IB Mathematics Analysis and Approaches SL: Syllabus

We are currently working on Study Notes for every Topic, and Sub-Topic. 

Topic 1

• Number operations in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

SL 1.1 : Sequences and Series

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 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

Topic 2

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

• 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)

Quadratic Solutions

• 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
• 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.

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 labelled 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 3.9/AHL 3.10: 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 3.7: 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 3.8: 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.

Topic 4

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 4.3: Central Tendency

Central Tendency

• Measures of central tendency (mean, median and mode). Estimation of 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 4.4: 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 4.5: Introduction to Probability

Introduction to Probability

• Concepts of 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 complementary 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 4.6: 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 4.7: Random variables

Discrete Variables

• Concept of discrete random variables and their probability distributions, expected value (mean), for discrete data, applications. 
• Expected value (mean), for discrete data.
• Variance of a discrete random variable. 
• Mean, variance and standard deviation of discrete random variables. 
• The effect of linear transformations of X.
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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 4.8: 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 are unknown. 

SL 4.10: Bivariate Statistics

Regression

• Equation of the regression line of x on y.
• Use of the equation for prediction purposes.

Topic 5

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 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

• 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 behaviour 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 inflexion 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. Optimisation 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 5.4: Integration

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

• 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 5.6: Kinematics

Kinematics Problems

• Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled.