Name: Justification with the intermediate value theorem: table | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T14:15:11.000Z
Duration: 5 min 29 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

- [Instructor] The table gives selected values

Can we use the intermediate value theorem

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

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

So, pause this video and see if you can think about this

Okay, well let's just visualize what's going on

and visually think about the intermediate value theorem.

and then let's say that this is my x-axis

We know when x is equal to zero, f of x is equal to zero.

So, we have a negative two right over there.

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

I'm doing it on a slightly different scale

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

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

So, one intuitive way of thinking about continuity

And it could have even wilder fluctuations

And here, we're picking the closed interval

from four to six, so let me look at that.

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

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

And so, if someone said hey, is there gonna be a solution

to f of x is equal to, say, five over this interval?

you're going to have f of x being equal to five.

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.

So, the intermediate value theorem does not apply.

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

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

but they're telling us it's continuous in general.

And then we can just look at what is the value

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

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

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

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

stuff

slightly

scale

negative