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  • - [Instructor] The table gives selected values

  • of the continuous function f.

  • All right, fair enough.

  • Can we use the intermediate value theorem

  • to say that the equation f of x equal is equal to zero

  • has a solution where four

  • is less than or equal to x is less than or equal to six?

  • 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 zero, f of x is equal to zero.

  • Let me draw those.

  • So, we have that point.

  • When x is equal to two,

  • y or f of x, y equals f of x

  • is gonna be equal to a negative two.

  • So, we have a negative two right over there.

  • When x is equal to four, so, three, four,

  • f of x is equal to three.

  • One, two, three.

  • I'm doing it on a slightly different scale

  • so that I can show everything.

  • And when x is equal to six, so, five, six,

  • f of x is equal to seven.

  • Three, four,

  • five, six, seven.

  • So, right over here.

  • Now, they also tell us that our function is continuous.

  • So, one 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 four to six, so let me look at that.

  • So, this is one, two, three, four here,

  • this is six 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 gonna take on every value

  • between f of four, which in this case,

  • so, this is f of four, is equal to three,

  • and f of six, which is equal to seven.

  • f of six,

  • which is equal to seven.

  • And so, if someone said hey, is there gonna 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 four and f of six,

  • 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

  • but zero is not

  • between f of four

  • and f of six.

  • So, the intermediate value theorem does not apply.

  • All right, 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 two is less than or equal to c

  • is less than or equal to four?

  • If so, write a justification.

  • We are given that f is continuous, so let write that down.

  • We are given

  • that f is continuous,

  • and if you wanna 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 end points?

  • Our interval goes from two to four,

  • so we're talking about this closed interval right over here.

  • We know that f of two

  • is going to be equal to negative two.

  • We see it on that table.

  • And what's f of four?

  • f of four is equal to three.

  • So, zero

  • is between

  • f of two and f of four.

  • And you can see it visually here.

  • There's no way to draw between this point and that point

  • without picking up your pen, without crossing the x-axis,

  • without having to point

  • where your function is equal to zero.

  • And so, we can say

  • according to the intermediate value theorem,

  • there is

  • a value c

  • such that

  • f of c is equal to zero

  • and two is less than or equal to c

  • is less than or equal to four.

  • So, all we're saying is hey, there must be a value c,

  • and the way I drew it here, that c value is right over

  • where c is between two and four,

  • where f of c is equal to zero.

  • And this seems all mathy

  • 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 two and f of four at least once.

- [Instructor] The table gives selected values

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Justification with the intermediate value theorem: table | AP Calculus AB | Khan Academy

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    yukang920108 に公開 2022 年 07 月 05 日
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