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Produce graphs of $ f $ that reveal all the important aspects of the curve. In particular, you should use graphs of $ f' $ and $ f" $ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.

$ f(x) = 6\sin x + \cot x $, $ - \pi \leqslant x \leqslant \pi $

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Campbell University

Harvey Mudd College

Baylor University

University of Nottingham

you want to produce Graphs of F of X is equal to six times sign of X plus code 100 x on the interval. Negative pi pi that were built all important aspects of the curve. And in particular, we want to use scraps of the first and second derivatives to estimate intervals, with functions increasing, decreasing. Look for any extreme values intervals column cavity as well as points of inflection. So first I just went ahead and plug in half into my graphing calculator. And when I did that, I ended up getting this first draft here on the left. And I mean, that looks pretty good. So maybe we really don't need to use calculus, too. Actually find a better viewing rectangle, but let's just go ahead and see. So I also went ahead and found the first and second derivatives and plotted does as well and then estimated what these X intercept going to be, and I chose my viewing windows. Well, in this case since the just negative pika pi, we really don't need to choose anything, but normally when you do this, you just want to make sure whatever your values or, uh, we always hit all the Ex intercepts, at least because those are the important pieces for the scraps. No one thing we might need toe have to think about is this function is actually not defined at negative pi zero or pie. So in their original function, let me just go ahead and drawn. We have vertical Assam totes, an extra zero and X is equal to pi and negative and likewise or are derivatives. We will have birth class in coats at those points, too, so we don't really have to worry about the endpoints. But for intervals of where we're figuring out if it's Kong Caleb for concrete down, we will care about those vertical Assam toots, right? Excessive. What is it, right? So now let's go ahead and find our intervals where it's increasing, decreasing first. So remember, a function is increasing when prime of X is strictly larger than zero. So it looks like it starts to become positive at negative 1.3 90 and then it's going to stay positive until negative 0.444 Then it's positive again. 0.444 to 1.398 now to find where it's decreasing. We want to find where F crime is strictly less than zero. So this is going to be from negative pie, because that's our point to negative 1.39 union negative 0.444 now to zero. Because remember, that's where are vertical ass into is going to be at union than zero 20.444 and then it's going to be one point one point 3982 pi being our other interval or other in point on this animal. Now we can go ahead and say whether these critical values are going to be Max's or mints. So our first point here, we're decreasing into it because the truth is negative to the left, and it's increasing after So we will have a local men at that point there, our next value were increasing into it and then decreasing after. So that would tell us this is a local Max. Let me write this above it like that, then the next one. So since we have that brutal class into that's going to start over, so it's decreasing into it than increasing after So this year is going to be, Amen. And our last point, it's increasing into and then decreasing after. So this will be a local Max. All right. Now that we have that, we can go ahead and look for where our functions calm, keep calm. Keep down. So remember, it will be con cave up when that double prime a strictly larger than zero. So this is going to be so we're gonna start from negative five, and we're going to go until we hit our first intercept, which is negative. 0.774 And then it starts to be positive on the right side of our burger Qassem too. So there will be 0 to 0.774 now for where it's concave down. We just look for where f double private, strictly less than zero. And that will just be the remaining the remainder of venerable. So negative 0.7 stone 4 to 0 union with 0.7742 And now to see if these are points and inflection. Remember, What we want to check is to see if we have changed con cavity around both of our ex intercepts. So to the left, It's conch, Ava, and to the right, it's Kong Kate down. So that means we have appointed inflection and or other one It is I can't give up and then concave down again. So also a point of reflection. So we know, at least for what X values are. Maxes and men's occur as well as our inflection pumps, so I'm gonna head and planted those values. So these 1st 4 are our maximum ins Max men's. Or at least I should say, local maximums. These next values rr inflection points and these last two are are in points. But in this case, the function is not defined. A negative pie for pie. Now, the next thing, once we have all those points, it is to decide what our viewing window should be. So we already know where X viewing window should be negative pie pie. So we don't really have to decide anything there, and we only need to decide on what our window for why should be so. What I did was, I just went through these values and look for the largest in the smallest one. So our smallest value is about negative six year, so I need to go something slightly under that. And our largest value was positive. Six. So we needed to go slightly larger than that one as well. And so we can see those six points for the local Max men. And our inflection points are all listed as those dots on our graph. And we can see now that we did actually miss some information from that first graph where, uh, we just pump it directly in two.