Calculus

Calculus is the branch of mathematics that deals with rates of change and motion . It grew out of a desire to understand various physical phenomena, such as the orbits of planets, and the effects of gravity. The immediate success of calculus in formulating physical laws and predicting their consequences led to development of a new division in mathematics called analysis, of which calculus remains a large part. Today, calculus is the essential language of science and engineering , providing the means by which physical laws are expressed in mathematical terms. As a scientific tool it is invaluable in the further analysis of physical laws, in predicting the behavior of electrical and mechanical systems governed by those laws, and in discovering new laws.

Calculus divides naturally into two parts, differential calculus and integral calculus. Differential calculus is concerned with finding the instantaneous rate at which one quantity changes with respect to another, called the derivative of the first quantity with respect to the second. For example, determining the speed of a falling body at a particular instant of time, say that of a skydiver or bungi jumper, is equivalent to calculating the instantaneous rate of change in his or her position with respect to time. In general, evaluating the derivative of a function , f(x), involves finding another function, f'(x), such that f'(x) is equal to the slope of the tangent to the graph of f(x) at each x. This is accomplished, for each 2, by determining the slope of an approximating line segment in the limit that its length approaches zero .

Integral calculus deals with the inverse of the derivative, namely, finding a function when its rate of change is known. For example, if a skydiver's velocity is a known function of time, then we may ask what is his or her position at any given time after jumping. Finding the original function, given its derivative, is called integration, and the function is called the indefinite integral. Evaluating the indefinite integral of any function between specific limits to definition of the definite integral, which is equal to the area under the graph of the function between the specified limits. The latter is developed as a natural consequence of approximating an area by summing the areas of a number of inscribed rectangles. The approximation becomes exact in the limit that the number of rectangles approaches infinity . Thus, both differential and integral calculus are based on the theory of limits.

The usefulness of calculus is indicated by its widespread application. For example, it is used in the design of navigation systems, particle accelerators , and synchrotron light sources. It is used to predict rocket trajectories, and the orbits of communications satellites. Calculus is the mathematical tool used to test theories about the origins of the universe, the development of tornadoes and hurricanes, and salt fingering in the oceans. It has even found extensive application in business, where it is used, among other things, to optimize production.

History

Calculus was invented, more or less simultaneously, by Isaac Newton and Gottfried Leibniz. Some of the essential ingredients, however, had their beginnings in ancient Greece. In the fourth century b.c., Eudoxus invented the so called method of exhaustion, in order to furnish proofs of certain geometric theorems without having to resort to arguments involving the infinite. Approximately a century later, Archimedes used the same method to find a formula for the area of a circle . Archimedes' method consisted of inscribing a polygon with n sides inside a circle, and circumscribing a similar polygon, again with n sides, outside the circle. Then, allowing n, in other words the number of sides, to get very large, he was able to show that the area of the circle was always greater than the area of the inscribed polygon and less than the area of the circumscribed polygon. As n grew very large, the areas of the two polygons tended to become equal, thus leading him to the area of a circle. The method of Archimedes persisted from the third century b.c. until the beginning of the seventeenth century a.d.when the work of Johannes Kepler, a German Astronomer, led to the discovery of general principles for the calculation of areas and volumes. Kepler's contribution was the notion of infinitesimals. He envisioned the inscribed polygons of Archimedes as being a collection of infinitely many, vanishingly small triangles. Thus, the area of a circle could be calculated by summing the areas of these triangles. While, Eudoxus and Archimedes had worked hard to avoid the infinite, Kepler embraced it. Simultaneous work on infinite sequences and sums of infinite sequences led the French mathematician Fermat, and others, to discover general methods for evaluating areas and volumesas the sums of infinite sequences rather than the sums of areas of common geometric figures. Finally, around the middle of the seventeenth century, Isaac Newton, in attempting to develop a universal theory of gravitation, discovered the derivative, a general method for determining the instantaneous rate of change of a function, based on the notion of infinitesimals. Though he did not explicitly define the integral at the time, Newton did recognize the need to solve differential equations. As a result, he invented methods of evaluating indefinite integrals very soon after introducing the derivative. It was Leibniz, however, whose work postdated that of Newton by some 10 years, who recognized and formulated the definite integral as an infinite sum of "lines," that is, as an area calculated by summing an infinite number of infinitely narrow rectangles.

Differential calculus

Differential calculus involves the analysis of functions, specifically, determining their instantaneous rates of change. An important feature of any function is its rate of change. Geometrically, rate of change is associated with the graph of a function. The rate of change of a straight line (the simplest kind of real valued function) is the slope of the line. The slope is defined as the ratio of the vertical change, or "rise," to the horizontal change, or "run," that occurs between any two points on the line. Because the slope is the same between any two points, the rate of change of such a function is said to be constant. In general, however, any function whose graph is not a straight line has a varying rate of change. The rate of change in the vicinity of a particular point on the graph of curve can be approximated by drawing a straight line through two points in the neighborhood of that point, and determining the slope of the line. Suppose we are interested in the rate of change at the point (x,f(x)). First, choose a second nearby point, say (x+h, f(x+h)). Then the slope of the line segment connecting these two points is [f(x+h) - f(x)] 4 [(x+h) - x]. The shorter the approximating line segment becomes, the more accurate the approximation of the rate of change at the point (x, f(x)) becomes. In the limit that h approaches zero the slope of the approximating line segment becomes exactly the rate of change of the function at the point (x,f(x)). Thus, the instantaneous rate of change of a function, called the derivative of the function, is defined by:

where the notation df(x) is intended to indicate that the derivative is the ratio of an infinitesimal change in f(x) (the rise) to the corresponding infinitesimal change in x (the run). The derivative of a function is itself a function, and so may also have a derivative. Often times the derivative of the derivative is an important quantity. Called the second derivative of the original function f, it is denoted by f"(x).

An important application of differential calculus involves using information about the first and second derivatives, and the appropriate geometric interpretations, to graph functions. For example, the first derivative of a function is its rate of change. The value of the first derivative at a given point is equal to the slope of the tangent to the graph of the function at that point. When the derivative is positive, the function is said to be increasing (the value of the function increases with increasing x). When the derivative is negative , the function is said to be decreasing (the value of the function decreases with increasing x). When the value of the derivative is zero at a point, the tangent is horizontal, and the function changes from increasing to decreasing, or vice versa, depending on the sign of the second derivative. The second derivative is the rate of change of the rate of change, and thus contains information about the curvature of the function. When the second derivative is positive, the function is concave upward (as though it would hold water ). When the second derivative is negative, the function is concave downward. With this knowledge, and a few points in the function, a reasonable graph can be drawn without having to plot hundreds of points.

Other applications of differential calculus include the solution of rate problems and optimization problems. In general, a rate is the ratio of change in one quantity to the simultaneous change in a second quantity. Thus, the derivative, being an instantaneous rate, is applicable to any problem in which the rate of change of one quantity with respect to another is of interest. Innumerable applications in engineering and science affect our daily lives. For instance, the instantaneous velocity of an orbiting communications satellite is calculated from knowledge of its position as a function of time. The acceleration of a falling body is calculated from knowledge of its velocity as a function of time, which in turn is calculated from knowledge of its position as a function of time. The force required to deliver natural gas through a pipeline, over large distances, is calculated using the derived of the gas pressure with respect to distance .

Optimization problems are problems that require knowledge of maximum or minimum values of functional relationships. For example, it can be shown that a sphere has the least surface area for a given volume of any geometric solid. Thus, the optimum shape for a raindrop is spherical because this shape contains the most water, but has the least amount of surface area, hence the least surface energy (a measure of the work required to form the drop).

Integral calculus

Integral calculus is the study of integration and methods for evaluating integrals. Integrals come in two kinds, definite and indefinite. The definite integral of a function, interpreted geometrically, corresponds to the area under the curve of the function between any two limits. Thus, it has a definite value depending on the limits chosen, The indefinite integral of a function is the inverse of the derivative of that function. That is, integrating (finding the integral) undoes differentiating (finding the derivative f'(x)). Integrating the derivative of a function returns the original function.

Indefinite Integral

The indefinite integral is the inverse of the derivative, that is, the integral of the derivative of a function is the original function. From the definition of derivative, f'(x) = df(x)/dx, we find f'(x)dx = df(x). To obtain the original function from this second equation, we "integrate" both sides, and write, ? f'(x)dx = ?df(x) = f(x) + C. The integral sign, ? , is intended to symbolize the summing, integrating, or putting together of the infinitesimal pieces df(x) to obtain the original function f(x). The constant, C, arises because functions that differ only by a constant are "parallel" to one another, and so have the same derivative (or slope) at each value of x. Defined in this manner, integrating amounts to guessing original functions based on prior knowledge of their derivatives. For example, if f(x) = x2 then f'(x) =2x. Thus, if asked to integrate the function g(x) = 2x, it is apparent that ?2xdx = x2 + C. In order to determine the value of C it is necessary to have an additional piece of information. Such information is referred to as an initial condition or a boundary condition, and is sufficient to determine which of the parallel curves is the desired one.

The primary application of indefinite integrals is in the solution of differential equations. A differential equation is any equation that contains at least one derived. The equation f'(x) = ax2+bx+c is an example of a differential equation. Many natural relationships are described by differential equations. For instance, heat conduction is related to the derivative of the temperature with respect to distance; the velocity of a fluid flowing through a pipe is related to the derivative of the pressure with respect to length of pipe; and the force on any massive body is related to the derivative of its momentum with respect to time.

Definite Integral

The definite integral corresponds to the area under the graph of a function, above the x-axis and between two vertical lines called the limits of integration. Consider approximating the area under the graph of a function f(x) by drawing a series of rectangles, and summing their areas to arrive at the total area, A(x) (see Figure 2). The height of each rectangle is the value of the function at x, namely f(x). The width of each rectangle is ?x = (b-a)/n, where n is the number of rectangles chosen. If we wish to know the area between x=a and x=b, then the area is given by the sum

(the Greek letter ? (sigma) is used to indicate that the n products f(xk) ?x corresponding to the n rectangles are to be summed). In the limit that n approaches infinity, ?x approaches 0, and the sum is exactly equal to the are. Since ?x approaches 0, it represents an infinitesimal change in the variable x, so the same notation used in defining the derivative is used to replace ?x with d. The product f(x) ?x becomes f(x)dx, and corresponds to an infinitesimal area, dA(x). The total area, then, is the sum of an infinite number of an infinite number of infinitesimal areas. Thus, the area A(x) between a and b is equal to the integral of f(x)dx, written,

The limits included above and below the integral sign indicate that the indefinite integral of f(x)d is to be evaluated at a and b and the values subtracted, that is,

This is interpreted as the area under the curve to the left of b minus the area under the curve to the left of a. Comparing this with the form of the indefinite integral we see that a function f(x) is the derivative of its "Area function," the constant C being evaluated by use of boundary conditions, namely the values a and b. There are many applications of definite integrals, among the most common are the determination of areas and volumes of revolution.

Resources

books

Abbot, P., and M.E. Wardle. Teach Yourself Calculus. Lincolnwood, IL: NTC Publishing, 1992.

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Silverman, Richard A. Essential Calculus With Applications. New York: Dover, 1989.

Swokowski, Earl W. Pre Calculus, Functions, and Graphs. 6th. ed. Boston, MA: PWS-KENT Publishing Co., 1990.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.

Periodicals

Moore, A.W. "A Brief History of Infinity." Scientific American 272 (1995): 112-116.

J. R. Maddocks

KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Derivative

—The derivative of a function is the instantaneous rate of change of the function's dependent variable with respect to its independent variable, interpreted geometrically as the slope of the tangent to the graph of the function.

Function

—A set of ordered pairs for which each of the first and second elements are related, no two first elements being equal.

Integral

—The integral is the inverse of the derivative, interpreted geometrically as the area under the graph of the function.

Limit

—A limit is a boundary which the value of a variable may come infinitely close to, but never reach.