here at Stanford University in Palo Alto, California. Actually, Stanford, California, it's its own town. I'm John Furrier, the founder of SiliconANGLE, and this is theCUBE, our flagship program. We go out to the events, extract a silver noise. I'm joined by my co-host, Jeff Kelly, this week, and Oprah Cahane, who's origami logic. You guys are stealth mode? Very stealthy. You can't see my face, but we've revealed that now. Welcome to theCUBE. So you can just watch the microphone there. It's a little noisy now, the session's getting out. I just saw Kara Schwisher from All Things D, Bud Collig, and all the luminaries are here, all the top Excel partners are here, and all the startups, you guys are startups. So talk a little bit about why you're here, and what are you guys talking about in the hallways? You haven't launched yet, or what's the deal? Yeah, so we're still very stealthy, haven't launched yet, so I can reveal a whole lot about what we do. Happy, of course, to talk about the things that we solve, the problem space that we tackle. And we're here, as this is certainly the event to be, we're one of Excel's recent investments in the big data space, and all this. How much did they put in? We raised it to around $9.3 million late last fall. Series A? Series A, with Excel being the primary investor. Awesome. So talk about the big data, what problem are you guys solving, what's the big challenge? So the challenge we're solving is how big data gets really applied to help marketing folks. Marketers are really struggling with the fact that they're surrounded by this ever-increasing number of customer touch points. Each touch point typically comes with some marketing system to manage the content, the workflow, the interaction with the audience across that channel. Whether it is web in the early days of digital marketing, down to advertising, search, social, which is now an explosion of the plethora of different types of channels and apps. And the more innovation we feed here in Silicon Valley and River Elf, driven by the axels of the world in funding, a lot of those next-gen marketing platform of social interaction platforms, the more touch points there are, the more data is generated by these systems, and the more complex is a marketer's world and around how do they touch their customers, where should they touch their customers, how can they make sense of the marketing landscape in and around them. And that complexity is not really effectively tackled by any existing platform tool. Thankfully, the underlying big data platform, such as the ones coming from Cloudera and Friends, are helping deal with how does one store the data, how does one process the data at scale. The next phase beyond that of verticalized big data apps is one where by this, how do business people actually monetize the data? And we're a perfect example of how does one apply that trend to marketing. We're building really, think of it as Splunk for marketers or Mint for marketers, that centralized cloud-based platform that lets folks in a marketing organization without IT involved, without data scientists involved, have all the data matters to them automatically collected into this big cloud-based platform, automatically integrated, because integration of data in any environment, and certainly in the marketing environment, which is super siloed, is extremely complex and difficult. So we apply data science to automatically integrate the data sets, and then provide beautiful visual marketer-optimized user experience for marketers to directly interact with the data and make sense of the world in and around them. So I definitely want to get into some of the issues around the challenges facing marketers and how they do their jobs. But I wonder if you could talk a little bit about, as an entrepreneur, you mentioned how the infrastructure layer where we're storing processing data is starting to mature. Talk about as an entrepreneur where the opportunity for yourself and others to actually start taking advantage of that infrastructure. Is it a lot easier today to start building a big data application than it was a year ago, two years ago? Talk about that transition and what it means as an entrepreneur with an idea for a big data application to actually have that underlying infrastructure and platform, whether it's Sadoop or something else, available to you and how that kind of makes it possible to actually turn these apps into reality. Yeah, so I think we are getting to that point that it's a whole lot easier to develop the application layer, riding it up with a platform. I'll say, much easier than it was a year ago, it's still a lot more difficult than it will be two or three years ahead as the platform is still evolving. When I say platform, I talk about the platform at large, Hadoop and the ecosystem and any of the other no-sequel and big data building blocks. There's still complexity around tying the individual pieces together into your underlying data platform, but that is getting less and less difficult over time. So it appears to me that this is a perfect time for entrepreneurs to think about not just what is that next potential piece that's missing at the underlying infrastructure layer. I think that's heavily crowded. A lot of this was started in place. The big players and big bits have been made. It's more about how do you help businesses or consumers depending on the nature of the application, monetize the value of the data. And as an entrepreneur, you don't need to think as much about solving the very fundamental basic computational and storage challenges that Hadoops of the world helped us all out of the box. But rather, what is the specific business workflow you can help tackle? Or the consumer need you can help tackle? And focus on that, and I think that's where a lot of the next generation value will come from big data or the next step in the evolution of the industry. Absolutely. Now the conversation is starting to turn to or include how do you monetize that data that all the players like Flutterers and Hortonworks and others of the world are actually helping you store a process. Now it's about entrepreneurs like yourselves building on top of those platforms and actually turning that data into something that can mean a business value. Exactly. So you're taking a kind of a function specific look at at the way you build your application who you're targeting marketers in your case. I wonder if you could talk about why you took that approach as opposed to building a more horizontal platform with visualizations on top of big data for maybe not just marketers with sales or finance or whatever the case may be. Why that specific focus on marketers or a specific domain? Yeah, so I think there are two questions in one here. First, why marketing? Our sense after thinking a lot about the application layer from an entrepreneurial point of view was that marketers are uniquely positioned in having and being really data and budget rich. They truly strategically impact the success of the companies they drive. On the other hand, they are skill and insight poor. Skill because to monetize the value from that data richness requires a lot of complexity, a lot of data science which does not exist in most organizations neither the IT nor the marketing department. Secondly, we felt that the value that we could provide in solving these pain points would be one which will have impact on the top line of companies versus potentially dealing with other potential verticals. Though functionally speaking, there are similar problems I'm sure that can be solved for any of the other functional departments that you've mentioned whether it's finance and are companies chasing that or whether it is sales or lots of companies chasing that and so on and so forth. Now, why vertically focus in this case functionally focus versus purely horizontal? There's another underlying theme here which I think is important and my bet is we'll see more and more of that over time. The complexity of any of the functional datasets marketing, sales, finance and so on and so forth is becoming so high and surely the marketing space given the data being so siloed and diffused and heterogeneous and in variety and form that makes any of the generic tools inherently ill at that and incapable to truly embrace the complexity of what needs to be solved in a way that is immediately valuable to the marketer and the business. You need to apply lots and lots of data science in order to automatically fuse those datasets. You need to apply lots and lots of data specific connector development to collect the data which is very complex in its form and normalize it into something that would be useful specifically for marketers. And certainly so the user experience I believe over time will end up being more and more verticalized in its value such that if you're building something for marketers just taking the generic ways to visualize data won't work. You need to visualize data in terms that are specific to marketing data types and specific to the marketing business processes which are different from sales or engineering or logistics or whatever the case is. Stay differently. Our thesis of over time generic BI while will be useful for certain use cases will diminish in value given the complexity of the data specific and function specific technology you need to slap on top of it in order for it to be useful out of the box for your business users. So my question to you is obviously you can't talk a lot about your company right now but talk about just as an industry participant what are the big misconceptions of big data? Is it easier than people think? Is it harder than people think? Obviously it's pretty hyped up right now even here at Excel where they're considered at the front edge of the world and they talk about consumerization of IT like it's a new thing. So you have this shadow IT, you have consumerization happening, been around for a while. What do you share about big data that you could clear up or just comment on? So a few different thoughts. First and foremost, the hype is certainly broad. I think over time the changes of big data is driving will be as massive and impactful on the entire industry as the evolution of the early relational database has been to enterprise software 20, 30 years ago. That being said, when folks use a term big data a lot of the complexities behind that are actually being obscured. First and foremost, it's not just about the size, it's about the variety of the data and a lot of folks seem to be missing that point. Another super critical point is the human aspect of it or the organizational aspect of it. In many organizations to really get to that optimal point where you can leverage the data in big data terms, you need to really break down silos which are not just technological in nature or rather organizational in nature. For example, when if you use CRM data with social data, with product inventory data and so on and so forth, it's not just a technical problem that's really difficult to tackle but rather organizationally, how do you get access to all these datasets? Who owns that within the organization and so on and so forth? In my earlier career, I've been more in the networking space and used to speak about the networking stack, layer one through layer seven and there are always difficult technology challenges there. You used to typically joke that the most problematic challenges are actually layer eight, the human layer, the organizational layer and I think it supplies to here in big data. Final point to be made, ultimately the value that folks drive out of big data, manifests itself in small data. It's those pieces of data within this massive haystack of silo data you mesh together that really matter and ultimately it's about the individual business user, how fast can they consume the value from that data, how fast can they find those nuggets of value that drive action and insights that drives ultimately some monetization versus some theoretical concept of here's value in the data. You know, we've seen that too. I was one of the first to co-intern fast data implying little data, because it moves around a lot, it's loose data, whatever you want to call it, but I would agree with you there. But I want to go back to some of the technology issues because even though layer eight, the human layer is challenging with mobile, which throws off a lot of small data. Gesture data, gestures here and there. But you mentioned relational databases. The database is a big part of the stack now where you have solidified networking, moving off that concept of virtualization now, in Naveler. What is the future of these stacks? There's stacks out there, there's technology stacks, and the database in particular.  Now you have a variety of data. So database got to fit into these stacks, a lot of multiple stacks, little dimensional issues. Can you comment on this whole database? Absolutely, so I think when folks talk about database in legacy terms, typically there was the relational database, a single source of truth, data warehouses, you know, dangling it around it and so on and so forth. But folks spoke about the database, and data was always pretty rigid and stuck into that template. I think reality of modern data is first and foremost super, super siloed, super fragmented, super diffused, so it's no longer a database.  Second point is the data is so varied in form. Unstructured data, structured data, temporal data, geospatial data, relationship data, graph data, all these things that certainly in our world marketers need to face, and traditionally they would need to use different tools to analyze different aspects. It takes analysis system to analyze unstructured takes from social. Some graph mapping software to figure out relationships between people and product and networks and whatever else, and so on and so forth. So I think we'll just see this plethora of more and more data types that need different special tools for them. And complexity ultimately will need to be dealt with, something we're looking at from a stack perspective by higher abstraction layers that help hide that complexity under something that's useful for a bitters person that doesn't need to and shouldn't care about whether some of the data is stored in one form to tackle unstructured data and some other form in order to tackle relationship or temporal or any other form of data set. So NetNet no longer the database, but many different types of data stores or different types of specialized form of data will need to be hidden at the application layer. Yeah, I think you're right on there. A business user doesn't care when they have a query or exploring data via visualizations if it's a relational database working into the covers or a dupe or anything else. It's just they want an answer to their question. They have a business problem, they want to solve it. They're not necessarily care about the underlying infrastructure. I want to go back though, you mentioned kind of the human problem and I was actually at the SAS Global Forum yesterday and there was a customer panel and one of the gentlemen there was a workshop financial services firm and he said, we have some long time employees, people who have over 30 years at this company. And he said, one of the biggest issues is of course the changing the way they think and do their job when it comes to using data to actually make decisions. Now of course, now SAS, interesting enough is one of those kind of more horizontal plays and they've got a big BI player which I'm sure you're quite aware of. But I wonder how do you, if you're talking to a CMO and they're saying, look, this sounds like a great way to get better insights into our business, but how do I go about convincing and all of my employees who've been doing things a certain way, getting answers a certain way to actually start adopting these new approaches and do it in a way that doesn't kind of, you don't want to go in and say, the analytics knows better than you. You want to go in and say, this is going to kind of complement what you do and what you know. How do you have that conversation and what are some tips for CMOs to make that transition with their employees? So I think it's a super critical point you're making and I totally agree with it. Ultimately, folks shouldn't have that presumption that big data is this kind of state that solves all of their marketing problems. It's a tool, it's just a means rather than the end. And in our mind, it's ultimately about bridging the value you can extract from data through automated technologies and big data infrastructure with intuition, certainly in the marketing space which is still critical because no machine, no machine learning engine can truly empathize with the consumer and the consumer journey that's being made as a marketer can with their intuition because there's a lot of emotion that's involved with Bridges process. Hence, at least the way we're taking this from a product development perspective and a market perspective, we're thinking about our solution is a way that helps bridge human intuition with underlying big data. First thing, second thing, when looking at evolving a marketing organization from driving purely on intuition or mostly intuition to having the appropriate balance between being data driven and intuition driven, it's all about measurement and KPIs or business objectives. How to tie ultimately measurable key performance indicators rather than just, activity metrics have to sort of be confined across the very social channels. How many likes, shares, retweets, more important for example for folks to measure what is the quality of the fan base they've created or if you're looking at folks coming in through some other acquisition channel, don't just measure the volumetric aspects of these things, measure the business impact of them. And as CMOs start driving objectives down in the organization as we're seeing with many of our early alpha customers, their grassroots folks, the worker B level folks don't need to be particularly driven to start using data because in order to be measured based on data driven metrics, and then they start using these sort of tools as part of their day to day. Interesting, so it's a combination of kind of incentivizing and kind of setting the tone from the top but also kind of creating that organic movement to adopt these kind of technologies that approach us, interesting. Okay, oh for thanks for coming on theCUBE, always great to get your perspective. Now we could have probed you, probed you a little bit more about your startup but you know it's not our style to, we'll find out for the back channels, everything and we'll report it on silkenangle.com but not only kidding, we respect your stealth mode. Congratulations on your financing. You guys didn't know your big data, you have the big data chops there so congratulations and thanks much. Thanks for being on theCUBE. This is Silkenangle theCUBE, our exclusive coverage, this is theCUBE, our flagship program, we go out to the events, extract the signal from the noise. I'm John Furrier with Jeff Kelly, we'll be right back with our next guest. Got a great lineup of entrepreneurs, industry leaders, old school guys, I have Bud Colligan, Sir Karrish Swisher, he wanted to get her on as well. This is theCUBE at Stanford Excel Symposium, 17th year, we'll be right back with our next guest.


Let us continue our discussion of spray formation and underlying the idea of spray formation is the concept of where you have bulk liquid inside the nozzle, bulk liquid is liquid that is still in some connected form in its entirety that breaks up into drops slightly outside the nozzle. Now, we want to see the process of the process and the forces involved that cause these cause this bulk liquid to break down into drops. So, this we are going to investigate using a theory called linear stability analysis. Before we go much further let us understand this word stability. What does stability mean?  We all know from common observation that if I leave a ball on the side of this trough this is going to roll down maybe even overshoot a little bit, but eventually it will come to rest right at the bottom. So, if I take a trough of oil then this is going to come to rest and this is what we call the equilibrium condition. Now, if I take another trough that is also and if I drop this ball here and let us say this is all liquid it is possible that it could come to rest here or if I am very lucky it could come to rest here where it overshoots and then you can clearly see it has to happen by chance or let us just say I gently placed it over there that is quite possible. Now these two are both equilibrium. So, both of these the previous was a case where there was only one equilibrium solution one equilibrium condition whereas, here is a case where there is more than one equilibrium condition. In fact, if I want to be precise there is three equilibrium conditions. So, what is the difference between this equilibrium condition and these two equilibrium conditions and the one where the ball is sitting on top. The fundamental difference is that if I was to take the I will number these equilibrium conditions as 1, 2 and 3 and I am asking this question what is different between 2 and 1. I am going to leave three out because three is the same as one for now. So, what is different between 2 and 1 as far as this between conditions 2 and 1. So, the difference is in the answer is in the response to small perturbations. If I took a ball at condition 1 and I will redraw the figure here just for clarity. If I displace it slightly what the ball will do is essentially trace its way back towards the equilibrium condition 1. If I apply the same logic to this if this is the direction of displacement the ball is going to or the ball is going to move away towards 3. If I displace the ball on this side it is going to come back towards 1. So, the nature of the perturbation is going to dictate the new equilibrium solution it is going to seek whereas, in case of 1 whether I perturb it to its left or to its right it is going to come back to 1 it is going to attempt to come back to 1. Now, the reason you have to understand two things here one is an equilibrium condition 2 is also an equilibrium condition and if I perturbed the system from its equilibrium position essentially the tendency to come back to the same equilibrium position or to go to a new equilibrium position is the only distinguishing factor between 1 and 2. Therefore, in the case of 1 1 is characterized by the feature 2 is characterized by the fact now it is very important one thing here these are only true when you are dealing with small perturbations. Think of what I mean by small if the perturbation let us say if I take this sort of an angle if that perturbation were to become large whether it is in one direction or the other let us say if I perturb it to where I move the ball to 2 exactly to 2 clearly it is not coming back to 1. So, what I had written in the previous slide will no longer be valid if delta theta is on the order of the angular displacement needed to take it from 1 to 2. If I go further than it goes to 3 which is also a case where the statement is not true whereas, for small enough perturbations. So, really speaking in a mathematical sense small means infinitely small. So, we are only looking at perturbations that are infinitely small in comparison to the mean state that we want to see if I just move it to its left or to its right slightly what will it do. So, this is if I take several equilibrium solutions of let us say in this case equilibrium is simply a force balance where the horizontal forces and the vertical forces are all in balance and that is they are all in balance at only 3 points in that trough and 2 of those 3 points are alike in their stability characteristics which in this case are 1 and 3 and the point 2 is different in its stability characteristics so mind you they are all equilibrium solutions it is just that 1 is 1 is stable 3 is stable 2 is unstable. So, what we are going to conclude from this 1 and 3 2 is an unstable equilibrium condition. So, now what does this mean for fluid mechanics and what does it mean for sprays and atomization if I take a condition again where let us say I will take a trough I will take a simple case where I will place oil above water. So, just I have a beaker of oil sitting above water in a general in a gravitational field just like that what do we learn from this oil is lighter than water. So, it pretty much this is an equilibrium condition now if I ask the question if I gently place water above oil and I ask you the question is this an equilibrium condition the answer is yes. So, both of these both of these both of these are equilibrium even though density of water is greater than the density of oil both of these are equilibrium conditions. How do I know that you can in fact the simplest ways you can actually experimentally do this you can take a little bit of oil in the bottom of beaker and very slowly pour water on top not really pour it is almost like your placing water on top and you can place a nice thick layer of water on top of oil this you will be able to see it. But if I was to take this beaker of oil and water and give it a slight impulse just give it a slight jerk the oil and when the oil is on water the oil and the water meniscus both sort of undergo some oscillations, but essentially come to rest in the same configuration whereas when you take water on top of oil and you give it a slight jerk over time the water would come down and the oil would go up you can see this experimentally those of you that have never done this it is a very simple experiment to do go into your kitchens today and do this experiment try to recreate this unstable equilibrium condition and this stable equilibrium condition and see for yourself that that is indeed an equilibrium condition unless the meniscus is perturbed. So, what is this little like shake or a jerk doing to the beaker essentially what I am doing is I am taking this meniscus and perturbing it just like that. Now, the top meniscus is also perturbed, but that is less important in comparison to the water oil meniscus. So, as soon as this is perturbed the fact that right around here I have an opportunity for the heavier fluid I have an opportunity for the heavier fluid to come down and the lighter fluid to go up is going to set up a motion in this direction that is going to cause it to become go to this stable condition of oil on top of water. So, every one of these little perturbations is going to set up a roll like that. Now, I did not draw these intermediate rolls here, but essentially there is also a roll there a roll there. So, as soon as I perturbed the meniscus slightly I will create these rolls what I mean by a roll is a condition where the heavier fluid wants to come back down to the bottom of the trough and the lighter fluid wants to rise up because that is the more favored equilibrium condition. So, the question of whether a fluid mechanic system is stable or unstable is determined by whether a perturbation of let us say in this case the meniscus is going to die out and cause the meniscus to return back to its original shape and condition or like in this condition will the meniscus grow. So, think of what would actually happen if you had the meniscus growing. So, if I impose a small perturbation on this meniscus over time like because the oil wants to rise up naturally and the water wants to come down the meniscus shape is now going to become altered and go to that condition. So, imagine I took this nice and flat meniscus with the water on top oil on the bottom and I put let us say in the simplest sense a sinusoidal perturbation a sinusoidal perturbation of this meniscus of some wavelength. So, let us say now I can also make this some wavelength lambda and that wavelength lambda remaining the same the amplitude of the perturbation will grow in the infinitesimally small time as soon as I have let go of this meniscus from this perturbation shape because water wants to come down and oil wants to go up in the immediate time following t. So, t slightly greater than 0 what is the behavior going to look like the shape of the meniscus is not going to be altered drastically in other words the functional form let us say in this case being sinusoidal except the amplitude will grow slightly. So, just by looking at whether the amplitude would grow or decay in a short interval of time following the perturbation itself we are able to characterize whether the perturbation is stable whether the system is stable to small perturbations or unstable to small perturbations. So, this is a case where the system is unstable to small perturbations how do I know that because the interface amplitude is going to grow with time. Now, this kind of an instability was is not something that is new it was discovered back in the mid-1800s and is and it goes by the name Rayleigh Taylor instability. Now, can you I mean we are only going to talk about this in a physical sense for now, but we will come to the mathematics shortly the rate of growth going from the red line to the green line that you see here the rate of growth is going to depend on lambda itself keeping everything else the same if lambda was short or long would cause the rate of growth to be different. And that is the one characteristic that makes the discussion of stability of fluid mechanic systems different from the stability of the mechanical system example that we discussed earlier there the only perturbation possible is in the angular coordinate theta that we discussed. So, a slight delta theta and the force as a function of delta theta is attempting to bring it back towards theta equal to 0 the one equilibrium condition that we are after right. So, whether whereas in the case of the Rayleigh Taylor instability of super post fluids I can impose an infinite now infinite different types of perturbations of different lambda of different wave lengths and if each wavelength has certain kind of growth characteristics then what sort of a perturbation do I really should I really be looking at. So, let me go back rephrase what I said in a more physical way in the previous example only one delta theta is possible whereas, here if I want to perturb this meniscus as a sinusoidal function there are an infinite number of sinusoidal functional perturbations possible of that flat initially flat meniscus, but each perturbation has a different growth rate associated with it the rate of growth of the meniscus is different for different lambda. So, what is it that is so in other words I could also have a certain lambda let us say for example, if I put a very very small lambda perturbation on it you will see from the mathematics later on that in fact it will be stable to that perturbation just because it is stable to one perturbation does not mean it is stable to all perturbations. So, in other words we have to sort of qualify the our discussion of stability from the earlier example and say that for a fluid mechanic system to be called a stable equilibrium system that system would have to be stable under all kinds of perturbations. So, whatever be lambda if the meniscus were to return back to this condition with water on top and oil on top then you can say that this is a stable equilibrium condition. So, with oil on top and with water at the bottom it does not matter how you perturb the meniscus as long as the perturbation is small in comparison to let us say the thickness of water and oil. As long as the amplitude of perturbation is small it will return back to the condition where oil is on top water is on bottom the meniscus is flat and everything is at rest that is the equilibrium condition that I know there and that will come it will come back to that condition irrespective of what kind of perturbation I give whereas for this particular situation there are there is a range of lambdas for which oil wants to reverse go back to the top water wants to come to the bottom. There are lambdas outside that range for which it will come to this condition but that does not mean that the system is stable it is it has to be called unstable because there is at least one lambda for which the system will go to another state. So, how do I know which lambda is equal is actually important. So, the way we approach the answer to that question is we say that I do not know what lambda is possible because if I take water on top oil at the bottom and I jerk this container slightly I do not know what sort of a lambda I imposed on the meniscus. So, we are going to assume that somehow the meniscus has a way of sampling all the possible values of lambda from 0 to infinity. Now, if the growth rate is different for different lambdas then there is at least one lambda which has the maximum growth rate that is going to somehow grow at the fastest rate in comparison to the rest. So, even though initially all values of lambda are possible we can look at at least one situation where the lambda causes the growth rate to be much higher than all it is all the other values of lambda there is one value of lambda for which the growth rate is higher than all the other values of lambda. This growth rate this lambda is eventually going to dominate as far as dictating the shape of the meniscus itself is concerned slightly later on after the perturbation has been introduced. So, what we will see is that there is one value of lambda that will survive this competition even though I start out with the meniscus which is purely random. So, I have like lots of different lambda superimposed on there one particular value of lambda would out compete the rest in the rate at which it grows. So, that is essentially the fun the sort of a physical understanding of the theory of linear instability analysis perturbations are small the meniscus retains its shape at least in the harmonic form. So, if the if in this case I impose sinusoidal sinusoidal perturbation the sinusoidal nature of the meniscus shape is not destroyed in the immediate times following t equal to 0, but the amplitude grows. So, if I have lots of different lambda each lambda is associated with the certain amplitude and the amplitude corresponding to that lambda grows at a rate as though it is the only superimposed disturbance. So, we are going to break this problem down into a linearized version of the problem wherein solutions to every given lambda can be superposed to find the solution to a general kind of a perturbation. So, let us go back now and see what are all the different kinds of instabilities that we see in a fluid mechanics system and we will try to give some physical arguments for the kinds of instabilities. The first one we just saw a moment ago is called Rayleigh Taylor instability this is essentially due to adverse density stratification that is a heavier fluid sitting on top of a lighter fluid is not completely the most stable of situations the lighter fluid wants to go up and the heavier fluid wants to come down therefore, this is cause for an instability. The second is what we will call Kelvin Helmholtz. So, what do we mean by this Kelvin Helmholtz instability let us say I have a fluid of density rho 1 and rho 2 the fluid here is moving at a velocity u 2 the fluid here is moving at a velocity u 1 and this is the interface. So, the interface because the because u 2 is different from u 1 there is a relative motion and that relative motion is cause for the meniscus the interface to become unstable. So, over time I could get I am going to just draw some little cartoons that are representative of what happens so if this meniscus is slightly perturbed then there is again a certain range of lambda for which this perturbation would want to grow. In fact, over time you can see some sort of a like waves where you know any this kind of a trough evolves into a wave that this is now fluid rho 1 and this is rho 2 this is rho 2 here. So, the meniscus can you can see some sort of a rolling up of rho 1 fluid into rho 2 region this is after a very long time we are not really concerned about that for now, but all I want to know is will the meniscus grow or decay a more generalized version of this u 2 u 1 situation is even is a situation where I have a smooth transition it does not have to be an abrupt step like I describe with u 2 and u 1. So, if I have some velocity profile u as a function of y what happens to the to this velocity profile if the fluids were to continue to flow the answer to that is that you will find that because this kind of a velocity profile has an inflection point that inflection point is sufficient to ensure that this velocity profile will be unstable. So, where do I actually see something like this? So, think of an atomizer. So, now I am injecting liquid through a little hole I have created a cylindrical column of liquid and this is air outside the air velocity is 0 the liquid velocity is some W L E Z. So, if I was to draw the velocity profile and the air is at 0 velocity. So, this meniscus here is subject to this relative velocity between the fluid moving fast and the outside air being at rest where did we encounter this kind of a physical situation this is your diesel injector. So, a diesel injector has this kind of a velocity profile we want to see that if this velocity profile if this was a nice thick pencil of liquid coming out of the diesel injector and into perfectly quiet air where there was no source of perturbations at all that pencil of liquid would remain exactly the way it is, but if I have some sort of a perturbation either in the air or in the nozzle or any other source I want to know if those perturbations will grow in amplitude or decay in amplitude. If they grow in amplitude that is a case where I could cause the interface to ultimately break up if they decay in amplitude that means I am going to create what looks like a firefighting nozzle. So, if I want if I am a firefighter linear instability analysis like this is extremely important for me to know the operating conditions under which any perturbation I introduced to this flow is not going to cause it to break up if there is such a possibility. So, I can just shoot a jet of liquid towards the target. So, this is a simple example of a Kelvin Helmholtz instability the third is what we call Rayleigh instability for this I will take a let us say a liquid column I am going to create now a liquid that is just to make the distinction with Kelvin Helmholtz instability it is easy to understand this in the context of nothing moving. So, if I somehow created a column of liquid in microgravity. So, I have a cylindrical column of liquid that is stretching that is infinitely long this infinite column of liquid is left like that would remain like that forever does everybody and look at I want you to imagine a long column of liquid like a pencil of liquid that is left in microgravity with no end caps. So, I am looking at an infinite column of liquid on either side if this infinite column had a nice smooth cylindrical interface there is nothing that causes it to a break up. In other words the pressure inside because of this curvature of the meniscus the pressure inside would be slightly greater than the pressure outside, but that inside pressure is the same everywhere inside this cylindrical column and the outside pressure is also equal everywhere outside. So, there is no net force acting on any element of fluid for it to transition to another state whereas, if I took this infinite column of liquid and perturbed it slightly just like that now I have created because the curvature here is smaller the radius is smaller curvature is larger because of that the pressure here would be higher than the pressure here assume for a moment initially the pressure everywhere was 0 inside the liquid and slightly negative outside just to make sure the force is balance out. We will see this mathematically in just a moment I just want to sort of introduce you to the physics of how these different kinds of instability work. If I take a column of liquid if there was no perturbations on the column of liquid it will essentially stay like a cylindrical column forever whereas, if I now impose small disturbances because I have introduced curvature differences on that cylindrical sheet the curvature here is different from the curvature there because of that the surface tension force here is different from the surface tension force on where I have a crest. So, where I have a crest versus where I have a trough I have different surface tension forces because of that under certain for certain lambda values this amplitude of the disturbance could grow and this growth is entirely determined only by surface tension forces nothing else it is because everything else is quiet no gravity no movement anywhere I could take a cylindrical column of liquid and if I impose all different kinds of lambda the rate of growth of one particular lambda would be faster than the rate of growth of all other kinds of lambda. So, you will see that quickly that value of lambda out competes everything else. So, assuming initially you had a chance to impose all the different values of lambda you will see that because of this competition helping one particular value of lambda that shape mode shape will show up as the shape of the cylindrical interface. This is the reason why when you have a faucet that is dripping it creates a very uniform drop size. So, anywhere that you see order anywhere that you see sort of a uniformity in a length scale the length scale is the easiest for us to perceive, but even time scale would be quite the same argument would also be applicable to time scales. So, any physical system that you see out there that has that shows sort of a repeated pattern either in length or time chances are that that sort of a system is governed by a state that was initially unstable and when several different values of lambda were several different perturbations each of a different lambda was given to that system one particular wavelength out competed everything what you see finally, with your naked eye is the result of that competition and having only one particular value of lambda. There is a very simple, but classic examples of this on some days when you look at the sky you see clouds arranged in almost like striations with the spacing being very uniform across the spacing between two rows of clouds you will see on some days is very repeatable. So, there is like a repeated pattern in the distance between like two rows of clouds that is a case of Kelvin-Helmholtz instability. There is also case like that is also in fact something like that is also governed by Rayleigh Taylor instability another case is waves on an ocean or waves on a ripples on a lake the spacing between a pair of ripples on a lake on a quiet nice day with where there is a gentle breeze flowing the ripples are very uniformly spaced right. The reason for that is there is no source out there that is exactly giving ripples at that wavelength this is that disturbances of all wavelengths are available in the air around in the water inside etcetera, but one particular value of lambda out competes in the rate at which it grows. So, what you see if I take if I took a perfectly flat meniscus of a lake with no ripples at all and gave all the different possible wavelengths one after the other the one that has the highest growth rate would show up quickly in time and that is what we perceive. There are many examples like this you know where like for example, if I took water on top of oil and you perturb it slightly it could be like a little tap on the side of a beaker. You will see that you will see holes formed on the meniscus through which the water is jetting downwards and between a pair of holes you will have oil rising upwards and the spacing between these is very repeatable and very uniform this is actually a reasonably simple experiment to do with a glass beaker water and oil in a kitchen it strongly recommend you do it. You can see how repeatable that pattern is without really having any kind of a control over it. So, it is not like I have to impose a certain kind of disturbance I have to just give a small impulse and impulse if you look at it in the four year space is nothing but a super position of every possible wavelength ok. So, I can take every possible wavelength and deliver it to the system and one particular wavelength is chosen by the system. We can actually go well beyond all of these mechanical systems take the spacing between leaves on a tree leaves on a little branch you will find some some kinds of trees have a very uniform spacing between the leaves ok on a simple branch it has nothing to do with some control control freak inside the branch that is causing leaves to sprout up exactly at that interval ok. So, a system like that wherever you see a natural system with with repeated patterns it is in general characterizable by this kind of a theory ok alright. So, then let us just quickly recap what we learnt today the first thing we did we is we introduced the idea of stability the ball in a trough is what in mechanics we call a single degree of freedom system there is only one degree of freedom for the ball ok. And then we extended the discussion and then finally we discussed a few different flow systems and the related instabilities an excellent reference in this regard is what is called hydrodynamic and hydromagnetic stability it is a book by S Chandrasekhar this would be a very good resource to read ok. Thank you.


Mm-hmm. Oh, it's you guys again You don't mind me.  What's up, man? Yep. I got on PC. I will carve them I guess I can sell the kestadon stuff. I don't plan on using it Oh Dog pile on the rabbit dog pile on the rabbit dog pile on the rabbit. I Got a head out. You have a good rest of your hunt. All right. I'll talk to man. Oh Toa cool. Yeah, I'm pronouncing that right No, I want to get down Okay, they're not gonna let me feel okay Whoa late nights just re-subscribe for 52 months in a row Good guy the money I'm showing some love in the chat. That is a long time man Thank you so much Like slicing ammo lot that and piercing. Yeah, my favorite is piercing at the moment.  I like sticky ammo a lot That's kind of expensive. Oh, I was supposed to make that jump chat. I Was lied to How you doing man? Welcome flies adjust Yeah, that's what I do man. It's like it's like dragon piercer That's how I use it. It's really cool. That's why I like it so much. I Go through a lot of it. Well, now I'm soloing. It's kind of tough Wait, this is the ones I got ran away from before it is I'll come over here. I'm out of the way now. These are respawning. I just left there There's a dude that hated me before Bye You don't remember me. Do you? I Need a ration and another one. No, no, no Do I need to eat food, let me see Let's do a There's a barbecue pit. Oh, I can't use it. I guess I don't have any meat. Oh, he's still on me chat He wants my blood No, thank you loves. All right. I'm gonna do these bones a couple more times And now I'm gonna probably do a hunt because I think I'll get the monster s-plus for my hunt That's where I got all the elements I had these guys are back and I happy Dun dun dun dun dun dun Still dropping frames. I'm gonna turn anti-aliasing back on Maybe that's causing Palico level in Greece. That's cool. Tell us where Zora Magdaros is. Yay armor points Honey, it's a honey. No, did they're wrong, flora? Yes, good uncapped the FPS V-sync is okay. I think I'm gonna turn the anti-aliasing back on though Because it wasn't like this yesterday. I could do the FXAA maybe. Oh, I'm gonna take a break guys Thanks for watching feel free to follow the stream if you enjoy yourself follow me on Twitter twitter.com I'm gonna restart the game and then take a little bit of a break get some water because I'm out of water So thanks for watching. It's been fun I'll be turned soon. Let me restart the game real quick So, thanks for watching it's been fun. I'll return soon. Let me restart the game real quick Well, I out of here that Starting up. No this means my own on session. I'd he's gonna change now So if you want to join I'll post this in chat. Tell us where Zora magóros is I forgot about these I'm way ahead of you real fire stone. What's up, buddy? This is why I need a monster bone plus so I can get this heavy shooter to so There's like They work Palamu Baroth. Well Yep, I'm trying to join somebody else's quest that did a SOS.   I'm too low for that. I Won't get in there and die I'm not wait I am No You lying wait a minute chat see It's details like that, you know, I Just I Oops Food now be ready. I'm here Something tells me I'm gonna like doing this going to SOS is he's so fast man. Wow I'm so coordinated. Oh I felt good Poison ammo. Yeah, I'm playing with keyboard mouse. Yep That's correct. Oh, they fight me They fight and chat I'll wipe him why ever why ever I barely miss me.  Oh They can climb up, but I can't that's cool That's cool. That's cool Dude when I land those those big hits, that's where I love the HBG That's oh, man It's so satisfying to land those and it's just like boom boom boom boom boom boom boom boom.  So I just got one shot. I thought I was right-clicking, but I think I don't think I was I Mean by Amy I'll call it right-clicking because that's why I rebound it Man, I hate messing up other people's hunts. It makes me feel really bad When you were that when you were the group man They just do more damage, I think because when I'm solo up that what that doesn't kill me. I'm play safe No No No, why'd he choose me? I didn't even hit him I'm running that Hit the side of the thing by the way chat I'm just staying up here to get stay out of the way because All right, let's go back to poison couple poison on them. Okay. I blocked that one. That was close missed Down the 30 frames Whoo, why why? Somebody's effects is having Magic effect on me. All right shield. All right. This this poison is starting to add up


What you would do is browse the site for the type of request that you might want to make to see if anyone's made it before. So I want to have a quick look at any requests that have been made to this council. It says that they have had no request, so in which case I need to make a request to this authority, so I press this button here and then it brings up this page which is where I can make my request. At the moment you'll see I've not logged in at all, so you can do this all without logging in and this is where I type a title for my request, so I'm going to say request for communications relating to peace treaty. Okay, making a request and you might not know what to write to start with. The ferret have done an amazing template FOI request which is quite easy to find on their site and if you do their online FOI course as well you have access to it. So this is the kind of the breakdown of the request and it's a good template to use if you want to kind of have some help writing your FOI request. I have used this template so I've written my FOI request out already in this Word document, Google document sorry and I'm just going to copy and paste it in. I've done that so you don't have to watch me type basically. So here we go. I have asked for specific information here between two parties. I've asked for documentary evidence specific terms and signatories related to that and I've specified what type of information I'm expecting to see relating to this and now all I need to do is sign my request there and now I can preview it. So this is a preview of the request. It gives me an opportunity to make sure that I've not put any personal information in. I'm not requesting personal information because you can't request personal information. That would be a subject access request and it shouldn't be public. So I'm happy with everything that I've written here and so instead of editing I can send and publish. This is the point where it's going to ask me to sign in. So if I hadn't already gotten the account which I do here obviously I would be able to create a new account here. Simply ask for name, email and a password. I'm going to sign in to my account. There we go. It's another preview of it just to make sure I'm happy with it. So basically you have three or four chances to edit the request to make sure that it's definitely asking for the things you want. It's including all the information and it's going to the right place. And then you finally get to send it. And here we go. I've sent the request and now I just need to wait for a response. Right here it tells me when I should expect the response by so December 29th 2020. And it says the status here is currently waiting for a response from the body I've made the request to. And this bit here says that the sending status is unknown. Once we know that the request has been delivered to the public body this will show a green tick. And that at that point you know that your request has been delivered to the public body and so the clock should start ticking essentially on your FOI request and you should then have 20 days in which the public body should respond. And there you go. That's that.


We all know the classic endings to a Bond film. Bond has just saved the day, defeated the villain, got the girl, followed by a signature O James or a grumpy 007. To me, this is what left you with a huge smile on your face like the cherry on the cake after a classic Bond adventure. Some may find it cringe-worthy, others love it, I fall in the latter. I think it's a real shame we never got to see Daniel Craig's Bond getting the girl on some floatable dinghy like in the old days, having saved the day, getting some winking to the audience. I mean, can you even imagine that stuff nowadays? With Craig, it always needed to be moody, walking off into the dark distance, dropping the necklace of his loved one with a dramatic I never left. Or driving off into the distance with his new love, or being blasted to pieces, it always needed to be different. I really hope they bring back a classic Bond ending in the future, but I think that's too much to ask in these days. In any case, today we are celebrating the way Bond films concluded as we are looking at my top 10 favorite Bond endings, starting with my honorable mention. Though the man with the golden guns ending is very run-of-the-mill with Bond having just defeated the villain, disposed of the henchmen, saved the day with a beautiful girl in his bed, and the joke of M trying to get in touch with Bond and Goodnight. But what makes it the honorable mention, to me, is Lulu's alternate version of the title song Ending the Film. To me, it's such a powerful way to end this film, because it's true, Bond should give you that reassurance that no matter what, he's always around to keep the world safe.  Killing off Bond in the end still holds up as the biggest sin the movie makers ever made, because this is the exact opposite of what I think any self-respecting Bond fan wanted to see. I did a whole two-part in-depth review on No Time to Die, talking about the ending in great lengths, voiced my opinions for both camps, but in short, it still leaves me with such a sour taste, which I think is exactly opposite of what I want to feel like walking out of a Bond film. Of course the real ending of the film is with Madeline and Bond's daughter talking about Bond almost in the shape of a legend, followed by the driving into the tunnel to Matera, ending with Louis Armstrong's classic, We Have All the Time in the World. But there is a lot of stuff that I did like in No Time to Die, but the ending certainly really wasn't for me. Now let's get to the 10 that I do really love. There is definitely a consensus within the Bond community that never says that Bond again is considered one of the worst Bond films. It's not even an official Eon produced Bond movie, and I dislike the title song Playing Over the Sea, but there is just something wholesome about Connery's true swan song, saying goodbye to the real 007, now going into retirement, getting the girl winking to the audience, it's incredibly cheesy, but God do I always have a major smile on my face when I see this dude do that. Even if you argue that by today's standards you could never get away with this kind of stuff anymore, but to me, Sean Connery is simply a winner. There is also something wholesome about the ending of GoldenEye, which I think has everything to do with the great chemistry between Brosnan and Skorupko. I don't think he had another Bond girl quite with this playful chemistry between them. The look of the pair, thinking no one is around, still holds up as a fun ending. It's all just good fun, and again, the playful chemistry between Bond and Natalia just really hits home. The day is saved, jokes are made, the smiling of the pair seems really genuine and it honestly just leaves you doing exactly the same, having just witnessed a fantastic Bond film. My only complaint is the song Playing Over the End credits, where I feel they really could have picked pretty much everything else to conclude the movie with. But other than that, a fun ending to a solid Bond film. Connery's Bond usually is the Bond to end the movies on the water, often in some inflatable rescue raft or on a boat. As a matter of fact, the only time he didn't end on the water was in Goldfinger where he gets pussy under the parachute. But my favorite of his Bond film endings is the one in Thunderbolt, where he once again finds himself in a rescue raft. Having just survived the explosion of the Disco Valente, prevented a nuclear war and now has the smoking hot villain's mistress in his arms, casually preparing the rope with Domino looking at him like what the hell is this man up to now, and he just looks effortlessly cool as if he's done all of this before. This is just routine to him. The plane comes in, the Bond theme blazes out and the pair is being dragged along the military plane. This is classic Bond ending at its finest. And this is badass. We all know the reason the Bond girl in the world is not enough was even cold Christmas to begin with is so that Bond could throw this cheesy line at her. But the reason I really like the ending of this film so much is because of this moment. The MI6 staff is zooming in through satellite images, figuring out the whereabouts of Bond after a nuclear submarine climax. They spot his DB5 through the satellite. They know where he is, then we get this moment. It's a great throwback to the classic Bond endings and I absolutely love that. This was 1999, a time I remember well, and a time where you could still get away with stuff like that. I mean this is the type of cheesiness that is just so much fun. And honestly, had you been in Bond's position, would you not totally crack that Christmas joke out too? You just gotta do it. Some may argue the ending to License to Kill is very quick in tying the story up. Felix Lighter's old theory like yeah the wife got killed, last leg too, but hey did I tell you I'm having french fries tonight? But I do think there is actually a lot going for this ending. It's celebrated at San Jazz's mansion for one. Throughout the movie there is a subtle, but clever subplot of loopy trying to provoke Pam into jealousy. Her female instinct is right on the mark. She knows that Pam likes Bond. The girl is a lot more manipulative towards men than she lets on using her Latina looks to get men to do exactly what she wants. And she succeeds in getting Pam over the edge multiple times. There's also a deleted scene engaging this dynamic further with Pam asking Bond who loopy is, expressing the jealousy that loopy is triggering. She's even checking if Pam is watching when she deliberately starts kissing Bond in front of her and she does trigger the desired reaction. He finally notices what is up, realizes he's been through a lot more with Pam, makes the choice to go for her, jumps into the pool and the pair make out. Granted it's far from correct, some women probably wouldn't have taken this from a man, but somehow it's heartwarming and it's right. And unlike GoldenEye I think Patty Labelle's song over the end credits starting here is so suitable and romantic and also adds as a fitting song that never fails to get me reflecting on this amazing film with a smile. I don't even care about the fish wiki, it never bothered me. This was the ending to the classic era of Bond films and the emotional song playing over the credits really just adds to that sentiment. Besides all the happy go-banging endings Bond usually gets on, it's not just Daniel Craig that gets to have the different endings. Lazenby obviously had a totally different one too and it's a really strong one. The assassination of Tracy still holds up as one of the most emotional, impactful and memorable endings in the entire series. I critique Lazenby often for his performance, but he's good in this particular moment. It helps that we've seen them develop their relationship throughout the film and to see Bond briefly in full happiness, only for it to be taken away by Blofeld and Irma Bunt. It's really heartbreaking. I never liked how they play the Bond theme over the credits though. Louis Armstrong's We Have All The Time In The World that was played for the end credits in No Time To Die totally should have been used in the actual film it was used in for the end credits right here. It just would have been so much more suitable. If only it was followed up by anything even mentioning the death of Bond's wife in Diamond Software Forever, still the biggest missed opportunity for me in the entire series. The ending I absolutely loved as a kid and still do just as much as an adult. Bond has just prevented a nuclear attack on two major cities. He has defeated Stromberg, taken care of Joss and saved the girl. The user Stromberg's escape pod ends up with the smoking hot agent XXX ready to celebrate saving the world once more, drifting into the British Navy ship only for this brilliant moment. Even if you find this cheesy, you have to admit, has this been your fantasy as a man? You have just saved the world, you're with a beautiful woman on your side, you're the winner. Would you really have done this any other way? Would you have said, oh sorry how unprofessional of me, please don't report me for inappropriate behavior? Of course not, you've saved the world, you're the man, time for consensual intercourse. I don't care, standards have changed. This is the fantasy and this is what James Bond is all about. Another example of a Bond ending in this exact same fashion is the ending of Moonraker, which though cheesy I've never been able to not grin from ear to ear with, Bond has prevented the extinction of the human race in outer space, and the British Minister of Defence and Q are all gathered with NASA, ready to talk to Bond who's just saved the world. The footage is about to be directly broadcasted to the White House and Buckingham Palace, when… And that double innuendo by Q, I mean come on, this is one of the best endings in all of Bond, no question. Even though I pretty much did a non-stop praising of Bond, celebrating with the Bond girls after saving the world, I have to admit there are different endings I really, really enjoy, and I don't think that any Bond fan can deny that Casino Royale has one of the best ones ever conceived. Bond has just lost the love of his life, committing a tragic suicide after being blackmailed into working for a mysterious criminal organization. Bond has just managed to track down one of the members of this group when… Apart from Majesties, this really was one of the first endings in my life to teach me you can end a Bond film without Bond sleeping with the Bond girl. I was fully prepared to see this movie end like the novel with… But instead, they made it come full circle with their ending. They showed us part of Bond's origin in an original way. He has now become the secret agent we know and love. This to me was a brilliant move and convinced me Craig put Bond in good hands after honestly having serious doubts beforehand. There was only one ending that triggered an even greater reaction for me. When we first watched Casino Royale, we had no queue and no money penny. There was just this random dude who I assumed was just supposed to be the new meal secretary working for M. Apart from Tanner, none of these MI6 regulars showed up in Quantum of Solace either. It had been a long time since Bond films felt like classic Bond films. Then Judy Dentist M died in Skyfall. It felt like the end of an era. I was absolutely expecting Ray Fiennes' Mallory beforehand to take over the mantle. However, I still don't know to this very day how I managed to dodge the news of Naomi Harris' agent Eve being the new money penny until seeing this film in cinema and I'm still so happy I got the surprise first hand. This whole scene to me was where the familiar feeling of a Bond film returned. You know, we've never formally been introduced. Oh. Well, my name's Eve. Eve Money Penny. I look forward to our time together, Miss Money Penny. I'm sure we'll have one or two close shaves. Morning, Double-O-Seven. Good morning, Tanner. You'll see you now. So, Double-O-Seven, lots to be done. Are you ready to get back to work? Pleasure, and the pleasure. This whole scene left me in a standing ovation in cinema, something I've never done in theater before and never done since. What a fantastic conclusion to an amazing Bond film. We finally have a cue again. Now it was revealed to be Money Penny all along. We not only have to fantasize Bond and Money Penny have history, we've seen it during the film. Now for the first time since the 80s, there was a male M again. Bond standing in that classic MI6 office with the classic letter door, another feature not seen since the old days, Bond receiving his next mission, obeying the new M, calling him Sir, all of this blending into the gun barrel and the 50-year anniversary logo. To me, this was the last time Bond was celebrated at an old time high. Skyfall's ending was pure class. It's been a while since I did a Ranking 007 episode, so if you enjoyed this one, please like and subscribe and check out the Ranking 007 playlist for the other Ranking episodes I made so far. Or check out my channel for a lot more Bond content. And if you want to go the extra mile in supporting the channel, please take a look at my Patreon page too. See you guys in the next video.


In this video, we're going to take a look at the solution to question six from the practice midterm exam for calculus to math 1220 So we're asked how large do we have to choose n so that the approximation using the trapezoidal rule tn to the integral We want to integrate from 0 to 1 cosine of x squared dx is accurate to within 0.001 Right there and so it also gives you a little bit of disclaimer here Notice if you take the the second derivative of cosine of x squared This will become negative two sine of x squared minus four x squared cosine of x squared And so on the interval on the interval zero to one one could ascertain that six is actually An upper bound for all of these numbers here and sort of the idea is sign You know sine of x squared it's gonna sit between one and negative one and so will cosine right cosine will also sit between these same bounds Right and so if you play around with that idea sine sits between one and negative one cosines as well The absolute value cannot exceed one So what could happen is this sign could go off towards one This sign could go off towards one and then for x squared the best it's gonna sit between zero and one as well So it could go off towards one so if you sort of think of worst-case scenarios everything becomes one one one one one one You're getting something like negative two minus four, which is a negative six if we take the absolute value of this You're gonna do with six it's not gives you some explanation where this six comes from is it the best bound not necessarily It's not but it's one what one could determine without a calculator whatsoever So how are we actually going to answer this one? So we have this idea about error and we need to work with that error here We're gonna use the error bound so we want to use the error bound associated to the trapezoid rule Which is a reminder the trapezoid rule error will be less than or equal to k times b minus a cubed over 12 in squared so that's the bound associated to the trapezoid rule and Some of these things we know the k value is given to you and you're gonna see that on the test as well I'm just gonna give you the k value Because what I want to see on this question is whether you can use correctly the error bound or not and Finding k turns out to be sort of an optimization problem something we saw in calculus one and although we probably should be able to do Optimization problems wink wink I'm gonna assume you can do that and so k will be given to you as a just as a time saver on this exam So then we plug in one minus zero cubed This it's above 12 times n squared and then again simplifying this we end up with 6 over 12 and squared Or more specifically 1 over 2 n squared like so now we want our error to be less than less than 0.0001 or in other words 1 out of 10,000 All right, so that's what I'm gonna do here And so we want to solve this inequality taking reciprocals We see that we get 2 n squared is greater than or equal to 10,000 and Divide both sides by two. We're gonna get the n squared is greater than or equal to 5,000 and so now we have to take the square root n is greater than or equal to 5,000 This is a step that you're gonna want to use your calculator And again, this shouldn't be a problem for most of us if we take the square root of 5,000 any scientific calculator can handle this You would get approximately 70.7 Now this n has to be an integer for which case we would round this up to the next highest integer and we would get 71 as our answer here now admittedly if you for some reason don't have a calculator I guess I mean honestly you should just use the the digital calculator will be provided to you during the exam So this is really sort of a mute issue But if you have your own physical calculator, you might want to use that instead Now a variation of this question of course is you could have a different function, of course given with different bounds But that mostly changes the fact how would one calculate k, which will be given to you The bounds of course are significant as they're part of the error bound right here You might you do need to know the error bound for the midpoint rule You also need to know the error bound for Simpsons rule because that does a that's a variation of this type of question here Another thing to be cautious about Simpsons rule is that if you're looking once you found your number Let's say you found 70.7 as it has to be good equal to that with Simpsons rule You do have to have an even number In which case you would then round that up to 72 and not 71 That is that that's a thing you want to watch out for on this question