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 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.

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, whom 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.

Important Topics

The following topics provide a good sample of the subject of calculus:

• Fundamental Theorem of Calculus
• Chain Rule
• Implicit differentiation

Calculus in Math Competitions

The use of calculus in pre-collegiate mathematics competitions is generally frowned upon, with the exception of the Harvard-MIT Mathematics Tournament. 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, will occasionally require it, but generally not.

Additional Note

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

See also

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