Maths AA terms: D

D

De Moivre's theoremHL only

For any integer $n$, $[r(\cos\theta + \mathrm{i}\sin\theta)]^n = r^n(\cos n\theta + \mathrm{i}\sin n\theta)$: to raise a complex number to a power, raise the modulus to that power and multiply the argument by it.

Introduced in 1.14 Complex roots and De Moivre’s theorem

Decimal placesalso d.p.

The number of digits written after the decimal point. Rounding to 2 decimal places keeps two digits after the point, as in $3.14159 \approx 3.14$.

Introduced in 1.1 Scientific notation

Decreasing functionalso decreasing, strictly decreasing

A function is decreasing on an interval if larger inputs give smaller outputs there: $x_1 < x_2$ implies $f(x_1) > f(x_2)$

Definite integralalso limits of integration

The number $\int_a^b f(x)\,dx$, the limit of sums of $f(x)\,\delta x$ from $x = a$

Dependent eventsalso dependence

Events that are not independent: knowing that one has occurred changes the probability of the other, so $P(A \cap B) \neq P(A)P(B)$.

Introduced in 4.11 Conditional probability and independence

Depreciation

The loss in value of an asset, such as a car or a machine, over time. With a fixed annual percentage rate the values form a geometric sequence with ratio less than 1.

Introduced in 1.4 Financial applications of geometric sequences

Derivativealso gradient function, f′(x), dy/dx

The function $f'(x)$, also written $\frac{dy}{dx}$, that gives the gradient of the curve $y = f(x)$

DifferentiabilityHL only · also differentiable

A function is differentiable at $x = a$ if $\lim_{h \to 0}\frac{f(a + h) - f(a)}{h}$

Differential equationHL only · also DE, first-order differential equation

An equation that relates an unknown function to one or more of its derivatives, such as $\frac{dy}{dx} = 2xy$. It is first-order when the highest derivative it contains is the first derivative.

Introduced in 5.18 First-order differential equations

Differentiationalso differentiate, differentiating

The process of finding the derivative of a function. Differentiating $y = f(x)$ with respect to $x$ gives $\frac{dy}{dx} = f'(x)$

Differentiation from first principlesHL only · also first principles

Finding a derivative directly from its definition $f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$

Direction vectorHL only

A non-zero vector parallel to a line, written $\boldsymbol{b}$ in $\boldsymbol{r} = \boldsymbol{a} + \lambda\boldsymbol{b}$. Any non-zero multiple of it is also a direction vector of the same line.

Introduced in 3.14 Vector equations of lines

Disc methodHL only · also disk method

Finding a volume of revolution by slicing the solid perpendicular to the axis of rotation into thin discs of volume $\pi r^2\,\delta x$ and adding them with an integral.

Introduced in 5.17 Areas and volumes of revolution

DiscontinuityHL only · also discontinuous, jump discontinuity

A point where a function is not continuous, such as a jump (the one-sided limits differ), a hole (the limit exists but is not $f(a)$) or a vertical asymptote.

Introduced in 5.12 Continuity, differentiability and first principles

Discrete data

Numerical data that can take only separate, distinct values (usually counts), such as the number of goals scored in a match.

Introduced in 4.1 Populations, samples and sampling

Discrete random variablealso random variable

A variable whose value is a numerical outcome of a random process and which can take only separate (countable) values, such as $0, 1, 2, \dots$. It is written with a capital letter, such as $X$.

Introduced in 4.7 Discrete random variables

Discrete uniform distributionalso uniform distribution

A probability distribution in which each of the $n$ possible values of a discrete random variable has the same probability $\frac{1}{n}$, such as the score on a fair die.

Introduced in 4.7 Discrete random variables

Discriminantalso Δ

The quantity $\Delta = b^2 - 4ac$ for the quadratic $ax^2 + bx + c$

Displacement

The position of a particle relative to a fixed origin, measured along a line with a sign for direction, usually written $s$. A change in displacement is final position minus initial position.

Introduced in 5.9 Kinematics

Displacement vectorHL only

The vector from one point to another, giving the change in position: $\overrightarrow{AB} = \boldsymbol{b} - \boldsymbol{a}$, where $\boldsymbol{a}$

Divergent seriesalso diverges, divergence

An infinite series whose partial sums do not approach a finite limit: they grow without bound or keep oscillating. A geometric series with $|r| \ge 1$ and $u_1 \neq 0$ diverges.

Introduced in 1.8 Infinite geometric series

Domain

The set of all input values for which a function is defined, or for which it is used in a context.

Introduced in 2.2 Functions, domain, range and inverse

Dominant termHL only · also leading term, highest power

The term of an expression that grows fastest as $x \to \infty$ and so decides its behaviour; in a polynomial it is the term with the highest power of $x$. For example, $3x^2$ is the dominant term of $3x^2 - 5x + 1$

Double angle identityalso double angle formula, double angle formulae

An identity expressing a trigonometric ratio of $2\theta$ in terms of ratios of $\theta$, such as $\sin 2\theta = 2\sin\theta\cos\theta$ and $\cos 2\theta = 1 - 2\sin^2\theta$