calculus - Encyclopædia Britannica

7/29/2019 calculus - Encyclopdia Britannica

1/5

calculus

Primary Contributor: John L. Berggren [1]

Encyclopdia Britannica

calculus, branch of mathematics concerned with the calculation [2] of instantaneous rates

of change (differential calculus) and the summation of infinitely many small factors to

determine some whole (integral calculus). Two mathematicians, Isaac Newton[3]

of

England and Gottfried Wilhelm Leibniz [4] of Germany, share credit for having

independently developed the calculus in the 17th century. Calculus is now the basic

entry point [5] for anyone wishing to study physics, chemistry, biology, economics,

finance, or actuarial science. Calculus makes it possible to solve problems as diverse as

tracking the position of a space shuttle [6] or predicting the pressure building up behind a

dam as the water rises. Computers have become a valuable tool for solving calculus

problems that were once considered impossibly difficult.

Calculating curves and areas under curves

The roots ofcalculus lie in some of the oldest geometry [7] problems on record. The

Egyptian Rhind papyrus [8] (c. 1650 bc) gives rules for finding the area [9] of a circle and

the volume[10]

of a truncated pyramid. Ancient Greek geometers investigated finding

tangents to curves, the centre of gravity [11] of plane and solid figures, and the volumes of

objects formed by revolving various curves about a fixed axis.

By 1635 the Italian mathematician Bonaventura Cavalieri[12]

had supplemented the

rigorous tools of Greek geometry [13] with heuristic methods that used the idea of

infinitely small segments of lines, areas, and volumes. In 1637 the French mathematician-

philosopher Ren Descartes[14]

published his invention of analytic geometry[15]

for

giving algebraic descriptions of geometric figures. Descartess method, in combination

http://www.britannica.com/EBchecked/topic/89161/cal

de 5 16/03/12 1


2/5

[16] with an ancient idea of curves being generated by a moving point, allowed

mathematicians such as Newton to describe motion algebraically. Suddenly geometers

could go beyond the single cases and ad hoc methods of previous times. They could see

patterns of results, and so conjecture new results, that the older geometric language had

obscured.

For example, the Greek geometer Archimedes [17] (c. 285212/211 bc) discovered as an

isolated result that the area of a segment of a parabola [18] is equal to a certain triangle.

But with algebraic notation [19], in which a parabola is written as y = x2, Cavalieri and

other geometers soon noted that the area between this curve and the x-axis from 0 to a is

a3/3 and that a similar rule holds for the curve y = x3namely, that the corresponding

area is a4/4. From here it was not difficult for them to guess that the general formula for

the area under a curve y = xn is an + 1/(n + 1).

Calculating velocities and slopes

The problem [20] of finding tangents to curves was closely related to an important

problem that arose from the Italian scientist Galileo Galileis investigations of motion,

that of finding the velocity [21] at any instant of a particle moving according to some law.

Galileo established that in tseconds a freely falling body falls a distance gt2/2, where g is

a constant [22] (later interpreted by Newton as the gravitational [23] constant). With the

definition of average velocity as the distance per time, the bodys average velocity over

an interval from tto t+ h is given by the expression [g(t+ h)2/2 gt2/2]/h. This

simplifies to gt+ gh/2 and is called the difference quotient of the function [24]gt2/2. As h

approaches 0, this formula approaches gt, which is interpreted as the instantaneous

velocity [25] of a falling body at time t.

This expression for motion is identical to that obtained for the slope of the tangent to theparabola f(t) =y = gt2/2 at the point t. In this geometric context, the expression gt+ gh/2

(or its equivalent [f(t+ h) f(t)]/h) denotes the slope of a secant line [26] connecting the

point (t, f(t)) to the nearby point (t+ h, f(t+ h)) (see figure). In the limit [27], with smaller

and smaller intervals h, the secant line approaches the tangent line and its slope at the

point t.


de 5 16/03/12 1


3/5

Thus, the difference quotient can be interpreted as instantaneous velocity or as the slope

of a tangent to a curve. It was the calculus that established this deep connection between

geometry and physicsin the process transforming physics and giving a new impetus to

the study of geometry.

Differentiation and integration[28]

Independently, Newton and Leibniz established simple rules for finding the formula for

the slope of the tangent to a curve at any point on it, given only a formula for the curve.

The rate of change of a function f(denoted by f) is known as its derivative [29]. Finding

the formula of the derivative [30] function is called differentiation [31], and the rules for

doing so form the basis of differential[32]

calculus. Depending on the context,

derivatives may be interpreted as slopes of tangent lines, velocities of moving particles,

or other quantities, and therein lies the great power of the differential calculus.

An important application of differential calculus is graphing a curve given its equation

[33]y = f(x). This involves, in particular, finding local maximum and minimum points on

the graph[34]

, as well as changes in inflection (convex to concave, or vice versa). When

examining a function used in a mathematical model [35], such geometric notions have

physical interpretations that allow a scientist or engineer to quickly gain a feeling for thebehaviour of a physical system.

The other great discovery of Newton and Leibniz was that finding the derivatives of

functions was, in a precise sense, the inverse [36] of the problem of finding areas under

curvesa principle now known as the fundamental theorem ofcalculus. Specifically,

Newton discovered that if there exists a function F(t) that denotes the area under the

curve y = f(x) from, say, 0 to t, then this functions derivative will equal the original curve

over that interval, F(t) =f(t). Hence, to find the area under the curve y = x2 from 0 to t, it

is enough to find a function Fso that F(t) = t2. The differential calculus shows that the

most general such function is x3/3 + C, where Cis an arbitrary constant. This is called the

(indefinite) [37] integral [38] of the function y = x2, and it is written as x2dx. The initial

symbol is an elongated S, which stands for sum, and dx indicates an infinitely small

increment of the variable, or axis, over which the function is being summed. Leibniz


de 5 16/03/12 1


4/5

introduced this because he thought of integration [39] as finding the area under a curve by

a summation of the areas of infinitely many infinitesimally [40] thin rectangles between

thex-axis and the curve. Newton and Leibniz discovered that integrating f(x) is

equivalent to solving a differential equation [41]i.e., finding a function F(t) so that

F(t) =f(t). In physical terms, solving this equation can be interpreted as finding the

distance F(t) traveled by an object whose velocity has a given expression f(t).

The branch of the calculus concerned with calculating integrals is the integral [42]

calculus, and among its many applications are finding work done by physical systems

and calculating pressure behind a dam at a given depth.

John L. Berggren

http://www.britannica.com/bps/user-profile/3484/John-L.-Berggren1.

http://www.britannica.com/EBchecked/topic/89152/calculation2.

http://www.britannica.com/EBchecked/topic/413189/Sir-Isaac-Newton3.

http://www.britannica.com/EBchecked/topic/335266/Gottfried-Wilhelm-Leibniz4.

http://www.britannica.com/EBchecked/topic/466303/point5.

http://www.britannica.com/EBchecked/topic/557444/space-shuttle6.

http://www.britannica.com/EBchecked/topic/229851/geometry7.

http://www.britannica.com/EBchecked/topic/501277/Rhind-papyrus8.

http://www.britannica.com/EBchecked/topic/33377/area9.

http://www.britannica.com/EBchecked/topic/632569/volume10.

http://www.britannica.com/EBchecked/topic/242556/centre-of-gravity11.

http://www.britannica.com/EBchecked/topic/100533/Bonaventura-Cavalieri12.

http://www.britannica.com/EBchecked/topic/229851/geometry13.

http://www.britannica.com/EBchecked/topic/158787/Rene-Descartes14.

http://www.britannica.com/EBchecked/topic/22548/analytic-geometry15.

http://www.britannica.com/EBchecked/topic/127320/combination16.

http://www.britannica.com/EBchecked/topic/32808/Archimedes17.

http://www.britannica.com/EBchecked/topic/442379/parabola18.


de 5 16/03/12 1


5/5

http://www.britannica.com/EBchecked/topic/14946/algebraic-notation19.

http://www.britannica.com/EBchecked/topic/477574/problem20.

http://www.britannica.com/EBchecked/topic/624901/velocity21.

http://www.britannica.com/EBchecked/topic/133764/constant22.

http://www.britannica.com/EBchecked/topic/242404/gravitation23.

http://www.britannica.com/EBchecked/topic/222041/function24.

http://www.britannica.com/EBchecked/topic/289234/instantaneous-velocity25.

http://www.britannica.com/EBchecked/topic/341961/line26.

http://www.britannica.com/EBchecked/topic/341417/limit27.

http://www.britannica.com/EBchecked/topic/289677/integration28.

http://www.britannica.com/EBchecked/topic/158518/derivative29.

http://www.britannica.com/EBchecked/topic/158518/derivative30.

http://www.britannica.com/EBchecked/topic/162982/differentiation31.

http://www.britannica.com/EBchecked/topic/162890/differential32.

http://www.britannica.com/EBchecked/topic/190622/equation33.

http://www.britannica.com/EBchecked/topic/241997/graph34.

http://www.britannica.com/EBchecked/topic/369135/mathematical-model35.

http://www.britannica.com/EBchecked/topic/292322/inverse36.

http://www.britannica.com/EBchecked/topic/284982/indefinite-integral37.

http://www.britannica.com/EBchecked/topic/289602/integral38.

http://www.britannica.com/EBchecked/topic/289677/integration39.

http://www.britannica.com/EBchecked/topic/287641/infinitesimal40.

http://www.britannica.com/EBchecked/topic/162910/differential-equation41.

http://www.britannica.com/EBchecked/topic/289602/integral42.


de 5 16/03/12 1

calculus - Encyclopædia Britannica

Documents

Transcript of calculus - Encyclopædia Britannica