WEBVTT
00:00:24.108 --> 00:00:26.110
PROFESSOR: Hi.
00:00:26.110 --> 00:00:28.725
Well, I hope you're ready
for second derivatives.
00:00:31.230 --> 00:00:34.560
We don't go higher than that
in many problems, but the
00:00:34.560 --> 00:00:36.400
second derivative is
an important--
00:00:36.400 --> 00:00:40.400
the derivative of the derivative
is an important
00:00:40.400 --> 00:00:45.180
thing to know, especially in
problems with maximum and
00:00:45.180 --> 00:00:49.200
minimum, which is the big
application of derivatives, to
00:00:49.200 --> 00:00:54.240
locate a maximum or a minimum,
and to decide which one it is.
00:00:54.240 --> 00:00:59.700
And I can tell you right away,
locating a maximum, minimum,
00:00:59.700 --> 00:01:02.690
is the first derivative's job.
00:01:02.690 --> 00:01:04.950
The first derivative is 0.
00:01:04.950 --> 00:01:08.710
If I have a maximum or a
minimum, and we'll have
00:01:08.710 --> 00:01:17.000
pictures, somewhere in the
middle of my function I'll
00:01:17.000 --> 00:01:20.220
recognize by derivative
equals 0.
00:01:20.220 --> 00:01:23.470
Slope equals 0, that the
function is leveling off,
00:01:23.470 --> 00:01:30.370
either bending down or bending
up, maximum or minimum.
00:01:30.370 --> 00:01:33.060
OK, and it's the second
derivative that
00:01:33.060 --> 00:01:34.840
tells me which it is.
00:01:34.840 --> 00:01:40.670
The second derivative tells me
the bending of the graph.
00:01:40.670 --> 00:01:47.870
OK, so we now will have
three generations.
00:01:47.870 --> 00:01:53.220
The big picture of calculus
started with two functions:
00:01:53.220 --> 00:01:54.930
the distance and the speed.
00:01:54.930 --> 00:02:01.720
And we discussed in detail the
connection between them.
00:02:01.720 --> 00:02:06.410
How to recover the speed if we
know the distance, take the
00:02:06.410 --> 00:02:09.030
derivative.
00:02:09.030 --> 00:02:13.590
Now comes the derivative of
the speed, which in that
00:02:13.590 --> 00:02:18.140
language, in the
distance-speed-time language,
00:02:18.140 --> 00:02:23.800
the second derivative is the
acceleration, the rate at
00:02:23.800 --> 00:02:26.760
which your speed is changing,
the rate at which you're
00:02:26.760 --> 00:02:30.510
speeding up or slowing down.
00:02:30.510 --> 00:02:33.320
And this is the way I
would write that.
00:02:33.320 --> 00:02:36.030
If the speed is the
first derivative--
00:02:36.030 --> 00:02:37.630
df dt--
00:02:37.630 --> 00:02:42.190
this is the way you write the
second derivative, and you say
00:02:42.190 --> 00:02:46.750
d second f dt squared.
00:02:46.750 --> 00:02:49.040
d second f dt squared.
00:02:49.040 --> 00:02:55.300
OK, so that's you could say the
physics example: distance,
00:02:55.300 --> 00:02:56.080
speed, acceleration.
00:02:56.080 --> 00:03:00.480
And I say physics because, of
course, acceleration is the a
00:03:00.480 --> 00:03:04.610
in Newton's Law f equals ma.
00:03:04.610 --> 00:03:10.400
For a graph, like these graphs
here, I won't especially use
00:03:10.400 --> 00:03:11.710
those physics words.
00:03:11.710 --> 00:03:13.860
I'll use graph words.
00:03:13.860 --> 00:03:16.850
So I would say function
one would be the
00:03:16.850 --> 00:03:20.280
height of the graph.
00:03:20.280 --> 00:03:25.690
And in this case, that height is
y equals x squared, so it's
00:03:25.690 --> 00:03:28.880
a simple parabola.
00:03:28.880 --> 00:03:31.020
Here would be the slope.
00:03:31.020 --> 00:03:35.660
I would use the word "slope"
for the second function.
00:03:35.660 --> 00:03:42.040
And the slope of y equals x
squared we know is 2x, so we
00:03:42.040 --> 00:03:44.300
see the slope increasing.
00:03:44.300 --> 00:03:49.100
And you see on this picture
the slope is increasing.
00:03:49.100 --> 00:03:53.340
As x increases, I'm going
up more steeply.
00:03:53.340 --> 00:03:55.670
Now, it's the second
derivative.
00:03:55.670 --> 00:03:57.540
And what shall I call that?
00:03:57.540 --> 00:03:58.390
Bending.
00:03:58.390 --> 00:04:01.930
Bending is the natural
word for the second
00:04:01.930 --> 00:04:04.320
derivative on a graph.
00:04:04.320 --> 00:04:07.310
And what do I--
00:04:07.310 --> 00:04:14.350
the derivative of 2x is 2, a
constant, a positive constant,
00:04:14.350 --> 00:04:18.730
and that positive constant tells
me that the slope is
00:04:18.730 --> 00:04:24.850
going upwards and that the
curve is bending upwards.
00:04:24.850 --> 00:04:30.240
So in this simple case,
we connect these three
00:04:30.240 --> 00:04:33.040
descriptions of our function.
00:04:33.040 --> 00:04:36.100
It's positive.
00:04:36.100 --> 00:04:38.490
It's slope is positive.
00:04:38.490 --> 00:04:40.860
And its second derivative--
00:04:40.860 --> 00:04:41.420
bending--
00:04:41.420 --> 00:04:42.980
is positive.
00:04:42.980 --> 00:04:47.020
And that gives us a function
that goes like that.
00:04:47.020 --> 00:04:53.790
Now, let me go to a different
function.
00:04:53.790 --> 00:05:02.720
Let me take a second example
now, an example where not
00:05:02.720 --> 00:05:03.970
everything is positive.
00:05:07.320 --> 00:05:09.290
But let's make it familiar.
00:05:09.290 --> 00:05:11.610
Take sine x.
00:05:11.610 --> 00:05:15.390
So sine x starts
out like that.
00:05:15.390 --> 00:05:22.660
So this is a graph of sine x up
to 90 degrees, pi over 2,
00:05:22.660 --> 00:05:25.680
so that's y equals sine x.
00:05:25.680 --> 00:05:31.580
OK, what do you think
about its slope?
00:05:31.580 --> 00:05:35.290
We know the derivative of sine
x, but before we write it
00:05:35.290 --> 00:05:38.240
down, look at the graph.
00:05:38.240 --> 00:05:40.550
The slope is positive, right?
00:05:40.550 --> 00:05:45.550
But the slope actually
starts out at 1.
00:05:45.550 --> 00:05:48.230
Better make it look a little
more realistic.
00:05:48.230 --> 00:05:50.600
That's a slope of 1 there.
00:05:50.600 --> 00:05:55.780
So the slope starts at 1 and the
slope drops to a slope of
00:05:55.780 --> 00:05:56.760
0 up there.
00:05:56.760 --> 00:05:58.660
So a slope of 1.
00:05:58.660 --> 00:06:00.860
I see here is a 1.
00:06:00.860 --> 00:06:04.170
Here I'm graphing y prime.
00:06:04.170 --> 00:06:09.370
dy dx I sometimes write as y
prime, just because it's
00:06:09.370 --> 00:06:12.780
shorter, and particularly, it'll
be shorter for a second
00:06:12.780 --> 00:06:13.620
derivative.
00:06:13.620 --> 00:06:23.380
So y prime, we know the
derivative of sine x is cos x,
00:06:23.380 --> 00:06:27.250
which is pretty neat actually,
that we start with a familiar
00:06:27.250 --> 00:06:33.400
function, and then we get its
twin, its other half.
00:06:33.400 --> 00:06:39.040
And the cosine is the slope of
the sine curve, and it starts
00:06:39.040 --> 00:06:43.860
at 1, a slope of 1, and it comes
down to 0, as we know
00:06:43.860 --> 00:06:45.110
the cosine does.
00:06:48.680 --> 00:06:52.100
So that's a graph
of the cosine.
00:06:52.100 --> 00:06:56.270
And now, of course, we have
three generations.
00:06:56.270 --> 00:06:59.620
I'm going to graph
y double prime.
00:06:59.620 --> 00:07:02.750
Let me put it up here. y double
prime, the second
00:07:02.750 --> 00:07:06.230
derivative, the derivative
of the cosine of x
00:07:06.230 --> 00:07:09.960
is minus sine x.
00:07:09.960 --> 00:07:11.850
OK, let's just--
00:07:11.850 --> 00:07:15.590
from the picture, what
am I seeing here?
00:07:15.590 --> 00:07:19.600
I'm seeing a slope of 0.
00:07:19.600 --> 00:07:22.960
I'm taking now the slope
of the slope.
00:07:22.960 --> 00:07:25.680
So here it starts at 0.
00:07:25.680 --> 00:07:28.990
The slope is downwards, so the
second derivative is going to
00:07:28.990 --> 00:07:30.220
be negative.
00:07:30.220 --> 00:07:33.570
Oh, and it is negative,
minus sign x.
00:07:33.570 --> 00:07:46.670
So the slope starts at 0 and
ends at minus 1 because that
00:07:46.670 --> 00:07:50.440
now comes down at a
negative slope.
00:07:50.440 --> 00:07:51.480
The slope is negative.
00:07:51.480 --> 00:07:59.140
I'm going downhill, and that's
a graph of the second
00:07:59.140 --> 00:08:01.330
derivative.
00:08:01.330 --> 00:08:04.950
And which way is our
function bending?
00:08:04.950 --> 00:08:07.360
It's bending down.
00:08:07.360 --> 00:08:10.890
As I go along, the slope
is dropping.
00:08:10.890 --> 00:08:14.440
And I see that in
the slope curve.
00:08:14.440 --> 00:08:15.590
It's falling.
00:08:15.590 --> 00:08:20.540
And I see it in the
bending curve
00:08:20.540 --> 00:08:25.530
because I'm below 0 here.
00:08:25.530 --> 00:08:36.640
This is bending down, where
that one was bending up.
00:08:39.690 --> 00:08:45.230
I could introduce the word
convex for something that
00:08:45.230 --> 00:08:48.960
bends upwards, and bending down,
I could introduce the
00:08:48.960 --> 00:08:56.010
word concave. But those
are just words.
00:08:56.010 --> 00:09:00.030
The graphs are telling us much
more than the words do.
00:09:00.030 --> 00:09:06.300
OK, so do you see that picture
bending down, but going up?
00:09:06.300 --> 00:09:12.750
So the slope is positive here,
but the second derivative, the
00:09:12.750 --> 00:09:14.290
slope is dropping.
00:09:14.290 --> 00:09:17.920
So the second derivative-- and
you have to pay attention to
00:09:17.920 --> 00:09:20.370
keep them straight.
00:09:20.370 --> 00:09:24.330
The second derivative is telling
us that the original
00:09:24.330 --> 00:09:25.640
one is bending down.
00:09:25.640 --> 00:09:33.030
OK, let me continue these graphs
just a little beyond 90
00:09:33.030 --> 00:09:38.090
degrees, pi over 2, because
you'll see something
00:09:38.090 --> 00:09:39.360
interesting.
00:09:39.360 --> 00:09:41.920
So what happens in the next
part of the graph?
00:09:41.920 --> 00:09:44.710
So this is going--
00:09:44.710 --> 00:09:46.140
the sine curve, of course,
00:09:46.140 --> 00:09:50.430
continues on its way downwards.
00:09:50.430 --> 00:09:55.250
So the slope is going negative,
as I know the cosine
00:09:55.250 --> 00:09:58.900
curve will do, as the cosine
curve will come like that.
00:09:58.900 --> 00:10:03.010
The slope down to minus
1, the slope--
00:10:03.010 --> 00:10:04.060
do you see here?
00:10:04.060 --> 00:10:11.320
The slope is negative, so on
this slope graph, I'm below 0.
00:10:11.320 --> 00:10:15.530
And the slope is 0.
00:10:15.530 --> 00:10:21.390
Let me put a little mark at
these points here, at these
00:10:21.390 --> 00:10:22.640
three points.
00:10:26.340 --> 00:10:28.940
Those are important points.
00:10:28.940 --> 00:10:35.250
In fact, that is a maximum,
of course.
00:10:35.250 --> 00:10:38.110
The sine curve hits
its maximum at 1.
00:10:41.600 --> 00:10:46.170
At that point when it hits its
maximum, what's its slope?
00:10:46.170 --> 00:10:52.780
When you hit a maximum, you're
not going up anymore.
00:10:52.780 --> 00:10:55.480
You haven't started down.
00:10:55.480 --> 00:10:58.320
The slope is 0 right there.
00:10:58.320 --> 00:10:59.900
What's the second derivative?
00:10:59.900 --> 00:11:02.260
What's the bending
at a maximum?
00:11:02.260 --> 00:11:05.550
The bending tells you that the
slope is going down, so the
00:11:05.550 --> 00:11:07.540
bending is negative.
00:11:07.540 --> 00:11:10.110
The bending is negative
at a maximum.
00:11:12.660 --> 00:11:14.620
Good.
00:11:14.620 --> 00:11:20.640
OK, now I'm going to continue
this sine curve for another 90
00:11:20.640 --> 00:11:25.910
degrees, the cosine curve, and
I'll continue the bending
00:11:25.910 --> 00:11:31.360
curve, so I have minus sine
x, which will go back up.
00:11:31.360 --> 00:11:34.130
OK, now what?
00:11:34.130 --> 00:11:35.380
Now what?
00:11:38.080 --> 00:11:41.420
And then, of course, it
would continue along.
00:11:41.420 --> 00:11:45.500
OK, there's something
interesting happening at 180
00:11:45.500 --> 00:11:47.990
degrees, at pi.
00:11:47.990 --> 00:11:50.030
Can I identify that point?
00:11:50.030 --> 00:11:52.640
So there's 180 degrees.
00:11:52.640 --> 00:11:54.150
Something's happening there.
00:11:54.150 --> 00:11:56.510
I don't see--
00:11:56.510 --> 00:11:58.890
I don't quite know how to say
what yet, but something's
00:11:58.890 --> 00:12:00.300
happening there.
00:12:00.300 --> 00:12:07.400
It's got to show up here, and
it has to show up here.
00:12:07.400 --> 00:12:13.450
So whatever is happening is
showing up by a point where y
00:12:13.450 --> 00:12:17.850
double prime, the second
derivative, is 0.
00:12:17.850 --> 00:12:23.060
That's my new little
observation, not as big a deal
00:12:23.060 --> 00:12:25.550
as maximum or minimum.
00:12:25.550 --> 00:12:26.990
This was a max here.
00:12:29.870 --> 00:12:34.320
And we identified it as a max
because the second derivative
00:12:34.320 --> 00:12:35.340
was negative.
00:12:35.340 --> 00:12:37.790
Now I'm interested
in this point.
00:12:37.790 --> 00:12:43.140
Can you see what's happening
at this point as far as
00:12:43.140 --> 00:12:45.500
bending goes?
00:12:45.500 --> 00:12:48.890
This curve is bending down.
00:12:48.890 --> 00:12:53.560
But when I continue, the
bending changes to up.
00:12:53.560 --> 00:12:58.640
This is a point where
the bending changes.
00:12:58.640 --> 00:13:03.260
The second derivative changes
sign, and we see it here.
00:13:03.260 --> 00:13:10.510
Up to this square point,
the bending is below 0.
00:13:10.510 --> 00:13:14.260
The bending is downwards
as I come to here.
00:13:14.260 --> 00:13:17.140
But then there's something
rather special that--
00:13:17.140 --> 00:13:20.160
you see, can I try to
blow that point up?
00:13:20.160 --> 00:13:23.540
Here the bending is down, and
there it turns to up, and
00:13:23.540 --> 00:13:26.360
right in there with the--
00:13:26.360 --> 00:13:29.140
this is called--
00:13:29.140 --> 00:13:32.110
so this is my final word
to introduce--
00:13:32.110 --> 00:13:35.310
inflection point.
00:13:35.310 --> 00:13:36.560
Don't ask me why.
00:13:40.920 --> 00:13:44.110
An inflection point is a
point where the second
00:13:44.110 --> 00:13:46.990
derivative is 0.
00:13:46.990 --> 00:13:49.050
And what does that mean?
00:13:49.050 --> 00:13:54.400
That means at that moment, it
stopped bending down, and it's
00:13:54.400 --> 00:13:57.000
going to start bending up.
00:13:57.000 --> 00:14:00.400
The second derivative is
passing through 0.
00:14:00.400 --> 00:14:02.660
The sign of bending
is changing.
00:14:02.660 --> 00:14:07.440
It's changing from concave
here to convex there.
00:14:07.440 --> 00:14:13.160
That's a significant
point on the graph.
00:14:13.160 --> 00:14:19.130
Not as big a thing as
the max or the min
00:14:19.130 --> 00:14:20.250
that we had over there.
00:14:20.250 --> 00:14:26.750
So let me draw one more example
and identify all these
00:14:26.750 --> 00:14:27.760
different points.
00:14:27.760 --> 00:14:30.790
OK, so here we go.
00:14:30.790 --> 00:14:35.190
I drew it ahead of time because
it's got a few loops,
00:14:35.190 --> 00:14:38.540
and I wanted to get
it in good form.
00:14:38.540 --> 00:14:40.350
OK, here it is.
00:14:40.350 --> 00:14:44.120
This is my function: x cubed
minus x squared.
00:14:44.120 --> 00:14:48.980
Well, before I look at the
picture, what would be the
00:14:48.980 --> 00:14:50.870
first calculus thing I do?
00:14:50.870 --> 00:14:52.660
I take the derivative.
00:14:52.660 --> 00:14:58.900
y prime is the derivative of
x cubed, is three x squared
00:14:58.900 --> 00:15:02.860
minus the derivative of x
squared, which is 2x.
00:15:02.860 --> 00:15:07.690
And now today, I take the
derivative of that.
00:15:07.690 --> 00:15:11.210
I take the second derivative,
y double prime.
00:15:11.210 --> 00:15:15.280
So the second derivative is
the derivative of this.
00:15:15.280 --> 00:15:18.880
x squared is going to give me
2x, and I have a 3, so it's
00:15:18.880 --> 00:15:21.070
all together 6x.
00:15:21.070 --> 00:15:27.730
And minus 2x, the slope of
that is minus 2, right?
00:15:27.730 --> 00:15:33.480
Cubic, quadratic, linear, and
if I cared about y triple
00:15:33.480 --> 00:15:36.750
prime, which I don't,
constant.
00:15:36.750 --> 00:15:39.640
And then the fourth derivatives
and all the rest
00:15:39.640 --> 00:15:41.810
would be 0 for this case.
00:15:41.810 --> 00:15:47.870
OK, now somehow, those
derivatives, those formulas
00:15:47.870 --> 00:15:52.820
for y, y prime, y double prime
should tell me details about
00:15:52.820 --> 00:15:54.950
this graph.
00:15:54.950 --> 00:15:58.040
And the first thing I'm
interested in and the most
00:15:58.040 --> 00:16:00.970
important thing is
max and min.
00:16:00.970 --> 00:16:04.940
So let me set y prime to be--
00:16:04.940 --> 00:16:09.070
which is 3x squared minus 2x.
00:16:09.070 --> 00:16:18.980
I'll set it to be 0 because I
want to look for max, or min.
00:16:18.980 --> 00:16:23.840
And I look for both at the same
time by setting y prime
00:16:23.840 --> 00:16:27.950
equals 0, and then I find out
which I've got by looking at y
00:16:27.950 --> 00:16:28.760
double prime.
00:16:28.760 --> 00:16:31.740
So let me set y prime to be 0.
00:16:31.740 --> 00:16:33.150
What are the solutions?
00:16:33.150 --> 00:16:39.530
Where are the points on the
curve where it's stationary?
00:16:39.530 --> 00:16:44.600
It's not climbing and
it's not dropping?
00:16:44.600 --> 00:16:48.020
Well, I see them on
the curve here.
00:16:48.020 --> 00:16:53.240
That is a point where
the slope is 0.
00:16:53.240 --> 00:16:55.560
And I see one down here.
00:16:55.560 --> 00:17:01.220
There is a point where the slope
is 0, but I can find
00:17:01.220 --> 00:17:02.580
them with algebra.
00:17:02.580 --> 00:17:08.670
I solve 3x squared equals
to 2x, and I see
00:17:08.670 --> 00:17:10.359
it's a quadratic equation.
00:17:10.359 --> 00:17:12.130
I expect to find two roots.
00:17:12.130 --> 00:17:17.480
One of them is x equals 0, and
the other one is what?
00:17:17.480 --> 00:17:24.990
If I cancel those x's to find
a non-zero, canceling those
00:17:24.990 --> 00:17:31.340
x's leaves me with 3x equals
2 or x equals 2/3.
00:17:31.340 --> 00:17:35.570
Yeah, and that's what
our graph shows.
00:17:35.570 --> 00:17:43.480
OK, now we can see on the
graph which is a max
00:17:43.480 --> 00:17:45.750
and which is a min.
00:17:45.750 --> 00:17:48.030
And by the way, let
me just notice, of
00:17:48.030 --> 00:17:49.475
course, this is the max.
00:17:52.530 --> 00:17:55.790
But let me just notice that
it's what I would
00:17:55.790 --> 00:17:59.150
call a local maximum.
00:17:59.150 --> 00:18:01.870
It's not the absolute top of
the function because the
00:18:01.870 --> 00:18:05.330
function later on is climbing
off to infinity.
00:18:05.330 --> 00:18:12.610
This would be way a maximum
in its neighborhood, so a
00:18:12.610 --> 00:18:16.910
maximum, and it's only
a local max.
00:18:16.910 --> 00:18:22.890
And what do I expect to see at
a maximum at x equals 0?
00:18:22.890 --> 00:18:30.220
I expect to see the slope 0 at
x equals 0, which it is.
00:18:30.220 --> 00:18:31.510
Check.
00:18:31.510 --> 00:18:37.420
And at a maximum, I need to know
the second derivative.
00:18:37.420 --> 00:18:39.460
OK, here's my formula.
00:18:39.460 --> 00:18:46.720
At x equals 0, I see y double
prime if x is 0 is minus 2.
00:18:46.720 --> 00:18:47.970
Good.
00:18:49.540 --> 00:18:55.990
Negative second derivative tells
me I'm bending down, as
00:18:55.990 --> 00:19:01.850
the graph confirms, and the
place where the slope is 0 is
00:19:01.850 --> 00:19:04.630
a maximum and not a minimum.
00:19:04.630 --> 00:19:06.990
What about the other one?
00:19:06.990 --> 00:19:08.460
What about at x equals 2/3?
00:19:12.320 --> 00:19:16.570
At that point, y double prime,
looking at my formula here for
00:19:16.570 --> 00:19:20.730
y double prime, is what?
00:19:20.730 --> 00:19:24.980
6 times 2/3 is 4 minus
2 is plus 2.
00:19:24.980 --> 00:19:26.680
4 minus 2 is 2.
00:19:26.680 --> 00:19:28.590
So this will be--
00:19:28.590 --> 00:19:33.120
this is positive, so I'm
expecting a min.
00:19:33.120 --> 00:19:36.510
At x equals 2/3, I'm
expecting a min.
00:19:36.510 --> 00:19:39.070
And, of course, it is.
00:19:39.070 --> 00:19:42.530
And again, it's only
a local minimum.
00:19:42.530 --> 00:19:45.490
The derivative can only tell
you what's happening very,
00:19:45.490 --> 00:19:48.070
very close to that point.
00:19:48.070 --> 00:19:51.640
The derivative doesn't know that
over here the function is
00:19:51.640 --> 00:19:53.670
going further down.
00:19:53.670 --> 00:19:58.360
So this is a min, and
again, a local min.
00:19:58.360 --> 00:20:02.370
OK, those are maximum
and minimum
00:20:02.370 --> 00:20:04.490
when we know the function.
00:20:04.490 --> 00:20:08.255
Oh yeah, I better do the
inflection point.
00:20:08.255 --> 00:20:11.140
Do you remember what the
inflection point is?
00:20:11.140 --> 00:20:16.310
The inflection point is when
the bending changes from--
00:20:16.310 --> 00:20:18.965
up to here I see that
bending down.
00:20:21.540 --> 00:20:24.860
From here, I see
it bending up.
00:20:24.860 --> 00:20:30.260
So I will not be surprised if
that's the point where the
00:20:30.260 --> 00:20:35.550
bending is changing, and 1/3
is the inflection point.
00:20:35.550 --> 00:20:37.860
And now how do we find
an inflection point?
00:20:37.860 --> 00:20:39.910
How do we identify this point?
00:20:39.910 --> 00:20:43.320
Well, y double prime
was negative.
00:20:43.320 --> 00:20:45.430
y double prime was positive.
00:20:45.430 --> 00:20:51.300
At that point, y double
prime is 0.
00:20:51.300 --> 00:20:53.790
This is an inflection point.
00:20:53.790 --> 00:20:55.580
And it is.
00:20:55.580 --> 00:21:03.800
At x equals to 1/3, I
do have 6 times 1/3.
00:21:03.800 --> 00:21:06.640
2 subtract 2, I have 0.
00:21:06.640 --> 00:21:10.870
So that is truly an
inflection point.
00:21:10.870 --> 00:21:13.900
And now I know all
the essential
00:21:13.900 --> 00:21:16.120
points about the curve.
00:21:16.120 --> 00:21:19.240
And these are the quantities--
00:21:19.240 --> 00:21:20.460
oh!
00:21:20.460 --> 00:21:24.860
Say you're an economist. You're
looking now at the
00:21:24.860 --> 00:21:30.070
statistics for the US economy
or the world economy.
00:21:30.070 --> 00:21:34.300
OK, I suppose we're in a--
00:21:34.300 --> 00:21:41.330
we had a local maximum there,
a happy time a little while
00:21:41.330 --> 00:21:46.870
ago, but it went downhill,
right?
00:21:46.870 --> 00:21:53.400
If y is, say, the gross product
for the world or gross
00:21:53.400 --> 00:21:55.500
national product,
it started down.
00:21:58.120 --> 00:22:03.110
The slope of that curve
was negative.
00:22:03.110 --> 00:22:04.740
The bending was even negative.
00:22:04.740 --> 00:22:08.350
It was going down faster
all the time.
00:22:08.350 --> 00:22:14.540
Now, at a certain moment, the
economy kept going down, but
00:22:14.540 --> 00:22:16.630
you could see some
sign of hope.
00:22:16.630 --> 00:22:18.320
And what was the sign of hope?
00:22:18.320 --> 00:22:23.290
It was the fact that it
started bending up.
00:22:23.290 --> 00:22:28.470
And probably that's where we are
as I'm making this video.
00:22:28.470 --> 00:22:33.250
I suspect we're still going
down, but we're bending up.
00:22:33.250 --> 00:22:40.090
And at some point, hopefully
tomorrow, we'll hit minimum
00:22:40.090 --> 00:22:42.330
and start really up.
00:22:42.330 --> 00:22:43.330
So I don't know.
00:22:43.330 --> 00:22:45.130
I would guess we're somewhere
in there, and
00:22:45.130 --> 00:22:46.880
I don't know where.
00:22:46.880 --> 00:22:50.830
If I knew where, mathematics
would be even more useful than
00:22:50.830 --> 00:22:53.620
it is, which would
be hard to do.
00:22:53.620 --> 00:22:58.690
OK, so that's an example
of how the second
00:22:58.690 --> 00:23:00.390
derivative comes in.
00:23:00.390 --> 00:23:09.600
Now, I started by giving this
lecture the title Max and Min
00:23:09.600 --> 00:23:14.900
and saying those are the biggest
applications of the
00:23:14.900 --> 00:23:16.070
derivative.
00:23:16.070 --> 00:23:19.880
Set the derivative
to 0 and solve.
00:23:19.880 --> 00:23:22.390
Locate maximum points,
minimum points.
00:23:22.390 --> 00:23:27.920
That's what calculus is most--
00:23:27.920 --> 00:23:33.620
many of the word problems, most
of the ones I see in use,
00:23:33.620 --> 00:23:37.790
involve derivative equals 0.
00:23:37.790 --> 00:23:41.895
OK, so let me take a
particular example.
00:23:44.730 --> 00:23:49.660
So these were graphs, simple
functions which I chose: sine
00:23:49.660 --> 00:23:52.920
x, x squared, x cubed
minus x squared.
00:23:52.920 --> 00:23:58.520
Now let me tell you the problem
because this is how
00:23:58.520 --> 00:24:00.210
math really comes.
00:24:00.210 --> 00:24:03.290
Let me tell you the problem, and
let's create the function.
00:24:03.290 --> 00:24:05.830
OK, so much it's the problem
I faced this
00:24:05.830 --> 00:24:09.080
morning and every morning.
00:24:09.080 --> 00:24:09.870
I live here.
00:24:09.870 --> 00:24:13.230
So OK, so here's home.
00:24:13.230 --> 00:24:20.500
And there is a-- the Mass Pike
is the fast road to MIT.
00:24:20.500 --> 00:24:26.360
So let me put in the Mass Pike
here, and let's say that's
00:24:26.360 --> 00:24:32.220
MIT, and I'm trying to get there
as fast as possible.
00:24:32.220 --> 00:24:40.010
OK, so for part of the time, I'm
going to have to drive on
00:24:40.010 --> 00:24:40.720
city streets.
00:24:40.720 --> 00:24:43.690
I do have to drive on city
streets, and then I get to go
00:24:43.690 --> 00:24:49.440
on the Mass Pike, which is,
let's say, twice as fast. The
00:24:49.440 --> 00:24:56.680
question is should I go directly
over to the fast road
00:24:56.680 --> 00:24:59.960
and then take off?
00:24:59.960 --> 00:25:01.470
Let's take off on
a good morning.
00:25:01.470 --> 00:25:04.540
The Mass Pike could be twice
as slow, but let's assume
00:25:04.540 --> 00:25:07.070
twice as fast. Should
I go straight over?
00:25:07.070 --> 00:25:09.440
Probably not.
00:25:09.440 --> 00:25:11.560
That's not the best way.
00:25:11.560 --> 00:25:17.050
I should probably pick up the
Mass Pike on some road.
00:25:17.050 --> 00:25:23.010
I could go directly to MIT on
the city streets at the slow
00:25:23.010 --> 00:25:29.150
rate, say 30 miles an hour or 30
kilometers an hour and 60,
00:25:29.150 --> 00:25:37.890
so speeds 30 and 60
as my speeds.
00:25:37.890 --> 00:25:44.550
OK, so now I should have
put in some measure.
00:25:44.550 --> 00:25:50.350
Let's call that distance
a, whatever it is.
00:25:50.350 --> 00:25:52.950
Maybe it's about three miles.
00:25:52.950 --> 00:25:54.420
And let me call--
00:25:54.420 --> 00:25:56.170
so that's the direct distance.
00:25:56.170 --> 00:25:59.650
If I just went direct to the
turnpike, I would go a
00:25:59.650 --> 00:26:02.380
distance a at 30 miles
an hour, and then
00:26:02.380 --> 00:26:03.700
I would go a distance--
00:26:03.700 --> 00:26:06.120
shall I call that b?--
00:26:06.120 --> 00:26:07.660
at 60.
00:26:07.660 --> 00:26:10.290
So that's one possibility.
00:26:10.290 --> 00:26:13.110
But I think it's not the best.
00:26:13.110 --> 00:26:15.410
I think better to--
00:26:15.410 --> 00:26:17.210
and you know better than me.
00:26:17.210 --> 00:26:19.380
I think I should probably
angle over
00:26:19.380 --> 00:26:23.170
here and pick up this--
00:26:23.170 --> 00:26:26.000
my question is where should
I join the Mass Pike.
00:26:26.000 --> 00:26:28.270
And let's--
00:26:28.270 --> 00:26:31.480
so we get a calculus problem,
let's model it.
00:26:31.480 --> 00:26:35.730
Suppose that I can join it
anywhere I like, not just at a
00:26:35.730 --> 00:26:39.930
couple of entrances.
00:26:39.930 --> 00:26:41.010
Anywhere.
00:26:41.010 --> 00:26:43.790
And the question is where?
00:26:43.790 --> 00:26:48.770
So calculus deals with the
continuous choice of x.
00:26:48.770 --> 00:26:52.180
So that is the unknown.
00:26:52.180 --> 00:26:54.830
I could take that as
the unknown x.
00:26:54.830 --> 00:26:57.510
That was a key step, of course,
deciding what should
00:26:57.510 --> 00:26:58.510
be the unknown.
00:26:58.510 --> 00:27:02.630
I could also have taken this
angle as an unknown, and that
00:27:02.630 --> 00:27:05.070
would be quite neat, too.
00:27:05.070 --> 00:27:07.530
But let me take that x.
00:27:07.530 --> 00:27:12.500
So this distance is
then b minus x.
00:27:12.500 --> 00:27:16.590
So that's what I travel
on the Mass Pike,
00:27:16.590 --> 00:27:20.340
so my time to minimize.
00:27:23.930 --> 00:27:26.480
I'm trying to minimize
my time.
00:27:26.480 --> 00:27:35.240
OK, so on this Mass Pike when
I travel at 60, I have
00:27:35.240 --> 00:27:40.770
distance divided by 60
is the time, right?
00:27:40.770 --> 00:27:42.860
Am I remembering correctly?
00:27:42.860 --> 00:27:44.310
Let's just remember.
00:27:44.310 --> 00:27:48.390
Distance is speed times time.
00:27:48.390 --> 00:27:50.370
That's the one we know.
00:27:50.370 --> 00:27:57.180
And then if I divide by the
speed, the time is the
00:27:57.180 --> 00:28:00.230
distance divided by the speed,
the distance divided by the
00:28:00.230 --> 00:28:02.470
speed on the pike.
00:28:02.470 --> 00:28:06.750
And now I have the distance
on the city streets.
00:28:06.750 --> 00:28:13.930
OK, so that speed is
going to be 30.
00:28:13.930 --> 00:28:17.700
So the time is going to be a bit
longer for the distance,
00:28:17.700 --> 00:28:19.310
and what is that distance?
00:28:19.310 --> 00:28:21.540
OK, that was a.
00:28:21.540 --> 00:28:22.790
This was x.
00:28:25.330 --> 00:28:30.390
Pythagoras is the great leveler
of mathematics.
00:28:33.580 --> 00:28:37.400
That's the distance on
the city streets.
00:28:37.400 --> 00:28:43.430
And now what do I do?
00:28:43.430 --> 00:28:45.440
I've got an expression
for the time.
00:28:45.440 --> 00:28:49.770
This is the quantity I'm
trying to minimize.
00:28:49.770 --> 00:28:53.820
I minimize it by taking its
derivative and set the
00:28:53.820 --> 00:28:55.670
derivative to 0.
00:28:55.670 --> 00:28:58.150
Take the derivative and set
the derivative to 0.
00:28:58.150 --> 00:29:01.980
So now this is where I use
the formulas of calculus.
00:29:01.980 --> 00:29:05.220
So the derivative, now I'm ready
to write the derivative,
00:29:05.220 --> 00:29:06.730
and I'll set it to 0.
00:29:06.730 --> 00:29:10.140
So the derivative of that, b is
a constant, so I have minus
00:29:10.140 --> 00:29:13.870
1/60; is that OK?
00:29:13.870 --> 00:29:17.220
Plus whatever the derivative
of this is.
00:29:17.220 --> 00:29:20.390
Well, I have 1/30.
00:29:20.390 --> 00:29:22.080
I always take the
constant first.
00:29:22.080 --> 00:29:24.980
Now I have to deal with
that expression.
00:29:24.980 --> 00:29:28.720
That is some quantity
square root.
00:29:28.720 --> 00:29:34.420
The square root is the 1/2
power, so I have 1/2 times
00:29:34.420 --> 00:29:37.600
this quantity to one
lower power.
00:29:37.600 --> 00:29:39.790
That's the minus 1/2 power.
00:29:39.790 --> 00:29:44.770
That means that I still have a
square root, but now it's a
00:29:44.770 --> 00:29:46.700
minus 1/2 power.
00:29:46.700 --> 00:29:48.560
It's down here.
00:29:48.560 --> 00:29:58.490
And then the chain rule says
don't forget the derivative of
00:29:58.490 --> 00:30:00.180
what's inside, which is 2x.
00:30:03.430 --> 00:30:08.050
OK, depending on what order
you've seen these videos and
00:30:08.050 --> 00:30:13.360
read text, you know the chain
rule, or you see it now.
00:30:13.360 --> 00:30:17.940
It's a very, very valuable rule
to find derivatives as
00:30:17.940 --> 00:30:19.910
the function gets complicated.
00:30:19.910 --> 00:30:23.460
And the thing to remember,
there will be a proper
00:30:23.460 --> 00:30:24.840
discussion of the chain rule.
00:30:24.840 --> 00:30:26.120
It's so important.
00:30:26.120 --> 00:30:29.800
But you're seeing it here that
the thing to remember is take
00:30:29.800 --> 00:30:33.780
also the derivative of what's
inside the a squared plus x
00:30:33.780 --> 00:30:36.850
squared, and the derivative of
the x squared is the 2x.
00:30:36.850 --> 00:30:39.120
OK, and that I have
to set to 0.
00:30:39.120 --> 00:30:43.180
And, of course, I'm going
to cancel the 2's, and
00:30:43.180 --> 00:30:44.250
I'll set it to 0.
00:30:44.250 --> 00:30:45.990
What does that mean
"set to zero"?
00:30:45.990 --> 00:30:47.300
Here's something minus.
00:30:47.300 --> 00:30:48.920
Here's something plus.
00:30:48.920 --> 00:30:51.450
I guess what I really want
is to make them equal.
00:30:55.120 --> 00:31:03.570
When the 1/60 equals this
messier expression, at that
00:31:03.570 --> 00:31:07.850
point the minus term cancels
the plus term.
00:31:07.850 --> 00:31:11.640
I get 0 for the derivative,
so I'm looking for
00:31:11.640 --> 00:31:14.450
derivative equals 0.
00:31:14.450 --> 00:31:17.000
That's my equation now.
00:31:17.000 --> 00:31:19.250
OK, now I just have
to solve it.
00:31:19.250 --> 00:31:22.260
All right, let's see.
00:31:22.260 --> 00:31:24.810
If I wanted to solve that,
I would probably multiply
00:31:24.810 --> 00:31:27.820
through by 60.
00:31:27.820 --> 00:31:29.750
Can I do this?
00:31:29.750 --> 00:31:32.600
I'll multiply both
sides by 60.
00:31:32.600 --> 00:31:35.670
That will cancel the 30 and
leave an extra 2, so
00:31:35.670 --> 00:31:38.910
I'll have a 2x here.
00:31:38.910 --> 00:31:42.600
And let me multiply also by
this miserable square root
00:31:42.600 --> 00:31:45.565
that's in the denominator
to get it up there.
00:31:51.180 --> 00:31:54.160
I think that's what I've got.
00:31:54.160 --> 00:31:57.020
That's the same equation as
this one, just simplified.
00:31:57.020 --> 00:31:59.010
Multiply through by 60.
00:31:59.010 --> 00:32:01.370
Multiply through by square
root of a squared plus x
00:32:01.370 --> 00:32:04.200
squared, and it's
looking good.
00:32:04.200 --> 00:32:07.070
All right, how am I going
to solve that?
00:32:07.070 --> 00:32:11.240
Well, the only mess up
is the square root.
00:32:11.240 --> 00:32:14.480
Get rid of that by squaring
both sides.
00:32:14.480 --> 00:32:18.690
So now I square both sides,
and I get a squared plus x
00:32:18.690 --> 00:32:23.440
squared, and the square
of 2x is 4x squared.
00:32:23.440 --> 00:32:28.855
All right, now I have an
equation that's way better.
00:32:32.550 --> 00:32:36.000
In fact, even better
if I subtract x
00:32:36.000 --> 00:32:37.080
squared from both sides.
00:32:37.080 --> 00:32:39.850
My equation is telling
me that a squared
00:32:39.850 --> 00:32:42.710
should be 3x squared.
00:32:42.710 --> 00:32:46.060
In other words, this
good x is--
00:32:46.060 --> 00:32:51.540
now I'm ready to take the square
root and find x itself.
00:32:51.540 --> 00:32:53.310
So put the 3 here.
00:32:53.310 --> 00:32:54.830
Take the square root.
00:32:54.830 --> 00:32:58.735
I'm getting a over the
square root of 3.
00:33:02.470 --> 00:33:08.920
So there is a word problem, a
minimum problem, where we had
00:33:08.920 --> 00:33:14.170
to create the function to
minimize, which was the time,
00:33:14.170 --> 00:33:18.810
trying to get to work as
quickly as possible.
00:33:18.810 --> 00:33:24.190
After naming the key quantity x,
then taking the derivative,
00:33:24.190 --> 00:33:29.150
then simplifying, that's where
the little work of calculus
00:33:29.150 --> 00:33:32.660
comes in, in the end getting
something nice, solving it,
00:33:32.660 --> 00:33:34.830
and getting the answer a
over square root of 3.
00:33:34.830 --> 00:33:40.120
So we now know what to do
driving in if there's an
00:33:40.120 --> 00:33:42.000
entrance where we
want to get it.
00:33:42.000 --> 00:33:47.100
And actually, it is a
beautiful answer.
00:33:47.100 --> 00:33:50.390
If this is a over the square
root of 3, this will turn out
00:33:50.390 --> 00:33:55.980
to be 30 degrees, pi over 6--
00:33:55.980 --> 00:33:57.640
I think.
00:33:57.640 --> 00:34:00.140
Yeah, I think that's right.
00:34:00.140 --> 00:34:03.260
So that's the conclusion
from calculus.
00:34:03.260 --> 00:34:04.990
Drive at a 30-degree angle.
00:34:04.990 --> 00:34:07.290
Hope that there's a road
going that way--
00:34:07.290 --> 00:34:09.000
sorry about that point--
00:34:09.000 --> 00:34:12.159
and join the turnpike.
00:34:12.159 --> 00:34:15.830
And probably the reason
for that nice
00:34:15.830 --> 00:34:19.239
answer, 30 degrees, came--
00:34:19.239 --> 00:34:23.860
I can't help but imagine that
because I chose 30 and 60
00:34:23.860 --> 00:34:30.060
here, a ratio of 1:2, and then
somehow the fact that the sine
00:34:30.060 --> 00:34:34.110
of 30 degrees is 1/2,
those two facts
00:34:34.110 --> 00:34:35.560
have got to be connected.
00:34:35.560 --> 00:34:39.860
So I change these 30 and 60
numbers, I'll change my
00:34:39.860 --> 00:34:43.260
answer, but basically, the
picture won't change much.
00:34:43.260 --> 00:34:51.949
And there's another little
point to make to really
00:34:51.949 --> 00:34:54.960
complete this problem.
00:34:54.960 --> 00:34:58.830
It could have happened that the
distance on the turnpike
00:34:58.830 --> 00:35:04.140
was very small and that
this was a dumb move.
00:35:04.140 --> 00:35:06.550
That 30-degree angle could
be overshooting
00:35:06.550 --> 00:35:11.070
MIT if MIT was there.
00:35:11.070 --> 00:35:16.310
So that's a case in which the
minimum time didn't happen
00:35:16.310 --> 00:35:21.210
where the derivative
bottomed out.
00:35:21.210 --> 00:35:27.701
If MIT was here, the good idea
would be go straight for it.
00:35:31.580 --> 00:35:36.180
Yeah, the extra part on the
turn-- you wouldn't drive on
00:35:36.180 --> 00:35:38.090
the turnpike at all.
00:35:38.090 --> 00:35:41.990
And that's a signal that somehow
in the graph, which I
00:35:41.990 --> 00:35:48.910
didn't graph this function, but
if I did, then this stuff
00:35:48.910 --> 00:35:52.580
would be locating the minimum
of the graph.
00:35:52.580 --> 00:35:57.000
But this extra example where you
go straight for MIT would
00:35:57.000 --> 00:36:00.110
be a case in which the minimum
is at the end.
00:36:00.110 --> 00:36:02.650
And, of course, that
could happen.
00:36:02.650 --> 00:36:06.520
You could have a graph that just
goes down, and then it
00:36:06.520 --> 00:36:09.380
ends, so the minimum is there.
00:36:09.380 --> 00:36:11.660
Even though the graph looks like
it's still going down,
00:36:11.660 --> 00:36:13.290
the graph ended.
00:36:13.290 --> 00:36:14.300
What can you do?
00:36:14.300 --> 00:36:16.500
That's the best point
there is.
00:36:16.500 --> 00:36:21.840
OK, so that is a--
00:36:21.840 --> 00:36:26.910
can I recap this lecture
coming first over here?
00:36:26.910 --> 00:36:30.870
So the lecture is about maximum
and minimum, and we
00:36:30.870 --> 00:36:35.310
learned which it is by the
second derivative.
00:36:35.310 --> 00:36:37.490
So then we had examples.
00:36:37.490 --> 00:36:40.980
There was an example of a
minimum when the second
00:36:40.980 --> 00:36:43.030
derivative was positive.
00:36:43.030 --> 00:36:47.110
Here was an example of a local
maximum when the second
00:36:47.110 --> 00:36:50.230
derivative was negative.
00:36:50.230 --> 00:36:55.240
Here with the sine and cosine,
those are nice examples.
00:36:55.240 --> 00:36:58.810
And it takes some patience
to go through them.
00:36:58.810 --> 00:37:03.750
I suggest you take another
simple function, like start
00:37:03.750 --> 00:37:06.100
with cosine x.
00:37:06.100 --> 00:37:07.290
Find its maximum.
00:37:07.290 --> 00:37:08.330
Find its minimum.
00:37:08.330 --> 00:37:11.700
Find its inflection points so
the inflection points are
00:37:11.700 --> 00:37:17.750
where the bending is 0 because
it's changing from bending one
00:37:17.750 --> 00:37:19.380
way to bending the other way.
00:37:19.380 --> 00:37:23.080
We didn't need an inflection
test--
00:37:23.080 --> 00:37:27.070
so actually, I didn't complete
the lecture, because I didn't
00:37:27.070 --> 00:37:30.140
compute the second derivative
and show that
00:37:30.140 --> 00:37:33.190
this was truly a minimum.
00:37:33.190 --> 00:37:34.270
I could have done that.
00:37:34.270 --> 00:37:37.040
I would have had to take the
derivative of this, which
00:37:37.040 --> 00:37:41.900
would be one level messier,
and look at its sign.
00:37:41.900 --> 00:37:43.610
I wouldn't have to
set it to 0.
00:37:43.610 --> 00:37:47.310
I would be looking at the sign
of the second derivative.
00:37:47.310 --> 00:37:52.910
And in this problem, it would
be safely come out positive
00:37:52.910 --> 00:37:57.210
sign, meaning bending upwards,
meaning that this point I've
00:37:57.210 --> 00:38:02.540
identified by all these steps
was truly the minimum time,
00:38:02.540 --> 00:38:04.230
not a maximum.
00:38:04.230 --> 00:38:10.120
OK, that's a big part of
important calculus
00:38:10.120 --> 00:38:11.670
applications.
00:38:11.670 --> 00:38:12.870
Thanks.
00:38:12.870 --> 00:38:14.650
NARRATOR: This has been
a production of MIT
00:38:14.650 --> 00:38:17.040
OpenCourseWare and
Gilbert Strang.
00:38:17.040 --> 00:38:19.310
Funding for this video was
provided by the Lord
00:38:19.310 --> 00:38:20.530
Foundation.
00:38:20.530 --> 00:38:23.660
To help OCW continue to provide
free and open access
00:38:23.660 --> 00:38:26.740
to MIT courses, please
make a donation at
00:38:26.740 --> 00:38:28.300
ocw.mit.edu/donate.