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1. Find Big 0 notations for the function f (2) 322as €as € -...

Question

1. Find Big 0 notations for the function f (2) 322as €as € -

1. Find Big 0 notations for the function f (2) 322 as € as € -



Answers

Find the function $f$ such that $f ^ { \prime } ( x ) = f ( x ) [ 1 - f ( x ) ]$ and
$f ( 0 ) = \frac { 1 } { 2 } .$

Yeah. So we have this differential equation but it's written in function notation and you could work with that but you're probably more used to seeing it written with X. And Y. So we can just do is replace this F. Prime of X with, do you have I. D. X. Since uh this is the derivative and and and replace the F. Of X with why? So basically we get um do you, Y by dx is equal to uh why? Times one minus why? Um And uh we're basically uh we're basically why is equal to F of X. And then in the end we can necessarily if necessary. A rewrite, why is F of X? But now we can work from here and this is separable equation. Um because we have anything that's in terms of why other than derivative on one side. But we also need to have whatever is in terms of why on the same side as um the derivative. So what we have to do is divide both sides by white times Y one minus Y. We do that, we get one over one minus why? Uh say Yeah, this I mean one over Y times one minus Y. Uh DY by DX is equal to one. We can we can essentially multiply both sides by um D X. Okay, and now we can integrate both sides. The right side is pretty obvious for integrating, it's going to be um that's gonna be X plus an arbitrary constancy. And on the left side on the left side we can do um the partial fractions method which you've learned before. Mhm. And if you were to do that um you find that this is this can be split into uh one over why and uh 1/1 -Y. Um And integrating that that you get one of absolute value i uh minus minus lawn of absolute value of one minus Y. No 1's getting people to ask, you know that equal to X plus plus C. And uh we can do log rules to combine the left side to get one of the absolute value of why? Uh Over one -Y. Yeah is equal to X. Let's see. Um and what that means is um absolute value of why over one -Y. We can raise both sides to E. And that means something that's going to be able to e. Uh the power of X plus C. And we can do is we want to take out sort of a we want to take out a constant from here. So we'll use experiment laws which tells us that's gonna be equal to E. To the sea times E. To the X. Um Okay and now no if if the absolute value of this is equal to this then that means that Uh the inside of that absolute value. So why over what one -Y. Um that's gonna be equal to plus or minus that right side. So we can take um that uh this part the plus or minus E. To the C. I need to see is a constant, right? So we can treat this is just one constant. Which we can call whatever you choose the textbook. You should call it A. So I do that and now we're gonna take the initial, we're gonna uh use the initial value given in the question which is fo zero equals 1/2 In place of why we put 1/2 in place of actual put zero. So uh we get 1/2 Over 1 -1/2 is equal to a times uh E. to the zero Each of the Zeros one. Yeah and 1 -1 is 1/2 and half over half of course is going to be one. So basically Is equal to one. Therefore we get that Y over one -Y. is equal to uh is equal to one times E. To the X. Which is just eat to the X. Yeah. Yeah. Remember we always want to um if possible uh rewrite this by isolating why? Um So now it's just a matter of rearranging to do to do so and often when we have a denominator early this we want to multiply both sides by that. And we get why is equal to eat the X times uh one minus y. Um expand and you get y. Is equal to E. To the X minus Y. E. To the X. I get why on the left side only. And we get why um Plus Y. E. To the X. Is equal to E. To the X. Factor out why on the left side why times one plus E. To the X. Because he could lead to the X. And our and then finally why is equal to ah E. To the X over one plus Z. T. X. And as I said before, questions originally written, sorry, that should be E to the X. Originally written in function notation. So we're going to just read for a final answer will replace. Why with that affects our final answer is F. Of X is equal to the to the X over one plus E. To the X.

So we have If the six people fixed into one minus effects on the condition, if you will. How ever these be upon there's off if X equal if x minus, it is blanks. And if zero is equal Therefore, we have the off FX over If it's one minus. If X equals good condition. Yes. So for that we have put affects people Way to use indication the way we're Why one minus one That is equal condition the X What we have won over why one minus y equals over when less be over one minutes away. And we have a sequel to when people a one minutes away That would be well on their way Have this morning when a B hence we have evolution one over. Well, glass integration went upon one minutes. Well, that it was too explicit. Si So Brady's Ellen less Ellen, one minute. Wait for word that expresses No. After this we have Ellen Welling minus Ellen one minus way that is given as explicit. Si Therefore me, why you were one minus y equals plus c for me, huh? Why, 11 minus y you will Do you know about express see that is even as you Parex into your body that is given as c you about us. No, What we have given the connection that is f D took over one minus f Gino equal to see you are zero that way. How over one minute half people to see our when Therefore it is 1/1 minus y people who see you know about us. Oh, yeah. When? Over one minus one. Equal your forex. So why does one minus y indeed Monix on one for minus my buttocks So far way have this equal going well, your parents over one place Politics This is answer to be given problem.

The question were given the f G thanks. It certainly could have. I'm the function g x and then we'll give e given any coach either one over x krampus one. And now we need to find a function evident that you hear here, notice that we see this on any stuff on DA rational function and it should think the farm the out of function here. So we should have the F accidents. A cogeneration and functioning under from one of our X. And now the inside is one. Go be the G X or again the GX Could you x squared plus one. That's gonna be one option for this time. I'm gonna completion.

Ah lg. Thanks. And it is dishonest. Cocina f I'm deficient. G ANC's and were given ico Judah one over X squared plus one. Yeah. We need to identify the function f action there from the ANC's Everything doesn't give me the summer The Jew functions so we can take this one with 1/2 here so we'll be the f X must be under form X plus one. And now this will be the function G act now. So have a G X. We called you one over X square. That's what we want option here.


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