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Use the guidelines of this section to sketch the curve.

$ y = \sin^3 x $

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Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

the domain for this function is X is all real. And if we didn't want to find the range, it would be because it's a sine function from negative ones worn. Um, the intercepts. The 1st 1 is 00 and we have another intercept. Um, every pie n we're in is a positive. Where is an integer zero? There is an odd symmetry to this graph. There are no Assam totes. And if we wanted to know the period, it would be to pie. So it's gonna repeat every two pi. It's Katherine. And if we wanted to find why, prime? Well, we know that this is the same thing as Sign X and bring the three on outside. And we can just do change rule from here. So that's three sign squared X CO sign X is equal to zero. All right, now, X said the derivative to zero to find critical points. Exes, Every pie over to end. No, my dad over to end. All right. And now, if we look at our number line, if you make one for the first derivative test, this is a prime, and we have pi over two, and I'm also gonna add in, um, three pi over too, because that's the next pi over to value on between zero and two pi. So if we test this in the first derivative test values between anything smaller than pie over two, it's gonna be positive between pi over two and three pi over to his negative and anything after, um, three pi over to up till two. Pi is going to be positive. So that is our first derivative test. Oh, it first, let's identify that this is because it's positive. So it's increasing and then it's decreasing here. So this is a local Max. This is a local man because it's decreasing in an increasing. So that's it for the first derivative test. It's gonna be useful for later and no, If we wanted to find the second derivative for, um testing for Akane cavity and inflection points, I would take a look at the inflection point because we know, like the general appearance of ah sangria. This is, you know, the general appearance. Ah, but we can find the inflection points by looking at why Double prime So six co sign squared X sine X minus three. Sign to the Power three X And if we said that equals zero Fine. You know, um we will see that the inflection points, the inflection points, um, exist that every hi And what that means is, um let's keep this isn't important information. We'll keep that and put it in the quarter. So we have inflection points, every pipe when I was trying to graph it. Okay, so let's say this is pi over, too. This is pie. Let's say this is through pi over, too. And this is to play. So this is one full period that we're graphing. And we said that we haven't intercepted 00 and we also said one at every pie end, so this would be high end, and this would be the next Titan. And now and now we said that we have a local Max here, a pie over to one and a local men at negative one. So the Max Pi over to one and the men for this period is right here. Then we said inflection points at every pie. So it's gonna change Con cavity. Well, first we know if this is the max right here, it's gonna be conch Eva, but the max and here inflection points and then it's gonna switch Con cavity. We'll switch again eventually. But here is the important part of this period here. We're going to see this is Khan gave up where the Max Max Local maximum is, and then it's going to slowly switch con cavity as it reaches the inflection point. And it's gonna switch con cavity again at the inflection point. So this is Kong caved out. Sorry. So this is concave down at this point, and then that's gonna be con cave up as it reaches the men and slowly going to go back to con came down and then back to conquer gave up. So this is a rough sketch of what the graph would look like.