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F() Verify Find the thhe 2x 8 - 7x U 2 function "and 1 or it in 1 the form Y {...

Question

F() Verify Find the thhe 2x 8 - 7x U 2 function "and 1 or it in 1 the form Y {

f() Verify Find the thhe 2x 8 - 7x U 2 function "and 1 or it in 1 the form Y {



Answers

Write the function in the form $y=f(u)$ and $u=g(x)$ Then find $d y / d x$ as a function of $x .$ $$y=\left(1-\frac{x}{7}\right)^{-7}$$

Rest of partial derivatives or the X and Y given Bigon right here. Let's do X first. That's the parts derivative with respect to X. Means we're treating lies a constant so drivable Why square that's gonna be zero derivative of two X to the sixth Little to use the power all it's gonna be 12 to the fifth. Now we have a plus two x y for treating. That's why the constant this is gonna be plus why now to take the derivative with respect to why we're going to use the power rule here that's gonna give us to why they ignore this to extra the sixth. Because at the constant, finally, we have two acts. Yeah, so these are two parcels.

Okay, Let's say we have the function f of x equals two X -7. What we can do here is take a look at the graph And decide if this is a 1-1 function from its graph. Look at the horizontal line that I'm swiping through this graph. Never does it intersect that horizontal line more than one time. Therefore, we would say this is a 1-1 function. Yes, this is a 1- one function.

So if we want to determine if this is 11 tau one or not, what we need to do is plug in some arbitrary values A and B, and then see if a and B must be equal to each other. Um, in this case is we have a quadratic. We know the shape of a quadratic is like a you, so it would fell the horizontal line test. So what you could do is really just find two values where they are different, but they have the same output. Eso just kind of often the side. All right, this But I'll do it more rigorous is Well, um, because we could essentially just do like f of negative one. And when we plug f of negative one in. So that would be, uh, one times two minus seven, which would be negative five. And then notice F one is also equal to negative five. So these two are equal to each other, but negative one is not equal toe one. And just from this, this implies ffx is not want one. So you could just do that. I mean, this is essentially what you would normally do. You just need to come up with one counter example. But if you have a hard time trying to find this, I'll show you another way how you may be able to come to the same conclusion as well. Um, so let's just go ahead and plug in A and B. We have f of A is equal to F A B, so that would give us two X squared minus seven is equal to hope not to x to a squared to a squared minus seven is equal to two B squared minus seven. So if we subtract seven and then divide by two on each side Um, notice we didn't know what a squared is it to be squared. And so now, at this point, if we were to take the square root on each side, then this is going to give a is equal to plus or minus B. So this here contradicts the fact that a should be equal to be because we can have where a is B or negative B. And so this is another way you can go ahead and say Okay, which implies AP of X is not 1 to 1. So either of these ways is a sufficient way to show it. It just kind of depends on which way you think is the better way. So I guess I should say like way one and then over here this is way too. But I mean, normally, if you confined to numbers way one is the better way to go about doing it.

Hello. So here we have artifacts is equal to. Well it's written as X squared minus seven X minus eight, divided by X plus one. Look at the numerator here, this is definitely a factor. Right? So this is going to be equal to x squared minus seven X minus eight. That factors as well. Um X squared and X times X eight. Well it could be four times too. But I mean to a seven in the middle of the day we do eight times one and then you get to a minus seven, we need a minus eight and a plus one that gives us X squared plus x minus eight, X is minus seven X and then minus eight. So X minus eight times X plus one is x squared minus seven, X minus eight. And then we're still divided by X plus one. But now we have a common factor here of X plus one. They're going to cancel it become one. So therefore the function really is just equal to ACS -8. So here's afro backs and then we find the first the derivatives for the first derivative f prime is going to be equal to just um just one. Second derivative of X is just one and derivative of negative 80. A constant threat of a constant is zero. The first derivative is one and that means the second derivative is going to be zero. The address of a constant is zero. And once we had zero, every derivative after that is while it's still gonna be zero. So the third derivative Is also going to be equal to zero.


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