Subject: Calculus

# Calculus

## Calculus.Calculus History

Hide minor edits - Show changes to output

April 06, 2011
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###[[~~Forms~~ ~~containing:~~ sqrt(a+bx)]]

###[[~~Forms~~ ~~containing:~~ ~~a'^2^'~~ ~~±~~ ~~x~~'^2^'~~)~~]]

###[[~~Forms~~ ~~containing:~~ ~~sqrt(x'^2^'~~ ~~±~~ ~~a~~'^2^')]]

###[[~~Forms~~ ~~containing:~~ ~~sqrt(a'^2^'~~ ~~-~~ ~~x~~'^2^')]]

###[[Forms containing: sqrt(2ax - x'^2^')]]

###[[

###[[

###[[

###[[Forms containing: sqrt(2ax - x'^2^')]]

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###[[List of integrals of irrational functions | Forms containing: sqrt(a+bx)]]

###[[List of integrals of irrational functions | Forms containing: a'^2^' ± x'^2^']]

###[[List of integrals of irrational functions | Forms containing: sqrt(x'^2^' ± a'^2^')]]

###[[List of integrals of irrational functions | Forms containing: sqrt(a'^2^' - x'^2^')]]

###[[List of integrals of irrational functions | Forms containing: sqrt(2ax - x'^2^')]]

###[[List of integrals of irrational functions | Forms containing: a'^2^' ± x'^2^']]

###[[List of integrals of irrational functions | Forms containing: sqrt(x'^2^' ± a'^2^')]]

###[[List of integrals of irrational functions | Forms containing: sqrt(a'^2^' - x'^2^')]]

###[[List of integrals of irrational functions | Forms containing: sqrt(2ax - x'^2^')]]

April 06, 2011
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###[[Forms containing ~~{$\~~sqrt~~{~~a+bx~~}$}~~]]

###[[Forms containing~~{$~~a^2 ~~\pm~~ x^2~~$}~~]]

###[[Forms containing~~{$\~~sqrt~~{~~x^2~~\pm a~~^2~~}$}~~]]

###[[Forms containing~~{$\~~sqrt~~{~~a^2 - x^2~~}$}~~]]

###[[Forms containing~~{$\~~sqrt~~{~~2ax - x^2~~}$}~~]]

###[[Forms containing

###[[Forms containing

###[[Forms containing

###[[Forms containing

to:

###[[Forms containing: sqrt(a+bx)]]

###[[Forms containing: a'^2^' ± x'^2^')]]

###[[Forms containing: sqrt(x'^2^' ± a'^2^')]]

###[[Forms containing: sqrt(a'^2^' - x'^2^')]]

###[[Forms containing: sqrt(2ax - x'^2^')]]

###[[Forms containing: a'^2^' ± x'^2^')]]

###[[Forms containing: sqrt(x'^2^' ± a'^2^')]]

###[[Forms containing: sqrt(a'^2^' - x'^2^')]]

###[[Forms containing: sqrt(2ax - x'^2^')]]

March 23, 2011
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In its modern sense, calculus includes several mathematical disciplines. Most commonly, it means the detailed analysis of rates of change in functions and is of utmost importance in all sciences. It is usually divided into two (closely related through the [[fundamental theorem of calculus]]) branches: [[differential calculus]] and [[integral calculus]].

The first, differential calculus, is concerned with finding the instantaneous rate of change (or derivative) of a function's value with respect to changes in its argument (roughly speaking, how much the value of a function changes with a small change in its argument). This derivative can also be interpreted as the slope of the function's graph at a specific point.

The second branch of calculus, integral calculus, studies methods for finding the integral of a function. An integral may be defined as the limit of a sum of terms which correspond to areas under the graph of a function. Considered as such, integration allows us to calculate the area under a curve and the surface area and volume of solids such as spheres and cones.

The first, differential calculus, is concerned with finding the instantaneous rate of change (or derivative) of a function's value with respect to changes in its argument (roughly speaking, how much the value of a function changes with a small change in its argument). This derivative can also be interpreted as the slope of the function's graph at a specific point.

The second branch of calculus, integral calculus, studies methods for finding the integral of a function. An integral may be defined as the limit of a sum of terms which correspond to areas under the graph of a function. Considered as such, integration allows us to calculate the area under a curve and the surface area and volume of solids such as spheres and cones.

January 09, 2011
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!! [[Limits]]

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!!! [[Limits]]

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!! [[Differential calculus]]

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!!! [[Differential calculus]]

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!!![[Integral calculus]]

#[[Antiderivative, Indefinite integral]]

#[[Simplest rules]]

#[[Sum rule in integration]]

#[[Constant factor rule in integration]]

#[[Linearity of integration]]

#[[Arbitrary constant of integration]]

#[[Fundamental theorem of calculus]]

#[[Integration by parts]]

#[[Inverse chain rule method]]

#[[Substitution rule]]

#[[Differentiation under the integral sign]]

#[[Trigonometric substitution]]

#[[Partial fractions in integration]]

#[[Quadratic integral]]

#[[Trapezium rule]]

#[[Integral of secant cubed]]

#[[Arclength]]

#[[Antiderivative, Indefinite integral]]

#[[Simplest rules]]

#[[Sum rule in integration]]

#[[Constant factor rule in integration]]

#[[Linearity of integration]]

#[[Arbitrary constant of integration]]

#[[Fundamental theorem of calculus]]

#[[Integration by parts]]

#[[Inverse chain rule method]]

#[[Substitution rule]]

#[[Differentiation under the integral sign]]

#[[Trigonometric substitution]]

#[[Partial fractions in integration]]

#[[Quadratic integral]]

#[[Trapezium rule]]

#[[Integral of secant cubed]]

#[[Arclength]]

Changed lines 73-78 from:

to:

!!![[Special functions and numbers]]

#[[Natural logarithm]]

#[[e]] (mathematical constant)

#[[Exponential function]]

#[[Stirling's approximation]]

#[[Bernoulli numbers]]

#[[Natural logarithm]]

#[[e]] (mathematical constant)

#[[Exponential function]]

#[[Stirling's approximation]]

#[[Bernoulli numbers]]

Changed lines 80-85 from:

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!!![[Numerical integration]]

#[[Rectangle method]]

#[[Trapezium rule]]

#[[Simpson's rule]]

#[[Newton–Cotes formulas]]

#[[Gaussian quadrature]]

#[[Rectangle method]]

#[[Trapezium rule]]

#[[Simpson's rule]]

#[[Newton–Cotes formulas]]

#[[Gaussian quadrature]]

Changed lines 87-105 from:

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!!! Lists and tables

#[[Table of common limits]]

#[[Table of derivatives]]

#[[Table of integrals]]

#[[List of integrals]]

##[[List of integrals of rational functions]]

##[[List of integrals of irrational functions]]

###[[Forms containing {$\sqrt{a+bx}$}]]

###[[Forms containing {$a^2 \pm x^2$}]]

###[[Forms containing {$\sqrt{x^2\pm a^2}$}]]

###[[Forms containing {$\sqrt{a^2 - x^2}$}]]

###[[Forms containing {$\sqrt{2ax - x^2}$}]]

##[[List of integrals of trigonometric functions]]

##[[List of integrals of inverse trigonometric functions]]

##[[List of integrals of hyperbolic functions]]

##[[List of integrals of exponential functions]]

##[[List of integrals of logarithmic functions]]

##[[List of integrals of area functions]]

#[[Table of mathematical symbols]]

#[[Table of common limits]]

#[[Table of derivatives]]

#[[Table of integrals]]

#[[List of integrals]]

##[[List of integrals of rational functions]]

##[[List of integrals of irrational functions]]

###[[Forms containing {$\sqrt{a+bx}$}]]

###[[Forms containing {$a^2 \pm x^2$}]]

###[[Forms containing {$\sqrt{x^2\pm a^2}$}]]

###[[Forms containing {$\sqrt{a^2 - x^2}$}]]

###[[Forms containing {$\sqrt{2ax - x^2}$}]]

##[[List of integrals of trigonometric functions]]

##[[List of integrals of inverse trigonometric functions]]

##[[List of integrals of hyperbolic functions]]

##[[List of integrals of exponential functions]]

##[[List of integrals of logarithmic functions]]

##[[List of integrals of area functions]]

#[[Table of mathematical symbols]]

Changed lines 107-117 from:

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!!![[Multivariable Calculus]]

#[[Partial derivative]]

#[[Disk integration]]

#[[Shell integration]]

#[[Gabriel's horn]]

#[[Jacobian matrix]]

#[[Hessian matrix]]

#[[Curvature]]

#[[Green's theorem]]

#[[Divergence theorem]]

#[[Stokes' theorem]]

#[[Partial derivative]]

#[[Disk integration]]

#[[Shell integration]]

#[[Gabriel's horn]]

#[[Jacobian matrix]]

#[[Hessian matrix]]

#[[Curvature]]

#[[Green's theorem]]

#[[Divergence theorem]]

#[[Stokes' theorem]]

Changed lines 119-124 from:

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!!![[Series]]

#[[Infinite series]]

#[[Maclaurin series]]

#[[Taylor series]]

#[[Fourier series]]

#[[Euler–Maclaurin formula]]

#[[Infinite series]]

#[[Maclaurin series]]

#[[Taylor series]]

#[[Fourier series]]

#[[Euler–Maclaurin formula]]

January 09, 2011
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!! [[Limits]]

# [[Limit of a function]]

# [[One-sided limit]]

# [[Limit of a sequence]]

# [[Indeterminate form]]

# [[Orders of approximation]]

# [[Definition of a Limit | (ε, δ)-Definition of a limit]]

# [[Limit of a function]]

# [[One-sided limit]]

# [[Limit of a sequence]]

# [[Indeterminate form]]

# [[Orders of approximation]]

# [[Definition of a Limit | (ε, δ)-Definition of a limit]]

Changed lines 18-52 from:

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!! [[Differential calculus]]

#[[Derivative]]

#[[Notations]]

##[[Newton's notation for differentiation]]

##[[Leibniz's notation for differentiation]]

#[[Simplest rules]]

##[[Derivative of a constant]]

##[[Sum rule in differentiation]]

##[[Constant factor rule in differentiation]]

##[[Linearity of differentiation]]

##[[Calculus with polynomials]]

#[[Derivative Examples]]

##[[Chain rule]]

##[[Product rule]]

##[[Quotient rule]]

#[[Derivatives of trigonometric functions]]

#[[Inverse functions and differentiation]]

#[[Implicit differentiation]]

#[[Higher Order Derivatives]]

#[[Stationary point]]

#[[Maxima and minima]]

#[[First derivative test]]

#[[Second derivative test]]

#[[Extreme value theorem]]

#[[Differential equation]]

#[[Differential operator]]

#[[Newton's method]]

#[[Taylor's theorem]]

#[[L'Hôpital's rule]]

#[[Leibniz's rule]]

#[[Mean value theorem]]

#[[Logarithmic derivative]]

#[[Differential (calculus)]]

#[[Related rates]]

#[[Regiomontanus' angle maximization problem]]

#[[Derivative]]

#[[Notations]]

##[[Newton's notation for differentiation]]

##[[Leibniz's notation for differentiation]]

#[[Simplest rules]]

##[[Derivative of a constant]]

##[[Sum rule in differentiation]]

##[[Constant factor rule in differentiation]]

##[[Linearity of differentiation]]

##[[Calculus with polynomials]]

#[[Derivative Examples]]

##[[Chain rule]]

##[[Product rule]]

##[[Quotient rule]]

#[[Derivatives of trigonometric functions]]

#[[Inverse functions and differentiation]]

#[[Implicit differentiation]]

#[[Higher Order Derivatives]]

#[[Stationary point]]

#[[Maxima and minima]]

#[[First derivative test]]

#[[Second derivative test]]

#[[Extreme value theorem]]

#[[Differential equation]]

#[[Differential operator]]

#[[Newton's method]]

#[[Taylor's theorem]]

#[[L'Hôpital's rule]]

#[[Leibniz's rule]]

#[[Mean value theorem]]

#[[Logarithmic derivative]]

#[[Differential (calculus)]]

#[[Related rates]]

#[[Regiomontanus' angle maximization problem]]

December 22, 2010
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Added lines 97-98:

****[[Forms containing {$\sqrt{a^2 - x^2}$}]]

****[[Forms containing {$\sqrt{2ax - x^2}$}]]

****[[Forms containing {$\sqrt{2ax - x^2}$}]]

December 20, 2010
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****[[Forms containing {$a^2 \pm x^2$}

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****[[Forms containing {$a^2 \pm x^2$}]]

December 20, 2010
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December 18, 2010
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Changed lines 90-91 from:

**[[Table of integrals]]

**[[Table of mathematical symbols]]

**[[Table of mathematical symbols]]

to:

**[[Table of integrals]]

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**[[Table of mathematical symbols]]

December 18, 2010
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**[[List of integrals of rational functions]]

**[[List of integrals of irrational functions]]

**[[List of integrals of trigonometric functions]]

**[[List of integrals of inverse trigonometric functions]]

**[[List of integrals of hyperbolic functions]]

**[[List of integrals of exponential functions]]

**[[List of integrals of logarithmic functions]]

**[[List of integrals of area functions]]

**[[List of integrals of irrational functions]]

**[[List of integrals of trigonometric functions]]

**[[List of integrals of inverse trigonometric functions]]

**[[List of integrals of hyperbolic functions]]

**[[List of integrals of exponential functions]]

**[[List of integrals of logarithmic functions]]

**[[List of integrals of area functions]]

to:

***[[List of integrals of rational functions]]

***[[List of integrals of irrational functions]]

***[[List of integrals of trigonometric functions]]

***[[List of integrals of inverse trigonometric functions]]

***[[List of integrals of hyperbolic functions]]

***[[List of integrals of exponential functions]]

***[[List of integrals of logarithmic functions]]

***[[List of integrals of area functions]]

***[[List of integrals of irrational functions]]

***[[List of integrals of trigonometric functions]]

***[[List of integrals of inverse trigonometric functions]]

***[[List of integrals of hyperbolic functions]]

***[[List of integrals of exponential functions]]

***[[List of integrals of logarithmic functions]]

***[[List of integrals of area functions]]

November 27, 2010
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Changed lines 29-32 from:

**[[Derivative ~~(examples)~~]]

**[[Chain rule]]

**[[Product rule]]

**[[Quotient rule]]

**[[Chain rule]]

**[[Product rule]]

**[[Quotient rule]]

to:

**[[Derivative Examples]]

***[[Chain rule]]

***[[Product rule]]

***[[Quotient rule]]

***[[Chain rule]]

***[[Product rule]]

***[[Quotient rule]]

November 11, 2010
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**[[Derivatives of ~~Trigonometric~~ ~~Functions~~]]

to:

**[[Derivatives of trigonometric functions]]

November 11, 2010
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**[[Derivatives of Trigonometric Functions]]

Added line 36:

**[[Higher Order Derivatives]]

November 07, 2010
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November 05, 2010
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Changed line 8 from:

Calculus is a massive subject, from modern economics, to ~~structural~~ engineering, to ~~fluid~~ dynamics, to space travel, to everyday problems in life calculus was a large step towards uniting life and math. Therefore the following list of items may seem intense, and even worthy of biting down your nails even a little further, but every journey starts with that initial step forward. So get started, click on the '''limits''' link below and see why it has been said that "Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time. (~Sophus Lie)."

to:

Calculus is a massive subject, from modern economics, to [[Engineer:Structural | structural engineering]], to [[Engineer:Water-Resources/BernoulliEquation | fluid dynamics]], to space travel, to everyday problems in life calculus was a large step towards uniting life and math. Therefore the following list of items may seem intense, and even worthy of biting down your nails even a little further, but every journey starts with that initial step forward. So get started, click on the '''limits''' link below and see why it has been said that "Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time. (~Sophus Lie)."

October 26, 2010
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Changed line 16 from:

** [[~~(ε,~~ ~~δ)-definition ~~of limit]]

to:

** [[Definition of a Limit | (ε, δ)-Definition of a limit]]

October 24, 2010
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October 23, 2010
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October 23, 2010
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Deleted line 0:

Added lines 4-117:

!! So, where do I start?

Calculus is a massive subject, from modern economics, to structural engineering, to fluid dynamics, to space travel, to everyday problems in life calculus was a large step towards uniting life and math. Therefore the following list of items may seem intense, and even worthy of biting down your nails even a little further, but every journey starts with that initial step forward. So get started, click on the '''limits''' link below and see why it has been said that "Among all of the mathematical disciplines the theory of differential equations is the most important... It furnishes the explanation of all those elementary manifestations of nature which involve time. (~Sophus Lie)."

* [[Limits]]

** [[Limit of a function]]

** [[One-sided limit]]

** [[Limit of a sequence]]

** [[Indeterminate form]]

** [[Orders of approximation]]

** [[(ε, δ)-definition of limit]]

*[[Differential calculus ]]

**[[Derivative]]

**[[Notations]]

***[[Newton's notation for differentiation]]

***[[Leibniz's notation for differentiation]]

**[[Simplest rules]]

***[[Derivative of a constant]]

***[[Sum rule in differentiation]]

***[[Constant factor rule in differentiation]]

***[[Linearity of differentiation]]

***[[Calculus with polynomials]]

**[[Derivative (examples)]]

**[[Chain rule]]

**[[Product rule]]

**[[Quotient rule]]

**[[Inverse functions and differentiation]]

**[[Implicit differentiation]]

**[[Stationary point]]

**[[Maxima and minima]]

**[[First derivative test]]

**[[Second derivative test]]

**[[Extreme value theorem]]

**[[Differential equation]]

**[[Differential operator]]

**[[Newton's method]]

**[[Taylor's theorem]]

**[[L'Hôpital's rule]]

**[[Leibniz's rule]]

**[[Mean value theorem]]

**[[Logarithmic derivative]]

**[[Differential (calculus)]]

**[[Related rates]]

**[[Regiomontanus' angle maximization problem]]

*[[Integral calculus]]

**[[Antiderivative, Indefinite integral]]

**[[Simplest rules]]

**[[Sum rule in integration]]

**[[Constant factor rule in integration]]

**[[Linearity of integration]]

**[[Arbitrary constant of integration]]

**[[Fundamental theorem of calculus]]

**[[Integration by parts]]

**[[Inverse chain rule method]]

**[[Substitution rule]]

**[[Differentiation under the integral sign]]

**[[Trigonometric substitution]]

**[[Partial fractions in integration]]

**[[Quadratic integral]]

**[[Trapezium rule]]

**[[Integral of secant cubed]]

**[[Arclength]]

*[[Special functions and numbers]]

**[[Natural logarithm]]

**[[e]] (mathematical constant)

**[[Exponential function]]

**[[Stirling's approximation]]

**[[Bernoulli numbers]]

*[[Numerical integration]]

**[[Rectangle method]]

**[[Trapezium rule]]

**[[Simpson's rule]]

**[[Newton–Cotes formulas]]

**[[Gaussian quadrature]]

*Lists and tables

**[[Table of common limits]]

**[[Table of derivatives]]

**[[Table of integrals]]

**[[Table of mathematical symbols]]

**[[List of integrals]]

**[[List of integrals of rational functions]]

**[[List of integrals of irrational functions]]

**[[List of integrals of trigonometric functions]]

**[[List of integrals of inverse trigonometric functions]]

**[[List of integrals of hyperbolic functions]]

**[[List of integrals of exponential functions]]

**[[List of integrals of logarithmic functions]]

**[[List of integrals of area functions]]

*[[Multivariable Calculus]]

**[[Partial derivative]]

**[[Disk integration]]

**[[Shell integration]]

**[[Gabriel's horn]]

**[[Jacobian matrix]]

**[[Hessian matrix]]

**[[Curvature]]

**[[Green's theorem]]

**[[Divergence theorem]]

**[[Stokes' theorem]]

*[[Series ]]

**[[Infinite series]]

**[[Maclaurin series]]

**[[Taylor series]]

**[[Fourier series]]

**[[Euler–Maclaurin formula]]