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(1) Find an equation of the tangent plane to the surface poiut Fith I = -2V-12p" +3r1 + Zy+35 at the...

Question

(1) Find an equation of the tangent plane to the surface poiut Fith I = -2V-12p" +3r1 + Zy+35 at the

(1) Find an equation of the tangent plane to the surface poiut Fith I = -2V-1 2p" +3r1 + Zy+35 at the



Answers

$1-6$ Find an equation of the tangent plane to the given surface at the specified point.
$$z=3(x-1)^{2}+2(y+3)^{2}+7, \quad(2,-2,12)$$

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.

This question asks us to find the tension line given a plane and a point on the plane. So as a refresher, the equation for a tangent line is T. Is equal to f sub X at a comma B times x minus a plus F sub Y at a comma b times y minus B plus Z at a comma B. So to start, I am going to factor out rz equation. Ze equation is x plus two squared minus two Times. Why -1? All squared minus five. So to start, I'm going to factor this out. So from here I have, Z is equal two, X squared plus four, X plus four minus two times Y squared minus two, Y plus one minus five. And if I simplify this even further, I have that Z is equal to X squared plus four, X -2, y squared Plus four, Y -3. So from here I can solve for F sub X and F sub Y. So my f sub x is two, x plus four is two, X plus four, two X plus four. And my F sub y is negative four. Why plus four? If we then plug in our point? So we have ffx at a comma B. We get that ffx at a comma B is eight and our f of y at a comma B is negative eight. It's taking all of these values. We can plug these in to our equation for a tangent plane. So we get that T is equal to eight times X -2 minus eight Times Y -3 plus three. And that is for our X. So from here we can simplify, We get that T is equal to eight X -16 minus eight. Why plus 24 Plus three. And if we add up all the like terms you get That T is equal to eight X -8, Y plus 11.

Were given a surface and appoint and rest to find an equation of the tangent plane to the surface. At this point, surface has the equation three y squared minus two x squared plus X. The point on the surface is too negative. One negative three. Yeah, well, right away. Noticed. Weaken right the as a function of X in y as Z equals three y squared minus two x squared plus X And therefore we have that the partial derivative f with respect to X is negative four x plus one and the partial derivative f with respect to why is six y so it follows that at our point to negative one negative three partial derivative with respect to X. Well, this is had to negative one which is negative. Seven. The partial derivative of F respect toe Why at two negative one is negative six. Therefore by equation two from this section. Vision for the tangent plane is Z minus and then rz value, which is negative three for this point, equals partial derivative of F with respect to x Evaluated A to negative one times X minus Our X value, which is to plus the partial derivative f with respect to why evaluated at two negative one times why minus or why value, which is negative one? Or why plus one and simplifying. We get Z plus three equals on the right hand side. We have negative seven times X minus two, minus six times Why plus one rearranging yet Z equals negative seven x minus six y and the constant term is simply five.

Discussions. Were you calling about the formula to compute the Denton plan. We have the formula C equal to the disease zero plus the first derivative, especially the X times x minus extra plus the first derivative, especially the Y times y minus Y zero. In this question were given the function she X Y E co two X squared plus Y square. And the front 1 -1, 2 here. The first time we need to compute the slope here so that she X. We go to the two X. Therefore evaluate. And the point We got a code 22 times one would be to something the first derivative respect. You know why we had to go to the to Y and therefore the G Y. If I were under point echo to two times a month wannabe months to and fine we will be able to computer attention plan. It will be C E co two. These are equity to Plus FX will be two times X -1. And then F Y. I could you manage to why manage uh y minus minus. Want to be plus one here. So and we signify we have the sequel to the two plus two x minus two minus two y minus two. So and we signify everything we shouldn't get uh two x minus two y minus C. Usually counselor buddhist. So we have -2 equals to zero. And this will be the question we are looking for


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