Connecting f, f', and f'' graphically | AP Calculus AB | Khan Academy

We have the graphs of three functions here, and what we know is that one of them is the function f, another is the first derivative of f, and then the third is the second derivative of f. Our goal is to figure out which function is which— which one is f, which is the first derivative, and which is the second. Like always, pause this video and see if you can work through it on your own before we do it together.

All right, now let's do this together. The way I'm going to tackle it is I'm going to try to sketch what we can about the derivatives of each of these graphs or each of the functions represented by these graphs.

So in this first graph here, in this orange color, we can see that the slope is quite positive here, but then it becomes less and less and less positive up until this point where the slope is going to be 0. Then it becomes more and more and more and more negative. So the derivative of this curve right over here, or the function represented by this curve, it's going to start off reasonably positive right over there. At this point, the derivative is going to cross 0 because our derivative is 0 there—slope of the tangent line. Then it's going to get more and more negative, or at least over the interval that we see. So it might look, I don't know, something like this. I don't know if it's a line or not; it might be some type of a curve, but it'll definitely have a trend something like that.

Now we can immediately tell that this blue graph is not the derivative of this orange graph. Its trend is opposite; over that interval, it's going from being negative to positive, as opposed to going from positive to negative. So we can rule out the blue graph as being the derivative of the orange graph.

But what about this magenta graph? It does have the right trend; in fact, it intersects the x-axis at the right place, right over there. At least over this interval, it seems it's positive from here to here. So it's positive; this graph is positive when the slope of the tangent line here is positive, and this graph is negative when the slope of the tangent line here is negative.

Now, one thing that might be causing some unease to immediately say that this last graph is the derivative of the first one is we're not used to situations where the derivative has more extreme points—more minima and maxima than the original function. But in this case, it could just be because we don't see the entire original function.

So, for example, if this last graph is indeed the derivative of this first graph, then what we see is our derivative is negative right over here, but then right around here it starts becoming less negative. So that point corresponds to roughly right over there. Then over here, our slope will become less and less and less negative, and then at this point, our slope would become 0, which would be right around there.

So for example, our graph might look something like this. We just didn't see it; it fell off of the part of the graph that we actually showed. So I would actually say that this is a good candidate for being the third function; it's a good candidate for being the derivative of the first function. So maybe we could say that this is f and that this is f prime.

Now let's look at the second graph. What would its derivative look like? So over here, our slope is quite negative, and it becomes less and less and less negative until we go right over here, where our slope is 0. So our derivative would intersect the x-axis right over there. It would start out negative and it would become less and less and less negative, and at this point, it crosses the x-axis and then it becomes more and more positive.

So we see here our derivative becomes more and more positive, but then right around here it seems like it's getting less positive again. So it might look something like this, where over here it's becoming less positive again, less positive, less positive, less positive. Right over here, our derivative would be zero, so our derivative would intersect the x-axis there, and then it just looks like the slope is getting more and more and more negative.

So our derivative is going to get more and more and more negative. Well, what I very roughly just sketched out looks an awful lot like the brown graph right over here. So this brown graph does indeed look like the derivative of this blue graph.

So what I would say is that this is actually f, and then this would be f prime. And then if this is f prime, the derivative of that is going to be f prime prime. So that looks good. I would actually go with this, and if you wanted, just for safe measure, you could try to sketch out what the derivative of this graph would be.

And actually, let's just do that. So over here, the derivative of this, so right now we have a positive slope of our tangent line that's getting less and less positive. It hits 0 right over there, so the derivative might look something like this over that interval.

Now the slope of the tangent line is getting more and more and more and more negative right until about that point. So it's getting more and more and more negative until about that point, and now it looks like it's getting less and less and less and less negative all the way until the derivative goes back to being zero. And then it looks like it's getting more and more and more and more positive.

So the derivative of this magenta curve looks like an upward-opening U, and we don't see that over here, so we could feel good that its derivative actually isn't depicted. So I feel good calling the middle graph f, the calling the left graph f prime, and calling the right graph the second derivative.