Calculus: What is the Intermediate Value Theorem?

Question

Can someone please explain to me the intermediate value theorem simply? I have read the definition in my book over and over again and it isn't clicking. I just need to know basically what it's saying. Nothing really long/extensive.

Answer 1

The simplest way of stating the IVT is that if you draw a continuous line on a graph, the line goes through every value between its endpoints. So, for any given value between the endpoints, there is some point where the graph passes through that value.

Answer 2

The simple way to think about it: If you go from point A to point B without teleporting, then for any point C between the two, you must have been at C at some time during the move.

This concept of a continuous function is so intuitive, that we really can think of it that way.

Applying it to math, we can see it with an example:

f(x) = x^2 - 2

At x = 0, the function is at f(0) = -2.  When x=2, the function is at f(2) = 2.

Since the function goes from from a y-value of -2 to a y-value of 2, it must have been at a y-value of 0 somewhere between those points.  Also, it must have had a y-value of 1 somewhere there too.