 # Comprehensive IB Maths AI SL & HL Syllabus

## Chapter 1: Number and Algebra

 Subtopic Subtopic Number IB Points to Understand Rounding 1.1 Approximations are a value or quantity that is nearly but not exactly correct. Estimations are rough calculations of the value, number, quantity, or extent of something. Percentage Error 1.2 Percentage error expresses as a percentage the difference between an approximate or measured value and an exact or known value. Significant figures are used to express it to the required degree of accuracy, starting from the first non-zero digit. Rules of Exponents 1.3 When same bases are multiplied, add the exponents When same bases are divided, subtract the exponents Multiply the powers when the numbers are raised by another number Rules of Logarithms 1.4 Product Rule Quotient Rule Power Rule Change of Base Rule Angles and Triangles 1.5 Equilateral: “equal”-lateral, so they have all equal sides. Isosceles: means “equal legs”, they have two equal “sides” joined by an “odd” side.  Scalene: means “uneven” or “odd”, so no equal sides. The term angle of elevation denotes the angle from horizontal upward to an object.  The term angle of depression denotes the angle from horizontal downward to an object. Circles 1.6 Length of Arc Area of the sector Volume and Surface area of figures 1.7 Formula for surface area and volume of geometric solids

## Chapter 2: Functions

 Subtopic Subtopic Number IB Points to Understand Function 2.1 A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. Domain 2.2 The domain of a function is the complete set of possible values of the independent variable. We determine the domain of each function by looking for those values of the independent variable Range 2.3 The range of a function is the complete set of all possible resulting values of the dependent variable, after we have substituted the domain. Finding the range Linear and Piecewise Functions 2.4 Linear Function: A linear function f(x) = mx+c where m and c are constants, represents a context with a constant rate of change. Piecewise functions: A piecewise function is a function where more than one formula is used to define the output. Graph of functions 2.5 The graph of a function f is the set of all points in the plane of the form (x, f(x)). We could also define the graph of f to be the graph of the equation y = f(x). Composite & Inverse functions 2.6 Function Composition is applying one function to the results of another. An inverse function or an anti-function is defined as a function, which can reverse into another function. The method of calculating an inverse is swapping of coordinates x and y. One-to-one functions 2.7 A function is said to be a one-to-one function only if every second element corresponds to the first value (values of x and y are used only once). Identity Functions 2.8 The identity function is a function which returns the same value, which was used as its argument. Transformations 2.9 Transformations before and after the original function

## Chapter 3: Sequences and Series

 Subtopic Subtopic Number IB Points to Understand Sequences and Series 3.1 A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. Series a number of events, objects, or people of a similar or related kind coming one after another. Arithmetic sequences 3.2 A sequence a1, a2, a3, …., an is an arithmetic sequence if there is a constant d for which an =an-1 +d  for all integers n > 1, d is called the common difference of the sequence, and d = an – an-1 for all integers n > 1. With simple interest, the key assumption is that you withdraw the interest from the bank as soon as it is paid and deposit it into a separate bank account. Geometric Sequences 3.3 A sequence a1, a2, a3, …, an is a geometric sequence if there is a constant r for which an = an-1 x r for all integers n > 1, r is called the common ratio of the sequence, and r = an ÷ an-1 for all integers n > 1. Compound Interest – At the end of each year, the interest will therefore be paid on the total balance earned so far Sigma Notation 3.4 The Summation Operator ∑ is used to denote the sum of a sequence. We call the sum of the terms in a sequence a series. Arithmetic Series 3.5 Arithmetic series is the sum of the terms of an Arithmetic sequence a1, a2, a3, …, an. Geometric Series 3.6 Geometric series is the sum of the terms of a geometric sequence a1, a2, a3, …, an Geometric mean is the central number in a geometric progression given by: a2 = √𝒂𝟏 × 𝒂𝟑 Annuities and Amortization 3.7 Annuities are a sequence of equal payments made at regular time intervals and amortization determines sequence of payments.

## Chapter 4: Geometry and Trigonometry

 Subtopic Subtopic Number IB Points to Understand Coordinate Geometry in a plane 4.1 Given the cartesian coordinates A(x1 , y1) and B(x2 , y2) the distance between A and B is given by: AB = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐  Lines and Intersections: Coincident lines, Parallel lines, Intersecting Lines If two lines are perpendicular to each other, one is said to be normal to the other at the point of intersection. Trigonometry 4.2 Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides Distances in three dimensions 4.3 Given the cartesian coordinates A(x1 , y1, z1) and B(x2 , y2, z2) the distance between A and B is given by: AB = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐 + (𝒛𝟐 − 𝒛𝟏)𝟐 Angles of rotation and radian measure 4.4 Angles of rotation are formed in the coordinate plane between the positive x-axis (initial side) and a ray The unit circle is the circle centered at the origin with radius equal to one unit.  Period: An interval containing values that occur repeatedly in a function. Even function: A continuous set of (x, f(x)) points in which f(−x) = f(x), with symmetry about the y-axis. Odd function: A continuous set of (x, f(x)) points in which f(−x) = −f(x), with symmetry about the origin. Periodic function: A continuous set of (x, f(x)) points that repeats at regular intervals. Graphical Analysis of trigonometric functions 4.5 The domain of the function y = sin(x) is all real numbers and the range is −1 ≤ y ≤ 1 The domain of the function y = tan(x) is all real numbers except the values where cos(x) is equal to 0, i.e., the values π/2+πn for all integers n. Voronoi diagrams 4.6 Voronoi diagram is a partition of a plane into regions close to each of a given set of objects.

## Chapter 5: Complex Numbers

 Subtopic Subtopic Number IB Points to Understand Imaginary Numbers 5.1 Real numbers are simply the combination of rational and irrational numbers, in the number system. Complex numbers cannot be represented on the number line, but are analytic solutions to equations whose solutions are not real numbers. These complex numbers have an imaginary unit. A number that is expressed in terms of the square root of a negative number is called an imaginary number.  |z| is the modulus of the complex number and the argument of z is defined as the angle between the positive real axis and the line from origin to z. Polar form and Euler form 5.2 The polar form of a complex number z = a + bi is z = r(cosθ + isinθ) Euler’s formula is the statement that 𝑒𝑖𝜃 = cos(𝜃) + isin(𝜃) Powers of complex numbers 5.3 DeMoivre’s Theorem states that zn = (r𝑒𝑖𝜃)n = rn 𝑒𝑖 𝑛𝜃 Applications of complex numbers 5.4 The voltage across the resistor is regarded as a real quantity, while the voltage across an inductor is regarded as a positive imaginary quantity, and across a capacitor we have a negative imaginary quantity. The impedance of a circuit is the total effective resistance to the flow of current by a combination of the elements of the circuit.

## Chapter 6: Matrix Algebra

 Subtopic Subtopic Number IB Points to Understand Matrix and operations 6.1 Matrix refers to an ordered rectangular arrangement of numbers which are either real or complex or functions.  Types of matrices: Row, Column, Square, Diagonal, Zero, Upper Triangular and Lower Triangular Adding and subtracting matrices Addition of matrices is commutative which means A+B = B+A Addition of matrices is associative which means A+(B+C) = (A+B)+C whereas subtraction is neither. Scalar multiplication refers to the product of a real number and a matrix.  Multiplication of matrices is non-commutative which means A*B ≠ B*A Multiplication of matrices is associative which means A*(B*C) = (A*B)*C Determinant and Inverses 6.2 If the determinant of the matrix ≠ 0, then the inverse of the matrix exists. Linear System 6.3 A given set of equations can be written in matrix form given by: AX = B where A contains the matrix of coefficients of x and y. Gauss-Jordan Elimination 6.4 Gauss-Jordan Elimination is an algorithm that can be used to solve systems of linear equations and to find the inverse of any invertible matrix. Row reduced Echelon form 6.5 The purpose of Gauss-Jordan Elimination is to use the three elementary row operations to convert a matrix into reduced-row echelon form. Eigenvectors and Eigenvalues 6.6 An eigenvector of a matrix A is a vector whose product when multiplied by the matrix is a scalar multiple of itself. The corresponding multiplier is often denoted as 𝜆 and referred to as an eigenvalue. A·v = λ·v Diagonalization 6.7 An n × n matrix A is diagonalizable if it is similar to a diagonal matrix: that is, if there exists an invertible n × n matrix P and a diagonal matrix Matrices and Geometric Transformations 6.8 The method we can use to find the matrix representing a transformation is given by the Matrix Basis theorem.

## Chapter 7: Vectors

 Subtopic Subtopic Number IB Points to Understand Vector Representation 7.1 Scalar is a quantity which only has magnitude but a vector has both magnitude and direction. When a vector is represented graphically, its magnitude is represented by the length of an arrow and its direction is represented by the direction of the arrow.  Unit vectors have a magnitude of 1. They are represented by 𝒗̂ = v/|v| where |v| represents the magnitude of vector v. Vector and parametric Equation of lines 7.2 The position vector of a point P, with coordinates (x, y, z), is the vector 𝑟⃗ with initial point on the origin and terminal point on P. Kinematics 7.3 The trajectory of a moving point can be determined at every moment t by an expression for the position vector of the point as a function of time. Scalar and Vector products 7.4 Scalar product or dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. Vector product or cross product is a binary operation on two vectors in three-dimensional space. Angles between two vectors can also be found using dot product and cross product.

## Chapter 8: Modeling real-life phenomena

 Subtopic Subtopic Number IB Points to Understand Polynomial functions 8.1 Linear models are used to describe situations where one quantity increases at a fixed rate relative to another quantity. Interpolations are predictions based on values within the range of known values of the independent variable. Extrapolations are predictions based on values outside the range of known values of the independent variable. If the a value is positive, the parabola opens up, and the function has a minimum. Similarly, if a < 0, the parabola opens down and the function will have a maximum.  Quadratic function using the vertex and the leading coefficient, and the resulting vertex form is f(x) = a(x – h)2 + k. The last form of a quadratic function that can be used to model a real-world scenario is factored form f(x) = a (x – r1)(x – r2), where r1 and r2 are the zeros (x-intercepts) of the function.  A Cubic Model uses cubic functions of the form ax3 + bx2 + cx + d can be used to model real-world situations. Exponential and logarithmic models 8.2 Exponential model arise in situations where the rate of change is a constant factor. Exponential decay and be used to model radioactive decay and depreciation. Exponential decay models of this form can model sales or learning curves where there is an upper limit.  The logarithmic model has a period of rapid increase, followed by a period where the growth slows, but the growth continues to increase without bound. Trigonometric models 8.3 Any motion that repeats itself in a fixed time period is considered periodic motion and can be modeled by a sinusoidal function. Logistic models 8.4 The logistics model begins with a slow growth, followed by a period of moderate growth, and then back to a period of slow growth. Direct and inverse variation 8.5 Direct variation describes a simple relationship between two variables. We say y varies directly with x if: y = kx Inverse variation describes another kind of relationship. We say y varies inversely with x if: xy = k

## Chapter 9: Descriptive statistics

 Subtopic Subtopic Number IB Points to Understand Collecting and organizing data 9.1 Qualitative data is non-numerical. Quantitative data is numerical. Discrete data takes specific values and continuous data can take a full range of values. A population includes all members of a defined group. A sample is a subset of the population i.e., a selection of individuals from the population. Biased sampling is where the method may cause you to draw misleading conclusions about the population. Simple random sampling is where every member of the population is equally likely to be chosen Stratified, Quota and Convenience sampling Statistical measures 9.2 The mode of a data set is the value that occurs most frequently. There can be no mode, one mode or several modes. The median of a data set is the value that lies in the middle when the data are arranged in size. If there are two values we take the mid-point of those points. The mean of a data set is the sum of all the values divided by the number of values. Range is found by subtracting the smallest number from the largest number.  Standard deviation gives an idea how the values are related to the mean. Variance is given by 𝜎2 Ways to present your data 9.3 A histogram is very similar to a bar chart but bar charts are used for graphing qualitative data and histograms are used for quantitative data. A box and whisker plot—also called a box plot—displays the five-number summary of a set of data. The cumulative frequency of a set of data or class intervals of a frequency table is the sum of the frequencies of the data up to a required level. Bivariate data has two variables and univariate data has one variable. Types of correlation Interpolation is where we find a value inside our set of data points. Extrapolation is where we find a value outside our set of data points.

## Chapter 10: Probability

 Subtopic Subtopic Number IB Points to Understand Concepts and definitions 10.1 Experiment: A process by which you obtain an observation.  Trials: Repeating an experiment a number of times.  Outcome: A possible result of an experiment. Event: An outcome or set of outcomes. Sample space: The set of all possible outcomes of an experiment, always denoted by U. Representing the sample space 10.2 A venn diagram represents mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set), common elements of the sets being represented by intersections of the circles. To solve problems involving independent events, it is often helpful to draw a tree diagram. Sample space is a term used in mathematics to mean all possible outcomes. Conditional probability 10.3 The conditional probability (which is basically Bayes’ theorem) of an event B is the probability that the event will occur given the knowledge that an event A has already occurred.

## Chapter 11: Differential Calculus

 Subtopic Subtopic Number IB Points to Understand Limits and derivatives 11.1 Suppose we have a function f(x). The value, a function attains, as the variable x approaches a particular value, say a, i.e., x → a is called its limit.  The derivative of a function f(x) at any point ‘a’ in its domain is given by: limh->0 [f(a+h) – f(a)]/h Chain rule 11.2 The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). Product and quotient rule 11.3 Product Rule: If the two functions f(x) and g(x) are differentiable then the product is differentiable and (f(x)g(x))′ = f′(x)g(x) + f(x)g′(x)  If the two functions f(x)f(x) and g(x)g(x) are differentiable (i.e. the derivative exist) then the quotient is differentiable Tangents and Normals 11.4 A tangent to a curve is a line that touches the curve at one point and has the same slope as the curve at that point. A normal to a curve is a line perpendicular to a tangent to the curve. Minima, Maxima and points of inflection 11.5 A function is increasing on an interval if for any x1 and x2 in the interval then, x1 < x2 implies f(x1) < f(x2) A function is decreasing on an interval if for any x1 and x2 in the interval then, x1 < x2 implies f(x1) > f(x2) An Inflection Point is where a curve changes from Concave upward to Concave downward or vice versa. To find absolute maximum or minimum we need to find local extrema and then compare them to see which one is the greatest or least value for the entire domain of f(x). Second derivative test 11.6 Suppose f(x) is a function of x that is twice differentiable at a stationary point x0. 1. If f’’(x0) > 0, then f has a local minimum at x0. 2. If f’’(x0) < 0, then f has a local maximum at x0. Optimisation 11.7 In optimization problems we are looking for the largest value or the smallest value that a function can take. Here we will be looking for the largest or smallest value of a function subject to some kind of constraint. Related Rates 11.8 Related rates look at the effect that a change in a particular rate has on another rate.

## Chapter 12: Integral Calculus

 Subtopic Subtopic Number IB Points to Understand Indefinite integrals 12.1 Given a function, f(x), an anti-derivative of f(x) is any function F(x) such that F′(x) = f(x) If F(x) is any anti-derivative of f(x) then the most general antiderivative of f(x) is called an indefinite integral and denoted by ∫ 𝑓(𝑥) 𝑑𝑥 = F(x) + c Substitution 12.2 Integration by Substitution is a method to find an integral, but only when it can be set up in a special way. Application to economics 12.3 Cost, Revenue and Profit function Area and the definite integral 12.4 The area problem will give us one of the interpretations of a definite integral and it will lead us to the definition of the definite integral. Given a function f(x) that is continuous on the interval [a,b] we divide the interval into n sub- intervals of equal width, Δx, and from each interval choose a point, xi.  Using substitution with definite integral Midpoint and trapezoidal rule Continuous money flow 12.5 TOTAL MONEY FLOW: If f(t) is the rate of money flow, then the total money flow over the time interval t = 0 to t = T is given by Total = ∫𝑇 𝑓(𝑡) 𝑑𝑡 PRESENT VALUE OF MONEY FLOW: If f(t) is the rate of continuous money flow at an interest rate r, compounded continuously for T years, ACCUMULATED AMOUNT OF MONEY FLOW AT TIME T: If f(t) is the rate of money flow at an interest rate r, at time t, the accumulated amount of money flow at time T is FV = erT ∫𝑇 𝑓(𝑡) 𝑒−𝑟𝑡 𝑑𝑡 Areas 12.6 When calculating the area between a curve and the x-axis, you should carry out separate calculations for the parts of the curve above the axis, and the parts of the curve below the axis. A similar technique to the one we have just used can also be employed to find the areas sandwiched between curves. Solids of revolution 12.7 To get a solid of revolution we start out with a function, y=f(x), on an interval [a, b]. The volume formula stays the same as the method of rings but the area differs. Kinematics and differential equations 12.8 A differential equation is an equation that contains a derivative. A separable differential equation is any differential equation that we can write in the following form: N(y) 𝑑𝑦 = M(x) Slope fields are visual representations of differential equations of the form dy/dx = f(x, y). Euler’s method 12.9 After approximations and substitutions Euler’s formula is given by: yn = yn-1 + h f(tn-1, yn-1).

## Chapter 13: Probability distributions

 Subtopic Subtopic Number IB Points to Understand Random variables 13.1 They can be discrete or continuous. All the probability distributions that we are going to discuss come under discrete random variables (DRV). The variance of a discrete random variable measures the spread or the variability of the distribution and is denoted by 𝜎2=∑𝑛 (𝑥−𝜇)2𝑃(X=xi) If a random variable takes all possible values between two limits or if it represents a real number then the random variable is called a continuous random variable. Binomial distribution 13.2 A random experiment which has only two possible outcomes is called the Bernoulli’s trail. Poisson Distribution 13.3 Poisson distribution gives the probability of happening of the number of events occurring in a fixed interval of time, space or any other parameter with a known average. Normal Distribution 13.4 A normal distribution is a binomial distribution for a very large n and the probability of success is close to half. Linear transformation 13.5 A linear transformation is a change to a variable characterized by one or more of the following operations. Random variable combinations and Markov chains 13.6 Effect on mean and variance A Markov chain is a mathematical system that experiences transitions from one state to another according to certain probabilistic rules.

## Chapter 14: Testing for validity

 Subtopic Subtopic Number IB Points to Understand Spearman’s rank correlation coefficient 14.1 Contingency tables (also called crosstabs or two-way tables) are used in statistics to summarize the relationship between several categorical variables. The Spearman’s Rank Correlation Coefficient is used to discover the strength of a link between two sets of data. The notation used is rs. Hypothesis testing 14.2 A statistical hypothesis is an assumption about a population parameter. A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution, is called a one-tailed test and a test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a two-tailed test. To hypothesis test with the binomial distribution, we must calculate the probability, p, of the observed event and any more extreme event happening.  Testing hypotheses with the Poisson distribution is very similar to testing them with the binomial distribution.  These values are obtained from the inverse of the cumulative distribution function of the standard normal distribution. i.e. we need to consider ∅-1x. The T-Test 14.3 The t-test is a statistical test which is widely used to compare the mean of two groups of samples.  One-sample t-test is used to compare the mean of a population to a specified theoretical mean (μ). Independent (or unpaired two sample) t-test is used to compare the means of two unrelated groups of samples Paired sample t-test: To compare the means of the two paired sets of data, the differences between all pairs must be, first, calculated. Chi squared test for independence 14.4 A chi-square test for independence is applied when you have two categorical variables from a single population. The p-value is the probability of observing a sample statistic as extreme as the test statistic. A chi-square goodness of fit test is applied when you have one categorical variable from a single population. Decision Errors 14.5 Type I error. A Type I error occurs when the researcher rejects a null hypothesis when it is true.  Type II error. A Type II error occurs when the researcher fails to reject a null hypothesis that is false.

## Chapter 15: Graph theor

 Subtopic Subtopic Number IB Points to Understand Graph theory 15.1 A graph is defined as a set of vertices and a set of edges. A vertex represents an object. An edge joins two vertices. A graph G is said to be connected if there exists a path between every pair of vertices. The in-degree of a vertex in a directed graph is the number of edges with that vertex as an end point. Graph representation 15.2 For a graph with |V| vertices, an adjacency matrix is a ∣V∣×∣V∣ matrix of 0s and 1s, where the entry in row i and column j is 1 if and only if the edge (i, j) is in the graph.  The transition matrix A associated to a directed graph is defined as follows. If there is an edge from i to j and the outdegree of vertex i is di, then on column i and row j we put 1/di. Minimum spanning tree 15.3 The cost of the spanning tree is the sum of the weights of all the edges in the tree.  Minimum spanning tree has direct application in the design of networks. Kruskal’s Algorithm builds the spanning tree by adding edges one by one into a growing spanning tree. In Prim’s Algorithm we grow the spanning tree from a starting position. The Chinese postman problem 15.4 A connected graph G is Eulerian if there exists a closed trail containing every edge of G. Such a trail is an Eulerian trail. In graph theory, an Euler cycle in a connected, weighted graph is called the Chinese Postman problem. Traveling sales problem 15.5 A path is a walk which does not pass through any vertex more than once. A cycle is a walk that begins and ends at the same vertex A Hamiltonian path or cycle is a path or cycle which passes through all the vertices in a graph. The classical traveling salesman problem (TSP) is to find the Hamiltonian cycle of least weight in a complete weighted graph. An upper bound can be found using the nearest neighbour algorithm Lower bounds can be found by using spanning trees.