we want to sketch the curve. Why is the X over X minus one now in Destructor? They give us this laundry list of steps we should follow any time we want to grab a function. So just go to the subs. So the first thing they tell us is to find our don't make so rational functions We need to make sure are not equal to zero in their domain, since we cannot divide by zero. And so the value that makes the denominator equals zero would be one. So our domain here would be negative. Infinity 21 Union one. John Kennedy. The next thing we're gonna go ahead and do is I've been to buy all of our intercepts intercepts. Let's first go ahead and find our Why interested? So why Interceptors went X is equal to zero. Just be why? Incident zero over zero minus one, which is your over negative one, which is just syrup. And for X intercept, we're going to set. Why go to zero and then solve for X so the dominator can ever equal zero or they could never make a function equals zero, so I could just multiply over the next. Nice one I get zero is equal to X. So the only value we have for an intercept is it? The next thing they tell us is to look for symmetry. So we know rational functions aren't really periodic. So we should check or even or odd. All right, So, Ephraim, negative X is going to be negative. X over negative X minus one, and we can go ahead and simplify those negatives to just give Exeter X Plus one. And so now this year does not equal to FX or negative aftereffects. So that tells us no symmetry early snow century, about the why access or origin. The next thing they tell us this Look for Assam pips and we know that where are denominator is equal to zero per rational function. We will have a as well as we possibly have an asset as excuse and then you're negative. So let's just go ahead and go through those. So we will have the limit as X approaches one from the right of FX. And I'm just going to call original function after Lex or why, So let's go ahead and see what we get. So in the numerator. It would be one from the right and then one from the right, minus one. Now one from the Web rate will be positive. Some simple sign of top here and one from the right. Minus one. Well, that means it'll be bigger than one. So subtracting something bigger problem from one will also be positive. So that tells us we're going to approach positivity. Now, we're going to do the same thing from the left. And now it's gonna be one from the left over, one from the left, minus one. So, just like before, the top will be positive. Just because if I get really close to one on the left, it also be a positive number. But when I approach one from the left, it's attract one. Well, I'm subtracting something, our instructing one from something smaller than one, so that's gonna be negative. So here, this is going to approach negative infinity. Just go ahead in below the biters between those sons. Next, we should check the end behavior of this. So the limit as X approaches ability. Uh, that looks and we know from chucker Q 2.6 that when we have a Russian polynomial that has the same degree in the top bottom. We would just take the leading coefficients and divide them so one over one would be one. And we know the same thing is going to happen for the other. And behavior as experts is negative. And Bernie don't still be one over one or one being next thing they want us to find is intervals of increasing and decreasing at any maximum minimum. So I'm going to do is go ahead and combine those into one step. So intervals of increasing slash decreasing and any local back slash men. That means we're gonna need to look at why Prime? So let's just go ahead and do that on another page. So why is it into X over X minus one? Now take the drifter the this we're gonna need to use questionable. So remember, questionable says early sort of way I remember is low D high minus hi below all of us were below For those D's are its meaning to take the derivative so X minus one times a driver, so that's minus. Then the opposite border over X minus one, and then what we have in the denominator squared. So we know the derivative of X is just going to be one. And, you know, the derivative of X is one again, and the derivative of zero are derivative of 10 So this just becomes one, so we can go ahead and simplify this year as X minus one my S X over X minus one. That's where these X's cancel out with each other and we're just left with negative one over X minus one square. Now, notice that this here is going to be strictly less than zero. Always because X minus one squared. We'll always be bigger than zero. Uh, and when we multiply that value by a negative one, then it will become negative. So what this tells us is, why always decreasing? Always decrease. Uh, as long as X does not equal to once, this is just not defined there. And we know we're not going to have any critical points since the only critical point that this would imply we would have this X equal to one. But X is equal to one is not defined for domain. So we have no critical points, so avoidable point here and Let's just go ahead and write that down again. So it's negative. One over X minus one Where and and so he found that decreasing interval wide crime is gonna be stripping lessons. Zero is going to be the interval negative 50 to 1 again. 121 30 and are increasing in trouble. Well, there was no increasing interval, but let's just go ahead and write that down anyways, so this would be the empty set. Or maybe I should just right, uh, and done all right. And from this we can just go ahead and concludes, It's it's always decreasing. No local maximum. So no, no. Cool. Um, max slash men's all right. The next thing they tell us to find is Kong cavity and inflection points on gravity on DDE inflection points. So that means we're gonna need look at why double time. Let's go back to the other page and find the second. So why double prime is going to be well, we can go ahead and rewrite this first as negative one x to the Mayas one for the negative second power by the X and we could go ahead and use the power rule to take this for evidence. So the negative one times negative too X minus one to the negative third power. And then we need to take the derivative of our inside functions. That's where you've been changeable and the derivative of X minus one. Just like when we were taking the first derivative. This is just going to be one so negative one times negative to is to. And then we could go ahead and read Reciprocate ex mice one gets and in the denominator again. So now we know that this will be strictly larger than zero when our denominator is positive, because the two will always be positive. And it'll just turn into what win is our denominator going to be positive divided something by possible possible. So we can really just ask what when it's nice one strictly larger, then zero. And that would just be X minus one or exit is strictly larger than one it will be can keep up. Until then. That tells us so why double prime will be less than zero on the rest of the interval. So just ex strictly less than one. All right, so I should say that this year is concave up, and this here is concave down Since why double time would be lessons Earl and greater than see? All right, So we have why double prime is too over X minus one Q. And we know it will be calm paid down or why double time sugar, less than zero on the interval. Negative. Infinity, too. Not just one. And it's going to be con cave up my double, my mystically larger than zero off the senator broken It should be one too. So since we had no critical values for this because the same reason before, the only value that we might want to look at from this is where it's undefined, which would be exit equal one. But our original function is not defined. That exceeded the one. So we're not going to end up having any inflection points. So no inflection points since X equal one not defined. All right, so now this is everything they suggested we do before we have to start dropping. So let's go ahead and grab now. So let's go ahead and plot her ass and cops first. So we know at X busy with one, we will have an Absolut A for Berta glasses. But this is exiting one. And at why is he the one you know? We will have a horizontal. So why is equal to one? All right now we know from the rite of our vertical ass in two, we should be coming from positive infinity. And so that means since we have our where is on plastic tote actual? Let's hold on. That was just pretty vertical first and to the left of our vertical ascent. Oh, it should be going to negative if any or negative. All right, now next. Go ahead and bought our interests, which is only zero burn 00 We have no symmetry. We have no Max is airmen's and we have no points of inflection. So now let's go ahead and go to our ask him to. And this is when we will need toe. Have it approach one so we can start on the left here. And so I need the first hit My intercepted do zero. And then that's going to tell me I'm going to approach my horizontal ass and hope from the bottom and likewise over here on the right of dysfunction we're gonna start from positivity, and then we're gonna need to go down one, since we have no points of interest here and we will perch y Z with the one from above. So this is a nice little sketch. You could maybe add some just Maur detail by the same, like maybe what? Negative 12 or some other points are. But since we're just trying to sketch it, I really don't think we need to do anything other than what we have listed here.
Discussion
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we want to sketch the curve. Why is the X over X minus one now in Destructor? They give us this laundry list of steps we should follow any time we want to grab a function. So just go to the subs. So the first thing they tell us is to find our don't make so rational functions We need to make sure are not equal to zero in their domain, since we cannot divide by zero. And so the value that makes the denominator equals zero would be one. So our domain here would be negative. Infinity 21 Union one. John Kennedy. The next thing we're gonna go ahead and do is I've been to buy all of our intercepts intercepts. Let's first go ahead and find our Why interested? So why Interceptors went X is equal to zero. Just be why? Incident zero over zero minus one, which is your over negative one, which is just syrup. And for X intercept, we're going to set. Why go to zero and then solve for X so the dominator can ever equal zero or they could never make a function equals zero, so I could just multiply over the next. Nice one I get zero is equal to X. So the only value we have for an intercept is it? The next thing they tell us is to look for symmetry. So we know rational functions aren't really periodic. So we should check or even or odd. All right, So, Ephraim, negative X is going to be negative. X over negative X minus one, and we can go ahead and simplify those negatives to just give Exeter X Plus one. And so now this year does not equal to FX or negative aftereffects. So that tells us no symmetry early snow century, about the why access or origin. The next thing they tell us this Look for Assam pips and we know that where are denominator is equal to zero per rational function. We will have a as well as we possibly have an asset as excuse and then you're negative. So let's just go ahead and go through those. So we will have the limit as X approaches one from the right of FX. And I'm just going to call original function after Lex or why, So let's go ahead and see what we get. So in the numerator. It would be one from the right and then one from the right, minus one. Now one from the Web rate will be positive. Some simple sign of top here and one from the right. Minus one. Well, that means it'll be bigger than one. So subtracting something bigger problem from one will also be positive. So that tells us we're going to approach positivity. Now, we're going to do the same thing from the left. And now it's gonna be one from the left over, one from the left, minus one. So, just like before, the top will be positive. Just because if I get really close to one on the left, it also be a positive number. But when I approach one from the left, it's attract one. Well, I'm subtracting something, our instructing one from something smaller than one, so that's gonna be negative. So here, this is going to approach negative infinity. Just go ahead in below the biters between those sons. Next, we should check the end behavior of this. So the limit as X approaches ability. Uh, that looks and we know from chucker Q 2.6 that when we have a Russian polynomial that has the same degree in the top bottom. We would just take the leading coefficients and divide them so one over one would be one. And we know the same thing is going to happen for the other. And behavior as experts is negative. And Bernie don't still be one over one or one being next thing they want us to find is intervals of increasing and decreasing at any maximum minimum. So I'm going to do is go ahead and combine those into one step. So intervals of increasing slash decreasing and any local back slash men. That means we're gonna need to look at why Prime? So let's just go ahead and do that on another page. So why is it into X over X minus one? Now take the drifter the this we're gonna need to use questionable. So remember, questionable says early sort of way I remember is low D high minus hi below all of us were below For those D's are its meaning to take the derivative so X minus one times a driver, so that's minus. Then the opposite border over X minus one, and then what we have in the denominator squared. So we know the derivative of X is just going to be one. And, you know, the derivative of X is one again, and the derivative of zero are derivative of 10 So this just becomes one, so we can go ahead and simplify this year as X minus one my S X over X minus one. That's where these X's cancel out with each other and we're just left with negative one over X minus one square. Now, notice that this here is going to be strictly less than zero. Always because X minus one squared. We'll always be bigger than zero. Uh, and when we multiply that value by a negative one, then it will become negative. So what this tells us is, why always decreasing? Always decrease. Uh, as long as X does not equal to once, this is just not defined there. And we know we're not going to have any critical points since the only critical point that this would imply we would have this X equal to one. But X is equal to one is not defined for domain. So we have no critical points, so avoidable point here and Let's just go ahead and write that down again. So it's negative. One over X minus one Where and and so he found that decreasing interval wide crime is gonna be stripping lessons. Zero is going to be the interval negative 50 to 1 again. 121 30 and are increasing in trouble. Well, there was no increasing interval, but let's just go ahead and write that down anyways, so this would be the empty set. Or maybe I should just right, uh, and done all right. And from this we can just go ahead and concludes, It's it's always decreasing. No local maximum. So no, no. Cool. Um, max slash men's all right. The next thing they tell us to find is Kong cavity and inflection points on gravity on DDE inflection points. So that means we're gonna need look at why double time. Let's go back to the other page and find the second. So why double prime is going to be well, we can go ahead and rewrite this first as negative one x to the Mayas one for the negative second power by the X and we could go ahead and use the power rule to take this for evidence. So the negative one times negative too X minus one to the negative third power. And then we need to take the derivative of our inside functions. That's where you've been changeable and the derivative of X minus one. Just like when we were taking the first derivative. This is just going to be one so negative one times negative to is to. And then we could go ahead and read Reciprocate ex mice one gets and in the denominator again. So now we know that this will be strictly larger than zero when our denominator is positive, because the two will always be positive. And it'll just turn into what win is our denominator going to be positive divided something by possible possible. So we can really just ask what when it's nice one strictly larger, then zero. And that would just be X minus one or exit is strictly larger than one it will be can keep up. Until then. That tells us so why double prime will be less than zero on the rest of the interval. So just ex strictly less than one. All right, so I should say that this year is concave up, and this here is concave down Since why double time would be lessons Earl and greater than see? All right, So we have why double prime is too over X minus one Q. And we know it will be calm paid down or why double time sugar, less than zero on the interval. Negative. Infinity, too. Not just one. And it's going to be con cave up my double, my mystically larger than zero off the senator broken It should be one too. So since we had no critical values for this because the same reason before, the only value that we might want to look at from this is where it's undefined, which would be exit equal one. But our original function is not defined. That exceeded the one. So we're not going to end up having any inflection points. So no inflection points since X equal one not defined. All right, so now this is everything they suggested we do before we have to start dropping. So let's go ahead and grab now. So let's go ahead and plot her ass and cops first. So we know at X busy with one, we will have an Absolut A for Berta glasses. But this is exiting one. And at why is he the one you know? We will have a horizontal. So why is equal to one? All right now we know from the rite of our vertical ass in two, we should be coming from positive infinity. And so that means since we have our where is on plastic tote actual? Let's hold on. That was just pretty vertical first and to the left of our vertical ascent. Oh, it should be going to negative if any or negative. All right, now next. Go ahead and bought our interests, which is only zero burn 00 We have no symmetry. We have no Max is airmen's and we have no points of inflection. So now let's go ahead and go to our ask him to. And this is when we will need toe. Have it approach one so we can start on the left here. And so I need the first hit My intercepted do zero. And then that's going to tell me I'm going to approach my horizontal ass and hope from the bottom and likewise over here on the right of dysfunction we're gonna start from positivity, and then we're gonna need to go down one, since we have no points of interest here and we will perch y Z with the one from above. So this is a nice little sketch. You could maybe add some just Maur detail by the same, like maybe what? Negative 12 or some other points are. But since we're just trying to sketch it, I really don't think we need to do anything other than what we have listed here.
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