Maths AA terms: L

L

L'Hôpital's ruleHL only · also L'Hopital, l'Hospital's rule

A method for limits of the indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$: if $\lim_{x \to a}\frac{f(x)}{g(x)}$ is of either form, then $\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}$, provided the second limit exists. It also holds as $x \to \infty$.

Introduced in 5.13 Limits and L’Hôpital’s rule

Law of large numbers

The result that, as the number of trials increases, the relative frequency of an event tends to get closer to its probability.

Introduced in 4.5 Concepts of probability

Law of total probabilityHL only · also total probability

For mutually exclusive events $B_1, B_2, \dots$ that cover the sample space, $P(A) = P(B_1)P(A \mid B_1) + P(B_2)P(A \mid B_2) + \dots$

LIATE ruleHL only · also LIATE

A guide for choosing $u$ in integration by parts: take as $u$ the factor that comes first in the list Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential.

Introduced in 5.16 Integration by substitution and by parts

Limitalso limiting value, tends to

The value $L$ that $f(x)$ gets as close as you like to as $x$ gets closer and closer to $a$ (without being equal to $a$), written $\lim_{x \to a} f(x) = L$

Limit at infinityHL only · also limits at infinity

The value $L$ that $f(x)$ approaches as $x$ increases without bound, written $\lim_{x \to \infty} f(x) = L$

Line of intersectionHL only

The line common to two non-parallel planes. Its direction vector is perpendicular to both normals, for example $\boldsymbol{n}_1 \times \boldsymbol{n}_2$.

Introduced in 3.18 Intersections of lines and planes

Linear differential equationHL only · also first-order linear differential equation

A first-order differential equation that can be written as $\frac{dy}{dx} + P(x)y = Q(x)$, in which $y$

Linear equationHL only

An equation in which each unknown appears only to the first power and is multiplied only by a constant, such as $2x - y + 3z = 7$.

Introduced in 1.16 Systems of linear equations

Linear model

A model of the form $y = mx + c$, in which one quantity changes at a constant rate $m$ with respect to another, starting from the initial value $c$.

Introduced in 2.1 Equations of straight lines

Local maximumalso maximum point, relative maximum

A point on a graph that is higher than all nearby points, where the graph changes from increasing to decreasing.

Introduced in 2.4 Key features of graphs

Local minimumalso minimum point, relative minimum

A point on a graph that is lower than all nearby points, where the graph changes from decreasing to increasing.

Introduced in 2.4 Key features of graphs

Logarithmalso log

The exponent to which a base $a$ must be raised to give a number $b$: $\log_a b = x$ exactly when $a^x = b$

Logarithmic differentiationHL only

Taking the natural logarithm of both sides of $y = f(x)$ and then differentiating implicitly. It is used for functions such as $x^x$, where the variable is in both the base and the exponent, and to simplify long products and quotients.

Introduced in 5.15 Derivatives of further functions

Logarithmic equationalso log equation

An equation in which the unknown appears inside a logarithm, such as $\log_2(x + 3) = 5$. Every solution must make the argument of each logarithm positive.

Introduced in 1.7 Laws of exponents and logarithms

Logarithmic function

A function of the form $f(x) = \log_a x$, $x > 0$, where $a > 0$

Logistic modelHL only · also logistic growth, logistic equation

A model of limited growth, $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$