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PROFESSOR: OK, we're ready to
start the eleventh lecture.
00:00:25.380 --> 00:00:29.390
We're still in the
middle of sketching.
00:00:29.390 --> 00:00:32.910
And, indeed, one of the
reasons why we did not
00:00:32.910 --> 00:00:37.040
talk about hyperbolic
functions is
00:00:37.040 --> 00:00:39.540
that we're running just
a little bit behind.
00:00:39.540 --> 00:00:41.720
And we'll catch up
a tiny bit today.
00:00:41.720 --> 00:00:46.000
And I hope all the way
on Tuesday of next week.
00:00:46.000 --> 00:00:58.640
So let me pick up where we
left off, with sketching.
00:00:58.640 --> 00:01:05.220
So this is a continuation.
00:01:05.220 --> 00:01:07.020
I want to give you
one more example
00:01:07.020 --> 00:01:08.390
of how to sketch things.
00:01:08.390 --> 00:01:10.380
And then we'll go through
it systematically.
00:01:10.380 --> 00:01:14.830
So the second example that we
did as one example last time,
00:01:14.830 --> 00:01:18.060
is this.
00:01:18.060 --> 00:01:21.970
The function is (x+1)/(x+2).
00:01:21.970 --> 00:01:25.780
And I'm going to save
you the time right now.
00:01:25.780 --> 00:01:28.629
This is very typical
of me, especially
00:01:28.629 --> 00:01:30.170
if you're in a hurry
on an exam, I'll
00:01:30.170 --> 00:01:32.920
just tell you what
the derivative is.
00:01:32.920 --> 00:01:34.490
So in this case,
it's 1 / (x+2)^2.
00:01:37.760 --> 00:01:42.327
Now, the reason why I'm
bringing this example up,
00:01:42.327 --> 00:01:44.660
even though it'll turn out
to be a relatively simple one
00:01:44.660 --> 00:01:49.150
to sketch, is that
it's easy to fall
00:01:49.150 --> 00:01:53.890
into a black hole
with this problem.
00:01:53.890 --> 00:01:57.470
So let me just show you.
00:01:57.470 --> 00:01:59.600
This is not equal to 0.
00:01:59.600 --> 00:02:01.180
It's never equal to 0.
00:02:01.180 --> 00:02:08.940
So that means there
are no critical points.
00:02:08.940 --> 00:02:13.710
At this point,
students, many students
00:02:13.710 --> 00:02:16.250
who have been
trained like monkeys
00:02:16.250 --> 00:02:18.710
to do exactly what
they've been told,
00:02:18.710 --> 00:02:21.170
suddenly freeze and give up.
00:02:21.170 --> 00:02:23.970
Because there's nothing to do.
00:02:23.970 --> 00:02:28.000
So this is the one thing that
I have to train out of you.
00:02:28.000 --> 00:02:32.260
You can't just give
up at this point.
00:02:32.260 --> 00:02:35.060
So what would you suggest?
00:02:35.060 --> 00:02:38.100
Can anybody get us
out of this jam?
00:02:38.100 --> 00:02:38.780
Yeah.
00:02:38.780 --> 00:02:47.410
STUDENT: [INAUDIBLE]
00:02:47.410 --> 00:02:48.150
PROFESSOR: Right.
00:02:48.150 --> 00:02:52.810
So the suggestion was
to find the x-values
00:02:52.810 --> 00:02:55.800
where f(x) is undefined.
00:02:55.800 --> 00:03:00.680
In fact, so now that's a
fairly sophisticated way
00:03:00.680 --> 00:03:03.090
of putting the point
that I want to make,
00:03:03.090 --> 00:03:05.380
which is that what we
want to do is go back
00:03:05.380 --> 00:03:07.770
to our precalculus skills.
00:03:07.770 --> 00:03:11.150
And just plot points.
00:03:11.150 --> 00:03:13.500
So instead, you go
back to precalculus
00:03:13.500 --> 00:03:16.934
and you just plot some points.
00:03:16.934 --> 00:03:18.350
It's a perfectly
reasonable thing.
00:03:18.350 --> 00:03:21.970
Now, it turns out that the
most important point to plot
00:03:21.970 --> 00:03:24.730
is the one that's not there.
00:03:24.730 --> 00:03:29.300
Namely, the value of x = -2.
00:03:29.300 --> 00:03:31.780
Which is just what
was suggested.
00:03:31.780 --> 00:03:38.710
Namely, we plot the points where
the function is not defined.
00:03:38.710 --> 00:03:41.220
So how do we do that?
00:03:41.220 --> 00:03:43.670
Well, you have to think
about it for a second
00:03:43.670 --> 00:03:46.470
and I'll introduce some
new notation when I do it.
00:03:46.470 --> 00:03:49.290
If I evaluate 2 at this
place, actually I can't do it.
00:03:49.290 --> 00:03:51.250
I have to do it from
the left and the right.
00:03:51.250 --> 00:03:57.840
So if I plug in -2 on the
positive side, from the right,
00:03:57.840 --> 00:04:03.500
that's going to be equal
to -2 + 1 divided by -2,
00:04:03.500 --> 00:04:06.590
a little bit more
than -2, plus 2.
00:04:06.590 --> 00:04:11.110
Which is -1 divided by -
now, this denominator is -2,
00:04:11.110 --> 00:04:12.730
a little more than that, plus 2.
00:04:12.730 --> 00:04:20.200
So it's a little more than 0.
00:04:20.200 --> 00:04:24.720
And that is, well we'll
fill that in in a second.
00:04:24.720 --> 00:04:25.620
Everybody's puzzled.
00:04:25.620 --> 00:04:26.120
Yes.
00:04:26.120 --> 00:04:30.670
STUDENT: [INAUDIBLE]
00:04:30.670 --> 00:04:36.105
PROFESSOR: No,
that's the function.
00:04:36.105 --> 00:04:37.980
I'm plotting points,
I'm not differentiating.
00:04:37.980 --> 00:04:38.250
I've already differentiated it.
00:04:38.250 --> 00:04:39.630
I've already got something
that's a little puzzling.
00:04:39.630 --> 00:04:41.250
Now I'm focusing
on the weird spot.
00:04:41.250 --> 00:04:42.166
Yes, another question.
00:04:42.166 --> 00:04:47.652
STUDENT: Wouldn't it be
a little less than 0?
00:04:47.652 --> 00:04:49.610
PROFESSOR: Wouldn't it
be a little less than 0?
00:04:49.610 --> 00:04:52.800
OK, that's a very good point
and this is a matter of notation
00:04:52.800 --> 00:04:53.310
here.
00:04:53.310 --> 00:04:55.690
And a matter of parentheses.
00:04:55.690 --> 00:04:57.430
So wouldn't this be
a little less than 2.
00:04:57.430 --> 00:05:01.900
Well, if the parentheses
were this way; that is, 2+ ,
00:05:01.900 --> 00:05:06.790
with a minus after I did the
2+ , then it would be less.
00:05:06.790 --> 00:05:09.830
But it's this way.
00:05:09.830 --> 00:05:10.670
OK.
00:05:10.670 --> 00:05:14.420
So the notation is,
you have a number
00:05:14.420 --> 00:05:17.190
and you take the
plus part of it.
00:05:17.190 --> 00:05:21.550
That's the part which is a
little bit bigger than it.
00:05:21.550 --> 00:05:26.840
And so this is what I mean.
00:05:26.840 --> 00:05:29.870
And if you like, here I can
put in those parentheses too.
00:05:29.870 --> 00:05:31.270
Yeah, another question.
00:05:31.270 --> 00:05:34.730
STUDENT: [INAUDIBLE]
00:05:34.730 --> 00:05:37.253
PROFESSOR: Why doesn't
the top one have a plus?
00:05:37.253 --> 00:05:39.378
The only reason why the
top one doesn't have a plus
00:05:39.378 --> 00:05:42.860
is that I don't need
it to evaluate this.
00:05:42.860 --> 00:05:45.410
And when I take the limit, I
can just plug in the value.
00:05:45.410 --> 00:05:48.250
Whereas here, I'm
still uncertain.
00:05:48.250 --> 00:05:49.430
Because it's going to be 0.
00:05:49.430 --> 00:05:51.740
And I want to know
which side of 0 it's on.
00:05:51.740 --> 00:05:55.190
Whether it's on the positive
side or the negative side.
00:05:55.190 --> 00:05:58.310
So this one, I could have
written here a parentheses 2+,
00:05:58.310 --> 00:06:01.410
but then it would have
just simplified to -1.
00:06:01.410 --> 00:06:04.410
In the limit.
00:06:04.410 --> 00:06:07.190
So now, I've got a
negative number divided
00:06:07.190 --> 00:06:10.024
by a tiny positive number.
00:06:10.024 --> 00:06:11.940
And so, somebody want
to tell me what that is?
00:06:11.940 --> 00:06:16.750
Negative infinity.
00:06:16.750 --> 00:06:21.460
So, we just evaluated this
function from one side.
00:06:21.460 --> 00:06:24.830
And if you follow
through the other side,
00:06:24.830 --> 00:06:30.700
so this one here, you get
something very similar,
00:06:30.700 --> 00:06:34.120
except that this should be--
whoops, what did I do wrong?
00:06:34.120 --> 00:06:37.690
I meant this.
00:06:37.690 --> 00:06:40.500
I wanted -2, the
same base point,
00:06:40.500 --> 00:06:43.750
but I want to go from the left.
00:06:43.750 --> 00:06:47.780
So that's going to be
-2 + 1, same numerator.
00:06:47.780 --> 00:06:51.840
And then this -2 on
the left, plus 2,
00:06:51.840 --> 00:06:57.580
and that's going to come
out to be -1 / 0-, -,
00:06:57.580 --> 00:07:00.670
which is plus infinity.
00:07:00.670 --> 00:07:10.520
Or just plain infinity, we don't
have to put the plus sign in.
00:07:10.520 --> 00:07:13.090
So this is the first
part of the problem.
00:07:13.090 --> 00:07:16.190
And the second piece, to
get ourselves started,
00:07:16.190 --> 00:07:18.260
you could evaluate this
function at any point.
00:07:18.260 --> 00:07:21.256
This is just the most
interesting point, alright?
00:07:21.256 --> 00:07:22.880
This is just the most
interesting place
00:07:22.880 --> 00:07:24.936
to evaluate it.
00:07:24.936 --> 00:07:26.560
Now, the next thing
that I'd like to do
00:07:26.560 --> 00:07:32.130
is to pay attention to the ends.
00:07:32.130 --> 00:07:34.290
And I haven't really
said what the ends are.
00:07:34.290 --> 00:07:37.399
So the ends are just all the
way to the left and all the way
00:07:37.399 --> 00:07:37.940
to the right.
00:07:37.940 --> 00:07:42.180
So that means x going to
plus or minus infinity.
00:07:42.180 --> 00:07:44.430
So that's the second thing
I want to pay attention to.
00:07:44.430 --> 00:07:49.450
Again, this is a little bit
like a video screen here.
00:07:49.450 --> 00:07:52.400
And we're about to discover
something that's really
00:07:52.400 --> 00:07:55.654
off the screen, in both cases.
00:07:55.654 --> 00:07:58.070
We're taking care of what's
happening way to the left, way
00:07:58.070 --> 00:07:59.360
to the right, here.
00:07:59.360 --> 00:08:01.170
And up above, we
just took care what
00:08:01.170 --> 00:08:05.280
happens way up and way down.
00:08:05.280 --> 00:08:11.550
So on these ends, I need
to do some more analysis.
00:08:11.550 --> 00:08:15.480
Which is related to
a precalculus skill
00:08:15.480 --> 00:08:18.310
which is evaluating limits.
00:08:18.310 --> 00:08:21.070
And here, the way to
do it is to divide
00:08:21.070 --> 00:08:23.010
by x the numerator
and denominator.
00:08:23.010 --> 00:08:27.360
Write it as (1 +
1/x) / (1 + 2/x).
00:08:27.360 --> 00:08:29.650
And then you can see
what happens as x
00:08:29.650 --> 00:08:30.980
goes to plus or minus infinity.
00:08:30.980 --> 00:08:33.590
It just goes to 1.
00:08:33.590 --> 00:08:37.690
So, no matter whether x
is positive or negative.
00:08:37.690 --> 00:08:42.810
When it gets huge, these two
extra numbers here go to 0.
00:08:42.810 --> 00:08:44.550
And so, this tends to 1.
00:08:44.550 --> 00:08:47.560
So if you like, you
could abbreviate this
00:08:47.560 --> 00:08:52.830
as f plus or minus
infinity is equal to 1.
00:08:52.830 --> 00:08:54.610
So now, I get to draw this.
00:08:54.610 --> 00:08:56.900
And we draw this
using asymptotes.
00:08:56.900 --> 00:09:01.800
So there's a level
which is y = 1.
00:09:01.800 --> 00:09:06.790
And then there's
another line to draw.
00:09:06.790 --> 00:09:11.180
Which is x = -2.
00:09:15.190 --> 00:09:18.520
And now, what information
do I have so far?
00:09:18.520 --> 00:09:20.740
Well, the information
that I have so far
00:09:20.740 --> 00:09:26.530
is that when we're coming in
from the right, that's to -2,
00:09:26.530 --> 00:09:28.570
it plunges down
to minus infinity.
00:09:28.570 --> 00:09:33.170
So that's down like this.
00:09:33.170 --> 00:09:39.780
And I also know that it goes up
to infinity on the other side
00:09:39.780 --> 00:09:41.850
of the asymptote.
00:09:41.850 --> 00:09:48.130
And over here, I know it's
going out to the level 1.
00:09:48.130 --> 00:09:53.410
And here it's also
going to the level 1.
00:09:53.410 --> 00:09:57.160
Now, there's an issue.
00:09:57.160 --> 00:09:59.490
I can almost finish
this graph now.
00:09:59.490 --> 00:10:01.460
I almost have enough
information to finish it.
00:10:01.460 --> 00:10:03.530
But there's one
thing which is making
00:10:03.530 --> 00:10:06.780
me hesitate a little bit.
00:10:06.780 --> 00:10:10.160
And that is, I don't know,
for instance, over here,
00:10:10.160 --> 00:10:14.230
whether it's going to maybe
dip below and come back up.
00:10:14.230 --> 00:10:16.930
Or not.
00:10:16.930 --> 00:10:20.140
So what does it do here?
00:10:20.140 --> 00:10:24.750
Can anybody see?
00:10:24.750 --> 00:10:25.250
Yeah.
00:10:25.250 --> 00:10:29.900
STUDENT: [INAUDIBLE]
00:10:29.900 --> 00:10:32.220
PROFESSOR: It can't
dip below because there
00:10:32.220 --> 00:10:33.178
are no critical points.
00:10:33.178 --> 00:10:34.540
What a precisely correct answer.
00:10:34.540 --> 00:10:36.730
So that's exactly right.
00:10:36.730 --> 00:10:43.390
The point here is that
because f' is not 0,
00:10:43.390 --> 00:10:45.020
it can't double back on itself.
00:10:45.020 --> 00:10:49.790
Because there can't be any
of these horizontal tangents.
00:10:49.790 --> 00:11:00.560
It can't double back,
so it can't backtrack.
00:11:00.560 --> 00:11:07.190
So sorry, if f' is not
0, f can't backtrack.
00:11:07.190 --> 00:11:09.410
And so that means that it
doesn't look like this.
00:11:09.410 --> 00:11:14.320
It just goes like this.
00:11:14.320 --> 00:11:15.690
So that's basically it.
00:11:15.690 --> 00:11:17.630
And it's practically
the end of the problem.
00:11:17.630 --> 00:11:19.500
Goes like this.
00:11:19.500 --> 00:11:21.870
Now you can decorate
your thing, right?
00:11:21.870 --> 00:11:24.660
You may notice that maybe it
crosses here, the axes, you can
00:11:24.660 --> 00:11:26.790
actually evaluate these places.
00:11:26.790 --> 00:11:27.500
And so forth.
00:11:27.500 --> 00:11:31.130
We're looking right now
for qualitative behavior.
00:11:31.130 --> 00:11:34.280
In fact, you can see
where these places hit.
00:11:34.280 --> 00:11:36.660
And it's actually a little
higher up than I drew.
00:11:36.660 --> 00:11:40.140
Maybe I'll draw it accurately.
00:11:40.140 --> 00:11:44.970
As we'll see in a second.
00:11:44.970 --> 00:11:47.710
So that's what happens
to this function.
00:11:47.710 --> 00:11:51.860
Now, let's just take a look
in a little bit more detail,
00:11:51.860 --> 00:11:56.470
by double checking.
00:11:56.470 --> 00:11:58.470
So we're just going to
double check what happens
00:11:58.470 --> 00:12:01.280
to the sign of the derivative.
00:12:01.280 --> 00:12:03.460
And in the meantime, I'm
going to explain to you
00:12:03.460 --> 00:12:05.510
what the derivative
is and also talk
00:12:05.510 --> 00:12:07.045
about the second derivative.
00:12:07.045 --> 00:12:11.820
So first of all, the trick
for evaluating the derivative
00:12:11.820 --> 00:12:13.570
is an algebraic one.
00:12:13.570 --> 00:12:16.080
I mean, obviously you can do
this by the quotient rule.
00:12:16.080 --> 00:12:24.250
But I just point out that this
is the same thing as this.
00:12:24.250 --> 00:12:27.330
And now it has, whoops,
that should be a 2
00:12:27.330 --> 00:12:28.690
in the denominator.
00:12:28.690 --> 00:12:33.050
And so, now this has
the form 1 - 1/(x+2).
00:12:35.950 --> 00:12:39.600
So this makes it easy to
see what the derivative is.
00:12:39.600 --> 00:12:42.730
Because the derivative of
a constant is 0, right?
00:12:42.730 --> 00:12:49.300
So this is, derivative, is just
going to be, switch the sign.
00:12:49.300 --> 00:12:53.410
This is what I wrote before.
00:12:53.410 --> 00:12:55.230
And that explains it.
00:12:55.230 --> 00:12:57.610
But incidentally,
it also shows you
00:12:57.610 --> 00:13:04.890
that that this is a hyperbola.
00:13:04.890 --> 00:13:09.620
These are just two
curves of a hyperbola.
00:13:09.620 --> 00:13:12.240
So now, let's check the sign.
00:13:12.240 --> 00:13:14.566
It's already totally
obvious to us
00:13:14.566 --> 00:13:15.940
that this is just
a double check.
00:13:15.940 --> 00:13:18.710
We didn't actually even have
to pay any attention to this.
00:13:18.710 --> 00:13:19.790
It had better be true.
00:13:19.790 --> 00:13:22.220
This is just going to
check our arithmetic.
00:13:22.220 --> 00:13:24.570
Namely, it's increasing here.
00:13:24.570 --> 00:13:26.610
It's increasing there.
00:13:26.610 --> 00:13:27.830
That's got to be true.
00:13:27.830 --> 00:13:30.970
And, sure enough,
this is positive,
00:13:30.970 --> 00:13:32.930
as you can see it's
1 over a square.
00:13:32.930 --> 00:13:34.070
So it is increasing.
00:13:34.070 --> 00:13:35.630
So we checked it.
00:13:35.630 --> 00:13:39.100
But now, there's one more
thing that I want to just
00:13:39.100 --> 00:13:40.720
have you watch out about.
00:13:40.720 --> 00:13:46.980
So this means that
f is increasing.
00:13:46.980 --> 00:13:51.830
On the interval minus
infinity < x < -2.
00:13:51.830 --> 00:13:56.700
And also from -2 all
the way out to infinity.
00:13:56.700 --> 00:14:02.260
So I just want to warn
you, you cannot say,
00:14:02.260 --> 00:14:10.370
don't say f is increasing on
(minus infinity, infinity),
00:14:10.370 --> 00:14:12.020
or all x.
00:14:12.020 --> 00:14:14.826
OK, this is just not true.
00:14:14.826 --> 00:14:16.700
I've written it on the
board, but it's wrong.
00:14:16.700 --> 00:14:18.660
I'd better get rid of it.
00:14:18.660 --> 00:14:19.200
There it is.
00:14:19.200 --> 00:14:20.970
Get rid of it.
00:14:20.970 --> 00:14:24.555
And the reason is, so first
of all it's totally obvious.
00:14:24.555 --> 00:14:25.390
It's going up here.
00:14:25.390 --> 00:14:28.700
But then it went
zooming back down there.
00:14:28.700 --> 00:14:35.860
And here this was true,
but only if x is not -2.
00:14:35.860 --> 00:14:37.400
So there's a break.
00:14:37.400 --> 00:14:39.280
And you've got to pay
attention to the break.
00:14:39.280 --> 00:14:51.060
So basically, the moral here is
that if you ignore this place,
00:14:51.060 --> 00:14:54.390
it's like ignoring Mount
Everest, or the Grand Canyon.
00:14:54.390 --> 00:14:56.640
You're ignoring the
most important feature
00:14:56.640 --> 00:14:58.174
of this function here.
00:14:58.174 --> 00:14:59.590
If you're going
to be figuring out
00:14:59.590 --> 00:15:01.980
where things are going up
and down, which is basically
00:15:01.980 --> 00:15:04.460
all we're doing, you'd
better pay attention
00:15:04.460 --> 00:15:07.330
to these kinds of places.
00:15:07.330 --> 00:15:09.330
So don't ignore them.
00:15:09.330 --> 00:15:13.140
So that's the first remark.
00:15:13.140 --> 00:15:17.340
And now there's just a little
bit of decoration as well.
00:15:17.340 --> 00:15:19.990
Which is the role of
the second derivative.
00:15:19.990 --> 00:15:21.990
So we've written down the
first derivative here.
00:15:21.990 --> 00:15:33.490
The second derivative is
now -2 / (x + 2)^3, right?
00:15:33.490 --> 00:15:36.300
So I get that from
differentiating this formula up
00:15:36.300 --> 00:15:39.490
here for the first derivative.
00:15:39.490 --> 00:15:43.970
And now, of course, that's
also, only works for x not equal
00:15:43.970 --> 00:15:47.680
to -2.
00:15:47.680 --> 00:15:54.820
And now, we can see that this
is going to be negative, let's
00:15:54.820 --> 00:15:56.650
see, where is it negative?
00:15:56.650 --> 00:15:58.540
When this is a
positive quantity,
00:15:58.540 --> 00:16:04.590
so when -2 < x <
infinity, it's negative.
00:16:04.590 --> 00:16:07.770
And this is where
this thing is concave.
00:16:07.770 --> 00:16:08.670
Let's see.
00:16:08.670 --> 00:16:12.650
Did I say that right?
00:16:12.650 --> 00:16:13.330
Negative, right?
00:16:13.330 --> 00:16:19.180
This is concave down.
00:16:19.180 --> 00:16:19.930
Right.
00:16:19.930 --> 00:16:22.570
And similarly, if I
look at this expression,
00:16:22.570 --> 00:16:27.830
the numerator is always negative
but the denominator becomes
00:16:27.830 --> 00:16:31.130
negative as well when x < -2.
00:16:31.130 --> 00:16:33.710
So this becomes positive.
00:16:33.710 --> 00:16:36.690
So this case, it was
negative over positive.
00:16:36.690 --> 00:16:40.950
In this case it was negative
divided by negative.
00:16:40.950 --> 00:16:46.300
So here, this is in the range
minus infinity < x < -2 And
00:16:46.300 --> 00:16:52.240
here it's concave up.
00:16:52.240 --> 00:16:54.962
Now, again, this is just
consistent with what
00:16:54.962 --> 00:16:55.920
we're already guessing.
00:16:55.920 --> 00:16:57.628
Of course we already
know it in this case
00:16:57.628 --> 00:16:59.680
if we know that
this is a hyperbola.
00:16:59.680 --> 00:17:01.600
That it's going
to be concave down
00:17:01.600 --> 00:17:04.270
to the right of the vertical
line, dotted vertical line.
00:17:04.270 --> 00:17:07.770
And concave up to the left.
00:17:07.770 --> 00:17:10.640
So what extra piece
of information
00:17:10.640 --> 00:17:16.100
is it that this is giving us?
00:17:16.100 --> 00:17:17.800
Did I say this backwards?
00:17:17.800 --> 00:17:18.520
No.
00:17:18.520 --> 00:17:19.222
That's OK.
00:17:19.222 --> 00:17:21.430
So what extra piece of
information is this giving us?
00:17:21.430 --> 00:17:23.400
It looks like it's giving
us hardly anything.
00:17:23.400 --> 00:17:25.680
And it really is giving
us hardly anything.
00:17:25.680 --> 00:17:28.840
But it is giving us something
that's a little aesthetic.
00:17:28.840 --> 00:17:34.280
It's ruling out the
possibility of a wiggle.
00:17:34.280 --> 00:17:37.590
There isn't anything
like that in the curve.
00:17:37.590 --> 00:17:39.790
It can't shift from
curving this way
00:17:39.790 --> 00:17:41.590
to curving that way
to curving this way.
00:17:41.590 --> 00:17:42.860
That doesn't happen.
00:17:42.860 --> 00:17:59.070
So these properties say there's
no wiggle in the graph of that.
00:17:59.070 --> 00:17:59.790
Alright.
00:17:59.790 --> 00:18:01.390
So.
00:18:01.390 --> 00:18:01.890
Question.
00:18:01.890 --> 00:18:05.570
STUDENT: Do we define the
increasing and decreasing based
00:18:05.570 --> 00:18:09.250
purely on the
derivative, or the sort
00:18:09.250 --> 00:18:13.390
of more general definition
of picking any two points
00:18:13.390 --> 00:18:14.310
and seeing.
00:18:14.310 --> 00:18:16.610
Because sometimes there
can be an inconsistency
00:18:16.610 --> 00:18:20.300
between the two definitions.
00:18:20.300 --> 00:18:26.110
PROFESSOR: OK, so the
question is, in this course,
00:18:26.110 --> 00:18:29.290
are we going to define
positive derivative as being
00:18:29.290 --> 00:18:31.360
the same thing as increasing.
00:18:31.360 --> 00:18:33.030
And the answer is no.
00:18:33.030 --> 00:18:36.210
We'll try to use these
terms separately.
00:18:36.210 --> 00:18:40.330
What's always true is
that if f' is positive,
00:18:40.330 --> 00:18:42.530
then f is increasing.
00:18:42.530 --> 00:18:45.060
But the reverse is
not necessarily true.
00:18:45.060 --> 00:18:47.190
It could be very flat,
the derivative can be 0
00:18:47.190 --> 00:18:50.050
and still the function
can be increasing.
00:18:50.050 --> 00:18:53.730
OK, the derivative can
be 0 at a few places.
00:18:53.730 --> 00:18:59.300
For instance, like some cubics.
00:18:59.300 --> 00:19:05.370
Other questions?
00:19:05.370 --> 00:19:09.860
So that's as much as I
need to say in general.
00:19:09.860 --> 00:19:11.390
I mean, in a specific case.
00:19:11.390 --> 00:19:13.240
But I want to get
you a general scheme
00:19:13.240 --> 00:19:16.500
and I want to go through a
more complicated example that
00:19:16.500 --> 00:19:22.750
gets all the features
of this kind of thing.
00:19:22.750 --> 00:19:34.710
So let's talk about a general
strategy for sketching.
00:19:34.710 --> 00:19:40.300
So the first part of this
strategy, if you like,
00:19:40.300 --> 00:19:40.800
let's see.
00:19:40.800 --> 00:19:42.430
I have it all plotted out here.
00:19:42.430 --> 00:19:45.930
So I'm going to make sure
I get it exactly the way
00:19:45.930 --> 00:19:47.780
I wanted you to see.
00:19:47.780 --> 00:19:51.440
So I have, it's plotting.
00:19:51.440 --> 00:19:52.720
The plot thickens.
00:19:52.720 --> 00:19:54.040
Here we go.
00:19:54.040 --> 00:19:57.710
So plot, what is it that
you should plot first?
00:19:57.710 --> 00:20:01.070
Before you even think
about derivatives,
00:20:01.070 --> 00:20:08.250
you should plot discontinuities.
00:20:08.250 --> 00:20:18.490
Especially the infinite ones.
00:20:18.490 --> 00:20:20.210
That's the first
thing you should do.
00:20:20.210 --> 00:20:27.160
And then, you should plot
end points, for ends.
00:20:27.160 --> 00:20:30.580
For x going to plus
or minus infinity
00:20:30.580 --> 00:20:35.640
if there don't happen to be
any finite ends to the problem.
00:20:35.640 --> 00:20:44.600
And the third thing you can
do is plot any easy points.
00:20:44.600 --> 00:20:49.000
This is optional.
00:20:49.000 --> 00:20:50.710
At your discretion.
00:20:50.710 --> 00:20:53.810
You might, for instance,
on this example,
00:20:53.810 --> 00:20:59.550
plot the places where the
graph crosses the axis.
00:20:59.550 --> 00:21:04.540
If you want to.
00:21:04.540 --> 00:21:05.810
So that's the first part.
00:21:05.810 --> 00:21:08.190
And again, this is
all precalculus.
00:21:08.190 --> 00:21:18.050
So now, in the second part we're
going to solve this equation
00:21:18.050 --> 00:21:29.360
and we're going to plot the
critical points and values.
00:21:29.360 --> 00:21:32.810
In the problem which we just
discussed, there weren't any.
00:21:32.810 --> 00:21:38.640
So this part was empty.
00:21:38.640 --> 00:21:49.685
So the third step is to decide
whether f', sorry, whether,
00:21:49.685 --> 00:22:01.120
f' is positive or
negative on each interval.
00:22:01.120 --> 00:22:17.210
Between critical
points, discontinuities.
00:22:17.210 --> 00:22:22.560
The direction of the sign, in
this case it doesn't change.
00:22:22.560 --> 00:22:24.900
It goes up here and
it also goes up here.
00:22:24.900 --> 00:22:27.770
But it could go up here
and then come back down.
00:22:27.770 --> 00:22:31.460
So the direction can change
at every critical point.
00:22:31.460 --> 00:22:33.850
It can change at
every discontinuity.
00:22:33.850 --> 00:22:35.350
And you don't know.
00:22:35.350 --> 00:22:39.310
However, this
particular step has
00:22:39.310 --> 00:22:47.230
to be consistent with 1
and 2, with steps 1 and 2.
00:22:47.230 --> 00:22:51.800
In fact, it will
never, if you can
00:22:51.800 --> 00:22:56.970
succeed in doing steps 1 and
2, you'll never need step 3.
00:22:56.970 --> 00:23:02.190
All it's doing is
double-checking.
00:23:02.190 --> 00:23:05.770
So if you made an arithmetic
mistake somewhere,
00:23:05.770 --> 00:23:09.160
you'll be able to see it.
00:23:09.160 --> 00:23:10.870
So that's maybe the
most important thing.
00:23:10.870 --> 00:23:13.050
And it's actually the most
frustrating thing for me
00:23:13.050 --> 00:23:17.700
when I see people working on
problems, is they start step 3,
00:23:17.700 --> 00:23:21.140
they get it wrong, and then they
start trying to draw the graph
00:23:21.140 --> 00:23:22.480
and it doesn't work.
00:23:22.480 --> 00:23:23.620
Because it's inconsistent.
00:23:23.620 --> 00:23:25.880
And the reason is
some arithmetic error
00:23:25.880 --> 00:23:27.630
with the derivative
or something like that
00:23:27.630 --> 00:23:29.880
or some other misinterpretation.
00:23:29.880 --> 00:23:31.930
And then there's a total mess.
00:23:31.930 --> 00:23:34.170
If you start with
these two steps,
00:23:34.170 --> 00:23:36.454
then you're going to know
when you get to this step
00:23:36.454 --> 00:23:37.620
that you're making mistakes.
00:23:37.620 --> 00:23:41.860
People don't generally make as
many mistakes in the first two
00:23:41.860 --> 00:23:42.360
steps.
00:23:42.360 --> 00:23:45.220
Anyway, in fact you can
skip this step if you want.
00:23:45.220 --> 00:23:49.470
But that's at risk of not
double-checking your work.
00:23:49.470 --> 00:23:51.280
So what's the fourth step?
00:23:51.280 --> 00:23:59.400
Well, we take a look at whether
f'' is positive or negative.
00:23:59.400 --> 00:24:02.300
And so we're deciding on things
like whether it's concave
00:24:02.300 --> 00:24:07.640
up or down.
00:24:07.640 --> 00:24:15.120
And we have these
points, f''(x) = 0,
00:24:15.120 --> 00:24:24.570
which are called
inflection points.
00:24:24.570 --> 00:24:31.550
And the last step is just
to combine everything.
00:24:31.550 --> 00:24:35.710
So this is this the
scheme, the general scheme.
00:24:35.710 --> 00:24:58.850
And let's just carry it
out in a particular case.
00:24:58.850 --> 00:25:02.280
So here's the function that
I'm going to use as an example.
00:25:02.280 --> 00:25:08.250
I'll use f(x) = x / ln x.
00:25:08.250 --> 00:25:11.460
And because the
logarithm-- yeah, question.
00:25:11.460 --> 00:25:11.960
Yeah.
00:25:11.960 --> 00:25:18.660
STUDENT: [INAUDIBLE]
00:25:18.660 --> 00:25:21.350
PROFESSOR: The question
is, is this optional.
00:25:21.350 --> 00:25:25.570
So that's a good question.
00:25:25.570 --> 00:25:26.390
Is this optional.
00:25:26.390 --> 00:25:31.516
STUDENT: [INAUDIBLE]
00:25:31.516 --> 00:25:37.590
PROFESSOR: OK, the question
is is this optional,
00:25:37.590 --> 00:25:38.660
this kind of question.
00:25:38.660 --> 00:25:48.620
And the answer is,
it's more than just--
00:25:48.620 --> 00:25:51.700
so, in many instances, I'm
not going to ask you to.
00:25:51.700 --> 00:25:55.170
I strongly recommend that
if I don't ask you to do it,
00:25:55.170 --> 00:25:57.050
that you not try.
00:25:57.050 --> 00:26:01.050
Because it's usually awful to
find the second derivative.
00:26:01.050 --> 00:26:02.940
Any time you can get
away without computing
00:26:02.940 --> 00:26:06.330
a second derivative,
you're better off.
00:26:06.330 --> 00:26:07.834
So in many, many instances.
00:26:07.834 --> 00:26:09.500
On the other hand,
if I ask you to do it
00:26:09.500 --> 00:26:13.060
it's because I want you
to have the work to do it.
00:26:13.060 --> 00:26:16.470
But basically, if
nobody forces you to,
00:26:16.470 --> 00:26:22.130
I would say never
do step 4 here.
00:26:22.130 --> 00:26:26.750
Other questions.
00:26:26.750 --> 00:26:27.610
All right.
00:26:27.610 --> 00:26:29.150
So we're going to
force ourselves
00:26:29.150 --> 00:26:31.810
to do step 4, however,
in this instance.
00:26:31.810 --> 00:26:35.010
But maybe this will be
one of the few times.
00:26:35.010 --> 00:26:39.140
So here we go, just for
illustrative purposes.
00:26:39.140 --> 00:26:43.240
OK, now.
00:26:43.240 --> 00:26:46.140
So here's the function
that I want to discuss.
00:26:46.140 --> 00:26:49.120
And the range has
to be x positive,
00:26:49.120 --> 00:26:55.500
because the logarithm is not
defined for negative values.
00:26:55.500 --> 00:26:57.660
So the first thing
that I'm going to do
00:26:57.660 --> 00:27:02.560
is, I'd like to follow
the scheme here.
00:27:02.560 --> 00:27:04.770
Because if I don't
follow the scheme,
00:27:04.770 --> 00:27:06.490
I'm going to get
a little mixed up.
00:27:06.490 --> 00:27:13.980
So the first part is to
find the singularities.
00:27:13.980 --> 00:27:17.060
That is, the places
where f is infinite.
00:27:17.060 --> 00:27:20.720
And that's when the logarithm,
the denominator, vanishes.
00:27:20.720 --> 00:27:25.630
So that's f(1+), if you like.
00:27:25.630 --> 00:27:32.000
So that's 1 / ln(1+),
which is 1 / 0,
00:27:32.000 --> 00:27:34.280
with a little bit of
positiveness to it.
00:27:34.280 --> 00:27:37.270
Which is infinity.
00:27:37.270 --> 00:27:39.830
And second, we do
it the other way.
00:27:39.830 --> 00:27:46.850
And not surprisingly, this comes
out to be negative infinity.
00:27:46.850 --> 00:27:51.980
Now, the next thing I
want to do is the ends.
00:27:51.980 --> 00:27:56.980
So I call these the ends.
00:27:56.980 --> 00:28:01.380
And there are two of them.
00:28:01.380 --> 00:28:06.250
One of them is f(0)
from the right.
00:28:06.250 --> 00:28:08.300
f(0+).
00:28:08.300 --> 00:28:21.120
So that is 0+ / ln(0+),
which is 0+ divided by, well,
00:28:21.120 --> 00:28:25.360
ln(0+) is actually
minus infinity.
00:28:25.360 --> 00:28:27.180
That's what happens
to the logarithm, goes
00:28:27.180 --> 00:28:28.170
to minus infinity.
00:28:28.170 --> 00:28:31.160
So this is 0 over infinity,
which is definitely 0,
00:28:31.160 --> 00:28:37.100
there's no problem about
what happens to this.
00:28:37.100 --> 00:28:42.910
The other side, so this is the
end, this is the first end.
00:28:42.910 --> 00:28:44.920
The range is this.
00:28:44.920 --> 00:28:48.019
And I just did
the left endpoint.
00:28:48.019 --> 00:28:49.810
And so now I have to
do the right endpoint,
00:28:49.810 --> 00:28:51.870
I have to let x go to infinity.
00:28:51.870 --> 00:28:53.591
So if I let x go
to infinity, I'm
00:28:53.591 --> 00:28:55.090
just going to have
to think about it
00:28:55.090 --> 00:28:57.690
a little bit by plugging
in a very large number.
00:28:57.690 --> 00:29:01.960
I'll plug in 10^10,
to see what happens.
00:29:01.960 --> 00:29:07.890
So if I plug in 10^10 into x /
ln x, I get 10^10 / ln(10^10).
00:29:11.730 --> 00:29:17.590
Which is 10^10 / (10 ln(10)).
00:29:17.590 --> 00:29:20.110
So the denominator,
this number here,
00:29:20.110 --> 00:29:23.180
is about 2 point something.
00:29:23.180 --> 00:29:25.130
2.3 or so.
00:29:25.130 --> 00:29:27.470
So this is maybe 230
in the denominator,
00:29:27.470 --> 00:29:31.900
and this is a number
with ten 0's after it.
00:29:31.900 --> 00:29:33.530
So it's very, very large.
00:29:33.530 --> 00:29:35.300
I claim it's big.
00:29:35.300 --> 00:29:38.080
And that gives us the
clue that what's happening
00:29:38.080 --> 00:29:40.540
is that this thing is infinite.
00:29:40.540 --> 00:29:42.150
So, in other words,
our conclusion
00:29:42.150 --> 00:29:52.120
is that f of
infinity is infinity.
00:29:52.120 --> 00:29:59.650
So what do we have so
far for our function?
00:29:59.650 --> 00:30:03.300
We're just trying to build the
scaffolding of the function.
00:30:03.300 --> 00:30:07.270
And we're doing it by taking
the most important points.
00:30:07.270 --> 00:30:09.147
And from a mathematician's
point of view,
00:30:09.147 --> 00:30:10.980
the most important
points are the ones which
00:30:10.980 --> 00:30:13.490
are sort of infinitely obvious.
00:30:13.490 --> 00:30:15.190
For the ends of the problem.
00:30:15.190 --> 00:30:19.730
So that's where we're heading.
00:30:19.730 --> 00:30:22.600
We have a vertical
asymptote, which is at x = 1.
00:30:22.600 --> 00:30:29.150
So this gives us x = 1.
00:30:29.150 --> 00:30:34.270
And we have a value which
is that it's 0 here.
00:30:34.270 --> 00:30:38.640
And we also know that when
we come in from the-- sorry,
00:30:38.640 --> 00:30:42.330
so we come in from
the left, that's
00:30:42.330 --> 00:30:46.060
f, the one from the left,
we get negative infinity.
00:30:46.060 --> 00:30:47.550
So it's diving down.
00:30:47.550 --> 00:30:52.460
It's going down like this.
00:30:52.460 --> 00:30:55.925
And, furthermore, on the other
side we know it's climbing up.
00:30:55.925 --> 00:30:58.560
So it's going up like this.
00:30:58.560 --> 00:31:00.270
Just start a little higher.
00:31:00.270 --> 00:31:00.770
Right, so.
00:31:00.770 --> 00:31:02.520
So far, this is what we know.
00:31:02.520 --> 00:31:05.810
Oh, and there's one
other thing that we know.
00:31:05.810 --> 00:31:12.420
When we go to plus infinity,
it's going back up.
00:31:12.420 --> 00:31:15.150
So, so far we have this.
00:31:15.150 --> 00:31:17.750
Now, already it should
be pretty obvious what's
00:31:17.750 --> 00:31:19.619
going to happen
to this function.
00:31:19.619 --> 00:31:21.160
So there shouldn't
be many surprises.
00:31:21.160 --> 00:31:23.160
It's going to come
down like this.
00:31:23.160 --> 00:31:27.090
Go like this, it's going to
turn around and go back up.
00:31:27.090 --> 00:31:29.290
That's what we expect.
00:31:29.290 --> 00:31:33.600
So we don't know that yet,
but we're pretty sure.
00:31:33.600 --> 00:31:36.800
So at this point, we can start
looking at the critical points.
00:31:36.800 --> 00:31:41.994
We can do our step 2 here -
we need a little bit more room
00:31:41.994 --> 00:31:45.490
here - and see what's
happening with this function.
00:31:45.490 --> 00:31:49.220
So I have to differentiate it.
00:31:49.220 --> 00:31:52.070
And it's, this is
the quotient rule.
00:31:52.070 --> 00:31:54.880
So remember the function
is up here, x / ln x.
00:31:54.880 --> 00:31:59.400
So I have a (ln x)^2
in the denominator.
00:31:59.400 --> 00:32:02.430
And I get here the derivative
of x is 1, so we get 1 *
00:32:02.430 --> 00:32:07.174
ln x minus x times
the derivative of ln
00:32:07.174 --> 00:32:07.840
x, which is 1/x.
00:32:10.560 --> 00:32:16.850
So all told, that's
(ln x - 1) / (ln x)^2.
00:32:20.160 --> 00:32:27.770
So here's our derivative.
00:32:27.770 --> 00:32:35.080
And now, if I set this equal to
0, at least in the numerator,
00:32:35.080 --> 00:32:40.970
the numerator is 0 when x = e.
00:32:40.970 --> 00:32:43.490
The log of e is 1.
00:32:43.490 --> 00:32:46.290
So here's our critical point.
00:32:46.290 --> 00:32:51.320
And we have a critical
value, which is f(e).
00:32:51.320 --> 00:32:55.610
And that's going to be e / ln e.
00:32:55.610 --> 00:32:57.240
Which is e, again.
00:32:57.240 --> 00:32:59.090
Because ln e = 1.
00:32:59.090 --> 00:33:01.890
So now I can also plot
the critical point,
00:33:01.890 --> 00:33:03.110
which is down here.
00:33:03.110 --> 00:33:07.680
And there's only one of
them, and it's at (e, e).
00:33:07.680 --> 00:33:09.530
That's kind of
not to scale here,
00:33:09.530 --> 00:33:12.520
because my blackboard
isn't quite tall enough.
00:33:12.520 --> 00:33:15.360
It should be over here
and then, it's slope 1.
00:33:15.360 --> 00:33:17.140
But I dipped it down.
00:33:17.140 --> 00:33:18.730
So this is not to
scale, and indeed
00:33:18.730 --> 00:33:20.490
that's one of the
things that we're not
00:33:20.490 --> 00:33:22.680
going to attempt to do
with these pictures,
00:33:22.680 --> 00:33:24.680
is to make them to scale.
00:33:24.680 --> 00:33:29.560
So the scale's a
little squashed.
00:33:29.560 --> 00:33:32.710
So, so far I have
this critical point.
00:33:32.710 --> 00:33:36.180
And, in fact, I'm going
to label it with a C.
00:33:36.180 --> 00:33:38.030
Whenever I have a
critical point I'll just
00:33:38.030 --> 00:33:41.490
make sure that I remember
that that's what it is.
00:33:41.490 --> 00:33:45.120
And since there's only one,
the rest of this picture
00:33:45.120 --> 00:33:49.900
is now correct.
00:33:49.900 --> 00:33:54.880
That's the same mechanism that
we used for the hyperbola.
00:33:54.880 --> 00:33:56.977
Namely, we know there's
only one place where
00:33:56.977 --> 00:33:57.810
the derivative is 0.
00:33:57.810 --> 00:33:59.980
So that means there
no more horizontals,
00:33:59.980 --> 00:34:01.950
so there's no more backtracking.
00:34:01.950 --> 00:34:03.330
It has to come down to here.
00:34:03.330 --> 00:34:03.970
Get to there.
00:34:03.970 --> 00:34:06.060
This is the only place
it can turn around.
00:34:06.060 --> 00:34:06.834
Goes back up.
00:34:06.834 --> 00:34:09.000
It has to start here and
it has to go down to there.
00:34:09.000 --> 00:34:10.600
It can't go above 0.
00:34:10.600 --> 00:34:13.600
Do not pass go, do
not get positive.
00:34:13.600 --> 00:34:20.230
It has to head down here.
00:34:20.230 --> 00:34:21.690
So that's great.
00:34:21.690 --> 00:34:25.080
That means that this picture is
almost completely correct now.
00:34:25.080 --> 00:34:27.520
And the rest is more
or less decoration.
00:34:27.520 --> 00:34:30.250
We're pretty much done
with the way it looks,
00:34:30.250 --> 00:34:34.570
at least schematically.
00:34:34.570 --> 00:34:37.700
However, I am going to punish
you, because I warned you.
00:34:37.700 --> 00:34:40.120
We are going to go over
here and do this step 4
00:34:40.120 --> 00:34:44.216
and fix up the concavity.
00:34:44.216 --> 00:34:45.840
And we're also going
to do a little bit
00:34:45.840 --> 00:35:00.770
of that double-checking.
00:35:00.770 --> 00:35:04.960
So now, let's
again-- just, I want
00:35:04.960 --> 00:35:10.660
to emphasize-- We're going
to do a double-check.
00:35:10.660 --> 00:35:12.240
This is part 3.
00:35:12.240 --> 00:35:16.870
But in advance, I already
have, based on this picture
00:35:16.870 --> 00:35:19.000
I already know what
has to be true.
00:35:19.000 --> 00:35:35.450
That f is decreasing on 0 to 1.
f is also decreasing on 1 to e.
00:35:35.450 --> 00:35:45.490
And f is increasing
on e to infinity.
00:35:45.490 --> 00:35:49.144
So, already, because we
plotted a bunch of points
00:35:49.144 --> 00:35:51.060
and we know that there
aren't any places where
00:35:51.060 --> 00:35:52.518
the derivative
vanishes, we already
00:35:52.518 --> 00:35:55.500
know it goes down, down, up.
00:35:55.500 --> 00:35:56.970
That's what it's got to do.
00:35:56.970 --> 00:35:59.550
Now, we'll just make sure that
we didn't make any arithmetic
00:35:59.550 --> 00:36:00.720
mistakes, now.
00:36:00.720 --> 00:36:02.980
By actually computing
the derivative,
00:36:02.980 --> 00:36:04.650
or staring at it, anyway.
00:36:04.650 --> 00:36:10.350
And making sure
that it's correct.
00:36:10.350 --> 00:36:17.600
So first of all, we just
take a look at the numerator.
00:36:17.600 --> 00:36:26.570
So f', remember, was
(ln x - 1) / (ln x)^2.
00:36:26.570 --> 00:36:28.480
So the denominator is positive.
00:36:28.480 --> 00:36:32.650
So let's just take a
look at the three ranges.
00:36:32.650 --> 00:36:37.330
So we have 0 < x < 1.
00:36:37.330 --> 00:36:40.310
And on that range, the
logarithm is negative,
00:36:40.310 --> 00:36:45.230
so this is negative divided by
positive, which is negative.
00:36:45.230 --> 00:36:47.220
That's decreasing, that's good.
00:36:47.220 --> 00:36:50.160
And in fact, that also
works on the next range.
00:36:50.160 --> 00:36:55.357
1 < x < e, it's negative
divided by positive.
00:36:55.357 --> 00:36:57.190
And the only reason why
we skipped 1, again,
00:36:57.190 --> 00:36:58.432
is that it's undefined there.
00:36:58.432 --> 00:37:01.040
And there's something
dramatic happening there.
00:37:01.040 --> 00:37:05.240
And then, at the last range,
when x is bigger than e,
00:37:05.240 --> 00:37:07.770
that means the logarithm
is already bigger than 1.
00:37:07.770 --> 00:37:09.145
So the numerator
is now positive,
00:37:09.145 --> 00:37:13.650
and the denominator's still
positive, so it's increasing.
00:37:13.650 --> 00:37:22.910
So we've just double-checked
something that we already knew.
00:37:22.910 --> 00:37:26.540
Alright, so that's
pretty much all
00:37:26.540 --> 00:37:29.200
there is to say about step 3.
00:37:29.200 --> 00:37:33.980
So this is checking the
positivity and negativity.
00:37:33.980 --> 00:37:35.660
And now, step 4.
00:37:35.660 --> 00:37:38.480
There is one small point which
I want to make before we go on.
00:37:38.480 --> 00:37:42.110
Which is that
sometimes, you can't
00:37:42.110 --> 00:37:45.445
evaluate the function or its
derivative particularly well.
00:37:45.445 --> 00:37:48.359
So sometimes you can't
plot the points very well.
00:37:48.359 --> 00:37:50.150
And if you can't plot
the points very well,
00:37:50.150 --> 00:37:52.440
then you might
have to do 3 first,
00:37:52.440 --> 00:37:55.290
to figure out what's
going on a little bit.
00:37:55.290 --> 00:37:59.150
You might have to skip.
00:37:59.150 --> 00:38:02.400
So now we're going to go on
to the second derivative.
00:38:02.400 --> 00:38:07.180
But first, I want to
use an algebraic trick
00:38:07.180 --> 00:38:08.470
to rearrange the terms.
00:38:08.470 --> 00:38:10.820
And I want to notice
one more little point.
00:38:10.820 --> 00:38:16.320
Which I-- as I say, this is
decoration for the graph.
00:38:16.320 --> 00:38:18.290
So I want to
rewrite the formula.
00:38:18.290 --> 00:38:22.320
Maybe I'll do it
right over here.
00:38:22.320 --> 00:38:27.270
Another way of writing this
is 1/(ln x) - 1/(ln x)^2.
00:38:31.440 --> 00:38:35.240
So that's another way of
writing the derivative.
00:38:35.240 --> 00:38:38.520
And that allows me
to notice something
00:38:38.520 --> 00:38:40.830
that I missed, before.
00:38:40.830 --> 00:38:47.410
When I solved the equation ln x
- 1 - this is equal to 0 here,
00:38:47.410 --> 00:38:48.590
this equation here.
00:38:48.590 --> 00:38:51.460
I missed a possibility.
00:38:51.460 --> 00:38:54.020
I missed the possibility
that the denominator
00:38:54.020 --> 00:38:58.690
could be infinity.
00:38:58.690 --> 00:39:02.040
So actually, if the
denominator's infinity,
00:39:02.040 --> 00:39:05.460
as you can see from the
other expression there,
00:39:05.460 --> 00:39:09.000
it actually is true that
the derivative is 0.
00:39:09.000 --> 00:39:16.710
So also when x = 0+, the
slope is going to be 0.
00:39:16.710 --> 00:39:19.050
Let me just
emphasize that again.
00:39:19.050 --> 00:39:23.500
If you evaluate using this
other formula over here,
00:39:23.500 --> 00:39:28.410
this is 1/(ln(0+))
- 1/(ln(0+))^2.
00:39:31.540 --> 00:39:36.500
That's 1 over -infinity - minus
1 over infinity, if you like,
00:39:36.500 --> 00:39:37.310
squared.
00:39:37.310 --> 00:39:40.630
Anyway, it's 0.
00:39:40.630 --> 00:39:42.330
So this is 0.
00:39:42.330 --> 00:39:43.410
The slope is 0 there.
00:39:43.410 --> 00:39:46.640
That is a little piece of
decoration on our graph.
00:39:46.640 --> 00:39:50.330
It's telling us, going
back to our graph here,
00:39:50.330 --> 00:39:53.560
it's telling us this is coming
in with slope horizontal.
00:39:53.560 --> 00:39:57.530
So we're starting out this way.
00:39:57.530 --> 00:40:01.013
That's just a little
start here to the graph.
00:40:01.013 --> 00:40:02.620
It's a horizontal slope.
00:40:02.620 --> 00:40:07.940
So there really were two places
where the slope was horizontal.
00:40:07.940 --> 00:40:11.470
Now, with the help of
this second formula
00:40:11.470 --> 00:40:16.972
I can also differentiate
a second time.
00:40:16.972 --> 00:40:19.430
So it's a little bit easier to
do that if I differentiate 1
00:40:19.430 --> 00:40:28.050
over the log, that's -(ln
x)^(-2) 1/x + 2 (ln x)^(-3)
00:40:28.050 --> 00:40:28.550
1/x.
00:40:34.270 --> 00:40:39.090
And that, if I put it
over a common denominator,
00:40:39.090 --> 00:40:48.690
is x (ln x)^3 times,
let's see here,
00:40:48.690 --> 00:40:55.340
I guess I'll have to
take the 2 - ln x.
00:40:55.340 --> 00:40:57.260
So I've now
rewritten the formula
00:40:57.260 --> 00:41:03.450
for the second
derivative as a ratio.
00:41:03.450 --> 00:41:09.610
Now, to decide the sign, you
see there are two places where
00:41:09.610 --> 00:41:11.910
the sign flips.
00:41:11.910 --> 00:41:16.030
The numerator crosses
when the logarithm is 2,
00:41:16.030 --> 00:41:18.620
that's going to be when x = e^2.
00:41:18.620 --> 00:41:22.400
And the denominator
flips when x = 1,
00:41:22.400 --> 00:41:28.020
that's when the log flips
from positive to negative.
00:41:28.020 --> 00:41:32.280
So we have a couple
of ranges here.
00:41:32.280 --> 00:41:37.580
So, first of all, we have
the range from 0 to 1.
00:41:37.580 --> 00:41:42.270
And then we have the
range from 1 to e^2.
00:41:42.270 --> 00:41:46.380
And then we have the range
from e^2 all the way out
00:41:46.380 --> 00:41:49.780
to infinity.
00:41:49.780 --> 00:41:57.210
So between 0 and
1, the numerator
00:41:57.210 --> 00:41:59.630
is, well this is a
negative number and this,
00:41:59.630 --> 00:42:01.300
so minus a negative
number is positive,
00:42:01.300 --> 00:42:04.650
so the numerator is positive.
00:42:04.650 --> 00:42:08.030
And the denominator is negative,
because the log is negative
00:42:08.030 --> 00:42:09.660
and it's taken to
the third power.
00:42:09.660 --> 00:42:12.170
So this is a negative
number, so it's positive
00:42:12.170 --> 00:42:15.190
divided by negative,
which is less than 0.
00:42:15.190 --> 00:42:18.770
That means it's concave down.
00:42:18.770 --> 00:42:26.040
So this is a concave down part.
00:42:26.040 --> 00:42:28.230
And that's a good thing,
because over here this
00:42:28.230 --> 00:42:29.120
was concave down.
00:42:29.120 --> 00:42:30.560
So there are no wiggles.
00:42:30.560 --> 00:42:34.260
It goes straight
down, like this.
00:42:34.260 --> 00:42:41.590
And then the other two pieces
are f'' is equal to, well
00:42:41.590 --> 00:42:43.260
it's going to switch here.
00:42:43.260 --> 00:42:44.856
The denominator
becomes positive.
00:42:44.856 --> 00:42:48.190
So it's positive over positive.
00:42:48.190 --> 00:42:56.670
So this is concave up.
00:42:56.670 --> 00:42:58.410
And that's going over here.
00:42:58.410 --> 00:43:02.715
But notice that it's not the
bottom where it turns around,
00:43:02.715 --> 00:43:07.740
it's somewhere else.
00:43:07.740 --> 00:43:09.930
So there's another
transition here.
00:43:09.930 --> 00:43:12.090
This is e^2.
00:43:12.090 --> 00:43:15.090
This is e.
00:43:15.090 --> 00:43:20.440
So what happens at the end is,
again, the sign flips again.
00:43:20.440 --> 00:43:26.580
Because the numerator, now,
when x > e^2, becomes negative.
00:43:26.580 --> 00:43:29.965
And this is negative divided
by positive, which is negative.
00:43:29.965 --> 00:43:35.630
And this is concave down.
00:43:35.630 --> 00:43:39.100
And so we didn't quite
draw the graph right.
00:43:39.100 --> 00:43:40.990
There's an inflection
point right here,
00:43:40.990 --> 00:43:45.670
which I'll label with an
I. And it makes a turn
00:43:45.670 --> 00:43:47.340
the other way at that point.
00:43:47.340 --> 00:43:49.480
So there was a wiggle.
00:43:49.480 --> 00:43:51.210
There's the wiggle.
00:43:51.210 --> 00:43:53.710
Still going up, still
going to infinity.
00:43:53.710 --> 00:43:57.040
But kind of the slope
of a mountain, right?
00:43:57.040 --> 00:44:01.240
It's going the other way.
00:44:01.240 --> 00:44:03.510
This point happens
to be (e^2, e^2 / 2).
00:44:09.350 --> 00:44:11.980
So that's as detailed
as we'll ever get.
00:44:11.980 --> 00:44:14.760
And indeed, the
next game is going
00:44:14.760 --> 00:44:18.730
to be avoid being-- is to
avoid being this detailed.
00:44:18.730 --> 00:44:21.760
So let me introduce
the next subject.
00:44:21.760 --> 00:44:48.310
Which is maxima and minima.
00:44:48.310 --> 00:45:04.160
OK, now, maxima and minima,
maximum and minimum problems
00:45:04.160 --> 00:45:06.550
can be described graphically
in the following ways.
00:45:06.550 --> 00:45:13.150
Suppose you have a
function, right, here it is.
00:45:13.150 --> 00:45:14.420
OK?
00:45:14.420 --> 00:45:24.630
Now, find the maximum.
00:45:24.630 --> 00:45:30.740
And find the minimum.
00:45:30.740 --> 00:45:31.410
OK.
00:45:31.410 --> 00:45:38.880
So this problem is done.
00:45:38.880 --> 00:45:51.625
The point being, that it is
easy to find max and the min
00:45:51.625 --> 00:45:58.670
with the sketch.
00:45:58.670 --> 00:46:00.330
It's very easy.
00:46:00.330 --> 00:46:05.130
The goal, the problem, is that
the sketch is a lot of work.
00:46:05.130 --> 00:46:10.180
We just spent 20 minutes
sketching something.
00:46:10.180 --> 00:46:12.637
We would not like to
spend all that time
00:46:12.637 --> 00:46:14.970
every single time we want to
find a maximum and minimum.
00:46:14.970 --> 00:46:19.576
So the goal is to do it
with-- so our goal is
00:46:19.576 --> 00:46:25.170
to use shortcuts.
00:46:25.170 --> 00:46:30.650
And, indeed, as I said
earlier, we certainly
00:46:30.650 --> 00:46:33.840
never want to use the second
derivative if we can avoid it.
00:46:33.840 --> 00:46:36.450
And we don't want to
decorate the graph
00:46:36.450 --> 00:46:38.550
and do all of these
elaborate, subtle,
00:46:38.550 --> 00:46:40.900
things which make the graph
look nicer and really,
00:46:40.900 --> 00:46:42.640
or aesthetically appropriate.
00:46:42.640 --> 00:46:46.700
But are totally unnecessary
to see whether the graph is
00:46:46.700 --> 00:46:54.290
up or down.
00:46:54.290 --> 00:46:57.030
So essentially,
this whole business
00:46:57.030 --> 00:47:00.030
is out, which is a good thing.
00:47:00.030 --> 00:47:03.640
And, unfortunately,
those early parts
00:47:03.640 --> 00:47:06.170
are the parts that
people tend to ignore.
00:47:06.170 --> 00:47:10.150
Which are typically,
often, very important.
00:47:10.150 --> 00:47:22.940
So let me first tell
you the main point here.
00:47:22.940 --> 00:47:32.760
So the key idea.
00:47:32.760 --> 00:47:39.650
Key to finding maximum.
00:47:39.650 --> 00:47:41.850
So the key point
is, we only need
00:47:41.850 --> 00:48:00.130
to look at critical points.
00:48:00.130 --> 00:48:01.980
Well, that's actually
what it seems
00:48:01.980 --> 00:48:05.490
like in many calculus classes.
00:48:05.490 --> 00:48:06.810
But that's not true.
00:48:06.810 --> 00:48:15.590
This is not the end
of the sentence.
00:48:15.590 --> 00:48:35.320
And, end points, and
points of discontinuity.
00:48:35.320 --> 00:48:37.980
So you must watch out for those.
00:48:37.980 --> 00:48:41.580
If you look at the example
that I just drew here,
00:48:41.580 --> 00:48:43.680
which is the one
that I carried out,
00:48:43.680 --> 00:48:50.120
you can see that there are
actually five extreme points
00:48:50.120 --> 00:48:51.250
on this picture.
00:48:51.250 --> 00:48:52.750
So let's switch.
00:48:52.750 --> 00:48:58.050
So we'll take a look.
00:48:58.050 --> 00:49:04.840
There are five places where
the max or the min might be.
00:49:04.840 --> 00:49:08.050
There's this important point.
00:49:08.050 --> 00:49:10.710
This is, as I say, the
scaffolding of the function.
00:49:10.710 --> 00:49:13.650
There's this point, there
down at minus infinity.
00:49:13.650 --> 00:49:18.370
There's this, there's
this, and there's this.
00:49:18.370 --> 00:49:23.620
Only one out of five
is a critical point.
00:49:23.620 --> 00:49:25.784
So there's more that you
have to pay attention to
00:49:25.784 --> 00:49:26.450
on the function.
00:49:26.450 --> 00:49:28.960
And you always have
to keep the schema,
00:49:28.960 --> 00:49:31.310
the picture of the function,
in the back of your head.
00:49:31.310 --> 00:49:33.760
Even though this may be
the most interesting point,
00:49:33.760 --> 00:49:36.520
and the one that you're
going to be looking at.
00:49:36.520 --> 00:49:41.180
So we'll do a few examples
of that next time.