WEBVTT 00:00:10.000 --> 00:00:16.000 Today we are going to do a last serious topic on the Laplace 00:00:14.000 --> 00:00:20.000 transform, the last topic for which I don't have to make 00:00:19.000 --> 00:00:25.000 frequent and profuse apologies. One of the things the Laplace 00:00:24.000 --> 00:00:30.000 transform does very well and one of the reasons why people like 00:00:29.000 --> 00:00:35.000 it, engineers like it, is that it handles functions 00:00:33.000 --> 00:00:39.000 with jump discontinuities very nicely. 00:00:47.000 --> 00:00:53.000 Now, the OR function with a jump discontinuity is-- Purple. 00:00:54.000 --> 00:01:00.000 Is a function called the unit step function. 00:01:00.000 --> 00:01:06.000 I will draw a graph of it. Even the graph is 00:01:03.000 --> 00:01:09.000 controversial, but everyone is agreed that it 00:01:07.000 --> 00:01:13.000 zero here and one there. What people are not agreed upon 00:01:12.000 --> 00:01:18.000 is its value at zero. And some people make it zero, 00:01:16.000 --> 00:01:22.000 some people make it one, and some equivocate like me. 00:01:20.000 --> 00:01:26.000 I will leave it undefined. It is u of t. 00:01:24.000 --> 00:01:30.000 It is called the unit step. 00:01:32.000 --> 00:01:38.000 Because that is what it is. And let's say we will leave u 00:01:35.000 --> 00:01:41.000 of zero undefined. 00:01:43.000 --> 00:01:49.000 If that makes you unhappy, get over it. 00:01:46.000 --> 00:01:52.000 Of course, we don't always want the jump to be at zero. 00:01:51.000 --> 00:01:57.000 Sometimes we will want to jump in another place. 00:01:55.000 --> 00:02:01.000 If I want the function to jump, let's say at the point a 00:02:00.000 --> 00:02:06.000 instead of jumping at zero, I am going to start doing what 00:02:05.000 --> 00:02:11.000 everybody does. You put in the vertical lines, 00:02:10.000 --> 00:02:16.000 even though I have no meaning, whatever, but it makes the 00:02:15.000 --> 00:02:21.000 graph look more connected and a little easier to read. 00:02:19.000 --> 00:02:25.000 So that function I will call u sub a is the function 00:02:24.000 --> 00:02:30.000 which jumps at the point a. How shall I give its 00:02:28.000 --> 00:02:34.000 definition? Well, you can see it is just 00:02:32.000 --> 00:02:38.000 the translation by a of the unit step function. 00:02:36.000 --> 00:02:42.000 So that is the way to write it, u of t minus a. 00:02:42.000 --> 00:02:48.000 Now I am not done. There is a unit box function, 00:02:46.000 --> 00:02:52.000 which we will draw in general terms like this. 00:02:51.000 --> 00:02:57.000 It gets to a, then it jumps up to one, 00:02:54.000 --> 00:03:00.000 falls down again at b and continues onto zero. 00:03:00.000 --> 00:03:06.000 This happens between a and b. And the value to which it 00:03:04.000 --> 00:03:10.000 arises is one. I will call this the unit box. 00:03:17.000 --> 00:03:23.000 It is a function of t, a very simple one but an 00:03:20.000 --> 00:03:26.000 important one. And what would be the formula 00:03:23.000 --> 00:03:29.000 for the unit box function? Well, in general, 00:03:26.000 --> 00:03:32.000 almost all of these functions, as you will see when you use 00:03:30.000 --> 00:03:36.000 jump discontinuity, the idea is to write them all 00:03:33.000 --> 00:03:39.000 cleverly using nothing but u of. 00:03:37.000 --> 00:03:43.000 Because it is that will have the Laplace transformer. 00:03:41.000 --> 00:03:47.000 The way to write this is (u)ab. 00:03:43.000 --> 00:03:49.000 And, if you like, you can treat this as the 00:03:47.000 --> 00:03:53.000 definition of it. Let's make it a definition. 00:03:50.000 --> 00:03:56.000 Okay, three lines. Or, better yet, 00:03:53.000 --> 00:03:59.000 a colon and two lines. I am defining this to be, 00:03:57.000 --> 00:04:03.000 what would it be? Make the unit step at a step up 00:04:02.000 --> 00:04:08.000 at a, but then I would continue at one all the time. 00:04:07.000 --> 00:04:13.000 I should, therefore, step down at b. 00:04:10.000 --> 00:04:16.000 Now, the way you step down is just by taking the negative of 00:04:15.000 --> 00:04:21.000 the unit step function. I step down at b by subtracting 00:04:20.000 --> 00:04:26.000 u sub b of t . In other words, 00:04:24.000 --> 00:04:30.000 it is u of t minus a minus u of t minus b. 00:04:29.000 --> 00:04:35.000 And now I have expressed it 00:04:32.000 --> 00:04:38.000 entirely in terms of the unit step function. 00:04:35.000 --> 00:04:41.000 That will be convenient when I want take the Laplace transform. 00:04:39.000 --> 00:04:45.000 What is so good about these things? 00:04:41.000 --> 00:04:47.000 Well, these functions, when you use them in 00:04:43.000 --> 00:04:49.000 multiplications, they transform other functions 00:04:46.000 --> 00:04:52.000 in a nice way. Not transform. 00:04:48.000 --> 00:04:54.000 That is not the right word. They operate on them. 00:04:51.000 --> 00:04:57.000 They turn them into other strange creatures, 00:04:53.000 --> 00:04:59.000 and it might be these strange creatures that you are 00:04:56.000 --> 00:05:02.000 interested in. Let me just draw you a picture. 00:05:01.000 --> 00:05:07.000 That will be good enough. Suppose we have some function 00:05:07.000 --> 00:05:13.000 like that, f of t, what would the function u sub 00:05:12.000 --> 00:05:18.000 ab, I will put in the variable, 00:05:16.000 --> 00:05:22.000 t times f of t, what function would that be? 00:05:21.000 --> 00:05:27.000 I am just going to draw its graph. 00:05:24.000 --> 00:05:30.000 What would its graph be? Well, in between a and b this 00:05:30.000 --> 00:05:36.000 function (u)ab of t has the value one. 00:05:34.000 --> 00:05:40.000 All I am doing is multiplying f of t by one. 00:05:39.000 --> 00:05:45.000 In short, I am not doing anything to it at all. 00:05:43.000 --> 00:05:49.000 Outside of that interval, (u)ab has the value zero, 00:05:47.000 --> 00:05:53.000 so that zero times f of t makes zero. 00:05:51.000 --> 00:05:57.000 And, therefore, outside of this it is zero. 00:05:55.000 --> 00:06:01.000 The effect of multiplying an arbitrary function by this unit 00:06:00.000 --> 00:06:06.000 box function is, you wipe away all of its graph 00:06:03.000 --> 00:06:09.000 except the part between a and b. Now, that is a very useful 00:06:10.000 --> 00:06:16.000 thing to be able to do. Well, that is enough of that. 00:06:14.000 --> 00:06:20.000 Now, let's get into the main topic. 00:06:17.000 --> 00:06:23.000 That is just preliminary. I will be using these functions 00:06:22.000 --> 00:06:28.000 all during the period, but the real topic is the 00:06:25.000 --> 00:06:31.000 following. Let's calculate the Laplace 00:06:29.000 --> 00:06:35.000 transform of the unit step function. 00:06:33.000 --> 00:06:39.000 Well, this is no very big deal. It is the integral from zero to 00:06:38.000 --> 00:06:44.000 infinity e to the minus s t times y of t, dt. 00:06:45.000 --> 00:06:51.000 But, look, when t is bigger than zero, 00:06:49.000 --> 00:06:55.000 this has the value one. So it is the same of the 00:06:53.000 --> 00:06:59.000 Laplace transform of one. In other words, 00:06:56.000 --> 00:07:02.000 it is one over s for positive values of s. 00:07:02.000 --> 00:07:08.000 Or, to make it very clear, the Laplace transform of one is 00:07:06.000 --> 00:07:12.000 exactly the same thing. As you see, the Laplace 00:07:09.000 --> 00:07:15.000 transform really is not interested in what happens when 00:07:13.000 --> 00:07:19.000 t is less than zero because that is not part of the 00:07:17.000 --> 00:07:23.000 domain of integration, the interval of integration. 00:07:21.000 --> 00:07:27.000 That is fine. They both have Laplace 00:07:24.000 --> 00:07:30.000 transform of one over s. 00:07:26.000 --> 00:07:32.000 What is the big deal? The big deal is, 00:07:30.000 --> 00:07:36.000 what is the inverse Laplace transform of one over s? 00:07:37.000 --> 00:07:43.000 Will the real function please 00:07:41.000 --> 00:07:47.000 stand up? Which of these two should I 00:07:45.000 --> 00:07:51.000 pick? Up to now in the course we have 00:07:49.000 --> 00:07:55.000 been picking one just because I never made a fuss over it and 00:07:56.000 --> 00:08:02.000 one was good enough. For today one is no longer 00:08:01.000 --> 00:08:07.000 going to be good enough. And we have to first 00:08:05.000 --> 00:08:11.000 investigate the thing in a slightly more theoretical way 00:08:09.000 --> 00:08:15.000 because this problem, I have illustrated it on the 00:08:13.000 --> 00:08:19.000 inverse Laplace transform of one over x, 00:08:17.000 --> 00:08:23.000 but it occurs for any inverse Laplace transform. 00:08:20.000 --> 00:08:26.000 Suppose I have, in other words, 00:08:23.000 --> 00:08:29.000 that a function f of t has as its Laplace transform capital F 00:08:27.000 --> 00:08:33.000 of s? And now, I ask what the inverse 00:08:32.000 --> 00:08:38.000 Laplace transform of capital F of s is. 00:08:37.000 --> 00:08:43.000 Well, of course you want to write f of t. 00:08:41.000 --> 00:08:47.000 But the same thing happens. I will draw you a picture. 00:08:45.000 --> 00:08:51.000 Suppose, in other words, that here is our function f of 00:08:50.000 --> 00:08:56.000 t. Well, one answer certainly is f 00:08:53.000 --> 00:08:59.000 of t. That is okay. 00:08:54.000 --> 00:09:00.000 That is the answer we have been using up until now. 00:09:00.000 --> 00:09:06.000 But, you see, I can complete this function in 00:09:04.000 --> 00:09:10.000 many other ways. Suppose I haven't told you what 00:09:08.000 --> 00:09:14.000 it was for s less than zero. Any of these possibilities all 00:09:14.000 --> 00:09:20.000 will produce the same Laplace transform. 00:09:17.000 --> 00:09:23.000 In fact, I can even make it this. 00:09:20.000 --> 00:09:26.000 That is okay. Each of these, 00:09:23.000 --> 00:09:29.000 f of t with any one of these tails, all have the same 00:09:28.000 --> 00:09:34.000 Laplace transform. 00:09:37.000 --> 00:09:43.000 Because the Laplace transform, remember the definition, 00:09:41.000 --> 00:09:47.000 integral zero to infinity, e to the negative s t, 00:09:44.000 --> 00:09:50.000 f of t, dt 00:09:49.000 --> 00:09:55.000 because the Laplace transform does not care what the function 00:09:53.000 --> 00:09:59.000 was doing for negative values of t. 00:09:56.000 --> 00:10:02.000 Now, if we have to have a unique answer -- 00:10:00.000 --> 00:10:06.000 And most of the time you don't because, in general, 00:10:03.000 --> 00:10:09.000 the Laplace transform is only used for problems for future 00:10:07.000 --> 00:10:13.000 time. That is the way the engineers 00:10:09.000 --> 00:10:15.000 and physicists and other people who use it habitually think of 00:10:13.000 --> 00:10:19.000 it. If your problem is starting now 00:10:15.000 --> 00:10:21.000 and going on into the future and you don't have to know anything 00:10:20.000 --> 00:10:26.000 about the past, that is a Laplace transform 00:10:23.000 --> 00:10:29.000 problem. If you also have to know about 00:10:25.000 --> 00:10:31.000 the past, then it is a Fourier transform problem. 00:10:30.000 --> 00:10:36.000 That is beyond the scope of this course, you will never hear 00:10:34.000 --> 00:10:40.000 that word again, but that is the difference. 00:10:37.000 --> 00:10:43.000 We are starting at time zero and going forward. 00:10:41.000 --> 00:10:47.000 All right. It does not care what f of t 00:10:44.000 --> 00:10:50.000 was doing for negative values of t. 00:10:46.000 --> 00:10:52.000 And that gives us a problem when we try to make the Laplace 00:10:51.000 --> 00:10:57.000 transform unique. Now, how will I make it unique? 00:10:54.000 --> 00:11:00.000 Well, there is a simple way of doing it. 00:10:59.000 --> 00:11:05.000 Let's agree that wherever it makes a difference, 00:11:02.000 --> 00:11:08.000 and most of the time it doesn't, but today it will, 00:11:06.000 --> 00:11:12.000 whenever it makes a difference we will declare, 00:11:09.000 --> 00:11:15.000 we will by brute force make our function zero for negative 00:11:14.000 --> 00:11:20.000 values of t. That makes it unique. 00:11:16.000 --> 00:11:22.000 I am going to say that to make it unique, now, 00:11:20.000 --> 00:11:26.000 how do I make f of t zero for negative values 00:11:24.000 --> 00:11:30.000 of t? The answer is multiply it by 00:11:27.000 --> 00:11:33.000 the unit step function. That leaves it what it was. 00:11:32.000 --> 00:11:38.000 It multiplies it by one for positive values but multiplies 00:11:37.000 --> 00:11:43.000 it by zero for negative values. The answer is going to be u of 00:11:42.000 --> 00:11:48.000 t times f of t. That will be the function that 00:11:48.000 --> 00:11:54.000 will look just that way that I drew, but I will draw it once 00:11:53.000 --> 00:11:59.000 more. It is the function that looks 00:11:56.000 --> 00:12:02.000 like this. And when I do this, 00:11:58.000 --> 00:12:04.000 it makes the inverse Laplace transform unique. 00:12:04.000 --> 00:12:10.000 Out of all the possible tails I might have put on f of t, 00:12:08.000 --> 00:12:14.000 it picks the least interesting one, 00:12:11.000 --> 00:12:17.000 the tail zero. 00:12:23.000 --> 00:12:29.000 That is a start. But what we have to do now is 00:12:26.000 --> 00:12:32.000 -- 00:12:35.000 --> 00:12:41.000 What I want is a formula. What we are going to need is, 00:12:38.000 --> 00:12:44.000 as you see right even in the beginning, if for example, 00:12:42.000 --> 00:12:48.000 if I want to calculate the Laplace transform of this, 00:12:46.000 --> 00:12:52.000 what I would like to have is a nice Laplace transform for the 00:12:50.000 --> 00:12:56.000 translate. If you translate a function, 00:12:52.000 --> 00:12:58.000 how does that effect this Laplace transform? 00:12:55.000 --> 00:13:01.000 In other words, the formula I am looking for is 00:12:58.000 --> 00:13:04.000 -- 00:13:08.000 --> 00:13:14.000 I want to express the Laplace transform of f of t minus a. 00:13:12.000 --> 00:13:18.000 In other words, 00:13:15.000 --> 00:13:21.000 the function translated, let's say a is positive, 00:13:19.000 --> 00:13:25.000 so I translate it to the right along the t axis by the distance 00:13:25.000 --> 00:13:31.000 a. I want a formula for this in 00:13:27.000 --> 00:13:33.000 terms of the Laplace transform of the function I started with. 00:13:34.000 --> 00:13:40.000 Now, my first task is to convince you that, 00:13:37.000 --> 00:13:43.000 though this would be very useful and interesting, 00:13:40.000 --> 00:13:46.000 there cannot possibly be such a formula. 00:13:43.000 --> 00:13:49.000 There is no such formula. 00:13:56.000 --> 00:14:02.000 Why not? Well, I think I will explain it 00:13:59.000 --> 00:14:05.000 over there since there is a little piece of board I did not 00:14:05.000 --> 00:14:11.000 use. Waste not want not. 00:14:07.000 --> 00:14:13.000 Why can't there be such a formula? 00:14:10.000 --> 00:14:16.000 What is it we are looking for? Let's take a nice average 00:14:15.000 --> 00:14:21.000 function f of t. It has a Laplace transform. 00:14:20.000 --> 00:14:26.000 And now I am going to translate it. 00:14:23.000 --> 00:14:29.000 Let's say this is the point negative a. 00:14:29.000 --> 00:14:35.000 And so the corresponding point positive a will be around here. 00:14:34.000 --> 00:14:40.000 I am going to translate it to the right by a. 00:14:38.000 --> 00:14:44.000 What is it going to look like? Well, then it is going to start 00:14:43.000 --> 00:14:49.000 here and is going to look like this dashy thing. 00:14:48.000 --> 00:14:54.000 That is f of t minus a. 00:14:51.000 --> 00:14:57.000 That is not too bad a picture. It will do. 00:14:54.000 --> 00:15:00.000 I just took that curve and shoved it to the right by a. 00:15:01.000 --> 00:15:07.000 Now, why is it impossible to express the Laplace transform of 00:15:09.000 --> 00:15:15.000 the dashed line in terms of the Laplace transform of the solid 00:15:18.000 --> 00:15:24.000 line? The answer is this piece. 00:15:23.000 --> 00:15:29.000 I will write it this way. The trouble is, 00:15:28.000 --> 00:15:34.000 this piece is not used for the Laplace transform of f of t. 00:15:37.000 --> 00:15:43.000 Why isn't it used? 00:15:41.000 --> 00:15:47.000 Well, because it occurs to the left of the vertical axis. 00:15:44.000 --> 00:15:50.000 It occurs for negative values of t. 00:15:47.000 --> 00:15:53.000 And the Laplace transform of f of t simply does not 00:15:51.000 --> 00:15:57.000 care what f of t was doing to the left of that line, 00:15:55.000 --> 00:16:01.000 for negative values of t. It does not enter into the 00:15:58.000 --> 00:16:04.000 integral. It was not used when I 00:16:01.000 --> 00:16:07.000 calculated this piece of the curve. 00:16:04.000 --> 00:16:10.000 It was not used when I calculated the Laplace transform 00:16:07.000 --> 00:16:13.000 of f of t. On the other hand, 00:16:10.000 --> 00:16:16.000 it is going to be needed. It occurs here, 00:16:13.000 --> 00:16:19.000 after I shift it to the right. It is going to be needed for 00:16:17.000 --> 00:16:23.000 the Laplace transform of f of t minus a, 00:16:21.000 --> 00:16:27.000 because I will have to start the integration here, 00:16:25.000 --> 00:16:31.000 and I will have to know what that is. 00:16:34.000 --> 00:16:40.000 In other words, when I took the Laplace 00:16:37.000 --> 00:16:43.000 transform, I automatically lost all information about the 00:16:41.000 --> 00:16:47.000 function for negative values of t. 00:16:44.000 --> 00:16:50.000 If I am later going to want some of that information for 00:16:48.000 --> 00:16:54.000 calculating this, I won't have it and, 00:16:51.000 --> 00:16:57.000 therefore, there cannot be a formula expressing one in terms 00:16:56.000 --> 00:17:02.000 of the other. Now, of course, 00:16:59.000 --> 00:17:05.000 that cannot be the answer, otherwise I would not have 00:17:02.000 --> 00:17:08.000 raised your expectations merely to dash them. 00:17:05.000 --> 00:17:11.000 I don't want to do that, of course. 00:17:07.000 --> 00:17:13.000 There is a formula, of course. 00:17:09.000 --> 00:17:15.000 It is just, I want to emphasize that you must write it my way 00:17:13.000 --> 00:17:19.000 because, if you write it any other way, you are going to get 00:17:17.000 --> 00:17:23.000 into the deepest trouble. The formula is-- the good 00:17:21.000 --> 00:17:27.000 formula, the right formula -- 00:17:30.000 --> 00:17:36.000 -- accepts the given. It says look, 00:17:32.000 --> 00:17:38.000 we have lost that pink part of it. 00:17:35.000 --> 00:17:41.000 Therefore, I can never recover that. 00:17:37.000 --> 00:17:43.000 Therefore, I won't ask for it. The translation formula I will 00:17:42.000 --> 00:17:48.000 ask for is not one for the Laplace transform of f of t 00:17:46.000 --> 00:17:52.000 minus a, but rather for the Laplace 00:17:50.000 --> 00:17:56.000 transform of this thing where I have wiped away that pink part 00:17:54.000 --> 00:18:00.000 from the translated function. In other words, 00:17:59.000 --> 00:18:05.000 the function I am talking about now is the formula for, 00:18:04.000 --> 00:18:10.000 I will put it over here to show you the function what we are 00:18:10.000 --> 00:18:16.000 talking about. It is the function f of, 00:18:13.000 --> 00:18:19.000 well, in terms of the pink function it is, 00:18:17.000 --> 00:18:23.000 I will have to reproduce some of that picture. 00:18:21.000 --> 00:18:27.000 There is f of t. f of t minus a, 00:18:26.000 --> 00:18:32.000 then, looked like this. And so the function I am 00:18:32.000 --> 00:18:38.000 looking for is, this is the thing translated, 00:18:36.000 --> 00:18:42.000 but when I get down to the corresponding, 00:18:39.000 --> 00:18:45.000 this is the point that corresponds to that one, 00:18:43.000 --> 00:18:49.000 I wipe it away and just go with zero after that. 00:18:48.000 --> 00:18:54.000 So this is u of t minus a times f of t minus a times f of t 00:18:53.000 --> 00:18:59.000 minus a. What is this Laplace transform? 00:19:00.000 --> 00:19:06.000 Now that does have a simple answer. 00:19:04.000 --> 00:19:10.000 The answer is it is e to the minus as, 00:19:10.000 --> 00:19:16.000 a funny exponential, times the Laplace transform of 00:19:16.000 --> 00:19:22.000 the original function. Now, this formula occurs in two 00:19:23.000 --> 00:19:29.000 forms. This one is not too bad 00:19:26.000 --> 00:19:32.000 looking. The trouble is, 00:19:29.000 --> 00:19:35.000 when you want to solve differential equations you are 00:19:33.000 --> 00:19:39.000 going to be extremely puzzled because the function that you 00:19:37.000 --> 00:19:43.000 will have to take to do the calculation on will not be given 00:19:41.000 --> 00:19:47.000 to you in the form f of t minus a. 00:19:44.000 --> 00:19:50.000 It will look sine t or t squared or some 00:19:48.000 --> 00:19:54.000 polynomial in t. It will not be written as t 00:19:50.000 --> 00:19:56.000 minus a. What do you do? 00:19:53.000 --> 00:19:59.000 If your function does not look like that but instead, 00:19:56.000 --> 00:20:02.000 in terms of symbols looks like this, you can still use the 00:20:00.000 --> 00:20:06.000 formula. Just a trivial change of 00:20:03.000 --> 00:20:09.000 variable means that you can write it instead. 00:20:06.000 --> 00:20:12.000 Now, this is one place, there is no way of writing the 00:20:10.000 --> 00:20:16.000 answer in terms of capital F of s. 00:20:13.000 --> 00:20:19.000 This is one of those cases where this notation is just no 00:20:17.000 --> 00:20:23.000 good anymore. I am going to have to write it 00:20:20.000 --> 00:20:26.000 using the L notation. The Laplace transform of f of, 00:20:24.000 --> 00:20:30.000 and wherever you see a t, you should write t plus a. 00:20:28.000 --> 00:20:34.000 Basically, this is the same 00:20:31.000 --> 00:20:37.000 formula as that one. But I will have to stand on my 00:20:35.000 --> 00:20:41.000 head for one minute to try to convince you of it. 00:20:38.000 --> 00:20:44.000 I won't do that now. I would like you just to take a 00:20:42.000 --> 00:20:48.000 look at the formula. You should know what it is 00:20:45.000 --> 00:20:51.000 called. There are a certain number of 00:20:48.000 --> 00:20:54.000 idiots who call this the exponential shift formula 00:20:51.000 --> 00:20:57.000 because on the right side you multiply by an exponential, 00:20:55.000 --> 00:21:01.000 and that corresponds to shifting the function. 00:21:00.000 --> 00:21:06.000 Unfortunately, we have preempted that. 00:21:02.000 --> 00:21:08.000 We are not going to call it this. 00:21:05.000 --> 00:21:11.000 I will call it what your book calls it. 00:21:08.000 --> 00:21:14.000 The difficulty is there is no universal designation for this 00:21:13.000 --> 00:21:19.000 formula, important as it is. However, your book calls this 00:21:17.000 --> 00:21:23.000 t-axis translation formula. Translation because I am 00:21:21.000 --> 00:21:27.000 translating on the t-axis. And that is what I do to the 00:21:25.000 --> 00:21:31.000 function, essentially. And this tells me what its new 00:21:30.000 --> 00:21:36.000 Laplace transform is. 00:21:38.000 --> 00:21:44.000 The other formula, remember it? 00:21:40.000 --> 00:21:46.000 The exponential shift formula, the shift or the translation 00:21:44.000 --> 00:21:50.000 occurs on the s-axis. In other words, 00:21:47.000 --> 00:21:53.000 the formula said that F of s minus a, 00:21:51.000 --> 00:21:57.000 you do the translation in s variable corresponded to 00:21:55.000 --> 00:22:01.000 multiplying this by e to the a t. 00:22:00.000 --> 00:22:06.000 In other words, the formulas are sort of dual 00:22:02.000 --> 00:22:08.000 to each other. This guy translates on the left 00:22:05.000 --> 00:22:11.000 side and multiplies by the exponential on the right. 00:22:08.000 --> 00:22:14.000 The formula that you know translates on the right and 00:22:12.000 --> 00:22:18.000 multiplies by the exponential on the left. 00:22:14.000 --> 00:22:20.000 What are we going to calculate? I am trying to calculate, 00:22:18.000 --> 00:22:24.000 so I am trying to prove this first formula. 00:22:20.000 --> 00:22:26.000 The second one will be an easy consequence. 00:22:23.000 --> 00:22:29.000 I am trying to calculate the Laplace transform of that thing. 00:22:27.000 --> 00:22:33.000 What is it? Well, it is the integral from 00:22:31.000 --> 00:22:37.000 zero to infinity of e to the minus st times u of t minus a, 00:22:35.000 --> 00:22:41.000 f of t minus a times dt. 00:22:42.000 --> 00:22:48.000 That is the formula for it. But I am trying to express it 00:22:46.000 --> 00:22:52.000 in terms of the Laplace transform of f itself. 00:22:49.000 --> 00:22:55.000 Now, it is trying to be the Laplace transform of f. 00:22:54.000 --> 00:23:00.000 The problem is that here, a (t minus a) occurs, 00:22:58.000 --> 00:23:04.000 which I don't like. I would like that to be just a 00:23:03.000 --> 00:23:09.000 t. Now, in order not to confuse 00:23:05.000 --> 00:23:11.000 you, and this is what confused everybody, I will set t1 equal 00:23:09.000 --> 00:23:15.000 to t minus a. I will change the variable. 00:23:13.000 --> 00:23:19.000 This is called changing the variable in a definite integral. 00:23:17.000 --> 00:23:23.000 How do you change the variable in a definite integral? 00:23:21.000 --> 00:23:27.000 You do it. Well, let's leave the limits 00:23:24.000 --> 00:23:30.000 for the moment. e to the minus s 00:23:27.000 --> 00:23:33.000 times -- Now, t, remember you can change 00:23:31.000 --> 00:23:37.000 the variable forwards, direct substitution, 00:23:33.000 --> 00:23:39.000 but now I have to use the inverse substitution. 00:23:37.000 --> 00:23:43.000 It's trivial, but t is equal to t1 plus a. 00:23:40.000 --> 00:23:46.000 To change this I must 00:23:42.000 --> 00:23:48.000 substitute backwards and make that t1 plus a. 00:23:45.000 --> 00:23:51.000 How about the rest of it? Well, this becomes u of t1. 00:23:49.000 --> 00:23:55.000 This is f of t1. 00:23:51.000 --> 00:23:57.000 I have to change the dt, too, but that's no problem. 00:23:55.000 --> 00:24:01.000 dt1 equals dt because a is a constant. 00:24:00.000 --> 00:24:06.000 That is dt1. And the last step is to put in 00:24:03.000 --> 00:24:09.000 the limits. Now, when t is equal to zero, 00:24:06.000 --> 00:24:12.000 t1 has the value negative a. 00:24:10.000 --> 00:24:16.000 So this has to be negative a when t is infinity. 00:24:13.000 --> 00:24:19.000 Infinity minus a is still infinity, so that is still 00:24:17.000 --> 00:24:23.000 infinity. In other words, 00:24:19.000 --> 00:24:25.000 this changes to that. These two things, 00:24:22.000 --> 00:24:28.000 whatever they are, they have the same value. 00:24:25.000 --> 00:24:31.000 All I have done is changed the variable. 00:24:30.000 --> 00:24:36.000 Make it change a variable. But now, of course, 00:24:32.000 --> 00:24:38.000 I want to make this look better. 00:24:34.000 --> 00:24:40.000 How am I going to do that? Well, first multiply out the 00:24:37.000 --> 00:24:43.000 exponential and then you get a factor e to the minus s(t1). 00:24:40.000 --> 00:24:46.000 That is good. 00:24:42.000 --> 00:24:48.000 That goes with this guy. Now I get a factor e to the 00:24:44.000 --> 00:24:50.000 minus s times a from the exponential law. 00:24:47.000 --> 00:24:53.000 But that does not have anything to do with the integral. 00:24:51.000 --> 00:24:57.000 It is a constant as far as the integral is concerned because it 00:24:54.000 --> 00:25:00.000 doesn't involve t1. And, therefore, 00:24:56.000 --> 00:25:02.000 I can pull it outside of the integral sign. 00:25:00.000 --> 00:25:06.000 And write that e to the minus s times a. 00:25:05.000 --> 00:25:11.000 Let's write it the other way. Times the integral of what? 00:25:10.000 --> 00:25:16.000 Well, e to the negative st1. 00:25:14.000 --> 00:25:20.000 Now, u of t1, f of t1 times dt1. 00:25:18.000 --> 00:25:24.000 Still integrated from minus a 00:25:22.000 --> 00:25:28.000 to infinity. And now the final step. 00:25:29.000 --> 00:25:35.000 This u of t1 is zero for negative values of t. 00:25:34.000 --> 00:25:40.000 And, therefore, it is equal to one for positive 00:25:38.000 --> 00:25:44.000 values of t. It is equal to zero for 00:25:42.000 --> 00:25:48.000 negative values of t, which means I can forget about 00:25:47.000 --> 00:25:53.000 the part of the integral that goes from negative a to zero. 00:25:52.000 --> 00:25:58.000 I better rewrite this. 00:25:56.000 --> 00:26:02.000 Okay, leave that. In other words, 00:26:00.000 --> 00:26:06.000 this is equal to e to the minus as times the 00:26:04.000 --> 00:26:10.000 integral from zero to infinity of e to the minus s -- 00:26:08.000 --> 00:26:14.000 Let me do the shifty part now. 00:26:17.000 --> 00:26:23.000 And this is since u of t1 is equal to zero for 00:26:23.000 --> 00:26:29.000 t1 less than zero. That is why I can replace this 00:26:29.000 --> 00:26:35.000 with zero. Because from negative a to 00:26:33.000 --> 00:26:39.000 zero, nothing is happening. The integrand is zero. 00:26:37.000 --> 00:26:43.000 And why can I get rid of it after that? 00:26:40.000 --> 00:26:46.000 Well, because it is one after that. 00:26:43.000 --> 00:26:49.000 And what is this thing? This is the Laplace transform. 00:26:47.000 --> 00:26:53.000 No, it is not the Laplace transform they said. 00:26:51.000 --> 00:26:57.000 Because you had t1 there, not t. 00:26:53.000 --> 00:26:59.000 It is the Laplace transform because this is a dummy 00:26:58.000 --> 00:27:04.000 variable. The t1 is integrated out. 00:27:01.000 --> 00:27:07.000 It is a dummy variable. It doesn't matter what you call 00:27:05.000 --> 00:27:11.000 it. It is still the Laplace 00:27:07.000 --> 00:27:13.000 transform if I make that wiggly t or t star or tau or u. 00:27:12.000 --> 00:27:18.000 I can call it anything I want and it is still the Laplace 00:27:16.000 --> 00:27:22.000 transform of f of t. 00:27:24.000 --> 00:27:30.000 What is the answer? That is e to the negative as 00:27:27.000 --> 00:27:33.000 times the Laplace transform of the function f. 00:27:33.000 --> 00:27:39.000 That is what I promised you in that formula. 00:27:35.000 --> 00:27:41.000 Now, how about the other formula? 00:27:37.000 --> 00:27:43.000 Well, let's look at that quickly. 00:27:39.000 --> 00:27:45.000 That is, as I say, just sleight of hand. 00:27:42.000 --> 00:27:48.000 But since that is the formula you will be using at least half 00:27:46.000 --> 00:27:52.000 the time you better learn it. This little sleight of hand is 00:27:50.000 --> 00:27:56.000 also reproduced in one page of notes that I give you, 00:27:53.000 --> 00:27:59.000 but maybe you will find it easy to understand if I talk it out 00:27:57.000 --> 00:28:03.000 loud. The problem now is for the 00:28:00.000 --> 00:28:06.000 second formula. I am going to have to recopy 00:28:03.000 --> 00:28:09.000 out the first one in order to make the argument in a form in 00:28:07.000 --> 00:28:13.000 which you will understand it, I hope. 00:28:09.000 --> 00:28:15.000 This goes to e to the minus as F s, 00:28:13.000 --> 00:28:19.000 except I am now going to write that not in F of s; 00:28:16.000 --> 00:28:22.000 since I will not be able to write the second formula using F 00:28:20.000 --> 00:28:26.000 of s, I am not going to write the first formula that way 00:28:24.000 --> 00:28:30.000 either. I will write it as the Laplace 00:28:26.000 --> 00:28:32.000 transform with f of t. 00:28:30.000 --> 00:28:36.000 Now, formally if somebody says, okay, how do I calculate the 00:28:34.000 --> 00:28:40.000 Laplace transform of this thing? I say put down this. 00:28:39.000 --> 00:28:45.000 Well, that has no t in it. It doesn't have the f in it 00:28:43.000 --> 00:28:49.000 either. Then write this. 00:28:45.000 --> 00:28:51.000 What formula did I do? I looked at that and changed t 00:28:50.000 --> 00:28:56.000 minus a to t. Now, how did I change t minus a? 00:28:55.000 --> 00:29:01.000 The way to say it is 00:28:58.000 --> 00:29:04.000 I changed t. Because the t is always there. 00:29:02.000 --> 00:29:08.000 t to t plus a. You get this by replace t by t 00:29:08.000 --> 00:29:14.000 plus a to get the right-hand side. 00:29:16.000 --> 00:29:22.000 I replace this t by t plus a, and that turns this 00:29:20.000 --> 00:29:26.000 into f of t. And that is the f of t 00:29:22.000 --> 00:29:28.000 that went in there. That is the universal rule for 00:29:26.000 --> 00:29:32.000 doing it. Now I am going to use that same 00:29:28.000 --> 00:29:34.000 rule for transforming u of t minus a times f of t. 00:29:32.000 --> 00:29:38.000 See, the problem is now I have 00:29:36.000 --> 00:29:42.000 a function like t squared or sine t, 00:29:39.000 --> 00:29:45.000 which is not written in terms of t minus a. 00:29:42.000 --> 00:29:48.000 And I don't know what to do with it. 00:29:44.000 --> 00:29:50.000 The answer is, by brute force, 00:29:46.000 --> 00:29:52.000 write it in terms of t minus a. 00:29:49.000 --> 00:29:55.000 What is brute force? Brute force is the following. 00:29:52.000 --> 00:29:58.000 I am going to put a t minus a there if it kills me. 00:29:56.000 --> 00:30:02.000 t minus a plus a. No harm in that, 00:30:01.000 --> 00:30:07.000 is there? Now there is a t minus a there, 00:30:04.000 --> 00:30:10.000 just the way there was up there. 00:30:06.000 --> 00:30:12.000 And now what is the rule? I am just going to follow my 00:30:11.000 --> 00:30:17.000 nose. What's sauce for the goose is 00:30:13.000 --> 00:30:19.000 sauce for the gander. Minus as, Laplace transform of 00:30:17.000 --> 00:30:23.000 f of, now what am I going to write here? 00:30:20.000 --> 00:30:26.000 Wherever I see a t, I am going to change it from t 00:30:24.000 --> 00:30:30.000 plus a. Here I see a t. 00:30:29.000 --> 00:30:35.000 I will change that to t plus a. 00:30:33.000 --> 00:30:39.000 What do I have? t plus a minus a plus a, 00:30:37.000 --> 00:30:43.000 well, if you can keep count, 00:30:41.000 --> 00:30:47.000 what does that make? It makes t plus a in 00:30:46.000 --> 00:30:52.000 the end. 00:31:02.000 --> 00:31:08.000 The peace that passeth understanding. 00:31:04.000 --> 00:31:10.000 Let's do some examples and suddenly you will breathe a sigh 00:31:08.000 --> 00:31:14.000 of relief that this all is doable anyway. 00:31:11.000 --> 00:31:17.000 Let's calculate something. I hope I am not covering up any 00:31:16.000 --> 00:31:22.000 crucial, yes I am. I am covering up the u of t's, 00:31:19.000 --> 00:31:25.000 but you know that by now. Let's see. 00:31:22.000 --> 00:31:28.000 What should we calculate first? What I just covered up. 00:31:27.000 --> 00:31:33.000 Let's calculate the Laplace transform of (u)ab of t. 00:31:31.000 --> 00:31:37.000 What is that going to be? 00:31:34.000 --> 00:31:40.000 Well, first of all, write out what it is in terms 00:31:38.000 --> 00:31:44.000 of the unit step function. 00:31:45.000 --> 00:31:51.000 Remember that formula? There. 00:31:47.000 --> 00:31:53.000 Now you see it. Now you don't. 00:31:50.000 --> 00:31:56.000 Its Laplace transform is going to be what? 00:31:54.000 --> 00:32:00.000 Well, the Laplace transform of t minus a, 00:31:59.000 --> 00:32:05.000 that is a special case here where this function is one. 00:32:05.000 --> 00:32:11.000 Well, that one. Either one. 00:32:07.000 --> 00:32:13.000 It makes no difference. It is simply going to be the 00:32:11.000 --> 00:32:17.000 Laplace transform of what f of t would have been, 00:32:15.000 --> 00:32:21.000 which is -- 00:32:26.000 --> 00:32:32.000 See, the Laplace transform of u of t is what? 00:32:30.000 --> 00:32:36.000 That's one over s, right? 00:32:31.000 --> 00:32:37.000 Because this is the function one, and we don't care the fact 00:32:36.000 --> 00:32:42.000 that it is zero or negative values of t. 00:32:39.000 --> 00:32:45.000 That is my f of s. And so I multiply it by e to 00:32:43.000 --> 00:32:49.000 the minus as times one over s. 00:32:46.000 --> 00:32:52.000 I am using this formula, e to the minus as times the 00:32:50.000 --> 00:32:56.000 Laplace transform of the unit step function, 00:32:53.000 --> 00:32:59.000 which is one over s. How about the translation? 00:32:59.000 --> 00:33:05.000 That was taken care of by the exponential factor. 00:33:03.000 --> 00:33:09.000 And it's minus because this is minus. 00:33:06.000 --> 00:33:12.000 The same thing with the b. This is the Laplace transform 00:33:10.000 --> 00:33:16.000 of the unit box function. It looks a little hairy. 00:33:14.000 --> 00:33:20.000 You will learn to work with it, don't worry about it. 00:33:19.000 --> 00:33:25.000 How about the Laplace transform of -- 00:33:23.000 --> 00:33:29.000 Okay. Let's use the other formula. 00:33:25.000 --> 00:33:31.000 What would be the Laplace transform of u of t minus one 00:33:29.000 --> 00:33:35.000 times t squared, for example? 00:33:33.000 --> 00:33:39.000 See, if I gave this to you and you only had the first formula, 00:33:38.000 --> 00:33:44.000 you would say, hey, but there is no t minus 00:33:41.000 --> 00:33:47.000 one in there. There is only t squared. 00:33:44.000 --> 00:33:50.000 What am I supposed to do? 00:33:46.000 --> 00:33:52.000 Well, some of you might dig way back into high school and say 00:33:51.000 --> 00:33:57.000 every polynomial can be written in powers of t minus one, 00:33:56.000 --> 00:34:02.000 that is what I will do. That would give the right 00:34:02.000 --> 00:34:08.000 answer. But in case you had forgotten 00:34:05.000 --> 00:34:11.000 how to do that, you don't have to know because 00:34:08.000 --> 00:34:14.000 you could use the other formula instead, which, 00:34:12.000 --> 00:34:18.000 by the way, is the way you do it. 00:34:15.000 --> 00:34:21.000 What are we going to do? It goes into e to the minus s. 00:34:19.000 --> 00:34:25.000 The a is one in this case, 00:34:22.000 --> 00:34:28.000 plus one. e to the minus s times the 00:34:25.000 --> 00:34:31.000 Laplace transform of what function? 00:34:30.000 --> 00:34:36.000 Change t to t plus one. 00:34:33.000 --> 00:34:39.000 The Laplace transform of t plus one squared. 00:34:38.000 --> 00:34:44.000 What is that? That is e to the minus s times 00:34:41.000 --> 00:34:47.000 the Laplace transform of t squared plus 2t plus one. 00:34:46.000 --> 00:34:52.000 What's that? 00:34:49.000 --> 00:34:55.000 Well, by the formulas which I am not bothering to write on the 00:34:54.000 --> 00:35:00.000 board anymore because you know them, it is e to the minus s 00:34:59.000 --> 00:35:05.000 times -- Laplace transform of t squared 00:35:04.000 --> 00:35:10.000 is two factorial over s cubed. 00:35:07.000 --> 00:35:13.000 Remember you always have to 00:35:09.000 --> 00:35:15.000 raise the exponent by one. This is two factorial, 00:35:12.000 --> 00:35:18.000 but that is the same as two. Plus two. 00:35:15.000 --> 00:35:21.000 This two comes from there. The Laplace transform of t is 00:35:19.000 --> 00:35:25.000 one over s squared. 00:35:21.000 --> 00:35:27.000 And, finally, the Laplace transform of one is 00:35:24.000 --> 00:35:30.000 one over s. You mean all that mess from 00:35:30.000 --> 00:35:36.000 this simple-looking function? This function is not so simple. 00:35:34.000 --> 00:35:40.000 What is its graph? What is it we are calculating 00:35:38.000 --> 00:35:44.000 the Laplace transform of? Well, it is the function t 00:35:42.000 --> 00:35:48.000 squared. But multiplying it by that 00:35:45.000 --> 00:35:51.000 factor u of t minus one means that the only part of 00:35:50.000 --> 00:35:56.000 it I am using is this part, because u of t minus one is one 00:35:55.000 --> 00:36:01.000 when t is bigger than one. 00:36:00.000 --> 00:36:06.000 But when t is less than one it is zero. 00:36:02.000 --> 00:36:08.000 That function doesn't look all that simple to me. 00:36:05.000 --> 00:36:11.000 And that is why its Laplace transform has three terms in it 00:36:09.000 --> 00:36:15.000 with this exponential factor. Well, it is a discontinuous 00:36:13.000 --> 00:36:19.000 function. And it gets discontinuous at a 00:36:16.000 --> 00:36:22.000 very peculiar spot. You have to expect that. 00:36:19.000 --> 00:36:25.000 Where in this does it tell you it becomes discontinuous at one? 00:36:23.000 --> 00:36:29.000 It is because this is e to the minus one times s. 00:36:29.000 --> 00:36:35.000 This tells you where the discontinuity occurs. 00:36:32.000 --> 00:36:38.000 The rest of it is just stuff you have to take because it is 00:36:37.000 --> 00:36:43.000 the function t squared. It's what it is. 00:36:41.000 --> 00:36:47.000 All right. I think most of you are going 00:36:44.000 --> 00:36:50.000 to encounter the worst troubles when you try to calculate 00:36:49.000 --> 00:36:55.000 inverse Laplace transforms, so let me try to explain how 00:36:53.000 --> 00:36:59.000 that is done. I will give you a simple 00:36:56.000 --> 00:37:02.000 example first. And then I will try to give you 00:37:01.000 --> 00:37:07.000 a slightly more complicated one. But even the simple one won't 00:37:06.000 --> 00:37:12.000 make your head ache. We are going to calculate the 00:37:11.000 --> 00:37:17.000 inverse Laplace transform of this guy, one plus e to the 00:37:16.000 --> 00:37:22.000 negative pi s -- 00:37:24.000 --> 00:37:30.000 -- divided by s squared plus one. 00:37:35.000 --> 00:37:41.000 All right. Now, the first thing you must 00:37:37.000 --> 00:37:43.000 do is as soon as you see exponential factors in there 00:37:40.000 --> 00:37:46.000 like that you know that these functions, the answer is going 00:37:44.000 --> 00:37:50.000 to be a discontinuous function. And you have got to separate 00:37:48.000 --> 00:37:54.000 out the different pieces of it that go with the different 00:37:52.000 --> 00:37:58.000 exponentials. Because the way the formula 00:37:54.000 --> 00:38:00.000 works, it has to be used differently for each value of a. 00:37:59.000 --> 00:38:05.000 Now, in this case, there is only one value of a 00:38:02.000 --> 00:38:08.000 that occurs. Negative pi. 00:38:03.000 --> 00:38:09.000 But it does mean that we are going to have to begin by 00:38:07.000 --> 00:38:13.000 separating out the thing into one over s squared plus one 00:38:11.000 --> 00:38:17.000 and this other factor e to the negative pi s 00:38:15.000 --> 00:38:21.000 divided by s squared plus one. 00:38:19.000 --> 00:38:25.000 Now all I have to do is take the inverse Laplace transform of 00:38:23.000 --> 00:38:29.000 each piece. The inverse Laplace transform 00:38:26.000 --> 00:38:32.000 of one over s squared plus one is -- 00:38:32.000 --> 00:38:38.000 Well, up to now we have been saying its sine t, 00:38:36.000 --> 00:38:42.000 right? If you say it is sine t you are 00:38:38.000 --> 00:38:44.000 going to get into trouble. How come? 00:38:41.000 --> 00:38:47.000 We didn't get into trouble before. 00:38:43.000 --> 00:38:49.000 Yes, but that was because there were no exponentials in the 00:38:47.000 --> 00:38:53.000 expression. When there are exponentials you 00:38:50.000 --> 00:38:56.000 have to be more careful. Make the inverse transform 00:38:54.000 --> 00:39:00.000 unique. Make it not sine t, 00:38:56.000 --> 00:39:02.000 but u of t sine t. 00:39:00.000 --> 00:39:06.000 You will see why in just a moment. 00:39:02.000 --> 00:39:08.000 If this weren't there then sine t would be perfectly okay. 00:39:07.000 --> 00:39:13.000 With that factor there, you have got to put in the u of 00:39:11.000 --> 00:39:17.000 t, otherwise you won't be able to get the formula to 00:39:16.000 --> 00:39:22.000 work right. In other words, 00:39:18.000 --> 00:39:24.000 I must use this particular one that I picked out to make it 00:39:23.000 --> 00:39:29.000 unique at the beginning of the period. 00:39:26.000 --> 00:39:32.000 Otherwise, it just won't work. Now, I know that is fine. 00:39:31.000 --> 00:39:37.000 But now what is the inverse Laplace transform of e to the 00:39:35.000 --> 00:39:41.000 minus pi? In other words, 00:39:38.000 --> 00:39:44.000 it is the same function, except I am now multiplying it 00:39:41.000 --> 00:39:47.000 by e to the negative pi s. 00:39:44.000 --> 00:39:50.000 Well, now I will use that formula. 00:39:46.000 --> 00:39:52.000 My f of s is one over s squared plus one, 00:39:50.000 --> 00:39:56.000 and that corresponds it to sine t. 00:39:54.000 --> 00:40:00.000 If I multiply it by e to the minus pi s, 00:39:57.000 --> 00:40:03.000 just copy it down. It now corresponds, 00:40:02.000 --> 00:40:08.000 the inverse Laplace transform, to what the left side says it 00:40:08.000 --> 00:40:14.000 does. u of t minus pi 00:40:12.000 --> 00:40:18.000 times, in other words, this corresponds to that. 00:40:17.000 --> 00:40:23.000 Then if I multiply it by e to the minus pi s, 00:40:23.000 --> 00:40:29.000 it corresponds to change the t to t minus pi. 00:40:38.000 --> 00:40:44.000 What is the answer? The answer is you sum these two 00:40:43.000 --> 00:40:49.000 pieces. The first piece is u of time 00:40:47.000 --> 00:40:53.000 sine t. The second piece is u of t 00:40:53.000 --> 00:40:59.000 minus pi sine t of minus pi. 00:41:00.000 --> 00:41:06.000 Now, if you leave the answer in that form it is technically 00:41:05.000 --> 00:41:11.000 correct, but you are going to lose a lot of credit. 00:41:09.000 --> 00:41:15.000 You have to transform it to make it look good. 00:41:13.000 --> 00:41:19.000 You have to make it intelligible. 00:41:16.000 --> 00:41:22.000 You are not allowed to leave it in that form. 00:41:20.000 --> 00:41:26.000 What could we do to it? Well, you see, 00:41:24.000 --> 00:41:30.000 this part of it is interesting whenever t is positive. 00:41:30.000 --> 00:41:36.000 This part of it is only interesting when t is greater 00:41:34.000 --> 00:41:40.000 than or equal to pi because this is zero. 00:41:37.000 --> 00:41:43.000 Before that this is zero. What you have to do is make 00:41:41.000 --> 00:41:47.000 cases. Let's call the answer f of t. 00:41:44.000 --> 00:41:50.000 The function has to be 00:41:47.000 --> 00:41:53.000 presented in what is called the cases format. 00:41:50.000 --> 00:41:56.000 That is what it is called when you type in tech, 00:41:54.000 --> 00:42:00.000 which I think a certain number of you can do anyway. 00:42:00.000 --> 00:42:06.000 You have to make cases. The first case is what happened 00:42:04.000 --> 00:42:10.000 between zero and pi? Well, between zero and pi, 00:42:08.000 --> 00:42:14.000 only this term is operational. The other one is zero because 00:42:12.000 --> 00:42:18.000 of that factor. Therefore, between zero and pi 00:42:16.000 --> 00:42:22.000 the function looks like, now, I don't have to put in the 00:42:21.000 --> 00:42:27.000 u of t because that is equal to one. 00:42:24.000 --> 00:42:30.000 It is equal to sine t between zero and pi. 00:42:30.000 --> 00:42:36.000 What is it equal to bigger than pi? 00:42:32.000 --> 00:42:38.000 Well, the first factor, the first term still obtains, 00:42:36.000 --> 00:42:42.000 so I have to include that. But now I have to add the 00:42:40.000 --> 00:42:46.000 second one. Well, what is the second term? 00:42:43.000 --> 00:42:49.000 I don't include the t minus pi, u of t minus pi 00:42:48.000 --> 00:42:54.000 anymore because that is now one. 00:42:51.000 --> 00:42:57.000 That has the value one. It is sine of t minus pi. 00:42:55.000 --> 00:43:01.000 But what is sine of t minus pi? 00:43:00.000 --> 00:43:06.000 You take the sine curve and you translate it to the right by pi. 00:43:07.000 --> 00:43:13.000 So what happens to it? It turns into this curve. 00:43:13.000 --> 00:43:19.000 In other words, it turns into the curve, 00:43:18.000 --> 00:43:24.000 what curve is that? Minus sine t. 00:43:30.000 --> 00:43:36.000 The other factor, this factor is one and this 00:43:33.000 --> 00:43:39.000 becomes negative sine t. 00:43:42.000 --> 00:43:48.000 And so the final answer is f of t is equal to sine t 00:43:50.000 --> 00:43:56.000 between zero and pi and zero for t greater than or equal 00:43:58.000 --> 00:44:04.000 to pi. That is the right form of the 00:44:03.000 --> 00:44:09.000 answer.