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Calculus, branch of mathematics concerned with rates of change, gradients of curves, maximum and minimum values of functions, and the calculation of lengths, areas, and volumes. It is widely used, especially in science and engineering, wherever continuously varying quantities occur.  

Calculus derives from ancient Greek geometry. Democritus calculated the volumes of pyramids and cones, probably by regarding them as consisting of infinitely many cross-sections of infinitesimal (infinitely small) thickness, and Eudoxus and Archimedes used the “method of exhaustion�?, finding the area of a circle by approximating it arbitrarily closely with inscribed polygons. However, difficulties with irrational numbers and the paradoxes of Zeno prevented a systematic theory developing. In the early 17th century Cavalieri and Torricelli extended the use of infinitesimals, while Descartes and Fermat used algebra for determining areas and tangents (integration and differentiation, in modern terms). Fermat and Barrow knew these two processes were closely related, and Newton (in the 1660s) and Leibniz (in the 1670s) proved the Fundamental Theorem of Calculus, that they are mutually inverse. Newton's discoveries, motivated by his theory of gravitation, preceded Leibniz's, but his delays in publishing them caused bitter priority disputes, and Leibniz's notation was eventually adopted.  

The 18th century saw widespread applications of calculus, but imprecise use of infinite and infinitesimal quantities and geometric intuition still caused confusion and controversy about its foundations, the philosopher Berkeley being a notable critic. The 19th-century analysts replaced these vague notions with firm foundations based on finite quantities: Bolzano and Cauchy defined limits and derivatives precisely, Cauchy and Riemann did likewise for integrals, and Dedekind and Weierstrass for real numbers. For instance, it was now understood that differentiable functions are continuous, and continuous functions integrable, but both converses fail. In the 20th century, non-standard analysis belatedly legitimized infinitesimals, while the development of computers increased the applicability of calculus.  

Differential calculus is concerned with rates of change. Suppose that two variables x and y are related by an equation y = f(x) for some function f, indicating how the value of y depends on the value of x. For instance, x could represent time, and y the distance travelled by some moving object at time x. A small change h in x, from a value x0 to x 0 + h, induces a change k in y from y 0 = f(x0) to y0 + k = f(x0 + h); thus k = f( x0 + h) - f(x0), and the ratio k/h represents the average rate of change of y as x increases 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 gradient of the line AB through the points A = (x 0,y0) and B = (x0 + h,y 0 + k) on this curve; this is shown in Figure 1, where h = AC and k = CB, so k/h is the tangent of the angle BAC.  

If h approaches 0, with x0 fixed, then k/h approaches the instantaneous rate of change of y at x0; geometrically, B approaches A along the graph of y = f(x), and the line AB approaches the tangent AT to the graph at A, so k/h approaches the gradient of the tangent (and hence of the curve) at A. We therefore define the derivative f′(x0) of the function y = f(x) at x0 to be the value (or limit) which k/h approaches as h approaches 0, written:

This represents both the rate of change of y and the gradient of the graph at A. When x is time and y is distance, for example, the derivative represents instantaneous velocity. Positive, negative, or zero values of f′(x0) respectively indicate that f(x) is increasing, decreasing, or stationary at x 0. The derivative is a new function f′(x) of x, sometimes denoted by dy/dx, df/dx or Df. For example, let y = f(x) = x2, so the graph is a parabola. Then

so k/h = 2x0 + h, which approaches 2x 0 as h→ 0. Thus the gradient when x = x0 is 2x0, and the derivative of f(x) = x2 is f′(x) = 2 x. Similarly xm has derivative mxm-1 for each fixed m. The derivatives of all commonly occurring functions are known: see the table for some examples.  

Using these rules, quite complicated functions can be differentiated: for instance x2 and x5 have derivatives 2 x and 5x4, so the function 3x2 - 4x5 has derivative (3x2 - 4x 5)′ = (3x2)′ - (4x5)′ = 3.( x2)′ - 4.(x5)′ = 3.(2x) - 4.(5 x4) = 6x - 20x4. More generally, any polynomial f(x) = a0 + a1x + ... + anxn has derivative f′(x) = a1 + 2a2 x + ... + nanxn-1; in particular, constant functions have derivative 0.

If y = u(z) and z = v(x), so that y depends on z and z depends on x, then y = u(v(x)), so y depends on x, written y = f(x) where f is the composition of u and v; the chain rule states that dy/dx = (dy/dz).(dz/dx), or equivalently f′(x) = u′(v(x)). v′(x). For instance, if y = ez where e = 2.718 ... is the exponential constant, and z = ax where a is any constant, then y = eax; now dy/dz = ez (see table) and dz/dx = a, so dy/dx = aeax.

Many problems can be formulated and solved using derivatives. For example, let y be the amount present in a sample of radioactive material at time x. According to theory and observation, the sample decays at a rate proportional to the amount remaining, that is, dy/dx = ay for some negative constant a. To find y in terms of x, we therefore need a function y = f(x) such that dy/dx = ay for all x. The most general such function is y = ce ax where c is a constant. Since e0 = 1 we have y = c when x = 0, so c is the initial amount present (at time x = 0). Since a<0 we have eax→ 0 as x increases, so y→ 0, confirming that the sample gradually decays to nothing. This is an example of exponential decay, shown in figure 3a. If a is a positive constant, we obtain the same solution y = ceax, but as time progresses y now increases rapidly (since eax does when a>0); this is exponential growth, shown in figure 3b and observed in nuclear explosions and certain animal communities, where growth-rate is proportional to population.

 

Integral calculus involves the inverse process to differentiation, called integration. Given a function f, we seek a function F with derivative F′ = f; this is an integral or antiderivative of f, written F(x) = ∫ f(x)dx or simply F = ∫f dx (a notation we will explain later). Tables of derivatives can be used for integration: thus x2 has derivative 2x, so 2x has x2 as an integral. If F is any integral of f, the most general integral of f is F + c, where c is an arbitrary constant called the constant of integration; this is because a constant has derivative 0, so ( F + c)′ = F′ + c′ = f + 0 = f. Thus ∫2xdx = x2 + c, for instance.

 

The basic rules for integrating compound functions resemble those for differentiation. The integral of a sum or difference is the sum or difference of their integrals, and likewise for multiplication by a constant. Thus x = y.2x has integral yx2, and similarly ∫xm dx = xm+1/(m + 1) for any m≠ -1. (We exclude m = -1 to avoid dividing by 0; the natural logarithm ln|x| is an integral of x-1 = 1/x for any x≠ 0.) Integration is generally harder than differentiation, but many of the more familiar functions can be integrated by these and other rules (see the table).