Solving for F in terms of C

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.730
We're told that the formula
for converting a Celsius

00:00:03.730 --> 00:00:08.400
temperature into Fahrenheit is
the Celsius temperature, or C,

00:00:08.400 --> 00:00:11.800
is equal to 5/9 times
the Fahrenheit

00:00:11.800 --> 00:00:14.340
temperature minus 32.

00:00:14.340 --> 00:00:17.320
Rewrite the formula so it
solves for Fahrenheit.

00:00:17.320 --> 00:00:21.940
So just like I mentioned, the
C is Celsius, is the Celsius

00:00:21.940 --> 00:00:22.780
temperature.

00:00:22.780 --> 00:00:25.045
And the F is the Fahrenheit
temperature.

00:00:27.830 --> 00:00:31.360
So right now, it's expressed in
terms of, we've solved for

00:00:31.360 --> 00:00:32.560
C in terms of F.

00:00:32.560 --> 00:00:34.830
Now we want to solve for
F in terms of C.

00:00:34.830 --> 00:00:36.480
So let's see how we're
going to do that.

00:00:36.480 --> 00:00:37.430
So I'll just rewrite it.

00:00:37.430 --> 00:00:44.060
C is equal to 5/9 times
F minus 32.

00:00:44.060 --> 00:00:46.510
And we want to solve for F.

00:00:46.510 --> 00:00:49.480
So the first useful thing to do
might be to get rid of this

00:00:49.480 --> 00:00:53.010
5/9 from both sides, or at least
from the right side of

00:00:53.010 --> 00:00:55.860
the equation, so we just
have an F minus 32.

00:00:55.860 --> 00:00:58.260
And the easiest way to do
that is to multiply by

00:00:58.260 --> 00:01:00.295
the inverse of 5/9.

00:01:00.295 --> 00:01:04.810
So if we multiply both sides of
this equation by 9/5-- and

00:01:04.810 --> 00:01:06.810
I could put it here, but it'll
squeeze a little tight, let me

00:01:06.810 --> 00:01:09.870
try it-- 9/5 in there.

00:01:09.870 --> 00:01:11.780
If I do it to the right-hand
side of the equation, I also

00:01:11.780 --> 00:01:14.600
have to do it to the left-hand
side of the equation.

00:01:14.600 --> 00:01:16.300
And what do I get?

00:01:16.300 --> 00:01:22.050
The left-hand side becomes 9/5
Celsius, or the Celsius

00:01:22.050 --> 00:01:25.480
temperature, 9/5 times the
Celsius temperature, is equal

00:01:25.480 --> 00:01:28.060
to-- the whole reason we wanted
to multiply the right

00:01:28.060 --> 00:01:31.330
side by 9/5 is these 9's cancel
out, this 5 cancels

00:01:31.330 --> 00:01:32.910
out, and this just
becomes a 1.

00:01:32.910 --> 00:01:35.820
So 9/5 times the Celsius
temperature is going to be

00:01:35.820 --> 00:01:38.720
equal to this expression:
the Fahrenheit

00:01:38.720 --> 00:01:40.430
temperature minus 32.

00:01:40.430 --> 00:01:42.560
We don't need the parentheses
anymore, because this is the

00:01:42.560 --> 00:01:45.410
only expression on the
right-hand side.

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Now, we want to solve for the
Fahrenheit temperature.

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So let's see if we can get
rid of this 32 from

00:01:51.580 --> 00:01:52.600
the right-hand side.

00:01:52.600 --> 00:01:56.350
The easiest way I know how
to do that is to add 32.

00:01:56.350 --> 00:01:59.920
Add 32 to both sides
of this equation.

00:01:59.920 --> 00:02:03.360
Add 32 to both sides.

00:02:03.360 --> 00:02:08.070
The left-hand side now becomes
9/5 times the Celsius

00:02:08.070 --> 00:02:16.060
temperature, plus 32.

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And that's going to be equal
to-- this negative 32 and

00:02:19.580 --> 00:02:22.630
positive 32 cancel out, that
was the whole point behind

00:02:22.630 --> 00:02:26.950
adding 32, so it's just going
to be equal to F.

00:02:26.950 --> 00:02:28.150
And we're done.

00:02:28.150 --> 00:02:32.430
We have rewritten this formula,
up here, so it solves

00:02:32.430 --> 00:02:33.770
for Fahrenheit.

00:02:33.770 --> 00:02:35.460
This is the Fahrenheit
right here.

00:02:35.460 --> 00:02:39.030
We've solved for Fahrenheit
in terms of the Celsius

00:02:39.030 --> 00:02:40.280
temperature.


Example: two-step equation with numerator x

WEBVTT
Kind: captions
Language: en

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We have the equation negative
16 is equal to x

00:00:06.210 --> 00:00:09.190
over 4, plus 2.

00:00:09.190 --> 00:00:11.880
And we need to solve for x.

00:00:11.880 --> 00:00:14.650
So we really just need to
isolate the x variable on one

00:00:14.650 --> 00:00:17.045
side of this equation, and the
best way to do that is first

00:00:17.045 --> 00:00:20.885
to isolate it-- isolate this
whole x over 4 term from all

00:00:20.885 --> 00:00:22.870
of the other terms. So in order
to do that, let's get

00:00:22.870 --> 00:00:23.700
rid of this 2.

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And the best way to get rid of
that 2 is to subtract it.

00:00:26.080 --> 00:00:28.470
But if we want to subtract it
from the right-hand side, we

00:00:28.470 --> 00:00:30.880
also have to subtract it from
the left-hand side, because

00:00:30.880 --> 00:00:32.530
this is an equation.

00:00:32.530 --> 00:00:35.370
If this is equal to that,
anything we do to that, we

00:00:35.370 --> 00:00:37.020
also have to do to this.

00:00:37.020 --> 00:00:39.540
So let's subtract 2
from both sides.

00:00:39.540 --> 00:00:44.140
So you subtract 2 from the
right, subtract 2 from the

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left, and we get, on the
left-hand side, negative 16

00:00:48.240 --> 00:00:51.300
minus 2 is negative 18.

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And then that is equal
to x over 4.

00:00:57.060 --> 00:01:00.650
And then we have positive 2
minus 2, which is just going

00:01:00.650 --> 00:01:02.410
to be 0, so we don't even
have to write that.

00:01:02.410 --> 00:01:04.610
I could write just a plus 0,
but I think that's a little

00:01:04.610 --> 00:01:05.660
unnecessary.

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And so we have negative 18
is equal to x over 4.

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And our whole goal here is
to isolate the x, to

00:01:12.020 --> 00:01:14.020
solve for the x.

00:01:14.020 --> 00:01:16.480
And the best way we can do
that, if we have x over 4

00:01:16.480 --> 00:01:18.770
here, if we multiply that
by 4, we're just

00:01:18.770 --> 00:01:19.860
going to have an x.

00:01:19.860 --> 00:01:23.520
So we can multiply that
by 4, but once

00:01:23.520 --> 00:01:24.680
again, this is an equation.

00:01:24.680 --> 00:01:26.290
Anything you do to the
right-hand side, you have to

00:01:26.290 --> 00:01:28.120
do to the left-hand side,
and vice versa.

00:01:28.120 --> 00:01:31.450
So if we multiply the right-hand
side by 4, we also

00:01:31.450 --> 00:01:34.850
have to multiply the left-hand
side by 4.

00:01:34.850 --> 00:01:39.470
So we get 4 times negative 18 is
equal to x over 4, times 4.

00:01:39.470 --> 00:01:42.010
The x over 4 times 4,
that cancels out.

00:01:42.010 --> 00:01:44.820
You divide something by 4 and
multiply by 4, you're just

00:01:44.820 --> 00:01:47.800
going to be left with an x.

00:01:47.800 --> 00:01:51.350
And on the other side,
4 times negative 18.

00:01:51.350 --> 00:01:52.760
Let's see, that's 40.

00:01:52.760 --> 00:01:53.990
Well, let's just write it out.

00:01:53.990 --> 00:01:56.490
So 18 times 4.

00:01:56.490 --> 00:02:01.450
If we were to multiply 18 times
4, 4 times 8 is 32.

00:02:01.450 --> 00:02:05.195
4 times 1 is 4, plus 1 is 72.

00:02:05.195 --> 00:02:09.660
But this is negative 18 times
4, so it's negative 72.

00:02:09.660 --> 00:02:12.370
So x is equal to negative 72.

00:02:12.370 --> 00:02:15.340
And if we want to check it, we
can just substitute it back

00:02:15.340 --> 00:02:16.380
into that original equation.

00:02:16.380 --> 00:02:17.390
So let's do that.

00:02:17.390 --> 00:02:22.800
Let's substitute this into
the original equation.

00:02:22.800 --> 00:02:27.110
So the original equation was
negative 16 is equal to--

00:02:27.110 --> 00:02:30.140
instead of writing x, I'm going
to write negative 72--

00:02:30.140 --> 00:02:36.840
is equal to negative
72 over 4 plus 2.

00:02:36.840 --> 00:02:38.260
Let's see if this is
actually true.

00:02:38.260 --> 00:02:41.640
So this right-hand side
simplifies to negative 72

00:02:41.640 --> 00:02:42.450
divided by 4.

00:02:42.450 --> 00:02:44.850
We already know that that
is negative 18.

00:02:44.850 --> 00:02:48.760
So this is equal to negative
18 plus 2.

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This is what the equation
becomes.

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And then the right-hand side,
negative 18 plus 2, that's

00:02:53.010 --> 00:02:54.500
negative 16.

00:02:54.500 --> 00:02:55.670
So it all comes out true.

00:02:55.670 --> 00:02:59.910
This right-hand side, when x is
equal to negative 72, does

00:02:59.910 --> 00:03:02.750
indeed equal negative 16.


Example 2: Variables on both sides

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:07.800
We have the equation 20 minus
7 times x is equal to 6

00:00:07.800 --> 00:00:10.650
times x minus 6.

00:00:10.650 --> 00:00:12.680
And we need to solve for x.

00:00:12.680 --> 00:00:15.180
So the way I like to do these
is we just like to separate

00:00:15.180 --> 00:00:17.950
the constant terms, which are
the 20 and the negative 6 on

00:00:17.950 --> 00:00:19.200
one side of the equation.

00:00:19.200 --> 00:00:20.710
I'll put them on the
right-hand side.

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And then we'll put all the x
terms, the negative 7x and the

00:00:23.610 --> 00:00:26.260
6x, we'll put it all on
the left-hand side.

00:00:26.260 --> 00:00:28.590
So to get the 20 out of the way
from the left-hand side,

00:00:28.590 --> 00:00:30.130
let's subtract it.

00:00:30.130 --> 00:00:32.030
Let's subtract it from
the left-hand side.

00:00:32.030 --> 00:00:33.690
But this is an equation,
anything you do to the

00:00:33.690 --> 00:00:35.960
left-hand side, you also have to
do to the right-hand side.

00:00:35.960 --> 00:00:38.930
If that is equal to that, in
order for them to still be

00:00:38.930 --> 00:00:41.160
equal, anything I do to the
left-hand side I have to do to

00:00:41.160 --> 00:00:42.350
the right-hand side.

00:00:42.350 --> 00:00:46.510
So I subtracted 20 from the
left, let me also subtract 20

00:00:46.510 --> 00:00:47.990
from the right.

00:00:47.990 --> 00:00:50.390
And so the left-hand side
of the equation, 20

00:00:50.390 --> 00:00:51.880
minus 20 is just 0.

00:00:51.880 --> 00:00:53.660
That was the whole point,
they cancel out.

00:00:53.660 --> 00:00:54.940
Don't have to write it down.

00:00:54.940 --> 00:00:58.510
And then I have a negative 7x,
it just gets carried down.

00:01:01.100 --> 00:01:03.530
And then that is equal to the

00:01:03.530 --> 00:01:05.489
right-hand side of the equation.

00:01:05.489 --> 00:01:07.420
I have a 6x.

00:01:07.420 --> 00:01:09.500
I'm not adding or subtracting
anything to that.

00:01:09.500 --> 00:01:13.000
But then I have a negative
6 minus 20.

00:01:13.000 --> 00:01:16.260
So if I'm already 6 below 0 on
the number line, and I go

00:01:16.260 --> 00:01:21.690
another 20 below that, that's
at negative 26.

00:01:21.690 --> 00:01:24.175
Now, the next thing we want to
do is we want to get all the x

00:01:24.175 --> 00:01:25.420
terms on the left-hand side.

00:01:25.420 --> 00:01:29.060
So we don't want this 6x here,
so maybe we subtract 6x from

00:01:29.060 --> 00:01:29.936
both sides.

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So let's subtract 6x from the
right, subtract 6x from the

00:01:34.860 --> 00:01:36.200
left, and what do we get?

00:01:39.110 --> 00:01:43.080
The left-hand side, negative
7x minus 6x,

00:01:43.080 --> 00:01:45.860
that's negative 13x.

00:01:45.860 --> 00:01:46.470
Right?

00:01:46.470 --> 00:01:49.220
Negative 7 of something minus
another 6 of that something is

00:01:49.220 --> 00:01:51.450
going to be negative 13
of that something.

00:01:51.450 --> 00:01:55.475
And that is going to be
equal to 6x minus 6x.

00:01:55.475 --> 00:01:56.400
That cancels out.

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That was the whole point by
subtracting negative 6x.

00:02:00.110 --> 00:02:03.520
And then we have just a negative
26, or minus 26,

00:02:03.520 --> 00:02:06.500
depending on how you want to
view it, so negative 13x is

00:02:06.500 --> 00:02:10.000
equal to negative 26.

00:02:10.000 --> 00:02:14.060
Now, our whole goal, just to
remember, is to isolate the x.

00:02:14.060 --> 00:02:16.470
We have a negative 13
times the x here.

00:02:16.470 --> 00:02:19.630
So the best way to isolate it is
if we have something times

00:02:19.630 --> 00:02:21.270
x, if we divide by
that something,

00:02:21.270 --> 00:02:22.480
we'll isolate the x.

00:02:22.480 --> 00:02:27.510
So let's divide by
negative 13.

00:02:27.510 --> 00:02:30.320
Now, you know by now, anything
you do to the left-hand side

00:02:30.320 --> 00:02:33.230
of an equation, you have to
do to the right-hand side.

00:02:33.230 --> 00:02:34.890
So we're going to have to
divide both sides of the

00:02:34.890 --> 00:02:37.880
equation by negative 13.

00:02:37.880 --> 00:02:39.730
Now, what does the left-hand
side become?

00:02:39.730 --> 00:02:42.770
Negative 13 times x divided
by negative 13, that's

00:02:42.770 --> 00:02:43.560
just going to be x.

00:02:43.560 --> 00:02:46.650
You multiply something times x,
divide it by the something,

00:02:46.650 --> 00:02:49.670
you're just going to
be left with an x.

00:02:49.670 --> 00:02:51.880
So the left-hand side
just becomes an x.

00:02:51.880 --> 00:02:58.240
x is equal to negative 26
divided by negative 13.

00:02:58.240 --> 00:03:00.260
Well, that's just positive
2, right?

00:03:00.260 --> 00:03:02.300
A negative divided by a negative
is a positive.

00:03:02.300 --> 00:03:05.240
26 divided by 13 is 2.

00:03:05.240 --> 00:03:07.740
And that is our answer.

00:03:07.740 --> 00:03:09.270
That is our answer.

00:03:09.270 --> 00:03:11.140
Now let's verify that
it really works.

00:03:11.140 --> 00:03:12.330
That's the fun thing
about algebra.

00:03:12.330 --> 00:03:14.470
You can always make sure that
you got the right answer.

00:03:14.470 --> 00:03:17.760
So let's substitute it back into
the original equation.

00:03:17.760 --> 00:03:28.580
So we have 20 minus 7 times x--
x is 2-- minus 7 times 2

00:03:28.580 --> 00:03:34.590
is equal to 6 times x--
we've solved for x, it

00:03:34.590 --> 00:03:38.310
is 2-- minus 6.

00:03:38.310 --> 00:03:40.940
So let's verify that this
left-hand side really does

00:03:40.940 --> 00:03:42.690
equal this right-hand side.

00:03:42.690 --> 00:03:46.750
So the left-hand side simplifies
to 20 minus 7 times

00:03:46.750 --> 00:03:48.390
2, which is 14.

00:03:48.390 --> 00:03:50.930
20 minus 14 is 6.

00:03:50.930 --> 00:03:53.270
That's what the left-hand
side simplifies to.

00:03:53.270 --> 00:03:58.400
The right-hand side, we have 6
times 2, which is 12 minus 6.

00:03:58.400 --> 00:04:00.580
12 minus 6 is 6.

00:04:00.580 --> 00:04:04.040
So they are, indeed, equal, and
we did, indeed, get the

00:04:04.040 --> 00:04:05.620
right answer.


Solving equations with the distributive property

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:05.120
We have the equation negative 9
minus this whole expression,

00:00:05.120 --> 00:00:08.720
9x minus 6-- this whole thing
is being subtracted from

00:00:08.720 --> 00:00:11.750
negative 9-- is equal to
3 times this whole

00:00:11.750 --> 00:00:15.500
expression, 4x plus 6.

00:00:15.500 --> 00:00:17.740
Now, a good place to start is
to just get rid of these

00:00:17.740 --> 00:00:18.400
parentheses.

00:00:18.400 --> 00:00:20.800
And the best way to get rid of
these parentheses is to kind

00:00:20.800 --> 00:00:21.940
of multiply them out.

00:00:21.940 --> 00:00:24.880
This has a negative 1-- you
just see a minus here, but

00:00:24.880 --> 00:00:27.660
it's just really the same thing
as having a negative 1--

00:00:27.660 --> 00:00:29.310
times this quantity.

00:00:29.310 --> 00:00:31.790
And here you have a 3
times this quantity.

00:00:31.790 --> 00:00:34.820
So let's multiply it out using
the distributive property.

00:00:34.820 --> 00:00:36.780
So the left-hand side of
our equation, we have

00:00:36.780 --> 00:00:38.250
our negative 9.

00:00:38.250 --> 00:00:41.710
And then we want to multiply the
negative 1 times each of

00:00:41.710 --> 00:00:47.980
these terms. So negative 1 times
9x is negative 9x, and

00:00:47.980 --> 00:00:54.040
then negative 1 times negative
6 is plus 6, or positive 6.

00:00:54.040 --> 00:00:57.230
And then that is going to be
equal to-- let's distribute

00:00:57.230 --> 00:01:01.430
the 3-- 3 times 4x is 12x.

00:01:01.430 --> 00:01:08.210
And then 3 times 6 is 18.

00:01:08.210 --> 00:01:10.850
Now what we want to do, let's
combine our constant

00:01:10.850 --> 00:01:11.780
terms, if we can.

00:01:11.780 --> 00:01:15.600
We have a negative 9 and a 6
here, on this side, we've

00:01:15.600 --> 00:01:17.800
combined all of our like terms.
We can't combine a 12x

00:01:17.800 --> 00:01:20.040
and an 18, so let's
combine this.

00:01:20.040 --> 00:01:24.550
So let's combine the negative 9
and the 6, our two constant

00:01:24.550 --> 00:01:26.260
terms on the left-hand
side of the equation.

00:01:26.260 --> 00:01:30.270
So we're going to have
this negative 9x.

00:01:30.270 --> 00:01:35.590
So we're going to have negative
9x plus-- let's see,

00:01:35.590 --> 00:01:40.030
we have a negative 9 and then
plus 6-- so negative 9 plus 6

00:01:40.030 --> 00:01:41.510
is negative 3.

00:01:41.510 --> 00:01:44.260
So we're going to have a
negative 9x, and then we have

00:01:44.260 --> 00:01:46.660
a negative 3, so minus
3 right here.

00:01:46.660 --> 00:01:50.430
That's the negative 9 plus
the 6, and that is

00:01:50.430 --> 00:01:54.850
equal to 12x plus 18.

00:01:54.850 --> 00:01:57.280
Now, we want to group all the
x terms on one side of the

00:01:57.280 --> 00:01:59.200
equation, and all of the
constant terms-- the negative

00:01:59.200 --> 00:02:02.150
3 and the positive 18 on the
other side-- I like to always

00:02:02.150 --> 00:02:04.650
have my x terms on the left-hand
side, if I can.

00:02:04.650 --> 00:02:07.250
You don't have to have them on
the left, so let's do that.

00:02:07.250 --> 00:02:09.410
So if I want all my x terms on
the left, I have to get rid of

00:02:09.410 --> 00:02:11.590
this 12x from the right.

00:02:11.590 --> 00:02:14.860
And the best way to do that is
to subtract 12x from both

00:02:14.860 --> 00:02:16.390
sides of the equation.

00:02:16.390 --> 00:02:21.620
So let me subtract 12x from the
right, and subtract 12x

00:02:21.620 --> 00:02:23.500
from the left.

00:02:23.500 --> 00:02:28.800
Now, on the left-hand side, I
have negative 9x minus 12x.

00:02:28.800 --> 00:02:32.880
So negative 9 minus 12,
that's negative 21.

00:02:32.880 --> 00:02:39.480
Negative 21x minus 3 is equal
to-- 12x minus 12x, well,

00:02:39.480 --> 00:02:40.240
that's just nothing.

00:02:40.240 --> 00:02:40.990
That's 0.

00:02:40.990 --> 00:02:42.250
So I could write a 0
here, but I don't

00:02:42.250 --> 00:02:42.960
have to write anything.

00:02:42.960 --> 00:02:45.830
That was the whole point of
subtracting the 12x from the

00:02:45.830 --> 00:02:46.910
left-hand side.

00:02:46.910 --> 00:02:49.420
And that is going to be equal
to-- so on the right-hand

00:02:49.420 --> 00:02:52.320
side, we just are
left with an 18.

00:02:52.320 --> 00:02:54.370
We are just left with
that 18 here.

00:02:54.370 --> 00:02:56.130
These guys canceled out.

00:02:56.130 --> 00:02:57.990
Now, let's get rid of
this negative 3 from

00:02:57.990 --> 00:02:58.840
the left-hand side.

00:02:58.840 --> 00:03:00.900
So on the left-hand side, we
only have x terms, and on the

00:03:00.900 --> 00:03:03.800
right-hand side, we only have
constant terms. So the best

00:03:03.800 --> 00:03:06.940
way to cancel out a negative
3 is to add 3.

00:03:06.940 --> 00:03:08.200
So it cancels out to 0.

00:03:08.200 --> 00:03:10.340
So we're going to add
3 to the left, let's

00:03:10.340 --> 00:03:12.320
add 3 to the right.

00:03:12.320 --> 00:03:16.490
And we get-- the left-hand side
of the equation, we have

00:03:16.490 --> 00:03:19.420
the negative 21x, no other x
term to add or subtract here,

00:03:19.420 --> 00:03:22.090
so we have negative 21x.

00:03:22.090 --> 00:03:25.320
The negative 3 and the plus 3,
or the positive 3, cancel

00:03:25.320 --> 00:03:28.060
out-- that was the whole
point-- equals--

00:03:28.060 --> 00:03:30.290
what's 18 plus 3?

00:03:30.290 --> 00:03:33.820
18 plus 3 is 21.

00:03:33.820 --> 00:03:36.730
So now we have negative
21x is equal to 21.

00:03:36.730 --> 00:03:38.880
And we want to solve for x.

00:03:38.880 --> 00:03:41.300
So if you have something times
x, and you just want it to be

00:03:41.300 --> 00:03:43.400
an x, let's divide by
that something.

00:03:43.400 --> 00:03:47.800
And in this case, that something
is negative 21.

00:03:47.800 --> 00:03:53.350
So let's divide both sides of
this equation by negative 21.

00:03:53.350 --> 00:03:55.670
Divide both sides
by negative 21.

00:03:55.670 --> 00:03:58.510
The left-hand side, negative
21 divided by negative 21,

00:03:58.510 --> 00:04:00.100
you're just left with an x.

00:04:00.100 --> 00:04:01.440
That was the whole point behind

00:04:01.440 --> 00:04:03.260
dividing by negative 21.

00:04:03.260 --> 00:04:05.820
And we get x is equal
to-- what's 21

00:04:05.820 --> 00:04:07.150
divided by negative 21?

00:04:07.150 --> 00:04:09.170
Well, that's just negative 1.

00:04:09.170 --> 00:04:09.450
Right?

00:04:09.450 --> 00:04:11.340
You have the positive version
divided by the negative

00:04:11.340 --> 00:04:13.850
version of itself, so it's
just negative 1.

00:04:13.850 --> 00:04:18.510
So that is our answer.

00:04:18.510 --> 00:04:21.060
Now let's verify that this
actually works for that

00:04:21.060 --> 00:04:21.850
original equation.

00:04:21.850 --> 00:04:25.320
So let's substitute negative 1
into that original equation.

00:04:25.320 --> 00:04:29.370
So we have negative 9-- I'll do
it over here; I'll do it in

00:04:29.370 --> 00:04:31.560
a different color than we've
been using-- we have negative

00:04:31.560 --> 00:04:35.400
9 minus-- that 1 wasn't there
originally, it's there

00:04:35.400 --> 00:04:40.320
implicitly-- minus 9
times negative 1.

00:04:40.320 --> 00:04:45.720
9 times-- I'll put negative 1
in parentheses-- minus 6 is

00:04:45.720 --> 00:04:47.720
equal to-- well, actually,
let me just solve for the

00:04:47.720 --> 00:04:51.600
left-hand side when I substitute
a negative 1 there.

00:04:51.600 --> 00:04:56.060
So the left-hand side becomes
negative 9, minus 9 times

00:04:56.060 --> 00:04:59.420
negative 1 is negative
9, minus 6.

00:04:59.420 --> 00:05:02.860
And so this is negative 9
minus-- in parentheses--

00:05:02.860 --> 00:05:06.740
negative 9 minus 6
is negative 15.

00:05:06.740 --> 00:05:10.110
So this is equal
to negative 15.

00:05:10.110 --> 00:05:13.450
And so we get negative 9-- let
me make sure I did that--

00:05:13.450 --> 00:05:16.080
negative 9 minus 6,
yep, negative 15.

00:05:16.080 --> 00:05:19.620
So negative 9 minus negative 15,
that's the same thing as

00:05:19.620 --> 00:05:24.590
negative 9 plus 15,
which is 6.

00:05:24.590 --> 00:05:26.980
So that's what we get on the
left-hand side of the equation

00:05:26.980 --> 00:05:29.000
when we substitute x is
equal to negative 1.

00:05:29.000 --> 00:05:30.680
We get that it equals 6.

00:05:30.680 --> 00:05:32.640
So let's see what happens when
we substitute negative 1 to

00:05:32.640 --> 00:05:34.640
the right-hand side
of the equation.

00:05:34.640 --> 00:05:35.680
I'll do it in green.

00:05:35.680 --> 00:05:43.750
We get 3 times 4 times
negative 1 plus 6.

00:05:43.750 --> 00:05:48.010
So that is 3 times negative
4 plus 6.

00:05:48.010 --> 00:05:49.530
Negative 4 plus 6 is 2.

00:05:49.530 --> 00:05:53.790
So it's 3 times 2,
which is also 6.

00:05:53.790 --> 00:05:56.460
So when x is equal to negative
1, you substitute here, the

00:05:56.460 --> 00:06:00.340
left-hand side becomes 6, and
the right-hand side becomes 6.

00:06:00.340 --> 00:06:03.610
So this definitely works out.


Linear systems word problem with substitution

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:04.920
So we have the question, Devon
is going to make 3 shelves for

00:00:04.920 --> 00:00:05.900
her father.

00:00:05.900 --> 00:00:09.440
She has a piece of lumber
that is 12 feet long.

00:00:09.440 --> 00:00:12.710
She wants the top shelf to be
half a foot shorter than the

00:00:12.710 --> 00:00:13.510
middle shelf.

00:00:13.510 --> 00:00:15.150
So let me do this in
different colors.

00:00:15.150 --> 00:00:22.280
She wants the top shelf to be
half a foot shorter than the

00:00:22.280 --> 00:00:23.150
middle shelf.

00:00:23.150 --> 00:00:25.870
So I'll just-- let's just read
the whole thing first. And the

00:00:25.870 --> 00:00:30.020
bottom shelf to be half a foot
shorter than twice the length

00:00:30.020 --> 00:00:31.120
of the top shelf.

00:00:31.120 --> 00:00:32.370
Let me do that in a
different color.

00:00:32.370 --> 00:00:34.040
I'll do that in blue.

00:00:34.040 --> 00:00:40.690
The bottom shelf to be half a
foot shorter than twice the

00:00:40.690 --> 00:00:43.300
length of the top shelf.

00:00:43.300 --> 00:00:46.340
How long will each shelf
be if she uses the

00:00:46.340 --> 00:00:49.440
entire 12 feet of wood?

00:00:49.440 --> 00:00:52.020
So let's define some variables
for our different shelves,

00:00:52.020 --> 00:00:53.000
because that's what we
have to figure out.

00:00:53.000 --> 00:00:55.200
We have the top shelf,
the middle shelf,

00:00:55.200 --> 00:00:56.530
and the bottom shelf.

00:00:56.530 --> 00:01:04.489
So let's say that t is equal to
length of top shelf-- t for

00:01:04.489 --> 00:01:08.150
top-- of top shelf.

00:01:08.150 --> 00:01:16.485
Let's make m equal the length
of the middle shelf.

00:01:20.170 --> 00:01:21.560
m for middle.

00:01:21.560 --> 00:01:29.120
And then let's make b equal to
the length of the bottom

00:01:29.120 --> 00:01:33.070
shelf-- b for bottom--
bottom shelf.

00:01:33.070 --> 00:01:35.520
So let's see what these
different statements tell us.

00:01:35.520 --> 00:01:37.920
So this first statement, she
says she wants the top shelf--

00:01:37.920 --> 00:01:40.450
and I'll do it in that same
color-- she wants the top

00:01:40.450 --> 00:01:44.120
shelf to be 1/2 a foot shorter
than the middle shelf.

00:01:44.120 --> 00:01:50.650
So she wants the length of the
top shelf to be-- so this is

00:01:50.650 --> 00:01:56.780
equal to 1/2 a foot shorter
than the middle shelf.

00:01:56.780 --> 00:01:59.590
So, if we're doing everything in
feet, it's going to be the

00:01:59.590 --> 00:02:02.640
length of the middle
shelf in feet minus

00:02:02.640 --> 00:02:05.010
1/2, minus 1/2 feet.

00:02:05.010 --> 00:02:08.060
So that's what that sentence
in orange is telling us.

00:02:08.060 --> 00:02:11.760
The top shelf needs to be 1/2 a
foot shorter than the length

00:02:11.760 --> 00:02:13.120
of the middle shelf.

00:02:13.120 --> 00:02:15.270
Now, what does the next
statement tell us?

00:02:15.270 --> 00:02:19.630
And the bottom shelf to be-- so
the bottom shelf needs to

00:02:19.630 --> 00:02:24.300
be equal to 1/2 a foot shorter
than-- so it's 1/2 a foot

00:02:24.300 --> 00:02:29.830
shorter than twice the length
of the top shelf.

00:02:29.830 --> 00:02:32.420
So it's 1/2 a foot shorter
than twice the

00:02:32.420 --> 00:02:36.680
length of the top shelf.

00:02:36.680 --> 00:02:38.620
These are the two statements
interpreted in

00:02:38.620 --> 00:02:39.820
equal equation form.

00:02:39.820 --> 00:02:42.910
The top shelf's length has to be
equal to the middle shelf's

00:02:42.910 --> 00:02:44.060
length minus 1/2.

00:02:44.060 --> 00:02:46.800
It's 1/2 foot shorter than
the middle shelf.

00:02:46.800 --> 00:02:50.750
And the bottom shelf needs to
be 1/2 a foot shorter than

00:02:50.750 --> 00:02:54.010
twice the length of
the top shelf.

00:02:54.010 --> 00:02:55.360
And so how do we solve this?

00:02:55.360 --> 00:02:57.100
Well, you can't just solve
it just with these two

00:02:57.100 --> 00:03:01.050
constraints, but they gave
us more information.

00:03:01.050 --> 00:03:04.040
They tell us how long will each
shelf be if she uses the

00:03:04.040 --> 00:03:06.480
entire 12 feet of wood?

00:03:06.480 --> 00:03:08.920
So the length of all of
the shelves have to

00:03:08.920 --> 00:03:10.880
add up to 12 feet.

00:03:10.880 --> 00:03:11.850
She's using all of it.

00:03:11.850 --> 00:03:20.380
So t plus m, plus b needs
to be equal to 12 feet.

00:03:20.380 --> 00:03:21.920
That's the length
of each of them.

00:03:21.920 --> 00:03:23.940
She's using all 12
feet of the wood.

00:03:23.940 --> 00:03:26.680
So the lengths have
to add to 12.

00:03:26.680 --> 00:03:28.450
So what can we do here?

00:03:28.450 --> 00:03:32.610
Well, we can get everything here
in terms of one variable,

00:03:32.610 --> 00:03:35.100
maybe we'll do it in terms of
m, and then substitute.

00:03:35.100 --> 00:03:39.590
So we already have
t in terms of m.

00:03:39.590 --> 00:03:41.960
We could, everywhere we see
a t, we could substitute

00:03:41.960 --> 00:03:44.220
with m minus 1/2.

00:03:44.220 --> 00:03:47.470
But here we have b
in terms of t.

00:03:47.470 --> 00:03:49.460
So how can we put this
in terms of m?

00:03:49.460 --> 00:03:52.190
Well, we know that t is
equal to m minus 1/2.

00:03:52.190 --> 00:03:55.720
So let's take, everywhere we see
a t, let's substitute it

00:03:55.720 --> 00:03:56.820
with this thing right here.

00:03:56.820 --> 00:03:58.770
That is what t is equal to.

00:03:58.770 --> 00:04:02.100
So we can rewrite this blue
equation as, the length of the

00:04:02.100 --> 00:04:06.150
bottom shelf is 2 times the
length of the top shelf, t,

00:04:06.150 --> 00:04:08.460
but we know that t is equal
to m minus 1/2.

00:04:15.930 --> 00:04:18.860
And if we wanted to simplify
that a little bit, this would

00:04:18.860 --> 00:04:23.990
be that the bottom shelf is
equal to-- let's distribute

00:04:23.990 --> 00:04:26.930
the 2-- 2 times m is 2m.

00:04:26.930 --> 00:04:29.670
2 times negative 1/2
is negative 1.

00:04:29.670 --> 00:04:32.100
And then minus another 1/2.

00:04:32.100 --> 00:04:37.710
Or, we could rewrite this as b
is equal to 2 times the middle

00:04:37.710 --> 00:04:40.390
shelf minus 3/2.

00:04:40.390 --> 00:04:40.740
Right?

00:04:40.740 --> 00:04:44.870
1/2 is 2/2 minus another
1/2 is negative

00:04:44.870 --> 00:04:46.800
3/2, just like that.

00:04:46.800 --> 00:04:49.140
So now we have everything in
terms of m, and we can

00:04:49.140 --> 00:04:51.260
substitute back here.

00:04:51.260 --> 00:04:56.020
So the top shelf-- instead of
putting a t there, we could

00:04:56.020 --> 00:04:58.070
put m minus 1/2.

00:04:58.070 --> 00:05:02.810
So we put m minus 1/2, plus
the length of the middle

00:05:02.810 --> 00:05:07.670
shelf, plus the length
of the bottom shelf.

00:05:07.670 --> 00:05:09.860
Well, we already put
that in terms of m.

00:05:09.860 --> 00:05:11.310
That's what we just did.

00:05:11.310 --> 00:05:14.250
This is the length of the bottom
shelf in terms of m.

00:05:14.250 --> 00:05:18.340
So instead of writing b there,
we could write 2m minus 3/2.

00:05:18.340 --> 00:05:25.460
Plus 2m minus 3/2, and
that is equal to 12.

00:05:25.460 --> 00:05:27.530
All we did is substitute
for t.

00:05:27.530 --> 00:05:31.110
We wrote t in terms of m, and
we wrote b in terms of m.

00:05:31.110 --> 00:05:35.590
Now let's combine the m terms
and the constant terms. So if

00:05:35.590 --> 00:05:39.730
we have, we have one m here, we
have another m there, and

00:05:39.730 --> 00:05:40.760
then we have a 2m there.

00:05:40.760 --> 00:05:41.550
They're all positive.

00:05:41.550 --> 00:05:44.530
So 1 plus 1, plus 2 is 4m.

00:05:44.530 --> 00:05:46.380
So we have 4m.

00:05:46.380 --> 00:05:48.490
And then what do our constant
terms tell us?

00:05:48.490 --> 00:05:54.590
We have a negative 1/2, and then
we have a negative 3/2.

00:05:54.590 --> 00:05:59.270
So negative 1/2 minus 3/2,
that is negative 4/2 or

00:05:59.270 --> 00:06:00.290
negative 2.

00:06:00.290 --> 00:06:02.670
So we have 4m minus 2.

00:06:02.670 --> 00:06:06.500
And, of course, we still
have that equals 12.

00:06:06.500 --> 00:06:09.446
Now, we want to isolate just the
m variable on one side of

00:06:09.446 --> 00:06:09.825
the equation.

00:06:09.825 --> 00:06:12.810
So let's add 2 to both sides
to get rid of this 2 on the

00:06:12.810 --> 00:06:14.430
left-hand side.

00:06:14.430 --> 00:06:18.850
So if we add 2 to both sides
of this equation, the

00:06:18.850 --> 00:06:23.000
left-hand side, we're just
left with 4m-- these guys

00:06:23.000 --> 00:06:29.330
cancel out-- is equal to 14.

00:06:29.330 --> 00:06:36.570
Now, divide both sides by 4, we
get m is equal to 14 over

00:06:36.570 --> 00:06:41.940
4, or we could call that 7/2
feet, because we're doing

00:06:41.940 --> 00:06:42.730
everything in feet.

00:06:42.730 --> 00:06:45.800
So we solved for m, but
now we still have to

00:06:45.800 --> 00:06:49.050
solve for t and b.

00:06:49.050 --> 00:06:49.720
So let's do that.

00:06:49.720 --> 00:06:51.230
Let's solve for t.

00:06:51.230 --> 00:06:53.420
t is equal to m minus 1/2.

00:06:53.420 --> 00:06:59.040
So it's equal to-- our m is 7/2
minus 1/2, which is equal

00:06:59.040 --> 00:07:03.410
to 6/2, or 3 feet.

00:07:03.410 --> 00:07:04.840
Everything is in feet,
so that's how we

00:07:04.840 --> 00:07:05.900
know it's feet there.

00:07:05.900 --> 00:07:08.060
So that's the top
shelf is 3 feet.

00:07:08.060 --> 00:07:11.930
The middle shelf is 7/2 feet,
which is the same thing as 3

00:07:11.930 --> 00:07:14.930
and 1/2 feet.

00:07:14.930 --> 00:07:17.480
And then the bottom shelf
is 2 times the top

00:07:17.480 --> 00:07:19.075
shelf, minus 1/2.

00:07:19.075 --> 00:07:21.480
So what's that going
to be equal to?

00:07:21.480 --> 00:07:25.090
That's going to be equal to 2
times 3 feet-- that's what the

00:07:25.090 --> 00:07:29.430
length of the top shelf is--
minus 1/2, which is equal to 6

00:07:29.430 --> 00:07:33.130
minus 1/2, or 5 and 1/2 feet.

00:07:36.270 --> 00:07:36.960
And we're done.

00:07:36.960 --> 00:07:40.310
And you can verify that these
definitely do add up to 12.

00:07:40.310 --> 00:07:44.480
5 and 1/2 plus 3 and 1/2 is 9,
plus 3 is 12 feet, and it

00:07:44.480 --> 00:07:46.440
meets all of the other
constraints.

00:07:46.440 --> 00:07:50.850
The top shelf is 1/2 a foot
shorter than the middle shelf,

00:07:50.850 --> 00:07:54.060
and the bottom shelf is 1/2
a foot shorter than 2

00:07:54.060 --> 00:07:55.410
times the top shelf.

00:07:55.410 --> 00:07:56.470
And we are done.

00:07:56.470 --> 00:07:59.470
We know the lengths of
the shelves that

00:07:59.470 --> 00:08:01.460
Devon needs to make.


A clever application of the distributive property to solve a multi-step equation

WEBVTT
Kind: captions
Language: en

00:00:00.420 --> 00:00:13.470
We have the equation 3/4x plus
2 is equal to 3/8x minus 4.

00:00:13.470 --> 00:00:16.129
Now, we could just, right from
the get go, solve this the way

00:00:16.129 --> 00:00:19.070
we solved everything else, group
the x terms, maybe on

00:00:19.070 --> 00:00:21.440
the left-hand side, group the
constant terms on the

00:00:21.440 --> 00:00:22.470
right-hand side.

00:00:22.470 --> 00:00:24.980
But adding and subtracting
fractions are messy.

00:00:24.980 --> 00:00:28.110
So what I'm going to do, right
from the start of this video,

00:00:28.110 --> 00:00:31.470
is to multiply both sides of
this equation by some number

00:00:31.470 --> 00:00:33.300
so I can get rid of
the fractions.

00:00:33.300 --> 00:00:35.880
And the best number to do it
by-- what number is the

00:00:35.880 --> 00:00:38.050
smallest number that if I
multiply both of these

00:00:38.050 --> 00:00:41.320
fractions by it, they won't be
fractions anymore, they'll be

00:00:41.320 --> 00:00:42.530
whole numbers?

00:00:42.530 --> 00:00:44.350
That smallest number
is going to be 8.

00:00:47.130 --> 00:00:50.080
I'm going to multiply 8 times
both sides of this equation.

00:00:50.080 --> 00:00:52.920
You say, hey, Sal, how
did you get 8?

00:00:52.920 --> 00:00:55.300
And I got 8 because I said,
well, what's the least common

00:00:55.300 --> 00:00:57.650
multiple of 4 and 8?

00:00:57.650 --> 00:01:01.600
Well, the smallest number that
is divisible by 4 and 8 is 8.

00:01:01.600 --> 00:01:03.580
So when you multiply
by 8, it's going to

00:01:03.580 --> 00:01:05.720
get rid of the fractions.

00:01:05.720 --> 00:01:07.300
And so let's see what happens.

00:01:07.300 --> 00:01:10.600
So 8 times 3/4, that's
the same thing as 8

00:01:10.600 --> 00:01:12.350
times 3 over 4.

00:01:12.350 --> 00:01:14.600
Let me do it on the
side over here.

00:01:14.600 --> 00:01:18.320
That's the same thing as 8 times
3 over 4, which is equal

00:01:18.320 --> 00:01:20.440
to 8 divided by 4 is just 2.

00:01:20.440 --> 00:01:23.160
So it's 2 times 3, which is 6.

00:01:23.160 --> 00:01:28.800
So the left-hand side becomes
8 times 3/4x is 6x.

00:01:28.800 --> 00:01:31.610
And then 8 times 2 is 16.

00:01:31.610 --> 00:01:34.800
You have to remember, when you
multiply both sides, or a

00:01:34.800 --> 00:01:37.070
side, of an equation by a
number, you multiply every

00:01:37.070 --> 00:01:38.040
term by that number.

00:01:38.040 --> 00:01:39.920
So you have to distribute
the 8.

00:01:39.920 --> 00:01:44.040
So the left-hand side is 6x plus
16 is going to be equal

00:01:44.040 --> 00:01:47.240
to-- 8 times 3/8, that's pretty
easy, the 8's cancel

00:01:47.240 --> 00:01:49.580
out and you're just
left with 3x.

00:01:49.580 --> 00:01:53.430
And then 8 times negative
4 is negative 32.

00:01:53.430 --> 00:01:57.130
And now we've cleaned up the
equation a good bit.

00:01:57.130 --> 00:01:59.690
Now the next thing, let's try to
get all the x terms on the

00:01:59.690 --> 00:02:01.890
left-hand side, and all the
constant terms on the right.

00:02:01.890 --> 00:02:05.220
So let's get rid of this
3x from the right.

00:02:05.220 --> 00:02:08.740
Let's subtract 3x from
both sides to do it.

00:02:08.740 --> 00:02:12.250
That's the best way I can think
of of getting rid of the

00:02:12.250 --> 00:02:14.320
3x from the right.

00:02:14.320 --> 00:02:19.180
The left-hand side of this
equation, 6x minus 3x is 3x.

00:02:19.180 --> 00:02:20.890
6 minus 3 is 3.

00:02:20.890 --> 00:02:26.380
And then you have a plus 16 is
equal to-- 3x minus 3x, that's

00:02:26.380 --> 00:02:28.530
the whole point of subtracting
3x, is so they cancel out.

00:02:28.530 --> 00:02:30.870
So those guys cancel out, and
we're just left with a

00:02:30.870 --> 00:02:34.220
negative 32.

00:02:34.220 --> 00:02:37.860
Now, let's get rid of the 16
from the left-hand side.

00:02:37.860 --> 00:02:40.550
So to get rid of it, we're going
to subtract 16 from both

00:02:40.550 --> 00:02:41.950
sides of this equation.

00:02:41.950 --> 00:02:45.200
Subtract 16 from both sides.

00:02:45.200 --> 00:02:47.600
The left-hand side of the
equation just becomes-- you

00:02:47.600 --> 00:02:51.650
have this 3x here; these 16's
cancel out, you don't have to

00:02:51.650 --> 00:02:57.420
write anything-- is equal to
negative 32 minus 16 is

00:02:57.420 --> 00:02:59.910
negative 48.

00:02:59.910 --> 00:03:02.325
So we have 3x is equal
to negative 48.

00:03:02.325 --> 00:03:05.820
To isolate the x, we can just
divide both sides of this

00:03:05.820 --> 00:03:08.350
equation by 3.

00:03:08.350 --> 00:03:11.940
So let's divide both sides
of that equation by 3.

00:03:11.940 --> 00:03:14.510
The left-hand side of the
equation, 3x divided by

00:03:14.510 --> 00:03:15.310
3 is just an x.

00:03:15.310 --> 00:03:16.750
That was the whole point
behind dividing

00:03:16.750 --> 00:03:18.480
both sides by 3.

00:03:18.480 --> 00:03:21.830
And the right-hand side,
negative 48 divided by 3 is

00:03:21.830 --> 00:03:25.710
negative 16.

00:03:25.710 --> 00:03:27.036
And we are done.

00:03:27.036 --> 00:03:30.910
x equals negative 16
is our solution.

00:03:30.910 --> 00:03:33.300
So let's make sure that this
actually works by substituting

00:03:33.300 --> 00:03:34.920
to the original equation
up here.

00:03:34.920 --> 00:03:36.030
And the original equation
didn't have

00:03:36.030 --> 00:03:37.260
those 8's out front.

00:03:37.260 --> 00:03:39.270
So let's substitute in the
original equation.

00:03:39.270 --> 00:03:50.710
We get 3/4-- 3 over 4-- times
negative 16 plus 2 needs to be

00:03:50.710 --> 00:04:02.010
equal to 3/8 times negative
16 minus 4.

00:04:02.010 --> 00:04:04.390
So 3/4 of 16 is 12.

00:04:04.390 --> 00:04:05.680
And you can think
of it this way.

00:04:05.680 --> 00:04:07.550
What's 16 divided by 4?

00:04:07.550 --> 00:04:08.670
It is 4.

00:04:08.670 --> 00:04:11.040
And then multiply that
by 3, it's 12,

00:04:11.040 --> 00:04:12.450
just multiplying fractions.

00:04:12.450 --> 00:04:14.390
So this is going to
be a negative 12.

00:04:14.390 --> 00:04:18.920
So we get negative 12 plus
2 on the left-hand side,

00:04:18.920 --> 00:04:21.800
negative 12 plus 2
is negative 10.

00:04:21.800 --> 00:04:23.650
So the left-hand side
is a negative 10.

00:04:23.650 --> 00:04:27.120
Let's see what the right-hand
side is.

00:04:27.120 --> 00:04:29.360
You have 3/8 times
negative 16.

00:04:29.360 --> 00:04:32.940
If you divide negative 16 by 8,
you get negative 2, times 3

00:04:32.940 --> 00:04:33.970
is a negative 6.

00:04:33.970 --> 00:04:36.710
So it's a negative 6 minus 4.

00:04:36.710 --> 00:04:40.060
Negative 6 minus 4
is negative 10.

00:04:40.060 --> 00:04:44.030
So when x is equal to negative
16, it does satisfy the

00:04:44.030 --> 00:04:44.730
original equation.

00:04:44.730 --> 00:04:47.560
Both sides of the equation
become negative 10.

00:04:47.560 --> 00:04:49.330
And we are done.


Challenge example: Sum of integers

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.770
We're told that the sum
of three consecutive

00:00:03.770 --> 00:00:07.080
odd integers is 231.

00:00:07.080 --> 00:00:09.470
What is the largest integer?

00:00:09.470 --> 00:00:11.340
So let's think about
this a little bit.

00:00:11.340 --> 00:00:13.920
Let's say that x is the
smallest integer.

00:00:13.920 --> 00:00:20.040
x is equal to the smallest
odd of these three.

00:00:20.040 --> 00:00:22.110
It's not the smallest odd
integer of all integers, it's

00:00:22.110 --> 00:00:24.270
is the smallest odd of
these three, the

00:00:24.270 --> 00:00:25.600
smallest odd integer.

00:00:25.600 --> 00:00:27.290
So what's the next
one going to be?

00:00:27.290 --> 00:00:29.380
Well, if I have one odd integer,
what's going to be

00:00:29.380 --> 00:00:30.650
the next odd integer?

00:00:30.650 --> 00:00:31.500
Let's think about this.

00:00:31.500 --> 00:00:34.050
If x was 3, what's the
next odd integer?

00:00:34.050 --> 00:00:34.920
It's 5.

00:00:34.920 --> 00:00:36.160
And then what's the next
one after that?

00:00:36.160 --> 00:00:36.820
It's 7.

00:00:36.820 --> 00:00:38.200
And the next one after that?

00:00:38.200 --> 00:00:39.350
It's 9.

00:00:39.350 --> 00:00:41.320
So every time we add 2.

00:00:41.320 --> 00:00:45.460
So if the smallest one is x, the
next smallest odd integer

00:00:45.460 --> 00:00:54.940
is x plus 2, is equal to the
next smallest odd integer--

00:00:54.940 --> 00:00:58.150
I'll write integer here--
odd integer.

00:00:58.150 --> 00:00:59.580
And then what would
be the next one?

00:00:59.580 --> 00:01:01.820
Well, we're going to add
2 to this one, right?

00:01:01.820 --> 00:01:04.379
So it's going to be x plus 4.

00:01:04.379 --> 00:01:04.860
Think about it.

00:01:04.860 --> 00:01:08.260
If the smallest is 3, then you
have x plus 2, which is 5.

00:01:08.260 --> 00:01:11.240
And then you have x plus
4, which is 7.

00:01:11.240 --> 00:01:18.260
So this will be the largest of
the consecutive odd integer in

00:01:18.260 --> 00:01:19.200
this group.

00:01:19.200 --> 00:01:21.940
And they tell us that the sum
of these consecutive odd

00:01:21.940 --> 00:01:23.910
integers is 231.

00:01:23.910 --> 00:01:25.600
What is the largest integer?

00:01:25.600 --> 00:01:29.190
So if I take x, x plus 2 and x
plus 4, and I sum them, they

00:01:29.190 --> 00:01:30.820
should be equal to 231.

00:01:30.820 --> 00:01:32.550
So let's do that.

00:01:32.550 --> 00:01:39.930
So we have x plus x plus 2, plus
x plus 4, and this needs

00:01:39.930 --> 00:01:41.880
to be equal to 231.

00:01:41.880 --> 00:01:43.865
And when they ask us what's the
largest one, we're going

00:01:43.865 --> 00:01:46.930
to have to tell them what
x plus 4 is equal to.

00:01:46.930 --> 00:01:48.790
So let's just solve
this equation.

00:01:48.790 --> 00:01:52.480
So let's add our x terms. We
have one x, two x, three x's,

00:01:52.480 --> 00:01:56.630
so we get 3x plus-- and then
what are our constants?

00:01:56.630 --> 00:01:58.890
We have a 2 and we have a 4.

00:01:58.890 --> 00:02:06.750
So 3x plus 6 is equal to 231.

00:02:06.750 --> 00:02:08.250
Now, let's get rid
of the 6 from the

00:02:08.250 --> 00:02:09.440
left-hand side of the equation.

00:02:09.440 --> 00:02:13.350
The best way to do that it is to
subtract 6 from both sides.

00:02:13.350 --> 00:02:15.910
So let's subtract 6
from both sides.

00:02:15.910 --> 00:02:19.990
The left-hand side, we're
just left with the 3x.

00:02:19.990 --> 00:02:21.470
The 6's cancel out.

00:02:21.470 --> 00:02:30.500
The right-hand side,
231 minus 6 is 225.

00:02:30.500 --> 00:02:32.895
We have 3x is equal to 225.

00:02:32.895 --> 00:02:36.030
To isolate the x, let's just
divide both sides by 3.

00:02:39.050 --> 00:02:41.720
The left-hand side, the 3's
cancel out-- that was the

00:02:41.720 --> 00:02:48.880
whole point behind dividing by
3-- we get just an x is being

00:02:48.880 --> 00:02:51.490
equal to-- and 225
divided by 3.

00:02:51.490 --> 00:02:52.320
Let me do it over here.

00:02:52.320 --> 00:02:56.630
So 3 goes into 225.

00:02:56.630 --> 00:02:59.750
It goes into 22 7 times.

00:02:59.750 --> 00:03:03.180
7 times 3 is 21.

00:03:03.180 --> 00:03:05.950
22 minus 21 is 1.

00:03:05.950 --> 00:03:07.810
Bring down the 5.

00:03:07.810 --> 00:03:10.540
3 goes into 15 5 times.

00:03:10.540 --> 00:03:12.700
5 times 3 is 15.

00:03:12.700 --> 00:03:14.860
Subtract, no remainder.

00:03:14.860 --> 00:03:20.150
So 225 divided by 3 is 75.

00:03:20.150 --> 00:03:25.840
So the smallest, the smallest
of the odd integers is 75.

00:03:25.840 --> 00:03:27.520
So this one is 75.

00:03:27.520 --> 00:03:29.430
What's x plus 2 going
to be equal to?

00:03:29.430 --> 00:03:33.670
Well, that's going
to be 2 more, 77.

00:03:33.670 --> 00:03:35.770
And what's x plus 4 going
to be equal to?

00:03:35.770 --> 00:03:37.550
Well, that's the largest
of them.

00:03:37.550 --> 00:03:43.130
x is 75 plus 4 is
going to be 79.

00:03:43.130 --> 00:03:45.470
And notice, we have three
odd integers.

00:03:45.470 --> 00:03:46.450
They're consecutive.

00:03:46.450 --> 00:03:49.110
They're the, you know, they're
the odd integers that come

00:03:49.110 --> 00:03:50.710
directly after each other.

00:03:50.710 --> 00:03:54.680
And let's verify that when we
add them up, we get 231.

00:03:54.680 --> 00:04:00.050
So if we get 75 plus-- let me
just write it like this-- 75

00:04:00.050 --> 00:04:04.850
plus 77, plus 79, want
to add them all up.

00:04:04.850 --> 00:04:06.920
5 plus 7 is 12.

00:04:06.920 --> 00:04:10.320
12 plus 9 is 21.

00:04:10.320 --> 00:04:11.780
Carry the 2.

00:04:11.780 --> 00:04:13.680
2 plus 7 is 9.

00:04:13.680 --> 00:04:16.209
9 plus 7 is 16.

00:04:16.209 --> 00:04:20.019
16 plus 7 is 23.

00:04:20.019 --> 00:04:20.990
So there you have it.

00:04:20.990 --> 00:04:23.210
The three consecutive odd
integers, when you add them

00:04:23.210 --> 00:04:25.830
up, you got 231.

00:04:25.830 --> 00:04:27.470
They're consecutive and odd.

00:04:27.470 --> 00:04:28.910
They ask, what's the largest?

00:04:28.910 --> 00:04:32.040
The largest is 79.


Overview of the Khan Academy Library

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.000

Overview of the Khan Academy Library

00:00:00.000 --> 00:00:02.080
There's now a lot of content on the Khan Academy

00:00:02.080 --> 00:00:05.610
so I thought I'd give your a quick overview video of how to think about the playlists

00:00:05.610 --> 00:00:09.500
so the most basic playlist is the arithmetic playlist

00:00:09.500 --> 00:00:11.880
arithmetic

00:00:11.880 --> 00:00:13.080
and this is

00:00:13.080 --> 00:00:16.010
it literally starts at one plus one equals two

00:00:16.010 --> 00:00:17.500
and it moves

00:00:17.500 --> 00:00:21.310
about as far as operations on decimals long division

00:00:21.310 --> 00:00:22.620
things like that

00:00:22.620 --> 00:00:25.330
the next most advanced playlist

00:00:25.330 --> 00:00:26.870
is pre-algebra

00:00:26.870 --> 00:00:29.530
if you're starting from scratch you might want to go in this order

00:00:29.530 --> 00:00:31.940
pre-algebra

00:00:31.940 --> 00:00:36.540
and pre-algebra as the name implies covers
a lot of the foundational concepts for algebra

00:00:36.540 --> 00:00:43.130
like adding, subtracting, multiplying, dividing negative numbers and fractions and exponents

00:00:43.130 --> 00:00:46.330
and then we have several playlists that address

00:00:46.330 --> 00:00:48.140
actual algebra so there's

00:00:48.140 --> 00:00:50.750
an algebra playlist

00:00:50.750 --> 00:00:52.870
there is an algebra playlist

00:00:52.870 --> 00:00:56.550
there it is an algebra one playlist

00:00:56.550 --> 00:00:59.400
algebra one playlist

00:00:59.400 --> 00:01:01.920
and then there is a

00:01:01.920 --> 00:01:03.920
there is a California

00:01:03.920 --> 00:01:06.930
California standards

00:01:06.930 --> 00:01:09.140
California standards and there's two playlists

00:01:09.140 --> 00:01:11.540
California standards algebra one

00:01:11.540 --> 00:01:14.340
and California standards algebra two

00:01:14.340 --> 00:01:19.170
this just algebra playlist this is the the I would
call this kind of the main

00:01:19.170 --> 00:01:21.000
playlists covering algebra

00:01:21.000 --> 00:01:23.950
when I just make an arbitrary algebra video I put it in this playlist

00:01:23.950 --> 00:01:25.470
and I try to put it in order of

00:01:25.470 --> 00:01:28.970
of of of of I guess advancement or of difficulty

00:01:28.970 --> 00:01:31.930
this starts at the most basic algebraic equations

00:01:31.930 --> 00:01:33.930
but with one caveat that

00:01:33.930 --> 00:01:38.340
the the first few videos in this playlist are old
videos these are some of the very first videos

00:01:38.340 --> 00:01:40.470
that I did when I started making  YouTube videos

00:01:40.470 --> 00:01:44.330
and so they're not in HD, they're kind of
rough I'm using Microsoft  paint

00:01:44.330 --> 00:01:47.730
they're they're not the the smoothest
videos on the planet

00:01:47.730 --> 00:01:49.760
but then obviously it also has a lot of new videos in this

00:01:49.760 --> 00:01:54.930
and this algebra playlist covers both algebra one and algebra two

00:01:54.930 --> 00:01:57.400
and I merged it into one playlist because frankly algebra two

00:01:57.400 --> 00:02:01.310
has a lot of overlap or a lot of review with algebra one

00:02:01.310 --> 00:02:03.930
this algebra one playerlist

00:02:03.930 --> 00:02:07.120
this is worked examples from ck12.org's

00:02:07.120 --> 00:02:11.120
ck12.org's open source algebra one book which

00:02:11.120 --> 00:02:16.220
is a fairly comprehensive for covering the
California standards and for those of you

00:02:16.220 --> 00:02:19.430
who aren't in California what I say is

00:02:19.430 --> 00:02:22.220
this is fairly comprehensive for algebra one and if you know algebra one by these standards

00:02:22.220 --> 00:02:26.180
you're probably going to know the great majority
of what you need to know

00:02:26.180 --> 00:02:28.810
wherever you might be taking algebra

00:02:28.810 --> 00:02:31.620
and frankly if you watch both of these playlists

00:02:31.620 --> 00:02:34.940
I'm pretty sure you're going to be able to
do any type of algebra problem that you might

00:02:34.940 --> 00:02:36.750
see in any curriculum

00:02:36.750 --> 00:02:37.810
math is math

00:02:37.810 --> 00:02:42.740
now this is worked examples that you can and
it's all done recently so all of these videos

00:02:42.740 --> 00:02:45.000
are of of fairly good quality

00:02:45.000 --> 00:02:47.150
and they're all in high definition

00:02:47.150 --> 00:02:50.650
and you could just watch this algebra one
playlist in order

00:02:50.650 --> 00:02:54.190
and i'll just give you a little bit of warning
the first sixteen videos

00:02:54.190 --> 00:02:55.840
sixteen videos

00:02:55.840 --> 00:02:59.220
the very first ones are kind of an overview
so they move a little bit faster so if you

00:02:59.220 --> 00:03:00.620
have a little bit

00:03:00.620 --> 00:03:02.120
if his you're seeing algebra for the first time

00:03:02.120 --> 00:03:04.340
and they seem like they're moving a little quickly

00:03:04.340 --> 00:03:07.030
that's just because they're giving a survey of
the course

00:03:07.030 --> 00:03:08.340
and that's why..

00:03:08.340 --> 00:03:12.210
and I did that just because that's the order that they were displayed in the book

00:03:12.210 --> 00:03:14.240
but after those first sixteen video

00:03:14.240 --> 00:03:16.030
starting with one step equation

00:03:16.030 --> 00:03:17.840
it actually goes at a very moderate pace

00:03:17.840 --> 00:03:18.840
and I do a lot of examples

00:03:18.840 --> 00:03:21.930
and explain the concepts as I do the examples

00:03:21.930 --> 00:03:24.620
so if you're just starting out want to learn
algebra

00:03:24.620 --> 00:03:28.530
what I'd recommend reviewing pre algebra, never hurts

00:03:28.530 --> 00:03:31.220
then these are the two core algebra playlists

00:03:31.220 --> 00:03:33.470
and you could start at either one

00:03:33.470 --> 00:03:34.840
and find where you

00:03:34.840 --> 00:03:36.160
where you feel comfortable

00:03:36.160 --> 00:03:37.460
I would recommend

00:03:37.460 --> 00:03:40.340
if you're some place in between you know a little
bit of algebra, start at the beginning

00:03:40.340 --> 00:03:44.150
watch every video and both of these playlists
and if there's something just

00:03:44.150 --> 00:03:47.440
super duper easy to you then you move on to
the next video

00:03:47.440 --> 00:03:49.840
but just make sure that you have all of your bases covered

00:03:49.840 --> 00:03:54.620
now, these these videos right these playlists right
here the California standards

00:03:54.620 --> 00:03:57.470
these are just worked examples

00:03:57.470 --> 00:03:59.560
based on the California standards

00:03:59.560 --> 00:04:01.550
so after watching these you want to see more examples

00:04:01.550 --> 00:04:04.340
it'll never hurt to watch these two playlists right there

00:04:04.340 --> 00:04:06.120
now once you're done with algebra

00:04:06.120 --> 00:04:09.000
you might want to look at the geometry playlist

00:04:09.000 --> 00:04:13.530
you could probably start on the geometry playlist just with

00:04:13.530 --> 00:04:17.930
pre-algebra but it's good to have that algebraic foundation there

00:04:17.930 --> 00:04:19.930
and in most school systems

00:04:19.930 --> 00:04:21.350
you take algebra one first

00:04:21.350 --> 00:04:22.550
then geometry

00:04:22.550 --> 00:04:23.470
then algebra two

00:04:23.470 --> 00:04:26.670
so this is just a recommended path you might want to
take

00:04:26.670 --> 00:04:28.600
and then after geometry

00:04:28.600 --> 00:04:32.500
you're ready to move into trigonometry

00:04:32.500 --> 00:04:35.120
trigonometry

00:04:35.120 --> 00:04:37.540
and after algebra you are also ready to move

00:04:37.540 --> 00:04:38.930
'cos this playlist as I said it

00:04:38.930 --> 00:04:40.940
goes through algebra two

00:04:40.940 --> 00:04:43.310
after this you're also ready for pre-calculus

00:04:43.310 --> 00:04:46.800
you're also ready for pre-calculus

00:04:46.800 --> 00:04:49.370
and I'll write it at the same level as trigonometry because
in most

00:04:49.370 --> 00:04:51.120
in most

00:04:51.120 --> 00:04:53.470
school systems you take trigonometry and pre-calculus

00:04:53.470 --> 00:04:54.550
in the same year

00:04:54.550 --> 00:04:58.060
usually your junior or your senior year

00:04:58.060 --> 00:04:59.560
and then obviously after  pre-calculus

00:04:59.560 --> 00:05:01.120
and trigonometry

00:05:01.120 --> 00:05:02.370
the next

00:05:02.370 --> 00:05:04.930
you now are ready for calculus

00:05:04.930 --> 00:05:06.930
and this is a fairly extensive playlist

00:05:06.930 --> 00:05:07.870
it covers

00:05:07.870 --> 00:05:11.350
calculus one, two and three

00:05:11.350 --> 00:05:14.740
so differential integral and multi-variable and vector calculus

00:05:14.740 --> 00:05:17.350
so in some places even you could almost say

00:05:17.350 --> 00:05:19.160
even through calculus four

00:05:19.160 --> 00:05:20.730
now we have another

00:05:20.730 --> 00:05:23.560
another playlist that has the word algebra in it

00:05:23.560 --> 00:05:26.160
but it is a much more advanced algebra

00:05:26.160 --> 00:05:27.750
even though it might sound like a simpler algebra

00:05:27.750 --> 00:05:29.870
it's called linear algebra

00:05:29.870 --> 00:05:31.360
linear algebra

00:05:31.360 --> 00:05:35.740
and if you've never seen algebra before
do not watch this playlist it might sound

00:05:35.740 --> 00:05:37.970
like "hey linear algebra is easier than non-linear algebra"

00:05:37.970 --> 00:05:40.750
No, this is a mathematically rigorous course

00:05:40.750 --> 00:05:43.940
that normally math and engineering majors take

00:05:43.940 --> 00:05:46.370
in their, probably their sophomore year of college

00:05:46.370 --> 00:05:47.750
and I would say that

00:05:47.750 --> 00:05:51.940
you could start off if you have just a basic
understanding of pre-calculus you could probably

00:05:51.940 --> 00:05:54.970
you could probably start the linear algebra
playlist

00:05:54.970 --> 00:05:58.750
but I would recommend being fairly sophisticated mathematically

00:05:58.750 --> 00:06:01.120
you probably want a little bit of your calculus out of the way, to start it

00:06:01.120 --> 00:06:04.750
even though it is really algebra

00:06:04.750 --> 00:06:07.720
it is mathematically more sophisticated than a lot of what goes on

00:06:07.720 --> 00:06:09.550
in the calculus playlist

00:06:09.550 --> 00:06:11.930
and in the calculus playlist once you're about halfway
through

00:06:11.930 --> 00:06:12.180
you don't have to go through the whole playlist

00:06:12.180 --> 00:06:16.730
you're then ready for another advanced playlist that we have

00:06:16.730 --> 00:06:20.120
and that's differential equations

00:06:20.120 --> 00:06:22.740
differential equations

00:06:22.740 --> 00:06:23.930
probably two-thirds of the way

00:06:23.930 --> 00:06:26.250
through the calculus playlist you're probably

00:06:26.250 --> 00:06:27.750
then ready for differential equations

00:06:27.750 --> 00:06:30.160
another usually sophomore level course

00:06:30.160 --> 00:06:31.870
for math and science and engineering majors

00:06:31.870 --> 00:06:34.310
now everything else we have some

00:06:34.310 --> 00:06:36.470
we have some probability and statistics

00:06:36.470 --> 00:06:39.350
algebra is all you really need for that,

00:06:39.350 --> 00:06:41.670
but I'll put it here because it's reasonably advanced

00:06:41.670 --> 00:06:42.930
probability

00:06:42.930 --> 00:06:46.370
there's a probability playlist

00:06:46.370 --> 00:06:48.370
probability

00:06:48.370 --> 00:06:50.740
and then there is a statistics playlist

00:06:50.740 --> 00:06:52.540
they're obviously related

00:06:52.540 --> 00:06:54.350
you might want to do the probability one first

00:06:54.350 --> 00:06:56.180
but you don't have to

00:06:56.180 --> 00:06:58.940
there's probability and there is a statistics

00:06:58.940 --> 00:07:01.560
statistics playlist

00:07:01.560 --> 00:07:02.750
we also have

00:07:02.750 --> 00:07:05.250
a California standards geometry playlist

00:07:05.250 --> 00:07:07.540
and that's worked examples just like that
one over there

00:07:07.540 --> 00:07:10.930
so i'll do that like that we also have

00:07:10.930 --> 00:07:12.550
California

00:07:12.550 --> 00:07:14.370
standards

00:07:14.370 --> 00:07:17.560
California standards geometry

00:07:17.560 --> 00:07:18.980
so you can

00:07:18.980 --> 00:07:22.350
you can look at that if you want to see more geometric

00:07:22.350 --> 00:07:24.180
worked examples

00:07:24.180 --> 00:07:26.680
and then when you think about the sciences

00:07:26.680 --> 00:07:27.750
we have physics

00:07:27.750 --> 00:07:29.980
and the physics this is mainly I call them

00:07:29.980 --> 00:07:31.730
applied algebra problems

00:07:31.730 --> 00:07:32.750
so you might want to watch

00:07:32.750 --> 00:07:34.940
physics right around after you finish algebra

00:07:34.940 --> 00:07:39.540
there's only a few videos that require calculus

00:07:39.540 --> 00:07:41.750
I'll eventually make an entire playlist

00:07:41.750 --> 00:07:43.750
on calculus based physics

00:07:43.750 --> 00:07:45.940
most of this it's just algebra based physics

00:07:45.940 --> 00:07:49.750
so you're ready to go if you have a good solid understanding of algebra

00:07:49.750 --> 00:07:51.940
and then the other sciences and the chemistry

00:07:51.940 --> 00:07:57.560
uh it'd be nice to have algebra out of the way, pre algebra is probably enough for most of it

00:07:57.560 --> 00:08:00.120
pre algebra is probably enough for most of it but
algebra

00:08:00.120 --> 00:08:01.980
if you have your algebra out of the way

00:08:01.980 --> 00:08:04.120
then you're going to have no trouble at all with

00:08:04.120 --> 00:08:05.980
the chemistry playlist

00:08:05.980 --> 00:08:06.930
chemistry

00:08:06.930 --> 00:08:08.740
and then biology does deal with a little bit of

00:08:08.740 --> 00:08:10.120
of probability

00:08:10.120 --> 00:08:11.540
and things like that so it wouldn't hurt

00:08:11.540 --> 00:08:13.310
but that's actually that doesn't even require

00:08:13.310 --> 00:08:15.750
too much mathematics or at least the mathematics

00:08:15.750 --> 00:08:17.160
that I've diagrammed here.

00:08:17.160 --> 00:08:18.870
we could put biology here

00:08:18.870 --> 00:08:20.370
I guess the big take away is that

00:08:20.370 --> 00:08:22.680
algebra's really important for

00:08:22.680 --> 00:08:23.750
pretty much anything you want to do.

00:08:23.750 --> 00:08:25.560
there's some genetics problems that require

00:08:25.560 --> 00:08:26.930
some basic level and

00:08:26.930 --> 00:08:28.930
and some population genetics of population

00:08:28.930 --> 00:08:32.620
genetics problems that have some biology in it and then

00:08:32.620 --> 00:08:34.120
all the finance and economics that also

00:08:34.120 --> 00:08:35.960
I think it assumes

00:08:35.960 --> 00:08:38.340
a base level of algebraic understandings

00:08:38.340 --> 00:08:39.500
so the things on banking

00:08:39.500 --> 00:08:43.960
things on banking you don't have to have the
super advanced if you just go maybe

00:08:43.960 --> 00:08:47.590
a halfway through one of these playlists you are
probably ready for these

00:08:47.590 --> 00:08:50.360
banking economics

00:08:50.360 --> 00:08:51.750
economics

00:08:51.750 --> 00:08:52.970
and finance

00:08:52.970 --> 00:08:55.480
so hopefully that gives you a good overview of everything
that's

00:08:55.480 --> 00:08:56.970
on the Khan Academy

00:08:56.970 --> 00:08:59.703
and let me know if you have any questions


L'Hôpital's rule example 3

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:08.060
We want to figure out the limit
as x approaches 1 of the

00:00:08.060 --> 00:00:14.570
expression x over x minus
1 minus 1 over the

00:00:14.570 --> 00:00:17.930
natural log of x.

00:00:17.930 --> 00:00:19.900
So let's just see what
happens when we just

00:00:19.900 --> 00:00:21.230
try to plug in the 1.

00:00:21.230 --> 00:00:24.630
What happens if we evaluate
this expression at 1?

00:00:24.630 --> 00:00:30.050
Well then, we're going to get
a one here, over 1 minus 1.

00:00:30.050 --> 00:00:35.040
So we're going to get something
like a 1 over a 0, minus 1

00:00:35.040 --> 00:00:37.520
over, and what's the
natural log of 1?

00:00:37.520 --> 00:00:40.250
e to the what power
is equal to one?

00:00:40.250 --> 00:00:43.140
Well, anything to the zeroth
power is equal to 1, so e to

00:00:43.140 --> 00:00:45.420
the zeroth power is going to
be equal to 1, so the natural

00:00:45.420 --> 00:00:49.350
log of 1 is going to be 0.

00:00:49.350 --> 00:00:51.820
So we get the strange,
undefined 1 over

00:00:51.820 --> 00:00:54.300
0 minus 1 over 0.

00:00:54.300 --> 00:00:56.370
It's this bizarre-looking
undefined form.

00:00:56.370 --> 00:00:58.820
But it's not the indeterminate
type of form that we looked

00:00:58.820 --> 00:00:59.880
for in l'Hopital's rule.

00:00:59.880 --> 00:01:02.625
We're not getting a 0 over
a 0, we're not getting an

00:01:02.625 --> 00:01:03.750
infinity over an infinity.

00:01:03.750 --> 00:01:06.640
So you might just say, hey,
OK, this is a non-l'Hopital's

00:01:06.640 --> 00:01:07.150
rule problem.

00:01:07.150 --> 00:01:09.910
We're going to have to figure
out this limit some other way.

00:01:09.910 --> 00:01:13.210
And I would say, well
don't give up just yet!

00:01:13.210 --> 00:01:16.880
Maybe we can manipulate this
algebraically somehow so that

00:01:16.880 --> 00:01:20.380
it will give us the l'Hopital
indeterminate form, and then

00:01:20.380 --> 00:01:23.040
we can just apply the rule.

00:01:23.040 --> 00:01:24.790
And to do that, let's just
see, what happens if we

00:01:24.790 --> 00:01:26.470
add these two expressions?

00:01:26.470 --> 00:01:29.865
So if we add them, so this
expression, if we add it, it

00:01:29.865 --> 00:01:32.160
will be, well, the common
denominator is going to be x

00:01:32.160 --> 00:01:36.850
minus 1 times the
natural log of x.

00:01:36.850 --> 00:01:38.740
I just multiplied
the denominators.

00:01:38.740 --> 00:01:43.420
And then the numerator is going
to be, well, if I multiply

00:01:43.420 --> 00:01:46.436
essentially this whole term by
natural log of x, so it's going

00:01:46.436 --> 00:01:51.317
to be x natural log of x, and
then this whole term I'm going

00:01:51.317 --> 00:01:52.930
to multiply by x minus one.

00:01:52.930 --> 00:01:54.955
So minus x minus 1.

00:01:58.510 --> 00:02:00.540
And you could break it apart
and see that this expression

00:02:00.540 --> 00:02:02.870
and this expression
are the same thing.

00:02:02.870 --> 00:02:07.000
This right here, that right
there, is the same thing as x

00:02:07.000 --> 00:02:10.310
over x minus 1, because the
natural log of x's cancel out.

00:02:10.310 --> 00:02:12.220
Let me get rid of that.

00:02:12.220 --> 00:02:18.430
And then this right here is the
same thing as 1 over natural

00:02:18.430 --> 00:02:21.510
log of x, because the x
minus 1's cancel out.

00:02:21.510 --> 00:02:23.630
So hopefully you realize,
all I did is I added

00:02:23.630 --> 00:02:25.120
these two expressions.

00:02:25.120 --> 00:02:29.110
So given that, let's see what
happens if I take the limit as

00:02:29.110 --> 00:02:31.600
x approaches 1 of this thing.

00:02:31.600 --> 00:02:33.010
Because these are
the same thing.

00:02:33.010 --> 00:02:35.320
Do we get anything
more interesting?

00:02:35.320 --> 00:02:36.360
So what do we have here?

00:02:36.360 --> 00:02:38.810
We have one times the
natural log of 1.

00:02:38.810 --> 00:02:43.650
The natural log of 1 is 0, so
we have 0 here, so that is a 0.

00:02:43.650 --> 00:02:47.200
Minus 1 minus 0, so that's
going to be another 0, minus 0.

00:02:47.200 --> 00:02:51.000
So we get a 0 in the numerator.

00:02:51.000 --> 00:02:55.570
And in the denominator we get a
1 minus 1, which is 0, times

00:02:55.570 --> 00:03:00.100
the natural log of 1, which is
0, so 0 times 0, that is 0.

00:03:00.100 --> 00:03:00.960
And there you have it.

00:03:00.960 --> 00:03:04.940
We have indeterminate form that
we need for l'Hopital's rule,

00:03:04.940 --> 00:03:07.110
assuming that if we take the
derivative of that, and put it

00:03:07.110 --> 00:03:09.360
over the derivative of that,
that that limit exists.

00:03:09.360 --> 00:03:11.130
So let's try to do it.

00:03:11.130 --> 00:03:15.340
So this is going to be equal
to, if the limit exists, this

00:03:15.340 --> 00:03:19.200
is going to be equal to the
limit as x approaches 1.

00:03:19.200 --> 00:03:22.490
And let's take the derivative
in magenta, I'll take

00:03:22.490 --> 00:03:26.190
the derivative of this
numerator right over here.

00:03:26.190 --> 00:03:28.590
And for this first term,
just do the product rule.

00:03:28.590 --> 00:03:32.970
Derivative of x is one, and
then so 1 times the natural log

00:03:32.970 --> 00:03:35.920
of x, the derivative of the
first term times

00:03:35.920 --> 00:03:36.930
the second term.

00:03:36.930 --> 00:03:39.570
And then we're going to have
plus the derivative of the

00:03:39.570 --> 00:03:43.820
second term plus 1 over
x times the first term.

00:03:43.820 --> 00:03:45.430
It's just the product rule.

00:03:45.430 --> 00:03:47.920
So 1 over x times x, we're
going to see, that's just 1,

00:03:47.920 --> 00:03:54.390
and then we have minus the
derivative of x minus 1.

00:03:54.390 --> 00:03:58.450
Well, the derivative of x minus
1 is just 1, so it's just

00:03:58.450 --> 00:04:01.090
going to be minus 1.

00:04:01.090 --> 00:04:08.710
And then, all of that is over
the derivative of this thing.

00:04:08.710 --> 00:04:11.340
So let's take the derivative
of that, over here.

00:04:11.340 --> 00:04:16.600
So the derivative of the first
term, of x minus 1, is just 1.

00:04:16.600 --> 00:04:20.330
Multiply that times the second
term, you get natural log of x.

00:04:20.330 --> 00:04:23.520
And then plus the derivative of
the second term, derivative

00:04:23.520 --> 00:04:28.350
of natural log of x is one
over x, times x minus 1.

00:04:32.140 --> 00:04:34.240
I think we can simplify
this a little bit.

00:04:34.240 --> 00:04:37.270
This 1 over x times
x, that's a 1.

00:04:37.270 --> 00:04:38.580
We're going to
subtract one from it.

00:04:38.580 --> 00:04:40.910
So these cancel
out, right there.

00:04:40.910 --> 00:04:45.710
And so this whole expression
can be rewritten as the limit

00:04:45.710 --> 00:04:51.260
as approaches 1, the numerator
is just natural log of x, do

00:04:51.260 --> 00:04:57.160
that in magenta, and the
denominator is the natural log

00:04:57.160 --> 00:05:03.600
of x plus x minus 1 over x

00:05:03.600 --> 00:05:05.250
So let's try to evaluate
this limit here.

00:05:05.250 --> 00:05:09.060
So if we take x approaches one
of natural log of x, that

00:05:09.060 --> 00:05:13.640
will give us a, well,
natural log of 1 is 0.

00:05:13.640 --> 00:05:19.720
And over here, we get natural
log of 1, which is 0.

00:05:19.720 --> 00:05:27.920
And then plus 1 minus 1 over
plus 1 minus 1 over 1, well,

00:05:27.920 --> 00:05:28.900
that's just going
to be another 0.

00:05:28.900 --> 00:05:29.810
1 minus 1 is zero.

00:05:29.810 --> 00:05:30.680
So you're going to
have 0 plus 0.

00:05:30.680 --> 00:05:34.140
So you're going to get
a 0 over 0 again.

00:05:34.140 --> 00:05:35.740
0 over 0.

00:05:35.740 --> 00:05:38.230
So once again, let's apply
l'Hopital's rule again.

00:05:38.230 --> 00:05:39.890
Let's take the derivative
of that, put it over

00:05:39.890 --> 00:05:41.240
the derivative of that.

00:05:41.240 --> 00:05:44.210
So this, if we're ever going to
get to a limit, is going to be

00:05:44.210 --> 00:05:51.950
equal to the limit as x
approaches 1 of the derivative

00:05:51.950 --> 00:05:56.320
of the numerator, 1 over x,
right, the derivative of ln of

00:05:56.320 --> 00:06:00.340
x is 1/x, over the derivative
of the denominator.

00:06:00.340 --> 00:06:01.160
And what's that?

00:06:01.160 --> 00:06:06.950
Well, derivative of natural
log of x is 1 over x plus

00:06:06.950 --> 00:06:09.590
derivative of x minus 1 over x.

00:06:09.590 --> 00:06:13.120
You could view it this way,
as 1 over x times x minus 1.

00:06:13.120 --> 00:06:16.730
Well, derivative of x to the
negative 1, we'll take the

00:06:16.730 --> 00:06:19.280
derivative of the first one
times the second thing, and

00:06:19.280 --> 00:06:20.670
then the derivative of the
second thing times

00:06:20.670 --> 00:06:21.610
the first thing.

00:06:21.610 --> 00:06:24.980
So the derivative of the first
term, x to the negative 1, is

00:06:24.980 --> 00:06:30.030
negative x to the negative 2
times the second term, times x

00:06:30.030 --> 00:06:34.830
minus 1, plus the derivative of
the second term, which is

00:06:34.830 --> 00:06:39.780
just 1 times the first
term, plus 1 over x.

00:06:39.780 --> 00:06:45.060
So this is going to be equal
to, I just had a random thing

00:06:45.060 --> 00:06:45.860
pop up on my computer.

00:06:45.860 --> 00:06:47.730
Sorry for that little
sound, if you heard it.

00:06:47.730 --> 00:06:48.780
But where was I?

00:06:48.780 --> 00:06:50.710
Oh, let's just simplify
this over here.

00:06:50.710 --> 00:06:52.210
We were doing our
l'Hopital's rule.

00:06:52.210 --> 00:06:58.010
So this is going to be equal
to, let me, this is going to be

00:06:58.010 --> 00:07:02.870
equal to, if we evaluate x as
equal to 1, the numerator is

00:07:02.870 --> 00:07:05.610
just 1/1, which is just 1.

00:07:05.610 --> 00:07:07.406
So we're definitely not going
to have an indeterminate or

00:07:07.406 --> 00:07:09.480
at least a 0/0 form anymore.

00:07:09.480 --> 00:07:12.080
And the denominator is going to
be, if you evaluate it at 1,

00:07:12.080 --> 00:07:18.180
this is 1/1, which is 1, plus
negative 1 to the negative 2.

00:07:18.180 --> 00:07:21.490
So, or you say, 1 to the
negative 2 is just 1, it's

00:07:21.490 --> 00:07:22.445
just a negative one.

00:07:22.445 --> 00:07:24.820
But then you multiply that
times 1 minus 1, which is

00:07:24.820 --> 00:07:27.100
0, so this whole term's
going to cancel out.

00:07:27.100 --> 00:07:29.890
And you have a plus
another 1 over 1.

00:07:29.890 --> 00:07:34.090
So plus 1 And so this is
going to be equal to 1/2.

00:07:34.090 --> 00:07:34.990
And there you have it.

00:07:34.990 --> 00:07:37.620
Using L'Hopital's rule and a
couple of steps, we solved

00:07:37.620 --> 00:07:39.050
something that at least
initially didn't look

00:07:39.050 --> 00:07:40.260
like it was 0/0.

00:07:40.260 --> 00:07:44.110
We just added the 2 terms, got
0/0, took derivatives of the

00:07:44.110 --> 00:07:46.460
numerators and the denominators
2 times in a row to

00:07:46.460 --> 00:07:49.180
eventually get our limit.


L'Hôpital's rule example 2

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:08.810
We need to evaluate the limit,
as x approaches infinity, of 4x

00:00:08.810 --> 00:00:18.350
squared minus 5x, all of that
over 1 minus 3x squared.

00:00:18.350 --> 00:00:20.120
So infinity is kind
of a strange number.

00:00:20.120 --> 00:00:22.940
You can't just plug in infinity
and see what happens.

00:00:22.940 --> 00:00:25.450
But if you wanted to evaluate
this limit, what you might try

00:00:25.450 --> 00:00:28.420
to do is just evaluate-- if you
want to find the limit as this

00:00:28.420 --> 00:00:31.020
numerator approaches infinity,
you put in really large numbers

00:00:31.020 --> 00:00:32.840
there, and you're going to see
that it approaches infinity.

00:00:32.840 --> 00:00:34.980
That the numerator
approaches infinity as

00:00:34.980 --> 00:00:36.200
x approaches infinity.

00:00:36.200 --> 00:00:38.300
And if you put really large
numbers in the denominator,

00:00:38.300 --> 00:00:40.600
you're going to see
that that also-- well,

00:00:40.600 --> 00:00:41.780
not quite infinity.

00:00:41.780 --> 00:00:45.230
3x squared will approach
infinity, but we're

00:00:45.230 --> 00:00:46.190
subtracting it.

00:00:49.100 --> 00:00:52.280
If you subtract infinity from
some non-infinite number, it's

00:00:52.280 --> 00:00:54.220
going to be negative infinity.

00:00:54.220 --> 00:00:57.610
So if you were to just kind
of evaluate it at infinity,

00:00:57.610 --> 00:01:00.260
the numerator, you would
get positive infinity.

00:01:00.260 --> 00:01:04.915
The denominator, you would
get negative infinity.

00:01:04.915 --> 00:01:06.260
So I'll write it like this.

00:01:06.260 --> 00:01:07.420
Negative infinity.

00:01:07.420 --> 00:01:09.800
And that's one of the
indeterminate forms

00:01:09.800 --> 00:01:11.770
that L'Hopital's Rule
can be applied to.

00:01:11.770 --> 00:01:13.040
And you're probably saying,
hey, Sal, why are we even

00:01:13.040 --> 00:01:14.400
using L'Hopital's Rule?

00:01:14.400 --> 00:01:16.450
I know how to do this
without L'Hopital's Rule.

00:01:16.450 --> 00:01:19.130
And you probably
do, or you should.

00:01:19.130 --> 00:01:20.160
And we'll do that in a second.

00:01:20.160 --> 00:01:22.220
But I just wanted to show you
that L'Hopital's Rule also

00:01:22.220 --> 00:01:24.150
works for this type of problem,
and I really just wanted to

00:01:24.150 --> 00:01:28.190
show you an example that had a
infinity over negative

00:01:28.190 --> 00:01:30.510
or positive infinity
indeterminate form.

00:01:30.510 --> 00:01:32.320
But let's apply
L'Hopital's Rule here.

00:01:32.320 --> 00:01:37.360
So if this limit exists, or if
the limit of their derivatives

00:01:37.360 --> 00:01:43.380
exist, then this limit's going
to be equal to the limit as x

00:01:43.380 --> 00:01:47.000
approaches infinity of the
derivative of the numerator.

00:01:47.000 --> 00:01:49.750
So the derivative of the
numerator is-- the derivative

00:01:49.750 --> 00:01:56.130
of 4x squared is 8x minus 5
over-- the derivative of the

00:01:56.130 --> 00:01:58.420
denominator is, well,
derivative of 1 is 0.

00:01:58.420 --> 00:02:03.200
Derivative of negative 3x
squared is negative 6x.

00:02:03.200 --> 00:02:06.200
And once again, when you
evaluated infinity, the

00:02:06.200 --> 00:02:08.930
numerator is going to
approach infinity.

00:02:08.930 --> 00:02:11.310
And the denominator is
approaching negative infinity.

00:02:11.310 --> 00:02:14.240
Negative 6 times infinity
is negative infinity.

00:02:14.240 --> 00:02:16.390
So this is negative infinity.

00:02:16.390 --> 00:02:18.240
So let's apply
L'Hopital's Rule again.

00:02:18.240 --> 00:02:22.250
So if the limit of these guys'
derivatives exist-- or the

00:02:22.250 --> 00:02:25.620
rational function of the
derivative of this guy divided

00:02:25.620 --> 00:02:28.870
by the derivative of that guy--
if that exists, then this

00:02:28.870 --> 00:02:33.010
limit's going to be equal to
the limit as x approaches

00:02:33.010 --> 00:02:38.280
infinity of-- arbitrarily
switch colors-- derivative

00:02:38.280 --> 00:02:41.570
of 8x minus 5 is just 8.

00:02:41.570 --> 00:02:45.710
Derivative of negative
6x is negative 6.

00:02:45.710 --> 00:02:48.040
And this is just going to be--
this is just a constant here.

00:02:48.040 --> 00:02:49.910
So it doesn't matter what limit
you're approaching, this is

00:02:49.910 --> 00:02:51.200
just going to equal this value.

00:02:51.200 --> 00:02:52.670
Which is what?

00:02:52.670 --> 00:02:55.540
If we put it in lowest
common form, or simplified

00:02:55.540 --> 00:02:57.670
form, it's negative 4/3.

00:03:01.810 --> 00:03:03.440
So this limit exists.

00:03:03.440 --> 00:03:04.920
This was an indeterminate form.

00:03:04.920 --> 00:03:11.060
And the limit of this
function's derivative over this

00:03:11.060 --> 00:03:13.680
function's derivative exists,
so this limit must also

00:03:13.680 --> 00:03:15.190
equal negative 4/3.

00:03:15.190 --> 00:03:17.610
And by that same argument,
that limit also must be

00:03:17.610 --> 00:03:20.520
equal to negative 4/3.

00:03:20.520 --> 00:03:21.950
And for those of you who
say, hey, we already

00:03:21.950 --> 00:03:23.540
knew how to do this.

00:03:23.540 --> 00:03:25.470
We could just factor
out an x squared.

00:03:25.470 --> 00:03:26.400
You are absolutely right.

00:03:26.400 --> 00:03:28.440
And I'll show you
that right here.

00:03:28.440 --> 00:03:30.580
Just to show you that it's
not the only-- you know,

00:03:30.580 --> 00:03:32.920
L'Hopital's Rule is not
the only game in town.

00:03:32.920 --> 00:03:35.630
And frankly, for this type of
problem, my first reaction

00:03:35.630 --> 00:03:38.380
probably wouldn't have been to
use L'Hopital's Rule first.

00:03:38.380 --> 00:03:42.360
You could have said that that
first limit-- so the limit as x

00:03:42.360 --> 00:03:49.630
approaches infinity of 4x
squared minus 5x over 1 minus

00:03:49.630 --> 00:03:55.830
3x squared is equal to the
limit as x approaches infinity.

00:03:55.830 --> 00:03:59.940
Let me draw a little line here,
to show you that this is equal

00:03:59.940 --> 00:04:02.010
to that, not to this
thing over here.

00:04:02.010 --> 00:04:03.790
This is equal to the limit
as x approaches infinity.

00:04:03.790 --> 00:04:06.020
Let's factor out an x squared
out of the numerator

00:04:06.020 --> 00:04:06.690
and the denominator.

00:04:06.690 --> 00:04:15.750
So you have an x squared
times 4 minus 5 over x.

00:04:15.750 --> 00:04:19.220
Right? x squared times 5
over x is going to be 5x.

00:04:19.220 --> 00:04:22.440
Divided by-- let's factor out
an x out of the numerator.

00:04:22.440 --> 00:04:29.750
So x squared times 1
over x squared minus 3.

00:04:29.750 --> 00:04:33.540
And then these x
squareds cancel out.

00:04:33.540 --> 00:04:37.920
So this is going to be equal
to the limit as x approaches

00:04:37.920 --> 00:04:45.990
infinity of 4 minus 5 over x
over 1 over x squared minus 3.

00:04:45.990 --> 00:04:47.760
And what's this going
to be equal to?

00:04:47.760 --> 00:04:50.610
Well, as x approaches
infinity-- 5 divided by

00:04:50.610 --> 00:04:52.050
infinity-- this term
is going to be 0.

00:04:52.050 --> 00:04:54.160
Super duper infinitely
large denominator,

00:04:54.160 --> 00:04:56.650
this is going to be 0.

00:04:56.650 --> 00:04:58.800
That is going to approach 0.

00:04:58.800 --> 00:04:59.900
And same argument.

00:04:59.900 --> 00:05:02.400
This right here is
going to approach 0.

00:05:02.400 --> 00:05:05.010
All you're left with is
a 4 and a negative 3.

00:05:05.010 --> 00:05:08.130
So this is going to be
equal to negative 4 over a

00:05:08.130 --> 00:05:10.610
negative 3, or negative 4/3.

00:05:10.610 --> 00:05:13.270
So you didn't have to do
use L'Hopital's Rule

00:05:13.270 --> 00:05:14.910
for this problem.


L'Hôpital's rule example 1

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:08.370
Let's say we need to evaluate
the limit as x approaches 0 of

00:00:08.370 --> 00:00:20.340
2 sine of x minus sine of 2x,
all of that over x

00:00:20.340 --> 00:00:24.210
minus sine of x.

00:00:24.210 --> 00:00:26.820
Now, the first thing that I
always try to do when I first

00:00:26.820 --> 00:00:29.160
see a limit problem is hey,
what happens if I just try to

00:00:29.160 --> 00:00:33.890
evaluate this function
at x is equal to 0?

00:00:33.890 --> 00:00:35.540
Maybe nothing crazy happens.

00:00:35.540 --> 00:00:37.090
So let's just try it out.

00:00:37.090 --> 00:00:40.340
If we try to do x equals
0, what happens?

00:00:40.340 --> 00:00:43.580
We get 2 sine of 0, which is 0.

00:00:43.580 --> 00:00:46.190
Minus sine of 2 times 0.

00:00:46.190 --> 00:00:49.960
Well, that's going to be sine
of 0 again, which is 0.

00:00:49.960 --> 00:00:53.510
So our numerator is
going to be equal to 0.

00:00:53.510 --> 00:00:54.750
Sine of 0, that's 0.

00:00:54.750 --> 00:00:56.440
And then we have another
sine of 0 there.

00:00:56.440 --> 00:00:58.340
That's another 0, so all 0's.

00:00:58.340 --> 00:00:59.590
And our denominator,
we're going to have

00:00:59.590 --> 00:01:02.590
a 0 minus sine of 0.

00:01:02.590 --> 00:01:05.200
Well that's also going to be 0.

00:01:05.200 --> 00:01:08.340
But we have that indeterminate
form, we have that undefined

00:01:08.340 --> 00:01:12.000
0/0 that we talked about
in the last video.

00:01:12.000 --> 00:01:15.720
So maybe we can use
L'Hopital's rule here.

00:01:15.720 --> 00:01:18.910
In order to use L'Hopital's
rule then the limit as x

00:01:18.910 --> 00:01:21.880
approaches 0 of the derivative
of this function over

00:01:21.880 --> 00:01:24.880
the derivative of this
function needs to exist.

00:01:24.880 --> 00:01:28.060
So let's just apply L'Hopital's
rule and let's just take the

00:01:28.060 --> 00:01:30.470
derivative of each of these and
see if we can find the limit.

00:01:30.470 --> 00:01:33.620
If we can, then that's going to
be the limit of this thing.

00:01:33.620 --> 00:01:38.750
So this thing, assuming that it
exists, is going to be equal to

00:01:38.750 --> 00:01:44.950
the limit as x approaches 0 of
the derivative of this

00:01:44.950 --> 00:01:46.600
numerator up here.

00:01:46.600 --> 00:01:48.380
And so what's the derivative
of the numerator going to be?

00:01:48.380 --> 00:01:49.315
I'll do it in a new color.

00:01:49.315 --> 00:01:51.050
I'll do it in green.

00:01:51.050 --> 00:01:55.370
Well, the derivative of 2
sine of x is 2 cosine of x.

00:01:57.920 --> 00:02:00.160
And then, minus-- well,
the derivative of sine

00:02:00.160 --> 00:02:04.150
of 2x is 2 cosine of 2x.

00:02:04.150 --> 00:02:07.440
So minus 2 cosine of 2x.

00:02:07.440 --> 00:02:09.210
Just use the chain rule
there, derivative of

00:02:09.210 --> 00:02:11.000
the inside is just 2.

00:02:11.000 --> 00:02:12.060
That's the 2 out there.

00:02:12.060 --> 00:02:15.405
Derivative of the outside is
cosine of 2x, and we had that

00:02:15.405 --> 00:02:16.790
negative number out there.

00:02:16.790 --> 00:02:19.610
So that's the derivative of
our numerator, maria, and

00:02:19.610 --> 00:02:20.500
what is the Derivative.

00:02:20.500 --> 00:02:21.920
of our denominator?

00:02:21.920 --> 00:02:25.430
Well, derivative of x is just
1, and derivative of sine

00:02:25.430 --> 00:02:26.870
of x is just cosine of x.

00:02:26.870 --> 00:02:29.920
So 1 minus cosine of x.

00:02:29.920 --> 00:02:31.950
So let's try to
evaluate this limit.

00:02:31.950 --> 00:02:33.150
What do we get?

00:02:33.150 --> 00:02:36.180
If we put a 0 up here we're
going to get 2 times cosine

00:02:36.180 --> 00:02:39.610
of 0, which is 2-- let
me write it like this.

00:02:39.610 --> 00:02:42.760
So this is 2 times cosine
of 0, which is 1.

00:02:42.760 --> 00:02:48.040
So it's 2 minus 2
cosine of 2 times 0.

00:02:48.040 --> 00:02:49.480
Let me write it this way.

00:02:49.480 --> 00:02:50.580
Actually, let me just
do it this way.

00:02:50.580 --> 00:02:53.150
If we just straight up evaluate
the limit of the numerator and

00:02:53.150 --> 00:02:54.270
the denominator, what
are we going to get?

00:02:54.270 --> 00:02:57.710
We get 2 cosine of
0, which is 2.

00:02:57.710 --> 00:03:01.760
Minus 2 times cosine of--
well, this 2 times 0 is

00:03:01.760 --> 00:03:02.620
still going to be 0.

00:03:02.620 --> 00:03:06.850
So minus 2 times cosine
of 0, which is 2.

00:03:06.850 --> 00:03:15.620
All of that over 1 minus the
cosine of 0, which is 1.

00:03:15.620 --> 00:03:17.295
So once again, we get 0/0.

00:03:22.100 --> 00:03:24.500
So does this mean that
the limit doesn't exist?

00:03:24.500 --> 00:03:27.420
No, it still might exist,
we might just want to do

00:03:27.420 --> 00:03:28.830
L'Hopital's rule again.

00:03:28.830 --> 00:03:30.610
Let me take the derivative
of that and put it over

00:03:30.610 --> 00:03:31.490
the derivative of that.

00:03:31.490 --> 00:03:34.610
And then take the limit and
maybe L'Hopital's rule

00:03:34.610 --> 00:03:35.700
will help us on the
next [INAUDIBLE].

00:03:35.700 --> 00:03:38.610
So let's see if it
gets us anywhere.

00:03:38.610 --> 00:03:43.030
So this should be equal to
the limit if L'Hopital's

00:03:43.030 --> 00:03:43.840
rule applies here.

00:03:43.840 --> 00:03:46.050
We're not 100% sure yet.

00:03:46.050 --> 00:03:49.950
This should be equal to the
limit as x approaches 0 of the

00:03:49.950 --> 00:03:53.690
derivative of that thing over
the derivative of that thing.

00:03:53.690 --> 00:03:58.240
So what's the derivative
of 2 cosine of x?

00:03:58.240 --> 00:04:00.485
Well, derivative of cosine
of x is negative sine of x.

00:04:00.485 --> 00:04:04.920
So it's negative 2 sine of x.

00:04:04.920 --> 00:04:11.320
And then derivative of cosine
of 2x is negative 2 sine of 2x.

00:04:11.320 --> 00:04:13.660
So we're going to have this
negative cancel out with the

00:04:13.660 --> 00:04:16.980
negative on the negative 2
and then a 2 times the 2.

00:04:16.980 --> 00:04:21.990
So it's going to be
plus 4 sine of 2x.

00:04:21.990 --> 00:04:23.790
Let me make sure I
did that right.

00:04:23.790 --> 00:04:26.640
We have the minus 2 or the
negative 2 on the outside.

00:04:26.640 --> 00:04:30.950
Derivative of cosine of 2x
is going to be 2 times

00:04:30.950 --> 00:04:32.590
negative sine of x.

00:04:32.590 --> 00:04:34.270
So the 2 times 2 is 4.

00:04:34.270 --> 00:04:36.780
The negative sine of x
times-- the negative

00:04:36.780 --> 00:04:38.060
right there's a plus.

00:04:38.060 --> 00:04:40.180
You have a positive sine,
so it's the sine of 2x.

00:04:40.180 --> 00:04:42.370
That's the numerator when
you take the derivative.

00:04:42.370 --> 00:04:45.260
And the denominator-- this
is just an exercise in

00:04:45.260 --> 00:04:45.830
taking derivatives.

00:04:45.830 --> 00:04:47.350
What's the derivative
of the denominator?

00:04:47.350 --> 00:04:49.310
Derivative of 1 is 0.

00:04:49.310 --> 00:04:52.790
And derivative negative
cosine of x is just-- well,

00:04:52.790 --> 00:04:53.900
that's just sine of x.

00:04:56.480 --> 00:04:57.950
So let's take this limit.

00:04:57.950 --> 00:05:00.050
So this is going to be equal
to-- well, immediately if I

00:05:00.050 --> 00:05:02.810
take x is equal to 0 in the
denominator, I know that

00:05:02.810 --> 00:05:05.190
sine of 0 is just 0.

00:05:05.190 --> 00:05:06.870
Let's see what happens
in the numerator.

00:05:06.870 --> 00:05:08.700
Negative 2 times sine of 0.

00:05:08.700 --> 00:05:10.740
That's going to be 0.

00:05:10.740 --> 00:05:13.880
And then plus 4 times
sine of 2 times 0.

00:05:13.880 --> 00:05:16.570
Well, that's still sine of 0,
so that's still going to be 0.

00:05:16.570 --> 00:05:19.210
So once again, we got
indeterminate form again.

00:05:19.210 --> 00:05:19.900
Are we done?

00:05:19.900 --> 00:05:20.820
Do we give up?

00:05:20.820 --> 00:05:22.730
Do we say that L'Hopital's
rule didn't work?

00:05:22.730 --> 00:05:26.550
No, because this could have
been our first limit problem.

00:05:26.550 --> 00:05:28.750
And if this is our first limit
problem we say, hey, maybe we

00:05:28.750 --> 00:05:30.680
could use L'Hopital's rule
here because we got an

00:05:30.680 --> 00:05:32.600
indeterminate form.

00:05:32.600 --> 00:05:34.990
Both the numerator and
the denominator approach

00:05:34.990 --> 00:05:36.660
0 as x approaches 0.

00:05:36.660 --> 00:05:40.110
So let's take the
derivatives again.

00:05:40.110 --> 00:05:42.870
This will be equal to--
if the limit exist, the

00:05:42.870 --> 00:05:45.730
limit as x approaches 0.

00:05:45.730 --> 00:05:47.685
Let's take the derivative
of the numerator.

00:05:47.685 --> 00:05:51.410
The derivative of negative
2 sine of x is negative

00:05:51.410 --> 00:05:54.100
2 cosine of x.

00:05:54.100 --> 00:05:57.850
And then, plus the
derivative of 4 sine of 2x.

00:05:57.850 --> 00:06:01.520
Well, it's 2 times
4, which is 8.

00:06:01.520 --> 00:06:04.330
Times cosine of 2x.

00:06:04.330 --> 00:06:08.320
Derivative of sine of
2x is 2 cosine of 2x.

00:06:08.320 --> 00:06:10.270
And that first 2 gets
multiplied by the

00:06:10.270 --> 00:06:11.840
4 to get the 8.

00:06:11.840 --> 00:06:17.030
And then the derivative of the
denominator, derivative of sine

00:06:17.030 --> 00:06:19.140
of x is just cosine of x.

00:06:19.140 --> 00:06:21.880
So let's evaluate
this character.

00:06:21.880 --> 00:06:24.620
So it looks like we've made
some headway or maybe

00:06:24.620 --> 00:06:27.160
L'Hopital's rule stop applying
here because we take the limit

00:06:27.160 --> 00:06:29.610
as x approaches 0
of cosine of x.

00:06:29.610 --> 00:06:30.650
That is 1.

00:06:30.650 --> 00:06:33.220
So we're definitely not going
to get that indeterminate form,

00:06:33.220 --> 00:06:36.120
that 0/0 on this iteration.

00:06:36.120 --> 00:06:38.160
Let's see what happens
to the numerator.

00:06:38.160 --> 00:06:41.950
We get negative 2
times cosine of 0.

00:06:41.950 --> 00:06:45.530
Well that's just negative 2
because cosine of 0 is 1.

00:06:45.530 --> 00:06:50.090
Plus 8 times cosine of 2x.

00:06:50.090 --> 00:06:53.590
Well, if x is 0, so it's going
to be cosine of 0, which is 1.

00:06:53.590 --> 00:06:54.690
So it's just going to be an 8.

00:06:54.690 --> 00:06:56.990
So negative 2 plus 8.

00:06:56.990 --> 00:06:59.500
Well this thing right here,
negative 2 plus 8 is 6.

00:06:59.500 --> 00:07:00.720
6 over 1.

00:07:00.720 --> 00:07:03.370
This whole thing is equal to 6.

00:07:03.370 --> 00:07:06.850
So L'Hopital's rule-- it
applies to this last step.

00:07:06.850 --> 00:07:09.630
If this was the problem we were
given and we said, hey, when we

00:07:09.630 --> 00:07:13.530
tried to apply the limit we get
the limit as this numerator

00:07:13.530 --> 00:07:15.010
approaches 0 is 0.

00:07:15.010 --> 00:07:19.790
Limit as this denominator
approaches 0 is 0.

00:07:19.790 --> 00:07:21.980
As the derivative of the
numerator over the derivative

00:07:21.980 --> 00:07:25.600
of the denominator, that
exists and it equals 6.

00:07:25.600 --> 00:07:28.590
So this limit must
be equal to 6.

00:07:28.590 --> 00:07:31.740
Well if this limit is equal to
6, by the same argument, this

00:07:31.740 --> 00:07:33.340
limit is also going
to be equal to 6.

00:07:33.340 --> 00:07:35.850
And by the same argument,
this limit has got to

00:07:35.850 --> 00:07:40.090
also be equal to 6.

00:07:40.090 --> 00:07:41.770
And we're done.