Whether or not the function moves around makes no difference.
From moment-to-moment the rate of increase is always equal to the height (the value of f).
This needs a picture: More precisely, if you have a function on the interval [A, B], then there’s a point c between A and B such that .
You can just as easily write this as or (since F’ =f).
However, there are some mathematicians who may take issue with mixing up the two terms.
This theorem is so important and widely used that it’s called the “fundamental theorem of calculus”, and it ties together the (opposite of the derivative) so tightly that the two words are essentially interchangeable.
As far as the function “knows”, at any particular moment it may as well be constant (dotted line in picture above).
One of the classic examples is the function Over any interval you pick, f still jumps around infinitely often, so the whole “things will get better as the number of rectangles increases” thing can never get off the ground.
: Say you’ve got a function f(x), and the area under f(x) (up to some value x) is given by A(x).
Then the statement “the area, A, is given by the anti-derivative of f” is equivalent to “the derivative of A is given by f”.
You can approximate the area under a function by dividing it up into a whole lot of tiny rectangles.
The area of each is the width times the height, where the height is any value of f in that particular interval.