LECTURE 2

TRANSLATION OF LIMITS

So last time, we defined the derivative as the slope of a tangent line. So that was our geometric point of view and we also did a couple of computations. We worked out that the derivative of 1 / x was -1 / x^2. And we also computed the derivative of x ^ nth power for n = 1, 2, etc., and that turned out to be x, I'm sorry, nx^(n-1). So that's what we did last time, and today I want to finish up with other points of view on what a derivative is. So this is extremely important, it's almost the most important thing I'll be saying in the class. But you'll have to think about it again when you start over and start using calculus in the real world. So again we're talking about what is a derivative and this is just a continuation of last time.

So, as I said last time, we talked about geometric interpretations, and today what we're gonna talk about is rate of change as an interpretation of the derivative. So remember we drew graphs of functions, y = f(x) and we kept track of the change in x and here the change in y, let's say. And then from this new point of view a rate of change, keeping track of the rate of change of x and the rate of change of y, it's the relative rate of change we're interested in, and that's delta y / delta x and that has another interpretation. This is the average change. Usually we would think of that, if x were measuring time and so the average and that's when this becomes a rate, and the average is over the time interval delta x. And then the limiting value is denoted dy/dx and so this one is the average rate of change and this one is the instantaneous rate.

Okay, so that's the point of view that I'd like to discuss now and give you just a couple of examples. So, let's see. Well, first of all, maybe some examples from physics here. So q is usually the name for a charge, and then dq/dt is what's known as current. So that's one physical example. A second example, which is probably the most tangible one, is we could denote know the letter s by distance and then the rate of change is that what we call speed. So those are the two typical examples and I just want to illustrate the second example in a little bit more detail because I think it's important to have some visceral sense of this notion of instantaneous speed. And I get to use the example of this very building to do that. Probably you know, or maybe you don't, that on Halloween there's an event that takes place in this building or really from the top of this building which is called the pumpkin drop. So let's illustrates this idea of rate of change with the pumpkin drop.

So what happens is this building, well let's see here's the building, and here's the dot, that's the beautiful grass out on this side of the building, and then there's some people up here and very small objects, well they're not that small when you're close to them, that get dumped over the side there. And they fall down. You know everything at MIT or a lot of things at MIT are physics experiments. That's the pumpkin drop.

So roughly speaking, the building is about 300 feet high, we're down here on the first usable floor. And so we're going to use instead of 300 feet, just for convenience purposes we'll use 80 meters because that makes the numbers come out simply. So we have the height which starts out at 80 meters at time and then the acceleration due to gravity gives you this formula for h, this is the height. So at time t = 0, we're up at the top, h is 80 meters, the units here are meters. And at time t = 4 you notice, (5 * 4^2) is 80. I picked these numbers conveniently so that we're down at the bottom. Okay, so this notion of average change here, so the average change, or the average speed here, maybe we'll call it the average speed since that's, over this time that it takes for the pumpkin to drop is going to be the change in h / the change in t. Which starts out at, what does it start out as? It starts out as 80, right? And it ends at 0. So actually we have to do it backwards. We have to take 0 - 80 because the first value is the final position and the second value is the initial position. And that's divided by 4 - 0; times 4 seconds minus times seconds. And so that of course is -20 meters per second. So the average speed of this guy is 20 meters a second.

Now, so why did I pick this example? Because, of course, the average, although interesting, is not really what anybody cares about who actually goes to the event. All we really care about is the instantaneous speed when it hits the pavement and so that's can be calculated at the bottom. So what's the instantaneous speed? That's the derivative, or maybe to be consistent with the notation I've been using so far, that's d/dt of h. All right? So that's d/dt of h. Now remember we have formulas for these things. We can differentiate this function now. We did that yesterday. So we're gonna take the rate of change and if you take a look at it, it's just the rate of change of 80 is 0, minus the rate change for this -5t^2, that's minus 10t.

So that's using the fact that d/dt of 80 is equal to and d/dt of t^2 is equal to 2t. The special case... Well I'm cheating here, but there's a special case that's obvious. I didn't throw it in over here. The case n = 2 is that second case there. But the case n = also works. Because that's constants. The derivative of a constant is 0. And then the factor n there's and that's consistent. And actually if you look at the formula above it you'll see that it's the case of n = -1. So we'll get a larger pattern soon enough with the powers.

Okay anyway. Back over here we have our rate of change and this is what it is. And at the bottom, at that point of impact, we have t = 4 and so h' which is the derivative is equal to -40 meters per second. So twice as fast as the average speed here, and if you need to convert that, that's about 90 miles an hour. Which is why the police are there at midnight on Halloween to make sure you're all safe and also why when you come you have to be prepared to clean up afterwards.

So anyway that's what happens, it's 90 miles an hour. It's actually the buildings a little taller, there's air resistance and I'm sure you can do a much more thorough study of this example. All right so now I want to give you a couple of more examples because time and these kinds of parameters and variables are not the only ones that are important for calculus. If it were only this kind of physics that was involved, then this would be a much more specialized subject than it. Is And so I want to give you a couple of examples that don't involved time as a variable. So the third example I'll give here is. The letter t often denotes temperature, and then dt/dx would be what is known as the temperature gradient. Which we really care about a lot when we're predicting the weather because it's that temperature difference that causes air flows and causes things to change. And then there's another theme which is throughout the sciences and engineering which I'm going to talk about under the heading of sensitivity of measurements.

So let me explain this. I don't want to belabor it because I just am doing this in order to introduce you to the ideas on your problem set which are the first case of this. So on problem set one you have an example which is based on a simplified model of GPS, sort of the Flat Earth Model. And in that situation, well, if the Earth is flat it's just a horizontal line like this. And then you have a satellite, which is over here, preferably above the earth, and the satellite or the system knows exactly where the point directly below the satellite is. So this point is treated as known. And I'm sitting here with my little GPS device and I want to know where I am. And the way I locate where I am is I communicate with this satellite by radio signals and I can measure this distance here which is called h. And then system will computer this horizontal distance which is L. So in other words what is measured, so h measured by radios, radio waves and a clock, or various clocks. And then L is deduced from h. And what's critical in all of these systems is that you don't know h exactly. There's an error in h which will denote delta h. There's some degree of uncertainty. The main uncertainty in GPS is from the ionosphere. But there are lots of corrections that are made of all kinds. And also if you're inside a building it's a problem to measure it. But it's an extremely important issue, as I'll explain in a second.

So the idea is we then get at delta L is estimated by considering this ratio delta L/delta h which is going to be approximately the same as the derivative of L with respect to h. So this is the thing that's easy because of course it's calculus. Calculus is the easy part and that allows us to deduce something about the real world that's close by over here. So the reason why you should care about this quite a bit is that it's used all the time to land airplanes. So you really do care that they actually know to within a few feet or even closer where your plane is and how high up it is and so forth.

All right. So that's it for the general introduction of what a derivative is. I'm sure you'll be getting used to this in a lot of different contexts throughout the course. And now we have to get back down to some rigorous details. Ok, everybody happy with what we've got so far? Yeah?

Student: How did you get the equation for height?

Professor: Ah good question question. The question was how did I get this equation for height? I just made it up because it's the formula from physics that you will learn when you take 8.01 and, in fact, it has to do with the fact that this is the speed if you differentiate another time you get acceleration and acceleration due to gravity is 10 meters per second. Which happens to be the second derivative of this. But anyway I just pulled it out of a hat from your physics class. So you can just say see 8.01 .

All right, other questions?

All right, so let's go on now. Now I have to be a little bit more systematic about limits. So let's do that now. So now what I'd like to talk about is limits and continuity. And this is a warm up for deriving all the rest of the formulas, all the rest of the formulas that I'm going to need to differentiate every function you know. Remember, that's our goal and we only have about a week left so we'd better get started.

So first of all there is what I will call easy limits. So what's an easy limit? An easy limit is something like the limit as x goes to 4 of (x 3 / x^2 1). And with this kind of limit all I have to do to evaluate it is to plug in x = 4 because, so what I get here is 4 3 / (4^2 1). And that's just 7 / 17. And that's the end of it. So those are the easy limits.

The second kind of limit, well so this isn't the only second kind of limit but I just want to point this out, it's very important is that: derivatives are are always harder than this. You can't get away with nothing here. So, why is that? Well, when you take a derivative, you're taking the limit as x goes to x0 of f(x), well we'll write it all out in all its glory. Here's the formula for the derivative. Now notice that if you plug in x = x0, always gives / 0. So it just basically never works. So we always are going to need some cancellation to make sense out of the limit.

Now in order to make things a little easier for myself to explain what's going on with limits I need to introduce just one more piece of notation. What I'm gonna introduce here is what's known as a left-hand and a right limit. If I take the limit as x tends to x0 with a sign here of some function, this is what's known as the right hand limit. And I can display it visually. So what does this mean? It means practically the same thing as x tends to x0 except there is one more restriction which has to do with this sign, which is we're going from the plus side of x0. That means x is bigger than x0. And I say right-hand, so there should be a hyphen here, right-hand limit because on the number line, if x zero is over here the x is to the right. All right? So that's the right-hand limit. And then this being the left side of the board, I'll put on the right side of the board the left limit, just to make things confusing. So that one has the minus sign here. I'm just a little dyslexic and I hope you're not. So I may have gotten that wrong. So this is the left-hand limit, and I'll draw it. So of course that just means x goes to x0 but x is to the left of x0 . And again, on the number line, here's the x0 and the x is on the other side of it.

Okay, so those two notations are going to help us to clarify a bunch of things. It's much more convenient to have this extra bit of description of limits than to just consider limits from both sides. Okay so I want to give an example of this. And also an example of how you're going to think about these sorts of problems. So I'll take a function which has two different definitions. Say it's x 1, when x > 0 and -x 2, when x 0, it's x 1. Now I can draw a picture of this. It's gonna be kind of little small because I'm gonna try to fit it down in here, but maybe I'll put the axis down below. So at height 1, I have to the right something of slope 1 so it goes up like this. All right? And then to the left of I have something which has slope -1, but it hits the axis at 2 so it's up here. So I had this sort of strange antennae figure here is my graph. Maybe I should draw these in another color to depict that. And then if I calculate these two limits here, what I see is that the limit as x goes to 0 from above of f(x), that's the same as the limit as x goes to of the formula here, x 1. Which turns out to be 1. And if I take the limit, so that's the left-hand limit. Sorry, I told you I was dyslexic. This is the right, so it's that right-hand. Here we go. So now I'm going from the left, and it's f(x) again, but now because I'm on that side the thing I need to plug is the other formula, -x 2, and that's gonna give us 2.

Now, notice that the left and right limits, and this is one little tiny subtlety and it's almost the only thing that I need you to really pay attention to a little bit right now, is that this, we did not need x = 0 value. In fact I never even told you what f(0) was here. If we stick it in we could stick it in. Okay let's say we stick it in on this side. Let's make it be that it's on this side. So that means that this point is in and this point is out. So that's a typical notation: this little open circle and this closed dot for when you include the. So in that case the value of f(x) happens to be the same as its right hand limit, namely the value is 1 here and not 2.

Okay, so that's the first kind of example. Questions? Okay, so now our next job is to introduce the definition of continuity. So that was the other topic here. So we're going to define. So f is continuous at x0 means that the limit of f(x) as x tends to x0 = f(x0) . Right? So the reason why I spend all this time paying attention to the left and the right and so on and so forth and focusing is that I want you to pay attention for one moment to what the content of this definition is. What it's saying is the following: continuous at x0 has various ingredients here. So the first one is that this limit exists. And what that means is that there's an honest limiting value both from the left and right. And they also have to be the same. All right, so that's what's going on here. And the second property is that f(x0) is defined. So I can't be in one of these situations where I haven't even specified what f(x0) is and they're equal. Okay, so that's the situation.

Now again let me emphasize a tricky part of the definition of a limit. This side, the left-hand side is completely independent, is evaluated by a procedure which does not involve the right-hand side. These are separate things. This one is, to evaluate it, you always avoid the limit point. So that's if you like a paradox, because it's exactly the question: is it true that if you plug in x0 you get the same answer as if you move in the limit? That's the issue that we're considering here. We have to make that distinction in order to say that these are two, otherwise this is just tautalogical. It doesn't have any meaning. But in fact it does have a meaning because one thing is evaluated separately with reference to all the other points and the other is evaluated right at the point in question. And indeed what these things are, are exactly the easy limits. That's exactly what we're talking about here. They're the ones you can evaluate this way. So we have to make the distinction. And these other ones are gonna be the ones which we can't evaluate that way. So these are the nice ones and that's why we care about them why we have a whole definitions associated with them. All right?

So now what's next? Well, I need to give you a a little tour, very brief tour, of the zoo of what are known as discontinuous functions. So sort of everything else that's not continuous. So, the first example here, let me just write it down here. It's jump discontinuities. So what would a jump discontinuity be? Well we've actually already seen it. The jump discontinuity is the example that we had right there. This is when the limit from the left and right exist, but are not equal. Okay, so that's as in the example. Right? In this example, the two limits, one of them was 1 and of them was 2. So that's a jump discontinuity. And this kind of issue, of whether something is continuous or not, may seem a little bit technical but it is true that people have worried about it a lot. Bob Merton, who was a professor at MIT when he did his work for the Nobel prize in economics, was interested in this very issue of whether stock prices of various kinds are continuous from the left or right in a certain model. And that was a very serious issue in developing the model that priced things that our hedge funds use all the time now. So left and right can really mean something very different. In this case left is the past and right is the future and it makes a big difference whether things are continuous from the left or continuous from the right. Right, is it true that the point is here, here, somewhere in the middle, somewhere else. That's a serious issue.

So the next example that I want to give you is a little bit more subtle. It's what's known as a removable discontinuity. And so what this means is that the limit from left and right are equal. So a picture of that would be, you have a function which is coming along like this and there's a hole maybe where, who knows either the function is undefined or maybe it's defined up here, and then it just continues on. All right? So the two limits are the same. And then of course the function in begging to be redefined so that we remove that hole. And that's why it's called a removable discontinuity. Now let me give you an example of this, or actually a couple of examples. So these are quite important examples which you will be working with in a few minutes. So the first is the function g(x), which is sin x / x, and the second will be the function h(x), which is 1 - cos x / x. So we have a problem at g(0) , g(0) is undefined. On the other hand it turns out this function has what's called a removable singularity. Namely the limit as x goes to of sin x / x does exist. In fact it's equal to 1. So that's a very important limit that we will work out either at the end of this lecture or the beginning of next lecture. And similarly, the limit of 1 - cos x / x, as x goes to 0 is 0. Maybe I'll put that a little farther away so you can read it. Okay, so these are very useful facts that we're going to need later on. And what they say is that these things have removable singularities, sorry removable discontinuity at x = 0. All right so as I say, we'll get to that in a few minutes.

Okay so are there any questions before I move on? Yeah?

Student: [INAUDIBLE]

Professor: The question is: why is this true? Is that what your question is? The answer is it's very, very unobvious, I haven't shown it to you yet, and if you were not surprised by it then that would be very strange indeed. So we haven't done it yet. You have to stay tuned until we do. Okay? We haven't shown it yet. And actually even this other statement, which maybe seems stranger still, is also not yet explained. Okay, so we are going to get there, as I said, either at the end of this lecture or at the beginning of next. Other questions?

All right, so let me just continue my tour of the zoo of discontinuities. And, I guess, I want to illustrate something with the convenience of right and left hand limits so I'll save this board about right and left-hand limits. So a third type of discontinuity is what's known as an infinite discontinuity. And we've already encountered one of these. I'm going to draw them over here. Remember the function y is 1 / x. That's this function here. But now I'd like to draw also the other branch of the hyperbola down here and allow myself to consider negative values of x. So here's the graph of 1 / x. And the convenience here of distinguishing the left and the right hand limits is very important because here I can write down that the limit as x goes to 0 of 1 / x. Well that's coming from the right and it's going up. So this limit is infinity. Whereas, the limit in the other direction, from the left, that one is going down. And so it's quite different, it's minus infinity. Now some people say that these limits are undefined but actually they're going in some very definite direction. So you should, whenever possible, specify what these limits are.

On the other hand, the statement that the limit as x goes to of 1 / x is infinity is simply wrong. Okay, it's not that people don't write this. It's just that it's wrong. It's not that they don't write it down. In fact you'll probably see it. It's because people are just thinking of the right hand branch. It's not that they're making a mistake necessarily, but anyway, it's sloppy. And there's some sloppiness that we'll endure and others that we'll try to avoid. So here, you want to say this, and it does make a difference. You know, plus infinity is an infinite number of dollars and minus infinity is and infinite amount of debt. They're actually different. They're not the same. So, you know, this is sloppy and this is actually more correct.

Okay, so now in addition, I just want to point out one more thing. Remember, we calculated the derivative, and that was -1/x^2. But, I want to draw the graph of that and make a few comments about it. So I'm going to draw the graph directly underneath the graph of the function. And notice what this graphs is. It goes like this, it's always negative, and it points down. So now this may look a little strange, that the derivative of this thing is this guy, but that's because of something very important. And you should always remember this about derivatives. The derivative function looks nothing like the function, necessarily. So you should just forget about that as being an idea. Some people feel like if one thing goes down, the other thing has to go down. Just forget that intuition. It's wrong. What we're dealing with here, if you remember, is the slope. So if you have a slope here, that corresponds to just a place over here and as the slope gets a little bit less steep, that's why we're approaching the horizontal axis. The number is getting a little smaller as we close in. Now over here, the slope is also negative. It is going down and as we get down here it's getting more and more negative. As we go here the slope, this function is going up, but its slope is going down. All right, so the slope is down on both sides and the notation that we use for that is well suited to this left and right business. Namely, the limit as x goes to of -1 / x^2, that's going to be equal to minus infinity. And that applies to x going to 0 and x going to 0-. So both have this property.

Finally let me just make one last comment about these two graphs. This function here is an odd function and when you take the derivative of an odd function you always get an even function. That's closely related to the fact that this 1 / x is an odd power and x^1 is an odd power and x^2 is an even power. So all of this your intuition should be reinforcing the fact that these pictures look right.

Okay, now there's one last kind of discontinuity that I want to mention briefly, which I will call other ugly discontinuities. And there are lots and lots of them. So one example would be the function y = sin 1 / x, as x goes to 0. And that looks a little bit like this. Back and forth and back and forth. It oscillates infinitely often as we tend to 0. There's no left or right limit in this case. So there is a very large quantity of things like that. Fortunately we're not gonna deal with them in this course. A lot of times in real life there are things that oscillate as time goes to infinity, but we're not going to worry about that right now. Okay, so that's our final mention of a discontinuity, and now I need to do just one more piece of groundwork for our formulas next time. Namely, I want to check for you one basic fact, one limiting tool. So this is going to be a theorem. Fortunately it's a very short theorem and has a very short proof. So the theorem goes under the name differentiable implies continuous. And what it says is the following: it says that if f is differentiable, in other words its the derivative exists at x0, then f is continuous at x0.

So, we're gonna need this is as a tool, it's a key step in the product and quotient rules. So I'd like to prove it right now for you. So here is the proof. Fortunately the proof is just one line. So first of all, I want to write in just the right way what it is that we have to check. So what we have to check is that the limit, as x goes to x0 of f(x) - f(x0) = 0. So this is what we want to know. We don't know it yet, but we're trying to check whether this is true or not.

So that's the same as the statement that the function is continuous because the limit of f(x) is supposed to be f(x0) and so this difference should have limit 0. And now, the way this is proved is just by rewriting it by multiplying and dividing by (x - x0). So I'll rewrite the limit as x goes to x0 of (f(x) - f(x0) / by x - x0) (x - x0). Okay, so I wrote down the same expression that I had here. This is just the same limit, but I multiplied and divided by (x - x0).

And now when I take the limit what happens is the limit of the first factor is f'(x0). That's the thing we know exists by our assumption. And the limit of the second factor is because the limit as x goes to x0 of (x - x0) is clearly 0 . So that's it. The answer is 0, which is what we wanted. So that's the proof. Now there's something exceedingly fishy looking about this proof and let me just point to it before we proceed. Namely, you're used in limits to setting x equal to 0. And this looks like we're multiplying, dividing by 0, exactly the thing which makes all proofs wrong in all kinds of algebraic situations and so on and so forth. You've been taught that that never works. Right? But somehow these limiting tricks have found a way around this and let me just make explicit what it is. In this limit we never are using x = x0. That's exactly the one value of x that we don't consider in this limit. That's how limits are cooked up. And that's sort of been the themes so far today, is that we don't have to consider that and so this multiplication and division by this number is legal. It may be small, this number, but it's always non-zero. So this really works, and it's really true, and we just checked that a differentiable function is continuous.

So I'm gonna have to carry out these limits, which are very interesting / limits next time. But let's hang on for one second to see if there any questions before we stop. Yeah, there is a question.

Student: [INAUDIBLE]

Professor: Repeat this proof right here? Just say again.

Student: [INAUDIBLE]

Professor: Okay, so there are two steps to the proof and the step that you're asking about is the first step. Right? And what I'm saying is if you have a number, and you multiply it by 10 / 10 it's the same number. If you multiply it by 3 / 3 it's the same number. 2 / 2, 1 / 1, and so on. So it is okay to change this to this, it's exactly the same thing. That's the first step. Yes?

Student: [INAUDIBLE]

Professor: Shhhh... The question was how does the proof, how does this line, yeah where the question mark is. So what I checked was that this number which is on the left hand side is equal to this very long complicated number which is equal to this number which is equal to this number. And so I've checked that this number is equal to because the last thing is 0. This is equal to that is equal to that is equal to 0. And that's the proof. Yes?

Student: [INAUDIBLE]

Professor: So that's a different question. Okay, so the hypothesis of differentiability I use because this limit is equal to this number. That that limit exits. That's how I use the hypothesis of the theorem. The conclusion of the theorem is the same as this because being continuous is the same as limit as x goes to x0 of f(x) = f(x0). That's the definition of continuity. And I subtracted f(x0) from both sides to get this as being the same thing. So this claim is continuity and it's the same as this question here. Last question.

Student: How did you get the 0 [INAUDIBLE]

Professor: How did we get the 0 from this? So the claim that is being made, so the claim is why is this tending to that. So for example, I'm going to have to erase something to explain that. So the claim is that the limit as x goes to x0 of x - x0 = 0. That's what I'm claiming. Okay, does that answer your question? Okay. All right. Ask me other stuff after lecture.

 

JONATHAN'S PARTY

Friday 13,2011. we celebrated the seventh birthday of Jonathan.
we bought pizza, bread and sodas for celebrated this wonderful moment; Jonathan was happy, so him sisters

NOT AFRAID

I like the songs of eminem, and I have choose not afraid for to share with you.
the lyric is:

I'm not afraid to take a stand
Everybody come take my hand
We'll walk this road together, through the storm
Whatever weather, cold or warm
Just let you know that, you're not alone
Holla if you feel that you've been down the same road

Yeah, It's been a ride...
I guess I had to go to that place to get to this one
Now some of you might still be in that place
If you're trying to get out, just follow me
I'll get you there

You can try and read my lyrics off of this paper before I lay 'em
But you won't take this thing out these words before I say 'em
Cause ain't no way I'm let you stop me from causing mayhem
When I say 'em or do something I do it, I don't give a damn
What you think, I'm doing this for me, so fuck the world
Feed it beans, it's gassed up, if a thing's stopping me
I'mma be what I set out to be, without a doubt undoubtedly
And all those who look down on me I'm tearing down your balcony
No if ands or buts don't try to ask him why or how can he
From Infinite down to the last Relapse album he's still shit'n
Whether he's on salary, paid hourly
Until he bows out or he shit's his bowels out of him
Whichever comes first, for better or worse
He's married to the game, like a fuck you for christmas
His gift is a curse, forget the earth he's got the urge
To pull his dick from the dirt and fuck the universe

I'm not afraid to take a stand
Everybody come take my hand
We'll walk this road together, through the storm
Whatever weather, cold or warm
Just let you know that, you're not alone
Holla if you feel that you've been down the same road

Ok quit playin' with the scissors and shit, and cut the crap
I shouldn't have to rhyme these words in the rhythm for you to know it's a rap
You said you was king, you lied through your teeth
For that fuck your fillings, instead of getting crowned you're getting capped
And to the fans, I'll never let you down again, I'm back
I promise to never go back on that promise, in fact
Let's be honest, that last Relapse CD was "ehhhh"
Perhaps I ran them accents into the ground

Relax, I ain't going back to that now
All I'm tryna say is get back, click-clack BLAOW
Cause I ain't playin' around
There's a game called circle and I don't know how
I'm way too up to back down
But I think I'm still tryna figure this crap out
Thought I had it mapped out but I guess I didn't
This fucking black cloud still follow's me around
But it's time to exercise these demons
These motherfuckers are doing jumping jacks now!

(Hook)

I'm not afraid to take a stand
Everybody come take my hand
We'll walk this road together, through the storm
Whatever weather, cold or warm
Just let you know that, you're not alone
Holla if you feel that you've been down the same road

(Bridge)

And I just can't keep living this way
So starting today, I'm breaking out of this cage
I'm standing up, Imma face my demons
I'm manning up, Imma hold my ground
I've had enough, now I'm so fed up
Time to put my life back together right now

(Verse 3)

It was my decision to get clean, I did it for me
Admittedly I probably did it subliminally for you
So I could come back a brand new me, you helped see me through
And don't even realise what you did, believe me you
I been through the ringer, but they can do little to the middle finger
I think I got a tear in my eye, I feel like the king of
My world, haters can make like bees with no stingers, and drop dead
No more beef flingers, no more drama from now on, I promise
To focus soley on handling my responsibility's as a father
So I solemnly swear to always treat this roof like my daughters and raise it
You couldn't lift a single shingle on it
Cause the way I feel, I'm strong enough to go to the club
Or the corner pub and lift the whole liquor counter up
Cause I'm raising the bar, I shoot for the moon
But I'm too busy gazing at stars, I feel amazing and

(Hook)

I'm not afraid to take a stand
Everybody come take my hand
We'll walk this road together, through the storm
Whatever weather, cold or warm
Just let you know that, you're not alone
Holla if you feel that you've been down the same road

TRANSLATION

I can't stay the three days during the event "flisol" but my classmates told me the homework and now I present you the translation of the look of love. Really I don't like.


The look of love is in your eyes
A look your smile can't disguise
The look of love is saying so much more than just words could ever say
And what my heart has heard, well it takes my breath away

I can hardly wait to hold you, feel my arms around you
How long I have waited
Waited just to love you, now that I have found you

You've got the
Look of love, it's on your face
A look that time can't erase
Be mine tonight, let this be just the start of so many nights like this
Let's take a lover's vow and then seal it with a kiss

I can hardly wait to hold you, feel my arms around you
How long I have waited
Waited just to love you, now that I have found you
Don't ever go
Don't ever go
I can hardly wait to hold you, feel my arms around you
How long I have waited
Waited just to love you, now that I have found you
Don't ever go
Don't ever go

My Experience

My experience during the event “flisol”

The last event celebrated in Chilpancingo Guerrero at April 9 2011, during this event we learned use and install linux, in effect was created flisol that motivate the use and the distribution of free software.

I think that linux is better that windows, in all aspects, I have windows in my machine but seldom use it, so I’ve linux and I prefer use it.

In the event named “flisol” because flisol mean “festival latinoamericano de instalacion de software libre”. I met many engineers that used many operative systems for example (windows, linux, mac, wifiway, etc.)  but they prefer use linux. Me too

Many students were interesting in linux, the engineer told us the advantages and the disadvantages that are few. The versions of linux are different and are adaptable to the activities’ person.  I learned use many applications in linux for example wine, compiz, kazam , gimp, loobi, virtual box, geogebra, mysql, php, apache and the most important wifiway. I couldn’t stay the there days just one day the Saturday, I couldn’t go to the disco and the zoo but one day was enough for knew more about linux and got the mails of the professors for give them my dudes about the free software.

Finally I was happy for this event and I wish go again, because was fantastic.

DIFERENCES BETWEEN MOODLE AND BLOGSPOT

MOODLE

In the conference of Flisol celebrated at April 9, 2011; We learned use moodle that is a software package for producing Internet-based courses and web sites. It is a global development project designed to support a social constructionist framework of education.

For me this software is very interesting and helpful for learn and development knowledge based on mathematics. So I will introduce you a short history about moodle.

Moodle is provided freely as Open Source software (under the GNU Public License). Basically this means Moodle is copyrighted, but that you have additional freedoms. You are allowed to copy, use and modify Moodle provided that you agree to: provide the source to others; not modify or remove the original license and copyrights, and apply this same license to any derivative work. Read the license for full details and please contact the copyright holder directly if you have any questions.

Moodle can be installed on any computer that can run PHP, and can support an SQL type database (for example MySQL). It can be run on Windows and Mac operating systems and many flavors of linux (for example Red Hat or Debian GNU). There are many knowledgeable Moodle Partners to assist you, even host your Moodle site.

The word Moodle was originally an acronym for Modular Object-Oriented Dynamic Learning Environment, which is mostly useful to programmers and education theorists. It's also a verb that describes the process of lazily meandering through something, doing things as it occurs to you to do them, an enjoyable tinkering that often leads to insight and creativity. As such it applies both to the way Moodle was developed, and to the way a student or teacher might approach studying or teaching an online course. Anyone who uses Moodle is a Moodler.

BLOGSPOT

A blog is a type of website or part of a website. Blogs are usually maintained by an individual with regular entries of commentary, descriptions of events, or other material such as graphics or video. Entries are commonly displayed in reverse-chronological order. Blog can also be used as a verb, meaning to maintain or add content to a blog.

Our homework is based on the differences between blogspot and moodle. Particularly I think that both are really good, because moodle and blogspot have different structure, but If you want teach or learn a topic of whatever subject you should use moodle; and if you want write or read descriptions of events, some pictures or videos you should use blogspot.

COLD PLAY

        Yo solía gobernar el mundo Los mares se alzaban cuando yo lo ordenaba Ahora en la mañana yo barro solo barro las calles que solía poseer Yo solía tirar el dado Sentir el miedo en los ojos de mi enemigo Escuchaba como la gente cantaba: "Ahora el viejo rey está muerto, ¡larga vida al rey!" Un minuto yo tenía la llave Al siguiente las paredes se cerraban en mí Y descubrí que mis castillos estaban construidos Sobre pilares de sal y pilares de arena Escucho las campanas de Jerusalen sonando Los coros del Calvario Romano están cantando Son mi espejo, mi espada y mi escudo Mis misioneros en un campo extranjero Por alguna razon que no puedo explicar Una vez que sabes que nunca hubo una palabra honesta Así era cuando yo gobernaba el mundo Fue el viento loco y salvaje Que tiró las puertas para dejarme entrar Ventanas rotas y el sonido de tambores La gente no podía creer en lo que me convertí Los revolucionarios esperan Mi cabeza en charola de plata Solo una marioneta en una cuerda solitaria Oh ¿Quien podría querer ser rey? Escucho las campanas de Jerusalen sonando Los coros del Calvario Romano están cantando Son mi espejo, mi espada y mi escudo Mis misioneros en un campo extranjero Por alguna razon que no puedo explicar Yo se que San Pedro dirá mi nombre Nunca hubo una palabra honesta Pero así era cuando yo gobernaba el mundo Escucho las campanas de Jerusalen sonando Los coros del Calvario Romano están cantando Son mi espejo, mi espada y mi escudo Mis misioneros en un campo extranjero Por alguna razon que no puedo explicar Yo se que San Pedro dirá mi nombre Nunca hubo una palabra honesta Pero así era cuando yo gobernaba el mundo.

RATE OF CHANGE

RATE OF CHANGE

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Professor: So, again welcome to 18.01. We're getting started today with what we're calling Unit One, a highly imaginative title. And it's differentiation. So, let me first tell you, briefly, what's in store in the next couple of weeks. The main topic today is what is a derivative. And, we're going to look at this from several different points of view, and the first one is the geometric interpretation. That's what we'll spend most of today on. And then, we'll also talk about a physical interpretation of what a derivative is.
And then there's going to be something else which I guess is maybe the reason why Calculus is so fundamental, and why we always start with it in most science and engineering schools, which is the importance of derivatives, of this, to all measurements. So that means pretty much every place. That means in science, in engineering, in economics, in political science, etc. Polling, lots of commercial applications, just about everything.
Now, that's what we'll be getting started with, and then there's another thing that we're gonna do in this unit, which is we're going to explain how to differentiate anything. So, how to differentiate any function you know. And that's kind of a tall order, but let me just give you an example. If you want to take the derivative - this we'll see today is the notation for the derivative of something - of some messy function like e ^ x arctanx. We'll work this out by the end of this unit.
All right? Anything you can think of, anything you can write down, we can differentiate it. All right, so that's what we're gonna do, and today as I said, we're gonna spend most of our time on this geometric interpretation. So let's begin with that.
So here we go with the geometric interpretation of derivatives. And, what we're going to do is just ask the geometric problem of finding the tangent line to some graph of some function at some point. Which is to say (x0, y0). So that's the problem that we're addressing here. Alright, so here's our problem, and now let me show you the solution. So, well, let's graph the function. Here's it's graph. Here's some point. All right, maybe I should draw it just a bit lower. So here's a point P. Maybe it's above the point x0. x0, by the way, this was supposed to be an x0. That was some fixed place on the x-axis. And now, in order to perform this mighty feat, I will use another color of chalk. How about red? OK. So here it is. There's the tangent line, Well, not quite straight. Close enough. All right? I did it.
That's the geometric problem. I achieved what I wanted to do, and it's kind of an interesting question, which unfortunately I can't solve for you in this class, which is, how did I do that? That is, how physically did I manage to know what to do to draw this tangent line? But that's what geometric problems are like. We visualize it. We can figure it out somewhere in our brains. It happens. And the task that we have now is to figure out how to do it analytically, to do it in a way that a machine could just as well as I did in drawing this tangent line.
So, what did we learn in high school about what a tangent line is? Well, a tangent line has an equation, and any line through a point has the equation y - y0 is equal to m the slope, times x - x0.
So here's the equation for that line, and now there are two pieces of information that we're going to need to work out what the line is. The first one is the point. That's that point P there. And to specify P, given x, we need to know the level of y, which is of course just f(x0). That's not a calculus problem, but anyway that's a very important part of the process. So that's the first thing we need to know. And the second thing we need to know is the slope. And that's this number m. And in calculus we have another name for it. We call it f prime of x0. Namely, the derivative of f. So that's the calculus part. That's the tricky part, and that's the part that we have to discuss now. So just to make that explicit here, I'm going to make a definition, which is that f '(x0) , which is known as the derivative, of f, at x0, is the slope of the tangent line to y = f (x) at the point, let's just call it p.
All right? So, that's what it is, but still I haven't made any progress in figuring out any better how I drew that line. So I have to say something that's more concrete, because I want to be able to cook up what these numbers are. I have to figure out what this number m is. And one way of thinking about that, let me just try this, so I certainly am taking for granted that in sort of non-calculus part that I know what a line through a point is. So I know this equation. But another possibility might be, this line here, how do I know - well, unfortunately, I didn't draw it quite straight, but there it is - how do I know that this orange line is not a tangent line, but this other line is a tangent line? Well, it's actually not so obvious, but I'm gonna describe it a little bit. It's not really the fact, this thing crosses at some other place, which is this point Q. But it's not really the fact that the thing crosses at two place, because the line could be wiggly, the curve could be wiggly, and it could cross back and forth a number of times. That's not what distinguishes the tangent line.
So I'm gonna have to somehow grasp this, and I'll first do it in language. And it's the following idea: it's that if you take this orange line, which is called a secant line, and you think of the point Q as getting closer and closer to P, then the slope of that line will get closer and closer to the slope of the red line. And if we draw it close enough, then that's gonna be the correct line. So that's really what I did, sort of in my brain when I drew that first line.
And so that's the way I'm going to articulate it first. Now, so the tangent line is equal to the limit of so called secant lines PQ, as Q tends to P. And here we're thinking of P as being fixed and Q as variable. All right? Again, this is still the geometric discussion, but now we're gonna be able to put symbols and formulas to this computation. And we'll be able to work out formulas in any example.
So let's do that. So first of all, I'm gonna write out these points P and Q again. So maybe we'll put P here and Q here. And I'm thinking of this line through them. I guess it was orange, so we'll leave it as orange. All right. And now I want to compute its slope. So this, gradually, we'll do this in two steps. And these steps will introduce us to the basic notations which are used throughout calculus, including multi-variable calculus, across the board. So the first notation that's used is you imagine here's the x-axis underneath, and here's the x0, the location directly below the point P. And we're traveling here a horizontal distance which is denoted by delta x. So that's delta x, so called. And we could also call it the change in x.
So that's one thing we want to measure in order to get the slope of this line PQ. And the other thing is this height. So that's this distance here, which we denote delta f, which is the change in f. And then, the slope is just the ratio, delta f / delta x. So this is the slope of the secant. And the process I just described over here with this limit applies not just to the whole line itself, but also in particular to its slope. And the way we write that is the limit as delta x goes to 0. And that's going to be our slope. So this is slope of the tangent line.
OK. Now, This is still a little general, and I want to work out a more usable form here, a better formula for this. And in order to do that, I'm gonna write delta f, the numerator more explicitly here. The change in f, so remember that the point P is the point (x0, f(x0)). All right, that's what we got for the formula for the point. And in order to compute these distances and in particular the vertical distance here, I'm gonna have to get a formula for Q as well. So if this horizontal distance is delta x, then this location is (x0 delta x). And so the point above that point has a formula, which is (x0 delta x, f(x0 and this is a mouthful, delta x)).
All right, so there's the formula for the point Q. Here's the formula for the point P. And now I can write a different formula for the derivative, which is the following: so this f'(x0) , which is the same as m, is going to be the limit as delta x goes to of the change in f, well the change in f is the value of f at the upper point here, which is (x0 delta x), and minus its value at the lower point P, which is f(x0), divided by delta x. All right, so this is the formula. I'm going to put this in a little box, because this is by far the most important formula today, which we use to derive pretty much everything else. And this is the way that we're going to be able to compute these numbers.
So let's do an example. This example, we'll call this example one. We'll take the function f(x) , which is 1/x . That's sufficiently complicated to have an interesting answer, and sufficiently straightforward that we can compute the derivative fairly quickly. So what is it that we're gonna do here? All we're going to do is we're going to plug in this formula here for that function. That's all we're going to do, and visually what we're accomplishing is somehow to take the hyperbola, and take a point on the hyperbola, and figure out some tangent line. That's what we're accomplishing when we do that. So we're accomplishing this geometrically but we'll be doing it algebraically. So first, we consider this difference delta f / delta x and write out its formula.
So I have to have a place. So I'm gonna make it again above this point x0, which is the general point. We'll make the general calculation. So the value of f at the top, when we move to the right by f(x), so I just read off from this, read off from here. The formula, the first thing I get here is 1 / x0 delta x. That's the left hand term. Minus 1 / x0, that's the right hand term. And then I have to divide that by delta x. OK, so here's our expression. And by the way this has a name. This thing is called a difference quotient. It's pretty complicated, because there's always a difference in the numerator. And in disguise, the denominator is a difference, because it's the difference between the value on the right side and the value on the left side here. OK, so now we're going to simplify it by some algebra.
So let's just take a look. So this is equal to, let's continue on the next level here. This is equal to 1 / delta x times... All I'm going to do is put it over a common denominator. So the common denominator is (x0 delta x)x0. And so in the numerator for the first expressions I have x0, and for the second expression I have x0 delta x. So this is the same thing as I had in the numerator before, factoring out this denominator. And here I put that numerator into this more amenable form.
And now there are two basic cancellations. The first one is that x0 and x0 cancel, so we have this. And then the second step is that these two expressions cancel, the numerator and the denominator. Now we have a cancellation that we can make use of. So we'll write that under here. And this is equals -1 / (x0 delta x)x0. And then the very last step is to take the limit as delta x tends to 0, and now we can do it. Before we couldn't do it. Why? Because the numerator and the denominator gave us / 0. But now that I've made this cancellation, I can pass to the limit. And all that happens is I set this delta x to 0, and I get -1/x0^2. So that's the answer. All right, so in other words what I've shown - let me put it up here- is that f'(x0) = -1/x0^2.
Now, let's look at the graph just a little bit to check this for plausibility, all right? What's happening here, is first of all it's negative. It's less than 0, which is a good thing. You see that slope there is negative. That's the simplest check that you could make. And the second thing that I would just like to point out is that as x goes to infinity, that as we go farther to the right, it gets less and less steep. So as x0 goes to infinity, less and less steep. So that's also consistent here, when x0 is very large, this is a smaller and smaller number in magnitude, although it's always negative. It's always sloping down. All right, so I've managed to fill the boards. So maybe I should stop for a question or two. Yes?
Student: [INAUDIBLE]
Professor: So the question is to explain again this limiting process. So the formula here is we have basically two numbers. So in other words, why is it that this expression, when delta x tends to 0, is equal to -1/x0^2 ? Let me illustrate it by sticking in a number for x0 to make it more explicit. All right, so for instance, let me stick in here for x0 the number 3. Then it's -1 / (3 delta x)3. That's the situation that we've got. And now the question is what happens as this number gets smaller and smaller and smaller, and gets to be practically 0? Well, literally what we can do is just plug in there, and you get (3 0)3 in the denominator. Minus one in the numerator. So this tends to -1/9 (over 3^2). And that's what I'm saying in general with this extra number here. Other questions? Yes.
Student: [INAUDIBLE]
Professor: So the question is what happened between this step and this step, right? Explain this step here. Alright, so there were two parts to that. The first is this delta x which is sitting in the denominator, I factored all the way out front. And so what's in the parentheses is supposed to be the same as what's in the numerator of this other expression. And then, at the same time as doing that, I put that expression, which is the difference of two fractions, I expressed it with a common denominator. So in the denominator here, you see the product of the denominators of the two fractions. And then I just figured out what the numerator had to be without really... Other questions? OK.
So I claim that on the whole, calculus gets a bad rap, that it's actually easier than most things. But there's a perception that it's harder. And so I really have a duty to give you the calculus made harder story here. So we have to make things harder, because that's our job. And this is actually what most people do in calculus, and it's the reason why calculus has a bad reputation. So the secret is that when people ask problems in calculus, they generally ask them in context. And there are many, many other things going on. And so the little piece of the problem which is calculus is actually fairly routine and has to be isolated and gotten through. But all the rest of it, relies on everything else you learned in mathematics up to this stage, from grade school through high school. So that's the complication. So now we're going to do a little bit of calculus made hard. By talking about a word problem.
We only have one sort of word problem that we can pose, because all we've talked about is this geometry point of view. So far those are the only kinds of word problems we can pose. So what we're gonna do is just pose such a problem. So find the areas of triangles, enclosed by the axes and the tangent to y = 1/x. OK, so that's a geometry problem. And let me draw a picture of it. It's practically the same as the picture for example one. We only consider the first quadrant. Here's our shape. All right, it's the hyperbola. And here's maybe one of our tangent lines, which is coming in like this. And then we're trying to find this area here. Right, so there's our problem. So why does it have to do with calculus? It has to do with calculus because there's a tangent line in it, so we're gonna need to do some calculus to answer this question. But as you'll see, the calculus is the easy part.
So let's get started with this problem. First of all, I'm gonna label a few things. And one important thing to remember of course, is that the curve is y = 1/x. That's perfectly reasonable to do. And also, we're gonna calculate the areas of the triangles, and you could ask yourself, in terms of what? Well, we're gonna have to pick a point and give it a name. And since we need a number, we're gonna have to do more than geometry. We're gonna have to do some of this analysis just as we've done before. So I'm gonna pick a point and, consistent with the labeling we've done before, I'm gonna to call it (x0, y0). So that's almost half the battle, having notations, x and y for the variables, and x0 and y0, for the specific point.
Now, once you see that you have these labellings, I hope it's reasonable to do the following. So first of all, this is the point x0, and over here is the point y0. That's something that we're used to in graphs. And in order to figure out the area of this triangle, it's pretty clear that we should find the base, which is that we should find this location here. And we should find the height, so we need to find that value there. Let's go ahead and do it. So how are we going to do this? Well, so let's just take a look. So what is it that we need to do? I claim that there's only one calculus step, and I'm gonna put a star here for this tangent line. I have to understand what the tangent line is. Once I've figured out what the tangent line is, the rest of the problem is no longer calculus. It's just that slope that we need. So what's the formula for the tangent line? Put that over here. it's going to be y - y0 is equal to, and here's the magic number, we already calculated it. It's in the box over there. It's -1/x0^2 ( x - x0). So this is the only bit of calculus in this problem. But now we're not done. We have to finish it. We have to figure out all the rest of these quantities so we can figure out the area.
All right. So how do we do that? Well, to find this point, this has a name. We're gonna find the so called x-intercept. That's the first thing we're going to do. So to do that, what we need to do is to find where this horizontal line meets that diagonal line. And the equation for the x-intercept is y = 0. So we plug in y = 0, that's this horizontal line, and we find this point. So let's do that into star. We get minus, oh one other thing we need to know. We know that y0 is f(x0) , and f(x) is 1/x , so this thing is 1/x0. And that's equal to -1/x0^2. And here's x, and here's x0. All right, so in order to find this x value, I have to plug in one equation into the other.
So this simplifies a bit. This is -x/x0^2. And this is plus 1/x0 because the x0 and x0^2 cancel somewhat. And so if I put this on the other side, I get x / x0^2 is equal to 2 / x0. And if I then multiply through - so that's what this implies - and if I multiply through by x0^2 I get x = 2x0.
OK, so I claim that this point weve just calculated it's 2x0. Now, I'm almost done. I need to get the other one. I need to get this one up here. Now I'm gonna use a very big shortcut to do that. So the shortcut to the y-intercept is to use symmetry. All right, I claim I can stare at this and I can look at that, and I know the formula for the y-intercept. It's equal to 2y0. All right. That's what that one is. So this one is 2y0. And the reason I know this is the following: so here's the symmetry of the situation, which is not completely direct. It's a kind of mirror symmetry around the diagonal. It involves the exchange of (x, y) with (y, x); so trading the roles of x and y. So the symmetry that I'm using is that any formula I get that involves x's and y's, if I trade all the x's and replace them by y's and trade all the y's and replace them by x's, then I'll have a correct formula on the other ways. So if everywhere I see a y I make it an x, and everywhere I see an x I make it a y, the switch will take place. So why is that? That's just an accident of this equation. That's because, so the symmetry explained... is that the equation is y= 1 / x. But that's the same thing as xy = 1, if I multiply through by x, which is the same thing as x = 1/y. So here's where the x and the y get reversed. OK now if you don't trust this explanation, you can also get the y-intercept by plugging x = into the equation star. OK? We plugged y = in and we got the x value. And you can do the same thing analogously the other way.
All right so I'm almost done with the geometry problem, and let's finish it off now. Well, let me hold off for one second before I finish it off. What I'd like to say is just make one more tiny remark. And this is the hardest part of calculus in my opinion. So the hardest part of calculus is that we call it one variable calculus, but we're perfectly happy to deal with four variables at a time or five, or any number. In this problem, I had an x, a y, an x0 and a y0. That's already four different things that have various relationships between them. Of course the manipulations we do with them are algebraic, and when we're doing the derivatives we just consider what's known as one variable calculus. But really there are millions of variable floating around potentially. So that's what makes things complicated, and that's something that you have to get used to. Now there's something else which is more subtle, and that I think many people who teach the subject or use the subject aren't aware, because they've already entered into the language and they're so comfortable with it that they don't even notice this confusion. There's something deliberately sloppy about the way we deal with these variables.
The reason is very simple. There are already four variables here. I don't wanna create six names for variables or eight names for variables. But really in this problem there were about eight. I just slipped them by you. So why is that? Well notice that the first time that I got a formula for y0 here, it was this point. And so the formula for y0, which I plugged in right here, was from the equation of the curve. y0 = 1 / x0. The second time I did it, I did not use y = 1 / x. I used this equation here, so this is not y = 1/x. That's the wrong thing to do. It's an easy mistake to make if the formulas are all a blur to you and you're not paying attention to where they are on the diagram.
You see that x-intercept calculation there involved where this horizontal line met this diagonal line, and y = represented this line here. So the sloppines is that y means two different things. And we do this constantly because it's way, way more complicated not to do it. It's much more convenient for us to allow ourselves the flexibility to change the role that this letter plays in the middle of a computation. And similarly, later on, if I had done this by this more straightforward method, for the y-intercept, I would have set x equal to 0. That would have been this vertical line, which is x = 0. But I didn't change the letter x when I did that, because that would be a waste for us. So this is one of the main confusions that happens. If you can keep yourself straight, you're a lot better off, and as I say this is one of the complexities.
All right, so now let's finish off the problem. Let me finally get this area here. So, actually I'll just finish it off right here. So the area of the triangle is, well it's the base times the height. The base is 2x0 the height is 2y0, and a half of that. So it's 1/2( 2x0)(2y0) , which is (2x0)(y0), which is, lo and behold, 2. So the amusing thing in this case is that it actually didn't matter what x0 and y0 are. We get the same answer every time. That's just an accident of the function 1 / x. It happens to be the function with that property.
All right, so we have some more business today, some serious business. So let me continue. So, first of all, I want to give you a few more notations. And these are just other notations that people use to refer to derivatives. And the first one is the following: we already wrote y = f(x). And so when we write delta y, that means the same thing as delta f. That's a typical notation. And previously we wrote f' for the derivative, so this is Newton's notation for the derivative. But there are other notations. And one of them is df/dx, and another one is dy/ dx, meaning exactly the same thing. And sometimes we let the function slip down below so that becomes d / dx (f) and d/ dx(y) . So these are all notations that are used for the derivative, and these were initiated by Leibniz. And these notations are used interchangeably, sometimes practically together. They both turn out to be extremely useful. This one omits - notice that this thing omits- the underlying base point, x0. That's one of the nuisances. It doesn't give you all the information. But there are lots of situations like that where people leave out some of the important information, and you have to fill it in from context. So that's another couple of notations.
So now I have one more calculation for you today. I carried out this calculation of the derivative of the function 1 / x. I wanna take care of some other powers. So let's do that.
So Example 2 is going to be the function f(x) = x^n. n = 1, 2, 3; one of these guys. And now what we're trying to figure out is the derivative with respect to x of x^n in our new notation, what this is equal to. So again, we're going to form this expression, delta f / delta x. And we're going to make some algebraic simplification. So what we plug in for delta f is ((x delta x)^n - x^n)/delta x. Now before, let me just stick this in then I'm gonna erase it. Before, I wrote x0 here and x0 there. But now I'm going to get rid of it, because in this particular calculation, it's a nuisance. I don't have an x floating around, which means something different from the x0. And I just don't wanna have to keep on writing all those symbols. It's a waste of blackboard energy. There's a total amount of energy, and I've already filled up so many blackboards that, there's just a limited amount. Plus, I'm trying to conserve chalk. Anyway, no 0's. So think of x as fixed. In this case, delta x moves and x is fixed in this calculation. All right now, in order to simplify this, in order to understand algebraically what's going on, I need to understand what the nth power of a sum is. And that's a famous formula. We only need a little tiny bit of it, called the binomial theorem. So, the binomial theorem which is in your text and explained in an appendix, says that if you take the sum of two guys and you take them to the nth power, that of course is (x delta x) multiplied by itself n times. And so the first term is x^n, that's when all of the n factors come in. And then, you could have this factor of delta x and all the rest x's. So at least one term of the form (x^(n-1))delta x. And how many times does that happen? Well, it happens when there's a factor from here, from the next factor, and so on, and so on, and so on. There's a total of n possible times that that happens. And now the great thing is that, with this alone, all the rest of the terms are junk that we won't have to worry about. So to be more specific, there's a very careful notation for the junk. The junk is what's called big O((delta x)^2). What that means is that these are terms of order, so with (delta x)^2, (delta x)^3 or higher. All right, that's how. Very exciting, higher order terms.
OK, so this is the only algebra that we need to do, and now we just need to combine it together to get our result. So, now I'm going to just carry out the cancellations that we need. So here we go. We have delta f / delta x, which remember was 1 / delta x times this, which is this times, now this is (x^n nx^(n-1) delta x this junk term) - x^n. So that's what we have so far based on our previous calculations. Now, I'm going to do the main cancellation, which is this. All right. So, that's 1/delta x( nx^(n-1) delta x this term here). And now I can divide in by delta x. So I get nx^(n-1) now it's O(delta x). There's at least one factor of delta x not two factors of delta x, because I have to cancel one of them. And now I can just take the limit. And the limit this term is gonna be 0. That's why I called it junk originally, because it disappears. And in math, junk is something that goes away. So this tends to, as delta x goes to 0, nx ^ (n-1). And so what I've shown you is that d/dx of x to the n minus - sorry -n, is equal to nx^(n-1).
So now this is gonna be super important to you right on your problem set in every possible way, and I want to tell you one thing, one way in which it's very important. One way that extends it immediately. So this thing extends to polynomials. We get quite a lot out of this one calculation. Namely, if I take d / dx of something like (x^3 5x^10) that's gonna be equal to 3x^2, that's applying this rule to x^3. And then here, I'll get 5*10 so 50x^9. So this is the type of thing that we get out of it, and we're gonna make more hay with that next time. Question. Yes. I turned myself off. Yes?
Student: [INAUDIBLE]
Professor: The question was the binomial theorem only works when delta x goes to 0. No, the binomial theorem is a general formula which also specifies exactly what the junk is. It's very much more detailed. But we only needed this part. We didn't care what all these crazy terms were. It's junk for our purposes now, because we don't happen to need any more than those first two terms. Yes, because delta x goes to 0. OK, see you next time.

ENGLISH NUMBERS

In this class, We learned the parts of speech, which are: subject, verb and predicate, the combination of two irregular verbs: to be and have ... pronouns: I, you, him, her, us, you ... which are singular pronouns (I, you, he, she) and plural (we, you and they)


Conjugation of the verb be in simple present:


I am
YOU are
HE is
IT is
WE are
YOU are
THEY are


Examples about this time:


I am looking for my keys.
you are next to me.
he is eating pozole.
she is seeing me.
it is under the table.
we are in the school.
they are doing their homework.


Conjugation of the verb be in past time:


I was
YOU were
HE was
SHE was
IT was
WE were
YOU were
THEY were


Examples about this time:


I was lost at the park.
YOU were swimming versus me.
HE was sick yesterday.
SHE was serious with me.
IT was lost.
WE were nervous about the exam.
THEY were hungry and upset.


Conjugation of the verb be in future time:


I will
YOU will
HE will
SHE will
IT will
WE will
THEY will


Note: Will or going to is used to express future time.
the use of shall with I or WE to express future time is possible but uncommon in American English. Shall is used more frequently in British than in American English.
In speech, going to is often pronounced "gonna".
the contracted form of will + not is won't.


Examples about this time:


I won't be here tomorrow.
YOU will help her.
Jack will finish his mathematic homework tomorrow.
She is going to graduate in June.
According to the whether report, IT is going to be cloudy tomorrow.
WE shall come soon.don't get impatient
THEY won't buy any beers.




Summary Chart Of Verb Tenses


*Simple Present:
Tom studies every day.


*Present Progressive:
Tom is studying right now.


*Present Perfect:
Tom has already studied chapter one.


*Present Perfect Progressive:
Tom has been studying for two hours.


*Simple Past:
Tom studied last night.


*Past Progressive:
Tom was studying when they came.


*Past Perfect:
Tom had already studied chapter one before he began studying chapter two.


*Past Perfect progressive:
Tom had been studying for two hours before his friends came.


*Simple Future:
Tom will study tomorrow.


*Future Progressive:
Tom will be studying when you come.


*Future Perfect:
Tom will already have studied chapter four before he studies chapter five.


*Future Perfect progressive:
Tom will have been studying for two hours by the time his roommate gets home.


conjugation of the verb and auxiliar have:
I have
you have
he has
she has
we have
you have
they have


Conjugation of the verb and auxiliar have in the past time:
I had
you had
he had
she had
we had
you had
they had