Difference between revisions of "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. | + | 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 [[rate]]s of change, accumulations of change, [[volume]]s of irregular [[solid]]s, and much more. |
| − | [[Limit]]s | + | [[Limit]]s are heavily used in calculus. The formal notion of a limit is what differentiates calculus from precalculus mathematics. |
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| + | The subject dealing with the rigorous foundations of calculus is called [[analysis]], specifically [[real analysis]]. | ||
==History== | ==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. | 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. | ||
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== Calculus in Math Competitions == | == 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 | physics competitions]] require it, as does the [[William Lowell Putnam Mathematical Competition|William Lowell Putnam competition]]. | 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 | physics competitions]] require it, as does the [[William Lowell Putnam Mathematical Competition|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 | + | None of the [[AMC]] 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. |
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== See also == | == See also == | ||
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* [[Limit]] | * [[Limit]] | ||
* [[Integral]] (It is suggested that you look at derivative before this) | * [[Integral]] (It is suggested that you look at derivative before this) | ||
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| + | === Topics === | ||
| + | * [[Chain Rule]] | ||
| + | * [[Implicit differentiation]] | ||
| + | * [[Fundamental Theorem of Calculus]] | ||
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[[Category:Calculus]] | [[Category:Calculus]] | ||
Revision as of 21:15, 6 December 2007
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 much more.
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, 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.
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 AMC 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.
See also
- Derivative
- Limit
- Integral (It is suggested that you look at derivative before this)
