Glossary

Every term, in one place

The same 19 definitions the lessons link to inline, wherever a sentence uses them — gathered here so a definition is never more than one page away.

A

algebraic long divisionpolynomial long division: Dividing one polynomial by another using the divide–multiply–subtract–bring-down layout of ordinary long division, producing a quotient and a remainder of lower degree than the divisor — fully creditable alongside comparing coefficients, though every subtraction is a fresh place for a sign error to enter.
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.

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.

chain ruledy/dx = (dy/du)(du/dx): The rule for differentiating a composite function, dy/dx = (dy/du)·(du/dx) — differentiate the outer function with respect to whatever was fed into it, then multiply by the derivative of that inner function itself. Also the mechanism behind dy/dx = 1/(dx/dy): the identity that a function and its own inverse relationship undo each other, differentiated.
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.
composite functionfg(x), composition of functions: fg(x) means f(g(x)) — apply g first, then feed its output into f. A genuinely different function from gf(x) in general, and a composite built from a domain-restricted inner or outer function inherits a domain restriction of its own, translated through whichever function acts first.
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.

definite integral∫ₐᵇf(x)dx: ∫ₐᵇf(x)dx = F(b) − F(a), where F is an antiderivative of f — equals the area between the curve and the x-axis ONLY where the curve stays on one side of the axis throughout [a,b]; where it crosses, the region has to be split at each root and the modulus of each piece added, or positive and negative area silently cancel inside one calculation.
domain: The set of inputs a function is actually applied to. Genuinely independent of the range: a domain can be completely unrestricted while the range is still bounded, because a quadratic's own turning point — not its domain — is what limits its outputs.

identity≡: An equation true for every value of the variable, marked with ≡ rather than = — licensing substitution of ANY value to derive a further result, unlike an ordinary equation, which is only true for the specific value(s) being solved for. Every trig identity, the division algorithm, and every double-angle or compound-angle formula are identities in exactly this sense.
integration by recognition: Spotting that an integrand is already the chain-rule result of differentiating ln|f(x)| or [f(x)]ⁿ⁺¹, so that ∫f'(x)/f(x) dx = ln|f(x)| + c and ∫f'(x)[f(x)]ⁿ dx = [f(x)]ⁿ⁺¹/(n+1) + c — both derived, not memorised as separate facts, by differentiating the claimed answer and checking it reproduces the original integrand.

logarithmlog, logₐ: logₐb = c means precisely that aᶜ = b — a logarithm is the power its base must be raised to, to produce a given number. It exists as a function only because aˣ never repeats an output, so 'what power gives this?' always has exactly one answer.

product rule(uv)' = u'v + uv': The rule for differentiating a product of two functions of x, (uv)' = u'v + uv' — both factors are changing as x changes, so the derivative has to account for each factor's own rate of change scaled by the other factor's current size, not the product of the two derivatives.
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.

rangerange of a function: The set of outputs a function actually produces, given its domain and rule — not a set chosen freely, but a consequence of the two together. Swapping x and y to find an inverse function exchanges these roles exactly: the domain of f⁻¹ is the range of f, and vice versa.
rational expression: One polynomial divided by another. Simplifying one means factorising numerator and denominator fully and cancelling a genuine shared FACTOR — never a term that is merely added, since only a true factor multiplies the whole of both top and bottom.

stationary pointturning point: A point on a curve where dy/dx = 0 — the gradient momentarily flat, found by differentiating, setting the result to zero, and solving; the matching y-coordinate then comes from substituting back into the ORIGINAL function, never into dy/dx.

tangenttangent line, tangent to a curve: A straight line that touches a curve at one point without crossing into it there. For a circle, a tangent always meets its radius at a right angle, so its gradient is the negative reciprocal of the radius's gradient; for a general curve, the tangent at a point shares that point's gradient, found by differentiation.
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.