I.

Introduction

Calculus (mathematics), branch of mathematics concerned with the study of such concepts as the rate of change of one variable quantity with respect to another, the slope of a curve at a prescribed point, the computation of the maximum and minimum values of functions, and the calculation of the area bounded by curves. Evolved from algebra, arithmetic, and geometry, it is the basis of that part of mathematics called analysis.

Calculus is widely employed in the physical, biological, and social sciences. It is used, for example, in the physical sciences to study the speed of a falling body, the rates of change in a chemical reaction, or the rate of decay of a radioactive material. In the biological sciences a problem such as the rate of growth of a colony of bacteria as a function of time is easily solved using calculus. In the social sciences calculus is widely used in the study of statistics and probability.

Calculus can be applied to many problems involving the notion of extreme amounts, such as the fastest, the most, the slowest, or the least. These maximum or minimum amounts may be described as values for which a certain rate of change (increase or decrease) is zero. By using calculus it is possible to determine how high a projectile will go by finding the point at which its change of altitude with respect to time, that is, its velocity, is equal to zero. Many general principles governing the behavior of physical processes are formulated almost invariably in terms of rates of change. It is also possible, through the insights provided by the methods of calculus, to resolve such problems in logic as the famous paradoxes posed by the Greek philosopher Zeno.

The fundamental concept of calculus, which distinguishes it from other branches of mathematics and is the source from which all its theory and applications are developed, is the theory of limits of functions of variables (see Function).

Let f be a function of the real variable x, which is denoted f(x), defined on some set of real numbers surrounding the number x₀. It is not required that the function be defined at the point x₀ itself. Let L be a real number. The expression

is read: “The limit of the function f(x), as x approaches x₀, is equal to the number L.” The notation is designed to convey the idea that f(x) can be made as “close” to L as desired simply by choosing an x sufficiently close to x₀. For example, if the function f(x) is defined as f(x) = x² + 3x + 2, and if x₀ = 3, then from the definition above it is true that

This is because, as x approaches 3 in value, x² approaches 9, 3x approaches 9, and 2 does not change, so their sum approaches 9 + 9 + 2, or 20.

Another type of limit important in the study of calculus can be illustrated as follows. Let the domain of a function f(x) include all of the numbers greater than some fixed number m. L is said to be the limit of the function f(x) as x becomes positively infinite, if, corresponding to a given positive number e, no matter how small, there exists a number M such that the numerical difference between f(x) and L (the absolute value |f(x) - L|) is less than e whenever x is greater than M. In this case the limit is written as

It is important to note that a limit, as just presented, is a two-way, or bilateral, concept: A dependent variable approaches a limit as an independent variable approaches a number or becomes infinite. The limit concept can be extended to a variable that is dependent on several independent variables. The statement “u is an infinitesimal” meaning “u is a variable approaching 0 as a limit,” found in a few present-day and in many older texts on calculus, is confusing and should be avoided. Further, it is essential to distinguish between the limit of f(x) as x approaches x₀ and the value of f(x) when x is x₀, that is, the correspondent of x₀. For example, if f(x) = sin x/x, then

The two branches into which elementary calculus is usually divided are differential calculus, based on the consideration of the limit of a certain ratio, and integral calculus, based on the consideration of the limit of a certain sum.

II.

Differential Calculus

Let the dependent variable y be a function of the independent variable x, expressed by y = f(x). If x₀ is a value of x in its domain of definition, then y₀ = f(x₀) is the corresponding value of y. Let h and k be real numbers, and let y₀ + k = f(x₀ + h). (Δx, read “delta x,” is used quite frequently in place of h.) When Δx is used in place of h,Δy is used in place of k. Then clearly

and

This ratio is called a difference quotient. Its intuitive meaning can be grasped from the geometrical interpretation of the graph of y = f(x). Let A and B be the points (x₀, y₀), (x₀ + h, y₀ + k), respectively, as in the Derivatives illustration. Draw the secant AB and the lines AC and CB, parallel to the x and y axes, respectively, so that h = AC, k = CB. Then the difference quotient k/h equals the tangent of angle BAC and is therefore, by definition, the slope of the secant AB. It is evident that if an insect were crawling along the curve from A to B, the abscissa x would always increase along its path but the ordinate y would first increase, slow down, then decrease. Thus, y varies with respect to x at different rates between A and B. If a second insect crawled from A to B along the secant, the ordinate y would vary at a constant rate, equal to the difference quotient k/h, with respect to the abscissa x. As the two insects start and end at the same points, the difference quotient may be regarded as the average rate of change of y = f(x) with respect to x in the interval AC.

If the limit of the ratio k/h exists as h approaches 0, this limit is called the derivative of y with respect to x, evaluated at x = x₀. For example, let y = x² and x = 3, so that y = 9. Then 9 + k = (3 + h)²; k = (3 + h)² - 9 = 6h + h²; k/h = 6 + h; and

If the derivative of y with respect to x is found for all values of x (in its domain) for which the derivative is defined, a new function is obtained, the derivative of y with respect to x. If y = f(x), the new function is written as y’ or f’(x), D_xy or D_xf(x), (dy)/(dx) or df(x)/dx. Thus, if y = x², y + k = (x + h)²; k = (x + h)² - x² = 2xh + h²; k/h = 2x + h, whence

As the derivative f’(x) of a function f(x) of x is itself a function of x, its derivative with respect to x can be found; it is called the second (order) derivative of y with respect to x, and is designated by any one of the symbols y” or f”(x), D_x²y or D_x²f(x), (d²y)/(dx²) or (d²f(x))/(dx²). Third- and higher-order derivatives are similarly designated.

Every application of differential calculus stems directly or indirectly from one or both of the two interpretations of the derivative as the slope of the tangent to the curve and as the rate of change of the dependent variable with respect to the independent variable. In a detailed study of the subject, rules and methods developed by the limit process are provided for rapid calculation of the derivatives of various functions directly by means of various known formulas. Differentiation is the name given to the process of finding a derivative.

Differential calculus provides a method of finding the slope of the tangent to a curve at a certain point; related rates of change, such as the rate at which the area of a circle increases (in square feet per minute) in terms of the radius (in feet) and the rate at which the radius increases (in feet per minute); velocities (rates of change of distance with respect to time) and accelerations (rates of change of velocities with respect to time, therefore represented as second derivatives of distance with respect to time) of points moving on straight lines or other curves; and absolute and relative maxima and minima.

III.

Integral Calculus

Let y = f(x) be a function defined for all x’s in the interval [a,b], that is, the set of x’s from x = a to x = b, including a and b, where a<b (suitable modifications can be made in the definitions to follow for more restricted ranges or domains). Let x₀, x₁, ..., x_n be a sequence of values of x such that a = x₀<x₁<x₂<...<x_n - 1 <x_n = b, and let h₁ = x₁ - x₀, h₂ = x₂ - x₁, ..., h_n = x_n - x_{n - 1}, in brief, h_i = x_i - x_{i - 1}, where i = 1, 2, ..., n. The x’s form a partition of the interval [a, b]; an h with a value not exceeded by any other h is called the norm of the partition. Let n values of x, for example, X₁, X₂, ..., X_n, be chosen so that x_{i - 1}<X_i<x_i, where i = 1, 2, ..., n. The sum of the area of the rectangles is given by

f(X1)h1 + f(X2)h2 + .... + f(Xn)hn

usually abbreviated to

(Σ is the Greek capital letter sigma.) Aside from the given function f(x) and the given a and b, the value of the sum clearly depends on n and on the choices of the x_i’s and X_i’s. In particular, if, after the x_i’s are chosen, the X_i’s are chosen so that f(X_i), for each i, is a maximum in the interval [x_{i - 1}, x_i] (that is, no ordinate from x_{i - 1} to x_i exceeds the ordinate at X_i), the sum is called an upper sum; similarly, if, after the x_i’s are chosen, the X_i’s are chosen so that f(X_i), for each i, is a minimum in the interval [x_{i - 1}, x_i], the sum is called a lower sum. It can be proved that the upper and lower sums will have limits,  and , respectively, as the norm approaches 0. If  and  are equal and have the common value S, S is called the definite integral of f(x) from a to b and is written

The symbol ∫ is an elongated S (for sum); the f(x) dx is suggested by a term f(X_i)h_i = f(X_i) Δx_i of the sum which is used in defining the definite integral.

If y = g(x), then by differentiation y’ = g’(x). Let g’(x) = f(x), and C be any constant. Then f(x) is also the derivative of g(x) + C. The expression g(x) + C is called the antiderivative of f(x), or the indefinite integral of f(x), and it is represented by

The dual use of the term integral is justified by one of the fundamental theorems of calculus, namely, if g(x) is an antiderivative of f(x), then, under suitable restrictions on f(x) and g(x),

The process of finding either an indefinite or a definite integral of a function f(x) is called integration; the fundamental theorem relates differentiation and integration.

If the antiderivative, g(x), of f(x) is not readily obtainable or is not known, the definite integral can be approximated by the trapezoidal rule, (b - a) [f(a) + f(b)]/2 or by the more accurate Simpson’s rule:

Integral calculus involves the inverse process of finding the derivative of a function, that is, it is the process of finding the function itself when its derivative is known. For example, integral calculus makes it possible to find the equation of a curve if the slope of the tangent is known at an arbitrary point; to find distance in terms of time if the velocity (or acceleration) is known; and to find the equation of a curve if its curvature is known. Integral calculus can also be used to find the lengths of curves, the areas of plane and curved surfaces, volumes of solids of revolution, centroids, moments of inertia, and total mass and total force.

IV.

Differential Equations

Calculus leads directly to the branch of mathematics called differential equations, which is extremely useful in engineering and in the physical sciences. An ordinary differential equation is an equation involving an independent variable, a dependent variable (one or both of these two may be missing), and one or more derivatives (at least one derivative must be present). Many physical laws or statements are initially expressed as differential equations. For example, the law that the acceleration of gravity is a constant g can be expressed mathematically by the differential equation d²x/dt² = g; the principle that the rate of disintegration of radium is proportional to the amount present is expressed as dR/dt = -kR. A differential equation is solved if an equivalent equation is found involving only the independent and dependent variables.

This article has considered functions of a single independent variable only. Partial derivatives, multiple integrals, and partial differential equations are defined and studied in investigating functions of two or more independent variables.

V.

Development of Calculus

The English and German mathematicians, respectively, Isaac Newton and Gottfried Wilhelm Leibniz invented calculus in the 17th century, but isolated results about its fundamental problems had been known for thousands of years. For example, the Egyptians discovered the rule for the volume of a pyramid as well as an approximation of the area of a circle. In ancient Greece, Archimedes proved that if c is the circumference and d the diameter of a circle, then 3‡d<c< 3d. His proof extended the method of inscribed and circumscribed figures developed by the Greek astronomer and mathematician Eudoxus. Archimedes used the same technique for his other results on areas and volumes. Archimedes discovered his results by means of heuristic arguments involving parallel slices of the figures and the law of the lever. Unfortunately, his treatise The Method was only rediscovered in the 19th century, so later mathematicians believed that the Greeks deliberately concealed their secret methods.

During the late middle ages in Europe, mathematicians studied translations of Archimedes’ treatises from Arabic. At the same time, philosophers were studying problems of change and the infinite, such as the addition of infinitely many quantities. Greek thinkers had seen only contradictions there, but medieval thinkers aided mathematics by making the infinite philosophically respectable.

By the early 17th century, mathematicians had developed methods for finding areas and volumes of a great variety of figures. In his Geometry by Indivisibles, the Italian mathematician F. B. Cavalieri, a student of the Italian physicist and astronomer Galileo, expanded on the work of the German astronomer Johannes Kepler on measuring volumes. He used what he called “indivisible magnitudes” to investigate areas under the curves y = xⁿ, n = 1 ...9. Also, his theorem on the volumes of figures contained between parallel planes (now called Cavalieri’s theorem) was known all over Europe. At about the same time, the French mathematician René Descartes’La Géométrie appeared. In this important work, Descartes showed how to use algebra to describe curves and obtain an algebraic analysis of geometric problems. A codiscoverer of this analytic geometry was the French mathematician Pierre de Fermat, who also discovered a method of finding the greatest or least value of some algebraic expressions—a method close to those now used in differential calculus.

About 20 years later, the English mathematician John Wallis published The Arithmetic of Infinites, in which he extrapolated from patterns that held for finite processes to get formulas for infinite processes. His colleague at the University of Cambridge was Newton’s teacher, the English mathematician Isaac Barrow, who published a book that stated geometrically the inverse relationship between problems of finding tangents and areas, a relationship known today as the fundamental theorem of calculus.

Although many other mathematicians of the time came close to discovering calculus, the real founders were Newton and Leibniz. Newton’s discovery (1665-66) combined infinite sums (infinite series), the binomial theorem for fractional exponents, and the algebraic expression of the inverse relation between tangents and areas into methods we know today as calculus. Newton, however, was reluctant to publish, so Leibniz became recognized as a codiscoverer because he published his discovery of differential calculus in 1684 and of integral calculus in 1686. It was Leibniz, also, who replaced Newton’s symbols with those familiar today.

In the following years, one problem that led to new results and concepts was that of describing mathematically the motion of a vibrating string. Leibniz’s students, the Bernoulli family of Swiss mathematicians (see Bernoulli, Daniel), used calculus to solve this and other problems, such as finding the curve of quickest descent connecting two given points in a vertical plane. In the 18th century, the great Swiss-Russian mathematician Leonhard Euler, who had studied with Johann Bernoulli, wrote his Introduction to the Analysis of Infinites, which summarized known results and also contained much new material, such as a strictly analytic treatment of trigonometric and exponential functions.

Despite these advances in technique, calculus remained without logical foundations. Only in 1821 did the French mathematician A. L. Cauchy succeed in giving a secure foundation to the subject by his theory of limits, a purely arithmetic theory that did not depend on geometric intuition or infinitesimals. Cauchy then showed how this could be used to give a logical account of the ideas of continuity, derivatives, integrals, and infinite series. In the next decade, the Russian mathematician N. I. Lobachevsky and German mathematician P. G. L. Dirichlet both gave the definition of a function as a correspondence between two sets of real numbers, and the logical foundations of calculus were completed by the German mathematician J. W. R. Dedekind in his theory of real numbers, in 1872.