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Find the equation ol the tangent plane t0 f (x,v) = In(x Zv) atthe point (3,1,0)...

Question

Find the equation ol the tangent plane t0 f (x,v) = In(x Zv) atthe point (3,1,0)

Find the equation ol the tangent plane t0 f (x,v) = In(x Zv) atthe point (3,1,0)



Answers

find the equation of the tangent plane at the given point. $z=y e^{x / y}$ at the point $(1,1, e)$

In this problem, we will cover the equation of a tangent plane. So to begin solving this problem, I have written out in green the general equation for a tangent plane. And we see that we will have to find F. Of a B. The partial derivative of F with respect to X. In the partial derivative of the function with respect to Y. So we will start by finding that F. A B, which in this case is going to be F of 118 and we just plug in one for why and we plug in one for X. And we will get that. This is going to be E. So now we want to find the partial derivative with respect to X. And we know that we're going to be holding the Y variable constant from that function Z. So we're just going to transfer that Y to the front and we know the derivative with respect to X. Of an E. Function is just the E function times the derivative of the exponents, which is X. Over why? And again we know why is being held constant. So we know the derivative of that is just going to be one over Y. And we have that times why times E. To the X over Y. And that just leaves us with E. To the X over Y. And the partial derivative with respect to X. At the 0.11 E yields us simply E. Because E to the power of one is just eat. Now we want to find the partial derivative with respect to why. And so we're going to hold the X variable constant in the functions E. But we see here that we'll be using the product rule because we have to smaller expressions with wide, which is the UAE itself and then each of the X over Y. So we do the derivative of the first function first, which is why? And that's going to give us one. So we're just left with E. To the X over Y plus the first function kept intact, times the derivative of the second one. So we have E. To the X over Y again multiplied by the derivative with respect to Y. Of the exponents. And this derivative is going to be negative X over Y squared. And we see that because we're multiplying my wife, this will just leave us with negative X over Y. So our partial derivative, it's going to be E. To the X over Y minus X over Y times each of the X over Y again. And when we plug in the 0.11 E. We will get that This partial derivative is actually going to be equal zero. So for our equation of the tangent plane, we will get Z equals E plus E, times X minus coordinate, just one. And because the partial derivative perspective, why is zero, there is nothing else. We can end it there, but we can also simplify this, so Z equals E to the X. Because we will have a minus eat, and therefore this is our equation for the tangent plane.

Hey, it's Thursday when you married here. We're gonna find the equation of detention plane here. We were given the equation of the 10 engine plane and Z equal to have it's called No. Why? And this is going to be equal to our equation. Well, and next minus to why P huh? Why zero is equal to three one zero. Look, that's X comma y You equal to one over X minus two. Why? And then hopefully d'oh three comma one says equal to one of why Ex con ally. Well, to native to over X minus two. Why? Why three comma one equal to negative too. If we substituted, we'll do it here, C minus zero, which is equal to one times. That's minus three minus two. I'll write it down below Go I minus one xy cools Z equals X minus two. Why minus one? No, I'm gonna box our answer

In this problem we will cover the equation of a tangent plane. So from the given differential in blue, we want to find the equation of a tangent plane At this .3 -2. So I have written in green first the formula for finding the differential. And we see from it and from the given differential in the problem that Are partial derivative with respect to X at the .3 -2 is going to be five And that the partial derivative with respect to why at 3 -2 is going to be one. So using this information and also using the general formula for a tangent plane equation given And green, we will begin writing our own equation. So we have Z Equals f of three negative two plus FX or FX of 3 -2 Times X -3 plus ever. Why FY. 3 -2 Times Y Plus two. And now we're going to start substituting in numerical values. So we have G is equal to eight Plus five times X -3 plus one Times Y-plus two. Which is just going to be just wipe list too, simplifying this further, we get eight plus five, X minus 15 plus Y. Last two. You know that eight minus 15 is negative seven negative seven plus two is negative five. So our final answer for the equation of this tangent plane is going to be, Z equals five x plus y minus five and there we have it.

In this problem of different station, we have to find the equation found in plain at the Point Tangent plane for the function given by as Wolf Extreme allies. It quinto Ln off floor experiments y squared that one from my one. So now we'll collected theater Morning creation of the tangent plane would say is that if if affects Goma Y is locally union or defensible, I take, um a B then puts Qianjin Plane is given by the equation music went to We're for free from a B plus FX all fake Amobi in two X minus eight Yes f y off a community into y minus. So we have to prove that it is different stability command be there is one called one So the difference ability definitions comparative. Therefore fixed tomorrow Y is defined in a disk d containing a coma B and but the fixed off they cannot being a fly off. It will not be exist then. If affects common, why is different political mopey? So we'll find the partial derivatives at this point. So if first let's find the function of the 0.1 come on, run. So that will be well toe Ln off four minus one that this l enough t now if eggs off, it's come all right. The greater differentiating partially with Inspector. Thanks. We get one bite for experiments. Rice well in differentiating Wow, this one. Getting a Texan to leaders think similarly, we can find a boy off. It's from home, right? Differentiating the function. Fashion with respect to y one. By full experiments. Vice with 43 minutes spice with differentiating partially with respectable minus two. Right? So now let's find the 0.1 comma one If it's off one comma, one will give us one bite. Four minus one into it. That is it. Culture eight by three f y off one comma one Give us one bite four minus one into minus. True, that is minus two. Bite we. This implies the partial generators our continuous which further implies that f off it's comma y. This different ship will bet one come up one so we can use the theater. So, using equation involved save one, there will be right You can write the equation off the tangent plane us She it was equaled. Their fourth one coming along was effects off one Kamala into X minus one less half way off when come on one and two by minus one. So substituting the values will get the fourth one comma Oneness Ellen off three less effects of one comma once a by three, Enjoying X minus one less Um, that's true by three that to run it with. So if you really like this equation will get eight by three in directs my stool by train toe. Why less Ellen downstream minus eight by three less to buy three. That would keep that he will get minus six party that it's minus two. It's safe, which not thinking something cool.


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