# Calculus

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 analyze instantaneous rates of change, accumulations of change, volumes of irregular solids, and many other types of problems in mathematics.

Limits are heavily used in calculus. The formal notion of a limit is what differentiates calculus from precalculus mathematics.

The subject dealing with the rigorous foundations of calculus is called analysis, specifically real analysis.

## History

Calculus was compiled into one mathematical science by Isaac Newton in 1665 and 1666. (Before this, some individual calculus ideas had been discovered by earlier mathematicians). However, Gottfried Leibniz, who did the same work independently a few years later, published his work earlier than Newton. This sparked an argument over who first discovered calculus. It is now known that Newton did discover calculus first, but Leibniz invented the majority of the notation we use today.

## 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.

There are a number of high school math contests that have a calculus round, or require calculus. These include:

- The Harvard-MIT Invitational Tournament (HMIT)

- The Johns-Hopkins Mathematics Tournament

- The Rocket City Math League

- The Rice University Mathematics Tournament

- The Chandler-Gilbert Community College Excellence in Mathematics High School Competition, precalculus and above division, team competition, in Arizona

None of the AMC competitions leading up to the IMO require it, nor does the ARML, even though calculus solutions are still permitted. Online high school competitions, such as the iTest, will occasionally require it, but generally not.

## See also

- Derivative
- Limit
- Integral (It is suggested that you look at derivative before this)