Gloss Ay

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9780521740531gsy.xml CUAU030-EVANS August 27, 2010 9:55

Glossary*

Aabsolute maximum: [p. 370] M is the absolutemaximum value of a continuous function f for aninterval [a, b] if f (x) ≤ M for all x ∈ [a, b].

absolute minimum: [p. 370] N is the absoluteminimum value of a continuous function f for aninterval [a, b] if f (x) ≥ N for all x ∈ [a, b].

absolute value function (or modulus function), |x|:[p. 20] |x| =

{x if x ≥ 0

−x if x < 0

acceleration: [p. 380] the rate of change of aparticle’s velocity with respect to time.

acceleration, average: [MM1&2] for the time

interval [t1, t2], is defined byv2 − v1

t2 − t1where v2 is the

velocity at time t2 and v1 is the velocity at time t1

acceleration, instantaneous: [MM1&2] a = dv

dtaddition rule (for probability): [p. 511] theprobability of A or B or both occurring, given by therule Pr(A ∪ B) = Pr(A) + Pr(B) − Pr(A ∩ B)

amplitude of circular functions: [p. 223] thedistance between the mean position and themaximum position, e.g. the graph of y = sin x has anamplitude of 1

antiderivative: [p. 447] the general antiderivative F,where F ′(x) = f (x), found using∫

f (x)dx = F(x) + c, where c is an arbitrary realnumber

arrangements: [p. 696] the number of arrangementsof n objects in groups of size r, given by

n!

(n − r )!= n × (n − 1) × (n − 2) . . . (n − r + 1)

* The glossary contains some terms which were introduced in Mathematical Methods CAS Units 1 and 2, but which are notexplicity mentioned in the Mathematical Methods CAS Units 3 and 4 study design. The reference for these is given as[MM1&2].

average value: [p. 475] The average value of afunction f with rule y = f (x) for an interval [a, b] is

defined as1

b − a

∫ ba f (x)dx .

BBernoulli sequence: [p. 542] describes therepetitions of an experiment where:

each trial results in one of two outcomesthe probability for success in a trial is a constantthe trials are independent

binomial distribution: [p. 543] the probability ofobserving X successes in n binomial trials each withprobability of success p, given by

Pr(X = x) =(

nx

)(p)x (1 − p)n−x x = 0, 1, . . . , n

where

(nx

)= n!

x!(n − x)!binomial expansion: [p. 699]

(x + a)n = xn +(

n

1

)xn−1a +

(n

2

)xn−2a2 + · · · + an

=n∑

k=0

(n

k

)xn−kak

The (r + 1)th term is

(n

r

)xn−r ar

binomial experiment: [p. 543] an experimentconsisting of a number, n, of identical trials. Eachtrial results in one of two outcomes, usuallydesignated either a success, S, or a failure, F. Theprobability of success on a single trial, p, is constantfor all trials. The trials are independent (so that theoutcome on any trial is not affected by the outcome ofany previous trial).

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698 Essential Mathematical Methods 3 & 4 CAS

Cchain rule: [p. 311] often used to differentiate somemore complicated functions by transforming theoriginal function into two simpler functions, e.g.,f (x) is transformed to h(x) and g(u) which are

‘chained’ together as xh−→ u

g−→ y. Using Leibniz

notation the chain rule is stated asdy

dx= dy

du· du

dx

change of base: [p. 195] loga b = logc b

logc a

circle, general equation of: [p. 5] the generalequation for a circle is (x – h)2 + (y – k)2 = r2. Thecentre of the circle is the point (h, k) and the radiusis r.

circular functions: [p. 213] the sine, cosine andtangent functions

complement: [p. 511] denoted, for example, A′, theset of points that are in ε but not in A. Similarly, B′ isthe set of points that are in ε but not in B.

composite functions: [p. 25] for functions f and gwith domains X and Y respectively, and such that therange of f ⊆ Y, the composite function of g with f isdefined as g ◦ f : X → R, where g ◦ f (x) = g( f (x))

compound event: [MM1&2] several outcomesfrom a random experiment, such as observing an evennumber when a die is rolled

conditional probability: [p. 512] the probability ofan event A occurring when it is known that someevent B has occurred, written as Pr(A|B)

constant function: [MM1&2] a function f given byf : R → R, f (x) = a, a ∈ R

continuous function: [p. 331] a function f is definedas continuous at the point x = a if the following threeconditions are met: f (x) is defined at x = a; lim

x→af (x)

exists; and limx→a

f (x) = f (a)

continuous random variable: [p. 594] a randomvariable that can take any value in an interval of thereal line

coordinates: [MM1&2] a unique ordered pair ofnumbers x, y that identifies a point on the coordinateplane. The first number in the ordered pair identifiesthe position with regard to the x-axis whereas thesecond number identifies the position with regard tothe y-axis.

cosine function: [p. 213] cosine �, or cos �, definedas the x-coordinate of the point P on the unit circlewhere OP forms an angle of � radians with thepositive ray of the x-axis.

x

y

1

1

0–1

–1

cos θ

sin θ

P(θ) = (cos θ, sin θ)

θ

cubic function: [p. 128] a third degree polynomialfunction, f, with rulef (x) = ax3 + bx2 + cx + d, a �= 0

cumulative distribution function: [p. 604] usuallydenoted F(x) and gives the probability that therandom variable X takes a value less than or equal to

x; that is, F(x) = Pr(X ≤ x) =∫ x

−∞f (t)dt

Ddefinite integral: [p. 444]

∫ b

af (x)dx is called the

definite integral from a to b

degree of polynomial: [p. 128] given by the value ofn, the highest power of x with a non-zero coefficient

dependent trials: [MM1&2] sampling withoutreplacement

derivative of cosine function: [p. 417] forf : R → R, f (�) = cos �, the derivative f ′ is givenby f ′ : R → R, f ′(�) = − sin �

derivative of exponential function: [p. 402] forf (x) = ekx, k ∈ R, the derivative f ′ is given byf ′(x) = kekx

derivative of logarithm function: [p. 406] forf : R+ → R, f (x) = loge (kx), the derivative f ′ is

given by f ′: R+ → R, f (x) = 1

xderivative of sine function: [p. 416] for f : R → R,f (�) = sin �, the derivative f ′ is givenby f ′ : R → R, f ′(�) = cos �

derivative of tangent function: [p. 418] forf (�) = tan k�, the derivative f ′ is given byf ′(�) = k sec2 k�

determinant of a 2 × 2 matrix: [p. 57] If

A =[

a bc d

], det(A) = ad − cb

difference of two cubes: [MM1&2] x3 − y3 =(x − y)(x2 + x y + y2)

difference of two squares: [MM1&2] x2 − y2 =(x − y)(x + y)

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Glossary 699

differentiability: [p. 333] A function f is said to bedifferentiable at a point (a, f (a)) if

limh→0

f (a + h) − f (a)

hexists.

differentiation rules: [p. 302] the general result forany non-zero real power gives: (1) for f (x) = xa, thederivative f ′ is given by f ′(x) = axa−1, for x > 0 anda ∈ R and (2) for f (x) = 1, f ′(x) = 0

dilation from the x-axis: [p. 87] in general adilation of a units from the x-axis is described by therule (x, y) → (x, ay). In general the curve withequation y = f (x) is mapped to the curve withequation y = a f (x) by the transformation with rule(x, y) → (x, ay).

dilation from the y-axis: [p. 87] in general adilation of a units from the y-axis is described by therule (x, y) → (ax, y). In general the curve withequation y = f (x) is mapped to the curve with

equation y = f( x

a

)by the transformation with rule

(x, y) → (ax, y).

dimension of a matrix: [p. 56] The size, ordimension, of the matrix is described by specifyingthe number of rows (horizontal lines) and columns(vertical lines) that occur in the matrix. A matrix withm rows and n columns is said to be an m × n matrix.

discontinuity at a point: [p. 331] a function is saidto be discontinuous at a point if it is not continuous atthat point

discrete random variable: [p. 519] a randomvariable X which can take only a countable number ofvalues, usually whole numbers

discriminant of a quadratic function: [p. 137] theexpression b2 – 4ac, which is part of the quadraticformula. For the quadratic equation0 = ax2 + bx + c,

if b2 − 4ac > 0, there are two solutionsif b2 − 4ac = 0, there is one solutionif b2 − 4ac < 0, there are no real solutions.

disjoint: [p. 2] two sets A and B are said to bedisjoint if they have no elements in common, i.e.A ∩ B = Ø

distance between two given points: [p. 51] thedistance between the points A (x1, y1) and B (x2, y2) is

given by AB = √(x2 − x1)2 + (y2 − y1)2

domain: [p. 5] the set of all the first elements ofordered pairs

Eelement (of a set): [p. 1] a member of, or an objectin, a set. If x is an element of a set A, this is writtenx ∈ A.

empirical probability: [MM1&2] the probabilityassigned to an event on the basis of repeated

experimentation, i.e.

Pr(A) ≈ number of times the event occurs

number of trialsfor large n

empty set, ø: [p. 2] the set that has no members

equating coefficients: [p. 129] if P(x) = a0 + a1x +a2x2 + · · · + anxn and Q(x) = b0 + b1x+ b2 x2

+ · · · + bmxm are equal, i.e. if P(x) = Q(x) for all x,then the degree of P(x) = degree of Q(x), anda0 = b0, a1 = b1, a2 = b2, . . . , an = bm

Euler’s number, e: [p. 180] e = limn→∞

(1 + 1

n

)n

even function: [p. 16] A function f is even iff (x) = f (−x) for all x in the domain of f.

expected value of a random variable, E(X) or �:[pp. 524, 608] the (true) mean value of the randomvariable X, where E(X) =

∑x

x ·Pr(X = x) =∑

x

x ·p(x) for a discrete random variable, and E(X)

=∫ ∞

−∞x f (x) dx for a continuous random variable

exponential function (or index function):[p. 174] the function, f (x) = kax, where k is anon-zero constant and the base a is a positive realnumber other than 1

Ffactor: [MM1&2] a number or expression thatdivides another number or expression withoutremainder

factor theorem: [p. 132] if ax + b is a factor of P(x)

then P

(− b

a

)= 0. Conversely, if P

(− b

a

)= 0

then ax + b is a factor of P

factorise: [MM1&2] express as a product offactors

formula: [MM1&2] an equation containingsymbols that states a relationship between two ormore quantities, e.g. A = lw (area = length × width).The value of A, called the ‘subject’ of the formula,can be found by substituting in given values of land w.

function: [p. 7] a relation for which each x-value ofan ordered pair has a unique y-value, i.e. if (a, b) and(a, c) are ordered pairs of a function, then b = c

function, many-to-one: [p. 14] a function for whichmore than one x-value maps onto the same y-value,e.g. y = x2

function, one-to-one: [p. 14] a function for whicheach y-value has a unique x-value, e.g. y = 2x



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fundamental theorem of calculus: [p. 456] the

theorem that states that∫ b

af (x)dx = G(b) − G(a),

where G is any antiderivative of f

Ggradient function: [p. 301] for a function f, thegradient or derived function is denoted by f ′:, where

f ′: R → R and f ′(x) = limh→0

f (x + h) − f (x)

h

gradient of a line, m: [p. 49] m = rise

run= y2 − y1

x2 − x1,

where (x1, y1) and (x2, y2) are coordinates of points onthe line. The gradient of a vertical line (parallel to they-axis) is undefined.

Hhorizontal line test: [p. 14] if a horizontal line canbe drawn anywhere on the graph of a function andonly ever intersects the graph a maximum of once,the function is one-to-one

hybrid function: [p. 15] a function that hasdifferent rules for different subsets of the domain

Iindependence: [p. 514] A and B are independentevents if Pr(A ∩ B) = Pr(A) × Pr(B)

independent trials: [p. 512] sampling withreplacement

index function (or exponential function):[p. 174] the function, f (x) = kax, where k is anon-zero constant and the base a is a positive realnumber other than 1

index laws: [p. 183]to multiply two numbers in exponent form withthe same base, add the exponents:am × an = am+n

to divide two numbers in exponent form withthe same base, subtract the exponents:am ÷ an = am–n

to raise the power of a to another power,multiply the exponents: (am)n = am×n

If ax = ay then x = y

for rational exponents: a1n = n

√a

inequation: [MM1&2] a mathematical statementthat contains an ‘inequality’ symbol rather than an‘equals’ sign, e.g. 2x + 1 < 4

integers: [p. 2] the elements of {. . ., −2, −1, 0,1, 2, . . .}

integration rules: [pp. 448–464]∫xr dx = xr+1

r + 1+ c, r ∈ Q\{−1}∫

f (x) + g(x)dx =∫

f (x)dx +∫

g(x)dx∫k f (x)dx = k

∫f (x) dx where k is a real number∫

ekx dx = 1

kekx + c∫

1

ax + bdx = 1

aloge (ax + b) + c∫

sin (kx + d)dx = −1

kcos (kx + d) + c∫

cos (kx + d)dx = 1

ksin (kx + d) + c

intersection of sets, ∩: [pp. 2, 510] the intersectionof sets A and B, written A ∩ B, represents theelements common to both A and B. Thus x ∈ A ∩ B ifand only if x ∈ A and x ∈ B.

inverse function: [p. 31] if f is a one-to-onefunction, then the inverse function f −1 is defined byf −1 (x) = y if f (y) = x, for x ∈ ran f, y ∈ dom f

irrational numbers: [p. 2] the real numbers that arenot rationals, e.g. � and

√2

KKarnaugh map: [p. 512] a probability table

Llaw of total probability: [MM1&2] in the case oftwo events, A and B: Pr(A) = Pr(A|B)Pr(B) +Pr(A|B′)Pr(B′)

limit of a function: [p. 328] the value that afunction with rule f (x) approaches as x approaches agiven value. lim

x→af (x) = p means that as x

approaches a, f (x) approaches p.

limits, properties of: [p. 329]limx→c

( f (x) + g(x)) = limx→c

f (x) + limx→c

g(x), i.e.

the limit of the sum is the sum of the limitslimx→c

(k f (x)) = k limx→c

f (x), k being a given

numberlimx→c

( f (x)g(x)) = limx→c

f (x) limx→c

g(x), i.e. the

limit of the product is the product of the limits

limx→c

f (x)

g(x)=

limx→c

f (x)

limx→c

g(x), provided lim

x→cg(x) �= 0

i.e. the limit of the quotient is the quotient of thelimits

linear equation: [pp. 45, 128] a polynomialequation of degree 1, e.g. 2x + 1 = 0

linear function: [p. 128] a function with rulef (x) = mx + c, e.g. f (x) = 3x + 1



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Glossary 701

literal equation: [MM1&2] an equation for thevariable x in which the coefficients of x, including theconstants, are pronumerals, e.g. ax + b = c

logarithm: [p. 184] if a ∈ R+\{1} and x ∈ Rthen the statements ax = n and loga n = x areequivalent

logarithm laws: [p. 185]loga (mn) = loga m + loga n

loga

(m

n

)= loga m − loga n

loga

(1

n

)= − loga n

loga (m p) = p loga m

MMarkov sequence (chain): [p. 569] describes therepetitions of an experiment where:

each trial results in one of two outcomesthe probability of each possible outcome for atrial is conditional only on the trial immediatelypreceding itthe conditional probabilities are the same foreach occasion.

matrices addition and subtraction: [p. 56]Addition will be defined for two matrices only whenthey have the same number of rows and the samenumber of columns. In this case the sum of twomatrices is found by adding corresponding elements,for example:[

1 23 2

]+

[0 −34 1

]=

[1 −17 3

]

matrices, equal: [p. 56] Two matrices A, B areequal, and can be written as A = B when:

each has the same number of rows and the samenumber of columnsthey have the same number or element atcorresponding positions.

matrix, inverse of a square matrix: [p. 57] B issaid to be the inverse of A when AB = BA = I. Theinverse of a square matrix A is denoted by A−1. Theinverse is unique. It does not exist for every squarematrix.

matrix multiplication: [p. 57] If A is an m × nmatrix and B is an n × r matrix, then the product ABis the m × r matrix whose entries are determined asfollows.To find the entry in row i and column j of AB singleout row i in matrix A and column j in matrix B.Multiply the corresponding entries from the row andcolumn and then add up the resulting products.Note: The product AB is defined only if the numberof columns of A is the same as the number of rowsof B.

matrix multiplication by a scalar: [p. 57] It isuseful to define multiplication of a matrix by a realnumber. If A is an m × n matrix, and k is a realnumber, then kA is an m × n matrix whose elementsare k times the corresponding elements of A, forexample:

3

[2 −20 1

]=

[6 −60 3

]

matrix, multiplicative identity: [p. 57] For squarematrices of a given dimension, e.g. 2 × 2, amultiplicative identity I exists.For example, for 2 × 2 matrices

I =[

1 00 1

], AI = IA = A, and this result holds for

any square matrix multiplied by the appropriatemultiplicative identity.

matrix, regular: [p. 57] A square matrix is said tobe regular if its inverse exists.

matrix, singular: [p. 57] a square matrix that doesnot have an inverse

matrix, square: [p. 57] a matrix with the samenumber of rows and columns

matrix, zero: [MM1&2] the m × n matrix with allelements equal to zero

maximal domain (implied domain): [p. 15] Whenthe rule for a relation is written and no domain isstipulated, then it is understood that the domain takenis the largest for which the rule has meaning. Thisdomain is called the maximal or implieddomain.

mean of a random variable, E(X) or �: [pp. 524,608] expected value

median of a random variable, m: [pp. 527, 610]the middle value of the distribution such that:∫ m

−∞f (x) dx = 0.5 for a continuous random variable

midpoint of a line segment: [p. 51] Let M(x, y) bethe midpoint of the line segment joining A (x1, y1)and B (x2, y2). The coordinates of M are(

x1 + x2

2,

y1 + y2

2

).

mode of a random variable, M: [pp. 528, 611] thevalue of the random variable which has thehighest probability of occurring such thatPr(X = M) ≥ Pr (X = x) for all other values of X for adiscrete random variable and f(M) ≥ f (x) for allother x for a continuous random variable

modulus function (or absolute value function), |x|:[p. 20] |x| =

{x if x ≥ 0

−x if x < 0

modulus function, properties of the: [p. 20]|ab| = |a| |b|∣∣∣a

b

∣∣∣ = |a||b|



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|a + b| ≤ |a| + |b|. If a and b are bothnon-negative or both non-positive then equalityholds.If a ≥ 0, |x| ≤ a is equivalent to −a ≤ x ≤ a.If a ≥ 0, |x − k| ≤ a is equivalent tok − a ≤ x ≤ k + a.

multiplication rule (for choices): [p. 695] whensequential choices are involved, the total number ofpossibilities is found by multiplying the number ofoptions at each successive stage

multiplication rule (for probability): [p. 513] therule to determine the probability of events A and Boccurring; Pr(A ∩ B) = Pr(A|B)·Pr(B)

multi-stage experiment: [p. 513] an experimentthat could be considered to take place in more thanone stage, such as tossing two coins

mutually exclusive: [p. 511] said of two sets withno elements in common

Nn factorial, n!: [p. 696] an abbreviation for theproduct of all the integers from n down to 1; n! =n × (n − 1) × (n − 2) × (n − 3) × · · · × 2 × 1

natural numbers: [p. 2] the elements of {1, 2,3, 4, . . .}nCr

(or

(n

r

)): [p. 696] nCr = n!

r !(n − r )!; the

number of combinations of n objects in groups ofsize r

normal distribution: [p. 633] the distribution of acontinuous random variable with probability density

function f (x) = 1

�√

2�e− 1

2

( x−��

)2

� ∈ R, � > 0,

where � is the mean and � the standard deviation ofthe random variable

normal, equation of the: [p. 348] Let P be thepoint on the curve y = f (x) with coordinates (x1, y1).Then, if f is differentiable for x = x1, the equation of

the normal at (x1, y1) is y − y1 = − 1

f ′ (x1)(x − x1)

not an element (of a set): [p. 1] the notationx /∈ A means x is not an element of A

Oodd function: [p. 16] A function f is odd iff (−x) = − f (x) for all x in the domain of f.

ordered pair, (x, y): [p. 5] a pair of elements x and yin which x is considered to be the first element and ythe second

Pperiod of a function: [p. 223] the period of afunction f with domain R is a positive number a such

that f(x + a) = f (x) for a in R, e.g. the period of thesine function is 2� as sin (x + 2�) = sin x. Forfunctions of the form y = a cos (nx + �) + c or y = a

sin (nx + �) + c the period is given by2�

n. The

period of a tangent is �.

polynomial: [p. 128] a rule of the typey = an xn + an−1xn−1 + · · · + a1x + a0 (n ∈ N )where a0, a1, . . . , an are numbers called coefficients

power function: [p. 81] a rule of the typef (x) = xn, where n is a rational number

probability: [p. 510] a numerical value assigned tothe likelihood of an event occurring. If the event A isimpossible then Pr(A) = 0, and if event A is certainthen Pr(A) = 1. Otherwise 0 < Pr(A) < 1

probability density function: [p. 596] usuallydenoted f (x); describes the probability distributionof a continuous random variable, X, such that

Pr(a ≤ X ≤ b) =∫ b

af (x)dx

probability distribution: [p. 521] denoted p(x) orPr(X = x), a function that assigns probabilities toeach value of X. It can be represented by a rule, atable or a graph, and must give a probability p(x) forevery value x that X can take.

probability table: [p. 511] a table used forillustrating a probability problem diagrammatically

product of functions: [p. 24] fg(x) = f (x) g(x) anddomain (fg) = dom f ∩ dom g

product rule: [p. 317] Let F(x) = f (x) · g(x). Iff ′(x) and g′(x) exist thenF ′(x) = f (x) · g′(x) + g(x) · f ′(x). The productrule may also be stated in Leibniz notation: if y = uvwhere u and v are functions of x, thendy

dx= u

dv

dx+ v

du

dx

Qqth percentile: [p. 610] the value of X, p, such thatPr(X ≤ p) = q%, given by the solution of the equation∫ p

−∞f (x) dx = q

100

quadratic formula: [p. 136] x = −b ± √b2 − 4ac

2ais the solution of the quadratic equationax2 + bx + c = 0

quadratic function, general equation of: [pp. 128,136] a second degree polynomial function, f, with rulef(x) = ax2 + bx + c, where a, b and c are constantsand a �= 0. The y-axis intercept is given by c.

quadratic function, turning point form of:[p. 136] a second degree polynomial function, f, withrule f (x) = a(x − h)2 + k, a �= 0, where (h, k) is theturning point



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Glossary 703

quartic function: [p. 128] a fourth degreepolynomial function, f, with rulef(x) = ax4 + bx3 + cx2 + dx + e, a �= 0

quotient rule: [p. 320] Let F(x) = f (x)

g(x),

g(x) �= 0. If f ′(x) and g′(x) exist, then

F ′(x) = g(x) · f ′(x) − f (x) · g′(x)

[g(x)]2. The quotient

rule may be also stated in Leibniz notation: if

y = u

v, v �= 0, where u and v are functions of x,

dy

dx=

vdu

dx− u

dv

dxv2

RR+: [p. 3] {x: x > 0}R−: [p. 3] {x: x < 0}R\{0}: [p. 3] the set of real numbers excluding 0

R2: [p. 3] {(x, y): x, y ∈ R}; the set of all orderedpairs of real numbers

radian: [p. 211] 1 radian (written 1c) is the anglesubtended at the centre of the unit circle by an arc oflength 1 unit

random experiment: [p. 510] an experiment, suchas the tossing of a die, where the outcome of a singletrial is uncertain but observable

random number tables: [MM1&2] a table thatcontains the digits 0, 1, . . . , 9 in random order, andthat can be used to generate random sequences ofnumbers

random variable: [p. 519] a variable that takes itsvalue from the outcome of a random experiment,such as the number of heads observed when a coin istossed three times

range: [p. 5] the set of all the second elements ofordered pairs of a relation

rational numbers: [p. 2] numbers of the formp

qwith p and q integers, q �= 0

rectangular hyperbola: [p. 81] the basic

rectangular hyperbola has equation y = 1

xreflection in the x-axis: [p. 90] In general areflection in the x-axis is described by the rule(x, y) → (x, −y). In general 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 the y-axis: [p. 90] In general areflection in the y-axis is described by the rule(x, y) → (−x, y). In general the curve with equationy = f(x) is mapped to the curve with equation

y = f(−x) by the transformation with rule(x, y) → (−x, y).

relation: [p. 5] a set of ordered pairs

relation, many-to-many: [p. 5] a relation for whichmore than one x-value maps onto more than oney-value, e.g. x2 + y2 = 4

relation, one-to-many: [p. 14] a relation for whichan x-value maps onto more than one y-value,

e.g. y = ±√

x

remainder theorem: [p. 132] when the polynomial

P(x) is divided by ax + b the remainder is P

(− b

a

)

run length: [p. 585] the number of times that thesame outcome is observed in a Bernoulli sequence

Ssample space, ε: [p. 509] the set of possibleoutcomes for an experiment

sampling with replacement: [p. 542] the process ofselecting individual objects sequentially from a groupof objects, and replacing the selected object, so thatthe probability of obtaining a particular object doesnot change with each successive selection.

sampling without replacement: [MM1&2] theprocess of selecting individual objects sequentiallyfrom a group of objects, and not replacing the selectedobject, so that the probability of obtaining a particularobject changes with each successive selection.

selections: [p. 696] the number of combinations of nobjects in groups of size r isnCr =

n Pr

r != n × (n − 1) × (n − 2) . . . (n − r + 1)

r !

= n!

r !(n − r )!set difference: [p. 2] the set difference of two sets Aand B is given by A\B = {x: x ∈ A, x ∈ B}simple event: [p. 510] a single outcome from arandom experiment, such as observing a two when adie is rolled

simulation: [MM1&2] the process of finding anapproximate solution to a probability problem byrepeated trials using a simulation model

simulation model: [MM1&2] a simple model thatis analogous to a real world situation. For example,the outcomes from a toss of a coin (head, tail) couldbe used as a simulation model for the sex of a child(male, female) under the assumption that in bothsituations the probabilities are 0.5 for each outcome.

simultaneous equations: [p. 45] equations of twoor more lines or curves in a cartesian plane, thesolution of which is the point of intersection of thelines or curves



P1: FXS/ABE P2: FXS



sine function: [p. 213] sine �, or sin �, defined asthe y-coordinate of the point P on the unit circlewhere OP forms an angle of � radians with thepositive ray of the x-axis

x

y

1

1

0–1

–1

cosθ

sinθ

P(θ) = (cosθ, sinθ)

θ

standard deviation of a random variable, � or sd(X ):[p. 529] a measure of the spread or variability suchthat sd(X) = √

Var(X )

standard normal distribution: [p. 633] a specialcase of the normal distribution where � =0 and� = 1

stationary point: [p. 358] a point with coordinates(a, g(a)) on a curve y = g(x) where g′(a) = 0

steady state: [p. 548] Many two-state Markovchains will converge to a steady state, independent ofthe initial state.

straight line, equation of (given two points):

[pp. 50] y − y1= m(x − x1), where m = y2 − y1

x2 − x1

straight line, equation of (gradient–intercept form):[p. 50] y = mx + c, where m is the gradient of theline

straight lines, perpendicular: [p. 50] If twostraight lines are perpendicular, the product of theirgradients is –1. Conversely, if the product of thegradients of two lines is –1, then the two lines areperpendicular.

Strictly decreasing: [p. 341] A function f is said tobe strictly decreasing on a given set if for all a and bin the set b > a implies f (b) < f (a).

Strictly increasing: [p. 341] A function f is said tobe strictly increasing on a given set if for all a and bin the set b > a implies f (b) > f (a).

subjective probability: [MM1&2] the probabilityassigned to an event on the basis of prior experience

subset: [p. 1] A set B is called a subset of set A, ifand only if x ∈ B implies x ∈ A. To indicate that B is asubset of A, we write B ⊆ A.

sum of functions: [p. 24] (f + g)(x) = f(x) + g(x),domain of (f + g) = dom f ∩ dom g

sum of two cubes: [MM1&2]x3 + y3 = (x + y) (x2 − xy + y2)

Ttangent, equation of: [p. 347] Let P be the point onthe curve y = f(x) with coordinates (x1, y1). Then, if fis differentiable for x = x1, the equation of the tangentat (x1, y1) is given by (y − y1) = f ′(x1)(x − x1)

tangent function: [p. 214] If a tangent to the unitcircle, at A, is drawn then the y-coordinate of C (thepoint of intersection of the extension of OP and thetangent) is called tangent �, or tan �.

x

y

1

1

0–1

–1

cosθ

θsinθ

P(θ)

tanθ

AD

C(1, y)

B

transition matrix: [p. 570] a square matrix thatgives the probability for each of the possibleoutcomes at each step of a Markov chain conditionalon each of the possible outcomes at the previous step

tree diagram: [p. 515] a diagram representing theoutcomes of a multi-stage experiment

Uunion of sets, ∪: [pp. 2, 511] the union of sets Aand B, written A ∪ B, is the set of elements that areeither in A or in B. This does not exclude objects thatare elements of both A and B.

Vvariance of a random variable, �2 or Var(X):[p. 529] a measure of the spread or variability suchthat Var(X) = E[(X − �)2]

Venn diagram: [MM1&2] a diagram showing setsand the relationships between sets

vertical line test: [p. 7] one way to identify if arelation is a function; draw a graph of the relation andif a vertical line can be drawn anywhere on the graphand only ever intersects the graph a maximum ofonce, the relation is a function



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