Glossary

Glossary

A

Acceleration [p.490]Theaccelerationof a particle is defined as the rate of change of its velocity with respect to time.

Acceleration, average [p.490]Theaverage accelerationof a particle for the time interval [t1,t2] is defined byv2−v1

t2−t1, wherev2is the velocity at timet2

andv1is the velocity at timet1.

Acceleration, instantaneous [p.542]a= dv dt Addition rule (for choices)[p.325]To determine the number of choices from disjoint alternatives, simply add up the available number for each alternative.

Addition rule for probability[p.279]the probability ofAorBor both occurring, given by the rule:

Pr(A∪B)=Pr(A)+Pr(B)−Pr(A∩B)

Amplitude of circular functions[p.430]The distance between the mean position and the maximum position is called theamplitude. The graph ofy=sinxhas an amplitude of 1. Antiderivative[p.601]To find the general antiderivativeF

IfF(x)= f(x), f(x)d x=F(x)+c,wherecis an arbitrary real number.

Arrangements[p.328]The number of arrangements ofnobjects in groups of sizeris given by:

n!

(n−r)!=n×(n−1)×(n−2). . .(n−r+1)

Average speed [p.541]

average speed= total distance travelled total time taken Average velocity[p.541]

average velocity= change in displacement total time taken

B

Binomial distribution [p.353]The probability of observingXsuccesses innbinomial trials each with probability of successp, given by

Pr(X=x)=

n x

(p)x_(1−p)n−x_{x=0,1, . . . ,n,}

where

n x

= n! x!(n−x)!

Binomial experiment [p.353]

The experiment consists of a number,n, of identical trials.

Each trial results in one of two outcomes, which are usually designated either asuccess,Sor a failure,F.

The probability of success on a single trial,p, is constant for all trials.

The trials are independent (so that the outcome on any trial is not affected by the outcome of any previous trial).

C

Chain rule [p.585]Thechain ruleis often used to differentiate some more complicated functions by transforming the original function into two simpler functions.

E.g.f(x) is transformed toh(x) andg(u), which are ‘chained’ together as

x−→h _u−→g _y

Using Leibniz notation, thechain ruleis stated as

d y d x =

d y d x ·

du d y.

Circle, general equation [p.140]The general equation for a circle is (x−h)2_+(y−k)2_=r2_. Thecentreof the circle is the point (h,k) and the radiusisr.

Complement, A [p.279]is the set of points that are in the sample space (ε) but not inA

Compound event [p.265]an event for which there is more than one outcome from a random experiment such as observing an even number when a die is rolled Conditional probability [p.294]the probability of an eventAoccurring when it is known that some eventBhas occurred; written as Pr(A|B)

Constant function [p.157]A function

f:R→R,f(x)=ais called aconstant function. Continuous function [p.525]A functionfis defined as continuous at the pointx=aif the following three conditions are met:

f(x) is defined atx=a lim

x→af(x) exists lim

x→a f(x)= f(a)

Coordinates [p.25]a unique ordered pair of numbers that identifies a point on the coordinate plane; the first number in the ordered pair identifies the position with regard to thex-axis, while the second number identifies the position with respect to they-axis

Cosine and sine [p.420] cosineis defined as thex-coordinate of the pointPon the unit circle whereOP forms an angle of

radians with the positive ray of thex-axis

0

θ

1 x

–1

–1 1

cos θ

sin θ

P (θ) = (cos θ, sin θ) y

sineis defined as they-coordinate of the point Pon the unit circle whereOPforms an angle of

radian with the positive ray of thex-axis Cubic function [p.190]A third degree polynomial is called a cubic and is a functionf, with rule

f(x)=ax3+bx2+cx+d,a=0

D

Degree of polynomial [p.85]is given by the value of n, the highest power ofxwith a non-zero coefficient.

Dependent trials or events [p.350]the same as sampling without replacement

The probability of one event is influenced by the outcome of another event.

Derived function seegradient function Determinant of a 2×2 matrix [p.75]If

A=

a b c d

, det(A)=ad−cd.

Difference of perfect squares [p.90] (x+a)(x−a)=x2_−a2

Difference of two cubes [p.202] x3_−y3_=(x−y)(x2_{+x y+y}2₎

Difference of two squares [p.90] x2_−y2_{=(x−y)(x+y)}

Differentiation rule [p.584]The general result for any non-zero real power gives:

Forf(x)=xa, f(x)=axa−1,forx>0 anda∈R.

Differentiation rules[p.510]The general rule of the derived function of f(x)=xn_{,n=1,2,3, . . .} For f(x)=xn_{, f(x)=nx}n−1_{,n=1,2,3, . . .} For f(x)=1,f(x)=0

For f(x)=g(x)+h(x), fx=g(x)+h(x) Forg(x)=k f(x),g(x)=k f(x)

Dilation from thex-axis [p.173]

In general, a dilation ofaunits from thex-axis is described by the rule (x,y)→(x,ay). In general, the curve with equationy= f(x) is mapped to the curve with equationy=a f(x) by the transformation with rule

(x,y)→(x,ay).

Dilation from they-axis [p.174]

In general, a dilation ofaunits from they-axis is described by the rule (x,y)→(ax,y) In general, the curve with equationy= f(x) is mapped to the curve with equationy= f

_x a

by the transformation with rule (x,y)→(ax,y).

Dimension of a matrix [p.61]The size, or dimension, of the matrix is described by specifying the number of rows (horizontal lines) and columns (vertical lines) that occur in the matrix. A matrix with mrows andncolumns is said to be anm×nmatrix.

Discontinuity at a point[p.525]The function is said to be discontinuous at a point if it is not continuous at that point. We say that a function is continuous everywhere if it is continuous for all real numbers.

Discrete random variable [p.344]a random variableXwhich can take only a countable number of values, usually whole numbers

Discriminant,, of a quadratic [p.109]the expressionb2_{−4ac, which is part of the quadratic} formula. For the quadratic equation

0=ax2_+bx+c:

Ifb2_{−4ac>0 there are two solutions.} Ifb2_{−4ac=0 there is one solution.} Ifb2_{−4ac<0 there are no real solutions.}

Disjoint sets [p.149]If setsAandBhave no elements in common, we sayAandB, aredisjoint and writeA∩B= ∅.

Distance between given points [p.48]The distance between the given pointsA(x1,y1) andB(x2,y2)

A B=(x2−x1)2+(y2−y1)2

E

Element[p.149]a member of or an object in a set

element of a set [p.149]Ifxis an element of a setAwe writex∈ A.

not an element of [p.149]The notationx∈/ A meansxisnotan element ofA.

Empirical probability[p.269]the probability assigned to an event on the basis of repeated experimentation; i.e.:

Pr (A)≈ number of times the event occurs number of trials for largen

Empty set,∅ [p.149]the set that has no members Exponential function (or index function) [p.382] the functionf(x)=kax_{, wherekis a non-zero} constant and the baseais a positive real number other than 1.

F

Factor [p.89]a number or expression that divides another number or expression without remainder

Factor theorem [p.200]Ifax+bis a factor ofP(x)

thenP

−b a

=0. Conversely, ifP

−b a

=0 thenax+bis a factor ofP.

Factorise [p.89]to express as a product of factors Formula[p.16]an equation containing symbols that states a relationship between two or more quantities

A=lw(area=length×width) is an example of a formula. The value ofA, called the subject of the formula, can be found by substituting in given values oflandw.

Function[p.156]a relation for which for each x-value of an ordered pair there is a uniquey-value of the ordered pair

This means if (a,b) and (a,c) are ordered pairs of a function thenb=c.

Function, many-to-one [p.161]More than one x-value maps onto the samey-value, e.g.y=x2_.

Function, one-to-one[p.161]Eachx-value maps onto a uniquey-value. e.g.y=2x.

Function, vertical line test [p.156]used to identify if a relation is afunction

If a vertical line can be drawn anywhere on the graph and only ever intersects the graph a maximum of once, the relation is afunction.

Fundamental theorem of integral calculus [p.613]states that_abf(x)d x=G(b)−G(a),where Gis any antiderivative off, and_abf(x)d xis the definite integral fromatob

G

Gradient (m) of a line [p.26]Gradient: m= rise

run = y2−y1 x2−x1

where (x1,y1) and (x2,y2) are

coordinates of points on the line

Gradient function [p.505]The gradient or derived function is denoted byf, where

f:R→Randf(x)=lim h→0

f(x+h)− f(x) h

Gradient of a vertical line (parallel to they-axis) [p.27]The gradient of avertical (parallel to the

y-axis)line is undefined.

H

Horizontal line test [p.161]If a horizontal line can be drawn anywhere on the graph of a function and only ever intersects the graph a maximum of once, the function isone-to-one.

Hybrid functions [p.164]functions which have different rules for different subsets of the domain

I

Implied domain seemaximal domain

Independence [p.301]AandBare independent events if:

Pr(A∩B)=Pr(A)×Pr(B) or Pr(A|B)=Pr(A) or Pr(B|A)=Pr(B)

Independent trials [p.353]the same as sampling with replacement

Index laws [p.389]

To multiply two numbers in exponent form with the same base,addthe exponents:

am_×an_=am+n

To divide two numbers in exponent form with the same base,subtractthe exponents:

am_÷an_=am−n

To raise the power ofato another power, multiplythe exponents:

(am₎n _=am×n

Ifax_=ay_thenx=y.

For rational exponentsa1n =√na

Inequation [p.14]a mathematical statement that contains an inequality symbol rather than an equals sign; e.g. 2x+1<4

step 0. The elements of the initial state matrix may be numbers, percentages, or the results of an individual trial.

Integers [p.149]the elements of Z = {. . . ,−2,−1,0,1,2, . . .} Integration, general results [p.602]

xrd x= x r+1

r+1+c,r∈ Q\{−1}

f(x)+g(x)d x=

k f(x)d x=k

Intersection of two sets [p.149]Theintersectionof setsAandBand is denoted byA∩Bandx∈ A∩B if and only ifx∈ Aandx∈B.

Inverse function [p.168]Iffis a one-to-one function a functionf, the inverse function, f−1_{is defined by,}

f−1_{(x)=yif f(y)=x, forx∈ran f,y∈dom f}

Irrational numbers [p.149]The real numbers which are not rational, are calledirrational (e.g.and√2).

K

Karnaugh map [p.282]a probability table

L

Law of total probability [p.297]In the case of two events,AandB:

Pr(A)=Pr(A|B)Pr(B)+Pr(A|B)Pr(B)

Limits, properties of [p.522] lim

x→c(f(x)+g(x))=limx→cf(x)+limx→cg(x) i.e. the limit of the sum is the sum of the limits lim

x→c(k f(x))=kxlim→cf(x),wherekis a given number

lim

x→c(f(x)g(x))=xlim→c f(x) limx→cg(x)

i.e. the limit of the product is the product of the limit

lim x→c

f(x) g(x) =

lim x→cf(x) lim x→cg(x)

,provided lim

x→cg(x)=0 i.e. the limit of the quotient is the quotient of the limit

Linear equation [p.1]a polynomial equation of degree 1; e.g. 2x+1=0

Linear function [p.157]a function

f:R→R, f(x)=mx+c; e.g. f(x)=3x+1 Literal equation [p.4]an equation for the variablex in which the coefficients ofx, including the constants are pronumerals; e.g.ax+b=c

Logarithm [p.397]Ifa∈R+\{1}andx∈R, then the statementsax_{=nand log}

an=xare equivalent.

Logarithm laws [p.398]Laws of logarithms: (1) log_a(mn)=log_am+log_an

(2) log_a

_m

n

=log_am−log_an

(3) log_a

1 n

= −log_an

(4) loga(m p

)=plogam

M

Many-to-many [p.161]More than onex-value maps onto more than oney-value; e.g.x2_+y2_=4. Many-to-one [p.161]More than onex-value maps onto the samey-value; e.g.y=x2_.

Markov chain [p.310]A sequence where the probability of each possible outcome is conditional only on the immediately preceding outcome and the conditional probabilities for each possible outcome are the same on each occasion.

Matrices, addition and subtraction [p.66] Addition willonlybe defined for two matrices when they have the same number of rows and the same number of columns. In such cases the sum of two matrices is found by adding corresponding elements. For example,

1 0 0 2

+

0 −3 4 1

=

1 −3 4 3

Matrices, equal [p.63]Two matricesA,B, are equaland can be written asA=Bwhen:

they have the same number of rows and the same number of columns

they have the same number or element at corresponding positions.

Matrices, multiplication of [p.70]IfAis anm×n matrix andBis ann×rmatrix, then the productAB is them×rmatrix whose entries are determined as follows.

To find the entry in rowiand columnjofAB single out rowiin matrixAand columnjin matrixB.

Multiply the corresponding entries from the row and column and then add up the resulting products.

Note:The productABis only defined if the number of columns ofAis the same as the number of rows ofB. Matrix multiplication by a scalar [p.66]It is useful to definemultiplication of a matrix by a real number. IfAis anm×nmatrix, andkis a real number, thenkAis anm×nmatrix whose elements arektimes the corresponding elements ofA. For example,

3

2 −2 0 1

=

Matrix, inverse of a square matrix[p.75]Bis said to be theinverseofAwhenAB=BA=I. The inverse of a square matrixAis denoted byA−1_{. The} inverse is unique. It does not exist for every square matrix.

Matrix, multiplicative identity[p.74]For square matrices of a given dimension (e.g. 2×2) a multiplicative identityIexists.

For example, for 2×2 matricesI=

1 0 0 1

.

AI=IA=A, and this result holds for any square matrix multiplied by the appropriate multiplicative identity.

Matrix, regular [p.75]A square matrix is said to beregularif its inverse exists.

Matrix, singular [p.75]Square matrices that do not have an inverse are calledsingularmatrices.

Matrix, square[p.74]A matrix with the same number of rows and columns is called a square matrix.

Matrix, zero [p.66]Them×nmatrix with all elements equal to zero is called thezero matrix. Maximal or implied domain [p.154]When the rule for a relation is written and no domain is stipulated then it is understood that the domain taken is the largest for which the rule has meaning.

Midpoint of a line segment [p.49]LetP(x,y) be the midpoint of the line segment joiningA(x1,y1) andB(x2,y2).

x= x1+x2

2 and y= y1+y2

2

Multiplication rule (for choices) [p.326]When sequential choices are involved, the total number of possibilities is found by multiplying the number of options at each successive stage.

Multiplication rule for probability:[p.296]the rule to determine the probability of eventsAandB occurring:

Pr(A∩B)=Pr(A|B)×Pr(B)

Multi-stage experiment[p.266]experiments that could be considered to take place in more than one stage; e.g. tossing two coins

Mutually exclusive[p.279]Two sets are said to be mutually exclusiveif they have no elements in common.

N

nFactorial,n![p.329]Denotedn! (and readn factorial) this is an abbreviation for the product of all the integers fromnto 1:

n!=n×(n−1)×(n−2)×(n−3)×. . .×2×1

Natural numbers [p.149]the elements of N= {1,2,3,4, . . .}

n_C

r (see also selection)[p.333]the number of

combinations ofnobjects in groups of sizer: n

Cr = n! (n−r)!r!

Normal, equation of [p.536]LetPbe the point on the curvey= f(x) with coordinates (x1,y1). Then, if fis differentiable forx=x1,the equation of the normal at (x1,y1) is

y−y1= − 1 f(x1)

(x−x1)

O

One-to-many [p.161]Onex-value maps onto more than oney-value; e.g.y= ±√x.

Ordered pair [p.25]Anordered pair, denoted (x,y), is a pair of elementsxandyin whichxis considered to be the first element andythe second.

P

Period of a function [p.430]The period of a functionfwith domainRis the smallest positive numberasuch that f(x+a)= f(x) forainR. For example, the period of the sine function is 2, as sin(x+2)=sinx.

Polynomial [p.85]A polynomial has a rule of the type

y=anxn+an−1xn−1+. . .+a1x+a0, (n∈N)

wherea0,a1, . . . ,anare numbers called coefficients.

Probability [p.264]A numerical value assigned to the likelihood of an event occurring.

If the eventAis impossible then Pr(A)=0, and if eventAis certain then Pr(A)=1, otherwise 0<Pr(A)<1.

Probability distribution [p.344]Denotedp(x) or Pr(X=x), this is a function which assigns probabilities to each value ofX. It can be represented by a rule, a table or a graph, and must give a probabilityp(x) for every valuexthatX can take.

Q

Quadratic formula [p.104]x= −b± √

b2−_4ac 2a is the solution of the quadratic equation 0=ax2_+bx+c

Quadratic relation [p.85]A quadratic is defined by the general ruley=ax2_{+bx+c, wherea, bandc} are constants anda=0.

Quadratic relation (turning point form) [p.99] turning pointform of a quadratic relation:

y=a(x−h)2_+k

Quartic function [p.191]a fourth degree polynomial is a functionf, with rule

f(x)=ax4_+bx3_+cx2_{+d x+e,a=0}

R

=the set of real numbers excluding 0

R2 _[p.171]R2_{= {(x,y):x,y∈ R}; i.e.R}2_{is the set} of all ordered pairs of real numbers

Radian [p.418]One radian (written 1c_{) is the angle} subtendedat the centre of the unit circle by an arc of length 1 unit.

Random experiment [p.265]an experiment, such as the rolling of a die, in which the outcome of a single trial is uncertain but observable

Random number tables [p.363]tables which contain the digits 0,1, . . . ,9 in random order, and which can be used to generate random sequences of numbers

Random variable [p.343]a variable that takes its value from the outcome of a random experiment; e.g. the number of heads observed which a coin is tossed three times

Range [p.151]the set of all the second elements of ordered pairs of a relation

Rational numbers [p.149]numbers of the form p q withpandqintegers,q=0

Rectangular hyperbola [p.133]The basic

rectangular hyperbola has equationy= 1 x. Reflection in thex-axis [p.175]In general

A reflection in thex-axis is described by the rule (x,y)→(x,−y).

The curve with equationy= f(x) is mapped to the curve with equationy= −f(x) by the transformation with rule (x,y)→(x,−y).

Reflection in they-axis [p.175]In general

A reflection in they-axis is described by the rule (x,y)→(−x,y).

The curve with equationy= f(x) is mapped to the curve with equationy= f(−x) by the transformation with rule (x,y)→(−x,y). Relation[p.151]a set of ordered pairs; e.g. {(x,y):y=x2_}

Remainder theorem [p.199]When the polynomial

P(x) is divided byax+bthe remainder isP

−b a

.

S

Sample space [p.265]the set of possible outcomes for the experiment, sometimes denotedε

Sampling with replacement [p.352]the process of selecting individual objects sequentially from a group of objects, and replacing the selected object, so that the probability of obtaining a particular object does not change with each successive selection

Sampling without replacement [p.349]the process of selecting individual objects sequentially from a group of objects, and not replacing the selected object, so that the probability of obtaining a particular object changes with each successive selection

Selections [p.333]The number of combinations of nobjects in groups of sizeris

n_C r =

n×(n−1)×(n−2). . .(n−r+1) r!

= n!

r!(n−r)!

Set difference [p.149]Theset differenceof two setsAandB:

A\B= {x:x∈ A,x∈/ B}

Simple event [p.265]a single outcome from a random experiment; e.g. observing a 2 when a die is rolled

Simulation [p.360]the process of finding an approximate solution to a probability problem by repeated trials using a simulation model

Simulation model [p.360]A simple model which is analogous to a real-world situation. For example, the outcomes from a toss of a coin (head, tail) could be used as a simulation model for the sex of a child (male, female) under the assumption that in both situations the probabilities are 0.5 for each outcome. Simultaneous equations [p.9]equations of two or more lines or curves in a Cartesian plane, the solution of which is the point of intersection of the lines or curves

State matrix [p.311]A matrix of dimensionm×1, denotedSn, which gives information about the Markov chain at stepn. The elements of the state matrix may be numbers, percentages, or probabilities.

Stationary point[p.547]A point with coordinates (a,g(a)) on a curvey=g(x) is said to be a stationary point ifg(a)=0.

Straight line, equation of given two points

[p.32]y−y1=m(x−x1), wherem=

y2−y1

x2−x1 Straight line, equation of, gradient–intercept form

[p.28]The general equation of a straight line is

y=mx+c, wheremis the gradient of the line. This form, expressing the relation in terms ofy, is called thegradient–intercept form.

Straight lines, perpendicular[p.45]If two straight lines are perpendicular, the product of their gradients is−1. Conversely, if the product of the gradients of two lines is−1 then the two lines are perpendicular.

Subjective probability [p.269]the probability assigned to an event on the basis of prior experience

Subset [p.149]A setBis called a subset of setA,if andonly ifx∈Bimpliesx∈A.

To indicate thatBis a subset ofA, we writeB⊆A. Sum of two cubes [p.202]

x3_+y3_=(x+y)(x2_{−x y+y}2₎

T

If a tangent to the unit circle atAis drawn, then the

ycoordinate ofC, the point of intersection of the extension ofOPand the tangent is called tangent(abbreviated to tan).

θ

x A B C (1, y)

D

1

–1

–1 0 _{cos θ} 1

tan θ sin θ

P(θ) y

Tangent, equation of [p.535]LetPbe the point on the curvey= f(x) with coordinates (x1,y1). Then, if

fis differentiable forx=x1, the equation of the tangent at (x1,y1) is given by

(y−y1)= f(x1)(x−x1).

Transition matrix [p.310]A matrix of conditional probabilities, denoted,T, which for a Markov chain that hasmoutcomes or states is a square matrix of dimensionm×m.

Translation [p.171]A translation ofhunits in the positive direction of thex-axis andkunits in the positive direction of they-axis is described by the rule (x,y)→(x+h,y+k), wherehand

kare positive numbers. In general, the curve of the image of the curve with equationy= f(x) is

y−k= f(x−h).

Tree diagram [p.297]a diagram representing the outcomes of a multi-stage experiment

U

Union of sets [p.149]The union of setsAandB, writtenA∪B, is the set of elements which are either inAor inB. This does not exclude objects which are elements of bothAandB.

V

Venn diagram [p.279]a diagram showing sets and the relationships between sets