5

9) Use the graph of flx) below to answer the following questions.(4 points) Draw your own axes and sketch a graph of f (x).(2 points) Find the solutions to the equa...

Question

9) Use the graph of flx) below to answer the following questions.(4 points) Draw your own axes and sketch a graph of f (x).(2 points) Find the solutions to the equation f' (x) = 0. Use your best estimate_

9) Use the graph of flx) below to answer the following questions. (4 points) Draw your own axes and sketch a graph of f (x). (2 points) Find the solutions to the equation f' (x) = 0. Use your best estimate_



Answers

$9-10$ Produce graphs of $f$ that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly.
$$f(x)=1+\frac{1}{x}+\frac{8}{x^{2}}+\frac{1}{x^{3}}$$

We have a whole sequence of problems. Um that are so some questions about this uh this graph here that I've tried to reproduce from the book, um it's obviously piecewise continuous. It's not smooth because it looks like we have a corner there. So linear here, linear here, but with a constant Y. And then some kind of probably more like a, like a cubic or cortical function. Um Over here. They don't give it, they don't tell us what these are. We could yeah, I think we have enough points. Let's see here, we have 12345 points. We could figure out a quartet polynomial. We probably want to find this slope here. So we could figure out a Quinton polynomial on that word. Have a zero slope here and go through all these points if we wanted to. But that's not what we were asked. So we're just asked a bunch of questions about this. So I'm just gonna go through all of them in this one video, because they're all very much related and there are, you know, I have to do with this this graph here. So the first question is fine. F zero and effort negative six. Well F zero, that's when X equals zero. So we come up here and we're at my at three and then at negative six we're out here and so they've labeled this point here, X is negative six, then why is minus three now they ask us is um is F of three positive or negative? So let's see here F of three is somewhere in this in this region here and so it's positive and it's probably, you know, just again, they didn't label the point but it looks like it's three for this whole region between zero and four. An f of minus minus four, so minus three were at zero minus five, we're at minus two. So if this is indeed a line then minus four were at minus one and then clearly f of my f one of X equals minus four is clearly negative. It's on it's to the left of here. No, they ask us, let's see here. Um for what values of X is F of X zero? Well f of X is why here? So we have zero here, zero here and zero here. So there's three points. And they are when X is minus three, when X is six and when X is 10 to those three points, then they ask us for what values of X. F greater than zero. Well we can see here that it's greater than zero here and also here. And I should probably say that this should probably change this to just assume that this doesn't get extended. So we have it's greater than equal to 10. Less than equal to 11. So from here to here, right, we have X equals minus 3 to 6. So and they said greater than all Right. So I should have I should just have greater than it's not equal to um And then from here to here, you know, it goes from 10 to 11 over this region. Little region here, we also have positive values. Then I asked for the what is the domain? Well, assuming that, you know, this doesn't continue on in any way that this is just, you know, it's only defined over this region. The domain goes from X equals minus 62 X equals 11. And they asked us for the range and the range is the span of y values. So it looks like the smallest Y value we have is minus three and the largest value we have is three. So that's the range. Then we'll ask for the X intercepts. The X intercepts are actually just the points where y equals zero. So we have the exact same points here minus three at six and 10. They ask us for what is the wiring rcep? Well that's when X zero and we see that in X zero. They tell us that why is three? Yeah. And how often does the line why it was gonna have intersected graph? Well, Y equals one half. If we go down um you can see that it's gonna be down something like this. So it looks like about three. And so we know it intersects somewhere here as it this crosses it, we know what intersects somewhere here as that crosses it. And over here at this point this goes from um Y equals zero to Y equals one. So one half is right in between there. So we know what crosses it there too. So that crosses three points. How often does the line cross? Um The graph crossed the line X equals five and I thought why it was five, so this is wrong. And so let's do that. Y equals five. Um No X X equals five. So it just crosses at once. And in fact if it's a function, if it's a function then it can only cross any any constant value of X at one at one point. Otherwise it's not by definition of function. So yeah, process somewhere here if this, well I don't know what kind of curve this is. So I would speculate on what the Y value is there then how for what values of X does F equal three? Well, it looks like again assuming this is all horizontal here, it looks like it's F equals three for this entire span here from 0 to 4. Mhm. Than for what values does F equal minus two? Well we have we have a point over here minus five minus two, so we're given that one and we're told this goes through eight minus two. And again this just this double cross comes back down here. So if we look at um for why it goes minus two, we just touched this point here and I'm assuming that's a minimum it appears to be, so it doesn't go beyond it somehow. Um you know, doesn't go below and then over here, so those are the two points X equals minus five and eight. And then for and what interval is the function increasing? What intervals? So obviously it's increasing here. Right. And it's increasing here. So they have yeah. Um And so this region here is minus 6 to 0. That's not increasing here. Nor is it decreasing? We'll come back to that later. And it looks like from here from 8 to 11. It's also increasing. Did they ask about when there's a decreasing? Well it's decreasing from here to here and that goes from 4 to 8. And they asked when is a constant? Well that is it's constant between zero and four. And then they asked, when is it non increasing? Okay, so that means constant or decreasing. So basically have the union of this set and this set which is 0 to 8. So from here, other way to here is non increasing. And then they ask if it when is it non decreasing? Well, again, that's the intersection of the constant region and this region here, so that winds up being minus 6 to 4. So here to here, it's non decreasing. And then over here, 11 or 8 to 11, it's also increasing or non decreasing. So those are all the questions they ask us about this this one graph. Um, so hopefully there should be pretty simple for you to do at this point, and hopefully none of this was kind of a surprise, um, to looking at this chart here and in asking all these questions.

This is a function doctors, therefore fixes equal toe X squared minus name. So we have to find that in trouble for which f off axis greater than or equal to zero. So now we already know the function like that's up. Shoot it in the condition it would become X squared minus 1,000,000 greater than or equal to zero. So it would be explained greater than or equal dough. Nine. After this, it would become X squared. Oh, come here Doesn't really x greater than or equal to three on X less than or equal to minus three as if we were on nine. The suitable values are plus or minus Terry, so blessed to the way we can put the same side as it does mine. STD design changes. So the interval will be from minus infinity to minus three, as it is less than or equal to union to the to the include er pull infinity. So this is that well, for which F or X is greater than or equal to zero. No letters Verify this by graphing calculator. This is the graphing calculator. Don't know that doesn't other function. The function ISS X squared minus nine. This is the function No legacy. The graph So the others that so the X intercepts are from to be toe minus today to enter the in between minus Syrian to the it is less than zero on after minus three. Toe minus. Infinity to minus three is positive on from three to infinity, it is again positive. So the indelible would be these two regions that is from minus infinity to ministry on from to the to infinitive. This is doing terrible for much of offense is greater than our equality.

We have a whole sequence of problems. Um that are so some questions about this uh this graph here that I've tried to reproduce from the book, um it's obviously piecewise continuous. It's not smooth because it looks like we have a corner there. So linear here, linear here, but with a constant Y. And then some kind of probably more like a, like a cubic or cortical function. Um Over here. They don't give it, they don't tell us what these are. We could yeah, I think we have enough points. Let's see here, we have 12345 points. We could figure out a quartet polynomial. We probably want to find this slope here. So we could figure out a Quinton polynomial on that word. Have a zero slope here and go through all these points if we wanted to. But that's not what we were asked. So we're just asked a bunch of questions about this. So I'm just gonna go through all of them in this one video, because they're all very much related and there are, you know, I have to do with this this graph here. So the first question is fine. F zero and effort negative six. Well F zero, that's when X equals zero. So we come up here and we're at my at three and then at negative six we're out here and so they've labeled this point here, X is negative six, then why is minus three now they ask us is um is F of three positive or negative? So let's see here F of three is somewhere in this in this region here and so it's positive and it's probably, you know, just again, they didn't label the point but it looks like it's three for this whole region between zero and four. An f of minus minus four, so minus three were at zero minus five, we're at minus two. So if this is indeed a line then minus four were at minus one and then clearly f of my f one of X equals minus four is clearly negative. It's on it's to the left of here. No, they ask us, let's see here. Um for what values of X is F of X zero? Well f of X is why here? So we have zero here, A zero here and zero here. So there's three points. And they are when X is minus three, when X is six and when X is 10 to those three points, then they ask us for what values of X. F greater than zero. Well we can see here that it's greater than zero here and also here. And I should probably say that this should probably change this to just assume that this doesn't get extended. So we have it's greater than equal to 10. Less than equal to 11. So from here to here, right, we have X equals minus 3 to 6. So and they said greater than all. Right. So I should have I should just have greater than it's not equal to um And then from here to here, you know, it goes from 10 to 11 over this region. Little region here, we also have positive values. Then I asked for the what is the domain? Well, assuming that, you know, this doesn't continue on in any way that this is just, you know, it's only defined over this region. The domain goes from X equals minus 62 X equals 11. And they asked us for the range and the range is the span of y values. So it looks like the smallest Y value we have is minus three and the largest value we have is three. So that's the range. Then we'll ask for the X intercepts. The X intercepts are actually just the points where y equals zero. So we have the exact same points here minus three at six and 10. They ask us for what is the wiring rcep? Well, that's when X zero and we see that in X zero. They tell us that why is three? Yeah. And how often does the line why it was gonna have intersected graph? Well, Y equals one half. If we go down um you can see that it's gonna be down something like this. So it looks like about three. And so we know it intersects somewhere here as it this crosses it, we know what intersects somewhere here as that crosses it. And over here at this point this goes from um Y equals zero to Y equals one. So one half is right in between there. So we know what crosses it there too, So that crosses three points. How often does the line cross? Um The graph crossed the line X equals five and I thought why it was five, so this is wrong. And so let's do that, Y equals five. Um No X X equals five. So it just crosses at once. And in fact if it's a function, if it's a function then it can only cross any any constant value of X at one at one point. Otherwise it's not by definition of function. So yeah, process somewhere here if this, well I don't know what kind of curve this is. So I would speculate on what the Y value is there then how for what values of X does F equal three? Well, it looks like, again, assuming this is all horizontal here, it looks like it's F equals three for this entire span here from 0 to 4. Mhm. Than for what values does F equal minus two? Well we have we have a point over here minus five minus two. So we're given that one and we're told this goes through eight minus two. And again this just just never cross comes back down here. So if we look at um for why it goes minus two, we just touched this point here and I'm assuming that's a minimum it appears to be so it doesn't go beyond it somehow. Um you know, doesn't go below. And then over here, so those are the two points X equals minus five and eight. And then for and what interval is the function increasing? What intervals? So obviously it's increasing here, Right. And it's increasing here. So they have yeah. Um And so this region here is minus 6 to 0. It's not increasing here, nor is it decreasing? We'll come back to that later and it looks like from here from 8 to 11. It's also increasing. Did they ask about when there's a decreasing? Well it's decreasing from here to here. And that goes from 4 to 8. And they asked when is a constant? Well that is it's constant between zero and four. And then they asked, when is it non increasing? Okay, so that means constant or decreasing. So basically have the union of this set and this set which is 0 to 8. So from here, other way to here is non increasing. And then they ask if it when is it non decreasing? Well, again that's the intersection of the constant region and this region here, so that winds up being minus 6 to 4. So here to here, it's non decreasing. And then over here, 11 or 8 to 11 it's also increasing or non decreasing. So those are all the questions they ask us about this, this one graph. Um, so hopefully there should be pretty simple for you to do at this point, and hopefully none of this was kind of a surprise, um, to looking at this chart here and in asking all these questions.

All right. Little explanation here. But the difference between two things finding F zero means finding the output of the function are our value of the function. When the input is zero finding the zeros of f ends, I'm being what input is an output of zero.


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