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# Intermediate value theorem

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.A (LO)
,
FUN‑1.A.1 (EK)

## Video transcript

what we're going to cover in this video is the intermediate value theorem which despite some of this math II language you'll see is one of the more intuitive theorems possibly the most intuitive theorem you will come across in a lot of your mathematical career so first I'll just read it out and then I'll interpret it and hopefully we'll all appreciate that it's pretty obvious I'm not going to prove it here but I think the conceptual underpinning so the theorem tells us that suppose F is a function continuous at every point of the interval the closed interval so we're including a and B so it's continuous at every point of the interval a B and so let me just draw a couple of examples of what F could look like just based on these first line so suppose F is a function continuous at every point of the interval a B so let me draw some axes here so that's my that's my y axis and this is my x axis so one situation if this is a and this is B F is continuous at every point of the interrupted interval a and B so that means it's got to be for sure defined at every point as well as B continues to be continuous you have to be defined at every point and the limit as u are the limit of the function as you approach that point should be equal to the value of the function of that point and so the function is definitely going to be defined at f of a so it's definitely going to have an f of a right over here this right over here is f of a maybe F of B is higher although we can look at different cases so that would be our F of B and they tell us it is a continuous function it is a continuous function so if you're trying to imagine continuous functions one way to think about it is if we're continuous over an interval we take the value of the function at one point of the interval and if it's continuous we need to be able to get to the other the the value of the function at the other point of the interval without picking up our pencil so I can do all sorts of things it's to be a function so I can't I can't do something like that but as long as I don't pick up my pencil this is a continuous function so there you go if I had to if I if I if the somehow the graph I had to pick up my pencil if I had to do something like this oops I got to pick up my pencil to do something like that well it's not continuous anymore if I had to do something like this and hoops pick up my pencil not continuous anymore if I had to do something like woop okay pick up my pencil go down here not continuous anymore so this is what a continuous function that a function that is continuous over the closed interval a B looks like I can draw some other examples in fact let me do that so let me draw one maybe where f of B is less than F of a so it's my y-axis then this is my x-axis and once again a and B don't both have to be positive they could both be negative one could be a could be negative B could be positive and maybe in this situation and F of a and F of B it could also be positive or negative but let's take a situation where this is f of a so that right over there is f of a this right over here is f of B f of B and once again we're saying f is a continuous function so I should be able to go from F of A to F be F of B draw a function without having to pick up my pencil so it could do something like this actually I want to make it go vertical it could go like this and then go down and then do something like that so these are both cases and I could draw an infinite number of cases where f is a function continuous at every point of the interval the closed interval from A to B now given that there's two ways to state the conclusion for the intermediate value theorem you'll see it written in one of these ways or something close to one of these ways and that's why I included both of these so one way to say it is well if the this first statement is true then F will take on every value between F of a and F of B over the interval and you see in both of these cases every interval for every every value between F of a and F of B so every value here is being taken on at some point you can pick some value you could pick some value in arbitrary value L right over here well look L happened right over there if you pick L well L happened right over there and actually it also happened there and it also happened there and the second bullet point describes the intermediate value theorem or in that way for any L between the values of F of a and F of B there exists the number C in the closed interval from A to B for which f of C equals L so there exists at least one C so in this case that would be our C over here there's potential there's multiple candidates for C that could be a candidate for C that could be the C so we could say there is there exists at least one number at least one number I'll throw that in there at least one number C in the interval for which this is true and you know something that might amuse you for a few minutes is try to draw a function where this first statement is true but somehow the second statement is not true so you say okay well let's say is let's assume that there's an L where there isn't two C in the interval well let me let me try to do that I'll draw it and I'll draw it big so that we can really see how obvious that we have to take on all of the values between F of a and F of B is so let me draw a big axis this time so that's my y-axis and that is my x-axis and I'll just do the case where just for simplicity that is a and that is B and let's say that this is f of a so that is f of a and let's say that this is f of B the dotted line alright F of B and we assume that we we have a continuous function here so the graph I could draw it from f of A to F of B from this point to this point without picking up my pencil from this coordinate a comma F of a to this coordinate B comma F of B without picking up my pencil well let's let's assume that there is some L that we don't take on let's say there's some value L right over here and and we never take on this value this continuous function never takes on this value as we go from X equaling a to X equal B let's see if I can draw that let's see if I can get from here to here without ever essentially crossing this dotted line well let's see I could let maybe I'll avoid it a little bit I'll avoid but up G how am I going to get there well without picking up my pencil well well I really need to cross that line all right well there you go I found we took on the value L and it happened at C which is in that closed interval so once again I'm not giving you a proof here but hopefully you have a good into intuition that the intermediate value theorem is it's kind of common sense the key is you're dealing with a continuous function if you were making its graph if you were to draw it between the coordinates a comma f of a and B comma F of B and you don't pick up your pencil which would be true of a continuous function well it's going to take on every value between F of a and F of B
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