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[-13 Points]DETAILSLARCALCET7 13.7.036 MY NOTESASK YOUR TEACHERPRACTICE ANOTHERFind the angle of inclination of the tangent plane to the surface at the given point:...

Question

[-13 Points]DETAILSLARCALCET7 13.7.036 MY NOTESASK YOUR TEACHERPRACTICE ANOTHERFind the angle of inclination of the tangent plane to the surface at the given point: v2 = 5, (2, 1, 5)Need Help?

[-13 Points] DETAILS LARCALCET7 13.7.036 MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the angle of inclination of the tangent plane to the surface at the given point: v2 = 5, (2, 1, 5) Need Help?



Answers

Find an equation of the plane tangent to the following surfaces at the given points (two planes and two equations). $$z=\frac{x-y}{x^{2}+y^{2}} ;\left(1,2,-\frac{1}{5}\right) \text { and }\left(2,-1, \frac{3}{5}\right)$$

Were given a surface and the point and were asked to find an equation of the tangent plane to this surface. At this point, surfaces equation Z equals X times the sign of X plus y, and the point is negative. 110 Right away we see that Z can be written as f a function of X and y therefore the partial derivative of F with respect to X by the product rule. This is X times the cosine of X plus why, plus the sign of X plus y the partial derivative f with respect to why is X times the co sign of X plus y therefore follows that the partial derivative of F with respect to X at our point which has coordinates negative one in one. This is going to be negative cosign of to We're sorry co sign of zero plus the sign of zero, which is negative one and likewise the partial derivative of F with respect. Why negative 11 says negative co sign of zero, which is also a negative one, and therefore in equation of the Tangent plane, his Z minus z coordinate of her 0.0 equals the partial derivative F with respect to X negative 11 times X minus the X coordinate, which is negative one plus the partial derivative with respect to why, at negative +11 times why minus the y coordinate, which is one. And this simplifies after substitution to Z equals. On the right hand side, we have negative X plus one minus. Why minus one and simplifying by moving all the variables. On one side we get X plus. Why plus Z and then was zero.

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.

Let's find the equation of the tangent plane to the surface given by this equation over here and at this given point. So the equation of attention plane is given by the following. So we have are partial derivatives evaluated at the X and Y values of the original point times X minus x not and then add in a similar looking for her. But this time involving the partial derivative with respect. So why and then multiply by. Why minus Why not. So in our problem, dizzy is given by that's just the variable that will be present in the operation of our equation of our plane, just like the ex and the y. And then we'll use f. That's why to just be this function here on the right hand side. So this is the function that will be taking the the partial derivatives off. And then we could see our ex not why not and see not is that given point, so we know these numbers as well. So before we start plugging this information into the formula, let's just go ahead and find those partial derivatives. So the first one is that partial derivative at any point x Y So in this case, we look at our polynomial X is the only variable here and for the first partial their images. So we just differentiate that with respect to X and then similarly with their Spencer why we would get negative for why minus one using the power rule. So now we can go ahead and plug in the point here. So FX now we plug in to common three. Hey, and you could see this on ly depends on X. So just go ahead and plug into into there and that'LL give us an eight so off two times for similarly plugging the point two three into F y. So this time you're getting negative for But now we have three minus one, which is too so minus eight. And also let's not forget that dizzy not is given to be three. So we'LL let him plug that in So that Z minus three partial derivative with respect to X we just found that to be eight and then x minus x not and then now we'll have a plus negative aids Delicious, right minus eight And then why minus y not so. That's why minus three. This may be simplified, but ultimately it'll still represent the equation of a plane. So in this case, maybe we can just do an extra step or two here, Okay? And then maybe we can go ahead and get all the ex wives and seize on one side. So I'll go ahead and subtract eight x and add the y es y over. And then on the right hand side what? Eight and then plus three, which is eleven. So we could stop right there. And that's the equation of our tangent plane. That's the final answer.

In discussion where we call about the formula to compute the pension plan. So it has a formula to echo disease zero plus the first derivative respected the X times x minus X zero plus the first derivative respect to the Y times y minus Y zero. In this question won't even function arch that's why ego to Ellen off square root, X squared plus Y square And upon 3, 4 and five. And the first time we need to compute the slope so much and respect to the X we should get echo to. Now doing the negativity. We have one on the square root on the X squared plus Y square. And by the general we have two times the X divided by the square root on the X squared plus Y square. Now evaluate the arch, X and the pond. You know Who should get the code to activate your three and then divided by X squared plus y square. So we have echoed to the 25 and similarly we do have in the first derivative especially why? Because of the symmetry. We will have the wire on the top, X squared plus y square. Therefore, if value and the point which we get a coach in the far over 25. As a result, we can express the attention plan. Now The Echo to is is there will be a line of five and then plus the 3/25 times x minus three. And then plus Far always 2025, why -4 it would signify we have in the Z ego to a line of five plus two hour 25 x minus nine hour 25 plus far our 25 y minus 16 hour 25. And it will simplify everything. We will have bring the city to the other side. We have the three hour 25 X mine plus for our 25 y minus Z these two. Here we ended up, we got to the minus 25 minus 25 minus one. So we have the plus and minus five minus one equal to zero. And this will be the question we are looking for.


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