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Use the guidelines of this section to sketch the curve.

$ y = \dfrac {\sin x}{1 + \cos x} $

see solution

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 5

Summary of Curve Sketching

Derivatives

Differentiation

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Missouri State University

Oregon State University

Baylor University

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So the domain for this one is going to be ex. Cannot be Hi. So x cannot be pie. Um, for every period of two pi. So it's important for every period of to play. And because the reason for that is that ask himto exists at X equals pi for every period of two pi. Okay, so our period is two pi obviously intercepts. We have 100 We have one to pi zero. We have one at four pi zero, so you could see the pattern here. And it also goes in the negative direction as well. So that's where our intercepts or symmetry isn't it be odd? So now, usually we would try to find the first derivative second derivative, um, said equal to zero and do the 1st 1st round of tests second derivative test. But both of them are inconclusive and just very difficult. Um, once we try to do them. So, um, instead of doing that, what we can do is we can see the limit as it approaches the ass. Santos is the function. Approaches are asking toads. What is our? Let me get a look like from both the right and the left. So let's try that. The limit the limit as X approaches Kai from the right side of sine X over one Post co sign X. So it's gonna be negative. Let me negative infinity sorted. And then the limits as X approaches. Hi. From the left. Remember this You're asking too of sign X over one plus coastline acts were fun is going to give us a positive infinity. All right. And that's important. Keep that over here. So this is the information that we have to graft. Um, So go ahead and try that. So obviously, this is a periodic function, so it's gonna you're gonna see your repeat some I'm gonna do. Um, first of all, let's put in our let's say this is two plays doesn't readings, and this is negative to pie. And we said that we have Assam toads at pie. So if this is too pie, if this is too pie than pie would be Hey here. So that's gonna be your ass in two, then. Here is negative. Clay brother asked me to. And again, we're also gonna have asked him to us here. The cynics tie for every period of two pie. We're going to see this and it's gonna, you know, his periodic. So it'll continue. But we're just gonna do a couple here and now we have our intercepts. 00 is one. He said that to pi zero is one. And so negative two by zero is one Great. Um, And now let's look it. Our limit as X approaches are asked him to from the right. So coming from the right, it's going to go to negative infinity. So it's gonna look something like this, going too negative infinity. And when it approaches from the left is gonna go up to infinity. And it's simple is that it's just gonna be the same over here as it approaches from the left, Infinity from the right and negative Infinity and Sora graph is going to look like this

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