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Ishee e grtpbt of of tlie derivative of continous function9(2.9) 9(2.1)shown belowother "SpIOM...

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Ishee e grtpbt of of tlie derivative of continous function9(2.9) 9(2.1)shown belowother "SpIOM

ishee e grtpbt of of tlie derivative of continous function 9(2.9) 9(2.1) shown below other "SpIOM



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For the graph in Figure 12.63, determine where the function is continuous/discontinuous and differentiable/not differentiable.

Now here we're told them, you know, HB greater than zero. Okay. And so we have F is continuous on some interval, some closed interval X not minus H two X not plus age. And it's differentiable on that on that open interval. And we want to show that if um we showed that if the limit of f prime f X as X approaches X not equals L, then F is differentiable at X. Not an F prime of X. Not because L So basically saying that we can just substitute next in the year and we'll be okay. Um So what we need to do is kind of show the limits from each side. So this would be the right sided right handed limit, right handed derivative. So as we approach X, not from the from the right. Mhm. So that's equal to um the limit of F prime uh X as we approach X. Not from from the right. And so that we we know that that is equal to L. All right. So we can then know that we can then rewrite this as F of X. Not um Right. Right. This right here is F of X not plus t minus f of x. Not all over T as T approaches zero from the right. But then we can use the um um we can write this in the form um that we had in the previous case. So the other form of the main value there. So we get this because this for some data that's between 01 and that's nice because we can then cancel the tease. So we know that then. Um F prime um F prime of X plus data. T um is with the limit as T approaches zero from the plus of this now equals L. Um uh So we got that. All right. Um and then we can do the same for the left hand. It basically just have to go through the whole same process. So and then we'll show and to get that this result for X zero as we approach zero, it's not from the left. So just go through the same thing, the same steps, the same reasoning to get this. And so that we know that the left hand and the right hand derivatives are both equal to each other. So that means that the derivative is and they're both equal to L. And so the derivative is equal to L at that point. So, it's kind of a kind of a strange problem here because it's kind of, you know, in some senses should be obvious. But you could find some weird functions um that uh that don't satisfy this so that, you know, you have weird things here. So if it's continuous and differentiable, then we know that we can basically if not if it's a nice function, we can just say, okay, well let's substitute X. Set a secret X not will be okay

In this problem we can find yes Bash at zero plus is equals two limit. It's tends to zero, half of it minus. Therefore zero by let this be okay because it's given that it's finite. Same after so you don't minus will be equals to limit. It's tends to zero fr zero minus effort because you don't mind a set they wanted by aah since it's an even function F minus it is effort. This is limit. That stands to zero. FH by catch, which is nothing but minus. Keep comparing it from here. So we see that F-0 plus & F-0 0 are not equal but both are finite, which means that every dish access containers had X equals to zero but not differential attacks. It was 20 correct option of options. What?

In discussion given in discussion, we have to find the value of LTD accidents to one. Limit accidents to infinity of one upon f. of one times half of X. Times into the power and plus G of X upon it. Is the power annex closed. What? So here given Half of one equal to 297. And here let f be continuous function and G abounded function. So first of all we take limit accidents to infinity. So here limit Accidents to one. All limit accidents to entreaty all one upon 297 because we know that the value of Apple one which is 297 times 1/2 of x. times into the power annex. Plus G of X upon into the power and X plus one. So now we have a divide numerator and denominator By 8 to the power annex. So here we get limit extents to one of limit accidents to infinity off one upon 297 times half of eggs blows G of X upon into the bowl and eggs upon one plus one upon italy part annex. So now we put your limits which is accidents to injury. So here began limit access to one one upon 297 times apple facts plus G. Of X. Upon infinity upon one plus one upon in three D. So here we get one upon 297 times limit accidents to one of apple X. So here we know that that is the value of limit Extents to one of Apple X. is 297 because here we know that limit accidents to one of apple X. Equal to Apple of one and here f of one value which is given 297. So we put here 297 upon 297 Equal to one. So it is our final answer.

Okay. And, uh, for this problem, we're given function here and we want to find the limit us. That's a porteous nine. And note that if you plug in nine, you're going to get zero down here, and that's not gonna work on. And they want us to use that table method. So all we have to do is plug in numbers. Ah, for instance, here point nine and then next one his point nine nine eight point nine nine. But nickels like, that s o one thing that you have to notice that we're doing it as we get closer to nine from the positive side, Meaning it's slightly larger than nine. And then from the negative side so you can go ahead and plug in numbers. For instance, ided eight point nine here and what I got wass about two hundred forty, uh, point three one. So go ahead and do the numbers. Find a numbers here, and then that will give you ah limit. But I'm gonna show you a second method that you were anyways going to be needing for, you know, doing limits, which is coming up. But since this is a good example, I'm going to try to show you the second method. So notice here how this is like, ah, like a cubic formula that you actually can simplify. Remember that. Okay. Do you remember a cue minus? No, my keep is a little bit off. Okay. Minus speak. You can be written as a minus being some basically packed. Arising it. This is a formula that you've seen before. I'm sure, but it comes in handy because I will simply by the actual equation. So here I'll have a square wass eighteen. Yes, he's square. Okay. All right. So using this formal outlets tried to re write X key minus seven twenty nine. So what I normally do is when I see something like those I just goes for Since I don't remember what nine key ways, but I figured it must be ninety must be seven twenty nine, and when I check, it's actually right. Nine. Keep a seven twenty nine. So in our case, it would be at a OK, so write it down here. So now K's a equals eggs amed fee would be nice. So now you can rewrite equifax slightly differently. X minus nine climes X square I'm just following the formula that it's right up here. So x square here waas a times B, which is, uh, nine times X owning ex here must be square B square is nine square, just eighty one. Right. Okay. And then I'm just going to write X minus nine at the bottom. Don't do anything down there. So now not that you have not X minus nine pose, um, at the numerator and, uh, down here so you can do as we can Cancel those out in this out play a given that we have to say x for X different from night. Because if we put nine, then it's gonna be a problem. Okay, so So you remember You don't need to do this for this problem, but I'm just showing you this that way. It would be simpler when we get to the next section. So now for FX, we have X square, class nine X plus eighty one. So when you do the limit as except Fortress nine Done, it would be easy to do it with this equation. So what? I found out using both the table in this simplified equation down here is that the limit of, uh, ever backs this acts of fortress nine you're going to find is going to be around two hundred forty three. So we did two different methods. Um, but if you want, all you have to do is just this section where you just but in numbers and then compare the number. You get the result. You get forever. Faxes the side of the table, aim the side of the table, which would be two hundred forty three, okay.


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