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Consider the equation Ty" + Ty = 10. For some values of (T; Yo) , this equation deter- mines y as function of near (To:. Yo); in other words; y is an implicitl...

Question

Consider the equation Ty" + Ty = 10. For some values of (T; Yo) , this equation deter- mines y as function of near (To:. Yo); in other words; y is an implicitly defined function near (Tu; Yo ) _Find the derivative d in general for the implicitly defined function: 2. Is (T,y) = (-1,2) one of the solutions to the equation?The point (T.y) = (1,-2) is a solution to the equation. Find the derivative of the implicitly defined function at the point (1,-2).Find the equation of the linc tangent to t

Consider the equation Ty" + Ty = 10. For some values of (T; Yo) , this equation deter- mines y as function of near (To:. Yo); in other words; y is an implicitly defined function near (Tu; Yo ) _ Find the derivative d in general for the implicitly defined function: 2. Is (T,y) = (-1,2) one of the solutions to the equation? The point (T.y) = (1,-2) is a solution to the equation. Find the derivative of the implicitly defined function at the point (1,-2). Find the equation of the linc tangent to the set of solutions to Ty" | ry = -10 at the point (1.



Answers

use implicit differentiation to find an equation of the tangent line to the graph at the given point. $$ x+y-1=\ln \left(x^{2}+y^{2}\right), \quad(1,0) $$

Questions attention line that for this equation here That also going through this point on the one. And so then if we take the derivative here, so we've got why prime is equal to 10 times one plus two X to the next power multiplied by the attributed the inside per chain role just times two. So putting that together here, we get that that's supposed to equal When we plug in zero in there. So we have my prime of zero equals 10 times one, The 9th Times two When she goes 20, that means that's our slope. That will consider here. So we have y minus one ankles, 20 times X minus cereal for the point slope formula. They're from here we Have one to both sides. We have 20 x plus one. So this is hard questions attention like

Hey, it's from someone. You rain here, so we have. Why is equal to X square minus one all over X square plus X plus one. We're gonna use the quotient role to find the derivative when we get X square minus one The derivative X square plus X plus one minus X square minus one and the derivative of X square plus X plus one This is all over X square plus X plus one square. When we simplify this, we get X square. It was for X plus one while over at square, less X plus one square. So to find a slope of the tangent line at one comma zero we're gonna plug in X is equal to one into our derivative when we get 2/3. So we're gonna create a line Why is equal to 2/3 X minus one plus zero? This becomes equal to two birds. X minus 2/3

Look So to get started with this. A big idea is just drawing a tangent line and estimating what that line looks like. So I drew the graph of technology. It's a Parabolas shifted up one unit On the graph is 2.102 which I've identified. So what does the tangent line look like? If I zoom in enough to this parabola at that .12, this little part of the graph will be extended and look something like this. So hopefully that tangent line makes sense to you that at that moment I have a function that's increasing. If I extend that little part of the graph to be a line, that's what the tangent line would look like. So what is the slope of the tangent line? Well, we can estimate it. It appears that another point on the tangent line graph is roughly two comma for So the slope is rise over run That's 4 -2/2 -1. to over one is to so roughly it appears that the slope of the tangent line is too. How could we think of slope a different way? Well, what we're encouraged to do is to just make a little table and see what happens when we consider The slope of a secret line between the .1, 2 and a point closer to that. So what if instead we considered The .1, 2 and then appoint At 1.01 And then 1.01 plugged in. Note, the function is X squared plus one. So if we plug in 1.1, We'd Square that and add one. So what's the slope among these two points? So what we're doing is picking two points super close together In considering the line between those two points. So the slope is the same will take the change in why? Over the change in X. So we get 1.01 squared plus one minus two. That's 1.1 squared minus one. Um That's practically zero, right? And one point oh one minus one Is also practically zero. 1.01 Squared Is very close to one. When we multiply that out, you get 1.0201. When we subtract one we get .0201 Over a .01. When we divide by .01, that's the same as multiplying by 100. So we get 2.01. So this second slope which just means the slope among two points. Is it essentially the tangent slope which we already estimated to be two? Lastly to finish this problem, let's consider a limit. So what is what are we really doing in the previous example? Let's consider the two points. 1, 2. And how about one plus a little amount. We'll call that little amount H and then that plugged into our function. So what we'd like to do is ask ourselves what happens when H goes to zero. So let's find the change in why? Over the change in X. So again each Justin notes Little run. How far away are we from that .12. So if we expand we get one plus two, H plus H squared minus one. Here are the ones cancel. So we get H. We notice these ones cancel. Leaving an H is a common factor in the numerator, which can cancel away with that ancient the denominator now is that H goes to zero, meaning it gets as small as we wish. The limit of this would be to which again is the canyon slope that we saw. So this calculation is essentially doing what we did above. But instead of 1.01, we have one plus any small amount. When that small amount goes to zero, we get to which is the tangent.

Fortune is given Y equals two X, lots X and points are given one and zero. We have to find the the engine client equation firstly we will differentiate this equation with respect to X and will use multiple hated differential rule. So according to the multiplication differentiation it is if we have to differentiate a multiplied by B. So it will be written as a. The ready mix of B plus B, divide the eggs of a no speak and write it in place office A. I will put eggs and in place of me I will put log excell videsh equals two eggs. The ready X off. In place of the means law X plus log eggs. And in place of B. A. S. E. X. No again different. Do you think it This is the identity. So separating this by using this box now it can be written as riders equals two X. Multiplied by as we know differentiation of log exists when upon X plus log X and we know differentiation of access or live one. So it can be written as videsh equal to X and X will be canceled out one plus only log X. This is the differentiation now we have given the points the value of X. S. One and the value of by zero. So in place of XI will put one in place. Software I report zero so it can be a tennis Videsh off one can is one because log X. And the value of log 110 So remaining my dishes request to zero. So the equation of tension decoration of the gentle Willoughby X minus excellent by minus violent equals two videsh. I am my x minus excellent. It means this is the point excellent and I even Subsequute the value via zero equals the value of our dishes Evan and x minus value of access but survive equals two X minus. But so we can say this is the equation of the change in if you want to write it in more easy for. So finally it can be re tennis y minus X. It was two minus. Well also, so we can say the sister, Also the equation of 13.


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