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Consider thc curve parametrizcx byrlt) (cos t,xint,e )0 < t <Compute 7"(t) xr"(t). Compute thc curvature of the curve the point r(t). Compute thc bi...

Question

Consider thc curve parametrizcx byrlt) (cos t,xint,e )0 < t <Compute 7"(t) xr"(t). Compute thc curvature of the curve the point r(t). Compute thc binormal vcrtor B(t) at the point r(t) . (Hint: Usc thc result fromFind # cquation of the oxculatiug plane of the FMTIMthe point (1,0,4).

Consider thc curve parametrizcx by rlt) (cos t,xint,e ) 0 < t < Compute 7"(t) xr"(t). Compute thc curvature of the curve the point r(t). Compute thc binormal vcrtor B(t) at the point r(t) . (Hint: Usc thc result from Find # cquation of the oxculatiug plane of the FMTIM the point (1,0,4).



Answers

Find the curvature $K$ of the plane curve at the given value of the parameter. $$\mathbf{r}(t)=t^{2} \mathbf{i}+\mathbf{j}, \quad t=2$$

In this question would record on the curvature K the X on the function F X You go to the F number, prime the X on, then taking the norm. Absolute here they wanted by the one plus F Graham on the X. How about you? Totally. About three on with you in this question were given the function f X because you of course I off the X and the ex coaches zero we see the first time you need to find a derivative of the function include U minus. I'm the ex from here we have the m prime on the zero because exactly zero here in court, you know, zero and then the second area. But they're on the function in called gender minus because I am the X. I found f double prime on the zero. It goes to the minus one, and therefore we can apply the formula to get the curvature on the value zero. It would go to the after prime minister. Absolute value equals to one divided by one plus zero. Power to power three over to then we can equal to one. That's gonna be the curvature we're looking for here.

In discussion what we call about a curvature family formula guy ego to the norm of uh empty. Yeah. Alright. Stay up from T. Comes with the end of a prompt E. And then they're running behind the numbers are empty square battery. Sorry here we are given the girl R. T. E. Co two T. I. Plus one of the nine deeper three. Train from the T. Able to to notice that we need to perform the grass products so we need to insult his. Okay here now the first time we need to compute the prime T. And we go to the uh I got you one here and Anderson gonna be uh huh The square on with three and 0 then the second derivative go to zero and then to our treaty is there. So from here we shall be able to compare the cost from the pitching them. That's all it will echo jew here we had the first time. We with zero the second term it will be the minus two of the treaty. The last time. Uh Sorry the minute time still you could choose their own and the last time it will be two and treaty and dan. Using the T. Equity to we should have a very little energy equity to So once you get to go to the Zozo four out of three and therefore we can find a curvature. Okay, Echo to the gnome of that ones will be square root on the phone square of a three square and then they want anybody know on the upper empty. So this one here it would echo to do. We need you here. And then we should have the number of the square root of 1-plus Jules Square. We fall so far over three square and then this will be about three. So yeah, I got you. The square root on the 16th over nine divided by the square root of One plus 16 over nights will be 25/9. Our three. So we have here will be four out of three divided by five out of three. About three. So we should get Geico june the 4/3 times the three power three the running by 53. So we have here the top will be in the cancel on the tree. So we have here will be three square will be nine items for 36 hours 1 25. And this could be the answer.

Okay we're going to find the curvature K. Of the plane curve traced out by the vector given. Now when you can write your vector in terms of why equals there is a different formula than um when you want to stay in parametric. So I'm going to go ahead and you know set my two T. Equal to X. So then I know that t. equals x divided by two. And so now I can go ahead and say that why equals that X divided by two to the third power. We can also say that's like a 1/8 X. To the third. This is going to make it a lot easier to take the derivative. My first derivative will be 3/8 and then I go down to power two X squared. My second derivative will be six states or what you can say three force times X. So now to actually place our values in the formula. Um I'm gonna first put them in terms of X. So I have that 3/4 X. Then I have one plus the square. So notice the three squares and the eight squares. Um and then I am to the X. to the 4th. And then that is to the 3/2 power. So we're now ready to take the excess out and replace them with two teas. So my X. To the fourth will be a two to the fourth time's A. T. To the fourth. So that will be 16 T. To the fourth. So that second fraction inside my apprentices. I'm gonna go ahead and clean that up. I also clean up my numerator as I do that. So now think of putting in common denominators, I'm gonna make my one a 4/4 and then the step I'm actually going to take that denominator four and take it to the three halves power. So remember square, it is two and then to the power of three is going to be eight. And that leaves me a four plus nine T to the fourth inside the parentheses. And then finally the eight and 2 can clean up Multiply Times three and then we can place in our denominator of four plus 92 the fourth all 23 helps power.

In this question would record about the curvature formula K. Ego. You know on the our prime T crossed with the aunt of a prime T. And then they running bind the norm on the our prime T. Taking about three. Now in this question when given the girl RT Echo two the T. I was one of a DJ Founded the Eco 2 1. Now the first time we try to computer uh notice that we have to perform the cost product so therefore we can insert the brazil came here. So it will help us to compute the cost product that found the opera empty. We go to the eye and then this will be plus the river to him will be minus one over the square tree. And then plus there. Okay And then on the part from T. and we Echo 20 I here we have the blast and we have the two hour TBA three train plus you're okay from here we will be able to compute the hospital between the our prime T. And end up empty. So therefore we shouldn't get ankle at you. This form the minus in the middle of term. So the first term will be zero. The middle term, it will be also zero. The last one will be the two. All the T x three. And then from here we should be able to find the curvature echo ju Now the time will be the norm. So will be the square root of that one. So we have only the far over the power six. And then they went invented gnome on the our prime T. So we have the square root of One last one of 41 of the tape four. And then this will be power three. So we evaluate this one ended the equal to one. So therefore we should get isn't very under the equal to one. So therefore we should get equal to the square it and far dividing by. He'll have the square root on the two hour three. So we have here will be to all the two squirrels or two. So counselor wanted to those who have equal to one over two And one I was credited to. And that's gonna be the answer.


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