AP Calculus AB and BC glossary

Intermediate Value Theorem

Also called: IVT

The Intermediate Value Theorem says that if a function is continuous on a closed interval, it attains every value between the two endpoint values somewhere on that interval. It is the standard tool for proving a solution exists without finding it.

f \text{ continuous on } [a,b], \ N \text{ between } f(a) \text{ and } f(b) \implies \exists c \in (a,b), \ f(c) = N

To show an equation has a root, rewrite it as $f(x) = 0$ , verify $f$ is continuous on a closed interval, and show $f(a)$ and $f(b)$ have opposite signs. Zero lies between them, so the IVT guarantees a $c$ with $f(c) = 0$ .

For full credit you must state continuity explicitly and show the two endpoint values. Naming the theorem without checking its hypothesis is the most common way to lose the point.

What it does not say

The IVT proves existence only. It never tells you how many solutions there are or where they are, and it says nothing when the function is discontinuous.

Appears in: Unit 1: Limits and Continuity