30th  Top calculus topics 
Question 1: The founders of calculus — Euler, Leibniz, Newton, ________, and many others — calculated with infinitesimals and achieved correct results even though no formal definition was available.  

Question 2: For example, there would have to be a ________ function that was well defined for infinite inputs.  

Question 3: ________, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals.  

Question 4: The extended axioms are IST, which stands either for ________ or for the initials of the three extra axioms: Idealization, Standardization, Transfer.  

Question 5: A ________ has all the firstorder properties of the real number system (regardless of whether they are usually taken as axiomatic) for statements involving the basic orderedfield relations +, ×, and ≤.  

Question 6: The word infinitesimal comes from a 17th century ________ coinage infinitesimus, which originally referred to the "infiniteth" item in a series.  

Question 7: For example, the ________ axiom x + y = y + x holds.  

Question 8: As an alternative to infinitesimals, the ________ in the 14th to 16th centuries made use of an intuitive notion of a limit to compute several series expansions.  

Question 9: ^{[1]} His ________ defines a number x as infinite if it satisfies the conditions x>1, x>1+1, x>1+1+1, ..., and infinitesimal if x≠0 and the same set of conditions holds for 1/x.  

Question 10: For comparison one should note that there was no formal definition of ________ at the time, either.  

