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