Introduction to metabolism: anabolism and catabolism

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What I want
to do in this video is talk

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about the processes that
make all life as we know it,

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life as we know it,

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and at it's essence, we
can call this metabolism.

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Metabolism.

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And this is the taking
energy in different forms,

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breaking it down into its
more fundamental components,

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and then building it up in
ways that we would find useful,

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useful for energy, useful for structure,

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so that we can actually live our lives,

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we can grow, we can reproduce,

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we can respond to our surroundings.

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So as I just said, metabolism,

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and we're gonna go into a
bunch of examples of this.

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Metabolism at it's heart is
really two different processes.

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There's the breaking
down of the substances

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for energy or for
structure to getting back

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to the building blocks,

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and we call that catabolism.

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Catabolism.

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So this is the breaking down of things

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and then once we've broken down things,

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we're ready to rebuild them in ways

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that we would find useful,

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and we call this anabolism.

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Anabolism or anabolism.

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Anabolism, just like that.

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And one way to think about it is

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imagine that someone had
built something with Legos

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and you want to build
something with Legos.

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Well you could go to those Legos

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and you'd want to break it down,

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but not break it down too much.

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You wouldn't melt the plastic.

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You would break it down into
the individual Lego pieces

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and then you would build it back up

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into whatever shape that
you actually cared about.

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And you might not actually
have to even break it down

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all the way to the basic Lego pieces.

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There might be structures
in that first Lego castle

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that was constructed that
you might find useful.

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So let's just think about
how all of this gets started.

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And what's exciting is that
all of this got started,

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or gets started, from stars,
from fusion reactions in stars.

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And this right over here
is a picture of a star,

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and a star that we are very familiar with.

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This is the sun.

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But you may or may not
realize that the sun

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is only one of probably several stars

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that have been involved
in life as we know it.

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The sun is our most
direct source of energy

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for most of life as we know it.

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There are some bacteria

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and things that are able
to live off of vents

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at the bottom of the ocean
because of the heat created,

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but the sun is our
primary source of energy.

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But when I say that other
stars might have been involved,

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including dead stars that
existed billions of years ago,

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it's because the heavier
elements that we're composed of,

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or that are around us in the environment,

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the carbon, the oxygen,
we could just keep going,

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pretty much everything
other than hydrogen,

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it was constructed in fusion
reactions from hydrogen

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inside of stars.

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So we really are made up
of the remnants of stars.

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And so here we are, we're on Earth.

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Earth is all this condensed matter

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from four and a half billion years ago.

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Probably some nearby
supernova got all of this dust

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that was constructed in a previous star

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to coalesce in that way,
and you have radiation.

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You have energy from the sun.

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And once again, that energy's
coming from fusion reactions,

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and it's what fusing lighter
elements into heavier elements,

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so the sun is also constructing
more heavy elements,

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but that energy,

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that energy makes its way to the Earth.

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And you have organisms,

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like plants, that are
able to use that energy

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to construct the material,

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the food, we could say,

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that is eventually going
to get around to us.

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And so this process you may or
may not be familiar with it,

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this is photosynthesis.

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And we're going to go
into a lot more detail.

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Synthesis.

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And as the word implies,
photo, it's photosynthesis,

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it's making things out of light,

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and one thing I like to ask people

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when they are first
exposed to photosynthesis,

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is like okay, we can
see this grass growing

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or we can see this wheat growing,

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or we can see a tree growing,

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but where is that material coming from?

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And the most common answer is like,

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"Oh, somehow it's coming from the ground,"

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and there are some nutrients
that are coming from the ground

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but it's really all about fixing carbon,

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and you're going to hear about this a lot

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especially as we talk
about the carbon cycle.

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But you have carbon dioxide
primarily in the air,

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so you have carbon,

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you have, I'll just write it this way.

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So you have carbon dioxide in the air

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and what photosynthesis
allows these plants to do

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is take the carbon in that carbon dioxide

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and form bonds with it,

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turn it from its gas
form into solid forms,

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into glucose molecules,
and then use that glucose

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to build up cellulose and to
build out other forms of starch

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and whatever else it might be.

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So it's taking these
molecules in the air...

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I'll just draw them as these little...

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It's taking these molecules

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that are in the air,

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and it's using the energy
of the sun to fix them,

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to actually form bonds

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between the carbons and with other things.

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As we said, we're mostly
carbon and hydrogen

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and we have some oxygen in there,

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but we're able to form these structures.

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Now from there other living organisms,

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and this is a huge oversimplification,

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it could involve bacteria,

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it could involve all sorts of things.

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And just a reminder, you
know, that photosynthesis,

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it isn't just light and it
isn't just the carbon dioxide.

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It also involves the water
and we talk about that.

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So you also have water involved.

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You also have the water involved.

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So you have the carbon dioxide,

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so CO2, light from the sun, and water.

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These things are able to grow
and nutrients from the Earth.

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And then from that, you're
able to construct things like,

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well, you can directly go to these plants

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that are taking energy from the sun

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and construct things like bread

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or you have other animals
that will eat things

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like the grass, and then break them down

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in their own way and they
will be assisted by bacteria

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and then rebuild themselves up

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into a cow, into milk.

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And so what this cow is doing,

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it's metabolizing this grass.

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It's able to break it down, it's able to

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catabolize the various molecules

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in the grass and break them
down into building blocks

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that can then used to build up the cow,

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to build up milk, and whatever else.

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And you might be saying, "What
are these types of molecules

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"that we keep breaking down

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"and then building back up?"

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Well, you have carbohydrates.

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Carbohydrates,

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and you're going to see
most of the molecules

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that I'm about to talk about.

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Frankly, all of them on the
back of nutritional package

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because it tells you what's inside of it.

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What is your body going to metabolize

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when it eats that

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whatever's inside of the package?

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So carbohydrates, these
are either simple sugars

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like glucose or fructose,

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or it could be polymers of these sugars,

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polysaccharides.

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It could be starches

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made up of many, many elements of the...

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Or many, many multiples
of these simple sugars.

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We could be talking about lipids.

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Lipids.

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So fatty acids, we could be
talking about cholesterols.

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These are essential structures,

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and they're also essential for,

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well, various metabolic pathways

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inside of, well, all of life,

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or it could be proteins.

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It could be

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proteins made up of amino acids.

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Sometimes people say the reason
why you want the proteins

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is because it's made up
of these amino acids.

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So you could break down these proteins

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and get the amino acids
and then build it up

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into new proteins.

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Proteins and amino acids.

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And all of these things, they are found

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in things like in the foods,

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in the foods we eat,

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which we will then consume,

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we will then metabolize.

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We will first catabolize
them and break them down

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and then we will use those building blocks

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to build it back up.

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And at the end of the day,
these provide the structures

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that make our bodies what they are.

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They allows us to interact
with our environment

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and they provide energy,

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and that energy, at the end of the day,

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the true molecular
currency for that energy

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is a molecule called ATP.

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That's a molecule of ATP right over there.

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Adenosine triphosphate.

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And the key for ATP, why it
is the molecular currency

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for energy, are the
three phosphate groups.

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So it has these three phosphate groups

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and we have a whole video on it,

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or a whole series of videos on it really,

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but the main key

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is that there's a lot of energy,

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especially beween I guess
on the last phosphate group.

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And this energy and that bond

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as the phosphate group
breaks off it can release it

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to provide all sorts of life mechanisms

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including being able to metabolize things.

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So ATP right over here,
this is the currency

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of energy in life as we know it.

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And it's actually an interesting molecule

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because it's essentially,
it's a piece of...

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If you were to just take this
section of it right over here,

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it really is, it looks
like a piece of RNA.

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It looks like you're taking a...

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You have ribose, you have the adenine,

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you have a phosphate group,

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and it can actually be used
to construct things like RNA

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and even DNA

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beyond just being the
molecular currency of energy.

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So it's fun to see how these
pieces all fit together,

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how they can be broken up,

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and you see these patterns
over and over again

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in biological systems, and
these biological systems

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are really just made up of breaking down

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and building back up
carbohydrates, lipids,

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fatty acids, cholesterols, other things,

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and proteins/amino acids.

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So this whole thing, you
know, look around you,

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look at your day, look at
the things you're consuming

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and you'll see this pattern
over and over and over again

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and it's all a little mind-blowing,

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because at the end of the day,

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the energy is coming from
the fusion reaction in stars

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and all of the heavy elements around you

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or even the not so heavy elements,

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all of the carbon, the oxygen,

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all of the really non-hydrogen elements,

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they've been made inside
of fusion reactions

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in stars that might have
existed billions and billions

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and billions of years ago,

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so hopefully that makes you
feel a little bit more connected

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to the universe.


How to interpret an expression with function notation

WEBVTT
Kind: captions
Language: en

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Let P of t denote the number of people P

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at the Copacabana beach in Rio de
Janeiro, t

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hours past midnight on a specific day.
What does the statement P of 5

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less than P of 9 mean? So before I
read these, let's think about this.

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So this is the number of people five
hours past midnight on a specific day. So

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this is saying that the number of people
five hours past midnight

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is less than the number of people nine
hours past midnight

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at Copacabana beach. So the number of people
five hours past midnight

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is less than the number people nine
hours past midnight. Let's get into these choices.

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The time at which there were five people
at the beach

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is earlier than the time there were nine
people.

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Who know? --  the five and the nine aren't
the number of people

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This is the time. This is the input into
the function. The

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input into the function is time. This
gives people as a function of time. You

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give a time.

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You give a time. This function P, it outputs

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P of t, which is equal to the number of
people

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number of people on the beach

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at time, at time

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t. So this five -- this is the input. This is
the time five hours after

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I shouldn't say time t. I guess you
could say

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a.m. time -- t hours past midnight at

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t hours past midnight, hours past

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past midnight. midnight.

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So we can definitely rule this out. This
isn't talking about five people or nine

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people. This is talking about

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five hours after midnight is input and
how many people we have five hours after

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midnight

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is how many people we have nine hours
after midnight. 5 a.m.

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is an earlier time the 9 a.m. Well,

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that is a true statement but that's not
what this statement is

00:01:56.590 --> 00:01:59.950
saying. It's not -- this isn't giving us
information which is no from,

00:01:59.950 --> 00:02:03.729
I guess, how we think about time or how
we measure time.

00:02:03.729 --> 00:02:06.810
the 5 a.m. is earlier than 9 a.m. but
that's not what this statement is trying

00:02:06.810 --> 00:02:07.320
to say.

00:02:07.320 --> 00:02:08.890
So rule that out.

00:02:08.890 --> 00:02:11.440
There were more people at the beach at
9 a.m.

00:02:11.440 --> 00:02:14.690
So to be clear, people at the beach at
9 a.m.

00:02:14.690 --> 00:02:17.890
that's that right over here -- than there were

00:02:17.890 --> 00:02:22.870
-- than there were at 5 a.m. -- than there were
at 5 a.m. Well, that's right over there.

00:02:22.870 --> 00:02:27.319
And we see -- we see this quantity here
is greater than those magenta quantity

00:02:27.319 --> 00:02:30.140
is indeed greater than this green quantity.
The green quantity

00:02:30.140 --> 00:02:33.940
is less than the magenta quantity, so we
would definitely pick that one.


Shifting and scaling parabolas

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.820
Here I've drawn the
most classic parabola,

00:00:02.820 --> 00:00:05.040
y is equal to x squared.

00:00:05.040 --> 00:00:09.710
And what I want to do is think
about what happens-- or how can

00:00:09.710 --> 00:00:12.660
I go about shifting
this parabola.

00:00:12.660 --> 00:00:14.860
And so let's think about
a couple of examples.

00:00:14.860 --> 00:00:16.940
So let's think about
the graph of the curve.

00:00:16.940 --> 00:00:19.790
This is y is equal to x squared.

00:00:19.790 --> 00:00:23.160
Let's think about what
the curve of y minus k

00:00:23.160 --> 00:00:24.550
is equal to x squared.

00:00:24.550 --> 00:00:26.770
What would this look like?

00:00:26.770 --> 00:00:29.330
Well, right over here, we
see when x is equal to 0,

00:00:29.330 --> 00:00:31.380
x squared is equal to 0.

00:00:31.380 --> 00:00:33.120
That's this yellow curve.

00:00:33.120 --> 00:00:36.470
So x squared is equal to y,
or y is equal to x squared.

00:00:36.470 --> 00:00:38.940
But for this one, x
squared isn't equal to y.

00:00:38.940 --> 00:00:41.270
It's equal to y minus k.

00:00:41.270 --> 00:00:44.090
So when x equals a
0, and we square it,

00:00:44.090 --> 00:00:45.770
0 squared doesn't get us to y.

00:00:45.770 --> 00:00:47.930
It gets us to y minus k.

00:00:47.930 --> 00:00:50.860
So this is going to
be k less than y.

00:00:50.860 --> 00:00:52.870
Or another way of thinking
about it, this is 0.

00:00:52.870 --> 00:00:57.590
If it's k less than y, y must
be at k, wherever k might be.

00:00:57.590 --> 00:01:00.460
So y must be at k,
right over there.

00:01:00.460 --> 00:01:02.640
So at least for this
point, it had the effect

00:01:02.640 --> 00:01:07.120
of shifting up the y value by k.

00:01:07.120 --> 00:01:09.670
And that's actually true
for any of these values.

00:01:09.670 --> 00:01:12.690
So let's think about x
being right over here.

00:01:12.690 --> 00:01:15.796
For this yellow curve,
you square this x value,

00:01:15.796 --> 00:01:16.670
and you get it there.

00:01:16.670 --> 00:01:18.490
And it's clearly not
drawn to scale the way

00:01:18.490 --> 00:01:19.906
that I've done it
right over here.

00:01:22.170 --> 00:01:23.930
But now for this
curve right over here,

00:01:23.930 --> 00:01:25.085
x squared doesn't cut it.

00:01:25.085 --> 00:01:26.950
It only gets you to y minus k.

00:01:26.950 --> 00:01:31.340
So y must be k higher than this.

00:01:31.340 --> 00:01:34.850
So this is y minus k. y
must be k higher than this.

00:01:34.850 --> 00:01:37.710
So y must be right over here.

00:01:37.710 --> 00:01:41.540
So this curve is essentially
this blue curve shifted up

00:01:41.540 --> 00:01:44.410
by k.

00:01:44.410 --> 00:01:47.950
So making it y minus k is equal
to x squared shifted it up

00:01:47.950 --> 00:01:48.770
by k.

00:01:48.770 --> 00:01:52.450
Whatever value this
is, shift it up by k.

00:01:52.450 --> 00:01:56.230
This distance is a constant
k, the vertical distance

00:01:56.230 --> 00:01:59.090
between these two parabolas.

00:01:59.090 --> 00:02:02.450
And I'll try to draw
it as cleanly as I can.

00:02:02.450 --> 00:02:05.370
This vertical distance
is a constant k.

00:02:05.370 --> 00:02:08.270
Now let's think about shifting
in the horizontal direction.

00:02:08.270 --> 00:02:12.710
Let's think about what happens
if I were to say y is equal to,

00:02:12.710 --> 00:02:19.070
not x squared, but
x minus h squared.

00:02:19.070 --> 00:02:20.070
So let's think about it.

00:02:20.070 --> 00:02:24.230
This is the value you would get
for y when you just square 0.

00:02:24.230 --> 00:02:26.160
You get y is equal to 0.

00:02:26.160 --> 00:02:28.830
How do we get y
equals 0 over here?

00:02:28.830 --> 00:02:32.470
Well, this quantity right
over here has to be 0.

00:02:32.470 --> 00:02:36.550
So x minus h has to be 0,
or x has to be equal to h.

00:02:36.550 --> 00:02:38.280
So let's say that h
is right over here.

00:02:38.280 --> 00:02:40.940
So x has to be equal to h.

00:02:40.940 --> 00:02:43.990
So one way to think about
it is, whatever value you

00:02:43.990 --> 00:02:46.890
were squaring here
to get your y,

00:02:46.890 --> 00:02:49.590
you now have to have
an h higher value

00:02:49.590 --> 00:02:50.810
to square that same thing.

00:02:50.810 --> 00:02:52.601
Because you're going
to subtract h from it.

00:02:52.601 --> 00:02:58.320
Just to get to 0,
x has to equal h.

00:02:58.320 --> 00:03:02.070
Here, if you wanted to square
1, x just had to be equal to 1.

00:03:02.070 --> 00:03:04.280
So here, let's just say,
for the sake of argument,

00:03:04.280 --> 00:03:07.080
that this is x is equal to 1.

00:03:07.080 --> 00:03:10.310
And this is 1 squared,
clearly not drawn to scale.

00:03:10.310 --> 00:03:12.170
So that would be 1, as well.

00:03:12.170 --> 00:03:15.590
But now to square 1, we don't
have to just get x equals 1.

00:03:15.590 --> 00:03:17.260
x has to be h plus 1.

00:03:17.260 --> 00:03:18.970
It has to be 1 higher than h.

00:03:18.970 --> 00:03:22.700
It has to be h plus 1 to
get to that same point.

00:03:22.700 --> 00:03:25.390
So you see the net
effect is that instead

00:03:25.390 --> 00:03:28.470
of squaring just x,
but squaring x minus h,

00:03:28.470 --> 00:03:31.480
we shifted the
curve to the right.

00:03:31.480 --> 00:03:36.330
So the curve-- let me do this in
this purple color, this magenta

00:03:36.330 --> 00:03:38.070
color-- will look like this.

00:03:38.070 --> 00:03:41.270
We shifted it to the right.

00:03:41.270 --> 00:03:48.620
And we shifted it
to the right by h.

00:03:48.620 --> 00:03:51.040
Now let's think of another
thought experiment.

00:03:51.040 --> 00:03:56.120
Let's imagine that-- let's
think about the curve

00:03:56.120 --> 00:04:01.390
y is equal to
negative x squared.

00:04:01.390 --> 00:04:03.415
Well, now whatever the
value of x squared is,

00:04:03.415 --> 00:04:05.040
we're going to take
the negative of it.

00:04:05.040 --> 00:04:07.250
So here, no matter what
x we took, we squared it.

00:04:07.250 --> 00:04:08.640
We get a positive value.

00:04:08.640 --> 00:04:10.670
Now we're always going
to get a negative value

00:04:10.670 --> 00:04:12.687
once we multiply it
times a negative 1.

00:04:12.687 --> 00:04:14.020
So it's going to look like this.

00:04:14.020 --> 00:04:15.530
It's going to be a
mirror image of y

00:04:15.530 --> 00:04:20.010
equals x squared reflected
over the horizontal axis.

00:04:20.010 --> 00:04:22.040
So it's going to look
something like that.

00:04:22.040 --> 00:04:26.480
So that's y is equal
to negative x squared.

00:04:26.480 --> 00:04:28.890
And now let's just imagine
scaling it even more.

00:04:28.890 --> 00:04:38.520
What would y equal
negative 2x squared?

00:04:38.520 --> 00:04:40.340
Well, actually, let
me do two things.

00:04:40.340 --> 00:04:43.220
So what would y equals
2x squared look like?

00:04:43.220 --> 00:04:45.060
So let's just take
the positive version,

00:04:45.060 --> 00:04:47.671
so y equals 2x squared.

00:04:47.671 --> 00:04:49.170
Well, now as we
square things, we're

00:04:49.170 --> 00:04:50.336
going to multiply them by 2.

00:04:50.336 --> 00:04:51.730
So it's going to
increase faster.

00:04:51.730 --> 00:04:53.480
So it's going to look
something like this.

00:04:53.480 --> 00:04:56.269
It's going to be
narrower and steeper.

00:04:56.269 --> 00:04:57.810
So it might look
something like this.

00:04:57.810 --> 00:04:59.880
And once again, I'm just
giving you the idea.

00:04:59.880 --> 00:05:01.890
I haven't really
drawn this to scale.

00:05:01.890 --> 00:05:06.110
So increasing it by a factor
will make it increase faster.

00:05:06.110 --> 00:05:12.000
If we did y equals
negative 2x squared,

00:05:12.000 --> 00:05:14.501
well, then it's going to get
negative faster on either side.

00:05:14.501 --> 00:05:16.250
So it's going to look
something like this.

00:05:16.250 --> 00:05:18.570
It's going to be the mirror
image of what I just drew.

00:05:18.570 --> 00:05:22.870
So it's going to be a narrower
parabola just like that.

00:05:22.870 --> 00:05:25.670
And similarly-- and I know that
my diagram is getting really

00:05:25.670 --> 00:05:27.810
messy right now--
but just remember

00:05:27.810 --> 00:05:29.580
we started with y
equals x squared,

00:05:29.580 --> 00:05:31.830
which is this curve
right over here.

00:05:31.830 --> 00:05:35.880
What happens if we did
y equals 1/2 x squared?

00:05:35.880 --> 00:05:37.760
I'm running out of
colors, as well.

00:05:37.760 --> 00:05:40.180
If we did y equals
1/2 x squared,

00:05:40.180 --> 00:05:42.180
well, then the thing's
going to increase slower.

00:05:42.180 --> 00:05:45.220
It's going to look the same,
but it's going to open up wider.

00:05:45.220 --> 00:05:46.580
It's going to increase slower.

00:05:46.580 --> 00:05:50.160
It's going to look
something like this.

00:05:50.160 --> 00:05:52.270
So this hopefully
gives you a sense

00:05:52.270 --> 00:05:55.360
of how we can shift
parabolas around.

00:05:55.360 --> 00:06:04.190
So for example, if I have-- and
I'm doing a very rough drawing

00:06:04.190 --> 00:06:05.940
here to give you the
general idea of what

00:06:05.940 --> 00:06:08.121
we're talking about.

00:06:08.121 --> 00:06:13.050
So if this is y
equals x squared,

00:06:13.050 --> 00:06:15.360
so that's the graph
of y equals x squared.

00:06:22.500 --> 00:06:25.240
Let me do this in a color
that I haven't used yet--

00:06:25.240 --> 00:06:34.991
the graph of y minus k is equal
to A times x minus h squared

00:06:34.991 --> 00:06:36.240
will look something like this.

00:06:36.240 --> 00:06:39.440
Instead of the vertex
being at 0, 0, the vertex--

00:06:39.440 --> 00:06:40.899
or the lowest, or
I guess you could

00:06:40.899 --> 00:06:42.440
say the minimum or
the maximum point,

00:06:42.440 --> 00:06:44.570
the extreme point in the
parabola, this point right

00:06:44.570 --> 00:06:47.069
over here, would be the maximum
point for a downward opening

00:06:47.069 --> 00:06:49.856
parabola, a minimum point for
an upward opening parabola--

00:06:49.856 --> 00:06:50.980
that's going to be shifted.

00:06:50.980 --> 00:06:56.630
It's going to be shifted
by h to the right and k up.

00:06:56.630 --> 00:06:58.740
So its vertex is going
to be right over here.

00:06:58.740 --> 00:07:03.674
And it's going to be scaled
by A. So if A is equal to 1,

00:07:03.674 --> 00:07:04.840
it's going to look the same.

00:07:04.840 --> 00:07:07.770
It's going to have
the same opening.

00:07:07.770 --> 00:07:09.310
So that's A equals 1.

00:07:09.310 --> 00:07:14.440
If A is greater than 1, it's
going to be steeper, like this.

00:07:14.440 --> 00:07:17.530
If A is less than 1
but greater than 0,

00:07:17.530 --> 00:07:20.240
it's just going to be
wider opening, like that.

00:07:23.370 --> 00:07:28.190
Actually, if A is 0, then it
just turns into a flat line.

00:07:28.190 --> 00:07:31.410
And then if A is negative
but less than negative 1,

00:07:31.410 --> 00:07:34.172
it's kind of a broad-opening
thing like that.

00:07:34.172 --> 00:07:36.100
Or I should say greater
than negative 1.

00:07:36.100 --> 00:07:37.580
If it's between
0 and negative 1,

00:07:37.580 --> 00:07:39.600
it will be a broad-opening
thing like that.

00:07:39.600 --> 00:07:41.580
At negative 1, it'll
look like a reflection

00:07:41.580 --> 00:07:44.110
of our original curve.

00:07:44.110 --> 00:07:49.287
And then if A is less
than negative 1--

00:07:49.287 --> 00:07:50.870
so it's even more
negative-- then it's

00:07:50.870 --> 00:07:54.130
going to be even a
steeper parabola that

00:07:54.130 --> 00:07:55.497
might look like that.

00:07:55.497 --> 00:07:57.080
So hopefully that
gives you a good way

00:07:57.080 --> 00:08:00.640
of how to shift and
scale parabolas.


Graphing a parabola with a table of values

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.420
We are asked to
graph the function

00:00:03.420 --> 00:00:08.370
f of x is equal to
negative 3x squared plus 8.

00:00:08.370 --> 00:00:11.420
So we'll do this by essentially
trying out different points

00:00:11.420 --> 00:00:13.400
for x, and seeing what
we get for f of x,

00:00:13.400 --> 00:00:14.720
and then graphing it.

00:00:14.720 --> 00:00:16.410
But the first question
I have for you

00:00:16.410 --> 00:00:18.810
is, just looking at
this function definition

00:00:18.810 --> 00:00:23.690
for f of x, what type
of graph will this be?

00:00:23.690 --> 00:00:24.710
Will just be a line?

00:00:24.710 --> 00:00:25.970
Will this be a parabola?

00:00:25.970 --> 00:00:28.580
Will this be something else,
a circle, something else,

00:00:28.580 --> 00:00:31.220
maybe something else
bizarre or strange?

00:00:31.220 --> 00:00:35.180
Well, this is pretty clearly
going to be a parabola here.

00:00:35.180 --> 00:00:37.400
You have the
function is defined,

00:00:37.400 --> 00:00:41.090
it's negative 3x squared, so
you have this second degree

00:00:41.090 --> 00:00:42.130
term here.

00:00:42.130 --> 00:00:44.880
You don't have any x thirds or
x to the fourths or anything

00:00:44.880 --> 00:00:48.084
else bizarre, so this is
going to be a parabola.

00:00:48.084 --> 00:00:50.000
Now, the other thing
that we could think about

00:00:50.000 --> 00:00:53.770
is whether the parabola is
going to open up like that

00:00:53.770 --> 00:00:55.966
or whether it's going
to open down like that.

00:00:55.966 --> 00:00:57.840
And just looking at this
function definition,

00:00:57.840 --> 00:01:00.660
do you have any intuition of
whether it's going to open up

00:01:00.660 --> 00:01:03.680
or it's going to open down?

00:01:03.680 --> 00:01:05.180
Well, if you look
at the coefficient

00:01:05.180 --> 00:01:08.300
on the x squared
term, the negative 3,

00:01:08.300 --> 00:01:11.930
that tells you that this
parabola is going to open down.

00:01:11.930 --> 00:01:14.550
It's going to open down.

00:01:14.550 --> 00:01:16.240
So with that
intuition now that we

00:01:16.240 --> 00:01:17.330
know it's going
to be a parabola,

00:01:17.330 --> 00:01:19.288
we know it's going to
open down, let's actually

00:01:19.288 --> 00:01:21.160
try to graph the thing.

00:01:21.160 --> 00:01:24.220
And let me draw some axes here.

00:01:24.220 --> 00:01:30.340
So let's say that this is my
x-axis, so that's my x-axis.

00:01:30.340 --> 00:01:36.317
And then let's make this right
over here, this is my y-axis.

00:01:39.450 --> 00:01:41.220
And let me make
a table of values

00:01:41.220 --> 00:01:45.000
and see what values
f of x takes on.

00:01:45.000 --> 00:01:49.640
So on one column, I'm
going to do my values

00:01:49.640 --> 00:01:51.770
for x and over on
the right I'm going

00:01:51.770 --> 00:01:53.354
to do my values for f of x.

00:01:53.354 --> 00:01:54.770
And then we can
plot these things.

00:01:54.770 --> 00:01:56.728
And actually I want to
take all of these values

00:01:56.728 --> 00:01:58.860
before I draw the
scale on these axes,

00:01:58.860 --> 00:02:01.650
so I know what might be
an appropriate scale.

00:02:01.650 --> 00:02:03.790
So I'm just going to
try a bunch of values.

00:02:03.790 --> 00:02:09.690
So let's try first what happens
when x is equal to negative 2.

00:02:09.690 --> 00:02:11.155
So when x is equal
to negative 2--

00:02:11.155 --> 00:02:12.530
and I'm just
picking numbers that

00:02:12.530 --> 00:02:15.140
will be relatively
easy to compute.

00:02:15.140 --> 00:02:17.170
When x is equal to
negative 2 what's f of x?

00:02:17.170 --> 00:02:19.850
Well, f of x is going
to be negative 3,

00:02:19.850 --> 00:02:25.700
this negative 3,
times negative 2

00:02:25.700 --> 00:02:30.230
squared plus 8, which is going
to be equal to, let's see.

00:02:30.230 --> 00:02:33.800
Negative 2 squared
is 4, positive 4,

00:02:33.800 --> 00:02:37.040
then we multiply that times
a negative 3, which gives us

00:02:37.040 --> 00:02:43.060
negative 12 plus 8,
gives us negative 4.

00:02:43.060 --> 00:02:44.510
Let's try another point.

00:02:44.510 --> 00:02:47.830
Let's see what happens when
x is equal to negative 1.

00:02:47.830 --> 00:02:50.920
What do we get for f of x then?

00:02:50.920 --> 00:02:54.820
Well, f of x is going to be
negative 3 times negative 1

00:02:54.820 --> 00:02:57.430
squared plus 8.

00:02:57.430 --> 00:03:02.160
So that's going to be-- see
negative 1 squared is just 1,

00:03:02.160 --> 00:03:04.040
and then that times
negative 3 is negative 3.

00:03:04.040 --> 00:03:07.880
Negative 3 plus 8 is 5.

00:03:07.880 --> 00:03:12.410
Now, what does f of x
equal when x is equal to 0?

00:03:12.410 --> 00:03:14.540
Well, this is pretty
easy to compute.

00:03:14.540 --> 00:03:19.510
When x is equal to
0, you get negative 3

00:03:19.510 --> 00:03:22.820
times 0 squared,
which is equal to--

00:03:22.820 --> 00:03:26.420
and we could write that
either way-- negative 3 times

00:03:26.420 --> 00:03:28.880
0 squared plus 8.

00:03:28.880 --> 00:03:34.560
Well, this just simplifies to 0,
and so you're just left with 8.

00:03:34.560 --> 00:03:37.480
Now, let's see what happens
when x is equal to 1.

00:03:37.480 --> 00:03:41.090
What do we get for f of x?

00:03:41.090 --> 00:03:47.520
Well, it's going to be negative
3 times 1 squared plus 8.

00:03:47.520 --> 00:03:52.840
So 1 squared is just 1,
negative 3 plus 8 is equal to 5.

00:03:52.840 --> 00:03:58.987
And then finally, what do we get
when x is equal to positive 2?

00:03:58.987 --> 00:04:01.070
What does f of x equal,
or another way of thinking

00:04:01.070 --> 00:04:03.592
about it, what is f of 2?

00:04:03.592 --> 00:04:05.080
Well, let's think about it.

00:04:05.080 --> 00:04:09.820
You get negative 3
times 2 squared plus 8.

00:04:09.820 --> 00:04:13.740
2 squared is 4, times
negative 3 is negative 12,

00:04:13.740 --> 00:04:16.320
plus 8 is equal to negative 4.

00:04:16.320 --> 00:04:18.019
So let's see if
we can plot this.

00:04:18.019 --> 00:04:21.680
So the x values that I picked go
from negative 2 to positive 2.

00:04:21.680 --> 00:04:25.505
So let's make this
negative 2, negative 1.

00:04:25.505 --> 00:04:27.080
This is 0.

00:04:27.080 --> 00:04:31.210
This is positive 1, and
that could be positive 2.

00:04:31.210 --> 00:04:35.330
And then our f of x values,
or we are essentially

00:04:35.330 --> 00:04:38.890
graphing y is equal to f of x,
so I can even say this is going

00:04:38.890 --> 00:04:40.610
to be the graph of y
is equal to f of x.

00:04:40.610 --> 00:04:44.750
Our f of x values take on
things between negative 4

00:04:44.750 --> 00:04:47.120
and positive 8.

00:04:47.120 --> 00:04:48.970
Let me try to draw that.

00:04:48.970 --> 00:04:56.150
So if this is positive 8, that's
positive 8, that is positive 4,

00:04:56.150 --> 00:04:58.640
and this is negative 4.

00:04:58.640 --> 00:05:00.770
This is negative 4.

00:05:00.770 --> 00:05:04.110
And if that's positive 4,
then this is positive 6,

00:05:04.110 --> 00:05:06.840
and then that right
there is 5, that is 7,

00:05:06.840 --> 00:05:10.820
this would be 2 that would be
3, and then that would be 1.

00:05:10.820 --> 00:05:13.210
Now, let's graph the points.

00:05:13.210 --> 00:05:17.370
When x is negative 2,
f of x is negative 4.

00:05:17.370 --> 00:05:18.840
And actually I
could say, this is

00:05:18.840 --> 00:05:21.260
the y is equal to f of x-axis.

00:05:21.260 --> 00:05:22.470
I'm going to plot f of x.

00:05:22.470 --> 00:05:24.360
I'm graphing, and
this is going to be

00:05:24.360 --> 00:05:27.380
the graph of y is
equal to this function.

00:05:27.380 --> 00:05:31.390
So let's graph
negative 2, negative 4.

00:05:31.390 --> 00:05:34.620
So that gets us,
when x is negative 2,

00:05:34.620 --> 00:05:36.905
f of x is negative 4.

00:05:36.905 --> 00:05:38.900
It's right over there.

00:05:38.900 --> 00:05:46.570
When x is equal to negative
1, f of x is equal to 5.

00:05:46.570 --> 00:05:51.360
And we're saying that y is
equal to f of x in this context.

00:05:51.360 --> 00:05:56.400
When x is 0, f of x or y-- I
could even write over here,

00:05:56.400 --> 00:05:59.210
I could say, y is
equal to f of x.

00:05:59.210 --> 00:06:03.080
When x is equal to
0, our f of x is 8.

00:06:03.080 --> 00:06:06.370
x is 0, f of x is 8.

00:06:06.370 --> 00:06:10.060
When x is 1, f of x is 5.

00:06:10.060 --> 00:06:14.090
When x is 1, y
equals f of x is 5.

00:06:14.090 --> 00:06:17.100
And then finally, when
x is equal to 2, f of x

00:06:17.100 --> 00:06:18.890
is equal to negative 4.

00:06:18.890 --> 00:06:22.290
So 2, negative 4,
gets us right there.

00:06:22.290 --> 00:06:23.760
And now we can connect the dots.

00:06:23.760 --> 00:06:26.160
We know this is going
to be a parabola.

00:06:26.160 --> 00:06:28.760
And I will do it in blue.

00:06:28.760 --> 00:06:31.730
So my best attempt-- I like
to draw it as a dotted line,

00:06:31.730 --> 00:06:35.750
just because it's
easier to not mess up--

00:06:35.750 --> 00:06:38.910
so it would look
something like that.

00:06:38.910 --> 00:06:41.536
And it keeps on
going just like that,

00:06:41.536 --> 00:06:46.180
and then I can actually make the
line a little bit more solid.

00:06:46.180 --> 00:06:48.890
So we see that we
definitely got a parabola,

00:06:48.890 --> 00:06:51.930
and just as our
intuition told us,

00:06:51.930 --> 00:06:53.660
our ability to inspect
the coefficient

00:06:53.660 --> 00:07:00.500
on the x-squared term told us,
that our parabola is indeed

00:07:00.500 --> 00:07:02.473
opening downwards.


Determining mistakes in steps example

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.290
Meredith is solving the
following problem for homework,

00:00:03.290 --> 00:00:07.730
2 times the quantity x plus
4 squared is equal to 242.

00:00:07.730 --> 00:00:09.980
She completes the problem
as seen in the steps below,

00:00:09.980 --> 00:00:11.772
and they give us the
steps right over here.

00:00:11.772 --> 00:00:13.313
When she gets to
school the next day,

00:00:13.313 --> 00:00:15.470
her teacher tells her that
the answer's x equals 7

00:00:15.470 --> 00:00:17.430
and x equals negative 15.

00:00:17.430 --> 00:00:19.400
She only got x equals 7 here.

00:00:19.400 --> 00:00:21.569
In what step did
she make an error?

00:00:21.569 --> 00:00:23.110
So this first step
right here-- and I

00:00:23.110 --> 00:00:24.200
encourage you to
pause this video

00:00:24.200 --> 00:00:25.783
and try to figure
this out on your own

00:00:25.783 --> 00:00:28.020
before I work through it.

00:00:28.020 --> 00:00:29.270
So this first step, let's see.

00:00:29.270 --> 00:00:31.630
She got rid of this 2 by
dividing the left-hand side

00:00:31.630 --> 00:00:33.440
by 2, and she
appropriately divided.

00:00:33.440 --> 00:00:34.530
Well, you can't just
do that to one side.

00:00:34.530 --> 00:00:35.905
You have to do
that to both sides

00:00:35.905 --> 00:00:37.390
in order to hold this equality.

00:00:37.390 --> 00:00:40.280
So she divided 242 by 2 as
well, so that is correct.

00:00:40.280 --> 00:00:41.990
Step one makes sense.

00:00:41.990 --> 00:00:43.960
And then she just
wanted-- instead

00:00:43.960 --> 00:00:45.440
of this being an
x plus 4 squared,

00:00:45.440 --> 00:00:47.020
she wanted it to be an x plus 4.

00:00:47.020 --> 00:00:49.520
So she attempted to take the
square root of both sides.

00:00:49.520 --> 00:00:50.760
She said hey, look.

00:00:50.760 --> 00:00:53.610
The square root of x plus
4 squared is x plus 4,

00:00:53.610 --> 00:00:57.500
and the square
root of 121 is 11.

00:00:57.500 --> 00:01:00.790
And this is where she made
a small but very, very,

00:01:00.790 --> 00:01:03.350
very, very important mistake.

00:01:03.350 --> 00:01:07.150
Because if something
squared is equal to 121,

00:01:07.150 --> 00:01:08.580
that means that
something could be

00:01:08.580 --> 00:01:13.030
the positive or negative
square root of 121.

00:01:13.030 --> 00:01:15.560
This thing that we're
squaring could be positive 11,

00:01:15.560 --> 00:01:18.320
because positive
11 squared is 121,

00:01:18.320 --> 00:01:21.900
or x plus 4 right over
here could be negative 11,

00:01:21.900 --> 00:01:25.669
because negative 11
squared is also 121.

00:01:25.669 --> 00:01:27.710
So this right over here,
this should say x plus 4

00:01:27.710 --> 00:01:31.790
is equal to the positive or
negative square root of 11,

00:01:31.790 --> 00:01:34.760
and so that's why she missed out
on one of the solutions right

00:01:34.760 --> 00:01:35.260
over here.

00:01:35.260 --> 00:01:36.930
So she messed up in step two.

00:01:36.930 --> 00:01:40.970
She should have taken
the positive and negative

00:01:40.970 --> 00:01:42.300
square root.

00:01:42.300 --> 00:01:45.130
So, we got that right.


Order of steps exercise example

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.800
Use the cards below to create
a list of steps in order

00:00:03.800 --> 00:00:05.980
that will solve the
following equation.

00:00:05.980 --> 00:00:10.020
3 times x plus 6
squared is equal to 75.

00:00:10.020 --> 00:00:12.340
And I encourage you to
pause this video now and try

00:00:12.340 --> 00:00:13.609
to figure it out on your own.

00:00:13.609 --> 00:00:15.650
Figure out which of these
steps and in what order

00:00:15.650 --> 00:00:19.290
you would do to
solve for x here.

00:00:19.290 --> 00:00:21.452
So I'm assuming
you've given it a go.

00:00:21.452 --> 00:00:23.160
So let's try to work
through it together.

00:00:23.160 --> 00:00:25.050
And first, let me just
rewrite the equation.

00:00:25.050 --> 00:00:29.820
So we have 3 times the
quantity x plus 6 squared

00:00:29.820 --> 00:00:33.080
is equal to 75.

00:00:33.080 --> 00:00:36.350
So what I want to do is I want
to isolate the x plus 6 squared

00:00:36.350 --> 00:00:37.390
on the left-hand side.

00:00:37.390 --> 00:00:38.931
Or another way of
thinking about it--

00:00:38.931 --> 00:00:41.000
I don't want this
3 here anymore.

00:00:41.000 --> 00:00:42.670
So how would I
get rid of that 3?

00:00:42.670 --> 00:00:46.060
Well, I could divide
the left-hand side by 3.

00:00:46.060 --> 00:00:48.877
But if I do that to only
one side of the equation,

00:00:48.877 --> 00:00:49.960
it won't be equal anymore.

00:00:49.960 --> 00:00:52.260
These two things in yellow
were equal to each other.

00:00:52.260 --> 00:00:54.274
If I want the equalities
to hold, anything

00:00:54.274 --> 00:00:55.690
that I do to the
left-hand side, I

00:00:55.690 --> 00:00:57.740
have to do the right-hand side.

00:00:57.740 --> 00:01:00.140
So let me divide
that by 3 as well.

00:01:00.140 --> 00:01:02.130
And so on the
left-hand side, I am

00:01:02.130 --> 00:01:09.840
left with x plus 6 squared
is equal to 75 divided by 3.

00:01:09.840 --> 00:01:13.067
So 75 divided by 3 is 25.

00:01:17.590 --> 00:01:20.100
So actually, let me just
pick out the first one I did.

00:01:20.100 --> 00:01:22.200
I divided both sides by 3.

00:01:22.200 --> 00:01:24.780
So that was my first step then.

00:01:24.780 --> 00:01:26.280
Let me write that
in a darker color.

00:01:26.280 --> 00:01:29.395
So that was my first
step right over there.

00:01:29.395 --> 00:01:31.020
Now let's think about
what we're doing.

00:01:31.020 --> 00:01:38.690
We're saying that something
squared is equal to 25.

00:01:38.690 --> 00:01:41.890
So this something could be
the positive or negative

00:01:41.890 --> 00:01:45.530
square root of 25.

00:01:45.530 --> 00:01:49.900
So we could write
this as x plus 6

00:01:49.900 --> 00:01:55.076
is equal to the plus or
minus square root of 25.

00:01:55.076 --> 00:01:57.200
So I'm essentially taking
the positive and negative

00:01:57.200 --> 00:01:59.870
square root of both sides.

00:01:59.870 --> 00:02:00.690
So, let's see.

00:02:00.690 --> 00:02:01.820
This looks like this step.

00:02:01.820 --> 00:02:03.770
I took the square
root of both sides.

00:02:03.770 --> 00:02:05.690
That's step number two.

00:02:05.690 --> 00:02:08.120
And so, let me
just rewrite this.

00:02:08.120 --> 00:02:13.190
This is the same thing as x plus
6 is equal to plus or minus 5.

00:02:13.190 --> 00:02:16.010
And now I want to just have
an x on the left-hand side.

00:02:16.010 --> 00:02:17.502
I want to solve for x.

00:02:17.502 --> 00:02:18.960
That's the goal
from the beginning.

00:02:18.960 --> 00:02:22.862
So I would like to
get rid of this 6.

00:02:22.862 --> 00:02:24.320
Well, the easiest
way to do that is

00:02:24.320 --> 00:02:26.200
to subtract 6 from
the left-hand side.

00:02:26.200 --> 00:02:27.750
But just like
before, I can't just

00:02:27.750 --> 00:02:29.350
do it from one side
of an equation.

00:02:29.350 --> 00:02:31.150
Then the equality
wouldn't be true.

00:02:31.150 --> 00:02:32.820
We're literally
saying that x plus 6

00:02:32.820 --> 00:02:35.870
is equal to plus or minus 5.

00:02:35.870 --> 00:02:38.710
So x plus 6 minus
6 is going to be

00:02:38.710 --> 00:02:41.790
equal to plus or
minus 5 minus 6.

00:02:41.790 --> 00:02:43.900
Or actually, let me
write it this way.

00:02:43.900 --> 00:02:46.350
So let me subtract
6 from both sides.

00:02:46.350 --> 00:02:49.410
On the left-hand side,
I'm left with an x.

00:02:49.410 --> 00:02:54.170
And on the right-hand side,
I could write it this way.

00:02:54.170 --> 00:02:55.810
Let me do it in
that green color.

00:02:55.810 --> 00:03:02.340
I have negative 6
plus or minus 5.

00:03:02.340 --> 00:03:04.160
So what are the
possible values of x?

00:03:04.160 --> 00:03:05.780
Or actually, I keep forgetting.

00:03:05.780 --> 00:03:08.247
We don't have to actually
give the value for x.

00:03:08.247 --> 00:03:09.830
We just have to say
what steps we did.

00:03:09.830 --> 00:03:10.621
So then, let's see.

00:03:10.621 --> 00:03:12.640
After we took the square
root of both sides,

00:03:12.640 --> 00:03:15.210
we then subtracted
6 from both sides.

00:03:15.210 --> 00:03:18.090
So that was step three
right over there.

00:03:18.090 --> 00:03:22.670
Then that got us to essentially
the two possible x's

00:03:22.670 --> 00:03:25.500
that would satisfy this
equation right over here.

00:03:25.500 --> 00:03:28.460
And just for fun, let's
actually solve it all the way.

00:03:28.460 --> 00:03:33.200
So if we solve it all the way,
so x is equal to negative 6

00:03:33.200 --> 00:03:37.690
plus 5 is negative 1, or x is
equal to negative 6 minus 5

00:03:37.690 --> 00:03:38.650
is negative 11.

00:03:38.650 --> 00:03:40.595
And you could verify
that both of these work.

00:03:40.595 --> 00:03:42.970
If you put either of them in
here-- if you put negative 1

00:03:42.970 --> 00:03:46.210
here, you get negative 1
plus 6 squared is 5 squared.

00:03:46.210 --> 00:03:48.850
If you put negative 11 here,
it's negative 11 plus 6

00:03:48.850 --> 00:03:49.960
is negative 5 squared.

00:03:49.960 --> 00:03:53.530
Obviously either plus or minus
5 squared is going to be 25.

00:03:53.530 --> 00:03:56.492
25 times 3 is 75.

00:03:56.492 --> 00:03:57.700
So these are our three steps.

00:03:57.700 --> 00:04:00.160
We divided both sides by 3.

00:04:00.160 --> 00:04:01.970
Then we took the square
root of both sides.

00:04:01.970 --> 00:04:03.990
Then we subtracted
6 from both sides.

00:04:03.990 --> 00:04:05.760
And then we were
essentially done.

00:04:05.760 --> 00:04:08.430
So let's input those steps.

00:04:08.430 --> 00:04:11.650
So the first thing we did,
we divide both sides by 3.

00:04:11.650 --> 00:04:13.610
That's the first thing we did.

00:04:13.610 --> 00:04:17.470
And then we took the
square root of both sides.

00:04:17.470 --> 00:04:22.820
And then we subtracted
6 from both sides.

00:04:22.820 --> 00:04:24.276
We got it right.


Practice computing and comparing rates

WEBVTT
Kind: captions
Language: en

00:00:00.194 --> 00:00:01.843
- [Voiceover] Let's get
some practice comparing

00:00:01.843 --> 00:00:03.862
and computing rates.

00:00:03.862 --> 00:00:06.997
So they tell us the pet
store has three fish tanks,

00:00:06.997 --> 00:00:09.667
each holding a different volume of water

00:00:09.667 --> 00:00:11.735
and a different number of fish.

00:00:11.735 --> 00:00:16.495
So Tank A has five fish, and
it has 40 liters of water,

00:00:16.495 --> 00:00:19.791
Tank B, 12 fish, 100 liters of water,

00:00:19.791 --> 00:00:23.914
and Tank C, 23 fish, and
it has 180 liters of water.

00:00:23.914 --> 00:00:27.035
Order the tanks by volume per fish

00:00:27.035 --> 00:00:29.033
from least to greatest.

00:00:29.033 --> 00:00:33.537
So let's think about what volume per fish,

00:00:33.537 --> 00:00:36.834
and we could think about this
as volume divided by fish.

00:00:36.834 --> 00:00:38.901
Volume per fish.

00:00:38.901 --> 00:00:40.897
All right, so here for Tank A,

00:00:40.897 --> 00:00:45.031
it's going to be 40 liters

00:00:45.031 --> 00:00:46.935
for every five fish.

00:00:46.935 --> 00:00:50.558
40 for every five fish, and let's see,

00:00:50.558 --> 00:00:52.458
40 over five is eight,

00:00:52.458 --> 00:00:55.818
so you have eight liters per fish,

00:00:55.818 --> 00:00:58.638
is the rate at which they
have to add water per fish

00:00:58.638 --> 00:01:00.247
for Tank A.

00:01:00.247 --> 00:01:05.146
Now, Tank B, you have 100 liters

00:01:05.146 --> 00:01:08.166
for 12 fish,

00:01:08.166 --> 00:01:09.574
so what is this going to be?

00:01:09.574 --> 00:01:12.964
This is going to be, 12
goes into 100 eight times,

00:01:12.964 --> 00:01:16.748
so eight times 12 is 96, and
then you have four left over.

00:01:16.748 --> 00:01:20.393
So this is going to be eight and 4/12,

00:01:20.393 --> 00:01:25.385
or eight and 1/3 liters per fish.

00:01:25.385 --> 00:01:28.846
And all I did is I converted
this improper fraction,

00:01:28.846 --> 00:01:31.679
100 over 12, to a mixed
number, and I simplified it.

00:01:31.679 --> 00:01:34.442
12 goes into 100 eight times
with a remainder of four,

00:01:34.442 --> 00:01:35.812
so it's eight and 4/12,

00:01:35.812 --> 00:01:38.109
which is the same thing as eight and 1/3.

00:01:38.109 --> 00:01:40.618
And then finally in Tank C,

00:01:40.618 --> 00:01:42.892
I have 180 liters

00:01:42.892 --> 00:01:47.482
for 23 fish.

00:01:47.482 --> 00:01:48.546
So what is this going to be?

00:01:48.546 --> 00:01:52.875
23 goes into 180. Let me
try to calculate this.

00:01:52.875 --> 00:01:57.286
23 goes into 180, it looks like it's going

00:01:57.286 --> 00:01:58.378
to be less than nine times.

00:01:58.378 --> 00:01:59.817
Is it eight times?

00:01:59.817 --> 00:02:01.628
Eight times, no, not eight times.

00:02:01.628 --> 00:02:03.230
Let's see, seven times.

00:02:03.230 --> 00:02:07.085
Seven times three is 21,
seven times two is 14,

00:02:07.085 --> 00:02:09.593
plus two is 16.

00:02:09.593 --> 00:02:12.216
When you subtract you
get a remainder of 19.

00:02:12.216 --> 00:02:15.119
So this is going to be seven
with a remainder of 19,

00:02:15.119 --> 00:02:16.818
or you could say this is seven

00:02:16.818 --> 00:02:21.188
and 19/13 liters

00:02:21.188 --> 00:02:22.898
per fish.

00:02:22.898 --> 00:02:24.291
So which one has,

00:02:24.291 --> 00:02:26.751
we're going to order the
tanks by volume per fish,

00:02:26.751 --> 00:02:28.377
from least to greatest,

00:02:28.377 --> 00:02:31.511
so Tank B has the largest volume per fish,

00:02:31.511 --> 00:02:34.460
has eight and 1/3 liters per fish,

00:02:34.460 --> 00:02:36.295
so this is in first place.

00:02:36.295 --> 00:02:38.687
And then Tank A is in second place.

00:02:38.687 --> 00:02:40.753
Tank A is in second place.

00:02:40.753 --> 00:02:45.188
And then Tank C is in third place.

00:02:45.188 --> 00:02:47.250
Oh, actually we wanna go
from least to greatest,

00:02:47.250 --> 00:02:49.268
so this is, let me write it this way.

00:02:49.268 --> 00:02:51.678
This is Tank C is the least,

00:02:51.678 --> 00:02:55.628
and Tank A is the greatest.

00:02:55.628 --> 00:02:57.749
So we really have to
swap this order around.

00:02:57.749 --> 00:02:59.305
Now, I just copied and pasted this

00:02:59.305 --> 00:03:00.513
from the Khan Academy exercises.

00:03:00.513 --> 00:03:03.020
Let me actually bring
the actual exercise up,

00:03:03.020 --> 00:03:04.436
and I can rearrange it.

00:03:04.436 --> 00:03:06.154
C was the least,

00:03:06.154 --> 00:03:09.804
and B was the most,

00:03:09.804 --> 00:03:11.564
and we check our answer
and we got it right.


Introduction to rates

WEBVTT
Kind: captions
Language: en

00:00:00.191 --> 00:00:01.702
- [Voiceover] What I want
to explore in this video

00:00:01.702 --> 00:00:04.925
is the notion of a "rate."

00:00:04.925 --> 00:00:07.180
So, let's look at some examples of rates

00:00:07.180 --> 00:00:10.018
that you've probably encountered
in your everyday life.

00:00:10.018 --> 00:00:11.893
So, if you're driving in
your car down the road,

00:00:11.893 --> 00:00:13.251
and you're looking at the speedometer,

00:00:13.251 --> 00:00:15.311
you might see that it
says that you are going

00:00:15.311 --> 00:00:18.452
35 M-P-H,

00:00:18.452 --> 00:00:20.020
where the M-P-H stands for

00:00:20.020 --> 00:00:23.085
35 miles

00:00:23.085 --> 00:00:25.859
per, per hour.

00:00:25.859 --> 00:00:26.587
Well, what's that saying?

00:00:26.587 --> 00:00:28.653
That's saying, well,
every hour, how many miles

00:00:28.653 --> 00:00:32.542
are you going if you were to
stay at that current rate.

00:00:32.542 --> 00:00:34.613
So, it could be a measure of speed.

00:00:34.613 --> 00:00:38.590
How much distance are you
covering per unit time?

00:00:38.590 --> 00:00:40.654
And, most typically, when
people talk about rates,

00:00:40.654 --> 00:00:41.622
that's what they're talking about.

00:00:41.622 --> 00:00:43.421
They're talking about
how much of something

00:00:43.421 --> 00:00:46.338
that is happening per unit time.

00:00:46.338 --> 00:00:48.933
And, it doesn't have to be
even distance per unit time,

00:00:48.933 --> 00:00:51.842
you might have a, you
might have your hourly rate

00:00:51.842 --> 00:00:54.366
for someone who is doing
some type of a job.

00:00:54.366 --> 00:00:56.153
They might say that they're making,

00:00:56.153 --> 00:00:59.613
they're making $10, so they're making $10.

00:00:59.613 --> 00:01:01.251
And, actually, let me
write the dollars out

00:01:01.251 --> 00:01:04.281
so the units become a
little bit more obvious,

00:01:04.281 --> 00:01:08.304
10 dollars, dollars per hour,

00:01:08.983 --> 00:01:11.814
dollars per,

00:01:11.814 --> 00:01:13.673
dollars per hour.

00:01:13.673 --> 00:01:15.545
And so, once again,
this is how much money.

00:01:15.545 --> 00:01:16.890
It's not talking about distance anymore.

00:01:16.890 --> 00:01:20.603
How much money is being
earned per unit time?

00:01:20.941 --> 00:01:23.089
And, so, even though rates
are often associated with

00:01:23.089 --> 00:01:25.639
how much something is
happening per some unit time,

00:01:25.639 --> 00:01:27.415
and it could be miles per hour,

00:01:27.415 --> 00:01:29.325
or it could be meters per second,

00:01:29.325 --> 00:01:30.977
or, in this case, it could be a wage,

00:01:30.977 --> 00:01:32.552
it could be dollars per hour.

00:01:32.552 --> 00:01:35.592
Rates don't have to be
just in those terms.

00:01:35.592 --> 00:01:38.072
In fact, you might say, "All right,

00:01:38.072 --> 00:01:40.713
"I have a dessert that I really enjoy,

00:01:40.713 --> 00:01:42.103
"but I'm very conscientious

00:01:42.103 --> 00:01:45.571
"about, about the number of
calories that I consume."

00:01:45.571 --> 00:01:48.985
And, you might, you
might see something like,

00:01:48.985 --> 00:01:53.155
there are 200 calories,

00:01:53.155 --> 00:01:57.083
calories per serving,

00:01:57.083 --> 00:01:58.633
per serving.

00:01:58.633 --> 00:02:01.051
And, so, this is telling
us the number of calories

00:02:01.051 --> 00:02:02.069
per a serving.

00:02:02.069 --> 00:02:03.246
And they'll tell us what a serving is.

00:02:03.246 --> 00:02:06.226
A serving might be a cup or
eight ounces or whatever else.

00:02:06.226 --> 00:02:08.899
And, so, I could say, "Okay,
look, if I have two servings,

00:02:08.899 --> 00:02:10.456
"then I'm gonna have 400 calories.

00:02:10.456 --> 00:02:12.902
"Same way, if I work two hours,
I'm gonna have 20 dollars.

00:02:12.902 --> 00:02:15.544
"If I, or if I go two hours,

00:02:15.544 --> 00:02:17.453
"I'm gonna go 70 miles."

00:02:17.453 --> 00:02:19.051
So, rates give you a sense.

00:02:19.051 --> 00:02:21.082
It's like, how fast is
something happening?

00:02:21.082 --> 00:02:23.995
Or how much of one thing is happening

00:02:23.995 --> 00:02:26.955
for every time something else happens?

00:02:26.955 --> 00:02:28.604
Now, I can write rates

00:02:28.604 --> 00:02:31.692
so they look an awful lot like a ratio.

00:02:31.692 --> 00:02:34.043
And, these words are,
actually, very related,

00:02:34.043 --> 00:02:35.583
'cause you see that even
how they're written.

00:02:35.583 --> 00:02:37.550
R-A-T, R-A-T.

00:02:37.550 --> 00:02:40.459
Their roots are coming
from the exact same idea.

00:02:40.459 --> 00:02:44.034
In fact, this rate over
here, 35 miles per hour,

00:02:44.034 --> 00:02:45.717
it could come from,

00:02:45.717 --> 00:02:49.448
"Hey, I just, I just went
35 miles in one hour,

00:02:49.448 --> 00:02:51.133
"what's the ratio?"

00:02:51.133 --> 00:02:53.577
So, the ratio

00:02:53.577 --> 00:02:56.385
of miles

00:02:56.385 --> 00:02:58.355
to hours.

00:02:58.355 --> 00:03:01.844
And, then, you could
say, "Well, I went 35,

00:03:01.844 --> 00:03:03.958
"the ratio miles to hours

00:03:03.958 --> 00:03:06.297
"was 35 to one."

00:03:06.297 --> 00:03:07.657
Or it could have been, maybe it was

00:03:07.657 --> 00:03:10.257
70 to two or something like that.

00:03:10.257 --> 00:03:11.840
But, that could have been reduced to

00:03:11.840 --> 00:03:13.357
35 to one.

00:03:13.357 --> 00:03:14.019
So, as a ratio,

00:03:14.019 --> 00:03:15.990
you would typically see
it written like this...

00:03:15.990 --> 00:03:19.624
Or maybe see it written like,
see it written like this...

00:03:19.624 --> 00:03:21.930
And, sometimes, you might
even see it written like

00:03:21.930 --> 00:03:25.876
this, 35 miles

00:03:25.881 --> 00:03:28.751
to one hour.

00:03:28.751 --> 00:03:30.333
But, now it's starting
to resemble more of the

00:03:30.333 --> 00:03:32.928
special case of a ratio,
which we call a "rate."

00:03:32.928 --> 00:03:35.482
Because, this is the same thing as 35.

00:03:35.482 --> 00:03:37.598
Instead of writing it
out "miles per hour,"

00:03:37.598 --> 00:03:39.800
you'll often see it written like this,

00:03:39.800 --> 00:03:44.338
miles per, miles per hour.

00:03:44.338 --> 00:03:46.794
So, these are very, very related ideas.

00:03:46.794 --> 00:03:50.007
If you find the ratio between
calories and servings,

00:03:50.007 --> 00:03:51.752
well, then, you're going
to be able to write,

00:03:51.752 --> 00:03:54.151
you're going to be able
to express it as a rate

00:03:54.151 --> 00:03:55.521
and vice versa.

00:03:55.521 --> 00:03:57.907
Now, why do we care about rates?

00:03:57.907 --> 00:04:00.730
Well, especially if we're
thinking about things like speed,

00:04:00.730 --> 00:04:03.913
without rates, it would
be hard to quantify

00:04:03.913 --> 00:04:05.360
how fast things are happening.

00:04:05.360 --> 00:04:06.626
Otherwise, we'd be in a
world where we're saying,

00:04:06.626 --> 00:04:07.926
"Hey, I'm faster than you," or

00:04:07.926 --> 00:04:09.818
"She's faster than me."

00:04:09.818 --> 00:04:11.084
But we wouldn't be able to quantify

00:04:11.084 --> 00:04:12.846
exactly how fast they are.

00:04:12.846 --> 00:04:15.702
But with rates, we can
say, "Hey, that person ran

00:04:15.702 --> 00:04:17.760
"a hundred meters in 10 seconds,

00:04:17.760 --> 00:04:19.745
"they run 10 meters per second."

00:04:19.745 --> 00:04:24.322
We can quantify exactly how
fast that thing is happening,

00:04:24.322 --> 00:04:26.357
the rate at which it is happening.

00:04:26.357 --> 00:04:28.987
Here, instead of saying,
"Hey, a cup of that

00:04:28.987 --> 00:04:33.567
"is gonna give you, is
gonna give you more energy,

00:04:33.567 --> 00:04:35.046
or, maybe, contribute more to your weight

00:04:35.046 --> 00:04:38.220
than a cup of that, and making
these relative comparisons,

00:04:38.220 --> 00:04:41.784
here, you can actually, you
can actually quantify things.

00:04:41.784 --> 00:04:42.799
And when we study rate,

00:04:42.799 --> 00:04:44.652
we're gonna study rate
a lot in mathematics.

00:04:44.652 --> 00:04:46.694
It's gonna be essential in algebra

00:04:46.694 --> 00:04:48.956
when we look at the rate of change

00:04:48.956 --> 00:04:52.271
of a line, how far it moves
in the vertical direction

00:04:52.271 --> 00:04:54.391
relative to the horizontal direction.

00:04:54.391 --> 00:04:56.318
We're gonna call that "slope".

00:04:56.318 --> 00:04:58.747
And you can even imagine
the slope of a hill as

00:04:58.747 --> 00:05:02.294
how fast is it climbing for
as much as you move forward.

00:05:02.294 --> 00:05:04.190
But we're also gonna study rates in detail

00:05:04.190 --> 00:05:05.351
when we go to calculus.

00:05:05.351 --> 00:05:08.251
In fact, the whole basis
of differential calculus,

00:05:08.251 --> 00:05:10.762
that you might see later in
high school and early college,

00:05:10.762 --> 00:05:13.530
is all about measuring instantaneous rate.

00:05:13.530 --> 00:05:16.752
How fast is something going right now?

00:05:16.752 --> 00:05:18.643
So, rates are really, really interesting,

00:05:18.643 --> 00:05:19.794
really, really important.

00:05:19.794 --> 00:05:21.467
And, I would guess that,

00:05:21.467 --> 00:05:22.847
if you just look around your life,

00:05:22.847 --> 00:05:24.287
even over the next few hours,

00:05:24.287 --> 00:05:25.494
you're going to encounter

00:05:25.494 --> 00:05:28.222
many, many, many rates.


Example 5: Adding polynomials with two variables

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.330
So let's get some practice
simplifying polynomials,

00:00:03.330 --> 00:00:04.830
especially in the
case where we have

00:00:04.830 --> 00:00:07.150
more than one
variable over here.

00:00:07.150 --> 00:00:11.720
So I have 4x squared y
minus 3x squared minus 2y.

00:00:11.720 --> 00:00:15.010
So that entire expression
plus the entire expression

00:00:15.010 --> 00:00:20.560
8xy minus 3x squared
plus 2x squared y plus 4.

00:00:20.560 --> 00:00:22.830
So the first thing
that jumps out at me

00:00:22.830 --> 00:00:25.410
is that I'm just adding this
expression to this expression.

00:00:25.410 --> 00:00:28.620
So to a large degree, these
parentheses don't matter.

00:00:28.620 --> 00:00:34.710
So I can just rewrite it as
4x squared y minus 3x squared

00:00:34.710 --> 00:00:43.080
minus 2y plus 8xy minus
3x squared plus 2x squared

00:00:43.080 --> 00:00:46.050
y plus 4.

00:00:46.050 --> 00:00:49.940
Now we can try to group
similar terms or like terms.

00:00:49.940 --> 00:00:52.060
So let's think about
what we have over here.

00:00:52.060 --> 00:00:56.900
So this first term right
here is a 4x squared y.

00:00:56.900 --> 00:00:59.760
So can I add this to any
of the other terms here?

00:00:59.760 --> 00:01:03.151
Do we have any other
x squared y terms?

00:01:03.151 --> 00:01:04.650
Well, sure, this
one right over here

00:01:04.650 --> 00:01:07.640
is another x squared y term.

00:01:07.640 --> 00:01:09.380
If I have 4 of
something-- in this case,

00:01:09.380 --> 00:01:13.360
I have 4x squared y's and
I add 2x squared y's to it,

00:01:13.360 --> 00:01:16.720
how many x squared
y's do I now have?

00:01:16.720 --> 00:01:24.990
Well, 4 plus 2-- I now
have 6x squared y's.

00:01:24.990 --> 00:01:27.560
Now let's move on to this term.

00:01:27.560 --> 00:01:30.590
So I have negative 3x squareds.

00:01:30.590 --> 00:01:35.050
Do I have any more x
squareds in this expression

00:01:35.050 --> 00:01:36.590
right over here?

00:01:36.590 --> 00:01:39.210
Well, sure, I have another
negative 3x squared.

00:01:39.210 --> 00:01:40.710
So if I have negative
3 of something

00:01:40.710 --> 00:01:45.140
and then I have another negative
3, I end up with negative 6

00:01:45.140 --> 00:01:45.890
of that thing.

00:01:45.890 --> 00:01:49.900
So it's negative 6x squared.

00:01:49.900 --> 00:01:53.830
Now let's think about
this negative 2y term.

00:01:53.830 --> 00:01:56.340
Are there any other
y's over here?

00:01:56.340 --> 00:01:58.380
Well, it doesn't
look like there are.

00:01:58.380 --> 00:01:59.570
This is an 8xy.

00:01:59.570 --> 00:02:00.200
This is a 4.

00:02:00.200 --> 00:02:01.830
There's no just y's.

00:02:01.830 --> 00:02:03.660
So I can't really
add that to anything.

00:02:03.660 --> 00:02:06.370
So I'll just rewrite
it, negative 2y.

00:02:06.370 --> 00:02:10.030
And then 8xy--
well, once again, it

00:02:10.030 --> 00:02:12.940
doesn't seem like that can
be added to anything else.

00:02:12.940 --> 00:02:15.320
So let's just write
that over again.

00:02:15.320 --> 00:02:19.220
And then finally, we just
have the constant term plus 4.

00:02:19.220 --> 00:02:21.520
And it pretty much
looks like we're done.

00:02:21.520 --> 00:02:28.430
We have simplified
this as much as we can.


How to determine the domain of a modeling function (example with a function)

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:05.290
Pooja had a beautiful plant. The plant
began sprouting 2 days before Pooja

00:00:05.290 --> 00:00:05.879
bought it,

00:00:05.879 --> 00:00:09.090
and she had it for 98 days
before it died.

00:00:09.090 --> 00:00:12.920
At its tallest, the plant was 30
centimetres tall.

00:00:12.920 --> 00:00:17.690
Alright, let h of t denote the height
of Pooja's plant h,

00:00:17.690 --> 00:00:22.000
measured in centimeters, t days from
the time she bought it.

00:00:22.000 --> 00:00:26.250
Which number type is more appropriate
for the domain of the function? They

00:00:26.250 --> 00:00:26.939
tell us whether it's

00:00:26.939 --> 00:00:30.260
integers or real numbers.
So we just need to remind ourselves.

00:00:30.260 --> 00:00:34.350
The domain of a function, that's a
set of all the inputs for which the

00:00:34.350 --> 00:00:35.880
function is defined.

00:00:35.880 --> 00:00:40.450
And the inputs here, this is t and
it represents days

00:00:40.450 --> 00:00:44.410
and if I input t, the number of days,
into the function h,

00:00:44.410 --> 00:00:48.590
it'll output the height of the plant. So let's
think about it. At first, you might

00:00:48.590 --> 00:00:52.579
say, in terms of, 0 days, 1 days,
2 days, and you might be tempted to say

00:00:52.579 --> 00:00:53.199
integers.

00:00:53.199 --> 00:00:59.030
But why not think about one and a half
days or 3.175 days?

00:00:59.030 --> 00:01:02.879
So I don't see any real -- not any reason
why t couldn't be a subset

00:01:02.879 --> 00:01:06.220
of real numbers. Why you can think about,

00:01:06.220 --> 00:01:10.100
you know, the 90 -- 97.99

00:01:10.100 --> 00:01:14.790
day. So I would say real numbers. And let's see.
Define the interval of the domain.

00:01:14.790 --> 00:01:18.530
So t. Let's see, the

00:01:18.530 --> 00:01:22.900
t would be defined up to 2 days
before she bought the plant. So I would say

00:01:22.900 --> 00:01:27.540
that t is equal to negative 2, all the way to
98 days.

00:01:27.540 --> 00:01:32.040
And t could be equal to negative 2. That would be
2 days before she bought it,

00:01:32.040 --> 00:01:36.549
or as high as 98. So let's think about it.
So the interval --

00:01:36.549 --> 00:01:39.990
I would include, I would include negative 2.
I would include

00:01:39.990 --> 00:01:44.270
the low point, so that's why I'm
using the bracket. If I wanted t to be

00:01:44.270 --> 00:01:47.619
greater than, not greater than or equal
to, I would do a parentheses. But since t

00:01:47.619 --> 00:01:50.189
could be equal to negative 2, I'm gonna use the brackets.

00:01:50.189 --> 00:01:53.590
And at the high end, it's 98,

00:01:53.590 --> 00:01:57.079
and we're going to include 98. So I'm
gonna put the brackets

00:01:57.079 --> 00:02:01.579
there as well. So t would be a member of
real numbers

00:02:01.579 --> 00:02:05.320
such that it is a member of this
-- of this interval right

00:02:05.320 --> 00:02:09.300
over here. So negative t-- t could
be negative 2

00:02:09.300 --> 00:02:12.960
but -- right I would say negative 2 is going to be less than
or equal to t, which is going to be

00:02:12.960 --> 00:02:14.980
less than or equal to 98.

00:02:14.980 --> 00:02:17.390
Let's check our answer. Got it right.


WEBVTT
Kind: captions
Language: en

00:00:00.429 --> 00:00:03.990
Arjun opened up a savings account last
year and put

00:00:03.990 --> 00:00:07.730
an initial sum in it. Let M of t
denote

00:00:07.730 --> 00:00:10.760
the account balance M measured in dollars

00:00:10.760 --> 00:00:14.730
t days since it was opened. What does
the statement

00:00:14.730 --> 00:00:17.859
M of 30 minus M

00:00:17.859 --> 00:00:22.130
of 0 equalling a hundred mean?
Before I look at the choices, let's think

00:00:22.130 --> 00:00:23.039
about what this means.

00:00:23.039 --> 00:00:27.340
When you input t equals 30 into your
function,

00:00:27.340 --> 00:00:31.820
you're going to get M of 30.
So let me make that clear. So if you say t

00:00:31.820 --> 00:00:36.550
is equal to 30, you input that into your
function

00:00:36.550 --> 00:00:39.920
M. You're going to get -- you're going to
get

00:00:39.920 --> 00:00:43.890
M of 30. So one way to think about it
is --

00:00:43.890 --> 00:00:48.719
This is the account balance 30 days
since it was opened. This is when t is

00:00:48.719 --> 00:00:51.800
equal to 30. This is the account balance
after 30 days

00:00:51.800 --> 00:00:55.260
Let's write that down. Balance -- balance

00:00:55.260 --> 00:01:02.260
after 30 days. Now, by the same logic, this
right over here this is when we put t

00:01:02.410 --> 00:01:06.650
where we said t equals 0. This is the balance
after 0 days, or you could say this is

00:01:06.650 --> 00:01:07.939
the initial balance.

00:01:07.939 --> 00:01:11.520
initial -- initial balance.

00:01:11.520 --> 00:01:15.100
So what they're doing -- they're taking our
balance after 30 days, and

00:01:15.100 --> 00:01:18.740
from that they're subtracting the initial
balance and they're saying that

00:01:18.740 --> 00:01:20.560
equal to 100.

00:01:20.560 --> 00:01:22.990
So there's a couple ways you can
interpret this and I haven't even looked

00:01:22.990 --> 00:01:25.460
at these choices yet. We'll see if
any of these match up.

00:01:25.460 --> 00:01:29.390
You could say that your balance after 30
days is a hundred dollars more than the

00:01:29.390 --> 00:01:29.930
initial

00:01:29.930 --> 00:01:33.979
balance. Or another way to think about is
you added a hundred dollars in the first

00:01:33.979 --> 00:01:35.049
30 days.

00:01:35.049 --> 00:01:38.280
Those are both legitimate ways to think
about it. I'll see which of these choices

00:01:38.280 --> 00:01:38.680
are

00:01:38.680 --> 00:01:44.159
consistent with that. 30 days after it was
opened, the balance of Arjun's account

00:01:44.159 --> 00:01:44.409
was

00:01:44.409 --> 00:01:47.500
equal to 100. No, that's not what that's
saying.

00:01:47.500 --> 00:01:51.829
This statement right over here -- the
balance 30 days after opening --

00:01:51.829 --> 00:01:54.829
this statement right over here -- this
would be equivalent.

00:01:54.829 --> 00:01:58.040
This is equivalent to saying that, M of

00:01:58.040 --> 00:02:01.259
30 -- this is the balance after 30
days after it was opened --

00:02:01.259 --> 00:02:05.250
is equal to 100. That's not what they tell us here.
They tell us that the difference

00:02:05.250 --> 00:02:09.300
between the balance after 30 days and
the initial balance -- that's a hundred.

00:02:09.300 --> 00:02:15.080
So we can rule that one out. Arjan had
the initial amount of money in his account

00:02:15.080 --> 00:02:20.990
30 days after he opened it. So if he had
the same amount -- if he had the initial

00:02:20.990 --> 00:02:21.810
amount -- let me

00:02:21.810 --> 00:02:25.520
write this down. So had the initial
amount of money -- the initial

00:02:25.520 --> 00:02:26.400
amount of money

00:02:26.400 --> 00:02:31.980
is M of 0. So they're saying he had the
initial amount of money in his account

00:02:31.980 --> 00:02:35.000
30 days after he opened it. Well, the amount
that he had in his account

00:02:35.000 --> 00:02:38.310
30 days after he opened it is M of 30.

00:02:38.310 --> 00:02:41.520
So these are the same amounts of money
then this --

00:02:41.520 --> 00:02:45.260
In order to be consistent with this,
you would have an equation like this. M of 0,

00:02:45.260 --> 00:02:46.080
the initial amount,

00:02:46.080 --> 00:02:50.920
is equal to the amount after 30 days.
That's not what they told us over here.

00:02:50.920 --> 00:02:54.250
We can rule that out. And then finally

00:02:54.250 --> 00:02:57.830
we have the choice Arjan made a profit
over a hundred dollars

00:02:57.830 --> 00:03:01.200
over the first 30 days since the account
was opened.

00:03:01.200 --> 00:03:06.000
That seems reasonable that his balance
is a hundred dollars higher. The

00:03:06.000 --> 00:03:09.550
difference between -- if you take the initial
balance and subtract it from his balance

00:03:09.550 --> 00:03:10.550
after 30 days,

00:03:10.550 --> 00:03:14.300
it's a hundred. And this right over here is a
hundred higher than his initial balance.

00:03:14.300 --> 00:03:17.470
So it makes sense that maybe he got the
profit out of an interest or

00:03:17.470 --> 00:03:18.239
something else

00:03:18.239 --> 00:03:20.860
that he got in his bank account over the
first 30 days.


How to create a function from an equation (example)

WEBVTT
Kind: captions
Language: en

00:00:00.429 --> 00:00:03.639
For a given input value b, the function

00:00:03.639 --> 00:00:09.420
f outputs a value a to satisfy the
following equation 4a plus 7b

00:00:09.420 --> 00:00:13.870
is equal to negative 52. So for a given input b,

00:00:13.870 --> 00:00:17.460
the function f, the function f
will

00:00:17.460 --> 00:00:23.359
output an a that satisfies this
relationship right over here for the a

00:00:23.359 --> 00:00:26.519
and the b. Write a formula for f of b

00:00:26.519 --> 00:00:30.720
in terms of b. So we want to do,
we just want to solve for --

00:00:30.720 --> 00:00:34.570
if we're given a "b", what "a" does that
imply that we have to

00:00:34.570 --> 00:00:38.290
output? Or another way to think about it is
-- let's just solve for

00:00:38.290 --> 00:00:41.950
a, or we could think about a as being a
function of

00:00:41.950 --> 00:00:45.100
b. So let's write this. So we have 4a

00:00:45.100 --> 00:00:48.129
plus 7b -- is equal to

00:00:48.129 --> 00:00:51.379
negative 52. So I can solve for a

00:00:51.379 --> 00:00:55.719
in terms of b, that any b that I have --
Let's say these b's are on the right hand side I

00:00:55.719 --> 00:00:56.020
can

00:00:56.020 --> 00:00:59.949
put it in. I can substitute that value
for b and I can just solve for a.

00:00:59.949 --> 00:01:02.239
I can solve for a that needs to be
outputted.

00:01:02.239 --> 00:01:06.090
So let's do that. Let's solve for a.
So I want to get all the a on

00:01:06.090 --> 00:01:09.490
I wanna just have an a leftover on the
left hand side, and have everything else on

00:01:09.490 --> 00:01:11.119
the right hand side including the b's.

00:01:11.119 --> 00:01:15.310
So let's get rid of this b on the left hand
side. And

00:01:15.310 --> 00:01:19.720
I can do that by subtracting 7b.
Of course I wanna do that to both sides.

00:01:19.720 --> 00:01:21.049
I can't just do an equation and

00:01:21.049 --> 00:01:24.340
do an operation only on one side like
that. So

00:01:24.340 --> 00:01:28.240
let's subtract, and we are left with
we are left with --

00:01:28.240 --> 00:01:32.909
the 7b's add up to zero. 7b minus 7b.
We're left with

00:01:32.909 --> 00:01:36.470
4a is equal to negative 52

00:01:36.470 --> 00:01:39.670
minus 7b, minus

00:01:39.670 --> 00:01:44.570
7b. Now, to isolate the a here,
just to have an a here instead of 4a,

00:01:44.570 --> 00:01:49.399
we can divide both sides by 4. We can
divide both sides by 4. So I'm gonna

00:01:49.399 --> 00:01:50.479
divide everything

00:01:50.479 --> 00:01:55.549
by 4. And on the left hand side, we got
our goal.

00:01:55.549 --> 00:02:00.119
We are left with an a is equal to -- Now
what's negative 54 divided by --

00:02:00.119 --> 00:02:04.229
What is the negative 52 divided by 4?
So let's think about it.

00:02:04.229 --> 00:02:09.039
52 is 40 plus 12. 40 divided by
4 is 10. 12 divided by

00:02:09.039 --> 00:02:12.910
4 is 3. So it's gonna be 13. Negative 13.
So it's negative

00:02:12.910 --> 00:02:17.280
13 minus 7/4 b

00:02:17.280 --> 00:02:21.190
minus 7/4 b. So

00:02:21.190 --> 00:02:25.120
given a "b", if you give me a "b", I can put
that value right over here,

00:02:25.120 --> 00:02:29.470
and I can calculate what the
corresponding a needs to be

00:02:29.470 --> 00:02:32.550
in order to satisfy this relationship.

00:02:32.550 --> 00:02:36.530
So if I want a formula for f of b in
terms of b, I can say, look,

00:02:36.530 --> 00:02:40.910
you give me a "b", the output of our
function, which is

00:02:40.910 --> 00:02:44.980
f of a. The output of our function is
going to be --

00:02:44.980 --> 00:02:48.510
is going to be negative 13 minus

00:02:48.510 --> 00:02:53.180
7/4 b, because the output of our
function needs to be an

00:02:53.180 --> 00:02:57.460
"a" that will satisfy -- that will satisfy

00:02:57.460 --> 00:03:00.570
this equation up here.

00:03:00.570 --> 00:03:01.420
So hopefully that helped.


How to match function input to output given the formula (example)

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:04.361
The function f is defined as follows:

00:00:04.361 --> 00:00:08.718
f of t is equal to negative
two t plus five.

00:00:08.718 --> 00:00:11.507
So whatever we input into this function,

00:00:11.507 --> 00:00:16.558
we multiply it times negative two,
and then we add five.

00:00:16.558 --> 00:00:21.453
So what is the input value
for which f of t is equal to 13?

00:00:21.453 --> 00:00:25.550
So if f of t is equal to 13, that
means that this thing over here

00:00:25.550 --> 00:00:30.258
is equal to 13 for some t, for some input.

00:00:30.258 --> 00:00:34.860
So we can just solve the equation,
negative two t plus five is equal to 13.

00:00:34.860 --> 00:00:36.181
So let's do that.

00:00:36.181 --> 00:00:41.100
Negative two t plus five is equal to 13.

00:00:41.100 --> 00:00:43.312
Well, we can subtract
five from both sides.

00:00:43.312 --> 00:00:46.741
I'm just trying to isolate the t
on the left hand-side.

00:00:46.741 --> 00:00:49.186
So, subtract negative five from the left,

00:00:49.186 --> 00:00:51.881
that's the whole reason why we
did that, so those disappear.

00:00:51.881 --> 00:00:53.759
But we have to do it
from the right as well.

00:00:53.759 --> 00:00:56.237
So you have 13 minus five is eight.

00:00:56.237 --> 00:00:59.210
And on the left hand-side
you still have your negative two t.

00:00:59.210 --> 00:01:02.232
So you have negative two t
is equal to eight.

00:01:02.232 --> 00:01:03.990
Now to just have a t on
the left hand-side,

00:01:03.990 --> 00:01:07.480
I want to divide both sides
by negative two.

00:01:07.480 --> 00:01:11.317
And I'm left with, t is equal to
eight divided by negative two,

00:01:11.317 --> 00:01:13.200
is equal to negative four.

00:01:13.200 --> 00:01:20.263
So you input negative four
into this function

00:01:20.268 --> 00:01:23.034
and it will output 13.

00:01:23.034 --> 00:01:28.377
Or we could write that f of
negative four is equal to 13.

00:01:28.377 --> 00:01:30.293
But this, is what they are looking for.

00:01:30.293 --> 00:01:31.373
This is the input value.

00:01:31.373 --> 00:01:34.850
Negative four is the input value
for which f of t is equal to 13.

00:01:34.850 --> 00:01:38.354
f of negative four is equal to 13.


How to find two function inputs with the same output given graph (example)

WEBVTT
Kind: captions
Language: en

00:00:00.220 --> 00:00:04.305
The graph of the function
f is shown below.

00:00:04.305 --> 00:00:07.727
What is the input value
other than negative five

00:00:07.727 --> 00:00:12.254
for which f of x is equal
to f of negative five?

00:00:12.254 --> 00:00:15.240
So, we have our x axis,
we have our y axis.

00:00:15.240 --> 00:00:19.742
And then in blue, they've
graphed y equals f of x.

00:00:19.742 --> 00:00:23.663
So for example when
x is equal to one, f of x,

00:00:23.663 --> 00:00:28.005
when y is going to be equal
to f of x --that's what this graph is--

00:00:28.005 --> 00:00:30.143
f of x is equal to one.

00:00:30.143 --> 00:00:36.418
When x is equal to seven,
f of seven, we see, is equal to five.

00:00:36.418 --> 00:00:41.580
When x is equal to nine, we see
that f of nine is equal to six.

00:00:41.580 --> 00:00:44.299
So what is the input value
other than negative five

00:00:44.299 --> 00:00:47.760
for which f of x is equal to
f of negative five?

00:00:47.760 --> 00:00:48.764
So, let's see.

00:00:48.764 --> 00:00:53.648
If x is equal to negative five,
f of negative five is...

00:00:53.648 --> 00:00:55.850
-- we move up here to the graph--

00:00:55.850 --> 00:00:57.043
It is equal to four.

00:00:57.043 --> 00:01:00.954
Cause this, once again, is the
graph y is equal to f of x.

00:01:00.954 --> 00:01:02.453
So the y coordinate here,

00:01:02.453 --> 00:01:06.568
this is what f of x, this is what
five of negative five is equal to.

00:01:06.568 --> 00:01:08.640
So f of negative five is equal to four.

00:01:08.640 --> 00:01:10.529
So where else does that happen?

00:01:10.529 --> 00:01:11.339
So let's see.

00:01:11.339 --> 00:01:13.983
Let's just move horizontally to the right.

00:01:13.983 --> 00:01:16.598
That happens here as well.

00:01:16.598 --> 00:01:18.630
So what input do we have to give?

00:01:18.630 --> 00:01:20.572
What's the x coordinate here to get

00:01:20.572 --> 00:01:25.573
y is equal to f of x or f of x is
equal to four right over here?

00:01:25.573 --> 00:01:27.946
Well, we see the x
coordinate is also four.

00:01:27.946 --> 00:01:31.728
So this tells us that f of four
is equal to four,

00:01:31.728 --> 00:01:35.084
which is the same thing as
f of negative five.

00:01:35.084 --> 00:01:39.521
So when x is four, the function
takes on the same value

00:01:39.521 --> 00:01:42.530
as when x is negative five.

00:01:42.530 --> 00:01:47.009
So x equals four. And we are done.


Linear models example 2

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.950
- [Voiceover] William has
a 26-liter glass tank.

00:00:03.950 --> 00:00:06.220
First, he wants to put some marbles in it,

00:00:06.220 --> 00:00:08.260
all of the same volume.

00:00:08.260 --> 00:00:10.450
Then, he wants to fill the tank with water

00:00:10.450 --> 00:00:12.190
until it's completely full.

00:00:12.190 --> 00:00:15.350
If he uses 85 marbles, he will have

00:00:15.350 --> 00:00:19.780
to add 20.9 liters of water.

00:00:19.780 --> 00:00:22.990
What is the volume of each marble?

00:00:22.990 --> 00:00:24.420
All right, so let's think about it.

00:00:24.420 --> 00:00:28.020
The volume of the marbles
plus the volume of the water

00:00:28.020 --> 00:00:30.330
are going to be equal to
the volume of the tank.

00:00:30.330 --> 00:00:31.210
They're going to fill up the tank.

00:00:31.210 --> 00:00:32.690
Let me write that down.

00:00:32.690 --> 00:00:36.320
So the volume, let me write V,

00:00:36.320 --> 00:00:38.310
V and I'll write a
little subscript M here.

00:00:38.310 --> 00:00:40.620
This is the volume of the marbles

00:00:40.620 --> 00:00:42.488
plus the volume of the water,

00:00:43.370 --> 00:00:45.952
plus the volume of the water,

00:00:45.952 --> 00:00:49.323
are going to be equal to
the volume of the tank.

00:00:49.323 --> 00:00:52.910
Are going to be equal to
the volume of the tank.

00:00:52.910 --> 00:00:55.720
Now, we don't know what the volume

00:00:55.720 --> 00:00:57.640
of 85 marbles is,

00:00:57.640 --> 00:01:00.520
but we know, this right over here,

00:01:00.520 --> 00:01:01.310
this is going to be

00:01:01.310 --> 00:01:05.110
85 times the volume of one marble.

00:01:05.110 --> 00:01:07.390
So this is volume of all the marbles,

00:01:07.390 --> 00:01:08.980
and maybe, I'll just write it out,

00:01:08.980 --> 00:01:11.700
volume, volume

00:01:11.700 --> 00:01:15.310
of one marble, one marble.

00:01:15.310 --> 00:01:16.090
I'll write it out.

00:01:16.090 --> 00:01:17.520
I could've put a variable in there,

00:01:17.520 --> 00:01:19.150
but just to make it clear.

00:01:19.150 --> 00:01:21.072
So 85 times the volume of one marble,

00:01:21.072 --> 00:01:23.590
that's gonna be the total
volume of the marbles,

00:01:23.590 --> 00:01:25.470
plus the volume of water.

00:01:25.470 --> 00:01:27.800
They tell us what the volume of water is.

00:01:27.800 --> 00:01:32.190
20.9 liters of water, plus 20.9

00:01:32.190 --> 00:01:34.770
is gonna be equal to the
volume of the entire tank.

00:01:34.770 --> 00:01:37.580
It's a 26-liter glass tank,

00:01:37.580 --> 00:01:40.440
so it's gonna be equal to 26.

00:01:40.440 --> 00:01:42.570
Remember, the whole thing
that's going on here is

00:01:42.570 --> 00:01:44.450
he wants to fill up the whole tank.

00:01:44.450 --> 00:01:47.950
He puts some marbles in
it that have some volume,

00:01:47.950 --> 00:01:49.960
and then whatever's left,
he fills it with water.

00:01:49.960 --> 00:01:51.680
So these two volumes combined

00:01:51.680 --> 00:01:54.120
have to add up to the
volume of the entire tank.

00:01:55.090 --> 00:01:57.890
Now, what is the, now we can just solve

00:01:57.890 --> 00:02:00.110
for the volume of one marble.

00:02:00.110 --> 00:02:02.220
And actually maybe I'll just,

00:02:02.220 --> 00:02:04.790
let's just call this the variable M.

00:02:04.790 --> 00:02:07.500
Let's just call this, let's call it M

00:02:07.500 --> 00:02:09.198
for the volume of one marble.

00:02:10.744 --> 00:02:12.997
So we get 85,

00:02:13.977 --> 00:02:17.037
we get 85 times M,

00:02:17.037 --> 00:02:19.947
maybe I'll just do this in one
color for the sake of time,

00:02:19.947 --> 00:02:23.687
plus 20.9 is equal to 26.

00:02:23.687 --> 00:02:27.417
Now, to solve for M we can
subtract 20.9 from both sides.

00:02:27.417 --> 00:02:30.017
So subtract 20.9,

00:02:30.017 --> 00:02:33.157
subtract 20.9 from both sides,

00:02:33.157 --> 00:02:36.697
and we get 85 times

00:02:36.697 --> 00:02:38.717
the volume of one marble

00:02:38.717 --> 00:02:42.577
is equal to, let's see, 26 minus 21

00:02:42.577 --> 00:02:43.987
would have been five,

00:02:43.987 --> 00:02:46.021
so this is gonna be five point...

00:02:49.037 --> 00:02:51.785
Right, this is gonna be 5.1,

00:02:51.785 --> 00:02:55.797
It's gonna be five, 5.1,

00:02:55.797 --> 00:02:57.887
or we can say the volume of each marble

00:02:57.887 --> 00:02:59.997
if we divide both sides by 85

00:02:59.997 --> 00:03:04.067
is 5.1 liters over 85.

00:03:04.067 --> 00:03:06.487
And I could calculate, let's see,

00:03:07.597 --> 00:03:08.987
let me just get a calculator out for this.

00:03:08.987 --> 00:03:10.957
We're allowed to use
calculators on this one.

00:03:10.957 --> 00:03:14.047
And so we get 5.1

00:03:14.047 --> 00:03:16.487
divided by 85,

00:03:16.487 --> 00:03:20.377
is equal to 0.06 liters,

00:03:20.377 --> 00:03:23.317
is equal to 0.06.

00:03:23.317 --> 00:03:24.917
So what is the volume of each marble?

00:03:24.917 --> 00:03:28.887
We just figured out 0.06 liters.

00:03:28.887 --> 00:03:30.727
And then how much water is necessary

00:03:30.727 --> 00:03:35.157
if William uses 200 marbles?

00:03:35.157 --> 00:03:36.757
So let's think about it.

00:03:36.757 --> 00:03:39.187
How much water would be necessary?

00:03:39.187 --> 00:03:42.337
Let's see, the volume, let's just say W

00:03:42.337 --> 00:03:44.577
is the amount of water that's necessary.

00:03:44.577 --> 00:03:47.590
And it's gonna be a function
of the number of marbles,

00:03:47.590 --> 00:03:51.067
the number of marbles, and let's see,

00:03:51.067 --> 00:03:53.387
if you have, if you have no marbles

00:03:53.387 --> 00:03:54.957
you're going to have to put 26 liters,

00:03:54.957 --> 00:03:57.351
you have to fill the whole
glass tank, the whole volume.

00:03:57.351 --> 00:03:59.414
So it's going to be equal to 26,

00:04:00.417 --> 00:04:02.247
minus the volume of the marbles.

00:04:02.247 --> 00:04:04.917
Well, what's the volume of
the marbles going to be?

00:04:04.917 --> 00:04:07.727
Well it's going to be the
volume of each marble.

00:04:07.727 --> 00:04:10.987
It's going to be 0.06

00:04:10.987 --> 00:04:13.517
times the number of marbles you have,

00:04:13.517 --> 00:04:16.177
times the number of marbles you have.

00:04:16.177 --> 00:04:18.457
Just like that we've
described a linear function

00:04:18.457 --> 00:04:20.277
that tells us how much
water we have to fill

00:04:20.277 --> 00:04:21.827
as a function of marbles.

00:04:21.827 --> 00:04:23.417
It's gonna be the total volume of the tank

00:04:23.417 --> 00:04:25.717
minus the volume of the marbles,

00:04:25.717 --> 00:04:27.197
and the volume of the marbles is the

00:04:27.197 --> 00:04:30.817
volume per marble times
the number of marbles.

00:04:30.817 --> 00:04:34.017
So if they're saying how much
water we need for 200 marbles,

00:04:34.017 --> 00:04:37.057
that's going to be W, the amount of water

00:04:37.057 --> 00:04:38.747
we have to fill as a function of marbles,

00:04:38.747 --> 00:04:41.097
where now the number of marbles is 200.

00:04:41.097 --> 00:04:45.579
That's going to be equal to 26 minus

00:04:45.579 --> 00:04:50.457
0.06 times 200,

00:04:50.457 --> 00:04:52.927
times 200.

00:04:52.927 --> 00:04:54.917
And what's that going to be?

00:04:54.917 --> 00:04:57.427
Well let's see, six times,

00:04:57.427 --> 00:05:01.087
this is six hundredths.

00:05:01.087 --> 00:05:04.357
Six hundredths times a hundred

00:05:04.357 --> 00:05:06.197
would be six,

00:05:06.197 --> 00:05:07.397
but now we're multiplying 200,

00:05:07.397 --> 00:05:11.057
so this is gonna be 12, so
this thing right over here,

00:05:11.057 --> 00:05:14.457
this thing right over here
is going to be equal to 12.

00:05:14.457 --> 00:05:18.757
And then 26 minus 12 is 14.

00:05:18.757 --> 00:05:22.017
So all of this, we need
to write it like this,

00:05:22.017 --> 00:05:25.187
all of this is going to be equal to 14.

00:05:25.187 --> 00:05:28.487
So how much water is necessary
if William uses 200 marbles?

00:05:28.487 --> 00:05:30.862
14 liters of water.

00:05:30.862 --> 00:05:35.523
26 minus 0.06 times 200.