Differential and Integral Calculus

The calculation is derived from the ancient Greek geometry. Democritus calculated the volume of pyramids and cones, is believed to be formed by considering an infinite number of infinitesimal thick sections (infinitely small) and Eudoxus and Archimedes used the "method of exhaustion" to find the area of ​​a circle with the accuracy required using inscribed polygons. However, the difficulties in working with irrational numbers and the paradoxes of Zeno of Eleaimpidieron formulate a systematic theory of computation. In the seventeenth century, Francesco B. Torricelli yEvangelista Cavalieri expanded the use of infinitesimals, and Descartes and Pierre de Fermat used the algebra to find the area and tangents (integration and differentiation in modern terms). Fermat and Isaac Barrow were certain that both calculations were related, although it was Isaac Newton (c. 1660) and Gottfried W. Leibniz (to 1670) who demonstrated that they are inverse, which is called the fundamental theorem of calculus. Newton's discovery, based on his theory of gravity, was prior to that of Leibniz, but the delay in its publication still causes disputes over who was first. However, eventually adopted the notation of Leibniz.

In the eighteenth century greatly increased the number of applications of calculus, but the imprecise use of infinite and infinitesimal quantities and geometric intuition, causing confusion and controversy still on the merits. One of his most notable critics was the Irish philosopher George Berkeley. In the nineteenth century mathematical analysts such vagueness replaced by a solid foundation based on finite quantities: Bernhard Bolzano and Augustin Louis Cauchy accurately defined the limits and derivatives, Cauchy and Bernhard Riemann did the same with the integrals, and Julius Dedekind and Karl Weierstrass with real numbers. For example, we learned that differentiable functions are continuous and continuous functions are integrable, although the reciprocals are false. In the twentieth century, unconventional analysis, legitimized the use of infinitesimals. At the same time, the advent of computers or computer applications has increased the calculation.

III. Differential calculus

Differential calculus studies the increases in the variables. Two variables x and y are related by the equation y = f (x), where the function f expresses the dependence of the value and the values ​​of x. For example, x and y may be time the distance traveled by a moving object at time x. A small increase in x h, a value x0 to x0 + h, an increase in k and passing of y0 = f (x0) y0 + a k = f (x0 + h), so that k = f (x0 + h) - f (x0). The ratio k / h represents the average increase as the x varies from x0 to x0 + h. The graph of the function y = f (x) is a curve in the xy plane and k / h is the slope of the line AB between the points A = (x0, y0) and B = (x0 + h, y0 + k) on this curve, this is shown in Figure 1, where h = AC = CB k, so k / h is the tangent of the angle BAC.

If h tends to 0, for a x0 fixed, then k / h is close to instantaneous change and x0, geometrically, B approaches A along the curve y = f (x), and the line AB tends to the tangent to the curve, AT, at point A. Therefore, k / h tends towards the slope of the tangent (and hence of the curve) in A. Thus, we define the derivative f '(x0) of the function y = f (x) at x0 as the limit which takes k / h when h tends to zero, which is written:

This value represents the magnitude of the variation of y and the slope of the curve at A. When, for example, x is time and y is the distance, the derivative is the instantaneous velocity. Positive, negative and zero of f '(x0) indicate that f (x) increases, decreases or is stationary relative to x0. The derivative of a function is in turn another function f '(x) of x, which is sometimes written as dy / dx, df / dx or Df. For example, if y = f (x) = x2 (parabola), then

so that k / h = 2x0 + h, which tends to 2x0 when h tends to 0. The slope of the curve when x = x0 is thus 2x0, and the derivative of f (x) = x 2 is f '(x) = 2x. Similarly, the derivative of xm is mxm m-1 for a constant. The derivatives of the most common functions are well known (see table below with some examples).

To calculate the derivative of a function, we must take into account a few details: First, take a very small h (positive or negative), but always different from zero. Second, not every function f has a derivative at all x0, for k / h may not have a limit when h 0, for example, f (x) = | x | has no derivative at x0 = 0, for k / h is 1 or -1 according as h> 0 O <0, geometrically, the curve has a vertex (and therefore has no tangent) in A = (0.0). Third, although the notation dy / dx suggests the ratio of two numbers dy and dx (indicating infinitesimal changes in y and x) is actually a single number, the limit of k / h when both quantities tend to zero.