use-newtons-method-section-47-where-needed-to-approximate-the-x-coordinates-of-the-intersections-o-2

Question

Use Newton's Method (Section 4.7), where needed, to approximate the $x$ -coordinates of the intersections of the curves to at least four decimal places, and then use those approximations
to approximate the area of the region.
$$
	ext { The region enclosed by the graphs of } y=x^{2} 	ext { and } y=\cos x
$$

Kenneth Kobos · Accepted Answer

we want to find the area of the region and closed by these two functions. The 1st 1 is in blue. That&#x27;s Y equals Coast X, and the second function is in green. Why equals X plus to sign next to the fore? So we find by using a graphing calculator that there is actually two enclosed regions the 1st 1 here and the 2nd 1 over here. So we want to find the total area. So we&#x27;re going to be adding these two regions area equals area one plus area two. So we&#x27;re going to determine each of these areas separately. Area one is equal to what we&#x27;re going to do. An integral. So I didn&#x27;t say which is area when we&#x27;re going to consider the the left most region to be area one. So we&#x27;re going to be integrating from X equals negative. 1.912 two X equals negative 1.224 It&#x27;s going to be our top function, minus our bottom function. The top function is the green one in this region. So this is X. You okay? Here we go. X plus two sign for X minus The bottom function, which is in Blue Coast X DX. So that&#x27;s area one area to will be computed similarly so for area to which is this region the right region? Um, we&#x27;re going to be into getting from our left most point which is now negative 1.224 to our right most point, which is 0.608 and we&#x27;re going to do our top function minus their bottom function. This time I&#x27;m the functions play the opposite rules. The top function is the blue one now, which is co sex and we subtract the bottom function which is green so minus X minus to sign for X d x. We&#x27;re allowed to use a into an integral calculator for this question. So after we compete these two areas and we add them together, we get her area. We are rounding to five decimal places, which means that our area that we obtain is 1.70 for 13 as the total area enclosed by both of these regions

Gregory Higby · Answer

Okay, Um So when heading, use a graphing calculator to show you, Um, the sign curve is the red one. That co sign curve is a black one. The vertical line x equal zero in the vertical lines. X equals two pi eso as you&#x27;re looking at this, what you have to be careful with if you&#x27;re not using a graphing calculator, is you want to make sure that you do co sign minus sign from zero to pi over four and then switch over Teoh do sign minus co sign from pirate for 25 pi over four and then switch back to co sign a minus sign from five power for 22 pi because that&#x27;s where they want you to stop. Otherwise, if you&#x27;re using a calculator, you can just, you know, find your function that has the in a role from zero to too high the way. This is where you find the high value in this calculator and what you want to do then is so you only have to do. One function is if you just do the absolute value, um, and makes everything positive. Um, now, the downside to this calculators, it does not give you an exact answer. Um, but if you checks with an exact answer and some calculus actually might let you get an exact answer, I have not seen this capability. It is exactly four route to, um But if you actually went through the process and you did zero to pi over four pi over 4 to 55 R four, then 5/4 to 2 pi and worked it allowed by hand finding values on the unit circle, you&#x27;ll get exactly that. Makes perfect sense for root twos. Your answer, by the way.

Sicelo Goqo · Answer

running. Uh, we trying to solve a problem than 11 under chancellor six application of Inter Cross. And we also looking at the areas between two kids. Okay, firstly, I&#x27;m going to switch Thio the white wood. So the question is divided into two parts. The first part. We are looking for the point of intersection off two kids. So I&#x27;m assuming that you have friend Ah, question the next part. We want to approve meet area off the region by bonded by two years. So let&#x27;s start with the part. But they okay, so what a we want to solve? So you want to solve simultaneously? You too inquisitions. Okay, just theories. Okay. We want to solve Metin your ex. Okay, since if you go back to the questions we want Thio approves meet X coordinate and also find the approximate area off region only by the two killers. So the fist kim is why you closed two x science experiment and wipe those extra before. So the quickest way of doing this to approximate is too brought to bloody function. I&#x27;ve really done so using this website. So all you need to do is just to enter the function. So why does too eggs? And in times of sign off x wing is will give you the fist gruff. So that&#x27;s will be read craft. And then why house Excellent before will be the blue Ashcroft. Okay, now, when we solve for X simultaneously firstly, we wait two kids So effort X equals to geo fix. So if it thinks we can see that&#x27;s the function of X sign, expect and g o fix is the function of X to the poor for so now all you need to do is to take out to bring the form on the other side. So you bring the four to the other side. Oh, you can take the X sign X square to the outside. Okay, just take X sign. Expected out of sight. We&#x27;re going to have ex trip off minus thanks. Thanks, sweetie. Remember X sign X way was posted. You take it to the other side to be negative. So what is left on the left inside from the previous step is zero. Now take out the common fiction. So the common factor here will be X. What we&#x27;re left with inside our kids, we have X Cube. Because if you divide X to the buffalo by execute excuse divide exciting square by ex weekends because you sign X squared. And this isn&#x27;t close to zero. Provided that X It&#x27;s not serious we can to disturb if X is not so okay, so no. From this x equals zero ordered him inside. Burkett supposed to zoom. So the situation here is X equals zero X cubed minus. Sign Exley, because zero. Okay, no, if we look it, Uh, okay. Now, if we look for export zero, we have extra classes and an X cubed minus sign. Extreme. Because it if you want to find an exit here, you can use new transmittal. So basically, you can set this to everything&#x27;s, but we are now. We&#x27;re not going to go to new transmitter. I&#x27;m just going to go straight to the car off and show you how to kids the next exit. So this is the graph. We got to the point way we had exoticism. So, of course, if you take X equals zero and substitute to any off two functions, you should get why he crosses, um, and those two functions. They also meet at this point. So exited a 0.896 And why? Video is little 0.6445 But you&#x27;re looking at the experience. So fx a considerable. Now that X is 0.896 So we we have to be already. So the eggs videos are ex across Israel and the other one is 0.896 But just for interesting, we can take more decimals. So depending on which software using, we can take more dismal. So eggs very here, using neutrons method any other metal. So just pick it from the craft. We get extra pills, too. Zero point AIDS. Nine five 9931 price. Okay, so now I&#x27;m going to go through the next page on do the second part off off the problem. So the first part was to find the X squad mate. Second qualities to find Aaron, which is probably by two kids. Some games for you. Good for your patient. And this is part of being it sees. Just erase. This is part. Okay, so maybe we&#x27;re looking for the area. Okay? I didn&#x27;t just see the area. So this is the area off the region, which is bonded by the two kids. Okay, so going back to our two kills So that area will be between X equals zero X equals 0.896 food. We have something like this. I&#x27;ll just pick that ricin on. So if what to functions. So you could lose two points for this point in the point looking at the exhibitors. So x 12 x two. Okay, so they get it between those two functions. Let&#x27;s call it a So call this area A This case is close to the inter grow between the two eggs. Various said our first things with zero and Arun is it? Open it. Nice food. Zero. I&#x27;ll take all the dismal so little point 8959931 by one. So that becomes the graph above minus the craft below. That&#x27;s cool. So the graph above is the Red Cross and the Red Cross is recruitment by X sign. Extreme and the craft below is the Blue Cross, which is extra powerful. So this means we&#x27;re going to have X sign X squared minus X to the poor fool. I didn&#x27;t think so. It&#x27;s agreed to two eggs. Okay, so we can separate the twins across. So a it close to the integral between zero and 0.89 by mine. Nine. Big one, which is thinks line. Expect D X minus other rays x to the bone. It&#x27;s also between zero and 0.89 by 9931 by four A. So, looking at it this integral easy to evaluate. So listens. A crowd becomes extra Ball five divided by five We evaluated between zero and 0.89599314 In order to do this assumption that the video on top which is this one minus that really blew. Is that okay? Just move on to the next page so that I can eat this crap. Okay, so we&#x27;re looking at the crow off eggs sign extreme. So Interco off eggs explain DX So here we can use Remember our limits given in terms of experience, we can use the u substitution, change the limit, or we can just use the use substation substituted, expect and we have a lead. The limits. So this dude Ah, second option. So firstly, let&#x27;s, uh, not to let you to be extreme. You to be expecting me. Just always, huh? So are you. Because you expect so many different sheets, do you? The X will give us two eggs, and then he must have made deals on the other side to Xbox. Then we can divide by two. Well, the saints. So if you divide by two, you have to you over to then substitutes for you the integral. Now we have extra Lex and his steaks. Phoenix So again tive minutes ago, we&#x27;ll sign. Explain close to you. Do you know how integral is in terms off you won? We can evaluate in scrub and then substitutes the limits. If you have only Dane&#x27;s club, you get millions here. One of two cause you between zero and eight, 9599315 Know firstly, substitutes that limits on top minus. Admit if you do, then you should Kids, huh? Zero coin zero. Uh, okay. It&#x27;s just different book in circles. Remember, we have an excellent into grow from here. Which is the second is Girl X executing ball 545 and then follow it using those two elements. So if you end up going to this function. We should end up with zero point. We should end the weed. Zittel Plane Zero. Who? So that&#x27;s gonna be Oh, fine. Alonso. Oh, you okay? Uh, that&#x27;s the end of the question. Uh, thank you.

Kenneth Kobos · Answer

we want to find the area of the enclosed region. Of these two equations, the 1st 1 is y equals X sign of X squared, which is in red. And the second is why equals X to the four, which is in blue. And we also are interested in X greater than or equal to zero. So using a graphing calculator, we can see that there are two intersection points, Um, which are at the X equals zero and also X is approximately 0.896 So what we want to find is the area which is enclosed. So this small area right here, So as usual, we&#x27;re going to determine this area by integrating eso area is equal to integral. We&#x27;re going from our left most point, which is X equals zero to write most point, which was zero point 896 And we&#x27;re going to do the top function minus the bottom function. So what we see here is that the top function is X sign of X squared, and the bottom function is X to the force. So we&#x27;re just going to subtract those x sign of X squared minus X to the fore DX and after solving with this integral will get the approximate area. There is approximate because we we have an approximation for the top people. Intersection point. So that&#x27;s what it&#x27;s going to be the approximate area. So let&#x27;s carry out the integration. The integral of X sign of X squared, uh, is negative half coasts of X squared. And if it&#x27;s not clear how we do that, you could do a U substitution of U equals X squared. So D&#x27;You equals two x d. X. So that&#x27;s how we get this first anti derivative and the second anti derivative is very clear. It&#x27;s 1/5 x to the five, and this is again going from X equals 0 to 0.896 So after we plug in everything, we get the approximate answer of 0.37

Fuzail Shakir · Answer

All right. So the area between the cruise, the area between the crew off recall this f call this G between zero to pie. So we label this as co sign a Becks close out of X minus one plus the coast side of X T X. This is equal to two co sign of X minus one, which is equal to two sign of X minus acts. And we&#x27;re gonna evaluate this from zero to pie. So in the end of final answer is negative two pi over three plus five, plus two times the square root of three.

Problem

Use Newton's Method (Section 4.7), where needed, …

Problem 40 Easy Difficulty

Answer

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1.09475
$$

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Calculus Early Transcendentals

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Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

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$$
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Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals