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Considerar Ia siguiente grafica Y f(x):Y = flx)iDonde ~es positiva?A. (_0,-2)U (1,3) B. (-2,1) U (3,0) C. (-2,1)U (3,~0) D. (~0,4) U (1,4)...

Question

Considerar Ia siguiente grafica Y f(x):Y = flx)iDonde ~es positiva?A. (_0,-2)U (1,3) B. (-2,1) U (3,0) C. (-2,1)U (3,~0) D. (~0,4) U (1,4)

Considerar Ia siguiente grafica Y f(x): Y = flx) iDonde ~es positiva? A. (_0,-2)U (1,3) B. (-2,1) U (3,0) C. (-2,1)U (3,~0) D. (~0,4) U (1,4)



Answers

Find the function values. $$ f(x, y)=\int_{x}^{y}(2 t-3) d t $$ $$ \begin{array}{ll}{\text { (a) } f(1,2)} & {\text { (b) } f(1,4)}\end{array} $$

We are given the equation f two X squared plus three, and we have to figure out the derivative of the function. We have to use the limit. We have to find the derivative using the limit definition of there is and and limit deaf initiative. Limit felt extends to zero of F of minus F over thanks, and we have to replace X in the equation with explosive belt living. Delta X tends to zero of X plus still toe squared, plus three times x plus l to act minus the entire equation. X Squared three sees over don't so we can expand this out. Exposed out X squared that is equal to limit belt extends to zero. We'll expand all this out, will distribute the three the X and dealt actual distribute this negative sign and will expand out this So the X plus Delta X squared is going to be where'd plus two plus Delta X squared. Then we have the plus three x us three felt minus X squared minus three over. Don't that and we have some similar terms. We have X squared and minus X squared. Have three x and minus three. So we're left with you're left with Limit Felt Extends to zero two x Delta three Felt plus Delta X Square over Delta X Notice how each term and the numerator has felt ECs weaken Factor that out. B zero limit belt extends to zero factor out Delta two X plus three Love So over felt we can cancel out the Delta and the Delta X is we could justify this because you might say this you're dividing by zero. Delta X is approaching zero, which implies were therefore felt X does not equal to zero. So we're left with Limit has felt extends to zero of two X plus three plus. And how do we evaluate this? Well, we have dealt access to zero, so Delta X is essentially is going to be zero, So our derivative is going to be two x plus three. We're also supposed to evaluate our function so F Prime X is equal to two x plus three. Supposed to be evaluated F prime at two numbers. First one's negative one. So that's just two times negative. One plus three is equal to minus two plus three equal to one and then evaluated at four. That's equal to two times four plus three vehicles to a plus three, which is

Okay, so we have this vector field. Uh huh. F is equal to x squared by cube as I hat plus X. Cute, Y squared times she hat. And so were given that in this problem, we already know that F. Is a vector uh field of a. It's a great field of dysfunction little. So when we want to use this fact to evaluate the integral oversee of F dot er where um this path is R. T. This C path is to find our T. Is equal to the factor T. -2 T. And then T cubed plus to see Where we know that she is between zero and 1. So just looking at what our vector, our position vector of our path is, it's pretty complicated. We have this take you back to T and then it's just X. Of T. And Y in T. We know that we're going to have to plug this into this equation and then get cubes Squares. This is a total of five powers. So we end up with um T to the 15th and that is just too complicated to have to deal with. It's gonna be a crazy polynomial. So we want to use to my advantage of the fact that we know that F is this gradient field. So this means that if F is a gradient field we can use this fundamental theorem. We know that the fundamental fundamental theorem of line intervals, fundamental theorem. I was writing that. So fundamental. You're for line any rules? This tells us this is what we're going to use. We're going to know that uhh since F is this gradient function we could take this integral over this path. We know that this is just Eagle two. Um The F evaluated appoint one. So if we call this P. R. Sorry, the function evaluated 10.2 minus didn't 0.1. The function evaluated 0.1 IFC is this path from 0.12 point two. So in this case we can use this to our advantage. If we just know what this function F is. So what we want to do is look at the vector field F is equal to x squared white cube. I had plus that's where executed Y squared J hat. And we already know that this is just partial F partial X times I had plus partial F partial Y I'm she had so we can set these equal, we know partial F partial X is equal to expert like cube and that partial F partial Y is just X. Q. Black square. And if you go ahead and integrate we can separate both of these to get the following. We have executed by squared partial of wine. We integrate both sides of both equations. Mhm. We end up doing the experts, we have F is equal to um we have X squared which were integrated as we have one third executed Y cubed plus a function G of Y. And then on the right we get A physical to now we're integrating wise we have 1/3 x cubed y cubed plus any function H of X. And like we've seen before these are identical except for this G and H function. So we just know F Is just 1/3 execute like you plus a constancy and now we don't need we don't need to know this constant because if we're doing this in a rule, we know that we're trying to solve this. So first we know that this is um the answer to part A. Because we want to find this function and then finally part B, we wouldn't evaluate um this integral F D. R oversee but we know that this is just the gradient function, the gradient field giving us simply with f of 0.2 minus f 4.1. And so we want to find out what P one p 2 r. So earlier like we said R. T. Is this vector field or this vector T. -2 T. And to T Q plus two T. And if we evaluate this from T +210 we know that th zero represents 00.1 TF one representing point represents point to. So then we know that um P one is just are of It's whatever the coordinates of our of zero R. And that's equal to zero comma zero. Just played in 0.2 is just looking in one. Sorry We are one. Yeah. And so we plug in one. We see this just 1 -2, It's a native one and then one plus 12 which is um positive three. So we know that .1 is just 00 mm and then point to its just negative 13 so we can go ahead and just do this. Uh The integral of F. D. R. Is just F. Of negative 13 minus F. Of 00 And if F is this 1/3 execute like Cube. Let's see. We'll see that we have um one third Native one Cube 3 cubed minus. Uh Don't forget the plus seed minus one third zero cube zero cubed. Um pussy. And we'll see that this is canceled so that we didn't have to know in the first place and this is also equal to zero. So we have, the distance is equal to negative one and then Three cubed over 3" negative as it's nice we have negative nine as the value of this integral.

Hi it's even after three dimensional in real value behalf the offense has given us, Let's hear from 0 to 1 ethics be it and find out that. So I just work on it. Just take here I speak yes do not to empty building we're going to find out uh if exposed and B. S 0 to 2 T. Activity technology we have affects the world war two plus. Mhm expert a plus week ignore him. So then I integrate from the door to yes you know to to They have to be now there are two FX years or they want to operate in the same thing. So that's gonna have to be a it will be have called it. There's going to be four plus eight. A simple integration over three plus to be this bad boy. Next. Uh So by solving this we get pipe Last six ft will negative. Next what we can do this multiplying or write the equation And x equals two plus liquid in one fx equals two plus. uh we have expert like the model I buy active So you never expect that person two weeks next time to then next time it wouldn't be. And we integrate now with respect to X. O. We go from the first that is the zero to uh be a political to be So integrating with respect to act we can be equals four plus four A plus to bangalore to indicate Do we get 40. Love be reported plus B. Is equal to from here for it to be implemented if all this were not here. So because for all their two equations. Question one, Question two, we have the values is negative. Well over 19 and these -28 or 19 Upopia. Now we have ethics coming out to be maybe not the value of the NV two negative negative like square Over 19 -28/19. There'll be 10/19 negative. Well explain over 1997 integration. Little to one FX to it. So You figured this year or 2. 1 10/19 negative. Well over 19. Expert, be it people integration with all of this and you're down to rest it. Forward 1990s Choice A Thank You.

We know that if we have a continuous function that we plug one into the top and bottom pieces of the function to get the same answer. So eight times one cubed minus six times one his B times one squared plus four we get a minus six equals B plus four. Now plug one into the derivatives of the talk and bottom minus. Sex equals to be times one. The three a minus six equals to be solving both equations. We end up with a equals negative 14 as our salute.


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