Applied Calculus for Engineers - Part 1: Single, Multivariable & Trig. Functions, Limits,Derivatives | Mark Misin | Skillshare

Applied Calculus for Engineers - Part 1: Single, Multivariable & Trig. Functions, Limits,Derivatives

Mark Misin, Aerospace & Robotics Engineer

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83 Lessons (11h 33m) View My Notes
    • 1. Promo Video

    • 2. Understand FUNCTIONS 1

    • 3. Understand FUNCTIONS 2

    • 4. SIM: Straight line functions in motion

    • 5. Python installation instructions - Ubuntu

    • 6. Python installation instructions - Windows 10

    • 7. AIRPLANE Application

    • 8. Abstract World VS Real World - 1

    • 9. Abstract World VS Real World - 2

    • 10. Manipulate STRAIGHT LINE FUNCTIONS

    • 11. Understand & find INVERSE FUNCTIONS

    • 12. Find INVERSE FUNCTIONS - example

    • 13. Manipulate PARABOLIC FUNCTIONS

    • 14. Manipulate CUBIC FUNCTIONS

    • 15. SIM: Car race exercise - who will win the race?

    • 16. SIM: Car race exercise - who will win the race - SOLUTION

    • 17. Functions VS Non-functions 1

    • 18. Functions VS Non-functions 2

    • 19. Find INVERSE of a PARABOLA


    • 21. Generalize FUNCTION MANIPULATION

    • 22. SIM: General functions in motion


    • 24. Learn to shift FUNCTIONS

    • 25. Learn about FUNCTION OPERATION

    • 26. Practice FUNCTION OPERATIONS



    • 29. Which terms DOMINATE

    • 30. SIM: 3 airplane race - who wins, and when?



    • 33. Origins of SINE FUNCTION




    • 37. AEROSPACE Example

    • 38. SIM: Avoid the crash - exercise

    • 39. SIM: Avoid the crash - solution 1

    • 40. SIM: Avoid the crash - solution 2

    • 41. SIM: Avoid the crash - solution 3

    • 42. SIM: Avoid the crash - solution 4

    • 43. SIM: Avoid the crash - solution 5

    • 44. SIM: Avoid the crash - solution 6

    • 45. SIM: Avoid the crash - solution 7

    • 46. SIM: Avoid the crash - solution 8

    • 47. SIM: Avoid the crash - solution 9

    • 48. SIM: Avoid the crash - solution 10

    • 49. 2 variable AIRPLANE example

    • 50. 3 variable AIRPLANE example

    • 51. Functions VS non-functions (MULTIVARIABLE)

    • 52. Equaling VS Approaching something

    • 53. LIMITS - Approaching Infinity

    • 54. SMART Financial Decision with LIMITS


    • 56. Model a WAVE

    • 57. From SINE to COSINE

    • 58. Practice LIMITS on Trig. Functions

    • 59. SIM: Limit simulation

    • 60. SIM: sin_limit simulation

    • 61. DERIVATIVES: Falling sphere 1

    • 62. DERIVATIVES: Falling sphere 2

    • 63. DERIVATIVES: Falling sphere 3

    • 64. SIM: Earth, Mars, Moon - exercise

    • 65. SIM: Earth, Mars, Moon - solution

    • 66. DERIVATIVES: Multidimensional point of view


    • 68. Chain Rule - Water Pipe Example 1

    • 69. Chain Rule - Water Pipe Example 2

    • 70. Differential Operators

    • 71. Multilevel Derivatives

    • 72. Derivation Rules 1

    • 73. Derivation Rules 2

    • 74. Implicit VS Explicit functions

    • 75. Derivation Rules - BIG exercise

    • 76. Trigonometric function derivatives 1

    • 77. Trigonometric function derivatives 2

    • 78. Minimize material used for can manufacturing 1

    • 79. Minimize material used for can manufacturing 2

    • 80. Minimize material used for can manufacturing 3

    • 81. Exponential Functions

    • 82. Logarithms, Log scale, and its Derivatives

    • 83. Maximize your profit


About This Class

Dear Students,

Welcome to the course! I am very happy to transfer my knowledge to you. This is Part 1 of the course (out of 3 parts + EXTRA).

WHY should you take this course? Fair question!

In Science & Engineering, literally EVERYTHING is based on Calculus. From Mechanical & Aerospace to Electrical & Computer to ARTIFICIAL INTELLIGENCE & MACHINE LEARNING and many more.

If you REALLY understand Calculus and you are trained to APPLY it to REAL LIFE problems, learning and understanding more advanced material will be much EASIER. Your life will be much EASIER!

In my course, I WILL DELIVER just that. I WILL make you understand Calculus INTUITIVELY & in terms of APPLICATION. Just give it a shot!

These are the topics that the course covers:

  1. Single variable functions

  2. Trigonometric functions

  3. Multidimensional functions

  4. Limits

  5. Derivatives - single variable

I wish you good luck in learning!



1. Promo Video: Welcome to the applied calculus course. My name is Mark, and I want to help you master calculus. After this course, you will walk away with strong intuition and understanding of calculus. And you will be trained to apply it in real life to the level not seen in other courses. You're going to learn about functions, including trigonometric functions, limits, and derivatives. And in addition to that, I'm going to create this very important bridge between theory and real-world applications. In addition, I have created all kinds of simulations in Python to backup my explanations. They will truly give you an amazing visual added value. No other course in calculus does that. This is really powerful stuff. My promise to you is this. If you give me a chance to teach Calculus, I will make it second nature to you. It will be a great investment. Take a look at some of my free videos, and if you like what you see, enroll in the course and let's get started. And looking forward to seeing you there. 2. Understand FUNCTIONS 1: hi there. And thank you for joining me for this course. We're going to start covering the most basic concept that you absolutely need to understand . And master, we're going to cover functions. I'm going to teach you what functions are what they're useful for, how to manipulate with them and how to apply them into real life. So the following lectures air going to be very interesting. I promise you that. However, before we start, I want to very quickly talk about how I'm going to be doing my lectures. So over the course of my lectures, I'm gonna be giving you certain small thought exercises. And then I'm gonna ask you to pause the video and try to solve them yourself. And then when you own post the video, I will solve them for you. Why do I want to do that? Well, I have been studying engineering for a long time, and from my experience, the best way to learn a concept is to first try a problem yourself and then look at the solutions because one day in the future, in real life, when you have to solve free life problems, you're not gonna have those solutions. So you need to have this ability. You know this this skill to to approach a problem for which you don't have solutions. And in order to train that skill, the best way to do that is to take a problem for which you do have solutions. And then you first give it a thought. You tried yourself and if you can do it or or you don't know how to do it, no problem. It's important that you give it a shot. And then and then you look at the solutions and you look at how other people did. Okay, let's get started now. In order to demonstrate the usefulness of functions, we're going to start with a little example. Consider a number line here that is for distance, and we measured distance in kilometers. And we also have time definitely measuring hours. And this example is about an airplane. This flies in this direction. You also know that the speed of the airplane he's 800 kilometers per hour and the speed is constant. What does it mean when I say constant? It means that it never changes. It's constant. It doesn't change. So here's your first exercise, and I would really recommend you to pose because because really, in this way you're gonna learn a lot more. In this way, the lectures air a lot more engaging, and you will think with me more. So it's really a great way to practice that skill to solve problems. The first, give it a shot by yourself and then look at the solutions. So the first exercise is so you know that at the beginning, at time equals zero hours the distance zero. What would be the airplanes? Distance at time equals 0.5 hours and at time equals one hour. At times it was 1.5 hours and the time equals two hours. So what would be the distance? Knowing the speed, which is constant. All right, see you soon. The best way to sold this exercise without memorizing any formulas is to look at the units and work with the units. So you have speed, which is 800 kilometers per hour. That means that in one hour, the airplane carver's 800 kilometers so you can simply write it like this. You have 100 kilometers in one hour, and then you can just multiply this ratio to your time. So, for instance, when time because 0.5, it would be something like this. 800 kilometers in one hour times 0.5 hours. You can also see from the units you cancel the units, right? Because these air time units hours is there time units and you're left with kilometers, which is that this which is a unit for distance. So in this case, it would be 400 kilometers and you kilometers, and you can put it here and here you would multiply 800 times one which is 800. In this case, it would be 1200 and here 800 kilometers in one hour times two hours you would have 1000 600 kilometers. So in two hours, the airplane will cover 1600 kilometers. But let's face it, drawing like that can be a little bit tedious. Right? So is there a better way, a more convenient and compact way to, let's say, to represent this information this time and distance information. Is there a better way to do it? Well, we could try to graphic, right. We could try to draw to access here. And I can put time here, and I can put distance here, and, uh, and then I could just say something like, Okay. Is Europe in 51? Okay, it's not to scale, though. Two. And then here you have the distance numbers, which is 400 800 kilometers, 1200 and 1600. So here's your second exercise. So pulls the video and, uh, try to graft this information. Tried to graph the graph. Tried to create the graph where you have the relationship between time and distance. All right, See? You know, well, I can very easily create this graph by, uh, by taking the information that I know, so I can. But time here I can put distance here, can create this little table, which would be, for the time zero 0.5 11.5. And to and then for the distance, I would have zero 400 800 1000 200 and 1600 in order to graph it. Well, I'll just first put the points on the graph. So for 00 it's here. 0.5 400. I put it here for one. I put it here 1.5. I put the points here and for two. I put the points here, so as you can see, it's a straight line. So there you have it. We have taken this information here and we have compacted into a graph. We can see the relationship between distance and time as a graph like that. But let me ask you a question. Now what if you want to know the distance of the airplane at a time equals 1.375 hours and let's assume that you don't know the speed. You don't know how fast the airplane goes. You only have information from this picture and you don't know the speed. So how would you find the distance? Just using this information? What one way would be to use the graph, right? You could go very precise, and you could find 1.375 here and then go to the graph and see the distance. But I think that you I think I think you would agree with me, that it would be a very painful process. So is there another way a more compact way to represent this information about the airplane and is there a way where we can just take whatever time we want and we put it in and we get the distance out? Well, how about we try to create a mathematical relationship? Let's try it. 3. Understand FUNCTIONS 2: in order to create a mathematical relationship between distance and time, we first need to understand how distance changes with respect the time. And for that I'm gonna introduce a new mathematical symbol called Delta and belting. Mathematics means nothing else but change. And I'm gonna give you a small example to illustrate two different scenarios. So the the formula for Delta is new situation miners, old situation, whatever it might be. So, for example, you have cards on the table, and in the first scenario, you first have three cars in the table and then you have one card on the table. When you have three cars in the table, that's the old situation. And then sometimes sometime later you have one card on the table. So dealt the cards or in English. The change of cards equals new miners, old one, which is a new situation minus three, which is all situation and the changes minus two. The changes negative to its negative because the new situation is smaller than the old situation. But on the contrary, in the second scenario, you have 1st 1 card on the table and then you have three cards on the table so the change of cars would be new miners old. The new situation is three miners, one which is all situation. So the change now it's positive and it's positive for two. It's plus two and thats so that symbol is going to be very important for this example and and in fact it's very important. Mathematics, the change of variables. So I have drawn the graph off time and distance here again, and in order to understand how how distance changes with respect to time, we would write it down like this. So Delta D Forward, Delta T and what that means. It means the change of distance for one unit change in time. So in our case, it would be how much the distance changes when the time changes one hour. But remember, and this is very important. So I really want to emphasize that is the change of distance for one unit change in time Onley. In our case, it's ours. If we had measured time in minutes, it would have bean the change of distance for one change in minute and obviously then the change of distance. This ratio would be a lot smaller because in one minute the distance would change a lot less. Dan in one hour. So in this example will stick with ours and to get the distance, what we can do, we can, for example, take the change of distance over the interval or over over the period off two hours. So let's see here for the distance. That whole situation is zero kilometers, but the new situation is 1600 kilometers. So I write 1600 which is new minus zero, which is old. And for the time again, the new situation is too two hours. The old situation is zero hours and we get 1600 over to equals 800 kilometers per hour. So in other words, the airplane covers went on 600 kilometers in two hours, or it's equivalent to 800 kilometers in one hour. And because the change here is constant, we can also take other intervals. We can, for instance, take this small interval here so we can have 1600 minus 1000 200 which is now the old situation and what that 600 would be, then use it. It would be the new situation, and for the time it would be too minus 1.5. So essentially, I'm doing the same thing. But I'm taking a different interval, and Aiken do that because in this case the change is constant and you can see from the straight line. If there are, If the line was something else, like a proble, then it wouldn't work, and we're gonna cover them or in derivatives. But in this case it would be 400 divided by 0.5, which is again 800 kilometers through our. And there you go. You have their relationship. You have the change off distance with respect the time. And like I said, what it means is the change of its the change of distance for one for one change unit of time. And it's very useful because now you can just take time. You can just take your time, and you can multiply it by this ratio and you get your distance. For example, in our case, we wanted to know when time equals 1.375 was the distance Well, what we can do is distance equals 800 which is our ratio here times 1.375 and that would give us 1100 kilometers. And this is your new function. In fact, this is your first function. It's D equals Delta D or dealt the tea times t. So, like you can see from this example, it's not a function is nothing else but the mathematical relationship between two variables . In our case, it's distance and time. But let me ask you a question. Do you remember when at the at the beginning of the lecture I asked you to do this little exercise where you had to calculate the distance and you know the speed of the airplane and you just had to multiplied by time? Well, as you can see from here, Delta D over Delta T, which is the change of distance with respect to time, it's nothing else but speed, right? It's kilometers per hour because that's what speed is. Speed is at a unit of distance in this case, kilometers. How much it changes per one unit of time in our cases one hour. So why did I Why did I go such a Why did I go through such a lengthy process in order to achieve the same result. Well, that's because I wanted you to see it as a mathematician because you don't only need to have times and distances. Let's say that you have ah, heater at home, right? And that heater gets power from electricity. And here you have power. And let's say it gives out temperature in the room. So this is your temperature in the room so you can have some kind of relationship between power and temperature. Yeah, temperature. And in this case, you would follow the same procedure. Now it would have nothing to do with speed. Then, however, the mathematical logic behind it is the same. It's Delta simp over per one unit Change in power, says Delta Temperature with respect to power, the change of temperature with respect to power so you can have as many variables as you want. And the final thing is, I want to make it a little bit more abstract because at school they don't even normally talk about specific variables. They just represent functions like this. Why equals Delta? Why, over Delta, X Times X or sometimes are actually many times they just say why times M Times X and they denote the change with an M and another notation is f as a function of X equals and times X , All right. 4. SIM: Straight line functions in motion: Welcome back. My goal in this course is to give you strong intuition and application skills in calculus. That's what I want for you. Therefore, I've created a series of Python simulations like this one to achieve this goal. These simulations will reinforce what I teach you on paper. And I believe that you will get enormous added value out of them. So with that being said, let's get started. What you're seeing here is the same airplane that we talked about before. And as discussed before, one way to represent this information of what's going on is like this. So you have an airplane flying. And you can see these big dots here. And these big dots, they appear every six minutes. So there are 20 dots here. So if you multiply them six by 20, then you will get two hours, which is our timeline. But of course, representing the information like that all the time can be quite tedious. So we are using functions. This is a graphical way of representing a function. And this is a mathematical way to represent the function. So here on the left side, as you can see, as time progresses, then the airplane covers distance. It covers the distance steadily, right? The rate of change of the distance covered with respect to time is constant and it's always 800 kilometers per hour. That means that if for example, I'm here and I've covered 400 kilometers in 0.5. hours. 400 divided by 0.5. I will get 800 kilometers per hour. Here I've called 800 kilometers in one hour. 800 kilometers per hour, one hundred, two hundred kilometers in 1.5 hours, 800 kilometers per hour. In fact, here you can see that as I go along the timeline, whatever point I take, whatever distance I've covered. If I divide it by the time covered or time spent, then I will always get 800 kilometers per hour. And you can also depict this rate of change with respect to time or slope as a function. So this is a graphical way. And because the rate of change is constant, you will have nothing but a horizontal line here. And a mathematical way to express that is very simple. Delta distance over delta time equals 800. All right, this is it for now. And I'll see you in the next lecture. Thank you very much. 5. Python installation instructions - Ubuntu: Welcome back. In this video, I would like to show you how to install Python and the necessary libraries in order for you to be able to run the simulations in Python are first start with Linux, Ubuntu. And then in the next video I'm going to do it in Windows. At the time of this recording. The Python version that you should install is Python 3.8.5. The newest out there is 3.9. However, I've checked that and the libraries such as numpy and matplotlib, they don't work well with Python 3.9 yet. I guess it's two new, perhaps they will later, but not at the time of this recording. So you should have Python 3.8.5. If you have Ubuntu 20.04, like I have it here, then Python 3.8.5 should come by default along with num pi. You can check the version in terminal like this, Python three and then two dashes and then version. So you see I have python 3.8.5 and you can check the version of Numpy like this, python three dash C. Then you put here a quotation mark, import, numpy, semicolon print. And then you're going to put here Num Py dot. Then you have to underlines here. And then you're gonna put version, then another two underlines, and then another quotation mark here. So here I have num pi 1.17.4. That's what I received by default. These two things I received by default when I installed my Ubuntu 20.04. If you don't have Python 3.8, then you can install it with these commands, sudo opt update. And you put your password here. So it will give you some upgrades. And then you will write sudo, which is superuser. Then OPT install software, dash properties, dash common. Then I'm going to write pseudo, add, dash, APT repository, PPA, colon, dead snakes than slash and PPA. So I'm going to press Enter and then I'm gonna write pseudo OPT and then update the pseudo APT. Install Python 3.8, not 3.93.8. And then to check the version, you write Python three dash, dash version. And then if you don't have num pi, you can add it like this. Pseudo APT install bison three dash NumPy. And so to check the version, you will have Python three, dash c, then quotation mark, import, numpy, print, parenthesis, num pi dot, then 200 lines, version 200 lines. And then you will have a quotation mark here. So that's your NumPy version here. And now you have to get the matplotlib library as well. At the time of this recording, the newest version is matplotlib 3.3.3. However, if you want to run all my simulations, then I recommend you to install matplotlib 3.2.2. You can do it with these commands. Sudo APT update. Then sudo APT install Python three dash PIP. Pip is a Python package installer. So you check the version of your PIP. So PIP for Python three would be PIP3, then dash, dash, and then version. So here it is. And now you write PIP3 install matplotlib. And now if I press Enter now then I should be able to get matplotlib 3.3.3. However, I want to get matplotlib 3.2.2. So I'm gonna put here two equal signs and then 3.2.2. And if you already have matplotlib 3.3.3, then this latest command will uninstall it automatically and replace it with 3.2.2. And now you can run the code, but this is very important. You have to run it in Linux terminal or Windows command prompt. Do not use IDEs like spider or Jupiter notebook. The animation does not want to work well in there. The most robust way is to run it from terminal or from Windows command prompt. So I have a file here, calculus sim, PID, train dot py, and I want to run it. So in my terminal I will go to my desktop. So change directory or CD Desktop enter. I can see all my stuff here with an ls command. And so in order to run this file, I'm going to write Python three and then Calculus sim PID trained up pie. And there you go, it works. So that has been in Linux. And now let's see how to do this in Windows. 6. Python installation instructions - Windows 10: Welcome back. So let us now try to install Python and its libraries for Windows. So this is the Python website here. The newest version is 3.9. However, the problem might be that at the time of this recording, the numpy and matplotlib libraries might not be ready for this version. At least it didn't work for me. So in order to run the simulations, we're going to take an earlier release, Python 3.8. And so here you have Python 3.8.6 at the time of this recording. So if you have a 64-bit machine, then you need to choose a 64 bit one. And if you have 32-bits, then you choose the 32 bit options. So we're going to choose the 64 bit web-based installer. And now we're going to open the file. What we're gonna do now we're going to add Python 3.8 to pass. I think it's important so that you could run your Python codes from whatever place in your computer. And now let's customize the installation. I personally don't need that documentation file, but you can leave it if you want. The next. Then we're gonna put here install for all users. And now, do you want to allow this app to make change to your device? Yes. And it's installing. And so we have finished installing Python and we're going to close it now. And so now we have our Python and let's check if it works. In our case, we will need to launch our programs from command prompt. So I want to test it from command prompts. The animation does not want to work well in IDEs like spider or Jupiter notebook. The most robust way is to run it from terminal or from Windows command prompt. So I've opened a notepad and let's see if I can write a program here. So I'm going to write here print and then hello world. And now I'm just going to save this somewhere. And I'm gonna save it like this. Hello dot py. So here I'm going to put all files and And there you go. Now I want my command prompt. Here it is. So now I need to find my way to desktop. So cd Desktop. Now I'm here. Ls. Well, unless is not working because it's a Ubuntu thing. But I think in Windows it's dear. And you see you have your hello dot pi here. So can we run it? We can. And so it works. Ok, but to run our simulations, we need a couple of libraries. We need numpy and we need matplotlib. When we installed our Python, then you can see that our PIP here was also included or PIP. So that's Python package installer. And in order to install num pi, we need to make sure that this PIP is the newest one. So we're gonna write here Python dash M, PIP install, dash, dash, upgrade PIP. Ok, so it now says successfully installed PIP 20.3. And now let's install Numpy. I'm going to write PIP install Numpy. Then two equals signs, 1.19.3. There is another version, 1.19.4. However, it didn't seem to work with Python 3.8. So I had 1.19.4 before and it didn't work. Now it got uninstalled. And now I have NumPy 1.19.3. Now let's check if it works. I'm going to read here Python. And then I'm going to write here Import num pi. It didn't give me an error, which is very good. So this is our code here. Import numpy as np, and then I'm going to write here print np times pi. So the pi value. And so now I'm going to write exit. Then I'm going to run this file and it gives me my PI value. So that's very good because it means that we also have working Numpy here. And now we need to install matplotlib. I would recommend you to install the matplotlib version 3.2.2. At the time of this recording, the newest version is matplotlib 3.3.3. However, if you want to run all my simulations, then I recommend you to install matplotlib 3.2.2. So I'm going to write here PIP install, and then matplotlib. Now if I run it now then I will most probably get the newest version, 3.3.3. But since I want to get 3.2.2, then I will put here two equal signs and then 3.2.2. Okay, so it says that successfully installed. So I have here this calculus simple PID train, and let's run it here. So calculus seemed PID train and it works. Now, it's a little bit slow here, but that's because I'm running my Windows ten virtual machine. At the moment, the IOM Linux user. So I use Ubuntu. And so in order to make this demonstration, I downloaded the Virtual Box machine. And that's why the simulation is a little bit slow. I'm sure it's going to be faster on your Windows machine. 7. AIRPLANE Application: you know the saying, Practice makes perfect. It's less practice a little bit. I propose an exercise here you have a car that go that goes in this direction and atomic ALS. Zero hours. The car has already traveled 300 kilometers, so try to create a time distance graph. Then try to answer this question. What is the change of distance with respect to time? What a speed. What is the change of distance with respect to time when time is measured in minutes? So the change of distance for one minute. And then what is the mathematical relationship of this entire situation in minutes? And what's the function in minutes? Ceasar. So let's try to solve it and see what we get. So in order to create the graph, we would first but the points here. So we have time and distance, and I will just create this little table and we have 03 six, nine and 12 and for the distance is 300. Because the car has already traveled 300 kilometers 600 here, 900 here, 1200 in 1500 so that the graph would be something like this time distance and then notice that that a time equals zero. You already have traveled 300 kilometers now. 369 12 369 12. Here you would have 300 600 900 1000 200 and 1500. Okay, so you have one point here than 363 and 600 This six and 900 this nine and 1200 this. And 12 and 1500 Kates. It's a straight line again, but notice in this case, it's not so precise. But look, in this case, this line function has shifted up. And that's because at zero hours you have already traveled 300 kilometers. Okay, so what would be the change of distance with respect the time And this is the case with ours, so I can just look at the change. So now I have to do it from here, not from here, but from here, because in this area there is no change. You want to measure the change. So let's say let's see. So in this case, the Delta D would be new miners old 1500 minus 300 which is 1200 kilometers and it happened over the interval off 12 hours. So Delta D would be 12 which is the new situation miners old, which is in you which is the old situation. So it's 12 hours and of so delta these 1200 over 12 which would be 100 kilometers for our okay and what is speed? Well, this is the speed, right? It's is the change of distance. We would respect the time kilometers per hour. Kilometers is distance ours time. So it's it's Ah, it is the speed that the car travels 100 kilometers per hour. That's the speed. OK, but let's look at the minutes now. In order to do that, we need to know how much How many minutes is 12 hours to get the minutes? We would simply say that OK, there 60 minutes in one hour and we have 12 hours. And in fact, if you see this is another function, this is minutes. This is a function between minutes and hours, so you can even have a graph that says, OK, I have hours here and I have minutes here, and the function would be minutes equals 60 times hours and this 60 would be the change of minutes with respect to with respect to ours, right, Delta minutes over Delta ours. So you see, their relationship between minutes and hours is also mathematical function. But okay, in this case, we would have 720 minutes. That's because 60 times 12 with 700 20. So the change of distance with respect to time if we measured in minutes, would be Delta D Delta. T minutes would be 1200 over 720 which is approximately 1.667 kilometers for one minute and notice. Now it's a lot less. Before, in one hour, the car travelled 100 kilometers, so the change was 100. And now you, one minute the car has traveled 1.667 kilometers, So it's logical, and you can see it from here that the changes a lot less. But it's still speed. All right, Still speed. Now the final thing is, is there a mathematical relationship? How would it be? Well, it would be D, which is the distance, and it's in my minutes by the way. So it would be Delta de or Delta T men Times time, which is also in minutes. Now. Don't forget. And there is a constant, which is plus 300 because remember, the graph was shifted up. And so the mathematical relationship would be this. And instead of Delta, the over Delta T, you would have 1.6 67 And, of course, mathematical relationship is the function. So it's also our function. Let me just write it down here as a formal thing. 1.667 times t in minutes plus 300. And that's your function. All right, so this concludes our first lecture. This lecture was more like a high school review. And starting from the next lecture, we're gonna dive deeper into calculus. We're gonna explore relationships between practical examples and the abstract mathematical world. We will also look at at more complicated functions. I will teach you how to manipulate them, and I think it's gonna be fun. So join me. See you later. 8. Abstract World VS Real World - 1: Welcome back. This lecture is called ab Surfers Israel. It's not a long lecture. However, I really wanted to include it because remember, this is not a calculus course. I simply want to give you some concepts from calculus that you would need in order to succeed in dynamics. So in this lecture, I'm gonna try to make some connections in between realistic examples and the abstract mathematical world, because at some point we're gonna be working with abstract functions. But I don't want it to be very absolute, very abstract to you. So I'm I'm gonna illustrate how people non mathematicians such as engineers and scientists , how they use mathematics, mathematics and calculus Hey, in their own realistic examples. So in our last lecture, we have this airplane example where we had a relationship between time and distance. That function illustrated how distance depends on time. How those two variables, how they relate to each other. However, is it the only way how distance can depend on time? Of course not. There are an infinite amount of ways how one variable can depend on another variable. For example, the distance off the airplane can dependent time like this distance equals one plus time squared. So posted video now and just graph how distance depends in time. How would the graph look like so you can take time? You can take time and distance, and for the time you can take zero 0.5 one 1.5 and to and just see what you get. So this what I got for the distances these air the valleys here, and I calculated them like this, and I graphed the function like this. And as you can see, it's not a straight line for relationship anymore. So you could have You could calculate the speed from this graph. But speed is not constant anymore, because the change here is it is like this, and here it's It's like this. In fact, here the speed he's higher, but it's a parabolic function graft from 0 to 2, and you can see that distance can depend on T on time. In many ways, however, that's not all. You don't only need to have distances and times like we mentioned in the previous lecture. We can have completely different variables. So this is another example where I want to apply functions. Consider that you have a tank here and you have a hose and water comes from the host and it is poured into the tank and we are interested in finding the mass of the tank. Now, the mass of the tank consists off the empty massive the tank when there is no water inside and also depends on how much water you have inside the tank. So the way we would use functions here, we would say that the told mass of the tank will equal the empty mass, plus the change off mass off the tank with respect to with respect to the volume, off water and from physics, we know that one cubic meter, no one cubic meter off water is 1000 kilograms. So the so the way to look at it is that the change, the mass off the time changes by 1000 kilograms. If you add one more cubic meter off water. So this function would be then, like this, the empty mass mass E. It's anti mass, plus 1000 times the volume off water, and you would graph it something like this. You have the volume off water and since you already have an empty mess. You don't start from zero. You start from here and then you would have some kind of straight line and that's straight line. If you were to calculate the the distances and the changes off of the total mass off the tank, then you would get that you would get that it would be 1000 kilograms per one cubic meters , so it would be something like this. Built a mass and dealt up volume water. So if you divide, if you make this ratio delta mass over delta volume, that would be 1000. You see another function 1000 kilograms for one volume for one cubic meter. So this is another function applied to a real life situation. But in mathematics, usually what they do in this case, they write it like this. Why equals M X plus B. And in this case, the graph would extend and be something like this, and it would go forever 9. Abstract World VS Real World - 2: So this is what mathematics does. It does not care about specific situations. It's signs off numbers and variables and how they're related Cheddar. It explores the relationships between them and creates tools that other engineers and scientists can use for their own specific situations. So, for example, in our airplane example, in the last lecture, we we would take this leaner function, which is this one, and we would apply it to that specific example. On the other hand, if you're a civil engineer and your task is to design a tank, then then you would apply this dysfunction to your specific situation. But you would only do that if you believe that this function accurately represents the physical situation. So in other words, it has to at least very closely approximate to what's really going on. So you can't have something like this that, let's say, the airplane. So again, the time distance graph that the airplane ah, that the relationship between time and distance is this. But you choose dysfunction so it doesn't make sense. So once you choose a tool from mathematics for your situation, it needs to represent the physical situation. Now, of course, it never represents the physical situation 100% because you can have a situation like this . Let me just show it to you very quickly. So let's say you have time and distance. And the real physical situation is something like this. So completely a straight line, then you you would have a straight line here, and there is a small error right between the physical situation and, uh, and your function. But then the engineer might think something like this. Okay, sure. Fair enough. It doesn't completely represent my physical situation, but should I should I create Should I choose a function that is super complicated in order to extremely precisely represent my function? Or I can just take an easy function that is easy to work with. And since the error is so small, it simply doesn't matter. And in many cases they're just gonna say, Yeah, I'm too lazy, and there's so small, I'm just gonna take the the easy way, which you should do, because you would save a lot of time by doing that, which is also a big issue in the world of engineering. In fact, you can have really bizarre functions that all existing mathematics, but never in the physical world. For example, we have this function. Why equals one over X if I graph it and I put X here and why? Here is the positives of the positive sides and these air the negative sides. So out my graph was look, something like this and these lines they would go to infinity so they would go to plus infinity here and minus infinity here, plus infinity here and mine is infinite to here. No, I don't know what in real world is infinity. Maybe you can approximate something with infinity. If something is very big, let's say universe is very big and then you can approximate it with a function with a function that goes to infinity. But an even more bizarre concept is minus infinity. I don't know what can be minus infinity in real life, but again, mathematics doesn't care about it. And it's not his job. It just investigates how variables relate to each other. And then you as an engineer or a scientist you choose, and you have to be skillful enough to choose what you take from this immense back off knowledge. So there again, the goal of this lecture was to create some kind of bridge between realistic examples and the abstract world. So in the next lecture, we're going to start manipulating those functions, and we're gonna b'more abstract, just working with wise and excess. But remember, always remember, they just represent relationships and and if you feel that they're too abstract, just try to think of an example all those functions that can be used for specific situations. I hope you'll stick around, see a 10. Manipulate STRAIGHT LINE FUNCTIONS: welcome back. So in this lecture, we're gonna generalize the topic of functions. We're gonna talk about different, different kinds of functions, and I'm also going to show you all the different ways you can manipulate and move around the functions. The main function that we have been using so far in our previous lectures has been the straight line function. And we could write it down like this. Why equals I m times X plus B. And we know that if B equals zero, then the graph of the function was something like this. We have x here. Why here And then we had some kind off straight line. Remember this theater, the positive size and these air the negative sides and m represented the change. How much the change of how much? Why change with respect to X when exchanges one unit. So if M is positive, then then this line is like this. However, what happens? What happens when and becomes negative? Well, if m becomes negative and I can say, for example, that M could be minus one In that case, how would the line be? Well, let's see if I have, for example, if I have five here at first, whatever quantity we're dealing with first, that quantity First X is three, and then it's five. But I mean, but in case of why first wise three and minus three and then it's minus five. So what would be the change of why? Well, it would be the new situation minus the old situation, right? So it would be minus five minus minus three, and it would be two minuses, G plush says minus five plus three. Would be miners to correct and built. The axe would be human is old five minus three equals two. So I m would be then minus 2/2. Would you mind is warm And in that case, the line would be like this. And it makes sense because now the change changes like this. So it's Ah, the new situation is smaller than the the old situation for the why and for the X. The new situation is still bigger than for. The new situation is still bigger than the old situation. All right, And that's why the line is negative now. And what would be a real life example for this kind of function? Well, what I can't think off is something like this. For example, if I have a lever here, right, and this is the pivot point so the lever can can rotate around this pivot point. And let's say it's connected here with the spring and spring here in In this position, it's in neutral position. It's not stretched in its not compressed. And let's say we measure distance in this way. So in this way, distance would be negative because it would go down right? But right now the distance is zero. So what happens if I put some kind of box that has a mass than this lever? It would go like this, right? And then you would have spring going like this, and then you would have some kind of distance, and that's how your function could look like. So you have mass here and then you have distance here. So the bigger than mass that you apply here, the bigger the mass, the lower the distance. So that would be a real life example for this for this fine function. But how about this? Be constant here. So what's the deal with this? Be constant right now. We assume that B equals zero. But what if it's not there? What if it's two or minus three or 1.5? What happens? Well, the way I think about it is if you look at this line right and forget for a second that it's a line and just imagine that you have an infinite amount of points here. Just imagine that for each possible X you have some kind of point here. And then if you choose, let's if you choose X equals two and you choose some kind off point at X equals to what happens when you when? When b equals, Let's say three. So what happens with this point? So let's say that if actually Kohstuh why equals also to right when B equals zero. So what happens when I add because three to this point? Well, it moves up, right? It moves off somewhere here and then why would be five now? The same thing The same logic would apply to allow the points on the line. So what would happen with the graphs then? Well, it would shift by three units. So if Why cause zero when B equals zero. If because three then and goes here so the entire graph would shift up. And, of course, if B is negative, if it's minus two, right, if it's a minor stew than the graph would shift down by two. So here it would be miners to, so this be constant here. It allows you to take whatever function you have, and it doesn't have to be a straight line function like we saw previously, but it takes whatever function we have, and it moves it up and down. So that's how you manipulate. That's how you can manipulate the function by moving it up and down. You just play around with this. Be constant, so I'm gonna give you a small exercise. Now, how would the graph look like you can just intuitively graph it? How would it look like if I say that I m equals zero? And we can be whatever you choose because be just moves functions up and down. How would the graph B when M equals zero? Well, let's see M equals zero. Then we know that M is nothing else but Delta y over Delta X. So a family equals zero then means Delta y has to be zero. So that means that there is no difference between a new and old situation. So so, whatever your exes. If if chemical zero and then there's no there is no difference ing why, then it must be a straight line. And that is, of course, if B is greater than zero. If B equals zero, then the straight line is lies along the X axis, and if it's negative, it lies somewhere here. But how about if if your graph is something like this? So you have X here and you have Why here and X equals three and then you have a vertical line. How would the function look like for this graph? So what would be the change in why? With this graph, the let's say, the new situation minus the old situation well could be either plus infinity new minus old . Oh, negative infinity, right? And Delta X. Well, it would be zero because the changes zero. So I m in this cage would be, let's say, negative plus or minus it, plus in plus or minus infinity divided by zero. But this thing is undefined, undefined and in fact it's not even a function and We're gonna talk about it a little bit later. Where we where we discuss grafts that are not functions. So in this case, you could write it down. Okay, It's just X equals three for a wise, but it's not a function. All right, so that's it about straight line functions. 11. Understand & find INVERSE FUNCTIONS: So now I want to cover another very important concept in functions and we're gonna look at inverse functions how to take an inverse off a function. So one way to look at functions is like this. You have X, which is an input or independent variable. And then you have why, which is output or dependent variable. And we call an independent variable because, well, we choose was we choose for X And why is a dependent variable? Because why depends on X, we can also illustrative function like this. I'm gonna take a complicated function. Let's say why equals 10 times X to the power of 5/3 plus seven and another way to look at it. Is that okay? We have X right. And then we apply some kind of Operation two X, which is called the function, and we get an output. Which is why so in this case, it would be X isn't input. And then our operator or function would be something like this. Plus seven. And then we would put X inside here, and then we get an output. Why, Okay, but let's take an easier example. Let's take, for example, why equals two X. Now, the graph for that function would be your do you know the the graph This So if I have X and why and I put here one and two, then, uh, one times two I would give me too. And Teoh two times two would give me four. So I would have graph which is like this where? Where the changes Delta y over Delta X would be to But now what? If so, our function is like this to parenthesis and why? And we put X inside these parenthesis and we get why What if we want to go back? What if what we want to do is to put why in front and we can write it down like this? So why old Oh stands for all equals X new. And then we want this X to go here. So ex old would be Why new So in verse means that we're gonna find a function that goes here in order to go in reverse direction from why two x so the way to do it. It's like this. You think this Why access and your rotated 90 degrees clockwise and then you take the X axis and you rotated 90 degrees counterclockwise. So, in other words, you take this number line here. Why old? And this is X old. You take the number line and you put it here. So now why old would be ex new. And then you take this line this ex old number line and you put it like this So x old would be Why new So OK, so in this function, if we go one unit X, we have to go two units up to get do why, Right? But what happens if we go? One unit up then How many units do we have to go to the right in order to get two X? Well, only 0.5. So if you go on this new on this X knew why old access If you kill one, then you get 0.5 right? So you can see from here you go one up and then you go to the right and then you're the meat. The line at X equals 0.5 and you just remap it here. So you move one and you move up by 0.5 the same thing here. If you move up by two then in order to get to the graph, you only need to move one unit off X old and then you're here. So you just re graphic here. So from to you, move up one and, well, the same thing here. If you go down by minus one, then you have to move to the left minus 0.5 and then minus two. It would be equivalent to mine is one. So you would get a graph like this with slope of the graph would be Well, why new? Which is eggs old would be 0.5 ex new or y old And it also makes sense Algebraic lee. So if if you why equals two X then if you use algebra than X one equals 1/2 why and then you just swap x two. Why and why two x and you will drive it down like this geezer. This is why new an ex new And there you go. We have found our inverse function. So inside this box, instead of the question mark, you will have 1/2 and that would be the operator and ex new would go inside this inside these parents is and then you get why new? And that's your inverse function. No, this algebraic manipulation that we did. It's not always that straightforward, though, And, uh, I'm going to go through one concept. First. I'm going to talk about which kind of graphs are not functions. You see, there are rules to being functions. There are certain rules that you need to follow in order to have function. When you wildly those rules and then you don't have a function. And after that, I'm going to cover inverse functions for parabolas. 12. Find INVERSE FUNCTIONS - example: So here have, ah small exercise for you. So let's go back to our airplane example. And we had this graph that represented the function where we had a relationship between distance and time and we knew that the relationship between them waas distance equals 800 time. So what would be the inverse off this function? Just pause video and I'll show it to you in a bit. Well, the inverse of it would be very easy. So here it would just be pure algebra. So now Well, first of all, our input was tea time, right? And then we had our function and then we put t inside here and then we got our distance. And in order to take the inverse really means that you just flip the variables. So now your input is distance. You have to find the function, then maps distance to time and you find it by in this case just by pure algebra which would be t equals one over 800 distance. And it is still a straight line function. But it has a way smaller slope now. So you're slope now would be, like, very, very small. And now you're you're independent. Variable would be this distance, and your dependent variable would be time. So this one over 800 what it means. It means the change of time with respect to distance for one unit off distance. So if before the change of distance with respect, the time or the change off distance for one hour of time was 800 kilometers, it now for one kilometer the time changes won over 800 hours. In other words, to cover one kilometer it took for the airplane. It took zero point 00125 hours or well, 0.125 times 60 minutes times 60 seconds would be 4.5 seconds. So it takes an airplane. The flies 800 kilometers per hour in order to cover one kilometer takes the airplane 4.5 seconds to do that. And just to show you how easy it is to work with the units and how useful it is to convert numbers from one from some units store the units. I just want to show you the process here. So I got 4.5 seconds like this, so I have 0.125 hours for one kilometre. So in one kilometer the airplane ah spends 0.0 went to five hours and then if I want seconds per kilometre, I just multiply this times 60 minutes in one hour times 60 seconds in one minute and notice You can cancel out the units of our and the units of minutes. And if you multiply all this together 0.125 times, 60 times 60 and then you would get 4.5 seconds per kilometre. So here also 4.5 seconds Burr kilometer. So you see now it's in reverse. Before, we had distance over time. So we had well, we could have had kilometers per second. We had kilometers per hour but could have had Guillain much per second. And now it's second per kilometre or 0.125 hours per kilometer. So taking an inverse of a function is like reversing the variable steam put unhelpful variables 13. Manipulate PARABOLIC FUNCTIONS: So now we're gonna talk about parabolic functions. The general equation for the parabolic function is why equals a X squared plus B, and let's assume that B equals zero and a equals one. So what you will get then is why equals X squared? So if we graph that ex and why? And let's say that we graphic for 12 miners one and minus two. Well, in this case, for the one, why would equal one squared equals one? Right. So we've we would also have one here. However, for for two, it would be why equals two squared equals four. And OK, I should have drawn in a little bit higher, so I would have four here and the same thing for the negative numbers. When it's minus one squared, it would be one and then minus two squared. It would be four. So we would have point here and we would have a point here, and then your graph would look something like this. So notice here to change and it is the same. It's symmetric, by the way, about the y axis. But notice that here the change is not the same anymore. Here, here if you go from zero to what zero toe one than the why Onley changes by one unit. But then, if you if you go from 1 to 2 on the X axis than the change of why would be three. So here the change itself changes. It's not constant anymore. We're gonna talk more about it in derivative, but not this notice that this function has an interesting property. Since it's symmetric you have when it's one equals one and miners one equals one. So what you have, you have a function which is at X and that equals the same function at minus X. These kind of functions have a special name. They're called even functions. And why that? Why is that? That's because here, when you're when you're on the positive side, you get a number in this case four. But if you're if you're on the negative side and you take minus two, the same number lay here only negative of it. You still have you still get four. So when once you have this kind of property, let's say this is too and this is minus two. If they give you the same number in this case for and four, then a function like that. It's called even. I'm gonna give you a small thought exercise now. So if we assume that be zero, then how would the graph look like relative to this graph when a equals one? How would the graph look like when a equals to a equals? 0.5 a equals zero and a equals minds one How would it look like? So I have drawn another graph here again, a equals one. So this line is for a equals one. But when a quiz to then if you think about it, what's gonna happen with all these points here, what will happen with them? Well, if you have this point in this point, if they are at four, why equals four? Then if you multiply them by two, that will move here, correct, They will move here and then these points here at X equals one. If you multiply them by two, they will move here. And if you take zero and you multiply it by two, it will still be zero. So your graph would be something like this to still go through zero. But you would see that it would hug the y axis more strongly. On the contrary, if we take a 0.5 and we take the same points, then at X equals one and miners one. If we multiplied by 0.5, then the points would go down here and here. Where at? Why? Where? Why is Europe on five? And then and then the function and the points at bike was four. They would move down to two because four times 0.5 equals two. And you can see that the graph still goes through zero. But now it hugs the y axes more in a more weaker way in a week away. Now, if a zero then this entire term become zero right and b zero, then you have a straight line, just like in the previous example. Straight line functions. How about if a equals miners one. And again, this is our principal graph here when a close one. Well, if acres miners one than thes two points, they move here because it because one times miners one is minus one. So I'm gonna have one point here and here. Well, if I take minus one squared, it's one minus one squared. It's one, but then one times mine is one. It's mine is one, so it would move here as well. And the same thing here. Four times miners one would be minus four and again here four times miners. One would be minus four. So you see that the graph now becomes symmetric about the X axis. It's the same graph like this, but it's upside down so you can see how a can manipulate this parable function. So if a is bigger than one, then it's like an amplifier. It hugs the y axis more, and if a is less than one, then it's It's like a week inner. It hugs the y axis less, and then if it's negative than it just changes its Ah, let's say shape. It's still a problem, but it's a parabola in another direction. So it's like this. And of course, if a was minus two, then it would hug the Y axis in a stronger way. Something like this. But notice? No, it is that no matter what we do with that a, the function still is even even if even if if a is negative and still If X is one, then why's miners one? And if access miners one? Then why's minds one? So you still have this symmetry and it's still but they still satisfies this even function requirement where you have, which is essentially this one. So no matter what you do with a, it still remains even another exercise. What would happen with this problem if B is not zero? If B is positive, let's say B equals one. And then what will happen if B equals minus one? Okay, just pause the video. And, uh, I'll soldiers for you in a sec. Well, remember, bees just a constant right? So it's just a number and a good way to imagine in your head what happens when you're at something toe A function is to forget about the line and think about the infinite points on the line and then pick a point and see what happens with the point if you add or subtract a number to it. So in this case, whatever point you take on this function, so if you have some kind of parabola, right, and instead of a line and just gonna put a lot of points here, So if I had one to all those points, all those points will shipped up by one. Right. So this point here will shift up by one this point. This point here will shift up by one. This point here will shift up by one. So you will have I probably like this. And the same thing is, if your prop Elise it's like this then again, whatever. Whatever he whatever number you add to it, it's either shifts up if you add b equals one because all those points will shift up. And if beak was miners one then this parabola would simply shift down and you would have miners one here so all those points would shit down by one unit. And also this parable here will shift down by one unit so you can see that this part in the pilot boy function is responsible for the shape off the function. But this part here it only shifts, functions, whatever functions you have up or down. By the way, one thing that I want to emphasize is that a here is not change. Remember, in straight in a straight line function, we had m equals delta. Why over Delta X. That's not the case here. We're gonna talk about it more in the derivatives. There is a special way to understand the change of functions like these, but the change here is not constant. So a a here is not a constant. I mean, a here is not a change off the function. It is just like an amplifier and amplifier or a week inner if it's less than one, and if it's negative than it's just changes the direction of the entire function. 14. Manipulate CUBIC FUNCTIONS: Welcome back. So we're going to continue with a cubic functions. And the general equation for the cubing function is Y equals a x cubed plus B. Now we're going to assume that a equals one and B equals 0. So we get y equals x cubed. And one thing that you need to notice here is that if you take a number cubed, let's say if you take two and you make cubed, then it's going to be two times two times two, and that would be eight. However, if you take minus two and you take it cubed, then it will be minus two times minus two times minus 22 minuses give a plus. But then that plus that you get from here would then minus will give you minus again. So it will be minus eight. So with that, pause the video and try to graph this function for x equals 012 and in fact minus1, minus2. So what you should get is something like this. 0. Then one cubed would be 12 cubed would be eight. Then minus one cubed should be minus1, and minus two cubed should be minus eight. So if you graph it and okay, it's now going to be up to scale, but it doesn't matter for now. So you're gonna get 12345678. And the same thing here, 12345678. And then you get 00, then you get 1128. And the same thing, minus1, minus2, minus eight. And this graph will be then something like this here and like this. Now notice that unlike with parabolic functions, where both a positive two, for instance in the negative two, gave the same y value. In this case, it gives the opposite value. So two gives eight and minus two gives minus eight. So when you have a situation where you have a function of X equals minus function of Minus x. So this is two and you get eight. And this is minus two. And you get minus eight. Those kind of functions. They are called odd functions. And in fact, I can tell you right now that all the functions that have even exponents, like two or four or six, they're all even functions. So they will give you a parabola, some kind of parabola. Of course, the bigger the exponent, the more the parabola will hug the y-axis. On the other hand, if you have odd exponents such as 135, etcetera, then the function will always looks like this. And with the bigger exponent, obviously it would hug the y-axis you more. But it would be an odd function. And here's another thought exercise. How would a influence the function? If a is bigger than one, smaller than one, then a is, is bigger than minus one and smaller than minus one. So just pause the video and just think intuitively how it would make this graph behave. Well, just like with the parabolic functions. A can be an amplifier and a weak learner. So if it's bigger than one, and we take this function when, when a is one, when it's bigger than one, it will hug the y-axis more. But when a is less than one, like 0.5. then it would act as a weak learners. So you would have a function like this maybe. However, if a is less than 0, then essentially the same logic applies. However, everything is in reverse. So let's say that if you're a is between 0 and minus one, then your function would hug the y-axis less. But then if your a is, let's say minus two, then it would hug the y-axis stronger. But of course, since a itself is negative, if you're negative, x variable cubed is also negative, then you would multiply negative a by negative x cubed and negative times negative equals positive. And that's why you would have a reversed function. So that's the intuitive way to look at it. 15. SIM: Car race exercise - who will win the race?: Welcome back. I have a small font exercise for you here. You can see five cars. And this is the starting line. And you have to finish lines at six kilometers and at 12 kilometers. And one of the cars is cheating. It's in front of the other cars. And you know the respective functions as well. So the blue one corresponds to the first car, read to the second, black to the third, green to the fourth, and purple to the fifth car. And knowing their functions, your task would be to determine which car would win the first trace at six kilometers and which car would win the second race at 12 kilometers? So just give it a try and the solution will be available in the next video. See you. 16. SIM: Car race exercise - who will win the race - SOLUTION: Welcome back. This is the solution for the proposed exercise. You can see that the first race is won by the green car and the second race by the blue car. Now, to calculate that, it's actually very easy. So if you look at these position values here, and let's say you just take six kilometers and then you replace this variable with six kilometers. And you, and you will do it here and here, and here, and here and here. And you can do the same thing with 12 kilometers. You simply replace the pos variable with a distance value. Once you have a number here instead of a variable, you will have equations with only one unknown. And then you can simply rearrange the equations and calculate the time. Each car needs to reach this point and this point. So if you put, for example, 12 kilometers here for the black car and you rearrange the equations and you calculate the time, then you will know how much time the car needs. The black car needs to reach 12 kilometers. And I've done that here. In Axle. You've got five cars here. The position values six kilometers and 12 kilometers. The rearranged equations for each car is different. So you rearrange the equations and then you calculate the time values for each car at six kilometers and at 12 kilometers. And then you just pick the smallest value, the smallest positive value for each of the cars. Because obviously, if you look at the last car, you look at the constant and its negative. That just tells you right away that this purple car simply will simply go backwards. But all other four cars, you just take the smallest positive value. So you can see that the green car reaches the six kilometer mark first in one hour. And the blue car is the first one that reaches the 12 kiloohm term mark. And it needs to hours for that. Therefore, the first race is won by the green car and the second race by the blue car. So that is it. Thank you very much and see you in the next lecture. 17. Functions VS Non-functions 1: welcome back. So in this section, I want to cover certain conditions that make a function of function. What do I mean by that? I want you to remember the very first example that we had on airplanes. Do you remember? We had this function where we had time, and then we had the distance, and then we had this function and then distance dependent on time like this d equals 800 t . Let me ask you a question. Now, what if I have a function like this? So I have time here, and I have distance here. And my graph, it's something like this. Does this function make sense? She's supposed video, and, uh, then think about it. And then, um, positive media. Well, no, it wouldn't make sense because essentially, what you're saying is that at time equals, let's say, two hours. The airplane has covered this distance, which is, I don't know, one kilometer, then also this distance, which is four kilometers, for example. I'm just throwing out numbers in this distance, which is six kilometers now. It's impossible for the airplane into our to cover one, four and six kilometers. And in fact, if you generalize it in the world of functions, then something like that cannot be a function. There's something called a vertical line test, which means that you take a vertical line and you go over overall the access. So in this case, is T. But let's say in general it's X we have X here and we have Why here? And then you take this verdict line and you go over ALS the access from minus infinity to infinity and, uh, their rule is that this that this vertical line needs to touch the graph Onley once. So if you have a graph something like this, right, and then you take this vertical line test and if you go over ALS the access the graph will always touch the line only once. Then it's a function. And in this case it's not a function Notice, however, that if I have function like this, let's say this line it's OK if many exes goto why, what do I mean by that? So if I take a random, why value here? Let's say why equals two and I draw a horizontal line like this and like this and you can see that this horizontal line cuts here, here, here and here. That means that many access, let's say X one x two and then x three and x four. You see, many excess go. Many exes are mapped to that one. Why? Which is why equals two. So a situation like that is okay, so something like that is okay. But something like that, where one X is mapped too many wise. This is not okay. This is not a function. Another way to look at it is like this. I'm going to introduce two new terms. One is domain. What is the main? The main is ALS, the possible X values, Orel, the possible input values that you can have all the possible independent values. So if I just say that this is a bag off, all the possible inputs that I can have and then another term would be a range ranges all the possible outputs, or, in our case, wise that you can have all the possible outputs. And I would say that you have certain wise here Now, for some functions, the domain would be from minus infinity to plus infinity like, for example, with the function like y equals X right. So you would have something like this, you would have a graph and this graph would go from minus infinity to plus infinity. And also why would go from minus infinity to plus infinitive? Because there is no reason that this graph would stop in this case you with say, that domain goes from minus infinity to plus infinity. However, in our airplane example our function was restricted because we started at time equals zero was it's right here we started at time equals zero. We did not measure what was before time equals zero. So our function started from here So our domain would be from zero to plus infinity So that would be our domain and our range the distance So in this case, domain would be alive. The possible times all the possible times and range would be all the possible distances. It was also vary from zero to plus infinity. And by the way, when we have this bracket, then that meets included That means zero is included. A parenthesis means not included So infinity and nor miners in fainted there never included If I had written like this zero to plus infinity then the function would look something like this. I would have an empty spot at equal AT T equals zero, and then I would have a function like this. So brackets included Prentice's not included, but the point that I want to make is that so in a function you can map multiple domain values into one range value so you can map this one here. You can also take another domain value and map it to the same place and the same place. So that is a function. Now. What is not a function is if you again say that this is a domain and range, and then you take one domain value or one X value, and you map it to one range value, and then you map it to another range value. So this is a general way to look at it. So this would be a function. This would not be a function because that because this picture would represent something like this, Or in fact, you probably own remember that there is. There is a circle function, so circle is represented. Radius squared equals X squared. Plus, why squared even though you can represent it like that. Graphically, it's not a function because if you apply in the vertical line test, you can see that it cuts the graph in two places, which means that it's not a function. 18. Functions VS Non-functions 2: now in the section world with disgust Whether some kind of graph is a function or not, we also briefly touched upon on the equation circle. So we had a circle, and that would be the graph. And the question of the circle was R squared equals X squared. Plus why square, Right. So this is X here, and this is why and that also comes from the Pythagorean theorem. That's because, say, if you draw a radios here, then you would have some kind of why here and then you would have some kind of X here and you would get this radius this one using a Pythagorean theory, which is the same thing. So x squared he would take this X square. He and you would add, Why squared And then you would get this quantity squared so radius would be square rude both x squared Plus why squared So that came also came from the from the triangle, right? So you have X and why here? And that's how you would get our with this equation. However, we also discussed that this graph cannot be a function because if you apply the vertical line test, you see that the vertical line touches. There's graphing to places, and therefore it doesn't qualify to being a function. However, what you can do, you can take that circle and you can divide it into two. You can choose two sides. You can either choose the lower side or the upper side. Let's first see the upper side. So I'm gonna draw a function now, hex. Why? And I'm just gonna Well, first of all, I'm gonna assume that the radios equals one Let's say one meter and then I can draw a circle here. Well, it's half of the circle, and the function off this half of the circle would be Why equals square root R squared minus X squared. So how did I get it? Well, I get it from this from this equation. So R squared equals X squared plus y squared. If I want to get why than what I do is this y squared equals r squared and then X squared goes to the other side says minus X squared. And then I just take the square root of it and I get why, right? So this equation, it represents this graph. And if radios is one. Then this is one. This is minus one, and why here at X equals zero? Why? Because one which makes sense because the radius is one. So if the Raiders is here excess one. If the radius points in this way, it's miners. One. If the radius goes up than X equals zero and why equals one? Let's just take an example. So it's if X equals zero than why equals one squared minus zero squared and you would get square root off one squared, which is one. So that's one way how you can represent a circle. You would just take one part of the circle the upper part. However, if you want to represent the other half, then it's very easy. What you do is very simply, let's say all the conditions are the same what you do, you simply put a minus sign in front of the original function. Why equals minus square root? R Squared miners X squared. Okay, and that's how you can represent the entire circle by choosing, let's say that's how you can represent the entire circle as a function, you just choose either one side or another side. You either choose the upper side and you use this equation or you choose the lower side, and then you choose this equation. Let's say a good way to understand why this graph will become this graph. If you put minor sign in front of it is like that if you drove the upper circle But instead of a line, you just imagine that there are a lot of points here. Well, if I put the miners in front and that means that I put miners one in front of this function and I multiply miners one by this function, if I do that that all these points, they will be mirrored about the X axis, so this point will be mirrored here. This point will be mirrored here. This point will be murdered here. This point will be married here. So if this is like 0.25 then it has to be minus 0.25 here because I simply multiplied this number by minus one. And that's how you can get this the lower circle. All right, there's one more thing that I want to cover, and that is the inverse off a cubic function 19. Find INVERSE of a PARABOLA: So I have a parabola here. I have a problem and I want to take any inverse of that parable. So what do I do? So I take why old access And I put it here and I rename it X new. And then I take the X old access. And I put it up like this and I recall it. Why new? Okay, So if we go on this wild access, if we go up by one, then what do we get? Well, we get one and minus one on the X old access. So if we re map it, we should get something like this one minus one. And then if I go to four, if I go to four here, then it would be too here, and it would be minus two here because if on this old access, I go to four, I need to go to to in order to get to the X old value. So, essentially, what just happened was that I took a square root of four and I got plus and minus two. So if you go from to, then you would get some kind of value here, which is around one point for something. So it's it would be equivalent the square root of two. And then for three you would also have ah, square root of trees somewhere here. So you would have a graph that would look something like this and the same thing would be here. So that would be the inverse graph of this function. Now it might be the inverse graph. However, Is it a function? No, it's not. Because remember, we apply the vertical line test and now it touches the graph in two places, so it cannot be the function. So that means that you have to choose. Decide that you want to work with. You're either choose this side and you go from zero to plus infinity. You choose your domain and then if you choose this domain, then you get this graph or you choose this side, which means that you work from from minus infinity 20 and remember, brackets are included, so you only take this part and then and then you get this graph. Both choices are valid. However, if you have a calculator than most kept, most calculators our program Teoh, choose this side and this one So that's why if you put if you take a square root off four in a calculator, you usually get to. But the reality is that both function both numbers are correct. So a calculator can also give you minus two and it would still be correct. So if it's just the upper function here, so you choose the domain from zero to plus infinity, then your general function would be Why equals square root off X. However, if you choose the other domain from minus infinity is zero and then you're inverse function would be this this line here then your function would be why equals minus square root of X . And by the way, remember that square root of X me is the same like X to the power of 1/2. Remember, So to goes here and one is the exponent off X, for example, he find half X to the power off 5/3. Then it's the same. If I write it like this. So three goes here and I put it here and then five goes here. So just a reminder now, another important part that I want to emphasize is this. So if you have a parable like function and you take the inverse. If it and then dean verse, as we so before can be either like this or it can be like this, then notice one thing. When you take the inverse, you never reach the negative values for why old or ex new, for that matter. You never reached this part, right? So you have the positive parts, but you don't have the negative parts, so you never reach it. And that's why when you put minus four into a calculator, it gives you an error. Well, in fact, if you take a miner's for here and you move here where you have minus four, then there is nothing here, absolutely nothing. 20. Find a CUBIC FUNCTION INVERSE: welcome back. So in this part, I would like to take an inverse function off One more function. I would like to show you how to take an inverse off a cubic function. So once draw cubic function here, why equals a Times X cute plus B. And let's just assume for the sake of simplicity that a equals one and B equals zero. That means that you're left with Why equals x cute. Now, what would be your graph? Well, if I just draw my X axis and my why access And I say that Okay, X is one to. And here it would be miners one and minus two. Well, if, uh if X is one, then one cubed would still be warm. Okay, so I'll just put one here. You. However, if I have too cute than two times two times two is eight. So I have eight over here, so I put it here. And if I have minus one cubed, then notice it's miners. One minus one, minus one. Now two minuses. Give a plus, but then you have one more minus and plus and minus will give you minus. So you are left with minors. One Here you have miners. One here and then minus two. Cubed would be would be minus eight. Because two times, two times two is eight, but then minus one minus one minus one would be minus. So you would have minus eight here. Soup plus eight. Here. Mine is eight here. And then you're function would be something like this. Okay, Now, in order to take the inverse function, remember all you do you take this access waxes and you turn it clockwise. 90 degrees. And then you take this, access the XX and you turn it up counterclockwise. So why do that? We have ex new equals. Why? Old right? This is why old now and then we take this X and we caught Why new equals X old. Okay. And then let's just take 1234578 Here is 11234567 and eight. And here it's enough. If you just put two and to Okay, So what happens if we go up by one? Well, if we go up by one or equivalently we go up by one here to the right. Here. If we go up by one and we go along the x old access we will reach one. So we just re mapped it here one and we get positive. One for the X old or wine Your Now, if we go on the Y, old access or ex new axis, If we go till eight, then we will get to right. We will get to hear and the same thing if we go down by miners one or to the left on this axis because these two are re mapped. So down here means do the left here. So if we go by miners one here we get minus one here and here we re map it to this place. And then if we go two miners eight to the left or on the old picture we get we go down by miners eight that on the old picture we get minus two as ex old and on the new picture we get minus two as why new? So we just re map it like this. And there you go. You can already graph. You can grab this inverse cubic function. It will be something like this and notice Now this is a big difference with with the square function, because with a square function, when you had this parabola right, you were not able to reach the negative values on the why. And that's why when you took the inverse off the cubic function, then okay, you have to choose. Either you take the upper side where you take the lower side, but you were never able to reach the negative values. And that's why when you put the minus four into a calculator and you took a square root of it, you've got an error. However, when you work with cubic functions, then it's a different story. Because you're inverse. Function now would be why new or ex old would be cube root off ex new or why old? And as you can see from here now, negative values are reachable. So, for example, if you insert minus eight into a calculator and you take a cube root of it, then you will get minus two. You will not get an error, and that's how you take the inverse of the cubic function. Now, in the next lecture, we're gonna take all of the functions that we have seen so far, and we're gonna wrap it'll up and generalized the entire thing. So season 21. Generalize FUNCTION MANIPULATION: welcome back. So so far we have seen quite many functions, right. We have seen straight line functions which look like this or like this, or shifted up or shifted down. And we have also seen parable it functions that looked more or less like this. We're see cubic functions. It looked like this. We have seen square root functions that looked like this, and we have seen cube root functions that looked like this. But if you think about it, then all those functions they can be generalized and one general formula can be created to represent all those functions. And that general formula is why equals a Times X to the power of and plus B. And you know that if n equals one, then it's a straight line function because then you have eight times X and then plus B and remember, be be shifts functions up and down. And ah, good way to visualize it is that whatever function you have, you just think off and in infinite amount of points and B is also a function in fact, be, let's save be equal seven. Then you can also think of it as an infinite amount of points that crosses the Y axis at seven, and then you. If you pick a point, let's say X equals minus three. Then you take this point and you add this point to this one, and then this point will shift up. And then, if it happens with all the points or the time graph will shift up. Now if n equals two, then it's a parabolic function because tense, a time squared, plus B cute and equals three. And if you have a square root function then and would equal 1/2 and a Couperin function and would equal 1/3 so you can see that you can take all those functions that we have seen so far and you congenital allies them, in fact, and can even be zero. What happens when an equal zero well X to the power of zero is one? So you are left with why equals eight times one plus B a plus B. So if a equals two and vehicles three, then you're left with five. So you will have some kind of graph at the But why equals five? Okay, there's one more category that we haven't seen so far and that is what happens if and is less than zero. What happens if and is let's say mine is one well X to the power of minors. One is nothing else than one over X. But if you think about it, then if you're function is why equals one over X and we assume that a equals one and B equals zero and then you have graph here. Why? What happens with with this expression? One over acts as X gets bigger and bigger. Well, if X gets bigger and bigger, then this entire expression gets smaller and smaller. So if X equals two, then it's 1/2 or 0.5. If X equals 10 there's 1/10 or zero point form, right? So your entire graph, if you take your time and graph it will be something like this. And also, if you have negative values like one over minus two, then you would have Maya's Europe in five and one over minus 10. Would be miners your 100.1. So it would be a symmetrical. In fact, what do you think would be, and even function or an odd function or nothing? It would be an all function because for access to you would get 0.5. And if actually minus two, you would get miners Europe on five and notice one thing. No matter how far you go along the x axis in either either sides, you will never reach a zero. So this line will never touch zero because you can always take an X that is always bigger than your previous X. So you would have to go to infinity. But before infinite, you can take whatever X that is as big or us a small on the negative side as you want, right? So I could take one over one million or 1,000,000,000 or trillion and still wouldn't be exactly zero. And the other thing is that this function here, it wouldn't even reach zero this line and this line, they wouldn't. They wouldn't reach X equals zero. They wouldn't touch this access. Why? Well, because you could always take a number that is smaller than your previous number. You can take one over, tend to the power of minus six or minus nine minus 12. So I'm gonna give you a small filed exercise. Now, what is the domain and range off this function The domain off this function starts like we discussed previously from minus infinity because the line never touches zero, it always goes to the left and the up the line always goes to the right So it has to go till plus infinity So it goes on forever. On the other hand, he'd never touches the Y axis, so it never goes where X equals zero. So you simply go from minus infinity and remembering. Think he's never included. Now you go to zero, but zero is not included. And then you put this you sign here, which is a joint. It joins two domains and then you go again. So from minus he fainted. 20 It's just this number line and you stop You stop at the number that is right next to zero. But you can always get a number that it that is, uh, smaller and smaller. So the line gets closer and closer to the y axis. But there is one. There is one number that the line can never reach. And that is zero and the same thing With this line, it never reaches zero. So again, zero not included. And then the line goes still plus in 30. And that would be the domain off this function and the range of the function. Well, it would be the same thing right now. I look at it like this. You start from here from down and you go up and there is one number that you never reach and that is zero. So from Maya's infinity, you go to zero and remember parenthesis, not brackets, because here is not included. The line cannot reaches zero and then you go from zero to plus infinity. And there you go. We have covers quite many functions so far, and we have been able to generalize it all into one formula, which is this one. So essentially the conclusion from here is that this term here is responsible for the shape of the function. So a and and they old determined the shape of the function and B which is just the constant . Just a number are an infinite amount of points on ah horizontal line that the just shifts functions up or down 22. SIM: General functions in motion: Here you have a similar case like in the first simulation. However, the distances that now the airplane is flying in the parabolic way, you can see these big dots here and they still appear every six minutes. However, because of the nature of the, of a parabolic function, these dots at the beginning are very close to each other. And that means that the airplane At the beginning flies very slowly. And in the end, as the airplane gradually picks up speed, the dots become further and further away from each other. You can see this in, in the function form as well. So this is a parabolic function here. And this is a mathematical version of it. Distance equals 800 times time squared. Now one thing that you can see here right away is that it's not a straight line function. And therefore, the rate of change of the function or of the airplane with respect to time. The rate of change of the airplanes distance with respect to time is not constant anymore. It changes and it makes sense because the airplane flies faster and faster all the time. And you can see this rate of change as a function here. Now if you're wondering how I got this function, don't worry. You need to know derivatives for that. And I will teach you all that in the derivative section. So once you've covered the derivative section, just come back here and take a look at it again and it will all make sense. So this is now the cubic version. And one thing that you can see is that when we had a parabolic function, then the final distance that the airplane AT covered in two hours was 3,200 kilometers. So the airplane reached two here. Now, since we have a cubic function, 800 times t cubed in two hours, the airplane covers 6,400 kilometers. And this effect of the airplane flying slowly at the beginning and faster in the end is even greater. When you have a cubic function. Which you need to notice is that at the beginning, at the very beginning, you will fly slower. And in the end, you will fly faster than in the parabolic version. So this is now to the power four. And you can see it really takes a long time for the airplane to pick up the speed. But in the end it goes really, really fast. And in two hours, it will cover much more distance. So in the case of power for it will cover 12,800 kilometers. When the power is less than one, then the nature of the function changes. And you can see that the airplane goes faster at the beginning and slower in the end. And here I've put the first, second, third power together for you so that you could see the difference. And you can clearly see that, OK, in the end, the cubic function achieves the highest speeds and it travels the biggest distance. However, pay attention to the fact that at the beginning, when the race starts, the straight-line function actually wins. And only then the parabolic and cubic functions, they pull ahead. And that comes from the nature of these functions. If you examine these functions here, you can see that at the beginning, the straight-line function has greater values. And then the parabolic functions has greater values, then the cubic functions, and then the cubic function has the lowest values. But then it all changes and the cubic function pulls ahead. Now, instead of a cubic function, I've replaced it with a square root function. And you can see that the nature of the function is completely different. At the beginning, you fly very fast and then you gradually lose speed. And that's why at the beginning, the square root function wins the race. However, then it becomes just too slow and other functions will pull ahead. And if you look at its rate of change, then it's also completely different. It has changed the shape. So now, because the speed is going down, the function looks like this. So I hope you've enjoyed the simulations and there will be more of them coming. So for now, that's it. And see you next time. Thank you. 23. Form PIECEWISE DEFINED FUNCTIONS: and there is one more function that I want to cover with you. And I'm going to introduce something called an absolute value. So an absolute value is two vertical lines like this. And then you put X inside. In fact, that general function could still be Why equals a times absolute value off X to the party and plus B. Now again, we assume that a equals one and B equals zero. But what is an absolute value? Well, an absolute value makes a positive number positive, so it doesn't change a positive number, but it changes the negative number. It makes it positive and it keeps 00 So if I have you fly half an absolute value of two, that would equal to If I have an absolute value of zero, that would be zero. If I have an absolute value of minus two, that would be to So how would the graph look like for this kind of function? Let's see. So I have X, why and well, if X equals zero and we assume that a quiz one and B zero So my functions why equals an absolute value off X? So if X equals zero. Then why would be zero if X equals one? Then why would also be one if X equals two? Why would be to so essentially on this side? It's the same like why equals X? So that's straight line. How? But on the other side, Well, if I have mine is one, then why would be an absolute value of minus one would be one so it would be here and minus two would be to minus two here, then I will have to hear. And again, it's a straight life function, and it's the same on this side. It's the same, like why equals minus X, which means this minus Well, it's miners, one which means that Delta y over Delta X is negative. And that's why you have this kind of line. So you have this small, let's say, the 1st 2 sides of a triangle, but you can't really go very creative with functions. You can define whatever functions you want. For example, I can define a function just like this, Why equals and then I write that one equals minus X, but only for the domain where X is less than zero, which means I'm talking about this domain here, not on the right side, not the polls. Decide but the negative side and notice that What do you think? What do you think? If you have smaller or bigger, then is your included or excluded? It's excluded, so zero is not included, which means these kind of signs they're the same like parenthesis right now. If I had another sign, let's say less than or equal to or bigger than or equal to. If I had written here, X is small than and equal to zero. Well, then it would be parenthesis because then zero would be included so less than are bigger than would be parenthesis and less than and equal to or bigger than the unequal to put the brackets because they represent numbers that are included. But in this case, you can see that you can see that zero is not included, so the function is minus X. When X is less than zero, then the function is X squared. If axes bigger for equal to then zero, but it's less than one. So what this sentence is saying is that X is included in zero. It's not included in one so it's in between zero and one, but one is not included. Then I can also say that that why equals two when X equals one. So only one only when X equals one. Then function. The function would be to why would be to And then I can say that why equals one when X is bigger than warm. So how would how would it be on a graph five x here? And I have Why here? So how would it be? So if X is less than zero and not included, so parenthesis, then you would have minus X. In fact, I think I lied for you as, ah as a small thought exercise. Just pause the video and and try to do it yourself and then I will show it to you. Okay, I hope you've posed the video. So the function here would be like this minus X. It would be it would go to zero, but then zero would not be included. Now you have one here and you have to hear you would have three here. So when exit goes from 0 to 1 and there is included, but one is not included, you draw it like this. So you start from zero. You make a point here and then and then you. You make this parable because X squared is a parabola. However, one is not included. So what you do, you write a circle, an empty circle and then you make this parabola and then you know that when X equals one, the wine is too. So what you do you put filled in circle here and then if X is bigger than one, then it's then why is simply one? So it's just straight line that goes still infinity. So notice if some things just like X is bigger than something like bigger than one. Then it goes to plus infinity. And in this case, when X is less than zero, it goes to minus infinity. So that would be your function. And then well, what would be the domain for this function? Well, it would be from minus infinity to plus infinity, right, because all the exes are represented here even at one you have something you something for Why, however, what about range? Well, if you look at it, then you have nothing bolos. You're right. Total emptiness so I guess you start from zero that it included. And you, But you do go until plus infinity. So you've read it telling this so that would be your range. Okay. Now we have covered many functions, and we have been able to generalize it. And in three ways, first of all we had, why equals a Times X. The par vent was be without the absolute sign values. Then the second function would be this where you did use the absolute values. And the 3rd 1 is you defined your function yourself and notes you can define whatever function you want. You have complete freedom to do that. Now, in the next lecture, I'm gonna teach you one more thing. One more manipulation that you can do with functions I'm gonna teach you to To shift functions to the right and to the left. So how do you take a function like this? Or any function for that matter? And how do you shifted to the right and to the left? You wouldn't know how to shift it up and down. Now I'm gonna show you how to do it to the right and to the left. All right. Thank you. And see you soon. 24. Learn to shift FUNCTIONS: Welcome back in this section, I'm gonna show you how to shift functions to the right and to the left, and you will notice that it's very easy to do. It only has two steps. And what are those? Well, I have one weird function here. Why? It goes 3.5 times X to the bar of 4.5 plus 2.5. So the numbers here quite weird. But I did it on purpose because I want to illustrate the point that no matter what numbers you have in order to shift the function, it doesn't matter. So there two steps that you need to do. The first step is that if you have dysfunction, you simply put parenthesis around X like this. All right, this is the first step. So if you if you have an absolute value of X, then you just put parenthesis around X not around absolute value effects. And when you have X squared, you don't put parenthesis like this. But just just like this around X, and then you take this part and you're right minus and then you choose the number until which you want to shift the function so if I want to shift the function to the right. If my function is here and this is my square root function, why equals X squared? And I want to shift it to positive too. So there's the positive sign. If you want to shifted still here, right, then what you need to do. You need to take that too. And you put it here. And then the entire function shifts to the right. If you want to shift to the left till minus two so that your problem would be here, then you have to put your minds to so you would have something like X minus minus two, which would give you X Plus two because two minuses and give a plush and what does work like that? Well, if you think about it, let's let's try with with this square function with this parable like function, why equals X squared? I take the parentheses. That's the first step. And the second step I subtract the number from the apprentices. Let's say to and then on Lee, other princes up put the square root sign. Well, what happens if you put two here for the X? What happens then well, this entire thing here become zero, right? So in other words, you can look at the Prentice's as another variable, but I'm gonna call it K. And then I would have Why equals K squared. So if you graph it, if I say that now, this axis is K and this is why. But then when Keiko's zero squared when? When Keiko zero, then the function would be zero so you would have a travel like dish. But Kay is nothing else, then X minus two. So that's why that's one thing to think about it, that if you put and X into in between parenthesis, then you can take this entire expression in the Prentice's and you can just say, give you the name, let's say K. And then the reason why subtraction works is because if you take X two, then that makes K zero. So as far as wise concerns, then the functions lies where it should lie when when the independent variable zero. So that's how you shifted. So how would you shift this function here? Well, you just follow the same procedure. You just right. Why equals 3.5 and I want to shift it. Let's say three units to the left to the negative side. So X minus minus three to the power of 4.5 plus 2.5. Now, I don't know what the graph would be for this function. However, you can check it out on Google. On Google, you can just type the type to function, and it will give you a graph. But that's how you shifted. And then, of course, it would be X plus three to the power of 4.5 because two minuses give a plus. So I have a small exercise here for you. But I have here is why equals X to the power of 3/5 plus 1.6. Now you know how to shift functions up and down, and you also know how to shift them to the right and to the left. So what happens? How would the function look like you don't have to graphic. But how would the function look like if you shifted down by 3.2 by 3.2 units and he shifted left so to the negative side, if you shifted left by 2.6? How the function looked like. All right, let's try to solve it. Well, if you shoot it to the left, then you follow the same procedure. Why equals? Take Xing to a parenthesis, and then you would have X minus minus 2.6, which will give you a plus to the power of 3/5. And then it would be plus 1.6 minus 3.2. So your function would be then why equals X plus 2.6 to the power of three or five. 1.6 miles, 3.2 would be minus 1.6. So that would be the solution for this exercise. All right, thank you very much for attending this lecture. Now you have learned quite a lot of things. You have learned how to shape the function because this part is responsible for the shape you know how to more fit. You can do many, many things with those functions. You know how to shift it up and down, you know, have to shift it to the right into the left. And you also know how to define your own functions. Remember, you can define piece wise defined functions and you can just create your own functions. You can define them the way you want, just like we did in the previous video. You are the grandmaster off functions now. All right. Thank you and see you soon. 25. Learn about FUNCTION OPERATION: Welcome back. I would like to start this lecture with an example. Do you remember the very first example that we had the airplane example in the first lecture where we had an airplane flying in a certain direction and we found out that that the best way to represent that information was is a mathematical function and we represented we represented the distance of the aircraft like this D, which is distance equals 800 which was the speed 800 kilometers per hour times time. So and we grafted like this. We had time and we had distance. And we have some some kind of graph, right? So that was our function. What about you had another airplane flying in the same direction and being exactly at there at the same distance with the with the first airplane. When time equals zero, we have two airplanes next to each other when time equals zero hours. But the second airplane is the first airplane. This is the second airplane. The second airplane flies with, uh, with this function, so 600 times T. And now you already know that 600 means speed, so go slower. And of course, if you graph it, then you know that that the function of the second aircraft would be with a smaller slope, something like this. So you can see that the first airplane will pull ahead and the the second aircraft will be behind the first aircraft and then the distance between them will get bigger and bigger and bigger as time goes on. So this is a small exercise for you. Just pause the video and try it out. How would you mathematically represent the distance between the between the two airplanes as they fly as they fly in this direction? The way you solve it is very simple. So this is D one D two and in order to find the distance between them as a mathematical relationship and simply take d one and use obstruct the two from it. So you what you will get is this 800 team minus 600 to antique affect her out T which is 800 minus 600 t, which is 200 to. And there you go by subtracting those two functions which you have. God is another function. But now the other function this one, I'm just gonna put distance between two airplanes with equal 200 team and you can grab it like this. T you have distance and you would have something like this. But now this graph it represents. How how much the first airplane pulls ahead with respect to the second airplane, right? So 200 now would be the change off distance between two airplanes with respect to time. So how fast their their distance. We have two airplanes here. How fast their distance increases and this example is is a good example to cover another point in functions. And that is operations because it's one thing to manipulate them, but you can do more things with them. In fact, you can have to rent them functions like why one equals something and why to equals something and then you can do many operations with them. For example, you can add them. Why one plus Why, too you can subtract them. Why one minus y two, you can multiply them. Why, one times why to you can divide them. Why one divided by white to In this case, you need to make sure that why too, never equal zero, because remember, you cannot divide by zero. So let's say if I have, If I have, why one which is X squared and then I have why to which is X, which is just X. If you want to divide them while one over why too, Then you would have X squared over X and then you would cancel them. You just get X. But remember this X here it cannot. It cannot be zero, so you just cannot be zero. So even though you get a straight line function, right, because why equals X would give you some kind of straight line function. So in this straight line example, the function would be simple. Right to the wire equals X. However, if you have X squared over X, you might think that it will give you the same thing. But it has an important distance. This X. Since it cannot be zero, your graph would be something like this. It would go like the straight line, but it would not include zero, and then it would just continue as a straight line. So in this case, in this case, the the the X in the denominator is not zero. So in this case, this function does not include zero. So I'm gonna give you an example using domains and ranges. So again, why one equals X squared. Why, too, equals X. Why one over? Why? To would be X squared over X. If you cancel them, you would have X. However, they're still not the same function. Because if you have, why equals X you're your function or your graph off the function. We'll view this. But if your function is like this, why equals X squared over X in your graph will be almost the same except when you reach zero. It's not included and then you have a straight line. So try toe pause the video and knowing that the domain off this function here goes from minus infinity to infinity. What will be the domain off this graph here off this function, what will be the domain? So the domain would go from minus infinity till zero, which is not included. And on the other side you have zero plus infinity and forgetting to join them is to create this link between the so that's the difference between them. But now talking about all these operators and by the way, you can also having a operator where you square the function, why one squared would be X squared, and then you square the entire thing, which would be X to the power four. But now how to get an intuition for for those kind of operations? Well, do you remember how to shift functions up and down? Well, if you think about it, then when you add or subtract or multiply functions, then essentially, it's the same thing you have. You have one function that is like this, and then you have another function that can be like this. And instead of a line, just think of them as an infinite amount of points for both functions and then choose some kind of X. That's a X equals three. And then you take this point and you take that point. And if it's in addition, you add those two points. Let's say if this is five and this is minus three than five plus minus three would give you five minus three. I would give you two, and then this point. Would they're the some would move somewhere here, and if it's obstruction than you would have to minuses five miners minus three would be eight. If it's a multiplication, you multiply lamb. If it's division, you divide them. If it's if it's just squared. If you have a parabola and I'm just gonna put points here, and if it's probably squared, then just take one point. Let's say if this point is four and you and then you square for, then you get 16. So that's a good way to get an intuition for for these kind of operations, it's just like with a straight line, because let's say if you have a straight line, it's also a function with infant points. But now you just have something else. Another line, but another shape. Just think of them as points. And then it's easier for you to grasp off what's going on when you when you add functions or subtract functions or multiply or divide them. All right, so I'm gonna give you a small set of exercises now 26. Practice FUNCTION OPERATIONS: So the homework is the following. You have to functions. You have. Why one and you have why? To. And then I ask you what is why one squared plus y two squared, divided by y two. And what cannot XB in order to avoid errors when your calculator or wherever you make your calculations. So you know that there is a certain restriction. What cannot XB All right, try to solve it and then I will show you the solutions. Well, I will start with the numerator, So why one squared? Okay, what is why one squared so you would have something like this. Two x squared plus one. And this is an algebra exercise. So just to get you warmed up with this a little bit, so it would be then four. Remember? First you take this squared says four x to the bar four plus two times this term and then this term. So plus four x squared and then plus one squared, which is one. Then the next one would be Why two squared and you would get it like this three x q plus two squared. That would be then. Okay. Three squared would be nine. Then X cubed squared would be six then plus six X cute times two. So, in fact, it would be 12 12 x cute plus two squared with before, so it would be two squared. The two scores would be four. Right? So I got this term like this that, uh, I multiply two times three x cubed times two because that's the formula. So if you have a plus B squared, then you have a squared bless to a B plus B squared, so you would have 12 x cubed here. And that's how I got this foot. That's how I got this term. And then b squared here with before. All right, so that's the main formula. I should have written it before, but now you have it. So I'm just gonna write down this entire thing again for So why two squared would be nine x to the part of six. And the reason for that is because three x cubed squared would be three squared times x cubed square. That would give you nine and three times still would be six. So it would be X to the part of six. So the conclusion is nine x to the bar of six. 12 x cubed plus four. So that's you're why two squared and why one squared We did it in the same way. Now why one squared Plus why two squared? Well, you would just take those two functions. One function was this and the other function was this well and you just add them, you end them. First of all, you take nine x to the part six and then there no more power of six in these two functions . So you move on, you take plus four x two the par four because you just add those two functions and you know that you cannot add extra the part of six and exit the part of four. Let you cannot do something like this and then get some kind of simply five version of it. You have to leave it in this form. So let's continue. We don't have any power for anymore in those two functions. Then we take plus 12 x cubed plus four x squared and the only thing that we can do, we can add four, 21 Right, so we have four plus one equals five, so That's our big function here and now we divided all by want. Why too. So we just divided all by by three x cubed plus two. Now remember And this is important this denominator here it cannot be zero. So in order for that not to be zero you have a first in order to find the X that in order for that not to be zero you have to find the X that would make it zero. And then you should avoid that X. So you approach it like this. You first make it zero just to find the X. So you have to make it there in order to find, you know, to know what what X should not be. So it will be three X cubed equals minus two. So ex cute would be then minus two, divided by three. I'm just gonna continue to over here X so that you would see what I write. X would be que brute minus two and cue prude or three. And there you have it. That's your answer. Like of course, I can put it in a calculator and have and have some kind off number there. However, it's perfectly fine to leave it in this form. So in conclusion you cannot have X. Which is this? Because if you have X, which is this than the denominator, become zero and you cannot divide by zero. So that's that concludes the ah, the Lecture on function operations like addition, subtraction, multiplication division taking functions to the power. Now, in the next lecture, we're gonna cover a very specific type of functions called Trigano Metric functions, and I'm going to spend some time with them because it's because it's very important. It's a very important topic, so I hope you stick around and see you soon. 27. Get QUADRATIC FUNCTION ROOTS 1: Welcome back in this lecture, I would like to generalize functions even more, and I specifically want to focus on quadratic functions. So you know that we had the straight line function, which was a X plus B. And then we had the parable it function, which was eight times X squared plus be now the general form off a quadratic function is a X squared plus b x plus c no. How did look like See here, whatever. Whatever you have here, see, would just shifted up and down. So let's just amid that and say that it zero. But let's look at those two terms separately. What does it mean? First of all, I'm gonna graph the first term. So I have X and I have why? And I'm just gonna graph a X squared. So let's say X squared is a parable like we saw before, right? So it's a X squared. But what is B X? Will be X is is a straight line function, right? And then, at some point, the problem would get become bigger than the straight line function. No. What? What are we doing here? Essentially, we're taking those two functions right we're taking those two functions and we're adding them up. And remember I told you in the previous video on the operations that a good way to visualize that is to imagine an infinite amount of points instead of a line. And then you just add up all those points. Now, of course, if we add zero and zero, then we would have zero. However, look here in this section if I had this and then he if I look at the straight line, I have a negative value here, right? The straight line is a negative value and the negative values bigger then the positive value caused by the parabola. Which means that if I graph a new function which is there some then even though I will have zero here. But let's see if I take this part here, then I will have something here. And only when these two are equal Thies to distance air equal Onley. Then I will have this this final Let's a minimum point, which means that if I add this term to this term, my proble will move in the diagonal. It will move diagonally. And of course, if I had the if I had a straight line function with with a negative be. So if it was something like this, why equals a X squared minus b X? Then I would have a normal parabola and then I would have some kind off. I would have some kind of straight line function like this, and then this parabola would ah, would still be a parabola. But then this point would move diagonally in this way and of course, Ah, In order to find the specific values, you would have to create this table off X and Y and look at the specific tables and look at the specific numbers for inputs. You choose your inputs and then you put your outputs inside it. But in order to have an intuition, that's how it works. You have, Ah, you have an initial parabola and then you have a straight line function, and as you add them, then it will cause the problem to shift either in this direction or in that direction, depending on the straight line. And of course, you can extend this concept toe bigger, uh, figure functions with higher power. Like for example, I can have a exe cute plus B X squared plus C X plus the and still the one with the highest power is the dominant. Which means if my cue function would be something like this than the shape was still be like this, but it would be shifted and those kind of functions, they're called pulling no meals. So this is a cubic polynomial, and in fact, this is a quadratic polynomial. And the reason why I want to focus on the quadratic polynomial is because when we have a karadic pull, Norman, sometimes we have to find the roots off this polynomial. I'm just gonna create space like that. I'm gonna leave it here so that you would see it. But the the point that I want to make is this if I have some kind of parabola, but it's it's like, let's say it's like that. We don't know the exact function, but it's like that. So this is X, and this is why. And then that's your parable. Sometimes you have to find the roots off that quadratic function, and the roots are when y equals zero. So you need to find these points here, and in order to do that you need to take this polynomial this quadratic polynomial. And now we're not gonna assume that c zero We're just gonna take the general thing. Why equals a X squared plus B x plus e and we're gonna make it equal to zero. Of course, we can make it equal to, for example, to if we make it equal to two than we would have to here and then And then we would find these points. But these are not roots. The roots of a quadratic function are those where why equals zero. So now we're interested in these points and there is a specific formula for that, and I'm gonna introduce it for you now, So forget about this one. We're just going to focus on these two points when Weyco zero, we're gonna find one X and another axe. And the formula for that is the following. It's called X, Want to equals minus B. So this minus B, we take B and we put minus in front plus minus. Sh and you're the reason why it's plus and minuses because you're gonna have two answers. You're gonna have some kind of central point and then and that is gonna be plus something in minus something, and that is B squared minus for a C. And both of them are divided by to a. So you see this part here it will give you some kind of central point. And then this part here will give you like, let's say that distance in one and in the other direction So you will get some kind of number here and then plus from the central point plus that number, you would move here and from the central Point minus that number, you would move there. And, uh, well, let's try an example. Let's safe. My quadratic function is to X squared, plus one x plus to Well, let's let's put four. It's better for then. Ah, my roots would be 1/2 so it would be miners be over to to a so it would be minus four two times to remember a now's to two times two is four plus minus, and then I would have square root. Four squared would be 16 minus four times to times to and all that divided by two times to now notice. There is a reason why I took four because Now, what will you get here? What will you get here? You will get 16 minus four times two is eight times two 16 16 minus 16. It it would be zero. So in this in this way you can see that. Okay, it would become zero here, and this entire this entire term would be zero. So in this rare case, you would have x one would equal X two equals miners one. Now what does it tell me? It tells me that if I have this function here and I don't need that anymore tells me that if I have X here and why here then I have minus one and that means that the proble is here . See? So I have the central point. But that's the only point. And in this case, that was the case when my bees four and and then to is ah, and the two was a and then see was also to. But let's take a bigger Bina 28. Get QUADRATIC FUNCTION ROOTS 2: So let's say that now let's say that now RB instead of four will be, I don't know. 55 Let's take it's five. So we have do. So this was our previous example. And now I'm gonna have another example here. So two X squared plus five X plus two equals zero. Okay, again, let's follow our formula, and our formula would be X 12 equals minus five. So we're following this formula minus five, which is B divided by two times A, which is two times two, which is four plus minus, plus minus. And then our square roots would be be squirt, which is five squared, which is 25 right? Five squared is 25 minus four times a ace too. Times see, which is to now that will still be 16 right? And then I will have to times a two times two, which is two times a which would be the same, like four. Now notice. What will I get here now? I will get here 25 minus four times. Two times two is 16 right? So I will have here nine, which is three. So this guy here will be minus 5/4, which is slightly above one plus minus. And then I would have 3/4. So it's ah 0.75. And in order to graph it, you will have it here. So I have my proble X. Why? And then, uh, my central point would be, let's say here minus 5/4. And then from that central point, I will move toe each side 0.75. So I will move here 0.75 and I will move here 0.75. So because 3/4 is your 0.75. So So I will move here and here, and then I have some kind of proble that goes to these two points. You see these two points? Okay, so what you have found you have found where the probable cuts the X axis. So this one gave you the central point, which is a little bit over one. And then from this one, you go 0.75 that way. So you add plus 0.75. And from this point, you go in that direction when you subtract 0.75. All right, and that's Why? That's why you have to do answers here. Minus five before blowsy open five and minor. 0.5. Okay. And now I want to cover one more case. Okay. Suppose that I have this quadratic function now and I'm gonna make it equal to zero, and I'm gonna Oh, and by the way, there is one easy trick that you can do in the when this quadratic function equals zero. You see, if I do it like this, two X plus two equals zero. Sometimes it's good to just divided all by two. And then you would have well, here you would have zero still. And then here you would cancel out these numbers and look, you will be left with an easier equation. Plus one equals zero. Right? It's easier. And now a would be one b would we want and see is one. So what would be the roots here? The roots here would be again minus B. Okay, I'm just gonna write it down here again. X 1/2 is minus B over to A. That will give you the central point and then plus minus square root B squared minus four times eight times see over two a can. That will be from the central point in either. In either direction, you will go this amount. And in this case, it would be then. Ah, well, miners one. Okay, It's two times one, which is two plus minus. And then here in the square root. I would have one squared minus four times one times one. And here I would have to times one. But notice. What will I get here? How much is one squared? Minus four. It would be minus three. Right. You cannot take a square root off a negative value. Because remember, from the inverse functions I showed you that. Okay, if you if you take an inverse function Oh, for a parabola. And then now this is the inverse then. Okay, You can either have this or you can have this, but you will never reach the negative values. So you cannot take the square root off miners three. Now what does that mean? It means that the solution does not exist. And graphically, it means this. It means that the proble is somewhere else. It means that the problem lies is somewhere above the x axis. But you see it never. It never touches the x axis. It never crosses the X axis. And therefore you can't have solution because the probable does not cross it. And this formula is is very useful. And at some point in the course, we're gonna be using it. So just keep that formula in mind. And now you also have the intuition for why, For some cases, you don't have solutions. That's because the proble does not cross the X axis. And sometimes you only have one solution because the problem only touches the X axis once barely touches it. All right, that's all. Thank you for attending the lecture. See you next time. 29. Which terms DOMINATE: there is one more case that I want to make. Earlier, I said that when I have a polynomial, let's say a X cubed plus B x squared plus C x plus plus de I said that this term is dominant, right? There's what I said that like in the end, in the end, if why equals a cubed, if that would be the function right then I would say that if I said that if I add the's terms then in the end, in the end, the function would still be something like this may be shifted, but still it would have the same shape. No, that only holds true. When you look at big, big values you see turn variables with huge powers, they place more importance to big numbers and less importance. Two smaller numbers. For example. If I have a straight line function here, then you can see that off course. When I have big numbers then, of course, this cube function would be the dominant force. But if I have this small numbers, then on the contrary, that on the contrary, this small of the number, the closer I get to zero the most dominant will be this one, and then it would be this one, and then it will be probable, which would be would be something like this. So some point it would be at some point it would be bigger, bigger than the Cuba function, right? But then, and but smaller than the than the straight line function. But then it will get bigger than the straight line function, and it would be, and it would become smaller than the cute function. So the general point that I want to make is this. If you have a big power here, the biggest power gives the largest dominance to big values and the smaller the numbers. If we're close to zero, in fact, then other terms will dominate. In fact, the straight line, the horizontal line would be the most dominant force then, and then a straight line with that has some kind of slope would be the dominant and then the proble. So in other words, if you have big values than dominance will go from here to here. So this is the most dominant, and this is the least dominant. If I have a very small values, then the dominance goes like this. So this would be the most dominant force. And this would be the least dominant force. And you can think about it That Okay, if I have 0.1 and if I take it cute and then I take the same value to square and then I just do it. 0.1 You see what I mean? Like here. In the first case, I would have 0.1 times 0.1 times 0.1 So that would give me a very, very, very small number. This number would be 0.1 time, 0.1 Still a small number, but bigger than this one. And this is of course, biggest. So that's what I want you to understand is that the closer you get to zero if you're if you're dealing with small numbers, then the dominance goes from here to here. And if you go and you have bigger numbers than then, of course, this term would be the most dominant warn. And that's why when you have this Cuba polynomial, in fact, what you have is that. Okay, If you look at very big values, then indeed it will be something like this. However, it will be something like this one, maybe to and three. So you see, So you see, like if you look at huge, let's say this is the number of years, right? So if you look at it hundreds of years, let's say we have hundreds of years like this is present miners 100 in the past, plus 100 in the past years. Then okay, you would see something like this, right? Indeed, you would see something like resembles into acute function. But when you go into on a one year or maybe half a year, then you'll see that Well, in fact, ah, if you're going to 0.5 years or I don't know, maybe days maybe, or 0.5 years. Like if we take thes years and we chop it in two days and we from the president, we only move a couple of days to the future or to the past, Then if we're in this region, you see that this term is not very dominant. So at some point, at some point, a dominant force would be the parabolic function. That's why you see something like a probable here. Then, at some point, when you get closer to zero than this term, the straight line function will become dominant. And then, at some point when you're very close to zero, right, when when you're very close to zero than even the straight line function is less dominant than, for example, ah, horizontal line. Because the horizontal line is bigger than a straight line function. So here this turn would be dominant. Then, as you move away from from zero, then this straight line function will be more dominant than at some points. This term becomes very dominant. And when you have big values, then of course, nothing can compete with this one. Nothing can compete with the first term. So I hope I've clarified it a little bit. All right. Thank you. And see you in the next lecture. 30. SIM: 3 airplane race - who wins, and when?: Welcome back. This is just another simulation to show you the fact that functions that have lower power are more dominant when the domain values are small. And functions with a higher power are more dominant when the domain values are large. So if you just look at this one, this is the square root function. It has bigger values when the domain values are small. And now the straight-line function is bigger. It has a bigger range when the domain values are bigger as well. And it's always cool to associate it with a real life example. So you can also imagine three airplanes racing. And the middle one follows the square root function. So it has a high initial acceleration. But then he just slows down heavily. And the two other airplanes with, with a straight-line function, they will just pull ahead. So that's it for now. And I'll see you in the next lecture. Thank you very much. 31. PI NUMBER & CIRCLE CIRCUMFERENCE: Welcome back in this section. I'm gonna teach you everything that you need to know about trigonometry functions. So let's get started. Let's imagine a circle. Let's say a ring or whatever Circle body, Right? So let's just drawn Ring a circle and what do we have with a circle? We have a radius and let's denote it within our and its length. Let's say that it can be one meters. It can be five meters. It can be as many meters as you choose Now the circle also has and diameter, you know, And in addition to that, we know that the diameter consists of two Radi I So if I put another radius here, then two times radius would give me a diameter. So this is it the amateur and then this is a damn Teoh and they're equal Now if I take this radius and I turn it like this, then it will be 300 and 60 degrees, right? So what we have in the universe? We have a lot of different circles with a lot of different radi I with a lot of difference diameters but they're all 306 degrees, so that is that one thing that unites old circles, they all have 360 degrees. What if we want to compute the circumference of a circle and what is the circumference? Well, sir, Conference is a unit of length as well, either in meters kilometers millimeters. And it's just if I start going along circle like that and I reach back here, then I get circumference and it's the length off the outer edge of the circle, and I would measure it in meters, kilometers, centimeters, etcetera. No, the fact that all circles have 306 degrees gives us nothing in order to calculate the circumference. So we need another way. Now. Is there something else that unites all the circles in the world and in the universe, for that matter? Well, turns out that there is because if I take a random circle and let's say I think it's a conference and I denoted with see one and I take the diameter of the circle and like to know with the one and then I take a small, a smaller circle, and then I have C two and then I have diameter Andy, too. And if I take a very big circle. And then I have C three circumference is three. No, I mean conference. We denoted with C three and can be with the length and in 1000 commenters, or maybe the the cross section of some kind of planet. And then it has its own diameter as well. Bamut of three I turns out that in our universe, all those circles are united by the fact that if you take there should conference and divide by their diameter. And they that ratio is equal to another circles the conference over its diameter. And again, another circles sin conference divided by its the amateur. In fact, you can take any circle in the universe. And if you calculated circumference and divide by its own diameter, then it would be equal to all other circles ratio and that ratio waas experimentally found to be three hundreds, three points, one four. And in fact, it has an infinite amount off decimal numbers. And of course, that number is pie. The magical pie? No. How will we calculate the circumference of a circle then? Well, by is circumference over diameter. So this is a ratio pie. So in order to get some conference already from the units or from the notations, you can see that all I need to know is the diameter of that circle. And then if I multiplied by pi, then if I imagine pie as a ratio circumference over diameter, I cancel out diameters and I get this a conference. So all I need to know for a circle is its diameter, and then I can calculate its two conference. So that's why Sinn Conference equals pie diameter now. Diameter, however, is two times radius, so I can also right circumference equals to pie are right to our pop to our pie. That would be our formula for circumference. And now you know where it's coming from. Now. Another thing that I want to mention is that pi is an irrational number. Now, what is an irrational number? An irrational number is a number that cannot be expressed with a ratio A and B. It's impossible to express an irrational number with the ratio. No, another irrational number would be square root of two. It means that you're you're not able. You're not gonna be able to to express this number as a ratio. You just can't do it, there is no way to do it. But the irrational number would be a number that you can express in a ratio for example, 0.333 And it would go in into infinity. An infinite number of threes. Well, you can expresses very conveniently, with one third but not square root of two in that pie. But wait a minute. They would just say that pie. He's a ratio off circumference over diameter. All right, so how can be an irrational number? Well, it is an irrational number, and the reason for that is very simple. You see, we can always go more precise. With this ratio, we can always take a more precise measurement device and and then calculate a more precise sue conference and also more precise diameter. And then this conference, this number will change slightly and diameter. We'll also change slightly, and then pie will also change slightly. So if I have a circle and then GAC would the circumference. And let's say that I use a device that is more precise than the device I used last year. Then I will have, let's say, more decimal numbers for the same conference and for the diameter. So if I think the ratio part is not something that is constant, it let's say these numbers here they ultimately they're not fixed. You see here they're fixed 1/3. That's always 1/3. It will never change. And if you divide them, you will get 0.333333 up until infinity. But with a pie. This ratio it's a conference of a diameter. Those two numbers that can always be measured more precisely and therefore they're not fix and therefore pie is not fixed. It's an irrational number. And an irrational number, of course, has an infinite amount off decimal numbers after its point. And that makes sense because there is no other way an irrational number has to have, and even in an amount of decimal numbers, because if you have some kind of number, let's say 3.1 for 75 and I and its precise and there's nothing after five, then I can express it as 31.475 over 10 so I can express it is the ratio, and in that case it's a rational number. So an irrational number, by the definition is a number that has an infinite amount off numbers after its decimal point. And maybe a good way to visualize the entire thing is that if I have, ah, some kind of number line and I measure pie on the number line and let's say zero here and well, it's not up to scale. So there's and gap here, so I'll just say that OK, here its 3.14 and here it's three point 15 and the logic behind it is this. The more precise have become with my devices. With my measurement devices, the more precisely I will be able to determine my pie in between 3.14 and 3.15. And right now I think the record is that you have 2.7 billion decimal numbers after 3.14. She can go very precise already. But of course, for real life applications, you will never need it now doesn't mean that we're unable to express a circle same conference with a 100% precision because pie is not the pilot we have. It's not 100% precise. Let's remember, pi is approximately 3.14 approximately. Well, no, we can express suit conference. We want her percent precision. If we say that radius is five meters, well, that's a conference would be five times two Hi will be 10 pi meters and we just leave it like that. We don't multiply 10 times 3.14 which would be 31.4. We don't do that. We just leave the expression having pie inside. And if we really need to know the entire number, we just multiply it by 3.14. But for duration, off equations and for scientific simplifying equations, we just lead as pie. 32. RADIANS VS DEGREES & ARC LENGTH: and now I want to draw your attention to something. We know that in order to calculate this conference of a circle, we had this formula two pi times are two pi times are But look, essentially. It's a straight line function, right? Because if I roll this actions here and I called are and here's a conference is C then Well , I have a straight line function here. And do you remember? You can get the change Delta psi over delta are would be to pie. So what does it mean? It means that if the radius of circle increases by one, then the circumference of a circle will increase by two pi, so it will be two pi meters more. So, for instance, let's say that if my first circle was with the radius of five meters than my first conference would be, too pie five, which is 10 pi meters. But then if I increase the reuse by by one meter, let's say now six meters, then it would be C two two pi times six would be 12 pi meters and you can see that there difference See to mind See you one would be to pie. All right, so that would be two pi meters. Okay, so that's one way to look at it. So we have a fixed circle, and no, we have a circle. And if we vary, the radius of the circle than the circumference off the circle will change. How about how about if I'm not interested in in? Then what? What if my circle is fixed, the radios is fixed, and what I want is is to calculate the length off a section of circles. Sick conference. In other words, I want to calculate some kind of arc. Let's see if I take 90 degrees here. I want to know what is the length of this ark? Well, in order to do that and let's denote the with the length hell, Oh, In this case, if that's 19 degrees than one way to do it would be 90 over 360 which is 14 times conference, So you would have 1/4 times coffers or through a conference divided by four. Right. But now I'm gonna show you another alternative to look at it, and I'm gonna introduce something called radiance, which is extremely important food so it's very important, very important to pay attention to that. Now I Look, what I can do now is that I can keep are fixed and I can say in this formula, I can say Now I keep are fixed and I let two pi renamed to pious feta. So if I very seater, then I will get length because he 50 Sita. Then I have zero length. I get our clinks. So the function here would be length equals our, which is now fixed. I'm just gonna put F here which fixed, let's say five meters and it's fixed times feta. So it's it's another function now. I very are very this part of the equation. And of course, if I get if I get ah two pi, then the length equals the circumference of the circle. Then if I have pie, then it's half of the circle. And if I have 90 degrees, which is pi over two, then I have 1/4 of the circle, and now this angle here. And of course, if I want to make mawr, let's say more circles around the circle, he found. If I want to make more than one rotation with the two rotations. Then I would have here later on four pi. And in fact, in fact, feta can be very not from zero to infinity but from minus infinity to plus infinity because I can take this radius and I can turn it in this way as well. So the convention is positive. Rotation is counterclockwise. So you go like this, you think of a clock and you go like this. But, uh, negative irritation is clockwise. It's like this. So as you rotate clockwise on this axis, you move to the left. And if you wrote the cara clockwise like this, then you move it to the right along this axis, you move to the right. But of course, physically, physically, you cannot have Ah, negative lengths. So we'll just well, just ah, start, uh, measuring length. Stunning from feet equals zero, and we go in this direction. So we so we will. In fact, you could have this absolute value function where you have length like this. And then as you as he wrote the clockwise, then your length would still be like, mirrored, mirrored, like this line over this length access. Okay, but Now this is an angle, but it's not degrees. It can be because two pi is six point. The something's explained to and its center, because pie itself was 3.14 was 3.1. So then two pi would be six point 6.28 right? And if we have degrees, then we would have something like Circumference equals to here in 60 times radius. But that's not the case. So for one circle for one, every time we complete a circle, it's two pi times radius do by six point to wait. Now it's just another type of angles, and we called radiance. So radiance is just another way to measure angles for a circle. And they are very useful because now, if we want to calculate the arc length off a circle, then we just take l equals and we take the radius, the fixed radius of circle, whatever circle we have, and we multiply it by that feta And then, for example, if you want 30 degrees, then that would be theater would be pi over six radiance that would be equivalent to sir 30 degrees, and then you would just multiply the length off 30 degrees and let's safe radius is five meters. It would be five times pi over six meters And how to get a relationship between radiance and degrees. Well, we have this function here. We have l equals 9/90 over 360 degrees, right time C. But C is two pi r, right? So we cannot apply to like this to buy our okay, But then we can make so this the degrees. This is the reason we're interested in. And we have just found out that the length also equals radius times. They're radiance. Now, if this and this are equal, which they are, then I can write it down like this are times feta equals 90 over 360 times two pi are so what I can do now I can cancel their radi i It's one radius on both sides so I can cancel them out because they're equal. And I can take this guy here and I can just make it as a variable And I let it vary just like I let very feta. And there you go. I have this equation feat that equals two pi over 360 times degrees, or FIFA equals pi over 180 because this is one and this would be 180 times degrees. And as you can see, it's also another function. It's also a straight line function. So I have radiance here and I have degrees here. And the relationship between them is pi over 180. It means that that the change off radiance with respect to one unit change of degree is pie over 180. 33. Origins of SINE FUNCTION: And now that you thoroughly understand where radiance come from when conference comes from , we're equipped to start covering the Trigano metric functions. Okay, so let's imagine that you have a bridge. They have some kind of bridge here. And Kate, it might have some trusses. And let's say let's just consider this triangle here, okay? And we're gonna do you know, this is why And we're gonna say that why equals three meters and and we're going to say that this length here. So from here till here, we're going to say that this is X, but we don't know the X. We don't know it, but then we also have radius. So we're well say that this distances radius here and we don't know radius either, but we would like to know it. What we do know is that the angle this angle here is pi over six radiance. So that's the bribe. That's the abbreviation for radiant rot. Our a D. Okay, so what can we do? But what People, Scientists, mathematicians, What they did in the past. They took a random circle, and they may they said Okay, this is X, and this is why and then they drew a random circle. Of course, you have radius here, which starts from here and you have radius. And then what they did They take that radius and then they rotated this radius around the circle and every time they rotated it and every time and I mean really like they just moved just a little bit. And then they made some measurements. Then they moved. The Raiders used a little bit more and they made some regimen measurements. So let's just say that you have a random angle here. And remember, if you go from here to here, then then ah, the angle would be pi over two radiance here. Radiance would be pie. So once the radius reaches here, it would be three pi over two. The reading Not the not the length of the radius, but the angle. All right, it's It's this STI angle theater. The theater here would be three power to And then finally, once we reach the entire circle that Sita would be to buy. So every time they rotated the circle just a little bit gonna draw it again, maybe bigger now to illustrate the point. So let's say my radius is here, then I have some kind of why here. Right then. I also have some kind of X here. And then I have radius, but radios is fixed. So what they did they made Okay, let's make a lying on access. And we called it radiance. And then I have another access here. And I'm just gonna say that this axis will be why dividers by radius and radios is fixed. It never changes. However, as I change the radius of the circle, you can see that Why always changes on We don't. We don't care about the X right now, so we just care about why. And you can see that radio stays the same. But why always changes. And here is well, why always changes. It is different in fact, here, why is negative And here why is negative as well. Okay, So as I rotate and you already know that if you wrote it in counterclockwise, then you go in this direction and I have to try here. Hi, pi. Over two three pie over to. And then if I go in the other direction in the core quiets I would have minus pyre for two minus pi minus three Pi over two and miners to pie. Now, if I measure it, if my feta is zero, then what is why? Well, then why zero and zero over radius must be zero. All right. And if my radius is here, If my radius is here, then what would be my Why? Well, it would be equal to our right. So in this case, why over arm would be one. Because why would be Why would have the same length like our particles to it would be one. And then with pie, it's again zero right, Because why zero three pi over two? I would have miners one, because why is now mine ish? Ah is a negative number and equal to our. So I would have Why over our is one. But since why is negative? It has to be minus war and two point again. Zero. And if I rotate in the same in the opposite direction core Khweis then it would be like this because now why is negative minus pi? Why's zero? I remember it. It's a ratio. So you divided by R. That's why it's mine is one here and that's Why? Once you reach minus three pi over two, then your why is a positive number divided by are their equal. So the ratio would be one and minus two pi here. And what they found out was that if you rotate this radius just a little bit and you and you make a point for each possible case for why over radius, they you will get a function something like like this. It's off course. All that should be equal. It's just it's my drawing. But this is all perfectly equal, and this function has a fancy name, which is a sign function. So every time you put a calculus, every time you take a calculator and you insert the 30 degrees inside the calculator and you take a sign of it with the calculator, does it converts 30 degrees into radiance, which would be, in this case, pi over six and then go somewhere here, Pi over six and checks out the ratio of while we're are and it will know it will get 0.5 in this case. That's why sign 30 degrees is Europe when five. But the 0.5 is nothing else but the ratio. Why you are over over the radius and 0.5 means that why's half of the radius? Okay, Makes sense, right? No, I'm gonna claim that. Okay, this was made for Random Circle, but I'm gonna claim that this function this relationship of this ratio and the radiance it holds for all circles in the universe and I'm gonna prove it to you and to prove it. It's very easy. And I'm gonna use an equation for a circle radius squared equals X squared. Plus, why squared now? That means that radio radius equals square root off X squared, plus y squared. Okay, so what if What if I'm just gonna multiply radius by a number? Whatever number 1.5 zero. I mean, 0.5 123 Whatever. Now, if I multiplied this radius by end, I will also have to put in here in order to make algebra work and satisfy the quality condition. Now, if I do that, then I can also do like that. So if I put in inside the square root, then it will be an squared. And the parents is X squared, plus y squared. Okay. And are equals and I will open the parenthesis and I will have n squared, X squared, plus and squared. Why squared and our equals? And I can write it like this and X squared plus and why square? And there you go. If I have some kind of circle, some kind of circle and I have radios are if I multiply that radius by a number, that's it too. Then you can see from this equation that also whatever x I have and whatever why I have, they will be multiplied automatically by the same number. So if Ennis to then X will also be multiplied by two and why will also be multiplied by two ? So now we can go back to our original example and we can find out what are is well, we know that if if the angle is pi over six radiance, then pi over six was is he opened five. So the ratio off why divided by R is 0.5 No, If I take that ratio while our and I multiplied but are then I will get why. But I'm interested in are so very simply I think are and then I simply divide. Why, by this ratio, which is why over our so the radius or in this case, just the length off this triangle would be three divided by 0.5, which means that it would be six meters. And there you go are here in our example it's six meters. So we use this function the sine function, to utilize the information about the angles and the information. But why to calculate are and we didn't have to know X and it will be super useful in dynamics, just like the concept of radiance. 34. ORIGINS OF COS & TAN FUNCTIONS: However, if I take my circle, I draw my circle here, then Okay, this is X. And this is why. And let's say if I take if I move this radius here, then Okay, we could take this. Why? And we could make a ratio of why over our but on the other hands, I can also take X and make a ratio out of it X over our so I can take the same procedure I can. I can take this line and call it feta, which is radiance. But now I can have and access here, which would be X over our And as I take this radius and turn it either counter clock ways or clockwise so counterclockwise would be right and moving to the left would be clockwise. In this case, you would see that if feta the rotation zero radiance. So there. So we start from here. This is our starting point. So we start from here once we start rotating the radius here, we see that once we haven't roped. When we haven't rotate anything yet then and then we'll exes X is equal to radius. Right, So here, X and radius is the same. So it has to be one. And if the radius is here than actual V zero and zero over arm would be zero pi over two, it would be here, and then you would have pie here. Three pi over two. And okay, I'm gonna extend it a little bit, and then I'm gonna have to play here and then minus pi over two minus pi minus three pi over two and minus two pi. And look, now I have dysfunction like this. This is minus one. This is one when the rotation is pie than the radius, he's here, so x is equal toe are but X is negative. So when I'm at Thai, then my point is here reply over 20 and to pile would be one again. And then there you go. My function is it's like this and not this. It's It's like the previous function, right? It's like the previous function, but shifted. So let me just, uh, mark our current function. This would be our current function and, as you can see, shifted but pi over two with respect with the previous function and this function when we take a ratio, If X over our well. This this function has also a fancy name, which is cause I So what you do in your calculator you put call sign and then either in degrees already in radiance, you put an angle inside it and let's say you put radiance and then it will give you this ratio, which is X over our. However, we can also have 1/3 alternative. We can also create this PSA graph, which would be it would be radiance here, right? And then as a ratio, I can have I can completely omit radius. And I can just say that, Okay, this racial will be why over acts. So as this radius rotates, then I will have a ratio of while Rex here. And of course, then if radius is a zero radiance, right, then why would be zero? So why over X would be zero now. The other thing is that once you have pi over four, which is 45 degrees, then let's say that this is 45 degrees. Then here, why an extra equal right, so at 45 degrees in this case, you would have one. However, what happens when you approach 90 degrees. What will you have them? Well, if you approach 90 degrees, then ex become zero. But you cannot divide by zero, right? So you would have some kind of why. But X would approach zero you can divide by zero. That means that every time the radius is either here or down here, which is three pi over two, you can't have a value there. And in fact, this is something called a tangent function. And with a tangent function, I mean, just writing down here with a tangent function, if you put 90 degrees in the calculator will give you an ever and that's why it will give you an error because you can't divide by zero. So every time you are pi over two or you are at three pie over to and also minus pi over two minus pi, etcetera. Every time you're here or here, you can't have a value there. So that's why the tangent function has this kind of shape. So the slope with this line. So if you're here and then you're closer here than the slope, the why over X would approach infinity because as X becomes smaller and smaller this entire thing would become bigger and bigger. So as X approaches zero why? Why Over X is a ratio with approach infinity. But it never touches power to you have nothing at PI over two and the same thing with the negative values and then at part it's the same thing. It's never touches three pi over two and pi over two. And here is Well, yeah, So this is a tension function. That's why you graphic like that because you're you're not gonna be able to reach power to . 35. INVERSE of TRIGONOMETRIC FUNCTIONS: And now I would like to go over the inverse functions off the trigger. Metric functions. So what I have done here, I have, uh, drawn all the three Trigano metric functions again. So you have the sine function that also lates like that. You have the coastline function that is simply shifted with the sine function by pi over two. All right, so this is just a difference between the sine and cosine function there shifted, but pi over two. You can see that here it zero when ah, radiance is zero. And when and in the course, in case when the radiance zero, then then, ah, the function has shifted by pi over two and then with the tension function, it was different because the tension function was the ratio of y over X and when you had an xer approaching zero and you would know it by looking at the circle. So when you have X here, why here? And then as you get closer and closer to this access as the radius as you turn it, good those rounds as it gets closer to this axis or to this access X approach zero. And since you can't divide by zero. Therefore, when when you approach the angle pi over two or 90 degrees, you simply cannot have a value there. So that's why you can see that an angle such as pi over 23 pi over two which is this one here. So this is three pi over two and the minus pi over to that is if you wrote it in the other direction in the clockwise direction and minds three power to would be like this. Then you can see that in these in these values, when radiance equals these values, you cannot have a value there. You cannot have ah ratio of why over X and therefore they do not touch it. Now, in order to find the inverse function off these three functions, will we follow the same procedure? We take this vertical axis and we turn it 90 degrees clockwise and this horizontal axis 90 degrees counterclockwise. So, for instance, for the sine function for the inverse sign function and by the way, the inverse functions they are denoted either like this and then you put and then you put the ratio in, like why over are here or you can also do it like this arc sine. And then why over our and the same thing with co sign and minus ec with coastline intentions, so would because signed miners one. And then you would put something here and then you can also write it as Arc co sign and same thing with 10 Jin tension miners one and Arc Tangent and inverse function simply means that you put that ratio in. So now this ratio is This ratio is like, uh, like, the independent value, and then you get the dependent value, which would be some kind of radiance. Okay, so how would we grab to sign the inverse sine function? Well, you you can already I guess that this access would be why, over our and then this access would be radiance. And as you can see from here, it also is between miners one and one. So therefore, you will have miners one here and plus one here and no, remember, ex new equals wild and why new equals x old. So as you go like this as you go up here, it's the same thing that going to the right from here and then you can see that the function up until pi over two would go. Something like this happened Teoh plus warm and here up until minus pi over two would be like this. Okay, so that would be the inverse function. And of course you might think like OK, but here it also is. Why doesn't it also it here? Well, here, you should just remember that. Okay, If you have vertical line tests, then remember that that line the vertical line needs touch the line of the function only once and not twice, not three times, and not more So therefore you have to just leave it here and same thing with a coastline function the inverse coastline function which would be X over our or here. And you would have ratings over here. And in fact, in fact, I should longer because because my angle pi over two would be here if you are at pi over too well, in fact, if you are, if the ratios at zero, then you're a pi over two. And if the ratios mine is one, then you will be at pi. All right? And so your function would be something like this and again. It cannot ossa late because otherwise it wouldn't be a function. And the final thing is, then the tangent. The inverse tangent function where you're ratio here would be why over X and here again you would have radiance. Now, unlike with the other functions here, it doesn't oscillate between miners one and one. Here it goes from minus infinity to plus infinity. And and what you can see is that it is bounded by minus pi over two and pi over two. So I can put some boundaries here. And as I go up or right, you're goes to the positive side as I go left here or down here, it goes to the negative side. And there you go. So this is the inverse tension function. And of course, you might think that. Okay, but I have these guys here as well. Well, when you take the inverse function, you just need to choose one of these and then your other graphic here or your shifted up. And of course, there is a natural question you might ask that. Okay, I have a circle and I have four quadrants. So when I have the ratio, how do I know which, which quadrants I'm talking about. Well, one thing is, of course, that purely from this you can know it because it just gives you the relationship between the ratio and the angle. But of course, the angle can be in this quadrant, or it can be in this quadrant or it can be in this quadrant, so you kind of have to know it from the context. It is easier with a inverse tension function because, well, over there, you know that if you're ratios, why over X and let's say both of them are positive. Then you know that you are in this quadrant because here both action. What are positive if you have a negative Y and positive X, then you know that you're in this quadrant so that the the angle that you really get is from 270 till 360. The grease, And if both of them are negative minus y and minus X, then you must be in this quarter because why and ex, they're both negative. And if only X is negative and why it's positive then. Then you know that you're in this quadrant, so yeah, you could. Knowing this information, you can know whether you are in between 0 90 degrees between 90 or 180 degrees or between 182 170 degrees or between 217 360 degrees. So then you know that. Okay, If I have some kind of ratio, then then in which quadrant I am, I can know that now, of course, with sine and cosine functions is a little bit more complicated here. This function, it only covers from minus pi over two, two pi over two. So So it's either if you rotate 90 degrees this way which like this or nine degrees this way, which is this. But it could also be this or it could be miners 100 Let's say 70 degrees. If your radius is like this, then it could be mine is 170 degrees. So from purely from the easy inverse functions, you can know it. You would have to know the context off your specific problem. In most cases, you do all right. And now, in order to finish up with the trigonometry functions, I have a small exercise for you. 36. Practice TRIGONOMETRIC FUNCTIONS: So I have a circle here and the radios has rotated. Caracol quiets 220 degrees and the radius is 10 meters. And the two questions are How much is 220 degrees in Ah radiance. And what is the Are clank So trying to do it yourself. And then I was sold for you. Okay, So in order to get the radiance we drive that formal which Waas Sita equals pie over 180 times the degrees. So theater would be pi over of 180 times. 220 it would be 11 hi over nine radiance. And now, if you want to get the are clanks of it, then you simply take the radius and your multiplied by Sita. So in this case, it would be 10 times 11 pi over nine and that would give you 110 pie over nine meters. And the number simple exercise is this one. So you have a triangle here, and, uh, you know that the rotation or the angle here is pi over four, which is 45 degrees and you know the why what would be the X? Just try to get yourself and then I'll show it to you. Okay, so here the most reasonable thing would be to use a tangent function. So we know that why over acts the ratio when it's pi over four or 45 degrees from tangent function, we would know that it's one. In other words, if you put inside your calculator, you put tangent pi over four or 45 degrees. You'll get one. Okay, since that's a ratio and you need X, then you know that why over x Times x I would give you why now If you want X that you simply need to se X equals why and you divided by this ratio. Why over X But why over X is one. So why would be why divided by one and it would be X so it makes sense. Why equals X and since why six meters X is also six meters. This has these have bean simple exercises just to refresh the concepts a little bit, the gold discourses. To give you an intuition, I want you to know what you're gonna magic functions are, and I want you to have an intuition told with those concepts. Now, the next section in the next section, I'm gonna show you one real life application from the IRS space engineering in the street where we take one bizarre function. In fact, its function where we had one over X and we had these kinds off grafts that went to infinity. I'm gonna show you how we use it in in the in the Irish based engineering industry. And after that, I will cover functions that are multi dimensional. And, uh, I know it might seem a little bit, uh, non intuitive now. No worries. I will make it intuitive for you. All right, take care and see you soon. 37. AEROSPACE Example: Welcome back in this section, I would like to revisit the topic that we had previously, which was about the abstract vs reality. And the reason why I won't include this topic is because I thought off a good example a very practical example in the aerospace engineering industry where one bizarre function could be used. Do you remember In the end of the abstract versus real section, we talked about a function that waas similar to to this one. However, it was in reverse. So I had the line over here and then I had the line over here and then those lines they would go to infinity, right? Well, if you have this function and this function was, why equals one over X now that gave us dysfunction. However, if you simply put the miners here, why equals miners one over X, Then everything will be turned around and you will have a line here and the line here because remember, if you just multiply something by one So if you have a number here, it would be just mirrored about the X axis and the same thing over here. No. Having said that, the goal of this lecture is to give you Ah, practical example Were a bizarre function like that could be used. Okay, so imagine that you have a wink. So this is a three d wing that I drew myself. And as the wing flashed through the air and this arrow here with V, these velocities stands for velocity and velocities speed with the direction. So speed would just be how fast something goes. But velocity would also indicate the direction in which direction you go do go north or south. So as this wink goes through the air, it generates some kind of lift. That's why airplanes fly because they have high velocity and their wings. They're shaped in such a way that those wings produced some kind off lift up boards and that keeps the air that keeps the airplane in the air. And the way the Irish based engineers calculate this force that we measured in Newtons. This left is with this formula 1/2 times roll times the square times this time seal. Okay. What is roll roll is air density and air density. The units of air density are kilograms per cubic meter. So that would be the units for for the air density and a good way to visualize it would be like this. So suppose that you were at the sea level. You're at your altitude a zero kilometers, and then you take one cubic meter off air, one cubic meter off air and you can see that inside this one. Too big meter. You have a love air molecules. However, if you're at an altitude off 10 kilometers and you take the same one cubic meter of air, you have a lot less air molecules inside that cube and therefore, the higher you go, the less mass one cubic meter of air has. Therefore, when you are closer to the sea level, your air is more dense. And then if you're higher than its less dense no, the more dense your air is, the higher your lift, the more force doing generates upwards. Now the is velocity. And here in this formula, V has to be squared. And, uh, you from this formula, you can see that since the ISS squared the lift, the force that keeps the airplane in the air, you see that a bigger contributor is the is the velocity because it's square so it has more important stand. For example, air density, that s is the wing area. So it's the entire area off this wing. And then finally, cl seal is just a number, so it's just a lift coefficient. It's a list coefficient that can be 0.511 point six and the list coefficient. It depends on the on a wing. It it's ah, it's a characteristic off a wing, so a lift confession depends on the shape of the wing, but it also depends on the angle of attack. Now the angle of attack is an angle that the wing makes relative to the air flow. So if the airplane flies in that direction, so the airflow hits the wing from this part from this side, however, the wing is tilted upwards from the cross from the Middle Cross section of the wing. Well, that's at that angle. Here is called the angle of attack, so C L also depends on that, and old and seal also depends on a Mac number. And what is a Mac number? Well, Mac number is just a ratio. It's a ratio of this speed off the airplane. Well, actually, velocity. I shouldn't be more correct. Here should be velocity. So the Mac number in the our space industry is, ah, velocity of the airplane. With respect to speed of sound and the speed of sound, the is more or less around. I don't know the exact number, but let's say it's 1000 kilometres per hour, so you can look up this information online. But the Mac number, by definition, is the velocity of the airplane, divided by the speed of sound. And, of course, if the Mac numbers one, then the airplane flies exactly at the speed of sound. So if you produce some kind of sound, let's say if you scream and if the airplanes Mac numbers one that your scream travels through the air as fast as the airplane. And and, of course, if it's if Mac number is more than one, if it's 1.4, then your airplane would fly faster than your scream. And if it's less than one than your screen with win the race. And so there are certain sections. Most passenger airplanes, they fly. They fly at Mac 0.8, and then the mag zero. When their plans were lost zero up until 0.8. That is considered the soft signing region. The supper supersonic region is considered when you fly faster than this speed of air. And, for example, that there was that airplane Concord, Concord was flying faster. I think it flew even faster than Mac two, which is twice the speed of sound and then from 0.8 till one point to this region is considered trans sonic. And so, what do you see from this graph? Well, first of all, you can see two types off lift coefficients here with a capital L. And with a small L Well, with a capital l ah, that would be the lift coefficient for the entire wing. However, if I take a cross section off that off that wing, let's say I just cut it in half and I take just the cross section of the wing. And I don't consider this dimension right. I only consider two dimensions. I don't consider the third dimension. So this is a three d wing, but I only take the cross section of it so I only consider two dimensions Then Then I denote this lift coefficient with a small L. And the reason for that is that they're not exactly equal because there are some certain, more complicated relationships between them. However, the list coefficient for this two dimensional cross section also depends on the shape off that cross section, the angle of attack off the cross section and then the Mac number. And and so on this graph on this graph you can see something like, Ah, corporate compress abilities effect. In other words, this graph shows you in other words, shape is obviously constant, but also, the angle of attack here is constant. So the only thing that we allow to vary is that Mac number. So we are investigating on this graph how this CL changes as I vary the Mac number and you can see that this dream like it's the rial physical thing. That's how we have experimentally established that how cl various was Mac number. However, now we want to mathematically model it. And earlier I told you that one way we could use bizarre functions like this is that we can just take certain sections of it. We don't have to go till infinity. We can take certain sections of it and then we can use those sections to mado a physical situation. And here we're taking this part and we're modeling this part of the compress ability effect or how how cl changes or behaves with respect to em. So how SEAL is related to M So, up from 0 to 0.8, we can use this function well. In fact, it's a modified function and I've shown you how I modified it because the real formula off cl is is this 12 pi times angle of attack over square root one minus Mac number squared, right? So what I did here, I simply took the denominator, and I and I just defined that it's minus X. And I've defined cl as why and essentially what I'm doing. I'm taking this bizarre function. I'm thinking one section of it, this section and I'm using this line here to model this part off the physical situation. And as you can see, as the Mac never becomes greater, the error becomes larger. So this is my mathematical situation, and this is the real physical situation. And you can see that as my mic number increases, the less precise my function becomes so I can just say that. Okay, I'm gonna finish using this function here and they hear I used something else to model this green line. And here I can also use something else tomorrow, this green line. And that's how it's done in real life. You you have a physical situation that you have that you have established using experiments , and then you can have different functions describing this physical phenomena in different places. So in this region you can use this function in this region. In the trans sonic region, you can use a different kind of function in the supersonic region. You would use a different kind of function, and then you take difficult of functions. You put it all together, and then you can describe a physical situation using different functions. So this is one example how how it is done in real life by engineers, many different functions, the take sections of it. And then they put it all together, and that's how it's done. So I hope it gave you a little bit off inside off how to how all that is used in practicality. See you soon. 38. SIM: Avoid the crash - exercise: Welcome back. I have a very interesting exercise for you here. What you're seeing is two green cars from the top view. And they start driving from here. And they follow some kind of function. So car one follows a parabolic function, 13 minus two times t squared. And Khartoum follows a straight line function, 30 minus 2x times time. And they drive in this dimension here, right? This dimension is not depicted on this function graph. On this function graph, this dimension is depicted this blue block, let's say its train that travels on the rails back and forth, right? And it follows a sinusoidal function. So this dimension here from minus eight to eight is this axis here from minus eight, 28. And this axis here is time. So if you look at the center of this block, then it is depicted here. So the center of this block moves along this dimension in a sinusoidal way. So you can see that, for example, when the center of it reaches seven, then it reaches seven here. Alright. So what's the challenge? So the challenge here is to avoid an accident. As you can see, the blue train crashes into the green car. And so the challenge for you is that knowing according to which functions the green cars drive and which function the blue train follows. Knowing all that information, you have to figure out how much of a delay you have to give to these green cars in order to start driving at the right moment so that they would pass through this point without crashing into the train. So right now, what you see is that the first car manages to pass through without a crash, and that's great, but not the second car. So you need to find like time intervals about when to start driving so that both cars would be able to pass this line without a crash. Alright, so I'm going to provide solutions in the next video. But really I think it's a very interesting exercise. So just give it a thought, even if you don't figure it out by yourself, that's okay. But I think it's important that you give it a try. Just think about it to drive those functions yourself. And try to figure out when you need, at what time you need to start driving. So that as you drive in this direction, according to these functions, so that you would avoid a crash. Alright, just give it a try and then I'll show you how I did it in the next video. See you there. 39. SIM: Avoid the crash - solution 1: Welcome back. This is the solution for the proposed exercise. I think it's a very good exercise because it covers a lot of the things that we have seen in this course, such as straight-line functions, parabolic sinusoidal functions. So I really hope that you tried to do this exercise by yourself first. And now I'm going to show you how I did it, right? So let let's get started. If you look at the motion of this blue train, right? And also at the motion of these two green cars, then you will notice that there are two very important squares. This one on the right and this one on the left. So if we look at this horizontal axis here, then this one is between 34 meters and this one between minus three and minus four metres. I call them the danger zones. Why are they important? Well, they're important because the lou train crosses them and also the green cars cross them. So if you want to avoid crashes, then you have to pay attention to them. And for now, since the red train is non-moving, then we don't care about this danger zone and that danger zone. Alright. So we're going to be working with these too dangerous zones. And we're gonna see how we are going to be able to avoid crashes between the cars and the train. 40. SIM: Avoid the crash - solution 2: So the important thing to see here right now is that the horizontal axis on the right graph is depicted vertically on the left plot. So you can even see that the blue trained and moves left and right here, moves up and down here. In other words, what you're seeing here is that the center of the blue train is being tracked as a function of time as it moves horizontally. So the blue train moves horizontally following a sinusoidal function. And the mathematical form of that sinusoidal function is here. Now it's important to understand that we're talking about the centre of the train, not the edges. This function is not tracking the position of the edges of the train. It is tracking the position of the center of the train horizontally. Now, the vertical axis on the right plot is not depicted on the left plot because it's not necessary. And then the horizontal axis on the left plot is the time axis. And the other thing that we have to pay attention to is what I mentioned earlier are the danger zones, the squares here. Now, here, this is the real graph, right? This is where the action happens, and this is the function graph. So on the real graph, in the real world, these danger zones, they have a shape of a square. However, if I want to depict them on the function graph, then they will look like this. You will have straight lines as a function of time that are constant. And it makes sense, right? Because if you think about it, this axis here, this is a time axis. So this axis here, it shows how something changes as a function of time. And if nothing changes as time progresses, then you will have a constant value here. So these dangerous zones, they're not moving right? They're not moving horizontally. If they were to move horizontally, let's say, just like this blue train is moving. Then you would also have a sinusoidal black function here. If, for example, the danger zones, the squares. If they moved horizontally in a parabolic function way, then these black lines, they would be a parabola here. However, since they're not moving horizontally, these black lines here, they are constant. Now one thing that I want to mention is that if those, what do you think? So this is a question to you. What do you think? If these squares, these danger zones, if they only moved vertically, like that, up and down, but not left-hand, right? How would these black lines look like? A small pause? The answer is, nothing would change here. Because remember this axis here, this vertical axis on the function graph represents the horizontal axis on the right graph, right? And the vertical axis on the right graph is not depicted on the function graph. So therefore, if these danger zones where to move vertically and not horizontally, then here they would still be just like they are constants, straight lines. All right. 41. SIM: Avoid the crash - solution 3: So let's focus on the blue train. Now. As it was mentioned earlier, this sinusoidal function is tracking the center of the train, right? It tracks how the center of the train moves horizontally. But I would say that it is much more useful to track the left and right edges of the train. So not the center, but the left and right edges. Why? Or because. You want to know when the train enters the danger zone and when the train leaves the danger zone. Because you need to make sure that the danger zone is completely empty and the train is not in the danger zone when the green car crosses it re so what's the point of tracking the center of the Train is better to track the left and the right edges of the train. Because then you know, when you enter the danger zone and when you leave the danger zone. And so how would you do it? How would you track the left and right edges of the train? Well, it's easy that the length of the train is two meters, right? So if you track the center of the train, then I mean, if this sinusoidal function tracks the center of the train, then the left edge of the train would be this function minus1. Right? So if the center is at seven, then the left would be at six. And the right side of the train would be this sine function plus one. So if the center of the train is seven, then the right side would be eight. So let me show it to you graphically. So there you have it. You have three function lines. Now. You have the center continues line. This is the function that tracks the center of the train. Then the upper line tracks the right side of the train. And the lower line here tracks the left side of the train. And now we're going to use these lines to record exactly the train enters and leaves the danger zone. 42. SIM: Avoid the crash - solution 4: So I've made it very slow for you so that it will be better for you to see what's happening. So let's wait until this function reaches the end and then I will continue talking. And you can watch it several times if you want to. Okay. Let's examine exactly what happened here. First of all, the blue train started moving towards the right side. And then you went to the left side and the negative to the right side. So when it started going towards the right side, first of all, it entered the danger zone with its right edge, right? And on the function graph, the right H is the upper line. Ok? So you can see that at this point here, when the upper line crosses the danger zone here at three meters, That's when the train enters the danger zone. And I've put here a vertical line and also a horizontal dashed line, which is not so good to see, but maybe you can see a little bit here. And so that's when the train enters the danger zone. And then the train leaves the danger zone when the left side of the train leaves this square. And so the left edge of the train is the lower line here. So when the lower line gets out of this danger zone, then it happens here where my cursor is. So I put here another vertical line. So that means that from here till here, if you look at the timeline, then this is a forbidden time interval. It means that the green car should not be in the danger zone. From, let's say, 0.4 seconds to 100 seconds. That's when the danger zone is occupied by the train. Now, after that. The train went to this side and then it started going back. While it was going there. It was in this period. And so when the train entered the danger zone again, it entered the danger zone with its left side. And that happened over here because the left side is the lower graph, right? So it entered again the danger zone here. So I put here another vertical line. And the danger zone. And the blue train left that danger zone when it's right side, left this square. And that happened here. See, when this upper line touches this black line, that means again that this is the forbidden time period. And after that, the train moves towards the other danger zone. And it's exactly the same thing. So it with its left side, it enters the danger zone. With its right side, it exits the danger zone. It happens here. And then again as it moves back, It's vice versa. And so if you look at this graph here, and remember this is a time axis. So from here you can see when the danger zones are occupied by the terrain and when they are free. And when they are free, then your cars can drive through the danger zone without crashing into the train. So from here to here, when the continuous line goes to a dashed line or a dashed line goes to the continuous line. That's when it's forbidden to enter the danger zone. And when you go from dashed line, two dashed line, or from continuous line, from continuous vertical line to continuous vertical line. That's when you are allowed to be in the danger zone because that's when the danger zone is free from the train. And of course, let's say when your train is here, then this danger zone is completely free. So obviously, in this time period, you're your green car on the right side can easily go through the danger zone on this side because your train is on the left side. So this is obviously free here and vice versa here. So when your trainees on this side, then you can see that this place is completely free. So this danger zone is completely free. 43. SIM: Avoid the crash - solution 5: Okay, now just let's observe the green cars for a second. And these black lines here. Just observed this black line and the green car and see what's happening. Alright, so now we're focusing on the green cars, right? And we are also focusing on these two lines here that are growing. So at the beginning they are black and then they become blue. And you can see that as this car enters the danger zone, this line here becomes blue. And when it exits the danger zone, this blue line stops increasing. So the black line, it's a time duration from when the car starts driving from its initial position up until it reaches the danger zone. Alright? And then when the car reaches the danger zone, then the line becomes blue. And one thing that you can notice here right away is that in this case, the blue line is a lot longer than in this case. And that's because this green car here is extremely slow, right? So the other car is a lot faster. So you're gonna see it now. So you see that this is a parabolic function at the beginning of the car. Slow not becomes faster and faster. And now its speed is very fast. So it crosses the danger zone very fast. And that's why you have a very short blue line here. And in this case is very slow. So it takes you a long time to get out of the danger zone, right? So the blue line shows you the time duration during which the green car is in the danger zone. And now you're just going to put all the information together, all the information that you know about the Green Corps and about the blue train. So if you know that this blue line here is the duration of the time during which the green car spends in the danger zone. And if you know, the forbidden time intervals that come from the trains, in this case from the blue train. And you also know the allowed time intervals during which the blue train is not in the danger zone. If you know all that information, then that means that we need to make sure is that these blue lines here that represent the green car, right? They have to be in the allowed time interval. So for example, in this case, the right car is okay. But not the left car. So the right car doesn't crash because it's blue line is in the allowed time interval. So you see it avoids the crash because the blue line is in the allow time interval. But in the left car case, it's not. The blue one here is in the forbidden time interval. And that's why you see the crash. So what can you do against it? Well, the right car is okay, right? So you have to maybe wait a little bit with a left car and add a little bit of a delay. If you look at it then, okay, if I go from four seconds to five seconds, then starting from here, I will be in the, in the allowed time interval. So if I wait for 1 second. So right now I entered the danger zone here. But if I enter the danger zone at five seconds instead of four seconds, then I should be okay, right? So maybe I should just wait for 1 second and then start driving. Alright, so I'm going to add a little bit of a delay for the left car. 44. SIM: Avoid the crash - solution 6: All right. I've added a 1.12 delay to the left car, and it's in purple. So when you see purple then that means that the car is at rest. So since it waited for 1.1 seconds, so now it should barely make it. And it does. So the allowed time interval starts here, and it finishes here. Alright? So again, the car is not moving the purple line. The car starts moving. The line turns black and it drives towards the danger zone. And then when it reaches the danger zone, then the line becomes blue. Alright? And in fact, you can add even a little bit of more delay because you see you have a little bit of space here. Of course, if you add too much delay, then you will just cross the dashed line and you will again be in this forbidden time interval. So in the next video, I will show you that you can apply exactly the same logic even when you have another function. In other words, even if your train moves left and right, according to a different kind of function, not just purely a sinusoidal function. You can still apply the same logic to it. Alright, see you in the next video. 45. SIM: Avoid the crash - solution 7: Welcome back. In this video, I just want to show you that you can apply exactly the same logic when you have a different function. I have a weird looking function here. However, if you want to avoid the crash, then the logic is exactly the same. You have this blue line here, which is the time duration of a green card being in the danger zone. And you have to make sure that this blue line and this one as well, that they stay inside the allowed time interval. All right? And if you're interested in seeing how this function looks like mathematically, then here this. So if you just examine the mathematical form of it, right? And if you then look at the shape, then you will see that in fact this function is also a sinusoidal function, but it has two terms. One term is here, the other, the other term is here. However, the second term has an amplitude that has, that is seven times less the amplitude of the first term, right? So you have a here and you have a divided by seven here. However, in the second term, the frequency seven times greater than in the first term. And so if you look at the form of this function now, then kinda make sense, right? When you have low frequency, you have high amplitude. So you see low-frequency here. You see one cycle. However, during this one big cycle, you have a huge amplitude. But then you see higher frequencies here, but low amplitudes. So that's why you see these waves here. These waves, they are because of this second term. Without the second term, it would just be a normal sinusoidal function. However, because you have this second term here, with high frequency and low amplitude, you see these ways, right? So you see that since you have high frequency here, you have these the sign small sinusoidal functions here, right? But because they have a small amplitude, then, then these waves are just very small compared to the huge amplitude of the first term. So again, you have low frequency and high amplitude. That's why you see this huge cycle. And then you have low amplitude and high frequency. That's why you see these small cycles. And if you add them together, then the entire function looks like this. Alright? So I have another exercise for you, which is, let's say, an extra bonus assignment for you. And I will introduce it in the next video. So, thank you very much. I hope you've enjoyed the, the exercise and I'll see you in the next one to tackle even a slightly harder problem. Alright. 46. SIM: Avoid the crash - solution 8: Welcome back. So I have a slightly harder problem here for you now. Now we are in a situation where even the red train is moving, not only the blue train, right? So now you have to take into account that you have two functions, blue and red. And so your task is to avoid crashes with both the blue train and red train, right? So both cars, they should be able to pass both the first line of danger zones and the second line of danger zones without getting into a crash. So good luck, and I will show you how I did it in the next video. Alright, but try it first on your own because then it really matters. See you in the next video. 47. SIM: Avoid the crash - solution 9: Welcome back. This is the solution for the more difficult exercise. However, if you look at it, then you'll see that the logic is exactly the same to solve the exercise. But of course, now you have to be a little bit more of an accountant because you have to keep track of so many things. And there are so many lines here, it can get messy. That's understandable. But still let's give it a try. So one thing that we can see is that both cars managed to avoid a collision with a blue train, but both cars crash into the red train. Alright? So if you remember, this blue line here, was the time duration of a green car being in one of these to danger zones, right? But in addition to that, you can also see that there is a, a dashed red line here, right? And now this dashed red line, it's the time duration of the green cars being in one of these danger zones. Because now you have two more danger zones, right? You also have a danger zone here and here, because the red train is now moving in going through these danger zones. So you have to take into account both danger zones. So you have four of them. And therefore you have now two blue lines here and two dashed red lines. And so you need to find a place where both of them are in a place that are somewhere in between allowed time intervals. We look at the first case. Then if we were to shift this black line forward a little bit up until, let's say here, into this location, then we should avoid a crash. So we should add some kind of delayed to the first car on the right side. So let's add a 2-second delay to this right-sided car. So I've added a 2 second delay to the right car. So it waits for two seconds, and then it starts going and it is able to avoid the crash with the blue and red train. Now one thing that I want to mention is that look, you have this blue line and then you have this red dotted line and they overlap. So what does it mean? It means that if you look now at this green car, right? And I want to show it to you again. It basically means that at some point in time, it's in both dangerous zones. So if you look at this front part of the car, it enters the second denture zone, but the back of the car is still in the first danger zone. And that's why you have this overlapping here. Alright? So again, see the front of the car enters the first danger zone. The second, but still it's in the first danger zone. And so you have this overlapping. But now if you look at this line, then it's a very long line. So it manages to avoid the crash with the blue train. But now with the red train in this entire line is just too long. So it seems that for the second car, it's kinda impossible to find a place that would allow it to avoid a crash with both cars. So maybe if I extend this time period, because right now we go from 0 to eight seconds. Maybe. If we go from 0 to 16 seconds, then because this is a repeating cycle, maybe we're going to be able to be somewhere here in this region. But after eight seconds. So let's give it a try. 48. SIM: Avoid the crash - solution 10: So I have added a 6 second delay to the car on the left. So after six seconds, it's going to start going. And now it has started going. So let's see. It is going to be able to avoid the blue train and also the red train. So now finally, we have found a solution where none of the cars crash into none of the trains. So that's a good thing. Problem solved and mission accomplished. So thank you very much. And I hope you've enjoyed this exercise and the simulations. I hope you've learned a lot. And I'll see you in the next video. 49. 2 variable AIRPLANE example: Welcome back in this lecture, we're going to expand on the knowledge of functions. So far, we have only seen two dimensional functions. We have only seen a function that has a now output That depends on Lee on one input. However, in real life, very often an output depends on multiple inputs. So I would like to start with an example. Do you remember the very first example that we had waas an airplane? So we had in their plane, the flu, and in this direction and what we did, we decided to graph its distance with respect to time. Right? And let me just resolve this graph again. So we have time. We had distance and we had some kind of relationship that we graft us a straight line and we had distance equals 800 times time And this 800 here it was the change off distance for one unit change of time. And from this example, you could see that distance depended on time. Okay, so that was one way to represent this information, however, is time the Onley factor that that influences distance. Well, how about if we have wind? Okay. How about if we have when Let me just draw this airplane again. So how about if we have wind in this direction? Well, if we have wind in this direction, that wouldn't it increase the airplanes distance covered in one hour? Or, if I have a wind in the other direction in the opposite direction, so the air molecules would hit the airplane from the front, then in one hour? Wouldn't the distance, in fact, that be smaller? So it would make sense to create another graph where I have distance and I have wind the wind speed and I could have something like this. I could have a straight line function like this, so positive wind would mean that the wind goes along with the airplane with the airplanes, velocity and and negative wind with be the direction against the airplane. So in this case that if when this zero is Iraq kilometers per hour, then uh, then we would have this situation. But if we have a negative went than in one hour, we would cover less this sense. If we have positive went, then we will cover more. Okay, and let's just put some kind of random mathematical relationship we're with respect the wind. So distance equals well, let's say 30 W plus 200. It's a random function. I just threw it out here just to illustrate the concept. Okay, so now what we have, we have a distance that depends both on time and wind, and another way to express it would be to draw this box in which we have a function, and then we would have two inputs now and then. One input would be time, and the other input would be W. And then you would have distance here and then both w and time. They would go into the function, and then you will operate on those inputs and you would get the output. And as a mathematical relationship, we could write it down like this so it's distance equals. That's the total total relationship now 800 t plus 30 w plus 200. So this is with respect to time and this with respect to win. Now, this function Dow. It's not a two dimensional function anymore. Now we have a two dimensional domain, so not only we have a line which is a domain, but in fact we have a plane, which is domain. So now we have now we have time and we have wind. So this is a two dimensional domain and the output would in fact go straight up fruit from the paper or straight down into the paper. So according to the convention up from the paper, it would be positive distance and down. It would be negative distance. So in other words, it's a three dimensional function with a two dimensional domain and then distance would be the output. So how would we graph it? Well, one way would be something like this that I draw a time here. And then I would draw wind here. And this is 90 degrees here, so you can think of it as a plane and then upwards. You would have distance. No, let's say that your your time is one hour and your wind is one kilometer per hour. It's not up to scale just to illustrate the concept. In that case, you would determine a point on the plane. So for both domains and then whatever you would get here, So in this case you would get distance equals 800 times. One is 800. Bless 30 times one is 30 plus 200 would be 1030 kilometers. Now, notice that if I take this, uh, the equals 30 W Then what units do we have for 30? Well, it's not speed anymore, because because wind already has the units of speed. Right? Wind has the units off kilometers for our and the distance has kilometers. So the change the Delta d With respect to Delta w, the units would be kilometer over kilometer per hour, which would be kilometers our over kilometer, so they would cancel out. So in fact, this sturdy here it would have a unit of time. So since now, the second dimension, which is the wind speed since it has different units. Also, that the change It's not speed anymore. In this case, 30. It would be a unit of time. So in the physical situation, you have to pay attention to the units. And now, once you have, uh, once you have determined you're Wnt okay, It's your distance is 1000 in 30 kilometers and you go up. You would determine 1030 kilometers and you can see it also, if you put this if you put these units here instead of 30 or if you put hours instead of 30 and then if you multiply hours times kilometers per hour, then it could give you a distance. So it's important to pay attention to units, so you have 1030 and you determine it on the distance access. So it's 1000 and 30 kilometers. And now since since it's a three d dimensional function and your domain is two dimensional , then you wouldn't have just a line that would represent the relationship between distance and and W and T. In fact, you would have some kind of surface, right? You would have something like this. So let me just try to draw surface the federal a parabola. Then I could drop rambling. This if I add another dimension here. Okay, so you see, this is one dimension. This is another dimension, and our ports you have ah, third dimension. Then you could have a surface here, so you see, it's a parabola, but it's in three D now, also adding another dimension to it. So So if you have a three dimensional function than than what you have, you would have a surface and here I am using screen cast to show you an example of it. See, So this is a three dimensional function. You have the red access which is X, and then you have the Y axis, which is green. And then you have 1/3 access, which is blue, which is the output so red and green, their inputs and th