Evaluating a source’s reasoning and evidence

WEBVTT
Kind: captions
Language: en

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- [David] Hello, readers.

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How do we know what is
true, and what isn't?

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My mama always told me, "Don't
believe everything you read.

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"Just because someone took the
time to write something down,

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"send it off to be typeset and designed

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"and printed in a book or
published on the internet,

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"doesn't mean they bothered
to tell you the truth."

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You should be skeptical of
all informational writing

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until the writer gives you
a reason to trust them.

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Honestly, you should be skeptical of me

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until I give you a reason to trust me.

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Who am I, after all?

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Just some stranger on the internet.

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Well, that's a little unfair, actually.

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You probably knew about Khan Academy

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well before you watched this video,

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which means you have a sense
that the information we teach

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is reputable or trustworthy.

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So, maybe I'm already at an advantage

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for you trustworthinesswise,

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because you associate
me with an institution

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you already have good feelings about.

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When an author writes for
informational purposes,

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they want you to believe that
they're telling the truth,

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but you can't just accept that on faith.

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No human being is so important

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that they can't be bothered
to explain themselves.

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Now, let's be clear,

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that doesn't mean that
everyone you disagree with

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personally owes you an explanation.

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It just means that you don't
have to swallow their arguments

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hook, line, and sinker.

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So, how do we move forward?

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We can't just go around

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in a nightmarish swirl of misinformation,

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never knowing whom to trust.

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That's no good.

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Society couldn't function.

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Imagine looking at the weather report

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and saying, "I don't believe you!"

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And then, walking outside
your home in shorts

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when the forecast calls for snow.

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That would be terrible, for you.

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The key is to ask yourself,

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"Why should you trust
the weather forecast?"

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Well,

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forecasters are scientists,
specifically meteorologists.

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Their job is to study weather
patterns and make predictions.

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And that doesn't mean
they always get it right,

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because they're making what
amounts to educated guesses

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about the weather.

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But their expertise

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and their access to weather
monitoring equipment

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make them likelier to know
what the weather's going to do

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than the average person would.

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And this is basically
the test you should bring

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to all informational writing.

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Good readers look for clues
to establish trustworthiness,

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to establish that the author knows

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what they're talking about,

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that they've done the research,

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that they're an expert worth listening to.

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Someone who hops on YouTube

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and says, "Here's why your
teachers are lying to you

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"about the goblins who live in the moon,"

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is someone that warrants some
skepticism, a little side-eye.

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Also, how awesome would it be
if there were moon goblins?

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And why would anyone want that information

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to be kept secret?

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Okay, when you look at a text,

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you have to wrap your head
around three big things.

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One, the main idea or the claim.

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What is the author trying to say,

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explain or convince you of?

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Two, the evidence for that claim?

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What are the facts, details,
or other pieces of information

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that tell us that that
claim is true or accurate?

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And three, the reasoning.

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How the evidence prove
that the claim is true?

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Okay, so let's imagine that
this conspiracy guy on YouTube

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is trying to make the
case about those goblins.

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Let's listen in on his argument.

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- [YouTuber] I firmly believe

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that there is a colony of
goblins living inside the moon

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and the government is keeping
this information from you.

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How do we know this?

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Look at the moon!

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See that little squiggle?

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You were taught that was
just a crater, but it isn't!

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It's actually the tip of an
enormous goblin moon base.

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How do I know that?

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The goblins told me so.

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I have a special phone line

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from my house to the moon goblins.

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And through me, they can answer
any questions you've got.

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Our moon door is always
open, the goblins tell me.

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- [David] All right, so, what's the claim?

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Goblins live inside the moon.

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What's his proof?

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What we think is a crater on the moon

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is actually a goblin moon base.

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What's his reasoning?

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What backs that up?

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He says the goblins told him so.

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Now, is that evidence?

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Or is that just something he's telling us?

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Well, one way we can know
he's probably full of baloney

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is that only he can talk
to the moon goblins.

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Right, he says he's got
a special phone line.

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They can answer questions through him.

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We can't talk to the goblins.

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We can't have access to
any of their information.

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He's behaving like the information

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is freely out there for
anyone, but it isn't.

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You have to go through him.

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So, can the stuff he's saying
be independently verified?

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That is to say, can anyone
other than this person

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prove that there are moon goblins

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or that that crater is a moon base?

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Are there photos or
video from those goblins?

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And if not, why not?

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This doesn't pass the smell test for me.

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There's proof, but only I
get to see it, is not proof.

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That's just made up.

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When it comes to making arguments,

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you have to show your work.

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A good writer has to earn
the trust of their audience.

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And a good reader has to make
a writer work for that trust.

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"What does that look
like?" I hear you asking.

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Excellent question.

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Let's take a look at a piece
that does its homework.

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Okay, let's read.

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"There are some conspiracy theorists

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"that claim that NASA's
moon landing missions

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"in the late 60s and 70s never happened,

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"despite the mountain of
evidence to the contrary.

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"They claim that the
photographs were faked,

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"the mission recordings were scripted,

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"and the moon rocks were nothing more

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"than ordinary earth rocks.

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"Perhaps you are among
these conspiracists.

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"If so, let me prove to you

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"that the moon landings took place.

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"Perhaps you are skeptical
of the US government,

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"which was in the middle of the Cold War

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"with the Soviet Union and
a shooting war in Vietnam.

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"Perhaps you believe that
any information from NASA

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"is therefore suspicious.

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"Fortunately for you,

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"there is plenty of third-party evidence

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"that Americans traveled
to and landed on the moon.

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"An observatory in the UK
that previously observed

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"the Russian launch of the Sputnik probe

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"used its radio telescope

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"to observe the 1969 moon landing.

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"The Bochum Observatory,
this one in Germany,

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"also confirms that mission.

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"Finally, present-day photos

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"from many different space
agencies around the world

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"have confirmed the presence
of footprints, rover tracks,

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"and the lunar rovers themselves.

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"In order for a moon
landing to have been faked,

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"the rocket scientist
James Michael Longuski

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"estimates that more than 400,000 people

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"who worked on the
projects in some capacity

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"would have had to keep
the conspiracy a secret.

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"In his 2006 book

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'The Seven Secrets of How to
Think Like a Rocket Scientist,'

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"Longuski suggests that given
how complicated it would be

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"to fake such a project,

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"it might actually be
easier to just go ahead

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"and do the moon landing
in the first place."

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So, if you don't believe
the US government,

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this writer is saying, let's
look at some other sources.

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How do we know those sources are real?

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Well, we have some names to look into

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to independently verify it.

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We've got this rocket scientist.

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We have the name of
this German observatory.

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We could look into and
verify these statements.

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It's not just one guy

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asserting that there's
goblins on the moon.

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It's someone saying here
are some non-NASA sources

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that can confirm that
Americans landed on the moon.

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Be careful out there.

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It's a wild world of people
who wanna bend your brains

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and turn you against your neighbors.

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Use your brain, your heart,

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and your reading skills to fend them off,

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as you develop your principles.

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You've got this, readers.

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You can learn anything.

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David out.


Reasoning with linear equations

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Kind: captions
Language: en

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- [Instructor] In many other videos,

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we've taken equations like this

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and tried to quote solve for X.

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What we're gonna do in
this video is deepen

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our understanding a little
bit about what's going on

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and really think about
the notion of equivalence

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or equivalence statements.

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So let me write that down, equivalence.

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Now, what do I mean by equivalence?

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Well, I'll use this equation here

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to essentially keep rewriting
it in equivalent ways

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and I'll talk a little bit
more about what that means.

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So one thing that I
could do to essentially

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write the same equivalence statement

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is I could distribute
this three onto the X

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or onto the X plus one
and then this part of it

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could be rewritten as three X plus three

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and then we have minus X is equal to nine.

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Now, what I will say or what
it might be obvious to you

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on some level is this top equation

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and this second equation are equivalent.

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What does that mean?

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It means if one of them
is true for given X,

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the other one will be true
for that same X as well,

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and vice versa and we can write other

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equivalence statements.

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For example, if we were
to combine X terms,

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if we were to take three
X and then minus X,

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right over there, I could
rewrite that as two X

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and then I have plus
three is equal to nine.

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Now all three of these
statements are equivalent.

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If there's an X where two times that X

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plus three is equal to nine,

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then it is also the case that
three times that X plus one

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minus X is equal to nine and vice versa.

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If there's some X that would
make this top equation true

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then it's going to make
this last equation true.

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And we can do other equivalence
preserving operations

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and you've seen them before.

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You could subtract three from both sides.

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In general, if you are adding
or subtracting the same value

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from both sides, it is
equivalence preserving.

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If you're distributing a value like we did

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in that first step that
is equivalence preserving.

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If you combine like terms so to speak,

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that is equivalence preserving

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and so here we'll do an
equivalence preserving operation.

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We'll subtract three from both sides

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and you would get two X is equal to six

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and once again, any X that
satisfies this last equation

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will satisfy any of the other
equations and vice versa.

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Any X that's satisfies any
of these other equations

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will satisfy this last one

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and so they are all
equivalent to each other.

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And then another equivalence
preserving operation

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is to multiply or divide both
sides by a non zero constant.

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And so here we could
divide both sides by two,

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two is not zero and it's constant

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and if we did that, we will get
another equivalent statement

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that X is equal to three.

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So any X that satisfies this
and there's one, X equals three

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would satisfy the other ones
and any X that satisfies

00:03:05.910 --> 00:03:07.970
any of the other ones
would satisfy this last one

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so these are all equivalent.

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So one way to think about it,

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adding the same number to
both sides of an equation

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that is equivalence preserving.

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Multiply or dividing both sides
by a non-zero constant value

00:03:23.030 --> 00:03:25.230
is equivalence preserving.

00:03:25.230 --> 00:03:27.310
Distributing like we
did in this first step

00:03:27.310 --> 00:03:29.410
equivalence preserving,
combining like terms

00:03:29.410 --> 00:03:31.140
equivalence preserving.

00:03:31.140 --> 00:03:32.010
Now you're probably saying,

00:03:32.010 --> 00:03:36.200
well, what are some non
equivalence preserving operations?

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Well, imagine something like this,

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lemme just start with
something very obvious.

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If I said that X is equal to two

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a non equivalence preserving operation

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is if I were to add or
subtract or multiply or divide

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only one side of this equation by a value

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let's say I only added
one to the left side.

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Then I would have X
plus one is equal to two

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and it is not the case that
anything that satisfies

00:03:59.593 --> 00:04:03.080
this second equation
satisfies the top equation

00:04:03.080 --> 00:04:04.860
or vice versa.

00:04:04.860 --> 00:04:07.660
X equals two clearly
satisfies the top equation

00:04:07.660 --> 00:04:09.140
but it doesn't satisfy the second one,

00:04:09.140 --> 00:04:11.310
that's because we did a non equivalence

00:04:11.310 --> 00:04:13.040
preserving operation.

00:04:13.040 --> 00:04:17.040
Likewise, if I only multiplied
the right hand side by three,

00:04:17.040 --> 00:04:19.980
I would get X is equal to six.

00:04:19.980 --> 00:04:23.660
Well, by only multiplying
the right hand side by value

00:04:23.660 --> 00:04:27.220
it's not the case that anything
that satisfies X equals six

00:04:27.220 --> 00:04:31.260
will satisfy X equals two,
that is somewhat obvious here.

00:04:31.260 --> 00:04:33.890
Now there's a little
bit trickier scenarios,

00:04:33.890 --> 00:04:38.890
let's say we have the equation
five X is equal to six X.

00:04:43.060 --> 00:04:47.510
Now, one temptation is, well,
I wanna do the same thing

00:04:47.510 --> 00:04:52.510
to both sides, I could just
divide both sides by X,

00:04:52.510 --> 00:04:54.580
what'll happen in that scenario?

00:04:54.580 --> 00:04:57.870
Well, if you divide both sides by X

00:04:57.870 --> 00:04:59.870
you could think that an
equivalence statement

00:04:59.870 --> 00:05:04.030
is that five is equal to six
and you know that there's no X

00:05:04.030 --> 00:05:05.710
for which five is equal to six.

00:05:05.710 --> 00:05:09.270
You can't make five equal
six or six equal five

00:05:09.270 --> 00:05:11.670
and so this would somehow
make you imply that okay,

00:05:11.670 --> 00:05:13.160
there's no X that can satisfy.

00:05:13.160 --> 00:05:15.460
If you assume that these
were equivalence statements,

00:05:15.460 --> 00:05:18.000
you'd say there's nothing that
could satisfy five equal six

00:05:18.000 --> 00:05:22.320
so maybe there's nothing that
satisfies this top equation.

00:05:22.320 --> 00:05:25.730
But this actually isn't an
equivalence preserving operation

00:05:25.730 --> 00:05:28.250
because you're actually
dealing with a scenario

00:05:28.250 --> 00:05:32.870
where X is equal to zero
and you're dividing by zero.

00:05:32.870 --> 00:05:34.780
And so you have to be very careful

00:05:34.780 --> 00:05:37.280
when you're dividing by a variable,

00:05:37.280 --> 00:05:40.400
especially if the variable
that makes that equation true

00:05:40.400 --> 00:05:42.350
happens to be zero.

00:05:42.350 --> 00:05:45.200
And so in order to be clear

00:05:45.200 --> 00:05:47.470
that you're preserving equivalence here,

00:05:47.470 --> 00:05:49.400
the way that I would tackle it is

00:05:49.400 --> 00:05:53.050
I would subtract five X from both sides

00:05:53.050 --> 00:05:54.980
and if you do that, and
that is an equivalence

00:05:54.980 --> 00:05:58.980
preserving operation, you
could subtract that expression

00:05:58.980 --> 00:06:01.230
from both sides or that
term from both sides

00:06:01.230 --> 00:06:05.410
and then you would be left
with zero is equal to X

00:06:05.410 --> 00:06:10.410
and now zero equals X
and five X equals six X,

00:06:10.430 --> 00:06:12.460
these are equivalence statements.

00:06:12.460 --> 00:06:14.250
They are equivalent equations

00:06:14.250 --> 00:06:16.800
anything that makes this
one true is going to make

00:06:16.800 --> 00:06:18.880
that one true and anyone
that makes that one true

00:06:18.880 --> 00:06:21.260
is going to make this one true.

00:06:21.260 --> 00:06:24.270
Now one last thing you
might have heard me say,

00:06:24.270 --> 00:06:28.700
you can multiply or
divide by non-zero value

00:06:28.700 --> 00:06:30.800
and that's going to be
equivalence preserving.

00:06:30.800 --> 00:06:32.170
And hopefully you've just got a sense

00:06:32.170 --> 00:06:33.960
of why dividing by zero
is not a good idea.

00:06:33.960 --> 00:06:36.170
In fact, dividing by zero
is always going to be

00:06:36.170 --> 00:06:38.860
a strange thing and it's undefined,

00:06:38.860 --> 00:06:41.070
but also multiplying by zero.

00:06:41.070 --> 00:06:44.560
For example, if I had, let's say, actually

00:06:44.560 --> 00:06:45.393
lemme start over here.

00:06:45.393 --> 00:06:49.630
If I had two X is equal to
six, and if I were to multiply

00:06:49.630 --> 00:06:54.630
both sides by zero, I would
get zero is equal to zero

00:06:54.960 --> 00:06:59.320
and zero equals zero is true for any X.

00:06:59.320 --> 00:07:03.670
Zero is always going to be equal to zero

00:07:03.670 --> 00:07:06.180
but the problem is, is that
first statement isn't true

00:07:06.180 --> 00:07:09.200
for all X it's only
true for X equals three.

00:07:09.200 --> 00:07:12.870
So these two are not
equivalent statements.

00:07:12.870 --> 00:07:17.360
They have a different set
of Xs that will satisfy them

00:07:17.360 --> 00:07:19.460
and so you have to be very
careful when you're dealing

00:07:19.460 --> 00:07:21.680
with things that are either are zero

00:07:21.680 --> 00:07:22.920
you can add or subtract zeros,

00:07:22.920 --> 00:07:25.930
obviously that's not
gonna change things much,

00:07:25.930 --> 00:07:27.920
but when you multiply both sides by zero,

00:07:27.920 --> 00:07:29.240
you can start getting things

00:07:29.240 --> 00:07:30.930
that are not equivalence statements

00:07:30.930 --> 00:07:33.030
and when you multiply or divide by things

00:07:33.030 --> 00:07:35.890
that could be zero, like variables,

00:07:35.890 --> 00:07:38.593
that also is a dangerous game to play.


Defining appropriate quantities for modeling

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:01.780
- [Instructor] So I have data here

00:00:01.780 --> 00:00:03.490
on two different websites,

00:00:03.490 --> 00:00:05.820
website A and website B.

00:00:05.820 --> 00:00:08.170
And my question to you is,

00:00:08.170 --> 00:00:12.620
which one is more productive?

00:00:12.620 --> 00:00:15.180
And some of you might be asking yourself,

00:00:15.180 --> 00:00:16.140
what does it mean

00:00:16.140 --> 00:00:17.720
to be productive?

00:00:17.720 --> 00:00:19.110
And at a very high level,

00:00:19.110 --> 00:00:21.820
you could view productivity
as how effective something

00:00:21.820 --> 00:00:23.400
is at producing.

00:00:23.400 --> 00:00:26.820
So for a given amount of input

00:00:26.820 --> 00:00:28.890
into the process,

00:00:28.890 --> 00:00:31.420
how much output are you getting?

00:00:31.420 --> 00:00:33.860
If you want to get a
little bit more exact,

00:00:33.860 --> 00:00:37.260
you could view it as the
rate of output per input.

00:00:37.260 --> 00:00:39.370
But that still makes us ask the question,

00:00:39.370 --> 00:00:42.290
what are the inputs and
what are the outputs?

00:00:42.290 --> 00:00:44.690
So pause this video and think about

00:00:44.690 --> 00:00:47.230
how you would measure productivity for or

00:00:47.230 --> 00:00:49.140
how you would compare the productivity of

00:00:49.140 --> 00:00:50.430
these two websites.

00:00:50.430 --> 00:00:51.840
And there's multiple ways

00:00:51.840 --> 00:00:52.673
to do it.

00:00:53.829 --> 00:00:55.830
All right, now let's
think through it together.

00:00:55.830 --> 00:00:56.970
What I'm first going to do

00:00:56.970 --> 00:00:58.620
is look at each of these lines and think

00:00:58.620 --> 00:01:01.130
about whether they are
an input or an output.

00:01:01.130 --> 00:01:02.930
So the number of writers, well,

00:01:02.930 --> 00:01:04.040
you need the writers

00:01:04.040 --> 00:01:06.020
to produce the website.

00:01:06.020 --> 00:01:09.880
So I would consider that an input.

00:01:09.880 --> 00:01:12.740
Although if you were the head of HR,

00:01:12.740 --> 00:01:14.550
the person who is hiring writers,

00:01:14.550 --> 00:01:15.860
the head of human resources,

00:01:15.860 --> 00:01:17.280
and you were hiring writers,

00:01:17.280 --> 00:01:18.600
then that could actually be an output.

00:01:18.600 --> 00:01:19.680
But if we're just thinking

00:01:19.680 --> 00:01:20.720
about creating a website,

00:01:20.720 --> 00:01:23.650
we need writers to create
it so that's an input.

00:01:23.650 --> 00:01:25.260
Number of posts posted,

00:01:25.260 --> 00:01:27.570
average number of words of post,

00:01:27.570 --> 00:01:28.900
average likes of post,

00:01:28.900 --> 00:01:30.790
average comments of post,

00:01:30.790 --> 00:01:32.380
number of new subscribers,

00:01:32.380 --> 00:01:35.230
those all feel like outputs.

00:01:35.230 --> 00:01:36.960
If the writers do a good job,

00:01:36.960 --> 00:01:38.620
they're going to produce a lot of posts,

00:01:38.620 --> 00:01:40.050
they're going to, I guess,
have a lot of words,

00:01:40.050 --> 00:01:40.883
I don't know if it's good

00:01:40.883 --> 00:01:42.070
to always be wordy.

00:01:42.070 --> 00:01:43.810
People will like their posts,

00:01:43.810 --> 00:01:45.110
they'll comment on those posts.

00:01:45.110 --> 00:01:46.890
We'll have new subscribers.

00:01:46.890 --> 00:01:48.663
So this is all output.

00:01:49.970 --> 00:01:51.340
What about revenue?

00:01:51.340 --> 00:01:53.490
That's how much money a site brings

00:01:53.490 --> 00:01:56.530
in before having to pay its expenses.

00:01:56.530 --> 00:02:00.620
So that also should be an output

00:02:00.620 --> 00:02:02.470
because if the writers do a good job,

00:02:02.470 --> 00:02:04.040
if the site is well run,

00:02:04.040 --> 00:02:06.860
they will generate a lot of revenue.

00:02:06.860 --> 00:02:09.140
What about expenses?

00:02:09.140 --> 00:02:11.780
Well, you need to use expenses.

00:02:11.780 --> 00:02:15.980
You need to spend expenses on
things like writer salaries in

00:02:15.980 --> 00:02:17.140
order to create your website,

00:02:17.140 --> 00:02:19.040
in order to generate that revenue.

00:02:19.040 --> 00:02:21.520
So I would consider that an input.

00:02:21.520 --> 00:02:22.430
And what about profit?

00:02:22.430 --> 00:02:24.511
Profit is all the money you bring in

00:02:24.511 --> 00:02:28.780
minus your expenses and
how much you get leftover.

00:02:28.780 --> 00:02:29.613
Well,

00:02:29.613 --> 00:02:31.630
some people would argue the
whole point of a business

00:02:31.630 --> 00:02:33.960
is to generate that profit.

00:02:33.960 --> 00:02:37.020
That is an outcome you want
to maximize, an output.

00:02:37.020 --> 00:02:39.083
And so I would consider that an output.

00:02:40.410 --> 00:02:43.890
And so the question of
productivity really just boils down

00:02:43.890 --> 00:02:46.330
to which output or what combinations

00:02:46.330 --> 00:02:49.330
of output do we want
to figure out the rate

00:02:49.330 --> 00:02:52.500
that we're generating
relative to some input or

00:02:52.500 --> 00:02:54.560
some combination of inputs?

00:02:54.560 --> 00:02:56.850
So a really simple way
of measuring productivity

00:02:56.850 --> 00:02:57.900
in this situation,

00:02:57.900 --> 00:03:00.810
let's define it, productivity.

00:03:00.810 --> 00:03:01.643
And once again,

00:03:01.643 --> 00:03:03.440
there's multiple ways of defining it.

00:03:03.440 --> 00:03:05.170
Let's say it is equal

00:03:05.170 --> 00:03:10.170
to the profit per number of writers.

00:03:10.770 --> 00:03:15.770
So it could be profit over
the number of writers.

00:03:18.150 --> 00:03:19.250
If we define it that way,

00:03:19.250 --> 00:03:21.403
which website is more productive?

00:03:22.360 --> 00:03:23.860
Well, website A,

00:03:23.860 --> 00:03:25.560
in that definition,

00:03:25.560 --> 00:03:27.490
A would have a productivity

00:03:27.490 --> 00:03:30.760
of $10,000 for five writers.

00:03:30.760 --> 00:03:35.760
So it would be $2,000 profit per writer.

00:03:39.150 --> 00:03:42.800
And what would be B's per
activity by this definition?

00:03:42.800 --> 00:03:47.020
Well, $28,000 divided by 11,

00:03:47.020 --> 00:03:51.640
28,000 divided by 11 writers,

00:03:51.640 --> 00:03:53.550
there's $28,000 in profit.

00:03:53.550 --> 00:03:55.200
That gives us a 2,000,

00:03:55.200 --> 00:03:58.620
and we'll round, 2,545.

00:03:58.620 --> 00:04:03.620
So approximately $2,545
of profit per writer.

00:04:06.620 --> 00:04:09.040
So based on this
measurement of productivity,

00:04:09.040 --> 00:04:11.000
it looks like website B

00:04:11.000 --> 00:04:12.773
is more productive.

00:04:13.730 --> 00:04:15.870
Now what if we did maybe
even a simpler one,

00:04:15.870 --> 00:04:17.250
or maybe one that you might think

00:04:17.250 --> 00:04:18.170
is even more natural?

00:04:18.170 --> 00:04:20.420
What if we thought about it as the number

00:04:20.420 --> 00:04:23.210
of posts per writer?

00:04:23.210 --> 00:04:24.540
So let's do it that way.

00:04:24.540 --> 00:04:27.860
So if we just defined productivity,

00:04:27.860 --> 00:04:28.693
I'll just do that,

00:04:28.693 --> 00:04:31.700
as being equal to the number

00:04:31.700 --> 00:04:36.700
of posts divided by the writers.

00:04:37.560 --> 00:04:38.850
Well, in this situation,

00:04:38.850 --> 00:04:40.654
website A would have a productivity

00:04:40.654 --> 00:04:44.690
of 110 divided by five,

00:04:44.690 --> 00:04:46.690
which would be 22,

00:04:46.690 --> 00:04:51.590
22 posts per writer,

00:04:51.590 --> 00:04:54.897
and website B 200 divided by 11 ,

00:04:55.990 --> 00:04:59.300
200 divided by 11 gets us,

00:04:59.300 --> 00:05:02.030
well, we could just
round that to maybe 18,

00:05:02.030 --> 00:05:04.290
since we want to get rough estimates,

00:05:04.290 --> 00:05:09.290
to 18 posts per writer.

00:05:10.510 --> 00:05:12.460
So by this measure of productivity,

00:05:12.460 --> 00:05:15.710
website A actually looks more productive.

00:05:15.710 --> 00:05:17.580
So hopefully this gives
you just a general sense

00:05:17.580 --> 00:05:18.890
of what productivity is,

00:05:18.890 --> 00:05:21.380
and an appreciation that
there's multiple ways

00:05:21.380 --> 00:05:22.290
of measuring it.

00:05:22.290 --> 00:05:24.320
And you could do much fancier things,

00:05:24.320 --> 00:05:25.340
you could say,

00:05:25.340 --> 00:05:27.580
you could make it a whole combination

00:05:27.580 --> 00:05:30.180
of outputs divided by a
combination of inputs.

00:05:30.180 --> 00:05:33.820
So you could define
productivity as the number

00:05:33.820 --> 00:05:36.570
of posts posted times the
average number of posts,

00:05:36.570 --> 00:05:37.820
you take that quantity,

00:05:37.820 --> 00:05:40.610
and then maybe add the
average likes per post

00:05:40.610 --> 00:05:42.910
or multiply by the average likes per post,

00:05:42.910 --> 00:05:46.160
and then add to that the comments per post

00:05:46.160 --> 00:05:48.050
or multiply by the comments per post

00:05:48.050 --> 00:05:50.040
and then divide by the number of writers

00:05:50.040 --> 00:05:51.870
or the number of writers and expenses

00:05:51.870 --> 00:05:53.630
or divide just by the expenses.

00:05:53.630 --> 00:05:58.170
But the general notion is you
wanna take some combination

00:05:58.170 --> 00:05:59.003
of the outputs,

00:05:59.003 --> 00:06:00.720
maybe even just one
output and divide it by

00:06:00.720 --> 00:06:04.630
some combination of the inputs
and you can actually decide

00:06:04.630 --> 00:06:06.440
how you even wanna combine,

00:06:06.440 --> 00:06:08.220
how you wanna mathematically combine

00:06:08.220 --> 00:06:09.763
those outputs and inputs.


Formulas and units: Volume of a pool

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:01.690
- [Instructor] We're told that Marvin

00:00:01.690 --> 00:00:04.200
has an inflatable wading
pool in his back yard.

00:00:04.200 --> 00:00:06.530
The pool is cylindrical with a base area

00:00:06.530 --> 00:00:09.920
of four square meters and
a height of 60 centimeters.

00:00:09.920 --> 00:00:13.590
What is the volume of
the pool in cubic meters?

00:00:13.590 --> 00:00:16.253
Pause this video and see
if you can figure that out.

00:00:17.110 --> 00:00:18.450
All right, now let's work
through this together.

00:00:18.450 --> 00:00:20.070
And let's just first visualize

00:00:20.070 --> 00:00:23.920
what this cylindrical
wading pool would look like.

00:00:23.920 --> 00:00:25.180
It would look something like this.

00:00:25.180 --> 00:00:26.710
A wading pool's kind of a small pool

00:00:26.710 --> 00:00:28.440
where you can just hang out a bit in it.

00:00:28.440 --> 00:00:31.620
You're not necessarily gonna
swim around too much in it.

00:00:31.620 --> 00:00:35.020
So it might look something like this.

00:00:35.020 --> 00:00:36.360
I know I'm not drawing it perfectly.

00:00:36.360 --> 00:00:38.090
It's kind of a hand-drawn situation,

00:00:38.090 --> 00:00:42.190
and I'm making it transparent
so that we can see the base.

00:00:42.190 --> 00:00:44.320
So the wading pool would
look something like that.

00:00:44.320 --> 00:00:48.750
They tell us that we have a
base area of four square meters.

00:00:48.750 --> 00:00:53.150
So this area right over
here, that's the base.

00:00:53.150 --> 00:00:55.490
That is four square meters.

00:00:55.490 --> 00:00:57.920
And it has a height of 60 centimeters,

00:00:57.920 --> 00:00:59.420
tell us that right over there.

00:00:59.420 --> 00:01:03.190
So this height is 60 centimeters.

00:01:03.190 --> 00:01:05.447
So the volume, our
reaction might be to say,

00:01:05.447 --> 00:01:08.636
"Okay, the volume of a cylinder
is the area of the base

00:01:08.636 --> 00:01:10.470
times the height."

00:01:10.470 --> 00:01:12.040
And so in this case,

00:01:12.040 --> 00:01:15.963
why wouldn't we just take four times 60,

00:01:16.840 --> 00:01:21.790
times 60, and we would
get a volume of 240.

00:01:21.790 --> 00:01:23.210
And we want it in cubic meters,

00:01:23.210 --> 00:01:25.580
so we just say 240 cubic meters.

00:01:25.580 --> 00:01:26.480
Is this true?

00:01:26.480 --> 00:01:28.476
Did I just do this correctly?

00:01:28.476 --> 00:01:30.850
Well, some of you might have realized

00:01:30.850 --> 00:01:31.990
that what I just multiplied,

00:01:31.990 --> 00:01:33.800
I didn't multiply four square meters

00:01:33.800 --> 00:01:37.450
times 60 meters to get 240 cubic meters.

00:01:37.450 --> 00:01:39.710
I just multiplied four square meters

00:01:39.710 --> 00:01:43.710
times 60, 60 centimeters.

00:01:43.710 --> 00:01:45.450
And if you multiply these two things,

00:01:45.450 --> 00:01:47.540
your actual units would
not be cubic meters.

00:01:47.540 --> 00:01:50.590
You would end up of units of
meters squared centimeters,

00:01:50.590 --> 00:01:51.710
which is not what they want

00:01:51.710 --> 00:01:54.320
and that is kind of a
bizarre set of units.

00:01:54.320 --> 00:01:56.890
So in order to get the
answer in cubic meters,

00:01:56.890 --> 00:02:01.820
we would wanna re-express 60
centimeters in terms of meters.

00:02:01.820 --> 00:02:05.450
Well, how many meters is 60 centimeters?

00:02:05.450 --> 00:02:08.480
Well, 100 centimeters make a meter.

00:02:08.480 --> 00:02:10.090
So I could write it this way.

00:02:10.090 --> 00:02:15.090
So 100 centimeters equal one meter.

00:02:17.280 --> 00:02:18.850
Or another way you could think about it is

00:02:18.850 --> 00:02:23.850
one centimeter is equal
to 1/100 of a meter.

00:02:25.870 --> 00:02:29.280
And so 60 centimeters is going to be equal

00:02:29.280 --> 00:02:34.210
to 60/100 of a meter.

00:02:34.210 --> 00:02:35.580
So now we can apply this,

00:02:35.580 --> 00:02:39.100
'cause we're dealing with
meters consistently now.

00:02:39.100 --> 00:02:41.530
So we can say, so this is actually wrong.

00:02:41.530 --> 00:02:43.350
We could say the volume
is going to be equal

00:02:43.350 --> 00:02:45.780
to the base in square meters,

00:02:45.780 --> 00:02:46.780
and I'm gonna write the units down

00:02:46.780 --> 00:02:48.380
and make sure we're doing the right thing,

00:02:48.380 --> 00:02:53.380
times the height, times
60 over 100 meters.

00:02:55.460 --> 00:02:57.570
And now everything works out.

00:02:57.570 --> 00:03:02.570
Four times 60 over 100 is
going to be 240 over 100.

00:03:04.260 --> 00:03:06.560
And then meters squared times meter,

00:03:06.560 --> 00:03:08.210
we are left with cubic meters,

00:03:08.210 --> 00:03:10.190
which is exactly what they asked us for.

00:03:10.190 --> 00:03:15.190
And of course, we could rewrite
this as 2.4 cubic meters.

00:03:16.290 --> 00:03:18.193
And we are done.


Formulas and units: Comparing rates

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.270
- [Instructor] We're told
that Hannah and Martine

00:00:02.270 --> 00:00:04.360
each got a plant for their home.

00:00:04.360 --> 00:00:06.460
Hannah measured that her plant grows,

00:00:06.460 --> 00:00:10.730
on average, 2 centimeters per week.

00:00:10.730 --> 00:00:13.570
Martine measured that her plant grows,

00:00:13.570 --> 00:00:18.330
on average, 3 millimeters per day.

00:00:18.330 --> 00:00:21.970
Which plant grows faster?

00:00:21.970 --> 00:00:23.260
Pause this video and see

00:00:23.260 --> 00:00:25.783
if you can figure that out on your own.

00:00:26.970 --> 00:00:28.960
All right, now let's go
through this together.

00:00:28.960 --> 00:00:30.020
So at first when you look at it,

00:00:30.020 --> 00:00:32.020
you might just compare 3 to 2

00:00:32.020 --> 00:00:34.460
and say, oh, 3 is larger than 2,

00:00:34.460 --> 00:00:37.330
therefore maybe Martine's
plant grows faster.

00:00:37.330 --> 00:00:39.890
And you would think that
until you look at the units.

00:00:39.890 --> 00:00:42.410
This is millimeters per day

00:00:42.410 --> 00:00:44.100
while for Hannah's plant

00:00:44.100 --> 00:00:45.830
it's in centimeters per week.

00:00:45.830 --> 00:00:47.620
So in order to really compare them

00:00:47.620 --> 00:00:49.870
we have to convert them to the same units

00:00:49.870 --> 00:00:52.410
in both length and time.

00:00:52.410 --> 00:00:54.480
So let's convert both of them,

00:00:54.480 --> 00:00:56.260
let's convert them both
to centimeters per week.

00:00:56.260 --> 00:00:57.740
You could just try to convert both of them

00:00:57.740 --> 00:00:58.840
to millimeters per day

00:00:58.840 --> 00:01:00.510
or I guess you could try
to convert both of them

00:01:00.510 --> 00:01:03.920
to meters per year, a third set of units,

00:01:03.920 --> 00:01:05.920
but centimeters per week seems reasonable

00:01:05.920 --> 00:01:08.870
since we already have
Hannah's plant rate at,

00:01:08.870 --> 00:01:09.720
so let me write this down.

00:01:09.720 --> 00:01:11.950
So Hannah, I'll just write H,

00:01:11.950 --> 00:01:16.340
grows at 2 centimeters per week.

00:01:16.340 --> 00:01:17.883
And then you have Martine,

00:01:21.120 --> 00:01:26.120
grows at an average of
3 millimeters per day.

00:01:28.640 --> 00:01:30.530
Now how do we convert
3 millimeters per day

00:01:30.530 --> 00:01:32.660
to centimeters per week?

00:01:32.660 --> 00:01:35.240
Well, first we could
convert the millimeters,

00:01:35.240 --> 00:01:37.630
actually, first, let's
convert the days into weeks.

00:01:37.630 --> 00:01:39.630
So how many days are there in a week?

00:01:39.630 --> 00:01:41.010
Well, there's 7 days in a week.

00:01:41.010 --> 00:01:43.320
So if we have how many
millimeters per day,

00:01:43.320 --> 00:01:45.960
if we wanted to know millimeters per week

00:01:45.960 --> 00:01:47.960
we would multiply times 7 days.

00:01:47.960 --> 00:01:49.030
So let me do that.

00:01:49.030 --> 00:01:52.230
So times 7 days

00:01:53.998 --> 00:01:55.823
in a week.

00:01:57.090 --> 00:01:58.790
That would get us,

00:01:58.790 --> 00:02:00.570
this would be equal to 3 times 7

00:02:00.570 --> 00:02:04.280
which is equal to 21 millimeters

00:02:06.230 --> 00:02:07.960
in a week.

00:02:07.960 --> 00:02:08.830
And you can see, actually,

00:02:08.830 --> 00:02:10.900
that the units cancel
out nicely like that,

00:02:10.900 --> 00:02:13.310
so you're left with millimeters a week.

00:02:13.310 --> 00:02:15.790
And that makes sense, 3 millimeters a day,

00:02:15.790 --> 00:02:17.810
you're able to do 7 times that in a week,

00:02:17.810 --> 00:02:19.400
21 millimeters a week.

00:02:19.400 --> 00:02:21.930
And then when you think
about 21 millimeters

00:02:21.930 --> 00:02:23.980
is how many centimeters?

00:02:23.980 --> 00:02:25.560
Well, we just have to remember

00:02:25.560 --> 00:02:28.980
that 1 centimeter is
equal to 10 millimeters,

00:02:28.980 --> 00:02:32.870
so if we wanna covert
this into centimeters,

00:02:32.870 --> 00:02:34.820
we essentially have to divide by 10.

00:02:34.820 --> 00:02:39.820
We could just say 1/10 of
a centimeter per millimeter

00:02:40.360 --> 00:02:41.480
and then that gets us,

00:02:41.480 --> 00:02:42.960
we could write it in different ways,

00:02:42.960 --> 00:02:43.900
but we could write this,

00:02:43.900 --> 00:02:46.360
and even here the units cancel out nicely,

00:02:46.360 --> 00:02:50.453
21 divided by 10 is 2.1
centimeters per week.

00:02:55.450 --> 00:02:57.300
Another way you could
have just thought about it

00:02:57.300 --> 00:03:02.300
is we could say 1 centimeter
is equal to 10 millimeters,

00:03:03.100 --> 00:03:05.577
or if you divide both sides by 10,

00:03:05.577 --> 00:03:10.570
1/10 of a centimeter is
equal to 1 millimeter,

00:03:10.570 --> 00:03:13.670
and if 1 millimeter is equal
to 1/10 of a centimeter

00:03:13.670 --> 00:03:18.310
then 21 millimeters is just
going to be 21 times this,

00:03:18.310 --> 00:03:21.700
21 times 1/10 is the same
thing as 21 divided by 10,

00:03:21.700 --> 00:03:26.063
it would be 2.1 centimeters.

00:03:27.340 --> 00:03:29.940
And so now we can compare
2.1 centimeters per week

00:03:29.940 --> 00:03:31.520
compared to 2 centimeters a week.

00:03:31.520 --> 00:03:32.353
Well, it turns out that

00:03:32.353 --> 00:03:36.100
when you actually compare
the appropriate units

00:03:36.100 --> 00:03:38.250
it still turns out that Martine's plant

00:03:38.250 --> 00:03:40.563
is growing just a little bit faster.


Using units to solve problems: Road trip

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:02.740
- [Instructor] We're told that
Ricky is going on a road trip

00:00:02.740 --> 00:00:04.690
that is 100 kilometers long.

00:00:04.690 --> 00:00:08.880
His average speed is
70 kilometers per hour.

00:00:08.880 --> 00:00:11.760
At that speed, he can
drive five kilometers

00:00:11.760 --> 00:00:14.900
for every liter of fuel that he uses.

00:00:14.900 --> 00:00:16.010
Fuel costs

00:00:18.117 --> 00:00:21.010
.60 dollars per liter.

00:00:21.010 --> 00:00:22.700
So the equivalent of 60 cents per liter,

00:00:22.700 --> 00:00:25.690
but they wrote it as
.60 dollars per liter.

00:00:25.690 --> 00:00:28.290
What is the cost of fuel for the trip?

00:00:28.290 --> 00:00:30.693
Pause the video and see if
you can figure that out.

00:00:32.240 --> 00:00:34.560
All right, so let's see what
information they gave us.

00:00:34.560 --> 00:00:38.247
They tell us that the trip
is 100 kilometers long.

00:00:39.903 --> 00:00:42.663
They tell us that the average speed

00:00:42.663 --> 00:00:44.830
is 70 kilometers per hour.

00:00:46.190 --> 00:00:48.357
So 70 kilometers per hour.

00:00:50.503 --> 00:00:53.170
They tell us that at that speed,

00:00:55.080 --> 00:00:56.550
he can drive five kilometers

00:00:56.550 --> 00:00:58.910
for every liter of fuel that he uses.

00:00:58.910 --> 00:01:03.910
So five kilometers per liter.

00:01:04.100 --> 00:01:05.990
And then they tell us that fuel costs

00:01:05.990 --> 00:01:09.890
0.60 dollars per liter.

00:01:09.890 --> 00:01:13.570
So then this last piece of
information right over here

00:01:13.570 --> 00:01:18.570
is that fuel costs 0.60 dollars per liter.

00:01:19.160 --> 00:01:21.640
Normally we would see that
written as 60 cents per liter,

00:01:21.640 --> 00:01:23.380
but let's just go with it this way.

00:01:23.380 --> 00:01:26.340
So what's going to be
useful for the total cost

00:01:26.340 --> 00:01:28.163
of the fuel for the trip?

00:01:29.100 --> 00:01:30.470
Well, we need to figure out

00:01:30.470 --> 00:01:32.030
how much fuel we're going to use,

00:01:32.030 --> 00:01:35.170
and then multiply that,
times the cost of the fuel.

00:01:35.170 --> 00:01:39.380
So how much fuel are we going to be using?

00:01:39.380 --> 00:01:41.800
Let's see, we're going 100 kilometers,

00:01:41.800 --> 00:01:43.250
that's the total distance.

00:01:43.250 --> 00:01:46.430
And then this tells us,
essentially how many liters

00:01:46.430 --> 00:01:49.380
we're going to have to use
over those 100 kilometers.

00:01:49.380 --> 00:01:51.930
Now you might say, how
exactly does that work?

00:01:51.930 --> 00:01:55.320
Well if I'm going five
kilometers per liter,

00:01:55.320 --> 00:01:58.850
if I were to take the
reciprocal of this information,

00:01:58.850 --> 00:02:03.170
I would get one fifth of a liter,

00:02:03.170 --> 00:02:04.450
of a liter,

00:02:04.450 --> 00:02:06.800
per kilometer.

00:02:06.800 --> 00:02:09.560
That's how much fuel I use per kilometer.

00:02:09.560 --> 00:02:11.290
One fifth of a liter.

00:02:11.290 --> 00:02:13.030
And so why is that useful?

00:02:13.030 --> 00:02:15.713
Well if I take 100 kilometers,

00:02:17.620 --> 00:02:18.790
and if I were to multiply,

00:02:18.790 --> 00:02:21.520
times 1/5th of a liter per kilometer,

00:02:21.520 --> 00:02:22.900
this is going to tell you

00:02:22.900 --> 00:02:24.700
that over the course of this trip,

00:02:24.700 --> 00:02:28.560
I am going to use 100 times 1/5th liters.

00:02:28.560 --> 00:02:29.790
Or this is going to tell us

00:02:29.790 --> 00:02:31.200
that over the course of the trip,

00:02:31.200 --> 00:02:33.810
we're are going to use 20 liters.

00:02:33.810 --> 00:02:35.720
And then if we were to multiply that,

00:02:35.720 --> 00:02:38.120
times the cost of fuel per liter,

00:02:38.120 --> 00:02:40.920
well then we know how much
the cost of our trip is.

00:02:40.920 --> 00:02:42.210
So let's do that.

00:02:42.210 --> 00:02:44.220
Then let's multiply this,

00:02:44.220 --> 00:02:49.220
times 0.60 dollars per liter,

00:02:51.880 --> 00:02:53.430
which is the same thing
as multiplying this,

00:02:53.430 --> 00:02:57.727
times 0.60 dollars per liter.

00:03:00.550 --> 00:03:01.710
The liters cancel out,

00:03:01.710 --> 00:03:03.050
so it's good that our units work out.

00:03:03.050 --> 00:03:04.970
We're left with just dollars here.

00:03:04.970 --> 00:03:08.640
So 20 times 0.60

00:03:08.640 --> 00:03:13.360
is going to get us to 12.

00:03:13.360 --> 00:03:16.540
So we are left with 12 dollars.

00:03:19.110 --> 00:03:20.930
And we're done, that's
the cost of our trip.

00:03:20.930 --> 00:03:21.970
And I know what you're thinking.

00:03:21.970 --> 00:03:26.160
Wait, we didn't use the
information right over here,

00:03:26.160 --> 00:03:28.440
that he's traveling an average speed

00:03:28.440 --> 00:03:29.810
of 70 kilometers per hour.

00:03:29.810 --> 00:03:32.060
It's true, we did not use
it in our calculation.

00:03:32.060 --> 00:03:34.840
Although it was kind of
useful because we had to know

00:03:34.840 --> 00:03:37.320
what his fuel efficiency
is, at that speed.

00:03:37.320 --> 00:03:38.270
So they're saying, they're traveling

00:03:38.270 --> 00:03:39.330
at 70 kilometers per hour,

00:03:39.330 --> 00:03:42.610
then at that speed, we
get this fuel efficiency.

00:03:42.610 --> 00:03:43.780
Now they could have just told us,

00:03:43.780 --> 00:03:44.840
they didn't even have to tell us this,

00:03:44.840 --> 00:03:45.900
they could have just told us,

00:03:45.900 --> 00:03:47.030
at whatever speed he's going,

00:03:47.030 --> 00:03:48.730
his fuel efficiency is this.

00:03:48.730 --> 00:03:50.150
And we still would have
been able to figure out

00:03:50.150 --> 00:03:53.023
the total cost of the fuel for the trip.


Using units to solve problems: Toy factory

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.250
- [Narrator] We're told
a factory makes toys

00:00:02.250 --> 00:00:04.280
that are sold for $10 apiece.

00:00:04.280 --> 00:00:05.930
The factory has 40 workers,

00:00:05.930 --> 00:00:09.770
and they each produce 25 toys a day.

00:00:09.770 --> 00:00:13.320
The factory is open five days a week.

00:00:13.320 --> 00:00:15.007
What is the total value of toys

00:00:15.007 --> 00:00:17.570
the factory produces in a day?

00:00:17.570 --> 00:00:20.580
Pause this video and see
if you can figure that out.

00:00:20.580 --> 00:00:22.860
All right, so let's
just think about a day,

00:00:22.860 --> 00:00:24.290
before I even look at this information.

00:00:24.290 --> 00:00:28.293
If I could figure out the value per toy,

00:00:29.780 --> 00:00:34.780
and then multiply that
times the number of toys,

00:00:34.900 --> 00:00:39.123
number of toys produced in a day,

00:00:43.290 --> 00:00:45.800
then we would have the total value.

00:00:45.800 --> 00:00:47.500
And let's see if they
give us that information.

00:00:47.500 --> 00:00:50.510
Well the value per toy,
they say the toys are sold

00:00:50.510 --> 00:00:53.910
for $10 apiece, so we
could write this this way.

00:00:53.910 --> 00:00:58.160
10 dollars per toy,

00:00:58.160 --> 00:01:00.130
and then they do tell us,

00:01:00.130 --> 00:01:02.500
or they give us the information
that we need to figure out

00:01:02.500 --> 00:01:04.110
how many are produced in a day.

00:01:04.110 --> 00:01:09.110
We have 40 workers, and they
each produce 25 toys a day.

00:01:09.410 --> 00:01:11.220
So the amount that's produced in a day,

00:01:11.220 --> 00:01:13.350
is going to be 40 workers

00:01:16.542 --> 00:01:19.990
times 25 toys per worker.

00:01:21.710 --> 00:01:25.300
Now I could say 25 toys
per worker per day,

00:01:25.300 --> 00:01:27.050
and that makes the units
a little complicated,

00:01:27.050 --> 00:01:29.790
or I could just realize
that this entire expression

00:01:29.790 --> 00:01:32.850
I'm creating is talking about one day.

00:01:32.850 --> 00:01:35.290
So the total number of
toys produced in a day

00:01:35.290 --> 00:01:37.720
is going to be the
product of these things.

00:01:37.720 --> 00:01:39.380
And we can say that the units work out,

00:01:39.380 --> 00:01:41.860
just to make sure that we're
getting in the right direction.

00:01:41.860 --> 00:01:43.730
A toy in the denominator cancels out

00:01:43.730 --> 00:01:46.310
with the toys in the numerator, workers,

00:01:46.310 --> 00:01:48.130
when you multiply it, this
would be in the numerator,

00:01:48.130 --> 00:01:49.120
this in the denominator.

00:01:49.120 --> 00:01:51.750
So workers, workers cancel out.

00:01:51.750 --> 00:01:56.750
And so I'm gonna be left with
10 times 40 times 25 dollars.

00:01:56.840 --> 00:01:59.700
And I do want it written in dollars.

00:01:59.700 --> 00:02:04.700
And so this is going to be
equal to 10 times 40 is 400,

00:02:06.140 --> 00:02:10.370
and then 400 times 25, let's
see, that's going to be

00:02:10.370 --> 00:02:14.530
4 times 25 times 100 so
that's 100 times 100,

00:02:14.530 --> 00:02:17.810
which is 10,000, and then the units

00:02:17.810 --> 00:02:19.803
we're left with is dollars.

00:02:20.980 --> 00:02:23.670
And now you might be
saying, wait, we didn't use

00:02:23.670 --> 00:02:26.250
all of the information and
that's true, we didn't use

00:02:26.250 --> 00:02:28.570
the fact that the factory
is open 5 days a week.

00:02:28.570 --> 00:02:30.600
We didn't need to use that information.

00:02:30.600 --> 00:02:32.340
That would have been useful if they said,

00:02:32.340 --> 00:02:33.677
what is the total value of toys

00:02:33.677 --> 00:02:36.390
the factory produces in a week.

00:02:36.390 --> 00:02:39.257
Then we would have said their
value per day is $10,000,

00:02:39.257 --> 00:02:42.310
and we could even write
it this way, per day,

00:02:42.310 --> 00:02:47.270
and then multiply that
times 5 days in a week,

00:02:47.270 --> 00:02:49.020
and that would have
given us the total value

00:02:49.020 --> 00:02:50.290
of the production in a week.

00:02:50.290 --> 00:02:51.480
But that's not what they're asking for,

00:02:51.480 --> 00:02:53.860
so we don't need that other information,

00:02:53.860 --> 00:02:55.440
and so we don't have to go to that step.

00:02:55.440 --> 00:02:57.870
And so this is really
just extra information,

00:02:57.870 --> 00:03:00.423
probably to distract you a bit.


Interpret quadratic models: Vertex form

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:02.410
- [Instructor] We're told that
Taylor opened a restaurant.

00:00:02.410 --> 00:00:05.050
The net value of the restaurant
in thousands of dollars,

00:00:05.050 --> 00:00:07.130
t months after its opening is modeled

00:00:07.130 --> 00:00:11.550
by v of t is equal to
two t squared minus 20t.

00:00:11.550 --> 00:00:12.520
Taylor wants to know

00:00:12.520 --> 00:00:16.640
what the restaurant's
lowest net value will be,

00:00:16.640 --> 00:00:21.640
underline that, and when
it will reach that value.

00:00:22.060 --> 00:00:24.190
So let's break it down step by step.

00:00:24.190 --> 00:00:26.240
The function which describes

00:00:26.240 --> 00:00:27.890
how the value of the restaurant,

00:00:27.890 --> 00:00:29.110
the net value of the restaurant,

00:00:29.110 --> 00:00:30.770
changes over time is right over here.

00:00:30.770 --> 00:00:33.130
If I were to graph it, I
can see that the coefficient

00:00:33.130 --> 00:00:35.030
on the quadratic term is positive,

00:00:35.030 --> 00:00:37.800
so it's going to be some form
of upward-opening parabola.

00:00:37.800 --> 00:00:39.320
I don't know exactly what it looks like,

00:00:39.320 --> 00:00:41.360
we can think about that in a second.

00:00:41.360 --> 00:00:44.830
And so it's going to have
some point, right over here,

00:00:44.830 --> 00:00:47.390
which really is the
vertex of this parabola,

00:00:47.390 --> 00:00:50.130
where it's going to hit
its lowest in that value,

00:00:50.130 --> 00:00:52.280
and that's going to happen at some time t,

00:00:52.280 --> 00:00:57.020
if you can imagine that this
right over here is the t-axis.

00:00:57.020 --> 00:01:00.390
So my first question
is, is there some form,

00:01:00.390 --> 00:01:02.050
is there some way that I can re-write

00:01:02.050 --> 00:01:05.030
this function algebraically
so it becomes very easy

00:01:05.030 --> 00:01:07.010
to pick out this low point,

00:01:07.010 --> 00:01:10.110
which is essentially the
vertex of this parabola?

00:01:10.110 --> 00:01:12.010
Pause this video and think about that.

00:01:13.310 --> 00:01:15.590
All right, so you can imagine the form

00:01:15.590 --> 00:01:17.820
that I'm talking about is vertex form,

00:01:17.820 --> 00:01:20.270
where you can clearly spot the vertex.

00:01:20.270 --> 00:01:21.620
And the way we can do that

00:01:21.620 --> 00:01:23.750
is actually by completing the square.

00:01:23.750 --> 00:01:25.810
So the first thing I will do is,

00:01:25.810 --> 00:01:27.390
actually let me factor out a two here,

00:01:27.390 --> 00:01:30.130
because two is a common
factor of both of these terms.

00:01:30.130 --> 00:01:35.010
So v of t would be equal to
two times t squared minus 10t.

00:01:35.010 --> 00:01:37.090
And I'm going to leave some space,

00:01:37.090 --> 00:01:41.330
because completing the square,
which gets us to vertex form,

00:01:41.330 --> 00:01:43.050
is all about adding and subtracting

00:01:43.050 --> 00:01:44.760
the same value on one side.

00:01:44.760 --> 00:01:47.160
So we're not actually changing
the value of that side,

00:01:47.160 --> 00:01:48.310
but writing it in a way

00:01:48.310 --> 00:01:51.430
so we have a perfect square expression,

00:01:51.430 --> 00:01:52.410
and then we're probably going

00:01:52.410 --> 00:01:55.110
to add or subtract some value out here.

00:01:55.110 --> 00:01:57.160
Now how do we make this a
perfect square expression?

00:01:57.160 --> 00:01:59.070
If any of this business

00:01:59.070 --> 00:02:01.100
about completing the square
looks unfamiliar to you,

00:02:01.100 --> 00:02:02.730
I encourage you to look
up completing the square

00:02:02.730 --> 00:02:04.530
on Khan Academy and review that.

00:02:04.530 --> 00:02:06.860
But the way that we complete the square

00:02:06.860 --> 00:02:10.560
is we look at this first degree
coefficient right over here,

00:02:10.560 --> 00:02:13.100
it's negative 10, and we say all right,

00:02:13.100 --> 00:02:15.680
well let's take half
of that and square it.

00:02:15.680 --> 00:02:19.490
So half of negative 10 is negative five,

00:02:19.490 --> 00:02:22.130
and if we were to square it, that's 25.

00:02:22.130 --> 00:02:24.960
So if we add 25 right over here,

00:02:24.960 --> 00:02:27.520
then this is going to become
a perfect square expression.

00:02:27.520 --> 00:02:29.890
And you can see that
it would be equivalent

00:02:29.890 --> 00:02:33.440
to this entire thing,
if we add 25 like that,

00:02:33.440 --> 00:02:38.440
is going to be equivalent
to t minus five squared,

00:02:38.470 --> 00:02:39.770
just this part right over here.

00:02:39.770 --> 00:02:43.580
That's why we took half
of this and we squared it.

00:02:43.580 --> 00:02:46.440
But as I alluded to a few seconds
ago, or a few minutes ago,

00:02:46.440 --> 00:02:50.269
you can't just willy
nilly add 25 to one side

00:02:50.269 --> 00:02:52.821
of an equation like this,

00:02:52.821 --> 00:02:55.960
that will make this
equality no longer true.

00:02:55.960 --> 00:02:58.590
And in fact we didn't just add 25.

00:02:58.590 --> 00:03:02.260
Remember we have this two out
here, we added two times 25.

00:03:02.260 --> 00:03:05.110
You can verify that if
you redistribute the two,

00:03:05.110 --> 00:03:09.590
you'd get two t squared minus 20t plus 50,

00:03:09.590 --> 00:03:11.550
plus two times 25.

00:03:11.550 --> 00:03:14.520
So in order to make the equality,

00:03:14.520 --> 00:03:17.360
or in order to allow it
to continue to be true,

00:03:17.360 --> 00:03:20.090
we have to subtract 50.

00:03:20.090 --> 00:03:20.970
So just to be clear,

00:03:20.970 --> 00:03:24.390
this isn't some kind of
strange thing I'm doing,

00:03:24.390 --> 00:03:27.370
all I did was add 50 and subtract 50.

00:03:27.370 --> 00:03:29.280
You're saying wait, you added 25, not 50.

00:03:29.280 --> 00:03:32.970
No look, when I added 25
here, it's in a parenthesis,

00:03:32.970 --> 00:03:35.980
and then the whole expression
is multiplied by two,

00:03:35.980 --> 00:03:39.670
so I really did add 50 here,
so then I subtract 50 here

00:03:39.670 --> 00:03:42.000
to get to what I originally had.

00:03:42.000 --> 00:03:43.840
And when you view it that way,

00:03:43.840 --> 00:03:48.840
now v of t is going to be equal
to two times this business,

00:03:49.150 --> 00:03:51.530
which we already established
is t minus five squared,

00:03:51.530 --> 00:03:55.860
and then we have the minus 50.

00:03:55.860 --> 00:03:57.440
Now why is this form useful?

00:03:57.440 --> 00:04:01.610
This is vertex form, it's very
easy to pick out the vertex.

00:04:01.610 --> 00:04:04.730
It's very easy to pick
out when the low point is.

00:04:04.730 --> 00:04:09.730
The low point here happens
when this part is minimized.

00:04:11.120 --> 00:04:12.620
And this part is
minimized, think about it,

00:04:12.620 --> 00:04:15.310
you have two times something squared.

00:04:15.310 --> 00:04:16.590
So if you have something squared,

00:04:16.590 --> 00:04:17.720
it's going to hit its lowest point

00:04:17.720 --> 00:04:20.330
when this something is zero,

00:04:20.330 --> 00:04:22.480
otherwise it's going
to be a positive value.

00:04:22.480 --> 00:04:25.440
And so this part right over here is going

00:04:25.440 --> 00:04:29.150
to be equal to zero
when t is equal to five.

00:04:29.150 --> 00:04:33.770
So the lowest value is
when t is equal to five.

00:04:33.770 --> 00:04:35.993
Let me do that in a different color,

00:04:35.993 --> 00:04:37.650
don't wanna reuse the colors too much.

00:04:37.650 --> 00:04:41.787
So if we say v of five

00:04:44.130 --> 00:04:45.810
is going to be equal

00:04:45.810 --> 00:04:50.810
to two times five minus five,

00:04:54.140 --> 00:04:55.450
trying to keep up with the colors,

00:04:55.450 --> 00:04:58.860
minus five squared minus 50.

00:04:58.860 --> 00:05:02.030
Notice this whole thing
becomes zero right over here.

00:05:02.030 --> 00:05:05.180
So v of five is equal to negative 50,

00:05:05.180 --> 00:05:06.710
that is when we hit our low point,

00:05:06.710 --> 00:05:09.490
in terms of the net
value of the restaurant.

00:05:09.490 --> 00:05:13.063
So t represents months,
so we hit our low point,

00:05:14.363 --> 00:05:17.150
we rewrote our function
in a form, in vertex form,

00:05:17.150 --> 00:05:18.440
so it's easy to pick out this value,

00:05:18.440 --> 00:05:21.860
and we see that this low point
happens at t equals five,

00:05:21.860 --> 00:05:24.460
which is at time five months.

00:05:24.460 --> 00:05:26.930
And then what is that lowest net value?

00:05:26.930 --> 00:05:28.020
Well it's negative 50.

00:05:28.020 --> 00:05:30.530
And remember, the function
gives us the net value

00:05:30.530 --> 00:05:34.843
in thousands of dollars,
so it's negative $50,000

00:05:36.280 --> 00:05:40.270
is the lowest net value of the restaurant.

00:05:40.270 --> 00:05:41.103
And you might say how do you

00:05:41.103 --> 00:05:43.440
have a negative value of something,

00:05:43.440 --> 00:05:46.630
well imagine if, say the
building is worth $50,000,

00:05:46.630 --> 00:05:49.333
but the restaurant owes $100,000,

00:05:49.333 --> 00:05:52.483
then it would have a
negative $50,000 net value.


Interpreting a parabola in context

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:01.270
- [Instructor] We're told that Adam flew

00:00:01.270 --> 00:00:04.630
his remote controlled
drone off of a platform.

00:00:04.630 --> 00:00:08.560
The function f models
the height of the drone

00:00:08.560 --> 00:00:10.940
above the ground, in meters,

00:00:10.940 --> 00:00:15.860
as a function of time, in
seconds, after takeoff.

00:00:15.860 --> 00:00:17.610
So what they want us
to do is plot the point

00:00:17.610 --> 00:00:20.410
on the graph of f that corresponds to each

00:00:20.410 --> 00:00:21.580
of the following things.

00:00:21.580 --> 00:00:23.030
So pause the video and
see if you can do that,

00:00:23.030 --> 00:00:25.440
and, obviously, you can't
draw on your screen.

00:00:25.440 --> 00:00:27.320
This is from an exercise on Khan Academy,

00:00:27.320 --> 00:00:28.720
but you can visually look at it,

00:00:28.720 --> 00:00:30.930
and even with your
finger, point to the part

00:00:30.930 --> 00:00:35.420
of the graph of f that
represents each of these things.

00:00:35.420 --> 00:00:36.830
All right, so the first thing here

00:00:36.830 --> 00:00:39.650
is the height of the platform.

00:00:39.650 --> 00:00:43.630
So the drone is at the
height of the platform

00:00:43.630 --> 00:00:44.890
right when it takes off,

00:00:44.890 --> 00:00:47.610
'cause it says Adam flew
his remote controlled drone

00:00:47.610 --> 00:00:49.410
off of a platform.

00:00:49.410 --> 00:00:52.640
So what is the time that he's taking off,

00:00:52.640 --> 00:00:54.180
the drone, or the drone is taking off?

00:00:54.180 --> 00:00:56.320
Well, that's going to
be at time t equals zero

00:00:56.320 --> 00:00:57.153
right over here.

00:00:57.153 --> 00:00:59.490
And what is the height of
the drone at that moment?

00:00:59.490 --> 00:01:01.150
It is 60 meters.

00:01:01.150 --> 00:01:03.690
So that must be the
height of the platform.

00:01:03.690 --> 00:01:05.670
So that point right over there tells us

00:01:05.670 --> 00:01:06.890
the height of the platform.

00:01:06.890 --> 00:01:09.310
And if they asked us what the
height of the platform is,

00:01:09.310 --> 00:01:11.240
it would be 60 meters.

00:01:11.240 --> 00:01:14.650
The next one is the
drone's maximum height.

00:01:14.650 --> 00:01:16.120
So then as time goes on,

00:01:16.120 --> 00:01:17.650
we can see the drone starts going

00:01:17.650 --> 00:01:19.420
to a higher and higher and higher height,

00:01:19.420 --> 00:01:21.070
gets as high as 80 meters.

00:01:21.070 --> 00:01:23.520
And then it starts going down.

00:01:23.520 --> 00:01:27.110
So it looks like 80
meters, at time 10 seconds,

00:01:27.110 --> 00:01:31.140
the drone hits a maximum
height of 80 meters.

00:01:31.140 --> 00:01:33.040
And then last but not least,

00:01:33.040 --> 00:01:36.870
they say the time when the
drone landed on the ground.

00:01:36.870 --> 00:01:38.480
Now, we can assume that the ground

00:01:38.480 --> 00:01:41.520
is when the height of the
drone is at zero meters,

00:01:41.520 --> 00:01:44.430
and we can see that that
happens right over here.

00:01:44.430 --> 00:01:47.950
And that happens at time
t equals 30 seconds.

00:01:47.950 --> 00:01:49.570
And so we've just marked it off,

00:01:49.570 --> 00:01:50.880
and I know what some of
you all are thinking.

00:01:50.880 --> 00:01:52.330
Wait, there's another time

00:01:52.330 --> 00:01:55.420
where the drone's height is at zero,

00:01:55.420 --> 00:01:56.470
and that's right over here.

00:01:56.470 --> 00:01:58.410
That's at negative 10 seconds.

00:01:58.410 --> 00:01:59.950
Couldn't we say that that's also a time

00:01:59.950 --> 00:02:01.530
when the drone landed on the ground?

00:02:01.530 --> 00:02:04.220
And this is a important point to realize,

00:02:04.220 --> 00:02:05.260
because if we're really trying

00:02:05.260 --> 00:02:06.910
to model the drone's behavior

00:02:06.910 --> 00:02:09.140
from time t equals zero,

00:02:09.140 --> 00:02:11.230
if t equals zero is
right when you take off

00:02:11.230 --> 00:02:12.870
all the way to it lands,

00:02:12.870 --> 00:02:15.770
then this parabola that we're
showing right over here,

00:02:15.770 --> 00:02:17.680
it actually, we would probably want

00:02:17.680 --> 00:02:20.660
to restrict its domain to positive times.

00:02:20.660 --> 00:02:22.950
And so this negative time
region right over here

00:02:22.950 --> 00:02:24.910
really doesn't make a lot of sense.

00:02:24.910 --> 00:02:28.940
We should probably consider
the non-negative values

00:02:28.940 --> 00:02:30.870
of time when we're trying to think

00:02:30.870 --> 00:02:32.783
about these different thins.


Sequences and domain

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.180
- [Instructor] The focus
of this video is going

00:00:02.180 --> 00:00:04.150
to be on sequences,

00:00:04.150 --> 00:00:05.980
which you have hopefully already seen.

00:00:05.980 --> 00:00:07.210
If you don't know what a sequence is,

00:00:07.210 --> 00:00:09.870
I encourage you to review
those videos on Khan Academy.

00:00:09.870 --> 00:00:11.630
But we're going to focus on

00:00:11.630 --> 00:00:13.980
how we can generate the same sequence

00:00:13.980 --> 00:00:18.220
with different functions
that have different domains.

00:00:18.220 --> 00:00:20.850
So let's just start with
an example sequence.

00:00:20.850 --> 00:00:22.780
Let's say we have a sequence.

00:00:22.780 --> 00:00:23.613
It's a six.

00:00:23.613 --> 00:00:24.850
You could call that the first term.

00:00:24.850 --> 00:00:27.100
Some people would call
that the zeroth term, six.

00:00:27.100 --> 00:00:28.430
And then if that's the first term,

00:00:28.430 --> 00:00:31.720
the second term is now a 12,

00:00:31.720 --> 00:00:34.530
then a 24,

00:00:34.530 --> 00:00:35.590
then a 48,

00:00:36.890 --> 00:00:38.420
and so on and so forth.

00:00:38.420 --> 00:00:41.530
And as we'll see, there's
multiple function definitions

00:00:41.530 --> 00:00:43.770
that could create the sequence.

00:00:43.770 --> 00:00:48.240
One way to think about it
is this is six times one.

00:00:48.240 --> 00:00:50.670
This is six times two.

00:00:50.670 --> 00:00:53.060
This is six times four.

00:00:53.060 --> 00:00:55.680
This is six times eight.

00:00:55.680 --> 00:00:57.540
So it looks like each term

00:00:57.540 --> 00:01:02.280
is six times a power of two.

00:01:02.280 --> 00:01:03.130
Let me make that clear.

00:01:03.130 --> 00:01:06.370
This one right over here is
six times two to the zero.

00:01:06.370 --> 00:01:07.910
That's six times one.

00:01:07.910 --> 00:01:11.910
This one over here is six
times two to the first.

00:01:11.910 --> 00:01:16.020
This one over here is
six times two squared,

00:01:16.020 --> 00:01:17.310
six times four.

00:01:17.310 --> 00:01:20.840
This right over here is six times two

00:01:20.840 --> 00:01:22.350
to the third.

00:01:22.350 --> 00:01:25.280
And so one way to view this is

00:01:25.280 --> 00:01:28.160
if you view this as the zeroth term.

00:01:28.160 --> 00:01:30.570
We could define a function,

00:01:30.570 --> 00:01:32.730
call it a of n,

00:01:32.730 --> 00:01:34.960
where n is referring to our index

00:01:34.960 --> 00:01:36.460
or which term in the sequence.

00:01:36.460 --> 00:01:41.060
And it's equal to six times two,

00:01:41.060 --> 00:01:43.640
six times two to the n,

00:01:43.640 --> 00:01:45.610
where n starts at zero,

00:01:45.610 --> 00:01:47.590
and then it keeps incrementing by one.

00:01:47.590 --> 00:01:51.690
So it's really all integers
greater than or equal to zero.

00:01:51.690 --> 00:01:54.493
And it's very important
to specify that domain,

00:01:58.978 --> 00:02:00.061
where n is an

00:02:03.837 --> 00:02:04.754
integer and

00:02:07.221 --> 00:02:10.720
n is greater than or equal to zero.

00:02:10.720 --> 00:02:13.300
You could see what happens
if n is not an integer.

00:02:13.300 --> 00:02:15.930
If you tried to put a 1.5 year

00:02:15.930 --> 00:02:17.380
or something like that,
then you're not going

00:02:17.380 --> 00:02:19.260
to get one of the terms in the sequence.

00:02:19.260 --> 00:02:23.520
And if we don't start at
zero, if we started at one,

00:02:23.520 --> 00:02:25.840
then this would be the
first term in the sequences,

00:02:25.840 --> 00:02:26.800
which is not what we want.

00:02:26.800 --> 00:02:28.190
We want to generate the sequence

00:02:28.190 --> 00:02:29.500
that I originally wrote down.

00:02:29.500 --> 00:02:32.210
And obviously, if you started
at n equals negative one,

00:02:32.210 --> 00:02:35.170
then you're gonna get a different
value for your first term.

00:02:35.170 --> 00:02:36.490
So this is one way

00:02:37.719 --> 00:02:40.230
to essentially define or create a function

00:02:40.230 --> 00:02:42.480
which generates this sequence.

00:02:42.480 --> 00:02:46.370
But as we'll see, there
are other ways to do it.

00:02:46.370 --> 00:02:48.720
For example, let me do another one.

00:02:48.720 --> 00:02:50.470
I'll do it in another color.

00:02:50.470 --> 00:02:53.030
Let's say I have b of n.

00:02:53.030 --> 00:02:55.313
And let's say I want to,

00:02:58.263 --> 00:03:01.160
instead of saying, okay, I'm
gonna start at n equals zero,

00:03:01.160 --> 00:03:03.680
and you could kind of view
this as the zeroth term,

00:03:03.680 --> 00:03:06.300
I want to start at n equals one.

00:03:06.300 --> 00:03:07.380
So what you could do is,

00:03:07.380 --> 00:03:11.700
is when you input a one, this
essentially becomes a zero.

00:03:11.700 --> 00:03:12.533
How do I do that?

00:03:12.533 --> 00:03:13.710
Well, I just subtract one from it.

00:03:13.710 --> 00:03:16.520
So I could say six times two

00:03:16.520 --> 00:03:19.890
to the n minus one power,

00:03:19.890 --> 00:03:24.010
where n is

00:03:24.010 --> 00:03:25.150
an integer

00:03:27.295 --> 00:03:30.190
and n is greater than

00:03:30.190 --> 00:03:31.870
or equal to one.

00:03:31.870 --> 00:03:34.720
Notice, now when we put
n equals one in here,

00:03:34.720 --> 00:03:37.040
we could maybe call this the first term.

00:03:37.040 --> 00:03:38.070
We want to generate a six.

00:03:38.070 --> 00:03:38.903
So what happens?

00:03:38.903 --> 00:03:41.160
One minus one, we get that zeroth power

00:03:41.160 --> 00:03:42.680
that we want right over there.

00:03:42.680 --> 00:03:45.880
And so six times two to
the zero is indeed six.

00:03:45.880 --> 00:03:47.210
Then when n is equal to two,

00:03:47.210 --> 00:03:49.660
it's six times two to the two minus one,

00:03:49.660 --> 00:03:51.400
which is just two to the first power.

00:03:51.400 --> 00:03:53.740
So it just becomes six times
two, which is equal to 12.

00:03:53.740 --> 00:03:56.230
So notice, these are
different function definitions

00:03:56.230 --> 00:03:57.800
with different domains,

00:03:57.800 --> 00:04:01.630
but they're generating
the exact same sequence.

00:04:01.630 --> 00:04:04.610
We could also do it recursively.

00:04:04.610 --> 00:04:06.130
And we've seen this in other videos.

00:04:06.130 --> 00:04:08.490
We can define a function recursively.

00:04:08.490 --> 00:04:09.590
We could say, all right, look,

00:04:09.590 --> 00:04:12.218
it looks like each of these terms

00:04:12.218 --> 00:04:15.860
in our sequence is
twice the previous term.

00:04:15.860 --> 00:04:16.700
So we could,

00:04:16.700 --> 00:04:19.290
if we want a recursive
definition for the sequence,

00:04:19.290 --> 00:04:21.240
we can define the first term,

00:04:21.240 --> 00:04:23.310
or, in this case, we
could say the zeroth term

00:04:23.310 --> 00:04:25.630
if we want to start at n equals zero.

00:04:25.630 --> 00:04:28.460
T of zero is equal to six.

00:04:28.460 --> 00:04:32.370
And then we could say t of n

00:04:33.280 --> 00:04:37.360
is equal to two times

00:04:37.360 --> 00:04:39.090
t of n minus one,

00:04:39.090 --> 00:04:41.340
t of n minus one.

00:04:41.340 --> 00:04:46.050
And then this is going to be for,

00:04:46.050 --> 00:04:49.290
or maybe I'll write it this way, where

00:04:50.170 --> 00:04:53.140
n is an integer

00:04:54.910 --> 00:04:58.020
and n is greater than

00:04:58.020 --> 00:04:59.670
or equal to zero.

00:04:59.670 --> 00:05:01.190
This would also generate the sequence.

00:05:01.190 --> 00:05:04.550
When you put n equals zero
here, you'll get that term.

00:05:04.550 --> 00:05:06.230
When you get n equals one,

00:05:06.230 --> 00:05:09.290
t of one is going to be two

00:05:09.290 --> 00:05:11.980
times t of one minus one, t of zero.

00:05:11.980 --> 00:05:13.890
In that case, it'd be t of,

00:05:13.890 --> 00:05:17.000
or, sorry, it would be two
times t of zero is six.

00:05:17.000 --> 00:05:19.380
So two times six, it would get you 12.

00:05:19.380 --> 00:05:22.090
Now if you wanted it so
that it generates the six

00:05:22.090 --> 00:05:24.523
when n equals one, you
could do it this way.

00:05:26.220 --> 00:05:27.670
You could write it,

00:05:27.670 --> 00:05:29.750
actually, maybe I should
have kept all of that,

00:05:29.750 --> 00:05:32.000
or I'm gonna have to rewrite all of that.

00:05:32.000 --> 00:05:35.020
But you could write it this way.

00:05:35.020 --> 00:05:38.030
Instead of saying t of
zero is equal to six,

00:05:38.030 --> 00:05:40.660
we could write t of one is equal to six.

00:05:40.660 --> 00:05:43.000
But now we'd have to
write a different domain,

00:05:43.000 --> 00:05:44.928
where n still has to be an integer.

00:05:44.928 --> 00:05:49.113
N is an integer.

00:05:50.100 --> 00:05:51.840
And now instead of saying n

00:05:51.840 --> 00:05:52.950
is greater than or equal to zero,

00:05:52.950 --> 00:05:56.510
now n is greater than or equal to one.

00:05:56.510 --> 00:05:58.940
So hopefully this video
hits the point home

00:05:58.940 --> 00:06:00.650
that there's multiple ways,

00:06:00.650 --> 00:06:02.090
either with a traditional,

00:06:02.090 --> 00:06:03.530
I guess you would say explicit function

00:06:03.530 --> 00:06:06.250
or a recursive function like this.

00:06:06.250 --> 00:06:08.560
And even in either of those cases,

00:06:08.560 --> 00:06:09.990
you could have different domains

00:06:09.990 --> 00:06:11.530
and different function definitions

00:06:11.530 --> 00:06:13.550
that generate the same sequence,

00:06:13.550 --> 00:06:15.723
but you really have to
think about the domain.


Using inequalities to solve problems

WEBVTT
Kind: captions
Language: en

00:00:00.310 --> 00:00:02.370
- [Instructor] We're told that
Kayla wants to visit a friend

00:00:02.370 --> 00:00:04.610
who lives eight kilometers away.

00:00:04.610 --> 00:00:07.120
She'll ride the subway as far as she can

00:00:07.120 --> 00:00:09.840
before walking the rest of the way.

00:00:09.840 --> 00:00:14.840
First, she needs to buy an
access pass that costs $5.50.

00:00:14.860 --> 00:00:18.970
There's also a fee of $1.25 per stop.

00:00:18.970 --> 00:00:20.960
This is an expensive subway.

00:00:20.960 --> 00:00:24.730
Kayla doesn't want to spend
more than $15 on the trip.

00:00:24.730 --> 00:00:26.660
So she wants to know the largest number

00:00:26.660 --> 00:00:28.650
of stops she can afford.

00:00:28.650 --> 00:00:33.650
Let S represent the number
of stops that Kayla buys.

00:00:34.220 --> 00:00:35.710
So first, pause this video

00:00:35.710 --> 00:00:38.350
and see if you can write an inequality

00:00:38.350 --> 00:00:41.210
that describes how many stops,

00:00:41.210 --> 00:00:43.730
or that describes the
situation that describes

00:00:43.730 --> 00:00:46.640
that she wants to take
as many stops as she can,

00:00:46.640 --> 00:00:49.103
but she doesn't wanna spend more than $15.

00:00:51.110 --> 00:00:52.910
All right, now let's do this together.

00:00:52.910 --> 00:00:54.280
So first, let's just
think about an expression

00:00:54.280 --> 00:00:56.350
for how much she spends.

00:00:56.350 --> 00:00:59.630
So no matter what, she's
going to spend $5.50,

00:00:59.630 --> 00:01:02.663
so we can write it like this, so $5.50,

00:01:04.180 --> 00:01:05.430
that's what she's going to spend,

00:01:05.430 --> 00:01:07.310
even if she doesn't take any stops.

00:01:07.310 --> 00:01:11.550
And it's $1.25 per stop, and
S is the number of stops.

00:01:11.550 --> 00:01:13.610
So the amount she's going
to spend just from the stops

00:01:13.610 --> 00:01:15.970
is going to be $1.25 times S.

00:01:15.970 --> 00:01:20.170
So it's going to be plus $1.25 S.

00:01:21.920 --> 00:01:23.580
This is the upfront she has to spend,

00:01:23.580 --> 00:01:25.290
and this is how much she's
going to spend on stop.

00:01:25.290 --> 00:01:26.730
So this is an S right over here

00:01:26.730 --> 00:01:28.070
and I wrote a five right next to it,

00:01:28.070 --> 00:01:29.490
they look kind of similar.

00:01:29.490 --> 00:01:34.150
And we know that she doesn't
want to spend more than $15.

00:01:34.150 --> 00:01:37.230
So she's willing to spend up to $15.

00:01:37.230 --> 00:01:39.410
So this total amount that she spends

00:01:39.410 --> 00:01:43.393
has to be less than or equal to $15.

00:01:44.970 --> 00:01:47.010
Or if we didn't write it
with the dollar symbols,

00:01:47.010 --> 00:01:51.777
we would write 5.50 plus 1.25 S

00:01:53.410 --> 00:01:55.160
is going to be less than or equal to,

00:01:55.160 --> 00:01:58.720
or needs to be less than or equal to 15.

00:01:58.720 --> 00:02:00.530
Now that we've written this inequality,

00:02:00.530 --> 00:02:05.460
what is the number of stops
that Kayla can afford?

00:02:05.460 --> 00:02:08.550
What's the largest number of
stops that she can afford?

00:02:08.550 --> 00:02:11.210
Pause this video and
try to figure that out.

00:02:11.210 --> 00:02:14.580
Well, to figure that out,
we just have to solve for S

00:02:14.580 --> 00:02:16.560
and then figure out what the largest S is

00:02:16.560 --> 00:02:20.260
that satisfies the inequality
once we've solved for S.

00:02:20.260 --> 00:02:21.320
So the first thing I would do

00:02:21.320 --> 00:02:25.453
is subtract 5.50 from both sides.

00:02:27.010 --> 00:02:31.933
When we do that, we are
left with 1.25 or $1.25 S

00:02:35.390 --> 00:02:40.210
is less than or equal to 9.50.

00:02:40.210 --> 00:02:43.120
And then I would divide
both sides by 1.25.

00:02:43.120 --> 00:02:45.660
And since I'm dividing both
sides by a positive value,

00:02:45.660 --> 00:02:48.420
it doesn't change the
direction of the inequality,

00:02:48.420 --> 00:02:53.373
1.25 and then divide this by 1.25,

00:02:56.876 --> 00:02:59.830
9.5 divide by 1.25

00:03:01.150 --> 00:03:04.870
is equal to 7.6.

00:03:04.870 --> 00:03:09.737
So we get that S needs to be
less than or equal to 7.6.

00:03:11.900 --> 00:03:15.820
So we can't take a
fractional number of stops.

00:03:15.820 --> 00:03:20.220
So the largest number of
stops that Kayla can take

00:03:20.220 --> 00:03:21.630
is going to be seven stops.

00:03:21.630 --> 00:03:22.463
She can't take eight,

00:03:22.463 --> 00:03:24.520
and she can't take 7 1/2 or 7.6.

00:03:24.520 --> 00:03:27.830
So the largest number she
can take is seven stops.

00:03:27.830 --> 00:03:32.830
So she can take as many

00:03:34.430 --> 00:03:38.280
as seven stops.

00:03:38.280 --> 00:03:39.523
And we are done.


Solve by completing the square: Non-integer solutions

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.420
- [Instructor] Let's say we're told

00:00:01.420 --> 00:00:06.420
that zero is equal to x
squared plus six x plus three,

00:00:07.300 --> 00:00:10.040
what is an x or what our x is

00:00:10.040 --> 00:00:11.920
that would satisfy this equation?

00:00:11.920 --> 00:00:14.800
Pause this video and try to figure it out.

00:00:14.800 --> 00:00:16.210
All right, now let's
work through it together.

00:00:16.210 --> 00:00:17.710
So the first thing that I would try to do

00:00:17.710 --> 00:00:19.950
is see if I could factor
this right hand expression,

00:00:19.950 --> 00:00:22.330
I have some expression,
it's equal to zero.

00:00:22.330 --> 00:00:25.620
So if I could factor it,
that might help solve.

00:00:25.620 --> 00:00:27.370
So let's see, can I think of two numbers

00:00:27.370 --> 00:00:30.020
that when I add them, I get six,

00:00:30.020 --> 00:00:34.000
and when I take their
product, I get positive three?

00:00:34.000 --> 00:00:36.330
Well, if I'm thinking
just in terms of integers,

00:00:36.330 --> 00:00:37.330
three is a prime number,

00:00:37.330 --> 00:00:40.790
it only has two factors one and three.

00:00:40.790 --> 00:00:43.820
And let's see one plus
three is not equal to six,

00:00:43.820 --> 00:00:47.480
so it doesn't look like factoring
is going to help me much.

00:00:47.480 --> 00:00:50.280
So the next thing I'll turn
to is completing the square.

00:00:50.280 --> 00:00:51.780
In fact, completing the square

00:00:53.816 --> 00:00:55.850
if there are x values that
would satisfy this equation,

00:00:55.850 --> 00:00:57.630
completing the square
will help us solve it.

00:00:57.630 --> 00:01:00.750
And the way I do it, I'll
say zero is equal to,

00:01:00.750 --> 00:01:04.000
let me rewrite the first
part, x squared plus six x,

00:01:04.000 --> 00:01:06.630
and then I'm gonna write
the plus three out here.

00:01:06.630 --> 00:01:10.500
And my goal is to add something

00:01:12.000 --> 00:01:13.690
to the right-hand
expression, right over here,

00:01:13.690 --> 00:01:15.220
and then I'm gonna
subtract that same thing,

00:01:15.220 --> 00:01:17.810
so I'm not really changing the
value of the right-hand side.

00:01:17.810 --> 00:01:19.990
And I wanna add something here

00:01:19.990 --> 00:01:21.150
that I'm later going to subtract,

00:01:21.150 --> 00:01:24.370
so that what I have in
parentheses is a perfect square.

00:01:24.370 --> 00:01:26.020
Well, the way to make it a perfect square,

00:01:26.020 --> 00:01:27.860
and we've talked about
this in other videos

00:01:27.860 --> 00:01:30.530
when we introduced ourselves
to completing the square,

00:01:30.530 --> 00:01:32.600
is we'll look at this first degree

00:01:32.600 --> 00:01:35.330
coefficient right over
here, this positive six,

00:01:35.330 --> 00:01:37.760
and say, okay, half of
that is positive three,

00:01:37.760 --> 00:01:41.750
and if we were to square
that, we would get nine.

00:01:41.750 --> 00:01:43.180
So let's add a nine there.

00:01:43.180 --> 00:01:45.120
And then we could also subtract a nine.

00:01:45.120 --> 00:01:47.500
Notice, we haven't changed the value

00:01:47.500 --> 00:01:48.850
of the right-hand side expression,

00:01:48.850 --> 00:01:51.170
we're adding nine and
we're subtracting nine.

00:01:51.170 --> 00:01:53.460
And actually, the parentheses
are just there to help it

00:01:53.460 --> 00:01:55.170
make a little bit more
visually clear to us,

00:01:55.170 --> 00:01:56.760
but you don't even need the parentheses,

00:01:56.760 --> 00:01:59.610
you would essentially get the same result.

00:01:59.610 --> 00:02:02.470
But then what happens if we
simplify this a little bit?

00:02:02.470 --> 00:02:04.540
Well, what I just showed you,

00:02:04.540 --> 00:02:07.100
let me do it in this green blue color,

00:02:07.100 --> 00:02:11.750
this thing can be rewritten
as x plus three squared.

00:02:11.750 --> 00:02:13.480
That's why we added nine there,

00:02:13.480 --> 00:02:15.790
we said, all right, we're
gonna be dealing with a three

00:02:15.790 --> 00:02:17.000
'cause three is half of six.

00:02:17.000 --> 00:02:19.070
And if we squared three,
we get a nine there.

00:02:19.070 --> 00:02:22.130
And then this second
part, right over here,

00:02:22.130 --> 00:02:25.520
three minus nine, that's
equal to negative six.

00:02:25.520 --> 00:02:26.900
So we could write it like this,

00:02:26.900 --> 00:02:31.110
zero is equal to x plus
three squared minus six.

00:02:31.110 --> 00:02:35.260
And now what we can do is
isolate this x plus three squared

00:02:35.260 --> 00:02:37.750
by adding six to both
sides, so let's do that.

00:02:37.750 --> 00:02:41.130
Let's add six there, let's add six there.

00:02:41.130 --> 00:02:44.880
And what we get on the left-hand side,

00:02:44.880 --> 00:02:48.210
we get six is equal to,
on the right-hand side,

00:02:48.210 --> 00:02:53.210
we just get x plus three squared.

00:02:53.370 --> 00:02:56.330
And now we can take the
square root of both sides,

00:02:56.330 --> 00:03:01.300
and we could say that the plus
or minus square root of six

00:03:04.090 --> 00:03:08.390
is equal to x plus three.

00:03:08.390 --> 00:03:09.930
And if this doesn't make full sense,

00:03:09.930 --> 00:03:11.870
just pause the video a little
bit and think about it.

00:03:11.870 --> 00:03:15.340
If I'm saying that something
squared is equal to six,

00:03:15.340 --> 00:03:17.070
that means that the something

00:03:17.070 --> 00:03:19.670
is either going to be the
positive square root of six,

00:03:19.670 --> 00:03:22.490
or the negative square root of six.

00:03:22.490 --> 00:03:25.320
And so now we can, if
we wanna solve for x,

00:03:25.320 --> 00:03:28.150
we can just subtract
three from both sides.

00:03:28.150 --> 00:03:32.653
So let's subtract three from
both sides, and what do we get?

00:03:33.900 --> 00:03:37.760
We get on the right-hand side,
we just are left with an x,

00:03:37.760 --> 00:03:41.220
and that's going to be
equal to negative three

00:03:41.220 --> 00:03:45.530
plus or minus the square root of six.

00:03:45.530 --> 00:03:47.750
And we are done.

00:03:47.750 --> 00:03:49.910
And obviously we could
rewrite this and say,

00:03:49.910 --> 00:03:51.760
x could be equal to negative three

00:03:51.760 --> 00:03:54.310
plus the square root of six,

00:03:54.310 --> 00:03:57.610
or x could be equal to negative three

00:03:57.610 --> 00:04:00.593
minus the square root of six.