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Elhuanhl; [0 nrpruxliata tha quandlty: (Rouno your enswer to four decinial places-)37155Plzesa axplain how YoU obtained the above approximation by answering tne fol...

Question

Elhuanhl; [0 nrpruxliata tha quandlty: (Rouno your enswer to four decinial places-)37155Plzesa axplain how YoU obtained the above approximation by answering tne followlng questions;In order to Use the differental, Ke need function What is the most approprlate function for this problem?What the differental of this function?What the most appropriate value for What Is the corresponding value of dx? What Is the corresponding value of dv?(Round your unswer t0 four decimal places:)[37u 2 Polnes]DETAI

Elhuanhl; [0 nrpruxliata tha quandlty: (Rouno your enswer to four decinial places-) 37155 Plzesa axplain how YoU obtained the above approximation by answering tne followlng questions; In order to Use the differental, Ke need function What is the most approprlate function for this problem? What the differental of this function? What the most appropriate value for What Is the corresponding value of dx? What Is the corresponding value of dv? (Round your unswer t0 four decimal places:) [37u 2 Polnes] DETAILS Use differentals to approximate the quantity (Round your Jnswer to four decimal places ) 474o,8 1zo.8 Please explain how You abtained the apove approximation by answering tne following questions; In order to Use tle differeritia need (unction, what the most eppropriate function for this problom? What differential of thts function? What I< the most #ppropriate valle for *? What correspondinq value of dx? What corresponding value of dy? (Round Your answer Tour declina places



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. $$\begin{aligned}&y^{\prime}=x+y, \quad y(0)=-7 / 10 ; \quad-4 \leq x \leq 4, \quad-4 \leq y \leq 4\\&b=1\end{aligned}$$

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

Okay. What we want to do is talk about or step through the process of taking a differential equation and plotting the slope field and then finding, um, several things related to that. So we're going to start with the temper into equation. Why? Prime is equal to negative X over. Why? And the first thing we do is to plot the slope failed over the interval of negative three to positive three for X and native three Teoh Positive Three for why? And so that's the first thing where you do is to clot to Slope Field. And of course, we're gonna do all of this on some kind of calculator or computer system. And so I'm gonna go ahead and change to a slope field generator, and I actually have it here already set up. It's a soap field generator by Dez knows. And so I have put in that, um, why prime is equal the negative x over why, and so we can go ahead and adjust. So here is negative. 3 to 3. And in, um, negative 3 to 3. I can actually move thistle bit down, okay? And so there is my slope fields. Um And then what we're gonna do is, um, come back and we Now, the second thing is to use a system to find the general solution to that differential equation. And so I'm gonna go ahead and switch to a, um, differential equation solver. Um, already have it in here on this is by symbol AB. There's multiple different ones available on the Internet or your t i t. I inspire. Um Well, actually do this. All you have to do is put in your different question, and then you hit go. And so it solved it to this process. So basically, it did a separation. Um, and so we have two solutions. Or we can write this as a why squared is equal to negative X squared, plus some constant number. And so come back here, and I'm actually gonna use my general as a y squared is equal. If you don't have, we're gonna eventually want to graph this solution. And so if you don't have an implicit graph for, then you might need to break it up into the two parts. So why squared is equal to negative X squared, Plus that constant number and the third thing like I said now is what What we want to do is graph the different solutions when see, one is equal to negative two negative one zero one. And to now remember, why is equal to the square root of negative X squared plus C one. And why's he also called a native the square root of negative X squared, plus his constant number. Now you notice if C one is native to or negative one and this there's gonna be no graph or no solution for these negative numbers because all of the square root would be native. And so I would be trying to take a square root or graphics grow it of a native number, which I cannot do it. Um And so really, um, even with zero, that still will not be a solution. Um, And so, um, the only two that are kind of viable would be one and two. And so let's go ahead and go back two or slow food generator. And I'll even show you that if I put in a Y squared equal to negative Rex squared for zero, I still go get a solution craft, so I may have to put in that plus one for one of them. And so there is my circle. And they were gonna do a equal to a negative lips and then that plus two as well. And so there is. So it's gonna be circles. And you can almost say that that why squared equal the negative X squared is probably gonna be just a doll in the center. Okay, so that is the third thing we want to do. Now. The fourth thing is, we actually want to find the solution at why of zero is equal to two for this, So we're gonna have, um, two squared is equal to see 12 That means see, one is equal to four. And so we actually have this. Why? Square is equal to negative X squared plus C one, four plus four. I'm sorry. Plus four. I meant to put him in there. And so we want to keep that in mind. We want to keep that in mind. And what we wanted, Teoh is actually find, um we're eventually going to graft this solution on on the float field as well. And so the fifth thing we want to do is actually, um, and when we grab this, we can actually graph him over this close interval from 0 to 2 itself. And now what we want to do is actually find some Mueller approximations, and we want to do it for in equal to four in equal to eight in equal to 16 and in equal to 32. So those are sub intervals, which means our steps they're going to be from 0 to 2 or steps are going to be 20.5. Um, 0.25 I believe. And then, of course, um, 0.1 to 5. And the reason I'm changing it to steps is because the system I'm going to get ready to use is has me just do steps, not the interval sub minerals. And then this was going to be zero point 06 to 5. And we want to have that. You're approximation, um, when we approach X equal to two and then, of course, we're going toe. Want to actually graph him over this interval. And so if exist to then my exact value according to the solution is going to be zero. Okay, so I'm a switch to a online calculator that does Euler approximations. And this is by planet couch. There's others available. I like this one because I can put in a bunch of different items. And so here is my different or equation my initial, um, values. Ah, point of approximation is that too? And in our first step waas um, 0.5. And then, unfortunately, my exact solution has to be put in as f of X is equal to a Y is equal to. So I can't do the y squared on dso. But I do know I'm going from, um I want to eventually graph from 0 to 2. So I'm doing the positive square rate that makes sense. So I'm just gonna put in that positive square root. Um, And then you hit calculate, and they it comes down here. It actually will craft it for you. Um, but we want to do all the approximations on the on the graph. And so here is my approximation, which is 1.14 with my four sub intervals. And then my exact solution is zero and in my absolute error, and so we're gonna go back here, and so this is 1.14 with an absolute error, um, equal to 1.14 20 Okay. And so we're gonna go back and do it several times this time. And I like this is well, because the only thing I would have to change really is this step size. And so that is 25 calculate its going to calculate it for me again. My approximate values 0.87 or can scroll down here and look at my approximate is 0.87 with an absolute error 0.8717 So this is 0.87 with an absolute error of 0.8717 Where you come back here, Um, and now do another step size of 0.1 to 5. Hey, calculate. And then we have 0.66 is our value. And of course, that's down here a swell, so we can come back here on this is webs 0.66 with an absolute their error 0.66 35 And in our last one is at 0.6 to 5. Error or step size, not error. I apologize. And then our values 0.5 scroll all the way down here. 0.5027 is our absolute terror. So this is 0.50 ham point 50 to 7. Okay, so we have all of those. You're approximations. So now what we want to do is actually graph all of the Euler approximations. And then, of course, we know that when, um, exes to my wife value is zero. So we're gonna come back here, I'm going to change to another. I never, um, Differential equation graph by Bluffton University. And the reason I'm changing is because it gave me the ability to not graft the particular solutions, but actually to do the, um the eggs, the the values, not the C values, but the values. And so here I have my differential equation. I'm going from 0 to 2, and I know all of my y values were positive. Um, and so, um, 20 was my actual. So that is going to be actually a dog in this year in the center. Um, And then the first solution is 1.14 and you notice it's a curve is going to be a circle out here. The 2nd 1 and I like Chris, because I can do them all together. Um, when I waas in that you're approximation. It would only do one at a time. And so there is my 2nd 1 And then this was 66 3rd 1 and then the 4th 1 was at 50 And so you noticed that all of these are now all of these are now approaching this 00 here. And so that is what it's doing. Um and so let's go back to here and talk about it, because what we really wanted to do was talk about the percent error, but, you know, percent error. In order for me to get percent error, it is actually the, um, actual or the absolute guy. Typically, we want to do the absolute value of the actual or the exact minus the observed, divided by the actual or the exact or the theoretical times 100 percent. But you notice if I would do that, I would be dividing by zero in this case because our or actual According to you, that was zero. So, um, in order to do divide by zero, Actually, there's a lot of things entailed. So what we're gonna do is just kind of look at the absolute error and you notice that the absolute error as my, um, sub intervals increase then my observed or my Euler approximation is actually approaching or getting closer, toothy actual of zero. And so, of course, your error is going to decrease as well.

Okay. What we want to do is to go ahead and step through the process of being able to start with a differential equation. Why? Prime is equal to sign of eggs. Times sign of why. And the first thing we want to do is to plot the soap fields. And we're going to do it over the interval from native six in the X two positive at six. And native, six in the Y two positive six. And so what? I'm gonna go ahead and do it. Switched to a slow field generator, actually using Dez most, um, found out it was a little bit easier to use. Um, and so I've already put him in. So sign of eggs time. Sign of why this is my two differential equation. And I'm going from negative six. Positive six and native six to positive six. So here is the slope fields for that differential equation. Then what we also want to do now is we want to find the general solution to that differential equation. And so what I'm going to go ahead into is change to a, um, differential equation solver. I'm going to go ahead and use simple lab. I like Symbol Lab because I can easily input sign of x times side of why hit Go. And here is my general solution. And also, I can go ahead and scroll on down and show my step so I can actually learn how they came up with that general solution. Now, I'm gonna go ahead and ignore this too pie in. They put the two pint end because it is repetitive. Um, so I'm really gonna focus on that two times The inverse tangent or arc tangent of e raised to the coastline of X plus C one. And so that is Excuse me. That is my general solution. So why is equal Teoh too? Times the arc tangent or in first handed have e raised to the negative coast on of X plus some constant number. Okay. And now, the third thing is, we want to actually graph the solutions for when See, one is equal to negative one negative to negative 101 and two. And that's another reason why, like Dez Moses, because I can actually go back and be able to graft those and Desmond Also, this local generated generator also does implicit crafting as well, which is helpful. So now what we're gonna do is dio why equals to two I'm times part can. And I think you don't have to be the do that multiplication sign of inverse tangent of e raised to. And I'm gonna go ahead and put that parentheses negative co sign. I'm yet Rex. Um, Plus and I would head, um, plus or no, actually, is gonna be minus que and that is all. And, um, And there I have your graph of when c is equal to native to now we're gonna do it forward and keep doing it now. And, um, I think I could go right with that inverse tangent of e raised to you. And I may go ahead and keep that in parentheses. Negative co sign and I think by default business as actually, um, actually, news in radiance, you should be able to go to the settings in here. And I think you can actually, um, change it two degrees by default, though it it is an radiance. And so that's something to make sure if you're doing this on a t, I inspire that you're in radian mode because we are working with Trigana metric functions. And so now we're going to do We can't keep doing it in this time or C value is zero. So it's too or can ah e raised to the negative co sign of X itself. So there is the other one. Um, yep. So there is the other one. It takes a while. It's kind of slow. So it takes a while to kind of catch up to you. Why is he going to or can he raised Teoh, And now we have native the sign of eggs and they were doing the plus one. And I also like grins because it actually gives me the different colors so I can distinguish. After a while it will start regenerating the colors. But for right now, I can actually see the different color. So, um, distinguish between my graph so our can of he raised who? The negative co sign of X and I believe that is going to be a plus two now. And so now I have all of the graphs graft onto my slope fields camp. So there is all the solutions to know what putting do is come back here and now what we wanted Teoh is actually we want to find an and eventually graph the solution that is specified by a specific initial condition. And the initial condition is going to be zero comma, three pop, zero ne 00 to. So now what we want to dio is to find that see value, um, for why evaluated at zero is equal to two for that particular initial condition. So now in have to is two times are tangent of e, raised to negative one plus that see one. So we have one is equal to the inverse tangent of E to the negative one plus at C one and now we're to solve for C once every take the tangent of both sides to that tangent of one is pi over four is equal to e to the negative one. Plus, uh yep, plus C one hope that see one has to be in the at. C one has got to be up there with that one. So plus C one So the natural log of pi over four is equal to negative one plus C one So see one is a whole toe one plus in natural Aga pi over four. And unfortunately for most of my, um, computer generating systems, I have unfortunately, we have to be decimal. So this is about 0.758 four. So now what we wanted to is that solution is y is equal to two arc tangent of e race to the negative co sign of X plus that 0.7584 and we could go ahead and grab that as well. But we're more interested and actually finding what? Why is when x is three pie over to okay? And so because that is going to be what we eventually are going to graph and also try to find Euler approximations for So now what we're gonna do is substitute in clips substitute in X equal to three pi over 43 pi over two. So wise it to two arc tangent of And then, of course, three pi over two is going to be zero. So this is E to the 00.7584 And so when I do that, we and ex his 0.7584 we get an approximate value of for my wife value of, um I had a calculated earlier of two point two point 2655 So when X is, um three pi over two, the According to this solution, when C is 0.7584 my wife value should be 2.2655 Okay, so that is critical. And eventually monograph all of these into into into the slope field. And so now what I want to do is find the Euler approximations and then compare the Euler approximations to this exact value. And so we want to do it for sub intervals of four, eight, 16 and 32. Now the the Euler approximate er I am using doesn't do something of old. It does step size. So basically, we're in. Take our interval from 0 to 3 pi over two and subdivided into 48 16 and 32 7 rolls. So that's gonna be a step size of well, the 1st 1 of 1.178 0.589 and point to nine 45 and then 0.1473 And now what we want to do is to find those who either a pox approximation. So three pi over two, three pi over two. And then we're also gonna be keeping an eye on the error, Aziz. Well, and then talk about it. Okay, so now I'm going to go to an online. You're approximate. Er, um, and I'm actually using one from Planet Cal. I like this one because and I was playing around with the earlier because I'm able to put in my differential equation my initial, um, conditions zero comma to, um, my point of approximation, which is three pyro to that. I can't put three Piper tune here. Unfortunately, we have to change to decimal. And then, of course, my first step size instead of myself in a role. But my step size is 1.178 And then, of course, I had to put it. I went ahead and put it in my exact solution. And this is where I'm going to have to if you knows I have to use that multiplication symbol. If you don't is not going to work correctly, I also can, um, tell it, um, how many decimal places? Okay. And so here, um and then I have to hear calculate. So you notice I'm taking my approximation out to four decimal places who want 3.1041 But unfortunately, when I come down here now, this will also graph it. So this is actually my this right here if you notice the exact solutions and orange on the approximation is in blue. Now, when I scroll down, it also gives me a table of a bunch of information. Now, unfortunately, over here, this is going to be my absolute error column, which I'm going to need to do you have in order to do my percent error. So it's out to four decimal places. It won't give me that last column. So unfortunately, to bring it out to two decimal places so I can see that last column. And so that last column is 0.8383 some a switch here. I'm gonna write my why value out to four decimals and then my percent error, of course, is going to be that absolutely out value error 40.8383 divided by that exact value of 2.265 and then, of course, multiplied by 100. And this gave me 37% error and Now I'm gonna keep doing it for my different sub intervals to me. Go back to my online calculator. I also like this one because I simply I don't have to report anything in. I simply change my step size and that is going to be zero 0.589 and then hit, Calculate. And of course, I'm going to, um, go to 44 decimal places to write my approximate approximate value of why should that's to put 2.9594 And then I'm gonna have to go back, unfortunately to to to get my error of 0.6936 So now I'm gonna go back here. That why value was 2.9594 and I'm gonna go ahead and calculate my percent error. This was 6936 and into ride by 2.2655 And I believe you get a 31% error. So it's decreasing, which you would think that it would. Right? And so now I'm gonna go back to my you a calculator and once again just changed my step size. So now meditate. My interval, which is 0 to 4.712 and then divided by I think it was sub intervals of 16. And so that would give you 29 45 and hit calculate. And then, of course, I want to change back to four decimal places for my value, which is 2.8430 and then change back to 22 can kind of come down here and get my absolute error, which is 0.5772 Okay. And you notice I've been scrolling by the graph. And the reason we're not really focused on the graph so much is because eventually I want to grab all of these solutions, including the exact And so I have another system that I'm going to be focused on in order graft those so I can see them all in one place because, unfortunately, unfortunately, this Euler approximation calculator will only graph one at a time based off of the exact. So now I'm gonna go back, and then I'm gonna do it for my last, um, sub interval of 32. And so that was a step size of no 0.1 for 73 approximately. And then, of course, hit Calculate And then once it can, I'm gonna take him out to four. You probably could leave him at two decimal places. It's always nice to have that fourth decimal. And so that's 2.7682 and then changing back to two. You never know when those decimal places air gonna come in handy. And the absolute errors 0.5037 some switch back here, right, these values down to 2.7682 and the percent error is going to be that 0.5037 divided by the 2.265 and that is a 22% error. So that makes sense that as we are increasing the sub intervals, which means I'm decrease in my sub size, that my percent error would get smaller. And, of course, maybe if I even make this to 64 clips that this would even get smaller. So now what I want to do is to graph my exact and my four approximations. So this one, I'm actually gonna go into another so field generator and this is one by bluffed hand, and I already have, and I'm gonna go ahead and clear this out. Um, I've already have. Am I different to equation? The X is going to go from zero to that three pi over two and then based off the just all of my y approximations, I'm going from 0 to 5. Now, The neat thing about this one is I can actually input the X and those y values. And so this was my three pi over two, which are probably could go ahead and do three pi over two in this one. But I went ahead and left it as my decimals. And so this is my exact solution, and I hit submit. So there is in red. And so now I can keep changing these to my approximations. And so and then it will graph all of them on the same slope build. And so this is three point one. Throw for one and hit, submit. Um, and so there is that one. And now I'm gonna go ahead. I'm hit him. Submit. He always looks pretty flat. And it just based off that soap field on. And then I'm going to go ahead and do that to point nine 594 and now you can actually see that one curving. And then we're gonna go on due to 84 30 I think with my next one are, you would expect these graphs to get closer and closer to the actual red one. And in that warmer 76 May 2 Fran great do right? And so then, of course, you can always take a snapshot of this is well and so I think we've kind of even talked about as the sub intervals increases, which means my step size decreases. You notice that my Y approximations get closer and closer, toothy, exact based off of those initial conditions. And then, of course, my percent error decreases as well.

Okay, so let's compute, do you? Why? Over today? So this is equal to the derivative of singe, which is kohsh times. It's inside an entire derivative of forex which is for and that gives us for Coche of forex.


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