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calculus





The branch of mathematics that deals with (1) the rate of change of quantities (which can be interpreted as the slopes of curves), known as differential calculus, and (2) the length, area, and volume of objects, known as integral calculus. It can be seen as an extension of analytical geometry.

Calculus was one of the most important developments in mathematics and also in physics, much of which involves studying how quickly one quantity changes with respect to another. It is no coincidence that one of the founders of calculus was the brilliant English physicist Isaac Newton; another was Gottfried Leibniz. William Fogg Osgood (1864–1943) said: "The calculus is the greatest aid we have to the application of physical truth in the broadest sense of the word." Although students nowadays learn the differential calculus first, the integral calculus has older roots.





Differential calculus

Consider the function y = f(x). This may be plotted as a curve on a set of Cartesian coordinates. Assuming that the curve is not a straight line, tangents to it at different points will have different gradients. Two points close together on the curve, (x, y) and (x+δx, y + δy), where δx and δy mean very small distances in the x- and y-directions, will usually have tangents of similar, though not identical, gradients. The gradient of the line passing through these two points is given by
[(y + δy) - y]/[(x + δx) - x],
and is the same as that of a tangent to the curve somewhere between the two points. The smaller δx is, the closer the two gradients will be; and if δx is infinitely small, they will be identical. This limit as x right arrow 0 is the derivative
dy/dx or f '(x)
of the function and is given by (putting f(x) in place of y):
f'(x) = lim (δxright arrow0) [f(x + δx) - f(x)] / δx.
This formula will give the gradient of the curve for f(x) at any value of x. For example, in the parabola y = x 2 the derivative at x = a is given by
f '(a) = lim (xright arrowa) (x 2 - a2)/(x - a) = lim (xright arrowa) (x + a)(x - a)/(x - a) = lim (xright arrowa) (x + a) = 2a.
Hence we can say that f '(x) = 2x for f(x) = x 2. This process is termed differentiation. In general if f(x) = x 2 then
f '(x) = z.xz - 1, and the second derivative
f ''(x) or d 2x/dx 2,
the result of differentiating again, is (z - 1).z.x z - 2, and so on.


Integral calculus

The derivative of f (x) gives the instantaneous rate of change of f(x) for a particular value of x. Now, consider the plot of y = f '(x), and assume that f '(x) = f(x) = 0 when x = 0. A very thin strip with one vertical side the f '(x)-axis and the other the line joining the points (δx, 0) and (δx, f '(δx)), will have an area of approximately δx, f '(δx). A second thin strip drawn next to it will have an area of roughly δx, f '(2δx), and so on. The area between the curve and the x-axis from x = 0 to x = a will therefore be roughly δx.f '(δx) + δx.f '(2δx) + ... + δx.f '(a + δx) + δx.f '(a). If one plots on a different graph x against (area of strip 1), (area of strips 1 + 2), area of strips 1 + 2 + 3), etc., one finds that one is plotting a close approximation to y = f(x). This reverse of differentiation is called integration, and f (x) is the integral of f '(x). We find, too, that the integral of x z is
x z + 1 / (z + 1)
which is what one would expect. This is the indefinite integral of x z since we have not specified how much of the curve we wish to consider, and we must add a constant, c, since the derivative of any constant is 0. As integration is the sum of the areas of the strips described, we symbolize it by an extended S. For example, the integral from 0 to 1 of x 2 is written as
integral x 2.dx.

Related category

   • CALCULUS AND ANALYSIS