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Studying for a test? Prepare with these 9 lessons on Antiderivatives and the fundamental theorem of calculus.
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Let's say that this right over here is the graph of lowercase f of x. That's lowercase f of x there. And let's say that we have some other function capital F of x. And if you were to take its derivative, so, capital F prime of x, that's equal to lowercase f of x, lower case f of x. So given that, which of these, which of these, could be the graph of capital, of capital F of x? And I encourage you to pause this video, and try and think about it on your own before we work through it. Well if, if this curve is going to be the derivative of one of them. That means that any, for any x value it's describing what the instantaneous rate of change. Or what the slope of the tangent line is, of which ever one of these is the possible capital F of x. So, let's just look at a couple of things right here so what, what do we know about lower case f of x? What do we know of which is the derivative of one of these? Well, one thing we know is it's always positive. It, it has as we go to negative infinity, it asymptotes towards 0. But it's always positive. So since this is describing the slope of one of these. That means that the slope of one of these always, or out of the candidates, has to always be positive. And, if we look at this, the slope of the tangent line here is, indeed, always positive. The slope of the tangent line here does look like it's positive. Every time we increase an x, we're increasing by y. Here it's positive. But here it's negative. When we increase by x, we decrease by y. So, we can rule, we can rule this one out. Now, what else, what else do we know? Well, this is the derivative. This is telling us the slope of the tangent line. So, for example, when x is equal to, when x is equal to negative 4, f of, f of negative 4 is pretty close to 0. It's pretty close to 0. So, it's slightly, slightly more than 0. So, that tells us that the slope of tangent line of capital F of x has to be pretty close to 0, when x equals negative 4. So, let's see, when x equals negative 4, the slope of tangent line, here, isn't close to 0, this actually looks closer to 1. So, we could rule this one out. Over here, when x is equal to negative 4, the slope of the tangent line, yeah, that actually does look pretty close to 0. So I won't rule that one out. And over here, the slope of the tangent line, when x is equal to negative 4, that also looks pretty close to 0. So these are still both in the running. So let's see how we can think of it different. So let's just pick another point. When x is equal to, when x is equal to 0, f of 0 looks like it's pretty close to 1. I don't know if it's exactly to 1. Actually it looks almost exactly. Almost exactly equal to 1. So when capital F of, so at capital F of 0, the slope of the tangent line needs to be pretty close to 1. So over here the slope of the tangent line, when x is equal to 0, that looks smaller than 1. So this slope is definitely not 1. While over here, when x is equal to 0. The slope of the tangent line does look, the slope of the tangent line does look pretty, pretty close, pretty close to 1. So this, right over here, looks like the best candidate for capital, for capital F of x. So that one right over there. Lemme, that is capital F of x. And you might say hey these look similar to each other. And fact they look almost or actually they do look almost identical. And you might remember from what you knew about differentiation, that these actually look like the basic exponential function. Were, were I didn't ask you to find out what the actual function was just the possible anti-derivative of this function would be. This is the derivative, lower case f is the, is the derivative of capital f, or you could say that capital f is an anti derivative of lower case f. And when you just inspect this, this looks like this, the, the function, both of these functions is, are e to the x. Because the derivative of e to the x is e to the x.