In the first lecture we looked at Proto calculus. It is a calculus which
predates even Archimedes. The derivative of f is defined there as Df(x) = f(x+1)-f(x)
and the integral is defined as Sf(x) = f(0)+f(1) + ... + f(x-1). We have seen in class
that DS f(x) = f(x) and S D f(x) = f(x)-f(0). This is the fundamental theorem of
discrete calculus. It is completely analogue to the fundamental theorem of calculus
∫0x (d/dt) f(t) dt
=f(x)-f(0) and (d/dx) ∫0x f(t) dt = f(x). You see the 20 second
proofs here:
Mathematicians have used tally sticks or pebbles
to compute at first. An other name for proto calculus is business calculus
or quantum calculus. There
are no limits taken yet.
You see something about Pebble calculus
the movie
"The Clan of the Cave Bear".
You can find something about the development of arithmetic here:
The concept of limits took a long time to develop. Zeno, the Greek philosopher
has thought a lot about it. He had asked himself questions like: if an arrow is at a fixed
position at any time, how can it move?
A bit more about the history of calculus is on these slides:
