Maths AA terms: E

E

End behaviourHL only

How the values of a function behave as $x \to \infty$ and as $x \to -\infty$. For a polynomial it depends only on the degree and the sign of the leading coefficient.

Introduced in 2.12 Polynomial functions

Equally likely outcomesalso equally likely

Outcomes that all have the same chance of occurring, such as the six faces of a fair die. Only then does $P(A) = \frac{n(A)}{n(U)}$ apply.

Introduced in 4.5 Concepts of probability

Equating coefficientsHL only · also comparing coefficients

Setting the coefficients of each power of $x$ on the two sides of an identity equal to each other, to find unknown constants.

Introduced in 1.11 Partial fractions

Equation

A statement that two expressions are equal, which may be true for only some values of the variable. Solving an equation means finding those values; for example $2x + 1 = 7$ is true only when $x = 3$.

Introduced in 1.6 Simple deductive proof

Equation in quadratic formalso hidden quadratic, disguised quadratic

An equation that becomes a quadratic equation after a substitution such as $u = x^2$, $u = e^x$ or $u = \sin x$

Euler formHL only · also exponential form

The way of writing a complex number with modulus $r$ and argument $\theta$ as $z = r\mathrm{e}^{\mathrm{i}\theta}$, using Euler's formula $\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta$

Euler's formulaHL only · also Euler's identity

The identity $\mathrm{e}^{\mathrm{i}\theta} = \cos\theta + \mathrm{i}\sin\theta$ for real $\theta$, linking the exponential function with the circular functions. With $\theta = \pi$

Euler's methodHL only

A numerical method that approximates the solution of $\frac{dy}{dx} = f(x, y)$ by short straight steps along the gradient: $y_{n+1} = y_n + h f(x_n, y_n)$

Euler's number ealso e

The irrational constant $e \approx 2.71828$, the limit of $\left(1 + \frac{1}{n}\right)^n$

Even functionHL only

A function for which $f(-x) = f(x)$ for every $x$ in its domain. Its graph is symmetric in the $y$-axis, for example $y = x^2$

Even numberalso even integer

An integer that is divisible by $2$, so it can be written as $2n$ for some integer $n$. Examples are $-4$, $0$ and $18$.

Introduced in

Eventalso events

A set of outcomes from the sample space, such as "an even number" when a die is rolled. An event occurs when the outcome of the trial is one of its outcomes.

Introduced in 4.5 Concepts of probability

Exact valuealso exact form, exact answer

A value written without rounding, using fractions, surds or $\pi$, such as $\sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$

Exhaustive eventsHL only · also partition

Events that together cover the whole sample space, so at least one of them must occur. Mutually exclusive, exhaustive events $B_1, B_2, B_3$ form a partition of $U$

Expected number of occurrencesalso expected frequency

The number of times an event is predicted to occur in $n$ trials when its probability is $p$, equal to $np$.

Introduced in 4.5 Concepts of probability

Expected valuealso E(X), expectation

The long-run mean value of a random variable over many repetitions, written $E(X)$ or $\mu$. For a discrete random variable, $E(X) = \sum x\,P(X = x)$

Exponentalso index, power, indices

The number written as a superscript in a power such as $a^n$, showing how many times the base is used as a factor when $n$ is a positive integer. For example, in $2^5$ the exponent is $5$.

Introduced in 1.5 Exponents and introduction to logarithms

Exponential decayalso half-life

Decrease in which a quantity is multiplied by the same factor between 0 and 1 over each equal time interval, as in $N = N_0 e^{kt}$ with $k < 0$. The time taken to halve,

Exponential equation

An equation in which the unknown appears in an exponent, such as $2^{x+1} = 5$. It is solved by writing both sides with the same base or by taking logarithms of both sides.

Introduced in 1.7 Laws of exponents and logarithms

Exponential function

A function of the form $f(x) = a^x$, where the base $a$ is a positive constant and $a \neq 1$

Exponential growthalso doubling time

Growth in which a quantity is multiplied by the same factor greater than 1 over each equal time interval, as in $N = N_0 e^{kt}$ with $k > 0$. The time taken to double,

Extended binomial theoremHL only · also binomial series

The infinite series $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots$

Extraneous solutionalso false solution

A value produced while solving an equation, for example after squaring both sides or splitting into cases, that does not satisfy the original equation and must be rejected.

Introduced in 1.7 Laws of exponents and logarithms

Extrapolationalso interpolation

Using a regression line to predict a value outside the range of the data used to find it. Predicting inside that range is interpolation, which is much more reliable.

Introduced in 4.4 Linear correlation and regression