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Use the guidelines of this section to sketch the curve.
$ y = \frac{1}{5} x^5 - \frac{8}{3} x^3 + 16x $
see graph
Calculus 1 / AB
Calculus 2 / BC
Chapter 4
Applications of Differentiation
Section 5
Summary of Curve Sketching
Derivatives
Differentiation
Volume
University of Michigan - Ann Arbor
University of Nottingham
Boston College
Lectures
01:11
In mathematics, integratio…
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we want to sketch the girl. Why is one accident my state? Third X cubed plus 16 X now in Destructor. It gives us a laundry list of how we should go about actually graphing a function. So let's go ahead and just follow. That was so the first thing they say is we need to determine what are domain should be and this is a polynomial, So the domain should be all real numbers. The next thing they say is we should figure out our intercepts should be Wow, our first in there. So let's go ahead and find the why, Anderson. So why intercept? Well, that's one X is equal to zero and my nose. All of those. How exits. So this would be why you deserve next. We can look for our ex intercepts six intercepts. So this is when why is it zero? So at least we found one of them in the last part. Since we know are y intercept is also 00 So we know one of those, um, so let's go ahead and talk to that excel when we do that. So we're going to have what? That fancy liquor So ex 1/5 X to the board. Why, yes, 1/3 X squared plus 60 now might wonder. Well, how can we figure out how this cortical function actually factors? So first we have zero is your ex. And now let's do this on another page. Just so we don't kind of congest this up So they have one worth X to the fourth minus 8/3 X square lost 60. So this here is that you're not the one who should be 1/5 store. One she might notice. This is in the form of a quadratic. So we can say that X squared is going to equal to negative be so positive. 8/3 plus or minus square root of B squared. There's gonna be 64 overnight and then minus or a c So is going to be 16 so, or time 16 is 64 and then divide that by five now 64 minus not six more overnight, minus 64 over five. This here is going to be strictly less than zero. So we don't even need to finish. Since our discriminate is less Do this tells us no riel solutions. What? So X equals zero will be our only X intercept. Next. We want to look at symmetry, so this is not a periodic function, or at least all know most are not periodic, but we can at least see if it's even or odd percent tree across the lie access or about the order. So what do we do that I look in a negative X? So let's just go ahead and plug that negative X in. And then we could go ahead and apply the powers to all of them. So negative one to the fifth will still be negative. One negative 12 The third would be negative one. And the negative X is just really multiply than anyone. So we can factor out the negative one from this. Thank you. Get the negative. Oh, our original question. No pressure. So then there's about one more time. So this is negative f of X, which implies odd function or symmetry about origin. The next thing we should find is our acid trips. Now we know Paul. No, no, you should not have any acid trips, but we can use the same logic to the girl fee and so limit as experts in 30 of athletics. Well, this is a odd degree, polynomial, but a positive coefficient. So this is going to go to infinity. And that means as X goes your negative infinity, it should go to negative before the same rationale before. And what I mean by FX is just our function. Why? The next thing they want to do is to determine where are interval is increasing, flash decreasing. And I'm just gonna go ahead and combine that with step six and say, Go hand in hand of finding our local. Max isn't mints. Oh, cool stash, men. So that tells us we need to find what Wide Prime minister. And let's just go ahead and do this. Another patient. We're not taking up space here, so why is one extra minus 8/3 X cubed plus 16? Thanks. Right. So to take the derivative of this, would I need to do a powerful So we get 1 50 times five, which is just going to be one times X now to the fourth power and then 8/3 minus for eight years Time Street. We just ate and then we subtract one from the powers announcing to the X squared and the derivative of 60. Next, just be 16. All right? Now you might notice, but this looks like a core drive again So you could go ahead and use contract for me. Or you might notice this is actually a perfect square. Oh, X squared minus four. So let's first figure out X squared square. Almost forgot that. So they always go ahead and set your secret zero. So that means we can set X squared minus four equals zero at the four over, and we get exited to plus or minus two. So we have our critical values, which will need in a second. And something else you might notice is when we factor it like this, this will always be greater than or equal to zero, which implies it will always be so. I should say non e creasy. So we'll use that in a minute, Abs. Well, all right, so we got frustrated. So let's go back and write that down. So why Crime is equal to X to the fourth minus eight x were lost 60. And so, like I was saying, we need to figure out intervals on increasing quitters. We're wide private, strictly greater than zero. And but we just go back were quick to look at what we have here. The only places where this will not dedicates are plus and minus two, since this function is always bigger than zero. So the interval for increasing will be negative. Infinity. You negative too. Union with negative, too. 22 union work to 200. And that's gonna tell us that or decreasing with. Forget where my problem will be. Strict lessons era, which will be nowhere. All right now, let's go ahead and put those two critical points that we found earlier, which was Z 10 negative too, and X is equal to two. So to the left of negative, too, it should be increasing at the function to the right of negative to. It should be increasing, and after two, it should be increasing. So But this tells us is by the first derivative tests that both of these are saddled once or in other words, they're not a maximum or minimum. And now the last thing we needed to grow is our con cavity, and our inflection points what was? Go ahead and figure out what our second derivative is coming back over here. So why Double prime is going to equal tube. So we need to take the derivative of this function right here so we could go ahead and write. Just using powerful is going to be or X cubed minus. I was going to be 16 x and then the derivative of 16 will just be zero. Now, let's go ahead in back through this. So the compactor on X out on before. So it's factual book knows get for X and then we would have X squared minus or so. If I said this you deserve here, we're going to get solutions. X is equal to zero and X squared minus four. Is he going to zero? And then that would tell us X here, B plus or minus. All right now for cavity. So we found why double prime should mean working picks huge minus 16 X now for con cave up. This means we want wider will find to be strictly larger than zero. And I just went ahead and already found this interval for so we wouldn't have to worry about on DDE. We know that it will be increasing. There will be conclave up on negative too. She's zero and to to infinity. Well, then we need to figure out where by double problem will be sickly less than zero. And this will be well, negative. Infinity negative too. And then 02 two. All right, so now let's go ahead and put our points that we found for zeros of this on the last page which were excessively negative too. X is a good dessert and exit. So we know on negative 2 to 0. It will be con cave, but And we know from two to infinity will be Conchita Now on negative too negative infinity or negative thing today, too. We'll be conquered down and likewise from zero to It will be cocky down. So we see that since we have a change of cavity for all three of these, all of these are inflection points and this was the last thing that they suggested we do. So let's go ahead and start graphing this now. So what's first pot are intercepts and we only have one intercept from part two, which was just 00 We know this function is going to be odd from our symmetry part. So what I'm gonna do is just grab the left hand side. Whatever I do under left hand side, I just need to reflect it across the origin. And since it just seems to be a sketch, it doesn't really matter. It's too accurate or not. And then for sm totes, we know it should go to negativity on the left. So she's going with that so negative Infinity and we know we had no local men's or local Max is. But we know we had saddle points of negative to two, so at least on the left, only negative to really matters. So let's go ahead and plot that so negative, too. So that means at negative to a kind of flattened out like this, and we know it's going to be below, since we're starting from negative infinity and our first point or first X intercept is over here at 00 All right, um, and we have inflection points that I get to end zero. So since we already know where 00 is, we also know that that should kind of flatten out at that point. That's well, all right, so let's go ahead and start cropping. So I start from negative spinning, and I'm gonna go until I hit my saddle point here. Negative too. And you see, we have a change of inflection at that point until we get to 00 and then it changes inflection again. And now we just need to reflect how this graph looks about the origin. It was gonna come up for a while and then it's gonna flatten out again, and then it's going to go straight. So you couldn't make this a little bit more detailed just by using the points we already know. We already know this point here should be, too, from what we did in part 56 But since we're just trying to sketch it all, I think that really matter. Is this squiggly line here and the ex intercepts So? So you could put a little bit more details, but I think this is good enough for just a sketch
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