Lesson 4.5

Intermediate Value Theorem

If you were below sea level and now you are above it, you MUST have crossed sea level. This simple idea is surprisingly powerful.

Introduction

Prerequisite Connection: Every polynomial function is continuous—you can draw it without lifting your pencil. This continuity is the secret ingredient for the IVT.

Today's Increment: The Intermediate Value Theorem (IVT) guarantees that if you have a point below a value and a point above a value, the continuous function connects them, crossing every value in between.

Why This Matters for Calculus: The IVT allows us to prove that solutions exist even when we can't find them precisely. It is the logic behind computational "root-finding" algorithms.

Explanation of Key Concepts

The Conditions

"If is continuous on , and is any number between and , then there is at least one number in such that ."

Condition 1: Continuous

Condition 2: Sign Change

Worked Examples

Level: Basic

Example 1: Does a Zero Exist?

Show has a zero between 1 and 2.

Step 1: Evaluate endpoints

(Negative).

(Positive).

Conclusion

Since

is polynomial (continuous) and changes sign from -1 to 5, there MUST be a zero where it crosses the x-axis.

Level: Intermediate

Example 2: Visualizing the Crossing

See how the function guarantees a crossing point.

Try to connect the dots without lifting your finger. You CANNOT avoid hitting

Level: Advanced

Example 3: Target Value isn't 0

Prove has a solution in .

Rewrite as

. We want

Evaluate at 1:

Evaluate at 2:

Proof

Sign change from negative to positive implies a root exists in

Common Pitfalls

Forgetting "Continuous":
IVT fails for rational functions like between -1 and 1. , , but it rarely hits 0 because of the vertical asymptote break! You MUST state continuity.
IVT doesn't find the answer:
IVT is an "Existence Theorem." It tells you the treasure is in the room, but it doesn't give you the map to find it. Don't try to solve for x unless asked!

Real-World Application

Geolocation and Time

There is a fun application of IVT called the "Antipodal Point Theorem."

At any given moment, there are two points on opposite sides of the Earth with the EXACT same temperature and pressure. IVT guarantees this because temperature varies continuously across the globe's surface.