Calculus
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The discovery of the branch of mathematics known as calculus was motivated by two classical problems: how to find the slope of the tangent line to a curve at a point and how to find the area bounded by a curve. What is surprising is that these two problems are fundamentally connected and, together with the notion of limits, can be used to analyse instantaneous rates of change, accumulations of change, volumes of irregular solids, and much more.
Limits are heavily used in calculus. The formal notion of a limit is what "differentiates" (hehe, pun) calculus from precalculus mathematics.
Student Guide to Calculus
The following topics provide a good introduction to the subject of calculus:
- Derivative
- Limit
- Integral (It is suggested that you look at derivative before this)
Calculus in Math Competitions
The use of calculus in pre-collegiate mathematics competitions is generally frowned upon. However, many physics competitions require it, as does the William Lowell Putnam competition.
None of the competitions leading up to the IMO require it, nor does the ARML. Online high school competitions, such as the iTest, which occasionally require it, but generally not.
Additional Note
The subject dealing with the rigorous foundations of calculus is called analysis, specifically real analysis.
