WEBVTT
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we want to sketch the curve. Why is the
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X over X minus one now in Destructor? They
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give us this laundry list of steps we should follow
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any time we want to grab a function. So
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just go to the subs. So the first thing
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they tell us is to find our don't make so
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rational functions We need to make sure are not equal
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to zero in their domain, since we cannot divide
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by zero. And so the value that makes the
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denominator equals zero would be one. So our domain
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here would be negative. Infinity 21 Union one.
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John Kennedy. The next thing we're gonna go ahead
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and do is I've been to buy all of our
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intercepts intercepts. Let's first go ahead and find our
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Why interested? So why Interceptors went X is equal
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to zero. Just be why? Incident zero over
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zero minus one, which is your over negative one
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, which is just syrup. And for X intercept
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, we're going to set. Why go to zero
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and then solve for X so the dominator can ever
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equal zero or they could never make a function equals
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zero, so I could just multiply over the next
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. Nice one I get zero is equal to X
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. So the only value we have for an intercept
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is it? The next thing they tell us is
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to look for symmetry. So we know rational functions
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aren't really periodic. So we should check or even
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or odd. All right, So, Ephraim,
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negative X is going to be negative. X over
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negative X minus one, and we can go ahead
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and simplify those negatives to just give Exeter X Plus
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one. And so now this year does not equal
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to FX or negative aftereffects. So that tells us
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no symmetry early snow century, about the why access
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or origin. The next thing they tell us this
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Look for Assam pips and we know that where are
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denominator is equal to zero per rational function. We
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will have a as well as we possibly have an
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asset as excuse and then you're negative. So let's
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just go ahead and go through those. So we
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will have the limit as X approaches one from the
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right of FX. And I'm just going to call
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original function after Lex or why, So let's go
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ahead and see what we get. So in the
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numerator. It would be one from the right and
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then one from the right, minus one. Now
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one from the Web rate will be positive. Some
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simple sign of top here and one from the right
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. Minus one. Well, that means it'll be
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bigger than one. So subtracting something bigger problem from
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one will also be positive. So that tells us
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we're going to approach positivity. Now, we're going
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to do the same thing from the left. And
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now it's gonna be one from the left over,
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one from the left, minus one. So,
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just like before, the top will be positive.
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Just because if I get really close to one on
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the left, it also be a positive number.
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But when I approach one from the left, it's
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attract one. Well, I'm subtracting something, our
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instructing one from something smaller than one, so that's
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gonna be negative. So here, this is going
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to approach negative infinity. Just go ahead in below
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the biters between those sons. Next, we should
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check the end behavior of this. So the limit
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as X approaches ability. Uh, that looks and
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we know from chucker Q 2.6 that when we have
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a Russian polynomial that has the same degree in the
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top bottom. We would just take the leading coefficients
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and divide them so one over one would be one
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. And we know the same thing is going to
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happen for the other. And behavior as experts is
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negative. And Bernie don't still be one over one
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or one being next thing they want us to find
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is intervals of increasing and decreasing at any maximum minimum
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. So I'm going to do is go ahead and
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combine those into one step. So intervals of increasing
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slash decreasing and any local back slash men. That
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means we're gonna need to look at why Prime?
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So let's just go ahead and do that on another
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page. So why is it into X over X
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minus one? Now take the drifter the this we're
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gonna need to use questionable. So remember, questionable
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says early sort of way I remember is low D
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high minus hi below all of us were below For
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those D's are its meaning to take the derivative so
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X minus one times a driver, so that's minus
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. Then the opposite border over X minus one,
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and then what we have in the denominator squared.
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So we know the derivative of X is just going
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to be one. And, you know, the
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derivative of X is one again, and the derivative
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of zero are derivative of 10 So this just becomes
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one, so we can go ahead and simplify this
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year as X minus one my S X over X
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minus one. That's where these X's cancel out with
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each other and we're just left with negative one over
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X minus one square. Now, notice that this
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here is going to be strictly less than zero.
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Always because X minus one squared. We'll always be
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bigger than zero. Uh, and when we multiply
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that value by a negative one, then it will
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become negative. So what this tells us is,
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why always decreasing? Always decrease. Uh, as
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long as X does not equal to once, this
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is just not defined there. And we know we're
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not going to have any critical points since the only
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critical point that this would imply we would have this
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X equal to one. But X is equal to
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one is not defined for domain. So we have
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no critical points, so avoidable point here and Let's
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just go ahead and write that down again. So
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it's negative. One over X minus one Where and
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and so he found that decreasing interval wide crime is
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gonna be stripping lessons. Zero is going to be
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the interval negative 50 to 1 again. 121 30
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and are increasing in trouble. Well, there was
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no increasing interval, but let's just go ahead and
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write that down anyways, so this would be the
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empty set. Or maybe I should just right,
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uh, and done all right. And from this
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we can just go ahead and concludes, It's it's
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always decreasing. No local maximum. So no,
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no. Cool. Um, max slash men's all
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right. The next thing they tell us to find
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is Kong cavity and inflection points on gravity on DDE
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inflection points. So that means we're gonna need look
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at why double time. Let's go back to the
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other page and find the second. So why double
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prime is going to be well, we can go
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ahead and rewrite this first as negative one x to
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the Mayas one for the negative second power by the
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X and we could go ahead and use the power
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rule to take this for evidence. So the negative
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one times negative too X minus one to the negative
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third power. And then we need to take the
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derivative of our inside functions. That's where you've been
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changeable and the derivative of X minus one. Just
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like when we were taking the first derivative. This
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is just going to be one so negative one times
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negative to is to. And then we could go
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ahead and read Reciprocate ex mice one gets and in
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the denominator again. So now we know that this
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will be strictly larger than zero when our denominator is
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positive, because the two will always be positive.
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And it'll just turn into what win is our denominator
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going to be positive divided something by possible possible.
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So we can really just ask what when it's nice
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one strictly larger, then zero. And that would
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just be X minus one or exit is strictly larger
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than one it will be can keep up. Until
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then. That tells us so why double prime will
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be less than zero on the rest of the interval
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. So just ex strictly less than one. All
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right, so I should say that this year is
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concave up, and this here is concave down Since
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why double time would be lessons Earl and greater than
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see? All right, So we have why double
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prime is too over X minus one Q. And
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we know it will be calm paid down or why
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double time sugar, less than zero on the interval
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. Negative. Infinity, too. Not just one
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. And it's going to be con cave up my
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double, my mystically larger than zero off the senator
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broken It should be one too. So since we
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had no critical values for this because the same reason
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before, the only value that we might want to
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look at from this is where it's undefined, which
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would be exit equal one. But our original function
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is not defined. That exceeded the one. So
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we're not going to end up having any inflection points
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. So no inflection points since X equal one not
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defined. All right, so now this is everything
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they suggested we do before we have to start dropping
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. So let's go ahead and grab now. So
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let's go ahead and plot her ass and cops first
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. So we know at X busy with one,
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we will have an Absolut A for Berta glasses.
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But this is exiting one. And at why is
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he the one you know? We will have a
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horizontal. So why is equal to one? All
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right now we know from the rite of our vertical
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ass in two, we should be coming from positive
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infinity. And so that means since we have our
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where is on plastic tote actual? Let's hold on
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. That was just pretty vertical first and to the
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left of our vertical ascent. Oh, it should
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be going to negative if any or negative. All
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right, now next. Go ahead and bought our
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interests, which is only zero burn 00 We have
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no symmetry. We have no Max is airmen's and
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we have no points of inflection. So now let's
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go ahead and go to our ask him to.
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And this is when we will need toe. Have
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it approach one so we can start on the left
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here. And so I need the first hit My
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intercepted do zero. And then that's going to tell
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me I'm going to approach my horizontal ass and hope
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from the bottom and likewise over here on the right
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of dysfunction we're gonna start from positivity, and then
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we're gonna need to go down one, since we
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have no points of interest here and we will perch
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y Z with the one from above. So this
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is a nice little sketch. You could maybe add
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some just Maur detail by the same, like maybe
206
00:14:54.799 --> 00:14:56.600 A:middle L:90%
what? Negative 12 or some other points are.
207
00:14:56.600 --> 00:14:58.090 A:middle L:90%
But since we're just trying to sketch it, I
208
00:14:58.090 --> 00:15:00.340 A:middle L:90%
really don't think we need to do anything other than
209
00:15:00.399 --> 00:15:01.649 A:middle L:90%
what we have listed here.