>> So, in this question, we have the following. We have a potential energy curve for a particle as a function of x and it looks like this. OK. And the values here are 0, 1, 2, 3, 4. And it goes from 0, 1, 2, 3, 4. And so this point matches with that point right there. OK. And x is, of course, in meters and potential energy is, of course, in joules. So, this is the potential energy function for a particle and what they want to know is what is f sub x at x equal 4. OK. See if you can work that out and I'll tell you what your options are. You've got A, B, C, D, E. And it is negative 4 newtons, negative 2 newtons, 0 newtons, 2 newtons, or 4 newtons. So, all right. What did you guys think? Table number 1 right here. What did you guys think? Did you come up with an answer for this one? >> E. >> E. OK. Table number 2, what'd you guys like? >> E. >> E. >> E. Table number 3? Do you-- feel like you want to be different? >> We'll go with D. >> D? As in dog? >> Yes. >> OK. Table number 4 right here. What'd you guys think? D? OK. Table number 5 right there. What do you guys think? >> E. >> D. >> E? >> D as in dog. >> D as in dog. Last table? >> E. >> E? OK. So, keep your mics on for a second. Let's talk about your answers and let's see what the people at home said. OK. People at home were completely all over the place. So, table number 1, you guys said E, right? Why did you say E? >> Just looking at the graph. >> OK. You looked at the graph and it made sense because I saw a 4 there and I saw a 4 there, so the answer's got to be 4, right? >> Yeah. >> Yeah. OK. Table number 2 over here. Did you guys answer D? This table right here? Did you guys answer D? So, what did you like about D? [ Inaudible Speaker ] You saw a 4 here and then you saw a 2 here. Right? So, you thought what? >> Something about the area under the curve. >> Something about the area under the curve. All right. Maybe. I don't know. Let's see. What's the area under this curve? It's going to be 2 times 4 and then a half of that, so that would be 4, so you would have said E. So, those that said D, what led you to D? [ Inaudible Speaker ] Yeah, 2 looks good because-- don't high five just yet. That's a little presumptuous, isn't it? They're in the back over here high fiving each other. Yeah. All right. That's good. Physics! [ Inaudible Speaker ] OK. So, you're trying to relate this to conservation of energy. We only have sort of half of the picture here, of course, because this is just the potential energy. We don't know what the kinetic energy is doing, right, but we know the kinetic energy must be doing enough to make this a constant. But we were of course asking about the force. So, you guys want to see the right answer? OK. Let's see what the right answer is. The right answer is B. >> Oh. OK. >> Right here. Which nobody seemed very interested in at all. Why is it B? What is the relationship between force and potential energy? Anybody remember what that is? Let's just tell you what it is. Force is the derivative of the potential energy, except there's a minus sign. OK? So, if I look at the slope right here, the slope of this thing is, of course, 2 because we've gone up 4 in a distance of 2. But I have to take the negative sign of that and that's where we get our negative 2. Now, when you look at potential energy curves, there's a very nice way to double check whether you're thinking about it correctly or not and that is put a ball on the hill. If this potential energy curve is a hill, the ball will tell you what's going to happen. What's going to happen to the ball on the hill? It's going to roll down the hill, right? It's going to head that way. Does that mean a positive force on it or a negative force? A negative because positive x is to the right. So, the force on it is not to the right, it's got to be to the left and that's where this negative sign came into it.