Squeeze theorem (sandwich theorem)

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:06.840
In this video I will prove
to you that the limit as

00:00:06.840 --> 00:00:15.290
x approaches 0 of sine of
x over x is equal to 1.

00:00:15.290 --> 00:00:18.530
But before I do that, before I
break into trigonometry, I'm

00:00:18.530 --> 00:00:22.650
going to go over another
aspect of limits.

00:00:22.650 --> 00:00:24.140
And that's the squeeze theorem.

00:00:24.140 --> 00:00:26.200
Because once you understand
what the squeeze theorem is,

00:00:26.200 --> 00:00:30.150
we can use the squeeze
theorem to prove this.

00:00:30.150 --> 00:00:33.510
It's actually a pretty involved
explanation, but I think you'll

00:00:33.510 --> 00:00:37.160
find it pretty neat and
satisfying if you get it.

00:00:37.160 --> 00:00:39.220
If you don't get it, maybe you
just want to memorize this.

00:00:39.220 --> 00:00:41.580
Because that's a very useful
limit to know later on when

00:00:41.580 --> 00:00:43.680
we take the derivatives
of trig functions.

00:00:43.680 --> 00:00:45.240
So what's the squeeze theorem?

00:00:45.240 --> 00:00:50.040
The squeeze theorem is
my favorite theorem in

00:00:50.040 --> 00:00:53.790
mathematics, possibly because
it has the word squeeze in it.

00:00:53.790 --> 00:00:56.560
Squeeze theorem.

00:00:56.560 --> 00:00:58.310
And when you read it in a
calculus book it looks

00:00:58.310 --> 00:00:59.740
all complicated.

00:00:59.740 --> 00:01:01.580
I don't know when you read
it, in a calculus book or

00:01:01.580 --> 00:01:02.500
in a precalculus book.

00:01:02.500 --> 00:01:05.080
It looks all complicated,
but what it's saying is

00:01:05.080 --> 00:01:07.440
frankly pretty obvious.

00:01:07.440 --> 00:01:08.410
Let me give you an example.

00:01:08.410 --> 00:01:16.710
If I told you that I
always-- so Sal always

00:01:16.710 --> 00:01:23.150
eats more than Umama.

00:01:23.150 --> 00:01:25.650
Umama is my wife.

00:01:25.650 --> 00:01:27.670
If I told you that this
is true, Sal always

00:01:27.670 --> 00:01:29.370
eats more than Umama.

00:01:29.370 --> 00:01:42.990
And I were also to say that Sal
always eats less than-- I don't

00:01:42.990 --> 00:01:45.190
know, let me make up a
fictional character--

00:01:45.190 --> 00:01:45.835
than Bill.

00:01:48.370 --> 00:01:52.020
So on any given day-- let's
say this is in a given day.

00:01:52.020 --> 00:01:58.120
Sal always eats more than Umama
in any given day, and Sal

00:01:58.120 --> 00:02:01.600
always eats less than
Bill on any given day.

00:02:01.600 --> 00:02:15.360
Now if I were tell you that on
Tuesday Umama ate 300 calories

00:02:15.360 --> 00:02:18.840
and on Tuesday Bill
ate 300 calories.

00:02:21.390 --> 00:02:25.820
So my question to you is, how
many calories did Sal eat,

00:02:25.820 --> 00:02:28.150
or did I eat, on Tuesday?

00:02:28.150 --> 00:02:33.380
Well I always eat more than
Umama-- well, more than or

00:02:33.380 --> 00:02:37.300
equal to Umama-- and I always
eat less than or equal to Bill.

00:02:37.300 --> 00:02:41.350
So then on Tuesday, I must
have eaten 300 calories.

00:02:41.350 --> 00:02:43.950
So this is the gist of the
squeeze theorem, and I'll do

00:02:43.950 --> 00:02:44.940
a little bit more formally.

00:02:44.940 --> 00:02:48.710
But it's essentially saying, if
I'm always greater than one

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thing and I'm always less than
another thing and at some point

00:02:52.190 --> 00:02:55.560
those two things are equal,
well then I must be equal

00:02:55.560 --> 00:02:57.120
to whatever those two
things are equal to.

00:02:57.120 --> 00:02:59.080
I've kind of been squeezed
in between them.

00:02:59.080 --> 00:03:01.600
I'm always in between Umama and
Bill, and if they're at the

00:03:01.600 --> 00:03:04.220
exact same point on
Tuesday, then I must be at

00:03:04.220 --> 00:03:05.000
that point as well.

00:03:05.000 --> 00:03:06.360
Or at least I must approach it.

00:03:06.360 --> 00:03:08.290
So let me write it
in math terms.

00:03:11.880 --> 00:03:18.730
So all it says is that, over
some domain, if I say that,

00:03:18.730 --> 00:03:25.300
let's say that g of x is less
than or equal to f of x, which

00:03:25.300 --> 00:03:29.310
is less than or equal to
h of x over some domain.

00:03:29.310 --> 00:03:38.720
And we also know that the limit
of g of x as x approaches a is

00:03:38.720 --> 00:03:45.070
equal to some limit, capital L,
and we also know that the limit

00:03:45.070 --> 00:03:52.140
as x approaches a of h of x
also equals L, then the squeeze

00:03:52.140 --> 00:03:55.200
theorem tells us-- and I'm not
going to prove that right

00:03:55.200 --> 00:03:57.540
here, but it's good to just
understand what the squeeze

00:03:57.540 --> 00:04:02.700
theorem is-- the squeeze
theorem tells us then the limit

00:04:02.700 --> 00:04:09.770
as x approaches a of f of x
must also be equal to L.

00:04:09.770 --> 00:04:11.230
And this is the same thing.

00:04:11.230 --> 00:04:14.090
This is example where f of x,
this could be how much Sal eats

00:04:14.090 --> 00:04:16.410
in a day, this could be how
much Umama eats in a

00:04:16.410 --> 00:04:17.330
day, this is Bill.

00:04:17.330 --> 00:04:19.980
So I always eat more than
Umama or less than Bill.

00:04:19.980 --> 00:04:25.190
And then on Tuesday, you could
say a is Tuesday, if Umama had

00:04:25.190 --> 00:04:28.650
300 calories and Bill had 300
calories, then I also had

00:04:28.650 --> 00:04:29.480
to eat 300 calories.

00:04:29.480 --> 00:04:32.350
Let me let me graph
that for you.

00:04:32.350 --> 00:04:36.470
Let me graph that, and I'll
do it in a different color.

00:04:36.470 --> 00:04:37.790
Squeeze theorem.

00:04:42.560 --> 00:04:44.050
Squeeze theorem.

00:04:44.050 --> 00:04:51.942
OK, so let's draw the
point a comma L.

00:04:51.942 --> 00:04:53.980
The point a comma L.

00:04:53.980 --> 00:04:55.840
Let's say this is a, that's
the point that we care

00:04:55.840 --> 00:04:59.900
about. a, and this is L.

00:04:59.900 --> 00:05:03.770
And we know, g of x, that's
the lower function, right?

00:05:03.770 --> 00:05:05.540
So let's say that this
green thing right

00:05:05.540 --> 00:05:07.540
here, this is g of x.

00:05:07.540 --> 00:05:10.030
So this is my g of x.

00:05:10.030 --> 00:05:14.110
And we know that as g of x
approaches-- so the g of x

00:05:14.110 --> 00:05:16.120
could look something
like that, right?

00:05:16.120 --> 00:05:18.910
And we know that the limit
as x approaches a of

00:05:18.910 --> 00:05:21.510
g of x is equal to L.

00:05:21.510 --> 00:05:23.590
So that's right there.

00:05:23.590 --> 00:05:26.860
So this is g of x.

00:05:26.860 --> 00:05:28.510
That's g of x.

00:05:28.510 --> 00:05:31.590
Let me do h of x in
a different color.

00:05:31.590 --> 00:05:33.570
So now h of x could look
something like this.

00:05:36.790 --> 00:05:38.740
Like that.

00:05:38.740 --> 00:05:41.870
So that's h of x.

00:05:41.870 --> 00:05:45.970
And we also know that the limit
as x approaches a of h of x --

00:05:45.970 --> 00:05:51.610
let's see, this is the
function of x axis.

00:05:51.610 --> 00:05:56.630
So you can call it h of
x, g of x, or f of x.

00:05:56.630 --> 00:06:00.350
That's just the dependent
access, and this is the x-axis.

00:06:00.350 --> 00:06:04.790
So once again, the limit as x
approaches a of h of x, well

00:06:04.790 --> 00:06:07.570
at that point right there,
h of a is equal to L.

00:06:07.570 --> 00:06:08.965
Or at least the limit
is equal to that.

00:06:11.470 --> 00:06:13.530
And none of these functions
actually have to even be

00:06:13.530 --> 00:06:17.210
defined at a, as long as these
limits, this limit exists

00:06:17.210 --> 00:06:18.110
and this limit exists.

00:06:18.110 --> 00:06:20.790
And that's also an important
thing to keep in mind.

00:06:20.790 --> 00:06:23.730
So what does this tell us?
f of x is always greater

00:06:23.730 --> 00:06:24.860
than this green function.

00:06:24.860 --> 00:06:27.350
It's always less
than h of x, right?

00:06:27.350 --> 00:06:29.910
So any f of x I draw,
it would have to be in

00:06:29.910 --> 00:06:31.080
between those two, right?

00:06:31.080 --> 00:06:34.800
So no matter how I draw it, if
I were to draw a function,

00:06:34.800 --> 00:06:38.600
it's bounded by those two
functions just by definition.

00:06:38.600 --> 00:06:40.500
So it has to go
through that point.

00:06:40.500 --> 00:06:41.910
Or at least it has to
approach that point.

00:06:41.910 --> 00:06:45.060
Maybe it's not defined at that
point, but the limit as we

00:06:45.060 --> 00:06:49.950
approach a of f of x also
has to be at point L.

00:06:49.950 --> 00:06:52.580
And maybe f of x doesn't have
to be defined right there, but

00:06:52.580 --> 00:06:54.940
the limit as we approach
it is going to be L.

00:06:54.940 --> 00:06:56.920
And hopefully that makes a
little bit of sense, and

00:06:56.920 --> 00:06:59.250
hopefully my calories
example made a little

00:06:59.250 --> 00:06:59.820
bit of sense to you.

00:06:59.820 --> 00:07:01.970
So let's keep that in
the back of our mind,

00:07:01.970 --> 00:07:03.860
the squeeze theorem.

00:07:03.860 --> 00:07:11.970
And now we will use that to
prove that the limit as x

00:07:11.970 --> 00:07:16.310
approaches 0 of sine of
x over x is equal to 1.

00:07:16.310 --> 00:07:17.940
And I want to do that,
one, because this is

00:07:17.940 --> 00:07:18.990
a super useful limit.

00:07:18.990 --> 00:07:20.600
And then the other thing is,
sometimes you learn the squeeze

00:07:20.600 --> 00:07:22.770
theorem, you're like, oh, well
that's obvious but

00:07:22.770 --> 00:07:23.890
when is it useful?

00:07:23.890 --> 00:07:25.090
And we'll see.

00:07:25.090 --> 00:07:26.520
Actually I'm going to do it in
the next video, since we're

00:07:26.520 --> 00:07:27.590
already pushing 8 minutes.

00:07:27.590 --> 00:07:29.340
But we'll see in the next video
that the squeeze theorem is

00:07:29.340 --> 00:07:32.500
tremendously useful when
we're trying to prove this.

00:07:32.500 --> 00:07:35.080
I will see you in
the next video.


Proof: lim (sin x)/x

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.970
Now that we have hopefully a
decent understanding of the

00:00:03.970 --> 00:00:09.280
squeeze theorem, we'll use that
to prove that the limit-- I'll

00:00:09.280 --> 00:00:17.190
do it in yellow-- the limit as
x approaches 0 of sine of

00:00:17.190 --> 00:00:22.280
x over x is equal to 1.

00:00:22.280 --> 00:00:25.090
And you must be bubbling over
with anticipation now, because

00:00:25.090 --> 00:00:26.535
I've said this so many times.

00:00:26.535 --> 00:00:29.410
So let's do it, and actually,
we have to go with-- obviously,

00:00:29.410 --> 00:00:32.220
they got our trigonometry-- and
it's actually a visual proof.

00:00:32.220 --> 00:00:35.640
So let me draw at least the
first and fourth quadrants

00:00:35.640 --> 00:00:37.440
of the unit circle.

00:00:37.440 --> 00:00:40.870
I'll do that in magenta.

00:00:40.870 --> 00:00:42.420
Let's see, let me see
if I can-- I should

00:00:42.420 --> 00:00:44.220
draw it pretty big.

00:00:44.220 --> 00:00:44.840
Let me see.

00:00:44.840 --> 00:00:47.795
I should draw it
like quite big.

00:00:47.795 --> 00:00:49.240
So I'll draw it like that.

00:00:54.570 --> 00:00:55.780
That's close enough.

00:00:55.780 --> 00:00:59.030
And then let me draw the axis.

00:00:59.030 --> 00:01:02.920
So this is the x-axis, would
look something like that.

00:01:02.920 --> 00:01:03.660
Sorry, that's the y-axis.

00:01:06.180 --> 00:01:07.100
There you go.

00:01:07.100 --> 00:01:10.790
And then the x-axis,
something like that.

00:01:10.790 --> 00:01:12.030
That's our unit circle.

00:01:14.810 --> 00:01:16.220
There you go.

00:01:16.220 --> 00:01:18.690
Now let me draw a couple
of other things.

00:01:18.690 --> 00:01:24.170
Let me draw a-- well, it is
a radius, but I'm going to

00:01:24.170 --> 00:01:26.580
go beyond the unit circle.

00:01:30.850 --> 00:01:34.970
So let's go like right there.

00:01:34.970 --> 00:01:39.690
Draw a couple of more things,
just to set up this problem.

00:01:39.690 --> 00:01:41.650
Nope, that's not what
I wanted to do.

00:01:41.650 --> 00:01:44.180
I wanted to do it right
from this point.

00:01:44.180 --> 00:01:45.340
Right like that.

00:01:45.340 --> 00:01:49.260
And then right from this
point, I want to do this.

00:01:52.720 --> 00:01:55.560
And then I want to draw another
right from that point.

00:01:55.560 --> 00:01:59.960
I'm going to do that.

00:01:59.960 --> 00:02:02.250
And now we are ready to go.

00:02:02.250 --> 00:02:03.640
So what did I say?

00:02:03.640 --> 00:02:05.860
This is the unit circle, right?

00:02:05.860 --> 00:02:07.570
So if that's the unit
circle, what does it mean?

00:02:07.570 --> 00:02:09.780
It's a circle with
a radius of 1.

00:02:09.780 --> 00:02:14.540
So the distance from
here to here is 1.

00:02:14.540 --> 00:02:21.040
And now if this is an angle x
in radians, what's the length

00:02:21.040 --> 00:02:23.260
of this line right here?

00:02:23.260 --> 00:02:24.310
What's the length of that line?

00:02:24.310 --> 00:02:27.430
Well, by definition, sine
of x is defined to be the

00:02:27.430 --> 00:02:30.830
y-coordinate of any point
on the unit circle.

00:02:30.830 --> 00:02:32.070
So this is sine of x.

00:02:35.270 --> 00:02:38.300
I'm going to run out of space,
so let me draw an arrow.

00:02:38.300 --> 00:02:41.320
So this is-- that right
there is sine of x.

00:02:44.300 --> 00:02:46.270
Now let me ask you a
slightly harder one.

00:02:46.270 --> 00:02:47.870
What's this length right here?

00:02:51.370 --> 00:02:52.970
Well, let's think about it.

00:02:52.970 --> 00:02:53.880
What is tangent?

00:02:53.880 --> 00:02:56.370
Let's go back to our SOHCAHTOA
definition of tangent.

00:02:56.370 --> 00:02:57.720
TOA.

00:02:57.720 --> 00:03:07.380
Tangent is equal to TOA:
opposite over adjacent.

00:03:07.380 --> 00:03:08.910
So what is a tangent of x?

00:03:14.460 --> 00:03:17.630
Well, it would be equal to-- we
could take this-- if we say

00:03:17.630 --> 00:03:20.450
that this is the right
triangle, it would be this

00:03:20.450 --> 00:03:26.110
length-- the opposite--
over the adjacent, right?

00:03:26.110 --> 00:03:28.440
So let's call this length
over here, let's call

00:03:28.440 --> 00:03:30.380
this o for opposite.

00:03:30.380 --> 00:03:31.620
But what's the adjacent length?

00:03:31.620 --> 00:03:34.600
What's this base of
this larger triangle?

00:03:34.600 --> 00:03:36.050
Well, it's the unit
circle, right?

00:03:36.050 --> 00:03:40.660
So the distance from here
to here-- that distance is

00:03:40.660 --> 00:03:41.830
also going to be 1, right?

00:03:41.830 --> 00:03:43.070
Because it's just
a radius again.

00:03:43.070 --> 00:03:44.400
That's 1.

00:03:44.400 --> 00:03:48.980
So opposite over adjacent is
equal to the tangent of x.

00:03:48.980 --> 00:03:52.350
But opposite over adjacent--
adjacent is just 1, right?

00:03:52.350 --> 00:03:56.280
So the opposite side, this side
right here, it's going to be

00:03:56.280 --> 00:03:57.655
equal to the tangent of x.

00:03:57.655 --> 00:04:00.312
Or another way of saying it,
tangent of x is equal to this

00:04:00.312 --> 00:04:02.930
side over 1, or tangent of
x is equal to this side.

00:04:02.930 --> 00:04:04.700
So let me write that down.

00:04:04.700 --> 00:04:08.080
That side is equal to
the tangent of x.

00:04:11.840 --> 00:04:17.770
Now, let's think about the area
of a couple of parts of this

00:04:17.770 --> 00:04:18.730
figure that I've drawn here.

00:04:18.730 --> 00:04:20.580
Maybe I should have drawn it
a little bigger, but I think

00:04:20.580 --> 00:04:22.560
we'll be able to do it.

00:04:22.560 --> 00:04:25.210
So first let me pick a
relatively small triangle.

00:04:25.210 --> 00:04:27.580
So let's do this
triangle right here.

00:04:27.580 --> 00:04:29.870
I'll trace it in green.

00:04:29.870 --> 00:04:34.250
So this triangle that I'm
tracing in green-- what is

00:04:34.250 --> 00:04:37.730
the area of that triangle?

00:04:37.730 --> 00:04:40.910
Well, that's going to be 1/2
times base times height.

00:04:40.910 --> 00:04:49.850
So it's 1/2 times the
base, which is 1.

00:04:49.850 --> 00:04:50.110
Right?

00:04:50.110 --> 00:04:51.600
It's this whole triangle.

00:04:51.600 --> 00:04:53.240
And then what's
the height of it?

00:04:53.240 --> 00:04:55.740
Well, we just figured out that
this height right here, that

00:04:55.740 --> 00:04:57.690
this height is sine of x.

00:04:57.690 --> 00:04:58.780
Times sine of x.

00:05:01.880 --> 00:05:04.650
So that's this green
triangle here, right?

00:05:04.650 --> 00:05:09.040
Now, what is the area of--
not that green triangle.

00:05:09.040 --> 00:05:11.180
Let me do it in another color.

00:05:11.180 --> 00:05:16.480
Let me do it in-- oh,
I'll do it in red.

00:05:16.480 --> 00:05:20.540
What is the area of this pi?

00:05:20.540 --> 00:05:23.470
This pi right here.

00:05:23.470 --> 00:05:24.270
That pi.

00:05:24.270 --> 00:05:26.620
Hope you see-- well, that's
not a different enough color.

00:05:26.620 --> 00:05:29.440
So, this pi right here.

00:05:29.440 --> 00:05:30.330
Or I'm going there.

00:05:30.330 --> 00:05:32.050
And then I'm going on the arc.

00:05:32.050 --> 00:05:34.630
So it's a little bit bigger
than the triangle we

00:05:34.630 --> 00:05:35.640
just figured out, right?

00:05:35.640 --> 00:05:37.130
It's always going to be a
little bit bigger, because it

00:05:37.130 --> 00:05:41.220
includes this area between that
triangle and the arc, right?

00:05:41.220 --> 00:05:42.890
What is the area of that arc?

00:05:45.600 --> 00:05:54.500
Well, if this angle is x-- it's
x radians-- what fraction

00:05:54.500 --> 00:05:56.660
of that is out of the
entire unit circle?

00:05:56.660 --> 00:05:59.920
Well, there are 2 pi radians in
a total unit circle, right?

00:05:59.920 --> 00:06:04.220
So this area right here is
going to be equal to what?

00:06:04.220 --> 00:06:09.690
It's going to be equal to the
fraction x is of the total

00:06:09.690 --> 00:06:14.270
radians in the unit
circle, right?

00:06:14.270 --> 00:06:17.060
So it's x radians over
2 pi radians in the

00:06:17.060 --> 00:06:18.450
entire unit circle.

00:06:18.450 --> 00:06:21.135
So that's kind of the fraction
that this is of-- you know, if

00:06:21.135 --> 00:06:23.770
you did it in degrees-- the
fraction that this is over 360

00:06:23.770 --> 00:06:26.350
degrees, times the area of
the whole circle, right?

00:06:26.350 --> 00:06:28.490
This tells us what fraction we
are of the circle, and we're

00:06:28.490 --> 00:06:30.280
going to want to multiply that
times the area of

00:06:30.280 --> 00:06:31.350
the whole circle.

00:06:31.350 --> 00:06:33.450
Well, what's the area
of the whole circle?

00:06:33.450 --> 00:06:38.660
Well, area is pi r squared,
the radius is 1, right?

00:06:38.660 --> 00:06:42.100
So the area of the entire
circle is just pi.

00:06:45.250 --> 00:06:47.700
Pi r squared, r is 1, so the
area of the circle-- so the

00:06:47.700 --> 00:06:51.960
area of this wedge right here,
is just going to be equal to--

00:06:51.960 --> 00:06:55.210
these pi's cancel out--
it's equal to x over 2.

00:06:55.210 --> 00:07:00.860
So that first small triangle,
that green triangle

00:07:00.860 --> 00:07:04.050
we did, is sine of x.

00:07:04.050 --> 00:07:07.630
1/2 sine of x, that's the
area of that green triangle.

00:07:07.630 --> 00:07:11.390
Then the slightly larger area
of this wedge is-- we figured

00:07:11.390 --> 00:07:12.640
out just now-- is x over 2.

00:07:12.640 --> 00:07:15.630
And now let's take the area
of that larger triangle,

00:07:15.630 --> 00:07:17.020
of this big triangle here.

00:07:17.020 --> 00:07:18.860
And that may be
the most obvious.

00:07:18.860 --> 00:07:20.130
So 1/2 base times height.

00:07:20.130 --> 00:07:24.950
So that's 1/2-- the base
is 1 again-- 1 times the

00:07:24.950 --> 00:07:28.720
height, is tangent of x.

00:07:28.720 --> 00:07:33.900
Equal to 1/2 tangent of x.

00:07:33.900 --> 00:07:36.640
Now, it should be clear just
looking at this diagram, no

00:07:36.640 --> 00:07:40.350
matter where I drew this top
line, that this green triangle

00:07:40.350 --> 00:07:43.690
has a smaller area than this
wedge, which has a smaller area

00:07:43.690 --> 00:07:45.330
than this large triangle.

00:07:45.330 --> 00:07:45.960
Right?

00:07:45.960 --> 00:07:48.510
So let's write an
inequality that says that.

00:07:48.510 --> 00:07:54.830
The green triangle-- the area
of the green triangle-- so 1/2

00:07:54.830 --> 00:07:58.900
the sine of x, that's the area
of the green triangle-- it's

00:07:58.900 --> 00:08:04.030
less than the area
of this wedge.

00:08:04.030 --> 00:08:06.360
So that's x over 2.

00:08:06.360 --> 00:08:09.540
And they're both less than
the area of this large

00:08:09.540 --> 00:08:11.200
triangle, right?

00:08:11.200 --> 00:08:13.860
Which is 1/2 tangent of x.

00:08:16.920 --> 00:08:18.010
Now when is this true?

00:08:18.010 --> 00:08:21.160
This is true as long as we're
in the first quadrant, right?

00:08:21.160 --> 00:08:23.400
As long as we're in
the first quadrant.

00:08:23.400 --> 00:08:25.960
It's also almost true if we go
into the fourth quadrant,

00:08:25.960 --> 00:08:28.310
except then the sine of x
becomes negative, the tangent

00:08:28.310 --> 00:08:31.130
of x becomes negative,
and x becomes negative.

00:08:31.130 --> 00:08:33.300
But if we take the absolute
value of everything, it still

00:08:33.300 --> 00:08:34.900
holds in the fourth quadrant.

00:08:34.900 --> 00:08:37.450
Because if you go negative, as
long as we take the absolute

00:08:37.450 --> 00:08:39.900
value, then the distance will
still hold and we still have

00:08:39.900 --> 00:08:42.120
positive areas and all
that kind of thing.

00:08:42.120 --> 00:08:47.060
So since my goal is to take the
limit as x approaches 0, and I

00:08:47.060 --> 00:08:49.540
want to take the limit-- in
order for this limit to be

00:08:49.540 --> 00:08:53.260
defined in general, it has to
be true from both the positive

00:08:53.260 --> 00:08:54.320
and the negative side.

00:08:54.320 --> 00:08:56.170
Let's take the absolute value
of both sides of this.

00:08:56.170 --> 00:08:57.750
And hopefully this
makes sense to you.

00:08:57.750 --> 00:08:59.950
If I were to draw the line down
here-- and this would be the

00:08:59.950 --> 00:09:02.720
sine of x, and that would be
the tangent of x-- as long as

00:09:02.720 --> 00:09:04.350
you took the absolute value of
everything, you're essentially

00:09:04.350 --> 00:09:06.370
just doing the same thing
as in the first quadrant.

00:09:06.370 --> 00:09:10.150
So let's take the absolute
value of everything.

00:09:10.150 --> 00:09:14.020
And that shouldn't change
anything, especially if you're

00:09:14.020 --> 00:09:14.750
in the first quadrant.

00:09:14.750 --> 00:09:16.190
And you might want to think
about it a little bit, why

00:09:16.190 --> 00:09:18.290
it doesn't change anything
in the second quadrant.

00:09:18.290 --> 00:09:19.510
So we have this inequality.

00:09:19.510 --> 00:09:21.270
Let's see if we can
play around with this.

00:09:21.270 --> 00:09:22.940
So first of all, let's just
multiply everything by 2

00:09:22.940 --> 00:09:25.180
and get rid of the 1/2's.

00:09:25.180 --> 00:09:33.110
So we get absolute value of
sine of x is less than absolute

00:09:33.110 --> 00:09:36.770
value of x, which is less than
the absolute value of

00:09:36.770 --> 00:09:38.500
the tangent of x.

00:09:38.500 --> 00:09:41.420
I hope I didn't confuse you by
taking the absolute value.

00:09:41.420 --> 00:09:44.940
That original inequality I
wrote was completely valid in

00:09:44.940 --> 00:09:48.310
the first quadrant, but since I
want this inequality to be true

00:09:48.310 --> 00:09:50.510
in the first and fourth
quadrants, because I'm taking

00:09:50.510 --> 00:09:53.260
the limit as x approaches 0
from both sides, I put that

00:09:53.260 --> 00:09:54.120
absolute value there.

00:09:54.120 --> 00:09:56.540
So you could draw the line down
there and do everything we did

00:09:56.540 --> 00:09:58.470
up there in the fourth
quadrant, but just take

00:09:58.470 --> 00:10:01.170
the absolute value and it
should work out the same.

00:10:01.170 --> 00:10:02.390
Anyway, back to the problem.

00:10:02.390 --> 00:10:03.730
So we have this inequality.

00:10:03.730 --> 00:10:06.630
And I'm running out of
space, so let me erase some

00:10:06.630 --> 00:10:07.710
of this stuff up here.

00:10:11.280 --> 00:10:12.570
Erase.

00:10:12.570 --> 00:10:12.760
Erase.

00:10:15.420 --> 00:10:17.940
Nope, that doesn't erase.

00:10:17.940 --> 00:10:18.960
OK.

00:10:18.960 --> 00:10:21.400
That should erase.

00:10:21.400 --> 00:10:23.230
OK.

00:10:23.230 --> 00:10:25.980
So we could erase everything
that took us so far.

00:10:25.980 --> 00:10:28.122
We can't forget this though.

00:10:28.122 --> 00:10:30.310
This gives a lot of space.

00:10:30.310 --> 00:10:32.430
OK.

00:10:32.430 --> 00:10:35.700
So let's take this, and let's
take that expression, and

00:10:35.700 --> 00:10:38.460
divide all of the sides.

00:10:38.460 --> 00:10:40.190
You know, and it has
three sides, a left,

00:10:40.190 --> 00:10:41.100
middle, and right.

00:10:41.100 --> 00:10:43.730
Let's divide them all by the
absolute value of sine of x.

00:10:43.730 --> 00:10:45.960
And since we know that the
absolute value of sine of x is

00:10:45.960 --> 00:10:49.640
a positive number, we know that
these less than signs

00:10:49.640 --> 00:10:51.155
don't change, right?

00:10:51.155 --> 00:10:52.260
So let's do that.

00:10:52.260 --> 00:10:55.210
So the absolute value of the
sine of x divided by the

00:10:55.210 --> 00:10:57.585
absolute value of the sine
of x, well, that equals 1.

00:11:00.440 --> 00:11:04.230
Which is less than the absolute
value of x divided by the

00:11:04.230 --> 00:11:05.870
absolute value of sine of x.

00:11:08.650 --> 00:11:12.420
Which is less than-- what's the
absolute value of tan-- so, all

00:11:12.420 --> 00:11:15.850
I'm doing is I'm taking the
absolute value of sine of x,

00:11:15.850 --> 00:11:20.910
absolute value of sine of x,
absolute value of sine of x.

00:11:20.910 --> 00:11:23.350
So what's the absolute value of
the tangent of x divided by the

00:11:23.350 --> 00:11:26.580
absolute value of
the sine of x?

00:11:26.580 --> 00:11:32.930
Well, tangent is just
sine over cosine.

00:11:32.930 --> 00:11:35.580
So that's equal to-- so, just
do this part right here.

00:11:35.580 --> 00:11:40.430
That's equal to sine over
cosine divided by sine.

00:11:40.430 --> 00:11:41.895
And you know, you could say
that that's the same thing

00:11:41.895 --> 00:11:42.680
as the absolute value.

00:11:42.680 --> 00:11:45.360
And the absolute value divided
by the absolute value.

00:11:45.360 --> 00:11:46.900
So what are you left with?

00:11:46.900 --> 00:11:50.620
Well, you're just left with 1
over-- this cancels out with

00:11:50.620 --> 00:11:54.550
this, that becomes a 1-- 1
over the absolute value

00:11:54.550 --> 00:11:57.720
of the cosine of x.

00:11:57.720 --> 00:12:00.630
So you might feel
we're getting close.

00:12:00.630 --> 00:12:03.700
Because this looks a lot like
this, it's just inverted.

00:12:03.700 --> 00:12:06.050
So to get to this,
let's invert it.

00:12:06.050 --> 00:12:07.750
And to invert it, what happens?

00:12:07.750 --> 00:12:10.090
Well, first of all, what
happens when you invert 1?

00:12:10.090 --> 00:12:13.880
Well, 1/1 is just 1.

00:12:13.880 --> 00:12:17.760
But when you invert both sides
of an inequality, you switch

00:12:17.760 --> 00:12:18.960
the inequality, right?

00:12:18.960 --> 00:12:22.480
And if that doesn't make sense
to you, think about this.

00:12:22.480 --> 00:12:26.870
You know, if I say 1/2 is less
than 2, and I invert both sides

00:12:26.870 --> 00:12:32.020
of that, I get 2 is
greater than 1/2.

00:12:32.020 --> 00:12:33.830
So that hopefully gives
you a little intuition.

00:12:33.830 --> 00:12:37.380
So if I'm inverting all of the
sides of this inequality, I

00:12:37.380 --> 00:12:40.210
have to switch the
inequality signs.

00:12:40.210 --> 00:12:48.130
So 1 is greater than absolute
value of sine of x, over the

00:12:48.130 --> 00:12:52.415
absolute value of x, which is
greater than absolute

00:12:52.415 --> 00:12:55.400
value of cosine of x.

00:12:55.400 --> 00:12:58.430
Now let me ask you a question.

00:12:58.430 --> 00:13:01.330
The absolute value of sine
of x over-- well, first

00:13:01.330 --> 00:13:04.030
of all, sine of x over x.

00:13:04.030 --> 00:13:10.140
Will there ever be a time when
sine of x over x is-- in the

00:13:10.140 --> 00:13:12.730
first or the fourth quadrant--
is there ever a time that

00:13:12.730 --> 00:13:17.110
sine of x over x is a
negative expression?

00:13:17.110 --> 00:13:19.680
Well, in the first quadrant,
sine of x is positive,

00:13:19.680 --> 00:13:20.910
and x is positive.

00:13:20.910 --> 00:13:22.960
So a positive divided
by a positive is

00:13:22.960 --> 00:13:24.090
going to be positive.

00:13:24.090 --> 00:13:29.610
And in the fourth quadrant,
sine of x is negative, y is

00:13:29.610 --> 00:13:32.390
negative, and the angle is
negative, so x is

00:13:32.390 --> 00:13:33.400
also negative.

00:13:33.400 --> 00:13:37.240
So in the fourth quadrant, sine
of x over x is going to be a

00:13:37.240 --> 00:13:39.010
negative divided by a negative.

00:13:39.010 --> 00:13:40.850
So it's going to be
a positive again.

00:13:40.850 --> 00:13:45.070
So sine of x over x is always
going to be a positive.

00:13:45.070 --> 00:13:48.120
So the absolute value signs
are kind of redundant.

00:13:48.120 --> 00:13:53.990
So we could write 1 is greater
than sine of x over x.

00:13:53.990 --> 00:13:57.250
And the same logic, in the
first and fourth quadrants--

00:13:57.250 --> 00:13:58.540
and that's where
we're dealing with.

00:13:58.540 --> 00:14:05.860
We're dealing with minus pi
over 2 is less than x, which

00:14:05.860 --> 00:14:08.000
is less than pi over 2.

00:14:08.000 --> 00:14:09.960
So we're going from
minus pi over 2 all

00:14:09.960 --> 00:14:11.310
the way to pi over 2.

00:14:11.310 --> 00:14:13.510
So we're in the fourth
and first quadrant.

00:14:13.510 --> 00:14:15.050
Is cosine of x ever negative?

00:14:15.050 --> 00:14:17.830
Well, cosine is the x value,
and the x-- by definition, in

00:14:17.830 --> 00:14:19.653
the first and fourth
quadrants-- the x value

00:14:19.653 --> 00:14:20.880
is always positive.

00:14:20.880 --> 00:14:23.480
So if this is always positive,
we can get rid of the

00:14:23.480 --> 00:14:27.940
absolute value signs there,
and just write that.

00:14:27.940 --> 00:14:33.430
And now, we are ready to
use the squeeze theorem.

00:14:33.430 --> 00:14:36.060
Let me erase all of
this down here now.

00:14:39.580 --> 00:14:42.500
So let me ask you a question.

00:14:42.500 --> 00:14:50.170
What is the limit, as
x approaches 0, of

00:14:50.170 --> 00:14:52.750
the function 1?

00:14:52.750 --> 00:14:55.510
Well, the function 1
is always equal to 1.

00:14:55.510 --> 00:14:57.460
So I can set the limit as x
approaches infinity, the limit

00:14:57.460 --> 00:14:59.310
as x approaches pi, anything.

00:14:59.310 --> 00:15:02.750
This is always going
to be equal to 1.

00:15:02.750 --> 00:15:05.620
So as x approaches 0,
this is equal to 1.

00:15:05.620 --> 00:15:14.570
And then what is the limit, as
x approaches 0, of cosine of x?

00:15:14.570 --> 00:15:15.620
Well, that's easy, too.

00:15:15.620 --> 00:15:18.170
As x approaches 0, cosine of 0
is just 1-- and as you get,

00:15:18.170 --> 00:15:22.620
you know, it's a continuous
function-- so the limit is 1.

00:15:22.620 --> 00:15:25.630
So we are ready to use
the squeeze theorem.

00:15:25.630 --> 00:15:31.990
As we approach 0, as
x approaches 0, this

00:15:31.990 --> 00:15:33.770
function approaches 1.

00:15:33.770 --> 00:15:35.650
This function approaches 1.

00:15:35.650 --> 00:15:38.350
And this function, this
expression, is in

00:15:38.350 --> 00:15:39.870
between the two.

00:15:39.870 --> 00:15:45.300
And if it's in between the two,
as we approach-- this is

00:15:45.300 --> 00:15:48.290
approaching 1 as we approach 0,
this is approaching 1 as we

00:15:48.290 --> 00:15:51.470
approach 0, and this is in
between them, so it also has to

00:15:51.470 --> 00:15:53.790
approach 1 as we approach 0.

00:15:53.790 --> 00:15:58.340
And so we are using the squeeze
theorem based on this and this.

00:15:58.340 --> 00:16:00.090
And you could say, you know,
therefore by the squeeze

00:16:00.090 --> 00:16:03.740
theorem, because this is true,
this is true, and this is true,

00:16:03.740 --> 00:16:11.010
sine of x over x, the limit as
x approaches 0, is equal to 1.

00:16:11.010 --> 00:16:13.310
So hopefully that gave
you the intuition.

00:16:13.310 --> 00:16:18.910
That another way to view it, as
this line gets smaller and

00:16:18.910 --> 00:16:24.410
smaller as it approaches 0, as
x approaches zero, that this

00:16:24.410 --> 00:16:27.300
area and this area converge, so
the area in between kind of has

00:16:27.300 --> 00:16:30.660
to converge to the
both of them.

00:16:30.660 --> 00:16:32.700
And if you want to see
it graphically, I've

00:16:32.700 --> 00:16:34.490
graphed it here.

00:16:34.490 --> 00:16:37.285
Let me see if I can
graph this thing.

00:16:37.285 --> 00:16:40.610
I'll show you the graph.

00:16:40.610 --> 00:16:42.850
Just so you believe me.

00:16:42.850 --> 00:16:47.270
So we said that 1 is always
greater than sine of x, which

00:16:47.270 --> 00:16:52.630
is always greater than cosine
of x, between negative pi

00:16:52.630 --> 00:16:53.440
over 2 and pi over 2.

00:16:53.440 --> 00:16:56.740
And of course, this isn't
defined at x is equal to 0.

00:16:56.740 --> 00:16:58.350
But we can figure
out the limit.

00:16:58.350 --> 00:16:59.530
So there we have it.

00:16:59.530 --> 00:17:03.130
This blue line right here,
that's the function 1.

00:17:03.130 --> 00:17:06.650
That's y is equal to 1.

00:17:06.650 --> 00:17:09.440
This light blue line right
here is cosine of x.

00:17:09.440 --> 00:17:11.430
And this is the graph
of sine of x over x.

00:17:11.430 --> 00:17:15.480
And you can see that I
actually typed it in.

00:17:15.480 --> 00:17:22.060
So sine of x over x, between
negative pi over 2 and pi over

00:17:22.060 --> 00:17:27.170
2, or the fourth and the first
quadrants, the red line

00:17:27.170 --> 00:17:29.100
is always in between.

00:17:29.100 --> 00:17:32.750
It's always in between the dark
blue and the light blue line.

00:17:32.750 --> 00:17:34.970
And so this is just an
intuition of what happens

00:17:34.970 --> 00:17:36.180
with the squeeze theorem.

00:17:36.180 --> 00:17:39.390
We know that the limit,
as this light blue line

00:17:39.390 --> 00:17:40.860
approaches 0, is 1.

00:17:40.860 --> 00:17:43.300
And we know the limit as
this top dark blue line

00:17:43.300 --> 00:17:45.500
approaches 0 is 1.

00:17:45.500 --> 00:17:47.780
And this red line is always
in between it, so it

00:17:47.780 --> 00:17:49.240
also approaches 1.

00:17:49.240 --> 00:17:50.600
So there you have it.

00:17:50.600 --> 00:17:53.370
The proof, using the squeeze
theorem, and a little bit of

00:17:53.370 --> 00:17:57.960
visual trigonometry, of why the
limit, as x approaches 0, of

00:17:57.960 --> 00:18:00.970
sine of x over x is equal to 1.

00:18:00.970 --> 00:18:03.900
I hope I haven't confused you.


Negative exponent intuition

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:05.450
I have been asked for some
intuition as to why, let's say,

00:00:05.450 --> 00:00:12.030
a to the minus b is equal
to 1 over a to the b.

00:00:12.030 --> 00:00:14.120
And before I give you the
intuition, I want you to

00:00:14.120 --> 00:00:17.420
just realize that this
really is a definition.

00:00:17.420 --> 00:00:17.920
I don't know.

00:00:17.920 --> 00:00:20.950
The inventor of mathematics
wasn't one person.

00:00:20.950 --> 00:00:23.120
It was, you know, a
convention that arose.

00:00:23.120 --> 00:00:27.580
But they defined this, and they
defined this for the reasons

00:00:27.580 --> 00:00:28.680
that I'm going to show you.

00:00:28.680 --> 00:00:30.760
Well, what I'm going to show
you is one of the reasons, and

00:00:30.760 --> 00:00:33.670
then we'll see that this is a
good definition, because once

00:00:33.670 --> 00:00:36.290
you learned exponent rules, all
of the other exponent rules

00:00:36.290 --> 00:00:39.440
stay consistent for negative
exponents and when you raise

00:00:39.440 --> 00:00:41.590
something to the zero power.

00:00:41.590 --> 00:00:44.740
So let's take the
positive exponents.

00:00:44.740 --> 00:00:47.180
Those are pretty
intuitive, I think.

00:00:47.180 --> 00:00:54.200
So the positive exponents, so
you have a to the 1, a squared,

00:00:54.200 --> 00:00:58.140
a cubed, a to the fourth.

00:00:58.140 --> 00:01:02.940
What's a to the 1? a to the 1,
we said, is a, and then to get

00:01:02.940 --> 00:01:06.060
to a squared, what did we do?

00:01:06.060 --> 00:01:08.200
We multiplied by a, right?

00:01:08.200 --> 00:01:10.650
a squared is just a times a.

00:01:10.650 --> 00:01:13.040
And then to get to a
cubed, what did we do?

00:01:13.040 --> 00:01:15.160
We multiplied by a again.

00:01:15.160 --> 00:01:17.420
And then to get to a to the
fourth, what did we do?

00:01:17.420 --> 00:01:18.920
We multiplied by a again.

00:01:18.920 --> 00:01:21.020
Or the other way, you could
imagine, is when you

00:01:21.020 --> 00:01:24.480
decrease the exponent,
what are we doing?

00:01:24.480 --> 00:01:29.560
We are multiplying by
1/a, or dividing by a.

00:01:29.560 --> 00:01:33.140
And similarly, you decrease
again, you're dividing by a.

00:01:33.140 --> 00:01:35.480
And to go from a squared
to a to the first,

00:01:35.480 --> 00:01:38.510
you're dividing by a.

00:01:38.510 --> 00:01:40.700
So let's use this progression
to figure out what

00:01:40.700 --> 00:01:41.720
a to the 0 is.

00:01:41.720 --> 00:01:43.900
So this is the first hard one.

00:01:43.900 --> 00:01:45.010
So a to the 0.

00:01:45.010 --> 00:01:49.990
So you're the inventor, the
founding mother of mathematics,

00:01:49.990 --> 00:01:52.170
and you need to define
what a to the 0 is.

00:01:52.170 --> 00:01:55.420
And, you know, maybe
it's 17, maybe it's pi.

00:01:55.420 --> 00:01:56.100
I don't know.

00:01:56.100 --> 00:01:58.860
It's up to you to decide
what a to the 0 is.

00:01:58.860 --> 00:02:02.140
But wouldn't it be nice if a to
the 0 retained this pattern?

00:02:02.140 --> 00:02:04.520
That every time you decrease
the exponent, you're

00:02:04.520 --> 00:02:07.270
dividing by a, right?

00:02:07.270 --> 00:02:11.700
So if you're going from a to
the first to a to the zero,

00:02:11.700 --> 00:02:14.160
wouldn't it be nice if
we just divided by a?

00:02:14.160 --> 00:02:15.220
So let's do that.

00:02:15.220 --> 00:02:18.520
So if we go from a to the
first, which is just a, and

00:02:18.520 --> 00:02:22.770
divide by a, right, so we're
just going to go-- we're just

00:02:22.770 --> 00:02:27.140
going to divide it by a,
what is a divided by a?

00:02:27.140 --> 00:02:29.730
Well, it's just 1.

00:02:29.730 --> 00:02:32.060
So that's where the
definition-- or that's one of

00:02:32.060 --> 00:02:35.220
the intuitions behind why
something to the 0-th

00:02:35.220 --> 00:02:37.420
power is equal to 1.

00:02:37.420 --> 00:02:40.210
Because when you take that
number and you divide it by

00:02:40.210 --> 00:02:43.190
itself one more time,
you just get 1.

00:02:43.190 --> 00:02:45.100
So that's pretty reasonable,
but now let's go into

00:02:45.100 --> 00:02:45.890
the negative domain.

00:02:45.890 --> 00:02:47.660
So what should a to
the negative 1 equal?

00:02:51.810 --> 00:02:54.400
Well, once again, it's nice if
we can retain this pattern,

00:02:54.400 --> 00:02:56.190
where every time we decrease
the exponent we're

00:02:56.190 --> 00:02:57.690
dividing by a.

00:02:57.690 --> 00:03:01.500
So let's divide by
a again, so 1/a.

00:03:01.500 --> 00:03:06.140
So we're going to take a to
the 0 and divide it by a.

00:03:06.140 --> 00:03:09.610
a to the 0 is one, so
what's 1 divided by a?

00:03:09.610 --> 00:03:12.090
It's 1/a.

00:03:12.090 --> 00:03:13.540
Now, let's do it one more
time, and then I think you're

00:03:13.540 --> 00:03:15.330
going to get the pattern.

00:03:15.330 --> 00:03:16.880
Well, I think you probably
already got the pattern.

00:03:16.880 --> 00:03:18.350
What's a to the minus 2?

00:03:18.350 --> 00:03:21.070
Well, we want-- you know,
it'd be silly now to

00:03:21.070 --> 00:03:21.990
change this pattern.

00:03:21.990 --> 00:03:25.130
Every time we decrease the
exponent, we're dividing by a,

00:03:25.130 --> 00:03:28.055
so to go from a to the minus 1
to a to the minus 2, let's

00:03:28.055 --> 00:03:30.470
just divide by a again.

00:03:30.470 --> 00:03:32.550
And what do we get?

00:03:32.550 --> 00:03:36.040
If you take 1/2 and divide by
a, you get 1 over a squared.

00:03:36.040 --> 00:03:38.700
And you could just keep doing
this pattern all the way to

00:03:38.700 --> 00:03:42.330
the left, and you would get a
to the minus b is equal

00:03:42.330 --> 00:03:44.790
to 1 over a to the b.

00:03:44.790 --> 00:03:48.790
Hopefully, that gave you a
little intuition as to why--

00:03:48.790 --> 00:03:51.090
well, first of all, you know,
the big mystery is, you know,

00:03:51.090 --> 00:03:53.590
something to the 0-th power,
why does that equal 1?

00:03:53.590 --> 00:03:55.970
First, keep in mind that
that's just a definition.

00:03:55.970 --> 00:03:58.000
Someone decided it should
be equal to 1, but they

00:03:58.000 --> 00:03:59.080
had a good reason.

00:03:59.080 --> 00:04:01.340
And their good reason
was they wanted to keep

00:04:01.340 --> 00:04:02.620
this pattern going.

00:04:02.620 --> 00:04:04.730
And that's the same reason
why they defined negative

00:04:04.730 --> 00:04:07.440
exponents in this way.

00:04:07.440 --> 00:04:10.300
And what's extra cool about it
is not only does it retain this

00:04:10.300 --> 00:04:13.150
pattern of when you decrease
exponents, you're dividing by

00:04:13.150 --> 00:04:15.430
a, or when you're increasing
exponents, you're multiplying

00:04:15.430 --> 00:04:18.580
by a, but as you'll see in the
exponent rules videos, all

00:04:18.580 --> 00:04:20.460
of the exponent rules hold.

00:04:20.460 --> 00:04:23.820
All of the exponent rules are
consistent with this definition

00:04:23.820 --> 00:04:26.610
of something to the 0-th power
and this definition of

00:04:26.610 --> 00:04:28.470
something to the
negative power.

00:04:28.470 --> 00:04:31.090
Hopefully, that didn't confuse
you and gave you a little bit

00:04:31.090 --> 00:04:34.010
of intuition and demystified
something that, frankly,

00:04:34.010 --> 00:04:37.130
is quite mystifying the
first time you learn it.


Quadratic formula (proof)

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.130
In the completing the square
video I kept saying that all

00:00:03.130 --> 00:00:05.810
the quadratic equation is
completing the square

00:00:05.810 --> 00:00:07.450
as kind of a short cut
of completing square.

00:00:07.450 --> 00:00:10.220
And I was under the impression
that I had done this

00:00:10.220 --> 00:00:12.120
proof already but now I
realize that I haven't.

00:00:12.120 --> 00:00:15.750
So let me prove the quadratic
equation to you, by

00:00:15.750 --> 00:00:16.690
completing the square.

00:00:19.820 --> 00:00:23.210
So let's say I have a
quadratic equation.

00:00:23.210 --> 00:00:25.870
I guess a quadratic equation is
actually what you're trying to

00:00:25.870 --> 00:00:28.760
solve, and what a lot of people
call the quadratic equation is

00:00:28.760 --> 00:00:29.960
actually the quadratic formula.

00:00:29.960 --> 00:00:33.100
But anyway I don't want to get
caught up in terminology.

00:00:33.100 --> 00:00:36.030
But let's say that I have
a quadratic equation that

00:00:36.030 --> 00:00:46.450
says ax squared plus bx
plus c is equal to 0.

00:00:46.450 --> 00:00:48.280
And let's just complete
the square here.

00:00:48.280 --> 00:00:49.480
So how do we do that?

00:00:49.480 --> 00:00:56.940
Well let's subtract c from both
sides so we get ax squared plus

00:00:56.940 --> 00:01:00.740
the bx is equal to minus c.

00:01:00.740 --> 00:01:03.040
And just like I said in the
completing the square video

00:01:03.040 --> 00:01:06.200
I don't like having this
a coefficient here.

00:01:06.200 --> 00:01:08.355
I like just having one
coefficient on my x squared

00:01:08.355 --> 00:01:10.980
term so let me divide
everything by a.

00:01:10.980 --> 00:01:21.440
So I get x squared plus b/a
x is equal to-- you have

00:01:21.440 --> 00:01:24.695
to divide both sides
by a --minus c/a.

00:01:27.940 --> 00:01:29.600
Now we are ready to
complete the square.

00:01:29.600 --> 00:01:30.890
What was completing the square?

00:01:30.890 --> 00:01:34.790
Well it's somehow adding
something to this expression so

00:01:34.790 --> 00:01:38.590
it has the form of something
that is the square

00:01:38.590 --> 00:01:39.140
of an expression.

00:01:39.140 --> 00:01:39.940
What do i mean by that?

00:01:39.940 --> 00:01:43.400
Well, I'll do a
little aside here.

00:01:43.400 --> 00:01:51.950
if I told you that x plus
a squared, that equals

00:01:51.950 --> 00:01:57.500
x squared plus two ax
plus a squared, right?

00:01:57.500 --> 00:02:01.330
So if I can add something here
so that this left hand side

00:02:01.330 --> 00:02:05.810
this expression looks like
this, then I could

00:02:05.810 --> 00:02:06.330
go the other way.

00:02:06.330 --> 00:02:09.690
I can say this is going to be
x plus something squared.

00:02:09.690 --> 00:02:11.590
So what do I have to
add on both sides?

00:02:11.590 --> 00:02:15.140
If you watched the completing
the square video this should be

00:02:15.140 --> 00:02:17.730
hopefully intuitive for you.

00:02:17.730 --> 00:02:21.510
What you do is you say well
this b/a, this corresponds to

00:02:21.510 --> 00:02:26.183
the 2a term, so a is going to
be half of this, is going to

00:02:26.183 --> 00:02:28.010
be half of this coefficient.

00:02:28.010 --> 00:02:29.100
That would be the a.

00:02:29.100 --> 00:02:31.620
And then what I need
to add is a squared.

00:02:31.620 --> 00:02:34.930
So I need to take half of
this and then square it and

00:02:34.930 --> 00:02:36.110
then add it to both sides.

00:02:36.110 --> 00:02:38.865
Let me do that in a
different color.

00:02:38.865 --> 00:02:40.810
Do it in this magenta.

00:02:40.810 --> 00:02:42.650
So I'm going to take half of
this-- I'm just completing

00:02:42.650 --> 00:02:45.100
square, that's all I'm doing,
no magic here --so

00:02:45.100 --> 00:02:47.450
plus half of this.

00:02:47.450 --> 00:02:50.230
Well half of that
is b/2a right?

00:02:50.230 --> 00:02:52.130
You just multiply by 1/2.

00:02:52.130 --> 00:02:54.240
And I have to square it.

00:02:54.240 --> 00:02:55.893
Well if I did it to the left
hand side of the equation, I

00:02:55.893 --> 00:02:57.660
have to do it to the
right hand side.

00:02:57.660 --> 00:03:01.206
So plus b/2a squared.

00:03:07.470 --> 00:03:10.670
And now I have this left hand
side of the equation in the

00:03:10.670 --> 00:03:13.750
form that it is the square of
an expression that is

00:03:13.750 --> 00:03:14.950
x plus something.

00:03:14.950 --> 00:03:15.880
And what is it?

00:03:15.880 --> 00:03:19.970
Well that's equal to-- let me
switch colors again --what's

00:03:19.970 --> 00:03:21.730
the left hand side of
this equation equal to?

00:03:21.730 --> 00:03:24.520
And you can just use this
pattern and go to the left.

00:03:24.520 --> 00:03:28.730
It's x plus what?

00:03:28.730 --> 00:03:32.960
Well we said a, you can do one
of two ways. a is 1/2 of this

00:03:32.960 --> 00:03:36.390
coefficient or a is the square
root of this coefficient or

00:03:36.390 --> 00:03:38.310
since we didn't even square it
we know that this

00:03:38.310 --> 00:03:40.970
is a. b/2a is a.

00:03:40.970 --> 00:03:49.060
So this is the same thing as x
plus b over 2a everything

00:03:49.060 --> 00:03:55.980
squared, and then that equals--
let's see if we can simplify

00:03:55.980 --> 00:04:00.230
this or make this a little
bit cleaner --that equals--

00:04:00.230 --> 00:04:04.760
See, if I were to have a common
denominator-- I'm just doing a

00:04:04.760 --> 00:04:07.600
little bit of algebra here
--see, when I square this it's

00:04:07.600 --> 00:04:10.780
going to be 4a squared--
let me let me write this.

00:04:10.780 --> 00:04:15.740
This is equal to b
squared over 4a squared.

00:04:15.740 --> 00:04:16.710
Right?

00:04:16.710 --> 00:04:19.860
And so if I have to add these
two fractions, let me make

00:04:19.860 --> 00:04:29.550
this equal to 4a squared.

00:04:29.550 --> 00:04:30.330
Right?

00:04:30.330 --> 00:04:31.750
And if the denominator is
4a squared, what does

00:04:31.750 --> 00:04:34.360
the minus c/a become?

00:04:34.360 --> 00:04:40.280
I See if I multiply the
denominator by 4a, I have to

00:04:40.280 --> 00:04:41.810
multiply the numerator by 4a.

00:04:41.810 --> 00:04:50.090
So this becomes
minus 4ac, right?

00:04:50.090 --> 00:04:53.030
And then b squared over
4a squared, well that's

00:04:53.030 --> 00:04:54.810
just still b squared.

00:04:54.810 --> 00:04:56.520
I'm just doing a little
bit of algrebra.

00:04:56.520 --> 00:04:57.520
Hopefully I'm not
confusing you.

00:04:57.520 --> 00:04:59.470
I just expanded this.

00:04:59.470 --> 00:05:02.330
I just took the square of this,
b squared over 4a squared.

00:05:02.330 --> 00:05:04.790
And then I added this to this,
I got a common denominator.

00:05:04.790 --> 00:05:09.710
And minus c/a is the same thing
as minus 4ac over 4a squared.

00:05:09.710 --> 00:05:11.570
And now we can take the
square root of both

00:05:11.570 --> 00:05:13.240
sides of this equation.

00:05:13.240 --> 00:05:15.760
And this should hopefully
start to look a little

00:05:15.760 --> 00:05:17.490
bit familiar to you now.

00:05:17.490 --> 00:05:19.290
So let's see, so we get x.

00:05:19.290 --> 00:05:21.080
So if we take the square root
of both sides of this equation

00:05:21.080 --> 00:05:29.780
we get x plus b/2a is equal to
the square root of this-- let's

00:05:29.780 --> 00:05:32.180
take the square root of the
numerator and the demoninator.

00:05:32.180 --> 00:05:35.950
So the numerator is-- I'm going
to put the b squared first, I'm

00:05:35.950 --> 00:05:38.110
just going to switch this
order, it doesn't matter --the

00:05:38.110 --> 00:05:43.660
square root of b squared
minus 4ac, right?

00:05:43.660 --> 00:05:46.440
That's just the numerator.

00:05:46.440 --> 00:05:48.240
I just the square root of it,
and we have to get the square

00:05:48.240 --> 00:05:49.770
root of the denominator too.

00:05:49.770 --> 00:05:51.970
What's the square
roof of 4a squared?

00:05:51.970 --> 00:05:54.020
Well it's just 2a, right?

00:05:54.020 --> 00:05:55.950
2a.

00:05:55.950 --> 00:05:56.800
And now what do we do?

00:05:56.800 --> 00:05:58.640
Oh, it's very important!

00:05:58.640 --> 00:06:00.400
When we're taking the square
root, it's not just the

00:06:00.400 --> 00:06:01.070
positive square root.

00:06:01.070 --> 00:06:03.450
It's the positive or
minus square root.

00:06:03.450 --> 00:06:06.600
We saw that couple of times
when we did the-- and you could

00:06:06.600 --> 00:06:09.090
say it's a plus or minus here
too, but if you look plus or

00:06:09.090 --> 00:06:10.800
minus on the top and a plus or
minus on the bottom, you can

00:06:10.800 --> 00:06:12.290
just write it once on the top.

00:06:12.290 --> 00:06:14.930
I'll let you think about why
you only have to write it once.

00:06:14.930 --> 00:06:17.560
If you had a negative an a
plus, or negative and a plus

00:06:17.560 --> 00:06:19.250
sometimes cancel out, or a
negative and a negative,

00:06:19.250 --> 00:06:20.790
that's the same thing as
just having a plus on top.

00:06:20.790 --> 00:06:22.210
Anyway, I think you get that.

00:06:22.210 --> 00:06:26.140
And now we just have to
subtract b/2a from both sides.

00:06:26.140 --> 00:06:33.680
and we get, we get-- and this
is the exciting part --we get x

00:06:33.680 --> 00:06:42.850
is equal to minus be over to 2a
plus or minus this thing, so

00:06:42.850 --> 00:06:51.790
minus b squared minus 4ac,
all of that over 2a.

00:06:51.790 --> 00:06:53.850
And we already have a common
denominator, so we can

00:06:53.850 --> 00:06:55.130
just add the fractions.

00:06:55.130 --> 00:06:58.880
So we got --and I'm going to do
this in a vibrant bold-- I

00:06:58.880 --> 00:07:02.570
don't know maybe not so much
bold, well green color --so we

00:07:02.570 --> 00:07:10.970
get x is equal to, numerator,
negative b plus or minus square

00:07:10.970 --> 00:07:19.470
root of b squared minus
4ac, all of that over 2a.

00:07:19.470 --> 00:07:23.010
And that is the famous
quadratic formula.

00:07:23.010 --> 00:07:25.480
So, there we go we proved it.

00:07:25.480 --> 00:07:28.410
And we proved it just from
completing the square.

00:07:28.410 --> 00:07:31.570
I hope you found that
vaguely interesting.

00:07:31.570 --> 00:07:33.450
See in the next Video.


Completing the square

WEBVTT
Kind: captions
Language: en

00:00:00.870 --> 00:00:03.610
Welcome to the video on
completing the square.

00:00:03.610 --> 00:00:04.440
What's completing the square?

00:00:04.440 --> 00:00:06.740
Well, it's a way to solve
a quadratic equation.

00:00:06.740 --> 00:00:09.700
And actually, let me just write
down a quadratic equation, and

00:00:09.700 --> 00:00:11.570
then I will show you how
to complete the square.

00:00:11.570 --> 00:00:13.460
And then we'll do another
example, and then maybe talk

00:00:13.460 --> 00:00:16.650
a little bit about why it's
called completing the square.

00:00:16.650 --> 00:00:27.770
So let's say I have this
equation: x squared plus 16x

00:00:27.770 --> 00:00:32.600
minus 57 is equal to 0.

00:00:32.600 --> 00:00:36.130
So what are the tools in our
toolkit right now that we

00:00:36.130 --> 00:00:36.970
could use to solve this?

00:00:36.970 --> 00:00:38.570
Well, we could try
to factor it out.

00:00:38.570 --> 00:00:41.770
We could say, what two numbers
add up to 16, and then when you

00:00:41.770 --> 00:00:44.060
multiply them they're minus 57?

00:00:44.060 --> 00:00:45.450
And you'd have to think
about it a little bit.

00:00:45.450 --> 00:00:47.360
And you might get whole
numbers, but you're not even

00:00:47.360 --> 00:00:49.050
sure if there are two whole
numbers that work

00:00:49.050 --> 00:00:49.540
out like that.

00:00:49.540 --> 00:00:50.630
This problem there are.

00:00:50.630 --> 00:00:53.510
But, you know, sometimes the
solution is a decimal number

00:00:53.510 --> 00:00:54.190
and you don't know it.

00:00:54.190 --> 00:00:58.150
So the only time you can really
factor is if you're sure that

00:00:58.150 --> 00:01:01.000
you could factor this into
kind of integer expressions.

00:01:01.000 --> 00:01:03.620
You know, x plus some integer
or x minus some integer

00:01:03.620 --> 00:01:05.920
times, you know, x plus
some other integer.

00:01:05.920 --> 00:01:06.990
Or likewise.

00:01:06.990 --> 00:01:09.240
The other option is to do
the quadratic equation.

00:01:09.240 --> 00:01:11.420
And what we're going to see is
actually the quadratic equation

00:01:11.420 --> 00:01:15.510
is just essentially a shortcut
to completing the square.

00:01:15.510 --> 00:01:18.410
The quadratic equation is
actually proven using

00:01:18.410 --> 00:01:19.420
completing the square.

00:01:19.420 --> 00:01:21.420
So what is completing
the square?

00:01:21.420 --> 00:01:23.340
So what do we do?

00:01:23.340 --> 00:01:27.080
Well, before we move into this
video let's see what happens

00:01:27.080 --> 00:01:30.930
if I square an expression.

00:01:30.930 --> 00:01:33.220
Let me do it in this down here.

00:01:33.220 --> 00:01:40.250
What is x plus a, squared?

00:01:40.250 --> 00:01:50.940
Well that equals x squared
plus 2ax plus a squared.

00:01:50.940 --> 00:01:51.680
Right?

00:01:51.680 --> 00:01:55.420
So if you ever see something in
this form, you know that it's

00:01:55.420 --> 00:01:57.740
x plus something squared.

00:01:57.740 --> 00:02:01.040
So wouldn't it be neat if we
could manipulate this equation

00:02:01.040 --> 00:02:05.900
so we can write it as x plus a
squared equals something,

00:02:05.900 --> 00:02:08.140
and then we could just
take the square root?

00:02:08.140 --> 00:02:11.580
And what we're going to do
is, actually, do just that.

00:02:11.580 --> 00:02:13.090
And that is completing
the square.

00:02:13.090 --> 00:02:15.010
So let me show you an example.

00:02:15.010 --> 00:02:16.515
I think an example will
make it a little clearer.

00:02:16.515 --> 00:02:17.620
Let me box this away.

00:02:17.620 --> 00:02:19.310
This is what you
need to remember.

00:02:19.310 --> 00:02:22.130
This is the whole rationale
behind competing the squares--

00:02:22.130 --> 00:02:25.650
to get an equation into this
form, onto one side of the

00:02:25.650 --> 00:02:27.940
equation, and just have a
number on the other side, so

00:02:27.940 --> 00:02:31.210
you could take the square
root of both sides.

00:02:31.210 --> 00:02:32.000
So let's see.

00:02:32.000 --> 00:02:33.970
First of all, let's just check
to make sure this isn't

00:02:33.970 --> 00:02:35.020
a perfect square.

00:02:35.020 --> 00:02:39.700
If this were, this coefficient
would be equivalent to the 2a.

00:02:39.700 --> 00:02:40.470
Right?

00:02:40.470 --> 00:02:44.440
So a would be 8, and
then this would be 64.

00:02:44.440 --> 00:02:48.270
This is clearly not 64, so
this right here is not

00:02:48.270 --> 00:02:50.840
a squared expression.

00:02:50.840 --> 00:02:51.680
So what can we do?

00:02:51.680 --> 00:02:55.990
Well let me get rid of the
57 by adding 57 to both

00:02:55.990 --> 00:02:57.200
sides of this equation.

00:02:57.200 --> 00:03:07.550
So I would get x squared
plus 16x is equal to 57.

00:03:07.550 --> 00:03:11.470
All I did is I added 57 to
both sides of this equation.

00:03:11.470 --> 00:03:16.300
Now, what could I add here so
that this, the left-hand side

00:03:16.300 --> 00:03:21.480
of this equation, becomes a
square of some expression

00:03:21.480 --> 00:03:24.820
like x plus a?

00:03:24.820 --> 00:03:28.790
If you just follow this pattern
down here, we have x squared

00:03:28.790 --> 00:03:37.880
plus 2ax-- so you could view
this right here as 2ax.

00:03:37.880 --> 00:03:39.090
Right?

00:03:39.090 --> 00:03:40.900
That's 2ax.

00:03:40.900 --> 00:03:43.520
And then we need to add
an a squared to it.

00:03:43.520 --> 00:03:44.040
Right?

00:03:44.040 --> 00:03:46.300
Plus a squared.

00:03:46.300 --> 00:03:48.010
And then we would
have the form here.

00:03:48.010 --> 00:03:50.510
But we know from basic algebra
that anything you do to one

00:03:50.510 --> 00:03:52.080
side of an equation you
have to do to another.

00:03:52.080 --> 00:03:54.230
So we added an a squared
here, so let's add an a

00:03:54.230 --> 00:03:56.840
squared here as well.

00:03:56.840 --> 00:04:01.350
And now you could essentially
rewrite this as a square

00:04:01.350 --> 00:04:02.260
of some expression.

00:04:02.260 --> 00:04:04.210
But before that we have to
figure out what a was?

00:04:04.210 --> 00:04:05.520
Well how do we do that?

00:04:05.520 --> 00:04:06.740
Well, what is a?

00:04:06.740 --> 00:04:10.720
If this expression right
here is 2ax, what is a?

00:04:10.720 --> 00:04:15.380
Well 2a is going to equal
16, so a is equal to 8.

00:04:15.380 --> 00:04:18.020
And you could usually do
that just by inspection;

00:04:18.020 --> 00:04:18.630
do it in your head.

00:04:18.630 --> 00:04:20.930
But if you wanted to see it
done algebraically you could

00:04:20.930 --> 00:04:25.690
actually write 2ax
is equal to 16x.

00:04:25.690 --> 00:04:29.090
And then divide both sides
by 2x, and you get a is

00:04:29.090 --> 00:04:31.430
equal to 16x over 2x.

00:04:31.430 --> 00:04:36.950
And assuming that x doesn't
equal 0 this evaluates to 8.

00:04:36.950 --> 00:04:38.130
So a is 8.

00:04:38.130 --> 00:04:42.430
So if a is 8 we could rewrite
that expression-- I'll switch

00:04:42.430 --> 00:04:49.030
colors arbitrarily-- as x
squared plus 16x

00:04:49.030 --> 00:04:50.470
plus a squared.

00:04:50.470 --> 00:04:54.180
Well, it's 64, because a is 8.

00:04:54.180 --> 00:04:59.170
Is equal to 57 plus 64.

00:04:59.170 --> 00:05:00.720
Right?

00:05:00.720 --> 00:05:04.600
I went through a fairly tedious
explanation here, but all we've

00:05:04.600 --> 00:05:08.890
really done to get from there
to there is we added 57 to both

00:05:08.890 --> 00:05:10.870
sides of this equation to kind
of get it on the right-hand

00:05:10.870 --> 00:05:14.320
side, and then we added 64 to
both sides of this equation.

00:05:14.320 --> 00:05:16.830
And why did I add 64 to both
sides of this equation?

00:05:16.830 --> 00:05:21.070
So that the left-hand side
expression takes this form.

00:05:21.070 --> 00:05:23.200
Now that the left-hand side
expression takes this form

00:05:23.200 --> 00:05:26.030
I can rewrite it as what?

00:05:26.030 --> 00:05:27.170
x plus a, squared.

00:05:27.170 --> 00:05:28.620
I can rewrite it in this form.

00:05:28.620 --> 00:05:35.550
And we know that a is 8, so it
becomes x plus 8, squared,

00:05:35.550 --> 00:05:39.730
is equal to-- and
what's 57 plus 64?

00:05:39.730 --> 00:05:43.090
It's 121.

00:05:43.090 --> 00:05:47.270
Now we have what looks like a
fairly straightforward-- it's

00:05:47.270 --> 00:05:48.960
still a quadratic equation,
actually, because if you

00:05:48.960 --> 00:05:50.350
were to expand this side
you'd get a quadratic.

00:05:50.350 --> 00:05:53.065
But we can solve this without
using the quadratic equation

00:05:53.065 --> 00:05:54.610
or without having to factor.

00:05:54.610 --> 00:05:57.390
We can just take the square
root of both sides of this.

00:05:57.390 --> 00:06:00.550
And if we take the square root
of both sides what do we get?

00:06:00.550 --> 00:06:03.610
We get-- once again,
arbitrarily switching colors--

00:06:03.610 --> 00:06:09.230
that x plus 8 is equal to, and
remember this, the plus or

00:06:09.230 --> 00:06:12.880
minus square root of 121.

00:06:12.880 --> 00:06:14.590
And what's the
square root of 121?

00:06:14.590 --> 00:06:15.960
Well it's 11, right?

00:06:15.960 --> 00:06:17.630
So then we come here.

00:06:17.630 --> 00:06:18.800
Let me box this away.

00:06:18.800 --> 00:06:20.620
This was just an aside.

00:06:20.620 --> 00:06:26.830
So we get x plus 8 is equal
to plus or minus 11.

00:06:26.830 --> 00:06:30.420
And so x is equal to-- subtract
8 from both sides-- minus

00:06:30.420 --> 00:06:33.860
8 plus or minus 11.

00:06:33.860 --> 00:06:41.590
And so x could equal-- so
minus 8 plus 11 is 3.

00:06:41.590 --> 00:06:41.970
Right?

00:06:44.800 --> 00:06:48.160
Let me make sure I
did that right.

00:06:48.160 --> 00:06:53.310
x is equal to minus
8 plus or minus 11.

00:06:53.310 --> 00:06:54.140
Yes.

00:06:54.140 --> 00:06:55.350
That's right.

00:06:55.350 --> 00:06:59.270
So x could be equal to 3.

00:06:59.270 --> 00:07:02.960
And then if I took minus
8 minus 11, x could

00:07:02.960 --> 00:07:10.416
also equal minus 19.

00:07:10.416 --> 00:07:11.350
All right.

00:07:11.350 --> 00:07:13.200
And let's see if
that makes sense.

00:07:13.200 --> 00:07:18.680
So in theory this should be
able to be factored as x

00:07:18.680 --> 00:07:23.770
minus 3 times x plus
19 is equal to 0.

00:07:23.770 --> 00:07:24.030
Right?

00:07:24.030 --> 00:07:26.160
Because these are the two
solutions of this equation.

00:07:26.160 --> 00:07:28.190
And that works out, right?

00:07:28.190 --> 00:07:31.340
Minus 3 times 19 is minus 57.

00:07:31.340 --> 00:07:36.920
And minus 3 plus
19 is plus 16x.

00:07:36.920 --> 00:07:39.120
We could have just immediately
factored it this way, but if

00:07:39.120 --> 00:07:41.030
that wasn't obvious to us--
because, you know, at least

00:07:41.030 --> 00:07:43.600
19 is kind of a strange
number-- we could do it

00:07:43.600 --> 00:07:46.800
completing the square.

00:07:46.800 --> 00:07:47.690
And so why is it called
completing the square?

00:07:47.690 --> 00:07:49.920
Because you get it in this form
and then you have to add this

00:07:49.920 --> 00:07:52.950
64 here to kind of complete the
square-- to turn this

00:07:52.950 --> 00:07:56.020
left-hand expression into
a squared expression.

00:07:56.020 --> 00:07:56.770
Let's do one more.

00:07:56.770 --> 00:07:59.920
And I'll do less explanation
and more just chugging through

00:07:59.920 --> 00:08:02.105
the problem, and that actually
might make it seem simpler.

00:08:04.800 --> 00:08:07.080
But this is going to
be a hairier problem.

00:08:07.080 --> 00:08:19.930
So let's say I have 6x squared
minus 7x minus 3 is equal to 0.

00:08:19.930 --> 00:08:22.980
You could try to factor it,
but personally I don't

00:08:22.980 --> 00:08:25.260
enjoy factoring things
when I have a coefficient.

00:08:25.260 --> 00:08:27.590
And you can say, oh well why
don't we divide both sides

00:08:27.590 --> 00:08:28.970
of this equation by 6?

00:08:28.970 --> 00:08:30.960
But then you'd get a fraction
here and a fraction here.

00:08:30.960 --> 00:08:33.580
And that's even worse to
factor just by inspection.

00:08:33.580 --> 00:08:35.190
You could do the
quadratic equation.

00:08:35.190 --> 00:08:37.310
And maybe I'll show you in a
future video, the quadratic

00:08:37.310 --> 00:08:39.500
equation-- and I think I've
already done one where I proved

00:08:39.500 --> 00:08:40.630
the quadratic equation.

00:08:40.630 --> 00:08:42.380
But the quadratic
equation is essentially

00:08:42.380 --> 00:08:43.170
completing the square.

00:08:43.170 --> 00:08:44.090
It's kind of a shortcut.

00:08:44.090 --> 00:08:46.280
It's just kind of
remembering the formula.

00:08:46.280 --> 00:08:48.320
But let's complete the square
here, because that's what the

00:08:48.320 --> 00:08:50.640
point of this video was.

00:08:50.640 --> 00:08:54.650
So let's add the 3 to both
sides of that equation.

00:08:54.650 --> 00:08:56.300
We could do-- well,
let's add the 3 first.

00:08:56.300 --> 00:09:04.820
So you get 6 x squared
minus 7x is equal to 3.

00:09:04.820 --> 00:09:06.770
I added 3 to both sides.

00:09:06.770 --> 00:09:09.470
And some teachers will leave
the minus 3 here, and then try

00:09:09.470 --> 00:09:11.050
to figure out what to add
to it and all of that.

00:09:11.050 --> 00:09:13.170
But I like to get it out of the
way so that I can figure out

00:09:13.170 --> 00:09:16.080
very clearly what number
I should put here.

00:09:16.080 --> 00:09:18.230
But I also don't
like the 6 here.

00:09:18.230 --> 00:09:19.550
It just complicates things.

00:09:19.550 --> 00:09:25.990
I like to have it x plus a
squared, not some square root

00:09:25.990 --> 00:09:27.450
coefficient on the x term.

00:09:27.450 --> 00:09:31.530
So let's divide both sides of
this equation by 6, and you get

00:09:31.530 --> 00:09:39.730
x squared minus 7/6 x is equal
to-- 3 divided by 6

00:09:39.730 --> 00:09:41.566
is equal to 1/2.

00:09:41.566 --> 00:09:43.190
And we could have made
that our first step.

00:09:43.190 --> 00:09:46.450
We could have divided by 6
right at that first step.

00:09:46.450 --> 00:09:49.250
Anyway, now let's try to
complete the square.

00:09:49.250 --> 00:09:51.800
So we have x squared-- I'm just
going to open up some space--

00:09:51.800 --> 00:09:59.530
minus 7/6 x plus something is
going to be equal to 1/2.

00:09:59.530 --> 00:10:02.400
And so we have to add something
here so that this left-hand

00:10:02.400 --> 00:10:05.290
expression becomes a
squared expression.

00:10:05.290 --> 00:10:06.620
So how do we do that?

00:10:06.620 --> 00:10:10.770
Well essentially we look at
this coefficient, and keep

00:10:10.770 --> 00:10:14.610
in mind this is not just
7/6 it's minus 7/6.

00:10:14.610 --> 00:10:17.460
You take 1/2 of it, and
then you square it.

00:10:17.460 --> 00:10:18.610
Right?

00:10:18.610 --> 00:10:19.690
Let me do it.

00:10:19.690 --> 00:10:25.290
x plus a, squared, is
equal to x squared plus

00:10:25.290 --> 00:10:28.820
2ax plus a squared.

00:10:28.820 --> 00:10:29.070
Right?

00:10:29.070 --> 00:10:30.750
This is what you have to
remember all the time.

00:10:30.750 --> 00:10:33.560
That's all completing the
square is based off of.

00:10:33.560 --> 00:10:34.980
So what did I say just now?

00:10:34.980 --> 00:10:37.260
Well, this term is going
to be 1/2 of this

00:10:37.260 --> 00:10:39.190
coefficient squared.

00:10:39.190 --> 00:10:40.190
And how do we know that?

00:10:40.190 --> 00:10:43.880
Because a is going to be 1/2 of
this coefficient if you just

00:10:43.880 --> 00:10:45.850
do a little bit of
pattern matching.

00:10:45.850 --> 00:10:48.760
So what's 1/2 of
this coefficient?

00:10:48.760 --> 00:10:54.050
1/2 of minus 7/6 is minus 7/12.

00:10:54.050 --> 00:10:56.640
So if you want you could
write a equals minus

00:10:56.640 --> 00:10:58.770
7/12 for our example.

00:10:58.770 --> 00:11:00.770
And I just multiplied
this by 1/2.

00:11:00.770 --> 00:11:01.980
Right?

00:11:01.980 --> 00:11:03.660
So what do I add to both sides?

00:11:03.660 --> 00:11:06.030
I add a squared.

00:11:06.030 --> 00:11:08.930
So what's 7/12 squared?

00:11:08.930 --> 00:11:13.220
Well that's going to be 49/144.

00:11:13.220 --> 00:11:15.000
If I did it to the left-hand
side I have to do it to

00:11:15.000 --> 00:11:16.630
the right-hand side.

00:11:16.630 --> 00:11:22.240
Plus 49/144.

00:11:22.240 --> 00:11:26.120
And now how can I simplify
this left-hand side?

00:11:26.120 --> 00:11:26.880
What's our next step?

00:11:26.880 --> 00:11:28.470
Well we now know it
is a perfect square.

00:11:28.470 --> 00:11:31.550
In fact, we know what a
is. a is minus 7/12.

00:11:31.550 --> 00:11:35.200
And so we know that this
left-hand side of this equation

00:11:35.200 --> 00:11:43.390
is x minus a-- or x plus a,
but a is a negative number.

00:11:43.390 --> 00:11:47.980
So x plus a, and a is
negative, squared.

00:11:47.980 --> 00:11:50.350
And if you want you can
multiply this out and confirm

00:11:50.350 --> 00:11:53.130
that it truly equals this.

00:11:53.130 --> 00:11:55.920
And that is going to be equal
to-- let's get a common

00:11:55.920 --> 00:11:58.360
denominator, 144.

00:11:58.360 --> 00:12:04.070
So 72 plus 49 equals 121.

00:12:04.070 --> 00:12:06.300
121/144.

00:12:06.300 --> 00:12:09.210
So we have x minus 7/12,
all of that squared

00:12:09.210 --> 00:12:13.180
is equal to 121/144.

00:12:13.180 --> 00:12:14.300
So what do we do now?

00:12:14.300 --> 00:12:15.570
Well now we just take
the square root of both

00:12:15.570 --> 00:12:17.700
sides of this equation.

00:12:17.700 --> 00:12:20.140
And I'm trying to
free up some space.

00:12:20.140 --> 00:12:22.215
Switch to green.

00:12:22.215 --> 00:12:25.320
Let me partition this off.

00:12:25.320 --> 00:12:33.310
And we get x minus 7/12 is
equal to the plus or minus

00:12:33.310 --> 00:12:33.940
square root of that.

00:12:33.940 --> 00:12:38.120
So plus or minus 11/12.

00:12:38.120 --> 00:12:38.390
Right?

00:12:38.390 --> 00:12:39.660
Square root of 121 is 11.

00:12:39.660 --> 00:12:42.420
Square root of 144 is 12.

00:12:42.420 --> 00:12:44.480
So then we could add 7/12 to
both sides of this equation,

00:12:44.480 --> 00:12:53.100
and we get x is equal to
7/12 plus or minus 11/12.

00:12:53.100 --> 00:12:58.660
Well that equals 7
plus or minus 11/12.

00:12:58.660 --> 00:13:00.050
So what are the two options?

00:13:00.050 --> 00:13:03.930
7 plus 11 is 18, over 12.

00:13:03.930 --> 00:13:08.210
So x could equal 18/12, is 3/2.

00:13:08.210 --> 00:13:11.010
Or, what's 7 minus 11?

00:13:11.010 --> 00:13:12.760
That's minus 4/12.

00:13:12.760 --> 00:13:15.370
So it's minus 1/3.

00:13:15.370 --> 00:13:16.630
There you have it.

00:13:16.630 --> 00:13:17.940
That is completing the square.

00:13:17.940 --> 00:13:20.220
Hopefully you found that
reasonably insightful.

00:13:20.220 --> 00:13:23.340
And if you want to prove the
quadratic equation, all you

00:13:23.340 --> 00:13:27.320
have to do is instead of having
numbers here, write a x squared

00:13:27.320 --> 00:13:29.820
plus bx plus c equals 0.

00:13:29.820 --> 00:13:34.130
And then complete the square
using the a, b, and c's

00:13:34.130 --> 00:13:35.060
instead of numbers.

00:13:35.060 --> 00:13:37.180
And you will end up with
the quadratic equation

00:13:37.180 --> 00:13:38.110
by this point.

00:13:38.110 --> 00:13:39.510
And I think I did
that in a video.

00:13:39.510 --> 00:13:41.600
Let me know if I didn't
and I'll do it for you.

00:13:41.600 --> 00:13:44.540
Anyway, I'll see you
in the next video.


Simple Differential Equations

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:04.030
Welcome to the video on
the introduction to

00:00:04.030 --> 00:00:05.020
differential equations.

00:00:05.020 --> 00:00:06.200
What's a differential?

00:00:06.200 --> 00:00:10.700
Well, we've been doing things
like if I said that y is

00:00:10.700 --> 00:00:12.900
equal to, I don't know, 3x.

00:00:12.900 --> 00:00:15.497
If we took the derivative of
both sides of this equation and

00:00:15.497 --> 00:00:18.720
we said -- one of the notations
we used is we said dy

00:00:18.720 --> 00:00:21.620
over dx is equal to 3.

00:00:21.620 --> 00:00:23.830
We took the derivative
of both sides of that.

00:00:23.830 --> 00:00:26.460
But this you could almost
use differential notation.

00:00:26.460 --> 00:00:31.640
dy is a differential and
dx is a differential.

00:00:31.640 --> 00:00:33.730
What does a differential mean?

00:00:33.730 --> 00:00:38.170
A differential just means an
infinitely small change in y,

00:00:38.170 --> 00:00:41.140
in this case, so that's a
differential in y, or an

00:00:41.140 --> 00:00:42.330
infinitely small change in x.

00:00:42.330 --> 00:00:46.570
So a differential it's like a
difference -- when we learned

00:00:46.570 --> 00:00:51.340
slope, we said slope is change
in y over change in x.

00:00:51.340 --> 00:00:53.950
Or you could say the difference
in y over difference in x.

00:00:53.950 --> 00:00:57.100
But once we started taking the
slope of curves we had to make

00:00:57.100 --> 00:01:01.130
this difference and this
difference approach 0.

00:01:01.130 --> 00:01:02.800
That's what a differential
essentially is.

00:01:02.800 --> 00:01:04.900
A differential is a really,
really, really, really

00:01:04.900 --> 00:01:06.810
infinitely small difference.

00:01:06.810 --> 00:01:08.910
So what is a
differential equation?

00:01:08.910 --> 00:01:11.680
Well it's an equation that
involves differentials.

00:01:11.680 --> 00:01:14.460
So, for example, this is
a differential equation.

00:01:14.460 --> 00:01:18.070
This will be the first
example that we solve.

00:01:18.070 --> 00:01:26.790
dy over dx is equal
to x squared plus 1.

00:01:26.790 --> 00:01:29.520
So how do we solve this
differential equation?

00:01:29.520 --> 00:01:31.750
Well, there's a couple of
ways you could say it.

00:01:31.750 --> 00:01:35.030
You say oh, well this is the
same thing as f prime of x.

00:01:35.030 --> 00:01:36.860
You know, y to the function
of x and you just take the

00:01:36.860 --> 00:01:38.530
antiderivative of both sides.

00:01:38.530 --> 00:01:40.700
But let's solve this properly
as a differential equation.

00:01:40.700 --> 00:01:44.330
So you can actually manipulate
these differentials the same

00:01:44.330 --> 00:01:47.780
way that you would manipulate
numbers for the most part.

00:01:47.780 --> 00:01:49.030
And so what could we do here?

00:01:49.030 --> 00:01:52.840
Let's multiply both sides
of this equation by dx.

00:01:52.840 --> 00:02:00.510
So you get, for a very small
change in y, dy, you get x

00:02:00.510 --> 00:02:05.550
squared plus 1 times a very
small change in the x.

00:02:05.550 --> 00:02:08.500
Or another way to say it is for
a very small change in x, if

00:02:08.500 --> 00:02:12.330
you want to figure out how much
does y change, you multiply

00:02:12.330 --> 00:02:15.760
times x squared plus 1 wherever
you are on the curve.

00:02:15.760 --> 00:02:17.190
So what does that do for us?

00:02:17.190 --> 00:02:20.240
Well let's take the
antiderivative of both

00:02:20.240 --> 00:02:21.810
sides or let's take the
integral of both sides.

00:02:21.810 --> 00:02:25.890
And if you watched all of the
videos on integration and the

00:02:25.890 --> 00:02:30.590
definite integral and area and
a curve, you realize that an

00:02:30.590 --> 00:02:34.950
integral is essentially a sum,
it's kind of an infinite

00:02:34.950 --> 00:02:38.400
sum of a bunch of these
infinitely small dy's.

00:02:38.400 --> 00:02:43.020
So if you take the sum of
all of the changes in dy's,

00:02:43.020 --> 00:02:45.070
you're left with a y on
this side of this equation.

00:02:45.070 --> 00:02:46.620
You might want to re-watch
all the videos that we

00:02:46.620 --> 00:02:49.010
did on integration.

00:02:49.010 --> 00:02:49.760
Then what is this?

00:02:49.760 --> 00:02:52.670
This is essentially just
-- this is the indefinite

00:02:52.670 --> 00:02:54.230
integral of x squared plus 1.

00:02:54.230 --> 00:02:56.020
So it's the
antiderivative of that.

00:02:56.020 --> 00:02:59.510
And what's the antiderivative
of x squared plus 1?

00:02:59.510 --> 00:03:04.500
Well it's x to the
third over 3.

00:03:04.500 --> 00:03:06.690
We're just taking the
derivative backwards.

00:03:06.690 --> 00:03:09.690
And once again, watch the
videos on the antiderivative.

00:03:09.690 --> 00:03:12.030
There's I think eight
or nine of them.

00:03:12.030 --> 00:03:15.120
Plus x and then plus c.

00:03:15.120 --> 00:03:16.890
And where does the c come from?

00:03:16.890 --> 00:03:18.760
Well we know that when we
take the derivative of a

00:03:18.760 --> 00:03:20.590
constant it goes to 0.

00:03:20.590 --> 00:03:22.540
So when you take the
antiderivative, we're like

00:03:22.540 --> 00:03:24.140
well, there could have
been a constant there.

00:03:24.140 --> 00:03:26.290
And that's where that
plus c comes from.

00:03:26.290 --> 00:03:33.530
So this is the general solution
to this differential equation.

00:03:33.530 --> 00:03:34.770
And that's something
interesting.

00:03:34.770 --> 00:03:37.910
So with traditional equations,
the solution tended to

00:03:37.910 --> 00:03:39.680
be a number, right?

00:03:39.680 --> 00:03:46.920
If I just told you y is
equal to 2y minus 1.

00:03:46.920 --> 00:03:49.820
Then you would get minus y
is equal to minus 1, you'd

00:03:49.820 --> 00:03:51.380
get y is equal to 1.

00:03:51.380 --> 00:03:54.390
So this is a traditional
equation and your solution

00:03:54.390 --> 00:03:58.360
was just a value -- you
solved for the variable.

00:03:58.360 --> 00:04:00.370
Differential equations
are something different.

00:04:00.370 --> 00:04:02.690
The solution is
actually a function.

00:04:02.690 --> 00:04:06.470
You're saying what
function satisfies this

00:04:06.470 --> 00:04:07.380
differential equation.

00:04:07.380 --> 00:04:08.910
So that's something
to keep in mind.

00:04:08.910 --> 00:04:10.670
Right now we're doing very
basic differential equations,

00:04:10.670 --> 00:04:12.490
but that's something to keep in
mind the whole time you learn

00:04:12.490 --> 00:04:13.210
differential equations.

00:04:13.210 --> 00:04:17.520
I think I'll eventually do
a play list on essentially

00:04:17.520 --> 00:04:19.600
an introductory course on
differential equations that

00:04:19.600 --> 00:04:21.000
you would take at college.

00:04:21.000 --> 00:04:22.880
And that applies even when
you start doing partial

00:04:22.880 --> 00:04:25.090
differential equations,
et cetera, et cetera.

00:04:25.090 --> 00:04:28.730
The solution to a differential
equation is not a number,

00:04:28.730 --> 00:04:30.660
it is a function.

00:04:30.660 --> 00:04:32.490
So anyway, this was the
general solution to this

00:04:32.490 --> 00:04:33.490
differential equation.

00:04:33.490 --> 00:04:36.370
And if you want the particular
solution, people normally give

00:04:36.370 --> 00:04:42.120
you initial conditions or they
give you points on the function

00:04:42.120 --> 00:04:43.740
and then you can
substitute back.

00:04:43.740 --> 00:04:48.150
In this problem they said that
dy dx equals x squared plus 1.

00:04:48.150 --> 00:04:51.230
And they said that -- let me
switch colors -- they said that

00:04:51.230 --> 00:04:56.980
y is equal to 1 at x is equal
to 1, or y is equal to 1

00:04:56.980 --> 00:04:58.690
when x is equal to 1.

00:04:58.690 --> 00:05:01.380
So we can use this information
now to solve for c.

00:05:01.380 --> 00:05:02.080
How do you do that?

00:05:02.080 --> 00:05:05.060
Well it says y is equal to 1.

00:05:05.060 --> 00:05:10.440
So 1 is equal to when x is
equal to 1, is equal to 1/3,

00:05:10.440 --> 00:05:18.170
right, 1 over 3, 1 to the third
power over 3, plus 1 plus c.

00:05:18.170 --> 00:05:19.130
And let's see what I can do.

00:05:19.130 --> 00:05:22.200
Subtract one from both
sides -- this comes 0.

00:05:22.200 --> 00:05:24.910
Subtract 1/3 from both
sides and you get c is

00:05:24.910 --> 00:05:27.420
equal to minus 1/3.

00:05:27.420 --> 00:05:31.860
So using these conditions, a
point where this function

00:05:31.860 --> 00:05:34.560
crosses through, we can now
give you the particular

00:05:34.560 --> 00:05:36.960
solution to this
differential equation.

00:05:36.960 --> 00:05:48.010
And that is y is equal to x to
the third over 3 plus x minus

00:05:48.010 --> 00:05:50.280
1/3, and then we just
solve for the c.

00:05:50.280 --> 00:05:54.670
And if you don't believe me,
take this expression and

00:05:54.670 --> 00:05:57.760
substitute it here and you will
see that it equals, if you were

00:05:57.760 --> 00:06:00.310
to take the derivative of y
with respect to x, you would

00:06:00.310 --> 00:06:02.670
see that it equals
x squared plus 1.

00:06:02.670 --> 00:06:03.550
Let's do another one.

00:06:08.410 --> 00:06:11.420
So it tells us -- so this is a
little bit more interesting --

00:06:11.420 --> 00:06:24.160
it says dy dx is equal to
x over y, and it has

00:06:24.160 --> 00:06:24.830
the same conditions.

00:06:24.830 --> 00:06:29.980
It says y equals
1 at x equals 1.

00:06:29.980 --> 00:06:31.640
Or when x equals 1, y equals 1.

00:06:31.640 --> 00:06:34.790
So let's find the function
that satisfies this equation.

00:06:34.790 --> 00:06:37.130
This one's interesting because
we have an x and a y on

00:06:37.130 --> 00:06:38.380
the right hand side
of the equation.

00:06:38.380 --> 00:06:40.910
So this kind of looks like
something we got after we

00:06:40.910 --> 00:06:42.680
did some type of implicit
differentiation.

00:06:42.680 --> 00:06:45.480
But let's see where we go.

00:06:45.480 --> 00:06:46.840
So let's do the same thing.

00:06:46.840 --> 00:06:51.280
Let's multiply both sides of
this equation by dx until you

00:06:51.280 --> 00:06:57.380
get dy is equal to x over y dx.

00:06:57.380 --> 00:07:01.290
And let's get this y over
onto the dy side, because

00:07:01.290 --> 00:07:03.280
it will be easy to take
the antiderivative then.

00:07:03.280 --> 00:07:11.990
So we get y dy is
equal to x dx.

00:07:11.990 --> 00:07:15.796
Now we can take the integral of
both sides of this equation, or

00:07:15.796 --> 00:07:18.070
take the antiderivative on this
side with respect to y on

00:07:18.070 --> 00:07:19.960
the side with respect to x.

00:07:22.630 --> 00:07:24.790
I don't know why I
did it in brown.

00:07:24.790 --> 00:07:27.650
So what's the
antiderivative here?

00:07:27.650 --> 00:07:34.500
Well, it is y squared over 2.

00:07:34.500 --> 00:07:38.060
And I could say plus
c, some constant.

00:07:38.060 --> 00:07:39.180
Let me do that.

00:07:39.180 --> 00:07:40.750
There could be a constant here.

00:07:40.750 --> 00:07:44.410
I'll call it c1.

00:07:44.410 --> 00:07:46.410
We don't know what
that constant is.

00:07:46.410 --> 00:07:57.590
And that equals x squared
over 2 plus c2, right?

00:07:57.590 --> 00:07:58.720
Or some other constant.

00:07:58.720 --> 00:08:00.630
Maybe it's the same
number, I don't know.

00:08:00.630 --> 00:08:02.350
But we don't know what
either of these are.

00:08:02.350 --> 00:08:07.010
I could re-write it as --
let's see, what could I do?

00:08:07.010 --> 00:08:08.820
Let me take the x
out of that side.

00:08:08.820 --> 00:08:18.450
So I could have y squared over
2 minus x squared over 2 is

00:08:18.450 --> 00:08:21.700
equal to -- and let me subtract
this from that side so I get

00:08:21.700 --> 00:08:24.070
the constants all on the right
hand side of the equation.

00:08:24.070 --> 00:08:26.840
c2 minus c1.

00:08:26.840 --> 00:08:29.960
I just took a c1, put it on the
right hand side, took the x

00:08:29.960 --> 00:08:31.530
squared over 2, put it on the
left hand side, that's

00:08:31.530 --> 00:08:32.430
why it's negative.

00:08:32.430 --> 00:08:33.500
And we didn't know.

00:08:33.500 --> 00:08:34.990
We said this could be
any constant and this

00:08:34.990 --> 00:08:35.760
could be any constant.

00:08:35.760 --> 00:08:39.200
So the difference between two
arbitrary constants could just

00:08:39.200 --> 00:08:40.610
be a third arbitrary constant.

00:08:40.610 --> 00:08:43.720
So I'll just
re-write that as c.

00:08:43.720 --> 00:08:48.620
So the general solution to this
differential equation is y

00:08:48.620 --> 00:08:56.080
squared over 2 minus x squared
over 2 is equal to c.

00:08:56.080 --> 00:08:57.810
Actually let's do something
else just to clean it up a

00:08:57.810 --> 00:09:00.090
little bit, because once again
this could be any constant.

00:09:00.090 --> 00:09:02.870
So let's multiply both sides
of this equation by 2 and

00:09:02.870 --> 00:09:08.740
you get y squared minus
x squared equal to 2c.

00:09:08.740 --> 00:09:11.280
Well now this is still any
constant number, so we could

00:09:11.280 --> 00:09:12.460
still write this as a c.

00:09:12.460 --> 00:09:17.070
So we have y squared minus
x squared is equal to c.

00:09:17.070 --> 00:09:22.570
Now let's use our initial
conditions to see what c is.

00:09:22.570 --> 00:09:29.110
So when y is 1, so 1 squared
minus when x is 1, 1 squared is

00:09:29.110 --> 00:09:34.770
equal to c, this is 1 minus 1,
so it's 0, right?

00:09:34.770 --> 00:09:36.590
c is equal to 0.

00:09:36.590 --> 00:09:39.570
So what is the particular
solution to this

00:09:39.570 --> 00:09:40.350
differential equation?

00:09:40.350 --> 00:09:42.480
I'll do it in green.

00:09:42.480 --> 00:09:48.350
It is y squared minus x
squared is equal to 0.

00:09:48.350 --> 00:09:50.270
Or we could add x squared
to both sides of that.

00:09:50.270 --> 00:09:55.510
We could also write it as y
squared is equal to x squared.

00:09:55.510 --> 00:09:58.870
Now you might be tempted to
take the square root of both

00:09:58.870 --> 00:10:02.490
sides of this and say
that y is equal to x.

00:10:02.490 --> 00:10:07.370
The reason why this would not
be accurate is because here x

00:10:07.370 --> 00:10:14.290
could be minus 2 and y could
be plus 2 or vice versa.

00:10:14.290 --> 00:10:16.560
So this would satisfy this
equation, but it would not

00:10:16.560 --> 00:10:17.710
satisfy this equation.

00:10:17.710 --> 00:10:20.380
So be careful when you
take that square root.

00:10:20.380 --> 00:10:22.180
You have to worry about
the plus or minus.

00:10:22.180 --> 00:10:23.020
Let's do one more.

00:10:29.750 --> 00:10:36.210
So this one says the derivative
of y with respect to x is equal

00:10:36.210 --> 00:10:42.130
to -- OK, so this is even a
little bit more interesting.

00:10:42.130 --> 00:10:47.775
This is equal to the
square root of x over y.

00:10:51.200 --> 00:10:56.890
And it says that y is equal
to 4 when is equal to 1.

00:10:56.890 --> 00:10:58.220
Yup, I think I'm
reading that right.

00:10:58.220 --> 00:11:00.670
We could do this one
very similarly.

00:11:00.670 --> 00:11:03.960
So let's just do the same step.

00:11:03.960 --> 00:11:06.660
So multiply both sides of
the equation times dx.

00:11:06.660 --> 00:11:11.940
So you get dy is equal to --
and just to skip a step, I'm

00:11:11.940 --> 00:11:15.630
going to re-write square root
of x over y is the square root

00:11:15.630 --> 00:11:18.210
of x over the square root
of y, and I multiply both

00:11:18.210 --> 00:11:21.180
sides of that times dx.

00:11:21.180 --> 00:11:23.010
Now let's multiply both
sides of this equation

00:11:23.010 --> 00:11:25.800
by the square root of y.

00:11:25.800 --> 00:11:29.530
And I'm just going to re-write
it as y to the 1/2 power.

00:11:29.530 --> 00:11:31.310
That's the same thing as
the square root of y.

00:11:31.310 --> 00:11:35.730
dy is equal to x to
the 1/2 power dx.

00:11:35.730 --> 00:11:37.400
I just multiplied that
there and re-wrote

00:11:37.400 --> 00:11:39.310
it as y to the 1/2.

00:11:39.310 --> 00:11:45.320
So what is the antiderivative
of y to the 1/2 power?

00:11:45.320 --> 00:11:54.780
Well it's just this plus 1, so
it's y the 3/2 power, and then

00:11:54.780 --> 00:11:56.960
times the inverse of this.

00:11:56.960 --> 00:11:57.940
So times 2/3.

00:12:00.730 --> 00:12:03.320
Or you could have said
divided by this, either way.

00:12:03.320 --> 00:12:05.610
And if you're not sure about
this, because sometimes it is

00:12:05.610 --> 00:12:08.560
confusing with the fractions
and whatever else,

00:12:08.560 --> 00:12:09.670
take the derivative.

00:12:09.670 --> 00:12:12.540
3/2 times 2/3 is 1, and then
you subtract this 1 from

00:12:12.540 --> 00:12:14.220
the exponent and you
get y to the 1/2.

00:12:14.220 --> 00:12:15.140
So that works.

00:12:15.140 --> 00:12:19.540
And of course, we kind of go
through the same drill, plus c1

00:12:19.540 --> 00:12:21.710
is equal to -- it's going to be
the same thing on this side

00:12:21.710 --> 00:12:29.460
-- 2/3 x to the 3/2 plus
c2, and what can we do?

00:12:29.460 --> 00:12:32.890
Well let's take the x to the
left hand side of this

00:12:32.890 --> 00:12:42.990
equation, so we get 2/3 y to
the 3/2 minus 2/3 x to the 3/2

00:12:42.990 --> 00:12:49.190
is equal to c2 minus c1, which
we can just re-write as c.

00:12:49.190 --> 00:12:52.820
And let's multiply both sides
of this equation by 3/2, so

00:12:52.820 --> 00:12:54.440
these two will become 1.

00:12:54.440 --> 00:13:03.980
So you get y to 3/2 minus x to
the 3/2 is equal to -- well

00:13:03.980 --> 00:13:05.970
what's 3/2 times
some constant c.

00:13:05.970 --> 00:13:07.800
But we didn't even know what
it was, we haven't solved for

00:13:07.800 --> 00:13:08.800
it, so we can still write c.

00:13:08.800 --> 00:13:10.060
I hope that doesn't
confuse you.

00:13:10.060 --> 00:13:12.100
3/2 c, we didn't
know what it was.

00:13:12.100 --> 00:13:14.480
We could call this c3
and now this is c4.

00:13:14.480 --> 00:13:17.440
It's a different constant, but
we still have to solve for it.

00:13:17.440 --> 00:13:19.790
And now let's use our
initial conditions.

00:13:19.790 --> 00:13:24.725
So the initial conditions tell
us that 4 -- and they don't

00:13:24.725 --> 00:13:25.640
have to be initial conditions.

00:13:25.640 --> 00:13:29.140
You can kind of say conditions
or points where we know that

00:13:29.140 --> 00:13:33.830
the particular solution to this
differential equation

00:13:33.830 --> 00:13:35.370
are satisfied.

00:13:35.370 --> 00:13:45.670
So 4 to the 3/2 minus 1 to
the 3/2 is equal to c.

00:13:45.670 --> 00:13:46.830
What's 4 to the 3/2?

00:13:46.830 --> 00:13:51.620
So 4 to the 1/2 is 2, and then
that to the 3 power, that's 8.

00:13:51.620 --> 00:13:54.130
Then 1 to any power, but
especially in this case, 1 to

00:13:54.130 --> 00:13:56.440
the square root of 1 is 1.

00:13:56.440 --> 00:13:58.440
Then to the third power is 1.

00:13:58.440 --> 00:14:02.890
So 8 minus 1 is equal to c.

00:14:02.890 --> 00:14:05.640
And so c is equal to 7.

00:14:05.640 --> 00:14:09.630
So the particular solution of
this differential equation I

00:14:09.630 --> 00:14:19.966
will do in a different color,
and it is y to the 3/2 minus

00:14:19.966 --> 00:14:24.620
x to the 3/2 is equal to 7.