BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry hyperbola – least upper bound axiom hyperbola   The locus of points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.Like the ellipse and parabola, the hyperbola is a conic section. Related article: Conics hypotenuse   On a right triangle, the side opposite the right angle. icosahedron   A polyhedron having twenty faces.The faces of a regular icosahedron are congruent, equilateral triangles.Cf. Platonic solid. identity function   A function that maps each domain element to itself. Also called the identity map. identity map   indeterminate form   A limit of an expression is said to be indeterminate, or in indeterminate form, if when evaluated directly it resolves to one of the formsSuch limits may often be evaluated by manipulating them algebraically before applying the limit, or, in the case of the first two indeterminate forms shown, by applying L'Hospital's Rule. Related article: Limits inf   Abbreviation of infimum. injection   An injective function, i.e., a function that is “one-to-one.” Equivalently, a function that maps exactly one element of its domain to each element of its range.Cf. surjection, bijection. injective   A function is injective, also called “one-to-one,” if to each element of the range at most one element of the domain is mapped by the function.Cf. surjective, bijective. integral   An antiderivative of a function. That is, if f(x) is a real-valued function, an antiderivative F(x) of f(x) has the property that the derivative of F with respect to x is f.The definite integral (called the Riemann integral) of a real-valued function f(x) from x = a to x = b is the limit of the Riemann sum:assuming this limit exists, where ci is in the i th subinterval of the partition of (a, b), and where a is the lower limit, and b is the upper limit, of the integral.Cf. fundamental theorem of calculus, Lebesgue integral. integral test   A test for the convergence of a series. See the related article for a complete description. Related article: Series integration   Obtaining an integral of a function. integration formulas    ARTICLE   See the article for a complete list of common integration formulas.Cf. integral. interval   The set of all real numbers lying between two given real numbers a and b. If both a and b are included in the set it is called a closed interval. If neither a nor b is included it is called an open interval. If only one of a or b is included the interval is called half-open (equivalently, half-closed). Intervals may be denoted using either interval notation or set notation, as follows: Although intervals are most commonly intervals on the real line, the definition carries over without modification to any totally ordered set. inverse function   If f is an injective (i.e., one-to-one) function, then f is said to be invertible, and its inverse is the function g satisfying g(f(x)) = f(g(x)) = x for every x in the domain of f. inverse image   Given a function f and a subset Y of the range of f, the inverse image (under f) of Y is the set of all x such that f(x) is in Y. Kepler’s Laws    ARTICLE   The laws of planetary motion discovered by Johannes Kepler. See the article for an exposition. laws of exponents   The following rules govern the behavior of exponents:Additionally, x0 = 1 for all x except 0, and 00 = 0 or is left undefined (i.e., it is an indeterminate form).Cf. rational exponent. laws of logarithms   The following rules governing the behavior of logarithms are easily derived, and very useful in calculations:log A + log B = log (A×B) log A – log B = log (A/B) log Ap = p × log A Students often confuse these rules, so it is worth memorizing them as “the sum of the logs is the log of the product – and not the product of the logs,” etc. least upper bound   An upper bound which is less than or equal to every other upper bound.Cf. greatest lower bound. least upper bound axiom   “Any subset of the real numbers which has an upper bound has a least upper bound.” This axiom, together with the field axioms, completely characterizes the set of real numbers.Cf. supremum. hyperbola – least upper bound axiom
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