 Let's do a quick recap of the major ideas from section 1.6 of active calculus on the second derivative. The first thing we learned in this section is a relationship between the sign of the derivative of a function and the kind of behavior the function exhibits. Namely that if f is a differentiable function, then f is increasing at x equals a if and only if f prime of a is bigger than 0, and f is decreasing at x equals a if and only if f prime of a is less than 0. This makes sense because if f is increasing at x equals a, then the tangent line to the graph of f at that point x equals a should have a positive slope, and since slope of a tangent line is by definition the derivative, then f prime of a is positive, similarly for a negative slope and for f decreasing. So in other words, the derivative of f tells us something about how the parent function f is behaving, but there's more to the behavior of a function than just whether it's increasing or decreasing. For example, here are three functions that are all increasing, but they are clearly all behaving very differently. To get at this level of behavior, we introduce the concept of the second derivative. This is just the derivative of the derivative of f. Since the derivative of a function f is just another function, we can reiterate the derivative taking process to get the second derivative. So the second derivative tells us the rate at which f prime is changing, the rate at which the slopes of f are changing, just as the first derivative tells us the rate at which the original parent function f is changing. That brings us back to the differences between the three curves that we just saw. The differences in these are captured by the concept of concavity. In the first curve, the function is increasing at an increasing rate. As x gets bigger, f not only gets higher, it also gets steeper. So not only is f increasing, but also f prime, its slopes are increasing. In the second graph, f is increasing, but its slopes appear not to be changing at all, because f is a linear function, so f prime is equal to zero. In the third graph, f is increasing, but it's getting less steep as it increases, and so therefore f prime, its slopes are decreasing. Based on this idea, we define the notion of concavity. Let's let f be a differentiable function on an interval from a to b. And we're going to say that f is concave up if and only if f prime is increasing on the interval a, b. And f is concave down on the interval a, b if and only if f prime is decreasing on a, b. Concavity can also be applied to functions whose graphs are decreasing, as is discussed in the text and in some of the subsequent videos. So we are learning that the derivative of a function yields information about the function's behavior in the form of increasing, decreasing behavior. And the second derivative yields information about f's concavity, whether f is increasing or decreasing at an increasing rate or a decreasing rate. In the next few videos, we're going to apply these basic concepts to specific examples.