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Find the Rate of Change of the function 10 flx,y)-x? +3xy pts pts at the point (3,2) in the direction of the vector v = 3i + 4j.Find the Maximum Rate of Change of f...

Question

Find the Rate of Change of the function 10 flx,y)-x? +3xy pts pts at the point (3,2) in the direction of the vector v = 3i + 4j.Find the Maximum Rate of Change of f(x,y) at the point (3,2). pts

Find the Rate of Change of the function 10 flx,y)-x? +3xy pts pts at the point (3,2) in the direction of the vector v = 3i + 4j. Find the Maximum Rate of Change of f(x,y) at the point (3,2). pts



Answers

Find the maximum rate of change of $ f $ at the given point and the direction in which it occurs.

$ f(x, y, z) = x/(y + z) $,
$ (8, 1, 3) $

Given these shown information and asked to provide the maximum rate of change of F and the direction in which that maximum rate of change occurs. And so our first step in doing this is going to be to find the partial of F with respect. You asks the partial respect a why in the partial, with respect dizzy so the pressure with respect to X it's going to be the following if we treat y and Z is are constants. And we think of this as x squared plus y squared plus C squared all in parentheses to the 1/2 the derivative is going to be one, huh? Times x squared. That's why squared closes e squared to the negative 1/2 times the derivative of the inside. There's gonna be two X and now my process. We're stuck to why and with respect Izzie are going to be the exact same thing. Except this is going to be a to why, for the partial, with respect to why and it's going to be a to Z for the partial, with respect dizzy. So I will just write those out and I pressure with respect, dizzy and what we're going to do with these is plug in our point into the partials and that, actually that Dr will give us the direction in which the maximum rate of change occurs. Okay, so if we plug in the 0.36 negative, too, we get the following 1/2 times three squared is nine six square is 36 36 plus nine is 45 z squared is four and 45 plus four is 49 so we get 1/2 times 49 to the negative one heart. That's one over the square root of 49 or 1/7. So we get 1/2 times 1/7. That's one of her. 14 times. Two times three. That's six over 14 which is equal to 3/7. Now for our partial with respect toe, why, we're gonna have the same thing in the parentheses. And so we're going to get 1/7 times 1/2, which is 1/14 we're going to get to time six. This time just 12/14 which is 6/7. And again, we're gonna get the same thing for Z At least the 1/14 part. So we're going to get 1/14 times, two times negative. Two. That's negative for over 14 or negative to over seven. And again, we're going about this in a vector. And that will be our direction in which the maximum rate of change occurs. Three sevens, 67 and negative, too. Seventh, another way in which we find the maximum would have changed is that we take the length of the vector we just found. And so we take the leg. Um, three sevens 67 negative. Two sevens. When we do that, we take the square root of the some of each of the component squared. And so we're gonna take three sevens squared plus six seven squared, plus negative to over seven squared. And so we get 9/49 plus 36/49 plus 4/49. And that's going to be 49 the square root, I should say, of 49/49 which is the square root of one, which is just one, and that is our maximum rate of change. And again, I got this by taking the Grady Inspector up here, which is the direction and taking that victor's length to get the maximum rate of change

Find the rate of change of f. at the .12 in the direction of this. You Okay, that's just asking the same thing. It's been asking you directional derivative. So first we have this you um if I wrote it as fractions, this is 6/10 which is 3/5 And this is 8/10 which is 4/5. And you can see that that has magnitude one. So it already is a unit vector. Next I have to find LF, which is two X. Comma to why? Now I have to find del f. at the .12. So it's 2 4. So the directional derivative of f in the direction of you at the .12 is to four Dotted with 3/5, 6/5 plus two times +364 times a large. 16 5th 22 5th Or 44 tents or 4.4. Any of those.

Were given a function F. And appoint and rest to find the maximum rate of change of this function at this point in the direction in which it occurs. The function is F. Of X. Y. Z equals X. Natural log of Y. Z. And the point is 1- one half. Win a bagel. That's stupid. Uh Do this by this. Uh We're gonna get a fucking fan or some shit. Dude. Now find the maximum rate of change. Let's find the greeting to our function F. This is the vector's components of the partial derivatives of F. So we have the natural log of Y. Z. We have X. Times Z. Over Y. Z. And X. Times Y over Y. Z. No dude, I can't wait to do mullen media's next big purchase. Dude, this simplifies to natural log of Y. Z. X. Over Y. And X. Over Z. Therefore the greeting to our function F. At the 0.1 to one half is equal to the natural log of one which is zero 1/2 or one half and 1/1 a half or two. The essex white bar stool, therefore, we know from the section at the maximum rate of change of our function at the 0.1 to one half is given by the magnitude of the gradient of our function at the 10.1 to 1 half, which is the square root of zero plus 1/4 plus four, which is Cleveland route 17 Over two. And this maximum rate of change occurs in the direction of the gradient of F at 1 to 1 a half, which we saw was equal to 01 half to. So a maximum rate of changes route 17/2 in the direction of 01 half to

Yeah, it was this function X squared y cubed. And the question is at the 0.1 negative negative one. To find a vector in the direction of maximum change of F. In minimum change of F and a zero change in F. All right. So that means we need to find the gradient DLF is two XY Cube and three X squared Y squared. So dell f at the 0.-12 is negative two times eight, which is negative 16 and three times four which is 12. Okay, So the vector in the direction of the maximum change is a vector in the same direction as Del F. So An answer to part A is -16. I plus 12. J. Okay. A vector in the same direction or as Del F. But you might want to make it into a unit victory. Don't have to I don't think. But here's what we have minus 16 square plus 12 square. I'm trying to find the magnitude 1st 12 squared. Okay, I'm not doing that. 16 is four times four. So it's four squared times four squared plus 12 squared is four squared times three squared. So that's four squared times four squared plus three squared. Can really I did all this in my head for square times five squared four times 5 20. Because I did the factoring first before I put it in the square and then I figured it out. So Okay, so the unit vector in this direction would be minus 16/20 which is minus 4/5 and 12/20 which is 3/5. Okay, I don't know. It might have been easier to get my calculator. What do you think? Okay, so that's the direction of maximum entropy increase. Okay, so let's say that's this way Then the direction of minimum increase is 180° from it. Okay, so Then the answer to that one would be 4 5th Common -3/5. Okay the first one went I did it backwards. I drew the pictures backwards. I got the right answer. 1st 1 went -4 5th um negative x. Positive way. So here's the original you that's the direction of greatest increase. And then the direction of minimum increase would be this way positive in the I direction, negative in the J. Direction. Okay And then see which direction would give. You know change. Well it would be whatever direction is that when you dot product with this you get zero. Okay because that's when that casa minus pi over two or three pi over two. Okay so it could be 3/5.4 5 Or it could be that would give you negative 12 plus 12 Or it could be negative 3 5th negative 4/5. Okay because I would give you 12 minus 12. Yeah.


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