Example visually evaluating discrete functions

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:02.940
- [Instructor] What we have
here is a visual depiction

00:00:02.940 --> 00:00:05.170
of a function.

00:00:05.170 --> 00:00:10.170
And this is a depiction
of Y is equal to H of X.

00:00:11.020 --> 00:00:13.616
Now when a lot of people see
function notation like this

00:00:13.616 --> 00:00:17.203
they see it as somewhat
intimidating until you realize

00:00:17.203 --> 00:00:18.790
what it's saying.

00:00:18.790 --> 00:00:22.000
All a function is, is
something that takes an input,

00:00:22.000 --> 00:00:26.160
in this case it's taking X as
an input and then the function

00:00:26.160 --> 00:00:30.050
does something to it and then
it spits out some other value

00:00:30.050 --> 00:00:33.450
which is going to be equal to Y.

00:00:33.450 --> 00:00:38.450
So for example what is H
of four based on this graph

00:00:39.489 --> 00:00:41.710
that you see right over here?

00:00:41.710 --> 00:00:44.230
Pause this video and think through that.

00:00:44.230 --> 00:00:48.050
Well all H of four means
is when I input four

00:00:48.050 --> 00:00:53.050
into my function H what
Y am I spitting out?

00:00:53.240 --> 00:00:56.289
Or another way to think about
it, when X is equal to four,

00:00:56.289 --> 00:00:59.452
what is Y equal to?

00:00:59.452 --> 00:01:04.452
Well when X is equal to four,
my function spits out that Y

00:01:04.932 --> 00:01:06.440
is equal to three.

00:01:06.440 --> 00:01:08.780
We know that from this
point right over here, so Y

00:01:08.780 --> 00:01:12.560
is equal to three, so H
of four is equal to three.

00:01:12.560 --> 00:01:14.810
Let's do another example.

00:01:14.810 --> 00:01:18.850
What is H of zero?

00:01:18.850 --> 00:01:21.530
Pause the video try to work that through.

00:01:21.530 --> 00:01:25.600
Well all this is saying is
if I input an X equals zero

00:01:25.600 --> 00:01:29.780
into the function what is going
to be the corresponding Y?

00:01:29.780 --> 00:01:34.780
Well when X equals zero we
see that Y is equal to four.

00:01:36.250 --> 00:01:38.160
So it's as simple as that.

00:01:38.160 --> 00:01:40.720
Given the input what is
going to be the output?

00:01:40.720 --> 00:01:43.460
And that's what these points
represent, each of these points

00:01:43.460 --> 00:01:47.680
represent a different
output for a given input.

00:01:47.680 --> 00:01:50.670
Now it's always good to keep
in mind one of the things

00:01:50.670 --> 00:01:53.370
that makes it a function is
that for given X that you input

00:01:53.370 --> 00:01:54.940
you only get one Y.

00:01:54.940 --> 00:01:59.380
For example if we had two dots
here, then all of a sudden

00:01:59.380 --> 00:02:03.510
or we have two dots for X
equals six, now all of a sudden

00:02:03.510 --> 00:02:06.370
we have a problem
figuring out what H of six

00:02:06.370 --> 00:02:09.153
would be equal to because
it could be equal to one

00:02:09.153 --> 00:02:11.760
or it could be equal to three.

00:02:11.760 --> 00:02:14.410
So if we had this extra dot
here, then this would no longer

00:02:14.410 --> 00:02:15.670
be a function.

00:02:15.670 --> 00:02:18.441
In order for it to be a
function for any given X,

00:02:18.441 --> 00:02:22.750
it has to output a unique
value, it can't output

00:02:22.750 --> 00:02:24.385
two possible values.

00:02:24.385 --> 00:02:27.190
Now the other way is possible.

00:02:27.190 --> 00:02:30.630
It is possible to have two
different X's that output

00:02:30.630 --> 00:02:31.890
the same value.

00:02:31.890 --> 00:02:36.890
For example, if this was circled
in what would H of negative

00:02:37.609 --> 00:02:39.049
four be?

00:02:39.049 --> 00:02:42.490
Well H of negative four when
X is equal to negative four,

00:02:42.490 --> 00:02:44.870
you put that into our function
it looks like the function

00:02:44.870 --> 00:02:46.500
would output two.

00:02:46.500 --> 00:02:49.160
So H of negative four
would be equal to two.

00:02:49.160 --> 00:02:53.550
But H of two is also equal
to we see very clearly there,

00:02:53.550 --> 00:02:56.710
when we input a two into the
function, the corresponding Y

00:02:56.710 --> 00:02:59.100
value is two as well.

00:02:59.100 --> 00:03:02.750
So it's okay for two
different X values to map

00:03:02.750 --> 00:03:05.200
to the same Y value that works.

00:03:05.200 --> 00:03:08.310
But if you had some type of
an arrangement, some type

00:03:08.310 --> 00:03:12.311
of a relationship where
for a given X value you had

00:03:12.311 --> 00:03:14.090
two different Y values,
then that would no longer

00:03:14.090 --> 00:03:15.080
be a function.

00:03:15.080 --> 00:03:17.720
But the example they gave
us is a function assuming

00:03:17.720 --> 00:03:18.803
I don't modify it.


Setting up systems of linear equations example

WEBVTT
Kind: captions
Language: en

00:00:00.370 --> 00:00:01.203
- [Instructor] In this video,

00:00:01.203 --> 00:00:02.036
we're going to get a little bit

00:00:02.036 --> 00:00:06.430
of practice setting up
systems of linear equations

00:00:06.430 --> 00:00:08.370
based on a word problem.

00:00:08.370 --> 00:00:10.130
We're not actually going
to end up solving it.

00:00:10.130 --> 00:00:12.370
You can do that if you
like just for kicks.

00:00:12.370 --> 00:00:15.630
But really we're going to
just focus on setting it up.

00:00:15.630 --> 00:00:19.890
So here we're told Lauren uses
a blend of dark roast beans

00:00:19.890 --> 00:00:23.710
and light roast beans to
make coffee at her cafe.

00:00:23.710 --> 00:00:28.420
She needs 80 kilograms of beans
in total for her next order.

00:00:28.420 --> 00:00:32.490
Dark roast beans cost $3 per kilogram,

00:00:32.490 --> 00:00:36.170
light roast beans cost $2 per kilogram,

00:00:36.170 --> 00:00:40.780
and someone wants to spend $220 total.

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Let D be the number of kilograms

00:00:43.110 --> 00:00:45.050
of dark roast beans she buys

00:00:45.050 --> 00:00:47.630
and L be the number of kilograms

00:00:47.630 --> 00:00:50.760
of light roast beans she buys.

00:00:50.760 --> 00:00:52.860
All right, so based on this information

00:00:52.860 --> 00:00:55.870
that we've been given, see
if you can pause this video

00:00:55.870 --> 00:00:58.020
and set up a system of equations.

00:00:58.020 --> 00:00:59.420
And it's going to have two equations

00:00:59.420 --> 00:01:03.300
with two unknowns, D and L,
that in theory we could solve

00:01:03.300 --> 00:01:05.630
to figure out the right
number of kilograms

00:01:05.630 --> 00:01:07.960
of dark roast beans and light roast beans

00:01:07.960 --> 00:01:10.070
that Karen should use.

00:01:10.070 --> 00:01:12.320
So pause this video and
try to work that out.

00:01:13.260 --> 00:01:14.640
All right, now let's do it together.

00:01:14.640 --> 00:01:16.610
And what I'm going to do
is I'm going to underline.

00:01:16.610 --> 00:01:20.850
So let's see, we know
that D is dark roast beans

00:01:20.850 --> 00:01:24.150
and the L is the number of
kilograms of light roast beans.

00:01:24.150 --> 00:01:25.610
And then they tell us here,

00:01:25.610 --> 00:01:30.610
they say she needs 80
kilograms of beans in total.

00:01:31.000 --> 00:01:33.610
So what we could say is,
hey, the number of kilograms

00:01:33.610 --> 00:01:37.150
of dark roast beans plus
the number of kilograms

00:01:37.150 --> 00:01:42.150
of light roast beans needs
to be equal to 80 kilograms.

00:01:42.230 --> 00:01:45.390
So the number of kilograms
of dark roast beans

00:01:45.390 --> 00:01:49.022
plus the number of kilograms
of light roast beans.

00:01:49.022 --> 00:01:52.300
(laughs) I'm having trouble
saying light roast beans.

00:01:52.300 --> 00:01:54.040
Well, this, what I just underlined here,

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it says needs to be 80 kilograms in total.

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So that needs to be 80.

00:01:58.690 --> 00:02:00.130
So this number of kilograms

00:02:00.130 --> 00:02:01.770
plus this number of kilograms
is going to be equal

00:02:01.770 --> 00:02:03.770
to your total number of kilograms.

00:02:03.770 --> 00:02:06.380
All right, so I have one
equation with two unknowns.

00:02:06.380 --> 00:02:08.670
Let's see if we can get another one.

00:02:08.670 --> 00:02:13.670
So next, they say dark roast
beans cost $3 per kilogram,

00:02:14.310 --> 00:02:17.350
light roast beans cost $2 per kilogram,

00:02:17.350 --> 00:02:21.060
and she wants to spend $220 total.

00:02:21.060 --> 00:02:24.430
So what I just underlined
in this aquamarine color

00:02:24.430 --> 00:02:25.890
we can set up another equation with.

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And if you haven't already set
up your system of equations,

00:02:28.620 --> 00:02:30.170
see if you can now do that.

00:02:30.170 --> 00:02:31.610
See if you can set up
this second equation.

00:02:31.610 --> 00:02:33.240
Pause the video.

00:02:33.240 --> 00:02:34.150
All right, well, the way to think

00:02:34.150 --> 00:02:36.130
about it is we just have
to have an expression

00:02:36.130 --> 00:02:38.220
for how much did she
spend on dark roast beans,

00:02:38.220 --> 00:02:40.280
how much did she spend
on light roast beans,

00:02:40.280 --> 00:02:42.080
and then we need to
add those two together,

00:02:42.080 --> 00:02:44.470
and that needs to be equal to $220

00:02:44.470 --> 00:02:46.470
'cause that's how much she
wants to spend in total.

00:02:46.470 --> 00:02:49.830
So how much does she
spend on dark roast beans?

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Well, it's going to be
the number of kilograms

00:02:52.390 --> 00:02:55.420
of dark roast beans that she buys,

00:02:55.420 --> 00:02:58.640
and it says that it cost $3 per kilogram,

00:02:58.640 --> 00:03:00.157
so we're gonna multiply it by three.

00:03:00.157 --> 00:03:02.640
$3 per kilogram times
the number of kilograms

00:03:02.640 --> 00:03:03.480
of dark roast beans.

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This is how much she
spends on dark roast beans.

00:03:06.300 --> 00:03:09.690
And so how much is she going
to spend on light roast beans?

00:03:09.690 --> 00:03:13.020
Well, she buys L kilograms
of light roast beans.

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They told us that there.

00:03:14.340 --> 00:03:16.650
And they cost $2 per kilogram,

00:03:16.650 --> 00:03:19.070
so $2 per kilogram times
the number of kilograms.

00:03:19.070 --> 00:03:21.390
This is how much she spends
on light roast beans.

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So you add how much she
spends on dark roast

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to how much she spends on light roast,

00:03:26.310 --> 00:03:30.930
and so this is going to be $220 in total.

00:03:30.930 --> 00:03:31.870
And there you have it.

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We have our two equations
with two unknowns,

00:03:35.540 --> 00:03:37.790
and so now we could go and solve it,

00:03:37.790 --> 00:03:39.850
but you can do that outside of this video.

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But the whole point of
this video is to understand

00:03:41.990 --> 00:03:45.740
how to construct these
based on the constraints,

00:03:45.740 --> 00:03:47.790
based on the information
that we see in this.

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So typically when you're
trying to set these up,

00:03:50.810 --> 00:03:52.480
there's often a sentence or two

00:03:52.480 --> 00:03:54.040
that will focus on one equation.

00:03:54.040 --> 00:03:57.440
So this first one is
saying, hey, the kilograms,

00:03:57.440 --> 00:03:59.690
let's add those up for the
total number of kilograms.

00:03:59.690 --> 00:04:01.940
And then there's another
sentence or two that'll focus on,

00:04:01.940 --> 00:04:04.160
in this case, some other equation.

00:04:04.160 --> 00:04:05.450
In this case, it's the price.

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So the price of the dark
plus the price of the light

00:04:08.130 --> 00:04:11.130
is going to be equal to the
total amount she wants to spend.


Interpreting points in context of graphs of systems

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:02.280
- [Instructor] We're told
that Lauren uses a blend

00:00:02.280 --> 00:00:04.610
of dark roast beans and light roast beans

00:00:04.610 --> 00:00:06.780
to make coffee at her cafe.

00:00:06.780 --> 00:00:11.580
She needs 80 kilograms of beans
in total for her next order.

00:00:11.580 --> 00:00:14.750
Dark roast beans cost $3 per kilogram,

00:00:14.750 --> 00:00:17.770
light roast beans cost $2 per kilogram,

00:00:17.770 --> 00:00:20.920
and she wants to spend $220 in total.

00:00:20.920 --> 00:00:22.960
And they tell us here's a graph

00:00:22.960 --> 00:00:26.060
that shows a system of
equations for this scenario

00:00:26.060 --> 00:00:28.680
where x is the number of kilograms

00:00:28.680 --> 00:00:30.790
of dark roast beans she buys

00:00:30.790 --> 00:00:33.390
and y is the number of kilograms

00:00:33.390 --> 00:00:35.880
of light roast beans she buys.

00:00:35.880 --> 00:00:39.180
All right, let me scroll down
so we can take a look at this.

00:00:39.180 --> 00:00:42.930
And so sure enough, so this blue line,

00:00:42.930 --> 00:00:44.900
and I'll write it out in blue,

00:00:44.900 --> 00:00:48.090
this x is the number of
kilograms of dark roast beans,

00:00:48.090 --> 00:00:51.170
y is the number of kilograms
of light roast beans,

00:00:51.170 --> 00:00:53.930
and she wants to buy a
total of 80 kilograms.

00:00:53.930 --> 00:00:56.300
That's what they told us up here.

00:00:56.300 --> 00:00:58.640
We can go back to look at that.

00:00:58.640 --> 00:01:01.340
She needs, I'll underline this in blue.

00:01:01.340 --> 00:01:05.000
She needs 80 kilograms of beans.

00:01:05.000 --> 00:01:08.050
So that constraint that
the sum of the kilograms

00:01:08.050 --> 00:01:10.490
of dark and light is equal to 80,

00:01:10.490 --> 00:01:12.270
that's represented by this equation.

00:01:12.270 --> 00:01:13.710
And if we were to graph it,

00:01:13.710 --> 00:01:16.610
that is this blue line right over here.

00:01:16.610 --> 00:01:20.350
And then this other constraint, three x,

00:01:20.350 --> 00:01:23.030
well, the dark roast beans
cost $3 per kilogram,

00:01:23.030 --> 00:01:25.730
so three x is how much
she spends on dark roast.

00:01:25.730 --> 00:01:27.920
Two y is how much she
spends on light roast

00:01:27.920 --> 00:01:30.190
'cause it's $2 per kilogram.

00:01:30.190 --> 00:01:32.680
And 220 is the amount
that she spends in total.

00:01:32.680 --> 00:01:35.310
And they tell us that up here.

00:01:35.310 --> 00:01:38.120
Dark roast beans cost $3 per kilogram,

00:01:38.120 --> 00:01:40.740
light roast beans cost $2 per kilogram,

00:01:40.740 --> 00:01:43.760
and she wants to spend $220.

00:01:43.760 --> 00:01:46.690
So this equation is
another way of expressing

00:01:46.690 --> 00:01:49.400
what I just underlined up here in green.

00:01:49.400 --> 00:01:53.170
And the green line shows all

00:01:53.170 --> 00:01:58.170
of the x y combinations that
would match these constraints.

00:01:59.130 --> 00:02:01.890
And so now let's do something interesting.

00:02:01.890 --> 00:02:05.510
They've labeled some points
here, point C, D, F, and E.

00:02:05.510 --> 00:02:07.020
And we're gonna think about what do each

00:02:07.020 --> 00:02:08.190
of these points represent?

00:02:08.190 --> 00:02:13.190
So for example, this point
C that is on the green line,

00:02:13.470 --> 00:02:17.600
but it sits above the blue line,

00:02:17.600 --> 00:02:19.030
what does this tell us?

00:02:19.030 --> 00:02:20.630
What does this point C represent?

00:02:20.630 --> 00:02:22.430
Pause this video and think about it.

00:02:24.020 --> 00:02:26.070
Well, if we're on the green line,

00:02:26.070 --> 00:02:29.430
that means that the amount
that she spends on dark roast

00:02:29.430 --> 00:02:33.080
plus the amount that she spends
on light roast is adding up

00:02:33.080 --> 00:02:35.790
to exactly $220.

00:02:35.790 --> 00:02:40.450
So she's definitely spending $220 at C,

00:02:40.450 --> 00:02:44.590
but how many total kilograms is she using?

00:02:44.590 --> 00:02:48.470
Well, the fact that for this given x,

00:02:48.470 --> 00:02:50.880
we're sitting above the line,

00:02:50.880 --> 00:02:54.917
that means that she's not
using exactly 80 kilograms.

00:02:54.917 --> 00:02:56.530
And we can see that over here.

00:02:56.530 --> 00:02:59.620
She's using, looks like
10 kilograms of dark,

00:02:59.620 --> 00:03:04.400
and it looks like something
like 95 kilograms of light.

00:03:04.400 --> 00:03:06.870
If you were to add those
two points together,

00:03:06.870 --> 00:03:10.340
it looks like she's using
something closer to 105 kilograms.

00:03:10.340 --> 00:03:11.820
So point C is a situation

00:03:11.820 --> 00:03:14.710
where she is spending exactly $220,

00:03:14.710 --> 00:03:18.750
but she's using more than 80 kilograms

00:03:18.750 --> 00:03:20.830
'cause it's not meeting
this second constraint.

00:03:20.830 --> 00:03:22.750
It's sitting above that line.

00:03:22.750 --> 00:03:23.750
Now let's think about point D.

00:03:23.750 --> 00:03:24.900
What does that represent?

00:03:24.900 --> 00:03:27.660
Pause the video and
try to figure that out.

00:03:27.660 --> 00:03:30.500
Well, because we sit on the blue line,

00:03:30.500 --> 00:03:33.140
that means that we are
meeting this constraint

00:03:33.140 --> 00:03:37.500
that the kilograms of
dark and light combined

00:03:37.500 --> 00:03:40.000
is equal to 80 kilograms.

00:03:40.000 --> 00:03:43.200
So she's using exactly 80 kilograms here,

00:03:43.200 --> 00:03:45.450
but what about her spending?

00:03:45.450 --> 00:03:49.440
Well, because this point
lies below the green line,

00:03:49.440 --> 00:03:52.200
that tells us that we are spending

00:03:52.200 --> 00:03:55.017
less than $220.

00:03:55.017 --> 00:03:56.730
And we could even try it out.

00:03:56.730 --> 00:04:01.730
Three times 20 plus two times 60 is what?

00:04:01.840 --> 00:04:05.150
60 plus 120 is $180.

00:04:05.150 --> 00:04:08.590
And so this is a point where
we're meeting this constraint,

00:04:08.590 --> 00:04:10.070
but we're not meeting this constraint.

00:04:10.070 --> 00:04:12.680
We're underspending right over here.

00:04:12.680 --> 00:04:14.810
Now, what about point F?

00:04:14.810 --> 00:04:18.080
Well, point F sits below
both of these lines.

00:04:18.080 --> 00:04:20.683
So pause your video and
think about what that means.

00:04:21.590 --> 00:04:24.930
Well, if we're sitting
below both of these lines,

00:04:24.930 --> 00:04:29.510
that means that neither
are we spending $220,

00:04:29.510 --> 00:04:32.500
nor are we using 80 kilograms.

00:04:32.500 --> 00:04:34.620
And you can see that if you
actually look at the numbers.

00:04:34.620 --> 00:04:35.610
You don't have to do this,

00:04:35.610 --> 00:04:37.860
but this is just to make
you feel good about it.

00:04:37.860 --> 00:04:41.020
It looks like she is
using 30 kilograms of dark

00:04:41.020 --> 00:04:45.480
and 30 kilograms of light,
so in total she is using,

00:04:45.480 --> 00:04:46.670
so this is a situation

00:04:46.670 --> 00:04:49.740
where she's using 60
kilograms in total, not 80.

00:04:49.740 --> 00:04:52.080
And so that's why we're not
sitting on this blue line.

00:04:52.080 --> 00:04:54.070
And if you look at how
much she's spending,

00:04:54.070 --> 00:04:55.740
she has 30 kilograms of each,

00:04:55.740 --> 00:04:58.140
so three times 30 plus two times 30,

00:04:58.140 --> 00:05:00.390
that's going to be 90 plus 60.

00:05:00.390 --> 00:05:02.057
That's also less than 220,

00:05:02.057 --> 00:05:06.000
and so that's why we see this
point is below these lines.

00:05:06.000 --> 00:05:09.283
And then last but not least,
what does point E represent?

00:05:10.470 --> 00:05:13.170
Well, point E sits on both of these lines,

00:05:13.170 --> 00:05:16.410
so that means that it meets
both of these constraints.

00:05:16.410 --> 00:05:20.740
This is a situation where
she is spending exactly $220,

00:05:20.740 --> 00:05:23.770
and the total number of
kilograms she's using

00:05:23.770 --> 00:05:26.490
of dark and light is exactly 80.

00:05:26.490 --> 00:05:29.250
And so if we wanted to
say, hey, what combination

00:05:29.250 --> 00:05:31.950
of dark and light would she need

00:05:31.950 --> 00:05:35.660
in order to meet both
constraints, E represents that,

00:05:35.660 --> 00:05:37.953
the intersection of these two lines.


Determining whether values are in domain of function

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:03.060
- [Instructor] We're asked
to determine for each x-value

00:00:03.060 --> 00:00:06.480
whether it is in the domain of f or not.

00:00:06.480 --> 00:00:09.530
And they have our definition
of f of x up here.

00:00:09.530 --> 00:00:11.740
So pause this video and see
if you can work through this

00:00:11.740 --> 00:00:13.063
before we do it together.

00:00:13.920 --> 00:00:16.030
All right, so just as a bit of a review,

00:00:16.030 --> 00:00:19.610
if x is in the domain of our function,

00:00:19.610 --> 00:00:23.530
that means that if we input
our x into our function,

00:00:23.530 --> 00:00:26.990
when we are going to get a
legitimate output f of x.

00:00:26.990 --> 00:00:30.140
But if for whatever reason
f isn't defined at x

00:00:30.140 --> 00:00:33.240
or it gets some kind of undefined state,

00:00:33.240 --> 00:00:36.560
well, then x would not be in the domain.

00:00:36.560 --> 00:00:38.240
So let's try these different values.

00:00:38.240 --> 00:00:42.410
Is x equal to negative
five in the domain of f?

00:00:42.410 --> 00:00:43.530
Well, let's see what happens

00:00:43.530 --> 00:00:46.970
if we try to evaluate f of negative five.

00:00:46.970 --> 00:00:50.880
Well, then in the numerator,
we get negative five plus five.

00:00:50.880 --> 00:00:52.030
Every place where we see an x,

00:00:52.030 --> 00:00:54.300
we replace it with a negative five.

00:00:54.300 --> 00:00:56.131
So it's negative five plus five,

00:00:56.131 --> 00:01:00.960
over negative five minus
three, which is equal to

00:01:00.960 --> 00:01:02.630
in our numerator, we get zero,

00:01:02.630 --> 00:01:06.130
and in our denominator,
we get negative eight.

00:01:06.130 --> 00:01:07.507
Now, at first you see the zero,

00:01:07.507 --> 00:01:08.740
and you might get a little bit worried,

00:01:08.740 --> 00:01:10.090
but it's just a zero in the numerator,

00:01:10.090 --> 00:01:12.640
so this whole thing just
evaluates to a zero,

00:01:12.640 --> 00:01:15.100
which is a completely legitimate output.

00:01:15.100 --> 00:01:19.690
So x equals negative
five is in the domain.

00:01:19.690 --> 00:01:21.910
What about x equals zero?

00:01:21.910 --> 00:01:22.743
Is that in the domain?

00:01:22.743 --> 00:01:23.576
Pause the video.

00:01:23.576 --> 00:01:25.290
See if you can figure that out.

00:01:25.290 --> 00:01:28.440
Well, f of zero is going to be equal to

00:01:28.440 --> 00:01:30.827
in our numerator, we have zero plus five,

00:01:30.827 --> 00:01:34.760
and in our denominator,
we have zero minus three.

00:01:34.760 --> 00:01:37.110
Well, that's just going to
get us five in the numerator

00:01:37.110 --> 00:01:38.970
and negative three in the denominator.

00:01:38.970 --> 00:01:40.940
This would just be negative 5/3.

00:01:40.940 --> 00:01:43.030
But this is a completely
legitimate output.

00:01:43.030 --> 00:01:45.240
So the function is
defined at x equals zero,

00:01:45.240 --> 00:01:47.760
so it's in the domain for sure.

00:01:47.760 --> 00:01:49.310
Now what about x equals three?

00:01:49.310 --> 00:01:51.690
Pause the video and
try to figure that out.

00:01:51.690 --> 00:01:52.950
Well, I'll do that up here.

00:01:52.950 --> 00:01:55.200
f of three is going to be equal to what?

00:01:55.200 --> 00:01:57.210
And you might already
see some warning signs

00:01:57.210 --> 00:01:59.510
as to what's going to happen
here in the denominator,

00:01:59.510 --> 00:02:00.890
but I'll just evaluate the whole thing.

00:02:00.890 --> 00:02:03.260
In the numerator, we get three plus five.

00:02:03.260 --> 00:02:06.670
In the denominator, we
get three minus three.

00:02:06.670 --> 00:02:10.230
So this is going to be
equal to eight over zero.

00:02:10.230 --> 00:02:12.760
Now what is eight divided by zero?

00:02:12.760 --> 00:02:13.880
Well, we don't know.

00:02:13.880 --> 00:02:16.960
This is one of those fascinating
things in mathematics.

00:02:16.960 --> 00:02:18.700
We haven't defined what happens

00:02:18.700 --> 00:02:21.010
when something is divided by zero.

00:02:21.010 --> 00:02:23.900
So three is not in the domain.

00:02:23.900 --> 00:02:27.860
The function is not defined
there, not in domain.

00:02:27.860 --> 00:02:29.203
Let's do another example.

00:02:30.860 --> 00:02:32.870
Determine for each x-value

00:02:32.870 --> 00:02:36.150
whether it is in the domain of g or not.

00:02:36.150 --> 00:02:36.983
So pause this video

00:02:36.983 --> 00:02:39.070
and try to work through
all three of these.

00:02:39.070 --> 00:02:41.790
So first of all, when x
equals negative three,

00:02:41.790 --> 00:02:44.710
do we get a legitimate g of x?

00:02:44.710 --> 00:02:45.570
So let's see.

00:02:45.570 --> 00:02:48.200
g of negative three, if
we try to evaluate this,

00:02:48.200 --> 00:02:49.400
that's going to be the square root

00:02:49.400 --> 00:02:52.520
of three times negative three,

00:02:52.520 --> 00:02:56.250
which is equal to the square
root of negative nine.

00:02:56.250 --> 00:02:59.689
Well, with just a principle
square root like this,

00:02:59.689 --> 00:03:02.130
we don't know how to evaluate this.

00:03:02.130 --> 00:03:05.220
So this is not in the domain.

00:03:05.220 --> 00:03:07.260
What about when x equals zero?

00:03:07.260 --> 00:03:10.760
Well, g of zero is going to be equal to

00:03:10.760 --> 00:03:12.870
the square root of three times zero,

00:03:12.870 --> 00:03:14.900
which is equal to the square root of zero,

00:03:14.900 --> 00:03:15.890
which is equal to zero,

00:03:15.890 --> 00:03:17.910
so that gave us a legitimate result.

00:03:17.910 --> 00:03:20.010
So that is in the domain.

00:03:20.010 --> 00:03:22.800
Now what about g of two, or x equals two?

00:03:22.800 --> 00:03:25.410
Does that give us a legitimate g of two?

00:03:25.410 --> 00:03:28.790
Well, g of two is going to
be equal to the square root

00:03:28.790 --> 00:03:32.330
of three times two, which is
equal to the square root of six

00:03:32.330 --> 00:03:34.280
which is a legitimate output.

00:03:34.280 --> 00:03:38.120
So x equals two is in the domain.

00:03:38.120 --> 00:03:39.933
Let's do one last example.

00:03:41.640 --> 00:03:44.680
So we're told, this h
of x right over here,

00:03:44.680 --> 00:03:46.470
and once again, we have
to figure out whether

00:03:46.470 --> 00:03:49.150
these x-values are in the domain or not.

00:03:49.150 --> 00:03:52.610
Pause this video and see if
you can work through that.

00:03:52.610 --> 00:03:53.920
All right, well, let's
just first think about

00:03:53.920 --> 00:03:55.540
h of negative one.

00:03:55.540 --> 00:03:57.490
What's that going to be equal to?

00:03:57.490 --> 00:03:59.690
Negative one, every place we see an x,

00:03:59.690 --> 00:04:01.860
we're going to replace
it with a negative one,

00:04:01.860 --> 00:04:04.450
minus five, squared.

00:04:04.450 --> 00:04:08.890
Well, this is going to be
equal to negative six squared,

00:04:08.890 --> 00:04:11.990
negative six squared, which
is equal to positive 36,

00:04:11.990 --> 00:04:13.800
which is a very legitimate output,

00:04:13.800 --> 00:04:16.853
and so this is definitely in the domain.

00:04:16.853 --> 00:04:18.430
What about five?

00:04:18.430 --> 00:04:21.320
So h of five is going to be equal to

00:04:21.320 --> 00:04:23.890
five minus five squared.

00:04:23.890 --> 00:04:25.270
Now you might be getting worried

00:04:25.270 --> 00:04:26.410
'cause you're seeing a zero here,

00:04:26.410 --> 00:04:28.240
but it's not like we're
trying to divide by zero.

00:04:28.240 --> 00:04:29.480
We're just squaring zero,

00:04:29.480 --> 00:04:31.190
which is completely legitimate.

00:04:31.190 --> 00:04:32.887
So zero squared is just a zero,

00:04:32.887 --> 00:04:36.410
and so h of five is very much defined.

00:04:36.410 --> 00:04:38.620
So this is in the domain.

00:04:38.620 --> 00:04:39.983
Now what about h of 10?

00:04:41.160 --> 00:04:45.540
Well, h of 10 is going to be
equal to 10 minus five squared,

00:04:45.540 --> 00:04:49.890
which is equal to five
squared, which is equal to 25.

00:04:49.890 --> 00:04:52.210
Once again, it's a very legitimate output.

00:04:52.210 --> 00:04:55.110
So the function is definitely
defined for x equals 10,

00:04:55.110 --> 00:04:55.943
and we're done.


Setting up a system of equations from context example

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.550
- [Instructor] In this
video, we're gonna get

00:00:01.550 --> 00:00:04.790
some more practice setting
up systems of equations.

00:00:04.790 --> 00:00:07.090
Not solving them, but
just setting them up.

00:00:07.090 --> 00:00:09.760
So we're told Sanjay's
dog weighs five times

00:00:09.760 --> 00:00:11.210
as much as his cat.

00:00:11.210 --> 00:00:15.470
His dog is also 20 kilograms
heavier than his cat.

00:00:15.470 --> 00:00:19.320
Let c be the cat's weight and
let d be the dog's weight.

00:00:19.320 --> 00:00:20.920
So pause this video and
see if you can set up

00:00:20.920 --> 00:00:24.040
a system of equation, two linear
equations with two unknowns

00:00:24.040 --> 00:00:27.040
that we could use to solve for c and d,

00:00:27.040 --> 00:00:28.790
but we don't have to in this video.

00:00:29.840 --> 00:00:31.520
All right so let's do it together.

00:00:31.520 --> 00:00:33.780
So, what I like to do is
usually there's a sentence

00:00:33.780 --> 00:00:37.010
or two that describes
each of the equations

00:00:37.010 --> 00:00:37.900
we wanna set up.

00:00:37.900 --> 00:00:42.900
So this first one tells us
Sanjay's dog weighs five times

00:00:43.010 --> 00:00:44.570
as much as his cat.

00:00:44.570 --> 00:00:46.290
So how much does his dog weigh?

00:00:46.290 --> 00:00:50.750
So his dog weighs d, so we
know d is going to be equal

00:00:50.750 --> 00:00:54.280
to five times as much as his cat weighs.

00:00:54.280 --> 00:00:58.200
So his cat weighs c, so d is
going to be equal to five times

00:00:58.200 --> 00:01:00.180
as much as his cat weighs.

00:01:00.180 --> 00:01:03.920
So that's one linear
equation using d and c.

00:01:03.920 --> 00:01:05.590
And so what's another one?

00:01:05.590 --> 00:01:10.340
Well, then we are told his dog
is also 20 kilograms heavier

00:01:10.340 --> 00:01:11.820
than his cat.

00:01:11.820 --> 00:01:16.640
So we could say that the
dog's weight is going

00:01:16.640 --> 00:01:19.500
to be equal to the cat's weight plus what?

00:01:19.500 --> 00:01:21.720
Plus 20 kilograms.

00:01:21.720 --> 00:01:23.190
We're assuming everything's in kilograms,

00:01:23.190 --> 00:01:24.710
so I don't have to write the units.

00:01:24.710 --> 00:01:27.780
But there you have it, I have
just set up two equations

00:01:27.780 --> 00:01:30.000
in two unknowns, two linear equations,

00:01:30.000 --> 00:01:33.520
based on the information
given in this word problem,

00:01:33.520 --> 00:01:35.191
which we could then
solve, and I encourage you

00:01:35.191 --> 00:01:37.460
to do so if you're curious.

00:01:37.460 --> 00:01:40.560
But sometimes, the difficult
part is just to find,

00:01:40.560 --> 00:01:43.430
is to re-express the
information that you're given

00:01:43.430 --> 00:01:44.580
in a mathematical form.

00:01:44.580 --> 00:01:46.240
But as you see, as you get practice,

00:01:46.240 --> 00:01:48.350
it becomes somewhat intuitive.

00:01:48.350 --> 00:01:50.390
What we see in blue is
just another way of writing

00:01:50.390 --> 00:01:52.710
what we underline in blue
and what we see in yellow

00:01:52.710 --> 00:01:55.480
is just another way of
writing or expressing

00:01:55.480 --> 00:01:57.743
wat we underlined in yellow up there.


Examples finding the domain of functions

WEBVTT
Kind: captions
Language: en

00:00:00.240 --> 00:00:01.100
- [Instructor] In this video,

00:00:01.100 --> 00:00:02.620
we're gonna do a few examples

00:00:02.620 --> 00:00:04.790
finding domains of functions.

00:00:04.790 --> 00:00:07.870
So let's say that we
have the function f of x

00:00:07.870 --> 00:00:11.610
is equal to x plus five

00:00:11.610 --> 00:00:14.490
over x minus two.

00:00:14.490 --> 00:00:16.380
What is going to be the domain

00:00:16.380 --> 00:00:17.760
of this function?

00:00:17.760 --> 00:00:20.460
Pause this video and
try to figure that out.

00:00:20.460 --> 00:00:21.410
All right, now let's do it together.

00:00:21.410 --> 00:00:24.530
Now the domain is the set of all x values

00:00:24.530 --> 00:00:26.530
that if we input it into this function,

00:00:26.530 --> 00:00:28.190
we're going to get a legitimate output.

00:00:28.190 --> 00:00:30.230
We're going to get a legitimate f of x.

00:00:30.230 --> 00:00:33.090
And so what's a situation
where we would not

00:00:33.090 --> 00:00:35.120
get a legitimate f of x?

00:00:35.120 --> 00:00:37.160
Well, if we input an x value

00:00:37.160 --> 00:00:39.910
that makes this denominator equal to zero,

00:00:39.910 --> 00:00:41.520
then we're going to divide by zero

00:00:41.520 --> 00:00:43.420
and that's going to be undefined.

00:00:43.420 --> 00:00:45.640
And so we could say that the domain,

00:00:45.640 --> 00:00:50.640
the domain here is all real values

00:00:53.030 --> 00:00:56.880
of x, such that

00:00:57.930 --> 00:01:01.720
x minus two does not equal zero.

00:01:01.720 --> 00:01:03.850
Now typically, people
would not want to just see

00:01:03.850 --> 00:01:06.570
that such that x minus
two does not equal zero,

00:01:06.570 --> 00:01:08.580
and so we can simplify this a little bit

00:01:08.580 --> 00:01:11.060
so that we just have an
x on the left hand side.

00:01:11.060 --> 00:01:13.910
So if we add two to both sides of this,

00:01:13.910 --> 00:01:16.300
we would get, actually,
let me just do that.

00:01:16.300 --> 00:01:18.580
Let me add two to both sides.

00:01:18.580 --> 00:01:20.450
So x minus two not equaling zero

00:01:20.450 --> 00:01:24.520
is the same thing as x not equaling two,

00:01:24.520 --> 00:01:26.100
and you could have done
that in your head, as well.

00:01:26.100 --> 00:01:28.150
If you wanted to keep x
minus two from being zero,

00:01:28.150 --> 00:01:30.090
x just can't be equal to two,

00:01:30.090 --> 00:01:32.550
and so typically, people
would say that the domain here

00:01:32.550 --> 00:01:34.150
is all real values of x

00:01:34.150 --> 00:01:38.370
such that x does not equal two.

00:01:38.370 --> 00:01:40.560
Let's do another example.

00:01:40.560 --> 00:01:43.060
Let's say that we're told that g of x

00:01:43.060 --> 00:01:48.060
is equal to the principle
root of x minus seven.

00:01:49.330 --> 00:01:51.230
What's the domain in this situation?

00:01:51.230 --> 00:01:52.550
What's the domain of g of x?

00:01:52.550 --> 00:01:55.300
Pause the video and
try to figure that out.

00:01:55.300 --> 00:01:57.173
Well, we could say that domain,

00:01:58.160 --> 00:02:02.770
the domain is going to
be all real values of x

00:02:06.110 --> 00:02:07.870
such that, are we going to have

00:02:07.870 --> 00:02:10.160
to put any constraints on this?

00:02:10.160 --> 00:02:14.150
Well when does a principle
root function break down?

00:02:14.150 --> 00:02:16.520
Well if we tried to
find the principle root,

00:02:16.520 --> 00:02:19.200
the square root of a negative number,

00:02:19.200 --> 00:02:21.550
well, that would then break down,

00:02:21.550 --> 00:02:23.120
and so x minus seven,

00:02:23.120 --> 00:02:25.710
whatever we have under the radical here

00:02:25.710 --> 00:02:28.470
needs to be greater than or equal to zero,

00:02:28.470 --> 00:02:31.350
so such that x minus seven

00:02:31.350 --> 00:02:35.030
needs to be greater than or equal to zero.

00:02:35.030 --> 00:02:37.620
Now another way to say
that is if we add seven

00:02:37.620 --> 00:02:39.150
to both sides of that,

00:02:39.150 --> 00:02:42.220
that would be saying that
x needs to be greater than

00:02:42.220 --> 00:02:44.030
or equal to seven,

00:02:44.030 --> 00:02:45.680
so let me just write it that way.

00:02:45.680 --> 00:02:48.760
So such that x is greater than

00:02:48.760 --> 00:02:50.540
or equal to seven.

00:02:50.540 --> 00:02:52.320
So all I did is I said, all right,

00:02:52.320 --> 00:02:53.870
where could this thing break down?

00:02:53.870 --> 00:02:57.230
Well, if I get x values
where this thing is negative,

00:02:57.230 --> 00:02:58.063
we're in trouble,

00:02:58.063 --> 00:03:01.270
so x needs to be greater,

00:03:01.270 --> 00:03:03.830
x minus seven, whatever we have in this,

00:03:03.830 --> 00:03:05.780
under the radical needs to be greater than

00:03:05.780 --> 00:03:07.180
or equal to zero,

00:03:07.180 --> 00:03:09.070
and so if you say that x minus seven

00:03:09.070 --> 00:03:10.540
needs to be greater than or equal to zero,

00:03:10.540 --> 00:03:12.150
you add seven to both sides.

00:03:12.150 --> 00:03:13.690
You get x needs to be greater than

00:03:13.690 --> 00:03:16.110
or equal to positive seven.

00:03:16.110 --> 00:03:18.400
Let's do one last example.

00:03:18.400 --> 00:03:20.660
Let's say we're told that h of x

00:03:20.660 --> 00:03:24.840
is equal to x minus five squared.

00:03:24.840 --> 00:03:26.163
What's the domain here?

00:03:27.270 --> 00:03:29.070
So let me write this down.

00:03:29.070 --> 00:03:34.033
The domain is all real values of x.

00:03:37.430 --> 00:03:40.470
Now are we going to have to
constrain this a little bit?

00:03:40.470 --> 00:03:43.060
Well, is there anything
that would cause this

00:03:43.060 --> 00:03:45.583
to not evaluate to a defined value?

00:03:46.700 --> 00:03:49.140
Well, we can square any value.

00:03:49.140 --> 00:03:51.180
To give me any real
number and if I square it,

00:03:51.180 --> 00:03:52.850
I'm gonna just get another real number,

00:03:52.850 --> 00:03:57.120
and so x minus five can
be equal to anything,

00:03:57.120 --> 00:04:00.010
and so x can be equal to anything.

00:04:00.010 --> 00:04:04.150
So here, the domain is
all real values of x.

00:04:04.150 --> 00:04:06.130
We didn't have to constrain it in any way

00:04:06.130 --> 00:04:06.990
like we did the other two.

00:04:06.990 --> 00:04:09.340
The other two, when
you deal with something

00:04:09.340 --> 00:04:11.230
in the denominator that
could be equal to zero,

00:04:11.230 --> 00:04:12.820
then you've got to make
sure that doesn't happen

00:04:12.820 --> 00:04:15.210
'cause that would get
you an undefined value

00:04:15.210 --> 00:04:18.360
and similarly, for a radical,

00:04:18.360 --> 00:04:20.860
you can't take the
square root of a negative

00:04:20.860 --> 00:04:22.840
and so we would, once again,

00:04:22.840 --> 00:04:24.323
have to constrain on that.


Performing a rotation to match figures

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:03.860
- [Instructor] Use one rotation
to map quadrilateral ABCD

00:00:03.860 --> 00:00:05.260
to the other quadrilateral.

00:00:05.260 --> 00:00:07.580
So to map this one to
this one right over here.

00:00:07.580 --> 00:00:11.070
Use a number between zero
and 360 degrees to describe

00:00:11.070 --> 00:00:11.903
the angle.

00:00:11.903 --> 00:00:13.720
Counter-clockwise is positive.

00:00:13.720 --> 00:00:15.820
So we're going to want to
move is counter-clockwise

00:00:15.820 --> 00:00:17.750
to try and get it to map there.

00:00:17.750 --> 00:00:19.830
And the only option they
give us 'cause they want us

00:00:19.830 --> 00:00:22.220
to do it with one rotation
is the rotation tool.

00:00:22.220 --> 00:00:24.450
And so we have to think about
what do you we want to rotate

00:00:24.450 --> 00:00:28.330
around what point and if
we put it right over here,

00:00:28.330 --> 00:00:32.390
it looks like this point point
A does correspond to this

00:00:32.390 --> 00:00:34.172
point right over here.

00:00:34.172 --> 00:00:36.570
So if we were to rotate this
around, if we were to rotate

00:00:36.570 --> 00:00:40.280
this around not 90 but
it looks like 180 degrees

00:00:40.280 --> 00:00:44.410
around this point, point
A would show up over here,

00:00:44.410 --> 00:00:49.410
it feels like point let's see
is that right is that right

00:00:49.670 --> 00:00:51.940
or let's actually just try it out.

00:00:51.940 --> 00:00:54.930
Point A would show up over,
no no no that's not right,

00:00:54.930 --> 00:00:56.600
that doesn't seem to...

00:00:56.600 --> 00:00:57.950
Let's try it out.

00:00:57.950 --> 00:01:02.650
'Cause if we rotated 180
degrees, oh actually I was right

00:01:02.650 --> 00:01:06.401
it did match up, that's why
this is interesting, it tests

00:01:06.401 --> 00:01:11.400
your visualization skills.

00:01:11.400 --> 00:01:14.060
So it did actually match
up and what I did is I put

00:01:14.060 --> 00:01:16.180
that point of rotation
exactly between those points

00:01:16.180 --> 00:01:19.092
but because it looked like
180 degrees around this point,

00:01:19.092 --> 00:01:23.610
so rotation by 180 degrees
about negative one negative one

00:01:23.610 --> 00:01:27.450
so the center of rotation is
negative one negative one,

00:01:27.450 --> 00:01:30.380
the angle of rotation is 180 degrees.

00:01:30.380 --> 00:01:35.380
Point A maps to this point
right over here, so point A

00:01:35.680 --> 00:01:40.680
maps to the point one negative
one, one negative one.

00:01:42.952 --> 00:01:47.320
And point C point C which is
diagonally opposite point A

00:01:47.320 --> 00:01:50.370
maps to this point right
over here, which is six

00:01:50.370 --> 00:01:55.370
negative six, six negative
six, six negative six

00:01:56.540 --> 00:01:58.003
and we got it right.


Solving equations and inequalities through substitution example 3

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.070
- [Instructor] Joey is training for a

00:00:02.070 --> 00:00:04.060
hot dog eating contest.

00:00:04.060 --> 00:00:05.850
The person who eats the most hot dogs

00:00:05.850 --> 00:00:08.040
in 10 minutes is the winner.

00:00:08.040 --> 00:00:10.970
If r is the number of hot
dogs that Joey can eat

00:00:10.970 --> 00:00:12.170
in a minute.

00:00:12.170 --> 00:00:14.890
And N is the total number
of hot dogs he eats

00:00:14.890 --> 00:00:16.100
in the contest.

00:00:16.100 --> 00:00:17.820
We can write the following equation.

00:00:17.820 --> 00:00:18.810
So this makes sense.

00:00:18.810 --> 00:00:20.830
He eats a total of N hot dogs.

00:00:20.830 --> 00:00:22.410
And the contest is 10 minutes.

00:00:22.410 --> 00:00:23.750
So you divide that by 10.

00:00:23.750 --> 00:00:26.080
So he's gonna eat N hot dog in 10 minutes.

00:00:26.080 --> 00:00:28.170
And so, yes, this will give you the number

00:00:28.170 --> 00:00:31.960
how many hot dogs on
average he ate per minute.

00:00:31.960 --> 00:00:35.290
If Joey can eat six and
four-fifths hot dogs per minute.

00:00:35.290 --> 00:00:38.560
So they're saying, r
is six and four-fifths.

00:00:38.560 --> 00:00:41.250
How many hot dogs does he eat during

00:00:41.250 --> 00:00:42.410
the 10 minute contest?

00:00:42.410 --> 00:00:45.467
So they're saying, "Hey,
what is N going to be,

00:00:45.467 --> 00:00:46.880
"right over here?"

00:00:46.880 --> 00:00:51.050
And so N divided by 10
is six and four-fifths.

00:00:51.050 --> 00:00:53.410
r in this situation is
six and four-fifths.

00:00:53.410 --> 00:00:55.970
That's how many hot dogs
he can eat per minute.

00:00:55.970 --> 00:00:58.670
Now there's a couple of ways
that you could tackle this.

00:00:58.670 --> 00:01:02.290
You could say, "Hey,
some number divided by 10

00:01:02.290 --> 00:01:04.280
is six and four-fifths."

00:01:04.280 --> 00:01:05.957
And so you say, "Well that number must be

00:01:05.957 --> 00:01:08.307
"six and four-fifths times 10.

00:01:08.307 --> 00:01:10.817
"And six and four-fifths times 10 would be

00:01:10.817 --> 00:01:14.030
"60 plus eight, or 68."

00:01:14.030 --> 00:01:16.360
So that's one way you could do it.

00:01:16.360 --> 00:01:18.190
And you're essentially
multiplying both sides

00:01:18.190 --> 00:01:19.680
of this equation by 10.

00:01:19.680 --> 00:01:20.980
The other way you could do it,

00:01:20.980 --> 00:01:22.500
is you could just try out numbers.

00:01:22.500 --> 00:01:23.980
And that usually takes a lot longer.

00:01:23.980 --> 00:01:25.790
But at least it will
give you the sense of how

00:01:25.790 --> 00:01:27.320
the equation relates.

00:01:27.320 --> 00:01:29.230
And, especially 'cause
they have choices here.

00:01:29.230 --> 00:01:31.120
It's not a bad idea.

00:01:31.120 --> 00:01:33.210
Obviously the ideal one is to just

00:01:33.210 --> 00:01:34.380
think logically through it.

00:01:34.380 --> 00:01:37.110
Hey, if something divided by
10 is six and four-fifths.

00:01:37.110 --> 00:01:39.370
Then 10 times six and
four-fifths is going to

00:01:39.370 --> 00:01:40.310
be that something.

00:01:40.310 --> 00:01:41.830
And that's how you can get 68.

00:01:41.830 --> 00:01:44.630
Or, if you want, you could
try out the different numbers.

00:01:44.630 --> 00:01:46.810
Look, 65 divided by 10,
is only gonna give you

00:01:46.810 --> 00:01:48.560
six point five.

00:01:48.560 --> 00:01:51.090
That's not the r we're looking for.

00:01:51.090 --> 00:01:54.180
56 divided 10 is gonna
give you five point six.

00:01:54.180 --> 00:01:55.743
Not the r we're looking for.

00:01:57.796 --> 00:02:01.400
68 divided by 10 is going
to give you six point eight.

00:02:01.400 --> 00:02:03.440
Or six and eight-tenths.

00:02:03.440 --> 00:02:06.030
And six and eight-tenths
is the same thing as

00:02:06.030 --> 00:02:07.430
six and four-fifths.

00:02:07.430 --> 00:02:08.970
So 68 works.

00:02:08.970 --> 00:02:11.620
45 hot dogs, you divide that by 10.

00:02:11.620 --> 00:02:14.120
You're gonna get four point
five hot dogs per minute.

00:02:14.120 --> 00:02:15.930
So all of these are completely legitimate

00:02:15.930 --> 00:02:16.763
ways of doing it.

00:02:16.763 --> 00:02:18.940
The important thing that you
understand this relationship.

00:02:18.940 --> 00:02:20.820
And you understand how
these variables work

00:02:20.820 --> 00:02:22.170
with respect to each other.


Visually determining vertical asymptotes

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:01.850
- [Instructor] Given the graph

00:00:01.850 --> 00:00:04.670
of y equals f of x pictured below,

00:00:04.670 --> 00:00:08.010
determine the equations of
all vertical asymptotes.

00:00:08.010 --> 00:00:09.960
Let's see what's going on here.

00:00:09.960 --> 00:00:12.550
So it looks like interesting
things are happening

00:00:12.550 --> 00:00:15.040
at x equals negative
four and x equals two.

00:00:15.040 --> 00:00:17.100
At x equals negative
four, as we approach it

00:00:17.100 --> 00:00:19.590
from the left the value of the function

00:00:19.590 --> 00:00:22.010
just becomes unbounded right over here.

00:00:22.010 --> 00:00:25.280
Looks like as we approach
x equals negative four

00:00:25.280 --> 00:00:29.310
from the left, the value of
our function goes to infinity.

00:00:29.310 --> 00:00:32.270
Likewise, as we approach
x equals negative four

00:00:32.270 --> 00:00:34.440
from the right, it looks like our,

00:00:34.440 --> 00:00:37.240
the value of our function
goes to infinity.

00:00:37.240 --> 00:00:40.160
So, I'd say that we definitely
have a vertical asymptote

00:00:40.160 --> 00:00:43.410
at x equals negative four.

00:00:43.410 --> 00:00:45.420
Now, let's look at x equals two.

00:00:45.420 --> 00:00:49.220
As we approach x equals two from the left,

00:00:49.220 --> 00:00:52.110
the value of our function once
again approaches infinity,

00:00:52.110 --> 00:00:54.610
or it becomes unbounded.

00:00:54.610 --> 00:00:56.850
Now, from the right, we
have an interesting thing.

00:00:56.850 --> 00:01:00.320
If we look at the limit from
the right right over here,

00:01:00.320 --> 00:01:02.740
it looks like we're
approaching a finite value.

00:01:02.740 --> 00:01:05.210
As we approach x equals
two from the right,

00:01:05.210 --> 00:01:07.560
it looks like we're approaching f

00:01:07.560 --> 00:01:09.940
of x is equal to negative four.

00:01:09.940 --> 00:01:13.820
But, just having a one-sided
limit that is unbounded,

00:01:13.820 --> 00:01:18.010
is enough to think about
this as a vertical asymptote.

00:01:18.010 --> 00:01:20.710
The function is not
defined right over here,

00:01:20.710 --> 00:01:23.470
and as we approach it from just one side,

00:01:23.470 --> 00:01:25.110
we are becoming unbounded.

00:01:25.110 --> 00:01:26.610
It looks like we're approaching infinity

00:01:26.610 --> 00:01:29.370
or negative infinity, so that by itself,

00:01:29.370 --> 00:01:34.040
this unbounded left hand limit,
or left-side limit by itself

00:01:34.040 --> 00:01:37.630
is enough to consider x equals
two a vertical asymptote.

00:01:37.630 --> 00:01:39.480
So, we can say that there's
a vertical asymptote

00:01:39.480 --> 00:01:43.843
at x equals negative
four and x equals two.


Cybersecurity and crime

WEBVTT
Kind: captions
Language: en

00:00:07.961 --> 00:00:09.567
- Hi. My name's Jenny Martin

00:00:09.567 --> 00:00:11.195
and I'm the Director of Cybersecurity

00:00:11.195 --> 00:00:13.633
Investigations at Symantec.

00:00:13.633 --> 00:00:16.211
Today cybercrime causes huge problems

00:00:16.211 --> 00:00:17.381
for society.

00:00:17.381 --> 00:00:19.804
Personally, financially, and even

00:00:19.804 --> 00:00:22.178
in matters of national security.

00:00:22.178 --> 00:00:23.857
Just in the last few years,

00:00:23.857 --> 00:00:25.754
hundreds of millions
of credit card numbers

00:00:25.754 --> 00:00:27.031
have been stolen.

00:00:27.031 --> 00:00:29.213
Tens of millions of
social security numbers

00:00:29.213 --> 00:00:31.881
and health care records were compromised.

00:00:31.881 --> 00:00:34.769
Even nuclear centrifuges have been hacked

00:00:34.769 --> 00:00:37.920
and unmanned aerial
drones have been hijacked.

00:00:37.920 --> 00:00:40.646
This is all done by
exploiting vulnerabilities

00:00:40.646 --> 00:00:42.588
in hardware and software.

00:00:42.588 --> 00:00:44.422
Or more often by taking advantage of

00:00:44.422 --> 00:00:46.565
unintentional decisions made by the people

00:00:46.565 --> 00:00:48.649
using the software.

00:00:51.764 --> 00:00:53.766
The people committing these cybercrimes

00:00:53.766 --> 00:00:56.542
don't fit a single profile or motivation.

00:00:56.542 --> 00:00:59.236
It could be anyone from an
international terrorist,

00:00:59.236 --> 00:01:02.254
to a teenager competing
for bragging rights.

00:01:02.524 --> 00:01:04.495
Today, the largest countries not only

00:01:04.495 --> 00:01:06.469
have a regular army, but also have

00:01:06.469 --> 00:01:08.918
a well-armed cyber army.

00:01:08.918 --> 00:01:10.676
In fact, the next world war may not

00:01:10.676 --> 00:01:12.248
be fought with traditional weapons

00:01:12.248 --> 00:01:14.096
but with computers, used to shut down

00:01:14.096 --> 00:01:17.141
national water supplies, energy grids,

00:01:17.141 --> 00:01:20.011
and transportation systems.

00:01:20.770 --> 00:01:22.100
- Hi. My name is Parisa

00:01:22.100 --> 00:01:24.684
and I'm Google's Security Princess.

00:01:24.684 --> 00:01:27.247
I've worked on a lot of
different Google products

00:01:27.247 --> 00:01:31.044
in a lot of different ways
to try and make our software

00:01:31.044 --> 00:01:33.427
as secure as possible.

00:01:33.758 --> 00:01:35.406
Now let's take a look at how cybercrime

00:01:35.406 --> 00:01:37.055
works under the hood.

00:01:37.055 --> 00:01:39.121
We'll learn about software viruses,

00:01:39.121 --> 00:01:42.790
denial-of-service attacks,
and phishing scams.

00:01:42.790 --> 00:01:45.328
In biology in life, a virus is an organism

00:01:45.328 --> 00:01:47.374
that is spread by coughing, sneezing,

00:01:47.374 --> 00:01:49.076
or physical contact.

00:01:49.076 --> 00:01:51.298
Viruses work by infecting cells,

00:01:51.298 --> 00:01:53.192
injecting their genetic material

00:01:53.192 --> 00:01:55.142
and using those cells to replicate.

00:01:55.142 --> 00:01:56.725
They can make people really sick

00:01:56.725 --> 00:01:59.038
and then spread to other people.

00:01:59.159 --> 00:02:02.085
A computer virus works a bit similarly.

00:02:02.085 --> 00:02:03.966
A virus is an executable program

00:02:03.966 --> 00:02:07.064
that gets installed,
usually unintentionally,

00:02:07.064 --> 00:02:09.678
and harms a user and their computer.

00:02:09.678 --> 00:02:12.050
It's also possible for
a virus to spread itself

00:02:12.050 --> 00:02:14.177
to other computers.

00:02:14.977 --> 00:02:16.527
Now how does a virus get on your computer

00:02:16.527 --> 00:02:17.970
in the first place?

00:02:17.970 --> 00:02:19.197
There are couple ways an attacker

00:02:19.197 --> 00:02:21.221
can infect someone's computer.

00:02:21.221 --> 00:02:23.559
They might lure a victim
into installing a program

00:02:23.559 --> 00:02:26.256
with deception about
the program's purpose.

00:02:26.256 --> 00:02:28.725
So for example, a lot
of viruses are disguised

00:02:28.725 --> 00:02:30.894
as security updates.

00:02:31.050 --> 00:02:33.269
It's also possible that the
software on your computer

00:02:33.269 --> 00:02:34.588
has a vulnerability.

00:02:34.588 --> 00:02:36.542
So an attacker can install itself

00:02:36.542 --> 00:02:39.509
without even needing explicit permission.

00:02:39.623 --> 00:02:41.604
Once a virus in on your computer

00:02:41.604 --> 00:02:44.227
it can steal or delete any of your files,

00:02:44.227 --> 00:02:46.961
control other programs,
or even allow someone else

00:02:46.961 --> 00:02:50.305
to remotely control your computer.

00:02:52.119 --> 00:02:54.424
Using computer viruses,
hackers can take over

00:02:54.424 --> 00:02:56.557
millions of computers worldwide.

00:02:56.557 --> 00:02:58.368
And then use them as a digital army,

00:02:58.368 --> 00:03:00.033
otherwise known as a botnet,

00:03:00.033 --> 00:03:03.028
to attack and take down websites.

00:03:03.028 --> 00:03:04.428
This kind of attack is called

00:03:04.428 --> 00:03:07.140
a distributed denial-of-service.

00:03:07.140 --> 00:03:08.962
A denial-of-service is when hackers

00:03:08.962 --> 00:03:11.668
overwhelm a website
with too many requests.

00:03:11.668 --> 00:03:13.665
We call it a distributed denial-of-service

00:03:13.665 --> 00:03:15.509
when the attack comes from many computers

00:03:15.509 --> 00:03:17.343
all at once.

00:03:18.308 --> 00:03:19.875
Most websites are ready to respond

00:03:19.875 --> 00:03:21.582
to millions of requests a day,

00:03:21.582 --> 00:03:23.716
but if you hit them with
billions or trillions

00:03:23.716 --> 00:03:26.067
of requests coming from different places,

00:03:26.067 --> 00:03:29.379
the computers are overloaded
and stop responding.

00:03:30.472 --> 00:03:32.346
- Another trick used by cybercriminals

00:03:32.346 --> 00:03:34.724
is to send large amounts of spam email

00:03:34.724 --> 00:03:36.419
in an attempt to trick people

00:03:36.419 --> 00:03:39.624
into sharing sensitive
personal information.

00:03:39.624 --> 00:03:42.526
This is called a phishing scam.

00:03:42.526 --> 00:03:44.029
A phishing scam is when you get

00:03:44.029 --> 00:03:46.380
what seems like a trustworthy email

00:03:46.380 --> 00:03:48.633
asking you to login to your account,

00:03:48.633 --> 00:03:51.966
but clicking the email
takes you to a fake website.

00:03:51.966 --> 00:03:54.328
If you login anyway, you've been tricked

00:03:54.328 --> 00:03:56.772
into giving your password away.

00:03:56.772 --> 00:03:58.826
Hackers can then use
your login credentials

00:03:58.826 --> 00:04:00.614
to access your real accounts

00:04:00.614 --> 00:04:02.764
to steal information, or maybe even

00:04:02.764 --> 00:04:04.799
to steal your money.

00:04:04.799 --> 00:04:06.534
Fortunately there are many companies,

00:04:06.534 --> 00:04:09.158
laws, and government organizations

00:04:09.158 --> 00:04:11.341
working to make the internet safer.

00:04:11.341 --> 00:04:14.214
But these efforts are not enough.

00:04:15.109 --> 00:04:17.285
You may think when a
computer system gets hacked,

00:04:17.285 --> 00:04:19.212
the problem was the security design,

00:04:19.212 --> 00:04:20.930
or the software.

00:04:20.930 --> 00:04:23.368
90% of the time a system
gets hacked however,

00:04:23.368 --> 00:04:25.103
it's not because of a security bug,

00:04:25.103 --> 00:04:29.144
but because of a simple
mistake made by a human.

00:04:31.379 --> 00:04:33.197
- It turns out there are
steps we can all take

00:04:33.197 --> 00:04:34.629
to protect ourselves.

00:04:34.629 --> 00:04:36.508
Often, your actions not only impact

00:04:36.508 --> 00:04:38.819
the security of your
own data and computer,

00:04:38.819 --> 00:04:40.465
but the security of everyone

00:04:40.465 --> 00:04:43.406
at your school, workplace, and home.

00:04:43.526 --> 00:04:46.435
With billions or trillions
of dollars at stake,

00:04:46.435 --> 00:04:48.468
cybercriminals get smarter each year,

00:04:48.468 --> 00:04:51.198
and we all need to keep up.


IP addresses and DNS

WEBVTT
Kind: captions
Language: en

00:00:00.901 --> 00:00:04.498
(upbeat piano and drum music)

00:00:10.024 --> 00:00:11.551
- Hi. My name is Paola,

00:00:11.551 --> 00:00:14.501
and I am a software engineer at Microsoft.

00:00:14.501 --> 00:00:17.077
Let's talk about how the internet works.

00:00:17.077 --> 00:00:19.933
My job relies on networks being able

00:00:19.933 --> 00:00:21.930
to talk with one another.

00:00:21.930 --> 00:00:24.774
But back in the 1970s, there was no

00:00:24.774 --> 00:00:26.956
standard method for this.

00:00:26.956 --> 00:00:29.821
It took the work of Vint Cerf and Bob Kahn

00:00:29.821 --> 00:00:32.126
to invent the Internetworking Protocol

00:00:32.126 --> 00:00:34.308
to make communication possible.

00:00:34.638 --> 00:00:36.570
This invention laid the groundwork

00:00:36.570 --> 00:00:38.975
for what we now call the internet.

00:00:39.566 --> 00:00:42.045
- The internet is a network of networks.

00:00:42.045 --> 00:00:45.503
It links billions of devices
together all around the globe.

00:00:46.723 --> 00:00:48.706
Maybe you're connected with a laptop

00:00:48.706 --> 00:00:50.903
or a phone through wifi.

00:00:50.903 --> 00:00:52.525
But then that wifi connection

00:00:52.525 --> 00:00:55.716
connects to an Internet
Service Provider, or ISP.

00:00:55.716 --> 00:00:58.363
And that ISP connects you
to billions and billions

00:00:58.363 --> 00:01:00.196
of devices around the world

00:01:00.196 --> 00:01:02.123
through hundreds of thousands of networks

00:01:02.123 --> 00:01:04.045
that are all interconnected.

00:01:05.465 --> 00:01:08.682
One thing that most
people do not appreciate,

00:01:08.682 --> 00:01:12.142
is that the internet is
really a design philosophy

00:01:12.142 --> 00:01:16.108
and an architecture expressed
in a set of protocols.

00:01:16.598 --> 00:01:19.629
A protocol is a well-known set
of rules and standards that,

00:01:19.629 --> 00:01:21.417
if all parties agree to use it,

00:01:21.417 --> 00:01:23.781
will allow them to
communicate without trouble.

00:01:24.927 --> 00:01:28.799
How the internet actually
physically works is less important

00:01:28.799 --> 00:01:31.761
than the fact that this
design philosophy has allowed

00:01:31.761 --> 00:01:33.920
the internet to adapt and absorb

00:01:33.920 --> 00:01:36.239
new communication technologies.

00:01:36.239 --> 00:01:38.464
This is because in order
for a new technology

00:01:38.464 --> 00:01:40.650
to use the internet in some fashion,

00:01:40.650 --> 00:01:43.455
it just needs to know which
protocols to work with.

00:01:44.763 --> 00:01:46.782
All the different devices on the internet

00:01:46.782 --> 00:01:48.489
have unique addresses.

00:01:48.489 --> 00:01:50.865
An address on the
internet is just a number,

00:01:50.865 --> 00:01:54.082
similar to a phone number,
or a sort of street address

00:01:54.082 --> 00:01:56.248
that's unique to each computer or device

00:01:56.248 --> 00:01:57.682
at the edge of the network.

00:01:59.314 --> 00:02:01.755
This is similar to how most
homes and businesses have

00:02:01.755 --> 00:02:03.099
a mailing address.

00:02:03.099 --> 00:02:04.499
You don't need to know a person

00:02:04.499 --> 00:02:05.829
to send them a letter in the mail,

00:02:05.829 --> 00:02:08.344
but you do need to know their address,

00:02:08.344 --> 00:02:10.141
and how to write the address properly,

00:02:10.141 --> 00:02:12.335
so the letter can be
carried by the mail system

00:02:12.335 --> 00:02:13.886
to its destination.

00:02:14.246 --> 00:02:15.854
The addressing system for computers

00:02:15.854 --> 00:02:17.724
on the internet is similar,

00:02:18.114 --> 00:02:20.645
and it forms part of one
of the most important

00:02:20.645 --> 00:02:23.354
protocols used in internet communication,

00:02:23.354 --> 00:02:26.235
simply called the
Internet Protocol, or IP.

00:02:26.958 --> 00:02:30.072
A computer's address, then,
is called its IP address.

00:02:30.514 --> 00:02:33.222
Visiting a website is really
just your computer asking

00:02:33.222 --> 00:02:35.299
another computer for information.

00:02:35.299 --> 00:02:37.627
Your computer sends a message
to the other computer's

00:02:37.627 --> 00:02:41.011
IP address, and it also sends
along its origin address

00:02:41.011 --> 00:02:43.673
so the other computer knows
where to send its response.

00:02:45.751 --> 00:02:47.895
- You may have seen an IP address.

00:02:47.895 --> 00:02:49.743
It's just a bunch of numbers.

00:02:50.043 --> 00:02:52.795
These numbers are
organized in a hierarchy.

00:02:53.425 --> 00:02:55.685
Just like a home address has a country,

00:02:55.685 --> 00:02:58.635
a city, a street, and a house number,

00:02:58.635 --> 00:03:01.242
an IP address has many parts.

00:03:01.472 --> 00:03:03.250
Just like all digital data,

00:03:03.250 --> 00:03:06.250
each of these numbers
is represented in bits.

00:03:06.655 --> 00:03:09.985
Traditional IP addresses are 32 bits long,

00:03:09.985 --> 00:03:13.540
with eight bits for each
part of the address.

00:03:13.540 --> 00:03:15.947
The earlier numbers usually identify

00:03:15.947 --> 00:03:18.653
the country and regional
network of the device.

00:03:18.653 --> 00:03:20.570
Then come the subnetworks.

00:03:20.570 --> 00:03:23.925
And then, finally, the address
of the specific device.

00:03:24.435 --> 00:03:28.485
This version of IP
addressing is called IPv4.

00:03:28.485 --> 00:03:30.968
It was designed in 1973,

00:03:30.968 --> 00:03:33.386
and widely adopted in the early '80s,

00:03:33.386 --> 00:03:36.729
and provides for more than
4 billion unique addresses

00:03:36.729 --> 00:03:39.527
for devices connecting to the internet.

00:03:39.797 --> 00:03:43.170
But the internet has turned
out to be much more popular

00:03:43.170 --> 00:03:45.339
than even Vint Cerf imagined,

00:03:45.339 --> 00:03:48.677
and 4 billion unique
addresses won't be enough.

00:03:48.677 --> 00:03:51.461
We're now in the middle
of a multi-year transition

00:03:51.461 --> 00:03:55.793
to a longer IP address format called IPv6,

00:03:55.793 --> 00:03:59.545
which uses 128 bits per address,

00:03:59.545 --> 00:04:04.545
and provides over 340
undecillion unique addresses.

00:04:05.130 --> 00:04:08.009
That's more than enough for
every grain of sand on Earth

00:04:08.009 --> 00:04:09.990
to have its own IP address.

00:04:10.805 --> 00:04:14.869
- Most users never see or
care about internet addresses.

00:04:14.869 --> 00:04:18.160
A system called the Domain
Name System, or DNS,

00:04:18.160 --> 00:04:22.065
associates names, like www.example.com,

00:04:22.065 --> 00:04:24.050
with the corresponding addresses.

00:04:24.050 --> 00:04:27.185
Your computer uses the DNS
to look up domain names

00:04:27.185 --> 00:04:30.206
and get the associated
IP address, which is used

00:04:30.206 --> 00:04:33.099
to connect your computer to the
destination on the internet.

00:04:33.099 --> 00:04:34.999
- [Voiceover] And it goes a
little something like this.

00:04:34.999 --> 00:04:35.619
- Hey, hi there.

00:04:35.619 --> 00:04:39.450
I want to go to www.code.org

00:04:41.313 --> 00:04:45.011
- Yeah, well I don't know the
IP address for that domain.

00:04:45.011 --> 00:04:46.729
Let me ask around.

00:04:46.859 --> 00:04:49.836
Hey, anyone know how to get to a...

00:04:50.396 --> 00:04:52.022
code.org?

00:04:53.292 --> 00:04:54.347
- Yeah, I got it right here.

00:04:54.347 --> 00:04:57.270
It's 174 dot 129

00:04:57.270 --> 00:05:00.184
dot 14 dot 120.

00:05:00.514 --> 00:05:01.842
- Oh, okay, great. Thanks.

00:05:01.842 --> 00:05:04.096
Yeah, I'm gonna write that
down and save it for later

00:05:04.096 --> 00:05:05.078
in case I need it.

00:05:05.078 --> 00:05:06.758
Hey, here's that address you wanted.

00:05:07.048 --> 00:05:08.574
- Awesome. Thank you.

00:05:10.527 --> 00:05:13.792
(piano flourish)

00:05:13.792 --> 00:05:17.359
- So how do we design a
system for billions of devices

00:05:17.359 --> 00:05:21.108
to find any one of billions
of different websites?

00:05:21.388 --> 00:05:24.097
There is no way one DNS server

00:05:24.097 --> 00:05:27.413
can handle all of the
requests from all devices.

00:05:27.633 --> 00:05:30.513
The answer is that DNS
servers are connected

00:05:30.513 --> 00:05:33.816
in a distributed hierarchy,
and are divided into zones,

00:05:33.822 --> 00:05:36.915
splitting up responsibility
for the major domains

00:05:36.915 --> 00:05:40.899
such as .org, .com, .net, et cetera.

00:05:41.829 --> 00:05:44.129
DNS was originally created to be

00:05:44.129 --> 00:05:47.116
an open and public communication protocol

00:05:47.116 --> 00:05:49.871
for government and
educational institutions.

00:05:49.871 --> 00:05:51.589
Because of its openness,

00:05:51.589 --> 00:05:54.706
DNS is susceptible to cyber attacks.

00:05:55.247 --> 00:05:58.399
An example attack is DNS spoofing.

00:05:58.399 --> 00:06:01.724
That's when a hacker
taps into a DNS server

00:06:01.724 --> 00:06:04.450
and changes it to match a domain name

00:06:04.450 --> 00:06:06.725
with the wrong IP address.

00:06:06.985 --> 00:06:11.588
This lets the attacker send
people to one imposter website.

00:06:11.588 --> 00:06:13.364
If this happens to you,

00:06:13.364 --> 00:06:15.694
you are vulnerable for more problems

00:06:15.694 --> 00:06:19.783
because you are using that
fake website as if it is real.

00:06:21.443 --> 00:06:25.597
The internet is huge, and
getting bigger every day.

00:06:25.597 --> 00:06:28.599
But the Domain Name System
and Internet Protocol

00:06:28.599 --> 00:06:30.485
are designed to scale,

00:06:30.485 --> 00:06:32.885
no matter how much the internet grows.


Encryption and public keys

WEBVTT
Kind: captions
Language: en

00:00:02.059 --> 00:00:06.859
(energetic music)

00:00:08.948 --> 00:00:10.826
- Hi. My name is Mia Gil Epner.

00:00:10.826 --> 00:00:13.466
I'm majoring in Computer
Science at UC Berkeley

00:00:13.466 --> 00:00:15.391
and I work for the Department of Defense

00:00:15.391 --> 00:00:17.655
where I try to keep information safe.

00:00:18.515 --> 00:00:21.701
The internet is an open and public system.

00:00:21.701 --> 00:00:24.007
We all send and receive information

00:00:24.007 --> 00:00:26.118
over shared wires and connections.

00:00:26.728 --> 00:00:28.327
Even though it's an open system,

00:00:28.327 --> 00:00:30.485
we still exchange a lot of private data,

00:00:30.485 --> 00:00:32.319
things like credit card numbers,

00:00:32.319 --> 00:00:35.384
bank information, passwords, and emails.

00:00:35.394 --> 00:00:38.707
So how is all this
private stuff kept secret?

00:00:38.747 --> 00:00:40.673
Data of any kind can be kept secret

00:00:40.673 --> 00:00:42.367
through a process known as encryption,

00:00:42.367 --> 00:00:44.410
descrambling or changing of the message

00:00:44.410 --> 00:00:46.410
to hide the original text.

00:00:46.410 --> 00:00:48.461
Now, decryption is the process

00:00:48.461 --> 00:00:51.710
of unscrambling that
message to make it readable.

00:00:51.710 --> 00:00:53.066
This is a simple idea,

00:00:53.066 --> 00:00:55.479
and people have been
doing it for centuries.

00:00:55.479 --> 00:00:58.256
One of the first well-known
methods of encryption

00:00:58.256 --> 00:01:01.447
was Caesar's cipher,
named after Julius Caesar,

00:01:01.447 --> 00:01:04.547
a Roman general who encrypted
his military commands

00:01:04.547 --> 00:01:07.254
to make sure that if a message
was intercepted by enemies,

00:01:07.254 --> 00:01:09.985
they wouldn't be able to read it.

00:01:10.045 --> 00:01:11.705
Caesar's cipher is an algorithm

00:01:11.705 --> 00:01:14.121
that substitutes each letter
in the original message

00:01:14.121 --> 00:01:17.024
with a letter a certain number
of steps down the alphabet.

00:01:17.024 --> 00:01:19.275
If the number is something only the sender

00:01:19.275 --> 00:01:22.368
and receiver know, then
it's called the key.

00:01:22.658 --> 00:01:26.776
It allows the reader to
unlock the secret message.

00:01:27.186 --> 00:01:30.424
For example, if your
original message is, "Hello",

00:01:30.524 --> 00:01:33.109
then, using the Caesar's cipher algorithm

00:01:33.109 --> 00:01:36.734
with a key of five, the
encrypted message would be this.

00:01:36.734 --> 00:01:38.464
(typrwriter keys clacking)

00:01:38.464 --> 00:01:39.614
(computer chime)

00:01:39.614 --> 00:01:41.486
To decrypt the message, the recipient

00:01:41.486 --> 00:01:45.880
would simply use the key
to reverse the process.

00:01:45.940 --> 00:01:48.116
But there's a big problem
with Caesar's cipher.

00:01:48.116 --> 00:01:52.261
Anybody can easily break or
crack the encrypted message

00:01:52.261 --> 00:01:54.234
by trying every possible key.

00:01:54.234 --> 00:01:57.229
In the English alphabet,
there are only 26 letters,

00:01:57.229 --> 00:01:59.877
which means you'd only
need to try, at most,

00:01:59.877 --> 00:02:02.775
26 keys to decrypt the message.

00:02:02.815 --> 00:02:05.904
Now, trying 26 possible
keys isn't very hard.

00:02:05.904 --> 00:02:07.993
It would take, at most, an hour to do.

00:02:07.993 --> 00:02:09.943
So let's make it harder.

00:02:09.943 --> 00:02:12.312
Instead of shifting every
letter by the same amount,

00:02:12.312 --> 00:02:15.284
let's shift each letter
by a different amount.

00:02:15.284 --> 00:02:17.977
In this example, a 10 digit key

00:02:17.977 --> 00:02:20.624
shows how many positions
each successive letter

00:02:20.624 --> 00:02:23.468
will be changed to
encrypt a longer message.

00:02:23.468 --> 00:02:26.135
(typewriter keys clacking)

00:02:26.135 --> 00:02:28.573
Guessing this key would be really hard.

00:02:28.593 --> 00:02:30.029
Using 10 digit encryption,

00:02:30.029 --> 00:02:33.973
there could be 10 billion
possible key solutions.

00:02:34.043 --> 00:02:37.027
Obviously, that's more than
any human could ever solve.

00:02:37.027 --> 00:02:41.161
It would take many centuries,
but an average computer today

00:02:41.161 --> 00:02:42.739
would take just a few seconds

00:02:42.739 --> 00:02:45.858
to try all 10 billion possibilities.

00:02:45.868 --> 00:02:48.116
So in a modern world, where the bad guys

00:02:48.116 --> 00:02:50.732
are armed with computers
instead of pencils,

00:02:50.732 --> 00:02:53.739
how can you encrypt messages so securely

00:02:53.739 --> 00:02:56.084
that they're too hard to crack?

00:02:56.084 --> 00:02:59.729
Now, "too hard" means that
there are too many possibilities

00:02:59.729 --> 00:03:02.756
to compute in a reasonable amount of time.

00:03:02.756 --> 00:03:04.832
Today's secure communications

00:03:04.832 --> 00:03:08.328
are encrypted using 256 bit keys.

00:03:08.328 --> 00:03:09.929
That means a bad guy's computer

00:03:09.929 --> 00:03:11.426
that intercepts your message,

00:03:11.426 --> 00:03:14.596
would need to try this
many possible options

00:03:14.596 --> 00:03:17.813
until they discover the
key and crack the message.

00:03:17.813 --> 00:03:18.408
(robot bleeps and beeps)

00:03:18.408 --> 00:03:20.118
(energetic music)

00:03:20.118 --> 00:03:23.154
Even if you had a hundred
thousand super computers,

00:03:23.154 --> 00:03:25.056
and each of them was able to try

00:03:25.056 --> 00:03:27.900
a million billion keys every second,

00:03:27.900 --> 00:03:30.733
it would take trillions
of trillions of trillions

00:03:30.733 --> 00:03:32.602
of years to try every option,

00:03:32.602 --> 00:03:34.309
just to crack a single message

00:03:34.309 --> 00:03:37.651
protected with 256 bit encryption.

00:03:37.651 --> 00:03:40.197
Of course, computer
chips get twice as fast,

00:03:40.197 --> 00:03:42.623
then half the size every year or so.

00:03:42.623 --> 00:03:45.410
If that pace of exponential
progress continues,

00:03:45.410 --> 00:03:48.242
today's impossible
problems will be solvable

00:03:48.242 --> 00:03:50.702
just a few hundred years in the future,

00:03:50.702 --> 00:03:53.983
and 256 bits won't be enough to be safe.

00:03:53.983 --> 00:03:55.887
In fact, we've already had to

00:03:55.887 --> 00:03:57.421
increase the standard key length

00:03:57.421 --> 00:04:00.966
to keep up with the speed of computers.

00:04:00.966 --> 00:04:03.167
The good news is, using a longer key

00:04:03.167 --> 00:04:05.315
doesn't make encrypting
messages much harder,

00:04:05.315 --> 00:04:08.484
but it exponentially increases
the number of guesses

00:04:08.484 --> 00:04:11.388
that it would to crack a cipher.

00:04:11.388 --> 00:04:14.393
When the sender and the
receiver share the same key

00:04:14.393 --> 00:04:16.251
to scramble and unscramble a message,

00:04:16.251 --> 00:04:19.129
it's called symmetric encryption.

00:04:19.149 --> 00:04:21.954
With symmetric encryption,
like Caesar's cipher,

00:04:21.954 --> 00:04:25.241
the secret key has to be
agreed on ahead of time

00:04:25.241 --> 00:04:27.569
by two people in private.

00:04:27.569 --> 00:04:28.995
That's great for people,

00:04:28.995 --> 00:04:31.304
but the internet is open and public,

00:04:31.304 --> 00:04:33.394
so it's impossible for two computers

00:04:33.394 --> 00:04:36.890
to meet in private to
agree on a secret key.

00:04:36.890 --> 00:04:40.047
Instead, computers use asymmetric keys,

00:04:40.047 --> 00:04:43.651
a public key that can be
exchanged with anybody

00:04:43.651 --> 00:04:46.623
and a private key that is not shared.

00:04:46.623 --> 00:04:49.593
The public key is used to encrypt data

00:04:49.593 --> 00:04:52.438
and anybody can use it to
create a secret message,

00:04:52.438 --> 00:04:55.606
but the secret can only be decrypted

00:04:55.606 --> 00:04:59.285
by a computer with access
to the private key.

00:04:59.335 --> 00:05:00.962
How it works is with some math

00:05:00.962 --> 00:05:02.813
that we won't get into right now.

00:05:02.863 --> 00:05:04.578
Think of it this way, imagine

00:05:04.578 --> 00:05:06.157
that you have a personal mailbox

00:05:06.157 --> 00:05:07.650
where anybody can deposit mail,

00:05:07.650 --> 00:05:09.554
but they need a key to do it.

00:05:09.554 --> 00:05:11.644
Now, you could make many copies

00:05:11.644 --> 00:05:13.838
of the deposit key, and
send one to your friend

00:05:13.838 --> 00:05:16.044
or even just make it publicly available.

00:05:16.044 --> 00:05:18.052
Your friend, or even a stranger,

00:05:18.052 --> 00:05:19.445
can use the public key

00:05:19.445 --> 00:05:22.139
to access your deposit
slot and drop a message in,

00:05:22.139 --> 00:05:25.924
but only you can open the
mailbox with your private key

00:05:25.924 --> 00:05:28.907
to access all of the secret
messages you've received.

00:05:28.907 --> 00:05:31.716
You can send a secure
message back to your friend

00:05:31.716 --> 00:05:34.525
by using the public deposit
key to their mailbox.

00:05:34.525 --> 00:05:37.533
This way, people can
exchange secure messages

00:05:37.533 --> 00:05:41.070
without ever needing to
agree on a private key.

00:05:41.100 --> 00:05:43.653
Public key cryptography is the foundation

00:05:43.653 --> 00:05:46.580
of all secure messaging
on the open internet

00:05:46.580 --> 00:05:51.131
including security protocols
known as SSL and TLS

00:05:51.131 --> 00:05:54.314
which protect us when
we're browsing the web.

00:05:54.324 --> 00:05:56.302
Your computer uses this today.

00:05:56.302 --> 00:05:57.960
Any time you see the little lock

00:05:57.960 --> 00:06:02.128
or the letters https in
your browser's address bar,

00:06:02.128 --> 00:06:05.599
this means your computer is
using public key encryption

00:06:05.599 --> 00:06:08.803
to exchange data securely
with the website you're on.

00:06:08.803 --> 00:06:10.183
(energetic music)

00:06:10.183 --> 00:06:12.241
As more and more people
get on the internet,

00:06:12.241 --> 00:06:14.806
more and more private
data will be transmitted,

00:06:14.806 --> 00:06:16.930
and the need to secure that data

00:06:16.930 --> 00:06:18.716
will be even more important.

00:06:18.756 --> 00:06:21.393
As computers become faster and faster,

00:06:21.393 --> 00:06:23.239
we'll have to develop new ways

00:06:23.239 --> 00:06:26.060
to make encryption too hard
for computers to break.

00:06:26.070 --> 00:06:29.470
This is what I do with my
work, and it's always changing.

00:06:29.470 --> 00:06:34.102
(energetic music)


Packet, routers, and reliability

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.815
- [Voiceover] Seven, six,
five, four, three, two, one.

00:00:02.815 --> 00:00:05.746
(funky music)

00:00:08.457 --> 00:00:10.906
- Hi, my name is Lynn Root.
I am the Software Engineer

00:00:10.906 --> 00:00:13.785
here at Spotify, and I will
be the first to admit that I

00:00:13.785 --> 00:00:17.467
often take for granted the
reliability of the Internet.

00:00:17.467 --> 00:00:19.374
The sheer amount of
information zooming around the

00:00:19.374 --> 00:00:22.717
Internet is astonishing. But
how's it possible for every

00:00:22.717 --> 00:00:26.253
piece of data to be
delivered to you reliably?

00:00:26.253 --> 00:00:28.554
Say you want to play a song from Spotify.

00:00:28.554 --> 00:00:30.493
It seems like your computer
connects directly to

00:00:30.493 --> 00:00:33.326
Spotify's servers, and
Spotify sends you a song

00:00:33.326 --> 00:00:35.892
on a direct, dedicated line.
But actually, that's not

00:00:35.892 --> 00:00:37.471
how the Internet works.

00:00:38.891 --> 00:00:41.750
If the Internet were made of
direct, dedicated connections,

00:00:41.750 --> 00:00:43.631
it would be impossible
to keep things working as

00:00:43.631 --> 00:00:46.289
millions of users join,
especially since there is no

00:00:46.289 --> 00:00:48.297
guarantee that every wire and computer is

00:00:48.297 --> 00:00:49.957
working all the time.

00:00:49.957 --> 00:00:53.034
Instead, data travels on
the Internet in a much less

00:00:53.034 --> 00:00:55.139
direct fashion.

00:00:55.139 --> 00:00:58.840
- Many, many years ago,
in the early 1970s,

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my partner, Bob Kahn,
and I began working on

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the design of what we
now call the Internet.

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Bob and I had the responsibility
and the opportunity

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to design the Internet's
protocols and its architecture.

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So, we persisted in participating
in the Internet's growth

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and evolution for all of this time,

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up to, and including, the present.

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The way information gets
transferred from one computer

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to another is pretty interesting.

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It need not follow a fixed path.

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In fact, your path may
change in the midst of a

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computer-to-computer conversation.

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Information on the Internet
goes from one computer

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to another in what we call
a packet of information,

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and a packet travels from
one place to another on

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the Internet a lot like how
you might get from one place

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to another in a car, depending
on traffic congestion or

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road conditions, you might
choose or be forced to take

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a different route to get
to the same place each time

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you travel.

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And just as you can
transport all sorts of stuff

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inside a car, many kinds of
digital information can be

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sent with IP packets, but
there are some limits.

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What if, for example, you
need to move a space shuttle

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from where it was built to
where it will be launched?

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A shuttle won't fit in one
truck, so it needs to be broken

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down into pieces, transported
using a fleet of trucks.

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They could all take different
routes, and might get to

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the destination at different
times, but once all the pieces

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are there, you can reassemble
the pieces into the

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complete shuttle, and
it'll be ready for launch.

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On the Internet, the
details work similarly.

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If you have a very large
image that you want to send

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to a friend or upload to a
website, that image might

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be made up of tens of billions
of bits of ones and zeroes,

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too many to send along in one packet.

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Since it's data on a
computer, the computer sending

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the image can quickly break
it into hundreds or even

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thousands of smaller parts called packets.

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Unlike cars or trucks, these
packets don't have drivers,

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and they don't choose their
route. Each packet has

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the internet address of
where it came from and where

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it's going. Special
computers on the Internet,

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called routers, act like
traffic managers to keep the

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packets moving through
the networks smoothly.

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If one route is congested,
individual packets may

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travel different routes
through the Internet,

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and they may arrive at the
destination at slightly

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different times, or even out of order.

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- So, let's talk about how this works.

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As part of the Internet
Protocol, every router

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keeps track of multiple
paths for sending packets,

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and it chooses the
cheapest available path for

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each piece of data, based on destination

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IP address for the packet.

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"Cheapest," in this
case, doesn't mean cost,

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but time and non-technical
factors such as politics

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and relationships between companies.

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Often the best route for data
to travel isn't necessarily

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the most direct. Having
options for paths makes the

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network fault tolerant,
which means the network can

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keep sending packets, even
if something goes horribly,

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horribly wrong. This is the
basis for a key principle of

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the Internet, reliability.

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Now, what if you want to request some data

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and not everything is delivered?

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Say you want to listen to a song.

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How can you bee 100%
sure all the data will be

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delivered so the song plays perfectly?

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Introducing your new best friend, TCP,

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Transmission Control Protocol.

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TCP manages the sending and
receiving of all your data

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as packets. Think of it like
a guaranteed mail service.

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When you request a song on your device,

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Spotify sends the song
broken up into many packets.

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When your packets arrive,
TCP does a full inventory

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and sends back acknowledgements
of each packet received.

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If all packets are there,
TCP signs for your delivery,

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and you're done.

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(upbeat music)

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If TCP finds some packets are missing,

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it won't sign. Otherwise, your
song wouldn't sound as good,

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or portions of the song could be missing.

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For each missing or incomplete packet,

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Spotify will resend them.
Once TCP verifies the delivery

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of many packets for that one song request,

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your song will start to play.

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(upbeat music)

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What's great about the
TCP and router systems is

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they're scalable. They can
work with eight devices or

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8,000,000,000 devices. In fact,
because of these principles

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of fault tolerance and redundancy,
the more routers we add,

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the more reliable the Internet becomes.

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What's also great is we
can grow and scale the

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Internet without interrupting
service for anybody using it.

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- The Internet is made of
hundreds of thousands of

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networks and billions
of computers and devices

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connected physically. These
different systems that make up

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the Internet connect to
each other, communicate with

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each other, and work together
because of agreed upon

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standards for how data is
sent around on the Internet.

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Computing devices, or
routers along the Internet,

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help all the packets make
their way to the destination

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where they're reassembled,
if necessary, in order.

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This happens billions of
times a day, whether you and

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others are sending an
email, visiting a webpage,

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doing a video chat, using a
mobile app, or when sensors

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or devices on the Internet
talk to each other.