Final High School Math: Lectures & Exercises
This page brings together 22 math lectures for final high school students preparing the Baccalaureate, A Level, Class 10, or equivalent academic milestones. Each lecture is available as a downloadable PDF, and most lectures are also accompanied by a video lecture.
The course covers sequences, limits and continuity, differentiation, logarithm and exponential functions, probability and random variables, convex functions, integration, trigonometry, complex numbers, differential equations, vectors and scalar products, arithmetic, combinatorics, matrices, logic and sets, and geometric transformations (isometries, displacements, similarities).
Cours et exercices de maths pour la Terminale et niveaux équivalents, en anglais. Voir Baccalauréat pour la version française.
Courses: PDF & Video
1 Numerical Sequences
Video and PDF of the math lesson « Numerical Sequences », for the final high school year.
We begin with the fundamental definitions and properties of sequences, examining their behavior as increasing or decreasing. We then study arithmetic sequences, defining them precisely and exploring the sum of their terms. We continue with geometric sequences, defining their specific nature and exploring the summation of their terms. Next, we address the limits of sequences, focusing on the limits of geometric sequences, the Sandwich theorem, and the limits of monotone bounded sequences. Finally, we examine the principle of mathematical induction, a powerful method for proving properties of numerical sequences.
Course Outline:
- Definition and properties of a sequence
- Increasing and decreasing sequences
- Bounded sequences
- Arithmetic sequences
- Definition of an arithmetic sequence
- Sum of terms of an arithmetic sequence
- Geometric sequences
- Definition of a geometric sequence
- Sum of terms of a geometric sequence
- Limits of sequences
- Limits of geometric sequences
- Sandwich theorem
- Limit of monotone bounded sequences
- Principle of mathematical induction
2 Limit of a Function
Video and PDF of the math lesson « Limit of a Function », for the final high school year.
We begin by establishing the fundamental concept of the limit of a function, exploring the idea of approaching values and the significance of limits in analyzing function behavior. We then focus on operations involving limits, learning techniques to handle indeterminate forms and the composition of functions. We introduce the Sandwich theorem, a powerful tool for determining limits by bounding a function between two others with known limits. We continue with right and left limits, exploring how limit behavior from different directions provides insights at specific points. Finally, we examine asymptotic behavior, including horizontal, vertical, and oblique asymptotes.
Course Outline:
- Defining the limit of a function
- Operations on the limits
- Remove indeterminacy in the limit
- Limit of composition of functions
- Sandwich theorem
- Right and left limits
- Asymptotes
- Horizontal and vertical asymptotes
- Oblique asymptote
3 Continuous Function
Video and PDF of the math lesson « Continuous Function », for the final high school year.
We begin by elucidating the concept of continuity, exploring the precise definition of a continuous function and how it relates to the behavior of functions at specific points and over intervals. We then explore the basic properties of continuous functions, including the preservation of limits, arithmetic operations, and the behavior of compositions. We unpack the significance of the Intermediate Value Theorem, illustrating its practical applications in solving equations and understanding function behavior. Finally, we explore continuous extension, allowing us to expand the domain of a function while preserving its continuity.
Course Outline:
- Definition of continuity
- Continuity basic properties
- Intermediate value theorem
- Continuous extension
4 Derivative Function
Video and PDF of the math lesson « Derivative Function », for the final high school year.
We begin with a rigorous examination of the definition of derivative, exploring how functions change at specific points through the concept of instantaneous rate of change and the formal definition using limits. We then delve into the essential properties of derivatives, including the sum, product, and ratio rules, and key derivatives to remember. We introduce the table of variations, a systematic approach for analyzing intervals of increase, decrease, and critical points. We continue with the tangent equation, exploring the geometric interpretation of derivatives in determining the equation of a tangent line. Finally, we explore the application of the derivative to calculate limits, offering an alternative method to evaluate indeterminate forms.
Course Outline:
- Definition of derivative
- Derivative properties
- Sum, product and ratio of differentiable functions
- Derivatives to remember
- Composition of two differentiable functions
- Table of variations
- Tangent equation
- Application of the derivative to calculate a limit
5 Logarithm Function
Video and PDF of the math lesson « Logarithm Function », for the final high school year.
We commence by clarifying the fundamental concept of logarithms, presenting a rigorous mathematical definition. We then focus on the key properties that govern logarithmic functions, including the product rule, quotient rule, and power rule, essential for solving logarithmic equations and simplifying expressions. We explore the limits of logarithmic functions, examining their asymptotic behavior and their role in calculus. Finally, we direct our attention to the common logarithm with base 10, which holds substantial importance across engineering, physics, and finance.
Course Outline:
- Definition of logarithm
- Properties of logarithm
- Limits of logarithm
- Base 10 logarithm
6 Exponential Function
Video and PDF of the math lesson « Exponential Function », for the final high school year.
We begin by establishing the fundamental definition of exponential functions, characterized by a constant base raised to a variable exponent. We then delve into the essential properties defining exponential functions, including their behavior under multiplication and division, and the concept of exponential growth rate. We explore the limits of exponential functions, shedding light on their behavior as inputs approach positive or negative infinity, and the concepts of horizontal asymptotes and exponential growth and decay. Finally, we examine the fascinating relationship between exponential functions and logarithms, uncovering how logarithms serve as powerful tools for solving exponential equations.
Course Outline:
- Definition of exponential function
- Properties of exponential function
- Limits of exponential function
- Relation of exponential function with the logarithm
7 Probability
Video and PDF of the math lesson « Probability », for the final high school year.
We begin with the fundamental definition of probability, which quantifies the likelihood of an event occurring and provides a numerical measure of uncertainty between 0 and 1. We then delve into the essential properties governing probabilities, unraveling the intricacies of uniform probability, where all outcomes are equally likely, and independent events, where the occurrence of one event does not influence the likelihood of another. Finally, we explore conditional probability, introducing essential tools such as the Bayes formula, the total probability formula, and probability trees, empowering students to calculate probabilities under varying conditions effectively.
Course Outline:
- Definition of a probability
- Properties of a probability
- Uniform probability (or equiprobable)
- Independent events
- Conditional probability
- Bayes formula
- Total probability formula
- Tree of probabilities
8 Convex Function
Video and PDF of the math lesson « Convex Function », for the final high school year.
We begin with a clear elucidation of what constitutes a convex function: a function whose graph lies below or on any chord connecting two of its points. We then delve into the characterization of convexity, exploring properties that define convex functions, including the concept of non-negative second derivatives, where curvature remains non-decreasing over the domain, and the convex combination property. Finally, we focus on inflection points, which mark the transition from convex to concave behavior and play a crucial role in determining function curvature.
Course Outline:
- Definition of a convex function
- Characterization of convexity of a function
- Inflection point
9 Primitive and Integral
Video and PDF of the math lesson « Primitive and Integral », for the final high school year.
We begin with an in-depth examination of primitive functions, also known as antiderivatives, defining a primitive as a function whose derivative equals a given function, and exploring its properties including the constant of integration and the linearity property. We then define the integral as the accumulation of quantities represented by a function over a given interval, and explore its properties including linearity, additivity, and the fundamental theorem of calculus. We introduce the integration by parts formula, a key method for evaluating certain integrals. Finally, we discuss the relationship between integrals and surface area, as well as their connection to the mean value of a function.
Course Outline:
- Primitive function
- Definition of the primitive
- Properties of the primitive
- Integral
- Definition of integral
- Properties of the integral
- Integration by parts formula
- Relation of integral to surface
- Relation of integral to mean
10 Random Variable
Video and PDF of the math lesson « Random Variable », for the final high school year.
We begin with a clear elucidation of what constitutes a random variable: a numerical outcome of a random phenomenon, with each possible outcome associated with a probability. We then delve into expectation and variance, essential measures of central tendency and dispersion, including the formula for the expectation of a function of a random variable. We examine the concept of independence among random variables, where the occurrence of one event does not affect another. We then focus on specific types: the Bernoulli random variable, representing a single trial with two possible outcomes, and the binomial random variable, representing the number of successes in a fixed number of independent Bernoulli trials. Finally, we explore the Law of Large Numbers, discussing related inequalities such as Markov, Chebyshev, and concentration inequalities.
Course Outline:
- Definition of a random variable
- Expectation and variance
- Expectation of a random variable
- Variance of a random variable
- Formula of expectation of a function of a random variable
- Independence of random variables
- Bernoulli random variable
- Binomial random variable
- Law of large numbers
- Markov and Chebychev inequalities
- Concentration inequality
11 Trigonometric Functions
Video and PDF of the math lesson « Trigonometric Functions », for the final high school year.
We begin by delving into the fundamental concepts of cosine and sine functions, establishing clear definitions and dissecting their properties, including their periodicity and relationships within the trigonometric realm. We then identify and elucidate the remarkable values embedded within cosine and sine functions, from the unit circle to key angles such as 0°, 30°, 45°, 60°, and 90°. We transition to the tangent function, defining it and elucidating its properties including periodicity, asymptotes, and its relationship with sine and cosine. Finally, we equip students with systematic methodologies for solving equations involving trigonometric functions.
Course Outline:
- Cosine and sine functions
- Definition of cosine and sine functions
- Properties of cosine and sine
- Remarkable values of cosine and sine
- Tangent function
- Definition of the tangent function
- Properties of the tangent function
- Solving equations with trigonometric functions
12 Complex Numbers
Video and PDF of the math lesson « Complex Numbers », for the final high school year.
We begin with a concise definition introducing students to the realm of complex numbers, which encompass real and imaginary components, and explore the properties of addition, subtraction, multiplication, and division on complex numbers. We then investigate various representations of complex numbers: the geometric representation on the complex plane, and the trigonometric and exponential forms. We explore techniques for solving equations involving complex numbers, focusing initially on second-degree equations and then on equations of the form z^n = a. Finally, we explore the transformative power of complex numbers in point transformations, including translations, rotations, dilations, and reflections.
Course Outline:
- Set of complex numbers
- Definition of complex numbers
- Properties of complex numbers
- Representations of a complex number
- Geometric representation of a complex number
- Trigonometric form of a complex number
- Exponential form of a complex number
- Solving equations in C
- Second degree equation
- Equation of type z^n = a
- Point transformations
- Transformation associated to an application in C
- Usual transformations
13 Differential Equations
Video and PDF of the math lesson « Differential Equations », for the final high school year.
We begin by defining what constitutes a differential equation, exploring through illustrative examples the diverse forms and applications of these equations, fundamental tools for modeling dynamic systems. We then focus on first-order differential equations, fundamental in many areas of science and engineering. We address homogeneous equations of the form y′ = ay, exploring their solutions and implications. We continue with equations with constant second members, y′ = ay + b, analyzing how the inclusion of a constant term alters system behavior. Finally, we extend our understanding to equations with general second members, y′ = ay + g(x), presenting methods to solve such equations efficiently.
Course Outline:
- Differential equations
- Definition of a differential equation
- Differential equation y′ = g
- First order differential equations
- Homogeneous equation y′ = ay
- Equation with constant second member y′ = ay + b
- Equation with general second member y′ = ay + g (x)
14 Vectors in Plane and Space
PDF of the math lesson « Vectors in Plane and Space », for the final high school year.
We begin with the foundation of vector concepts, starting with the construction of vectors and distinguishing between linearly independent and dependent vectors, and grasping the concept of direction vectors of lines and planes. We then delve into the vector plane, exploring its bases and coordinates, and learning to express lines in the plane using Cartesian and parametric equations. We extend the discussion to three-dimensional vector space, including bases and coordinates in three dimensions, and the Cartesian and parametric equations of lines in space. We explore the significance of orthonormal bases and coordinate systems in establishing consistent frameworks for representing vectors. Finally, we examine the concept of affine or vector planes and spaces, offering insights into their geometric properties.
Course Outline:
- Vectors in plane and space
- Construction of vectors
- Linearly independent or dependent vectors
- Direction vector
- Vector plane
- Base and coordinates in the plane
- Cartesian and parametric equations of a line in the plane
- Vector space
- Base and coordinates in the space
- Cartesian and parametric equations of a line in the space
- Orthonormal basis and coordinate system
- Affine or vector plane and space
15 Scalar Product of Vectors
PDF of the math lesson « Scalar Product of Vectors », for the final high school year.
We begin by defining the scalar product and its computation from vector coordinates, exploring how it encapsulates the notion of angle between vectors and investigating its properties including commutativity, distributivity, and the orthogonality condition. We examine the geometric interpretation of the scalar product and its role in determining projections and magnitudes of vectors. We then delve into the concept of a normal vector, exploring vectors normal to lines in the plane and to planes in three-dimensional space, and the relationship between normal vectors and the equations defining lines and planes. Finally, we examine the intersections of lines and planes, investigating the conditions under which two planes intersect and the geometric insights behind the intersection of a line and a plane.
Course Outline:
- Scalar product in the plane and in space
- Scalar product from coordinates
- Properties of the scalar product
- Remarkable identities of the scalar product
- Relation of the scalar product to geometry
- Normal vector
- Vector normal to a line in the plane
- Vector normal to a plane in space
- Intersections of lines and planes
- Intersection of two planes
- Intersection of a line and a plane
16 Arithmetic
PDF of the math lesson « Arithmetic », for the final high school year.
We begin by delving into the foundational concept of divisibility within the set of integers Z, exploring the definition and properties of divisibility, and understanding the Euclidean division algorithm for finding quotients and remainders efficiently. We discuss prime numbers, the building blocks of the integers, and congruence, which relates to divisibility in a modular arithmetic context. We then focus on the greatest common divisor (GCD), learning how to compute it and understanding Euclid’s theorem and algorithm. Finally, we explore coprime numbers, discussing Bézout’s identity and theorem, Gauss theorem, and the Bézout-Euclide algorithm for efficiently computing coefficients in Bézout’s theorem.
Course Outline:
- Divisibility in Z
- Definition and properties of divisibility
- Euclidean division
- Prime numbers
- Congruence
- Greatest common divisor
- GCD of 2 integers
- Euclid’s theorem and algorithm
- Coprime numbers
- Identity and Bézout’s theorem
- Gauss theorem
- Bézout-Euclide algorithm
17 Enumeration and Combinatorics
PDF of the math lesson « Enumeration and Combinatorics », for the final high school year.
We begin by exploring the fundamental enumeration techniques: lists, arrangements, combinations, and permutations, learning to distinguish between them based on whether order and repetition matter. A summary table consolidates these distinctions for quick reference. We then delve into the properties of binomial coefficients, unraveling the patterns in Pascal’s triangle through its formula, and exploring Newton’s binomial formula with its applications in expanding binomials to any power.
Course Outline:
- Enumeration cases
- Lists
- Arrangements
- Combinations
- Permutations
- Summary table: order and repetition
- Properties of binomial coefficients
- Pascal’s triangle formula
- Newton’s binomial formula
18 Matrices
PDF of the math lesson « Matrices », for the final high school year.
We begin with an introduction to matrices, learning about their structure, the addition of matrices, multiplication by a scalar, the product of matrices, and the inverse of a matrix. We then learn how to raise a square matrix to a given power, uncovering the underlying patterns and properties, with special attention to diagonal matrices and diagonalization. We explore the application of matrices in solving linear systems of equations using matrix notation. Finally, we delve into the concept of a matrix associated with a linear application, understanding how matrices represent linear transformations between vector spaces.
Course Outline:
- Introduction to matrices
- Definition of matrices and first properties
- Addition of matrices
- Multiplying a matrix by a number
- Product of matrices
- Inverse of a matrix
- Power of a square matrix
- Definition and properties of the power of a matrix
- Diagonal matrix
- Diagonalization
- Linear systems
- Matrix associated with a linear application
19 Logic and Sets
PDF of the math lesson « Logic and Sets », for the final high school year.
We begin by exploring the fundamental concepts of logic, starting with assertions and predicates and learning to formulate statements and determine their truth values. We delve into logical connectors such as conjunction, disjunction, and negation, and into logical quantifiers, including existential and universal quantifiers. We then transition to set theory, learning about sets as collections of objects, exploring various types including empty sets, finite sets, and infinite sets. We delve into operations on sets, such as union, intersection, and complement. Finally, we explore finite sets, learning how to count elements and analyze the cardinality of sets.
Course Outline:
- Basics of logic
- Assertions and predicates
- Logical connectors
- Logical quantifiers
- Basics of set theory
- Sets
- Operations on sets
- Finite sets
20 Isometries of the Plane
PDF of the math lesson « Isometries of the Plane », for the final high school year.
We begin by defining what constitutes an isometry of the plane and exploring its key properties, examining how isometries interact with the scalar product in relation to distance preservation, and investigating the inverse of an isometry and methods for characterizing these transformative operations. We then delve into the composition of isometries, exploring general properties governing their combination, particularly the intriguing dynamics involved in composing two orthogonal symmetries. Finally, we explore the concept of invariant points under isometries, classifying isometries based on their behavior towards invariant points, and analyzing isometries as compositions of symmetries.
Course Outline:
- Definition and properties of an isometry
- Definition of an isometry of the plane
- Isometries and scalar product
- Inverse of an isometry
- Characterization of an isometry
- Composition of isometries
- General properties of isometries composition
- Composition of two orthogonal symmetries
- Isometries and invariant points
- Isometries having invariant points
- Isometries having no invariant point
- Isometries as compositions of symmetries
21 Displacements and Antidisplacements
PDF of the math lesson « Displacements and Antidisplacements », for the final high school year.
We begin by defining what constitutes a displacement or an antidisplacement and examining their key properties, delving into the classification of isometries and exploring the characterization of displacements and antidisplacements. We then investigate the angle of a displacement, shedding light on how these transformations affect the orientation of geometric objects in the plane, and we explore the composition of displacements as well as their representation in complex numbers. Finally, we turn our attention to antidisplacements, which reverse the orientation of geometric objects while preserving distances, contrasting their unique properties with those of displacements.
Course Outline:
- Definition and properties
- Classification of isometries
- Characterization of a displacement or an antidisplacement
- Displacements
- Angle of a displacement
- Composition of displacements
- Representation of a displacement in C
- Antidisplacements
22 Similarities
PDF of the math lesson « Similarities », for the final high school year.
We begin with the basic concepts of similarities, exploring the fundamental transformations of homotheties and translations as building blocks, defining similarities as transformations that preserve shape while possibly altering size or orientation, and distinguishing between direct and indirect similarities. We then delve deeper into direct similarities, examining their characteristic elements, the reduced form of direct similarities, and their representation in the complex plane (C). Finally, we explore indirect similarities, which involve transformations that preserve shape but reverse orientation, contrasting their unique properties with those of direct similarities.
Course Outline:
- Introduction to Similarities
- Homotheties and translations
- Definition of similarities
- Direct or indirect similarities
- Direct similarities
- Characteristic elements of a direct similarity
- Reduced form and representation in C
- Indirect similarities
About These Topics
These final high school mathematics resources are designed for students preparing the French Baccalaureate, the British A Level, the Indian Class 10, or any equivalent academic level. The materials provide a comprehensive coverage of the topics typically taught in the final year of high school, including calculus (limits, continuity, differentiation, integration), analysis (sequences, logarithm and exponential functions, differential equations), probability and statistics (random variables, Bernoulli and binomial distributions, law of large numbers), algebra (complex numbers, matrices, vectors), geometry (scalar products, isometries, similarities), and discrete mathematics (arithmetic, combinatorics, logic and sets).
Each lecture combines rigorous mathematical definitions, worked examples, and a course outline summarizing the key concepts. Most lectures include both a downloadable PDF and a video lecture, while some are available as PDF only. The material is suitable for students, teachers seeking supplementary resources, and self-learners aiming to consolidate their understanding of foundational mathematics before moving on to undergraduate studies.
For the French-language version of these courses, designed for the Baccalauréat français and including additional resources (exercises, problem sets, and revision sheets), see Baccalauréat.
Last Updated on 24 mai 2026 by Mohamed Kadhem KARRAY