Maths HL 3rd Ed 2nd Imprint

1. Theory of Knowledge

1.1 Pure and Applied Mathematics

1.2 Axioms

1.3 Proof

1.3.1 Rules of Inference

1.3.2 Proof by Exhaustion

1.3.3 Direct Proof

1.3.4 Proof by Contradiction

1.4 Paradox

1.4.1 What is a Paradox?

1.4.2 Russell's Paradox

1.5 Mathematics and Other Disciplines

1.6 The Extended Essay

2. Algebra of Linear and Quadratic Expressions

2.1 The Real Number Line

2.1.1 The Real Number Line

2.1.2 Set Builder Notation

2.1.3 Interval Notation

2.1.4 Number Systems

2.1.5 Irrational Numbers

2.1.6 The Absolute Value

2.2 Linear Algebra

2.2.1 Review of Linear Equations

2.2.2 Linear Inequations

2.3 Linear Functions

2.3.1 Graph of the Linear Function

2.3.2 Simultaneous Linear Equations in Two Unknowns

2.3.3 Simultaneous Linear Equations in Three Unknowns

2.4 Quadratics

2.4.1 Quadratic Equation

2.4.2 Quadratic Function

2.4.3 Quadratic Inequalities

2.4.4 Simultaneous Equations Involving Linear-Quadratic Equations

3. Polynomials

3.1 Algebra of Polynomials

3.1.1 Definition

3.1.2 Addition and Multiplication of Polynomials

3.1.3 Division of Polynomials

3.2 Synthetic Division

3.3 The Remainder Theorem

3.4 The Factor Theorem

3.5 Equations and Inequations

3.5.1 Polynomial Equations

3.5.2 Polynomial Inequations

3.6 Sketching Polynomials

3.6.1 Graphical Significance of Roots

3.6.2 Cubic Functions

4. The Binomial Theorem

4.1 The Binomial Theorem

4.1.1 The Binomial Theorem

4.1.2 The General Term

4.2 Proof

5. Functions and Relations

5.1 Relations

5.1.1 Relations

5.1.2 The Cartesian Plane

5.1.3 Implied Domain

5.1.4 Types of Relations

5.1.5 Sketching with the Graphics Calculator

5.2 Functions

5.2.1 Some Definitions

5.3 Some Standard Functions

5.3.1 Hybrid Functions and Continuity

5.3.2 The Absolute Value Function

5.3.3 The Exponential Function

5.3.4 The Logarithmic Function

5.3.5 Equations of the Form y = xⁿ, n = −1, −2

5.4 Algebra of Functions

5.4.1 Basic Operations and Composite Functions

5.4.2 Identity and Inverse Functions

6. Transformations of Graphs

6.1 Translations

6.1.1 Horizontal Translation

6.1.2 Vertical Translation

6.2 Dilations

6.2.1 Dilation from the x-axis

6.2.2 Dilation from the y-axis

6.3 Reflections

6.4 Reciprocal of a Function

7. Exponential and Logarithmic Functions

7.1 Exponents

7.1.1 Basic Rules of Indices

7.1.2 Indicial Equations

7.1.3 Equations of the Form b^f(x) = b^g(x)

7.1.4 What if the Base is Not the Same?

7.1.5 A Special Base e

7.2 Exponential Modelling

7.3 Logarithms

7.3.1 What are Logarithms

7.3.2 Can we find the Logarithm of a Negative Number?

7.4 The Algebra of Logarithms

7.5 Logarithmic Modelling

Revision Set A, Paper 1 and Paper 2 Style Questions

8. Sequences and Series

8.1 Arithmetic Sequences and Series

8.1.1 Arithmetic Sequences

8.1.2 Arithmetic Series

8.1.3 Sigma Notation

8.2 Geometric Sequences and Series

8.2.1 Geometric Sequences

8.2.2 Geometric Series

8.2.3 Combined A.P.s and G.P.s

8.2.4 Convergent Series

8.3 Compound Interest and Superannuation

8.3.1 Compound Interest

8.3.2 Superannuation

9. Mensuration

9.1 Trigonometric Ratios

9.1.1 Review of Trigonometric Functions for Right-angled Triangles

9.1.2 Exact Values

9.2 Applications

9.2.1 Angle of Elevation and Depression

9.2.2 Bearings

9.3 Right Angles in 3-Dimensions

9.4 Area of a Triangle

9.5 Non Right-Angled Triangles

9.5.1 The Sine Rule

9.5.2 The Ambiguous Case

9.5.3 Applications of the Sine Rule

9.5.4 The Cosine Rule

9.5.5 Applications of the Cosine Rule

9.6 More Applications in 3-D

9.7 Arc, Sectors and Segments

9.7.1 Radian Measure of an Angle

9.7.2 Arc Length

9.7.3 Area of Sector

10. Circular Trigonometric Functions

10.1 Trigonometric Ratios

10.1.1 The Unit Circle

10.1.2 Angle of Any Magnitude

10.2 Trigonometric Identities

10.2.1 The Fundamental Identity

10.2.2 Compound Angle Identities

10.3 Trigonometric Functions

10.3.1 The Sine, Cosine and Tangent Functions

10.3.2 Transformations of Trigonometric Functions

10.4 Inverse Trigonometric Functions

10.4.1 The Inverse Sine Function

10.4.2 The Inverse Cosine Function

10.4.3 The Inverse Tangent Function

10.5 Trigonometric Equations

10.5.1 Solution to sin x = a, cos x = a and tan x = a

10.6 Applications

11. Complex Numbers

11.1 Complex Numbers

11.1.1 Introduction

11.1.2 Notations and i² = −1

11.1.3 Using the TI-83 with Complex Numbers

11.1.4 The Algebra of Complex Numbers

11.2 Geometrical Representation of Complex Numbers

11.2.1 The Argand Diagram

11.2.2 Geometrical Properties of Complex Numbers

11.3 Polar Form of Complex Numbers

11.4 Polynomials over the Complex Field

11.4.1 Quadratic Equations

11.4.2 Polynomial Equations of order ≥ 3

11.4.3 Solutions to Equations of the Form zⁿ = x + iy

11.5 Miscellaneous Proofs

11.5.1 The Triangle Inequality

11.5.2 De Moivre's Theorem

11.5.3 Fundamental Theorem of Algebra

11.5.4 Conjugate Root Theorem

12. Mathematical Induction

12.1 Mathematical Induction

12.1.1 The Principles of Mathematical Induction and Proofs

12.2 Further Examples Part 1

12.3 Further Examples Part 2

12.4 Forming Conjectures

12.4.1 The nth Term of a Sequence

12.4.2 The Sum to n Terms of a Sequence

Revision Set B, Paper 1 and Paper 2 Style Questions

13. Statistics

13.1 Describing Data

13.1.1 Data Collection

13.1.2 Types of Data

13.1.3 Discrete and Continuous Data

13.2 Frequency Diagrams

13.2.1 Producing a Frequency Diagram

13.2.2 Using a Graphics Calculator

13.3 Statistical Measures 1

13.3.1 Measure of Central Tendency

13.3.2 Mode

13.3.3 Mean

13.3.4 Median

13.4 Statistical Measures 2

13.4.1 Measures of Spread

13.4.2 Variance and Standard Deviation

13.4.3 Using a Graphics Calculator

13.5 Statistical Measures 3

13.5.1 Quartiles

13.5.2 Box Plot

14. Counting Principles

14.1 Multiplication Principle

14.1.1 Definition

14.1.2 Multiplication Principle

14.1.3 Permutations

14.2 Combinations

15. Probability

15.1 Probability

15.1.1 Probability as a Long-Term Relative Frequency

15.1.2 Theoretical Probability

15.1.3 Laws of Probability

15.1.4 Definition of Probability

15.1.5 Problem Solving Strategies in Probability

15.2 Probability and Venn Diagrams

15.3 Conditional Probability

15.3.1 Informal Definition of Conditional Probability

15.3.2 Formal Definition of Conditional Probability

15.3.3 Independence

15.4 Bayes' Theorem

15.4.1 Law of Total Probability

15.4.2 Bayes' Theorem for Two Elements

15.5 Using Permutations and Combinations in Probability

16. Discrete Random Variables

16.1 Discrete Random Variables

16.1.1 Concept of a Random Variable

16.1.2 Discrete Random Variable

16.1.3 Probability Distributions

16.1.4 Properties of the Probability Function

16.1.5 Constructing Probability Functions

16.2 Mean and Variance

16.2.1 Central Tendency and Expectation

16.2.2 So, what exactly does E(X) measure?

16.2.3 Properties of the Expectation Function

16.2.4 Variance

16.2.5 Properties of the Variance

16.3 The Binomial Distribution

16.3.1 The Binomial Experiment

16.3.2 Bernoulli Trials

16.3.3 Properties of the Binomial Experiment

16.3.4 Binomial Distribution

16.3.5 Expectation, Mode and Variance for the Binomial Distribution

16.4 Hypergeometric Distribution

16.4.1 Hypergeometric Distribution Function

16.4.2 Mean and Variance of the Hypergeometric Distribution

16.5 Poisson Distribution

16.5.1 Poisson Distribution Function

16.5.2 Poisson Recurrence Formula

16.5.3 Mean and Variance of the Poisson Distribution

16.5.4 Poisson as a Limit of Binomial

17. The Normal Distribution

17.1 The Normal Distribution

17.1.1 Why the Normal Distribution?

17.1.2 The Standard Normal Curve

17.1.3 Using the Standard Normal Table

17.2 Formalising the Definition of the Normal Distribution

17.2.1 The Normal Distribution

17.2.2 Properties of This Curve

17.2.3 Finding Probabilities Using the Normal Distribution

17.2.4 The Standard Normal Distribution

17.2.5 Finding Probabilities

17.2.6 Standardising any Normal Distribution

17.2.7 Inverse Problems

17.2.8 Finding Quantiles

Revision Set C, Paper 1 and Paper 2 Style Questions

18. Rates of Change

18.1 Quantitative Measure

18.1.1 Functional Dependence

18.1.2 Quantitative Aspects of Change

18.1.3 Average Rate of Change

18.1.4 Determining the Average Rate of Change

18.1.5 Velocity as a Measure of Rate of Change of Displacement

18.2 Qualitative Measure

18.2.1 Qualitative Aspects of Change

18.2.2 Describing the Behaviour of a Graph

18.2.3 Producing a Graph from a Physical Situation

18.3 Instantaneous Rate of Change

18.3.1 Informal Idea of Limits

18.4 Differentiation Process

18.4.1 The Derivative and the Gradient Function

18.4.2 Notation and Language

19. Differential Calculus

19.1 Differentiation

19.1.1 Review

19.1.2 Power Rule for Differentiation

19.1.3 Derivative of a Sum or Difference

19.2 Graphical Interpretation of the Derivative

19.2.1 The Value of the Derivative at a Particular Point on a Curve

19.2.2 Gradient Function from a Graph

19.2.3 Differentiating with Variables other than x and y

19.3 Derivative of Transcendental Functions

19.3.1 Derivative of Circular Trigonometric Functions

19.3.2 Derivative of the Exponential Functions

19.3.3 Derivative of the Natural Log Functions

19.3.4 Derivative of a Product of Functions

19.3.5 Derivative of a Quotient of Functions

19.3.6 The Chain Rule

19.3.7 Derivative of Reciprocal Circular Functions

19.4 Derivative of Inverse Trigonometric Functions

19.4.1 Derivative of Sin⁻¹(x)

19.4.2 Derivative of Cos⁻¹(x)

19.4.3 Derivative of Tan⁻¹(x)

19.4.4 Generalisations for the Derivative of the Inverse Circular Functions

19.5 Derivative of aˣ and logₐx

19.5.1 Differentiating y = aˣ

19.5.2 Differentiating y = logₐx

19.5.3 Generalisation of Exponential and Logarithmic Derivatives

19.6 Second Derivative

19.7 Implicit Differentiation

19.7.1 Implicit Relations

19.7.2 Second Derivative

19.8 Proofs

19.8.1 Derivative of y = xⁿ where n is a Negative Integer

19.8.2 Derivative of y = xⁿ where n is a Fraction

19.8.3 Product Rule and Quotient Rule

19.8.4 Derivative of Some Trigonometric Functions

19.8.5 Exponential and y = xⁿ where n is Real

20. Differential Calculus and Curve Sketching

20.1 Tangents and Normals

20.1.1 Equation of Tangent

20.1.2 Equation of Normal

20.2 Curve Sketching

20.2.1 Increasing and Decreasing Functions

20.2.2 Stationary Points

20.2.3 Global Maxima and Minima

20.2.4 Vertical Tangents and Cusps

20.2.5 Summary

20.3 The Second Derivative and its Application

20.3.1 Definition

20.4 Rational Functions

20.4.1 Sketching the Graph of x ↦ (ax+b)/(cx+d), cx+d ≠ 0

20.4.2 Other Rational Functions

20.4.3 Oblique and Curved Asymptotes

21. Applications of Differential Calculus

21.1 Rates of Change

21.1.1 Definitions

21.1.2 Rates of Change and Their Sign

21.2 Applied Rates of Change

21.2.1 Application to Economics

21.2.2 Application to Kinematics

21.3 Kinematics

21.3.1 Motion Along a Straight Line

21.4 Related Rates

21.5 Applied Maxima and Minima Problems

21.5.1 Maxima-Minima Problems

21.5.2 End point Maxima-Minima Problems

21.5.3 Optimisation for Integer Valued Variables

Revision Set D, Paper 1 and Paper 2 Style Questions

22. Integration and its Applications

22.1 Integration

22.1.1 Antidifferentiation and the Indefinite Integral

22.1.2 Language and Notation

22.1.3 Determining the Indefinite Integral

22.1.4 Properties of the Indefinite Integral

22.2 Solving for c

22.3 Standard Integrals

22.4 The Definite Integral

22.4.1 Why the Definite Integral?

22.4.2 Language and Notation

22.4.3 Properties of the Definite Integral

22.5 Applications of Integration

22.5.1 Introduction to the Area Beneath a Curve

22.5.2 In Search of a Better Approximation

22.5.3 Towards an Exact Value

22.5.4 The Definite Integral and Areas

22.5.5 Further Observation about Areas

22.5.6 The Signed Areas

22.5.7 Steps for Finding Areas

22.5.8 Area Between Two Curves

22.6 Applications to Kinematics

22.7 Applications to Probability

22.7.1 Continuous Random Variables

22.7.2 Mode, Mean, Median and Variance

22.8 Volumes, Solid of Revolution

23. Further Integration

23.1 Integration by Substitution

23.1.1 Indefinite Integrals

23.1.2 Substitution Rule

23.2 Integration by Parts

23.2.1 The Basics

23.2.2 Repeated Integration by Parts

23.3 Applications

24. Differential Equations

24.1 Differential Equations

24.1.1 What are Differential Equations?

24.1.2 Verifying a Solution

24.2 Solving Differential Equations

24.3 Applications

25. Matrices

25.1 Introduction to Matrices

25.1.1 Definitions

25.1.2 Matrix Multiplication

25.2 Inverses and Determinants

25.2.1 Inverse and Determinant of 2 by 2 Matrices

25.2.2 Inverse and Determinant of 3 by 3 Matrices

25.3 Simultaneous Equations

26. Vectors

26.1 Introduction to Vectors

26.1.1 Scalar and Vector Quantities

26.2 Representing Vectors

26.2.1 Directed Line Segment

26.2.2 Magnitude of a Vector

26.2.3 Equal Vectors

26.2.4 Negative Vectors

26.2.5 Zero Vector

26.2.6 Orientation and Vectors

26.3 Algebra and Geometry of Vectors

26.3.1 Addition of Vectors

26.3.2 Subtraction of Vectors

26.3.3 Multiplication of Vectors by Scalars

26.3.4 Angle Between Vectors

26.3.5 Applications to Geometry

26.4 Cartesian Representation of Vectors in 2-D and 3-D

26.4.1 Representation in 2-D

26.4.2 Unit Vector and Base Vector Notation

26.4.3 Representation in 3-D

26.4.4 Vector Operations

26.5 Further Properties of Vectors in 2-D and 3-D

26.5.1 Magnitude of a Vector

26.5.2 Unit Vectors

26.6 Scalar Product of 2 Vectors

26.6.1 Definition of the Scalar Product

26.6.2 Properties of the Scalar Product

26.6.3 Special Cases of the Scalar Product

26.6.4 Direction Cosine

26.6.5 Using a Graphics Calculator

26.7 Vector Equation of a Line

26.7.1 Vector Equation of a Line in 2-D

26.7.2 Lines in Three Dimensions

27. 3-D Geometry

27.1 Vector Product

27.1.1 Vector Product

27.1.2 Vector Form of the Vector Product

27.1.3 Applications of the Vector Product

27.2 Planes in 3-Dimensions

27.2.1 Vector Equation of a Plane

27.2.2 Cartesian Equation of a Plane

27.2.3 Normal Vector Form of a Plane

27.2.4 The Normal Form

27.3 Intersecting Lines and Planes

27.3.1 Intersection of Two Lines

27.3.2 Intersection of a Line and a Plane

27.3.3 Intersection of Two Planes

27.3.4 Intersection of Three Planes

Revision Set E, Paper 1 and Paper 2 Style Questions

Answers