Data Mining: Identifying Functions From Derivative Graphs

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  • 0:10 Quick Review
  • 1:02 Match the Graphs
  • 1:26 Graph #1
  • 3:48 Graph #2
  • 7:06 Graph #3
  • 8:54 Lesson Summary
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Lesson Transcript
Instructor: Eric Garneau
If you saw the graph of speed as a function of time for a bicycle, a jet, and a VW bug, could you pick which vehicle produced which graph? In this lesson, try it as we match functions with their derivatives.

Quick Review

What can we learn from the derivative of a function? Well, let's review. The derivative of a function is the slope of that function - really the slope of the tangent of the function. If the derivative is positive, then the function's increasing. If the derivative is negative, the function's decreasing. If the derivative is zero, then your function might be at a minimum or a maximum, but either way it's not changing.

The second derivative is the slope of the first derivative, and the second derivative tells you a little bit about the concavity of your function. If the second derivative is positive, you have a cup (something that's concave up). If the second derivative is negative, you have a frown (something that's concave down). If the second derivative is zero, then you might be at an inflection point if you're going between some region that's concave up and one that's concave down.

Match the Graph

Let's try to put these into practice. Let's say that you have three different derivative graphs, all without numbers, and they look like this. You have three possible graphs of f. Say you have these here. Now let's make it our goal to match the derivative with its original function, or at least a possibility for its original function.

When y is increasing, a region of the function is concave up
Graph One

Graph #1

Let's take a look at this first graph. There are two points of this graph that might stick out at you as being important. We've got y` as a function of x. For small values of x, y` is constant. Not only is it constant, but it's equal to zero. For large values of x, y` is positive and increasing. What do these two things mean? Well, when y` is constant at zero, then my function won't be changing. When y` is increasing, then I know that my second derivative is going to be positive, and I've got a region where my function is going to be concave up; it's going to look like a cup.

So let's take a look at our three possibilities for our function, and let's see how they match with each of these two points that we've brought out. First, let's see if any of them have a constant value for smaller values of x, because y` is zero for smaller values of x. This first function: No, that is obviously not a constant. This second function, well, that's a constant. I don't know what the value is, but it's constant; it's not changing, so that's a possibility. The third function: No; again, this is not a constant. It's increasing, so this can't work either. So right now it's looking like we're going to choose this second graph. But just to be sure, let's take a look at the second point, where we have that y` is increasing for larger values of x. We've got something that's going to be concave up for larger values of x.

Our first graph is concave up for larger values of x; this looks like a cup here, so that would've been fine. Our second graph is also increasing; this is also a cup for larger values of x. Our third graph is constant; that's not a cup. I don't know who put that graph in here. Our first possibility for f matched one of the two things that we needed. Our second graph matched both, though. It had a constant value for smaller values of x, and it was concave up for larger values of x. So I'm going to say that our first derivative graph, here, matches our second function, here.

Graph #2

Let's take a look at our second graph. Our second graph has a few more points that might be of interest. First, we have two places on this graph where y` equals zero, so we have two places where we might have a minimum or a maximum. Okay, that's something good to keep in mind. Second, for small values of x, y` is decreasing. That means that we have something that's concave down, because y``, our second derivative, is going to be less than zero - we've got a frown. For larger values of x, y` (our derivative, this graph here) is increasing, and since the derivative is increasing, the second derivative will be positive for all these values. This means that our original function is going to be concave up. So we're looking for something that's concave down in the first half and concave up in the second half - so it looks like a frown in the first half and a cup in the second half - and might possibly have a maximum or minimum at these two intermediate points.

The two points where y equals zero means there may be a minimum or a maximum
Graph Two

So let's look. First let's look at these points where y` is equal to zero. In our first graph, y` would be equal to zero at roughly these two points. Well, that's a maximum and a minimum, so indeed our tangent here is going to be zero at both of these points, so this might work. The second one - okay, well, y` is zero at this first one, I'll agree with that. The tangent here is going to be zero. But for this second one the tangent is not zero, so that one doesn't really work. For the third one, the tangent isn't zero at the first point, but it is zero at the second point. So what we've got is that the first graph has two points that match, the second graph has one of two points and the third graph has one of two points.

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