斜率场与可分离微分方程

Any questions from the entire logistic model?Of a calculus AB before we go on to the BC material, and here, and in fact, we have been solving differential equations, but there is a topic called the slope field. We're gonna spend one or two sessions on this, and then say the more things about the slope field in the future when we do comprehensive problem solving. And the slope field itself, it's really nothing but a graphical representation of the kind of differential.Equation we've been solving so far. For example, if I give you this differential equation, that the y prime uh times the x squared equal to let's say, um, sin x, and then times y. Just very quickly, how how do we solve? Ah, oh, actually, this is not a very nice example. Let me give you.One second. I'm actually thinking how how I could make this actually solvable. Let's actually do. Alright.How do we solve something like this? This is a quick review. That's not even the purpose of the session, but I want you to recognize what kind of differential equation this is and how do we solve it. I know it's a trig function because it's got sine x.What part of that would be, but that's not very helpful when you just recognize the overall because when you're finding anti derivative, maybe knowing the type would help, but that science would be bundled up in very complicated ways. The first thing you do, you you think about when you see a differential equation is what isolating the differential.嗯， meaning. Oh, we get all the y's to one side. Yeah, yeah, very good. Separable. You want to test out whether it's separable. Meaning, you want to write the derivative in terms of the differential, which is very good, actually. You like that's the dy over the x squared dx, would equal to the y sine x. Excellent. So we isolate the differentials. In fact, what we're really doing is.It's separate variables. We move the y to one side, and this is to the other, the x now. And I do hope you recognize the integrability of both sides of the equation. We know for sure it falls into a good type. We know for sure the antiderivatives are findable. Findable. Hey, Eddie, it's good to see you. I'm turning on your recording. We're doing a bit of review. But today, major job is to introduce a new concept. It's a very last major topic of our calculus A B part, and beyond that, it will be B C material. So look at this. What this topic is called a slope field. The so called slope field is a graphical representation of a differential equation, and you see what it means. So I give you this differential equation as a bit of reveal, and we notice it separable. We have already successfully.Separated, and now we're looking at it this year. Now go ahead and integrate both sides.That's SyneX, right?The left side is just natural log of y, absolute natural log of absolute value of y. Very good. On the left side, that's obvious. Very good. And I like your specifically mentioning the absolute value.For the right-hand side, uh, here, uh, would be equal to negative x squared.Cosine x integral to x cosine x dx. Ah, very good. You're saying if you want to, ah, have you given me the entire integral, or are you doing integration by part on the right? Ah, integration by part. Right, so which part you put behind the differential first.明明有 integrated 的 sin x，毕竟呢。Yeah, we're getting a x. Really again, please. What exactly are you getting? Uh, negative x squared cosine x. Yes, very good. Plus, uh, integral to x cosine x dx. I credit.Degree, and continue. Ah, so we can do the same thing again. Yes. We can like take out the two. Yes. And then that would be. Ah. X nine x. Minus. No, it would be just be x sine x plus cosine x. Hold on. Would be. I thought you would nicely do the same again. By the way, where did that sine come from? Like we can, take out.The two. No, hold on. Just in the previous step, where did we get the sine x here? Oh no, that's a cosine. Okay, sorry, no one there. So that is a cosine, and now we're gonna actually put it behind now. Yeah, right.Do that. Do that again. Meaning we're getting minus a two x of the sine x. That's what you're getting, isn't it? Yeah. Yeah. Yeah.Yeah, yeah, that's that's what I'm getting. And now we're finishing.Obviously, they just are adding twice of the cosine x. Excellent procedure, it's entirely wonderful. And apparently, you know your integration by part. Elaine, are you getting exactly the same? Yeah, excellent. So we do know the solution to this. Now, eventually, when we want to isolate y, we have to take off the absolute value. So the answer to this differential equation has. I'll just ask you a friend here: is that the even function or odd function?When you finally solve what is e s, sorry, what is a y s, because but this is going to be the e to whatever is entire thing on the on the exponent, right? We actually have that, and add on twice of the x of the sine x, and add twice of the cosine x. Now the question would be: Is this even function or odd function, and are we completely done?Should I write in this manner, or should I write to address this absolute value of the Y issue? Eventually, that tiny bit of a detail actually matters. The exponent will always be positive, so I don't think you need.To worry about that absolute value. We don't. The whole point of having the absolute value there is that the y doesn't have to be positive. Y can well be negative. So tell me what's going to happen there.I'm kind of confused. Like, are we trying to like see how we account for when y is negative? Yes, I'm asking you whether we should give account for the case where.All want to be negative.````````````Would you just like add on an arbitrary constant? We could do that. We're yet to answer. In fact, the adding arbitrary constant should happen here. So when you exponentiate, the arbitrary constant would actually go in the front. And that object constant would be absolutely positive because it comes from e to some constant power. This is a translation of that. So the c could be negative, but that's a rename the c. It's a different constant, yeah. So the time by time you're bundling up, it's e to the c as a constant now. It's really positive. So that doesn't really handle the case where the y could be negative.If the y is negative, can we use imaginary numbers? Oh, here we don't because we're explicitly solving the y as a real function, and that's a default. People don't state that assumption here. That's a requirement from the AP. Period. So we should know what this possibility of imaginary numbers is for. Okay, it's confined to our.Own tracks.I think we have to put an absolute value sign on the y. Right, but eventually the goal is to really isolate it. Why don't we just add plus or minus here? Because if you want to take off the absolute value sign, there are definitely two possibilities. I don't see anything wrong with the possibility that the y is simply negative. Now coming.Out of the absolute value sign, basically this would itself equal to a negative y, so that negative appears on the right. This is the correct way to write your final answer. So remember, you took so much trouble to keep all the rigor along the way to have that absolute value of the y. Just don't miss how exactly it pans out in terms of the the real way we write our y function. So this is how we could write our y function.Does this make sense? Okay. Well, finally, we do know this is going to be some constants. See here, and that means I be you could indeed just write a c in the front and say the c is an arbitrary real number. That's actually fine. Okay. So it basically like includes the plus minus inside the c. Yes, right. But this is a ready.Fine, you see, from the very exponentializing process here. Excellent, that. Now, given this kind of differential equation, we know how to solve it. But then, if I want to make, I want you to make a prior judgment, meaning whether this is going to be even function or odd function. What would you see? What would you say? Is that even or odd? May I go to the restroom? Yes.We're about to actually draw that thought field now, but I want you to have some just from your own reasoning process, some prior knowledge of the symmetry we're going to observe. ```中文嗯。你来，tell me your mental process. How do we judge whether the function's even or odd? Uh huh. Uh, because of the chain rule, we know the rate of change, um, is is a trigonometric uh, uh, is like a trigonometric function, and.Those are symmetric about the y-axis, so I think it's even. And without the graph here, can you tell from the shape of that function? Your reasoning backward, meaning we need to know what is going to be how the graph looks like when we actually look at the function itself. Maybe you did have your reasoning forward.You just didn't say it that way. You told me from the result of the graph. I wouldn't know what the graph looks like until we analyze the symmetry first.```中文嗯。Can you talk. Oh, I was using the rate of change. There's no need to use a rate of change. If I give you a function, how do you tell it even function or odd function? You try to plug in positive negative x and see whether the function will come out the same or the opposite or there's no strict pattern. For example, if I ask you, x squared cosine x is this even function or odd function? It's even. Good. What about the two x sine x?There are so many ways to answer that. You can think about the sine x as its power expansion. Power expansion is the quickest way to judge a function is even or odd.Is it odd? Let's try it. When you do the power expansion, well, in fact, the better way is to why don't we just plug in a power of half of the pi and negative half of the pi. This pair of numbers.Plug it into twice like sine x, see what you're getting.πi and negative π. You get in π and negative π. It's πx itself, even or odd.It's even. Okay, so this is actually an even function because you're multiplying odd function with odd function. Or you could see sine x is a power expansion that has only the odd terms in it. When you multiply by x, now it's it becoming all even terms, right? So that's even function. What's going to happen when you add them together? In fact, we're getting two x sine x subtracted by the x radical sine x. What about this whole thing? This combination is that even function or odd function?Welcome back, Eddie. We're trying to tease out whether this function given in such a manner, it's an even function or odd function.Is it even? Yes, plus twice of the cosine x. Now I'm adding in even the last term here. Is this whole bundle even or odd? It's even. Good, that whole thing is even. Ah, you can see what. We're doing. We're trying to go from inner function to the outer. Look at the presence of x on the exponent. The amount to altogether even function, but after that, we exponentiated. Then overall, are we getting an even function or odd function? It's still even. Yes, it is still even. I.Well, which actually means you described the symmetry of the graph. Now, when you draw the entire thing, it looks like we need only do so for one semi-plane on the positive side, and then the negative side takes care of itself. You can just reflect it over across one axis, and that's a definition of even function. But if we're dealing with a separable differential equation to begin with, and look at the process of solving, we give equal opportunity or some kind of symmetrical credence.To x and y now, in fact, we integrated them separately. Up to this point, I would say the two variables are really symmetrical. There's no difference between the two rows of the independent and dependent. Only in the last step, we choose to isolate one and write it in terms of the other. Now it is sensible or it's meaningful to ask whether the function will be symmetrical about the x-axis. So now the way to formulate it would be.Is this function even or odd about the y? What does that mean? It's biographically symmetrical about the x-axis. You know, for sure. In fact, no function could be actually symmetrical about the x-axis and still stay a function. Because if it is now, it's not a function. It doesn't pass the vertical line test. It just means it's more appropriate for us to actually write x as a function of y. Now that could work, but if you actually have this kind of symmetry about both, and then you.It will not have a function whatsoever. What you have is a relation such as a circle that could be the solution to something, but not never quite really the same branch. You have to have two separate equations for these two upper and the lower branches. Are you guys with me so far? And if you understand what I just said, let's think.Think about whether the the solution here is magical or why. Meaning, is it even or out in terms of why? Meaning, is this magical about the accesses. And as you're thinking about it, why don't we actually start introducing the concept of the slope field? What does that actually mean? The slope field is a graph of the derivative. So first of all, we're going to isolate the derivative. Why derivative actually equal to the y x squared.Let's say x, right, and then if this is the entire y derivative now, and then why don't we actually think about the meaning of the derivative? It actually means the slope, and then we're gonna simply drop down the slope here in such a manner. Now here is my coordinate system, and notice here that the x and that's the y. If I'm picking an arbitrary point in space now, for example, I'll just look at the one one here, and we plug in.This is an x equal one y equal one. The two coordinates are already given. So what else are we graphing? And I picked it without checking whether it really fit into that known equation. In fact, it doesn't. We just don't know. Nevertheless, when you graph the slope field, we're going to actually graph it at every single point on the plane by plugging in the coordinates here and the value we have for the y derivative. You give me an estimate with a one significant digit. When we find x equals to one, y equals to one. That's the point we have chosen. What's gonna come out of the y derivative.```````````````What are we thinking?I'm calculating the sine one using the power expansion. Oh, please don't. Oh, this is going to take a long time. Okay, you will. You can. You can. Why should I say, please don't? Okay, actually, Elaine is actually calculating what sine one is. She's saying we have a power expansion. That's a very good thought, you know. You learn the tool, you use it. It's going to be one minus one six, and then plus the one twentieth, and it doesn't converge too slowly. I mean, it converges. Pretty fast. We convert to radians. And well, the the radius of convergence is infinity, but you could just observe it does converge pretty fast. Oh, the one is already in radians. It's already in radians. I didn't write one degree there. We're plugging a value of the one. By the way, it always means in radians. Otherwise, how would you square up one degree? So it's fine.The next number is really too small. I ask only for one significant digit. In fact, you could give me even two now, which is going to be. And go quickly and finish this. However, what I meant for you to realize is that one radian is approximately how many degrees? It's very close to sixty. Please graph it on your unit circle. So in fact, you would know that sine x is pretty close to root three over two. So the answer will be about zero point eight or zero point nine, actually.It would be closer to that. Oh, but that's a little less. Yeah, I got like zero point eight four one five. Okay, so it's actually eight four one five, huh? So it's barely it's zero point eight. If that's the case, and well, I just want you to have a sense of what this value is, and then we're going to actually graph that value here. Remember, it means a slope. What does that mean? It's a rise over run. It's a local and slope. So, and zero point eight is pretty.Close to one, but a little less than one, so this is going to be zero point eight. Usually, they draw you a little dash, which is not as long as what I'm drawing, but I want to illustrate it just so that the picture looks clear to you. And then we do likewise. For example, if I plug in the one, the two now, what are we getting? You can see which is double that, right? So by the time we're getting here, and that's will be doubled into one point six is now, but.If you plug in a three, and then you have to double that again, it's getting steeper, doesn't it? Apply to the same point, a same value of the x here, but if I reduce it to one half, that I'm also bring down the slope to one half. If I reduce it to one fourth, and I'm further reducing the slope to still one fourth, so this is sort of a the how you create a slope field. Meaning, you just draw the little dashes. Eventually, this little dash.It's here would actually represent what's gonna happen to the curve at that point here. It's the orientation of the curve, it's the slope of the curve. Instantaneously, then you plot your. What about x equal to zero? That everywhere x equal to zero, you're always getting zero. On the y plot, right? That's represented by this little horizontal line, meaning the slope is zero. And we notice the whole function is even function regarding.X, but see what's going to happen when you plug in this virtual point. For example, X equal to ninety-one, X equal to ninety-two. What do we observe about the slopes? Give me a description.中文嗯。 ``````How can you manage the process? Negative eight zero point eight. Yeah, indeed, all of them become negative. So when you graph them, you're gonna actually have a graph that looks like this, rightDo we know? Very good. Does that surprise you? Well, this contradicts with the fact that actually y itself is an even function, as we just carefully analyzed earlier. When we solve it after we solve everything here, we look at the final answer. We analyze, oh, this is an even function. But when we look at the slope field now, this is an even function.But then this the way we wrote it, that's an odd function. Well, you wouldn't know until you actually plug in the numbers now. But to be an odd function, it just means when you plug in the positive or negative x, we're definitely getting the same um, ah, whether the same value of the y now. The y value is sort of fixed as a control, and this is obviously odd because you're multiplying x squared, which is even function, with a sine x, which is odd function. But look at the graph.The derivatives are the. Does that actually match with the fact that the y itself is even?嗯。It doesn't contradict anything. That looks like even. In fact, we should notice every single year and function. Definitely would have a the corresponding point here. Out of the slope. Yeah. One side is increasing, the other side would be exactly decreasing. So remember here's a little feature and meaning. If the derivative of the slope field here, it's all the, and then defined by. In fact, at the opposite side. If the polynomial negative x, you actually have opposite slopes. The graph itself is even. Now it's meaningful to ask whether this function is even or odd regarding y. And it's easy to eyeball because when I say regarding y now, I'm holding the x as a constant value as a control. Now we're really saying at a certain specific chosen value of the x. Now we don't change it. We're going to really just change our y as we did earlier. We increase the y from a polynomial to polynomial two, and.Reduce it to one half and one fourth. Sorry, now the question is: What's going to happen if I continue the this pattern here by plugging in the x equal to one? Hold that one now and make the one negative. What do we see on the other side of the graph? You're even welcome to put it on my board. For example, if I plug in the one negative one, and one negative two, what do we have for the slopes?So the slope feels like, the vice versa also be true. Like if the slope field's derivative is odd. I mean, even then, would the original function be odd? Very good. Though, because all of this is just a coefficient that we have. I why only why? So that just means it's odd. While we plug in the.Negative two that what we get or negative one we're getting exactly the opposite of what we got here in the first quadrant, right? You just flip them over. So this is going to be slightly negative, slightly negative, and this double in negative, and that's a quadruple, triple in negative, etc. So we can see this whole graph. It's odd. The graph is odd. The field is odd in terms of both the even, but both the x and the y. But the graph looks like symmetrical about the x axis.And the y-axis. By the way, that's reflected in the fact you have a y absolute value. The minute we come out with a solution now, just look at over here. Both sides of the equation are even. The left-hand side, by the very fact that you were taking the absolute value to begin with, surely it's even function regarding y, and this side it's also even function regarding x. Now, what does that mean? That means when you graph a slope field, it's sufficient to graph only one.We can just graph it isolated in the first quadrant now, and then just map it, flip it over, mirror it over to get to the graph in all of the fourth quadrants. You guys are with me, huh? Makes sense. All right. Well, knowing this much now, my real question is, and I'm not going to tell you—you really have to think about it. Well, if the entire.It's a, it's even graph regarding the x-axis and regarding the y-axis here. Eventually, why do we take only one branch? And because the way you write it here, you choose a c, although we do permit a c to be positive or negative. But for example, if I choose a c equal to a negative zero point two now, then are we getting both branches of the graph?Or are we getting only one? In fact, let's see what would be your answer if I simply choose the c equals to zero point two. And apparently, this is our one here. And what does that actually mean? For example, immediately I want to find out where would be my y-intercept. 零点二e的平方，说一遍。零点二e的平方，说一遍。Now, ah, that's zero point two. That's now we're plugging the I C loop zero, we're getting exactly that. And but roughly speaking, what numerical value would that give us? This graph would give us something close to eight seven point eight-ish, so we're getting about one point six. And the graph would actually go from here if this is one, and then that's where the graph goes from. But then eventually, you have to meet here, right? It's going to actually go.Maybe like that, but my question would be: Does it fluctuate in the middle? It might. After we pass actually half of the pi, now then the cosine will become negative. Although the first term is still positive, so I don't exactly know how this is going to go for the future. Maybe it's going to go waved, but roughly speaking, we do know when x is approaching infinity, and these terms would actually be approaching infinity. But knowing that.Otherwise, once we fix the c is positive now, it stays positive. Then I wonder, did we lose the this mathematical part of the curve? Meaning would be actually something like this. According to Stillfield, if I actually have a graph on the top like that, we should be having a similar mirror image on the bottom, right? Do we pick both, or do we automatically?周围人。The plus minus we wrote earlier is the C, right? I know. Well, I chose the C at zero point two, which actually gives us a y-intercept. Meaning, when you plug in zero, you're getting something close to one point six.Alright, remember I gave I actually sent to the group of file.And which gives us the theoretical minimum of the calculus BC material. Do you guys still remember that file? It's not the MIT one. I'll send you guys two links. But one of that is a book focused on the AP preparation. Do you remember? I think I remember. Check your WeChat. Is there? In case you can't find that, you have to let me know immediately so that I can resend it as soon as possible. But I did send you guys a file, and you can. See the last two chapters are differential equations and solfields, and for once I want to do read about the solfield because there's some rudimentary conceptual teaching you really don't need to depend on me to do. And next session, I'm gonna actually start by resolving this issue of whether we take both sides of the mirror image or only one side. It's about the definition of the function. And all right, keep up your good work. Try to hand your homework a little earlier because if you.Send the translation for this dialogue.

斜率场与可分离微分方程

斜率场与可分离微分方程

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