Derivative as a function
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Matching functions & their derivatives graphically (old)
The function f of x is shown in green. The sliding purple window may contain a section of an antiderivative of the function, F of x. So, essentially it's saying, this green function, or part of this green function, is potentially the derivative of this purple function. And what we need to do is-- it says, where does the function in the sliding window correspond to the antiderivative of our function? The antiderivative of f of x, usually, write as big F of x. This is just saying that, lowercase f of x is just the derivative of big F of x. So, at what point could the derivative of the purple function-- and I'm going to move the purple function around-- where can the derivative of that be the green stuff. So let's just focus on the purple stuff first. So the derivative-- we can just view it as the slope of the tangent line-- between this point and this point, we see that we have a constant negative slope, and then we have a constant positive slope. So let's see, where here do we have a constant negative slope? Well, now here the slope is 0, and it gets more negative. Here we have a constant positive slope, not a constant negative slope. Here we have a constant negative slope, so maybe it matches up over there. So here we have a constant negative slope, but then on the purple function, we have a positive slope, but where the potential derivative is here, we just have a slope of 0. So, this doesn't match up either. So it looks like in this case, there's actually no solution. Let's see if this works out. Yes, correct. Next question. Let's do another one. A function f of x is shown purple. The sliding green window may contain a section of its derivative. So now we're trying to say, at what point of this purple function might the derivative look like this green function? So in this green function, if this is the function's derivative, here the slope is very negative. It goes to 0, and then the slope gets positive. So let's think about it. So over here, the slope is just a constant negative, so that won't work. If we shift it over here, our slope is very steep in the negative direction and then it gets less and less steep in the negative direction, and it goes all the way, and then over here the slope is 0. And over here, if this is the derivative, it seems to match up, the slope is 0. And then it gets more and more steep in the positive direction. So this matches up. It looks like over this interval, the green the function is indeed the derivative of this purple function. So let's see. Let's check our answer. Correct. Next question. Let's do another one. This is exciting. A function f of x is shown in green. The sliding purple window may contain a section of an antiderivative of the function, F of x. So, now we say, let's match up this little purple section to its derivative. So the green is the derivative, the purple function is the thing we're taking the derivative of. So if we just look at the purple, we see that we have a constant negative slope in the first part of it, then our slope-- so let me just look for where I can find a constant negative slope. So here, this is a constant positive slope. This is not a constant slope. This is a constant positive. Here's a constant negative slope. Let's see if this works. So over this interval, between here and here, my slope is a constant negative, and indeed, it looks like a constant negative. And you see it's a constant negative 1. And over here, you see the derivative is right at negative 1, and it's constant, so that part looks good. And then when I look at the purple function, my slope is 0 starting off, then it gets more and more steep in the negative direction. And so my slope is 0, and it gets more and more negative, so this is indeed seems to match up. So, let's check our answer. Yes, got it right. I could keep doing this. This is so much fun.