Problem 49 Medium Difficulty

Use the guidelines of this section to sketch the curve.

$ y = \ln(\sin x) $

Answer

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Video Transcript

the domain for this function is, um X cannot be pie end. All right, intercepts. Um, there is every Scylla strait, this every, um, pie over to that. We see. Um, there's gonna be one in each to pie, period, you know? All right, symmetry. There is none. Assam totes at X equals pi n. So the reason why the domain cannot have this is because is because there isn't asking too. Um, at this value. And the reason for that is because sign pi is zero and Ellen zeros undefined, so we cannot have that. So now let's do increasing decreasing intervals. So why prime is co sign Oh, CO sign of X over a sign of X. Is that that equal to zero? And we see, um, ex every pie over too, plus pi n. So if we do the first derivative tests for pi over too high over to, we're going to see, it's ah positive here. So increasing here and decreasing there. So it's gonna be a increasing to decreasing. So this is a local max at every pie over to cause this is a periodic function or has repeats. So that's for of prime. It's right that in this is of prime. Okay. And now we can look at f double prime to see Con cavity. So why double prime? The second derivative is negative. Co sign squared X over science. Weird X. All right. Minus one equal zero eso We see an X at every pi over too goes to plan and x three pi over too. Plus two pi n and is an integer Okay, so this is for a double prime f double prime. This is the con cavity test. This is pi over too. This is three Pi over too. It's always gonna be con cave down, down, down, always down. Okay, so let's graft now. So, um, we have asked me, Toto, every pie. So if this is negative pie, this is pie. If this is to pie, this is negative to pie. Negative. Okay. And this is where our asking totes are as well as here. All right, so now we can see, um, a local max that pie over too. So pi over to zero. Is that point? So pi over too. This is negative player too. I'm sorry. It's not here. Here. Okay, so it's gonna be Kong cave down, so see how it's increasing and then decreasing. And then also here, it's gonna be conquered down con caved in and notice the space. There is nothing here, nothing here.