IB Mathematics Analysis and Approaches HL

Topic 5: Calculus

A structured path through the finalized Radford Mathematics calculus lessons.

AA HL · Topic 5

Calculus learning path

Work through the lessons in syllabus order. Each page contains an online lesson, worked example, practice prompts and relevant video tutorials. Premium printable handouts are available separately through the Radford Mathematics Store.

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SL 5.1Limits and the DerivativeApproach limits numerically and graphically, then interpret the derivative as gradient and rate of change.SL 5.2Increasing and Decreasing FunctionsRead the sign of the derivative to identify where a function rises, falls or is stationary.SL 5.3Power Rule DifferentiationDifferentiate polynomial terms efficiently and connect the algebra to gradient.SL 5.4Tangents and NormalsUse derivatives to construct tangent and normal lines at a point on a curve.SL 5.5Anti-Differentiation and Boundary ConditionsReverse the power rule, include the constant of integration and use a point to determine a particular function.SL 5.6Differentiation RulesDifferentiate standard functions and combinations using chain, product and quotient rules.SL 5.7Second Derivative and Graphical BehaviourUse the second derivative to describe concavity and identify changes of curvature.SL 5.8Stationary Points and Derivative TestsClassify stationary points reliably with first- and second-derivative tests.SL 5.8OptimisationBuild and analyse an objective function, including closed-domain endpoint checks.SL 5.9KinematicsMove fluently between displacement, velocity and acceleration and distinguish displacement from total distance.SL 5.10Standard IntegralsRecognise standard antiderivatives and integrate linear composites with correct scale factors.SL 5.10Integration by SubstitutionRecognise an inner function and its derivative, then change variable to simplify the integral.SL 5.11Definite Integrals: Evaluation and PropertiesEvaluate definite integrals analytically and interpret them as accumulated signed change.SL 5.11Areas Using Definite IntegralsTurn graph geometry into one or more positive area integrals.AHL 5.12Limits, Continuity and First PrinciplesConnect limits, continuity, differentiability and the derivative from first principles.AHL 5.13L’Hôpital’s RuleRecognise indeterminate 0/0 and ∞/∞ forms and evaluate limits with L’Hôpital’s rule when appropriate.AHL 5.14Implicit DifferentiationDifferentiate relations where y is not isolated, using the chain rule on every y-term.AHL 5.14Related RatesDifferentiate a geometric/physical constraint with respect to time and connect simultaneous rates of change.AHL 5.14AA HL OptimisationOptimise models with implicit constraints and endpoint possibilities.AHL 5.15Further Derivatives and AntiderivativesExtend the standard derivative/integral toolkit and use partial fractions to create integrable forms.AHL 5.16Advanced Integration: Substitution and Integration by PartsChoose between substitution and integration by parts, including repeated/cyclic applications.AHL 5.17Areas with the y-Axis and Volumes of RevolutionUse horizontal or vertical strips to model area and volume about either coordinate axis.AHL 5.18Differential EquationsSolve separable, homogeneous and linear first-order equations and approximate solutions with Euler’s method.AHL 5.19Maclaurin SeriesBuild and manipulate series near x=0 for approximation, limits and differential equations.

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