Presidential oversight of the bureaucracy

WEBVTT
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Language: en

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- [Instructor] You can view this diagram

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as an org chart of the government

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of the United States, and what we have

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highlighted in this blue-green color,

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that is the executive branch,

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the things that fall under
the executive branch.

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And you can't even see everything,

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a lot of it falls off of the screen.

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When you see something like this

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with roughly 2.6 million people,

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part of that executive branch,

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part of the government's bureaucracy,

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and with the president at the top

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of the executive branch,
you might get the idea

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that the president is quite powerful.

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And you would be correct.

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The president has a powerful role,

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but there're some factors about how

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the executive branch is structured

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and its size that might keep the president

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from being quite as powerful

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as you might first believe.

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First of all, managing 2.6 million people

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is no easy task.

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Ask anyone who even has to manage

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an organization of 20 or 30 people,

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much less 2,000 people or 200,000 people,

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it can get quite difficult.

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You have to manage through
many layers of management.

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The other interesting thing

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that makes the president's role

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in the executive branch different

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than, say, the CEO of a corporation,

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is that yes, the president can make

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many appointments, in fact, the president

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can make several thousand appointments

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and most of those appointments

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sit atop the various departments.

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But most of the bureaucracy,

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the federal bureaucracy,
most of this 2.6 million,

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they are not appointed by the president.

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Those are professionals in their field,

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they are part of the civil service system.

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So, as they make various judgements

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and interpret laws in certain ways

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and execute the machinery
of the bureaucracy,

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they might not be completely in lockstep

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with the goals of the president,

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the goals that the president has

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for their administration.

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Related to that, the senior appointments,

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especially the cabinet secretaries,

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those serve at the
pleasure of the president.

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If the president becomes
unhappy with them,

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if the president finds
that they're disloyal

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in some way, the president

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can choose to have them leave,

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essentially fire them,
although in the news

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you'll sometimes see as the president

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accepts their resignation.

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But as you get further
down the bureaucracy,

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the president does not have that control.

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The Supreme Court has
ruled that for officers

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the president can fire them.

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But as you go down into
the regular ranking file,

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the majority of this 2.6 million people,

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the president can't
fire them without cause.

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The president can't fire them just because

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they don't like those employees

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or they feel that those employees

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don't completely agree with the president.

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It has to be for a just cause.

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This is why you'll see that some parts

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of the executive branch, say,

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from the Department of Justice,

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they might have investigations

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into things that the president

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does not want them to investigate into.

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Often times they will
investigate the president.

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This happened during the
Nixon administration,

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part of the Watergate investigations.

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This happened under the
Clinton administration.

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This is happening at
the time of this video

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as part of the Trump administration,

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the Mueller investigation.

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And the president can't just willy-nilly

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fire folks if they say,

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"Hey, why are you
looking into my matters?"

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because of this principle.

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So they can't fire just anyone

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within the executive branch.

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Now, what leverage does the president have

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to maybe exert more control

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than the things that
I have just described?

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Well, of course, I've mentioned

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the appointments that
the president can make,

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and this could number

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in the several thousands of appointments.

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I've seen estimates as high as 4,000

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appointments that the president can make.

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They can fire, the
president can fire officers.

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So these would be the senior
political appointments,

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things like the cabinet secretaries,

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the people at the head

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of these various departments.

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But even that could have

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some significant political consequences.

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The president could try to shift

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responsibilities from one part

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of the bureaucracy to another,

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or shift budget from one part

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of the bureaucracy to another.

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But this is also not so easy,

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'cause once again it's a large

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and complex bureaucracy.

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The president would have to do it

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through many layers of management.

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And at the end of the
day things like budget,

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which department gets how much money,

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this is, at the end of the day,

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something that the Congress will decide

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as once again part of the separation

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of powers that we've talked about

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in multiple videos.

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Another thing that the
president can try to do,

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and we've talked about
this in other videos,

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are the idea of executive orders.

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Executive orders often times can

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be a president trying to clarify

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how a law should be interpreted,

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how it should be executed upon

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by the bureaucracy, if the president feels

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that its bureaucracy is
not interpreting laws

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in the way that meets
the president's goals.

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So I will leave you there.

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The president clearly the most powerful

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office in the United States,

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on top of a 2.6 million person bureaucracy

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that's part of the executive branch.

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But the president is not a king,

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the president is not a
chief executive officer

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of a for-profit company
where they can have

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a lot more control.

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They're limited by just

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the inertia of the bureaucracy.

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They're limited by the ability

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to hire and fire people.

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They're limited by the complexity

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of the bureaucracy.

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But they do have some
tools at their disposal,

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where they can try to shape the direction

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and the goals of these

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2.6 million federal employees.


Gupta Dynasty

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- [Instructor] In previous videos,

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we talk about the emergence
of the Maurya Empire,

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around 322 BCE,

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shortly after the invasion
of Alexander the Great,

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as the first truly great Indian empire

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that unifies most of
the Indian subcontinent.

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That empire eventually falls,

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and the next significant empire to emerge,

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especially if we talk about

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influence on India and the world,

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is the Gupta Empire, which
emerges over 500 years later.

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Let's zoom in on our timeline

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to get a deeper appreciation
of the Gupta Empire.

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It's believed that its
start was with Sri Gupta,

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he started the Gupta Dynasty around 240,

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and it's disputed where they emerged,

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it might have been in that
region or in that region.

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There's different accounts of where

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the Gupta Dynasty initially emerged,

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but even in the early 300's,

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they really had a control
of a few small kingdoms.

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It wasn't until the
reign of Chandragupta I

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that it becomes a significant dynasty.

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We need to be careful.

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Don't confuse this
Chandragupta, the First,

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with Chandragupta Maurya, who
founded the Maurya Dynasty

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over 600 years before the
time we're talking about.

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Chandragupta I, it's interesting because

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he is really able to gain power,

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not initially through conquest,
but through a marriage.

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He has a marriage with
the princess Kumaradevi,

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and as a dowry, he's given control over

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much of this region of northeast India.

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This region of Magadha or Magadha,

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I am always having
trouble pronouncing that,

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so my apologies,

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including the famous city of Pataliputra,

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which even at the time
of the Maurya Empire

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and before the Maurya Empire,

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this was a famous seat of power.

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But once he's in control of this region,

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then he and his successors are able

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to have increased conquests over India.

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You see in this light blue color

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what his son Samudragupta was able to do,

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and then one of Samudragupta's sons,

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Chandragupta II, is able
to conquer even more.

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But what makes the
Gupta Empire distinctive

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isn't just that they
were able to unify much,

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or conquer much, of India again.

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What really makes them distinctive

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is because of that unification,

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and especially the
wealth that began to flow

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into the capitol,

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they were able to be sponsors
of significant culture,

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and science, and the arts.

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That's why historians
view the Gupta Empire

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as the golden age of India.

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Just to get an appreciation for this,

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the Gupta Empire was during
the time of Kalidasa,

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and he is considered to be
the greatest writer ever

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in the Sanskrit language.

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He is the William Shakespeare of Sanskrit.

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Beyond literature and writing,

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you have significant
contributions to science,

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most notably, Aryabhata.

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He's known for a very
accurate approximation of pi,

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but even more important, a recognition

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that was an approximation,

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and that he potentially recognized
the irrationality of pi,

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one of the first to do so.

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The word sin, the trig function,

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is derived from Aryabhata's
word for that function,

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so he established some of the
early ideas of trigonometry.

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He did work in summation,

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he did significant astronomical work,

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recognizing the rotation of the Earth

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versus the rotation of the heavens.

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He had an early concept of gravity.

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Even some of these

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notions of the place value system,

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and zero, and decimal notation,

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many of our modern notions of it

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are traced back to Aryabhata.

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In other videos on the Islamic golden age,

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when we talk about
folks like al-Khwarizmi,

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a lot of his work was based on

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what he learned from Aryabhata.

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Beyond the sciences, and once again,

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this is just a sample of all that happened

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during this period,

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you have the significant
Hindu epics, the Mahabharata,

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the Ramayana, the Periyas, get
written down and formalized.

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You can say they were canonized.

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The game of chess,

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or the early version of the game of chess,

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was invented, called chaturanga,

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and they had horsemen,
which were the knights,

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they had infantry, which were the pawns,

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they had elephants, which
eventually turned into bishops,

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but as it migrated into
Persia, the Muslim world,

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and then into Europe, it became
our modern game of chess.

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Famously, there is this iron pillar,

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that is now in New Delhi,

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that is traced back to the
time of the Gupta Empire,

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and is believed to the
reign of Chandragupta II.

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What's amazing about this,

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this is a pillar that's over 20 feet high,

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made out of raw iron, and
over 1500, 1600 years,

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it hasn't corroded.

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It has inscriptions on
it that help historians

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point to the Gupta Empire.

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This is some of the coinage
of the Gupta Empire.

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So the big takeaway here is,
this was India's golden age,

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the classical period of India.

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A lot of modern Hinduism
and Indian culture

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can be traced back to this time period.

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But it isn't just its influence on India.

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In other videos, we talk
about the Islamic golden age,

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and much of that golden age,

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which emerges two, three,
400 years after the time,

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after the Gupta Empire falls,

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much of that work in
based on the discoveries

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and the work that is collected

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during the time of the Guptas,

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and then that becomes
a bridge, eventually,

00:06:06.920 --> 00:06:09.120
to the European Renaissance.

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Now, like all empires,

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the Gupta Empire does eventually fall,

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in the mid-sixth century,
it's believed around 540, 550,

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and one of the main causes,

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there is invasions from
people called the Hunas,

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who historians believe
are either the Huns,

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or a group that are closely
related to the Huns.

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It's considered to be one of the causes

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of the eventual decline
of the Gupta Dynasty.


Marginal and conditional distributions

WEBVTT
Kind: captions
Language: en

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- [Instructor] Let's
say that we are trying

00:00:01.240 --> 00:00:04.640
to understand a relationship
in a classroom of 200 students

00:00:04.640 --> 00:00:09.080
between the amount of time
studied and the percent correct.

00:00:09.080 --> 00:00:11.050
What we could do is we
could set up some buckets

00:00:11.050 --> 00:00:14.230
of time studied and some
buckets of percent correct

00:00:14.230 --> 00:00:16.240
and then we could survey the students

00:00:16.240 --> 00:00:19.530
and/or look at the data
from the scores on the test.

00:00:19.530 --> 00:00:22.800
And then we can place
students in these buckets.

00:00:22.800 --> 00:00:25.390
So what you see right over
here, this is a two-way table.

00:00:25.390 --> 00:00:28.000
And you can also view this
as a joint distribution

00:00:28.000 --> 00:00:30.130
along these two dimensions.

00:00:30.130 --> 00:00:34.070
So one way to read this is that 20

00:00:34.070 --> 00:00:36.880
out of the 200 total students got

00:00:36.880 --> 00:00:40.220
between a 60 and 79% on the test

00:00:40.220 --> 00:00:44.290
and studied between 21 and 40 minutes.

00:00:44.290 --> 00:00:46.040
So there's all sorts of interesting things

00:00:46.040 --> 00:00:48.240
that we could try to glean from this,

00:00:48.240 --> 00:00:49.770
but what we're going
to focus on this video

00:00:49.770 --> 00:00:52.080
is two more types of distributions

00:00:52.080 --> 00:00:55.110
other than the joint distribution
that we see in this data.

00:00:55.110 --> 00:00:58.300
One type is a marginal distribution.

00:00:58.300 --> 00:01:02.060
And a marginal distribution
is just focusing

00:01:02.060 --> 00:01:04.800
on one of these dimensions.

00:01:04.800 --> 00:01:07.760
And one way to think about
it is you can determine it

00:01:07.760 --> 00:01:09.260
by looking at the margin.

00:01:09.260 --> 00:01:10.550
So, for example, if you wanted

00:01:10.550 --> 00:01:13.140
to figure out the marginal distribution

00:01:13.140 --> 00:01:16.180
of the percent correct,

00:01:16.180 --> 00:01:19.290
what you could do is look
at the total of these rows.

00:01:19.290 --> 00:01:21.400
So these counts right over here

00:01:21.400 --> 00:01:25.090
give you the marginal distribution
of the percent correct.

00:01:25.090 --> 00:01:28.840
40 out of the 200 got
between 80 and a hundred.

00:01:28.840 --> 00:01:32.950
60 out of the 200 got between
60 and 79, so on and so forth.

00:01:32.950 --> 00:01:35.170
Now, a marginal distribution
could be represented

00:01:35.170 --> 00:01:37.450
as counts or as percentages.

00:01:37.450 --> 00:01:38.960
So if you represent it as percentages,

00:01:38.960 --> 00:01:41.220
you would divide each of
these counts by the total,

00:01:41.220 --> 00:01:42.310
which is 200.

00:01:42.310 --> 00:01:45.200
So 40 over 200, that would be 20%.

00:01:45.200 --> 00:01:48.120
60 out of 200, that would be 30%.

00:01:48.120 --> 00:01:51.550
70 out of 200, that would be 35%.

00:01:51.550 --> 00:01:53.540
20 out of 200 is 10%.

00:01:53.540 --> 00:01:56.110
And 10 out of 200 is 5%.

00:01:56.110 --> 00:01:58.270
So this right over here
in terms of percentages

00:01:58.270 --> 00:02:01.610
gives you the marginal
distribution of the percent correct

00:02:01.610 --> 00:02:03.040
based on these buckets.

00:02:03.040 --> 00:02:07.520
So you can say 10% got
between a 20 and a 39.

00:02:07.520 --> 00:02:08.370
Now, you could also think

00:02:08.370 --> 00:02:10.530
about marginal
distributions the other way.

00:02:10.530 --> 00:02:11.990
You could think about
the marginal distribution

00:02:11.990 --> 00:02:14.040
for the time studied in the class.

00:02:14.040 --> 00:02:18.690
Then you would look at these
counts right over here.

00:02:18.690 --> 00:02:21.330
You would say a total
of 14 students studied

00:02:21.330 --> 00:02:23.580
between zero and 20 minutes.

00:02:23.580 --> 00:02:26.020
You're not thinking about
the percent correct anymore.

00:02:26.020 --> 00:02:28.680
A total of 30 studied
between 21 and 40 minutes.

00:02:28.680 --> 00:02:31.470
And likewise, you could
write these as percentages.

00:02:31.470 --> 00:02:33.270
This would be 7%.

00:02:34.120 --> 00:02:35.980
This would be 15%.

00:02:35.980 --> 00:02:38.200
This would be 43%.

00:02:38.200 --> 00:02:42.420
And this would be 35% right over there.

00:02:42.420 --> 00:02:45.140
Now, another idea that
you might sometimes see

00:02:45.140 --> 00:02:45.973
when people are trying

00:02:45.973 --> 00:02:48.390
to interpret a joint
distribution like this

00:02:48.390 --> 00:02:51.900
or get more information or
more realizations from it

00:02:51.900 --> 00:02:53.680
is to think about something known

00:02:53.680 --> 00:02:56.023
as a conditional distribution.

00:02:57.030 --> 00:03:01.290
Conditional distribution.

00:03:01.290 --> 00:03:04.680
And this is the
distribution of one variable

00:03:04.680 --> 00:03:07.350
given something true
about the other variable.

00:03:07.350 --> 00:03:09.840
So, for example, an example
of a conditional distribution

00:03:09.840 --> 00:03:14.840
would be the distribution
of percent correct

00:03:16.510 --> 00:03:19.770
given that students

00:03:24.670 --> 00:03:29.670
study between, let's
say, 41 and 60 minutes.

00:03:30.390 --> 00:03:34.310
Between 41

00:03:34.310 --> 00:03:36.810
and 60 minutes.

00:03:36.810 --> 00:03:37.950
Well, to think about that,

00:03:37.950 --> 00:03:40.020
you would first look at your condition.

00:03:40.020 --> 00:03:41.290
Okay, let's look at the students

00:03:41.290 --> 00:03:43.900
who have studied between
41 and 60 minutes.

00:03:43.900 --> 00:03:46.230
That would be this column right over here.

00:03:46.230 --> 00:03:48.470
And then that column,
the information in it,

00:03:48.470 --> 00:03:51.060
can give you your
conditional distribution.

00:03:51.060 --> 00:03:52.430
Now, an important thing to realize

00:03:52.430 --> 00:03:54.450
is a marginal distribution
can be represented

00:03:54.450 --> 00:03:57.430
as counts for the various
buckets or percentages,

00:03:57.430 --> 00:03:59.980
while the standard practice
for conditional distribution

00:03:59.980 --> 00:04:02.150
is to think in terms of percentages.

00:04:02.150 --> 00:04:04.790
So the conditional distribution
of the percent correct

00:04:04.790 --> 00:04:07.910
given that students study
between 41 and 60 minutes,

00:04:07.910 --> 00:04:09.450
it would look something like this.

00:04:09.450 --> 00:04:11.140
Let me get a little bit more space.

00:04:11.140 --> 00:04:16.140
So if we set up the various
categories, 80 to 100,

00:04:16.290 --> 00:04:21.290
60 to 79, 40 to 59,

00:04:21.590 --> 00:04:25.110
continue it over here, 20 to 39,

00:04:25.110 --> 00:04:27.470
and zero to 19,

00:04:27.470 --> 00:04:29.450
what we'd wanna do is
calculate the percentage

00:04:29.450 --> 00:04:31.750
that fall into each of these buckets given

00:04:31.750 --> 00:04:34.020
that we're studying
between 41 and 60 minutes.

00:04:34.020 --> 00:04:35.700
So this first one, 80 to a hundred,

00:04:35.700 --> 00:04:39.140
it would be 16 out of the 86 students.

00:04:39.140 --> 00:04:43.380
So we would write 16 out of 86,

00:04:43.380 --> 00:04:48.220
which is equal to 16 divided by 86

00:04:48.220 --> 00:04:50.950
is equal to, I'll just
round to one decimal place.

00:04:50.950 --> 00:04:53.723
It's roughly 18.6%.

00:04:53.723 --> 00:04:55.600
18.6.

00:04:55.600 --> 00:04:58.640
Approximately equal to 18.6%.

00:04:58.640 --> 00:05:01.250
And then to get the full
conditional distribution,

00:05:01.250 --> 00:05:02.270
we would keep doing that.

00:05:02.270 --> 00:05:03.870
We would figure out the percentage.

00:05:03.870 --> 00:05:07.620
60 to 79, that would 30 out of 86.

00:05:07.620 --> 00:05:11.440
30 out of 86, whatever percentage that is,

00:05:11.440 --> 00:05:12.620
and so on and so forth

00:05:12.620 --> 00:05:15.403
in order to get that entire distribution.


Limits of combined functions: piecewise functions

WEBVTT
Kind: captions
Language: en

00:00:00.280 --> 00:00:01.440
- [Instructor] We are asked to find

00:00:01.440 --> 00:00:04.310
these three different limits.

00:00:04.310 --> 00:00:05.520
I encourage you like always,

00:00:05.520 --> 00:00:07.260
pause this video and try to do it yourself

00:00:07.260 --> 00:00:09.330
before we do it together.

00:00:09.330 --> 00:00:10.590
So when you do this first one,

00:00:10.590 --> 00:00:11.990
you might just try to find the limit

00:00:11.990 --> 00:00:13.800
as x approaches negative two of f of x

00:00:13.800 --> 00:00:16.120
and then the limit as x
approaches negative two of g of x

00:00:16.120 --> 00:00:18.510
and then add those two limits together.

00:00:18.510 --> 00:00:20.530
But you will quickly find a problem,

00:00:20.530 --> 00:00:21.520
'cause when you find the limit

00:00:21.520 --> 00:00:23.850
as x approaches negative two of f of x,

00:00:23.850 --> 00:00:25.490
it looks as we are
approaching negative two

00:00:25.490 --> 00:00:27.700
from the left, it looks
like we're approaching one.

00:00:27.700 --> 00:00:30.020
As we approach x equals
negative two from the right,

00:00:30.020 --> 00:00:31.640
it looks like we're approaching three.

00:00:31.640 --> 00:00:32.690
So it looks like the limit

00:00:32.690 --> 00:00:35.270
as x approaches negative
two of f of x doesn't exist,

00:00:35.270 --> 00:00:37.020
and the same thing's true of g of x.

00:00:37.020 --> 00:00:38.010
If we approach from the left,

00:00:38.010 --> 00:00:39.610
it looks like we're approaching three.

00:00:39.610 --> 00:00:40.930
If we approach from the right,

00:00:40.930 --> 00:00:42.790
it looks like we're approaching one.

00:00:42.790 --> 00:00:45.540
But it turns out that
this limit can still exist

00:00:45.540 --> 00:00:50.150
as long as the limit as
x approaches negative two

00:00:50.150 --> 00:00:52.520
from the left of the sum,

00:00:52.520 --> 00:00:54.473
f of x plus g of x,

00:00:55.820 --> 00:00:59.420
exists and is equal to the limit

00:00:59.420 --> 00:01:04.160
as x approaches negative two
from the right of the sum,

00:01:04.160 --> 00:01:06.870
f of x plus g of x.

00:01:06.870 --> 00:01:08.490
So what are these things?

00:01:08.490 --> 00:01:11.380
Well, as we approach
negative two from the left,

00:01:11.380 --> 00:01:14.780
f of x is approaching, looks like one,

00:01:14.780 --> 00:01:16.710
and g of x is approaching three.

00:01:16.710 --> 00:01:18.910
So it looks like we're
approaching one and three.

00:01:18.910 --> 00:01:20.170
So it looks like this is approaching.

00:01:20.170 --> 00:01:22.240
The sum is going to approach four.

00:01:22.240 --> 00:01:24.110
And if we're coming from the right,

00:01:24.110 --> 00:01:26.600
f of x looks like it's approaching three

00:01:26.600 --> 00:01:31.513
and g of x looks like
it is approaching one.

00:01:32.690 --> 00:01:34.750
Once again, this is equal to four.

00:01:34.750 --> 00:01:36.900
And since the left and right handed limits

00:01:36.900 --> 00:01:38.210
are approaching the same thing,

00:01:38.210 --> 00:01:42.100
we would say that this limit
exists and it is equal to four.

00:01:42.100 --> 00:01:45.700
Now let's do this next
example as x approaches one.

00:01:45.700 --> 00:01:48.090
Well, we'll do the exact same exercise.

00:01:48.090 --> 00:01:50.260
And once again, if you look
at the individual limits

00:01:50.260 --> 00:01:54.020
for f of x from the left and
the right as we approach one,

00:01:54.020 --> 00:01:55.410
this limit doesn't exist.

00:01:55.410 --> 00:01:58.470
But the limit as x approaches
one of the sum might exist,

00:01:58.470 --> 00:01:59.820
so let's try that out.

00:01:59.820 --> 00:02:03.020
So the limit as x approaches one

00:02:03.020 --> 00:02:08.020
from the left hand side
of f of x plus g of x,

00:02:08.210 --> 00:02:10.053
what is that going to be equal to?

00:02:11.160 --> 00:02:14.380
So f of x, as we approach
one from the left,

00:02:14.380 --> 00:02:16.420
looks like this is approaching two.

00:02:16.420 --> 00:02:17.850
I'm just doing this for shorthand.

00:02:17.850 --> 00:02:20.840
And g of x, as we approach
one from the left,

00:02:20.840 --> 00:02:23.120
it looks like it is approaching zero.

00:02:23.120 --> 00:02:26.150
So this will be approaching
two plus zero, which is two.

00:02:26.150 --> 00:02:27.750
And then the limit,

00:02:27.750 --> 00:02:31.270
as x approaches one
from the right hand side

00:02:31.270 --> 00:02:36.270
of f of x plus g of x
is going to be equal to.

00:02:36.330 --> 00:02:38.080
Well, for f of x as we're approaching one

00:02:38.080 --> 00:02:39.040
from the right hand side,

00:02:39.040 --> 00:02:41.090
looks like it's approaching negative one.

00:02:41.930 --> 00:02:44.280
And for g of x as we're approaching one

00:02:44.280 --> 00:02:45.140
from the the right hand side,

00:02:45.140 --> 00:02:47.770
looks like we're approaching zero again.

00:02:47.770 --> 00:02:49.880
Here it looks like we're
approaching negative one.

00:02:49.880 --> 00:02:51.900
So the left and right hand limits

00:02:51.900 --> 00:02:53.300
aren't approaching the same value,

00:02:53.300 --> 00:02:58.210
so this one does not exist.

00:02:58.210 --> 00:03:00.010
And then last but not least,

00:03:00.010 --> 00:03:03.430
x approaches one of f of x times g of x.

00:03:03.430 --> 00:03:04.840
So we'll do the same drill.

00:03:04.840 --> 00:03:09.840
Limit as x approaches one
from the left hand side

00:03:09.860 --> 00:03:13.303
of f of x times g of x.

00:03:14.240 --> 00:03:16.770
Well, here, and we can
even use the values here.

00:03:16.770 --> 00:03:19.030
We see it was approaching
one from the left.

00:03:19.030 --> 00:03:22.620
We are approaching two, so this is two.

00:03:22.620 --> 00:03:26.300
And when we're approaching
one from the left here,

00:03:26.300 --> 00:03:27.723
we're approaching zero.

00:03:30.310 --> 00:03:32.860
We're gonna be approaching
two times zero, which is zero.

00:03:32.860 --> 00:03:35.250
And then we approach from the right.

00:03:35.250 --> 00:03:38.010
X approaches one from the right

00:03:38.010 --> 00:03:42.770
of f of x times g of x.

00:03:42.770 --> 00:03:44.360
Well, we already saw when
we're approaching one

00:03:44.360 --> 00:03:46.410
from the right of f of x,

00:03:46.410 --> 00:03:48.900
we're approaching negative one.

00:03:48.900 --> 00:03:51.510
But g of x, approaching
one from the right,

00:03:51.510 --> 00:03:53.150
is still approaching zero,

00:03:53.150 --> 00:03:55.670
so this is going to be zero
again, so this limit exists.

00:03:55.670 --> 00:03:56.550
We get the same limit

00:03:56.550 --> 00:03:58.600
when we approach from
the left and the right.

00:03:58.600 --> 00:04:01.160
It is equal to zero.

00:04:01.160 --> 00:04:03.100
So these are pretty interesting examples,

00:04:03.100 --> 00:04:04.310
because sometimes when you think

00:04:04.310 --> 00:04:06.380
that the component limits don't exist

00:04:06.380 --> 00:04:07.450
that that means that the sum

00:04:07.450 --> 00:04:09.010
or the product might not exist,

00:04:09.010 --> 00:04:10.950
but this shows at least two examples

00:04:10.950 --> 00:04:12.300
where that is not the case.


Alternating series test

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:02.280
- [Narrator] Let's now expose ourselves

00:00:02.280 --> 00:00:04.940
to another test of convergence,

00:00:04.940 --> 00:00:08.230
and that's the Alternating Series Test.

00:00:08.230 --> 00:00:10.040
I'll explain the Alternating Series Test

00:00:10.040 --> 00:00:13.220
and I'll apply it to an
actual series while I do it

00:00:13.220 --> 00:00:14.450
to make the...

00:00:15.990 --> 00:00:18.088
Explanation of the Alternating Series Test

00:00:18.088 --> 00:00:20.900
a little bit more concrete.

00:00:20.900 --> 00:00:25.730
Let's say that I have some
series, some infinite series.

00:00:25.730 --> 00:00:30.730
Let's say it goes from N equals
K to infinity of A sub N.

00:00:30.940 --> 00:00:33.010
Let's say I can write it as

00:00:33.010 --> 00:00:34.580
or I can rewrite A sub N.

00:00:34.580 --> 00:00:37.260
So let's say A sub N, I can write.

00:00:37.260 --> 00:00:42.080
So A sub N is equal to
negative one to the N,

00:00:42.080 --> 00:00:43.770
times B sub N

00:00:43.770 --> 00:00:48.770
or A sub N is equal to
negative one to the N plus one

00:00:49.830 --> 00:00:53.185
times B sub N

00:00:53.185 --> 00:00:57.210
where B sub N is greater
than or equal to zero

00:00:57.210 --> 00:00:58.680
for all the Ns we care about.

00:00:58.680 --> 00:01:03.350
So for all of these integer
Ns greater than or equal to K.

00:01:03.350 --> 00:01:06.860
If all of these things, if
all of these things are true

00:01:06.860 --> 00:01:09.160
and we know two more things,

00:01:09.160 --> 00:01:14.160
and we know number one, the
limit as N approaches infinity

00:01:16.610 --> 00:01:19.893
of B sub N is equal to zero.

00:01:21.370 --> 00:01:26.370
Number two, B sub N is
a decreasing sequence.

00:01:27.970 --> 00:01:30.410
Decreasing...

00:01:30.410 --> 00:01:33.090
Decreasing sequence.

00:01:33.090 --> 00:01:38.090
Then that lets us know that
the original infinite series,

00:01:38.430 --> 00:01:40.890
the original infinite series,

00:01:40.890 --> 00:01:42.950
is going to converge.

00:01:42.950 --> 00:01:45.630
So this might seem a little
bit abstract right now.

00:01:45.630 --> 00:01:49.610
Let's actually show, let's
use this with an actual series

00:01:49.610 --> 00:01:51.260
to make it a little bit more,

00:01:51.260 --> 00:01:53.454
a little bit more concrete.

00:01:53.454 --> 00:01:56.121
Let's say that I had the series,

00:01:57.167 --> 00:02:02.167
let's say I had the series
from N equals one to infinity

00:02:02.602 --> 00:02:05.269
of negative one to the N over N.

00:02:08.895 --> 00:02:10.751
We could write it out
just to make this series

00:02:10.751 --> 00:02:11.899
a little bit more concrete.

00:02:11.899 --> 00:02:15.000
When N is equal to one, this
is gonna be negative one

00:02:15.000 --> 00:02:17.150
to the one power.

00:02:17.150 --> 00:02:18.680
Actually, let's just
make this a little bit,

00:02:18.680 --> 00:02:20.606
let's make this a little
bit more interesting.

00:02:20.606 --> 00:02:21.800
Let's make this negative
one to the N plus one.

00:02:21.800 --> 00:02:23.710
When N is equal to one,

00:02:23.710 --> 00:02:26.910
this is gonna be negative
one squared over one

00:02:26.910 --> 00:02:28.590
which is gonna be one.

00:02:28.590 --> 00:02:30.950
Then when N is two, it's
gonna be negative one

00:02:30.950 --> 00:02:31.940
to the third power

00:02:31.940 --> 00:02:34.630
which is gonna be negative one half.

00:02:34.630 --> 00:02:36.560
So it's minus one half

00:02:36.560 --> 00:02:38.430
plus one third

00:02:38.430 --> 00:02:40.150
minus one fourth

00:02:40.150 --> 00:02:44.280
plus minus and it keeps going
on and on and on forever.

00:02:44.280 --> 00:02:47.980
Now, can we rewrite
this A sub N like this.

00:02:47.980 --> 00:02:48.813
Well sure.

00:02:48.813 --> 00:02:50.190
The negative one to the N plus one is

00:02:50.190 --> 00:02:52.150
actually explicitly called out.

00:02:52.150 --> 00:02:55.990
We can rewrite our A sub N,

00:02:55.990 --> 00:02:57.540
so let me do that.

00:02:57.540 --> 00:03:02.529
So negative, so A sub N which
is equal to negative one

00:03:02.529 --> 00:03:06.090
to the N plus one over N.

00:03:06.090 --> 00:03:10.170
This is clearly the same
thing as negative one

00:03:10.170 --> 00:03:13.880
to the N plus one times one over N

00:03:13.880 --> 00:03:16.580
which is, which we can then say

00:03:16.580 --> 00:03:19.943
this thing right over
here could be our B sub N.

00:03:19.943 --> 00:03:23.650
This right over here is our B sub N.

00:03:23.650 --> 00:03:26.050
We can verify that our
B sub N is going to be

00:03:26.050 --> 00:03:29.060
greater than or equal to zero
for all the Ns we care about.

00:03:29.060 --> 00:03:32.754
So our B sub N is equal to one over N.

00:03:32.754 --> 00:03:35.070
Clearly this is gonna be
greater than or equal to zero

00:03:35.070 --> 00:03:38.470
for any, for any positive N.

00:03:38.470 --> 00:03:39.640
Now what's the limit?

00:03:39.640 --> 00:03:41.021
As B sub N,

00:03:41.021 --> 00:03:45.000
What's the limit of B sub
N as N approaches infinity?

00:03:45.000 --> 00:03:48.300
The limit of, let me
just write one over N,

00:03:48.300 --> 00:03:52.480
one over N, as N approaches
infinity is going

00:03:52.480 --> 00:03:53.440
to be equal to zero.

00:03:53.440 --> 00:03:55.920
So we satisfied the first constraint.

00:03:55.920 --> 00:03:58.336
Then this is clearly a decreasing sequence

00:03:58.336 --> 00:04:02.630
as N increases the denominators
are going to increase.

00:04:02.630 --> 00:04:03.720
With a larger denominator,

00:04:03.720 --> 00:04:05.450
you're going to have a lower value.

00:04:05.450 --> 00:04:09.256
We can also say one
over N is a decreasing,

00:04:09.256 --> 00:04:13.130
decreasing sequence

00:04:13.130 --> 00:04:15.460
for the Ns that we care about.

00:04:15.460 --> 00:04:20.460
So this satifies, this
is satisfied as well.

00:04:20.560 --> 00:04:24.650
Based on that, this thing is always,

00:04:24.650 --> 00:04:26.510
this thing right over here is always

00:04:26.510 --> 00:04:28.410
greater than or equal to zero.

00:04:28.410 --> 00:04:31.823
The limit, as one over
N or as our B sub N,

00:04:31.823 --> 00:04:34.680
as N approaches infinity,
is going to be zero.

00:04:34.680 --> 00:04:35.920
It's a decreasing sequence.

00:04:35.920 --> 00:04:38.950
Therefore we can say
that our originial series

00:04:38.950 --> 00:04:40.870
actually converges.

00:04:40.870 --> 00:04:45.053
So N equals 1 to infinity of negative one

00:04:45.053 --> 00:04:47.540
to the N plus over N.

00:04:47.540 --> 00:04:49.100
And that's kind of interesting.

00:04:49.100 --> 00:04:50.250
Because we've already seen

00:04:50.250 --> 00:04:51.680
that if all of these were positive,

00:04:51.680 --> 00:04:53.220
if all of these terms were positive,

00:04:53.220 --> 00:04:55.750
we just have the Harmonic Series,

00:04:55.750 --> 00:04:57.860
and that one didn't converge.

00:04:57.860 --> 00:05:00.820
But this one did, putting these
negatives here do the trick.

00:05:00.820 --> 00:05:03.630
Actually we can prove this
one over here converges

00:05:03.630 --> 00:05:05.460
using other techniques.

00:05:05.460 --> 00:05:06.780
Maybe if we have time,

00:05:06.780 --> 00:05:09.080
actually in particular
the limit comparison test.

00:05:09.080 --> 00:05:12.210
I'll just throw that out
there in case you are curious.

00:05:12.210 --> 00:05:14.170
So this is a pretty powerful tool.

00:05:14.170 --> 00:05:17.010
It looks a little bit about
like that Divergence Test,

00:05:17.010 --> 00:05:19.090
but remember the
Divergence Test is really,

00:05:19.090 --> 00:05:22.010
is only useful if you want
to show something diverges.

00:05:22.010 --> 00:05:25.490
If the limit of, if
the limit of your terms

00:05:25.490 --> 00:05:27.330
do not approach zero,

00:05:27.330 --> 00:05:29.600
then you say okay, that
thing is going to diverge.

00:05:29.600 --> 00:05:30.433
This thing is useful

00:05:30.433 --> 00:05:33.220
because you can actually
prove convergence.

00:05:33.220 --> 00:05:36.010
Once again, if something does not pass

00:05:36.010 --> 00:05:38.010
the alternating series test,

00:05:38.010 --> 00:05:40.860
that does not necessarily
mean that it diverges.

00:05:40.860 --> 00:05:42.470
It just means that you couldn't use

00:05:42.470 --> 00:05:46.623
the Alternating Series Test
to prove that it converges.


Continuity at a point

WEBVTT
Kind: captions
Language: en

00:00:01.080 --> 00:00:02.640
- [Instructor] What we're
going to do in this video

00:00:02.640 --> 00:00:07.263
is come up with a more rigorous
definition for continuity.

00:00:08.540 --> 00:00:10.420
And the general idea of continuity,

00:00:10.420 --> 00:00:12.980
we've got an intuitive idea of the past,

00:00:12.980 --> 00:00:15.660
is that a function is
continuous at a point,

00:00:15.660 --> 00:00:18.350
is if you can draw the
graph of that function

00:00:18.350 --> 00:00:20.240
at that point without
picking up your pencil.

00:00:20.240 --> 00:00:21.790
So what do we mean by that?

00:00:21.790 --> 00:00:22.730
And this is...

00:00:22.730 --> 00:00:24.520
What I just said is not that rigorous,

00:00:24.520 --> 00:00:26.520
or not rigorous at all,

00:00:26.520 --> 00:00:29.930
is that well, let's think about
the point right over here.

00:00:29.930 --> 00:00:31.690
Let's say that's RC.

00:00:31.690 --> 00:00:34.320
If I can draw the graph at that point,

00:00:34.320 --> 00:00:35.560
the value of the function at that point

00:00:35.560 --> 00:00:38.390
without picking up my pencil, or my pen,

00:00:38.390 --> 00:00:39.430
then it's continuous there.

00:00:39.430 --> 00:00:40.770
So I could just start here,

00:00:40.770 --> 00:00:42.590
and I don't have to pick up my pencil,

00:00:42.590 --> 00:00:43.440
and there you go.

00:00:43.440 --> 00:00:45.710
I can go through that point,

00:00:45.710 --> 00:00:48.750
so we could say that our
function is continuous there.

00:00:48.750 --> 00:00:50.750
But if I had a function that looked

00:00:50.750 --> 00:00:52.440
somewhat different that that,

00:00:52.440 --> 00:00:54.850
if I had a function that looked like this,

00:00:54.850 --> 00:00:58.330
let's say that it is
defined up until then,

00:00:58.330 --> 00:00:59.990
and then there's a bit of a jump,

00:00:59.990 --> 00:01:02.120
and then it goes like this,

00:01:02.120 --> 00:01:05.373
well this would be very hard to draw at...

00:01:05.373 --> 00:01:07.750
This function would be very hard to draw

00:01:07.750 --> 00:01:11.580
going through x equals c
without picking up my pen.

00:01:11.580 --> 00:01:13.160
Let's see, my pen is touching the screen,

00:01:13.160 --> 00:01:15.150
touching the screen, touching the screen.

00:01:15.150 --> 00:01:17.530
How do I keep drawing this function

00:01:17.530 --> 00:01:20.060
without picking up my pen?

00:01:20.060 --> 00:01:21.540
I would have to pick it up,

00:01:21.540 --> 00:01:23.090
and then move back down here.

00:01:23.090 --> 00:01:25.620
And so that is an intuitive sense

00:01:25.620 --> 00:01:29.200
that we are not continuous
in this case right over here.

00:01:29.200 --> 00:01:30.330
Well let's actually come up

00:01:30.330 --> 00:01:34.190
with a formal definition for continuity,

00:01:34.190 --> 00:01:37.590
and then see if it feels intuitive for us.

00:01:37.590 --> 00:01:41.630
So the formal definition of
continuity, let's start here,

00:01:41.630 --> 00:01:43.930
we'll start with continuity at a point.

00:01:43.930 --> 00:01:48.260
So we could say the
function f is continuous...

00:01:49.100 --> 00:01:54.100
Continuous at x equals c.

00:01:54.940 --> 00:01:56.580
If, and only if...

00:01:56.580 --> 00:02:00.470
I'll draw this two-way arrow
to show if, and only if,

00:02:00.470 --> 00:02:03.950
the two-sided limit of f of x,

00:02:03.950 --> 00:02:06.880
as x approaches c,

00:02:06.880 --> 00:02:09.690
is equal to f of c.

00:02:09.690 --> 00:02:11.550
So this seems very technical.

00:02:11.550 --> 00:02:13.900
But let's just think
about what it's saying.

00:02:13.900 --> 00:02:14.820
It's saying look,

00:02:14.820 --> 00:02:17.680
if the limit as we approach c

00:02:17.680 --> 00:02:20.180
from the left and the right of f of x,

00:02:20.180 --> 00:02:22.680
if that's actually the
value of our function there,

00:02:22.680 --> 00:02:25.750
then we are continuous at that point.

00:02:25.750 --> 00:02:27.420
So let's look at three examples.

00:02:27.420 --> 00:02:30.270
Let's look at one
example, we are, we're...

00:02:30.270 --> 00:02:33.390
By our picking-up-the-pencil idea,

00:02:33.390 --> 00:02:35.460
it feels like we are
continuous at a point.

00:02:35.460 --> 00:02:37.190
And then let's think
about a couple of examples

00:02:37.190 --> 00:02:39.270
where it doesn't seem like
we're continuous at a point,

00:02:39.270 --> 00:02:43.460
and see how this more
rigorous definition applies.

00:02:43.460 --> 00:02:46.410
So, let's say that my function...

00:02:46.410 --> 00:02:48.680
So let's say this right over here

00:02:48.680 --> 00:02:52.810
is y is equal to f of x.

00:02:52.810 --> 00:02:55.260
And, we care about the
behavior right over here

00:02:55.260 --> 00:02:56.390
when x is equal to c.

00:02:56.390 --> 00:02:59.280
This is my X-axis, that's my Y-axis.

00:02:59.280 --> 00:03:03.100
So we care about the behavior
when x is equal to c.

00:03:03.100 --> 00:03:05.010
And so, notice,

00:03:05.010 --> 00:03:07.180
from our first intuitive sense,

00:03:07.180 --> 00:03:09.720
I can definitely draw this function

00:03:09.720 --> 00:03:13.110
as we go through x equals c
without picking up my pencil,

00:03:13.110 --> 00:03:14.610
so it feels continuous there.

00:03:14.610 --> 00:03:19.000
There's no jumps or
discontinuities that we can tell.

00:03:19.000 --> 00:03:20.570
It just kind of keeps on going.

00:03:20.570 --> 00:03:23.830
It seems all connected, is
one way to think about it.

00:03:23.830 --> 00:03:25.960
But let's think about this definition.

00:03:25.960 --> 00:03:30.960
Well, the limit as x approaches
c from the left, it is...

00:03:31.640 --> 00:03:33.580
As we approach from the left,

00:03:33.580 --> 00:03:36.570
it looks like it is approaching...

00:03:36.570 --> 00:03:40.760
It looks like it is approaching f of c.

00:03:40.760 --> 00:03:45.760
So this is the value,
f of c right over here.

00:03:45.820 --> 00:03:48.920
And as we approach from the right,

00:03:48.920 --> 00:03:51.530
as we approach from the right,

00:03:51.530 --> 00:03:55.410
it also looks like it's
approaching f of c.

00:03:55.410 --> 00:03:57.520
And we are defined right at x equals c.

00:03:57.520 --> 00:04:00.150
And it is the value
that we are approaching

00:04:00.150 --> 00:04:01.910
from both the left or the right.

00:04:01.910 --> 00:04:04.140
So this seems good in this scenario.

00:04:04.140 --> 00:04:05.480
So now let's look at some scenarios

00:04:05.480 --> 00:04:07.370
that we would have to pick up the pencil

00:04:07.370 --> 00:04:11.010
as we draw the function through
that point, through that...

00:04:11.010 --> 00:04:11.998
Through that...

00:04:11.998 --> 00:04:14.840
When x is equal to c.

00:04:14.840 --> 00:04:16.933
So let's look at a scenario.

00:04:17.850 --> 00:04:19.830
Let's look at a scenario where we have

00:04:19.830 --> 00:04:21.870
what's often called a point discontinuity,

00:04:21.870 --> 00:04:24.000
although you don't have
to know at this point,

00:04:24.000 --> 00:04:27.540
no pun intended, the
formal terminology for it.

00:04:27.540 --> 00:04:30.970
So let's say we have a function that...

00:04:30.970 --> 00:04:32.737
Let's see, this is c.

00:04:32.737 --> 00:04:35.690
Now let's say our function
looks something...

00:04:35.690 --> 00:04:37.500
Something like this.

00:04:37.500 --> 00:04:39.260
So we go like this,

00:04:39.260 --> 00:04:42.630
and at c let's say it's equal to that.

00:04:42.630 --> 00:04:46.660
So, f of c is right over here.

00:04:46.660 --> 00:04:48.610
F of c would be that value.

00:04:48.610 --> 00:04:51.300
But what's the limit as x approaches c?

00:04:51.300 --> 00:04:54.020
So the limit as x approaches c,

00:04:54.020 --> 00:04:57.610
this would be a two-sided limit of f of x.

00:04:57.610 --> 00:05:00.270
Well, this is, as we
approach from the left,

00:05:00.270 --> 00:05:03.860
it looks like we are approaching
this value right over here.

00:05:03.860 --> 00:05:04.890
And from the right,

00:05:04.890 --> 00:05:08.460
it looks like we are
approaching that same value.

00:05:08.460 --> 00:05:10.073
And so, we could call that L.

00:05:11.160 --> 00:05:13.960
And L is different than f of c.

00:05:13.960 --> 00:05:16.740
And so, in this case, by
our formal definition,

00:05:16.740 --> 00:05:19.350
we will not be continuous at, for...

00:05:19.350 --> 00:05:21.930
F will not be continuous for x is equal...

00:05:21.930 --> 00:05:23.530
Or at the point x...

00:05:23.530 --> 00:05:26.780
Or when x is equal to c.

00:05:26.780 --> 00:05:27.770
And you can see that there.

00:05:27.770 --> 00:05:29.160
If we try to draw this, okay,

00:05:29.160 --> 00:05:30.380
my pencil is touching the paper,

00:05:30.380 --> 00:05:32.080
touching the paper, touching the paper.

00:05:32.080 --> 00:05:34.740
Uh oh, if I needed to keep
drawing this function,

00:05:34.740 --> 00:05:37.200
I'd have to pick up my
pencil, move it over here,

00:05:37.200 --> 00:05:39.730
then pick it up again and
then jump right back down.

00:05:39.730 --> 00:05:41.580
And but this rigorous definition

00:05:41.580 --> 00:05:43.720
is giving us the same conclusion.

00:05:43.720 --> 00:05:46.160
The limit as we approach x equals c

00:05:46.160 --> 00:05:47.370
from the left and the right,

00:05:47.370 --> 00:05:49.640
it's a different value than f of c.

00:05:49.640 --> 00:05:52.340
And so, this is not continuous.

00:05:52.340 --> 00:05:53.900
Not...

00:05:53.900 --> 00:05:56.203
Not continuous.

00:05:57.330 --> 00:05:59.140
And let's think about another scenario.

00:05:59.140 --> 00:06:00.370
Let's think about a scenario...

00:06:00.370 --> 00:06:01.810
And actually, maybe let's
think about a scenario

00:06:01.810 --> 00:06:02.980
where the limit...

00:06:02.980 --> 00:06:05.600
The two-sided limit doesn't even exist.

00:06:05.600 --> 00:06:09.830
So, there are my axes, x and y.

00:06:09.830 --> 00:06:11.670
And let's say it's doing
something like this.

00:06:11.670 --> 00:06:14.560
Let's say it's doing something like this,

00:06:14.560 --> 00:06:17.490
and that it does something
like this and goes like that.

00:06:17.490 --> 00:06:22.190
And let's say that this
right over here is our c.

00:06:22.190 --> 00:06:26.183
And so let's see, this is
f of c right over here.

00:06:27.310 --> 00:06:28.740
That is...

00:06:28.740 --> 00:06:30.290
Lemme draw a little bit neater.

00:06:31.270 --> 00:06:34.010
That is f of c.

00:06:34.010 --> 00:06:37.220
And it does look like the limit,

00:06:37.220 --> 00:06:41.500
as x approaches c from the left,

00:06:41.500 --> 00:06:44.130
so from values less than c,

00:06:44.130 --> 00:06:46.483
it does look like that
is approaching f of c.

00:06:48.530 --> 00:06:50.360
But if we look at the limit

00:06:50.360 --> 00:06:54.310
as x approaches c from the right,

00:06:54.310 --> 00:06:57.130
that looks like it's
approaching some other value.

00:06:57.130 --> 00:06:59.240
That looks like it's approaching
this value right over here,

00:06:59.240 --> 00:07:00.370
let's call it L.

00:07:00.370 --> 00:07:01.370
That's approaching L,

00:07:01.370 --> 00:07:04.270
and L does not equal f of c.

00:07:04.270 --> 00:07:06.430
And so in this situation,

00:07:06.430 --> 00:07:08.370
the two-sided limit doesn't even exist.

00:07:08.370 --> 00:07:10.410
We're approaching two different values

00:07:10.410 --> 00:07:12.500
when we approach from the
left and from the right.

00:07:12.500 --> 00:07:16.510
And since so the limit
doesn't even exist at c,

00:07:16.510 --> 00:07:19.890
this is definitely not
going to be continuous.

00:07:19.890 --> 00:07:22.370
And this matches up to our expectations

00:07:22.370 --> 00:07:24.820
with our little
do-we-have-to-pick-up-the-pencil test.

00:07:24.820 --> 00:07:26.880
If I have to draw this,
I can leave my pencil,

00:07:26.880 --> 00:07:28.150
it's on the paper, it's on the paper,

00:07:28.150 --> 00:07:29.590
it's on the paper, it's on the paper.

00:07:29.590 --> 00:07:32.610
How am I going to continue
to draw this function,

00:07:32.610 --> 00:07:35.500
this graph of the function,
without picking up my pencil?

00:07:35.500 --> 00:07:37.737
Pick it up, put it back down,
and then keep drawing it.

00:07:37.737 --> 00:07:42.737
And so once again, this right
over here is not continuous.

00:07:43.300 --> 00:07:45.920
Both intuitively, by our
pick-up-the-pencil definition,

00:07:45.920 --> 00:07:49.310
and also by this more
rigorous definition where,

00:07:49.310 --> 00:07:50.510
in this case the limit,

00:07:50.510 --> 00:07:53.240
the two-sided limit at x
equals c doesn't even exist,

00:07:53.240 --> 00:07:54.920
so we're definitely not
gonna be continuous.

00:07:54.920 --> 00:07:57.570
But even when the
two-sided limit does exist,

00:07:57.570 --> 00:08:00.010
but the limit is a different value

00:08:00.010 --> 00:08:01.190
than the value of the function,

00:08:01.190 --> 00:08:03.020
that will also not be continuous.

00:08:03.020 --> 00:08:05.340
The only situation that
it's going to be continuous

00:08:05.340 --> 00:08:09.300
is if the two-sided limit
approaches the same value

00:08:09.300 --> 00:08:10.640
as the value of the function.

00:08:10.640 --> 00:08:12.310
And if that's true, then we're continuous.

00:08:12.310 --> 00:08:14.610
If we're continuous,
that is going to be true.


Manifest Destiny

WEBVTT
Kind: captions
Language: en

00:00:00.280 --> 00:00:01.120
- [Instructor] This is a print

00:00:01.120 --> 00:00:04.850
showing the San Francisco Harbor in 1848.

00:00:04.850 --> 00:00:06.750
There's a little smattering of houses,

00:00:06.750 --> 00:00:08.480
and a few boats in the water.

00:00:08.480 --> 00:00:11.420
It looks pretty peaceful, and it was.

00:00:11.420 --> 00:00:15.160
San Francisco only had
about a thousand residents,

00:00:15.160 --> 00:00:19.500
and California had only
newly become a U.S. territory

00:00:19.500 --> 00:00:22.730
at the close of the Mexican-American War.

00:00:22.730 --> 00:00:27.580
And this is a photograph of
San Francisco Harbor in 1850.

00:00:27.580 --> 00:00:29.490
The water is crowded with ships,

00:00:29.490 --> 00:00:31.740
and the land is crowded with houses.

00:00:31.740 --> 00:00:33.970
Less than two years later,

00:00:33.970 --> 00:00:37.250
San Francisco had 30-thousand residents,

00:00:37.250 --> 00:00:41.040
mainly young men who had
come from all over the world,

00:00:41.040 --> 00:00:44.100
making the city perhaps the
most culturally-diverse place

00:00:44.100 --> 00:00:46.360
on earth at that time period.

00:00:46.360 --> 00:00:47.860
What happened?

00:00:47.860 --> 00:00:50.340
The short answer is, gold.

00:00:50.340 --> 00:00:54.440
In January 1848, gold was
discovered in California,

00:00:54.440 --> 00:00:56.320
near the Sierra Nevada mountains,

00:00:56.320 --> 00:00:57.660
right about here.

00:00:57.660 --> 00:01:00.600
San Francisco was the
gateway to that gold.

00:01:00.600 --> 00:01:02.810
The nearest harbor where ships could land

00:01:02.810 --> 00:01:06.040
with prospective gold
miners from the East Coast,

00:01:06.040 --> 00:01:09.020
Europe, South America, and Asia.

00:01:09.020 --> 00:01:12.550
Before the gold rush,
the non-Indian population

00:01:12.550 --> 00:01:16.270
of the State of California
was about 15-thousand people.

00:01:16.270 --> 00:01:20.570
By 1860, it was more than 350-thousand.

00:01:20.570 --> 00:01:22.180
And, in the same time period,

00:01:22.180 --> 00:01:27.000
the Native-American population
decreased from 150-thousand

00:01:27.000 --> 00:01:28.890
to only 30 thousand.

00:01:28.890 --> 00:01:32.330
The gold rush, and its
impact on California,

00:01:32.330 --> 00:01:36.700
is one very dramatic
illustration of the causes

00:01:36.700 --> 00:01:39.720
and effects of westward migration

00:01:39.720 --> 00:01:42.260
in the years surrounding the Civil War.

00:01:42.260 --> 00:01:44.650
This drive to expand the United States

00:01:44.650 --> 00:01:48.760
West to the Pacific is often
called manifest destiny,

00:01:48.760 --> 00:01:50.270
based on a phrase that was coined

00:01:50.270 --> 00:01:52.750
by New York journalist John O'Sullivan,

00:01:52.750 --> 00:01:56.080
who wrote in 1845 that westward expansion

00:01:56.080 --> 00:01:59.820
would be "The fulfillment
of our manifest destiny

00:01:59.820 --> 00:02:03.370
"to overspread the continent
allotted by Providence

00:02:03.370 --> 00:02:04.640
"for the free development

00:02:04.640 --> 00:02:07.440
"of our yearly multiplying millions."

00:02:07.440 --> 00:02:10.130
The word 'manifest,' in this sense,

00:02:10.130 --> 00:02:12.980
means clear, or obvious,

00:02:12.980 --> 00:02:16.220
and 'Providence' is another
word for God's help.

00:02:16.220 --> 00:02:20.840
So O'Sullivan was saying that
God had provided the continent

00:02:20.840 --> 00:02:22.760
for the United States to expand,

00:02:22.760 --> 00:02:24.900
and it was obviously the destiny

00:02:24.900 --> 00:02:26.750
of the United States to do so.

00:02:26.750 --> 00:02:29.120
But despite the prevailing idea

00:02:29.120 --> 00:02:32.740
that the American West was an empty land

00:02:32.740 --> 00:02:35.380
full of limitless resources,

00:02:35.380 --> 00:02:37.680
there were in fact a lot of Native people

00:02:37.680 --> 00:02:39.310
already living in the West.

00:02:39.310 --> 00:02:42.360
And the arrival of people
not only from the East Coast,

00:02:42.360 --> 00:02:44.100
but from all over the world

00:02:44.100 --> 00:02:46.180
in the second half of the 19th Century,

00:02:46.180 --> 00:02:50.110
would have enormous effects
on both people and politics.

00:02:50.110 --> 00:02:54.010
So let's start by talking
about what drew immigrants

00:02:54.010 --> 00:02:55.540
to the West in this era.

00:02:55.540 --> 00:02:56.800
First and foremost,

00:02:56.800 --> 00:02:58.940
they were drawn by economic opportunities

00:02:58.940 --> 00:03:00.260
available in the West.

00:03:00.260 --> 00:03:03.230
Before there were gold miners
flooding San Francisco,

00:03:03.230 --> 00:03:06.090
most people who went to
the West were farmers.

00:03:06.090 --> 00:03:08.530
As land became scarcer in the East,

00:03:08.530 --> 00:03:10.750
a trickle of farming families headed

00:03:10.750 --> 00:03:13.260
to the fertile Willamette
Valley of Oregon,

00:03:13.260 --> 00:03:14.890
through the Oregon Trail.

00:03:14.890 --> 00:03:17.300
After the discovery of gold in California,

00:03:17.300 --> 00:03:19.010
and later in Montana,

00:03:19.010 --> 00:03:21.930
westward migration
increased exponentially,

00:03:21.930 --> 00:03:25.230
but only a few miners
actually struck it rich,

00:03:25.230 --> 00:03:27.310
mainly those who were already in the area

00:03:27.310 --> 00:03:29.100
before gold was discovered.

00:03:29.100 --> 00:03:32.890
Mining and farming weren't the
only economic opportunities

00:03:32.890 --> 00:03:34.380
available in the west.

00:03:34.380 --> 00:03:36.340
Many people found work in the industries

00:03:36.340 --> 00:03:39.690
that served the miners,
like hardware stores,

00:03:39.690 --> 00:03:41.840
boarding houses, and restaurants.

00:03:41.840 --> 00:03:43.330
There was also the railroad.

00:03:43.330 --> 00:03:45.550
Between 1860 and 1880,

00:03:45.550 --> 00:03:49.400
the miles of railroad track
in the United States tripled.

00:03:49.400 --> 00:03:50.970
And, as the railroad expanded,

00:03:50.970 --> 00:03:54.110
so did opportunities for
work on the railroad.

00:03:54.110 --> 00:03:56.470
The expansion of the railroad was one way

00:03:56.470 --> 00:03:59.990
that the Federal Government
facilitated westward migration.

00:03:59.990 --> 00:04:03.380
In 1862, in the midst of the Civil War,

00:04:03.380 --> 00:04:06.280
Congress passed the Pacific Railway Act,

00:04:06.280 --> 00:04:10.360
which granted railroad companies
more than 100-million acres

00:04:10.360 --> 00:04:13.310
in order to complete a
transcontinental railroad,

00:04:13.310 --> 00:04:15.660
which they did in 1869.

00:04:15.660 --> 00:04:18.140
The transcontinental
railroad reduced the time

00:04:18.140 --> 00:04:21.590
it took to get across the
country from five months,

00:04:21.590 --> 00:04:25.200
so just six days, which made traveling,

00:04:25.200 --> 00:04:29.250
and transporting goods to and
from the West, much easier.

00:04:29.250 --> 00:04:33.660
In 1862, Congress also
passed the Homestead Act,

00:04:33.660 --> 00:04:37.760
which grated 160 acres of land, for free

00:04:37.760 --> 00:04:40.160
to anyone over the age of 21

00:04:40.160 --> 00:04:43.010
who had never taken up arms
against the U.S. Government,

00:04:43.010 --> 00:04:46.060
so no one who was affiliated
with the Confederacy,

00:04:46.060 --> 00:04:47.830
as long as they made improvements

00:04:47.830 --> 00:04:50.640
to the land within five years.

00:04:50.640 --> 00:04:55.310
And this included women,
immigrants, and African-Americans.

00:04:55.310 --> 00:04:58.530
The Homestead Act was
the wartime extension

00:04:58.530 --> 00:05:01.210
of the ideas of the free soil movement:

00:05:01.210 --> 00:05:05.730
to populate Western lands with
small, independent farmers,

00:05:05.730 --> 00:05:08.750
rather than slaveholders
on giant plantations.

00:05:08.750 --> 00:05:13.010
More than 1.5 million people
acquired land this way.

00:05:13.010 --> 00:05:15.700
The last reason that
Americans headed West,

00:05:15.700 --> 00:05:17.140
that I'll talk about here,

00:05:17.140 --> 00:05:19.860
was cultural messaging of the time period.

00:05:19.860 --> 00:05:23.260
I mentioned earlier this
notion of manifest destiny,

00:05:23.260 --> 00:05:25.590
that the United States
had a divine mission

00:05:25.590 --> 00:05:27.720
to spread across North America.

00:05:27.720 --> 00:05:30.700
Closely related to that
was a widespread belief

00:05:30.700 --> 00:05:33.450
among whites that American civilization

00:05:33.450 --> 00:05:35.580
was superior to other cultures,

00:05:35.580 --> 00:05:38.710
and that any barriers to U.S. expansion,

00:05:38.710 --> 00:05:41.850
like Native Americans
and Mexican Americans

00:05:41.850 --> 00:05:44.790
who possessed the land onto
which settlers flooded,

00:05:44.790 --> 00:05:48.100
were obstacles to
progress and civilization.

00:05:48.100 --> 00:05:51.370
This painting, which was painted in 1872

00:05:51.370 --> 00:05:53.640
by the artist John Gast,

00:05:53.640 --> 00:05:56.050
is called American Progress.

00:05:56.050 --> 00:05:59.510
In it, you can see an
allegorical figure of America

00:05:59.510 --> 00:06:03.370
holding a schoolbook, and
helping to lay telegraph wire.

00:06:03.370 --> 00:06:06.470
She brings with her symbols
of American civilization:

00:06:06.470 --> 00:06:09.010
railroads, and covered wagons,

00:06:09.010 --> 00:06:11.200
and farmers with log cabins.

00:06:11.200 --> 00:06:13.840
And she drives away
symbols of what the artist

00:06:13.840 --> 00:06:16.850
portrays as wilderness, or savagery:

00:06:16.850 --> 00:06:21.160
Native Americans, buffalo,
even an angry bear down here.

00:06:21.160 --> 00:06:23.020
You can even see how the artist painted

00:06:23.020 --> 00:06:27.250
the right side of the painting
with a bright, clear sky,

00:06:27.250 --> 00:06:30.350
and the left side with
dark shadows and clouds

00:06:30.350 --> 00:06:32.190
so that this central figure of America

00:06:32.190 --> 00:06:34.930
seems to be driving out the darkness.

00:06:34.930 --> 00:06:36.620
I encourage you to pause the video

00:06:36.620 --> 00:06:40.200
and see how many symbols of civilization,

00:06:40.200 --> 00:06:42.000
and symbols of wilderness,

00:06:42.000 --> 00:06:44.040
you can identify in this painting.

00:06:44.040 --> 00:06:47.480
Now that we've discussed the
causes of westward expansion,

00:06:47.480 --> 00:06:49.640
let's talk about some of its effects.

00:06:49.640 --> 00:06:53.480
A major one is an increase
in sectional conflict.

00:06:53.480 --> 00:06:56.490
As new Western states joined the union,

00:06:56.490 --> 00:06:59.190
it inflamed tensions
over the balance of power

00:06:59.190 --> 00:07:02.090
between free and slave states in Congress,

00:07:02.090 --> 00:07:04.740
which ultimately would
lead to the Civil War.

00:07:04.740 --> 00:07:06.650
Another effect was an increase

00:07:06.650 --> 00:07:08.890
in racial conflict in the West.

00:07:08.890 --> 00:07:11.860
As people from all over
the world came to the West,

00:07:11.860 --> 00:07:14.680
and competed for land and gold,

00:07:14.680 --> 00:07:17.340
there was a surge in racial violence.

00:07:17.340 --> 00:07:19.570
Before, and after the Civil War,

00:07:19.570 --> 00:07:23.160
as white settlers crowded onto
the lands of Plains Indians,

00:07:23.160 --> 00:07:25.370
the U.S. Army sought to exterminate them,

00:07:25.370 --> 00:07:27.840
or confine them to reservations.

00:07:27.840 --> 00:07:32.400
In California, white miners
sought to expel foreign miners,

00:07:32.400 --> 00:07:35.500
and Native Americans,
from regions with gold.

00:07:35.500 --> 00:07:40.200
Vigilantes killed or expelled
80% of the Native population

00:07:40.200 --> 00:07:42.340
of the region in just over a decade.

00:07:42.340 --> 00:07:43.670
Also, in California,

00:07:43.670 --> 00:07:46.980
vigilante groups attacked
Chinese communities,

00:07:46.980 --> 00:07:49.460
and even tried to destroy Chinatown

00:07:49.460 --> 00:07:51.900
in San Francisco in 1877.

00:07:51.900 --> 00:07:53.680
The State Government in California

00:07:53.680 --> 00:07:57.460
also imposed high taxes on foreign miners,

00:07:57.460 --> 00:07:59.040
especially the Chinese.

00:07:59.040 --> 00:08:02.060
These discriminatory laws
would lay the groundwork

00:08:02.060 --> 00:08:05.180
for the first race-based
immigration restriction

00:08:05.180 --> 00:08:09.190
in U.S. history, the Chinese
Exclusion Act of 1882.

00:08:09.190 --> 00:08:12.630
The enormous increase in
westward expansion in this era

00:08:12.630 --> 00:08:16.250
led to unprecedented prosperity for some,

00:08:16.250 --> 00:08:19.110
and unprecedented misery for others.

00:08:19.110 --> 00:08:23.140
But in 1877, at the end
of the Reconstruction Era,

00:08:23.140 --> 00:08:27.230
the process of westward
expansion was not yet complete.

00:08:27.230 --> 00:08:31.540
Many of the political, social,
and economic consequences

00:08:31.540 --> 00:08:33.330
of the events in this time period

00:08:33.330 --> 00:08:35.630
would become even more pronounced

00:08:35.630 --> 00:08:37.880
in the last years of the 19th Century.


Water potential example

WEBVTT
Kind: captions
Language: en

00:00:01.090 --> 00:00:02.790
- [Instructor] We're
told that six identical

00:00:02.790 --> 00:00:06.600
potato core cubes were
isolated from a potato.

00:00:06.600 --> 00:00:09.680
The initial weight of
each cube was recorded.

00:00:09.680 --> 00:00:13.640
Each cube was then placed
in one of six open beakers,

00:00:13.640 --> 00:00:16.840
each containing a
different sucrose solution.

00:00:16.840 --> 00:00:19.510
The cubes remained in
the beakers for 24 hours

00:00:19.510 --> 00:00:23.080
at a constant temperature
of 23 degrees Celsius.

00:00:23.080 --> 00:00:25.670
After 24 hours, the cubes were removed

00:00:25.670 --> 00:00:28.810
from the beakers, blotted, and reweighed.

00:00:28.810 --> 00:00:30.830
The percent change in mass,

00:00:30.830 --> 00:00:33.490
due to a net gain or loss of water,

00:00:33.490 --> 00:00:35.500
was calculated for each cube,

00:00:35.500 --> 00:00:38.460
and the results are shown
in the graph to the right,

00:00:38.460 --> 00:00:40.370
so, this graph right over here.

00:00:40.370 --> 00:00:41.990
A straight line is drawn

00:00:41.990 --> 00:00:43.957
on the graph to help estimate results

00:00:43.957 --> 00:00:47.453
from other sucrose
concentrations not tested.

00:00:48.810 --> 00:00:51.220
Using the straight line on the graph,

00:00:51.220 --> 00:00:54.530
calculate the water potential, in bars,

00:00:54.530 --> 00:00:58.400
of the potato core cubes
at 23 degrees Celsius.

00:00:58.400 --> 00:01:01.220
Give your answer to one decimal place.

00:01:01.220 --> 00:01:04.660
So, pause this video and see
if you can work that out.

00:01:04.660 --> 00:01:06.020
All right, so, first, let's just make sure

00:01:06.020 --> 00:01:08.100
we're understanding what's going on here.

00:01:08.100 --> 00:01:10.640
So, there was a potato.

00:01:10.640 --> 00:01:14.330
We took six cubes from that potato

00:01:14.330 --> 00:01:16.650
and we stuck those six cubes

00:01:16.650 --> 00:01:20.960
into six different sucrose
molarity solutions,

00:01:20.960 --> 00:01:24.000
and so, this data point right over here,

00:01:24.000 --> 00:01:26.470
this was the situation where
we took one of the cubes,

00:01:26.470 --> 00:01:28.250
so, this was a sucrose solution.

00:01:28.250 --> 00:01:31.670
This was a solution, actually,
that contained no sucrose,

00:01:31.670 --> 00:01:35.140
and so, when we put the
cube in that solution,

00:01:35.140 --> 00:01:37.930
we saw a net gain of mass.

00:01:37.930 --> 00:01:41.170
It looks like it's about 22% gain in mass,

00:01:41.170 --> 00:01:42.310
and so, that would have happened

00:01:42.310 --> 00:01:47.190
because water would have
flowed into the cube.

00:01:47.190 --> 00:01:50.570
Now, at the other
extreme, right over here,

00:01:50.570 --> 00:01:53.650
this is a solution that
has a lot of sucrose.

00:01:53.650 --> 00:01:56.140
It had a very high sucrose concentration,

00:01:56.140 --> 00:01:58.350
and when we put a cube in there,

00:01:58.350 --> 00:02:02.630
we see that the mass of
that cube went down by 25%,

00:02:02.630 --> 00:02:03.590
and that would have been

00:02:03.590 --> 00:02:08.590
because of the net outflow
of water from that cube.

00:02:08.700 --> 00:02:11.330
So, how do we figure
out the water potential

00:02:11.330 --> 00:02:15.910
of the core cubes at 23 degrees Celsius?

00:02:15.910 --> 00:02:17.940
Well, we could think about a situation

00:02:17.940 --> 00:02:21.980
where there's some sucrose
concentration where,

00:02:21.980 --> 00:02:24.900
if the cube and the sucrose solution

00:02:24.900 --> 00:02:27.320
have the same water potential,

00:02:27.320 --> 00:02:30.700
then you're not going to have
any net inflow or outflow,

00:02:30.700 --> 00:02:33.370
and so, where do we see that on the graph?

00:02:33.370 --> 00:02:35.750
Well, what we'd wanna
do, we have that line

00:02:35.750 --> 00:02:37.780
where they're trying
to fit the data points,

00:02:37.780 --> 00:02:41.610
and so, where would we expect
to see 0% change in mass?

00:02:41.610 --> 00:02:44.560
So, we would go right over
here to 0% change in mass.

00:02:44.560 --> 00:02:47.510
We would go to the line right over there,

00:02:47.510 --> 00:02:52.090
and then, we see that this line would say

00:02:52.090 --> 00:02:54.710
that there's a 0% change in mass.

00:02:54.710 --> 00:02:57.820
See, if this is 0.4 right over here,

00:02:57.820 --> 00:03:02.770
this is 0.5 right over here,

00:03:02.770 --> 00:03:07.770
so, this is about a 0.44
molar sucrose solution,

00:03:07.960 --> 00:03:11.500
0.44 molar solution.

00:03:11.500 --> 00:03:14.000
So, if we can figure
out the water potential

00:03:14.000 --> 00:03:17.930
of this 0.44 molar sucrose solution,

00:03:17.930 --> 00:03:18.970
well, that's also going to be

00:03:18.970 --> 00:03:22.480
the water potential of the potato cubes.

00:03:22.480 --> 00:03:24.200
Well, how do we do that?

00:03:24.200 --> 00:03:25.810
Well, we've seen the equation before

00:03:25.810 --> 00:03:27.140
where we introduced ourselves

00:03:27.140 --> 00:03:28.900
to the idea of water potential,

00:03:28.900 --> 00:03:31.980
that water potential,
using the Greek letter psi,

00:03:31.980 --> 00:03:35.200
is going to be equal
to the solute potential

00:03:35.200 --> 00:03:37.680
plus the pressure potential.

00:03:37.680 --> 00:03:39.530
Now, we're dealing with
all open containers.

00:03:39.530 --> 00:03:41.370
We don't have anything that's some piston

00:03:41.370 --> 00:03:43.960
or something that's pressing
down on these containers,

00:03:43.960 --> 00:03:45.380
and so, because of that,

00:03:45.380 --> 00:03:50.370
the pressure potential is
going to be equal to zero,

00:03:50.370 --> 00:03:53.210
and so, we just have to figure
out the solute potential.

00:03:53.210 --> 00:03:56.690
So, the solute potential,
we have introduced ourselves

00:03:56.690 --> 00:03:58.670
to this formula in previous videos.

00:03:58.670 --> 00:04:03.670
It's negative i times C times R times T.

00:04:05.000 --> 00:04:08.660
This i right over here, this
is our ionization constant.

00:04:08.660 --> 00:04:10.720
This is, since we're dealing
with sucrose solutions,

00:04:10.720 --> 00:04:12.900
it says okay, if I took
sucrose and put it into water,

00:04:12.900 --> 00:04:14.700
every one of those sucrose molecules,

00:04:14.700 --> 00:04:17.670
does it stay one molecule,
or does it disassociate?

00:04:17.670 --> 00:04:20.420
Well, sucrose doesn't disassociate at all.

00:04:20.420 --> 00:04:23.130
It just stays one molecule,
so, this would be one.

00:04:23.130 --> 00:04:25.020
If we were dealing with,
say, sodium chloride,

00:04:25.020 --> 00:04:27.410
each sodium chloride
molecule would disassociate

00:04:27.410 --> 00:04:30.360
into a sodium ion and a chloride ion,

00:04:30.360 --> 00:04:32.500
and so, then, this would be two,

00:04:32.500 --> 00:04:34.980
but this was one for sucrose.

00:04:34.980 --> 00:04:39.480
C is the molarity of our solution,

00:04:39.480 --> 00:04:43.690
and so, we estimated that to be 0.44,

00:04:43.690 --> 00:04:45.590
so, let me write this down.

00:04:45.590 --> 00:04:49.870
Our solute water potential is going

00:04:49.870 --> 00:04:54.260
to be equal to negative one times 0.4,

00:04:54.260 --> 00:04:59.260
0.44, I should say, and
that's going to be moles.

00:04:59.440 --> 00:05:01.410
I'll write out all the units.

00:05:01.410 --> 00:05:04.810
Moles per liter times,
it's sometimes called

00:05:04.810 --> 00:05:09.250
the pressure constant in this context,

00:05:09.250 --> 00:05:11.680
but this is also the
universal gas constant,

00:05:11.680 --> 00:05:13.420
and if you were doing
something like the AP exam,

00:05:13.420 --> 00:05:15.260
they would give you what this is.

00:05:15.260 --> 00:05:18.823
So, this is 0.0831 liters times bars,

00:05:24.370 --> 00:05:27.900
all of that over mole Kelvin.

00:05:27.900 --> 00:05:29.820
If you're used to seeing
other values of this,

00:05:29.820 --> 00:05:31.180
it's probably because they're dealing

00:05:31.180 --> 00:05:32.700
with other units right over here,

00:05:32.700 --> 00:05:34.550
but this is the universal gas constant.

00:05:34.550 --> 00:05:36.490
And then, we have to multiply that times

00:05:36.490 --> 00:05:39.750
the temperature that we're
dealing with in Kelvin.

00:05:39.750 --> 00:05:42.420
Now, it's 23 degrees Celsius.

00:05:42.420 --> 00:05:46.080
To convert to Kelvin, we just add to 273,

00:05:46.080 --> 00:05:51.080
so, 273 plus 23 is going
to be 296, 296 Kelvin,

00:05:57.910 --> 00:05:59.740
and so, this is going to be equal to,

00:05:59.740 --> 00:06:03.090
we have a negative here, and
we could look at the units.

00:06:03.090 --> 00:06:04.820
We have the liter canceling out liters,

00:06:04.820 --> 00:06:06.360
moles canceling out with moles,

00:06:06.360 --> 00:06:08.090
Kelvin canceling out with Kelvin,

00:06:08.090 --> 00:06:11.030
so, we're going to get something
in bars, which makes sense.

00:06:11.030 --> 00:06:14.740
That is the unit for our water potential.

00:06:14.740 --> 00:06:17.463
And then, we get the calculator out.

00:06:18.630 --> 00:06:23.630
So, we have 0.44 times
0.831 times 296, 296,

00:06:30.680 --> 00:06:32.630
is equal to, and they want us

00:06:32.630 --> 00:06:34.640
to round our answer to one decimal place,

00:06:34.640 --> 00:06:36.990
so, approximately 10.8,

00:06:36.990 --> 00:06:38.860
and we already had that
negative out front,

00:06:38.860 --> 00:06:43.503
so, negative 10.8 bars, and we're done.


Molecular variation

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:02.320
- [Instructor] We are now going to discuss

00:00:02.320 --> 00:00:06.120
molecular variation in cells.

00:00:06.120 --> 00:00:08.380
You are probably familiar with the idea

00:00:08.380 --> 00:00:10.110
that you have a variation

00:00:10.110 --> 00:00:13.150
of genetic makeups in a population.

00:00:13.150 --> 00:00:16.260
But even within an organism you have

00:00:16.260 --> 00:00:19.370
variation in the types of molecules

00:00:19.370 --> 00:00:21.810
that an organism can produce

00:00:21.810 --> 00:00:23.960
and when they produce them.

00:00:23.960 --> 00:00:28.140
So, for example, we know
that we all have DNA,

00:00:28.140 --> 00:00:30.360
all organisms, living organisms

00:00:30.360 --> 00:00:33.100
that we know about, they have DNA.

00:00:33.100 --> 00:00:36.000
I'll just do this as a
quick drawing of DNA.

00:00:36.000 --> 00:00:39.270
We know that we have genes in our DNA

00:00:39.270 --> 00:00:42.230
that code eventually, they go from DNA

00:00:42.230 --> 00:00:45.180
to messenger RNA, and then
they go to the ribosomes

00:00:45.180 --> 00:00:47.663
to be translated into proteins.

00:00:48.900 --> 00:00:51.380
And these proteins are a major way

00:00:51.380 --> 00:00:54.930
of expressing what is encoded in our DNA.

00:00:54.930 --> 00:00:57.200
Now, it turns out that our DNA

00:00:57.200 --> 00:00:59.900
will encode for not only multiple proteins

00:00:59.900 --> 00:01:03.180
but multiple types of the same protein,

00:01:03.180 --> 00:01:05.870
and it can encode for
some of these proteins

00:01:05.870 --> 00:01:07.680
more under certain circumstances

00:01:07.680 --> 00:01:10.540
and other proteins more
in other circumstances

00:01:10.540 --> 00:01:12.130
based on environmental factors.

00:01:12.130 --> 00:01:15.060
Those environmental
factors might influence

00:01:15.060 --> 00:01:19.300
what part of the DNA is being transcribed

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into mRNA, which then is translated

00:01:21.690 --> 00:01:24.330
into proteins at different times.

00:01:24.330 --> 00:01:25.760
And there's several

00:01:25.760 --> 00:01:28.740
very interesting examples of this.

00:01:28.740 --> 00:01:30.490
It turns out that hemoglobin,

00:01:30.490 --> 00:01:31.650
which you might recognize

00:01:31.650 --> 00:01:33.930
as the protein complex that binds

00:01:33.930 --> 00:01:36.350
to oxygen in our red blood cells,

00:01:36.350 --> 00:01:40.700
that the type of predominant
hemoglobin changes

00:01:40.700 --> 00:01:43.250
from when we are inside our mothers' wombs

00:01:43.250 --> 00:01:46.640
to when we become independent beings.

00:01:46.640 --> 00:01:49.420
So, this right over here is a picture

00:01:49.420 --> 00:01:51.200
of a hemoglobin molecule.

00:01:51.200 --> 00:01:52.860
You see here four heme groups

00:01:52.860 --> 00:01:55.110
that each bind two oxygen,

00:01:55.110 --> 00:01:57.640
and when you're a fetus the primary type

00:01:57.640 --> 00:02:02.080
of hemoglobin is hemoglobin F,

00:02:02.080 --> 00:02:04.930
and then once we come
out of our mothers' wombs

00:02:04.930 --> 00:02:07.740
the hemoglobin F stops getting produced

00:02:07.740 --> 00:02:10.750
and we go to hemoglobin A.

00:02:10.750 --> 00:02:12.440
Now you might say, well, why do we have

00:02:12.440 --> 00:02:15.710
this variation in the type of hemoglobin?

00:02:15.710 --> 00:02:17.820
And the answer is, that those are

00:02:17.820 --> 00:02:19.620
two different environments.

00:02:19.620 --> 00:02:21.810
When a fetus is in the mother's womb,

00:02:21.810 --> 00:02:23.810
it's not directly breathing.

00:02:23.810 --> 00:02:26.750
It's getting its oxygen
from the mother's blood.

00:02:26.750 --> 00:02:28.180
The mother's blood does not mix

00:02:28.180 --> 00:02:29.920
directly with the baby's blood,

00:02:29.920 --> 00:02:32.240
but there's a boundary where you have

00:02:32.240 --> 00:02:34.260
the mother's blood here, and I'll say

00:02:34.260 --> 00:02:37.120
this is the baby's blood right over here,

00:02:37.120 --> 00:02:41.800
and you have the gas
exchange of the oxygen

00:02:41.800 --> 00:02:45.150
going through that boundary,

00:02:45.150 --> 00:02:46.170
and then of course the release

00:02:46.170 --> 00:02:49.650
of the carbon dioxide going the other way.

00:02:49.650 --> 00:02:51.060
And this environment

00:02:51.060 --> 00:02:53.660
where the baby's red blood cells

00:02:53.660 --> 00:02:55.410
have to bind to the oxygen

00:02:55.410 --> 00:02:58.360
is a relatively low oxygen environment

00:02:58.360 --> 00:03:00.800
compared to, say, our lungs,

00:03:00.800 --> 00:03:04.320
because it has oxygenated
and deoxygenated blood

00:03:04.320 --> 00:03:05.730
mixing in that same place,

00:03:05.730 --> 00:03:07.600
and it does not have direct access

00:03:07.600 --> 00:03:09.770
to, say, the lungs.

00:03:09.770 --> 00:03:11.870
And so in this low oxygen environment,

00:03:11.870 --> 00:03:14.370
the hemoglobin molecules have to be

00:03:14.370 --> 00:03:18.400
really really really good
at binding to oxygen.

00:03:18.400 --> 00:03:21.040
And we can see that from this diagram

00:03:21.040 --> 00:03:23.670
right over here, where the horizontal axis

00:03:23.670 --> 00:03:26.080
is the partial pressure of oxygen,

00:03:26.080 --> 00:03:28.690
and the vertical axis is how saturated

00:03:28.690 --> 00:03:30.450
with oxygen these different

00:03:30.450 --> 00:03:32.990
hemoglobin molecules can become.

00:03:32.990 --> 00:03:36.170
And you can see that the fetal hemoglobin,

00:03:36.170 --> 00:03:38.210
which is depicted by this blue curve,

00:03:38.210 --> 00:03:41.790
it gets 50% saturated at
a lower partial pressure

00:03:41.790 --> 00:03:44.700
of oxygen than the adult hemoglobin.

00:03:44.700 --> 00:03:45.720
So one way to think about it,

00:03:45.720 --> 00:03:48.420
it is stickier, it binds with that oxygen.

00:03:48.420 --> 00:03:50.900
It can pull that oxygen out of the blood

00:03:50.900 --> 00:03:52.590
far better, which makes sense

00:03:52.590 --> 00:03:54.850
for the environment that the fetus is in.

00:03:54.850 --> 00:03:56.940
But once it comes out
of the mother's womb,

00:03:56.940 --> 00:03:58.510
it doesn't need that stickiness

00:03:58.510 --> 00:04:00.570
and there're some drawbacks
of that stickiness as well

00:04:00.570 --> 00:04:02.890
because it makes it hard for that oxygen

00:04:02.890 --> 00:04:05.880
to go into as many of the body's tissues,

00:04:05.880 --> 00:04:07.940
and so that's why you have this transition

00:04:07.940 --> 00:04:10.723
from hemoglobin F to hemoglobin A.

00:04:11.570 --> 00:04:13.750
It's not just hemoglobin where we see

00:04:13.750 --> 00:04:15.343
this molecular variation.

00:04:16.860 --> 00:04:20.060
Plants and other organisms
that conduct photosynthesis

00:04:20.060 --> 00:04:22.390
contain multiple types of chlorophyll.

00:04:22.390 --> 00:04:25.160
Remember chlorophyll is a very important

00:04:25.160 --> 00:04:27.660
molecule in capturing light energy

00:04:27.660 --> 00:04:30.080
which can then be used to help synthesize

00:04:30.080 --> 00:04:32.630
carbohydrates in things like plants.

00:04:32.630 --> 00:04:35.950
And here we see how two
different chlorophyll molecules,

00:04:35.950 --> 00:04:37.740
both that would be found in plants,

00:04:37.740 --> 00:04:40.490
how well they absorb light
of different frequencies.

00:04:40.490 --> 00:04:42.880
So you can see chlorophyll
a is really good

00:04:42.880 --> 00:04:46.270
at absorbing the violet
bordering on blue light,

00:04:46.270 --> 00:04:48.610
while chlorophyll b is better

00:04:48.610 --> 00:04:50.920
at the blue-green type of light.

00:04:50.920 --> 00:04:52.260
And then you have another peak here

00:04:52.260 --> 00:04:55.000
where chlorophyll b is better at absorbing

00:04:55.000 --> 00:04:57.750
an orangish red, while chlorophyll a

00:04:57.750 --> 00:05:01.180
is better at absorbing,
I guess you could say,

00:05:01.180 --> 00:05:04.960
a red bordering on infrared wavelength.

00:05:04.960 --> 00:05:06.410
And the reason why it is valuable

00:05:06.410 --> 00:05:08.550
is that the light that the plant gets,

00:05:08.550 --> 00:05:09.980
especially at different times of day,

00:05:09.980 --> 00:05:11.490
at different times of year,

00:05:11.490 --> 00:05:13.540
is going to have different wavelengths,

00:05:13.540 --> 00:05:15.510
and so this just lets the plant

00:05:15.510 --> 00:05:18.640
capture more energy that it can use

00:05:18.640 --> 00:05:20.010
in photosynthesis.

00:05:20.010 --> 00:05:21.920
And these were just two examples

00:05:21.920 --> 00:05:23.830
of molecular variation.

00:05:23.830 --> 00:05:25.330
In our cellular membranes,

00:05:25.330 --> 00:05:28.030
there's multiple types of phospholipids

00:05:28.030 --> 00:05:32.180
that are forming the phospholipid bilayer,

00:05:32.180 --> 00:05:34.720
and those multiple types are,

00:05:34.720 --> 00:05:37.250
they have different levels of how fluid

00:05:37.250 --> 00:05:39.130
they are at different temperatures.

00:05:39.130 --> 00:05:41.040
And there're animal studies that show

00:05:41.040 --> 00:05:43.610
that the variations change

00:05:43.610 --> 00:05:44.900
depending on the conditions.

00:05:44.900 --> 00:05:46.660
For example, a cold-blooded animal

00:05:46.660 --> 00:05:49.610
might have more of the fluid phospholipids

00:05:49.610 --> 00:05:53.010
when it is very cold so that the membranes

00:05:53.010 --> 00:05:55.120
don't become overly rigid.

00:05:55.120 --> 00:05:56.470
But I will leave you there.

00:05:56.470 --> 00:05:58.660
This is just to appreciate this idea

00:05:58.660 --> 00:06:01.370
that we have all sorts
of molecular variation

00:06:01.370 --> 00:06:04.980
inside organism cells, and it allows

00:06:04.980 --> 00:06:07.560
those organisms to better adapt

00:06:07.560 --> 00:06:10.410
to their environment or different stages

00:06:10.410 --> 00:06:11.653
of their development.


ATP synthase

WEBVTT
Kind: captions
Language: en

00:00:00.530 --> 00:00:02.110
- [Instructor] In this video,
we're going to talk about

00:00:02.110 --> 00:00:04.170
what is arguably my favorite enzyme,

00:00:04.170 --> 00:00:06.430
and that is ATP synthase.

00:00:06.430 --> 00:00:09.340
And you might be able to predict
from its name what it does.

00:00:09.340 --> 00:00:11.920
It synthesizes ATP.

00:00:11.920 --> 00:00:14.040
Now you've probably seen it before.

00:00:14.040 --> 00:00:16.100
We saw it when we looked at respiration,

00:00:16.100 --> 00:00:18.450
or you will see it when
you look at respiration,

00:00:18.450 --> 00:00:21.510
which is going on in most
of the cells of your body.

00:00:21.510 --> 00:00:25.290
And you also see it when
you study photosynthesis.

00:00:25.290 --> 00:00:27.240
The general thing that it does is,

00:00:27.240 --> 00:00:31.250
is it sits across a phospholipid membrane.

00:00:31.250 --> 00:00:33.970
And through other processes,

00:00:33.970 --> 00:00:37.340
you will have hydrogen
ion concentration increase

00:00:37.340 --> 00:00:38.810
on one side of the membrane,

00:00:38.810 --> 00:00:41.450
have a higher hydrogen ion concentration

00:00:41.450 --> 00:00:43.290
on one side than on the other side.

00:00:43.290 --> 00:00:46.630
You still might have a few over here.

00:00:46.630 --> 00:00:50.100
And a hydrogen ion is
essentially a proton.

00:00:50.100 --> 00:00:53.360
So on this side of the membrane,
it'll be more positive,

00:00:53.360 --> 00:00:55.670
so there will be a electromotive force

00:00:55.670 --> 00:00:56.720
to go to the other side.

00:00:56.720 --> 00:00:58.600
And also, you just have
a higher concentration,

00:00:58.600 --> 00:01:01.800
so there's a chemical gradient,
a concentration gradient,

00:01:01.800 --> 00:01:03.780
where if there was some
way for these protons

00:01:03.780 --> 00:01:06.140
to get to this side, they
would want to get there.

00:01:06.140 --> 00:01:08.500
So there's an electrochemical gradient

00:01:08.500 --> 00:01:09.900
that they would want to go down.

00:01:09.900 --> 00:01:14.900
And ATP synthase provides a
channel for those protons.

00:01:15.100 --> 00:01:19.000
But as those protons travel
through the ATP synthase,

00:01:19.000 --> 00:01:23.630
they turn this part of it,
which drives this axle,

00:01:23.630 --> 00:01:27.140
and then this axle nudges
these parts of the protein

00:01:27.140 --> 00:01:30.010
so that they jam together an ADP

00:01:30.010 --> 00:01:33.590
with a phosphate group to produce ATP.

00:01:33.590 --> 00:01:38.030
So down here, you, going into
this part of the complex,

00:01:38.030 --> 00:01:41.270
you'll have an ADP and a phosphate group.

00:01:41.270 --> 00:01:43.410
And then that rotation force

00:01:43.410 --> 00:01:45.990
that's provided by that
electrochemical gradient,

00:01:45.990 --> 00:01:48.220
that then produces

00:01:48.220 --> 00:01:50.380
our ATP.

00:01:50.380 --> 00:01:53.190
And that's going to be the
case both in respiration,

00:01:53.190 --> 00:01:54.940
which occurs in the mitochondria,

00:01:54.940 --> 00:01:58.940
and in photosynthesis,
which occur in chloroplasts.

00:01:58.940 --> 00:02:01.430
Now there's a few differences.

00:02:01.430 --> 00:02:05.880
In mitochondria, the
hydrogen ions, these protons,

00:02:05.880 --> 00:02:07.700
the concentration builds up

00:02:07.700 --> 00:02:10.273
in the intermembrane space right over here

00:02:10.273 --> 00:02:13.080
because of the electron transport chain.

00:02:13.080 --> 00:02:14.730
And we studied that in other videos.

00:02:14.730 --> 00:02:18.790
And then the protons travel
through the ATP synthase.

00:02:18.790 --> 00:02:20.920
You could see a little mini
version right over here.

00:02:20.920 --> 00:02:23.500
You could imagine that
what we see really big,

00:02:23.500 --> 00:02:26.860
that is a blown-up version of
this part of the mitochondria,

00:02:26.860 --> 00:02:29.360
and this is, of course, is not to scale.

00:02:29.360 --> 00:02:31.010
So in the case of a mitochondria,

00:02:31.010 --> 00:02:33.110
this would be the inner membrane.

00:02:33.110 --> 00:02:35.410
Right over here would be
the intermembrane space

00:02:35.410 --> 00:02:38.241
between the inner and the outer membrane,

00:02:38.241 --> 00:02:42.140
intermembrane space.

00:02:42.140 --> 00:02:47.140
And right over here would be
the matrix of the mitochondria.

00:02:48.060 --> 00:02:50.950
And so as the protons go through,

00:02:50.950 --> 00:02:54.923
they're able to produce ATP in the matrix.

00:02:55.880 --> 00:02:57.430
Now in chloroplasts,

00:02:57.430 --> 00:03:01.910
the hydrogen protons build
up inside the thylakoids,

00:03:01.910 --> 00:03:05.660
which are these parts of the chloroplast.

00:03:05.660 --> 00:03:07.480
That space inside the thylakoid

00:03:07.480 --> 00:03:09.440
is often called the thylakoid space,

00:03:09.440 --> 00:03:11.070
sometimes called the lumen.

00:03:11.070 --> 00:03:14.830
That proton buildup inside
the thylakoids happens

00:03:14.830 --> 00:03:16.890
because of the light reactions,

00:03:16.890 --> 00:03:19.430
the first phase of photosynthesis.

00:03:19.430 --> 00:03:22.160
But then those protons will travel

00:03:22.160 --> 00:03:26.450
through the thylakoid
membrane, through to this area,

00:03:26.450 --> 00:03:30.510
which is known as the
stroma in chloroplasts,

00:03:30.510 --> 00:03:33.900
and they produce the ATP in the stroma.

00:03:33.900 --> 00:03:36.830
But then the ATP is
used in the second phase

00:03:36.830 --> 00:03:41.600
of photosynthesis to synthesize
carbohydrates, which is,

00:03:41.600 --> 00:03:45.370
you could view as one of the
end products of photosynthesis.

00:03:45.370 --> 00:03:47.790
So the big takeaway of this video is, one,

00:03:47.790 --> 00:03:50.310
ATP synthase is incredibly cool.

00:03:50.310 --> 00:03:51.760
If you look up on the internet,

00:03:51.760 --> 00:03:55.150
you can find some simulations
that show ATP synthase

00:03:55.150 --> 00:03:59.780
and how it acts like a motor
to jam the phosphate group

00:03:59.780 --> 00:04:02.580
to the ADP to produce ATP.

00:04:02.580 --> 00:04:06.030
And ATP synthase in
mitochondria and chloroplasts

00:04:06.030 --> 00:04:07.840
are remarkably similar,

00:04:07.840 --> 00:04:12.080
although they sit in different
parts of these organelles.

00:04:12.080 --> 00:04:14.930
And the ATP in mitochondria,

00:04:14.930 --> 00:04:17.590
you can view as the end
product of respiration,

00:04:17.590 --> 00:04:20.630
while the ATP produced in chloroplasts

00:04:20.630 --> 00:04:22.470
is an intermediary store of energy,

00:04:22.470 --> 00:04:26.033
which is then used to
synthesize carbohydrates.