Calculus:
The Fundamental Theorem
August 1666
Isaac Newton
Introduction
In this note we prove a theorem in two parts connecting differentiation and integration. We modestly propose to call this the Fundamental Theorem of Calculus. We note that certain other mathematicians have not yet proved this result. We mention fleetingly that Gregory [BBB] and Barrow [AAA] proved results that can be interpreted as being the Fundamental Theorem of Calculus.
The two parts of the Fundamental Theorem of Calculus are laid out in Theorem XXX and Theorem YYY in the next section. A (non-existent) proof will be given, with a diagram.
The Theorem
Theorem. Fundamental Theorem of Calculus. Part 1
Let f be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by
F(x) = integral from a to x f(t) dt.
Then, F is continuous on [a, b], differentiable on the open interval (a, b), and
F'(x) = f(x),
for all x in (a, b).
Theorem. Fundamental Theorem of Calculus. Part 2
Let f be a real-valued function defined on a closed interval [a, b] that admits an antiderivative g on [a, b]. That is, f and g are functions such that for all x in [a, b], f(x) = g'(x).
If f is integrable on [a, b] then
integral from a to b f(x) dx = g(b) - g(a).
Proof
Any proof would have a diagram...
[Diagram here. Name of file is FTC.pdf]
References
AAA Isaac Barrow, Some Paper, Some Journal 2 (1662), pp 12--15.
BBB James Gregory, Geometriae Par Universalis, Chapter 6, Royal Society of London, 1660.