Calculus I
The Mean Value Theorem
Or: somewhere, your speedometer read exactly your average speed
Companion notebook. Play with this chapter’s ideas — sliders, animations, and drills — in
notebooks/ch6_mvt.ipynb.
By the end of this chapter you will have the single most useful theorem in all of Calculus 1 — the one that quietly justifies almost everything in the second half of this book.
You will be able to state and use Rolle’s theorem and the Mean Value Theorem, and picture what they say: if you average miles an hour over a trip, then at some instant your speedometer read exactly . Average rate of change, achieved as an instantaneous rate of change, somewhere in between.
But the theorem itself is not the point. Its three consequences are, and they are the load-bearing beams of the rest of the book. You will prove that a function with zero slope everywhere is constant — finally repaying a debt we took out back in Chapter 5. You will prove that the sign of the derivative controls whether a function rises or falls — the engine that will let you find maxima and minima in Chapter 8. And you will prove a bound that says a function can’t change faster than its steepest slope allows — the idea that makes every numerical method in engineering trustworthy.
Here is the theme, and it connects to the last chapter. In Chapter 2 you saw that continuity forces structure: a continuous function can’t get from one value to another without passing through everything in between. This chapter is the sequel: differentiability forces structure too, and more of it. A function smooth enough to have a derivative everywhere is far more constrained than it looks — it cannot wander freely, and the Mean Value Theorem is the leash.
(One dependency, named honestly: this chapter leans on the Extreme Value Theorem — a continuous function on a closed interval actually attains a highest and a lowest value somewhere on it. That belongs to the family of “continuity forces structure” results from Chapter 2; here we’ll use it as a tool.)