What does infinitesimal calculus mean?

Answer 1

Infinitesimal calculus pretty much means non-rigorous calculus, i.e. calculus without the notion of limits to prove its validity. When Newton and Leibniz originally formulated calculus, they used derivatives and integrals in the same manner that they're still used today, but they provided no formalism as to how those techniques were mathematically valid, therefore causing quite a debate as to their worth. The infinitesimals themselves simply had to be accepted as valid, in and of themselves, for the theory to work.