WEBVTT
00:00:00.000 --> 00:00:08.700
PROFESSOR: Welcome
back to recitation.
00:00:08.700 --> 00:00:10.620
Today what we're
going to do is use
00:00:10.620 --> 00:00:12.710
what we know about first
and second derivatives
00:00:12.710 --> 00:00:15.040
and what we know about
functions from way
00:00:15.040 --> 00:00:18.770
back in algebra and
precalculus, to sketch a curve.
00:00:18.770 --> 00:00:21.990
So I want you to sketch
the curve y equals x over 1
00:00:21.990 --> 00:00:23.099
plus x squared.
00:00:23.099 --> 00:00:24.890
Doesn't have to be
perfect, but try and use
00:00:24.890 --> 00:00:26.944
what you know about
these derivatives,
00:00:26.944 --> 00:00:28.860
first and second derivatives
of this function,
00:00:28.860 --> 00:00:32.372
and what you've talked
about in the lecture to get
00:00:32.372 --> 00:00:33.580
a pretty good sketch of this.
00:00:33.580 --> 00:00:35.120
I'll give you a little
time to work on it
00:00:35.120 --> 00:00:37.300
and then I'll be back and
I'll work on it for you.
00:00:46.070 --> 00:00:47.060
Welcome back.
00:00:47.060 --> 00:00:49.540
So hopefully you feel good
about the sketch you've drawn.
00:00:49.540 --> 00:00:52.050
But just to check everything,
we can go through it together.
00:00:52.050 --> 00:00:54.341
And what I'm going to do,
just to keep track of things,
00:00:54.341 --> 00:00:56.770
is I'm going to put
an axis in this region
00:00:56.770 --> 00:00:59.820
and then I'm going to do all
my work sort of off to the side
00:00:59.820 --> 00:01:00.790
and come back slowly.
00:01:00.790 --> 00:01:03.170
So we'll try and keep track
of everything that way.
00:01:03.170 --> 00:01:06.540
So before I do
anything else I'm just
00:01:06.540 --> 00:01:10.740
going to draw myself
a nice axis here.
00:01:10.740 --> 00:01:20.110
And I'll give myself even a
little bit of-- oops, that's
00:01:20.110 --> 00:01:23.730
maybe a little off, but-- so
we'll assume every hash mark
00:01:23.730 --> 00:01:24.930
is one unit.
00:01:24.930 --> 00:01:28.610
I'll just put a 1 there so
we know every hash mark here
00:01:28.610 --> 00:01:30.760
is going to represent one unit.
00:01:30.760 --> 00:01:32.810
And I won't write
the rest of them.
00:01:32.810 --> 00:01:35.017
Now one of the things
you always do first,
00:01:35.017 --> 00:01:36.850
is you want to make
sure that you understand
00:01:36.850 --> 00:01:38.454
where the function is defined.
00:01:38.454 --> 00:01:40.620
So we have to check right
away, are there any values
00:01:40.620 --> 00:01:43.030
of x for which this
function is not defined?
00:01:43.030 --> 00:01:45.070
Well, how can that happen?
00:01:45.070 --> 00:01:48.100
If it were a logarithm or if
it were a square root function
00:01:48.100 --> 00:01:49.665
we would have
problems in the domain
00:01:49.665 --> 00:01:51.290
where would have to
check and make sure
00:01:51.290 --> 00:01:53.690
that the input was positive.
00:01:53.690 --> 00:01:55.780
In this case, because we
have a rational function,
00:01:55.780 --> 00:01:59.490
we have to make sure that
the denominator is never
00:01:59.490 --> 00:02:00.740
equal to 0.
00:02:00.740 --> 00:02:02.190
But if you notice,
the denominator
00:02:02.190 --> 00:02:03.249
is 1 plus x squared.
00:02:03.249 --> 00:02:05.415
Well, x squared is always
bigger than or equal to 0,
00:02:05.415 --> 00:02:07.300
and once I add 1,
I'm in the clear.
00:02:07.300 --> 00:02:09.650
I'm always positive
in the denominator.
00:02:09.650 --> 00:02:11.670
So the denominator
is always positive,
00:02:11.670 --> 00:02:14.392
so I don't have to put
any vertical asymptotes.
00:02:14.392 --> 00:02:16.350
Some other things we
think about before we even
00:02:16.350 --> 00:02:18.980
start taking
derivatives, or anything
00:02:18.980 --> 00:02:21.660
I can find out about this
function, like end behavior.
00:02:21.660 --> 00:02:23.530
When we say end
behavior we mean,
00:02:23.530 --> 00:02:26.100
what happens as x goes
to positive infinity
00:02:26.100 --> 00:02:28.320
and as x goes to
negative infinity?
00:02:28.320 --> 00:02:30.120
And from what
you've seen before,
00:02:30.120 --> 00:02:32.630
as x goes to positive
infinity, because this
00:02:32.630 --> 00:02:35.440
is a rational function,
the higher power
00:02:35.440 --> 00:02:36.590
is going to win out.
00:02:36.590 --> 00:02:38.250
The higher power
always wins out.
00:02:38.250 --> 00:02:41.360
So the higher power here
is in the denominator,
00:02:41.360 --> 00:02:43.140
so as x goes to
positive infinity
00:02:43.140 --> 00:02:46.110
this whole expression
is going to head to 0.
00:02:46.110 --> 00:02:48.610
For large values of x the
x squared is significantly
00:02:48.610 --> 00:02:49.930
bigger than the x.
00:02:49.930 --> 00:02:52.890
And so the denominator
is significantly bigger
00:02:52.890 --> 00:02:54.010
than the numerator.
00:02:54.010 --> 00:02:55.850
That's how we can
think about this.
00:02:55.850 --> 00:02:58.210
So when x goes to
plus or minus infinity
00:02:58.210 --> 00:03:00.520
we know that our function
is going to be headed to 0,
00:03:00.520 --> 00:03:02.740
so it has a
horizontal asymptote.
00:03:02.740 --> 00:03:03.240
OK.
00:03:03.240 --> 00:03:05.990
And then another thing we
would-- we should notice
00:03:05.990 --> 00:03:07.200
is the sign of the graph.
00:03:07.200 --> 00:03:10.020
Notice where the
sign will change.
00:03:10.020 --> 00:03:12.320
This denominator
is always positive
00:03:12.320 --> 00:03:14.250
so the sign of the
function depends completely
00:03:14.250 --> 00:03:15.569
on the numerator.
00:03:15.569 --> 00:03:17.110
And so when the
numerator is positive
00:03:17.110 --> 00:03:18.590
this function will be positive.
00:03:18.590 --> 00:03:21.930
When the numerator is negative
this function will be negative.
00:03:21.930 --> 00:03:26.130
So that's a little bit that
we should keep in mind.
00:03:26.130 --> 00:03:30.180
And now let's go to using
our derivatives to figure
00:03:30.180 --> 00:03:33.490
out a little bit more.
00:03:33.490 --> 00:03:35.860
So obviously, first I
should take some derivatives
00:03:35.860 --> 00:03:39.960
and then we'll look at what
we can get out of them.
00:03:39.960 --> 00:03:46.460
So let's let f of x equal
x over 1 plus x squared.
00:03:46.460 --> 00:03:50.400
So then f prime of
x, what do we get?
00:03:50.400 --> 00:04:00.580
We get 1 plus x squared minus x
times 2x over 1 plus x squared,
00:04:00.580 --> 00:04:01.590
squared.
00:04:01.590 --> 00:04:03.961
So I'm just going to
continue that straight below.
00:04:03.961 --> 00:04:04.460
Let's see.
00:04:04.460 --> 00:04:08.870
I can keep this x squared
minus 2x squared, gives me a 1
00:04:08.870 --> 00:04:11.890
minus x squared, in
the numerator, over 1
00:04:11.890 --> 00:04:14.570
plus x squared,
quantity squared.
00:04:14.570 --> 00:04:15.070
OK.
00:04:15.070 --> 00:04:16.860
I'm going to keep
that right here.
00:04:16.860 --> 00:04:19.160
We're going to do a
little bit of calculation
00:04:19.160 --> 00:04:22.910
below in a moment, but I'm going
to record the second derivative
00:04:22.910 --> 00:04:24.920
just to the right.
00:04:24.920 --> 00:04:26.990
So the second
derivative, remember,
00:04:26.990 --> 00:04:28.960
is the derivative of
the first derivative.
00:04:28.960 --> 00:04:30.668
So now I'm going to
take this derivative,
00:04:30.668 --> 00:04:33.850
again using the quotient
rule, which I used here.
00:04:33.850 --> 00:04:37.720
So the derivative of
the top is minus 2x
00:04:37.720 --> 00:04:43.040
and then times 1 plus
x squared squared
00:04:43.040 --> 00:04:44.460
and then I subtract
the derivative
00:04:44.460 --> 00:04:46.770
of the bottom times the top.
00:04:46.770 --> 00:04:50.149
So I'll keep the top
here, 1 minus x squared.
00:04:50.149 --> 00:04:51.690
And then the derivative
of the bottom
00:04:51.690 --> 00:04:55.850
has a little chain rule on it,
so I'm going to get a times 2
00:04:55.850 --> 00:05:00.570
times 1 plus x squared times 2x.
00:05:00.570 --> 00:05:06.350
And then this whole thing is
over 1 plus-- whoa-- x plus 1.
00:05:06.350 --> 00:05:09.130
We'll write x squared
plus 1 to the fourth.
00:05:09.130 --> 00:05:12.570
Sorry to switch the direction
or the order of those.
00:05:12.570 --> 00:05:13.600
OK.
00:05:13.600 --> 00:05:16.560
Now I'm going to
pull out a 1 plus x
00:05:16.560 --> 00:05:18.310
squared from the
numerator to simplify it.
00:05:21.770 --> 00:05:23.780
And then I'm going to
see what I have left.
00:05:23.780 --> 00:05:26.420
Here I have a 1 plus x
squared times a negative 2x.
00:05:26.420 --> 00:05:30.640
That's going to be negative
2x minus 2 x cubed.
00:05:30.640 --> 00:05:35.650
Here I'm going to have-- 2
times 2 is 4x times this 1
00:05:35.650 --> 00:05:37.180
minus x squared.
00:05:37.180 --> 00:05:47.764
So I have a minus 4x plus
4 x squared-- cubed, sorry.
00:05:47.764 --> 00:05:48.430
Let's make sure.
00:05:48.430 --> 00:05:53.220
So I should have a 4x here
and then an x squared times
00:05:53.220 --> 00:05:55.490
4x, which is 4 x cubed.
00:05:55.490 --> 00:05:58.680
And that sign
should be positive.
00:05:58.680 --> 00:06:01.055
And then I still
have to divide by 1
00:06:01.055 --> 00:06:03.350
plus x squared to the fourth.
00:06:03.350 --> 00:06:04.840
To make this much
simpler I'm just
00:06:04.840 --> 00:06:07.900
going to divide out one
of the 1 plus x squareds,
00:06:07.900 --> 00:06:10.250
simplify what's inside, and
we'll leave it that way.
00:06:10.250 --> 00:06:15.620
Actually, let me move this down
so there's a little more room.
00:06:15.620 --> 00:06:27.070
So the numerator will now
be 2 x cubed minus 6x over 1
00:06:27.070 --> 00:06:31.200
plus x squared to the third.
00:06:31.200 --> 00:06:33.060
So these were some
tools that we needed.
00:06:33.060 --> 00:06:35.200
Now we're going to
try and use them.
00:06:35.200 --> 00:06:37.640
So let's recall what we know.
00:06:37.640 --> 00:06:40.580
We know that when the
derivative is equal to 0,
00:06:40.580 --> 00:06:43.590
we have a maximum or
minimum for the function.
00:06:43.590 --> 00:06:46.170
And we know that when the
second derivative is equal to 0,
00:06:46.170 --> 00:06:47.980
we have changes in concavity.
00:06:47.980 --> 00:06:49.370
So let's find those places.
00:06:49.370 --> 00:06:51.140
Let's find where the
first derivative is 0
00:06:51.140 --> 00:06:53.240
and let's find where the
second derivative is 0.
00:06:53.240 --> 00:06:58.290
So I'm going to work under each
individual function to do that.
00:06:58.290 --> 00:07:00.350
So where is f prime equal to 0?
00:07:00.350 --> 00:07:02.110
Well, f prime is
only equal to 0 when
00:07:02.110 --> 00:07:03.690
the numerator is equal to 0.
00:07:03.690 --> 00:07:08.069
So let's solve 1 minus
x squared equals 0.
00:07:08.069 --> 00:07:10.610
Well that's-- there's a couple
ways you can think about that.
00:07:10.610 --> 00:07:12.132
You could factor
it and then solve,
00:07:12.132 --> 00:07:13.840
or you could see right
away this is going
00:07:13.840 --> 00:07:15.660
to be x is plus or minus 1.
00:07:15.660 --> 00:07:17.640
You get the same
thing if you factor.
00:07:17.640 --> 00:07:21.109
But we see x is equal
to plus or minus 1.
00:07:21.109 --> 00:07:23.150
So those are our maximum
values or minimum values
00:07:23.150 --> 00:07:24.220
for the function.
00:07:24.220 --> 00:07:24.720
OK.
00:07:24.720 --> 00:07:27.500
So we know that this is an
important spot for the x-value
00:07:27.500 --> 00:07:30.200
and that's an important
spot for the x-value.
00:07:30.200 --> 00:07:32.680
Now let's just come
over here and look at,
00:07:32.680 --> 00:07:35.605
when is the second
derivative equal to 0?
00:07:35.605 --> 00:07:37.230
So the second derivative
is equal to 0,
00:07:37.230 --> 00:07:40.370
again, when the
numerator is equal to 0.
00:07:40.370 --> 00:07:41.600
So let's look at what we get.
00:07:41.600 --> 00:07:48.770
Well, if we factor that we
get 2x times x squared minus 3
00:07:48.770 --> 00:07:50.460
equals 0.
00:07:50.460 --> 00:07:52.705
So this has three places
it's going to be equal to 0.
00:07:52.705 --> 00:07:56.140
It's going to be equal
to 0 at 0, x equals 0,
00:07:56.140 --> 00:07:59.020
and it's going to be equal to 0
at plus or minus root 3, which
00:07:59.020 --> 00:08:01.520
is sort of unfortunate that we
don't know exactly where that
00:08:01.520 --> 00:08:04.050
is, but we know it's
between 1 and 2.
00:08:04.050 --> 00:08:06.626
I think it's about 1.7
or something like this.
00:08:06.626 --> 00:08:08.375
So we know we're
interested in the point x
00:08:08.375 --> 00:08:14.870
equals 0 and the points x equal
plus or minus square root of 3.
00:08:14.870 --> 00:08:16.370
So these are our
places of interest.
00:08:16.370 --> 00:08:19.550
And so let's evaluate at
least a couple of these places
00:08:19.550 --> 00:08:21.360
and see what's going on.
00:08:21.360 --> 00:08:23.910
Let's go back to the
graph to do this.
00:08:23.910 --> 00:08:26.110
Now I want to
point out something
00:08:26.110 --> 00:08:29.220
I didn't say earlier, which
is, if you know the function is
00:08:29.220 --> 00:08:30.830
defined everywhere,
what you might
00:08:30.830 --> 00:08:32.785
want to do is evaluate
the function at x
00:08:32.785 --> 00:08:34.434
equals 0 right away.
00:08:34.434 --> 00:08:35.850
It's an easy place
to evaluate it.
00:08:35.850 --> 00:08:37.780
It gives you sort of
a launching point.
00:08:37.780 --> 00:08:41.460
So if I evaluate this
at x equals 0 I get 0.
00:08:41.460 --> 00:08:45.940
So I know the point
(0, 0) is on the graph.
00:08:45.940 --> 00:08:47.240
So I know that's one point.
00:08:47.240 --> 00:08:48.830
And now what I'm
interested in, if you
00:08:48.830 --> 00:08:51.090
think about-- we know
where maxes or mins occur,
00:08:51.090 --> 00:08:53.902
we know a max or min occurs
at x equals plus or minus 1.
00:08:53.902 --> 00:08:55.610
Or we have a hope for
a max or min there.
00:08:55.610 --> 00:08:57.540
It's a critical point, at least.
00:08:57.540 --> 00:08:59.240
So I can evaluate the
function-- sorry--
00:08:59.240 --> 00:09:01.840
I can evaluate the function
at 1 and at negative 1
00:09:01.840 --> 00:09:04.950
and I can then
plot those points.
00:09:04.950 --> 00:09:09.700
So when x is 1, I get 1 over 1
plus 1 squared, so I get 1/2.
00:09:09.700 --> 00:09:11.695
So with input 1
I get output 1/2.
00:09:11.695 --> 00:09:14.380
I'm going to erase
that 1 now so we don't
00:09:14.380 --> 00:09:16.540
lose track of what's happening.
00:09:16.540 --> 00:09:18.120
That looks potentially
like it could
00:09:18.120 --> 00:09:22.530
be a maximum, given sort
of what's happening here,
00:09:22.530 --> 00:09:24.280
to the left.
00:09:24.280 --> 00:09:26.365
So let's plug in
negative 1 for x.
00:09:26.365 --> 00:09:30.520
I get a negative 1 over 1 plus
quantity negative 1 squared.
00:09:30.520 --> 00:09:33.810
So I get negative 1 over
2, so I get negative 1/2.
00:09:33.810 --> 00:09:37.540
So at x equals negative
1, I get negative 1/2.
00:09:37.540 --> 00:09:40.450
And let's recall what we know
about the end behavior, which
00:09:40.450 --> 00:09:41.590
we said at the beginning.
00:09:41.590 --> 00:09:44.790
The end behavior of this is as
x goes to positive infinity,
00:09:44.790 --> 00:09:47.490
the function's outputs go to 0.
00:09:47.490 --> 00:09:50.240
Which tells you that, in fact,
this has to be a maximum.
00:09:50.240 --> 00:09:52.880
There are the only two places
where the function can change
00:09:52.880 --> 00:09:55.830
direction from going
up to going down,
00:09:55.830 --> 00:09:57.680
or from going down to going up.
00:09:57.680 --> 00:10:00.650
So it has to be that
this is a maximum.
00:10:00.650 --> 00:10:04.040
It has to be that
this is a minimum.
00:10:04.040 --> 00:10:06.330
So, and also notice
0, based on what
00:10:06.330 --> 00:10:07.830
we know about the
second derivative,
00:10:07.830 --> 00:10:09.700
is one of the inflection points.
00:10:09.700 --> 00:10:14.660
So that's also representing a
place where the derivative is
00:10:14.660 --> 00:10:15.870
changing sign.
00:10:15.870 --> 00:10:18.470
So maybe the derivative
was increasing
00:10:18.470 --> 00:10:22.186
and then it's going
to start decreasing.
00:10:22.186 --> 00:10:24.310
So let's look-- I think I
might have said something
00:10:24.310 --> 00:10:26.430
a little off there, so I'm
going to maybe come back and see
00:10:26.430 --> 00:10:28.140
if I have to fix
anything in a moment--
00:10:28.140 --> 00:10:31.990
but let me draw a rough
sketch of what's happening.
00:10:31.990 --> 00:10:34.090
Very rough, very roughly
we know we're going up
00:10:34.090 --> 00:10:36.030
and then we're going down.
00:10:36.030 --> 00:10:37.830
We're going down
here and then we
00:10:37.830 --> 00:10:41.420
have to go back up
because the end behavior.
00:10:41.420 --> 00:10:43.129
So we have three
inflection points--
00:10:43.129 --> 00:10:44.670
this is what I want
to point out-- we
00:10:44.670 --> 00:10:46.070
have three inflection points.
00:10:46.070 --> 00:10:49.760
We have an inflection point at
0 and at plus or minus root 3.
00:10:49.760 --> 00:10:53.190
So we said root 3 is bigger
than 1, it's less than 2.
00:10:53.190 --> 00:10:55.249
So I know somewhere
in here I have
00:10:55.249 --> 00:10:56.790
an inflection point,
which represents
00:10:56.790 --> 00:10:57.873
a change in the concavity.
00:10:57.873 --> 00:10:58.550
Right?
00:10:58.550 --> 00:11:00.680
Which represents
how the derivative
00:11:00.680 --> 00:11:05.030
is going to change the
direction, whether it's
00:11:05.030 --> 00:11:08.720
continuing to get more negative
and then getting more positive
00:11:08.720 --> 00:11:09.884
than it was previously.
00:11:09.884 --> 00:11:11.800
So yeah, that's where--
we're looking at where
00:11:11.800 --> 00:11:13.740
the derivative changes sign.
00:11:13.740 --> 00:11:14.620
As I said before.
00:11:14.620 --> 00:11:17.850
So let me point out-- this
is a change in concavity.
00:11:17.850 --> 00:11:21.030
Maybe right about
in this x region
00:11:21.030 --> 00:11:23.800
we want to change concavity,
and then this x region
00:11:23.800 --> 00:11:25.500
we want to change concavity.
00:11:25.500 --> 00:11:29.620
So the graph will look something
like going up, going down,
00:11:29.620 --> 00:11:30.120
going down.
00:11:30.120 --> 00:11:34.180
And then I've tried to represent
the change in concavity
00:11:34.180 --> 00:11:35.510
changing that direction there.
00:11:38.880 --> 00:11:43.260
And I'm doing something
that I didn't tell you yet.
00:11:43.260 --> 00:11:47.250
But if you notice, this looks
highly symmetric, doesn't it?
00:11:47.250 --> 00:11:49.900
And in fact, one thing I didn't
tell you about this function--
00:11:49.900 --> 00:11:51.940
that maybe you picked
up on already--
00:11:51.940 --> 00:11:54.410
is that when I take
the right-hand side
00:11:54.410 --> 00:11:58.050
and I rotate it about the
origin I get the left hand side.
00:11:58.050 --> 00:11:59.040
Why is that?
00:11:59.040 --> 00:12:00.770
That's because this
is an odd function.
00:12:00.770 --> 00:12:02.510
Why is it an odd function?
00:12:02.510 --> 00:12:05.020
Because the numerator is an odd
function and the denominator
00:12:05.020 --> 00:12:06.180
is an even function.
00:12:06.180 --> 00:12:09.200
And so the quotient
is an odd function.
00:12:09.200 --> 00:12:12.840
So this is, I would say,
a fairly good sketch
00:12:12.840 --> 00:12:15.400
of the curve y equals x
over 1 plus x squared.
00:12:15.400 --> 00:12:17.920
So hopefully yours looked
something like this.
00:12:17.920 --> 00:12:19.507
And that's where we'll stop.