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Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | I get K is equal to negative one half, negative one half, natural log, natural log of two thirds. I just swap sides. Natural log of two thirds. Let me do that since I kept the colors going so long. Let me keep it that way. Natural log of two thirds. And so we have solved for all of the constants. |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | Let me do that since I kept the colors going so long. Let me keep it that way. Natural log of two thirds. And so we have solved for all of the constants. So now we can rewrite this thing right over here. We can rewrite it as, so we decided for a mini drum roll here. We're not completely done yet. |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | And so we have solved for all of the constants. So now we can rewrite this thing right over here. We can rewrite it as, so we decided for a mini drum roll here. We're not completely done yet. We get T of T is equal to, is equal to 60 E, E to the negative K. Well negative K, the negative and negative is going to be posi... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | We're not completely done yet. We get T of T is equal to, is equal to 60 E, E to the negative K. Well negative K, the negative and negative is going to be positive. It's going to be one half natural log of two thirds. So one half natural log of two thirds. Negative K, so negative of a negative. So one half natural log ... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | So one half natural log of two thirds. Negative K, so negative of a negative. So one half natural log of two thirds. Which actually will be a negative value. Two thirds is less than E. So you're going to have a natural log of it is going to be negative. So it makes you feel good that the temperature is going to be goin... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | Which actually will be a negative value. Two thirds is less than E. So you're going to have a natural log of it is going to be negative. So it makes you feel good that the temperature is going to be going down over time. And so plus, oh times T, times T plus 20. Plus 20. And so now all we have to do is we have to figur... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | And so plus, oh times T, times T plus 20. Plus 20. And so now all we have to do is we have to figure out what T gets us to a temperature of 40 degrees Celsius. So let's solve for that. So if we want this to be 40, 40 is equal to, and actually now I'm just going to stick to one color as we march through this part. So 40... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | So let's solve for that. So if we want this to be 40, 40 is equal to, and actually now I'm just going to stick to one color as we march through this part. So 40 is going to be equal to 60 E to the one half natural log of two thirds T power plus 20. Now let's see, we can subtract 20 from both sides. We get 20 is equal t... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | Now let's see, we can subtract 20 from both sides. We get 20 is equal to 60 E, all that crazy business. One half natural log of two thirds times T. Now we can divide both sides by 60 and we get one third, 20 divided by 60 is one third, is equal to E to the one half natural log of two thirds times T. Now let's see, we c... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | So we get the natural log of one third is equal to one half natural log of two thirds times T. And then home stretch to solve for T, you just divide both sides by one half natural log of two thirds. So we get T is equal to this, which is the natural log of one third divided by one half natural log of two thirds. Well i... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | Two thirds. Now let's see if this actually makes a sensical answer. So let me get a calculator out. Actually I could just use Google here. So I had, actually I already forgot what it was. Natural one, so I have natural log of one third over natural log of two thirds and the whole thing times two. So I can type, I can t... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | Actually I could just use Google here. So I had, actually I already forgot what it was. Natural one, so I have natural log of one third over natural log of two thirds and the whole thing times two. So I can type, I can type two, two times the natural log of, natural log of one third, natural log of one third divided by... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | So I can type, I can type two, two times the natural log of, natural log of one third, natural log of one third divided by, divided by the natural log of two thirds. Let's see what Google gets us. Alright, it didn't, what did I, where, how did I mess up? So this is equal to two times the natural log, oh okay, I messed ... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | So this is equal to two times the natural log, oh okay, I messed up the parentheses. Let me make this clear. The natural log of one third divided by the natural log of two thirds. And then we're, there you go. If we were to round to the nearest hundredth, it would be 5.42. So 5.42 minutes. Remember, everything we were ... |
Applying Newton's Law of Cooling to warm oatmeal First order differential equations Khan Academy.mp3 | And then we're, there you go. If we were to round to the nearest hundredth, it would be 5.42. So 5.42 minutes. Remember, everything we were doing were in minutes. So this right over here, this is approximately equal to 5.42, 5.42 minutes. And we are done. That's how long it will take us to cool to 40 degrees. |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | For example, if you know that y is equal to f of x, you might write this as y prime. You might write this as dy dx, which you'll often hear me say is the derivative of y with respect to x, and that you could use the derivative of f with respect to x, because y is equal to our function. But then later on, especially whe... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | For example, you will learn, or you might have already seen, if you're trying to solve the differential equation, the derivative of y with respect to x is equal to y, so the rate of change of y with respect to x is equal to the value of y itself. This is one of the most basic differential equations you might see. You'l... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | So you multiply both sides by dx, and then you have, so that would cancel out algebraically. And so you see people treat it like that. So you have dy is equal to y times dx, and then they'll say, okay, let's divide both sides by y, which is a reasonable thing to do. Y is an algebraic expression. So if you divide both s... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | Y is an algebraic expression. So if you divide both sides by y, you get one over y dy is equal to dx. And then folks will integrate both sides to find a general solution to this differential equation. But my point on this video isn't to think about how do you solve a differential equation here, but to think about this ... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | But my point on this video isn't to think about how do you solve a differential equation here, but to think about this notion of using what we call differentials, so a dx or a dy, and treating them algebraically like this, treating them as algebraic expressions where I can just multiply both sides by just dx or dy or d... |
Addressing treating differentials algebraically AP Calculus AB Khan Academy.mp3 | It's not super mathematically rigorous, but it has proven to be a useful tool for us to find these solutions. And conceptually, the way that I think about a dy or a dx is this is the super small change in y in response to a super small change in x. And that's essentially what this definition of the limit is telling us,... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | I'll now introduce you to the concept of the Laplace transform, and this is truly one of the most useful concepts that you'll learn, not just in differential equations, but really in mathematics. And especially if you're going to go into engineering, you'll find that the Laplace transform, besides helping you solve dif... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | Now I'll just teach you what it is, Laplace transform. I'll teach you what it is, make you comfortable with the mathematics of it, and then in a couple of videos from now, I'll actually show you how it is useful to use it to solve differential equations. We'll actually solve some of the differential equations we did be... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So what is the Laplace transform? Well, the Laplace transform, the notation is the L, like Laverne from Laverne and Shirley. Might be before many of your times, but I grew up on that. So actually, I think it was even reruns when I was a kid. Maybe, well anyway. So Laplace transform of some function, and here the conven... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So actually, I think it was even reruns when I was a kid. Maybe, well anyway. So Laplace transform of some function, and here the convention, instead of saying f of x, people say f of t. And the reason is, is because in a lot of the differential equations or a lot of engineering, you actually are converting from a func... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | But Laplace transform of a function of t, it transforms that function into some other function of s. And how does it do that? Well actually, let me just do some mathematical notation that probably won't mean much to you. So what is a transform? Well, the way I think of it is kind of a function of functions. A function ... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | Well, the way I think of it is kind of a function of functions. A function will take you from one set of, well in what we've been dealing with, one set of numbers to another set of numbers. A transform will take you from one set of functions to another set of functions. Let me just define this. The Laplace transform, f... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | Let me just define this. The Laplace transform, for our purposes, is defined as the improper integral, and I know I haven't actually done improper integrals just yet, but I'll explain them in a few seconds, the improper integral from 0 to infinity of e to the minus st times f of t. So whatever's between the Laplace tra... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So what is a Laplace transform? Well, let's say that f of t is equal to 1. So what is the Laplace transform of 1? So if f of t is equal to 1, it's just a constant function of time, well that will be equal to, and let me rewrite it, well actually let me just substitute exactly the way I wrote it here. So that's the impr... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So if f of t is equal to 1, it's just a constant function of time, well that will be equal to, and let me rewrite it, well actually let me just substitute exactly the way I wrote it here. So that's the improper integral from 0 to infinity of e to the minus st times 1 here. Well, I don't have to rewrite it here, but the... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | And I know that infinity is probably bugging you right now, but we'll deal with that shortly. Actually, let's deal with that right now. This is the same thing. This is the same thing as the limit. And let's say as a approaches infinity of the integral from 0 to a, e to the minus st dt. So this is, just so you feel a li... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | This is the same thing as the limit. And let's say as a approaches infinity of the integral from 0 to a, e to the minus st dt. So this is, just so you feel a little bit more comfortable with it, you might have guessed that this is the same thing, because obviously you can't evaluate infinity, but you can take the limit... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So anyway, let's take the antiderivative and evaluate this improper definite integral or this improper integral. So what's the antiderivative of e to the minus st with respect to dt? Well, it's equal to minus 1 over s e to the minus st. If you don't believe me, take the derivative of this. You take minus s times that, ... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | If you don't believe me, take the derivative of this. You take minus s times that, that would all cancel out and you'd just be left with e to the minus st. Fair enough. And we're going to take, actually let me delete this here, this equal sign, because I could actually use some of that real estate. We are going to take... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | We are going to take the limit as a approaches infinity. You don't always have to do this, but this is the first time we're dealing with improper integrals, so I figured I might as well remind you that we're taking a limit. And we're going to evaluate, we took the antiderivative, now we have to evaluate it at a minus t... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So this is equal to the limit as a approaches infinity. OK, if we substitute a in here first, we get minus 1 over s. Remember, we're dealing with t. We took the integral with respect to t. e to the minus s a, right? That's what happens when I put a in here. Minus, now what happens if when I put t equals 0 in here? So w... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | Minus, now what happens if when I put t equals 0 in here? So when t equals 0, it becomes e to the minus s times 0, this whole thing becomes 1, and I'm just left with minus 1 over s. So I have minus 1 over s. Fair enough. And then let me scroll down a little bit. I wrote a little bit bigger than I wanted to, but that's ... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | I wrote a little bit bigger than I wanted to, but that's OK. So this is going to be the limit as a approaches infinity of minus 1 over s e to the minus s a minus minus 1 over s, so plus 1 over s. So what's the limit as a approaches infinity? Well, what's this term going to do? As a approaches infinity, the exponent is ... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | As a approaches infinity, the exponent is going to get, if we assume that s is greater than 0, and we'll make that assumption for now. Let me write that down explicitly. Let's assume that s is greater than 0. So if we assume that s is greater than 0, then as a approaches infinity, what's going to happen? Well, this ter... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So if we assume that s is greater than 0, then as a approaches infinity, what's going to happen? Well, this term is going to go to 0, right? e to the minus a googol is a very, very small number, and e to the minus googolplex is a very even smaller number. So then this e to the minus infinity approaches 0, so this term ... |
Laplace transform 1 Laplace transform Differential Equations Khan Academy.mp3 | So then this e to the minus infinity approaches 0, so this term approaches 0. This term isn't affected because it has no a in it, so we're just left with 1 over s. So there you go. This is a significant moment in your life. You have just been exposed to your first Laplace transform. Eventually, and I'll show you in a f... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Let's keep doing some Laplace transforms. And one, it's good to see where a lot of those Laplace transform tables you'll see later on actually come from. And it just makes you comfortable with the mathematics, which is really just kind of your second semester calculus mathematics. But it makes you comfortable with the ... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | But it makes you comfortable with the whole notion of what we're doing. So first of all, let me just rewrite the definition of the Laplace transform. So it's the L from Laverne and Shirley. So Laplace transform of some function of t is equal to the improper integral from 0 to infinity of e to the minus st times our fun... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | So Laplace transform of some function of t is equal to the improper integral from 0 to infinity of e to the minus st times our function. Times our function of t and with respect to dt. So let's do another Laplace transform. Let's say that we want to take the Laplace transform. And now our function f of t, let's say it ... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Let's say that we want to take the Laplace transform. And now our function f of t, let's say it is e to the at. Laplace transform of e to the at. Well, we just substituted it into this definition of the Laplace transform. So that is equal to. And this is all going to be really good integration practice for us, especial... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Well, we just substituted it into this definition of the Laplace transform. So that is equal to. And this is all going to be really good integration practice for us, especially integration by parts. Almost every Laplace transform problem turns into an integration by parts problem. Which as we learned long ago, integrat... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Almost every Laplace transform problem turns into an integration by parts problem. Which as we learned long ago, integration by parts is just the reverse product rule. So anyway, this is equal to the integral from 0 to infinity, e to the minus st times e to the at. That's our f of t. dt. Well, this is equal to, just ad... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | That's our f of t. dt. Well, this is equal to, just adding the exponents, because we're going to use the exponents. Equal to, just adding the exponents, because we have the same base, the integral from 0 to infinity of e to the what? a minus s. e to the a minus st dt. And what's the antiderivative of this? Well, that's... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | a minus s. e to the a minus st dt. And what's the antiderivative of this? Well, that's equal to what with respect to t? So we're at, it's equal to 1 over a minus s. That's just going to be a constant, right? So we can just leave it out on the outside. 1 over a minus s times e to the a minus st. And we're going to evalu... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | So we're at, it's equal to 1 over a minus s. That's just going to be a constant, right? So we can just leave it out on the outside. 1 over a minus s times e to the a minus st. And we're going to evaluate that from t is equal to infinity, or the limit as t approaches infinity, to t is equal to 0. And I could have put th... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | And I could have put this inside the brackets, but it's just a constant term, right? a, none of them have t's in them, so I can just pull them out. And so this is equal to 1 over a minus s times. Now, what is the limit? We essentially have to evaluate t at infinity. So what is the limit at infinity? Well, we have two c... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Now, what is the limit? We essentially have to evaluate t at infinity. So what is the limit at infinity? Well, we have two cases here, right? If this exponent, if this a minus s is a positive number, if a minus s is greater than 0, what's going to happen? Well, as we approach infinity, e to the infinity just gets bigge... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Well, we have two cases here, right? If this exponent, if this a minus s is a positive number, if a minus s is greater than 0, what's going to happen? Well, as we approach infinity, e to the infinity just gets bigger and bigger and bigger, right? Because it's e to an infinitely positive exponent. So it actually, we don... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Because it's e to an infinitely positive exponent. So it actually, we don't get an answer. And when you do improper integrals, when you take the limit to infinity and it doesn't come to a finite number, the limit doesn't approach anything, that means that the limit, or actually the improper integral, diverges. And so t... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | And so there is no limit. And to some degree, we can say that the Laplace transform is not defined when a minus s is greater than 0, or when a is greater than s. Now, what happens if a minus s is less than 0? So if a minus s is less than 0, well then, this is going to be some negative number here, right? And then if we... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | And then if we take e to an infinitely negative number, well then, that does approach something. That approaches 0. And we saw that in the previous video. And I hope you understand what I'm saying, right? e to an infinitely negative number is 0, or it approaches 0, while e to an infinitely positive number is just infin... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | And I hope you understand what I'm saying, right? e to an infinitely negative number is 0, or it approaches 0, while e to an infinitely positive number is just infinity, so it doesn't really converge on anything. So anyway, so if I assume that a minus s is less than 0, or a is less than s, and this is the assumption I ... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | So if a minus s is less than 0, then this is a negative number, e to the a minus s times, well, t, where t approaches infinity, will be 0, minus this integral evaluated at 0. So when you evaluate this at 0, what happens? t equals 0, this whole thing becomes 0, e to the 0 is 1. e to the 0 is 1, and we are left with what... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Minus 1 over a minus s, and that's just the same thing as 1 over s minus a. So we have our next entry in our Laplace transform table. We have our next entry, and that is the Laplace transform of e to the a t is equal to 1 over s minus a, as long as we make the assumption that s is greater than a. This is true when s is... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | This is true when s is greater than a, or a is less than s. Either way, you could view it either way. So that's our second entry in our Laplace transform table. Fascinating. And actually, let's relate this to our previous entry in our Laplace transform table. What was our first entry in our Laplace transform table? It ... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | And actually, let's relate this to our previous entry in our Laplace transform table. What was our first entry in our Laplace transform table? It was Laplace transform of 1 is equal to 1 over s, right? Well, isn't 1 just the same thing as e to the 0? So we could have said that this is the Laplace, I know I'm running ou... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | Well, isn't 1 just the same thing as e to the 0? So we could have said that this is the Laplace, I know I'm running out of space, but I'll do it here in purple. We could have said Laplace transform of 1 is the same thing as e to the 0 times t, right? And that equals 1 over s. And luckily, it's good to see that that is ... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | And that equals 1 over s. And luckily, it's good to see that that is consistent. And actually, remember, we even made the condition when s is greater than 0, right? We assumed that s is greater than 0 in this example. And lo and behold, here again, you say s is greater than 0. This is completely consistent with this on... |
Laplace transform 2 Laplace transform Differential Equations Khan Academy.mp3 | And lo and behold, here again, you say s is greater than 0. This is completely consistent with this one, right? Because if a is equal to 0, then the Laplace transform of e to the 0 is just 1 over s minus 0. That's just 1 over s. And we have to assume that s is greater than 0. So really, these are kind of the same entry... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | And I defined it to be, and I'll do the shifted version of it, you're already hopefully reasonably familiar with it. If I shift it, so Dirac delta of t minus c, we can say that it equals 0 when t does not equal c, so it equals 0 everywhere, but it essentially pops up to infinity. And we have to be careful with this inf... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | I'm going to write it in quotes. It pops up to infinity, and we even saw in the previous video, it's kind of different degrees of infinity, because you can still multiply this by other numbers to get larger Dirac delta functions when t is equal to c. But more important than this, and this is kind of a pseudo-definition... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | So this will be equal to 2. So if I put a 2 out here, this infinity will have to be twice as high so that the area is now 2. That's why I put that infinity in parentheses. But it's an interesting function. I talked about it at the end of the last video, that it can help model things that kind of jar things all of a sud... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | But it's an interesting function. I talked about it at the end of the last video, that it can help model things that kind of jar things all of a sudden, but they impart a fixed amount of impulse on something, a fixed amount of change in momentum. And we'll understand that a little bit more in the future. But let's kind... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | But let's kind of get the mathematical tools completely understood, and let's try to figure out what the Dirac delta function does when we multiply it, what it does to the Laplace transform when we multiply it times some function. So let's say I have my Dirac delta function, and I'm going to shift it. That's a little b... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | And if you want to unshift it, you just say, okay, well, c equals 0. What happens when c equals 0? And I'm going to shift it and multiply it times some arbitrary function f of t. If I wanted to figure out the Laplace transform of just the delta function by itself, I could say f of t is equal to 1. So let's take our Lap... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | So let's take our Laplace transform of this, and we can just use the definition of the Laplace transform. So this is equal to the area from 0 to infinity underneath, or we could call it the integral from 0 to infinity, of e to the minus st. That's just part of the Laplace transform definition, times this thing. And I'l... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Now, here I'm going to make a little bit of an intuitive argument. A lot of the math we do is kind of, especially if you want to be very rigorous and formal, the Dirac delta function starts to break down a lot of tools that you might have not realized it would break down. But I think intuitively we can still work with ... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | So I'm going to solve this integral for you intuitively, and I think it'll make some sense. So let's draw this. Let me draw this, what we're trying to do. So let me draw what we're trying to take the integral of. And we only care from 0 to infinity, so I'll only do it from 0 to infinity. And I'll assume that c is great... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | So let me draw what we're trying to take the integral of. And we only care from 0 to infinity, so I'll only do it from 0 to infinity. And I'll assume that c is greater than 0, that the delta function pops up someplace in the positive t-axis. So what is this first part going to look like? What is that going to look like... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | So what is this first part going to look like? What is that going to look like? e to the minus st times f of t. I don't know. It's going to be some function. e to the minus st starts at 1 and drops down, but we're multiplying it times some arbitrary function. So I'll just draw it like this. Maybe it looks something lik... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | It's going to be some function. e to the minus st starts at 1 and drops down, but we're multiplying it times some arbitrary function. So I'll just draw it like this. Maybe it looks something like this. This right here is e to the minus st times f of t. And the f of t is what kind of gives it its arbitrary shape. Fair e... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Maybe it looks something like this. This right here is e to the minus st times f of t. And the f of t is what kind of gives it its arbitrary shape. Fair enough. Now let's graph our Dirac delta function. It's 0 everywhere except right at c, it pops up infinitely high. But we only draw an arrow that is of height 1 to sho... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Now let's graph our Dirac delta function. It's 0 everywhere except right at c, it pops up infinitely high. But we only draw an arrow that is of height 1 to show that its area is 1. Normally when you graph things you don't draw arrows, but this arrow shows that the area under this infinitely high thing is 1. So we do a ... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Normally when you graph things you don't draw arrows, but this arrow shows that the area under this infinitely high thing is 1. So we do a 1 there. So if we multiply this, we care about the area under this whole thing. When we multiply these two functions, when we multiply this times this times the delta function, let ... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | When we multiply these two functions, when we multiply this times this times the delta function, let me write this, this is the delta function shifted to c. If I multiply that times that, what do I get? This is kind of the key intuition here. Let me redraw my axes. Let me see if I can do it a little bit straighter. Don... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Let me see if I can do it a little bit straighter. Don't judge me by the straightness of my axes. So that's t. So what happens when I multiply these two? Everywhere, when t equals anything other than c, the Dirac delta function is 0. So it's 0 times anything. I don't care what this function is, it's going to be 0. So i... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Everywhere, when t equals anything other than c, the Dirac delta function is 0. So it's 0 times anything. I don't care what this function is, it's going to be 0. So it's going to be 0 everywhere except something interesting happens at t is equal to c. At t equals c, what's the value of the function? Well, it's going to... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | So it's going to be 0 everywhere except something interesting happens at t is equal to c. At t equals c, what's the value of the function? Well, it's going to be the value of the Dirac delta function times whatever height this is. This is going to be this point right here, or this right there, that point, this is going... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | This is going to be e to the minus sc times f of c. All I'm doing is I'm just evaluating this function at c. So that's the point right there. So if you take this point, which is just some number, it could be 5, 5 times this, you're just getting 5 times the Dirac delta function. Or, in this case, it's not 5, it's this l... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | I could just draw it like this. When I multiply this thing times my little delta function there, I get this. The height, it's a delta function, but it's scaled now. It's scaled, so now my new thing is going to look like this. If I just multiply that times that, I essentially get e to the minus sc times f of c. This mig... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | It's scaled, so now my new thing is going to look like this. If I just multiply that times that, I essentially get e to the minus sc times f of c. This might look like some fancy function, but it's just a number when we consider it in terms of t. S becomes something when we go into the plus world, but from t's point of... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | So it's this constant times my Dirac delta function, times delta of t minus c. When I multiply that thing times that thing, all I'm left with is this thing. This height is still going to be infinitely high, but it's infinitely high scaled in such a way that its area is going to be not 1, and I'll show it to you. So wha... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Taking the integral of this thing from minus infinity to infinity, since this thing is this thing, it should be the same thing as taking the integral of this thing from minus infinity to infinity. So let's do that. Actually, we only have to do it from minus infinity. I said from 0 to infinity. So if we take from 0 to i... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | I said from 0 to infinity. So if we take from 0 to infinity, what I'm saying is taking this integral is equivalent to taking this integral. So e to the minus sc f of c times my delta function, t minus c dt. Now, this thing right here, let me make this very clear. I'm claiming that this is equivalent to this, because ev... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Now, this thing right here, let me make this very clear. I'm claiming that this is equivalent to this, because everywhere else the delta function zeros out this function, so we only care about this function, or e to the minus st f of t, when t is equal to c. And so that's why we were able to turn it into a constant. Bu... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | And so this is equal to, I'm going to go backwards here just to kind of save space and still give you these things to look at. If we take out the constants from inside of the integral, we get e to the minus sc times f of c times the integral from 0 to infinity of f of t minus c dt. Oh, sorry, not f of t minus c. Let me... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | This is not an f. I have to be very careful. This is a delta. Let me do that in a different color. I took out the constant terms there, and it's going to be of delta of t minus c dt. Let me get the right color, dt. Now, what is this thing by definition? This thing is 1. |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | I took out the constant terms there, and it's going to be of delta of t minus c dt. Let me get the right color, dt. Now, what is this thing by definition? This thing is 1. I mean, we could put it from minus infinity to infinity. It doesn't matter. The only time where it has any error is right under c. So this thing is ... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | This thing is 1. I mean, we could put it from minus infinity to infinity. It doesn't matter. The only time where it has any error is right under c. So this thing is equal to 1. So this whole integral right there has been reduced to this right there because this is just equal to 1. So the Laplace transform of our shifte... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | The only time where it has any error is right under c. So this thing is equal to 1. So this whole integral right there has been reduced to this right there because this is just equal to 1. So the Laplace transform of our shifted delta function times some other function is equal to e to the minus sc times f of c. Let me... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | Let me write it all at once. So the Laplace transform of our shifted delta function, t minus c, times some function f of t, it equals e to the minus c. Essentially, we're just evaluating e to the minus sc evaluated at c. So e to the minus cs times f of c. We're essentially just evaluating these things at c. This is wha... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | What is the Laplace transform? Actually, what is the Laplace transform of just the plain vanilla delta function? Well, in this case, we have c is equal to 0 and f of t is equal to 1. It's just a constant term. So if we do that, then the Laplace transform of this thing is just going to be e to the minus 0 times s times ... |
Laplace transform of the dirac delta function Laplace transform Khan Academy.mp3 | It's just a constant term. So if we do that, then the Laplace transform of this thing is just going to be e to the minus 0 times s times 1, which is just equal to 1. So the Laplace transform of our delta function is 1, which is a nice, clean thing to find out. Then if we wanted to just figure out the Laplace transform ... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Let's keep building our table of Laplace transforms. And now we'll do a fairly hairy problem, so I'm going to have to focus so that I don't make a careless mistake. But let's say we want to take the Laplace transform, and this is a useful one. Actually, all of them we've done so far are useful. I'll tell you when we st... |
L{sin(at)}) - transform of sin(at) Laplace transform Differential Equations Khan Academy.mp3 | Actually, all of them we've done so far are useful. I'll tell you when we start doing not so useful ones. Let's say we want to take the Laplace transform of the sine of some constant times t. Well, our definition of the Laplace transform that says that it's the improper integral. Remember, the Laplace transform is just... |
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