Estimating derivatives of a function at a point
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- [Instructor] So we're told that this table gives select values of the differentiable function F. So it gives us the value of the function at a few values for X, in particular, five different values for X and it tells us what the corresponding f(x) is. And they say, what is the best estimate for f'(4)? So this is the derivative of our function F when X is equal to four. Or another way to think about it, what is the slope of the tangent line when X is equal to four for f(x)? So what is the best estimate for f'(4) we can make based on this table? So, let's just visualize what's going on before we even look at the choices. So let me draw some axes here. And let me plot these points. We know that these would sit on the curve of Y is equal to f(x). When X is zero, f(x) is 72. So this is the point (0,72). This is the point (3,95) clearly, two different scales on the X and Y axes. This is the point (5,112). This is (6,77). This is (9,54), actually, let me write out the this is one, two, three, four, five, six, seven, eight, nine, and ten. Now they want us to know, they want to what is the derivative of our function when F is equal to four? Well, they haven't told us even what the value of F is at four. We don't know what that point is. But what they're trying to do is, well, we're trying to make a best estimate. And using these points, we don't even know exactly what the curve looks like. It could look like all sorts of things. We could try to fit a reasonably smooth curve. The curve might look something like that. But it might be wackier. It might do something like this. Let me try to do it. It might look something like this. So we don't know, for sure. All we know is that it needs to go through those points. 'Cause they've just sampled to the function at those points. But let's just, for the sake of this exercise, let's assume the simplest, let's say it's a nice smooth curve without too many twists and turns that goes through these points, just like that. So what they're asking, okay, when X is equal to four if this yellow curve were the actual curve then what is the slope of the tangent line, at that point? So we would be visualizing that. Now to be clear, this tangent line that I just drew this would be for this version of our function that I did connecting these points. That does not have to be the actual function. We know that the actual function has to go through those points. But I'm just doing this for visualization purposes. One of the whole ideas here is that all we do have is the sample and we're trying to get a best estimate. We don't know if it's even gonna be a good estimate. It's just going to be a best estimate. So what we generally do when we just have some data around a point, is, let's use a data points that are closest to that point and find slopes of secant lines pretty close around that point. And that's going to give us our best estimate for the slope of the tangent line. So what points do we have near F of four, or near the point (4,f(4))? Well, they give us what F is equal to when X is equal to three. They give us this point, right over here. Let me do this in another color. So, (3,95) that is that right over there. And they also give us (5,112). That is that point right over there. And so what we could do we could say, well, what is the average rate of change between these two points? Another way to think about it is, what is the slope of the secant line between those two points. And, that would be our best estimate for the slope of the tangent line at X equals four. Do we know that it's a good estimate? Do we know that it's even close? No, we don't know for sure, but that would be the best estimate. It would be better than trying to take the the average rate of change between When X equals three and X equals six. Or between when X equals zero and X equals nine. These are pretty close around four. And so, let's do that. Let's find the average rate of change between when X goes from three to five. So, we can see here our change in X. Let me do this in a new color. So our change in X here is equal to plus two. And I can draw that out. My change in X here is plus two. And, my change in Y is going to be when my X increased by two, my change in Y is plus, let's see, this is if I add ten I get to I get to 105. If I add another seven so this is plus 17. So this is plus 17 right over here. Plus 17. And so my change in Y over change in X. Change in Y over my change in X. For this secant line between when X is equaling three and X is equaling five is going to be equal to 17 over two. Seventeen over two. Which is equal to 8.5. So the slope of this green line here is 8.5. And that would be our best estimate for the slope of the tangent line when X equals four of the curve Y is equal to f(x). And so, lucky for us the people who wrote this question had the exact same logic, and they did it right over there. So you wouldn't have to graph it the way I did. I did it just to help us visualize what's going on. In general, when you see a question like this they're really saying, look, you don't have all the data you need to figure out exactly what f'(4) is. But if you can find points close to, or around f'(4) and find the secant line, the average rate of the slope of the secant line. Or the average rate of change between those points that's going to be our best estimate for the instantaneous rate of change when X equals four. Or the derivative when X equals four.
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