| Start | Ok, so a couple things. There is another math online homework thing that will be available at noon, after today.
Then there are 2 paper home-works that will be available at noon. If you should, you can either hand them to me Friday in class, or Monday in class. Or some of you, may not show up to class, which is a shock. Given that many of you are already not here. You can give them to the main office, and the assistant there will take them. |
| 0:33 | They're due at noon on Monday. There's probably not going to be any recitations next week, I don't know if your TA's have canceled them yet.
But I told them they had permission to cancel, and generally, you give people the option, you know what they choose. Right? So there you go. We are having class Monday, at 11:00 where I will give out the final exam questions in advance for anybody who shows up. That actually won't happen, but one of you probably believed that just now. Which is sad, there's also no Easter Bunny. |
| 1:03 | But if 3 of you show up, I might actually do it, we'll see.
I have been known to reward people when there's very low attendance. Ok, anyway, not a lot to do in today's class. So, we'll see what we can get done. Look at this, this isn't washing the board! Come on. That's just sort of pushing chalk around. Wow I'm so disappointed. |
| 1:30 | Anyway, um, hi. So we did all of that adding up rectangle stuff. Isn't that fun? Then it started to get not fun. I have to say, it's very tedious, which is why we invented integration.
And if you don't have to do these, these approximations, as I said last class, there are functions where there is no way to do the integral. For example, you could take the derivative of e^x^2, pretty easy. You cannot integrate over the reals e^x^2 |
| 2:00 | You can integrate e^x, you can't integrate e^x^2.
It's because theres no function whose derivative is e^x2 So it has to be a real function, Ok? As far as you're concerned. The good news is, very few of the functions you will deal with cannot be integrated, almost all of them can. So, that whole approximation thing isn't really important, it's more important to understand it conceptually, so you know what's going on. Um, so now, let's just do the integrals for real. So remember, and I did this the other day, but I'll do it again, so if you have some zone, |
| 2:35 | and you want to find that area from a to b, you could do the rectangle thing that's really annoying, so what you do instead is you integrate the function from a-b.
So you put those what you call limits of integration. |
| 3:00 | And what this means is you do the anti-derivative at B - the anti-derivative at A.
That is the anti-derivative. And there's no +C. This is called a definite integral, the indefinite integral is called indefinite because you have a +C term, you don't know what the actual function is, but you can find it if we give you an initial value. |
| 3:39 | But here what would happen is the C's would cancel. That's why we don't need it. So for example,
So when we did the approximation of the integral of x^2+3 from 0-4,
Now we can find the actual value from 0-4.
|
| 4:06 | So what's the anti-derivative of x^2?
Yup, x^3/3, What's the anti-derivative of 3? 3x, you guys are good. You draw this line, that just sort of stands for what the limits are. You plug in the top number, you plug in the bottom number and you subtract. |
| 4:46 | So this was 64/3+27-0,
So that's 48 and 1/3, and I believe we guessed it was 48.
|
| 5:03 | Oh that should be 4, I'm sorry.
Good catch. And actually, right, I think this came out to 32 when we were guessing. And this comes out to 32 1/3. Ok? Remember we did the approximation and underneath we got 24 and above we got 48, and we said well maybe it's 32, and it's 32 and 1/3. |
| 5:30 | Ok? And that's exact. And again, you get to that integral by doing the rectangles, doing an infinite number of infinitesimally thin rectangles.
And when you add up all of this infinitely small rectangles, you get this. The dx, is what was the delta x, so that is the width of each rectangle. This is the heights. So it's each height* each width. And you add them all up from 0-4 so that the giant S stands for sum. |
| 6:02 | And that's integration, ok? So if we said,
By the way, you're going to need a calculator for the homework. Feel free to use a calculator on the homework.
Feel free to use a calculator on anything but the exam. |
| 6:44 | Alright, let's see if you can do that.
Alright that's long enough. So all this says is take the integral from 1-2 of 6x^2+8x-5dx. |
| 7:07 | So what is 6x^2 anti-derivative? It's 6x^3/3+8x^2/2-5x
Let's simplify that, that's 2x^3+4x^2-5x and we're going to evaluate that from 1-2.
|
| 7:31 | Ok? So you want to simplify these things as much as possible, before you plug in.
Alright, so plug in 2 you get (2(2)^3+4(2)^2-5(2)) - (2(1)^3+4(1)^2-5(1)) So let's see, that's... |
| 8:01 | (16+16-10)-(2+4-5)=
Which is 21.
Ok? Know how you get to 21? Most of you? Good. So you know, when you say the land of partial credit, well you get partial credit for getting here, |
| 8:31 | and you'll get more partial credit for getting here,
And you get that last bit of credit for getting to there.
Ok? So there's sort of multiple steps to a problem like this. Alright, let's say we practice a couple more of these. |
| 9:08 | Alright, so let's integrate this, so what's the integral of e^x? e^x!
From 0-2 that's e^2-e^0, which is e^2-1, that's it. We don't have to figure out what e^2 is. Ok? e^2 should be about 8, a little less than 8, 7.8, something like that. e^3 is very close to 20, but as I said, you're never going to be in a situation where you have to use e. You'll have a calculator, so you get an e, you figure out what e is. |
| 9:42 | Alright, this one, is the ln|x| from 2-5. Which is ln|5|-ln|2|, which again we'd just be done. You could make that ln|5/2| if you wanted but it's not necessary.
Supposed I gave you the integral from -2 to -5 of 1/x. Remember, it's absolute value. |
| 10:08 | Ok? So this would be ln|x| from -2- -5, so again it would be ln|5|-ln|2|/
Ok? Yes?
Can you leave the |5|? Sure. It's not wrong. Yeah. |
| 10:32 | You should know that the absolute value of 5 is 5, but you can leave the bars there if you want.
I understand the way people work on exams and stuff. And they're nervous that maybe it's not 5 this time, so just in case they leave the absolute values on it, it's fine. We're going to have a shorter class today, I know that makes you sad. But there's a little stuff more that we have to do. So suppose we're told that the marginal profit is P'(x)*(sqrt)x-6 |
| 11:11 | and this is for the sale of x seats on our jet.
Where x is in hundreds of dollars. Ok? So when x is 5, it means 500, not 5. Ok? |
| 12:21 | Ok, see if you can figure that out.
And that's sort of a real world example. Wait, x is in seats, P is in hundreds of dollars. |
| 12:32 | We need to know what the graph looks like.
(sqrt)x-6, let's do a better graph than that. (sqrt)x-6 looks something like that, ok, where this is 36. |
| 13:02 | So this profit is a negative number.
Remember we're doing the area between the curve and the x axis, so you're losing money certainly up to 36. But beyond 36, you're still losing money until this area, equals this area. And then at some magic point, you'll break even and then beyond that, you'll make money. So you can guess what question c is going to be. Ok, so part A. |
| 13:31 | My profit is missing a t it says profi.
So I go to find the integral from 0-60 of (x^1/2 -6)dx That is x^3/2/(3/2) - 6x from 0-60. 64, sorry. |
| 14:08 | Which is ((2/3)(64)^3/2 -6(64)) - 0, when we plug in 0 to both of these terms, it's 0.
So let's see thats 1024/3 -384, that's a negative number. |
| 14:36 | That is um -128/3? Is that right?
-$42.66 And then times 100, so you're losing about $4000 |
| 15:00 | You guys get that number? Well one person is saying yes, anybody else? I'm doing it in my head, it's possible I'm messing up, you know I'm old.
Those of you who agreed with the I am old sentence, might as well just leave class now, you're not going to pass, it's over. Ok, what if I increase the 64 to 100 seats? Well you could 2 ways. You could go from 0-100, and then the 0-64 you could subtract, or you could just do from 64-100. |
| 15:31 | So we're gonna do
(2/3)x^3/2-6x
from 64-100, to find out if they're making money at 100.
Well let's see, that is ((2/3)(100)^3/2 -6(100))- (-128/3), because that's what you get when you plug in 64. |
| 16:07 | 2000/3 - 600, comes out around 66 and 1/3, ok?
Plus, you're making more money, you have minus, minus this. |
| 16:32 | I think you get
109 and 1/3. Anybody else get 109 and 1/3?
So you're making money. The question is, where is the break even point? What number breaks even? |
| 17:01 | Did I say how I did the second one again [?]
So we want to know, what's the number going from 0-k of the integral of x^1-2 -6 dx that equals 0.
Yeah so part c is what's the break even number. |
| 17:44 | Do you guys remember the 7 bus? Any of you here last year when we had the 7 bus running?
It was a bus from the SAC to NYC that ran maybe 5 times a day, each way, and was $7 each way if you reserved it in advance. That's pretty good. So I sat on the bus, you know sometimes I got on that bus at 11:00 and there were maybe a total of 15 people for the bus to the city. |
| 18:09 | So 7*15 = $105, so I said to myself there's no way they're breaking even, if that's how much they spend on gas and the bus driver, easily.
Not counting the bus. So I sat there and said to myself maybe different rides, maybe it's full if you take the bus at 5:00. No. No, there were very few times where that bus was really full, which was too bad because the people who took it really liked the idea of $7 from the SAC. |
| 18:38 | It took you to grand central station basically. So it turns out they we're making money,
because they ran enough of these, but the company itself was losing money so that's why they stopped.
But if any of you want to get in the entrepreneurial business, I recommend a shuttle bus that runs from the SAC to NYC there's obviously a high demand for that, making a stop in Queens on the way in, you could probably charge $10 easily |
| 19:03 | and fill that bus at certain times of day.
That's what I would do. You know there's what they call the china town bus? It runs from Chinatown to Boston and there's one that runs from Washington D.C. Charges nothing, maybe $20 but they pack that bus. So the break even number is really important If you go into a restaurant, there we're these restaurants in Manhattan I used to go into, we'd go in and there was nobody there. So you say to yourself, maybe it's just a bad day, so you go in a different day, no-one there, maybe 3 people. Plus staff and food, you say to yourself, this restaurant will not survive. |
| 19:35 | You can sit there and you can think of how many people you need to do to survive, this is a real business calculation.
So there's a sushi place I like to go to around here, so I say what do I think is their rent, air conditioning, food, how many people are working there, blah blah blah, And say to myself, probably about $30,000 a month. Sounds like a big number, but there's 30 days in a month, $1000 a day. That's to break even. |
| 20:00 | So you need to make $1000 a day. If each person is spending $20, which they're not, you need to get 50 people every day.
Ok? That's just to break even. You say, well we really need to get 100 people everyday or why are you bothering to have your sushi restaurant. This isn't counting the fact that you can steal cash and things like that. You can't make that much money if you're crooked. You sit there and you go for 3/4 times, you say well, 4 people at lunch, 20 people at dinner, how many take outs are they doing? |
| 20:32 | And you pretty quickly say to yourself, well, this restaurant won't be here a year from now.
And sure enough, a year from now, they're gone. Ok? So if you guys want to get into business, that's the kind of stuff you want to do. That's why the break even number is so important. See if they're making it. There's this new place that's opening any day now, across the from the train station, the calzone place? Have you guys seen the signs for it? Well that's going to be a tea place, the dog grooming place is going to be a tea place. |
| 21:00 | Next to it, right. Next to the 7-11, is going to be a calzone place. It's going to be open late, so you can go to the bench and then go get your food, American food tastes really good,
so you go in there and say to yourself are they going to make it? Go in and look at all these various shops.
Ask yourself, how much do I think they need to make and are they breaking even? That's real business stuff ok? And you're not gonna use calculus to figure that out, you can just make a rough guess. You could ask if you get the owners in a good mood, and he/she will tell you what the rent is, and the electric bill, you know air conditioning is expensive, |
| 21:32 | you got your employees, taxes, but if you don't make any money you don't pay taxes.
Usually you can avoid taxes for a couple of years but at some point you have to pay them. When you're losing money you don't pay taxes, so in the beginning, the IRS kind of give you a little time to [?]. You got the cost of all the supplies, and you say to yourself, alright, how many customers do I need to get in here every day? So think in reverse. The starbucks in the bookstore? That place is a gold mine. |
| 22:01 | Think of how much money, think of how many people go through that starbucks every day.
I mean, I have trouble calculating but they're making thousands of dollars a day so that's what you want. They want to have that line out the door. When they say sorry we're so crowded, they're not sorry they're crowded they're thrilled about it. They're just sorry you have to wait. But anyway, these are the kind of calculations you want to do. So I will give you the break even. If you're thinking about the formula, you say, well let's see, (2/3)x^(3/2)-6x |
| 22:35 | from 0-k, pick any letter you want.
So since the 0 part doesn't matter, it's (2/3)k^(3/2)-6k=0 So remember what k^(3/2) is thats k*k^(1/2). |
| 23:06 | So you can factor out k, and you're left with (2/3)k^(1/2)-6 = 0
So at 0, when k=0, we're certainly going to get 0. Duh.
Ok, we're not integrating anything, or this. So we just need to solve that. So that is (2/3)*(sqrt)k=6, so (sqrt)k = 9, |
| 23:44 | So k = 81.
So if you get 81 seats, you get 0 money for the flight. And then you start to make money, so you need at least 81 seats, and airlines are very complex because they're constantly changing the seat prices, and who sits where and all that stuff. |
| 24:02 | They spend a lot of money, the major cost to airlines is jet fuel and maintenance, you know, you have to keep the plane from crashing and you have to fill it with fuel.
And fuel is very expensive. So that's why they pack you in. They came up with this clever idea about 15 years ago, a lot of businesses use 9/11 as an excuse to take money from you. So one of the things they did was say 'Oh wow. We're going to have to check all these bags now. They're very cynical remember, so they said, well we're gonna charge you a fee to put your bags on the plane. So you guys are used to that. There never used to be a fee. You could put as many bags as you want, up to some reasonable limit, at 8 bags they charged you. |
| 24:36 | Now they just charge you for everything. Some of them say, "wow we won't charge you"
Trust me, they're building it into the price. Airlines used to lose lots of money, this is where they start to make money. They just make money on bag fees.
Or they charge you for peanuts. And they say "oh well people couldn't stand the food anyway, so we might as well just give you a bag of peanuts, and if you want a second bag of peanuts they charge you a dollar. Ok? And you know, you laugh, but you know if you guys get jobs in the business world, some of you are going to have these kinds of jobs. |
| 25:02 | Right? What should we charge for sprite on the next airline flight to make more money?
And you'll be the person who says if we raise it from $1.10 to $1.20, you'll make this many more million dollars this year. So, you know, these are real world problems. And often you don't really have an equation. If that makes any sense. If you can have equation, this is the way you figure it out. Ok? Hope that was entertaining. |
| 25:30 | Alright well I got through everything today, we've got 8 minutes to spare, I will see you on Friday, and for those of you are already on vacation... |