Pure core: algebra, calculus and trigonometry: every key term you need (+ practice quiz)
16 flashcard terms for A-Level Mathematics Topic 1, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 8-question quiz — free, no account needed.
b squared minus 4ac. Positive gives two real roots, zero gives one repeated root, negative gives no real roots.
Completing the square
Rewriting a quadratic as a(x + p) squared + q. Immediately gives the turning point at (-p, q) and proves the minimum value.
Factor theorem
If f(a) = 0 then (x - a) is a factor of f(x). The standard route into factorising cubics.
Binomial expansion
Expands (1 + x) to the power n using binomial coefficients. For non-integer n the expansion is infinite and valid only for |x| < 1.
Differentiation from first principles
The limit of [f(x + h) - f(x)] / h as h tends to zero. The formal definition the standard rules are derived from.
Chain rule
Differentiates a composite function: multiply the derivative of the outer function by the derivative of the inner one.
Product rule
For y = uv, dy/dx = u(dv/dx) + v(du/dx). Used whenever two functions of x are multiplied.
Stationary point
Where dy/dx = 0. The second derivative classifies it: negative is a maximum, positive a minimum, zero needs further testing.
Definite integral
The signed area between a curve and the x-axis over an interval, found by evaluating the antiderivative at both limits and subtracting.
Integration by substitution
Reverses the chain rule by replacing part of the integrand with a new variable — remember to change the limits too.
Trigonometric identity
sin squared plus cos squared equals 1, and tan equals sin over cos. The two identities most other trigonometry work is built from.
Radian
The angle subtended at the centre by an arc equal in length to the radius. Required for calculus with trigonometric functions.
Small angle approximation
For small theta in radians, sin(theta) is approximately theta, tan(theta) is approximately theta, and cos(theta) is approximately 1 minus half theta squared.
Logarithm law
log(a) + log(b) = log(ab), and n log(a) = log(a to the n). Used to linearise exponential relationships for graphing.
Vector equation of a line
r = a + t*d, where a is a position vector on the line and d is the direction vector. Two lines with non-parallel directions may still be skew.
Proof by contradiction
Assume the negation of the statement, derive a logical impossibility, and conclude the original statement must be true.