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Bonus) Let c(t) = (t" 2t) and f (x,Y, 2) = (x y , 2xy, 2) (a) Find ( f c)t): (b) Find a parametrization for the tangent Iine t0 the curve f 0 € at t =...

Question

Bonus) Let c(t) = (t" 2t) and f (x,Y, 2) = (x y , 2xy, 2) (a) Find ( f c)t): (b) Find a parametrization for the tangent Iine t0 the curve f 0 € at t =

Bonus) Let c(t) = (t" 2t) and f (x,Y, 2) = (x y , 2xy, 2) (a) Find ( f c)t): (b) Find a parametrization for the tangent Iine t0 the curve f 0 € at t =



Answers

Let $\mathbf{c}(t)=\left(t^{3}, t^{2}, 2 t\right)$ and $f(x, y, z)=\left(x^{2}-y^{2}, 2 x y, z^{2}\right).$ (a) Find $(f \circ \mathbf{c})(t).$ (b) Find a parametrization for the tangent line to the curve $f$ o $\mathbf{c}$ at $t=1.$

Problem is, find an equation with attendant to the curve at a given point. And then Graff the curve and the tenant first that we compute slope of the tangent. It's assist you. Why the axe? It is you got to be. Why he over yaks. So why did he is to keep us lan on the axe. Titty is two minus one and that is a point. There are three. We have his square minus t is equal to zero. On that, he square past Ruslan. Yes, the photo three. So I write this's one. This is two two minus line. We have to be plus one city. Code three key is equal to one. Planning the snow. Here we have you. Why, yes, it's the culture three. So the equation was a tenant into the curve. At that point, there are three. Is why minus three issue called toe read Hans Sachs. Or you can write like, why you going to three Ice Class three. And now if we use some graphing tours to graph the curl on his attendant So here the blue one is a graph was a curve. And is it right? One is a graph of his attendant

The problem is find an equation of the tenant into the curve. That is a given point on graph the Cruyff and is a tenant. Is he going to sign party? Why is he going to square capacity? The point is Cyril too. First of a computer. Salo Paulo Determinant! Do you know why Yaks that since he called toe delighted He over yaks. Titty? Why did he is to keep us Juan on DH. Jax did. He is high times, Signe Hi Tio. And one x Y is that is the point zero to we have a sign. I is little too. Cyril. Andi He squired last year. And the culture too certainly have. He is Nico to Juan war. He is in control. Negative two on one. We're plugging. He got one and t go too negative to the wide. The acts we can't find Also the ICO too negative. Three or four high? No. For example, Monti's called One thing is three on Cus. I'm pies next he wants So this is next to the aerial up. Hi, Auntie, is you go to negative too. The numerator as next to three. But Kasai negative too high is one, since it's also negative. Really? Pie is that your creation was a tenant. It's why you can't tow negative three o war. Hi, I'm Saks and plus two. Now this is a graph is a fluid Juan. No one is to graph of the curve and the right one is the graph after tangent.

Okay in this video we're going to be looking at the tangent of the curve X. Is equal to co sign T plus co sign to T. And why is equal to sci fi Plus side of two T. And we're gonna be looking at the 0.-1 comma one. And since you're going to find the tangent line and we know we know that why X. And Y. Is negative 11. We need to find the slope which is the derivative dy dx at this point negative 11. And we need to find the time T. Because this time T will tell us when we take the derivative, how we can plug in to solve. So we need to solve the system of equations. We know that co signed, he plus co signed to T. Is equal to -1 and then sci fi The sign of two T. Is equal to positive one. So here's how we can think about this. So the maximum value Of Sine or Cosine is one. So theoretically if we have something like this, let's go ahead and draw the unit circle. We know that our X values, You have a max of it 1 -1 and our live values have a max of one -1. So the blues would be the signs and the greens would be the coincides. So with theoretically we should be able to find its pointy where one of these Is equal to -1 And the other zero or Vice Versa. Um And then also you can find one where this is one and this is zero, I actually believe that it will be the other way. So we'd have um If we start say we start co sign of pi over two. So co sign of pi over two. This is what this this would be equal to zero. I put the red dot but it should be on top of the blue. And if we do cosine pi which is two times prior to we get here co sign of pi is equal to negative one. So this work. So this is zero and this is negative one that would correspond to She is equal to Pi over two. Now we know we found the right time if the same is true for the science. So if we see sine of pi or to sign Pi over two is equal to zero is a good one. And then here sign of pie It's equal to zero. So it actually does work. So we see that this is one and this zero if time is pirate over to and so we have a valid solution T is equal to pi or two Results in the points -1 to come a one. And so now we can take the derivative DVD X and see what we get and plug in power or two. So we know that we have. Why is scientific lists I into T. I haven't directed appreciating that just gives us I'm sorry let me go ahead and write the whole thing. We know this is dy DT over the X. City. So we see that we have courtside seats close to co science. Ut as ry derivative. And since our X derivatives are the same thing with Co Science we have negative sci fi My s to sign up to two and then so we want to plug in so we want to plug in T. is equal to pi or two which gives us the following we see coastline power to Is we know it's zero and then we have plus two times cosine of pi which is negative one. And then we have negative sine of pi over two which is negative one -2 times zero. And so our slope just ends up being negative two of a negative one or just positive too. And now it's easy to write are tangent line because we know dy dx had negative one one is just equal to two. And so are tangent line is just Y. Is equal to two times X plus one plus one. And so this is a valid solution. This is a tangent line to the curve. And so now we want to grab this alongside our uh original curve. So let me go ahead and pull up graph so we can see here that we know that this blue curve is our parametric. So that's our parametric curve and his purple line is our tangent. This is Why is equal to two times x plus one plus one. And you can see that this point here is negative one comma one and it it's perfectly tangent at that point. So we know that we found a valid solution Why is equal to two times x plus one plus one.

The problem is, find an equation of the tangent to the curve at a point corresponding to the cumin. Wallows off the pipe permit. Er axe. Is it going to root of T plus one? Why is the equal to the square minus Tootie? The T is the photo. For first we compute the slope of the tangent. The snuff. Asti Y Yeah, this is a culture. Why did he over yaks, You hear? Why did he is to be minus two on that? The X ditty is half terms you two make tive one half on. One key is equal to war. Axe is equal to three. Why you got to sixteen minus eight. The six to eight and it's a slow. But why, Jax? It's the control. Eight minus two over one half times thiss one over. You tell forthe one over two since Lee Culture twenty four. So the equation of the tenant is why minus paid. Yes, he got to on before Ham's X minus three simplified this equation. This is why he called too. Twenty four times sacks minus sixty four. This is, uh, equation of the tenant into the curve.


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