Question
0.1. (30 points) Consider the parametric curve in the xy plane defined by the following parametric equations.x =t3 _ t,y = 3t - 6.Find an equation for the line tangent to the curve at the point where the curve crosses theaxis
0.1. (30 points) Consider the parametric curve in the xy plane defined by the following parametric equations. x =t3 _ t, y = 3t - 6. Find an equation for the line tangent to the curve at the point where the curve crosses the axis
Answers
Parametric equations and a value for the parameter $t$ are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of $t.$
$x=t^{2}+3, y=6-t^{3} ; t=2$
Parametric curve CFT where this tangent line has sloped M equals three. Okay well remember the slope of the tangent line is equal to the derivative of why with respect to X. Okay so your choices are you have two ways to do it? You can use equation six or you can try and eliminate the parameter and then take the drift of the regular way. Okay. Both of these are going to be hard to solve her T. So we want to just use equation six. It's the best way anyway. And it says if you're doing dy dx then you take the derivative of Y with respect to the parameter and then the derivative of X with respect to the parameter. So the derivative of Y. Three T squared minus six, the derivative of X 16 minus two. And they want to know when does that equal to 3? So let's make that 3/1 cross multiply and you get three T squared minus six equals 18 T -6. So three t square minus 18 t Plus six. Those go away equals zero Factor out of three T. So at T equals zero and T Equal six. So at the points in tangent equals three. At When T0 We're at 00 And when T is six were at we get stopped and calculate CF six is 636 times 308 -12. So yeah um 88 96 And then six cubed -6 squared. That's six square times 6 -1. That's 36 times 5 1 8, 210. So at the zero and the .96 to 10 Um the derivative is equal to three.
You just couldn't very calling about attention. Line which on the formula X y z, it will go thio the X 00 z zero plus the experiment on the T zero time stay plus the white prime under T zero time stay plus busy Bram and the T zero time Stay in this question were given the X y Z E Co two that goes I t Y co TT sequel Charity Society and and the pond here Aunt the four minus pi pi zero. Now look at the middle term here, see, why could you d so means Then why could you d equals two pi? And now the next time we need to find a derivative so explain why primacy prime for the experiment going to go to the course I t and then minus d society, the rivers and the TV what you want for the Z prime get to go to the society and then plus the Ghost city now using this value on the t looking under t equal to pi Then we get the express ico to, uh here on the price cycle to zero. So we're gonna minus one. Why? Prime echoed you want the prime echo Jew on the prime? We get equal June the minus one. So will be minus pi here. And if we have attention line, it would be X Y z you go to where the point here will be minus pi pi zero. Then we have a direction here. Dynasty blessed e one spi T.
This question A parameter question on the tension line. I suppose that we have the parameter question on the from X'd Y t and a Z t. Then we have the tension line. It will be on the form X y is the G. It would equal with you the X and attempt and upon t zero why and the time Uh, t zero and see end up on t zero plus the ex prime t zero times 30 and then plus why Prime under T zero times D plus the Z prime and the T zero times d and now in discussion were given vector function XP co +21 plus two squared up T. Why ico it you? The the It's three months d is they go to the deep dreamless day and the pond 302 So here we get this here will be that x t zero white t zero edition with a Z t zero here therefore which need to fight the ex prime y prime in Z prime. So we should get equal to you. They give them the ex prime and we get Geico tune the, uh, one I was carrying the day for the white prime Getting close to the treaty square minus one for the Z prime Going to Treaty Square plus one. And from here we can look in the t e. Go to, uh here. We see that in Lebanon that the, uh we need to find a validity. Now, to find a validity, we noticed that we have the ah X equal to one plus two squared tria equals squared t on equal to three. It means Then the two squared the equal to the two squared the equal to one. And it means that to him is equal to one. And one thing that you want to get the, uh, ex Graham. It will equal to the one when I prime equal to the two and see prime get equal to the farm on our own. Get the tent in line. It will be on the form X. They were equal to the three. And then plus day. Why equal to the zero plus two d and the equal to the two plus 40
Parametric equations of the tangent line to this curve X. Y. Z. At the 0.21. So for sonny dX DT which will be one over T. An dy DT, which will be two times one half T. To the minus one half Which is one over the square root of T. And then DZ DT equals two teeth. Okay? So at the 0 to 1 um we have t squared equals one. So T. Is plus or minus juan. But why is two squirts of tea? So T. Must be positive one. So these slopes become one one and what's two? So x equals zero plus 1. T. Y. equals two plus 1. T. Z equals I can't see what that oh two plus one cheek. Wait X's wine. Okay. See is one uh just got all confused there. One because that the uh in the point the z coordinate is one and then plus two T. Because that's the slope. So one T. Two plus one T. One plus two T.