| Start | Who runs a big economic advisory firm he's like one of my oldest friends.
We were talking about what he's looking for when he recruits. So starting pay at this firm for college graduates is $120,000 a year. Just so you understand this is like one of these big shot Wall Street economic advisory firms. They do very very well, however they only hire a couple of people a year. They don't care at all where you went to college. It's called Cornerstone Macro. And they do economic research and financial stuff for Wall Street blah blah blah. |
| 0:32 | None of you are in the position yet you haven't graduated college but they pay very well.
You should get a bonus on top of that so you could hit the $200,000 mark pretty fast there. But it's a small place. But anyways he said two things that he looks for that he finds are lacking one is skills with Excel. So in my business statistics class almost every class is sort of a partial Excel lesson. It's not really gonna show up here. My advice to all of you over the next couple of years is get good with Excel. |
| 1:02 | Wouldn't matter where you're gonna be in your business jobs. They're going to sit you in front of a computer. It's going to be a Windows computer
not an Apple and they're going to ask you to use Excel.
The reason they like Windows rather than Apple is there's some things that don't work properly in the Apple version of Microsoft Office. I don't know why maybe because Microsoft hates Apple. Maybe it's because they make millions and millions off this maybe not that much but for some reason it works much better on Windows. |
| 1:31 | At these firms. It's only in movies you see them sitting at Apple computers. In real life it's not gonna be an Apple.
I wouldn't worry about it in college but be prepared to have to make the transition. You just can't. A lot of the business productivity tools were really in the 80's and 90's and they were all Microsoft. Nobody used Apple stuff- they're updated it but they haven't really transferred it. So he said he really looks for Excel skills. Stony Brook has free Excel learning stuff. |
| 2:00 | I don't know when the next Excel seminar will be scheduled.
Just go to the Stony Brook website and do some searching. It's all the division of information and technology. They have some free online stuff. I mean you could just go and teach yourself Excel skills. You don't have to be a master but he wants to feel that you can do stuff. That you don't just view Excel as a word processor with boxes. The other thing he said which I know I've commented to you guys before is he agreed with me he said he wants to know that you know what's going on in the world. |
| 2:31 | And that when he interviews you and says what do you think of the news last night from Seoul that you should understand that something happened in Korea last night
that you should know about because it's effecting the markets this morning.
Usually it's out there in the news you know they announce that Samsung had a terrible quarter or something like that. But you should know these things so he agrees with he he likes the Wall Street Journal app more. I like the Bloomberg app a little more but I don't think it really matters. It's five minutes of work and you should start practicing that. |
| 3:02 | Okay and you're here for a few years. Most of you are freshman so don't get too excited.
But that's the kind of things he's looking for. He and I used to work together at Banker's Trust back in the '80s. And we would have a stack of newspapers that we would work through every morning. Literally just skimming the headlines. We weren't looking for too much depth you just wanna know you know that the London central bank has raised interest rates by 4%. You just need to know that and of course you need to know what that means. But that's part of why you get the education over the next few years is so you know what that means. |
| 3:31 | When central banks raise interest rates that tends to slow down the economy.
Just incase you were wondering why. Why would high interest rates slow the economy? It's more expensive to borrow and you're more likely to put your money in the bank and not use your money in other ways. The more you can get for putting your money away into interest rate bearing interests |
| 4:01 | objects the less likely you are to use your money for something else like food.
So when I was in grad school interest rates were 18%. You could put your money in the bank and get 18%. Right now you get .1% just so you can do this comparison. So 18% means why would I do anything other than stick my money in the bank? Which was the idea because we had runaway inflation at that point. And it just slowed the economy down into this huge recession. |
| 4:32 | Anyway enough of that.
Time to learn something new. So we're actually a little ahead on the syllabus. That's why I won't feel so bad if we end up not having a Friday class. I don't generally cancel them but you know the doctors they aren't always on time. Wanna find out how you can get a job with my friend's company? Go here, study math and applied economics, do really well, ask me in 3 years. But it's small so they don't hire that much but that's what they're looking for and they're looking for sales and analysis and things like that. |
| 5:05 | And they've done very well. Very well.
His wife is actually the real managing partner. He is the backup managing partner. She does this stuff this woman Nancy. She's insane so she'll- typical week so yesterday she flew to Frankfurt, today she's going to Moscow, Thursday she flies to Hong Kong, that's gotta take a day. Right? Then she's gonna spend 4 hours at some meeting and then fly to Los Angeles and come back and so she'll go around the globe like this. |
| 5:36 | All the time she does these crazy things. I don't know how anybody can do that and not lose their mind.
But George Clooney did that in movies so apparently it's possible. And he's George Clooney so. Some people confuse the two of us a lot. That wasn't funny. Alright time to get into calculus a bit. Now I don't know how much of that you recorded Shannon. All of it. All of it. Awesome! |
| 6:01 | Future people will look at it and be like what the heck was that?
What does that have to do with math? It has nothing to do with math. This class is business calculus. It's not math. Right? I first taught business calculus when I was in grad school, kinda fun. We didn't have any computers or calculators then. There was no Excel. They had a very primitive early version but it wasn't available then to students. So everything was paper and pencil so of course we couldn't give you any fun numbers |
| 6:30 | I think you were allowed to use calculators.
I forget. Alright so when we're talking about limits so I don't know if any of you have done anything with limits so limits are a major factor in calculus. They are the key mental step that enables mathematicians to make the transition from you know static slope, a line, to curves. So the Greeks used to talk a lot about this math stuff and they ran into an interesting puzzle which was Achilles versus the tortoise. |
| 7:02 | It essentially says let's say I wanna get from here to the podium there.
Well in order to get from here to the podium first I have to get half way to the podium. And then I have to go half way from here to here. And I have to go another half way from here to here. I keep having to go half ways. But there's always going to be a little bit left over so how can I actually get there? You say of course you can get there but that doesn't make any sense. So they wrestled with this for hundreds of years. |
| 7:30 | Because they knew that there had to be a way to say that you could get there.
But they of course are mathematicians they have to be justified mathematically. So eventually they did what they call limits and limits basically say well look you get so close that you might as well say that you're there. The difference between almost there and actually zero is infinitely small, an infinite decimal. And at some point it's so small that we can throw it away. Because you can make it as small as you want. Then they spent the next 200 years sort of refining this mathematically. |
| 8:04 | And these involve what are called limits so the idea of a limit
is when you get really close to a number but you don't actually get to the number.
So infinitesimally close to a number. |
| 8:33 | And a lot of times it's just the same as if you were actually at the number.
So let's take something simple like f(x) = x^2 So what happens when you plug in 3 is you get 3^2 = 9. But let's say this function is not defined at 3. It was defined anywhere but 3. |
| 9:08 | You say it doesn't have a value at 3. So let's redefine this now and say
Okay and we're not gonna give it a value when x is 3.
So then you'd say well where is the hole? |
| 9:31 | Well it's obvious on a problem like this you say the hole is at 9.
Because when you plug in 3 you get 9. Because 3^2. But how can we say that mathematically? What we want to say is that the limit Limit (lim) when x gets really close to 3 of f(x) what's that? What you say mathematically is you say well what happens when it's really close to 3 but at < 3. |
| 10:04 | So 2.9999999 you just keep going with the 9's.
Well I'm just under 9 but I'm so close to 9 that I might as well pretend I'm at 9. Okay because this is 9. And what happens when I'm just a little bit bigger than 3? I'm also very very close to 9. So at 3.0000001 or 2.9999999 I'm essentially at 3 on either side but I'm not actually there. |
| 10:34 | And this was the big intellectual thing it's like you know you're essentially here but you're not actually there you say you're so close you might as well say that you're there.
Because it doesn't matter. And the key with calculus was to show that it doesn't matter. Once you did that you could do all sorts of cool stuff. So the idea is that the limit as x approaches 3 of f(x) is 9. So you're not saying that f(3) = 9. |
| 11:00 | There is no f(3). But what you're saying is that when you get really close to 3 you get really close to 9.
And as I said mathematically you could be as close as you want. So far so good? This sounds kind of fancy. Okay what questions are you gonna ask me? Well we have lets see... First we take something very simple. |
| 11:34 | And we would say...
So limit when x approaches 2 with a little negative symbol.
That little negative symbol is minus. It means when you approach it from a number just less than 2. So from the negative side of 2. So if you imagine the x axis, this is x^3 + 1 |
| 12:03 | and 2 when you're coming at it from this direction that's the 2-.
Also known as the lefthand side. When you look at it from that side you say well it looks pretty close to 2^3 + 1. Because this is a very simple function. Nothing fun is going on with this type. Now what happens when I go from the other side? So what symbol do you think we use for the other side? |
| 12:33 | We use minus for the left so anyone wanna guess?
A plus sign. Oh you guys are all geniuses it's amazing. A plus symbol. So this is the minus side. This is the plus also known as the righthand side. So you call them the lefthand limit and the righthand limit. Okay this is the left limit, this is the right limit. It's also 2^3 + 1 and because they agree then you can say |
| 13:06 | the limit when x -> 2 altogether is 9.
Because sometimes they're not going to agree. We'll give you an example in a second. So that's what you have to guard against. Because if they don't match then you have a problem. So you pick an equation. |
| 13:43 | This equation really kinda looks more like this.
That's what's called a piece wise function. So the trick with this function. This says when x ≤ 2 use this for f(x). |
| 14:06 | And when x > 2 use this for f(x). They're not the same equation.
How and do they meet? Well it turns out they don't actually meet there. I really shouldn't have drawn the picture so fast it looks more like that. So let's look at this and say okay what's the limit when you look at x from the minus (-) side. 2 from the - side. |
| 14:39 | Well if we look at 2 from the - side this is < 2.
So we use this branch of the piece wise function. So we get 2^3 + 1 And if we look at 2 from the plus (+) side |
| 15:05 | We're now looking at the other branch of the equation.
That's this one. Notice they don't match. So 2^2 + 1 is 5. So those two don't equal each other. So that's fine so you've got the lefthand limit you say when it's <2 you get 9. When it's >2 you do 5 but what happens when it's 2? |
| 15:36 | So what's the limit when x -> 2?
I don't specify a sign. Have you ever seen Mean Girls? The limit does not exist. Actually in that problem in Mean Girls the limit did exist but that's okay. It's a movie. So we write DNE because we like to- yes I've seen Mean Girls too when it came out. I was younger then. |
| 16:00 | I was still inappropriately old for it but what the heck it was a fun movie.
Pretty good so far. So we write DNE like I said. That's the lazy way of writing does not exist. But the first general idea behind limits so now we could look at the graph and you could tell we didn't even really need the numbers. You look at this graph and say well when you're on this side of 2 you're over here. But when you're on this side of 2 you're down here. You're not close to each other so you remember on this graph the closer you get to 3 the closer you get to 9. |
| 16:36 | Here, as you get closer and closer to 2 you still are stuck with the gap on the one side of the function and the other side.
You can't close that so because you can't close that gap there isn't a limit at the actual value. Okay so far so good? |
| 17:01 | Of course the book gets all technical and then you can do this in the calculator.
But we don't really like to do it in the calculator because it's annoying. Alright so let's do some practice. |
| 17:33 | So let's say this is our f(x).
We'll use a nice simple graph like this. So I could ask you 3 things. I could ask you what's the limit when x ->4- ? |
| 18:26 | Five, right. You look at this and you say what happens when we're just <4?
|
| 18:31 | So limit 4- means a number <4 but very very close to 4.
Infinitesimally close is going to equal 5. You say okay what happens when I approach 4 from the other side? What's the limit gonna be? 7, right. You guys are smart. It's not a complicated problem. Therefore the limit x -> 4 DNE. The limit does not exist. |
| 19:07 | Don't confuse that with infinity because infinity is a whole different thing.
Infinity exists. It's just infinite. What if instead I give you a graph that looks like that? |
| 19:41 | So it's two different functions.
We got a lefthand function and a righthand function. The only interesting thing that's going on here is what's happening at 4. So again I ask you what's the limit as x -> 4- ? |
| 20:03 | Any ideas?
5. You guys are geniuses. Usually when I ask a question in class the answer is going to be simple. I will try not to ask questions where it really hurts because that's what I'm here for right? But every once in a while if I want to torture you make you cry and go home to your parents and say "I hate my life" I'll give you one of those. What about when we go from the other side of 4? |
| 20:35 | Well even though we're on a different function we're still at 5.
So the limit from the right side is still also 5. Therefore, the limit as x -> 4 is 5. So remember what i said is if the lefthand limit and the righthand limit match, that's the limit. |
| 21:04 | That's how you figure out what a limit is. If they don't match that's when it does not exist.
Oh well for anything else the function behaves okay so we go back to this. Let's say what happens at you know 10. So the left side and the right side are going to basically agree. |
| 21:33 | So then it'll be whatever that y-value is when I get to 10.
So the only place where you could have a problem in right here at 4. So the rest of the time the function is pretty well behaved. It's only interesting right there at 4. So if I were to do that algebraically, |
| 22:14 | so say I give you f(x) |
| 22:44 | And I give you this and I say alright what happens when we're looking at both sides of 4?
You say well the limit when x -> 4 - of f(x), which branch is the minus side the top or the bottom? |
| 23:03 | It's the top. And the reason it's the top is because that's <4.
Okay because some of you are going to come up to me later and say which one is the minus side and which one is the plus side? The minus side is less than the number and the plus side is greater than the number. So for less than the number I get 8+3= 11 |
| 23:30 | And here I get 20-9 which is also 11.
So this is one of these graphs like that. Actually it looks kind of more like this. And at 4 we're at 11. Figure not drawn to scale. That's what they say on the SATs. So even though it's two different functions they're meeting at 11 so we can say that lim x -> 4 is 11. |
| 24:10 | How are we doing on this limit stuff so far?
I don't know how many of you have seen limits before. Some of you probably did. Sometimes it's a pre-calculus topic sometimes it's not. Depends on your teacher. Some of the pre-calculus teachers just wanna get right into calculus. They get ahead of themselves which annoys the teacher who's teaching after them. |
| 24:30 | But that's the way it goes.
I haven't been around high school in a few years but a lot of that. Alright suppose instead, |
| 25:06 | Suppose I give you this function.
So again, nothing terribly interesting is going on with this function except at exactly x = 5. Now you want to find the limits. So say okay what happens when x -> 5 from the minus side? |
| 25:32 | Well when x approaches 5 from the negative side I get 2.
And what happens when I approach 5 from the plus side? I also get 2. Because you said both sides you're squeezing in x approaches 5 and y approaches 2. But there is no value at 5. |
| 26:01 | There is no f(5). So how can I say the limit is going to be 2? Because remember the definition of how we would find the limit
is what happens when x is exactly 5.
I'm sorry, is NOT exactly 5. When it's infinitesimally close to 5. Well we can get closer and closer to 2 we can get as close to 2 as we want. As they say in math, arbitrarily close. So the limit isn't telling you what f(2) is because f(2) does not exist. |
| 26:40 | So there's no value for f(2) but again we don't care. We're just asking for the limit.
So far so good? Alright let's give you a practice problem. |
| 27:00 | Here's a fun one.
|
| 28:35 | Okay have fun with that.
I meant f(5) yeah you knew I meant that. Thank you. f(2) exists. It's something. Alright take a minute. See if you can figure that out. |
| 32:15 | So when you ask for the limit you basically go to the graph and you're asking for the y-value when x is at a certain value.
A lot of times you say well what's the difference between this and just plugging in the number. Most of the time there is no difference. The limit is a tool that enables us to then allow us to do the derivative. |
| 32:37 | So what happens when we get really close to 1 but we're a negative value?
Well when I get really close to 1, not a negative value I'm sorry we're less than 1. So 0.99999. Well there you get really close to 1 and you get really close to 3. Because I'm going to be on this graph. I'm less than 1. So this is gonna be 3. That make sense? |
| 33:00 | How do I know I'm on this graph? Because the lim x -> 1- means values <1 and I look at the y-values and they're always on this graph.
And for 1+ I'm looking from this side. So I'm always looking on this graph so this is gonna be 1. Now f of exactly 1 happens to be 2. That's this dot. |
| 33:36 | And the limit as x approaches 1 well remember what I told you. The limit as x approaches 1 is you take a look at the lefthand limit
and the righthand limit and see if they match.
Since they don't match that limit does not exist. So far so good? |
| 34:01 | How about the limit as x approaches -2 from the negative side?
Well if you're looking at <-2 that's here. So we're looking at this part of the line. It kinda looks like 0. It's close to the x axis. What about when x approaches -2 from the other side? So this way. Well again, nothing funny is happening it's just getting to 0. |
| 34:52 | Alright then at -2 f(x) = 0.
Because as I said there's no jumps or holes or anything funny going on there. |
| 35:02 | Alright and the limit as x approaches -2 you just look at these two limits and since they match this is 0.
So far so good? How'd we do on this? Easy? Medium? Wanna practice another one? Okay we'll do one more. It's mostly just getting the terminology right. |
| 35:36 | This is a good one.
Where are my magic glasses? |
| 36:47 | Okay let's find |
| 37:46 | Okay. There's a few.
|
| 39:54 | I'll give you one more minute. That way we'll get it all in before class is over.
|
| 40:43 | Alright that's probably enough time. I want to get it all in.
Again, what do I mean when I say the limit as x approaches -2 from the negative side? I mean I'm imagining numbers less than -2 so I could get any value I want. |
| 41:00 | But then as I get very close to -2 then I start getting closer and closer to 4.
At exactly -2 I would be at 4 but remember it's the limit so a tiny bit less than -2 is a tiny bit less than 4 so we just call it 4. Now I say what about values when I'm just a little bit bigger than -2? -1.999, -1.9999999, and so on. I get closer and closer to positive 2 here. |
| 41:30 | So the fact that at -2 exactly I'm not at 2 doesn't matter.
It's very close. Because I'm very close to -2 I'm going to get 2. And since these 2 don't match this one, this does not exist. What about positive 2? Well let's see. I didn't really give you a value here but let's call that 5. Sorry about that. When the limit gets really close to 2 from the left side so just under 2. |
| 42:05 | So as I get closer to 2 I can get closer and closer to 5.
So exactly 5 I'm going to get 5, it doesn't matter. Just on the left side I'll get 5. And from the right side again I'm going to get 5. Because we have these values closer and closer to 5. So you could do 2.1, 2.01, 2.001, 2.0001, you just get closer and closer. |
| 42:30 | Arbitrarily close to 2, you're arbitrarily close to 5.
So this one is 5. Now for 4 I look at this and I say when x gets really close to 4 from the left side I can get really close to 2. And when x gets really close to 4 from the right side I also can get really close to 2. So as I said the hole doesn't matter. The fact that f(4) isn't 2 doesn't matter because I want to know what happens when I'm really close to 4. |
| 43:00 | When I'm really close to 4 I get really close to 2.
Okay we'll do a homework set on this you guys can reinforce it some more. Alright that's enough for one day. I'll see you on Friday I hope. |