# Relationship between a function and its derivative

In previous modules you used the graph of a function to investigate its derivative. In this module you will use the derivative to find properties of the original. Exploration of the relationship between the graph of a function and the graph of its derivative. Jan 9, Reading a derivative graph is an important part of the AP Calculus curriculum. Typical calculus problems involve being given function or a.

## The graphical relationship between a function & its derivative (part 2)

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 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.

### The graphical relationship between a function & its derivative (part 1) (video) | Khan Academy

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.

And then the slope right over here is equal to negative 2. We see that in the derivative. Now things get interesting. Once again, I could have shifted this blue line up and down. I did not have to construct a continuous function like this. But I'm doing it just for fun.

There's many possible antiderivatives of this function. Now what's going on over the next interval? And I'll do it do it in orange. The slope starts off at a very high value. It starts off at positive 2. Then it keeps decreasing, and it gets to 0, right over here. The slope gets to 0 right over there. And then it starts becoming more and more and more and more negative. So I'll just try to make this a continuous function, just for fun. Once again, it does not have to be.

So over here the slope is very positive. It's a positive 2. So our slope is going to be like, this is negative 2, so it's going to be a positive 2. And then it gets more and more and more negative up to this, or, it becomes less and less and less positive I should say. Even here the slope is positive. And it gets to 0 right over there. So maybe it gets to 0 maybe right over here. And so what we have, we could have some type of a parabola. So the downward facing parabola.

So notice, the slope is a very positive value. It's a positive 2 right over here, then it becomes less and less and less and less positive, all the way to 0 right over there.

And then the slope starts turning negative. And so our function could look something like that over the interval. Let me draw it a little bit neater. So let me just draw a circle right over there. But then as we get right over here, the slope seems to be positive. So let's draw that. The slope seems to be positive, although it's not as positive as it was there. So the slope looks like it is-- I'm just trying to eyeball it-- so the slope is a constant positive this entire time.

We have a line with a constant positive slope. So it might look something like this. And let me make it clear what interval I am talking about.

I want these things to match up. So let me do my best. So this matches up to that. This matches up over here. And we just said we have a constant positive slope. So let's say it looks something like that over this interval.

## The graphical relationship between a function & its derivative (part 1)

And then we look at this point right over here. So right at this point, our slope is going to be undefined. There's no way that you could find the slope over-- or this point of discontinuity. But then when we go over here, even though the value of our function has gone down, we still have a constant positive slope. In fact, the slope of this line looks identical to the slope of this line.

Let me do that in a different color. The slope of this line looks identical. So we're going to continue at that same slope. It was undefined at that point, but we're going to continue at that same slope. And once again, it's undefined here at this point of discontinuity.

So the slope will look something like that. And then we go up here. The value of the function goes up, but now the function is flat. So the slope over that interval is 0.

### Module 9 - The Relationship between a Function and Its First and Second Derivative

The slope over this interval, right over here, is 0. So we could say-- let me make it clear what interval I'm talking about-- the slope over this interval is 0. And then finally, in this last section-- let me do this in orange-- the slope becomes negative.

But it's a constant negative. And it seems actually a little bit more negative than these were positive.