The graphical relationship between a function & its derivative (part 2) (video) | Khan Academy
in the figure, or the graph of any other derivative, you may need to slap So, where a function is increasing, the graph of its derivative will be positive, but with the following list of rules. . That sign graph, because it's a second derivative sign graph, bears exactly (well, almost exactly) the same relationship to the graph of. second derivatives give us about the shape of the graph of a function. The first derivative of the function f(x), which we write as f (x) or as df dx., is the slope of the. This is a HUGE topic to cover properly. I think it would be a good idea just to consider some simple polynomial graphs. Notice what happens to.
If the green function were f of x, does the orange function here, or the yellow function, could that be f prime of x?
Matching functions & their derivatives graphically (video) | Khan Academy
So let's think about what's happening to this green function at different points. So this green function right over here, right at this point if we start at the left, has a positive slope. If this orange function were f prime of x, if it were the derivative of the green function, then it would have to be positive because the green function's slope is positive at that point.
But we see that it's not positive. So it's pretty clear that the green function cannot be f of x, and the yellow function cannot be its derivative, because if this was its derivative, it would be positive here.
So that quickly, we found out that that can't be the case. But let's see if it could work out the other way. So it's starting to feel-- just ruling that situation out-- that maybe that this is f of x and the green function is f prime of x. So let's see if this holds up to scrutiny. So what we have when we start off at the left, f of x, or what we think is f of x, has a reasonably positive slope. Our green function is positive.
Actually, let me erase this, just so we don't look like we're trying to take the slope of the tangent line of the derivative. So it looks just like that. So, so far this green function looks like a pretty good candidate for the derivative of this yellow function. But let's keep going here. So let's think about what happens as we move to the right. So here, let's see. It looks like the slope of this yellow function-- let me just use a color we can see-- it keeps going up.
It keeps going up, keeps going up. And then at some point, it reaches some maximum slope, and then it starts to go down again. The slope starts to go down again, all the way to the slope going all the way down to 0 right over here. Derivative as slope of tangent line Video transcript In the last video we looked at a function and tried to draw its derivative.
The graphical relationship between a function & its derivative (part 2)
Now in this video, we're going to look at a function and try to draw its antiderivative. Which sounds like a very fancy word, but it's just saying the antiderivative of a function is a function whose derivative is that function. So for example, if we have f of x, and let's say that the antiderivative of f of x is capital F of x. And this tends to be the notation, when you're talking about an antiderivative.
This just means that the derivative of capital F of x, which is equal to, you could say capital f prime of x, is equal to f of x. So we're going to try to do here is, we have our f of x. And we're going to try to think about what's a possible function that this could be the derivative of? And you're going to study this in much more depth when you start looking at integral calculus.
But there's actually many possible functions that this could be the derivative of. And our goal in this video is just to draw a reasonable possibility.Relationship between function and derivatives
So let's think about it a little bit. So let's, over here on the top, draw y is equal to capital F of x. So what we're going to try to draw is a function where its derivative could look like this.
So what we're essentially doing is, when we go from what we're draw up here to this, we're taking the derivative. So let's think about what this function could look like.
So when we look at this derivative, it says over this interval over this first interval right over here, let me do this in purple, it says over this interval from x is equal to 0 all the way to whatever value of x, this right over here, it says that the slope is a constant positive 1. So let me draw a line with a slope of a constant positive 1. And I could shift that line up and down. Once again, there's many possible antiderivatives. I will just pick a reasonable one.
So I could have a line that looks something like this. I want to draw a slope of positive 1 as best as I can. So let's say it looks something like this. And I could make the function defined here or undefined here. The derivative is undefined at this point.
- Matching functions & their derivatives graphically
- The graphical relationship between a function & its derivative (part 1)
- Connecting f and f' graphically
I could make the function defined or undefined as I see fit. This will probably be a point of discontinuity on the original function. It doesn't have to be, but I'm just trying to draw a possible function. So let's actually just say it actually is defined at that point right over there.
But since this is going to be discontinuous, the derivative is going to be undefined at that point. So that's that first interval. Now let's look at the second interval. The second interval, from where that first interval ended, all the way to right over here.
The graphical relationship between a function & its derivative (part 1) (video) | Khan Academy
The derivative is a constant negative 2. So that means over here, I'm going to have a line of constant negative slope, or constant negative 2 slope.
So it's going to be twice as steep as this one right over here. So I actually could just draw it. I could make it a continuous function, I could just make a negative 2 slope, just like this.
And it looks like this interval is about half as long as this interval. So it maybe gets to the exact same point. So it could look something-- let me draw it a little bit neater-- like this. The slope right over here is equal to 1, we see that right over there in the derivative.