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# 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 three-dimensional objects, it becomes clear that the integral is also a useful tool for determining the volume of a three-dimensional 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 (16461716) and Sir Isaac Newton (16421727).

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.

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.st-and.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, Δyx, 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 Δyx is called the derivative of y with respect to x and is indicated by dy/dx or Dxy: 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 d2y/dx2. 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=xn, then y′=nxn - 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 real-valued function of a real variable, the ratio Δyx=(y2 - y1)/(x2 - x1) represents the slope of a straight line through the two points P (x1,y1) and Q (x2,y2) on the graph of the function. If P is taken closer to Q, then x1 will approach x2 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=x0 through x1, x2, x3, … xi - 1, xi, … , up to xn=b. The width of a given subinterval is equal to the difference between the adjacent values of x, or Δxi=xi - xi - 1, where i designates the typical, or ith, subinterval. On each Δxi a rectangle can be formed of width Δxi, height yi=f(xi) (the value of the function corresponding to the value of x on the right-hand side of the subinterval), and area ΔAi=f(xixi. 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 Dxz and Dyz. Three different second derivatives are possible, ∂2z/∂x2, ∂2z/∂y2, and ∂2z/∂xy=∂2z/∂yx. 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 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 (16421727) and German mathematician Gottfried Wilhelm Leibniz (16461716). 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 ∫ba f(x)dx = g(b)−g(a), where g is any function whose derivative is the function f.

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## calculus

calculus (kal-kew-lŭs) n. (pl. calculi)
1. a stone: a hard pebble-like 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

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