Intermediate value theorem

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What we're
gonna cover in this video

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is the intermediate value theorem.

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Which, despite some of this
mathy language you'll see

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is one of the more intuitive theorems

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possibly the most
intuitive theorem you will

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come across in a lot of your

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mathematical career.

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So first I'll just read it out

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and then I'll interpret it and hopefully

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we'll all appreciate
that it's pretty obvious.

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I'm not going to prove it here.

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But, I think the conceptual
underpinning here is

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it should be straightforward.

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So the theorem tells us
that suppose F is a function

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continuous at every point of the interval

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the closed interval, so
we're including A and B.

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So it's continuous at every
point of the interval A, B.

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Let me just draw a couple of examples

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of what F could look like just

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based on these first lines.

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Suppose F is a function
continuous at every point

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of the interval A, B.

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So let me draw some axes here.

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So that's my Y axis.

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And this is my X axis.

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So, one situation

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if this is A.

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And this is B.

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F is continuous at every
point of the interval

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of the closed interval A and B.

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So that means it's got
to be for sure defined

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at every point.

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As well, as to be continuous you have

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to defined at every point.

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And the limit of the function
that is recorded at that point

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should be equal to the value
of the function of that point.

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And so the function is
definitely going to be defined

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at F of A.

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So it's definitely going to have an F of A

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right over here.

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That's right over here

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is F of A.

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Maybe F of B is higher.

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Although we can look at different cases.

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So that would be our F of B.

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And they tell us it is
a continuous function.

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It is a continuous function.

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So if you're trying to
imagine continuous functions

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one way to think about it is

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if we're continuous over an interval

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we take the value of the function at

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one point of the interval.

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And, if it's continuous
we need to be able to

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get to the other, the
value of the function

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at the other point of the interval

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without picking up our pencil.

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So, I can do all sorts of things

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and it still has to be a function.

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So, I can't do something like that.

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But,

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as long as I don't pick up my pencil

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this is a continuous function.

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So, there you go.

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If the somehow the graph

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I had to pick up my pencil.

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If I had to do something like this

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oops, I got to pick up my
pencil do something like that,

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well that's not continuous anymore.

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If I had to do something like this

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and oops, pick up my pencil

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not continuous anymore.

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If I had to do something like

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wooo.

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Whoa, okay, pick up my
pencil, go down here,

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not continuous anymore.

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So, this is what a continuous function

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that a function that is continuous

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over the closed interval A, B looks like.

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I can draw some other examples, in fact,

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let me do that.

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So let me draw

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one.

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Maybe where F of B is less than

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F of A.

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So it's my Y axis.

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And this is my

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X axis.

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And once again, A and B don't both

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have to be positive,

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they can both be negative.

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One could be, A could be negative.

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B could be positive.

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And maybe in this situation.

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And F of A and F of B

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it could also be a positive or negative.

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But let's take a situation where this is

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F of A.

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So that, right over there,

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is F of A.

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This right over here

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is F of B.

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F of B.

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And once again we're saying F is a

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continuous function.

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So I should be able to go from F of A

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to F of B

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F of B draw a function

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without having to pick up my pencil.

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So it could do something like this.

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Actually I want to make it go vertical.

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It could go like this

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and then go down.

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And then

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do something

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like that.

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So these are both cases

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and I could draw an
infinite number of cases

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where F is a function
continuous at every point

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of the interval.

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The closed interval, from A to B.

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Now, given that

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there's two ways to state the conclusion

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for the intermediate value theorem.

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You'll see it written in one of these ways

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or something close to one of these ways.

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And that's why I included both of these.

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So one way to say it is, well

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if this first statement is true

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then F will take on every value

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between F of A and F of B

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over the interval.

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And you see in both of these cases

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every interval, sorry, every

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every value between F of A

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and F of B.

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So every value here

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is being taken on at some point.

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You can pick some value.

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You can pick some value,
an arbitrary value

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L, right over here.

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Oh look.

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L happened right over there.

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If you pick L

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well, L happened right over there.

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And actually it also happened there

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and it also happened there.

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And this second bullet point describes

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the intermediate value
theorem more that way.

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For any L between the values of F and A

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and F of B

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there are exists a number C

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in the closed interval from A to B

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for which F of C equals L.

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So there exists at least one C.

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So in this case

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that would be our C.

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Over here, there's potential

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there's multiple candidates for C.

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That could be a candidate for C.

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That could be a C.

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So we could say there exists at least

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one number.

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At least

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one number, I'll throw that in there,

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at least one number C

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in the interval for which this is true.

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And, something that might amuse you

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for a few minutes is

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try to draw a function where this first

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statement is true.

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But somehow the second statement is

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not true.

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So, you say, okay, well let's say

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let's assume that there's an L

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where there isn't a C in the interval.

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Let me try and do that.

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And I'll draw it big so that

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we can really see how obvious

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that we have to take on

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all of the values between F and A

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and F of B is.

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So,

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let me draw a big axis this time.

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So that's my Y axis.

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And,

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that is my X axis.

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And I'll just do the case where

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just for simplicity, that is A

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and that is B.

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And let's say

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that this is F of A.

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So that is

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F of A.

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And let's say that this

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is F of B.

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Little dotted line.

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All right.

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F of B.

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And we assume that we

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we have a continuous function here.

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So the graph, I could draw it from

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F of A to F of B from this point

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to this point

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without picking up my pencil.

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From this coordinate A comma F of A

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to this coordinate B comma F of B

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without picking up my pencil.

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Well, let's assume

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that there is some L
that we don't take on.

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Let's say there's some
value L right over here.

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And,

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and we never take on this value.

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This continuous function
never takes on this value

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as we go from X equaling A to X equal B.

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Let's see if I can draw that.

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Let's see if I can get

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from here

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to here

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without ever essentially
crossing this dotted line.

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Well let's see, I could, wooo,

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maybe I would a little bit.

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But gee, how am I gonna get there?

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Well, without picking up my pencil.

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Well,

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well, I really need to
cross that line,all right.

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Well, there you go.

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I found, we took on the value L

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and it happened at C

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which is in that closed interval.

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So once again, I'm not
giving you a proof here.

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But hopefully you have a good intuition

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that the intermediate value theorem

00:07:45.124 --> 00:07:46.447
is kind of common sense.

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The key is you're dealing
with a continuous function.

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If you make its graph

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if you were to draw it between

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the coordinates A comma F of A

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and B comma F of B

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and you don't pick up your pencil,

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which would be true of
a continuous function.

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Well, it's going to take on every value

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between F of A and F of B.


Analyzing functions for discontinuities (discontinuity example)

WEBVTT
Kind: captions
Language: en

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- [Voiceover] So we've
got this function f of x

00:00:03.260 --> 00:00:05.768
that is a piecewise continuous.

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It's defined over several intervals here

00:00:09.231 --> 00:00:11.238
for x being,

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or for zero less than x,

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and being less than or equal to two.

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F of x is natural log of x.

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For any x's larger than two, well then,

00:00:20.237 --> 00:00:21.510
f of x is going to be x squared

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times the natural log of x.

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And what we wanna do is

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we wanna find the limit of f of x

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as x approaches two.

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Now, what's interesting
about the value two

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is that that's essentially
the boundary between

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these two intervals.

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If we wanted to evaluate it at two,

00:00:36.202 --> 00:00:38.095
we would fall into this first interval.

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F of two, well, two is
less than or equal to two,

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and it's greater than zero.

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So f of two would be
pretty straightforward.

00:00:46.652 --> 00:00:48.390
That would just be natural log of two.

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But that's not necessarily
what the limit is going to be.

00:00:51.981 --> 00:00:53.431
To figure out what the
limit is going to be,

00:00:53.431 --> 00:00:54.264
we should think about, well,

00:00:54.264 --> 00:00:56.047
what's the limit as we
approach from the left,

00:00:56.047 --> 00:00:57.887
what's the limit as we
approach from the right,

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and do those exist, and if they do exist,

00:01:00.028 --> 00:01:00.910
are they the same thing,

00:01:00.910 --> 00:01:02.868
and if they are the same thing, well then,

00:01:02.868 --> 00:01:04.726
we have a well-defined limit.

00:01:04.726 --> 00:01:06.840
So let's do that.

00:01:06.840 --> 00:01:10.499
Let's first think about the limit.

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The limit of f of x as we approach two

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from the left, from values lower than two.

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Well, this is gonna be the case

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where we're gonna be
operating in this interval

00:01:24.004 --> 00:01:25.027
right over here.

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We're operating from values less than two,

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and we're going to be approaching two

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from the left.

00:01:32.231 --> 00:01:35.224
And so we'll fall under this clause,

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and so, since this clause or case

00:01:38.726 --> 00:01:40.688
is continuous over the interval

00:01:40.688 --> 00:01:41.809
in which we're operating,

00:01:41.809 --> 00:01:45.252
and for sure between, or for
all values greater than zero

00:01:45.252 --> 00:01:46.893
and less than or equal to two,

00:01:46.893 --> 00:01:49.716
this limit is going to be equal to

00:01:49.716 --> 00:01:51.991
just this clause evaluated at two,

00:01:51.991 --> 00:01:54.267
because it's continuous over the interval.

00:01:54.267 --> 00:01:58.268
So this is just going to
be the natural log of two.

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All right, so now let's
think about the limit

00:02:00.773 --> 00:02:02.171
from the right hand side,

00:02:02.171 --> 00:02:04.741
from values greater than two.

00:02:04.741 --> 00:02:06.574
So the limit of f of x

00:02:09.512 --> 00:02:13.716
as x approaches two from
the right hand side.

00:02:13.716 --> 00:02:16.142
Well, even though two
falls into this clause,

00:02:16.142 --> 00:02:18.518
as soon as we go anything
greater than two,

00:02:18.518 --> 00:02:19.876
we fall in this clause.

00:02:19.876 --> 00:02:21.750
So we're gonna be approaching two

00:02:21.750 --> 00:02:24.276
essentially using this case.

00:02:24.276 --> 00:02:26.753
And once again, this case here

00:02:26.753 --> 00:02:29.503
is continuous for all x values

00:02:31.979 --> 00:02:33.513
not only greater than two,

00:02:33.513 --> 00:02:36.572
actually, you know, greater
than or equal to two.

00:02:36.572 --> 00:02:39.556
And so, for this one over here,

00:02:39.556 --> 00:02:42.013
we can make the same
argument that this limit

00:02:42.013 --> 00:02:45.680
is going to be this
clause evaluated at two,

00:02:46.986 --> 00:02:49.503
because once again, if we just evaluated

00:02:49.503 --> 00:02:51.366
the function at two, it
falls under this clause,

00:02:51.366 --> 00:02:53.680
but if we're approaching from the right,

00:02:53.680 --> 00:02:54.766
if we're approaching from the right,

00:02:54.766 --> 00:02:56.383
those are x values greater than two,

00:02:56.383 --> 00:02:58.498
so this clause is what's at play.

00:02:58.498 --> 00:03:01.853
So we'll evaluate this clause at two.

00:03:01.853 --> 00:03:04.474
So, because it is continuous.

00:03:04.474 --> 00:03:06.823
So this is going to be two squared

00:03:06.823 --> 00:03:09.973
times the natural log of two.

00:03:09.973 --> 00:03:12.358
And so this is equal to four

00:03:12.358 --> 00:03:14.611
times the natural log of two.

00:03:14.611 --> 00:03:17.256
Four times the natural log of two.

00:03:17.256 --> 00:03:21.004
So, the right hand limit does exist.

00:03:21.004 --> 00:03:23.387
The left hand limit does exist.

00:03:23.387 --> 00:03:25.587
But the thing that might jump out at you

00:03:25.587 --> 00:03:27.874
is that these are two different values.

00:03:27.874 --> 00:03:29.934
We approach a different
value from the left

00:03:29.934 --> 00:03:31.755
as we do from the right.

00:03:31.755 --> 00:03:32.914
If you were to graph this,

00:03:32.914 --> 00:03:36.954
you would see a jump in the actual graph.

00:03:36.954 --> 00:03:41.116
You would see a discontinuity
occurring there.

00:03:41.116 --> 00:03:43.308
And so for this one in particular,

00:03:43.308 --> 00:03:46.016
you have that jump discontinuity,

00:03:46.016 --> 00:03:48.034
this limit would not exist

00:03:48.034 --> 00:03:50.255
because the left hand limit
and the right hand limit

00:03:50.255 --> 00:03:51.925
go to two different values.

00:03:51.925 --> 00:03:53.842
So this does not exist.


Analyzing functions for discontinuities (continuous example)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.469
- [Voiceover] So we have
g of x being defined

00:00:02.469 --> 00:00:06.636
as the log of 3x when zero is
less than x is less than three

00:00:08.163 --> 00:00:11.389
and four minus x times the log of nine

00:00:11.389 --> 00:00:15.056
when x is greater than or equal to three.

00:00:16.555 --> 00:00:19.421
So based on this definition of g of x,

00:00:19.421 --> 00:00:21.279
we want to find the limit of g of x

00:00:21.279 --> 00:00:24.415
as x approaches three, and once again,

00:00:24.415 --> 00:00:27.735
this three is right at
the interface between

00:00:27.735 --> 00:00:30.722
these two clauses or these two cases.

00:00:30.722 --> 00:00:34.709
We go to this first case when
x is between zero and three,

00:00:34.709 --> 00:00:37.359
when it's greater than
zero and less than three,

00:00:37.359 --> 00:00:39.908
and then at three, we hit this case.

00:00:39.908 --> 00:00:42.318
So in order to find the
limit, we want to find

00:00:42.318 --> 00:00:44.315
the limit from the left hand side

00:00:44.315 --> 00:00:46.303
which will have us dealing
with this situation

00:00:46.303 --> 00:00:49.282
'cause if we're less than
three we're in this clause,

00:00:49.282 --> 00:00:51.897
and we also want to find a
limit from the right hand side

00:00:51.897 --> 00:00:54.911
which would put us in this
clause right over here,

00:00:54.911 --> 00:00:57.179
and then if both of those limits exist

00:00:57.179 --> 00:00:59.891
and if they are the same,
then that is going to be

00:00:59.891 --> 00:01:02.560
the limit of this, so let's do that.

00:01:02.560 --> 00:01:05.957
So let me first go from
the left hand side.

00:01:05.957 --> 00:01:09.615
So the limit as x
approaches three from values

00:01:09.615 --> 00:01:12.572
less than three, so we're
gonna approach from the left

00:01:12.572 --> 00:01:16.322
of g of x, well, this
is equivalent to saying

00:01:18.564 --> 00:01:22.758
this is the limit as x approaches three

00:01:22.758 --> 00:01:24.444
from the negative side.

00:01:24.444 --> 00:01:27.936
When x is less than three,
which is what's happening here,

00:01:27.936 --> 00:01:29.946
we're approaching three from the left,

00:01:29.946 --> 00:01:32.022
we're in this clause right over here.

00:01:32.022 --> 00:01:33.889
So we're gonna be
operating right over there.

00:01:33.889 --> 00:01:37.568
That is what g of x is when
we are less than three.

00:01:37.568 --> 00:01:38.651
So log of 3x,

00:01:42.272 --> 00:01:46.439
and since this function
right over here is defined

00:01:47.274 --> 00:01:50.089
and continuous over the
interval we care about,

00:01:50.089 --> 00:01:53.903
it's defined continuous for
all x's greater than zero,

00:01:53.903 --> 00:01:56.142
we can just substitute three in here

00:01:56.142 --> 00:01:58.016
to see what it would be approaching.

00:01:58.016 --> 00:02:02.183
So this would be equal to
log of three times three,

00:02:03.228 --> 00:02:06.759
or logarithm of nine, and once again

00:02:06.759 --> 00:02:09.240
when people just write log
here within writing the base,

00:02:09.240 --> 00:02:12.823
it's implied that it
is 10 right over here.

00:02:14.094 --> 00:02:15.963
So this is log base 10.

00:02:15.963 --> 00:02:17.853
That's just a good thing to know

00:02:17.853 --> 00:02:20.688
that sometimes gets missed a little bit.

00:02:20.688 --> 00:02:22.892
All right, now let's think
about the other case.

00:02:22.892 --> 00:02:25.218
Let's think about the
situation where we are

00:02:25.218 --> 00:02:27.440
approaching three from
the right hand side,

00:02:27.440 --> 00:02:29.764
from values greater than three.

00:02:29.764 --> 00:02:33.222
Well, we are now going
to be in this scenario

00:02:33.222 --> 00:02:36.576
right over there, so
this is going to be equal

00:02:36.576 --> 00:02:39.646
to the limit as x approaches three

00:02:39.646 --> 00:02:42.994
from the positive direction,
from the right hand side

00:02:42.994 --> 00:02:45.660
of, well g of x is in this clause

00:02:45.660 --> 00:02:47.854
when we are greater than three,

00:02:47.854 --> 00:02:50.937
so four minus x times log of nine,

00:02:53.013 --> 00:02:56.193
and this looks like some type
of a logarithm expression

00:02:56.193 --> 00:02:58.503
at first until you
realize that log of nine

00:02:58.503 --> 00:03:00.848
is just a constant, log base 10 of nine

00:03:00.848 --> 00:03:03.546
is gonna be some number close to one.

00:03:03.546 --> 00:03:06.710
This expression would
actually define a line.

00:03:06.710 --> 00:03:09.708
For x greater than or equal
to three, g of x is just

00:03:09.708 --> 00:03:12.175
a line even though it looks
a little bit complicated.

00:03:12.175 --> 00:03:16.030
And so this is actually
defined for all real numbers,

00:03:16.030 --> 00:03:20.197
and it's also continuous for
any x that you put into it.

00:03:21.600 --> 00:03:23.681
So to find this limit, to think about

00:03:23.681 --> 00:03:25.278
what is this expression approaching

00:03:25.278 --> 00:03:27.824
as we approach three from
the positive direction,

00:03:27.824 --> 00:03:29.950
well we can just evaluate a three.

00:03:29.950 --> 00:03:32.852
So it's going to be four minus three

00:03:32.852 --> 00:03:35.876
times log of nine, well that's just one,

00:03:35.876 --> 00:03:39.126
so that's equal to log base 10 of nine.

00:03:40.082 --> 00:03:44.091
So the limit from the left
equals the limit from the right.

00:03:44.091 --> 00:03:47.758
They're both log nine,
so the answer here is

00:03:49.454 --> 00:03:50.787
log log of nine,

00:03:53.746 --> 00:03:55.079
and we are done.


Diode

WEBVTT
Kind: captions
Language: en

00:00:00.287 --> 00:00:03.143
- [Voiceover] The diode is our
first semi-conductor device,

00:00:03.143 --> 00:00:04.883
and it's a really important one.

00:00:04.883 --> 00:00:07.311
Every other semi-conductor
is basically made

00:00:07.311 --> 00:00:09.545
from combinations of diodes.

00:00:09.545 --> 00:00:12.059
And here's a picture of
a diode that you can buy.

00:00:12.059 --> 00:00:16.240
This is a, just a small
little glass package,

00:00:16.240 --> 00:00:20.117
and that distance right there
is about four millimeters.

00:00:20.117 --> 00:00:23.135
And inside here, right inside here,

00:00:23.135 --> 00:00:25.218
is a little silicon chip,

00:00:26.306 --> 00:00:29.428
and it's manufactured to be a diode.

00:00:29.428 --> 00:00:32.368
So the question is, what is a diode?

00:00:32.368 --> 00:00:36.927
A diode is something that
conducts current in one direction,

00:00:36.927 --> 00:00:41.042
and does not conduct current
in the other direction.

00:00:41.042 --> 00:00:45.209
And the symbol we use for
a diode looks like this.

00:00:50.317 --> 00:00:52.815
It has this big arrow here,

00:00:52.815 --> 00:00:56.982
that points in the direction
of the forward current.

00:00:58.005 --> 00:01:00.431
One way to understand how a diode works

00:01:00.431 --> 00:01:02.688
is to draw an IV curve for it.

00:01:02.688 --> 00:01:05.855
So let's draw an IV curve for a diode.

00:01:10.436 --> 00:01:13.267
If it was a perfect diode, made
in some unknown technology,

00:01:13.267 --> 00:01:17.061
what would happen is in
the reverse direction,

00:01:17.061 --> 00:01:20.811
if the voltage across
the diode was negative,

00:01:22.268 --> 00:01:25.595
we'll label the voltage this way,

00:01:25.595 --> 00:01:28.534
if the voltage across the
diode was negative, that is,

00:01:28.534 --> 00:01:32.701
this terminal is at a higher
voltage than this terminal,

00:01:33.836 --> 00:01:35.439
there would be zero current flowing.

00:01:35.439 --> 00:01:38.272
And then for any positive voltage,

00:01:40.791 --> 00:01:44.188
basically the diode
would look like a wire.

00:01:44.188 --> 00:01:45.919
So I can call that, that's essentially

00:01:45.919 --> 00:01:48.336
model number zero of a diode.

00:01:51.475 --> 00:01:52.982
Now when we build real diodes,

00:01:52.982 --> 00:01:57.383
what happens is we don't quite
get that perfect behavior,

00:01:57.383 --> 00:02:01.402
so in particular, if we
build a diode out of silicon,

00:02:01.402 --> 00:02:05.235
we can go to a, I'll go
to a number one model.

00:02:08.681 --> 00:02:11.495
And a silicon diode
actually doesn't conduct

00:02:11.495 --> 00:02:13.745
to a slight positive voltage,

00:02:13.745 --> 00:02:16.196
and then it would go up like that,

00:02:16.196 --> 00:02:19.279
where this is around point six volts.

00:02:20.210 --> 00:02:22.530
For a lot of simple
circuits that we build,

00:02:22.530 --> 00:02:26.030
this is a pretty good IV model of a diode.

00:02:27.339 --> 00:02:31.814
Just as a reminder, when we
have the IV curve of resistors,

00:02:31.814 --> 00:02:35.061
a resistors IV curve
looks something like this,

00:02:35.061 --> 00:02:38.266
it was a line that went through zero,

00:02:38.266 --> 00:02:39.809
and had a constant slope,

00:02:39.809 --> 00:02:44.148
so a diode is a really
different kind of device,

00:02:44.148 --> 00:02:48.315
it's a non-linear device,
as we can see from this.

00:02:49.768 --> 00:02:53.264
Let me move up here and now
we'll go to a next level model,

00:02:53.264 --> 00:02:57.347
that is actually the one
I wanna talk about most.

00:02:59.218 --> 00:03:02.738
This is the model of diode
that we use most of the time,

00:03:02.738 --> 00:03:05.655
so I'll call this model number two.

00:03:06.706 --> 00:03:08.938
This is the model that you use
when you'll simulate circuits

00:03:08.938 --> 00:03:11.799
or simulate diodes and we're gonna talk

00:03:11.799 --> 00:03:13.826
about this a little bit more.

00:03:13.826 --> 00:03:17.993
When you have a diode, if I
gave you a diode like this,

00:03:20.419 --> 00:03:22.629
and I said what's the IV curve of it?

00:03:22.629 --> 00:03:26.796
So what I would do is I
would find some sort of box

00:03:27.980 --> 00:03:31.397
that made voltage for me, a power supply,

00:03:32.696 --> 00:03:33.906
with an adjustment on it,

00:03:33.906 --> 00:03:38.073
and then I would also have
something that read current.

00:03:40.895 --> 00:03:45.062
So this is an ammeter, and
this is a voltage supply.

00:03:46.224 --> 00:03:48.385
And we hook that up like that.

00:03:48.385 --> 00:03:50.693
What we're gonna do is we're
gonna generate this IV curve

00:03:50.693 --> 00:03:54.110
by making actual measurements of I and V.

00:03:57.149 --> 00:03:59.176
So my first V setting is zero,

00:03:59.176 --> 00:04:01.676
that gives me this point here,

00:04:02.512 --> 00:04:05.034
I hope I measure a current of zero,

00:04:05.034 --> 00:04:06.538
otherwise this thing
would be generating power,

00:04:06.538 --> 00:04:08.538
which it's not gonna do.

00:04:09.849 --> 00:04:11.220
And then I turn up the voltage slightly,

00:04:11.220 --> 00:04:12.757
and what I notice is there's no current,

00:04:12.757 --> 00:04:15.512
there's no current when
it's at point one volts,

00:04:15.512 --> 00:04:16.880
or point two volts.

00:04:16.880 --> 00:04:20.880
And then when it gets to
around point six volts,

00:04:22.225 --> 00:04:25.225
on the diode, here's VD, and here's,

00:04:32.524 --> 00:04:34.092
when the voltage on the
diode is around point six volts

00:04:34.092 --> 00:04:35.619
what I notice is the current goes up.

00:04:35.619 --> 00:04:37.559
So it goes up to five milliamps,

00:04:37.559 --> 00:04:40.284
and then a little bit higher,
it goes up to ten milliamps,

00:04:40.284 --> 00:04:43.761
like that, and I can
plot out all these points

00:04:43.761 --> 00:04:46.374
along this part of the curve.

00:04:46.374 --> 00:04:49.899
Now, I go back here and
I change the voltage here

00:04:49.899 --> 00:04:52.828
to read the other way around,

00:04:52.828 --> 00:04:56.995
and that means I'm traveling
this way on the voltage axis.

00:04:59.073 --> 00:05:03.112
And what I'll read, my ammeter,
will read zero milliamps.

00:05:03.112 --> 00:05:05.710
Zero, zero, zero, zero, zero.

00:05:05.710 --> 00:05:09.046
And so they plot in this
part of the line here.

00:05:09.046 --> 00:05:12.202
Now if I make this voltage
really large and really negative,

00:05:12.202 --> 00:05:15.299
say I make this like minus 50 volts,

00:05:15.299 --> 00:05:17.216
that's this point here,

00:05:18.263 --> 00:05:21.938
what happens is I see a really
sharp increasing current,

00:05:21.938 --> 00:05:25.438
like that right there, and it keeps going.

00:05:26.863 --> 00:05:30.874
And that is called the
breakdown, VBR is breakdown.

00:05:30.874 --> 00:05:32.289
And for silicon diodes,

00:05:32.289 --> 00:05:36.400
minus 50 volts is a
typical value for that.

00:05:36.400 --> 00:05:38.846
This graph here shows
a break in the scale,

00:05:38.846 --> 00:05:41.706
so this is minus one
volt, minus two volts,

00:05:41.706 --> 00:05:45.516
and then we go all the way out
to 50 volts, minus 50 volts,

00:05:45.516 --> 00:05:47.113
and that's where the breakdown occurs.

00:05:47.113 --> 00:05:49.231
And most of the time
when we're using diodes,

00:05:49.231 --> 00:05:52.079
we're using them between
plus or minus one volt

00:05:52.079 --> 00:05:53.996
across their terminals.

00:05:56.599 --> 00:06:00.813
That's how we know what the IV
characteristic of a diode is.

00:06:00.813 --> 00:06:02.778
And what we can do is actually,

00:06:02.778 --> 00:06:05.312
for this section of the curve right here,

00:06:05.312 --> 00:06:07.142
for this part of the curve,

00:06:07.142 --> 00:06:09.243
I can model this with an equation.

00:06:09.243 --> 00:06:11.993
And the equation looks like this.

00:06:13.119 --> 00:06:15.039
This is the IV equation for a diode,

00:06:15.039 --> 00:06:19.206
so this is sort of like
the Ohm's law for a diode.

00:06:20.363 --> 00:06:23.113
I equals IS, this is the current,

00:06:25.783 --> 00:06:29.950
times e to the q, that's
the charge on an electron,

00:06:32.511 --> 00:06:36.883
times V on the diode, that's
the voltage on the diode,

00:06:36.883 --> 00:06:38.883
divided by kT minus one.

00:06:44.506 --> 00:06:46.043
K is Boltzmann's constant,

00:06:46.043 --> 00:06:48.935
and T is the temperature of the device,

00:06:48.935 --> 00:06:50.518
measured in Kelvin.

00:06:52.562 --> 00:06:56.603
So this equation actually
fits this part of this curve

00:06:56.603 --> 00:06:59.853
for a real diode, it's a fitting curve.

00:07:02.224 --> 00:07:03.939
We'll look at these
constants one at a time.

00:07:03.939 --> 00:07:06.939
IS is called the saturation current.

00:07:11.350 --> 00:07:12.933
Saturation current.

00:07:17.926 --> 00:07:20.938
And for silicon, for
silicon that's a value

00:07:20.938 --> 00:07:23.938
of about 10 to the minus 12 amperes,

00:07:26.262 --> 00:07:30.713
which is one picoampere,
that's how much IS is.

00:07:30.713 --> 00:07:33.296
Q is the charge on an electron,

00:07:35.988 --> 00:07:40.155
and that equals 1.602 times
10 to the minus 19 coulombs.

00:07:47.176 --> 00:07:50.926
That's q, VD is the
voltage across the diode,

00:07:52.529 --> 00:07:56.696
K is Boltzmann's constant,
that's a small k, usually,

00:08:00.449 --> 00:08:02.866
and that equals 1.38 times 10

00:08:06.790 --> 00:08:09.623
to the minus 23 Joules per Kelvin.

00:08:18.123 --> 00:08:20.373
And the last variable is T,

00:08:21.395 --> 00:08:22.938
and that's the temperature,

00:08:22.938 --> 00:08:26.605
and that's measured in
Kelvin, with a big K.

00:08:28.274 --> 00:08:30.388
Kelvin is the absolute temperature scale,

00:08:30.388 --> 00:08:34.388
so zero Kelvin equals
minus 273 degrees Celsius.

00:08:41.934 --> 00:08:43.267
Very, very cold.

00:08:44.683 --> 00:08:48.100
So this right here is the diode equation,

00:08:49.755 --> 00:08:52.172
that's the diode IV equation.

00:08:53.458 --> 00:08:57.135
And it has this exponential shape in it,

00:08:57.135 --> 00:08:59.058
it has this exponential term in it,

00:08:59.058 --> 00:09:00.198
but when we look over here,

00:09:00.198 --> 00:09:02.073
maybe this doesn't look
like an exponential curve,

00:09:02.073 --> 00:09:03.636
you haven't seen a curve like that.

00:09:03.636 --> 00:09:05.136
But that actually is just a trick

00:09:05.136 --> 00:09:07.022
of the scale of this drawing,

00:09:07.022 --> 00:09:09.633
so what I wanna do now
is I'm gonna zoom in

00:09:09.633 --> 00:09:12.352
really super close, right
on this origin right here,

00:09:12.352 --> 00:09:16.519
and we're gonna see how this
exponential term shows up,

00:09:17.828 --> 00:09:21.161
and we'll see what the meaning of IS is.

00:09:28.092 --> 00:09:31.509
I equals IS times e qV over KT minus one,

00:09:39.238 --> 00:09:41.769
and here's a close up,
here's an extreme close-up

00:09:41.769 --> 00:09:44.151
on the origin of the diode curve.

00:09:44.151 --> 00:09:46.728
The voltage scale is blown
up by about a factor of 10,

00:09:46.728 --> 00:09:49.958
so here's 1/10th of a volt
forward across the diode,

00:09:49.958 --> 00:09:53.859
and the current scale is super blown up,

00:09:53.859 --> 00:09:55.610
this is in picoamperes now,

00:09:55.610 --> 00:09:58.094
so this is in 10 to the minus 12th amperes

00:09:58.094 --> 00:10:01.334
instead of 10 to the minus three.

00:10:01.334 --> 00:10:02.332
And you can see here,

00:10:02.332 --> 00:10:05.068
this is a more familiar
looking exponential curve.

00:10:05.068 --> 00:10:07.734
And over here there's a
little bit of an offset,

00:10:07.734 --> 00:10:10.686
there's a little tiny current
in the reverse direction

00:10:10.686 --> 00:10:12.662
when the voltage is negative.

00:10:12.662 --> 00:10:15.329
And this amount here, that's IS,

00:10:17.954 --> 00:10:21.555
flowing in the negative
direction in the diode.

00:10:21.555 --> 00:10:24.832
If we look at the diode equation,
and you let V go negative,

00:10:24.832 --> 00:10:28.999
what happens is this term
here in the diode equation

00:10:29.914 --> 00:10:33.061
becomes very, very small compared to one.

00:10:33.061 --> 00:10:35.728
And what's left is IS times one,

00:10:37.166 --> 00:10:39.580
and that's what we're
looking at right here.

00:10:39.580 --> 00:10:41.503
This is a really small current,

00:10:41.503 --> 00:10:43.477
as you can see from the scale here,

00:10:43.477 --> 00:10:47.122
it's down in the low picoamps area.

00:10:47.122 --> 00:10:48.948
Almost all the time you
can ignore this current,

00:10:48.948 --> 00:10:51.115
and just treat it as zero.

00:10:52.001 --> 00:10:54.489
Whenever I wanna use a diode in a circuit,

00:10:54.489 --> 00:10:57.867
and we'll see how we solve
circuits that include

00:10:57.867 --> 00:10:59.396
these non-linear diodes in them.


Brexit and European Union primer

WEBVTT
Kind: captions
Language: en

00:00:00.190 --> 00:00:01.929
- [Voiceover] Given all
of the recent talk about

00:00:01.929 --> 00:00:05.962
the United Kingdom deciding
to leave the European Union,

00:00:05.962 --> 00:00:10.156
often referred to as Brexit,
short for British exit

00:00:10.156 --> 00:00:12.300
from the European Union.

00:00:12.300 --> 00:00:14.862
I thought it would be
interesting to do a primer on

00:00:14.862 --> 00:00:17.966
what exactly is the European Union?

00:00:17.966 --> 00:00:21.028
This is gonna be a high
level primer, the topic can

00:00:21.028 --> 00:00:23.590
get quite complex, quite fast.

00:00:23.590 --> 00:00:27.557
But to understand what the
European Union even is,

00:00:27.557 --> 00:00:31.777
I will draw a little lines
to represent the spectrum

00:00:31.777 --> 00:00:35.172
of, I guess you could say
national organizations,

00:00:35.172 --> 00:00:36.908
or organizations of nations.

00:00:36.908 --> 00:00:39.229
So at this end right over here, I'll draw,

00:00:39.229 --> 00:00:43.155
this is a trade block, this
is when a group of nations

00:00:43.155 --> 00:00:46.257
get together and say,
"Hey, let's have free trade

00:00:46.257 --> 00:00:47.164
"between our countries".

00:00:47.164 --> 00:00:49.144
Let's make it, if we make
something in one country,

00:00:49.144 --> 00:00:52.195
we want to sell in the next,
that there's no tariff on them.

00:00:52.195 --> 00:00:54.553
And it might include things
like, it's easy to move money

00:00:54.553 --> 00:00:56.885
between one country to another,
and sometimes it will even

00:00:56.885 --> 00:00:59.408
include it's easy to move
people between one country to

00:00:59.408 --> 00:01:01.720
another, and a good
example of a trade block,

00:01:01.720 --> 00:01:05.527
especially from a North
American point-of-view is NAFTA,

00:01:05.527 --> 00:01:08.582
North American Free Trade Agreement.

00:01:08.582 --> 00:01:11.018
This is the United States,
Mexico, and Canada having

00:01:11.018 --> 00:01:12.140
a free trade agreement.

00:01:12.140 --> 00:01:13.340
You make something in
one of those countries,

00:01:13.340 --> 00:01:15.160
you want to sell it in the
other one, you don't have to

00:01:15.160 --> 00:01:19.284
have, you don't have any
tariffs, it makes it reasonably

00:01:19.284 --> 00:01:22.144
easy to transport goods
from one country to another.

00:01:22.144 --> 00:01:24.542
There's even some provisions
around investments

00:01:24.542 --> 00:01:27.272
and the movement of people.

00:01:27.272 --> 00:01:29.255
Why would a country ant to do this?

00:01:29.255 --> 00:01:32.685
There's many economic
arguments and theories that it

00:01:32.685 --> 00:01:35.750
increases the prosperities
of all of the parties.

00:01:35.750 --> 00:01:38.118
Now some folks will disagree
with that, but they might

00:01:38.118 --> 00:01:40.108
say on average it might, but
there's definitely going to be

00:01:40.108 --> 00:01:42.252
some losers in different
countries depending on which

00:01:42.252 --> 00:01:45.891
industries go where, but
that's the rational for why

00:01:45.891 --> 00:01:47.673
free trade agreements exist.

00:01:47.673 --> 00:01:50.192
That it would increase the
economic growth of all involved

00:01:50.192 --> 00:01:52.840
and it'll make them all
be able to participate in

00:01:52.840 --> 00:01:55.507
a larger economic shared market.

00:01:56.560 --> 00:01:59.578
Now at this side of the spectrum,
I'm gonna talk about when

00:01:59.578 --> 00:02:02.760
states get really close to
each other to really form

00:02:02.760 --> 00:02:06.069
a meta-state, to really
form a country made up of

00:02:06.069 --> 00:02:09.997
those states, and I will
call this a federation.

00:02:09.997 --> 00:02:13.681
Maybe the most famous
federation, especially from

00:02:13.681 --> 00:02:17.239
an American point-of-view
would be the United States.

00:02:17.239 --> 00:02:21.589
United States of America,
it is made up of 50 state.

00:02:21.589 --> 00:02:25.018
It literally stands for
United States of America.

00:02:25.018 --> 00:02:28.780
Those 50 states have become
so integrated that they act as

00:02:28.780 --> 00:02:33.248
one country, they act as one
nation, they act as one people.

00:02:33.248 --> 00:02:35.978
Each of those states,
they are semi-autonomous,

00:02:35.978 --> 00:02:38.866
they do have their own
governor, their own legislator,

00:02:38.866 --> 00:02:42.533
their own courts their
own laws, but they...

00:02:44.690 --> 00:02:48.501
There's also a federal government,
there's a federal army,

00:02:48.501 --> 00:02:50.522
federal courts, there's a president,

00:02:50.522 --> 00:02:53.873
there is a federal legislature.

00:02:53.873 --> 00:02:57.593
They are so integrated
as to be one country.

00:02:57.593 --> 00:03:00.153
Now in between you'll
sometimes hear words like

00:03:00.153 --> 00:03:03.403
confederation, or confederacy, con-fed,

00:03:05.524 --> 00:03:06.691
confederation.

00:03:09.328 --> 00:03:13.919
And this is more of a lose
term, there's not a strict,

00:03:13.919 --> 00:03:16.276
this is a confederation, or
that is not a confederation,

00:03:16.276 --> 00:03:20.504
but it's generally viewed
as looser than federation.

00:03:20.504 --> 00:03:22.602
That the states, or the
countries that are members

00:03:22.602 --> 00:03:25.701
of a confederation can opt-in or opt-out.

00:03:25.701 --> 00:03:27.938
That they might be
integrated in some ways,

00:03:27.938 --> 00:03:30.749
and not integrated in other ways.

00:03:30.749 --> 00:03:34.708
When we think about the
European Union, which involves,

00:03:34.708 --> 00:03:37.606
which for sure involves
aspects of a trade block.

00:03:37.606 --> 00:03:39.879
It has free trade, it involves free trade

00:03:39.879 --> 00:03:42.042
between its members.

00:03:42.042 --> 00:03:44.517
It talks about the movement of people,

00:03:44.517 --> 00:03:46.017
so free, movement.

00:03:47.536 --> 00:03:50.558
But it goes further than
a traditional trade block,

00:03:50.558 --> 00:03:53.001
it goes further than something like NAFTA.

00:03:53.001 --> 00:03:56.846
It has, in Brussels, it
has rotating leadership,

00:03:56.846 --> 00:04:01.388
it has regulations and laws
that often apply to things

00:04:01.388 --> 00:04:04.526
like free trade, and free
movement, but it can also apply

00:04:04.526 --> 00:04:06.302
to things outside of that.

00:04:06.302 --> 00:04:09.779
There's even some integration
between militaries,

00:04:09.779 --> 00:04:12.660
or I guess at minimum you
could say cooperation.

00:04:12.660 --> 00:04:14.939
Each of the members have their
own military, and they are

00:04:14.939 --> 00:04:17.868
independent states in there
own right, but there is more

00:04:17.868 --> 00:04:21.340
integration than you would
see in something like NAFTA.

00:04:21.340 --> 00:04:25.518
So the European Union, I would
put it some place in this

00:04:25.518 --> 00:04:29.156
spectrum right over here, and
depending on whom you talk

00:04:29.156 --> 00:04:31.467
to you will get different
options, and different people

00:04:31.467 --> 00:04:34.191
will debate, some people
say, "It's more about trade,

00:04:34.191 --> 00:04:37.574
"and having a large shared
common market, and maybe being

00:04:37.574 --> 00:04:39.478
"able to have a little
more clout on the world,

00:04:39.478 --> 00:04:41.627
"even from a foreign
policy point-of-view."

00:04:41.627 --> 00:04:44.315
While some people will say,
"No, it seems to be moving

00:04:44.315 --> 00:04:46.545
"in this direction where
it's becoming a tighter

00:04:46.545 --> 00:04:50.757
"confederation of some kind,
albeit with still a lot of

00:04:50.757 --> 00:04:53.955
"independence from each
of the member states."

00:04:53.955 --> 00:04:56.395
And the people who think it's
more on the right hand side,

00:04:56.395 --> 00:04:58.911
they'll point things like,
look its got its own flag.

00:04:58.911 --> 00:05:02.634
It's got it's own anthem,
you have some military,

00:05:02.634 --> 00:05:05.691
more than cooperation, you
even have some integration.

00:05:05.691 --> 00:05:08.418
The trend seems to be
moving from left to right.

00:05:08.418 --> 00:05:12.389
It's not only is it about
trade, it's also about currency.

00:05:12.389 --> 00:05:14.788
That's where I'm going to
introduce this other concept,

00:05:14.788 --> 00:05:18.473
called the Eurozone, which
is often confused with

00:05:18.473 --> 00:05:21.244
the European Union, that's
because there's a lot of overlap

00:05:21.244 --> 00:05:22.279
between the two.

00:05:22.279 --> 00:05:26.407
The Eurozone is a subset of the
European Union that uses the

00:05:26.407 --> 00:05:30.747
same currency, and then that
currency is called the Euro.

00:05:30.747 --> 00:05:35.328
This is about currency, and
this is, about the Euro.

00:05:35.328 --> 00:05:38.095
Once again, this would
move it a little bit more

00:05:38.095 --> 00:05:40.040
to the right, not only
do you have free trade,

00:05:40.040 --> 00:05:43.672
and free movement of people,
and you have some regulations.

00:05:43.672 --> 00:05:45.700
And of course the member
states have representation

00:05:45.700 --> 00:05:47.600
in determining what those regulations are,

00:05:47.600 --> 00:05:51.614
but you also have most of the
European Union has a shared

00:05:51.614 --> 00:05:54.670
currency, which of course
facilitates free trade and free

00:05:54.670 --> 00:05:57.353
movement, but this is more
than what you see in NAFTA.

00:05:57.353 --> 00:06:00.079
Mexico, the United States
and Canada do not have

00:06:00.079 --> 00:06:02.110
a shared currency.

00:06:02.110 --> 00:06:04.878
What this map shows us is both the total,

00:06:04.878 --> 00:06:07.608
the European Union, but it
also shows that it's not

00:06:07.608 --> 00:06:10.334
exactly the same thing as the Eurozone.

00:06:10.334 --> 00:06:15.127
Everything we see in blue
and this pink color combined,

00:06:15.127 --> 00:06:17.606
that is the European Union.

00:06:17.606 --> 00:06:20.164
The difference between
the blue and the pink,

00:06:20.164 --> 00:06:23.261
is that the blue are European
members that also use

00:06:23.261 --> 00:06:25.831
the Euro as their currency,
so they are members of

00:06:25.831 --> 00:06:29.427
the Eurozone, so Spain, and
France, and Germany and Italy,

00:06:29.427 --> 00:06:31.817
and the Republic of Ireland.

00:06:31.817 --> 00:06:35.829
The are members of both the
European Union and the Eurozone.

00:06:35.829 --> 00:06:39.587
While countries like the
United Kingdom, and Denmark,

00:06:39.587 --> 00:06:42.318
and Sweden, and Poland, and
I could keep going on and on.

00:06:42.318 --> 00:06:46.485
They do not currently have
the Euro as their currency.

00:06:47.485 --> 00:06:48.848
They have their own currency,

00:06:48.848 --> 00:06:51.320
they have their own central banks.

00:06:51.320 --> 00:06:53.919
The UK, which is where we
started this conversation,

00:06:53.919 --> 00:06:56.935
they have the British Pound sterling.

00:06:56.935 --> 00:07:00.575
Now in terms of whether, or
how, or whether they're gonna

00:07:00.575 --> 00:07:03.351
move to the Euro, it's different
from country to country.

00:07:03.351 --> 00:07:05.370
Countries like,

00:07:05.370 --> 00:07:09.140
countries like the UK and
Denmark have official opt-outs.

00:07:09.140 --> 00:07:12.200
They've never had any intention,
they've opt-outs of going

00:07:12.200 --> 00:07:13.235
to the Euro.

00:07:13.235 --> 00:07:16.332
Countries like Sweden, it's
not as official, but it doesn't

00:07:16.332 --> 00:07:19.228
look like they have any
intention of going to the Euro,

00:07:19.228 --> 00:07:21.588
while other countries, it's
just a matter of time before

00:07:21.588 --> 00:07:24.277
they do go to the Euro, and
it's more of them meeting

00:07:24.277 --> 00:07:27.379
certain economic and fiscal
thresholds so that they

00:07:27.379 --> 00:07:30.151
can enter into the Eurozone.

00:07:30.151 --> 00:07:32.172
Once again, why would you want to become

00:07:32.172 --> 00:07:33.333
part of the Eurozone?

00:07:33.333 --> 00:07:35.803
If you're a country
with a smaller economy,

00:07:35.803 --> 00:07:38.154
or a smaller currency,
or less stable currency,

00:07:38.154 --> 00:07:40.631
you say, "Well, I could become
part of a larger economic

00:07:40.631 --> 00:07:42.781
"zone, I'm already part of
free trade and all that,

00:07:42.781 --> 00:07:46.672
"but the currency makes
it even a stronger thing."

00:07:46.672 --> 00:07:47.956
Why would you not want to do it?

00:07:47.956 --> 00:07:50.390
Well you say, "I have a strong
enough current, I don't want

00:07:50.390 --> 00:07:54.484
"to give monetary policy
to folks in Frankfurt."

00:07:54.484 --> 00:07:58.784
Once again, good people can
argue very good arguments

00:07:58.784 --> 00:08:01.056
on either side of that.

00:08:01.056 --> 00:08:03.581
When you look at this map
here, you might also notice

00:08:03.581 --> 00:08:06.435
that Kosovo and Montenegro
are colored in in this

00:08:06.435 --> 00:08:09.232
little pink color, and
this is interesting because

00:08:09.232 --> 00:08:12.575
they are not members of the
European Union, but they have

00:08:12.575 --> 00:08:16.251
decided to use the Euro as
their official currency anyway.

00:08:16.251 --> 00:08:18.779
So they're not even official
members of the Eurozone,

00:08:18.779 --> 00:08:20.476
but they still use the Euro.

00:08:20.476 --> 00:08:23.877
So that's why they get their
own special color here.

00:08:23.877 --> 00:08:26.976
So if we go back to Brexit,
if we go back to the

00:08:26.976 --> 00:08:30.894
United Kingdom leaving, what
are the implications here?

00:08:30.894 --> 00:08:33.747
Well, the first clear implication
is that their currency

00:08:33.747 --> 00:08:36.561
is not going to change, and the
future of their currency is,

00:08:36.561 --> 00:08:38.591
is not going to change.

00:08:38.591 --> 00:08:40.993
They were never planning,
they don't use the Euro,

00:08:40.993 --> 00:08:43.015
and they were never planning
on going to the Euro.

00:08:43.015 --> 00:08:46.450
And so, they are using the
Pound sterling, and they will

00:08:46.450 --> 00:08:48.559
continue to use the Pound sterling.

00:08:48.559 --> 00:08:50.542
That's kind of a no-brainer
and the Brexit doesn't

00:08:50.542 --> 00:08:51.533
impact that.

00:08:51.533 --> 00:08:52.700
Same currency.

00:08:55.833 --> 00:08:59.300
They're using the Pound
regardless of which way the

00:08:59.300 --> 00:09:01.082
Brexit vote went.

00:09:01.082 --> 00:09:03.765
What the Brexit vote does
by taking them out of the

00:09:03.765 --> 00:09:07.487
European Union, is it puts
these things in question,

00:09:07.487 --> 00:09:09.802
things like free trade and free movement,

00:09:09.802 --> 00:09:13.071
and what regulations they
will adopt or not adopt.

00:09:13.071 --> 00:09:15.054
It doesn't mean that
there won't be free trade,

00:09:15.054 --> 00:09:17.078
or there won't be free
movement, or that regulations

00:09:17.078 --> 00:09:18.607
will change dramatically.

00:09:18.607 --> 00:09:20.833
It also doesn't mean that
that's not going to happen.

00:09:20.833 --> 00:09:25.001
What it means is it's all going
to be called into question.

00:09:25.001 --> 00:09:27.933
These are now going to have
to be separate negotiations

00:09:27.933 --> 00:09:29.586
between the United Kingdom

00:09:29.586 --> 00:09:32.503
and the rest of the European Union.


History of the Republican Party

WEBVTT
Kind: captions
Language: en

00:00:00.234 --> 00:00:01.067
- [Voiceover] Hey, Kim!

00:00:01.067 --> 00:00:01.900
- [Voiceover] Hi, David!

00:00:01.900 --> 00:00:04.852
- [Voiceover] So, with the
Republican National Convention

00:00:04.852 --> 00:00:07.471
coming up in just a couple of
weeks as we're recording this,

00:00:07.471 --> 00:00:09.206
you thought it would be
like a really good idea

00:00:09.206 --> 00:00:13.148
to sit down and examine the
history of the Republican Party.

00:00:13.148 --> 00:00:15.936
So what's, what's going
on in the country in 1854

00:00:15.936 --> 00:00:18.648
that leads to this party forming?

00:00:18.648 --> 00:00:20.244
- [Voiceover] Well,
there're growing discussions

00:00:20.244 --> 00:00:21.478
over slavery,

00:00:21.478 --> 00:00:23.981
and why the slavery
should expand to the West.

00:00:23.981 --> 00:00:26.497
Now, all throughout the 19th century

00:00:26.497 --> 00:00:28.701
the citizens of the United States had been

00:00:28.701 --> 00:00:30.924
kind of compromising on
the issue of slavery.

00:00:30.924 --> 00:00:33.146
First, they had a line
between North and South,

00:00:33.146 --> 00:00:35.709
said only slave states
could be below this line.

00:00:35.709 --> 00:00:40.151
Now the Kansas-Nebraska Act
overturns that compromise.

00:00:40.151 --> 00:00:43.424
The Kansas-Nebraska Act,
which says that the citizens

00:00:43.424 --> 00:00:46.749
of a territory when applying for statehood

00:00:46.749 --> 00:00:50.237
can themselves decide
whether or not that state

00:00:50.237 --> 00:00:51.702
should have slavery.

00:00:51.702 --> 00:00:54.016
- [Voiceover] So, even
though Kansas and Nebraska

00:00:54.016 --> 00:00:57.713
are north of this parallel in Missouri

00:00:57.713 --> 00:01:00.329
above which slavery couldn't exist,

00:01:00.329 --> 00:01:03.686
this new law kind of overturns
that, that agreement?

00:01:03.686 --> 00:01:04.519
- [Voiceover] Exactly.

00:01:04.519 --> 00:01:08.848
So a number of US citizens,
who are anti-slavery,

00:01:08.848 --> 00:01:11.447
which means that they don't want slavery

00:01:11.447 --> 00:01:13.961
to spread into western territories,

00:01:13.961 --> 00:01:15.978
mainly because they want those territories

00:01:15.978 --> 00:01:18.288
free from white farmers

00:01:18.288 --> 00:01:22.098
to not have to compete
with wealthy slaveholders

00:01:22.098 --> 00:01:25.662
who have free labor to
farm and ship their goods

00:01:25.662 --> 00:01:27.238
and sell their crops.

00:01:27.238 --> 00:01:29.250
- [Voiceover] What about
people that hate slavery

00:01:29.250 --> 00:01:32.457
and think it's immoral
and want to abolish it?

00:01:32.457 --> 00:01:34.443
- [Voiceover] Those people
are called abolitionists.

00:01:34.443 --> 00:01:35.580
- [Voiceover] Yeah,
that's a convenient name.

00:01:35.580 --> 00:01:36.413
- [Voiceover] Yes.

00:01:36.413 --> 00:01:40.007
And the abolitionists,
really before the 1850s

00:01:40.007 --> 00:01:42.175
they were kind of considered
the lunatic fringe,

00:01:42.175 --> 00:01:45.416
only those sorts of people would imagine

00:01:45.416 --> 00:01:49.149
that you would want to end
slavery right now everywhere

00:01:49.149 --> 00:01:50.885
that exists in the United States.

00:01:50.885 --> 00:01:52.933
So, they don't want to
just not have slavery

00:01:52.933 --> 00:01:53.864
out in the West,

00:01:53.864 --> 00:01:55.881
they want slavery to be
ended where it exists

00:01:55.881 --> 00:01:57.339
already in the South.
- [Voiceover] Right.

00:01:57.339 --> 00:01:59.419
- [Voiceover] So, those
who believed in abolition,

00:01:59.419 --> 00:02:01.268
those who believed in anti-slavery

00:02:01.268 --> 00:02:04.017
went to a new party, the Republican Party.

00:02:04.017 --> 00:02:07.575
- [Voiceover] So even
within the Republican Party

00:02:07.575 --> 00:02:10.533
abolitionism was still on the fringe

00:02:10.533 --> 00:02:12.474
of the party plank?

00:02:12.474 --> 00:02:13.531
- [Voiceover] Yeah, I would say so.

00:02:13.531 --> 00:02:17.234
So, the new Republican
Party which really comes out

00:02:17.234 --> 00:02:18.908
to an extremely strong start,

00:02:18.908 --> 00:02:21.156
they run their first candidate in 1856,

00:02:21.156 --> 00:02:23.849
he gets second place in national election,

00:02:23.849 --> 00:02:25.654
which is amazing.
- [Voiceover] Not bad.

00:02:25.654 --> 00:02:27.921
- [Voiceover] But their
second candidate for President

00:02:27.921 --> 00:02:29.399
is Abraham Lincoln.

00:02:29.399 --> 00:02:31.980
And Lincoln himself is
actually kind of considered

00:02:31.980 --> 00:02:35.230
a moderate, because he is anti-slavery,

00:02:36.237 --> 00:02:38.600
he is not an abolitionist.

00:02:38.600 --> 00:02:41.514
But nonetheless, the
South perceives Lincoln

00:02:41.514 --> 00:02:45.549
to be an abolitionist and
white Southerners revolt

00:02:45.549 --> 00:02:47.334
and start the Civil War.

00:02:47.334 --> 00:02:48.788
- [Voiceover] So, because he is perceived

00:02:48.788 --> 00:02:51.096
as an abolitionist,
because he is a Republican,

00:02:51.096 --> 00:02:52.832
that's why South Carolina secedes?

00:02:52.832 --> 00:02:53.665
- [Voiceover] Exactly.

00:02:53.665 --> 00:02:55.305
So, the Civil War ensues.

00:02:55.305 --> 00:02:57.355
This is a four year long battle.

00:02:57.355 --> 00:02:59.188
620,000 Americans die.

00:03:00.320 --> 00:03:03.479
And at the end of the day, the North,

00:03:03.479 --> 00:03:06.239
the United States of America
led by the Republican Party

00:03:06.239 --> 00:03:07.930
is victorious.

00:03:07.930 --> 00:03:09.228
- [Voiceover] So the
victory of the United States

00:03:09.228 --> 00:03:11.442
in the Civil War kind of assures

00:03:11.442 --> 00:03:14.757
dominion of the Republican
Party for a generation.

00:03:14.757 --> 00:03:16.854
- [Voiceover] Yeah, I would
say even more than that.

00:03:16.854 --> 00:03:19.016
So, for the rest of the 19th century

00:03:19.016 --> 00:03:21.695
and really into the early 20th century

00:03:21.695 --> 00:03:24.553
the Republican Party is the
stronger political party

00:03:24.553 --> 00:03:26.595
in the United States.

00:03:26.595 --> 00:03:28.911
- [Voiceover] So, from
the end of the Civil War,

00:03:28.911 --> 00:03:32.212
from 1865, until about when would you say?

00:03:32.212 --> 00:03:33.635
- [Voiceover] I would
say the Great Depression.

00:03:33.635 --> 00:03:36.189
- [Voiceover] So, it's
an almost unbroken string

00:03:36.189 --> 00:03:37.673
of republican presidencies.

00:03:37.673 --> 00:03:38.715
- [Voiceover] Yeah, there are only three

00:03:38.715 --> 00:03:40.678
Democratic presidents in this time period.

00:03:40.678 --> 00:03:44.862
So it's 72 years of
pretty much uninterrupted

00:03:44.862 --> 00:03:46.436
Republican rule.

00:03:46.436 --> 00:03:50.259
And the Republican Party is
the party of anti-slavery.

00:03:50.259 --> 00:03:52.761
During the Civil War they were the party

00:03:52.761 --> 00:03:55.414
of the emancipation
proclamation under Lincoln.

00:03:55.414 --> 00:03:59.960
So, it's during their rule
that the 13th Amendment,

00:03:59.960 --> 00:04:02.353
abolishing slavery, is passed.

00:04:02.353 --> 00:04:05.740
The 14th Amendment, which
guarantees equal citizenship

00:04:05.740 --> 00:04:07.861
to African Americans, is passed.

00:04:07.861 --> 00:04:10.670
And the 15th Amendment,
guaranteeing the right to vote

00:04:10.670 --> 00:04:12.600
for African Americans, is passed.

00:04:12.600 --> 00:04:13.516
- [Voiceover] So in the period

00:04:13.516 --> 00:04:16.254
immediately following the Civil
War called Reconstruction,

00:04:16.254 --> 00:04:18.363
when we see the election
of some of the first

00:04:18.363 --> 00:04:20.919
African American senators
and representatives

00:04:20.919 --> 00:04:22.019
to the Congress.

00:04:22.019 --> 00:04:22.852
- [Voiceover] Exactly.

00:04:22.852 --> 00:04:26.517
So, during this time period
quite a few African American men

00:04:26.517 --> 00:04:30.343
were elected to US Congress,
and many more served

00:04:30.343 --> 00:04:33.235
in appointed roles, like postmaster.

00:04:33.235 --> 00:04:35.048
- [Voiceover] So this is
when we get the election

00:04:35.048 --> 00:04:38.063
of Senator Hiram Revels from Mississippi.

00:04:38.063 --> 00:04:39.654
- [Voiceover] Exactly.

00:04:39.654 --> 00:04:43.171
So Hiram Revels was one of the first two

00:04:43.171 --> 00:04:45.551
African American senators.

00:04:45.551 --> 00:04:48.568
So after the Civil War the
Republican Party was really

00:04:48.568 --> 00:04:50.999
kind of this party of the Gilded Age.

00:04:50.999 --> 00:04:54.863
They believed in modernizing
the infrastructure

00:04:54.863 --> 00:04:56.121
of the United States.

00:04:56.121 --> 00:04:57.839
They built lots of railroads.

00:04:57.839 --> 00:04:59.582
They enacted policies

00:04:59.582 --> 00:05:01.654
that would protect American business.

00:05:01.654 --> 00:05:04.387
And it's really in this early period

00:05:04.387 --> 00:05:07.092
of the turn of the century
that the Republican Party

00:05:07.092 --> 00:05:10.362
becomes associated with
protections of business there.

00:05:10.362 --> 00:05:12.410
- [Voiceover] Is that
what the elephant's about?

00:05:12.410 --> 00:05:13.378
- [Voiceover] Kind of, yes.

00:05:13.378 --> 00:05:16.045
So, the elephant was popularized

00:05:17.119 --> 00:05:19.952
in an 1870 cartoon by Thomas Nast.

00:05:20.968 --> 00:05:22.415
- [Voiceover] Oh, the
same guy that gave us

00:05:22.415 --> 00:05:23.608
Santa Claus, right?

00:05:23.608 --> 00:05:24.758
- [Voiceover] Yes.

00:05:24.758 --> 00:05:29.351
And Nast depicted the
Republican Party as an elephant

00:05:29.351 --> 00:05:31.445
because it was a party of strength,

00:05:31.445 --> 00:05:34.109
a really big consequential party.

00:05:34.109 --> 00:05:35.502
- [Voiceover] That's so
fascinating to have gone

00:05:35.502 --> 00:05:37.860
from this, like, insurgency party to this,

00:05:37.860 --> 00:05:40.109
like, to being perceived as the elephant

00:05:40.109 --> 00:05:42.554
of electoral politics in 30 years.

00:05:42.554 --> 00:05:43.563
- [Voiceover] Yeah, it's amazing.

00:05:43.563 --> 00:05:46.044
Unfortunately, it kinda
all comes crashing down

00:05:46.044 --> 00:05:47.233
with the Great Depression.

00:05:47.233 --> 00:05:48.066
- [Voiceover] Sure.

00:05:48.066 --> 00:05:49.569
- [Voiceover] So, the
pro-business policies,

00:05:49.569 --> 00:05:51.885
the lack of regulation in the 1920s

00:05:51.885 --> 00:05:55.218
leads to the stock market crash of 1929.

00:05:56.614 --> 00:06:00.053
And it was a Republican
president, Herbert Hoover,

00:06:00.053 --> 00:06:04.038
who was in the presidency
at the time of the crash.

00:06:04.038 --> 00:06:08.454
And so, in 1932, Democrat
Franklin Roosevelt

00:06:08.454 --> 00:06:09.993
is elected president.

00:06:09.993 --> 00:06:14.009
And the following 40 years, more or less,

00:06:14.009 --> 00:06:18.176
are going to be the time
of Democratic ascendancy.

00:06:19.163 --> 00:06:22.252
But, in the meantime, there is one notable

00:06:22.252 --> 00:06:24.074
Republican president.

00:06:24.074 --> 00:06:26.124
- [Voiceover] Ike for
President, Ike for President.

00:06:26.124 --> 00:06:28.444
- [Voiceover] I like Ike, you like Ike,

00:06:28.444 --> 00:06:29.304
Everybody likes Ike.

00:06:29.304 --> 00:06:30.307
- [Voiceover] For President.

00:06:30.307 --> 00:06:32.485
- [Voiceover] (laughs) So, we were quoting

00:06:32.485 --> 00:06:35.324
one of Ike's campaign commercials.

00:06:35.324 --> 00:06:37.015
- [Voiceover] Who is Ike, Kim?

00:06:37.015 --> 00:06:40.070
- [Voiceover] Ike was
General Dwight Eisenhower

00:06:40.070 --> 00:06:43.049
who was a World War II hero.

00:06:43.049 --> 00:06:46.252
He was so popular he
would have been elected

00:06:46.252 --> 00:06:47.381
had his...

00:06:47.381 --> 00:06:50.115
- [Voiceover] Could have been
from the Martian party, right?

00:06:50.115 --> 00:06:52.540
- [Voiceover] Yes (laughs).

00:06:52.540 --> 00:06:54.643
- [Voiceover] Like any party
expect the Communist Party

00:06:54.643 --> 00:06:57.112
would have propelled
Eisenhower to the presidency.

00:06:57.112 --> 00:06:57.985
- [Voiceover] Yes. Exactly.

00:06:57.985 --> 00:07:01.435
And you're right that one
of the, you know, key themes

00:07:01.435 --> 00:07:03.861
of this time period was anti-Communism,

00:07:03.861 --> 00:07:06.138
and both Republicans and Democrats

00:07:06.138 --> 00:07:08.242
had an anti-Communist bent.

00:07:08.242 --> 00:07:11.602
But Eisenhower was elected in 1952,

00:07:11.602 --> 00:07:13.988
and he was really the first president

00:07:13.988 --> 00:07:17.174
to use commercial spots to get elected.

00:07:17.174 --> 00:07:19.262
He had these cute little jingles

00:07:19.262 --> 00:07:20.401
that were very memorable.

00:07:20.401 --> 00:07:21.339
- [Voiceover] So catchy.

00:07:21.339 --> 00:07:23.089
- [Voiceover] And he
really used the medium

00:07:23.089 --> 00:07:24.201
of television well.

00:07:24.201 --> 00:07:27.782
So he's kind of the father of TV ads.

00:07:27.782 --> 00:07:28.615
- [Voiceover] Yeah, it's interesting,

00:07:28.615 --> 00:07:29.866
because I think we think of Kennedy

00:07:29.866 --> 00:07:31.595
as being the first television president.

00:07:31.595 --> 00:07:33.419
But I would, I think we
could both make the claim

00:07:33.419 --> 00:07:35.015
that it's, it's really Eisenhower.

00:07:35.015 --> 00:07:36.140
- [Voiceover] That is a great point.

00:07:36.140 --> 00:07:39.732
So, Eisenhower is kind
of a Republican moment

00:07:39.732 --> 00:07:42.438
in a much larger Democratic era.

00:07:42.438 --> 00:07:43.271
- [Voiceover] Okay.

00:07:43.271 --> 00:07:44.487
- [Voiceover] And this is the era

00:07:44.487 --> 00:07:46.521
when the Civil Rights Movement happens.

00:07:46.521 --> 00:07:49.243
This is the era of the
Great Society programs,

00:07:49.243 --> 00:07:51.442
which were Lyndon Johnson's programs

00:07:51.442 --> 00:07:54.484
to try to attack poverty,
and New Deal programs.

00:07:54.484 --> 00:07:57.260
So this is really the
birth of the welfare state.

00:07:57.260 --> 00:07:59.245
So in this time period
the Republicans begin

00:07:59.245 --> 00:08:01.353
to experience a demographic shift.

00:08:01.353 --> 00:08:04.360
So first, they had been the
party that was most known

00:08:04.360 --> 00:08:06.326
for representing African Americans,

00:08:06.326 --> 00:08:08.243
because they were the party of Lincoln.

00:08:08.243 --> 00:08:10.186
But during the New Deal,

00:08:10.186 --> 00:08:13.912
when most people really
needed economic help,

00:08:13.912 --> 00:08:15.425
the African American constituency

00:08:15.425 --> 00:08:16.968
moved over to the Democrats.

00:08:16.968 --> 00:08:20.524
They actually had a campaign
saying to African Americans

00:08:20.524 --> 00:08:24.041
"turn your picture of
Abraham Lincoln to the wall"

00:08:24.041 --> 00:08:26.590
so that he can't see you changed parties.

00:08:26.590 --> 00:08:28.500
But this is really the time period

00:08:28.500 --> 00:08:30.281
when the Democratic Party

00:08:30.281 --> 00:08:32.788
begins to pick up the
votes of African Americans.

00:08:32.788 --> 00:08:37.372
And so, over the course of
the 1930s through the 1960s,

00:08:37.372 --> 00:08:40.846
as the Democratic Party begins to advocate

00:08:40.846 --> 00:08:43.759
bigger and bigger government,
a larger welfare state,

00:08:43.759 --> 00:08:46.304
and more and more social progress,

00:08:46.304 --> 00:08:50.402
the Republicans develop a
conservative response to that.

00:08:50.402 --> 00:08:52.652
And in the 1970s and 1980s,

00:08:53.521 --> 00:08:55.641
in the aftermath of the
Civil Rights Movement,

00:08:55.641 --> 00:08:59.558
many whites in the South
felt that the social chaos

00:08:59.558 --> 00:09:02.198
of the Civil Rights
Movement had gone too far.

00:09:02.198 --> 00:09:05.198
And so, they left the Democratic Party,

00:09:05.198 --> 00:09:08.131
which had been traditionally
a party in the South,

00:09:08.131 --> 00:09:10.129
and joined the Republican Party,

00:09:10.129 --> 00:09:13.185
which was presenting a
more conservative face

00:09:13.185 --> 00:09:14.592
towards social change.

00:09:14.592 --> 00:09:18.549
And so, in the 1980s this
new conservative movement

00:09:18.549 --> 00:09:21.644
really came together in the
person of Ronald Reagan.

00:09:21.644 --> 00:09:23.255
And Ronald Reagan brought together

00:09:23.255 --> 00:09:25.121
a number of constituencies.

00:09:25.121 --> 00:09:28.747
He brought together business interests,

00:09:28.747 --> 00:09:32.090
who wanted less government
regulation of business.

00:09:32.090 --> 00:09:36.090
He also brought together
Christian evangelicals,

00:09:37.017 --> 00:09:39.517
who wanted a more conservative

00:09:40.717 --> 00:09:43.634
social value program in government.

00:09:45.563 --> 00:09:48.165
And he also brought
together anti-Communists,

00:09:48.165 --> 00:09:50.138
who felt that the Democratic presidents

00:09:50.138 --> 00:09:53.592
had been too soft on
Communism during their tenure.

00:09:53.592 --> 00:09:54.500
- [Voiceover] So this
is interesting to me,

00:09:54.500 --> 00:09:57.778
because it seems to be
around the era of Reagan

00:09:57.778 --> 00:09:58.957
that we started to see the beginnings

00:09:58.957 --> 00:10:03.257
of ideological polarization
within the parties.

00:10:03.257 --> 00:10:06.509
- [Voiceover] I would say
that's kind of been around

00:10:06.509 --> 00:10:08.791
since the beginning, more or less.

00:10:08.791 --> 00:10:11.791
You know, the two
original political parties

00:10:11.791 --> 00:10:13.034
in the United States,

00:10:13.034 --> 00:10:15.787
the Federalists and the Anti-Federalists,

00:10:15.787 --> 00:10:17.711
or the Democratic-Republicans.

00:10:17.711 --> 00:10:19.761
This is the party led by Thomas Jefferson

00:10:19.761 --> 00:10:22.406
versus the party led
by Alexander Hamilton.

00:10:22.406 --> 00:10:26.259
You know, they had the
same idea of the sort

00:10:26.259 --> 00:10:27.724
of large central government

00:10:27.724 --> 00:10:29.614
versus the small central government.

00:10:29.614 --> 00:10:33.449
In many ways we're still
debating the same issues

00:10:33.449 --> 00:10:37.599
that Hamilton and Jefferson
were debating in 1800.

00:10:37.599 --> 00:10:39.467
- [Voiceover] So, okay, so we're seeing

00:10:39.467 --> 00:10:41.857
this conservative coalition

00:10:41.857 --> 00:10:45.429
coalesce around the election of Reagan,

00:10:45.429 --> 00:10:47.429
and his election was like a sweep, right?

00:10:47.429 --> 00:10:48.262
- [Voiceover] Yes.

00:10:48.262 --> 00:10:51.991
Yeah, he deregulates a lot of industries.

00:10:51.991 --> 00:10:56.074
He defends conservative
social family values,

00:10:56.074 --> 00:10:58.344
like prayer in school, for example.

00:10:58.344 --> 00:11:03.054
And he takes a very hard
line against Communism.

00:11:03.054 --> 00:11:04.471
And George W. Bush,

00:11:04.471 --> 00:11:07.006
who was the most recent
Republican president,

00:11:07.006 --> 00:11:08.689
had a fairly similar agenda,

00:11:08.689 --> 00:11:11.554
although less emphasis on anti-Communism

00:11:11.554 --> 00:11:14.237
and instead an emphasis on anti-terrorism.

00:11:14.237 --> 00:11:16.760
- [Voiceover] So we're seeing
this shift over last 150 years

00:11:16.760 --> 00:11:18.997
of party priorities for the Republicans

00:11:18.997 --> 00:11:21.713
as the country changes and
as its demographics change.

00:11:21.713 --> 00:11:22.546
- [Voiceover] And we'll find out

00:11:22.546 --> 00:11:24.773
how the story of the
Republican Party continues

00:11:24.773 --> 00:11:26.190
in this election.


United Kingdom vs Great Britain vs England primer

WEBVTT
Kind: captions
Language: en

00:00:00.275 --> 00:00:01.366
- [Voiceover] For someone who lives

00:00:01.366 --> 00:00:02.916
outside of the United Kingdom,

00:00:02.916 --> 00:00:06.997
the terms United Kingdom and
Great Britain and England

00:00:06.997 --> 00:00:09.174
often feel interchangeable,

00:00:09.174 --> 00:00:11.329
and they feel like they're
referring to the same thing,

00:00:11.329 --> 00:00:12.967
but as we'll see in this video,

00:00:12.967 --> 00:00:17.007
they aren't referring to
exactly the same thing.

00:00:17.007 --> 00:00:19.146
So the United Kingdom
is actually short for

00:00:19.146 --> 00:00:23.226
the United Kingdom of Great
Britain and northern Ireland.

00:00:23.226 --> 00:00:26.887
So the United Kingdom
includes everything here,

00:00:26.887 --> 00:00:30.244
on the right, on the
island of Great Britain,

00:00:30.244 --> 00:00:32.130
so it includes all of that,

00:00:32.130 --> 00:00:36.424
and it includes part of
the island of Ireland,

00:00:36.424 --> 00:00:39.450
it includes northern
Ireland, right over there.

00:00:39.450 --> 00:00:41.864
That is the United Kingdom,

00:00:41.864 --> 00:00:46.287
the United Kingdom of Great
Britain and northern Ireland.

00:00:46.287 --> 00:00:49.134
Now, what are they talking about

00:00:49.134 --> 00:00:50.793
when people say Great Britain,

00:00:50.793 --> 00:00:52.424
well they're just referring
to the large island,

00:00:52.424 --> 00:00:54.636
here on the right.

00:00:54.636 --> 00:00:55.959
And what about England?

00:00:55.959 --> 00:00:58.876
Well, England is a
subset of Great Britain.

00:00:58.876 --> 00:01:02.534
England is this region
in this peachy color,

00:01:02.534 --> 00:01:05.142
right over here on the island,

00:01:05.142 --> 00:01:07.725
on the island of Great Britain.

00:01:08.596 --> 00:01:11.981
Now the reason why some people
will interchange these words

00:01:11.981 --> 00:01:14.289
is because London is in England,

00:01:14.289 --> 00:01:17.832
it's the seat of power
for the United Kingdom.

00:01:17.832 --> 00:01:19.347
But there are other regions,

00:01:19.347 --> 00:01:21.754
and some people would even say nations,

00:01:21.754 --> 00:01:24.232
because the folks in these
regions, historically,

00:01:24.232 --> 00:01:27.556
have shared common traditions
and common dialects,

00:01:27.556 --> 00:01:30.152
sometimes bordering on common language,

00:01:30.152 --> 00:01:33.699
and the other regions, or
you could even say nations,

00:01:33.699 --> 00:01:36.982
of people who have a common
affinity, include the Welsh,

00:01:36.982 --> 00:01:41.513
and so, they are in Wales,
also part of Great Britain,

00:01:41.513 --> 00:01:43.133
which is part of the United Kingdom.

00:01:43.133 --> 00:01:45.699
You have the Scots, also,

00:01:45.699 --> 00:01:48.805
or the people who live in Scotland,

00:01:48.805 --> 00:01:51.055
also part of Great Britain,

00:01:52.202 --> 00:01:53.635
which is part of the United Kingdom.

00:01:53.635 --> 00:01:56.680
So that's Great Britain, and as we said,

00:01:56.680 --> 00:01:58.231
the United Kingdom is
the United Kingdom of

00:01:58.231 --> 00:02:00.482
Great Britain and northern Ireland.

00:02:00.482 --> 00:02:04.524
So it also includes this, it
includes this northern part,

00:02:04.524 --> 00:02:07.846
right over here, of the island of Ireland.

00:02:07.846 --> 00:02:09.544
And so some of you might be saying,

00:02:09.544 --> 00:02:12.371
well what is right here in green,

00:02:12.371 --> 00:02:15.263
what is the rest of the island of Ireland?

00:02:15.263 --> 00:02:19.430
Well that is the Republic,
the Republic of Ireland.

00:02:22.551 --> 00:02:25.152
So hopefully that clarifies a little bit,

00:02:25.152 --> 00:02:26.665
the difference between the United Kingdom,

00:02:26.665 --> 00:02:29.996
which includes Wales, England, Scotland,

00:02:29.996 --> 00:02:32.625
and northern Ireland,
it's the United Kingdom

00:02:32.625 --> 00:02:35.084
of Great Britain and northern Ireland.

00:02:35.084 --> 00:02:36.717
The difference between
that, Great Britain,

00:02:36.717 --> 00:02:39.368
which is the large
island here on the right,

00:02:39.368 --> 00:02:42.504
England, which is a part of Great Britain,

00:02:42.504 --> 00:02:46.671
and the Republic of Ireland,
which we just talked about.


Trig limit using double angle identity

WEBVTT
Kind: captions
Language: en

00:00:00.817 --> 00:00:02.204
- [Voiceover] All right, let's
see if we can find the limit

00:00:02.204 --> 00:00:05.184
of 1 over the square
root of 2 sine of theta

00:00:05.184 --> 00:00:06.626
over cosine of 2 theta,

00:00:06.626 --> 00:00:09.876
as theta approaches negative pi over 4.

00:00:11.011 --> 00:00:13.759
And like always, try to give
it a shot before we go through

00:00:13.759 --> 00:00:14.899
it together.

00:00:14.899 --> 00:00:17.125
Well, one take on it is well, let's just,

00:00:17.125 --> 00:00:19.153
let's just say that this is
going to be the same thing

00:00:19.153 --> 00:00:20.397
as the limit,

00:00:20.397 --> 00:00:23.742
as theta approaches negative pi over 4

00:00:23.742 --> 00:00:26.825
of 1 plus square root of 2 sine theta

00:00:28.731 --> 00:00:29.898
over the limit

00:00:31.295 --> 00:00:34.545
as theta approaches negative pi over 4.

00:00:35.505 --> 00:00:37.296
Make sure we can see that negative there,

00:00:37.296 --> 00:00:39.046
of cosine of 2 theta,

00:00:40.107 --> 00:00:43.384
and both of these expressions are,

00:00:43.384 --> 00:00:44.969
if these were function definitions

00:00:44.969 --> 00:00:47.493
or if we were to graph y equals

00:00:47.493 --> 00:00:50.203
1 plus square root of sine,

00:00:50.203 --> 00:00:51.989
square root of 2 times sine theta,

00:00:51.989 --> 00:00:54.325
or y equals cosine of 2 theta,

00:00:54.325 --> 00:00:55.978
we would get continuous functions,

00:00:55.978 --> 00:00:59.389
especially at theta is
equal to negative pi over 4,

00:00:59.389 --> 00:01:01.072
so we could just substitute in.

00:01:01.072 --> 00:01:03.532
We'll see well this is
going to be equal to

00:01:03.532 --> 00:01:05.941
this expression evaluated
at negative pi over 4,

00:01:05.941 --> 00:01:08.006
so 1 plus square root of 2

00:01:08.006 --> 00:01:10.756
times sine of negative pi over 4,

00:01:12.301 --> 00:01:13.218
over cosine

00:01:14.886 --> 00:01:17.386
of 2 times negative pi over 4.

00:01:19.156 --> 00:01:21.156
Now, negative pi over 4,

00:01:23.031 --> 00:01:26.140
sine of negative pi over 4 is going to be

00:01:26.140 --> 00:01:28.793
negative square root of 2 over 2,

00:01:28.793 --> 00:01:32.886
so this is negative
square root of 2 over 2,

00:01:32.886 --> 00:01:34.241
we're assuming this is in radians,

00:01:34.241 --> 00:01:35.760
if we're thinking in
degrees, this would be

00:01:35.760 --> 00:01:38.106
a negative 45-degree angle,
so this is one of the,

00:01:38.106 --> 00:01:41.065
one of the trig values
that it's good to know

00:01:41.065 --> 00:01:43.077
and so if you have,

00:01:43.077 --> 00:01:44.751
if you have 1,

00:01:44.751 --> 00:01:45.922
so let's see,

00:01:45.922 --> 00:01:46.879
actually, let me just rewrite it,

00:01:46.879 --> 00:01:50.190
so this is going to be equal to 1 plus

00:01:50.190 --> 00:01:52.324
square root of 2 times that

00:01:52.324 --> 00:01:55.730
is going to be negative 2 over 2,

00:01:55.730 --> 00:01:58.038
so this is going to be

00:01:58.038 --> 00:01:59.331
minus 1,

00:01:59.331 --> 00:02:00.369
that's the numerator over here.

00:02:00.369 --> 00:02:02.958
All of this stuff simplifies to negative 1

00:02:02.958 --> 00:02:03.791
over,

00:02:04.781 --> 00:02:07.943
this is going to be cosine of

00:02:07.943 --> 00:02:10.809
negative pi over 2, right?

00:02:10.809 --> 00:02:13.489
This is negative pi over 2,

00:02:13.489 --> 00:02:15.866
cosine of negative pi over
2, if you thought in degrees,

00:02:15.866 --> 00:02:17.444
that's going to be negative 90 degrees.

00:02:17.444 --> 00:02:22.015
Well, cosine of that is
just going to be zero,

00:02:22.015 --> 00:02:24.512
so what we end up with

00:02:24.512 --> 00:02:27.294
is equal to zero over zero,

00:02:27.294 --> 00:02:28.868
and as we've talked about before,

00:02:28.868 --> 00:02:31.066
if we had something
non-zero divided by zero,

00:02:31.066 --> 00:02:32.564
we'd say, okay, that's undefined.

00:02:32.564 --> 00:02:33.543
We might as well give up,

00:02:33.543 --> 00:02:35.464
but we have this indeterminate
form, it does not mean

00:02:35.464 --> 00:02:36.985
the limit does not exist.

00:02:36.985 --> 00:02:39.292
It's usually a clue that
we should use some tools

00:02:39.292 --> 00:02:41.923
in our toolkit, one of which
is to do some manipulation

00:02:41.923 --> 00:02:44.537
here to get an expression
that maybe is defined

00:02:44.537 --> 00:02:46.049
at theta is equal to,

00:02:46.049 --> 00:02:47.014
or does not,

00:02:47.014 --> 00:02:48.263
is not an indeterminate form,

00:02:48.263 --> 00:02:50.231
that theta is equal to pi over 4

00:02:50.231 --> 00:02:53.401
and we'll see other tools in
our toolkit in the future.

00:02:53.401 --> 00:02:56.678
So let me algebraically
manipulate this a little bit.

00:02:56.678 --> 00:03:00.007
So if I have 1 plus the square root of 2,

00:03:00.007 --> 00:03:02.674
sine theta, over cosine 2 theta,

00:03:05.493 --> 00:03:06.811
as you can imagine,

00:03:06.811 --> 00:03:10.612
the things that might be useful
here are our trig identities

00:03:10.612 --> 00:03:14.040
and in particular, cosine of
2 theta seems interesting.

00:03:14.040 --> 00:03:16.381
Let me write some trig
identities involving cosine

00:03:16.381 --> 00:03:17.546
of 2 theta.

00:03:17.546 --> 00:03:18.650
I'll write it over here.

00:03:18.650 --> 00:03:21.063
So we know that cosine

00:03:21.063 --> 00:03:21.896
of 2 theta

00:03:23.156 --> 00:03:24.073
is equal to

00:03:25.344 --> 00:03:28.023
cosine squared of theta

00:03:28.023 --> 00:03:31.094
minus sine squared of theta

00:03:31.094 --> 00:03:32.677
which is equal to 1

00:03:34.152 --> 00:03:36.569
minus 2 sine squared of theta

00:03:38.055 --> 00:03:39.472
which is equal to

00:03:40.562 --> 00:03:43.145
2 cosine squared theta minus 1,

00:03:44.794 --> 00:03:47.662
and you can go from this
one to this one to this one

00:03:47.662 --> 00:03:49.668
just using the Pythagorean identity.

00:03:49.668 --> 00:03:52.553
We proved that in earlier
videos in trigonometry

00:03:52.553 --> 00:03:54.485
on Khan Academy.

00:03:54.485 --> 00:03:56.608
Now, do any of these look useful?

00:03:56.608 --> 00:03:58.819
Well, all of these three
are going to be differences

00:03:58.819 --> 00:04:02.034
of squares, so we can factor
them in interesting ways,

00:04:02.034 --> 00:04:04.682
and remember, our goal
at the end of the day

00:04:04.682 --> 00:04:07.258
is maybe cancel things out
that are making us get this

00:04:07.258 --> 00:04:08.740
zero over zero,

00:04:08.740 --> 00:04:11.704
and if I could factor this

00:04:11.704 --> 00:04:14.465
into something that involved
a 1 plus square root

00:04:14.465 --> 00:04:15.388
of 2 sine theta,

00:04:15.388 --> 00:04:16.356
then I'm going to be in business,

00:04:16.356 --> 00:04:17.656
and it looks like,

00:04:17.656 --> 00:04:19.591
it looks like this right over here,

00:04:19.591 --> 00:04:21.342
that can be factored as

00:04:21.342 --> 00:04:24.175
1 plus square root of 2 sine theta

00:04:25.982 --> 00:04:27.368
times

00:04:27.368 --> 00:04:30.368
1 minus square root of 2 sine theta,

00:04:31.812 --> 00:04:33.119
so let me use this.

00:04:33.119 --> 00:04:36.051
Cosine of 2 theta is the same thing,

00:04:36.051 --> 00:04:37.483
cosine of 2 theta is the same thing

00:04:37.483 --> 00:04:39.782
as 1 minus 2 sine squared theta,

00:04:39.782 --> 00:04:41.663
which is just a difference of squares.

00:04:41.663 --> 00:04:43.197
We can rewrite that as,

00:04:43.197 --> 00:04:44.986
this is a-squared minus b-squared,

00:04:44.986 --> 00:04:47.736
this is a plus b times a minus b,

00:04:48.953 --> 00:04:51.818
so I can just replace this with

00:04:51.818 --> 00:04:54.651
1 plus square root of 2 sine theta

00:04:56.923 --> 00:05:00.423
times 1 minus square root of 2 sine theta,

00:05:02.480 --> 00:05:04.384
and now, we have some nice cancelling,

00:05:04.384 --> 00:05:06.773
or potential cancelling that can occur,

00:05:06.773 --> 00:05:08.301
so we could say

00:05:08.301 --> 00:05:09.637
that cancels with that

00:05:09.637 --> 00:05:12.869
and we could say that
that is going to be equal,

00:05:12.869 --> 00:05:14.511
and let me do this in a new color,

00:05:14.511 --> 00:05:16.198
this is going to be equal to,

00:05:16.198 --> 00:05:17.413
in the numerator we just have 1,

00:05:17.413 --> 00:05:19.486
in the denominator we just are left with

00:05:19.486 --> 00:05:22.486
1 minus square root of 2 sine theta,

00:05:23.679 --> 00:05:26.941
and if we want these
expressions to truly be equal,

00:05:26.941 --> 00:05:29.833
we would have to have
them to have the same,

00:05:29.833 --> 00:05:31.844
if you view them as function definitions,

00:05:31.844 --> 00:05:33.408
as having the same domain,

00:05:33.408 --> 00:05:34.621
so this one right over here,

00:05:34.621 --> 00:05:36.108
this one we already saw is

00:05:36.108 --> 00:05:39.914
not defined at theta is
equal to negative pi over 4,

00:05:39.914 --> 00:05:41.238
and so this one,

00:05:41.238 --> 00:05:42.611
in order for these to be equivalent,

00:05:42.611 --> 00:05:44.867
we have to say that this one is also not,

00:05:44.867 --> 00:05:47.077
and actually, other
places, but let's just,

00:05:47.077 --> 00:05:48.472
let's just say theta

00:05:48.472 --> 00:05:49.802
does not,

00:05:49.802 --> 00:05:52.153
does not equal negative,

00:05:52.153 --> 00:05:53.736
negative pi over 4,

00:05:55.095 --> 00:05:56.942
and we could think about
all of this happening

00:05:56.942 --> 00:06:00.377
in some type of an open interval
around negative pi over 4

00:06:00.377 --> 00:06:02.149
if we wanted to get very precise,

00:06:02.149 --> 00:06:03.284
but if we wanted to,

00:06:03.284 --> 00:06:05.413
for this particular case,

00:06:05.413 --> 00:06:06.638
well, let's just say,

00:06:06.638 --> 00:06:09.019
everything we're doing
is in the open interval,

00:06:09.019 --> 00:06:09.852
so in,

00:06:11.403 --> 00:06:12.820
in open interval,

00:06:15.926 --> 00:06:17.926
in open interval between

00:06:19.177 --> 00:06:20.381
theta,

00:06:20.381 --> 00:06:22.734
or, say, negative 1 and 1,

00:06:22.734 --> 00:06:24.081
and I think that covers it

00:06:24.081 --> 00:06:25.805
because if we have pi,

00:06:25.805 --> 00:06:27.938
if we have pi over 4

00:06:27.938 --> 00:06:30.260
that is not going to get us the

00:06:30.260 --> 00:06:31.927
zero over zero form,

00:06:33.059 --> 00:06:34.142
and pi over 4

00:06:34.996 --> 00:06:37.261
would make this denominator equal to zero

00:06:37.261 --> 00:06:39.006
but it also makes,

00:06:39.006 --> 00:06:41.873
let's see, pi over 4 also
will make this denominator

00:06:41.873 --> 00:06:43.022
equal to zero,

00:06:43.022 --> 00:06:45.156
'cause we would get 1 minus 1,

00:06:45.156 --> 00:06:46.069
so I think,

00:06:46.069 --> 00:06:47.559
I think we're good if we're just assuming,

00:06:47.559 --> 00:06:48.844
if we're restricted to this open interval

00:06:48.844 --> 00:06:50.346
and that's okay because
we're taking the limit

00:06:50.346 --> 00:06:53.777
as it approaches something
within this open interval,

00:06:53.777 --> 00:06:56.047
and I'm being extra precise
because I'm trying to explain it

00:06:56.047 --> 00:06:57.748
to you and it's important to be precise,

00:06:57.748 --> 00:06:59.295
but obviously, if you're
working this out on a test

00:06:59.295 --> 00:07:00.930
or notebook, you wouldn't be taking,

00:07:00.930 --> 00:07:01.821
putting,

00:07:01.821 --> 00:07:03.328
or taking as much trouble

00:07:03.328 --> 00:07:06.643
to be putting all of these caveats in.

00:07:06.643 --> 00:07:09.220
So, what we've now realized

00:07:09.220 --> 00:07:10.413
is that, okay,

00:07:10.413 --> 00:07:11.552
this expression,

00:07:11.552 --> 00:07:12.681
actually, let's think about this.

00:07:12.681 --> 00:07:15.716
Let's think about the limit,

00:07:15.716 --> 00:07:20.110
the limit as theta
approaches negative pi over 4

00:07:20.110 --> 00:07:21.094
of this thing,

00:07:21.094 --> 00:07:22.316
without the restriction,

00:07:22.316 --> 00:07:25.684
of 1 over 1 minus the square root of 2

00:07:25.684 --> 00:07:26.851
sine of theta.

00:07:28.222 --> 00:07:29.532
If we're dealing with this over,

00:07:29.532 --> 00:07:31.421
you know, with this open interval,

00:07:31.421 --> 00:07:33.748
wait, actually, even disregarding that,

00:07:33.748 --> 00:07:34.667
theta,

00:07:34.667 --> 00:07:35.570
this theta,

00:07:35.570 --> 00:07:38.389
or this expression is continuous at,

00:07:38.389 --> 00:07:40.445
it is defined and it is continuous

00:07:40.445 --> 00:07:43.434
at theta is equal to negative pi over 4

00:07:43.434 --> 00:07:45.045
so this is just going to be equal to

00:07:45.045 --> 00:07:48.327
1 over 1 minus the square root of 2

00:07:48.327 --> 00:07:49.410
times sine of

00:07:51.362 --> 00:07:53.691
negative pi over 4.

00:07:53.691 --> 00:07:55.941
Sine of negative pi over 4.

00:07:58.240 --> 00:07:59.955
Sine of negative pi over 4,

00:07:59.955 --> 00:08:03.693
we've already seen is negative
square root of 2 over 2,

00:08:03.693 --> 00:08:07.068
and so this is going to be equal to 1 over

00:08:07.068 --> 00:08:10.134
1 minus square root of 2 times

00:08:10.134 --> 00:08:12.081
the negative square root of 2 over 2,

00:08:12.081 --> 00:08:13.421
so negative, negative,

00:08:13.421 --> 00:08:14.403
you get a positive,

00:08:14.403 --> 00:08:16.347
square root of 2 times square root of 2

00:08:16.347 --> 00:08:18.182
is 2, over 2 is going to be 1.

00:08:18.182 --> 00:08:21.139
So this is going to be equal to 1/2.

00:08:21.139 --> 00:08:22.423
And so,

00:08:22.423 --> 00:08:23.695
I want to be very clear.

00:08:23.695 --> 00:08:26.697
This expression is not the same thing

00:08:26.697 --> 00:08:28.496
as this expression.

00:08:28.496 --> 00:08:31.371
They are the same thing
at all values of theta,

00:08:31.371 --> 00:08:33.608
especially if we're dealing
in this open interval

00:08:33.608 --> 00:08:35.763
except at theta equals

00:08:35.763 --> 00:08:36.967
negative pi over 4.

00:08:36.967 --> 00:08:38.320
This one is not defined

00:08:38.320 --> 00:08:40.191
and this one is defined,

00:08:40.191 --> 00:08:42.094
but as we've seen multiple times before,

00:08:42.094 --> 00:08:43.940
if we find a function

00:08:43.940 --> 00:08:46.831
that is equal to our original

00:08:46.831 --> 00:08:47.664
or an expression,

00:08:47.664 --> 00:08:49.480
is equal to our original expression,

00:08:49.480 --> 00:08:51.342
and all values of theta

00:08:51.342 --> 00:08:52.444
except,

00:08:52.444 --> 00:08:55.216
except where the original
one was not defined

00:08:55.216 --> 00:08:56.049
at a certain point,

00:08:56.049 --> 00:08:59.427
but this new one is defined
and is continuous there,

00:08:59.427 --> 00:09:02.818
well then these two limits
are going to be equal,

00:09:02.818 --> 00:09:04.881
so if this limit is 1/2,

00:09:04.881 --> 00:09:06.898
then this limit is going to be 1/2,

00:09:06.898 --> 00:09:08.405
and I've said this in previous videos.

00:09:08.405 --> 00:09:09.867
It might be very tempting to say, well,

00:09:09.867 --> 00:09:11.339
I'm just going to algebraically
simplify this in some way

00:09:11.339 --> 00:09:12.347
to get this,

00:09:12.347 --> 00:09:14.006
and I'm not going to
worry about too much about

00:09:14.006 --> 00:09:15.678
these constraints, and then
I'm just going to substitute

00:09:15.678 --> 00:09:17.303
negative pi over 4,

00:09:17.303 --> 00:09:18.909
and you will get this answer

00:09:18.909 --> 00:09:20.202
which is the correct answer

00:09:20.202 --> 00:09:22.093
but it's really important to recognize

00:09:22.093 --> 00:09:24.553
that this expression and this expression

00:09:24.553 --> 00:09:26.285
are not the same thing

00:09:26.285 --> 00:09:29.012
and what allows you to do this is,

00:09:29.012 --> 00:09:30.553
is the truth

00:09:30.553 --> 00:09:32.758
that if you have two functions,

00:09:32.758 --> 00:09:34.425
if you have f and g,

00:09:37.097 --> 00:09:39.522
two functions equal,

00:09:39.522 --> 00:09:40.558
let me write it this way,

00:09:40.558 --> 00:09:41.391
equal,

00:09:42.696 --> 00:09:43.529
equal for

00:09:44.481 --> 00:09:45.314
all x,

00:09:46.384 --> 00:09:47.634
except for all,

00:09:49.059 --> 00:09:50.197
wait, let me just write this this way,

00:09:50.197 --> 00:09:52.364
for all x except for a,

00:09:53.407 --> 00:09:54.657
then the limit,

00:09:55.764 --> 00:09:58.527
then, and let me write it this way,

00:09:58.527 --> 00:10:00.753
equal for all except,

00:10:00.753 --> 00:10:02.208
for all x except a

00:10:02.208 --> 00:10:03.625
and f continuous,

00:10:06.362 --> 00:10:07.695
continuous at a,

00:10:10.083 --> 00:10:10.916
then,

00:10:11.777 --> 00:10:13.777
then the limit of f of x

00:10:15.028 --> 00:10:16.642
as x approaches a

00:10:16.642 --> 00:10:19.790
is going to be equal
to the limit of g of x

00:10:19.790 --> 00:10:21.290
as x approaches a,

00:10:21.290 --> 00:10:22.434
and I said this in multiple videos

00:10:22.434 --> 00:10:23.923
and that's what we are doing right here,

00:10:23.923 --> 00:10:27.194
but just so you can make
sure you got it right,

00:10:27.194 --> 00:10:29.861
the answer here is 1/2.