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Current time:0:00Total duration:5:29

Justification with the intermediate value theorem: table

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

Video transcript

the table gives selected values of the continuous function f alright fair enough can we use the intermediate value theorem to say that the equation f of X is equal to zero has a solution where 4 is less than or equal to X is less than or equal to 6 if so write a justification so pause this video and see if you can think about this on your own before we do it together okay well let's just visualize what's going on and visually think about the intermediate value theorem so if that's my y-axis there and then let's say that this is my x-axis right over here we've been given some points over here we know when X is equal to 0 f of X is equal to 0 let me draw those so I mean we have that point when X is equal to 2y or f of X y equals f of X is going to be equal to negative 2 so we have a negative 2 right over there when X is equal to 4 so 3 for f of X is equal to 3 1 2 3 I'm doing on a slightly different scale so that I can show everything and when X is equal to 6 so 5 6 f of X is equal to 7 so 3 4 5 6 7 so right over here now they also tell us that our function is continuous so what intuitive way of thinking about continuity is I can connect all of these dots without lifting my pencil so the function might look I'm just gonna make up some stuff it might look something anything like what I just drew just now and it could have even Wilder fluctuations but that is what my F looks like now the intermediate value theorem says hey pick a closed interval and here we're picking the closed interval from 4 to 6 so let me look at that so this is 1 2 3 4 here this is 6 here so we're gonna look at this closed interval and the intermediate value theorem tells us that look if we're continuous over that closed interval our function f is going to take on every value between F of 4 f of 4 which in this case so this is f of 4 is equal to 3 and f of 6 and f of 6 which is equal to 7 f of six which is equal to seven and so if someone said hey is there going to be a solution to f of X is equal to say five over this interval yes over this interval for some X you're going to have f of X being equal to five but they're not asking us for an f of X equaling something between these two values they're asking us for an f of X equaling zero zero isn't between F of 4 and F of 6 and so we cannot use the intermediate value theorem here and so if we wanted to write it out we could say f is continuous is continuous but but zero is not between between F of 4 and F of 6 so the intermediate value theorem does not apply alright let's do the second one so here they say can we use the intermediate value theorem to say that there is a value C such that f of C equals zero and 2 is less than or equal to C is less than or equal to 4 if so write that write a justification so we are given that f is continuous so let me write that down we are given that f is continuous and if you want to be over that interval but they're telling us it's continuous in general and then we can just look at what is the value of the function at these endpoints so our interval goes from 2 to 4 so we're talking about this closed interval right over here we know that F of 2 f of 2 is going to be equal to negative 2 we see it in that table and what's F of 4 f of 4 is equal to 3 f of 4 is equal to 3 so 0 is between is between F of 2 and F of 4 and you can see it visually here there's no way to draw between this point and that point and without picking up your pen without crossing the x-axis without having a point where your function is equal to zero and so we can say so according according to the intermediate value theorem there is a value C such that F of C is equal to zero and 2 is less than or equal to C is less than or equal to 4 so all we're saying is hey there must be a value C and the way I drew it here that Vout that C value is right over here where C is between 2 & 4 where f of C is equal to zero and this seems all mathy and and a little bit confusing sometimes but it's saying something very intuitive if I had to go from this point to that point without picking up my pen I am going to at least cross every value between F of 2 and F of 4 at least once