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JIt is worthwhile noting that the sequence (IT"Il"") in (10) need not be monotone. To illustrate this, consider T: defined byX= (61, €2, 63,(0,...

Question

JIt is worthwhile noting that the sequence (IT"Il"") in (10) need not be monotone. To illustrate this, consider T: defined byX= (61, €2, 63,(0, 51,262, 53,284, $s,

JIt is worthwhile noting that the sequence (IT"Il"") in (10) need not be monotone. To illustrate this, consider T: defined by X= (61, €2, 63, (0, 51,262, 53,284, $s,



Answers

In Exercises $61-64$, use graphs to determine whether the equa. tion could possibly be an identity or is definitely not an identity. $$\frac{\sec t+\csc t}{1+\tan t}=\csc t$$

Maybe a good problem. Number 61. Yeah. Good for you on that, then. D is a little card off. Bye bye. Do minus speak. So we're first a car, just artistic. Go to court stuff by my do my nasty physical do one by bad bye bye to my nasty Danny. When it's because too, Danny, my understand people. One plus 10 a 10 p that I do minus that the doing it by one plus bad bye bye to in the time of day. So if a day in the denominator, what will bigger thing is, huh? Right. Plus done by do in good time day doing it by on my radio my just dandy. And if you thought this one weren't by Dan Hueber do my last Dante. Bless that I do. You do, Dante. Do it anyway. And I do minus stacked on 19. Anything that needed in front businesses invented one day a d minus Banque de plus Dan fiber Do beautiful day outside here in your Dante. They were by your dad by, But we will take on sir one minus. I be you do todo time type. I do. Okay. So why didn't they getting it? Yeah. Real big getting us one. Bring me the zero plus this stand by. But this endeavor to get cancer therefore time be doing it by. Why buy in three days? Zero have learned minus little. Just go to that did. He's using Good. And let us thank you.

Let's solve this initial value problem by integrating first. So integrate dxy team begin X equals in a row of one over t being ln of tea and the interval of minus one over t squared just being ah plus one over tea. And we could really just checked about taking the derivative of liberty. It is negative one over T Square plus 60 plus a constant will call. See, So to find this constancy, let's plug in the information were given here that one X equals zero t equals one. If we plug in X equals zero, we get Ellen of T. A lot of one is just zero as well plus one over one, which is one plus six times one which is six close. See, and we get that C equals negative seven. So let's write this on the next page. We have our ex weaken right X equals Ellen of tea were just copying everything over plus one over tea. Close 60 and then plus C to write plus c, we just substitute. We found that sea is negative. Seven. We write negative seven and one thing we do have to note, though, is that he has to be greater than zero. And how do we know this? Because when we have Ellen of tea and we see that, of course, there is no absolute value here that if we plug in a value less than T r. Sorry, less than zero like T equals negative one. We can't have a lot of negative one. So X equals this when tea is greater than zero, and that's a solution.

Hello and welcome. We are looking at Chapter two, Section one Problem 55. Now we're given a new type of equation. Well, new to me, maybe not new to you. The Ricker equation. It looks like this. The general form at least. All right, so we're just introducing an exponential. Really? This is be here and were given some parameters values to plug in here, this specific case. So this is what specific to this question here we are asked to plot some terms to see how it behaves. And then kind of in further convergence, Convergence s O. I did use a spreadsheet to graft this. And so, uh, it looks like this. Ah, sequence is converging. Thio this decimal value here Something to do. 0.6931 The last six terms here are all 6.6931 So it's getting closer and closer to some value. This 0.6931 fish. Um, so you can say this appears to converge? We're not sure exactly. Toe what? It's just near 0.6931 All right. Uh, so moving forward from there. It says if so, if it appears to be convergent, estimate the limit and then assuming the limit exists, that is its exact value. So we're gonna estimate 0.6931 and we're gonna assume the limit exists. So just like we've done with a few different problems here, we're gonna make an assumption if we assume the limit exists, we're assuming that the limit as C goes to infinity of except e converges to some value. We're assuming the limit exists. So it goes to l just call it hell. And if we make that assumption that it follows logically that this also except E plus one the next term, uh, the limit as to goes to infinity of that same sequence, basically just one term down the line, it's gonna obviously converge to the same value. All right, So what we do then, is we take the limit of both sides of our equation so I could plug in my see value of two, plug everything in. I'm taking the limit as he goes to infinity of both sides. So the left hand side is definitely getting go to l. That was part of my assumption. My right hand side, it's going to, um Well, I cannot hold the two out of the limit on the right hand side is a little bit more complicated, so let's focus on that first. Yeah. So let's continue this on. Ah, the next page here. So I can sure the work more clearly. All right. So, uh, limits using properties of limits, they're pretty flexible things. Um, So I can write this as the limit s t goes to infinity of except E. And then the limit of a product, you can take the limit of each of the factors. That's one of our linen properties. So by our assumption, we have l equals two. And then this here goes to el by our assumption. And then what weaken do, uh, with this limit here, we can actually move. The limit is just and a constant value 2.7 something so I can actually move the limit into the exponents. That's one of our limit properties as well. So little squished here, but, um, the the limit is in the exponents with the negative x two x sub d. So, after we done resulting all of the limits, it just looks like this. L equals two times at all times E to the negative, Al. So remembering what we know about negative exponents this we could also write this. I would suggest it is running this times one over easy bl or that's the same is over e to be out. So if I want to get, um, l by itself, you could cross, multiply, could do different things. We just want, um, everything in the new greater so I'll go ahead and do it and more steps than I need to. Just to be clear, I just multiply e the lt other side. Now I'm gonna divide both sides by l get you to the l equals two. And then to get rid of that e the exponential I need to use natural logs or natural log cancels with E. If I do it to one side, I have to do it to both sides. And so then I end up with l equals natural law of two s o weaken. Check to see if this works with our our graph by typing in natural log of two into our calculator. If you type in natural, log on to that's what it's approximately 0.6931 So, this, um, this is consistent with our graph. I'll just show you that one more time. This is about 0.6931 from our grasp. All right, so, uh, double checking the question, make sure we answer the whole thing. Um, if it seems to be convergent, estimate the limit. Assuming Dylan just calculate its exact value. And that's exactly what we did. We are.

Problem decided whether usually hide what? So this is given to us the last one. So we're gonna find the first different shoes. First of its I say that happen under a lot of that second differences model. If neither rivers first we have six. I can already tell that ports well, 22 before the second difference. Six. But you two times 2 4 to 18 between old guilty. Since you have the same second difference. Is this it right now that we've determined it's erratic, You know what? Oh, that's cool. Plus a few times. And the $3 we know by the fall it's one. It's yeah, must many times one squared or just a plus tires people See you. Yeah, me too. Second time to see what people say Lost people. Time to square, just for a plus what we know to be to tend to see what and then you also know they said 16 and this would be called free spirit types us. Just just I would never use three so minus sampling and see what turns up or else thanks is agreement. So let's start off with you when you said Cap six minus 06 for eight months. Plus to the spirit that suit. So from this six nights? No, I know that when I do it. So, yeah, um, if you wanted to do so long as he could, something just Well, I mean, you can also do. Insect realizes it, but I don't think it's that. Do you get tempted clothes? I was to Elsa. Was time minus five? That's what you do in your life too. We could do either in this case, so want to have this? You're probably thinking, How did I perceive do you think next stuff. Wow, I can I c c and it's freshly. Please, You're right. So I have speakers Negative c make this year. Bus goes. I see. That's what a step one. Just talking. And then I would have and two both expressed, you know, both beings expressed in terms of that. Then we have a simple variable. I'm by day I said this would be cost down in a problem. Value would be in terms of some lose six points number. Put emphasis on that. That's all I know. That negative must be 486 minutes theory, which was negative too. We want to find possible to most. It's becomes too. This. So now we have. We'll see past trips up since only Somalia, too. Except you six. So you would have way bust two times, which we have six months to thank you two times 6 12 No. Plus two times negative. Who's six? That prostitute Ian Posse. Plus this week Possible. Sometimes doing my 60 two's without this value. Other pursuits. So have a single very trips. This working class, too. And they got more. It's a TV equal sex check. My work. Yeah. So, dance. I'm going to die a cheat. You checked your That's your doctor. A TV program, Be those 10 minutes fired. Over What a plus, two times 10. Just one. And then you get by this fire to Taipei and honey And then I tiu What's that? This is all so Yeah, this made with that? Don't do that. Yeah. Past funny over on minus six. That passport. So I would add negative 60 plus way. Do you have for a walls? I checked into that 82. One thing I ate was too. I thought that was you. Which is why I have to work in the first place. Mom here, once I know equals forever for C two times two Percy. Which is it? So it was You know, I'm gonna have a guy using this form here this year for my answers equals a times and swears that you once were plus a few times on, but use you So you have two more that you're just Patsy. Just adding to search is mine too. There you have it. Two and two is vision, and you have a quadratic model here. Thanks, guys.


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