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Question 3If 4z? + 4z + %y = 5 and y(5)23,find y' (5) by implicit differentiation:Submit QuestionQuestionFind the slope of the tangent line to the curvelz? + 4...

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Question 3If 4z? + 4z + %y = 5 and y(5)23,find y' (5) by implicit differentiation:Submit QuestionQuestionFind the slope of the tangent line to the curvelz? + 4ty 3y3~7at the point2,1).Submit Question

Question 3 If 4z? + 4z + %y = 5 and y(5) 23,find y' (5) by implicit differentiation: Submit Question Question Find the slope of the tangent line to the curve lz? + 4ty 3y3 ~7 at the point 2,1). Submit Question



Answers

Find an equation for the tangent plane to $z=f(x, y)$ at (3,-2) if the differential at (3,-2) is $d f=5 d x+d y$ and $f(3,-2)=8$

For the given problem, we want to consider the equation of the tangent line to the curve at the given point. So this is going to be Y equals to execute five X. And that is where we have um the point we're gonna be looking at is negative 13. So the first thing we want to do is determine the slope at this given point. So thankfully there's local linearity, which means we can zoom in and see that the slope at this particular point is going to be um won we see we go up 1/1, So because of this we see the slip is going to be X. So Y equals X, abruptly. We know the point negative 13 is a point. So if you plug in a negative one here and the three, and for y we end up saying that our be valuable before. And sure enough, this does actually produce a tangent line to the curve at the given point. So this is our final attention equation.

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.

We want to find a different station to dysfunction using implicit transition. Let's differentiate. We respect to X on both sides. So for the first time, execute When we differentiate, we get three X square for the class. For the second time we get three wide square. Don't forget to differentiate. The wine will be white prime because to one moment differentiate, we get zero. So bring all the terms without the white plan to the right side. So we have three wide square white prime is equal to minus three X square. So what do we have here? Why prime will be equals toe The three kings canceled will be able to minus X squared over Why square?

So we need to find the tangent line to the equation. Rmc. At uh C. Equal to three. So we want to end up finding something like Y equals mx plus B. Or in this case are equals M. Z plus B. Um Where the slope M. Could be found by taking the derivative our prime Um evaluated at the point we're interested in three. So let's go do this first. Uh So to find the derivative our prime of Z. When you need to remember how to take derivatives of logs. And um to do that we take one over uh the natural log of the base. So one over L. N. Five. And then the bottom we also multiply by the argument, which in our case is to Z squared plus seven. Um But we also have the chain rule and that is to multiply by the derivative of the insect. So what's the derivative of two Z squared? That becomes four Z. And a review of seven is 0. Uh So what is this? We've got four z over Ellen five times the quantity two Z squared plus seven. Um So now we're ready to go ahead and find our slope, our prime of three. So if we plug in three for Z, We get 12 on top over Ellen five times. What is the C squared is three squared which is nine, nine times two is 18 and 18 plus seven. Uh is 25. So let's go ahead and look at this in here. So 12 divided by Ln five times 25. And if we go in around this to the nearest 10th, This is approximately equal to 0.3. So this is our slope. And um so in order to find B, we need a point on the line and we're interested at when Z is equal to three, So when Z is equal to three, what is our? Um so we've got log base five of uh It's our function here too. Times is he squared plus 7? We just had this quantity was 25. Alright so 25 And long based five of 25. Remember this means five raised to what power gives me 25. So this is equal to two. Um so this means that a point on the line is 32 and we have everything but are Y intercept so we can go ahead and put those in. So R. Equals M. Z plus B. Are we said is to Um is .3, Z is three and B. We need to solve for So two is equal to .9 plus b. Subtracting the .9 from both sides. We get Uh 1.1 is equal to be. Therefore our final answer is going to be for the tangent line. Uh Let's keep the same notation here. It's going to be our equals 0.3 Z Plus 1.1. So that's the equation of our tangent line.


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