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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
See identity function.
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 forms
Such 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.
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