Factoring quadratics as (x+a)(x+b) (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.959
To better understand how
we can factor second degree

00:00:02.959 --> 00:00:05.500
expressions like this, I'm going
to go through some examples.

00:00:05.500 --> 00:00:08.302
We'll factor this expression and
we'll factor this expression.

00:00:08.302 --> 00:00:10.010
And hopefully it'll
give you a background

00:00:10.010 --> 00:00:12.974
on how you could generally
factor expressions like this.

00:00:12.974 --> 00:00:14.390
And to think about
it, let's think

00:00:14.390 --> 00:00:18.370
about what happens if I were
to multiply x plus something

00:00:18.370 --> 00:00:21.790
times x plus something else.

00:00:21.790 --> 00:00:25.330
Well, if I were to multiply
this out, what do I get?

00:00:25.330 --> 00:00:30.520
Well, you're going to get
x squared plus ax plus bx,

00:00:30.520 --> 00:00:37.730
which is the same thing as
a plus bx plus a times b.

00:00:37.730 --> 00:00:41.150
So if you wanted to go
from this form, which

00:00:41.150 --> 00:00:44.922
is what we have in these
two examples, back to this,

00:00:44.922 --> 00:00:46.630
you really just have
to think about well,

00:00:46.630 --> 00:00:48.940
what's our coefficient
on our x term,

00:00:48.940 --> 00:00:53.530
and can I figure out two numbers
that when I take their sum,

00:00:53.530 --> 00:00:56.369
are equal to that coefficient,
and what's my constant term,

00:00:56.369 --> 00:00:58.660
and can I think of two numbers,
those same two numbers,

00:00:58.660 --> 00:01:01.410
that when I take the product
equal that constant term?

00:01:01.410 --> 00:01:02.980
So let's do that over here.

00:01:02.980 --> 00:01:06.760
If we look at our
coefficient on x,

00:01:06.760 --> 00:01:09.610
can we think of
an a plus ab that

00:01:09.610 --> 00:01:12.730
is equal to that
number negative 14?

00:01:12.730 --> 00:01:15.690
And can we think
of the same a and b

00:01:15.690 --> 00:01:17.400
that if we were to
take its product,

00:01:17.400 --> 00:01:19.440
it would be equal to 40?

00:01:22.010 --> 00:01:26.660
So what's an a and a b
that would work over here?

00:01:26.660 --> 00:01:28.970
Well, let's think about
this a little bit.

00:01:28.970 --> 00:01:33.790
If I have 4 times 10
is 40, but 4 plus 10

00:01:33.790 --> 00:01:35.260
is equal to positive 14.

00:01:35.260 --> 00:01:36.980
So that wouldn't quite work.

00:01:36.980 --> 00:01:39.040
What happens if we make
them both negative?

00:01:39.040 --> 00:01:43.410
If we have negative
4 plus negative 10,

00:01:43.410 --> 00:01:45.680
well that's going to be
equal to negative 14.

00:01:45.680 --> 00:01:53.280
And negative 4 times
negative 10 is equal to 40.

00:01:53.280 --> 00:01:55.710
The fact that this number
right over here is positive,

00:01:55.710 --> 00:01:57.610
this number right
over here is positive,

00:01:57.610 --> 00:02:00.045
tells you that these are
going to be the same sign.

00:02:05.537 --> 00:02:07.370
If this number right
over here was negative,

00:02:07.370 --> 00:02:09.600
then we would have
different signs.

00:02:09.600 --> 00:02:11.050
And so if you have
2 numbers that

00:02:11.050 --> 00:02:12.840
are going to be the same
sign and they add up

00:02:12.840 --> 00:02:14.548
to a negative number,
then that tells you

00:02:14.548 --> 00:02:16.672
that they're both
going to be negative.

00:02:16.672 --> 00:02:18.130
So just going back
to this, we know

00:02:18.130 --> 00:02:20.800
that a is going to be negative
4, b is equal to negative 10,

00:02:20.800 --> 00:02:22.050
and we are done factoring it.

00:02:22.050 --> 00:02:31.670
We can factor this expression
as x plus negative 4 times x

00:02:31.670 --> 00:02:34.090
plus negative 10.

00:02:34.090 --> 00:02:35.600
Or another way to
write that, that's

00:02:35.600 --> 00:02:40.020
x minus 4 times x minus 10.

00:02:40.020 --> 00:02:42.400
Now let's do the
same thing over here.

00:02:42.400 --> 00:02:47.240
Can we think of
an a plus b that's

00:02:47.240 --> 00:02:51.360
equal to the coefficient
on the x term?

00:02:51.360 --> 00:02:53.810
Well, the coefficient
on the x term here

00:02:53.810 --> 00:02:55.900
is essentially
negative 1 times x.

00:02:55.900 --> 00:02:58.270
So we could say the
coefficient is negative 1.

00:02:58.270 --> 00:03:00.830
And can we think of
an a times b where

00:03:00.830 --> 00:03:04.545
it's going to be
equal to negative 12?

00:03:09.270 --> 00:03:11.340
Well, let's think about
this a little bit.

00:03:11.340 --> 00:03:13.346
The product of the 2
numbers is negative,

00:03:13.346 --> 00:03:15.220
so that means that they
have different signs.

00:03:20.240 --> 00:03:23.640
So one will be positive
and one will be negative.

00:03:23.640 --> 00:03:26.900
And so when I add the two
together, I get to negative 1.

00:03:26.900 --> 00:03:29.570
Well, just think about the
factors of negative 12.

00:03:29.570 --> 00:03:33.239
Well, what about if one is 3
and maybe one is negative 4.

00:03:33.239 --> 00:03:34.280
Well, that seems to work.

00:03:34.280 --> 00:03:36.690
And you really just have
to try these numbers out.

00:03:36.690 --> 00:03:40.820
If a is 3 plus
negative 4, that indeed

00:03:40.820 --> 00:03:42.500
turns out to be negative 1.

00:03:42.500 --> 00:03:45.740
And if we have 3
times negative 4,

00:03:45.740 --> 00:03:48.150
that indeed is equal
to negative 12.

00:03:48.150 --> 00:03:49.340
So that seems to work out.

00:03:49.340 --> 00:03:51.173
And it's really a matter
of trial and error.

00:03:51.173 --> 00:03:53.982
You could try negative
3 plus 4, but then that

00:03:53.982 --> 00:03:55.440
wouldn't have worked
out over here.

00:03:55.440 --> 00:03:56.830
You could have
tried two and six,

00:03:56.830 --> 00:04:00.165
but that wouldn't have
worked out on this number.

00:04:00.165 --> 00:04:01.540
Or 2 and negative
6, you wouldn't

00:04:01.540 --> 00:04:04.160
have gotten the sum to
be equal to negative 1.

00:04:04.160 --> 00:04:06.720
But now that we've figured
out what the a and b are,

00:04:06.720 --> 00:04:10.070
what is this
expression factored?

00:04:10.070 --> 00:04:16.089
Well, it's going to be x plus
3 times x plus negative 4,

00:04:16.089 --> 00:04:19.005
or we could say x minus 4.


Constructing an exponential equation example

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:04.710
Liam opened a savings
account and put $6,250 in it.

00:00:04.710 --> 00:00:08.330
Each year, the account
increases by 20%.

00:00:08.330 --> 00:00:13.540
How many years will it take
the account to reach $12,960?

00:00:13.540 --> 00:00:15.600
Write an equation that
models the situation.

00:00:15.600 --> 00:00:17.590
Use t to represent
the number of years

00:00:17.590 --> 00:00:21.207
since Liam opened the account.

00:00:21.207 --> 00:00:22.790
So I encourage you
to pause this video

00:00:22.790 --> 00:00:24.430
and actually try to do
it on your own first.

00:00:24.430 --> 00:00:26.710
Try to write this equation
that models the situation

00:00:26.710 --> 00:00:29.487
using the variable t in
the way they described.

00:00:29.487 --> 00:00:31.070
And then actually
answer the question,

00:00:31.070 --> 00:00:36.297
how many years will it take for
the account to reach $12,960?

00:00:36.297 --> 00:00:37.630
Well, let's just think about it.

00:00:37.630 --> 00:00:39.750
So t is to represent
the number of years

00:00:39.750 --> 00:00:41.260
since Liam opened the account.

00:00:41.260 --> 00:00:44.159
So let's just say it's
been 0 years since Liam

00:00:44.159 --> 00:00:44.950
opened the account.

00:00:44.950 --> 00:00:46.290
How much is he going to have?

00:00:46.290 --> 00:00:49.590
Well, he's just going
to have $6,250 in it.

00:00:49.590 --> 00:00:51.480
That's how much he starts with.

00:00:51.480 --> 00:00:54.219
Now, let's say it's
been one year since he

00:00:54.219 --> 00:00:55.010
opened the account.

00:00:55.010 --> 00:00:56.510
How much will he have?

00:00:56.510 --> 00:01:01.060
Well, he's going to
have $6,250 times--

00:01:01.060 --> 00:01:08.880
or let's write it this
way-- plus 20% 6,250.

00:01:08.880 --> 00:01:11.180
It grows 20% every year.

00:01:11.180 --> 00:01:13.900
So this is how much he
started the year with,

00:01:13.900 --> 00:01:18.180
and then he gets another
20% of that 6,250.

00:01:18.180 --> 00:01:20.880
If we factor out
a 6,250, this is

00:01:20.880 --> 00:01:29.730
equal to 6,250 times 1 plus 20%,
or we could write that as 0.2.

00:01:29.730 --> 00:01:38.795
Which is equal to
6,250 times 1.2.

00:01:38.795 --> 00:01:42.135
Now, how much is he going to
have at the end of two years?

00:01:42.135 --> 00:01:44.385
Well, he's going to have the
same amount that he still

00:01:44.385 --> 00:01:48.890
had at the end of
one year, times 1.2.

00:01:48.890 --> 00:01:50.774
Because it grew by 20% again.

00:01:50.774 --> 00:01:52.440
So he's going to have
the amount that he

00:01:52.440 --> 00:01:57.490
had at the end of
one year times 1.2,

00:01:57.490 --> 00:02:10.360
which is equal to 6,250
times 1.2, times 1.2.

00:02:10.360 --> 00:02:19.010
Which is equal to 6,250
times 1.2 squared.

00:02:19.010 --> 00:02:21.987
I think you might see
where this is going.

00:02:21.987 --> 00:02:24.320
Or I could even write it like
this, order of operations,

00:02:24.320 --> 00:02:26.160
you do the exponent first.

00:02:26.160 --> 00:02:28.530
So what about after three years?

00:02:28.530 --> 00:02:31.360
So after three years, well,
we're just going to compound.

00:02:31.360 --> 00:02:34.140
We're going to multiply
by 1.2 once again.

00:02:34.140 --> 00:02:42.930
So then he's going to have 6,250
times 1.2 to the third power.

00:02:42.930 --> 00:02:44.340
And so after t
years, we're going

00:02:44.340 --> 00:02:47.200
to multiply by 1.2
that many times.

00:02:47.200 --> 00:02:55.850
So after t years in his account,
he's going to have 6,250 times

00:02:55.850 --> 00:03:00.880
1.2 to the t'th power.

00:03:00.880 --> 00:03:06.400
1.2 to the t'th power,
or to the t power.

00:03:06.400 --> 00:03:09.080
I don't want get
confused with the thing

00:03:09.080 --> 00:03:10.430
that you use to take bites with.

00:03:10.430 --> 00:03:11.550
Anyway.

00:03:11.550 --> 00:03:14.220
So they say, write an equation
that models the situation.

00:03:14.220 --> 00:03:16.178
So we want to figure out
how many years will it

00:03:16.178 --> 00:03:18.210
take the account
to reach 12,960?

00:03:18.210 --> 00:03:19.760
So we essentially
want to say, when

00:03:19.760 --> 00:03:22.780
is the account
going to be $12,960?

00:03:22.780 --> 00:03:27.490
Or we could write
12,960, when is

00:03:27.490 --> 00:03:38.510
that going to be equal to
6,250 times 1.2 to the t power?

00:03:38.510 --> 00:03:40.600
So that's the equation
right over there

00:03:40.600 --> 00:03:42.390
that models the situation.

00:03:42.390 --> 00:03:46.960
And then we need to think
about how we can actually

00:03:46.960 --> 00:03:49.330
go about solving this thing.

00:03:49.330 --> 00:03:51.970
Well, a natural thing is
to isolate the t variable.

00:03:51.970 --> 00:03:55.240
Let's divide both
sides by 6,250.

00:03:55.240 --> 00:03:57.920
So we could get-- and if
we flip the two sides,

00:03:57.920 --> 00:04:06.770
we could get 1.2 to the t power
is equal to-- well let me write

00:04:06.770 --> 00:04:16.630
this, 12,960 divided by 6,250.

00:04:16.630 --> 00:04:18.399
And since they're
both divisible by 10,

00:04:18.399 --> 00:04:21.230
why don't we divide
them both by 10?

00:04:21.230 --> 00:04:26.770
So it's 1,296 divided by 625.

00:04:26.770 --> 00:04:28.450
And there's several
ways that you

00:04:28.450 --> 00:04:30.320
could solve this
problem at this point.

00:04:30.320 --> 00:04:32.160
One way, if you feel
confident that this

00:04:32.160 --> 00:04:34.545
is going to have an integer
answer right over here,

00:04:34.545 --> 00:04:36.670
you could literally just
try to use your calculator

00:04:36.670 --> 00:04:40.440
and multiply 1.2 enough times
to get whatever number this is.

00:04:40.440 --> 00:04:42.240
And so we could do it that way.

00:04:42.240 --> 00:04:44.240
And as we'll see, there's
a more systematic way

00:04:44.240 --> 00:04:45.960
of doing it once you
learn about logarithms,

00:04:45.960 --> 00:04:47.126
and I'll do that at the end.

00:04:47.126 --> 00:04:48.560
But I'll do that
last just in case

00:04:48.560 --> 00:04:50.840
you haven't been exposed
to logarithms yet.

00:04:50.840 --> 00:04:55.350
So you could literally
say-- so let me just

00:04:55.350 --> 00:04:57.250
exit out of everything.

00:04:57.250 --> 00:05:03.550
So you could literally say OK,
let's see, 1,296 divided by 625

00:05:03.550 --> 00:05:04.720
is this value.

00:05:04.720 --> 00:05:07.270
So let's see how many times
we have to multiply by 1.2.

00:05:07.270 --> 00:05:11.460
1.2 times 1.2 gets
us-- well, that

00:05:11.460 --> 00:05:12.770
doesn't get us close enough.

00:05:12.770 --> 00:05:14.670
So let's try it three times.

00:05:14.670 --> 00:05:17.930
So let's take that same number.

00:05:17.930 --> 00:05:22.190
Let's just take
1.2, let's raise it.

00:05:22.190 --> 00:05:24.832
Let's raise 1.2.

00:05:24.832 --> 00:05:26.040
Let's just do it three times.

00:05:26.040 --> 00:05:30.397
Times 1.2 times 1.2.

00:05:30.397 --> 00:05:31.730
That still doesn't get us there.

00:05:31.730 --> 00:05:35.460
What if we were to multiply
by 1.2 one more time?

00:05:35.460 --> 00:05:37.990
Well, that actually
gets us there.

00:05:37.990 --> 00:05:40.020
And we just did
this by brute force.

00:05:40.020 --> 00:05:42.930
1.2 to the fourth power
will give us this value.

00:05:42.930 --> 00:05:45.585
So that's one way, kind
of a brute-force way,

00:05:45.585 --> 00:05:48.700
of figuring out that
t is equal to 4.

00:05:48.700 --> 00:05:51.439
Another way, this just might be
a little bit less intuitive, it

00:05:51.439 --> 00:05:52.980
might jump out of
it, gee, this looks

00:05:52.980 --> 00:05:54.530
like some type of a power of 5.

00:05:54.530 --> 00:06:00.400
We know that 5 to the first
is 5, 5 squared is 25,

00:06:00.400 --> 00:06:07.612
5 to the third is 125,
5 to the fourth is 625.

00:06:07.612 --> 00:06:09.570
And so you might recognize
this right over here

00:06:09.570 --> 00:06:11.349
is 5 to the fourth.

00:06:11.349 --> 00:06:12.890
And it's actually
a little bit harder

00:06:12.890 --> 00:06:14.560
to recognize that
this right over here

00:06:14.560 --> 00:06:16.360
is 6 to the fourth power.

00:06:16.360 --> 00:06:18.690
And this right over here is 6/5.

00:06:18.690 --> 00:06:26.440
So we could rewrite this as
6/5 to the t is equal to 6

00:06:26.440 --> 00:06:31.320
to the fourth over
5 to the fourth.

00:06:31.320 --> 00:06:40.790
Which is the same thing as
6/5 to the fourth power.

00:06:40.790 --> 00:06:44.790
So here you'd say, well, 6/5 to
the t needs to be equal to 6/5

00:06:44.790 --> 00:06:45.980
to the fourth power.

00:06:45.980 --> 00:06:48.070
t must be equal to 4.

00:06:48.070 --> 00:06:50.740
Now, this is nice
when you can recognize

00:06:50.740 --> 00:06:53.740
that this is something raised
to the fourth power, which

00:06:53.740 --> 00:06:54.680
isn't easy to do.

00:06:54.680 --> 00:06:56.526
Or if you know
this is an integer,

00:06:56.526 --> 00:06:58.150
and you can just keep
multiplying 1.2--

00:06:58.150 --> 00:06:59.750
if you know it's a low integer.

00:06:59.750 --> 00:07:01.670
But the systematic
way of doing it

00:07:01.670 --> 00:07:03.156
is to actually use logarithms.

00:07:03.156 --> 00:07:04.780
And there's many
videos on Khan Academy

00:07:04.780 --> 00:07:05.977
about how to use logarithms.

00:07:05.977 --> 00:07:07.810
But if you're more
concerned with, well gee,

00:07:07.810 --> 00:07:10.710
if I just want to figure
out 1.2 to what power

00:07:10.710 --> 00:07:13.660
is equal to this thing,
what you would do--

00:07:13.660 --> 00:07:16.720
and we prove this
in other videos--

00:07:16.720 --> 00:07:20.940
is if you say, look, let's
take the thing that we want 1.2

00:07:20.940 --> 00:07:22.730
to some power to be.

00:07:22.730 --> 00:07:24.105
Let's take the
logarithm of that.

00:07:24.105 --> 00:07:26.188
And actually, you could
take a logarithm any base.

00:07:26.188 --> 00:07:28.980
Your calculator tends to have
a natural log, which is base e,

00:07:28.980 --> 00:07:30.040
and a log base 10.

00:07:30.040 --> 00:07:32.210
We could just take
a log base 10.

00:07:32.210 --> 00:07:33.420
So let's do that.

00:07:33.420 --> 00:07:40.700
So we'll take the logarithm of
what we want to get to, 2.0736,

00:07:40.700 --> 00:07:42.620
and divide that by
the thing that we're

00:07:42.620 --> 00:07:44.940
trying to take the power
of to get to this number.

00:07:44.940 --> 00:07:49.000
So divided by the
logarithm of 1.2.

00:07:49.000 --> 00:07:51.940
And once again, we prove this--
actually, I wanted to divide.

00:07:51.940 --> 00:07:56.239
So let me insert
division symbol.

00:07:56.239 --> 00:07:58.280
So once again, it might
look like a little voodoo

00:07:58.280 --> 00:07:58.825
right here.

00:07:58.825 --> 00:08:00.450
We prove it in other
videos, but if you

00:08:00.450 --> 00:08:02.940
wanted to use a calculator to
calculate things like this,

00:08:02.940 --> 00:08:05.610
because sometimes it won't be
a nice integer number of years.

00:08:05.610 --> 00:08:07.297
It might be 3 and
1/2 years, or it

00:08:07.297 --> 00:08:11.820
might be 7.1234 years,
whatever it might be.

00:08:11.820 --> 00:08:13.980
This will give you a
more precise answer.

00:08:13.980 --> 00:08:15.790
So what do you want to get to?

00:08:15.790 --> 00:08:18.130
You want to get to 2.0736.

00:08:18.130 --> 00:08:19.660
What are you raising
to some power?

00:08:19.660 --> 00:08:20.594
1.2.

00:08:20.594 --> 00:08:22.010
Divide the log of
the thing you're

00:08:22.010 --> 00:08:26.030
trying to get to divided
by the log of what the base

00:08:26.030 --> 00:08:30.010
that you're trying to raise to
a power, and you click Enter.

00:08:30.010 --> 00:08:31.680
And then you get--
so this is literally

00:08:31.680 --> 00:08:34.679
another way of saying that
1.2 to the fourth power

00:08:34.679 --> 00:08:36.451
is going to be 2.0736.

00:08:36.451 --> 00:08:38.159
So once again, if this
looks like voodoo,

00:08:38.159 --> 00:08:39.617
you don't know what
logarithms are,

00:08:39.617 --> 00:08:41.406
we have videos on
Khan Academy on that.

00:08:41.406 --> 00:08:43.030
But there's multiple
ways to tackle it,

00:08:43.030 --> 00:08:45.140
especially this problem
where the answer was

00:08:45.140 --> 00:08:47.400
a little bit simpler.


Finding a piecewise function definition from graph

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.877
Select the piecewise function
whose graph is shown below.

00:00:04.877 --> 00:00:06.460
Or I guess we should
say to the right.

00:00:06.460 --> 00:00:08.660
I copied and pasted it
so it's on the right now.

00:00:08.660 --> 00:00:11.250
So we have this piecewise
continuous function.

00:00:11.250 --> 00:00:15.960
So it's not defined for x
being negative 2 or lower.

00:00:15.960 --> 00:00:18.340
But then starting at x
greater than negative 2,

00:00:18.340 --> 00:00:19.870
it starts being defined.

00:00:19.870 --> 00:00:22.210
It's continuous all
the way until we

00:00:22.210 --> 00:00:26.130
get to the point x equals 2 and
then we have a discontinuity.

00:00:26.130 --> 00:00:29.750
And then it starts getting
it defined again down here.

00:00:29.750 --> 00:00:33.090
And then it is continuous for
a little while all the way.

00:00:33.090 --> 00:00:39.990
And then when x is greater than
6, it's once again undefined.

00:00:39.990 --> 00:00:42.330
So let's think about
which of these functions

00:00:42.330 --> 00:00:44.300
describe this one over here.

00:00:44.300 --> 00:00:50.120
So this one looks like a
radical function shifted.

00:00:50.120 --> 00:00:57.490
So square root of x
would look like this.

00:00:57.490 --> 00:00:59.640
Let me do it in a color
that you could see.

00:00:59.640 --> 00:01:04.800
Square root of x
would look like this.

00:01:04.800 --> 00:01:06.610
So the square root
of x looks like this.

00:01:06.610 --> 00:01:08.110
And this just looks
like square root

00:01:08.110 --> 00:01:11.280
of x shifted 2 to the left.

00:01:11.280 --> 00:01:14.480
So this looks like
square root of x plus 2.

00:01:14.480 --> 00:01:19.040
This one right over here looks
like square root of x plus 2.

00:01:19.040 --> 00:01:20.910
And you could verify that.

00:01:20.910 --> 00:01:24.320
When x is negative 2, negative
2 plus 2, square root of that

00:01:24.320 --> 00:01:25.680
is going to be 0.

00:01:25.680 --> 00:01:27.970
And it's not defined
there, but we

00:01:27.970 --> 00:01:29.650
see that if we
were to continue it

00:01:29.650 --> 00:01:31.070
would have been defined there.

00:01:31.070 --> 00:01:32.480
Let's try some other points.

00:01:32.480 --> 00:01:37.950
When x is negative 1,
negative 1 plus 2 is 1.

00:01:37.950 --> 00:01:41.310
Principal root of 1 is 1.

00:01:41.310 --> 00:01:42.410
Let's try 2.

00:01:42.410 --> 00:01:44.900
When x is equal to
2, 2 plus 2 is 4.

00:01:44.900 --> 00:01:47.950
Principal root of
4 is positive 2.

00:01:47.950 --> 00:01:50.670
So this just looks like
a pretty good candidate.

00:01:50.670 --> 00:01:52.580
So it looks like our function.

00:01:52.580 --> 00:01:55.639
So let's see, it looks like
our function would be--

00:01:55.639 --> 00:01:57.180
and I'm not going
to call it anything

00:01:57.180 --> 00:01:59.250
because it could
be p, h, g, or f.

00:01:59.250 --> 00:02:02.120
But our function, if I
were to write it out,

00:02:02.120 --> 00:02:03.890
it looks like over
this first interval.

00:02:03.890 --> 00:02:11.120
So it looks like it's the square
root of x plus 2 for negative 2

00:02:11.120 --> 00:02:15.350
being less than x-- it's
not defined at negative 2,

00:02:15.350 --> 00:02:18.750
but as long as x is greater than
negative 2 and x is less than

00:02:18.750 --> 00:02:20.000
or equal to 2.

00:02:20.000 --> 00:02:22.910
And x is less than
or equal to 2.

00:02:22.910 --> 00:02:26.690
So that's this part
of the function.

00:02:26.690 --> 00:02:29.160
And then it jumps down.

00:02:29.160 --> 00:02:31.900
Now, this looks like
x to the third. x

00:02:31.900 --> 00:02:33.910
to the third looks
something like this.

00:02:33.910 --> 00:02:40.990
So x to the third power
looks something like that.

00:02:40.990 --> 00:02:44.310
So let's see, negative 2 to
the third power is negative 8.

00:02:44.310 --> 00:02:45.700
So it looks something like that.

00:02:45.700 --> 00:02:48.470
That's what x to the
third looks like.

00:02:48.470 --> 00:02:50.656
So 2 to the third power is 8.

00:02:53.580 --> 00:02:55.480
So x to the third looks
something like that.

00:02:55.480 --> 00:02:59.930
This looks like x to the
third shifted over 4.

00:02:59.930 --> 00:03:06.410
So I'm guessing that this is
x minus 4 to the third power.

00:03:06.410 --> 00:03:08.200
But we can verify that.

00:03:08.200 --> 00:03:10.880
When x is equal to 4, 4
minus 4 to the third power,

00:03:10.880 --> 00:03:13.430
we do indeed get the value
of this expression being

00:03:13.430 --> 00:03:14.690
equal to 0.

00:03:14.690 --> 00:03:20.820
When x is 6, 6 minus 4 is 2 to
the third power is indeed 8.

00:03:20.820 --> 00:03:25.710
When x is 2, 2 minus 4 is
negative 2 to the third power

00:03:25.710 --> 00:03:27.410
is indeed negative 8.

00:03:27.410 --> 00:03:29.590
So this over this
interval, it is

00:03:29.590 --> 00:03:31.690
x minus 4 to the third power.

00:03:31.690 --> 00:03:39.210
So we could say so this part of
the function right over here,

00:03:39.210 --> 00:03:50.260
we could say is x minus 4
to the third power for 4x

00:03:50.260 --> 00:03:55.050
being greater than 2, or we
could say 2 is less than x.

00:03:55.050 --> 00:03:58.926
And it's defined all the
way to x being equal to 6,

00:03:58.926 --> 00:04:00.050
but not being greater than.

00:04:00.050 --> 00:04:03.610
So x is less than or equal to 6.

00:04:03.610 --> 00:04:05.960
So which of these choices
are what we just put in?

00:04:05.960 --> 00:04:09.000
So the square root of x plus 2--
well, that's not-- let's see.

00:04:09.000 --> 00:04:11.140
Square root of x
plus 2 for negative 2

00:04:11.140 --> 00:04:13.760
is less than x is less
than or equal to 2.

00:04:13.760 --> 00:04:16.899
X minus 4 to the third
for 2 is less than x,

00:04:16.899 --> 00:04:18.420
which is less than
or equal to 6.

00:04:18.420 --> 00:04:21.791
So I would go with that one.


Matching expressions to define intervals of functions example

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.950
Match each function
to its interval

00:00:02.950 --> 00:00:04.789
as it appears on
the graph below.

00:00:04.789 --> 00:00:06.330
There will be
functions and intervals

00:00:06.330 --> 00:00:07.663
left over when you are finished.

00:00:07.663 --> 00:00:10.100
So we have this
crazy-looking graph here.

00:00:10.100 --> 00:00:13.400
And what we want to do is think
about what expressions define

00:00:13.400 --> 00:00:15.050
the different parts
of this graph.

00:00:15.050 --> 00:00:17.790
So if we thought,
for example, that

00:00:17.790 --> 00:00:21.300
in the interval between where
x is greater than or equal to 3

00:00:21.300 --> 00:00:25.210
and x is less than 7, that
this is being defined by 1/x,

00:00:25.210 --> 00:00:28.120
we would drag this into
that just like that.

00:00:28.120 --> 00:00:30.170
So to help us think
through it, I've

00:00:30.170 --> 00:00:32.490
copy and pasted this thing
onto our scratch pad.

00:00:32.490 --> 00:00:35.260
And let's just go
interval by interval.

00:00:35.260 --> 00:00:39.300
So this starts at x is
greater than negative 4,

00:00:39.300 --> 00:00:40.720
not greater than or equal to.

00:00:40.720 --> 00:00:44.560
Or we could say it starts at
negative 4 is less than x--

00:00:44.560 --> 00:00:47.014
not less than or equal to,
because the function is not

00:00:47.014 --> 00:00:47.930
defined at negative 4.

00:00:47.930 --> 00:00:49.820
We have an open circle here.

00:00:49.820 --> 00:00:52.250
But let's see what a
good candidate expression

00:00:52.250 --> 00:00:54.750
is for this part
of the function.

00:00:54.750 --> 00:00:57.460
So to do that, let's just find
some nice, well-defined points

00:00:57.460 --> 00:00:57.960
here.

00:00:57.960 --> 00:01:00.140
So it looks like
we have that point

00:01:00.140 --> 00:01:05.420
over there is the point negative
2 comma-- looks like 1/2.

00:01:05.420 --> 00:01:10.320
We have the point here,
which is negative 1 comma 1.

00:01:10.320 --> 00:01:12.410
And those look
like two good ones.

00:01:12.410 --> 00:01:14.607
So let's see which of
these seem to work,

00:01:14.607 --> 00:01:15.690
at least for those points.

00:01:15.690 --> 00:01:17.450
Maybe we can rule out some.

00:01:17.450 --> 00:01:20.160
So 1/x-- so let's see.

00:01:20.160 --> 00:01:26.460
If x is negative 2, 1/x would be
negative 1/2, not positive 1/2.

00:01:26.460 --> 00:01:30.020
So this isn't being used to
define this first section.

00:01:30.020 --> 00:01:33.090
So we can rule that out for
at least the first section.

00:01:33.090 --> 00:01:34.510
Now let's see.

00:01:34.510 --> 00:01:36.640
f of x is equal to negative 1/x.

00:01:36.640 --> 00:01:38.170
Now, this looks pretty good.

00:01:38.170 --> 00:01:42.850
Negative 1 over negative
2 is positive 1/2,

00:01:42.850 --> 00:01:43.900
so it works for that.

00:01:43.900 --> 00:01:47.930
Negative 1 over negative 1 is 1.

00:01:47.930 --> 00:01:51.410
So this right over
here seems to work

00:01:51.410 --> 00:01:53.370
for at least this interval.

00:01:53.370 --> 00:01:54.330
So it seems to work.

00:01:54.330 --> 00:01:56.190
We're doing negative 1/x.

00:01:56.190 --> 00:01:58.100
And it seems to work
even after that,

00:01:58.100 --> 00:02:02.172
where we have the point right
over here, 1, negative 1.

00:02:02.172 --> 00:02:04.340
Negative 1 over 1 is negative 1.

00:02:04.340 --> 00:02:08.169
And over here, you
have 2, negative 1/2,

00:02:08.169 --> 00:02:10.639
which is also consistent
with this right over here.

00:02:10.639 --> 00:02:16.260
So I would throw this
one-- and so this function

00:02:16.260 --> 00:02:20.280
works from the interval
negative 4 is less than x,

00:02:20.280 --> 00:02:22.730
which is less than
or equal to 2.

00:02:22.730 --> 00:02:26.820
So that's this interval
right over here.

00:02:26.820 --> 00:02:29.150
So let's actually fill that in.

00:02:29.150 --> 00:02:34.030
So that is-- actually,
you can't see it.

00:02:34.030 --> 00:02:36.240
It's off the screen right now.

00:02:36.240 --> 00:02:40.670
That is that interval
right over there.

00:02:40.670 --> 00:02:44.110
Now let's get back to
my little scratch pad.

00:02:44.110 --> 00:02:49.450
Now, our function is not
defined for x between 2 and 3.

00:02:49.450 --> 00:02:51.830
But then starting at 3, the
function gets defined again.

00:02:51.830 --> 00:02:52.470
And let's do the same thing.

00:02:52.470 --> 00:02:54.450
Let's look at some
interesting points

00:02:54.450 --> 00:02:55.860
that might prove valuable.

00:02:55.860 --> 00:02:57.831
So this point right
over here-- actually,

00:02:57.831 --> 00:02:58.830
let me do it in magenta.

00:02:58.830 --> 00:03:02.240
You have this point, which
is the point 3, negative 2.

00:03:02.240 --> 00:03:05.180
You have this point right over
here, which is interesting,

00:03:05.180 --> 00:03:08.687
5 comma 2.

00:03:08.687 --> 00:03:10.270
So let's test some
of these points out

00:03:10.270 --> 00:03:12.040
and see which of these work.

00:03:12.040 --> 00:03:14.100
Well, 1/x definitely
doesn't work.

00:03:14.100 --> 00:03:16.170
1/5 is not 2.

00:03:16.170 --> 00:03:17.610
Negative 1 over x doesn't work.

00:03:17.610 --> 00:03:20.590
Negative 1 over 5 is not 2.

00:03:20.590 --> 00:03:25.170
1 over 5 squared is not 2.

00:03:25.170 --> 00:03:27.122
1/2 times 5 plus 5
to the third power--

00:03:27.122 --> 00:03:28.330
that's 10 to the third power.

00:03:28.330 --> 00:03:29.330
That's 1,000 times 1/2.

00:03:29.330 --> 00:03:29.920
That's 500.

00:03:29.920 --> 00:03:31.872
This goes to 502, not 2.

00:03:31.872 --> 00:03:33.580
So we're going to rule
this one out, too.

00:03:33.580 --> 00:03:36.090
So let me rule it
out for the interval

00:03:36.090 --> 00:03:38.190
that we now care about.

00:03:38.190 --> 00:03:42.490
We now care about this interval,
or this part of the function.

00:03:42.490 --> 00:03:43.390
So let's see.

00:03:43.390 --> 00:03:47.590
If we have 5 here-- 5 minus 5 is
0, to the third power is still

00:03:47.590 --> 00:03:50.090
0, times 1/2 is
still 0, plus 2 is 2.

00:03:50.090 --> 00:03:54.280
So this point is consistent with
this function right over here.

00:03:54.280 --> 00:03:54.920
Now, let's see.

00:03:54.920 --> 00:03:56.295
Does that work
for 3, negative 2?

00:03:56.295 --> 00:04:00.240
3 minus 5 is negative 2, to
the third power is negative 8,

00:04:00.240 --> 00:04:02.990
times 1/2 is negative
4, plus 2 is negative 2.

00:04:02.990 --> 00:04:06.990
So this looks like a
pretty good candidate.

00:04:06.990 --> 00:04:09.570
Let's just make sure we
can rule this one out, too.

00:04:09.570 --> 00:04:10.070
Let's see.

00:04:10.070 --> 00:04:11.520
5 comma 2-- let's see.

00:04:11.520 --> 00:04:14.740
5 plus 5 is 10, squared
is 100, times 1/2 is 50.

00:04:14.740 --> 00:04:18.300
52, not 2, so we can rule
this one out, as well.

00:04:18.300 --> 00:04:21.430
So it looks pretty clear that
we could use this expression

00:04:21.430 --> 00:04:23.880
to define the
interval x is greater

00:04:23.880 --> 00:04:27.040
than or equal to 3 and less than
7, not less than or equal to.

00:04:27.040 --> 00:04:28.620
It's not defined at 7.

00:04:28.620 --> 00:04:33.720
So we could use this for this
interval right over here.

00:04:33.720 --> 00:04:36.770
So let's go back to
the actual exercise.

00:04:36.770 --> 00:04:40.390
And so we said this one for
this interval right over here.

00:04:40.390 --> 00:04:45.800
And let's make sure that we got
it right So check our answer,

00:04:45.800 --> 00:04:47.530
and we did.


Estimating a solution to nonlinear system with calculator part 2

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.000
In the last video, we estimated
the solution to e to the x

00:00:04.000 --> 00:00:07.340
is equal to 1 over x times
x minus 1 times x minus 2

00:00:07.340 --> 00:00:08.260
using a calculator.

00:00:08.260 --> 00:00:10.910
We got a first rough estimate
by just looking at this graph,

00:00:10.910 --> 00:00:15.050
and then we tried values out to
truly zero in on, or get close

00:00:15.050 --> 00:00:17.637
to the x value
where this is true.

00:00:17.637 --> 00:00:19.220
What I now want to
do is actually just

00:00:19.220 --> 00:00:22.710
use the graphing functionality
of this calculator

00:00:22.710 --> 00:00:25.880
to try to estimate the
solution graphically.

00:00:25.880 --> 00:00:29.710
So let's go to Graph, and
what I'm going to try to do

00:00:29.710 --> 00:00:31.271
is graph both of
these functions.

00:00:31.271 --> 00:00:32.770
So the first
function-- let me clear

00:00:32.770 --> 00:00:35.270
this-- the first
one is e of x, which

00:00:35.270 --> 00:00:37.780
on the graphing
calculator will be y1.

00:00:37.780 --> 00:00:42.750
And that's going to
be e to the x power,

00:00:42.750 --> 00:00:46.720
and then the second
one, y2, will

00:00:46.720 --> 00:00:50.090
be r of x, which is
going to be 1 divided

00:00:50.090 --> 00:01:02.350
by x times x minus
1 times x minus 2.

00:01:07.110 --> 00:01:08.040
And so, let's see.

00:01:08.040 --> 00:01:09.830
I have to close this
parentheses as well.

00:01:09.830 --> 00:01:11.170
So I've entered it in the graph.

00:01:11.170 --> 00:01:13.410
And I care about
where we zoom in,

00:01:13.410 --> 00:01:16.590
so I really want to zoom
into this part of the graph

00:01:16.590 --> 00:01:17.490
right over here.

00:01:17.490 --> 00:01:19.710
So let me go to the Range.

00:01:19.710 --> 00:01:23.960
So, actually there's also a Zoom
functionality that I could use,

00:01:23.960 --> 00:01:26.200
but let's-- actually,
let me do that.

00:01:26.200 --> 00:01:29.872
That could be fun, so
let's just graph it.

00:01:29.872 --> 00:01:31.330
Actually, let's
just see what range

00:01:31.330 --> 00:01:33.110
it's graphed at right now.

00:01:33.110 --> 00:01:33.850
Let's see.

00:01:33.850 --> 00:01:36.220
What we would care
about, let's start

00:01:36.220 --> 00:01:39.200
with a rough
approximation, just to see

00:01:39.200 --> 00:01:41.290
that this is indeed
the same graph.

00:01:41.290 --> 00:01:45.082
So let's start with x
going from 0 up to-- I

00:01:45.082 --> 00:01:49.070
don't know-- 3, so this would
be this part of the graph

00:01:49.070 --> 00:01:50.250
right over here.

00:01:50.250 --> 00:01:51.600
And then the x scale is 1.

00:01:51.600 --> 00:01:54.000
That's what they'll
mark off, every 1.

00:01:54.000 --> 00:01:57.180
We could even mark off
every 0.5 if we want.

00:01:57.180 --> 00:01:59.660
Like this one is
marked off every 0.5.

00:01:59.660 --> 00:02:04.820
And the y minimum, let's go
from 0 to-- on this range,

00:02:04.820 --> 00:02:07.330
actually it goes pretty high,
and the way this is graphed,

00:02:07.330 --> 00:02:10.800
it goes all the way up to,
looks like 10, so I'll go 10.

00:02:10.800 --> 00:02:12.610
I'll leave the y scale as 1.

00:02:12.610 --> 00:02:15.200
They mark it off every
1 right over here.

00:02:15.200 --> 00:02:17.750
And now let's graph this thing.

00:02:17.750 --> 00:02:23.700
And that was e of x, and
now it's graphing r of x.

00:02:23.700 --> 00:02:25.580
And you see it
indeed looks pretty

00:02:25.580 --> 00:02:27.380
similar to what we have here.

00:02:27.380 --> 00:02:30.320
Now what we care about is this
point, or on our calculator,

00:02:30.320 --> 00:02:31.560
this point right over here.

00:02:31.560 --> 00:02:33.540
We want to figure
out what x value--

00:02:33.540 --> 00:02:35.900
what is the x-coordinate of
this point of intersection?

00:02:35.900 --> 00:02:38.950
This is when our two functions
are equal to each other.

00:02:38.950 --> 00:02:44.170
So let me zoom in on this.

00:02:44.170 --> 00:02:46.880
I think I can use this
Box functionality.

00:02:46.880 --> 00:02:50.010
So it essentially lets me
construct a box around this,

00:02:50.010 --> 00:02:52.260
and it's going to
zoom into that box.

00:02:52.260 --> 00:02:56.140
So I'm going to get as tight
in on this as I can go.

00:02:56.140 --> 00:02:59.610
So if I press, I can
get even tighter on it.

00:02:59.610 --> 00:03:04.160
So if I press Enter, now I
can define the other corner

00:03:04.160 --> 00:03:06.700
of the box, so
that's pretty good.

00:03:06.700 --> 00:03:08.640
I'm going to zoom
in, press Enter,

00:03:08.640 --> 00:03:12.220
and that's zoomed in on
to that little teeny box.

00:03:12.220 --> 00:03:16.090
So that was e of x, and now
it's going to graph r of x.

00:03:16.090 --> 00:03:19.860
So now let me try
to trace the graph.

00:03:19.860 --> 00:03:21.230
So let's see.

00:03:21.230 --> 00:03:24.500
Trace, so it's letting
me trace e of x.

00:03:24.500 --> 00:03:25.510
And let's see.

00:03:25.510 --> 00:03:29.280
If I look at my x values
decrease-- so at this point,

00:03:29.280 --> 00:03:31.530
e of x is still
higher than r of x.

00:03:31.530 --> 00:03:39.790
And if we get right over here--
so 2.056, we see that r of x

00:03:39.790 --> 00:03:44.270
is above e of x.

00:03:44.270 --> 00:03:46.670
We just see that
graphically, and then we're

00:03:46.670 --> 00:03:49.430
left of the point
of intersection.

00:03:49.430 --> 00:03:53.020
And then we're still left of
the point of intersection.

00:03:53.020 --> 00:03:54.910
Now we're right of the
point of intersection,

00:03:54.910 --> 00:03:56.660
so it looks like the
point of intersection

00:03:56.660 --> 00:04:02.760
is between 2.057 and 2.059.

00:04:02.760 --> 00:04:04.260
And so in the
previous video when

00:04:04.260 --> 00:04:09.490
we said where our
estimate was 2.06,

00:04:09.490 --> 00:04:13.640
we were definitely within 0.01
of the point of intersection.

00:04:13.640 --> 00:04:18.038
If we did want to get even more
precise, we could zoom in more,

00:04:18.038 --> 00:04:20.329
and I encourage you if you've
got a graphing calculator

00:04:20.329 --> 00:04:23.463
like this to actually
try that out.


Estimating a solution to nonlinear system with calculator

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:04.290
Graphs of e of x
equals e to the x and r

00:00:04.290 --> 00:00:08.910
of x is equal to 1 over x
times x minus 1 times x minus 2

00:00:08.910 --> 00:00:10.730
are shown below.

00:00:10.730 --> 00:00:14.960
Estimate the solution to e
to the x-- so that's e of x--

00:00:14.960 --> 00:00:19.260
being equal to essentially
r of x within 0.01.

00:00:19.260 --> 00:00:21.790
So we want to
figure out for what

00:00:21.790 --> 00:00:27.457
value does e of x equal r of x?

00:00:27.457 --> 00:00:28.790
And they want us to estimate it.

00:00:28.790 --> 00:00:31.310
We can either just try
to get as close as we can

00:00:31.310 --> 00:00:32.439
from this graph.

00:00:32.439 --> 00:00:35.610
They want us to be within 0.01.

00:00:35.610 --> 00:00:37.140
And we can also
use a calculator,

00:00:37.140 --> 00:00:40.320
kind of try numbers out
to hopefully zero in

00:00:40.320 --> 00:00:43.390
on this point right
over here, where e of x

00:00:43.390 --> 00:00:46.960
is equal to r of x.

00:00:46.960 --> 00:00:50.220
So what I want to do is, let
me draw a little table here.

00:00:50.220 --> 00:00:52.510
Let's try out some x values.

00:00:52.510 --> 00:00:54.300
And then for each
of these x values,

00:00:54.300 --> 00:01:04.110
let's see where we land on e of
x and where we land on r of x,

00:01:04.110 --> 00:01:07.125
and then we can decide whether
we are too high or too low.

00:01:07.125 --> 00:01:09.500
And I encourage you to pause
this video before I actually

00:01:09.500 --> 00:01:12.100
go ahead and do this, and
try to do this on your own.

00:01:12.100 --> 00:01:15.190
But I do suggest using
some form of calculator

00:01:15.190 --> 00:01:18.179
or, well, probably a calculator.

00:01:18.179 --> 00:01:19.720
I'm assuming you've
given a go at it,

00:01:19.720 --> 00:01:20.870
and now I will attempt it.

00:01:20.870 --> 00:01:22.690
Now just eyeballing
it-- and eyeballing it

00:01:22.690 --> 00:01:25.350
is helpful, because that'll
give us kind of our first order

00:01:25.350 --> 00:01:29.350
approximation of at what x value
are these two functions equal.

00:01:29.350 --> 00:01:32.120
If I just look at this
graph the way it's drawn,

00:01:32.120 --> 00:01:40.130
it looks like this is
pretty close to 2.1.

00:01:40.130 --> 00:01:43.580
It looks like when x is
2.1, both of those functions

00:01:43.580 --> 00:01:45.960
look pretty close
to-- I don't know.

00:01:45.960 --> 00:01:51.140
This looks like about 7.7 or
7.8 or something like that.

00:01:51.140 --> 00:01:53.310
But let's figure out
what they're doing.

00:01:53.310 --> 00:01:57.310
So let's see, when
x is equal to 2.1--

00:01:57.310 --> 00:02:04.210
get my calculator out-- when
x is equal to 2.1, well,

00:02:04.210 --> 00:02:06.880
e of x is just e to the x power.

00:02:06.880 --> 00:02:15.430
So e to the 2.1 power
is equal to 8.166.

00:02:15.430 --> 00:02:20.320
Let me write that down, 8.166.

00:02:20.320 --> 00:02:22.006
And what is r of x?

00:02:22.006 --> 00:02:29.700
r of x is 1 divided by x,
so that's going to be 2.1.

00:02:29.700 --> 00:02:31.020
Times x minus 1.

00:02:31.020 --> 00:02:36.460
Well, that's going to be
1.1, so that's times 1.1.

00:02:36.460 --> 00:02:38.660
Times x minus 2.

00:02:38.660 --> 00:02:42.660
Well, that's just going
to be 0.1, times 0.1.

00:02:42.660 --> 00:02:46.770
And that is equal to
4-- did I do that right?

00:02:46.770 --> 00:02:48.080
No, that can't be.

00:02:48.080 --> 00:02:58.006
2.1 over-- 2.1
times 1.1 times 0.1.

00:02:58.006 --> 00:03:00.090
1 over all of that.

00:03:00.090 --> 00:03:01.810
4.32?

00:03:01.810 --> 00:03:06.380
Let's see, 2.1 r
of x is 4 point-- I

00:03:06.380 --> 00:03:07.440
guess that's possible.

00:03:07.440 --> 00:03:09.190
Actually, that looks
right, because r of x

00:03:09.190 --> 00:03:11.850
declines so sharply
right over here.

00:03:11.850 --> 00:03:14.830
So it's actually, at 2.1
where actually r of x

00:03:14.830 --> 00:03:22.290
is actually closer to right
over here, give or take.

00:03:22.290 --> 00:03:26.520
So it's equal to 4.32.

00:03:26.520 --> 00:03:31.370
So 2.1 e of x is actually a
much larger value than r of x.

00:03:31.370 --> 00:03:36.250
So e of x is clearly
too high. r of x

00:03:36.250 --> 00:03:38.960
is already dropped
a good bit by then.

00:03:38.960 --> 00:03:41.860
If I were to go all
the way down to 2, at 2

00:03:41.860 --> 00:03:46.470
it looks like actually r
of x kind of spikes up.

00:03:46.470 --> 00:03:48.780
It just goes to infinity
as we approach 2.

00:03:48.780 --> 00:03:50.940
So we're not going to go
all the way down to 2,

00:03:50.940 --> 00:03:52.648
but why don't we lower
this a little bit.

00:03:52.648 --> 00:03:55.360
Why don't we try 2.05?

00:03:55.360 --> 00:04:00.390
So 2.05, what is e of x?

00:04:00.390 --> 00:04:02.670
e of x is e to the x, right?

00:04:02.670 --> 00:04:11.380
So e to the 2.05 power gets
us 7.76-- I'll round it,

00:04:11.380 --> 00:04:15.070
8-- 7.768.

00:04:15.070 --> 00:04:18.160
Approximately 7--
actually, all these

00:04:18.160 --> 00:04:23.860
are approximate, so
I'll just write 7.768.

00:04:23.860 --> 00:04:24.790
And what is r of x?

00:04:24.790 --> 00:04:26.980
I'll just keep rounding
to the thousands.

00:04:26.980 --> 00:04:29.570
Here, well, we didn't have
to round too much just

00:04:29.570 --> 00:04:33.560
because that was so far
off, but I'll put it there.

00:04:33.560 --> 00:04:37.080
Actually, it was
329, so I could--

00:04:37.080 --> 00:04:39.660
let me write it this way-- 3290.

00:04:39.660 --> 00:04:41.160
So let me throw
that 9 here, just so

00:04:41.160 --> 00:04:44.800
everything-- we evaluate
the function to thousandths.

00:04:44.800 --> 00:04:48.900
So let's evaluate r of
x, when we're at 2.05,

00:04:48.900 --> 00:04:54.420
it's going to be 1 divided by
x, which is now 2.05, times x

00:04:54.420 --> 00:05:02.290
minus 1, which is 1.05, times
x minus 2, which is 0.05.

00:05:02.290 --> 00:05:14.270
And that gets us to 9.29-- I'll
round to 2-- 9.292, so 9.292.

00:05:14.270 --> 00:05:17.450
So now we're on this
side, where r of x

00:05:17.450 --> 00:05:20.840
is roughly right over
here and it's more than e

00:05:20.840 --> 00:05:24.930
of x, which is at 7.7,
which is right around here.

00:05:24.930 --> 00:05:28.050
So now our x value is too low.

00:05:28.050 --> 00:05:30.670
So maybe let's see if we
can go a little bit higher.

00:05:30.670 --> 00:05:35.182
And let's try to go roughly
halfway between these two,

00:05:35.182 --> 00:05:37.390
but I don't want to get too
precise, because you have

00:05:37.390 --> 00:05:39.170
to get to the nearest hundredth.

00:05:39.170 --> 00:05:45.730
So let's go to 2.07.

00:05:45.730 --> 00:06:00.740
So e to the 2.07 is equal to
7.925 if I round it, 7.925.

00:06:00.740 --> 00:06:04.970
I want to do all this in green
just to be color consistent.

00:06:04.970 --> 00:06:08.300
And now let's evaluate r
of x at that same value.

00:06:08.300 --> 00:06:14.880
So 1 divided by x, which is
2.07, times that minus 1,

00:06:14.880 --> 00:06:21.170
which is 1.07, times that
minus 2, which is 0.07,

00:06:21.170 --> 00:06:34.120
which gives us 6.44, I
guess we could say 6.450.

00:06:34.120 --> 00:06:38.130
So at 2.05 that was too
low, 2.07 is too high.

00:06:38.130 --> 00:06:43.100
Now, r of x has
dropped below e of x.

00:06:43.100 --> 00:06:46.870
So we know the right answer is
in between these two numbers,

00:06:46.870 --> 00:06:49.820
and so if we select
2.06 that's definitely

00:06:49.820 --> 00:06:53.940
going to be within 0.01
of the right answer.

00:06:53.940 --> 00:07:01.380
So I would go with
2.06 is definitely

00:07:01.380 --> 00:07:06.429
going to be within the 0.01 of
the correct solution to this.

00:07:06.429 --> 00:07:08.470
But just for fun, let's
actually just try it out.

00:07:08.470 --> 00:07:16.460
So e to the 2.06 is 7.84--
I guess we could round to 6.

00:07:16.460 --> 00:07:18.380
And if we were to
evaluate r of x,

00:07:18.380 --> 00:07:23.380
it's 1 divided by 2.06
times that minus 1,

00:07:23.380 --> 00:07:29.120
which is 1.06, times 0.06.

00:07:29.120 --> 00:07:33.500
It gets us to 7.632.

00:07:33.500 --> 00:07:36.410
So we're also
getting pretty close,

00:07:36.410 --> 00:07:38.120
but our precision
that they gave,

00:07:38.120 --> 00:07:41.420
they don't say that they have
to be within each other of that,

00:07:41.420 --> 00:07:43.070
they say, let's
estimate the solution.

00:07:43.070 --> 00:07:47.110
So there's some actual
precise solution

00:07:47.110 --> 00:07:50.710
to this right over here, some x
value, where these are actually

00:07:50.710 --> 00:07:51.690
equal to each other.

00:07:51.690 --> 00:07:54.590
That's the x value, which gives
us this point of intersection.

00:07:54.590 --> 00:07:57.990
We just have to get within
0.01 of that x value,

00:07:57.990 --> 00:08:02.900
and 2.06 definitely
does the trick.


Saturated fats, unsaturated fats, and trans fats

WEBVTT
Kind: captions
Language: en

00:00:00.284 --> 00:00:02.687
- So I have drawn four different

00:00:02.687 --> 00:00:05.125
triglyceride molecules over here.

00:00:05.125 --> 00:00:06.064
And some of you might be saying,

00:00:06.064 --> 00:00:09.013
"wait I thought triglycerides they involve

00:00:09.013 --> 00:00:11.952
"all of these carbons and hydrogens and oxygens".

00:00:11.952 --> 00:00:14.158
Well I see the oxygens over here

00:00:14.158 --> 00:00:16.654
but where's all of the carbons and hydrogens?

00:00:16.654 --> 00:00:18.593
And my answer to you is that they will be implicit.

00:00:18.593 --> 00:00:20.857
This is a short-hand way of diagramming out

00:00:20.857 --> 00:00:22.053
these large molecules.

00:00:22.053 --> 00:00:23.388
And you'll see this many times

00:00:23.388 --> 00:00:26.128
when you take chemistry, biology, organic chemistry.

00:00:26.128 --> 00:00:27.289
And what's happening here is that

00:00:27.289 --> 00:00:30.516
each of these pointy parts of this chain,

00:00:30.516 --> 00:00:33.024
each of these vertices there's implicitly a carbon.

00:00:33.024 --> 00:00:34.791
There's a carbon there, there's a carbon there.

00:00:34.791 --> 00:00:35.753
There's a carbon there.

00:00:35.753 --> 00:00:36.588
And there's also,

00:00:36.588 --> 00:00:39.386
if we don't put a letter at the end of the chain,

00:00:39.386 --> 00:00:41.790
there's also a carbon right over there.

00:00:41.790 --> 00:00:43.345
So implicitly there's a carbon there,

00:00:43.345 --> 00:00:45.656
there's a carbon there, there's a carbon there.

00:00:45.656 --> 00:00:46.724
Now you might also be wondering

00:00:46.724 --> 00:00:47.827
what about the hydrogens?

00:00:47.827 --> 00:00:52.726
Well we assume that every carbon has four covalent bonds.

00:00:52.726 --> 00:00:55.222
And so if there's extra covalent bonds

00:00:55.222 --> 00:00:56.221
for each of these carbons,

00:00:56.221 --> 00:00:59.089
we assume that those covalent bonds are with hydrogen.

00:00:59.089 --> 00:01:01.445
So for example, this carbon right over here,

00:01:01.445 --> 00:01:03.408
it has two covalent bonds.

00:01:03.408 --> 00:01:06.252
So the other two covalent bonds must be

00:01:06.252 --> 00:01:07.819
must be with hydrogens.

00:01:07.819 --> 00:01:10.188
Same thing for this carbon right over there.

00:01:10.188 --> 00:01:11.987
It must be bonded to two hydrogens.

00:01:11.987 --> 00:01:13.508
This carbon, the way its drawn, we only

00:01:13.508 --> 00:01:15.761
explicitly see one covalent bond.

00:01:15.761 --> 00:01:18.222
So there must be three covalent bonds.

00:01:18.222 --> 00:01:21.124
There must be three covalent bonds to hydrogens.

00:01:21.124 --> 00:01:22.881
So the carbons and hydrogens are all there.

00:01:22.881 --> 00:01:25.025
They're just implicitly there.

00:01:25.025 --> 00:01:26.953
Now you might notice, and actually, if you don't notice,

00:01:26.953 --> 00:01:28.195
I encourage you to pause

00:01:28.195 --> 00:01:29.518
and think about what the difference is

00:01:29.518 --> 00:01:31.318
between this triglyceride, this triglyceride

00:01:31.318 --> 00:01:33.350
this triglyceride and that triglyceride.

00:01:33.350 --> 00:01:36.067
And it might jump at you pretty quickly.

00:01:36.067 --> 00:01:38.609
What jumps at you is this triglyceride has

00:01:38.609 --> 00:01:40.258
no double bonds.

00:01:40.258 --> 00:01:42.289
This one has one double bond.

00:01:42.289 --> 00:01:44.960
This one has several double bonds.

00:01:44.960 --> 00:01:47.583
And this one also has several double bonds.

00:01:47.583 --> 00:01:48.884
But these two are also different

00:01:48.884 --> 00:01:50.590
and we're gonna think about the way that they're different.

00:01:50.590 --> 00:01:52.983
You might immediately see that this one

00:01:52.983 --> 00:01:54.654
kinda kinds and curves

00:01:54.654 --> 00:01:56.755
while this one is able to stay relatively straight.

00:01:56.755 --> 00:01:58.183
And actually that is the main difference.

00:01:58.183 --> 00:01:59.542
And we're gonna talk about it in a second

00:01:59.542 --> 00:02:01.446
why that is.

00:02:01.446 --> 00:02:02.793
But first lets talk about this one.

00:02:02.793 --> 00:02:05.242
When you have all single bonds

00:02:05.242 --> 00:02:06.217
one way to think about it is that

00:02:06.217 --> 00:02:08.551
you have put as many hydrogens onto

00:02:08.551 --> 00:02:10.652
these carbon chains as you can.

00:02:10.652 --> 00:02:11.953
Or another way of thinking about it is

00:02:11.953 --> 00:02:15.889
we have saturated this fat with hydrogens.

00:02:15.889 --> 00:02:18.870
And that's why this is called a saturated fat.

00:02:19.360 --> 00:02:22.286
A saturated, this is a saturated fat.

00:02:22.286 --> 00:02:23.958
This is something that you might have heard of.

00:02:23.958 --> 00:02:26.651
This is referring to things like butter.

00:02:26.651 --> 00:02:28.718
They've a lot of different fats that we tend to

00:02:28.718 --> 00:02:31.492
associate with being solid at room temperature.

00:02:31.492 --> 00:02:33.489
And the reason why they're solid at room temperature

00:02:33.489 --> 00:02:37.193
is because this are all bonded to these hydrogens.

00:02:37.193 --> 00:02:39.724
There's no kinks for them to do to the double bonds.

00:02:39.724 --> 00:02:42.150
These are able to be relatively dense.

00:02:42.150 --> 00:02:43.787
Which allows it to be solid

00:02:43.787 --> 00:02:47.189
or typically solid at room temperature.

00:02:47.189 --> 00:02:48.559
And these are sometimes associated with,

00:02:48.559 --> 00:02:50.393
I don't want to get into the whole nutritional battles

00:02:50.393 --> 00:02:52.750
about saturated fat and fat's good or bad.

00:02:52.750 --> 00:02:54.178
But these are sometimes associated

00:02:54.178 --> 00:02:56.384
these are sometimes called your bad fats.

00:02:56.384 --> 00:02:57.952
But as we'll see they aren't the worse fats.

00:02:57.952 --> 00:03:00.517
The worse fats are actually this one right over here.

00:03:00.517 --> 00:03:02.619
But we'll talk about that in a second.

00:03:02.619 --> 00:03:04.627
So this is saturated fat because a way to think about it

00:03:04.627 --> 00:03:07.727
is it's saturated with as many hydrogens as possible.

00:03:07.727 --> 00:03:09.620
Now this one of the right,

00:03:09.620 --> 00:03:10.595
and I don't know anything about it

00:03:10.595 --> 00:03:11.686
it has all single bonds.

00:03:11.686 --> 00:03:14.925
Now over here, or all single bonds between the carbons.

00:03:14.925 --> 00:03:17.421
Over here we see a double bond.

00:03:17.421 --> 00:03:20.661
We see a double bond between that carbon and that carbon.

00:03:20.661 --> 00:03:21.717
And because of that,

00:03:21.717 --> 00:03:25.223
this carbon already has three covalent bonds

00:03:25.223 --> 00:03:27.325
so it's only gonna be bonded with one hydrogen.

00:03:27.325 --> 00:03:28.881
This one already has three covalent bonds

00:03:28.881 --> 00:03:31.121
so it's only gonna be bonded with one hydrogen

00:03:31.121 --> 00:03:34.326
as opposed to two like these characters over here.

00:03:34.326 --> 00:03:35.452
And because of that we don't have

00:03:35.452 --> 00:03:37.391
as many hydrogens on the chain as possible.

00:03:37.391 --> 00:03:40.596
So we consider this to be an unsaturated fat.

00:03:41.336 --> 00:03:46.888
Unsaturated fat.

00:03:46.898 --> 00:03:49.651
We don't have as many hydrogens as possible.

00:03:49.651 --> 00:03:51.323
And because of that unsaturated fat,

00:03:51.323 --> 00:03:53.424
and specially polyunsaturated fats,

00:03:53.424 --> 00:03:55.827
these double bonds they tend to form

00:03:55.827 --> 00:03:58.056
these kinks in the structure

00:03:58.056 --> 00:04:00.483
which keep the molecules from getting really, really dense

00:04:00.483 --> 00:04:03.989
which tends to make them liquid or more likely to be liquid

00:04:03.989 --> 00:04:05.754
at room temperature.

00:04:05.754 --> 00:04:07.553
So this is an unsaturated fat.

00:04:07.553 --> 00:04:09.295
This right over here, we have multiple,

00:04:09.295 --> 00:04:13.591
we have multiple double bonds in play.

00:04:13.591 --> 00:04:16.383
This is called a polyunsaturated fat.

00:04:16.923 --> 00:04:22.002
Polyunsaturated fat.

00:04:22.112 --> 00:04:23.170
And since this actually only has,

00:04:23.170 --> 00:04:24.817
they're both unsaturated fats.

00:04:24.817 --> 00:04:27.151
This is a polyunsaturated fat.

00:04:27.151 --> 00:04:28.718
And since this only has one double bond

00:04:28.718 --> 00:04:33.353
we can call this a monounsaturated fat.

00:04:33.353 --> 00:04:34.651
So these are both unsaturated fat.

00:04:34.651 --> 00:04:36.660
This is many times happening so poly.

00:04:36.660 --> 00:04:40.108
This is happening once, so we call it monounsaturated fat.

00:04:40.108 --> 00:04:41.513
Now one question you might have is

00:04:41.513 --> 00:04:44.849
why are the kinks forming for

00:04:44.849 --> 00:04:46.656
this molecule and this molecule.

00:04:46.656 --> 00:04:48.827
And why are they not forming with this molecule

00:04:48.827 --> 00:04:51.242
even though this one has double bonds as well.

00:04:51.242 --> 00:04:52.890
And this goes to our good friends

00:04:52.890 --> 00:04:55.119
cis and trance.

00:04:55.119 --> 00:04:56.420
And so you might remember,

00:04:56.420 --> 00:04:59.520
if you have a double bond between carbons.

00:04:59.520 --> 00:05:01.482
And lets say that you have,

00:05:01.482 --> 00:05:06.358
lets say that you have one configuration.

00:05:07.398 --> 00:05:08.250
Let me just do it this way.

00:05:08.250 --> 00:05:09.620
Lets say you've one configuration where

00:05:09.620 --> 00:05:12.093
this is attached to some type of a carbon chain,

00:05:12.093 --> 00:05:14.495
and lets say that this is a hydrogen.

00:05:15.019 --> 00:05:17.120
Then on the other side the rest of the carbon chain

00:05:17.120 --> 00:05:19.326
it could be in one of two configurations.

00:05:19.326 --> 00:05:20.789
The rest of the carbon

00:05:20.789 --> 00:05:22.983
the rest of the carbon chain could

00:05:22.983 --> 00:05:25.793
be on the same side as the carbon chain on the left.

00:05:25.793 --> 00:05:28.443
So I'll put R' there.

00:05:28.507 --> 00:05:31.067
And, let me do that in the same color.

00:05:31.067 --> 00:05:33.904
So you have your hydrogen.

00:05:33.904 --> 00:05:35.594
So this is one configuration.

00:05:35.594 --> 00:05:38.000
Now another configuration would be

00:05:38.320 --> 00:05:40.189
carbon double bonded to carbon.

00:05:40.189 --> 00:05:42.093
You have your hydrogen but now the hydrogens

00:05:42.093 --> 00:05:43.951
are on opposite sides.

00:05:43.951 --> 00:05:48.780
And your chains,

00:05:48.780 --> 00:05:50.360
the rest of the carbon chains

00:05:50.360 --> 00:05:52.469
are on opposite sides.

00:05:53.123 --> 00:05:54.261
And as we talked about before,

00:05:54.261 --> 00:05:55.793
this is because double bonds are rigid.

00:05:55.793 --> 00:05:56.885
You can't rotate around it.

00:05:56.885 --> 00:05:58.452
So it matters, these are actually different,

00:05:58.452 --> 00:06:00.960
these are actually different isomers right over here.

00:06:00.960 --> 00:06:02.422
Depending on whether this chain is on

00:06:02.422 --> 00:06:03.850
the same side as this chain.

00:06:03.850 --> 00:06:05.395
Or whether is on the same side.

00:06:05.395 --> 00:06:06.486
When it's on the same side

00:06:06.486 --> 00:06:08.785
we call this a cis configuration

00:06:10.395 --> 00:06:13.858
and this is a trans, trans configuration.

00:06:13.858 --> 00:06:15.681
And it turns out that most naturally

00:06:15.681 --> 00:06:19.582
produced unsaturated fats are in the cis configuration.

00:06:19.582 --> 00:06:21.428
And because they're on the cis configuration

00:06:21.428 --> 00:06:23.053
whenever you have these double bonds,

00:06:23.053 --> 00:06:25.759
it forms, it makes the chain actually bend.

00:06:25.759 --> 00:06:27.361
And if you have many of these double bonds

00:06:27.361 --> 00:06:28.789
it makes it bend a lot.

00:06:28.789 --> 00:06:30.751
And so polyunsaturated fats are even more

00:06:30.751 --> 00:06:34.060
likely to be liquid because it's very hard to pack them.

00:06:34.060 --> 00:06:35.557
So what's going on over here?

00:06:35.557 --> 00:06:37.786
Well these are actually the configuration

00:06:37.786 --> 00:06:39.981
where the, I guess you could say,

00:06:39.981 --> 00:06:41.525
the rest of the carbon chains

00:06:41.525 --> 00:06:43.789
are on opposite sides of the double bond.

00:06:43.789 --> 00:06:45.287
Notice this carbon chain it might be

00:06:45.287 --> 00:06:46.517
a little bit hard to see

00:06:46.517 --> 00:06:48.061
is above the double bond.

00:06:48.061 --> 00:06:50.685
We formed a covalent bond going upward there.

00:06:50.685 --> 00:06:54.551
While this carbon chain,

00:06:54.551 --> 00:06:57.558
right over here, is below the double bond.

00:06:57.558 --> 00:06:59.288
And because of that it doesn't form a kink

00:06:59.288 --> 00:07:01.355
and these are in the trans configuration.

00:07:01.355 --> 00:07:02.585
And so this right over here,

00:07:02.585 --> 00:07:04.919
this is called a trans fat.

00:07:04.919 --> 00:07:08.571
This is called a trans fat.

00:07:08.657 --> 00:07:10.745
And they aren't typically found,

00:07:10.745 --> 00:07:12.512
they aren't typically found in nature.

00:07:12.512 --> 00:07:14.843
Now it's interesting about trans fats

00:07:14.843 --> 00:07:16.355
is a lot of folks say "hey you know, okay,

00:07:16.355 --> 00:07:18.410
"maybe saturated fat is bad for us.

00:07:18.410 --> 00:07:20.418
"Maybe we shouldn't eat as much butter and all of that".

00:07:20.418 --> 00:07:23.518
And some of that's even up for debate these days.

00:07:23.518 --> 00:07:27.361
And they say, "what if we started with unsaturated fats?".

00:07:27.361 --> 00:07:29.927
Which are typically viewed as more healthy,

00:07:29.927 --> 00:07:32.249
our polyunsaturated fat.

00:07:32.249 --> 00:07:34.350
And were just throw a bunch of hydrogens on them,

00:07:34.350 --> 00:07:37.949
may be not to fully saturate it,

00:07:37.949 --> 00:07:39.656
but enough hydrogen so that some of these

00:07:39.656 --> 00:07:41.084
double bonds disappear.

00:07:41.084 --> 00:07:42.860
And so it's more solid at room temperature which

00:07:42.860 --> 00:07:46.622
might make it a good replacement for butter in cooking.

00:07:46.622 --> 00:07:49.455
And a lot of the shortening that you might have seen

00:07:49.455 --> 00:07:52.138
even 10, 15 years ago, or even today in a lot of places,

00:07:52.138 --> 00:07:54.482
they're essentially trans fat.

00:07:54.482 --> 00:07:56.583
And so what happened is is that,

00:07:56.583 --> 00:07:58.395
yes, you can replace a lot of these,

00:07:58.395 --> 00:08:00.821
you can start to saturate it more with hydrogens.

00:08:00.821 --> 00:08:05.828
But that process also turned some of the

00:08:05.828 --> 00:08:07.528
cis double bonds

00:08:07.528 --> 00:08:09.517
into trans double bonds.

00:08:09.517 --> 00:08:10.924
And at first people said, "oh that's harmless,

00:08:10.924 --> 00:08:14.428
"this is probably good for you, it's still unsaturated".

00:08:14.428 --> 00:08:17.690
But it has some of the properties of a saturated fat.

00:08:17.690 --> 00:08:20.291
It's nice and solid and buttery and all of that.

00:08:20.291 --> 00:08:22.183
And maybe is even cheaper to produce.

00:08:22.183 --> 00:08:23.855
You can take it from other oils.

00:08:23.855 --> 00:08:26.293
But it turns out that this is very unhealthy.

00:08:26.293 --> 00:08:27.884
There's a lot of debates in nutrition,

00:08:27.884 --> 00:08:30.287
but this is unequivocally unhealthy.

00:08:30.287 --> 00:08:31.460
This does not exist in nature

00:08:31.460 --> 00:08:33.120
and it has all sorts of bad impacts.

00:08:33.120 --> 00:08:35.488
And that's why a lot of states, and even countries

00:08:35.488 --> 00:08:38.391
have now banned trans fats.

00:08:38.391 --> 00:08:40.783
They actually, I've even heard people

00:08:40.783 --> 00:08:43.116
go so far and say "this is actually poisonous

00:08:43.116 --> 00:08:44.684
"to your body" in certain ways.

00:08:44.684 --> 00:08:48.190
And it really affects you in extremely negative ways.

00:08:48.190 --> 00:08:50.152
That might be extra strong language

00:08:50.152 --> 00:08:52.061
but I'm, I guess I'm trying to scare you a little bit.

00:08:52.061 --> 00:08:53.334
Don't eat trans fats.

00:08:53.334 --> 00:08:55.372
Anyway, hopefully you enjoyed that.


Molecular structure of triglycerides (fats)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.050
- [Voiceover] Let's study the molecular structure

00:00:02.050 --> 00:00:04.393
of triglycerides.

00:00:07.483 --> 00:00:11.320
And in everyday language we often call these fats.

00:00:11.320 --> 00:00:14.500
We often call these fats.

00:00:14.500 --> 00:00:16.700
Which is often kind of a feared word

00:00:16.700 --> 00:00:19.360
but as we'll see it's essential for life.

00:00:19.360 --> 00:00:23.360
Fats in a liquid form is sometimes referred to as oils.

00:00:23.360 --> 00:00:25.920
And a more technical term for triglycerides--

00:00:25.920 --> 00:00:27.550
Triglyceride is the word that you might hear

00:00:27.550 --> 00:00:29.291
when you go to the doctor's office.

00:00:29.291 --> 00:00:32.430
But it can also be referred to as tri--

00:00:32.430 --> 00:00:34.650
And I'm going to do the different parts in different colors.

00:00:34.650 --> 00:00:38.520
Triacylglycerol.

00:00:40.899 --> 00:00:43.659
I actually like this second name more

00:00:43.659 --> 00:00:46.489
because it makes sense when you break down

00:00:46.489 --> 00:00:51.159
the molecule, triacylglycerol.

00:00:51.159 --> 00:00:53.760
But triglycerides, this is literally fat.

00:00:53.760 --> 00:00:55.950
Or where what most people typically refer to as fat,

00:00:55.950 --> 00:00:57.160
they're talking about triglycerides.

00:00:57.160 --> 00:00:58.490
If you were to get your cholesterol checked

00:00:58.490 --> 00:01:00.290
or if your parents get their cholesterol checked,

00:01:00.290 --> 00:01:02.090
they'll probably get a report on triglycerides

00:01:02.090 --> 00:01:04.029
which is really a measure of the fat

00:01:04.029 --> 00:01:06.590
concentration in their blood.

00:01:06.590 --> 00:01:08.930
But let's think about what a triglyceride is.

00:01:08.930 --> 00:01:11.499
And so here I have the constituent molecules

00:01:11.499 --> 00:01:13.977
of a triglyceride.

00:01:13.977 --> 00:01:18.479
This molecule on the left, this is glycerol.

00:01:18.479 --> 00:01:19.750
Let me do it in that same blue

00:01:19.750 --> 00:01:22.129
that I just wrote glycerol in.

00:01:22.129 --> 00:01:25.649
This is glycerol.

00:01:25.649 --> 00:01:28.160
We can see we have a three carbon chain right over here.

00:01:28.160 --> 00:01:29.830
Each of those carbons is attached

00:01:29.830 --> 00:01:32.529
to a hydroxyl group, an O-H group.

00:01:32.529 --> 00:01:33.960
If you just get attached to one of them,

00:01:33.960 --> 00:01:34.870
this makes you an alcohol

00:01:34.870 --> 00:01:37.090
so this definitely makes it an alcohol.

00:01:37.090 --> 00:01:38.520
This is sometimes called a triol.

00:01:38.520 --> 00:01:41.690
It has three hydroxyl groups right over here.

00:01:41.690 --> 00:01:44.700
And this is considered to be a sugar alcohol.

00:01:45.490 --> 00:01:47.300
A sugar alcohol.

00:01:47.300 --> 00:01:49.800
We see why it's an alcohol, it's got its hydroxyl groups.

00:01:49.800 --> 00:01:50.892
And it's also sweet.

00:01:50.892 --> 00:01:53.459
If you were to drink a solution of

00:01:53.459 --> 00:01:56.050
glycerol it will actually be sweet.

00:01:56.050 --> 00:01:58.480
And you might already be familiar with the word glycerol

00:01:58.480 --> 00:02:01.090
from the famous Bush song Glycerine.

00:02:01.090 --> 00:02:04.250
Glycerine is referring to glycerol.

00:02:04.250 --> 00:02:06.360
Glycerine.

00:02:06.360 --> 00:02:07.830
♫ Glycerine ♫

00:02:07.830 --> 00:02:11.019
So there you go, that is glycerol right over there.

00:02:11.019 --> 00:02:14.560
And you form a triglyceride when a glycerol molecule

00:02:14.560 --> 00:02:18.330
reacts with three fatty acid molecules.

00:02:18.330 --> 00:02:20.650
So these characters on the right,

00:02:20.650 --> 00:02:23.789
these are each a fatty acid.

00:02:23.789 --> 00:02:26.090
These are each a fatty acid.

00:02:26.090 --> 00:02:28.750
This is one fatty acid, this is another fatty acid,

00:02:28.750 --> 00:02:30.560
this is another fatty acid.

00:02:30.560 --> 00:02:32.460
Now why do we call them "fatty" and in particular,

00:02:32.460 --> 00:02:34.630
why do we call them a "fatty acid"?

00:02:34.630 --> 00:02:36.050
Well, you have one part that has these

00:02:36.050 --> 00:02:37.920
big carbon chains here.

00:02:37.920 --> 00:02:39.960
And I drew these dot dot dots 'cause it can actually have

00:02:39.960 --> 00:02:41.670
different numbers of carbons.

00:02:41.670 --> 00:02:45.180
They can have anywhere between--

00:02:45.180 --> 00:02:47.020
Based on my research you can have into the

00:02:47.020 --> 00:02:48.990
high 20's numbers of carbons.

00:02:48.990 --> 00:02:50.760
In most biological systems you tend to have an

00:02:50.760 --> 00:02:52.760
even number of carbons.

00:02:52.760 --> 00:02:55.050
And in most animal biology

00:02:55.050 --> 00:02:57.232
you tend to have 16 to 20 carbons.

00:02:57.232 --> 00:03:00.220
And if it's an even number, you're talking about 16, 18, 20

00:03:00.220 --> 00:03:02.290
carbons involved in these fatty acids.

00:03:02.290 --> 00:03:05.290
All the ones I've drawn have all single bonds.

00:03:06.470 --> 00:03:09.400
We have as many hydrogens bonded to it as possible.

00:03:09.400 --> 00:03:10.860
But you could also have double bonds there.

00:03:10.860 --> 00:03:12.720
And we'll talk about that more when we talk about

00:03:12.720 --> 00:03:16.420
saturated and unsaturated fats.

00:03:16.420 --> 00:03:19.630
So you have this carbon chain here.

00:03:19.630 --> 00:03:22.030
That part is hydrophobic, it won't dissolve well in water.

00:03:22.030 --> 00:03:25.976
And that's why people tend to refer to it as kind of fatty.

00:03:25.976 --> 00:03:27.630
And then you have on the left-hand,

00:03:27.630 --> 00:03:29.530
or at least the way I've drawn it on the left-hand side,

00:03:29.530 --> 00:03:31.300
you have a carboxyl group.

00:03:31.300 --> 00:03:35.460
So this right over here, this is a carboxyl group.

00:03:35.460 --> 00:03:38.600
Carboxyl group.

00:03:38.600 --> 00:03:39.760
And this is acidic.

00:03:39.760 --> 00:03:42.460
It's very easy for it to donate a hydrogen

00:03:42.460 --> 00:03:43.920
because if it donates a hydrogen

00:03:43.920 --> 00:03:45.660
or a hydrogen proton I should say.

00:03:45.660 --> 00:03:47.760
Because if this oxygen takes those electrons

00:03:47.760 --> 00:03:50.230
that negative charge can be spread between

00:03:50.230 --> 00:03:53.430
this oxygen and this oxygen over here.

00:03:53.430 --> 00:03:56.070
And so it's very willing to give away a hydrogen

00:03:56.070 --> 00:03:57.900
so that makes it acidic.

00:03:57.900 --> 00:03:59.600
So that's where the "acid" comes from.

00:03:59.600 --> 00:04:01.220
That's a fatty acid.

00:04:01.220 --> 00:04:02.760
And when you have these three molecules

00:04:02.760 --> 00:04:04.090
they don't have to be identical.

00:04:04.090 --> 00:04:06.260
This one could have 18 carbons.

00:04:06.260 --> 00:04:08.900
This one over here could have 16 carbons.

00:04:08.900 --> 00:04:10.630
This one could have 20 carbons,

00:04:10.630 --> 00:04:12.360
it might have a couple of double bonds in it.

00:04:12.360 --> 00:04:14.590
So they could all be different.

00:04:14.590 --> 00:04:17.350
But let's think about how they actually form together.

00:04:17.350 --> 00:04:21.600
Like those robot movies from the 80's like Voltron.

00:04:21.600 --> 00:04:22.860
So what you have happening,

00:04:22.860 --> 00:04:24.360
and this is gonna be a little bit of organic chemistry

00:04:24.360 --> 00:04:27.800
and this really just a review of dehydration synthesis.

00:04:27.800 --> 00:04:31.530
You have a lone pair

00:04:31.530 --> 00:04:33.260
in each of these oxygens.

00:04:33.260 --> 00:04:35.790
These oxygens that are part of the hydroxyl group.

00:04:35.790 --> 00:04:37.130
So let me draw that.

00:04:37.130 --> 00:04:38.190
So you have a lone pair there

00:04:38.190 --> 00:04:39.750
or actually you have two lone pairs.

00:04:39.750 --> 00:04:42.860
You have two lone pairs over there.

00:04:42.860 --> 00:04:46.470
You have two lone pairs over there.

00:04:46.470 --> 00:04:48.040
And so you can imagine a world where

00:04:48.040 --> 00:04:50.650
if they just bump past each other in just the right way

00:04:50.650 --> 00:04:51.960
this carbon over here is gonna

00:04:51.960 --> 00:04:53.620
have a partially positive charge.

00:04:53.620 --> 00:04:55.760
It has three bonds to two oxygens.

00:04:55.760 --> 00:05:00.390
Oxygens that are very elctro negative.

00:05:00.390 --> 00:05:03.510
And so this guy, if he just bumps into it in the right way,

00:05:03.510 --> 00:05:05.450
these electrons could be used

00:05:05.450 --> 00:05:08.550
to form a bond with this carbon,

00:05:08.550 --> 00:05:10.600
the one that has that partially positive charge.

00:05:10.600 --> 00:05:15.260
And just as that is happening this bond can be let go.

00:05:15.260 --> 00:05:18.878
This oxygen say, "Ok look, you're sharing in another pair?

00:05:18.878 --> 00:05:19.990
"I've had enough of that.

00:05:19.990 --> 00:05:22.190
"I'm going to take these electrons for myself

00:05:22.190 --> 00:05:24.760
"or even better I'm going to take them for myself

00:05:24.760 --> 00:05:27.160
"or to eventually capture or maybe I capture first,

00:05:27.160 --> 00:05:30.370
"it can happen in all different ways, a hydrogen proton.

00:05:30.370 --> 00:05:33.070
"I'm gonna capture a hydrogen proton and then I can become

00:05:33.070 --> 00:05:35.820
"this collection right over here

00:05:35.820 --> 00:05:38.790
"could become a water molecule."

00:05:38.790 --> 00:05:41.360
I'm doing just a very high level overview of the mechanism.

00:05:41.360 --> 00:05:42.660
You can go to more depth of this

00:05:42.660 --> 00:05:45.320
when you go into organic chemistry.

00:05:45.320 --> 00:05:47.630
But this could happen three times.

00:05:47.630 --> 00:05:49.820
So it can happen again over here.

00:05:49.820 --> 00:05:51.670
It could happen again over here.

00:05:51.670 --> 00:05:53.860
This guy takes this--

00:05:53.860 --> 00:05:56.320
Let me do that in that same white color.

00:05:56.320 --> 00:05:58.090
So he takes both of these electrons

00:05:58.090 --> 00:06:00.960
that he so badly wants to take back and maybe uses it

00:06:00.960 --> 00:06:05.200
to grab a hydrogen proton.

00:06:05.200 --> 00:06:05.900
It could happen the other way.

00:06:05.900 --> 00:06:08.000
Maybe grabs a hydrogen proton

00:06:08.000 --> 00:06:10.560
and then just as he's leaving, this guy comes back.

00:06:10.560 --> 00:06:12.970
So there's different ways that all of these could happen

00:06:12.970 --> 00:06:14.560
but this is the general idea.

00:06:14.560 --> 00:06:17.000
And then you have it happening a third time down here.

00:06:17.000 --> 00:06:18.640
One of these lone pairs

00:06:20.420 --> 00:06:22.630
come and form a bond with this carbon.

00:06:22.630 --> 00:06:24.160
This carbon in the carbonyl group,

00:06:24.160 --> 00:06:26.130
part of this carboxyl group.

00:06:26.130 --> 00:06:29.460
And so once again,

00:06:29.460 --> 00:06:32.690
this guy can take those two electrons away

00:06:32.690 --> 00:06:37.220
and maybe share that pair with a hydrogen proton.

00:06:37.220 --> 00:06:39.620
Once again, this is forming a water molecule,

00:06:39.620 --> 00:06:41.180
this is forming a water molecule.

00:06:41.180 --> 00:06:43.075
So three water molecules are going to be produced.

00:06:43.075 --> 00:06:44.830
This is why we call it dehydration synthesis.

00:06:44.830 --> 00:06:46.750
We're losing three water molecules

00:06:46.750 --> 00:06:49.650
in order to form these bonds.

00:06:49.650 --> 00:06:52.790
So what's it going to look like after this has happened?

00:06:52.790 --> 00:06:56.290
Well, let me scroll down here.

00:06:56.290 --> 00:06:57.760
And actually let me just zoom.

00:06:57.760 --> 00:06:59.530
Actually just let me scroll down here.

00:06:59.530 --> 00:07:03.560
So this green bond over here is going to now--

00:07:03.560 --> 00:07:06.220
So this green bond that's this big long curvy thing

00:07:06.220 --> 00:07:08.420
is this green bond.

00:07:08.420 --> 00:07:11.050
And this second green bond

00:07:11.050 --> 00:07:14.330
is this green bond.

00:07:14.330 --> 00:07:19.120
And this third green bond is that green bond.

00:07:21.188 --> 00:07:23.091
The way I've drawn it right now,

00:07:23.091 --> 00:07:27.321
each of these oxygens haven't let go of its hydrogens.

00:07:27.321 --> 00:07:28.821
And that could actually happen before

00:07:28.821 --> 00:07:30.611
or after or all at the same time.

00:07:30.611 --> 00:07:32.990
Chemistry actually is not a clean thing.

00:07:32.990 --> 00:07:37.621
But I could, if I want, I could draw the hydrogens here.

00:07:41.774 --> 00:07:45.011
I could draw the hydrogens.

00:07:45.011 --> 00:07:47.881
I could draw the hydrogens over here

00:07:47.881 --> 00:07:50.761
and then these would have a positive charge.

00:07:50.761 --> 00:07:52.461
These would have a positive charge.

00:07:52.461 --> 00:07:55.701
But then you could imagine another water molecule coming by

00:07:55.701 --> 00:07:59.631
and kind of taking one of these hydrogen protons,

00:07:59.631 --> 00:08:02.787
taking the hydrogen protons away from each of those oxygens.

00:08:02.787 --> 00:08:05.331
And then you would be left with

00:08:09.690 --> 00:08:12.291
this molecule right over here.

00:08:12.291 --> 00:08:14.830
And remember we produced three water molecules.

00:08:14.830 --> 00:08:16.920
So that's one,

00:08:19.060 --> 00:08:20.860
two

00:08:20.860 --> 00:08:24.401
and three water molecules.

00:08:24.401 --> 00:08:25.821
And now this molecule,

00:08:25.821 --> 00:08:27.240
if you ignore the water molecules out there,

00:08:27.240 --> 00:08:29.381
this is a triglyceride.

00:08:30.041 --> 00:08:31.451
Let me write it again.

00:08:31.451 --> 00:08:33.821
Actually, let me write the slightly more technical term.

00:08:33.821 --> 00:08:37.231
Sometimes referred to as

00:08:37.231 --> 00:08:40.231
triacylglycerol.

00:08:42.891 --> 00:08:45.661
Well we know where the glycerol comes from.

00:08:45.661 --> 00:08:48.991
It has this glycerine or this glycerol

00:08:48.991 --> 00:08:50.591
backbone right over here.

00:08:50.591 --> 00:08:52.121
Now what is "acyl" mean?

00:08:52.121 --> 00:08:54.891
Well acyl is a functional group

00:08:54.891 --> 00:08:59.791
where you have a carbon that's part of a carbonyl group.

00:08:59.791 --> 00:09:03.721
It can be bound to a kind of an organic chain

00:09:03.721 --> 00:09:05.651
right over here, I'll just call that R.

00:09:05.651 --> 00:09:08.621
And then it's bound to something else.

00:09:08.621 --> 00:09:12.191
And so we have three acyl groups so "triacyl".

00:09:12.191 --> 00:09:15.961
So this right over here, this right here is an acyl group.

00:09:15.961 --> 00:09:18.491
This right over here is an acyl group.

00:09:18.491 --> 00:09:20.011
This right over here is an acyl group.

00:09:20.011 --> 00:09:24.551
Each of them, they're bound to an oxygen right over here.

00:09:24.551 --> 00:09:26.121
And actually that gives us more practice

00:09:26.121 --> 00:09:27.761
with another functional group.

00:09:27.761 --> 00:09:31.461
When you have a situation--

00:09:31.461 --> 00:09:32.991
Let me get another color out.

00:09:32.991 --> 00:09:37.621
When you have a carbonyl group

00:09:37.621 --> 00:09:41.441
and let's say you have just an organic chain right over here

00:09:41.441 --> 00:09:43.071
and then you have an oxygen

00:09:43.071 --> 00:09:45.973
and then you have another organic chain right over there.

00:09:45.973 --> 00:09:48.171
This thing

00:09:49.851 --> 00:09:51.981
is called an ester.

00:09:53.801 --> 00:09:56.561
And you actually see three esters right over here.

00:09:56.561 --> 00:09:59.301
So you see this ester.

00:10:00.061 --> 00:10:01.431
This ester right over here.

00:10:01.431 --> 00:10:04.191
You see this ester right over here.

00:10:04.191 --> 00:10:06.621
And then this ester right over here.

00:10:06.621 --> 00:10:09.051
So this wasn't just practice in learning about

00:10:09.051 --> 00:10:12.451
fats or triacylglycerol or triglycerides.

00:10:12.451 --> 00:10:14.291
These are all words for the same thing.

00:10:14.291 --> 00:10:17.031
It's giving us good practice with these functional groups.

00:10:17.031 --> 00:10:18.911
But now you see where triacylglycerol comes from.

00:10:18.911 --> 00:10:20.291
You have glycerol backbone

00:10:20.291 --> 00:10:23.161
and you have these three acyl groups

00:10:23.161 --> 00:10:26.791
that allow it to attach to the fatty acids.

00:10:26.791 --> 00:10:30.611
So that's what fat is when people are referring to it.

00:10:30.611 --> 00:10:31.611
And this is just pictures of it.

00:10:31.611 --> 00:10:34.731
This is coconut oil right over here.

00:10:34.731 --> 00:10:37.621
It's right here, it's below its melting point

00:10:37.621 --> 00:10:39.376
so it's in its solid form.

00:10:39.376 --> 00:10:41.051
And this right over here, I'm not sure which type

00:10:41.051 --> 00:10:43.511
of oil this is but it's past its melting point

00:10:43.511 --> 00:10:44.651
and so it's in its liquid form.

00:10:44.651 --> 00:10:48.231
So people would typically refer to this as an oil.

00:10:48.231 --> 00:10:51.521
But this is made up of a bunch of triglycerides.

00:10:51.521 --> 00:10:54.078
This is a big mixture of triglycerides here.

00:10:54.078 --> 00:10:56.051
And if you were to see their molecules,

00:10:56.051 --> 00:10:57.691
they would all have generally this shape

00:10:57.691 --> 00:11:00.361
but the fatty acids would have different numbers of carbons

00:11:00.361 --> 00:11:02.891
and might have different numbers of double bonds.


Molecular structure of fructose

WEBVTT
Kind: captions
Language: en

00:00:00.226 --> 00:00:02.223
- [Voiceover] We've been giving a lot of attention to our

00:00:02.223 --> 00:00:05.961
old friend glucose, the monosaccharide glucose

00:00:05.961 --> 00:00:09.526
that is the building block of things like glycogen

00:00:09.526 --> 00:00:11.824
and starch and chitin,

00:00:11.824 --> 00:00:14.146
and what I want to do in this video is give a little

00:00:14.146 --> 00:00:17.525
attention to another fairly prominent monosaccharide,

00:00:17.525 --> 00:00:20.114
and that is fructose.

00:00:20.114 --> 00:00:23.678
That is fructose, often known as fruit sugar,

00:00:23.678 --> 00:00:25.756
and on its own, as a monosaccharide,

00:00:25.756 --> 00:00:29.727
it is actually the sweetest of all the sugars, fructose,

00:00:29.727 --> 00:00:31.573
and so let's compare the two.

00:00:31.573 --> 00:00:33.187
So the first thing that might jump out at you,

00:00:33.187 --> 00:00:34.487
and I encourage you to do this actually.

00:00:34.487 --> 00:00:37.123
These two molecules, these would be helpful ones

00:00:37.123 --> 00:00:39.584
to actually just be able to draw on your own because

00:00:39.584 --> 00:00:41.987
you will see these many, many, many times

00:00:41.987 --> 00:00:44.681
in your biological and in your chemistry and your

00:00:44.681 --> 00:00:47.061
organic chemistry careers,

00:00:47.061 --> 00:00:49.882
but if we just compare these two, we see that they both

00:00:49.882 --> 00:00:52.761
have six carbons, so for both of them,

00:00:52.761 --> 00:00:54.491
the chemical formula, you would have

00:00:54.491 --> 00:00:58.520
six carbons and then you see that they both have,

00:00:58.520 --> 00:01:01.852
they both have one, two, three, four, five, six, seven,

00:01:01.852 --> 00:01:03.825
eight, nine, 10, 11, 12.

00:01:03.825 --> 00:01:05.927
One, two, three, four, five, six, seven,

00:01:05.927 --> 00:01:07.912
eight, nine, 10, 11, 12.

00:01:07.912 --> 00:01:10.919
They both have 12 hydrogens,

00:01:10.919 --> 00:01:14.356
and to round it all out, they both have six oxygens,

00:01:14.356 --> 00:01:16.295
one, two, three, four, five, six.

00:01:16.295 --> 00:01:18.826
One, two, three, four, five, six.

00:01:18.826 --> 00:01:22.692
So they both have six oxygens.

00:01:22.692 --> 00:01:25.919
So they are, they have the same chemical formula

00:01:25.919 --> 00:01:28.834
but they are different molecules, so that makes them

00:01:28.834 --> 00:01:31.550
isomers, and as we will see, they are structural isomers,

00:01:31.550 --> 00:01:33.187
the different constituents, these,

00:01:33.187 --> 00:01:36.055
the constituent carbons, hydrogens and oxygens

00:01:36.055 --> 00:01:37.576
are bonded in a different way,

00:01:37.576 --> 00:01:39.352
which makes them structural isomers,

00:01:39.352 --> 00:01:41.384
and when you inspect it a little bit, there's a,

00:01:41.384 --> 00:01:44.054
there's one main thing that jumps out at you.

00:01:44.054 --> 00:01:46.225
On the glucose molecule, when it's in its

00:01:46.225 --> 00:01:47.525
straight chain form,

00:01:47.525 --> 00:01:48.686
we're gonna think about it,

00:01:48.686 --> 00:01:51.322
think about them both in their cyclical forms in a second,

00:01:51.322 --> 00:01:54.526
but when they're in the straight chain form,

00:01:54.526 --> 00:01:57.185
the glucose molecule has a carbonyl group

00:01:57.185 --> 00:02:00.517
at the number one carbon, if we start numbering up here.

00:02:00.517 --> 00:02:04.917
So if we start numbering one, two, three, four,

00:02:04.917 --> 00:02:08.818
five, six, the glucose straight chain has a carbonyl group

00:02:08.818 --> 00:02:12.127
at its one carbon while the fructose molecule

00:02:12.127 --> 00:02:14.159
has a carbonyl group, let's see,

00:02:14.159 --> 00:02:18.060
one, two, three, four, five, six,

00:02:18.060 --> 00:02:19.952
at the number two carbon.

00:02:19.952 --> 00:02:22.158
Remember, carbonyl group is just a carbon double bonded

00:02:22.158 --> 00:02:25.850
to an oxygen, so it has a carbonyl group bonded at its

00:02:25.850 --> 00:02:27.417
number two oxygen,

00:02:27.417 --> 00:02:29.252
and that's the main structural difference,

00:02:29.252 --> 00:02:31.852
even if you look at the way that I've drawn these,

00:02:31.852 --> 00:02:34.811
a hydroxyl group, you have three hydroxyl groups on,

00:02:34.811 --> 00:02:35.951
on the right side,

00:02:35.951 --> 00:02:37.855
then you have one on the left, three on the right,

00:02:37.855 --> 00:02:38.923
one on the left,

00:02:38.923 --> 00:02:43.126
and so the main difference is just what we did over here.

00:02:43.126 --> 00:02:45.726
And just to get a little bit more experience with our,

00:02:45.726 --> 00:02:48.420
with our names for our different functional groups,

00:02:48.420 --> 00:02:50.626
when we studied glucose we knew that, look, if you have

00:02:50.626 --> 00:02:53.284
a carbonyl group that's at the end of a carbon chain,

00:02:53.284 --> 00:02:55.688
so, this carbon in the carbonyl group is bonded

00:02:55.688 --> 00:02:59.728
to a hydrogen and then the rest of the carbon chain,

00:02:59.728 --> 00:03:01.957
you would call this, right over here,

00:03:01.957 --> 00:03:04.616
you would call this an aldehyde functional group.

00:03:04.616 --> 00:03:08.209
Aldehyde, aldehyde functional group and it makes,

00:03:08.209 --> 00:03:09.283
it makes the entire molecule,

00:03:09.283 --> 00:03:11.918
it allows it to be categorized as an aldehyde.

00:03:11.918 --> 00:03:15.199
Over here, where the carbon in the carbonyl group isn't

00:03:15.199 --> 00:03:17.723
at the end of the chain, it's in the middle of the chain,

00:03:17.723 --> 00:03:19.755
or its in the midst of the chain, so to speak,

00:03:19.755 --> 00:03:21.589
it's not exactly in the middle, but it's,

00:03:21.589 --> 00:03:23.842
this carbon is bonded to two different carbons.

00:03:23.842 --> 00:03:25.781
We call this molecule a ketone,

00:03:25.781 --> 00:03:27.452
and this would be a ketone functional group

00:03:27.452 --> 00:03:28.579
because it's bonded,

00:03:28.579 --> 00:03:31.922
it's bonded to two carbons, so this fructose,

00:03:31.922 --> 00:03:33.919
fructose, would be,

00:03:33.919 --> 00:03:36.984
fructose would be considered a ketone,

00:03:36.984 --> 00:03:38.784
and now let's think about how it might form

00:03:38.784 --> 00:03:42.128
a cyclical form, and as we've talked about before,

00:03:42.128 --> 00:03:44.345
the straight chain and the cyclical form, they can be

00:03:44.345 --> 00:03:47.120
in equilibrium if they are in an aqueous solution.

00:03:47.120 --> 00:03:48.954
And what I'm drawing here, these aren't the only

00:03:48.954 --> 00:03:51.323
forms that glucose takes on, but if you were to look it up

00:03:51.323 --> 00:03:52.925
in a text book or on the internet,

00:03:52.925 --> 00:03:55.293
you will most typically, this is what is most prevalent

00:03:55.293 --> 00:03:57.046
in nature as well when you're talking about the cyclical

00:03:57.046 --> 00:04:00.494
forms, find glucose in a six,

00:04:00.494 --> 00:04:02.921
in a six-member ring right over here,

00:04:02.921 --> 00:04:05.022
where five of them are carbons and then you have

00:04:05.022 --> 00:04:06.125
an oxygen,

00:04:06.125 --> 00:04:09.817
while fructose is most typical in a five-member ring,

00:04:09.817 --> 00:04:12.406
where four of them are carbons and then one of them

00:04:12.406 --> 00:04:13.521
is oxygen.

00:04:13.521 --> 00:04:15.750
Glucose can form a six-member ring,

00:04:15.750 --> 00:04:16.957
when you have a six-member ring,

00:04:16.957 --> 00:04:19.684
when you have a carbohydrate that involves a six-member

00:04:19.684 --> 00:04:22.809
ring where one of the items is an oxygen like this,

00:04:22.809 --> 00:04:25.549
you call this a pyranose.

00:04:25.549 --> 00:04:28.951
Pyranose, so that's a carbohydrate ring where it has

00:04:28.951 --> 00:04:31.180
six members, one of which is oxygen,

00:04:31.180 --> 00:04:32.259
and when you have five members,

00:04:32.259 --> 00:04:34.860
one of which is oxygen, this is called a furanose.

00:04:34.860 --> 00:04:37.554
Furanose ring.

00:04:37.554 --> 00:04:41.350
And actually, fructose can be found in either a furanose

00:04:41.350 --> 00:04:44.322
form or a pyranose form, but the furanose form

00:04:44.322 --> 00:04:46.424
is the one that's most typical and when people think

00:04:46.424 --> 00:04:49.419
of fructose as a ring, they most typically think of it

00:04:49.419 --> 00:04:50.812
in this form.

00:04:50.812 --> 00:04:52.054
Well how does this form,

00:04:52.054 --> 00:04:53.947
how does this form form?

00:04:53.947 --> 00:04:57.047
Especially in comparison to what's happening in glucose?

00:04:57.047 --> 00:04:58.254
Well when we saw,

00:04:58.254 --> 00:04:59.984
what we saw in glucose,

00:04:59.984 --> 00:05:02.387
in both cases, what you're going to do is have an attack

00:05:02.387 --> 00:05:05.150
on the carbon that's part of the carbonyl group,

00:05:05.150 --> 00:05:07.925
so, over here, that is the number one carbon.

00:05:07.925 --> 00:05:09.794
This is the number one carbon here,

00:05:09.794 --> 00:05:12.186
and over here, this is the number two carbon.

00:05:12.186 --> 00:05:14.288
This is the number two carbon.

00:05:14.288 --> 00:05:16.981
The number two carbon, right over there,

00:05:16.981 --> 00:05:19.489
and in both cases, you have,

00:05:19.489 --> 00:05:21.091
if you want to form these forms,

00:05:21.091 --> 00:05:22.693
over here we have the,

00:05:22.693 --> 00:05:25.341
we had the oxygen that's in,

00:05:25.341 --> 00:05:27.558
the oxygen in the hydroxyl group attached to

00:05:27.558 --> 00:05:28.974
the number five carbon.

00:05:28.974 --> 00:05:30.658
The oxygen in the hydroxyl group attached to

00:05:30.658 --> 00:05:31.981
the number five carbon.

00:05:31.981 --> 00:05:36.451
This is the one, two, three, four, five carbon.

00:05:36.451 --> 00:05:38.959
This oxygen is what does,

00:05:38.959 --> 00:05:40.851
is what forms the bond,

00:05:40.851 --> 00:05:44.148
is what forms this bond up here when you were talking

00:05:44.148 --> 00:05:46.447
about glucose, and over here,

00:05:46.447 --> 00:05:49.559
on fructose, once again, we look at the number five,

00:05:49.559 --> 00:05:52.203
the oxygen on the hydroxyl group on the number five

00:05:52.203 --> 00:05:54.841
carbon, so we look at this hydroxyl group right over here,

00:05:54.841 --> 00:05:56.826
that's this hydroxyl group right over here,

00:05:56.826 --> 00:05:58.487
and this oxygen,

00:05:58.487 --> 00:06:00.623
this oxygen does,

00:06:00.623 --> 00:06:01.993
forms a bond,

00:06:01.993 --> 00:06:04.582
forms a bond with the carbon in the carbonyl group,

00:06:04.582 --> 00:06:07.775
and when that happens, that allows one of these,

00:06:07.775 --> 00:06:10.747
that allows one of these bonds in these double bonds

00:06:10.747 --> 00:06:11.989
or one of the two double bonds

00:06:11.989 --> 00:06:15.187
to instead be released from the carbon and then essentially

00:06:15.187 --> 00:06:16.726
form a bond,

00:06:16.726 --> 00:06:19.385
form a bond with a proton floating around.

00:06:19.385 --> 00:06:22.125
That proton is actually likely to be involved

00:06:22.125 --> 00:06:24.481
in a hydronium molecule.

00:06:24.481 --> 00:06:26.293
And the same thing would've happened in glucose

00:06:26.293 --> 00:06:27.256
right over here.

00:06:27.256 --> 00:06:30.286
This, one of the two double bonds could now be used

00:06:30.286 --> 00:06:32.086
to form a bond,

00:06:32.086 --> 00:06:35.488
to form a bond with a hydrogen proton,

00:06:35.488 --> 00:06:38.019
and so it's essentially the same mechanism.

00:06:38.019 --> 00:06:41.223
The reason why this is a five-member ring involving four

00:06:41.223 --> 00:06:44.474
carbons versus a six-member ring involving five carbons,

00:06:44.474 --> 00:06:46.854
is just the carbonyl group is on a,

00:06:46.854 --> 00:06:50.395
is kind of one further down the chain than it is

00:06:50.395 --> 00:06:54.036
in the glucose, and so that's why this forms this,

00:06:54.036 --> 00:06:57.187
this pentagonal shape right over here,

00:06:57.187 --> 00:06:58.684
and we've talked about it in other videos.

00:06:58.684 --> 00:07:00.405
Fructose will, I've already talked about it in this video,

00:07:00.405 --> 00:07:03.619
fructose is the sweetest of all of the monosaccharides,

00:07:03.619 --> 00:07:06.719
the sweetest sugar, and when we talk about table sugar,

00:07:06.719 --> 00:07:09.760
the sugar that you might put into your tea or your coffee,

00:07:09.760 --> 00:07:12.489
that table sugar, that's called sucrose.

00:07:12.489 --> 00:07:15.287
Sucrose, sucrose,

00:07:15.287 --> 00:07:20.221
and it's a disaccharide made up of glucose and fructose,

00:07:20.221 --> 00:07:23.054
and the way that you form the bond, it's still happening

00:07:23.054 --> 00:07:26.885
through our dehydration synthesis,

00:07:26.885 --> 00:07:28.719
where you have the,

00:07:28.719 --> 00:07:30.554
where you would have the oxygen

00:07:30.554 --> 00:07:33.491
on the number one carbon in glucose,

00:07:33.491 --> 00:07:34.594
this one over here,

00:07:34.594 --> 00:07:37.218
let me do this in a color that I have not used yet.

00:07:37.218 --> 00:07:40.155
See I've used almost every color, so this is difficult.

00:07:40.155 --> 00:07:42.222
But I will, well I'll just use this green.

00:07:42.222 --> 00:07:45.519
This oxygen, essentially, can form a bond,

00:07:45.519 --> 00:07:48.422
can form a bond with the,

00:07:48.422 --> 00:07:52.787
with the number two carbon on the fructose,

00:07:52.787 --> 00:07:56.189
and when it forms that bond, then this guy,

00:07:56.189 --> 00:07:58.058
this bond right over here, can be,

00:07:58.058 --> 00:08:00.461
let me do that in a different color,

00:08:00.461 --> 00:08:03.155
this bond can be then,

00:08:03.155 --> 00:08:06.127
it can then go back to the oxygen or if you really want

00:08:06.127 --> 00:08:07.857
to think about it in another way,

00:08:07.857 --> 00:08:10.225
it can then be used to form a bond,

00:08:10.225 --> 00:08:14.474
it can be used to form a bond with some hydrogen proton

00:08:14.474 --> 00:08:16.251
floating around, so essentially,

00:08:16.251 --> 00:08:18.851
it's gonna be a carbon bonded to an oxygen

00:08:18.851 --> 00:08:22.311
which is then going to be bonded to another,

00:08:22.311 --> 00:08:24.587
which is then going to be bonded to another carbon,

00:08:24.587 --> 00:08:26.328
and then someone else can, you know,

00:08:26.328 --> 00:08:29.022
some other water molecule

00:08:29.022 --> 00:08:32.052
might be able to sop up this hydrogen proton right over here

00:08:32.052 --> 00:08:33.294
and then you would have,

00:08:33.294 --> 00:08:34.827
you would have your bond,

00:08:34.827 --> 00:08:36.452
that type of bonds that we're familiar with,

00:08:36.452 --> 00:08:39.552
these glycosidic linkages that we have when we have

00:08:39.552 --> 00:08:42.420
dehydration synthesis, these linkages that are really

00:08:42.420 --> 00:08:45.287
the basis for how we start with monosaccharides

00:08:45.287 --> 00:08:47.052
and we, and we create,

00:08:47.052 --> 00:08:49.119
and we create disaccharides, and if we keep going,

00:08:49.119 --> 00:08:52.868
we could create polysaccharides, so sucrose,

00:08:52.868 --> 00:08:55.318
sucrose is, one way to think about it,

00:08:55.318 --> 00:08:56.619
if I have my,

00:08:56.619 --> 00:08:58.894
if I have my glucose up here,

00:08:58.894 --> 00:09:01.425
if my glucose,

00:09:01.425 --> 00:09:03.155
actually let me just draw it this way,

00:09:03.155 --> 00:09:05.291
so you have a glucose,

00:09:05.291 --> 00:09:06.255
whoops,

00:09:06.255 --> 00:09:09.123
you have your glucose,

00:09:09.123 --> 00:09:12.350
glucose forms,

00:09:12.350 --> 00:09:15.218
forms a glycosidic bond

00:09:15.218 --> 00:09:17.250
with your fructose,

00:09:17.250 --> 00:09:21.194
with your fructose,

00:09:21.194 --> 00:09:23.984
with your fructose right over here

00:09:23.984 --> 00:09:25.260
and that's what sucrose is,

00:09:25.260 --> 00:09:27.791
and to really visualize it properly, you kind of have to

00:09:27.791 --> 00:09:30.880
flip this fructose over and then move it up here

00:09:30.880 --> 00:09:31.808
and then they could,

00:09:31.808 --> 00:09:32.528
they could,

00:09:32.528 --> 00:09:35.524
this character can then form a bond with this carbon

00:09:35.524 --> 00:09:37.434
right over there.