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Let f(-) =2Uaetthc Sccond Drritarive Test ta lind alllucal €lrcI Fin ahcinicni"hen (iconchve Lnd Ihc itctvul "hric( D (UIAJC Latn Utcecng Vour mnten...

Question

Let f(-) =2Uaetthc Sccond Drritarive Test ta lind alllucal €lrcI Fin ahcinicni"hen (iconchve Lnd Ihc itctvul "hric( D (UIAJC Latn Utcecng Vour mnten incuu:lity notatn mecralnetntion nic lulct Intolv Hntmen nrcunl Uc Meilucet incutcicet Mud MI iniecchoi Mtiuaitu Fuuplui, cureiu [Mce Itauutaiawci cuctt elut explalns tuw > [ now Ihat Ihuse Inllectiol mulnte Usc Ihc [Xenxn (haphing Cakculat wcbiile to Inakc # pruph 084 Fut (h eexyeli nule ' Ilie lechl ettieini Mkl unllacIl mallla

Let f(-) =2 Uaetthc Sccond Drritarive Test ta lind alllucal €lrcI Fin ahcinicni"hen (iconchve Lnd Ihc itctvul "hric( D (UIAJC Latn Utcecng Vour mnten incuu:lity notatn mecralnetntion nic lulct Intolv Hntmen nrcunl Uc Meilucet incutcicet Mud MI iniecchoi Mtiuaitu Fuuplui, cureiu [Mce Itauutaiawci cuctt elut explalns tuw > [ now Ihat Ihuse Inllectiol mulnte Usc Ihc [Xenxn (haphing Cakculat wcbiile to Inakc # pruph 084 Fut (h eexyeli nule ' Ilie lechl ettieini Mkl unllacIl mallla



Answers

Given $f(x)=2 x^{2}-3 x,$ determine: $f(-1), f\left(\frac{1}{3}\right), f(a)$ and $f(a+h)$

Okay, so we're given the falling rational function and were asked to find for part a f evaluated at zero. So in this case are X value is zero. So what we do is for every X file in our right hand side we replaced, it was there also have two times zero plus 1/0 squared plus 3 10 0 minus four. So that used me one over negative floor as my solution. Okay, so we also have part B b is f evaluated at two, which is equal to two times two plus one over you have X. Here's that's two squared close three times to minus four. Okay, so two times two is equal to four, and then we add one. So that's five on top. Now we have two squared. That gives me a four and in three times to six. So that's four plus six minus four. He's to Ford's cancel, and enough with a six. So a solution in this case is 5/6, and lastly, we have part See, which is F evaluated at one. So we get two times one plus one over X, which is again one squared plus three times one minus four. All right, so we have two plus one on the top, so that gives me three. You have one plus three minus four. So we have What is this? This is four mon ish for which is equal to drill. So we have 3/0. That means that our function of elevated out one is on the find.

Always consider this function. It affects T. Which is uh that -2 Squared of T over one plus. So I want to find those derivative. I mean just like the last two terror before this one, we're just gonna consider anything that is not X as a constant and anything that has an X. As a variable. So this one is just gonna be, you know, square root of X can be written as excellent power one half. And the derivative is going to be one half excellent, negative one half. So that is a derivative of any squared effects. So this one is going to be just X to the negative 1/2 because this is the derivative of the screwed effects. And then whenever you multiply by two, it just takes away this one half. Right? So this is what you have then minus. Oh, I beg your pardon? Uh I'm going to use a coaching role. Okay, I got carried away by this derivative thing. So it's just gonna be one plus. In fact we're not going to use a caution rule because the denominator here does not have a annex. So it's just a constant. Right? So what we're gonna do here is separate. This this uh fraction. We're going to do This over one plus The -2 T. Over that. Yes, this one is a total constant because it does not contain an X. So this one's derivative, this is going to go to zero. Yeah, I mean that is still this right? This this one is still this one. I just separated it. Okay, so in the eyes in the context of uh partial derivative of the function of respect to X. This one does not contain an X. Does not have any X. Here. So it's just a constant. So the derivative is going to go to zero. So we're just going to concentrate on this one. And I, like I said, the numerator contains the next and its derivative is excellent. Negative one half over one plus. Right? Because uh square root of X has this Esther as the derivative. And I want to really multiply by two. You have justice. Right? So this is what you have. Another person can want to put it in this form. Yeah. Right, That is entirely up to you, right? Or you can make it this way. Again, one over square root of X plus two squared of X. T. There are many forms you can put it right, This is correct. This is correct. This is correct. Right? Any which way? No, I want to find a partial derivative with respect to uh T Okay, now, since the denominator contains a T, it would not be too wise to split it up like that. You can just consider it in this compact form. So since the denominator has T. We can now use a question rule. So the question rule has says to have the denominator squared. So this is gonna be one plus two squared of T. Square right? Because the denominator contains city and we're taking partial to a bit of a respected T. And so loaded high is going to be one Plus 2 Description of T. Now this one is saying is completely a constant. So it's going to go to zero what is to part of what is a derivative of this one? Well you know this squared of T. Behaves like this thing as soon as I'm gonna put team place effects. So whenever I multiply by negative two all that I have is negative T. To the negative one half. That's all I have then minus loaded. High. Hi. Hello actually so you maintain the numerator and then you differentiate the denominator right? So that is also gonna be T to the 1/2. Right? So you got to clean it up a little bit. So this is gonna be okay. Uh You know negative one plus That teach the negative 1/2 in minus. Uh You know Now this teacher 1/2 is the same as the square root of tea. So I'm just going to have XT right? Because this square root and this is square root of X. So it's just gonna be squared of X. T. Then minus plus because this is a negative. So plus do now this is screwed of tea. Time squared of tea is actually gonna be T. Right. You have a script of T. Times square root of tea is the same as squaring of T. Square. And the square council cancels the the square root and you have t. Okay that is what is happening. So you have this one then over The denominator which is one plus. That's okay. So uh yeah you can pretty much leave it like this. This is not square root of tea because it is not t to the power one half. It is t to the power negative one half. So it's basically going to be this over that. Okay, do not get confused. T to the power of one half is square root of T T to the power of negative one half is one over the square root of T. They're not the same. Okay. That is why I couldn't take this one to foil. Right? Because technically is going to be under is going to be over square root of tea. That is what this one means. Okay so you can just multiply Yeah uh but what you can do is I mean whatever you foil, you're going to have negative t to the negative one half minus two because this is teaching at one half and this is teaching the negative one half. So whatever you multiply, you're gonna have just to to the zero which is just one in minus XD plus two T. Okay. Yeah, so basically this is what you have and the notably one plus two squared of T uh squared. Ok, so you have this one? Yeah, you can make it, aye, aye sir or you can live it this way, any which way is fine. Let me ride those two here. Uh You know whatever you try to make it nicer Yeah, let me I want to make it nicer. A little more compact and easy to understand this thing right here. Okay, is the same as one over square root of two? Like I said, so one over square root of T. What I'm gonna do is bring this one to the bottom so I bring it here. But when I do that then I got to do a few things and I got to multiply each and every term here by square root of T. So this is gonna be negative. Okay. Yeah negative one because there's gonna be this negative one on top. I just brought this script of teacher denominator. So I have negative one. But here is gonna be negative two squared of tea. Right? That is the small thing I gotta do when I bring this thing to the denominator. And then this thing is gonna be negative primitive X. Because yeah I want to write multiply uh this by screwed of T. You're gonna have square root of X. Squared of tea time squared of T. Multiplying this thing by square root of T. And this spirit of T. And the spirit of T. Is going to be T. Actually. So uh what's you gonna have is justice this tea? Okay then. Plus two. Whenever I multiply this one by squared of T. I get T. To the three of the two. Right? Uh There's a little mistake. This is supposed to be negative one half, you know? Uh Yeah let's go through it again. This is protecting the partial derivative respected T. Okay so we're gonna use a caution rule. The question rule is going to have this denominator here as squared. So that's what you have here. Now the numerator is going to be. You maintain this one as against taking the derivative of this one. So that is what you have here. And the derivative of the numerator is just negative T. To the one half. And then minus you maintain the numerator and to the review of the denominator. So the numerator is this guy. And then the derivative of the denominator is supposed to be negative T. T. to the negative one half. Right? Because the denominator here is going to be the derivative of the derivative of this. Denominator is just gonna be T to the negative one half. Because chest this one is one half T to the negative one half. And I'm multiplied by two. Then it takes away the so it's T to the negative one half. Okay, so that is a little correction I wanted to make here. So yeah that is an important development. So then we just have to uh Revised this one a little bit. So this is going to be okay, You know, two square root of X -2 squared of T. Uh Let me put it this way, one over square root of tea. This tea to the negative one half is the same as one over square root of T. So this is gonna be the same thing. One of the spirit of tea. Okay, so now I can safely bring this thing this script of T to the denominator without having to make any fuss. So when I bring it to the denominator then the numerator is gonna stay the same. The numerator is gonna be negative one plus square root of T. Two squared of T. And then minus. Okay I can bring the denominator down because this and this are common. So I can just factor it out and then bring it down. That is why I have this one. Now the new, what is left is gonna be just this portion and just this portion right, descriptive. These now at the bottom here and now I'm going to foil right when I foil this is going to be negative one minus two when I'm for this one is gonna be negative two to the X. Plus three to the Uh two times screwed of T. Right? So two times creative teeth negative negative is gonna foil. So you're gonna distribute this negative to these two terms and you're gonna get this right and then you distribute in this negative to these two terms as well. So you have this one no negative to to the script to times square root of T. Plus The positive one is going to go to zero right? It's like having two times creative team minus two times creative T. So you just have negative one minus two squared effects and over screwed of T. One plus twos. Yeah, this is what you have. So that is the partial derivative with respect to T.

Okay, so given falling function and rest defines FMLA, that negative one that's two times thinking one squared minus three times they won. What does that give me? That's too plus three, which is equal to five. And then for next one. We went after Bali to God's well today. That's two times 1/3 squared, minus three guys, one of our great. So we get to overnight minus one or one is fine overnight. So that, um, negative seven overnight. And for the next one, we went after Valerie to. That's a So we're just gonna replace, excavate and then after Valley that a plus H that's two times a plus h squiffy. When is three times a plus age? Okay, so let's multiply out our A plus h. So we get to times a a squared plus two h and n plus eight square one is three a minus the age that's playing out to get falling. Okay, so let's simplify. Um, yeah, it seems like you can't simplify anything further, so we'll just leave it. That's it.

So for this question, we are given a function, and we just have to find that function at different X values. So let's start with this. All the way to get have to do is plug in a negative one for X for the first problem right here. So we have two times negative. One squared minus four times negative. One plus two. Right. So you have two times native one squared minus four times a night of one plus two. When we do that, we're just going to get about eight. Next, we have two times zero squared, minus four times zero plus two. Right? Said the zero. Anything Look by my zero is just gonna be zero. So that's going to have a value of two. Then we have a value of one. So we have two times one squared minus four times one, which is for plus two. Right, so we have two minus four plus two. That's just gonna be equipment to zero. So we just go ahead and just double check that on my calculator, and in fact, it wouldn't see it. So the next thing we have is where to plug in the values for the value of to write or wherever. Exes. So you have two times two squared, minus four times two plus two Dustin unequal shoes and two times two squared is 44 times to his 88 Minus eight is zero and some Cavs era post to which is to last week, would have to fucking about Europe. Three for X. And when we do that, we're going to catch two times three squared minus four times three plus two until we get three squares. Nine million times two is 18 18 minus 12 6 and six plus two is eight. So here we have now our values are you hear the x and y gone. So when we graph it, you can go ahead on and use a graphing calculator or we can just grab the plates. So I'm just going to insert an image of how the graph is until look. So here we have a craft that was slower, smaller. So this is how the broth looks right. So if we want to find, all really have to do is plugging the Prince of X and Y's, for example, when we have negative one right, we have negative one. Um, ate some is going to go ahead and draw a little bit more. Here we have negative one. We're gonna kind of drug a little bit more here. So at eight, we're going to have a negative one, right? Sure. That'll be like right here and then for zero. We have that. It's too, which is right here. Then we have when it's one right, we have a value of zero when X is too. We have again. That is too. And when X is three right here again, it's going to H just kind of draw that. So this is how the graph is gonna look using a plot. The point. So the delay in and breach right? If you wanna look at the dome Ian and Range of Onley our table Ah, we're gonna use thes bodies, right? Negative. 123 But if you look at the domain and range for the entire function, let's look at how it works over the domain X right. It's going to be negative. Infinity to positive infinity because X is always going to continue going in the right direction and the left direction. It just keeps on going now for the why, right? For the why we have that the highest. It doesn't go lower than the X axis, right? You know that why it's going to be equal to zero or it's going to be greater than.


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