Glossary

Every term, in one place

The same 25 definitions the lessons link to inline, wherever a sentence uses them — gathered here so a definition is never more than one page away.

A

antiderivativeindefinite integral: A function F(x) whose derivative is a given f(x), i.e. F'(x) = f(x) — found by reversing a standard differentiation result term by term. Every antiderivative of the same f(x) differs only by a constant, which is exactly why that constant cancels out of a definite integral's subtraction.
arithmetic sequencearithmetic series, arithmetic progression: A sequence generated by adding the same fixed common difference d at every step, uₙ = a + (n−1)d — fully determined by just two numbers, a and d, which is why a question giving any two terms is secretly two simultaneous equations. Its sum formula, Sₙ = ½n[2a+(n−1)d], is proved by writing the series forwards and in reverse and adding, since every matched pair then totals the same constant.
asymptotehorizontal asymptote, vertical asymptote: A line a curve approaches arbitrarily closely without ever touching or crossing it — a horizontal asymptote where a function settles as x → ±∞ (as eˣ does at y = 0), or a vertical one where a function diverges because a denominator or an argument hits zero (as ln x does at x = 0, or tan θ does at θ = 90°).

B

binomial expansionbinomial theorem: The expansion of (a+bx)ⁿ for a positive integer n into n+1 terms, ∑ C(n,r)a^(n−r)(bx)^r — each term's binomial coefficient is a headcount of how many ways r of the n brackets can contribute the bx, not a formula to memorise separately from that counting argument.

C

change of base formula: logₐx = log_b(x) / log_b(a), converting a logarithm in one base into logs in whatever base a calculator actually offers — the one log law printed in the exam formula booklet, since the other four have to be recalled from memory.
common differenced (common difference): The fixed amount added to each term of an arithmetic sequence to reach the next, uₙ₊₁ − uₙ = d — constant by definition, the two-second check that tells an arithmetic sequence apart from a geometric one, which has a constant RATIO instead.
common ratior (common ratio): The fixed multiplier between consecutive terms of a geometric sequence, uₙ₊₁ / uₙ = r — constant by definition, unlike an arithmetic sequence's constant DIFFERENCE, and the quantity whose size (not sign) decides whether the sequence's sum to infinity exists.
completing the square: Rewriting a quadratic ax² + bx + c as a(x − p)² + q by building the perfect square matching its x² and x terms, then correcting the constant. Reads off a turning point, a range, or — for x² + y² + 2gx + 2fy + c = 0 — a circle's centre and radius directly, with the coordinate always the NEGATIVE of the value inside the bracket.
counter-exampledisproof by counter-example, counterexample: A single case that fails a 'for all' claim, which is enough on its own to disprove it. A claim true for every case in a set is really one AND-statement per case, and one false part makes the whole AND false — which is why disproof never needs more than one case, however large or unbounded the set is.

D

discriminantb² − 4ac: The quantity b² − 4ac for a quadratic ax² + bx + c, whose sign decides how many real roots it has — positive for two, zero for one repeated, negative for none. Checked whenever a quadratic newly appears (from a substitution, a division, or a factorisation search) before spending time hunting for real roots that may not exist.

E

exponential functiony = aˣ: A function y = aˣ with a fixed base a (a > 0, a ≠ 1) and the variable in the exponent — always positive, always passing through (0,1), with a horizontal asymptote at y = 0 and no x-intercept, increasing for a > 1 and decreasing for 0 < a < 1.

G

geometric sequencegeometric series, geometric progression: A sequence generated by multiplying by the same fixed common ratio r at every step, uₙ = ar^(n−1) — fully determined by a and r. Its finite-sum formula, Sₙ = a(1−rⁿ)/(1−r), is proved by multiplying the whole series by r and subtracting, since that shift makes every term but two cancel.

L

laws of logarithmslog laws, product law (logarithms), quotient law (logarithms), power law (logarithms): Five identities following from the laws of indices applied to a logarithm's own definition: logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, logₐ(xᵏ) = k·logₐx, logₐ(1/x) = −logₐx, and logₐa = 1. A log of a SUM never splits apart, and subtracting two logs divides their arguments — it never divides the logs themselves.

O

ordinate: A y-value read off a curve at a specified x-value, used as one of the heights in a trapezium-rule calculation — n trapezium-rule strips always need n+1 ordinates, since each interior ordinate marks the shared edge of two adjoining strips.

P

Pascal's triangle: A triangular array in which each entry is the sum of the two entries above it, whose nth row (counting the top '1' as row 0) gives exactly the n+1 binomial coefficients of (a+bx)ⁿ — the same headcount as ⁿCᵣ, built up one bracket at a time instead of computed by one factorial calculation.
periodic sequence: A sequence that repeats its exact pattern of values every k terms forever, uₙ₊ₖ = uₙ for every n — genuinely different from being constant, since the individual terms can keep changing as long as the whole pattern comes back around.
point of inflectioninflexion point, stationary point of inflection: A point where a curve's concavity changes — where d²y/dx² changes sign — which does not require the gradient to be zero there at all. The special case where a genuine stationary point (dy/dx = 0) also happens to be one is exactly what makes the second derivative test return zero: the gradient's own rate of change is momentarily zero without deciding whether it was increasing or decreasing.
proof by exhaustionexhaustion, proof by cases: Proving a 'for all' claim true by checking every case in a finite, explicitly bounded domain — available only when the set of cases can actually be listed. Needs the bound stated, every case listed, each one checked, and a closing conclusion; a completed table with no closing sentence is not yet a finished proof.

R

recurrence relationrecurrence formula: A rule of the form xₙ₊₁ = f(xₙ) that generates each term from the one before it, given a starting value — genuinely different from a formula for the nth term, since reaching term n from term 1 always takes (n−1) applications of the rule, not n.
reference angle: The acute angle whose trig ratio has the same SIZE as the one actually being solved for — found first from the calculator's principal value, then used together with which quadrant a function is positive or negative in to generate every other solution in the required interval.

S

second derivative test: Substituting a stationary point's own x-value into d²y/dx² to classify it: positive means a minimum, negative means a maximum, and exactly zero is inconclusive — the fallback then being to test the sign of dy/dx itself just below and just above the point.
sum to infinityS∞: The finite limit a geometric series' partial sums approach as more terms are added, S∞ = a/(1−r), which exists only when |r| < 1 — the condition under which rⁿ shrinks to zero as n grows, since rⁿ is the only part of the finite-sum formula that depends on n at all.

T

trapezium rule: A method for approximating ∫ₐᵇy dx by replacing the curve with a chain of straight-line trapezia, ∫ₐᵇy dx ≈ ½h{(y₀+yₙ) + 2(y₁+…+yₙ₋₁)} with h = (b−a)/n — every interior ordinate is doubled because it is shared between two neighbouring trapezia, while the two endpoints belong to only one each. Whether the estimate over- or undershoots the true area depends on which way the curve bends between the ordinates, not on any property of the rule itself.
trigonometric identitytan θ = sin θ / cos θ, sin²θ + cos²θ = 1: A relationship between trig functions true for every angle, not just some — most centrally tan θ ≡ sin θ/cos θ and sin²θ + cos²θ ≡ 1, both of which fall out of reading the coordinates of a point on the unit circle two different ways. Used to collapse an equation mixing sin, cos and tan into one function, which usually then becomes an ordinary quadratic.

U

unit circle: A circle of radius 1 centred at the origin, whose point at angle θ (measured anticlockwise from the positive x-axis) is, by definition, (cos θ, sin θ) — the extension of sine and cosine to any real angle, and the single picture every trig identity and every solution-generation rule in this course reduces to.