5

2) Find a funetion f (x) describing the given griphf(x) =...

Question

2) Find a funetion f (x) describing the given griphf(x) =

2) Find a funetion f (x) describing the given griph f(x) =



Answers

Find the function $ f $ such that $ f'(x) = xf(x) - x $ and $ f(0) = 2. $

Okay, so we want to find f prime of a given after back. So we're gonna need after today for our equation. That's one is to a squared. Okay, so throughout this as the limits as X approaches a of a flex times ar minus upper basin, it's plus two a squared all over X minus a. Let's notice. Here in our new Morita, we can factor out, um, a negative to to get X squared minus eight, squared all over X minus a Let's note. In first term, that's the difference of two squares. So we get excellent. It's a time to expose if we factored out on let's cancel out our like terms. I know we can use decks up, so we get to minus two times a pills. A. So this is equal to negative two times to aim, which is equal to negative for a for our primal bay

So in this problem we are asked to find f prime at a when half of x is X to the -2. Well, by definition, if private A Is the limit as H approaches zero of F. Of a plus H minus F of a over H. All right. So that means that this is now the limit as h approaches zero F of a plus age and in this function F of X. So that is one over a plus H squared minus one over a squared all that is over eight. All right. So that means I have a limit as H goes to zero then of what? When I get a common denominator in the top. All right. And so I'll have a squared coming on top and then combine the fractions. That means I'll have a squared minus a squared plus two. A H plus H squared all over a squared times A plus H squared times H from the from the denominator previously. All right. So like this a squared minus at a square is but those two are gonna be gone, aren't they? So that means I have the limit as h goes to zero of minus to a H plus H squared over a squared times A plus h squared H. But I have an H in each term in the numerator so I can cancel one of those out. Can't I like that? So, that means I now have the limit As a joke goes to zero of minus to a plus H over a squared times A plus H squared. Okay, so taking now the limit as H goes to zero in the numerator, I'm just left with minus two way, aren't I? And the denominator. I have a squared times A squared, right? Because H goes to zero. So A plus H just goes to a square that let's just a square. So that means I'm left with -2 A over eight of the 4th. Well I can cancel 1, 1 of those AIDS out. and so that means I'm left with -2 over a cute is our derivative If prime it ain't.

Okay, so we're giving our falling function Selects a new way to find a prime of it. So looking at this equation here, we're gonna also need enough of experts age. So that's going to be exposed. Age to depart to. And let's expand this. We get X squared plus two times x times h h word. So let's rewrite this as the limits as H approaches their own of X squared plus two times x engage post each word minus off of exes. Riches minus X squared all over h looking council water like tens and all its notice. In our numerator, we have an ancient Coleman. So let's factor that out of age and in two X um, plus h all over age. That's cancel out that h who is he here that are derivative is going to be two x plus h Get on. Wait, Let me just double check this. I think I wrote or derivative wrong. Oh, no, actually, it's OK. Okay. So this here is our driven

Okay, so it's find f prime of a given effort back. So for our equation, we're going to need a vein so magical to eight squared. So to rewrite this as the limits as X approaches a, uh, X squared minus a squared all over X minus a. Now, let's know, in our new murdering we have a difference of two squares so we can rewrite this as X minus a times expose, aim. Let's cancel out our like terms and I'll use Indrek sub. We have a plus A so f prime of a is equal to two a.


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