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The graphs of the polar curves: r =4 r =3+2 cos are shown in the figure to the right: The curves ande 5T intersect at 0Let R be the shaded region that is inside the...

Question

The graphs of the polar curves: r =4 r =3+2 cos are shown in the figure to the right: The curves ande 5T intersect at 0Let R be the shaded region that is inside the graph of r = 4 and also outside the graph of r = 3 + 2cos 0_ as shown in the figure above. Write an expression involving an integral for the area of R:(b) Find the slope of the line tangent to the graph of r = 3 + 2cos € at 0 = 2(c)particle moves along the portion of the curve r = 3 + 2cos € for 0 < 0 2 The particle moves in s

The graphs of the polar curves: r =4 r =3+2 cos are shown in the figure to the right: The curves ande 5T intersect at 0 Let R be the shaded region that is inside the graph of r = 4 and also outside the graph of r = 3 + 2cos 0_ as shown in the figure above. Write an expression involving an integral for the area of R: (b) Find the slope of the line tangent to the graph of r = 3 + 2cos € at 0 = 2 (c) particle moves along the portion of the curve r = 3 + 2cos € for 0 < 0 2 The particle moves in such a way that the distance between the particle and the origin increases at a constant rate of 3 units per second. Find the rate at which the angle 0 changes with respect to time at the instant when the position of the particle corresponds to 0 = Indicate units of measure.



Answers

Give the position vectors of particles moving along various curves in the $x y$ -plane. In each case, find the particle's velocity and acceleration vectors at the stated times, and sketch them as vectors on the curve. Motion on the circle $x^{2}+y^{2}=16$ $$\mathbf{r}(t)=\left(4 \cos \frac{t}{2}\right) \mathbf{i}+\left(4 \sin \frac{t}{2}\right) \mathbf{j} ; \quad t=\pi \text { and } 3 \pi / 2$$

This is Chapter four. Problem number 114 were given a position makes for particle on the X Y plane position. Victor goes as to tea. I had plus to sign Pi ti over four j hat now in part they were asked the X and Y components at Time T equals zero Juan 234 seconds S o X component is you know is gonna be to t here. And the white component is gonna be to sign high tea over floor. So we'll only a we need to do is just playing in the the time here for you know, t zero people want people studio because three Teeples four seconds now, extent is gonna be when? Let's use this sort of notation The eggs, then at time equals zero seconds is gonna be cool to zero right x time equals one. Then two times one, he's gonna be two seconds. Two meters? Probably. These air, the position strike in the extraction T equals two seconds. Gonna be four meters. T equals three. Second is gonna be six mirrors, ext Eagles. Four second is gonna be eight meters now for why we're getting the same. It's just we have the sine function here. Why? He holds t zero second. Um, science zero. Brian's hand 00 Why? T equals one second. Regan is one here. Pi over four. Sign pi over four attempts to he's a spurt of tooth. And then why, tinkles? Two seconds is gonna give us. I have to, um, Times too. So power or two is one. So we're gonna get to from here. Why? It goes t three seconds. We're still getting square root of to break White was t four seconds we're getting for these are gonna cancel out Sign Pi zero. So we're getting back to zero now. We're asked to sketch the particles trajectory based on these positions in both x and y directions that became so you need to do is just draw If you want to put this on, you know, on a software that's fine. I'm just gonna sketch it because that's what we're asked. At the end, it's X valleys or 02 46 eight. So when where? Zero wise also zero. And then it goes to square root of two, then goes up to two and then comes back. So, um, when when we are at this is where time people zero. Right. So then the second point is actually here. And then the part made the turn point is here. And then we going back to spirit of two, right? And then we going back to zero action. So particles trajectory. Let me draw. Bad is the line that could extend these thoughts to each other. Right? All right. So again, this point is where t equals one teeth falls two seconds. He was three. TV goes four seconds right now in part B. This concludes part a party were asked Frank was velocity et velocity at T equals 123 seconds. So in order to get to the velocity, what we need to do is to take a derivative off the position vector that is given to us. If you do that, you're gonna find two. I had so distributable that signed term is gonna come out to me. Pi over two whole sign Pie t over four J hat. Right. So then then velocity When Teeples one is gonna be too. I had less 1.11 J hat velocity is gonna be 20 equals two seconds to I had because that's independent, right? Plus zero j hat because I'll sign 90 0 And then let's write 1/3 1 here. The t those three seconds. What we're getting is, too. I had now the opposite side. And why direction? And we're also said sketch that ensure that these velocities are tangential to the path. So that's easy, right? So t equals one second we're here, and you see that the both components are positive, So yes, and we might as well show it this one right handed me here across. Um, the cur here at this 0.2020 equals three. We have the negative of the same vector, Which means while the exes still positive. But why is negative? So then it's in this direction, right? So, yes, indeed, it's tangential. And look at this top point here, too. I purely in the extraction. Right. Can usual so that you can show. Um bye. Drawing vectors This way. Now in part. See? Well, next page artsy were asked what? The acceleration, the components of acceleration is in order to get through the acceleration. Um, really take one more derivative of the velocity vector is you know to get to the acceleration if we do that. Well, the I had terms disappears because going back here to the dirt off to zero derivative off this term comes out to be negative. Pi squared over eight sign I t over four j hat. So then X components, whatever time is asked which TV pulls one second people's a X. Teeples too of equals to a except T equals three seconds. They're all zero now, eh? Why, though, 20 was one comes out. Well, you need to do is just a plug in here, right? Signed by a pirate report and then times pi squared. Just put 3.14 pork, pork pie. What's gonna come out is negative 0.87 So sicken after what we're getting is and they get a 1.2. And the second after what we're getting is negative 0.87 of these air hand meters per second. Squared off course

Question. Even Verma t question Fonda X incorrigible for go side day. Why, you could You liked you said they quality in the interval between user in a troop. I hear energy to find the equation. Excellent. Why? We need to eliminate a permit, Barometer, the inner to do so. No. This that from the first question, we can get the angst over from a square. It was acquitted. Of course. I square something. Why are we to square incredulous? I square t now we can Buddha. Lila, we can pull to do the a submission. So we should get X number. Far square. That's why over two square and we go June this way and we get number 21 It isn't excellently with the founder lips. And then we should be much scandal graph now. And here we have the Vertex on DA for No for to go to here. So does will be the lips form here and because actually good, because I so it will be the counterclockwise here

So we have the position vector our equaling to T. I hat plus to sign of high key over four J hat. And so of course the length and meters, time and seconds and the angles and radiance. And so to find the velocity vector, we're simply taking the derivative of the position vector with respect to time. And so we have D R DT equaling then too. I have plus then pi over two co sign of pie T over four J. At. And then for the acceleration vector, this would simply be equal to the derivative of the velocity vector with respect to time. And so this would then be equal to negative Pi squared over eight. multiplied by sign of the pie T over four J hat. Of course there isn't any X component, the derivative of a constant with respect to time or with respect to any variable. The derivative of a constant will be zero. And so the X component then is eliminated. These are the equations that will be setting up our graph for our various times. We have uh Time equaling zero, 2.03.0 and 4.0 seconds. And so constructing our little graph here would be our position vector, velocity vector and acceleration vector evaluated at 0123 and four seconds. That is the end of the solution. Thank you for watching.

Hello. The objective of this problem is to find a cart ation equation on graph it and show its But, um, the old is given that X is equal to four. Sign. Okay, team, why is equal to course by course lt where d is between Cyril and Dubai. Then given these or next step, he's too, uh, sold for sign of tea and for co Schefty. Why get why we're doing this? We're going to see it like so let's so for sign of tears so angsty by before Because we divide both sides before and we get the peace. Eggs are for a sign of team and we get that. Why you But by five is equipped to coarsen. Oh too. Then we're going to square both terms since, or objectives different importation questions. So let's square both terms. We have the X Square over 16. Yes, he put science quarter Okay. Similarly wise square or 25 is he quit to Coast San square. Okay, team And we know that saying squared o t. You must co signs quarter t you see what you want. Therefore, it is equivalent to say, uh X square over 16 plus what's where Over 25 is equal to one. Notice that and this would be your quotation question. Now notice, too that this is the equation. We're on the leaps. Therefore, ones refined or equation No, we have to graph it. And the way I mean, we can do this. You see? Not for or we can make a table. Ah, similar to this one for different values of t. In this case, we're going to you zero by every two I three by your tomb and Dubai X because who are using? Because we're sorry because we're using a quotation. We need to find buns for X and y so anxious he could we know that X is equal to four sign of team. And that why is equal to five course entity. So Bergen deceptive to these bodies. 14. So zero sandals. Zero is 00 port. Um 00 Uh, goes on of zero is 11 since price five, seemingly for pirate to buy Sign A pirate two is 11 this for his four. Because I don't buy were 20 growth in this 50 And so on this one with the Cyril negative five. It will be no great for zero. And this one is going to back to Sorry. Zero in five. Therefore, these are the points that we're going to use in order, uh, to find the to grab or a leaps. So in this graph it. So here is there by access. Yours are exact. Sees not that this are reduced or throat. You, um then 12345 1234 Fire. 12345 12345 Okay, so let's graph it. Oh, our first point is zero full of fight. We're next point is four comma zero, then zero Coleman Negative five, then negative for 10 And finally we go back to Syria. If I Therefore this graph coast lying something in this, then from here to here from 40 to 0 negative fine from steer O negative fight to four, Cyril. And finally, from narrative or zero to serial fire. And as you notice, since we're talking in terms of D, it goes from 0 to 2 pi. So from zero tool? No, by over two. It goes this way. Isn't our quarter course this way? Rovira to pine. What was Theis Wing? From high throughput Your to it goes this way and for on through priority to to buy was this way. So we consider that the bad is, isn't it leaps that goes club waste. Where in formula is X square. You gotta by 16. That's why Square 25 is he going to want? And that would be Thank you.


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