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University of North Texas



Problem 5 Medium Difficulty

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) = \dfrac{x}{x^3 + x^2 + 1} $


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Video Transcript

we want to produce a graph of F equal to X over X cubed plus X squared plus one that reveals all the important aspects of the curb. And in particular, we want to use the graphs of F prime in double time to estimate the intervals for the function is increasing, decreasing. Find any extreme values, intervals of con cavity and any possible inflection points. So I went ahead and found the versus Aiken derivatives and went ahead and plotted those already and found all of our possible critical points just to save a little bit of time. Because otherwise this video would take probably well over an hour to do by hand and over on the left. This graph was just what happened when I plunged it into my calculator right away. So maybe this is a good curve of it. I mean, it looks pretty nice. Um, but using when the information we get from the first and second derivative grafts we may be able to see Oh, we needed to zoom out a little bit more. Or maybe there's an area of the graphic. Details have been. So just go ahead and get started by estimating our intervals of where the function is increasing and decreasing. So we know that a function will be increasing when F prime is strictly larger than is there. And so maybe I should give a little bit of rationale behind why I chose these windows for each of these graphs for F prime in F double because I actually didn't choose these. So the first thing I did was I just zoomed super far out and make sure that I found all of my ex intercepts because I don't want to lose any of those. Uh, so that's just pretty much how I went about choosing both with intervals just to make sure that all of our ex intercepts are there, because those will be important for us and also something else. Um, notice that around one point, far negative 1.5. It looks like we have a vertical ass into that is happening, at least from the sketches of the graph. And if you zoom really far out, it looks like it's getting really close to negative 1.5 for that. Berg lost. So I just went ahead and I included that as well. It's not exactly that value, but it's somewhere around there just estimating, using the breath. So now we know is going to be increasing from negative 30 up until or vertical s into, which is going to be around negative 1.5. Then we're on a union. This would Well, it looks like it's positive on the other side of our birth class income. So negative 1.5. And it looks like it's positive until we hit our actual first X intercept, 0.657 and then it approaches the horizontal ass until at Why's he got zero? Because you might notice that our denominator has a larger power than her numerator. So we know it should just keep on going to, uh, zero as we get as we get a larger in order for our excise. So this would be our interval of increase. Now let's go ahead and find where the function is decreasing, and so remember that's going to be where private strictly lessons there. And that only occurs on 0.6572 and Kennedy. Now, something we might need to do for a possible maximum is to look at any undefined values as well as just where the function is equal to zero. But since we have the same thing, are essentially the same thing in the denominator of our original potion F as well as our derivative of that, we won't have to worry about that, cause we'll know that whatever values are undefined for f will also be undefined for F Right now, you may notice, over here at our zero of F prime Well, it is increasing after it. Er, it's increasing into the point, and then it's decreasing after the point. So that tells us at X is equal to zero. Wait 657 that this should be a local Max. Okay. And now we can go ahead and look for con cavity and possible points of inflection. So just like before, we're gonna have this for glass in tow around negative 1.5. And here we actually do have a little bit more interesting values. All right, so Kong Cave up, remember, is going to be where an F double prime is strictly larger than zero. So, again are denominator should have a larger power than our numerator. So we know that there's this war's on the last photo at Liza Zero. And so that means we're going to go from negative affinity to our vertical ask skip at about negative 1.5 and then union. This will will. Then we become positive again. Around negative 0.49 until zero union and been after 1.1 to infinity. Next for not decreasing, we should be down. Oh, and I also said increasing their they should actually be Khan. Caged up is what we found. So go ahead on my lips. So this should be Khan gave up. Cock it down. All right, so this is when f double private strictly lessons there. So we are negative for our second derivative graph to the right of are vertical. Ask him to, and this will go on until about 0.49 And we're gonna union this with than 0 to 1.1. So now all we need to do is go to our critical values and check to see that each of these are points of inflection. So it just happens to be that all of these change cavity. So we start to the left of negative 0.49 Khan came down, and then to the right of that, it's conch Ava. Then at zero, we changed the com cave down. And then at 1.1, we changed to Con Cave. So all three of those will be points of inflection on our graph. All right, now that we have all that information, I went ahead and took those three critical values that we had before and plotted them. And so now my new view viewing Breck table that I chose so just a little bit of rational behind why I chose it was so I looked at my smallest X value here and my largest X value. And I knew I needed to at least be a little bit too left a little bit of right of those. And I decided to go from negative 43 just because I also remembered that I had this vertical ask him to at about negative 1.5. So I just went slightly to the left of that toe, also capture all that information, and then I chose negative 2 to 3 for my why values just due to the fact of how are values of importance that we got here arranged, so I just went slightly outside of that. So it wasn't all squished together. So you this pretty much looks like the exact same graph that we started lift for the viewing rectangle. So it looks like we just got lucky this time.

University of North Texas
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