calculus

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An illustration of the difference between average and instantaneous rates of changeThe graph of f(t) shows the secant between (t, f(t)) and (t + h, f(t + h)) and the tangent to f(t) at t. As the time interval h approaches zero, the secant (average speed) approaches the tangent (actual, or instantaneous, speed) at (t, f(t)).

The transformation of a circular region into an approximately rectangular regionThis suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

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An illustration of the difference between average and instantaneous rates of changeThe graph of f(t) shows the secant between (t, f(t)) and (t + h, f(t + h)) and the tangent to f(t) at t. As the time interval h approaches zero, the secant (average speed) approaches the tangent (actual, or instantaneous, speed) at (t, f(t)).

The transformation of a circular region into an approximately rectangular regionThis suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

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Topics

integration differentiation integral calculus fundamental theorem of calculus differential calculus L’Hôpital’s rule length of a curve quotient rule derivative integral probability theory cryptology computer science combinatorics algebra statistics geometry automata theory set theory information theory

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Key People

Understanding Newton's theory of universal gravitation

Isaac Newton

English physicist and mathematician

Leonhard Euler

Leonhard Euler

Swiss mathematician

John Wallis, oil painting after a portrait by Sir Godfrey Kneller; in the National Portrait Gallery, London

John Wallis

English mathematician

Augustin-Louis Cauchy

French mathematician

Johann Bernoulli, oil painting by Johann Jakob Meyer, 1720; in a private collection

Johann Bernoulli

Swiss mathematician

Seki Takakazu

Japanese mathematician

Colin Maclaurin, engraving by S. Freeman; in the British Museum.

Colin Maclaurin

Scottish mathematician

Brook Taylor, detail of a gouache miniature by Joseph Goupy; in the National Portrait Gallery, London.

Brook Taylor

British mathematician

Vito Volterra

Italian mathematician

Carathéodory, Constantin

Constantin Carathéodory

Greek-German mathematician

Gilbert Ames Bliss

American mathematician

Quiz

Equations written on blackboard

Numbers and Mathematics