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University of North Texas



Problem 15 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \frac{x^2}{x^2 + 3} $


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Video Transcript

we want to sketch the bottle of wine ex way over. Exploited cluster. So in the structure, we are given this laundry list of steps we should probably source gonna go into accepts the 1st 1 is to find the Dominion over function. Well, national functions only we have have to worry about is when our denominators, but in this case explode plus three lonely, be greater than or equal to three. So our domain is just going to be all grill numbers. The next thing we want to find our intercepts. So let's go ahead and find our accidents that first. So X intercept will be where why is seriously get X were over. Now there's something I could never look into the denominator to make secret zero. So I just let the numerous equal to zero. And so that tells me physically. And since we know this, we already know why Interceptors Well, since exited zero would be where our why herself. So we don't even need to find out why intercepts is the same as our X. The next thing we want to do is look for symmetry. Well, this was fun about rational functions are not known for being periodic, so we should check to see even or odd using of negative X, so plugging in negative get negative X square over negative x squared. And that would simplify X squared over extra closely, which is equal to X. And that's going to imply that we haven't even function, and even functions are symmetric about why access symmetry across one access. So with that, if we just crap one side over function, we could just reflected across the y axis. Now, the next thing we're going to want to look for is acid trips. So since the denominator can ever equal zero on this rational country, we're not gonna happen. Your brother glass in tubes. But since the power is the Satan in or the largest power should say is the same, we know that as X approaches positive or negative, entity that Mr B, where we just divide the coefficients o r Parliament are of the largest degree of beach. So we have one over one, which is one. And when you really just know this from chapter 2.6, so we have a horizontal us until out why is it so what the next thing it also do is to look for intervals where the functions increasing or decreasing. So I'm gonna go ahead and quit that end with finding our local max. And then since those two things go hand in hand So let's go ahead and find a derivative of this first. So we have Why is it X squared over square three? Now, take the derivative bets. We're going to need to use questionable questionable status. Well, jihad minus high, low, all over. That was a load him. So X squared behind, minus I all over this. Where of what's over X where? Plus three. And then we need to square. All right, Now we know the derivative of experience is just going to be two bucks. Then the derivative on expert is going to be two X, and the derivative of three is just zero. Now, let's go ahead and do a little bit of algebra. And once we simplify everything, we should just be left with six x over X squared. Plus, Lee squared. Now we can go ahead and surface equal to zero to find our critical value. So again, nothing. I have a blood into the Dominator will make this function equals zero. But we can set the numerator from zero to find a critical points. So I do six ecstasy with zero, which would just imply X is equal to zero will be our critical point. So let's go ahead and write that we're here now. So y prime is equal to six sites over exploited plus three squint. So now, to figure out where our function is increasing, we want to find worldwide climate strictly largely zero, and this is going to be on from zero to infinity. Then we want to find where this function is decreasing. Which would be where? Why, private? Strictly less than zero. And that's gonna be from later 20 to 0. Now, let's go ahead and put our critical point down here and use the Crystal River to test it. Figure out if this will be a max man or meter. So to the right of excessive zero function is increasing to the left of excessive zero. The function is decreasing, so that's all those at X is equal to zero. We have a local men, so we know our origin is going to be a local mint and the last thing they tell us, which side is the con cavity and any inflection points we may have. So we're going to need a second derivative to do this. So let's come over here. So why Double prime is going to be equal to So I'm gonna factor that six up start and then to take the road of exposure X squared plus B squared, we're gonna have to use caution. Will do so again. X squared, plus three. Where times the derivative x minus, then the optimal. So x Times directive. Oh, x squared Plus, Lee, where All over what we have in the Dominator Scream which would now be the fourth power Now the derivative of X is just going to be one the derivative of excellent plus three squared will first draft to use power rules that b two times expert plus three. And then you have to take the driver on the inside where it's gonna be two X And now if you were to go through and do that algebra you should end up with 18 one minus x with over X squared plus three And now Cute. All right, so begin we could go ahead and set. The secret is there to traffic. The girls are possible points of inflection. So that would tell us we need to make one nice X squared equals zero or acceptable to plus or minus one. All right, so we're gonna need those values in a second, right? So we have why Double prime is equal to 18 times one minus X over X squared, plus three. And we're going to want to find with this con cave, which is where white double bind strictly grave and do and this is going to be on the interval. Negative one. Then we want to figure out where this is, Khan Cave down or where? Why, that will prime the strictly less things are gonna be negative. Infinity to negative one Union one two. And if we go ahead and put those two points that we found earlier and then see if we have a change of heart cavity, we can determine if these were going to be in collection points or not. So on negative one. Her negative feeling too negative. One for Fokker is decreasing on negative 1 to 1. The fiction is concave up. Uh, I meant to say calm, paid down to the left of the month when not decreasing and to the right of excessive auto won the conference on cave down. So we have a change of Kong cavity. Both So we know both these points should be points of inflection. And so that was the last thing they said you should do before if you start graphing this. So look ahead in start plotting some points. So we know our intercept tribute 00 So we'll see about that. Um, some, our sm toe would be at Why is a good one. So is one of our people we know at X equals zero. We're gonna have a minimum, so we don't need to figure out what that value provides for it. It's above or below. Are whores on pasta? Chose this. We already know with zero, and we know at negative one and one, we should have a change in con cabinet. So let's go ahead and start now so we can start from this minimum window and we can keep on going until we hit negative one, and then we should have a change in contact ity and then we'll go and get as close to our wars on Fastow's we can from the bomb. And then we could just go ahead and work like this across and I should let me move. One will be closer. So it actually does look a little bit more cement. So just reflecting across, why access it should now look something kind of like this. So from negative 1 to 1 that is deathly calm cave Oh, and to the left, it is concrete down and to the right of one that is concave down as well. So looks like we have everything we need. You could go on and maybe say what negative one and one are if you really want to of the specific. But as far as I'm concerned, all we really need to know is the shape at each of those points and just adding anything more. Just extra information. At this point,