 So can we refine our graph any more beyond the stick figure? Well, I can use the stick to do that if we use the second derivative. So let's consider what happens when we graph using the second derivative. So again, the first derivative gives us the slope of the line tangent to the graph of y equals f of x, and it also tells us how the function itself is changing. So again, remember that if the derivative is positive, then our function is increasing. Likewise, if our derivative is negative, then our function is decreasing. The thing to recognize here is that this is going to be true for any derivative. So if I consider the derivative of the derivative, that second derivative, if the derivative of the derivative is positive, then the derivative itself is increasing. So if a derivative is positive, something's increasing. Now remember that the derivative is the slope of the line tangent to the graph of y equals f of x, which means that the slope of the tangent lines are increasing. And here's a very, very, very, very important idea to keep in mind. The slope of a line is constant. If the slope of a line is increasing, you're not dealing with a line. So this phrase, the slope is increasing, you should be careful to remember that in every statement of that form, there is an unstated and implied as x increases. So the slope is increasing as x increases, as we move from left to right. So it's not that the slope of the tangent line itself is increasing, but as I move left to right on the graph, the slope of the tangent line that I draw at those points will be increasing. Again, similarly, if the derivative is negative, then that tells me the derivative is decreasing. But again, the derivative is the slope of the line tangent to the graph. So the slope of the tangent lines are decreasing as we move from left to right. Well, here's one where the algebraic statement is actually easier to visualize than the geometric statement. I can imagine very easily what it means for a function to be increasing. It's getting greater and greater values as x increases. Again, it's easy to picture what it means when a function is decreasing. It means that for larger and larger values of x, the function, in this case the derivative, is going to be smaller and smaller. But what does it mean when we're dealing with slope? And the only good way of getting a nice geometric intuition for what this looks like is to do problems. So let's take a look at it. So suppose I have a function, value at 2 equals 5, the derivative is 3, the second derivative is 4, and let's take a look at what our graph looks like near x equal to 2. So let's consider our graph of y equals f of x. So I know that f of 2 equals 5. So if I'm graphing y equals f of x, the y value is the same as the function value. My function value is 5, so I know what the y value is 5. The x value is the argument of the function, so I know the x value is 2. So this bit of information, f of 2 equals 5, tells us that our graph passes through the point 2, 5. So we'll go ahead and plot that point. Looks something like there. And one important point about graphing. The purpose of the sketch of a graph is to organize our thoughts on a problem. We are never going to read any information directly off of the sketch of a graph, because we can't get our sketch to be as accurate as it needs to be to be able to read information usefully off of it. The purpose of a sketch of a graph is to organize our information. So what really matters is it'd be nice to have the right relative positions. So here the point 2, 5 is positive x value someplace to the right, and positive y value someplace above. So 2, 5 someplace around there. And if you plot it over here, over here, over here, that's all fine. It doesn't really matter where we plot it, as long as it's someplace in this vicinity. We don't want to plot it down here, because that implies positive x, but negative y value. And while we could count out 1, 2, 1, 2, 3, 4, 5, and then plot the point, it's a lot more work than is warranted by what we're trying to do. So relative position, very important. Accurate position, it's kind of nice, but in general not absolutely required. Let's take a look at that second piece of information. f' of 2 is equal to 3. So remember that this is the derivative of the function that corresponds to the slope of the line tangent to the graph. So this tells me add x equals 2. The slope of the line tangent to the graph is 3. So I know where I am, and I also know what my tangent line looks like. It has slope 3. And again, we don't have to get too precise with what that looks like. Anything that has a slope that is going in the right relative direction, slope positive means it's going that way somehow. So I'll go ahead and sketch that line. So there's my slope 3 line. Just as a note, when you're trying to sketch the graph, it doesn't pay to make these tangent lines extend too far. So we just want to draw a short little bit of the tangent line. The tangent line extends indefinitely in both directions, but for purposes of sketching the graph, we don't want to draw that tangent line as going too far away from the point where we started. Let's take a look at that. Second derivative equal to 4. So add x equals 2. Second derivative is equal to 4. So that's positive. So that tells me that the first derivative is increasing. And again, the implied directionality here is as I go to the right. As I go to the right, I would expect to see a greater slope of the tangent line. Well, if this is slope 3, at a later point, I have slope more than 3. So I'm going to be looking at something that's a little bit steeper than the original slope. It also helps to go backwards. If I'm increasing as I go to the right, it follows that I must be decreasing as I go to the left. So if I go backwards from this point, here's slope 3. A little bit earlier, I should see slope less than 3. So what does less than 3 look like? Well, one possibility is that it might be something that looks a little bit flatter like that. So here's my sketch of the graph near x equals 2. I'm going through the point 2.5 because that's what my function value tells me. The tangent line looks like that because that's what the first derivative tells me. And if I keep going, I'll have steeper tangent lines later on, shallower tangent lines earlier, because that's what the second derivative tells me. Well, let's take a look at another problem. So once again, I have function value, first derivative, second derivative, and I want to sketch the graph. So again, from the function value, I know a point on the graph 2.5. From the first derivative value, I know the slope of the line tangent to the graph. So first derivative is 3. So the slope of the tangent line is 3. Looks something like that. Again, draw a short version of the tangent. And now the second derivative is negative. And what that tells me is that the derivative itself is going to be decreasing. Second derivative negative says that the thing that it's the derivative of, the first derivative is going to be decreasing. So if I follow the graph to the right, the slope of the tangent lines are going to be less. Well, if this is 3, then what does less than 3 slope look like? Something a little shallower. And if I go in the other direction, if I go backwards, the slope is going to be greater. So if this is slope 3, slope more than 3 looks like that. And so there's my sketch of the graph near the point. Well, again, let's take another example. Again, function, first derivative, second derivative, and what does that look like? So again, I know a point on the graph. So my function gives me a point on the graph to 5. x equals 2, y equals 5. First derivative, negative 3. That tells me the slope of the line tangent to the graph at this point has slope negative 3. And I'll draw that line. Again, short line. I don't want to go too far on it. Second derivative, positive. So that tells me that the first derivative is increasing, which means if I follow the graph to the right, the slope of the line tangent to the right will be more here's where we have to be a little bit careful. It's going to be more than what it was. It was negative 3, but if a graph has a slope of more than negative 3, that could be something like negative 2, negative 1. What that actually means is it's something a little bit less steep than before. And that is entirely because our original slope was negative. So if I increase from a negative number, I get closer to 0, closer to slope 0, closer to a horizontal line. And if I keep increasing, then I might start to get an up slope. But from a negative to more than that amount, it's going to be a little bit shallower. Likewise, if I go backwards, my slope is going to be less, but less than negative 3, something like negative 5, negative 200, visually that's going to look like something that is steeper. And so my graph, before I hit the point, I'm going to have a much steeper graph at that. And here's a rough connect the dots straight line sketch of our graph through x equals 2, given the information.