Efnisyfirlit

• Contents
• Introduction
• Chapter 1: Algebra and function basics
  • 1.1: Equations and formulae
    • Equations, identities and formulae
    • Equations and graphs
    • Equations of lines
    • Distances and midpoints
    • Systems of linear equations
  • 1.2: Definition of a function
    • Domain and range of a function
    • Function notation
  • 1.3: Composite functions
    • Composition of functions
    • Decomposing a composite function
  • 1.4: Inverse functions
    • Pairs of inverse functions
    • The existence of an inverse function
    • Finding the inverse of a function
  • 1.5: Transformations of functions
    • Graphs of common functions
    • Vertical and horizontal translations
    • Reflections
    • Non-rigid transformations: stretching and shrinking
    • Reciprocal and absolute value graphs
• Chapter 2: Functions
  • 2.1: Polynomial functions
    • Graphs of polynomial functions
  • 2.2: Quadratic functions
    • The graph of f(x) = a(x - h)2 + k
    • Completing the square
    • Zeros of a quadratic function
    • The quadratic formula and the discriminant
  • 2.3: Zeros, factors and remainders
    • Polynomial division
  • 2.4: Rational functions
  • 2.5: Solving equations and inequalities
    • Equations involving a radical
    • Equations involving fractions
    • Equations in quadratic form
    • Equations involving absolute value (modulus)
    • Solving inequalities
    • Quadratic inequalities
    • Absolute value (modulus) inequalities
    • Algebraic and graphical methods
  • 2.6: Partial fractions
• Chapter 3: Sequences and series
  • 3.1: Sequences
  • 3.2: Arithmetic sequences
  • 3.3: Geometric sequences
    • Compound interest
  • 3.4: Series
    • Sigma notation
    • Arithmetic series
    • Geometric series
    • Applications of series to compound interest calculations
  • 3.5: The binomial theorem
    • Using the binomial theorem
  • 3.6: Counting principles
    • Simple counting problems
    • Permutations
    • Combinations
• Chapter 4: Exponential and logarithmic functions
  • 4.1: Exponential functions
    • Characteristics of exponential functions
  • 4.2: Exponential growth and decay
    • Mathematical models of growth and decay
  • 4.3: The number e
  • 4.4: Logarithmic functions
  • 4.5: Exponential and logarithmic equations
    • Solving exponential equations
    • Solving logarithmic equations
    • Exponential and logarithmic inequalities
• Chapter 5: Proofs
  • 5.1: Basic laws and simple proofs
    • Logic basics
    • Valid arguments
  • 5.2: Direct proofs
  • 5.3: Indirect proofs
    • Proof by contradiction
    • Proof by contrapositive
  • 5.4: Mathematical induction
• Chapter 6: Trigonometric functions and equations
  • 6.1: Angles, circles, arcs and sectors
    • Measuring angles: degree measure and radian measure
    • The unit circle
    • Arc length
    • Geometry of a circle
    • Sector of a circle
  • 6.2: The unit circle and trigonometric functions
    • Trigonometric functions
    • Evaluating trigonometric functions
  • 6.3: Graphs of trigonometric functions
    • Graphs of the sine and cosine functions
    • Graphs of transformations of the sine and cosine functions
    • Graph of the tangent function
  • 6.4: Trigonometric equations
    • The unit circle and exact solutions to trigonometric equations
    • Graphical solutions to trigonometric equations
    • Analytic solutions to trigonometric equations
  • 6.5: Trigonometric identities
    • Compound angle identities (sum and difference identities)
    • Double angle identities
  • 6.6: Inverse trigonometric functions
    • Defining the inverse sine function
    • Defining the inverse cosine and inverse tangent functions
• Chapter 7: Geometry and trigonometry
  • 7.1: Measurements in three dimensions
    • 3-dimensional solids: volumes and surface areas
  • 7.2: Right-angled triangles and trigonometric functions of acute angles
    • Right-angled triangles
    • Trigonometric functions of an acute angle
  • 7.3: Trigonometric functions of any angle
    • Defining trigonometric functions for any angle in standard position
    • Areas of triangles
    • Equations of lines and angles between two lines
  • 7.4: The sine rule and the cosine rule
    • Possible triangles constructed from three given parts
    • The sine rule
    • Finding unknowns given two angles and any side (ASA or AAS)
    • Two sides and a non-included angle (SSA) – the ambiguous case
    • The cosine rule
    • Finding unknowns given two sides and the included angle (SAS)
    • Finding unknowns given three sides (SSS)
• Chapter 8: Complex numbers
  • 8.1: Complex numbers
    • Algebraic structure of complex numbers
    • Addition, subtraction and multiplication
    • Division
    • Conjugate
    • Properties of conjugates
    • Conjugate zeros of polynomials
  • 8.2: The complex plane
    • Modulus-argument (trigonometric or polar) form of a complex number
    • Multiplication
    • Division of complex numbers
  • 8.3: Powers and roots of complex numbers
    • Uses of de Moivre’s theorem in problem solving
    • nth roots of a complex number
    • nth roots of unity
    • Euler’s formula
• Chapter 9: Vectors, lines, and planes
  • 9.1: Vectors from a geometric viewpoint
    • Addition and subtraction of vectors
    • Unit vectors
  • 9.2: Scalar (dot) product
    • Direction angles, direction cosines
  • 9.3: Vector (cross) product
    • Properties of the vector product
    • The scalar triple product
  • 9.4: Lines in space
    • Vector equation of a line
    • Line segments
    • Intersecting, parallel, and skew straight lines
    • Application of lines to motion
    • Distance from a point to a line (optional)
  • 9.5: Planes
    • Equations of a plane
    • Vector equation of a plane
    • Unit vector equation of a plane
    • Parametric form for the equation of a plane
    • Distance between a point and a plane
    • The angle between two planes
    • The angle between a line and a plane
    • Line of intersection of two planes
• Chapter 10: Statistics
  • 10.1: Graphical tools
    • Classification of variables
    • Frequency distribution (table)
  • 10.2: Measures of central tendency
  • 10.3: Measures of variability
    • Range
    • Variance and standard deviation
    • The interquartile range and measures of non-central tendency
    • Shape, centre, and spread
  • 10.4: Linear regression
    • Correlation and covariance
    • What to look for in a scatter plot
    • Covariance
    • Some facts worth knowing about covariance
    • Correlation
    • Least squares regression
    • Why the least-squares regression line?
    • Features of the regression line
• Chapter 11: Probability
  • 11.1: Randomness and probability
    • Basic definitions
    • Tree diagrams, tables and grids
  • 11.2: Probability assignments
    • Probability rules
    • Equally likely outcomes
    • Geometric probability
    • Probability calculation for equally likely outcomes using counting principles
  • 11.3: Operations with events
    • Conditional probability
    • Independence
  • 11.4: Bayes’ theorem
    • Bayes’ theorem – simple case
    • General rule
• Chapter 12: Differential calculus 1
  • 12.1: Limits of functions
  • 12.2: The derivative of a function: definition and basic rules
    • Tangent lines and the slope (gradient) of a curve
    • Differentiating from first principles
    • Basic differentiation rules
    • Differentiating sin x and cos x using the limit definition for derivatives
    • Continuity and differentiability
  • 12.3: Maxima and minima: first and second derivatives
    • The relationship between a function and its derivative
    • The first derivative test
    • Change in displacement and velocity
    • A function and its second derivative
    • The second derivative test
  • 12.4: Tangents and normals
    • Finding equations of tangents
    • The normal to a curve at a point
• Chapter 13: Differential calculus 2
  • 13.1: Derivatives of composite functions, products and quotients
    • The chain rule
    • The product rule
    • The quotient rule
    • Higher derivatives
  • 13.2: Derivatives of trigonometric and exponential functions
    • Derivatives of trigonometric functions
    • Derivatives of exponential functions
  • 13.3: Implicit differentiation, logarithmic functions and inverse trigonometric functions
    • Implicit differentiation
    • Derivatives of logarithmic functions
    • Derivatives of inverse trigonometric functions
  • 13.4: Related rates
  • 13.5: Optimisation
  • 13.6: l’Hôpital’s rule
• Chapter 14: Integral calculus 1
  • 14.1: Antiderivative
    • Notation
    • Integration by simple substitution – change of variables
  • 14.2: Integration by parts
    • Using integration by parts to find unknown integrals
  • 14.3: More methods of integration
  • 14.4: Area and the definite integral
    • Basic properties of the definite integral
    • Average value of a function
    • Max–min inequality
    • The first fundamental theorem of integral calculus
    • The second fundamental theorem of integral calculus
    • Using substitution with the definite integral
  • 14.5: Integration by method of partial fractions
  • 14.6: Areas
    • Areas between curves of functions of the form y = f(x) and the x-axis
    • Areas along the y-axis
  • 14.7: Volumes with integrals
    • Washers
    • An alternative method: volumes by cylindrical shells
  • 14.8: Modelling linear motion
    • Displacement and total distance travelled
    • Position and velocity from acceleration
    • Uniformly accelerated motion
• Chapter 15: Probability distributions
  • 15.1: Random variables
    • Discrete probability distribution
    • Probability distribution functions of discrete random variables
    • Expected values
    • Variance and standard deviation
  • 15.2: The binomial distribution
    • The binomial distribution
  • 15.3: Continuous distributions
    • Probability density function
    • Cumulative distribution functions
    • Measures of centre, position, and spread of a continuous distribution
  • 15.4: The normal distribution
    • The normal distribution
    • The inverse normal distribution
  • 15.5: Expectation algebra
    • The expected value of a linear function of X
    • Linear combinations of random variables
    • Interesting application I
    • Interesting application II
    • Linear combinations of normally distributed random variables
• Chapter 16: Integral calculus 2
  • 16.1: Differential equations
    • Solution of a differential equation
    • Separable differential equations
    • Logistic differential equations
    • Homogeneous differential equations
  • 16.2: First order linear differential equations – use of integrating factor
  • 16.3: Numerical solutions: Euler’s method
    • Slope fields
    • Euler’s method
    • Euler’s numerical method
  • 16.4: Power series: Maclaurin’s series
    • Local linear approximation
    • Maclaurin and Taylor polynomials
    • Maclaurin and Taylor series
    • Differentiating and integrating power series
    • Algebraic operations on power series
    • Power series solutions of differential equations
• Internal assessment
  • Mathematical exploration
  • Internal assessment criteria
• Theory of knowledge
  • Perspectives
    • Mathematics and number
    • Purpose: mathematics for its own sake
    • Purpose: mathematical models
    • Constructivist view of mathematics
    • Platonic view of mathematics
  • The methods and tools of mathematics
    • The language and concepts of mathematics
    • Notation
    • Algebra
    • Proof
    • Sets
    • Mappings between sets
    • Infinite sets
  • Mathematics and the knower
    • Beauty by the numbers
    • Beauty in numbers
    • Mathematics and personal intuitions
    • Mathematics and personal qualities
  • Conclusion
• Answers
• Index
• Back Cover

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