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Find the direction of maximum rate of change of the function f(&,y) x2 ey y3 In(c) at the point (1, 0) _(1,0)35(2,1) J2(1,1) (0,1)...

Question

Find the direction of maximum rate of change of the function f(&,y) x2 ey y3 In(c) at the point (1, 0) _(1,0)35(2,1) J2(1,1) (0,1)

Find the direction of maximum rate of change of the function f(&,y) x2 ey y3 In(c) at the point (1, 0) _ (1,0) 35(2,1) J2(1,1) (0,1)



Answers

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

$ f(x, y) = \sin (xy) $,
$ (1, 0) $

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

Were given a function and a point. And we're asked 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 over Y plus Z. And overnight actually, And the point has coordinates. 813. Mhm. Find the maximum rate of change will find the gradient of F at this point. This is the vector's components are the partial derivatives of F. So we have one over Y plus Z minus X. Over Y plus Z squared and minus X over Y plus Z squared. And therefore the gradient of f at the .813. Uh huh. This is equal to 1/4 1 4th -8 over four square to 16. This is negative one half and negative eight over Yeah. 16, which is negative one half. When that was that was the original being. And therefore from this section we know the maximum rate of change of our function F at the 0.813 is given by the magnitude of the gradient of F at 813 which is the square root of 1/16 plus 1/4 plus 1/4 and this simplifies to 3/4. Moreover, the maximum rate of change occurs in the direction of the gradient of f at 813, which we already saw was vector 1/4 really negative one half negative one half. So the maximum rate of change occurs well right in the direction of any scalar multiple of 1/4 negative one half negative one half. For example, one negative two negative two, and a maximum rate of change is 3/4.

The given state Mindy's If off X comma y is equal to sign off, it's why on the beach is given ask one Angelo to Venice states that if as different civil Fransson on and the little men off if then the maximum value has do you f Mama B is equal. Do Mar del f If Mama B so they'll have. That's why. Itwas if off now these events. That's why on if why x y so we will find these two bottom It does. First we have events. Let's go away. This is given as white boss. It's why on if why it's going away just given as ex costs ex wife now Well right dlf x comma y that iwas Why? Cause that's why on if Sonny, this is ex cost ex wife So we have del f one Kamaji tau equals d Oh, come on! And according to till then the maximum rate of change off f is the F one comma Jiro, this is given as more you look about one which is equal to one on it are gods in the direction off day love hands along d'oh! On one. This is the answer to the problem

We're told it's Herman. The maximum rate of change of F at one comma zero in the direction that this change occurs in now the direction that the change of Curzon is going to be equal to the radiant vector. And so to find that vector, we're going to take the partial of effort. Expected X and the partial respect a Why it's the pressure. With respect to X, it's going to be equal to co sign of X. Why times why and the partial with respect to why is going to be equal to again coastline and ice. Why this time? Times X And we want to evaluate both of these at the 0.1 common zero. And when we do that, we get co sign of one time zero, which is zero time zero. So we just get zero. And for the partial respect a lie, we're going to get coastline of one times here, which again is co sign of zero, which is one times X, which is once you get one times one and get one. Okay, so the direction in which it occurs is going to be the doctor. Zero come a one and to find the maximum rate of change. We're going to take the length of the spectre and so we're going to take the square root. So again, we're taking build wing of the vector 01 and we're gonna take the square root of the some of each component squared. And so we're going to take zero squared plus one squared Nazi gold zero plus one. So we're gonna get the square root of one, and that is just equal to one. And again, doctor is the maximum rate of change.


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