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Current time:0:00Total duration:6:58

Video transcript

what we're going to talk about in this video are three theorems that are sometimes collectively known as existence theorems so the first that we're going to talk about is the intermediate value theorem and the common thread here all of the existence theorem say hey we're looking for something over an interval there exists an x value between a and B where something interesting happens in the intermediate value theorem we assume that if we're continuous over the closed interval from A to B and in fact all of these existence theorems assume that our function is continuous over the closed interval from A to B then we take on every value between F of a and F of B or another way to think about it is take a value from F between F of a and F of B including F of a or F of B let's call that value L the intermediate value theorem tells us that there if we make you assume the assumption that F is continuous over this interval that there must be a value between a and B that takes on the value L and I challenge you try to draw a continuous function that goes from a comma F of A to B comma f of B that does not go through L if your continuous you've got to go through L the only way that you can't go to the other way the only way you could avoid going through L is if you are discontinuous if you had a discontinuity right there then you could avoid going through L but we're assuming we are continuous over the interval and so here it's pretty intuitive that if you're continuous there exists C between a and B including it could be a or b that takes on any of those values in this case in this particular value that sits between F of a and F of B so that's the first existence theorem the second is what's often known as the extreme value theorem and this one is similarly intuitive I think once again it assumes that F is continuous over a comma over this it closed interval but here we say look if it is continuous over that closed interval then there exists that's why should called existence theorem there exist values between a and B and it could happen at a or B where the function takes on a maximum and there's a value between a and B where the function takes on a minimum value over that interval so once again try to draw a function that does not take on a minimum and maximum value over that interval if I just draw a straight line here well then the maximum value happens when X is B the minimum value happens when X is a if I do something like that the maximum value occurs once again it is occurring in this interval it's occurring right here at C and then the minimum value is occurring at a if I draw something like let me draw it like this I draw something like this then the minimum value is occurring at this x value and then the maximum value is occurring at that value x value this is the saying that they exist they exist over that interval and it might happen a day it might happen at B and once again the only way I can construct something where these maximum or minimum values these extreme values don't exist is if I make it discontinuous so for example what if I had a graph that looked like right at where we thought we were going to have a maximum value or discontinuous well now you don't have a clear maximum value similarly I could do something like this where now we do not have a clear minimum value and so there does not exist an X where the function takes on a moment a minimum value in that interval but once again we are assuming that we are continuous and so we will find these extreme values and we have other videos where we go into much more depth on it last but not least to complete the trifecta we have the mean value theorem and this one is also intuitive it starts going into differentiability in the derivative and it adds an extra constraint force above and beyond saying that we're continuous over the closed interval we also assume that we are differentiable over the open interval if you're differentiable over an interval it does mean that you're continuous but if you're continuous it does not necessarily mean that you're differentiable and all this tells us is if I have a continuous and differentiable function over this interval differentiable over the open interval continuous over the closed interval so let me draw that so let me draw something like this and if I were to compute the average rate of change from a comma f of A to B comma F of B so the average rate of change I will do in this pink color so that would be the slope of this line right over here that would be the average rate of change the slope of that line they exist the mean value theorem tells us that there exists a point C where the derivative of our function at that point the slope of the tangent line at that point is the same as the average rate of change and we could eyeball that here that looks like for this curve there's actually several points where the derivative looks the same as the average rate of change maybe right over here the slope of the tangent line looks like it has the exact same slope as the average rate of change so that could be our C it exists we feel good it exists but there's there could be more than one value right over there it looks like the slope is the same as the average rate of change that could also be our C right over there so how could we construct something where this isn't true well if we don't assume differentiability over the interval we can actually find a continuous function where it isn't true where you can find a point whose derivative is the same as the average rate of change so for example here's my counter case maybe something like an absolute value function where the average rate where the average rate of change let me draw a little bit better so this was an AB some type of an absolute value function where this is linear up to this point and then linear up to this point we're not going to be differentiable over here so not the derivative isn't defined at this point right over here well now you can't at no point over this interval is the slope of the tangent line the same as the slope connect of the or this is the same as the average rate of change you might try to make an argument that don't maybe right over there but we're not differentiable over there there isn't a well-defined tangent line or there's the well-defined tangent line and a well-defined derivative or slope of a tangent line so in other videos we'll go into more depth but it's nice to look at them all together see what they are all talking about they're all talking about the distance of an X value in the interval where something interesting happens where we take on the value between F of a and F of B where we take on extreme values or where the derivative at that point is the same as the average rate of change over the interval