｢(F(x+h)−F(x))/h :h⨠0｣ = F(x)▽x
Suppose a function A(x) that maps x to the area under the graph of f(x) from a sufficiently far point on the left side to the x on the right side.
Such a function is called the integral of f(x).
When x increases by sufficiently small h, the change in A(x) is between f(x)·h and f(x+h)·h.
So the ratio of a change in A(x) and the change in x that caused it is between f(x) and f(x+h).
As h approaches 0, the limit of the ratio becomes f(x).
｢(A(x+h)−A(x))/h :h⨠0｣ = f(x)
The differentiation makes the constant term zero. So the antidifferentiation cannot recover the constant term of the original function from the derivative.
Let F(x) be the antiderivative of f(x).
Then F(x) and A(x) differ only by a constant c.
The area under the graph of f(x) from a to b is as follows:
A(b)−A(a)
= (F(b)+c)−(F(a)+c)
= F(b)−F(a)
= f(x)△x｢a,b｣
This is called the fundamental theorem of calculus.
