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Current time:0:00Total duration:6:45

The graphical relationship between a function & its derivative (part 2)

Video transcript

in the last video we looked at a function and tried to draw its derivative now in this video we're going to look at a function and try to draw its antiderivative antiderivative which sounds like a very fancy word but it's just saying the antiderivative of a function is a function whose derivative is that function so for example if we have f of X and let's say that the antiderivative anti derivative of anti derivative of f of X is capital f of X and this is tends to be the notation when you're talking about an antiderivative this just means this literally just means that the derivative that the derivative of capital f of X which is equal to could say capital F prime of X is equal to f of X so what we're going to try to do here is we have our f of X and we're going to try to think about what's a possible function that this could be the derivative of and you're going to study this in much more depth when you when you start looking at integral calculus but there's actually many possible functions that this could be the derivative of and our goal in this video is just to draw a reasonable possibility so let's think about it a little bit so let's over here on the top draw Y is equal to capital f of X so what we're going to try to draw is a function where it's derivative could look like this so we're essentially doing is we are when we go from what we draw up here to this we're taking the derivative so let's think about what this function could look like so when we look at its derivative it says over this interval over this first interval right over here let me do this in purple it says over this interval from X is equal to zero all the way to whatever value of x this right over here it says that the slope is a constant positive one so let me draw a line with a slope of a constant positive one and I could shift that line up and down once again there's many possible anti derivatives I will just pick a reasonable one so I could have a line that looks something like this I could have a that looks something like this I want to draw a slope of a slope of positive one as best as I can so let's say it looks something like this and I could make the function defined here undefined here the derivative is undefined at this point I can make the function defined or undefined as I as I see fit this will probably be a point of discontinuity on their original function it doesn't have to be but I'm just trying to draw a possible function so let's actually just let's just say it actually is defined at that point right over there but since this is going to be discontinuous the derivative is going to be undefined at that point so that's that first interval now let's look at the second interval the second rule from where that first interval ended all the way to right over here the derivative is a constant negative two so that means over here I'm going to have a line of constant negative slope or constant negative two slope so it's going to be twice as steep as this one right over here so I actually could just draw it I could make it a continuous function I could just make a negative two slope just like this and it looks like this interval is about half as long as this interval so it maybe gets to the exact same point so it could look it could look something let me draw it let me draw it a little bit neater it could look something like it could look something like this could look something like this the slope this slope right over here slope right over here is equal to one we see that right over there in the derivative and then the slope right over here the slope right over here is equal to negative two we see that in the derivative now things get interesting once again I could have shifted this blue line up and down and I did not have to construct a continuous function like this but I'm doing it just for fun there's many possible anti derivatives of this function now what's going on over the next interval what is going on over this next interval and I'll do Lin do it in orange the slope starts off at a very high value or it started starts off at positive two then it keeps decreasing and it gets to zero right over here the slope gets to zero right over there and then it starts becoming more and more and more and more negative so I'll just try to make this a continuous function just for fun once again it does not have to be so over here the slope is very positive it's a positive two so our slope is going to be like this is negative two so it's going to be a positive two and then it gets more and more and more negative up to this more it becomes less and less and less positive I should say even here the slope is positive and it gets to zero right over there so maybe it gets to zero it gets to zero maybe right over here and so what we have we could have some type of a parabola so on a downward facing parabola so notice the slope is a very positive value it's a positive two right over here then it becomes less and less and less and less positive all the way to zero right over there and then the slope starts turning negative then the slope starts turning negative and so our function could look something like that over the interval let me draw a little bit let me draw it a little bit neater this is this is symmetric however positive our slope was here it's equally negative here so our curve should probably be symmetric as well so let me draw it let me draw it like this so over that interval it could look like that and then finally over this last interval this last interval or I guess we could say it keeps going our slope is zero our slope is zero right over here once again I don't have to draw a continuous function but when the slope is zero that just means that just means that I have a line with slope zero I have a horizontal line and I could draw that horizontal line up here in which case I'd have to say it's it's discontinuous or or I could draw that horizontal line right over here and try to make my antiderivative a continuous function so once again I could have shifted any of these segments up or down and gotten the exact same derivative but then I would not have gotten a continuous function like this but we have been able to construct a possible antiderivative for f of X and just as a reminder that's just saying if this f of X is that the antiderivative is a function that f of X be the derivative of