 Let's have a quick review over the main ideas of section 3.2 in active calculus on using derivatives to describe families of functions. By a family of functions, we mean a function that includes one or more parameters, which are constants that can change from one instance to the next. For example, f of x equals ax squared plus b is a family of functions using two parameters a and b that describe a certain class of parabolas. Here I'm in geogibra and you can see an interactive diagram that shows this family and you can change members of the family by moving the sliders. Every choice of a and b gives another parabola in the family. But notice that no matter what member of the family we're observing, they all have certain things in common. For example, each member of the family appears to have a critical value at x equals zero. Now, we might be interested in knowing what the y value of that critical number is and how it depends on the choice of a and b. Doing so would be an example of what this section is all about, using the derivative to understand a general property that is common to all members of this family. Other than this idea, there's nothing new in this section. To find this critical number, for instance, we would need to take the derivative of f and find out where f prime is zero and where it's undefined. The computational aspect of this task would involve one slightly different idea that we would have to have parameters that show up in the function as constants because they are constants and then take all derivatives with respect to x only. As a quick example, the derivative of f here is two times a times x because a and b are parameters. That is, they are numbers or constants, not variables. So they behave the same as actual numbers for derivative purposes. So we see that f prime of x would be two a x and that's a linear function with slope two a. And so it's never undefined. But on the other hand, f prime of x is equal to zero whenever x is zero. That means that every member of this family has a critical number at x equals zero, which is what we suspected. And at this point, the y value is f of zero equals a times zero squared plus b and that's equal to b. So in other words, we've just found out that every parabola in this family has a critical value at x equals zero, y equals b, where b is the parameter from the function definition. And we can see this in the Geodraber sketch. In the section, we'll use this idea to identify many other aspects of the behavior of a family of functions, including increasing, decreasing behavior, relative extreme values, concavity, inflection points, and more.