Lectures>Practice

# Concavity and Curve Sketching

Concavity is a measure of the rate of change of the rate of change of a function. If that sounds a little over the top, it can be but it's also a useful tool in several physical applications and in getting a detailed sketch of a function's graph.

# Videos

We've seen how we can get several different useful pieces of information from the first derivative. Next, we'll try the same kind of analysis with the second derivative.
Concavity gives us a second way, called the Second Derivative Test, for determining whether or not a critical value is a local extrema.
You've already learned a variety of techniques for graphing polynomials. In this lesson, we're going to expand on that by adding finding the location of the maximum and minimum values to our toolbox.
Calculus techniques didn't have a lot to add to graphing polynomials. When we step up the complexity to rational, their usefulness is going to increase. Knowing where a function is increasing vs. decreasing will usually eliminate the need to pick points to determine where a function is in a specific interval.
This is where our calculus techniques really come to the foreground. In this lecture, we're going to look at functions with exponential, logarithmic and trigonometric components that would be difficult if not impossible to do without our new tools.