Hello, Osleigh to sync here. I think when the mod wave was first released, a lot of attention was given to the wave part of the name. The last few years have kind of seen this resurgence in interesting wavetable synthesis and rightly so, I think it's a great way to explore sounds. But I think really where the real power of this synth lies actually is in the mod part of the name because the way that the modulators are laid out, the types of modulation sources that we have, the way that we can mangle them, the way that we can have them interact, and the very, very extensive mod matrix on this synth. Actually, the fact that it's a wavetable synth is always not that important. The way that you can interact with that synthesis engine with this extensive set of modulation options is really where the excitement lies for me personally anyway. To that end over the course of the next few videos, I'd like to explore each of the different modulation sources on the mod wave and talk about why they're interesting and what sort of creative things you can do with them. Because the level of flexibility in some of these modulation sources really are only otherwise available in the in the world of modular and even then selling on every module that you're going to come across. So there aren't many synths that I feel like I could spend an entire video talking about its envelopes, but the mod wave is one of them and that is what we're going to do in this video today. So the first thing we should probably note before we talk about any particular features of the envelopes is that we have four of them on the mod wave and they are labeled filter amp oscon and os2. What is important to note about each of these envelopes is that you can send them anywhere. The only one that has to go to the place that's listed here is the amp. The amp will always go to the amp, so the loudness of each note that's being played, but filter os1 and os2, they're kind of suggested that that's where they're going and there are sort of pre-rooted knobs for you to turn to send them there, but you don't have to send them there and you can send them to any other modulatable parameter on the synth. So the next thing that we should probably look at here is what type of envelope we are dealing with and it is for all intents and purposes our good old familiar ADSR attack, decay, sustain and release envelope. And if you'd like to learn a little bit more about how envelopes work then I'll put a link in the description to a video that I did about envelope recipes, but this is probably the envelope type that we're most familiar with seeing on most synths. One thing that is definitely worth noting about the envelopes that aren't the envelope is that our sustain can be negative and that's a really really interesting thing. Let's take a look at how that might be used. So I've just got an initialised patch here with just a little bit of release on my amp envelope here and if I will just stick with the filter envelope I think as a starting point we'll just darken the filter a little bit like that and our filter envelope is currently set up, we'll have it in a sort of standard way that you would maybe expect to see kind of a slightly plucky patch here and if we give that some some envelope amount there on the filter this is kind of the standard filter set up isn't it really. A little bit of resonance we've got that pluck it's sitting at a particular point and then we have that sort of dying away aspect at the end there it's sort of typical of a standard sort of filter set up. The nice thing with the envelope on the filter here is that we can set our sustain to be lower than its starting point. Let me open up the filter overall a little bit so it goes down to a really low place and then swoops up as you release which is kind of a different feel than you're able to easily get on a lot of synths which I quite like and it's just one of those things that through the nature of this envelope set up we can achieve so rather than dying down we die which is very nice. So each of our envelopes has three pages the first is where we can set the envelope shape with the ADS up and I'll just set this to be like a little kind of sound maybe like a second decay and same on the release something like that. On page two we have this ability to alter the curve so what does this actually do and what does it do to our patch sonically. So we're going to start with the amp envelope here because it's probably where we can most easily hear what's going on here so the curve allows us to change each of our the movement parts of our envelope our attack our decay and our release between being linear and exponential or logarithmic as it is on the attack portion. So let's just listen to what that does before we talk about why and what it might be used for so I'm just going to my decay portion of my envelope here I'm going to set it back to linear I want you to listen to the way that the tail end of the note kind of goes. Can you hear the very end of that note there it's kind of unnatural right it kind of just feels like it dies very quickly at the end there despite the fact that it's got a smooth transition to zero and that's because in the real world the volume of things don't tend to die off like that in a linear fashion. Now if we contrast that with say setting it halfway so this is halfway between linear and exponential and listen again to the very end of the note sounds a lot more natural right the very end of the note seems to die off a lot more sort of sensibly you might think that also if you compare that with linear if we listen to the starter note now when we go halfway here that very first part of the note is now sort of pluckier and a bit more tacky so what's happening here is as we move towards exponential at the very end here with exponential the start of the decay will be very steep it will go through a sort of more or less linear portion in the middle and then the end of it where it's already very very low is going to be very very smooth linear on the other hand obviously just goes down in a linear fashion so here the start is a bit more abrupt and the end is a bit more smooth and if we go right the way to the end the start is very abrupt very plucky and the end is it's very smooth although we can't necessarily hear all of it because we're so low by the time that we get there again in the middle and then quite unnatural there when you hear it with the linear so generally for the amp envelope we probably do want it somewhere in the middle somewhere from sort of five upwards and we can still even with our 10 where it feels like it's dying off a lot faster we can kind of compensate by lengthening the decay and we still get that much more plucky start out of it and so sort of balancing the decay time and the curve can get you slightly different characters of both the onset and end of our sound it's a really really nice way to find chain what's going on in your patch so on the other hand we have our attack here so I'll just send this fairly long like not 20 seconds but maybe like 10 seconds somewhere around there yeah and then we'll come into the second page here where we have our curve here we'll start what we're doing start with a linear attack and listen to the very start of the note on this can you see how it almost felt like there was a lag to the onset of the sound like actually when I play the note it almost doesn't start straight away if we turn this up maybe to halfway again what this is going to do is the kind of the opposite to the exponential and that is that the very start of the note is going to go and the very start of the curve is going to increase quite quickly go to a kind of linear bit and then sort of slow down at the end and now so this was the linear onset if we go to a logarithmic or sorry a halfway can you hear there that the start of the note happens quite quickly but then we still get all of that movement throughout the rest of the range so for very very long envelopes being able to switch to something approaching a logarithmic envelope especially for attack that can make sure that your patch still kind of starts when you play a note and right the way at the other end at at at 10 comes on real quick and then a lot of movement happens in the middle and then when it slows down towards the top of the attack you don't hear so much movement because it's moving so slowly by that point but this means that you can have obscenely long like 90 second envelopes and still start as soon as you hit the key but then still have all of that movement happen up there so generally with our amp envelopes we're going to want to have somewhere sort of you can probably a bit lower than five but somewhere between sort of like three and ten would generally be a good place for our amp envelopes and as you go longer and longer you might want to consider going more logarithmic to accommodate that so if my advice is for amp envelopes to stick towards the exponential logarithmic side of things what's the point of having the range really so it's nice for fine tuning on the other modulation sources we can be a bit more creative with our curves and not necessarily stick so hard and fast to this rule so i'm just going to set this back to be like a gate with a bit of a release something like that maybe let's move over to the filter for a second and i'll just shut the filter down a bit and give us some mount there so here i want to set this like a plucky thing like that something like that so at the moment the way this is set what are we doing the way this is set is that we're sort of halfway again if i set this back to linear for our decay that's still a very useful sound compared to the halfway and compared to full exponential very plucky now that first bit feels a lot more abrupt and sharp so for maybe trying to do like a really sort of acidic plucky attack to a to a bass line or something we might want to go towards the exponential but i think like the the linear portion here sounds almost more like someone smoothly moving a novel right it's a bit more synthetic but synthetic is good on a synth you know that's what that's kind of what we want sometimes so here it's a much more creative choice much more of a static choice rather than something that's sort of like you probably want exponential most of the time that's not the case on modulating other parameters other than amplitude again also on the attack if we give this a bit of a longer attack so here's a linear attack and frankly that's perfectly nice again it kind of sounds to me like someone turning the filter on which is which is nice if we go to the extreme within logarithmic we kind of get that swoop in a lot sooner which again is really nice and it's a purely aesthetic choice the other side effect of having this set to logarithmic instead you might be able to hear is that it feels like we've got an attack portion and then it kind of chills out more or less at the top because that last part of our curve is so gentle that it's almost a way of delaying our decay portion so it feels like it's finished but the decay only just kicked in there right so we can also use this to almost give us a pseudo hold section on our attack envelope as well which is cool and again it's going to vary from parameter to parameter and patch to patch well what you don't want to do here but like I say unlike the amp envelope this curve fine tuning gives us a lot more of a creative license to do stuff that's interesting I think on other parameters okay so having those really flexible curves on the envelope a really nice way of fine tuning things but the final page of our envelope actually gives us something that's really quite exciting so let me just set this back to be like a plucky friend again and I'll do something like that great okay so page two we have our curve page three we have this trigger page and this is the same on all of them the difference with amp compared to the other pages is that on the other page we have this trigger at note on setting and I'll talk about that in a second but they all basically function in the same way otherwise and what we're able to do on this page is say that I want this envelope to not necessarily only trigger because I press the note and so I want this to trigger because of something else and that list of other things is basically any modulation source in the entire synth so anything that we can use to modulate another thing we can also use to trigger this envelope and this is really really exciting so as a basic example here if I set the trigger source to be our um let's use our our amp LFO in fact there we go um our amp LFO is sat here it's going at um two hertz and if I now play a note you can hear that my amp envelope is being triggered over and over again to be clear this isn't the amp LFO that's actually doing this if I was to change the shape of my envelope it's absolutely the envelope that's being triggered here the LFO is just being used as a source to trigger it that's really cool so I'm just going to get rid of that for a second move that back over to off there we go so back to our normal implementation and I just want to show you a few different applications of this envelope trigger that are I think particularly interesting and creative and hopefully that will spark some other ideas for you to look at as well so the first example I want to give you is related to the sequencer so I'm just going to set this um amp envelope just to be a little plucky friend again so something like I'll do yeah okay so um let me just give it some release as well there's a little plucky sound so um we'll talk about the sequencer in depth in another video because there's a lot to talk about about the sequencer but the thing that you need to know about the sequencer is that it's not a normal sequencer it's not kind of a sequence that you play things into and then they sort of play back I'll show you what what I mean um so if I come across to the pitch lane here and I'll just make our sequence a little bit shorter maybe just like six notes or something and um let's just sequence some notes in here or rather sequence some offsets because you sequence the transposition of the note that's playing rather than um the actual note that's playing which allows you to repitch the sequence on different notes by playing different notes on the keyboard as I say there's a whole other video about this please don't see this as a limitation what I'm about to show you because the flexibility that this gives you to do other stuff outside of normal sequencing I'd much rather have this on this synth in particular than a conventional sequencer anyway so step one we'll just have to play the root note then we'll have it do a fifth and then uh like a minor third and then an octave above and then like another minor third and then a fifth again or something like that um doesn't really matter um for this illustrative purpose so if I play a note well we can kind of hear that the sequence is playing in the background there but it's just kind of been faded out by our envelope and if we turn up the sustain on our amp envelope you can kind of see what's going on here the sequence is playing and it's changing the pitch of our notes but it's not it's not doing a key thing that we would expect a sequencer to do which is to retrigger our envelope right um so can we have our sequence retrigger our envelope the answer is yes certainly we can so if we come back into that third page here with our trigger here and we're going to change our trigger source to be on generators down here somewhere we have the step pulse here so we've told our envelope to um retrigger on each step pulse of the sequencer and now if I hold down a note we get what we would expect to have with our sequencer now the interesting thing is this is going to be working per note that I play that's something to discuss on the sequencer video as I say there's a lot to talk about with this sequence there's a lot of things that we can do but one thing I do want to show you um before we um move away from the sequencer is that at the moment we've coupled our notes um our envelope being played to the notes uh on every step on the pitch and one of the brilliant things that you can do with the mod way which I love to do in the world of modular and I can't only think of another um sort of normal hardware synth way you can do this is decouple our gate and our um our pitch so um we've got these abcd lanes of sequencing here as well so I'm going to come in here and I'm going to set the trigger instead to be step sequence a we see we've got our threshold here of plus 50 and that means that um any um step on this um sequencer lane which is above 50 it's going to trigger um I'm just going to make this short a bit a different length to the other one so I'm going to go with seven and I'm just going to send set the transition to be um I'll set it to be uh off actually uh individual work as well but we'll just set it off this just means that we're not getting slew between the which can mess with the envelope triggering but now if I play a note go back to our original situation because we're not getting uh any more trigger information because none of the uh steps here have anything other than zero on them so if I set this one to anything above 50 it doesn't matter you can now hear that every seven steps we're getting a uh trigger happening there we choose another one like the sixth one and now we can decouple these two uh different parts of our uh pitch sequence we could turn our sustain out a bit so we still hear the bit in the background and we get these cool rhythms happening and if we hold that uh we could turn our filter down and have this set as a nice little uh plucky thing as well it's not going to trigger at the moment because we're not playing any notes but we could set uh this envelope our filter envelope to trigger uh using the next sequence perhaps um step sequence b uh we can come down to step sequence b again set it maybe to another length as well maybe like four this time and maybe just on that first step we set it to be so now we have our filter envelope being fired based on a pattern defined by one of our sequences and then uh amp by a different one and by changing the length of the sequence so maybe we can have as five instead we can get some really nice things going on that are quite complex sequencing ideas and I say there's a lot more to talk about with the sequencer which we will get to don't worry but you can see how having this uh trigger is something that isn't the key can really get us to some very interesting places so this next idea is kind of related but um a little bit different so we're going to start again by setting our amp envelope to being a plucky friend just because it's easiest to hear what's going on that way okay and I'm going to come across to my amp helipho and I'm going to tempo sync it maybe run a little bit faster um or a lot faster and I'm going to set its shape to be a random one which is a uh sample and halt kind of thing going on there so I'm not using this to modulate anything I'm just going to use this as a trigger source so coming back to my amp envelope here and we'll come it into page three and I'm going to set my trigger source for this envelope to be that LFO so generators uh amp LFO there we go and if I now hold down we can hear that that is being triggered in time but not on every step and if we think about the shape of our um amp LFO here it's whizzing up and down and every time it crosses that 50 threshold it's going to trigger the note and it's going to be doing that about um 25 percent of the time because um it's also going negative it's about 25 percent of the time we're going to be getting a trigger happening now we can come back to our amp envelope here and we can change how often that's going to be triggering by lowering our threshold so we should get more now which we certainly are and if we put the threshold up like as high as like 95 it's going to be barely happening very very rarely uh based upon that tempo that we've got defined there so it's been a while yep okay there's another one there now uh unfortunately um this is not modulatable but we can go around that but this idea here of creating um timed but um semi-random triggers is like a cornerstone for generative patches if we want to create something that's chance-based but controllable this is a great way of doing it a great way of approaching it now ideally i'd kind of want this to be performable uh but we can't um assign this to a modulation source like the knobs because it um it just can't so we can't do it but it's kind of another control which um we can kind of do the same thing that is modulatable so if we come across to our LFO here we have this offset parameter here and what this is going to do is shift our waveform up or down and we'll talk about this in a bit more uh depth um when we come to the LFO video but uh it allows us to to basically do the same sort of thing but it's modulatable which means we could assign it to like a knob that we could have as a macro here so if we wanted to have this happen more regularly we could turn our offset up a bit and if we wanted it to happen less often we could turn the offset down right so if we wanted to we could add a modulation target for that control amp LFO offset to this knob here we could set it uh 50 yesh in either direction like that and now we have a nice sort of control over how often that's going to be happening to basically nothing there to more there but we probably also want to change the threshold around a little bit to make it fine tune but yeah we can make that perform well there so if we want um this sort of generative thing we'd probably want to slow it down a little bit but you know semi-random but still in time and controllable triggers which is cool so this final idea I want to show you is a little bit different to the other two we've been looking at because this is a way of us being able to get more complex envelope shapes than we can usually get with just a single ADSR by combining two different envelopes via that trigger source so let me just turn down one of these oscillators okay and to make this really really obvious I'm going to work with the pitch of the oscillators just because it's easiest to hear you probably know what you may want to do with pitch but you might want to experiment with some more idea with other parameters that we can modulate let's start with this first oscillator here and we'll set it um sorry this first envelope here this oscillator one envelope we'll set it to have like a an onset like that and then it could just have a hold and release something like that that's fine um and we'll assign it to the tuning of this first oscillator here so mod we want it to be assigned to the tuning and I want to use this envelope here enter and we'll make it really obvious we'll jump up by like a fifth so if I play a note now we get that raise up to a fifth right so that gives us our sort of basic pitch shape to this um note well if we wanted to do something a little bit more interesting with like a little flourish here so hopefully we want to go uh like someone sort of bent a whammy bar on a guitar right at the end of that ascension that we had there with our um pitch now we can't do that with a normal a dsr but we can do it by combining two different envelopes so let's think about that kind of shape what that is is um this sort of triangle shape that we already have set up here on oscillator uh two envelope um so let's make it a little bit shorter we have a little wow wow kind of sound uh shape going on there if we apply that negatively to our pitch we're going to get um kind of happening now um let's assign that to pitch first of all so we're going to say yep we want it to be the tuning of this envelope we want to use this here press enter and we'll just maybe go down by three semi tones i'll do so if i now play a note there's it's not doing what we want it to do that's kind of a weird delay to start to the um uh uh sweep up there so what we really want to happen is we want oscillator one envelope to get all the way to the top and then when we get to the top we want it to go okay so what we can do is come into page three of this um envelope here the oscillator two envelope and we can send its trigger source to be the other envelope we're using so oscillator one envelope and we're going to say that when it gets to not quite 100% because for some reason it doesn't work if it's 100% but if we say 99 it's going to um trigger this envelope and we don't want this to happen when we play the night we only want it to happen when it's triggered so we can turn trigger add to note on off so now if i play this note we should have this rise up as defined by this envelope it gets to the top and it's going to do the little wiggle fingers crossed hear that so we've actually added two stages to our envelope here i'm going to make it a bit shorter um whether or not that in particular is a useful sound your mileage may vary um but hopefully that demonstrates this idea that um we can combine different envelopes by having them trigger at different times and we could we could chain four of them together having each one triggered by the the next envelope so this one triggers this one which then triggers this one which could then trigger that one again uh we can create quite complex envelope shapes by um having them trigger when they hit their peak each time i think probably just having two changes is is enough most of the time um nevertheless it is interesting that you could take this to it's a logical conclusion and do something bonkers make a bit more obvious might be diving down further and of course these two envelopes don't have to be affecting the same thing either we could say we get to the top of our pitch swoop and then you do a little filter dip or something rather than a tuning dip or whatever it happens to be um we could uh configure this to a wealth of different things um but i just wanted to highlight that you can trigger one envelope to fire at the end or at the apex of another envelope so anyway um that's it for today uh with the envelopes i hope that demonstrated how flexible the envelopes can be uh in terms of how they can be combined how they can be triggered and how they can be fine tuned using that really really useful curve control um as i say it's a level of control that i haven't really seen uh for uh envelopes on on many other sort of hardware since uh and it's kind of a testament to the ideas that the mod wave envelopes kind of came up with if you did enjoy the video um as always if uh you wanted to leave a thumbs up uh that would always be massively appreciated and make sure you subscribe to the channel so you don't miss out on any upcoming synth fun especially if you're interested in the mod wave quite a lot of mod wave stuff planned to turn up in the next few weeks and months other than that um as always thank you so much for joining me and until next time take care bye


Hello everyone. I'm Bani.  I feel very strongly about this. I think this is a skill that a lot of us need to continuously learn and improve on and that's why I'm here to share some of my knowledge and some of my learnings and again this is just directional because a lot of the recommendations I make here may differ made from case to case but I'm here and happy to share whatever I know. So let's get started. Before I dive deep into the topic, I just want to introduce myself. I'll give you a little bit of background. I was the ex-product manager lead for user engagement for Flipkart. For those who are not familiar with it, it's a $13 billion unicorn based out of India.  I'm glad I was a part of that trajectory and growth. I was their first mobile product manager and by the end of it I was leading their user engagement charter as a product manager. As of today, I've worked in Silicon Valley for the last couple of years. I've worked with a lot of startups. Recently I've worked with Shift which is an on-demand grocery service. I managed their entire member experiences across mobile apps and web. I specifically look at creating the life for experiences for users, retention and engagement. I've had the opportunity to work and this is mostly when I was back in India across a variety of startups. So I've worked with B2B, B2C. I've seen most of them fail so I have that sort of learnings as well. I worked across mobile apps, desktop, again developing and developed markets and I'm an architect by education so I'm very, very picky about my pixels. So I have a lot of focus on product design and business. I actually started my career as an investment banker so I moved from investment banking to venture capital so I know that side of the ecosystem on what it takes to be sort of on page with investors, what do investors look for from founders and product specialists so on and so forth. I have four topics that I want to cover during the course of this presentation very briefly at that. The first is why do I believe and this is my core belief why is prioritization a must-have skill in today's product management world. Is there and if yes, is there a right or a better time to prioritize during your or in your scheme of every day to do things to do as a product manager. How do you prioritize? So what are some of the popular frameworks that I have used or I continue to use and this is sort of the crux of this presentation I would say which is there are so many challenges along the way and anybody who's worked as a product manager would know that it's not always that rosy picture to say oh I know what I want to build and it'll go out there and we'll be able to build it. There are tons of roadblocks and I'm going to try and talk about some of those and how we can circumvent some of those. So why is this important and why did I choose this topic? Even today I'm very very sort of I love the product management community here and back in India and this is a common theme. I don't know how to pick the right thing. I don't know whether I should invest in an existing feature or build a new feature. I don't know whether I should pivot. What does pivoting mean? Why is this more important than that? How do I funnel in this stakeholder's request versus that? How do I prioritize? This is something that's a common theme even if you go online and check Google you will see that there is so much content around this because there is demand for this content. So every day this seems to be the biggest challenge that we face as a community and this is again a cross. This is an epidemic problem. This is a cross PTOB PMs enterprise consumer PMs, platform PMs, startup PMs, unicorn PMs, established company PMs and so I just wanted to reiterate that this is something we must do in order to make the right kind of impact at work and what I mean by right impact is you can either prioritize to make like Ken Norton says 10% impact or you can make a 10x impact and if you're good at prioritization you can do the latter or at least strive to do it. This is something that I chanced upon when I was reading something and I think it articulates this core belief very well which is that a lot of us as product managers we focus on making our product successful. So we are constantly trying to look at dashboards and trying to measure product KPIs which are how do we make the product successful but there is also something and a concept called product manager KPIs which is how well are you enabling yourself to make the product successful. So what I mean by that is of course the first thing is as a product manager my KPI would be how are my product KPIs doing which is the product that I handle which is the product that I manage but there are also more layers of abstraction which mean how well am I building the vision and does it take into account all my stakeholders am I keeping all my stakeholders aligned on that vision does everybody speak the same language and you would go back to that famous diagram where product is supposed to be at the intersection of design and business and tech right but are we keeping all those stakeholders aligned on that common vision and then this is sort of the summary of what this presentation is all about can you keep all those stakeholders working on the most important thing which is prioritization right. So again prioritization is not only key to the success of your product but also for you as a product manager to go to the next level if you can't do this well enough you will never make that 10x impact this is again something that I have come to learn by success or by failure I have seen that and you'll sort of agree with me on this product managers are continuously context switching high level backlog high level backlog backlog high level we are doing this every day every hour where we have to keep our eye on the 30 thousand feet view and the nitty gritties of the project you are executing right and in this context switching and going back and forth we need to realize where we should prioritize first so once you have a roadmap don't forget to go back to it to the high level themes to keep and to continuously keep prioritizing the themes because your themes inside a roadmap will map to sort of what I call the big herodicious goals of the company so keep going back to this to not lose sight of your roadmap or your vision in pursuit of the backlog and so what I'm the point I'm trying to make is when you funnel in everything and you make a roadmap and from the roadmap you make a backlog don't forget to go back and keep prioritizing this to churn out the right list there how to prioritize again this is a more sort of educational thing I don't want to dive too deep into it there are more than 20 frameworks available for prioritization I use a couple I have used a couple and there's a lot of content on this but the ones that I particularly find useful or I have used in the past this is a new one so this is called opportunity scoring and the premise of this framework is that every user on your service uses you to accomplish a job are you under serving them over serving them or just about serving them right for that job you can read up more about this or I'm happy to answer questions post but this is called opportunity scoring where you basically try to find areas in your roadmap or your backlog where you are either under serving somebody and hence it should be prioritized or you're over serving them and hence it should be deprived this is called opportunity scoring there's another one which has been around for a long while and we may not use it in this particular form but I think we all use it in some way which is the KNO model which basically says what do I absolutely need in the product for it to work just the must have what will I build and nobody will want it which is the indifferent what seems really attractive which is the blue line there which means that every time you increase functionality in that feature or for that theme you will get higher and higher satisfaction from the users and then there's performance which means that's something you need to keep maintaining and keep supporting for example your app shouldn't crash as you scale users and so these are two sort of frameworks that help you prioritize those tricky ones there are also a couple of other ones which you will which you're probably using already or you would have seen other pms using now one that I particularly like to use is the in is Ian's framework basically what it says is take your company goals have product themes around those for example retention is a theme virality is a theme or user engagement is a theme have those themes have your project ideas under those themes rate those project ideas by impact by cost whatever your prioritization criteria is for example burn for example resources whatever your prioritization criteria may be and then order them in there that is Ian's framework it's very popularly used by a feature is again one where you have you're trying to decide between two different approaches to solving the same problem or two different concepts and you get your users to say where will they put their money so you have limited currency and you say between these five things where will you put your currency and you basically so I have used this for example in user testing where I say I'm to improve search via this or via this these are my three ideas in search where are my users putting their currency to say this is what I want Moscow is again must have should have could have won't do again these are different names to sort of the same premise which is what is going to move the needle up the most and what is what what are those features which nobody is really going to care about or which is not going to move the needle again my idea of introducing this was not to say that you should pick one and commit to it it is to say that there is a variety of tools out there I keep moving between some of these for example when I'm really trying to scratch my brain to get out innovative things or trying to think of things that push myself to think of things where I'm we are probably not serving our users and we can do better I go for this I'm trying to do this example now at ship but the point again the point is that different tools that different frameworks do what's best for you and don't be afraid to oscillate between them and try each of them depending on your project or your team or the stage of your startup I wanted to do this and tell me if it's a good idea I wanted to actually do like a sample exercise with the group here just to get the energy up and this is a live example and it put me in a really tough spot when I was in Flipkart so I'm happy to share my journey of of how I managed it so there was this big feature that we wanted to release this is way back in 2015 or 14 I forget but we wanted to do this really new idea which had failed miserably in the West and so obviously India is leaps behind in terms of e-commerce trajectory from the West and so we always look to the West for best practices or ideas or inspiration and all the competitive research that and I this was like my baby and I come up with this idea and I was excited about it but obviously to my dismay I saw that it had failed miserably miserably in the West and it's a different story that we did end up building it but the exercise I want to do with you is there were two use cases of this feature we could go down this route or we could go down the other route and it was not either or it was which one should we do first which one should be prioritized first and so the feature was basically a messaging interface inside an e-commerce app and the use cases were should we allow Flipkart users to talk to each other via the messaging interface so you and I are Flipkart users can we talk to each other or should we allow Flipkart user to talk to the seller whose product I'm buying both are very very different use cases both seem very powerful and there was sort of a gridlock for the upper management to say no we want to do user to seller and me and my team had a different viewpoint which was to say we wanted a user to user and so I went we went back to we took a step back and said how would we prioritize this if this if we were to be unbiased parties to this exercise the first thing I did was what matters to the company right now we were at that phase where Flipkart had gone from when I joined them we had less than 5 million users across the mobile apps by the time we were launching this feature we I think we were already a 20 million so which was pretty much most of the Indian population that transacted online so what mattered to the company was not really acquisition it was engagement we wanted our 25 million users to come back to the app so that was the one company goal that we wanted the feature to align to the second was put the like I mentioned before what is your prioritization criteria what is your hurdle what is what is that thing that is going to block you for us was it dollars no for us was it engineering bandwidth no for us it was what kind of data are we going to get if two Flipkart users talk to each other about deals about categories that are trending about the kind of questions that I would ask my husband before I purchase something that is very very rich data for Flipkart to mine so I I use some of those criteria to score those two versus each other right another thing will this give us strategic advantage will this make us first to market will this be a differentiator for us because we were at that stage where we needed to do things strategically so it was again the company goal was not acquisition as much as it was engagement we had different criteria and so basically I scored all of those those two against each other on that the other thing was scaling with this I haven't mentioned it here but the other thing that stood out to me on the user to users versus user to seller was that user to seller will be restricted to the scale of the orders that we have on Flipkart which is I place an order once a month but and that will that will be my conversation with the seller so I bought this TV from you I want to know what is the warranty and it'll be very product specific but a user to user will scale those conversations will be about shirts about diapers about categories about sales about hey what is our budget for I don't know Amy's gift for her birthday or a lot of other things so we looked at we did the math to say this is going to be x number of sort of we are going to impact x number of conversations or x number of users this is going to be 10 times of that so like I said a lot of different prioritization criteria the second thing we worked with teams to make sure that they we had their buy-in in these criteria for example I ticked off that sort of tech cost was not a concern it was because I had the implicit buy-in of the engineering manager of the engineering director so the other thing that I wanted to highlight is again aligned to the company goal have your prioritization criteria put in place score your different ideas according to the criteria and do that not in silos but working with teams again this is what I said we did it because we wanted engagement not acquisition we wanted a strategic advantage and we were we wanted to bet on in-house capabilities so we wanted to build that infrastructure we wanted to push our teams to build to build that so our criteria was different and we went ahead with user-to-user because we were able to do this prioritization okay this is like I said sort of the this is the not the rosy picture of prioritization this is actually putting it into action there are tons of challenges I've highlighted three which might be most common which most of us would have faced first is resources I've faced this in a 20 member team at flipkart and a 200 member team at flipkart resources will always be limited especially because of our roles we are product managers but who are we kidding we manage our products it's not like we have teams dedicated to us we will never have designers 20 designers or 200 engineers completely allocated to our product so we will always have a shared pool of resources and they're always limited multiple stakeholders again that's just a virtue of our job we work with business with marketing with analytics operations with other fellow product managers who are managing other parts of the product say inside the mobile app leadership team and who not like every of course and customers so you have so many so many stakeholders and they all have their voice and you need to hear all of them and the last is and again this is something I have faced and I'm sure we have faced absence of data it's something that you really believe in you know it's going to be a hit it's something that's never been done before right but there's no data how do you sell it you know maybe because you can't user test maybe because you don't have the budget for it you don't have the tool for it you don't have time for it you don't have data for it so on and so forth and so then with these challenges and there are a couple of more how do you really execute on your priority list how do you actually get the most important thing to be worked on and according to my definition prioritization or the process of prioritization doesn't end up making that list it is constantly making some all your stakeholders working on that most important item which is the number one on your priority priority list right that to me is prioritization so how can you sort of overcome these there is hope like the slide title says limited resources cut back on your requirements and what I mean by that is make the perfect MVP which is your minimum viable product so even for example features that I've built a flipkart or features that I'm building at ship or other startups try to scale back on what is going to make the most impact out of your product idea right there will be 20 percent of it which will make 80 percent of the impact try to carve out that 20 percent and leave the riffraff that is called versioning or making the MVP or phasing it out whatever you may like to call it this is a very very good pushback to limited resources where you cut down on what's going to make the most impact multiple stakeholders what I have seen working in this particular challenge is align people it is very much a part of your job to keep everybody on your team aligned internally and up and across so up the ladder and across the verticals right it is very much our job and again whenever you feel the need fall back to your company goals another sort of recommendation here is try to get the bank for the buck so when you're trying to funnel so many different requests from so many different stakeholders like your customers and your leadership team or whoever try to cut out the noise and as you gain experience as a product manager you will very easily be able to cut off the not cut out the noise and try to keep the meaty stuff absence of data what you can do about that stick your neck out um I've done this once a couple of times but the ones that I really stuck my neck out it worked out well and it didn't work out well I'll tell you why it worked out well because I gained tons of experience trying to do this whole process over and over and over again with different stakeholders and different parts of the value chain and it taught me that if you really believe in something you need to stick your neck out own it so there will be many many times where you won't have data or you won't have time to test you won't have tools for it whatever the case may be own it list out the risks sort of have a plan for the risks say I know this is a known risk this is the associated risk and this is how I'm going to tackle it if it happens but sometimes listen to your gut and let your gut guide you um again always focus on making that 10x impact there will be some quick wins down the way and you must do those always keep your eye on the price which is how can I take this product from here to here and so I just wanted to sort of I threw in a lot of points here and there I just wanted to sort of later than wrap it up be an advocate for your stakeholders while you are prioritizing do not forget your stakeholders but be very very impact focused cut out the noise and ask the right questions this is how you will build a good base for your priority list things that you need to prioritize always when you have a priority list visualize it always keep it up to date and be very very transparent about it and remember your job doesn't end at visualizing a priority list it should be about you evangelizing that list you need to go from door to door get everybody together and have everybody motivated and eager to work on that thing that you think is the most promising thing and I've seen that shift when you have your engineering team motivated and kicked about the the feature versus not there is a world of a difference in the speed at which they work or the thought that they put in in just testing it just makes a world of a difference if your team is on page so evangelize your priority list continuously and like I said align them align everybody and drive your execution be accountable for execution of your priority list think of that as a part of the prioritization process and again I've mentioned this point because I've seen a lot of this happening where we make a list and you stick to it it's like our bible don't hold on to your roadmap or your backlog or your priority list go back and adjust it if your company goals change go back and adjust it if your prioritization criteria change go back and adjust it if now you have data versus before you didn't and do that continuously and again this is just a summary don't work in silos prioritization will not work that way if you want people to work on that next big thing you need to keep them aligned be open to how you failed in the past and what you can do better next this is both in terms of how you prioritized or which framework you use keep learning from the past and definitely adapt to changing goals of the company your startup or your company always be in a particular stage and that stage will warrant different company goals always align yourself to those goals thank you and open to any questions that you may have sure and thanks for the example that was very interesting sure and I was wondering if you had an example of a failed prioritization or time the time when not necessarily in your own career but the time when a bad prioritization happened and what kind of consequences it led to? I think this happens every day my memory is failing me I'm sure I've done a couple of those but every time and this is again my notion every time something that you did did not move the needle it is bad prioritization it means you did not do your user research you did not find the right metrics or you did not align to the right goal or you didn't execute it right it was poor prioritization forgive me because I can't think of a specific example I forget what Evernote did but they launched a specific feature I think last year which had terrible terrible feedback yeah work chat or something I forget it was poor prioritization because they picked the wrong thing and they prioritized it for business goals not forgetting completely the customer advocacy part of prioritization so they didn't have that criteria of how is this going to be met with from our users which are millions of users right so they dropped the ball in terms of fact is my assumption so I would think that is an example of poor prioritization sure right right sure right right right right right sure as a product manager again I think the onus is on us to be very in sync with what is our technology infrastructure enable us for example there may be a feature request for you to say um this is just a hypothetical example to say why don't you say this why don't your email say this as a subject line this is something 50 000 of your user set but if that does not go well with the branding of your company um you're going to hear that you're going to funnel it but you're not going to execute on it till it matches with your branding so whether it's tech capabilities or just things that and there are tons of these requests um in fact just to add to that especially users will continuously give you our solutions my take to that is and this is something which the jobs to be done I don't know if others are familiar with it basically what it says is the customer so my hypothesis or my theory on that is don't listen to the solution that the customer is giving you which is what they will do 99% of the time they'll say do this do this do this try to get to the heart of it what are they trying to they are trying to get you to change the subject line because maybe the subject line you're using makes them think it's spam so it doesn't have to be the subject line they are asking you to change to but when you get to the root of the problem you can you can work your way around to solve that problem for whichever stakeholder it will always be a mix it will always be a mix and that's where I specifically wanted to call out the gut prioritization there will be times and that's why we have facebook's of the world and google's of the world or slacks of the world these are people and teams and ideas that have been you know done and they've taken the world by storm it is because there was gut and some fair bit of quantitative data to back it so I think it has to go hand in hand there will be sometimes where data will just shout out to you to say oh we need to do this and there will be times when you will not have that and so you will have a mix and a you know pinch off your gut or your qualitative feedback sure interesting question I think my gut has to be strong for both of them um even if there's tons of data and I as a product manager have from learnings in the past or just best practice practices that I know or something similar I've done in other companies if I know that you know again like users can ask and ask and five million users can ask for something for flipkart that is very very strong data still if you think as a product manager um it is your job to to be the wall in between to say I have very strong data but this is the way I should slice and dice it maybe all the for example all those five million users are users from a particular state and so that means that the problem or the solution is specific to that state it doesn't mean it has to be on all of flipkart so to answer your question I don't know if I've done that well but what I'm trying to say is as a product manager you need to have your heart and soul to ever in a in a particular project to even imagine to get everybody's alignment on it you cannot be I have low confidence on this but it has high data and I have high confidence on this but this has no no I think the the competition would be this doesn't have data but this has data I have more instincts that this will work and I have less instincts that this will work but I still think both will work um did that answer your question to some degree yes of course you will need to validate it so there is that whole process of you take an approach you don't have data to back whether the approach is right or not but then you validate it there is post factor validation and that's where you get your qualitative and quantitative feeder you feed data into something that you think might work vis-a-vis you have data on something that should work yes yes and do a better job of of it hey so you talked about prioritization and how it is extremely important to prioritize tasks that help serve underserved means how exactly do you guys well sort of the process that you guys perform to identify underserved so I'll actually um that was one framework that is available to you so that slide was to give you just exposure to so many types of frameworks that are available that is one particular framework not everybody uses it and what that framework does I can talk to you about the after the presentation about it it's something that's called ODI basically it means that the framework maps satisfaction versus importance and it says the more important something is the more satisfaction it will drive and if you're not serving that quadrant it's basically underserved and like I I think I mentioned that I'm trying to do it more and more because I find value in it to to find those really hidden innovative things that we are not doing today so it's just a framework you can choose whichever framework works for you sorry the opportunity scoring framework sure so um usually there's a high level roadmap where you will have things like strategic advantage or user engagement or virality or growth or whatever your themes may be or user acquisition or activation or whatever your themes may be and then beneath below that whatever your visualization technique is you basically have this is what I want to do short term this is what I want to do medium term this is my long term goal and then when you're double click on one of those ideas they will they can be a sort of mini roadmap inside that to say this is the feature but this is my mvp this is my v1 my v2 my v3 depending on the scope of the idea it actually that's very dependent on the tool you use to visualize the ideal way and I think there are tools out there that help you do that is you have the high level which is this is what I want to do in quarter one this is what I want to do in quarter two which is like high level statements I want to solve for user retention in quarter four and then when you click on that the tool will allow you to see actual project ideas again cordoned by I want to do this first I want to do this second I want to do this third I mean double click on one particular idea there are tools like product port there are tools like some people use JIRA some people use clubhouse there are tons of tools out there but essentially you always start from the high level and then you can double click and double click and double click because when you actually get a feature executed there are tons of details on what you want the functionality to be and so on and so forth so sure hey oh oh so so we again that depends on the company and the culture within the company so yes I do and this is not a step on any team stores but it's generally a good idea to know where your team is spending its time and resources and it particularly goes back to the alignment for example if your qa team is testing a feature which doesn't need to go out for the next three weeks they're spending time and effort and overtime on that just because they do not know that we need to roll x feature out right now because we are fighting some fire this it is for the purpose of alignment and so that there is no communication gap so we we use slack constantly to make sure that everybody is focused on doing the right things great last question I think sure sure sure I think that's a great question so there are certain things where either those ideas amount to just customer delight or they just amount to better experience for example making sure that your app is not buggy it's not a it's not a metric that most of us have on our dashboards to say so many crashes happen today it's not something and all of us will be guilty of that but that is something that is an integral part of customer experience so yes to answer your question we should always strive to have five percent seven percent ten percent of our engineering bandwidth working on those things for example at ship we've just recently our qa team just recently made a couple of fixes for being accessible for people who are colorblind and people exactly and so we do try to carve out time for these things I mean these are not our top business metrics but we care about these things go ahead there is no method to that madness no to be fair I think it's just being persistent there's no math that you can do to say I want seven percent every one week and because it's seven percent I want so many hours spent on this it may not always work like that it is that your team believes that we need to spend time on this and you have that ready for them to work on so two things you can do as a product manager is to make your team believe that this is important enough for them to spend time on and for them to be willing to spend that seven percent of their time on it and second is to keep it available and in the right order so that when they have their bandwidth and they have a few hours to spend on it they have everything ready for them to get going and that sort of sets them up for success and that's something you can do as a product manager thank you for being such a great crowd I hope this helped


I'm a recovering TV meteorologist, I guess you might say. Co-evolution might not be the right perspective. Nimbus evolved, and then TV weather evolved after that, hanging on to maybe the antennas of the solar panels, or at least the data feed. As forecasters saw the data coming down and trying to figure out, that could really help us. I started out in Seattle, and any help we had in the 70s with trying to figure out what was coming in from the Pacific would have been a huge help, and obviously very important to us. So, co-evolution, and the thing I really wanna emphasize here is more the educational aspects that the Nimbus program has really provided us. Okay, we advance here, there we go. So, informal science learning and public awareness of science I think really has been enhanced by the Nimbus projects and the various projects that have come out of the Nimbus project. I sort of found my way here at NASA as a result of trying to do a story about the rescue system about seven years ago or so, and so that really sort of comes from Nimbus tracking the reindeer back in those early days. Have volume on this one? No sound? Anyway, this is what TV looked like before satellite imagery. Look at the set. This is right after Camille, the Katrina of the 1960s. There are no satellite loops, no radar loops. It's magnetic hurricane symbols on a map, dials on the side. Those are magnetic words you'd spend an hour putting up magnetic words on a weather chart. We evolved, started out in Seattle. I was particularly interested in the early images. I would go down to the National Weather Service, take a four by five Polaroid of the latest satellite over the Pacific, mostly clouds all the way from Seattle to Japan, and go back to the station and show this little four by five image on the TV camera, and kept hoping that we'd see a clear spot sometime, but between September and June in Seattle, forget about it. Most of the time it was just great cloudy skies all the way, but sort of a Roshek ink block test. In the early days, Rory Leep was one of our real heroes. He had a staff of six TV meteorologists, and he had in 1966 some of the first images, but that receiver cost his station $11,000. That was above our budget in Seattle, and think now of the quarter of a million that they spend on Doppler radar at the present time. Even by 1980, look how crude the satellite imagery was. We've come a long ways. This is the Weather Channel, 1982. I was fortunate enough to do a pilot for this about a year ahead of their actual launch to see if they could sell the advertising. We were thinking in 1982, who's gonna watch weather 24-7? But look how crude it was. There was the radar, there was their set, there was their map. Even in 1982, the satellite imagery was just very minimal. Let's take an example of what we can show now. Sea ice, we have now sea ice in the can really from 1964, as I understand it, from Nimbus. We're looking down on part of the Arctic Ocean here. I might be there somewhere on T3, but maybe not, but by 1983, thanks to Nimbus, in 1987, we're able to really begin to monitor the sea ice. And look what we can do now today with letting the public know about the science of the world. We do live shots now out of NASA Goddard once a month, basically, with morning TV forecasters, and we give them two minutes or so to ask questions of our scientists, whether it be about ozone hole or Arctic sea ice, and basically part of, oh. There was a movie there, can we play that? Here we go. Part of the informal science education process is show me. If you can show me, I can tend to believe it maybe a little quicker. We can actually take people now, thanks to the imagery that we have, that started with the Nimbus program to the Arctic and show them it is melting. We can show them elements of the environment now that we never could before, and we can also bring it home in a sense that the how and the why of what's going on, now that we know about the three-dimensional view of the earth and the melting ice and the open ocean, changing the heat budget of different areas of the world, and perhaps affecting the jet stream flow and how that may affect storms like Sandy hitting the east coast or the polar vortex. A very complicated process, but now we have a way. My work really started out about seven years ago at George Mason Climate Change Education, which is really earth science education, and the imagery and the data, the data-based imagery we can show now with NASA and NOAA data is really helpful. In informal science education, what works in museum zoos or aquariums, aquarium, I guess it is, is that aha moment of science, the facts, the concepts, the method, how do scientists do, how do they propose these theories, and then number four, science is a bunch of people working together, it's a social system which sometimes goes wrong. Nimbus B crashes into the ocean, but we learn from that and we go on. Is TV news still important? Yes, it's still the way most people learn about science. After 12 years of school, where do we learn about science? We probably learned first on TV. And our general model at George Mason is we have communication practitioners, the broadcast meteorologists, and then you have the science experts, and then you have the communication scientists, those people who understand how people, really the psychology or social science of how people perceive risk and why they take action during tornadoes or hurricanes. But the content experts, of course, are key to this. Here's our perfect example of B, informal science education. Science is not perfect, it is a process, peer review, goes back and forth, and through news reports, sometimes we can show people how human it is, right? And there's the recovery coming in from the ocean. Another example, ocean temperatures. This was the data coming in from Nimbus, or this is Nimbus 7, I think, about ocean temperatures. Look what we can do now, and can we play this movie? We can show height, plus or minus seven inches, I think it is on this side, and then we've got Rossby and Kelvin waves on the other side, sea height and temperature differences. All this visualization now, we can show people and give them a sense of, wow, this is a complicated and very interesting world. We can show the world salinity. Who would have thought? With the satellites that have evolved out of Nimbus that we can show, and Paul will talk more about this, I'm sure, the ozone hole, the fact that we could show them in the early days, there's a hole, something's missing. People can sense that, they can get it, and now look at the data, all the kinds of data that we can show them, and we can even do it on our iPhone. I can take people and say, hey, let's look at sea surface height around the world, and I can make it, it's in your pocket. All this coming in from people who were here at NASA thinking there's data that we can get via a satellite. Look at the wonderful informal science education, and do we have sound here or not? Not at all? Okay, this is too bad because Mike uses an iPad to show his folks wonderful science, often it's NASA science, and he makes a point at the end of this, here's some wonderful science education and also inspiration. I couldn't have written those words better for him the way he said it, but that's a legacy I think of what the Nimbus engineers, scientists, meteorologists, infrared experts, all of you have done.  It's really exciting and look where we are now, we can see it all on our iPhone, on our iPad. So thank you very much.


Namaste, Myself, Mr. Berardar Bala Saheb, Assistant Professor, Department of Humanities and Sciences, Walchen Institute of Technology, Solapur. In this video lecture, we will discuss Laplace transform using property effect of multiplication by T. Learning outcome, at the end of this session, students will be able to apply effect of multiplication by T property to find Laplace transform of U1 function. Let us pause the video for a while and write the answer to the question. Find the Laplace transform of cos of 3T. Come back, I hope you have written answer to the question. Here I will going to give the solution for this question. Question is, find the Laplace transform of cos of 3T. We know that Laplace of cos of 8 equal to S upon S square plus A square. This is from Laplace transform table. In this, replace A equal to 3 so that Laplace of cos of 3T equal to S upon S square plus 3 square. That is Laplace of cos of 3T equal to S upon S square plus 9. Now, let us start with property effect of multiplication by T. It states that if Laplace of f of T equal to f of S, then Laplace of T raise to N into f of T is equal to minus 1 raise to N into nth order derivative of f of S with respect to 2S, where N is positive integer. Let us see the proof of this statement. By definition of Laplace transform, we have Laplace of f of T equal to f of S which is equal to integration of e raise to minus S T into f of T dt with the limit T equal to 0 to infinity. Now, differentiating both this side with respect to 2S using differentiation under integral sign rule, we get d by dS of f of S equal to d by dS of integration of e raise to minus S T into f of T dt where T equal to 0 to infinity. By DUIS rule, when we take this ordinary derivative inside the integration, it converts to partial derivative with respect to 2S. Therefore, d by dS of f of S equal to integration 0 to infinity dou by dou S of e raise to minus S T into f of T dt. Now, differentiate this integrand with respect to S partially keeping T constant, we get which is equal to integration 0 to infinity derivative of e raise to minus S T with respect to 2S is minus T into e raise to minus S T into f of T as it is dt which is equal to minus integration e raise to minus S T in bracket T into f of T dt where T equal to 0 to infinity. Now, this definite integration is nothing but Laplace transform of T into f of T. Therefore, d by dS of f of S equal to minus into Laplace of T into f of T. From this result, we can write Laplace of T into f of T equal to minus 1 into d by dS of f of S. Similarly, Laplace transform of T square into f of T equal to minus 1 raise to 2 into second order derivative of f of S with respect to 2S and Laplace transform of T cube into f of T equal to minus 1 raise to cube into third order derivative of f of S with respect to 2S. And so on in general Laplace transform of t raise to n into f of T equal to minus 1 raise to n into n third order derivative of f of S with respect to 2S where n is the positive integer, this is the problem. Now, let us consider example, example 1 find Laplace transform of t into sin of a t, here f of t nothing but sin a t, first we find it is Laplace transform, therefore Laplace of sin a t equal to a upon s square plus a square denoted by f of s. Now, by property Laplace transform of t into f of t is equal to minus 1 first order derivative of f of s with respect to 2 s. In this f of t is sin a t and f of s means Laplace of sin a t. After substituting we get Laplace of t into sin a t equal to minus 1 into d by d s of a upon s square plus a square. Now, differentiate this right hand side a upon s square plus a square with respect to t s using cosine true we get which is equal to minus 1 as it is keeping denominator s square plus a square in numerator into derivative of numerator a is 0 minus a into derivative of denominator with respect to t s is 2 s whole divided by square of s square plus a square which is equal to minus 1 into numerator minus 2 s upon in denominator s square plus a square bracket square. Therefore, Laplace transform of t into sin a t equal to 2 a s upon s square plus a square bracket square this is the required answer. Now, consider example 2 find the Laplace transform of t square into cos of 2 t. Here f of t means cos of 2 t and now we have to find its Laplace transform. Laplace transform of cos 2 t equal to s upon s square plus 2 square which is s upon s square plus 4 and denoted by f of s. This is due to formula Laplace of cos a t equal to s upon s square plus a square. Now, by property of effect of multiplication by t that is Laplace of t square into f of t equal to minus 1 bracket square into second order derivative of f of s with respect to t s.  Now, keeping one d by d s as it is and first find first order derivative of s upon s square plus 4 using cos n true we get which is equal to d by d s as it is and in numerator write denominator s square plus 4 and into derivative of numerator s with respect to t s is 1 minus s into derivative of denominator is 2 s upon s square plus 4 bracket square that is equal to d by d s in bracket after simplification numerator becomes 4 minus s square and denominator s square plus 4 bracket square. Again differentiate this RHS with respect to t s using cos n true we get in bracket keeping s square plus 4 bracket square in numerator as it is into derivative of numerator with respect to t s is minus 2 s minus keeping numerator 4 minus s square as it is and derivative of this denominator with respect to t s is 2 in bracket s square plus 4 bracket close into 2 s whole divided by square of that denominator gives s square plus 4 bracket rest to 4 which is equal to. Now, taking s square plus 4 common factor in numerator in bracket minus 2 s into bracket s square plus 4 minus 4 into s in bracket 4 minus s square bracket close and curly bracket close whole divided by s square plus 4 bracket rest to 4. Now, one factor s square plus 4 gets cancelled from numerator and denominator and it is equal to we get after opening the bracket in numerator minus 2 s cube minus 8 s minus 16 s plus 4 into s cube whole divided by s square plus 4 bracket cube. Therefore, law plus transform of t square into cos 2 t is equal to now in numerator minus 2 s cube plus 4 s cube use plus 2 s cube minus 8 s minus 16 s use minus 24 s whole divided by s square plus 4 bracket cube this is the required answer. Now, let us consider example 3 find the law plus transform of t into under root 1 plus sin of 2 t here f of t equal to under root 1 plus sin 2 t now find it is law plus transform before that we have to simplify to remove square root sin. Therefore, root of 1 plus sin 2 t is equal to under root of 1 you can write as sin square t plus cos square t from the trigonometric function and plus sin 2 t you can write 2 into sin t into cos t which is equal to under root sin square t plus cos square t plus 2 into sin t cos t you can write as sin t plus cos t bracket square and now in right side square root and bracket square gets cancelled it gives sin t plus cos t. Now, taking law plus operator on both the side so that law plus transform of under root 1 plus sin 2 t equal to law plus of sin t plus cos t we know that law plus transform of sin t is 1 upon s square plus 1 and plus law plus of cos of t is s upon s square plus 1. Therefore, law plus of under root 1 plus sin 2 t equal to 1 upon s square plus 1 plus s upon s square plus 1 which is equal to now taking L c m in the right hand side we get numerator 1 plus s upon s square plus 1 which is equal to f of s. Now, by effect of multiplication by t property law plus of t into f of t equal to minus 1 into d by d s of f of s. Therefore, law plus of t into under root 1 plus sin 2 t equal to minus 1 into d by d s of f of s means 1 plus s upon s square plus 1. Now, by differentiating with respect to t s using quotient rule we get which is equal to minus 1 into numerator keeping denominator s square plus 1 as it is into derivative of numerator with respect to t s is 1 minus numerator 1 plus s as it is and derivative of denominator is 2 s whole divided by s square plus 1 bracket square which is equal to minus 1 into when we open the bracket in numerator we get s square plus 1 minus 2 s minus 2 s square upon s square plus 1 bracket square that is equal to minus 1 into numerator 1 minus 2 s minus s square upon s square plus 1 bracket square. Hence, law plus of t into under root 1 plus sin 2 t equal to now s square plus 2 s minus 1 upon s square plus 1 bracket square this is the required answer. To prepare this video lecture I refer these two books as references. Thank you.


What's up everyone it's your girl Kenzie here from Layton Sports Cards and I'm ripping a couple personals here for Tobias B. We've got a 2021 wild card illumination football hobby as well as a 2021 Panini Chronicles football H2 box. All right best of luck thanks again for hopping in. Tree Sermon. Got Givante Williams on the comments. Smashing Zach Wilson nice one there not numbered on that. Trevor Lawrence protostar and E. Gerrit Patterson Otto. Solid first pack. Mike Jones base. Got Ernest Joan Blue. Smashing Kyle Pitts. Protostar, Kaderius Tony. Otto of Brandon Stevens. And Kyle Pitts on the comics. Trevor Lawrence sparkle out of $199. Comics Kyle Pitts. A PS2 Otto nice. Fleepe a Frank Smashing and Photostar to $249. Ian Book. Last pack of wild card. I'll get into your Chronicles H2. Got Sage Serrat to $199 on the speckle. Otto a Givante Williams nice. That is out of $99. Kyle Trask Photostar. Brandon Stevens Otto. Two autos out of this last one. And GC Horn Smashing. All right Chronicles H2 time. Moe Debo. Got Trey Lance Rookie. Crown Row DeVonte Smith. Givante Williams. Killil Herbert Rookie. Givante Williams. Jalen Waddle base. Montgomery. Got Trey Lance Rookie base. Justin Fields out of Legacy. Crown Row Elijah Moore. Brevin Jordan on the purple foil. Rookie to $49. Micah Parsons Rookie. Kaderius Tony and Rondell Moore. Michael Collins. Jamar Chase. Killil Herbert. We've got Redfall to $399. Mike Evans. Rondell Moore. PS2 and Rondell Moore. Here a car. Silver Rondell Moore from Score Updates. Crown Rows. Adoffe Oway and Kenneth Gainwell. Givante Williams Rookie. Davis Mills. Jack Prescott. Corey Davis. Aaron Donald. Michael Carter on their prison Black Rookie. Crown Rows. Ian Book and Quitty Pay. We've got Rashad Bateman out of 99 from Elites. Najee Harris and A. Kellen Mons. Last pack. Here for you Tobias. Tuva Givante. Davis Mills Rookie. We've got Ramon Dre Stevenson. Terrace Marshall Jr. Kellen the Mon Rookie. Trey Lance Rookie and A. JC Horn Rookie Acetate. All right eat Tobias. That'll do it for your boxes. Thanks again for hopping in. We'll get these right out to you.


The U.S. central bank has announced its biggest interest rate increase in more than two decades as it toughens its fight against fast rising prices. The Federal Reserve said it was lifting its benchmark interest rate by half a percentage point to a range of 0.75% to 1% after a smaller rise in March, where its U.S. inflation at a 40-year high further hikes are expected. The push marks the latest effort to contain spike in costs being felt by households around the world.


Hi everyone, Joe for Jaspy's casebreaks.com with our last filler. We always write last filler when we're down to our last filler. 2021 rookies and stars, number 15. Now we got a few things happening here. Six extra spots here being given away. Then we're going to do the break itself with the second dice roll. Then we'll give away those mixer spots. Yes, Chad, I did. And a big thanks to this group here. So now let's give away those first spots first. So top six after 10. Six into four, 10 times. One, two, three, four, five, six, seven, eight, nine, and 10. So after 10, I got Chad, John, Oliver, Chad, Adam, and Chad with extra spots going your way. So a little buy one, get one scenario for you. All right, now that gives you a little extra chances here for maybe a mixer spot that's coming up in the next video. Now let's randomize you a team. One, two, three, four times. We're going to do the break itself. One, two, three, and four. Got Chad down to Chad. And one, two, three, four times for the teams. One, two, three, and four. Falcons down to the Rams. All right, so there's the first half of the list right here. And there's the second half of the list right over here. And let's print and rip. Who's dissecting the grates? You like that? Where's my knife? There's the final printout. We've got Trevor Lawrence, rookie card, Davis Mills, and Heinz Ward, Chad with the Jags. All right, so nothing too wild here, but it's going to be wild as who's going to get those final six mixer spots. Let's flip back to the list for our third and final dice roll. And it's going to be the top six after four or two, six times. One, two, three, four, five. And good luck, sixth and final time. So from seven on down, it's going to be sad times. Sad times are Chad. Although he's got a lot of spots. He's got to be up in the top six somewhere. Seven out of 32, seven down to 32, and everyone in between. Thanks for giving it a shot. We'll get him next time. Top six, though. Happy times. You're in the mixer. Al is happy. There's Chad. Chad again. Adam, Chad, and Adam. So congrats to the top six. We'll see you in the next video for the break itself, jaspyscasebreaks.com.