Problem 40 Hard Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \dfrac {\sin x}{2 + \cos x} $

Answer

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Video Transcript

the domain for this one is exes, all riel. The intercepts are 00 Z z. One to see. In addition, we have pi zero, and then we have to pie zeros. And the pattern continues to reply. Zero Except symmetry is odd. There are no Assam totes. All right. And now let's take a look at increasing and decreasing intervals. So why prime First derivative test my prime. Ms. Sign speared hex over co sign. Thanks. Plus two squared plus co sign next over co sign exposed to equals zero. To find the critical points, X equals two pi over three and X equals for by over three for every period of two pi. Now, this is periodic, so obviously it will repeat. Okay, So first derivative test to pi over three here and for pi over three, it's gonna be increasing here between these two values, it will decrease. And then it will increase from then on. Hurry. And just so we know, um, f of to pi over three gives us 0.58 and f four pi over three gives us negative 0.58. All right, so this is the first derivative test which tells us increasing and decreasing intervals. And now we can solve for the second derivative to test for Akane Cavity. All right, so why in double Prime is equal to two sign thanks Times sign Square Dex plus Co sign Critics plus co sign next minus two. That's all over co sign X Plus two. And that's gonna be cubed. All right, so we have X equals zero X equals tie and X equals two pi for the region that I'm crafting. All right, so here's a double prime for a second derivative test. I have zero pie. And to play here is going to become cave up here. It's gonna become cave down here is gonna be up and go back to down. So just as a reminder, sign science, Which is means these are all inflection points, inflection, point inflection point an inflection point. No, we can graph Hajric. So we have no assam toes. Um but we do have intercepts a 00 here 00 pies. You're so saying this is to pie. So, you know, half of that is pie and yours negative to pie. All right, so here we have a tie zero intercept. So that means atnegative pies here it will have an intercept to pie will have any respect. That means that pot negative to pie will have an intercept, all right. And we know that it's increasing and then decreasing. So an increase and then a decrease. It means that this is a local maximum value. So let's say that this is one and this is negative one. So we have, um, to pi over three and 0.58. So that's approximately to cry over three, you're appointed tailed stays, right? There is interest, a rough sketch. And then we have our minimum value for pi over three four pi over three and negatives your 0.28 I say this is our minimum value local minimum. So no, it increases. Hits are max decreases, and then it will continue decreasing until his sermon, and then it will increase again, and then hurry. And it's gonna do the same thing on the other side because it's God. But, you know, showing all symmetry all right. And let's test for Akane Cavity gonna continue. So, Khan, Kevin, we were going to see its Khan cave up from before zero it'll become keep up its correct. After a 00 it'll be. Khan came down until it hits pi correct, and then I will become king.