Topic 5 is the largest topic in Maths AA by weighting (~40% of HL, ~30% of SL). SL covers differentiation, integration, and applications. HL extends to advanced techniques, differential equations, Maclaurin series, and L\'Hopital\'s rule.
First principles: f\'(x) = lim(h→0) [f(x+h)−f(x)]/h. Rules: power (nxⁿ⁻¹), chain (f(g(x)))\' = f\'(g(x))·g\'(x), product (fg)\' = f\'g + fg\', quotient (f/g)\' = (f\'g − fg\')/g². Derivatives of sin, cos, tan, eˣ, ln x. Higher derivatives. HL: implicit differentiation, related rates.
Equations of tangent and normal lines. Increasing/decreasing intervals using sign of f\'(x). Local maxima/minima: f\'(x) = 0 and second derivative test. Points of inflexion: f\'\'(x) = 0 and sign change. Optimisation problems: maximising/minimising real-world quantities subject to constraints.
Anti-differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Integration of sin, cos, eˣ, 1/x. Definite integral: ∫ₐᵇ f(x)dx = F(b) − F(a). Area under curve. Area between curves: ∫(upper − lower)dx. HL: integration by substitution, by parts, partial fractions.
Displacement s, velocity v = ds/dt, acceleration a = dv/dt. Given v(t), displacement s = ∫v dt; distance = ∫|v| dt. HL: separable differential equations dy/dx = f(x)g(y) solved by ∫dy/g(y) = ∫f(x)dx. Slope fields. Euler\'s method for numerical approximation.
Calculus is the largest single topic in Maths AA. At SL, it accounts for approximately 30% of the course. At HL, it represents roughly 40% — the biggest topic by far. You should expect at least one full long-form question on calculus in each paper, and calculus concepts frequently appear integrated with other topics (e.g., optimisation with functions, kinematics with vectors).
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