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Use the guidelines of this section to sketch the curve.$$y=\frac{\sin x}{1+\cos x}$$

$2 \pi n<x<\pi(2 n+1)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 4

Curve Sketching

Derivatives

Differentiation

Applications of the Derivative

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So the goal of this problem is to sketch the graph and analyze its properties. Um that kind of give the graph its shape that we know. So the graph in this case is sine of X over one Plus Cosine X. And there's all these properties that are wanting to analyze. So we see that the domain, first of all is going to be all real numbers except for these vertical ascent of values. And those are gonna occur at all the odd pi values if that makes sense. So this is this value right here. It's going to is just pie one pie that is and this is three pi five pi so on and so forth. So two N plus one times pi where N is an integer. That's going to be the domain. Those values do not exist on the ground but all other real values do why intercept? We see there is one of the origins zero. That's also going to be an X intercept. But then there's clearly a lot more X intercepts. That's all the even high values. So um that would be too and high. Um where N is an integer value. Again, those are going to be the X intercepts. Then we see that there is symmetry about the origin um based on this right here. So that means that based on that, we do have some symmetry about the origin, which is also odd symmetry. The vertical assam tote as we already discussed, is going to be at all those odd pi values. And then we want to look at the intervals of increase. So this graph is going to be increasing from here to here. Here. Here, here here. So we see is that the graph is always increasing. Um There's never a point at which it's decreasing. We see it's concave up. If we look, for example, it's derivative graph, we see that it is always increasing baits on this and then if we look at its double derivative, we see that it's common cavity is something that's changing. So it's concave up and then it goes concave down and then it keeps constantly changing. Um It's concave up between the positive pi or the even pi values and the occupy values. And then it's concave down from the occupy values to the even pi values. So keep switching back and forth. Um And then that will give us our inflection points as a result. Which then allows us to view our final graph which looks like this.

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