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Current time:0:00Total duration:5:05

The graphical relationship between a function & its derivative (part 1)

Video transcript

so I've got this crazy discontinuous function here which we'll call f of X and my goal is to try to draw its derivative right over here so what I need to think about is the slope of the tangent line or the slope at each point in this curve and then try my best to draw that slope so let's let's try to tackle it so right over here right over here at this point the slope is positive and it's actually it's a good bit it's a good bit positive and then as we get larger and larger X's the slope is still positive but it's less positive it's less positive and all the way up to this point right over here where it becomes where it becomes zero so let's see how I could draw that over here so over here we know that the slope must be equal to zero right over here remember over here I'm going to try to draw Y is equal to F prime F prime of X and I'm going to assume that this is this is some type of a parabola and you'll learn shortly why I had to make that assumption but let's say that so let's see here the slope is quite positive so let's say the slope is right over here and then it gets less and less and less positive and I'll assume it does it in a linear fashion that's why I had assumed that it's some type of a parabola so it gets less and less and less positive notice here the slope here for example the slope is still positive and so when you look at the derivative the slope is still a positive value but as we get larger and larger X's up to this point the slope is getting less and less positive all the way to zero and then the slope is getting more and more negative and at this point it seems like the slope is just as negative as it was positive there so at this point right over here the slope is just as negative just as negative as it was positive right over there so it seems like this would be a reasonable a reasonable view of the slope of the slope of the tangent line over this interval now let's think about as we get to this point here the slope seems constant our slope is a constant positive value so once again our slope here is a constant positive line let me be careful here because at this point our slope won't really be defined because our slope you could draw multiple you can draw multiple tangent lines at this little point this little pointy point so let me just draw a circle let me draw a circle right over there but then as we get right over here the slope the slope seems to be the slope seems to be positive so let's draw that the slope seems to be positive and although it's not as positive as it was there so the slope the slope looks like it is I'll make it I'll just I'm just trying to eyeball it so the slope is a constant positive this entire time we have a line with the constant positive slope so it might look something something like this and let me make we could clear what interval let me make it clear what interval I am talking about I want these things to match up so let me do my best so this this matches up this matches up to that this matches up over here and we just said we have a constant positive slope so let's say it looks something it looks something like that over this interval and then we look at this point right over here so right at this point our slope is going to be undefined there's no way that you could find the slope over or this point of discontinuity but then when we go over here even though the value of our function has gone down we still have a constant positive slope in fact the slope of this line looks identical to the slope of this line let me do that in a different color the slope of this line looks identical so we're going to continue at that same slope it was undefined at that point but we're going to continue at that same slope we're going to continue at that same slope and once again it's undefined here at this point of discontinuity so the slope will look something like that and then we go up here the value of the function goes up but now the function is flat so the slope over that interval is 0 the slope over this interval right over here is 0 so we could say the slope over that interval let me make it clear what interval I'm talking about the slope over this interval is 0 and then finally in this last section in this last section the slope let me do this in orange in this last section the slope becomes negative but it's a constant negative it's a constant negative and it seems actually a little bit more negative than these were positive so I would draw it I would draw it right over there so it's a weird looking function but the whole point of this video is to give you an intuition for thinking about what the slope of this function might look at it again at any point and by doing so we have essentially drawn the derivative over that interval