Key definitions

The terms the course expects you to state, written to be quoted rather than paraphrased: precise wording that answers the question as asked, with a link to the lesson each belongs to. Search for a term, or jump to its letter.

131 terms

A B C D E F G H I L M N O P R S T U V W

AAcceleration, Actual significance level, Amplitude…6 terms

Acceleration: The rate of change of velocity with time. — Kinematics with variable acceleration

Actual significance level: The probability of landing in the critical region when the null hypothesis is true. Because the binomial is discrete you rarely hit 5 per cent exactly, so the real false-alarm rate usually sits a little under the nominal one. — Hypothesis testing with the binomial

Amplitude: Half the difference between the maximum and minimum values of a periodic function. — Trigonometric graphs and equations

Apollonius locus: The set of points whose distances from two fixed points are in a fixed ratio other than 1. It is a circle, not a line; equal distances give the perpendicular bisector instead. — Further loci and regions in the Argand diagram

Arithmetic sequence: A sequence in which the difference between consecutive terms is constant, the common difference d. — Arithmetic series

Asymptote: A line the distance from the curve to which tends to zero as x or y grows without bound. A curve may cross an asymptote; what makes it one is where the curve ends up, not whether it ever touches. — Graphs, proportion and transformations

BBellman's principle of optimality, Bezout's identity, Binomial coefficient3 terms

Bellman's principle of optimality: Any part of an optimal path is itself optimal between its own two ends. That is what licenses working backwards from the final stage, quoting values already found. — Dynamic programming

Bezout's identity: The highest common factor of a and b can be written as ax + by for integers x and y, found by unwinding the Euclidean algorithm. When that factor is 1, x is the inverse of a modulo b. — The Euclidean algorithm and Bezout's identity

Binomial coefficient: The number of ways of choosing r objects from n, written nCr, which is the coefficient of the corresponding term in the expansion of (a + b)ⁿ. — The binomial expansion

CCapacity of a cut, Cartesian form, Cayley-Hamilton theorem…18 terms

Capacity of a cut: The total capacity of the arcs directed from the source side to the sink side only. Arcs crossing back towards the source contribute nothing at all. — Flows in networks: cuts and capacity

Cartesian form: The equation of a curve in x and y alone, obtained from parametric equations by eliminating the parameter. — Parametric equations

Cayley-Hamilton theorem: Every square matrix satisfies its own characteristic equation, which turns any power of it into a linear combination of lower powers and, when det M ≠ 0, hands over the inverse without cofactors. — Diagonalisation and the Cayley-Hamilton theorem

Change of sign: If a continuous function takes opposite signs at the ends of an interval, it has a root inside; continuity is what makes the argument work. — Locating roots and iteration

Coefficient of friction: The constant μ for which the maximum friction available between two surfaces is μ times the normal reaction. — Friction and inclined planes

Collinear points: Points lying on one straight line, shown by two of the joining vectors being parallel and sharing a point. — Vectors in three dimensions

Common difference: The constant amount added to each term of an arithmetic sequence to reach the next. — Arithmetic series

Common ratio: The constant factor multiplying each term of a geometric sequence to reach the next. — Geometric series

Completed square form: A quadratic written as a(x + p)² + q, which exhibits the vertex at (−p, q) and so the maximum or minimum value directly. — Quadratic functions

Composite function: The function fg, meaning apply g first and then f, defined wherever the output of g lies in the domain of f. — Functions, inverses and the modulus

Conditional probability: The probability of A given B, that is the probability of A within the cases where B has occurred. — Conditional probability

Conjecture: A statement believed to be true but not yet proved. — The structure of proof: deduction and exhaustion

Conservation of momentum: In a collision the internal forces are equal and opposite for the same time, so the impulses cancel and the total momentum before equals the total after. It holds whatever happens to the energy. — Momentum and impulse

Convergent geometric series: A geometric series has a sum to infinity exactly when |r| < 1, and that sum is a/(1 − r). — Geometric series

Counterexample: A single specific case for which a general statement fails, which is enough to disprove the statement. — Disproof and proof by contradiction

Critical activity: One whose earliest and latest start times agree, so it has zero float. The critical activities form a continuous path whose durations total the project duration. — Critical path analysis

Critical region: The set of outcomes for which the null hypothesis is rejected, chosen so that its probability under the null hypothesis is at most the significance level. — Hypothesis testing with the binomial

Critical values: The values at which a quadratic inequality changes sign, that is the roots of the corresponding equation; the solution is read off between or outside them according to the direction of the inequality. — Simultaneous equations and inequalities

DDecreasing function on an interval, Definite integral, Derivative…9 terms

Decreasing function on an interval: A function whose derivative is negative throughout the interval. — Tangents, turning points and curve behaviour

Definite integral: The signed area between a curve and the x-axis over an interval, evaluated as the change in any antiderivative across that interval. — Definite integrals and areas

Derivative: The instantaneous rate of change of a function with respect to its variable, equal to the gradient of the tangent at that point. — Differentiating powers of x

Differential equation: An equation relating a function to one or more of its derivatives. — Rates of change and building differential equations

Differentiation from first principles: The derivative of f(x) is the limit as h tends to zero of (f(x + h) − f(x))/h, provided the limit exists. — The derivative from first principles

Discriminant: For ax² + bx + c, the quantity b² − 4ac, whose sign gives the number of real roots: positive two, zero one repeated, negative none. — Quadratic functions

Displacement: The change in position, a vector, as distinct from distance travelled, which is a scalar and never decreases. — Kinematics with constant acceleration

Domain: The set of inputs on which a function is defined. — Functions, inverses and the modulus

Dominance: A row is dominated when another is at least as good in every column; a column is dominated when another is at least as small in every row. Dominated options are deleted outright, often reducing a game to a two by two. — Game theory: play safe and stable solutions

EEquilibrium, Eulerian and semi-Eulerian, Expected monetary value…4 terms

Equilibrium: A body is in equilibrium when the resultant force on it is zero, so it stays at rest or moves with constant velocity. — Statics of a particle

Eulerian and semi-Eulerian: In a connected graph, no odd nodes means a closed trail exists using every edge once. Exactly two odd nodes means such a trail exists but has to start at one of them and finish at the other. More than two means neither. — Graphs: order, Eulerian paths and planarity

Expected monetary value: The probability-weighted average of the outcomes reachable from a chance node, so the long-run average pay-off of taking that branch. — Decision analysis

Exponential growth and decay: A quantity changing at a rate proportional to its current size, giving growth when the constant is positive and decay when it is negative. — Exponential functions and e

FFlow-augmenting path, Frequency density, Function3 terms

Flow-augmenting path: A route from source to sink using spare capacity forwards or existing flow backwards. It carries the smallest of those numbers along it, and at least one arc becomes saturated. — Maximum flow and the labelling procedure

Frequency density: Frequency divided by class width, plotted as the height of a histogram bar so that area rather than height represents frequency. — Representing and interpreting data

Function: A mapping that sends every element of the domain to exactly one output. Two inputs may land on the same output. One input may never have two. — Functions, inverses and the modulus

GGeometric sequence, Gradient of a line, Group3 terms

Geometric sequence: A sequence in which the ratio between consecutive terms is constant, the common ratio r. — Geometric series

Gradient of a line: The increase in y per unit increase in x, constant along a straight line. — Straight lines

Group: A set with a binary operation satisfying closure, associativity, the existence of an identity, and an inverse for every element. Commutativity is not required. — Groups and their axioms

HHarmonic form, Hungarian reduction2 terms

Harmonic form: a sin θ + b cos θ written as R sin(θ + α), which exposes the maximum R and minimum −R without differentiating. — Compound angles and the harmonic form

Hungarian reduction: Subtract each row's minimum from that row, then each column's from that column. Every complete allocation loses the same total, so the cheapest one is unchanged and the zeros mark the bargains. — Allocation and the Hungarian algorithm

IIdentity, If and only if, Implicit differentiation…11 terms

Identity: An equation true for every value of the variable for which both sides are defined, as distinct from an equation true only at particular values. — Trigonometric modelling

If and only if: Two statements are connected by if and only if when each implies the other, so a proof of an equivalence has two directions to establish. — The structure of proof: deduction and exhaustion

Implicit differentiation: Differentiating an equation in x and y as it stands, applying the chain rule to every term in y, rather than first solving for y. — Implicit and parametric differentiation

Improper integral: An integral with an infinite limit or an integrand unbounded at a limit, defined as the limit of an ordinary integral, and said to converge when that limit is finite. — Mean values and improper integrals

Increasing function on an interval: One whose outputs follow the order of its inputs: for x₂ beyond x₁ in the interval, f(x₂) is at least f(x₁), and strictly increasing means always greater. Say which you mean, then test it: f'(x) ≥ 0 throughout gives the first, f'(x) > 0 the second. — Tangents, turning points and curve behaviour

Indefinite integral: The family of all functions with a given derivative, written with an arbitrary constant because the constant is lost on differentiating. — Integration as antidifferentiation

Independent events: Events A and B are independent when P(A ∩ B) = P(A) × P(B); equivalently, when P(A|B) = P(A), so B carries no news about A. — Probability and Venn diagrams

Integration by parts: The reverse of the product rule: the integral of u dv/dx equals uv minus the integral of v du/dx. — Integration by substitution and by parts

Interpolation and extrapolation: Using a regression line inside the range of the data is interpolation, which the data support; using it outside that range is extrapolation, which they do not. Interpolation removes that particular risk rather than guaranteeing a good prediction, since the fit and the scatter still matter. — Correlation and regression

Interquartile range: The upper quartile minus the lower quartile, a measure of spread unaffected by extreme values. — Measures of location and spread

Invariant line: A line whose every point maps to a point of the same line under the transformation. Points may move along it; none leaves it. — Systems of equations and invariance

LL'Hospital's rule, Lagrange's theorem, Limiting equilibrium…5 terms

L'Hospital's rule: For an indeterminate form 0/0 or ∞/∞, the limit of f/g equals the limit of f'/g', provided that second limit exists. — Limits and L'Hospital's rule

Lagrange's theorem: The order of any subgroup of a finite group divides the order of the group, so the order of every element divides it too. — Subgroups, Lagrange's theorem and isomorphism

Limiting equilibrium: The state in which an object is still in equilibrium but on the point of slipping, so friction has reached its maximum value μR. — Friction and inclined planes

Linearising a model: Taking logarithms of a power or exponential model so that the relationship plots as a straight line, whose gradient and intercept give the constants. — Log graphs and exponential models

Logarithm: The power to which a must be raised to give x. So a^y = x and y = log base a of x say the same thing, with a > 0, a ≠ 1 and x > 0. — Logarithms and their laws

MMinimum spanning tree, Moment of a force, Mutually exclusive3 terms

Minimum spanning tree: A set of edges joining all n nodes, using n − 1 edges and closing no cycle, of the least total weight any such set can have. — Minimum spanning trees: Prim and Kruskal

Moment of a force: The turning effect of a force about a point: the magnitude of the force times the perpendicular distance from the point to its line of action. — Moments

Mutually exclusive: Two events are mutually exclusive when they cannot both occur, so the probability of their intersection is zero. — Probability and Venn diagrams

NNewton's first law, Newton's second law, Newton's third law…6 terms

Newton's first law: A body remains at rest or moves with constant velocity unless acted on by a resultant force. — Forces and Newton's laws

Newton's second law: The resultant force on a body equals its mass times its acceleration, and acts in the direction of that acceleration. — Forces and Newton's laws

Newton's third law: If one body exerts a force on another, the second exerts a force on the first that is equal in magnitude and opposite in direction. — Forces and Newton's laws

Normal reaction: The component of the contact force perpendicular to the surface. — Connected particles and pulleys

North-west corner method: Fill the top-left cell as far as supply and demand allow, cross off whichever runs out, repeat. It never looks at a cost, so the plan it produces is usually poor. Its job is to be feasible, not to be good. — Transportation problems

Null and alternative hypotheses: The null hypothesis is the statement about the population being tested; the alternative is what is accepted if the evidence is strong enough against it. — Hypothesis testing with the binomial

OOne-tailed and two-tailed, One-to-one, Order of an algorithm…4 terms

One-tailed and two-tailed: A one-tailed test looks for a change in a stated direction; a two-tailed test looks for a change in either, and splits the significance level between the two tails. — Hypothesis testing: correlation and the normal

One-to-one: A function is one-to-one when distinct inputs give distinct outputs, which is exactly the condition for an inverse function to exist. — Functions, inverses and the modulus

Order of an algorithm: How the number of operations grows with the size n of the problem. An order n² algorithm takes four times as long on a list twice as long. — Algorithms, sorting and bin packing

Outlier: A value far enough from the rest to be treated separately, judged against a stated rule such as more than 1.5 interquartile ranges beyond a quartile. — Representing and interpreting data

PParameter, Partial fractions, Particle…18 terms

Parameter: A third variable in terms of which x and y are each given, so that letting it vary traces the curve. — Parametric equations

Partial fractions: A single algebraic fraction rewritten as a sum of fractions with simpler denominators, one for each factor of the original denominator. — Partial fractions

Particle: A body modelled as having mass but no size, so that its rotation and any variation across it can be ignored. — Modelling, quantities and units

Particular solution: The single solution of a differential equation picked out from the general family by a given condition. — Solving differential equations

Period: The smallest positive interval after which a periodic function repeats. — Trigonometric graphs and equations

Periodic sequence: A sequence that repeats itself after a fixed number of terms, so u of n plus k equals u of n for every n. Its order is the smallest k that does this, since every multiple of that k repeats as well. — Sequences and sigma notation

Perpendicular gradients: Two non-vertical lines are perpendicular exactly when the product of their gradients is −1. — Straight lines

Piecewise function: A function given by different expressions on different parts of its domain. — Functions in modelling

Point of inflection: A point where the curve changes its sense of concavity, so the second derivative changes sign. — Tangents, turning points and curve behaviour

Population and sample: The population is every member of the group under study; a sample is the subset actually measured. — Sampling and the large data set

Position vector: The vector from a fixed origin to a point, so its components are the coordinates of that point. — Vectors in two dimensions

Positive skew: A long tail running to the right, with the bulk of the data to its left. In the single-peaked cases on this paper mode sits below median below mean, and quoting that order with the tail named earns the mark. It is evidence rather than proof: three equal averages do not force symmetry. — Mean, variance and skewness of continuous variables

Power of a test: The probability of correctly rejecting the null hypothesis at a stated alternative, equal to one minus the probability of a Type II error there. — The quality of tests

Principal argument: For z ≠ 0, the angle the position vector of z makes with the positive real axis, measured anticlockwise, in the interval (−π, π]. The origin has modulus 0 and no argument at all. — Modulus, argument and loci

Principal value: The single output an inverse trigonometric function returns, chosen from a restricted range so that the inverse is a function at all. — Reciprocal and inverse trigonometric functions

Projectile: A body moving freely under gravity alone, so its horizontal velocity is constant and its vertical acceleration is g downwards. — Projectiles

Proof by contradiction: A proof that assumes the negation of the statement and derives a contradiction, so the negation is false and the statement is true. — Disproof and proof by contradiction

Proof by exhaustion: Splitting the statement into a finite set of cases that between them cover every possibility, then checking each one. It only proves anything if the cases genuinely leave nothing out, so say why they do. — The structure of proof: deduction and exhaustion

RRadian, Range, Rationalising the denominator…5 terms

Radian: The angle subtended at the centre of a circle by an arc equal in length to the radius. — Radians, arcs and small angles

Range: The set of outputs a function actually takes over its domain. — Functions, inverses and the modulus

Rationalising the denominator: Rewriting a quotient so that no surd remains in the denominator, by multiplying above and below by a suitable factor. — Indices and surds

Recurrence relation: A rule defining each term of a sequence from earlier terms, together with a starting value. — Sequences and sigma notation

Residual: The observed y minus the y the regression line predicts at that same x, so a vertical gap and never a perpendicular one. Least squares makes the sum of their squares as small as possible, and the residuals themselves always add to zero. — Least squares regression and residuals

SSampling frame, Scalar and vector, Separating the variables…12 terms

Sampling frame: The list of the population from which a sample is drawn; a member missing from the frame can never be sampled. — Sampling and the large data set

Scalar and vector: A scalar has magnitude only; a vector has magnitude and direction. — Vectors in two dimensions

Separating the variables: Rearranging a differential equation so each variable appears on its own side with its own differential, after which both sides are integrated. — Solving differential equations

Significance level: The level a test is set to, fixed before the data are seen. The actual probability of rejecting a true null hypothesis is the test's size, which for a discrete test sits at or below it. — Hypothesis testing with the binomial

Simple random sample: A sample in which every member of the population is equally likely to be chosen, and every possible sample of that size equally likely. — Sampling and the large data set

Slack, surplus and artificial: A ≤ constraint gains a slack variable; a ≥ constraint loses a surplus variable and then gains an artificial one so that a starting solution exists at all. — Linear programming: formulation and graphical solution

Small angle approximations: For θ in radians and small, sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2; the results are false if θ is in degrees. — Radians, arcs and small angles

Smooth and light: Smooth means friction is modelled as zero; light means mass is modelled as zero, so tension is the same throughout a light string. — Modelling, quantities and units

Staircase and cobweb diagrams: Sketches of an iteration on the curve and the line y = x, converging in steps to a root or spiralling about it. — Locating roots and iteration

Standard deviation: The root mean square deviation from the mean, in the same units as the data. — Measures of location and spread

Stationary point: A point where the derivative is zero. — Tangents, turning points and curve behaviour

Surd: An irrational root left in exact form rather than written as a decimal. — Indices and surds

TTangent to a circle, The ambiguous case, The binomial conditions…14 terms

Tangent to a circle: A line meeting the circle exactly once; it is perpendicular to the radius at the point of contact. — Circles

The ambiguous case: When the sine rule is used to find an angle, two angles in the range have the same sine, so both the acute and the obtuse possibility must be tested against the triangle. — Triangles and the sine and cosine rules

The binomial conditions: A fixed number of independent trials, each with two outcomes and the same probability of success throughout. — The binomial distribution

The chain rule: For a function of a function, dy/dx = (dy/du)(du/dx). — The product, quotient and chain rules

The equation of a circle: (x − a)² + (y − b)² = r² is the circle of centre (a, b) and radius r. — Circles

The factor theorem: (x − a) is a factor of the polynomial f(x) if and only if f(a) = 0. — Polynomials and the factor theorem

The limit of a sum: The definite integral as the limit of the total area of rectangles under the curve as their width tends to zero. — Areas, parametric curves and the limit of a sum

The Newton-Raphson method: An iteration taking the next estimate as the point where the tangent at the current one crosses the x-axis, which fails if that derivative is zero. — Newton-Raphson and the trapezium rule

The normal distribution: A continuous distribution, symmetric about its mean, in which about 95 per cent of values lie within two standard deviations of the mean. — The normal distribution

The number e: The base for which the gradient of y = e^x equals its own value at every point. — Exponential functions and e

The product moment correlation coefficient: A measure of linear association between −1 and 1, where 0 indicates no linear relationship and never that the variables are unrelated. — Correlation and regression

The remainder theorem: When a polynomial f(x) is divided by (x − a), the remainder is f(a). — Polynomials and the factor theorem

The trapezium rule: An estimate of a definite integral that replaces the curve by straight chords, overestimating where the curve is concave up and underestimating where it is concave down. — Newton-Raphson and the trapezium rule

The validity condition: The expansion of (1 + x)ⁿ for n negative or non-integer is an infinite series that converges only for |x| < 1, so a valid answer states that range. — The general binomial expansion

UUniform beam, Unit vector2 terms

Uniform beam: A beam whose mass is evenly spread, so its whole weight can be modelled as acting at its midpoint. — Moments

Unit vector: A vector of magnitude 1, obtained from any non-zero vector by dividing it by its own magnitude. — Vectors in two dimensions

VVelocity and speed1 terms

Velocity and speed: Velocity is the rate of change of displacement and is a vector; speed is its magnitude. — Kinematics with constant acceleration

WWeight, Work-energy principle2 terms

Weight: The force of gravity on a body, of magnitude mg, directed vertically downwards. — Forces and Newton's laws

Work-energy principle: The total work done by all the forces equals the change in kinetic energy. The normal reaction never contributes, since it acts at right angles to the motion. — Work, energy and power

Terms that come up often enough to drill also run on the flashcards, in the same wording. For the one big idea per topic, see the key ideas.