 Welcome back we're at lecture 27 math 241. I mentioned the other day that why Or ask you to go look at why a thousand dollars is called a grand and then I reported back the next day And the look on your faces was that of disbelief so here is a picture of a $1,000 bill from late 1700s See the zeros you can zoom in on that a little bit more on one of the zeros Apparently that's why it was called a grand. It was originally called the grand watermelon note And if you look at the background here in the background of these three zeros not only is it not a very Round-looking zero. It's kind of a watermelon like zero that background Got at the nickname of the grand watermelon note and then later she got shortened to that which was a grand So if you're like me You haven't seen too many of these in your lifetime $1,000 bills, but this was the the first one And that's why it's called a grand so that makes it hopefully a little more believable that it is a That is historically now there also is one for why a dollar is called a buck And I don't remember that one, but I was thinking about that today walking over here So I think I have to look that one up again We did the logistic population growth model Certainly not to do it again, but to show you where we started with a differential equation So initially it's kind of exponential We changed the lettering a little different differently from the book So there was a capital K here and this is supposed to be a lowercase k. So we call this L, which is the limiting Population or the carrying capacity So the rate of change of population is directly proportional to the population In in addition to that we have this kind of limiting factor as p gets close to L The population growth rate gets very very close to zero so we went through the whole thing decomp decomposed into partial fractions and Came up with there's a whole lot here where the dots are That you're not responsible for As far as developing that in a test But we came up with this equation So we can use that I think we did use that right we predicted we took some data from 1970 I didn't bring this with me. So I'm having to guess somebody correctly if I'm wrong 70 to 1990 Is that the data we used population of the US and we kind of made up a limiting value or a carrying capacity of 500 million and Predicted the population in 2010 to be just shy of 300 million sound right and I think the population of the US is actually already in 2009 over 300 million so it under predicted so it's not a foolproof prediction model by any means but it does use the limiting value or the carrying capacity Rather than just an uninhibited growth model We also did some web assigned questions, but we didn't have time for enough of them So let's do that a couple of other things from the text to finish up 7.5 and Tomorrow will be dedicated to reviewing the test Looking at an old test that has been given on this and Wednesday in here for this group of students that are in the studio classroom this semester your test is Wednesday We did number three on web assigned Also number four was mentioned. What is that? Find the solution of the differential equation that satisfies the given initial condition and that's x plus 2 y Square root x squared plus one It's not plus it's just applied by a square Y 2 y times. Yeah, okay. We'll just make that a big dot here the times x squared plus So Right here X equals zero y is equal to one That concerned me a little bit that we had Addition but the fact that we've got equals zero means that we can move one term to the other side And it will simply be the negative of that. So let's do that. Let's move the X term to the other side, which would now be negative x So what is the process from here? This is all at the end of the problem this information here And that'll help us all for C or K or whatever we have in the problem at that point in time What's going to work here, okay, well, let's get the DX's where they're going to go and divide by x squared plus one Square root it is a square root, right? I Think that's it So 2 y dy equals So the left side looks pretty benign as far as trying to integrate that how about the right side what? Looks like it might work on the right side You substitution now if this were One plus x squared and we didn't have the square root then we might be possibly thinking Inverse tangent, but I think if we call this x squared plus one to the negative one-half Times negative x times DX then I think we're in business for substitution Let you equal x squared plus one Do is to x dx we're going to integrate to y dy So what do we need and how do we accomplish that? Okay, well, we don't need the negative right so we can kind of slide that out in front or We can multiply by negative one There and also here And we need a two So we need a one-half out in front So that ought to be du which it looks like du and that ought to be u to the negative one-half All right, so left side is what integral of 2 y dy Possibility of a constant which we'll throw in on the other side Just kind of stuck with that out in front And we've got u to the negative a half du which integrates to Right you to the one-half over one-half or to you to the one-half So to you To the one-half that work at an arbitrary constant that we're putting together from both sides We'll just put it on the right side So we can knock out these and we have negative Is that a little bizarre looking to you? I guess we're going to have to compensate for that with the C, right? Because I'm not liking the fact that y squared is the negative of some Principle square root so we do have negative here. We're going to have to correct that with the constant How does it? Say the answer blank for the web assign. What does it say? Does it say y equals? Y equals okay Let's go ahead and sub now for X equals zero and y equals one so that would be one X is zero so that gives us C as what? to To so our equation is that and I guess if we want to solve for y Since we're doing the taking of the square root. Let me see if we've got Anything that's going to tell us any differently here in this problem What I'm Deliberating right now is do we need both of those plus or minus? Okay, how can we not include both of them? Because you already solved for your C To make sure that we're dealing with something positive. Well, we put Kelly on the spot. What do you think Kelly? Do you think we ought to have plus or minus? It's that fair is that not fair to ask him as soon as he comes in sorry Plus or minus about them. Yeah, he says yes. I don't see any reason to not I Mean what this part is taking care of we know that what's under the radical is positive We know that otherwise we can't do the problem Now to get from this Equation to this equation. I think we have to allow for both possibilities now web assign might take just the principal square root But I don't see anything that says we can eliminate The negative at this point. I mean we do have a specific point x equals zero y equals one But that's not to say that y can't be negative Y equals a negative Possible that work if I had questions is it are your questions answered on this? Okay Six was another problem That was asked about now. I think we set up six and I'm not sure how far we got with six in class as an example We did the whole thing six all the way through does anybody still have a question on six I got an email that said we had a question We didn't do it as the first example because It had two different kind of spickets coming into the tank So I think we did that when we were doing tank problems at the end of 7.3 Yeah, we did an example from the book and I didn't know if it was a web assign question and it turned out that it was So we did problem 38 from the textbook, which is number six. I mean just in case somebody has an issue with the Setup let's get it set up So we've got rate of change of salt See if I have that What we wrote down that day. What did I call salt a? a was the amount of salt So the rate of change of salt is The rate at which salt is coming in we have two things coming in point oh five Kilograms of salt per liter And that is entering the tank at a rate of five liters per minute We also so I know this wasn't the first example we did but I we did the problem that same day we Started out doing tank problems Also coming in point oh four Kilograms of salt per liter and that's 10 liters per minute so you can see the leaders Knocking out and it should sound like an amount of salt coming in which it's kilograms per minute kilograms per minute And now leaving the tank We have a kilograms of salt at any point in time Equally distributed in the 1,000 liters Which is what we started with the stuff coming in and the stuff going out is the same So we ought to have a thousand liters at any point in time 15 liters So one more step and then if we have any more further questions we can go further, but We can add these two together Their product added to their product. What was that if I have that written down point six five And then we could reduce 15 over a thousand. What did we come up with there? 200 and this Is that the first problem we had to do that Maybe the first problem where we had to actually factor out the lead coefficient of A so that makes the separation of variables a little bit easier So let's write down one more step. So if we factor out negative three over 200 We got a minus got that somewhere 130 over three So we separated integrated both sides any Further issues with this one So if you had questions on number six check out what we did on 38 because it is Exactly the same problem any other web of sign questions anything as you look through there that You thought was a question. I don't have a hard copy of it. What's five look like? That one orthogonal trajectories sounds like it might be a test question Find the orthogonal trajectories We're okay, we're good on web of sign Okay, and I think I changed that deadline my thinking right ahead 5,000 things to do Friday. I think that was one of the things I did So that's due tonight. I believe all right, so let's wrap up 7.5 and Look at some other models. We're not going to have test questions on this But just so you can see how you could take the models that we have here are a couple that we have And these look like Population growth models, but they could be anything that's growing exponentially So we did that one and we got that population That's kind of our standard Exponential population growth or it could be decay actually depending on the sign of K And we've seen that and you probably ought to know that coming into the test that that's kind of Good old-fashioned regular Exponential growth. We've seen that as a lot of different models Either the CT or e to the KT Actually, this is exactly The same mathematical model, which is what is that? Continuously compounded interest so if we know the interest rate We know the time we know the amount that we're throwing in there. This is the most That that money could make at that interest rate for that length of time Compounding it more than every minute more than every second compounded continuously Though kind of the main thing That we took the time to develop the development of it is not a test question but we got Limiting value in the numerator. We got that equation If we have a logistic population growth problem That will be given to you Okay, that equation will be given to you We did an example. So we took two data points. Whatever we decided was our first data point. We call that time zero Second data point is so many years After time zero and then where we're trying to use that to predict is also so many years After time zero. I think our second data point was 20 years After time zero and our prediction point was 40 years. Is that right after time zero? So we can plug those things in Find a from our first data point Let me back up L is given to us in the problem or we in fact made up in our problem I think it's probably better for me to give you that in the problem first consistency of answers a you find with the first data point K and then negate it plug it into the formula you find with the second data point Some other models that are shown here That I think are worth a couple of minutes of class time So we've got this limiting value in this one now. Tell me what you think this term Does to that same model that we looked at before which has this Asymptotic effect when the population is very close to capital L What do you think that would do like a boundary on the bottom kind of a boundary on the bottom? So it's almost like an extinction type term in here that is if the population is very close to M So whatever M is this Kind of magic number that as the population gets close to that it can potentially go to extinction So this gives us a kind of an upper limit for the population This gives us a lower limit an extinction value, but both have the same effect in this in the sense that as P approaches M. This term approaches zero which kind of flat lines the population Here as P approaches L This term is zero which says the rate of growth is zero which kind of again flat lines the the growth of the population So that's kind of the same model with a lower limit or an extinction term put in See what you think this model would do which is also mentioned in the book, but Not a whole lot is done with it at this point in time. You'll probably see it again in your Differential equations course to actually solve for it. What do you think would cause that? So this would be if you were going to harvest some of the population like Fish, let's say for example That are getting too densely populated in a confined area pond or lake They would harvest some to have the water then would be more oxygenated. There's more food for the existing fish In fact, I have fished I love fishing and I fished in some ponds that were so loaded with brim that the owner of the pond Sounds kind of cruel But told me if I catch some smaller brim to not put them back in to just throw them on the bank for the turtles Because it was too loaded with brim take so that they weren't the bass and the brim weren't growing at the rate They should have been growing there wasn't enough food so Seems kind of cruel, but I harvesting goes on I guess we all probably had some chicken this weekend unfortunately that was you know kind of a Chicken that was probably grown up for that purpose. It seems kind of sad, but that would be a harvest model this next section In the book is something that we're not going to cover But let me spend two minutes on it to show you because you'll probably see it again in your differential equations course It's called a predator prey system Kind of coexisting populations that Have very interesting growth rates one with respect to the other so there's a good Example in there on rabbits and wolves So let's say that the wolf population is Growing well in order for it to grow there must be which this assumed that this is one of their main food sources That there must be a lot of rabbits in order for the wolf population to be growing But as the wolf population is growing what's happening to the rabbit population It's declining because they're getting their nourishment that way and at a certain point the rabbit population is Not completely made extinct, but it's to the point where it can't support that number of wolves that have Grown in population because there were an ample amount of rabbits So now the rabbit population is down what happens to the wolf population has a result It starts decreasing and if it decreases to the point where the rabbits and you know rabbits are Increasing fairly rapidly So then the rabbit population comes up while the wolf population is declining Then after the rabbit population gets up there to a point where it's higher than normal There's more out there for the wolves to eat so therefore they can have more wolves and it supports more wolves Do you see how the two populations interact? so if you would sketch them instead of Looking like this which is exponential growth or this which is logistic Population growth you would see and they call this a phase plane That while one is growing the other is decaying and then kind of vice versa And you end up back here and depending upon the value that you Establish for your starting point you can have kind of Differing results here, but they all kind of go back to where you started and they're somewhat elliptical in shape Kind of interesting interaction of two populations where? One is the predator the other is the prey and as one is population is growing the others declining And so on so you will visit that again. Unfortunately. It's not a part of this course anymore All right, well we have some time so I thought tomorrow would be kind of crunched if we did all the review tomorrow So let's go back and Start to talk about some of the things for the test Don't think I have an old test with me today Like we can make sure we do that tomorrow All right, so our test one ended with six point four Average value of function So we will begin with six point five Which was loaded in fact a lot of these sections that We are including on this test Have multiple topics within that section so test two first topic looks like Chapter six section five we had in their Spring problems is that our first problem? What else did we have there other applications to physics and engineering Moment Center of Mass regular old work Problems where we took force times distance and integrated that over the Normally the kind of start time to the end time we had hydrostatic pressure Where we took the kind of the end of the tank or the submerged plate and we Took little pieces of area little elements of area parallel to the surface of the liquid And we integrated what the density of the stuff doing the pressing The depth of each individual Piece right each little rectangle Times the area right From the kind of top to the bottom of that particular Might be a good thing if you've got your web assign caught up and done to take a look back at those things I don't know that we're going to have a time enough time tomorrow to look at an example of each type But as you look back and we can do maybe two of the four here in an example problem Or at least set it up to the point where you could take it from there What I think is important on a lot of these is to get a diagram so you can look at the little skinny little rectangles parallel to the surface of the water Center of mass that'll probably involve a little bit of memorization on your part What's the center of mass in terms of the x value the centroid? What's the center of mass as far as the y value? We did not do six six or six seven So we then went to chapter seven This is kind of an odd section in a sense to find a test question from Because this was just introductory. What are these things called slope fields? If we have a differential equation and we're handed a solution can we validate a solution? What else did we have in seven one? They talked about Logistic growth, but we didn't do anything with it really other than look at the differential equation They talked about spring problems. We didn't really solve any Motion of spring problems. They had a second derivative in them. So that was a Second-order differential equation talked about initial value problems and how that enters in so Just about everything that was introduced in seven point one we looked at Later or we will look at as we continue kind of outside the book in the using second-order differential equations slope fields again a little more specifically and Approximating solutions using Euler's method So you might want to take a look at that before coming to class tomorrow. We can talk about How it is we would generate a slope field I probably will not Time-wise be asking you to generate a slope field, but maybe at a couple of points What would the slope look like the tiny little line segment that kind of serves as a little guide post or sign post What would a couple of them look like? What would I think there's a web of sign question on what would the nature of solutions look like? In a certain slope field if the y of zero was Four or if the y of zero was negative three Then you have one that was a picture and you had to picture what different solutions would be like that went through different points in the plane Take a look back at that web of sign. I'm pretty sure there's one of those on the web of sign and Euler's method Euler's method we're going to take the tangent line to the curve As long as it's the delta x is fairly small and we're going to go along the tangent line to the curve and find the next point Even though the tangent line is not exactly on The curve itself we're going to use the tangent and we're going to have The differential equation given to us so that is the slope so we are then able to find A description of the slope in terms of x's and y's and numbers So we would start with an x zero That's going to be given to us and a y zero and then it's our task to find X one that's usually the easier part. I hope it is How do you find the new x value in terms of the old x value? Old x Plus the change in x to get the new y we would start with the old y And then what kind of arithmetic do we do? Right so we would have this Differential equation right and if it's in terms of x's and y's and numbers we'll plug in x zero and y zero because we're trying to generate x one and y one we use the predecessor Times delta x the change in x along the tangent to the curve and the change in x Along the curve itself are exactly the same so it's kind of old y plus the change in y along the tangent to the curve and Then that becomes our data to generate our next one right x2 found the same way Y2 found in a similar fashion so Euler's method be prepared on the test to have an Euler's method problem from 7.3 You can help me prioritize these things tomorrow that if we need to look at a work problem We can set one up Or if we don't then we need to do a spring problem If not that then maybe a hydrostatic pressure problem. We're not going to have time to do all these 7.3 was loaded with different things just the concept of separable differential equations we've done pretty much every type That you could possibly do in some way shape or form with examples from 7.3 and 7.4 and also 7.5 We did some circuit problems More than likely at this point in time I know we'll revisit circuit problems this point in time I would probably provide what it is that you need as far as the the laws that govern Voltage drops and so on Orthogonal trajectories let's talk for a few seconds about what are those orthogonal trajectories? That was a long time ago. I don't I have a short memory. I'm an old man Tell me what orthogonal trajectories are. I'm an old man that painted all weekend I'm so glad to be back here and have the paintbrush at home Okay So curves that intersect other curves and they are mutually perpendicular right actually the curves themselves It's kind of difficult for them to be forming right angles But the tangent lines at the points of intersection are perpendicular So we've got a curve here and another curve here. So actually this tangent line And this tangent line are Perpendicular so if we know the slope of one set of curves What do we do to generate? This new family of curves that serve as the orthogonal trajectories. What do we do with the original slope? Take the negative reciprocal right and then use the separable differential equation technique and Kind of get out from under the differential equation and what is the family of curves? For which their derivatives are negative reciprocals of the original set of derivatives in the fourth type would be the tank problems Just In case you were thinking you know cash this test is loaded with stuff He might not put a tank problem in that will not happen This I've never given a test over this material and not included a tank problem And if you want to make a note for yourself That this will also be a problem for the final exam. There will be a tank problem on the final exam as well So it's a I think it's one of the major types of problems that we do over this material And it in an actually encompasses quite a bit Pardon the salt thing. Yeah the salt the brine the pesticide I don't know what else we did the room with the co2 all those are tank problems fourth section Regular exponential growth and decay So that's kind of the standard version. You need to know that Version coming into the test and that's going to work for both growth and decay if it's growth K is positive And if it's decay K is negative. So what did we do here? we did Population did we do a population growth? I can't remember the example we did here Population of Raleigh, maybe oh you did my little town. Oh, that's right. We did your town, which was moon, right? Which brought to mind? I don't know if I mentioned it that day my college roommate went to moon high school Did I bring that up? That's bizarre, but it's in Pennsylvania moon township the moon tigers So it wasn't the first moon that I've heard of we also did a decay. We did the Carbon dating problem where the C-14 was decaying Half-life of five thousand seven hundred fifty years and we determined the Linen wrapping in the Dead Sea Scrolls how old they were approximately Let's see So that's regular population growth And decay we did Newton's law of cooling I think that's probably fair if I ask you to know that coming in because it's very similar to just regular Exponential growth, how does that look different now? I don't know if this book has negative K here You know really need the negative K K turns out to be negative right if it's cooling so I'll just put K K is a negative number It's got a little extra baggage tacked on to the end surrounding medium temperature of the surrounding medium So it looks very similar to just exponential decay With this guy added on to the end We also did continuously compounded interest which really kind of comes into this model. It's the same exact model From the logistic model Good so tomorrow we can just do problems from an old test and kind of other problems that you want us to do and What does that look like? It looks like the limiting value in the numerator Again, you don't have to have the negative KT. It turns out that K is negative So whether you have the negative sign or not there. It's going to be negative anyway So it's a number in that exponent position There is some if you're looking through seven five they do Euler's method again. We've already done Euler's method We've got a an exact solution to this so we don't need to approximate it I Told you you're not responsible for the development of this And you're also not responsible for that harvesting model We looked at and the extinction model, but you can change Existing models to kind of make it Make it what you want it to be based on the extinction or the harvesting portion of that model Okay, we're in good shape to just do kind of problems all day tomorrow and Wednesday in here will take the test So if you have problems that you want to make sure we do You need to bring those in Or at least the type that you want to do and we'll find one