Mathematics IB Standard Level (3rd ed)

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

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. Modelling

3.1. Modelling — An Introduction

3.1.1. Modelling — Phase 1

3.1.2. Modelling — Phase 2

3.1.3. Modelling — Phase 3

3.2. Mathematical Models

3.2.1. Mathematical Investigation and Modelling

3.2.2. Open and Closed Problem Solving

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. 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 $y=x^n$, $n=-1,-2$

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

A. Revision Set — 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. Trigonometric Equations

10.4.1. Solution to $\sin x=a$, $\cos x=a$ and $\tan x=a$

10.5. Applications

B. Revision Set — Paper 1 and Paper 2 Style Questions

11. Matrices

11.1. Introduction to Matrices

11.1.1. Definitions

11.1.2. Matrix Multiplication

11.2. Inverses and Determinants

11.2.1. Inverse and Determinant of $2\times2$ Matrices

11.2.2. Inverse and Determinant of $3\times3$ Matrices

11.3. Simultaneous Equations

12. Vectors

12.1. Introduction to Vectors

12.1.1. Scalar and Vector Quantities

12.2. Representing Vectors

12.2.1. Directed Line Segment

12.2.2. Magnitude of a Vector

12.2.3. Equal Vectors

12.2.4. Negative Vectors

12.2.5. Zero Vector

12.2.6. Orientation and Vectors

12.3. Algebra and Geometry of Vectors

12.3.1. Addition of Vectors

12.3.2. Subtraction of Vectors

12.3.3. Multiplication of Vectors by Scalars

12.3.4. Angle Between Vectors

12.3.5. Applications to Geometry

12.4. Cartesian Representation of Vectors in 2-D and 3-D

12.4.1. Representation in 2-D

12.4.2. Unit Vector and Base Vector Notation

12.4.3. Representation in 3-D

12.4.4. Vector Operations

12.5. Further Properties of Vectors in 2-D and 3-D

12.5.1. Magnitude of a Vector

12.5.2. Unit Vectors

12.6. Scalar Product of 2 Vectors

12.6.1. Definition of the Scalar Product

12.6.2. Properties of the Scalar Product

12.6.3. Special Cases of the Scalar Product

12.6.4. Direction Cosine

12.6.5. Using a Graphics Calculator

12.7. Vector Equation of a Line

12.7.1. Vector Equation of a Line in 2-D

12.7.2. Application of the Vector Equation $\mathbf{r}=\mathbf{a}+t\mathbf{b}$

12.7.3. Lines in Three Dimensions

C. Revision Set — 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

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

D. Revision Set — 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. Second Derivative

19.5. Proofs

19.5.1. Derivative of $y=x^n$ Where $n$ Is a Negative Integer

19.5.2. Derivative of $y=x^n$ Where $n$ Is a Fraction

19.5.3. Product Rule and Quotient Rule

19.5.4. Derivative of Some Trigonometric Functions

19.5.5. Exponential and $y=x^n$ 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 $x\mapsto\frac{ax+b}{cx+d}$, $cx+d\ne0$

20.4.2. Other Rational Functions

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. Applied Maxima and Minima Problems

21.4.1. Maxima–Minima Problems

21.4.2. End Point Maxima/Minima Problems

21.4.3. Optimisation for Integer Valued Variables

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. Volumes: Solids of Revolution

E. Revision Set — Paper 1 and Paper 2 Style Questions

23. Portfolio Work

23.1. Introducing the Portfolio

23.1.1. Purpose of the Portfolio

23.1.2. Problem Solving

23.2. Portfolio Work Type I — Mathematical Investigation

23.2.1. All Is Fine and Well… but Where Do I Start?

23.3. Portfolio Work Type II — Mathematical Modelling

23.3.1. Carrying Out a Modelling Investigation

23.3.2. Modelling Investigation Tasks

24. Answers to Exercises

25. Answers to Revision Sets