,360 --> 00:18:10,690
Well, if you remember from plane
geometry, there was a
362
00:18:10,690 --> 00:18:13,230
particular format that
had to be followed.
363
00:18:13,230 --> 00:18:17,970
It was called the statement
reason format.
364
00:18:17,970 --> 00:18:19,930
For every statement that
you wrote down, you
365
00:18:19,930 --> 00:18:21,220
had to give a reason.
366
00:18:21,220 --> 00:18:23,890
And the reason couldn't be
things like because, or why
367
00:18:23,890 --> 00:18:25,910
not, or obvious.
368
00:18:25,910 --> 00:18:27,290
The reasons had to be what?
369
00:18:27,290 --> 00:18:32,610
Either definitions, or rules, or
previously proven theorems.
370
00:18:32,610 --> 00:18:35,080
In other words, notice that
even though we never
371
00:18:35,080 --> 00:18:38,820
emphasized it, back in plane
geometry when you were drawing
372
00:18:38,820 --> 00:18:42,400
this diagram and getting the
result, that was the geometric
373
00:18:42,400 --> 00:18:43,650
intuition part.
374
00:18:43,650 --> 00:18:45,320
In other words, this was
where you showed
375
00:18:45,320 --> 00:18:47,100
the result was plausible.
376
00:18:47,100 --> 00:18:48,300
The logic part--
377
00:18:48,300 --> 00:18:51,690
and this is why geometry is
being stressed particularly in
378
00:18:51,690 --> 00:18:52,880
the modern curriculum.
379
00:18:52,880 --> 00:18:56,060
In terms of logic, notice that
once you had your intuitive
380
00:18:56,060 --> 00:19:00,660
picture, the statement reason
part followed independently of
381
00:19:00,660 --> 00:19:02,110
the picture.
382
00:19:02,110 --> 00:19:06,090
You used the picture to set
yourself up, but the final
383
00:19:06,090 --> 00:19:08,660
proof hinged on what?
384
00:19:08,660 --> 00:19:11,730
Having the result follow
purely from the axioms
385
00:19:11,730 --> 00:19:12,250
themselves.
386
00:19:12,250 --> 00:19:13,530
From the assumptions.
387
00:19:13,530 --> 00:19:16,080
And by the way, this is the
basic difference between
388
00:19:16,080 --> 00:19:18,550
traditional and modern
geometry.
389
00:19:18,550 --> 00:19:22,961
In modern geometry, let's go
back to the same proof.
390
00:19:22,961 --> 00:19:25,370
And it's a rather interesting
point and I think you'll see a
391
00:19:25,370 --> 00:19:28,400
connection between what's
happening in geometry and
392
00:19:28,400 --> 00:19:30,540
what's happening in calculus.
393
00:19:30,540 --> 00:19:32,360
You see, remember how
we proved this.
394
00:19:32,360 --> 00:19:36,120
We said draw the angle bisector
and call this point
395
00:19:36,120 --> 00:19:40,600
D. And then we went through
this and got this result.
396
00:19:40,600 --> 00:19:42,550
In modern geometry
they say look it.
397
00:19:42,550 --> 00:19:44,790
Without looking at the picture,
how do you know that
398
00:19:44,790 --> 00:19:46,750
D falls between B and C?
399
00:19:46,750 --> 00:19:49,190
It's obvious in the picture
that it does.
400
00:19:49,190 --> 00:19:52,280
But if everything has to follow
inescapably from your
401
00:19:52,280 --> 00:19:59,850
rules when you go through the
statement reason part, then
402
00:19:59,850 --> 00:20:02,980
unless you have some rule or
definition that tells you that
403
00:20:02,980 --> 00:20:06,800
D must fall between B and C,
you can't use this result.
404
00:20:06,800 --> 00:20:09,430
In other words, your result
will be plausible from a
405
00:20:09,430 --> 00:20:12,340
picture, but not provable
analytically
406
00:20:12,340 --> 00:20:15,590
So in modern geometry, we
add a few axioms, a few
407
00:20:15,590 --> 00:20:16,820
rules of the game.
408
00:20:16,820 --> 00:20:19,790
They're called the axioms of
betweenness, the axioms of
409
00:20:19,790 --> 00:20:20,740
separation.
410
00:20:20,740 --> 00:20:23,570
How one point separates other
points and what this thing
411
00:20:23,570 --> 00:20:26,710
means analytically, so that we
can continue on this way.
412
00:20:26,710 --> 00:20:29,570
Now you see what I'm driving
at is simply this.
413
00:20:29,570 --> 00:20:32,670
The Utopian way I think of
learning is to first have an
414
00:20:32,670 --> 00:20:35,160
intuitive picture of
what's going on.
415
00:20:35,160 --> 00:20:39,360
Then you proceed gradually to
learn what rigor means.
416
00:20:39,360 --> 00:20:41,780
As I may have said to you
before, in the language of
417
00:20:41,780 --> 00:20:44,880
functions, rigor is a function
of the 'rigoree'.
418
00:20:44,880 --> 00:20:47,250
In other words, if a person is
perfectly willing to accept a
419
00:20:47,250 --> 00:20:50,200
result, and he's not going to
get into any trouble using it,
420
00:20:50,200 --> 00:20:52,380
let him use the intuitive
result.
421
00:20:52,380 --> 00:20:55,010
On the other hand, if you wanted
to teach him later that
422
00:20:55,010 --> 00:20:58,620
there are pitfalls using his
intuition as a background,
423
00:20:58,620 --> 00:21:01,150
then we can come ahead and start
to do things in a little
424
00:21:01,150 --> 00:21:03,280
bit more of a sophisticated
manner.
425
00:21:03,280 --> 00:21:06,110
Now, you see what I'm driving
at I guess is this.
426
00:21:06,110 --> 00:21:09,290
If a youngster survives this
procedure of going from the
427
00:21:09,290 --> 00:21:12,670
intuitive approach to the
rigorous approach to the
428
00:21:12,670 --> 00:21:16,080
logical difference between
analysis and geometry, he'll
429
00:21:16,080 --> 00:21:19,320
be in great shape when
he gets to calculus.
430
00:21:19,320 --> 00:21:21,090
You see, the idea is
exactly the same.
431
00:21:21,090 --> 00:21:23,570
What we're saying in the
calculus is simply this.
432
00:21:23,570 --> 00:21:26,070
Given the derivative of an
inverse function, we do it
433
00:21:26,070 --> 00:21:29,380
first in a way that makes good
geometric sense to us.
434
00:21:29,380 --> 00:21:32,100
Then to make sure that the
results do not depend on our
435
00:21:32,100 --> 00:21:34,860
picture, and that our results
can be generalized to more
436
00:21:34,860 --> 00:21:37,650
variables or to tougher analytic
situations where we
437
00:21:37,650 --> 00:21:40,530
can't draw the picture, then
we tried to pick up the
438
00:21:40,530 --> 00:21:45,190
sophistication that allows us to
remove the picture proceed
439
00:21:45,190 --> 00:21:47,060
purely by analysis.
440
00:21:47,060 --> 00:21:49,590
What I'm telling you as you read
the text is if you can do
441
00:21:49,590 --> 00:21:51,100
both, fine.
442
00:21:51,100 --> 00:21:53,570
Learn the proofs and
the intuition.
443
00:21:53,570 --> 00:21:56,410
If you have trouble with the
proofs, at least satisfy
444
00:21:56,410 --> 00:21:58,220
yourself that they're there.
445
00:21:58,220 --> 00:22:00,960
That they do seem to follow
from the basic axioms and
446
00:22:00,960 --> 00:22:02,050
other assumptions.
447
00:22:02,050 --> 00:22:06,320
But meanwhile, rely heavily
on the intuitive results.
448
00:22:06,320 --> 00:22:08,710
And the important thing being
that you have a picture as to
449
00:22:08,710 --> 00:22:10,060
what's going on.
450
00:22:10,060 --> 00:22:11,930
The second aside that
I'd like to make--
451
00:22:11,930 --> 00:22:14,600
and in fact, I would like to
conclude our lesson for today
452
00:22:14,600 --> 00:22:16,650
with this very important
aside--
453
00:22:16,650 --> 00:22:17,840
is the following.
454
00:22:17,840 --> 00:22:20,530
You remember when we were first
learning limits and we
455
00:22:20,530 --> 00:22:21,540
talked about--
456
00:22:21,540 --> 00:22:23,590
well, let me just make
an aside right at
457
00:22:23,590 --> 00:22:24,400
the beginning here.
458
00:22:24,400 --> 00:22:26,320
Hate to do that, but it
just occurred to me.
459
00:22:26,320 --> 00:22:28,900
Remember when we said, let's
compute the limit of 'f of x'
460
00:22:28,900 --> 00:22:30,390
as 'x' approaches 'a'.
461
00:22:30,390 --> 00:22:33,370
And our first approach was to
say, OK, let's just replace
462
00:22:33,370 --> 00:22:35,690
'x' by 'a'.
463
00:22:35,690 --> 00:22:38,800
And you said, OK, this is fine,
but what happens if you
464
00:22:38,800 --> 00:22:40,890
get a 0/0 form?
465
00:22:40,890 --> 00:22:43,490
And the counterargument to that
was, well, if 'f' and 'a'
466
00:22:43,490 --> 00:22:46,120
are chosen at random, how likely
is it that we're going
467
00:22:46,120 --> 00:22:47,810
to get a 0/0 form?
468
00:22:47,810 --> 00:22:50,600
Answer's well, it's not
too likely at all.
469
00:22:50,600 --> 00:22:52,480
And then the answer to that
was, well, but look it.
470
00:22:52,480 --> 00:22:54,700
Every time you take a
derivative, you're going to
471
00:22:54,700 --> 00:22:58,120
get a 0/0 form.
472
00:22:58,120 --> 00:23:00,300
And so the question
was, in calculus
473
00:23:00,300 --> 00:23:02,370
0/0 was very important.
474
00:23:02,370 --> 00:23:05,090
Now we're going to ask the same
kind of a question about
475
00:23:05,090 --> 00:23:06,390
1:1 functions.
476
00:23:06,390 --> 00:23:08,610
I guess the best way to
state it is bluntly.
477
00:23:08,610 --> 00:23:13,690
How likely is it that a given
function f is 1:1?
478
00:23:13,690 --> 00:23:18,840
For example, if I draw a curve
like this, is this graph 1:1?
479
00:23:18,840 --> 00:23:20,400
The answer, of course, is no.
480
00:23:20,400 --> 00:23:23,990
For example, if I pick the point
'y1' over here and come
481
00:23:23,990 --> 00:23:27,380
across here, I find at least
in this picture, three
482
00:23:27,380 --> 00:23:31,230
different candidates, three
different "x's".
483
00:23:31,230 --> 00:23:35,180
'x1', 'x2', and 'x3'
for which what?
484
00:23:35,180 --> 00:23:43,810
'f of x1' equals 'f of x2',
equals 'f of x3' equals 'y1'.
485
00:23:43,810 --> 00:23:46,910
And at first glance, you might
be tempted to say, oops, we
486
00:23:46,910 --> 00:23:50,070
can't apply any of our theory
to this particular function.
487
00:23:50,070 --> 00:23:52,300
But here's the very
important point.
488
00:23:52,300 --> 00:23:57,560
Very often in calculus we do not
start with 'y1' and look
489
00:23:57,560 --> 00:24:01,250
to see whether we have
'x1', 'x2', or 'x3'.
490
00:24:01,250 --> 00:24:04,560
Very often in calculus we're
starting at something like,
491
00:24:04,560 --> 00:24:06,860
oh, for the sake of
argument, 'x3'.
492
00:24:06,860 --> 00:24:09,640
And we say, hey, I wonder
what's going on in a
493
00:24:09,640 --> 00:24:11,920
neighborhood of 'x3'.
494
00:24:11,920 --> 00:24:14,440
If you want a fancy word to take
care of that, it's what
495
00:24:14,440 --> 00:24:17,900
the mathematician calls the
difference between local and
496
00:24:17,900 --> 00:24:19,270
global properties.
497
00:24:19,270 --> 00:24:22,440
And those words are exactly
what they sound like.
498
00:24:22,440 --> 00:24:26,050
Local means in a neighborhood
of a point and global means
499
00:24:26,050 --> 00:24:30,210
let's look at the curve
in the large.
500
00:24:30,210 --> 00:24:31,250
And the point is this.
501
00:24:31,250 --> 00:24:33,920
That very, very often in
calculus, we are not
502
00:24:33,920 --> 00:24:36,070
interested in what's
happening globally.
503
00:24:36,070 --> 00:24:39,680
For example, when you're driving
in a car and you're
504
00:24:39,680 --> 00:24:42,980
driving along say, the New York
Thruway, and you're near
505
00:24:42,980 --> 00:24:46,670
Albany and somebody says, what's
our gas situation?
506
00:24:46,670 --> 00:24:49,710
Somehow or other, what your
gas situation was when you
507
00:24:49,710 --> 00:24:52,830
were near Buffalo has no bearing
on the problem here.
508
00:24:52,830 --> 00:24:56,002
How full the gas tank is and the
problems involved with a
509
00:24:56,002 --> 00:24:58,500
full gas tank are local
properties.
510
00:24:58,500 --> 00:25:01,640
And whether this gets more
abstract or not is irrelevant.
511
00:25:01,640 --> 00:25:04,830
All we're saying is that in
calculus, very often you're
512
00:25:04,830 --> 00:25:07,830
dealing with a neighborhood
of a point.
513
00:25:07,830 --> 00:25:10,330
And notice this, and we'll
do is intuitively.
514
00:25:10,330 --> 00:25:12,640
The book again, supplies
the rigorous proof.
515
00:25:12,640 --> 00:25:17,635
Notice that if 'f prime
of x3' here is not 0.
516
00:25:17,635 --> 00:25:19,670
Well, for the sake of argument,
in this case, we
517
00:25:19,670 --> 00:25:21,250
notice that the curve
is always rising.
518
00:25:21,250 --> 00:25:24,370
Notice that with a smooth curve,
if it's rising at a
519
00:25:24,370 --> 00:25:27,970
particular point, obviously it's
going to be rising in a
520
00:25:27,970 --> 00:25:31,370
neighborhood of that point.
521
00:25:31,370 --> 00:25:32,820
Just look at the picture here.
522
00:25:32,820 --> 00:25:34,160
And what we're saying is this.
523
00:25:34,160 --> 00:25:36,790
What does it mean for a
function to be 1:1?
524
00:25:36,790 --> 00:25:41,170
For a function to be 1:1 on an
interval, it's sufficient that
525
00:25:41,170 --> 00:25:44,290
either 'f prime' never be
negative, or 'f prime' never
526
00:25:44,290 --> 00:25:46,270
be positive.
527
00:25:46,270 --> 00:25:48,510
In other words, what
it means is this.
528
00:25:48,510 --> 00:25:52,360
That as long as the derivative
is not 0, we can find a
529
00:25:52,360 --> 00:25:55,510
neighborhood, a local
neighborhood, that will make
530
00:25:55,510 --> 00:25:59,400
that function 1:1 on that
particular neighborhood.
531
00:25:59,400 --> 00:26:01,240
In other words, once I'm working
with this particular
532
00:26:01,240 --> 00:26:03,570
neighborhood, and I can't
stress this point enough
533
00:26:03,570 --> 00:26:06,100
because it's going to come
up over and over again.
534
00:26:06,100 --> 00:26:08,780
It's going to come up in more
sophisticated forms when we
535
00:26:08,780 --> 00:26:10,860
deal with functions of
several variables.
536
00:26:10,860 --> 00:26:12,970
But the idea is what?
537
00:26:12,970 --> 00:26:15,510
That in a neighborhood of a
point where the derivative is
538
00:26:15,510 --> 00:26:19,390
not 0, the function may be
viewed as being 1:1.
539
00:26:19,390 --> 00:26:22,890
You see, the tough part is that
if you start with 'y1',
540
00:26:22,890 --> 00:26:26,580
you have no way of knowing just
from that whether you
541
00:26:26,580 --> 00:26:30,100
want a neighborhood near
'x1', or 'x2', or 'x3'.
542
00:26:30,100 --> 00:26:33,150
But if you know what
neighborhood you want, in a
543
00:26:33,150 --> 00:26:35,420
sufficiently small neighborhood,
the function
544
00:26:35,420 --> 00:26:37,400
always behaves like it's 1:1.
545
00:26:37,400 --> 00:26:41,470
In fact, the only problem that
one runs into is the case
546
00:26:41,470 --> 00:26:46,540
where at the point in question,
the derivative is 0.
547
00:26:46,540 --> 00:26:48,580
And I think you can see
pictorially what happens in
548
00:26:48,580 --> 00:26:50,000
that case right away.
549
00:26:50,000 --> 00:26:53,090
As soon as the derivative is 0,
notice that no matter how
550
00:26:53,090 --> 00:26:56,750
small an interval-- well, I
shouldn't say this is a
551
00:26:56,750 --> 00:26:57,790
possibility.
552
00:26:57,790 --> 00:26:58,780
Let me show you what
I mean by that.
553
00:26:58,780 --> 00:27:01,500
Suppose the curve has
a low point here.
554
00:27:01,500 --> 00:27:04,640
What I'm saying is if the
curve does this, then no
555
00:27:04,640 --> 00:27:09,130
matter how small an interval
we choose surrounding 'x1',
556
00:27:09,130 --> 00:27:12,480
the function will not be
1:1 in that interval.
557
00:27:12,480 --> 00:27:14,690
No matter what interval you pick
here, if you look at the
558
00:27:14,690 --> 00:27:16,990
image for every point
in the image, there
559
00:27:16,990 --> 00:27:18,860
are going to be what?
560
00:27:18,860 --> 00:27:22,890
Two back mappings, two points
that come from here.
561
00:27:22,890 --> 00:27:26,010
The reason I say you have
to be careful is this.
562
00:27:26,010 --> 00:27:29,330
You see, you can have a case
where the curve does this.
563
00:27:29,330 --> 00:27:33,310
It comes in, gets tangent
to the x-axis, and then
564
00:27:33,310 --> 00:27:34,650
goes down like this.
565
00:27:34,650 --> 00:27:36,510
You see, at this particular
point, the
566
00:27:36,510 --> 00:27:39,140
derivative at 0 is 0.
567
00:27:39,140 --> 00:27:41,860
Yet, the curve is still never
falling in this area.
568
00:27:41,860 --> 00:27:44,500
This particular curve is 1:1.
569
00:27:44,500 --> 00:27:46,650
I guess what we're saying here
is that when a derivative is
570
00:27:46,650 --> 00:27:48,990
0, be careful because something
571
00:27:48,990 --> 00:27:50,150
like this can happen.
572
00:27:50,150 --> 00:27:57,160
If the derivative is not 0,
then we know that in the
573
00:27:57,160 --> 00:27:59,545
neighborhood of the point in
question, as long as the curve
574
00:27:59,545 --> 00:28:03,400
is smooth, it represents
a 1:1 function.
575
00:28:03,400 --> 00:28:05,210
And this is what we'll be doing
very, very often in
576
00:28:05,210 --> 00:28:08,100
calculus, is using neighborhoods
of points at
577
00:28:08,100 --> 00:28:10,400
which the derivative is not 0.
578
00:28:10,400 --> 00:28:11,960
Now, what this leads
to is this.
579
00:28:11,960 --> 00:28:14,280
When you start talking about
things like a derivative being
580
00:28:14,280 --> 00:28:18,500
0, and intervals and
one-to-oneness, I think you
581
00:28:18,500 --> 00:28:23,510
can see that this suggests a
rather powerful means or need
582
00:28:23,510 --> 00:28:26,760
for doing the geometry
of curve plotting.
583
00:28:26,760 --> 00:28:29,780
This will be the topic of
our next investigation.
584
00:28:29,780 --> 00:28:31,350
And so until next
time, goodbye.
585
00:28:31,350 --> 00:28:34,480
586
00:28:34,480 --> 00:28:37,010
ANNOUNCER: Funding for the
publication of this video was
587
00:28:37,010 --> 00:28:41,730
provided by the Gabriella and
Paul Rosenbaum Foundation.
588
00:28:41,730 --> 00:28:45,900
Help OCW continue to provide
free and open access to MIT
589
00:28:45,900 --> 00:28:50,100
courses by making a donation
at ocw.mit.edu/donate.
590
00:28:50,100 --> 00:28:54,847