Calculus

:For other uses of the term calculus see calculus (disambiguation)

Differential calculus

Main article: Derivative

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The derivative measures the sensitivity of one variable to small changes in another variable. Consider the formula:

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:mathrm{Speed} = rac{mathrm{Distance}}{mathrm{Time}}

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for an object moving at constant speed. The speed of a car, as measured by the speedometer, is the derivative of the car's displacement as a function of time. Calculus is a mathematical tool for dealing with this complex but natural and familiar situation.

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Differential calculus determines the instantaneous speed at any given specific instant in time, not just average speed during an interval of time. The formula Speed = Distance/Time applied to a single instant is the meaningless quotient "zero divided by zero". This is avoided, however, because the quotient Distance/Time is not used for a single instant (as in a still photograph). Rather a formula is developed for the quotient Distance/Time in which division by zero can be avoided, by a method called "taking the limit".

Related Topics:
Zero divided by zero - Photograph - Limit

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The derivative answers the question: as the elapsed time approaches zero, what does the average speed computed by Distance/Time approach? In mathematical language, this is an example of "taking a limit."

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More formally, differential calculus defines the instantaneous rate of change (the derivative) of a mathematical function's value, with respect to changes of the variable. The derivative is defined as a limit of a difference quotient.

Related Topics:
Mathematical function - Value - Variable - Difference quotient

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The derivative of a function, if it exists, gives information about small pieces of its graph. It is useful for finding the maxima and minima of a function — because at those points the graph is flat (i.e. the slope of the graph is zero). Another application of differential calculus is Newton's method, an algorithm to find zeroes of a function by approximating the graph of the function by tangent lines. Differential calculus has been applied to many questions that are not first formulated in the language of calculus.

Related Topics:
Maxima and minima - Newton's method - Algorithm - Zeroes - Tangent

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The derivative lies at the heart of the physical sciences. Newton's law of motion, Force = Mass × Acceleration, involves calculus because acceleration is a derivative. (See Differential equation.) Maxwell's theory of electromagnetism and Einstein's theory of gravity (general relativity) are also expressed in the language of differential calculus, as is the basic theory of electrical circuits and much of engineering. It is also applied to problems in biology, economics, and many other areas.

Related Topics:
Physical science - Differential equation - Electromagnetism - Einstein - General relativity - Electrical circuit - Engineering - Biology - Economics

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