Pixar in a Box

WEBVTT
Kind: captions
Language: en

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- Welcome to Pixar In A Box

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a collaboration between
Pixar Animation Studios

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- and the Khan Academy.

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- I'm Fran, I work here on
the technical side of things.

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- And I'm Alex, I work here

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on the art and story side of things.

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- (Fran) Pixar In A Box will introduce you

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to some of the fundamental
skills we use to make our movies.

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- (Alex) There's examples
of real life film challenges

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that show how an artist's idea can create

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a tidal wave of technical creativity.

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And how technical advances can inspire

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artists to think of new
ideas to bring to the screen.

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You'll meet artists, scientists,

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animators, coders, sculptors,

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all kinds of different people.

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- And the thing is, most of them use

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skills you may already
be learning in school.

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But here in the Box, you can see how

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we use those skills to make cool stuff.

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Before you dive into these lessons,

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you should see how it all
fits together inside Pixar,

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the real Pixar.

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Come on, check it out.

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(light, flittering music)

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- At Pixar on any given day

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we're working on various
stages of many films.

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- But it all begins with an idea.

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For example, Toy Story started with the

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notion that when kids leave the room
their toys come to life.

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And honestly I was convinced that my toys did that.

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(scampering instrumental music)

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- Our films start in story,

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along with the director and the writer

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we figure out what happens
using simple drawings.

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- (Fran) It's kind of like a comic book.

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- (Alex) Exactly, and while we're drawing,

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production designers and their team start

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designing the world and the characters.

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(smooth, cheerful orchestral music)

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While they're painting
and drawing and scultping

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our storyboards go to editorial,

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where they string together all

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the drawings that we've created.

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- We time them out, add music,
dialogue and sound effects.

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(clicking)

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- He's the only one who
knew what the heck was going

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aaaaaaahhhhhhh!

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- And as these shots go through
each stage in production,

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we'll update the scene
over and over again.

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And this is where the
science and the math come in.

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- Who knew what the heck was going

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aaaaaaahhhhhhh!

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(mischevious music)

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- What's next?

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Pipeline, we're ready to start
making the film you'll see

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in the theater and this
happens in a particular order.

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It's time for our technical
artist to figure out

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how we're going to create
the movie in the computer.

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- I've been here for nine
years and I still have no idea

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how you guys turn our drawings
into the finished film.

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- To be honest, something
stumps us on every movie.

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- Hey Gailen.

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- Hey.

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- You've been here since Toy Story,

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what's been hard on each film?

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- Well on Toy Story everything.

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Here let me show you.

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On Toy Story we were inventing

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the entire process from scratch.

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Monsters Incorporated, fur and clothing.

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On Cars, reflective metal surfaces.

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On every new film there's
a new technical challenge.

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On Inside Out we had to
deal with a character

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made of glowing particles and it took lots

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of people to figure out
just that one thing.

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- (Alex) Whoa, what is she made of?

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- Her shape will be made of
points and particles of light.

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She was tough, this is the 17th version.

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- And the characters have to move,

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so someone has to add
controls to the model.

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- Like a puppet, right?

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- Yes, except instead of strings,

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our animators will use a computer program

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to move the characters in a digital world.

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(bemused music)

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Next in the pipeline, sets.

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(playful music)

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- (Alex) So for Cars 2 you
built the entire city of London?

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- We needed a huge chunk of the city

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because Mayor and
McQueen speed through it.

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So we figured out to grow buildings

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with enough variation
for them to look real.

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- (Fran) And we move through that set

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with our virtual cameras.

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Next up, animation.

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- I do know what the animators do.

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They bring the characters to life.

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- (Fran) You see how she's moving but

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her clothes and hair are missing?

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Adding and moving those
elements is going to be

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someone else's job
further down the pipeline.

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And by somebody else, I mean me

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and about 20 other
simulation technical artists.

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Where are you getting all these shirts?

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We have to build everything you see,

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including the textures and surfaces

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which help make the world
and characters believable.

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Next stop, lighting.

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- Ironically it's really dark
in the lighting department.

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(playful music)

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So when you start, there are no lights?

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- No any source of light is something

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we have to add into the scene.

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In this shot alone there are 230 lights.

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- (Fran) Last stop, the Renderfarm.

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- A film is really a
series of images or frames.

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There's 24 of them every second.

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(light, moving music)

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- (Fran) This is where we make the frames.

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Everything comes together here,

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all the art, math and science.

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A single frame can take more
than 24 hours to render,

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and that's just one frame,

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and that's assuming we
don't run into any snags.

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- Whoa! (thuds)

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(soaring orchestral music)

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- Wow, it's incredible seeing final shots.

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And now that I know how
they're made, even cooler.

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- So that's the tour, our next step is to

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jump back into the Box
and choose a lesson.

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What do you want to try?

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- Well this one on building
robots using combinatorics,

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that looks pretty cool.

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- What about the one on sets and staging?

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- Yeah or the one on
animating The Incredibles.

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- (Fran) Awesome, I
always wanted to animate.

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(silly trumpet music)


Alcohol or ethanol fermentation

WEBVTT
Kind: captions
Language: en

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- [Voiceover] We've
already seen multiple times

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that glycolysis is the process

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where we start with a glucose molecule

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which has six carbons and
we're able to break it down

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into two pyruvate molecules

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which each have three carbons.

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And in the process of doing so,

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we produce a net of two ATPs,

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we reduce NAD+ to NADH,

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we're adding a hydride
anion, so this is reduction,

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this is reduction going
on right over there

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and then we say, "Well, what happens?"

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and, of course, we have the
water, some water produced

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and some hydrogen ions.

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But then the next question
is, well what happens next?

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And we could talk about
cellular respiration

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and that's what will happen
in many organisms including...

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It's happening in my body right now

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in order for me to generate
as much ATP as possible.

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That's why I inhale oxygen.

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Because that oxygen is needed
for cellular respiration.

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But sometimes there isn't oxygen around

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or I'm the type of organism
that just doesn't use oxygen

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or doesn't conduct cellular respiration.

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Now we've already seen one example of that

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and that was lactic acid fermentation.

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It's a way of recycling these NADH's,

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oxidizing them back.

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Oxidizing them back to NAD+.

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So this oxidation,

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this oxidation from NADH to NAD+,

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this is what we do in fermentation.

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Actually, both variations of fermentation

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that we're going to talk about.

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And lactic acid
fermentation, we talked about

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taking the pyruvates to oxidize the NADH

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and in doing so the pyruvate
turns into lactic acid

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and that's used in yogurt
production, in sauerkraut,

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it's also used in your muscles,

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when your muscles run out of oxygen.

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If you're kind of sprinting
or your lifting some

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really really really heavy weights,

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you probably have some lactic
acid fermentation going on.

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But now we're gonna talk about
another type of fermentation

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and that is alcohol fermentation.

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Alcohol.

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Alcohol or sometimes called
ethanol fermentation.

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Ethanol fermentation.

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Because when we talk
about drinking alcohol,

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we're talking about ethanol.

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Ethanol

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fermentation.

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And as you might guess,
this is the process

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by which alcohol is produced
and things like bread.

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It's used in baking a lot.

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But let's first think about the reaction

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and then we'll talk about

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maybe where you might
have encountered it last.

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So we saw that pyruvate...

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When we finish glycolysis, we have,

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for each molecule of glucose,

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we have two molecules of pyruvate

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and so each of these
molecules of pyruvate...

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The first step is they
lose their carboxyl group

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right over here and that's facilitated by

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pyruvate decarboxylase.

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This name is all about getting rid of

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a pyruvate's carboxyl group.

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And you can see, when you
remove a carboxyl group,

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it's a carbon and two oxygens,

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that is going to be,

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and I'm not showing you the
detailed mechanism here,

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but that's going to be a carbon dioxide

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that is released and then what's leftover,

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what's leftover, and once again

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I'm not showing you
the detailed mechanism,

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but you can account at
least for the various atoms,

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what's leftover is acetaldehyde.

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Acetaldehyde, I have trouble saying that.

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And then the next step,

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the acetaldehyde can be reduced.

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One way you can think about it,

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you can say it's reduced by the NADH

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to turn into ethanol.

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It's a similar mechanism

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to what we saw in lactic acid fermentation

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where we went from
pyruvate to lactic acid,

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but now it's acetaldehyde
going to ethanol.

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So this right over here,

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this is, this is reduced.

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Reduced.

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It gains a hydride.

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One way to think about it is

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it's gaining a hydride anion

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and in the process that
this is being reduced,

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the NADH is being oxidized.

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So this, this is being oxidized.

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And the whole point of this,

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the whole reason why it occurs

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is so that you can oxidize the NADH

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to have more fresh NAD+ for
glycolysis to occur again.

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And as I mentioned in the video
on lactic acid fermentation,

00:04:06.692 --> 00:04:08.004
it's a little bit of a shame,

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because the pyruvate
still has energy into it.

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If you had oxygen around,

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you could have cellular respiration,

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you could go into the Krebs cycle,

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the citric acid cycle,

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and derive more energy from it.

00:04:17.709 --> 00:04:20.844
The NADH also, the
process of oxidizing it,

00:04:20.844 --> 00:04:23.490
in theory, you can use it
to generate more energy.

00:04:23.490 --> 00:04:24.605
But we're not doing that

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when we do either type of fermentation

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whether ethanol fermentation

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or we're talking about
lactic acid fermentation.

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But this is a process that has

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proven very useful for human civilization.

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This right over here,
the most famous actor

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when we're talking about
ethanol fermentation

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or alcohol fermentation is
this character right over here.

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This is a yeast cell.

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Yeasts are categorized as fungi.

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That is a yeast cell.

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It's a handful of micometers in diameter,

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although they can vary.

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And variations of yeast are used

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in things like bread
making and wine making

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or alcohol production.

00:05:03.437 --> 00:05:05.259
Beer, whatever you wanna talk about.

00:05:05.259 --> 00:05:07.123
Because what the yeast does,

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it uses, it digests the
sugar, it performs glycolysis

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and then it performs alcohol fermentation.

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What makes bread so fluffy
is this step right over here.

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When the carboxyl group gets
stripped off of the pyruvate

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facilitated by the pyruvate decarboxylase,

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and the carbon dioxide gets released,

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it makes the bread fluffy.

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It bubbles through the dough

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and gives it its nice spongy flavor.

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But you also have ethanol being produced.

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So it's actually an interesting fact

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that a lot of bread...

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You know, I've been reading up on it,

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a lot of the ethanol might get baked off,

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but bread will usually have

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trace amounts, trace
amounts of alcohol in it.

00:05:44.799 --> 00:05:47.289
So bread will have trace amounts,

00:05:47.289 --> 00:05:49.470
trace amounts of ethanol in it

00:05:49.470 --> 00:05:50.630
because the whole process,

00:05:50.630 --> 00:05:53.173
you're using yeast to leaven the bread.

00:05:53.173 --> 00:05:54.858
To give it this fluffiness,

00:05:54.858 --> 00:05:56.706
that yeast is performing glycolysis

00:05:56.706 --> 00:05:59.430
and alcohol fermentation
is producing ethanol

00:05:59.430 --> 00:06:00.905
and in the process the carbon dioxide

00:06:00.905 --> 00:06:02.926
that gives the fluffiness
but there's still going to be

00:06:02.926 --> 00:06:07.163
some trace amounts of ethanol.

00:06:07.163 --> 00:06:08.961
Now, of course, if you're
talking about wine production,

00:06:08.961 --> 00:06:12.437
your goal is to get to the actual ethanol

00:06:12.437 --> 00:06:15.625
which we view as traditional
drinking alcohol.

00:06:15.625 --> 00:06:20.625
So yeast plays a fairly
important role in our society.


Lactic acid fermentation

WEBVTT
Kind: captions
Language: en

00:00:00.186 --> 00:00:02.260
- When we first learned about glycolysis,

00:00:02.260 --> 00:00:05.255
we saw that if you start
with a molecule of glucose,

00:00:05.255 --> 00:00:08.125
and you carry forward with glycolysis,

00:00:08.125 --> 00:00:10.904
that glucose, which is a six carbon sugar,

00:00:10.904 --> 00:00:12.592
it's got oxygens and hydrogens as well,

00:00:12.592 --> 00:00:14.110
but it's a six carbon sugar,

00:00:14.110 --> 00:00:16.643
it gets split into two pyruvate molecules,

00:00:16.643 --> 00:00:19.713
and each pyruvate has three carbons.

00:00:19.713 --> 00:00:22.679
And in the process of doing
so, we're able to produce

00:00:22.679 --> 00:00:25.584
a net of two ATPs.

00:00:25.584 --> 00:00:27.816
We use two ATPs in the investment phase,

00:00:27.816 --> 00:00:30.777
then we produce four
ATPs in the payoff phase

00:00:30.777 --> 00:00:34.166
for a net of two ATPs.

00:00:34.166 --> 00:00:36.186
But that's not all that happens.

00:00:36.186 --> 00:00:39.679
You also have two NAD molecules,

00:00:39.679 --> 00:00:42.831
nicotinamide adenine dinucleotide,

00:00:42.831 --> 00:00:45.872
getting reduced to NADH.

00:00:45.872 --> 00:00:47.081
And why is it getting reduced?

00:00:47.081 --> 00:00:50.167
Well we see it's a positive,
it's a positive molecule here,

00:00:50.167 --> 00:00:53.116
it becomes a neutral
molecule, it gains electrons.

00:00:53.116 --> 00:00:57.300
So this over here becomes reduced.

00:00:57.300 --> 00:00:59.079
Now, the next question you might have is,

00:00:59.079 --> 00:01:01.009
"Well what happens next?"

00:01:01.009 --> 00:01:04.374
Well if you're me or you,
you might continue on

00:01:04.374 --> 00:01:06.910
and if you have enough
oxygen you'll continue on

00:01:06.910 --> 00:01:08.742
with cellular respiration.

00:01:08.742 --> 00:01:10.454
Things move on to the mitochondria

00:01:10.454 --> 00:01:13.594
because the pyruvates and the NADHs can be

00:01:13.594 --> 00:01:16.243
can also be used to produce more ATP.

00:01:16.243 --> 00:01:18.656
The pyruvate gets more broken down

00:01:18.656 --> 00:01:22.140
in the Krebs cycle, also known
as the citric acid cycle.

00:01:22.140 --> 00:01:25.984
And also and that produces ATPs and NADHs

00:01:25.984 --> 00:01:27.745
and the NADHs can participate

00:01:27.745 --> 00:01:29.947
in the electron transport
chain which eventually leads

00:01:29.947 --> 00:01:32.751
to even more ATPs being produced.

00:01:32.751 --> 00:01:34.019
But what if you're in a situation

00:01:34.019 --> 00:01:35.412
that maybe we don't have oxygen,

00:01:35.412 --> 00:01:38.615
or maybe you're just like,
you're just the type of organism

00:01:38.615 --> 00:01:40.980
that doesn't like to use
oxygen or doesn't know

00:01:40.980 --> 00:01:44.034
how to use oxygen, what happens next?

00:01:44.034 --> 00:01:45.844
Well what we're gonna
talk about in this video

00:01:45.844 --> 00:01:47.736
is one potential pathway,

00:01:47.736 --> 00:01:50.816
and that's lactic acid fermentation,

00:01:50.816 --> 00:01:53.177
which is one of the two
major forms of fermentation.

00:01:53.177 --> 00:01:57.128
Lactic acid fermentation.

00:01:57.128 --> 00:02:00.158
Fermen,

00:02:00.158 --> 00:02:02.363
lactic acid fermenation.

00:02:02.363 --> 00:02:06.241
And lactic acid fermentation
isn't so much about producing

00:02:06.241 --> 00:02:09.044
more ATPs, it's more about recycling

00:02:09.044 --> 00:02:10.857
the pyruvate and the NADH.

00:02:10.857 --> 00:02:13.140
Even though the pyruvate and NADH can,

00:02:13.140 --> 00:02:17.533
it has free energy to give
that could be converted to ATP,

00:02:17.533 --> 00:02:19.711
if we're gonna be doing
lactic acid fermentation,

00:02:19.711 --> 00:02:21.713
we kind of give up on that,

00:02:21.713 --> 00:02:23.940
and then we actually use the pyruvate

00:02:23.940 --> 00:02:26.374
to oxidize the NADH

00:02:26.374 --> 00:02:28.704
to become NAD+

00:02:28.704 --> 00:02:32.654
so that we have more NAD+ for
glycolysis to occur again.

00:02:32.654 --> 00:02:35.201
So organisms that do fermentation,

00:02:35.201 --> 00:02:38.171
their main energy source
is the glycolysis,

00:02:38.171 --> 00:02:41.246
and then the fermentation
is all about recycling

00:02:41.246 --> 00:02:44.670
what it views as waste
materials, pyruvate and NADH,

00:02:44.670 --> 00:02:46.483
so that you can have more NAD

00:02:46.483 --> 00:02:48.979
to have glycolysis occurring again.

00:02:48.979 --> 00:02:51.950
Now, you might say, "Oh
is this some strange thing

00:02:51.950 --> 00:02:54.676
"that we don't encounter much in life?"

00:02:54.676 --> 00:02:57.110
Probably everyday or
maybe at least every week,

00:02:57.110 --> 00:02:59.717
you probably consume some organisms

00:02:59.717 --> 00:03:02.330
that perform lactic acid fermentation.

00:03:02.330 --> 00:03:04.400
This right over here, this
is a picture of yogurt.

00:03:04.400 --> 00:03:08.377
Yogurt is what we get when you have

00:03:08.377 --> 00:03:12.745
species of lactobacillus
digesting the sugars in the milk

00:03:12.745 --> 00:03:15.177
and then they're performing glycolysis

00:03:15.177 --> 00:03:18.097
and then they perform
lactic acid fermentation,

00:03:18.097 --> 00:03:21.512
converting the pyruvate into lactate.

00:03:21.512 --> 00:03:25.002
Or if you view the conjugate
acid version of it,

00:03:25.002 --> 00:03:28.451
lactic acid, you could say
pyruvic acid, lactic acid.

00:03:28.451 --> 00:03:32.047
Pyruvate is the conjugate
base for pyruvic acid.

00:03:32.047 --> 00:03:35.666
Lactate is the conjugate
base for lactic acid.

00:03:35.666 --> 00:03:38.418
But that's what's giving it

00:03:38.418 --> 00:03:40.311
its uniquely yogurt taste,

00:03:40.311 --> 00:03:43.747
it's this bacteria here,

00:03:43.747 --> 00:03:44.612
the lactobacillus,

00:03:44.612 --> 00:03:46.886
this is just one variation of it,

00:03:46.886 --> 00:03:49.597
and there's slightly different
variations of lactobacillus

00:03:49.597 --> 00:03:51.457
that do each of these foods.

00:03:51.457 --> 00:03:53.346
But this is yogurt, this right over here,

00:03:53.346 --> 00:03:55.582
if you're into Korean
food, this is kimchi,

00:03:55.582 --> 00:03:58.107
this uses a variation of lactobacillus

00:03:58.107 --> 00:04:01.493
to once again perform
lactic acid fermentation

00:04:01.493 --> 00:04:03.356
on the sugars in the vegetables.

00:04:03.356 --> 00:04:05.480
This is sauerkraut, once again,

00:04:05.480 --> 00:04:09.067
a variation of lactobacillus,
a species of lactobacillus

00:04:09.067 --> 00:04:12.144
performing lactic acid fermentation

00:04:12.144 --> 00:04:13.660
on the sugars in the cabbage.

00:04:13.660 --> 00:04:16.037
Sauerkraut literally
means, "sour cabbage."

00:04:16.037 --> 00:04:17.819
That's what it is.

00:04:17.819 --> 00:04:20.752
So let's think a little bit
more about what's going on.

00:04:20.752 --> 00:04:24.445
So as I mentioned, it's
all about taking your

00:04:24.445 --> 00:04:26.619
pyruvate or pyruvic acid,

00:04:26.619 --> 00:04:29.712
the way I've drawn it right
over here, this is pyruvic acid.

00:04:29.712 --> 00:04:34.311
Pyruvic, pyruvic acid,

00:04:34.311 --> 00:04:35.687
right over here.

00:04:35.687 --> 00:04:38.133
Because we have our hydrogen proton,

00:04:38.133 --> 00:04:39.515
if we lose our hydrogen proton,

00:04:39.515 --> 00:04:41.648
this is the same thing drawn again,

00:04:41.648 --> 00:04:43.708
but now we don't have
the hydrogen proton here,

00:04:43.708 --> 00:04:45.852
this oxygen kept that electron,

00:04:45.852 --> 00:04:48.443
and all of the other hydrogens,
all the hydrogens here,

00:04:48.443 --> 00:04:49.510
they are implicit.

00:04:49.510 --> 00:04:50.919
So the three hydrogens here,

00:04:50.919 --> 00:04:52.771
they're implicit on this carbon.

00:04:52.771 --> 00:04:54.673
I'm just drawing it with
a different notation.

00:04:54.673 --> 00:04:57.282
And so this one, where
we've lost the proton,

00:04:57.282 --> 00:04:59.321
we would call this pyruvate.

00:04:59.321 --> 00:05:03.062
Pyruvate.

00:05:03.062 --> 00:05:04.696
And what we have happening is

00:05:04.696 --> 00:05:07.222
that the pyruvic acid, or the pyruvate

00:05:07.222 --> 00:05:10.907
is used to oxidize the NADH,

00:05:10.907 --> 00:05:14.886
take away a hydride, take
away an electron from,

00:05:14.886 --> 00:05:18.707
actually more than just electron but net,

00:05:18.707 --> 00:05:21.634
you have the NADH losing electrons.

00:05:21.634 --> 00:05:25.138
And so if it's losing electrons,
it's getting oxidized.

00:05:25.138 --> 00:05:27.522
So it's oxidized.

00:05:28.581 --> 00:05:32.392
So that the NAD+ can be
reused in glycolysis.

00:05:32.392 --> 00:05:35.608
And when pyruvic acid
does this to the NADH,

00:05:35.608 --> 00:05:38.041
it gets reduced,

00:05:39.191 --> 00:05:40.696
it gets reduced,

00:05:41.946 --> 00:05:44.483
it gets reduced, it gains electrons

00:05:44.483 --> 00:05:46.724
and, if we're thinking
about the acid forms,

00:05:46.724 --> 00:05:50.060
it would turn into this right
over here is lactic acid.

00:05:50.060 --> 00:05:51.595
Lactic,

00:05:51.595 --> 00:05:53.686
lactic,

00:05:53.686 --> 00:05:55.177
lactic acid.

00:05:55.177 --> 00:05:57.952
And that's why we call it
lactic acid fermentation,

00:05:57.952 --> 00:05:59.377
'cause you're taking that pyruvate,

00:05:59.377 --> 00:06:01.230
if you had oxygen around,
or if you knew how to do it,

00:06:01.230 --> 00:06:03.507
use the oxygen, you might continue on

00:06:03.507 --> 00:06:05.756
with cellular respiration
and use that for energy.

00:06:05.756 --> 00:06:07.485
But lactic acid fermentation,

00:06:07.485 --> 00:06:10.693
we use it to oxidize the NADH

00:06:10.693 --> 00:06:12.908
so we get more NAD+.

00:06:12.908 --> 00:06:14.820
And let's just now get
a better appreciation

00:06:14.820 --> 00:06:16.070
for all of this happens.

00:06:16.070 --> 00:06:18.609
So the first thing that I want to show you

00:06:18.609 --> 00:06:20.267
because a lot of times in biology classes,

00:06:20.267 --> 00:06:22.725
you just learn NAD, NADH,

00:06:22.725 --> 00:06:25.140
and it just seems like this
somewhat abstract molecule.

00:06:25.140 --> 00:06:26.252
But this is a picture of it.

00:06:26.252 --> 00:06:29.250
This is nicotinamide adenine dinucleotide.

00:06:29.250 --> 00:06:31.416
And it's kind of a mouthful,

00:06:31.416 --> 00:06:33.608
but when you break it down,
you see these patterns

00:06:33.608 --> 00:06:35.200
that you see repeatedly in biology.

00:06:35.200 --> 00:06:37.895
This is, this right over here is

00:06:37.895 --> 00:06:40.039
what gives us the nicotinamide.

00:06:40.039 --> 00:06:42.550
This right over here is
our good friend adenine.

00:06:42.550 --> 00:06:45.016
We see that in ATP, we see that also

00:06:45.016 --> 00:06:48.389
as one of the nitrogenous
bases in DNA and RNA.

00:06:48.389 --> 00:06:50.684
You have ribose right over here,

00:06:50.684 --> 00:06:52.074
this is derived from ribose.

00:06:52.074 --> 00:06:54.867
You have a phosphate group,
you have a phosphate group.

00:06:54.867 --> 00:06:57.651
So, nicotinamide, adenine,

00:06:57.651 --> 00:07:00.581
you have a nucleotide right over there,

00:07:00.581 --> 00:07:03.640
you have another nucleotide
right over there,

00:07:03.640 --> 00:07:07.036
so it's nicotine adenine dinucleotide.

00:07:07.036 --> 00:07:08.336
So the name makes a lot of sense.

00:07:08.336 --> 00:07:09.514
But I wanted to show this to you

00:07:09.514 --> 00:07:10.962
to get an appreciation that it's

00:07:10.962 --> 00:07:15.518
a fairly involved molecule over here.

00:07:15.518 --> 00:07:17.683
You know sometimes when you
just see the letters, NAD,

00:07:17.683 --> 00:07:19.544
you don't get a full appreciation for it.

00:07:19.544 --> 00:07:22.275
And it's a coenzyme, and we learned

00:07:22.275 --> 00:07:24.281
about coenzymes in other videos.

00:07:24.281 --> 00:07:27.058
Where the enzyme that catalyzes this

00:07:27.058 --> 00:07:29.403
is lactic acid dehydrogenase.

00:07:29.403 --> 00:07:31.924
And remember enzymes

00:07:31.924 --> 00:07:36.610
are for the most part just
these big protein structures,

00:07:36.610 --> 00:07:40.938
so all folded up in all different ways.

00:07:40.938 --> 00:07:43.445
And then you have the NAD,

00:07:43.445 --> 00:07:46.010
the NAD,

00:07:46.010 --> 00:07:50.168
or in the case of lactic acid
fermentation right over here,

00:07:50.168 --> 00:07:54.540
you have the NADH, so this
is the NADH right over here,

00:07:54.540 --> 00:07:56.664
and I'm just kind of drawing
what it could look like.

00:07:56.664 --> 00:07:58.425
This isn't actually what it looks like.

00:07:58.425 --> 00:08:02.352
It's going to react with the pyruvate.

00:08:02.352 --> 00:08:04.032
And let me do that in a...

00:08:04.971 --> 00:08:09.203
It's going to react with the pyruvate

00:08:09.203 --> 00:08:11.618
and by doing so, even though the pyruvate

00:08:11.618 --> 00:08:12.848
you might formally consider

00:08:12.848 --> 00:08:14.579
to be the substrate of the enzyme,

00:08:14.579 --> 00:08:19.315
the whole purpose is to get
your NADH to be oxidized,

00:08:19.315 --> 00:08:22.381
to lose a hydrogen and an extra electron.

00:08:22.381 --> 00:08:24.375
So a hydrogen proton, a hydrogen electron,

00:08:24.375 --> 00:08:25.644
and another electron.

00:08:25.644 --> 00:08:27.557
So really lose a hydride.

00:08:27.557 --> 00:08:29.088
So how does that happen?

00:08:29.088 --> 00:08:32.652
Well it happens because you
have this nitrogen here,

00:08:32.652 --> 00:08:34.978
it has an extra lone pair of electrons,

00:08:34.978 --> 00:08:38.743
so it can form, that lone
pair can form a double bond

00:08:38.743 --> 00:08:40.073
right over there.

00:08:40.073 --> 00:08:42.942
Well if that carbon, if
this now has a double bond,

00:08:42.942 --> 00:08:45.176
this carbon has to let
go of this double bond,

00:08:45.176 --> 00:08:47.651
so that goes over there.

00:08:47.651 --> 00:08:51.182
That goes over there.

00:08:51.182 --> 00:08:54.044
And then if there's now
a double bond over here,

00:08:54.044 --> 00:08:55.548
is which we see in the end product,

00:08:55.548 --> 00:08:57.409
this carbon's gonna have
to let go of a bond.

00:08:57.409 --> 00:09:00.318
And it lets go of the entire covalent bond

00:09:00.318 --> 00:09:02.917
with this hydrogen, so
both of the electrons.

00:09:02.917 --> 00:09:06.884
So it's gonna let go of
both of these electrons,

00:09:06.884 --> 00:09:08.211
right over here.

00:09:08.211 --> 00:09:10.221
And then both of those electrons

00:09:10.221 --> 00:09:14.812
can attach to this
carbon right over there.

00:09:14.812 --> 00:09:17.748
Now that carbon forms a new covalent bond,

00:09:17.748 --> 00:09:20.004
it has to let go of one
of its covalent bonds

00:09:20.004 --> 00:09:23.315
and so it could let of one of these

00:09:23.315 --> 00:09:25.060
double bonds with this oxygen.

00:09:25.060 --> 00:09:27.402
And those could either
go back to that oxygen,

00:09:27.402 --> 00:09:30.283
or more likely, they can be
used to grab a hydrogen proton

00:09:30.283 --> 00:09:34.341
maybe from a passing water
molecule or hydronium molecule.

00:09:34.341 --> 00:09:37.179
And so, I can draw it like this.

00:09:37.179 --> 00:09:40.950
Could grab a hydrogen, a hydrogen proton.

00:09:40.950 --> 00:09:42.843
And so what do we end up with?

00:09:42.843 --> 00:09:47.344
Well, that lone pair of
electrons is now there,

00:09:47.344 --> 00:09:49.695
we now have a double
bond right over there,

00:09:49.695 --> 00:09:51.247
now we've lost one of these hydrogens,

00:09:51.247 --> 00:09:52.785
I obviously haven't drawn
all of the hydrogens

00:09:52.785 --> 00:09:54.042
in these molecules.

00:09:54.042 --> 00:09:57.716
So now we have, now we are back to NAD.

00:09:57.716 --> 00:10:01.336
And since this nitrogen has essentially,

00:10:01.336 --> 00:10:03.768
it was neutral before, but now it is

00:10:03.768 --> 00:10:05.377
instead of keeping these two electrons,

00:10:05.377 --> 00:10:06.908
it's sharing these two electrons,

00:10:06.908 --> 00:10:09.644
so now it's going to
have a positive charge.

00:10:09.644 --> 00:10:13.370
So this is why we call it NAD+.

00:10:13.370 --> 00:10:16.144
It lost a hydrogen and an electron.

00:10:16.144 --> 00:10:19.146
A hydrogen, the hydrogen's
electron, and another electron

00:10:19.146 --> 00:10:23.064
so now it has a positive charge.

00:10:23.064 --> 00:10:27.207
And the pyruvate is now the conjugate,

00:10:27.207 --> 00:10:29.086
with the way I've drawn
it here, since we've,

00:10:29.086 --> 00:10:30.964
I show it deprotonated,

00:10:30.964 --> 00:10:33.852
I would say that this is now lactate.

00:10:33.852 --> 00:10:35.074
This is now lactate.

00:10:35.074 --> 00:10:37.006
If we had our proton over here,

00:10:37.006 --> 00:10:40.847
we would call it lactic acid.

00:10:40.847 --> 00:10:43.777
So anyway, hopefully you
get a kick out of this,

00:10:43.777 --> 00:10:45.908
I know I do, it's kind of interesting

00:10:45.908 --> 00:10:47.142
that all of this can happen.

00:10:47.142 --> 00:10:50.603
Lactobacillus isn't the only
organism that does this,

00:10:50.603 --> 00:10:53.009
but this is a fairly useful
organism for all sorts

00:10:53.009 --> 00:10:56.105
of delicious food that we have.

00:10:56.105 --> 00:10:57.350
I just get an appreciation,

00:10:57.350 --> 00:10:59.338
my mind's always blown that these

00:10:59.338 --> 00:11:01.824
fairly complex processes
are constantly occurring

00:11:01.824 --> 00:11:02.891
in nature all around us,

00:11:02.891 --> 00:11:04.582
sometimes even in our own bodies.

00:11:04.582 --> 00:11:07.641
And these organisms that we
would consider quite small,

00:11:07.641 --> 00:11:10.053
for example, this
lactobacillus right over here,

00:11:10.053 --> 00:11:13.244
this is on the order of

00:11:13.244 --> 00:11:15.349
five to 10 micrometers.

00:11:15.349 --> 00:11:20.258
Five micrometers, so five
millionths of a meter.


Steps of glycolysis

WEBVTT
Kind: captions
Language: en

00:00:00.341 --> 00:00:01.723
- [Voiceover] So let's give ourselves

00:00:01.723 --> 00:00:03.568
an overview of glycolysis.

00:00:03.568 --> 00:00:04.962
and glycolysis is an incredibly

00:00:04.962 --> 00:00:07.585
important biochemical pathway.

00:00:07.585 --> 00:00:10.441
It occures in practically
all life as we know it

00:00:10.441 --> 00:00:14.249
and it's all about taking
glucose as a fuel and,

00:00:14.249 --> 00:00:16.060
in the process of breaking it up,

00:00:16.060 --> 00:00:18.544
lycing the glucose, glycolysis,

00:00:18.544 --> 00:00:22.979
breaking it up into
two pyruvate molecules.

00:00:22.979 --> 00:00:25.846
Glucose is a six carbon molecule.

00:00:25.846 --> 00:00:29.306
Each of the pyruvates are
three carbon molecules.

00:00:29.306 --> 00:00:30.908
In the process of doing that,

00:00:30.908 --> 00:00:33.508
you produce two ATPs net.

00:00:33.508 --> 00:00:36.387
It actually turns out that
you need to use two ATPs

00:00:36.387 --> 00:00:38.339
and then you produce four.

00:00:39.409 --> 00:00:42.235
So you use two ATPs.

00:00:42.235 --> 00:00:43.930
That's often called the investment phase

00:00:43.930 --> 00:00:45.289
and we'll talk about that in a second.

00:00:46.939 --> 00:00:51.258
And then you produce four ATPs

00:00:51.258 --> 00:00:54.717
for a net of

00:00:54.717 --> 00:00:57.411
plus two ATPs

00:00:57.411 --> 00:00:59.152
and that's what we see right over here.

00:00:59.152 --> 00:01:03.656
You see a net of two ATPs
being produced directly

00:01:03.656 --> 00:01:08.149
by glycolysis, and then you also have

00:01:08.149 --> 00:01:11.736
the reduction of NAD to NADH.

00:01:11.736 --> 00:01:15.509
Remember, reduction is all
about gaining electrons,

00:01:15.509 --> 00:01:18.528
and over here, NAD, that's nicotinamide

00:01:18.528 --> 00:01:21.390
adinine dinucleotide, we
have other videos on that,

00:01:21.390 --> 00:01:23.747
it's an interesting
molecule, it's actually a

00:01:23.747 --> 00:01:26.974
fairly decent-sized molecule, we see this

00:01:26.974 --> 00:01:29.493
positive charge, but
then we see that not only

00:01:29.493 --> 00:01:32.036
does it gain a hydrogen, but
it loses its positive charge.

00:01:32.036 --> 00:01:34.915
It gains a hydrogen and an electron.

00:01:34.915 --> 00:01:37.051
You can think on a net basis it's gaining

00:01:37.051 --> 00:01:38.606
a hydride.

00:01:40.766 --> 00:01:44.412
Now a hydride anion's
not going to typically be

00:01:44.412 --> 00:01:47.036
all by itself, but on a net
basis, you can think about

00:01:47.036 --> 00:01:48.952
that's what's happening.

00:01:48.952 --> 00:01:52.968
And so it's gaining a
hydrogen and an extra electron

00:01:52.968 --> 00:01:57.488
and so this, the NAD+, this
is going to get reduced.

00:01:57.488 --> 00:02:00.831
That is going to get reduced to NADH.

00:02:00.831 --> 00:02:03.443
So this is getting reduced to NADH.

00:02:03.443 --> 00:02:06.763
And that NADH, it can then

00:02:06.763 --> 00:02:09.747
be oxidized in the
electron transport chain.

00:02:09.747 --> 00:02:11.581
We'll study that later
on when we think about

00:02:11.581 --> 00:02:14.240
oxidative phosphorylation, to produce

00:02:14.240 --> 00:02:15.900
even more ATPs.

00:02:15.900 --> 00:02:18.988
But on a very high-level, simple basis.

00:02:18.988 --> 00:02:21.101
Glucose being broken down in pyruvate,

00:02:21.101 --> 00:02:23.968
six carbons, three carbons each of these

00:02:23.968 --> 00:02:25.605
pyruvates, now there's
other things attached

00:02:25.605 --> 00:02:27.602
to the carbons, and we'll
see that in a little bit.

00:02:27.602 --> 00:02:31.885
Two ATPs net generated,
and you have the reduction

00:02:31.885 --> 00:02:36.506
of two NADs to two NADHs,
and those can be used

00:02:36.506 --> 00:02:39.706
later on to produce more ATPs.

00:02:39.706 --> 00:02:42.724
Now, glycolysis is
typically just the beginning

00:02:42.724 --> 00:02:44.140
of cellular respiration.

00:02:44.140 --> 00:02:48.207
If oxygen is around, then
you have these products,

00:02:48.207 --> 00:02:51.968
some of these moving into the mitochondria

00:02:51.968 --> 00:02:53.872
where you can have the citric acid cycle,

00:02:53.872 --> 00:02:58.226
Krebs cycle, and the oxidative
phosphorylation occur.

00:02:58.226 --> 00:03:00.629
If you don't have oxygen
around, then you're

00:03:00.629 --> 00:03:03.229
going to do anaerobic
respiration, or you're

00:03:03.229 --> 00:03:05.191
going to go into fermentation.

00:03:05.191 --> 00:03:07.014
We'll talk about that in
a future video, and that's

00:03:07.014 --> 00:03:09.428
really about figuring
out what to do with these

00:03:09.428 --> 00:03:13.863
products, and especially
replenishing your NAD+.

00:03:13.863 --> 00:03:15.871
Now that we have a very
high-level overview

00:03:15.871 --> 00:03:18.553
of glycolysis, let's get
a better appreciation

00:03:18.553 --> 00:03:20.968
for exactly what's going on.

00:03:20.968 --> 00:03:23.914
And whenever I look at these more detailed

00:03:25.417 --> 00:03:27.576
processes, the one
thing to just appreciate

00:03:27.576 --> 00:03:30.617
is how much complexity is occurring

00:03:30.617 --> 00:03:32.417
in all of your cells right now.

00:03:32.417 --> 00:03:35.145
This is fairly abstract,

00:03:35.145 --> 00:03:37.699
to even imagine these things, but this

00:03:37.699 --> 00:03:39.800
is happening throughout your body

00:03:39.800 --> 00:03:42.064
gazillions of times, right now.

00:03:42.064 --> 00:03:43.260
This isn't something that is somehow

00:03:43.260 --> 00:03:44.479
distant from you.

00:03:44.479 --> 00:03:46.498
And it's also fun to appreciate, well how

00:03:46.498 --> 00:03:48.356
all of this was discovered by scientists.

00:03:48.356 --> 00:03:51.119
That's a whole other
fascinating discussion.

00:03:51.119 --> 00:03:52.756
But the whole point of
this video is just to

00:03:52.756 --> 00:03:55.855
give us an appreciation
for the actual mechanism

00:03:55.855 --> 00:03:57.724
or the reaction by which it occurs.

00:03:57.724 --> 00:03:58.978
I'm not gonna go into the detailed

00:03:58.978 --> 00:04:01.381
organic chemistry mechanism.

00:04:01.381 --> 00:04:04.353
So over here, this is a
glucose molecule over here,

00:04:04.353 --> 00:04:07.894
you see one, two, three,
four, five, six carbons.

00:04:07.894 --> 00:04:11.295
And then the first step
is, it gets phosphorylated

00:04:11.295 --> 00:04:13.234
and we have a whole video
on the phosphorylation

00:04:13.234 --> 00:04:16.183
of glucose, and all of
these steps are facilitated

00:04:16.183 --> 00:04:17.216
with enzymes.

00:04:17.216 --> 00:04:20.374
The phosphorylation is
facilitated with the hexokinase.

00:04:20.374 --> 00:04:23.035
Kinase is a general term for an enzyme

00:04:23.035 --> 00:04:25.705
that either facilitates phosphorylation or

00:04:25.705 --> 00:04:29.211
dephosphorylates, it's dealing with

00:04:29.211 --> 00:04:31.150
phosphorylation, I guess you could say.

00:04:31.150 --> 00:04:33.007
And enzymes are all about lowering

00:04:33.007 --> 00:04:34.505
the activation energy.

00:04:34.505 --> 00:04:36.745
And the way that
hexokinases do, or part of

00:04:36.745 --> 00:04:39.369
how they do it, is they
involve the cofactor,

00:04:39.369 --> 00:04:40.843
a magnesium ion.

00:04:40.843 --> 00:04:43.003
And we've talked about
that in other videos,

00:04:43.003 --> 00:04:46.207
how cofactors can help an enzyme lower the

00:04:46.207 --> 00:04:47.553
activation energy.

00:04:47.553 --> 00:04:51.210
And to do the phosphorylation,
we use an ATP.

00:04:51.210 --> 00:04:54.670
So this is minus one ATP.

00:04:54.670 --> 00:04:58.205
So we are in the investment phase.

00:04:58.205 --> 00:05:01.147
But this reaction strongly goes from

00:05:01.147 --> 00:05:05.164
left to right, it's a
coupled reaction that,

00:05:05.164 --> 00:05:08.228
phosphorylating the glucose, that

00:05:08.228 --> 00:05:11.746
requires free energy, but
the ATP releases free energy

00:05:11.746 --> 00:05:13.905
you couple these reactions,
it strongly goes from

00:05:13.905 --> 00:05:15.298
left to right.

00:05:15.298 --> 00:05:17.899
Now, and just to be clear what happened,

00:05:17.899 --> 00:05:21.242
this over here got replaced, or maybe

00:05:21.242 --> 00:05:25.387
I should say this over here got replaced

00:05:25.387 --> 00:05:27.882
with that over there.

00:05:27.882 --> 00:05:29.690
Just to keep track of what's happening.

00:05:29.690 --> 00:05:32.430
Now, another enzyme-catalyzed reaction,

00:05:32.430 --> 00:05:34.171
this one is actually
an equilibrium, it can

00:05:34.171 --> 00:05:36.992
go both ways, but as we'll see, the right

00:05:36.992 --> 00:05:39.639
side or the things that
are further into the

00:05:39.639 --> 00:05:41.810
glycolysis process, these are constantly

00:05:41.810 --> 00:05:44.190
being turned into further
products, so their

00:05:44.190 --> 00:05:45.792
concentrations are
going to go down, and so

00:05:45.792 --> 00:05:48.439
the reaction will tend to go that way.

00:05:48.439 --> 00:05:51.608
Although this particular
reaction, going from

00:05:51.608 --> 00:05:54.255
glucose 6-phosphate to
fructose 6-phosphate,

00:05:54.255 --> 00:05:56.275
this could be an equilibrium.

00:05:56.275 --> 00:05:58.457
But the enzyme that facilitates this,

00:05:58.457 --> 00:06:02.915
phosphoglucose isomerase, these are

00:06:02.915 --> 00:06:06.080
enzymes that help go from one isomer of

00:06:06.080 --> 00:06:07.775
a molecule to another isomer.

00:06:07.775 --> 00:06:09.040
And that's what's happening here.

00:06:09.040 --> 00:06:12.360
Instead of this oxygen
being bound to this carbon,

00:06:12.360 --> 00:06:14.659
this bond forms with this carbon.

00:06:14.659 --> 00:06:16.899
So you have fructose,
you have the five-carbon

00:06:16.899 --> 00:06:19.453
ring over here, or you
have the five-element

00:06:19.453 --> 00:06:21.102
ring, you have four carbons in it,

00:06:21.102 --> 00:06:23.268
versus a six-element ring
where right over here

00:06:23.268 --> 00:06:24.905
you have five carbons.

00:06:24.905 --> 00:06:27.146
So this bond goes to this
carbon right over here

00:06:27.146 --> 00:06:28.887
and that's the main difference.

00:06:28.887 --> 00:06:31.209
And then you have another, very strong

00:06:31.209 --> 00:06:33.508
forward reaction, once again facilitated

00:06:33.508 --> 00:06:37.931
by ATP, and this is done
by phosphofructokinase.

00:06:37.931 --> 00:06:39.684
It has the word kinase in it.

00:06:39.684 --> 00:06:41.947
And it's using up the ATP, you can guess

00:06:41.947 --> 00:06:42.911
what's going to happen.

00:06:42.911 --> 00:06:44.884
We're going to attach another phosphate

00:06:44.884 --> 00:06:48.878
group to the fructose 6-phosphate, and now

00:06:48.878 --> 00:06:50.712
you have two of these phosphate groups.

00:06:50.712 --> 00:06:54.497
So this hydrogen right over here is now

00:06:54.497 --> 00:06:58.978
replaced with another phosphate group.

00:06:58.978 --> 00:07:00.499
And once again it's facilitated by the

00:07:00.499 --> 00:07:03.575
magnesium cofactor, it helps stabilize

00:07:03.575 --> 00:07:05.432
some of the negative
charge associated with

00:07:05.432 --> 00:07:08.149
the phosphate groups, we talk
about that in other videos.

00:07:08.149 --> 00:07:10.773
But the important thing
is, it uses another ATP.

00:07:10.773 --> 00:07:13.545
We're still in the investment phase,

00:07:13.545 --> 00:07:16.389
negative one ATP.

00:07:16.389 --> 00:07:17.805
And every time I look at
this it's just fascinating

00:07:17.805 --> 00:07:20.290
that all of this stuff is
happening in your cells

00:07:20.290 --> 00:07:21.671
as we speak.

00:07:21.671 --> 00:07:24.213
In fact, in order for me to
speak this has to happen,

00:07:24.213 --> 00:07:26.454
because my body needs to
take glucose and come up

00:07:26.454 --> 00:07:30.064
with some energy to turn
into ATPs so that my muscles

00:07:30.064 --> 00:07:32.769
can actually move and I can
actually inhale and exhale

00:07:32.769 --> 00:07:34.267
and all the things that
I need to do for speech.

00:07:34.267 --> 00:07:37.065
So appreciate what's going on over here.

00:07:37.065 --> 00:07:38.806
Now the next step we talk about, the whole

00:07:38.806 --> 00:07:43.008
process of glycolysis is lysing glucose.

00:07:43.008 --> 00:07:45.493
And over here this is derived from

00:07:45.493 --> 00:07:47.269
glucose and some phosphates, and the

00:07:47.269 --> 00:07:49.707
next step, we're actually
going to break it up.

00:07:49.707 --> 00:07:52.029
And we're going to break
it up using the enzyme

00:07:52.029 --> 00:07:54.269
fructose biphosphate aldolase.

00:07:54.269 --> 00:07:57.787
Aldolase enzymes facilitate
the aldol reaction.

00:07:57.787 --> 00:08:00.909
And this one, the aldol reaction could be

00:08:00.909 --> 00:08:05.449
to merge two molecules or in this case,

00:08:05.449 --> 00:08:07.503
we're going to break them up.

00:08:07.503 --> 00:08:10.963
And we break them up into
two three-carbon chains.

00:08:10.963 --> 00:08:12.507
Now these two three-carbon chains,

00:08:12.507 --> 00:08:14.782
glyceraldehyde 3-phosphate,
and this character

00:08:14.782 --> 00:08:17.104
right over here, they
can be converted between

00:08:17.104 --> 00:08:19.461
the two with another isomerase, this

00:08:19.461 --> 00:08:22.479
triosephosphate isomerase right over here.

00:08:22.479 --> 00:08:25.602
So at this point in glycloysis,
we can think of ourselves

00:08:25.602 --> 00:08:27.424
as really having two of these.

00:08:27.424 --> 00:08:30.988
So let's say two times
glyceraldehyde 3-phosphate.

00:08:30.988 --> 00:08:32.521
So as we go further on, just imagining

00:08:32.521 --> 00:08:36.886
this happening twice for
every glucose molecule.

00:08:36.886 --> 00:08:38.441
And any time you get
confused, I encourage you

00:08:38.441 --> 00:08:39.683
to pause the video.

00:08:39.683 --> 00:08:42.319
See how these pieces and these pieces

00:08:42.319 --> 00:08:45.140
put together, can form that over there.

00:08:45.140 --> 00:08:46.661
So now we have another reaction,

00:08:46.661 --> 00:08:49.667
it's facilitated by a dehydrogenase.

00:08:49.667 --> 00:08:52.906
Dehydrogenases usually are involved in

00:08:52.906 --> 00:08:56.342
this case, this is the reduction of NAD.

00:08:56.342 --> 00:08:58.444
We saw that in the overview video.

00:08:58.444 --> 00:09:02.286
So NAD is being reduced.

00:09:02.286 --> 00:09:04.736
And this can be used, this NADH later on

00:09:04.736 --> 00:09:06.698
can be used in the
electron transport chain

00:09:06.698 --> 00:09:09.577
to potentially produce some more ATP,

00:09:09.577 --> 00:09:12.699
but in that process we also add

00:09:12.699 --> 00:09:14.278
another phosphate group to the

00:09:14.278 --> 00:09:16.205
glyceraladehyde 3-phosphate.

00:09:16.205 --> 00:09:19.456
So you see this phosphate group

00:09:19.456 --> 00:09:23.710
right over here that wasn't there before.

00:09:26.966 --> 00:09:29.984
And actually this right over here is

00:09:29.984 --> 00:09:33.641
I should have arrows on both sides,

00:09:33.641 --> 00:09:35.278
this right over here, that reaction

00:09:35.278 --> 00:09:36.926
could actually go both directions.

00:09:36.926 --> 00:09:39.063
Actually, that reaction can as well.

00:09:39.063 --> 00:09:43.265
And then, we are now

00:09:43.265 --> 00:09:45.900
going to be in the payoff phase.

00:09:45.900 --> 00:09:48.303
So this right over here, we're starting

00:09:48.303 --> 00:09:50.033
with this molecule that has these

00:09:50.033 --> 00:09:52.448
two phosphate groups, and then

00:09:52.448 --> 00:09:55.350
using the phosphoglycerate kinase, we're

00:09:55.350 --> 00:09:57.323
able to pop one of those
phosphate groups off

00:09:57.323 --> 00:10:01.131
and in the process, produce ATP.

00:10:01.131 --> 00:10:02.942
Now we might want to say plus one ATP,

00:10:02.942 --> 00:10:03.906
but we have to remember, this is now

00:10:03.906 --> 00:10:05.171
happening twice.

00:10:05.171 --> 00:10:07.946
Cuz we had two of those
glyceraldehyde 3-phosphates,

00:10:07.946 --> 00:10:09.606
so now we could say,
if we're talking about

00:10:09.606 --> 00:10:13.866
this happening twice, plus two ATPs.

00:10:13.866 --> 00:10:16.142
We are now in the payoff phase.

00:10:16.142 --> 00:10:18.138
Then you have, facilitated by the

00:10:18.138 --> 00:10:21.691
phosphoglycerate mutase, a mutase is a

00:10:21.691 --> 00:10:23.235
class of isomerases.

00:10:23.235 --> 00:10:24.396
I have trouble saying that.

00:10:24.396 --> 00:10:26.021
That'll take a functional
group from one place

00:10:26.021 --> 00:10:28.157
to another, or take one part of a molecule

00:10:28.157 --> 00:10:31.162
to another part, and you
see this phosphate group

00:10:33.315 --> 00:10:37.192
moving on from this carbon
to the middle carbon.

00:10:37.192 --> 00:10:38.725
And so that's what that's doing.

00:10:38.725 --> 00:10:41.731
Then we use an enloase to get over here

00:10:41.731 --> 00:10:46.469
and then the pyruvate kinase, and here

00:10:46.469 --> 00:10:47.888
the kinase is going to be used to

00:10:47.888 --> 00:10:50.802
dephosphorylate this
molecule right over here,

00:10:50.802 --> 00:10:52.601
and it gets us to the way I've drawn it is

00:10:52.601 --> 00:10:55.829
pyruvic acid, since I've
drawn the hydrogen here,

00:10:55.829 --> 00:10:57.837
and if the hydrogen is
let go and this oxygen

00:10:57.837 --> 00:11:01.203
hogs the electron, we
would call this pyruvate.

00:11:01.203 --> 00:11:04.419
And this is considered to be the end of,

00:11:04.419 --> 00:11:06.996
I guess you could say,
mainstream glycolysis.

00:11:06.996 --> 00:11:09.179
But what happened, and I don't want to

00:11:09.179 --> 00:11:11.327
glaze over what happened over here,

00:11:11.327 --> 00:11:14.287
this ADP got converted
to another ATP, but it's

00:11:14.287 --> 00:11:15.796
going to happen twice.

00:11:15.796 --> 00:11:19.418
So this is another plus two ATPs.

00:11:19.418 --> 00:11:22.436
So hopefully you see the investment phase,

00:11:22.436 --> 00:11:25.032
we use an ATP right over
here to phosphorylate

00:11:25.032 --> 00:11:28.418
the glucose, we use
another ATP right over here

00:11:28.418 --> 00:11:30.461
to throw that second phosphate group

00:11:30.461 --> 00:11:34.002
on what was the fructose 6-phosphate,

00:11:34.002 --> 00:11:36.045
but then we get the payoff phase.

00:11:36.045 --> 00:11:38.239
So we're able to produce this NADH, and

00:11:38.239 --> 00:11:39.899
this is actually going to be two NADHs,

00:11:39.899 --> 00:11:41.478
because everything here's going to happen

00:11:41.478 --> 00:11:43.928
twice now, we can assume
that this character

00:11:43.928 --> 00:11:45.959
over here also gets converted to a

00:11:45.959 --> 00:11:49.326
glyceraldehyde 3-phosphate, and now

00:11:49.326 --> 00:11:51.787
we've produced two ATPs,
cuz this is happening twice,

00:11:51.787 --> 00:11:54.445
and we've produced two
ATPs right over there.

00:11:54.445 --> 00:11:57.965
So hopefully everything we
talked about in the beginning

00:11:57.965 --> 00:12:00.237
actually makes sense.


Evaluating sequences in recursive form

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:02.305
- [Voiceover] So we've
got the function g here,

00:00:02.305 --> 00:00:04.789
and what I want you to
do is pause the video

00:00:04.789 --> 00:00:06.763
and figure out what g of one is,

00:00:06.763 --> 00:00:08.968
figure out what g of two is,

00:00:08.968 --> 00:00:13.334
g of three, and g of four.

00:00:13.334 --> 00:00:16.143
Figure out what all four of these are.

00:00:16.143 --> 00:00:18.303
All right, now let's work
through this together.

00:00:18.303 --> 00:00:22.134
So g of one, if n is equal to one,

00:00:22.134 --> 00:00:23.922
well, then we're gonna hit
this case right over here,

00:00:23.922 --> 00:00:27.242
if n is equal to one, g is equal to four.

00:00:27.242 --> 00:00:29.332
So that was pretty straightforward.

00:00:29.332 --> 00:00:31.816
Now, g of two,

00:00:31.816 --> 00:00:33.837
if n is equal to two,

00:00:33.837 --> 00:00:36.902
well, two is greater than
one and it's a whole number,

00:00:36.902 --> 00:00:38.828
and so we would use this case.

00:00:38.828 --> 00:00:39.688
And this is interesting,

00:00:39.688 --> 00:00:41.847
because it's defined in
terms of the function,

00:00:41.847 --> 00:00:43.728
but it's not defined in terms of g of n,

00:00:43.728 --> 00:00:45.864
but g of n minus one.

00:00:45.864 --> 00:00:48.859
So if n is two, 'cause we're
evaluating g of two here,

00:00:48.859 --> 00:00:51.715
this is going to be g of two minus one,

00:00:51.715 --> 00:00:55.845
or g of one, plus 3.2.

00:00:55.845 --> 00:00:58.215
Plus 3.2.

00:00:58.215 --> 00:00:59.793
Well, what's that going to be?

00:00:59.793 --> 00:01:02.695
Well, g of one, we know is equal to four.

00:01:02.695 --> 00:01:04.298
We just figured that out.

00:01:04.298 --> 00:01:08.969
So four plus 3.2, that is 7.2.

00:01:09.359 --> 00:01:10.690
All right, let's keep going.

00:01:10.690 --> 00:01:13.472
G of three, we're gonna
fall into this case again,

00:01:13.472 --> 00:01:16.266
because three is greater than
one and it's a whole number.

00:01:16.266 --> 00:01:20.005
So this is going to be
g of three minus one,

00:01:20.005 --> 00:01:23.451
or g of two, plus 3.2.

00:01:23.451 --> 00:01:24.721
Plus 3.2.

00:01:24.721 --> 00:01:27.794
Well, we know what g of two is. It's 7.2.

00:01:27.794 --> 00:01:30.964
We just figured that out. It is 7.2.

00:01:30.964 --> 00:01:34.083
7.2 plus 3.2 is going

00:01:34.083 --> 00:01:37.183
to be equal to 10.4.

00:01:37.183 --> 00:01:40.321
And then g of four,
well, we fall here again.

00:01:40.321 --> 00:01:43.041
This is gonna be g of three plus 3.2.

00:01:43.041 --> 00:01:47.174
G of three plus 3.2.

00:01:47.174 --> 00:01:49.014
What is that going to be equal to?

00:01:49.014 --> 00:01:51.700
Well, g of three we just
figured out is 10.4.

00:01:51.700 --> 00:01:54.392
10.4 plus 3.2 is going

00:01:54.392 --> 00:01:57.352
to be 13.6.

00:01:57.352 --> 00:01:58.456
And so what you have here,

00:01:58.456 --> 00:01:59.942
this is actually quite interesting.

00:01:59.942 --> 00:02:03.449
You can think of this function g,

00:02:03.449 --> 00:02:06.955
and we see that it's defined
over all positive integers.

00:02:06.955 --> 00:02:08.766
Because it's defined
over positive integers,

00:02:08.766 --> 00:02:11.041
we could think of it
as defining a sequence,

00:02:11.041 --> 00:02:12.598
and we see what the sequence here is.

00:02:12.598 --> 00:02:16.335
The first term is four,

00:02:16.335 --> 00:02:19.625
second term is 7.2,

00:02:19.625 --> 00:02:23.035
next term is 10.4,

00:02:23.035 --> 00:02:25.965
next term is 13.6,

00:02:25.965 --> 00:02:27.968
and it could keep going on and on and on.

00:02:27.968 --> 00:02:30.986
And what's happening? What's
happening in this sequence?

00:02:30.986 --> 00:02:32.867
Well, we're starting with four.

00:02:32.867 --> 00:02:34.121
We're starting with four,

00:02:34.121 --> 00:02:36.629
and this case of the
function gave us that.

00:02:36.629 --> 00:02:38.436
If n is equal to four,

00:02:38.436 --> 00:02:40.204
if n is equal to one,

00:02:40.204 --> 00:02:42.086
the function is going to be equal to four,

00:02:42.086 --> 00:02:44.524
and then for each term after that,

00:02:44.524 --> 00:02:47.425
you take the previous
term and you add 3.2.

00:02:47.425 --> 00:02:51.419
So we add 3.2 for the
second term, we add 3.2,

00:02:51.419 --> 00:02:53.473
so we just keep adding,

00:02:53.473 --> 00:02:58.293
we just keep adding 3.2,

00:02:58.293 --> 00:03:02.982
not .32, 3.2 to get to the next term.

00:03:02.982 --> 00:03:04.676
Now, we could've defined it
that way, we could've said,

00:03:04.676 --> 00:03:06.546
"Hey, let's have a sequence
where the first term is four

00:03:06.546 --> 00:03:10.120
"and then we keep adding
3.2 to get each next term."

00:03:10.120 --> 00:03:12.781
But this is another
interesting way of defining it.

00:03:12.781 --> 00:03:14.243
And this way of defining it,

00:03:14.243 --> 00:03:16.589
where we defined it as
an algebraic function,

00:03:16.589 --> 00:03:19.793
a function that's defined
over all positive integers,

00:03:19.793 --> 00:03:21.302
where we have a base case.

00:03:21.302 --> 00:03:22.745
And a base case, really, in this case,

00:03:22.745 --> 00:03:24.483
gave us our first term,

00:03:24.483 --> 00:03:25.876
and then we have this other case

00:03:25.876 --> 00:03:28.082
that's defined in terms of the function.

00:03:28.082 --> 00:03:30.079
Then you have to recurse backwards

00:03:30.079 --> 00:03:32.099
to eventually get to a base case.

00:03:32.099 --> 00:03:34.816
We call this a recursive function.

00:03:34.816 --> 00:03:38.786
Recursive function.

00:03:38.786 --> 00:03:40.366
So with this example, we're seeing

00:03:40.366 --> 00:03:42.827
how a recursive function can be used

00:03:42.827 --> 00:03:45.636
to define an actual sequence.

00:03:45.636 --> 00:03:47.656
And we went in order here,

00:03:47.656 --> 00:03:48.910
but you could've gone
the other way around.

00:03:48.910 --> 00:03:52.499
If I said, "Oh, well, what's g of,

00:03:52.499 --> 00:03:54.629
"what's g of six?"

00:03:54.629 --> 00:03:55.945
Well, you'd go into this case, you'd say,

00:03:55.945 --> 00:04:00.008
"OK, that's going to
be g of five plus 3.2."

00:04:00.008 --> 00:04:03.398
It's gonna be the previous term plus 3.2

00:04:03.398 --> 00:04:05.650
if we view it as a sequence.

00:04:05.650 --> 00:04:06.533
Well, then we're gonna have to figure out

00:04:06.533 --> 00:04:07.438
what the previous term is.

00:04:07.438 --> 00:04:11.138
G of five is going to be g of four,

00:04:11.138 --> 00:04:14.278
g of four plus 3.2,

00:04:14.278 --> 00:04:16.262
and you would keep going
back and back and back,

00:04:16.262 --> 00:04:18.119
but we've already figured
out what g of four is.

00:04:18.119 --> 00:04:21.487
It's 13.6,

00:04:21.487 --> 00:04:25.357
so this is 16.8,

00:04:25.357 --> 00:04:29.496
and then if g of five is 16.8,

00:04:29.496 --> 00:04:33.095
16.8, you add 3.2 there, you would get 20.

00:04:33.095 --> 00:04:34.980
So you could start at g of
six and keep backing up,

00:04:34.980 --> 00:04:36.350
all the way until you get to g of one,

00:04:36.350 --> 00:04:37.719
and then you figure out what that is.

00:04:37.719 --> 00:04:40.274
You recurse back to your base case,

00:04:40.274 --> 00:04:43.292
and then you're able to
fill in all of the blanks.

00:04:43.292 --> 00:04:45.241
Let's do a few more examples of this.

00:04:46.311 --> 00:04:48.191
So we have this function here.

00:04:48.191 --> 00:04:50.606
So let's say that this defines a sequence.

00:04:50.606 --> 00:04:52.603
Let's think about what
the first four terms

00:04:52.603 --> 00:04:54.089
of that sequence are, and once again,

00:04:54.089 --> 00:04:56.389
I encourage you to pause the
video and figure that out.

00:04:57.409 --> 00:04:59.453
All right, let's work through it.

00:04:59.453 --> 00:05:03.562
So h of one is, well,
they very clearly tell us

00:05:03.562 --> 00:05:04.747
that's going to be 14.

00:05:04.747 --> 00:05:07.811
If n is equal to one, h is 14.

00:05:07.811 --> 00:05:11.172
H of two,

00:05:11.172 --> 00:05:13.242
well, now we're falling into this case,

00:05:13.242 --> 00:05:15.659
'cause two is greater than
one and it's a whole number,

00:05:15.659 --> 00:05:19.336
so it's gonna be 28 over h of one,

00:05:19.336 --> 00:05:21.246
over h of one.

00:05:21.246 --> 00:05:22.973
Well, we know h of one is 14,

00:05:22.973 --> 00:05:26.738
so it's gonna be 28 over
14, which is equal to two.

00:05:26.738 --> 00:05:29.572
Now h of three.

00:05:29.572 --> 00:05:32.358
H of three, we're gonna
fall into this case again.

00:05:32.358 --> 00:05:34.053
It's gonna be 28,

00:05:34.053 --> 00:05:38.083
28 divided by h of two,

00:05:38.083 --> 00:05:39.462
if we're thinking of this as a sequence,

00:05:39.462 --> 00:05:42.388
divided by the previous
term in the sequence.

00:05:42.388 --> 00:05:43.990
So 28 divided by h of two,

00:05:43.990 --> 00:05:47.875
we know that h of two is equal to,

00:05:47.875 --> 00:05:49.215
is equal to two.

00:05:49.215 --> 00:05:50.050
We just figured that out.

00:05:50.050 --> 00:05:52.674
So we go back to 14,
something very interesting.

00:05:52.674 --> 00:05:54.440
I think you see where this is going.

00:05:54.440 --> 00:05:58.943
H of four is gonna be 28
divided by h of three,

00:05:58.943 --> 00:06:02.301
28 divided by h of three,

00:06:02.301 --> 00:06:04.251
which is 28 divided by,

00:06:04.251 --> 00:06:05.414
this is h of three right over here,

00:06:05.414 --> 00:06:06.386
we just figured that out,

00:06:06.386 --> 00:06:09.276
divided by 14, which is back to two.

00:06:09.276 --> 00:06:12.387
If we were to think of this as a sequence,

00:06:12.387 --> 00:06:16.033
we'd say, "All right, let's
see, the first term is 14,

00:06:16.033 --> 00:06:19.245
"then we get to two,

00:06:19.245 --> 00:06:21.055
"then we go to 14,

00:06:21.055 --> 00:06:22.235
"then we get to two."

00:06:22.235 --> 00:06:24.020
So one way to think about this sequence is

00:06:24.020 --> 00:06:27.317
that we just keep alternating
between 14 and twos.

00:06:27.317 --> 00:06:31.009
All of the odd terms
of the sequence are 14,

00:06:31.009 --> 00:06:32.890
all of the even terms
of the sequence are two.

00:06:32.890 --> 00:06:34.817
That's one way to think about it.

00:06:34.817 --> 00:06:36.328
Or another way to think about it is,

00:06:36.328 --> 00:06:38.346
we're starting with 14,

00:06:38.346 --> 00:06:42.015
and each successive term is
the previous term divided,

00:06:42.015 --> 00:06:45.126
is 28 divided by the previous term.

00:06:45.126 --> 00:06:48.284
So here, 28 divided by 14 is two.

00:06:48.284 --> 00:06:51.813
28 divided by two is 14.
28 divided by 14 is two.

00:06:51.813 --> 00:06:53.206
And we keep going on and on and on,

00:06:53.206 --> 00:06:57.223
and that's what was actually
going on right over there.

00:06:57.223 --> 00:06:58.582
Let's do one more of these.

00:06:59.592 --> 00:07:00.776
And this one is interesting,

00:07:00.776 --> 00:07:02.026
because we now have,

00:07:02.026 --> 00:07:06.636
we now have two base cases.

00:07:06.636 --> 00:07:08.397
So let's think about this.

00:07:08.397 --> 00:07:10.716
This is, and actually let's just say

00:07:10.716 --> 00:07:14.450
we wanted to figure out, we
wanted to figure out what,

00:07:14.450 --> 00:07:17.050
what f of four is.

00:07:17.050 --> 00:07:20.436
F of four, well, we're
gonna fall into this case.

00:07:20.436 --> 00:07:22.410
Four is greater than two
and it's a whole number.

00:07:22.410 --> 00:07:24.732
It's gonna be f of four minus two,

00:07:24.732 --> 00:07:29.440
so it's gonna be f of two
plus f of four minus one,

00:07:29.440 --> 00:07:31.460
plus f of three.

00:07:31.460 --> 00:07:32.830
So f of four is gonna be the sum

00:07:32.830 --> 00:07:35.338
of the preceding two numbers.

00:07:35.338 --> 00:07:36.360
All right, so let's figure out

00:07:36.360 --> 00:07:38.728
what f of three is going to be.

00:07:38.728 --> 00:07:40.725
F of three, we fall into this case again,

00:07:40.725 --> 00:07:43.987
it's gonna be f of three
minus two is f of one,

00:07:44.927 --> 00:07:48.642
plus f of three minus one, plus f of two,

00:07:48.642 --> 00:07:51.243
the sum of the preceding two numbers.

00:07:51.243 --> 00:07:53.727
So let's figure out what
f of two is going to be.

00:07:53.727 --> 00:07:54.772
Well, now we're not doing the sum

00:07:54.772 --> 00:07:56.212
of the preceding two numbers anymore.

00:07:56.212 --> 00:07:59.277
We fall into this base
case. N is now equal to two.

00:07:59.277 --> 00:08:01.274
It's going to be equal to negative four.

00:08:01.274 --> 00:08:02.435
And we're gonna have to figure out

00:08:02.435 --> 00:08:04.408
what f of one is as well.

00:08:04.408 --> 00:08:06.010
And we see when n is equal to one,

00:08:06.010 --> 00:08:07.891
f is equal to negative six.

00:08:07.891 --> 00:08:10.306
We have two base cases right over here.

00:08:10.306 --> 00:08:13.928
Base cases, cases that aren't defined

00:08:13.928 --> 00:08:15.344
in terms of the function itself.

00:08:15.344 --> 00:08:16.296
And you need that,

00:08:16.296 --> 00:08:18.061
'cause otherwise you'd
just be recursing forever.

00:08:18.061 --> 00:08:20.615
You would never get to actual numbers.

00:08:20.615 --> 00:08:23.502
But now we can use these to
fill in the values up here.

00:08:23.502 --> 00:08:26.954
So the sequence is negative six,

00:08:26.954 --> 00:08:29.833
then we go to negative
four as the second term,

00:08:29.833 --> 00:08:32.016
and then the third term is
the sum of the previous two.

00:08:32.016 --> 00:08:36.613
Negative six plus negative
four is negative 10.

00:08:36.613 --> 00:08:40.119
Negative six plus negative
four is negative 10.

00:08:40.119 --> 00:08:42.371
And then the fourth term is
the sum of the previous two.

00:08:42.371 --> 00:08:43.416
We see it right over here.

00:08:43.416 --> 00:08:45.854
The second term, f of
two, plus f of three.

00:08:45.854 --> 00:08:50.173
Negative four plus
negative 10 is negative 14.

00:08:50.173 --> 00:08:53.122
And we could keep going on
and on and on like that.

00:08:53.122 --> 00:08:55.815
So this right over here is negative 14.

00:08:55.815 --> 00:08:57.324
So the whole point of this video,

00:08:57.324 --> 00:08:59.809
you're a little bit familiar
with recursive functions now,

00:08:59.809 --> 00:09:01.573
and also you can see how these can be used

00:09:01.573 --> 00:09:04.638
to define actual sequences.


Recursive formulas for arithmetic sequences

WEBVTT
Kind: captions
Language: en

00:00:00.375 --> 00:00:01.769
So we see here in this table,

00:00:01.769 --> 00:00:03.580
that for given "n's", when "n" is one,

00:00:03.580 --> 00:00:06.482
"f" of "n" is 12, when
"n" is 2, "f" of "n" is 5,

00:00:06.482 --> 00:00:08.812
and when "n" is three, "f"
of "n" is negative two,

00:00:08.822 --> 00:00:10.476
when "n" is four, "f"
of "n" is negative nine.

00:00:11.536 --> 00:00:12.681
And so one way to think about it is

00:00:12.681 --> 00:00:15.445
this function "f" is defining a sequence

00:00:15.445 --> 00:00:19.392
where the first term
of this sequence is 12.

00:00:19.392 --> 00:00:22.712
The second term of this sequence is five.

00:00:22.712 --> 00:00:25.939
The third term of this
sequence is negative two.

00:00:25.939 --> 00:00:29.097
The fourth term of the
sequence is negative nine.

00:00:29.097 --> 00:00:30.514
And it goes on, and on, and on.

00:00:30.514 --> 00:00:33.834
And you might notice that
it's a arithmetic sequence.

00:00:33.834 --> 00:00:36.411
We start with a 12, and
then the next term...

00:00:36.411 --> 00:00:38.942
What have we done? We've subtracted seven.

00:00:38.942 --> 00:00:41.860
Now to go from the second
to the third term...

00:00:41.860 --> 00:00:43.915
What do we do? We subtract seven.

00:00:43.915 --> 00:00:47.185
So each term is seven less
than the term before it.

00:00:47.185 --> 00:00:50.436
Now with that out of the
way, see if you can define

00:00:50.436 --> 00:00:55.358
this function of "n." If you
can define it explicitly.

00:00:55.358 --> 00:00:57.958
So figure out a function definition.

00:00:57.958 --> 00:01:00.907
So I want to figure out
"f" of "n" is equal.

00:01:00.907 --> 00:01:03.183
I want you to figure out
what this needs to be so that

00:01:03.183 --> 00:01:05.249
if you input "n" here, it gives you

00:01:05.249 --> 00:01:06.898
the appropriate "f" of "n."

00:01:08.888 --> 00:01:10.195
So let's think about it a little bit.

00:01:10.195 --> 00:01:12.679
It's going to be, we could think
of it as we're starting at,

00:01:12.679 --> 00:01:14.815
The first term is going to be 12.

00:01:16.115 --> 00:01:21.115
But then, we are going to subtract seven.

00:01:22.906 --> 00:01:23.941
And what are we going to subtract seven?

00:01:23.941 --> 00:01:25.287
How many times are we
going to subtract seven?

00:01:25.287 --> 00:01:28.608
So for the first term, we
subtract seven zero times.

00:01:28.608 --> 00:01:29.745
And so we just get 12.

00:01:29.745 --> 00:01:32.996
For the second term,
we subtract seven once.

00:01:32.996 --> 00:01:37.925
For the third term, we subtract
seven twice. One, two times.

00:01:38.345 --> 00:01:41.657
For the fourth term, we
subtract seven three times.

00:01:41.657 --> 00:01:44.373
So it looks like whatever term we're on,

00:01:44.373 --> 00:01:47.345
we're subtracting seven "n" minus one,

00:01:47.345 --> 00:01:50.433
we're subtracting seven
whatever term we're on,

00:01:50.433 --> 00:01:52.520
that term minus one time.

00:01:52.520 --> 00:01:55.509
So it's "n" minus one times.

00:01:55.509 --> 00:01:57.028
And let's see if this actually works out.

00:01:57.028 --> 00:02:00.905
So "f" of one is going to
be 12 minus seven times

00:02:00.905 --> 00:02:02.461
one minus one, that's a zero.

00:02:02.461 --> 00:02:04.365
So that's all just going to be 12.

00:02:04.365 --> 00:02:06.756
"f" of two is going to be 12 minus seven

00:02:06.756 --> 00:02:08.637
times two minus one.

00:02:08.637 --> 00:02:12.236
So it's going to be 12
minus seven times one.

00:02:12.236 --> 00:02:14.442
We're just going to subtract seven once,

00:02:14.442 --> 00:02:15.556
which is exactly the case.

00:02:15.556 --> 00:02:17.785
We started at 12, we subtract seven once.

00:02:17.785 --> 00:02:20.641
"f" of three, you can keep testing this.

00:02:20.641 --> 00:02:23.660
12 minus, and we should have
to subtract seven twice.

00:02:23.660 --> 00:02:26.841
And we see three minus one, is two times.

00:02:26.841 --> 00:02:29.000
So we're going to
subtract seven two times.

00:02:29.000 --> 00:02:31.369
So this looks right on.

00:02:31.369 --> 00:02:33.412
We've defined the function explicitly.

00:02:33.412 --> 00:02:37.150
We've defined "f" explicitly
for this sequence.

00:02:37.150 --> 00:02:39.170
Let's do another example, here.

00:02:40.600 --> 00:02:43.535
So in this case, we have some function

00:02:43.535 --> 00:02:45.579
definitions already here.

00:02:45.579 --> 00:02:47.204
So you have your sequence,

00:02:47.204 --> 00:02:48.643
it's kind of viewed in this table.

00:02:48.643 --> 00:02:51.592
You could view it as the
first term is negative 100.

00:02:51.592 --> 00:02:54.541
The next term is negative
50, next term is zero,

00:02:54.541 --> 00:02:56.143
next term is 50.

00:02:56.143 --> 00:02:57.722
And it's very clear that this is

00:02:57.722 --> 00:02:59.022
also an arithmetic sequence.

00:02:59.022 --> 00:03:04.022
We're starting at negative
100, and then, we're adding 50.

00:03:04.436 --> 00:03:07.747
And then we're adding 50,

00:03:07.747 --> 00:03:10.437
and then we are adding 50.

00:03:10.437 --> 00:03:12.118
So each term is 50 more
than the term before it.

00:03:12.118 --> 00:03:14.184
And what I want you to
do is pause the video,

00:03:14.184 --> 00:03:16.529
and think about which
of these definitions of

00:03:16.529 --> 00:03:18.619
the function "f" are correct.

00:03:18.619 --> 00:03:20.012
And it might be more than one.

00:03:21.862 --> 00:03:23.542
Alright, so let's think about it.

00:03:23.542 --> 00:03:26.212
So this definition right over here,

00:03:26.222 --> 00:03:27.232
one way to think about it is by

00:03:27.232 --> 00:03:30.186
saying I'm going to start at negative 100.

00:03:30.186 --> 00:03:33.109
And I'm going to add
50, "n" minus one times.

00:03:33.109 --> 00:03:34.814
Does this make sense?

00:03:34.844 --> 00:03:38.542
Well, for the first term,
if we start at negative 100,

00:03:38.542 --> 00:03:40.260
we don't want to add 50 at all,

00:03:40.260 --> 00:03:42.884
we want to add 50 zero
times, and it works out.

00:03:42.884 --> 00:03:45.925
Because one minus one is going to be zero.

00:03:45.925 --> 00:03:48.154
So it checks out for "n" equals one.

00:03:48.154 --> 00:03:50.755
Let's see, for "n" equals two,
you start at negative 100,

00:03:50.755 --> 00:03:55.144
I want to add 50 once.

00:03:56.464 --> 00:03:57.854
So this should be a one.

00:03:57.854 --> 00:03:59.880
Two minus one, yup, it's a one!

00:03:59.880 --> 00:04:02.504
We're adding 50. Whatever this number is,

00:04:02.504 --> 00:04:04.106
whatever "n" is, we're adding 50

00:04:04.106 --> 00:04:07.310
one less that number of times.

00:04:07.310 --> 00:04:09.539
So for here, we're adding 50 twice.

00:04:09.539 --> 00:04:12.372
When "n" is four, we're
adding 50 three times.

00:04:12.372 --> 00:04:14.253
And this one checks out.

00:04:14.253 --> 00:04:16.830
When "n" is four, we're adding 50,

00:04:16.830 --> 00:04:18.989
four minus one, three times.

00:04:18.989 --> 00:04:22.844
Negative 100, plus 50 times three.

00:04:22.844 --> 00:04:24.655
We're adding 50 three times, adding 50

00:04:24.655 --> 00:04:26.860
one, two, three times.

00:04:26.860 --> 00:04:30.413
Well that gives you
50. So I like this one.

00:04:30.413 --> 00:04:32.038
Now let's see about this one over here.

00:04:32.038 --> 00:04:34.964
Negative 150 plus 50 "n."

00:04:35.994 --> 00:04:37.982
Alright, that's one way
of saying, so let's see

00:04:37.982 --> 00:04:41.233
If "n" is equal to one,
it's going to be negative...

00:04:41.233 --> 00:04:43.392
Actually, let me draw
a table for this one.

00:04:43.392 --> 00:04:46.736
So if we have "n" and we have "f" of "n."

00:04:47.816 --> 00:04:49.336
This is going to be for this
character right over here.

00:04:49.336 --> 00:04:54.336
So if "n" is one, it's going
to negative 150, plus 50.

00:04:54.886 --> 00:04:59.493
Which is negative 100,
yup that checks out.

00:04:59.493 --> 00:05:02.548
When "n" is two, we get negative 150,

00:05:02.548 --> 00:05:07.354
plus 50 times two, which is going to be...

00:05:07.354 --> 00:05:10.451
This is 100, and there's negative 150,

00:05:10.451 --> 00:05:12.138
this is going to be negative 50.

00:05:12.138 --> 00:05:15.179
When "n" is three, and
that checks out, of course.

00:05:15.179 --> 00:05:18.792
When "n" is three, you get negative 150,

00:05:18.792 --> 00:05:21.959
plus 50, times three,
which is equal to zero.

00:05:21.959 --> 00:05:22.911
This checks out.

00:05:22.911 --> 00:05:24.513
This one over here is going to work.

00:05:24.513 --> 00:05:25.930
And you might say, "Well, hey,

00:05:25.930 --> 00:05:27.160
these formulas look different."

00:05:27.160 --> 00:05:29.041
Well you can algebraically manipulate them

00:05:29.041 --> 00:05:33.100
to see that they're the exact same thing.

00:05:33.100 --> 00:05:34.405
If you were to take this first one, it's

00:05:34.405 --> 00:05:38.189
negative 100, plus,
let's distribute this 50,

00:05:38.189 --> 00:05:42.926
plus 50 "n", plus 50 "n", minus 50.

00:05:44.966 --> 00:05:49.966
Well, negative 100, minus
50, that's negative 150.

00:05:50.844 --> 00:05:52.399
And then you have plus 50 "n."

00:05:52.399 --> 00:05:54.535
So these are algebraically the exact same

00:05:54.535 --> 00:05:56.717
definition for our function.

00:05:56.717 --> 00:05:58.018
Now what about this one here?

00:05:58.018 --> 00:06:01.060
Negative 100 plus 50
"n", does this one work?

00:06:01.060 --> 00:06:03.475
Let's see, when "n" is equal to one,

00:06:03.475 --> 00:06:08.304
this would be negative 100
plus 50, which is negative 50.

00:06:08.304 --> 00:06:09.744
Well no, this doesn't work.

00:06:09.744 --> 00:06:11.230
We need to get a negative 100 here.

00:06:11.230 --> 00:06:14.527
So this one is not, not correct.


Converting recursive & explicit forms of arithmetic sequences

WEBVTT
Kind: captions
Language: en

00:00:00.063 --> 00:00:02.583
- So I have a function here, h of n,

00:00:02.583 --> 00:00:04.547
and let's say that it explicitly defines

00:00:04.547 --> 00:00:05.890
the terms of a sequence.

00:00:05.890 --> 00:00:06.842
Let me make a little...

00:00:06.842 --> 00:00:09.986
Let me make a quick table here.

00:00:11.736 --> 00:00:15.318
We have n, and then we have h of n.

00:00:15.318 --> 00:00:19.430
When n is equal to one,
h of n is negative 31,

00:00:19.430 --> 00:00:22.752
minus seven times one minus one,

00:00:22.752 --> 00:00:23.876
which is going to be...

00:00:23.876 --> 00:00:25.234
This is just going to be zero,

00:00:25.234 --> 00:00:27.440
so it's going to be negative 31.

00:00:27.440 --> 00:00:31.640
When n is equal to two,
it's going to be negative 31

00:00:31.640 --> 00:00:36.054
minus seven times two minus one,

00:00:36.054 --> 00:00:38.385
so two minus one.

00:00:38.385 --> 00:00:40.407
This is just going to be one,

00:00:40.407 --> 00:00:42.241
so it's negative 31 minus seven,

00:00:42.241 --> 00:00:45.994
which is equal to negative 38.

00:00:45.994 --> 00:00:47.662
When n is equal to three,

00:00:47.662 --> 00:00:51.296
it's gonna be negative 31 minus seven

00:00:51.296 --> 00:00:52.910
times three minus one,

00:00:52.910 --> 00:00:54.059
which is just two,

00:00:54.059 --> 00:00:56.988
so we're gonna subtract seven twice.

00:00:56.988 --> 00:01:00.467
It's gonna be negative 31 minus 14,

00:01:00.467 --> 00:01:03.323
which is equal negative 45.

00:01:03.323 --> 00:01:05.678
What do we see happening here?

00:01:05.678 --> 00:01:07.828
We're starting at negative 31,

00:01:07.828 --> 00:01:10.228
and then we keep subtracting,

00:01:10.228 --> 00:01:12.947
we keep subtracting negative seven.

00:01:12.947 --> 00:01:15.907
We keep subtracting
negative seven from that.

00:01:15.907 --> 00:01:19.561
In fact we subtract negative
seven one less than the term...

00:01:19.561 --> 00:01:22.026
We subtract negative seven one less times

00:01:22.026 --> 00:01:23.198
than the term we're dealing with.

00:01:23.198 --> 00:01:24.556
If we're dealing with the third term

00:01:24.556 --> 00:01:26.576
we subtract negative seven twice.

00:01:26.576 --> 00:01:28.074
If we're dealing with the second term

00:01:28.074 --> 00:01:30.274
we subtract negative seven once.

00:01:30.274 --> 00:01:31.498
This is all nice,

00:01:31.498 --> 00:01:33.437
but what I want you to
do now is pause the video

00:01:33.437 --> 00:01:36.223
and see if you can define
this exact same sequence.

00:01:36.223 --> 00:01:38.777
The sequence here is you
start at negative 31,

00:01:38.777 --> 00:01:40.658
and you keep subtracting negative seven,

00:01:40.658 --> 00:01:43.943
so negative 38, negative 45.

00:01:43.943 --> 00:01:47.050
The next one is gonna be negative 52,

00:01:47.050 --> 00:01:48.210
and you go on and on and on.

00:01:48.210 --> 00:01:50.311
You keep subtracting negative seven.

00:01:50.311 --> 00:01:52.830
Can we define this sequence

00:01:52.830 --> 00:01:55.321
in terms of a recursive function?

00:01:55.321 --> 00:01:56.872
Why don't you have a go at that.

00:01:57.512 --> 00:01:59.761
Let's try to define it in
terms of a recursive function.

00:01:59.761 --> 00:02:01.968
Let's just call that g of n,

00:02:01.968 --> 00:02:03.290
so g of n.

00:02:03.290 --> 00:02:05.199
In some ways a recursive
function is easier,

00:02:05.199 --> 00:02:06.297
because you can say okay look.

00:02:06.297 --> 00:02:08.619
The first term when n is equal to one,

00:02:08.619 --> 00:02:11.184
if n is equal to one,
let me just write it,

00:02:11.184 --> 00:02:12.990
If n is equal to one,

00:02:12.990 --> 00:02:16.067
if n is equal to one,

00:02:16.067 --> 00:02:17.680
what's g of n gonna be?

00:02:17.680 --> 00:02:19.568
It's gonna be negative 31,

00:02:19.568 --> 00:02:20.943
negative 31.

00:02:20.943 --> 00:02:22.760
And if n,

00:02:22.760 --> 00:02:26.387
if n is greater than
one and a whole number,

00:02:26.387 --> 00:02:28.883
so this is gonna be defined
for all positive integers,

00:02:28.883 --> 00:02:32.441
and whole, and whole number,

00:02:32.441 --> 00:02:34.989
it's just going to be the previous term,

00:02:34.989 --> 00:02:37.764
so g of n minus one

00:02:37.764 --> 00:02:39.100
minus seven,

00:02:39.100 --> 00:02:40.469
minus seven.

00:02:40.469 --> 00:02:41.528
We're saying hey

00:02:42.738 --> 00:02:44.764
if we're just picking an arbitrary term

00:02:44.764 --> 00:02:46.285
we just have to look at the previous term

00:02:46.285 --> 00:02:49.628
and then subtract, and
then subtract seven.

00:02:49.628 --> 00:02:51.764
It all works out nice and easy,

00:02:51.764 --> 00:02:52.763
because you keep looking at

00:02:52.763 --> 00:02:54.051
previous, previous, previous terms

00:02:54.051 --> 00:02:56.652
all the way until you
get to the base case,

00:02:56.652 --> 00:02:58.103
which is when n is equal to one,

00:02:58.103 --> 00:02:59.984
and you can build up back from that.

00:02:59.984 --> 00:03:02.886
You get this exact same sequence.

00:03:02.886 --> 00:03:03.989
Let's do another example,

00:03:03.989 --> 00:03:05.742
but let's go the other way around.

00:03:05.742 --> 00:03:08.645
Here we have a, we have a

00:03:08.645 --> 00:03:10.722
sequence defined recursively,

00:03:10.722 --> 00:03:12.382
and I want to create a function

00:03:12.382 --> 00:03:15.132
that defines a sequence explicitly.

00:03:15.132 --> 00:03:16.781
Let's think about this.

00:03:16.781 --> 00:03:17.664
One way to think about it,

00:03:17.664 --> 00:03:19.431
this sequence,

00:03:19.431 --> 00:03:22.679
when n is equal to one it starts at 9.6,

00:03:22.679 --> 00:03:25.967
and then every term is the previous term

00:03:25.967 --> 00:03:28.274
minus 0.1.

00:03:28.274 --> 00:03:30.576
The second term is gonna
be the previous term

00:03:30.576 --> 00:03:33.963
minus 0.1, so it's gonna be 9.5.

00:03:33.963 --> 00:03:35.928
Then you're gonna go to 9.4.

00:03:35.928 --> 00:03:37.752
Then you're gonna go to 9.3.

00:03:37.752 --> 00:03:40.326
We could keep going on and on and on.

00:03:40.326 --> 00:03:42.794
If we want, we could
make a little table here,

00:03:43.524 --> 00:03:45.503
and we could say

00:03:45.503 --> 00:03:48.823
this is n, this is h of n,

00:03:48.823 --> 00:03:50.727
and you see when n is equal to one,

00:03:50.727 --> 00:03:52.864
h of n is 9.6.

00:03:52.864 --> 00:03:54.105
When n is equal to two,

00:03:54.105 --> 00:03:55.382
we're now in this case over here,

00:03:55.382 --> 00:03:57.402
it's gonna h of two minus one,

00:03:57.402 --> 00:04:01.687
so it's gonna be h of one minus 0.1.

00:04:02.357 --> 00:04:04.542
It's just gonna be this minus 0.1,

00:04:04.542 --> 00:04:07.862
which is going to be 9.5.

00:04:07.862 --> 00:04:11.953
When h is three, it's gonna be h of two,

00:04:11.953 --> 00:04:15.436
h of two minus 0.1,

00:04:15.436 --> 00:04:17.538
minus 0.1.

00:04:17.538 --> 00:04:19.001
H of two is right over here.

00:04:19.001 --> 00:04:22.600
You subtract a tenth you're gonna get 9.4,

00:04:22.600 --> 00:04:24.504
exactly what we saw over here.

00:04:24.504 --> 00:04:26.529
Let's see if we can pause the video now

00:04:26.529 --> 00:04:28.144
and define this...

00:04:28.144 --> 00:04:30.761
Create a function that
constructs or defines

00:04:30.761 --> 00:04:33.698
this arithmetic sequence explicitly.

00:04:33.698 --> 00:04:34.522
Here it was recursively.

00:04:34.522 --> 00:04:36.735
We wanna define it explicitly.

00:04:37.695 --> 00:04:38.888
So let's just call it, I don't know,

00:04:38.888 --> 00:04:41.254
let's just call it f of n.

00:04:41.254 --> 00:04:43.939
We can say look, it's gonna be 9.6,

00:04:44.709 --> 00:04:46.272
but we're gonna subtract,

00:04:46.272 --> 00:04:48.257
we're gonna subtract 0.1

00:04:48.257 --> 00:04:49.696
a certain number of times

00:04:49.696 --> 00:04:51.716
depending on what term
we're talking about.

00:04:51.716 --> 00:04:53.817
We're gonna subtract 0.1,

00:04:53.817 --> 00:04:55.849
but how many times are
we gonna subtract it

00:04:55.849 --> 00:04:57.753
as a function of n?

00:04:58.463 --> 00:04:58.890
Let's see.

00:04:58.890 --> 00:05:00.028
If we're talking about the first term

00:05:00.028 --> 00:05:02.127
we subtract zero times.

00:05:02.127 --> 00:05:04.968
The second term we subtract one time.

00:05:04.968 --> 00:05:07.354
The third term we subtract two times.

00:05:07.354 --> 00:05:09.907
The fourth term we subtract three times.

00:05:09.907 --> 00:05:11.533
Whatever term we're talking about

00:05:11.533 --> 00:05:15.468
we subtract that term minus one times.

00:05:15.468 --> 00:05:16.977
If we're talking about the nth term,

00:05:16.977 --> 00:05:21.273
we subtracted this
value n minus one times.

00:05:21.273 --> 00:05:23.420
You can verify that this is going to work.

00:05:23.420 --> 00:05:25.738
When n is equal to one

00:05:25.738 --> 00:05:27.623
this term here is going to be zero,

00:05:27.623 --> 00:05:28.679
so this whole thing is gonna be zero.

00:05:28.679 --> 00:05:30.003
You get 9.6.

00:05:30.003 --> 00:05:32.314
When n is equal to two,

00:05:32.314 --> 00:05:33.619
two minus one,

00:05:33.619 --> 00:05:36.573
you subtract 0.1 one time.

00:05:36.573 --> 00:05:39.290
9.6 minus 0.1 is 9.5.

00:05:39.290 --> 00:05:40.033
You could keep doing that.

00:05:40.033 --> 00:05:41.171
You could draw a table,

00:05:41.171 --> 00:05:42.633
and evaluate these if you want to.

00:05:42.633 --> 00:05:45.531
The key thing is your'e starting at 9.6

00:05:45.531 --> 00:05:47.882
and you're subtracting 0.1

00:05:47.882 --> 00:05:50.411
one fewer times than the
term you're looking at.

00:05:50.411 --> 00:05:52.849
If you're looking at the if n is equal to,

00:05:52.849 --> 00:05:55.090
this is n is equal to four,

00:05:55.090 --> 00:05:57.922
well you're gonna subtract 0.1 three times,

00:05:57.922 --> 00:05:58.886
and you see that.

00:05:58.886 --> 00:06:00.603
Subtract 0.1 once,

00:06:00.603 --> 00:06:02.206
subtract 0.1 twice,

00:06:02.206 --> 00:06:06.271
subtract 0.1 three times.