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# 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

the graph of function f is given below it has a vertical tangent at the point 3 comma 0 so 3 comma 0 has a vertical tangent let me draw that so it has a vertical tangent right over there and a horizontal tangent at the points 0 comma negative 3 0 comma negative 3 so as a horizontal tangent right over there and also has a horizontal tangent at 6 comma 3 so 6 comma 3 let me draw the horizontal tangent just like that select all the X values for which F is not differentiable select all that apply so f prime F prime like I'll write in a shorthand so 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 a vertical tangent why is a vertical tangent to place where it's hard to define our derivative well remember our derivative is we're really still trying to find a rate of change of Y with respect to X but when you when you have a vertical tangent you change your X a very small amount you have a infinite change in Y either in the positive or the negative direction so that's one situation where you have no at no derivative and they tell us where we have a vertical tangent in here where X is equal to 3 so we have no we F is not differentiable at x equals 3 because of the vertical tangent you might see what about horizontal tangents don't horizontal tangents are completely fine horizontal tangents are places where the derivative is equal to 0 so f prime of 6 is equal to 0 f prime of 0 is equal to 0 what are other scenarios well another scenario where you're not going to have a defined derivative is where the graph is not continuous not continuous and we see right over here at x equals negative 3 our graph is not continuous so x equals negative 3 it's not continuous and those are the only places where f is not differentiable that they should that they're giving us options on we don't know what what the graph is doing to the left or the right these are I guess would be intra in cases but they haven't they haven't given us those choices here and we already said at x equals 0 the derivative is 0 it's defined it's differentiable there and at x equals 6 the derivative is 0 we have a flat flat tangent so once again it's defined there as well let's do another one of these oh and 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 and 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 or like one that doesn't look too sharp or like this and the reason why I think 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 the this point as we approach this point from either side we have different slopes notice our slope is positive right over here where as x increases Y is increasing well our slope is negative here so as you try 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 so 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 and actually this one does have some sharp turns so this could be 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 3 we see that and horizontal asymptotes at y equals 0 yep this this end of the curve as X approaches negative infinity it looks like Y is approaching zero and it has another horizontal asymptote at y equals 4 is X approaches infinity it looks like our graph is trending down to Y is equal to 4 select the X values for which F is not differentiable so first of all we could think about vertical tangents don't not doesn't seem to have any vertical tangents that we can think about where we are not continuous well we're definitely not continuous where we have this vertical asymptote right over here so we're not continuous at x equals negative 3 we're also not continuous at X is equal to 1 and then the a situation where we are not going to be differentiable is where we have a sharp turn or you can kind of view it as a sharp point on our graph and I see a sharp point right over there notice as we approach from the left hand side the slope looks like a looks like a constant I don't know it's like a positive three-halves well right as we go to the right side of that it looks like our slope turns negative and so if you were trying 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 on either side so we also the F is also not differentiable at this at the x value that gives us that little sharp that sharp point right over there and 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 so let me mark that off and then we could check x equals 0 x equals 0 is 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 so we're completely cool at x equals 0