Mathematics: Analysis and Approaches HL

Calculus

Higher-level differentiation, integration, optimization, rates, concavity, and differential equations.

Differentiate with product quotient and chain rules Choose and apply higher-level differentiation rules accurately. Use implicit differentiation Differentiate implicit equations and find gradients or tangent information. Solve related rates problems Connect changing variables and use derivatives to calculate related rates. Use second derivative and concavity Classify stationary points and interpret concavity using the second derivative. Integrate by substitution Use substitution to evaluate integrals and explain the change of variable. Integrate by parts Apply integration by parts to products of functions and definite integrals. Solve first order differential equations Solve separable or first-order differential equations and interpret constants. Differentiate exponential logarithmic and trigonometric functions Differentiate common transcendental functions and interpret derivative values. Use optimization with constraints Build and optimize constrained models using derivatives and endpoint checks. Analyse points of inflection Use second derivatives and sign changes to identify and interpret inflection points. Integrate exponential logarithmic and trigonometric functions Integrate common HL function types and interpret constants or definite values. Find area between curves Set up and evaluate definite integrals for regions between curves. Use kinematics with variable acceleration Connect displacement, velocity, and acceleration using differentiation and integration. Solve separable differential equations Separate variables, integrate, and apply initial conditions in differential equations. Interpret slope fields and solution curves Use differential equation slope information to interpret solution curve behaviour. Use L Hopital rule Evaluate indeterminate limits using derivative comparisons where valid. Use Taylor polynomial approximations Build and interpret local polynomial approximations to functions. Use Maclaurin series Use standard Maclaurin expansions to approximate and identify coefficients. Analyse convergence of series Use convergence tests and conditions for infinite series. Use differential equations in growth models Solve and interpret differential equation models for growth and decay. Use Euler method Approximate differential equation solutions with step-by-step Euler updates. Use volumes of revolution Set up and evaluate volume integrals for solids of revolution. Use arc length and surface area integrals Set up and interpret integrals for arc length or surface area. Use improper integrals Evaluate improper integrals and determine convergence in context. Use partial fractions in integration Decompose rational functions and integrate using partial fractions. Use reduction formulae Apply and interpret reduction formulae for repeated integration patterns. Analyse logistic differential equations Solve and interpret logistic differential equation models. Use coupled differential models Interpret simple systems of differential equations and equilibrium behaviour. Use parametric differentiation Find gradients and tangent information from parametric equations. Use polar differentiation Interpret gradients and rates in polar curve contexts. Use curvature concepts Interpret curvature, concavity, and local bending from derivatives. Optimise multivariable expressions Use partial derivative reasoning or constraints to optimize two-variable expressions. Use numerical integration error estimates Interpret trapezoidal, Simpson, and error-bound information for numerical integration. Use Leibniz notation in related rates Set up related-rate equations using Leibniz notation and variable dependencies. Differentiate inverse trigonometric functions Apply inverse trigonometric derivative rules with domain restrictions. Integrate inverse trigonometric forms Recognise and integrate forms leading to inverse trigonometric functions. Use hyperbolic differentiation and integration Differentiate and integrate hyperbolic functions in AA HL contexts. Solve second order differential equations Solve simple second-order differential equations and interpret constants. Analyse equilibrium and stability in differential equations Classify equilibrium points and stability from differential equations. Use power series representations Build and use power series expansions for functions. Estimate errors in Taylor approximations Use remainder bounds and context to estimate Taylor approximation errors. Use differential equations in motion models Model motion with differential equations and interpret velocity or displacement. Optimise with implicit constraints Use implicit constraints and derivative reasoning in optimization. Use logarithmic differentiation Differentiate products, quotients, and variable powers using logarithmic differentiation. Use implicit second derivatives Find and interpret second derivatives from implicit relationships. Evaluate limits with series expansions Use series expansions to evaluate limiting expressions. Use comparison tests for improper integrals Determine convergence of improper integrals using comparison reasoning. Analyse parametric curve concavity Find concavity and local behaviour from parametric derivatives. Use polar area integrals Calculate and interpret areas enclosed by polar curves. Solve differential equations with integrating factors Solve first-order linear differential equations using integrating factors. Analyse phase lines for differential equations Use phase lines to classify intervals and equilibrium behaviour. Use numerical solutions to differential equations Interpret numerical differential equation methods and step-size effects. Optimise rates with parameter constraints Use derivative reasoning to optimize rates subject to parameter constraints. Use differentiation under a parameter Differentiate parameter-dependent expressions and interpret the resulting rate. Analyse families of solution curves Interpret constants and parameters in families of differential equation solution curves. Use separable models with initial conditions Solve separable differential equations and apply initial conditions. Evaluate improper integrals with parameters Determine parameter values for convergence and evaluate improper integrals. Use arc length for parametric curves Calculate and interpret arc length for parametric curves. Use surface area of revolution models Set up and interpret surface area of revolution integrals. Analyse convergence of power series Find radius or interval of convergence for power series. Use Taylor polynomials in local modelling Use Taylor polynomials to approximate and interpret local function behaviour. Solve constrained optimization with Lagrange style reasoning Use constraint equations and multiplier-style reasoning to optimize quantities. Analyse curvature in parametric models Use derivative relationships to interpret curvature in parametric models.