Efnisyfirlit

• Contents
• Introduction
• Chapter 1: Number and algebra basics
• 1.1: Estimation and approximation
• Rounding answers
• Percentage error
• Significant figures (s.f.)
• Percentage error revisited
• 1.2: Rules of exponents
• 1.3: Scientific notation
• 1.4: Exponents and logarithms
• 1.5: Rules of logarithms
• Chapter 2: Functions
• 2.1: Concept of a function, domain and range
• Domain and range of a function
• 2.2: Linear and piecewise functions
• Linear functions
• Piecewise functions
• Graphs of functions
• 2.3: Composite functions
• Composition of functions
• Finding the domain of a composite function
• 2.4: Inverse functions
• Pairs of inverse functions
• The existence of an inverse function
• Finding the inverse of a function
• 2.5: Transformations of graphs
• Chapter 3: Sequences and series
• 3.1: Sequences
• 3.2: Arithmetic sequences
• Simple interest
• 3.3: Geometric sequences
• Compound interest
• Exponential growth and decay
• 3.4: Series
• Arithmetic series
• Geometric series
• Sum to infinity of a geometric series
• 3.5: Annuities and amortisation
• Chapter 4: Geometry and trigonometry 1
• 4.1: Coordinate geometry in a plane – lengths of segments and midpoints
• Coordinate geometry in a plane – lines and intersections
• Coordinate geometry in a plane – perpendicular lines
• 4.2: Trigonometry
• Basic definitions
• Modelling problems with trigonometry
• 4.3: Areas of triangles, sine rule and cosine rule
• Sine rule
• Revisiting the area of a triangle with the sine rule
• Cosine rule
• 4.4: Measurements in 3 dimensions
• Volumes and surface areas
• Distances in 3 dimensions
• Midpoints of line segments in 3-dimensional space
• Chapter 5: Geometry and trigonometry 2
• 5.1: Arc length and area of a sector
• Area of a sector
• Length of an arc
• 5.2: Angles of rotation and radian measure
• Angles
• Measuring angles: degree measure and radian measure
• The unit circle
• Arc length
• Area of a sector
• 5.3: The unit circle and trigonometric functions
• The trigonometric functions in the unit circle
• Trigonometric functions of real numbers
• Trigonometric identities
• 5.4: Graphical analysis of trigonometric functions
• Graphs of the sine and cosine functions
• Graph of the tangent function
• Graphical solutions to trigonometric equations
• 5.5: Voronoi diagrams
• Constructing Voronoi diagrams
• Largest empty circle
• Chapter 6: Complex numbers
• 6.1: Imaginary numbers
• 6.2: Operations with complex numbers
• 6.3: The complex plane
• 6.4: Powers and roots of complex numbers
• The polar form of complex numbers
• Powers of complex numbers
• Roots of complex numbers
• The Euler form of complex numbers
• 6.5: Applications of complex numbers
• Multiple sinusoidal functions
• Impedance – complex variables used in electrical theory
• Impedance in parallel circuits
• Chapter 7: Matrix algebra
• 7.1: Matrix definitions and operations
• What is a matrix?
• Vectors
• Matrix operations
• 7.2: Applications to systems
• Systems of linear equations
• Matrix inverse
• 7.3: Further properties and applications
• 7.4: Eigenvectors and eigenvalues
• Diagonalisation
• Markov chains
• 7.5: Matrices and geometric transformations
• Chapter 8: Vectors
• 8.1: Vector representation
• Vector properties
• Vector addition and scalar multiplication
• Unit vectors
• Vector operations illustrated
• 8.2: Vector and parametric equations of lines
• 8.3: Kinematics
• Minimum distance between a point and an object in motion
• Minimum distance between two objects in motion
• Minimum distance between two moving objects in 3 dimensions
• 8.4: Scalar and vector products
• The scalar product
• The vector product
• 8.5: Angles between vectors
• Chapter 9: Modelling real-life phenomena
• 9.1: Polynomial functions
• Developing and testing a linear model
• Extending and revising models
• Models don’t often capture reality perfectly
• Interpreting and evaluating linear models
• Quadratic models
• Cubic models
• Piecewise models
• 9.2: Exponential and logarithmic models
• Just how fast is exponential growth?
• Developing exponential models
• Interpreting exponential models
• Graphical interpretation
• 9.3: Trigonometric models
• Exploration
• Developing trigonometric models
• 9.4: Logistic models
• 9.5: Direct and inverse variation
• Direct variation
• Inverse variation
• 9.6: Further modelling skills
• Choosing a model
• Testing a model
• Interpolation versus extrapolation
• Chapter 10: Descriptive statistics
• 10.1: Data and variables
• Variables
• Random and non-random sampling
• Random sampling
• Non-random sampling
• 10.2: Displaying distributions using graphs
• 10.3: Measures of central tendency and spread
• Measures of location and centre
• Measures of variability and spread
• Range and quartiles
• Chapter 11: Probability of events
• 11.1: Concepts and definitions
• 11.2: Representing the sample space
• Venn diagrams
• Tree diagrams
• 11.3: Conditional probability
• Independence
• Chapter 12: Graph theory
• 12.1: Graphs: definitions
• Proof of the handshaking theorem
• Some special graphs
• Graph representation
• 12.2: Paths, walks and trails
• Adjacency matrices and walks
• Properties of connected graphs
• Eulerian graphs
• Hamiltonian graphs
• 12.3: Planar graphs
• Euler’s formula
• Homeomorphic graphs
• Transition matrix for a strongly connected graph
• 12.4: Trees
• Spanning trees
• 12.5: Weighted graphs and greedy algorithm
• Representation
• 12.6: Shortest path, route inspection and the travelling salesman problem
• The route inspection problem
• The travelling salesman problem
• Chapter 13: Introduction to differential calculus
• 13.1: Limits and instantaneous rate of change
• The idea of a limit
• Instantaneous rate of change
• 13.2: Derivative of a function
• Interpretations of the derivative
• 13.3: Derivatives of functions of the form f (x) = axn
• Exploration of the power rule
• 13.4: Derivatives of composite functions, products and quotients
• The chain rule
• The product rule
• The quotient rule
• Higher derivatives
• 13.5: Derivatives of sin x, cos x, tan x, ex and ln x
• Derivatives of trigonometric functions
• Derivative of ex
• Derivative of ln x
• Chapter 14: Further differential calculus
• 14.1: Minima, maxima and points of inflection
• First derivative test
• Second derivative test
• 14.2: Tangents and normals
• 14.3: Optimisation
• 14.4: Related rates
• Implicit differentiation
• Related rates
• Chapter 15: Probability distributions
• 15.1: Random variables
• Discrete data
• Continuous data
• 15.2: Binomial distribution
• 15.3: Poisson distribution
• The Poisson distribution as an estimator for small values of p
• The Poisson distribution as a probability model for the rate of occurrences
• 15.4: Normal distribution
• The empirical rule
• The standardised score
• The inverse normal
• 15.5: Transformations and combinations of data
• Transformed data with the uniform and normal distributions
• Combinations of normal distributions
• Combinations of Poisson distributions
• 15.6: Matrix applications (Markov chains)
• Chapter 16: Integral calculus 1
• 16.1: Antiderivative
• Notation
• Basic integration formulae
• Integration by simple substitution – change of variables
• Some applications to economics
• 16.2: Area and the definite integral
• Basic properties of the defi nite integral
• Average value of a fuction
• Max-min inequality
• The first fundamental theorem of integral calculus
• The second fundamental theorem of integral calculus
• Using substitution with the definite integral
• Numerical integration: the trapezoidal rule
• Continuous money flow
• 16.3: Areas
• Areas between curves of functions of the form y = f (x) and the x-axis
• Area between curves
• Areas along the y-axis (optional)
• Some applications of area in economics
• Some applications in probability
• 16.4: Volumes with integrals
• Washers
• 16.5: Modelling linear motion
• Displacement and total distance travelled
• Position and velocity from acceleration
• Uniformly accelerated motion
• Chapter 17: Inferential statistics
• 17.1: Statistical inference, reliability, and validity
• Arriving at a generalisation
• 17.2: Unbiased estimators
• 17.3: Distribution of sample means
• 17.4: Probability intervals and confidence intervals
• Confidence intervals made to any specification
• Margin of error
• Confi dence interval for means
• Chapter 18: Statistical tests and analyses
• 18.1: The Student t-test
• Confidence interval for means revisited
• Using the t-test with a single sample
• 18.2: Hypothesis testing of means
• 18.3: Type I and type II errors
• The power of the test
• 18.4: t-test of two means
• Paired t-tests
• Unpaired t-tests (2-sample t-test)
• 18.5: Chi-squared test of the goodness of fit (GOF)
• 18.6: Chi-squared test of independence
• Degrees of freedom
• Calculating the chi-squared statistic
• Chapter 19: Bivariate analysis
• 19.1: Scatter diagrams
• Form
• Direction
• Strength
• Unusual features
• Estimating the line of best-fit
• 19.2: Measures of correlation
• Pearson’s product-moment correlation coeffi cient ( r )
• Spearman’s rank correlation coefficient (rs)
• Differences between Pearson’s r and Spearman’s rs
• 19.3: Linear regression
• Least-squares regression line
• Coefficient of determination
• 19.4: Non-linear regression and models
• Getting started
• Using a GDC
• Linearising data using logarithms
• Interpreting graphs with logarithmic axes
• Chapter 20: Integral calculus 2
• 20.1: Differential equations
• Solution of a differential equation
• Separable differential equations
• Logistic differential equations
• Differential equations reducible to variables separable
• 20.2: More applications of differential equations
• Electric circuits
• Mixture problems
• 20.3: Numerical solutions: slope fields and Euler’s method
• Slope fields
• Euler’s method
• Euler’s numerical method
• 20.4: Coupled differential equations
• Analytical solution of coupled systems
• Numerical solution of coupled systems
• Solution of second-order differential equations using coupled systems
• Internal assessment
• Mathematical exploration
• Internal assessment criteria
• Mathematical exploration – HL student checklist
• 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