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5) Find an equation for the tangent plane to the graph of g(1,y) = IOz Vy - sin (4x + y) at (-1,4) . Write your answer in the form z = ax + by + €...

Question

5) Find an equation for the tangent plane to the graph of g(1,y) = IOz Vy - sin (4x + y) at (-1,4) . Write your answer in the form z = ax + by + €

5) Find an equation for the tangent plane to the graph of g(1,y) = IOz Vy - sin (4x + y) at (-1,4) . Write your answer in the form z = ax + by + €



Answers

find the equation of the tangent plane at the given point. $z=\sin (x y)$ at $x=2, y=3 \pi / 4$

You see in court Dunton inversion the X qy and a boned 11 by phone Hey, we re going down the dental plan formula in his f x x minus X zero point times going minutes Quiet zero plus f c Didn't see when it's these, they're all just zero. And when we first them we need to find out actually. Really? Well, do you know the riveted the attention of really what you want Novo on August Next? Why? Square and times would, uh, dry on top Now something for the everybody Coach in the legs off. I want us next one square. Ever see in coaching, minus one on and four. From here they can find that plan. It would be f X. Now it will be 11 So we have a one off Jew for the FX and explain this one. Now everyone would be assenting implicit half Why minus one minus one can see my ass. I have far a coaching. Is there, uh, within the financial again 1/2 of the banks plus 1/2 time job I wanna see So it would have a minus one. You go to no one greenness full

Question will come about a pension plan formula. So we have the especially with respect to the x Times x minus x zero plus f y comes choir minutes wise there are something for the easiest well, this dream honesty zero go to zero And now in this question, were given Does they go to the tension in verse? I'm the exploit I onda 00.11 on the far in the first time we see we can and you find a function f x y z culture the tension in verse on the X y and Z And from here we can find a trip over the river Dio f x and Y M c A and the f x Gennaco June the why over one plus x squared y square. And if why we have the X over one plus x squared y square I never said they were gonna go to minus one. Therefore, we have the pension plan. Formula will Bay can we would respond into this one and then we get echo June 1 half and then x minus one. Something from the F white Could you want half a swell? Why minus one and minus one for the the finest by far, coaches zero. And with him, if I have the one half blessed one half why and then we have a go to this one being one minus by out of far

This question asks us to solve for the tangent plane given a point and the plane to do this, we first need to know how to find a tangent plane. The equation for a tangent plane is T. Is equal to F sub X. At a comma B times x minus a plus F sub Y. At a Cumbie times y minus B plus z at a comma B. So from here we can solve. So our F sub X is for X under F sub Y Is two, Y -5. Well or point is 1:02 -4. So if we plug in or point, we get the F sub X at a comma B is four. Never F sub Y at a comma B is negative one. So now we have that. Plus we have our playing so we can plug it into the equation. We have T is equal 24 times x minus a. And a is one plus negative one Times Why -7. & B is too plus Hersey at a comma B. Well rz at a column B is just value given to us at the point and the value given to us at the point is negative for so from here we can simplify so we'll bring this up here and so to simplify, we can bring out Or we can multiply out our four. So we have four X -4 -Y plus two minus four. And if we simplify even further, we get the T. Is equal to four X minus y minus six.

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. Of Y. And that just leaves us with E. To the X. Over why. 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 Y. 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 2nd 1. So we have the 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 way, this will just leave us with negative X over Y. So our partial derivative is 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 .1, 1 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- 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 to Z equals E to the X. Because we will have a minus eat, and therefore this is our equation for the tangent plane.


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