Difference between revisions of "Calculus"
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::The word "calculus" has historically had a much more general meaning than at present. It means "a method or 'trick' for calculating something". (The word means "little stone" in Latin, referring to the use of pebbles for counting.) Among the types of calculus are: | ::The word "calculus" has historically had a much more general meaning than at present. It means "a method or 'trick' for calculating something". (The word means "little stone" in Latin, referring to the use of pebbles for counting.) Among the types of calculus are: | ||
| − | :::*'''Propositional calculus'''—a notation for proving theorems about logical expressions. | + | :::*'''[[Propositional calculus]]'''—a notation for proving theorems about logical expressions. |
:::*'''Predicate calculus'''—a notation for proving theorems about functions in [[set theory]]. | :::*'''Predicate calculus'''—a notation for proving theorems about functions in [[set theory]]. | ||
:::*'''Lambda calculus'''—a notation for proving theorems about recursive functions and computability in [[computer science]]. | :::*'''Lambda calculus'''—a notation for proving theorems about recursive functions and computability in [[computer science]]. | ||
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:::*and, of course, '''Differential''' and '''Integral''' calculus—the common modern meaning of the term, and the subject of this article. These two forms are sometimes collectively called '''infinitesimal calculus'''. | :::*and, of course, '''Differential''' and '''Integral''' calculus—the common modern meaning of the term, and the subject of this article. These two forms are sometimes collectively called '''infinitesimal calculus'''. | ||
| − | '''Calculus''' is the [[mathematics|mathematical]] subject that studies rates of change of functions. There are two main branches of calculus—[[derivative|differential]] calculus, and [[integral]] calculus. There are subfields of these: single [[variable]] calculus, [[exterior calculus]], and multi [[variable]] calculus. | + | '''Calculus''' (that is, the "infinitesimal calculus", see above) is the [[mathematics|mathematical]] subject that studies rates of change of functions. There are two main branches of calculus—[[derivative|differential]] calculus, and [[integral]] calculus. There are subfields of these: single [[variable]] calculus, [[exterior calculus]], and multi [[variable]] calculus. |
Calculus has numerous applications in many fields, such as [[engineering]], [[physics]], [[chemistry]], [[biology]], and [[economics]]. Many physical properties can be explained and/or modeled through the use of calculus. | Calculus has numerous applications in many fields, such as [[engineering]], [[physics]], [[chemistry]], [[biology]], and [[economics]]. Many physical properties can be explained and/or modeled through the use of calculus. | ||
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===Indefinite Integrals=== | ===Indefinite Integrals=== | ||
{{main|Indefinite integral}} | {{main|Indefinite integral}} | ||
| − | The antiderivative of a function is often called the ''indefinite integral''. | + | The antiderivative of a function is often called the ''indefinite integral''. It is called indefinite because the limits of integration are not specified. So, for example, the derivative of <math>\frac{x^3}{3}+7</math> is <math>x^2</math>. From this it follows that the antiderivative of <math>x^2</math> could be <math>\frac{x^3}{3}+7</math>. But note that the "7" in that formula was a [[red herring]]. Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it. This arbitrary constant is usually written '''C''' and is called the "constant of integration". The indefinite integral could be written: |
| − | <math>x^2</math> could be <math>\frac{x^3}{3}+7</math>. But note that the "7" in that formula was a [[red herring]]. Adding any constant to a function doesn't change its derivative, so the antiderivative of <math>x^2</math> could have any constant added to it. This arbitrary constant is usually written '''C''' and is called the "constant of integration". The indefinite integral could be written: | + | |
:<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>, | :<math>\int x^2\ \mathrm{d}x = \frac{x^3}{3} + C\,</math>, | ||
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The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b. Note that the constant of integration cancels out. | The Fundamental Theorem of Calculus says that the area under the graph of <math>x^2</math> between a and b is the difference in the values of <math>\frac{x^3}{3}+C</math> between a and b. Note that the constant of integration cancels out. | ||
| − | ===Definite | + | ===Definite Integrals=== |
{{main|Definite integral}} | {{main|Definite integral}} | ||
This kind of integral is called a ''definite integral'', written with the limits <math>[a,b]</math>: | This kind of integral is called a ''definite integral'', written with the limits <math>[a,b]</math>: | ||
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The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article. Definite integrals are strongly connected to the [[Fundamental Theorem of Calculus]]. | The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article. Definite integrals are strongly connected to the [[Fundamental Theorem of Calculus]]. | ||
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==Derivatives== | ==Derivatives== | ||
| − | {{Main| | + | {{Main|Derivative}} |
Derivatives are defined as the instantaneous rate of change of differentiable functions. Derivatives themselves can be functions that give the slope of the instantaneous rate of change of a function at any differentiable point. The slope of a tangent line is another way of defining a derivative. A tangent line touches a graph at only one point (locally). So its slope can be interpreted as the instantaneous rate of change of that graph. | Derivatives are defined as the instantaneous rate of change of differentiable functions. Derivatives themselves can be functions that give the slope of the instantaneous rate of change of a function at any differentiable point. The slope of a tangent line is another way of defining a derivative. A tangent line touches a graph at only one point (locally). So its slope can be interpreted as the instantaneous rate of change of that graph. | ||
Revision as of 21:52, March 30, 2015
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This article/section deals with mathematical concepts appropriate for late high school or early college. |
- The word "calculus" has historically had a much more general meaning than at present. It means "a method or 'trick' for calculating something". (The word means "little stone" in Latin, referring to the use of pebbles for counting.) Among the types of calculus are:
- Propositional calculus—a notation for proving theorems about logical expressions.
- Predicate calculus—a notation for proving theorems about functions in set theory.
- Lambda calculus—a notation for proving theorems about recursive functions and computability in computer science.
- Residue calculus—a trick for evaluating integrals of complex functions around a closed loop, by examining the singularities of the function inside the loop.
- and, of course, Differential and Integral calculus—the common modern meaning of the term, and the subject of this article. These two forms are sometimes collectively called infinitesimal calculus.
- The word "calculus" has historically had a much more general meaning than at present. It means "a method or 'trick' for calculating something". (The word means "little stone" in Latin, referring to the use of pebbles for counting.) Among the types of calculus are:
Calculus (that is, the "infinitesimal calculus", see above) is the mathematical subject that studies rates of change of functions. There are two main branches of calculus—differential calculus, and integral calculus. There are subfields of these: single variable calculus, exterior calculus, and multi variable calculus.
Calculus has numerous applications in many fields, such as engineering, physics, chemistry, biology, and economics. Many physical properties can be explained and/or modeled through the use of calculus.
Isaac Newton was one of the most famous inventors of calculus, however there are several others who developed calculus simultaneously including Gottfried Leibniz.[1] The disagreement over the originator of calculus created rifts within the European mathematical community for years.
Contents
Integrals
For a more detailed treatment, see Integral.
Integration is primarily defined as the method to calculate the area in an xy-plane above the x-axis. There are two fundamentally different kinds of integrals.
Indefinite Integrals
For a more detailed treatment, see Indefinite integral.
The antiderivative of a function is often called the indefinite integral. It is called indefinite because the limits of integration are not specified. So, for example, the derivative of 
is 
. From this it follows that the antiderivative of could be . But note that the "7" in that formula was a red herring. Adding any constant to a function doesn't change its derivative, so the antiderivative of could have any constant added to it. This arbitrary constant is usually written C and is called the "constant of integration". The indefinite integral could be written:
,
The Fundamental Theorem of Calculus says that the area under the graph of between a and b is the difference in the values of 
between a and b. Note that the constant of integration cancels out.
Definite Integrals
For a more detailed treatment, see Definite integral.
This kind of integral is called a definite integral, written with the limits 
:
The above is a simplified "intuitive" treatment of calculus and of this theorem. The actual "rigorous" proof, "rigorous" definitions of derivative and integral, and statement of the conditions under which the theorem is true, are beyond the scope of this article. Definite integrals are strongly connected to the Fundamental Theorem of Calculus.
Derivatives
For a more detailed treatment, see Derivative.
Derivatives are defined as the instantaneous rate of change of differentiable functions. Derivatives themselves can be functions that give the slope of the instantaneous rate of change of a function at any differentiable point. The slope of a tangent line is another way of defining a derivative. A tangent line touches a graph at only one point (locally). So its slope can be interpreted as the instantaneous rate of change of that graph.
Power Functions
External Links
See also
References
- ↑ Calculus, 5th edition, by James Stewart, Nelson Canada
