Area of a kite

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.557
What is the area of this figure?

00:00:03.557 --> 00:00:05.390
And this figure right
over here is sometimes

00:00:05.390 --> 00:00:07.225
called a kite for
obvious reasons.

00:00:07.225 --> 00:00:08.600
If you tied some
string here, you

00:00:08.600 --> 00:00:10.054
might want to fly
it at the beach.

00:00:10.054 --> 00:00:11.970
And another way to think
about what a kite is,

00:00:11.970 --> 00:00:20.900
it's a quadrilateral that is
symmetric around a diagonal.

00:00:20.900 --> 00:00:24.350
So this right over here is the
diagonal of this quadrilateral.

00:00:24.350 --> 00:00:25.910
And it's symmetric around it.

00:00:25.910 --> 00:00:29.955
This top part and this bottom
part are mirror images.

00:00:29.955 --> 00:00:32.080
And to think about how we
might find the area of it

00:00:32.080 --> 00:00:34.200
given that we've been
given essentially

00:00:34.200 --> 00:00:36.770
the width of this
kite, and we've also

00:00:36.770 --> 00:00:39.120
been given the
height of this kite,

00:00:39.120 --> 00:00:41.090
or if you view this
as a sideways kite,

00:00:41.090 --> 00:00:42.890
you could view this
is the height and that

00:00:42.890 --> 00:00:44.950
the eight centimeters
as the width.

00:00:44.950 --> 00:00:46.660
Given that we've got
those dimensions,

00:00:46.660 --> 00:00:49.400
how can we actually
figure out its area?

00:00:49.400 --> 00:00:52.570
So to do that, let
me actually copy

00:00:52.570 --> 00:00:53.920
and paste half of the kite.

00:00:53.920 --> 00:00:56.180
So this is the bottom
half of the kite.

00:00:56.180 --> 00:01:01.530
And then let's take the
top half of the kite

00:01:01.530 --> 00:01:03.450
and split it up into sections.

00:01:03.450 --> 00:01:06.560
So I have this little
red section here.

00:01:06.560 --> 00:01:08.410
I have this red section here.

00:01:08.410 --> 00:01:15.560
And actually, I'm going to try
to color the actual lines here

00:01:15.560 --> 00:01:18.010
so that we can keep
track of those as well.

00:01:18.010 --> 00:01:23.910
So I'll make this line green
and I'll make this line purple.

00:01:23.910 --> 00:01:27.290
So imagine taking this little
triangle right over here--

00:01:27.290 --> 00:01:31.020
and actually, let me do
this one too in blue.

00:01:31.020 --> 00:01:34.630
So this one over here is blue.

00:01:34.630 --> 00:01:35.510
You get the picture.

00:01:35.510 --> 00:01:37.695
Let me try to color it
in at least reasonably.

00:01:37.695 --> 00:01:38.780
So I'll color it in.

00:01:38.780 --> 00:01:44.620
And then I could make this
segment right over here,

00:01:44.620 --> 00:01:46.840
I'm going to make orange.

00:01:46.840 --> 00:01:49.540
So let's start focusing
on this red triangle here.

00:01:49.540 --> 00:01:57.800
Imagine flipping it over and
then moving it down here.

00:01:57.800 --> 00:01:59.380
So what would it look like?

00:01:59.380 --> 00:02:04.230
Well then the green side is
going to now be over here.

00:02:04.230 --> 00:02:08.340
This kind of mauve colored
side is still on the bottom.

00:02:08.340 --> 00:02:12.580
And my red triangle is going
to look something like this.

00:02:12.580 --> 00:02:15.940
My red triangle is
going to look like that.

00:02:15.940 --> 00:02:19.090
Now let's do the same thing
with this bigger blue triangle.

00:02:19.090 --> 00:02:22.750
Let's flip it over and
then move it down here.

00:02:22.750 --> 00:02:28.050
So this green side, since we've
flipped it, is now over here.

00:02:28.050 --> 00:02:32.615
And this orange side
is now over here.

00:02:35.480 --> 00:02:42.680
And we have this
blue right over here.

00:02:42.680 --> 00:02:45.160
And the reason that we know
that it definitely fits

00:02:45.160 --> 00:02:48.360
is the fact that it is
symmetric around this diagonal,

00:02:48.360 --> 00:02:50.560
that this length
right over here is

00:02:50.560 --> 00:02:54.420
equivalent to this
length right over here.

00:02:54.420 --> 00:02:57.220
That's why it fits
perfectly like this.

00:02:57.220 --> 00:02:59.510
Now, what we just
constructed is clearly

00:02:59.510 --> 00:03:06.800
a rectangle, a rectangle that
is 14 centimeters wide and not

00:03:06.800 --> 00:03:10.490
8 centimeters high, it's
half of 8 centimeters high.

00:03:10.490 --> 00:03:14.890
So it's 8 centimeters times
1/2 or 4 centimeters high.

00:03:14.890 --> 00:03:17.030
And we know how to
find the area of this.

00:03:17.030 --> 00:03:20.490
This is 4 centimeters
times 14 centimeters.

00:03:20.490 --> 00:03:25.460
So the area is equal
to 4 centimeters

00:03:25.460 --> 00:03:30.240
times 14 centimeters which
is equal to-- let's see,

00:03:30.240 --> 00:03:35.640
that's 40 plus 16--
56 square centimeters.

00:03:35.640 --> 00:03:37.590
So if you're taking
the area of a kite,

00:03:37.590 --> 00:03:41.370
you're really just
taking 1/2 the width

00:03:41.370 --> 00:03:45.410
times the height, or 1/2
the width times the height,

00:03:45.410 --> 00:03:48.540
any way you want
to think about it.


Area of a trapezoid

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:03.240
So right here, we have
a four-sided figure,

00:00:03.240 --> 00:00:06.530
or a quadrilateral,
where two of the sides

00:00:06.530 --> 00:00:08.520
are parallel to each other.

00:00:08.520 --> 00:00:10.636
And so this, by
definition, is a trapezoid.

00:00:14.530 --> 00:00:16.570
And what we want to do
is, given the dimensions

00:00:16.570 --> 00:00:20.630
that they've given us, what
is the area of this trapezoid.

00:00:20.630 --> 00:00:22.660
So let's just think through it.

00:00:22.660 --> 00:00:26.250
So what would we get if we
multiplied this long base

00:00:26.250 --> 00:00:28.670
6 times the height 3?

00:00:28.670 --> 00:00:33.750
So what do we get if
we multiply 6 times 3?

00:00:33.750 --> 00:00:35.900
Well, that would be the
area of a rectangle that

00:00:35.900 --> 00:00:39.790
is 6 units wide
and 3 units high.

00:00:39.790 --> 00:00:42.530
So that would give us
the area of a figure that

00:00:42.530 --> 00:00:44.980
looked like-- let me do
it in this pink color.

00:00:44.980 --> 00:00:49.940
The area of a figure that looked
like this would be 6 times 3.

00:00:49.940 --> 00:00:53.790
So it would give us this
entire area right over there.

00:00:53.790 --> 00:00:55.760
Now, the trapezoid is
clearly less than that,

00:00:55.760 --> 00:00:58.770
but let's just go with
the thought experiment.

00:00:58.770 --> 00:01:04.980
Now, what would happen if
we went with 2 times 3?

00:01:04.980 --> 00:01:07.910
Well, now we'd be finding
the area of a rectangle that

00:01:07.910 --> 00:01:10.260
has a width of 2
and a height of 3.

00:01:10.260 --> 00:01:14.810
So you could imagine that being
this rectangle right over here.

00:01:14.810 --> 00:01:18.240
So that is this rectangle
right over here.

00:01:18.240 --> 00:01:22.130
So that's the 2
times 3 rectangle.

00:01:22.130 --> 00:01:26.160
Now, it looks like the
area of the trapezoid

00:01:26.160 --> 00:01:28.910
should be in between
these two numbers.

00:01:28.910 --> 00:01:32.490
Maybe it should be exactly
halfway in between,

00:01:32.490 --> 00:01:36.050
because when you look at the
area difference between the two

00:01:36.050 --> 00:01:39.240
rectangles-- and let
me color that in.

00:01:39.240 --> 00:01:43.030
So this is the area difference
on the left-hand side.

00:01:43.030 --> 00:01:48.980
And this is the area difference
on the right-hand side.

00:01:48.980 --> 00:01:51.090
If we focus on
the trapezoid, you

00:01:51.090 --> 00:01:56.480
see that if we start with the
yellow, the smaller rectangle,

00:01:56.480 --> 00:01:59.610
it reclaims half
of the area, half

00:01:59.610 --> 00:02:03.030
of the difference between
the smaller rectangle

00:02:03.030 --> 00:02:05.240
and the larger one on
the left-hand side.

00:02:05.240 --> 00:02:07.920
It gets exactly half of
it on the left-hand side.

00:02:07.920 --> 00:02:10.050
And it gets half the
difference between the smaller

00:02:10.050 --> 00:02:12.290
and the larger on
the right-hand side.

00:02:12.290 --> 00:02:17.260
So it completely makes
sense that the area

00:02:17.260 --> 00:02:20.420
of the trapezoid, this
entire area right over here,

00:02:20.420 --> 00:02:22.310
should really just
be the average.

00:02:22.310 --> 00:02:25.420
It should exactly be
halfway between the areas

00:02:25.420 --> 00:02:28.172
of the smaller rectangle
and the larger rectangle.

00:02:28.172 --> 00:02:30.130
So let's take the average
of those two numbers.

00:02:30.130 --> 00:02:38.160
It's going to be 6 times 3 plus
2 times 3, all of that over 2.

00:02:38.160 --> 00:02:40.230
So when you think about
an area of a trapezoid,

00:02:40.230 --> 00:02:44.940
you look at the two bases, the
long base and the short base.

00:02:47.840 --> 00:02:50.410
Multiply each of those times
the height, and then you

00:02:50.410 --> 00:02:51.720
could take the average of them.

00:02:51.720 --> 00:02:53.680
Or you could also
think of it as this

00:02:53.680 --> 00:02:57.440
is the same thing as 6 plus 2.

00:02:57.440 --> 00:02:59.490
And I'm just factoring
out a 3 here.

00:02:59.490 --> 00:03:12.760
6 plus 2 times 3, and
then all of that over 2,

00:03:12.760 --> 00:03:14.274
which is the same
thing as-- and I'm

00:03:14.274 --> 00:03:15.690
just writing it
in different ways.

00:03:15.690 --> 00:03:17.690
These are all different
ways to think about it--

00:03:17.690 --> 00:03:25.450
6 plus 2 over 2, and
then that times 3.

00:03:25.450 --> 00:03:27.820
So you could view
it as the average

00:03:27.820 --> 00:03:30.560
of the smaller and
larger rectangle.

00:03:30.560 --> 00:03:32.790
So you multiply each of
the bases times the height

00:03:32.790 --> 00:03:34.180
and then take the average.

00:03:34.180 --> 00:03:37.540
You could view it as-- well,
let's just add up the two base

00:03:37.540 --> 00:03:41.360
lengths, multiply that times the
height, and then divide by 2.

00:03:41.360 --> 00:03:43.710
Or you could say, hey, let's
take the average of the two

00:03:43.710 --> 00:03:46.481
base lengths and
multiply that by 3.

00:03:46.481 --> 00:03:48.230
And that gives you
another interesting way

00:03:48.230 --> 00:03:48.980
to think about it.

00:03:48.980 --> 00:03:52.850
If you take the average of these
two lengths, 6 plus 2 over 2

00:03:52.850 --> 00:03:54.660
is 4.

00:03:54.660 --> 00:03:57.770
So that would be a width
that looks something

00:03:57.770 --> 00:03:59.690
like-- let me do this in orange.

00:03:59.690 --> 00:04:03.080
A width of 4 would look
something like this.

00:04:03.080 --> 00:04:05.000
A width of 4 would look
something like that,

00:04:05.000 --> 00:04:07.050
and you're multiplying
that times the height.

00:04:07.050 --> 00:04:11.440
Well, that would be a rectangle
like this that is exactly

00:04:11.440 --> 00:04:14.190
halfway in between
the areas of the small

00:04:14.190 --> 00:04:16.089
and the large rectangle.

00:04:16.089 --> 00:04:18.420
So these are all
equivalent statements.

00:04:18.420 --> 00:04:20.010
Now let's actually
just calculate it.

00:04:20.010 --> 00:04:21.176
So we could do any of these.

00:04:21.176 --> 00:04:24.120
6 times 3 is 18.

00:04:24.120 --> 00:04:28.630
This is 18 plus 6, over 2.

00:04:28.630 --> 00:04:31.501
That is 24/2, or 12.

00:04:31.501 --> 00:04:32.750
You could also do it this way.

00:04:32.750 --> 00:04:38.090
6 plus 2 is 8, times 3 is
24, divided by 2 is 12.

00:04:38.090 --> 00:04:42.430
6 plus 2 divided by 2
is 4, times 3 is 12.

00:04:42.430 --> 00:04:47.600
Either way, the area of this
trapezoid is 12 square units.


Conditional statements and deductive reasoning

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.603
I have a statement
here in yellow,

00:00:02.603 --> 00:00:04.019
which, for the
sake of this video,

00:00:04.019 --> 00:00:08.240
we will assume is true,
that eating spinach makes

00:00:08.240 --> 00:00:09.872
you stronger.

00:00:09.872 --> 00:00:11.330
And then after
this, in white, I've

00:00:11.330 --> 00:00:12.840
written a bunch of
other statements.

00:00:12.840 --> 00:00:15.200
And I want to figure out if
these are reasonable things

00:00:15.200 --> 00:00:18.950
to say, given the assumption
that this first statement is

00:00:18.950 --> 00:00:21.610
true, that eating spinach
makes you stronger.

00:00:21.610 --> 00:00:23.270
And we could even
put it over here.

00:00:23.270 --> 00:00:29.830
"Eating spinach
makes you stronger."

00:00:29.830 --> 00:00:31.530
So let's read this
first sentence.

00:00:31.530 --> 00:00:35.152
"If you eat spinach, then
you will get stronger."

00:00:35.152 --> 00:00:37.360
So this is just really a
rephrasing of the first one.

00:00:37.360 --> 00:00:39.590
This is really just
putting it in if-then form.

00:00:39.590 --> 00:00:43.110
You have your "if" and
your "then," but it's

00:00:43.110 --> 00:00:44.640
really saying the same thing.

00:00:44.640 --> 00:00:47.810
If you eat spinach-- that's
the same thing as eating

00:00:47.810 --> 00:00:52.220
the spinach-- then you
will become stronger,

00:00:52.220 --> 00:00:55.630
then you will be stronger.

00:00:55.630 --> 00:00:56.720
You get stronger.

00:00:56.720 --> 00:00:58.650
So this is saying
the exact same thing.

00:00:58.650 --> 00:01:02.200
I will say that if we assume
this first statement is true,

00:01:02.200 --> 00:01:05.230
then the second statement
is definitely true.

00:01:05.230 --> 00:01:06.420
Now, this third statement.

00:01:06.420 --> 00:01:11.800
"If you don't eat spinach, then
you will not get stronger."

00:01:11.800 --> 00:01:13.500
So this is interesting.

00:01:13.500 --> 00:01:16.520
This is kind of assuming that
the only way to get stronger

00:01:16.520 --> 00:01:17.530
is by eating spinach.

00:01:17.530 --> 00:01:19.590
But that's not what that
first statement says.

00:01:19.590 --> 00:01:21.350
That first statement
says eating spinach

00:01:21.350 --> 00:01:23.001
is a way to get you stronger.

00:01:23.001 --> 00:01:24.750
So if we think about
ways to get stronger,

00:01:24.750 --> 00:01:26.800
there are other ways
to get stronger.

00:01:26.800 --> 00:01:30.140
You could lift weights.

00:01:30.140 --> 00:01:33.370
And maybe you don't want
to eat any spinach at all.

00:01:33.370 --> 00:01:34.660
Maybe you eat eggs.

00:01:38.400 --> 00:01:38.900
Who knows?

00:01:38.900 --> 00:01:43.585
There's a million, you know--
maybe you have a-- drink milk.

00:01:43.585 --> 00:01:44.210
Who knows what?

00:01:44.210 --> 00:01:49.270
There are a lot of options
that will get you stronger.

00:01:49.270 --> 00:01:50.690
Drink milk.

00:01:50.690 --> 00:01:53.320
So eating spinach
is not the only way.

00:01:53.320 --> 00:01:56.260
So if you don't eat
spinach, you still

00:01:56.260 --> 00:01:58.220
might be able to get stronger.

00:01:58.220 --> 00:02:00.510
They're not ruling out that
you can't get stronger.

00:02:00.510 --> 00:02:03.970
So I would say, I would
not go with this statement

00:02:03.970 --> 00:02:04.760
right over there.

00:02:04.760 --> 00:02:07.560
Let me do it in red so
that I can mark it off.

00:02:07.560 --> 00:02:10.289
This one is not
necessarily true.

00:02:10.289 --> 00:02:11.804
You can get stronger other ways.

00:02:11.804 --> 00:02:14.220
Now, if you eat spinach, you
will definitely get stronger,

00:02:14.220 --> 00:02:15.770
if you believe that statement.

00:02:15.770 --> 00:02:17.270
But you can get
stronger other ways.

00:02:17.270 --> 00:02:19.907
So just by not eating spinach,
if you lift weights, eat eggs,

00:02:19.907 --> 00:02:21.990
who knows what else, you
could still get stronger.

00:02:21.990 --> 00:02:24.440
So we'll rule out
the second statement.

00:02:24.440 --> 00:02:26.030
Now, this third statement.

00:02:26.030 --> 00:02:28.030
"You are not stronger.

00:02:28.030 --> 00:02:31.510
Therefore, you must not
have eaten spinach."

00:02:31.510 --> 00:02:34.080
Now, this is where we really
have to put our logician hats.

00:02:34.080 --> 00:02:36.100
And a logician is just
someone who has really,

00:02:36.100 --> 00:02:38.170
really, really good logic.

00:02:38.170 --> 00:02:39.610
"You are not stronger.

00:02:39.610 --> 00:02:41.520
Therefore, you must not
have eaten spinach."

00:02:41.520 --> 00:02:44.020
And we have to be very
careful to separate

00:02:44.020 --> 00:02:45.970
our everyday
experience versus what

00:02:45.970 --> 00:02:47.970
we're trying to
deduce logically.

00:02:47.970 --> 00:02:53.220
So in our everyday
experience, I'm not stronger,

00:02:53.220 --> 00:02:54.844
but maybe I did eat spinach.

00:02:54.844 --> 00:02:57.010
And we'd say, hey, you know,
maybe sometimes spinach

00:02:57.010 --> 00:02:59.230
doesn't make me stronger,
et cetera, et cetera.

00:02:59.230 --> 00:03:01.970
But we have to remember, we're
assuming this first statement

00:03:01.970 --> 00:03:02.740
is true.

00:03:02.740 --> 00:03:05.350
We're assuming this first
statement is absolutely true.

00:03:05.350 --> 00:03:08.260
Eating spinach
makes you stronger.

00:03:08.260 --> 00:03:10.900
If you had eaten
spinach, by definition,

00:03:10.900 --> 00:03:13.710
this first statement says
you will be stronger.

00:03:13.710 --> 00:03:15.280
If you're not
stronger, then there's

00:03:15.280 --> 00:03:17.870
no way that you
ate some spinach.

00:03:17.870 --> 00:03:19.670
So this is actually true.

00:03:19.670 --> 00:03:21.920
This is actually
true if we assume

00:03:21.920 --> 00:03:24.360
this first statement is true.

00:03:24.360 --> 00:03:25.560
Now, this last statement.

00:03:25.560 --> 00:03:27.190
"You are stronger.

00:03:27.190 --> 00:03:30.160
Therefore, you must
have eaten spinach."

00:03:30.160 --> 00:03:32.690
Well, this goes
back to this idea.

00:03:32.690 --> 00:03:35.290
That first statement didn't
say spinach is the only way

00:03:35.290 --> 00:03:39.430
to get stronger, or the
second statement wasn't if

00:03:39.430 --> 00:03:43.002
and only if you eat spinach,
then you'll get stronger.

00:03:43.002 --> 00:03:44.460
There's other ways
to get stronger.

00:03:44.460 --> 00:03:47.270
You could lift weights,
eat eggs, drink milk.

00:03:47.270 --> 00:03:49.400
So this last statement--
you are stronger.

00:03:49.400 --> 00:03:51.190
Therefore, you must
have eaten spinach--

00:03:51.190 --> 00:03:52.580
isn't necessarily true.

00:03:52.580 --> 00:03:55.840
You could have drank--
lifted weights,

00:03:55.840 --> 00:03:58.700
or ate eggs, or
anything like that.

00:03:58.700 --> 00:04:01.620
So I will rule this
one out, as well.

00:04:01.620 --> 00:04:08.463
You cannot make this conclusion
given this first statement.


Logical argument and deductive reasoning exercise example

WEBVTT
Kind: captions
Language: en

00:00:02.330 --> 00:00:05.060
Use the given information to
make a logical conclusion,

00:00:05.060 --> 00:00:06.250
if possible.

00:00:06.250 --> 00:00:08.280
If a logical conclusion
is not possible,

00:00:08.280 --> 00:00:11.580
choose "no logical
conclusion possible."

00:00:11.580 --> 00:00:12.110
All right.

00:00:12.110 --> 00:00:13.401
So let's see what they tell us.

00:00:13.401 --> 00:00:16.300
So the statement is, if
I go to practice today,

00:00:16.300 --> 00:00:18.760
then I will play in
the game tomorrow.

00:00:18.760 --> 00:00:20.910
So this statement
is saying, look,

00:00:20.910 --> 00:00:22.730
if I go to practice
today, then I'm

00:00:22.730 --> 00:00:25.610
definitely going to play
in the game tomorrow.

00:00:25.610 --> 00:00:29.950
And then they say, I will not
play in the game tomorrow.

00:00:29.950 --> 00:00:32.809
Well, I just said that if
I go to practice today,

00:00:32.809 --> 00:00:34.100
I'm going to play in that game.

00:00:34.100 --> 00:00:36.141
And they just said, I'm
not playing in that game.

00:00:36.141 --> 00:00:38.850
So I must not be going
to practice today.

00:00:38.850 --> 00:00:41.490
Because if I went, I would play.

00:00:41.490 --> 00:00:45.030
So I did not go to practice.

00:00:45.030 --> 00:00:49.070
So let's do a couple
more of these.

00:00:49.070 --> 00:00:51.330
OK, so the same instructions.

00:00:51.330 --> 00:00:56.760
If William misses the bus, then
he will be late for school.

00:00:56.760 --> 00:01:00.560
William is late for school.

00:01:00.560 --> 00:01:02.780
So let's think about
this a little bit.

00:01:02.780 --> 00:01:05.910
If William misses the bus, then
he will be late for school.

00:01:05.910 --> 00:01:07.710
William is late for school.

00:01:07.710 --> 00:01:10.150
So missing the bus
is one way that you

00:01:10.150 --> 00:01:11.490
could be late for school.

00:01:11.490 --> 00:01:13.390
But he might have
not missed the bus.

00:01:13.390 --> 00:01:18.950
He might have caught the bus,
but then kind of sat in the bus

00:01:18.950 --> 00:01:19.930
once he got to school.

00:01:19.930 --> 00:01:22.130
So missing the bus
is one of the ways

00:01:22.130 --> 00:01:24.270
that he could be
late for school.

00:01:24.270 --> 00:01:26.650
But it's not the only way
he could be late for school.

00:01:26.650 --> 00:01:28.510
They just say, if
he misses the bus,

00:01:28.510 --> 00:01:31.290
then he is late for school.

00:01:31.290 --> 00:01:34.270
Now, if they said, he
is not late for school,

00:01:34.270 --> 00:01:36.390
then you know that he
didn't miss the bus.

00:01:36.390 --> 00:01:39.170
Because if he did, he
would have been late.

00:01:39.170 --> 00:01:42.210
But there's many ways to be
late, not just missing the bus.

00:01:42.210 --> 00:01:45.180
So we don't know for a fact
that William definitely

00:01:45.180 --> 00:01:47.270
missed the bus.

00:01:47.270 --> 00:01:49.600
Let's do a few more of these.

00:01:49.600 --> 00:01:53.920
If a ray bisects an angle, so
it splits that angle in two,

00:01:53.920 --> 00:01:57.700
then it creates two
congruent angles, all right?

00:01:57.700 --> 00:02:00.880
If there are two
congruent angles,

00:02:00.880 --> 00:02:04.860
then the two angles
have the same measure.

00:02:04.860 --> 00:02:05.770
All right.

00:02:05.770 --> 00:02:07.740
So this one's a
little bit more hairy.

00:02:07.740 --> 00:02:09.560
So let's see, if a
ray bisects an angle,

00:02:09.560 --> 00:02:11.101
then it creates two
congruent angles.

00:02:11.101 --> 00:02:12.800
Yep, OK, I can buy that.

00:02:12.800 --> 00:02:14.630
In fact, that is
definitely true.

00:02:14.630 --> 00:02:16.840
If there are two
congruent angles,

00:02:16.840 --> 00:02:18.910
then the two angles
have the same measure.

00:02:18.910 --> 00:02:21.330
OK, I can buy that, as well.

00:02:21.330 --> 00:02:24.130
If a ray bisects an
angle, then the two angles

00:02:24.130 --> 00:02:26.140
have the same measure.

00:02:26.140 --> 00:02:26.900
So let's see.

00:02:26.900 --> 00:02:29.910
So this first sentence says,
if a ray bisects an angle,

00:02:29.910 --> 00:02:32.617
then it creates two
congruent angles.

00:02:32.617 --> 00:02:35.200
But then the next one says, if
there are two congruent angles,

00:02:35.200 --> 00:02:37.620
then the two angles
have the same measure.

00:02:37.620 --> 00:02:39.790
So if a ray bisects
an angle, there's

00:02:39.790 --> 00:02:44.260
two congruent angles, which
the next sentence tells us

00:02:44.260 --> 00:02:47.010
that the two angles
have the same measure.

00:02:47.010 --> 00:02:48.280
And that's the statement here.

00:02:48.280 --> 00:02:50.540
If a ray bisects an
angle, then the two angles

00:02:50.540 --> 00:02:51.930
have the same measure.

00:02:51.930 --> 00:02:54.390
And you're going to say, wait,
wait, how is this different?

00:02:54.390 --> 00:02:57.990
Look, in the first sentence, it
says the ray bisecting an angle

00:02:57.990 --> 00:03:00.170
creates two congruent angles.

00:03:00.170 --> 00:03:03.370
So we don't know yet what
congruent angle means.

00:03:03.370 --> 00:03:07.020
The next sentence says, if
there are two congruent angles,

00:03:07.020 --> 00:03:09.220
then the two angles
have the same measure.

00:03:09.220 --> 00:03:10.710
So the second
sentence lets us know

00:03:10.710 --> 00:03:14.450
that two congruent angles
must have the same measure.

00:03:14.450 --> 00:03:17.460
So now, we can kind of merge
these two together to say,

00:03:17.460 --> 00:03:19.530
if a ray bisects
an angle, then it

00:03:19.530 --> 00:03:22.070
creates two angles that
have the same measure, which

00:03:22.070 --> 00:03:24.740
is this choice right over here.


Conditional statements exercise examples

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.370
Write the conditional
statement in if-then form.

00:00:04.370 --> 00:00:06.920
And the statement is,
dogs have four legs, which

00:00:06.920 --> 00:00:07.800
we know is true.

00:00:07.800 --> 00:00:09.300
So another way of
thinking about it,

00:00:09.300 --> 00:00:13.860
you say, well, if you have a dog
or if an animal is a dog then

00:00:13.860 --> 00:00:15.280
it's going to have four legs.

00:00:15.280 --> 00:00:16.820
That's an equivalent
statement to this here.

00:00:16.820 --> 00:00:19.070
So let's see if we can
construct a sentence like that.

00:00:19.070 --> 00:00:31.270
So if an animal is a dog
then it has four legs.

00:00:31.270 --> 00:00:33.140
So that was pretty
straightforward.

00:00:33.140 --> 00:00:35.510
Let's do a few more of these.

00:00:35.510 --> 00:00:36.480
So here, same thing.

00:00:36.480 --> 00:00:38.250
We want to write this
as in if-then form,

00:00:38.250 --> 00:00:43.080
I will win a gold medal
when I go to the Olympics.

00:00:43.080 --> 00:00:43.760
So let's see.

00:00:43.760 --> 00:00:46.490
This is essentially saying,
if I go to the Olympics

00:00:46.490 --> 00:00:48.450
then I will win a gold medal.

00:00:48.450 --> 00:00:59.157
So if I go to the Olympics
then I win a gold medal.

00:00:59.157 --> 00:01:00.990
Now, you might be tempted
to say, hey, well,

00:01:00.990 --> 00:01:03.640
I saw gold medal first here
then I saw Olympics later

00:01:03.640 --> 00:01:05.310
in the sentence, but
this isn't saying

00:01:05.310 --> 00:01:09.340
that if you win a
gold medal then you

00:01:09.340 --> 00:01:12.030
have gone to the Olympics or
then you will go the Olympics.

00:01:12.030 --> 00:01:14.204
You might win a gold medal
at the local track meet

00:01:14.204 --> 00:01:15.620
or something, and
that's not going

00:01:15.620 --> 00:01:17.870
to ensure that you're
going to the Olympics.

00:01:17.870 --> 00:01:20.332
This is saying that if
you go the Olympics--

00:01:20.332 --> 00:01:22.040
this person is very
confident who is ever

00:01:22.040 --> 00:01:24.235
saying this-- I will
win a gold medal when

00:01:24.235 --> 00:01:25.110
I go to the Olympics.

00:01:25.110 --> 00:01:28.870
So that's the same
statement as saying,

00:01:28.870 --> 00:01:33.360
if I go to the Olympics, I'm
going to win a gold medal.

00:01:33.360 --> 00:01:36.637
So if I go to the Olympics
then I win a gold medal.

00:01:36.637 --> 00:01:38.470
And actually, this
statement is a little bit

00:01:38.470 --> 00:01:40.220
more confident than this
one because here they say,

00:01:40.220 --> 00:01:41.880
if I go to the Olympics
while here they

00:01:41.880 --> 00:01:43.540
say when I go to the Olympics.

00:01:43.540 --> 00:01:45.550
But I think you get the idea.

00:01:45.550 --> 00:01:47.890
Let's do a couple more of these.

00:01:47.890 --> 00:01:51.160
I will go to Disneyland
when I win a gold medal.

00:01:51.160 --> 00:01:53.280
So now, winning
the gold medal is

00:01:53.280 --> 00:01:55.440
what's driving stuff to happen.

00:01:55.440 --> 00:02:00.130
So if I win a gold medal
then I will go to Disneyland.

00:02:00.130 --> 00:02:12.574
So if I win a gold medal
then I go to Disneyland.

00:02:12.574 --> 00:02:13.490
Oh, that wasn't right.

00:02:13.490 --> 00:02:15.180
Oh, I didn't put this.

00:02:15.180 --> 00:02:16.670
I go to Disneyland.

00:02:16.670 --> 00:02:17.510
I got too careless.

00:02:17.510 --> 00:02:20.350
I got to make sure I do
if I win a gold medal then

00:02:20.350 --> 00:02:22.280
I go to Disneyland.

00:02:22.280 --> 00:02:23.901
And now we got it right.

00:02:23.901 --> 00:02:24.650
Let's do one more.

00:02:24.650 --> 00:02:25.774
I can't do enough of these.

00:02:25.774 --> 00:02:28.150
These are incredibly fun.

00:02:28.150 --> 00:02:32.030
Isosceles triangles
have two equal sides.

00:02:32.030 --> 00:02:34.300
So if I have an
isosceles triangle

00:02:34.300 --> 00:02:37.224
then I have two equal sides.

00:02:37.224 --> 00:02:45.380
So if a shape is an
isosceles triangle

00:02:45.380 --> 00:02:50.570
then it has two equal sides.

00:02:50.570 --> 00:02:53.020
And we got it right.


Absolute value equation with no solution

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:02.320
So we're asked to
solve for x, and we

00:00:02.320 --> 00:00:05.060
have this equation with
absolute values in it.

00:00:05.060 --> 00:00:06.750
So it's 4 times
the absolute value

00:00:06.750 --> 00:00:12.580
of x plus 10 plus 4 is equal to
6 times the absolute value of x

00:00:12.580 --> 00:00:14.382
plus 10 plus 10.

00:00:14.382 --> 00:00:16.090
And at first, this
looks really daunting,

00:00:16.090 --> 00:00:19.680
but the key is to just solve for
this absolute value expression

00:00:19.680 --> 00:00:21.039
and then go from there.

00:00:21.039 --> 00:00:23.080
Let me just rewrite it so
that the absolute value

00:00:23.080 --> 00:00:24.860
expression really jumps out.

00:00:24.860 --> 00:00:28.240
So this is 4 times
the absolute value

00:00:28.240 --> 00:00:37.450
of x plus 10 plus 4 is equal to
6 times the absolute value of x

00:00:37.450 --> 00:00:39.500
plus 10 plus 10.

00:00:42.090 --> 00:00:47.580
So let's get all of the
absolute values of x plus 10

00:00:47.580 --> 00:00:48.620
on the left-hand side.

00:00:48.620 --> 00:00:50.100
So I want to get
rid of the 6 times

00:00:50.100 --> 00:00:52.340
the absolute value of
x plus 10 on the right.

00:00:52.340 --> 00:00:53.720
Well, how would I do that?

00:00:53.720 --> 00:00:59.360
Well, I could subtract 6 times
the absolute value of x plus 10

00:00:59.360 --> 00:01:01.150
from the right,
but we've already

00:01:01.150 --> 00:01:02.295
seen this multiple times.

00:01:02.295 --> 00:01:03.920
If these two things
are equal, and if I

00:01:03.920 --> 00:01:05.601
want to keep them
equal, if I subtract 6

00:01:05.601 --> 00:01:07.600
from the right-hand side,
I've got to subtract--

00:01:07.600 --> 00:01:10.222
or if I subtract 6 times the
absolute value of x plus 10

00:01:10.222 --> 00:01:12.680
from the right-hand side, I
have to subtract the same thing

00:01:12.680 --> 00:01:14.310
from the left-hand side.

00:01:14.310 --> 00:01:17.230
So we're going to
have minus 6 times

00:01:17.230 --> 00:01:21.190
the absolute value of x plus 10.

00:01:21.190 --> 00:01:23.385
And likewise, I want to
get all my constant terms,

00:01:23.385 --> 00:01:26.230
I want to get this 4 out
of the left-hand side.

00:01:26.230 --> 00:01:29.340
So let me subtract
4 from the left,

00:01:29.340 --> 00:01:32.090
and then I have to also
do it on the right,

00:01:32.090 --> 00:01:33.980
otherwise my equality
wouldn't hold.

00:01:33.980 --> 00:01:38.310
And now let's see
what we end up with.

00:01:38.310 --> 00:01:42.680
So on the left-hand side,
the 4 minus 4, that's 0.

00:01:42.680 --> 00:01:46.099
You have 4 of something
minus 6 of something,

00:01:46.099 --> 00:01:48.140
that means you're going
to end up with negative 2

00:01:48.140 --> 00:01:49.700
of that something.

00:01:49.700 --> 00:01:54.405
Negative 2 of the absolute
value of x plus 10.

00:01:54.405 --> 00:01:56.280
Remember, this might
seem a little confusing,

00:01:56.280 --> 00:01:58.850
but remember, if
you had 4 apples

00:01:58.850 --> 00:02:01.860
and you subtract 6 apples, you
now have negative 2 apples,

00:02:01.860 --> 00:02:03.590
I guess you owe
someone the apples.

00:02:03.590 --> 00:02:06.280
Same way, you have 4
of this expression,

00:02:06.280 --> 00:02:08.080
you take away 6 of
this expression,

00:02:08.080 --> 00:02:10.230
you now have negative
2 of this expression.

00:02:10.230 --> 00:02:11.730
Let me write it a
little bit neater.

00:02:11.730 --> 00:02:14.710
So it's negative 2 times the
absolute value of x plus 10

00:02:14.710 --> 00:02:20.620
is equal to, well the whole
point of this, of the 6 times

00:02:20.620 --> 00:02:23.010
the absolute value of
x plus 10 minus 6 times

00:02:23.010 --> 00:02:25.690
the absolute value of x plus
10 is to make those cancel out,

00:02:25.690 --> 00:02:29.610
and then you have 10 minus
4, which is equal to 6.

00:02:29.610 --> 00:02:33.320
Now, we want to solve for the
absolute value of x plus 10.

00:02:33.320 --> 00:02:34.820
So let's get rid
of this negative 2,

00:02:34.820 --> 00:02:38.420
and we can do that by dividing
both sides by negative 2.

00:02:38.420 --> 00:02:40.670
You might realize,
everything we've done so far

00:02:40.670 --> 00:02:42.782
is just treating
this red expression

00:02:42.782 --> 00:02:44.490
as almost just like
a variable, and we're

00:02:44.490 --> 00:02:46.073
going to solve for
that red expression

00:02:46.073 --> 00:02:47.950
and then take it from there.

00:02:47.950 --> 00:02:50.250
So negative 2 divided
by negative 2 is 1.

00:02:50.250 --> 00:02:52.490
6 divided by negative
2 is negative 3.

00:02:52.490 --> 00:02:56.660
So we get the absolute
value of x plus 10

00:02:56.660 --> 00:03:01.980
is equal to negative 3.

00:03:01.980 --> 00:03:05.930
Now, this gets us to a
very interesting situation.

00:03:05.930 --> 00:03:08.180
You might say maybe this
could be the positive version

00:03:08.180 --> 00:03:12.060
or the negative, but
remember, absolute value

00:03:12.060 --> 00:03:16.660
is always non-negative.

00:03:16.660 --> 00:03:18.906
If you took the absolute
value of 0, you would get 0.

00:03:18.906 --> 00:03:20.530
But the absolute
value of anything else

00:03:20.530 --> 00:03:21.850
is going to be positive.

00:03:21.850 --> 00:03:23.760
So this thing right
over here is definitely

00:03:23.760 --> 00:03:25.900
going to be greater
than or equal to 0.

00:03:25.900 --> 00:03:27.625
Doesn't matter what
x you put in there,

00:03:27.625 --> 00:03:29.000
when you take its
absolute value,

00:03:29.000 --> 00:03:30.416
you're going to
get a value that's

00:03:30.416 --> 00:03:31.990
greater than or equal to 0.

00:03:31.990 --> 00:03:35.600
So there's no x that you
could find that's somehow--

00:03:35.600 --> 00:03:38.010
you put it there, you add 10,
you take the absolute value

00:03:38.010 --> 00:03:40.450
of it, you're actually
getting a negative value.

00:03:40.450 --> 00:03:46.640
So this right over here
has absolutely no solution.

00:03:46.640 --> 00:03:49.910
And I'll put some exclamation
marks there for emphasis.


Absolute value equation example

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:01.830
We're asked to solve for x.

00:00:01.830 --> 00:00:03.560
Let me just rewrite
this equation

00:00:03.560 --> 00:00:05.550
so that the absolute
values really pop out.

00:00:05.550 --> 00:00:12.210
So this is 8 times the
absolute value of x plus 7

00:00:12.210 --> 00:00:19.140
plus 4-- in that same color--
is equal to negative 6 times

00:00:19.140 --> 00:00:23.475
the absolute value
of x plus 7 plus 6.

00:00:26.250 --> 00:00:28.965
Now the key here-- at first
it looks kind of daunting.

00:00:28.965 --> 00:00:30.900
It's this complex equation.

00:00:30.900 --> 00:00:32.664
You have these
absolute values in it.

00:00:32.664 --> 00:00:34.080
But the way to
think about this is

00:00:34.080 --> 00:00:36.642
if you could solve for the
absolute value expression,

00:00:36.642 --> 00:00:39.100
you could then-- it then turns
into a much simpler problem,

00:00:39.100 --> 00:00:40.710
then you can take it from there.

00:00:40.710 --> 00:00:43.500
So you could almost treat this
expression-- the absolute value

00:00:43.500 --> 00:00:46.130
of x plus 7, you can just
treat it as a variable,

00:00:46.130 --> 00:00:48.610
and then once you
solve for that,

00:00:48.610 --> 00:00:50.940
it becomes a simpler
absolute value problem.

00:00:50.940 --> 00:00:51.940
So let's try to do that.

00:00:51.940 --> 00:00:54.910
Let's try to solve
for not x first.

00:00:54.910 --> 00:00:57.632
We're just going to solve for
the absolute value of x plus 7.

00:00:57.632 --> 00:00:58.590
You'll see what I mean.

00:00:58.590 --> 00:01:01.309
So I want to get all of the
absolute values of x plus 7

00:01:01.309 --> 00:01:03.100
on the left-hand side,
so I want to get rid

00:01:03.100 --> 00:01:04.970
of this one on the
right-hand side.

00:01:04.970 --> 00:01:08.910
Easiest way to get
rid of it is to add

00:01:08.910 --> 00:01:14.606
6 times the absolute value of x
plus 7 to the right-hand side.

00:01:14.606 --> 00:01:16.980
We can't, of course, only do
that to the right-hand side.

00:01:16.980 --> 00:01:18.650
If these two things
are equal and we

00:01:18.650 --> 00:01:20.644
are being told
that they are, then

00:01:20.644 --> 00:01:22.060
if you add something
on this side,

00:01:22.060 --> 00:01:23.726
the only way that the
equality will hold

00:01:23.726 --> 00:01:25.660
is if you still do it
on the left-hand side.

00:01:25.660 --> 00:01:31.650
So let's do that, so plus 6
times the absolute value of x

00:01:31.650 --> 00:01:33.220
plus 7.

00:01:33.220 --> 00:01:35.740
And I want to get all
of these constant terms

00:01:35.740 --> 00:01:36.920
on to the right-hand side.

00:01:36.920 --> 00:01:39.410
So I want to get rid
of this positive 4.

00:01:39.410 --> 00:01:41.797
Easiest way is to subtract
4 right over there,

00:01:41.797 --> 00:01:43.380
but if we do it on
the left-hand side,

00:01:43.380 --> 00:01:46.860
we have to do it on the
right-hand side as well.

00:01:46.860 --> 00:01:51.710
And so what does this get us?

00:01:51.710 --> 00:01:54.620
So our left-hand side, if
I have 8 of something--

00:01:54.620 --> 00:01:57.580
and in this case the something
is absolute values of x plus

00:01:57.580 --> 00:02:01.430
7's-- but if I have 8 of
something and I add 6 of that

00:02:01.430 --> 00:02:04.720
same something, I now
have 14 of that something.

00:02:04.720 --> 00:02:08.360
So that's going to be
14 absolute values of x

00:02:08.360 --> 00:02:12.596
plus 7, 14 times the
absolute value of x plus 7.

00:02:12.596 --> 00:02:16.330
The 4 and the negative 4 cancel
out, and that was intentional.

00:02:16.330 --> 00:02:19.930
The negative 6 and the
6 x plus 7's cancel out,

00:02:19.930 --> 00:02:22.250
or absolute values of
x plus 7's cancel out,

00:02:22.250 --> 00:02:23.650
and that was intentional.

00:02:23.650 --> 00:02:27.080
And then we're left with 6
minus 4, which is just 2.

00:02:27.080 --> 00:02:30.720
So that's going
to be equal to 2.

00:02:30.720 --> 00:02:32.350
Now just as promised,
we want to solve

00:02:32.350 --> 00:02:34.590
for the absolute
value of x plus 7,

00:02:34.590 --> 00:02:36.970
so let's divide both
sides by 14 to get rid

00:02:36.970 --> 00:02:39.459
of that coefficient
there, or that factor,

00:02:39.459 --> 00:02:41.500
or whatever you want to
call it, the thing that's

00:02:41.500 --> 00:02:44.200
multiplying the absolute
value of x plus 7.

00:02:44.200 --> 00:02:46.930
So we'll divide
both sides by 14,

00:02:46.930 --> 00:02:49.860
and we are left with
the absolute value

00:02:49.860 --> 00:02:55.410
of x plus 7 is equal to 2/14.

00:02:55.410 --> 00:02:59.500
They're both divisible by 2, so
this is the same thing as 1/7.

00:02:59.500 --> 00:03:01.140
So just as promised,
we've now solved

00:03:01.140 --> 00:03:03.180
for the absolute
value of x plus 7,

00:03:03.180 --> 00:03:05.050
but we really need
to solve for x.

00:03:05.050 --> 00:03:07.300
So how can we
reason through this?

00:03:07.300 --> 00:03:11.460
So if I take the absolute value
of something and I got you 1/7,

00:03:11.460 --> 00:03:14.320
there's two possible things that
I took the absolute value of.

00:03:14.320 --> 00:03:17.850
I could have taken the
absolute value of positive 1/7,

00:03:17.850 --> 00:03:20.770
or I could've taken the
absolute value of negative 1/7.

00:03:20.770 --> 00:03:22.900
So this thing that we're
taking the absolute value

00:03:22.900 --> 00:03:30.360
of-- so x plus 7-- could
be equal to positive 1/7,

00:03:30.360 --> 00:03:35.600
or x plus 7 could be
equal to negative 1/7.

00:03:39.300 --> 00:03:42.580
And just think about
that for a second.

00:03:42.580 --> 00:03:45.410
If this thing right over
here were equal to 1/7,

00:03:45.410 --> 00:03:47.870
you take its absolute
value, it'd be 1/7.

00:03:47.870 --> 00:03:52.090
If this thing was negative 1/7,
you take its absolute value,

00:03:52.090 --> 00:03:53.590
it would be positive 1/7.

00:03:53.590 --> 00:03:55.390
So that's how we got this.

00:03:55.390 --> 00:03:57.470
So now let's just solve for x.

00:03:57.470 --> 00:04:00.550
So if we subtract
7 from both sides

00:04:00.550 --> 00:04:02.160
for this left-hand
equation, we get

00:04:02.160 --> 00:04:08.570
x is equal to 1/7
minus-- and 7 we

00:04:08.570 --> 00:04:17.640
can rewrite as 49/7, which
is equal to negative 48/7.

00:04:17.640 --> 00:04:19.829
So that's one possibility for x.

00:04:19.829 --> 00:04:23.600
And then the other
possibility we

00:04:23.600 --> 00:04:30.680
would get x is equal to-- so we
have negative 1/7 minus 49/7.

00:04:30.680 --> 00:04:32.670
We're just subtracting
7 from both sides.

00:04:32.670 --> 00:04:34.770
That's what 49/7 is.

00:04:34.770 --> 00:04:40.170
And then this gets
us to negative 50/7.

00:04:40.170 --> 00:04:43.640
So the two solutions
to this what we thought

00:04:43.640 --> 00:04:45.850
was a complicated
equation are negative

00:04:45.850 --> 00:04:49.130
48/7 and negative 50/7.


Mixed number or improper fraction on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:05.214
Move the orange dot to
9/4 on the number line.

00:00:05.214 --> 00:00:07.130
Let's think about this
number line right here.

00:00:07.130 --> 00:00:08.360
This is 0.

00:00:08.360 --> 00:00:09.230
This is 3.

00:00:09.230 --> 00:00:11.870
So this must be 1 and 2.

00:00:11.870 --> 00:00:13.660
We need to go to 9/4.

00:00:13.660 --> 00:00:15.690
So there's two ways
of thinking about 9/4.

00:00:15.690 --> 00:00:18.350
One way is to try to convert
it to a mixed number.

00:00:18.350 --> 00:00:20.810
4 goes into 9 two times.

00:00:20.810 --> 00:00:22.840
And you'll have 1 left over.

00:00:22.840 --> 00:00:25.340
So this you could
do as 2 and 1/4.

00:00:25.340 --> 00:00:29.410
So you could say, this is 1, 2.

00:00:29.410 --> 00:00:32.409
And then, we have to go 1/4
of the way between 2 and 3.

00:00:32.409 --> 00:00:34.840
So that looks like
about 2 and 1/4.

00:00:34.840 --> 00:00:36.440
So you could think
about it that way.

00:00:36.440 --> 00:00:38.960
Or you could even view it
as an improper fraction.

00:00:38.960 --> 00:00:40.790
So let's go by 1/4's.

00:00:40.790 --> 00:00:49.740
So that's 1/4, 2/4, 3/4, 4/4,
which is a whole, 5/4, 6/4,

00:00:49.740 --> 00:00:53.730
7/4, 8/4, which is
equal to 2 wholes.

00:00:53.730 --> 00:00:55.980
8 divided by 4 is 2.

00:00:55.980 --> 00:00:59.150
And then, we go from 8/4 to 9/4.

00:00:59.150 --> 00:01:02.410
Either way, we get to that exact
same point on the number line.

00:01:02.410 --> 00:01:04.800
Let's do a few more of these.

00:01:04.800 --> 00:01:08.720
Move the orange dot to
11/5 on the number line.

00:01:08.720 --> 00:01:10.680
So we can do it both ways again.

00:01:10.680 --> 00:01:12.989
5 goes into 11 two times.

00:01:12.989 --> 00:01:14.280
And then, you have 1 left over.

00:01:14.280 --> 00:01:16.660
So you could call
this 2 and 1/5.

00:01:16.660 --> 00:01:19.969
So we could go 1,
2, and then 1/5.

00:01:19.969 --> 00:01:21.510
Or we could just
think of it in terms

00:01:21.510 --> 00:01:23.580
of 1/5's as an
improper fraction.

00:01:23.580 --> 00:01:35.740
1/5, 2/5, 3/5, 4/5, 5/5 is
a whole, 6/5, 7/5, 8/5, 9/5,

00:01:35.740 --> 00:01:37.050
10/5.

00:01:37.050 --> 00:01:38.480
10 divided by 5 is 2.

00:01:38.480 --> 00:01:40.530
10/5 is two wholes.

00:01:40.530 --> 00:01:44.440
10/5 and 11/5--
I'm having more fun

00:01:44.440 --> 00:01:46.790
than I expected to
with this exercise.

00:01:46.790 --> 00:01:48.460
Let's do one more.

00:01:48.460 --> 00:01:51.010
Orange dot to 5/2
on the number line.

00:01:51.010 --> 00:01:52.300
So this is 1/2.

00:01:52.300 --> 00:01:54.480
This is 2/2, which is a whole.

00:01:54.480 --> 00:01:56.050
This is 3/2.

00:01:56.050 --> 00:01:58.010
This is 4/2, which
is two wholes.

00:01:58.010 --> 00:01:59.262
This is 5/2.

00:01:59.262 --> 00:02:01.220
Another way you could
have thought about it-- 2

00:02:01.220 --> 00:02:03.890
goes into 5 two times
with a remainder 1.

00:02:03.890 --> 00:02:06.240
So this is 2 and 1/2
as a mixed number--

00:02:06.240 --> 00:02:10.640
1, 2, and 1/2 as a mixed number.


Multiplying two fractions: an explanation

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.840
Let's think about what it
means to multiply 2 over 3,

00:00:04.840 --> 00:00:11.904
or 2/3, times 4/5.

00:00:11.904 --> 00:00:13.320
In a previous
video, we've already

00:00:13.320 --> 00:00:15.070
seen how we can
actually compute this.

00:00:15.070 --> 00:00:17.490
This is going to be equal
to-- in the numerator,

00:00:17.490 --> 00:00:19.380
we just multiply the numerators.

00:00:19.380 --> 00:00:20.970
So it's going to be 2 times 4.

00:00:25.050 --> 00:00:27.880
And in the denominator, we
just multiply the denominator.

00:00:27.880 --> 00:00:31.130
So it's going to be 3 times 5.

00:00:34.670 --> 00:00:38.400
And so the numerator
is going to be 8,

00:00:38.400 --> 00:00:40.504
and the denominator
is going to be 15.

00:00:40.504 --> 00:00:42.420
And this is about as
simple as we can make it.

00:00:42.420 --> 00:00:48.490
8 and 15 don't have any factors
common to each other, than 1,

00:00:48.490 --> 00:00:49.490
so this is what it is.

00:00:49.490 --> 00:00:50.380
It's 8/15.

00:00:50.380 --> 00:00:52.880
But how, why does that
actually makes sense?

00:00:52.880 --> 00:00:54.360
And to think about
it, we'll think

00:00:54.360 --> 00:00:56.910
of two ways of visualizing it.

00:00:56.910 --> 00:00:57.966
So let's draw 2/3.

00:01:01.090 --> 00:01:02.430
I'll draw it relatively big.

00:01:02.430 --> 00:01:07.390
So I'm going to draw 2/3, and
I'm going to take 4/5 of it.

00:01:07.390 --> 00:01:10.440
So 2/3, and I'm going
to make it pretty big.

00:01:16.600 --> 00:01:17.590
Just like this.

00:01:17.590 --> 00:01:20.140
So this is 1/3.

00:01:20.140 --> 00:01:21.990
And then this would be 2/3.

00:01:21.990 --> 00:01:24.770
Which I could do a little bit
better job making those equal,

00:01:24.770 --> 00:01:27.330
or at least closer
to looking equal.

00:01:27.330 --> 00:01:28.170
So there you go.

00:01:28.170 --> 00:01:30.000
I have thirds.

00:01:30.000 --> 00:01:32.400
Let me do it one more time.

00:01:32.400 --> 00:01:34.910
So here I have drawn thirds.

00:01:34.910 --> 00:01:37.630
2/3 represents 2 of them.

00:01:37.630 --> 00:01:40.720
It represents 2 of them.

00:01:40.720 --> 00:01:44.880
One way to think about
this is 2/3 times 4/5

00:01:44.880 --> 00:01:48.720
is 4/5 of this 2/3.

00:01:48.720 --> 00:01:54.630
So how do we divide
this 2/3 into fifths?

00:01:54.630 --> 00:01:57.600
Well, what if we divided each
of these sections into 5.

00:01:57.600 --> 00:01:59.000
So let's do that.

00:01:59.000 --> 00:02:01.512
So let's divide each into 5.

00:02:01.512 --> 00:02:05.958
1, 2, 3, 4, 5.

00:02:05.958 --> 00:02:09.430
1, 2, 3, 4, 5.

00:02:09.430 --> 00:02:11.800
And I could even divide
this into 5 if I want.

00:02:11.800 --> 00:02:16.330
1, 2, 3, 4, 5.

00:02:16.330 --> 00:02:19.180
And we want to take 4/5
of this section here.

00:02:19.180 --> 00:02:20.760
So how many fifths
do we have here?

00:02:20.760 --> 00:02:24.924
We have 1, 2, 3, 4,
5, 6, 7, 8, 9, 10.

00:02:24.924 --> 00:02:26.090
And we've got to be careful.

00:02:26.090 --> 00:02:27.520
These really aren't fifths.

00:02:27.520 --> 00:02:29.421
These are actually
15ths, because the whole

00:02:29.421 --> 00:02:30.420
is this thing over here.

00:02:30.420 --> 00:02:33.130
So I should really say
how many 15ths do we have?

00:02:33.130 --> 00:02:35.080
And that's where we
get this number from.

00:02:35.080 --> 00:02:40.965
But you see if 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14,

00:02:40.965 --> 00:02:41.840
15.

00:02:41.840 --> 00:02:43.180
Where did that come from?

00:02:43.180 --> 00:02:45.219
I had 3, I had thirds.

00:02:45.219 --> 00:02:46.760
And then I took each
of those thirds,

00:02:46.760 --> 00:02:48.410
and I split them into fifths.

00:02:48.410 --> 00:02:50.710
So then I have five
times as many sections.

00:02:50.710 --> 00:02:52.930
3 times 5 is 15.

00:02:52.930 --> 00:02:56.150
But now we want 4/5 of
this right over here.

00:02:56.150 --> 00:02:58.220
This is 10/15 right over here.

00:02:58.220 --> 00:03:00.530
Notice it's the
same thing as 2/3.

00:03:00.530 --> 00:03:03.730
Now if we want to
take 4/5 of that,

00:03:03.730 --> 00:03:06.700
if you have 10 of something,
that's going to be 8 of them.

00:03:06.700 --> 00:03:08.280
So we're going to
take 8 of them.

00:03:08.280 --> 00:03:18.440
So 1, 2, 3, 4, 5, 6, 7, 8.

00:03:18.440 --> 00:03:24.310
We took 8 of the
15, so that is 8/15.

00:03:24.310 --> 00:03:27.220
You could have thought about
it the other way around.

00:03:27.220 --> 00:03:29.230
You could have
started with fifths.

00:03:29.230 --> 00:03:31.690
So let me draw it that way.

00:03:31.690 --> 00:03:34.250
So let me draw a whole.

00:03:34.250 --> 00:03:35.220
So this is a whole.

00:03:38.840 --> 00:03:40.860
Let me cut it into
five equal pieces,

00:03:40.860 --> 00:03:43.440
or as close as I can
draw five equal pieces.

00:03:43.440 --> 00:03:48.770
1, 2, 3, 4, 5.

00:03:48.770 --> 00:03:53.030
4/5, we're going to
shade in 4 of them.

00:03:53.030 --> 00:03:55.670
4 of the 5 equal pieces.

00:03:55.670 --> 00:03:58.660
3, 4.

00:03:58.660 --> 00:04:00.980
And now we want to
take 2/3 of that.

00:04:00.980 --> 00:04:02.500
Well, how can we do that?

00:04:02.500 --> 00:04:06.280
Well, let's split each
of these 5 into 3 pieces.

00:04:06.280 --> 00:04:09.650
So now we have
essentially 15ths again.

00:04:09.650 --> 00:04:19.610
So 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15.

00:04:19.610 --> 00:04:21.580
We want to take 2/3
of this yellow area.

00:04:21.580 --> 00:04:23.330
We're not taking 2/3
of the whole section.

00:04:23.330 --> 00:04:25.910
We're taking 2/3 of the 4/5.

00:04:25.910 --> 00:04:27.440
So how many 15ths
do we have here?

00:04:27.440 --> 00:04:33.660
We have 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12.

00:04:33.660 --> 00:04:35.610
So if you have 12 of
something, and you

00:04:35.610 --> 00:04:40.040
want to take 2/3 of that, you're
going to be taking 8 of it.

00:04:40.040 --> 00:04:50.970
So you're going to be taking
1, 2, 3, 4, 5, 6, 7, 8 or 8

00:04:50.970 --> 00:04:52.940
of the fifteenths now.

00:04:52.940 --> 00:04:56.290
So either way, you get
to the same result.

00:04:56.290 --> 00:05:00.360
One way, you're thinking
of taking 4/5 of 2/3.

00:05:00.360 --> 00:05:05.143
Another way you could think of
it as you're taking 2/3 of 4/5.


Multiplying negative and positive fractions

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.070
Let's do a few examples
multiplying fractions.

00:00:04.070 --> 00:00:09.045
So let's multiply
negative 7 times 3/49.

00:00:12.520 --> 00:00:15.400
So you might say, I don't
see a fraction here.

00:00:15.400 --> 00:00:16.840
This looks like an integer.

00:00:16.840 --> 00:00:19.850
But you just to remind yourself
that the negative 7 can

00:00:19.850 --> 00:00:26.610
be rewritten as
negative 7/1 times 3/49.

00:00:26.610 --> 00:00:28.710
Now we can multiply
the numerators.

00:00:28.710 --> 00:00:31.950
So the numerator is going
to be negative 7 times 3.

00:00:31.950 --> 00:00:34.700
And the denominator is
going to be 1 times 49.

00:00:34.700 --> 00:00:36.590
1 times 49.

00:00:36.590 --> 00:00:40.650
And this is going to be
equal to-- 7 times 3 is 21.

00:00:40.650 --> 00:00:42.975
And one of their
signs is negative,

00:00:42.975 --> 00:00:45.350
so a negative times a positive
is going to be a negative.

00:00:45.350 --> 00:00:47.030
So this is going
to be negative 21.

00:00:47.030 --> 00:00:49.090
You could view this as
negative 7 plus negative 7

00:00:49.090 --> 00:00:50.460
plus negative 7.

00:00:50.460 --> 00:00:54.470
And that's going to be over 49.

00:00:54.470 --> 00:00:57.320
And this is the correct
value, but we can simplify it

00:00:57.320 --> 00:01:02.150
more because 21 and 49
both share 7 as a factor.

00:01:02.150 --> 00:01:03.793
That's their greatest
common factor.

00:01:03.793 --> 00:01:05.209
So let's divide
both the numerator

00:01:05.209 --> 00:01:07.400
and the denominator by 7.

00:01:07.400 --> 00:01:10.810
Divide the numerator and
the denominator by 7.

00:01:10.810 --> 00:01:13.470
And so this gets us
negative 3 in the numerator.

00:01:13.470 --> 00:01:15.120
And in the
denominator, we have 7.

00:01:15.120 --> 00:01:16.960
So we could view it
as negative 3 over 7.

00:01:16.960 --> 00:01:20.070
Or, you could even do
it as negative 3/7.

00:01:20.070 --> 00:01:21.720
Let's do another one.

00:01:21.720 --> 00:01:30.040
Let's take 5/9 times-- I'll
switch colors more in this one.

00:01:30.040 --> 00:01:32.280
That one's a little monotonous
going all red there.

00:01:32.280 --> 00:01:36.790
5/9 times 3/15.

00:01:40.550 --> 00:01:42.220
So this is going
to be equal to-- we

00:01:42.220 --> 00:01:43.970
multiply the numerators.

00:01:43.970 --> 00:01:46.340
So it's going to be 5 times 3.

00:01:46.340 --> 00:01:50.170
5 times 3 in the numerator.

00:01:50.170 --> 00:01:53.123
And the denominator is
going to be 9 times 15.

00:01:53.123 --> 00:01:58.530
9 times 15.

00:02:02.050 --> 00:02:04.700
We could multiply them out,
but just leaving it like this

00:02:04.700 --> 00:02:06.760
you see that there is
already common factors

00:02:06.760 --> 00:02:08.500
in the numerator
and the denominator.

00:02:08.500 --> 00:02:10.220
Both the numerator
and the denominator,

00:02:10.220 --> 00:02:12.440
they're both divisible
by 5 and they're

00:02:12.440 --> 00:02:14.700
both divisible by 3,
which essentially tells us

00:02:14.700 --> 00:02:17.600
that they're divisible by 15.

00:02:17.600 --> 00:02:21.920
So we can divide the numerator
and denominator by 15.

00:02:21.920 --> 00:02:24.270
So divide the
numerator by 15, which

00:02:24.270 --> 00:02:27.020
is just like dividing by
5 and then dividing by 3.

00:02:27.020 --> 00:02:29.090
So we'll just divide by 15.

00:02:29.090 --> 00:02:30.510
Divide by 15.

00:02:30.510 --> 00:02:33.910
And this is going to be equal
to-- well, 5 times 3 is 15.

00:02:33.910 --> 00:02:36.480
Divided by 15 you get
1 in the numerator.

00:02:36.480 --> 00:02:39.700
And in the denominator,
9 times 15 divided by 15.

00:02:39.700 --> 00:02:43.160
Well, that's just going to be 9.

00:02:43.160 --> 00:02:45.090
So it's equal to 1/9.

00:02:45.090 --> 00:02:46.150
Let's do another one.

00:02:46.150 --> 00:02:54.860
What would negative 5/9
times negative 3/15 be?

00:02:54.860 --> 00:02:56.360
Well, we've already
figured out what

00:02:56.360 --> 00:03:00.309
positive 5/9 times
positive 3/15 would be.

00:03:00.309 --> 00:03:02.100
So now we just have to
care about the sign.

00:03:02.100 --> 00:03:05.090
If we were just multiplying the
two positives, it would be 1/9.

00:03:05.090 --> 00:03:07.110
But now we have to
think about the fact

00:03:07.110 --> 00:03:09.409
that we're multiplying by a
negative times a negative.

00:03:09.409 --> 00:03:10.825
Now, we remember
when you multiply

00:03:10.825 --> 00:03:14.120
a negative times a
negative, it's a positive.

00:03:14.120 --> 00:03:15.690
The only way that
you get a negative

00:03:15.690 --> 00:03:17.690
is if one of those two
numbers that you're

00:03:17.690 --> 00:03:20.290
taking the product of
is negative, not two.

00:03:20.290 --> 00:03:22.090
If both are positive,
it's positive.

00:03:22.090 --> 00:03:24.912
If both are negative,
it's positive.

00:03:24.912 --> 00:03:25.995
Let's do one more example.

00:03:29.020 --> 00:03:32.620
Let's take 5-- I'm using
the number 5 a lot.

00:03:32.620 --> 00:03:40.100
So let's do 3/2, just
to show that this

00:03:40.100 --> 00:03:42.450
would work with
improper fractions.

00:03:42.450 --> 00:03:49.270
3/2 times negative 7/10.

00:03:53.720 --> 00:03:55.940
I'm arbitrarily picking colors.

00:03:55.940 --> 00:03:58.910
And so our numerator is going
to be 3 times negative 7.

00:03:58.910 --> 00:04:02.500
3 times negative 7.

00:04:02.500 --> 00:04:06.790
And our denominator is
going to be 2 times 10.

00:04:06.790 --> 00:04:07.890
2 times 10.

00:04:12.550 --> 00:04:14.150
So this is going to
be the numerator.

00:04:14.150 --> 00:04:18.480
Positive times a
negative is a negative.

00:04:18.480 --> 00:04:22.010
3 times negative
7 is negative 21.

00:04:22.010 --> 00:04:23.660
Negative 21.

00:04:23.660 --> 00:04:25.550
And the denominator, 2 times 10.

00:04:25.550 --> 00:04:27.120
Well, that is just 20.

00:04:27.120 --> 00:04:29.710
So this is negative 21/20.

00:04:29.710 --> 00:04:33.110
And you really can't
simplify this any further.


Multiplying unit fractions and whole numbers

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:07.610
Let's think about how or
what 1/2 times 5 represents.

00:00:07.610 --> 00:00:11.600
So one way to think about it is
that this could be five 1/2's

00:00:11.600 --> 00:00:12.370
added together.

00:00:12.370 --> 00:00:17.050
So you could view this
as 1/2 plus 1/2 plus 1/2

00:00:17.050 --> 00:00:22.520
plus 1/2 plus 1/2, which is the
same thing as 1 plus 1 plus 1

00:00:22.520 --> 00:00:27.753
plus 1 plus 1, over 2,
which is equal to 5/2.

00:00:32.810 --> 00:00:34.900
The other way to
think about this

00:00:34.900 --> 00:00:36.695
is that you start with 5 things.

00:00:41.670 --> 00:00:42.960
So let's say, that's 1 thing.

00:00:42.960 --> 00:00:47.100
Let me copy and paste that
so they all look the same.

00:00:47.100 --> 00:00:48.930
So then let me paste it.

00:00:48.930 --> 00:00:51.340
So that's 2 things.

00:00:51.340 --> 00:00:53.890
That's 3 things.

00:00:53.890 --> 00:00:56.970
That's 4 things.

00:00:56.970 --> 00:00:58.500
And that's 5 things.

00:00:58.500 --> 00:01:01.850
So the other way to think about
it is you start with 5 things,

00:01:01.850 --> 00:01:03.880
and you take 1/2 of them.

00:01:03.880 --> 00:01:06.070
So what would be 1/2 of this?

00:01:06.070 --> 00:01:07.180
Well, let's see.

00:01:07.180 --> 00:01:11.290
You have 5 things, so you
would get-- 5 divided by 2

00:01:11.290 --> 00:01:13.610
would be 2 and 1/2.

00:01:13.610 --> 00:01:16.630
So you would get this far.

00:01:16.630 --> 00:01:18.450
Let me make it like this.

00:01:18.450 --> 00:01:20.630
So you would get this one.

00:01:20.630 --> 00:01:22.260
You would get this one.

00:01:22.260 --> 00:01:24.210
And you would get this one.

00:01:24.210 --> 00:01:27.200
Now, is this the
same thing as 5/2?

00:01:27.200 --> 00:01:29.980
Well, what happens if we
divide each of these wholes

00:01:29.980 --> 00:01:30.900
into halves?

00:01:30.900 --> 00:01:32.350
So let's do that.

00:01:32.350 --> 00:01:36.080
So if we just multiplied-- so
we just divide each of these

00:01:36.080 --> 00:01:37.330
into 2.

00:01:37.330 --> 00:01:41.670
So instead of having 5
wholes, we now have 10 halves.

00:01:44.640 --> 00:01:47.100
How many of those halves
have we filled in?

00:01:47.100 --> 00:01:52.790
Well, we have filled
in 1, 2, 3, 4, 5.

00:01:52.790 --> 00:01:57.986
So this is also equal to 5/2.

00:02:02.410 --> 00:02:04.480
So far we just did it
thinking about what

00:02:04.480 --> 00:02:06.760
multiplication actually means.

00:02:06.760 --> 00:02:09.937
But if you said, well,
how did I compute this?

00:02:09.937 --> 00:02:11.562
Well, the way you
could think about it,

00:02:11.562 --> 00:02:13.450
and multiplying
fractions is actually

00:02:13.450 --> 00:02:15.820
straightforward from
that point of view,

00:02:15.820 --> 00:02:19.550
is as long as you can express
both of them as fractions,

00:02:19.550 --> 00:02:24.410
and 5 we already know is
the same thing as 5 ones,

00:02:24.410 --> 00:02:27.925
so this we can just
multiply times 5/1.

00:02:32.450 --> 00:02:35.570
So now that I've expressed
both of them as fractions,

00:02:35.570 --> 00:02:37.960
I can just multiply
the numerator.

00:02:37.960 --> 00:02:46.560
So 1 times 5 over 2 times 1.

00:02:51.550 --> 00:02:53.860
And what's that
going to be equal to?

00:02:53.860 --> 00:02:57.700
Well, 1 times 5 is 5.

00:02:57.700 --> 00:03:03.120
2 times 1 is 2.

00:03:03.120 --> 00:03:06.675
So once again, we get 5/2.


Dividing a decimal by a whole number

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:09.150
Let's see if we can divide
5.005 by-- let's divide it by 7,

00:00:09.150 --> 00:00:11.430
and see what we get.

00:00:11.430 --> 00:00:21.390
So we can rewrite this
as 5.005 divided by 7.

00:00:21.390 --> 00:00:23.704
And the key here is to
keep track of the decimal.

00:00:23.704 --> 00:00:25.120
But other than
that, you're really

00:00:25.120 --> 00:00:27.920
treating it like a traditional
long division problem.

00:00:27.920 --> 00:00:30.300
So you want to put
the decimal, it's

00:00:30.300 --> 00:00:32.250
to the right of the ones place.

00:00:32.250 --> 00:00:33.770
So it's right over there.

00:00:33.770 --> 00:00:36.002
I just put the decimal
right above the decimal.

00:00:36.002 --> 00:00:37.460
And now we would
treat it just like

00:00:37.460 --> 00:00:39.720
a traditional long
division problem.

00:00:39.720 --> 00:00:42.625
So how many times
does 7 go into 5?

00:00:42.625 --> 00:00:45.580
Well, it goes into
5 zero times, and we

00:00:45.580 --> 00:00:47.849
could write 0 times 7 is 0.

00:00:47.849 --> 00:00:49.390
Actually, let's do
that, just to show

00:00:49.390 --> 00:00:52.710
that it works to kind of do the
process all the way through.

00:00:52.710 --> 00:00:54.540
0 times 7 is 0.

00:00:54.540 --> 00:00:57.450
You subtract, you get 5.

00:00:57.450 --> 00:00:59.980
So you get this 5
right over here.

00:00:59.980 --> 00:01:03.830
So you get 5, and then we
can bring down another 0.

00:01:03.830 --> 00:01:06.530
We bring down this
0 right over here.

00:01:06.530 --> 00:01:11.260
So now we ask ourselves, how
many times does 7 go into 50?

00:01:11.260 --> 00:01:14.760
Well, 7 times 7 is 49, so
it's going to go seven times.

00:01:14.760 --> 00:01:17.110
7 times 7, 49.

00:01:17.110 --> 00:01:19.770
Subtract, you get 1.

00:01:19.770 --> 00:01:24.140
And now we can
bring down this 0.

00:01:24.140 --> 00:01:25.570
Let's do that.

00:01:25.570 --> 00:01:27.540
We bring down that 0.

00:01:27.540 --> 00:01:30.280
7 goes into 10 one time.

00:01:30.280 --> 00:01:32.100
1 times 7 is 7.

00:01:32.100 --> 00:01:34.320
Subtract, get a 3.

00:01:34.320 --> 00:01:37.860
And now finally, we
can bring down this 5.

00:01:37.860 --> 00:01:41.320
We can bring down the 5.

00:01:41.320 --> 00:01:43.940
And we ask ourselves, how
many times does 7 go into 35?

00:01:43.940 --> 00:01:45.740
Well, 5 times 7 is 35.

00:01:45.740 --> 00:01:47.760
This goes 5 times.

00:01:47.760 --> 00:01:50.270
5 times 7 is 35.

00:01:50.270 --> 00:01:52.030
Subtract, and we are done.

00:01:52.030 --> 00:01:54.370
We have divided it completely.

00:01:54.370 --> 00:02:02.360
And so 5.005 divided
by 7 is equal to 0.715.


Formal and alternate form of the derivative for ln x

WEBVTT
Kind: captions
Language: en

00:00:00.980 --> 00:00:05.260
Let's say that f of x is
equal to the natural log of x,

00:00:05.260 --> 00:00:08.240
and we want to figure out what
the slope of the tangent line

00:00:08.240 --> 00:00:13.030
to the curve f is when x
is equal to the number e.

00:00:16.180 --> 00:00:18.550
So here, x is equal
to the number e.

00:00:18.550 --> 00:00:22.080
The point e comma
1 is on the curve.

00:00:22.080 --> 00:00:23.720
f of e is 1.

00:00:23.720 --> 00:00:25.680
The natural log of e is 1.

00:00:25.680 --> 00:00:28.240
And I've drawn the slope
of the tangent line,

00:00:28.240 --> 00:00:30.144
or I've drawn the tangent line.

00:00:30.144 --> 00:00:32.060
And we need to figure
out what the slope of it

00:00:32.060 --> 00:00:34.380
is, or at least come up
with an expression for it.

00:00:34.380 --> 00:00:36.421
And I'm going to come up
with an expression using

00:00:36.421 --> 00:00:39.510
both the formal definition
and the alternate definition.

00:00:39.510 --> 00:00:41.940
That will allow us to
compare them a little bit.

00:00:41.940 --> 00:00:44.096
So let's think about first
the formal definition.

00:00:44.096 --> 00:00:45.470
So the formal
definition wants us

00:00:45.470 --> 00:00:48.000
to find an expression for the
derivative of our function

00:00:48.000 --> 00:00:48.630
at any x.

00:00:48.630 --> 00:00:51.890
So let's say that this is some
arbitrary x right over here.

00:00:51.890 --> 00:00:56.170
This would be the
point x comma f of x.

00:00:56.170 --> 00:01:02.180
And let's say that this is--
let's call this x plus h.

00:01:02.180 --> 00:01:06.170
So this distance right
over here is going to be h.

00:01:06.170 --> 00:01:08.100
This right over
here is going to be

00:01:08.100 --> 00:01:14.460
the point, x plus
h f of x plus h.

00:01:14.460 --> 00:01:18.990
Now, the whole underlying
idea of the formal definition

00:01:18.990 --> 00:01:22.500
of limits is to find the
slope of the secant line

00:01:22.500 --> 00:01:26.960
between these two
points, and then

00:01:26.960 --> 00:01:31.320
take the limit as
h approaches 0.

00:01:31.320 --> 00:01:33.540
As h gets closer and
closer, this blue point

00:01:33.540 --> 00:01:36.600
is going to get closer and
closer and closer to x.

00:01:36.600 --> 00:01:39.380
And this point is going to
approach it on the curve.

00:01:39.380 --> 00:01:41.240
And the secant line
is going to become

00:01:41.240 --> 00:01:45.320
a better and better and
better approximation

00:01:45.320 --> 00:01:47.640
of the tangent line at x.

00:01:47.640 --> 00:01:49.030
So let's actually do that.

00:01:49.030 --> 00:01:50.880
So what's the slope
of the secant line?

00:01:50.880 --> 00:01:53.550
Well, it's the change in
your vertical axis, which

00:01:53.550 --> 00:02:01.270
is going to be f of x
plus h minus f of x--

00:02:01.270 --> 00:02:04.040
over the change in
your horizontal axis.

00:02:04.040 --> 00:02:06.210
And that's x plus h minus x.

00:02:06.210 --> 00:02:08.570
And we see here the
difference is just h.

00:02:08.570 --> 00:02:09.889
Over h.

00:02:09.889 --> 00:02:13.530
And we're going to
take the limit of that

00:02:13.530 --> 00:02:16.570
as h approaches 0.

00:02:16.570 --> 00:02:19.250
So in the case when f of
x is the natural log of x,

00:02:19.250 --> 00:02:24.340
this will reduce to the
limit as h approaches 0.

00:02:24.340 --> 00:02:29.120
f of x plus h is
the natural log of x

00:02:29.120 --> 00:02:36.200
plus h minus the natural log
of x, all of that over h.

00:02:39.010 --> 00:02:42.620
So this right over here,
for our particular f of x,

00:02:42.620 --> 00:02:46.410
this is equal to f prime of x.

00:02:46.410 --> 00:02:49.460
So if we wanted to evaluate
this when x is equal to e,

00:02:49.460 --> 00:02:51.180
then everywhere we
see an x we just

00:02:51.180 --> 00:02:52.530
have to replace it with an e.

00:02:52.530 --> 00:02:54.900
This is essentially
expressing our derivative

00:02:54.900 --> 00:02:56.659
as a function of x.

00:02:56.659 --> 00:02:58.450
It's kind of a
crazy-looking function of x.

00:02:58.450 --> 00:03:00.140
You have a limit
here and all of that.

00:03:00.140 --> 00:03:03.050
But every place you see an x,
like any function definition,

00:03:03.050 --> 00:03:04.460
you can replace
it now with an e.

00:03:04.460 --> 00:03:08.650
So we can-- let me just do that.

00:03:08.650 --> 00:03:09.500
Whoops.

00:03:09.500 --> 00:03:11.560
I lost my screen.

00:03:11.560 --> 00:03:12.520
Here we go.

00:03:12.520 --> 00:03:20.390
So we could write
f prime of e is

00:03:20.390 --> 00:03:27.410
equal to the limit as h
approaches 0 of natural log--

00:03:27.410 --> 00:03:28.910
let me do it in the
same color so we

00:03:28.910 --> 00:03:34.520
can keep track of things--
natural log of e plus h--

00:03:34.520 --> 00:03:37.040
I'll just leave
that blank for now--

00:03:37.040 --> 00:03:41.960
minus the natural log of
e, all of that over h.

00:03:46.280 --> 00:03:48.380
So just like that.

00:03:48.380 --> 00:03:50.860
This right over here, if
we evaluate this limit--

00:03:50.860 --> 00:03:52.710
if we're able to and
we actually can--

00:03:52.710 --> 00:03:54.500
if we are able to
evaluate this limit,

00:03:54.500 --> 00:03:57.180
this would give us the slope
of the tangent line when

00:03:57.180 --> 00:03:58.270
x equals e.

00:03:58.270 --> 00:04:00.760
This is doing the
formal definition.

00:04:00.760 --> 00:04:02.370
Now let's do the
alternate definition.

00:04:02.370 --> 00:04:04.720
The alternate
definition-- if you

00:04:04.720 --> 00:04:08.442
don't want to find a
general derivative expressed

00:04:08.442 --> 00:04:10.150
as a function of x
like this and you just

00:04:10.150 --> 00:04:12.470
want to find the slope
at a particular point,

00:04:12.470 --> 00:04:15.430
the alternate definition kind of
just gets straight to the point

00:04:15.430 --> 00:04:16.209
there.

00:04:16.209 --> 00:04:19.070
So what they say is
hey, look, let's imagine

00:04:19.070 --> 00:04:22.240
some other x value here.

00:04:22.240 --> 00:04:25.280
So let's imagine
some other x value.

00:04:25.280 --> 00:04:31.210
This right over here is
the point x comma-- well,

00:04:31.210 --> 00:04:36.222
we could say f of x or we could
even say the natural log of x.

00:04:36.222 --> 00:04:38.805
What is the slope of the secant
line between those two points?

00:04:44.220 --> 00:04:47.510
Well, it's going to be
your change in y values.

00:04:47.510 --> 00:04:54.480
So it's going to be
natural log of x minus 1--

00:04:54.480 --> 00:05:01.140
let me do that red color--
over your change in x values.

00:05:01.140 --> 00:05:02.468
That's x minus e.

00:05:07.140 --> 00:05:09.810
So that's the slope of the
secant line between those two

00:05:09.810 --> 00:05:10.432
points.

00:05:10.432 --> 00:05:12.390
Well, what if you want
to get the tangent line?

00:05:12.390 --> 00:05:15.490
Well, let's just take the
limit as x approaches e.

00:05:15.490 --> 00:05:17.280
As x gets closer and
closer and closer,

00:05:17.280 --> 00:05:19.750
these points are going to get
closer and closer and closer,

00:05:19.750 --> 00:05:21.833
and the secant line is
going to better approximate

00:05:21.833 --> 00:05:22.870
the tangent line.

00:05:22.870 --> 00:05:27.530
So we're just going to take
the limit as x approaches e.

00:05:27.530 --> 00:05:28.780
So either one of this.

00:05:28.780 --> 00:05:31.860
This is using the formal
definition of a limit.

00:05:31.860 --> 00:05:35.530
Let me make it clear that that
h does not belong part of it.

00:05:35.530 --> 00:05:38.798
So we could either do it
using the formal definition

00:05:38.798 --> 00:05:44.408
or the alternate definition
of the derivative.


Formal and alternate form of the derivative example 1

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:02.950
The alternate form of the
derivative of the function

00:00:02.950 --> 00:00:05.600
f, at a number a,
denoted by f prime of a,

00:00:05.600 --> 00:00:08.640
is given by this stuff.

00:00:08.640 --> 00:00:10.632
Now this might look a
little strange to you,

00:00:10.632 --> 00:00:12.590
but if you really think
about what it's saying,

00:00:12.590 --> 00:00:13.965
it's really just
taking the slope

00:00:13.965 --> 00:00:18.260
of the tangent line
between a comma f of a.

00:00:18.260 --> 00:00:21.735
So let's imagine some
arbitrary function like this.

00:00:21.735 --> 00:00:23.360
Let's say that that
is-- well I'll just

00:00:23.360 --> 00:00:27.020
write that's our function f.

00:00:27.020 --> 00:00:28.620
And so you could
have the point when

00:00:28.620 --> 00:00:31.680
x is equal to a-- this is our
x-axis-- when x is equal to a,

00:00:31.680 --> 00:00:35.450
this is the point a, f of a.

00:00:35.450 --> 00:00:38.570
You notice a, f of a.

00:00:38.570 --> 00:00:41.910
And then we could take the slope
between that and some arbitrary

00:00:41.910 --> 00:00:43.910
point, let's call that x.

00:00:43.910 --> 00:00:47.286
So this is the point x, f of x.

00:00:50.500 --> 00:00:53.290
And notice, the numerator
right here, this

00:00:53.290 --> 00:00:55.860
is just our change in the
value of our function.

00:00:55.860 --> 00:00:58.830
Or you could view that as the
change in the vertical axis.

00:00:58.830 --> 00:01:01.690
So that would give you this
distance right over here.

00:01:01.690 --> 00:01:04.379
That's what we're doing
up here in the numerator.

00:01:04.379 --> 00:01:06.410
And then in the
denominator, we're

00:01:06.410 --> 00:01:09.680
finding the change in
our horizontal values,

00:01:09.680 --> 00:01:11.131
horizontal coordinates.

00:01:11.131 --> 00:01:12.630
Let me do that in
a different color.

00:01:15.980 --> 00:01:19.791
So the change in the horizontal,
that's this right over here.

00:01:19.791 --> 00:01:22.290
And then they're trying to find
the limit as x approaches a.

00:01:22.290 --> 00:01:25.730
So as x gets closer and
closer and closer and closer

00:01:25.730 --> 00:01:29.950
to a, what's going to happen
is, is that when x is out here,

00:01:29.950 --> 00:01:31.840
we have this secant line.

00:01:31.840 --> 00:01:35.800
We're finding the slope
of this secant line.

00:01:35.800 --> 00:01:38.950
But as x gets closer and
closer, the secant lines

00:01:38.950 --> 00:01:41.230
better and better and
better approximate

00:01:41.230 --> 00:01:42.960
the slope of the tangent line.

00:01:42.960 --> 00:01:47.290
Where the limit, as x approaches
a, but doesn't quite equal a,

00:01:47.290 --> 00:01:48.920
is going to be--
this is actually

00:01:48.920 --> 00:01:50.780
our definition of
our derivative.

00:01:50.780 --> 00:01:53.600
Or I guess the alternate form
of the derivative definition.

00:01:53.600 --> 00:01:56.994
And this would be the slope of
the tangent line, if it exists.

00:01:56.994 --> 00:01:58.410
So with that all
that out the way,

00:01:58.410 --> 00:01:59.868
let's try to answer
their question.

00:01:59.868 --> 00:02:02.300
With the Alternative Form
of the Derivative as an aid,

00:02:02.300 --> 00:02:04.300
make sense of the
following limit expression

00:02:04.300 --> 00:02:08.560
by identifying the function
f and the number a.

00:02:08.560 --> 00:02:12.090
So right here, they want to find
the slope of the tangent line

00:02:12.090 --> 00:02:13.150
at 5.

00:02:13.150 --> 00:02:15.930
Here they wanted to find the
slope of the tangent line at a.

00:02:15.930 --> 00:02:21.650
So it's pretty clear
that a is equal to 5.

00:02:21.650 --> 00:02:27.340
And that f of a is equal to 125.

00:02:27.340 --> 00:02:30.780
Now what about f of x?

00:02:30.780 --> 00:02:33.740
Well here, it's a limit
of f of x minus f of a.

00:02:33.740 --> 00:02:38.375
Well here it's the limit as
x to the third minus 125.

00:02:38.375 --> 00:02:39.250
And this makes sense.

00:02:39.250 --> 00:02:43.860
If f of x is equal
to x to the third,

00:02:43.860 --> 00:02:49.940
then it makes sense that f of 5
is going to be 5 to the third,

00:02:49.940 --> 00:02:52.924
is going to be 125.

00:02:52.924 --> 00:02:55.340
And we're also taking up here
the limit as x approaches a.

00:02:55.340 --> 00:03:00.210
Here we're taking the
limit as x approaches 5.

00:03:00.210 --> 00:03:02.200
So this is the derivative
of the function

00:03:02.200 --> 00:03:04.960
f of x is equal
to x to the third.

00:03:04.960 --> 00:03:07.380
Let me write that down
in the green color.

00:03:07.380 --> 00:03:13.610
x to the third at the
number a is equal to 5.

00:03:13.610 --> 00:03:15.465
And so we can imagine this.

00:03:15.465 --> 00:03:18.090
Let's try to actually graph it,
just so that we can imagine it.

00:03:18.090 --> 00:03:20.510
Actually, I'll do
it out here, where

00:03:20.510 --> 00:03:24.050
I have a little bit better
contrast with the colors.

00:03:24.050 --> 00:03:27.410
So let's say that is my y-axis.

00:03:27.410 --> 00:03:31.250
Let's say that
this is my x-axis.

00:03:31.250 --> 00:03:33.470
I'm not going to quite
draw it to scale.

00:03:33.470 --> 00:03:36.140
Let's say this right
over here is the 125.

00:03:36.140 --> 00:03:38.950
Or y, this is when y equals 125.

00:03:38.950 --> 00:03:41.680
This is when x is equal
to 5, so they're clearly

00:03:41.680 --> 00:03:43.870
not at the same scale.

00:03:43.870 --> 00:03:48.466
But the function is going
to look something like this.

00:03:48.466 --> 00:03:50.090
We know what x to
the third looks like,

00:03:50.090 --> 00:03:57.480
it looks something like this.

00:04:01.050 --> 00:04:05.300
So here, our a is equal to 5.

00:04:05.300 --> 00:04:14.950
This point right
over here is 5, 125.

00:04:14.950 --> 00:04:17.680
And then we're taking the
slope between that point

00:04:17.680 --> 00:04:21.260
and an arbitrary x-value.

00:04:21.260 --> 00:04:23.650
Or I should say an arbitrary
other point on the curve.

00:04:23.650 --> 00:04:25.690
So this right over here
would be the point,

00:04:25.690 --> 00:04:29.130
we could call that
x, x to the third.

00:04:29.130 --> 00:04:32.699
We know that f of x is
equal to x to the third.

00:04:32.699 --> 00:04:33.740
And let me make it clear.

00:04:33.740 --> 00:04:40.260
This is a graph of y is
equal to x to the third.

00:04:40.260 --> 00:04:43.675
And so this expression, right
over here, all of this, this

00:04:43.675 --> 00:04:45.390
is the slope between
these two points.

00:04:48.700 --> 00:04:51.770
And as we take the
limit as x approaches 5,

00:04:51.770 --> 00:04:55.120
so right now this
is our x, as x gets

00:04:55.120 --> 00:04:59.440
closer and closer
and closer to 5,

00:04:59.440 --> 00:05:01.760
the secant lines are
going to better and better

00:05:01.760 --> 00:05:08.120
approximate the slope of the
tangent line at x equals 5.

00:05:08.120 --> 00:05:16.460
So the slope of a tangent line
would look something like that.


Formal and alternate form of the derivative

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.070
So let's think
about how we could

00:00:02.070 --> 00:00:04.170
find the slope of
the tangent line

00:00:04.170 --> 00:00:05.930
to this curve right
over here, so what

00:00:05.930 --> 00:00:09.230
I have drawn in red, at
the point x equals a.

00:00:09.230 --> 00:00:11.300
And we've already seen
this with the definition

00:00:11.300 --> 00:00:12.252
of the derivative.

00:00:12.252 --> 00:00:14.460
We could try to find a
general function that gives us

00:00:14.460 --> 00:00:16.770
the slope of the tangent
line at any point.

00:00:16.770 --> 00:00:19.030
So let's say we have
some arbitrary point.

00:00:19.030 --> 00:00:22.960
Let me define some arbitrary
point x right over here.

00:00:22.960 --> 00:00:26.930
Then this would be the
point x comma f of x.

00:00:26.930 --> 00:00:29.730
And then we could
take some x plus h.

00:00:29.730 --> 00:00:33.890
So let's say that this
right over here is the point

00:00:33.890 --> 00:00:35.700
x plus h.

00:00:35.700 --> 00:00:42.350
And so this point would be
x plus h, f of x plus h.

00:00:42.350 --> 00:00:45.610
We can find the slope
of the secant line that

00:00:45.610 --> 00:00:47.860
goes between these two points.

00:00:47.860 --> 00:00:50.350
So that would be your
change in your vertical,

00:00:50.350 --> 00:00:56.000
which would be f
of x plus h minus f

00:00:56.000 --> 00:01:02.850
of x, over the change
in the horizontal, which

00:01:02.850 --> 00:01:04.637
would be x plus h minus x.

00:01:10.740 --> 00:01:12.680
And these two x's cancel.

00:01:12.680 --> 00:01:15.290
So this would be the
slope of this secant line.

00:01:15.290 --> 00:01:20.350
And then if we want to find the
slope of the tangent line at x,

00:01:20.350 --> 00:01:24.260
we would just take the
limit of this expression

00:01:24.260 --> 00:01:26.550
as h approaches 0.

00:01:26.550 --> 00:01:30.810
As h approaches 0, this
point moves towards x.

00:01:30.810 --> 00:01:33.500
And that slope of the secant
line between these two

00:01:33.500 --> 00:01:36.680
is going to approximate the
slope of the tangent line at x.

00:01:36.680 --> 00:01:39.560
And so this right over
here, this we would say

00:01:39.560 --> 00:01:44.130
is equal to f prime of x.

00:01:44.130 --> 00:01:45.980
This is still a function of x.

00:01:45.980 --> 00:01:50.860
You give me an arbitrary x
where the derivative is defined.

00:01:50.860 --> 00:01:53.750
I'm going to plug it into this,
whatever this ends up being.

00:01:53.750 --> 00:01:55.870
It might be some nice,
clean algebraic expression.

00:01:55.870 --> 00:01:57.420
Then I'm going to
give you a number.

00:01:57.420 --> 00:01:59.430
So for example, if
you wanted to find--

00:01:59.430 --> 00:02:00.920
you could calculate
this somehow.

00:02:00.920 --> 00:02:02.600
Or you could even
leave it in this form.

00:02:02.600 --> 00:02:07.730
And then if you
wanted f prime of a,

00:02:07.730 --> 00:02:10.729
you would just substitute a
into your function definition.

00:02:10.729 --> 00:02:12.270
And you would say,
well, that's going

00:02:12.270 --> 00:02:17.590
to be the limit as h approaches
0 of-- every place you see

00:02:17.590 --> 00:02:20.340
an x, replace it with
an a. f of-- I'll

00:02:20.340 --> 00:02:31.980
stay in this color for now--
blank plus h minus f of blank,

00:02:31.980 --> 00:02:36.670
all of that over h.

00:02:36.670 --> 00:02:41.040
And I left those blanks so
I could write the a in red.

00:02:41.040 --> 00:02:44.300
Notice, every place where I
had an x before, it's now an a.

00:02:44.300 --> 00:02:47.300
So this is the derivative
evaluated at a.

00:02:47.300 --> 00:02:50.500
So this is one way to find
the slope of the tangent line

00:02:50.500 --> 00:02:51.720
when x equals a.

00:02:51.720 --> 00:02:54.310
Another way-- and
this is often used

00:02:54.310 --> 00:02:56.130
as the alternate form
of the derivative--

00:02:56.130 --> 00:02:59.140
would be to do it directly.

00:02:59.140 --> 00:03:01.820
So this is the point
a comma f of a.

00:03:01.820 --> 00:03:05.990
Let's just take another
arbitrary point someplace.

00:03:05.990 --> 00:03:09.270
So let's say this
is the value x.

00:03:09.270 --> 00:03:12.480
This point right over here on
the function would be x comma

00:03:12.480 --> 00:03:13.970
f of x.

00:03:13.970 --> 00:03:18.370
And so what's the slope of the
secant line between these two

00:03:18.370 --> 00:03:19.080
points?

00:03:19.080 --> 00:03:21.260
Well, it would be change
in the vertical, which

00:03:21.260 --> 00:03:29.980
would be f of x minus f of a,
over change in the horizontal,

00:03:29.980 --> 00:03:31.344
over x minus a.

00:03:31.344 --> 00:03:33.260
Actually, let me do that
in that purple color.

00:03:33.260 --> 00:03:36.560
Over x minus a.

00:03:36.560 --> 00:03:39.600
Now, how could we get a better
and better approximation

00:03:39.600 --> 00:03:42.460
for the slope of the
tangent line here?

00:03:42.460 --> 00:03:45.645
Well, we could take the
limit as x approaches a.

00:03:45.645 --> 00:03:48.540
As x gets closer and
closer and closer to a,

00:03:48.540 --> 00:03:51.510
the secant line slope is
going to better and better

00:03:51.510 --> 00:03:54.320
and better approximate the
slope of the tangent line,

00:03:54.320 --> 00:03:56.370
this tangent line that
I have in red here.

00:03:56.370 --> 00:04:04.430
So we would want to take the
limit as x approaches a here.

00:04:04.430 --> 00:04:09.370
Either way, we're doing
the exact same thing.

00:04:09.370 --> 00:04:12.760
We have an expression for
the slope of a secant line.

00:04:12.760 --> 00:04:15.790
And then we're bringing those
x values of those points

00:04:15.790 --> 00:04:19.040
closer and closer together.

00:04:19.040 --> 00:04:21.700
So the slopes of those secant
lines better and better

00:04:21.700 --> 00:04:24.510
and better approximate that
slope of the tangent line.

00:04:24.510 --> 00:04:27.390
And at the limit, it does become
the slope of the tangent line.

00:04:27.390 --> 00:04:31.550
That is the definition
of the derivative.

00:04:31.550 --> 00:04:34.382
So this is the more standard
definition of a derivative.

00:04:34.382 --> 00:04:36.590
It would give you your
derivative as a function of x.

00:04:36.590 --> 00:04:40.900
And then you can then input
your particular value of x.

00:04:40.900 --> 00:04:44.129
Or you could use the alternate
form of the derivative.

00:04:44.129 --> 00:04:45.670
If you know that,
hey, look, I'm just

00:04:45.670 --> 00:04:47.530
looking to find the
derivative exactly at a.

00:04:47.530 --> 00:04:49.772
I don't need a
general function of f.

00:04:49.772 --> 00:04:50.730
Then you could do this.

00:04:50.730 --> 00:04:53.260
But they're doing
the same thing.


Approximating instantaneous rate of change word problem

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:04.030
The table shows the number of
stores of a popular US coffee

00:00:04.030 --> 00:00:06.750
chain from 2000 to 2006.

00:00:06.750 --> 00:00:08.930
The number of stores
recorded is the number

00:00:08.930 --> 00:00:11.000
at the start of each
year on January 1.

00:00:11.000 --> 00:00:16.230
So in 2000, there was 1,996
stores, in 2005, 6,177,

00:00:16.230 --> 00:00:17.680
so on and so forth.

00:00:17.680 --> 00:00:20.060
Determine a reasonable
approximation

00:00:20.060 --> 00:00:25.190
for the instantaneous
rate of change,

00:00:25.190 --> 00:00:29.710
in coffee stores per year
at the beginning of 2003--

00:00:29.710 --> 00:00:32.360
so we care about
2003-- by taking

00:00:32.360 --> 00:00:35.500
the average of two
nearby secant slopes.

00:00:35.500 --> 00:00:38.070
So let's visualize this.

00:00:38.070 --> 00:00:41.740
So this right over here, I've
plotted all of the points.

00:00:41.740 --> 00:00:44.210
Now let me make sure
that the axes are clear.

00:00:44.210 --> 00:00:48.990
This horizontal axis, this is my
t-axis that tells us the year.

00:00:48.990 --> 00:00:51.820
And then the vertical axis
is the number of stores.

00:00:51.820 --> 00:00:55.200
And we could even say that
it is a function of time.

00:00:55.200 --> 00:00:57.110
So you see in the
year 2000, there

00:00:57.110 --> 00:01:02.510
was 1,996 stores-- 2003, 4,272.

00:01:02.510 --> 00:01:06.460
2003-- let me do that
in that blue color.

00:01:06.460 --> 00:01:10.560
2003, 4,272 stores.

00:01:10.560 --> 00:01:12.990
Now if you could imagine
that they're constantly

00:01:12.990 --> 00:01:14.930
adding stores, you could
even imagine minute

00:01:14.930 --> 00:01:16.710
by minute they're adding stores.

00:01:16.710 --> 00:01:18.670
So this is just
sampling the number

00:01:18.670 --> 00:01:20.140
of stores they had on January 1.

00:01:20.140 --> 00:01:21.810
But if you were
to really plot it

00:01:21.810 --> 00:01:23.350
as a more continuous
function, it

00:01:23.350 --> 00:01:24.790
might look something like this.

00:01:24.790 --> 00:01:26.480
I'll do my best
to approximate it.

00:01:26.480 --> 00:01:29.350
It might be more of
some type of curve

00:01:29.350 --> 00:01:31.015
that looks something like this.

00:01:33.570 --> 00:01:35.490
And once again, I'm
just approximating

00:01:35.490 --> 00:01:38.140
what it might
actually look like.

00:01:38.140 --> 00:01:40.520
So when they're saying the
instantaneous rate of change

00:01:40.520 --> 00:01:43.490
in coffee stores
per year, so this

00:01:43.490 --> 00:01:45.810
is the change, the
instantaneous rate

00:01:45.810 --> 00:01:48.830
of change of stores per time.

00:01:48.830 --> 00:01:51.770
They're really saying, we
need to approximate the slope

00:01:51.770 --> 00:01:56.120
of the tangent line in
2003, when time is 2003.

00:01:56.120 --> 00:02:01.702
So the tangent line might
look something like that.

00:02:01.702 --> 00:02:04.160
I want to draw it so you see
that this line right over here

00:02:04.160 --> 00:02:05.350
is tangent.

00:02:05.350 --> 00:02:06.628
Now, they say approximate.

00:02:06.628 --> 00:02:08.919
We don't have the information
to figure it out exactly.

00:02:08.919 --> 00:02:11.440
But we have some data around
it, and we can figure out

00:02:11.440 --> 00:02:13.760
the slopes of the secant
lines between this point

00:02:13.760 --> 00:02:14.500
and those points.

00:02:14.500 --> 00:02:16.130
And then we can take the
average of the slopes

00:02:16.130 --> 00:02:17.620
of the secant lines
to approximate

00:02:17.620 --> 00:02:19.350
the slope of this tangent line.

00:02:19.350 --> 00:02:23.810
So for example, we could find
the slope of this secant line

00:02:23.810 --> 00:02:27.930
right over here as we
go from 2002 to 2003.

00:02:27.930 --> 00:02:31.010
And then we can find the
slope of this secant line

00:02:31.010 --> 00:02:34.430
as we go from 2003 to 2004.

00:02:34.430 --> 00:02:35.930
And if we average
those, that should

00:02:35.930 --> 00:02:39.390
be a pretty good approximation
for the instantaneous rate

00:02:39.390 --> 00:02:40.860
of change in 2003.

00:02:40.860 --> 00:02:42.510
So let's do that.

00:02:42.510 --> 00:02:45.470
So the slope of this
pink secant line,

00:02:45.470 --> 00:02:49.320
as we go from 2002
to 2003, that's

00:02:49.320 --> 00:02:53.460
going to be the number
of stores in 2003

00:02:53.460 --> 00:02:57.270
minus the number
of stores in 2002.

00:02:57.270 --> 00:02:59.840
So that's the change
in our number of stores

00:02:59.840 --> 00:03:02.230
over the change in years,
or the change in time.

00:03:02.230 --> 00:03:07.670
So this is going to
be 2003 minus 2002.

00:03:07.670 --> 00:03:09.830
And so what is this
going to be equal to?

00:03:09.830 --> 00:03:13.630
Let' see, in 2003 we
have 4,272 stores.

00:03:13.630 --> 00:03:17.092
In 2002, we have 3,501.

00:03:17.092 --> 00:03:19.300
It's going to be that over--
well, the denominator is

00:03:19.300 --> 00:03:20.370
just 1.

00:03:20.370 --> 00:03:22.510
So this is going to
simplify to, let's see.

00:03:22.510 --> 00:03:25.406
I can do a little bit
of math on the side

00:03:25.406 --> 00:03:27.280
here just so I don't
make a careless mistake.

00:03:27.280 --> 00:03:36.480
Minus 3501, 2 minus 1 is 1, 7
minus 0 is 7, 42 minus 35 is 7.

00:03:36.480 --> 00:03:39.950
So this is equal to 771.

00:03:39.950 --> 00:03:43.770
So their average rate of
change from 2002 to 2003

00:03:43.770 --> 00:03:47.130
was 771 stores per year.

00:03:52.380 --> 00:03:54.750
Now let's do the same
thing for this red secant

00:03:54.750 --> 00:03:57.350
line between 2003 and 2004.

00:03:57.350 --> 00:03:58.990
The slope of that
secant line is going

00:03:58.990 --> 00:04:03.060
to be the number of stores
in 2004 minus the number

00:04:03.060 --> 00:04:12.271
of stores in 2003 over 2004
minus 2003, which is equal to,

00:04:12.271 --> 00:04:12.770
let's see.

00:04:12.770 --> 00:04:16.649
They had 5,239 stores in 2004.

00:04:16.649 --> 00:04:21.709
They have 4,272 stores in 2003.

00:04:21.709 --> 00:04:26.240
And then only one year changes,
so 2004 minus 2003 is 1.

00:04:26.240 --> 00:04:29.320
So this is equal to-- I'll
do the subtraction problem up

00:04:29.320 --> 00:04:29.820
here.

00:04:29.820 --> 00:04:37.930
So 5,239 minus 4,272.

00:04:37.930 --> 00:04:38.860
So this, let's see.

00:04:38.860 --> 00:04:40.540
9 minus 2 is 7.

00:04:40.540 --> 00:04:42.040
Let's see, it looks
like we're going

00:04:42.040 --> 00:04:44.400
to do a little bit
of regrouping here.

00:04:44.400 --> 00:04:47.940
So let's say we take one of
the thousands from the 5,000,

00:04:47.940 --> 00:04:48.990
so it's 4.

00:04:48.990 --> 00:04:50.740
And then that
becomes 10 hundreds.

00:04:50.740 --> 00:04:54.000
So this becomes 1,200.

00:04:54.000 --> 00:04:57.000
And now let's take one of those
100s and give it to the 10.

00:04:57.000 --> 00:05:01.110
So now this becomes 1,100,
and we give that 10.

00:05:01.110 --> 00:05:04.210
So then this becomes 13.

00:05:04.210 --> 00:05:08.949
We took 100 from here,
and that gave us 10 tens.

00:05:08.949 --> 00:05:10.240
So now we're ready to subtract.

00:05:10.240 --> 00:05:16.060
13 minus 7 is 6, 11 minus 2
is 9, and then 4 minus 4 is 0.

00:05:16.060 --> 00:05:17.820
So it's 967.

00:05:17.820 --> 00:05:21.720
So this is equal to
967 stores per year.

00:05:24.300 --> 00:05:26.050
So we really just have
to take the average

00:05:26.050 --> 00:05:28.900
of this number and that
number to approximate

00:05:28.900 --> 00:05:30.590
the instantaneous
rate of change.

00:05:30.590 --> 00:05:31.650
So let's do that.

00:05:31.650 --> 00:05:41.210
So let's take 967 plus 771,
and then we can divide by 2.

00:05:41.210 --> 00:05:43.040
So let's actually
figure out what that is.

00:05:43.040 --> 00:05:45.850
So let me add 771 over here.

00:05:45.850 --> 00:05:47.450
7 plus 1 is 8.

00:05:47.450 --> 00:05:53.100
6 plus 7 is 13.

00:05:53.100 --> 00:05:56.900
And then 1, plus this
is 17 right over here.

00:05:56.900 --> 00:05:58.970
And then we're going
to divide that by 2.

00:05:58.970 --> 00:06:08.860
So 2 goes into 1,738,
it goes into 17 8 times.

00:06:08.860 --> 00:06:11.670
8 times 2 is 16.

00:06:11.670 --> 00:06:15.240
Subtract, we get a
13, it goes six times.

00:06:15.240 --> 00:06:16.200
12.

00:06:16.200 --> 00:06:18.300
Subtract, we get an 18.

00:06:18.300 --> 00:06:19.660
It goes nine times.

00:06:19.660 --> 00:06:21.860
And then we don't
have any remainder.

00:06:21.860 --> 00:06:24.370
So 18, no remainder.

00:06:24.370 --> 00:06:27.260
So our approximation for
the instantaneous rate

00:06:27.260 --> 00:06:30.830
of change for the slope of our
approximation for the slope

00:06:30.830 --> 00:06:40.461
of the tangent line right over
here is 869 stores per year.