Citizens United v. Federal Election Commission

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- [Sal] This is Sal here with Rick Hasen,

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who's a Professor of Law
at UC Irvine School of Law,

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specializing in election law.

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I'm here with Bradley
Smith, who's former Chairman

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of the Federal Election Commission.

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He's also a Professor of Law

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at Capital University in Columbus, Ohio,

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and runs the Center for
Competitive Politics.

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So, we're here to talk about

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what seems like a very
important Supreme Court case,

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Citizens United versus Federal
Election Commission in 2010.

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Could you give a little
bit of background on

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what got us to this
Supreme Court decision?

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- [Rick] Well, the federal government,

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specifically Congress, gets
to pass laws that regulate

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the financing of federal elections,

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and then, for state and local elections,

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that would be an issue for each state,

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and so, Congress, going
back more than a century,

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has passed laws limiting
the amount of money,

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kinds of money that can be raised,

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what has to be disclosed, they
passed a whole bunch of laws,

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and these laws are sometimes challenged

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as being unconstitutional.

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The usual challenge is
that the law violates

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part of the First Amendment,

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which protects freedom of
speech and association,

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and the specific issue in
the Citizens United case

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had to do with the First
Amendment rights of corporations

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to spend money to support
or oppose candidates

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for federal office.

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- [Bradley] Well, it is an important case,

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but it is a widely misunderstood case.

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The case, for example,
is not about the question

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of whether, quote,
corporations are people.

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We sometimes hear that.

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Not a single justice on the
court, in any of the opinions,

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addresses that issue, because
it's a longstanding doctrine

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of corporate personhood
that corporations have

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certain rights that people have

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when they join together to associate.

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The case is not about the question

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of whether money equals speech.

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The court has long held,

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not that money is speech, but
that you can't try to limit

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an activity which would be
protected under the Constitution,

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like speech, by limiting money.

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For example, you could not
say it shall be illegal

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to spend any money in the
United States to publish a book

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or to pay an author or to
operate a bookstore, right?

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- [Rick] So, if you were,
say, General Motors or Google,

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and you wanted to put
an ad online or on TV

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or in a newspaper that says

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support President Smith for reelection,

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that would be illegal if it was paid for

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with money from the corporation.

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Corporations could still participate,

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but they'd have to set
up a separate committee,

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called a political action
committee or a PAC,

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and they could only solicit certain people

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to put money into that pack,

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and they couldn't take
money that they made

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from selling cars or software
or something like that

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and use it for political purposes,

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and in Citizens United,
reversing some earlier cases,

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the Supreme Court said that corporations,

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like individuals, have
a First Amendment right

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to spend, independently,

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supporting or opposing
candidates for office,

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and that it would be
unconstitutional to limit this

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because it would be too
much of an infringement

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on the speech rights of corporations.

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- [Bradley] It's about
what rights people have

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to engage in speech when they have chosen

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to organize as a corporation,

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so the court makes the
holding that corporations

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and unions have constitutional protection

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under the First Amendment

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when they make independent expenditures

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to advocate for the election
or defeat of candidates,

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and by independent expenditures,

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we don't say that they
can contribute money

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directly to candidate campaigns,

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but they are free to spend their own money

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to communicate with the public at large,

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and say we think you should
vote for this candidate

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or against this candidate
for these reasons,

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and that, I think, in a nutshell,

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is what the case is about
at that highest level.

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- [Sal] So, just to get a
little bit more into the case,

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it's really all around this
Bipartisan Campaign Reform Act,

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the McCain-Feingold Act of 2002.

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What was all that about,
and then how did that lead,

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eventually, to this Supreme Court decision

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of Citizens United versus
Federal Election Commission?

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- [Bradley] Well, in some
ways, it goes back much further

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than the Bipartisan Campaign Reform Act.

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We should note that Congress, in 1907,

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passed the first federal
statute that limited

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corporate contributions
to campaigns, right?

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But corporations quickly found out, well,

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we can just do expenditures.

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We don't contribute to the campaign.

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We just spend our own money,
but within this framework,

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both corporations and unions
could do a lot of things.

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For example, they could run
ads that said things like,

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you know, Congressman Jones
is a no-good, anti worker,

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worker line person who wants
to destroy American industry.

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Call Congressman Jones and tell him

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we don't need his agenda in Washington,

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and that didn't count as
a campaign expenditure,

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because it didn't specifically urge people

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to vote for or against a candidate.

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- [Rick] So, President Bush signed a law,

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President George W.
Bush, signed a law called

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the Bipartisan Campaign Reform Act,

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and this was in 2002, I believe it was,

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and it's commonly known as
the McCain-Feingold Law,

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because those are the
two big Senate sponsors,

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and the National Rifle Association,

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Senator McConnell, a bunch of people sued

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and argued it was unconstitutional.

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While the case was going
through the courts,

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the National Rifle Association,
which definitely has

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a political point of view, set
up a satellite radio station

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called NRA News, and they said oh, well,

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if news organizations can
engage in this unlimited

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kind of electoral activity,

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we'll just make ourselves
a news organization,

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and many people thought this
was going to be a kind of

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sham way of just running
political ads all the time.

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Well, what happened was that the NRA News

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actually started having news
programs, opinion programs,

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call-in programs, not
just on the election,

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but on issues that would matter to people

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who were sympathetic to the NRA,

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and it actually became a
bonafide news organization.

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- [Bradley] So, the McCain-Feingold Law,

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or the Bipartisan Campaign Reform Act,

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as it's officially known,
attempted to close this alleged

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or perceived loophole by saying okay,

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a corporation or a union
can't even mention a candidate

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in a broadcast ad within 60
days of a general election,

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or within 30 days of a primary election.

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So, enter a group called Citizens United.

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- [Rick] Citizens United
is an ideological group

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that is doing political
activities, still around,

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and they had made a documentary
called Hillary: The Movie.

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This was coming out at the time,

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not of the past election that we just had,

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but back in 2008,

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in that period when Hillary Clinton

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and Barack Obama were
going to be competing

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for the Democratic
nomination to be President,

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in the 2008 period,

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and Citizens United is
a nonprofit corporation.

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That is, they're not selling shoes.

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They're just using the corporate form

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to be able to engage in
associational activity,

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and there had already been a rule,

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well before the Supreme
Court had established,

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which said if you're an
ideological corporation,

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and you don't take money
from for-profit companies,

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you can already spend
whatever money you have

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in your treasury on political
ads and political activity,

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but Citizens United
deliberately took money

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from for-profit corporations,
and they did it because

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they were trying to set up a test case.

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What they wanted to do was
to take this documentary

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they had made, Hillary: The Movie,

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which said very negative
things about Hillary Clinton,

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and they wanted to pay Comcast
Cable a million dollars

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to make this movie available
as a video on demand,

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which anyone could watch
by choosing it from,

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you know, their cable guide.

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They'd be able to watch it,

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and under the federal statute,

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the Federal Election
Commission said this looks like

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it's going to be on television,

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broadcast in a period
close to the election,

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and it's going to feature
the name Hillary Clinton,

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who's a candidate for President,

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and that would make it an election ad.

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It cannot be paid for by corporate funds.

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There were lots of ways
that the Supreme Court

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could have gone, aside from
overturning earlier cases,

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but the court was bold here,

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and on a five to four vote,

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the liberals on the court
against the conservatives,

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the conservatives won, and the

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longstanding federal
prohibition on corporations

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and labor unions as well spending money

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in federal elections fell.

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- [Sal] So, just as kind
of a lay citizen, you know,

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on the face of it, I understand why

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there's all of this
Congressional legislation.

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You're afraid of undue
influence from corporations,

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but you know, we were able
to talk to a few experts who,

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I think, took a more sympathetic view

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to the Supreme Court
ruling, and their point was

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Citizens United was this organization.

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It definitely had an
ideological point of view,

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and it was making this content

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that was clearly anti-Hillary Clinton,

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but there's many organizations,
media organizations,

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you know, I think we could
point to news channels,

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websites, which most people would consider

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to be news organizations
or media organizations,

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that do definitely lean one way
or the other, left or right,

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and they're arguing that the reason why

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the Supreme Court had to rule this way is

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if what Citizens United wasn't allowed,

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then whether you're talking
about an MSNBC or a FOX News,

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that that would start
putting them under scrutiny.

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- [Rick] Well, the law that
limited what corporations can do

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created an exemption
for press organizations

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that were engaged in
journalistic kinds of activities,

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and indeed, the majority
on the courts said

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we have to treat them the same.

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- [Bradley] The justices
expressed real concern about

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the nature of the government's argument.

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They said, you know, you're saying

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you can limit broadcast ads,
but what about internet?

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Could you limit communications
about candidates

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over the internet?

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And the government attorneys
eventually said yes, we can.

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We can limit anything that
involves a corporation

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speaking about a candidate for office,

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whether it's on the internet or whatever,

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and by the time the
oral argument was over,

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the government had asserted
that it would have the right,

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for example, to limit
the publication of a book

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that included one line of advocacy

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urging people to vote for a candidate

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at the very end of a long,
you know, 500 page book.

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They had suggested that
they could prohibit a union

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from publishing a pamphlet that advocated

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the election of a candidate.

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You could envision, you know, a candidate,

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something like why working Americans

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should support, you know,
Hillary Clinton for President,

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or something like that.

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They had suggested that

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books over Kindle could be
banned under existing law,

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and so, the Supreme Court said alright,

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this is very problematic to us.

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They scheduled it for
a second oral argument,

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heard the case again, and
at the end of the case,

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said we don't think that this is right,

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at least a majority did,

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said you cannot limit the
speech of a group of people

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or a body simply because
it is incorporated.

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- [Rick] But there was an
argument of the dissenters

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that said the press is different.

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The press serves an educational
and informative function,

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which is different than the
function of other corporations,

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and it's permissible, maybe in
part because the Constitution

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does provide separate
protection for the press.

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There was a big debate
about what that means,

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to carve this out.

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Now, that does create an inconsistency.

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We'd be saying we're
gonna treat Google one way

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and FOX News and The
New York Times another,

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but the current law is also inconsistent,

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so let me give you an
example from the other side,

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which is think is very hard for those

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who support Citizens United to answer.

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In a case that came a couple
of years after Citizens United,

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there was a guy who was
a Canadian law student,

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then he became a lawyer in New York.

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He went to Harvard Law
School, worked in New York,

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and he was here on a work
visa, and he wanted to,

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according to his complaint that he filed,

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he wanted to go to
Kinko's and make flyers,

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spend 50 cents making flyers,

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saying re-elect President Obama,

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and he wanted to hand those
flyers out in Central Park.

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Well, because he was not a citizen,

00:13:00.910 --> 00:13:04.150
even though he's in the United
States and has other kinds

00:13:04.150 --> 00:13:05.960
of First Amendment rights,
even though he's not a citizen,

00:13:05.960 --> 00:13:06.970
he would violate federal law.

00:13:06.970 --> 00:13:10.100
He could go to jail for
years, face a $10,000 fine,

00:13:10.100 --> 00:13:12.050
if he hands that piece
of paper to one person,

00:13:12.050 --> 00:13:15.110
because then he'd be engaging
in election-related activity

00:13:15.110 --> 00:13:18.100
as a non-citizen, and this case went up

00:13:18.100 --> 00:13:20.990
and the Supreme Court had
a chance to explain why,

00:13:20.990 --> 00:13:24.070
as in Citizens United, if
the identity of the speaker

00:13:24.070 --> 00:13:25.770
doesn't matter in the
First Amendment, really,

00:13:25.770 --> 00:13:29.616
it's always about more speech
all the time in elections,

00:13:29.616 --> 00:13:32.790
why is it that this guy, Benjamin
Blumen, could be shut up?

00:13:32.790 --> 00:13:36.440
And the Supreme Court
didn't even hear the case.

00:13:36.440 --> 00:13:39.450
It came up on a special kind of appeal

00:13:39.450 --> 00:13:41.980
from a three-judge
court, which means that,

00:13:41.980 --> 00:13:43.470
when the court declined to hear the case,

00:13:43.470 --> 00:13:46.340
it agreed that the lower court was right,

00:13:46.340 --> 00:13:48.490
that this law is constitutional.

00:13:48.490 --> 00:13:50.570
So, we now have this new anomaly,

00:13:50.570 --> 00:13:54.510
which is that we say
that non-human entities,

00:13:54.510 --> 00:13:58.010
these corporations that
only exist because the state

00:13:58.010 --> 00:14:00.970
allows it to create this fake thing,

00:14:00.970 --> 00:14:03.760
non-human entities have
these First Amendment rights,

00:14:03.760 --> 00:14:07.130
but yet a human being who
was living in New York

00:14:07.130 --> 00:14:10.150
who wanted to spend 50
cents, he couldn't do that,

00:14:10.150 --> 00:14:11.496
and so, what's the difference,

00:14:11.496 --> 00:14:14.070
if it's really true that the identity

00:14:14.070 --> 00:14:15.840
of the speaker doesn't matter,

00:14:15.840 --> 00:14:18.240
which the Supreme Court
said to Citizens United?

00:14:19.378 --> 00:14:20.700
Well, certainly, the identity
of the speaker matters

00:14:20.700 --> 00:14:22.840
when that speaker is a foreign individual,

00:14:22.840 --> 00:14:25.940
or a foreign government,
or a foreign corporation.

00:14:25.940 --> 00:14:27.960
- [Sal] So, it seems like
the majority opinion,

00:14:27.960 --> 00:14:31.330
they're not saying that large corporations

00:14:31.330 --> 00:14:35.790
being able to funnel tens or
hundreds of millions of dollars

00:14:35.790 --> 00:14:39.095
for or against a campaign,
independently of the candidate,

00:14:39.095 --> 00:14:41.790
that that won't necessarily corrupt,

00:14:41.790 --> 00:14:44.060
but it's more of by trying to regulate it

00:14:44.060 --> 00:14:46.600
in the way that the
McCain-Feingold Act does,

00:14:46.600 --> 00:14:49.480
it actually has far-reaching consequences,

00:14:49.480 --> 00:14:51.730
or trying to regulate it in the way that

00:14:51.730 --> 00:14:53.610
the Federal Election Commission
was trying to regulate it

00:14:53.610 --> 00:14:56.610
with Citizens United, it has
far-reaching consequences

00:14:56.610 --> 00:14:58.810
that touch on just general
free speech issues,

00:14:58.810 --> 00:15:00.300
like what you just talked about, you know,

00:15:00.300 --> 00:15:03.040
a sentence in a book that
happens to be published or sold

00:15:03.040 --> 00:15:06.970
during the election, or a movie
made by a random filmmaker.

00:15:06.970 --> 00:15:08.990
- [Bradley] I think, for
one thing, for example,

00:15:08.990 --> 00:15:10.336
a lot of people, again,

00:15:10.336 --> 00:15:12.740
they resent to say that
corporations aren't people,

00:15:12.740 --> 00:15:14.610
but they don't realize
corporations have always,

00:15:14.610 --> 00:15:17.750
again, had some of these
rights, acted as corporations,

00:15:17.750 --> 00:15:21.300
and acted in the political
arena in a variety of ways,

00:15:21.300 --> 00:15:23.390
and I think that what these
people should ask themselves is,

00:15:23.390 --> 00:15:26.740
well, if Exxon were not incorporated,

00:15:26.740 --> 00:15:29.480
would they go oh, well,
Exxon's not incorporated.

00:15:29.480 --> 00:15:30.313
Well, that's fine,

00:15:30.313 --> 00:15:32.320
let them spend all the
money they want, right?

00:15:32.320 --> 00:15:34.820
And conversely, I find
most of these people,

00:15:34.820 --> 00:15:37.594
if you say, well, should
we be able to censor

00:15:37.594 --> 00:15:38.830
The New York Times?

00:15:38.830 --> 00:15:39.980
It's a corporation?

00:15:39.980 --> 00:15:42.020
They say well, no, no, no, not that,

00:15:42.020 --> 00:15:45.140
and when you see that, I
think it makes people realize

00:15:45.140 --> 00:15:46.620
the question is not really whether

00:15:46.620 --> 00:15:49.240
it's a corporation speaking or not.

00:15:49.240 --> 00:15:52.550
The question is whether or not we think

00:15:52.550 --> 00:15:56.650
the First Amendment protects
the expenditure of money.

00:15:56.650 --> 00:15:58.470
- [Rick] We actually had to have rules

00:15:58.470 --> 00:16:01.000
before Citizens United to differentiate,

00:16:01.000 --> 00:16:04.010
because, actually, this was the law, that

00:16:05.010 --> 00:16:07.570
Google and General Motors
could not spend their funds

00:16:07.570 --> 00:16:09.210
directly on election speech,

00:16:09.210 --> 00:16:12.600
but The New York Times and FOX News could,

00:16:12.600 --> 00:16:13.900
and so we had tests,

00:16:13.900 --> 00:16:16.904
and the Federal Election
Commission had tests,

00:16:16.904 --> 00:16:18.470
and there are just some
disagreements about

00:16:18.470 --> 00:16:19.620
how to best do it.

00:16:19.620 --> 00:16:23.110
I would point to the work
of Professor Sonja West

00:16:23.110 --> 00:16:25.400
at the University of Georgia,

00:16:25.400 --> 00:16:26.650
who has written extensively on this,

00:16:26.650 --> 00:16:28.710
and she said the test should be people

00:16:28.710 --> 00:16:31.100
who are regularly engaged in journalism.

00:16:31.100 --> 00:16:33.080
That doesn't mean it has to be objective

00:16:33.080 --> 00:16:34.400
and you can't have a point of view.

00:16:34.400 --> 00:16:36.850
It doesn't mean there's a
problem with FOX News or MSNBC,

00:16:36.850 --> 00:16:38.970
because they might have a point of view,

00:16:38.970 --> 00:16:40.620
but it means you're doing something

00:16:40.620 --> 00:16:42.540
other than electioneering.

00:16:42.540 --> 00:16:45.088
- [Bradley] I do think that,
yes, that's what I'm saying,

00:16:45.088 --> 00:16:48.280
that these are some of the
costs of having a vibrant,

00:16:48.280 --> 00:16:49.830
robust First Amendment,

00:16:49.830 --> 00:16:52.670
and if you want to give
the government the power

00:16:52.670 --> 00:16:55.050
to regulate, you know, books to movies,

00:16:55.050 --> 00:16:58.110
you should think about who's
the politician you most hate?

00:16:58.110 --> 00:16:59.290
Is it Hillary Clinton?

00:16:59.290 --> 00:17:00.480
Is it Donald Trump?

00:17:00.480 --> 00:17:02.150
Is it somebody else?

00:17:02.150 --> 00:17:04.880
Do you want them in charge of deciding

00:17:04.880 --> 00:17:07.450
who gets to speak out in
politics and in what way

00:17:07.450 --> 00:17:10.038
and in what fashion and what manner?

00:17:10.038 --> 00:17:12.930
And I think you should think
about it in those terms,

00:17:12.930 --> 00:17:15.360
and that sometimes makes people realize

00:17:15.360 --> 00:17:19.710
maybe it's best to let things go.

00:17:19.710 --> 00:17:22.040
Now, I would say,
however, having said that,

00:17:22.040 --> 00:17:24.320
that a lot of the arguments
about corporate influence

00:17:24.320 --> 00:17:26.560
are vastly overblown.

00:17:26.560 --> 00:17:28.560
There's quite a bit of debate, really,

00:17:28.560 --> 00:17:30.310
about whether or not
corporations, in fact,

00:17:30.310 --> 00:17:34.607
benefit from spending money on politics.

00:17:34.607 --> 00:17:37.970
Big corporations, you know,
the Fortune 500 types,

00:17:37.970 --> 00:17:40.600
what they really like to do
is spend money on lobbying.

00:17:40.600 --> 00:17:42.520
That's much more direct.

00:17:42.520 --> 00:17:44.450
They can talk to candidates directly.

00:17:44.450 --> 00:17:45.700
They can let other people elect folks.

00:17:45.700 --> 00:17:47.350
There's almost certain
to be some folks elected

00:17:47.350 --> 00:17:49.700
who they will like and they
can go work with those people

00:17:49.700 --> 00:17:51.930
and craft messages and bills,

00:17:51.930 --> 00:17:53.510
and they can do this
more behind the scenes,

00:17:53.510 --> 00:17:55.512
and that way, they don't have to offend

00:17:55.512 --> 00:17:56.770
any of their customers who don't like

00:17:56.770 --> 00:17:58.470
the party that they're supporting,

00:17:58.470 --> 00:18:00.680
but it's also clear
that just spending money

00:18:00.680 --> 00:18:02.558
doesn't automatically win the elections.

00:18:02.558 --> 00:18:04.560
You can look at the last
presidential election.

00:18:04.560 --> 00:18:06.450
I mean, Donald Trump spent, all totaled,

00:18:06.450 --> 00:18:09.360
about a quarter of what the
Hillary Clinton forces spent.

00:18:09.360 --> 00:18:10.690
It certainly helps to have money,

00:18:10.690 --> 00:18:13.960
but in the end, it's not dispositive.

00:18:13.960 --> 00:18:17.770
What the money does is it's
spent to try to inform voters,

00:18:17.770 --> 00:18:20.710
and those voters go out
and vote at the polls.

00:18:20.710 --> 00:18:22.880
- [Sal] So we've learned
that, in Citizens United,

00:18:22.880 --> 00:18:25.070
the Supreme Court ruled
that political spending

00:18:25.070 --> 00:18:28.430
by corporations,
associations, and labor unions

00:18:28.430 --> 00:18:32.880
is a form of protected speech
under the First Amendment,

00:18:32.880 --> 00:18:36.790
but whether or how the
government can limit free speech

00:18:36.790 --> 00:18:40.480
in campaign contributions
is still a matter of debate.

00:18:40.480 --> 00:18:43.640
As Rick Hasen notes, one
test for the difference

00:18:43.640 --> 00:18:46.560
between the free speech
rights accorded to journalists

00:18:46.560 --> 00:18:50.410
and those or corporations might
be whether they are entities

00:18:50.410 --> 00:18:52.690
regularly engaged in journalism,

00:18:52.690 --> 00:18:54.600
but as Bradley Smith points out,

00:18:54.600 --> 00:18:56.720
allowing the government to impose limits

00:18:56.720 --> 00:19:00.000
on how people or corporations
speak out in politics

00:19:00.000 --> 00:19:02.990
may lead to unintended consequences.

00:19:02.990 --> 00:19:05.450
To learn more about Citizens United,

00:19:05.450 --> 00:19:07.720
visit the National Constitution Center's

00:19:07.720 --> 00:19:11.130
Interactive Constitution,
and Khan Academy's resources

00:19:11.130 --> 00:19:13.953
on U.S. government and politics.


Introduction to genetic engineering

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:03.940
- [Instructor] The idea
of genetic engineering

00:00:03.940 --> 00:00:07.530
is something that we associate
with the 20th century.

00:00:07.530 --> 00:00:08.910
We didn't even know

00:00:08.910 --> 00:00:12.660
that genes were actually
the mechanism of heredity

00:00:12.660 --> 00:00:15.590
until the middle of the 20th century.

00:00:15.590 --> 00:00:19.620
And the direct modification
of genes for some purpose

00:00:19.620 --> 00:00:23.180
really didn't even start
happening until the 1970s.

00:00:23.180 --> 00:00:25.580
But it's worth noting that human beings

00:00:25.580 --> 00:00:29.090
have been, in some ways,
influencing the genetics

00:00:29.090 --> 00:00:33.830
of organisms for a very,
very, very long time.

00:00:33.830 --> 00:00:37.940
For example, in nature you have
things that look like this.

00:00:37.940 --> 00:00:39.740
We call them wolves.

00:00:39.740 --> 00:00:41.350
But today in our lives,

00:00:41.350 --> 00:00:44.440
we also have things that look like this.

00:00:44.440 --> 00:00:47.890
Now these dogs did not
come about naturally.

00:00:47.890 --> 00:00:52.430
They came about from many
generations of breeding,

00:00:52.430 --> 00:00:54.690
of human beings taking things

00:00:54.690 --> 00:00:57.400
that started off looking
like wolves or foxes,

00:00:57.400 --> 00:00:59.740
and selecting for certain traits.

00:00:59.740 --> 00:01:01.420
They might have selected for traits

00:01:01.420 --> 00:01:04.020
that maybe they're small, and
they look more like puppies,

00:01:04.020 --> 00:01:05.540
even when they're full grown.

00:01:05.540 --> 00:01:07.440
Or traits that they are more docile.

00:01:07.440 --> 00:01:08.560
They're more likely to listen

00:01:08.560 --> 00:01:10.800
to what a human being says to do.

00:01:10.800 --> 00:01:13.210
Or traits that they're
good at killing rodents,

00:01:13.210 --> 00:01:14.730
or whatever else it might be,

00:01:14.730 --> 00:01:18.410
and over time, repeated
selection of those traits

00:01:18.410 --> 00:01:22.450
led to what we see as these
different breeds of dogs.

00:01:22.450 --> 00:01:24.560
So even though human
beings were doing this

00:01:24.560 --> 00:01:26.860
for hundreds and thousands of years,

00:01:26.860 --> 00:01:30.230
they were influencing
what DNA gets passed on

00:01:30.230 --> 00:01:32.290
from one generation to another.

00:01:32.290 --> 00:01:35.220
They didn't know about
the actual genetic code,

00:01:35.220 --> 00:01:39.400
but it was a form of genetic
manipulation nonetheless.

00:01:39.400 --> 00:01:43.310
Similarly, if you look
into the plant kingdom,

00:01:43.310 --> 00:01:46.560
when you go to the store and
you see that sweet apple,

00:01:46.560 --> 00:01:49.080
things like that might not have existed

00:01:49.080 --> 00:01:50.960
in the form that you see them today.

00:01:50.960 --> 00:01:54.160
It is very likely, in fact
most agricultural products,

00:01:54.160 --> 00:01:56.190
people might have found wild apples,

00:01:56.190 --> 00:01:58.610
and we could be talking
thousands of years ago.

00:01:58.610 --> 00:02:01.350
They might have found wild apple trees.

00:02:01.350 --> 00:02:04.060
Let me draw a quick apple tree here.

00:02:04.060 --> 00:02:05.124
And they might've found

00:02:05.124 --> 00:02:07.740
that the apples were a little bit sour,

00:02:07.740 --> 00:02:11.140
and small, and hard to
eat, and hard to digest,

00:02:11.140 --> 00:02:14.190
but over time, people selected the trees

00:02:14.190 --> 00:02:17.070
that had sweeter apples,
that had larger apples,

00:02:17.070 --> 00:02:18.020
and made the conditions

00:02:18.020 --> 00:02:20.360
so that they were more
likely to reproduce.

00:02:20.360 --> 00:02:23.130
So over time, you got
larger and sweeter apples

00:02:23.130 --> 00:02:25.670
like the type that you see at your store.

00:02:25.670 --> 00:02:26.503
And as I mentioned,

00:02:26.503 --> 00:02:29.000
most agricultural products
that we have today

00:02:29.000 --> 00:02:32.360
are the result of hundreds
or thousands of years

00:02:32.360 --> 00:02:34.430
of this type of breeding.

00:02:34.430 --> 00:02:37.340
So even though that might
not be the formal definition

00:02:37.340 --> 00:02:39.760
of genetic engineering,
which typically is referring

00:02:39.760 --> 00:02:41.670
to something like gene modification,

00:02:41.670 --> 00:02:44.190
which we've only been able to
do in the last few decades,

00:02:44.190 --> 00:02:46.750
human beings have been
doing some type of influence

00:02:46.750 --> 00:02:51.070
on organisms' DNA for
a very, very long time.

00:02:51.070 --> 00:02:53.880
But with that said, in
the last few decades,

00:02:53.880 --> 00:02:57.540
we've been able to become
much, much more precise

00:02:57.540 --> 00:03:01.920
with influencing DNA
through genetic engineering.

00:03:01.920 --> 00:03:02.753
In other videos,

00:03:02.753 --> 00:03:05.720
we'll go more detail
about how that is done,

00:03:05.720 --> 00:03:08.953
but you have this idea of recombinant DNA,

00:03:12.680 --> 00:03:14.950
where you could take
genes from one organism

00:03:14.950 --> 00:03:16.800
and put them in another organism.

00:03:16.800 --> 00:03:19.210
And you might say, why is this useful?

00:03:19.210 --> 00:03:21.700
Well let's say there is
a tree you wanna grow.

00:03:21.700 --> 00:03:23.200
Let's say it's an apple tree,

00:03:23.200 --> 00:03:25.900
but it's very susceptible to
a certain type of disease,

00:03:25.900 --> 00:03:28.400
and if that disease hits,
you lose all of your crop.

00:03:28.400 --> 00:03:32.910
But what if you could insert
into the DNA of that apple tree

00:03:32.910 --> 00:03:37.910
maybe a gene that makes it
more resistant to that disease?

00:03:38.170 --> 00:03:40.790
And this is what people actually do today.

00:03:40.790 --> 00:03:45.210
So they will insert DNA,

00:03:45.210 --> 00:03:47.500
and it could be from
some pretty wild things.

00:03:47.500 --> 00:03:51.250
I've read stories about
inserting insect DNA into a plant

00:03:51.250 --> 00:03:54.650
so that it will be more
robust in some way or another.

00:03:54.650 --> 00:03:58.070
Now this idea of recombinant DNA

00:03:58.070 --> 00:04:00.420
and genetically modifying food,

00:04:00.420 --> 00:04:04.640
this is often known as a
genetically modified organism,

00:04:04.640 --> 00:04:06.550
this is somewhat controversial.

00:04:06.550 --> 00:04:08.480
Many people say, hey this is good.

00:04:08.480 --> 00:04:11.510
It allows us to produce more robust foods.

00:04:11.510 --> 00:04:14.070
In fact, part of this recombinant DNA,

00:04:14.070 --> 00:04:16.380
inserting DNA into something else,

00:04:16.380 --> 00:04:17.840
it might make it more nutritious.

00:04:17.840 --> 00:04:20.290
It might provide for more vitamins.

00:04:20.290 --> 00:04:21.530
But other people would argue that,

00:04:21.530 --> 00:04:24.450
hey we don't know exactly
what all the repercussions

00:04:24.450 --> 00:04:26.280
of what we're doing will happen.

00:04:26.280 --> 00:04:27.530
We might think it's helpful,

00:04:27.530 --> 00:04:30.150
but when you're taking
DNA from one organism

00:04:30.150 --> 00:04:31.550
and putting it into another,

00:04:31.550 --> 00:04:33.070
how does that affect the nutrition,

00:04:33.070 --> 00:04:36.370
or the long-lasting effects
of eating that over time?

00:04:36.370 --> 00:04:38.750
So this is something
for you to think about.

00:04:38.750 --> 00:04:40.110
And we're starting to go,

00:04:40.110 --> 00:04:41.890
when we start asking these questions,

00:04:41.890 --> 00:04:44.520
into the field of bioethics.

00:04:44.520 --> 00:04:47.660
As we have more and more
control over the genome,

00:04:47.660 --> 00:04:50.140
and especially, as we'll
see, the human genome,

00:04:50.140 --> 00:04:51.610
there's questions that we have to ask

00:04:51.610 --> 00:04:54.520
about is it good, or is it bad?

00:04:54.520 --> 00:04:57.520
But going back to this
idea of genetic engineering

00:04:57.520 --> 00:04:59.410
and recombinant DNA,

00:04:59.410 --> 00:05:01.890
other things that you
could do is you could,

00:05:01.890 --> 00:05:05.800
let's say that we need to
produce insulin for diabetics.

00:05:05.800 --> 00:05:09.010
Well, maybe you can take a bacteria cell,

00:05:09.010 --> 00:05:11.650
and insert into the bacteria cell

00:05:11.650 --> 00:05:14.380
the gene that helps produce for insulin.

00:05:14.380 --> 00:05:17.210
And then, all of a sudden,
that bacteria cell can become

00:05:17.210 --> 00:05:19.950
an human-insulin-producing factory,

00:05:19.950 --> 00:05:22.500
so that we could have more
insulin for diabetics,

00:05:22.500 --> 00:05:24.790
that this insulin could then be harvested.

00:05:24.790 --> 00:05:27.770
That is a use of recombinant DNA.

00:05:27.770 --> 00:05:29.500
And so we really are going into

00:05:29.500 --> 00:05:32.070
an interesting period in humanity.

00:05:32.070 --> 00:05:35.490
For many thousands of years
we were breeding things,

00:05:35.490 --> 00:05:37.740
but now we're learning
to manipulate things

00:05:37.740 --> 00:05:40.260
at a very fine-grain level.

00:05:40.260 --> 00:05:43.070
And it makes us ask
all sorts of questions.

00:05:43.070 --> 00:05:45.740
There are likely to be some
very good things we can do.

00:05:45.740 --> 00:05:47.100
Produce new medicines.

00:05:47.100 --> 00:05:49.610
Produce more robust crops.

00:05:49.610 --> 00:05:51.320
But there are also questions about

00:05:51.320 --> 00:05:52.580
what are the side effects?

00:05:52.580 --> 00:05:55.820
And the bioethics get really
interesting when we start

00:05:55.820 --> 00:05:58.900
thinking about modifying
the human genome itself.

00:05:58.900 --> 00:06:00.240
Because when you're modifying genetics,

00:06:00.240 --> 00:06:02.080
you're not just modifying one organism,

00:06:02.080 --> 00:06:04.200
you're modifying all of their offspring.

00:06:04.200 --> 00:06:07.080
You're modifying the genes
that exist in the gene pool.

00:06:07.080 --> 00:06:08.880
And so some very good arguments

00:06:08.880 --> 00:06:10.700
for modifying the human genome are

00:06:10.700 --> 00:06:13.810
correcting diseases, some things
that could be very serious.

00:06:13.810 --> 00:06:15.250
But over time, things might,

00:06:15.250 --> 00:06:17.024
people might say, hey I
wanna be a little bit taller,

00:06:17.024 --> 00:06:19.770
or I want a certain hair color,
or I want straighter teeth.

00:06:19.770 --> 00:06:20.603
Or they might say,

00:06:20.603 --> 00:06:23.730
I want a child who has
this trait or that trait.

00:06:23.730 --> 00:06:25.860
So things get very, very interesting

00:06:25.860 --> 00:06:30.553
the more we think about how
genetic engineering may be used.


Introduction to experimental design

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.860
- [Instructor] What we are
going to do in this video

00:00:02.860 --> 00:00:06.620
is talk a little bit about
experiments in science

00:00:06.620 --> 00:00:09.060
and experiments are really the heart

00:00:09.060 --> 00:00:12.050
of all scientific progress.

00:00:12.050 --> 00:00:13.080
If you think about,

00:00:13.080 --> 00:00:16.540
let's just say this represents
just baseline knowledge

00:00:16.540 --> 00:00:18.560
and then people have hunches in the world

00:00:18.560 --> 00:00:19.810
and for a lot of times people say, hey,

00:00:19.810 --> 00:00:22.230
I have a hunch that that
thing is good for you

00:00:22.230 --> 00:00:24.120
or that thing is good for you

00:00:24.120 --> 00:00:26.820
but they really had no way of measuring

00:00:26.820 --> 00:00:28.233
how confident they were.

00:00:28.233 --> 00:00:30.560
They really had no good way of proving it

00:00:30.560 --> 00:00:32.930
and even more, because they
had no good way of proving it,

00:00:32.930 --> 00:00:37.110
it was hard for people to
build on top of that knowledge

00:00:37.110 --> 00:00:41.070
but with the scientific
method and experiments,

00:00:41.070 --> 00:00:44.620
people were able to say, hey,
we have a hypothesis here

00:00:44.620 --> 00:00:47.150
and we were able to do some
well-designed experiments

00:00:47.150 --> 00:00:50.020
and so we feel pretty
good that this is true

00:00:50.020 --> 00:00:51.640
and then future people are going to say,

00:00:51.640 --> 00:00:54.190
hey, since we feel pretty
good that this is true,

00:00:54.190 --> 00:00:57.170
maybe we can design experiment
to see whether that is true.

00:00:57.170 --> 00:01:00.510
Hey, that actually is true and
then they can build on that

00:01:00.510 --> 00:01:03.630
and we end up having scientific progress

00:01:03.630 --> 00:01:06.460
that can accumulate over hundreds of years

00:01:06.460 --> 00:01:08.260
and this is really important

00:01:08.260 --> 00:01:09.930
that the experiments are well-designed

00:01:09.930 --> 00:01:12.470
because in the future and
this happens all the time,

00:01:12.470 --> 00:01:14.810
we might realize that hey,
actually there was a little,

00:01:14.810 --> 00:01:17.880
a few assumptions baked in
here that weren't accurate

00:01:17.880 --> 00:01:22.170
that allowed us to make
essentially misleading conclusions.

00:01:22.170 --> 00:01:24.180
So our conclusion wasn't quite right there

00:01:24.180 --> 00:01:26.910
and then we will have to
rebuild from that point

00:01:26.910 --> 00:01:30.230
in order to make sure that
we are truly making progress.

00:01:30.230 --> 00:01:31.340
So the key question is

00:01:31.340 --> 00:01:34.440
how do we set up well-designed experiments

00:01:34.440 --> 00:01:36.020
and it's a whole field of study

00:01:36.020 --> 00:01:37.440
but the whole purpose of this video

00:01:37.440 --> 00:01:39.450
is to really give an introduction to it.

00:01:39.450 --> 00:01:43.370
So let's just start with a hypothesis.

00:01:43.370 --> 00:01:45.860
Let's say that you have a hypothesis

00:01:45.860 --> 00:01:50.860
that some pill that is made up
of the petals of some flower,

00:01:52.010 --> 00:01:55.210
that this pill right
over here, it improves,

00:01:55.210 --> 00:01:59.663
it improves running, running speed.

00:02:01.640 --> 00:02:05.360
It improves running speed
if someone were to take it.

00:02:05.360 --> 00:02:07.720
So the important thing of any hypothesis,

00:02:07.720 --> 00:02:09.800
it has to be testable

00:02:09.800 --> 00:02:11.160
and so what you do is you have to think

00:02:11.160 --> 00:02:13.320
well, how am I going to test it?

00:02:13.320 --> 00:02:14.570
Well, what you can do,

00:02:14.570 --> 00:02:17.660
so how are you going to
test your hypothesis?

00:02:17.660 --> 00:02:19.840
At first, you might say, give the pill,

00:02:19.840 --> 00:02:24.840
so give the pill to some runners,

00:02:26.120 --> 00:02:29.330
to some runners and test their time,

00:02:29.330 --> 00:02:31.400
test their 100 meter time,

00:02:31.400 --> 00:02:36.400
test their 100 meter time before the pill,

00:02:39.370 --> 00:02:44.320
before the pill and after

00:02:44.320 --> 00:02:46.830
and you might say, hey,
maybe if, I don't know,

00:02:46.830 --> 00:02:50.990
their times improve after,
maybe my hypothesis is correct.

00:02:50.990 --> 00:02:53.200
Pause this video and see
if you feel comfortable

00:02:53.200 --> 00:02:57.580
with this test right over
here, this experiment.

00:02:57.580 --> 00:02:59.490
Well, actually, there's several problems

00:02:59.490 --> 00:03:00.750
with this experiment.

00:03:00.750 --> 00:03:04.230
How are you selecting these runners

00:03:04.230 --> 00:03:07.440
and if you give them the pill
and their speed improves,

00:03:07.440 --> 00:03:09.870
did it truly improve because of the pill

00:03:09.870 --> 00:03:11.780
or did it improve because
of some other thing

00:03:11.780 --> 00:03:12.613
that they are doing?

00:03:12.613 --> 00:03:13.850
Maybe they got new shoes

00:03:13.850 --> 00:03:16.130
or maybe their diet improved in some way

00:03:16.130 --> 00:03:18.890
or maybe they just had a
psychological improvement.

00:03:18.890 --> 00:03:21.240
This is often known as the placebo effect.

00:03:21.240 --> 00:03:22.340
If people are taking something

00:03:22.340 --> 00:03:26.610
that they think will help
them, it often will help them

00:03:26.610 --> 00:03:29.650
even if that thing is
just an empty capsule

00:03:29.650 --> 00:03:31.440
or just a sugar tablet.

00:03:31.440 --> 00:03:34.750
So how do you avoid these types of errors?

00:03:34.750 --> 00:03:37.590
Well, what you could do
is you can find runners

00:03:37.590 --> 00:03:40.030
and put them into two groups.

00:03:40.030 --> 00:03:43.030
So let's say this is one
group right over here

00:03:43.030 --> 00:03:46.050
and then this is another group
and what you would wanna do

00:03:46.050 --> 00:03:50.000
is you'd wanna go into
the population of people

00:03:50.960 --> 00:03:53.930
and you would want to randomly select

00:03:53.930 --> 00:03:57.660
whether someone goes into
one group or another group.

00:03:57.660 --> 00:03:58.830
Why random?

00:03:58.830 --> 00:04:01.120
Because if you don't randomly select,

00:04:01.120 --> 00:04:03.690
there's a chance that there
might be some implicit bias

00:04:03.690 --> 00:04:06.360
that you might just happen
to be picking people

00:04:06.360 --> 00:04:10.640
who maybe their running speed
is on an upward trajectory

00:04:10.640 --> 00:04:12.570
and they just happen to go into the group

00:04:12.570 --> 00:04:15.060
that will eventually get the pill.

00:04:15.060 --> 00:04:19.770
So you randomly, randomly
put them in those groups

00:04:19.770 --> 00:04:22.250
and what you wanna do is
you'll have a control group

00:04:22.250 --> 00:04:25.240
and you'll have a group
that gets your pill

00:04:25.240 --> 00:04:30.240
and so this group gets
the pill, gets the pill

00:04:30.300 --> 00:04:33.540
and now you might be tempted
for this group to say,

00:04:33.540 --> 00:04:37.090
oh, they don't get a pill and
then after a few months of it

00:04:37.090 --> 00:04:38.950
and it should be the same amount of time,

00:04:38.950 --> 00:04:41.830
you say, hey, did this
group's times improve

00:04:41.830 --> 00:04:42.880
over the 100 meters?

00:04:42.880 --> 00:04:44.890
How did that compare to this group?

00:04:44.890 --> 00:04:46.200
But be very careful.

00:04:46.200 --> 00:04:49.480
If this group gets the pill
and this group gets nothing

00:04:49.480 --> 00:04:52.940
then the pill might be providing
that placebo effect again

00:04:52.940 --> 00:04:54.860
just making people think
they're getting something

00:04:54.860 --> 00:04:55.700
that's making 'em faster.

00:04:55.700 --> 00:04:58.520
It might actually be a
self-fulfilling prophecy.

00:04:58.520 --> 00:04:59.920
So it's actually important

00:04:59.920 --> 00:05:03.290
that you also give these people a pill

00:05:03.290 --> 00:05:05.270
although this pill would
just look like a pill

00:05:05.270 --> 00:05:09.980
so this would be just an
empty, empty, empty pill

00:05:11.030 --> 00:05:12.480
that looks the same.

00:05:12.480 --> 00:05:14.050
Now, there's another idea

00:05:14.050 --> 00:05:16.240
when you're designing
scientific experiments

00:05:16.240 --> 00:05:18.530
that it needs to be double blind.

00:05:18.530 --> 00:05:20.440
Let me write this down.

00:05:20.440 --> 00:05:22.680
Double blind.

00:05:22.680 --> 00:05:23.950
So as you could imagine,

00:05:23.950 --> 00:05:25.970
it implies that two things are blind here.

00:05:25.970 --> 00:05:27.520
So the first thing that needs to be blind

00:05:27.520 --> 00:05:29.380
is the people themselves

00:05:29.380 --> 00:05:32.110
should not know which group
they're getting put into.

00:05:32.110 --> 00:05:34.370
They should not know
which pill they are taking

00:05:34.370 --> 00:05:36.640
because obviously, if you
put someone in this pill,

00:05:36.640 --> 00:05:38.390
in this group and you say, hey,
you're in the control group,

00:05:38.390 --> 00:05:40.210
we're just gonna give you an empty pill,

00:05:40.210 --> 00:05:42.810
well, then the placebo
effect might not be,

00:05:42.810 --> 00:05:44.300
it might not work.

00:05:44.300 --> 00:05:47.150
It's also important
'cause it's double blind

00:05:47.150 --> 00:05:49.100
that the people who are
working with the runners,

00:05:49.100 --> 00:05:50.330
the people who are measuring them

00:05:50.330 --> 00:05:52.030
so that the researcher is right over here

00:05:52.030 --> 00:05:54.070
so I'll draw someone with a clipboard.

00:05:54.070 --> 00:05:57.140
So the researchers who
are observing these people

00:05:57.140 --> 00:05:58.500
and maybe administering the pill

00:05:58.500 --> 00:06:00.100
and telling them about the experiment

00:06:00.100 --> 00:06:01.940
that they too do not know

00:06:01.940 --> 00:06:03.860
which group they are administering it to

00:06:03.860 --> 00:06:07.020
because if they did, they might
be able to signal somehow.

00:06:07.020 --> 00:06:09.730
They might be able to even subconsciously

00:06:09.730 --> 00:06:12.970
give a sense of which group folks are in

00:06:12.970 --> 00:06:14.890
and so let's say we do all of these things

00:06:14.890 --> 00:06:16.190
and so we're getting in the direction

00:06:16.190 --> 00:06:17.920
of a well-designed experiment

00:06:17.920 --> 00:06:22.920
and we find that the
10 people in this group

00:06:23.140 --> 00:06:26.900
versus the 10 people in this
group after three months,

00:06:26.900 --> 00:06:31.130
these folks had a 5%
improvement in running speed

00:06:31.130 --> 00:06:35.260
and these people had a 10%
improvement in running speed.

00:06:35.260 --> 00:06:39.650
Is that enough to conclude
that our hypothesis is correct?

00:06:39.650 --> 00:06:42.660
Well, you might be tempted,
it seems suggestive

00:06:42.660 --> 00:06:44.910
but that's where
statistics come into order

00:06:44.910 --> 00:06:48.110
because there's just some random
chance that you got lucky,

00:06:48.110 --> 00:06:49.820
that you happened to pick the people that,

00:06:49.820 --> 00:06:50.890
your pill does nothing

00:06:50.890 --> 00:06:52.660
but you just happened to pick the people

00:06:52.660 --> 00:06:54.510
who are going to improve more

00:06:54.510 --> 00:06:57.160
and there's a whole field
of inferential statistics

00:06:57.160 --> 00:06:58.640
when you take a statistics course

00:06:58.640 --> 00:07:01.260
that will go in more depth into this

00:07:01.260 --> 00:07:02.880
but essentially, what you're
gonna do is you're gonna say,

00:07:02.880 --> 00:07:05.320
hey, assume that your pill does nothing,

00:07:05.320 --> 00:07:10.290
what's the probability of
getting this result for 10 people

00:07:10.290 --> 00:07:11.360
or what's the probability

00:07:11.360 --> 00:07:13.740
of getting this difference in result

00:07:13.740 --> 00:07:16.890
and if that probability is very low,

00:07:16.890 --> 00:07:18.900
well, you say, hey, that would suggest

00:07:18.900 --> 00:07:22.670
that my pill actually does do something.

00:07:22.670 --> 00:07:27.670
Now, another important principle
of an experiment like this

00:07:27.910 --> 00:07:31.460
is it needs to be replicable, replicable

00:07:32.380 --> 00:07:35.040
because even though you
thought you did a good job,

00:07:35.040 --> 00:07:36.620
people might not wanna
take your word for it

00:07:36.620 --> 00:07:39.750
and it's important in science
for people to be skeptical.

00:07:39.750 --> 00:07:42.650
When people do experiments,
they want to have a result

00:07:42.650 --> 00:07:44.880
and that bias might creep in

00:07:44.880 --> 00:07:46.920
and so if someone else does an experiment,

00:07:46.920 --> 00:07:48.640
you need to say how
you did that experiment

00:07:48.640 --> 00:07:52.060
so other people can see if
they get the same results

00:07:52.060 --> 00:07:54.360
because even though you
think you randomly selected,

00:07:54.360 --> 00:07:56.830
you might only do it with
people from a certain country

00:07:56.830 --> 00:07:58.780
or under certain weather conditions

00:07:58.780 --> 00:08:00.900
or assuming other constraints

00:08:00.900 --> 00:08:03.620
and then these people might
do it slightly differently

00:08:03.620 --> 00:08:06.410
or in a different country or
under different constraints

00:08:06.410 --> 00:08:08.660
and realize that hey,
the explanation for this

00:08:08.660 --> 00:08:11.020
maybe was something else.

00:08:11.020 --> 00:08:12.350
Another thing to keep in mind

00:08:12.350 --> 00:08:15.070
is the larger your
samples right over here,

00:08:15.070 --> 00:08:18.080
the larger groups that
you're able to do this with,

00:08:18.080 --> 00:08:22.330
the stronger that the
statistics actually become

00:08:22.330 --> 00:08:23.840
and I would say not just larger

00:08:23.840 --> 00:08:25.970
but the more diverse across genders,

00:08:25.970 --> 00:08:28.880
across ethnicities, across geographies.

00:08:28.880 --> 00:08:33.840
So the big picture here
is all scientific progress

00:08:33.840 --> 00:08:37.180
is based on us designing good experiments

00:08:37.180 --> 00:08:38.670
and being very rigorous

00:08:38.670 --> 00:08:41.040
about how we think about those experiments

00:08:41.040 --> 00:08:43.820
and what I've highlighted
here is just the beginning

00:08:43.820 --> 00:08:46.780
of how we might think about
designing those experiments

00:08:46.780 --> 00:08:49.520
and as you go into your
scientific careers,

00:08:49.520 --> 00:08:51.620
look at other people's experiments

00:08:51.620 --> 00:08:53.870
and see whether they've done these things

00:08:53.870 --> 00:08:55.600
because many times you will find

00:08:55.600 --> 00:08:58.723
that it is not as rigorous
as it might seem at first.


Negative definite integrals

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:02.040
- [Instructor] We've already thought about

00:00:02.040 --> 00:00:03.600
what a definite integral means.

00:00:03.600 --> 00:00:06.970
If I'm taking the definite
integral from a to b

00:00:06.970 --> 00:00:09.550
of f of x dx,

00:00:09.550 --> 00:00:13.190
I can just view that as the area

00:00:13.190 --> 00:00:15.420
below my function f.

00:00:15.420 --> 00:00:19.190
So if this is my y-axis,
this is my x-axis,

00:00:19.190 --> 00:00:22.150
and y is equal to f of x,
so something like that.

00:00:22.150 --> 00:00:24.980
Y is equal to f of x.

00:00:24.980 --> 00:00:26.570
And if this is a

00:00:26.570 --> 00:00:28.420
and if this is b,

00:00:28.420 --> 00:00:31.060
I could just view

00:00:31.060 --> 00:00:34.990
this expression as being
equal to this area.

00:00:35.980 --> 00:00:39.910
But what if my function
was not above the x-axis?

00:00:39.910 --> 00:00:42.260
What if it was below the x-axis?

00:00:42.260 --> 00:00:44.380
So these are going to be equivalent.

00:00:44.380 --> 00:00:47.320
Let's say, let me just draw that scenario.

00:00:47.320 --> 00:00:51.270
So let me draw a scenario where

00:00:51.270 --> 00:00:53.000
that's my x-axis,

00:00:53.000 --> 00:00:55.030
that is my y-axis.

00:00:55.030 --> 00:00:57.120
And let's say I have,

00:00:58.000 --> 00:01:01.150
let's say I have a function
that looks like that.

00:01:01.150 --> 00:01:04.740
So that is y is equal to g of x.

00:01:04.740 --> 00:01:08.360
And let's say that this
right over here is a,

00:01:08.360 --> 00:01:10.490
and this right over here is b.

00:01:10.490 --> 00:01:12.970
And let's say that this area

00:01:12.970 --> 00:01:16.320
right over here is equal to five.

00:01:17.290 --> 00:01:18.270
Well, if I were to ask you

00:01:18.270 --> 00:01:22.570
what is the definite integral from a to b

00:01:22.570 --> 00:01:27.570
of g of x dx, what do you
think it is going to be?

00:01:28.010 --> 00:01:29.730
Well, you might be
tempted to say, hey, well,

00:01:29.730 --> 00:01:32.710
it's just the area again
between my curve and the x-axis.

00:01:32.710 --> 00:01:34.640
You might be tempted to say, hey,

00:01:34.640 --> 00:01:36.750
this is just going to be equal to five.

00:01:36.750 --> 00:01:38.830
But you have to be very careful.

00:01:38.830 --> 00:01:42.720
Because if you're looking
at the area above your curve

00:01:42.720 --> 00:01:44.400
and below your x-axis,

00:01:44.400 --> 00:01:47.780
versus below your curve
and above the x-axis,

00:01:47.780 --> 00:01:49.620
this definite integral
is actually going to be

00:01:49.620 --> 00:01:53.030
the negative of the area.

00:01:53.030 --> 00:01:56.380
Now, we'll see later on why
this will work out nicely

00:01:56.380 --> 00:01:59.520
with a whole set of
integration properties.

00:01:59.520 --> 00:02:02.230
But if you want to get
some intuition for it,

00:02:02.230 --> 00:02:05.490
let's just think about
velocity versus time graphs.

00:02:05.490 --> 00:02:06.500
So,

00:02:06.500 --> 00:02:08.060
if I,

00:02:08.060 --> 00:02:10.620
in my horizontal axis, that is time.

00:02:10.620 --> 00:02:13.200
My vertical axis, this is velocity.

00:02:13.200 --> 00:02:16.450
And velocity is going to be
measured in meters per second.

00:02:16.450 --> 00:02:19.760
Time is going to be measured in seconds.

00:02:19.760 --> 00:02:20.710
Time is measured in seconds.

00:02:20.710 --> 00:02:23.180
And actually, I'm going
to do two scenarios here.

00:02:23.180 --> 00:02:27.480
So let's say that I have a
first velocity time graph.

00:02:27.480 --> 00:02:31.450
Let's just call it v one of
t, which is equal to three.

00:02:31.450 --> 00:02:35.980
And it would be three meters
per second, so one, two, three.

00:02:35.980 --> 00:02:37.500
So it would

00:02:37.500 --> 00:02:39.200
look like that.

00:02:39.200 --> 00:02:40.840
That is v one of t.

00:02:40.840 --> 00:02:43.170
And if I were to look
at the definite integral

00:02:43.170 --> 00:02:46.300
going from time equals
one to time equals five

00:02:46.300 --> 00:02:51.300
of v sub one of t dt, what
would this be equal to?

00:02:51.440 --> 00:02:54.000
Well, here my function is above my t-axis.

00:02:54.000 --> 00:02:56.980
So I'll just go from one to

00:02:56.980 --> 00:02:59.030
five, which will be around there.

00:02:59.030 --> 00:03:01.120
And I could just think
about the area here,

00:03:01.120 --> 00:03:03.720
and this area is pretty easy to calculate.

00:03:03.720 --> 00:03:07.900
It's going to be three meters
per second times four seconds.

00:03:07.900 --> 00:03:09.710
That's my change in time.

00:03:09.710 --> 00:03:12.410
And so this is going to be 12 meters.

00:03:12.410 --> 00:03:15.000
And so this is going to be equal to 12.

00:03:15.000 --> 00:03:16.750
And one way to conceptualize this

00:03:16.750 --> 00:03:19.440
is this gives us our change in position.

00:03:19.440 --> 00:03:22.030
If my velocity is three meters per second,

00:03:22.030 --> 00:03:23.940
and since it's positive
you can conceptualize that

00:03:23.940 --> 00:03:26.710
as it's going to the right
at three meters per second.

00:03:26.710 --> 00:03:28.310
What is my change in position?

00:03:28.310 --> 00:03:31.450
Well, I would have gone
12 meters to the right.

00:03:31.450 --> 00:03:33.920
And you don't need calculus
to figure that out.

00:03:33.920 --> 00:03:36.410
Three meters per second times four seconds

00:03:36.410 --> 00:03:38.240
would be 12 meters.

00:03:38.240 --> 00:03:40.470
But what if it were the other way around?

00:03:40.470 --> 00:03:42.300
What if I had another velocity function,

00:03:42.300 --> 00:03:45.390
;et's call that v sub two of t,

00:03:45.390 --> 00:03:49.650
that is equal to negative
two meters per second?

00:03:49.650 --> 00:03:51.820
And it's just a constant
negative two meters per second.

00:03:51.820 --> 00:03:52.950
So this is

00:03:52.950 --> 00:03:56.450
v sub two of t right over here.

00:03:56.450 --> 00:03:58.570
What would or what should

00:03:58.570 --> 00:04:03.570
the definite integral from one
to five of v sub two of t be,

00:04:04.010 --> 00:04:05.660
dt be equal to?

00:04:06.630 --> 00:04:09.840
Well, it should be equal
to my change in position.

00:04:09.840 --> 00:04:11.650
But if my velocity is negative,

00:04:11.650 --> 00:04:14.210
that means I'm moving to the left.

00:04:14.210 --> 00:04:18.050
That means my change in
position should be to the left,

00:04:18.050 --> 00:04:19.460
as opposed to to the right.

00:04:20.320 --> 00:04:23.370
And so we can just look at
this area right over here.

00:04:23.370 --> 00:04:24.710
When if you just look
at it as the rectangle,

00:04:24.710 --> 00:04:28.130
it's going to be two times
four, which is equal to eight.

00:04:28.130 --> 00:04:29.760
But you have to be very careful.

00:04:29.760 --> 00:04:34.760
Since it is below my horizontal
axis and above my function,

00:04:34.990 --> 00:04:36.730
this is going to be negative.

00:04:36.730 --> 00:04:39.210
And this should make a lot of sense.

00:04:39.210 --> 00:04:42.290
If I'm going two meters
per second to the left

00:04:42.290 --> 00:04:44.960
for four seconds, or another
way to think about it,

00:04:44.960 --> 00:04:47.140
if I'm going negative
two meters per second

00:04:47.140 --> 00:04:49.570
for four seconds, then
my change in position

00:04:49.570 --> 00:04:51.350
is going to be negative eight meters.

00:04:51.350 --> 00:04:54.050
I would have moved eight
meters to the left,

00:04:54.050 --> 00:04:57.890
if we say the convention is
negative means to the left.

00:04:57.890 --> 00:04:59.660
So the big takeaway is,

00:04:59.660 --> 00:05:03.920
if it's below your function
and above the horizontal axis,

00:05:03.920 --> 00:05:06.900
the definite integral, and
if your a is less than b,

00:05:06.900 --> 00:05:09.210
then your definite integral
is going to be positive.

00:05:09.210 --> 00:05:11.620
If your a is less than b,

00:05:11.620 --> 00:05:13.840
but your function over that interval is

00:05:13.840 --> 00:05:16.400
below the horizontal axis,

00:05:16.400 --> 00:05:18.450
then your definite integral
is going to be negative.

00:05:18.450 --> 00:05:19.780
And in the future, we'll also look

00:05:19.780 --> 00:05:21.520
at definite integrals
that are a mix of both,

00:05:21.520 --> 00:05:23.450
but that's a little bit more complicated.


Verifying solutions to differential equations

WEBVTT
Kind: captions
Language: en

00:00:00.120 --> 00:00:01.820
- [Instructor] So let's write
down a differential equation,

00:00:01.820 --> 00:00:04.880
the derivative of y with respect to x

00:00:04.880 --> 00:00:08.580
is equal to four y over x.

00:00:08.580 --> 00:00:11.100
And what we'll see in
this video is the solution

00:00:11.100 --> 00:00:13.210
to a differential equation isn't a value

00:00:13.210 --> 00:00:14.700
or a set of values.

00:00:14.700 --> 00:00:17.520
It's a function or a set of functions.

00:00:17.520 --> 00:00:20.020
But before we go about
actually trying to solve this

00:00:20.020 --> 00:00:21.840
or figure out all of the solutions,

00:00:21.840 --> 00:00:24.080
let's test whether certain equations,

00:00:24.080 --> 00:00:25.120
certain functions,

00:00:25.120 --> 00:00:28.280
are solutions to this
differential equation.

00:00:28.280 --> 00:00:30.030
So for example,

00:00:30.030 --> 00:00:33.370
if I have y is equal to four x,

00:00:33.370 --> 00:00:37.020
is this a solution to this
differential equation?

00:00:37.020 --> 00:00:40.010
Pause the video and see
if you can figure it out.

00:00:40.010 --> 00:00:41.940
Well to see if this is a solution,

00:00:41.940 --> 00:00:43.770
what we have to do is figure
out the derivative of y

00:00:43.770 --> 00:00:46.400
with respect to x and
see is that truly equal

00:00:46.400 --> 00:00:48.540
to four times y over x.

00:00:48.540 --> 00:00:51.530
And I'm gonna try to express
everything in terms of x

00:00:51.530 --> 00:00:53.670
to see if I really have an equality there.

00:00:53.670 --> 00:00:56.240
So first let's figure
out the derivative of y

00:00:56.240 --> 00:00:57.890
with respect to x.

00:00:57.890 --> 00:01:00.000
Well that's just going
to be equal to four.

00:01:00.000 --> 00:01:02.150
We've seen that many times before.

00:01:02.150 --> 00:01:03.820
And so what we need to test is,

00:01:03.820 --> 00:01:06.880
is four, the derivative
of y with respect to x,

00:01:06.880 --> 00:01:10.550
equal to four times, I could write y,

00:01:10.550 --> 00:01:12.140
but instead of y let's write four x.

00:01:12.140 --> 00:01:14.620
I'm gonna put everything in terms of x.

00:01:14.620 --> 00:01:15.960
So y is equal to four x,

00:01:15.960 --> 00:01:18.660
so instead of four y I could
write four times four x,

00:01:18.660 --> 00:01:20.390
all of that over x.

00:01:20.390 --> 00:01:21.900
Is this true?

00:01:21.900 --> 00:01:23.790
Well that x cancels with
that and I'm gonna get

00:01:23.790 --> 00:01:28.480
four is equal to 16,
which it clearly is not.

00:01:28.480 --> 00:01:31.170
And so this is not a solution.

00:01:31.170 --> 00:01:34.140
Not a solution

00:01:34.140 --> 00:01:36.330
to our differential equation.

00:01:36.330 --> 00:01:38.780
Let's look at another equation.

00:01:38.780 --> 00:01:42.890
What about y is equal to
x to the fourth power?

00:01:42.890 --> 00:01:46.010
Pause this video and see
if this is a solution

00:01:46.010 --> 00:01:48.307
to our original differential equation.

00:01:48.307 --> 00:01:50.650
Well we're going to do the same thing.

00:01:50.650 --> 00:01:54.150
What's the derivative
of y with respect to x?

00:01:54.150 --> 00:01:56.210
This is equal to, just
using the power rule,

00:01:56.210 --> 00:01:59.030
four x to the third power.

00:01:59.030 --> 00:02:00.420
And so what we have to test is,

00:02:00.420 --> 00:02:03.070
is four x to the third power,

00:02:03.070 --> 00:02:04.790
that's the derivative
of y with respect to x,

00:02:04.790 --> 00:02:08.010
equal to four times y,

00:02:08.010 --> 00:02:10.230
instead of writing a y I'm gonna
write it all in terms of x,

00:02:10.230 --> 00:02:13.280
so is that equal to four
times x to the fourth,

00:02:13.280 --> 00:02:15.810
because x to the fourth
is the same thing as y,

00:02:15.810 --> 00:02:18.600
divided by x?

00:02:18.600 --> 00:02:21.470
And so let's see, x to
the fourth divided by x,

00:02:21.470 --> 00:02:23.660
that is going to be x to the third.

00:02:23.660 --> 00:02:25.790
And so you will indeed
get four x to the third

00:02:25.790 --> 00:02:27.500
is equal to four x to the third.

00:02:27.500 --> 00:02:30.150
So check, this is a solution.

00:02:30.150 --> 00:02:33.310
So is a solution.

00:02:33.310 --> 00:02:34.950
It's not necessarily the only solution,

00:02:34.950 --> 00:02:38.940
but it is a solution to
that differential equation.

00:02:38.940 --> 00:02:41.190
Let's look at another
differential equation.

00:02:41.190 --> 00:02:43.650
Let's say that I had,

00:02:43.650 --> 00:02:46.040
and I'm gonna write it
with different notation,

00:02:46.040 --> 00:02:49.580
f prime of x is equal to

00:02:49.580 --> 00:02:52.550
f of x minus x.

00:02:52.550 --> 00:02:55.100
And the first function that I wanna test,

00:02:55.100 --> 00:02:59.620
let's say I have f of x is equal to two x.

00:02:59.620 --> 00:03:02.900
Is this a solution to this
differential equation?

00:03:02.900 --> 00:03:05.900
Pause the video again and
see if you can figure it out.

00:03:05.900 --> 00:03:06.850
Well to figure that out,

00:03:06.850 --> 00:03:09.620
you have to say well what is f prime of x?

00:03:09.620 --> 00:03:14.220
f prime of x is just
going to be equal to two.

00:03:14.220 --> 00:03:16.300
And then test the equality.

00:03:16.300 --> 00:03:20.740
Is two, is f prime of x, equal to f of x,

00:03:20.740 --> 00:03:23.500
which is two x, minus x,

00:03:23.500 --> 00:03:25.160
minus x?

00:03:25.160 --> 00:03:30.160
And so let's see we are going
to get two is equal to x.

00:03:30.250 --> 00:03:31.840
So you might be tempted to say oh hey

00:03:31.840 --> 00:03:34.600
I just solved for x or
something like that.

00:03:34.600 --> 00:03:37.670
But this would tell you
that this is not a solution

00:03:37.670 --> 00:03:40.853
because this needs to be true for any x

00:03:40.853 --> 00:03:43.970
that is in the domain of this function.

00:03:43.970 --> 00:03:46.790
And so this is, I'll just put an x there,

00:03:46.790 --> 00:03:50.320
or I'll put a incorrect there to say not,

00:03:50.320 --> 00:03:55.320
not a, not a solution.

00:03:55.460 --> 00:03:56.930
Just to be clear again,

00:03:56.930 --> 00:04:00.130
this needs, in order for a
function to be a solution

00:04:00.130 --> 00:04:01.610
of this differential equation,

00:04:01.610 --> 00:04:03.490
it needs to work for any x

00:04:03.490 --> 00:04:05.510
that you can put into the function.

00:04:05.510 --> 00:04:06.940
Let's look at another one.

00:04:06.940 --> 00:04:11.940
Let's say that we have f of
x is equal to x plus one.

00:04:12.500 --> 00:04:15.140
Pause the video and
see, is this a solution

00:04:15.140 --> 00:04:17.280
to our differential equation?

00:04:17.280 --> 00:04:18.520
Well same drill.

00:04:18.520 --> 00:04:21.500
f prime of x is going to be equal to one.

00:04:21.500 --> 00:04:23.460
And so we have to see is f prime of x,

00:04:23.460 --> 00:04:26.240
which is equal to one,
is it equal to f of x,

00:04:26.240 --> 00:04:30.220
which is x plus one, x plus one, minus x?

00:04:30.220 --> 00:04:32.940
And so here, you see no matter what x is,

00:04:32.940 --> 00:04:35.080
this equation is going to be true.

00:04:35.080 --> 00:04:37.530
So this is a solution,

00:04:37.530 --> 00:04:40.223
is a solution.

00:04:41.130 --> 00:04:43.160
Let's do a few more of these.

00:04:43.160 --> 00:04:46.000
Let me scroll down little bit
so I have a little bit more,

00:04:46.000 --> 00:04:47.797
a little bit more space,

00:04:47.797 --> 00:04:50.480
but make sure we see our
original differential equation.

00:04:50.480 --> 00:04:55.480
Let's test whether, I'm
gonna do it in a red color,

00:04:55.960 --> 00:04:59.100
let's test whether f of x

00:04:59.100 --> 00:05:03.060
equals e to the x plus x plus one

00:05:03.060 --> 00:05:05.970
is a solution to this
differential equation.

00:05:05.970 --> 00:05:08.320
Pause the video again and
see if you can figure it out.

00:05:08.320 --> 00:05:10.210
All right well let's figure
out the derivative here.

00:05:10.210 --> 00:05:12.520
f prime of x is going to be equal to,

00:05:12.520 --> 00:05:15.910
derivative of e to the x with
respect to x is e to the x,

00:05:15.910 --> 00:05:17.560
which I always find amazing.

00:05:17.560 --> 00:05:20.200
And so and then plus one
and the derivative of this

00:05:20.200 --> 00:05:21.830
with respect to x is just zero.

00:05:21.830 --> 00:05:23.250
And then let's substitute this

00:05:23.250 --> 00:05:24.810
into our original differential equation.

00:05:24.810 --> 00:05:28.330
So f prime of x is e to the x plus one.

00:05:28.330 --> 00:05:30.080
Is that equal to f of x,

00:05:30.080 --> 00:05:35.080
which is e to the x plus x
plus one, minus x, minus x?

00:05:37.070 --> 00:05:39.130
And if that x cancels out with that x,

00:05:39.130 --> 00:05:41.880
it is indeed, they are indeed equal.

00:05:41.880 --> 00:05:44.170
So this is also a solution.

00:05:44.170 --> 00:05:48.943
So this, this is a solution.

00:05:50.180 --> 00:05:51.013
And we're done.


Finding definite integrals using area formulas

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:02.870
- [Instructor] We're told to
find the following integrals,

00:00:02.870 --> 00:00:06.620
and we're given the graph
of f right over here.

00:00:06.620 --> 00:00:08.390
So this first one is the definite integral

00:00:08.390 --> 00:00:11.460
from negative six to
negative two of f of x dx.

00:00:11.460 --> 00:00:13.477
Pause this video and see if
you can figure this one out

00:00:13.477 --> 00:00:15.750
from this graph.

00:00:15.750 --> 00:00:18.830
All right we're going
from x equals negative six

00:00:18.830 --> 00:00:21.760
to x equals negative two,

00:00:21.760 --> 00:00:24.470
and the definite integral
is going to be the area

00:00:24.470 --> 00:00:27.980
below our graph and above the x-axis.

00:00:27.980 --> 00:00:30.890
So it's going to be this
area right over here.

00:00:30.890 --> 00:00:32.560
And how do we figure that out?

00:00:32.560 --> 00:00:34.730
Well this is a semicircle,

00:00:34.730 --> 00:00:36.490
and we know how to find
the area of a circle

00:00:36.490 --> 00:00:37.660
if we know its radius.

00:00:37.660 --> 00:00:41.160
And this circle has radius two,

00:00:41.160 --> 00:00:43.000
has a radius of two.

00:00:43.000 --> 00:00:45.410
No matter what direction
we go in from the center,

00:00:45.410 --> 00:00:47.660
it has a radius of two.

00:00:47.660 --> 00:00:51.370
And so the area of a
circle is pi r squared.

00:00:51.370 --> 00:00:54.760
So it would be pi times our
radius which is two squared,

00:00:54.760 --> 00:00:56.100
but this is a semicircle,

00:00:56.100 --> 00:00:57.510
so I'm gonna divide by two.

00:00:57.510 --> 00:01:00.230
It's only 1/2 the area of the full circle.

00:01:00.230 --> 00:01:02.200
So this is going to be four pi over two,

00:01:02.200 --> 00:01:04.540
which is equal to two pi.

00:01:04.540 --> 00:01:06.210
All right let's do another one.

00:01:06.210 --> 00:01:07.670
So here we have the definite integral

00:01:07.670 --> 00:01:10.050
from negative two to one of f of x dx.

00:01:10.050 --> 00:01:12.920
Pause the video and see if
you can figure that out.

00:01:12.920 --> 00:01:13.990
All right let's do it together.

00:01:13.990 --> 00:01:17.610
So we're going from negative two to one,

00:01:17.610 --> 00:01:20.980
and so we have to be a
little bit careful here.

00:01:20.980 --> 00:01:21.850
So the definite integral,

00:01:21.850 --> 00:01:25.150
you could view it as the area below the,

00:01:25.150 --> 00:01:28.590
below the function and above the x-axis.

00:01:28.590 --> 00:01:31.800
But here the function is below the x-axis.

00:01:31.800 --> 00:01:33.440
And so what we can do is,

00:01:33.440 --> 00:01:35.640
we can figure out this area,

00:01:35.640 --> 00:01:38.520
just knowing what we know about geometry,

00:01:38.520 --> 00:01:40.510
and then we have to
realize that this is going

00:01:40.510 --> 00:01:43.840
to be a negative value
for the definite integral

00:01:43.840 --> 00:01:46.830
because our function is below the x-axis.

00:01:46.830 --> 00:01:48.880
So what's the area here?

00:01:48.880 --> 00:01:50.650
Well there's a couple of
ways to think about it.

00:01:50.650 --> 00:01:54.050
We could split it up into a few shapes.

00:01:54.050 --> 00:01:55.660
So you could just view it as a trapezoid

00:01:55.660 --> 00:01:57.830
or you can just split
it up into a rectangle

00:01:57.830 --> 00:01:59.270
and two triangles.

00:01:59.270 --> 00:02:01.200
So if you split it up like this,

00:02:01.200 --> 00:02:03.920
this triangle right over here has an area

00:02:03.920 --> 00:02:06.433
of one times two times 1/2.

00:02:07.320 --> 00:02:09.550
So this has an area of one.

00:02:09.550 --> 00:02:12.240
This rectangle right over here has an area

00:02:12.240 --> 00:02:14.280
of two times one,

00:02:14.280 --> 00:02:15.960
so it has an area of two.

00:02:15.960 --> 00:02:18.530
And then this triangle
right over here is the same

00:02:18.530 --> 00:02:20.050
area as the first one.

00:02:20.050 --> 00:02:22.890
It's going to have a base
of one, a height of two,

00:02:22.890 --> 00:02:26.030
so it's one times two times 1/2.

00:02:26.030 --> 00:02:28.500
Remember the area of a triangle
is 1/2 base times height.

00:02:28.500 --> 00:02:29.810
So it's one.

00:02:29.810 --> 00:02:33.640
So if you add up those areas,
one plus two plus one is four,

00:02:33.640 --> 00:02:35.910
and so you might be tempted
to say oh is this going to be

00:02:35.910 --> 00:02:37.200
equal to four?

00:02:37.200 --> 00:02:40.880
But remember our function
is below the x-axis here,

00:02:40.880 --> 00:02:44.790
and so this is going
to be a negative four.

00:02:44.790 --> 00:02:47.470
All right let's do another one.

00:02:47.470 --> 00:02:51.440
So now we're gonna go from
one to four of f of x dx.

00:02:51.440 --> 00:02:54.350
So pause the video and see
if you can figure that out.

00:02:54.350 --> 00:02:57.020
So we're gonna go from here to here,

00:02:57.020 --> 00:03:01.400
and so it's gonna be this
area right over there.

00:03:01.400 --> 00:03:02.640
So how do we figure that out?

00:03:02.640 --> 00:03:05.340
Well it's just the formula
for the area of a triangle,

00:03:05.340 --> 00:03:08.823
base times height times 1/2.

00:03:09.760 --> 00:03:13.200
So or you could say 1/2 times our base,

00:03:13.200 --> 00:03:15.730
which is a length of,

00:03:15.730 --> 00:03:17.530
see we have a base of
three right over here,

00:03:17.530 --> 00:03:19.236
go from one to four,

00:03:19.236 --> 00:03:21.280
so 1/2 times three times our height,

00:03:21.280 --> 00:03:24.240
which is one, two, three, four,

00:03:24.240 --> 00:03:25.680
times four.

00:03:25.680 --> 00:03:28.320
Well this is just going to get us six.

00:03:28.320 --> 00:03:30.650
All right last but not least,

00:03:30.650 --> 00:03:35.540
if we are going from
four to six of f of x dx.

00:03:35.540 --> 00:03:38.450
So that's going to be
this area right over here,

00:03:38.450 --> 00:03:39.310
but we have to be careful.

00:03:39.310 --> 00:03:41.270
Our function is below the x-axis,

00:03:41.270 --> 00:03:43.250
so we'll figure out this
area and then it's going

00:03:43.250 --> 00:03:44.750
to be negative.

00:03:44.750 --> 00:03:48.920
So this is a half of a
circle of radius one.

00:03:48.920 --> 00:03:53.440
And so the area of a circle
is pi times r squared,

00:03:53.440 --> 00:03:55.930
so it's pi times one squared.

00:03:55.930 --> 00:03:58.270
That would be the area if
we went all the way around

00:03:58.270 --> 00:04:00.300
like that, but this is
only half of the circle,

00:04:00.300 --> 00:04:01.710
so divided by two.

00:04:01.710 --> 00:04:06.710
And since this area is
above the function and below

00:04:06.750 --> 00:04:09.550
the x-axis, it's going to be negative.

00:04:09.550 --> 00:04:12.898
So this is going to be equal
to negative pi over two.

00:04:12.898 --> 00:04:14.983
And we are done.


Limits of piecewise functions

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:01.350
- [Instructor] Let's
think a little bit about

00:00:01.350 --> 00:00:04.910
limits of piecewise functions
that are defined algebraically

00:00:04.910 --> 00:00:07.020
like our f of x right over here.

00:00:07.020 --> 00:00:08.890
Pause this video and see
if you can figure out

00:00:08.890 --> 00:00:10.640
what these various limits would be,

00:00:10.640 --> 00:00:11.890
some of them are one-sided,

00:00:11.890 --> 00:00:15.500
and some of them are regular
limits, or two-sided limits.

00:00:15.500 --> 00:00:16.980
Alright, let's start with this first one,

00:00:16.980 --> 00:00:19.120
the limit as x approaches four,

00:00:19.120 --> 00:00:22.180
from values larger than equaling four,

00:00:22.180 --> 00:00:24.400
so that's what that plus tells us.

00:00:24.400 --> 00:00:27.050
And so when x is greater than four,

00:00:27.050 --> 00:00:29.750
our f of x is equal to square root of x.

00:00:29.750 --> 00:00:33.750
So as we are approaching
four from the right,

00:00:33.750 --> 00:00:37.400
we are really thinking about
this part of the function.

00:00:37.400 --> 00:00:41.710
And so this is going to be
equal to the square root of

00:00:41.710 --> 00:00:43.950
four, even though right at four,

00:00:43.950 --> 00:00:45.630
our f of x is equal to this,

00:00:45.630 --> 00:00:48.260
we are approaching from
values greater than four,

00:00:48.260 --> 00:00:51.380
we're approaching from
the right, so we would use

00:00:51.380 --> 00:00:54.320
this part of our function definition,

00:00:54.320 --> 00:00:57.470
and so this is going to be equal to two.

00:00:57.470 --> 00:01:00.010
Now what about our limit of f of x,

00:01:00.010 --> 00:01:02.420
as we approach four from the left?

00:01:02.420 --> 00:01:06.440
Well then we would use this
part of our function definition.

00:01:06.440 --> 00:01:10.460
And so this is going to
be equal to four plus two

00:01:10.460 --> 00:01:12.580
over four minus one,

00:01:12.580 --> 00:01:16.140
which is equal to 6 over three,

00:01:16.140 --> 00:01:18.210
which is equal to two.

00:01:18.210 --> 00:01:20.450
And so if we wanna say
what is the limit of f of x

00:01:20.450 --> 00:01:24.160
as x approaches four, well
this is a good scenario here

00:01:24.160 --> 00:01:27.440
because from both the left and the right

00:01:27.440 --> 00:01:29.760
as we approach x equals
four, we're approaching

00:01:29.760 --> 00:01:32.340
the same value, and we
know, that in order for

00:01:32.340 --> 00:01:35.120
the two side limit to have
a limit, you have to be

00:01:35.120 --> 00:01:38.200
approaching the same thing
from the right and the left.

00:01:38.200 --> 00:01:41.460
And we are, and so this is
going to be equal to two.

00:01:41.460 --> 00:01:44.920
Now what's the limit as x
approaches two of f of x?

00:01:44.920 --> 00:01:47.560
Well, as x approaches
two, we are going to be

00:01:47.560 --> 00:01:50.360
completely in this
scenario right over here.

00:01:50.360 --> 00:01:53.270
Now interesting things do
happen at x equals one here,

00:01:53.270 --> 00:01:57.770
our denominator goes to
zero, but at x equals two,

00:01:57.770 --> 00:01:59.930
this part of the curve
is gonna be continuous

00:01:59.930 --> 00:02:01.640
so we can just substitute
the value, it's going to be

00:02:01.640 --> 00:02:06.050
two plus two, over two minus
one, which is four over one,

00:02:06.050 --> 00:02:08.010
which is equal to four.

00:02:08.010 --> 00:02:09.633
Let's do another example.

00:02:10.760 --> 00:02:13.770
So we have another piecewise function,

00:02:13.770 --> 00:02:18.030
and so let's pause our video
and figure out these things.

00:02:18.030 --> 00:02:19.670
Alright, now let's do this together.

00:02:19.670 --> 00:02:22.670
So what's the limit as x
approaches negative one

00:02:22.670 --> 00:02:24.660
from the right?

00:02:24.660 --> 00:02:26.640
So if we're approaching from the right,

00:02:26.640 --> 00:02:28.920
when we are greater than
or equal to negative one,

00:02:28.920 --> 00:02:31.740
we are in this part of
our piecewise function,

00:02:31.740 --> 00:02:35.490
and so we would say, this
is going to approach,

00:02:35.490 --> 00:02:40.150
this is gonna be two, to
the negative one power,

00:02:40.150 --> 00:02:42.550
which is equal to one half.

00:02:42.550 --> 00:02:44.920
What about if we're
approaching from the left?

00:02:44.920 --> 00:02:46.430
Well, if we're approaching from the left,

00:02:46.430 --> 00:02:48.130
we're in this scenario right over here,

00:02:48.130 --> 00:02:51.130
we're to the left of
x equals negative one,

00:02:51.130 --> 00:02:54.080
and so this is going to
be equal to the sine,

00:02:54.080 --> 00:02:57.010
'cause we're in this case,
for our piecewise function,

00:02:57.010 --> 00:03:00.330
of negative one plus one,
which is the sine of zero,

00:03:00.330 --> 00:03:02.440
which is equal to zero.

00:03:02.440 --> 00:03:04.860
Now what's the two-sided
limit as x approaches

00:03:04.860 --> 00:03:06.690
negative one of g of x?

00:03:06.690 --> 00:03:08.630
Well we're approaching
two different values

00:03:08.630 --> 00:03:09.760
as we approach from the right,

00:03:09.760 --> 00:03:11.220
and as we approach from the left.

00:03:11.220 --> 00:03:13.190
And if our one-sided
limits aren't approaching

00:03:13.190 --> 00:03:16.640
the same value, well then
this limit does not exist.

00:03:16.640 --> 00:03:20.030
Does not exist.

00:03:20.030 --> 00:03:21.740
And what's the limit of g of x,

00:03:21.740 --> 00:03:25.080
as x approaches zero from the right?

00:03:25.080 --> 00:03:28.070
Well, if we're talking
about approaching zero

00:03:28.070 --> 00:03:30.960
from the right, we are
going to be in this case

00:03:30.960 --> 00:03:34.650
right over here, zero is
definitely in this interval,

00:03:34.650 --> 00:03:37.570
and over this interval,
this right over here

00:03:37.570 --> 00:03:41.660
is going to be continuous,
and so we can just substitute

00:03:41.660 --> 00:03:44.470
x equals zero there, so it's
gonna be two to the zero,

00:03:44.470 --> 00:03:48.293
which is, indeed, equal
to one, and we're done.


Power rule (with rewriting the expression)

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:01.280
- [Instructor] What we're
going to do in this video

00:00:01.280 --> 00:00:05.250
is get some practice taking
derivatives with the power rule.

00:00:05.250 --> 00:00:06.565
So let's say we take the derivative

00:00:06.565 --> 00:00:10.660
with respect to x of one over x.

00:00:10.660 --> 00:00:12.410
What is that going to be equal to?

00:00:12.410 --> 00:00:14.563
Pause this video and try to figure it out.

00:00:15.460 --> 00:00:16.347
So at first, you might say,

00:00:16.347 --> 00:00:18.520
"How does the power rule apply here?"

00:00:18.520 --> 00:00:20.730
The power rule, just to remind ourselves,

00:00:20.730 --> 00:00:21.950
it tells us that if we're taking

00:00:21.950 --> 00:00:25.780
the derivative of x to
the n with respect to x,

00:00:25.780 --> 00:00:27.150
so if we're taking the derivative of that,

00:00:27.150 --> 00:00:28.500
that that's going to be equal to,

00:00:28.500 --> 00:00:30.200
we take the exponent, bring it out front,

00:00:30.200 --> 00:00:32.540
and we've proven it in other videos,

00:00:32.540 --> 00:00:35.460
but this is gonna be n times x to the,

00:00:35.460 --> 00:00:37.120
and then we decrement the exponent.

00:00:37.120 --> 00:00:39.700
So, n minus one.

00:00:39.700 --> 00:00:41.760
But this does not look like that,

00:00:41.760 --> 00:00:44.390
and the key is to
appreciate that one over x

00:00:44.390 --> 00:00:47.000
is the same thing as
x to the negative one.

00:00:47.000 --> 00:00:48.380
So, this is going to be the derivative

00:00:48.380 --> 00:00:52.890
with respect to x of
x to the negative one.

00:00:52.890 --> 00:00:54.540
And now, this looks a lot more

00:00:54.540 --> 00:00:56.150
like what you might be used to,

00:00:56.150 --> 00:00:57.960
where this is going to be equal to,

00:00:57.960 --> 00:01:00.370
you take our exponent, bring it out front,

00:01:00.370 --> 00:01:01.840
so it's negative one,

00:01:01.840 --> 00:01:05.690
times x to the negative one minus one,

00:01:05.690 --> 00:01:07.230
negative one minus one.

00:01:07.230 --> 00:01:08.610
And so, this is going to be equal

00:01:08.610 --> 00:01:12.870
to negative x to the negative two,

00:01:12.870 --> 00:01:14.020
and we're done.

00:01:14.020 --> 00:01:16.330
Let's do another example.

00:01:16.330 --> 00:01:18.720
Let's say that we're told that f of x

00:01:18.720 --> 00:01:21.750
is equal to the cube root of x

00:01:21.750 --> 00:01:25.890
and we wanna figure out what
f prime of x is equal to.

00:01:25.890 --> 00:01:29.170
Pause the video and see if
you can figure it out again.

00:01:29.170 --> 00:01:30.357
Well, once again, you might say,

00:01:30.357 --> 00:01:33.427
"Hey, how do I take the
derivative of something like this,

00:01:33.427 --> 00:01:37.067
"especially if my goal or
if I'm thinking that maybe

00:01:37.067 --> 00:01:39.430
"the power rule might be useful?"

00:01:39.430 --> 00:01:42.820
And the idea is to rewrite
this as an exponent,

00:01:42.820 --> 00:01:47.820
if you can rewrite the cube
root as x to the 1/3 power.

00:01:49.090 --> 00:01:51.780
And so, the derivative, you take the 1/3,

00:01:51.780 --> 00:01:54.080
bring it out front, so it's 1/3

00:01:54.080 --> 00:01:57.840
x to the 1/3 minus one power.

00:01:57.840 --> 00:01:59.950
And so, this is going to be 1/3

00:01:59.950 --> 00:02:03.340
times x to the 1/3 minus one

00:02:03.340 --> 00:02:05.500
is negative 2/3,

00:02:05.500 --> 00:02:09.140
negative 2/3 power, and we are done.

00:02:09.140 --> 00:02:11.140
And hopefully through these examples,

00:02:11.140 --> 00:02:13.700
you're seeing that the power
rule is incredibly powerful.

00:02:13.700 --> 00:02:17.560
You can tackle a far
broader range of derivatives

00:02:17.560 --> 00:02:19.470
than you might have initially thought.

00:02:19.470 --> 00:02:20.500
Let's do another example,

00:02:20.500 --> 00:02:23.060
and I'll make this one
really nice and hairy.

00:02:23.060 --> 00:02:25.210
Let's say we wanna
figure out the derivative

00:02:25.210 --> 00:02:29.176
with respect to x of the cube root

00:02:29.176 --> 00:02:33.040
of x squared.

00:02:33.040 --> 00:02:34.550
What is this going to be?

00:02:34.550 --> 00:02:35.383
And actually,

00:02:35.383 --> 00:02:36.800
let's just not figure out
what the derivative is,

00:02:36.800 --> 00:02:41.480
let's figure out the
derivative at x equals eight.

00:02:41.480 --> 00:02:45.010
Pause this video again and see
if you can figure that out.

00:02:45.010 --> 00:02:45.930
Well, what we're gonna do

00:02:45.930 --> 00:02:47.810
is first just figure out what this is

00:02:47.810 --> 00:02:50.530
and then we're going evaluate
it at x equals eight.

00:02:50.530 --> 00:02:54.010
And the key thing to appreciate
is this is the same thing,

00:02:54.010 --> 00:02:55.730
and we're just gonna
do what we did up here

00:02:55.730 --> 00:02:58.350
as the derivative with respect to x.

00:02:58.350 --> 00:03:00.140
Instead of saying the
cube root of x squared,

00:03:00.140 --> 00:03:04.810
we can say this is x
squared to the 1/3 power,

00:03:04.810 --> 00:03:06.830
which is the same thing as the derivative

00:03:06.830 --> 00:03:09.610
with respect to x of,

00:03:09.610 --> 00:03:11.758
well, x squared, if I raise
something to an exponent

00:03:11.758 --> 00:03:13.480
and then raise that to an exponent,

00:03:13.480 --> 00:03:15.560
I can just take the
product of the exponents.

00:03:15.560 --> 00:03:19.120
And so, this is gonna be x
to the two times 1/3 power

00:03:19.120 --> 00:03:21.150
or to the 2/3 power.

00:03:21.150 --> 00:03:23.750
And now, this is just
going to be equal to,

00:03:23.750 --> 00:03:26.930
I'll do it right over here,
bring the 2/3 out front,

00:03:26.930 --> 00:03:31.560
2/3 times x to the, what's 2/3 minus one?

00:03:31.560 --> 00:03:33.590
Well, that's 2/3 minus 3/3

00:03:33.590 --> 00:03:36.443
or it would be negative 1/3 power.

00:03:36.443 --> 00:03:39.210
Now, we wanna know what
happens at x equals eight,

00:03:39.210 --> 00:03:40.810
so let's just evaluate that.

00:03:40.810 --> 00:03:45.650
That's going to be 2/3
times x is equal to eight

00:03:45.650 --> 00:03:48.660
to the negative 1/3 power.

00:03:48.660 --> 00:03:51.690
Well, what's eight to the 1/3 power?

00:03:51.690 --> 00:03:56.580
Eight to the 1/3 power is
going to be equal to two,

00:03:56.580 --> 00:04:00.830
and so, eight to the
negative 1/3 power is 1/2.

00:04:00.830 --> 00:04:03.630
Actually, let me just
do that step-by-step.

00:04:03.630 --> 00:04:07.330
So, this is going to
be equal to 2/3 times,

00:04:07.330 --> 00:04:12.080
we could do it this way, one
over eight to the 1/3 power.

00:04:12.080 --> 00:04:14.910
And so, this is just one over two,

00:04:14.910 --> 00:04:16.443
2/3 times 1/2,

00:04:17.590 --> 00:04:22.533
well, that's just going to be
equal to 1/3, and we're done.


Example identifying roles in a food web

WEBVTT
Kind: captions
Language: en

00:00:00.210 --> 00:00:03.280
- [Instructor] We are asked,
who is a secondary consumer

00:00:03.280 --> 00:00:04.970
in this diagram?

00:00:04.970 --> 00:00:07.973
So pause this video, and see
if you can figure that out.

00:00:09.060 --> 00:00:11.650
Alright, now let's work
through this together.

00:00:11.650 --> 00:00:13.930
So let's just make sure we
understand this diagram.

00:00:13.930 --> 00:00:17.230
When we have an arrow
from grasses to mouse,

00:00:17.230 --> 00:00:20.890
it means that the grasses
go into the mouse's mouth.

00:00:20.890 --> 00:00:23.190
It means that the mouse eats the grasses.

00:00:23.190 --> 00:00:26.400
Similarly, the grasshopper
eats the grasses.

00:00:26.400 --> 00:00:28.780
And then we have these
arrows from the mouse,

00:00:28.780 --> 00:00:31.140
to the coyote, the hawk,
the snake, the vulture.

00:00:31.140 --> 00:00:35.100
That means that all of these
characters may eat the mouse.

00:00:35.100 --> 00:00:38.120
Likewise, a hawk eats a grasshopper.

00:00:38.120 --> 00:00:39.990
So now that we understand this diagram,

00:00:39.990 --> 00:00:44.990
let's label where these various
folks fit in the food chain.

00:00:45.080 --> 00:00:47.080
So the grasses right over here,

00:00:47.080 --> 00:00:50.870
they are a primary producer.

00:00:50.870 --> 00:00:53.880
They are using photosynthesis
in order to take

00:00:53.880 --> 00:00:57.750
light energy from the sun in
conjunction with carbon dioxide

00:00:57.750 --> 00:01:02.750
in the air and water, in order
to store energy in its bonds.

00:01:03.050 --> 00:01:06.670
Now the grasshopper and
the mouse eat that grass

00:01:06.670 --> 00:01:08.490
for that energy.

00:01:08.490 --> 00:01:11.630
And so, the grasshopper and the mouse,

00:01:11.630 --> 00:01:14.950
since they directly eat
that primary producer,

00:01:14.950 --> 00:01:17.130
they would be primary...

00:01:17.130 --> 00:01:20.400
Primary consumers.

00:01:20.400 --> 00:01:22.120
Consumers.

00:01:22.120 --> 00:01:24.410
And I think you see where this is going.

00:01:24.410 --> 00:01:27.720
The folks who then eat
the primary consumers,

00:01:27.720 --> 00:01:30.190
so this would be the hawk,
the coyote, the vulture,

00:01:30.190 --> 00:01:35.170
and the snake, these would
all be secondary consumers.

00:01:35.170 --> 00:01:36.890
Secondary...

00:01:36.890 --> 00:01:40.160
Secondary consumers.

00:01:40.160 --> 00:01:40.993
And we're done.

00:01:40.993 --> 00:01:42.910
Who is a secondary
consumer in this diagram?

00:01:42.910 --> 00:01:44.840
We could say the coyote's
a secondary consumer.

00:01:44.840 --> 00:01:47.440
The hawk is also a secondary consumer.

00:01:47.440 --> 00:01:49.350
The vulture is a secondary consumer.

00:01:49.350 --> 00:01:51.500
And so is the snake.

00:01:51.500 --> 00:01:54.410
And as you can see, that's
okay, even in a situation

00:01:54.410 --> 00:01:56.520
where some secondary consumers eat

00:01:56.520 --> 00:01:58.227
other secondary consumers.

00:01:58.227 --> 00:02:01.640
The coyote might eat a hawk,
or a vulture might eat a snake,

00:02:01.640 --> 00:02:04.160
and a coyote might eat a
vulture, which eats a snake,

00:02:04.160 --> 00:02:07.250
which might eat a mouse,
which eats the grass.

00:02:07.250 --> 00:02:10.813
But any of these could be
considered secondary consumers.