Increment of a Function
Big Idea: Before we can understand derivatives and the heart of calculus, we need to understand how functions change. The increment of a function measures exactly that — how much the output of a function changes when we change the input.
1. Increment of the Argument
Let $ f(x) $ be a function defined on some interval, and let $ x_0 $ be a point in its domain. If we change the argument (input) from $ x_0 $ to a new value $ x_1 $, the increment of the argument is:
$$ \Delta x = x_1 - x_0 $$
Here, $ \Delta x $ (read "delta x") can be positive (moving right), negative (moving left), or zero. The Greek letter $ \Delta $ (capital delta) stands for "change in" or "increment of."
Note that $ x_1 = x_0 + \Delta x $, so the new point is the original point plus the increment.