EDEXCEL A LEVEL MATHS: GLOSSARY

The absolute value is the magnitude or non-negative value of a number or expression.

acceleration

Acceleration is the rate of change of velocity. The SI unit for acceleration is metres per second per second.

acceptance region

The acceptance region is the range of values of a random variable for which the null hypothesis is not rejected.

addition formulae or compound angle formulae
The addition formulae, or compound angle formulae, are relationships between the trigonometric functions of sums or differences of angles, such as A + B or A − B, and trigonometric functions of individual angles A and B
The addition formulae that you need to know (and are given in the formula booklet) are:

\(
\sin (\textit{A} \pm \textit{B})=\sin \textit{A} \cos \textit{B} \pm \cos \textit{A} \sin \textit{B}\\
\cos (\textit{A} \pm \textit{B})=\cos \textit{A} \cos \textit{B} \mp \sin \textit{A} \sin \textit{B} \\
\tan (\textit{A} \pm \textit{B})=\frac{\tan \textit{A} \pm \tan \textit{B}}{1-\tan \textit{A} \tan \textit{B}}
\)

alternative hypothesis

In a hypothesis test, the alternative hypothesis is the claim that the value of a population parameter has changed.

ambiguous

The ambiguous case refers to the situation when there are two possible answers when working out a missing angle using the sine rule.

anomaly

An anomaly is a piece of data that includes an error.

arithmetic sequence

An arithmetic sequence is a pattern of terms that increase or decrease by a common difference.

asymptote

An asymptote is a line that approaches a curve but does not meet it.

base
In mathematics, in particular in the context of exponential functions, a base refers to the number that is raised to a power in an exponential expression.
For example, in the exponential function
f(x) = ab^x
the base is the number b
Common bases include:
• e, the base of the natural logarithm (approximately equal to 2.71828)
• 10, the base of the common logarithm (used in the decimal system)
• 2, the base often used in computer science and binary systems.

binomial coefficient
The binomial coefficients are numbers that appear before the variable in a binomial expansion.

binomial distribution
The binomial distribution is a discrete probability distribution where there are two possible outcomes that occur with constant probability.

binomial expansion
A binomial expansion is the written-out expression when a two-term expression is raised to a power.

boundary condition
A boundary condition is the information necessary to determine a unique (particular) solution to a differential equation.

box plot
A box plot is a diagram showing the quartiles, maximum value, minimum value and outliers of a data set.

chain rule
The chain rule describes how to differentiate a composite function. It allows you to find the derivative of a composite function by differentiating its outer and inner functions separately.
If y = g[f(x)], then, by letting u = f(x) and y = g(u), the derivative of y with respect to x is given by the formula:

\(
\frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}}=\frac{\textsf{d} \textit{y}}{\textsf{d} \textit{u}} \times \frac{\textsf{d} \textit{u}}{d \textit{x}}
\)

chord
A chord is a line segment that joins two points on the circumference of a circle.

circumcircle
A circumcircle is a circle that passes through the vertices of a polygon, such as a triangle.

cobweb diagram
A cobweb diagram is a diagram resembling a spider’s web that shows convergence to a root when using the x = g(x) method of iteration.

column vector
A column vector is a mathematical object that represents a list of numbers or elements arranged vertically in a single column.
In two dimensions, a column vector is written as \(\left(\begin{array}{l}
\textit{x} \\
\textit{y}
\end{array}\right)
\)

In three dimensions, a column vector is written as \(\left(\begin{array}{l}
\textit{x} \\
\textit{y} \\
\textit{z}
\end{array}\right)
\)

common difference
The common difference is the fixed difference between terms in an arithmetic sequence.

common ratio
The common ratio is the fixed multiplier that the terms in a geometric sequence increase or decrease by.

completing the square
Completing the square means expressing a quadratic in the form p(x + q)² + r, where p, q and r are constants.

component form
Component form is the representation of a vector as either a column vector or using i, j, k notation.

composite function
A composite function is a function made up of two or more other functions; it has the form y = g[f(x)], where function f is applied to x and then function g is applied to the result.

compound angle formulae
See addition formulae for definition.

compression
Compression occurs when a force presses inward on an object, such as a rod or spring, so that it becomes compacted.

concave
A graph is concave if, for any two points on the graph, the line segment joining those points lies below or on the graph itself; the graph curves downward or remains flat between any two points. If a graph is concave for a given interval [a, b], the second derivative is negative for all x in that interval.

conditional probability
Conditional probability is the chance of one event occurring given information about whether another event has occurred.

connected rate of change
A connected rate of change describes how the rate of change of one variable, in a given context, affects changes in another variable.

constant of integration
A constant of integration is an arbitrary constant term that is added to the result of an indefinite integral.

constant of proportionality
The constant of proportionality is the ratio between two quantities that are in proportion.

continuous random variable
A continuous random variable is a random variable that can take any value in a range.

converge
To converge means to tend towards a finite value.

convex
A graph is convex if, for any two points on the graph, the line segment joining those points lies above or on the graph itself; the graph curves upwards or remains flat between any two points. If a graph is convex for a given interval [a, b], the second derivative is positive for all x in that interval.

cosecant
The cosecant, usually denoted by cosec, is a trigonometric function that is the reciprocal of the sine function. It is defined as \(\operatorname{cosec} \theta=\frac{1}{\sin\theta}\)

cosine rule
The cosine rule is a formula used in trigonometry to find the length of a side or the size of an angle in a triangle. The cosine rule states:

\(
\textit{a}^{\scriptsize{2}} =\textit{b}^{\scriptsize{2}} +\textit{c}^{\scriptsize{2}} -2 \textit{b} \textit{c} \cos \textit{A}
\)

where a, b and c are the sides of the triangle and A is the angle opposite side a

The cosine rule can be rearranged to solve for any of the variables given the other variables. For example, if you know the lengths of sides a and b and the size of angle C, you can find the length of side c using this formula:

\(
\textit{c}^{\scriptsize{2}} =\textit{a}^{\scriptsize{2}} +\textit{b}^{\scriptsize{2}} -2 \textit{a} \textit{b} \cos \textit{C}
\)

You can also rearrange the cosine rule to find an angle, for example:

\(
\cos {A}=\frac{{b}^{\scriptsize{2}}+{c}^{\scriptsize{2}}-{a}^{\scriptsize{2}}}{2{b}{c}}
\)

cotangent
The cotangent, usually denoted by cot, is a trigonometric function that is the reciprocal of the tangent function. It is defined as \(\cot \theta=\frac{1}\tan \theta\)

counter example
A counter example is an example that satisfies the conditions of a statement but not its conclusion, and so disproves the statement.

critical region
The critical region is the range of values of a random variable for which the null hypothesis is rejected.

critical value (statistics)
The critical value is the threshold for rejecting the null hypothesis.

critical values (pure)
The critical values are the values of a variable for which a function is equal to zero or undefined.

cubic
A cubic is an algebraic expression involving a power of 3 of a variable and with no higher powers.

cumulative frequency
The cumulative frequency is the total frequency up to a certain point.

cumulative probability
The cumulative probability is the probability of a random variable being up to and including a particular value.

decreasing function
A function f(x) defined on an interval [a, b] is a decreasing function if, for any two values x₁ and x₂ in the interval such that x₁ < x₂, the corresponding function values satisfy f(x₁) > f(x₁); the first derivative of the function is negative for all x in that interval.

decreasing sequence
A decreasing sequence is a sequence in which every term is strictly less than the term before.

definite integral
The definite integral of a function y = f(x) over the interval [a, b] is denoted by \(\int_{\scriptsize{a}}^{\scriptsize{b}} \textit{y} \textsf{ d} \textit{x} \)

The result of a definite integral represents the signed area between the graph of the function and the x-axis over the specified interval. If the function is non-negative over the interval, the definite integral represents the area under the curve; otherwise, it represents the net accumulation, accounting for areas above and below the x-axis.

definite integration
Definite integration is the process of finding the numerical value of a definite integral over a specified interval.

denominator
The denominator is the number or expression on the bottom of a fraction.

dependent variable
The dependent variable, normally shown on the vertical axis of a graph, is the variable that is potentially affected by the independent variable; it is also called the response variable.

diameter
A diameter is a chord that passes through the centre of a circle; the greatest length across a circle.

differential equation
A differential equation is an equation that describes how a function is related to its derivative. First order differential equations with separable variables are of the form \( \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}}=\textsf{f}(\textit{x}) \textsf{g}(\textit{y}) \)

differentiation from first principles
Differentiation from first principles is a method used to find the derivative of a function by applying the definition of the derivative directly. The derivative of a function at a point is defined as the limit of the difference quotient as the interval over which it is calculated approaches zero.
For a function f(x), the derivative f′(x) at a point x is given by:

\(
\textsf{f}^{\prime}(\textit{x})=\lim_{h \to 0}\frac{\textsf{f}(\textit{x}+\textit{h})-\textsf{f}(\textit{x})}{\textit{h}}
\)

To find the derivative of a function using first principles, follow these steps:
1. Start with the definition of the derivative:

\(
\textsf{f}^{\prime}(\textit{x})=\lim_{h \to 0}\frac{\textsf{f}(\textit{x}+\textit{h})-\textsf{f}(\textit{x})}{\textit{h}}
\)

2. Substitute the function f(x) into the formula.
3. Expand and simplify the expression.
4. Evaluate the limit as h approaches zero.
5. The result of this limit is the derivative of the function at the given point x

discrete random variable
A discrete random variable is a random variable that can only take certain specific values.

discrete uniform distribution
The discrete uniform distribution is a distribution in which the probability of every possible value of the random variable is equal.

discriminant
The discriminant is the value given by b² − 4ac for the quadratic function ax² + bx + c

displacement
Displacement is the change in position of an object. It is a vector quantity with both magnitude and direction.

distance
Distance is the magnitude of displacement. The SI unit for distance is metres.

domain
A domain is the set of possible inputs of a mapping.

double angle formulae
The double angle formulae are relationships that express trigonometric functions of double angles, 2A, in terms of trigonometric functions of the original angle A
The double angle formulae that you need to know are:

\(
\begin{aligned}
\sin 2 \textit{A} & =2 \sin \textit{A} \cos \textit{A} \\
\cos 2 \textit{A} & =\cos ^{\scriptsize{2}} \textit{A}-\sin ^{\scriptsize{2}} \textit{A} \\
& =2 \cos ^{\scriptsize{2}} \textit{A}-1 \\
& =1-2 \sin ^{\scriptsize{2}} \textit{A} \\
\tan 2 \mathrm{~\textit{A}} & =\frac{2 \tan \textit{A}}{1-\tan ^{\scriptsize{2}} \textit{A}}
\end{aligned}
\)

elimination method
The elimination method is a method of solving simultaneous equations by adding or subtracting multiples of the equations to eliminate one of the variables.

equilibrium
If the resultant force on an object is zero, then the object is in equilibrium.

event
An event is one or more outcomes of an experiment.

exponential function
An exponential function is a function of the form f(x) = ab^x, where a and b are constants, and b is the base of the exponential function (b > 0)

When b > 1, the function exhibits exponential growth. When 0 < b < 1, the function exhibits exponential decay. Exponential functions arise in various natural phenomena and have wide-ranging applications in fields such as physics, biology, economics, and engineering. Some common examples of exponential functions include population growth, radioactive decay, compound interest, and the growth of bacterial colonies.

exponential model
An exponential model is used when a variable increases or decreases at a rate proportional to its current value. The general form of this type of relationship is y = ab^x

extrapolation
Extrapolation involves making an estimate outside the range of available data; this is generally considered unreliable.

factor theorem
The factor theorem states that, for a polynomial f(x), if \( \textsf{f}\left(\frac{\textit{b}}{\textit{a}}\right)=0 \), then (bx − a) is a factor of f(x)

first derivative
The first derivative represents the rate of change of the function with respect to its independent variable; it measures how the value of the function changes as the input variable changes. If y = f(x) is a function of the variable x, then its first derivative is written as \( \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}} \) or f′(x)

Geometrically, the first derivative represents the slope of the tangent line to the graph of the function at a specific point.

frequency polygon
A frequency polygon is a statistical diagram where the midpoint of each class is plotted against the frequency of the class.

friction
Friction is the force that an object experiences when trying to move over another object or surface. It acts in the opposite direction to the motion.

function
A function is a mapping where every input is mapped to exactly one output.

fundamental theorem of calculus
The fundamental theorem of calculus states, in simple terms, that integration is the reverse process of differentiation.

general solution
A general solution of a differential equation is a solution that encompasses all possible solutions. It represents a family of solutions that satisfy the equation, typically containing one or more arbitrary constants.

geometric sequence
A geometric sequence is a pattern of terms that increase or decrease by a common ratio.

gradient
The gradient is the slope of a graph; it is the rate of change of y with respect to x at a point on a curve or line.

gradient function
The gradient function is the derivative of a function of one variable. For a function f(x), the gradient function is denoted by f′(x), and it represents the rate of change of the function with respect to x at each point.

histogram
A histogram is a statistical diagram where the area of each bar is proportional to the frequency of the class it represents; it is used with continuous data.

horizontal stretch
A horizontal stretch is a graph transformation in which the y-coordinates of points are fixed and the x-coordinates are divided by a scale factor.

The graph of y = f(kx) is a horizontal stretch with scale factor 1/ k of the graph of y = f(x); the x-coordinates are divided by k

hypothesis test
A hypothesis test is a statistical procedure for using a sample to determine whether there has been a change in a population parameter.

i
i is a unit vector in the positive x-direction.

implicit differentiation
Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation, where y cannot be easily expressed explicitly in terms of x

To find the derivative of an implicitly defined function y with respect to x, you differentiate it with respect to y and then multiply the result by\( \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}} \)

Two useful formulae are:

\( \frac{\textsf{d}}{\textsf{d} \textit{x}} \textsf{f}(\textit{y})=\textsf{f}^{\prime}(\textit{y}) \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}} \)

and

\( \frac{\textsf{d}}{\textsf{d} \textit{x}}(\textit{x} \textit{y})=\textit{x} \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}}+\textit{y} \)

implicit form
An equation is in implicit form when neither x nor y is explicitly isolated on one side of the equation. An example of an equation given in implicit form is x² + y² = 1

improper algebraic fraction
An improper algebraic fraction is a fraction where the expression in the numerator has order equal to or greater than the order of the expression in the denominator.

increasing function
A function f(x) defined on an interval [a, b] is an increasing function if, for any two values x₁ and x₂ in the interval such that x₁ < x₂, the corresponding function values satisfy f(x₁) < f(x₂); the first derivative is positive for all x in that interval.

increasing sequence
An increasing sequence is a sequence where every term is strictly more than the term before.

independent
Two events are independent if the occurrence of one event does not affect the likelihood of the other.

indefinite integral
An indefinite integral is a mathematical operation that represents the set of all possible functions whose derivative equals a given function.

Given a function f(x), the indefinite integral of f(x) with respect to x, is denoted by \( \int \textsf{f}(\textit{x}) \textsf{d} \textit{x}=\textsf{F}(\textit{x})+\textit{c} \), where c is a constant of integration.

independent variable
The independent variable, normally shown on the horizontal axis of a graph, is a variable that potentially affects another; it is also called the control variable.

index
An index is the power of a base.

inextensible
An inextensible string does not stretch.

integer
An integer is a whole number.

integration
Integration is the reverse process to differentiation.

integration by parts
Integration by parts is a technique used to evaluate the integral of a product of two functions. It is based on the product rule for differentiation and provides a method for transforming certain types of integrals into simpler forms that are easier to evaluate.

The integration by parts formula states that the integral of the product of two functions, u and \( \frac{\textsf{d} \textit{v}}{\textsf{d} \textit{x}} \), with respect to x, can be expressed as:

\( \int \textit{u} \frac{\textsf{d} \textit{v}}{\textsf{d} \textit{x}} \textsf{d} \textit{x}=\textit{u} \textit{v}-\int \textit{v} \frac{\textsf{d} \textit{u}}{\textsf{d} \textit{x}} \textsf{d} \textit{x} \)

integration by substitution
Integration by substitution is a technique used to simplify and evaluate integrals by introducing a new variable, typically denoted as u, and its corresponding differential, du, and substituting both u and du into the integral. This process transforms the original integral into a simpler form that is often easier to evaluate.

The general steps for integration by substitution are:
1. Identify a part of the integrand that can be written as a function of a single variable.
2. Choose a suitable substitution such that the new variable u simplifies the integral.
3. Compute the differential du by differentiating u with respect to the original variable.
4. Rewrite the integral in terms of u and du
5. Evaluate the new integral using the substitution.
6. If necessary, rewrite the result in terms of the original variable.

If you are computing a definite integral using substitution, you can also substitute the limits in terms of the new variable u.

interpolation
Interpolation involves making an estimate inside the range of available data.

inverse trigonometric function
The inverse trigonometric functions are:

\(
\begin{aligned}
& \sin ^{\scriptsize{-1}} \textit{x}=\arcsin \textit{x} \\
& \cos ^{\scriptsize{-1}} \textit{x}=\arccos \textit{x} \\
& \tan ^{\scriptsize{-1}} \textit{x}=\arctan \textit{x}
\end{aligned}
\)

inverse function
An inverse function ‘undoes’ or ‘reverses’ the effect of another function. For a function f(x), the inverse function is denoted by f⁻¹ (x); therefore ff⁻¹ (x) = x

iteration
Iteration is when a rule, function, or process is applied repeatedly.

iterative method
An iterative method is a computational technique used to approximate solutions to mathematical problems through a repeated process of refinement or improvement. Instead of directly solving a problem in one step, iterative methods involve performing a series of computations, with each iteration refining the approximation to the desired solution. Two methods of iteration are the x = g(x) rearrangement method and the Newton–Raphson method.

j
j is a unit vector in the positive y-direction.

k
k is a unit vector in the positive z-direction.

large data set
The large data set is the data set provided by the exam board with which you need to be familiar.

laws of logarithms
There are several fundamental laws of logarithms, including:

• The product law: \( \log _\textit{b}(\textit{x} \textit{y})=\log _\textit{b} \textit{x}+\log _\textit{b} \textit{y} \)

This states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the individual numbers.

• The quotient law: \( \log _\textit{b}\left(\frac{\textit{x}}{\textit{y}}\right)=\log _\textit{b} \textit{x}-\log _\textit{b} \textit{y} \)

This states that the logarithm of the quotient of two numbers is equal to the difference of the logarithms of the individual numbers.

• The power law: \( \log _\textit{b}\left(\textit{x}^\textit{n}\right)=\textit{n} \log _\textit{b} \textit{x} \)

This states that the logarithm of a number raised to a power is equal to the product of the exponent and the logarithm of the base.

light
A light object is treated as having no mass.

limit
A limit is a finite value that a series tends towards.

limiting equilibrium
An object on the point of moving is in limiting equilibrium.

linear model
A linear model is an equation of the form y = mx + c used to represent a real-world scenario.

linear interpolation
Linear interpolation is a method of estimating the value of a statistic between two known points by assuming that the data is equally spaced between the points.

logarithm
A logarithm is a mathematical function that gives the power or exponent to which a given base must be raised to produce a certain number; it is the inverse of exponentiation.

The logarithm of a number x with respect to a base b, is denoted by log_bx = y

This equation means that b raised to the power of y equals x

magnitude
The magnitude of a vector is its size, or length.

In two dimensions, a vector v = xi + yj has magnitude \( |\mathbf{\textsf{v}}|=\sqrt{\textit{x}^{\scriptsize{2}} +\textit{y}^{\scriptsize{2}} } \)

In three dimensions, a vector v = xi + yj + zk has magnitude \( |\mathbf{\textsf{v}}|=\sqrt{\textit{x}^{\scriptsize{2}} +\textit{y}^{\scriptsize{2}} +\textit{z}^{\scriptsize{2}} } \)

many-to-many
A many-to-many mapping is a mapping where inputs can map to more than one output, and two or more outputs can result from the same input.

many-to-one
A many-to-one mapping is a mapping where two or more inputs can map to the same output.

mapping
A mapping is a rule that associates an output with an input.

maximum point
The maximum point is the highest value attained by a function within a specific domain; graphically, it corresponds to the highest point of the graph of the function within the specified domain.

mean
The mean is the sum of all the values in a data set divided by the number of values.

median
The median is the middle value in an ordered set of data.

minimum point
The minimum point is the lowest value attained by a function within a specific domain; graphically, it corresponds to the lowest point of the graph of the function within the specified domain.

mode/modal class
The mode or modal class is the most commonly occurring value or class.

modulus function
A modulus function is a mapping involving the absolute value of an expression.

moment
A moment is the turning effect of a force about a point.

mutually exclusive
When two events cannot occur at the same time, they are mutually exclusive.

natural logarithm
A natural logarithm is a logarithmic function with a base of the mathematical constant e; it is denoted by ln x. The natural logarithm of a positive real number x is defined as ln x = y, which means that e raised to the power of y equals x.

natural number
A natural number is a positive whole number.

negation
A negation is the opposite of a statement; if the statement is true, then the negation must be false.

negative correlation
Negative correlation occurs when an increase in one variable is associated with a decrease in another.

Newton’s first law
Newton’s first law of motion states that an object will remain stationary or continue to move with a constant velocity unless acted upon by an external force.

Newton’s second law
Newton’s second law of motion states that if a resultant force acts upon an object then it will accelerate.

Newton’s third law
Newton’s third law of motion states that if object A exerts a force on object B, then object B exerts a force of the same magnitude on object A, but in the opposite direction.

Newton–Raphson method
The Newton–Raphson method is a numerical method used for finding successively better approximations to the roots of a function. The formula for the Newton–Raphson method, for a given initial value x₀, is:

\( \textit{x}_{\scriptsize{n+1}} =\textit{x}_{\scriptsize{n}}-\frac{\textsf{f}\left(\textit{x}_{\scriptsize{n}}\right)}{\textsf{f}^{\prime}\left(\textit{x}_{\scriptsize{n}}\right)}
\)

normal
A normal is a line that is perpendicular to a given curve at a specific point; it is perpendicular to the tangent line at that point.

Normal distribution
The Normal distribution is a continuous distribution that is symmetrical about its mean.

normal reaction
The normal reaction is the force exerted on an object by a surface with which it is in contact. It acts perpendicular to the surface.

nth term
The nth term is a rule for defining a term of a sequence according to its position in the sequence.

null hypothesis
In a hypothesis test, the null hypothesis is the claim that the value of a population parameter has not changed.

numerator
The numerator is the number or expression on the top of a fraction.

one-to-many
A one-to-many mapping is a mapping where inputs can map to more than one output.

one-to-one
A one-to-one mapping is a mapping where inputs map to at most one output, and outputs result from at most one input.

opportunity sampling
Opportunity sampling is a method of non-random sampling where the sample is the most convenient individuals.

order
The order of a sequence is the period over which it repeats.

outcome
An outcome is a possible result of an experiment involving chance.

outlier
An outlier is a value that lies significantly outside the usual trend of the data.

parabola
A parabola is a U-shaped curve with a line of symmetry.

parallel
Parallel lines are lines that have the same gradient.

parameter
A parameter is a variable that other variables are defined in terms of.

parametric equations
Parametric equations are equations in which x and y are both defined in terms of another variable instead of the relationship between them being given directly.

partial fractions
Partial fractions are the fractions used in the decomposition of an algebraic fraction.

particle
An object modelled as a particle has no size or shape, only mass acting at a single point.

particular solution
A particular solution is a specific solution to a differential equation that satisfies the equation and any additional constraints or conditions imposed on the problem (the boundary conditions). Unlike the general solution, which contains arbitrary constants representing a family of solutions, a particular solution is unique and is fully determined once the appropriate conditions are applied.

Pascal’s triangle
Pascal’s triangle is a triangle formed by 1s at the start and end of each row and the sum of the two numbers above elsewhere; it can be used to find binomial coefficients.

peg
A peg is a dimensionless support.

period
In mathematics, a period is the length of the shortest interval over which the values of a function repeat. It is the smallest positive value p such that, for all x, the function satisfies f(x + p) = f(x)
The concept of period is applied to trigonometric functions such as sine, cosine, tangent and their reciprocal functions (cosecant, secant, cotangent), as well as to periodic functions in general. For example, the sine and cosine functions have a period of 2π radians (or 360°) and the tangent function has a period of π radians (or 180°).

periodic
A sequence is periodic if it has a repeating pattern.

perpendicular
Perpendicular lines are lines that meet at 90 degrees.

perpendicular bisector
A perpendicular bisector is a line that cuts a line segment into two equal parts and intersects it at 90 degrees.

piecewise function
A piecewise function is a function defined by separate rules for different intervals in the domain.

plane
A plane is a completely flat surface. Walls and floors are modelled as planes.

point of inflection
A point of inflection is a point on the graph of a function where the curvature changes direction; it changes from being concave to convex, or vice versa. At a point of inflection, the tangent line may cross the curve.

point of intersection
A point of intersection is a common point that two lines or curves pass through.

population
A population is all the individuals of interest in a survey.

population parameter
A population parameter is a number that describes a property of a whole population.

position vector
A position vector is a vector that represents the position of a point in space relative to a fixed reference point, usually the origin.

positive correlation
Positive correlation occurs when an increase in one variable is associated with an increase in another.

probability distribution
A probability distribution is a function or table showing the probability of every possible value of a random variable.

product moment correlation coefficient (PMCC)
The product moment correlation coefficient is a measure of the strength of the linear relationship between two variables.

product rule
The product rule is a formula used to find the derivative of the product of two functions.

If \( \textit{y}=\textsf{f}(\textit{x}) \textsf{g}(\textit{x}) \), then \( \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}}=\textsf{f}(\textit{x}) \textsf{g}^{\prime}(\textit{x})+\textsf{f}^{\prime}(\textit{x}) \textsf{g}(\textit{x}) \)

This formula can be remembered by the phrase ‘the derivative of the first times the second, plus the first times the derivative of the second’.

projectile
A projectile is an object that is subject to an initial force that causes it to move through the air, and then moves freely under gravity.

proof by contradiction
Proof by contradiction is a method of proving that a statement is true by showing that its negation is false.

proof by exhaustion
Proof by exhaustion is a method of proving that a statement is true by considering all the possible cases separately.

quadratic
A quadratic is an algebraic expression involving a power of 2 of a variable and with no higher powers.

quadratic formula
For the quadratic equation ax² + bx + c = 0, the quadratic formula gives the solutions:

\(\textit{x}=\frac{-\textit{b} \pm \sqrt{\textit{b}^{\scriptsize{2}} -4 \textit{a} \textit{c}}}{2 \textit{a}} \)

quartic
A quartic is an algebraic expression involving a power of 4 of a variable and with no higher powers.

quota sampling
Quota sampling is a method in which a certain number of individuals with specific characteristics are sampled; it is used in conjunction with non-random sampling.

quotient
A quotient is the result when one number or expression is divided by another.

quotient rule
The quotient rule is a formula used to find the derivative of a quotient of two functions that is particularly useful when dealing with functions that are expressed as the ratio of one function to another.

If \( \textit{y}=\frac{\textsf{f}(\textit{x})}{\textsf{g}(\textit{x})} \), then \( \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}}=\frac{\textsf{f}^{\prime}(\textit{x}) \textsf{g}(\textit{x})-\textsf{f}(\textit{x}) \textsf{g}^{\prime}(\textit{x})}{[\textsf{g}(\textit{x})]^{\scriptsize{2}} } \)

The formula can be remembered by the phrase ‘the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared’.

radian
A radian is a unit of angular measure defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. As the circumference of a circle is 2π times the radius, there are 2π radians in a full turn.

radius
A radius is a line segment from the centre to the circumference of a circle.

random sample
A random sample is a sample for which every member of a population has an equal chance of being selected; it is used with a sampling frame.

random variable
A random variable is a variable whose value is determined by chance.

range (mechanics)
The range is the maximum horizontal displacement of a projectile.

range (pure)
The range is the set of possible outputs of a mapping.

range of validity
The range of validity is an inequality that describes the values of a variable for which an expansion can be used.

rational number
A rational number is a number that can be expressed as a fraction with an integer numerator and denominator.

real number
All rational and irrational numbers are real numbers.

reciprocal
The reciprocal of a number or function is its multiplicative inverse; when a number or function is multiplied by its reciprocal the result is 1.

reciprocal function
A reciprocal function is a rule that involves dividing 1 by some function.

recurrence relation
A recurrence relation is a rule that defines each term of a sequence using the previous term or terms.

reflection
A reflection is a graph transformation in which a graph is flipped over a line.

The graph of y = f(−x) is a horizontal reflection in the line x = 0 (the y-axis) of the graph of y = f(x)

The graph of y = −f(x) is a vertical reflection in the line y = 0 (the x-axis) of the graph of y = f(x)

region
A region is an area on a graph.

regression line
A regression line is a straight line that can be drawn if there is a correlation between two variables plotted on a scatter graph; the regression line of y on x is written as y = a + bx

remainder
The remainder is a number or expression left over after a division.

repeated factor
A repeated factor is a factor raised to a power.

resolve
To resolve a force means to write it as two perpendicular components.

resultant force
A resultant force is a single force that can replace all of the forces acting on a system.

resultant vector
A resultant vector is a single vector that represents the combination or sum of two or more vectors. When multiple vectors act on a point or object simultaneously, their effects can be combined to form a single resultant vector that represents the net effect of all the individual vectors.

The resultant vector is often depicted as the vector that closes the triangle (or polygon) formed by the original vectors; this is known as the triangle law of vector addition.

The resultant vector can be calculated algebraically by adding or subtracting the components of the individual vectors.

reverse chain rule
The reverse chain rule describes the ‘reverse’ or ‘backwards’ application of the chain rule to find integrals. It allows you to simplify integrals by recognising a composite function and replacing it with a single variable, effectively ‘undoing’ the chain rule. It is a special case of integration by substitution.

rigid body
A rigid body is an object made up of particles, all of which remain at the same fixed distances from each other whether the object is at rest or in motion.

rod
A rod is an object that has mass and length, but no thickness. A rod does not bend; it is used for modelling tow bars and beams.

root
A root of a function is a value or values of the independent variable (often denoted by x) for which the function equals zero; it is a solution to the equation f(x) = 0

If a function has multiple roots, each of these values represents a point where the function crosses the x-axis.

rough
Rough is the opposite of smooth; friction is present.

sample
A sample is a smaller group of individuals that is used to represent the whole population in a statistical survey.

sample space
A sample space is the set of all possible outcomes.

scalar
A scalar is a quantity that is fully described as having magnitude without any associated direction. Scalars are distinguished from vectors, which are quantities that have both magnitude and direction.

secant
The secant, usually denoted by sec, is a trigonometric function that is the reciprocal of the cosine function. It is defined as \(\sec \theta=\frac{1}{\cos \theta} \)

second derivative
A second derivative is a measure of how the rate of change of the slope of a function varies with respect to its independent variable. It provides information about the curvature of the graph of the function.

If y = f(x) is a function of the variable x, then its second derivative is written as f′′(x) or \( \frac{\textsf{d}^2 \textit{y}}{\textsf{d} \textit{x}^2} \)

separating the variables
Separating the variables is a technique used to solve first order differential equations of the form \( \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}}=\textsf{f}(\textit{x}) \textsf{g}(\textit{y}) \)

The equation is rewritten so that each variable appears on only one side of the equation, which allows the variables to be ‘separated’ and then integrated separately.

If \( \frac{\textsf{d} \textit{y}}{\textsf{d} \textit{x}}=\textsf{f}(\textit{x}) \textsf{g}(\textit{y}) \), then \( \int \frac{1}{\textsf{g}(\textit{y})} \textsf{d} \textit{y}=\int \textsf{f}(\textit{x}) \textsf{d} \textit{x} \)

series
A series is the sum of a sequence.

significance level
In a hypothesis test, a significance level is an indication of the strength of evidence required to reject a null hypothesis; the actual significance level of a test is the probability of incorrectly rejecting the null hypothesis.

simple random sampling
Simple random sampling is a sampling method in which each member of the population has an equal chance of being chosen, such as by using a random number generator.

simultaneous equations
Simultaneous equations are two or more equations that must be satisfied by the same values of the variables.

sine rule
The sine rule is a formula used in trigonometry to find the relationship between the angles and sides of a triangle. It states:

\( \frac{\textit{a}}{\sin \textit{A}}=\frac{\textit{b}}{\sin \textit{B}}=\frac{\textit{c}}{\sin \textit{C}} \)

where a, b and c are the sides of the triangle and A, B and C are the angles opposite the corresponding sides.

You can rearrange the sine rule to find an angle:

\( \frac{\sin \textit{A}}{\textit{a}}=\frac{\sin \textit{B}}{\textit{b}}=\frac{\sin \textit{C}}{\textit{c}} \)

small angle approximation
Small angle approximation is a technique used to simplify trigonometric functions involving angles that are very small. It is based on the idea that, for small angles, the sine, cosine and tangent of the angle can be approximated by the angle itself, expressed in radians.

If θ is a small angle measured in radians, then the small angle approximations are:

\(
\begin{aligned}
& \sin \theta \approx \theta \\
& \cos \theta \approx 1-\frac{\theta^{\scriptsize{2}}}{2} \\
& \tan \theta \approx \theta
\end{aligned}
\)

smooth
A smooth surface, hinge or pulley has no friction.

speed
Speed is the magnitude of velocity. The SI unit for speed is metres per second.

staircase diagram
A staircase diagram is a diagram resembling a staircase that shows convergence to a root when using the x = g(x) method of iteration.

standard deviation
The standard deviation is a measure of the average dispersion of data from the mean; it is the square-root of the variance.

standard Normal distribution
The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1

stationary point
A stationary point is a point on the graph of a function where the derivative is zero.

stratified sampling
Stratified sampling is a sampling method by which the population is split into strata (layers) so that the proportion of each strata in the sample is the same as in the population.

stratify
To stratify means to split a population into strata (layers) according to particular characteristics.

strict inequality
A strict inequality is an inequality for which the boundary value or values are not included; it uses < or >

substitution method
The substitution method is a method of solving simultaneous equations where one equation is rearranged and then substituted into the other.

subtend
Two lines subtend an angle when they meet at a point.

sum to infinity
The sum to infinity is the limit of a sequence as the number of terms tends to infinity.

surd
A surd is the square root of a non-square number.

systematic random sampling
Systematic random sampling is a sampling method in which individuals are chosen according to a rule.

tangent
A tangent is a line that touches a curve exactly once without crossing it.

tension
Tension is a pulling force that is transmitted along a string or rod when it is pulled tight.

test statistic
A test statistic is a result from a sample that is used to determine the conclusion of a hypothesis test.

thrust
Thrust is the opposite of tension. It occurs along a rod or spring when a force acts at both ends to push inwards.

translation
A translation is a graph transformation in which a graph is moved horizontally and/or vertically.

The graph of y = f(x + a) is a translation of the graph of y = f(x) by the vector \( \binom{-\textit{a}}{0} \)

The graph of y = f(x) + a is a translation of the graph of y = f(x) by the vector \( \binom{0}{\textit{a}} \)

trapezium rule
The trapezium rule is a numerical method used to approximate the definite integral of a function over a given interval. It works by approximating the area under the curve of the function by dividing the interval into small trapezia and summing up their areas. The trapezium rule is particularly useful when the function is difficult or impossible to integrate analytically.

The formula for the trapezium rule with n trapezia is:

\( \int_{\scriptsize{a}} ^{\scriptsize{b}} \textit{y} \textsf{ d} \textit{x} \approx \frac{1}{2} ~\textit{h}\left\{\left(\textit{y}_{\scriptsize{0}} +\textit{y}_{\scriptsize{n}} \right)+2\left(\textit{y}_{\scriptsize{1}} +\textit{y}_{\scriptsize{2}} +\cdots+\textit{y}_{\scriptsize{n-1}} \right)\right\} \), where \( \textit{h}=\frac{\textit{b}-\textit{a}}{\textit{n}} \)

The accuracy of the trapezium rule increases as the number of strips (trapezia) increases, approaching the exact value of the integral as this number approaches infinity.

triangle law
The triangle law is a fundamental principle in vector addition that states that if two vectors are represented by two sides of a triangle, then their resultant vector is represented by the third side of the triangle, taken from the initial point of the first vector to the terminal point of the second vector.

If u and v are two vectors, then the resultant vector r obtained by adding these vectors is represented by the vector that completes the triangle formed by u and v

turning point
A turning point is a point on the graph of a function where the direction of the curve changes. It is where the function changes from increasing to decreasing (local maximum), or from decreasing to increasing (local minimum). At a turning point, the tangent line is horizontal, indicating that the rate of change of the function is zero.

two-tailed test
A two-tailed test is a hypothesis test to see whether a population parameter has changed (without specifying an increase or a decrease).

uniform
An object is uniform if its mass is evenly distributed. A uniform rod is modelled with the weight acting at the centre.

unit circle
A unit circle is a circle with a radius of 1 unit, typically centred at the origin. In trigonometry, the unit circle plays a crucial role in understanding the relationships between the angles and trigonometric ratios. Since the radius of the unit circle is 1, any point P(x, y) on the circle can be represented in terms of the cosine and sine of an angle θ, measured anticlockwise from the positive x-axis, as P(cos θ, sin θ)

unit vector
A unit vector is a vector with a magnitude of 1

You can find the unit vector, \( \mathbf {\hat {a} } \), in the direction of a given vector a by dividing a by its magnitude:

\( \mathbf {\hat{a} }=\frac{\mathbf{a}}{|\mathbf{a}|} \)

variance
The variance is a measure of the spread of data; it is the square of the standard deviation.

vector
A vector is a quantity that has both magnitude and direction. Vectors are distinguished from scalars, which are quantities that have only magnitude.

vector cosine
The vector cosine is the cosine of the angle between a vector and a coordinate axis; it is also known as the directional cosine.

For vector a = xi + yj + zk that makes angles of θ_x, θ_y and θ_z with the x-, y– and z-axes respectively,

\( \cos \theta_{\scriptsize{x}} =\frac{\textit{x}}{|\mathbf{a}|} \), \(\cos \theta_{\scriptsize{y}} =\frac{\textit{y}}{|\mathbf{a}|} \) and \( \cos \theta_{\scriptsize{z}} =\frac{\textit{z}}{|\mathbf{a}|} \)

velocity
Velocity is the rate of change of displacement. The SI unit for velocity is metres per second.

vertical stretch
A vertical stretch is a graph transformation in which the x-coordinates of points are fixed and the y-coordinates are multiplied by a scale factor.

The graph of y = kf(x) is a vertical stretch with scale factor k of the graph of y = f(x)

weight

Weight is the force due to gravity acting vertically downwards.