Abstract World VS Real World - 1

Mark Misin
A free video tutorial from Mark Misin
Aerospace & Robotics Engineer
4.6 instructor rating • 2 courses • 2,767 students

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INTUITION MATTERS! - Applied Calculus for Engineers-Complete

Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

34:53:37 of on-demand video • Updated October 2020

  • You will develop very strong intuition & understanding in Calculus
  • You will learn how to apply Calculus in real life to the level not seen in other courses
English [Auto] Welcome back this lecture is called absolute versus real. So in this lecture I'm going to try to make some connections in between realistic examples and the abstract mathematical world because at some point we're going to be working with abstract functions but I don't want it to be very abstract to you. So I'm I'm going to illustrate how people non-mathematicians such as engineers and scientists how they use mathematics and calculus in their own realistic examples. So in our last lecture we had this airplane example where we had a relationship between time and distance that functioned 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 of the airplane can depend on time like this distance equals one plus time squared. So post the video now and just graph how distance depends on time how the graph look like. It can take time and distance and for the time it can take zero 0.5 one one point five and two and just see what you get. So this one good for the distances. These are the values 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 but relationship anymore. So you could calculate the speed from the graph. But speed is not constant anymore because the change here is like this. And here it's it's like this. In fact here the speed is higher but it's a parabolic function graph 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 hose 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 of the empty mass of 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 total mass of the tank would equal the empty mass plus the change of mass of the tank with respect to with respect to the volume of water and from physics we know that one cubic metre one cubic metre of water is 1000 kilograms. So the way to look at it is that the mass of the time changes by 1000 kilograms. If you add one more cubic metre of water. So this function would be then like this. The empty mass mass e it's empty mass plus 1000 times the volume of water and you would graph it something like this you have the volume of water and since you're the have an empty mass you don't start from zero. You start from here and then you would have some kind of straight line and a straight line. If you were to calculate the distances and the changes of the total mass of the tank then you would get that you would get that it would be 1000 kilograms per 1 cubic metres. So it would be something like this built on mass and Delta 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 per 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 y equals m x plus B. And in this case the graph would extend and be something like this and it would go forever.