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00:59

Andy S.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

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Felicia S.

00:56

Greninjack D.

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is our direct comparison test. So for this we're answering the question. Does a Siri's and we're going to use beasts of then converge or diverge? So we're looking at Visa. Been were given Visa Ben as our Siri's that were asked the converge or diverge question for? So what? The direct comparison test says that let's say we have a sub in less than or equal to be seven less than or equal toe some C seven. So we have this inequality. So if c seven, which is over on the right hand side, converges. So what this is saying is that you find so you find a larger than be seven, a larger Siri's. And you know that this Siri's is convergent. So you have to know how Ace abandon c seven Behave. You find a larger convergent Siri's Okay, so if Sea Saban converges, you have found a larger convergent Siri's. Then the Siri's beasts of been also converges. So if you can find a larger convergent Siri's than beasts, event converges. The flip side of the direct comparison test is that if a sub in diverges, so what this means is that you you you're trying to answer this question. You find a smaller Siri's than Bisa Ben, and you know the behavior of a suburban. You find a smaller and you know that it is divergent, so you find a smaller, divergent Siri's. Then what you can say is that the Siris of B seven also diverges. So another way to look at this in summary is that you are given. So we're given the Siri's be Sabet, and you need to find either and a seven, which needs to be less than or a C seven, which could be a greater than Siri's. So you need to find a neither in a seven or a C seven, but you need to know how these Siri's behave. So when you're picking these Siri's, when we're picking what were directly comparing beasts of n two When picking Siri's to compare to Bisa Ben, we want to look for a couple of things you wanna look for common. Siri's soapy Siri's are often very good. We wanna look for peace. Siri's the harmonic. Siri's is used often, and you just in general wanna look for common. Siri's so common Siri's and you you have to know how they behave. So the reason we use P Siris in Harmonic Siri's is because we know whether or not these diverge or converge quickly so we can tell whether a Siri's is diverging or converging quickly. Again, let's look at the direct comparisons. So remember, we need to look for these common Siri's. So if we have the Siri's piece of Ben, you need to compare Bisa been to another Siri's that you know the behavior of so that you know the behavior. What I mean by behavior of a Siri's is whether or not it converges or diverges that you know the behavior of and again you have to situations. Bc event is less than or equal to C seven and C seven converges, then beast event converges these air the Onley to situations that are possible in the direct comparison test or the flip side is that you find another Siri's that is less than or equal to be Saban, and that s have been diverges. Then we are able to say that Bisa Ben diverges in one. So if we're looking at one IFC seven diverges, we cannot draw a conclusion in two. If the smaller Siri's converges. We cannot draw a conclusion on what we need. So these air the Onley to situations you have. You find one bigger that converges, or you find a Siri's that smaller and diverges. And again we're given this beast event, and we need to look for common Siri's because we need to know the behavior of what we're comparing. Two. So now we'll look at some examples, and it'll probably help solidify this idea of needing to know how these behave. But first, we're going to do some fraction review and how to compare fraction fraction comparing because when you use the direct comparison, test your most often comparing fractions. So a sub NBC Ben A. Seven usually fractions. So the first tip is we need to look at change numerator or the denominator. And I say this because I'm you Onley wannabe, changing either the top of the fraction or the bottom of the fraction. So the numerator, that's the top, the denominator that's the bottom of your fraction. Okay, so you either you want to do or you want to do one or the other. So for looking at the numerator. So for comparing the numerator of a fraction, Um, and increasing numerator. So an increasing numerator. So this means the denominator staying the same Right now you have the same denominator and increasing numerator. If we increase the numerator, we increase the fraction, which means the whole fraction gets bigger. If we have one over end and we're comparing it to two over in the bottom is saying the same. The numerator is increasing, the fraction is increasing. That's one example. If you have in over and squared and we're comparing Thio End plus one over and squared denominators are the same. But the top of the fraction got bigger because n plus one is greater than just a nen, so the fraction gets bigger. So that's how we're gonna look at the numerator if we're comparing if we're changing denominators denominators. So if we're changing denominators using the direct comparison, let's say we have one over to to the end and we're comparing it toe one over end. The easiest way for me to look at this fraction is to go off to the side and on Lee. Look at the denominators to end and end to end is greater than end. But when you put this infraction form, the sign flips. So if you were to rewrite this as a fraction each of these as a fraction, your sign flips. So when I compare denominators, I like to go off to the side. So if I have something like one over N minus seven and I'm comparing it toe one over N minus three, I go off to the side and I say okay, and minus seven compared to end minus three. Well, if I'm taking away seven, that is going to be less than a number. If I'm only taking away three. So because if I have this inequality pointing to the left when I put everything over infraction form I just slipped the sign so I would have won over and minus seven. Flipped the sign one over and minus three. One other example for comparing denominators. Let's say I have one over N compared to one over and plus one. I again would go off to the side. I would look at N and I would look at and plus one and plus one is greater, which means infraction. For him, the Fraction one over in is greater than one over and plus one. So this is how I'm going to do the comparison with denominators again when we're using the direct comparison tests were trying to find a Siri's a seven or C seven that's similar to be seven, but that we know the behavior off. So this is why being able to efficiently compare your fractions is going to come in handy. When we go through these examples, we're gonna look at this Siri's, and we want to know if it converges or diverges. So the first thing for the direct comparison test is identify you're a sub in, and that's your original Siri's. And then we need to find a beast event that it's similar to our ace event, but that we know how it behaved. It needs to be a common Siri's. So with one over and squared plus 31 over and Squared, is the natural choice for a beast event. It is similar to a C event. Also, this is a piece Erie's. So if we have one over and squared, this is going to converge. This converges because P is equal to two, which is greater than one, So we have a P Siri's here. So now that we've identified the beasts of Ben, we know how it behaves. We need to make the comparison. So for comparing a seven with our beasts event again, I like to go off to the side and compare my denominator. So I have in squared plus 30 which is going to be greater than and squared. But when I put these on the fraction, I need to flip the sign. So I flipped the sign and I have that a seven is less than B seven and Bisa Ben converges. So if I have a bigger Siri's that is converging, something smaller than that is also going to converge. So then we can say by the D. C t by the direct comparison test Bisa Ben converges. So our Siris of one over and squared plus 30 which is our ace a been also is going to converge, and that's how you're going to use the direct comparison test. So I identified RBC been made our comparison and then because we had a bigger Siri's that converged, we were able to draw the conclusion that made sense here. Direct comparison test. So we need our race event which is one over the square root of end minus one. The natural B seven, which is the Siri's to compare it to, is one over the square root event, one over the square root event. Here we have that P is equal toe one half, which is less than one because we have a P Siri's with P being less than one. This Siri's diverges. So here we have a beast of Ben That Diverges was identified are a seven. Now we need to compare the Siri's so we have one over the square root of end minus one. We need to compare it to one over the square root of N. I like to go off to the side to make my comparison with the denominators. Square root of n minus one is less than just the square root event. And when I use make thes denominators into fractions and put the one on top, I need to flip the sign. So here I have that one over. The square root of end is less than one over the square root of end minus one one over the square root event was my bisa Ben, So I be seven less than a seven and might be seven diverged. So now I have a smaller Siri's that is diverging. And now I can say that by the direct comparison test, because my smaller Siri's is diverging. I can say that my original Siri's from N equals two to Infinity also is diverging. Okay, so there you have it. Another example of the direct comparison test Again find you're a sub in finding the beasts of and the similar Siri's. It is usually taking away the minus one another plus when it's usually making it just more simple. And then you go through the comparison to see if you can draw the conclusion again. So here we're going to take a look at a couple different options of how to do this. So we have. We're using. The direct comparison tests are a sub en is one divided by end three to the end, our beast event. It's going to be natural to use one over in or to use the 1/3 to the end. Okay, well, one over end, we know is the harmonic Siri's, and this does diverge. Okay, so we know this harmonic. Siri's diverges So let's compare. So are a cement is one over in three to the end, comparing it toe one over n again. When I'm doing the comparison, I like to go off to the side. I have n three to the end compared to just end this end. Three to the end is larger, and then I just have end. But when I make it a fraction, I flipped the sign. So here what I have is a Siri's where the larger Siri's is diverging and there's nothing I can say because I have a sub in Is less than Visa been Bisa Ben Diverges. But there's nothing I can say here, so I can't draw a conclusion if I'm comparing if I'm letting Visa then be won over in. But I had another option to compare, which is 1/3 2 then. So if I let a sub in, which has to stay the same because it's my original Siri's, and I let my beasts of N B 1/3 to the end well, 1/3 2, then hopefully you can recognize. This is a geometric. Siri's right. We're looking at a Siri's. Our would be one third, which is less than one. And because one third is less than one, this converges. So now we have a beast event that converges. So now when I compare one over and three to the end with 1/3 to the end, I go off to the side and three to the end is going to be greater than three to the end when I make it a fraction I flipped my sign and I have a NASA been less than Bisa been. But now my bisa been, which is equal to the Siris of 1/3 to the end, converges because it is a geometric Siri's. So now my larger Siri's is convergent, and I can make the conclusion that by the direct comparison test, my original Siri's and through the in also converges. So this is a very good example of what I mean when you need to be able to draw the correct conclusion. Yes, we know that the Siri's one over N diverges, but when we make the comparison here, this doesn't make sense. If the bigger one diverges, the smaller one could go either way. But by using and comparing to something else that was similar. We were able to draw the correct conclusion that are Siri's does converge by the direct comparison test. So if you don't immediately pick the right beasts of men, but another one's is Justus, Easy toe look at or easy to tell you can use that, but we do know a lot of other tests, and we will know a lot of other tests. So there's other options. If the direct comparison test doesn't immediately come naturally, I did try to to the end, so any time you have signs and co signs, it is a good I use a good idea to use the direct comparison tests because we know that the absolute value of sign is always less than one less than or equal toe one. So what this means is that if we square sign that's going to make this positive. So science squared of n will always be less than or equal toe one. So we almost naturally this way confined RBC event RB Cement is going to be one over to to the end, and this is because when we go to make our comparison sine squared of end divided by two to the end and we're gonna compare it toe one over to to the end will sign square. We already showed you up here is always less than or equal toe one. And because the denominators air the same, this is just a comparison of the numerator. So whichever direction the numerator um equality goes the fraction equality will go if the denominators air saying the same So now we just have to look at our one divided by two to the end, which is our B seven b seven is a geometric Siri's with our equal toe one half which is less than one. So Bisa ben converges And because the absolute value of sign or sign is always less than or equal toe one, we have that a seven is listener equal to a bisa been we're Bisa Ben does converge so because we have a B seven that converges and a seven is less than that by the direct comparison test our original Siris of sine squared and over to to the end from an equals one to infinity also converges Great. So what we've shown here is we're using this property of signs this is why the direct comparison always works very good, with co signs as well. The best trick is to make the beast of Ben make the numerator of your Visa Bed one and keep the bottom the same and then make the comparison and test your beast event again. We're always checking to make sure we can draw the correct conclusion with a direct comparison test. So make sure you think through this a Saban is less than something that converges. So if something that's larger converges, something that's smaller must also converge. And that's the conclusion we've got to minus, and we want to use the direct comparison test are a Saban is n post too divided by and squared minus in. But are beasts of Ben is gonna take a little bit more work to find. So usually the natural thing is to just get rid of our constant to and let beasts of NBN divided by and squared minus in. This is equal to one over n minus one. If you do some subtraction here, right? So for comparing in plus two and and so a race A been in this B seven we have in plus two divided by in squared minus in compared Toa one over and minus one. So our bottoms air almost exactly the same. So we're just gonna look at our top and plus two, and then we just have a one on top. So that means we're gonna have this inequality we have that a sub in is greater than RBS event. But then this one over n minus one. There's not a whole lot you can tell about this. This isn't a common Siri's. We don't immediately know what this Siris of visa and does. So we're going to do is we're gonna pick another Siri's. And now we're gonna let Bisa Ben be won over in. If we had won over in into the mix that inequality continues. So now we're gonna let our B seven immediately be won over end. The reason for the intermediate step is so you can more clearly tell what you're inequality is doing. So now we're gonna have this inequality and the beast of Ben. We've kind of deduced from a couple steps, and we're going to compare it with one over in. We've already said this is the direction or sign goes because we've looked at this step down here. If we have n minus one compared to end end minus one, we're subtracting is less than an end. But when we put it as a fraction, we flipped the sign and we see our end squared are a seven here and are one minus in here. I'm just copying that this direction went over in. That's the harmonic Siri's. And what we know about the harmonic Siri's is that it does, in fact, diverge. So now what we have is a beast event that is less than in a seven with B seven diverging. So are smaller. Siri's diverges. We can draw our conclusion by the direct comparison test. This Siris of n Plus two, divided by in squared minus in, also diverges. So sometimes it could be a little bit tricky to find that, um, comparison. And if it is tricky to find the comparison, don't use the direct comparison test use a different way. This one wasn't too bad, especially right off the bat here. If you got rid of the two in the end, you would end up with an over and squared, which would be one over in, so you could directly get to this piece event. However, making sure you're inequality went the right way. It could be a little tricky if you don't take this middle step to get the correct inequality. So this is just an example of how it could get a little bit more complicated. But in the end, you're conclusion Is that your Siri's diverges again? Anytime you're using the direct comparison test, make sure you're drawing a logical conclusion. Your smaller Siri's must diverge. A larger Siri's must converge to be able to draw a conclusion. You can review those guidelines and the lecture video of the direct comparison test. Harrison trick. So again, if we're looking at a seven and we're saying a seven is less than B seven Bisa Ben Convergence, I wanna look at why we're able to draw the conclusions we can if we look on a graph and we have some Siri's beasts event. So it's just a function, and we're saying that this beast event converges so beast event does converge. It goes to zero. If a Saban is less than bisa been, that means a Saban is less than B some of everywhere So if a Saban is this dotted line, there's nowhere that this a sub in will ever be bigger. Then be seven. So eventually a seven also has to converge because it cannot go past B seven. That's why we say that the bigger Siri's needs to converse. So if you get confused, feel free to draw the graph. The other situation with a direct comparison test is that if RBC Ben is less than a Saban, then we need to say that our visa been diverges. So what this looks like on a graph or if you need to prove it to yourself, is that my best event, which is the saw? The line is something that goes on forever. My a seven dash line is another function that is always greater than he said Ben, which means that this Asa Ben will never dipped below piece of n and that and turns me a Saban must also converged because it will forever be greater than visa event. So this is why in review, when we do the direct comparison test, if we have the inequality, a sub in less than Bisa been to draw a conclusion. The Siris of Visa, Ben must converge to prove that a PSA Ben converges bisa been converges. Then we're allowed to say a Sibon also converges and you can think of that graph for draw the graph if you need Thio and then if we have an ace, a been and B seven is less than a seven. Then we have to say that Bisa been diverges to be able to draw the conclusion that are Siri's Assab in also diverges. So again, you can't say the opposite in any of these situations. These air the Onley two scenarios where you're able to talk about the A seven Siri's. So keep these in mind as we go through the examples that will help you understand how the conclusions are drawn and feel free to draw the graphs. If you ever forget what needs to convert your diverge

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