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U(x,Y, 2) is a scalar field_ How is the directional derivative dU/ds related to VU? Unit vector d lies along the line x/3-y/4-2/5, pointing away from the origin: Fi...

Question

U(x,Y, 2) is a scalar field_ How is the directional derivative dU/ds related to VU? Unit vector d lies along the line x/3-y/4-2/5, pointing away from the origin: Find d_ and hence find the directional derivative of U = x2+y2+z2 at the point [1,2,3] along & Find the maximum rate of increase of U with respect to distance at the point [1,2,3]. Show that VU is perpendicular to any surface U-constant_ Find the unit normal to the level surfaces of U x2 y2 + 22 in the direction of increasing U at t

U(x,Y, 2) is a scalar field_ How is the directional derivative dU/ds related to VU? Unit vector d lies along the line x/3-y/4-2/5, pointing away from the origin: Find d_ and hence find the directional derivative of U = x2+y2+z2 at the point [1,2,3] along & Find the maximum rate of increase of U with respect to distance at the point [1,2,3]. Show that VU is perpendicular to any surface U-constant_ Find the unit normal to the level surfaces of U x2 y2 + 22 in the direction of increasing U at the point [1,2,3].



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. $$f(x, y, z)=4-x^{2}+3 y^{2}+\frac{z^{2}}{2} ; P(0,2,-1) ;\left\langle 0, \frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right\rangle$$

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.

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.

You were given a function. Three variables, or are these four things first we compute greeting at that point. Well, you know, so speaking to you that squared plus two squared b squared and we have quite TV 10 for look, you first people greeting up we get like they're like or why me? Where you compute this ingredient as a point p We got two years ago part Secondly, they asked, justifying the vector in the direction of maximum. So the way worded use weird at first, but so in the direction of national interest in the direction of maximum repeat ingredient, we were really optimistic Point you elected representation of the radiance of at we just the green p divided by Maggie. Do you get one over 1028 times Director to use their own 32 in the third part that I find great change in that direction again, this is just no matter between ingredient. We know from part two. That's just 1028. The last part were asked if I direction of derivative at P in the direction given back to you. Will you like your one of arrival here? euro one of variety. So the prosecutor ingredient of that about you know, collector you We have huge. They're very to you. One medical heretical to 01 over. Radical where you get heretical to zero a radical. Que? Yeah, theoretical. Two or three years, A lot of years over you here. You know that The same thing. I was two times radical, too. 17 radical here. Over here. No, sorry. 30 people. 34 over. Radical here. And that just 17.

Given this function. They were asked to do these four things here. Look forward to first base. Compete, Ingredient. At this point, he what I find ingredient. So you have years X V minus one. There were you change rules. So the X y Z's ex wife the bringing wise here have exploit the minus one. Are you thinking fly and the That's why we get at the eyes. Exploit the minus one. It is you yet, x y. I need to live the life. No playing always we got so no, actually zero. So this is gonna be here and there as well. And why times you should give us negative. And e Teoh we know zero 40 0 euro minus one. Use the minus one. Whatever you need, we have minus one next to ask. You find the unit vector in the direction and lack of decreased. So we're just gonna find the unit vector in the direction of the radiant. So over here they grading up, waited at p divided by faggoty. What's the value of this? We have whatever he weird we take the square root That's just wanna written. We have whatever e inspector, my one over e euro zero yet minus one over E squared euros. And we're asked, you find the rate of change in the direction back, please. Magnitude of the green. We know the value of ingredients, but he just you whatever. And finally we're buying direction to be rooted of at that point p in the direction of you, we just take the green of at P. I don't expect you three yet by 1/0 0 I thought bias to every three to over three and monitor. We get here over three e 100 retreating.


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