Calculus
Calculus
As students first begin to study calculus in high school or college, many may be unsure about what calculus is. What are the fundamental concepts that underlie calculus? Who has been credited for the discovery of calculus and how is calculus used today?
What is Calculus?
Calculus was invented as a tool for solving problems. Prior to the development of calculus, there were a variety of different problems that could not be addressed using the mathematics that was available. For example, scientists did not know how to measure the speed of an object when that speed was changing over time. Also, a more effective method was desired for finding the area of a region that did not have straight edges. Geometry , algebra , and trigonometry , which were well understood, did not provide the necessary tools to adequately address these problems.
At the time in which calculus was developed, automobiles had not been invented. However, automobiles are an example of how calculus may be used to describe motion. When the driver pushes on the accelerator of a car, the speed of that car increases. The rate at which the car is moving, or the velocity , increases with respect to time. When the driver steps on the brakes, the speed of the car decreases. The velocity decreases with respect to time.
As a driver continues to press on the accelerator of a car, the velocity of that car continues to increase. "Acceleration" is a concept that is used to describe how velocity changes over time. Velocity and acceleration are measured using a fundamental concept of calculus that is called the derivative .
Derivatives can be used to describe the motion of many different objects. For example, derivatives have been used to describe the orbits of the planets and the descent of space shuttles. Derivatives are also used in a variety of different fields. Electrical engineers use derivatives to describe the change in current within an electric circuit. Economists use derivatives to describe the profits and losses of a given business.
The concept of a derivative is also useful for finding a tangent line to a given curve at a specific point. A tangent line is a straight line that touches a curve at only one point when restricted to a very small region. An example of a tangent line to a curve is shown in the figure. The straight line and the curve touch at only one point. The straight line is the tangent line to the curve at that point.
Tangent lines are useful tools for understanding the angle at which light passes through a lens. Derivatives and tangent lines were useful tools in the development and continued improvement of the telescope. Derivatives are also used today by optometrists or eye doctors to develop more effective methods for correcting vision. Physicists use tangent lines to describe the direction in which an object is traveling, and chemists use tangent lines to predict the outcomes of chemical reactions. These are only a few examples of the many uses of tangent lines in science, engineering, and medicine.
Derivatives along with the concept of a tangent line can be used to find the maximum or minimum value for a given situation. For example, a business person may wish to determine how to maximize profit and minimize expense. Astronomers also use derivatives and the concept of a tangent line to find the maximum or minimum distance of Earth from the Sun.
The derivative is closely related to another important concept in calculus, the integral . The integral, much like the derivative, has many applications. For example, physicists use the integral to describe the compression of a spring. Engineers use the integral to find the "center of mass" or the point at which an object balances. Mathematicians use the integral to find the areas of surfaces, the lengths of curves, and the volumes of solids.
The basic concepts that underlie the integral can be described using two other mathematical concepts that are important to the study of calculus— "area" and "limit." Many students know that finding the area of a rectangle requires multiplying the base of the rectangle by the height of the rectangle. Finding the area of a shape that does not have all straight edges is more difficult.
The area between the curve and the x axis is colored in (a) of the figure on the following page. One way to estimate the area of the portion of the figure that is colored is to divide the region into rectangles as is shown in (b). Some of the rectangles contain less area than is contained in the colored region. Some of the rectangles contain more area than is contained in the colored region. To estimate the area of the colored region, the area of the six rectangles can be added together.
If a better estimate is desired, the colored region can be divided into more rectangles with smaller bases, as shown in (c). The areas of these
rectangles can then be added together to acquire a better approximation to the area of the colored region.
If an even better estimate of the colored region is desired, it can be divided into even more rectangles with smaller bases. This process of dividing the colored region into smaller and smaller rectangles can be continued. Eventually, the bases of the rectangles are so small that the lengths of these bases are getting close to zero.
The concept of allowing the bases of the rectangles to approach zero is based on the limit concept. The integral is a mathematically defined function that uses the limit concept to find the exact area beneath a curve by dividing the region into successively smaller rectangles and adding the areas of these rectangles. By extending the process described here to the study of threedimensional objects, it becomes clear that the integral is also a useful tool for determining the volume of a threedimensional object that does not have all straight edges.
An interesting relationship in calculus is that the derivative and the integral are inverse processes. Much like subtraction reverses addition, differentiation (finding the derivative) reverses integration. The reverse of this statement, integration reverses differentiation, is also true. This relationship between derivatives and integrals is referred to as the "Fundamental Theorem of Calculus." The Fundamental Theorem of Calculus allows integrals to be used in motion problems and derivatives to be used in area problems.
Who Invented Calculus?
Pinpointing who invented calculus is a difficult task. The current content that comprises calculus has been the result of the efforts of numerous scientists. These scientists have come from a variety of different scientific backgrounds and represent many nations and both genders. History, however, typically recognizes the contributions of two scientists as having laid the foundations for modern calculus: Gottfried Wilhelm Leibniz (1646–1716) and Sir Isaac Newton (1642–1727).
Leibniz was born in Leipzig, Germany, and had a Ph.D. in law from the University of Altdorf. He had no formal training in mathematics. Leibniz taught himself mathematics by reading papers and journals. Newton was born in Woolsthorpe, England. He received his master's degree in mathematics from the University of Cambridge.
The question of who invented calculus was debated throughout Leibniz's and Newton's lives. Most scientists on the continent of Europe credited Leibniz as the inventor of calculus, whereas most scientists in England credited Newton as the inventor of calculus. History suggests that both of these men independently discovered the Fundamental Theorem of Calculus, which describes the relationship between derivatives and integrals.
The contributions of Leibniz and Newton have often been separated based on their area of concentration. Leibniz was primarily interested in examining methods for finding the area beneath a curve and extending these methods to the examination of volumes. This led him to detailed investigations of the integral concept.
Leibniz is also credited for creating a notation for the integral, ∫. The integral symbol looks like an elongated "S." Because finding the area under a curve requires "summing" rectangles, Leibniz used the integral sign to indicate the summing process. Leibniz is also credited for developing a notation for finding a derivative. This notation is of the form . Both of these symbols are still used in calculus today.
Newton was interested in the study of "fluxions." Fluxions refers to methods that are used to describe how things change over time. As discussed earlier, the motion of an object often changes over time and can be described using derivatives. Today, the study of fluxions is referred to as the study of calculus. Newton is also credited with finding many different applications of calculus to the physical world.
It is important to note that the ideas of Leibniz and Newton had built upon the ideas of many other scientists, including Kepler, Galileo, Cavalieri, Fermat, Descartes, Torricelli, Barrow, Gregory, and Huygens. Also, calculus continued to advance due to the efforts of the scientists who followed. These individuals included the Bernoulli brothers, L'Hôpital, Euler, Lagrange, Cauchy, Cantor, and Peano. In fact, the current formulation of the limit concept is credited to Louis Cauchy. Cauchy's definition of the limit concept appeared in a textbook in 1821, almost 100 years after the deaths of Leibniz and Newton.
Who Uses Calculus Today?
Calculus is used in a broad range of fields for a variety of purposes. Advancements have been made and continue to be made in the fields of medicine, research, and education that are supported by the methods of calculus. Everyone experiences the benefits of calculus in their daily lives. These benefits include the availability of television, the radio, the telephone, and the World Wide Web.
Calculus is also used in the design and construction of houses, buildings, bridges, and computers. A background in calculus is required in a number of different careers, including physics, chemistry, engineering, computer science, education, and business. Because calculus is important to so many different fields, it is considered to be an important subject of study for both high school and college students.
It is unlikely the Leibniz or Newton could have predicted the broad impact that their discovery would eventually have on the world around them.
see also Limit; Measurements, Irregular.
Barbara M. Moskal
Bibliography
Boyer, Carl. A History of Mathematics. New York: John Wiley & Sons, 1991.
———. "The History of Calculus." In Historical Topics for the Mathematics Classroom, 31st Yearbook, National Council of Teachers of Mathematics, 1969.
Internet Resources
O'Connor, J. J., and E. F. Robertson. "The Rise of Calculus." In MacTutor History of Mathematics Archive. St Andrews, Scotland: University of St Andrews, 1996. <http://www~groupsdcs.stand.ac.uk/~history/HistTopics/The_rise_of_calculus.html>.
Walker, Richard. "Highlights in the History of Calculus." In Branches of Mathematics. Mansfield, PA: Mansfield University, 2000. <http://www.mnsfld.edu/~rwalker/Calculus.html>.
WHO WAS FIRST?
Gottfried Wilhelm Leibniz began investigating calculus 10 years after Sir Isaac Newton and may not have been aware of Newton's efforts. Yet Leibniz published his results 20 years before Newton published. So although Newton discovered many of the concepts in calculus before Leibniz, Leibniz was the first to make his own work public.
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calculus
calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value. The English physicist Isaac Newton and the German mathematician G. W. Leibniz, working independently, developed the calculus during the 17th cent. The calculus and its basic tools of differentiation and integration serve as the foundation for the larger branch of mathematics known as analysis. The methods of calculus are essential to modern physics and to most other branches of modern science and engineering.
The Differential Calculus
The differential calculus arises from the study of the limit of a quotient, Δy/Δx, as the denominator Δx approaches zero, where x and y are variables. y may be expressed as some function of x, or f(x), and Δy and Δx represent corresponding increments, or changes, in y and x. The limit of Δy/Δx is called the derivative of y with respect to x and is indicated by dy/dx or D_{x}y:
The symbols dy and dx are called differentials (they are single symbols, not products), and the process of finding the derivative of y=f(x) is called differentiation. The derivative dy/dx=df(x)/dx is also denoted by y′, or f′(x). The derivative f′(x) is itself a function of x and may be differentiated, the result being termed the second derivative of y with respect to x and denoted by y″, f″(x), or d^{2}y/dx^{2}. This process can be continued to yield a third derivative, a fourth derivative, and so on. In practice formulas have been developed for finding the derivatives of all commonly encountered functions. For example, if y=x^{n}, then y′=nx^{n  1}, and if y=sin x, then y′=cos x (see trigonometry). In general, the derivative of y with respect to x expresses the rate of change in y for a change in x. In physical applications the independent variable (here x) is frequently time; e.g., if s=f(t) expresses the relationship between distance traveled, s, and time elapsed, t, then s′=f′(t) represents the rate of change of distance with time, i.e., the speed, or velocity.
Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative f′(t)=ds/dt, however, gives the velocity for any particular value of t, i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If y=f(x) is a realvalued function of a real variable, the ratio Δy/Δx=(y_{2}  y_{1})/(x_{2}  x_{1}) represents the slope of a straight line through the two points P (x_{1},y_{1}) and Q (x_{2},y_{2}) on the graph of the function. If P is taken closer to Q, then x_{1} will approach x_{2} and Δx will approach zero. In the limit where Δx approaches zero, the ratio becomes the derivative dy/dx=f′(x) and represents the slope of a line that touches the curve at the single point Q, i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.
The Integral Calculus
The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes. For example, consider the problem of determining the area under a given curve y=f(x) between two values of x, say a and b. Let the interval between a and b be divided into n subintervals, from a=x_{0} through x_{1}, x_{2}, x_{3}, … x_{i  1}, x_{i}, … , up to x_{n}=b. The width of a given subinterval is equal to the difference between the adjacent values of x, or Δx_{i}=x_{i}  x_{i  1}, where i designates the typical, or ith, subinterval. On each Δx_{i} a rectangle can be formed of width Δx_{i}, height y_{i}=f(x_{i}) (the value of the function corresponding to the value of x on the righthand side of the subinterval), and area ΔA_{i}=f(x_{i})Δx_{i}. In some cases, the rectangle may extend above the curve, while in other cases it may fail to include some of the area under the curve; however, if the areas of all these rectangles are added together, the sum will be an approximation of the area under the curve.
This approximation can be improved by increasing n, the number of subintervals, thus decreasing the widths of the Δx's and the amounts by which the ΔA's exceed or fall short of the actual area under the curve. In the limit where n approaches infinity (and the largest Δx approaches zero), the sum is equal to the area under the curve:
The last expression on the right is called the integral of f(x), and f(x) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.
An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if F(x) is a function whose derivative is f(x), then the area under the graph of y=f(x) between a and b is equal to F(b)  F(a). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols:
The function F(x), which is equal to the integral of f(x), is sometimes called an antiderivative of f(x), while the process of finding F(x) from f(x) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, a and b, are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function F(x) resulting from integration is determined only to within the addition of an arbitrary constant C, since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of f(x) is
The value of the constant C must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., x or t, but also functions of several variables. For example, if z=f(x,y) is a function of two independent variables, x and y, then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂z/∂x and ∂z/∂y or by D_{x}z and D_{y}z. Three different second derivatives are possible, ∂^{2}z/∂x^{2}, ∂^{2}z/∂y^{2}, and ∂^{2}z/∂x∂y=∂^{2}z/∂y∂x. Such derivatives are called partial derivatives. In any partial differentiation all independent variables other than the one being considered are treated as constants.
Bibliography
See R. Courant and F. John, Introduction to Calculus and Analysis, Vol. I (1965); M. Kline, Calculus: An Intuitive and Physical Approach (2 vol., 1967); G. B. Thomas and R. L. Finney, Calculus and Analytic Geometry (7th ed. 2 vol., 1988).
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Calculus
Calculus
Calculus is a field of mathematics that deals with rates of change and motion. Suppose that one nation fires a rocket carrying a bomb into the atmosphere, aimed at a second nation. The first nation must know exactly what path the rocket will follow if the attack is to be successful. And the second nation must know the same information if it is to protect itself against the attack. In this example, calculus is used by mathematicians in both nations to study the motion of the rocket.
Calculus was originally developed in the late 1600s by two great scientific minds, English physicist Isaac Newton (1642–1727) and German mathematician Gottfried Wilhelm Leibniz (1646–1716). Both scholars presented their ideas at about the same time, so credit for the invention of calculus must go to both. The debate over credit at the time, however, reached intense levels and sparked bad feelings between the two countries involved (Great Britain and Germany). Over the past 300 years, calculus has become an absolutely essential mathematical tool in every field of science, mathematics, and engineering.
To illustrate the basic principles of calculus, imagine that you are studying changes in population in your hometown over the past 100 years. As you graph the data you collected, you can see that population increased for a number of years, then decreased for a period of time before beginning a second increase. One question you might want to ask is what the rate of change in the population was at any given time, such as any given year. For example, was population increasing at the same rate in 1980 that it was in 1890? One way to answer that question is to locate two points on the curve. The rate of change for this part of the graph, then, is how steeply the curve rises between these two points.
Differential and integral calculus
Calculus can be subdivided into two general categories: differential and integral calculus. Differential calculus deals with problems of the type above, in which some mathematical function (such as population change) is known. From the graphical or mathematical representation of that function, the rate of change can be calculated.
The reverse process can also be performed. For example, it may be possible to find the rate of change for some function. From that rate of change, then, it may be possible to determine the original function itself. This field of mathematics is known as integral calculus.
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calculus
cal·cu·lus / ˈkalkyələs/ • n. 1. (pl. lus·es ) (also infinitesimal calculus) the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus. 2. (pl. lus·es) Math. & Logic a particular method or system of calculation or reasoning. 3. (pl. li / ˌlī; ˌlē/ ) Med. a concretion of minerals formed within the body, esp. in the kidney or gallbladder. ∎ another term for tartar.
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calculus
calculus Branch of mathematics dealing with continuously changing quantities. Differential calculus is used to find slopes of curves and rates of change of a given quantity with respect to another. Integral calculus is used to find the areas enclosed by curves. Gottfried Leibniz and Sir Isaac Newton independently discovered the fundamental theorem of calculus. This is ∫^{b}_{a} f(x)dx = g(b)−g(a), where g is any function whose derivative is the function f.
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calculus
calculus (kalkewlŭs) n. (pl. calculi)
1. a stone: a hard pebblelike mass formed within the body, particularly in the gall bladder (see gallstone) or anywhere in the urinary tract. Calculi may also occur in the ducts of the salivary glands.
2. a calcified deposit that forms on the surface of a tooth as it is covered with plaque.
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calculus
calculus stone in an animal body; †gen. (system of) calculation XVII; spec. in differential, integral (etc.) calculus XVIII. — L. calculus pebble, etc.
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calculus
calculus
•Callas, callous, callus, Dallas, Pallas, phallus
•Nablus • manless
•hapless, mapless
•atlas, fatless, hatless
•braless, parlous
•armless • artless
•jealous, zealous
•endless • legless • sexless • airless
•talus • bacillus • windlass • Nicklaus
•obelus • strobilus
•acidophilus, Theophilus
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•mimulus, stimulus
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Calculus
Calculus
Calculus is that branch of mathematics that deals with instantaneous rates of change of quantities (differentiation) and with the accumulation of quantities (integration). 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 bunjee 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 (1642–1727) and Gottfried Leibniz (1646–1716). Some of its essential concepts, however, had their beginnings in ancient Greece. In the fourth century BC, Eudoxus (408–355 BC) 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 (287?–212 BC) 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 BC until the beginning of the seventeenth century AD, when the work of Johannes Kepler (1571–1630), 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 volumes as 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, the determination of their instantaneous rates of change. A rate of change is an intuitively familiar concept: an object’s speed, for example, is the rate of change of its position. An object’s speed may be changing (as when a moving car is braked to a halt, or accelerated), and its speed at any given moment is the “instantaneous” rate of change of its position. 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 (Figure 1).
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 derivative 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) = x^{2} then f’(x) = 2x. Thus, if asked to integrate the function g(x) = 2x, it is apparent that 2xdx=x^{2} + 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 function. The equation f ’(x) = ax^{2} +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 xaxis 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) (Figure 2). The height of each rectangle is the value of the function at x, namely f (x). The width of each rectangle is Δx = (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 following sum:
Here, the Greek letter ζ (sigma) is used to indicate that the n products f(x_{k}) Δx corresponding to the n rectangles are to be summed. In the limit that n approaches infinity, Δx approaches 0, and the sum is exactly equal to the area. Since Δx approaches 0, it represents an infinitesimal change in the variable x, so the same notation used in defining the derivative is used to replace Δx with d. The product f(x) Δx 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
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.
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.
Calculus remains, and will remain, the fundamental mathematical language of all the sciences. It is used in biology, physics, social science, medicine, and every other field where mathematical functions are used to describe phenomena.
Resources
BOOKS
Larson, Ron, et al. Calculus with Analytic Geometry. Boston: Houghton Mifflin Company, 2005.
Sullivan, Michael. Precalculus. Upper Saddle River, NJ: Prentice Hall, 2004.
Verberg, Dale, et al. Calculus (9th ed.). Upper Saddle River, NJ: Prentice Hall, 2006.
PERIODICALS
Moore, A.W. “A Brief History of Infinity.” Scientific American 272 (1995): 112116.
J. R. Maddocks
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Calculus
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 xaxis 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 Δx = (ba)/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 Σ (sigma) is used to indicate that the n products f(xk) Δx corresponding to the n rectangles are to be summed). In the limit that n approaches infinity, Δx approaches 0, and the sum is exactly equal to the are. Since Δx approaches 0, it represents an infinitesimal change in the variable x, so the same notation used in defining the derivative is used to replace Δx with d. The product f(x) Δx 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: PWSKENT 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): 112116.
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.
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