| Start | We're going to do application to economics and biology
Which more of you will be interested in than applications in physics.
Even though I think the physics stuff is kinda cool, the problem is most of you haven't taken physics yet as I said before, so it doesn't really work. By the way, a lot of you haven't taken economics and you won't see this stuff in bio so fast either, well, it's tricky. Consumer surplus. |
| 0:37 | Have any of you taken economics? High school? Wow not many of you, ok.
So in economics one of the things you have to think about, you have to think about demand and supply. So you're manufacturing iPads, and the question is how many iPads do you make and how much should you charge? So what you can do is you can set up a couple of curves, |
| 1:02 | You have price here and quantity here.
Quantity is also known as amount, ok? So, we're selling tablets, actually we're going to sell Microsoft Surface tablets since they're way better than iPads anyway, Do you agree with that do you use a surface? Anyway, so the question is what does demand look like? Well, if the price is high, you won't sell many microsoft surfaces. But if the price is low, you'll sell a lot. |
| 1:40 | So you get a curve, it turns out to be curved inward for complicated reasons which I won't worry about now. You could think of it as a straight line but then it wouldn't be fun.
And as the price comes down, you sell more of them. Ok. Now you have the other side though, supply. If the price is low, it doesn't make sense to make very many of them, because you're not making very much money. |
| 2:03 | As you raise price, you have incentive to sell more of them. So the general idea of economics, is this is the demand, and this is supply
And this is the magic price.
it kinda make sense, and you don't have to be too mathematical to make sense out of that. In other words, if I can't charge very much, why should I make a lot of these things. |
| 2:32 | Right, if I can charge a lot, if I can make a lot of money every time I sell one, then I should supply a lot of these things.
The problem is the higher the price the lower the demand, so there has to be a spot right in the middle where you say that will get me the most number of people who will both buy and sell. So ignoring the supply part of the curve for a moment, you pick a price, like there, |
| 3:02 | call that p, and this, is called consumer surplus.
Ok, so remember, you're charging some price, this is the number you're gonna sell, and what does this money mean? What does that consumer surplus mean? |
| 3:41 | Think about consumer surplus, a person who buys the microsoft surface.
No, no, no, no. We sold this many, ok, we sold q at price p. So what does this amount represent? It represents the money that consumers saved. They would have spent more money, right? Theres a whole area under the curve thing. |
| 4:07 | This is how much profit or money the seller could have made but this is how much money the seller actually made.
Well if you want to be all economics, yes. But yes thats correct, basically, you could have sold it for more but you wanted a bigger quantity so you sold it for less. |
| 4:30 | Ok? So you kinda gave up a certain amount of profit/money by selling it, ok, so we call that consumer surplus.
It's much more technical than this because there's a lot of other stuff going on, there isn't just one curve, this is just all we care about at the moment. So we want to calculate that area, ok. For example, I'll read this example out of the book so I don't mess it up. |
| 5:00 | The demand for a product in dollars, which most of you are running out of at this point in the semester, is, or are there meal points you have? Have you gotten special meal points?
By the way theres talk that next year you're going to pay a set amount for your meal plan and then you can eat as much as you want. Right, is that gonna happen? Good. |
| 5:35 | Thats the way a lot of schools do it now, they charge you more, and count on the fact that some of you won't eat a lot.
And you know, others of you will eat more, so it balances out. Obviously, they have people who are math-y, who will figure out what price they should charge you, based on the average amount that you eat, they're not really going to do you a favor but it simplifies things. Um, anyway, lets say this price is p=1200-.2x-.0001x^2 with this parabola, |
| 6:11 | Ok, so we wanna know, so x is the quantity that you sell, and p is the price you charge, So find the consumer surplus when the sales=500 |
| 6:45 | So the formula this is what you guys really care about, you are going to do, the integral from 0-sale level |
| 7:14 | of p(x)-p
Ok? So first we're gonna have to figure out what the price is which is what we'll figure out in a second.
And this is the commodity, this is the amount, the demand, this is the price and this is dx. |
| 7:33 | Ok, so how do we figure out what p is? Well p is found by plugging 500 into here.
So P = 1200-.2(500)-.0001(500)^2 Which is 1075? Yes. |
| 8:06 | So our integral becomes the integral from 0-500(1200-.2x-.0001x^2-1075)dx. Just that straight forward, ok?
|
| 8:37 | So we can simplify that a bit, let's see. 1200-1075 is 125-.2x-.0001x^2dx
So that's 125x-.1x^2-(.0001x^3)/3 from 0-500. Ok, and that gives you this area.
|
| 9:15 | Ok, because you're really doing the difference between the area between the curve and the line, we know how to find the area between 2 things, right? The difference between the curve and the line.
So I don't care what you get when you plug that in, you get ugh, $33,333.33, that's the savings to you. |
| 9:37 | Ok, so the real idea was you want to find the area between the demand curve and your sales price.
You find your sales price by taking the number you're going to sell, plugging it into the quantity formula, ok, that gives you the price, And then you take the curve minus that price, and you integrate over your sales total. |
| 10:04 | Can we do an example? Sure, let's do a nice easy one The demand for a certain commodity P=20-.05x, find the consumer surplus when the sales level is 300.
|
| 10:33 | So it's just like that problem before. So let's work on this problem for a minute,
The demand for the product is what? So you do the same thing as before, yes?
Well, you have scrap paper, so, I don't know if you will have a problem like this and I don't know what the arithmetic level will be for this one, |
| 11:00 | we will find out. I am not doing it to be annoying, I am not in charge. So first we find, P,
20-.05(300)=P, I know you guys hate fractions, but it is easier if you do this in fractions.
20-(1/2)*300, which is 20-15=5. |
| 11:31 | So you have satisfaction, you only need to use the fingers on one hand so far. Ok, so the integral from 0-300 of 20-.05x+5 dx. I am sympathetic to the no calculator issue, don't think I'm not.
It's just, thats the just the way it is, some things never change. |
| 12:02 | -5! yeah you're right. 20-.05x-5dx. So that's from 0-300, 15-.05x dx
You take that, you get 15x-.05x^2/2 from 0-300, so look, if you get to there, you're going to get most of the points. I am sympathetic.
|
| 12:33 | If you want to do that last step you can also say this is 15x-x^2/40 from 0-300
That's 4500-90,000/40, which is 2250, which equals 2250.
|
| 13:07 | If you had to do this, how would you do that? Because you might have to do arithmetic on the test.
Don't do 300^2, so 300*300/40 cancel a 0, 4 can cancel out that and get 15, and now you have 300*15/2, cancel another 2, now it's 150*15, it's easier that way. |
| 13:46 | Oh I don't know, that could be, that's just the way I would do it.
I always do reduction before I do arithmetic. Alright, so that's one kind of thing you might have to do. |
| 14:01 | Let's look at another type of practice problem.
Ok, so let's do a disease problem because we know you guys love disease. |
| 14:32 | Suppose you want to find the number of cases of a disease in some certain period of time. So lets use the equation, 2500+e^.6t, so this is the number of people who are going to come down with the zombie flu per hour.
|
| 15:03 | So you'll get a lot of zombies later. So this is the rate of increase in zombies when the zombies come I'm standing behind him.
Let them get you first they're gonna be busy munching for a while so we can kinda run away. So we want to find how many more zombies we get between I don't know, 10 hours and 14 hours. We just integrate from 10-14. |
| 15:44 | It's just that simple. It's gonna happen one of these days. I'm gonna see the zombies, I am hiding down here.
Below ground, closed doors, you guys are out of luck. Alright so this is an easy integral, right, this is 2500t+e^.6t/.6 from 10-14, and that's messy. |
| 16:21 | Where did I get the what?
Oh because I said I was doing between 10 and 14 hours, so the increase in zombies between 10 and 14 hours |
| 16:32 | When we do the increase, we're finding the number that the population has gone up, in total you're going to integrate the change.
Because what happens when you do the integral of the derivative, well, you get z right? Because that's derivative is z, that's the change in z, so this is going to give you the number of zombies in the 14th hour, the number of zombies in the 10th hour, take the difference and you get the number of zombies between the 10th and 14th hour. |
| 17:03 | Ok, that's how much zombie population went up.
We see them on campus every year, we tag them with nerf guns. In real life that's not how it would work. Ok, I don't care what that comes out to, do we care? Alright so we can do another example from the book A hot wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitos is increasing at a rate of dm/dt = 2200+10e^.8t. |
| 18:19 | How much does the mosquito population increase between week 5 and week 9? I believe that's what the book was asking for, yes.
|
| 19:05 | So the rate of increasing mosquitos is dm/dt, which is 2200+10e^.8t
By how much does the mosquito population increase between week 5 and week 9?
Alright, let's do this one. This is a very straight forward one, much easier than all that physics stuff, right? |
| 19:36 | So if we want to know how much the population increases between week 5 and week 9, we're going to integrate the rate the mosquitos are changing. And that'll give us the total number of mosquitos between the 5th and 9th week, so we just have to integrate this.
|
| 20:01 | And for those of you who forget, the integral of e^kx, is just e^kx/k.
The reason it's divided by k is to take the derivative, when you take the derivative of this, you get e^kx *k/k, and that cancels the ks out, so that's why you have to divide by that number when you're doing the anti-derivative. |
| 20:33 | Ok, so here we get, 2200t+10e^.8t/.8 from 5-9.
.8 = 4/5, right? |
| 21:01 | So, you should be able to do that, let's see. That is (19800+12.5e^7.2)-(11000+12.5e^4), I don't know those are big numbers. Lots of mosquitos, lots of zika, what happens if you get the zika virus?
|
| 21:39 | You just feel cold right, we're not even sure yet if there's actually a limit, they're working on it. There seems to be a limit though.
So I'm not sure which problems we should do. |
| 22:03 | I'm gonna do one more pressure problem and then I'm moving into review. We have water,
and the dimensions of this tank are 4x5x2
it's filled with water. And you have a little hole in the top of the tank and you're now going to attach a hose to the top of the tank and you're going to pump the water out, suck it out of the tank.
|
| 22:38 | How much work would you be doing?
So you say to yourself, what are you doing? Well I'm lifting the water. Remember work is the force you have to exert * the distance over which the force is exerted. ok? |
| 23:07 | Not actually ending a sentence in a preposition when you say exerted over, over is actually an adverb its modifying exert, so you can say it that way, you can say over which the force is exerted, you just sound [?]
Well you had the english lesson thrown in there for those of you who didn't take advanced grammar.
So next time somebody says 'the person whom I'm talking to' Wrong! it's 'the person I am talking to.' |
| 23:33 | Talking about is different than talking to, talking with, so the preposition is actually modified. But I digress, so we have,
you imagine that you have a little slice of the water.
If the width of that slice is dx, ok, it's a box, it's a square shape, or rectangular shape, more like this, good enough. |
| 24:05 | So what's the volume of that water, well the volume of that water would be 4x5xdx
The density of water well let's put this in meters, the density of water is what?
|
| 24:30 | 1000, very good ok, so the mass of the water is 1000*20dx
it's really (delta)x but when we do the actual calculation it turns into dx.
Ok, and then the weight of the water is the force of gravity acting on that water. And the force is mg, where g=9.8, when I teach physics I use 10, I don't bother with 9.8 because it makes math easier. |
| 25:13 | So, and those of you who got excited, no I am not teaching physics at stony brook any time soon, sorry.
Alright, so here the force is 20,000dx*9.8 |
| 25:35 | Ok, which is, 196,000dx, so that's the force we're gonna need.
Then the work is the integral of Fx And we're going to lift it how far? 2m. |
| 26:01 | So it's going to be the integral from 0-2 of 196000, I put an extra 0 in there didn't I. Yes, thanks Obama. anyway.
and then I multiply that by x, dx. That's the work it'll take to pump all of that water out. |
| 26:31 | By the way if it wasn't water, what if it was oil what would happen? Well all that would happen is that instead of multiplying by 1000, you'd multiply by the appropriate density, rho.
You'd use the density of oil. What's the density of oil? Water is more dense, oil floats on top. Hense the phrase oil slick. Whats more dense, cream or milk? |
| 27:00 | Milk. if you ever saw old fashion milk, the cream sits on top.
if you put cream in your coffee it stays on top you have to stir it in. yeah? Why is x next to work because it's F*x. So this now becomes 98000x^2 from 0-2 so thats 98000*4-0, which is 292,000J |
| 27:44 | Sure its not F(x) it's F*x. I understand. Its F multiplied by x.
That would be hydrostatic force, just hydro because it's water. But it's essentially, well I'll give you an example, we're gonna open the swimming pool here at stony brook in a little while. |
| 28:10 | The swimming pool has a floor that you're able to raise and lower which is really cool so they can have aqua-therapy where you have the water stay here and they can just raise the water level so people can swim.
Ok, so what would you do? I mean it's an olympic sized swimming pool, theres's a lot of water inside the swimming pool, it weighs a lot. It |
| 28:32 | It also takes a long time to fill up a swimming pool like that. So what would you do to raise and lower the floor?
So you sit there and you say look if I want to raise the floor I'm pumping all of the water out of the tank. Thats an incredible amount of work. Or you could have a second floor that sits on top of the floor that would hold some force and then you just have to lift basically a screen. |
| 29:00 | Which still weighs a lot, because it's going to be a screen the size of an olympic swimming pool, and it's going to be fighting the water as it comes up but the water goes through the holes.
So what do you think [?] engineers? Well what are the problems when you lift that screen? What do you want to keep it from doing? Bend, twist, you want to make it rigid, it could sag in the middle it's a complicated problem. Ok, so you could have it in pieces, lift it in multiple stages. |
| 29:31 | You could lift it very slowly, but it's a tricky problem.
But that's where hydrostatic pressure, well hydrostatic pressure really shows up in things like you know you're pumping oil, gas into your gas tank. Out of a gas tank thing, station, and into your car. Right or your suctioning water or gas out of the car, your car in this case when the zombies come. Ok, so on that note, we're going to shift into review for the final. |
| 30:07 | Now we have not yet circulated the final, but the final is comprehensive,
cumulative, whatever you call it, it's going to be comprehensive and cumulative, so you have to ask yourself, what have we done in class?
|
| 30:30 | Well first thing we did was we just kinda learned what an anti-derivative is. What an integral is. An integral is just a derivative backwards.
That means we could anti-differentiate things. Very simple things like polynomials, trig functions, e, 1/x, etc. thats the first thing you have to be able to do. Then, we talk about what an integral really is we did Riemann sums. Those were fun. No? Those were not fun? No it was painful? More painful than 2nd grade? Really? Tell us about your pain. |
| 31:05 | Then we did a technique of integration for so long, that we just lost track of time it was so much fun.
And now we did applications, area, volume, arc length, average value, physics, business and biology. I think that's pretty much everything we did. Did I leave anything out? The volume, the area, |
| 31:33 | we did in depth improper integrals, those are always entertaining,
We skipped approximate integration because I think we had snow, it's fun when you lose like 5 classes for snow.
I pretty much factor that into my syllabus these days because, you know. We did lots and lots of integrals. Let's go back to the beginning. Lets be sure we can just do some basic integration. |
| 32:03 | So once again by the way, I will put up some review problems at some point after I've seen the exam.
However, the truth is the review packets for the 1st 2 midterms, you can just do them again. You should redo your midterm want to know what the exams look like, you have 2 of them. You should redo your midterms you should read through the chapters so that you can understand some of the concept stuff, some of that true/false stuff. |
| 32:33 | I would try to read through it, after all you all bought the book, right?
I'll see if I can find some additional pages to put up that might be helpful, also looking at the stuff from by book is much more mechanical, my book is not a theoretical book. It's a calculus study guide. Although astonishingly there are classes that use the text book I'm happy but they didn't ask me. Every time they buy a book a get a dollar so I'm not complaining. |
| 33:04 | All of a sudden I got much more attractive didn't I? I never noticed how handsome professor Kahn is. He's single isn't he?
Anyway, but that's what a I recommend, how to do odd problems from the text book, you know you have lots of spare time it's not like you have work for other classes. So you could sit here now and just do math for a week. That's 168 hours! |
| 33:32 | No you have more than that you have 210 hours.
Ok, 210 hours to get cracking. The nightmare is almost over folks. |
| 34:08 | Ok, you should be able to differentiate that.
That's a pi at the end in case you can't tell, so you should be able to do that. This problem is easy but at the beginning you were having trouble with this. So this is. 5x^4/4 -x^3+2x^2+(pi)x+C |
| 34:44 | Are the TA's and I going to subtract if you don't write +C? No? I hope not.
I mean you should write it, you're wrong if you don't, but you kinda don't need to beat people up for that. You should all be able to do an integral of the form x^n, you should know that that's x^n+1/(n+1)+ C |
| 35:10 | So far so good? Now of course I can make this messier,
I'll give you some more annoying ones.
|
| 35:30 | Ok, so what's the integral of sin(4x)?
Well the derivative of sinx is cosx, so the integral is -(cosx). Divide by 4, what about 2/x, it's 2log|x|, don't forget your absolute value bars. and e^pi(x) is just e^pi(x)/pi + C, don't forget that those are my exact words. |
| 36:04 | So as a reminder, the integral of sin(kx)dx = -cos(kx)/k +C The integral of the cos(kx)= sin(kx)/k +C |
| 36:35 | The integral of dx/(ax+b) is ln|ax+b| / a
How are we doing so far?
Good? We can do these basic integrals right? These seemed hard at the time but now you've got them, right? |
| 37:01 | Let's do a couple more, just to make sure.
What if we wanted to do the integral of the (sqrt)x, that's the same as x^(1/2), so that becomes x^(3/2)/(3/2), the integral of sec^2x is tanx, and the integral of (1/(1+x^2)) is tan^-1(x) |
| 37:38 | The same as arctan(x), +C, get that one?
Ok let's do another messy one. We're getting into good stuff on Wednesday, I'm just doing basics now. |
| 38:18 | Ok, let's try those 2.
Ok,. the integral of dx/(5x+1) is ln|5x+1|/5 +C |
| 38:33 | Ok? Make sure you can do something like that.
Now what about the integral of tanx? The derivative is sec^2x, but the integral is not. Give it a shot. Well, right so what we're gonna have to do is figure out how to do this, so you should make this sinx/cosx, this is one of the ones you have to memorize. |
| 39:03 | So you know how to do it all the time.
Then we use u substitution. So why don't we let u=cosx, du= -sin(x)dx |
| 39:32 | So that means that is integral now becomes -(1/u)du You know how to integrate du/u. That;'s ln|u| And now you subsitiute back in and now you get -ln|cos(x)|+ C. That's also by the way, the ln(sec(x))+C because you take the -1 |
| 40:07 | you put it up and you get cos^-1(x) and cos^-1(x) is 1/cos(x), which is sec(x), so sometimes you'll see this, sometimes you'll see this.
Some more ones to memorize. |
| 40:33 | the integral of sec(x), we have a little trick for this one.
That is the ln|sec(x)+tan(x)|+C That's 2 more for your collection. The one thing I'll recommend you do in the next few days, is try to make a page with, pretend you can take a bigger page into the exam. |
| 41:06 | And start working on what you want to take. And of course you just do reduction, to get it down that 1/4 piece of paper.
So far so good? And do you remember where this came from? You multiply top and bottom by secx+tanx and then you can do a u substitution. |
| 41:31 | You doing ok so far? Let's give you one slightly messier one.
Ok, let's do these 2. Let's do these. So the first one, I would do u substitution. Let u = 1+x^2, you guys are getting the hang of this, you've had problems before, but now you're comfortable with it. |
| 42:08 | du is 2xdx.
Which means 1/2 du = xdx. So this now becomes (1/2)(integral of ) u^(1/2)du |
| 42:30 | So far so good?
And this becomes (1/2)*u^(3/2)/(3/2) and with the magic of algebra we get (1/3)u^(3/2)+C Which we now substitute back and we have (1/3)(x^2+1)^(3/2) +C |
| 43:02 | So far so good, alright. Now what about the second one? Well, let u = 1+x.
du = dx, right? You're thinking yeah, but what about this x here? Well u-1=x. So this now becomes the integral of (u-1)(sqrt)(u) du. and this is the integral of (u(3/2)-u^(1/2))du |
| 43:48 | So again if u is 1+x, then u-1 is x, so now you can go and for x you substitute u-1, for 1+x you substitute u, and then you distribute the sqrt through with u, you get U^3/2)-u^(1/2).
|
| 44:07 | That's u^(5/2)/(5/2)-u^(3/2)/(3/2)+C.
Which is (2/5)(1+x)^(5/2)-(2/3)(x+1)^(3/2), ok? |
| 44:37 | How are we doing on u substitution?
Alright I think that's enough for today, I'll see everyone on Wednesday and we'll do a lot more review. |