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Topics in a Pre-CalculusCourse

To learn more about a topic listed below, click the topic name to go to thecorrespondingMathWorldclassroom page.

General

AsymptoteA line or curve that approaches a given curvearbitrarily closely.

Curve

A continuous map from a one-dimensionalspace to an n-dimensional space. Looselyspeaking, the word "curve" is often used tomean the function graph of a two- or three-

dimensional curve.

eterminant

!# A function that assigns a scalar to as$uare matri% !or, e$uivalently, its lineartransformation#. ! The value of this functionfor a particular matri%. The matri% has aninverse if and only if its determinant isnon'ero.

(arametric )$uations

A set of e$uations that together e%press a setof $uantities as e%plicit functions of a numberof independent variables, which are knownas parameters.

(laneA two-dimensional surface defined by lineare$uations.

(lane CurveA curve that lies in a single plane. A planecurve may be closed or open.

(olar CoordinatesA two-dimensional coordinate system inwhich points in two dimensions are given byan angle and a distance from the origin.

*ational +unctionA function that can be written as the $uotientof two polynomials.

*eflectionThe operation of e%changing all points of amathematical obect with their mirror images.

*otationThe turning of an obect or coordinate systemabout a fi%ed point.

*otation atri% A matri% that corresponds to the lineartransformation of a rotation.

calar

A value !such as a measurement# that hasonly magnitude but not direction. Thiscontrasts with a vector, which has directionas well as magnitude.

pherical CoordinatesA coordinate system in which points in three-dimensional space are given by two anglesand a distance from the origin.

Tangent LineA line that touches but does not cross a curveat a given point.

TranslationA transformation consisting of a constant shiftwith no rotation or stretching.

Complex Numbers

Comple% Conugate/The result of changing the sign of theimaginary part of a comple% number.

Comple% 0umber/A number consisting of a real part and animaginary part. A comple% number is anelement of the comple% plane.

Comple% (lane/The set of all comple% numbers. 1ust as allreal numbers can be imagined as sitting on aline, all comple% numbers can be thought ofas points in a plane.

i/The symbol used to denote the s$uare root of-.

2maginary 0umber/A multiple of the imaginary unit i!the s$uareroot of -#.

Conic Sections

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Conic ection/

The nondegenerate curves generated by theintersections of a plane with one or twonappes of a cone. A conic section can alsobe reali'ed as the 'ero set of a $uadratice$uation in two variables.

)llipse/ A conic section with eccentricity less thanone. 2t resembles a s$uashed circle.

3yperbola/A conic section with eccentricity greater thanone. A hyperbola consists of two separatebranches.

Locus/

The set of all points !usually forming a curveor surface# satisfying some condition. +ore%ample, the locus of points in a plane thatare e$uidistant from a given point is a circle.

(arabola/A conic section with eccentricity e$ual to one.(arabolas appear as the graphs of $uadratic

e$uations and the traectories of proectiles.

Exponents and Logarithms

e/The base of the natural logarithm,appro%imately e$ual to &.45. After pi,eisthe most important constant in mathematics.

)%ponential +unction/The function consisting of the base of thenatural logarithm etaken to the power of thevariable.

Logarithm/The power to which a number !called thebase# is raised to produce a given number6e.g., the logarithm of 77 to the base 7 is &.

0atural Logarithm/ The logarithm having base e.

Functions

omain/!# The set of values for which a function isdefined. ! 2n topology, a connected, openset.

+unction/

A relation that uni$uely associates membersof one set with members of another set. Theterm "function" is sometimes implicitlyunderstood to mean continuous function,linear function, or function into the comple%numbers.

2nverse +unction/+or a function f, the function f-for which f!f-!x## 8 xfor any x.

*ange/

!# The set of all values that a function cantake. ! The difference between theminimum and the ma%imum values of a dataset.

Vectors

Cross (roduct/A product of two vectors that results in a

vector perpendicular to both.

ot (roduct/A product of two vectors, which results in ascalar.

0ormal 9ector/ A vector perpendicular to a surface.

9ector/!# A mathematical entity that has bothmagnitude !which can be 'ero# and direction.! An element of a vector space.

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