AP Calculus Review: Limits, Inverses and Tabular Problems

Alright, thank you, Elaine. I'm gonna turn down your recording in a second. What's really happening is that it's easy to eyeball if part of the integrand turns out to be the derivative of the so-called inner function, but it wouldn't be so nice now if we're looking at this guy. That's actually what we're integrating, right? Now here, none is the others' derivative inner function because we got the second power of the sine, third power of the cosine.There is a trick. Well, actually, there is a technique. My class is here now, and Jonathan, let's go today. However, stay for a minute, Elaine and Ivy. Please turn on your video for me now. And Ivy, did you turn on your recording? I'm giving you the permission now. We're getting started. So, could you guys show me? And this is integrant. How do we find the antiderivative of this function? I'm writing it down here.One second. This is a cosine x cubed times a sine x squared. What is the antiderivative of this guy? Elaine, where are you? Hey. Eddie, good to see you.Where are you, Elaine? Usually, Elaine is the person giving us the answer. But Ivy, could you help us? How do we find the antiderivative of this? Could we like? Addy, you are more than welcome to talk to.I like, I forget what the rule is called, but I think you could like choose sine x and take the derivative. Of that, first. Take the derivative of sine, we're finding the anti derivative. Oh, find the.Cosine squared is equal to like sine one minus sine squared x, but we could like split cosine cubed up into like cosine squared x and cosine x. Beautiful, that's right on target. Idea is noticing, and we could still look at the cosine x. I'm going to write it in terms of the cosine squared cosine x. Now, this cosine can be understood as because of the chain.The derivative of the inner function, which is sine. But then you might, Jonathan, you might wonder, the remaining part is not only a function of sine. Oh, but it is, because we actually have our, I, this, a periodic identity. Meaning, Jonathan, we could write it in terms of one minus sine x squared times sine x squared, and then followed by the cosine x. Now, would you finish taking the anti-derivative? That's that's homework, Jonathan. Would you?Beautiful, and look forward to seeing you on Wednesday. Bye, keep up a good work. Thank you. Of course. Bye. Yep, and here, my guys. This is you are my one of my favorite groups. Elaine is asking for the recording today. She's not here, but we do want to discuss. We're done with the total material now. Elaine will be taking calc BC test. Ivy, you'll be taking the calc AB test. Johnson is also the one. He's the guy that you guys. So a moment earlier, it's also planning on taking the CACBC test. I want to reorganize the two groups, so and I want to split it up. But Eddie, it's up to you because you're not taking any tests this year, are you? Uh, I I don't think so because I'm. Can you take A B test in eighth grade? You can, and you can Elaine. It's also in eighth grade. He's taking the B C test, but it does take some special arrangement.You can't take it at the school, but you can take it with some kind of independent organization in California. But right now it's too too late. Well, for you, there's no there's no hurry next year. You can take it. But my question would be, my tentative my tentative plan is to resplit the group here so that Ivy and Eddie you both go for the calc AB so that I just keep focusing on the AB techniques to the point of providing it so that you could go to the test with confidence and that.I'm gonna pull Elaine out so that she'll be joining Jonathan to be prepared for the BC test. Um, I want to talk to the parents, but I I just finish work every every day very very late. So would you tell your parents here? There must be some kind of a reorganization. And if this is gonna happen, it's gonna happen pretty soon now immediately this week. So I will talk to your mom and dad. And I still want to keep the same meeting time for Eddie and Eddie because it's hard.To find a common time for Edna, so we're keeping this meeting time, and I'm trying to actually make Edna join Johnson's group. So this is my tentative idea. But if you have any alternative idea, for example, I mean, you just want to learn the whole nine yards of the BC this year, you can stay. Then there's no need to do the split. And however, then for the BC group here, my focus of the comprehensive problem solving training will be leaning more toward the BC material because.Actually, three more categories of the material haven't even covered. We have even begun to cover those are unique to BC. So that's still a significant chunk of material, including polar coordinates, including parametric functions, at great depth, and including the convergence and the divergence of the McLaurin expansion. So I would say I need to go on to at least one more month of a conceptual teaching, very likely six weeks. That's what I'm planning for the BC class.It's a picture clear. So think about it. Tell me your decision. Are we okay? Yeah. Okay. Now we're gonna actually go on to look at the test prep now. I want to be sure that we cover every single problem and solving technique. We don't really have an archive for the past multiple choice problems here. That's unfortunate, but we do have the five steps to the AP.Which has some multiple choice problem. I consider those as a not so difficult, and so if you have your McGraw Hill, and I believe at the very end there are actually two sample tests. Would somebody share your screen for the McGraw Hill Five Step to the Five that AP Prime Book? If you don't have it downloaded, go for the older copy, and they're all free. It's two thousand eleven or two thousand twelve. They're even better than the current editions because they.The editions are just getting too easy. What is it called again? It's called "Five Steps to a Five," McGraw Hill, and they that's the whole series. And they have an AP Calculus BC. Five steps. Five steps. You've never used that. At the very beginning, I sent a link to everybody. 泰拼音ไทปินเมกราเฮล คุณรู้ไหม ทำไมคุณไม่แสดงหน้าจอให้ฉันดู ฉันจะดูว่าคุณกำลังดาวน์โหลดไฟล์ที่ถูกต้องหรือไม่ ฉันรู้ ฉันคิด ขอโทษ นะWait. Ah, should should I share my screen or? Oh, please. You go ahead. Either one of you, as long as we can get a copy. Hey Miguel, you need to remember the time. It's eight forty to nine forty. Let's add the exam. Miguel, I'm talking to you. It's eight forty to nine forty. Oh okay. Yeah.I'm seeing a lot. So go ahead, Eddie. What are you doing? Are you sharing your screen? We're not seeing anything. Yeah, I'm trying.Thank you. So I do. You have that book. Download it. I have the physical copy. Oh okay, you can bring which edition is it? Do you see it? Yeah, I do. Is it this one? No, no. Ah, yeah, yeah, that is that is. Ah, however, go to the older copy. Just type in the five sep to the five twenty eleven, digital free, and type in.PDF as well. In the Google or into this website. I into Google because I don't believe the website wants to give you the free copy. They want you to ask you to buy them. A P C A B C A C A B. Very good, then that's the good one. Oh, the first one, the first one.嗯，哦，the second one is also a very good link. I highly encourage you to get that book too. Very good. And can you get you the content table of content now? I believe there are actually two sample tests at the very bottom, at very last. So chapter. Oh, hold on, scrub. Really? They don't give you sample problems.Can you try go back to Google. And then type in, uh, Baron Calculus, uh, AP Calculus, AB, Baron. PDF. But they do give you sample tests. It's not five steps to a five anymore.It's a different series. Baron. Review yes, and and the PDF. Ah, cancel the five steps to a five now. Yes. Twenty eleven. Copy. We want a free copy. There's no need to get it. Let me see, Aaron. Can you see that? Try that. The one I'm hovering over right. Because that's Internet Archive, it's a well-credited site. They give you the free files. Yes, that is the one. And can we go to the table of contents to see whether they give you samples? I believe they do. One second. Initial review session. Go forward, please. Practice. Yeah. Oh, they only give you one. Now looks like they give you.Three sample tests. I okay. Let's go to page five hundred twenty-one. I do have two. In fact, the past two files here twenty-eight, twelve, and I think that's posted online here for everybody. Hey, what's happening? Oh, they only give you several pages. Shit. Okay, well.Later, and we're gonna actually find more exercises. Well, today my major point is so that we I I want you to see the format. Oh, that I'm gonna actually find you the file. They shouldn't stop sharing. Yes, please. Yeah, there's a public.Exam here. You know, go to it. Start sharing. I'll tell you where to find it. Go to College Board. In fact, you don't need to. You can just type in your Google, the twenty twelve public practice exam AP Calc AB. Twenty twelve public practice exam. This is one for everybody's access, and because it's old.It's actually a bit more challenging. I like it. Recent years, it's just more and more dum dum dum. Um, I found it beautiful. Would you share it, please? Yeah. Thank you. Sorry for the time delay.Is it this one? Exactly. Just get to the multiple choices. I want you to see the structure. That section one, you got actually two sections. One is with calculator, the other sections without the calculator. There are the other four sections for the multiple choice. There are only two sections, with or without the calculator. Oh, it's very long. You have to get into the middle of the file.For that to even begin, there's a lot of overhead in the beginning. Oh, good, good, good. Okay, so you see the total—it's a the entire length of the test is three and a half hours, I believe. It's just very long now. So, section one part A, there's a part B, which I believe. Could we quickly go to part B? And that's the. This is a calculation section. Part B is no calculator. I think that's thirty-five minutes.Yes, oh, that's also fifty minutes. Wait a second, is that free response or multiple choice? Could we scroll down a bit? Oh, that's your multiple choice, right? So, in fact, the between the multiple choice to two sessions, it's almost two hours now. It's actually an hour and forty minutes. Another two free response session would be altogether, I think, an hour and twenty minutes or something like that. Alright, let's get to the very beginning. In fact, let's do part B first. This is without calculus. It's highly conceptual, and can I see how many questions there are altogether? I think it will be thirty or twenty-five or something. Part B is a graphing. Oh, that's required, right? So the part B is a calculator required. So let's get to part A first. Part A is the longer section. You can see it takes only two minutes to solve each problem. Therefore, speed is necessary.You can't ever spend. You see, this is on every chair. Two minutes a problem here. You are never going to spend more than, let's say, five minutes on a single problem here. Now, if you find yourself spending more than four minutes, I would say cut the loss. Really, just mark it and come back to it if you have time. Okay, otherwise you could easily lose track of time. Um, oh, you are just okay. Let's continue. And we're going to do it together for today. Later.I'm gonna divide them up into different topics here and treat each topic at a time. Now we just want to get a feel for what we do. For each problem, I'm gonna give you one minute, and then I'll give you my quick way of doing it. So what is a derivative here? Just feel free. You want to compete. You really want to buzz as fast as you can.嗯。I'm getting sine x plus x cosine x. Brilliant B. There's no need. I can I need to say anything. I know. I mean you know it. A, look at number two. That is the product rule. Here, there's a moral lesson. Okay, don't jump into real calculation. There's no calculator. Meaning, very much. Ah, very often the problem can be solved just by eyeballing the conceptual picture. It takes me.Two seconds to know what is the correct answer. There's no need to take the derivative, although conceptually that's the thing you do in order to identify where is a turning point. But just with common sense.Guys, what do you say? I will share with you all without your mental process.Could you like factor that affects? Increasing or decreasing, right? Well, the overarching shape of this cubic now. You know the leading coefficient is negative, so it's going like that, right? So we know whatever the interval, and that's the increasing, must be in the middle. It's all the function. Therefore, this is the only possible answer because it's the only interval that's symmetrical and it's in the middle. And knowing that's all the function, we know the whole graph will be.So, what about the origin? If you do factor it, yes, you can do that. Then you're getting actually ten minus x and ten plus x. Yeah, it gets you the same answer, but it takes at least I would say thirty seconds. But then just knowing the symmetry, I know in the over arching trend that it takes only two seconds. Are we good? More or less, and know the over arching behavior before you worry about the details.Hello, Eddie. Do we feel comfortable? Yeah. Okay, uh, scroll down please. Let's get number three. This is how you can save time. Alright, we're finding the anti derivative. Keep your.Personalize a formula sheet on the side. I don't fault you for cheating. In fact, even on the AP, you are also given a limited number of formulas. You should get used to what formulas are given to you by printing it out from the college board. That's a standard formula sheet given to every single test.Could be like rewrite the C G X and the tangent X. Yes, we recognize it. During the C G X, it's just C G tangent. It's not much to it. It's one of the things you just need to own your mental library. In case you don't remember that, and basically you just choose the simple one, take the derivative, try it out one by one.And it won't be too difficult. If you get the second, it's basically the cosine inverse, and by using the triangle, you're getting cosine x second negative of the negative there's a sine x, and they rewrite into the second times the tangent. I don't need to say much. Number four, please.Fast, really. Remember, we're trying to be fast. I think it's a. Yes, and basically, the roof is seven and the plus one over the x, and one implies one. That's it. Continue. Most of these are very easy. Good, this is purely conceptual. Notice here, fourth. Grab your pencil, marker, the keywords.Um, I think C is false. Yes, at four, this is obviously discontinuous, where the derivative is undefined. Get me all the points where the derivative is undefined.嗯，like on the boundaries. Yes, one six. Uh, and four. Only four.嗯。Two. Very good. Two and three and four. These are sharp turning points. Although the function is continuous, the derivative is undefined because there is no slope. You get a left derivative at that.The left slope doesn't equal to the right slope. Obviously, that's where the function is not differentiable, right? Yeah. Can I apply mean value theorem to the interval from one to three?二，有一个尖锐的拐点。二个，所以它不满足，因为这个函数是不可分的，但它是开区间。可以应用介值定理到区间从四到六。嗯，是。Yes, would it concern us because the function is discontinuous at the four? Because it does require function is continuous on the closed interval. No, because it's outside, meaning halfway continuity. We do have when you look at the interval. The so-called continuity on the boundary only requires it's continuous, meaning that point must be long. Had I changed that into, for example, is the mean value theorem applicable to the closed interval between three.Four. Hmm. No. No, very good. Because in fact, this is a four that's not continuous. That's the this continuity that matters for the interval chosen. Eddie, you're good. Yeah, yeah, next one, please. Oh, this is a distance traveled. Very good. Notice they ask for the distance, not the displacement. So what do we do here? I'm super glad this comes up. We just reviewed the technique lately.所以。We take the, we take the entire derivative. But. We integrate. Yeah, but we also the three. From what? Any what did you say. We integrate from zero to.Yes, absolutely. What's the integral? Do we put absolute value? Brilliant, that's very good. I fortunately in this case it doesn't change the sign. So actually, even if you forget, we're lucky enough not to matter. But I think you're absolutely right. We put the absolute value, because in case there's a turn, okay.But we don't cancel the distant travel. Because we can hold a distance now. Idee, I'm very proud. That's excellent. It's right. Finish it, please.呃，I got D. You got it. I you. Uh, I'm getting.Eighteen. Yes, seven. I'm also using this to really just probe a little: what would be your weak points that we need to review further, and which one would be its derivative.I'm getting E. Brilliant. That's it. The train will immediately help you recognize number eight. Roman sum. Notice here you always want to mark. They specify which kind of Roman sum were supposed to use. And also notice these are not the even intervals. The distance interval are different. So, talk me through how we're gonna do this.There's the.Could we like do something about the intervals?What's happening? So that's the four seven. Then you got a big integral that's fifteen, and the values are six point five, six point two, five point nine, five point six. What values I will using to calculate the total area.嗯。We. Red, red woman, son. For the first interval, the bass is a three, and the high.The height is what. The height is six point two. Yeah, you're just using the right end of the interval times of three, and then plus right end of the interval times of five, plus right end of the interval times another three, and add them together. You're done. But that gives you the change. How do you get the final answer? If you do so, roughly speaking, you'll be.Approximately getting a sixty-six-ish, because it's about eleven times a six-ish, a little less. But then we're finding what is the number of a what's the value at time equal to fifteen. Apparently, that number is less. When you just add those together, I double check the ending digit, and not really. The ending digit would be five, and then plus a four. So it is, it is that exactly. But then, what's your final answer though? That's.艾迪，我求所问。对。Did we? We didn't include the, um, the fifty liters from t equals four. Yeah, indeed. Whatever the integral, the Riemann sum gives you by fundamental theorem of calculus. If you go from the integral of a and b, f of x dx, now what you're getting is antiderivative of a b minus antiderivative of a. Remember, it gives you the change of the function.It doesn't give you the final function value. So indeed, we add on to the fifty here. Show your answer is actually this. We're good. Yeah. Okay, number nine. Eddie, are you crystal clear on this? Yeah. Okay, next one.Um, I got E. By doing. Um, you just, um, I cancelled out the. Yeah, it's a removable discontinuity. You just plug it through in. Exactly, but if you define it that way, basically it's fine now. Is it differentiable?I know the function is undefined. You have a hole there, but I can fill up the hole. Oh, we did fill up the hole here. Now it's a function differentiable at that location.哎，对。 So that points like part of the function. Yeah. Therefore. It could. Yes, in fact, there is nothing different between the one I feel linear and the simply that linear function two s plus one. They're identical, so it's differentiable.As long as you know that, continue. Number ten. Oh, good. Let's write the integral before we evaluate it. So it would be from zero to two.Well, how do we know for sure? It's from zero because that's in the first quadrant, right? Yeah. If we do not limit it to the first quadrant, then in fact, we wouldn't have a closed region, because in fact, this exponential function goes.Like this, and I equal two like that. So you are good. And then what's the integrand? Oh, why do I draw a horizontal line there? I mean this. So what's the integrand? E to x over two power. You got it.And what's the answer?So, when we take the antiderivative, it would be. I'm getting a.Good. So the antiderivative is twice of the e of that one, right? That's the antiderivative. I will plug in the boundary. Are you getting the same? Yeah. Good. Next one. Go as fast as you can because remember in the real testing situation, you have only two minutes to work out each problem.嗯，我 like this。 I think D is not an option. The D.Is the only option. Meaning the limit. No, you're saying D is not the option. Are you eliminating D or are you choosing D? Eliminating. Good, I agree. They are saying I just approaching to a D. Indeed, the limit is actually zero. I agree.嗯，I think the whole function is continuous.I agree. That is indeed continuous. It's also symmetrical about the point x minus two, right? You were taking the absolute value, and the domain are all reals. What is the shape of the graph? You think about it: the shifted version, reflected version of the square root of x. What's the shape of the square root of x, except for a shifted at two.哎，Who is drawing that? No, that's not accurate. Is this square like this? Yeah, that's right. The square root is the mirror image of the, quadratic, correct very.Good. So knowing having a mental picture of the graph, it save a thousand words. Then A B C E, which one is correct?I think it's not differentiable at x equals two. Yes, so it's it's continuous but not differentiable because of the sharp turning point. What would you call that? Is that the vertical asymptote? It's not the asymptote because the function is approaching zero, not approaching infinity. I don't know what's happening. Nevertheless, though, that's a vertical tangent line—a instantaneous vertical tangent line.It's hanging to the curve, except for it's not the derivative. Is that crystal clear? Yeah. Okay, let's get number twelve. That is a good problem. I'll be back in thirty seconds. I need to grab my charger.哦。嗯。I'm back. Since you're substituting the U.You've also to change the boundaries. Very good. Three parts now: integrant, differential, and the boundary. So, which one are we getting?嗯。Why does that take so long? Why D? Are you carefully substituting each part? So the U first boundary, let's just move. Eddie, what do you get?Anywhere are you? What did you get? That has taken too long. Here, since you set y into equal to square root of x, the boundary becomes actually from one to two, right? Because you just plug in the boundary of the four into that square root here. Then the e to the u, I just e u now divided by the u, and then what is the differential dx? You rewrite your x here as u squared, so the dx actually equals twice of the u du, and you you plug it in to u du after cancellations. Be. What went wrong, Ali? When I was differentiating the dx, I did something wrong. What did you do wrong?Like, I. Um, I took u equals square of x, and I did du. But you can do that. Ah, the result is still written in terms of dx, isn't it? Yeah. Well, even that it's not going to get you the wrong answer because the du really equal to one half. And then go ahead. I'm getting C, I think.Yeah, yeah, I meant to see. I meant to see. I actually wrote down which is the C. I just mean circled. Yeah, you're right, Eddie. So, and then you do, you would actually equal to the one over because you're taking the one half, and then the square of the x, correct. Yeah, but then you have to rewrite the dx in terms of the du though, which gets you twice of the du multiplied by another square of x, which is the u itself. So.It would be identical. But remember, whenever you're doing the substitution, I'd write a note to yourself: first, you want to write the d whatever the original dummy variable in terms of the new substituted variable before you differentiate, because you actually the goal is to write everything in terms of the y. So the first thing you do is to reverse it, and then you differentiate. Okay, so everything clear, huh?Number thirteen, please. Oh, this is easy. I don't even there's no need. You guys, you don't have to do that. I trust you can. Look at number fourteen. There are several steps to it, and I need you to talk me through all the.Yes. Oh, do we need to? If you do that, it's not going to really save you anything. Can we just keep that composition and then just do the chain rule for the derivative of the outer times the derivative of the inner? Instead of plugging it in.And work out what's that composite function. Do you see what I'm talking about? If we do that, what you're getting is this is a whole function which is a three x minus a two squared minus so forth, and you just take the derivative of this, which is fine. But I think it's easier for us to do this equal to the derivative of the outer function at whatever the g of x. Now when you plug in a three, what you get is a seven, and then derivative of the inner function at three. Well, derivative inner function just linear.So just equal to three always. Derivative of the outer function would equal to one over basically the x cancels out, right? And you plug in your your seven. Remember, you're not plugging in the three, you're plugging in seven, because the number to plug in is the g here. Makes sense. So you're getting another seven over of forty five, and after a bit of simplification, you're getting seven over five. I think it's faster this way. Okay. Eddie, do you confirm? Yeah. Beautiful. I just want to remind you: the any calculation regarding the order function, you have to plug in, uh, correct independent variable. That's not x; it's a g of x. Go to the next one, please.嗯，这是 definitely worth doing。Why don't you guys tell me the sign of each of the three numbers: h six, h derivative six, h double derivative.What is p? What is n? So, h derivative six would just be zero. Yes, because the derivative of the h now is simply the f of x, which at six exactly is zero.So we do know that just zero, that's the zero, that's the zero. So these inequalities are really just a comparison of the zeros. I mean, we're looking for their signs. So what do we know about the second derivative? It's positive. Very good because this graph is increasing, correct. So we know that's positive, and I'm going to immediately eliminate the ones; those are not correct.We're left with the only A and C now. Then what do we know about each of the six? Each of the six is negative. Very good. Why? Because it's under. Yeah, this part of the area. Well done. So we know that's A. Eddie, do you agree? Yeah. Very good. Continue. And this is also worth.对。可以容易，take the road. You can. But let's recognize what's going on conceptually. The particle is at rest, meaning one meaning the derivative simply equal to zero, right? Yeah. Okay, and by the way, doesn't really matter whether it's positive or negative. I mean, the this function itself, it's increasing or decreasing at the moment here. It's a turning point. But I think it's actually easier if you just.Go by the graph. This is apparently a quadratic equation, isn't it? And the two roots are a and b, and with a leading coefficient which is positive. I think knowing it's quadratic, it's easier to eyeball what would be the zero, the horizontal tangent line. I got B. Yes, half only half way in between, but that's only because it's just so compellingly.I think the medical about that point. Makes sense. Next one, please.嗯。嗯，I think the slopes constant the whole time.嗯，absolute end. Therefore. I think it's a. That's it, because why should.And the integrated digits of positive events increasing. You got it. Number eighteen. This is really a conceptual problem.对，是，嗯，对，嗯。If you plug in zero, it's like h equal zero, it's zero over zero. Meaning, absolute it's in determinant form. Otherwise, it's not worth doing. Yeah, it is. Indeterminant, but actually conceptually, it means something. Does it look like you see? That's a.Tell me what the expected outcome.So the answer is one over x with a four plug in. And I'd, in case you don't want to use the mental power to think about it, the means of derivative. You realize that zero over zero. We can use L'Hôpital's rule. And the function is h. So if we take the derivative on the bottom, we have one. On the top, this is the only appearance of that variable, so it's just going to be that, right? Yeah. And it's no longer zero over zero. You can just plug in the h. It's approaching zero. So L'Hôpital's rule.Really, it's a great labor saver. Except for it doesn't tell you the why, but it's very good to use. I want to nine nineteen, please. I was actually hoping we could finish the set. I'm in the section. Let me say, what's the quicker way? We want to.And the tangent line has a slope, so fundamentally we're taking the derivative. But if you take the derivative, that's eighty miles. The quotient rule, it's not very convenient. I prefer. Remember, it's not proper. You always do long division first. I prefer to write it that in this manner. Right, then it's a lot easier to take the derivative of this because it's just the it's a negative power now. Basically, what you're getting is a derivative of this. The result.It's simply twice of a x plus two squared. Correct. Yeah. And if you want that to be one half, what are the two solutions? It's multiple choice, so in fact, the options are revealing. But if they don't give you the option, you're solving it yourself. I want you to be careful to get both of the solutions now. So we need the denominator before, meaning we need the x plus two to be positive or.Native two. Careful, watch out for it. Therefore, what are the two solutions? Hmm. ABC. Yes. I want to number twenty, please.嗯，I'm glad this comes up. You surely don't want to find that inverse function by solving that cubic, because it will be as well. It's not worth your time. We don't need it too. 那，好，我们如何找到导数的逆函数，而没有写那个逆函数？Conceptually, if we want to find the derivative of inverse function, we need only reciprocalize the derivative of the original function.So instead, we're taking the derivative of the original function, which gives us, by the chain rule, six times a further two. What you're getting is actually, or when I said six, it's a three times a two. It's six times of the two x plus one squared, right? Yeah. And then we just need to plug in, recognizing this y must be one. That means that this whole cube must be one. That just means inside the cube, two x plus one must be one. We don't even need to say.It's equal to zero, although it is zero, so the answer just one six. It's a process crystal. Just keep in mind that all of these are conceptual. You don't need to really do the plug-in or find the inverse function. Whenever you're going down that road, probably you're making it a lot more complicated than it has to. Homework could be: Would you guys finish the remaining section? That's the non calculator, and record your time.Because if you take too long, that probably you could improve the different methods of solving it. They're supposed to be quick. Priso. Yeah. Alright, take care. Bye. Yep. Alright, thank you. Of course. Hey, truly good to see you guys. And as you come in, do ask me for the recording. I want to investigate more about the elastic road. I want to consolidate our constitutive laws as well as look at more application.In particular, addressing. Oh, I'm gonna actually invite Elaine to go. Thank you. Sorry, there was a student belonging to you. Thank you, thank you. Come on.

AP Calculus Review: Limits, Inverses and Tabular Problems

AP Calculus Review: Limits, Inverses and Tabular Problems

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