Intermediate value theorem facts for kids

The Intermediate Value Theorem is a cool idea in math that helps us understand how continuous functions work. Imagine you're drawing a line on a piece of paper without lifting your pencil. If your line starts at one height and ends at another, it has to pass through every single height in between!

This theorem says that if you have a function, let's call it f, that is smooth and unbroken (we call this "continuous") over a certain range of numbers (called a "closed interval"), then it must hit every value between its starting point and its ending point.

For example, if you know that at the beginning of the interval, the function's value is 10, and at the end, it's 20, then the function must have taken on every value like 11, 15, 19.5, and so on, somewhere in that interval.

What is the Intermediate Value Theorem?

The Intermediate Value Theorem (often shortened to IVT) is a basic rule in calculus. It tells us something important about functions that are "continuous."

Imagine you are walking up a hill. If you start at a height of 10 meters and end at a height of 50 meters, you must have passed through every height between 10 and 50 meters at some point during your walk. You can't jump from 10 meters to 50 meters without being at 20, 30, or 40 meters first! The IVT is like this for numbers.

Understanding "Continuous Function"

A continuous function is like a line or curve you can draw without lifting your pencil from the paper. There are no sudden jumps, breaks, or holes in the graph of the function.

Think of the temperature outside. It changes smoothly. It doesn't suddenly jump from 10 degrees to 20 degrees without passing through 11, 12, or 15 degrees. That's a continuous change.

What are "Intervals" and "f(x)"?

In math, an interval is just a set of numbers between two specific points. For example, the interval from 1 to 5 includes all numbers like 1, 2, 3, 4, 5, and all the numbers in between, like 1.5 or 3.7.

When we talk about f(x), we mean the value of the function f at a specific number x. So, if f(x) is a function that tells you the temperature at a certain time x, then f(1) would be the temperature at 1 o'clock.

Bolzano's Theorem: A Special Case

A very useful part of the Intermediate Value Theorem is called Bolzano's Theorem. This theorem is a special case of the IVT.

It says that if a continuous function changes its sign (meaning it goes from being negative to positive, or positive to negative) within an interval, then it must have crossed zero somewhere in that interval.

For example, if you have a continuous function where f(1) is -1 (a negative number) and f(2) is 2 (a positive number), then there must be some number between 1 and 2 where the function's value is exactly 0. This point where f(x) = 0 is called a root of the function. Finding roots is very important in many areas of math and science.

Real-Life Examples

The Intermediate Value Theorem might seem abstract, but it helps explain many things in the real world:

• Temperature: If the temperature outside was 5°C in the morning and 20°C in the afternoon, then at some point during the day, it must have been exactly 10°C, 15°C, or any temperature between 5°C and 20°C.
• Climbing a Mountain: If you climb a mountain from a starting height of 100 meters to a peak of 500 meters, you must have passed through every height between 100 and 500 meters.
• Growing Taller: If a child was 140 cm tall at age 12 and 160 cm tall at age 15, then at some point between those ages, they must have been exactly 150 cm tall.

These examples work because temperature, height, and growth are generally continuous processes.

See also

In Spanish: Teorema del valor intermedio para niños