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PreviousProblem ListNextpoint) Suppose f (T,y) = %,P = (-4,-3) and v = 4i + 2j A. Find the gradient of f. Vf Note: Your answers should be expressions of x and y; e....

Question

PreviousProblem ListNextpoint) Suppose f (T,y) = %,P = (-4,-3) and v = 4i + 2j A. Find the gradient of f. Vf Note: Your answers should be expressions of x and y; e.g: "3x - 4y" Find the gradient of f at the point P (Vf) (P) iq Note: Your answers should be numbersC. Find the directional derivative of f at P in the direction of Duf Note: Your answer should be a numberFind the maximum rate of change of f at PNote: Your answer should be numberE. Find the (unit) direction vector in which th

Previous Problem List Next point) Suppose f (T,y) = %,P = (-4,-3) and v = 4i + 2j A. Find the gradient of f. Vf Note: Your answers should be expressions of x and y; e.g: "3x - 4y" Find the gradient of f at the point P (Vf) (P) iq Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of Duf Note: Your answer should be a number Find the maximum rate of change of f at P Note: Your answer should be number E. Find the (unit) direction vector in which the maximum rate of change occurs at P Note: Your answers should be numbers



Answers

Consider the following functions $f,$ points $P,$ and unit vectors $\mathbf{u}$. a. Compute the gradient of $f$ and evaluate it at $P$. b. Find the unit vector in the direction of maximum increase of $f$ at $P$. c. Find the rate of change of the function in the direction of maximum increase at $P$ d. Find the directional derivative at $P$ in the direction of the given vector. $$f(x, y, z)=e^{x y z-1} ; P(0,1,-1) ;\left\langle-\frac{2}{3}, \frac{2}{3},-\frac{1}{3}\right\rangle$$

In this problem it is asked to compute the great yet off a given function s and then he valued it at a given point. B Grady it off f that this DLF is it? Why do and fix? It's like have said that this part should anybody off have respect to eggs? Partial derivative off F with respect to why I'm partially native off f with respect Tuesday the given function f is X minus a divided by by minus said So affects will be one divided by by Mina said at five will be minus X minus. Is it divided by by minus? Said squid, And I have seven b X minus y divided by by minus dead squid This company, Dennis, we can take one divided by by minus eight common outside. So this will be I head minus X minus it divided by the square. No, we have taken by line. Is that Koeman outside? So this will be on Lee by minus, said Jay ahead plus X minus by divided by by my nose ed G head. So this is our required grading. Now we have any value it at a given point. B three to minus one. So we will substitute X equal to three. Why will do? And they're equal to minus one so that they left egg three to minus one. Do you think will do one divided from my likeness. Say that this one divided by two minus minus one. So one divided by two plus one that this day I hit minus X minus their means three minus minus 123 plus word food divided by by minus eight is three j head less X minus Y means three minus two. Like this one divided by my line is it is three that this one divided by three gay head that this they left back to give that point is one divided by three common outside my reply but I had minus 40 my day by three j heck, that's one divided by three k head. No, The next question asked ISS to find the unit vector in the direction off. Maximum increase off act be and we know the direction of maximum in Greece. Off f isn't the direction off del f that this? We have to find the unique factor in the direction off day left so unique factor is equal. Do del f divided by the magnitude off the left to find this first of every fine, develop the value off Meg. Did you love the lift that this magnitude off the left is equal to? I'm gonna root coefficient off. I had that This one battery. It's square. So it this one by name coefficient of J head is minus four by nine. So it's square is 16 by B one. Bless coefficient off head is one by nine. And so it's square is one by 81. So this is a quite do. I'm done. I would 26 by eight people. I'm so this is equal to 26 by route 26 by nine. This is the man need Yudof. They left. So now we find the required unit Vector. Our day left is one by three I minus four by three j plus one by three Gay divided by my nature of Dell If we have found Toby Route 26 divided by nine. So this will be three ones on three on three trees are nine. So this is equal to three divided About what? 26 I hit minus four by three j head Last one Pipe leak A head This is the required unit vector in the direction off maximum Increase off f and think evil point B Now the next Kocian asked is 25 The rate of change off the function in the direction off maximum in Greece would be the date off. Tinder function in the direction of maximum increasing B is the magnitude off they left the tree found is equal to root. Prentice's by name So rude buying this expert nine is direct off. Change off the function in the direction off maximum increase on the given point B Now the next question asked is to find the directional very effective at B in the direction off. Given factor one by three to buy three minus go by t This is sticky bun meto let us the note this given vector by uba So now we find direction or Daddy will do as we know the formula for directional Daddy Wake do 40 function, I think given point there The key one point is three to minus one So and dickey 1.3 to minus one in the direction off Piggy went back to you. He's sick. Will do. They left at that point B that this left a three to minus one dark project meeting given back to you north that this given back then you must be a unit vector. If not, be herto transform it to the unit Vector. But here, the given actor you is itself a unit vector. So no need to change it. So Viva substitue now venues They left a three to minus one. We have found hit in this but by three minus four by nine. But by night now it's dark product. Big our back to you. But that this one battery No, by three and minus two by three. So this is a quite do. But by night minus eight by playing piece event minus two by 27. So this is equal do three minus eight minus two, divided by 27. That basic will do mind the seven divided by 27 which is our require directional directive And think you are going to be in the direction All think even better

In this problem this asked to find Think they get off the given function f and then evaluating it. Did you that point B vino deck degraded off f that this day left? Is it going to do? And six f by I am sick. The best impartial, innovative off effort respect to X, then partial derivative off F with respect to buy and then partially defective off f with respect. To say so. This is a will do and fixed these two X divided by one plus X squared plus by square plus zigs good as F is given Toby Natural Law off. But plus X squared plus y squared plus X squared, It's dead even is. First we find the daily with their off l. A. That this one divided by X squared plus y squared plus takes square and then by chain rule it will be my people. And by the directive off one plus x squared plus y squared plus X quit we just two weeks. So in this two weeks, divided by bike plus X squared plus y squared plays a squid. Similarly, if what is so why do I date by one plus x squared plus by square plus six quid, huh? I've said is Does that divided by one plus x Quick bliss by square, plus the square. Now we will right into it In the form off you need directors. I j get so this would be first we can take Komen, don't divided by one plus x squared plus y squared plus x square from all the three components. So it went before divided by one plus x scripless Why scriptures x squared and my deep light but x why they so that can be done s thanks. I bless by j bless Zak Gay. This is the required Grady. And now we have devalued this executable 0.1 bun minus one. Then this We have to find the left egg but but minus one So it will be going do so divided by but plus X squared will also be one by square will also be one. And so expect that this minus one square will also be one. So denominator will be by plus one plus one plus one that this four my d blind by excited that this one I'm so I plus Bun J that this Jay less, minus one into OK, that this mine escape. So this is a going do one divided by do I bless day mine escape. Now the next question asked is to find the unit vector in the direction off Maximum increase off f and the give a point b vino there the direction off Maximum increase off F is in the direction off the great being off f that this day left Then this. We have to find the unit vector off then F unit Director off the left can be found us They left divided by magnitude Off they left We will first find the magnitude off they left be Just think will do another would be coefficients off i j k with this quest that this quest off the coefficient off I'm JK so coefficient of ice one by do and it's square is one by phone less coefficient off. J is one by two and saw its square is one by foot. Bless coefficient off case my nessman by do And so it's squared is bun by four. So this is a well do another route three by four are it Does it will do Route three divided by two. So the unit vector off the left will be going toe. The left is but if I did, I do I plus J minus k divided but magnitude. Off the left is rubes three divided by two. So this is it will do one divided by three as this tool. And this Toobin began. So so. One divided by 123 My team blind by I plus J minus cape. So the unit vector in the direction off maximum increase off F and B is one divided by fruit tea. I best day minus K. Now the next question asked is 25 later. Change off the function in the direction off Maximum Ingres egg be They have changed off the function in the direction off maximum in Greece is the magnitude off the great get off F that this magnitude off Della which we have found Toby quite withdrew. Three. Divided by Duke that this the rate of change of the function in the direction off maximum in please is Group three divided by two. Now the next question asked is to find the directional daily with direct T in the direction of the G one factor you'll by three Dubai three minus one by three This is think e one back toe like this big will do you first. We will check where there is a unique vector or not. Yes, it is a unique vector. So the directional daddy Wait too. In the direction off that you and give my 0.11 minus one. Is it quite do? By formula physique will do the great get off f at the same point that this 11 minus one dot product of it did You need vector in which we have to find the directional derivative like this you had. So this isn't will do Della at 11 minus one b have found is this one. So this is a quarto one divided by two. What coefficient off ice. One coefficient of jay's also and coefficient off case minus one. So if this but divided by two into one that is one divided by dough again one divided by do and then one divided by doing do minus one. That this minus one divided by two dark product bit. You had that. This our unit director, You bitches so divided by three so divided by three and minus one, divided by three. Now this is a quite do but divided by to my people. I two, divided by three, is so divided by six yes, but divided by two months. People have a two. Divided by three is, I think so. Divided by six less minus one, divided by two into minus one. Divided by 30 is one divided by six. So this is a goingto five, divided by six. That this the required directional daily rate is five divided by six, which is our required answer.

So were given a function we were asked to be sporting. First, we're told Compute greeted us at a great immigrated and we for Why we for the and for that's what now where you compete the great end up at a point. P both reading off a p. Just here First he has four times my one on minus one. That's just for we have four times one money money for maybe have four tires. 114 That's right. Buying do you vector in the direction of luck The direction of maximum just ingredient were buying the unit vector organization of the greediest of look fine magnitude off ingredient about. We have four squared by four squared minus four Where thank you for radical three. Now we just divide the green at P. By this magnitude, we have one of the four radical three times collector for money for, like to ask, you fall in the rate of change in the direction of black men that we bag of ingredients off peak which is for radical now, actually compute direction of derivative at P in the direction of the given about you. So here, use the doctor here we have ingredient of empty. Not defector. You. We have four minus four minus four. Thought one of a radical three. What if a radical three in minus one over Radical do you get for over product with three money for radically three. Four of radical three. Jesse before over. Radical three. Quick is equal here before Operatically Creek Radical three. Let me for radically 3/3.

Were given a function. Three variables were asked to do these four things. First ingredient at a quick P. Here he has dysfunction F and you have a point. Compete ingredient of up. So a wise person here we have, they have Why? What? Me We're gonna have what we there would have. Why here we now compete ingredient off at that point, you like he won, we have. But if you one month one we have to three. Then we have a few body here. We should get Vector my one three. Now we're up to find the vector in the direction of maximum increase at P to the direction of Lex of increased t simply ingredient P. We have to find the unit vector ingredient. All we knew magnitude up the sector which is radical and divided by. Are you all ingredient which is minus 13? We got minus one over radical, the radical 10 actually defying the rate of change of us in that direction. That's simply the magnitude of the green in the direction of P. Just radical. Finally, we keep the direction and derivative at P the direction of giving back to you hearing you defector Here look fine. The direction of the river. We take the greeting of us at P. I thought the vector new. We have won 30 No narrow minded radical U minus one to we have. You're my yoga radical here and yet it just might be all right with you. You go, you my be over here water.


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