| Start | You have an exam in a couple weeks.
We haven't done that much so you shouldn't worry too much. A lot of the first exam is sort of a review of previous semesters of math. But we're getting into the new stuff so we'll get you there. The exam is 2 weeks from today I think. October 3rd. That's a very special day. I wrote that in my diary. I did. |
| 0:32 | For those of you who aren't paying attention. Today is 98 days from Christmas.
That's good. Keep counting we're getting there. Very soon it'll be winter vacation and you'll be over with the semester. If you're a junior there's just not a lot left which is good. Then you have to get out and work. We'll see how much you really like that. Other stuff before I move on to new material. No? We were all able to handle MyMathLab? That's all going smoothly? |
| 1:02 | Okay good. If you're not speaking then the assumption is that the answer is yes.
Just so later when you complain I'll say you shouldn't spoken in class. Okay so other types of limit stuff. We've been doing a little bit of limits. We did numbers, we did what happens when you go to 0 in the denominator. Now let's think of what happens when you have limits that go to infinity. Shannon did you start recording at any point? Yeah. Okay. |
| 1:38 | Anyway so now let's do limits to ∞.
So we did a little bit of that the other day. We're going to do some more of that. So one of the common limits that you see is infinite end behavior. So functions usually do one of a couple things. Either as x gets very large the equation just sort of heads off. |
| 2:01 | Up towards infinity. Sometimes it heads down towards negative infinity.
And sometimes as x gets very large it flattens out. And since you imagine like you have the pond filled with bacteria or algae or something. And you get more and more algae and at some point you've kind of filled the pond. There's no more pond for the algae to go. I mean some of it has to start dying off but you get a flat spot. And you get a horizontal asymptote. Sometimes that horizontal asymptote is 0. |
| 2:31 | But that's still a horizontal asymptote. So these are sort of different scenarios of what happens what they call end behavior.
What happens when x becomes a very big number. Okay so how could we analyze that? Well we could say well let's say we have a function that consists of a polynomial with a bunch of terms. As many as you want, it could be zero. And a polynomial on the bottom so you have a rational expression. So this is some set of x's and other things on top and on the bottom. |
| 3:08 | We'll give you a concrete example in a second.
And then what you want to do is you want to figure out what happens as we head up towards infinity so let's do an example. Let's say we have |
| 3:37 | something simple like that so we have a quadratic polynomial on top and a quadratic on the bottom.
So what happens when x becomes bigger and bigger? Well when x is say 100 x^2 is 10,000. When x is 1,000 x^2 is 1,000,000. |
| 4:07 | ??
So as x gets very large the x^2 term becomes so big that the x or the -4x^10 will stop being important in terms of the overall behavior of the function. So if x is 1,000 this is 3,000,000 + 2,001. Okay the 2,001 is just the small little bit that's left over. |
| 4:32 | If you have $3,000,000 and I have $3,002,001 I don't really have anything more than you do I have $2,000 more but that's not a lot.
If you get to $1,000,000 then that's 3 trillion + 2 million. This 2 million becomes a rounding error. Same thing with the denominator. Furthermore these terms, whether they're added or subtracted, aren't important. The fact that one is 2 and one is 4, that's not going to matter because they're very small compared to the first term. |
| 5:00 | So the highest power term dominates the function.
Okay? That's the math word it dominates. It's the only one that really matters. Same with the top and bottom. The only thing that matters is the highest term on top and the highest term on bottom. Not necessarily the one on the left because of course I could write these in any order I want. You just care about the fact that the top is squared and the bottom is squared. So then there's 3 possibilities. The top is bigger than the bottom, the top is equal to the bottom, or the top is less than the bottom. |
| 5:43 | So if the top power is bigger than the bottom power okay the degree of the numerator is higher than the degree of the denominator then this limit as x -> ∞ of f(x) is ∞.
|
| 6:01 | It could also be -∞ it depends on the signs of what's going on there.
So we'll do one in a second but so if this is a negative coefficient then you'd get -∞. So in other words if the top power is bigger than the bottom power then it dominates the expression so much that the other terms don't really matter and this heads off towards ∞. So that's this one because it's going up. |
| 6:30 | It might be going down but same idea.
So the other function can't really control it. Now if they're the same then the limit as x -> ∞ of f(x) becomes a/b. And if the bottom is bigger than the top |
| 7:00 | then the limit becomes 0.
So in this example because these don't matter than the x^2's would cancel at some point right and this would just become 3/5. So the lim x- > ∞ of f(x) = 3/5. |
| 7:30 | Because let's get rid of- put these back.
Okay because as x becomes very large the 2x and the 1 are not important, the 4x and the 10 are not important so all you really have is 3x^2/5x^2. The x^2's would cancel and then you would just have 3/5. Suppose instead |
| 8:18 | So you know when x is 1,000 this is 3 billion, this is only 5 million.
A billionaire is a lot richer than a millionaire. So at some point this 3x^3 becomes way bigger than 5x^2. You could think of this as behaving really like 3x^3/5x^2. |
| 8:42 | Then they x^2's would cancel and then you'd just get 3x/5. Well fine then when x gets really big the whole thing gets really big.
Okay it's not really equal so. It's more like arrows. So the limit as x heads off to ∞ of f(x) is ∞. |
| 9:14 | And how would this be -∞? If that was -3x^3.
Okay? Or if this was going to -∞. So that says that the function goes up and down you know the function does something fun in the middle but essentially has that shape. |
| 9:35 | As x gets very big it goes up and as x gets very negative it goes down.
This one is what happens when the function, that's 3/5, is behaving like that. |
| 10:02 | I'm not sure if that's the asymptote there so that's not really accurate.
But it's doing something like that. Of course it might be doing this. I'd have to play with the graph a couple of minutes to figure it out. But either way as x gets very large it just starts to look like a horizontal asymptote at 3/5. So far so good? Alright. Let's do another type of example. |
| 11:00 | Okay now we look at this and we say well the denominator is bigger than the numerator.
So just like the other one but now the bottom is a lot bigger than the top. So when x goes to infinity that's going to look like 0. How do we feel about these? So far so good? |
| 11:40 | Okay so let's have you do a couple to make sure everybody gets it.
|
| 12:44 | Okay try those limits for a minute.
|
| 15:13 | Okay. You shouldn't need too long for these.
You can do these more or less by inspection as they say. In other words you don't have to do any calculating. You can just sort of figure it out. So you look at the top one. So all you care about is what's the highest degree term on top. |
| 15:30 | 8x^3. And what's the highest degree term on bottom. Don't worry about these. They stopped being important as you go towards infinity.
Because the bottom power is x^4 and the top power is only x^3 the bottom is bigger than the top and you get 0. Okay remember the n is bigger than the m. That's when you put a 0. |
| 16:00 | Okay you look at the second one. Again all you care about is the top versus the bottom.
Notice it doesn't matter what order I write them in. The most important term on top, the biggest term, will be the x^10 and the bottom will be x^7. x^10 is a much higher power than x^7. So it's going to go to ∞ but it's negative x^10 so it's going to go to -∞. Two for two. Get one more right you get to go to college. Oh wait. |
| 16:33 | Alright the third one. Again order doesn't matter.
So all you care about is the 6x^5 and then -2x^5. Since they're the same power 6/-2 is -3 so that's going to look like -3. You do okay on these? I would do more examples but I think you get this. |
| 17:00 | We'll put some in the homework.
So those are the limits ?? as you go out towards infinity. Okay so I talked a little bit last time and I'll talk a little bit more this time on continuity. So I was doing some of those graphing examples last class and the limits. I sort of sneaked around continuity without actually doing it. But now we'll talk about it. |
| 17:32 | So continuity again this is a math concept that matters more to mathematicians than it probably does to you guys but
in the business world many many things are discontinuous.
You do need to know that you can work with discontinuous functions. The only problem is right at the break. In-between where the function becomes continuous and where it doesn't. |
| 18:00 | In other words what's an example of a discontinuous function in the business world?
Well as I said some of these payment plans where we say your first 1,000 text messages, well the data limits plan. Okay so you get 8 GB of data for $29.99 also known as $30. And if you go over 8 GB now you pay for the data although companies are stopping that. |
| 18:34 | So that's discontinuous because it breaks at that spot.
Continuity refers to more or less when the function has a break in it does it resume where it stopped. Or does it jump? Is there a hole in it? |
| 19:03 | Is there an asymptote? It's the 3 main things alright you get jumps, you get holes, and you get asymptotes.
So a jump discontinuity is a function that does something like that. It jumps. Instantly. |
| 19:30 | So you're on the left equation right up until you get to point A and then all the sudden at A all of the sudden you're at a different equation.
As I said typical phenomenon of that are pricing plans. The parking garage that says the first hour is $9.99 and then every hour after that is $2.99. Or in Manhattan you know 30 bucks for the first hour and then 20 bucks an hour after that or something horrible. |
| 20:00 | I don't know the last time any of you tried to park a car in Manhattan. Don't.
A hole is just as it sounds. There's a hole. It just doesn't exist at that one spot. Sometimes the function has a different value at that spot. Sometimes it just has a hole. This is also sometimes called a removable discontinuity. Because you could fix the problem by just plugging the appropriate value in the hole. |
| 20:33 | But if you imagine that you're writing this on paper so one of the things is you have your pencil, you get to this spot,
you have to leave the paper to get to that spot.
Again you have to leave the paper because there's a hole and then you can get back in there. Medicaid, Medicare. I think everybody in this room is too young for Medicare although I'm closer than the rest of you. There's a drug benefit so when you go to the pharmacy and you get drugs, those kind of drugs |
| 21:02 | you get reimbursed. Medicare covers that up to some number like $3,000 worth of drugs.
And then all the sudden it doesn't cover you again. You're on your own until you get to something like $6,000. And then you're covered from there on in so there's a hole in the middle there. Where you have to pay for some intermediate amount of drugs so if you do $3,500 worth of drug expenses and they cover $3,000 you have to pay for $500 of them. Does that make any sense? I don't have it exactly right because I haven't gotten on Medicare yet I still have a few years left. |
| 21:33 | But they have this hole in there which is really annoying but they didn't ask me.
And asymptotes well you know what asymptotes are. In this case this would be the vertical asymptote type where a function just kind of does something like that. And there's nothing you can do with a vertical asymptote to make the function continuous, other than redefine the function. |
| 22:01 | At that spot it's jumping up to ∞ or -∞.
Don't confuse vertical asymptotes with horizontal asymptotes. The horizontal asymptote stuff we just did. That's really end behavior. The vertical asymptote you cannot cross that line. There is no value at that line. You can have functions with a horizontal asymptote where you cross the line. So you can have a function that can cross as many times as you want. You could have a function that does something like this. |
| 22:34 | So it has a horizontal asymptote right there.
But it crosses it an infinite number of times. Okay so don't confuse the two. You cannot have a value at the vertical asymptote you can have values on the horizontal asymptote. Sometimes this causes people issues. Alright so continuous. This is probably the most theoretical thing we're going to do for a while. Which I know makes you happy. |
| 23:08 | Say you have a function like f(x) = 8/x-1.
If you were to graph that See on video nobody's going to hear your question. Her question was what's an example of a vertical asymptote. Okay so the horizontal asymptote really the word asymptote is a bit deceptive. |
| 23:35 | It just goes out towards 0 when x gets very large because the bottom power is 1 and the top power is 0.
So the bottom is a higher power than the top. You realize that's x^0 right? Okay because anything to the zero is 1. But at x=1 there's no value. At x=0.9999 you keep going down there so it's -∞. And 1.0001 you go up so it's positive ∞. So there's a break there. By the way the asymptotes, the function can go the same way on both sides. |
| 24:03 | It doesn't have to split. They could both go up or both go down.
The key is there is no value at 1. I'd have to think about it to come up with a business example for a vertical asymptote. There's things that aren't defined at 0. There's just no value at 0. But we're not too worried about those. Alright so let's be mathematical. I think we can come up with a definition of what we need to guarantee that a function is continuous because we're going to do derivatives. |
| 24:39 | And in order to be able to do the derivative of a function at a point the function has to be continuous at that point.
Or it doesn't work mathematically. So we're going to need to know a couple of things. So first of all we can't have- the function has to have a value at the point. So we might say f(x) is continuous at x=a if, it's really if and only if, but not important |
| 25:15 | 1. f(a) exists.
In other words, it has to have a value. It has to have a value at a. On these two it does on this one it doesn't. |
| 25:32 | 2. The limit as x approaches a of f(x) exists.
So it has to be approaching a from both sides. Another way to do this you could say the limit from the left side has to be the same as the limit from the right side. |
| 26:14 | 3. The first and the second have to equal each other.
In other words f(a) has to equal the limit when x approaches a of f(x). Alright enough theory. Let's do some examples. |
| 26:37 | Typical test question. I ask you if something is continuous.
Nobody's asked their hand and said "Are you going to do practice problems for the exam?". Because what did I say at the first day? |
| 27:04 | Right. I'll make a practice test.
Or something. I'll think of some way to have you guys practice. Then the real exam will look nothing like it and I'll laugh really hard. Because that's how I get pleasure. Alright suppose we have f(x) |
| 27:42 | So two slightly different lines.
So question is is this function continuous at x=1? |
| 28:02 | Other than that, notice it's a line so 6x+8 is a line, 5x+9 is a line or in other words a polynomial.
And polynomials are always continuous. So any polynomial is continuous. So the only problem here is what happens at x=1. Well notice it's defined for x<1 it's defined for x>1 but it's not defined at x=1. So this fails condition one so this is not continuous at x=1 because f(1) does not exist. |
| 29:02 | Alright let's do a slightly different function.
So by the way what would this one look like? This would be a function where it's kind of going something like that. So it's going up with a slope of 6 and then it's going up with a slope of 5, this is not very accurate. |
| 29:33 | A slope of 6 would actually be steep but whatever.
And at 1 it doesn't have a value. Now at less than 1 notice the limit at less than 1 is 14 and at greater than 1 is 14 so they're both zooming in on 14 there. But they don't actually get to it because there's no value at 1. |
| 30:00 | Alright let's do a slightly different one.
Okay so now we have 6x+8 when x ≤ 1 and 5x+7 when x>1. |
| 30:32 | So let's see does f(1) exist?
Remember I said the other things aren't that interesting. Sure you have a value of 1 you plug it in and you get whatever you get, 14. But the limit when x approaches 1 from the minus side is 14 because for numbers less than 1 when you plug in 1 you get 6+8. |
| 31:04 | But if you do from the plus side you get 5+7 is only 12.
These two don't match so the limit as x approaches 1 of f(x) does not exist. |
| 31:31 | And this as a function this has a jump.
So you get to there... figure not drawn to scale as they say. So far so good? |
| 32:02 | What you can do so you can eyeball this you just look at this you say if I plug in 1 here I get 14 if I plug in 1 there I get 12.
Those don't match so that's how I know it's not continuous. Here I plug in 1 I get 14 and plug in 1 I get 14 so the limits match but there is no value at 1. Because I didn't write less than or equal to I just wrote less than. I didn't write greater than or equal to I just wrote greater than. Okay? So slightly different. Alright the third type. |
| 32:32 | So this passed condition one but didn't pass condition two.
Alright let's try |
| 33:09 | well let's do something slightly different |
| 33:33 | Alright so clearly the interesting stuff is going on at 4.
Because with 4 you get that 0 in the denominator. Otherwise you're just looking at a polynomial divided by a polynomial and a polynomial. Not a lot going on there. And let's check so let's see does f(4) exist? Sure. f(4) = 5. So condition one is satisfied. f(x) exists. |
| 34:09 | Okay let's do that limit.
So the limit when x -> 4- (from the minus side) is- how would we do that limit by the way? Well what you would do is you would say this top can be factored. |
| 34:48 | Now remember when you were in algebra class and you canceled the (x-4)'s and your teacher took off?
And you said I don't understand I thought we were allowed to cancel things. And your teacher said well how do you know x is not 4? |
| 35:00 | Because at 4 you have 0/0 so you're not going to be allowed to cancel because it's not always true.
But the point is here we're doing the limit as x approaches 4 from the minus side but we're approaching 4 we are not equal to 4. Since we're not equal to 4 we can cancel. And you'd get 1. So when you get those kind of rational limits that's what you do you factor the top and bottom and cancel. |
| 35:32 | Remember you're allowed to cancel because x is not 4.
And by the way it doesn't really matter if this is a minus or a plus. You're going to get the same answer. So the limit exists. The limit is one. But f(4) is and the limit is 1 so this is a function that sorta does that. |
| 36:06 | Right up to 4 it's at 1 on either side but at 4 itself it jumps up to 5.
So because the limit does not equal f(4) because the limit does not equal f(4) this is not continuous at 4. It's continuous everywhere else. |
| 36:40 | So far so good?
|
| 37:34 | Okay let's look at this.
Well f(1) is 9-2=7. So f(1) exists. I just plugged in 1 into the top equation. (In reference to the last problem) Yes you can say it's not continuous because something like that. |
| 38:09 | If you wrote that kind of stuff I would give you full credit I'd say you obviously understood the question.
I'll try to make sure the TA's are on their game. How are we doing with the TA's so far are we happy? We're learning things? They're all okay? |
| 38:35 | Okay so you plug in f(1), f(1) is 7.
Then the limit when x -> 1- of f(x) is also 9-2 which is 7. And then the limit as x -> 1+ of f(x) is also 7. |
| 39:10 | And the limit when x -> 1 of f(x) = f(1). They're both equal to 7.
Therefore, the 3 dots mean therefore, f(x) is continuous, that's what cts means, at x=1. So continuous everywhere. |
| 39:39 | And this is a graph that's sort of doing that.
So when you leave the steeper line and you go to the flatter line you're still at the same spot. You're at 7. |
| 40:01 | Everybody got the idea?
I'll give you one to practice. |
| 40:54 | Okay take 2 minutes and see how you do.
|
| 42:56 | Let's do this one.
So first, what happens when x is not 9? |
| 43:02 | Well you get 10. I'm sorry when x is 9. I apologize.
So f(9) = 10. Now let's do the limit. So the limit when x approaches 9 for this function. The denominator should give you a clue on how to factor. Okay so there's gotta be an x-9 term in there. |
| 43:44 | Okay now you cancel, you're left with (x+1) and that's going to come out 10.
And oh look f(9) was 10. So f(9) = lim x -> 9. Therefore f(x) is continuous at x=9. |
| 44:17 | How'd we do on this one? Happy?
There are examples of when it doesn't cancel in real life but I wouldn't bother. |
| 44:30 | I mean yes would there be an example where they don't cancel? That takes all the fun out of it so sure.
I want them to cancel. But the test is whether- what we really want to know is just can you test if something is continuous? That's really what I want to know is do you know what the 3 things are to test. This says x can't be 9. So when it's not 9 it's this function and when it is 9 it just has the value of 10. |
| 45:05 | So when I do the limit I can cancel these x-9's.
Then when I plug in 9+1 I get 10. Alright well that's enough for one day. I'll see you on Wednesday I hope. |