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Find three different surfaces that contain the curve

$ r(t) = 2t i + e^t j + e^{2t} k $

$y=e^{x / 2} \quad z=e^{x} \quad z=y^{2}$

Vector Functions

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Johns Hopkins University

Oregon State University

Harvey Mudd College

Baylor University

in this problem, It is us that find three different surfaces that contained the cove. R. T equals two 28. Right? Let's eat the poverty J close it to the power to T keith. Yeah. Yeah. Yes. So this is a vector function, right? So vector function is given to us and we need to find the three different surfaces. So how can we do? So let's see. So the objective is to find the three surface. Right? And the parametric equation will be bored. The parametric, okay, equations from the given vector function, right? Is what? This is. X is equal to two T. Yes. Because this isn't the palm X. I plus Y. J plus Z. K. Right. Can we write in this way? Yes. So X is equal to two. Tv assumed this is 40 and this is why. And this is that right? So why equals through into the poverty coefficient of J and coefficient of gay will be value of that. That is into the public toti. So now we've been sold for the equations were unknown. Very even right. So solving equation what unknown variable? So X will be duty. This implies that He is nothing but this is X x two. So we will put this value in by NZ can we put in the violins? Yes, value of T in terms of X. So no By becomes what by becomes into the Power X x two. Hmm. That becomes mm E to the power two. And to expose to that is into the power eggs only. Right? So what can be done now? So for this we can say we can square this value by value. We can inquire it. So what it will come. Mhm squaring, you know, Yeah, boat. These things we get by square it was to into the power X. Because after squaring to two cuts off, that is into the power X. That is equal to that only because that is the value of said yes. So now we can stay here. Yeah. Therefore why is equal to each of the power expert to here? And that is equal to eat the power X. And also we can see that is equal to vice. But yes, this venue came from here after squaring d value by from here. Right? So this is very simple. We have taken the value of T. Uh and we have put the value in by NZ. And we have got this equation. So these are the three surfaces. Right? So we can see these are the three surfaces. Okay, Okay. So this is how we solve this problem. I hope your new strategic concept. Thanks for watching