IB Maths AA
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Unit 3 — Partial Fractions
IB Mathematics: Analysis and Approaches
Unit 1 — Exponents and Logarithms
1.1 Laws of Exponents and Exponential Equations
1.2 Exponential Functions, Growth and Decay
1.3 Standard Form
1.4 Logarithms and the Laws of Logarithms
1.5 Logarithmic Equations and Graphs
Review exercises
Unit 2 — Sequences and Series
2.1 Sequences, Recurrence Relations and Series
2.2 Arithmetic Sequences and Series
2.3 Geometric Sequences and Series
2.4 Sum to Infinity
2.5 Financial Applications
2.6 Sigma Notation
Review exercises
Unit 3 — Partial Fractions
3.1 Partial Fractions
Review exercises
Unit 4 — The Binomial Theorem
4.1 Factorials, Binomial Coefficients and (1 + b)ⁿ
4.2 Expanding (a + b)ⁿ and the General Term
4.3 Fractional and Negative Indices
Review exercises
Unit 5 — Counting Principles
5.1 Counting Principles and Permutations
5.2 Permutations with Restrictions
5.3 Combinations
Review exercises
Unit 6 — Mathematical Proofs
6.1 The Language of Proof
6.2 Deductive Proof
6.3 Proof by Contradiction and Counterexamples
6.4 Proof by Mathematical Induction
Review exercises
Unit 7 — Linear Functions
7.1 Length, Gradient and Parallel Lines
7.2 Equations of Straight Lines
7.3 Midpoints and Perpendicular Bisectors
Review exercises
Unit 8 — Quadratic Functions
8.1 Completing the Square; Maximum and Minimum Values
8.2 Graphs of Quadratic Functions
8.3 The Discriminant
8.4 Applications of Quadratic Functions
8.5 Quadratic Inequalities
Review exercises
Unit 9 — Functions
9.1 The Idea of a Function
9.2 Domain and Range
9.3 One-to-One and Inverse Functions
9.4 Composite Functions
9.5 Restricting the Domain; Even, Odd and Self-Inverse Functions
9.6 Key Features of Graphs
9.7 Solving Equations Analytically and Graphically
Review exercises
Unit 10 — Further Functions
10.1 Reciprocal and Rational Functions
10.2 Further Rational Functions
10.3 Modulus Functions and Equations
Review exercises
Unit 11 — Polynomials and Inequalities
11.1 Polynomials, Identities and Division
11.2 The Remainder and Factor Theorems
11.3 Polynomial Equations; Sum and Product of Roots
11.4 Polynomial and Rational Inequalities
11.5 Modulus Inequalities and Graphical Methods
Review exercises
Unit 12 — Graph Transformations
12.1 Translations, Stretches and Reflections
12.2 Composite Transformations
12.3 Graphs of y = f(ax + b)
12.4 Graphs of |f(x)| and f(|x|)
12.5 Graphs of 1/f(x) and [f(x)]²
Review exercises
Unit 14 — Trigonometric Functions
14.1 Three-Dimensional Solids
14.2 Right-Angled Trigonometry and Exact Values
14.3 The Sine Rule, Cosine Rule and Area of a Triangle
14.4 Bearings and Applications
14.5 Radians, Arc Length and Sector Area
14.6 The Unit Circle and Trigonometric Ratios of Any Angle
14.7 Graphs of Sine and Cosine
14.8 The Tangent Graph and General Trigonometric Functions
14.9 Reciprocal and Inverse Trigonometric Functions
14.10 Modelling Periodic Behaviour
14.11 Trigonometric Identities
14.12 Solving Trigonometric Equations
14.13 Compound and Double Angle Formulae
Review exercises
Unit 15 — Statistics
15.1 Populations, Samples and Data
15.2 Measures of Central Tendency
15.3 Measures of Dispersion
15.4 Stem-and-Leaf Diagrams and Histograms
15.5 Cumulative Frequency and Box Plots
15.6 Symmetry, Skew and Outliers
Review exercises
Unit 16 — Probability
16.1 The Language of Probability
16.2 Conditional Probability and Independence
16.3 Bayes’ Theorem
16.4 Venn Diagrams and Tree Diagrams
16.5 Probability Using Counting Principles
Review exercises
Unit 17 — Correlation and Regression
17.1 Scatter Diagrams and Correlation
17.2 Regression Lines
17.3 Making Predictions
Review exercises
Unit 18 — Discrete Random Variables and the Binomial Distribution
18.1 Discrete Random Variables and Expected Value
18.2 Variance and Linear Transformations
18.3 The Binomial Distribution
18.4 Mean and Variance of a Binomial Distribution
Review exercises
Unit 19 — Continuous Random Variables and the Normal Distribution
19.1 Continuous Random Variables
19.2 Variance, Mode and Median of a Continuous Random Variable
19.3 The Normal Distribution
19.4 Standardising and Inverse Normal Calculations
Review exercises
Unit 20 — Vectors I
20.1 Scalars, Vectors and Vector Operations
20.2 Vectors in Component Form
20.3 The Ratio Theorem and Collinear Points
20.4 The Scalar Product
20.5 The Vector Product
Review exercises
Unit 21 — Vectors II
21.1 Equations of a Line
21.2 A Point and a Line
21.3 Two Lines
Review exercises
Unit 22 — Vectors III
22.1 Vector Equations of a Plane
22.2 The Cartesian Equation of a Plane
22.3 A Line and a Plane
22.4 A Point and a Plane
22.5 Two Planes
22.6 Three Planes
Review exercises
Unit 23 — Complex Numbers
23.1 Complex Numbers in Cartesian Form
23.2 Conjugates and Polynomial Equations
23.3 The Argand Diagram, Modulus and Argument
23.4 Polar and Euler Forms
23.5 De Moivre’s Theorem
23.6 Roots of Complex Numbers
23.7 Geometric Interpretation
Review exercises
Unit 24 — Differentiation Techniques
24.1 The Concept of a Limit
24.2 The Derivative and Basic Rules
24.3 The Chain, Product and Quotient Rules
24.4 Derivatives of Trigonometric Functions
24.5 Derivatives of Exponential and Logarithmic Functions
24.6 Implicit Differentiation
24.7 Higher Derivatives
Review exercises
Unit 25 — Applications of Differentiation
25.1 Tangents and Normals
25.2 Increasing and Decreasing Functions; Concavity
25.3 Stationary Points and Points of Inflexion
25.4 The Graph of the Derivative
25.5 Rates of Change and Related Rates
25.6 Optimisation
Review exercises
Unit 26 — Limits
26.1 Limits of Functions
26.2 Trigonometric Limits and Limits at Infinity
26.3 Continuity, First Principles and Differentiability
26.4 L’Hôpital’s Rule
Review exercises
Unit 27 — Integration Techniques
27.1 Anti-Derivatives and Indefinite Integrals
27.2 Exponential and Logarithmic Integrals
27.3 Trigonometric Integrals
27.4 Integrating Rational Functions
27.5 Integration by Substitution
27.6 Integration by Parts
Review exercises
Unit 28 — Applications of Integration
28.1 Definite Integrals
28.2 The Definite Integral as a Limit of a Sum
28.3 Area between a Curve and an Axis
28.4 Area between Curves
28.5 Volumes of Revolution
Review exercises
Unit 29 — Kinematics
29.1 Kinematics with Differentiation
29.2 Kinematics with Integration
Review exercises
Unit 30 — Maclaurin Series
30.1 Maclaurin Series of Standard Functions
30.2 Convergence
30.3 Building New Series
Review exercises
Unit 31 — Differential Equations
31.1 Differential Equations and dy/dx = f(x)
31.2 Euler’s Method
31.3 Separable Equations and Logistic Growth
31.4 Homogeneous Equations
31.5 The Integrating Factor Method
31.6 Maclaurin Series from Differential Equations; Modelling
Review exercises
Unit 32 — Systems of Linear Equations
32.1 Systems of Linear Equations
Unit 3 — Partial Fractions
Topic 1: Number and Algebra
Syllabus content:
AHL 1.11
Section
Syllabus
3.1 Partial Fractions
AHL 1.11
Review exercises
Review exercises
3.1 Partial Fractions