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Given d ={(ex _ y), y(0.3) = 5, h 0.3.By using Runge-Kutta Fourth dx Order Method to approximate y(0.6), we know that J1 = Yo +h(k1 + 2kz + 2k3 +k4)What is k1 value...

Question

Given d ={(ex _ y), y(0.3) = 5, h 0.3.By using Runge-Kutta Fourth dx Order Method to approximate y(0.6), we know that J1 = Yo +h(k1 + 2kz + 2k3 +k4)What is k1 value?Select one:A -7.8833B. 49.4411C-23.6501D. 54.2314

Given d ={(ex _ y), y(0.3) = 5, h 0.3.By using Runge-Kutta Fourth dx Order Method to approximate y(0.6), we know that J1 = Yo +h(k1 + 2kz + 2k3 +k4) What is k1 value? Select one: A -7.8833 B. 49.4411 C-23.6501 D. 54.2314



Answers

If $\int_{-1}^{4} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=4$ and $\int_{2}^{4}[7-\mathrm{f}(\mathrm{x})] \mathrm{d} \mathrm{x}=7$, then the value of $\int_{-1}^{2} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$ is (a) 5 (b) 4 (c) 3 (d) $-3$

So this is a function and refined integral that which is the area and which of these four values. Becks, approximate that area so we can sketch it to see how it's gonna work. Visibly. So does this one. Does the studio. There's the three. This is for okay. We're only interested between Syrian four because there is an ornament and for the upper limit. And this is one. This is to you. All right, this is three. And so when you sketch this crew of eggs, you get in something like that if that. Okay, so, uh, this is the area we're tryingto gentrifying, so we try and fix approximate, but best approximate. I mean, you can approximated by some extended triangles, right? Like that. Like that. Okay, so let me make it solid. Maybe this one solid like this, and this is like that. And then this one. So So I think that so if you're approximately like that, it can see that, uh, the with is gonna be one throughout. So you could bring it out. And then a functional values here is one. The function of about here, you can see approximately is 1.5 the function of about a year approximately we can say is like one point, maybe six or seven on a functional value Here is is to write. So when you add all of these things, you get in something like, uh uh, 6612 which is closer to five than anything here because this is five. This is negative. Three, this is 10. This is to just 86 going to it's closer to five. So approximately this area, which is this definite integral Okay, is approximately five square units approximately. We have to use a rectangles approximately approximated than a problem is this is fine square units.

We're going to approximate the value of the integral from 3- eight of the Square Root of One to Sex. By the traps. All the rule with five sub intervals. So let's start by defining the function F of x equal Squares of one plus x. On the closed interval 38. So because One plus X is positive for all the numbers on the table 38 function is well defined. We now calculate the step size age which is too common Distance between any two consecutive notes. And this is defined as the length of the interval of integration. 8 -3 Over the numbers of intervals five. So we get five or 5 equals one. So age is one. And with this we determine the notes, we will be used in here. And the first note X zero is three. Then we have X one is 4. X two is 5 X three is 6 Takes four is 7 and X five his age. And now we can say that the integral from 3-8 of the spirit of one for sex is approximately equal to 35. There is a trap. So the rule using five serve intervals which is age half times the image of the first note if zero plus to the image of the second note X one plus two times the image of the third note. X two was two times the image of the fourth note X three Plus two times the image of ex war plus Plus the image of the last note X five. So is equal to h remember is one. So we get one half times The image at three Plus two. f. at four Was to F at five. This too F at six was to F at seven plus F at eight There is 1/2 times now F is squared of one plus X. So here is the square to four to those two times squares of five plus. To score to six plus two square to seven Plus two skirts of eight Plus skirt of nine which is three. So you get one half times. And to hear through here we have five Last two times a quarter to five. The skirt of six plus Curtis seven plus security page we had taken to common factor. And here we use a calculator. We have found that these have approximately equal to yeah 12 0.6 5973 6156 0937 48. So the interval prom debate of discourage of one plus eggs separate ultimately equal to 12.6 5973 6156 09 37 for eight. And that was calculated his interrupt. So as a rule with ah fives of intervals and now we calculate the exact value of the integral. And for that we need to calculate Injury from 3 to 8 of discovery of one plus x differential effects. And for that we use here a change of variable U. Equals one plus eggs. Which implies that differential of U. Is equal to differential of X. And when you is three. Sorry I meant that mhm. When X equals three which is the lower limit of integration In the original integral which is about X. So when X is three you is four and when X is eight you is nine. Because we use this equation here. The value of you is the value of X plus one. So for X equals three us four and for a X equals eight us nine. So these are the new limits of integration when we found the interest in terms of you so this is the change of variable and the limits of integration With that we say that this is interesting sequel to the drill for 4-9 of the square root of you because one plus X is you. And differential of exes differential of you. So we get this and when we are calculated in a different interval we don't have to go back to the original interval we are going to obtain here the numerical value of the definite integral directly. That is because the interest debt said definite into then the integral of this is due to the one half plus one over one have Plus one evaluated between you equal four and U equals nine. And we know this is opponent is 2,3 house 3/2 and because it will be in the denominator It goes to the numerator as 2/3. Thank you to the three House and that between you equal four and U equals nine Then this is 2/3 times then at nine we get nine to the three half minus four to the three. How's this is 2/3 times nine to the three halfs the same as squares of nine to the third, that is three to the minus and this is the square root of four to the third, that is two to the third, That is 2/3 times 27 -8 and 27 -8 is 19 Times two is 38, 38 3rd. And then we can say that the integral from 3 to 8 of the square root of one plus x Is exactly equal to 38/3. And if we calculated tess Imo Representation of this number is 12 experience and if we compare this with the approximation Calculated by the transcendent rule with fights of intervals we see that we have the integral part 12 and we have this first decimal six but we can say we have the following decimal because The next two decimals the approximations are five and 9 Which is close to six. So we can say that we have three figures 12 And to the decimals equal to six in the approximation which is not bad. So we have found the approximation to the given integral is in terms of the rule with myself intervals and then the we calculated the exact value and compared to the approximation.

Okay, So for this problem, we have n equals for and then f of X equals one over X squared. And we're looking at the interval between one and five. So what I want to do is I want to create a table of my exes in my f of X is that way because I'm gonna use those in my formulas. And what I'm gonna do is I'm gonna take five minus four, which is my interval. So when I find the length of that interval and divided by the number of increments that I want, five minus one is 4/4 is one. So I'm going to go from go increased by one until I get to five. So then, if I plug in X each one of these, I'm going to get one 0.25 1/9 1/16 and one 25th. So these were going to be my first through last f of X is them would be using. So if I look at the trapezoidal rule, we'll be looking at the change in X, which is what we found to be one divide by two. And then I'm gonna do f of my first ffx and I'm a multiply the three middle um f of X is that I have because I have five total but my multiply those middle three by two and then the last one is just going to be multiplied by one. So then what I'm going to do is my change in X. Like I said, it was gonna be one. So it's gonna be 1/2, and then I have one plus two times 20.25 plus two times 1/9 plus two times 1/16 plus 1/25. So then if I multiply and add everything together when I have 0.5 times 1.8872 So this is going to be approximately 0.9436 okay, and then for Simpson's rule is going to be a very similar formula. So when I have s four equals that same change in X, But this time over three and again like the last time, I'm gonna have my middle three green multiplied by something. It's gonna be alternating between four and two, starting with four. Then my last one will not have anything multiplied by it. So Then we said the X is the change in X is one. So then one and then four times 40.25 plus two times 1/9 plus four times 1/16 plus 1/25 1 25th. So then one third is gonna be multiplied times 2.51 to 2. That's the number I get after you multiplying at everything divided by three or multiplied by one third, whichever you prefer get 0.8374 And then the last step I have to dio is I need to integrate this from 5 to 1, and this is one over X squared DX. So what I know is that I'm gonna have one over X because this is the same thing actually going to rewrite this, um, as saying x to the negative, too. So I'm adding an exponents to this, so it's gonna be X to the negative one. And then I got to multiply by the reciprocal of negative one, which is just negative. So this is the same thing as negative one over X. So I'm looking at this in terms of five, so it's gonna be negative 1/5 minus negative one. So I've got my double negative. So this is going to be one minus of fifth, which is 4/5 or 0.8.

Okay, so for this problem were given that in equals four and f of X equals 3/5. 5 minus X, and it's given to us in the interval between negative one and three. So what I want to do is I want to figure out my interval spaces so we can utilize them to find ffx and then plug him into the formulas which I'll provide as we go. So to go ahead and create the table of X and ffx and then for the increments, I want to take the interval and subtract thumb and divide by my end value. So this is now going to be three plus one, which is 4/4, which is one. So this is going to go and increments of one from negative 123 So I have negative 1012 and three. So when X is negative, one f of X is going to be 10.5. What x zero f of X will be 00.6. What X is one? Ffx will be 10.75 when X is to ffx is one and what X is three. We plug it in and find out f of X equals 1.5. So now I'm going to utilize thes into trapezoidal rule. So since Ennis four. So I'm gonna take the, um, change in X value. So we all we incremental these by one over, over to. So it's actually going to be the change in X over to and then we're gonna have f of x zero, which is our initial value plus two f of x one plus two f of x two plus two f of x three plus f of x four. So when I plug in the values I have, it's going to be changing. X is 1/2 my initial values 0.5 plus two times 20.6 plus two times 20.75 plus two times one plus 1.5. So if I were to add and multiply all of these together using order of operations, I get 6.7. So this is going to end up being 3.35 Okay, So now for Simpson's rule, Simpson drills Very similar. So s a four. Since in is four change of x over three f of x zero, plus four times f of X one plus two f of x two plus four f of x three plus f of x four Remember the outer to do not get anything multiplied by them. So now I'm gonna have my change in excess by ones. So I'm gonna have 0.5 plus four times 40.6 plus two times 20.75 plus four times one plus 1.5. So if I were toe add and multiply all of those values together, I would end up with 9.9, which is going to be 3.3 once we have multiplied by the one third and then the last part asked us to find the used the integral so between negative one and three of 3/5 minus x d x and since three is a constant, what I'm going to do is I'm gonna move it to the outside and I am left with 1/5 minus x d x. And so anything with one over itself is going to now be a log. Ah, five minus x. And then I have to multiply this by negative because X is negative, has a coefficient of negative one. But I'm looking at this in terms of between three and negative one. So for my first value, I'm gonna have a negative three natural log of to because five minus three is two minus negative. Three natural log of six because five minus negative oneness. Five plus one, which is six. So when you add all this together, it's gonna be approximately 3.2958


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