Calculus is the branch of mathematical analysis concerned with the rates of change of continuous functions as their arguments change. Two men are now credited with discovering calculus, Sir Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany. For almost a century, development of the subject was inhibited by a bitter controversy over priority between supporters of Newton and those of Leibniz.
A basic concept of calculus is limit, an idea applied by the early Greeks in geometry. Archimedes inscribed equilateral polygons in a circle. Upon increasing the number of sides, the areas of the polygons (which he could calculate) approach the area of the circle as a limit. Using this result together with a similar idea involving circumscribed polygons, he was able to find the area of the circle as , in which is the radius of the circle and is a constant that has a value between and .
The area of an irregularly shaped plate also can be found by subdividing it into rectangles of equal width. If the number of rectangles is made larger and larger, the sum of their areas (found by multiplying base by height) approaches the required area as a limit. The same procedure can be used to find volumes of spheres, cones, and other solid objects. The beauty and importance of calculus is that it provides a systematic way for the exact calculation of many areas, volumes, and other quantities that were beyond the methods of the early Greeks.
Newton's discovery of calculus, legend says, may very well have been inspired
by an apple falling from a tree. As an apple falls, it moves faster and faster;
that is, it has not only a velocity but an acceleration. Newton expressed
this mathematically by supposing that at any stage of its motion the apple
drops a small additional distance
during a brief additional time interval
. Then the velocity is very nearly equal to the distance
divided by the time
i.e.,
. The exact velocity v would be the limit of
as
gets closer and closer to zero or, as we say, approaches zero. That is,
The quantity is called the derivative of s with respect to , or the rate of change of with respect to . It is possible to think of and as numbers whose ratio is equal to ; is called the differential of , and the differential of .
Just as velocity is the rate of change, or derivative, of the distance
with respect to time, so the acceleration is the rate of change, or derivative,
of the velocity with respect to time. Therefore a, the acceleration, would
be
To find derivatives of with respect to , the dependence of on must be known; in other words, must be expressed as a function of . Usually this functional dependence is stated as a formula relating and . That part of calculus dealing with derivatives is called differential calculus.
Given
as a function of
, the derivative (that is,
) of
can be found. Conversely, if
is known it is possible to work backward to get
. This process of finding what is called the anti-derivative of
is begun by rewriting the equation
as
. The quantity
is here regarded as the anti-differential of
, denoted by a special symbol called an integral sign:
The last equation specifies the integral of with respect to . That part of calculus dealing with integrals is called integral calculus. Applications of integral calculus involve finding the limit of a sum of many small quantities, such as the rectangular slices of an irregular plane figure.