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  • - [Voiceover] Is the function given

  • below continuous slash differentiable at x equals three?

  • They've defined it piece-wise,

  • and we have some choices.

  • Continuous, not differentiable.

  • Differentiable, not continuous.

  • Both continuous and differentiable.

  • Neither continuous not differentiable.

  • Now one of these we can knock out right from the get go.

  • In order to be differentiable you need to be continuous.

  • You cannot have differentiable but not continuous.

  • So let's just rule that one out.

  • And now let's think about continuity.

  • So let's first think about continuity.

  • And frankly, if it isn't continuous,

  • then it's not going to be differentiable.

  • So let's think about it a little bit.

  • So in order to be continuous, f of ...

  • Using a darker color.

  • F of three needs to be equal to

  • the limit of f of x as x approaches three.

  • Now what is f of three?

  • Well let's see, we've fallen to this case right over here,

  • because x is equal to three, so six times three

  • is 18, minus nine is nine, so this is nine.

  • So the limit of f of x as x approaches three

  • needs to be equal to nine.

  • So let's first think about the limit

  • as we approach from the left hand side.

  • The limit as x approaches three.

  • X approaches three from the left hand side of f of x.

  • Well when x is less than three

  • we fall into this case, so f of x

  • is just going to be equal to x squared.

  • And so this is defined and continuous

  • for all real numbers, so we can just

  • substitute the three in there.

  • So this is going to be equal to nine.

  • Now what's the limit of as we approach

  • three from the right hand side of f of x?

  • Well as we approach from the right,

  • this one right over here is f of x

  • is equal to six x minus nine.

  • So we just write six x minus nine.

  • And once again six x minus nine is defined

  • and continuous for all real numbers,

  • so we could just pop a three in

  • there and you get 18 minus nine.

  • Well this is also equal to nine,

  • so the right and left hand, the left

  • and right hand limits both equal nine,

  • which is equal to the value of the function there,

  • so it is definitely continuous.

  • So we can rule out this choice right over there.

  • And now let's think about differentiablity.

  • So in order to be differentiable ...

  • So differentiable, I'll just diff-er-ent-iable.

  • In order to be differentiable the limit

  • as x approaches three of f of x minus f of three

  • over x minus three needs to exist.

  • So let's see if we can evaluate this.

  • So first of all we know what f of three is.

  • F of three, we already evaluated this.

  • This is going to be nine.

  • And let's see if we can evaluate this limit,

  • or let's see what the limit is as we approach

  • from the left hand side or the right hand side,

  • and if they are approaching the same thing

  • then that same thing that they are approaching is a limit.

  • So let's first think about the limit

  • as x approaches three from the left hand side.

  • So it's over x minus three, and we have f of x minus nine.

  • But as we approach from the left hand side,

  • this is f of x, as x is less than three,

  • f of x is equal to x squared.

  • So this would be instead of f of x minus 9

  • I'll write x squared minus nine, and x squared minus nine.

  • This is a difference of squares,

  • so this is x plus three times x minus three,

  • x plus three times x minus three.

  • And so these would cancel out.

  • We can say that is equivalent to x plus three

  • as long as x does not equal three.

  • That's okay because we're approaching from the left,

  • and as we approach from the left

  • x plus three is defined for all real numbers,

  • it's continuous for all real numbers,

  • so we can just substitute the three in there.

  • So we would get a six.

  • So now let's try to evaluate the limit

  • as we approach from the right hand side.

  • So once again it's f of x, but as we approach

  • from the right hand side, f of x is six x minus nine.

  • That's our f of x.

  • And then we have minus f of three, which is nine.

  • So it's six x minus 18.

  • Six x minus 18.

  • Well that's the same thing as six times x minus three,

  • and as we approach from the right,

  • well that's just going to be equal to six.

  • So it looks like our derivative exists there,

  • and it is equal to limit as x approaches three

  • of all of this because this is equal to six,

  • because the limit is approached from the left

  • and the right is also equal to six.

  • So this looks like we are both

  • continuous and differentiable.

- [Voiceover] Is the function given

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04-3 一點的可微性:代數(函數是可微的)(Differentiability at a point: algebraic (function is differentiable) | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 12 日
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