Lesson 16: The Second Derivative, Revisited; Distance as Area

Here's the on-ramp for this lesson.

What Will We Learn?

• This lesson marks the end of our study of differentiation (for now, at least; we will later return to the topic when we study the calculus of transcendental functions). As such, the lesson is divided into two main topics. The first is what information the second derivative gives us about the underlying function's graph. (The answers are what was promised in Lesson 11's Lesson Preview.) We'll learn that the second derivative tells us about the curvature of the function's graph -- whether the graph curves up, curves down, doesn't curve (i.e., is linear). The second topic of the lesson launches us into a new direction: the study of integration. We'll start off that journey by inverting the investigations that led us to derivatives in the first place -- our study of instantaneous speed -- and asking instead: Given the speed of an object, can we determine how far it travels? And was true of our study of differentiation, the geometric approach we'll take to answering that question will lay the foundation for deep insights later about integration and its relation to differentiation.

Why Do We Need to Learn This?

• The second derivative, as a concept, gives us information about the curvature of the graph of a function. It therefore helps us sketch functions' graphs and also understand why they curve in the ways they do. Also, in many applications -- most notable in the sciences -- the second derivative describes important quantities we can measure in real-world contexts (e.g., force/acceleration in the physical and life sciences). Finally, the second half of the lesson -- where we learn to extract distance traveled from speed -- is the introduction to integration, the final branch in the Tree of Calculus.