5

(1 point) Suppose that f(z,y,2) = r"yz Iyz* is a function of three variables Find the 'gradient of f(z,y,2) . Answer: Vf(z,y, 2)2. Evaluate the gradient a...

Question

(1 point) Suppose that f(z,y,2) = r"yz Iyz* is a function of three variables Find the 'gradient of f(z,y,2) . Answer: Vf(z,y, 2)2. Evaluate the gradient at the point P(2,~2, 2) Answer: V f(2, 2)3. Find the rate of (change of f(€,%,2) at P In the direction of (he vector u (0, Answer: Du f(2,

(1 point) Suppose that f(z,y,2) = r"yz Iyz* is a function of three variables Find the 'gradient of f(z,y,2) . Answer: Vf(z,y, 2) 2. Evaluate the gradient at the point P(2,~2, 2) Answer: V f(2, 2) 3. Find the rate of (change of f(€,%,2) at P In the direction of (he vector u (0, Answer: Du f(2,



Answers

(a) Find the gradient of $f$ .
(b) Evaluate the gradient at the point $P$ .
(c) Find the rate of change of $f$ at $P$ in the direction of the
vector \mathbf{u} .
$$f(x, y)=y^{2} / x, \quad P(1,2), \quad \mathbf{u}=\frac{1}{3}(2 \mathbf{i}+\sqrt{5} \mathbf{j})$$

In this problem, you're given the information shown and first asked to provide ingredient of A to do this. First we have to find the partial derivative of F with respect to X in the person with respect to why and so the partial of f with respect to X It's going to be the following is going to be signed. Excuse me is going to be co sign a two x most three y and then we want to treat why, as our constant and exes are variable and therefore the derivative of the inside is going to be too. And so that is going to be equal to two co sign of two X plus three. Why and then are partial with respect to Why is going to be the following again. The derivative of sign of whatever is in the parentheses is going to be co sign. Have two x plus three y and then if we treat excess er constant and wiser variable, the derivative of the inside is going to be three and so that is going to be equal to three co sign of two x plus three y. And now we just have to put this into a vector, and we found ingredient. The partial with respect to X goes where X would normally go in a bacteria. The pressure with respect to why goes where why would normally go. And so we get to co sign of two X Plus three wine comma three co sign two x plus three y. So that is our answer to part A. And then per p asks to her for us to plug in the point p into this vector. And when we do that, we get the following to co sign of two times negative six with his negative 12 plus three times, for which is 12. So you need a 12 plus 12 that zero, and then we get three times co sign and it's going to be negative 12 plus 12 again. And so we're going to get too close on of +03 coastline of zero and coast on it. Zero is equal to warm because here degrees is over. Here is your obedience and the adjacent and the hypotheses. They're both equal toe one and so close icicle toe one and so we get two times for come with three times one. So it's going to be two comma three and that is our answer to party. No part C asks us to determine the rate of change of ah, at the point given in the direction of the vector that they give us now we always have to make sure that the length of the vector, the of the direction we're using is one. And so we're going to do that. I'm going to distribute the 1/2 in order to make it easier on us. So when I do that, I get Route three over to common negative 1/2 for a vector because the vector in normal form would be through 3/2 comma, negative 1/2. And if we want to make sure the length is equal to one, we're gonna take the square root of each component square. Does everyone take the square root of Route 3/2? Squared Those 1/2 squared. We're 3/2 squared is 3/4. Those 1/2 square that's one over. For that, I get 3/4, plus one of her force for over four, which is one, and so is he going to one like we want it now we want to take the dot Product of what? We've thrown a party with our direction. Vector. So we want to take to calm a three dot Route three over to I'm a negative 1/2. And when we do this, we got the following two times. Route three over to the twos. Cancel out. We just get Route three. Been three times negative. 1/2. It's just minus three halves and you're more than welcome to simplify this. If you would like, however, this is the answer.

Yeah. If off x comma wine it is People, sign off two X plus three wives on dhe. We have given equals minus six on Ford on you. Iwas rude tree over two on minus one of our You know about it off the question Hill is del F off X. Come on. Why? These people do ethics. It's from a white on white. It's going first of all, we will find all those Jones so fester fall. If it's thanks, come away. So this is equal False two x plus three while in to do dizzy polluted myself Cost two x plus three But no if Why explore why people do in the because To express through Ianto treatises equal free course to its place Feeling on DDE del f off. Come on. Why, Nikolas myself, boss two x plus three Wine on three gas to express freely. This is defensive about it. Now we have bought Be off the question. Well, well done. Minuses! Come before this is equal. Do dry myself cause Manus tree last 12 and do this comma three cause minus duality Blast. This is equal to myself. Course judo degree and the choice of clothes, judo degree, Just evil to country. No park. Si off the question states, do you If off do come on three keepers to come on three Indu I don't t 02 on mine is gonna want to This is equal to good three my as three by two This is that answer to the given from

Hey, it's clear. So when you read here so far apart A. There's three parts. The first part asked us to find the Grady int, But just given by this formula. So for a case, you know that that X it's come a why was equal to negative wise square over X squared And but why Ex con a life was equal to to buy over X So for this we're gonna get negative Waas Square over X square comet to lie over. And that's the first part for the second part. Put it down below. So for part A, we got negative wife square Overact square called a to y over ends. Now, if we plug in one comma two, we get negative for comma for part, See us is to find the rate of change. So we know that this represents the rate of change and from exercise being we got negative for common negative for and we're gonna use equation nine, which is Joe I'm writing currently calma y. Come on. That's why that's cool or so. If we plug in one comment too, we get for skirt five minus a over three. After recent

For this problem, we are given a function of three variables X, Y&Z. And we are asked to compute the gradient, Evaluate the gradient at our .302 and then evaluate, the rate of change in the direction of our of our unit director up here, you and so the first thing you want to do is compute the gradients as we were asked to do. The gradient is given by the vector of the partial derivatives, F sub X, F sub Y and F sub Z. So all we do is we take the derivative of this function with respect to X, Y and Z. And we get the derivative with respect to X is just going to be equal to eat of the two Y. Z. Because X is our only variable that we have there with X. So E to the two Y. Z. For with respect to why we have to kind of do the chain rule with this function E. Here. So we leave everything as is and then we multiply by the derivative of the experiment with respect to Y, which is equal to two times Z. And lastly we do the same thing I think with respect to Z, we get x to the east of the two Y. Z. But this time we take the drift of the exponent with respect to Y with respect to Z. And we get to y. So more formally we can see that we get, you want to be a little bit more um simplify simplify a little bit more. I'll show you what to do. We get two xz Either the two IZ and we get to X Y E to the two Y Z. And if you're feeling like this is still a little bit too clunky. You can see that every single one of these values had every single of these components has heated the too easy. So get factor that app. So you have one two X Z two X Y E to the two White C. You don't have to do this. Maybe just a little bit helpful for evaluation later. The next part we want to evaluate this um this gradient at our 0.30 to so to do that we plugged in the values of three for x zero for y and two for Z. And we get one Here we would get two times three times two here we get two times three times zero and we multiply all of these by E to the two times zero times two. Well, first off, see that's going to be Eaton zero which is one. That's very helpful. And we see this is one this first part here is 12 and this is zero. And so this is our gradient, which means that if I'm at the 0.30 to I want to move in the direction 1 12 0. And that will be my fastest rate of change my function value will be changing the fastest. Lastly here you want to compute how fast is F changing at our point when I move in the direction of you. We can find that this is in fact a unit vector, which we do need, we need to find that the magnitude inspector is equal to one because we need that magnitude to be equal to one. In order to use our directional derivative formula which says that the the the directional derivative of F in the direction of you is you go to the gradient of F at our point X. Y dot product with you. And this must be a unit factor in this case we do have one so we can just apply it and so we can just apply this here we get 1 12 0 dot product with two thirds, negative two thirds, one third. And we did adopt the X multiplies together. So we just get two thirds the wise multiplied together, we get 24/3 and this is then just zero. And so we get negative 22/3. And these are function values changing at a rate of 22/3 what is decreasing at that rally. And so this is just another way of doing it. So you you take the derivatives with respect to x, y and z to get the gradient plug those values in and you take the dot plot, making sure that are a unit vector of you is in fact a unit vector


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