Matrix multiplication (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.380
Well, after the last video,
hopefully, we're a little

00:00:02.380 --> 00:00:04.330
familiar with how you
add matrices.

00:00:04.330 --> 00:00:07.150
So now let's learn how
to multiply matrices.

00:00:07.150 --> 00:00:11.450
And keep in mind, these are
human-created definitions for

00:00:11.450 --> 00:00:12.840
matrix multiplication.

00:00:12.840 --> 00:00:15.000
We could have come up with
completely different ways to

00:00:15.000 --> 00:00:15.450
multiply it.

00:00:15.450 --> 00:00:18.510
But I encourage you to learn
this way because it'll help

00:00:18.510 --> 00:00:19.810
you in math class.

00:00:19.810 --> 00:00:22.100
And we will see later that
there's actually a lot of

00:00:22.100 --> 00:00:24.730
applications that come out
of this type of matrix

00:00:24.730 --> 00:00:25.290
multiplication.

00:00:25.290 --> 00:00:26.340
So let me think of
two matrices.

00:00:26.340 --> 00:00:30.320
I will do two 2 by 2 matrices,
and let's multiply them.

00:00:30.320 --> 00:00:34.500
Let's say-- let me pick some
random numbers: 2,

00:00:34.500 --> 00:00:40.530
minus 3, 7, and 5.

00:00:40.530 --> 00:00:42.980
And I'm going to multiply that
matrix, or that table of

00:00:42.980 --> 00:00:56.440
numbers, times 10, minus 8--
let me pick a good number

00:00:56.440 --> 00:01:03.980
here-- 12, and then minus 2.

00:01:03.980 --> 00:01:07.400
So now there might be a strong
temptation-- and you know in

00:01:07.400 --> 00:01:11.330
some ways it's not even an
illegitimate temptation-- to

00:01:11.330 --> 00:01:13.880
do the same thing with
multiplication that we did

00:01:13.880 --> 00:01:17.520
with addition, to just multiply
the corresponding

00:01:17.520 --> 00:01:20.580
terms. So you might be tempted
to say, well, the first term

00:01:20.580 --> 00:01:23.160
right here, the 1, 1 term, or
in the first row and first

00:01:23.160 --> 00:01:25.070
column, is going to
be 2 times 10.

00:01:25.070 --> 00:01:26.970
And this term is going
to be minus 3 times

00:01:26.970 --> 00:01:28.230
minus 8 and so forth.

00:01:28.230 --> 00:01:30.330
And that's how we added matrices
so maybe it's a

00:01:30.330 --> 00:01:33.510
natural extension to multiply
matrices the same way.

00:01:33.510 --> 00:01:36.180
And that is legitimate.

00:01:36.180 --> 00:01:38.530
One could define it that way,
but that's not the way it is

00:01:38.530 --> 00:01:39.410
in the real world.

00:01:39.410 --> 00:01:40.380
And the way in the real world,

00:01:40.380 --> 00:01:42.470
unfortunately, is more complex.

00:01:42.470 --> 00:01:44.980
But if you look at a
bunch of examples I

00:01:44.980 --> 00:01:45.670
think you'll get it.

00:01:45.670 --> 00:01:47.460
And you'll learn that
it's actually fairly

00:01:47.460 --> 00:01:47.980
straightforward.

00:01:47.980 --> 00:01:48.985
So how do we do it?

00:01:48.985 --> 00:01:53.290
So this first term that's in
the first row and its first

00:01:53.290 --> 00:01:58.050
column, is equal to essentially
this first row's

00:01:58.050 --> 00:02:01.380
vector-- no, this first
row vector--

00:02:01.380 --> 00:02:04.790
times this column vector.

00:02:04.790 --> 00:02:08.229
Now what do I mean
by that, right?

00:02:08.229 --> 00:02:11.320
So it's getting it's row
information from the first

00:02:11.320 --> 00:02:14.170
matrix's row, and it's getting
it's column information from

00:02:14.170 --> 00:02:16.320
the second matrix's column.

00:02:16.320 --> 00:02:17.190
So how do I do that?

00:02:17.190 --> 00:02:18.860
If you're familiar with dot
product, it's essentially the

00:02:18.860 --> 00:02:20.930
dot product of these
two matrices.

00:02:20.930 --> 00:02:24.820
Or without saying it that fancy,
it's just this: it's 2

00:02:24.820 --> 00:02:31.890
times 10, so 2-- I'm going to
write small-- times 10, plus

00:02:31.890 --> 00:02:39.640
minus 3 times 12.

00:02:39.640 --> 00:02:42.630
I'm going to run out of space.

00:02:42.630 --> 00:02:45.600
And so what's this second
term over here?

00:02:45.600 --> 00:02:49.170
Well, we're still on the first
row of the product vector but

00:02:49.170 --> 00:02:50.350
now we're on the
second column.

00:02:50.350 --> 00:02:51.910
We get our column information
from here.

00:02:51.910 --> 00:02:58.800
So let's pick a good color--
this is a slightly different

00:02:58.800 --> 00:03:00.530
shade of purple.

00:03:00.530 --> 00:03:04.110
So now this is going to be--
I'll do that in another

00:03:04.110 --> 00:03:10.580
color-- 2 times minus 8-- let me
just write out the number--

00:03:10.580 --> 00:03:18.810
2 times minus 8 is minus 16,
plus minus 3 times minus 2--

00:03:18.810 --> 00:03:21.160
what's minus 3 times minus 2?

00:03:21.160 --> 00:03:26.400
That is plus 6, right?

00:03:26.400 --> 00:03:28.870
So that's in row 1 column 2.

00:03:28.870 --> 00:03:30.590
It's minus 16 plus 6.

00:03:30.590 --> 00:03:31.830
And then let's come down here.

00:03:31.830 --> 00:03:33.920
So now we're in the
second row.

00:03:33.920 --> 00:03:35.550
So now we're going to use--
we're getting our row

00:03:35.550 --> 00:03:37.930
information from the first
matrix-- I know this is

00:03:37.930 --> 00:03:41.380
confusing and I feel bad for you
right now, but we're going

00:03:41.380 --> 00:03:45.230
to a bunch of examples and
I think it'll make sense.

00:03:45.230 --> 00:03:49.060
So this term-- the bottom left
term-- is going to be this row

00:03:49.060 --> 00:03:50.310
times this column.

00:03:50.310 --> 00:03:58.010
So it's going to be 7 times
10, so 70, plus 7 times 10

00:03:58.010 --> 00:04:05.650
plus 5 times 12, plus 60.

00:04:05.650 --> 00:04:09.290
And then the bottom right term
is going to be 7 times minus

00:04:09.290 --> 00:04:19.540
8, which is minus 56 plus
5 times minus 2.

00:04:19.540 --> 00:04:22.900
So that's minus 10.

00:04:22.900 --> 00:04:31.290
So the final product is going to
be 2 times 10 is 20, minus

00:04:31.290 --> 00:04:41.040
36, so that's minus 16
plus 6, that's 10.

00:04:41.040 --> 00:04:42.220
90-- was that what I said?

00:04:42.220 --> 00:04:46.390
No, it was-- 70, plus
60, that's 130.

00:04:46.390 --> 00:04:55.580
And then minus 56 minus
10, so minus 66.

00:04:55.580 --> 00:04:56.400
So there you have it.

00:04:56.400 --> 00:04:59.020
We just multiplied this matrix
times this matrix.

00:04:59.020 --> 00:05:00.360
Let me do another example.

00:05:00.360 --> 00:05:03.490
And I think I'll actually
squeeze it on this side so

00:05:03.490 --> 00:05:06.106
that we can write this side out
a little bit more neatly.

00:05:06.106 --> 00:05:18.630
So let's take the matrix and
now 1, 2, 3, 4, times the

00:05:18.630 --> 00:05:27.710
matrix 5, 6, 7, 8.

00:05:27.710 --> 00:05:30.200
Now we have much more space to
work with so this should come

00:05:30.200 --> 00:05:32.760
out neater.

00:05:32.760 --> 00:05:37.460
OK, but I'm going to do the
same thing, so to get this

00:05:37.460 --> 00:05:40.350
term right here-- the top left
term-- we're going to take--

00:05:40.350 --> 00:05:43.070
or the one that has row 1 column
1-- we're going to take

00:05:43.070 --> 00:05:52.050
the row 1 information from
here, and the column 1

00:05:52.050 --> 00:05:54.440
information from here.

00:05:54.440 --> 00:05:56.070
So you can view it as
this row vector

00:05:56.070 --> 00:05:57.120
times this column vector.

00:05:57.120 --> 00:06:01.680
So it results, 1 times
5 plus 2 times 7.

00:06:07.730 --> 00:06:08.380
Right?

00:06:08.380 --> 00:06:09.610
There you go.

00:06:09.610 --> 00:06:12.520
And so this term, it'll be this
row vector times this

00:06:12.520 --> 00:06:15.740
column vector-- let me do that
in a different color-- will be

00:06:15.740 --> 00:06:20.550
1 times 6 plus 2 times 8.

00:06:20.550 --> 00:06:21.370
Let me write that down.

00:06:21.370 --> 00:06:28.590
So it's 1 times 6
plus 2 times 8.

00:06:33.270 --> 00:06:34.720
Now we go down to
the second row.

00:06:34.720 --> 00:06:38.130
And we get our row information
from the first vector-- let me

00:06:38.130 --> 00:06:44.150
circle it with this color--
and it is 3 times 5

00:06:44.150 --> 00:06:45.520
plus 4 times 7.

00:06:52.800 --> 00:06:54.760
And then we are in the bottom
right, so we're in the bottom

00:06:54.760 --> 00:06:57.070
row and second column.

00:06:57.070 --> 00:06:59.350
So we get our row information
from here and our column

00:06:59.350 --> 00:07:00.930
information from here.

00:07:00.930 --> 00:07:03.860
So it's 3 times 6
plus 4 times 8.

00:07:10.300 --> 00:07:13.680
And if we simplify,
this is 5 plus--

00:07:13.680 --> 00:07:15.950
Well actually, let me just
remind you where all the

00:07:15.950 --> 00:07:16.630
numbers came from.

00:07:16.630 --> 00:07:18.200
So we have that green
color, right?

00:07:18.200 --> 00:07:25.890
This 1 and this 2, that's
this 1 and this 2,

00:07:25.890 --> 00:07:28.620
this 1 and this 2.

00:07:28.620 --> 00:07:28.755
Right?

00:07:28.755 --> 00:07:30.330
And notice, these were in the
first row and they're in the

00:07:30.330 --> 00:07:31.660
first row here.

00:07:31.660 --> 00:07:33.640
And this 5 and this 7?

00:07:33.640 --> 00:07:39.920
Well, that's this 5 and this
7, and this 5 and this 7.

00:07:39.920 --> 00:07:42.800
So, interesting.

00:07:42.800 --> 00:07:45.410
This was in column 1 of the
second matrix and this is in

00:07:45.410 --> 00:07:47.780
column 1 in the product
matrix.

00:07:47.780 --> 00:07:51.030
And similarly, the
6 and the 8.

00:07:51.030 --> 00:07:55.610
That's this 6, this 8, and then
it's used here, this 6

00:07:55.610 --> 00:07:57.540
and this 8.

00:07:57.540 --> 00:08:00.250
And then finally this 3 and the
4 in the brown, so that's

00:08:00.250 --> 00:08:03.980
this 3, this 4, and
this 3 and this 4.

00:08:03.980 --> 00:08:05.290
And we could of course
simplify all of it.

00:08:05.290 --> 00:08:10.180
This was 1 times 5 plus 2 times
7, so that's 5 plus 14,

00:08:10.180 --> 00:08:15.410
so this is 19.

00:08:15.410 --> 00:08:19.200
This is 1 times 6 plus 2
times 8, so it's 6 plus

00:08:19.200 --> 00:08:22.330
16, so that's 22.

00:08:22.330 --> 00:08:26.380
This is 3 times 5
plus 4 times 7.

00:08:26.380 --> 00:08:32.909
So 15 plus 28, 38, 43-- if my
math is correct-- and then we

00:08:32.909 --> 00:08:36.250
have 3 times 6 plus 4 times 8.

00:08:36.250 --> 00:08:44.220
So that's 18 plus
32, that's 50.

00:08:44.220 --> 00:08:45.980
So now let me ask you-- just so
you know that the product

00:08:45.980 --> 00:08:47.690
matrix-- just write
it neatly-- is

00:08:47.690 --> 00:08:53.610
19, 22, 43, and 50.

00:08:53.610 --> 00:08:55.100
So now let me ask
you a question.

00:08:55.100 --> 00:08:58.810
When we did matrix addition we
learned that if I had two

00:08:58.810 --> 00:09:03.210
matrices-- it didn't matter what
order we added them in.

00:09:03.210 --> 00:09:06.970
So if we said, A plus B-- and
these are matrices; that's why

00:09:06.970 --> 00:09:09.340
I'm making them all bold-- we
said this is the same thing as

00:09:09.340 --> 00:09:12.330
B plus A, based on how
we define matrix

00:09:12.330 --> 00:09:15.520
addition, B plus A.

00:09:15.520 --> 00:09:17.060
So now let me ask
you a question.

00:09:17.060 --> 00:09:22.970
Is multiplying two matrices,
is AB-- that's just means

00:09:22.970 --> 00:09:26.460
we're multiplying A and B-- is
that the same thing as BA?

00:09:30.090 --> 00:09:30.820
Does it matter?

00:09:30.820 --> 00:09:33.800
Does the order of the matrix
multiplication matter?

00:09:33.800 --> 00:09:36.310
And so, I'll tell you right
now, it actually matters a

00:09:36.310 --> 00:09:37.090
tremendous amount.

00:09:37.090 --> 00:09:39.550
And actually there are certain
matrices that you can add in

00:09:39.550 --> 00:09:42.040
one direction that you can't
add in the other-- oh, that

00:09:42.040 --> 00:09:47.300
you can multiply in one way but
you can't multiply in the

00:09:47.300 --> 00:09:48.200
other order.

00:09:48.200 --> 00:09:50.755
And well, I'll show you that
in an example-- but just to

00:09:50.755 --> 00:09:52.800
show that this isn't even equal
for most matrices, I

00:09:52.800 --> 00:09:56.640
encourage you to multiply these
two matrices in the

00:09:56.640 --> 00:09:57.110
other order.

00:09:57.110 --> 00:09:58.950
Actually let me do that.

00:09:58.950 --> 00:10:00.630
Let me do that really
fast just to prove

00:10:00.630 --> 00:10:01.440
the point to you.

00:10:01.440 --> 00:10:02.830
So let me delete all
this top part.

00:10:06.040 --> 00:10:13.830
Let me delete all of it, and
actually I can delete to this.

00:10:13.830 --> 00:10:15.960
So hopefully, you know that when
I multiply this matrix

00:10:15.960 --> 00:10:17.850
times this matrix, I got this.

00:10:17.850 --> 00:10:20.250
So let me switch the order--
and I'll do it fairly fast

00:10:20.250 --> 00:10:23.140
just so as to not bore you-- so
let me switch the order of

00:10:23.140 --> 00:10:24.150
the matrix multiplication.

00:10:24.150 --> 00:10:26.755
This is good as this is another
example-- so I'm going

00:10:26.755 --> 00:10:36.740
to multiply this matrix: 5, 6,
7, 8, times this matrix-- and

00:10:36.740 --> 00:10:40.400
I just switched the order; and
we're testing to see whether

00:10:40.400 --> 00:10:46.700
order matters-- 1, 2, 3, 4.

00:10:46.700 --> 00:10:49.010
Let's do it-- and I won't do all
the colors and everything,

00:10:49.010 --> 00:10:49.690
I'll just do it systematically.

00:10:49.690 --> 00:10:54.330
I think you just have to see a
lot of examples here-- So this

00:10:54.330 --> 00:10:56.410
first term gets its row
information from the first

00:10:56.410 --> 00:10:58.810
matrix, column information
from the second matrix.

00:10:58.810 --> 00:11:06.340
So it's 5 times 1 plus 6 times
3, so it's 5 times 1--

00:11:06.340 --> 00:11:09.130
Let me just write,
actually edit.

00:11:09.130 --> 00:11:17.650
I'm going to skip a step here--
OK so it's 5 times 1

00:11:17.650 --> 00:11:23.480
plus 6 times 3, plus 18.

00:11:23.480 --> 00:11:25.110
What's the second term here?

00:11:25.110 --> 00:11:29.650
It's going to be 5 times
2 plus 6 times 4.

00:11:29.650 --> 00:11:40.190
So 5 times 2 is 10, plus
6 times 4 is 24.

00:11:40.190 --> 00:11:42.390
Right, now we just took
this row times this

00:11:42.390 --> 00:11:44.680
column right here.

00:11:44.680 --> 00:11:48.030
OK now we're down here for the
set-- so then we're doing this

00:11:48.030 --> 00:11:51.095
row, this element right here at
the bottom left is going to

00:11:51.095 --> 00:11:52.960
use this row, and this column.

00:11:52.960 --> 00:12:00.410
So it's 7 times 1
plus 8 times 3.

00:12:00.410 --> 00:12:02.980
8 times 3 is 24.

00:12:02.980 --> 00:12:05.430
And then finally, to get this
element we're essentially

00:12:05.430 --> 00:12:11.820
multiplying this row times this
column, so it's 7 times 2

00:12:11.820 --> 00:12:21.810
is 14, plus 8 times
4, plus 32.

00:12:21.810 --> 00:12:30.020
So this is equal to 5
plus 18 is 23, 34.

00:12:30.020 --> 00:12:31.100
What's 7 plus 24?

00:12:31.100 --> 00:12:35.530
That's 31, 46.

00:12:35.530 --> 00:12:44.140
So notice, if we called
this matrix A and this

00:12:44.140 --> 00:12:47.080
is matrix B, right?

00:12:47.080 --> 00:12:57.560
In the last example, we showed
that A times B is equal to 19,

00:12:57.560 --> 00:13:02.640
22, 43, 50.

00:13:02.640 --> 00:13:06.550
And we just showed that, well,
if you reverse the order, B

00:13:06.550 --> 00:13:10.225
times A is actually this
completely different matrix.

00:13:10.225 --> 00:13:12.230
So the order in which
you multiply

00:13:12.230 --> 00:13:15.140
matrices completely matters.

00:13:15.140 --> 00:13:16.330
So I'm actually running
out of time.

00:13:16.330 --> 00:13:19.210
In the next video I going talk
a little bit more about the

00:13:19.210 --> 00:13:22.260
types of matrix-- well, one, we
know that order matters--

00:13:22.260 --> 00:13:25.110
and in the next video I'll
show that what type of

00:13:25.110 --> 00:13:27.460
matrices can be multiplied
by each other.

00:13:27.460 --> 00:13:29.820
When we added or subtracted
matrices, we just said, well

00:13:29.820 --> 00:13:31.850
they have to have the same
dimensions because you're

00:13:31.850 --> 00:13:33.855
adding or subtracting
corresponding terms. But

00:13:33.855 --> 00:13:37.040
you'll see with multiplication
it's a little bit different.

00:13:37.040 --> 00:13:38.253
And we'll do that in
the next video.

00:13:38.253 --> 00:13:39.790
See you soon.


Implicit Differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.293
Welcome to the video on
implicit differentiation.

00:00:07.640 --> 00:00:09.560
Let's just explain the
difference between implicit

00:00:09.560 --> 00:00:10.590
and explicit first.

00:00:10.590 --> 00:00:16.280
So if I had a function that
was, let's say, y is equal to

00:00:16.280 --> 00:00:21.090
x squared plus 2x plus 3.

00:00:21.090 --> 00:00:25.230
In this situation, y, the
variable y, is defined

00:00:25.230 --> 00:00:26.710
explicitly in terms of x.

00:00:26.710 --> 00:00:27.730
How do I know that?

00:00:27.730 --> 00:00:30.650
Well, if you give me an x, I
can just input it into this

00:00:30.650 --> 00:00:34.220
expression right there,
and I explicitly get y.

00:00:34.220 --> 00:00:34.440
Right?

00:00:34.440 --> 00:00:36.830
It just pops out
of the equation.

00:00:36.830 --> 00:00:38.800
But on the other hand, what if
I had something like this.

00:00:38.800 --> 00:00:46.220
What if I had x squared plus
y squared is equal to 100.

00:00:46.220 --> 00:00:50.470
Well now, if I give you an x,
you still can figure out a y,

00:00:50.470 --> 00:00:52.180
but it's just not as easy.

00:00:52.180 --> 00:00:53.200
Right?

00:00:53.200 --> 00:00:58.260
If I said, x is equal to, I
don't know, x is equal to 9,

00:00:58.260 --> 00:01:03.310
you would put 9 in here, you
would get 81, plus y squared is

00:01:03.310 --> 00:01:07.570
equal to 100, you'd subtract
that, you'd get y squared

00:01:07.570 --> 00:01:08.780
is equal to, what, 19?

00:01:08.780 --> 00:01:09.950
Those numbers are weird.

00:01:09.950 --> 00:01:11.980
And then you'd get y is
equal to the plus or

00:01:11.980 --> 00:01:15.790
minus square root of 19.

00:01:15.790 --> 00:01:21.470
So you still could solve for
y, but this expression right

00:01:21.470 --> 00:01:25.890
here, or this equation,
defines y implicitly.

00:01:25.890 --> 00:01:27.610
And that's a different, you
know, when people use the

00:01:27.610 --> 00:01:30.730
word explicit or implicit
in everyday language,

00:01:30.730 --> 00:01:31.360
it's the same thing.

00:01:31.360 --> 00:01:33.670
When you say, he
explicitly said this,

00:01:33.670 --> 00:01:36.020
it means, I am hungry.

00:01:36.020 --> 00:01:38.230
I explicitly stated
that I'm hungry.

00:01:38.230 --> 00:01:39.840
But you could implicitly
state that you're hungry.

00:01:39.840 --> 00:01:43.040
You could say, oh well, you
know, I haven't eaten for a

00:01:43.040 --> 00:01:47.300
long time, and you know, boy,
it would be nice to maybe eat

00:01:47.300 --> 00:01:49.290
something, and that would
be kind of an implicit

00:01:49.290 --> 00:01:50.500
expression of your hunger.

00:01:50.500 --> 00:01:53.480
Because someone would have to
kind of do a little bit of work

00:01:53.480 --> 00:01:55.580
to come to the conclusion that
you're hungry, instead

00:01:55.580 --> 00:01:57.140
of you telling them.

00:01:57.140 --> 00:01:58.600
So with that out of the
way, let's actually do

00:01:58.600 --> 00:02:01.120
some differentiation.

00:02:01.120 --> 00:02:03.160
So everything you've learned
about differentiation right

00:02:03.160 --> 00:02:05.260
now has been explicit
differentiation.

00:02:05.260 --> 00:02:07.070
So example.

00:02:07.070 --> 00:02:13.030
For this function, we know
how to take the explicit

00:02:13.030 --> 00:02:13.940
derivative here.

00:02:13.940 --> 00:02:15.730
Well, it's an explicitly
defined function, so we

00:02:15.730 --> 00:02:16.430
just take the derivative.

00:02:16.430 --> 00:02:18.210
We apply the
derivative operator.

00:02:18.210 --> 00:02:21.390
We take the rate of change of
both sides of this equation

00:02:21.390 --> 00:02:22.420
with respect to x.

00:02:22.420 --> 00:02:24.050
You could do it like this.

00:02:24.050 --> 00:02:26.540
We're taking the derivative
operator, I know this is

00:02:26.540 --> 00:02:29.180
getting messy, we're taking the
derivative operator on that

00:02:29.180 --> 00:02:32.130
side, and we're also taking the
derivative operator

00:02:32.130 --> 00:02:33.640
on this side.

00:02:33.640 --> 00:02:34.940
I didn't plan to write
that there that's

00:02:34.940 --> 00:02:37.090
why-- so it happens.

00:02:37.090 --> 00:02:40.500
On the left side, the rate
of change at y changes

00:02:40.500 --> 00:02:42.770
with respect to x, well
we just write that.

00:02:42.770 --> 00:02:45.110
The rate of change of y,
for every small change in

00:02:45.110 --> 00:02:47.720
y, what is change in x.

00:02:47.720 --> 00:02:50.900
That equals-- and then,
we know how to do this

00:02:50.900 --> 00:02:51.630
type of derivative.

00:02:51.630 --> 00:02:53.650
This is just a
simple quadratic.

00:02:53.650 --> 00:02:58.620
So it equals 2x plus 2 plus 0.

00:02:58.620 --> 00:02:58.960
So there.

00:02:58.960 --> 00:03:02.310
We just did some explicit
differentiation.

00:03:02.310 --> 00:03:04.090
So how do we do the
same thing here?

00:03:04.090 --> 00:03:05.270
Well, we're going to
do the same thing.

00:03:05.270 --> 00:03:07.150
We're going to apply the
derivative operator to both

00:03:07.150 --> 00:03:08.515
sides of this equation.

00:03:08.515 --> 00:03:10.740
So let me do that, I'll do
that in a different color.

00:03:10.740 --> 00:03:12.650
Let me erase all of this,
too, because this is kind

00:03:12.650 --> 00:03:16.310
of obvious things that
you might already know.

00:03:16.310 --> 00:03:18.060
Let me erase all of that.

00:03:20.950 --> 00:03:23.110
OK, so I'll take the derivative
operator to both sides

00:03:23.110 --> 00:03:24.090
of this equation.

00:03:24.090 --> 00:03:26.630
So this is no different
than what we did up here.

00:03:26.630 --> 00:03:28.905
And as you learned in algebra,
anything you do to one side of

00:03:28.905 --> 00:03:30.920
the equation, you have to do to
the other side of the equation.

00:03:30.920 --> 00:03:34.580
Otherwise it would no
longer be an equation.

00:03:34.580 --> 00:03:36.830
Take the derivative operator--
because we're taking the

00:03:36.830 --> 00:03:37.703
derivative with respect to x.

00:03:37.703 --> 00:03:41.510
We could've taken it
with respect to y.

00:03:41.510 --> 00:03:47.150
x squared plus y squared is
equal to, have to take the

00:03:47.150 --> 00:03:48.480
derivative of both
sides, right?

00:03:50.990 --> 00:03:54.440
Derivative of 100 with
respect to x, right?

00:03:54.440 --> 00:04:01.212
So this equals the derivative
with respect to x, let

00:04:01.212 --> 00:04:02.090
me put it in brackets.

00:04:02.090 --> 00:04:04.270
I was going to do parentheses,
but I think brackets.

00:04:04.270 --> 00:04:07.000
Of x squared, plus the
derivative with respect

00:04:07.000 --> 00:04:12.440
to x of y squared.

00:04:12.440 --> 00:04:15.470
And what's the derivative
with respect to x of 100?

00:04:15.470 --> 00:04:19.040
For every change in x, how
much does this change?

00:04:19.040 --> 00:04:19.960
Well, it doesn't change at all.

00:04:19.960 --> 00:04:21.830
It's going to stay 100, right?

00:04:21.830 --> 00:04:22.690
So this is 0.

00:04:22.690 --> 00:04:24.870
The derivative of a constant,
really with respect to any

00:04:24.870 --> 00:04:26.840
changing variable,
is going to be 0.

00:04:26.840 --> 00:04:29.570
So that was just 0.

00:04:29.570 --> 00:04:32.595
And what's the derivative of
x squared with respect to x?

00:04:32.595 --> 00:04:34.530
Well, we did that
up here, right?

00:04:34.530 --> 00:04:36.500
When we took this derivative,
we just took the derivative

00:04:36.500 --> 00:04:38.400
of each of these terms, the
derivative of x squared with

00:04:38.400 --> 00:04:40.620
respect to x, that's just 2x.

00:04:40.620 --> 00:04:49.030
So we get 2x plus-- now this
term here is where you might

00:04:49.030 --> 00:04:51.050
get a little bit confused.

00:04:51.050 --> 00:04:57.500
I am taking the derivative of
y squared with respect to x.

00:05:00.335 --> 00:05:01.795
So let me take you
[? on the side. ?]

00:05:01.795 --> 00:05:04.480
You know you shouldn't do this
every time that you do one of

00:05:04.480 --> 00:05:07.210
these implicit differentiation
problems, but I'm doing this

00:05:07.210 --> 00:05:09.950
because I want you to
understand what we're doing.

00:05:09.950 --> 00:05:16.890
So let's say I just defined g
as equal to y squared, right?

00:05:16.890 --> 00:05:18.750
So let's say this was g.

00:05:18.750 --> 00:05:21.200
And I want to take
the derivative of g

00:05:21.200 --> 00:05:22.220
with respect to x.

00:05:22.220 --> 00:05:23.890
So I apply the derivative
operator on both sides of

00:05:23.890 --> 00:05:26.070
this, so let me do that.

00:05:26.070 --> 00:05:33.130
So the derivative, let me
do a d of x on that side,

00:05:33.130 --> 00:05:36.530
d of x on that side.

00:05:36.530 --> 00:05:38.720
That's a d, there you go.

00:05:38.720 --> 00:05:44.677
So what we get is the
derivative of g with respect to

00:05:44.677 --> 00:05:50.760
x is equal to the derivative
with respect to x of y squared.

00:05:50.760 --> 00:05:52.080
And so far, we haven't
really done much.

00:05:52.080 --> 00:05:54.190
I've just rewrote things.

00:05:54.190 --> 00:05:56.970
But what does the
chain rule tell us?

00:05:56.970 --> 00:06:00.660
The chain rule tells us that
the derivative, let me

00:06:00.660 --> 00:06:01.550
stay in the same color.

00:06:01.550 --> 00:06:03.620
I'm always switching colors.

00:06:03.620 --> 00:06:08.510
The derivative of any variable
with respect to x, or with

00:06:08.510 --> 00:06:12.520
respect to another variable, is
equal to the derivative of g

00:06:12.520 --> 00:06:17.340
with respect to y times
the derivative of y

00:06:17.340 --> 00:06:19.080
with respect to x.

00:06:19.080 --> 00:06:23.010
And if you don't have the full
intuition of the chain rule

00:06:23.010 --> 00:06:24.740
yet, there's a couple
of ways to do it.

00:06:24.740 --> 00:06:29.450
You know, all of these dg's and
dy's that you see throughout

00:06:29.450 --> 00:06:32.460
integral and differential
calculus, all these really mean

00:06:32.460 --> 00:06:34.830
are really, really, really,
really, small change in g, a

00:06:34.830 --> 00:06:36.910
really, really, really, small
change in y, a really,

00:06:36.910 --> 00:06:38.420
really, small change in x.

00:06:38.420 --> 00:06:43.290
So you can almost view these,
if you multiply this out, this

00:06:43.290 --> 00:06:45.920
really, really small change in
y is the same as this really,

00:06:45.920 --> 00:06:46.870
really small change in y.

00:06:46.870 --> 00:06:49.390
So these are
essentially fractions.

00:06:49.390 --> 00:06:53.130
So you could cancel these out,
and that's why you would get

00:06:53.130 --> 00:06:56.320
the change in g with respect to
x, or the derivative of

00:06:56.320 --> 00:06:58.260
g with respect to x.

00:06:58.260 --> 00:07:00.540
So this is what the
chain rule tells us.

00:07:00.540 --> 00:07:04.430
So given that this is what the
chain rule tells us, how does

00:07:04.430 --> 00:07:07.160
that help us figuring out this?

00:07:07.160 --> 00:07:10.985
Well, we know that the
derivative of g with respect to

00:07:10.985 --> 00:07:14.230
x is equal to the derivative
of g with respect to y.

00:07:14.230 --> 00:07:16.770
But we already know that
g is equal to y squared.

00:07:16.770 --> 00:07:17.890
I just made that definition.

00:07:17.890 --> 00:07:19.340
That's where we started from.

00:07:19.340 --> 00:07:22.800
So what's the derivative
of g with respect to y?

00:07:22.800 --> 00:07:27.810
Well, that's just 2y, right,
just using our derivative

00:07:27.810 --> 00:07:30.800
rules, or our polynomial rules,
which hopefully you believe

00:07:30.800 --> 00:07:33.230
actually work, because there
are several videos

00:07:33.230 --> 00:07:34.680
where we prove it.

00:07:34.680 --> 00:07:39.780
And then, times the derivative
of y with respect to x.

00:07:39.780 --> 00:07:42.650
Which we don't know, and that's
actually kind of what we're

00:07:42.650 --> 00:07:45.470
going to try to solve for.

00:07:45.470 --> 00:07:48.700
So the derivative of g with
respect to x is 2y times

00:07:48.700 --> 00:07:50.800
the derivative of y
with respect to x.

00:07:50.800 --> 00:07:53.850
Well, g is just the same
thing is y squared, right?

00:07:53.850 --> 00:07:56.280
So the derivative
of, this is g.

00:07:56.280 --> 00:07:58.450
And normally when you do this,
you don't have to go through

00:07:58.450 --> 00:08:00.510
the whole problem of
substituting for g.

00:08:00.510 --> 00:08:02.010
I just wanted to give
you that intuition.

00:08:02.010 --> 00:08:04.270
So what's this
derivative right here?

00:08:04.270 --> 00:08:06.870
Well, it's going to be the
derivative of this expression

00:08:06.870 --> 00:08:10.920
with respect to y, from the
chain rule, so that's 2y.

00:08:10.920 --> 00:08:20.050
Times the derivative of y with
respect to x is equal to 0.

00:08:20.050 --> 00:08:22.450
When you do it mechanically,
it's actually quite simple.

00:08:22.450 --> 00:08:24.710
You just say, oh, well, let me
pretend like y is like an x,

00:08:24.710 --> 00:08:26.130
take a derivative, and
I multiply it times

00:08:26.130 --> 00:08:27.630
the derivative of y
with respect to x.

00:08:27.630 --> 00:08:30.060
Mechanically it's very simple,
but I really want you to

00:08:30.060 --> 00:08:32.640
understand that intuition that
this is coming out

00:08:32.640 --> 00:08:33.175
of the chain rule.

00:08:33.175 --> 00:08:34.660
And actually, the chain
rule itself is not

00:08:34.660 --> 00:08:36.760
this bizarre concept.

00:08:36.760 --> 00:08:39.230
It's really, you're just
multiplying fractions.

00:08:39.230 --> 00:08:41.870
So let me clear up some
space here, and we can

00:08:41.870 --> 00:08:42.850
finish this problem.

00:08:42.850 --> 00:08:44.670
And then I'll do a problem
where you have to do

00:08:44.670 --> 00:08:45.850
implicit differentiation.

00:08:45.850 --> 00:08:47.950
And even in this one that I
just gave, it's really useful

00:08:47.950 --> 00:08:50.320
to do implicit differentiation.

00:08:50.320 --> 00:08:57.680
So let me, I could clear this
guy out up here, clear that

00:08:57.680 --> 00:09:02.080
out, clear all of this
stuff out, there we go.

00:09:02.080 --> 00:09:04.070
Clear all of that out.

00:09:04.070 --> 00:09:06.470
Ready to do some more work.

00:09:06.470 --> 00:09:08.390
So now we can solve this
for the derivative of

00:09:08.390 --> 00:09:09.630
y with respect to x.

00:09:09.630 --> 00:09:11.960
So we get, let's subtract
2x from both sides.

00:09:11.960 --> 00:09:21.050
We get 2y dy dx is equal to--
I'm just subtracting 2x

00:09:21.050 --> 00:09:24.560
from both sides-- minus 2x.

00:09:24.560 --> 00:09:25.730
And then what do we get?

00:09:25.730 --> 00:09:31.925
We get dy over dx is equal to,
divide both sides by 2y, the

00:09:31.925 --> 00:09:38.340
2's cancel out, you
get minus x over y.

00:09:38.340 --> 00:09:40.650
And this is interesting,
just as it is, right?

00:09:40.650 --> 00:09:41.760
Because what was this?

00:09:41.760 --> 00:09:46.620
This was the equation for a
circle with radius 10, right?

00:09:46.620 --> 00:09:48.110
You know, r squared.

00:09:48.110 --> 00:09:50.540
And if you know your
trigonometry and all of that,

00:09:50.540 --> 00:09:55.730
this tells you that the slope
of a circle, at any point, is

00:09:55.730 --> 00:09:57.950
equal to the negative
of x over the y.

00:09:57.950 --> 00:10:01.440
Well, we know from trigonometry
that the x is the cosine, and

00:10:01.440 --> 00:10:04.490
the y is the sine, and so that
tells you something interesting

00:10:04.490 --> 00:10:06.280
about how the derivative is
related to the tangent

00:10:06.280 --> 00:10:06.790
and all of that.

00:10:06.790 --> 00:10:07.790
But I won't go into that.

00:10:07.790 --> 00:10:09.390
We're just learning implicit
differentiation here.

00:10:09.390 --> 00:10:12.270
But it's sometimes it's fun
to just think about things.

00:10:12.270 --> 00:10:13.730
So how's this different
from everything else

00:10:13.730 --> 00:10:14.720
we learned so far?

00:10:14.720 --> 00:10:17.690
Well, here we have the
derivative, the derivative is

00:10:17.690 --> 00:10:21.800
actually a function of both
the x and the y value.

00:10:21.800 --> 00:10:25.080
And if we feel really
uncomfortable having this y

00:10:25.080 --> 00:10:29.140
here, if we don't like that,
what we could do, is

00:10:29.140 --> 00:10:30.030
we can substitute.

00:10:30.030 --> 00:10:31.300
So let's solve for y here.

00:10:31.300 --> 00:10:37.930
We get y squared is equal to
100, minus x squared, and then

00:10:37.930 --> 00:10:41.610
you get why is equal to the
plus or minus square root

00:10:41.610 --> 00:10:44.830
of 100 minus x squared.

00:10:44.830 --> 00:10:47.460
So you just substitute
that in here.

00:10:47.460 --> 00:10:49.060
Another way of writing
the derivative of y with

00:10:49.060 --> 00:10:51.710
respect to x, sometimes
you'll see y prime.

00:10:51.710 --> 00:10:53.010
I'm just switching
notation, just so you're

00:10:53.010 --> 00:10:53.660
familiar with it.

00:10:53.660 --> 00:11:02.050
Is equal to minus x over the
plus or minus square root

00:11:02.050 --> 00:11:05.170
of 100 minus x squared.

00:11:05.170 --> 00:11:08.170
And to some degree, since we
have plus or minus down here,

00:11:08.170 --> 00:11:10.190
this minus sign kind of
doesn't matter, right?

00:11:10.190 --> 00:11:12.240
Because when it's a minus here
it becomes a plus, when it's

00:11:12.240 --> 00:11:14.150
plus it's minuses, so you
can kind of ignore this.

00:11:14.150 --> 00:11:17.090
That's something interesting
for you to think about as well.

00:11:17.090 --> 00:11:18.420
Anyway.

00:11:18.420 --> 00:11:21.170
And this is interesting,
because our traditional way of

00:11:21.170 --> 00:11:23.950
solving derivatives, we had
never seen a plus

00:11:23.950 --> 00:11:24.580
or minus here.

00:11:24.580 --> 00:11:26.480
So that might be a little
bit interesting to you.

00:11:26.480 --> 00:11:29.500
So now let's do a problem
that is actually a bit

00:11:29.500 --> 00:11:30.940
of a trick problem.

00:11:30.940 --> 00:11:32.920
It's not a trick problem but
it's actually something, that I

00:11:32.920 --> 00:11:38.050
think, I first saw it either
when I was practicing for the

00:11:38.050 --> 00:11:40.850
AP Calculus exam, or actually
saw it on the AP Calculus exam.

00:11:40.850 --> 00:11:42.080
So it's something that
you might encounter.

00:11:42.080 --> 00:11:43.870
But it's really neat.

00:11:43.870 --> 00:11:50.060
Let's say I have y is
equal to x to the x.

00:11:50.060 --> 00:11:51.600
So first, you say,
oh, this looks easy.

00:11:51.600 --> 00:11:53.690
Maybe I could use the
power rule, but no.

00:11:53.690 --> 00:11:56.700
Chain rule, you have 2 x's
here, and I mean, you might

00:11:56.700 --> 00:11:58.600
want to just pause and think
about this for a little bit,

00:11:58.600 --> 00:12:00.620
because this actually isn't
that straightforward

00:12:00.620 --> 00:12:01.890
of a problem.

00:12:01.890 --> 00:12:05.780
And even though right here, y
is explicitly defined in terms

00:12:05.780 --> 00:12:09.560
of x, right, if I give you x,
if I say x is 3, well then y

00:12:09.560 --> 00:12:11.200
is 3 to the third, y is 27.

00:12:11.200 --> 00:12:12.730
So this is an
explicit function.

00:12:12.730 --> 00:12:15.100
I explicitly defined
y in terms of x.

00:12:15.100 --> 00:12:18.340
But it's very hard to
differentiate explicitly.

00:12:18.340 --> 00:12:20.520
And here's the trick.

00:12:20.520 --> 00:12:22.610
Let's take the natural log, or
actually you could take any

00:12:22.610 --> 00:12:24.510
logarithm, but let's take
the natural log on both

00:12:24.510 --> 00:12:26.060
sides of this equation.

00:12:26.060 --> 00:12:31.190
So you get the natural log of
y is equal to the natural

00:12:31.190 --> 00:12:34.370
log of x to the x.

00:12:34.370 --> 00:12:36.420
And why did I do that?

00:12:36.420 --> 00:12:38.450
Because now I can use
what we know about

00:12:38.450 --> 00:12:39.590
logarithm properties.

00:12:39.590 --> 00:12:46.310
We know that log of a to the
b is equal to b log of a.

00:12:46.310 --> 00:12:48.430
And you can watch the
exponent properties videos,

00:12:48.430 --> 00:12:49.790
if you forgot that.

00:12:49.790 --> 00:12:53.650
So let's rewrite this
using this property.

00:12:53.650 --> 00:12:55.340
And I'm going to switch colors.

00:12:55.340 --> 00:13:00.410
So we get the natural log
of y is equal to x times

00:13:00.410 --> 00:13:03.740
the natural log of x.

00:13:03.740 --> 00:13:07.890
And now we can implicitly
differentiate this.

00:13:07.890 --> 00:13:10.440
So let's take the derivative
with respect to x of both

00:13:10.440 --> 00:13:11.500
sides of this equation.

00:13:11.500 --> 00:13:12.765
And I'll switch colors again.

00:13:16.040 --> 00:13:17.500
d dx.

00:13:17.500 --> 00:13:20.750
And I'm writing this just so
you know what I'm doing.

00:13:20.750 --> 00:13:27.710
is equal to d dx of
x natural log of x.

00:13:27.710 --> 00:13:31.690
So the derivative of this
expression with respect to x is

00:13:31.690 --> 00:13:35.210
equal to the derivative of this
expression with respect to y,

00:13:35.210 --> 00:13:37.260
times the derivative of
y with respect to x.

00:13:37.260 --> 00:13:40.000
So what's the derivative of
this with respect to y?

00:13:40.000 --> 00:13:41.295
What's 1/y?

00:13:41.295 --> 00:13:45.360
1/y times the derivative, just
the chain rule, the derivative

00:13:45.360 --> 00:13:47.520
of y with respect to x.

00:13:47.520 --> 00:13:52.410
dy dx is equal to-- and
remember, just mechanically, if

00:13:52.410 --> 00:13:54.140
you want to make sure you're
doing it right, you just take

00:13:54.140 --> 00:13:55.900
the derivative with respect to
y, and you multiply

00:13:55.900 --> 00:13:56.530
it times dy dy.

00:13:56.530 --> 00:13:59.340
That's just the chain room.

00:13:59.340 --> 00:14:01.160
Equals, and now we take
the derivative of this.

00:14:01.160 --> 00:14:03.130
Well, now we can just
use the product rule.

00:14:03.130 --> 00:14:05.000
The derivative of
the first, right?

00:14:05.000 --> 00:14:06.280
It's this times this.

00:14:06.280 --> 00:14:07.610
So what's the derivative of x?

00:14:07.610 --> 00:14:08.880
It's 1.

00:14:08.880 --> 00:14:13.450
So it's 1 times the natural log
of x plus the derivative of

00:14:13.450 --> 00:14:18.420
the second term, 1/x, times
the first term, times x.

00:14:18.420 --> 00:14:20.400
Let me clear that out up there.

00:14:20.400 --> 00:14:23.190
I'm already over time, but I've
become a YouTube partner,

00:14:23.190 --> 00:14:24.875
so my videos can be longer.

00:14:24.875 --> 00:14:26.410
But I don't want to make
them too long, because

00:14:26.410 --> 00:14:28.030
you'll get bored.

00:14:28.030 --> 00:14:30.210
All right.

00:14:30.210 --> 00:14:31.370
So let's simplify this.

00:14:31.370 --> 00:14:43.840
We get 1/y dy dx is equal to
the natural log of x, plus,

00:14:43.840 --> 00:14:46.580
well, that just cancels
out and becomes 1, right?

00:14:46.580 --> 00:14:49.900
And so we have the derivative
of y with respect

00:14:49.900 --> 00:14:52.510
to x is equal to y.

00:14:52.510 --> 00:14:55.220
I'm just multiplying both sides
of this equation times y.

00:14:55.220 --> 00:15:00.270
y times the natural
log of x plus 1.

00:15:00.270 --> 00:15:02.520
Now, you might say you're done,
but some people just don't like

00:15:02.520 --> 00:15:06.040
it, the change of y with
respect to x, they don't like

00:15:06.040 --> 00:15:07.810
it being defined in terms of y.

00:15:07.810 --> 00:15:08.620
So let's substitute.

00:15:08.620 --> 00:15:12.820
We know that y is
equal x to the x.

00:15:12.820 --> 00:15:21.900
So that equals x to the x times
the natural log of x plus one.

00:15:21.900 --> 00:15:25.380
So another way of saying it is
that y prime, or the change in

00:15:25.380 --> 00:15:28.180
y with respect to x, or the
slope at any point of this

00:15:28.180 --> 00:15:34.690
curve, is equal to x to the x,
natural log of x,

00:15:34.690 --> 00:15:36.020
plus x to the x.

00:15:36.020 --> 00:15:37.530
I just distributed
the x to the x.

00:15:37.530 --> 00:15:38.770
But that's pretty neat.

00:15:38.770 --> 00:15:41.430
This is kind of a, you know, a
five-star problem that you

00:15:41.430 --> 00:15:43.560
might see in your books, and
you might see it in a math

00:15:43.560 --> 00:15:45.690
competition, and people
will be really impressed.

00:15:45.690 --> 00:15:47.390
And they'd be even more
impressed if you kind of

00:15:47.390 --> 00:15:49.635
recognized this pattern, and
you're able to find the

00:15:49.635 --> 00:15:51.290
antiderivative of it, and
recognize that it

00:15:51.290 --> 00:15:52.460
is x to the x.

00:15:52.460 --> 00:15:54.230
Anyway, I've talked
for too long.

00:15:54.230 --> 00:15:56.580
I will see you in
the next video.


Fun Trig Problem

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:02.540
I received a problem
from Bradley.

00:00:02.540 --> 00:00:04.220
I don't know his last name.

00:00:04.220 --> 00:00:05.300
I'm assuming it's a he.

00:00:05.300 --> 00:00:06.050
I don't know where he lives.

00:00:06.050 --> 00:00:07.670
But the problem he
gave is interesting.

00:00:07.670 --> 00:00:08.980
And I don't think I've
covered this before.

00:00:08.980 --> 00:00:10.090
So I think it's worth covering.

00:00:10.090 --> 00:00:13.730
So the problem he gave, if I
read his note properly, is

00:00:13.730 --> 00:00:25.310
this: 3 sine squared of x is
equal to 1 plus cosine of x.

00:00:25.310 --> 00:00:27.730
So at first cut, this seems
like a difficult problem.

00:00:27.730 --> 00:00:29.270
How do I-- You
can't solve for x.

00:00:29.270 --> 00:00:31.350
You would have arcsines
and the square roots

00:00:31.350 --> 00:00:32.950
and cosines et cetera.

00:00:32.950 --> 00:00:33.500
Et cetera.

00:00:33.500 --> 00:00:36.070
So the way I approached this
is-- Any time that if I see a

00:00:36.070 --> 00:00:39.880
cosine x here but then I see a
sine squared x here, I start

00:00:39.880 --> 00:00:42.860
thinking of what trig
identities are at my disposal.

00:00:42.860 --> 00:00:45.970
And what trig identities
involve a sine squared x?

00:00:45.970 --> 00:00:48.280
Well the most basic trig
identity, and this comes out

00:00:48.280 --> 00:00:51.420
of the unit circle definition
of trig functions, is that

00:00:51.420 --> 00:00:58.530
sine squared x plus cosine
squared x is equal to 1.

00:00:58.530 --> 00:01:01.510
And that comes out of the fact
that the equation of a circle

00:01:01.510 --> 00:01:04.500
is x squared plus y squared is
equal to the radius squared.

00:01:04.500 --> 00:01:05.100
But it's the unit circle.

00:01:05.100 --> 00:01:06.650
It's equal to 1 squared.

00:01:06.650 --> 00:01:07.170
But anyway.

00:01:07.170 --> 00:01:09.200
Hopefully you have this
memorized if you've already

00:01:09.200 --> 00:01:11.410
been watching the trig videos.

00:01:11.410 --> 00:01:13.070
So what does sine
squared x equal?

00:01:13.070 --> 00:01:14.280
Well let's solve for it.

00:01:14.280 --> 00:01:23.270
So sine squared x is equal to 1
minus cosine squared x, right?

00:01:23.270 --> 00:01:27.700
So we could substitute this
term right here with this.

00:01:27.700 --> 00:01:28.630
And what does that get us?

00:01:28.630 --> 00:01:30.880
Well we're just playing around
at this point, but at least

00:01:30.880 --> 00:01:34.620
that way, everything is
in terms of cosine of x.

00:01:34.620 --> 00:01:35.310
So let's do that.

00:01:35.310 --> 00:01:35.860
Let's substitute.

00:01:35.860 --> 00:01:39.680
So we get 3 times
sine squared of x.

00:01:39.680 --> 00:01:41.520
We just showed that sine
squared of x is the same thing

00:01:41.520 --> 00:01:43.075
as 1 minus cosine squared of x.

00:01:49.120 --> 00:01:53.730
Is equal to 1 plus cosine of x.

00:01:53.730 --> 00:01:55.560
We can simplify a little bit.

00:01:55.560 --> 00:02:05.480
3 minus 3 cosine squared of x
is equal to 1 plus cosine of x.

00:02:05.480 --> 00:02:05.910
I don't know.

00:02:05.910 --> 00:02:07.990
Just for kicks, let's put
everything onto the right

00:02:07.990 --> 00:02:09.550
side of the equation.

00:02:09.550 --> 00:02:11.620
And you'll see it
wasn't just for kicks.

00:02:11.620 --> 00:02:12.870
0-- right?

00:02:12.870 --> 00:02:15.230
I'm just going to --is equal
to-- let's put this onto

00:02:15.230 --> 00:02:20.150
the right side --3
cosine squared x.

00:02:20.150 --> 00:02:21.810
And then-- Let's see.

00:02:21.810 --> 00:02:23.940
We have to subtract
3 from this side.

00:02:23.940 --> 00:02:25.570
Well let's just
write the cosine x.

00:02:25.570 --> 00:02:27.260
Plus cosine x.

00:02:27.260 --> 00:02:30.400
And then 1 minus 3 is minus 2.

00:02:30.400 --> 00:02:31.910
Let me make sure I didn't
make a careless mistake.

00:02:31.910 --> 00:02:32.870
We have negative 3 here.

00:02:32.870 --> 00:02:35.493
We added 3 cosine x--
3 cosine squared of x

00:02:35.493 --> 00:02:36.960
to both sides, right?

00:02:36.960 --> 00:02:38.500
We subtracted 3
from both sides.

00:02:38.500 --> 00:02:41.360
Minus 2 and this cosine
x is this cosine x.

00:02:41.360 --> 00:02:43.950
Now what can we do?

00:02:43.950 --> 00:02:46.410
Well this is where it
gets interesting.

00:02:46.410 --> 00:02:50.740
Because this looks an awful lot
like a quadratic equation

00:02:50.740 --> 00:02:57.920
except for the fact that
instead of having ax squared

00:02:57.920 --> 00:03:02.400
plus bx plus c, we have
a cosine squared x.

00:03:02.400 --> 00:03:04.360
So instead of just having an
x squared, we have a whole

00:03:04.360 --> 00:03:05.940
cosine of x squared.

00:03:05.940 --> 00:03:07.470
So what do I mean by that?

00:03:07.470 --> 00:03:08.720
Let me make a substitution.

00:03:08.720 --> 00:03:12.760
And then I think it'll
all become clear to you.

00:03:12.760 --> 00:03:16.030
Let's make the substitution
that a-- and I'm just picking

00:03:16.030 --> 00:03:21.430
the letter a arbitrarily
--is equal to cosine of x.

00:03:21.430 --> 00:03:23.836
So if we were to take the
cosine x's of this and replace

00:03:23.836 --> 00:03:25.450
them with a, what do we get?

00:03:25.450 --> 00:03:26.760
And I'm just going to
switch it around.

00:03:26.760 --> 00:03:28.790
So I want to put the
0 on that side.

00:03:28.790 --> 00:03:30.340
Equal 0.

00:03:30.340 --> 00:03:32.875
So we get 3-- Well
cosine squared x.

00:03:32.875 --> 00:03:36.040
That's the same thing as
cosine of x squared, right?

00:03:36.040 --> 00:03:47.840
So we get 3a squared plus
a minus 2 is equal to 0.

00:03:47.840 --> 00:03:49.870
Well now we have a
pure quadratic.

00:03:49.870 --> 00:03:54.500
And we can solve it using
the quadratic equation.

00:03:54.500 --> 00:03:56.690
So what's the
quadratic equation?

00:03:56.690 --> 00:03:58.760
Let me write it up here.

00:03:58.760 --> 00:04:02.080
Negative b plus or minus
the square root of b

00:04:02.080 --> 00:04:04.700
squared minus 4ac.

00:04:04.700 --> 00:04:06.560
All of that over 2a.

00:04:06.560 --> 00:04:08.550
So what are the roots
of this equation?

00:04:08.550 --> 00:04:10.400
Well what's minus-- And
remember this a is

00:04:10.400 --> 00:04:11.440
different than this a.

00:04:11.440 --> 00:04:13.040
Maybe I shouldn't have
used a as a letter.

00:04:13.040 --> 00:04:15.410
But these-- a, b, and c in
the quadratic equation

00:04:15.410 --> 00:04:16.460
represent the coefficients.

00:04:16.460 --> 00:04:19.360
So this is a.

00:04:19.360 --> 00:04:21.500
b is 1.

00:04:21.500 --> 00:04:23.340
And c is just minus 2.

00:04:23.340 --> 00:04:26.080
So what are the roots of this?

00:04:26.080 --> 00:04:29.780
So the a's that solve
this. a can equal--

00:04:29.780 --> 00:04:30.780
And I know I confused you.

00:04:30.780 --> 00:04:32.190
I could-- Let me actually
write it different.

00:04:32.190 --> 00:04:35.630
Let's make this, instead of
a is equal to cosine x, let

00:04:35.630 --> 00:04:37.610
me say that-- I don't know.

00:04:37.610 --> 00:04:39.540
Let me pick a good letter
that's not involved in

00:04:39.540 --> 00:04:41.540
the-- Let me say d.

00:04:41.540 --> 00:04:45.920
So 3d squared plus
d minus is 2.

00:04:45.920 --> 00:04:49.250
So now the a's, b's and c's are
definitely the coefficients.

00:04:49.250 --> 00:04:53.390
So the solutions to this are
d-- because I didn't want to

00:04:53.390 --> 00:04:57.060
use a, b, or c --d is
equal to minus b.

00:04:57.060 --> 00:04:58.350
Well, b is 1.

00:04:58.350 --> 00:04:59.730
Minus 1.

00:04:59.730 --> 00:05:01.600
And if this is completely
foreign to you, you should

00:05:01.600 --> 00:05:05.010
review the videos on
the quadratic equation.

00:05:05.010 --> 00:05:07.560
Minus b squared.

00:05:07.560 --> 00:05:09.690
Well that's 1 squared.

00:05:09.690 --> 00:05:12.030
Minus 4ac.

00:05:12.030 --> 00:05:18.220
So minus 4 times a,
times 3, times c.

00:05:18.220 --> 00:05:20.590
Well c is minus 2, right?

00:05:20.590 --> 00:05:22.630
So we get a-- That
minus cancels there.

00:05:22.630 --> 00:05:23.780
And we have a 2 there.

00:05:23.780 --> 00:05:28.550
All of that over 2 times a. a
is 3, so we have it over 6.

00:05:28.550 --> 00:05:31.700
So this equals minus 1
plus or minus the square

00:05:31.700 --> 00:05:32.250
root-- What is this?

00:05:32.250 --> 00:05:33.480
4 times 3 times 2.

00:05:33.480 --> 00:05:35.240
24 plus 1.

00:05:35.240 --> 00:05:35.900
25.

00:05:35.900 --> 00:05:35.950
Oh.

00:05:35.950 --> 00:05:37.820
This works out cleanly.

00:05:37.820 --> 00:05:39.290
Over 6.

00:05:39.290 --> 00:05:46.160
So that equals minus 1
plus or minus 5 over 6.

00:05:46.160 --> 00:05:47.480
And so what are the roots?

00:05:47.480 --> 00:05:51.760
The roots are-- What's
minus 1 minus 5?

00:05:51.760 --> 00:05:53.670
That's minus 6 over 6.

00:05:53.670 --> 00:05:55.510
So it's minus 1.

00:05:55.510 --> 00:05:57.420
What's the other one?

00:05:57.420 --> 00:05:59.850
Minus 1 plus 5 is 4.

00:05:59.850 --> 00:06:03.840
4 over 6 is 2/3.

00:06:03.840 --> 00:06:05.260
So the solution is to
the equation-- Let me

00:06:05.260 --> 00:06:06.490
clear up some space.

00:06:06.490 --> 00:06:12.210
Hopefully it'll let me
clear up some space here.

00:06:15.150 --> 00:06:15.600
Let me see.

00:06:15.600 --> 00:06:18.170
What was I doing?

00:06:18.170 --> 00:06:18.580
Oh.

00:06:18.580 --> 00:06:21.780
Maybe I want to leave--
I can get rid of this.

00:06:21.780 --> 00:06:24.210
You know the identity.

00:06:24.210 --> 00:06:28.380
And you also know the
quadratic formula.

00:06:28.380 --> 00:06:28.840
Let's see.

00:06:28.840 --> 00:06:30.310
Actually, let me get
rid of this too.

00:06:30.310 --> 00:06:33.860
Clear up a bunch of space.

00:06:33.860 --> 00:06:36.410
I wanted to leave this here
because this showed how this

00:06:36.410 --> 00:06:39.380
turned into a quadratic, but
instead of having it in terms

00:06:39.380 --> 00:06:43.560
of just a variable, we have
it in terms of cosine of x.

00:06:43.560 --> 00:06:45.550
And then we made this d
is equal to cosine of x.

00:06:45.550 --> 00:06:47.030
Anyway.

00:06:47.030 --> 00:06:50.970
So the solution to this
equation is that quadratic.

00:06:50.970 --> 00:06:57.080
Is d is equal to minus
1 or 2/3, right?

00:06:57.080 --> 00:06:59.700
But, of course, we made the
substitution long ago that

00:06:59.700 --> 00:07:01.500
d is equal to cosine of x.

00:07:01.500 --> 00:07:06.240
So the solution to this
equation, in terms of x, is the

00:07:06.240 --> 00:07:07.840
solution to this equation.

00:07:07.840 --> 00:07:17.500
Cosine of x is equal to minus 1
or cosine of x is equal to 2/3.

00:07:17.500 --> 00:07:18.850
Well this one's easy, right?

00:07:18.850 --> 00:07:23.590
x is equal to
arccosine of minus 1.

00:07:23.590 --> 00:07:25.690
I always forget if there's two
c's when you do arccosine.

00:07:25.690 --> 00:07:30.330
Anyway, so what-- At what
degree or radian value does the

00:07:30.330 --> 00:07:33.020
cosine of x equal minus 1?

00:07:33.020 --> 00:07:35.040
Well it's at pi, right?

00:07:35.040 --> 00:07:39.895
So x could equal pi, which
is also or 180 degrees.

00:07:42.650 --> 00:07:44.390
This one is not as easy.

00:07:44.390 --> 00:07:47.010
I think I will have to use
a calculator for this.

00:07:47.010 --> 00:07:49.330
Unless I'm-- whoops.

00:07:49.330 --> 00:07:52.520
So you may not realize it,
but Google is actually

00:07:52.520 --> 00:07:55.110
a calculator.

00:07:55.110 --> 00:07:59.540
And a far more advanced
calculator than most.

00:07:59.540 --> 00:08:01.555
So we could use Google
to figure out the

00:08:01.555 --> 00:08:03.660
arccosine of 2/3.

00:08:03.660 --> 00:08:05.790
Let's do that.

00:08:05.790 --> 00:08:08.090
Arccosine-- and I don't
know if I'm spelling it

00:08:08.090 --> 00:08:11.230
right --of 2 over 3.

00:08:11.230 --> 00:08:21.370
Google tells us that it's
0.841 and a bunch of numbers.

00:08:21.370 --> 00:08:27.160
So x is equal to
arccosine of 2 over 3.

00:08:27.160 --> 00:08:35.070
So x is equal to 0.84106.

00:08:35.070 --> 00:08:35.950
Let's see if they work.

00:08:35.950 --> 00:08:38.660
Let's, just for fun, let's
just see if this one works.

00:08:38.660 --> 00:08:42.180
Let's see if we substitute
pi into this equation we

00:08:42.180 --> 00:08:44.320
get the correct answer.

00:08:44.320 --> 00:08:46.810
Well what's sine of pi?

00:08:46.810 --> 00:08:49.240
Let me erase all of this
is so we can check it.

00:08:49.240 --> 00:08:50.140
I'm only going to check pi.

00:08:50.140 --> 00:08:51.680
The 0.84.

00:08:51.680 --> 00:08:52.220
And I don't know.

00:08:52.220 --> 00:08:52.800
That's messy.

00:08:52.800 --> 00:08:54.870
But you could do that
in your own time.

00:08:54.870 --> 00:08:58.318
So let's check pi.

00:08:58.318 --> 00:08:59.480
x equals-- No.

00:08:59.480 --> 00:09:00.680
That's not what I wanted to do.

00:09:06.590 --> 00:09:09.030
So what is-- Let's make
sure this works with pi.

00:09:09.030 --> 00:09:17.510
3 sine squared of pi is equal
to 1 plus cosine of pi.

00:09:17.510 --> 00:09:18.900
Well what's sine of pi?

00:09:18.900 --> 00:09:24.570
This is equal to 3
sine of pi squared.

00:09:24.570 --> 00:09:27.760
This is equal to 1
plus cosine of pi.

00:09:27.760 --> 00:09:29.940
Well sine of pi is 0, right?

00:09:29.940 --> 00:09:32.320
The y-value when you
go 180 degrees is 0.

00:09:32.320 --> 00:09:34.180
So this is 0.

00:09:34.180 --> 00:09:36.360
And what's cosine of pi?

00:09:36.360 --> 00:09:37.890
Cosine of pi is negative 1.

00:09:37.890 --> 00:09:39.100
So 1 plus minus 1.

00:09:39.100 --> 00:09:40.400
Well this is true.

00:09:40.400 --> 00:09:41.790
So pi worked in that equation.

00:09:41.790 --> 00:09:46.600
I think if you substitute
that 0841068 whatever, you'd

00:09:46.600 --> 00:09:47.990
also find that that works.

00:09:47.990 --> 00:09:49.550
So thanks Bradley
for sending this.

00:09:49.550 --> 00:09:51.430
I thought this was a neat
problem because it looks

00:09:51.430 --> 00:09:52.660
like it's trigonometry.

00:09:52.660 --> 00:09:54.190
And it was trigonometry but
you had to know a little

00:09:54.190 --> 00:09:54.840
bit of identities.

00:09:54.840 --> 00:09:58.020
And then you had to recognize
it as a quadratic equation.

00:09:58.020 --> 00:10:00.590
I will see you in
a future video.


Example: Analyzing a more complex resistor circuit

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.600
Let's see if we can apply
what we've learned to a

00:00:03.600 --> 00:00:06.890
particularly hairy problem
that I have constructed.

00:00:06.890 --> 00:00:09.600
So let me see how I can
construct this.

00:00:09.600 --> 00:00:15.780
So let's say in parallel, I have
this resistor up here.

00:00:15.780 --> 00:00:17.590
And I try to make it
so the numbers work

00:00:17.590 --> 00:00:19.760
out reasonably neat.

00:00:19.760 --> 00:00:28.000
That is 4 ohms. Then I have
another resistor right here.

00:00:28.000 --> 00:00:36.340
That is 8 ohms. Then I have
another resistor right here.

00:00:36.340 --> 00:00:43.760
That is 16 ohms. And then, I
have another resistor here,

00:00:43.760 --> 00:00:48.640
that's ohms. Actually, I'm now
making it up on the fly.

00:00:48.640 --> 00:00:50.990
I think the numbers
might work out OK.

00:00:50.990 --> 00:00:54.710
16 ohms.

00:00:54.710 --> 00:00:59.750
And let's say that now here in
series, I have a resistor that

00:00:59.750 --> 00:01:07.720
is 1 ohm, and then in parallel
to this whole thing-- now you

00:01:07.720 --> 00:01:12.080
can see how hairy it's getting--
I have a resistor

00:01:12.080 --> 00:01:22.420
that is 3 ohms. And let's say
I have a resistor here.

00:01:26.640 --> 00:01:29.650
Let's just make it
simple: 1 ohm.

00:01:29.650 --> 00:01:36.010
And just to make the numbers
reasonably easy-- I am doing

00:01:36.010 --> 00:01:38.850
this on the fly now-- that's
the positive terminal,

00:01:38.850 --> 00:01:41.140
negative terminal.

00:01:41.140 --> 00:01:47.370
Let's say that the voltage
difference is 20 volts.

00:01:47.370 --> 00:01:52.050
So what I want us to do is,
figure out what is the current

00:01:52.050 --> 00:01:54.540
flowing through the wire
at that point?

00:01:54.540 --> 00:01:56.010
Obviously, that's going to be
different than the current at

00:01:56.010 --> 00:01:58.090
that point, that point, that
point, that point, all of

00:01:58.090 --> 00:01:59.710
these different points, but it's
going to be the same as

00:01:59.710 --> 00:02:01.880
the current flowing
at this point.

00:02:01.880 --> 00:02:03.370
So what is I?

00:02:03.370 --> 00:02:05.920
So the easiest way to do this
is try to figure out the

00:02:05.920 --> 00:02:07.040
equivalent resistance.

00:02:07.040 --> 00:02:09.199
Because once we know the
equivalent resistance of this

00:02:09.199 --> 00:02:12.940
big hairball, then we can just
use Ohm's law and be done.

00:02:12.940 --> 00:02:16.460
So first of all, let's just
start at, I could argue, the

00:02:16.460 --> 00:02:17.090
simplest part.

00:02:17.090 --> 00:02:19.310
Let's see if we could figure out
the equivalent resistance

00:02:19.310 --> 00:02:23.550
of these four resistors
in parallel.

00:02:23.550 --> 00:02:26.160
Well, we know that that
resistance is going to be

00:02:26.160 --> 00:02:34.800
equal to 1/4 plus 1/8
plus 1/16 plus 1/16.

00:02:34.800 --> 00:02:37.450
So that resistance-- and
now it's just adding

00:02:37.450 --> 00:02:39.100
fractions-- over 16.

00:02:39.100 --> 00:02:48.740
1/4 is 4/16 plus 2/16 plus 1
plus 1, so 1/R is equal to 4

00:02:48.740 --> 00:02:53.170
plus 2 is equal to 8/16-- the
numbers are working out-- is

00:02:53.170 --> 00:02:57.260
equal to 1/2, so that equivalent
resistance is 2.

00:02:57.260 --> 00:02:59.490
So that, quickly, we just
said, well, all of these

00:02:59.490 --> 00:03:04.110
resistors combined is equal to
2 ohms. So let me erase that

00:03:04.110 --> 00:03:05.620
and simplify our drawing.

00:03:09.020 --> 00:03:10.510
Simplify it.

00:03:10.510 --> 00:03:29.310
So that whole thing could now
be simplified as 2 ohms. I

00:03:29.310 --> 00:03:30.240
lost some wire here.

00:03:30.240 --> 00:03:33.000
I want to make sure that
circuit can still flow.

00:03:33.000 --> 00:03:36.910
So that easily, I turned that
big, hairy mess into something

00:03:36.910 --> 00:03:39.540
that is a lot less hairy.

00:03:39.540 --> 00:03:41.950
Well, what is the equivalent
resistance of this resistor

00:03:41.950 --> 00:03:43.330
and this resistor?

00:03:43.330 --> 00:03:46.290
Well, they're in series, and
series resistors, they just

00:03:46.290 --> 00:03:47.710
add up together, right?

00:03:47.710 --> 00:03:50.770
So the combined resistance of
this 2-ohm resistor and this

00:03:50.770 --> 00:03:53.510
1-ohm resistor is just
a 3-ohm resistor.

00:03:53.510 --> 00:03:57.580
So let's erase and simplify.

00:03:57.580 --> 00:04:02.280
So then we get that combined
resistor, right?

00:04:02.280 --> 00:04:05.360
We had the 2-ohm that we
had simplified and

00:04:05.360 --> 00:04:06.610
then we had a 1-ohm.

00:04:12.200 --> 00:04:15.250
So we had a 2-ohm and a 1-ohm
in series, so those simplify

00:04:15.250 --> 00:04:16.880
to 3 ohms.

00:04:16.880 --> 00:04:19.300
Well, now this is getting
really simple.

00:04:19.300 --> 00:04:23.670
So what do these two resistors
simplify to?

00:04:23.670 --> 00:04:27.710
Well, 1 over their combined
resistance is

00:04:27.710 --> 00:04:31.040
equal to 1/3 plus 1/3.

00:04:31.040 --> 00:04:31.870
It equals what?

00:04:31.870 --> 00:04:32.770
2/3.

00:04:32.770 --> 00:04:41.560
1/R is equal to 2/3, so R is
equal to 3/2, or we could say

00:04:41.560 --> 00:04:45.960
1.5, right?

00:04:45.960 --> 00:04:51.100
So let's erase that and
simplify our drawing.

00:04:51.100 --> 00:04:57.905
So this whole mess, the 3-ohm
resistor in parallel with the

00:04:57.905 --> 00:05:00.980
other 3-ohm resistor is equal
to one resistor with a 1.5

00:05:00.980 --> 00:05:02.020
resistance.

00:05:02.020 --> 00:05:03.600
And actually, this is actually
a good point to give you a

00:05:03.600 --> 00:05:05.320
little intuition, right?

00:05:05.320 --> 00:05:07.550
Because even though these are
3-ohm resistors, we have two

00:05:07.550 --> 00:05:10.880
of them, so you're kind of
increasing the pipe that the

00:05:10.880 --> 00:05:14.790
electrons can go in by a
factor of two, right?

00:05:14.790 --> 00:05:17.150
So it's actually decreasing
the resistance.

00:05:17.150 --> 00:05:19.590
It's giving more avenues for the
electrons to go through.

00:05:19.590 --> 00:05:21.500
Actually, they're going to be
going in that direction.

00:05:21.500 --> 00:05:23.880
And that's why the combined
resistance of both of these in

00:05:23.880 --> 00:05:27.160
parallel is actually half
of either one of these

00:05:27.160 --> 00:05:27.565
resistances.

00:05:27.565 --> 00:05:30.090
I encourage you to think about
that some more to give you

00:05:30.090 --> 00:05:31.610
some intuition of what's
actually going on with the

00:05:31.610 --> 00:05:36.100
electrons, although I'll do a
whole video on resistivity.

00:05:36.100 --> 00:05:41.180
OK so we said those two
resistors combined-- I want to

00:05:41.180 --> 00:05:43.010
delete all of that.

00:05:43.010 --> 00:05:47.210
Those two resistors combined
equal to a 1.5-ohm resistor.

00:05:49.850 --> 00:05:53.540
That's 1.5 ohms. And now all
we're left with is two

00:05:53.540 --> 00:05:59.270
resistors in parallel, so the
whole circuit becomes this,

00:05:59.270 --> 00:06:00.860
which is the very basic one.

00:06:00.860 --> 00:06:04.270
This is a resistor: 1.5
ohms, 1 ohm in series.

00:06:04.270 --> 00:06:05.300
Did I say parallel just now?

00:06:05.300 --> 00:06:06.510
No, they're in series.

00:06:06.510 --> 00:06:13.210
1.5 plus 1, that's 2.5 ohms.
The voltage is 20

00:06:13.210 --> 00:06:14.710
volts across them.

00:06:14.710 --> 00:06:16.920
So what is the current?

00:06:16.920 --> 00:06:17.860
Ohm's law.

00:06:17.860 --> 00:06:19.920
V is equal to IR.

00:06:19.920 --> 00:06:24.500
Voltage is 20 is equal to
current times our equivalent

00:06:24.500 --> 00:06:31.290
resistance times 2.5 ohms. Or
another way to write 2.5 five

00:06:31.290 --> 00:06:33.410
is 5/2, right?

00:06:33.410 --> 00:06:38.060
So 20 is equal to I times 5/2.

00:06:38.060 --> 00:06:45.570
Or I is equal to 2/5 times
20, and what is that?

00:06:45.570 --> 00:06:49.950
2/5 is equal to I
is equal to 8.

00:06:49.950 --> 00:06:51.470
8 amperes.

00:06:51.470 --> 00:06:54.590
That was not so bad,
I don't think.

00:06:54.590 --> 00:06:56.850
Although when you saw it
initially, it probably looked

00:06:56.850 --> 00:06:58.540
extremely intimidating.

00:06:58.540 --> 00:07:01.190
Anyway, if you understood that,
you can actually solve

00:07:01.190 --> 00:07:04.920
fairly complicated circuit
problems. I will see you in

00:07:04.920 --> 00:07:06.510
future videos.


Resistors in parallel

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:02.400
Last video, we saw what
happens when we have

00:00:02.400 --> 00:00:03.970
resistors in series.

00:00:03.970 --> 00:00:05.390
Now let's see what happens
when we have

00:00:05.390 --> 00:00:07.600
resistors in parallel.

00:00:07.600 --> 00:00:09.860
All right, let me pick
a new color.

00:00:09.860 --> 00:00:12.960
New color will be magenta.

00:00:12.960 --> 00:00:16.720
There's my battery: positive,
negative.

00:00:16.720 --> 00:00:21.140
There's my ideal conducting
wire.

00:00:21.140 --> 00:00:26.000
Here's my ideal conducting wire,
but then-- and this is

00:00:26.000 --> 00:00:30.460
new-- it branches off, and
I have two resistances.

00:00:30.460 --> 00:00:31.710
I have one here.

00:00:41.910 --> 00:00:43.140
And that's another resistance.

00:00:43.140 --> 00:00:46.320
And let's say that this has a
resistance R1, this has a

00:00:46.320 --> 00:00:49.130
resistance R2.

00:00:49.130 --> 00:00:52.220
And, of course, the
non-intuitive convention is

00:00:52.220 --> 00:00:55.180
that the current flows from the
positive to the negative

00:00:55.180 --> 00:00:57.410
terminal, but we know that the
electrons are actually flowing

00:00:57.410 --> 00:00:58.370
in the other direction.

00:00:58.370 --> 00:01:00.910
And I want to keep saying that,
because I think it's so

00:01:00.910 --> 00:01:03.700
important to understand what's
actually happening as opposed

00:01:03.700 --> 00:01:05.870
to the convention.

00:01:05.870 --> 00:01:08.450
Well, anyway, in the previous
video, we said, well, when we

00:01:08.450 --> 00:01:12.610
have devices or components in
series, that the current

00:01:12.610 --> 00:01:15.540
through the entire circuit
is constant.

00:01:15.540 --> 00:01:18.770
But let's think about
what happens here.

00:01:18.770 --> 00:01:20.660
So we have these electrons--
actually, let's think about it

00:01:20.660 --> 00:01:21.300
from the electron flow.

00:01:21.300 --> 00:01:23.240
The electrons are flowing,
flowing, flowing at a given

00:01:23.240 --> 00:01:27.770
rate, and then here they
have a choice.

00:01:27.770 --> 00:01:31.070
Some of them can take this top
path, some of them can take

00:01:31.070 --> 00:01:33.160
this bottom path.

00:01:33.160 --> 00:01:38.110
So if you think about it, the
flow of electrons in this

00:01:38.110 --> 00:01:41.920
branch plus the flow of
electrons in that branch have

00:01:41.920 --> 00:01:44.130
to add up to the flow
of the electrons in

00:01:44.130 --> 00:01:45.190
this branch, right?

00:01:45.190 --> 00:01:47.610
And then they're going to meet
back up, and then the flow of

00:01:47.610 --> 00:01:50.490
electrons here-- so if we think
of it this way, and now

00:01:50.490 --> 00:01:53.510
I'm going to go back to the
convention, this is I1.

00:01:56.150 --> 00:01:59.110
So you have these electrons
flowing at a given rate.

00:01:59.110 --> 00:02:01.160
This is the current
right here.

00:02:01.160 --> 00:02:05.560
They're going to branch off,
and maybe half of them go--

00:02:05.560 --> 00:02:09.800
we'll see if the resistances
are equal, if both of these

00:02:09.800 --> 00:02:13.280
branches have an equal amount
of capacity in terms of how

00:02:13.280 --> 00:02:15.030
fast the electrons
can flow through.

00:02:15.030 --> 00:02:17.220
If they're equal, or since we're
going to current in this

00:02:17.220 --> 00:02:21.390
direction, let's talk about
positrons or positive charges.

00:02:21.390 --> 00:02:23.950
If the positive charges--
although I just want to keep

00:02:23.950 --> 00:02:25.350
saying that it is
not the positive

00:02:25.350 --> 00:02:26.170
charges that are moving.

00:02:26.170 --> 00:02:26.900
It's the electrons.

00:02:26.900 --> 00:02:30.680
But if you say that the lack of
electrons can flow equally

00:02:30.680 --> 00:02:33.950
easily between both paths,
that's if the resistances were

00:02:33.950 --> 00:02:36.500
the same, we could imagine that
the current, the flow,

00:02:36.500 --> 00:02:38.270
would split itself up,
and then over here

00:02:38.270 --> 00:02:39.180
would meet back up.

00:02:39.180 --> 00:02:43.000
And then we would say that the
current here would also be I1.

00:02:43.000 --> 00:02:45.970
But let's figure out where
the current's going.

00:02:45.970 --> 00:02:51.120
Let's call this current I2 and
let's call this current I3.

00:02:54.260 --> 00:02:57.660
So I think it is reasonable, and
you can imagine with water

00:02:57.660 --> 00:03:02.550
pipes or anything, that the
current going into the branch

00:03:02.550 --> 00:03:07.590
is equal to the current
exiting the branch.

00:03:07.590 --> 00:03:10.830
Or you could even think of it
that the current entering--

00:03:10.830 --> 00:03:15.110
when the currents I2 and I1
merge, that they combine and

00:03:15.110 --> 00:03:16.520
they become Current 1, right?

00:03:16.520 --> 00:03:17.450
I mean, think about it.

00:03:17.450 --> 00:03:21.800
In a given second, if this is
5 coulombs per second-- I'm

00:03:21.800 --> 00:03:31.660
just making up numbers-- and
this is 6 coulombs per second,

00:03:31.660 --> 00:03:34.320
in a given second right here,
you're going to have 5

00:03:34.320 --> 00:03:38.760
coulombs coming from this branch
and 6 coulombs coming

00:03:38.760 --> 00:03:41.850
from this branch, so you're
going to have 11 coulombs per

00:03:41.850 --> 00:03:45.190
second coming out once they've
merged, so this would be 11

00:03:45.190 --> 00:03:45.800
coulombs per second.

00:03:45.800 --> 00:03:49.420
So I think hopefully that makes
sense to you that this

00:03:49.420 --> 00:03:52.480
current is equal to the
combination of this current

00:03:52.480 --> 00:03:54.120
and that current.

00:03:54.120 --> 00:03:55.900
Now, what do we also know?

00:03:55.900 --> 00:04:03.090
We also know the voltage along
this entire ideal wire is

00:04:03.090 --> 00:04:06.800
constant, so then voltage-- let
me draw that in another

00:04:06.800 --> 00:04:09.260
color, in blue.

00:04:09.260 --> 00:04:13.900
So, for example, the voltage
anywhere along this blue that

00:04:13.900 --> 00:04:19.350
I'm filling in is going to be
the same, because this wire is

00:04:19.350 --> 00:04:23.220
an ideal conductor, and you can
almost view this blue part

00:04:23.220 --> 00:04:24.830
as an extension of the positive

00:04:24.830 --> 00:04:26.810
terminal of the battery.

00:04:26.810 --> 00:04:30.850
And very similarly-- I'll do it
in yellow-- we could draw

00:04:30.850 --> 00:04:33.430
this wire as an extension
of the negative

00:04:33.430 --> 00:04:36.750
terminal of the battery.

00:04:36.750 --> 00:04:38.280
This is an extension
of the negative

00:04:38.280 --> 00:04:39.530
terminal of the battery.

00:04:45.150 --> 00:04:48.800
So the voltage difference
between here and here-- so

00:04:48.800 --> 00:04:51.440
let's call that the total
voltage, or let's just call it

00:04:51.440 --> 00:04:53.480
the voltage, right?

00:04:53.480 --> 00:04:55.980
The voltage difference between
that point and that point is

00:04:55.980 --> 00:04:58.600
the exact same thing as the
voltage difference between

00:04:58.600 --> 00:05:02.290
this point and this point, which
is the exact same thing

00:05:02.290 --> 00:05:04.180
as the voltage difference
between this

00:05:04.180 --> 00:05:07.200
point and this point.

00:05:07.200 --> 00:05:08.510
So what can we say?

00:05:08.510 --> 00:05:10.890
What is the total current
in the system?

00:05:10.890 --> 00:05:12.880
If we just viewed this as a
black box, that this is some

00:05:12.880 --> 00:05:17.880
type of total resistance, well,
the total current in the

00:05:17.880 --> 00:05:22.260
system would be the total
voltage, the voltage divided

00:05:22.260 --> 00:05:25.590
by-- let's call this our total
resistance, right?

00:05:25.590 --> 00:05:27.580
Let's say we couldn't see this
and we just said, oh, that's

00:05:27.580 --> 00:05:32.840
just some total resistance,
and that is equal to the

00:05:32.840 --> 00:05:35.140
current going through R1.

00:05:35.140 --> 00:05:37.350
This is I1.

00:05:37.350 --> 00:05:39.100
This is a 1 right here.

00:05:39.100 --> 00:05:40.350
This is current I1.

00:05:42.730 --> 00:05:43.560
What's current I1?

00:05:43.560 --> 00:05:47.160
Well, it's going to be the
voltage across this resistor

00:05:47.160 --> 00:05:48.990
divided by the resistance,
right?

00:05:48.990 --> 00:05:53.210
That's what Ohm's law tells
us: V is equal to IR, or

00:05:53.210 --> 00:05:57.770
another way we could say it is V
over R is equal to I, right?

00:05:57.770 --> 00:06:01.080
So I1 is equal to the voltage
across this resistor, but we

00:06:01.080 --> 00:06:02.590
just said that voltage
is the same thing as

00:06:02.590 --> 00:06:04.350
this voltage, right?

00:06:04.350 --> 00:06:06.330
The voltage here is the same
thing as the voltage here.

00:06:06.330 --> 00:06:08.090
The voltage here is the same
thing as the voltage here.

00:06:08.090 --> 00:06:11.840
So the voltage across that
resistor is still V, and so

00:06:11.840 --> 00:06:15.980
the current flowing across that
resistor is V over R1.

00:06:15.980 --> 00:06:19.400
And the same logic:
what is I2?

00:06:19.400 --> 00:06:21.160
I2 is this current.

00:06:21.160 --> 00:06:23.900
What is the voltage across
this device?

00:06:23.900 --> 00:06:25.930
Well, that's just
V again, right?

00:06:25.930 --> 00:06:28.400
It's the same thing as the
voltage across this device, so

00:06:28.400 --> 00:06:32.450
it's V over R2 by Ohm's law.

00:06:32.450 --> 00:06:34.870
Well, all these V's are the
same, so we can divide both

00:06:34.870 --> 00:06:38.100
sides of that equation by V,
and we get 1 over the total

00:06:38.100 --> 00:06:46.160
resistance is equal to 1
over R1 plus 1 over R2.

00:06:46.160 --> 00:06:49.300
And you could make that argument
if we had an R3 here.

00:06:49.300 --> 00:06:54.520
Let's say that we had another
device, and that is R3.

00:06:54.520 --> 00:06:56.430
You could use the exact same
argument, and you would have a

00:06:56.430 --> 00:06:58.590
plus 1 over R3.

00:06:58.590 --> 00:07:02.260
And if you had Rn or 10 of them,
you'd just keep that 1

00:07:02.260 --> 00:07:05.350
over R4, R5, et cetera.

00:07:05.350 --> 00:07:08.170
So let's see if we can use
this information we have

00:07:08.170 --> 00:07:10.100
learned to actually solve a
problem, and I actually find

00:07:10.100 --> 00:07:12.760
that it's always easier to
solve a problem than to

00:07:12.760 --> 00:07:15.740
explain the theory
behind a problem.

00:07:15.740 --> 00:07:17.750
You'll see that with most of
these circuit problems, it's

00:07:17.750 --> 00:07:19.860
actually very basic
mathematics.

00:07:19.860 --> 00:07:23.510
So let's say I have a 16
volt battery plus,

00:07:23.510 --> 00:07:27.380
minus, it's 16 volts.

00:07:27.380 --> 00:07:29.660
And just to hit the point home
that you always don't have to

00:07:29.660 --> 00:07:32.090
draw circuits the same, although
it is nice if you're

00:07:32.090 --> 00:07:33.900
actually drawing complicated
circuits, I could

00:07:33.900 --> 00:07:34.950
draw it like this.

00:07:34.950 --> 00:07:38.650
I could draw the circuit like
this, and let's say that

00:07:38.650 --> 00:07:39.806
there's a resistor here.

00:07:39.806 --> 00:07:43.230
And then let's say there's a
wire and then there's another

00:07:43.230 --> 00:07:47.460
resistor here, and that this
decides to do some random

00:07:47.460 --> 00:07:51.200
loopy thing here and that they
connect here, and that they

00:07:51.200 --> 00:07:52.590
come back here.

00:07:52.590 --> 00:07:55.710
This strange thing that I have
drawn, which you will never

00:07:55.710 --> 00:07:57.970
see in any textbook, because
most people are more

00:07:57.970 --> 00:08:02.080
reasonable than me, is the exact
same-- you can almost

00:08:02.080 --> 00:08:05.440
view it topologically as the
exact same circuit as what I

00:08:05.440 --> 00:08:07.740
drew in the previous diagram,
although now I will assign

00:08:07.740 --> 00:08:08.730
numbers to it.

00:08:08.730 --> 00:08:16.810
Let's say that this resistance
is 20 ohms and let's say that

00:08:16.810 --> 00:08:23.590
this resistance is 5 ohms. What
I want to know is, what

00:08:23.590 --> 00:08:26.690
is the current through
the system?

00:08:26.690 --> 00:08:28.040
First, we'll have to figure
out what the equivalent

00:08:28.040 --> 00:08:30.460
resistance is, and then we could
just use Ohm's law to

00:08:30.460 --> 00:08:31.720
figure out the current
in the system.

00:08:31.720 --> 00:08:34.070
So we want to know what the
current is, and we know that

00:08:34.070 --> 00:08:37.240
the convention is that current
flows from the positive

00:08:37.240 --> 00:08:39.400
terminal to the negative
terminal.

00:08:39.400 --> 00:08:41.679
So how do we figure out the
equivalent resistance?

00:08:41.679 --> 00:08:44.890
Well, we know that we just
hopefully proved to you that

00:08:44.890 --> 00:08:49.950
the total resistance is equal to
1 over this resistor plus 1

00:08:49.950 --> 00:08:52.930
over this resistor.

00:08:52.930 --> 00:08:57.300
So 1 over-- I won't
keep writing it.

00:08:57.300 --> 00:08:58.880
What's 1 over 20?

00:08:58.880 --> 00:09:01.750
Well, actually, let's just
make it a fraction.

00:09:01.750 --> 00:09:06.130
So it's 1/20 plus-- 1/5
is what over 20?

00:09:06.130 --> 00:09:07.880
That's 4/20, right?

00:09:07.880 --> 00:09:14.060
So 1 over our total resistance
is equal to 5/20, which is

00:09:14.060 --> 00:09:14.650
equal to what?

00:09:14.650 --> 00:09:15.900
1/4.

00:09:18.210 --> 00:09:22.520
So if 1/R is equal to 1/4,
R must be equal to 4.

00:09:22.520 --> 00:09:25.850
R is equal to 4 ohms.

00:09:25.850 --> 00:09:30.260
So we could redraw this
crazy circuit as this.

00:09:30.260 --> 00:09:32.100
I'll try to draw it
small down here.

00:09:36.840 --> 00:09:40.950
We could redraw this where this
resistance is 4 ohms and

00:09:40.950 --> 00:09:43.600
this is 16 volts.

00:09:43.600 --> 00:09:47.140
We could say that this whole
thing combined is really just

00:09:47.140 --> 00:09:52.770
a resistor that is 4 ohms. Well,
if we have a 16-volt

00:09:52.770 --> 00:09:55.340
potential difference, current
is flowing that way, even

00:09:55.340 --> 00:09:56.910
though that's not what the
electrons are doing.

00:09:56.910 --> 00:09:58.890
And that's what our
resistance is, 4

00:09:58.890 --> 00:10:00.270
ohms. What is the current?

00:10:00.270 --> 00:10:02.870
V equals IR, Ohms law.

00:10:02.870 --> 00:10:05.020
The voltage is 16 volts.

00:10:05.020 --> 00:10:11.330
It equals the current times 4
ohms. So current is equal to

00:10:11.330 --> 00:10:15.460
16 divided by 4, is
equal to 4 amps.

00:10:15.460 --> 00:10:17.150
So let's do something
interesting.

00:10:17.150 --> 00:10:20.640
Let's figure out what the
current is flowing through.

00:10:20.640 --> 00:10:21.470
What's this?

00:10:21.470 --> 00:10:27.330
What's the current I1 and
what's this current I2?

00:10:27.330 --> 00:10:29.430
Well, we know that the potential
difference from here

00:10:29.430 --> 00:10:31.330
to here is also 16
volts, right?

00:10:31.330 --> 00:10:35.530
Because this whole thing is
essentially at the same

00:10:35.530 --> 00:10:37.420
potential and this whole thing
is essentially at the same

00:10:37.420 --> 00:10:40.990
potential, so you have 16
volts across there.

00:10:40.990 --> 00:10:46.610
16 volts divided by 20 ohms,
so let's call this I1.

00:10:46.610 --> 00:10:54.830
So I1 is equal to 16 volts
divided by 20 ohms, which is

00:10:54.830 --> 00:10:55.300
equal to what?

00:10:55.300 --> 00:10:56.660
4/5.

00:10:56.660 --> 00:11:02.820
So it equals 4/5 of an ampere,
or 0.8 amperes.

00:11:02.820 --> 00:11:05.440
And similarly, what is the
amount of current flowing

00:11:05.440 --> 00:11:06.810
through here?

00:11:06.810 --> 00:11:07.590
I2?

00:11:07.590 --> 00:11:09.110
I'm going to do this in
a different color.

00:11:09.110 --> 00:11:11.546
It's getting confusing.

00:11:11.546 --> 00:11:13.030
I'll do it in the
vibrant yellow.

00:11:13.030 --> 00:11:15.090
So the current flowing through
here, once again, the

00:11:15.090 --> 00:11:17.680
potential difference from here--
that's not different

00:11:17.680 --> 00:11:19.890
enough-- the potential
difference from here to here

00:11:19.890 --> 00:11:22.190
is also 16 volts, right?

00:11:22.190 --> 00:11:26.240
So the current is going to be
I2, is going to be equal to

00:11:26.240 --> 00:11:35.390
16/5, which is equal
to 3 1/5 amps.

00:11:35.390 --> 00:11:37.730
So most of the current is
actually flowing through this,

00:11:37.730 --> 00:11:40.530
and that makes sense because the
resistance is less, right?

00:11:40.530 --> 00:11:42.230
So that should hopefully give
you a little bit of intuition

00:11:42.230 --> 00:11:42.980
of what's going on.

00:11:42.980 --> 00:11:45.910
And less current is flowing
through here, so I2 through

00:11:45.910 --> 00:11:53.630
the 20-ohm resistor is 0.8 amps
is I1, and I2 through the

00:11:53.630 --> 00:11:58.250
5-ohm resistor is equal
to 3.2 amps.

00:11:58.250 --> 00:12:00.740
And it makes sense that when
you add these two currents

00:12:00.740 --> 00:12:04.560
together, the 3.2 amperes
flowing through here and the

00:12:04.560 --> 00:12:07.000
0.8 amperes flowing through
here, that when they merge,

00:12:07.000 --> 00:12:10.840
they merge and then you have 4
amperes flowing through there.

00:12:10.840 --> 00:12:14.280
Anyway, hopefully, I have given
you some intuition on

00:12:14.280 --> 00:12:17.850
what happens when we put
resistors in parallel.

00:12:17.850 --> 00:12:19.100
I will see you in
the next video.