IB Mathematics Analysis and Approaches HL Syllabus Checklist
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What is the Math Analysis and Approaches HL Syllabus?
The IB Mathematics Analysis and Approaches HL syllabus contains everything that the IB requires you to know for your IB Examinations.
We advise students to use study notes for new concepts, practice their understand through IB questions, and to make sure they remember their Internal Assessment.
Syllabus Overview
The IB Mathematics Analysis and Approaches HL Syllabus is divided into five main topics.
- Number and Algebra
- Functions
- Geometry and Trigonometry
- Statistics
- Calculus
Detailed Syllabus
We are currently working on Study Notes for every Topic, Sub-Topic, and Sub-Sub-Topic.
Topic 1: Numbers and Algebra
SL 1.0: Scientific Notation
- Number operations in the form a × 10^k where 1 ≤ a < 10 and k is an integer.
Arithmetic/Geometric Series
- Use of the formulae for the nth term and the sum of the first n terms of the sequence.
- Applications
- Use of the formulae for the nth term and the sum of the first n terms of the sequence.
- Use of the formulae for the nth term and the sum of the first n terms of the sequence
- Sum of infinite convergent geometric sequences
Sigma Notation
- Use of sigma notation for the sums of geometric sequences
- Applications
SL 1.2: Exponents and Logarithms
Laws of Exponents and Logarithms
- Laws of exponents with integer exponents
- Laws of exponents with rational exponents.
- Laws of logarithms.
- Change of base of a logarithm.
- Introduction to logarithms with base 10 and e
Applications
- Solving exponential equations, including using a logarithm
- Numerical evaluation of logarithms using technology
- Exponential Equations
- compound interest
- annual depreciation.
SL 1.3: Proofs
Introduction to Proofs
- Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof
- The symbols and notation for equality and identity.
Methods of Proof
- Proof by mathematical induction
- Proof by contradiction
- Use of a counterexample to show that a statement is not always true
SL 1.4: Binomial Theorem
The Binomial Theorem
- The binomial theorem
- Use of Pascal’s triangle
AHL 1.5: Counting Principles
Counting Principles
- Counting principles, including permutations and combinations
- Extension of the binomial theorem to fractional and negative indices
AHL 1.6 Complex Numbers
Complex Numbers
- Complex numbers introduction
- Cartesian form
AHL 1.7: Complex Plane
Forms in the Complex Plane
- Modulus–argument (polar) form Euler form
- Euler form
Sums, Products, and Quotients
- Sums, products, and quotients in Cartesian, polar, or Euler forms and their geometric interpretation.
AHL 1.8: Complex roots
Complex Roots
- Complex conjugate roots of quadratic and polynomial equations with real coefficients
- De Moivre’s theorem and its extension to rational exponents
- Powers and roots of complex numbers.
AHL 1.9: System of Equations
Systems of Linear Equations
- Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions, or no solution.
Topic 2: Functions
SL 2.1: Basic Functions
Intro to Functions
- Different forms of the equation of a straight line.
- Gradient; intercepts.
Lines
- Lines with gradients m1 and m2
- Parallel Lines
- Perpendicular Lines
Function Concepts
- Concept of a function, domain, range, and graph.
Function Notations
- Function notation
- The concept of a function as a mathematical model
Inverse Functions
- An informal concept that an inverse function reverses or undoes the effect of a function
- Inverse functions
SL 2.2: Graph of Functions
Graph of Functions
- The graph of a function
- Creating a sketch from information given or a context, including transferring a graph from screen to paper
- Determine key features of graphs. Maximum and minimum values; intercepts
Graphs Using Technology
- Using technology to graph functions including their sums and differences.Using technology to graph functions including their sums and differences.
- Finding the point of intersection of two curves or lines using technology
SL 2.3: Special Functions
Different Functions
- Composite Functions
- Identity function. Finding the inverse function
- The quadratic function
Function Forms
- The form f(x) = a(x − p)(x − q), x-intercepts (p, 0) and (q, 0).
- The form f(x) = a (x − h)^2 + k, vertex (h,k)
- A solution of quadratic equations and inequalities
- The quadratic formula.
- The discriminant, and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
Other Functions
- The reciprocal function and its graph
- Rational functions
- Equations of vertical and horizontal asymptotes.
- Exponential functions and their graphs f(x) = az , a > 0, f(x) = ex
- Logarithmic functions and their graphs:f(x) = logax, x > 0, f(x) = lnx, x > 0
- Odd and even functions
- Finding the inverse function, including domain restrictions
- Self-inverse functions.
- Rational functions and its forms
- The graphs of the functions, y = | f(x)|
- y = f(|x|), y = 1 f(x) , y = f(ax + b), y = [f(x)]2
- The solution of modulus equations and inequalities
SL 2.4: Solving Equations
Introduction to Solving Equations
- Solving equations, both graphically and analytically
- Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach.
Inequalities
- Solutions of g(x) ≥ f(x), both graphically and analytically.
Applications
- Applications of graphing skills and solving equations that relate to real-life situations.
Graph Transformations
- Transformations of graphs
- Translations: y = f(x) + b; y = f(x − a)
- Reflections (in both axes): y = − f(x); y = f( − x).
- Vertical stretch with scale factor p: y = p f(x).
- Horizontal stretch with scale factor 1\ q : y = f(qx).
Composite Transformations
- Composite transformations.
AHL 2.6: Polynomial Functions
Introduction to Polynomial Functions
- Polynomial functions, their graphs, and equations; zeros, roots, and factors.
The Factor and Remainder Theorems
Solving Polynomial Equations
- Sum and product of the roots of polynomial equations.
Topic 3: Geometry and Trigonometry
SL 3.1: Introduction to Geometry
Distance and Midpoint
- The distance between two points in three-dimensional space, and their midpoint.
Surface Area and Volume
- Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere, and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane.
SL 3.2: Sine and Cosine Rules
Right Triangles
- Use of sine, cosine, and tangent ratios to find the sides and angles of right-angled triangles.
Sine and Cosine Rules
- The sine rule: a/sinA = b/sinB = c/sinC. The cosine rule: c2 = a2 + b2 − 2abcos(C); cos(C) = a2 + b 2 − c2 2ab. Area of a triangle as 1/2absin(C).
SL 3.3: Triangles
Triangle Trigonometry
- Applications of right and non-right angled trigonometry, including Pythagoras’s theorem. Angles of elevation and depression.
- Construction of labeled diagrams from written statements.
SL 3.4: The Circle
The Circle
- The circle: radian measure of angles; length of an arc; area of a sector.
SL 3.5: Trigonometry
Definition of Trig Functions
- Definition of cosθ, sinθ in terms of the unit circle.
- Definition of tanθ as sinθ/cosθ
- Exact values of trigonometric ratios of 0, π/6 , π/4 , π/3 , π/2, and their multiples.
Extension of Sine Rule
- Extension of the sine rule to the ambiguous case.
SL 3.6/AHL 3.7/AHL 3.8: Identities
Pythagorean Identity
- The Pythagorean identity cos2θ + sin2θ = 1. Double angle identities for sine and cosine.
- The relationship between trigonometric ratios.
Reciprocal Trigonometric ratios
- Definition of the reciprocal trigonometric ratios secθ, cosecθ, and cotθ.
- Pythagorean identities: 1 + tan2θ = sec2θ, 1 + cot2θ = cosec2θ
- The inverse functions f(x) = arcsinx, f(x) = arccosx, f(x) = arctanx; their domains and ranges; their graphs.
- Compound angle identities. Double angle identity for tan.
SL 3.9: Trig Composites
Circular Functions
- The circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs Composite functions of the form f(x) = asin(b(x + c)) + d.
Transformations
Real-life Situations
SL 3.10: Solving Trig Equations
Solving Trig Equations
- Solving trigonometric equations in a finite interval, both graphically and analytically.
- Equations leading to quadratic equations in sinx, cosx, or tanx.
Trig Relations
- Relationships between trigonometric functions and the symmetry properties of their graphs.
AHL 3.11: Vector Concepts
Introduction to Vectors
- Concept of a vector; position vectors; displacement vectors. Representation of vectors using directed line segments. Base vectors i, j, k. Components of a vector: v = v1i + v2 j + v3k.
Algebraic and geometric approaches
- the sum and difference of two vectors
- the zero vector 0, the vector −v
- multiplication by a scalar, kv, parallel vectors
- the magnitude of a vector, |v|; unit vectors, v /|v|
- position vectors OA → = a, OB → = b
- displacement vector AB → = b − a
Proofs of geometrical properties using vectors.
AHL 3.12: Multiple Vectors
Multiple Vectors
- The definition of the scalar product of two vectors. The angle between two vectors. Perpendicular vectors; parallel vectors.
AHL 3.13: Vector Equations
Equation of a Line
- Vector equation of a line in two and three dimensions: r = a + λb.
Angle Between Two Lines
- The angle between two lines.
Applications
- Simple applications to kinematics.
AHL 3.14: Points of Intersections
Points of Intersection
- Coincident, parallel, intersecting, and skew lines, distinguishing between these cases. Points of the intersection.
AHL 3.15: Dot Product
Dot Product
- The definition of the vector product of two vectors.
- Properties of the vector product
- Geometric interpretation of | v × w |
AHL 3.16: Vector Planes
Vector Equations
- Vector equations of a plane: r = a + λb + μc, where b and c are non-parallel vectors within the plane. r · n = a · n, where n is normal to the plane and a is the position vector of a point on the plane. Cartesian equation of a plane ax + by + cz = d
AHL: 3.17 Intersecting Planes
Intersections
Angles
- Intersections of: a line with a plane; two planes; three planes. The angle between a line and a plane; two planes.
Topic 4: Statistics
SL 4.1: Population and Data
Population
- Concepts of population, sample, random sample, discrete and continuous data.
- Reliability of data sources and bias in sampling.
- Interpretation of outliers
- Sampling techniques and their effectiveness.
Presentation of Data
- Presentation of data (discrete and continuous): frequency distributions (tables).
- Histograms; cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range, and interquartile range (IQR).
- Production and understanding of box and whisker diagrams.
SL 4.2: Central Tendency
Central Tendency
- Measures of central tendency (mean, median, and mode). Estimation of the mean from grouped data.
Modal Class
- Modal class.
- Measures of dispersion (interquartile range, standard deviation, and variance).
- Effect of constant changes on the original data.
- Quartiles of discrete data.
SL 4.3: Linear Correlation
Scatter diagrams
- Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient, r
- Scatter diagrams; lines of best fit, by eye, passing through the mean point.
Regression
- Equation of the regression line of y on x.
- Use of the equation of the regression line for prediction purposes. Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.
SL 4.4: Introduction to Probability
Introduction to Probability
- Concepts of the trial, outcome, equally likely outcomes, relative frequency, sample space (U), and event. The probability of an event A is P(A) = n(A)/n(U). The complimentary events A and A′ (not A).
- Expected number of occurrences.
Venn Diagrams
- Use of Venn diagrams, tree diagrams, sample space diagrams, and tables of outcomes to calculate probabilities.
SL 4.5: Types of Probabilities
Combined Events
- Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B), mutually exclusive events: P(A ∩ B) = 0.
Conditional Events
- Conditional probability: P(A|B) = P(A ∩ B)/P(B).
- Formal definition and use of the formulae: P(A|B) = P(A ∩ B)/P(B) for conditional probabilities
Independent Events
- Independent events: P(A ∩ B) = P(A)P(B).
- Formal definition and use of the formulae: P(A|B) = P(A) = P(A|B′) for independent events.
Bayes Theorem
- Use of Bayes’ theorem for a maximum of three events.
SL 4.6: Random variables
Discrete Variables
- Concept of discrete random variables and their probability distributions, the expected value (mean), for discrete data, applications.
- Expected value (mean), for discrete data.
- The variance of a discrete random variable.
- Mean, variance, and standard deviation of discrete random variables.
- The effect of linear transformations of X.
Applications
- Applications
Continuous Variables
- Mode and median of continuous random variables.
- Continuous random variables and their probability density functions.
- Mean, variance, and standard deviation of continuous random variables.
- The effect of linear transformations of X.
SL 4.7: Distributions
Binomial Distribution
- Binomial distribution.
- Mean and variance of the binomial distribution.
Normal Distribution
- The normal distribution and curve, properties of the normal distribution, diagrammatic representation.
- Normal probability calculations.
- Inverse normal calculations
- Standardization of normal variables (z– values).
- Inverse normal calculations where mean and standard deviation is unknown.
SL 4.8: Bivariate Statistics
Regression
- Equation of the regression line of x on y.
- Use of the equation for prediction purposes.
Topic 5: Calculus
Sl 5.1: Limits
Limits
- Introduction to the concept of a limit.
- Informal understanding of continuity and differentiability of a function at a point.
- Understanding of limits (convergence and divergence). Definition of derivative from first principles f ′(x) = lim h → 0 f(x + h) − f(x) h .
L’Hopital
- The evaluation of limits of the form lim x → a f(x) g(x) and lim x → ∞ f(x) g(x) using l’Hôpital’s rule or the Maclaurin series.
- Repeated use of l’Hôpital’s rule.
SL 5.2: Derivatives
Introduction to Derivatives
- Derivative interpreted as gradient function and as the rate of change.
- Increasing and decreasing functions. Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0.
- Tangents and normals at a given point, and their equations
- Derivative of f(x) = axn is f ′(x) = anxn − 1 , n ∈ ℤ The derivative of functions of the form f(x) = axn + bxn − 1 . . . . where all exponents are integers.
Advanced Derivatives
- A derivative of x n (n ∈ ℚ), sinx, cosx, e x, and lnx. Differentiation of a sum and a multiple of these functions.
- The chain rule for composite functions. The product and quotient rules.
- The second derivative. Graphical behavior of functions, including the relationship between the graphs of f, f ′, and f ″. Local maximum and minimum points. Testing for maximum and minimum.
- Points of inflection with zero and non-zero gradients.
- Higher derivatives.
- Derivatives of tanx, secx, cosecx, cotx, a x , logax, arcsinx, arccosx, arctanx.
Implicit Differentiation
- Implicit differentiation. Related rates of change. Optimization problems
- Derivative of f(x) = axn is f ′(x) = anxn − 1 , n ∈ ℤ The derivative of functions of the form f(x) = axn + bxn − 1 + . . . where all exponents are integers.
- Implicit differentiation. Related rates of change.
SL 5.3: Differentiation
First-order differentials
- Variables are separable.
- First-order differential equations. Numerical solution of dy dx = f(x, y) using Euler’s method.
Homogeneous differentials
- Homogeneous differential equation dy dx = f( y x ) using the substitution y = vx.
- Solution of y′ + P(x)y = Q(x), using the integrating factor.
Introduction to Integration
- Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn − 1 + …., where n ∈ ℤ, n ≠ − 1
- Anti-differentiation with a boundary condition to determine the constant term.
Advanced integrals
- The indefinite integral of x n (n ∈ ℚ), sinx, cosx, 1 x, and e x
- The composites of any of these with the linear function ax + b.
Definite Integrals
- Definite integrals, including analytical approach.
- Definite integrals using technology. Area of a region enclosed by a curve y = f(x) and the x -axis, where f(x) > 0
SL 5.5: Further Integration
Optimization
- Optimization
Integration by Substitution
- Integration by inspection (reverse chain rule) or by substitution for expressions of the form: ∫ kg′(x)f(g(x))dx
- Integration by substitution.
Integration by Parts
- Integration by parts.
- Repeated integration by parts
Integration of two functions
- Areas of a region enclosed by a curve y = f(x) and the x-axis, where f(x) can be positive or negative, without the use of technology. Areas between curves.
- Area of the region enclosed by a curve and the y axis in a given interval. Volumes of revolution about the x-axis or y-axis.
- Indefinite integrals of the derivatives of any of the above functions. The composites of any of these with a linear function.
SL 5.6: Kinematics
Kinematics Problems
- Kinematic problems involving displacement s, velocity v, acceleration a, and total distance traveled.
AHL 5.7: Maclaurin Series
Maclaurin Series
- Maclaurin series to obtain expansions for e x , sinx, cosx, ln(1 + x), (1 + x) p , p ∈ ℚ.
- Use of simple substitution, products, integration, and differentiation to obtain other series.
- Maclaurin series developed from differential equations.
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