| Start | To sort of the real world.
So am I in the right spot Shannon? So I guess if we're recording this, we hadn't quite gotten the whole thing yet where we download the recordings and find a place to put them. We're working on that. But some time in the next few days I should be able to give you a link on Blackboard and say this is the video for the class that you missed. Most of these classes you may not care. But soon they'll get more complicated and you'll care. |
| 0:30 | At least I hope so. Maybe not.
If you show up you don't really need to see those videos, but you can always rewatch it and go through the pain twice. Or multiple times. You can watch it again and again. You can do one of those snap things where it does the back and forth boomerang kind of thing. That'd be fun. Or is that Instagram? I think Instagram does the boomerang. I don't know. We can do me dancing with a reindeer head with antlers and stuff. Anyway um so last time I talked about linear models. |
| 1:02 | I said linear models are pretty unrealistic because in real life very few things are actually linear.
Growth isn't linear no business stuff. But what you do is you say well it's sort of linear and then because linear models are easier to work with because they're lines. Right remember you learned lines in 7th grade those were easy. It's the harder ?? of linear parts so often a linear model is good what they call first approximation. And then you go with a better model to do better approximations so |
| 1:33 | quadratic functions for a second one so we know what quadratics look like.
So a quadratic is a parabola so if you have so a quadratic has a squared term. So a quadratic, which you know because you all know the quadratic equation is of this form. So it contains a squared term so what happens is you look at what's going on in your model |
| 2:04 | and you say well it's not really behaving like a line.
It seems to be going like something that goes up and then comes back down. So it follows a quadratic model or a piece of it follows a quadratic model. You know, you zoom in on it so you have some big thing going on but for here it may look like a quadratic and here it may look like a quadratic. That's actually cubic |
| 2:32 | when it has that kind of a shape.
But so sometimes you only care about a piece of what's going on because I don't know this could have to do with sales and you could reach some part of the equation where you say well that's just not going to happen so we won't worry about way out there but I'm worried about this zone. So a quadratic model would be very nice Quadratics again are very useful because there's only a small number of things that can go on in a quadratic equation. |
| 3:00 | It's very well behaved.
It has either a maximum or a minimum. In calculus you learn how to find the maximum and minimum it's pretty easy. If it has zeros that cross the x-axis You can find those two zeros. Very easily because what you do is you say well they would be at -b + or - √(b^2-4ac)/2a |
| 3:32 | In other words if you plug that stuff in
you'll get two answers. You'll get a positive, -b + this thing and -b - this thing.
The minus is the left one and the positive is the right one. You can find the vertex The vertex will be where x is So there's a lot that you can do with quadratic equations. |
| 4:02 | So they're very useful for modeling.
You actually see them a lot in physics. The other thing that's nice about a quadratic is it's symmetric. So everything that happens as it goes up reoccurs as it goes down so you throw something up in the air it comes back down it hits the same heights on the way up as it does on the way down. So if we have a quadratic equation you can draw a line right through the middle |
| 4:32 | that's you axis of symmetry or your line of symmetry.
And the line of symmetry is x = -b/2a. So every value here so if you have an x value here and some corresponding y value, if you go the same distance to the other side of the axis |
| 5:02 | you get the same y value.
And that's very nice as I said we come down and come back up you retrace the same numbers. And that's a very useful asset to parabolas. So when you do modeling you will find that you can often do a quadratic model and it'll give you a good approximation. So where would that show up in business context? Where would you say well this is likely to have a quadratic model. |
| 5:35 | Truth is probably not very much.
I can't really think of good examples of quadratics in the business world. I mean this is likely, you have some you know you're raising a price and for a while you say no because it'll come back down. |
| 6:01 | I can't think of good obvious examples more or less most likely quadratics is just something you work with
because they're easy to work with and they're good for a piece of the model not the whole model.
There's a good example so something that's a fad so so you know you're all added, what was that thing that was big a couple years ago Angry Birds? So lots of people getting Angry Birds and then they had enough birds and they're not angry anymore and they |
| 6:31 | it goes back down again.
That's a good example. But as I said those don't show up too much. Unfortunately what you see a lot of are polynomials and exponentials. So polynomial has as many powers as you want where the powers are, well let's give you a definition of polynomial. So a polynomial usually looks something like this. |
| 7:14 | The a's are coefficients.
They can be integers or not. And the powers are always integers, positive integers. Or nonnegative integers so n has to be ≥ 0 integer. |
| 7:37 | And you string them together so
this is a constant
if you have a1x+0 that's a line.
If you had the squared term you'd get a quadratic. If you throw in a cubed term you'd get a cubic and so on. So you could have y=3 as one, y=3+2x, |
| 8:00 | y=3+2x+x^2,
y=3+2x+x^2-x^3
The cubics actually start to show up more often in mathematical modeling.
So those are all polynomials. Cubics show up because you say for example in manufacturing you'll manufacture to be able to make the optimal amount and you reach a point where you need a new machine. |
| 8:30 | If you think about, say costs
you'll get something that looks like that.
So this is the profit you make so you start manufacturing you don't make much profit then you start to make a lot of profit now you have to buy a new machine. So it drops because you have to get rid of the cost of the second machine. But then at some point you've paid for the second machine your profit starts going up even more. So that's why a cubic term is often very useful. Of course then you buy a 3rd machine and a 4th machine and so on, but the cubic is very useful for modeling. |
| 9:02 | So one of the things that you'll do a lot of
in business especially if you get into the quantitative side of business
you do a lot of mathematical modeling.
It's basically like physics except it's not quite real world it's business world. So you say well what would this look like? As you get more quantitative you start to find a set of equations as you model things. When you're doing manufacturing processes there's optimization so |
| 9:30 | let's say that we're making iPhones
How many of each type of iPhone should we make?
Of each color? So let's say the iPhone 25 is coming out. How many white ones and blue ones and purple ones and gray ones and bigger ones and small ones and things like that. You could have a company and you're making furniture. How many tables? How many chairs? Because let's say you're making tables and chairs. |
| 10:01 | And sofas and things like that.
You only have so much lumber so how many of each should we make? You can sell a sofa for more than you can sell a chair, but it takes more materials to make a sofa. And you might sell fewer of them so how many of each type of thing do we want to sell? These are actually not that complicated of a set of equations to set up then you can plug them into excel or a more powerful computer program and it can solve it for you. Usually pretty quickly. |
| 10:31 | You wanna do say scheduling so you have UPS
has to route where all those packages are going to show up everyday.
So you click on Amazon and two days later your package shows up at your door. How do they figure out the optimal way to get it from whatever warehouse it's in wherever it is in America to your door in two days? That's a complicated routing pattern. First you have to get it out of the factory to some point for it to be picked up. Then you have to figure out where to transport it to then you have to have it delivered at the other end. |
| 11:02 | But there's, you know, the equations aren't really complicated.
There's just a lot of them and you take all those equations say you can solve them with some high powered computer power. But think about how we would do that so let's think about the mail. So the mail shows up at your local post office and then they deliver it into your local zip code so just your mail itself they have to figure out how they're going to leave the post office everyday, assuming once it's been sorted and then you have I don't know 9 deliverers for your zone. |
| 11:35 | They do the same route everyday so somebody had to figure out what that route is.
And that's why your mail generally shows up at the same time. The mail is this thing where you get pieces of paper that people write on in a thing called an envelope. There's something called a stamp and then it shows up in a mail box. It's not electronic but anyway the post office is fantastic at delivering that last stage of delivering. They're really very very efficient at taking it from the central point and getting the box to your door |
| 12:05 | at approximately 4 o'clock every afternoon.
So they optimize the group and you get it at 4 and your neighbor gets it at 4:01 and then 4:02 and so on. It's just like the school bus. So these are not complex equations there's just a bunch of them and in mathematical modeling you do that. So those equations are mostly linear. But you might have a set of 25 or 30 of them. And you have to find, you only have 2 equations and 2 variables, you have to solve it. |
| 12:32 | This would be 25 equations and 25 variables.
Which takes a little bit more computing power, you wouldn't want to do that by hand. So quadratics and cubics, they show up more often in like I said cubics show up more often in things like manufacturing. As you add each new cost component you get a drop and then an increase as you pay off the cost. Assuming of course that you pay it off. So we work with a lot of polynomials in this class. |
| 13:03 | Polynomials are also very useful for approximating other kinds of functions.
Many of which we won't deal with but for trigonometric functions of sines and cosines you could approximate sines and cosines quite well with quadratics and cubics. You remember sine. You hated that sine cosine tangent stuff but you remember that stuff. You could approximate sine very easily with a cubic equation. |
| 13:31 | For certain values so if you want to find the sine of you know 23 degrees
to put it into radians you could get a cubic equation that'll get you a very very accurate
approximation within say 8 decimal places without a lot of work.
So polynomials are much easier to work with than trigonometric functions. So that's where they show up a lot. Another type of function that we deal with are rational functions. |
| 14:01 | Polynomials are going to be the one we spend a lot of time on in class.
So rational functions come from the word ratio not as in rational this way ,but it might be related. |
| 14:30 | A rational function will have a denominator so you'll have a
well we'll mostly deal with something like this.
But that's an example so rational functions you'll get a numerator and a denominator and that often means you'll get a vertical asymptote. So one of the things that shows up in a rational equation is this is undefined where it's zero. |
| 15:00 | So for example if you had
something like that
when x=5
the y value matches, it becomes infinite it doesn't matter.
So this equation has approximately that graph. Don't worry about that for now. |
| 15:31 | So for rational functions you'll often get this, well for these type of rational functions you'll get a pair of hyperboles.
Then you'll get an asymptote. I don't know where those show up in business ?? into a rational equation. But we do them. Just to keep you on your toes. And you may not all go in to business. We'll see, but let's see. Ah, places where they do show up. Inverse relationships. When something goes up and something else goes down. |
| 16:02 | So if you have yeah you might have um an inflation might look like this where this I = inflation |
| 16:31 | the G = the price of gold
Does that make sense? Something like that works.
Well we'll pretend it works. It doesn't really matter as long as you get an inverse relationship. In inverse relationships as one thing goes up the other thing goes down. I'm going to change this a little bit. |
| 17:01 | It's not that important, but since it's on video
I wouldn't use inflation.
Let's use stock prices instead. So stocks are inversely related to the price of gold. And you'll notice that. Every time the stock market has a big drop the price of gold goes up. So that would graph something like this. Of course you're looking for that spot. So you can figure out how to go in either direction. |
| 17:32 | You know why the gold goes up when stocks drop?
Yeah so there's a higher demand for gold. Why? That's right, trying to buy gold. So what happens is you're losing money. You have all your stocks, they're starting to lose value. So you get nervous you say well these stocks, the whole thing is just made up anyway. So you put your money in gold because golds always got value. |
| 18:00 | Right? Gold so far, we see the value of gold. Have no idea really why.
But golds very valuable and therefore if the world is coming to an end you wanna have gold. Because dollar bills won't be of any value unless people actually think they have value. But gold with always be worth something so you can always trade whatever you've got for gold. Or the other way around. So as people get nervous about stocks when stock prices drop the price of gold goes up. And vice versa. |
| 18:30 | On a day to day basis you don't see much of this but when we have periods where
the market really goes into a bare market gold will really explode.
It also goes up with inflation fears. As people become worried about inflation, which will erode the value of your money, paper money, or your electronic money these days they switch to gold because they say well gold will be stable. You know, when the apocalypse comes and we all have to grab our shot guns and head up to the mountains you wanna take whatever gold you've got with you. You know you take your dog, your food, your gold, and you head up to the mountains and |
| 19:04 | barricade yourself in a cabin.
That won't work for long because the zombies will come and get you, but for a little while it seems to work. Of course if you're a zombie then you don't have to worry about the zombies anymore. Because they're not gonna come eat you. You're a zombie. But anyway so gold is often the fear index so you watch with you saw what's going on with North and South Korea that's scaring a lot of people. |
| 19:30 | Maybe justifiably.
The price of gold is going up. People say well we're gonna have a nuclear holocaust and the worlds gonna come to an end but I'll have my gold and my shotgun and my dog. And my cat and mouse. I don't know if anybody else is going to be up there but there you go. So that's a model so you want to understand these things so what happens with calculus you want to understand how they behave. As I said with calculus what's useful for calculus is to figure out this spot and this spot. |
| 20:01 | What's useful for calculus is to figure out this right here.
That's sort of the natural thing you want to know like I was asking the other day about break even. The first question you ask yourself in the business when you're starting your business up is what's the minimum thing I need to do to make enough money for this thing to be worth while? You know you open your restaurant if you don't make any money you're going to close your restaurant so how much do I need to make to keep going? It's a good test to do for yourselves as you become sort of businessy. |
| 20:30 | Go out to some restaurant that you like
and look at how crowded it is. And if it's not crowded ask yourself
how much money do they need to make everyday to stay open?
Then figure out how much you make per customer and say to yourself am I seeing this many customers? And if not then at some point you're going to have to find a new restaurant. You don't really know about their costs but you'll go to some place and say geez we're the only ones in here every time I eat here. You know that's a bad time. Or you eat at a weird time. |
| 21:00 | Entirely possible.
You can look at bars and see how much money they make so last night at the Bench they make a lot of money right? Well actually are they making a lot of money? I don't know. There are a lot of people who are there for free. So you want to figure out how much do they need to make to be able to stay open another day. Are they paying rent? They have to pay the electricity bill. So these are sort of the magic questions and These are things you can model with equations. It sounds really fancy but the truth is you can model them with relatively simple equations. |
| 21:33 | Other functions that are going to show up.
I don't really care too much but you should know them. The big one's coming but let's see. We'll draw their graphs in a second. By the way there's a homework assignment that'll be going up today. |
| 22:02 | But you'll see on Blackboard first homework assignment but we haven't done that much so don't worry it's not a hard assignment.
So we've got square roots these haven't ?? So square root function looks like this. Approximately. And that's y equals something times the √x so it's a sideways parabola. |
| 22:32 | Then you've got power functions.
That's y times x to a power where n is greater than 1. And that generally looks like that. That's when things are growing. And they're growing and they're not stopping growing they just keep growing. That's a power function. |
| 23:08 | And then the most important variant in this
will be the exponential functions.
Which is the same thing. But in our case the specific will be involving e. I'm sorry. I just wrote that backwards. |
| 23:32 | That's not x. I apologize. That's a constant to the n.
Not x to the n. X to the n is a polynomial. So specifically it looks something like that. That is your exponential function. This shows up all over the place. |
| 24:00 | So compound interest is a typical example.
When you put your money in the bank or anywhere that earns a return it compounds. So when you get compound interest from the bank you can get annual interest quarterly interest, your loans are angle interest. The more frequently the money compounds you know you have different compound periods I'm sorry you can have money that compounds once a year so that means you put your money in the bank and a year later they pay you interest. |
| 24:35 | So you have two banks and one says I'll pay you interest once a year of 6%.
So another bank comes along and says well I'll pay you interest of 6% but I'll compound it twice a year. Which means I'll give you 3% at the end of six months and 3% again at the end of the second six months. That's a little better because the second set of interest, the second 3% will be based on the number that had already compounded once. |
| 25:02 | So you'll get a little bit more money.
So now somebody else says well I'll do every month. I'll give you 0.5% every month. Rather than 6% once a year. It looks like the same number but it's even better. Somebody else says I'll do it everyday. I'll compound everyday. I'll give you a little bit of interest So you earn more and more. So then somebody says well I'll just compound it all the time. So what's called continuous compounding. Most of the interest rates that you see quoted are converted to continuous compounding. |
| 25:32 | Continuous growth. So that uses that model. So exponential growth in general
is also related to continuous compounding.
That is something that is always growing. The idea of continuous compounding is the amount |
| 26:01 | as time, t = 1
depends on amount at time t = 0, etc.
So in other words, however much you have at any particular moment depends on how much you had at the previous moment. So you think about bacteria. You have bacteria in your body it's growing. |
| 26:31 | You know you get a cold. The amount that you have at 8 am
will determine how much bacteria you have in your body at 9 am.
and at 10 am and at 10:01 and 10:01:01. So the bacteria is always adding to itself so these always have that chain. They have this kind of shape. So the amount that you have at any moment depends on the amount that you had at the previous moment. So cancer grows this way. And that's part of the problem with cancer is it can grow unlimited. |
| 27:02 | It's only limited by you.
And at some point there's too much cancer and that's that. But usually you can cut it off. Because remember I said another type of growth that you often see was that S curve. So this part looks like the exponential growth. But then something causes it to flatten out. You know you have bacteria and they're in a petri dish. But at some point the petri dish, you run out of petri dish. |
| 27:31 | You run out of food for the bacteria so they can't grow forever.
They need more food or they just start to die off and that's why it flattens back out. You can only have so much at any moment but continuous compounding is a very important growth thing that you'll see all sorts of places in business modeling. Because we're going to try to do a bit of modeling in this class. And see if we're successful. Isn't this fun? So let's see. |
| 28:12 | I don't want to do it with equations because we haven't really done the calculus part of it yet
but we've got the climate
which seems to be going out of control at the moment. Or the weather is going out of control
around there and then. The Caribbean right?
My mothers in Boca so I said are you worried about the storm and she said no. You gotta admire that. So you say why? Well my mother |
| 28:33 | was in Israel during desert storm
and she said I was, they were shooting missiles at us and I was wearing my gas mask and going into shelter
so if that didn't scare me a hurricanes not going to scare me.
I was like go mom. That's pretty good. So um of course a hurricane is not a missile. It's a little more destructive but hey, if I don't have to go down there and get her that's fine with me. But what happens with the greenhouse effect |
| 29:02 | is the heat starts to compound
and that's part of the problem that we're having right now
with global warming or climate change or however you wanna call it is
it may not cap so Venus the planet Venus is a runaway greenhouse model.
I mean eventually it stopped but it stopped at the uh I forget how hot it is down there at the surface of Venus. But it's hot. Hundreds of degrees or thousands of degrees so you have |
| 29:31 | the gas and it's heating up the Earth and the Earth heats up more
because it's got more heat and then it heats up even more
so it's continuously compounding. So one of the problems we have is
the population is growing, heat is growing, you may reach a point where you can't turn it off.
Or it may just get too hot and then we reach that point. Some people think we've already passed that point. I do. I think that we're not going backwards. We're not going to cool the Earth back down. It would take a miracle. Like we're going to have a zombie plague. The zombies don't use as much energy. |
| 30:00 | We just see them sort of eat each other and then die.
When is the zombie game on campus? Anybody know? It's soon. There's a humans vs. zombies, well you guys are mostly freshman There's a humans vs. zombies thing on campus. You'll see them in about a week. The people who are doing it take it very seriously. It's like the quidditch game. The quidditch team takes it very seriously. |
| 30:30 | The rest of us are like but you're not actually flying on a broomstick.
But anyway so compounding show up a lot with things like global warming I mean we had the hurricane you can keep feeding that if you get the water warm enough you can get 500 mph winds you can't actually but you can get 180 mph winds and then the Pacific they get upwards of the 200s so their typhoons are really powerful. Okay let's see what else do we want to talk about today? Ah one other thing right. |
| 31:02 | So let's just make a little table for everybody to save.
C'mon just give us some kind of equation. Let's just do some math already. Let's get a bachelors in math You should make sure you know the certain sort of family of graphs. |
| 31:35 | The function that looks something sort of like this
will have that graph. That's a line obviously.
A quadratic will look something like that. it could be upside down. It could go through the x axis. |
| 32:10 | Cubic looks something like this.
It does not have to go through the origin. It could be upside down again. But you're going to want to be comfortable with this family of graphs. |
| 32:32 | Square root
will look like that.
I didn't really spend any time on it but and absolute value graph will look like that. Absolute value really just shows up because you say I don't care if something is positive or negative. I just care about the quantity. |
| 33:07 | So notice by the way a quadratic equation has one maximum minimum a cubic equation has 2 maxima per minima If you had a forth power a quartic it would be 3 maxima per minima so if you had an n a polynomial of degree n then the number of maximums and minimums it has |
| 33:30 | is gonna be n-1. So if you had something to the tenth
then you can have at most 9 maximums and minimums.
There's 9 places where it reaches a bottom or a top. You cannot have 10. You have have fewer than 9. But you cannot have more than 9. And then we have exponential. |
| 34:04 | Any by the way
this is exponential growth. Exponential decay
this goes the other direction.
So exponential decay just goes down. So just are there are things that go up |
| 34:30 | continually forever, there are things that come down and just get closer and closer to zero.
So radioactivity decays exponentially. There's less radiation at the time than at any time previously. It drops this way. Sound is, as you go farther away from sound it drops exponentially. Electric discharge, so a lightning bolt the farther you are from lightning the less electricity there is. In the business world things that decay |
| 35:05 | there's what's called depreciation. You guys know what depreciation is?
So as the values of assets depreciate we model them in terms of accounting rules, usually theres 2 or 3 types of depreciation. Right? There's linear depreciation. What else do we use? |
| 35:31 | There's accelerated depreciation but it's all basically modeled as linear but the reality is
their asset decays exponentially. So you buy your car
and your car is worth less every single day.
Everyday you drive your car it's worth a little bit less. Somebody goes in your car and says what's the mileage? Which in your group I guess would be the kilometer age? I think they still use the word mileage. I'm pretty sure of that. |
| 36:02 | You would be beat for these things. You need exponential models which are messy so often again in the real world
you just approximate them, which you can approximate them with polynomials
to figure out values.
So let's see so I'm giving you guys a homework assignment for 2 reasons I don't want you to freak out. I think it's a very straightforward assignment. You'll see it's in Blackboard. Or it'll be up in a little while. And it'll go through mymathlab. First I want to make sure you got the whole mymathlab thing under control |
| 36:31 | because this is our first year using it in math.
I'm the experiment let's say. That means you guys are the experiment. So I want to make sure it works correctly. So at about noon it'll pop up on Blackboard and then you'll be able to click on mymathlab and you'll be able to do the assignment. I asked to make it maybe less than a half hour. I don't think it's a lot of work but remember I wrote it I know what I'm doing. So I want to make sure everyone can do it and I want you to all be able to successfully access it. So you have a week I think to play with it. |
| 37:02 | That's it. Enjoy your weekend.
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