Invalid conclusions from studies example

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- [Instructor] Jerry was reading

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about a study that
looked at the connection

00:00:02.930 --> 00:00:06.380
between smartphone usage
and happiness based on data

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from approximately 5,000
randomly selected teenagers.

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The study found that on average

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the teens who spent
more time on smartphones

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were significantly less happy

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than those who spent
less time on smartphones.

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Jerry concluded that spending more time

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on smartphones makes teens less happy.

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All right this is interesting.

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So what I want you to do is think

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about whether Jerry is making
a valid conclusion or not.

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And why or why don't you think

00:00:34.667 --> 00:00:36.823
he's making a valid conclusion?

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All right now let's work on this together.

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This is really important to understand

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because you will see things like this

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in the popular media all the time

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that try to establish a causality

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when there might not be causality.

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Or at least where the study might

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not be able to show causality.

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So right now Jerry is
saying he's concluding

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that smartphone usage, smartphone usage

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makes teens less happy, so he's

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assuming there's a causal connection.

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Smartphone usage causes
teens to be less happy.

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Less happy, can he actually
make that conclusion

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from this study based
on how it was designed?

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Well the first thing to ask ourselves

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is is this an experimental study

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that is designed to establish causality

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or is it an observational study

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where we really can just
say there's an association

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but we really can't make a
statement about causality?

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Well in experimental
study he would have had

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to have a control group
and then a treatment group

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sometimes called a experimental group.

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So I'll say that's control group,

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that's the treatment or
the experimental group.

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And then you randomly assign folks

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to one of those two groups and then

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you would make that treatment group

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use a cell phone more and
see if they are less happy.

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That's not what happened here.

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What happened here was
an observational study.

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In this study we are
looking at two variables.

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So you have your smartphone usage

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and then you have the teen happiness.

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And they took these 5,000
randomly selected teenagers

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and they figured out
their smartphone usage

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and their happiness, maybe
with a survey of some kind.

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And then you could plot those data points.

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You would have 5,000 data points.

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So this data point right over here

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would be a very happy teenager

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that doesn't use a smartphone much.

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This would be a not so happy teenager

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that uses a smartphone a lot.

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And so you would plot those data points

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and there might be a teenager

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who's unhappy and doesn't use a smartphone

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or one that is happy and
that uses a smartphone a lot.

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But you can see there's a trend,

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there's an association that in general

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the teenagers who use the smartphones more

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seem to be less happy and the teenagers

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who use the smartphones
less seem to be more happy.

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But it's important to realize

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that the causality could
go the other way around.

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Maybe less happy teenagers
use their smartphones more

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and maybe more happy teenagers

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don't find a need to use a smartphone.

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Or there could be some variable

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that's not even being
observed in this study

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that has a causal relationship
with both of these.

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So there could be some other variable

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that might cause someone to be less happy

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and use their smartphone more.

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So in an observational
study, you can really

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just say there is an association.

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You wouldn't be able to say
that there is causality.

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So Jerry is not making a valid conclusion,

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it's an observational study, we've only

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established an association not causality.


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WEBVTT
Kind: captions
Language: en

00:00:02.210 --> 00:00:04.040
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Generalizabilty of survey results example

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Language: en

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- [Instructor] First-year
students at a certain

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large university are
required to live on campus

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in one of the 24
available residence halls.

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After their first year,

00:00:10.780 --> 00:00:13.980
students have the option
to live away from campus,

00:00:13.980 --> 00:00:17.774
but many choose to continue
living in the residence halls.

00:00:17.774 --> 00:00:22.080
Estella oversees 12 of
these residence halls.

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Her department surveyed a
large simple random sample

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of first-year students who live
in those 12 residence halls

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about their overall
satisfaction with campus living.

00:00:34.670 --> 00:00:38.120
Estella can safely generalize the results

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of the survey to which population?

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So pause this video, and see
if you could figure it out.

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All right, so let's do this together.

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So Estella has done,

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it's a large simple random sample

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of first-year students.

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So let's see,

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choice A is only those
students who were surveyed.

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Well, no, this was a simple random sample,

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and it was a large sample, so
it's meant to be indicative

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of all first-year students.

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You can generalize more
than just making statements

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about just the students who were surveyed.

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All first-year students,

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but only those who live in
these 12 residence halls.

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Yeah, I think this one looks fair

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because you can't generalize to people

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who don't live in those residence halls.

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Maybe Estella oversees the
12 best residence halls

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or the 12 worst residence halls,

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and so you wouldn't get,
if that were the case,

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you would not be able to
generalize beyond that.

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Or these might be the 12
that are closest to campus

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or the 12 that are furthest from campus.

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So you can only generalize to people

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who live in those residence halls.

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All students, first-year and not,

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but only those who live in
these 12 residence halls.

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Well, the issue here is,

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is that a second-year student
or a third-year student

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might just have a different perspective,

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even if they live in those,
in that same building

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and we did a large random
sample of first-year students.

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We didn't do a large random
sample of all people in that,

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in that, those 12 residence
halls, so rule that out.

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All first-year students
at the entire university,

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but not students beyond their first year.

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Well, this wouldn't make sense

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because there would be first-year students

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who would be outside of
those residence halls.

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And as I already mentioned,
the ones that Estella manages

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might not be indicative of where

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all of the first-year students
in this university live.


Ideology and social policy

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- [Instructor] In this off-white color,

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I have a handful of statements

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that you might hear folks say especially

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in the United States,

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and what we're going to think
about is are these statements

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that you would typically
hear from a liberal?

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And I'm gonna make a little key here,

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so I'm gonna use a light blue for liberal.

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Conservative, I'll use this salmon color

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for conservative, or a libertarian?

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So I'll do this white
color for libertarian.

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Libertarian.

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And I've pre-grouped these
statements into categories.

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They could be views on social
issues or economic issues

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and within each of those dimensions,

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they could be views that
might want more social

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or economic control or ones

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that want less social or economic control,

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and this isn't a perfect full sampling of

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all of the major issues
that folks will talk about

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and some of these things
kind of border on social

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and economic, or some of them

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could be viewed as more
control depending on

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how you view it or less control,

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but this is a first take on things.

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So let's just go from the
top left to the bottom right,

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and I encourage you to
keep pausing this video

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and coming to your own conclusion.

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So this first statement,
affirmative action is a good thing.

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Would you expect that to come
from someone with a liberal,

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conservative, or libertarian ideology?

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Well, this would typically come

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from someone with a liberal ideology.

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Much of the ideas and policies

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that liberals would favor
might be around curing

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social injustice, so they might say,

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hey, we don't have the proper
diversity in certain walks

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of life right now, we don't
have a level playing field,

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so we need to make sure

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that people are properly represented,

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and so that would be something

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that you would hear from liberals,

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and it would be something

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that in general involves more control.

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A libertarian or a conservative would say,

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hey, we shouldn't try to
do social engineering here,

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we should maybe make sure
people have equal opportunities,

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but that doesn't mean

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that you have to have equal
representation and outcome.

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This next statement, marriage should be

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between a man and woman.

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Pause this video.

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Who is likely to say that?

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Well, that would typically come

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from someone with a conservative ideology.

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This would be around a theme that

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many folks with conservative
ideologies would point

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to values that are worth
preserving in a society,

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that should not change,

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and so that would
typically come from them,

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while a liberal or libertarian
is more likely to say,

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hey, what goes on between two people,

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well, that's their decision

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and really the government should

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have no say on what they do.

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We need tougher penalties

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and policing to deal with crime.

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Well, once again, that would
be something that you would

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likely hear from someone
with a conservative ideology.

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A liberal or libertarian
would be on the opposite side.

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They would say, hey, I'm
afraid of too strict penalties,

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too strict policing, I'm afraid

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that the government is putting
too many people in jail.

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So now let's go to fewer social controls.

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So marijuana should be legalized.

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Pause this video.

00:03:00.353 --> 00:03:01.690
Who is more likely to say that?

00:03:02.800 --> 00:03:06.450
Well, many liberals would
be likely to say that,

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and also libertarians.

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They would say, hey, once again,

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what someone does within
their own privacy,

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why should the government have a say,

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while many conservatives,

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and this goes back to values

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and kind of social fiber, might say,

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well no, some of these drugs,

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they break down people's value system.

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Maybe they lead to crime and
so a conservative would be more

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likely to be against legalizing
something like marijuana.

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The next statement.

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Laws restricting gun
rights are unconstitutional

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and won't have the intended impact.

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Pause this video.

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Who would say that?

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Well, you would often hear a statement

00:03:43.700 --> 00:03:47.120
like this from someone with
a conservative ideology,

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but also from someone with
a libertarian ideology,

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so libertarian ideology would
also say something like that,

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and they would both cite

00:03:55.400 --> 00:03:58.740
that people have a right
to protect themselves,

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they might cite, hey, people have a right

00:04:00.520 --> 00:04:02.550
to use guns for things like hunting,

00:04:02.550 --> 00:04:05.330
and especially libertarians
might make the argument

00:04:05.330 --> 00:04:07.160
that people need to be armed in order

00:04:07.160 --> 00:04:10.330
to protect themselves from
an overly strong government,

00:04:10.330 --> 00:04:12.173
from an overly strong police state.

00:04:13.290 --> 00:04:14.170
So the next statement.

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Same-sex couples should
have full marriage rights.

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Who's most likely to say that?

00:04:18.800 --> 00:04:20.730
Well, this is kind of
the opposite statement

00:04:20.730 --> 00:04:22.550
of this one we saw up here,

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and for the reasons we mentioned,

00:04:24.250 --> 00:04:25.350
this is more likely to come

00:04:25.350 --> 00:04:30.350
from someone with a liberal
ideology or a libertarian,

00:04:30.380 --> 00:04:32.370
libertarian ideology.

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And for them, it would be about

00:04:33.480 --> 00:04:35.960
personal liberty and equal rights.

00:04:35.960 --> 00:04:38.810
And so you might already see some trends

00:04:38.810 --> 00:04:41.810
that the conservative
and liberals are kind

00:04:41.810 --> 00:04:44.740
of mixed on the social when
it comes to more control

00:04:44.740 --> 00:04:46.810
or less control, and don't read

00:04:46.810 --> 00:04:49.030
too much into how many there are of these.

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This is just, you could view

00:04:50.250 --> 00:04:51.840
as a random sampling of some issues.

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If I sampled more, you might
have more liberal boxes up here

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or more conservative boxes down here,

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but it seems like a mix.

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But one clear thing that comes out

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is that libertarians
on the social dimension

00:05:02.260 --> 00:05:04.740
are for less control.

00:05:04.740 --> 00:05:07.400
But what about the economic dimension?

00:05:07.400 --> 00:05:08.430
So this first statement,

00:05:08.430 --> 00:05:11.410
everyone has a right
to quality healthcare.

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Who's more likely to say that?

00:05:14.280 --> 00:05:17.490
Well, that would be a liberal.

00:05:17.490 --> 00:05:20.410
Liberals, once again, they
think a lot about injustice

00:05:20.410 --> 00:05:23.060
and inequality and they
want the government

00:05:23.060 --> 00:05:25.140
to be proactive about it.

00:05:25.140 --> 00:05:26.950
And so they would say,
hey, the government should

00:05:26.950 --> 00:05:29.800
somehow supply universal
healthcare to folks.

00:05:29.800 --> 00:05:31.970
Conservatives or libertarians,

00:05:31.970 --> 00:05:34.720
not that they don't care
about injustice or inequality,

00:05:34.720 --> 00:05:36.130
but they would argue that, hey,

00:05:36.130 --> 00:05:38.440
when the government kind
of intervenes in things,

00:05:38.440 --> 00:05:41.320
they don't, they have all sorts
of unintended consequences

00:05:41.320 --> 00:05:44.690
and oftentimes do more harm than good.

00:05:44.690 --> 00:05:45.580
The next statement.

00:05:45.580 --> 00:05:48.210
Rent control is a good thing.

00:05:48.210 --> 00:05:51.470
Well, once again, this is
likely to come from a liberal,

00:05:51.470 --> 00:05:53.480
probably motivated from the idea

00:05:53.480 --> 00:05:56.330
that if rents or living
costs are getting too high,

00:05:56.330 --> 00:05:59.580
that this is a way to avoid
people being displaced

00:05:59.580 --> 00:06:01.740
or being priced out of their neighborhood.

00:06:01.740 --> 00:06:03.900
Now, conservatives and
libertarians would argue

00:06:03.900 --> 00:06:06.200
that rent control distorts the market,

00:06:06.200 --> 00:06:07.630
it actually makes the properties

00:06:07.630 --> 00:06:10.670
that aren't on rent
control overly expensive,

00:06:10.670 --> 00:06:14.000
and so it's probably
doing more harm than good.

00:06:14.000 --> 00:06:14.833
The next statement.

00:06:14.833 --> 00:06:17.450
Social safety nets are crucial.

00:06:17.450 --> 00:06:19.140
Once again, this is likely

00:06:19.140 --> 00:06:22.810
to come from someone
with a liberal ideology.

00:06:22.810 --> 00:06:25.060
They believe that someone,

00:06:25.060 --> 00:06:28.150
that the government has
a role in making sure

00:06:28.150 --> 00:06:30.860
that everyone has a
basic standard of living,

00:06:30.860 --> 00:06:34.240
so programs like food stamps or welfare.

00:06:34.240 --> 00:06:36.840
Every worker has the
right to a living wage.

00:06:36.840 --> 00:06:38.040
And living wage is sometimes,

00:06:38.040 --> 00:06:40.790
you'll hear this in the context
of minimum-wage discussions.

00:06:40.790 --> 00:06:42.540
Who would say that?

00:06:42.540 --> 00:06:43.610
Well, once again, you're

00:06:43.610 --> 00:06:47.410
likely to hear this from
someone with a liberal ideology.

00:06:47.410 --> 00:06:50.460
They believe that higher
minimum wages are a good thing,

00:06:50.460 --> 00:06:52.750
while conservatives or
libertarians would argue that,

00:06:52.750 --> 00:06:54.930
no, that distorts the market

00:06:54.930 --> 00:06:56.910
and actually you might
increase unemployment

00:06:56.910 --> 00:06:58.450
and you might hurt the very people

00:06:58.450 --> 00:07:00.640
that you're trying to help with some

00:07:00.640 --> 00:07:02.990
of these efforts around minimum ages.

00:07:02.990 --> 00:07:04.170
And what about here when we go

00:07:04.170 --> 00:07:06.200
to less control on the economy?

00:07:06.200 --> 00:07:07.270
Government intervention

00:07:07.270 --> 00:07:10.850
in the free market will slow
the economy, hurting everyone.

00:07:10.850 --> 00:07:12.140
Well, as you could imagine,

00:07:12.140 --> 00:07:14.260
this is kind of the opposite
statement as these here,

00:07:14.260 --> 00:07:15.490
because these once at the top

00:07:15.490 --> 00:07:17.030
are all the government intervening

00:07:17.030 --> 00:07:18.720
in a free economy in some way,

00:07:18.720 --> 00:07:20.970
and so this statement is more

00:07:20.970 --> 00:07:23.690
likely to come from a conservative,

00:07:23.690 --> 00:07:28.250
some of the conservative
ideology or a libertarian.

00:07:28.250 --> 00:07:31.460
And so when you look in
the economic dimension,

00:07:31.460 --> 00:07:33.520
the distinction between
conservative and liberal,

00:07:33.520 --> 00:07:35.040
at least when it comes to more control

00:07:35.040 --> 00:07:37.210
and less control, become more clear.

00:07:37.210 --> 00:07:41.160
On economics, liberals
tend to favor more control,

00:07:41.160 --> 00:07:43.700
more government intervention with a focus

00:07:43.700 --> 00:07:45.260
of trying to address social

00:07:45.260 --> 00:07:49.010
or economic inequality or
environmental concerns,

00:07:49.010 --> 00:07:51.840
while conservatives and
libertarians will often say,

00:07:51.840 --> 00:07:53.660
well, that's gonna do more harm than good,

00:07:53.660 --> 00:07:55.550
the government does have some functions

00:07:55.550 --> 00:07:57.970
but it should really
be as small as possible

00:07:57.970 --> 00:07:59.640
and free markets are what's,

00:07:59.640 --> 00:08:03.000
in the end of the day, going
to make people better off.

00:08:03.000 --> 00:08:05.020
So I will leave you there,

00:08:05.020 --> 00:08:06.940
this is a very simple model,

00:08:06.940 --> 00:08:09.150
but it helps us start to
think about these issues,

00:08:09.150 --> 00:08:10.420
and so when you think about issues

00:08:10.420 --> 00:08:12.130
and you read about it in the newspaper,

00:08:12.130 --> 00:08:14.630
try to think about it in
this type of a framework.


Introduction to limits at infinity

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:03.290
- [Instructor] We now have
a lot of experience taking

00:00:03.290 --> 00:00:07.330
limits of functions, if
I'm taking limit of f of x.

00:00:07.330 --> 00:00:10.690
What we're gonna think about,
what does f of x approach as x

00:00:10.690 --> 00:00:13.520
approaches some value a?

00:00:13.520 --> 00:00:16.420
And this would be equal to some limit.

00:00:16.420 --> 00:00:18.640
Now everything we've done up till now

00:00:18.640 --> 00:00:21.690
is where a is a finite value.

00:00:21.690 --> 00:00:23.380
But when you look at the graph

00:00:23.380 --> 00:00:25.040
of the function f right over here,

00:00:25.040 --> 00:00:27.210
you see something interesting happens.

00:00:27.210 --> 00:00:31.500
As x gets larger and larger,
it looks like our function f

00:00:31.500 --> 00:00:33.680
is getting closer and closer to two.

00:00:33.680 --> 00:00:35.120
It looks like we have a horizontal

00:00:35.120 --> 00:00:37.370
asymptote at y equals two.

00:00:37.370 --> 00:00:40.370
Similarly, as x gets
more and more negative,

00:00:40.370 --> 00:00:43.600
it also seems like we have
a horizontal asymptote

00:00:43.600 --> 00:00:45.430
at y equals two.

00:00:45.430 --> 00:00:49.130
So is there some type of notation
we can use to think about

00:00:49.130 --> 00:00:52.790
what is the graph approaching
as x gets much larger

00:00:52.790 --> 00:00:55.090
or as x gets smaller and smaller?

00:00:55.090 --> 00:00:58.770
And the answer there
is limits at infinity.

00:00:58.770 --> 00:01:01.220
So if we want to think
about what is this graph,

00:01:01.220 --> 00:01:02.780
what is this function approaching

00:01:02.780 --> 00:01:04.680
as x gets larger and larger,

00:01:04.680 --> 00:01:08.950
we can think about the limit of f of x

00:01:08.950 --> 00:01:12.620
as x approaches positive infinity.

00:01:12.620 --> 00:01:14.610
So that's the notation, and I'm not going

00:01:14.610 --> 00:01:16.760
to give you the formal
definition of this right now.

00:01:16.760 --> 00:01:18.930
There, in future videos, we might do that.

00:01:18.930 --> 00:01:22.220
But it's this idea, as x gets
larger and larger and larger,

00:01:22.220 --> 00:01:24.420
does it look like that our function

00:01:24.420 --> 00:01:27.540
is approaching some finite value,

00:01:27.540 --> 00:01:29.470
that we have a horizontal asymptote there?

00:01:29.470 --> 00:01:31.880
And in this situation,
it looks like it is.

00:01:31.880 --> 00:01:34.870
It looks like it's
approaching the value two.

00:01:34.870 --> 00:01:38.990
And for this particular
function, the limit

00:01:38.990 --> 00:01:43.990
of f of x as x approaches
negative infinity

00:01:44.040 --> 00:01:47.490
also looks like it is approaching two.

00:01:47.490 --> 00:01:49.920
This is not always going to be the same.

00:01:49.920 --> 00:01:52.750
You could have a situation, maybe we had,

00:01:52.750 --> 00:01:53.940
you could have another function.

00:01:53.940 --> 00:01:55.240
So let me draw a little horizontal

00:01:55.240 --> 00:01:57.290
asymptote right over here.

00:01:57.290 --> 00:01:59.790
You could imagine a function
that looks like this.

00:01:59.790 --> 00:02:01.410
So I'm going to do it like that,

00:02:01.410 --> 00:02:05.490
and maybe it does
something wacky like this.

00:02:05.490 --> 00:02:08.790
Then it comes down, and it
does something like this.

00:02:08.790 --> 00:02:12.580
Here, our limit as x approaches
infinity is still two,

00:02:12.580 --> 00:02:15.200
but our limit as x
approaches negative infinity,

00:02:15.200 --> 00:02:17.280
right over here, would be negative two.

00:02:17.280 --> 00:02:19.170
And of course, there's
many situations where,

00:02:19.170 --> 00:02:21.560
as you approach infinity
or negative infinity,

00:02:21.560 --> 00:02:24.970
you aren't actually
approaching some finite value.

00:02:24.970 --> 00:02:27.520
You don't have a horizontal asymptote.

00:02:27.520 --> 00:02:29.330
But the whole point of this video is just

00:02:29.330 --> 00:02:32.070
to make you familiar with this notation.

00:02:32.070 --> 00:02:33.230
And limits at infinity

00:02:33.230 --> 00:02:35.170
or you could say limits
at negative infinity,

00:02:35.170 --> 00:02:37.360
they have a different formal definition

00:02:37.360 --> 00:02:39.430
than some of the limits that
we've looked at in the past,

00:02:39.430 --> 00:02:41.460
where we are approaching a finite value.

00:02:41.460 --> 00:02:43.410
But intuitively, they make sense,

00:02:43.410 --> 00:02:45.243
that these are indeed limits.


Introduction to infinite limits

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.800
- [Instructor] In a previous
video, we explored the graphs

00:00:00.000 --> 00:00:00.880
of Y equals one over X
squared and one over X.

00:00:00.880 --> 00:00:02.980
In a previous video we've
looked at these graphs.

00:00:02.980 --> 00:00:05.350
This is Y is equal to one over X squared.

00:00:05.350 --> 00:00:07.840
This is Y is equal to one over X.

00:00:07.840 --> 00:00:09.700
And we explored what's the limit

00:00:09.700 --> 00:00:13.850
as X approaches zero in
either of those scenarios.

00:00:13.850 --> 00:00:15.820
And in this left scenario we saw

00:00:15.820 --> 00:00:18.340
as X becomes less and less negative,

00:00:18.340 --> 00:00:22.916
as it approaches zero
from the left hand side,

00:00:22.916 --> 00:00:26.180
the value of one over
X squared is unbounded

00:00:26.180 --> 00:00:27.560
in the positive direction.

00:00:27.560 --> 00:00:30.930
And the same thing happens as
we approach X from the right,

00:00:30.930 --> 00:00:32.530
as we become less and less positive

00:00:32.530 --> 00:00:34.170
but we are still positive,

00:00:34.170 --> 00:00:35.990
the value of one over X squared becomes

00:00:35.990 --> 00:00:38.010
unbounded in the positive direction.

00:00:38.010 --> 00:00:39.727
So in that video, we just said, "Hey,

00:00:39.727 --> 00:00:43.150
"one could say that this
limit is unbounded."

00:00:43.150 --> 00:00:45.270
But what we're going
to do in this video is

00:00:45.270 --> 00:00:47.410
introduce new notation.

00:00:47.410 --> 00:00:49.150
Instead of just saying it's unbounded,

00:00:49.150 --> 00:00:51.160
we could say, "Hey, from
both the left and the right

00:00:51.160 --> 00:00:53.590
it looks like we're going
to positive infinity".

00:00:53.590 --> 00:00:55.697
So we can introduce
this notation of saying,

00:00:55.697 --> 00:00:58.320
"Hey, this is going to infinity",

00:00:58.320 --> 00:01:00.240
which you will sometimes see used.

00:01:00.240 --> 00:01:01.700
Some people would call this unbounded,

00:01:01.700 --> 00:01:03.160
some people say it does not exist

00:01:03.160 --> 00:01:05.730
because it's not approaching
some finite value,

00:01:05.730 --> 00:01:07.630
while some people will use this notation

00:01:07.630 --> 00:01:10.220
of the limit going to infinity.

00:01:10.220 --> 00:01:11.760
But what about this scenario?

00:01:11.760 --> 00:01:14.220
Can we use our new notation here?

00:01:14.220 --> 00:01:18.310
Well, when we approach zero from the left,

00:01:18.310 --> 00:01:21.120
it looks like we're unbounded
in the negative direction,

00:01:21.120 --> 00:01:23.310
and when we approach zero from the right,

00:01:23.310 --> 00:01:26.260
we are unbounded in
the positive direction.

00:01:26.260 --> 00:01:28.760
So, here you still could not say

00:01:28.760 --> 00:01:30.710
that the limit is approaching infinity

00:01:30.710 --> 00:01:32.440
because from the right
it's approaching infinity,

00:01:32.440 --> 00:01:34.660
but from the left it's
approaching negative infinity.

00:01:34.660 --> 00:01:39.660
So you would still say
that this does not exist.

00:01:39.760 --> 00:01:42.340
You could do one sided limits here,

00:01:42.340 --> 00:01:43.670
which if you're not familiar with,

00:01:43.670 --> 00:01:45.780
I encourage you to review
it on Khan Academy.

00:01:45.780 --> 00:01:49.010
If you said the limit of one over X

00:01:49.010 --> 00:01:53.340
as X approaches zero
from the left hand side,

00:01:53.340 --> 00:01:55.810
from values less than zero,

00:01:55.810 --> 00:01:57.627
well then you would look at
this right over here and say,

00:01:57.627 --> 00:01:59.544
"Well, look, it looks like we're going

00:01:59.544 --> 00:02:00.790
unbounded in the negative direction".

00:02:00.790 --> 00:02:04.270
So you would say this is
equal to negative infinity.

00:02:04.270 --> 00:02:09.270
And of course if you said the
limit as X approaches zero

00:02:09.670 --> 00:02:12.700
from the right of one over X, well here

00:02:12.700 --> 00:02:14.500
you're unbounded in the positive direction

00:02:14.500 --> 00:02:17.650
so that's going to be
equal to positive infinity.

00:02:17.650 --> 00:02:19.760
Let's do an example
problem from Khan Academy

00:02:19.760 --> 00:02:22.493
based on this idea and this notation.

00:02:23.610 --> 00:02:27.540
So here it says, consider
graphs A, B, and C.

00:02:27.540 --> 00:02:30.470
The dashed lines represent asymptotes.

00:02:30.470 --> 00:02:33.260
Which of the graphs agree
with this statement,

00:02:33.260 --> 00:02:36.160
that the limit as X approaches 1 of H of X

00:02:36.160 --> 00:02:37.480
is equal to infinity?

00:02:37.480 --> 00:02:39.980
Pause this video and see
if you can figure it out.

00:02:40.940 --> 00:02:42.350
Alright, let's go through each of these.

00:02:42.350 --> 00:02:44.850
So we want to think about
what happens at X equals one.

00:02:44.850 --> 00:02:47.860
So that's right over here on graph A.

00:02:47.860 --> 00:02:49.880
So as we approach X equals one,

00:02:49.880 --> 00:02:52.120
so let me write this, so the limit,

00:02:52.120 --> 00:02:53.860
let me do this for the different graphs.

00:02:53.860 --> 00:02:58.753
So, for graph A, the
limit as x approaches one

00:02:59.680 --> 00:03:02.360
from the left, that looks like

00:03:02.360 --> 00:03:04.160
it's unbounded in the positive direction.

00:03:04.160 --> 00:03:07.091
That equals infinity and the limit

00:03:07.091 --> 00:03:11.530
as X approaches one from the right,

00:03:11.530 --> 00:03:14.020
well that looks like it's
going to negative infinity.

00:03:14.020 --> 00:03:15.970
That equals negative infinity.

00:03:15.970 --> 00:03:18.770
And since these are going
in two different directions,

00:03:18.770 --> 00:03:19.860
you wouldn't be able to say that

00:03:19.860 --> 00:03:21.420
the limit as X approaches one

00:03:21.420 --> 00:03:23.450
from both directions is equal to infinity.

00:03:23.450 --> 00:03:25.700
So I would rule this one out.

00:03:25.700 --> 00:03:27.710
Now let's look at choice B.

00:03:27.710 --> 00:03:32.710
What's the limit as X
approaches one from the left?

00:03:33.220 --> 00:03:36.250
And of course these are of H of X.

00:03:36.250 --> 00:03:37.610
Gotta write that down.

00:03:37.610 --> 00:03:40.970
So, of H of X right over here.

00:03:40.970 --> 00:03:43.589
Well, as we approach from the left,

00:03:43.589 --> 00:03:47.390
looks like we're going
to positive infinity.

00:03:47.390 --> 00:03:50.740
And it looks like the limit of H of X

00:03:50.740 --> 00:03:54.220
as we approach one from the right is

00:03:54.220 --> 00:03:56.860
also going to positive infinity.

00:03:56.860 --> 00:03:58.710
And so, since we're
approaching you could say

00:03:58.710 --> 00:04:02.630
the same direction of infinity,
you could say this for B.

00:04:02.630 --> 00:04:04.490
So B meets the constraints, but

00:04:04.490 --> 00:04:06.730
let's just check C to make sure.

00:04:06.730 --> 00:04:09.890
Well, you can see very
clearly X equals one,

00:04:09.890 --> 00:04:11.230
that as we approach it from the left,

00:04:11.230 --> 00:04:12.490
we go to negative infinity,

00:04:12.490 --> 00:04:14.712
and as we approach from the right,

00:04:14.712 --> 00:04:15.545
we got to positive infinity.

00:04:16.401 --> 00:04:18.740
So this, once again,
would not be approaching

00:04:18.740 --> 00:04:19.880
the same infinity.

00:04:19.880 --> 00:04:22.293
So you would rule this one out, as well.


Unbounded limits

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:01.620
- [Instructor] So right over here,

00:00:01.620 --> 00:00:06.620
we have the graph of y is
equal to one over x squared.

00:00:06.680 --> 00:00:10.810
And my question to you
is what is the limit

00:00:10.810 --> 00:00:15.610
of one over x squared
as x approaches zero?

00:00:15.610 --> 00:00:18.283
Pause this video, and see
if you can figure that out.

00:00:19.340 --> 00:00:20.890
Well, when you try to figure it out,

00:00:20.890 --> 00:00:22.700
you immediately see something interesting

00:00:22.700 --> 00:00:24.010
happening at x equals zero.

00:00:24.010 --> 00:00:27.270
The closer we get to zero from the left,

00:00:27.270 --> 00:00:28.980
you take one over x squared,

00:00:28.980 --> 00:00:31.170
it just gets larger and larger and larger.

00:00:31.170 --> 00:00:33.450
It doesn't approach some finite value.

00:00:33.450 --> 00:00:35.840
It's unbounded, has no bound.

00:00:35.840 --> 00:00:37.310
And the same thing is happening

00:00:37.310 --> 00:00:38.850
as we approach from the right.

00:00:38.850 --> 00:00:42.640
As we get values closer and
closer to zero from the right,

00:00:42.640 --> 00:00:44.260
we get larger and larger values

00:00:44.260 --> 00:00:47.390
for one over x squared without bound.

00:00:47.390 --> 00:00:50.200
So terminology that
folks will sometimes use,

00:00:50.200 --> 00:00:52.720
where they're both going
in the same direction,

00:00:52.720 --> 00:00:57.530
but it's unbounded, is they'll
say this limit is unbounded.

00:00:57.530 --> 00:00:59.520
In some context, you
might hear teachers say

00:00:59.520 --> 00:01:01.310
that this limit does not exist or,

00:01:01.310 --> 00:01:03.460
and it definitely does not
exist if you're thinking about

00:01:03.460 --> 00:01:05.370
approaching a finite value.

00:01:05.370 --> 00:01:08.930
In future videos, we'll start
to introduce ideas of infinity

00:01:08.930 --> 00:01:11.330
and notations around limits and infinity,

00:01:11.330 --> 00:01:13.870
where we can get a
little bit more specific

00:01:13.870 --> 00:01:16.140
about what type of limit this is.

00:01:16.140 --> 00:01:16.973
But with that out of the way,

00:01:16.973 --> 00:01:18.480
let's look at another scenario.

00:01:19.770 --> 00:01:22.510
This right over here, you might recognize

00:01:22.510 --> 00:01:26.710
as the graph of y is equal to one over x.

00:01:26.710 --> 00:01:28.270
So I'm going to ask you the same question.

00:01:28.270 --> 00:01:30.090
Pause this video, and think about

00:01:30.090 --> 00:01:34.210
what's the limit of one
over x as x approaches zero?

00:01:34.210 --> 00:01:36.010
Pause this video, and figure it out.

00:01:37.240 --> 00:01:40.810
All right, so here, when
we approach from the left,

00:01:40.810 --> 00:01:43.260
we get more and more and
more negative values.

00:01:43.260 --> 00:01:45.170
While we, when we approach from the right,

00:01:45.170 --> 00:01:47.560
we're getting more and
more positive values.

00:01:47.560 --> 00:01:48.720
So in this situation,

00:01:48.720 --> 00:01:52.140
where we're not getting
unbounded in the same direction,

00:01:52.140 --> 00:01:53.690
the previous example, we were both,

00:01:53.690 --> 00:01:56.040
we were being unbounded
in the positive direction.

00:01:56.040 --> 00:01:57.420
But here, on the, from the left,

00:01:57.420 --> 00:01:59.460
we're getting unbounded
in the negative direction.

00:01:59.460 --> 00:02:01.160
While from the right,
we're getting unbounded

00:02:01.160 --> 00:02:02.580
in the positive direction.

00:02:02.580 --> 00:02:04.070
And so when you're
thinking about the limit

00:02:04.070 --> 00:02:07.100
as you approach a point, if
it's not even approaching

00:02:07.100 --> 00:02:09.790
the same value or even the same direction,

00:02:09.790 --> 00:02:12.890
you would just clearly say
that this limit does not exist,

00:02:12.890 --> 00:02:14.230
does

00:02:14.230 --> 00:02:15.870
not

00:02:15.870 --> 00:02:17.120
exist.

00:02:17.120 --> 00:02:19.160
So this is a situation,
where you would not even say

00:02:19.160 --> 00:02:20.660
that this is an unbounded limit

00:02:20.660 --> 00:02:22.160
or that the limit is unbounded.

00:02:22.160 --> 00:02:24.580
Because you're going in two
different directions when you

00:02:24.580 --> 00:02:27.180
approach from the right and
when you approach from the left,

00:02:27.180 --> 00:02:30.563
you would just clearly say does not exist.


Evolution through variation and natural selection

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:03.570
- [Instructor] In this
video, we are going to focus

00:00:03.570 --> 00:00:06.900
even more on the idea of evolution.

00:00:06.900 --> 00:00:08.470
We introduce it in other videos,

00:00:08.470 --> 00:00:10.450
but here were really
gonna focus on what it is

00:00:10.450 --> 00:00:11.380
and what it isn't.

00:00:11.380 --> 00:00:14.710
And as I've mentioned before,
it's a super important idea.

00:00:14.710 --> 00:00:17.850
If you were to try to understand
life on another planet,

00:00:17.850 --> 00:00:21.140
well you would have to
understand how evolution works.

00:00:21.140 --> 00:00:23.850
And so evolution as we know it is based

00:00:23.850 --> 00:00:26.260
on two underlying principles.

00:00:26.260 --> 00:00:29.060
It's based on a principle
that in a population

00:00:29.060 --> 00:00:32.313
of a species, that there is variation.

00:00:33.180 --> 00:00:36.120
That there's variation in the phenotype,

00:00:36.120 --> 00:00:39.560
in the traits that that population has.

00:00:39.560 --> 00:00:42.330
Now the other idea that's
related to that variation

00:00:42.330 --> 00:00:44.593
is that you have natural selection.

00:00:45.510 --> 00:00:48.300
Natural selection, if
someone tries to explain

00:00:48.300 --> 00:00:51.090
evolution without variation
and natural selection,

00:00:51.090 --> 00:00:53.470
they might not be explaining evolution,

00:00:53.470 --> 00:00:55.910
they might be explaining something else.

00:00:55.910 --> 00:00:59.500
So this idea, so if I have
a population of circles,

00:00:59.500 --> 00:01:02.680
some of the circles are blue circles,

00:01:02.680 --> 00:01:05.593
some are yellow circles,

00:01:06.580 --> 00:01:09.850
yellow circles, some are pink circles;

00:01:09.850 --> 00:01:12.770
and there are, for sure, traits,

00:01:12.770 --> 00:01:15.990
there are phenotypes that
might not have any bearing

00:01:15.990 --> 00:01:19.940
on an organism's ability
to survive or reproduce.

00:01:19.940 --> 00:01:23.730
In that case, that variation
would probably continue.

00:01:23.730 --> 00:01:27.340
But there will be some
variations, some phenotypes;

00:01:27.340 --> 00:01:29.170
remember, phenotypes is what's expressed

00:01:29.170 --> 00:01:32.470
versus genotypes which
are the actual genes,

00:01:32.470 --> 00:01:35.710
the actual alleles that an organism has.

00:01:35.710 --> 00:01:38.090
But evolution occurs because,

00:01:38.090 --> 00:01:41.230
depending where we are in our environment

00:01:41.230 --> 00:01:43.980
and the time and space of our environment,

00:01:43.980 --> 00:01:47.020
certain phenotypes might be slightly more

00:01:47.020 --> 00:01:48.700
advantageous than others.

00:01:48.700 --> 00:01:50.730
They might make it slightly more likely

00:01:50.730 --> 00:01:53.440
that the organism will
survive and reproduce,

00:01:53.440 --> 00:01:55.840
slightly less likely that it will die

00:01:55.840 --> 00:01:57.560
or not be able to reproduce.

00:01:57.560 --> 00:02:01.770
And so if we said these
circles were living things,

00:02:01.770 --> 00:02:04.830
and let's say the environment
that they happen to be in

00:02:04.830 --> 00:02:08.140
right now, it's that much more likely

00:02:08.140 --> 00:02:10.670
for them to reproduce and survive

00:02:10.670 --> 00:02:14.230
if they're warmer colors, if
they are an orange or a pink

00:02:14.230 --> 00:02:15.610
verus a blue.

00:02:15.610 --> 00:02:18.040
So then in your next generation,

00:02:18.040 --> 00:02:22.060
the yellows and the pinks
are more likely to reproduce,

00:02:22.060 --> 00:02:25.040
and so they might reproduce
a little bit more.

00:02:25.040 --> 00:02:27.950
Let me put some of the pinks in there.

00:02:27.950 --> 00:02:32.050
They're going to reproduce
a little bit more.

00:02:32.050 --> 00:02:34.840
And the blues might have
a lower probability,

00:02:34.840 --> 00:02:37.340
or would have, based on
this situation I described,

00:02:37.340 --> 00:02:38.210
of reproducing.

00:02:38.210 --> 00:02:40.090
So you'll still have some blues,

00:02:40.090 --> 00:02:42.760
but you might have fewer blues.

00:02:42.760 --> 00:02:45.210
And then this process could keep going on

00:02:45.210 --> 00:02:47.590
for a very, very, very, very long time.

00:02:47.590 --> 00:02:51.230
And at some point you might
see only pinks and yellows.

00:02:51.230 --> 00:02:53.640
And this is only one
phenotype that we're talking

00:02:53.640 --> 00:02:54.900
about being selected for.

00:02:54.900 --> 00:02:56.340
There might be another
one, if we're talking about

00:02:56.340 --> 00:02:59.270
circles it might be
roundness or whatever else.

00:02:59.270 --> 00:03:02.190
So the key thing is is
you need that variation.

00:03:02.190 --> 00:03:05.280
And depending on where
in time and space you are

00:03:05.280 --> 00:03:08.210
in that environment, that
some of those variations,

00:03:08.210 --> 00:03:10.720
some of those phenotypes,
the exhibited traits,

00:03:10.720 --> 00:03:14.000
might make you more
successful at reproducing

00:03:14.000 --> 00:03:17.070
and passing on that
trait to your offspring.

00:03:17.070 --> 00:03:19.540
And I really wanna contrast
this with what some

00:03:19.540 --> 00:03:23.720
people think of when
they think of evolution.

00:03:23.720 --> 00:03:27.110
Some people think that
evolution is a process

00:03:27.110 --> 00:03:31.940
of going from basic to advanced.

00:03:31.940 --> 00:03:34.600
And the reason why we
sometimes think this is

00:03:34.600 --> 00:03:37.610
we think of many of the
early organisms on Earth

00:03:37.610 --> 00:03:40.110
maybe being unicellular organisms.

00:03:40.110 --> 00:03:43.930
And now we have these complex
animals like ourselves

00:03:43.930 --> 00:03:47.393
that we consider much more
advanced than those things,

00:03:47.393 --> 00:03:49.430
we have trillions of cells.

00:03:49.430 --> 00:03:52.470
But remember, unicellular
organisms continue

00:03:52.470 --> 00:03:54.910
to be very successful even today.

00:03:54.910 --> 00:03:56.903
And some of them aren't
all to more advanced

00:03:56.903 --> 00:03:59.310
than things that might've existed, say,

00:03:59.310 --> 00:04:02.900
500 million years ago, or
even a billion years ago.

00:04:02.900 --> 00:04:04.920
And so even though these things might be

00:04:04.920 --> 00:04:07.880
the product of evolution, all it says is

00:04:07.880 --> 00:04:10.620
that overtime you have this variation,

00:04:10.620 --> 00:04:13.260
and then based on which
variations, which phenotypes,

00:04:13.260 --> 00:04:15.160
were more suited to their environment,

00:04:15.160 --> 00:04:18.960
those are the ones that were
able to pass on their traits.

00:04:18.960 --> 00:04:21.220
Another place where this
misconception comes from

00:04:21.220 --> 00:04:24.010
is you've seen these
diagrams where you have

00:04:24.010 --> 00:04:27.720
these apes where the first apes are

00:04:27.720 --> 00:04:30.480
kind of hunched over, and
then they get upright,

00:04:30.480 --> 00:04:32.530
or more and more upright.

00:04:32.530 --> 00:04:34.680
And eventually you see a walking person,

00:04:34.680 --> 00:04:37.010
and we are officially apes as well.

00:04:37.010 --> 00:04:39.670
And we tend to think that
we are pretty advanced,

00:04:39.670 --> 00:04:41.600
we do things like wear hats and things.

00:04:41.600 --> 00:04:43.390
And so we say, oh yeah, it's going from

00:04:43.390 --> 00:04:44.810
primitive to advanced,

00:04:44.810 --> 00:04:46.600
or from basic to advanced.

00:04:46.600 --> 00:04:48.100
But that is not the case.

00:04:48.100 --> 00:04:50.620
Remember, even today's world, we aren't

00:04:50.620 --> 00:04:52.560
the end of evolution.

00:04:52.560 --> 00:04:55.090
Even in our world, there's many animals

00:04:55.090 --> 00:04:59.120
in the animal kingdom that
are at all different levels

00:04:59.120 --> 00:05:01.870
of complexity or if we wanna think about

00:05:01.870 --> 00:05:04.200
intelligence or sophistication.

00:05:04.200 --> 00:05:07.780
And there could, for
sure, be a reality where

00:05:07.780 --> 00:05:10.820
you could imagine a human
colony goes to another planet,

00:05:10.820 --> 00:05:14.520
and it's actually suitable
to be more hunched over.

00:05:14.520 --> 00:05:17.690
In which case, the human
beings, the variant,

00:05:17.690 --> 00:05:20.500
the variations of humans who
are better at hunching over

00:05:20.500 --> 00:05:21.670
might be more successful.

00:05:21.670 --> 00:05:22.880
Maybe they need to go into these tunnels

00:05:22.880 --> 00:05:25.470
to find some animals to
eat or whatever else,

00:05:25.470 --> 00:05:26.303
or to farm.

00:05:27.147 --> 00:05:29.460
And then on that planet,
that variation might be

00:05:29.460 --> 00:05:31.470
the one that wins out.

00:05:31.470 --> 00:05:33.800
Now another misconception that people have

00:05:33.800 --> 00:05:35.530
with evolution is, when they hear it,

00:05:35.530 --> 00:05:37.610
they kinda think, hey, this might be due

00:05:37.610 --> 00:05:41.120
to the effort of an
animal over its lifetime.

00:05:41.120 --> 00:05:44.060
Some of the early
explanations that folks gave

00:05:44.060 --> 00:05:47.900
before Darwin of why a
giraffe, for example,

00:05:47.900 --> 00:05:50.580
has a long neck,

00:05:50.580 --> 00:05:53.230
some people believe that, hey,
some of the early giraffes

00:05:53.230 --> 00:05:57.350
might've been straining to
get leaves on tall trees,

00:05:57.350 --> 00:06:00.350
and that straining somehow
stretched out their necks,

00:06:00.350 --> 00:06:02.630
and somehow they might've
been able to pass that on

00:06:02.630 --> 00:06:04.080
to their offspring.

00:06:04.080 --> 00:06:06.090
This is not the case.

00:06:06.090 --> 00:06:08.690
The explanation consisted
with our understanding

00:06:08.690 --> 00:06:11.210
of evolution is that you
would've had a population

00:06:11.210 --> 00:06:13.470
of giraffe ancestors,

00:06:13.470 --> 00:06:16.399
and they would've had variation in their,

00:06:16.399 --> 00:06:19.940
in their neck length.

00:06:19.940 --> 00:06:22.800
And because of the
environment that they happen

00:06:22.800 --> 00:06:25.130
to be in for that population,

00:06:25.130 --> 00:06:28.370
the variance that had longer neck lengths

00:06:28.370 --> 00:06:30.250
would have been able to have a slightly

00:06:30.250 --> 00:06:33.680
higher probability of
reproducing and surviving.

00:06:33.680 --> 00:06:36.540
And so overtime, those longer necks

00:06:36.540 --> 00:06:40.710
became more predominant in the population.

00:06:40.710 --> 00:06:42.600
And at some point, you might say,

00:06:42.600 --> 00:06:45.030
well why don't giraffe neck lengths

00:06:45.030 --> 00:06:47.000
become indefinitely tall?

00:06:47.000 --> 00:06:48.810
Well that's because at some the variations

00:06:48.810 --> 00:06:50.930
that had super long
necks, even longer than

00:06:50.930 --> 00:06:53.690
the giraffes we had today,
that might've created

00:06:53.690 --> 00:06:55.720
some other difficulties
making it harder for them

00:06:55.720 --> 00:06:57.070
to survive and reproduce.

00:06:57.070 --> 00:06:58.860
Maybe their heats weren't strong enough

00:06:58.860 --> 00:07:00.970
and so they died of heart attacks.

00:07:00.970 --> 00:07:04.610
Or maybe it was more
difficult to escape predators

00:07:04.610 --> 00:07:07.920
for the variations where
the necks were too long.

00:07:07.920 --> 00:07:10.240
And so it's really important to recognize

00:07:10.240 --> 00:07:13.750
that evolution is a byproduct

00:07:13.750 --> 00:07:16.340
of variation that happens in populations

00:07:16.340 --> 00:07:19.080
and then natural selection,
due to the environment,

00:07:19.080 --> 00:07:20.920
on those variations.

00:07:20.920 --> 00:07:22.750
And they act on the phenotype,

00:07:22.750 --> 00:07:26.340
they act on the exhibited
traits that are shown.

00:07:26.340 --> 00:07:29.720
Evolution is not necessarily a process

00:07:29.720 --> 00:07:32.060
going from basic to advanced.

00:07:32.060 --> 00:07:35.290
And it's not due to an individual organism

00:07:35.290 --> 00:07:38.610
somehow exercising or training

00:07:38.610 --> 00:07:41.133
itself to the environment that it's in.


Impact of mass on orbital speed

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:03.700
- [Instructor] A satellite of
mass lowercase m orbits Earth

00:00:03.700 --> 00:00:07.270
at radius capital R and speed
v naught as shown below.

00:00:07.270 --> 00:00:09.480
So this has mass lowercase m.

00:00:09.480 --> 00:00:12.880
An aerospace engineer decides
to launch a second satellite

00:00:12.880 --> 00:00:15.840
that is double the mass
into the same orbit.

00:00:15.840 --> 00:00:18.280
So the same orbit, so this radius is still

00:00:18.280 --> 00:00:19.760
gonna be capital R.

00:00:19.760 --> 00:00:22.510
And so this satellite,
the second satellite,

00:00:22.510 --> 00:00:25.220
has a mass of two m.

00:00:25.220 --> 00:00:27.210
The mass of Earth is M.

00:00:27.210 --> 00:00:29.480
So this is Earth right here, capital M.

00:00:29.480 --> 00:00:32.650
What is the speed, lowercase
v, of the heavier satellite

00:00:32.650 --> 00:00:33.940
in terms of v naught?

00:00:33.940 --> 00:00:36.840
And speed you can view as
the magnitude of velocity,

00:00:36.840 --> 00:00:38.380
and so that's why it's lowercase v

00:00:38.380 --> 00:00:39.870
without a vector symbol on it.

00:00:39.870 --> 00:00:42.150
And so what we're trying to figure out,

00:00:42.150 --> 00:00:44.740
the magnitude of its velocity,
in order to stay in orbit.

00:00:44.740 --> 00:00:48.440
What is lowercase v going to be equal to?

00:00:48.440 --> 00:00:50.400
So pause this video and see
if you can figure it out

00:00:50.400 --> 00:00:51.742
on your own.

00:00:51.742 --> 00:00:54.413
All right so to tackle this,
remember the whole reason

00:00:54.413 --> 00:00:58.060
why something stays in
orbit instead of just going

00:00:58.060 --> 00:01:01.100
in a straight line through
space is because there is going

00:01:01.100 --> 00:01:05.090
to be a constant magnitude
centripetal acceleration

00:01:05.090 --> 00:01:08.560
towards the center of
Earth that keeps turning

00:01:08.560 --> 00:01:12.400
I guess you could say the
satellite in this circular path.

00:01:12.400 --> 00:01:15.890
And we've seen from other
videos that the magnitude

00:01:15.890 --> 00:01:18.600
of our centripetal acceleration
is going to be equal

00:01:18.600 --> 00:01:20.200
to the magnitude of our velocity,

00:01:20.200 --> 00:01:22.570
and I'll just use this first satellite,

00:01:22.570 --> 00:01:25.876
so the magnitude of its
velocity squared divided

00:01:25.876 --> 00:01:29.863
by our radius, which in
this case is a capital R.

00:01:30.720 --> 00:01:34.740
But what determines our
centripetal acceleration?

00:01:34.740 --> 00:01:38.340
Well we can explore Newton's
law of gravitation there.

00:01:38.340 --> 00:01:43.340
So if we think about the
magnitude of the force of gravity,

00:01:43.780 --> 00:01:46.720
well that's going to be equal to G,

00:01:46.720 --> 00:01:49.210
which is the universal
gravitational constant,

00:01:49.210 --> 00:01:51.278
times the product of the two masses

00:01:51.278 --> 00:01:53.340
that have the force between them,

00:01:53.340 --> 00:01:55.830
so the product of the
mass of Earth, capital M,

00:01:55.830 --> 00:01:58.010
and the mass of this satellite,

00:01:58.010 --> 00:01:59.777
I'll just focus on this satellite for now,

00:01:59.777 --> 00:02:04.010
divided by the distance
between their center

00:02:04.010 --> 00:02:05.120
of masses squared.

00:02:05.120 --> 00:02:08.730
In this case that is
R, capital R, squared.

00:02:08.730 --> 00:02:11.170
And if you wanted the
centripetal acceleration,

00:02:11.170 --> 00:02:14.350
you would just divide
force divided by mass.

00:02:14.350 --> 00:02:16.070
Remember from Newton's second law,

00:02:16.070 --> 00:02:18.733
we know that F is equal to ma.

00:02:20.190 --> 00:02:22.990
And so if we're talking about
centripetal acceleration,

00:02:22.990 --> 00:02:26.500
it's the force of gravity
that is causing it.

00:02:26.500 --> 00:02:29.390
And so if you wanna solve
for centripetal acceleration,

00:02:29.390 --> 00:02:33.680
you just divide both sides of these by m.

00:02:33.680 --> 00:02:35.760
And so our centripetal acceleration here,

00:02:35.760 --> 00:02:39.170
if you divide our force
of gravity by lowercase m,

00:02:39.170 --> 00:02:40.730
by the mass of the satellite,

00:02:40.730 --> 00:02:42.130
our centripetal
acceleration is going to be

00:02:42.130 --> 00:02:45.700
the universal gravitational
constant times the mass of Earth

00:02:45.700 --> 00:02:48.750
divided by the radius squared.

00:02:48.750 --> 00:02:51.270
And so we could then
take this and substitute

00:02:51.270 --> 00:02:53.600
it back over here and solve

00:02:53.600 --> 00:02:56.130
for the magnitude of our velocity.

00:02:56.130 --> 00:02:59.280
So what you're going to
have is the universal

00:02:59.280 --> 00:03:01.350
gravitational constant
times the mass of Earth

00:03:01.350 --> 00:03:05.200
divided by the radius squared is equal to

00:03:05.200 --> 00:03:07.927
the magnitude of our
velocity or the speed squared

00:03:07.927 --> 00:03:10.130
divided by capital R.

00:03:10.130 --> 00:03:12.940
Now you can multiply both sides by R,

00:03:12.940 --> 00:03:14.590
and I'll swap sides as well,

00:03:14.590 --> 00:03:17.380
and you're gonna get v
naught squared is going to be

00:03:17.380 --> 00:03:21.320
equal to capital G times capital M over R.

00:03:21.320 --> 00:03:22.980
Or if you take the square
root of both sides,

00:03:22.980 --> 00:03:27.600
you get v naught is
equal to the square root

00:03:27.600 --> 00:03:30.060
of the universal gravitational constant

00:03:30.060 --> 00:03:33.070
times the mass of Earth
divided by the distance

00:03:33.070 --> 00:03:35.190
between the center of masses.

00:03:35.190 --> 00:03:36.960
Now what's interesting here is,

00:03:36.960 --> 00:03:41.410
we see this speed we need in
order to maintain this orbit

00:03:41.410 --> 00:03:45.670
in no way is it a function of
the mass of this satellite.

00:03:45.670 --> 00:03:49.440
I don't see a lowercase m
anywhere in this expression

00:03:49.440 --> 00:03:50.960
on the right-hand side.

00:03:50.960 --> 00:03:54.750
And since this is independent of the mass

00:03:54.750 --> 00:03:56.480
of the thing that is in orbit,

00:03:56.480 --> 00:03:59.940
if you double the mass,
if you go from lowercase m

00:03:59.940 --> 00:04:01.880
to two times lowercase m,

00:04:01.880 --> 00:04:05.290
it does not change the
needed orbital speed.

00:04:05.290 --> 00:04:07.830
And so what is the speed
v of the heavier satellite

00:04:07.830 --> 00:04:08.663
in terms of v naught?

00:04:08.663 --> 00:04:10.120
It's gonna be the same thing.

00:04:10.120 --> 00:04:13.730
We could write v is going
to be equal to v naught.

00:04:13.730 --> 00:04:15.820
Doesn't matter what you
do to the mass here,

00:04:15.820 --> 00:04:18.313
you're going to need
the same orbital speed.


Factorial and counting seat arrangements

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.510
- [Instructor] In this video
we are going to introduce

00:00:02.510 --> 00:00:05.900
ourselves to the idea of permutations,

00:00:05.900 --> 00:00:09.220
which is a fancy word for a
pretty straight forward concept,

00:00:09.220 --> 00:00:11.120
which is what are the number of ways

00:00:11.120 --> 00:00:12.740
that we can arrange things?

00:00:12.740 --> 00:00:14.960
How many different
possibilities are there?

00:00:14.960 --> 00:00:17.200
And to make that a little bit tangible,

00:00:17.200 --> 00:00:20.630
let's have an example with say a sofa.

00:00:20.630 --> 00:00:23.700
My sofa can seat exactly three people.

00:00:23.700 --> 00:00:26.140
I have seat number one
on the left of the sofa,

00:00:26.140 --> 00:00:28.180
seat number two in the middle of the sofa,

00:00:28.180 --> 00:00:31.420
and seat number three on
the right of the sofa.

00:00:31.420 --> 00:00:33.570
And let's say we're going
to have three people

00:00:33.570 --> 00:00:35.350
who are going to sit in these three seats,

00:00:35.350 --> 00:00:39.750
person A, person B, and person C.

00:00:39.750 --> 00:00:43.300
How many different ways
can these three people

00:00:43.300 --> 00:00:45.760
sit in these three seats?

00:00:45.760 --> 00:00:47.440
Pause this video and see if you can

00:00:47.440 --> 00:00:48.853
figure it out on your own.

00:00:50.010 --> 00:00:52.550
Well, there's several
ways to approach this.

00:00:52.550 --> 00:00:54.010
One way is just try to think

00:00:54.010 --> 00:00:56.040
through all of the possibilities.

00:00:56.040 --> 00:00:57.670
You could do it systematically.

00:00:57.670 --> 00:00:59.320
You could say alright,

00:00:59.320 --> 00:01:02.630
if I have person A in seat number one,

00:01:02.630 --> 00:01:04.960
then I could have person
B in seat number two,

00:01:04.960 --> 00:01:07.790
and person C in seat number three.

00:01:07.790 --> 00:01:09.770
And I could think of another situation.

00:01:09.770 --> 00:01:11.650
If I have person A in seat number one,

00:01:11.650 --> 00:01:13.430
I could then swap B and C.

00:01:13.430 --> 00:01:16.320
So it could look like that.

00:01:16.320 --> 00:01:18.700
And that's all of the situations,

00:01:18.700 --> 00:01:21.610
all of the permutations where
I have A in seat number one.

00:01:21.610 --> 00:01:23.880
So now let's put someone
else in seat number one.

00:01:23.880 --> 00:01:26.150
So now let's put B in seat number one,

00:01:26.150 --> 00:01:28.940
and I could put A in the
middle and C on the right.

00:01:28.940 --> 00:01:30.870
Or I could put B in seat number one,

00:01:30.870 --> 00:01:32.180
and then swap A and C.

00:01:32.180 --> 00:01:34.310
So C and then A.

00:01:34.310 --> 00:01:38.040
And then if I put C in seat number one,

00:01:38.040 --> 00:01:40.660
well I could put A in the
middle and B on the right.

00:01:40.660 --> 00:01:42.680
Or with C in seat number one,

00:01:42.680 --> 00:01:45.680
I could put B in the
middle and A on the right.

00:01:45.680 --> 00:01:47.980
And these are actually
all of the permutations

00:01:47.980 --> 00:01:49.500
and you can see that there are

00:01:49.500 --> 00:01:51.470
one, two, three,

00:01:51.470 --> 00:01:53.860
four, five, six.

00:01:53.860 --> 00:01:55.620
Now this wasn't too bad.

00:01:55.620 --> 00:01:56.950
And in general, if you're thinking about

00:01:56.950 --> 00:01:59.097
permutations of six things

00:01:59.097 --> 00:02:01.720
or three things in three spaces,

00:02:01.720 --> 00:02:03.430
you can do it by hand.

00:02:03.430 --> 00:02:05.240
But it could get very
complicated if I said,

00:02:05.240 --> 00:02:08.040
hey, I have 100 seats
and I have 100 people

00:02:08.040 --> 00:02:08.930
that are going to sit in them.

00:02:08.930 --> 00:02:11.690
How do I figure it out mathematically?

00:02:11.690 --> 00:02:13.070
Well the way that you would do it,

00:02:13.070 --> 00:02:14.850
and this is going to be a
technique that you can use

00:02:14.850 --> 00:02:17.960
for really any number of
people and any number of seats

00:02:17.960 --> 00:02:22.020
is to really just build off
of what we just did here.

00:02:22.020 --> 00:02:24.720
What we did here is we
started with seat number one

00:02:24.720 --> 00:02:28.100
and we said alright, how many
different possibilities are,

00:02:28.100 --> 00:02:30.090
how many different people
could sit in seat number one

00:02:30.090 --> 00:02:32.960
assuming no one has sat down before.

00:02:32.960 --> 00:02:35.950
Well, three different people
could sit in seat number one.

00:02:35.950 --> 00:02:36.950
You could see it right over here.

00:02:36.950 --> 00:02:39.540
This is where A is sitting
in seat number one,

00:02:39.540 --> 00:02:41.750
this is where B is sitting
in seat number one,

00:02:41.750 --> 00:02:45.130
and this is where C is
sitting in seat number one.

00:02:45.130 --> 00:02:48.410
Now for each of those three possibilities,

00:02:48.410 --> 00:02:51.300
how many people can
sit in seat number two?

00:02:51.300 --> 00:02:53.840
Well, we saw when A
sits in seat number one,

00:02:53.840 --> 00:02:56.500
there's two different
possibilities for seat number two.

00:02:56.500 --> 00:02:58.120
When B sits in seat number one,

00:02:58.120 --> 00:03:00.850
there's two different
possibilities for seat number two.

00:03:00.850 --> 00:03:03.380
When C sits in seat number
one, this is a tongue-twister,

00:03:03.380 --> 00:03:06.310
there's two different
possibilities for seat number two.

00:03:06.310 --> 00:03:09.200
And so, you're gonna have two
different possibilities here.

00:03:09.200 --> 00:03:10.770
Another way to think about it is,

00:03:10.770 --> 00:03:12.470
one person has already sat down here,

00:03:12.470 --> 00:03:14.030
there's three different
ways of getting that,

00:03:14.030 --> 00:03:16.240
and so there's two
people left who could sit

00:03:16.240 --> 00:03:18.420
in the second seat and we
saw that right over here,

00:03:18.420 --> 00:03:22.030
where we really wrote
out the permutations.

00:03:22.030 --> 00:03:24.330
And so how many different
permutations are there

00:03:24.330 --> 00:03:26.880
for seat number one and seat number two?

00:03:26.880 --> 00:03:27.920
Well, you would multiply.

00:03:27.920 --> 00:03:29.960
For each of these three you have two,

00:03:29.960 --> 00:03:31.500
for each of these three
in seat number one,

00:03:31.500 --> 00:03:33.600
you have two in seat number two.

00:03:33.600 --> 00:03:36.570
And then what about seat number three?

00:03:36.570 --> 00:03:38.480
Well, if you know who's in seat number one

00:03:38.480 --> 00:03:41.300
and seat number two,
there's only one person who

00:03:41.300 --> 00:03:43.180
can be in seat number three.

00:03:43.180 --> 00:03:44.570
And another way to think about it,

00:03:44.570 --> 00:03:46.190
if two people have already sat down,

00:03:46.190 --> 00:03:48.790
there's only one person who
could be in seat number three.

00:03:48.790 --> 00:03:51.530
And so mathematically, what
we could do is just say

00:03:51.530 --> 00:03:53.660
three times two times one.

00:03:53.660 --> 00:03:54.890
And you might recognize the

00:03:54.890 --> 00:03:57.240
mathematical operation factorial,

00:03:57.240 --> 00:03:59.980
which literally just means
hey, start with that number,

00:03:59.980 --> 00:04:02.700
and then keep multiplying
it by the numbers

00:04:02.700 --> 00:04:04.760
one less than that and
then one less than that

00:04:04.760 --> 00:04:06.840
all the way until you get to one.

00:04:06.840 --> 00:04:08.460
And this is three factorial,

00:04:08.460 --> 00:04:11.360
which is going to be equal to six,

00:04:11.360 --> 00:04:13.200
which is exactly what we got here.

00:04:13.200 --> 00:04:15.960
And to appreciate the power of this,

00:04:15.960 --> 00:04:18.270
let's extend our example.

00:04:18.270 --> 00:04:19.940
Let's say that we have five seats.

00:04:19.940 --> 00:04:22.480
One, two, three,

00:04:22.480 --> 00:04:24.210
four, five.

00:04:24.210 --> 00:04:26.160
And we have five people,

00:04:26.160 --> 00:04:28.780
person A, B, C,

00:04:28.780 --> 00:04:30.590
D, and E.

00:04:30.590 --> 00:04:32.570
How many different ways
can these five people

00:04:32.570 --> 00:04:33.700
sit in these five seats?

00:04:33.700 --> 00:04:35.450
Pause this video and figure it out.

00:04:36.300 --> 00:04:37.610
Well, you might immediately say

00:04:37.610 --> 00:04:40.530
well that's going to be five factorial,

00:04:40.530 --> 00:04:41.690
which is going to be equal to

00:04:41.690 --> 00:04:44.500
five times four times three

00:04:44.500 --> 00:04:46.690
times two times one.

00:04:46.690 --> 00:04:49.130
Five times four is 20.

00:04:49.130 --> 00:04:52.450
20 times three is 60.

00:04:52.450 --> 00:04:55.080
And then 60 times two is 120.

00:04:55.080 --> 00:04:58.340
And then 120 times one is equal to 120.

00:04:58.340 --> 00:05:01.869
And once again, that makes a lot of sense.

00:05:01.869 --> 00:05:04.340
If no one's sat down, there's five

00:05:04.340 --> 00:05:06.480
different possibilities
for seat number one.

00:05:06.480 --> 00:05:08.830
And then for each of those possibilities,

00:05:08.830 --> 00:05:11.120
there's four people who
could sit in seat number two.

00:05:11.120 --> 00:05:13.580
And then for each of
those 20 possibilities

00:05:13.580 --> 00:05:15.560
in seat numbers one and two,

00:05:15.560 --> 00:05:17.480
well there's gonna be
three people who could

00:05:17.480 --> 00:05:19.110
sit in seat number three.

00:05:19.110 --> 00:05:21.620
And for each of these 60 possibilities,

00:05:21.620 --> 00:05:24.430
there's two people who can
sit in seat number four.

00:05:24.430 --> 00:05:26.590
And then once you know who's
in the first four seats,

00:05:26.590 --> 00:05:29.490
you know who has to
sit in that fifth seat.

00:05:29.490 --> 00:05:31.563
And that's where we got that 120 from.


Invertible and noninvertibles matrices

WEBVTT
Kind: captions
Language: en

00:00:00.110 --> 00:00:01.690
- [Instructor] So let
me just write a general

00:00:01.690 --> 00:00:03.500
two-by-two matrix A.

00:00:03.500 --> 00:00:08.500
So let's just say its
elements are a, b, c, and d.

00:00:09.760 --> 00:00:11.900
Now, from previous videos, we have learned

00:00:11.900 --> 00:00:16.080
how to find the inverse of our matrix A,

00:00:16.080 --> 00:00:18.410
the formula that we went over.

00:00:18.410 --> 00:00:22.060
The inverse of our matrix
A is going to be equal to

00:00:22.060 --> 00:00:26.370
one over the determinant of our matrix A

00:00:26.370 --> 00:00:30.490
times what is often called
the adjoint of our matrix A.

00:00:30.490 --> 00:00:33.630
And that's you are
swapping the a and the d.

00:00:33.630 --> 00:00:36.050
So you have a d here and an a here.

00:00:36.050 --> 00:00:37.620
You're making the b and the c negative.

00:00:37.620 --> 00:00:40.000
And this works for a
two-by-two when you do a,

00:00:40.000 --> 00:00:41.510
when you go to a three-by-three case,

00:00:41.510 --> 00:00:43.480
it gets a little bit more complicated.

00:00:43.480 --> 00:00:45.270
So we're gonna have a negative b here

00:00:45.270 --> 00:00:48.190
and a negative c here.

00:00:48.190 --> 00:00:49.990
Or another way to think about it,

00:00:49.990 --> 00:00:51.890
the determinant of a two-by-two matrix

00:00:51.890 --> 00:00:55.779
is going to be one over ad minus bc,

00:00:55.779 --> 00:00:58.160
ad minus bc,

00:00:58.160 --> 00:01:02.650
and then you multiply
that times the adjoint.

00:01:02.650 --> 00:01:04.610
So that is fancy word,

00:01:04.610 --> 00:01:05.660
that's a fancy word for it.

00:01:05.660 --> 00:01:10.310
So we have d, negative
b, negative c, and a.

00:01:10.310 --> 00:01:11.890
So that is all review.

00:01:11.890 --> 00:01:13.490
If it's not, I encourage you to review it

00:01:13.490 --> 00:01:15.810
in other places on Khan Academy.

00:01:15.810 --> 00:01:18.890
But what we're going to ask
ourselves in this video is,

00:01:18.890 --> 00:01:23.320
is there a situation where
a matrix is not invertible?

00:01:23.320 --> 00:01:24.620
So pause this video,

00:01:24.620 --> 00:01:26.860
and see if you can figure out a scenario

00:01:26.860 --> 00:01:28.830
or a general set of scenarios

00:01:28.830 --> 00:01:31.473
where a matrix is not
going to be invertible.

00:01:33.000 --> 00:01:36.550
So one way to think about a
matrix that is not invertible

00:01:36.550 --> 00:01:38.140
is that, for that matrix,

00:01:38.140 --> 00:01:42.000
we would somehow not be able
to make this calculation.

00:01:42.000 --> 00:01:44.830
Now, no matter what a, b, c, and d are,

00:01:44.830 --> 00:01:47.730
we can always find an adjoint matrix.

00:01:47.730 --> 00:01:51.010
But what's a situation where
one over the determinant

00:01:51.010 --> 00:01:53.750
is we're not going to
be able to calculate it?

00:01:53.750 --> 00:01:56.340
Well, a situation where the determinant,

00:01:56.340 --> 00:02:00.230
the determinant of our
matrix A is equal to zero,

00:02:00.230 --> 00:02:02.520
or I could write the
determinant of our matrix A

00:02:02.520 --> 00:02:04.070
is going to be equal to zero.

00:02:04.070 --> 00:02:06.480
In that situation, we're not
going to be able to calculate

00:02:06.480 --> 00:02:10.020
one over the determinant
because that is undefined.

00:02:10.020 --> 00:02:11.590
And so we say,

00:02:11.590 --> 00:02:13.140
we say A

00:02:13.140 --> 00:02:15.300
is not invertible,

00:02:16.497 --> 00:02:17.913
invertible,

00:02:18.950 --> 00:02:22.310
if and only if, I'll do this
two-way arrow right over here,

00:02:22.310 --> 00:02:26.140
our determinant for our
matrix A is equal to zero.

00:02:26.140 --> 00:02:28.090
Because if our determinant
is equal to zero,

00:02:28.090 --> 00:02:30.620
we're going to have to divide by zero

00:02:30.620 --> 00:02:33.460
when we find the inverse.

00:02:33.460 --> 00:02:35.410
Now, let's explore what that

00:02:35.410 --> 00:02:37.120
would look like a little bit more.

00:02:37.120 --> 00:02:38.990
If our determinant is
going to be equal to zero,

00:02:38.990 --> 00:02:41.030
that means that this
expression right over here

00:02:41.030 --> 00:02:42.770
is going to be equal to zero.

00:02:42.770 --> 00:02:47.230
So that means that ad
minus bc is equal to zero.

00:02:47.230 --> 00:02:49.030
We can add bc to both sides.

00:02:49.030 --> 00:02:53.950
That means that a times
d is equal to b times c.

00:02:53.950 --> 00:02:55.870
And then we can algebraically manipulate

00:02:55.870 --> 00:03:00.870
this a little bit more if we
divided both sides by b and d.

00:03:01.680 --> 00:03:02.560
So let me just do that.

00:03:02.560 --> 00:03:06.450
If I divide both sides by b and d,

00:03:06.450 --> 00:03:09.880
then the d's cancel out here,
and the b's cancel out here.

00:03:09.880 --> 00:03:12.830
So this could be a over b

00:03:12.830 --> 00:03:15.790
is equal to c over d.

00:03:15.790 --> 00:03:18.320
So we have a situation where the,

00:03:18.320 --> 00:03:21.800
you could think of it as
the ratio between a and b

00:03:21.800 --> 00:03:25.350
is equivalent to the
ratio between c and d.

00:03:25.350 --> 00:03:27.150
You could also think
about it the other way.

00:03:27.150 --> 00:03:28.240
You could divide,

00:03:28.240 --> 00:03:32.233
let's start with that original
equation, ad is equal to bc.

00:03:33.350 --> 00:03:36.600
If you divided both sides by

00:03:36.600 --> 00:03:38.380
c and d,

00:03:38.380 --> 00:03:41.600
so you divide both sides by cd,

00:03:41.600 --> 00:03:45.910
what you're going to be
left with is that a over c

00:03:45.910 --> 00:03:49.050
is going to be equal to b over d.

00:03:49.050 --> 00:03:51.700
And so the ratio between a and c

00:03:51.700 --> 00:03:53.760
is equal to the ratio between b and d.

00:03:53.760 --> 00:03:55.540
So all of those situations,

00:03:55.540 --> 00:03:57.710
you're going to have
a noninvertible matrix

00:03:57.710 --> 00:04:00.650
because the determinant is
going to be equal to zero.

00:04:00.650 --> 00:04:03.190
And so let's look at a few examples.

00:04:03.190 --> 00:04:06.100
I want you to tell me whether
the following matrices

00:04:06.100 --> 00:04:07.693
are invertible or not.

00:04:08.530 --> 00:04:12.860
So let's say I have the matrix five, one,

00:04:12.860 --> 00:04:14.730
three, two.

00:04:14.730 --> 00:04:16.430
Is this an invertible matrix?

00:04:16.430 --> 00:04:18.593
Pause this video, and
try to figure it out.

00:04:19.530 --> 00:04:23.270
So the determinant of this matrix is going

00:04:23.270 --> 00:04:27.030
to be equal to five times two, 10,

00:04:27.030 --> 00:04:28.370
minus three times one,

00:04:28.370 --> 00:04:30.370
minus three, which is equal to seven.

00:04:30.370 --> 00:04:33.700
And since this is not equal to zero,

00:04:33.700 --> 00:04:37.060
we know that this matrix is invertible.

00:04:37.060 --> 00:04:39.240
We can find an inverse.

00:04:39.240 --> 00:04:42.700
Now, what about this matrix?

00:04:42.700 --> 00:04:45.670
So 10, 20,

00:04:45.670 --> 00:04:49.900
and 250 and 500,

00:04:49.900 --> 00:04:52.620
is this matrix invertible?

00:04:52.620 --> 00:04:54.390
The determinant here,

00:04:54.390 --> 00:04:56.121
of 10, 20,

00:04:56.121 --> 00:04:59.220
250, and 500,

00:04:59.220 --> 00:05:04.220
this is going to be equal to
10 times 500, which is 5,000,

00:05:04.240 --> 00:05:09.240
minus 20 times 250, which is
5,000, which is equal to zero.

00:05:09.510 --> 00:05:12.270
And once again, if you
wanted to find its inverse,

00:05:12.270 --> 00:05:14.300
you would take your adjoint matrix

00:05:14.300 --> 00:05:16.400
and multiply it times
one over the determinant.

00:05:16.400 --> 00:05:20.600
But one over zero is not defined,
so this is not invertible.


Determinant when multiplying a matrix by a constant

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.880
- [Instructor] So let's
say that I have matrix A

00:00:02.880 --> 00:00:03.720
and its elements.

00:00:03.720 --> 00:00:07.460
It's a two-by-two, a, b, c, d.

00:00:07.460 --> 00:00:09.540
We have a lot of practice
taking determinants

00:00:09.540 --> 00:00:11.610
of matrices like this.

00:00:11.610 --> 00:00:14.560
The determinant of this matrix,

00:00:14.560 --> 00:00:18.350
same thing as determinant of a, b, c, d.

00:00:18.350 --> 00:00:23.350
It's going to be equal to a
times d, ad, minus b times c.

00:00:24.760 --> 00:00:28.150
Now what would happen
if we multiply one row

00:00:28.150 --> 00:00:30.830
of this matrix by a constant?

00:00:30.830 --> 00:00:33.140
What would happen to its determinant?

00:00:33.140 --> 00:00:34.860
Well, let's try it out.

00:00:34.860 --> 00:00:35.693
So you have your original,

00:00:35.693 --> 00:00:40.693
the determinant of your
original matrix, a, b, c, d.

00:00:41.030 --> 00:00:43.150
I'm just rewriting what
I just did up here.

00:00:43.150 --> 00:00:43.983
That's ad

00:00:44.957 --> 00:00:47.000
minus bc.

00:00:47.000 --> 00:00:48.380
Now if I were to multiply,

00:00:48.380 --> 00:00:51.750
let's say, this first row by a constant k,

00:00:51.750 --> 00:00:53.880
how would that change the determinant?

00:00:53.880 --> 00:00:57.470
Well, instead of this being
just an a, this is now a ka.

00:00:57.470 --> 00:01:00.343
And instead of this being
just a b, it is now a kb.

00:01:01.660 --> 00:01:05.521
And so this is equal to k times

00:01:05.521 --> 00:01:08.610
ad minus bc,

00:01:08.610 --> 00:01:11.450
which is the same thing as

00:01:11.450 --> 00:01:12.930
it's equal to k

00:01:12.930 --> 00:01:17.000
times our original
determinant of our matrix A.

00:01:17.000 --> 00:01:18.470
So that seems pretty interesting,

00:01:18.470 --> 00:01:20.620
and I encourage you to see that if you,

00:01:20.620 --> 00:01:22.380
instead of doing the first
row, you did the second row,

00:01:22.380 --> 00:01:24.410
you would have gotten the same outcome.

00:01:24.410 --> 00:01:26.560
And then you could also verify that look,

00:01:26.560 --> 00:01:30.040
if I multiply both of these
rows by that constant k,

00:01:30.040 --> 00:01:31.623
how would that change things?

00:01:32.800 --> 00:01:36.450
Well, then I'm gonna have ka times kd,

00:01:36.450 --> 00:01:38.500
so you're gonna have
a k squeezed in there.

00:01:38.500 --> 00:01:43.130
And then you're gonna have kb times kc.

00:01:43.130 --> 00:01:44.880
And so this would actually be,

00:01:44.880 --> 00:01:47.770
you could factor out a k squared.

00:01:47.770 --> 00:01:49.750
And so this is going to be k squared

00:01:49.750 --> 00:01:51.870
times the determinant of A.

00:01:51.870 --> 00:01:55.080
And that can be extended to
a generalized property that,

00:01:55.080 --> 00:01:56.730
if I have some

00:01:56.730 --> 00:02:00.340
n-by-n matrix A,

00:02:00.340 --> 00:02:03.060
then the determinant,

00:02:03.060 --> 00:02:05.943
the determinant of k

00:02:05.943 --> 00:02:08.700
times that n-by-n

00:02:08.700 --> 00:02:10.163
matrix A,

00:02:11.140 --> 00:02:12.130
the determinant of this,

00:02:12.130 --> 00:02:15.080
when I multiply a constant
times all, entire matrix,

00:02:15.080 --> 00:02:17.390
I'm multiplying that
constant times all the rows,

00:02:17.390 --> 00:02:19.070
we could say all of the elements.

00:02:19.070 --> 00:02:20.620
Well, this is going to be equal to,

00:02:20.620 --> 00:02:21.453
pause this video.

00:02:21.453 --> 00:02:23.180
See if you can intuit

00:02:23.180 --> 00:02:25.960
what this general formula is going to be.

00:02:25.960 --> 00:02:27.700
You might be tempted to say it's k

00:02:27.700 --> 00:02:28.650
times the determinant of A,

00:02:28.650 --> 00:02:32.270
but remember that's only
if I multiply one row by k.

00:02:32.270 --> 00:02:35.970
But if I multiply the entire matrix by k,

00:02:35.970 --> 00:02:37.020
well, then this is going,

00:02:37.020 --> 00:02:40.730
this determinant is going
to be the constant k

00:02:40.730 --> 00:02:43.420
to the nth power

00:02:43.420 --> 00:02:46.200
times the determinant

00:02:46.200 --> 00:02:48.890
of our n-by-n

00:02:48.890 --> 00:02:50.680
matrix A.

00:02:50.680 --> 00:02:53.820
And you could see this play
out in a three-by-three case.

00:02:53.820 --> 00:02:55.440
In fact, I encourage you to try it out

00:02:55.440 --> 00:02:56.570
with some three-by-threes.

00:02:56.570 --> 00:02:58.640
And you could also do a generalized proof

00:02:58.640 --> 00:03:01.860
for an n-by-n case, but
I won't do that now.

00:03:01.860 --> 00:03:03.910
This is really just to give you the idea.


LearnStorm and Ceiba College Prep

WEBVTT
Kind: captions
Language: en

00:00:00.920 --> 00:00:02.040
- [Narrator] Watsonville, California

00:00:02.040 --> 00:00:03.830
is an agricultural community.

00:00:03.830 --> 00:00:06.880
We have 98% of our students are Latino

00:00:06.880 --> 00:00:09.330
and about 90% of our
students speak Spanish

00:00:09.330 --> 00:00:10.990
as their first language.

00:00:10.990 --> 00:00:13.180
Teachers cannot do their job alone.

00:00:13.180 --> 00:00:14.600
They need to have resources

00:00:14.600 --> 00:00:16.720
because every class that they receive,

00:00:16.720 --> 00:00:18.890
which is usually between 25 and 30,

00:00:18.890 --> 00:00:21.361
has a variety of learners
in that classroom.

00:00:21.361 --> 00:00:24.150
Teachers use a wide variety of tech tools

00:00:24.150 --> 00:00:27.600
in their classrooms and Kahn
Academy is one of those tools,

00:00:27.600 --> 00:00:30.828
'cause it allows for differentiation
amongst the learners.

00:00:30.828 --> 00:00:33.128
- I want to subtract
four here, whatcha do?

00:00:33.128 --> 00:00:36.103
(students respond)

00:00:36.103 --> 00:00:39.320
- Miss Kotto takes teaching as a skill

00:00:39.320 --> 00:00:41.193
and turns it into an art form.

00:00:41.193 --> 00:00:43.610
- This year was amazing with LearnStorm.

00:00:43.610 --> 00:00:45.095
I had a group of students
that couldn't add,

00:00:45.095 --> 00:00:48.200
subtract, multiply, or divide fractions.

00:00:48.200 --> 00:00:50.470
They avoided it at all costs.

00:00:50.470 --> 00:00:52.290
But since it was an assignment,

00:00:52.290 --> 00:00:53.580
they would try it and they would

00:00:53.580 --> 00:00:55.780
eventually do much better.

00:00:55.780 --> 00:00:58.788
And what was amazing was that
the lower-achieving students

00:00:58.788 --> 00:01:03.400
were able to see that they
were part of something huge.

00:01:03.400 --> 00:01:05.170
They were able to contribute,

00:01:05.170 --> 00:01:08.480
and that, I thought, boosted
their self-confidence,

00:01:08.480 --> 00:01:10.620
their self-esteem, and I didn't realize

00:01:10.620 --> 00:01:13.310
that would happen after doing LearnStorm.

00:01:14.421 --> 00:01:16.050
- LearnStorm really made me realize

00:01:16.050 --> 00:01:19.130
how much you can grow in
just a short amount of time.

00:01:19.130 --> 00:01:21.360
If you put that same amount of effort,

00:01:21.360 --> 00:01:23.510
you can grow even more.

00:01:23.510 --> 00:01:24.810
- I'm impressed.

00:01:24.810 --> 00:01:26.210
Many of you guys are getting it right.

00:01:26.210 --> 00:01:28.710
Some of you guys are getting stuck.

00:01:28.710 --> 00:01:30.300
I will show you where
you're getting stuck,

00:01:30.300 --> 00:01:32.090
and I love that you're not giving up.

00:01:32.090 --> 00:01:34.849
- All of our classmates
work so well together,

00:01:34.849 --> 00:01:39.060
and it really showed that
we really did win as a team.

00:01:39.060 --> 00:01:41.040
It was really breathtaking

00:01:41.040 --> 00:01:44.110
'cause I think our goal
was just to reach Level 12,

00:01:44.110 --> 00:01:47.095
and then we ended up reaching Level 36.

00:01:47.095 --> 00:01:51.163
We completed, I think, 8,080 assignments.

00:01:52.230 --> 00:01:53.320
- First thing in the morning,

00:01:53.320 --> 00:01:56.290
I had everybody start doing
some mastery challenges.

00:01:56.290 --> 00:01:58.110
After I took roll I said, okay guys,

00:01:58.110 --> 00:02:00.006
let's look at the progress tracker.

00:02:00.006 --> 00:02:01.650
And it just kept going up.

00:02:01.650 --> 00:02:02.820
The students loved it.

00:02:02.820 --> 00:02:04.530
The growth mindset activities,

00:02:04.530 --> 00:02:08.010
we watched one every
single week as a group,

00:02:08.010 --> 00:02:10.770
and they would share their
ideas with their partners.

00:02:10.770 --> 00:02:12.747
I can see towards the end of the year,

00:02:12.747 --> 00:02:13.890
they don't give up.

00:02:13.890 --> 00:02:15.740
They work harder.

00:02:15.740 --> 00:02:18.532
- Keep going, keep growing
means to push your limits

00:02:18.532 --> 00:02:21.680
and to, even if you make a mistake,

00:02:21.680 --> 00:02:24.320
learn from them and keep
growing when you're learning.

00:02:24.320 --> 00:02:25.730
- Thinking about LearnStorm

00:02:25.730 --> 00:02:27.270
and the students going to college

00:02:27.270 --> 00:02:28.668
almost makes me want to cry

00:02:28.668 --> 00:02:30.760
because I think at the beginning,

00:02:30.760 --> 00:02:33.340
so many of the students
felt I just can't do it.

00:02:33.340 --> 00:02:35.690
But because they were
able to do LearnStorm

00:02:35.690 --> 00:02:37.750
and their skills are so much higher now,

00:02:37.750 --> 00:02:40.880
they know that they can
keep going and keep growing,

00:02:40.880 --> 00:02:43.093
and nothing will stand in their way.


LearnStorm at Wewahitchka High School

WEBVTT
Kind: captions
Language: en

00:00:01.135 --> 00:00:03.230
- Wewahitchka High School is
a great little school here.

00:00:03.230 --> 00:00:08.200
It's a rural area, about
350 kids, seven through 12.

00:00:08.200 --> 00:00:10.260
A great school though,
our kids are interested

00:00:10.260 --> 00:00:12.270
in bettering themselves.

00:00:12.270 --> 00:00:16.020
Ms. Cameron Totman is a
reading ELA teacher here,

00:00:16.020 --> 00:00:17.490
English Language Arts teacher.

00:00:17.490 --> 00:00:19.390
Particularly what
impressed me about Cameron

00:00:19.390 --> 00:00:23.470
is she wants to individualize
education for each student.

00:00:23.470 --> 00:00:25.920
She's jumping on the LearnStorm

00:00:25.920 --> 00:00:28.600
at the Kahn Academy as one of her tools.

00:00:28.600 --> 00:00:29.640
- The activity that you guys

00:00:29.640 --> 00:00:34.640
are gonna do next is talking
about the growth mind-set.

00:00:34.760 --> 00:00:36.860
I was introduced to LearnStorm this year

00:00:36.860 --> 00:00:40.630
by one of our math teachers
who was participating.

00:00:40.630 --> 00:00:42.920
She thought that we would benefit

00:00:42.920 --> 00:00:45.580
from the growth mindset content

00:00:45.580 --> 00:00:48.830
and activities that came with LearnStorm.

00:00:48.830 --> 00:00:49.860
- One thing that I learned from

00:00:49.860 --> 00:00:52.200
the growth mindset activities is that

00:00:52.200 --> 00:00:55.330
your intelligence can grow,
just like your muscles can grow.

00:00:55.330 --> 00:00:57.630
And that's something
that really stuck with me

00:00:57.630 --> 00:00:59.705
'cause I didn't know that before.

00:00:59.705 --> 00:01:02.288
(upbeat music)

00:01:04.380 --> 00:01:07.170
- This is my first year
actually using Kahn Academy

00:01:07.170 --> 00:01:11.780
in a classroom and I've found
it extremely valuable tool.

00:01:11.780 --> 00:01:14.580
I was teaching a class for students

00:01:14.580 --> 00:01:16.650
who were trying to
improve their test score

00:01:16.650 --> 00:01:19.970
on the SAT and ACT and
so we found some specific

00:01:19.970 --> 00:01:22.743
lessons that worked really well with them.

00:01:23.670 --> 00:01:27.930
Definitely the LearnStorm
portion of the program

00:01:27.930 --> 00:01:30.110
was more enjoyable to students I think,

00:01:30.110 --> 00:01:33.270
then just the regular skills
practice that they were doing.

00:01:33.270 --> 00:01:36.370
They were motivated by seeing their daily

00:01:36.370 --> 00:01:38.104
progress on that progress tracker.

00:01:38.104 --> 00:01:41.104
(students cheering)

00:01:43.424 --> 00:01:47.140
- LearnStorm really helps me
because it really helped me

00:01:47.140 --> 00:01:50.623
realize that I had more
potential than I thought I did.

00:01:51.780 --> 00:01:55.731
- LearnStorm made everything
fun, made learning fun.

00:01:55.731 --> 00:01:58.020
LearnStorm helped me build more confidence

00:01:58.020 --> 00:02:01.390
and believing in myself
that I can do this.

00:02:01.390 --> 00:02:02.300
I don't really think I would have

00:02:02.300 --> 00:02:04.063
got through high school without it.

00:02:04.950 --> 00:02:08.600
- [Cameron] I think the biggest
impact that LearnStorm had

00:02:08.600 --> 00:02:12.610
on my students this year
was for those students

00:02:12.610 --> 00:02:15.120
who really took the time and engaged

00:02:15.120 --> 00:02:17.600
in the self reflection activities.

00:02:17.600 --> 00:02:19.760
They've learned the skill of goal setting.

00:02:19.760 --> 00:02:21.100
They've learned things that are going

00:02:21.100 --> 00:02:23.030
to affect them their whole lives

00:02:23.030 --> 00:02:26.233
and they're going to be able
to keep going and keep growing.

00:02:27.110 --> 00:02:29.300
- [Jay] The world is
going digital and we need

00:02:29.300 --> 00:02:33.520
to start up finding good
digital resources in schools.

00:02:33.520 --> 00:02:35.830
Kahn Academy has been the first one that

00:02:35.830 --> 00:02:39.780
I felt like could be a significant,

00:02:39.780 --> 00:02:41.690
a very significant part of a curriculum

00:02:41.690 --> 00:02:43.099
here at Wewahitchka high school.

00:02:43.099 --> 00:02:45.682
(upbeat music)


LearnStorm at Pine Hill Middle School

WEBVTT
Kind: captions
Language: en

00:00:01.050 --> 00:00:03.600
- Here at Pine Hill Middle,
we have a diverse group

00:00:06.199 --> 00:00:09.310
- Raise your hand if you have
at least three lessons passed.

00:00:09.310 --> 00:00:10.143
Good job.

00:00:10.143 --> 00:00:13.660
- When it comes to Ms.
Grubbs, she is so creative.

00:00:13.660 --> 00:00:15.760
One of the resources she has used

00:00:15.760 --> 00:00:18.730
is Khan Academy with the LearnStorm.

00:00:18.730 --> 00:00:22.210
- This class mainly focuses on students

00:00:22.210 --> 00:00:24.370
with learning needs so we have

00:00:24.370 --> 00:00:27.740
to come up with alternative strategies

00:00:27.740 --> 00:00:29.941
to kind of try to help those students.

00:00:29.941 --> 00:00:34.941
LearnStorm is an excellent
resource for differentiation.

00:00:35.260 --> 00:00:37.360
Differentiation, it's one of the most

00:00:37.360 --> 00:00:39.480
difficult things to accomplish.

00:00:39.480 --> 00:00:42.320
They expect you to have one person

00:00:42.320 --> 00:00:45.940
to service thirty people individually.

00:00:45.940 --> 00:00:47.047
Is everybody ready?

00:00:47.047 --> 00:00:48.891
- [Whole Class] Yes, ma'am.

00:00:48.891 --> 00:00:51.520
- Alright, he got three over here.

00:00:51.520 --> 00:00:56.240
I don't necessarily grade
accuracy, I grade progress.

00:00:56.240 --> 00:01:00.040
So, LearnStorm is a good
way to differentiate

00:01:00.970 --> 00:01:05.400
and as well as motivate and
with the growth mindset pieces

00:01:05.400 --> 00:01:09.810
you can also educate them about
why it's important to learn,

00:01:09.810 --> 00:01:13.030
how they learn, how their
body is set up to learn,

00:01:13.030 --> 00:01:15.350
which strategies help them learn.

00:01:15.350 --> 00:01:18.330
It's amazing, it's amazing and it's free.

00:01:19.190 --> 00:01:20.700
It's free!

00:01:20.700 --> 00:01:22.650
- The best part of doing LearnStorm,

00:01:22.650 --> 00:01:25.020
it built up my confidence to do more.

00:01:25.020 --> 00:01:26.740
To do better for myself.

00:01:26.740 --> 00:01:28.540
Be a better person.

00:01:28.540 --> 00:01:31.740
- I think one of the most
inspirational stories

00:01:31.740 --> 00:01:33.390
was one of my students.

00:01:33.390 --> 00:01:37.080
He was the first one to
complete all 60 assignments

00:01:37.080 --> 00:01:39.600
and after he completed
those 60 assignments,

00:01:39.600 --> 00:01:41.160
he asked for more.

00:01:41.160 --> 00:01:44.030
- I did over 300 lessons in one week.

00:01:44.030 --> 00:01:45.400
LearnStorm is good.

00:01:45.400 --> 00:01:48.473
I felt proud of myself
getting all these good grades.

00:01:48.473 --> 00:01:51.056
(upbeat music)

00:01:53.900 --> 00:01:57.640
- End of each week, I assess my students.

00:01:57.640 --> 00:02:00.440
Every single one of 'em
shows significant progress

00:02:00.440 --> 00:02:03.710
and their energy improved significantly

00:02:03.710 --> 00:02:05.890
because they believe they can do it now.

00:02:05.890 --> 00:02:08.870
So, they're very, very, very happy

00:02:08.870 --> 00:02:10.770
and pleased with themselves.

00:02:10.770 --> 00:02:15.770
I would encourage any
teacher to utilize LearnStorm

00:02:16.410 --> 00:02:20.220
as a motivational tool for
students to get started.

00:02:20.220 --> 00:02:23.070
Kinda get that jump started
at the beginning of the year.

00:02:23.070 --> 00:02:25.260
I mean, you want students to
get excited about learning.

00:02:25.260 --> 00:02:27.942
What better way than the LearnStorm

00:02:27.942 --> 00:02:30.027
to get them excited about learning.

00:02:30.027 --> 00:02:34.277
- [Whole Class] LearnStorm,
LearnStorm, LearnStorm!