Proof of the Fundamental Theorem of Calculus

and we did not use any Epsilon's and deltas you'll see what I mean ladies and gentlemen dr. pan everyone and by popular demand today I will prove the fundamental theorem of calculus says hi am I like pie and then here we have the fundamental theorem calculus part two I will not prove it in full generality but I will prove a slightly more special case maybe the following and as I said this is slightly more specific than the regular FTC because regular FTC says integral from A to B of f of X you know is equal to an antiderivative of f at B minus an antiderivative of F at a this assumes that F is differentiable and we also need to assume that F prime is continuous that says with this formulation of a very very elegant proof that does not use any Epsilon's and deltas and to do this we'll use a you know a little trick and you'll see what I mean help me okay trick but for this also by popular than man let me remind you how everyone integral works so recall maybe the ironic sense integration is and it just follows in a bunch of steps the first step is to divide up the interval for me from A to B into n steps so fix that some positive integer and chop so think of a very linear cake any one of those each piece has the same size so the point is because you're chopping off the interval a to from A to B into n pieces well each piece will have length as the total length of the interval B minus a but divided by n because you're splitting up into n pieces and we'll call this Delta X like a small change in X here your one piece of length Delta X you have another piece of that length X etc etc and before we continue I need to introduce a little bit of notation and I'm sorry result so some of the patient is problem first of all the first point day we call this X naught Y naught X naught X naught is a and then the next point I'll call it X 1 and what is X 1 is a plus this little piece so a plus X the next point we call it X 2 what is in suit a plus Delta X plus a plus two dollars etc etc the five-point movie X I is a I Delta X okay and just as a sanity check well the last will call him because so the point is the rightmost point actually corresponds to eat that's good news okay that's the first thing we chopped off the enjoy going to end pieces and now this is X I X I and again you you also have to grab or F prime what do we want to do on each piece we just could choose a random point X I star of your liking so you think you can choose whatever you want so you know some people they like the left so they put choose this point to be X I minus one other people they choose the right you know to choose upon X I some people they like a balance they just reach the midpoint and other people they're like two-thirds here then choose this one you can choose any point that's fine but the point no pun intended is that once you choose the X I star you consider a certain rectangle a rectangle of height and prime XA star and a with this this so maybe this to choose a point X I star and consider of height and width the width is precisely the length of this piece but remember the length of this piece is just Delta X so we get this rectangle which it's supposed to be a very good approximation of the area under F prime but I'm just a bad drawer so it doesn't look like this one but that says remember that this Reckling was very small each piece what we had to consider the rectangle but the point is we did it on each piece and the point is in the end we can the point this is suppose this is supposed to be a very good approximation of you know the area under F so would you really want to approximate the area is you want to take the sum of the areas of each so now to get the total area what you wanna do is you sum all those angles then you get in summation so as I said each rectangular area good approximation and to get a better and better approximation you just take money you chop more more pieces in other words you let the width go to zero which is mathematically expressed as you this is by definition so all this time I've spent on defining the integral but it is not that I'm lost because this is actually a very crucial to prove the fundamental theorem of calculus you might say I object and I overruled this objection because you might say that what doesn't this limit depend on the choice of the X I start remember I said you can choose any X I start you want in turns out that if this function f prime is continuous that limit is independent of your choice of X I star so you can choose nearly choose any point in that little interval and still get the same limit and you may say hey this is weird but it's not very weird because it's this function f prime is continuous then on each little rectangle it doesn't move very much so the values of F prime don't change much on this interval so if you just choose another point nearby that the answer in the end wouldn't change either of course this is a very informal statement not today today is not an epsilon Delta day but you know if you want to prove the fundamental theorem of calculus we want to have some relationship between F prime in health that's said now if you remember your calculus and go all the way back there's actually another theorem the talks about the relationship F Prime and F I don't mean to be rude but I value your time because we will talk about the mean value theorem by the mpg so it's like the VMP what do we have we actually mounted on each little interval X I can actually compare maybe what does the mean value theorem say it says that there's some point C in that interval such that the slope of the line connecting this point in this point is actually equals to F prime of C let's clean this up a little bit well this X I minus X I minus 1 it's precisely the length of that interval Delta X equals F from a now remembering here comes the most important part of the proof well this point C is actually depends on I because it's in that interval but this point C is very valuable it gives us a lot of information relating F and F prime but remember in our definition of the interval we are allowed to choose any point in the interval to define our integral so because that point is too special let's just choose that point C so let s start be that C very important [Music] in other words remember this song involves Delta X and prime of X I start then actually equals to this little thing and now that we have this nice and entity which looks almost like FTC Reich let's just plug this in into this formula so this educational you learned about this and you learn let's see if I can write this on one line and so let's put a bit of spice so what do we have well look we have this magic formula so let's just calculate what this song is so the next no he was a really big explosion happening because this F of x1 cancels this F of x2 cancels and if you continue that F of x3 cancel and everything cancels except for the first term in the last row so sorry I didn't write it in one line but almost it's too exciting to build on that for it so after all this exposure here's the nice thing we know what X is X is just X so we have limit n goes to infinity of F of Z minus F of a but this does not depend on n it's like a constant and in fact will enthalpy and here we are fundamentally proved the second part of the fundamental theorem of calculus because if we go back in this line we've shown an integral from A to B of F prime of X DX equals to F of B minus F of a and we did not use any extra ones and the others Wow and then the box is much bigger today why this because it's very exciting fine say something about the price oh my god oh yeah oh my god I'd use me he thought it'd be done

Channel: Dr Peyam

Views: 21,080

Rating: 4.8578949 out of 5

Keywords: calculus, fundamental theorem of calculus, ftc, antiderivatives, integration, Riemann, integral, area, proof, math

Id: vf5f_SuZ6XE

Length: 18min 58sec (1138 seconds)

Published: Fri Oct 06 2017