solve-the-given-problems-by-using-implicit-differentiation-show-that-two-tangents-to-the-curve-x2x-y

Question

Solve the given problems by using implicit differentiation.
Show that two tangents to the curve $x^{2}+x y+y^{2}=7$ at the points where it crosses the $x$ -axis are parallel.

Tom Greenwood · Accepted Answer

right. We have the equation X squared, plus X y plus y squared equals seven, and they want us to find the points where this crosses the X axis. And so if it&#x27;s crossing the X axis, that means why equals zero. And so if we plug in Y equals zero, obviously the X Y and the Y squared term become zero, so we have X squared equal seven or X equals plus or minus the square root of seven Now, next thing they asked us to do was to show that the tangent lines at those two points are parallel to each other. Well, parallel lines have the same slope, and the first derivative, when it&#x27;s evaluated at a point, gives us the slope of the tangent line. So let&#x27;s find the first derivative here. So driven about X squared is gonna be two x plus, doing implicit differentiation and using the product rule. We have X times, the directive of why we should be R D Y d Ekstrom and use the white prime notation, plus why times the derivative of X, which is just one and then plus the derivative of Y squared, will be to why times why prime and that&#x27;ll equal zero. And so now we can solve this for why crime or we just plug in our values of plus and minus square root of seven as is. And that&#x27;s what I&#x27;m gonna do. First, I&#x27;m gonna plug in X equals Positive Square root seven. So that would give us to square with seven plus the square root of seven. Why crying plus R Y value is zero. So why is zero? And that means this is also zero. So I don&#x27;t even need my plus sign. I can just say that to square root of seven plus the square root of seven white prime equals zero And then we subtract the two square root of seven from both sides and we get square with a seven y prime equals negative to square of seven. And then when we divide, we get why Prime equals negative too. And now for X equals the negative square root of seven. Same thing putting that in. We have negative two square root of seven minus square to seven times by prime and our Y values were still zero, and that equals zero. So negative square with a seven. Why prime all equal, positive two square root of seven. So that means that why prime equals negative to when we divide by the negative square root of seven. So we have the same first derivative. So that means the changing lines have the same slope. So, yes, they are parallel to each other.

Robert Daugherty · Answer

okay, section 3.7 problem number 41 were given a curve X squared plus x y plus y squared equals seven were asked to find the two points where there&#x27;s curved crosses the X axis and show that the changes of these points are parallel. So if this thing crosses the X axis, that&#x27;s just code for why is equal to zero. Where is why equal to zero in this equation right here. If I set, why equal to zero? I get X squared equal seven, which means that X equal plus or minus the square root of seven. So what this boils down to is I need to look at the point. Negative square 270 and at the point square root of 70 And I need to find the tangent line to What was this equation X squared plus X y plus y squared equal seven. So to find the change, that line that&#x27;s this code for any defined the derivative. So this is going to be too X plus Okay, then use the product rule the derivative of X times. Why, plus x times, the derivative of why plus to why do I D X is equal to zero. So this means that X plus two I do I. D. X is equal to negative to x minus. Why? So this means that D Y d X is equal to negative two x minus. Why, over X plus to why so now it just comes out. I need evaluate this derivative at both of those points. So again, D Y d X is equal to negative y minus two x divided by X plus, too. Why? So let&#x27;s evaluate. Do I d. X at the square root of seven. So the square root of 70 So that&#x27;s evaluated at this point. So that is going to be zero minus two square root of seven over the square root of seven plus zero. So this is minus two route seven over route seven, which is minus two. Now let&#x27;s evaluate this same derivative. Do I D. X at the point. Negative route 70 So that gives me zero minus two negative. Route seven over negative Route seven minus zero. So this gives me to route seven over negative route seven, which is negative too. So what have I done? I found the points where I cross the X axis. That was wise it with a zero. Those points were a route 70 and negative room 70 I found the derivative evaluated at those two points and found that the slope was the same. So if the slope

Bahar Tehranipoor · Answer

So this problem wants us to find points on the curve X squared. His ex wife&#x27;s pi squared equals seven, eh? Where the tenant is parallel to the axe access and be the tangent is parallel to the y axis. So first, let&#x27;s solve for do idea so we can find the slope of our tangent line. We do this using implicit differentiation. The derivative of X squared is too X Then we use the product rule here to find X d Y d X plus one times Why? So why? Who use the product rule here and then Plus two. Why do you Why d X equals zero? Because a derivative seven it&#x27;s just syrup. So now we&#x27;re going to take all the terms that have do ideas and leave them on the side and subtract the rest the other side. So we get rid of two X and move it here. We&#x27;re going to get rid of this. Why movie here? So now we have X D Y d X plus two. Why do you lie? The X equals minus two x minus y we&#x27;re gonna factor out this DVD X this and then we&#x27;re going to divide both sides by this expose to eye. So we&#x27;re left with D Y D. X equals minus two x minus y over X for us to live. So to solve for this problem where it says a and B, we&#x27;re gonna look at a first. This is where the tangent is parallel to the X axis. And of course, if the line is parallel to the X axis, it would look like this. And that would mean it&#x27;s why equals something. And that means that our Sloper, because a slope equals zero. So we&#x27;re going to solve for where do your ideas equals zero for part A. We&#x27;re going to say negative two x minus y overexpose tau equals to row If we moved on both sides, but it&#x27;s exposed to equal zero, we get minus two eggs minus y equals zero like that and we could do is we could just make it negative. Two x equals y and we have Why equals negative 1/2 X or actually, no. Sorry about that. Why equals negative two x It&#x27;s all we have. We have y equals negative to X is where our slope equals zero. We could plug it back in and we see that&#x27;s correct. And now that&#x27;s where it&#x27;s perilously exactly. And then where we have B that says, Where&#x27;s the pearls? The Y axis? That means its X equals something. And so that means that do idea isn&#x27;t defined and do idea exciting to find with the denominator equal zero So explosive, too. I should equal zero. That gives us two. Why equals Native X, And this is where why equals negative 1/2 x means that our slope isn&#x27;t defined in that it&#x27;s too. It&#x27;s parallel to the why accents. So this is why I chose to act is the points where attention is probably exactly as in y equals negative 1/2 facts of the points where the tenants pearls relaxes.

Clarissa Noh · Answer

Hey, it&#x27;s clear. So when you re here so you have X minus two. Why is equal to two and this continue Y is equal to 1/2 X minus one. This is the M, which is the slope of the line. So here we&#x27;re gonna find the derivative of the curve. Why is equal to X alertness one over X plus one. The derivative is equal to one times next, plus one minus one term X minus one over X plus worn square which becomes equal to two over X plus one square. So this is the slope of the tension line to Wyatt X. So when we plug in, make this equal to one. Have we get when half is equal to two over X plus one square that&#x27;s can be equal to negative three or one. So we&#x27;re gonna get the Y coordinates. So when X is equal to negative three Well, why is equal to two when X is equal to one? Why is equal to zero? We&#x27;re gonna pluck the points and slope into the point slope line equation. So for a negative three korma too, we get why is equal to 1/2 x minus and negative. Three less, too, which is equal to one. Have X less stubborn has for one comma zero we have. Why is equal to one, huh X minus one plus zero, which is equal to 1/2 X minus one her.

Mike Gaerlan · Answer

for part A. We need to verify that this point lies on the curve. So to do that, we just plug in our X and Y into this and verify that this equation is still satisfied. So for X, we&#x27;re going to plug in to and then for why we&#x27;re gonna plug in one. So we&#x27;ll have this is too squared. Plus and then to times one and then plus, why squared? So one squared right? And we&#x27;re trying to check to see if this is equal to seven. When we take this, this becomes four plus two plus one, and it&#x27;s not equal to seven. Well, four plus two is six plus one is seven. So we do get seven is equal to seven. So we have verified that this point does lie on the curve. Okay, Next, we need to determine equation of the tangent line. So to do so, we need to first find d y t X. So to find d y t X, we&#x27;re gonna take the derivative with respect to X on both sides of that equation. So yes, a a plus bi square. It&#x27;s gonna be equal to the derivative of seven. So Let&#x27;s take this site first derivative of X squared by power rule, That&#x27;s two X and then plus this derivative here, we&#x27;re gonna need to use product will. So we&#x27;ll do First times derivative second. So that&#x27;s gonna be X plus and then D y DX and then plus second. So why times the derivative of first? Just just one. Now we&#x27;re gonna take the derivative of y squared. That&#x27;s gonna be, too. Why, by Power rule and then by a chain rule, we need to multiply by D Y t X. And then on the right hand side, we&#x27;re just taking the room of the constant, which is this zero. Next, we&#x27;re gonna leave all of our d y. The X terms here and then we&#x27;re gonna move are non B y the X terms over to the other side. So we&#x27;ll have x d Y d x plus to why d Y. The X is gonna equal to negative two x minus y. But next I&#x27;m going to factor out of b y the x from both of these terms. So I get X Plus two. Why is equal to negative two x minus y? And then now to find the wide e X. I&#x27;m going to divide by this here, so we&#x27;ll get that D Y d x Move this up here D Y d x is gonna be equal to negative two x minus y all over X plus two y Now to actually get the slope. At this point, we&#x27;re gonna plug in. I wear variables so d y r d x at two comma one this end of equal to So, um, I&#x27;ll do this into different colors again. So two comma one here, there&#x27;s gonna be good. Too negative too. Times two and then minus We have one here and then one was our X, which is to plus two times one. So here this becomes negative for minus one. So we get a negative five at the top, and then at the bottom, we have two plus two, which is four. So are slow. It&#x27;s gonna be negative. Five force. And then because we have a point, we&#x27;re going to use point slope for him this point. So formula remember the point. So formula is why equals or why minus why not is equal to m times X minus X Not so plugging in our numbers. We have y minus one. It&#x27;s gonna be equal to our slope. So our slope, which is negative five force and then times X minus ex lot is too. So let&#x27;s simplify this We have Why is gonna be able to, um y minus one first is equal to negative five Force X and then this times, this is going to be able to positive five halves. And then now we&#x27;re gonna add one over to this side. So we get our final equation is going to be equal to why is equal to negative 5/4 X plus seven halves.

Problem

Solve the given problems by using implicit differ…

Problem 31 Easy Difficulty

Solve the given problems by using implicit differentiation.
Show that two tangents to the curve $x^{2}+x y+y^{2}=7$ at the points where it crosses the $x$ -axis are parallel.

Answer

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

Anna Marie V.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus