European Baccalaureate Mathematics (5-period course) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Discriminant of a quadratic

For ax² + bx + c = 0, Δ = b² − 4ac. Positive Δ gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Sum and product of roots

If x₁ and x₂ are roots of ax² + bx + c = 0, then x₁ + x₂ = −b/a and x₁x₂ = c/a. Useful for building equations from known roots.

Change of base formula

log_b(x) = ln x / ln b for b > 0, b ≠ 1, x > 0. Allows evaluation of any logarithm using natural or common logarithms on a calculator.

Binomial coefficient

C(n, k) = n!/(k!(n − k)!) counts the k-element subsets of an n-element set and gives the coefficient of aⁿ⁻ᵏbᵏ in the expansion of (a + b)ⁿ.

Absolute value inequality

|x − a| < r means a − r < x < a + r, a single interval. |x − a| > r means x < a − r or x > a + r, two separate rays.

Inverse function condition

f has an inverse only if it is injective on its domain, so each output corresponds to exactly one input. The graphs of f and f⁻¹ are mirror images in y = x.

Even and odd functions

Even: f(−x) = f(x), symmetric about the y-axis. Odd: f(−x) = −f(x), symmetric about the origin. Polynomials with only even or only odd powers are examples.

Asymptotes of rational functions

Vertical asymptotes occur where the denominator vanishes but the numerator does not. Horizontal asymptotes follow from the limit of the function as x tends to plus or minus infinity.

Domain restrictions

Exclude x where a denominator is zero, where an even root's radicand is negative, and where a logarithm's argument is zero or negative. These conditions determine the domain.

Graph transformation rules

y = f(x − h) + k shifts the graph h units right and k units up. y = a·f(bx) stretches vertically by factor a and compresses horizontally by factor 1/b.

Existence of a limit

lim f(x) as x approaches a exists exactly when the left-hand and right-hand limits both exist and are equal. The value f(a) does not affect the limit.

Derivative of a product

The product rule states (uv)′ = u′v + uv′, where u and v are differentiable functions of x. The quotient rule is (u/v)′ = (u′v − uv′)/v².

Chain rule

For a composite function, (f ∘ g)′(x) = f′(g(x))·g′(x). Differentiate the outer function at the inner value, then multiply by the inner derivative.

Fundamental theorem of calculus

If F is an antiderivative of f, then the definite integral of f from a to b equals F(b) − F(a). It links areas under curves with antiderivatives.

First derivative test

Where f′ changes from positive to negative, f has a local maximum; from negative to positive, a local minimum. No sign change means no extremum there.

Integration by parts

The rule ∫u dv = uv − ∫v du reverses the product rule. Choose u to simplify on differentiation and dv to be easily integrable.

Dot product and perpendicularity

u·v = u₁v₁ + u₂v₂ + u₃v₃ = |u||v|cos θ. Two nonzero vectors are perpendicular exactly when their dot product equals zero.

Vector equation of a line

A line is written r = a + t·d, where a is the position vector of a point on the line, d is a direction vector, and t is a real parameter.

Distance from point to plane

For plane ax + by + cz + d = 0 and point (x₀, y₀, z₀), the distance is |ax₀ + by₀ + cz₀ + d| divided by √(a² + b² + c²).

Cross product magnitude

For vectors u and v in space, |u × v| = |u||v|sin θ, which equals the area of the parallelogram spanned by u and v. The direction is perpendicular to both.

Cosine rule

In a triangle, a² = b² + c² − 2bc·cos A. Use it when two sides and the included angle, or all three sides, are known.

Sine rule

a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. Use it when a side and its opposite angle are known.

Double angle identities

sin 2x = 2 sin x cos x and cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1. These allow reduction of powers and angle doubling.

Amplitude and period

For y = A sin(Bx + C) + D, the amplitude is |A|, the period is 2π/|B|, the phase shift is −C/B, and D is the vertical shift.

General solution of sin x = sin α

The solutions are x = α + 2kπ or x = π − α + 2kπ for integer k. The second family comes from the symmetry of the sine graph.

Conditional probability

P(A|B) = P(A ∩ B)/P(B) for P(B) > 0. Events A and B are independent exactly when P(A ∩ B) = P(A)·P(B).

Binomial distribution parameters

For X ~ B(n, p), P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ, with E(X) = np and Var(X) = np(1 − p). Trials must be independent with constant p.

Variance of a random variable

Var(X) = E(X²) − [E(X)]². It measures spread around the mean; the standard deviation is the positive square root of the variance.

Standard normal standardisation

Any normal variable is converted by z = (x − μ)/σ. About 68% of values lie within one standard deviation of the mean and about 95% within two.

Sum of geometric series

Sₙ = u₁(1 − rⁿ)/(1 − r) for r ≠ 1. The infinite sum converges to u₁/(1 − r) only when |r| < 1.

Principle of induction

Prove the statement for n = 1, assume it for n = k, then use that assumption to prove it for n = k + 1. This establishes it for all positive integers.

Sum of first n squares

Σ_{k=1}^{n} k² = n(n + 1)(2n + 1)/6, while Σ_{k=1}^{n} k = n(n + 1)/2. Both are proved by induction and used in series problems.