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HOW DO YOU SEE IT? Using the graph of $f,$
(a) determine whether $d y / d t$ is positive or negative given that $d x / d t$ is negative, and (b) determine whether $d x / d t$ is positive or negative given that $d y / d t$ is positive.

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.

In Problem 38. We have two graphs for the function. If the first graph here in the second graffiti for birdie, we want to determine whether D y by D. T is most of or negative. Given that the X by DT is negative, let's go for birthday. We can notice that from the graph given here as X increases why is decreases? This means here we have the oy by the X equals negative. But what is the relation between D. Y by D. X and D. Y by D. T From the general we have deRoy by DT equals de Roy by the X Multiply it by the X by DT the X Boy deity and we have here Dear boy by the X is negative given from the graph and the X by DT is negative given from the problem then negative by a negative which makes the oy. Why did he is posted for the second graph? We cannot desert as X increases. Why is increases which makes doi by the X equals most of value from the chin rule we have doi by DT equals newly by the X but applied by the x by DT from the graph devoid by the X is positive and from the given The X by DT is negative. Then boast of by negative gives a negative value for Bharti we want Do determine bar TV whether the XB GT is negative or busted. Given that doi boy DT equals a positive, we will do the same procedure from the chain rule we have new toy by DT equals divide by the X multiplied by the X by ditty given from the graph D y by D. X is negative and or we can change now because we have we want to get the X from the General The XB GT equals the X by deRoy Multiply it, boy the ex boy Dear boy, What the bloody boy D y by d? T The x by the boy will be also negative for the first graph here because as why increases as well increases exit decrease, then the boy by the X equals negative body. Then, from the graph, we have the x by device, Serie the ex boy deroy equals negative one. The X, by the way, is negative from the graph and d y by D. t s positive given from the problem, then it will be negative. The same concept applies here. The X by DT from the chin rule equals the X by D boy multiplied by D y by D. T. From the grave as y increases X increases, which makes the X by doi is boosted. Then this is boosted from the graph. This is most of from the given. Then the answer will be positive and this is the final answer off our problem.

So we're looking at the graph shown, we want to know um if the points at which points this is true. So we have that the first derivative and second derivatives are positive. Well considering our graph looks something like this, we see that the first derivative is going to be positive at point B. Because the slope is positive there at point C. The slope is positive too. Um at Point D it looks to be zero. The second derivative, though we see it can be concave down here so it can be negative. However, A and B are concave up, so those would both be positive A. And B. And then um they're gonna be negative the opposite one, so A is negative this one right here and E is going to be negative over the second derivative is negative only for both right here. So we would say E. Is our answer for that one. And for the first one our answer was B. So it's our final answer.

All right. See you of different relationship. Got a year? Thanks. Why? According to them, you have some function that looks like these. So here we're assuming that since Izzy the plane. Why does this mean? Why use equal to if I fix so we know. Well, if, uh, what happens with wide? Who is the white team? Positive or negative for but the every city. So how do you know that? How do you know that? So Well, we can, since we have this relationship. Come differentiate both sides equations. We have that divided. We even buy. It's prime on them. But the chill terms? Yeah, X, um said at the y Begin my exit. Them's then the office. But as you can see here in this drawing, they take his thing. Stop off the time line in each point. But he's always Hey, if you said the Chrissy function, so if prime music here. So that, uh, well, already so that if Bryant changes the sign. So for the x City get you, Um oh. If prime changes the side there, it's the the ex. Why? What's this from, uh or? Yeah, 40 years of the what city damn it. Why do people you nearly with you? I know that. No, if, uh, you know why it's positive we start from this being positive. So that means that prosthetic is going to be able to negative to that school. You can vote. See that if Brian is never zero this time for T Rex city Well, well, I want a renegade number. It's a negative number. So positive times negative. It's you. Why did this positive then? Extinct? No, it seemed that we had a something this form. So your help here, X exes? Why Exes? I'm asking that you had something like this. Why she ableto ever fix where oh is? He said So in that case, upon initiation, why we even buy Brian the ex with you? But you know well, if Brian from that drawing if it's always increasing, it's about me is low. But each point it's positive. So if Brian So it's positive. So this sign for Turgut beside for G X city, you'll always be the same. So then if you had the exiting said I work for years. DX city is negative. The negative is a positive number. It's so why keep this thing. Why is that? You know if, uh if possibly why the velocity? Well again, it seems here, toe if crime is there were zero. So we can have that we can divide. Yeah. So if you promise it was the number one of her friends, it's always also posted also was it so Why did he is positive? Positive? Positive. So exit e b. Well, he was so because, all right.