Pancyprian Advanced Math key terms

A rule assigning to each element x of a domain exactly one element f(x) of a codomain. For every input there is a single output, and the domain, codomain and rule together define it.

The domain is the set of all inputs for which a function is defined; the range is the set of all outputs it actually takes. Finding them requires checking restrictions such as division by zero and square roots.

The function that reverses another: if f(a) = b then f⁻¹(b) = a. It exists only when f is one-to-one, and its graph is the reflection of the graph of f in the line y = x.

The function formed by applying one function to the output of another, written (f∘g)(x) = f(g(x)). Its domain consists of those x in the domain of g for which g(x) lies in the domain of f.

A property describing whether a function consistently increases or decreases. A function is strictly increasing if a < b always gives f(a) < f(b); strictly decreasing if a < b always gives f(a) > f(b).

A line that the graph of a function approaches arbitrarily closely as x or y tends to infinity, or as the function tends to a point where it is undefined. It may be vertical, horizontal or oblique.

A function expressible as the quotient of two polynomials, f(x) = P(x)/Q(x) with Q not identically zero. Its behaviour near the zeros of Q produces vertical asymptotes or removable holes.

Functions of the form f(x) = aˣ with a > 0, a ≠ 1, and their inverses f(x) = logₐ x. They are mutual inverses, and the natural base e gives the functions whose derivatives are themselves.

An expression of the form aₙxⁿ + … + a₁x + a₀ with real coefficients and non-negative integer powers. Its degree is the highest power with non-zero coefficient, and it is defined for all real x.

Dividing a polynomial P(x) by (x − a) leaves remainder P(a); (x − a) is a factor exactly when P(a) = 0. These theorems link polynomial division with roots and factorisation.

The value L that f(x) approaches as x approaches a point a, written lim f(x) = L as x → a. It concerns behaviour near a, not the value at a, and may be one-sided.

A function is continuous at a point a when lim f(x) as x → a equals f(a): the limit exists, the value exists, and they agree. Continuity on an interval means continuity at every point of it.

The instantaneous rate of change of a function, defined as the limit of the difference quotient [f(x+h) − f(x)]/h as h → 0. Geometrically it is the slope of the tangent line at a point.

A function is differentiable at a point when its derivative exists there, which requires the difference quotient to have the same limit from both sides. Differentiability at a point implies continuity there, but not conversely.

The straight line touching a curve at a point with slope equal to the derivative there, with equation y − f(a) = f′(a)(x − a). It gives the best linear approximation to the curve near that point.

The line through a point of a curve perpendicular to the tangent there. Its slope is the negative reciprocal of the derivative, −1/f′(a), provided the derivative is non-zero.

The rule for differentiating a composite function: if y = f(g(x)) then dy/dx = f′(g(x))·g′(x). It is the most frequently used differentiation rule in compound expressions.

Rolle's theorem states that a function continuous on [a, b], differentiable on (a, b) and equal at the endpoints has a point with zero derivative inside. The Mean Value Theorem generalises this to a tangent parallel to the chord.

A local maximum or minimum of a function, where the value is largest or smallest in some surrounding interval. It is located by solving f′(x) = 0 and checking the sign change of the derivative.

The derivative of the derivative, f″(x), describing how the slope changes. Its sign determines concavity: f″ > 0 means convex (concave upward), f″ < 0 concave downward, and a sign change gives an inflection point.

If f is continuous on a closed interval [a, b], it takes every value between f(a) and f(b) at least once inside. It is the standard tool for proving that an equation has a root in an interval.

The family of all antiderivatives of a function, written ∫f(x)dx = F(x) + c where F′ = f. The arbitrary constant c reflects that antiderivatives differ only by a constant.

The signed area between a curve and the x-axis from x = a to x = b, defined as a limit of Riemann sums and evaluated as F(b) − F(a) by the Fundamental Theorem of Calculus.

It links differentiation and integration: if f is continuous then the area function F(x) = ∫f(t)dt from a to x satisfies F′(x) = f(x), and the definite integral equals G(b) − G(a) for any antiderivative G.

The integration analogue of the product rule: ∫u dv = uv − ∫v du. It is used when an integrand is a product, choosing u to simplify on differentiation and dv to be easily integrated.

An integration technique reversing the chain rule: replace a compound part of the integrand by a new variable u, convert dx to du, integrate, then substitute back. For definite integrals the limits must also be converted.

The region enclosed between two graphs from x = a to x = b has area ∫|f(x) − g(x)|dx, taking the upper curve minus the lower. Intersection points usually determine the limits of integration.

Rotating a region about an axis produces a solid whose volume is found by integrating cross-sectional areas. For rotation about the x-axis, V = π∫[f(x)]²dx between the appropriate limits.

An ordered list of numbers a₁, a₂, a₃, … indexed by the positive integers, defined either by a general term aₙ or by a recurrence relation. Its behaviour as n grows determines convergence.

A sequence converges to L if the terms get arbitrarily close to L for all sufficiently large n. Otherwise it diverges, either to infinity or by oscillation.

An arithmetic progression has a constant difference d, with nth term a + (n−1)d. A geometric progression has a constant ratio r, with nth term arⁿ⁻¹ and sum a(1 − rⁿ)/(1 − r) for r ≠ 1.

An infinite geometric series with ratio r converges only when |r| < 1, and then its sum is a/(1 − r). If |r| ≥ 1 the series diverges and no finite sum exists.

The sum of the terms of a sequence, written Σaₙ. A series converges when the sequence of its partial sums converges to a finite limit, and diverges otherwise.

It expands (a + b)ⁿ as the sum of terms C(n, k)aⁿ⁻ᵏbᵏ for k from 0 to n, where C(n, k) = n!/[k!(n−k)!]. The coefficients form the binomial (Pascal) triangle.

The number C(n, k), read n choose k, counting the ways of selecting k objects from n without regard to order. In the binomial expansion it is the coefficient of the term containing bᵏ.

The formula for the k-th term of a binomial expansion, Tₖ₊₁ = C(n, k)aⁿ⁻ᵏbᵏ. It lets you find a specific power or coefficient without expanding the whole expression.

A number of the form z = a + bi with real a and b and i² = −1. The real part is a and the imaginary part is b; complex numbers extend the real line to a plane.

The conjugate of z = a + bi is z̄ = a − bi, the reflection of z in the real axis. Products z·z̄ are real, which makes conjugates essential for dividing complex numbers.

The modulus |z| is the distance from the origin, √(a² + b²); the argument is the angle θ the vector makes with the positive real axis. Together they give the polar form z = |z|(cos θ + i sin θ).

Writing z = r(cos θ + i sin θ), De Moivre's theorem states that zⁿ = rⁿ(cos nθ + i sin nθ). It simplifies powers and roots, and the nth roots are equally spaced around a circle.

A quantity with both magnitude and direction, represented by a directed segment or by components. Two vectors are equal when they have the same magnitude and direction, regardless of position.

The scalar a·b = |a||b|cos θ, also computed as the sum of products of corresponding components. It measures alignment: it is zero exactly when the vectors are perpendicular.

A vector product defined for vectors in space, with magnitude |a||b|sin θ and direction perpendicular to both by the right-hand rule. It gives the area of the parallelogram spanned by the vectors.

Points are collinear when the vectors between them are parallel, i.e. one is a scalar multiple of another. Points or lines are coplanar when a scalar triple product of spanning vectors equals zero.

A line is given by a point plus a direction vector, r = a + λb, or in Cartesian form by equal ratios. A plane is given by a point and normal vector, (r − a)·n = 0.

A curve obtained by cutting a cone with a plane: circle, ellipse, parabola or hyperbola. Each has a standard Cartesian equation with characteristic foci, directrix and eccentricity.

The set of points at fixed distance r from a centre (a, b) satisfies (x − a)² + (y − b)² = r². The general form x² + y² + Dx + Ey + F = 0 is completed to the square to reveal centre and radius.

The number e describing the shape of a conic, equal to the ratio of distance to focus over distance to directrix. For a circle e = 0, an ellipse 0 < e < 1, a parabola e = 1, a hyperbola e > 1.

A number between 0 and 1 measuring how likely an event is, equal to favourable outcomes divided by total equally likely outcomes in a classical sample space. The probabilities of all outcomes sum to 1.

The probability of event A given that B has occurred, written P(A|B) = P(A ∩ B)/P(B) with P(B) > 0. It restricts the sample space to the outcomes in B.

Events A and B are independent when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Knowing one occurred gives no information about the other.

A random variable assigns a numerical value to each outcome of an experiment; its distribution lists those values with their probabilities. The sum of probabilities must equal 1.

The expected value E(X) = Σx·P(X = x) is the long-run mean of a random variable. The variance Var(X) = E(X²) − [E(X)]² measures the spread about that mean.

The distribution of the number of successes in n independent trials each with success probability p, giving P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. Its mean is np and variance np(1 − p).

The positive square root of the variance, expressed in the same units as the data. It measures typical distance from the mean and is the basis of standardising values.

A continuous bell-shaped distribution symmetric about its mean μ with spread σ, denoted N(μ, σ²). Probabilities are found by standardising to z = (x − μ)/σ and using tables of the standard normal.