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MethcmnbeiPuitiDO NOT WRITE YOUR NAME ON THIS PAGE6. [15 points] For each of the following questions; fill in the blank with the letter corresponding the aliswer Tr...

Question

MethcmnbeiPuitiDO NOT WRITE YOUR NAME ON THIS PAGE6. [15 points] For each of the following questions; fill in the blank with the letter corresponding the aliswer Troni the bottom of the page that correctly completes the sentence_ No credit will be given for unclear answers_ You do not need to show Your work:points] The limit . lim(+2)"[3 points] The value of the integraldT: IlZ[3 points] The value of the integral[3 points] The value of for which the differential equation y = Ay is satisfied

Meth cmnbei Puiti DO NOT WRITE YOUR NAME ON THIS PAGE 6. [15 points] For each of the following questions; fill in the blank with the letter corresponding the aliswer Troni the bottom of the page that correctly completes the sentence_ No credit will be given for unclear answers_ You do not need to show Your work: points] The limit . lim (+2)" [3 points] The value of the integral dT: IlZ [3 points] The value of the integral [3 points] The value of for which the differential equation y = Ay is satisfied by the function f (t) 07 points] The length of the polar curve > = cos(0) between -"/4 and 0 = "/4 (A)is 2' > (F)-is (B) _is e. (G)-is 0 (C)_is e2 (H)_is 2. (D) _is 2e_ (I)-_does not exist . (E)__is 3e. (J)__diverges:



Answers

Use a CAS to explore graphically each of the differential equations. Perform the following steps to help with your explorations.
a. Plot a slope field for the differential equation in the given $x y$ -window. b. Find the general solution of the differential equation using your CAS DE solver. c. Graph the solutions for the values of the arbitrary constant $C=-2,-1,0,1,2$ superimposed on your slope field plot. d. Find and graph the solution that satisfies the specified initial condition over the interval $[0, b]$ e. Find the Euler numerical approximation to the solution of the initial value problem with 4 subintervals of the $x$ -interval, and plot the Euler approximation superimposed on the graph produced in part (d). f. Repeat part (e) for $8,16,$ and 32 subintervals. Plot these three Euler approximations superimposed on the graph from part (e). g. Find the error ( $y$ (exact) $-y$ (Euler)) at the specified point $x=b$ for each of your four Euler approximations. Discuss the improvement in the percentage error. $$\begin{array}{l}y^{\prime}=(\sin x)(\sin y), \quad y(0)=2 ; \quad-6 \leq x \leq 6, \quad-6 \leq y \leq 6 \\b=3 \pi / 2\end{array}$$

Okay. What we're gonna do is we're gonna step through the process of, um, doing multiple things. So we're going to start with, um, a differential equation. Why? Prime is equal to X plus why? And the first thing we're gonna do is we want to plot the slope field, and we want to do it over, um, the X interval of negative four to positive four. And the UAE interval of negative four to positive four as well. And so, um, we're gonna go ahead. I'm gonna switch to, um, a slow field generator, which is actually Dez Mose. Um and so I have so field generator by Dez knows I already have X plus Y, um, in my differential equation place. So here is the slope field, Um, and you consume in if you want to. Um, I'm gonna keep it from basically negative 66 but you can zoom in if you would like to. Um, it's a little bit harder with Dismas, but if you have a t I inspire, you can set your window a swell. So that's the first thing. So this is the soap field of X Y. Prime equals x plus. Why And then the second thing we want to do is we want to find, um, the general solution. Okay, So what I'm gonna do is I'm a switch to ah, differential equation solver. This happens to be a simple lab. And so, as you can see, I've already put in there's multiple ones on the on the Internet. Why prime is equal to X plus y. And so when I did that, this one you since it's free, you're gonna have to deal with some advertisements. And then here is the solution right here and in below it. You can also see, um, the showing the steps of how the symbol lab actually calculated that general solution. So here's our generous elation. A negative x minus one plus some constant number times E to the X. So there is our general solution of Let's see why equal to negative X minus one plus some constant number times E to the X and then the second. The third thing we want to dio is we want to graph on our slow filled each solution for C one equal to negative to negative one, 01 and two. So that's what the third thing we want to do. So I'm gonna switch back to my slope field generator and let's see if we can get these in. Um, and so I have, um Why equal to negative Hex Mine looks. It's not what I wanted to dio. So there we go. Um, negative X minus one. And then we had No, it's gonna be a minus two minus to miss the 1st 1 and then e raised the X. So there is the first solution craft. Then we're gonna do why equals negative eggs minus one minus one. He raised Toothy X. And so there is the second solution, and then we're in. Keep going. Um, and the next one is just negative x on this one. Those the, um C one is zero. And then we do Why equals on this one. And then it was a plus. He raised the eggs. There is, so I haven't As you can see, these are all color coded. Um, and then they last one is negative. Ags minus one plus to you to the X. And so we'll wrap pop, generate. And there we have the five particular solutions graft in our slow field. Okay. And then, of course, you can always take a snapshot. Okay, Now, what we want to do is we want to find and graph the solution for with our initial condition of why of zero is equal to native seven tents. So we want to know what that solution is. And so we have Negative. 7/10 is equal to negative one plus C one e raised to the x o raised to the zero. I'm sorry, cause x zero. And so this is going to give me negative? No. We're gonna add the one over. So it's three tents is equal to see one. So that particular solution is why is equal to negative X minus one plus 3/10 mm to the X. Okay, so there we have that one. And so and we're also going to, um, graph the solution that satisfies on this space. Specified initial condition over the interval over the interval. Um, from zero to one this time. Okay. Now what I'm gonna go ahead and do is I'm gonna go ahead and not do that yet because I want to be able to graph in these, um, for a bunch of the next. Another step. So we have a bunch of Nixon other steps. Um And so, um, and what we want to do is, since it's over that interval, I actually need to know what he is at one. So we wanted at one. And so if I put in one four X, we're going to get a negative 1.18 so I need to keep that in mind. Okay, So now what we want to do is we want to find the Euler approximations so we can keep going on this, and we're gonna find the Euler approximations with several different sub intervals. So the first time is four sub intervals and there were do eight and then 16 and then 32 and number grand actually superimposed each of these on to our slope field. Okay, so that's something to remember. Um And so what we're gonna do is I'm gonna go ahead and switch to an online Euler Approximation calculator, and so I have one Here. It's to Planet Cal cook. Um and so how you do this is you putting your different your equation. Um, actually, go up here and put in the differential equation you put in the differential equation the initial X and Y values the point of approximation, which is that be value. And, of course, your step size. And so the first time we want to do it is at and 47 rolls, which would be, um, a 0.25 And then, of course, you devalue your exact solution, which is that negative X minus one plus 10.3 or three tents. E raised the X and then you hit calculate. And so we're to write down a couple things. So as I scroll on down one, the approximation appear is the approximate y value. So when X is one my approximate why values negative 1.27 So that's going to be critical to remember. And then also, we're gonna also look at kind of the, um, absolute error, which is this 0.831 time. And so where you come back and we're gonna add the's an so for sub for four four sub intervals. My approximation is one comma negative 1.27 and then I have an absolute error of 0.0 eight for you on. But I want the percent error. So the percent error is equal to that 0.831 divided by the actual which is or the absolute value of the actual, which is 1.18 So this is going to give me an error. Uh, 7.4%. Okay. And so now we're gonna go back into 8 16 and 32. Um, so we're gonna go back to the all online calculator, and I may come back here, and the only thing I really need to change is this Step size. So one divided by eight is point 1 to 5 it calculate again and you notice now my why value approximation is negative. 1.23 and my absolute errors 0.457 So I may come back, and I'm gonna write that down. So this is one common negative. 1.23 and that percent error now, is that point? What did we say? It was? 0.457 Divided by that 1.18 which is going to give me 3.87%. So that's a big decrease. And so now we're gonna go back and keep doing it for step size of 16 which is 160.6 to 5 for my step size. It calculates, and then we come down. Now it's negative 1.21 for my approximate and in absolutely era points he or two for one. So this would be it. One common negative 1.21 percent here. Equal two point 0241 divided by that 1.18 which is equal to 2.4%. And then we're gonna do the 4th 1 of believe it 0.0, 3125 it calculate. And now it's negative 1.20 with an absolute of error of 0.124 So this is one comma and negative 1.20 with a percent error equal two point 01 to 4, divided by the 1.18 which is 1.5 Percy. Okay. And so now what we want to do is go ahead and over the inter fall from zero toe one is to actually graph in or plot these Euler approximations. So I'm gonna actually switch to a another, um, slope. Um, field. Um which happens to be through a bluffed in. It's just gonna be able to help me plot in these specific points as supposed to the general solution or the Jialu Shin particular solution. So what? I'm gonna go ahead and do. I'm gonna go ahead and clear all curves. Um, and we actually wanted it from 0 to 4. So, um I mean, zero co one holly guys. Um and then I think this is actually gonna be in the negative side. Um, let me go, um, native to 20 Okay. And then what you do here is you put in all your solution. So with the 1st 1 we have put in is the actual workers at 18 negative. 1.18 Um, the next one. We had waas for that force of interval, which is that negative 1.27 Then we had a negative one point 23 lips and submit brat. And as you can see, the green is further away than your exact solution. And so each time I believe it's going to get closer and closer. Um And then, of course, that purple got closer to the red and then Of course, we had the last one, uh, zero. So there are my four that's in the orange so you can go ahead. And, as you can tell, as your sub intervals increased your approximate Euler approximation approach the exact value, Um and so let's go ahead and go back. And of course, you can take a snapshot of this, um and so let's go back to here. And then let's go ahead and discuss the improvement in the error, which makes sense. You would believe you would think that as, um, based off a calculus as the number of sub intervals as the number of sub intervals increased, then the, um approximation would approach the actual value or the exact solution. And so you would expect your percent error two d quick decrease, which it does

Okay. What we want to step through is to take a differential equation. And our differential equation is why prime is equal to why times two minus y, um And we want to actually, um, graph or plot the slope field. And we wanted over the interval from X values from 0 to 4 and the why values from zero 23 Okay, so I'm gonna go ahead and change to a slope field generator, which is called gizmos on. I was working on it earlier, so let me clear this all. And so the slope field generator is kind of nice because it provides me the ability toe actually generate the slope builds. And so this waas, um, your number to go ahead and write it as a to why minus why cleared. And so it gives me the ability to generate the slope field, and I'm actually wanting to go from 0 to 4, so I can actually move this and we're going from 0 to 4 in the X and then 0 to 3 in the UAE. So we're kind of focused on this area right here. And so there is my slope field and then the second thing we want to dio is to actually find the general solution to that differential equation. So I'm gonna switch gears and I'm actually going to go to a different to equation solver. And so I've already pull it up. I'm using simple lab and I like this one because it actually you all you have to do is put why prime is equal to that. Why times two months, Why hit go and then what it does is it automatically gives me my solution in my general solution. And then if you needed to, it all actually also gives you the steps. So that's general solution is wise equal to over E to the negative two x minus some constant number plus one. So here we go. Or a general solution is why is equal to two divided by E raised the native to ex, um, minus some constant number plus one. So there's our general solution, and now what we want to do is back on. Our Sofield is we want to graft, um, particular solutions when our constant number is negative. Two negative, 101 and two. So I'm gonna switch back over to our slow food generator. And I'm gonna grab these in so wise, equal to two divided by you know, we had e raised to that negative two x plus a two because we had minus, so plus whips, I need him back up here, plus books. I'm probably gonna have to put him in parentheses. So, um, native to X plus a to, um and then plus a one. So there is our 1st 1 and then we're gonna keep doing this. So why is equal to two divided by and then, um, we had e and then raised to And of course, we could have that print parentheses of native to eggs, plus one and then a plus one. So there's our second solution, and then we had zero. So this is why is equal to two divided by e, um, raised to the negative to eggs this time? Because, see, 10 plus the one and we have that solution, and then we can keep going. Um, I'm going to do another one. Why is equal to two? We've got a couple more to do raised to the hooks. I need a good ground here. Um e raised. Who's in the course of parentheses. Negative to eggs, this time minus one in Andhra plus one. And in our last one where we had that, um, positive to which will make it a negative, too. So why is equal to two divided by he raised to the parentheses made of two eggs minus two and in the plus one. And so And I like this a swell, because all color codes. So here are my five particular solutions for those constants guys. So we've done that. And of course, you can always take a snapshot, Um, and put it in your, um, your work. So we've craft those. So now what we want to do is we want a solution that satisfies. So now we want to find the solution that satisfies why of zero is equal to 1/2. So when Why is 1/2 two over? And then, of course, this is going to be, um when X is zero. Basically, we have e raised to the negative C one, which is the constant we're looking for. So four is equal to e to the negative C one plus one. So three is equal to e to the negative, C one. So that means see, one is equal to, um, negative. Natural log of three or negative 1.1. Okay. And so there is our solution. Um four. Um, so we have Why is equal to two divided by may Expand this e to the negative to eggs plus 1.1 plus one so we can actually go back in to our slope generator and graft. That one in a swell. So why is equal to two delighted by e raised to the parentheses? Negative to eggs. And I think it was plus 1.1 this way and that, um, graph lips. He did not. Oh, no. There it is. The blue one, I was gonna say, And it's a second, um, because we had an additional blue on. So it's this one right here, the second blue one right here. Okay. And so we have that. And now what we wanted to is, we actually want to, um, later on, graph him over the close interval from zero 23 and really, what I want to do, um, used to find the y value when X is three. So why is equal to two over E to the negative six plus and I went ahead and changed this. That to the natural Auger three plus one. So that is going to give me about 1.98 five. So when x His three my y values 1.985 and we're gonna have to remember this because now what we wanted to you is to, um do Euler approximations at that X equal to three. Okay. And and so what I want to do now is to some Euler approximations. And of course, we're not gonna do it by hand. We're gonna use a, um, another system to calculate those Euler approximations. And what we want to do is for sub intervals of four, eight, 16 and 32. So being equal to 32 so the system I'm using actually cannot do sub intervals. It needs to do the CEP size. So we're intake or interval, which is 0 to 3 and three divided by the number of seven rolls. So this is gonna be a step size of 70.75 a step size of 0.375 then 0.1875 and then the 2nd 1 the 4th 1 of 10.909375 And so we actually want to find out what? That why value is approximated to for each of those, Okay. And so now I'm gonna go back to a online, you're calculator by planet Talc, and there's several different. I like this one because actually will also graph it, even though we're not going to use this system. And so what we do? Plus, you can put in a bunch of the point of approximation step sizes instead of subbing and rolls. And, of course, you can always put in that exact solution. Um, the only thing about this one is you're gonna have to you. You also use your multiplication symbol. Does it work? You just put two. Why? So here is my different to equation. It won't let you put infraction. So you have to put in decimals and then the first step sizes 0.75 And so we do that. And right here it tells me it is 1.96 or I can scroll down. And here is my exact solution is in the orange. The blue solution is in the is theater approximation. So we strolled down here the approximations 1.96 the exact values 1.99 And I had enlisted his 1.985 with an absolute terror 0.2 to 7. So we're write all those down. So this is 1.96 and we are actually wanting the percent error. And so remember, Percy error is gonna be that that absolutely air divided by the actual value. So this is point 0 to 27 divided by that 1.985 or 1.99 which is going to give me about a 1.14% error. And now we're gonna do it for each of the step size. So we're gonna go back now we're gonna change. And I like this, too, because I don't have to re type things in. I can just change that step size. And so that is 0.375 hit. Calculate. My approximate value is to this time. So I go down here, my approximate values to 1.99 and the absolute of 0.1 points here. 135 So this is going to be, too, with a percent here. And we're writing the Percy error because when we come back and actually talk about this, and so that is about and of course, we multiply it by 100%. Um is your 1000.68% which, what, you should expect that it would be decreasing. So we come back changed except size once again. And this time it is point 1875 Hey, Calculate. And we have 1.99 So here is 1.99 scroll all the way down to get our absolute error of 0.8 zero. So this is one point 1.99 and percent error equal 2.80 Divided by that 1.985 and, of course, multiplied by hundreds. So that is a 0.4% error last one. So come down here. And then this time it iss um zero nine b 75 calculate. And this the same 1.99 a swell. But, you know, this is rounding to two decimal places, but, you know, our absolute error changed a little bit, so, you know, that this is really 1.9985 and this is gonna be 1.9 something something. So, um, I just have it set to two decimal places because you can actually change it appear. But when I change it to do extend it out, I lose some of my columns. And I really wanted to see all of those columns. So if I actually roll scroll down here. This is 1.989 five and this is 1.9852 So there is, um there's a difference. Kind of give or take on the decimal places. So came over here, and this was once again 1.99 and a percent error equal 2.43 I believe waas divided by that 1.985 So 0.22% hair. OK, so now what we want to do is we want to graph the, um we want to graph the exact solution and the approximate the Euler approximations all in the same craft, um, over the interval from 0 to 3. And so what we're gonna do is I'm actually going to change to a different slow field generator by Bluffton University. And I like this one because I can actually put in my my values here. My point values, not my constant. So I may change this, and it waas to why minus Why squared? We're going from 0 to 3 here and we noticed, or why values increase to about two, so we can actually leave it from 0 to 2. Possibly. Maybe we can even do 0 to 3. So let's go ahead into that. On that. Clear all of those out. Um And so now what we're gonna do is wearing actually do the exact one, which is three and one point 985 and I hit Submit. So there is the fact curve, and now what we're gonna do is just change. Thes two are probably either approximations to the 1st 1 was 1.96 Say it well, with actually 2.0 and then the next one waas 1.99 and we had several of those that were 1.99 So, um, if you notice, um, this one is at 32 which doesn't quite make sense. So it might have let me do that again. Out, let's say and staying out there. So, um, we probably need to go back and actually find Tune this instead around into two decimal places if we if we needed to. In fact, let's go ahead and do that. Let's go back here to our online dealer. And that, too, with at that 0.375 So let me do that. 0.375 again. And then let's go ahead and leave it out for four decimal places and see, um and that is a 1.9987 So let's change that to 1.9987 and then let me go ahead. Let me go ahead into the other ones. Let's get mawr decimal places out there. Um, it would probably a little bit better. Um, one meet temper. 1.9 33 And then we had that last one, uh, 09 threesome. Brive the four clean or to get the error. Um, this is 1.9895 Um and so I m forcefully to get the error, we're gonna have to leave it in two decimals, so let me Go ahead and clear all the curves and let's start again. So the actual waas one point No. 185 which are probably did. And three since I had that out to three. Um, and then 1.96 We never went back to do that one. That's okay in the 9987 So he's way it there. And then, um 9933 And the under last fall was 9895 looks no, i ng 895 That gives us a little better curves. And you notice the Red Bull was our exact one. And so you noticed that, um as our sub intervals decreases, why's our sub intervals? Increases are step size decreases. The approximation gets closer to our exact on dso. You can all course take a snapshot of this. So that's what we want to talk about. So if you notice as as our step size or sub interval increases, which means our step size decreases, then of course, our percent error is going to decrease as well, because the closer we get to the exact based off of our calculations, the exact solution


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