Calculus is the branch of mathematics concerned with the study of rates of change, slopes of curves at given points, areas and volumes bounded by curves, and similar problems. Scientists apply calculus to numerous problems in physics, astronomy, mathematics, and engineering. In recent years calculus has also been applied to problems in business, the biological sciences, and the social sciences. The development of calculus in the 17th century made possible the solution of many problems that had been insoluble by the methods of arithmetic, algebra, and geometry. These problems include the determination of Newton’s three laws of motion (see Mechanics) and the theory of electromagnetism.
Calculus consists of two main branches: differential calculus and integral calculus. Differential calculus deals with the rate at which quantities change. Integral calculus develops methods for finding the areas enclosed by curved boundaries. In both branches two concepts are central: function and limit.
Many relationships in nature and in mathematics can be expressed by functions. For example, a car moving at a speed of 50 mph travels a distance that changes constantly, depending on how long the car has traveled. Both distance and time are variables, but because the distance covered depends on the time of travel, distance can be represented as a function of time. A coal mine grows hotter as one descends, and so temperature can be expressed as a function of depth. A mathematical curve takes on new values of y as the value of x changes. When the value of y is determined by the value of x, we say that “y is a function of x” and we write y = f(x). A function is a rule, or equation, that tells us how to compute the y values given the x values (or vice versa). But unlike in algebra where the variables are static, the variables in calculus are constantly changing.
A key characteristic of calculus is that its solutions involve the idea of a limit. If you start out with a whole pie and repeatedly give away half of what is left, the sum of the amounts given away can never exceed 1 (the whole pie). At the same time, no matter how much you give away, a small amount will remain. Thus, you can never give away 1 entire pie. The sum of the series of pieces given away—y, ‚, ˆ, w, and so forth—approaches but never reaches 1, and so 1 represents the limit. If we call the sum of the series of pieces S, then S is a function of the number of pieces (n) in the series, or S = f(n), and the limit of f(n) as n approaches infinity is 1. Solutions in integral calculus involve breaking irregular areas and volumes into ever-smaller parts, where the notion of limit proves useful. Sir Isaac Newton was the first to clarify the notion of a limit and apply it to calculus.
