So good afternoon everyone. Welcome to the final session on differential equations So in today's session, I am planning to cover up this topic so that next class we can start with vectors and 3d So in today's session, we are going to primarily cover Exact differential equations. We are going to talk about Orthogonal trajectories We are also going to talk about equations of first order and higher degree and We are also going to take up some application of differential equations in the field of physics Okay, and chemistry also And math of course, okay So let us begin with the concept of exact differential equation So let us start with exact differential equation Exact Differential equation Okay, as the name itself suggests These kind of questions are basically of such a type where Let's say you have a function m of x y dx Plus a function n of x y. So both are multivariate functions of x and y And these functions are such that They Can be written like a exact differential of some element Okay, so they can be written as exact differential of let's say a function u x comma y equal to zero. That means u x comma y equal to constant becomes the direct solution for this If you remember towards the end of the previous class, we actually talked about Method of inspection right where we learned how to write a particular Expression as an exact differential. So that was actually a part of this particular method So those questions were under the concept of reducible to exact differential equation So in today's session, we are going to talk on a broader sense That if a differential equation is of this nature Okay, how do you figure out number one? Whether it is an exact differential equation. So first of all, how what are the necessary and sufficient conditions? What are the necessary slash? Necessary and I should say not slash What's the necessary and sufficient condition? for it to be for any od to be An exact differential equation. So how do you identify? How do you identify that it is an exact differential equation? Okay Second thing, how do you find your? How to find out how to find out Your u x comma y function as you can see in the solution. This is your No final expression whose differential was given to you in this form Okay, so basically you are reducing this guy to a differential of a function And from there you are going to get the final solution. So this is going to be the general solution Okay, so how do you first know whether it is an exact differential equation? Okay And secondly, how do you find out your u of x comma y? So the necessary and the sufficient condition The necessary and the sufficient condition For any differential equation to be an exact differential equation Is If you have been given your function your differential equation to be of this nature And you realize that and you realize that Doh m by doh y is equal to doh n by doh x Okay If your condition if your differential equation satisfies this condition Right, so whatever function is given to you here and here If this condition is satisfied by these two functions Then it implies that it's a case of an exact differential equation Okay, so before you proceed You have to check whether this test is satisfied or not Now just a word of advice over here Please do not use this When do you have a format different than this? Are you getting my point here? If your format is different than what is written over here, this condition may not work So please do not try to apply this to reducible to exact differential equation which we had done in the last class Are you getting my point? So this is for those cases where this entire function boils down to just differential of a single function Okay, so it boils down to this form. It will work only in this case Are you getting my point here? Okay, so There has been some mistakes done by students in the past Where they tried to bring the previous class concept that we had done under this periphery See many of them may not satisfy this Okay, but they are the cases of reducible to exact differential This is actually a case of exact differential There's a difference between both the cases do not mix them Okay, I'll give you an example little later on But before that we need to understand why is this condition required See this is again a slightly higher level concept When you say there is a Let's say I call mdx. Let's say I drop the word x comma y. I don't want to write too much Okay, let's say you want to claim that this is differential of some element u Okay, so m is a function of x y n is a function of x y u is also a function of x y So remember we had done something when we are learning implicit differentiation that du is actually doh u by doh x into dx Doh u by doh y into dy Do you recall this This is how we actually got the formula for our implicit function differentiation. Do you remember everybody? Okay, yes, if you compare it to this guy, let's compare it to Let's compare this with So you're claiming that both these things are same Okay, that means indirectly you're trying to claim that m is nothing but doh u by doh x Okay, and n is nothing but doh u by doh y Okay, now If this is the case I can say one thing from surety over here that if I differentiate this again with respect to y that means I I do doh to u by doh x doh y It would be Same as doing differentiating this partially with respect to x Okay, oh, sorry. So this this is y over here. This is x over here Okay, and if you differentiate this with respect to Partially with respect to x. This is what I will get with when I do doh m by doh y And when I do doh n by doh x And you would realize that for any function in x and y which of course should be differentiable There is no difference between these two terms. Actually, both are same This you learn in your higher grades also in partial derivative You will realize that this result and this result would be the same results Okay So since these two are equal it implies that doh m by doh y Will be equal to doh n by doh x and that's how this condition comes out which I had written over here Okay, so please make a note of this Now you will not find this proof in your regular textbooks This is something which I am telling you from my highest studies of calculus Well, you'll learn this in your first year or second year of Your engineering Okay, so this is applicable. I'm again repeating it. This is applicable when You are expecting an exact function differential to come from here Are you getting my point? Okay Now how to find the solution in this case? Okay, let's say this condition is met fine I understood that it is going to be I realize that it is going to be an exact differential equation So what is going to be the solution for it? So again without going giving you any proof the solution for this is given by Okay, so please make a note of this So the solution of this differential equation is going to be The integral of m function with respect to x Keeping y as constant plus integral of Those terms in n which do not contain x so you can say terms of the n function Okay, so remember you will be given you will be given this kind of a Expression and you realize that this was a exact differential equation Then how would you find the solution by the use of this formula which i'm giving you so m is this function Integral of this function with respect to x keeping y as constant. This is important. You have to treat y as constant Plus integral of those terms of n not containing x Not containing x What happened to my spelling not containing x With respect to y and just put a c on the right side Okay, so whatever comes out from here. This would be your u x comma y Yeah, actually we got this condition by solving that given set of conditions Actually, I don't I don't want to get into the proof of it. But since you have asked it see When you have compared this your m is actually Doh u by doh x correct Yes or no correct And your n was doh u by doh y correct So what do we normally say is that If you look at this expression If you look at this expression, it is basically trying to say u is Integral of m with respect to x keeping y as a constant Yes or no Correct. Yes Keeping y as a constant and from this condition you get u as integral of n Okay, and you'll end up getting such functions Which are function you'll get you'll end up getting such, you know expressions which are just completely functions of y Okay, so if you integrate this Basically, you will end up getting something like this I just write it down over here So n will become du by dy Plus some functions of y Getting the point.  Yes So that's why f y means those terms Which do not contain an x Like this Okay, you don't have to go into details of this because it requires your understanding of partial derivative also So please make a note of this so two things you need to understand over here is one How do you identify a differential equation? So if you realize that your differential equation satisfies this condition Okay, then it's the case of an exact differential equation And secondly your solution is going to be integral of m dx Where y is treated as a constant Plus integral of those terms in n Not containing x Or you can say not containing x terms in n Integral with respect to y and just put a constant on the right side. Is it fine? So Yes, sir if we have the m dx plus n dy equal to zero type of equation And dou m by dou y is not equal to dou n by dou x Then it is not an exact differential equation. So but it could be reducible to exact It could be reducible to exact but you cannot directly use this formula that you have discussed over here Okay, sir What's the point? Yes, sir Okay, so let's take few questions on it Let us first start with this question Uh, let's take um, yeah One plus two x y Cos x square minus two x y dx Plus sine x square Minus x square dy equal to zero everybody. Please try this out So first thing is this is your m This is your n Okay, just check dou m by dou y And dou n by dou x are they giving you the same expression? Let us check Dou m by dou y means you have to only treat y as a variable x will be treated like a constant So this will become two x cos x square minus two x Okay, here You have to differentiate partially with respect to x. So this is obvious that will be cos x square into two x minus two x Yes, they are same They both are same So the moment you realize both are same you can directly go with the final formula that we have discussed Integral of m with respect to x treating y as a constant Treating y as a constant plus terms Not containing x Okay dy And write a c on the other side Okay. Oh, so sorry Now terms, uh, sorry m m was integral of One plus two x y cos x square minus two x y dx Okay, now remember there is All the terms in n All the terms in n have an x in them, right? So you have to write a zero over here Okay, equal to constant. So the other term will not appear at all Yes, so we need to just integrate this. So this is x Yes, how would you integrate this? Remember you have to take x square as a t So it is something like two y or you can say just a y And you have two x cos of x square dx integral Okay equal to a c minus two x y dx Okay, so this is going to be x This is just like cos t integral cos t integral is sign t so it'll be This and this is going to be minus y x square Equal to c. So this is going to be a final answer Is that clear? Any questions here any concerns? Correct correct param Okay, let's take another one dy by dx plus y cos x plus sine y plus y by sine x plus x cos y plus x equal to zero The first attempt should always be whether it is variable separable if that doesn't work. Is it reducible to variable separable? Okay Homogeneous you can directly come to see from the expression that you don't have to even try it out because looking at the Expression itself you can come to know that whether expression is homogeneous or non-homogeneous Then try lde if you feel lde can be applied If nothing works, then probably it may be a case of an exact differential equation So so can they ask this stuff in means in advance? See, uh, they can ask this stuff in uh Advance not in means Okay, so In advance they can give you See the uh, I think you may have seen the paper this year. Did you did you get a chance to see the paper advance paper? Only physics so far Oh, okay. See math also the paper was not very easy. Okay, and I felt that uh The paper were much difficult as compared to what we used to get previous years One of our very good students. He's just getting 50 in the second paper So like for uh, and if there's a term containing both x and y we still don't consider it right Sorry, yeah, yes if if there is a uh term with both x and y you have to drop that term Okay, so for example, if you write this You will be writing y cos x plus sine y plus y dy sorry, uh dx plus sine x plus x cos y plus y sorry plus x dy equal to zero correct So this is your m and this is your n So first we'll find dou m by dou y dou m by dou y will give you a cos x plus cos y Plus a one dou n by dou x dou n by dou x dou n by dou x Yeah, this will give you cos x Plus cos y Plus one. Yes, both are same both are coming out to be the same Okay, so your solution will be Integral of m with respect to x but treating y as a constant So this integral but treating y as a constant In reality, it is not but it is just for the process to follow Okay, plus Integral of those terms here, which are not containing x so I don't see there is any such term So you'll have to write zero Okay, of course it will give you some constant which will club with the constant over here So you don't have to put no multiple terms Okay, so this is uh easy. So this will give you y sine x plus Plus this will give you x sine y Plus x y This is going to be your answer Is that fine any questions here clear? Yes, so try one more y square e to the power x y square Plus four x cube dx Plus Two x y e to the power x y square minus three y square dy Done. Uh, could you use chain rule in these kind of problems? Yeah, I mean, yeah The observation is always there, uh, Ruchir observation Is a special case or you can say it is a I think I should say it's a broader case of these kind of problems this is actually You know no brainer when when you realize that your doh m by doh y is equal to doh n by doh x It works even for those cases where observation works Okay, so observation is a broader way to solve it And when it fits into the formula, then it actually becomes a no brainer Then you can solve it by the use of this formula blindly Okay, so if you can solve it by observation Nothing like that. I mean that is the most robust way that you should you know apply to solve a question So method of inspection is a broader method Of this particular type Correct So it can apply even to those cases which were exact differential equations But it was not covered under that necessary condition Try to get me that Okay, so I want to say inspection Inspection is a broader methodology to solve And within that there are some functions which will which will satisfy doh m by doh y is equal to doh n by doh x Are you getting it? This becomes no brainer That means you can solve it without You know trying to do anything but for these problems. Yes, you have to apply your inspection method Which is actually reducible to exact differential. So these are the ones which are reducible to Exact do you get what I want to say? Yes Okay, so I think In this case you would realize That let's say this is your m This is your n Correct So what was doh m by doh y? That would be two y e to the power x y square So so we don't differentiate the x y square part here plus y square e to the power x y square, correct? This will be zero So I'm applying product rule on this. So this part I differentiated first this I kept it as it is And then I differentiated the e to the power x y square part over here Oh, sorry. What about the y square x y? Yeah, sorry Wait a minute. What about the y square? Ah, that's what I did know Uh, this part I differentiated over here This term I copied as it is Then y square I copied as it is derivative of e to the power y square e to the power x y square into derivative of this with respect to y so it'll be two x y no Okay. Yeah. Yes. I got What about doh m by doh? Sorry doh n by doh x Doh n by doh x Okay, same thing. Let's apply over here. So this will become two y e to the power x y square Okay, and then it will be two x y e to the power x y square into y square So that will give you two x y cubes. So I think this term matches with this term matches with this and this term matches with this And three y square minus three y square anyways is zero. So yes, both are same both will give me the same results Okay, so when you realize that it is fulfilling your criteria, then it becomes a no-brainer question So you just have to integrate y square e to the power x y square plus four x cube dx treating y as a constant plus The terms in n which do not have any x. So only minus three y square is such a term which doesn't have any x dy and just write a c on the right hand side correct So let us complete this. So this is just like y square e to the power x y square divided by y square Remember you have to follow the reverse chain rule over here So this is a constant and this is some constant multiplied to x. You have to divide by that constant correct This term integration is x to the power four here You'll get minus y cube and this is equal to c y square y square can be cancelled off Your final solution would look like this This is your final solution Okay, this you can also do with inspection again. Don't get me wrong This is not beyond your inspection inspection is a much broader way to solve questions Okay, so these are those few instances Within inspection, which can be solved by the use of this formula So if you don't want to apply your brain That means you check first of all, is it an exact if is it satisfying that exact differential equation scenario? If yes, then you can go ahead with the use of the formula. You don't have to figure out by inspection anymore Okay, but let's say you don't want to use a formula. You want to go by inspection then you can go for the inspection Are you getting my point here? What I'm trying to say Okay, let's start this question x square dy minus y square dx Plus x y square x minus y dy is equal to zero So if you see you have just minus y square dx as your Minus y square is your m And rest of the term that is x square x y square x minus y is your n term Yes or no? now If you do dou m by dou y You'll end up getting minus 2y And if you do dou n by dou x What do you get? You get 2x You get 2x y square And you get minus y cube They don't match They are not the same Right, so in this case It is reducible to exact differential. You cannot solve it by the use of the formula Okay, so how will I solve this? If you remember, this is the question that I had taken with you last class If you just turn back to the notes of the last class, you realize that this question I had done in the class Do you recall or should I do it again? Okay, so basically what I want to explain through this question is that here The formula will not work. The formula fails here Okay, because this condition is not met So you cannot solve it even though it is an even though it is reducible to exact differential Okay, you cannot solve it by the formula Right, so you have to go by inspection method. So as we have discussed the last class again, I'll I'll repeat the Question once again over here because it is hidden So let's say I want to solve this anyhow If you turn back to your previous class notes, this question was actually solved So we take x square y square common. So it becomes dy by y square. Now, I think you will recall this Okay And if you take x square y cube common, it becomes one by y minus one by x dy equal to zero, okay Let's say you divide throughout with x square y square. So it becomes dy by y square minus dx by dx by x square And this will become y One by y minus one by x dy Okay, so you divide by so this is nothing but negative differential of one by y minus one by x Okay divide throughout with one by y minus one by x. So this term will come down and this is nothing but negative ln One by y minus one by x plus y square y2 equal to a constant so this question was A case where you could actually not use the formula directly, but it is a case of reducible to Exact differential, okay.  Yes or no But we could not figure this out. It was because probably They were inclusion and exclusion of certain terms from here You dropped certain terms out of here. You could not rearrange it as Was required by the formula Right here your m and n were corrupted because see ultimately what you did you dropped this term, right? Did you see that you dropped x square y square term? this term It was unnecessarily included in your m and n and because of that your doh m by doh y equal to doh n by Doh x condition got distorted Right, so you were not able to identify That it was actually a case of exact differential equation because your m and n functions got corrupted Are you getting my point here?  Yes If the formula works you are lucky Go ahead apply the sorry if the condition works doh m by doh y equal to doh n by doh x works You are lucky you can apply your formula and get your job done If it doesn't work then unfortunately you have to rely on your problem solving instincts to get the question done Okay So with this I'm going to I'm going to something which we call as Higher order derivatives But sorry higher degree derivatives, but still limited to the first order now. This is something which is Not a very very important concept from j main point of view, but you never know in j advance they can give you as a comprehension So now we are going to talk about


She was the Philippines' worst nightmare. We were like inside a washing machine with the winds very strong. Experts knew the place, the date, and the time she was attacking. By around seven, the first wave of the storm surge hit us, followed by two more waves. But despite the warning, her target was still caught on repair. The third surge is the worst. It's like a tsunami. We were just hanging by the window, by the air convent. Others call her Hayan. But in the Philippines, we call her Yolanda, the world's strongest typhoon to make landfall. No one foresaw that it would be that bad. Everything was flat. Everything was completely destroyed. Yeah, the piles of bodies and the debris, like ground zero. You cannot even talk about your physical losses because all around you, people are dead. Two years after the most deadly super storm, we take an in-depth look at Tacloban cities we're building experience. Government officials, NGOs, and experts reveal the lessons learned after the disaster and uncover what they did to recover, rebuild, and come back a better city. This is Tacloban City today, one of the highly urbanized cities in the Philippines. Viewed from above, hardly any physical trace of the devastation can be seen. The recovery experience of Tacloban from Typhoon Hayan has caught the attention of numerous researchers from around the world. Half of the world now lives in cities and urban centers. And that urbanization process, more people living in cities and urban centers, is accelerating. But in the Philippines, urbanization is fastest in secondary cities like Tacloban. What Tacloban experience can happen anywhere in the Philippines, especially with climate change. Of the onslaught of the disaster, Tacloban suffered from a deadly tsunami-like storm surge that killed almost 2,700 people and caused major damage to infrastructure. Right after the typhoon, it was like wartime. Everybody's houses were destroyed, so we started to just map out who's alive, who's dead. I think the major problem there is, what are we going to do on a daily basis? Should we focus on the relief first, or should we focus on the debris? But the problem is, we lack the resources to help us. In fact, it's the mayor who is driving our truck during that time. We don't have people, we don't have resources, we don't have augmentation. The sheer magnitude of the disaster also wiped out any existing framework for disaster management, pushing local officials to rethink their strategy and role as first responders. Known for so, that it would be that bad. It would be so widespread that all of our, even our department heads could not function. And of course, the lack of a public broadcast system created panic. And then later on, ultimately, the sense of, you know, the loss of security. What really killed Tacloban was when the economy stopped. When there was no more business, there was no more supplies coming in. That's a problem because the entire region depends on us here. We're the center of economic activity in the region. All the warehouses were damaged. Few of them were able to survive. But the problem now was, since there was no police visibility, most of them got looted. The local government should have some kind of a broadcast system. If not, actually they can do the per-region thing. You know, then it'll do away with all of the confusion, the panic, and the franticness that, you know, and the hysteria that led to the looting. Like that, that's actually dangerous. 40,000 houses in Tacloban were also destroyed by Haiyan. Most of these non-jewel houses were built along coastal areas. Right after Typhoon Haiyan, we automatically started work with the master plan on recovery and rehabilitation. We call this the Tacloban City Recovery and Rehabilitation Plan. Essentially, it maps out all the strategies and projects of Tacloban City from right after Haiyan until like 10 years after. It is an inclusive, holistic kind of plan. So it deals with several key result areas for most of which, of course, is housing, which is shelter, infra, livelihood, business, and commerce among many other critical areas. The task is overwhelming, relocating thousands of informal settlers away from danger zones. This is the map of Tacloban City. This is where the surge was. And then this is the north of Tacloban City. And we will be transferring the 15,000 families to this Tacloban North where there are residential places, there are commercial places, there are institutional places. What we did was to have a Tacloban North Technical Working Group to see that every social, economic, infrastructure, services and necessities are there. But that takes time. But among the planned 14,500 households to be relocated, only 7% or 1,000 families have been relocated as of March 2016. Most affected families are poor fisherfolk and so relocation can mean restricted access to their livelihood. At first, they didn't want to relocate. The first problem was to find a place to live. Because most of the people here are fishing. If they return, for the meantime, for daily needs. The national government thinks that relocation is by far the most durable solution to their post-disaster resettlement concern. But experts believe alternative housing solutions should also be explored. People stay in a certain place for a reason. You just can't relocate a certain part of the city to another part of the city that for now is not even prepared to accept that number of people. I know there are issues in terms of water, in terms of power, in terms of connectivity. I think one of the main strategies is really because we had a lot of danger areas. We have to find safer areas. One of our main strategies is really to have a new township to build a city extension in the north of Tacloban City, where the hazards are almost not present except the wind. So the mindset now has also changed. Now people understand that when you don't move from high-risk areas that you're going to die before it was just a concept. Now it's reality. Over the course of two years, Tacloban has received a downpour of support from all corners of the world. More than 150 international organizations set up base in the city for affected communities. For Tacloban, Haiyan provided the opportunity to really build back better. There's a long way to go. There's a lot of work to be done, but I think more or less they have acquired certain capacities that they can use for their own development, but they can also share it with other cities and probably the world. And with the lessons learned, we just don't want to build back but definitely better. This is a good laboratory where the world could learn from. One very critical learning which we can share to other cities or LGUs is to go back to hardcore scientific data. The thing about Tacloban City before Typhoon Yolanda, there was an inconsistency between our hazard maps and what actually happened on the ground. Number two, the planning process is a negotiation process, meaning you just have to be very clear about your priorities in the long run. The Tacloban LGU felt it had innovated or experimented by not really following the steps in total, but finding which will work for us. And then third is to speed things up. Tacloban City averages 20 Typhoons per year. Rebuilding Tacloban is a daunting task for a developing nation like the Philippines. But the disaster opened a lot of opportunities for development and gave the government the chance to once more take the lead, not only in restoring a devastated city, but also in rebuilding a sustainable Tacloban.


STEAM DRUNK! You can make a solid argument that ActRazor is a top 20 game on the Super Nintendo. For those unfamiliar, ActRazor was a very early SNES title made by Quintet, and it was otherwise known as the goofy game that combined action platforming with town simulation stuff. The game got a sequel a few years later, ActRazor 2, and while it's definitely got some of the best graphics and art direction on the Super Nintendo, the game was a disappointment because Quintet decided to do away with the town simulation mode entirely. It's just a regular old action platformer and a dreadfully difficult one at that, and it's just not all that fun to play. Fast forward to 2019, and we get a trailer for a game called SoulSeraf, and the reaction was unanimous. Hey, look it's Modern ActRazor! It's a side-scroller with town sim stuff, only with more fleshed out real-time simulation mechanics. Yeah, I know I'm a bit late getting to this one, but my backlog is ridiculous right now, and I'm sorry I didn't get to it sooner, only so I could have been more timely when I tell people that this game is a total stay-away. Sure, it's nice to see that games like ActRazor haven't been forgotten, but SoulSeraf completely misses the point of what made ActRazor great. See, ActRazor was unique not just because of the gimmick of having two genres mesh together. It was the rhythm between the two gameplay modes that worked so well. The pacing was spot-on. SoulSeraf does not have anything close to a similar rhythm. The time you're in RTS mode goes on seemingly forever, and some of the quote-unquote stages in the side-scrolling mode are like 15 seconds long. And that's before I even get to what a mess these stages are. The first standard introductory level is whatever. It's pretty standard stuff, and it's clearly wearing its ActRazor influence on its sleeve with the same proportions and a similar layout, only with modern addendums like a double jump and a charged-up attack, as well as a ranged attack with your bow and arrow. But jeez, Louise, everything just comes across as thrown together, like all this side-scrolling stuff is just there for the sake of being there. I don't know, maybe I'm too far in my own head, but it really comes across like blatant pandering, like hey folks, remember ActRazor? If it were remade today, I think it'd look a little something like this. And that's the end of it, there's no actual substance here. The worst of it are these abrupt stages where enemies just materialized from nowhere. I mean, come on, really?  The enemy patterns in general here are just generic, and like I said, thrown together. The town simulation mode, or the real-time strategy mode, boils down to a simpler tower defense game. You set up your town structure, making sure you have enough food and wood to supply your army, and it's the same as about a gazillion other tower defense games, you just sit and wait out all the waves. Throw in some deity powers that are mostly cosmetic, and you go build a temple by the enemy spawn points, which activates the side-scrolling levels, and you've got something that can kinda sort of call back to ActRazor. But as you can see, the overall presentation here is really no different than a mobile game. The fact that they're charging $15 for a half-baked platformer combined with a simple tower defense game, you'd find for like a dollar on your phone, is pretty inexcusable. It all comes across as just being thrown together in a very short time, and as a result, I can't help but feel like this game is just pandering. I will say, I think the RTS stuff is fine for what it is, but again, it's like, why am I paying $15 for this? I think the most positive thing to glean from Sol Seraph is the lesson that games like ActRazor don't grow on trees, and there's a reason that they're more than the sum of their parts. It's because they get stuff like pacing and flow down correctly. Sol Seraph doesn't even come close to getting the individual elements correct, let alone big picture stuff like pacing. If you're hoping to find something close to a quote-unquote spiritual sequel to ActRazor that's actually worth playing, I think the best you'll find is one of the Dark Cloud games, because unfortunately, Sol Seraph is a stay away. Alright, I want to thank you for watching, and I hope you have a great rest of your day.


I have settled down to the task of writing these lectures and have drawn up my chairs to my two tables. Two tables? Yes, there are duplicates of every object about me. Two tables. Two chairs. Two pens. This is not a very profound beginning to a course which ought to read transcendent levels of scientific philosophy. But we cannot touch bedrock immediately. We must scratch a bit at the surface of things first. And whenever I begin to scratch, the first thing I strike is my two tables. One of them has been familiar to me from earliest years.  How shall I describe it? It has extension. It is comparatively permanent. It is colored. Above all, it is substantial. By substantial, I do not merely mean that it does not collapse when I lean upon it. I mean that it is constituted of substance. And by that word, I am trying to convey to you some conception of its intrinsic nature. It is a thing, not like space, which is a mere negation, nor like time, which is heaven knows what. But that will not help you to my meaning because it is the distinctive characteristic of a thing to have this substantiality. And I do not think substantiality can be described better than by saying that it is an ordinary table. And so we go round in circles. After all, if you are a plain common sense man, not too much worried with science, you will be confident that you understand the nature of an ordinary table. Table number two is my scientific table. It is a more recent acquaintance, and I do not feel so familiar with it. It does not belong to the world previously mentioned, that world which spontaneously appears around me when I open my eyes. It is part of a world which has forced itself on my attention. My scientific table is mostly emptiness. Sparsely scattered in that emptiness are numerous electrical charges rushing about with great speed. But their combined bulk amounts to less than a billionth of the bulk of the table itself. Notwithstanding its strange construction, it turns out to be an entirely efficient table. It supports my writing paper as satisfactorily as table number one. For when I lay the paper on it, the little electrical particles with their headlong speed keep on hitting the underside, so that the paper is maintained in a ping-pong-like fashion at a nearly steady level. If I lean upon this table, I shall not go through.  Reviewing their properties one by one, there seems to be nothing to choose between the two tables for ordinary purposes. But when abnormal circumstances come about, then my scientific table shows itself superior. If the house catches fire, my scientific table will dissolve quite naturally into scientific smoke, whereas my familiar table undergoes a metamorphosis of substance which I can only regard as miraculous. There is nothing substantial about my second table. It is nearly all empty space, space pervaded, it is true, by fields of force. But these are assigned to the category of influences, not of things. Even in the minute part, which is not empty, we must not assume there is substance. In dissecting matter into electrical charges, we have traveled far from the picture which first gave rise to the conception of substance. The whole trend of modern scientific views is to break down the separate categories of things, influences, forms, etc., and to substitute a common background of all experience. Whether we are studying a material object, a magnetic field, a geometrical figure, or a duration of time, our scientific information is summed up in measures. Neither the apparatus of measurement nor the mode of using it suggests that there is anything essentially different in these problems. The measures themselves afford no ground for a classification by categories. We feel it necessary to concede some background to the measures, an external world. But the attributes of this world, except insofar as they are reflected in the measures, are outside scientific scrutiny. Science has revolted against attaching the exact knowledge contained in these measurements to a traditional picture gallery of conceptions which convey no information and obtrude irrelevancies into the scheme of knowledge. I will not here stress further the non-substantiality of electrons, since it is scarcely necessary to the present line of thought. Conceive them as you will. There is a vast difference between my scientific table, with its substance thinly scattered in specks in a region mostly empty, and the table of everyday life which we regard as solid reality. It makes all the difference in the world whether the paper before me is poised on a swarm of flies and sustained by a series of tiny blows from the swarm underneath, or whether it is supported because there is substance below it. All the difference in conception at least, but no difference to my practical task in the lectures. Modern physics has by delicate test and remorseless logic assured me that my second scientific table is the only one which is really there, wherever there may be. On the other hand, I need not tell you that modern physics will never succeed in exercising that first table, strange compound of external nature, mental imagery, and inherited prejudice, which lies visible to my eyes and tangible to my grasp. However, we must bid goodbye to it for the present, for we are about to turn from the familiar world to the scientific world revealed by physics. To speak paradoxically of two worlds, are they not really two aspects or two interpretations of one in the same world? Yes, no doubt they are ultimately to be identified after some fashion, but the process by which the external world of physics is transformed into a world of familiar is outside the scope of physics. And so the world studied according to the methods of physics remain detached from the world familiar to consciousness. Provisionally, we regard the table, which is the subject of physical research, as altogether separate from the familiar table. It is true that the whole scientific inquiry starts from the familiar world, and in the end it must return to the familiar world. But the part of the journey over which the physicist has charged is in foreign territory. Until recently there was a much closer linkage. The physicist used to borrow the raw materials of his world from the familiar world, but he does so no longer. His raw materials are electrons, quanta, potentials, Hamiltonian functions, etc., and he is nowadays careful to guard these from contamination by conceptions from the common world. There is a familiar table parallel to the scientific table, but there is no familiar electron or quantum parallel to these scientific entities. We do not even desire to manufacture a familiar counterpart to these things. After the physicist has quite finished his world building a linkage or identification is allowed, but premature attempts at linkage have been found to be entirely mischievous. Science aims at constructing a world which is a symbolic counterpart of the world of commonplace experience. It is not necessary that every individual symbol should represent something in common experience. The man in the street making this demand for concrete explanations of the things referred to in science will be disappointed. It is like our experience in learning to read. That which is written in a book is symbolic of a story in real life. The whole intention of the book is that ultimately a reader will identify some symbol, say bread, with one of the conceptions of familiar life. But it is mischievous to attempt such identifications prematurely before the letters are strung into words and the words into sentences. The symbol E is not the counterpart of anything in familiar life. Letters are abstract and sooner or later one has to realize it. The external world of physics has thus become a world of shadows. In removing our illusions we have removed the substance of the world. Later perhaps we may inquire whether in our zeal to cut out all that is unreal we may not have used the knife too ruthlessly. Perhaps reality is a child which cannot survive without its nurse's illusion. But if so that is of little concern to the scientist who has good reasons for pursuing his investigations in the world of shadows and is content to leave to the philosopher the determination of its status in regard to reality. In the world of physics we watch a shadow-graphed performance of the drama of familiar life. The shadow of my elbow rests on the shadow table as the shadow ink flows over the shadow paper. It is all symbolic and as a symbol the physicist leaves it. Then comes the alchemist Mind who transmutes the symbols. The sparsely spread nuclei of electric force become a tangible solid. Their restless agitation becomes the warmth of summer, a gorgeous rainbow. In the transmuted world new significances arise which are scarcely to be traced in the world of symbols so that it becomes a world of beauty and purpose and the last suffering and evil. The frank realization that physical science is concerned with a world of shadows is one of the most significant of recent advances. I do not mean that physicists are to any extent preoccupied with the philosophical implications of this. From their point of view it is not so much a withdrawal as an assertion of freedom. But I am convinced that a just appreciation of the physical world as it is understood by science today carries with it a feeling of open-mindedness towards a wider significance. The path of science must be pursued for its own sake, irrespective of the views it may afford of a wider landscape. In this spirit we must follow the path whether it leads to the hill of vision or the tunnel of obscurity. Thanks for watching.


Welcome back to our lecture series, Math 1050, College Algebra for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misildine. In lecture 37, we are beginning, essentially our final unit, unit six about Exponential and Logarithmic Functions. We'll see later on in this series that our last unit, seven, which we'll entitle pre-calculus, we're going to review most of the things we've seen in this series. Unit six offers the last new content for this lecture series on College Algebra. Like I said, this unit is going to be about Exponential and Logarithmic Functions. Instead of just jumping into the deep end of the pool here, in this chapter, I actually wanted to motivate the Exponentials and Logarithmic Functions a little bit more than I have in other topics. In this lecture, we're going to focus on the idea of money. That is, let's talk a little bit about some financial mathematics and how these naturally lead to questions about Exponential Functions. Say we open a new savings account, I want to track the growth of the account. Suppose that we have some initial deposit, we're going to call it P, and this is referred to as the principal of our investment. Now, just remind ourselves that earlier in this series, when we talked about linear functions, we introduced the idea of simple interest. What I want to do right now is develop the notion of compounded interest, which while simple interest leads to linear functions, compounded interest leads very naturally to Exponential Functions. So we have this principal P, which is the initial deposit we made into the savings account. And then there's going to be an interest rate, and which for the lack of a better name, I'm going to call it I% right now. Now, I'm just using I here as a variable. This is not the imaginary unit. That would be kind of weird to take I% if that was an imaginary number. I just mean, there's some interest rate. We're going to call it I right now, just for the sake of label. And this is going to be an interest unit, an interest rate per unit of time, right? So this might be like, oh, you make 5% annually or 1% monthly, something like that. So it's typically in finance though, when you talk about time, time is usually measured in years. And so we can talk about interest rate per year annually, something like that. So after the time period, whatever that turns out to be has elapsed, we are going to gain P times I dollars, right? And then the total balance would look like the following. So after one unit of time, we would get that A equals P plus PI. So as you dissect this, what does it mean? Well, you're going to get back your original principle, right? If you don't get back your original principle with your investment, that's not a very good investment. We should get our principle back. But then in addition to principle, we collect interest on that principle. So you're going to get a percentage of the original investment P added to that. Well, as a mathematician, I cannot help myself. I see an expression like P plus PI. I have to factor it. And this turns out to be P times one plus I, right? And this right here is essentially just the simple interest formula we saw previously. That the amount will equal the principle times one plus I, which although we didn't call it I at the time, we called it R, the rate. And then we also put a unit of time. So as multiple units of time elapsed, we get this simple interest rate. Well, for this consideration, we're doing time equals one at the moment, all right? And so that's why you don't see a T and we're just using I instead of R for a moment for interest rate, okay? So that's how much we'd have after one unit of time. And I do want to kind of keep track of that. When we have one unit of time, that's how much money we'd have. Well, what if we don't withdraw anything from the account? If we wait another time unit, our balance will grow from the interest of the initial principle and from the interest gathered on the principle itself. So this is the idea behind compounded interest. What if the interest starts collecting interest? So after a second unit of time, which we'll just say after two years have elapsed, then we're gonna have the amount we had previously. So we're gonna have our principle back again. We're gonna get back the interest we gathered off the principle. So again, factoring this, we have one plus I again. So we get the amount back we had previously, but then we're gonna collect interest on this, P times one plus I times I. Basically how I want you to think about it, it's the second unit of time. It's as if we were doing the first unit of time, but we had a bigger principle this time. But again, as a mathematician, I can't help myself. I notice there's a common divisor. P is common to both, but we also have a one plus I that's common to both. If I factor those out, this will give me that the amount is equal to P times one plus I. That's what I factored out. But then what's left behind when you factor these things out, that'd give me a one, and this one would give me an I. So you get P times one plus I times one plus I. We can factor that even better. We get the amount is equal to the principle times one plus I squared. And so after two units of time, the amount in the account would be P times one plus I squared, all right? What if we were to do this for another year? What if we wait another unit of time? Well, for this third time unit, how much would we have in the account? Well, we're gonna have the same amount we had the previous year, but then we collected interest on this entire amount, not just interest on the principle P, but we got interest off of every amount that's in there. So interest is collecting interest, so the interest is compounding. So we're gonna take the amount from the previous year, P times one plus I squared, and then we're gonna times that by I, for which then you should notice there's a common divisor of P times one plus I squared. When we factor that out, we end up with P times one plus I squared, but then the first group leaves behind a one, the second group leaves behind an I, and when you factor that further, you get P times one plus I cubed, all right? And we start to see a pattern that emerges. The next time around, when we look at, okay, four units of time have elapsed now, we still haven't made any withdrawals here, then the amount is gonna equal P times one plus I cubed plus the amount from previously, plus the interest we collect, P times one plus I cubed times I, and when you factor that, you're gonna get the amount is equal to P times one plus I to the fourth. And so then we have to ask, well, what's the pattern here if we continue down in this trajectory, right? If we continue down this trajectory, what happens? What happens when we equal the nth moment in time, right? Well, and again, I'm just gonna actually leave it just unspecified as T, like what if we just come to the teeth moment, right? We're not gonna specify what it is. Well, looking at our pattern we've seen so far, notice we see the following. The amount is equal to P times one plus I, and I'm gonna write this to the first power. T squared, right? Is gonna give us this one right here, and T squared, I wanna talk about T equals two. You'll get A equals P times one plus I squared. When T equals three, you get A equals P times one plus I cubed. For the fourth time, you're gonna get A equals P plus I to the fourth. This seems to suggest where we're going here. If we don't specify the time unit, it looks like the amount is gonna equal P times one plus I to the teeth power. So whatever T turns out to be, you're just gonna take an exponent with that regard. I do wanna add one more to our list. What about T equals zero, right? What about this start of the experiment? At the start of the experiment, we should only have the principle we've made no interest yet, but that actually agrees with our formula, oops, because if we take a number to the zero of power, that's just equal to one, right? So A to the zero is equal to one with an exponent law we've seen previously. So even when we take T equals zero, that seems to match up. And so you see that with this compounded interest scenario we see right here, what is happening is that this growth, the growth of this interest is actually growing with respect to the exponent here, T. All right, so now let's make, let's make another change to this conversation here. So instead of the interest rate I, I'm going to make a substitution where we're gonna say I is equal to R divided by N. So what do these numbers mean here? R is gonna be our annual percentage rate or sometimes called APR for short. So this would be the interest rate for the entire year. And the symbol N, I'm gonna introduce now, we're gonna call this the number of compounds per year. So what I mean by a compound. So when it comes to finance, although interest rates are usually always given as annual percentage rate, APR, when you collect interest could happen more frequently. So for example, we could have, if you had an annual interest rate, of course, that would suggest that N equals one. You could collect interest on an investment semi-annually in which case that means the number of compounds is two. If you had quarterly interest, whoops, quarterly interest, we'd be saying that your compound is four. For your usual consumer, if you have like a savings account or something, your interest is probably monthly. So you get 12 compounds per year, okay? And then you could have like bi-weekly, weekly, whatever. You know, the number of compounds tells us how frequently we are collecting interest here. And so then this number right here, I equals R over N, this is gonna tell you the percentage per compound, right? So what's the interest rate per compound? So if you had, for example, 12% annual interest, then if you got interest monthly, right? Then that actually would be, your monthly interest rate would be 1%. You just take 12 divided by 12. And I should apologize that the examples we're gonna be using in these videos are examples that came from mathematical textbooks we've written prior to 2008. And so they might not, you know, for the year 2020 when this video was recorded, they don't look like realistic interest rates. Now, we're not doing this from a financial investment. I'm not giving any strategy on how you should invest. I'm just gonna teach you the mathematics of these equations. So the actual interest rates are really somewhat irrelevant, but I don't want anyone to get depressed by these things that we'll see in examples in just a moment. But when you take this formula we see right here and combined it with this one right here, we see the following formula. We're gonna get A of t. So we're treating this as a function with respect to time.  And this right here is referred to as our compounded interest formula. And so again, how do you interpret this formula? p is the principle, this is the initial investment. r is gonna be the APR, the annual percentage rate. n is the number of compounds per year. So we take the annual rate and divide it by how frequently interest is collected. That'll then give you the interest per compound. And then we're gonna have up here nt where t is gonna be measuring years. In finance problems time will always be measured in years. Growth in investments is often slow going years as an appropriate measurement of time. Now, if you take t and you times it by n, n is the number of compounds per year. So nt is gonna be the number of compounds that have occurred. So for example, if a usual car loan is done for a five year time period, after a 12 months per year, that means for your typical car loan, there'll be 60 installments. That is 60 payments you would pay because you take 12 times five years there. Okay, so this is our compounded interest formula. It's a beefed up version of the simple interest formula we saw before and it more accurately measures how usual interest problems are computed in finance. Much more realistic. Simple interest, basically mom and dad might even give you a simple interest loan but that's about it nowadays. So let's see an example of how one can do some computations using this compounded interest formula. Suppose that a credit union pays an interest rate of 8% per annum, that just means per year, right? And annum here is just a Latin word for year, kind of like why we say annual. 8% APR compounded quarterly on a certain savings plan. If $1,000 is deposited in such a plan and the interest is left to accumulate, how much is the account after one year? So we're gonna use our compounded interest formula. We have to compute the amount. That's what they want us to know. What is, how much is in the account, right? That's the amount. This will equal P times one plus R over N times NT. So we need to know all of these variables here. So principle is the initial investment which we see that was $1,000, okay? Then the one is a constant. We need the rate. This is the annual percentage rate and we see that's gonna be 8% per annum per year. So when it comes to these formulas, don't put in an 8, you're gonna put 0.08 because it's 8%, we should write it as a decimal in the formula. N is gonna be the number of compounds which we see it's compounded quarterly. Like we mentioned in the previous slide that means N should equal four. And then the last thing that's determined is T equals years, right? And so we want to invest for one year, so T equals one. So then we just have to compute A by plugging in these values. We're gonna take 1,000 times one plus 0.08 divided by four raised to the four times one. So essentially in a year, four quarters will have occurred. So computing some of these things, we see for example that four goes into 8%, well, 0.02 times. That is the quarterly interest rate would be 2%. Four times one is four.  That's gonna be 1.02, right? So we see this, we see there's gonna be a 2% growth per month of, excuse me, per quarter and there's gonna be four quarters. So we have to compute this. Now be aware, we could compute this by hand. Taking the fourth power just means 1.02 times 1.02 times 1.02 times 1.02. But oftentimes these calculations are gonna be very messy at best and darn near impossible all the time. So these are type of calculations. You're gonna wanna use a calculator, preferably a scientific calculator that has both exponential and logarithmic buttons. We'll talk about logarithms later on in this unit here. So don't be a hero, use a calculator on this one. Now you wanna try to keep as many decimals in the expansion as possible on your calculator. So if you take 1.02 and take it to the fourth power, your typical scientific calculator will give you 1.082431, excuse me, 3216, like so. And so notice that you have 1, 2, 3, 4, 5, 6, 7, 8. You have eight decimal places, that's very good. We won't have a lot of decimal places because even these small decimal places when you start raising them to large powers potentially could have an impact on the final result. Cause after all, when you time something by 1,000, that moves the decimal place by three locations, right? So we wanna make sure that we have lots of decimal places. As these financial problems in this series will focus around US dollars. So our official currency in this conversation will be USD. Whenever you're working with US dollars, we always write round the final answer to the nearest penny that is to the nearest 100th, that's two decimal places. But to make sure that our round is not incorrect, we don't wanna round off by a penny or something like that. People get mad when you round incorrectly their money cause that often means they lose more. So we will, we're gonna have lots of decimal. So eight decimal places is fairly safe since we're gonna be multiplying things by 1,000. But if we invested like a million dollars, we need a lot more decimals to make sure that interest is calculated correctly, okay? Which admittedly on this one, I do believe this isn't, I didn't round this one. If you actually take 1.02 to the fourth, you get exactly this decimal expansion. So times in it by 1,000, like I said, that's just gonna move the decimal over by three locations. So we go 1, 2, 3. And so we end up with 1,082 and 43 cents. So the next decimal would be a two, I would round that down. And so this is how much the account should be worth. So notice you get back the original thousand dollars. The interest that we would have collected over this year is gonna be $82 and 43 cents. And that's pretty nice for not even, I didn't do anything, just I let the bank invest my money and it gave me about $83 back in interest. Well, that's of course with an 8% interest rate, which again, that's difficult to find in the year 2020. Let's look at another example here. So what if, let's see, what if annual interest rate compounded annually, excuse me, what annual interest rate compounded annually should you seek if you wanna double your investment in five years? So if you expect it to double, right? What does the interest rate have to be? Assuming you could actually shop around for such a thing. By all means, I would love to find a bank that would double my investment in five years. But let's just kinda keep things simple here. Just treat it as a mathematical exercise. What can we do? Well, some things we can unravel here. Compounded annually, that would say that the number of compounds is equal to one, okay? Five years, that's the timeframe, five here. This, it's asking what rate, right? So we actually don't know what the rate is. That's what we're trying to solve for. And then we want the investment to double, okay? So we don't necessarily know what the principle is either, but we can say we want the amount to be 2P, whatever the principle is, we wanna double it, okay? So looking at our formula, A equals P times one plus R over N to the NT. Let's plug in what we know. So A will become a 2P. P, we don't know what it is. That seems a little bit concerning to us, but actually it turns out we don't need to know what it is. We're gonna go one plus the rate, we don't know. That's actually what we need to solve for. The compounded annually is one. And then we get one times five. So what we can do is the following, we get 2P equals P times one plus R to the fifth power. Great, this is the equation we have to solve. And it seems a little bit concerning because we have two variables, right? Principle and rate. Rates what we need to solve for both our principle. Well, the good news is the principle doesn't actually matter because it's a fair assessment, a fair assumption to say that the principle is not zero. If you want your investment to double, you need something to double. If I deposit zero dollars in the bank, then voila, it doubled instantaneously because twice to zero is zero, right? So we have to invest something, whether that's a dollar, a million dollars, five dollars. We have to invest something if we expected to double. So we can assume that P is not zero, which means we can actually divide both sides by P. And then we end up with the equation two equals one plus R to the fifth. So the cool thing about this one is that the time it takes to double your investment doesn't matter. Whether you invest $1, $100 or $10,000, the time it takes to double is the same. And so this is actually a good advice when it comes to investments, right? That if you're trying to save up for retirement, it's a good idea to start putting lots. Even if it's a small amount, you want to put money into your account even when you're young because its ability to grow interest is irrelevant of how much is in there, but the more that's in there, the more interest you'll collect. It'll double irregardless of how much principle you have. So then we solving for R right here, we're going to take the fifth root of both sides to get rid of the power of five. So we end up with one plus R equals the fifth root of two. Now this is another smart suggestion when you're doing your problems on a calculator that even though I could approximate the fifth root of two right now, I want to keep the exact answer exact until the very end. Because if I start approximating things too early, then the rounding errors might get out of hand. So then to solve for R, we're going to subtract one. And so the exact answer will be the fifth root of two minus one for which then I'm going to put that in my calculator and I'm going to end up with, well, the fifth root of two is 1.148698. So if you subtract one from me, you're going to get 0.148698. And so let's round this to two decimal places. Our interest rate R should be approximately 14.87% if we want to double our investment in five years.