Increasing and Decreasing Functions
AA HL · Topic 5 · SL 5.2
Read the sign of the derivative to identify where a function rises, falls or is stationary.
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Power Rule Differentiation
AA HL · Topic 5 · SL 5.3
Differentiate polynomial terms efficiently and connect the algebra to gradient.
Tangents and Normals
AA HL · Topic 5 · SL 5.4
Use derivatives to construct tangent and normal lines at a point on a curve.
Anti-Differentiation and Boundary Conditions
AA HL · Topic 5 · SL 5.5
Reverse the power rule, include the constant of integration and use a point to determine a particular function.
Differentiation Rules
AA HL · Topic 5 · SL 5.6
Differentiate standard functions and combinations using chain, product and quotient rules.
Second Derivative and Graphical Behaviour
AA HL · Topic 5 · SL 5.7
Use the second derivative to describe concavity and identify changes of curvature.
Stationary Points and Derivative Tests
AA HL · Topic 5 · SL 5.8
Classify stationary points reliably with first- and second-derivative tests.
Optimisation
AA HL · Topic 5 · SL 5.8
Build and analyse an objective function, including closed-domain endpoint checks.
Kinematics
AA HL · Topic 5 · SL 5.9
Move fluently between displacement, velocity and acceleration and distinguish displacement from total distance.
Standard Integrals
AA HL · Topic 5 · SL 5.10
Recognise standard antiderivatives and integrate linear composites with correct scale factors.
Integration by Substitution
AA HL · Topic 5 · SL 5.10
Recognise an inner function and its derivative, then change variable to simplify the integral.
Definite Integrals: Evaluation and Properties
AA HL · Topic 5 · SL 5.11
Evaluate definite integrals analytically, then use reversing limits, interval splitting, linearity and symmetry.
Areas Using Definite Integrals
AA HL · Topic 5 · SL 5.11
From signed area and its applications to total geometrical area and area between two curves.
Limits, Continuity and First Principles
AA HL · Topic 5 · AHL 5.12
Understand limits and informal continuity/differentiability, derive polynomial derivatives from first principles, and interpret higher derivatives.
L’Hôpital’s Rule
AA HL · Topic 5 · AHL 5.13
Evaluate indeterminate limits with l’Hôpital’s rule, repeated use, algebraic rewriting and a Maclaurin-series check.
Implicit Differentiation
AA HL · Topic 5 · AHL 5.14
Differentiate relations where y is not isolated, using the chain rule on every y-term.
Related Rates
AA HL · Topic 5 · AHL 5.14
Differentiate a geometric/physical constraint with respect to time and connect simultaneous rates of change.
AA HL Optimisation
AA HL · Topic 5 · AHL 5.14
Optimise models with implicit constraints and endpoint possibilities.
Further Derivatives and Antiderivatives
AA HL · Topic 5 · AHL 5.15
Extend the standard derivative/integral toolkit and use partial fractions to create integrable forms.
Advanced Integration: Substitution and Integration by Parts
AA HL · Topic 5 · AHL 5.16
Choose between substitution and integration by parts, including repeated/cyclic applications.
Areas with the y-Axis and Volumes of Revolution
AA HL · Topic 5 · AHL 5.17
Use horizontal or vertical strips to model area and volume about either coordinate axis.
Differential Equations
AA HL · Topic 5 · AHL 5.18
Solve separable, homogeneous and linear first-order equations and approximate solutions with Euler’s method.
Maclaurin Series
AA HL · Topic 5 · AHL 5.19
Build and manipulate series near x=0 for approximation, limits and differential equations.