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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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A2 初級

Differentiability at a point: algebraic (function is differentiable) | AP Calculus AB | Khan Academy

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yukang920108 に公開 2022 年 07 月 12 日