Why is calculus going to be so much fun?
Welcome to calculus one. I'm glad you're in this course, or maybe quest for understanding calculus. Alright, we really are on a quest, you know, we're taking thousands of years of human ingenuity. Right? The sort of triumph of humankind to be able to understand numbers and functions. All of these insights that humans have had for thousands of years, and we're just distilling all those down to short videos and exercises. You know, and it's not going to be easy, all right. To take thousands of years of insight, and try to cram that in to short online learning modules, it's going to be challenging to understand all this stuff. But I think the payoff is worth it. You know? Anf for me the payoff is worth it, because the concepts in calculus are just really cool.
Calculus is a huge subject, and it combines just a lot of different topics. You know, a calculus course and this course, introduces things like functions and limits, talk about infinity, talk about derivative, talk about area and integrals. And the big surprise is that all of these seemingly unrelated topics, end up being related. Right? There's connections between all of these different concepts that are appearing in calculus. And I think those kinds of connections are just very exciting.
Another cool thing about calculus, is that we get to do a lot of neat computations. Alright, we're going to learn how to compute derivatives, compute areas. You know, but don't get me wrong. Right? At the end of this whole process, the goal isn't really to be able to compute things. Right? The goal is insights, it's understanding, it's some appreciation of the concepts, and how these concepts connect together. And doing the computations is certainly a prerequisite to being able to understand those concepts. But I really hope to, to dig deeper, you know, and to be able to see how all of these different ideas within calculus are related.
What is a function?
Hàm số biểu thị mối tương quan (quan hệ) giữa 2 đại lượng một cách toán học. Giả sử, bạn mua một cây bút thì trả 5 ngàn, 2 cây thì 10 ngàn, vậy thì bạn có thể suy ra số tiền bạn phải trả gấp 5 lần số cây bút bạn mua. Ở đây ta có một hàm số y = 5x với x là số bút bạn mua, và y là số tiền bạn phải trả. Tại sao phải dùng toán học ở đây? Rõ ràng, với ví dụ đơn giản như trên, ta không cần dùng toán học. Tuy nhiên, đối với các mối quan hệ phức tạp hơn, toán học là một công cụ hữu hiệu để thể hiện mối quan hệ. Chẳng hạn, nếu bạn mua 1 laptop với giá 9 triệu, 4 chiếc với giá 32 triệu, 9 chiếc với giá 63 triệu. Nếu bạn mua 16 laptop thì cần phải trả bao nhiêu? Mối quan hệ giữa số tiền phải trả và số laptop cần mua là gì? Một khả năng về mối quan hệ ở đây là $y = 10x - \sqrt{x^3}$, do đó số tiền phải trả cho 16 laptop là 116 triệu.
Calculus study functions and this course is no exception. We're going to be studying functions in this course, alright? Functions like f of x or the sine function or the square root function. And that raises a question what are functions?If If we're going to be studying functions, we should know what they are. And that question is pretty metaphysical. I mean what are numbers? What are numbers for? I mean that question doesn't make a lot of sense does it? I, you know, it's not like a number four is a physical object that I can go look at it and see if it's got polka dots or it's striped or something, right? But I know four objects when I see them, right? I sort understand what numbers do more than what they are in some metaphysical sense. The same is true about functions, right? I'm not really going to tell you what a function is. What I'm going to tell you is what a function does. This is what a function does. A function assigns to each number in its domain another number. And this definition doesn't say anything about how the function does that assignment. Let's see an example. Just make up some example. Suppose I've got some function, I'll call it f, f for function. Maybe this function assigns to the number two, the number four. So, I'll write f(2) is 4 or f(3) is 9 Or f(4) is 16 of f(5) is 25. I'm just making this up. This is a function. And I'm telling you what number it assigns to each number, right? f assigns the number two, the number four. f(3) is 9, right? So, f assigns to the number three, the number nine. Well, look, I'm not going to just list off every single assignment that f makes. So instead, one way to talk about these assignments is to use a rule, like f(x) = x^2. And this single rule explains how all of these assignments are made, alright? This rule said that f assigns to the number x, the number x^2. So in particular, f assigns the number five, five squared or 25, or f assigns the number four, four squared which is sixteen or f assigns the number three, three squared which is nine, alright? So, a lot of times, when you actually want to talk about how these assignments are being made, use some sort of rule like this: and you write f of x is something, to compute the output value, right? So, a function assigns to each number in its domain another number. One way to do that is with a rule. Of course, this definition of function involves another word, domain. What is a domain? Unless, I say otherwise, the domain consists of all numbers for which the rule makes sense, right? A function assigns to each number in its domain another number. And the domain is just the numbers that I can plug in. Let's take a look to see what I mean by this. Suppose I've got a function f(x) = 1 / x. So, that's a function given by a rule. It takes an input x and produces the output 1 / x, right? It assigns the number x, the number 1 / x. But this rule doesn't always make sense, right? I'm dividing by x, alright? And to divide by x, what do I need? Well, I must have that x is not zero, not permitted to divide by zero. So, I can plug in any number for x except for zero. And that's when this rule makes sense. So, I'll summarize that by saying that the domain of f is all real numbers except zero. So, a function takes some input and produces some output. That's what it does. But how is that supposed to make us feel, you know? How are we supposed to imagine that? Well, here's one metaphor that you could use to try to think about what a function is actually doing, right? You could imagine a, a conveyor belt with the function, you know, And you could imagine the numbers coming in. Boom. Being hit by the function and then going out transformed somehow, by whatever the rule is of the function. Here's what I'm talking about. I've got a conveyor belt here, I got this big box, and imagine if that big box is the function. And in here, I've got the input, imagine the number five. Let's see what this machine is going to do. Maybe this is the function f(x) = x^2 + x and I've got this number five here. And I'll start the conveyer belt going so the number five starts moving through the machine and when the machine is done processing it, it comes out the other side and now, it's the number 30, alright. Because f(5) is 5^2 + 5, which is 25 + 5 is 20.
So, we have seen how you can write down a function by using a, a rule involving these, these mathematical symbols, right? x^2 + x. You can also write down a function just using English words. So, let's see an example of that, just make up a new function here. I'll call it R(x) and I'll define ¡ R(x) to be thrice, that means three times, thrice the square root of x. And here, I've computed some, some values of the function, you know, like the function that four is six. Well, why is that? Well, R(4) would be three times the square root of four, which is three times two which is six. Okay. Now, of course, when I do the calculation this way, you know, it's not too surprising that instead of writing this out in words, thrice the square root of x, I mean I could have just written three times the square root of x. So, maybe you're not too impressed with this. The point is that you can define a function just by writing down what the function is supposed to do using English words. So, we've seen an example of a function that we defined just in terms of algebra, f(x) = x^2 + x. I've also seen an example of a function that I define entirely with just English words. We can kind of combine those two things, alright? We can use the English to sort of pick out different kinds of of algebra. Let's see an example of that now. So, here's another function I just made up. g(x) is x^22 if x is bigger than or equal to five. And twice x if x is less than five. The point here is that I'm using just a little bit of English. So, this word, if, in order to select based on how big x is, a different algebraic rule for calculating the function and this is a little abstract. Let me just do with some calculations and that might convince you, you know, how, how this notation is, this so-called piecewise notation works. So, what is g(1)? Well, let's take a look. One is less than five so that means I use this second rule for calculating g, so it's two times one which is two. What's g of, say, four? Well, four is still less than five so I use the second rule again. Twice the input, two times four and that means the output is eight. But what is, say, g(5)? Ooh. Five is not less than five. Five is greater than or equal to five. So, this if is telling me to use the first of the two rules for calculating g so I'm going to use five squared as the output and that will be 25. Or g(10). Well, ten is definitely bigger than or equal to five and it's bigger than five. So, I again use the first rule and the output to this function is computed as ten squared or 100. So, this is, you know, really more complicated than just using some algebra, right? I'm using these if statements to select which of these two rules g will use to compute its output. in principle, functions can be really complicated. I mean, all the examples we've seen are just doing various kinds of algebra. I mean, maybe different algebra, depending upon which value of x I'm plugging in. But it's all, you know, adding, subtracting, multiplying, dividing that sort of thing. But you can do really complicated things with, with functions. let's see an example of something that is you know really different than doing straight up algebra. I'll do a much crazier example. Alright. C(x) is the number of even digits in the number x, when x is a whole number. It is zero if x isn't a whole number, so otherwise. And it is a really different and crazy function, right? So, C of, say, 7236 is two, and why is that? This is a whole number so I use the first part of the rule. And the number of even digits in this number, well, I count them. That's odd, even, odd, even. So, two and six are the two even digits. The value of that function is two. Well, let's take a look at this number here, 60,202. That again is a whole number, so the value of the function at 60,202 is the number of even digits in x. And this number has how many even digits? One, two, three, four, five, they're all even digits so that value is five. Here's another example. Here's another 5-digit number. 53,531. Again, it's a whole number so I use the first part of the rule, the number of even digits. I count how many of these digits are even. Five is odd, odd, odd, odd, odd. There are no even digits. So, the value of this function is zero at this point. Now, if I plug in a number that's not a whole number like this, this is not a whole number, then I use the otherwise part of the definition. And the function at that point is just a zero. So, we can use English to define a function but you got to be careful, alright? Sometimes, when you write another definition of a function, you might write down something that's more ambiguous than, than you intended. So, let's try to define a function. Just making this stuff up, I'll call the function B(x) for bad. And I'll say that the value of B(x) is some rearrangement of the digits of x. And it's okay that I'm using English to define my functions. There's nothing wrong with that. But this definition is too ambiguous to be the definition of a function. here's a definition of what goes wrong. What is the function's value at 352? Well, the function takes its input and rearranges the digits somehow, alright? So, you might think that the functions value at 352 is 325. Because 325 is some rearrangement of the digits of 352, right, I took the five and the two, swapped their positions. But then, the functions value at 352 should also be 235 because 235 is also a rearrangement of 352. The function's value with 352 should also be 532 because 532 is some rearrangement of the digits of 352. This is terrible, alright? A function is suppose to take its input and produce unambiguously a single output value. But this so-called function takes this single input value and purportedly produces all these possible outputs. This thing here is not a function, alright? A function takes one input and produces one output.
When are two functions the same?
So, who knows what a function is, right? But I know what it does. It takes an input value, and produces an output value. And we've got a whole bunch of functions, right? And we can take these functions and start asking questions about them. What happens when you plug in a really big number, or a really small number? Or, or what happens when you plug in two numbers that are nearby each other? How are the outputs related, right? Those are the kinds of questions that are going to occupy us for the rest of the term. But even before we start thinking about questions like that, right? There are some things that we can still ask about functions. Like, how do you know when two functions are the same function? For instance, here's two functions. f(x) = (1+x)^2, g(x) = x^2 + 2x + 1. Are these the same function? Now, let's try. Look at the value like f(2). f(2) = (1+2)^2, start replacing the x by two, 1 + 2 = 3. 3^2 = 9. Well, what's what's g(2)? Well, g(2) would be 2^2 + 2 * 2 + 1. 2^2 = 4, 2 * 2 = 4 + 1, 4 + 4 = 8 + 1 = 9. Look, f and g, when I plug in x = 2 give me the same output value of nine. And that should be a little bit surprising, right? Because the way that f and g are telling me to compute their output is totally different. f takes the input two, adds one to it and squares it to get nine. g takes two, squares it, doubles it, adds those two numbers to one, to get nine. So, the method by which f and g are doing the calculations is totally different, right? This sequence of operations is not the same as this sequence of operations. The, the rules are different. And yet, look at this. f(x), for any value of x, right? Is 1 + x * 1 + x, right? That's 1 + x^2. Well, I could expand this out, right? 1 * 1 + x, and then x * 1 + x. I could combine some of these terms, right? 1 + x + x = 2x. x * x = x^2. Look, 1 + 2x + x^2, that's g(x). this is really quite surprising. f and g don't compute their output in the same way, right? This one is doing something different than this function, and yet, for any input value, f's output value is this, which is the same by expanding out as g(x). Now, how we're going to deal with this? We're going to say that f and g are at the same function, right? Not because they have the same rule, right? But because for every input value, they have the same output value. Here's a much more subtle example. Again, I got two functions. f is defined like this. f(x) = x^2 / x, and g is defined like this, g(x) is just x, the identity function. Same question, is f the same as g? Are these the same function? Now, they're not the same rule, right? This is not the same as this. So, you know, it's a little more subtle, you know? But that's okay, right? Two functions are the same if they have the same output for each input. So, let's see if that happens here. let's just pick some value to get a first test. Let's take a look at f(5), right? f(5) would be 5^2 / 5, that's 25. 5^2 / 5, that's 5. Well, that's the same as g(5), right? If I plug anything into g, I just get the same thing out. So, plug in five, you get five. So, at least at the value five, f and g agree. You might think this always works, right? Because of something like this. You might want to say, well, f(x) that's x^2 / x, no matter what x is. You might rewrite this x^22 as x * x / x. And then, you'd be tempted to say, cancel one of these xes with the x in the denominator. And then, you'd write equals x. And x, well that's, that's g(x). So, this looks like a pretty convincing argument, right? Over here, I've got f of x, I've got a bunch of equal signs. And over here, I've got g(x). So maybe that means F and G are the same function. Ha, but not so fast. What happens if you plug in zero? What's f(0)? Well, I know what g(0) is. g(0) is zero, right? Zero is in the domain of g because zero makes sense for this rule. But, what's f(0)? Well, that would be zero squared over zero, whoa. Okay. You see this is terrible, right? I cannot divide by zero. This rule, x^2 / x doesn't make sense when x is equal to zero. So, zero is not in the domain of f, but it is in the domain of g. So, I'm going to say that these are not the same function. They don't have the same domain, right? f isn't defined at zero, and g is defined at zero. In that sense, these are really different functions. This example suggests that there's a real richness to this theory of functions, right? And we're going to be studying it a lot more this term.
How can more functions be made?
Functions are going to be the main star of the course. So we should be building up in sort of a repertoire or library of functions that we might be interested in studying. Here's the first function in our library f(x) = x, the identity function. Whatever you plug in, this function outputs that same thing. Here's another function, a constant function. You pick some number c, c stands for constant. Alright? And you can define this function, f(x) = c. Whatever you plugin for x, f just ignores that and then outputs the original value, c.
Here's a function, f(x) = 3x + 2. And if you're thinking about stuff like that, why not stuff like this? Pick two numbers, a and b, and then you can define a function like this, f(x) = ax + b. You can think about fifth f(x) = x^5 or nth power, f(x) = x^n for some fixed value of n. And about polynomials, like this complicated looking polynomial, f(x) = 2x^3 + 5x^2 - 2x + 1. If you're thinking about polynomials, you might want to think about roots, f(x) equals, say the square root of x. You might remember the absolute value function f(x) equals the absolute value of x. You might have some experience with trig functions, like sines, cosines, and tangents, or with other transcendental functions like logarithms and exponentials. So now we've got our small library of functions, the identity function, constant functions, polynomials, some trig functions, and I want more functions. I, I want some way to be able to take two functions and produce a new function out of them. Okay. So in this setup I've got a conveyor belt and I've got two functions, a function here and a function here. Let's pick out what these functions should be. Maybe the first function I'll call f and f(x) would be 2X + 1. So I'll call this function f and maybe the second function I'll call g and g will take its input and square it, so g(x) will be x^2. So I'll label this function, g. And now here, I've got a number 3 and I am going to run that number through the first function and whatever comes out of the first function, I'm going to plug in to the second function to see what comes out. So let's take that number 3, let's start moving the conveyor belt. It's going to go through the function f, f(3) is 2 * 3 + 1 to 6 + 1, which is 7. So now we've got a 7 right there. So the 3 went into the function and came out as a 7. Now I'm going to take the output to f and put it in to the input of g. So g(7), well it's going to be 7^2 and that'll be 49. So here now, coming out of the function g, is the number 49. And I could have written this in a little bit a little bit of a shorthand way. I could have just written g(f(4)), right, f(3) is 7 and g(7) is 49. So once I've got this sort of conveyor belt metaphor going on in my head, I could do the following trick. I can take two functions. I can take the output to the first function and plug it in to the input of the second function.
What are some real-world examples of functions?
So, there's all these different ways of taking two functions and producing new functions. You could add, subtract, multiply, divide two functions, and could take two functions and compose them, meaning that the output for the one becomes the input for the other. In light to this, I'd encourage you just to pick up your pen and just write down some extraordinarily complicated functions, alright? The function that you write down has probably never been written down in the history of humankind. I mean, there's just so many different choices that you could make when you are combining all the algebraic operations. And that's part of what makes Calculus so amazing, right? There's just a huge variety of functions out there. But not, not every function really has its source in just combinations of algebraic symbols, right? A lot of the functions that we want to study are really functions that are somehow coming from the real world. So, I want to see some real world examples of, of functions right now. So, here's one unit conversion from Celsius to Fahrenheit. These are two different temperature scales. So, the function would be f of x, it's 9 * x / 5 + 32. So, this is just a linear function. It's a number times x plus a number. let's take a look what's f of zero. And that would be 9 * 0 / 5 + 32. Well, that's zero plus 32, that just 32 and, of course, zero degrees Celsius is the same thing as 32 degrees Fahrenheit. Here's another example. What's f of, say, 37? Well, that's nine * 37 / 5 + 32. 9 * 37 is 333 / 5 + 32. 333 / 5 is 66.6, so 66.6 + 32 is 98.6. And indeed, 37 degrees Celsius is the same thing as 98.6 degrees Fahrenheit. So, this function takes in something in Celsius and spits out something in Fahrenheit. Unit conversion is an example of a function, but hardly the coolest example. This is a much cooler example from the real world. What is this thing? Well, this thing here is a microcontroller. So, a very small computer and it's attached to a couple light emitting diodes, LEDs. With a name like light emitting diode, you might think that they light up, and they could. But in this circuit, I'm using the light emitting diodes in reverse. I'm using them as light sensors. This one happens to be a red one. This one happens to be a green one. So, what this circuit does is let me detect how much red and green light is falling on these sensors. At the other end is a USB cable and it plugs into my computer so I can record the results. The data that I gathered from the real world using the microcontroller. It's really two different functions. A function for the red LED and a function for the green LED. Along the x-axis, I've plotted the number of seconds that have elapsed since June 5th, 2012 at 6:0303 p.m. And on the y-axis, I'm plotting the number of clock cycles it took to discharge the LED. So, what is the red function do? It's input is a number a number of seconds that have elapsed since this particular moment in time. It's output is how many clock cycles it takes at that particular moment in time to discharge the red LED. Now, this thing was sitting in my windowsill, right? And the sun was rising. And as the sun rises, there's more light shining on the sensors which means fewer clock cycles are necessary to discharge the LED. And you can see that in this graph, right? The red function is decreasing as the sun rises. There's tons more examples of functions coming from the real world. Here's one, human population. It's a function. The input is a year, the output is the number of people alive during that year. If you want to see this function just take a look at Wikipedia and their article on population growth. There's a graph of that function, along the x-axis is years and along the y-axis is human population. And as long as researching the Internet, here's another example of a real world function. It's a function I'll call f of n, and it'll be defined by the rule f of n equals the number of Google hits when we search for the number n. Let's try it out. Let's figure out some values of this function like f of 188. So I plug 188 into Google, and I find that there's about 1.08 billion hits. So, the function at 188 is about a billion, alright, the input is 188 and the output of this function is the number of Google hits. let's try about 4 * 188, that's 752. And if I search for that, there's 308 million hits, alright? So, f, the function, at 752 is about 300 million. if we're persistent, we can plug inn lots of numbers, and make a really nice-looking chart like this. Now, you do this for hundreds of numbers, right? You type them into Google, you see how many Google hits you get. And you can plot them, right? It's a function so you can the graph of the function. Along the x-axis is the number that I typed into Google. On the y-axis is the millions of Google hits that I get. And when you look at the graph of this function, it's not random. There's real structure here, right? The function is decreasing, right? Larger values get smaller outputs because, you know, there is fewer webpages that talk about really large numbers than about popular small numbers. But even more dramatically, when you plot this on this special log, log graph paper, the graph looks like it's sitting near a straight line. I mean, that's, that's really amazing when you think about it. I mean, this is some sort of pattern that's just hidden in the number of pages that talk about numbers. I mean why? Where is this coming from, right? It's a function from the real world. Input is a number. The output is a number. We want to understand that function. Calculus is part of the tool kit for analyzing problems like this. So, we've seen what functions do. They take their input and they transform in into some output. And we've even sort of got this mental image now, this metaphor of a machine. A conveyor belt that's transforming the input into the output. We've seen how to build a lot of new functions using algebra. or say, composing two functions. And we've thought about some real world examples. Now, we're going to be thinking more about functions for the rest of the term. But if you've got questions right now, I encourage you to contact me as soon as possible. And I encourage you to get started on the homework right away. Good luck.
Độ Celsius (°C hay độ C) là đơn vị đo nhiệt độ được đặt tên theo nhà thiên văn học người Thụy Điển Anders Celsius(1701–1744). Ông là người đầu tiên đề ra hệ thống đo nhiệt độ căn cứ theo trạng thái của nước với 100 độ C (212 độFahrenheit) là nước sôi và 0 độ C (32 độ Fahrenheit) là nước đá đông ở khí áp tiêu biểu (standard atmosphere) vào năm1742. Hai năm sau nhà khoa học Carolus Linnaeus đảo ngược hệ thống đó và lấy 0 độ là nước đá đông và 100 là nước sôi.[1] Hệ thống này được gọi là hệ thống centigrade tức bách phân và danh từ này được dùng phổ biến cho đến nay mặc dù kể từ năm 1948, hệ thống nhiệt độ này đã chính thức vinh danh nhà khoa học Celsius bằng cách đặt theo tên của ông.[2]Một lý do nữa Celsius được dùng thay vì centigrade là vì thuật ngữ "bách phân" cũng được sử dụng ở lục địa châu Âu để đo một góc phẳng bằng phần vạn của góc vuông. Ở Việt Nam, độ C được sử dụng phổ biến nhất.
Fahrenheit phát triển thang nhiệt độ của ông sau khi viếng thăm nhà thiên văn học người Đan Mạch Ole Rømer ở Copenhagen. Rømer đã tạo ra chiếc nhiệt kế đầu tiên mà trong đó ông sử dụng hai điểm chuẩn để phân định. Trong thang Rømer thì điểm đóng băng của nước là 7,5 độ, điểm sôi là 60 độ, và thân nhiệt trung bình của con người theo đó sẽ là 22,5 độ theo phép đo của Rømer.
Fahrenheit chọn điểm số không trên thang nhiệt độ của ông là nhiệt độ thấp nhất của mùa đông năm 1708/1709, một mùa đông khắc nghiệt, ở thành phố Gdansk (Danzig)quê hương ông. Bằng một hỗn hợp „nước đá, nước và Amoni clorid (NH4Cl)" (còn gọi là hỗn hợp lạnh) sau đó ông có thể tạo lại điểm số không cũng như là điểm chuẩn thứ nhất (−17,8 °C) này. Fahrenheit muốn bằng cách đó tránh được nhiệt độ âm, như thường gặp ở thang nhiệt độ Rømer-Skala trong hoàn cảnh đời sống bình thường.
Năm 1714, ông xác định điểm chuẩn thứ hai là nhiệt độ đóng băng của nước tinh khiết (ở 32 °F) và điểm chuẩn thứ ba là "thân nhiệt của một người khỏe mạnh" (ở 96 °F).[1]
Theo các tiêu chuẩn hiện nay thì các điểm chuẩn trên và dưới khó có thể tạo lại một cách thực sự chính xác được. Vì thế mà thang nhiệt độ này về sau đã được xác định lại theo hai điểm chuẩn mới là nhiệt độ đóng băng và nhiệt độ sôi của nước, tức là 32 °F và 212 °F. Theo đó, thân nhiệt bình thường của con người sẽ là 98,6 °F (37 °C), chứ không phải là 96 °F (35,6 °C) như Fahrenheit đã xác định nữa.
Thang nhiệt độ Fahrenheit đã được sử dụng khá lâu ở Châu Âu, cho tới khi bị thay thế bởi thang nhiệt độ Celsius. Thang nhiệt độ Fahrenheit ngày nay vẫn được sử dụng rộng rãi ở Mỹ và một số quốc gia nói tiếng Anh khác.
Sử dụng[sửa | sửa mã nguồn]
Thang nhiệt độ Fahrenheit từng được sử dụng chủ yếu trong đo đạc thời tiết, công nghiệp và y tế ở hầu hết các nước nói tiếng Anh cho đến những năm 1960. Vào nửa cuối những năm 1960 và 1970, thang nhiệt độ Celsius dần dần được các chính phủ đưa vào sử dụng trong kế hoạch chuẩn hóa hệ thống đo lường.
Những người ủng hộ thang nhiệt độ Fahrenheit cho rằng sự phổ biến của nó trước kia là do yếu tố tiện dụng. Đơn vị của nó chỉ bằng 5⁄9 của một độ Celsius, cho phép thể hiện chính xác hơn các đo đạc nhiệt độ mà không cần sử dụng đến các đơn vị lẻ. Ngoài ra, nhiệt độ không khí môi trường ở hầu hết các vùng cư dân trên thế giới thường cũng không vượt xa giới hạn từ 0 °F đến 100 °F, vì thế mà thang nhiệt độ Fahrenheit được cho là thể hiện nhiệt độ mà con người có thể cảm nhận được, thể hiện theo từng cấp 10 độ một trong hệ thống Fahrenheit. Hơn nữa, đồng thời mức thay đổi nhiệt độ nhỏ nhất có thể cảm nhận được là một độ Fahrenheit, nghĩa là một người bình thường có thể nhận biết nếu có chênh lệch nhiệt độ ở mức chỉ một độ.
Nhưng cũng có những người ủng hộ thang nhiệt độ Celsius lập luận rằng hệ thống của họ cũng rất tự nhiên; ví dụ như họ có thể nói rằng nhiệt độ từ 0–10 °C là lạnh, 10–20 °C là mát mẻ, 20–30 °C là ấm áp và 30–40 °C là nóng.
Ở Mỹ, hệ thống Fahrenheit vẫn là hệ thống được chấp nhận là chuẩn cho mục đích phi khoa học. Mọi quốc gia khác đã áp dụng thang nhiệt độ chính là Celsius. Fahrenheit đôi khi vẫn được thế hệ cũ sử dụng, đặc biệt là để đo nhiệt độ ở các mức cao.
Celsius (thang nhiệt độ bách phân)[sửa | sửa mã nguồn]
| từ Celsius | sang Celsius | |
|---|---|---|
| Fahrenheit | [°F] = [°C] × 9⁄5 + 32 | [°C] = ([°F] − 32) × 5⁄9 |
| Kelvin | [K] = [°C] + 273.15 | [°C] = [K] − 273.15 |
| Rankine | [°R] = ([°C] + 273.15) × 9⁄5 | [°C] = ([°R] − 491.67) × 5⁄9 |
| Delisle | [°De] = (100 − [°C]) × 3⁄2 | [°C] = 100 − [°De] × 2⁄3 |
| Newton | [°N] = [°C] × 33⁄100 | [°C] = [°N] × 100⁄33 |
| Réaumur | [°Ré] = [°C] × 4⁄5 | [°C] = [°Ré] × 5⁄4 |
| Rømer | [°Rø] = [°C] × 21⁄40 + 7.5 | [°C] = ([°Rø] − 7.5) × 40⁄21 |
What is the domain of square root?
So, before we talk about the domain of the square root function, we just want to remind ourselves what the square root function even is. So here, I've made a graph of the square root function. And along the x-axis, I plot the numbers one to sixteen and in the y-axis I've got the numbers one through four. And then in this green curve here, I've plotted the the, the square root function. What is the square root, right? Well, here's an example. Here, I've got the square root of four. And I'm saying the square root of four is two. What that means, is if I take the number two and I square it, I get back four. I don't know, if I move over to, say, the square root of nine, I get three. And that's because three squared is nine, alright, or if I move over a little bit further, the square root of sixteen is four. And that's because four squared is sixteen. I know there's some crazier values, too. If I move over here to the square root of two, well, the square root of two is this sort of crazy number 1.414213 blah, blah, blah. And maybe it's a little bit surprising, that if I take that number and square it, I get back two. So, what's going on here, alright? The square root function takes a number and spits out a new number, that new number when you multiply it by itself and you square it, you get back your original number. Now, here's the question, what sorts of numbers can I take the square root of? That's asking the question, what's the domain of the square root function? Now, that we've seen the graph, let's try to write down in words a definition of the square root function. So, in light of what we just seen, you might think that the definition of the function f(x) equals square root of x is a number which squares to x. There's a problem with this though. Take a look at say, f(9). What would f(9) be? Well, if you're thinking the square root of x is a number which squares to x, then you might think that f(9) would be -3, alright? Because -3^2 is 9. But then, you might also think that half of nine should be three, right? Because 3^2 is also 9. This is bad, alright? A function is supposed to be unambiguous. It's supposed to have one output for each input. If you take this as the definition of the square root function, just any number which squares to x, you've introduced some ambiguity, alright? What's the square root of nine? Is it -3 or is it +3? Both of those numbers square to nine. So, this is, this is bad, alright? The solution is to change the definition. Instead of having the the square root function be just a number which squares to x, you're going to take it to be the nonnegative number which squares to x. This is better, alright? In our example here, if I only am allowed to choose the nonnegative number, which squares to x, then f(9) equals -3, well, -3 is not nonnegative, -3 is negative. So, that means that this isn't the case, right? All I'm left with is f(9) = 3, right? Three is the nonnegative number which squares to nine. Alright. So, this will be our definition for for, for the square root function. The square root of x is the nonnegative number which squares to x. There's one particular place where this plays out and it's extraordinarily important. So, let's take a look at that now. We've got our definition. The squared of x is the nonnegative number which squares to x. Now, there's one popular misconception that comes up because of this definition. So, in light of the definition of the square root, right, the square root of a number being the nonnegative number which squares the number to the radical, you might be tricked into thinking that the square root of x squared is x. That's not true and let's see why. Let's do a specific example where say, x is -4. So, if I replace the x's here by -4, the left hand side is the square root of -4 squared, right? Square root of x squared but with x replaced with -4. Now, - 4 * -4 is 16. This is the square root of 16 and the square root of sixteen, the definition of the square root is the nonnegative number which squares to 16. There's two numbers that square to 16, +4 and -4. But the square root is by convention, the nonnegative one, so this is equal to 4. Duh, look at what happened. -4, square root of -4^2 + 4, that's the x over here. This is not true, right. You should not be tricked into thinking that that's the case. Instead, something else is true, right? What is true is this. The square root of x squared is the absolute value of x. And that works in this specific case, right? When x is -4, the square root of -4 squared, the square root of 16 is 4. And 4 really is the absolute value of -4. Alright. So, this is a mistake that comes up quite a bit. People are often tricked into thinking that the square root of x squared is just x, alright? They're just trying to cancel the square roots in the squaring. That's not possible. Instead, what is true is the square root of x^2 is the absolute value of x. So, we've got a definition of the square root function and we've seen that the square root of x^2 is not just x, it's the absolute value of x. Now, that doesn't actually address the original question, right? The original question is, what's the domain of this square root function? What sorts of numbers can I take root of? For instance, can I take the square root of a negative number? Let's see why not. Very concretely. Does it make sense, say, to talk about the square root of -16? Well, if it did that would be some number. So, I'll call that number k for crazy, alright? And what do I know about that number k? Well, k^22 would have to be -16. Remember, the definition of the square root function? It's a number that I square to get back the original number. So, if there were a square root of -16, when I square it, I get back -16. And imagining here that k is some real number. And that means there's three possibilities. Either k is positive, k is zero, or k is negative. If k is positive, then k squared would also be positive because a positive number times a positive number is still positive. But that can't be, because k squared is supposed to be -16. So, this first possibility doesn't happen. Now, if k were zero, then k squared would be zero, but k squared is supposed to be -16. So, k isn't zero. Is k negative? Well then, what's k squared? That would be a negative number times a negative number, and that would still be positive. And that can't be because k squared is supposed to be -16. So, this possibility also doesn't happen. So, all of our possibilities have been eliminated, alright? There can't be a real number k, which is the square root of -16. Because if k were positive, k squared would be positive but k squared has to be negative. k can't be zero because then k squared isn't negative and k can't be negative because then k squared is positive but k squared is supposed to be negative, alright? The upshot is that it just doesn't make any sense to talk about the square root of a negative number. In contrast, it does make sense to talk about the square root of zero, which is just zero, zero squared is zero. And it also makes sense to talk about the square root of positive numbers. So, to summarize the situation, we can say that the domain of the square root function is all the numbers between zero and infinity, including zero. So, I'm using the square bracket. But, of course, not including infinity because infinity is not a number. Sometimes, you're asked to calculate the domain of a function that's more complicated than, than just the square root of x. Let's see an example of that. So, let's try this. Let's try to find the domain of this function g, which is the square root of 2x + 4. And remember, the domain consists of all the inputs for which the rule makes sense. So, I just have to think which x values makes sense for this rule? Well, in order to take the square root of 2x + 4, I'm going to need that 2x + four is not negative because I can't take the square root of a negative number so I need to guarantee that 2x + 4 is not negative, meaning greater than or equal to zero. Now, I can subtract four from both sides and I get that 2x is at least -4. Then, I can divide both sides by two. Two is positive, so it doesn't change the inequality. x is bigger than or equal to -2. So, as long as x is at least -2, then 2x + 4 is at least zero, which means it makes sense to take the square root. So, I can summarize the situation, the domain of g consists of all numbers greater than or equal to -2. This is our notation for that. I used a square bracket to include the -2 and the round bracket here on the infinity, because infinity is not number, it's not part of the domain. So, that example was a little bit harder. Let's do an even harder example where I've got multiple square roots, all right, the square root of something plus the square root of something. And let's figure out the domain of this function that has two separate square roots. This is the function T(x) equals the square root of 1 - x plus the square root of 1 + x. Now, in order for this rule to make sense, I have to be able to take this square root and also take this square root. In other words, in order to do this first square root, I'm going to need that 1 - x is bigger than or equal to zero, alright? I need the thing under the square root to be nonnegative in order to do a square root. In order to take this square root, I need 1 + x to be bigger than or equal to zero. And both of these things have to be true in order to take both of these square roots and then add them together. So, I'll put an and between them. Now, I go to x to both sides and this inequality and I get one is bigger than or equal to x. And I can subtract one from both sides of this and I'll get x is bigger or equal to -1. And again, both of these things have to happen, right? I need x less than one and x bigger than or equal to -1 in order to evaluate this function. Let me write this in in a more reasonable way, right? Instead of writing one bigger than or equal to x, I can write what I just said, x less than one. And here, I'll write, this is x bigger than or equal to -1. Now, I could write these inequalities as something about an interval. I could say that x is in the interval -1 to 1, alright? To say that x is less than one and bigger than or equal to -1, exactly means that your inside this interval. And I'm using square brackets here, because I've got greater than or equal to, less than or equal to. And then, I can summarize the situation by writing the domain of T is this interval, alright? And this is describing the values of x for which this rule makes sense at the domain of the function T. Let's do one more example. some square root problem where I've also got an x squared term. Let's calculate the domain of this function C. C of x is the square root of 1 - x^2. So, the domain consists of all the inputs for which the rule makes sense. So, I'm looking for which values of x make the thing under the square root nonnegative. There's lots of different ways to think about which values of x make this true. one way is to factor 1 - x^2. So, I could factor 1 - x^2 as 1 + x * 1 - x, alright? That is equal to 1 - x^2. I'm looking for when that's nonnegative. This is a little bit easier to think about because now, I just got to figure out when these two terms have the same sign, alright? When they're both positive or they're both negative, then their product is bigger than or equal to zero. So, to think about that, I'll draw a number line. And I'll first think about when 1 + x is positive and negative. So, something special happens at -1, alright? When x is minus one, 1 + x is zero. When x is less than -1, 1 + x is negative. And when x is bigger than -1, 1 plus x is positive.
Alright. Now, compare this with 1 - x, alright? 1 - x, something exciting happens at one, alright When X is less then one, 1 - x is positive.
And when x is bigger than one, 1 - x is negative. Now, I'm not trying really to understand 1 + x or 1 - x, I'm trying to understand their product. So, when I multiply those two together, I get 1 - x^2 and I want to know, you know, when is that positive or or negative.
Let me mark down the special points -1 and 1. And now, 1 - x^2 is the product of these so I can think about various values of x. So, when x is less than -1, then 1 + x is negative and 1 - x is positive, and a negative number times a positive number is negative. When x is between -1 and 1, then 1 + x is positive and 1 - x is also positive in that region, so their product is positive. And when x is bigger than one, 1 + x is positive and 1 - x is negative, so their product is negative. Now, this gets me most of the way there, alright? Because what I'm trying to understand is when this product is nonnegative and I can see that it's positive in this region, I could also think about what happens when I plug in -1 and 1. When I plug in -1, I get 1 - 1 which is zero. And when I plug in one, I get 1 - 1 which is zero. So, the function is, in fact, is zero in between here at -1 and 1. So, I'm just trying to figure out which values of x make 1 - x^2 nonnegative. Well, -1, 1, and anything in between. So, one way to summarize the situation is to say that the domain of my function C consists of all real numbers between -1 and 1, including -1 and 1. So, I'm using the square brackets. As long as x is inside here, then 1 - x^2 is nonnegative. That means it makes sense to take the square root and that's the domain of C.
Morally, what is the limit of a sum?
There are subtleties even to things that appear as simple as addition. Here, I've got some addition problems. 4279 + 1202, 4279 + 1190, 4269 + 1207, 42731202. + 1191, and 4270 + 1100 You'll notice that they're all close to this problem. The numbers that I've been listing off are all hovering around this problem. Anyway, I'm going to give out these problems to some people and have them try to do them. Hi, I'm and I'm Math junior undergraduate at Ohio State. Yeah. My name is Jacob Turner. I'm a graduate TA here at OSU. Alright.
People are finished doing the arithmetic problems. Let's record the answers. So here, 4270 + 1200 was 5470. 4279 + 1202 was 5481. The next one is 5476, I've got 5467, and the last one was 5464. what do all these numbers have in common? They're all really close together. Is that just an accident? Of course, it's not an accident, right? Here's the fact. Near the sum of two numbers is the sum of two nearby numbers. These arithmetic problems are not just random arithmetic problems. Look at the numbers I'm asking them to add. 4277 and 1190, those Those numbers are really close to 4273 and 1191. Which is really close to 4270 and 1200. Which is really close to 4269 and 1207, alright? Near the sum of two numbers is the sum of two nearby numbers, all of the answers are nearby as well. How does this relate to limits? Let's take a look. Here's how it relates to limits. The limit of f of x plus g of x as x approaches a, is the limit of f of x as x approaches a plus the limit of g of x, as x approaches a. How is this related to those arithmetic problems? Well, remember what this limit is saying. This is saying, what can I make f of x plus g of x close to, if I'm willing to make x sufficiently close to a. Well, it's going to be close to whatever I can make f of x close to added to whatever I can make g of x close to, right? It's the same kind of setup, right? Near the sum of two values is the sum of the nearby values. For the limit of a sum is the sum of the limits. We can use this fact to do some calculations. Let's see how. So, here's a limit problem. The limit of x squared plus x as x approaches two. I really want you to resist the temptation to just plug in two. We're going to be using our limit laws to try to evaluate this limit. Now, this is the limit of a sum, and the limit of the sum is the sum of the limits provided the limits exist. So, this limit of x squared plus x is equal to the limit of x squared as x approaches 2 plus the limit of x as x approaches 2. Now, what's the limit of x squared? Because the limit of the products is also the product of the limits provided the limits exist, this is the limit of a product. This is the limit of x times x. That's what x squared means, it's x times x. So, I could rewrite this as the limit of x times x, as x approaches 2+. Now what's the limit of x as x approaches 2? This is asking, what does x get close to when x gets close to 2? Or, some more precisely, what can I guarantee that x is close to if I'm willing to make x sufficiently close to 2? The limit of x as x approaches 2 is 2, alright? So, this limit is just two. Now, this a limit of a product, and the limit of a product is the product of the limits provided the limits exist. So, this limit is the limit of x as x approaches 2 times the limit of x as x approaches to +2. Now again, the limit of x as x approaches two, right? What can I guarantee that x is close to if x is close to 2? Well, two. So, this is just 2. This limit is the same thing, it's again just 2. And here, I have +2. 2 * 2 + 2 is 6, which is the value of the limit of x squared plus x as x approaches 2.
The takeaway message here is ask not what your country can do for you but what you can do for your country. Or in other words, that the limit of a sum is the sum of the limits provided the limits exist. It's the same rhetorical device right, x-y, y-x. Alright, the limit of a sum is the sum of the limits provided the limits exist. I hope this is very memorable because these kinds of chiastic rules are going to be used throughout our time together in order to evaluate limits. Soon, we're going to see that the same sort of pattern holds not just for sums, but for differences, for products and almost for quotients. Good luck.
Morally, what is the limit of a sum?
There are subtleties even to things that appear as simple as addition. Here, I've got some addition problems. 4279 + 1202, 4279 + 1190, 4269 + 1207, 42731202. + 1191, and 4270 + 1100 You'll notice that they're all close to this problem. The numbers that I've been listing off are all hovering around this problem. Anyway, I'm going to give out these problems to some people and have them try to do them. Hi, I'm and I'm Math junior undergraduate at Ohio State. Yeah. My name is Jacob Turner. I'm a graduate TA here at OSU. Alright.
People are finished doing the arithmetic problems. Let's record the answers. So here, 4270 + 1200 was 5470. 4279 + 1202 was 5481. The next one is 5476, I've got 5467, and the last one was 5464. what do all these numbers have in common? They're all really close together. Is that just an accident? Of course, it's not an accident, right? Here's the fact. Near the sum of two numbers is the sum of two nearby numbers. These arithmetic problems are not just random arithmetic problems. Look at the numbers I'm asking them to add. 4277 and 1190, those Those numbers are really close to 4273 and 1191. Which is really close to 4270 and 1200. Which is really close to 4269 and 1207, alright? Near the sum of two numbers is the sum of two nearby numbers, all of the answers are nearby as well. How does this relate to limits? Let's take a look. Here's how it relates to limits. The limit of f of x plus g of x as x approaches a, is the limit of f of x as x approaches a plus the limit of g of x, as x approaches a. How is this related to those arithmetic problems? Well, remember what this limit is saying. This is saying, what can I make f of x plus g of x close to, if I'm willing to make x sufficiently close to a. Well, it's going to be close to whatever I can make f of x close to added to whatever I can make g of x close to, right? It's the same kind of setup, right? Near the sum of two values is the sum of the nearby values. For the limit of a sum is the sum of the limits. We can use this fact to do some calculations. Let's see how. So, here's a limit problem. The limit of x squared plus x as x approaches two. I really want you to resist the temptation to just plug in two. We're going to be using our limit laws to try to evaluate this limit. Now, this is the limit of a sum, and the limit of the sum is the sum of the limits provided the limits exist. So, this limit of x squared plus x is equal to the limit of x squared as x approaches 2 plus the limit of x as x approaches 2. Now, what's the limit of x squared? Because the limit of the products is also the product of the limits provided the limits exist, this is the limit of a product. This is the limit of x times x. That's what x squared means, it's x times x. So, I could rewrite this as the limit of x times x, as x approaches 2+. Now what's the limit of x as x approaches 2? This is asking, what does x get close to when x gets close to 2? Or, some more precisely, what can I guarantee that x is close to if I'm willing to make x sufficiently close to 2? The limit of x as x approaches 2 is 2, alright? So, this limit is just two. Now, this a limit of a product, and the limit of a product is the product of the limits provided the limits exist. So, this limit is the limit of x as x approaches 2 times the limit of x as x approaches to +2. Now again, the limit of x as x approaches two, right? What can I guarantee that x is close to if x is close to 2? Well, two. So, this is just 2. This limit is the same thing, it's again just 2. And here, I have +2. 2 * 2 + 2 is 6, which is the value of the limit of x squared plus x as x approaches 2.
The takeaway message here is ask not what your country can do for you but what you can do for your country. Or in other words, that the limit of a sum is the sum of the limits provided the limits exist. It's the same rhetorical device right, x-y, y-x. Alright, the limit of a sum is the sum of the limits provided the limits exist. I hope this is very memorable because these kinds of chiastic rules are going to be used throughout our time together in order to evaluate limits. Soon, we're going to see that the same sort of pattern holds not just for sums, but for differences, for products and almost for quotients. Good luck.
What is the limit of sin (1/x)?
So here's a table of values of the function F of X equals sin 1 / X. F of one is really sin of one, it's like 8.. F of 1. which is really sin of one over 1. sin of 10 -.5. F of 01.. This will be sine of 100, it's also about -0.5. F of 0.001, which is like sine of 1000, well that's 0.8 and some more, right? So the question is these numbers aren't really getting close to anything in particular. Can you really say that if you evaluate f at values which are close to but not equal to zero, that the outputs are actually getting close to anything in particular. I mean this is positive, negative, negative, positive, negative, positive, negative, it's looking pretty bad.
Instead of a table, let's look at a graph. Here, I've got a graph of the funtcion f (x) = sin 1 / x. And you see the middle of this graph is just that horrible green blob. Right? It's really hard to make out any detail. You might think that's just a consequence of the fact that I'm drawing this graph with such thick lines. You know, and if I used thinner lines to draw my graph, maybe I could, you know get rid of this green blob and really see some detail. Even if I dial down the size of the lines that I'm using to draw this graph, the blob thing is still there, you know. And it's really there in the graph of the function. Even if these lines were true lines, zero thickness, it wouldn't be possible to fit even a single atom next to the Y axis without touching the graph of this function. The graph is oscillating wildly near zero. Even if your input is very close to zero your output could be anything between -1 and 1. So in light of this evidence, the limit of sine 1 / x as x approaches 0 does not. exist. Which sometimes I'll abbreviate DNE, for does not exist. what does it even mean to say it doesn't exist? What do we mean by the definition of limit? To say the limit equals something means that I can make the output as close as I want to l by making x close to a. So when I say this limit doesn't exist, I mean it's not the case that this limit is equal to anything, okay? If you tell me this limit is some positive number, well look. When I evaluate the function at a number very close to 0, the output is negative. So the limit is probably not some positive number but there's also inputs very close to zero that give positive outputs so that the limit is pulling out a negative number either. Limit is pulling out zero either cause none of these numbers are getting close to zero. So in this sense. This limit just doesn't exist because it's not the case that this limit is equal to anything in particular. If you tell me this limit is equal to l, I'm going to show you numbers close to zero which aren't close to l. Let's see another example along the same lines. This is a particularly confusing example because in the function f(x)x) = sine pi / x. The function evaluated at 1 is 0. That's pretty clear because that is sine of pi and sine of pi is zero. About the function at 0.1, I'm counting that's also equal to zero. The function at 0.01, that is also zero. This can be kind of confusing. When you take a look here, I typed in sin pi divided by 0.01 on to my calculator. This is calculating the function's value at 01.. If I ask my calculator to do this, it is not telling me the answer zero, right? The calculator's giving me this, admittedly, a very small number, right, E -11 here. But it's still not actually zero. So, can I convince you that this is even true? That the functions value at 01. actually is equal to zero.
What is f (0.01)? Well it's the same as sine, of pi / 0.01. Now here I'm taking pi and I'm dividing it by 0.01.
That's the same thing as what? That's the same thing as multiplying a 100. I'm dividing by a hundredth, that's the same as multiplying by a 100. So this function at.01 is sine of 100 pi. What's sine of 100 pi? Well, think back to what the graph of sine looks like. Here's a graph of sine, zero to two pi. If I do it again. Here it is at four pi, and I drew it again. Here it is at six pi, and I'm going to keep on going. And eventually, I'm going to get to 100 pi. And at that point, sign really is going to be equal to zero. Calculators are great, they're also terrible. This calculator can't really calculate with pi, all it can do is calculate with some approximation to pi. We can use our human mind to evaluate this function exactly. In light of this evidence, you might be tricked into believing that the limit of f(x) as x approaches 0 is equal to 0. After all these points are approaching zero, and function evaluating each of these points is zero. So maybe that means that this is true. So it looks like the limit is equal to zero. But, what happens if I look at some other points? We'll take a look at this example. Here's the same function, f of x equals sign of pi over x. This function, if I evaluate it at 75. is this, maybe a little bit mysterious number, negative 866. and so forth. If I evaluate this function at 075. you get the same thing. If I evaluate the function at 0075,. I get the same thing. At 00075. I get the same thing, .000075, I get the same thing. So, what's going on here.
Well what is this number? I mean 0.8666. This isn't just some sort of random number. Right? This is in fact negative the square root of three over two. And it looks like this function at all of these points has the same value, negative the square root of three over two. So does that mean that the limit as X approaches zero of F of X, is equal to negative the square root of three over two? I mean again all of these values, .75.075.0075, these input values are approaching zero, and the functions value at all of those inputs is the same. So, what gives, is the limit zero, is a negative point A, which is it?. Okay, okay, I've been little bit too tricky in picking my input points. Since the same function, F of X equals Sin pie over X, and here, I am picking a collection of points, again approaching zero, .7, .07, .007, .0007. Its getting closer and closer to zero. But now, my output values are looking pretty random. I mean, they are not over the same, for instance. So this is, maybe some evidence, that, the limit. Of sine pi over x as x approaches zero. Doesn't exist. Here I've got a bunch of input points that are getting closer and closer to zero but my output values at least don't appear to be getting close to. We're not just learning. We're exploring. I encourage you to cook up you own examples. We've seen a couple examples now of where limits don't exist but can you come up with more?
What is the limit of (sin x)/x?
So here's a great function to look at. The function is going to be defined by f(x) = sine x / x. My question is what's the limit of this function, as x approaches zero. Let's try to guess the limit by looking at a table of function values. So here's a bunch of input values that are getting closer and closer to zero, right.1, .01, .001. It's getting closer and closer to zero. Then looking at my output values from my function, right? So f(1) is sine of 1 / 1. It's the sine of 1. f(.1), that's sign of 1. over 1,. it's 99.. A little bit more. f(.01) is 9999. a little bit more. And you keep looking down here and these numbers seem to be getting close to something, alright. 999999. this is really, really close to one. So based on this table of values your tempted to guess that the limit of f(x) as x approaches 0 is 1. Another way to gain some insight about this limit will be to look at the graph. Here's the graph. This is the graph of sign x over x. And you can see that when x equals zero, functions not defined there because I can't divide by zero, so I got this little hole in the graph. Nevertheless, I'm claiming that the limit as x approaches 0 is equal to 1 which actually means that I can make the output as close to 1 as you like, if you're willing to have the input be close enough to 0. Instead of talking about closeness, push this red button and turn on this red interval. So when I say close to one, what I really mean is the output is inside this, this red interval. And that red interval might be really big or it might be really small. But to be close to one is going to mean inside the red interval. The point is that, can turn on this blue interval. And as close as you want the output to be the one, I can promise you that the output is within the red interval if the input is within this blue interval. When the red interval is really big, well that's not much of a challenge. I can have a really wide blue interval and anything inside the blue interval has output landing inside the red interval. But even when the red interval is very, very small there's still some tiny blue interval so that whenever x is within the blue interval, the output is within the tiny red interval. In other words, even if you want the output to be really close to one. I can promise you that the output is that close to one, if you're willing to have the input be close enough to zero. So, we've looked at the function values, we've looked at the graph. We've got this idea that the limit of sine x over x as x approaches zero is equal to one. But it's just that, it's just an idea. We don't yet have a rigorous argument that this limit is equal to one. Here's a sketch of a more rigorous argument that the limit of sine x / x, as x approaches 0 is equal to one. It turns out that for values of x which are close to but not equal to zero, this is true. Cosine of x is less than sine x over x, and sine x over x is less than one. Now why would you care about this? Note, the limit of cosine x as x approaches zero is one and the limit of 1 is 1 because the limit of a constant function is just that constant. So I know that the limit of this side is one and the limit of this side is one and what I'm trying to conclude is that the limit of the thing in between is also one. And it turns out there's a way to do this. Let's take a look. Here's what we're going to use, the squeeze theorum. Suppose you've got three functions, I'm calling them GF and H. G(x) is less than equal to f(x) and f(x) is less than equal to H(x). For values of x that are near A, but maybe these inner qualities don't hold at the point A. Also, suppose that the limit of G(x) as x approaches A, is equal to the limit of H(x) as x approach A, is equal to sum L. So the limit of G(x), the limit of H of X are the same value, L. The, you get to conclude the limit of f as x approaches a exists and it equals l.
Why is this thing called the Squeeze Theorem or some people call it the Sandwich Theorem or the Pinching Theorem? Let's take a look. Just pictorially, why is this called the squeeze theorem? I've got an example here. Three functions. G, F, and H. And again, G(x) is less than F(x), F(x) less than H(x). Now, note, the limit of G(x) as x approaches A is L. And the limit of H(x) as x approaches A is L. F is squeezed, or sandwiched, between H and G. And consequently, the limit of f as x approaches A is also equal to L. Now, we're going to use the squeeze theorem to try to understand the limit of sin x over x. So we've got the Squeeze Theorem. And what do I know? I know that cosine X is less than sine x / x is less than 1 for values of x that are close to but not equal to 0. And the limit of cosine x as x approaches 0 is equal to 1. If you like, because cosines continuous and cosine of 0 is 1. Also the limit of 1 as x approaches 0 is equal to 1 because the limit of a constant function is that constant. So the limit of this function is one, the limit of this function is one as x approaches zero. And that means by the Squeeze Theorem, the limit of sine x / x is also equal to 1
What is the limit of (x^2 - 1)/(x-1)?
Limits are probably the most important concept in this course. So we should really have a definition of what we mean by limit. Now here is what we mean by limits. To say that the limit of f of x as x approaches a is equal to L means that f of x can be as close to L as desired by making x close enough to a. There is a tons of subtlety to this definition so it's worth to look at an example. So let's take a look at this function. This is the function that takes an input x and spits out x^second minus one divided by by x minus one. So let's try plugging in number three into this function. So I plug in number three into this function and I have to just compute, right? Three squared minus one over three minus one well, that's three squared is nine minus one is eight three minus one is two and nine divided by two is four And sure enough, out of this function comes the number four Let's look at that example again but with a little bit more detail. this is actually a pretty complicated function. Alright? But I can open up the function. Alright. And take a look at how the functions actually doing its calculations. You can think of this function as having three different steps. Alright. One of the steps squares its input and subtracts one, and so I calculate the numerator. Another step just subtracts one from its input. The outputs of those two steps then get plugged into the division. And that's how I get the output of this big complicated function. Now, something like x^two - one, you could also think of that as having some, you know separate steps as well. But this is good for right now. Okay. Now let's see what happens. I take the number three and I plug it into the function. Alright? Now I'm going to be calculating the numerator and the denominator separately, so I'll take those 3s, and up here, I'll look at three^two - one and I'll get out eight. And down here, three - one became two Now the eight and the two get plugged into the division, and eight divided by two is four and that becomes the output of the function, right? Input's three, output is four but when I look at it this way, I can see how all the steps are, are playing out. Okay. I evaluate the function at three, but who cares? Well, let's try to evaluate the function at one instead of at three. So what happens when we plug in the number one into this function? I got the number one here. I'm going to look inside. I'm going to open up this function. Now imagine I've got this number one. I'm going to plug it into the function. All right. Now I'm going to be evaluating the numerator and denominator separately, so I'm going to take this one and split it up, and plug it into the numerator and the denominator. The numerator sends its input to its input squared minus one. So one^two minus one is zero and the same thing down here, one - one is zero Now I've got 0 and 0 which I'm going to be plugging in to the. Okay, very bad. Right? I'm dividing by zero and I can not proceed, so this function is not defined at one. So I can't plug one into the function. But if I wanted to figure out what the function's value was that inputs near one, I could do that. So let's try to plug in one point one instead so let's plug one point one into this function. I can't plug in one because I need to divide them by zero, but let's try plugging in one point one I'm going to open up the function again and take one point one plug it into the function. Now one point one is going to to be evaluated in the numerator and the denominator. one point one^second minus one is twenty one. And one point one minus one became one. Now twentyone and one are going into the division. And twenty one divided by one is two point one So when I evaluate the function at one pint one I get out two point one. Instead of just plugging in one value, let's plug in a whole bunch of values. We'll make a table. So use that same function again. F of x is x^second minus one divided by x minus one Now, I can't plug 1 into the function, 'because if I plug in one, I'd be dividing by zero, and I can't divide by zero. One isn't in the domain of this function. But I can plug in numbers near one, right? And we saw that one point one if I plug in that, I get two point one. Right? And if I plug in one point zero one I get two point zero one If I plug in 1 point zero zero one I get two point zero zero one. Right? And so on. If I plug in 1.000001 I get 2.000001. Right. Well, what's going on here? I could summarize this situation by saying the following.
The limit of x squared minus one over x minus one as x approaches one is equal to two. Why is that? Well, this is because. I can make x^2-1 over x minus one. As close to two as I want. If. I make x. Close enough.
To one. Lets see. Here's my table alright if you want the output of this function to be within a billionth of two all you need to do is to make sure that your input is within a trillionth of one alright. As long as your input is close enough to one you can guarantee that your output is as close to two as you like. This is just looking at a table of values. You know, maybe a dozen values and seeing what they're getting close to. It would be a lot better if there were a more convincing argument. So let's go back to our definition of limit. To say the limit of f of x equals l means that f of x can be made as close to l as you desire by making x close enough to a. And let me emphasize something. Close enough. But not equal. To a. Why does something like this matter? Well, let's go back to our example. In our example the function wasn't defined at one. But the limit doesn't depend upon the function's value at one. It only depends on the function's value near one. So x squared minus one over x minus one is equal to x plus one as long as x isn't equal to one right. As long as x isn't one this is a true statement. So now what's the limit as x goes to one of x squared minus one over x - one.
Well, this is the limit as x approaches one of x. one = one. because the limit doesn't depend upon the value of the function at one. It only depends upon the values of the function near one. And as a result, these two things have the same limit. Even better the limit of x plus one, as x approaches one, well that's the limit of a sum. And the limit of a sum is the sum of the limits. So I can rewrite this limit as the limit as x goes to one of x plus the limit as x goes to one of one. And what the limit of x as x goes to one? Well that's asking what can I make x close to if I make x close enough to one? Well that's one. And the limit of one as x goes to one is asking me what's one close to when x is close to well there's not even an x in this right wiggling x doesn't affect this at all so that limits also one. And one plus one is two so indeed the limit. x^twenty two minus one divided by x - one as x approaches one is two. Limits provide information about what a functions values are approaching, alright. It's a way of accessing otherwise forbidden information. I might not be able to plug in the value one, because that would have entailed dividing by zero. And yet I know, that the functions output is as close to two as I like. As long as the input is close to but not equal to one.
What is the limit of a product?
Here is an arithmetic problem, 660 * 310. I'm going give this exercise to Bart. So, the product of 660 and 310 is 204,600. But, what if I perturb these inputs a little bit? Instead of assigning this multiplication exercise, I could have assigned this multiplication exercise, 664 * 311. Let's give this to somebody else. Hi. My name is Vadranna. So, 664 * 311 is 206,504. Which isn't so far off of the answer that Bart got when he multiplied 660 times 310 and got 204,600. So, look at this. We've got two different problems. The input to these multiplication problems are similar, the outputs are also similar. Let's do some more, very similar multiplication problems. Hello, my name is Sean Gory. Oh, this is great. Look. 204,600, 206,504, 206,926, 204,702. We multiplied all of these pairs of nearby numbers, and the result of the multiplications were also nearby. There's a limit lesson hiding at all of this. If the limit of f of x as x approaches a is L, and the limit of g of x as x approaches a is M, then the limit of f of x times g of x as x approaches a is equal to L times M. In other words, the limit of a product is the product of the limits provided those limits exist
What is the limit of a quotient?
Fundamentally, limits are promises. When I tell you the limit of F of X equals L, as X approaches A; I'm promising you something. I'm promising you that I can get F of X as close to L as you like as long as X is close enough to A. Thinking of limits as promises helps us to understand statements like these. Let's suppose that you tell me that you know the limit of f of x as x approaches a is equal to something, maybe this is equal to l. And maybe the limit of g of x as x approaches a is equal to m. What that really is, is a promise that you can make F of X as close to L as I like. And it's a promise that you can make G of X as close to M as I like, as long as I'm willing to make X close enough to A. But if you can promise me that you can make F of X close to L, and G of X close to M. Then I can turn back and promise you that F of X plus G of X is as close to L plus M as you like. Alright? If I want to make this close to something, I just ask you to make F of X close enough to L, and G of X close enough to M. So that F of X plus G of X, is as you like to L plus M. In other words, the limit of a sum is the sum of the limits. And the limit of a difference is the difference of the limits. And the limit of a product is the product of the limits. And what about quotient. And something similar is true for division. If the limit of F of X as X approaches A is L. And the limit of G of X as X approaches A is M, which isn't zero, then the limit of F of X over G of X, as X approaches A is L over M. In other words, the limit of the quotient is the quotient of the limits, provided those limits exist and, the limit of the denominator is non-zero. Let us do something with our new found knowledge about limits of quotients. Here's a limit problem: I'm going to limit x-squared over x plus one as x approaches two, alright? I'll promise you that x-squared over x plus one is close to something whenever x is close enough to two. This is the limit of a quotient, and the limit of the quotient's the quotient of the limits, provided the limit of the denominator is not zero, and in this case, it's not. So the limit of the quotient is the quotient of the limits. Here's the limit of the numerator. The limit is X approaches two of X plus one. This is the limit of the denominator. Now this is the limit of x squared. X squared is X times X. This is a limit of a product. And the limit of a products the product to the limits so I can replace the limit of the numerator with a limit of X as X approaches two times the limit of X as X approaches two. because this is the limit of X times X and here's the product limits. Limit of X times the limit of X as X approaches two. The denominator here is the limit of X plus one as X approaches two but that's a limit of a sum and the limit of the sum is the sum of the limits. So the limit of X plus one is the limit of X plus the limit of one, as X approaches two. Lets keep going. So I've got the limit of X times the limit of X over the limit of X plus the limit of one. And all of these limits are being taken, as X approaches two. What's the limit of X? That's asking what can you guarantee X is close to if you're willing to have X be close enough to two? Two is the limit of X as X goes to two. So limit of X as X goes to two is two. The limit of X as X goes to two is doubt is for multiplying, divided by limit of X as X approaches two plus with the limit of one as X approaches two. This is asking, what can I guarantee one is close to two if I am willing to have X be close enough to two. Well, one is already close to one, right. The limit of a constant is that constant. So this is just one. Two times two is four. Two plus one is three. And so the limit of this expression is four-thirds. At this point you are asking yourself why is the rule for limits of quotients different than limits of the products. A limit of a product is a product of the limits. Laws of limits exist. Why do I have to worry about the limit of a nominator being non-zero, when I'm taking the limit of a quotient? Most basically the problem is that you can't divide by zero. You can't go around telling people that the limit of a quotient is a quotient of the limits because the limit of the denominator might be zero and then you'd be telling people to divide by zero, which they can't do. You can't divide by zero. But you can think about it even a little more subtlety. You know, let's kind of unpack this a bit. Here's an example to think about: the limit of x over x minus three as x approaches six. This is no problem, alright? The numerator is a number close to six, it's how we're thinking about it, and the denominator is a number close to six minus three. A number close to six minus three, that means the denominator is a number close to three. Now, we've got a number close to six divided by a number close to three. Well, that's a number close to two. And, indeed, I mean, this limit is equal to two. I can make this quotient as close to two as I like. Because I can make the numerator as close to six as I need, the denominator is as close to three as I need, to guarantee that this ratio is as close to two as you like. So that limit is two. But what if instead of asking about the limit as x approaches six, I'd ask about the limit as x approaches three. Well, then what would I know? Then I'd know that the numerator was a number close to three, and the denominator was a number close to three minus three. The denominators aren't close to zero. The limited denominator is zero. That's exactly the scenario that the rule for taking limits of quotients is forbidding us from considering. We're not allowed to use the rule for limits of quotients here because the limited denominator is zero. But what really goes wrong? I mean, yeah, I can't divide by zero. Fine, I'm not going to divide by zero, I'm just dividing by numbers close to zero. But what happens when I divide by numbers close to zero? A number close to three divided by a number close to zero. Is that close to anything? If a number's close to zero it might be positive, and very small. Three divided by a small positive number is a huge positive number. What if the denominator were a number close to zero but negative? Very small, negative number close to zero. Three divided by a small but negative number? That would be a hugely negative number. Every negative. Well, this means this thing should be getting close to both a gigantic positive number and a very, very negative number. This thing isn't getting close to anything. Fundamentally that's the problem but not all is lost. Think about this slightly harder example: the limit of x-squared minus one over x minus one as x approaches one. It's a limit of a quotient. So your first temptation is to replace the limit of a quotient, by the quotient of the limits. But you can't replace it without quotient of limits, because the limit of the denominator is zero. The limit of X-1 as X approaches one is equal to zero. So it seems like our limit laws have failed us. And then we have got one more trick up our sleeve. Look at the numerator. X squared - one that factors as X+1 times, X-1. First record that fact. And I am going to record the fact that the numerator factor is X+1 times X-1. Now I've still got a limit of a quotient and the limit of the denominator is still zero. So it seems like we're stuck. But now I've got a factor of X minus one in the numerator and a factor of X minus one in the denominator, and I can use that fact. What I want to do is imagine canceling these, right. I'd like to write that this is equal to the limit of just X plus one, as X goes to one. But note, these are not actually the same function. Alright? This thing up here is not defined at one, this thing is defined at one. And yet the limit doesn't care. The limit only depends upon values of the function near one. And near one, this and this are exactly the same. I'm not allowed to plug one into this, I am allowed to plug one into this. They're different functions. But those functions are equal if you're only considering values that are near and not equal to one. As a result, these limits are the same. This is another limit of a sum, and the limit of a sum is the sum of the limits, so this is the limit of X plus the limit of one as X approaches one. The limit of X as X approaches one is one. The limit of one, which is a constant, as X approaches one is one. This is one plus one. This is two. And so that limit is equal to two and our limit laws have again saved the day.
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