Jacaranda Maths Quest Units 1&2 Mathematical Methods 11 for Queensland

Title page
Copyright page
Contents
About this resource
About eBookPLUS and studyON
Acknowledgements
CHAPTER 1 Arithmetic sequences
1.1 Overview
1.1.1 Introduction
1.2 Arithmetic sequences
1.2.1 Defining mathematical sequences
1.2.2 Sequences expressed as functions
1.2.3 Arithmetic sequences
1.2.4 The recursive definition of arithmetic sequences
1.3 The general form of an arithmetic sequence
1.3.1 The general term of an arithmetic sequence
1.3.2 Graphical display of sequences
1.4 The sum of an arithmetic sequence
1.4.1 Arithmetic sequences and series
1.4.2 A visual explanation of Sn
1.5 Applications of arithmetic sequences
1.5.1 Simple interest
1.5.2 Depreciating assets
1.6 Review: exam practice
Answers
REVISION UNIT 1 Algebra, statistics and functions
TOPIC 1 Arithmetic and geometric sequences and series 1
CHAPTER 2 Functions
2.1 Overview
2.1.1 Introduction
2.2 Functions and relations
2.2.1 Set and interval notation
2.2.2 Relations
2.2.3 Functions
2.3 Function notation
2.3.1 Domain and range
2.3.2 Function notation
2.3.3 Formal mapping notation
2.4 Transformations of functions
2.4.1 Dilations
2.4.2 Dilation from the x-axis by factor a
2.4.3 Dilation from the y-axis by factor b
2.4.4 Reflections
2.4.5 Translations
2.4.6 Combinations of transformations
2.5 Piece-wise functions
2.5.1 Piece-wise functions
2.5.2 Modelling with piece-wise functions
2.6 Review: exam practice
Answers
CHAPTER 3 Quadratic relationships
3.1 Overview
3.1.1 Introduction
3.2 Graphs of quadratic functions
3.2.1 The graph of y = x2 and transformations
3.2.2 Sketching parabolas from their equations
3.2.3 The general, or polynomial form, y = ax2 + bx + c
3.2.4 Turning point form, y = a(x − b)2 + c
3.2.5 Factorised, or x-intercept, form y = a(x − b)(x − c)
3.2.6 Determining the rule of a quadratic polynomial from a graph
3.2.7 Using simultaneous equations
3.3 Solving quadratic equations with rational roots
3.3.1 Quadratic equations and the Null Factor Law
3.3.2 Using the perfect square form of a quadratic
3.3.3 Equations that reduce to quadratic form
3.4 Factorising and solving quadratics over R
3.4.1 Factorisation over R
3.4.2 The quadratic formula
3.5 The discriminant
3.5.1 Defining the discriminant
3.5.2 The role of the discriminant in quadratic equations
3.5.3 The discriminant and the x-intercepts
3.5.4 Intersections of lines and parabolas
3.6 Modelling with quadratic functions
3.6.1 Quadratically related variables
3.6.2 Maximum and minimum values
3.7 Review: exam practice
Answers
CHAPTER 4 Inverse proportions and graphs of relations
4.1 Overview
4.1.1 Introduction
4.2 The hyperbola
4.2.1 The graph of y =1x
4.2.2 General equation of a hyperbola
4.2.3 Finding the equation of a hyperbola
4.2.4 Modelling with the hyperbola
4.3 Inverse proportion
4.4 The circle
4.4.1 Equation of a circle
4.4.2 Semicircles
4.5 The sideways parabola
4.5.1 The relation y2 = x
4.5.2 Transformations of the graph of y2 = x
4.5.3 Determining the rule for the sideways parabola
4.6 Review: exam practice
Answers
CHAPTER 5 Powers and polynomials
5.1 Overview
5.1.1 Introduction
5.2 Polynomials
5.2.1 Classification of polynomials
5.2.2 Polynomial notation
5.2.3 Identity of polynomials
5.2.4 Expansion of cubic and quadratic polynomials from factors
5.2.5 Operations on polynomials
5.2.6 Division of polynomials
5.3 Graphs of cubic polynomials
5.3.1 The graph of y = x3 and transformations
5.3.2 Cubic graphs with three x-intercepts
5.3.3 Cubic graphs with two x-intercepts
5.3.4 Cubic graphs in the general form y = ax3 + bx2 + cx + d
5.3.5 Determining the equation of cubic graph
5.4 The factor and remainder theorems
5.4.1 The remainder theorem
5.4.2 The factor theorem
5.4.3 Factorising polynomials
5.5 Solving cubic equations
5.5.1 Polynomial equations
5.5.2 Solving cubic equations using the Null Factor Law
5.5.3 Intersections of cubic graphs with linear and quadratic graphs
5.6 Cubic models and applications
5.7 Graphs of quartic polynomials
5.7.1 Graphs of quartic polynomials of the form y = a(x − b)4 + c
5.7.2 Quartic polynomials which can be expressed as the product of linear factors
5.8 Solving polynomial equations
5.8.1 The method of bisection
5.8.2 Using the intersections of two graphs to estimate solutions to equations
5.8.3 Estimating coordinates of turning points
5.9 Review: exam practice
Answers
REVISION UNIT 1 Algebra, statistics and functions
TOPIC 2 Functions and graphs
PRACTICE ASSESSMENT 1
Mathematical Methods: Problem solving and modelling task
CHAPTER 6 Counting and probability
6.1 Overview
6.1.1 Introduction
6.2 Fundamentals of probability
6.2.1 Notation and fundamentals: outcomes, sample spaces and events
6.2.2 Venn diagrams
6.2.3 Probability tables
6.2.4 Tree diagrams
6.3 Relative frequency
6.3.1 Relative frequency
6.4 Conditional probability
6.4.1 Introduction
6.4.2 Formula for conditional probability
6.4.3 Multiplication of probabilities
6.4.4 Probability tree diagrams
6.5 Independence
6.5.1 Introduction
6.5.2 Test for mathematical independence
6.5.3 Independent trials
6.6 Permutations and combinations
6.6.1 Arrangements or permutations
6.6.2 Arrangements in a circle
6.6.3 Arrangements with objects grouped together
6.6.4 Arrangements where some objects may be identical
6.6.5 Combinations or selections
6.7 Pascal’s triangle and binomial expansions
6.7.1 Pascal’s triangle
6.7.2 Formula for binomial coefficients
6.7.3 Pascal’s triangle with combinatoric coefficients
6.7.4 Extending the binomial expansion to probability
6.8 Review: exam practice
Answers
REVISION UNIT 1 Algebra, statistics and functions
TOPIC 3 Counting and probability
CHAPTER 7 Indices
7.1 Overview
7.1.1 Introduction
7.2 Index laws
7.2.1 Introduction
7.2.2 Review of the index laws
7.2.3 Products and quotients
7.3 Negative and rational indices
7.3.1 Negative indices
7.3.2 Fractional indices
7.4 Indicial equations and scientific notation
7.4.1 Indicial equations
7.4.2 Method of equating indices
7.4.3 Indicial equations which reduce to quadratic form
7.4.4 Scientific notation (standard form)
7.4.5 Significant figures
7.5 Review: exam practice
Answers
REVISION UNIT 1 Algebra, statistics and functions
TOPIC 4 Exponential functions 1
CHAPTER 8 Geometric sequences
8.1 Overview
8.1.1 Introduction
8.2 Recursive definition and the general term of geometric sequences
8.2.1 Common ratio of geometric sequences
8.2.2 The recursive definition of a geometric sequence
8.2.3 The general term of the geometric sequence
8.3 The sum of a geometric sequence
8.3.1 Limiting behaviour as n → ∞
8.3.2 The sum of the first n terms of a geometric sequence
8.3.3 The sum of an infinite geometric sequence
8.4 Geometric sequences in context
8.4.1 Growth and decay in the real world
8.4.2 Compound interest
8.4.3 Reducing balance depreciation
8.5 Review: exam practice
Answers
REVISION UNIT 1 Algebra, statistics and functions
TOPIC 5 Arithmetic and geometric sequences and series 2
PRACTICE ASSESSMENT 2
Mathematical Methods: Unit 1 examination
CHAPTER 9 Exponential and logarithmic functions
9.1 Overview
9.1.1 Introduction
9.2 Exponential functions
9.2.1 The graph of y = ax where a > 1
9.2.2 The graph of y = ax where 0 < a < 1
9.2.3 Translations of exponential graphs
9.3 Logarithmic functions
9.3.1 Defining logarithms
9.3.2 Logarithm laws
9.3.3 The graph of y = loga(x) for a > 1
9.3.4 Extension: Transformations of logarithmic graphs
9.4 Modelling with exponential functions
9.4.1 Exponential growth and decay models
9.4.2 Analysing data
9.5 Solving equations with indices
9.5.1 Logarithms as operators
9.5.2 Equations containing logarithms
9.6 Review: exam practice
Answers
REVISION UNIT 2 Calculus and further functions
TOPIC 1 Exponential functions 2
TOPIC 2 The logarithmic function 1
CHAPTER 10 Trigonometric functions
10.1 Overview
10.1.1 Introduction
10.2 Trigonometry review
10.2.1 Right-angled triangles
10.2.2 Exact values for trigonometric ratios of 30°, 45°, 60°
10.2.3 Deducing one trigonometric ratio from another
10.2.4 Area of a triangle
10.3 Radian measure
10.3.1 Definition of radian measure
10.3.2 Extended angle measure
10.3.3 Using radians in calculations
10.4 Unit circle definitions
10.4.1 Trigonometric points
10.4.2 Unit circle definitions of the sine, cosine and tangent functions
10.4.3 Unit circle definition of the tangent function
10.4.4 Domains and ranges of the trigonometric functions
10.5 Exact values and symmetry properties
10.5.1 The signs of the sine, cosine and tangent values in the four quadrants
10.5.2 The sine, cosine and tangent values at the boundaries of the quadrants
10.5.3 Trigonometric points symmetric to [?] where? ∈ {30°, 45°, 60°,?6,?4,?3}
10.5.4 Symmetry properties
10.6 Graphs of the sine, cosine and tangent functions
10.6.1 The graphs of y = sin(x) and y = cos(x)
10.6.2 One cycle of the graph of y = sin(x)
10.6.3 One cycle of the graph of y = cos(x)
10.6.4 Guide to sketching the graphs on extended domains
10.6.5 The graph of y = tan (x)
10.7 Transformations of sine and cosine graphs
10.7.1 Transformations of the sine and cosine graphs
10.7.2 Amplitude changes
10.7.3 Period changes
10.7.4 Equilibrium (or mean) position changes
10.7.5 Phase changes
10.7.6 The graphs of y = A sin(B(x + C)) + D and y = A cos(B(x + C)) + D
10.7.7 Forming the equation of a sine or cosine graph
10.8 Solving rigonometric equations
10.8.1 Solving trigonometric equations on finite domains
10.8.2 Symmetric forms
10.8.3 Trigonometric equations with boundary value solutions
10.8.4 Further types of trigonometric equations
10.8.5 Solving trigonometric equations which require a change of domain
10.9 Modelling with trigonometric functions
10.9.1 Maximum and minimum values
10.10 Review: exam practice
Answers
REVISION UNIT 2 Calculus and further functions
TOPIC 3 Trigonometric functions 1
CHAPTER 11 Rates of change
11.1 Overview
11.1.1 Introduction
11.2 Exploring rates of change
11.2.1 Constant and variable rates of change
11.2.2 Average rates of change
11.2.3 Instantaneous rates of change
11.3 The difference quotient
11.3.1 The limit
11.3.2 The gradient as a limit
11.4 Differentiating simple functions
11.5 Interpreting the derivative
11.5.1 Interpreting the derivative as the instantaneous rate of change
11.5.2 Interpreting the derivative as the gradient of a tangent line
11.6 Review: exam practice
Answers
CHAPTER 12 Properties and applications of derivatives
12.1 Overview
12.2 Differentiation by formula
12.3 The derivative as a function
12.3.1 Derivative notation
12.3.2 Differentiability of a function
12.4 Properties of the derivative
12.5 Differentiation of power and polynomial functions
12.6 Review: exam practice
Answers
CHAPTER 13 Applications of derivatives
13.1 Overview
13.1.1 Introduction
13.2 Gradient and equation of a tangent
13.2.1 Tangents
13.2.2 Normals
13.3 Displacement–time graphs
13.3.1 Definitions
13.4 Sketching curves using derivatives
13.4.1 Sketching curves with the first derivative test
13.4.2 Sketching curves with the second derivative test
13.4.3 Global maxima and minima
13.4.4 End behaviour of a function
13.5 Modelling optimisation problems
13.5.1 When the rule for the function is known
13.5.2 When the rule for the function is not known
13.6 Review: exam practice
Answers
REVISION UNIT 2 Calculus and further functions
TOPIC 4 Introduction to differential calculus
CHAPTER 14 Differentiation rules
14.1 Overview
14.2 The product rule
14.3 The quotient rule
14.4 The chain rule
14.5 Applications of the product, quotient and chain rules
14.6 Review: exam practice
Answers
REVISION UNIT 2 Calculus and further functions
RevisionUnit2Topic05
TOPIC 5 Further differentiation and applications 1
CHAPTER 15 Discrete random variables 1
15.1 Overview
15.1.1 Introduction
15.2 Discrete random variables
15.3 Expected values
15.4 Variance and standard deviation
15.4.1 Introduction
15.4.2 Properties of the variance
15.5 Applications of discrete random variables
15.6 Review: exam practice
Answers
REVISION UNIT 2 Calculus and further functions
TOPIC 6 Discrete random variables 1
PRACTICE ASSESSMENT 3
Mathematical Methods: Unit 2 examination
PRACTICE ASSESSMENT 4
Mathematical Methods: Units 1 & 2 examination
GLOSSARY
INDEX