IB Math AA SL flashcards

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

Arithmetic sequence nth term

uₙ = u₁ + (n − 1)d, where u₁ is the first term and d is the constant common difference; equivalently d = u₂ − u₁.

Sum of a finite geometric series

Sₙ = u₁(rⁿ − 1)/(r − 1) for r ≠ 1, where u₁ is the first term and r the common ratio; when r = 1 simply Sₙ = nu₁.

Sum to infinity

For a geometric series with |r| < 1 the sum converges to S∞ = u₁/(1 − r); if |r| ≥ 1 the series diverges and no finite sum exists.

Change of base for logarithms

log_a x = log_b x / log_b a, which converts any logarithm to base 10 or base e; for example log₂ 7 = ln 7 / ln 2.

General term of binomial expansion

In the expansion of (a + b)ⁿ, the term containing b^r is C(n, r) a^(n−r) b^r, where C(n, r) = n!/(r!(n − r)!).

Modulus and conjugate

For z = a + bi, the modulus |z| = √(a² + b²) is the distance from the origin, and the conjugate is z* = a − bi; z z* = |z|².

Discriminant of a quadratic

For ax² + bx + c = 0, Δ = b² − 4ac: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots but a complex conjugate pair.

Vertex form of a quadratic

f(x) = a(x − h)² + k has vertex (h, k) and axis of symmetry x = h; it is obtained by completing the square, and a > 0 opens upward.

Composite function

(f ∘ g)(x) = f(g(x)): apply g first, then f. It is defined only for x in the domain of g with g(x) in the domain of f.

Inverse function

f⁻¹ reverses f, so f(a) = b means f⁻¹(b) = a. Its graph is the reflection of y = f(x) in y = x, and domain and range are interchanged.

Horizontal stretch y = f(bx)

For b > 0 the graph of y = f(bx) is stretched parallel to the x-axis by factor 1/b; values b > 1 compress the graph, values 0 < b < 1 widen it.

Remainder and factor theorems

Dividing a polynomial f(x) by (x − a) leaves remainder f(a); consequently (x − a) is a factor of f(x) exactly when f(a) = 0.

Horizontal asymptote of rational function

Compare degrees: numerator degree lower gives y = 0; equal degrees give the ratio of leading coefficients; numerator degree higher gives no horizontal asymptote.

Cosine rule

a² = b² + c² − 2bc cos A links three sides to the angle opposite a; use it for SAS and SSS triangles, or rearranged as cos A = (b² + c² − a²)/(2bc).

Sine rule

a/sin A = b/sin B = c/sin C relates sides to opposite angles; use it with two angles and a side, or in the ambiguous case where two triangles fit.

Triangle area using sine

Area = ½ab sin C, where a and b are two sides and C is the angle included between them; it works for any triangle, not just right-angled ones.

Arc length in radians

For a circle of radius r and central angle θ measured in radians, arc length l = rθ; convert from degrees first using 180° = π.

Sector area in radians

Sector area A = ½r²θ with θ in radians; this equals the fraction θ/(2π) of the full circle area πr².

Double angle formula for cosine

cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; pick the version that matches the other expressions in the equation you are solving.

Scalar product of two vectors

a·b = |a||b| cos θ = a₁b₁ + a₂b₂; the vectors are perpendicular exactly when a·b = 0, and θ = arccos(a·b/(|a||b|)).

Conditional probability

P(A | B) = P(A ∩ B)/P(B) is the probability of A given that B has already occurred; it requires P(B) > 0 and restricts the sample space to B.

Independent events

A and B are independent when P(A ∩ B) = P(A)P(B), equivalently P(A | B) = P(A); knowing one occurred does not change the probability of the other.

Expected value of discrete variable

E(X) = Σ x P(X = x) is the long-run mean of a discrete random variable; it need not be a value that X can actually take.

Binomial mean and variance

For X ~ B(n, p): E(X) = np and Var(X) = np(1 − p), where n is the number of independent trials and p the constant success probability.

Standardising a normal variable

If X ~ N(μ, σ²), then z = (x − μ)/σ follows the standard normal distribution; a negative z means the value lies below the mean.

Interquartile range and outliers

IQR = Q3 − Q1 describes the spread of the middle half of the data; a value is treated as an outlier if it lies below Q1 − 1.5 IQR or above Q3 + 1.5 IQR.

Chain rule

dy/dx = dy/du · du/dx for y a function of u and u a function of x; used for composite functions such as (3x + 1)⁵.

Product rule

If y = uv then dy/dx = u′v + uv′, where u′ and v′ are the derivatives of u and v taken with respect to x.

Quotient rule

If y = u/v then dy/dx = (u′v − uv′)/v², valid where v ≠ 0; the numerator order matters, so subtracting the wrong way flips the sign.

Second derivative test

At a stationary point where f′(x) = 0: f″(x) > 0 gives a local minimum, f″(x) < 0 a local maximum, and f″(x) = 0 leaves the test inconclusive.

Integrating powers of x

∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1; the separate case ∫x⁻¹ dx = ln|x| + C is needed because that rule fails there.

Definite integral as net area

∫_a^b f(x) dx = F(b) − F(a) gives the signed area between the curve and the x-axis; regions below the axis contribute negatively, so split the interval.