Maths glossary template
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absolute value 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: 102 \(\begin{aligned} & \sin (A \pm B)=\sin A \cos B \pm \cos A \sin B \\ & \cos (A \pm B)=\cos A \cos B \mp \sin A \sin B \\ & \tan (A \pm B)=\frac{\tan A \pm \tan B}{1-\tan A \tan B} \end{aligned} \) 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. |
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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) = abx the base is the number b Common bases include:
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. |
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census A census is a survey in which information is collected from every member of a population. 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: \( \frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x}\)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 \(\binom{x}{y}\) In three dimensions, a column vector is written as \(\left(\begin{array}{l} x \\ y \\ z \end{array}\right) \)
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: a2 = b2 + c2 – 2bc cos 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: c2 = a2 + b2 – 2ab cos C You can also rearrange the cosine rule to find an angle, for example: \(\cos A=\frac{b^2+c^2-a^2}{2 b c}\)cotangent The cotangent, usually denoted by cot, is a trigonometric function that is the reciprocal of the tangent function. It is defined as \(\cot \theta=\frac{1}{\tan \theta}\) 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. |
| D |
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(x2); 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 \(\int_a^b y d x\) 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 \(\frac{d y}{d x}=f(x) g(y)\) 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:
To find the derivative of a function using first principles, follow these steps:
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: sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A = 2 cos2 A − 1 = 1 – 2 sin2 A
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