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## Connecting differentiability and continuity: determining when derivatives do and do not exist

# Differentiability at a point: graphical

AP.CALC:

FUN‑2 (EU)

, FUN‑2.A (LO)

, FUN‑2.A.1 (EK)

, FUN‑2.A.2 (EK)

## Video transcript

- [Voiceover] The graph of
function f is given below. It has a vertical tangent at
the point three comma zero. Three comma zero has a vertical
tangent, let me draw that. It has a vertical
tangent right over there, and a horizontal tangent at the point zero comma negative three. Zero comma negative three, so it has a horizontal
tangent right over there, and also has a horizontal
tangent at six comma three. Six comma three, let me draw the horizontal tangent, just like that. Select all the x-values for
which f is not differentiable. Select all that apply. F prime, f prime, I'll
write it in short hand. We say no f prime under it's going to happen
under three conditions. The first condition you could say well we have a vertical tangent. Vertical tangent. Why is a vertical tangent a place where it's hard to define our derivative? Well, remember, our derivative is we're really trying to find
our rate of change of y with respect to x, but when you have a vertical tangent, you change your x a very small amount, you have an infinite change in y, either in the positive or
the negative direction. That's one situation where
you have no derivative. They tell us where we have
a vertical tangent in here, where x is equal to three. We have no ... F is not differentiable at x equals three because of the vertical tangent. You might say what about
horizontal tangents? No, horizontal tangents
are completely fine. Horizontal tangents are
places where the derivative is equal to zero. F prime of six is equal to zero. F prime of zero is equal to zero. What are other scenarios? Well another scenario
where you're not gonna have a defined derivative is where
the graph is not continuous. Not continuous. We see right over here at
x equals negative three, our graph is not continuous. X equals negative three
it's not continuous. Those are thee only places
where f is not differentiable that they're giving us options on. We don't know what the graph is doing to the left or the right. These there I guess would
be interesting cases. They haven't given us those choices here. We already said, at x equals
0, the derivative is zero. It's defined. It's differentiable there. At x equals six, the derivative is zero. We have a flat tangent. Once again it's defined there as well. Let's do another one of these. Actually, I didn't include, I think that this takes care of this problem, but there's a third scenario in which we have, I'll
call it a sharp turn. A sharp turn. This isn't the most mathy
definition right over here, but it's easy to recognize. A sharp turn is something like that, or like, well no, that
doesn't look too sharp, or like this. The reason why where you
have these sharp bends or sharp turns as opposed to something that looks more smooth like that. The reason why we're not
differentiable there is as we approach this point, as we approach this
point from either side, we have different slopes. Notice our slope is
positive right over here, as x increases, y is increasing, While the slope is negative here. As you're trying to find
the limit of our slope as we approach this point, it's not going to exist
because it's different on the left hand side
and the right hand side. That's why the sharp turns, I
don't see any sharp turns here so it doesn't apply to this example. Let's do one more examples. Actually this one does
have some sharp turns. This could be interesting. The graph of function f is
given to the left right here. It has a vertical asymptote
at x equals negative three, we see that, and horizontal asymptotes
at y equals zero. This end of the curve as x
approaches negative infinity it looks like y is approaching zero. It has another horizontal
asymptote at y equals four. As x approaches infinity, it looks like our graph is trending down
to y is equal to four. Select the x values for which
f is not differentiable. First of all, we could think
about vertical tangents. Doesn't seem to have
any vertical tangents. Then we could think about
where we are not continuous. Well, we're definitely not continuous where we have this vertical
asymptote right over here. We're not continuous at
x equals negative three. We're also not continuous
at x is equal to one. Then the last situation where we are not going to be differentiable is where we have a sharp turn, or you could kind of view it
as a sharp point, on our graph. I see a sharp point right over there. Notice as we approach
from the left hand side, the slope looks like a
constant, I don't know, it's like a positive three halves, while as we go to the right side of that it looks like our slope turns negative. If you were to try to find
the limit of the slope as we approach from either side, which is essentially
what you're trying to do when you try to find the derivative, well it's not going to be defined because it's different from either side. F is also not differentiable
at the x value that gives us that little
sharp point right over there. If you were to graph the derivative, which we will do in future videos, you will see that the derivative is not continuous at that point. Let me mark that off. Then we can check x equals zero. X equals zero's completely cool. We're at a point that our tangent line is definitely not vertical. We're definitely continuous there. We definitely do not have
a sharp point or edge. We're completely cool at x equals zero.

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