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Glossary
Chapter 1: Number systems and arithmetic
- Overview
- 1.1 The number systems ℕ, ℤ, ℚ and ℝ
- 1.2 Irrational numbers and surds
- 1.3 Geometric construction of √2 and √3
- 1.4 Absolute value
- 1.5 Indices and the rules of indices
- 1.6 Solving problems with indices
- 1.7 Scientific notation and accuracy
- Review
Chapter 2: Arithmetic in context
- Overview
- 2.1 Percentages and percentage change
- 2.2 Profit, loss and mark-up
- 2.3 Income tax, USC and net pay
- 2.4 Simple and compound interest
- 2.5 Present value
- 2.6 Checking whether a result is reasonable
- Review
Chapter 3: Length, area and volume
- Overview
- 3.1 Perimeter and area of plane figures
- 3.2 The circle: circumference and area
- 3.3 Volume of prisms and cylinders
- 3.4 Nets of prisms, cylinders and spheres
- 3.5 Surface area and curved surface area
- 3.6 Similar figures: length, area and volume ratios
- 3.7 The volume and surface area of a sphere
- Review
Chapter 4: Algebra: expressions and equations
- Overview
- 4.1 Generating expressions from patterns
- 4.2 Representing relations in tables and graphs
- 4.3 Finding a formula from a pattern
- 4.4 Features of linear relationships
- 4.5 Evaluating and simplifying expressions
- 4.6 Solving linear equations
- 4.7 Simultaneous equations in two variables
- 4.8 Solving quadratic equations
- 4.9 Linear inequalities
- 4.10 Inequalities and the modulus
- Review
Chapter 5: Complex numbers
- Overview
- 5.1 The imaginary unit and complex numbers
- 5.2 Arithmetic with complex numbers
- 5.3 The complex conjugate and division
- 5.4 The Argand diagram and modulus
- 5.5 Cubic equations and the conjugate root theorem
- Review
Chapter 6: Functions
- Overview
- 6.1 Relations and functions
- 6.2 Domain, range and notation
- 6.3 Composite and inverse functions
- 6.4 Injective, surjective and bijective functions
- 6.5 Graphing functions and transformations
- 6.6 Exponential and logarithmic functions
- Review
Chapter 7: Differentiation
- Overview
- 7.1 The derivative from first principles
- 7.2 Differentiating powers and polynomials
- 7.3 The product rule
- 7.4 The quotient rule
- 7.5 The chain rule
- 7.6 The second derivative and concavity
- 7.7 Rates of change in context
- 7.8 Stationary points, maxima and minima
- 7.9 Curve sketching
- 7.10 Differentiating exponential and trigonometric functions
- Review
Chapter 8: Integration
- Overview
- 8.1 Antiderivatives
- 8.2 The definite integral and area
- 8.3 The fundamental theorem of calculus
- 8.4 Area between a curve and the x-axis
- 8.5 Integration by substitution
- 8.6 Average value and accumulated change
- Review
Chapter 9: Sequences and series
- Overview
- 9.1 Arithmetic sequences
- 9.2 Arithmetic series
- 9.3 Geometric sequences and series
- 9.4 Infinite geometric series
- 9.5 Recurrence relations
- 9.6 Proof by induction
- Review
Chapter 10: Counting and probability
- Overview
- 10.1 Counting outcomes and the fundamental principle
- 10.2 Permutations
- 10.3 Combinations
- 10.4 Sample spaces and set theory
- 10.5 Axioms of probability
- 10.6 Conditional probability
- 10.7 Independent events
- 10.8 Bernoulli trials and the binomial distribution
- Review
Chapter 11: Statistics
- Overview
- 11.1 Populations and samples
- 11.2 Sampling methods and bias
- 11.3 Representing data: charts and histograms
- 11.4 Measures of centre and spread
- 11.5 The standard deviation
- 11.6 The normal distribution
- 11.7 Sampling variability and the margin of error
- 11.8 The central limit theorem
- 11.9 Hypothesis testing and confidence intervals
- 11.10 Correlation and regression
- 11.11 Statistical reasoning and misused statistics
- Review
Chapter 12: Trigonometry
- Overview
- 12.1 The trigonometric ratios in a right triangle
- 12.2 Pythagoras' theorem
- 12.3 Radians and degrees
- 12.4 The unit circle and trigonometric functions of any angle
- 12.5 Graphs of the trigonometric functions
- 12.6 The sine and cosine rules
- 12.7 Area of a triangle
- 12.8 Trigonometric identities
- 12.9 Solving trigonometric equations
- 12.10 Bearings and three-dimensional problems
- Review
Chapter 13: Coordinate geometry
- Overview
- 13.1 Distance, midpoint and slope
- 13.2 The equation of a line
- 13.3 Parallel and perpendicular lines
- 13.4 The equation of a circle
- 13.5 Tangents to a circle
- 13.6 Intersection of a line and a circle
- 13.7 Dividing a line segment in a given ratio
- Review
Chapter 14: Transformation geometry
- Overview
- 14.1 Translations
- 14.2 Axial symmetry and reflection
- 14.3 Central symmetry and rotation
- 14.4 Enlargements and the scale factor
- 14.5 Transformations and symmetry in the plane
- 14.6 Composing transformations
- Review
Chapter 15: Synthetic geometry: foundations
- Overview
- 15.1 Axioms and the rules of deduction
- 15.2 The first theorems on angles
- 15.3 Congruent triangles
- 15.4 Parallel lines
- 15.5 The isosceles triangle and its converse
- 15.6 Perpendicular lines and right angles
- 15.7 Constructions 1–6
- 15.8 Writing a formal proof
- Review
Chapter 16: Synthetic geometry: quadrilaterals, similarity and area
- Overview
- 16.1 Quadrilaterals and parallelograms
- 16.2 Corollaries of the parallelogram theorems
- 16.3 Similar triangles
- 16.4 Proportionality and ratios in triangles
- 16.5 Pythagoras' theorem proved
- 16.6 Area and the area of a triangle
- 16.7 Constructions 7–15
- Review
Chapter 17: Synthetic geometry: circles
- Overview
- 17.1 Chords and the centre
- 17.2 The angle in a semicircle and related theorems
- 17.3 Cyclic quadrilaterals
- 17.4 Tangents to a circle
- 17.5 The special points of a triangle
- 17.6 Constructions 16–22
- 17.7 Proving a circle theorem
- 17.8 Enlargements and similar triangles in the circle
- Review
Chapter 18: Vectors
- Overview
- 18.1 Vectors and their representation
- 18.2 Vector arithmetic
- 18.3 The dot product
- 18.4 Perpendicularity and the angle between vectors
- 18.5 Vectors in three dimensions
- 18.6 Geometric applications of vectors
- Review
About this book

8.5 Integration by substitution — Practise with help

Stub. Scaffolded practice with progressive exposure: the student attempts each step, then reveals a hint or a fuller worked step only if needed. Tier: H · refs: 5.2