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Let $f(x, y, z)=e^{-x^{2}-y^{2}-z^{2}}=e^{-r^{2}},$ with $r$ as in Exercise $31 .$ Compute $\nabla f$ directly and using Eq. (9)

So for the first part, we need h crime of three, which is going to be given by F crime G of three multiplied with G prime of feet and G of three from the table is given us one. So we need to find f prime of one and multiply that with G prime of three f Prime of one is five and do prime of three is 20. So our final answer is 100. Yeah. For the second part, H. Pryor of two will be given by of prime give to into G prime too. And from the table, we see that GF too is equal to five and now we just need to multiply. That would g programming too. So if prime five is given as minus 10 and G prime of two was given this 10. So our final answer comes out to me and I give 100 for the third part. Where and we need to find Pierre for that is gonna be given by G prime efforts for multiplied with F prime for and from the table. Once again, effort for is given as one, and we just need to multiply that with private floor. Sergey, Prime of one is to and f prime of four is minus eight. Your final answer is my ass. 16 for part D, we need to find p prime of tool, which is going to be given by G Prime ever to multiplied with I'm trying to and f of two from the table is three. So we have the final three multiplied. With that, Brian was too g Prime of three is given us 20. And if crime of two was given us too. So our final answer is 40. For the last part, we need to find a judge Prime five. So we need f prod f prime Do your five and we multiply that with G prime of five. June 5 from the table has given us to on And she promised life as it is f crime of two from the table is too and g prime of to do you from the five. I'm sorry. This 20 sore final answer That's 40

Okay. Sinners hear from parts a T E. That we need to find a derivative of H and pizza. So we'll start for finding that So each prime of X well that's equal to will use a general. So that's F prime of gov time to derivative of G of X or G, and all four p p. Prime of X. That's equal to G prime of of of x times the derivative over inside, which is of prime at X Rocket. So we want each prime of all the way to that three. So let's see, let me write this out in terms of three. Okay, so it's start. We're finding g of three, so that gives me one. And then we have g prime at three. That's 20. I know we need to find f prime at one, and that's five. So we have five times 20 and that's equal to, um, 100. Okay. And now for a party we have, um, I'll just plug and exited with two. Okay, so we first need to find G evaluated up to It's about supplies and in g prime about the way to that, too. That's equal to 10 okay, and we also need to find f prime at, and that's equal to negative. So I get negative 10 times 10 and that's equal to negative 100 for Part C. We have P prime. I'll just write that out in terms of X is equal to four. Okay, so it's find F at four off a war that's one in that F climate for that's negative eight. And then we also need to find you part of that one, and that gives it to So we have two times negative AIDS, which is equal to, um, it's you, the people to negative 16. Okay, again, we have p prime, so I'll just plug that in in terms of X is equal to Okay, so you have g prime of half of all the weight of that, too. That's 3.5 prime at two. That is. Actually I've got to is three. Oh, that's what we have there. And that's prime at you. That's too. And now finding G party, my three. We yet 20 do you have 20 times to, which is equal to 40. And now for our last one inch prime evaluated at X is equal to a point. We have the following. So starting with G at five that you could too and then g climate five etc. With 20 finding F crime. That two that is two. So we have two times 20 and that's equal to 40. Okay, so we get the falling solutions for a B C D E.

Hello. So here we have a function F of X is equal to two X squared minus three X. So for part A we are evaluating F. Of A plus H. So F of A plus H means go ahead and plug in A plus X. For X. So we get two times A plus H squared minus three times A plus H. Then we just um square out eight plus eight and 53. And we get this is going to be equal to a to a squared plus two eight squared plus four A H minus three times A minus three H. So there we have F. Of A plus H. And then party we have F. Of A plus H minus everyday. All divided by H. Whoa. That's crazy. Right so F B. We have here F F. Of A plus H minus F of a. All divided by H. Also explosion here. So therefore this is going to be equal to well just plugging in this here for X. In our function we get this is going to be equal to what A. To a squared plus a +28 squared Plus A four A. H minus three A minus three H. Uh Well we have minus to a squared plus three A. All divided awful divided by age which is going to be equal to a combine like terms two X squared plus four H minus three age. All divided by age which is gonna give us uh once the ages cancel out, we just get a two H. Plus uh for a minus three A. So there we have a different question of F. Of A plus H minus F. Of a. All divided by H. Take care.

In the question. We have to find the immunizations for the given on effects on the white. Understand, which is a close to excess Where less vice well plus z squared at the point, eh? About +11 form of be part still move on. Steve went on, moving forwards dissolution or give us a function on one on one with three at exactly 11 moment. One which would tie it to buy that for the given function, but a less too x minus home less to buy my next one less to finance. It will make us less wine, less tools. My next three now moving works of art. So the unionization for they will be one plus zero x minus. Really blessed to why minus one less zero minus which by my next one, see one way effects that one and that one for a given for parts B wets minus one. And this one final question. Thank you


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