The Intermediate Value Theorem

Section 6.1 The Intermediate Value Theorem

After our brief exploration of integrals, we now go back to our study of functions and their properties. In Section 3.5 we introduced the concept of “continuous functions”. In this section we explore a fundamental property that is satisfied by all continuous functions, known as the “Intermediate Value Theorem” (IVT). While the statement of theorem may seem rather obvious, its proof is quite involved, and its applications are far-reaching, in fact even surprising, as we will see! In particular, one application of the IVT is to locate the roots of complicated functions: repeated applications of the IVT gives rises to a numerical root finding method, known as the “bisection method”.

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Objectives

You should be able to:

Explain and illustrate the Intermediate Value Theorem.
Apply the Intermediate Value Theorem to locate the roots of a function.
Solve simple problems using the Intermediate Value Theorem.

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Subsection 6.1.1 Instructional video

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Subsection 6.1.2 Key concepts

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Concept 6.1.1. Intermediate Value Theorem (IVT).

Let $f$ be a continuous function over the interval $[a,b]\text{,}$ and let $N$ be any number between $f(a)$ and $f(b)\text{,}$ with $f(a) \neq f(b)\text{.}$ Then the Intermediate Value Theorem states that there must exist a $c \in (a,b)$ such that $f(c) =N\text{.}$

Equivalently, for any $N$ between $f(a)$ and $f(b)\text{,}$ the horizontal line $y=N$ must intersect the graph of $f$ at least once.

Two things to note:

$c$ may not be unique;
If $f$ is not continuous, then the statement of the Intermediate Value Theorem may not hold.

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Concept 6.1.2. Using the IVT to locate roots of functions.

To locate the roots of a continuous function $f$ using the Intermediate Value Theorem, pick two values of $x\text{,}$ say $x=a$ and $x=b\text{,}$ such that $f(a)$ is negative and $f(b)$ is positive. Then the Intermediate Value Theorem implies that $f$ must have at least one root between $x=a$ and $x=b\text{.}$

You can repeat the process using the midpoint of the previous interval to get a better numerical approximation for the location of the root: this is known as the bisection method.

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MATH 144: Calculus for the Physical Sciences I