Mathematics: Analysis and Approaches

Describe arithmetic and geometric patterns with closed-form formulas, sum any finite series, write sums compactly with sigma notation, and decide when an infinite geometric series converges.

Master integer and rational exponent laws, define the logarithm as the inverse of exponentiation, apply log laws and change of base, and solve exponential equations.

Compress numbers of any scale with standard form, model money over time with compound interest, depreciation, and annuities, and learn the LHS-to-RHS structure of a deductive proof.

Use Pascal's triangle and the binomial theorem to expand (a+b)^n and target specific terms; apply counting principles, permutations, combinations, and the extended binomial theorem (HL).

(HL only) Complex numbers in Cartesian, polar, and Euler form; De Moivre and complex roots; proof by induction, contradiction, and counterexample; systems of linear equations.

Equations of straight lines (forms, gradient, parallel and perpendicular); the concept of a function with domain, range, and notation; sketching and interpreting graphs; key features (intercepts, max/min, asymptotes, intersections); composite, identity, and inverse functions.

Sketching parabolas (vertex, axis of symmetry, intercepts) and converting between general, factored, and vertex forms; solving quadratic equations and inequalities (factoring, completing the square, quadratic formula); the discriminant and the nature of roots; reciprocal y = 1/x and general rational functions (ax + b)/(cx + d) with their asymptotes.

Sketching exponential functions f(x) = a^x and f(x) = e^x and their logarithmic inverses log_a x and ln x; identifying domain, range, and asymptotes; solving f(x) = g(x) graphically (with technology) and analytically (algebraic manipulation); applying these techniques to real-life modelling — population growth, compound interest, radioactive decay.

Five atomic transformations of y = f(x) — translations (vertical and horizontal), stretches (vertical and horizontal), reflections — and their composite y = a f(b(x − h)) + k. INSIDE-FIRST, OUTSIDE-SECOND ordering, anchor-point tracking, and the sign-flip / reciprocal traps that catch most students.

HL-only extensions to the function toolkit: polynomial roots via the factor / remainder theorems and Vieta's formulas; rational functions including oblique asymptotes; even, odd, and self-inverse functions; solving inequalities g(x) ≥ f(x) and modulus equations |x − a|; sketching y = |f(x)|, y = f(|x|), and y = 1/f(x) from y = f(x).

3D Cartesian distance and midpoint formulas (Pythagoras-twice); volume and surface area of standard solids; angles in 3D (between two lines, between a line and a plane, between two planes — all via right-triangle decomposition); sine and cosine rules including the ambiguous case; the area formula ½ ab sin C; bearings and angles of elevation/depression in navigation and surveying.

Radian measure (the natural angular unit), arc length s = rθ, sector area A = ½r²θ. Unit-circle definitions of sin θ (y-coordinate), cos θ (x-coordinate), tan θ = sin θ / cos θ — generalising right-triangle ratios to ANY angle. Exact values for special angles 0, π/6, π/4, π/3, π/2 with quadrant sign rules (ASTC). Pythagorean identity sin²θ + cos²θ = 1 and double-angle identities for sin 2θ and cos 2θ.

The base curves y = sin x, y = cos x, y = tan x — period, amplitude, range, and asymptotes. Transformations y = a sin(b(x − c)) + d with amplitude |a|, period 2π/|b|, phase shift c, and vertical shift d. Modelling real-world periodic data (tides, daylight, oscillations). Solving linear trigonometric equations in a finite interval (reference angle + ASTC + period extension) and quadratic-in-sin/cos/tan equations (substitute, factor, split into linear cases).

HL only. Reciprocal trig ratios sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = cos θ/sin θ, and the derived Pythagorean identities 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ. Inverse trig functions arcsin, arccos, arctan with their principal-value domains and ranges. Compound-angle identities for sin(A ± B), cos(A ± B), tan(A ± B); double-angle for tan; symmetry properties (evenness/oddness, periodicity); co-function relations like sin(π/2 − θ) = cos θ.

HL only. Vectors as directed quantities — position vectors anchored at the origin, displacement vectors free to translate, base vectors i, j, k. Magnitude formula |v| = √(x² + y² + z²) (Pythagoras in 3D). Vector arithmetic (sums, differences, scalar multiples) component-wise. Midpoint formula m = (a + b)/2 and applications to collinearity proofs. Subsequent lessons cover scalar product, vector equation of a line, cross product and planes, and intersections of lines/planes.

Statistics begins with two questions: WHO are you measuring (population vs sample) and WHAT are you measuring (discrete vs continuous data). Five named sampling techniques (simple random, convenience, systematic, stratified, quota) with their bias risks. Frequency tables, histograms (touching bars), cumulative-frequency S-curves, box-and-whisker plots. Quartiles Q1/Q2/Q3 and IQR captured graphically and numerically.

Summary numbers for distributions: mean, median, mode for centre; range, IQR, variance, standard deviation for spread. Effect of constant translation (shifts centre, leaves spread alone) and scaling (centre + spread × |k|, variance × k²) on every statistic. Two-variable analysis: scatter plots, Pearson's r ∈ [−1, +1] for linear association strength + direction (with the famous r ≠ causation warning). Both regression lines: y on x (predicts y) and x on y (predicts x), distinguishing them and warning against extrapolation.

Probability is counting: P(A) = (favourable outcomes) / (total outcomes in the sample space). The complement rule P(A') = 1 − P(A) shortcuts hard counts. Expected number of occurrences = n × P(A) — the long-run average over n trials. For combined events, the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) avoids double-counting; Venn, tree, and sample-space diagrams organise problems. Mutually exclusive events satisfy P(A ∩ B) = 0; independent events satisfy P(A ∩ B) = P(A) P(B). Conditional probability P(A | B) = P(A ∩ B) / P(B) restricts the sample space to B — at the heart of the base-rate fallacy in medical testing.

Discrete and continuous random variables. A discrete random variable X has a probability distribution: a table of values with probabilities summing to 1. Expected value E(X) = Σ x · P(X = x) is the long-run average. The BINOMIAL distribution X ~ B(n, p) models n independent identical Bernoulli trials — mean = np, variance = np(1−p). The NORMAL distribution N(μ, σ²) is the bell curve: symmetric about μ, inflection points at μ ± σ, total area = 1, with the GDC (normCDF / invNorm) handling probability calculations. Standardisation Z = (X − μ)/σ converts any normal X to the standard normal N(0, 1) — letting you compare scores across distributions and find unknown μ or σ from probability statements.

HL-only extension to D.3 (probability) and D.4 (distributions). Bayes' theorem flips a conditional: P(A | B) = P(B | A) · P(A) / P(B), letting you update a prior belief once you see new evidence. The IB caps Bayes problems at THREE mutually exclusive priors. The shortcut formula Var(X) = E(X²) − [E(X)]² avoids squaring deviations one-by-one. Linear transformations Y = aX + b shift the mean by +b and scale the variance by a² (the +b drops out — translations don't change spread). CONTINUOUS random variables replace probability mass with probability DENSITY: P(X = c) = 0 for any single point, and probability lives in the AREA under a pdf f(x) where ∫ f(x) dx = 1. Calculate mean μ = ∫ x·f(x) dx, median m via ∫_{−∞}^m f(x) dx = 0.5, mode by setting f'(x) = 0, and Var(X) = ∫ x²·f(x) dx − μ². Throughout: integration from E.2 is required for Lesson 3.

The foundations of calculus. The DERIVATIVE f'(x) is a NEW function whose value at any x equals the slope of the tangent to y = f(x) at that x. Geometrically: tangent slope. Physically: instantaneous rate of change (velocity, growth rate, marginal cost). Defined formally as the LIMIT of the difference quotient: f'(x) = lim_{h→0} [f(x+h) − f(x)] / h — set up the quotient, simplify so h CANCELS, then take h → 0. Computing every derivative this way is slow, so the POWER RULE provides a shortcut for any polynomial term: d/dx(axⁿ) = an·x^(n−1) — bring the exponent down, subtract 1. Constants disappear (slope of horizontal line = 0); negative exponents work the same way (sign flips). With f'(x) in hand, READ off behaviour: f'(x) > 0 → f increasing; f'(x) < 0 → f decreasing; f'(x) = 0 → stationary point (max, min, or inflexion). And USE f'(x₀) as a slope to write the TANGENT line y − y₀ = f'(x₀)(x − x₀) and the NORMAL line (perpendicular: m_n = −1/f'(x₀)) at any point on the curve.

The foundations of integration. The INDEFINITE integral ∫f(x) dx = F(x) + C is the antiderivative — F'(x) = f(x) — with the +C capturing the family of antiderivatives that all share the same derivative (because differentiation discards constants). A single boundary condition (x₀, y₀) PINS DOWN the +C: substitute, set equal to y₀, solve. The DEFINITE integral ∫_a^b f(x) dx returns a NUMBER — the SIGNED area between the curve and the x-axis from x = a to x = b (no +C, since limits eliminate it). Above the axis counts positive; below counts negative. Standard antiderivatives: power rule reverse ∫xⁿ dx = x^(n+1)/(n+1) + C (for n ≠ −1), special case ∫(1/x) dx = ln|x| + C, exponential ∫e^x dx = e^x + C, trig ∫sin x dx = −cos x + C and ∫cos x dx = sin x + C. The REVERSE CHAIN RULE handles composites of the form f(g(x))·g'(x). The FUNDAMENTAL THEOREM OF CALCULUS computes any definite integral via F(b) − F(a). Areas between two curves: ∫_a^b [top − bottom] dx, where a and b are the x-coordinates of the intersection points.

Advanced differentiation: the five standard derivatives (x^n, sin x, cos x, e^x, ln x) plus the three combination rules — chain for nested f(g(x)), product for u·v, and quotient for u/v. The SECOND derivative f''(x) reveals SHAPE: positive ⇒ concave up, negative ⇒ concave down, sign change ⇒ point of inflection. Optimisation locates maxima and minima by solving f'(x) = 0 and classifying via f''(x).

Calculus IS motion. The FORWARD chain differentiates displacement s(t) once for velocity v(t) = ds/dt and again for acceleration a(t) = dv/dt = d²s/dt². The REVERSE chain integrates acceleration back to velocity (+ C₁ pinned by v(0)) and then to displacement (+ C₂ pinned by s(0)). Signed displacement ∫v dt under-counts the path when v changes sign — the absolute total distance is ∫|v| dt, found by splitting the interval at the zeros of v.

The HL-only rigour-and-techniques chapter of calculus. Lesson 1 builds the foundation: continuity at a point (three-condition test), differentiability via the limit definition (with the canonical failure modes — corner, vertical tangent, prior discontinuity), and higher-order derivatives f^(n)(x) (polynomial termination, e^x as a fixed point, sin/cos period-4 cycle). Future lessons cover l'Hôpital's rule, implicit differentiation, related rates, advanced integration techniques, differential equations, Maclaurin series, and volumes of revolution.