Things like that if you have a continuously differentiable function, take the difference in value for two different points and divide by their distance, there must be a point in bewteen those two points at which the derivative equals the calculated value.
As a professor at my university once told, this is the basis for the speed control method used in the Netherlands.
Anyway, I am just making the distinction between applied calculus and math because a lot of people do not know that math is a totally different beast than they think.
Before you have a class on it at the university, you have no idea.
Even quite a lot of math students do not get the difference in the first semesters.
They still think that it is about learning the techniques to calculate different things and the results of theorems, while it is actually about the way of thinking involved in proving the theorems.