IB Mathematics: Analysis and Approaches Study Guide (HL and SL)

IB Mathematics: Analysis and Approaches is the closest thing to a university-preparatory pure mathematics course available at secondary level. It demands mathematical rigour — the ability to construct and follow proofs, to work with abstract structures, and to extend techniques to unfamiliar situations — alongside computational fluency in calculus, algebra, and statistics.

For HL students, the course culminates in Paper 3, which presents genuinely open-ended mathematical problems and rewards sustained mathematical reasoning under time pressure. This is the part of the course that most closely resembles university mathematics.

Algebra: the toolkit

Sequences and series: Arithmetic sequences (common difference d): nth term = a₁ + (n−1)d; sum = n(a₁ + aₙ)/2. Geometric sequences (common ratio r): nth term = a₁rⁿ⁻¹; sum = a₁(1−rⁿ)/(1−r); infinite sum (|r| < 1): a₁/(1−r).

Binomial theorem: (a + b)ⁿ = Σ C(n,r) aⁿ⁻ʳ bʳ where C(n,r) = n!/(r!(n−r)!). The binomial coefficient C(n,r) gives the coefficient of each term. For finding a specific term: the (r+1)th term is C(n,r) aⁿ⁻ʳ bʳ.

Proof by induction (HL): Used to prove statements about positive integers. Structure: (1) Base case — verify the statement holds for n = 1; (2) Inductive step — assume true for n = k, prove for n = k + 1; (3) Conclusion. The inductive step requires: writing what you need to prove explicitly, using the inductive hypothesis to transform the expression, and completing the algebraic manipulation to reach the required form.

Complex numbers (HL): Cartesian form z = a + bi. Modulus: |z| = √(a² + b²). Argument: arg(z) = arctan(b/a) (adjust for quadrant). Polar form: z = r(cosθ + i sinθ) = re^(iθ) (Euler's form). De Moivre's theorem: zⁿ = rⁿ(cos nθ + i sin nθ). Finding nth roots: the n roots of zⁿ = w lie at equal angular spacing of 2π/n in the complex plane.

Functions: the unified framework

Composite and inverse functions: f(g(x)) applies g first, then f. For f⁻¹ to exist, f must be one-to-one. Inverse function: swap x and y, solve for y. Domain of f⁻¹ = range of f. Graphically: f⁻¹ is the reflection of f in y = x.

Logarithms and exponentials: Change of base: log_a(x) = ln(x)/ln(a). Laws: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a. Natural exponential: d/dx(e^x) = e^x; d/dx(e^(f(x))) = f'(x)e^(f(x)). Natural log: d/dx(ln x) = 1/x.

Trigonometric identities to know completely:

• Pythagorean: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
• Double angle: sin 2θ = 2 sinθ cosθ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ
• Compound angle: sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB ∓ sinA sinB

Calculus: the dominant topic

Differentiation: All standard rules (power, chain, product, quotient). Implicit differentiation: treat y as a function of x, apply chain rule to y terms. Related rates: differentiate an equation involving two variables with respect to time, substitute known rate values.

Applications: Tangent and normal lines, stationary points and their classification (first/second derivative test), concavity and inflection points, optimisation on closed intervals (candidates test), L'Hôpital's rule for 0/0 and ∞/∞ indeterminate forms.

Integration: All standard antiderivatives. U-substitution. Integration by parts: ∫u dv = uv − ∫v du. Areas between curves. Volumes of revolution about the x-axis (disk method: V = π∫[f(x)]² dx) or y-axis.

Differential equations (HL): Separable equations (separate and integrate both sides). First-order linear ODEs using integrating factor (multiply both sides by e^(∫P(x)dx), recognise as d/dx of a product). Euler's method.

Maclaurin series (HL): f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + ... Standard series to know: sin x = x − x³/3! + x⁵/5! − ...; cos x = 1 − x²/2! + x⁴/4! − ...; e^x = 1 + x + x²/2! + x³/3! + ...; ln(1+x) = x − x²/2 + x³/3 − ... (|x| ≤ 1). Use these to find limits (replace functions with their series), to approximate values, and to solve differential equations.

Statistics and probability

Probability distributions: Binomial: X ~ B(n, p), P(X = x) = C(n,x) pˣ(1−p)^(n−x), E(X) = np, Var(X) = np(1−p). Normal: X ~ N(μ, σ²), use calculator for probabilities. Poisson (HL): X ~ Po(λ), P(X = x) = e^(−λ)λˣ/x!, E(X) = λ, Var(X) = λ.

Hypothesis testing: The structure is the same across all tests: state H₀ and H₁, check conditions, calculate test statistic, find p-value (or critical region), compare to significance level, state conclusion in context. For the t-test (testing a mean), chi-squared test (testing independence or goodness of fit), and correlation (testing whether r ≠ 0).

The Internal Assessment: mathematical exploration

The IA is a mathematical essay exploring a topic of your choice. The key evaluation criteria are:

• Use of mathematics: Is the mathematics correct? Is it at an appropriate level? Does it go beyond routine textbook application?
• Personal engagement: Is there evidence of genuine interest, personal voice, or original investigation?
• Reflection: Are there reflective comments that go beyond description — questioning limitations, making connections, extending results?

Strong IA topics have a clear mathematical question, use techniques from the course, and have an element of investigation rather than just explanation. Examples: investigating the relationship between Fibonacci numbers and the golden ratio beyond what the textbook covers; exploring the mathematics of cryptographic techniques; investigating a physical phenomenon using calculus.

Use the Spaced Repetition Flashcard Tool for calculus derivative and integral rules, series formulas, and probability distributions. The Pomodoro Timer is valuable for past-paper practice sessions. For the sciences that depend most on Mathematics AA HL, see the IB Physics study guide.