| Start | And some time this week we'll start getting the video lectures up.
Things are still uncoordinated there the first week of school so, just getting it under control. Okay so now we're gonna do something that you also know which is slope. Now part of what happens when you're dealing with functions so in business particularly you wanna look at how things are changing, remember I told you you had that S curve. So you're gonna look at slope and calculus is just all about slopes the whole course is really dealing with slopes of things |
| 0:30 | which means, the slope tells you how fast something is changing.
So you have some point of information and you have some later point of information and you say good I can connect those two points and make a line which is, you know, obvious you learned this in third grade or something. But what you can do then is you can make a projection, you can say well if I have this piece of information and this piece of information then later I should expect one here and one there and so on. Assuming that whatever we're doing moves in a straight line. |
| 1:03 | In calculus you no longer have to assume things are in a line.
Okay, theres a fudge that calculus uses that works fabulously. But for now we just wanna say well okay so I have a line. So how could I, instead of always having to graph and looking at the lines how can I turn this into some sort of formula? So you do that by saying well what I'm saying what's going on with this line what I'm really saying is every time I move a certain amount in this direction |
| 1:32 | how much do I go up? If I go the same amount
I'll always go up the same amount.
So we call this slope, right? So slope is basically every time I go to the right a certain amount how much do I go up or down? Right, because you can have things that go down. And that's what slope is. And what you do is then you create, you make what is called a unit conversion or a unit measurement. And the unit meaning the word one |
| 2:00 | you say well then how much does it go up every time I move one to the right?
And that's what slope is really all about is converting whatever data you have to something over 1. And we have a very simple formula for that you say well how much do I change in the x direction? How much do I change in the y direction? And then that change |
| 2:30 | which we could represent shorthand
is called the slope.
So the delta, delta stands for change. So you say when I change x by a certain amount how much do I change y by? This is fairly obvious right? So if you say we have inventory and I have um inventory |
| 3:02 | on day one = 100 units
and inventory on day 8 = 163 units.
How much can I expect, how many units on day 30? |
| 3:38 | So sort of a typical 6th grade math question,
depending on where you went to school, but
of course we'll try to play with that a little more.
So there's some assumptions in a problem like this. So the first assumption would be that the inventory is growing linearly, it's growing like a line. Why is that a terrible assumption? Why is that unrealistic? |
| 4:06 | Okay so it may not, it could vary in amounts for certain periods of time
what's the other problem with it?
That's correct. What's another issue you could have assuming that the inventory is going to grow in a line? Think about it, look at what's happening, okay? You have to do some thinking it's not very hard at all the answer is fairly, when I tell you you'll say yeah well I knew that. |
| 4:34 | I'm just not raising my hand.
Okay, how many units would we have on day 1,000? A lot. Day 10,000? You're going to start to run out of places to store all your inventory. Okay, it can't go up in a line forever. That can't make any sense because you'd end up with an infinite amount of inventory. Okay, so a line doesn't really make sense, your inventory probably grows like this. |
| 5:00 | And then flattens out. It may even flatten out right away.
Okay, but it certainly doesn't just keep growing. Because that would be nonsense. However, for a short period of time a line might be very useful. So if you say my inventory, I start with no inventory and I accumulate inventory and then I reach the size of the warehouse and that's when I don't want anymore inventory anymore. At small intervals these all look like lines. |
| 5:30 | And so what calculus does is it enables you to break apart to look at
the microscopic level of how it's changing.
Okay, but how would we do this? Well we'd say okay let's use our formula. On day one I have 100 units and day 8 I have 163 units. So change in y would be 163 - 100 change in x would be 8 - 1 because this units |
| 6:04 | per day.
I could do it the other way around that would be days per unit, but that's just silly. That would be if I wanted to know how long I keep inventory for. Like how many days do I have a particular unit. So this comes out 63/7 = 9 units in a day. So my inventory is growing 9 units per day. |
| 6:32 | And I started with 100.
So I'm gonna have 100 units and I'm gonna add 9 units times 30 days and get 370 units. Is that 29 days or is it 30 days? It might only be 29 days. But we'll fudge it for now. So you want a more general formula than this. Okay, so you guys know what formula we use. |
| 7:01 | Okay, let's say the more general formula
would be I take that slope
and I multiply it by the number of days
and I add my starting amount.
Now you guys have seen this formula you've all memorized this. There's actually another version of this. It looks like, you take the slope formula and you cross multiply. |
| 7:33 | And that's actually the equation of a line that you want to become more comfortable with.
This is what point slope formula is, whoever just said that. This is easier to memorize the slope intercept form, but this is the form that you're going to run into most of the time in calculus because in calculus you'll do two things. You're going to be given some piece of information that'll be the point you'll figure out what the slope is and then you use this to come up with the equation of the line. |
| 8:06 | So in general you can say
y = 9x + 100
where x is the number of days.
And so once you have this formula now you don't need to play with the information anymore because then you can give any number of days you want and as I said now it's obvious that this can't really work because at 1,000 days you have 9,000 units |
| 8:32 | and at 1,000,000 days you have 9,000,000 units and a million takes a long time.
A million days ago is about how long ago? Was Moses around one million days ago? Okay so it wasn't Moses. Probably about 1,000 B.C. give or take 100 years. It's a long time so I don't think we have to worry about 1,000,000 days, but but it's a big number it doesn't make much sense to just inquire forever so the truth is the equation is more complicated. |
| 9:04 | But, for a small period of time that might be very useful.
So one that happens is you want to be comfortable with what slopes are all about. So as we do this in calculus we'll run into 4 types of slopes. We'll run into lines that are going upward, that's a positive slope. |
| 9:34 | Down, that's a negative slope.
The most important one, horizontal, which is a zero slope, which means it doesn't change. So as you change the number of units in the x direction the y units doesn't change at all. |
| 10:01 | Why is that really important? Well once you know when
something gets to zero that means once you know a point where it doesn't change at all
that's very useful let's say you're throwing a ball up in the air and at the top it stops moving so
its position doesn't change at all.
So if you're looking at say, the growth of your company when you get to the top it stopped growing now it's gonna go negatively it's going to go back down. |
| 10:32 | So you want to know where it's going to stop growing
so calculus is very useful for exactly that for finding the maximums and the minimums.
Okay and that point will be where the slope is zero. Then we have infinite slope which isn't really very useful but you should know what it is. So infinite slope is something that doesn't make much sense because it says you make no change and it climbs up an infinite amount. |
| 11:18 | For example
let's take a good real world example.
Okay so you all get a printing quota at Stony Brook right? How many pages do you get per semester? |
| 11:33 | 2,000?
Let's do 2,500 because that's 5 packs of xerox paper. So there's 500 in a package. 5 packs of xerox paper. In a toner cartridge I think it is about 10,000. If we know that um |
| 12:02 | When we have 1,000 students that's gonna be 400 cartidges and if we have 1,300 students |
| 12:30 | that's gonna be 550 cartridges. I'm making these numbers up.
Well and there's 16,000 undergrads How many cartridges do we need? Feel free to use your calculator. |
| 14:55 | This is a very primitive example, but you actually kind of need to know this.
Stony Brook has to supply the toner cartridges so they probably buy these bigger cartridges |
| 15:03 | And they probably refill them themselves, you know
they get some drums of toner and then refill the cartridges themselves, but they may not.
It might be cheaper to do some contract, but you can figure out how many cartridges you'd need to buy a semester. And if each student gets 2,500 pages, you figure most students hit their quota you kind of want to know if they do or not so that's probably one cartridge for every 2 to 3 students. |
| 15:34 | So figure you need like 5,000 cartridges per semester, that's not counting graduate students and professors.
So you might need to buy, let's say, 10,000 cartridges. So let's think business for a second So 10,000 cartridges for laser printing or printing on campus. 10,000, that's a lot. What do you think we pay per cartridge? Any ideas? |
| 16:01 | Yeah we probably pay, what do you think we pay per cartridge?
We're going to do a poll ?? Have you bought toner cartridges? C'mon you guys own printers you must've bought something like it. No we don't pay $400 per cartridge. Yeah 10 bucks maybe let's say $10 it's just a guess. So that's $100,000 per semester. So that's $200,000 per year so that starts to add up to real money. You have other problems with the cartridges. You have to store them somewhere. So 10,000 cartridges is a lot of cartridges. Each cartridge is about this big. |
| 16:34 | So you can figure how big the room needs to be to put,
so do you want all 10,000 at once? You need a warehouse.
Right? You'd need a room like this. Assuming they'll all be identical cartridges because every printer on campus is the same, but it's not. The other thing is you just have a continual supply so you basically have a cartridge dude who shows up everyday on campus in a van and drives around and says who needs cartridges? Which is more realistically what happens. When you have people who their entire job is just to drive around campus |
| 17:03 | fixing the copy machines, supplying toner cartridges,
bringing in palettes of paper, I mean think how many sheets of paper that is that you've got to be going through a day.
It's a small city you've got how many students undergraduates at Stony Brook? 16,000 right about 8,000 grad students. Then you have about 15,000 people who work here. Plus you have a few thousand more over there in the hospital side, but they're like a separate universe. |
| 17:30 | So you've got about 40,000 people.
Okay, so that's a lot. So you're always having things breaking and being replaced so you wouldn't want to store all these cartridges at once. But you'd set up a contract with your supplier assuming that you will buy a certain number of cartridges every year. Where you go through then he or she has like probably 5 or 6 people who are just on campus all the time moving this stuff in and out. So since you guys want to go into the business world these are things you actually think about. Okay this all falls under decision sciences or operations research. |
| 18:04 | So for you Business 220 is your first exposure to that.
But you just sort of say to yourself, what do I need to do? Like if you think about a bank, you have a branch of a bank and they have money everyday at the banks, you kind of want to figure how much cash do we need for our daily operations? You need some reserve, so the armored car shows up with the guys with the guns. And then they bring you the money, okay, and you figure well how much do I need everyday? |
| 18:30 | Well it fills a truck so it's a lot of money.
Probably $20,000 on hand at the bank, it's not really very much. Think of the movies. No one keeps millions and millions of dollars at the bank. It just would be silly. Okay, there's also ?? have a very big truck or a moving van to show up to move all that money. Okay but, you're going to have to move these things in and out so if you go into business say you're at the Apple store How many iPhones do you keep on hand? I mean there's lots of different types of iPhones. And how many chargers? You guys go through chargers like I go through razor blades. |
| 19:01 | I mean how often do you lose your chargers? How many do you own?
They're in the charger business if you haven't noticed. Every phone has a new plug so you have to get a new thingy. They're not stupid they do that on purpose. The linear model helps when you're doing a small version of planning for this kind of stuff. So the heart of why you're using lines all the time. How did we do on this? Let's see. So we would need slope |
| 19:36 | the slope is going to be 550-400.
And the students is 1300-1000. So that comes out 150/300 which is 1/2. So we have a half of a cartridge per student or |
| 20:01 | to flip it two students = one cartridge.
So you should have gotten 8,000 cartridges. I get 8,000, but I could be wrong. But let's see. Well when we started off with 400 for 1,000 students so let's see. What you would do is you would say |
| 20:37 | y - 400 = 1/2x - 1000
So you get y - 400 = 1/2x - 500
So y = 1/2x - 100
So it's unrealistic because I really didn't do zero students at zero cartridges.
So this doesn't really make sense when you're doing a half per student |
| 21:03 | I should really be starting with 500 but
I was just sort of whipping up numbers.
Alright, now you plug in 16,000 students. You see what I did? Okay this is this equation. So y minus the 400 cartridges is 1/2, that's our slope, x - 1000. I could've used 550 and 1,500. I'd end up in the same spot. And now you plug in 16,000 and you get 7,900 cartridges. |
| 21:45 | Does that make sense?
Let's do another example make sure everybody really gets it. This is really the point where we have 1,000 students. |
| 22:00 | We start with 400 cartridges and then we go up to 1,300 students with 550 cartridges.
So that's our x's and y's. Okay so now we get our equation so we have y - 400 is this so when we plug in 16,000 students we get 7,900 cartridges. As I said it's slightly messed up because it shouldn't be 1,000 and 100 but it doesn't matter. It's just the idea. |
| 22:37 | Okay let's do this another way.
So where it more likely shows up in the business context. You guys have any business classes yet? |
| 23:00 | You've had that first intro to business class.
No accounting or finance or any of those things right? |
| 23:36 | It should be the same formula.
Okay, but I didn't do it on that one so I have to check so this shouldn't give you 8,400 you should get something 900. I don't know I have to look. I approached it from a different point of view. But it's possible. What I really want you to get out of this right now isn't so much the formula as what's going on, but now I'll do one that's more realistic. |
| 24:05 | So what you really get in business
is you get what are called fixed costs and variable costs.
So you'll see this right away in accounting and finance class. Fixed costs and variable costs in any busniess fixed costs are no matter what you want to do you you're stuck with those costs. Like if you were manufacturing iPhones |
| 24:32 | the fixed costs are the factory the equipment. You can't make any iPhones until you've built the factory.
So the fixed cost is the cost that's independent of how many units you make. And then the variable cost is the cost per unit. And in the simplest form this looks like a line so the fixed cost would be your y-intercept it's what you start with. |
| 25:00 | and your variable cost would be your slope.
So since you get out here in the business world like my company, I have a tutoring company okay so I ?? the tutoring business. And so what are my variable costs? My variable costs are things like what I have to pay per tutor, you know taxes, pencils, calculators, okay because you don't know what you're going to need |
| 25:33 | per tutor, but each time I add another tutor I have to add a certain amount of cost.
My fixed costs are what I have no matter what so a phone, if you had a website Cost of marketing, but marketing cost can be a variable cost, it doesn't have to be fixed. But as I said so two things so one is I think about what do I need just to get going? So let's say you get a desk, you need a place for people to come, you get a phone line. |
| 26:01 | You need a printer, a computer, you pay some rent.
So all that stuff is fixed costs you start with that and then for ten tutors it'll cost me a certain amount of money if I add an eleventh tutor how much more, that's called a marginal cost. How much per additional person. So tutoring is a very service business it doesn't really work that much, but you know you're Subway and you have to have the kitchen prep, you have to have the counter |
| 26:32 | and the cash register and the sign and all that stuff and then your variable costs
are costs per employee, the price of the buns, you know the heros because that goes up and down.
Okay so that varies, so in business you often have these two things. And fixed costs are essentially unavoidable because you have to fix them and they're fixed, they don't move. But you can bargain on them so some person says we'll rent this to you for $8 per sq ft per month, that's usually how it's done. |
| 27:00 | Or sq ft per year. So if you want a 1,000 sq ft place that's $8,000
And then somebody else says we'll do it for $7.50 so now you've dropped your rent, right? That's good.
So you negotiate as much as you can under fixed cost and you maybe try to hold variable cost down as much as possible. That's why you pay your employees as little as possible. So when you become a boss now you'll suddenly understand why minimum wage is too high. Right you say wow I can't believe we're paying them $7.25 can we pay them 1 penny? Can we do that? |
| 27:30 | Can we just sort of lock them in and make them work?
I'm joking, but in other words that's a variable cost. Health benefits and pension and snacks you know all sorts of stuff are variable costs so you run your little shop with all your little employees and you have you know free donuts for them every morning you're paying for the donuts. So that's a variable cost. You don't have to bill that in. So when you're doing the equation of the linear version of this |
| 28:01 | remember that's our equation of a line
so this is your variable cost and this is your fixed cost.
What would the example be? Okay so we're going to have a ski shop. I guess it's the rental and |
| 28:32 | it says to get the ski shop going up for the season
the fixed cost is $45,000.
And then the costs to rent the skis is $80 per skis or per pair of skis. I've never skied but it seems you put one on each foot as opposed to what is that? Snowboarding? |
| 29:21 | And then you can rent them, did it really say $450 a day? Is that realistic?
Who goes skiing? |
| 29:31 | What does it cost to rent a pair of skis?
Or sell? What does a pair of skis cost? $100 for a pair of skis? Really? I can't believe they're that cheap. I would've guessed more like $500. Well they say $450. So let's say you can sell them $100? Really? |
| 30:08 | Okay so now you've got your profile for what you think you could make.
So if it costs you $80 to buy a pair of skis then you cal sell them for $450. Then your profit |
| 30:31 | will be $370 per pair, but not really so don't write that.
Your total cost so it's the y = mx + b it is $45,000 + 80x (per pair of skis). Your total revenue |
| 31:08 | is $450 times the number of skis (x).
So profit is gonna be revenue minus cost. |
| 31:38 | That's going to be $450 times your skis (x)
- $45,000 + 80x
Do you know what they use to symbolize profit in economics?
They could use a p but they use pi. Which is really confusing because we're in math. Pi means something totally different. |
| 32:00 | So we'll just say p.
So your profit is then going to be -45,000 + 370x So to interpret that Before you do anything the day you open your ski shop you spend $45,000. So that's you fixed cost right? You walk in the door where skis are for sale 45,000 bucks in the hole right away. And then we're making money back at the rate of $370 every time we sell a pair of skis. |
| 32:34 | So what's the first thing you wanna know then? What's the magic number you're looking for?
What do you wanna know first? If you say I'm starting off $45,000 in debt and I get $370 every time someone buys a pair of skis what question am I going to ask you? |
| 33:02 | I gotta sell some skis.
I gotta get up to what? What number is important to me? So how many skis I need to sell just to get back to zero. Your break even number. You guys understand that right? So first thing I would ask you if I were the bank right, you're borrowing money from me to get going I'd say well how many pairs of skis do we need to sell? Well we need to get back to zero. So you can figure that out right that's just 45,000 divided by 370. |
| 33:31 | 122 pairs of skis.
Then the question you ask next if you're the banker if you're evaluating this is is that realistic. And by the way we haven't really factored in some other costs. Because you have employee costs and all that kind of stuff, but we'll pretend that's just all part of the magic $45,000. It's a very primitive example. So I don't know can we sell 122 pairs of skis? Well where are we? Are we in Tampa? Not gonna sell 122 pairs of skis. Are we in Buffalo? Probably. Colorado? I'm sure. |
| 34:04 | Okay we might sell 122 in a week ora day.
Hard to know. So there's a big emphasis on what we call critical thinking out here in the business school. If you just think about these things and ask yourself would they make any sense. I know you're just getting going in the analysis, but alright so that's another way to think about lines and linear growth. So Friday I'm going to put up a problem set for the following week. Don't be intimidataed |
| 34:34 | it's going to be nice and straight forward.
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