So final of few vets first you need to to think about are increasing indeed increasing functions and station reporand. Just say if you can't hear me at any point. So Yeah, it's a bit noisy. There's a lot of background noisy. I wonder if I can. Some platforms have like background noise remothat's a bit quite. Let me just see if I can move some request. Okay. That's anybody. So a little better. Sorry, I wasn't muted better. Yeah, it's a lot better musomehow sorry. Yes, quite arranat home this time, increasing, decreasing. So that's exactly what it sounds like. So an increasing function is if we have numbers getting bigger than the function, it's getting bigger too. So it's kind of it's going something like that. Yeah. So this is a function, if you not a guess of a property for the derivative, if it's an increase in function. So it's positive, absolutely positive. And simildecreasing, what do you recommend on servative doing? Always negative? Absolutely. And then the other interesting feature is what in a stationary point, that's something you've probably across before. Wait, sorry, could you repeat again a stationary point? No, but I might have seen it. I'm not sure. So station reflis kind of when we turn around, so these are all three, okay? When the gradient is zero? Absolutely. Yeah. So we have one of these kind of flat points and they have three types, maximum, minimum. And the one it's easy to forget about it can also do this kind of thing and drama very well, but sort of slow down to flat and then get going again. So we need to be able to tell where our stapoints are and which type of thatbe really important if we're looking to sketch a unfamiliar function. So I'm going to introduce a new concept. So we have our derivative, our first derivative, and that's going to be zero. Then now think about the second derivative, which is just the derivative of the derivative. So that's the rate of change and the derivative. So if you think about what the derivatives do, these first two cases, so this one here, this is on the left of the maximum, it's increasing. We have a positive gradient. And on the other side, would secrerease would have a negative rate. So getting a better sort of brain stretch here, the rates of change of the gradient, the derivative of the gradient, is it positive or negative? Positive same. So it's zero there one. We know to. It should be negative then. But when it comes from positive, so if we're going positive, zero negatively decreasing. So the gradient is decreasing. So second derivative here, similarly from one of man. Now the negative rate of here we go from zero and then we're positive. So what's the second means of doing. Positive. So that is our first go to technique figuring out type. We look at the second tuitive at that point, negative to maximum positive a if you have equal to zero. Then unfortunately, we just don't know that test doesn't work. So if we observe it, we need to use a different technique, which is kind of a less neat one. We just look at the derivative a little bit to the left and a little bit to the right, because that's really been looking at here. That tells us what we can. So the maximum she asked before it's going opstart the negative, or minimum just before negative, after positive. And our point of inflection over here is the point of inflection. You have positive, positive, positive, positive, or because we could have it negative, well, negative. So that's our last results. We can't use the neat secondary. So think we look at the sign of the gradient a bit to the left and into the words, what makes it super far? Yeah, excellent. Let's try. Yes. Is ent. Could you find the station we points and identify what April station points are? We access the word please. Yes, one there's only one point. This. One. Let's next thing is, what kind of stationary point is this? So remember, whether it's positive or negative and if it's equal to zero. And ck, we have to do something else. So what's the second way to doing that? Equal to zero. Yeah. So do we move straight away? That's what I'm in. Sorry, can you repeat, please? So absolutely it's zero at that point. So does the second drito test tell us what type of station we might be at? Sorry, there's too much noise. I couldn't hear you. So the second derivative equal to zero. So does the second derivative test tell us what type of statreport you have or are we going to need to do something else? Yeah, quite right. Do something else. Yeah, absolutely. So can you remember them for what do you do when the second derivative test pails find use a different point that's very close to it? Yeah, we look what's happening a bit to the left and look into the right. See brait's doing okay. So pick your favorite point a bit to the left and a bit to the right up to you. But how would I well, we just want something. It's kind of up to you. We have our point 11, and we want to know just a little bit over here and a little bit over here, what's the grade of so you just need to pick a number a little bit less than one little bit. That's easy to calculate. So I would have one in mind, but I'll see what you reon. So I'll be thinking maybe point 911.1 to the lessson, but that's not a right to wrong answer. We'll discuss that. Maybe point zero 911. Yeah, that seems sensible to go forward to me. So remember what we do with these expore that fits the left and bits and right. We work out the gradient of those. I don't have a calculator, though we could probably just adopt it. And I had itprobably make our life easier by taking the three out and then actually. We're dealing with that, right? And then Erion, that's pretty easy to put ug them in target. Yeah. So. Yes, so the same absolutely the type of station where. Sorry, could you repeat again? So what type of stationary point do we have if we have positive gradients on both sides? So it also must be positive. So we're looking for what type of station point we have. Is it a maximum? Positively. So that's not really the type thing I'm looking for. I'm looking for whether this is a stationary point, sorry, a maximum, a minimum or point of inflection. That's what we're looking for here. What type of stationary point do we have? So we have minimums, we have maximums and we have points of infflection. Now picture draw in one of those diagrams where the gradient is positive and where it's negative. Because I think it was my Internet. So what we're looking for as what type of station point is there at one point? Is it a maximum? Is it a minimum? Or is it a point of inflection? And we can work that out using the fact that we know the radidient is positive to the left and to the right of this point, that is. So it's inflinflection absolutely. Because here we can see positive, positive slijust to wrap it up. Could you therefore sketch. Yeah but could you give me the Yeah again, so try and put that all together. Something like that. And for the whole thing, we could notice that there's a roof x in there. So it's going to go through. Sca day should be Orito with something like. So started with the hard one. I'm not sure why, but that's what we do when the secondary to tests fails us or more example. Could you find the station point and determine what type of station point? Always positive. So always. Is that contribuvato? Give me. Yeah, Yeah, absolutely. That's actually our second job. Before we thought about the second derivative need to find our stapoint. Please quickly, please. Yeah, so it's always busted. Yeah, so finish shing the question there. I want to know where the statpoints are and also know what kind of statstream point they are. So it's said to a negative 40. And what type of station point is it? It's. Does graph. So can you tell me what second derivatives doing. Second with like about value and then it's lower times. So it's positive, absolutely. So it's kind of just something to remember this Candy little test, our second derivative, it's positive that's a in and if it's negative, that's a matter. We can work it out using the technique we talked about. It's kind of just once that I would say, so could you put that something like that? So could you put it all together and sketch this graph for me? There's definitely a feature that I like. You've got the minimum. But if you notice that we can factorize it. Did you want to change your graph at all? The minimal point ight shouldn't change though, right? It's at 48. So you quite back of a minimum of 2:48. But what do we know because there's a factor of x it. So could repeat. So notice we have a factor of x in our equation. What does that tell us about the graph? It's not to do with the minimum, but another important feature. Y intercept is negative 30 to x point. Think about how youcalculate the the y intereffect it's to do with the intercepts. Yeah. See, we can spot how we need to tweak the intercepts, specifically the. So describe to me, how did you calculate the y set when x equals zero? So just and check that, calculate what the right idea. Check the plugging into the right equation. It's about what you need to tweak. I don't know this, so you're plgging in zero here, then it x equals zero. That doesn't give us -32. You just need to double check your calculation showing you'll you'll pick yourself. Which calculation do I need to return? So you've said the y intercept is -32. That isn't quite right. I might x equals zero, but just check your calculatwhen. You want to know what does y equal to when x equals it? It's not -32. Yeah, it's zero and we can swap that straight away because there's a factor zero. Not quite so this is something like that yet. So we might need to would go like that. Yeah, you got the right idea, something like that kind of thing. Yeah, excellent. So this is a really important skill, especially if you're wanting to do either maths or something. That's sy University one day University admissions really put a lot of emphasis on grass keso. You could start getting hirit's kind of thing. It was just one more slightly interesting question on this. Could you find the set of values to which the following function is dereasing? So here's a function. I'd like you to find a set of values that this function is decreasing. Sorry, could you repeat it? So here's a function. I'd like you to find a set of values for which this function is increasing. So let's say there which explain why done that. Deffind the minimum point, not the minimum definfind the stationary Yeah the question also, where is this decreasing? So when six x plus four is less than zero. So think through what we said about through the pictures, said an increasing one. Is something like that decreasing or something like that we actually don't need to get into second dervative. And you remember what are are property of derivatives, property of the gradient is increasing versus ing the gradient. This is always positive. This is just this interesting. Absolutely. Where is the gradient negative? The analysis to solve the inequality practice, make sure you've written out like ke you've jupped the next there. So a little bit of quadratic economties practically thing. Do you remember how a deep gracal inequaliis Yeah, I'm just doing it on paper. Oh. Yes, that's not your final answer, but that's the involof file yet. Not quite. So you've got the two important numbers. And how do we turn that into a final answer? So for values that are smaller than effects that are smaller than. Negative or quite. So explain to me your method. Let's check that. Not doing anything just by wait. No automatic pilot floreceived the quadrative formula. Absolutely. Yeah. That gives us the root of the wait. That gives us the answers. If the question was, when is this equal to zero? So if this was a question. These two groups. But the thing is that was a question. Question is, when is it less zero? So can you remember how do we use these two values to work out when it's more less. What's your method? Once you found the roots, how do we use them for quality? You described your net. So you said less than -43. Could you explain why was that in a little bit of a guess? No, was a bit of a guess. So my opfor that tic inequality is to draw it. So we have a drastic positive x squared pericient. So it's this made up. We have two roots. I'm just going to put an x axis in there. There's our two reasons. Now we looking for where it's less than zero. So I reckon you can just look at this time, but just read out what exile is, is the coratic lesson zero. 对。Also when x is between. So so you've popped there your sigthe wrong way and we don't want included here. But Yeah, we've got the idea. We just find the area that we're interested in and we read it after. That is how I would approach quadratic employees ties. Always just give some super quick sketch, but avoid picking the wrong region. Excellent. And that's our fine larger. So actually, we don't need the second derivative. If we're just talking about whether a function is increasing or decreasing, so there's a couple of kind of quite similar concepts. We have to keep distinct ct. If they're on stattionpoints, we look for where is the radiant zero and decide what type stationary point we have. Then we can look at the second derivative. If the second deritive telnothing, if you wait zero, then we do first derivative a bits the left and the right. So that's one kind of harithat's our stage. Similarly, if we're talking about increasing, decreasing, we just look at make gradients and whether that's positive activity. So it's a lot of quite similar ideas. We just need to keep them kind of separated in our life. Does that make sense? Yeah. Okay. Final example of this. Could you tell me where this function is increasing? Where is this increasing? For which values? So again, think really we want to set the sequence ts of it. To find the quadratic. I poin it the way that people work dyfunction in degrees. I think of which of our areas were what tells us the. X intercept. So think about what this increasing function looks like. Increasing looks something like this. Decreasing look something like this. So what's the difference between two? What's the gradient doing here? What's the gradient doing here? One is always positive. One is always negative. So one of our key ideas to really into to memory and increase it, option positive Readus decrease their aspects. Wait, I'm gonna find the increasing so this is absolutely great. They've set up the right inequalifor I to solve it. Yes. X is between negative one. And. I would suggest so kill and you slisomewhere. Either way, it's definitely overkill. Can you mind for this particular stic? So you repeat, so what if I did this and you see a much simpler way of proceeding? Yeah. So again, always worth drawing the diagram. So x squares, nice and see what that just looks like this. So if we're looking at what x squared is, greater ter than four. So you're right that two is the Key Number that can afform. So that's I don't think that's quite what you mean, but are you trying to say between two and my two? Let's have another good at that. That says less than minus two and greater than two. So that one doesn't quite make it. So that makes sense, but that says between two and minus two. Is that what we want if we want x squared greater than four? And look on the diagram, see if you can circle the areas where x squared is greater informed hidden placset areas for Yeah can you see how they're two disjoint areas? So the thing is, we can't express this all as one inequality. There are no numbers that are simultaneously greater than two, less than minus two. That also doesn't really make sense because what you're saying here is that it's greater than two and it's greater than minus two. And that isn't quite right. We don't want it in the middle. So my hintage, you're going to need two separate from Yeah that works to be fair or much simpler. We just chuan all in there. So quadratic inequalities are really worth getting really awon that because that comes a later Yeah. So lots of similar ideas goes around. We're doing some practice to know exactly when we're using White. It's not conceptually super tricky. I think maybe just a little bit of practice so you can keep the ideas really the same. What we do when we looking for decreasing what the station points, what we're doing for station bikes. Yeah. Some of a topic, but I've now had to a bit of all. It's more like problem solving questions. I going look at these sheets I've sent in the chat and happen if you question what. Yeah, the same I did, but kind of framing it different, showing four the real worlside of this. I have a question. Correct right miss Yeah that's we could simplify down a bit but Yeah basically happy with that. Nice. What about part b? So the total area of the metal sheet would be the surface area of the be the Yeah absolutely remembering that the top is empty from over the top one. So Yeah, we should have to add up. What does it mean for which is a minimum? So just when it's a decreasing function or Oh no, sorry, when it's three dient equal als zero, a is a function a is a function of x and we want to find minimum of a. Just have a think about notation there. Sorry, you know we're there. Excellent members. Do I need the catheator for that once? It a bit nicer. So a three is pretty. The cufruto thousit's kind of nice, typica bit nicer. Sorry, miss what torii saying. I didn't hear we could write it in a slightly nicform. Let's not reach reccalculator. Just see if we could pop that in a nice form using the fact that while the key roof three is pretty nasty, the key root of a thousand is pretty nice. Can we use that fact to write a bit better? Almost. That down. So turns a more interesting method. So talk me through and do d. I don't know. So this is exactly what we're were doing earlier, but it does form different with different betas that I understand. So a is this function. We could even call it a of x. So it's some, if you imagine what the graph looks like, it's some kind of squiggly thing with a minute. See, perfect. We're trying to find where is this minimum value. So to do that, we're finding where we got this flat point is 0.0 derivative. We've done that meeting. We've found the derivative you set equal to zero, but we found that this point happens here. Oh, we need to allow you then. So so does the y value then. Yeah, absolutely. So can you just tell me what you're going to plug it to? To squared plus. Yeah, I'm going plug it into. That. Excellent. And let's maybe we not do the nasty numbers, but just tell me how you would approach. Finding this and this. To do what's the nicer, easier method that's easier when it works to the second deritive? Absolutely. So the second dervative memory test, can you remember what the test is? What tells us we have a minimum. If the change is. Hate to no, I can't remember is a minimum from positive to negative or from negative to positive. So not negative positive that would be sorry. Yeah minimum what? Negative only from negative to positive capso. What does that mean about the second negative vapositive or negative. Should be positive. Nice. So these ideas, inststationpoints ideas, really useful in modeling things like reward scenarios, but not mamatter. Okay, that makes sense. Excellent. I think let's do a tiny bit of second geometernow practice. Let's get a few more derivatives. So we've only actually encountered a handful of things we can differentiate in this point, basically just exterior. So let's add a few more to our brands. Well, first really important one. You to the so really the key result here is that e to the x differentiates to itself. E to the x, that's the key result. We can expand that. We can generalize it and put any number out the front here. This is that to be a special case of chawe'll encounter zobut. For now, the things to remember are e to the x differentiates to itself. And if we have a number out the front of that x, then we just gain that nudown the bottom as well. So that's our first important new results. Another than important one is. The next the French Chione, I'm afraid we don't really have any means of Cruthis would be with the nice you've got at this wiso, slightly boring just going through results. But the third one is kind of an interesting one. We can prove this one. So this third one is when we have a number on the bottom. And x is in the power that differentiates to a to the x learning and again, a more general altervariation. Is if we have a number out the front of the decthat, just slightly changes it. There. Yeah. So a couple more types of functions that we're now unable to. We can actually include this last one using the first two and it's part neactually. All I'm going to do is I'll show you the trick and then see how far you can get either. So we're interested in we have. Y being a to power x, what's divide by x? So the trick is I've going to write y as e of to the a to the x. Does that make sense? Because e and are inverse to each other. You can't across that. Yes, because the natural log is log b. Yeah. So are you happy that are you happy that these are balancing each other out? So a to the X, E, A to the x. Now, I just realized we probably haven't covered this yet. We have a lot of it. I don't know. We don't have. We've said that if you have a power inside the, that's the same. Yeah, it would move to the front here as it would go x and excellent. So is this proof making sense so far? Yeah, excellent. So I'm going to hand over to you. Can you now, in this type form, differentiate it? And I proit be all you need is this first result here. Let me scroll down and correct it. So feed the care. Absolutely, because that Lena is just a number. So we're using this rule and then can you rewrite it in a more head? You get. Could you repethat was great. Can you rewrite it so we end up with this on so you see how we get to that final. How would I cancel it out those? So what we kind of want to do, it's kind of unpeel what we've done. So what if I get you started? We're got to put it back inside. Can you see how the rest goes? See the final week, we're kind of unpealing what we did earlier. So that be I've learned of something what's very equal to. It's just canceled out. Absolutely. Then we can just say those two. So we have a to the x. That makes sense. Yeah, excellent final few checking each other ones. And then maybe a few more in our own twelve. It is next to cosine almost sine goes to plus goes to minus sine. Yep. So it sounds like you think you've across them before. Just got to get a minus, right? There are so of repertoiis. Great. You can now do extra power. You can do e, you can do that's not you can do X, E power. And we got some basic, but now we've kind of funcwe don't have to differentiate. It's all about, can we confinthem into more interesting combinations? So most important one products, what if we have again? No, sorry, there's nothing. So I want to do if we have a function that's made up of two familiar functions times together, again, arenreally free. Yeah. We don't really have the left at this point that I'm satisand slightly enough. Still want to memorize. We come across the polfloor. Yeah, it's the same. But if I val effects. Or not quite if only it's sorry, it's quite a fitting one. Not it's a fiti'll tell you how I remember it, square the bottom. Put it on top, differentiate the other one minus the other way round. I would remember it as that little al kind of line. So otherwise, it's just a bit be honest, it's actually given to an ada center. Worth having it, obviously. So let's do. Some practice. How about? So could you differentiate that? It looks like didn't need it. 是你。What about? Make it's desperatness or something, the roll back. The instantly memory is denso. That's a. Always so be limited. So this is just that sister. Thanks, guys. It's a natural logical too. So it be sine negative sine. Yeah be that we'll just put out time to time I might finally be a for lesson Yeah should we keep going this for now or is there anything more pracing come up at school particularly you want to look at? No, no, it's good to finish the syllabus sentence I finished the school entry exam so I'm waiting for the ibcse in April. I'm actually you've done you 60 pressing excellent. Lovely. So Yeah, I think we'll cover a few more topics and then I thought maybe we start to get to which are a University admissions test sts, but they're really good for just see familiar understanding of the first year and p level material. Yeah, great. I've got his asthought. I will see you next time. Great stuff, right? Yeah, thank you, miss.