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IE34: 2uzu-06-30 23.00Z255 i(u) = ue[0,1] 7 () = ue[12] Zu(l - u)' I(u-1)? A planar curve consists of two segments as and (1) How is the parametric continuity ...

Question

IE34: 2uzu-06-30 23.00Z255 i(u) = ue[0,1] 7 () = ue[12] Zu(l - u)' I(u-1)? A planar curve consists of two segments as and (1) How is the parametric continuity Ck in the segment = (u) ?How is the parametric continuity Ck of the whole curve?(7.05)T WEfepitt

IE34: 2uzu-06-30 23.00 Z25 5 i(u) = ue[0,1] 7 () = ue[12] Zu(l - u)' I(u-1)? A planar curve consists of two segments as and (1) How is the parametric continuity Ck in the segment = (u) ? How is the parametric continuity Ck of the whole curve? (7.05) T W Efepitt



Answers

Find parametric equations for the given curve.
Segment joining $(1,1)$ and $(2,3)$

And the problem. Who has to find the parametric curve joining the segments 11 and 23? Well, that's just the line. So we can find the slope of the lines of the change in the mom. Why is, too? And the change in excess wants of the slope is too. Then we can take a point that's on that line and with the slope weaken right, the equation for the line, which is then why equals two times a quantity X minus one plus one. We can then set X equal the tea. And if we do that and simplify this, we get why equal? It's two to t minus one and then where T would range from 1 to 2.

We're given the equation for the track tricks curved where L. Is greater than zero. And we want to show that it has the following properties that for all T the segment from CFT two T comma zero is tangent to the curve and has a length. Yeah so we have our track drinks curve there and here I we produced a figure a little bit. So we have our curve and then a tangent coming off of that curve at some point. He not. And that tangent intersects the X. Axis at a value Tina. And the length distance from this point at this point is that L. So first we need to get the tangent of this track trig function. And so the tangent vector is simply this. Okay so we can use some trig identities and just take the derivative of these hyperbolic hyperbolic functions. Get this value here. So what we wanna do is you want to say well the line this line here has a slope of why component of the tangent divided by the X. Component of the tangent. And that turns out to be the negative. Um Co hyperbolic co second or one over the hyperbolic sine of tea? Not over L. And then we know that we need to take the slope times X minus X. Not an X. Not. Is this guy evaluated a tina? And then we need to add on why not? Is this guy evaluated a peanut? Yeah we can figure out if we do some simplification and use some hyperbolic identities. We get this thing is actually the quantity X. 14 at minus X times the hyperbolic cosi can of tangent of tonight over L. So that's what this line is. Yeah. So what we can see pretty clearly is that um when X equals t not Y equals zero. So in fact we cross zero at this point. So that is true. And so that would mean to find the distance. All right. And so we get this result. Obviously when X is t not why is zero? And now we want to find the distance. So we have two points. We have a point on the track tricks and our point on the act on the on the X axis. And then we just need to figure out the distance between those two points. And we have those two points. And if we go through, yeah, if we plug those, plug that in, we get that. We need we take the difference and things. Something is cancel out. And we get that we need the magnitude of this vector. And we can using some hyperbolic identities. We can show that the magnitude of this isn't just L.

We're giving the equation for the track tricks curve where hell is greater than zero. And we want to show that it has the following properties that for all t the segment from C A T to T comma zero is tangent to the curve. It has a length? No. So you have our track drinks curve there and here I he produced the figure a little. So we have curve on a tangent coming off of that curve at some point, Tina and that tangent intersex, the X axis value and the length distance from this point at this point, is that so? First, we need to get the tangent of this track tricks function. And so the tangent rector is simply this. Okay, so we can use a trick. Identities, Just take the derivative of these hyperbolic metabolic functions and get this value here. So what we want to do is you want to say well, the line This line here has the slope of why component of the tangent divided by the ex comport Qianjin on That turns out to be the negative Um Cole Hyperbolic Oh, seeking one over the hyperbolic sign of tea not over and then we know that we need to take the slope times X minus X not It's not ISS. This guy evaluated Aquino and then we need to add on Why not? Is this guy evaluated? And you can figure out if we do some simplification and you some hyperbolic identities, you get that this thing is actually the quantity x 14 at minus x times the hyperbolic coast seeking of tangent of Tina over l. So that's what this this line is. So what we can see pretty clearly is that, um when x equals t not why you go zero So, in fact, meet cross zero at this point so that it's true and said I would lean the find the distance all right. And so we get this result. Obviously, when X is Tina, why is zero now? We want to find the distance. So we have two points. We have a point on the track tricks and our point on the x Acto y it on the X axis. And then we just need to figure out the distance between those two points and we have those two points. And if we go through, no. If we plug those plug that in. You get that. We need to take the difference. I think something is cancel out. When we get that. We need the magnitude of this record. Um, and we can using some hyperbolic identities, we can show that the magnitude that this isn't just out.


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