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BONUS: Find the slope of the line tangent to the parametric equations below (known as the butterfly curve) at the point (x,y) =(,e - X= sin(t) ecos(t ) 2 cos(4t) - ...

Question

BONUS: Find the slope of the line tangent to the parametric equations below (known as the butterfly curve) at the point (x,y) =(,e - X= sin(t) ecos(t ) 2 cos(4t) - sin' (#)) cos(t) cus( ) ~2cos(4t) - sin (z te"

BONUS: Find the slope of the line tangent to the parametric equations below (known as the butterfly curve) at the point (x,y) =(,e - X= sin(t) ecos(t ) 2 cos(4t) - sin' (#)) cos(t) cus( ) ~2cos(4t) - sin (z te"



Answers

Find the equations of the tangent lines at the point where the curve crosses itself. $$x=2 \sin 2 t, \quad y=3 \sin t$$

Were given Parametric equations for a curve in the point lying on this curve, we were asked to find Parametric equations for the tangent line to this curve at this point. And then we were asked to graph both the curve in the tangent line together. Parametric equations for the curve are X equals t co sign T. Why equals T and Z equals t signed And the point lying on the curve is negative pie high zero from y equals t you have at this point T is equal to pi you have express empty is equal to by the crowd True negative t sigh Inti plus co sign of tea Why Premise t is equal to one and z prime of tea is equal to ti ko sai nt plus sine of t So if teasing Europe I we have the ex prime a pie equals negative pipe and 00 plus co sign a pious negative one white kind of pie is one and Z prime pie is equal to pi times co signing pie This is negative pie plus zero mega pie So we obtained that a vector parallel to the line Just hand it to the point is the vector negative one one negative pie. And so now we have It was a point on the line and effective parallel to the line so that we can write the parents of equations for the line as X equals negative pi minus t. Why equals hi plus T from Z equals zero minus pi t. Using Dismas is three d parametric curve graph for I graphed both curve and blue in the tangent line in orange and you can see that this is the point on the curve here in green, which is where the line is tangent to and we see that points on a curve close to this Greenpoint. The tangent line is a decent approximation, but as we get further and further away in the curve, the tangent line becomes a worse in worse approximation.

This question. Will even the Berman three questions thanks Nico to one breast Jew goes off the and why ankle? Two months to us, too, citing. And we noticed that we can write. Write a stare into the X minus one nickeled unit, your core society and why plus two according to society and every square on everything. Then we should get ICO Jew the X minus one square It could uniform course square t something. Why plus two square Get a good you know for So I squared. They follow size square to you can register, Angela One man us go size squared the and then we get echo June, uh, for minus phone goes I square t Then we say far causes. Quit the equity. This one citizen implies them while I'm blessed to square echoed zero for minus thanks minus one square How we get this one will be X minus one square Plus why bust to square Eager to for this one with a circle was this center You go to one month's Jew and a radius Go to the square root of fun It could you too so it would try to a scant it will be. Just go now. Center it. Could you one minus two. So it will be the center will be. Yeah. And then the writers it could. You do. So we have this circle now.

Today we're going to solve problem number 32. He had given the question Is X equals 70 bye equals saying to the so eat exa quince 70 y equals find toe equals zero. It could be zero on by will be zero and vehicles by before 0.7. What a D equals pi bento Learn zero right? The vehicles three. By before it will be 0.7 minus luck because by zero and still fight by before it's minus 0.7 and what it's prepared to it will be mine. A smart. It's a little at seven by before it's minus 0.7. I don't minus one. It's bye zero A little. Yeah. Kerr will be Make a good one. Thank you.

In this question, we want to find the equation of tensions at the point where the curve crosses itself. Now this is a pair of parametric function. So our task today is to grass to find where the curve crosses itself. Two, we need to establish what is the dy dx dy dx is actually the wine over the T divided by the X over T. T. Entirely. The equation of attention. You will use this y minus Y. One equals two. M x minus X one. Where M is the gradient X one, Y 1 is the point where the code crosses itself. So, if we were to graph using any graphing utility, the cup itself actually looks something like this. So the point it crosses itself is actually at the origin. Yeah. So at the origin is 00. So the point of interest here will be X0. Why is zero? So this will be zero And this one will be zero. Now, let's find our D Y D T and R D. S. D. T. First. All right. So the X. D. T. Differentiating extra respected T. Since two is multiplied to the function here. So too is kept aside Now sign when I differentiate we respect here. I'll get co sign booty and don't forget to differentiate the two T. Itself duty. The two can be kept aside. Now when you differentiate T. Itself with respect to T. You just get one. So that will be all. So the X. DT will be four co sign to T. Now for dy DT Trees multiplied to 70. So trees kept aside not sign when I differentiate I'll get called society. So let's take a look at our D. Y. The X. It will be Dy DT. So that would be trico society divided by the X. D. T. And that will be four Hussein duty. So at the point interests of 00. So at 00 where is our T. Now? We can find the TAT 00. We can just stop into over here. So when X equals zero I have zero equals two to sign duty. So my t. is going to be zero and the other T. Will just be hi So at T goes to zero. Okay? Or when he goes to zero. Yeah West. My dy dx dy dx will be just some zero into the T. Over here. You will get three quarter. And so the tension would be why minus Y. 10 equals two. Now the tension street quarter ingredients you called over X minus X. Y. Is also zero. So one of my tension will be white goes to cheek water. X. Thanks. Now when he equals two pi my dy dx again you t goes to pie into here. Your dy Dx will be minus three quarter. So my tension will be Y minus zero equals two minus creek water X zero. So my other tension line is y equals 2- Trick Water X. Mhm.


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