Hydrogen bonding in water

WEBVTT
Kind: captions
Language: en

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- [Voiceover] I don't think
it's any secret to anyone

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that water is essential to life.

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Most of the biological,
or actually in fact

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all of the significant
biological processes

00:00:09.665 --> 00:00:11.437
in your body are dependent on water

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and are probably
occurring inside of water.

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When you think of cells in
your body, the cytoplasm

00:00:15.860 --> 00:00:19.120
inside of your cells,
that is mainly water.

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In fact, me, who is
talking to you right now,

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I am 60% to 70% water.

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You could think of me as kind of this big

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bag of water making a video right now.

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And it's not just human
beings that need water.

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Life as we know it is dependent on water.

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That why when we have the search for

00:00:36.125 --> 00:00:37.422
signs of life on other planets

00:00:37.422 --> 00:00:40.206
we're always looking for signs of water.

00:00:40.206 --> 00:00:44.224
Maybe life can occur in
other types of substances,

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but water is essential
to life as we know it.

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And to understand why water is so special

00:00:50.518 --> 00:00:52.578
let's start to understand
the structure of water

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and how it interacts with itself.

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And so water, as you
probably already know,

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is made up of one oxygen
atom and two hydrogen atoms.

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That's why we call it H2O.

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And they are bonded with covalent bonds.

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And covalent bonds, each of these bonds

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is this pair of electrons
that both of these

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atoms get to pretend like they have.

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And so you have these two pairs.

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And you might be saying,
"Well, why did I draw

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"the two hydrogens on this end?

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"Why didn't I draw them on
opposite sides of the oxygen?"

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Well that's because oxygen also
has two lone electron pairs.

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Two lone electron pairs.

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And these things are always
repelling each other.

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The electrons are repelling
from each other, and so,

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in reality if we were looking
at it in three dimensions,

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the oxygen molecule is kind
of a tetrahedral shape.

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I could try to, let me try
to draw it a little bit.

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So if this is the oxygen right over here

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then you would have, you could have

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maybe one lone pair of electrons.

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I'll draw it as a little
green circle there.

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Another lone pair of electrons back here.

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Then you have the covalent bond.

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You have the covalent bond to

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one hydrogen atom right over there.

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And then you have the covalent bond

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to the other hydrogen atom.

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And so you see it forms
this tetrahedral shape,

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It's pretty close to a tetrahedron.

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Just like this, but the
key is that the hydrogens

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are on one end of the molecule.

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And this is, we're going
to see, very very important

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to the unique properties, or to the,

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what gives water its special properties.

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Now, one thing to realize
is, it's very, in chemistry

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we draw these electrons very
neatly, these dots up here.

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We draw these covalent bonds very neatly.

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But that's not the way
that it actually works.

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Electrons are jumping around constantly.

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They're buzzing around, it's actually

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much more of a, even when
you think about electrons,

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it's more of a probability
of where you might find them.

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And so instead of thinking
of these electrons as

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definitely here or
definitely in these bonds,

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They're actually more of in this cloud

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around the different atoms.

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They're in this cloud that
kind of describes a probability

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of where you might find them as they buzz

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and they jump around.

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And what's interesting about water

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is oxygen is extremely electronegative.

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So oxygen, that's oxygen
and that's oxygen,

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it is extremely electronegative,
it's one of the more

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electronegative elements we know of.

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It's definitely way more
electronegative than hydrogen.

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And you might be saying, "Well, Sal,

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"what does it mean to be electronegative?"

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Well, electronegative
is just a fancy way of

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saying that it hogs electrons.

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It likes to keep electrons for itself.

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Hogs electrons, so that's what's going on.

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Oxygen like to keep the
electrons more around itself

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than the partners that it's bonding with.

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So even in these covalent bonds, you say,

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"Hey, we're supposed to be
sharing these electrons."

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Oxygen says, "Well I still want them to

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"spend a little bit more time with me."

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And so they actually do spend more time

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on the side without the hydrogens

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than they do around the hydrogens.

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And you can imagine what
this is going to do.

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This is going to form a
partial negative charge at the,

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I guess you could say,
the non-hydrogen end

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that is the end that has, that's
well I guess this top end,

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the way I've drawn it right over here.

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And this Greek letter
delta, this is to signify

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a partial charge, and it's
a partial negative charge.

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Because electrons are negative.

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And then over here,
since you have a slight

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deficiency of electrons, because they're

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spending so much time around the oxygen,

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it forms a partial positive
charge right over there.

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So right when you just
look at one water molecule,

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that doesn't seem so interesting.

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But it becomes really
interesting when you look at

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many water molecules interacting together.

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So let me draw another water
molecule right over here.

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So it's oxygen, you have two hydrogens,

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and then you have the bonds between them.

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You have a partially
negative charge there.

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Partially positive charge on that end.

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And so you can imagine the partial,

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the side that has a partially
negative charge is going to be

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attracted to the side that has
a partially positive charge.

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And that attraction between these two,

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this is called a hydrogen bond.

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So that right over there
is called a hydrogen bond.

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And this is key to the behavior of water.

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And we're going to see
that in future videos.

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All the different ways that hydrogen bonds

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give water its unique characteristics.

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Hydrogen bonds are weaker
than covalent bonds,

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but they're strong enough to
give water that kind of nice

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fluid nature when we're
thinking about kind of normal,

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or you could say, normal
temperatures and pressures.

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This nice fluid nature, it
allows these things to be

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attracted to each other,
to have some cohesion,

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but also to break and reform
and flow past each other.

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So you can imagine another
hydrogen bond with another

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water molecule right over here.

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So put my hydrogens over there.

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Put my hydrogens, your
bonds, partial negative,

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partial positive right over there.

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And so we'll see in future
videos, hydrogen bonds,

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key for water flowing past itself.

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Key for its properties to

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its ability to take in heat.

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Key for its ability to
regulate temperature.

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The key for its ability is
why lakes don't freeze over.

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It's key for some of
its properties around

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evaporative cooling and surface tension

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and adhesion and cohesion,
and we'll see that.

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And probably most important,

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and it's hard to rank of these things,

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if we're thinking about
biological systems,

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this polarity that we
have in water molecules

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and these hydrogen bonding,

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it's key for its ability to be a solvent,

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for it to be able to have

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polar molecules be
dissolved inside of water.

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And we'll see that in future videos.


Overview of the Bolshevik Revolution

WEBVTT
Kind: captions
Language: en

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As we enter into the 1900s

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imperial Russia is under the absolute control of the czar and [a] [czar] an analogous term would be emperor he has

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absolute control over Russia, and he's part of the Romanov dynasty

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Romanov Dynasty

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That has had control that has ruled over imperial Russia since the early 1600s, so for Roughly

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300 years, but as we see from the late 1800s through through the early 1900's really over the course of TSAr

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nicholas II rule you see ever increasing tensions between the ruling class between the royalty and

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the aristocracy and the labor class sometimes called referring themselves as the proletariat the proletariat comes from an

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Enrollment Society it was the the lowest class the people who [were] kind of doing the labor in Roman society?

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But they proudly referred to themselves as a proletariat and as we'll see the the revolution that occurs in

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1917 that's really the culmination of the struggle between the laborers the ones that are working under ever increasingly harsh conditions in

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factories in Urban centers or the ones that are working in the fields and and

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The ruling class and this tension just keeps getting worse and worse and worse [as] we [get] into

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1905 and 1905 is one of the more notable events that show that things aren't going all that well for the tSAr in

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January of that year you have a an orthodox priest

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Leading a a strike leading a protest

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Trying to get this is in St.. Petersburg trying to March to the TSAr's palace, and it's an unarmed

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It's a peaceful protest

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But they want to be heard they want to explain to the tSAr. How hard the working conditions are for laborers under in in

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Imperial Russia and the Tsar isn't actually in town at the [time]

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But his force [is] actually opened fire on these peaceful protesters and that that that event in which

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There's all these casualties of these peaceful protesters is called Bloody Sunday

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Bloody and this was in January

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Bloody Sunday, which is in January of 1905 and this sparks an attempted revolution?

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Where [are] it people who are sympathetic to those protesters they start taking [to] the street throughout Russia?

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You start having the government forces suppressing them wherever they are but eventually that that revolution is put down

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It's an unsuccessful

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revolution but it increases the tension even further and frankly even folks who might have been

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Somewhat sympathetic to the TSAr before almost viewed him as a bit of a father figure now are thinking hey

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You know this guy's forces open open open fire on peaceful protesters

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maybe he's really not on our side and those tensions continue to increase as we go all the way into

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1914 which we've seen in [other] videos is the start of world war one and

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like happens

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And we've seen it in both world war one and world war two russia tends to take the brunt of these things and world war

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one is Incredibly difficult [on] Russia first of all all of the a

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Good fraction of the young men have to go fight, then a lot of the economy has to be

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repointed at

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supporting the war effort at the supplies for the troops the food for the troops and so it starts to really hurt the conditions for

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the average person

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Which [were] already bad remember they were protesting in 1905 and now things are getting worse as we go through World war one

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For the average individual now to make a bit of a concession [to] the general public and to move somewhat into a more representative

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world after the revolution of 1905 the TSAr does create the Duma which is a

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representative assembly

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Representative assembly that didn't have a ton of power

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But it did start to give the people voice and every time

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They assembled to him whether the TSAr was actually not too pleased with it

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He's like who are all of these people who seem to be very aggressively anti me and he kept getting the Duma

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dissolved and then a new Duma would be elected and this kept going on and on and on all the way until we get to

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The beginning of World war [ron] now as I said World war one

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conditions start to get worse people start to get even more

00:04:06.140 --> 00:04:11.290
Skeptical of Tsar rule to make things [even] worse when [TSAr] [nicholas] II has left, st.

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Petersburg to really help

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Lead the Troops or in his mind lead the troops near the front of the war the country

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Or the empire is really goes under the control of his wife the CzArina, Alexandra

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And she is heavily under the influence of this gentle right over here

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Gregory Rasputin and Rasputin is one of these fascinating

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Characters from history because he was this mystical faith healer type guy who was able to really?

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Manipulate the the Czarina and to some degree the TSAr and so people started getting suspicious of well

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Who is this guy this?

00:04:45.830 --> 00:04:51.609
Mysterious guy who's hanging out in the palace and seeming to have very strong influence over our?

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rulers and there were even suspicious of the Czarina at this time [the] lot of the royalty they married between

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countries and the Czarina

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She was actually born in

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Germany and her father was German she grew up in Germany and her mug rand mother was actually Queen Victoria of England and you might

00:05:08.680 --> 00:05:14.139
Remember that that Russia is now at war with Germany. So they're like hey. We have this German

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Czarina essentially controlling the country under the the sway of this mysterious shady figure gregory

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rasputin people do not like the way that this is going and so by

00:05:26.749 --> 00:05:32.619
1917 things come to a head and in February and especially march of 1917 the

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They are able to force the tsAr to abdicate to say hey enough is enough

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We're taking it straight to you you have to

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You have to give up your power, and it's called the February revolution

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But the TsAr doesn't actually abdicate until march so you could argue that you could call it the March

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Revolution now once the [tSAr] abdicates you have a provisional government

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You have a provisional government

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Take control and all of this is happening in

00:06:01.399 --> 00:06:08.409
1917 but people are also not happy with this provisional government remember the war is continuing to go on it's not really looking that well

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For Russia they're continuing to lose people the economy is continuing to tank and what you have is workers start to to

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Start to assemble start to organize in the various cities

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And they form these workers councils these councils and the word and russian for them is soviets

00:06:26.719 --> 00:06:34.659
These soviets start to form and you fast forward into october of that year and these soviets are able to

00:06:34.759 --> 00:06:39.338
along with the Bolsheviks, which is a party are able to Force the

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provisional government

00:06:41.080 --> 00:06:42.080
to

00:06:42.080 --> 00:06:49.419
To abdicate its power or to give up its power and then you have the control by a soviet [Council] and in the soviet Council

00:06:49.750 --> 00:06:50.710
the Bolsheviks

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Led by Vladimir Lenin are the ones who are [actually] able to take power now revolutions?

00:06:57.699 --> 00:06:59.799
I mentioned you know what revolutions are happen

00:06:59.800 --> 00:07:06.609
Or they're the culmination of decades of increasing tension and as we've seen multiple times in history revolutions are not clean things

00:07:06.610 --> 00:07:12.520
it's not like the tSAr abdicate or the government abdicates and [another] government takes over and then everyone's all nice the

00:07:12.949 --> 00:07:15.699
Bolshevik Revolution which installs which it now has this

00:07:15.919 --> 00:07:19.209
council of soviets which [we're] the Bolsheviks aren't control led by

00:07:19.340 --> 00:07:24.489
Vladimir Lenin you can imagine the people who are royal who are loyal to the king the monarchists?

00:07:24.490 --> 00:07:27.759
They're not happy you can imagine people who are more aligned with

00:07:28.490 --> 00:07:34.210
Capitalist models of government because the soviets and the the bolsheviks they are strongly strongly

00:07:34.639 --> 00:07:39.429
Socialist they they're followers of Marx. They imagine an ideal state that has a

00:07:40.099 --> 00:07:42.099
That has very limited

00:07:42.259 --> 00:07:49.089
private property rights and on and on and on and so this sparks [a] the russian [Civil] [war] which lasts from

00:07:49.849 --> 00:07:51.710
1917 to

00:07:51.710 --> 00:07:58.269
1922 and like Most Civil Wars it is incredibly Bloody you have two to three million casualties

00:07:58.909 --> 00:08:06.069
Two to three million casualties as part of that [civil] war and casualties those could be deaths those could be injuries

00:08:06.169 --> 00:08:08.169
but in the end the

00:08:08.180 --> 00:08:12.819
Bolshevik Red Army is victorious and sometimes that Civil war is viewed as a

00:08:12.979 --> 00:08:19.269
conflict between the red Army of the Bolsheviks the one that favoured Socialism extreme socialism Communism really and

00:08:19.490 --> 00:08:24.519
The white Army the army that was either on the side of the monarchists or they're more capitalist

00:08:24.520 --> 00:08:26.199
But they're definitely against the red army

00:08:26.199 --> 00:08:30.309
And there's also another group the green army which is you know not aligned with either one of them

00:08:30.310 --> 00:08:34.779
So let me let me write this down. It's really the civil war is red Army

00:08:35.300 --> 00:08:36.680
versus

00:08:36.680 --> 00:08:38.680
versus white Army

00:08:38.959 --> 00:08:45.039
Versus white army and then there's even the notion of a green army which is not aligned with either of these verses

00:08:45.620 --> 00:08:48.729
The Green Army and this continues all [the] way through

00:08:49.130 --> 00:08:56.839
1922 until the Red Army is able to be victorious, and that's really the official of the soviet Union in

00:08:57.570 --> 00:09:04.879
1922 and Lenin is the first premier of the soviet union, So he is the premier of [the] soviet union?

00:09:04.880 --> 00:09:11.029
But he does not last long he eventually dies from from natural causes [a] Few years later in

00:09:12.660 --> 00:09:18.320
1924 where he is followed by Stalin and you might be asking well, what happened to the TSAr and his?

00:09:18.450 --> 00:09:23.539
Family after they abdicated well, it's not that great of an outcome and it's it's at least for them

00:09:23.540 --> 00:09:27.230
And it's one of these famous things in history where his entire family

00:09:27.390 --> 00:09:34.909
His entire family actually gets executed the summer after he abdicates and so this was an entire family every member of his [family] was

00:09:35.040 --> 00:09:41.300
Was executed there were some theories that that one of that his daughter right over here, anastasia might have actually escaped

00:09:41.300 --> 00:09:49.070
But it was later lee later conclusively proven that she too was was also executed so you see that right over here the [tSAr]

00:09:49.350 --> 00:09:54.709
Abdicating in nineteen abdicating in 1917 and then the next summer his entire family is

00:09:55.230 --> 00:09:56.370
actually

00:09:56.370 --> 00:10:03.260
Executed so it's a bloody beginning. It's a bloody period But this this is kind of what happens when you have

00:10:03.779 --> 00:10:05.779
significant revolutions


Chemical reactions introduction

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.050
- [Voiceover] Let's talk a little bit

00:00:02.050 --> 00:00:04.690
about chemical reactions.

00:00:04.690 --> 00:00:07.720
And chemical reactions
are a very big deal.

00:00:07.720 --> 00:00:11.520
Without chemical reactions,
you or I would not exist.

00:00:11.520 --> 00:00:14.650
In your body right now,
there are countless

00:00:14.650 --> 00:00:17.250
chemical reactions going on every second.

00:00:17.250 --> 00:00:20.080
Without chemical reactions,
we would have no life,

00:00:20.080 --> 00:00:23.220
we would not even have the
universe as we know it.

00:00:23.220 --> 00:00:25.420
So what are chemical reactions.

00:00:25.420 --> 00:00:28.790
Well, they're any time that
you have bonds being formed

00:00:28.790 --> 00:00:31.420
or broken between atoms or molecules.

00:00:31.420 --> 00:00:34.320
So what are we talking about there?

00:00:34.320 --> 00:00:36.590
Well this is maybe one of the most

00:00:36.590 --> 00:00:38.190
fundamental chemical reactions.

00:00:38.190 --> 00:00:40.220
Once again if one never
occurred, we'd be in trouble,

00:00:40.220 --> 00:00:43.010
we would not have, we
would not have any water.

00:00:43.010 --> 00:00:45.620
But let's think about what
it is actually describing.

00:00:45.620 --> 00:00:49.220
So over here on the left-hand
side we have the reactants.

00:00:49.220 --> 00:00:50.990
Let me write that down.

00:00:50.990 --> 00:00:53.490
So here we have the reactants.

00:00:53.490 --> 00:00:56.750
These are the molecules
that are going to react.

00:00:56.750 --> 00:01:00.550
And then we have an arrow
that moves us to the product.

00:01:00.550 --> 00:01:02.290
So let me do that in a different color.

00:01:02.290 --> 00:01:05.620
So we have an arrow that
moves us to the product,

00:01:05.620 --> 00:01:07.820
or we could say the products.

00:01:07.820 --> 00:01:09.920
And so what are the reactants here?

00:01:09.920 --> 00:01:12.250
Well we have molecular hydrogen

00:01:12.250 --> 00:01:14.250
and we have molecular oxygen.

00:01:14.250 --> 00:01:16.310
Now why did I say molecular hydrogen?

00:01:16.310 --> 00:01:18.290
Because molecular hydrogen,
which is the state

00:01:18.290 --> 00:01:20.190
that you would typically find hydrogen in

00:01:20.190 --> 00:01:22.120
if you just have it by itself,

00:01:22.120 --> 00:01:24.820
it is actually made up
of two hydrogen atoms.

00:01:24.820 --> 00:01:28.250
You see it right over here,
one, two hydrogen atoms.

00:01:28.250 --> 00:01:30.990
And what we have in order
to have this reaction,

00:01:30.990 --> 00:01:33.690
you don't just need one
molecular hydrogen and one,

00:01:33.690 --> 00:01:36.990
or one molecule of hydrogen
and one molecule of oxygen.

00:01:36.990 --> 00:01:39.720
For every, for this reaction to happen,

00:01:39.720 --> 00:01:43.650
you actually have two molecules
of molecular hydrogen.

00:01:43.650 --> 00:01:46.050
So this is actually made
up of four hydrogen atoms.

00:01:46.050 --> 00:01:47.490
So let me make this clear.

00:01:47.490 --> 00:01:49.610
So this right over here,

00:01:49.610 --> 00:01:52.950
this is two molecules
of molecular hydrogen.

00:01:52.950 --> 00:01:54.350
And that's why we have the two

00:01:54.350 --> 00:01:57.120
right out front of the H sub-two.

00:01:57.120 --> 00:02:00.180
This little subscript
two tells us there's two

00:02:00.180 --> 00:02:02.690
of the hydrogen atoms in this molecule.

00:02:02.690 --> 00:02:04.650
And then this big, this big white two

00:02:04.650 --> 00:02:06.490
that we have right over here,
that tells us that we're

00:02:06.490 --> 00:02:08.220
dealing with two of those molecules

00:02:08.220 --> 00:02:09.880
for this reaction to
happen, that we need two

00:02:09.880 --> 00:02:12.520
of these molecules for every,

00:02:12.520 --> 00:02:14.990
for every molecule of molecular oxygen.

00:02:14.990 --> 00:02:16.790
And molecular oxygen, once again,

00:02:16.790 --> 00:02:19.720
this is composed of two oxygen atoms.

00:02:19.720 --> 00:02:21.180
One, two.

00:02:21.180 --> 00:02:23.390
So under the right conditions,
so you need a little bit

00:02:23.390 --> 00:02:25.590
of energy to make this happen.

00:02:25.590 --> 00:02:26.790
If under the right conditions these

00:02:26.790 --> 00:02:27.920
two things are going to react.

00:02:27.920 --> 00:02:29.880
And actually it's very, very reactive,

00:02:29.880 --> 00:02:31.520
molecular hydrogen and molecular oxygen.

00:02:31.520 --> 00:02:34.250
So much so that it's actually
used for rocket fuel.

00:02:34.250 --> 00:02:38.350
You are going to produce
two molecules of water.

00:02:38.350 --> 00:02:40.420
We see that right over here.

00:02:40.420 --> 00:02:43.820
And look, I did not create
or destroy any atoms.

00:02:43.820 --> 00:02:47.550
I had one, I had one,

00:02:47.550 --> 00:02:49.990
I had one oxygen atom here.

00:02:49.990 --> 00:02:53.110
It was part of the oxygen
molecule right here,

00:02:53.110 --> 00:02:55.460
then I have the second
one right over here now.

00:02:55.460 --> 00:02:57.750
Now they are part of separate molecules.

00:02:57.750 --> 00:03:00.050
I had, I had a,

00:03:00.050 --> 00:03:02.250
I had one two, three, four hydrogens.

00:03:02.250 --> 00:03:05.920
I now have one, two, three,

00:03:05.920 --> 00:03:08.620
four hydrogens, just like that.

00:03:08.620 --> 00:03:09.950
And actually this produces a...

00:03:09.950 --> 00:03:11.920
So we could say some energy,

00:03:11.920 --> 00:03:14.250
and I'm being inexact right over here.

00:03:14.250 --> 00:03:18.290
Some energy and then we
could say a lot of energy.

00:03:18.290 --> 00:03:20.520
A lot of energy.

00:03:20.520 --> 00:03:22.690
So this is a reaction that
you just give it a little bit

00:03:22.690 --> 00:03:25.650
of a kick-start and it
really wants to happen.

00:03:25.650 --> 00:03:29.590
A lot, a lot of energy.

00:03:29.590 --> 00:03:31.290
So one thing that you might wonder,

00:03:31.290 --> 00:03:32.920
and this is something
that I first wondered

00:03:32.920 --> 00:03:34.050
when I learned about reactions,

00:03:34.050 --> 00:03:36.180
well how do, how does this happen?

00:03:36.180 --> 00:03:38.590
You know, is this a very organized thing?

00:03:38.590 --> 00:03:40.920
You know, do these molecules somehow

00:03:40.920 --> 00:03:42.780
know to react with each other?

00:03:42.780 --> 00:03:44.450
And the answer's no.

00:03:44.450 --> 00:03:47.590
Chemistry is a incredibly messy thing.

00:03:47.590 --> 00:03:50.850
You have these things bouncing
around, they have energy.

00:03:50.850 --> 00:03:52.220
They're bouncing around all over the place

00:03:52.220 --> 00:03:53.850
and actually when you provide energy,

00:03:53.850 --> 00:03:56.010
they're gonna bounce around
even more rigorously,

00:03:56.010 --> 00:03:59.890
enough so that they
collide in the right ways

00:03:59.890 --> 00:04:02.182
so that they break their old bonds

00:04:02.182 --> 00:04:04.020
and then they form these new bonds.

00:04:04.020 --> 00:04:06.850
So whenever you see these
reactions in biology

00:04:06.850 --> 00:04:08.290
or chemistry class, keep that in mind.

00:04:08.290 --> 00:04:11.820
It looks all neat and
organized but in a real system,

00:04:11.820 --> 00:04:13.550
these are all of these things just

00:04:13.550 --> 00:04:15.520
bouncing around in all
different crazy ways.

00:04:15.520 --> 00:04:17.220
And that's why energy's
an important thing here.

00:04:17.220 --> 00:04:19.320
Because the more energy
you apply to the system,

00:04:19.320 --> 00:04:21.180
the more that they're
going to bounce around,

00:04:21.180 --> 00:04:23.120
the more that they're going
to interact with each other.

00:04:23.120 --> 00:04:25.850
The more reactants you put in,

00:04:25.850 --> 00:04:27.920
the more chance they're
going to bounce around

00:04:27.920 --> 00:04:30.750
and be able to react with each other.

00:04:30.750 --> 00:04:32.050
Now I'm gonna introduce another word

00:04:32.050 --> 00:04:33.890
that you're gonna see in chemistry a lot.

00:04:33.890 --> 00:04:36.250
This water, these two...

00:04:36.250 --> 00:04:39.890
We see we have two water molecules here.

00:04:39.890 --> 00:04:42.590
We could call them molecules,
but since they are actually

00:04:42.590 --> 00:04:45.420
made up of two or more different elements,

00:04:45.420 --> 00:04:48.220
we could also call this a compound.

00:04:48.220 --> 00:04:52.050
So water, water is, you
could call it a molecule,

00:04:52.050 --> 00:04:54.220
or you could call it a compound.

00:04:54.220 --> 00:04:56.250
So this is a molecule or compound,

00:04:56.250 --> 00:04:57.690
while this molecular hydrogen,

00:04:57.690 --> 00:04:59.650
you would not call this a compound.

00:04:59.650 --> 00:05:01.950
And this molecular oxygen,
of course it's a molecule,

00:05:01.950 --> 00:05:05.290
but you would not call
it a compound either.

00:05:05.290 --> 00:05:07.190
And just to get an appreciation

00:05:07.190 --> 00:05:08.990
of how much energy this produces,

00:05:08.990 --> 00:05:11.120
let me just show you this
picture right over here.

00:05:11.120 --> 00:05:12.920
That's the space shuttle and this,

00:05:12.920 --> 00:05:15.820
this big tank right over here, let me...

00:05:15.820 --> 00:05:18.640
This big tank contains a bunch

00:05:18.640 --> 00:05:20.950
of liquid oxygen and hydrogen.

00:05:20.950 --> 00:05:25.190
And to create this
incredible amount of energy,

00:05:25.190 --> 00:05:26.420
it actually just...

00:05:26.420 --> 00:05:29.650
You mix the two together
with a little bit of energy

00:05:29.650 --> 00:05:31.520
and then you produce a ton of energy

00:05:31.520 --> 00:05:34.520
that makes the rocket, that
makes the space shuttle.

00:05:34.520 --> 00:05:36.210
Well, space shuttle's
been discontinued now,

00:05:36.210 --> 00:05:39.550
but back when they did it, to make it

00:05:39.550 --> 00:05:44.350
get it's necessary,
it's necessary velocity.

00:05:44.350 --> 00:05:45.750
Now let's talk about the idea.

00:05:45.750 --> 00:05:48.910
So, you know, this reaction,
strongly goes in this,

00:05:48.910 --> 00:05:51.380
in the direction of going to water.

00:05:51.380 --> 00:05:53.420
But it can actually go the other way,

00:05:53.420 --> 00:05:55.720
but it's very, very hard
for it to go the other way.

00:05:55.720 --> 00:05:57.020
So in general we would consider

00:05:57.020 --> 00:06:00.750
this to be an irreversible
reaction, even though it is.

00:06:00.750 --> 00:06:02.120
You know irreversible sounds like,

00:06:02.120 --> 00:06:03.620
hey you can't go the other way.

00:06:03.620 --> 00:06:05.120
It just really means that it's very

00:06:05.120 --> 00:06:06.220
unlikely to go the other way.

00:06:06.220 --> 00:06:08.550
You have to supply a lot of
energy to go the other way.

00:06:08.550 --> 00:06:10.650
To make this reaction go the other way,

00:06:10.650 --> 00:06:12.690
you would have to do
something called electrolysis,

00:06:12.690 --> 00:06:14.520
you provide energy, etcetera, etcetera.

00:06:14.520 --> 00:06:16.620
But in general, the way
that this is written,

00:06:16.620 --> 00:06:18.690
because the arrow is only
pointing in one direction,

00:06:18.690 --> 00:06:21.750
this is implying that it is irreversible.

00:06:21.750 --> 00:06:24.490
Irreversible.

00:06:24.490 --> 00:06:25.820
Irreversible.

00:06:25.820 --> 00:06:26.920
Which probably makes you think,

00:06:26.920 --> 00:06:29.390
well what about reversible reactions?

00:06:29.390 --> 00:06:31.850
And I have an example

00:06:31.850 --> 00:06:36.320
of a reversible reaction, right over here.

00:06:36.320 --> 00:06:40.880
I have a one bicarbonate ion.

00:06:40.880 --> 00:06:44.150
And the word ion, that's
just used to describe

00:06:44.150 --> 00:06:47.490
any molecule or atom that has either,

00:06:47.490 --> 00:06:49.620
has an imbalance of electrons or protons

00:06:49.620 --> 00:06:52.250
that cause it to have a net charge.

00:06:52.250 --> 00:06:55.320
So this makes this an ion,
and actually right over here,

00:06:55.320 --> 00:06:58.850
this is a hydrogen, this is a
hydrogen ion right over here.

00:06:58.850 --> 00:07:00.550
Both of these are charged.

00:07:00.550 --> 00:07:03.820
One has a positive charge,
one has a negative charge.

00:07:03.820 --> 00:07:07.050
But they are both, they are both ions.

00:07:07.050 --> 00:07:08.320
And this reactions right over here,

00:07:08.320 --> 00:07:11.420
you have the bicarbonate ion
that looks something like this.

00:07:11.420 --> 00:07:14.020
This is just my hand-drawing of it.

00:07:14.020 --> 00:07:16.880
Reacting with a hydrogen ion,
it's really a hydrogen atom

00:07:16.880 --> 00:07:19.320
that has lost it's electron,
so some people would even say

00:07:19.320 --> 00:07:22.220
this is a proton right over here.

00:07:22.220 --> 00:07:24.120
This is an equilibrium reaction,

00:07:24.120 --> 00:07:26.050
where it can form carbonic acid.

00:07:26.050 --> 00:07:28.120
And notice all that's
happening is this hydrogen

00:07:28.120 --> 00:07:31.880
is attaching to one of
the oxygens over here.

00:07:31.880 --> 00:07:34.790
And this is an equilibrium
because if in an actual,

00:07:34.790 --> 00:07:37.920
in an actual solution,
it's going back and forth.

00:07:37.920 --> 00:07:39.850
If you actually provide more reactants,

00:07:39.850 --> 00:07:41.450
you're gonna go more in that direction.

00:07:41.450 --> 00:07:43.920
If you provide more of
the products over here,

00:07:43.920 --> 00:07:47.090
then you're gonna go in that direction.

00:07:47.090 --> 00:07:50.450
And so in an actual, in
an actual environment,

00:07:50.450 --> 00:07:53.050
in an actual system, it's constantly going

00:07:53.050 --> 00:07:55.350
back and forth between these two things.

00:07:55.350 --> 00:07:57.090
And different reversible reactions might

00:07:57.090 --> 00:07:58.920
tend to one side or the other.

00:07:58.920 --> 00:08:01.450
If you provide more of
the stuff on one side,

00:08:01.450 --> 00:08:03.490
it might go more in the other
directions because these

00:08:03.490 --> 00:08:05.750
are gonna, they're gonna be
more likely to interact,

00:08:05.750 --> 00:08:06.950
Or if you provide more of this,

00:08:06.950 --> 00:08:08.450
it might go in the other direction

00:08:08.450 --> 00:08:12.190
because these might more likely react

00:08:12.190 --> 00:08:16.150
with their surroundings or
disassociate in some way.

00:08:16.150 --> 00:08:18.020
Now just to get a sense of, you know,

00:08:18.020 --> 00:08:20.020
it's nice to kind of, you
know, are these just some

00:08:20.020 --> 00:08:21.550
random letters that I wrote here?

00:08:21.550 --> 00:08:25.150
Carbonic acid is actually an
incredibly important molecule,

00:08:25.150 --> 00:08:26.920
or we could call it a
compound because it's made up

00:08:26.920 --> 00:08:29.090
of two or, two or more elements,

00:08:29.090 --> 00:08:31.250
in living systems and in fact,

00:08:31.250 --> 00:08:33.390
you know, even in the environment.

00:08:33.390 --> 00:08:36.090
And even when you go out
to get some fast food.

00:08:36.090 --> 00:08:38.650
When you have carbonated
drinks, it has carbonic acid

00:08:38.650 --> 00:08:41.620
in it that disassociates
into carbon dioxide

00:08:41.620 --> 00:08:44.150
and that carbon dioxide is
what you see bubbling up.

00:08:44.150 --> 00:08:47.520
Carbonic acid is incredibly
important in how your body

00:08:47.520 --> 00:08:52.420
deals with excess carbon
dioxide in its bloodstream.

00:08:52.420 --> 00:08:55.760
Carbonic acid is involved in the oceans

00:08:55.760 --> 00:08:58.720
taking up carbon dioxide
from the atmosphere.

00:08:58.720 --> 00:09:00.280
So when you're studying chemistry,

00:09:00.280 --> 00:09:03.220
especially in the context of
biology, these aren't just,

00:09:03.220 --> 00:09:05.920
you know, interesting things
that seem very academic.

00:09:05.920 --> 00:09:09.890
These are effecting your
real life and your body

00:09:09.890 --> 00:09:11.340
and your environment.


Dividing decimals completely

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:02.100
- [Voiceover] Let's figure out what 6.3

00:00:02.100 --> 00:00:05.620
divided by 0.25 is.

00:00:05.620 --> 00:00:07.232
And when I look at something like this,

00:00:07.232 --> 00:00:08.832
the first thing that
I think about is well,

00:00:08.832 --> 00:00:10.792
is there a way that I can rewrite this?

00:00:10.792 --> 00:00:13.492
Instead of dividing it by 25 hundredths,

00:00:13.492 --> 00:00:15.952
instead of dividing it by 0.25,

00:00:15.952 --> 00:00:19.552
I could divide it by a
whole number, maybe 25.

00:00:19.552 --> 00:00:23.212
And so how do I turn
25 hundredths into 25?

00:00:23.212 --> 00:00:24.432
Well, if I multiply it by,

00:00:24.432 --> 00:00:26.892
if I multiply it by 10 once,

00:00:26.892 --> 00:00:29.133
then. the decimal's one spot to the right.

00:00:29.133 --> 00:00:31.592
And if I multiply it by 10 twice,

00:00:31.592 --> 00:00:33.992
then I'm gonna move
two spots to the right.

00:00:33.992 --> 00:00:37.752
And 25 hundredths is going to become 25.

00:00:37.752 --> 00:00:41.512
I multiplied by 10 twice,
which is the same thing

00:00:41.512 --> 00:00:43.852
as multiplying by a hundred.

00:00:43.852 --> 00:00:46.972
Now of course, I can't only do it to 0.25,

00:00:46.972 --> 00:00:49.912
that would change the value
of this entire expression.

00:00:49.912 --> 00:00:52.113
I also have to do it to 6.3.

00:00:52.113 --> 00:00:54.852
If I move the decimal to
the right twice over here,

00:00:54.852 --> 00:00:56.332
to get a whole number, I gotta move

00:00:56.332 --> 00:00:58.593
the decimal twice to the right over here.

00:00:58.593 --> 00:01:01.072
So let's see, if I move
it to the right once,

00:01:01.072 --> 00:01:06.033
I get 63, so that's one
multiplication by 10.

00:01:06.033 --> 00:01:07.672
And now I wanna move
it to the right again,

00:01:07.672 --> 00:01:09.412
and you might say, "Wait,
wait, there doesn't look like

00:01:09.412 --> 00:01:10.612
"there's any other digits here

00:01:10.612 --> 00:01:12.652
"that I can move to the right of."

00:01:12.652 --> 00:01:14.252
And we just have to remind ourselves,

00:01:14.252 --> 00:01:18.832
that 6.3, well that's going
to be the same thing as 6.30,

00:01:18.832 --> 00:01:20.052
or we could actually as many zeroes

00:01:20.052 --> 00:01:21.352
as we want to the end of this,

00:01:21.352 --> 00:01:24.082
and not change the value, it's still 6.3.

00:01:24.082 --> 00:01:26.002
Six and three tenths is the same thing

00:01:26.002 --> 00:01:28.222
as six and 30 hundredths.

00:01:28.222 --> 00:01:31.232
So now, I can move the decimal
one more spot to the right.

00:01:32.192 --> 00:01:34.382
And, so I'm multiplying by 10 again.

00:01:34.382 --> 00:01:37.582
And it becomes 630.

00:01:37.582 --> 00:01:40.982
So 6.3 divided by 0.25 is the same thing

00:01:40.982 --> 00:01:44.502
as 630 divided by 25.

00:01:44.502 --> 00:01:45.263
And let me do that again,

00:01:45.263 --> 00:01:48.322
because this is really, the
tricky part, or the artful part,

00:01:48.322 --> 00:01:49.942
when you're dividing decimals.

00:01:49.942 --> 00:01:51.812
If I were to write,

00:01:51.812 --> 00:01:56.772
0.25, being divided into,

00:01:56.772 --> 00:01:59.002
being divided into,

00:01:59.002 --> 00:02:00.782
6.3.

00:02:00.782 --> 00:02:03.422
Six, let me do it.

00:02:03.422 --> 00:02:07.112
Six, 6.3.

00:02:07.112 --> 00:02:09.612
Now, what I care about is moving
this decimal to the right,

00:02:09.612 --> 00:02:11.512
far enough so that this
becomes a whole number.

00:02:11.512 --> 00:02:12.852
I don't have to move it any further,

00:02:12.852 --> 00:02:15.372
just far enough so that
this becomes a whole number.

00:02:15.372 --> 00:02:16.632
And then I have to move this,

00:02:16.632 --> 00:02:18.812
the same number of times to the right.

00:02:18.812 --> 00:02:20.712
But the goal is to make
this a whole number,

00:02:20.712 --> 00:02:22.512
not to make the 6.3 a whole number,

00:02:22.512 --> 00:02:25.732
although that is going to
happen for this particular case.

00:02:25.732 --> 00:02:28.292
So if I move it two spaces to the right,

00:02:28.292 --> 00:02:30.122
so, one, two.

00:02:31.152 --> 00:02:33.812
0.25 becomes 25,

00:02:33.812 --> 00:02:35.872
and then 6.3 becomes,

00:02:35.872 --> 00:02:38.512
move it one space, two
spaces to the right,

00:02:38.512 --> 00:02:40.612
it becomes 630.

00:02:40.612 --> 00:02:43.372
And so let me, let me clarify that,

00:02:43.372 --> 00:02:45.952
let me just clear all of this
stuff away out of the way,

00:02:45.952 --> 00:02:49.352
to just make that clear what I just did.

00:02:49.352 --> 00:02:50.392
So we don't have to deal with all

00:02:50.392 --> 00:02:52.532
of this messiness when we're actually

00:02:52.532 --> 00:02:54.912
doing our long division now.

00:02:54.912 --> 00:02:57.312
And so this one over here,

00:02:57.312 --> 00:02:59.652
let me erase all of this business.

00:02:59.652 --> 00:03:02.472
And we are ready to do some,

00:03:02.472 --> 00:03:06.653
we're ready to do some,
some long division, alright.

00:03:06.653 --> 00:03:08.372
So now this is just straightforward,

00:03:08.372 --> 00:03:11.032
dividing a two digit number
into a three digit number.

00:03:11.032 --> 00:03:12.592
And so we could say,

00:03:12.592 --> 00:03:14.892
well, 25 goes into six zero times,

00:03:14.892 --> 00:03:15.612
so let's keep going.

00:03:15.612 --> 00:03:18.672
25 goes into 63, let's see,

00:03:18.672 --> 00:03:20.373
two times 25 is 50,

00:03:20.373 --> 00:03:22.813
three times 25 is 75, so that's too much,

00:03:22.813 --> 00:03:24.912
so we wanna go two times.

00:03:24.912 --> 00:03:27.642
Two times 25 is going to be 50,

00:03:28.712 --> 00:03:30.255
and we can subtract.

00:03:30.255 --> 00:03:33.255
Or, if you didn't know two
times 25 is going to 50,

00:03:33.255 --> 00:03:36.375
you could have said two times five is 10,

00:03:36.375 --> 00:03:38.335
carry the one or regroup the one.

00:03:38.335 --> 00:03:41.975
Two times two is four, plus one is five.

00:03:41.975 --> 00:03:43.595
So we got 50.

00:03:43.595 --> 00:03:45.915
And now we subtract, and we get

00:03:45.915 --> 00:03:47.595
three minus zero is three,

00:03:47.595 --> 00:03:50.175
six minus five is one.

00:03:50.175 --> 00:03:53.675
And now we bring down the next digit.

00:03:53.675 --> 00:03:56.335
We can bring down this zero.

00:03:56.335 --> 00:03:57.785
And we could say, 25, and let me

00:03:57.785 --> 00:04:00.565
scratch this out, so I
don't confuse myself later.

00:04:00.565 --> 00:04:03.505
25 goes into 130 how many times?

00:04:03.505 --> 00:04:06.225
Let's see, 25 times four is 100,

00:04:06.225 --> 00:04:08.585
25 times five is 125,

00:04:08.585 --> 00:04:11.085
25 times six would be
150, so that's too big.

00:04:11.085 --> 00:04:12.985
So I'm gonna go five times.

00:04:12.985 --> 00:04:15.825
So five times,

00:04:15.825 --> 00:04:16.737
And I write, it's very important to

00:04:16.737 --> 00:04:18.677
keep track of my places right here,

00:04:18.677 --> 00:04:20.157
I brought down the zero to make

00:04:20.157 --> 00:04:21.857
the 13 into 130,

00:04:21.857 --> 00:04:24.317
so I say 25 goes into 130 five times,

00:04:24.317 --> 00:04:26.237
I write it above the zero,

00:04:26.237 --> 00:04:28.517
I write it right above
the zero right over there.

00:04:28.517 --> 00:04:31.257
And then five times 25, five times five,

00:04:31.257 --> 00:04:32.177
we already know what it is,

00:04:32.177 --> 00:04:33.217
but I'll just figure it out.

00:04:33.217 --> 00:04:36.838
Five times five is 25, regroup the two,

00:04:36.838 --> 00:04:39.798
five times two is 10, plus two is 12.

00:04:40.798 --> 00:04:44.310
And now we can, now we can subtract.

00:04:44.310 --> 00:04:45.830
And you might know
offhand, well (mumbles),

00:04:45.830 --> 00:04:48.910
130 minus 125 is going to be five,

00:04:48.910 --> 00:04:50.930
or if you like, you could
do a little regrouping,

00:04:50.930 --> 00:04:53.190
can make this, you could
take a 10 from there,

00:04:53.190 --> 00:04:54.590
and then put the 10 in the ones place,

00:04:54.590 --> 00:04:57.810
and this becomes 10,
10 minus five is five.

00:04:57.810 --> 00:04:58.950
Now, we aren't done yet.

00:04:58.950 --> 00:05:01.050
We wanna divide this completely,

00:05:01.050 --> 00:05:03.911
so we can bring down another zero.

00:05:03.911 --> 00:05:05.550
We can bring down another zero,

00:05:05.550 --> 00:05:07.250
and I'm picking an appropriate color,

00:05:07.250 --> 00:05:09.290
to, for the next, so I'll use,

00:05:09.290 --> 00:05:11.350
well I already used the green once,

00:05:11.350 --> 00:05:12.650
we'll use yellow.

00:05:12.650 --> 00:05:15.150
So once again, I can
bring down another zero,

00:05:15.150 --> 00:05:16.170
but I have to be very careful here,

00:05:16.170 --> 00:05:17.670
I can't just, I can't just throw

00:05:17.670 --> 00:05:19.130
a zero right over there.

00:05:19.130 --> 00:05:21.990
That's gonna turn 630 into 6300.

00:05:21.990 --> 00:05:23.990
I have to be very particular,
say no, look, look.

00:05:23.990 --> 00:05:25.650
This next zero I'm gonna put,

00:05:25.650 --> 00:05:27.710
that's gonna be after the decimal.

00:05:27.710 --> 00:05:32.090
Now 630 is the same thing as 630.0,

00:05:32.090 --> 00:05:33.510
and if I'm gonna add a decimal there,

00:05:33.510 --> 00:05:35.530
I wanna throw the decimal right above it.

00:05:35.530 --> 00:05:37.350
Need to be very careful with that.

00:05:37.350 --> 00:05:39.250
But now we're ready to start
bringing down the zero.

00:05:39.250 --> 00:05:41.790
So let's bring down this zero.

00:05:41.790 --> 00:05:44.051
And we have 50,

00:05:44.051 --> 00:05:45.850
how many times does 25 go into 50?

00:05:45.850 --> 00:05:47.010
Let me scratch this out,

00:05:47.010 --> 00:05:48.350
so we don't confuse ourselves later.

00:05:48.350 --> 00:05:51.110
25 goes two times into 50,

00:05:51.110 --> 00:05:54.201
two times 25 is in fact 50,

00:05:54.201 --> 00:05:55.741
is in fact 50.

00:05:55.741 --> 00:05:57.801
And so now we can subtract.

00:05:57.801 --> 00:06:01.242
And we now have a remainder of zero,

00:06:01.242 --> 00:06:03.721
and there's nothing left to
bring down other than zeros,

00:06:03.721 --> 00:06:04.801
and we have a remainder of zero,

00:06:04.801 --> 00:06:07.741
so we are done dividing completely.

00:06:07.741 --> 00:06:12.581
You take 630, divided by 25, it's 25.2.

00:06:12.581 --> 00:06:14.321
Or if you look at our original problem,

00:06:14.321 --> 00:06:16.091
if you look at our original problem,

00:06:16.091 --> 00:06:19.212
which was 6.30 divided by 0.25,

00:06:19.212 --> 00:06:22.211
that also is 25.2.

00:06:22.211 --> 00:06:25.391
So this is also going to be 25.2.

00:06:25.391 --> 00:06:27.931
Let me rewrite it, let me rewrite it.

00:06:27.931 --> 00:06:30.931
So our original problem was 6.3

00:06:30.931 --> 00:06:34.571
divided by 0.25,

00:06:34.571 --> 00:06:36.482
which we just figured out,

00:06:36.482 --> 00:06:37.322
what we just figured out

00:06:37.322 --> 00:06:41.082
is 25.2.

00:06:41.082 --> 00:06:42.522
And we're done.


Figuring out coordinate of bended figure using trig

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.490
Kinematics is the
geometry of motion.

00:00:03.490 --> 00:00:06.150
To simplify the motion
of a pointer finger,

00:00:06.150 --> 00:00:09.310
the three finger bones are
modeled as narrow cylinders.

00:00:09.310 --> 00:00:11.870
The two joints are
modeled as small spheres

00:00:11.870 --> 00:00:14.820
each 0.1 centimeters
in diameter--

00:00:14.820 --> 00:00:16.980
so they show this
right over here--

00:00:16.980 --> 00:00:18.890
and rotating from the center.

00:00:18.890 --> 00:00:22.260
The finger is mapped on a
Cartesian coordinate plane.

00:00:22.260 --> 00:00:24.749
So here's the first joint
of the pointer finger,

00:00:24.749 --> 00:00:26.290
and while I'm doing
this I'm actually

00:00:26.290 --> 00:00:29.940
looking at the side of
my left pointer finger

00:00:29.940 --> 00:00:31.270
and it's pointing straight up.

00:00:31.270 --> 00:00:35.247
And this is what it looks like
although my-- this top section

00:00:35.247 --> 00:00:36.830
right over here is
much longer for me.

00:00:36.830 --> 00:00:38.496
It's much closer to
this middle section.

00:00:38.496 --> 00:00:40.820
But anyway let's see
what they have to say.

00:00:40.820 --> 00:00:43.740
They say the first finger
bone is 2.4 centimeters long,

00:00:43.740 --> 00:00:46.510
which actually looks about
right for mine as well.

00:00:46.510 --> 00:00:48.980
The middle finger bone is
2/3 the length of the first,

00:00:48.980 --> 00:00:51.120
so that's this length
right over here.

00:00:51.120 --> 00:00:55.530
And 2/3 of 2.4 is
going to be, what, 1.6.

00:00:55.530 --> 00:00:58.690
So this distance right over--
if we do that in a color

00:00:58.690 --> 00:01:02.140
we can actually see-- this
distance right over here

00:01:02.140 --> 00:01:04.890
is 2.4 centimeters.

00:01:04.890 --> 00:01:08.080
This distance right over
here is 2/3 of that,

00:01:08.080 --> 00:01:10.390
so that's 1.6 centimeters.

00:01:10.390 --> 00:01:12.940
And the top finger bone is 1/3
the length of the first, which

00:01:12.940 --> 00:01:15.490
does seem short, but I'll
just go with their numbers.

00:01:15.490 --> 00:01:21.560
So 1/3 of 2.4 , this
is 0.8 centimeters--

00:01:21.560 --> 00:01:23.270
everything we're doing
is in centimeters.

00:01:23.270 --> 00:01:24.810
The center of the
tip of the finger

00:01:24.810 --> 00:01:27.790
goes from open at 0.5-- so
that point right over there--

00:01:27.790 --> 00:01:28.659
to a closed fist.

00:01:28.659 --> 00:01:30.950
So when the fist is closed--
and I'm looking at my hand

00:01:30.950 --> 00:01:33.090
doing it right over
here-- the finger

00:01:33.090 --> 00:01:35.250
will look something like
this-- the pointer finger.

00:01:35.250 --> 00:01:38.080
And so the tip
goes to x comma y.

00:01:38.080 --> 00:01:42.930
They want us to figure out
the x-coordinate of x comma y.

00:01:42.930 --> 00:01:46.760
So maybe we don't even have to
worry about the y-coordinate.

00:01:46.760 --> 00:01:49.000
So let me draw this
a little bit larger

00:01:49.000 --> 00:01:51.520
just so that we can
make sense of it.

00:01:51.520 --> 00:01:54.080
So we have this first
section of the finger.

00:01:54.080 --> 00:01:56.030
It's 2.4.

00:01:56.030 --> 00:02:01.870
So this is 2.4 centimeters and
it's completely horizontal.

00:02:01.870 --> 00:02:03.890
I'll try to draw it
as large as I can

00:02:03.890 --> 00:02:06.890
so that we can make
sense of all of this.

00:02:06.890 --> 00:02:10.669
So that right over there
is 2.4 centimeters.

00:02:10.669 --> 00:02:13.000
And then it hits this joint.

00:02:13.000 --> 00:02:18.250
This joint has a radius-- or has
a diameter of 0.1 centimeters.

00:02:20.760 --> 00:02:28.160
So it's radius right over
there is 0.05 centimeters.

00:02:28.160 --> 00:02:31.490
So the entire length of this
section you see here is 2.45.

00:02:31.490 --> 00:02:38.590
So this point right over
here is at 2.45 comma 0.

00:02:38.590 --> 00:02:40.430
Then you have the
next section, which

00:02:40.430 --> 00:02:42.730
has a length of 1.6
centimeters, and they

00:02:42.730 --> 00:02:45.530
tell us this is a
45 degree angle.

00:02:45.530 --> 00:02:54.510
So this right over here
is 1.6 centimeters.

00:02:54.510 --> 00:02:58.650
And of course you have
that 0.05 radius again.

00:02:58.650 --> 00:03:03.310
So this entire length
is going to be 1.65.

00:03:03.310 --> 00:03:04.870
But then we have
another joint here

00:03:04.870 --> 00:03:06.070
which we can't forget about.

00:03:06.070 --> 00:03:07.670
We have this joint
right over here.

00:03:07.670 --> 00:03:09.220
So let's draw that.

00:03:09.220 --> 00:03:11.420
So you have this
joint right over here.

00:03:11.420 --> 00:03:11.920
Draw it.

00:03:11.920 --> 00:03:15.590
It has a diameter
of 0.1 centimeters,

00:03:15.590 --> 00:03:19.170
and I'm definitely not
drawing this joint to scale.

00:03:19.170 --> 00:03:22.730
I'm just trying to make sure
we pay attention to the joint.

00:03:22.730 --> 00:03:25.910
So let me draw a
little bit better.

00:03:25.910 --> 00:03:28.370
Something like this.

00:03:28.370 --> 00:03:31.040
And once again we
have another 0.5.

00:03:31.040 --> 00:03:34.490
So this entire length, if we
were to go from this center

00:03:34.490 --> 00:03:38.520
to that center, is
going to be 1.6.

00:03:38.520 --> 00:03:41.300
Plus these two radii, which
add up to 0.1 diameter,

00:03:41.300 --> 00:03:45.250
which is going to
be plus 0.1 which

00:03:45.250 --> 00:03:48.310
is equal to 1.7 centimeters.

00:03:48.310 --> 00:03:52.020
So this entire side
is 1.7 centimeters.

00:03:52.020 --> 00:03:56.120
This entire side is 2.45.

00:03:56.120 --> 00:04:01.060
So plus 0.05 is equal to 2.45.

00:04:01.060 --> 00:04:03.450
And then finally
this last section--

00:04:03.450 --> 00:04:07.080
we have a right angle here.

00:04:07.080 --> 00:04:09.560
They tell us that
this is a right angle.

00:04:09.560 --> 00:04:15.590
And this section right
over here is 0.8.

00:04:15.590 --> 00:04:18.240
This is going to be--
let me make it clear--

00:04:18.240 --> 00:04:20.570
so this section
right over here--

00:04:20.570 --> 00:04:25.040
if we don't count
the joint--is 0.8.

00:04:25.040 --> 00:04:33.140
But then we have of course
this 0.05 centimeters.

00:04:33.140 --> 00:04:39.620
So plus 0.05 you get this
whole length right over here.

00:04:39.620 --> 00:04:40.900
I don't want to clutter it.

00:04:40.900 --> 00:04:48.900
This whole length to
the tip right is 0.857.

00:04:48.900 --> 00:04:51.151
Now how do we figure out the
coordinate of this thing?

00:04:51.151 --> 00:04:53.233
And I encourage you to
pause the video at any time

00:04:53.233 --> 00:04:55.000
to try to figure
out this and really

00:04:55.000 --> 00:04:57.150
just to save us work
let's see if we can just

00:04:57.150 --> 00:04:58.440
focus on the x-squared.

00:04:58.440 --> 00:05:00.939
And I'm just going try to come
up with an expression for it.

00:05:00.939 --> 00:05:03.900
I don't feel like using
my calculator just now.

00:05:03.900 --> 00:05:06.640
So the first thing is, let's try
to figure out an x-coordinate.

00:05:06.640 --> 00:05:08.790
Let's try to figure out the
x-coordinate for that point

00:05:08.790 --> 00:05:09.520
right over there.

00:05:09.520 --> 00:05:14.900
So where does that point
intersect the x-axis?

00:05:14.900 --> 00:05:18.980
So essentially what is this
distance right over here?

00:05:22.460 --> 00:05:31.060
What's going to be 2.45 minus
whatever this distance is?

00:05:31.060 --> 00:05:33.980
And we should be able to
use this distance given

00:05:33.980 --> 00:05:36.510
the fact that this
is a 45 degree angle.

00:05:39.250 --> 00:05:40.790
We know this length
right over here,

00:05:40.790 --> 00:05:42.310
and this is a right triangle.

00:05:42.310 --> 00:05:45.480
We could use a little
bit of trigonometry.

00:05:45.480 --> 00:05:49.030
We know that the cosine--
relative to this angle

00:05:49.030 --> 00:05:51.950
this is the hypotenuse
and this side is adjacent.

00:05:51.950 --> 00:05:55.710
So we know the
cosine of 45 degrees

00:05:55.710 --> 00:05:57.574
is equal to the length
of the adjacent side.

00:05:57.574 --> 00:05:59.740
So let's say that's what
we're trying to figure out.

00:05:59.740 --> 00:06:02.525
Let's just say that's
going to be a for adjacent.

00:06:02.525 --> 00:06:05.040
Now let me do this in a color
that I haven't used yet.

00:06:05.040 --> 00:06:08.540
So it's going to be equal
to the length of that side A

00:06:08.540 --> 00:06:10.790
over the length
of the hypotenuse.

00:06:10.790 --> 00:06:12.790
Over adjacent, over hypotenuse.

00:06:12.790 --> 00:06:17.065
The hypotenuse we already
figured out is 1.7 centimeters.

00:06:20.140 --> 00:06:30.540
Or we could say that a is equal
to 1.7 times the cosine of 45

00:06:30.540 --> 00:06:31.040
degrees.

00:06:34.370 --> 00:06:35.560
That's this length.

00:06:35.560 --> 00:06:38.550
So if you wanted this
x-coordinate right over here,

00:06:38.550 --> 00:06:41.260
that x-coordinate is going
to be this x-coordinate,

00:06:41.260 --> 00:06:45.650
which is 2.45 minus a.

00:06:45.650 --> 00:06:53.700
So minus 1.7 times the
cosine of 45 degrees-- and we

00:06:53.700 --> 00:06:58.570
don't know what the y-coordinate
is we haven't figured that out.

00:06:58.570 --> 00:07:01.512
But the question doesn't ask us
to.-- But let's see if we can

00:07:01.512 --> 00:07:02.970
just, going with
the x-coordinates,

00:07:02.970 --> 00:07:05.344
if we can get all the way to
this x-coordinate right over

00:07:05.344 --> 00:07:06.030
here.

00:07:06.030 --> 00:07:10.480
Well this x-coordinate, is that
x-coordinate right over there.

00:07:10.480 --> 00:07:13.360
And so let's see, if we
can figure out this length,

00:07:13.360 --> 00:07:16.170
we could subtract that
length from this x-coordinate

00:07:16.170 --> 00:07:17.557
and get to this x-coordinate.

00:07:17.557 --> 00:07:19.140
And how can we figure
out that length?

00:07:19.140 --> 00:07:22.240
Well that length's going to be
the same thing as this length

00:07:22.240 --> 00:07:24.290
down here.

00:07:24.290 --> 00:07:29.530
And this is a completely
a horizontal line.

00:07:29.530 --> 00:07:33.020
So this is going to be parallel
to that line over there.

00:07:33.020 --> 00:07:36.430
If you view this as
a transversal of two

00:07:36.430 --> 00:07:39.830
parallel lines if this angle
is 45 degrees I'm running out

00:07:39.830 --> 00:07:42.440
of colors here-- if this
angle is 45 degrees,

00:07:42.440 --> 00:07:45.970
then this angle is also
going to be 45 degrees.

00:07:45.970 --> 00:07:49.020
If this is a right angle,
which it is, that's a 90--

00:07:49.020 --> 00:07:51.680
so then this is going to
have to be 45 as well.

00:07:51.680 --> 00:07:55.844
Because 45 plus 90 plus 45
are going to be 180 degrees.

00:07:55.844 --> 00:07:57.010
We've gone completely round.

00:07:57.010 --> 00:08:00.090
We're forming a
completely straight line.

00:08:00.090 --> 00:08:01.890
So this is a 45 degree angle.

00:08:07.360 --> 00:08:13.950
How do we figure out this
length right over here?

00:08:13.950 --> 00:08:20.260
Well once again, the
cosine of 45 degrees

00:08:20.260 --> 00:08:23.530
is going to be equal to the
length of the adjacent side,

00:08:23.530 --> 00:08:25.190
which is what we
want to figure out.

00:08:25.190 --> 00:08:27.010
We don't want-- it I know
this thing's getting messy--

00:08:27.010 --> 00:08:29.300
We want to figure out this
length right over here.

00:08:29.300 --> 00:08:32.039
It's going to be equal--
let's call that length x--

00:08:32.039 --> 00:08:34.640
it's going to be equal to x over
the length of the hypotenuse.

00:08:34.640 --> 00:08:37.080
That's what this 0.85 is.

00:08:37.080 --> 00:08:39.820
So that 0.85, that is the
hypotenuse right there.

00:08:39.820 --> 00:08:42.520
And my apologies
for this messiness.

00:08:42.520 --> 00:08:44.140
0.85.

00:08:44.140 --> 00:08:53.700
Or we get x is equal to 0.85
times the cosine of 45 degrees.

00:09:01.780 --> 00:09:04.840
So that's this length
right over here.

00:09:04.840 --> 00:09:07.440
So if we want this
x-coordinate right over here,

00:09:07.440 --> 00:09:11.185
we just subtract this
distance from this distance.

00:09:11.185 --> 00:09:12.560
So the x-coordinate--
that's what

00:09:12.560 --> 00:09:14.480
the whole problem's
going to be about.

00:09:14.480 --> 00:09:16.770
The x-coordinate is going
to be-- let me just write it

00:09:16.770 --> 00:09:21.850
in one color-- 2.45 minus
1.7 times the cosine of 45

00:09:21.850 --> 00:09:23.780
degrees minus this.

00:09:23.780 --> 00:09:28.840
Minus 0.85 times the
cosine of 45 degrees.

00:09:28.840 --> 00:09:31.320
And so this is going
to be equal to-- we

00:09:31.320 --> 00:09:32.850
deserve a drum
roll-- this is going

00:09:32.850 --> 00:09:38.650
to be equal to 2.45 minus--
what is this-- 2.55 times

00:09:38.650 --> 00:09:40.880
the cosine of 45 degrees.

00:09:40.880 --> 00:09:42.550
That's going to be
the x-coordinate.

00:09:42.550 --> 00:09:44.540
And we don't know what
the y-coordinate is,

00:09:44.540 --> 00:09:47.890
but all they asked us
for is the x-coordinate.


Long division with decimals

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:05.160
- [Voiceover] Let's figure out
what 2.211 divided by 6.7 is.

00:00:05.529 --> 00:00:07.248
So the first thing I like to do is,

00:00:07.248 --> 00:00:09.906
I don't like to divide by a decimal.

00:00:09.906 --> 00:00:13.342
So I'm going to multiply 6.7,

00:00:13.342 --> 00:00:15.769
I'm going to multiply 6.7 by 10,

00:00:15.769 --> 00:00:19.167
which is essentially
just taking this decimal

00:00:19.167 --> 00:00:22.038
and moving it one space to the right.

00:00:22.038 --> 00:00:24.442
Well I can't just do that to the 6.7,

00:00:24.442 --> 00:00:26.837
I also have to do that to the 2.21.

00:00:26.837 --> 00:00:28.400
So let me move its decimal

00:00:28.400 --> 00:00:31.128
one space to the right.

00:00:31.128 --> 00:00:34.112
And so now I can rewrite the problem.

00:00:34.112 --> 00:00:35.494
I can rewrite it as

00:00:35.494 --> 00:00:37.164
22,

00:00:37.164 --> 00:00:38.943
22 point,

00:00:38.943 --> 00:00:42.253
22.11

00:00:42.253 --> 00:00:43.746
divided by,

00:00:43.746 --> 00:00:44.976
divided by,

00:00:44.976 --> 00:00:46.553
instead of writing it as 6.7

00:00:46.553 --> 00:00:48.813
I can now write it as 67.

00:00:48.813 --> 00:00:51.896
67, let me do it in that same color.

00:00:51.896 --> 00:00:54.088
I can now write it as 67.

00:00:54.088 --> 00:00:55.613
I could write a decimal right over here

00:00:55.613 --> 00:00:56.861
but that's not going to change the fact

00:00:56.861 --> 00:00:59.660
that this is just now the whole number 67.

00:00:59.660 --> 00:01:02.121
And so now I'm ready to
do some long division

00:01:02.121 --> 00:01:03.939
now that I'm dividing by a whole number.

00:01:03.939 --> 00:01:05.976
You might say, "Wait, I still
have a decimal over here,"

00:01:05.976 --> 00:01:06.974
but that's okay

00:01:06.974 --> 00:01:09.354
and you're going to see that in a second.

00:01:09.354 --> 00:01:11.038
So let's try to do some long division.

00:01:11.038 --> 00:01:12.291
So we're going to take,

00:01:12.291 --> 00:01:16.102
we're going to take 22.11

00:01:16.102 --> 00:01:17.910
or 22 and 11 hundredths,

00:01:17.910 --> 00:01:19.152
and we are going to divide,

00:01:19.152 --> 00:01:20.754
let me write that a little bit bigger

00:01:20.754 --> 00:01:22.774
just so we have some space.

00:01:22.774 --> 00:01:27.026
22.11,

00:01:27.026 --> 00:01:29.770
and we are going to divide that by...

00:01:29.770 --> 00:01:32.617
We are going to divide that by 67.

00:01:33.036 --> 00:01:35.820
67.

00:01:35.820 --> 00:01:38.023
Alright, let's do some long division now.

00:01:38.023 --> 00:01:39.539
And actually, before we even do that,

00:01:39.539 --> 00:01:41.304
I like to keep track of my decimal.

00:01:41.304 --> 00:01:42.860
So my decimal is over here,

00:01:42.860 --> 00:01:45.204
I'm going to write my
decimal right over here

00:01:45.204 --> 00:01:46.458
in the answer.

00:01:46.458 --> 00:01:48.281
And when you're doing these
long division problems

00:01:48.281 --> 00:01:50.580
it's really important
to write things neatly

00:01:50.580 --> 00:01:52.379
and keep things in nice columns

00:01:52.379 --> 00:01:53.679
and keep track of your place value

00:01:53.679 --> 00:01:55.479
because if you don't write
things in neat columns

00:01:55.479 --> 00:01:58.137
then frankly you'll probably
lose track of your place value.

00:01:58.137 --> 00:01:59.159
But let's do this.

00:01:59.159 --> 00:02:03.280
So 67 goes into 2 zero
times, so let's move on.

00:02:03.280 --> 00:02:06.461
67 goes into 22 zero times.

00:02:06.461 --> 00:02:10.242
67 goes into 221.

00:02:10.242 --> 00:02:12.363
So let's think about that a little bit.

00:02:12.363 --> 00:02:14.579
This is pretty close to 70,

00:02:14.579 --> 00:02:18.428
70 times 3 would be 210.

00:02:18.428 --> 00:02:20.508
So this looks like maybe three times,

00:02:20.508 --> 00:02:22.423
three times seems about right.

00:02:22.423 --> 00:02:24.849
So let's try, let's put a 3 up here.

00:02:24.849 --> 00:02:28.290
3 times 7 is 21,

00:02:28.290 --> 00:02:29.740
carry the 2,

00:02:29.740 --> 00:02:34.574
3 times 6 is 18, plus 2 is 20,

00:02:34.574 --> 00:02:35.564
is 20.

00:02:35.564 --> 00:02:38.537
So we get 201 and the difference between

00:02:38.537 --> 00:02:42.312
221 and 201 is going to be...

00:02:42.312 --> 00:02:44.723
Well you just get a 0
here, you get a 2 here,

00:02:44.723 --> 00:02:46.860
then you get a 0 here, I
don't have to write it.

00:02:46.860 --> 00:02:47.916
It's 20.

00:02:47.916 --> 00:02:52.408
So 3 was right, if 3 times
67 were higher than 221

00:02:52.408 --> 00:02:53.826
then we'd be in trouble.

00:02:53.826 --> 00:02:57.863
And then if 3 times 67 were
lower but it was so low

00:02:57.863 --> 00:03:00.962
that when you subtracted you
got a number higher than 67

00:03:00.962 --> 00:03:02.302
then that means you could have thrown in

00:03:02.302 --> 00:03:04.820
more 67s in there, but
this number's just right.

00:03:04.820 --> 00:03:08.624
It's lower but our remainder is less than,

00:03:08.624 --> 00:03:12.063
is less than 67, so let's keep going.

00:03:12.063 --> 00:03:14.617
So now we can bring down,

00:03:14.617 --> 00:03:17.477
we can bring down the 1

00:03:17.477 --> 00:03:19.568
and we see 67 goes into 201,

00:03:19.568 --> 00:03:20.793
well we just figured that out.

00:03:20.793 --> 00:03:24.695
67 goes into 201 three times, three times.

00:03:24.695 --> 00:03:27.678
3 times 7 is one, carry the 2,

00:03:27.678 --> 00:03:28.723
gotta make sure we don't...

00:03:28.723 --> 00:03:30.824
Well, if we're doing the
same problem as we did before

00:03:30.824 --> 00:03:34.257
but it's good to not have
the previous 2 there,

00:03:34.257 --> 00:03:36.002
sometimes you might want to erase it.

00:03:36.002 --> 00:03:39.449
Then you have 3 times
6 is 18 plus 2 is 20.

00:03:39.449 --> 00:03:42.435
And now you subtract, and we are done

00:03:42.435 --> 00:03:44.581
because we have no remainder

00:03:44.581 --> 00:03:48.185
and there's nothing
here left to bring down.

00:03:48.185 --> 00:03:49.550
And so

00:03:49.550 --> 00:03:52.754
22.11 divided by 67 is...

00:03:52.754 --> 00:03:55.075
And we can throw a 0 here
just for good measure.

00:03:55.075 --> 00:03:57.757
It's 0.33.

00:03:57.757 --> 00:04:02.598
0.33, this is equal to 0.33.

00:04:02.598 --> 00:04:06.786
Or this is equal to 0.33.

00:04:06.786 --> 00:04:08.026
And we're done.


What is the range of a function?

WEBVTT
Kind: captions
Language: en

00:00:00.249 --> 00:00:01.851
- As a little bit of
a review, we know that

00:00:01.851 --> 00:00:04.754
if we have some function,
let's call it "f".

00:00:04.754 --> 00:00:06.461
We don't have to call it
"f", but "f" is the letter

00:00:06.461 --> 00:00:08.655
most typically used for functions,

00:00:08.655 --> 00:00:11.221
that if I give it an input, a valid input,

00:00:11.221 --> 00:00:14.785
if I give it a valid input,
and I use the variable "x"

00:00:14.785 --> 00:00:19.785
for that valid input, it is
going to map that to an output.

00:00:19.995 --> 00:00:21.992
It is going to map that, or produce,

00:00:21.992 --> 00:00:24.453
given this x, it's going
to product an output

00:00:24.453 --> 00:00:26.926
that we would call "f(x)."

00:00:26.926 --> 00:00:28.087
And we've already talked a little bit

00:00:28.087 --> 00:00:29.921
about the notion of a domain.

00:00:29.921 --> 00:00:32.882
A domain is the set of all of the inputs

00:00:32.882 --> 00:00:35.088
over which the function is defined.

00:00:35.088 --> 00:00:39.291
So if this the domain here,
if this is the domain here,

00:00:39.291 --> 00:00:42.657
and I take a value here,
and I put that in for x,

00:00:42.657 --> 00:00:45.386
then the function is
going to output an f(x).

00:00:45.386 --> 00:00:47.917
If I take something that's
outside of the domain,

00:00:47.917 --> 00:00:49.693
let me do that in a different color...

00:00:49.693 --> 00:00:52.723
If I take something that
is outside of the domain

00:00:52.723 --> 00:00:55.161
and try to input it into this function,

00:00:55.161 --> 00:00:56.322
the function will say, "hey, wait wait,"

00:00:56.322 --> 00:00:57.994
"I'm not defined for that thing"

00:00:57.994 --> 00:01:00.061
"that's outside of the domain."

00:01:00.061 --> 00:01:04.081
Now another interesting
thing to think about,

00:01:04.081 --> 00:01:06.258
and that's actually what
the focus of this video is,

00:01:06.258 --> 00:01:08.661
okay, we know the set of
all of the valid inputs,

00:01:08.661 --> 00:01:10.960
that's called the domain,
but what about all,

00:01:10.960 --> 00:01:13.293
the set of all of the
outputs that the function

00:01:13.293 --> 00:01:14.861
could actually produce?

00:01:14.861 --> 00:01:16.591
And we have a name for that.

00:01:16.591 --> 00:01:19.551
That is called the range of the function.

00:01:19.551 --> 00:01:20.771
So the range.

00:01:21.659 --> 00:01:23.690
The range, and the most typical,

00:01:23.690 --> 00:01:25.386
there's actually a couple
of definitions for range,

00:01:25.386 --> 00:01:27.359
but the most typical definition for range

00:01:27.359 --> 00:01:30.018
is "the set of all possible outputs."

00:01:30.018 --> 00:01:32.514
So you give me, you input
something from the domain,

00:01:32.514 --> 00:01:34.894
it's going to output
something, and by definition,

00:01:34.894 --> 00:01:36.856
because we have outputted
it from this function,

00:01:36.856 --> 00:01:39.016
that thing is going to be in the range,

00:01:39.016 --> 00:01:40.722
and if we take the set
of all of the things

00:01:40.722 --> 00:01:42.115
that the function could output,

00:01:42.115 --> 00:01:44.821
that is going to make up the range.

00:01:44.821 --> 00:01:49.755
So this right over here is
the set of all possible,

00:01:49.755 --> 00:01:53.284
all possible outputs.

00:01:53.284 --> 00:01:55.188
All possible outputs.

00:01:55.188 --> 00:01:57.023
So let's make that a
little bit more concrete,

00:01:57.023 --> 00:01:58.590
with an example.

00:01:58.590 --> 00:02:02.317
So let's say that I have the function f(x)

00:02:02.317 --> 00:02:06.195
defined as, so once again,
I'm gonna input x's,

00:02:06.195 --> 00:02:10.990
and I have my function f,
and I'm gonna output f(x).

00:02:10.990 --> 00:02:12.359
And let's say this def...

00:02:12.359 --> 00:02:13.950
The function definition
here, the thing that tries to

00:02:13.950 --> 00:02:16.620
figure out, "okay, given an
x, what f(x) do I produce?",

00:02:16.620 --> 00:02:20.359
the definition says "f(x)
is going to be equal"

00:02:20.359 --> 00:02:23.284
"to whatever my input is, squared."

00:02:23.284 --> 00:02:24.562
Well, just as a little bit of review,

00:02:24.562 --> 00:02:26.988
we know what the domain
here is going to be.

00:02:26.988 --> 00:02:30.959
The domain is the set of all valid inputs.

00:02:30.959 --> 00:02:33.118
So what are the valid inputs here?

00:02:33.118 --> 00:02:34.894
Well, I could take any real number

00:02:34.894 --> 00:02:37.251
and input into this, and I
could take any real number

00:02:37.251 --> 00:02:39.388
and I can square it, there's
nothing wrong with that,

00:02:39.388 --> 00:02:42.151
and so the domain is all real numbers.

00:02:42.151 --> 00:02:47.151
All, all real, all real numbers.

00:02:48.153 --> 00:02:49.825
But what's the range?

00:02:49.825 --> 00:02:50.855
Maybe I'll do that in a different color

00:02:50.855 --> 00:02:52.260
just to highlight it.

00:02:52.260 --> 00:02:54.025
What is going to be the range here,

00:02:54.025 --> 00:02:57.209
what is the set of all possible outputs?

00:02:58.190 --> 00:02:59.490
Well if you think about, actually,

00:02:59.490 --> 00:03:02.160
to help us think about, let
me actually draw a graph here.

00:03:02.160 --> 00:03:04.157
Of what this looks like.

00:03:04.157 --> 00:03:06.224
What this looks like.

00:03:06.224 --> 00:03:08.546
So the graph of "f(x)
is equal to x squared"

00:03:08.546 --> 00:03:10.287
is going to look something like this.

00:03:10.287 --> 00:03:15.024
So, it's gonna look, it's going
to look something like this.

00:03:15.024 --> 00:03:17.056
I'm obviously hand-drawing
it, so it's not perfect.

00:03:17.056 --> 00:03:19.320
It's gonna be a parabola with a,

00:03:19.320 --> 00:03:22.153
with a vertex right here at the origin.

00:03:22.153 --> 00:03:24.591
So this is the graph, this is the graph,

00:03:24.591 --> 00:03:28.921
"y is equal to f(x)," this
of course is the x-axis,

00:03:28.921 --> 00:03:31.592
this of course is the y-axis.

00:03:31.592 --> 00:03:33.089
So let's think about it, what is the set

00:03:33.089 --> 00:03:35.052
of all possible outputs?

00:03:35.052 --> 00:03:36.723
Well in this case, the set
of all possible outputs

00:03:36.723 --> 00:03:38.952
is the set of all possible y's here.

00:03:38.952 --> 00:03:43.295
Well, we see, y can take
on any non-negative value.

00:03:43.295 --> 00:03:47.857
y could be zero, y could one,
y could be pi, y could be e,

00:03:47.857 --> 00:03:51.027
but y cannot be negative.

00:03:51.027 --> 00:03:54.324
So the range here is, the range...

00:03:54.324 --> 00:03:56.124
We could, well we could
say it a couple of ways,

00:03:56.124 --> 00:03:59.189
we could say, "f(x)",
let me write it this way.

00:03:59.189 --> 00:04:03.252
"f(x) is a member of the real numbers"

00:04:03.252 --> 00:04:08.252
"such that, is such that
f(x) is greater than"

00:04:09.823 --> 00:04:11.391
"or equal to zero."

00:04:11.391 --> 00:04:13.051
We could write it that
way, if we wanted to

00:04:13.051 --> 00:04:15.559
write it in a less mathy notation,

00:04:15.559 --> 00:04:19.355
we could say that "f(x) is going to be"

00:04:19.355 --> 00:04:21.921
"greater than or equal to zero."

00:04:21.921 --> 00:04:24.185
f(x) is not going to be negative,

00:04:24.185 --> 00:04:27.621
so any non-negative number, the set of all

00:04:27.621 --> 00:04:31.824
non-negative numbers, that is our range.

00:04:31.824 --> 00:04:33.055
Let's do another example of this,

00:04:33.055 --> 00:04:35.981
just to make it a little bit,
just to make it a little bit,

00:04:35.981 --> 00:04:37.420
a little bit clearer.

00:04:37.420 --> 00:04:42.250
Let's say that I had,
let's say that I had g(x),

00:04:42.250 --> 00:04:45.153
let's say I have g(x),
I'll do this in white,

00:04:45.153 --> 00:04:49.820
let's say it's equal
to "x squared over x."

00:04:49.820 --> 00:04:52.595
So we could try to
simplify g(x) a little bit,

00:04:52.595 --> 00:04:53.825
we could say, "look, if I have x squared"

00:04:53.825 --> 00:04:55.590
"and I divide it by x, that's gonna,"

00:04:55.590 --> 00:04:59.653
"that's the same thing as
g(x) being equal to x."

00:04:59.653 --> 00:05:02.347
"x squared over x" is x,
but we have to be careful.

00:05:02.347 --> 00:05:04.657
Because right over here, we have to,

00:05:04.657 --> 00:05:08.094
in our domain, x cannot be equal to zero.

00:05:08.094 --> 00:05:10.393
If x is equal to zero,
we get zero over zero,

00:05:10.393 --> 00:05:11.786
we get indeterminate form.

00:05:11.786 --> 00:05:14.828
So in order for this function
to be the exact same function,

00:05:14.828 --> 00:05:16.987
we have to put that,
'cause it's not obvious now

00:05:16.987 --> 00:05:21.204
from the definition, we have to say,

00:05:21.204 --> 00:05:23.988
"x cannot be equal to zero."

00:05:23.988 --> 00:05:25.985
So g(x) is equal to x for any x

00:05:25.985 --> 00:05:28.284
as long as x is not equal to zero.

00:05:28.284 --> 00:05:31.291
Now these two function
definitions are equivalent.

00:05:31.291 --> 00:05:32.710
And we could even graph it.

00:05:33.551 --> 00:05:35.455
We could graph it, it's going to look,

00:05:35.455 --> 00:05:38.961
I'm gonna do a quick and
dirty version of this graph.

00:05:38.961 --> 00:05:40.784
It's gonna look something like, this.

00:05:40.784 --> 00:05:43.222
It's gonna have a slope of one,

00:05:43.222 --> 00:05:44.859
but it's gonna have a hole right at zero,

00:05:44.859 --> 00:05:47.019
'cause it's not defined at zero.

00:05:47.019 --> 00:05:48.958
So it's gonna look like this.

00:05:48.958 --> 00:05:53.625
So the domain here, the domain of g

00:05:53.625 --> 00:05:57.027
is going to be, "x is a
member of the real numbers"

00:05:57.027 --> 00:06:00.486
"such that x does not equal zero,"

00:06:00.486 --> 00:06:03.853
and the range is actually
going to be the same thing.

00:06:03.853 --> 00:06:06.407
The range here is going to be,

00:06:06.407 --> 00:06:10.761
we could say "f(x) is a
member of the real numbers"

00:06:10.761 --> 00:06:14.860
"such that f(x) does not equal zero."

00:06:14.860 --> 00:06:17.692
"f(x) does not equal zero."

00:06:17.692 --> 00:06:19.852
So the domain is all real
numbers except for zero,

00:06:19.852 --> 00:06:24.852
the range is all real
numbers except for zero.

00:06:24.983 --> 00:06:28.118
So the big takeaway here is
the range is all the pos...

00:06:28.118 --> 00:06:32.054
The set of all possible
outputs of your function.

00:06:32.054 --> 00:06:35.061
The domain is the set of all valid inputs

00:06:35.061 --> 00:06:36.472
into your function.


Interpreting graphs of proportional relationships (examples)

WEBVTT
Kind: captions
Language: en

00:00:00.214 --> 00:00:02.396
- [Voiceover] Let's get some
practice interpreting graphs

00:00:02.396 --> 00:00:04.509
of proportional relationships.

00:00:04.509 --> 00:00:07.458
This says the proportion
relationship between the distance

00:00:07.458 --> 00:00:10.786
driven and the amount of time driving

00:00:10.786 --> 00:00:13.549
shown in the following graph,
so we have the distance driven

00:00:13.549 --> 00:00:16.591
on the vertical axis, it's
measured in kilometers,

00:00:16.591 --> 00:00:19.447
then we have the time driving
and it's measured in hours

00:00:19.447 --> 00:00:21.606
along the horizontal axis.

00:00:21.606 --> 00:00:24.137
We can tell just visually
that this, indeed,

00:00:24.137 --> 00:00:27.109
is a proportional relationship,
how do we know that?

00:00:27.109 --> 00:00:30.139
Well, the point (0, 0) is on this graph,

00:00:30.139 --> 00:00:32.113
the graph goes through the origin.

00:00:32.113 --> 00:00:35.930
If we have zero time, then we
have zero distance, and we can

00:00:35.930 --> 00:00:38.536
also see that it's a line, then
it's a linear relationship.

00:00:38.536 --> 00:00:40.569
If you have a linear
relationship that goes through

00:00:40.569 --> 00:00:43.538
the origin, you're dealing with
a proportional relationship.

00:00:43.538 --> 00:00:46.411
You could also see that by
taking out some points here.

00:00:46.411 --> 00:00:49.189
Let's see, I'm just eyeballing,
so I'm gonna look at where

00:00:49.189 --> 00:00:52.510
does the graph kind of hit
a very well-defined point.

00:00:52.510 --> 00:00:56.331
Actually, point A right
over here, we see that when

00:00:56.331 --> 00:00:59.787
our time is five hours,

00:00:59.787 --> 00:01:03.191
our distance travelled or driven

00:01:04.563 --> 00:01:07.042
is 400 km.

00:01:08.882 --> 00:01:12.799
Then if we look at time,
this point right over here,

00:01:12.799 --> 00:01:17.233
when our time is 2.5,

00:01:17.233 --> 00:01:20.683
we see that our distance driven is 200 km.

00:01:20.683 --> 00:01:23.248
And notice, the ratio
between these variables

00:01:23.248 --> 00:01:25.912
at any one of these points is the same.

00:01:25.912 --> 00:01:30.028
400 divided by five is 80,

00:01:30.028 --> 00:01:33.410
and 200 divided by 2.5
is also going to be 80.

00:01:33.410 --> 00:01:35.705
Or if you wanna go the other way around,

00:01:35.705 --> 00:01:38.285
to go from time to distance,

00:01:40.265 --> 00:01:44.173
we're always multiplying by 80.

00:01:44.173 --> 00:01:48.004
In fact, we can say that
distance divided by time,

00:01:48.004 --> 00:01:52.366
our proportionality
constant is going to be 80.

00:01:52.366 --> 00:01:54.183
Or if we wanted to
include the units there,

00:01:54.183 --> 00:01:56.040
it might be a little
more obvious than dealing

00:01:56.040 --> 00:01:59.616
with the rate, distance is in
kilometers, time is in hours,

00:01:59.616 --> 00:02:03.537
80 km per hour,

00:02:03.537 --> 00:02:05.673
this is the rate at which we are driving.

00:02:05.673 --> 00:02:09.016
We are going at this
speed, 80 km per hour.

00:02:09.016 --> 00:02:13.031
This is also the proportionality constant.

00:02:14.011 --> 00:02:15.398
Anyway, with all of that out of the way,

00:02:15.398 --> 00:02:17.540
let's actually answer the questions.

00:02:17.540 --> 00:02:19.738
Which statements about the graph are true?

00:02:19.738 --> 00:02:21.131
Select all that apply.

00:02:21.131 --> 00:02:23.879
The vertical coordinate
of point A represents

00:02:23.879 --> 00:02:26.850
the distance driven in four hours.

00:02:26.850 --> 00:02:29.938
The vertical coordinate.

00:02:29.938 --> 00:02:33.820
So point A is at the coordinate (5, 400).

00:02:34.810 --> 00:02:37.968
The vertical coordinate
tells us how high to go up,

00:02:37.968 --> 00:02:39.802
how far to move in the vertical direction.

00:02:39.802 --> 00:02:41.683
That's gonna be the second
coordinate right over here,

00:02:41.683 --> 00:02:43.796
so this is the vertical coordinate.

00:02:43.796 --> 00:02:46.838
This right over here tells
us the distance we've driven

00:02:46.838 --> 00:02:49.996
in four hours, so yes, the
vertical coordinate of point A

00:02:49.996 --> 00:02:52.527
represents the distance
driven in four hours.

00:02:52.527 --> 00:02:56.656
We've driven 400 km, I like
that one, I'll check that one.

00:02:56.656 --> 00:03:00.698
The distance driven in one hour is 80 km.

00:03:03.455 --> 00:03:06.553
When you just try to eyeball
it off of the graph here,

00:03:06.553 --> 00:03:08.978
I see, after one hour, it
looks like it's a little bit

00:03:08.978 --> 00:03:13.217
at more than 75, so yeah, 80
seems like a reasonable one.

00:03:13.217 --> 00:03:14.918
But we see it even more
clearly when we look

00:03:14.918 --> 00:03:16.560
at the calculation that we did.

00:03:16.560 --> 00:03:20.491
If you take a distance,
any one of these distance,

00:03:20.491 --> 00:03:24.996
400 km over five hours,

00:03:26.156 --> 00:03:29.219
tells you that your rate

00:03:29.219 --> 00:03:32.329
is 80 km per hour,

00:03:32.329 --> 00:03:34.079
and you know this is going
to happen for any point

00:03:34.079 --> 00:03:36.447
on this line, they tell us it's
a proportional relationship,

00:03:36.447 --> 00:03:39.101
and we can tell, visually, it's
a proportional relationship.

00:03:39.101 --> 00:03:42.385
It goes through the
origin and it is a line.

00:03:42.385 --> 00:03:46.598
So the distance travelled in
one hour is going to be 80 km

00:03:47.698 --> 00:03:50.339
'cause we're going 80 km per hour.

00:03:50.339 --> 00:03:53.411
So in one hour, we're gonna go 80 km.

00:03:53.411 --> 00:03:56.058
So I like that choice as
well, of course, I won't pick

00:03:56.058 --> 00:03:59.541
none of the above because I
found two choices that I liked.

00:03:59.541 --> 00:04:02.235
Let's do another one
of these, this is fun.

00:04:02.235 --> 00:04:05.475
A grocery story sells cashews.

00:04:05.475 --> 00:04:08.763
The relationship between
weight and cost of cashews

00:04:08.763 --> 00:04:11.575
is shown in the following
graph, and once again,

00:04:11.575 --> 00:04:13.357
we see it is a proportional relationship.

00:04:13.357 --> 00:04:16.096
It goes through the
origin, and it is a line.

00:04:16.096 --> 00:04:19.579
Which statements about the graph are true?

00:04:19.579 --> 00:04:21.762
Select all that apply.

00:04:21.762 --> 00:04:24.416
The point (0, 0) shows the cost.

00:04:24.416 --> 00:04:27.783
It's $0 for 0 kg of cashews.

00:04:27.783 --> 00:04:30.758
Yeah, 0 kg of cashews, it's $0.

00:04:30.758 --> 00:04:32.702
Yeah, that makes sense.

00:04:32.702 --> 00:04:36.208
The point (2, 60), that's
this point right over here,

00:04:36.208 --> 00:04:40.371
shows the cost is $2 for 60 kg

00:04:40.371 --> 00:04:44.784
of cashews, the cost is
$2 for 60 kg of cashews.

00:04:45.814 --> 00:04:49.586
No, this is showing us
that the cost is $60

00:04:49.586 --> 00:04:52.396
the cost is $60

00:04:52.396 --> 00:04:56.563
for 2 kg of cashews.

00:04:57.633 --> 00:05:00.213
This axis right over here is
weight, actually, if you're

00:05:00.213 --> 00:05:02.694
measuring kilograms, you're
really talking about mass,

00:05:02.694 --> 00:05:05.504
but I don't wanna get too particular here.

00:05:05.504 --> 00:05:09.550
You have a mass of 2 kg,
I guess you could say

00:05:09.550 --> 00:05:12.035
the horizontal coordinate
tells you the mass.

00:05:12.035 --> 00:05:16.127
The vertical coordinate tells
you the cost, and that is $60.

00:05:16.127 --> 00:05:19.908
So it's $60 for 2 kg, not $2 for 60 kg.

00:05:19.908 --> 00:05:23.328
That would be a deal, if you could get,

00:05:23.328 --> 00:05:26.743
this is what a medium-sized grown man

00:05:26.743 --> 00:05:30.772
would weigh 60 kg, and
$2 for that much cashews,

00:05:30.772 --> 00:05:32.254
that would be an incredible deal.

00:05:32.254 --> 00:05:33.996
So you definitely wanna rule that one out.

00:05:33.996 --> 00:05:35.970
And I'm not gonna pick none
of the above because I already

00:05:35.970 --> 00:05:39.963
figured that I liked the
first choice up there.

00:05:39.963 --> 00:05:42.680
Let's just do one more,
just for good measure.

00:05:42.680 --> 00:05:45.652
An employee earns an hourly
wage shown in the graph below.

00:05:45.652 --> 00:05:48.325
Find the hourly wage.

00:05:48.325 --> 00:05:51.921
We see here in the horizontal
axis, we have hours worked,

00:05:51.921 --> 00:05:53.964
and in the vertical axis, earnings.

00:05:53.964 --> 00:05:57.686
We see it's a proportional relationship.

00:05:57.686 --> 00:06:00.237
And that makes sense, the ratio
between earnings and hours

00:06:00.237 --> 00:06:02.280
should always be constant,
if we're earning money

00:06:02.280 --> 00:06:04.227
at a constant rate, we see it visually,

00:06:04.227 --> 00:06:07.341
we go through the origin and
we're dealing with a line.

00:06:07.341 --> 00:06:09.647
So there's a bunch of ways
that we could look at it.

00:06:09.647 --> 00:06:12.185
We could say hours,

00:06:15.565 --> 00:06:17.961
hours, then dollars.

00:06:17.961 --> 00:06:20.707
After one hour, we've made $40.

00:06:20.707 --> 00:06:22.346
And just the fact, visually, that we know

00:06:22.346 --> 00:06:24.219
it's a proportional
relationship says we're making

00:06:24.219 --> 00:06:26.994
$40 per hour.

00:06:26.994 --> 00:06:29.156
If we take dollars divided by hours,

00:06:29.156 --> 00:06:32.442
I'll write out the units,

00:06:32.442 --> 00:06:35.705
40 dollars divided by one hour.

00:06:35.705 --> 00:06:40.261
This right over here
is 40 dollars per hour.

00:06:41.351 --> 00:06:43.917
You could view this as our
proportionality constant,

00:06:43.917 --> 00:06:46.004
or you could use the rate in
which we're earning money,

00:06:46.004 --> 00:06:48.070
$40 per hour, and you see that.

00:06:48.070 --> 00:06:51.622
Two hours, $80.

00:06:51.622 --> 00:06:54.246
80 divided by two is 40.

00:06:54.246 --> 00:06:57.706
Three hours, $120.

00:06:57.706 --> 00:07:00.210
120 divided by three is 40.

00:07:00.210 --> 00:07:02.648
Our proportionality constant is 40.

00:07:02.648 --> 00:07:06.050
That your earnings

00:07:07.550 --> 00:07:10.444
divided by hours

00:07:10.444 --> 00:07:12.909
is always going to be equal to 40,

00:07:12.909 --> 00:07:17.434
or you could say that your earnings

00:07:18.604 --> 00:07:21.194
is equal to, if you multiply
both sides by hours,

00:07:21.194 --> 00:07:24.561
is going to be equal
to 40 times your hours.

00:07:24.561 --> 00:07:26.760
Your earnings in dollars
is going to be equal

00:07:26.760 --> 00:07:29.517
to 40 times your hours.