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Find the points of horizontal tangency to the polar curve_ r = a sin 0 0 < 0 < T, a > 0(r, 0) =0,0(smaller value)(r, 0) =(larger r value)Find the points of...

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Find the points of horizontal tangency to the polar curve_ r = a sin 0 0 < 0 < T, a > 0(r, 0) =0,0(smaller value)(r, 0) =(larger r value)Find the points of vertical tangency to the polar curve_ (r, 0) = (smaller 0 value)(r, 0) =(larger value)Need Help?Rcad ItTalk to a TutorSubmit AnswerSave Progress

Find the points of horizontal tangency to the polar curve_ r = a sin 0 0 < 0 < T, a > 0 (r, 0) = 0,0 (smaller value) (r, 0) = (larger r value) Find the points of vertical tangency to the polar curve_ (r, 0) = (smaller 0 value) (r, 0) = (larger value) Need Help? Rcad It Talk to a Tutor Submit Answer Save Progress



Answers

Find the points of horizontal and vertical tangency (if any) to the polar curve. $$ r=1-\sin \theta $$

Our discussion on polar coordinates and polar equations. Um It's all kind of leading to this general important feature um and it's ultimately the derivatives. That's what calculus is kind of all about derivatives and integral. So we're gonna be looking at the derivatives of a polar curve. And what that's gonna look like is dy dx not the RD data but the rectangular derivative. And we know that that's the same thing as D Y D. Theta over. Um put those inferences over. B X. D. Theta. So we have the white teeth data is going to be our prime same theater. Okay. Plus are costing data because it's going to be buy the product rule and then it's gonna be divided by the XD data is going to be our prime coastline data minus are signed data. With this now constructed we will be able to calculate the derivative Dy dx for any polar equation. We just take the derivative of the R value. We have our assigned data. So once we take the derivative are are with respect to the with respect to theta will be able to end up getting our equation as a result. So that would end up giving us our final answer once we calculate this. Um And then it's also important to note that if we want to find when there's horizontal tangent, C keep my horizontal tangent C is when the slope is equal to zero. So it doesn't matter what goes on here. As long as the numerator equal zero, we have horizontal tangent C. So we can just set this whole thing on the bottom or we can set the thing on the top equal to zero and get rid of the stuff on the bottom. If however we want to find vertical tangent C. Points, then we get rid of what's on the top. We know it's going to be vertical when the slope is undefined, so it's going to be when this is equal to zero. And that will allow us to finish solving a lot of these different problems.

Our discussion on polar coordinates and polar equations. Um It's all kind of leading to this general important feature um and it's ultimately the derivatives. That's what calculus is kind of all about derivatives and integral. So we're gonna be looking at the derivatives of a polar curve. And what that's gonna look like is dy dx not the RD data but the rectangular derivative. And we know that that's the same thing as D Y D. Theta over. Um put those inferences over. B X. D. Theta. So we have the white teeth data is going to be our prime same theater. Okay. Plus are costing data because it's going to be buy the product rule and then it's gonna be divided by the XD data is going to be our prime coastline data minus are signed data. With this now constructed we will be able to calculate the derivative Dy dx for any polar equation. We just take the derivative of the R value. We have our assigned data. So once we take the derivative are are with respect to the with respect to theta will be able to end up getting our equation as a result. So that would end up giving us our final answer once we calculate this. Um And then it's also important to note that if we want to find when there's horizontal tangent, C keep my horizontal tangent C is when the slope is equal to zero. So it doesn't matter what goes on here. As long as the numerator equal zero, we have horizontal tangent C. So we can just set this whole thing on the bottom or we can set the thing on the top equal to zero and get rid of the stuff on the bottom. If however we want to find vertical tangent C. Points, then we get rid of what's on the top. We know it's going to be vertical when the slope is undefined, so it's going to be when this is equal to zero. And that will allow us to finish solving a lot of these different problems.

Our discussion on polar coordinates and polar equations. Um It's all kind of leading to this general important feature um and it's ultimately the derivatives. That's what calculus is kind of all about derivatives and integral. So we're gonna be looking at the derivatives of a polar curve. And what that's gonna look like is dy dx not the RD data but the rectangular derivative. And we know that that's the same thing as D Y D. Theta over. Um put those inferences over. B X. D. Theta. So we have the white teeth data is going to be our prime same theater. Okay. Plus are costing data because it's going to be buy the product rule and then it's gonna be divided by the XD data is going to be our prime coastline data minus are signed data. With this now constructed we will be able to calculate the derivative Dy dx for any polar equation. We just take the derivative of the R value. We have our assigned data. So once we take the derivative are are with respect to the with respect to theta will be able to end up getting our equation as a result. So that would end up giving us our final answer once we calculate this. Um And then it's also important to note that if we want to find when there's horizontal tangent, C keep my horizontal tangent C is when the slope is equal to zero. So it doesn't matter what goes on here. As long as the numerator equal zero, we have horizontal tangent C. So we can just set this whole thing on the bottom or we can set the thing on the top equal to zero and get rid of the stuff on the bottom. If however we want to find vertical tangent C. Points, then we get rid of what's on the top. We know it's going to be vertical when the slope is undefined, so it's going to be when this is equal to zero. And that will allow us to finish solving a lot of these different problems.

In this question we want to find horizontal and vertical tensions to these para parliamentary functions. If there are. So speaking of tension, we want to find a dy dx we know that the ideas is the wife of a dictator over the ex dictator for horizontal tensions. You can it will look like this now. It means the slope is flat. So the dy dx zero over here. So we were going to set the dy dx zero, find the day to day, stuff it into the X. M. One. You get the coordinates. So the vertical tension. Yeah. Okay. It will look like this particular line, particular line. The Dvds is infinity. So only if your the idea is is a fraction. So supposes you over V where you and we are functions of teacher for it to be infinity. That means we're gonna set the denominator to zero, right? Because divide by zero you get infinity. So find the data there stop into X and Y. And you get the coordinates. So let's find our the X. T. T. To A. And B. Y. D. Tito. First. Yeah. So yes the tita. Second when I differentiate with respect to data I will get second to 10 22. For why When I differentiate one respect to teeter for tangent I will get second square tita. So my dy dx will be D. Y. And D Tito. Which is this seconds creditor divided by the extent to which is this. You can see that these councils. So I'm going to have second is actually one over co sign and one of attention is gonna be Hussein oversight. Now you can see that the chosen cancels. So I'm just left with one over science teacher. So for horizontal tangent. Yeah. Mhm. I'm gonna set my dy dx 20 It means one of a scientist to is equal zero. Now to get zero. My denominator cannot be zero. Right? But my numerator is not zero or so. So there is no solution. So that means there is no horizontal engine. For vertical engine. I'm gonna set the denominator. My dy dx 20 that his one sets to zero. Immense data is zero or Opie. No something to to go zero into the X. And Y. Over here. My ex will be one. My wife is zero subtitle goes to pie into X. And Y. Over here my X. Is minus one. My wife is zero. So for vertical tension the two points are 10 and minus 10 Let's verify the grab. I'm using the smallest to graph the graph. You can see this is how we grab. Parametric. So this is the X. X. Is equal to second T. Now instead of the time using T. And the wind is tension T. You can see that this is the graph. There are no horizontal tension, but the vertical tensions are over here and over here. At this point, at this point excess minus 110 This point X is one and 10 and that correspond to what we have just found. So we are right.


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