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9 Is there quadratic function P, which takes the pre-dlefined values p (0) p"' (0) , p"" (0)2 Is there quadratic function p which takes the pre...

Question

9 Is there quadratic function P, which takes the pre-dlefined values p (0) p"' (0) , p"" (0)2 Is there quadratic function p which takes the pre-defined values fl1p () = dx, p (0) and p' (0)?

9 Is there quadratic function P, which takes the pre-dlefined values p (0) p"' (0) , p"" (0)2 Is there quadratic function p which takes the pre-defined values fl1p () = dx, p (0) and p' (0)?



Answers

Suppose that a polynomial function $P$ is defined in such a way that $P(3)=-4$ and $P(4)=-10 .$ Can we be certain that there is no zero between 3 and $4 ?$ Explain, using a graph.

So in this problem, we're looking at the integral of a rational function. X over X squared plus B explicit E where a is not equal to zero. We haven't least a quadratic in the Dinu and that will we have a quadratic in the denominator. And the question wants to know if this integral could ever produce a function with no log rhythmic terms. So what would it What would we have to do to do that? So I suppose that I suppose that B and C where equals zero so we'd have X over X squared would be some constant over a X, so that would get us a log term. So we can't just have either. I want to consider. So how do we get something that's not a large term from here? So we would have to have if we if x squared plus b x plus e had real roots. Suppose, um, was due P and Q, so we can rewrite this as X over X minus p times X minus cube, and that would be some constant over X minus p, plus another constant over X minus que These would get us long term so it has to be some kind of, uh, way need some kind of imaginary or complex roots. So I suppose we could right x where X squared plus B x plus e as like X squared plus p squared would have x over X squared plus B squared. So if this was the case that we could do the use substitution, you equal to X square plus p squared that d you would be to acts DX and we could rewrite are integral as one half do you over you, which we get us again. A large terms. We have Ellen, one half Ellen of you, which is X squared plus p squared so it can't take this form. So this is an imaginary route. So let's look at a complex root So complex root means that we can write this as X over X minus p squared plus cues word. So from here we what? We want to dio iss convert this into something where we can dio au substitution and something where we got a tangent crease. So the problem with that is so here's my DX So my break up this fraction I get X minus p over X minus p. Squared plus Q squared and P over X minus p squared plus u squared. So when I use now for this piece, I have to use the U substitution. So you equal to X minus p squared plus cues work then to you is to x minus P d x b a minus there. So then my intern roll, I'm going to get a one over you again and I'll give me a large piece. So even though this component got me a change in inverse one overpay in front o never Hugh Tangent of X minus p over Q. I still have a large pizza. So in conclusion, um, this intern role is always going to result in a function with some kind of log rhythmic composed.

Okay, so we want to know if there's a value K such that they told for all values of X. Well, let's solve for K. So let's take the natural log of both sides. So we get that X is equal to que X and then the natural log of to divide both sides by X and a natural law go to on. We see that this issue when K is equal to one over the natural, log on to

Okay. And number 61 we have one extra step that we need to do. They've already factored out of a common factor of nine P minus two, and we're left with this. Try Nobile, This train. Oh, Bill still needs to be factored for us to use your product property. So I'm gonna bring this up to the side. Nine p minus two. Could, uh, the past equals zero for one of the solutions. Then we're going to Ah, factor This and this will help us to find the other two answers. So to make peace squared would multiply. Peters he to make 11. You can only most by one times 11 but it's negative. So one of them is negative, and one is positive for us to have a negative 10. That means that the bigger number has to be negative. So this is one p negative 11 p which together makes negative 10 p. So these are the two factors for this train a meal. Then we can set each of those equals zero. So p plus one equals zero or P minus 11 equals zero. Over here, I can add to to both sides and then all divide all terms by nine. So that would mean that P is equal to about about his one. This is zero, so we're only looking at the other side, which is two nights. That's what solution. Another solution comes from subtracting one from both sides. We get P is equal to negative one, and finally, we can add 11 to both sides in this equation and we get the P is equal to 11. And so these are three solutions for what the value P could be to make the overall equation true.

We are asked. We are told that the minute as X approaches zero of F x over G of X equals 10 we're also told that both functions F and G r Paulino wheels of the fix this quickly are Pollina meals. Lastly, we're told that G of zero equals two, but we're supposed to find out what f zero equals. That's the problem for today. The person that we need to do for this promise me that we need to realize that the function that's within the limits or the two functions that are within the limit. This expression right here will be a rational polynomial. Irrational polynomial is just anything that can be expressed as a ratio of two polynomial in this case. But first column we will be f of s and the second polynomial will be g of X. So, using that property, we know that the limits as X approaches a of ration as long of rational Polito meals will equal the, uh are a for our for some rational polynomial will know that this is a true fact. If denominator of the irrational polynomial does not equal zero and we know this is true because G of zero equals two. Given that we can then replace everything here with, uh, with with plumbing in our A Basically, in this case, eight equals zero. So we plug in the A on the top. This becomes f zero we to solve for on the bottom we get geo zero and this will equal 10. So now we know that this equals choose. We can multiply both sides by two. I'll get f zero equals but 20 So that is our problem solved. I should say something here, though, that we know for a fact that when I when I did when I told you that we can simply assume that the bottom equals is not zero we knew when we know that the bottom is not zero. The reason why is if you analyze is a little bit more closely and you break this apart using the limit law of division, you'll get the limit as X approaches zero of FX over the limit as X approaches zero over g of axe. So how did I immediately know that this right here was equal to this right here? Well, the reason is because normally, if I just gave you these two piece of information. I told you this. And this. Those two things would not actually necessarily have to go hand in hand with each other. For example, if I told you that G of X, where a piece wise function and I told you that, you know, from here it equals two X plus one. If X is not equal zero and equals two in this case when X equals zero, this would present a dilemma because the limit as X approaches zero geo backs would actually be reliant on this top part right here. This was the limit would actually be one. But we know that the limit does not equal the actual value at that point. So how do we know that these two are equal to each other? Well, the reason is because both f n g r polynomial is, and there's an aspect about holla new mules that makes it so that we know that the limit s ex approaches of value a will be G or the function at that value. A. The reason is because polynomial is have this aspect that they are continuous. So because of that, we know that there is no such thing as a G is not a piece. Once function like this, it's a continuous function. We don't need to worry about anything like that, and we can indeed make the substitution.


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