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What Is the approrlmute direction of the gradient vector Jt each ot the following polnts?TAt the point €~ T+T T-)At the polnt AAt the point 02. _7+7 e _t-)...

Question

What Is the approrlmute direction of the gradient vector Jt each ot the following polnts?TAt the point €~ T+T T-)At the polnt AAt the point 02. _7+7 e _t-)

What Is the approrlmute direction of the gradient vector Jt each ot the following polnts? T At the point € ~ T+T T-) At the polnt A At the point 0 2. _7+7 e _t-)



Answers

Find the directional derivative of the function at the given point in the direction of the vector $v$ .
$V(u, t)=e^{-u t}, \quad(0,3), \quad \mathbf{v}=[2,-1]$

This time they give us the function. H of X Y is equal to e to the negative X minus one, and they're gonna want us to evaluate a directional derivative. So let's compete the grating and reach. Let's see, partial X. What does the negative XY But see minutes, Why and then the same thing, actually, for the white component. That's nice, because when we evaluate the radiant at our point natural law, go to natural law, go free, we're gonna get the same value in both. So we have negative e to the let's see negative natural log of two plus natural lager three. You know, properties. Ah, longer than this out can turn into a multiplication of the butts. So anyone natural log of six. We can pull that negative in there, so we end up with negative. So the natural love of 1/6. But these air increases. They cancel out all the Senate with the negative 1/6. Unless same thing for the second component. So it's negative. 1/6 common negative 1/6. What about the direction they give us? Well, they give us the direction 11 Actually, you directed that we're going to be considering is actually one over the spirit of 21 over this girl. So let's see that action under video in the year direction of H is going to be equal to you. Frantic negative 1/6. What's actually now It's one of six and take the dot product with the Vector one over the square root of two, one of the spirit of two, and we'll find that the direction on their video of H at that point is negative. 1/6. Square it of to on the negative one over its X squared, okay?

Think of us The function he of excellent is equal to the natural long of four plus x squared plus y squared. And they're gonna want a computer directional derivative. So it's compete, ingredient or natural Law given input is one over that input times the derivative of that input. But we're impartial. Your suspect X We're actually gonna have to Exxon's up for the why. It'll be similar, but we'll have a to y on top when we do the chain rollers suspect blind. Okay, so now let's evaluate at the point they want us to take our ingredient. It's at the point. Negative one come on to Well, it's fine. Muncie on the bottom like that. I mean, it is the same. Every right. We'll end up with the four plus one plus for so dividing by nine on the top. We have a negative too. So they have to overnight with Chris. Component on the right will be four overnight. All right? And they're gonna want us to compute this directional derivative in the direction. Teoh, comma one. Well, we know the unit vector that corresponds to that will be. Do you over the square defined one over is very violent. So when we're going to compute the directional derivative and that you direction of P, or you need to compute the dot product of our little unit back there and are given ingredient at the point, so what we're gonna end up with is, let's see negative for over a nine square root of by plus for over nine scripture to five that just comes out zero. So the direction already about that boy is actually zero. That's pretty cool traveling along the level curves still at that point in that direction.

Mother in this problem were asked to find the directional derivative in the direction of our vector. 111 Let me have a function, G. And we'll take this at the 0.1 to 0. So it's used Equation four here. So for a directional derivative will have to do one over the norm or the length of our vector 111 Now we'll multiply this by a doubt products. So the first part of the dot product will be the Grady int of G. We'll evaluate that at the point. We're at 120 dot product of that with the vector 111 Okay, so it's filling what we can. The length of the vector is not too bad. Ah, one squared plus one squared is one squared is three. So that line of being the squared of three Grady int of Gene Let's be careful about this so we'll get a vector with the partial of GS. Respect to X partial of juice. Respect. Why a partial G with respect to Z and rebellion. But that one too. Oh, still dotted with 111 Okay, so at this point, that doesn't change. Let's get these partial derivatives Taken care of partial of G with respect to X. So if y and Z are just Constance than the partial of with With with respect to X will be just that e to the minus y z part partial of g with respect to why so the exits just a constant that will stay there. We'll still have e to the minus y z, but then, by the chain rule, we need to multiply by at minus a Z. So this is all the partial with respect. Why so multiply all this by minus Z. So we'll put the minus in front there in the Z there. And then finally, with respect to Z, very similar excess Still a constant well, e the minus y z And then with respect, Izzie, the derivative is minus y so minus where I am that work and still at 120 Okay. And the dot product is with 11 Wow. All right, so this one over the square to three will not change it all this just now. Plug in one for X two for wind, zero for Z and see what we get. So since disease. Zero This whole first coordinate e to the minus y Z That'll be eat of zero, which is one for the second part will get, see negative one times while 00 So that makes the whole thing zero And then finally here let's see negative x times Why, it's negative one times too. So negative too. And then in the e to the y z z a zero. So that whole thing becomes one. So you shouldn't be getting this for our vector that it was 111 and you want over the skirt of three The dark products Not too bad here at all. One times on this 10 times 10 negative. Two times one is negative too. Actually have one over the square of three one minus two is negative one. So final answer. Negative one over the skirt of three. Hopefully, that helped

His problem. They want us to see They want us to compute a directional derivative. Uh, given point, it's gonna be able to function for excrement minus one 0.1 for direction. Negative tuning. Either one. First things first. Going to the radio. We're going to need This is going to be partial derivative with respect, X for the first component eight x e to the four Experian minutes when? But then the y component is going to be and see negative e to the negative. Why? Bus for excursions can't is not Katrina. And now we're gonna take the dot product. Was this back to our unit? Director? I'm gonna end up with Well, aren't seen the why is just gonna be positive, Inspector out the Kameni to the four x sprint when it's wet. When we add the X component, not product ways. Negative to win. That one is minus eight, 1916 16. Thanks. You know, we can evaluate this. That's the point. One for it on it was Nancy. This is going to be negative. 15 in the front and Ito for minus four is, uh, zero. So we end up with negative 15. Awesome


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