repeat-exercise-47-for-the-curve-y-x2-1-1-x4

Name: repeat exercise 47 for the curve y x2 1 1 x4
Uploaded: 2019-12-04
Duration: 1 min 28 s
Channel: Numerade
Description: Repeat Exercise 47 for the curve $ y = (x^2 + 1)^{-1} - x^4 $.

Question

Repeat Exercise 47 for the curve $ y = (x^2 + 1)^{-1} - x^4 $.

Amrita Bhasin · Accepted Answer

Okay, say, this is a graph. We know we have two zeros. 0.869 and negative 0.8. 600 again, These are approximates. We&#x27;re using the graft to estimate thumb. Okay, No area but set up are integral using these as our bounds. Okay, remember, we can actually consider this to be the inverse tangent of acts. Because remember, the integral from A to B of one over one plus x scored detox is inverse tangent of X. And then for the second part of the fourth, we use the power rule increased the exponent by one divide by the new exponents. OK, plug in. We have to time 0.7154 minus 0.99 which gives us approximately 1.232 secs as the area below the curve and above the X axis.

Wen Zheng · Answer

The problem is sketch of the Curve X Square plus y Square two three&#x27;s Power and see it Go to or time. Sex Square. I am so blessed square We know Axe is equal to our times. Sign Fada and why seek Auteuil? Our times sine theta So X square plus y square is equal to our squire. This equation is a square shows Reese Power is sick. Culture War Times Square Co sign Jada Square Times Square Sign. They&#x27;d ask White. This&#x27;s to force terms. You&#x27;LL signs or sign Data choir. This&#x27;s at or sign Who&#x27;s there? It&#x27;s going we have. Ah, it&#x27;s equal to Cyril. Or are you going to plus or minus? Sign to data things. One third eyes equal to zero R is equal to zero, so we only need needed two jobs. A graph of our iko too, plus or minus sign data two times. In fact, the graph of our U go to sign tooth better is the same as a graph of our physical too negative sign to theta, so we only need two job. The graph of our yuko, too. Sign true hams data. Now let&#x27;s see how to judge the graph of thiss you question the one fatal goes from zero to pi over four. Our goes from Cyril toe one silicon jaw rough is like this And when they have those from pi over four t o hi o r two our goes from one to their Oh, so it&#x27;s like this. And one day that goes from hi, we&#x27;re to tow three pi over four. Our goes from zero to the one three power forces Repair over to this is a no. What are one thing goes from pi over two, two, three pi over four. Our goes from zero two negative one. So if our goes from zero two plus two one as the graph is like this But since our goes from zero to nick to one So Graff iss thiss on the one data goes from re pile. Where to? To a pie Our goes from Nick to want to Cyril If our goes from one to zero, it&#x27;Ll Look stock this since eyes negative. So the graph is this similarly we can job the other part. The final result is luck. This This is a graph of this curve

Andrew Madea · Answer

we&#x27;re given the equation. F event is equal to X four minus four. Then we have to sketch first. We can figure out the domain of this function and the domain of this functions and doesn&#x27;t have an irrational or square root aspects of it. It is just negative. Infinity to infinity. There are no restrictions. You can also say that it is for all real numbers. What are the intercepts we could talk about? First is the X interest. So we can set this equation equal to zero, and we can factor out minus four. So we have an X intercept already. And then for this part, we have X cubed minus 400 And that is equal to the cube root of four. So the cube root four zero. So what about X intercept? Well, if I plug in zero for X, then we get zero minus zero. So the Y intercept is also 00 or the origin. We can perform the symmetry test so we can see if it is. Even if the original equation is the same thing as if we were placed acts with negative. We have negative X to the fourth power minus four times negative. We left with to the fourth less for this is not this original equation. It is not even. So. What about the Well, if we negate the entire function, Is that the same thing as what we got from the from the even? So what happens when we negate the entire function? We get negative X to the fore plus four X And these two functions are not the same thing because this is negative for negative four. This is positive. So it is not odd as well. So there is not even not on. And we could, um the next step is asking toes, but there are no ask him to its any polynomial function next step, since we were all this is from original function. To find more information, we have to find the first derivative, which is four X cubed minus four. And we could set this equal to zero to find any minimum or maximum values. So four X cubed minus four equal to zero we solve for X, we get remove This four over here, divided by four x cubed is equal to one or X is just equal to one And if we plug this back into the original equation so we have our X value are y value is one minus four. So that&#x27;s negative. Three. Since we have this value of one, we can figure out being intervals we contest. So we have negative infinity to one and then we have one to infinity to find out if it is increasing or decreasing. So we have our we can have a test value. We can have our sign. So for this test volume, you can say that&#x27;s a zero for this one. We can just say to So if we plug in zero to our the first derivative equation yet a value of zero minus four So that is negative to get too cute is a and that&#x27;s obviously going to be positive. So it will be decreasing on this interval, decreasing and they&#x27;ll be increased on this. So we said it was increasing on one to infinity and decreasing on negative infinity to one So and next thing we can check is concave ity. So we need to figure out the second derivative one of the river of this function which is just 12 x squared. We could find points of inflection. So we set this equal to zero. But then it is just going to be X is equal to zero. As a potential points of inflection, we could figure out intervals length negative infinity 20 to figure out the concave ity of these intervals. But notice that the second derivative is 12 x squared. This function will always be greater than or equal to zero for all X. No matter what value of extra potency here, it will be greater than equal to zero. So that means the concave ity the sign of whenever test value will be positive here, and it will be positive here. So if you chose negative one for this and if we chose that, say one for here, it would be positive, and this will be positive as well. So this is actually not a point of inflection. Oh, so you can just list here No point of inflection. The last thing we have to do is actually sketch the ground. So you know a few things about this craft? No, it&#x27;s X centers up. I&#x27;ll draw it in a blue. Oh, this one is at 00 this one is at the cube root of four zero. And then we have our what looks like a a minimum because it is decreasing from here from negative infinity toe. One, 21 and you&#x27;ll be increasing. So try to draw. That would look like he thought that like this, that is our graph f of

Bobby Barnes · Answer

we want to sketch a graph of curve. Why is it X equal with minus or ex? So in this chapter, they give us this laundry list of steps. We should follow an order to graph something, so it&#x27;s just kind of go through them step by step. So the first step is they want to figure out what the domain of dysfunction. Well, this is a polynomial, and we know polynomial is have a domain of all real numbers or negative infinity. The next part is one to figure our intercepts, so intercepts. So let&#x27;s go ahead and find our Why intercept first, just because that&#x27;s a little bit quicker. So why intercept is when X equals zero? So that would give us why is equitable? Extra support is zero and then four x zero. So we get zero for why interested and now for our X intercept. Remember, he said, why equal does aerial? So you get zero is equal to X to the fourth minus for X. Let&#x27;s go ahead and pull that or or the X out, and we get X Times X huge minus or so This tells us either X is equal to zero or X visible to the cube root of four. And this year is approximate the one point by. All right. Next, I did not go next. We want to see if there&#x27;s any symmetry. Intergraph, which is we want to look at of negative X and so doing this year. And so by EPA, Becks, I just mean our original function here. Why? So we can go ahead now and plug in negative X into here and doing that so plugging negative X and X support will just be extra aboard and then plug in negative x in for X with the negative X and that it would become plus four X. Now, this here does not equal to FX, Nor is it negative FX. So what this tells us is no symmetry court, so we got the first little bit done. Now, the next thing they suggest we buy is any acid trips. So we know since this is a polynomial, there should be no acid tips. But we can at least follow the similar steps to figure out it. What kind of in behavior we should have for this. So what we&#x27;re going to look at is the limit as X approaches Infinity of affects. Well, this is a court IQ function, and we know those kind of look like x squared. So since I was a positive coefficient, we know this is going to go to infinity. And likewise, we know what we have been. Even highest power for a polynomial. We will have the same. So the limit as X approaches negative penny of after Max also goes to the next thing they want. Supply is where this function is increasing and decreasing, so increasing slash decreasing intervals. So that means we want to find why pride. So why prime is going to be so be derivative with respect to X Oh X to the fourth minus war it&#x27;s and to take the derivative of both of these weeds powerful. So first we will get or X now Q. Because we need instruct all that power and then the derivative before X a jest or so. Let&#x27;s go ahead and factor that four out. So four X cubed, minus one. So if we were to just kind of look at what X cubed minus one looks like we know it&#x27;ll look something along the lines of this now it might kind of curve, but at least we know it&#x27;ll look something like It&#x27;ll fly an island in, Go back again so well that toeses this here. So why prime will be less than zero win. So let&#x27;s sexually first step that equals zero to figure out. But that should be so if we said it equals zero that just tell those X should be one So too the left of that point We know the function would be negative. So this will be less than zero when we&#x27;re on interval. Negative Infinity Q zero. And to remember this here tells us that the function is decreasing and when wide prime is strictly larger than zero. So this should be zero should be infinity too. The value What? So because of that our necks, then it would be one to infinity. And this here is our interval where it is increasing. Oh, so normally you would just set this equal to it are put into a inequality to figure out where it&#x27;s bigger than equal to zero. But we&#x27;re just gonna take the cheap way out so you don&#x27;t have to do more algorithm with you all right. The next step is to find our local Max is slash men&#x27;s and what we need to find those are critical points and we already found that are critical. Point is exiting the one because we want to find a wide prime is zero and we already know that sex is equal One last step. Now we can use the fact that we know on the interval of negative finito won, the function is decreasing and then it&#x27;s increasing after. So this tells us we have a local men at exit one. The next step is to look for con cavity and any inflection points we might have. So that means we need to look at the second derivative. So why double price? So we&#x27;re gonna go ahead and take more X cubed minus four and take the derivative of that that would be derivative with respect to X or excuse me in this war, so would go ahead to use power would take the derivative of X cubes, love the three x squared. We have well X squared and in the derivative of or we&#x27;ll just be zero. Now notice that this function here, we&#x27;ll always be greater than or equal to zero. And because of that, we know this wall be con cave up so con a Oh, and also there will be no inflection points Now that we have all this information, this was the last thing they thought was important so we could go ahead and try to grab our option. So let&#x27;s go ahead and first just locked down our exit. My inner sense. So we have our X intercept being or a wire set being 00 And we have our X intercept 10 and the other one is about one point. Just go ahead and put you grew Oh, four. All right, so we have those intercepts style, So there&#x27;s no symmetry our ass in tokes we know Well, there were no acid totes, but we do have the end behavior being on both sides, going to infinity. We know at X is equal to one. We have a local men, so now we can just go ahead and connect our graphs and everything. So it&#x27;s going to come from infinity, and the first point it needs to hit is due. And then, well, we said they were supposed to be a local men at 11 So it needs to curve up again like this, and then the next point need you need to make sure we get would be our other X intercept And then this is just gonna go off until like this. Now it&#x27;s not drawing very well, but at least you kind of get the idea. That should be more smooth. But unfortunately for me, I was not an artist. Now the only thing I might also say he should find is what value is here. Ah, local men. But I really don&#x27;t think it&#x27;s too important since we&#x27;re just trying to sketch graph with it. But this is all I think we really need to show for our graph. Just maybe draw a little bit.

Bobby Barnes · Answer

we want to sketch the graph of why is he going to x over expert mind sport Now in the structure they give us this laundry list of steps we should follow to sketch a graph. And one of the things we should always do for any rational function is to go ahead and factor in your area. The Dominator, Since we need to be ableto find any kind of holes or vertical ascent Oops, as well as our ex intercepts. So memory doesn&#x27;t factor, since just but the denominator is different square. So it&#x27;s gonna be X plus two X minus two and X plus two all over X. All right now, we can&#x27;t cancel anything else. Let&#x27;s go to the first step, which is our domain. Well, we know we need to make sure that our denominator does not equal to zero. So that&#x27;s gonna be an exit with the two and negative too. So the domain is gonna be negative. Infinity too Negative to union. Negative too. 22 Union to To infinity. Let me go ahead and drop this down. What? The next thing they tell us that the court is our intercepts. So let&#x27;s go ahead and find our X intercept First X intercept. Well, that&#x27;s going to be where the function is equal to zero. Or why is it so? Zero is equal to X over X squared minus four. And I&#x27;m not factoring that it on there because remember, no matter what we plug into the dominator, you never make this equals zero. So Soling for zero, we would just get X is equal to zero and since exited desirable. That would also tell us our Winer said it would be at the same point. So bye intercept is also going to just be why is zero? So the origin is our only interest. The next thing they suggest we find is any kind of symmetry we might have. So national functions aren&#x27;t really known for being periodic. So what we want to check him said, is whether the function is even or odd. And we do that, remember, by looking at of negative X, so we&#x27;ll get the negative X we get negative X and negative. X squared is just going to be ex squared minus four. Now notice that this here is equal to negative of necks. So this here implies that we have an odd function. So we have symmetry about our origin. All right, the next thing we need to chart core is our ass in Toots. Now we know that the limit as X approaches. So plus and minus infinity of our function at the backs, which is really just why so since our denominator has the larger power going back to dr 2.6, this tells us we&#x27;ll have a war zone toss until at, why? Easy, which is zero. All right, now we need to figure out our vertical acid vote since we just found our horizontal using that information. So the limit as X approaches. So we said our Berta classic. Oh, it&#x27;s going to be Where are the dominators? Zero. So X is equal to an X equals negative. So start with negative, too. So negative, too. From the right of Ethel bex. So let&#x27;s go ahead and plug that in. This could be negative too. From the right. Oh, negative too. From the right. Minus two times. Negative too. From the right. Plus two. Uh, let me go screw that over a little bit now. Negative, too. From the right is still going to be negative if I subtract so negative too from the right is gonna be negative again. And then if I subtract something that&#x27;s still gonna be negative and then if I do negative too. To the right, that will be something like negative 1.99 If I add that to two would be positive, so is gonna be overall positive. So that tells us that this should go to positive infinity. And we don&#x27;t even need to do the same steps are looking for as we approach it from the left because we know both. These are Ligier terms. That means they should have opposite behavior on either side. So just to save a little bit of time, we know that this here should go too negative, and Bennett and you could go to the same logic that we didn&#x27;t know before, but just to save a little bit of time. Now, the next thing we want to do is look at our other interesting, which is going to our other verte class, which is gonna be as X approaches to. So let&#x27;s look for right again. Let&#x27;s start so it&#x27;s ready to from the right over two from the right, minus two times two from the right, plus two and using a similar logic that we did before. So the new mayor is going to be positive. Slightly to the right. We&#x27;ll still be a positive number now. Something slightly to the right of two will be like 2.1 and subtracting, too. From that well, that would still be positive and tune from the right will be all positive number. So addict, something possibly possibly as well. So this here will approach positivity and using similar logic that we did before. We can find out that as we approach to formula, this should be negative. Infinity already. The next step is we want to find our intervals of where this function is increasing and decreasing and any kind of local Max&#x27;s ornaments. So this is That&#x27;s five and six, and I think they should be combined into one step. So we want to find what, why prime ms. So let&#x27;s go ahead and do that on another page. So if y is equal to X over X squared my sport now take the derivative of this. We&#x27;re going to need to use questionable. So question rule is low d high minus high d low all over the square. What&#x27;s below? And these just represent derivative. So going in filling that out, we will get the following expression and X squared minus for And then we need to square that now the derivative of X is going to be one, and the derivative of X squared should be to excusing the power rule and in the derivative of, um, negative forward, just be zero. So that should be two X and we can go ahead and rewrite this. Ah X squared minus four minus two x squared All over X squared minus four, I swear. Then, in the numerator, shove negative X squared minus four. Unless factor that negative out. And I will leave us with X squared plus four in the numerator over X squared minus four squared. Now notice that expert plus four will always be strictly greater than zero. And expert minus four squared should also always be greater than or equal to zero. So that means since we had that negative out here, this function will always be less than zero or decreasing. So go ahead and use that. So let&#x27;s go ahead and write. What? Why? Prime is real fast. Well, wide prime was negative, X squared plus four over X weird minus four square. And we already said that this year is going to be strictly less than zero. So our interval of decreasing will be everywhere that the function is just defied. So it&#x27;s always decreasing since why Prime Mister Crewe? Us these air. So it would just be negative infinity to make it up to union. Negative too, to do union to to infinity. All right. And we have no critical values for this since the numerator can ever equal zero. So the function can never actually equals zero and are derivative is undefined at X is equal to two and negative too. But those are not in our domain, so we don&#x27;t need to check those for critical ones. All right, The next step is we need to find our con cavity, so call on cavity and any kind of inflection points. So that means we need to figure out what? Why double prime Ms actually go ahead and move this suicide over here instead. So let&#x27;s just go ahead and little barrier there. So, like I&#x27;m saying, we need to go find what? Why? Double promise first. So let&#x27;s go over to hear again and start taking the derivative. So why double prime? So we can factor that negative out front and then just apply bullshit rule again. So we are going to do low d high, so X squared minus for squared times, the derivative of what&#x27;s above, which is X square plus four minus. Then the opposite order. So x squared plus four times the derivative with respect to X of what we have in the denominator x squared minus for squared and then all over what we have in the denominator squared. So X squared minus four squared But squaring that would give us to the power. And so don&#x27;t forget this You negative out here now, taking the derivative ot to these well expert plus four is just going to be two x and to take the drilled of X squared minus sportswear do enough to use Powerball and chain will. So the derivative of the outside, his first going to be two x squared minus four and then the derivative on the inside is going to be two x part. And just so we don&#x27;t have to do all that algebra when we simplify this, we should be left with two x x squared plus 12 over X squared minus four. Keep. So let&#x27;s go ahead and use that. Actually, before we do that, let&#x27;s figure out where this function is equal to zero. So again the denominator can never eat with zero, so we don&#x27;t need to worry about that. So we just need to worry about the numerator. Well, X squared plus 12 can ever equal to zero. And we know two x can. So we get to X zero or exes just is there. So this is one possible point of inflection. All right, so it was the hand right down that derivative or second, Really, we just found you&#x27;re going to be two x x squared lost well over X, where minus four Q. And just for the sake of brevity, to find where this option strictly larger than zero, I already went ahead and solved. That is, well, why Double prime greater than zero tells us what the function is comfortable and that&#x27;s going to be on negative, too. 20 Union too. Then, fans, Why double prime theatrically less than zero. Well, this stuff this is going to be conch aid down. And this is just the rest of the using interval from our domain. So negative many to negative to union with zero toe too. So we found that ecstasy. Coke zero is a possible point of inflection. So let&#x27;s go ahead and see if we have a change in Kong cabin here. So to the right of zero, we will be conch eight down and to the left of zero, We&#x27;re going to be con cave. So this here tells us we do have a point of inflection at X is equal to zero. All right, now, let&#x27;s go ahead and actually grab this function. So we know we have our intercept at only the origin. We know the function is odd, so it should be symmetric about the origin. And now we can go ahead and plot our ass in toots. So we know we haven&#x27;t asked. Toe X is equal to negative too, and X is equal to two. So excited, too accident negative too. And we have a horizontal ask himto at why is equal to zero. Um, so we have no local men&#x27;s or maxims from 0.5 and six, but we know at zero at X is equal zero wish of a change in con cabinet. What now? For horizontal Herbert glasses tips. So let&#x27;s start there to the right of negative too. We said that this should be going to positive infinity until the left of negative, too. We should be going too negative infinity. And we also said to the right of two we should be going to positivity until after to wish you were going to negative infinity. Now we can go ahead and start grabbing, so I&#x27;m just gonna start in this middle section, so we&#x27;re gonna start to the right of negative too, and we need to go to our origin. And at that point, it&#x27;s one to switch con cavity. So to the left of this, it&#x27;s complicated. But to the right, is Kong cave down now to the right of two? Well, we have no points of interest, so we&#x27;re just going to go until we hit our horizontal awesome too. And we will need to approach it from above as we can see there and now you can go ahead and go to the left of negative two. We have no points of interest after that. So we&#x27;re just going to go toe was awful, Jacinto and approach it from below, so you could maybe plot some actual numbers to make it look a little bit prettier. But since we&#x27;re just sketching it, I think this is a good place to stop.

Linh Vu · Answer

function y equals ju X minus or not far thanks square and the question asked her to find it curvature onda value up to so we know done the curvature k we know in the game of the exit with Iko June the absolute no wide about graham the Veniamin one plus y prime square Totally about under a tree and a Jew And therefore we need to find the Boston Mubi, the white Bram where it goes You were given the one here. The father What chemical? To one minus 1/2 on the X. And then the one of a family could humanise half. And therefore we have everything If when we have the guy next week Oh, joke. What about removing 1/2? Dividing by, uh is absolutely so we have again the 1/2 now Now we will have one breast the one minus 1/2 square Everything about our ambulatory under two. So we on, uh, the nominee they nominated. We can certainly find this one and then we get go We don&#x27;t need to signify because from here we just need to look in a very okay Joe him fo kept you. We go tell 1/2 inviting by. We haven&#x27;t had to be one. Plus, you go with her to inside. Here we get the one minus one B zero B, one breath. So, you know, about 300 to 4. We get equal to one on with Julie over one and go to another to we know final answer here.

Amrita Bhasin · Answer

so for the for this one are similar to the previous problem. You first find the X intercept and these are the value. If you use a calculator to numerically solve the equation and or you can just look at the graph together, approximate value and this world was integral and to evaluate is integral. Yunos Antiterror ripped you off This one eyes attention and that&#x27;s why you use power rule. So I&#x27;ll live remaining detail to you.

Amrita Bhasin · Answer

Alright based off of the problem, I&#x27;m trusting that are allowed to use a graphing calculator plus the equation is kind of awkward. X squared plus one to the negative first power. And then we have minus X. To the fourth hole account. I don&#x27;t know how you would do this in your head and maybe there&#x27;s some fancy math going on. But once you do that you can zoom in and we actually only care about the X coordinates. This must be an even function because it looks symmetric with the Y. Axis. And we care about negative .8692 positive .869. So that&#x27;s what I&#x27;m gonna do. I&#x27;m gonna pull up my tool under my functions and sometimes you have to click around before you find where your integral is. Mhm. And I believe I typed that I remember that. So negative .8692 positive .869. Yeah. And uh you unfortunately I might have to re type this equation otherwise you can copy and paste it down. Um And then you also might want to tell the calculator that X. Is your independent variable. I&#x27;m sure some other calculators might do that for you. Um and I rounded these answers these bounds, so my answer might be off a little bit, but if I were rounding my answer I would write down 1.233. Rounding to three decim, otherwise you can copy down the whole thing. Um And it should make sense that that answer is correct because if you look at trying to zoom around and it&#x27;s not letting me and alright, there we go. Uh That that looks about right, the area looks like it should be a little bit bigger than one but smaller than two. Um things like that. So 1.233 is what I would write down.

Problem

The area of the region that lies to the right of …

Problem 48 Hard Difficulty

Repeat Exercise 47 for the curve $ y = (x^2 + 1)^{-1} - x^4 $.

Answer

Area below the given curve and above the $x$ -axis is $\approx 1.2326$

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Area below the given curve and above the $x$ -axis is $\approx 1.2326$

Calculus

Grace H.

Catherine R.

Kayleah T.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartCalculus

Grace H.

Catherine R.

Kayleah T.

Joseph L.

James Stewart

Calculus