The Differential Calculus

The differential calculus arises from the study of the limit of a quotient, Δy/Δx, as the denominator Δx approaches zero, where x and y are variables. y may be expressed as some function of x, or f(x), and Δy and Δx represent corresponding increments, or changes, in y and x. The limit of Δy/Δx is called the derivative of y with respect to x and is indicated by dy/dx or D_xy:The symbols dy and dx are called differentials (they are single symbols, not products), and the process of finding the derivative of y=f(x) is called differentiation. The derivative dy/dx=df(x)/dx is also denoted by y', or f'(x). The derivative f'(x) is itself a function of x and may be differentiated, the result being termed the second derivative of y with respect to x and denoted by y″, f″(x), or d²y/dx². This process can be continued to yield a third derivative, a fourth derivative, and so on. In practice formulas have been developed for finding the derivatives of all commonly encountered functions. For example, if y=xⁿ, then y'=nx^n - 1, and if y=sin x, then y'=cos x (see trigonometry). In general, the derivative of y with respect to x expresses the rate of change in y for a change in x. In physical applications the independent variable (here x) is frequently time; e.g., if s=f(t) expresses the relationship between distance traveled, s, and time elapsed, t, then s'=f'(t) represents the rate of change of distance with time, i.e., the speed, or velocity.

Everyday calculations of velocity usually divide the distance traveled by the total time elapsed, yielding the average velocity. The derivative f'(t)=ds/dt, however, gives the velocity for any particular value of t, i.e., the instantaneous velocity. Geometrically, the derivative is interpreted as the slope of the line tangent to a curve at a point. If y=f(x) is a real-valued function of a real variable, the ratio Δy/Δx=(y₂ - y₁)/(x₂ - x₁) represents the slope of a straight line through the two points P (x₁,y₁) and Q (x₂,y₂) on the graph of the function. If P is taken closer to Q, then x₁ will approach x₂ and Δx will approach zero. In the limit where Δx approaches zero, the ratio becomes the derivative dy/dx=f'(x) and represents the slope of a line that touches the curve at the single point Q, i.e., the tangent line. This property of the derivative yields many applications for the calculus, e.g., in the design of optical mirrors and lenses and the determination of projectile paths.

The Integral Calculus

The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes. For example, consider the problem of determining the area under a given curve y=f(x) between two values of x, say a and b. Let the interval between a and b be divided into n subintervals, from a=x₀ through x₁, x₂, x₃, … x_i - 1, x_i, … , up to x_n=b. The width of a given subinterval is equal to the difference between the adjacent values of x, or Δx_i=x_i - x_i - 1, where i designates the typical, or ith, subinterval. On each Δx_i a rectangle can be formed of width Δx_i, height y_i=f(x_i) (the value of the function corresponding to the value of x on the right-hand side of the subinterval), and area ΔA_i=f(x_i)Δx_i. In some cases, the rectangle may extend above the curve, while in other cases it may fail to include some of the area under the curve; however, if the areas of all these rectangles are added together, the sum will be an approximation of the area under the curve.

This approximation can be improved by increasing n, the number of subintervals, thus decreasing the widths of the Δx's and the amounts by which the ΔA's exceed or fall short of the actual area under the curve. In the limit where n approaches infinity (and the largest Δx approaches zero), the sum is equal to the area under the curve:The last expression on the right is called the integral of f(x), and f(x) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.

An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if F(x) is a function whose derivative is f(x), then the area under the graph of y=f(x) between a and b is equal to F(b) - F(a). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols:The function F(x), which is equal to the integral of f(x), is sometimes called an antiderivative of f(x), while the process of finding F(x) from f(x) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, a and b, are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function F(x) resulting from integration is determined only to within the addition of an arbitrary constant C, since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of f(x) isThe value of the constant C must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., x or t, but also functions of several variables. For example, if z=f(x,y) is a function of two independent variables, x and y, then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂z/∂x and ∂z/∂y or by D_xz and D_yz. Three different second derivatives are possible, ∂²z/∂x², ∂²z/∂y², and ∂²z/∂x∂y=∂²z/∂y∂x. Such derivatives are called partial derivatives. In any partial differentiation all independent variables other than the one being considered are treated as constants.

Formal system of propositions and their logical relationships. As opposed to the predicate calculus, the propositional calculus employs simple, unanalyzed propositions rather than predicates as its atomic units. Simple (atomic) propositions are denoted by lowercase Roman letters (e.g., p, q), and compound (molecular) propositions are formed using the standard symbols ∧ for “and,” ∨ for “or,” ⊃ for “if . . . then,” and ¬ for “not.” As a formal system, the propositional calculus is concerned with determining which formulas (compound proposition forms) are provable from the axioms. Valid inferences among propositions are reflected by the provable formulas, because (for any formulas A and B) A ⊃ B is provable if and only if B is a logical consequence of A. The propositional calculus is consistent in that there exists no formula A in it such that both A and ¬ A are provable. It is also complete in the sense that the addition of any unprovable formula as a new axiom would introduce a contradiction. Further, there exists an effective procedure for deciding whether a given formula is provable in the system. Seealso logic, predicate calculus, laws of thought.

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Mass of minerals and organic matter that may form in a kidney. Urine contains many salts in solution, and low fluid volume or high mineral concentration can cause these salts to precipitate and grow, forming stones. Large stones can block urine flow, be a focus for infection, or cause renal colic (painful spasms). They can obstruct the urinary system at various points. Treatment deals with any underlying problem (e.g., infection or obstruction), tries to dissolve stones with drugs or ultrasound (lithotripsy), or removes large ones surgically.

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Part of modern symbolic logic which systematically exhibits the logical relations between propositions involving quantifiers such as “all” and “some.” The predicate calculus usually builds on some form of the propositional calculus and introduces quantifiers, individual variables, and predicate letters. A sentence of the form “All F's are either G's or H's” is symbolically rendered as (∀x)[Fx ⊃ (Gx ∨ Hx)], and “Some F's are both G's and H's” is symbolically rendered as (∃x)[Fx ∧ (Gx ∧ Hx)]. Once conditions of truth and falsity for the basic types of propositions have been determined, the propositions formulable within the calculus are grouped into three mutually exclusive classes: (1) those that are true on every possible specification of the meaning of their predicate signs, such as “Everything is F or is not F”; (2) those false on every such specification, such as “Something is F and not F”; and (3) those true on some specifications and false on others, such as “Something is F and is G.” These are called, respectively, the valid, inconsistent, and contingent propositions. Certain valid proposition types may be selected as axioms or as the basis for rules of inference. There exist multiple complete axiomatizations of first-order (or lower) predicate calculus (“first-order” meaning that quantifiers bind individual variables but not variables ranging over predicates of individuals). Seealso logic.

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In calculus, the process of finding a function whose derivative is a given function. The term, sometimes used interchangeably with “antidifferentiation,” is indicated symbolically with the integral sign ∫. (The differential math.dmath.x usually follows to indicate math.x as the variable.) The basic rules of integration are: (1) ∫(math.f + math.g)math.dmath.x = ∫math.fmath.dmath.x + ∫math.gmath.dmath.x (where math.f and math.g are functions of the variable math.x), (2) ∫math.kmath.fmath.dmath.x = math.k∫math.fmath.dmath.x (math.k is a constant), and (3) (math.C is a constant). Note that any constant value may be added onto an indefinite integral without changing its derivative. Thus, the indefinite integral of 2math.x is math.x² + math.C, where math.C can be any real number. A definite integral is an indefinite integral evaluated over an interval. The result is not affected by the choice for the value of math.C. Seealso differentiation.

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International association of Latin American countries originally dedicated to improving its members' economic well-being through free trade. At its founding in 1960 LAFTA included Argentina, Brazil, Chile, Mexico, Paraguay, Peru, and Uruguay; by 1970 Ecuador, Colombia, Venezuela, and Bolivia had joined. The organization aimed to remove all trade barriers over 12 years, but its members' geographic and economic diversity made that goal impossible. LAFTA was superseded in 1980 by the LAIA, which established bilateral trading agreements between members, which were divided into three groups according to their level of economic development. Cuba was admitted with observer status in 1986, and it became a full member in 1999. Seealso Inter-American Development Bank.

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Mathematical process of finding the derivative of a function. Defined abstractly as a process involving limits, in practice it may be done using algebraic manipulations that rely on three basic formulas and four rules of operation. The formulas are: (1) the derivative of math.x^math.n is math.nmath.x^{math.n − 1}, (2) the derivative of sin math.x is cos math.x, and (3) the derivative of the exponential function math.e^math.x is itself. The rules are: (1) (math.amath.f + math.bmath.g)' = math.amath.f' + math.bmath.g', (2) (math.fmath.g)' = math.fmath.g' + math.gmath.f', (3) (math.f/math.g)' = (math.gmath.f' − math.fmath.g')/math.g², and (4) (math.f(math.g))' = math.f'(math.g)math.g', where math.a and math.b are constants, math.f and math.g are functions, and a prime (') indicates the derivative. The last formula is called the chain rule. The derivation and exploration of these formulas and rules is the subject of differential calculus. Seealso integration.

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Field of mathematics that analyzes aspects of change in processes or systems that can be modeled by functions. Through its two primary tools—the derivative and the integral—it allows precise calculation of rates of change and of the total amount of change in such a system. The derivative and the integral grew out of the idea of a limit, the logical extension of the concept of a function over smaller and smaller intervals. The relationship between differential calculus and integral calculus, known as the fundamental theorem of calculus, was discovered in the late 17th century independently by Isaac Newton and Gottfried Wilhelm Leibniz. Calculus was one of the major scientific breakthroughs of the modern era.

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