Discrete VS Continuous functions - what's the difference

Mark Misin
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INTUITION MATTERS! - Applied Calculus for Engineers-Complete

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

34:53:36 of on-demand video • Updated October 2020

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English [Auto] Welcome back in this video we will start making our way into the world of integrals. My explanation will be slightly different from the standard explanation. To properly explain to you what they are I first need to show you the difference between discrete and continuous functions that will make the explanation of integrals a lot more intuitive. So let's get started. So I have a typical function here a parabola Weikel 6 squared. And here it is. And this kind of function is called a continuous function and you can know it because if you look at the line it's a continuous line. You don't have any discrete steps. Say that I want to discretized this function now. Well I could do it for example with Resolution 1 and I could do it. For example like this. So I can take the middle point 0 0 and then I can go 0.5 to the right and zero point five to the left or right. So I would have zero point five here and minus zero point five here. So the total distance of this red line here is 1. So the zero point five on one side and minus 0.5 to the other side. So I could just say that instead of having discontinues blackline here I can say that. OK since at 0 0 my function my y is zero. Then when I consider the interval from minus 0.5 to 0.5 then all values from minus 0.5 to 0.5 will be zero. Now the next interval that I will consider is from 0.5 to 1.5 right. So I have one point five here and then I'm going to take the middle point of it and I'm going to see what the values in the continuous function from the middle point which now is 1. So it's here. So if my X is 1 then my continuous function tells me the why is one. So I'm going to say that from 0.5 to 1.5 my value my y will be 1 like this. You see. So it's not a continuous line anymore. It's a step it's like a stair. So from 0.5 to 1.5 values one and the same thing from minus 0.5 to minus 1.5 so the same thing here minus 1 is the middle point and that will give me why cause 1. So I can say that from minus 0.5 to minus one point five for all these x's. My wife will have only one value which is 1 and then if we go from 1.5 to 2.5 and then I can take the value which is to the middle X. And then why would before so I can say that from 1.5 to 2.5. I will have for all values for these Xs from 1.5 to 2.5 will be 4 and the same thing on the other side from minus 1.5 to minus two point five point from minus 2 and give me 4 and for minors 1.5 to 2.5 when X equals four miners 1.5 to 2.5. All my wife will be 4. So you see now you have these kind of steps here in red. You don't have a continuous line. You have some kind of steps and some can discontinues changes from my 0.5 to 0.5 unit zero and then all of a sudden you jump on to the next level and then again you jump on to the next level. And this is called a discrete function. There you go in red. You have now discrete function no continuous line just certain levels and then they change this continuously. So its discrete function in black. Its a continuous function but in red its discrete function. And since my resolution here was one for example from 0.5 to zero point five so that distance here was one and again from 0.5 to 1.5 it was one I can say that one of these intervals I could call them Delta X.. Right. So I have an equally interval I chopped my Exene to equal intervals and I can call it delta x and in this example my delta x would be 1. However what if I decide to increase my resolution. What if I want to chop my X into smaller pieces. What if I want my delta x to be 0.5 Well if my delta x equals 0.5 then you can see that I will have more levels here. So again the blackline is the continuous function. But since my delta x is smaller then I chop my X into into a bigger amount of intervals and there for example here this interval would span from minus 0.25 up until 0.25. All right so the total resolution is 0.5. So that told distance the tool that the X is 0.5 and then if I go from 0.5 to 0.75 I would have this level from 0.75 to one point twenty five my level would be here and et cetera et cetera. And you can see that now compared to this function here I will have a lot more levels. So here I have one two three four five and here one two three four five six seven eight nine. So you can see that if I decrease my delta x then I increase my precision. And notice one interesting thing the smaller delta x becomes the more precise I become. And remember the the most precise scenario is a continuous function continuous function gives you the most precision right with discrete functions. You lose a little bit of precision because in reality when you're at zero point five here if your ex 0.5 then your Y would be 0.25. But then if you discretized it then you're zero point five would be zero. So you see you with discrete functions you lose that kind of precision. However if you discretized your function to smaller and smaller pieces then you will have more and more of those pieces. And your function becomes more and more precise and so the smaller your delta x is the more precise your function becomes. And if your delta x approaches zero. Right. So the closer it gets to zero the more precise your function becomes. So if you're delta x approaches zero then your discreet function I'm going to call it if you're discrete function approaches. Your continuous function. Does that make sense. Yes it does. So the smaller your Daltex is the more intervals you will have on the x axis the more levels you will have the more precision you will have. The more closely you approximate this continuous function because in the end this with functions the approximate continuous functions and then depending on the resolution then you can choose your precision of how you approximate your continuous function. So the higher your resolution the more closely you approximate your your continuous function. And so if you're dealt x approaches zero then you're discrete function approaches. Your continuous function. In other words if your delta x approaches zero then something like this something which is discontinuous like this set of stairs would approach this continues. Line.