Online Math Courses for Final High School, A level, and Class 10 (French Baccalaureate)

We believe in making mathematics accessible and enjoyable for everyone. We offer a comprehensive collection of PDFs and videos covering a wide range of topics crucial for your final high school year and beyond.

Looking for comprehensive math resources to ace your final high school year or to deepen your understanding of key mathematical concepts? Our webpage offers an extensive array of math courses, resources, and video tutorials designed to cater to various learning needs. Whether you’re a student seeking free online math courses, a teacher searching for supplementary materials, or someone in need of free math tutoring online, our platform has you covered. Our PDF math resources and video math tutorials provide a dynamic and interactive learning experience, accessible anytime, anywhere.

Our comprehensive math courses and resources are not only tailored for high school curriculums but are also suited for students pursuing A Level Mathematics in the UK. Whether you’re studying for A Level math exams, seeking additional support for A Level math concepts, or looking for A Level math tutoring online, this Webpage provides the ideal platform for your academic journey. Our materials cover a wide range of topics, including pure mathematics, calculus, trigonometry, and more, aligning seamlessly with the A Level math syllabus.

Our math lessons are also suited for students in Class 10 in India. Whether you’re preparing for Class 10 math exams, seeking to enhance your understanding of advanced math concepts, or looking for online resources for 10th class maths, this Webpage has everything you need to succeed. Our materials cover a wide range of topics, including algebra, geometry, and arithmetic, aligning seamlessly with the Class 10 math syllabus in India. For French lessons, see Baccalauréat.

1 Numerical sequences

PDF, Video of Math lesson « Numerical sequences » (i.e. sequences of real numbers) for final High school year.

In this lecture on Numerical Sequences, we will delve into the fundamental concepts and properties that underpin the study of sequences, a crucial aspect of mathematical analysis. Starting with the foundational definitions and properties, we will progress through arithmetic and geometric sequences, exploring their sums and limits, before concluding with the powerful principle of mathematical induction.

Definition and Properties of a Sequence: We begin by establishing a solid understanding of what constitutes a sequence and exploring its fundamental properties. We delve into the classification of sequences as increasing, decreasing, or bounded, providing insights into their behavior and characteristics.
Arithmetic Sequences: Building upon the foundational knowledge from the previous section, we introduce arithmetic sequences, defining them and exploring their properties. By examining the formula for the nth term and the sum of terms, students will develop a deep understanding of how arithmetic sequences progress and evolve.
Geometric Sequences: Transitioning from arithmetic to geometric sequences, we define geometric sequences and analyze their properties. Students will learn about the geometric ratio and the formula for the sum of terms in a geometric sequence.
Limits of Sequences: This section delves into the concept of limits within sequences, a pivotal aspect of mathematical analysis. We explore the limit of a geometric sequence, introduce the Sandwich theorem to evaluate limits, and discuss the convergence of monotone bounded sequences.
Induction Principle: Concluding our exploration of numerical sequences, we introduce the powerful principle of mathematical induction. Students will learn how to apply mathematical induction to prove statements about sequences, paving the way for more sophisticated mathematical reasoning and problem-solving techniques.

Discover comprehensive insights into numerical sequences, including arithmetic and geometric progressions, and delve into their respective formulas for sum calculation. Explore the definition and properties of arithmetic sequences, alongside the limits of sequences, to deepen your understanding of mathematical analysis. Unravel the geometric sequence equation and series formula, unlocking the mysteries of geometric progressions. With clear explanations and practical examples, master the summation of arithmetic progressions and geometric series, empowering yourself with essential mathematical tools for academic success. Whether you’re seeking to grasp the arithmetic sequence formula or comprehend the intricacies of geometric progression, this lecture provides invaluable knowledge tailored to your learning needs.

Course Outline:
1 Definition and properties of a sequence
1.1 Increasing and decreasing sequences
1.2 Bounded sequences
2 Arithmetic sequences
2.1 Definition of an arithmetic sequence
2.2 Sum of terms of an arithmetic sequence
3 Geometric sequences
3.1 Definition of a geometric sequence
3.2 Sum of terms of a geometric sequence
4 Limits of sequences
4.1 Limit of a geometric sequence
4.2 Sandwich theorem
4.3 Limit of monotone bounded sequences
5 Induction principle

2 Limit of a function

PDF, Video of Math lesson « Limit of a function » for final High school year.

Welcome to a comprehensive lecture on the limit of a function, designed for final year high school students. In this lecture, we will explore fundamental concepts essential for understanding calculus, ranging from defining the limit to exploring various theorems and applications.

Defining the limit of a function: We begin by establishing the fundamental concept of the limit of a function. Through intuitive explanations and concrete examples, students will grasp the idea of approaching values and the significance of limits in analyzing the behavior of functions.
Operations on the limits: This section focuses on various operations involving limits. Students will learn techniques to handle indeterminate forms and explore how limits behave under the composition of functions, enhancing their problem-solving skills and understanding of limit evaluations.
Sandwich theorem: Introducing a powerful tool in calculus, the Sandwich theorem provides a systematic approach to determining the limit of a function by « sandwiching » it between two other functions with known limits. This theorem offers valuable insights into challenging limit problems and strengthens students’ understanding of limit concepts.
Right and left limits: We delve into the nuances of limit behavior from different directions, exploring the concepts of right and left limits. Through graphical representations and analytical techniques, students will discern how these limits provide insights into the behavior of functions at specific points.
Asymptotes: In the final section, we explore asymptotic behavior, including horizontal, vertical, and oblique asymptotes. Through graphical analysis and algebraic reasoning, students will uncover how functions behave as they approach infinity, gaining valuable insights into the behavior of functions in the long run.

Our lecture provides comprehensive insights into key topics such as the sandwich theorem, asymptotes, and the determination of horizontal and vertical asymptotes. Students will learn practical techniques, including rules for horizontal asymptotes and methods for determining asymptotes through equations. With illustrative examples showcasing the application of the sandwich theorem, students will grasp its significance in evaluating limits effectively. Whether you’re seeking to understand the definition of limits or aiming to master the intricacies of asymptotic behavior, our lecture offers invaluable resources and explanations to aid your comprehension.

Course Outline:
1 Defining the limit of a function
2 Operations on the limits
2.1 Remove indeterminacy in the limit
2.2 Limit of composition of functions
3 Sandwich theorem
4 Right and left limits
5 Asymptotes
5.1 Horizontal and vertical asymptotes
5.2 Oblique asymptote

3 Continuous function

PDF, Video of Math lesson on « Limit of a function » for final High school year.

In this lecture on « Continuous Function, » we delve into the fundamental concepts and properties of continuity within functions, an essential topic in high school mathematics. Understanding continuity lays the groundwork for deeper comprehension of calculus and real analysis. This lecture is structured into five sections, each focusing on key aspects of continuous functions.

Definition of Continuity: The lecture begins by elucidating the concept of continuity, which forms the bedrock of analysis in calculus. Here, students will grasp the precise definition of a continuous function, exploring how it relates to the behavior of functions at specific points and over intervals.
Continuity Basic Properties: Building upon the foundational definition, this section delves into the basic properties that continuous functions exhibit. Students will explore properties such as the preservation of limits, arithmetic operations on continuous functions, and the behavior of compositions of continuous functions.
Intermediate Value Theorem: One of the most powerful consequences of continuity, the Intermediate Value Theorem (IVT) asserts that a continuous function takes on every value between two given values. In this section, we unpack the significance of IVT, illustrating its practical applications in solving equations and understanding the behavior of functions.
Continuous Extension: The concept of continuous extension allows us to expand the domain of a function while preserving its continuity. Here, students will explore techniques for extending functions across intervals or to the closure of their domains without introducing discontinuities.

Explore in-depth discussions and practical examples covering a wide array of topics crucial for understanding calculus, including the definition of continuity, intermediate value theorem, and the intricate relationship between limits and continuity. Delve into real-world applications and problem-solving strategies, empowering yourself with the knowledge necessary to master these fundamental concepts. Whether you’re seeking clarity on continuity in composite functions or tackling intermediate value theorem problems, this comprehensive lecture offers insights and explanations tailored to meet your educational needs. With clear explanations and illustrative examples, our content demystifies complex calculus concepts, making them accessible and engaging for high school students and mathematics enthusiasts alike.

Course Outline:
1 Definition of continuity
2 Continuity basic properties
3 Intermediate value theorem
4 Continuous extension

4 Derivative function

PDF, Video of Math lesson on « Derivative function » for final High school year.

Welcome to the comprehensive lecture on « Derivative Function, » a pivotal topic in high school mathematics that forms the backbone of calculus. This lecture delves into the fundamental concepts and applications of derivatives, providing students with a solid foundation for understanding rates of change, slopes of curves, and optimization problems. Structured into five sections, this lecture aims to equip students with the necessary tools and techniques to master the intricacies of derivative functions.

Definition of Derivative: The lecture commences with a rigorous examination of the definition of derivative, laying the groundwork for understanding how functions change at specific points. Students will delve into the concept of instantaneous rate of change and explore the formal definition of derivative using limits. Through intuitive examples and graphical representations, the notion of slope as a derivative will become clear, paving the way for deeper exploration.
Derivative Properties: This section delves into the essential properties of derivatives, providing insights into the behavior of functions under differentiation. Subsections will cover topics such as the sum, product, and ratio rules for differentiable functions, highlighting how these rules facilitate the differentiation of more complex functions. Additionally, students will familiarize themselves with key derivatives to remember, aiding in the efficient calculation of derivatives.
Table of Variations: A crucial tool in understanding the behavior of functions, the table of variations provides a systematic approach to analyzing the intervals of increase, decrease, and critical points of a function. Through step-by-step explanations and illustrative examples, students will learn how to construct and interpret a table of variations, enabling them to make informed conclusions about the behavior of functions.
Tangent Equation: The tangent equation section focuses on the geometric interpretation of derivatives, particularly in determining the equation of a tangent line to a curve at a given point. Through geometric reasoning and algebraic manipulation, students will learn how to find the slope of the tangent line and use it to derive the equation of the tangent line.
Application of the Derivative to Calculate a Limit: In the final section, students will explore how derivatives can be utilized to calculate limits, offering an alternative method to evaluate indeterminate forms and tackle challenging limit problems. Through a series of guided examples, students will gain confidence in applying derivative concepts to limit calculations.

Unlock the secrets of calculus with our comprehensive lecture on the derivative function, designed to equip high school students with the essential tools and techniques for mastering calculus concepts. Explore topics ranging from the fundamental definition of derivatives to advanced differentiation rules, including the quotient rule, product rule, and chain rule. Delve into practical applications such as finding tangent lines and calculating derivatives to solve real-world problems. With clear explanations and step-by-step guides, students will gain a deep understanding of continuity, differentiability, and the intricate relationship between functions and their derivatives.

Course Outline:
1 Definition of derivative
2 Derivative properties
2.1 Sum, product and ratio of differentiable functions
2.2 Derivatives to remember
2.3 Composition of two differentiable functions
3 Table of variations
4 Tangent equation
5 Application of the derivative to calculate a limit

5 Logarithm function

PDF, Video of Math lesson on « Logarithm function » for final High school year.

In this lecture on the logarithm function, we will delve into the fundamental concepts and properties of logarithms, crucial for understanding various mathematical and scientific principles. The lecture is structured as follows:

Definition of Logarithm: We will commence by clarifying the fundamental concept of logarithms. This will be followed by presenting a rigorous mathematical definition, ensuring a comprehensive understanding of this essential mathematical function.
Properties of Logarithm: This section will focus on the key properties that govern logarithmic functions. We’ll discuss properties such as the product rule, quotient rule, and power rule, which play a vital role in manipulating logarithmic expressions. Understanding these properties will enable students to effectively solve logarithmic equations and simplify expressions with ease.
Limits of Logarithm: Delving deeper into the mathematical underpinnings, we’ll explore the limits of logarithmic functions. Through rigorous analysis and graphical representations, students will grasp the behavior of logarithmic functions as they approach certain values, providing insights into their asymptotic behavior and the role of limits in calculus.
Base 10 Logarithm: In this section, our attention will be directed towards the common logarithm, which operates with a base of 10. This specific logarithmic function holds substantial importance across multiple disciplines such as engineering, physics, and finance. Acquiring a profound understanding of base 10 logarithms empowers students to decipher logarithmic scales effectively.

Explore the intricacies of logarithmic functions, including the natural logarithm ln(x), base 10 logarithms, and their derivatives. Dive into the properties of logarithms, their graphical representations, and the fundamental formulas governing their behavior. Uncover the power of logarithmic differentiation and its application in calculus. Whether you’re seeking to understand the derivative of log(x) or the graph of ln(x), this lecture provides a comprehensive guide to mastering logarithmic concepts. With detailed explanations and practical examples, grasp the significance of logarithms in mathematical analysis and scientific computations. Delve into logarithmic formulae, understand logarithmic differentiation techniques, and gain insights into the logarithmic functions that underpin various mathematical disciplines.

Course Outline:
1 Definition of logarithm
2 Properties of logarithm
3 Limits of logarithm
4 Base 10 logarithm

6 Exponential function

PDF, Video of Math lesson on « Exponential function » for final High school year.

Welcome to the lecture on the Exponential Function, a fundamental concept in mathematics with wide-ranging applications. Throughout this session, we will delve into the definition, properties, limits, and the intriguing relationship between exponential functions and logarithms.

Definition of Exponential Function: We begin by establishing a solid understanding of what constitutes an exponential function. This section will cover the fundamental definition of exponential functions, exploring how they are characterized by a constant base raised to a variable exponent.
Properties of Exponential Function: In this segment, we delve into the essential properties that define exponential functions. From their behavior under multiplication and division to the concept of exponential growth rate, students will gain insight into the core characteristics that distinguish exponential functions from other mathematical functions.
Limits of Exponential Function: As we delve deeper into the intricacies of exponential functions, understanding their limits becomes crucial. This section will explore the behavior of exponential functions as their input approaches positive or negative infinity, shedding light on concepts such as horizontal asymptotes and the behavior of exponential growth and decay over time.
Relation of Exponential Function with the Logarithm: The final section of our lecture explores the fascinating relationship between exponential functions and logarithms. By examining the inverse nature of these functions, we uncover how logarithms serve as powerful tools for solving exponential equations and vice versa. Through practical examples, students will witness firsthand how these seemingly distinct mathematical concepts intertwine, enriching their understanding of both exponential and logarithmic functions.

Our comprehensive coverage of the exponential function encompasses its definition, properties, and intricate relationships with logarithms. Additionally, we explore the limits of exponential functions, shedding light on their behavior as inputs approach positive or negative infinity. Through detailed explanations and illustrative examples, students will grasp the fundamental concepts of exponential growth and decay, preparing them for advanced mathematical applications. Whether you’re seeking to understand the derivative of an exponential function or graphing an exponential curve, our lecture provides invaluable insights into this essential mathematical topic. Dive deeper into exponential formulas, properties, and graphs, and uncover the profound connections between exponential and logarithmic functions.

Course Outline:
1 Definition of exponential function
2 Properties of exponential function
3 Limits of exponential function
4 Relation of exponential function with the logarithm

7 Probability

PDF, Video of Math lesson on « Probability » for final High school year.

Welcome to the lecture on probability designed for final year high school students. Probability, the study of uncertainty and chance, plays a crucial role in various aspects of our lives, from predicting outcomes in games to making informed decisions in everyday situations. In this lecture, we will embark on a journey to unravel the mysteries of probability, exploring its definition, properties, and practical applications that will empower students to navigate through uncertain situations with confidence and precision.

Definition of Probability: Our exploration begins with the fundamental definition of probability. Probability, in its essence, quantifies the likelihood of an event occurring, providing a numerical measure of uncertainty. Through insightful explanations and illustrative examples, students will develop a clear understanding of probability as a concept, learning to interpret probabilities as fractions between 0 and 1.
Properties of Probability: Moving forward, we delve into the essential properties that govern probabilities. Within this section, we unravel 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. By dissecting these properties, students will gain valuable insights into how probabilities behave in different contexts, sharpening their analytical skills and critical thinking abilities.
Conditional Probability: Here, students will delve into the concept of probability under given conditions. We will introduce essential tools such as the Bayes formula and the total probability formula, empowering students to calculate probabilities under varying conditions effectively. Additionally, we will explore the concept of probability trees, providing students with a visual aid to navigate through complex probabilistic scenarios with clarity and precision.

For those seeking comprehensive insights into probability and its applications, our lecture serves as an invaluable resource. Whether you’re delving into the fundamental definition of probability or unraveling the complexities of conditional probability and its associated formulas such as Bayes theorem, our content provides a thorough exploration. From understanding the laws governing probability to grasping the significance of independent events and uniform probability, equip yourself with the tools needed to navigate uncertainty with confidence.

Course Outline:
1 Definition of a probability
2 Properties of a probability
2.1 Uniform probability (or equiprobable)
2.2 Independent events
3 Conditional probability
3.1 Bayes formula
3.2 Total probability formula
3.3 Tree of probabilities

8 Convex function

PDF, Video of Math lesson on « Convex function » for final High school year.

Welcome to this lecture on convex functions tailored for final year high school students. Convex functions, a fundamental concept in mathematics and optimization, play a pivotal role in various fields, from economics to engineering. In this lecture, we will delve into the definition, properties, and characterization of convex functions, offering students a comprehensive understanding of this essential mathematical concept.

Definition of a Convex Function: Our exploration begins with a clear elucidation of what constitutes a convex function. A function is considered convex if the line segment connecting any two points on its graph lies above or on the graph itself. Through intuitive explanations and graphical representations, students will grasp the notion of convexity and its significance in mathematical analysis.
Characterization of Convexity of a Function: Moving forward, we delve into the characterization of convexity and explore various properties that define convex functions. These properties include the concept of non-negative second derivatives, where a function’s curvature remains non-decreasing over its domain, and the convex combination property, which ensures that the function lies below the chords connecting its points.
Inflection Point: In the final section of our lecture, we focus on inflection points, which play a crucial role in determining the convexity of a function. An inflection point is a point on the graph of a function where the curvature changes sign, indicating a transition from convex to concave or vice versa. Through interactive examples and visual aids, students will learn to identify inflection points and understand their significance in analyzing the behavior of functions.

Whether you’re seeking to understand the fundamental properties of convex functions or unravel the mysteries of inflection points and concavity, our content offers in-depth insights to fuel your mathematical exploration. From identifying inflection points using calculus techniques to discerning the concavity of functions, we provide practical guidance and intuitive explanations to enhance your understanding. Join us as we delve into the intricacies of convex and concave functions, equipping you with the knowledge and skills to tackle complex mathematical concepts with confidence.

Course Outline:
1 Definition of a convex function
2 Characterization of convexity of a function
3 Inflection point

9 Primitive and integral

PDF, Video of Math lesson on « Primitive and integral » for final High school year.

Welcome to an insightful lecture on integral calculus tailored for final year high school students. Integral calculus, a cornerstone of mathematical analysis, provides powerful tools for quantifying quantities such as area, volume, and average value. In this lecture, we will delve into the concepts of primitive functions and integrals, exploring their definitions, properties, and practical applications.

Primitive Function: Our exploration begins with an in-depth examination of primitive functions. A primitive, also known as an antiderivative, is a function whose derivative is equal to a given function. In this section, we define the primitive function and delve into its properties, including the constant of integration and the linearity property.
Integral: We start by defining the integral as the accumulation of quantities represented by a function over a given interval. Students will explore the properties of the integral, including linearity, additivity, and the fundamental theorem of calculus. Additionally, we introduce integration by parts formula, which provides a method for evaluating certain types of integrals. Furthermore, we discuss the relationship between integrals and surface area, as well as their connection to the concept of the mean value of a function.

Whether you’re delving into the intricacies of integration by parts formula or seeking to understand the fundamental properties of integrals, our content offers invaluable insights to fuel your mathematical exploration. From defining integrals and primitive functions to evaluating integrals using established techniques, we provide practical guidance and intuitive explanations to enhance your understanding. Join us as we unravel the mysteries of integral calculus, equipping you with the knowledge and skills to tackle complex mathematical concepts with confidence and precision.

Course Outline:
1 Primitive function
1.1 Definition of the primitive
1.2 Properties of the primitive
2 Integral
2.1 Definition of integral
2.2 Properties of the integral
2.3 Integration by parts formula
2.4 Relation of integral to surface
2.5 Relation of integral to mean

10 Random variable

PDF, Video of Math lesson on « Random variable » for final High school year.

Welcome to this lecture on random variables tailored for final year high school students. Random variables are fundamental concepts in probability theory, serving as mathematical representations of uncertain outcomes. In this lecture, we will explore the definition, properties, and applications of random variables, providing students with a solid foundation in probability theory and statistical analysis.

Definition of a Random Variable: Our exploration begins with a clear elucidation of what constitutes a random variable. A random variable is a numerical outcome of a random phenomenon, with each possible outcome associated with a probability. Through intuitive explanations and illustrative examples, students will grasp the concept of random variables and understand their significance in modeling uncertain events.
Expectation and Variance: Moving forward, we delve into the concepts of expectation and variance, which are essential measures of central tendency and dispersion for random variables. Within this section, we explore the calculation of the expectation and variance of a random variable, providing students with the tools to analyze the distribution of random outcomes effectively. Additionally, we introduce the formula for calculating the expectation of a function of a random variable, extending their understanding to more complex scenarios.
Independence of Random Variables: In the following section, we examine the concept of independence among random variables. Two random variables are considered independent if the occurrence of one event does not affect the likelihood of the other. Through illustrative examples and interactive exercises, students will discern the implications of independence in probabilistic scenarios, laying the groundwork for more advanced analyses.
Bernoulli Random Variable: We then focus on specific types of random variables, starting with the Bernoulli random variable. A Bernoulli random variable represents a single trial with two possible outcomes, typically denoted as success or failure.
Binomial Random Variable: Building upon the foundation laid by the Bernoulli random variable, we delve into the binomial random variable. A binomial random variable represents the number of successes in a fixed number of independent Bernoulli trials.
Law of Large Numbers: In the final section of our lecture, we explore the Law of Large Numbers, a fundamental principle in probability theory. This law states that as the number of trials or observations increases, the sample mean converges to the population mean. We discuss related inequalities such as Markov and Chebyshev inequalities, as well as concentration inequalities, providing students with valuable tools for analyzing probabilistic phenomena.

Whether you’re exploring the intricacies of Bernoulli random variables or unraveling the fundamental concepts behind the Law of Large Numbers, our content offers invaluable insights to fuel your mathematical exploration. From understanding the definition and properties of random variables to calculating expectations and variances, we provide practical guidance and intuitive explanations to enhance your understanding.

Course Outline:
1 Definition of a random variable
2 Expectation and variance
2.1 Expectation of a random variable
2.2 Variance of a random variable
2.3 Formula of expectation of a function of a random variable
3 Independence of random variables
4 Bernoulli random variable
5 Binomial random variable
6 Law of large numbers
6.1 Markov and Chebychev inequalities
6.2 Concentration inequality

11 Trigonometric functions

PDF, Video of Math lesson on « Trigonometric functions » for final High school year.

Welcome to our comprehensive lecture on Trigonometric Functions, tailored specifically for final-year high school students. We will embark on a journey to explore the intricate world of cosine, sine, and tangent functions, unraveling their definitions, properties, and applications.

Cosine and Sine Functions: In this section, we delve into the fundamental concepts of cosine and sine functions. Initially, we establish a clear definition of both functions, laying the groundwork for deeper exploration. Subsequently, we dissect the properties inherent in cosine and sine, uncovering their behavior, periodicity, and relationships within the trigonometric realm.
Remarkable Values of Cosine and Sine: Building upon the foundation laid in the previous section, we 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 unravel the significance of these values and their implications in various mathematical contexts.
Tangent Function: Transitioning to the tangent function, we explore its definition and intricacies. Through a systematic approach, we define the tangent function and elucidate its properties, including its periodicity, asymptotes, and relationship with sine and cosine.
Solving Equations with Trigonometric Functions: Concluding our lecture, we equip students with the essential tools to solve equations involving trigonometric functions. Drawing upon the concepts covered throughout the session, we demonstrate systematic methodologies for tackling trigonometric equations.

This lesson offers an extensive array of trigonometry tools, including trig tables, formulas for sine, cosine, and tangent, as well as differentiation techniques for trigonometric functions. Whether you’re a high school student exploring the basics or a seasoned mathematician delving into advanced concepts, our platform provides valuable insights and practical guidance. Dive into the depths of trigonometry, from differentiation of trig functions to the derivation of sine, cosine, and tangent formulas.

Course Outline:
1 Cosine and sine functions
1.1 Definition of cosine and sine functions
1.2 Properties of cosine and sine
2 Remarkable values of cosine and sine
3 Tangent function
3.1 Definition of the tangent function
3.2 Properties of the tangent function
4 Solving equations with trigonometric functions

12 Complex numbers

PDF, Video of Math lesson on « Complex numbers » for final High school year.

Welcome to our comprehensive lecture on Complex Numbers, designed specifically for final-year high school students. Throughout this session, we will unravel the fascinating world of complex numbers, exploring their definitions, properties, representations, and practical applications.

Set of Complex Numbers: Beginning with a concise definition, we introduce students to the realm of complex numbers, which encompass real and imaginary components. Subsequently, we explore the properties inherent in complex numbers, including addition, subtraction, multiplication, and division, laying the groundwork for deeper exploration and understanding.
Representations of a Complex Number: Building upon the foundational knowledge established in the previous section, we investigate various representations of complex numbers. From the geometric representation on the complex plane to the trigonometric and exponential forms, students will learn how to visualize and manipulate complex numbers in different contexts.
Solving Equations in C: Transitioning to practical applications, we explore techniques for solving equations involving complex numbers. Focusing initially on second-degree equations, students will learn systematic approaches to find solutions in the complex plane. Additionally, we delve into equations of the form z^n = a, equipping students with the tools to tackle complex exponential equations efficiently.
Point Transformations: Concluding our lecture, we explore the transformative power of complex numbers in point transformations. Beginning with an exploration of transformations associated with applications in the complex plane, students will learn how complex numbers can be used to map points and shapes onto the complex plane. Additionally, we delve into the realm of usual transformations, including translations, rotations, dilations, and reflections, showcasing the versatility and applicability of complex numbers in geometric transformations.

Delve into the foundational concepts of complex numbers, including their definition, properties, and representations, such as the geometric, trigonometric, and exponential forms. Unravel the mysteries of solving second-degree equations and mastering techniques like De Moivre’s theorem, all while gaining insights into practical applications in point transformations and beyond. Whether you’re a high school student navigating the intricacies of algebra or a math enthusiast delving into advanced concepts, our platform provides invaluable resources and guidance. Discover the power and versatility of complex numbers in mathematics and beyond, from fundamental algebraic manipulation to complex polar and standard forms.

Course Outline
1 Set of complex numbers
1.1 Definition of complex numbers
1.2 Properties of complex numbers
2 Representations of a complex number
2.1 Geometric representation of a complex number
2.2 Trigonometric form of a complex number
2.3 Exponential form of a complex number
3 Solving equations in C
3.1 Second degree equation
3.2 Equation of type z^n = a
4 Point transformations
4.1 Transformation associated to an application in C
4.2 Usual transformations

13 Differential equations

PDF, Video of Math lesson on « Differential equations » for final High school year.

In this lecture on « Differential Equations, » we will delve into the fundamental concepts and applications of these mathematical tools. Differential equations play a crucial role in modeling various phenomena across diverse fields, from physics to economics. Throughout this lecture, we will explore the key principles and techniques involved in solving differential equations, starting from their basic definitions to tackling first-order equations with varying complexities.

Differential Equations: Differential equations serve as a cornerstone in understanding dynamic systems and their behaviors. We will begin by defining what constitutes a differential equation. Through illustrative examples, we will elucidate the diverse forms and applications of these equations, paving the way for deeper exploration.
First Order Differential Equations: Within this section, we will focus on first-order differential equations, which are fundamental in many areas of science and engineering. We will first address homogeneous equations of the form y′ = ay, exploring their solutions and implications. Following this, we will tackle equations with constant second members, y′ = ay + b, analyzing how the inclusion of a constant term alters the behavior of the system. Finally, we will extend our understanding to equations with general second members, y′ = ay + g(x), where the second member is a function of the independent variable, presenting methods to solve such equations efficiently.

To enhance your mastery of differential equations, our lesson covers topics such as solving differential equations, understanding differential equation homogeneity, and tackling linear equations. Delve into detailed explanations on the degree and order of differential equations, and explore practical examples elucidating concepts like exact equations and first-order linear equations. Whether you’re seeking clarity on elementary differential equations or guidance on finding general solutions, our course offers a wealth of insights tailored to cater to your learning journey.

Course Outline:
1 Differential equations
1.1 Definition of a differential equation
1.2 Differential equation y′ = g
2 First order differential equations
2.1 Homogeneous equation y′ = ay
2.2 Equation with constant second member y′ = ay + b
2.3 Equation with general second member y′ = ay + g (x)

14 Vectors in plane and space

PDF of Math lesson on « Vectors in plane and space » for final High school year.

This lecture on « Vectors in Plane and Space, » is designed to equip high school students with a solid understanding of vector concepts and their applications in two-dimensional and three-dimensional spaces. Vectors play a crucial role in various fields, including physics, engineering, and computer science. Through this structured course, students will explore the fundamental principles of vectors, ranging from their construction to their utilization in defining lines and coordinate systems in both plane and space.

Vectors in Plane and Space: This section serves as the foundation for understanding vectors, beginning with the construction of vectors and distinguishing between linearly independent and dependent vectors. Students will grasp the concept of direction vectors of lines and planes.
Vector Plane: Delving into the application of vectors in the plane, this section elucidates the notion of a vector plane, exploring its bases and coordinates. Students will learn how to express lines in the plane using both Cartesian and parametric equations, gaining proficiency in representing geometric entities with vector notation.
Vector Space: Extending the discussion to three-dimensional space, this section introduces vectors in space, including bases and coordinates in three dimensions. Students will delve into the Cartesian and parametric equations of lines in space, enhancing their ability to analyze spatial relationships and geometric configurations.
Orthonormal Basis and Coordinate System: Building upon the concepts introduced in earlier sections, this segment explores the significance of orthonormal bases and coordinate systems in vector spaces. Students will understand the importance of establishing a consistent framework for representing vectors and geometric objects accurately.
Affine or Vector Plane and Space: Concluding the lecture, this section delves into the concept of affine or vector planes and spaces, offering insights into their geometric properties and applications. Students will explore how affine spaces extend the notion of vector spaces, providing a broader perspective on spatial relationships and transformations.

For in-depth insights into vectors and their applications in both plane and space, our lesson offers a comprehensive resource tailored to meet the educational needs of high school students. Explore fundamental concepts such as vector construction, direction vectors, and the linear dependence and independence of vectors. With a focus on clarity and practicality, we delve into the intricacies of vector spaces and vector planes, providing clear explanations and illustrative examples to enhance understanding. Whether you’re seeking to clarify the definition of vector space or understand the relationship between vectors and planes, our webpage offers a wealth of resources to support your learning journey in linear algebra and beyond.

Course Outline:
1 Vectors in plane and space
1.1 Construction of vectors
1.2 Linearly independent or dependent vectors
1.3 Direction vector
2 Vector plane
2.1 Base and coordinates in the plane
2.2 Cartesian and parametric equations of a line in the plane
3 Vector space
3.1 Base and coordinates in the space
3.2 Cartesian and parametric equations of a line in the space
4 Orthonormal basis and coordinate system
5 Affine or vector plane and space

15 Scalar product of vectors

PDF of Math lesson on « Scalar product of vectors » for final High school year.

Welcome to our lecture on the scalar product of vectors, a fundamental concept in both mathematics and physics. Understanding the scalar product not only aids in solving problems involving vectors but also provides insights into various geometrical phenomena. In this lecture, we will delve into the intricacies of the scalar product, exploring its applications in the plane and in three-dimensional space.

Scalar Product in the Plane and in Space: In this section, we will begin by defining the scalar product and its computation from vector coordinates. We’ll explore how the scalar product encapsulates the notion of angle between vectors and investigate its properties, including commutativity, distributivity, and the orthogonality condition. Remarkable identities derived from the scalar product will be presented, shedding light on its algebraic properties. Moreover, we’ll examine the geometric interpretation of the scalar product, elucidating its role in determining projections and magnitudes of vectors in both the plane and in space.
Normal Vector: The concept of a normal vector plays a pivotal role in geometry and physics, particularly in the study of lines and planes. In this section, we’ll delve into the notion of a vector normal to a line in the plane and extend our understanding to vectors normal to planes in three-dimensional space. We’ll explore the relationship between normal vectors and the equations defining lines and planes, elucidating how normal vectors provide crucial information about orientation and directionality in geometric contexts.
Intersections of Lines and Planes: The intersection of lines and planes serves as a fundamental topic in analytical geometry with numerous real-world applications. In this final section, we will examine the intersection of two planes, investigating the conditions under which they intersect and determining the nature of their intersection. Additionally, we’ll explore the intersection of a line and a plane, unraveling the geometric insights behind such intersections and providing methods for determining their coordinates and characteristics.

For students seeking comprehensive resources on the scalar product of vectors, our lecture provides an in-depth exploration of this fundamental concept. Delving into the intricacies of scalar dot products, inner products, and their properties, our content offers clarity and depth for high school learners. From understanding the multiplication of vectors by scalars to unraveling the geometric implications of scalar products, our lecture equips students with the tools to excel in linear algebra and analytical geometry. Whether you’re searching to find the scalar product of two vectors or seeking to comprehend the inner product of vectors, our webpage serves as a valuable resource, offering practical examples and problem-solving strategies to enhance your understanding.

Course Outline:
1 Scalar product in the plane and in space
1.1 Scalar product from coordinates
1.2 Properties of the scalar product
1.3 Remarkable identities of the scalar product
1.4 Relation of the scalar product to geometry
2 Normal vector
2.1 Vector normal to a line in the plane
2.2 Vector normal to a plane in space
3 Intersections of lines and planes
3.1 Intersection of two planes
3.2 Intersection of a line and a plane

16 Arithmetic

PDF of Math lesson on « Arithmetic » for final High school year.

Welcome to the lecture on Arithmetic, designed for final year High School students. This lecture aims to provide a deep understanding of fundamental concepts in number theory, particularly focusing on divisibility, greatest common divisor (GCD), and coprime numbers.

Divisibility in Z: In this section, we delve into the foundational concept of divisibility within the set of integers. We explore the definition and properties of divisibility, understanding how one integer can divide another without leaving a remainder. Euclidean division, a crucial algorithm for finding quotients and remainders efficiently, will be thoroughly explained. Additionally, we’ll discuss prime numbers, the building blocks of the integers, and examine their significance in number theory. Congruence, which relates to divisibility in a modular arithmetic context, will also be covered, providing students with a holistic understanding of divisibility within Z.
Greatest Common Divisor: Moving forward, we focus on the concept of the greatest common divisor (GCD), a fundamental tool in number theory. Students will learn how to compute the GCD of two integers and understand its significance in various mathematical contexts. Euclid’s theorem and algorithm for finding the GCD, dating back to ancient Greece, will be elucidated, providing insight into the timeless nature of mathematical principles.
Coprime Numbers: In the final section, we explore coprime numbers, which play a crucial role in many areas of mathematics, including cryptography and number theory. Students will understand the concept of coprimality and its implications. Identity and Bézout’s theorem will be discussed, shedding light on the relationship between coprime numbers and integer linear combinations. Gauss theorem, offering a powerful criterion for determining coprimality, will also be covered. Finally, students will be introduced to the Bézout-Euclide algorithm, which efficiently computes coefficients for Bézout’s theorem, empowering them with practical tools for solving problems involving coprime numbers.

From understanding divisibility and prime numbers to mastering the Euclidean division algorithm and exploring the significance of coprime numbers, this lecture covers essential topics in number theory essential for high school mathematics. Delve into the nuances of arithmetic meaning, the beauty of congruence in math, and the power of the greatest common divisor (GCD). Discover the elegance of Euclid’s division lemma and gain insights into basic arithmetic principles. Whether you’re seeking to define arithmetic, unravel the mysteries of GCD in maths, or grasp the significance of congruence in mathematics, our lecture provides a holistic understanding of these fundamental concepts.

Course Outline:
1 Divisibility in Z
1.1 Definition and properties of divisibility
1.2 Euclidean division
1.3 Prime numbers
1.4 Congruence
2 Greatest common divisor
2.1 GCD of 2 integers
2.2 Euclid’s theorem and algorithm
3 Coprime numbers
3.1 Identity and Bézout’s theorem
3.2 Gauss theorem
3.3 Bézout-Euclide algorithm

17 Enumeration and Combinatorics

PDF of Math lesson on « Enumeration and Combinatorics » for final High school year.

Divisibility in Z: In this section, we delve into the foundational concept of divisibility within the set of integers. We explore the definition and properties of divisibility, understanding how one integer can divide another without leaving a remainder. Euclidean division, a crucial algorithm for finding quotients and remainders efficiently, will be thoroughly explained. Additionally, we’ll discuss prime numbers, the building blocks of the integers, and examine their significance in number theory. Congruence, which relates to divisibility in a modular arithmetic context, will also be covered, providing students with a holistic understanding of divisibility within Z.
Greatest Common Divisor: Moving forward, we focus on the concept of the greatest common divisor (GCD), a fundamental tool in number theory. Students will learn how to compute the GCD of two integers and understand its significance in various mathematical contexts. Euclid’s theorem and algorithm for finding the GCD, dating back to ancient Greece, will be elucidated, providing insight into the timeless nature of mathematical principles.
Coprime Numbers: In the final section, we explore coprime numbers, which play a crucial role in many areas of mathematics, including cryptography and number theory. Students will understand the concept of coprimality and its implications. Identity and Bézout’s theorem will be discussed, shedding light on the relationship between coprime numbers and integer linear combinations. Gauss theorem, offering a powerful criterion for determining coprimality, will also be covered. Finally, students will be introduced to the Bézout-Euclide algorithm, which efficiently computes coefficients for Bézout’s theorem, empowering them with practical tools for solving problems involving coprime numbers.

From exploring enumeration techniques such as arrangements, permutations, and combinations to unraveling the mysteries behind Pascal’s triangle formula and the properties of binomial coefficients, this lecture equips high school students with essential tools for tackling complex counting problems. Discover the elegance of Newton’s binomial formula and its applications in expanding binomials to any power. Whether you’re intrigued by the intricacies of permutations, fascinated by the patterns in Pascal’s triangle, or seeking to understand the algebraic foundations of combinatorial mathematics, our lecture provides a thorough exploration of these fundamental concepts.

Course Outline:
1 Enumeration cases
1.1 Lists
1.2 Arrangements
1.3 Combinaisons
1.4 Permutations
1.5 Summary table: order and repetition
2 Properties of binomial coefficients
2.1 Pascal’s triangle formula
2.2 Newton’s binomial formula

18 Matrices

PDF of Math lesson on « Matrices » for final High school year.

This lecture serves as an introduction to the fundamental concepts and operations involving matrices, providing students with a solid understanding of their properties and applications in various mathematical contexts.

Introduction to Matrices: In this foundational section, students will be introduced to the concept of matrices and their basic properties. They will learn about the structure of matrices, addition of matrices, multiplication of a matrix by a scalar, and the product of matrices.
Power of a Square Matrix: Moving forward, students will learn how to raise a square matrix to a given power, uncovering the underlying patterns and properties. Special attention will be given to diagonal matrices and diagonalization, providing students with insights into the role of eigenvalues and eigenvectors in matrix algebra.
Linear Systems: In this section, students will explore the application of matrices in solving linear systems of equations. They will learn how to represent a system of linear equations using matrix notation and apply matrix operations to solve these systems efficiently.
Matrix Associated with a Linear Application: Concluding the lecture, students will delve into the concept of a matrix associated with a linear application. They will understand how matrices can represent linear transformations and mappings between vector spaces.

From understanding the basics of matrix operations such as matrix multiplication and addition to exploring advanced concepts like the inverse of a matrix, diagonalization, and elementary matrices, our lecture equips high school students with essential tools for mastering matrix algebra. Learn how to efficiently multiply matrices, find the inverse of a 2×2 matrix, and analyze the properties of diagonal matrices.

Course Outline:
1 Introduction to matrices
1.1 Definition of matrices and first properties
1.2 Addition of matrices
1.3 Multiplying a matrix by a number
1.4 Product of matrices
1.5 Inverse of a matrix
2 Power of a square matrix
2.1 Definition and properties of the power of a matrix
2.2 Diagonal matrix
2.3 Diagonalization
3 Linear systems
4 Matrix associated with a linear application

19 Logic and sets

PDF of Math lesson on « Logic and sets » for final High school year.

This lecture on Logic and Sets aims to provide students with a solid foundation in both logic and set theory, essential branches of mathematics that underpin various fields of study.

Basics of Logic: In this foundational section, students will explore the fundamental concepts of logic, starting with assertions and predicates. They will learn how to formulate statements and determine their truth values. Moving on, students will delve into logical connectors such as conjunction, disjunction, and negation, understanding how these connectives operate to form compound statements. Furthermore, students will be introduced to logical quantifiers, including existential and universal quantifiers, and learn how to express statements about sets using quantified expressions.
Basics of Set Theory: Transitioning to set theory, students will learn about the concept of sets, which represent collections of objects, and explore various types of sets, including empty sets, finite sets, and infinite sets. Next, students will delve into operations on sets, such as union, intersection, and complement, gaining insights into how these operations manipulate set elements. Finally, students will explore finite sets, learning how to count elements and analyze the cardinality of sets.

From mastering the basics of logic, including assertions, predicates, and logical connectors, to exploring the intricacies of set theory, such as operations on sets and finite sets, our lecture equips high school students with essential tools for understanding mathematical reasoning and problem-solving. Learn about the logic behind mathematical operations, such as set union and intersection, and discover how these concepts intertwine to form the backbone of mathematical logic.

Course Outline:
1 Basics of logic
1.1 Assertions and predicates
1.2 Logical connectors
1.3 Logical quantifiers
2 Basics of set theory
2.1 Sets
2.2 Operations on sets
2.3 Finite sets

20 Isometries of the plane

PDF of Math lesson on « Isometries of the plane » for final High school year.

This lecture on « Isometries of the Plane, » presents transformations that preserve distances and shapes in the plane. Isometries play a crucial role in geometry, offering insights into symmetry, composition, and the fundamental properties of geometric objects.

Definition and Properties of an Isometry: In this section, we lay the groundwork by defining what constitutes an isometry of the plane and exploring its key properties. We begin with a precise definition of isometries and examine how they interact with the scalar product, shedding light on their behavior in relation to distance preservation. Additionally, we investigate the inverse of an isometry and delve into methods for characterizing these transformative operations.
Composition of Isometries: Building upon the foundation laid in the previous section, we delve into the composition of isometries, exploring the general properties that govern their combination. Specifically, we investigate the intriguing dynamics involved in composing two orthogonal symmetries, unraveling the intricate interplay between different types of transformations in the plane.
Isometries and Invariant Points: In this final section, we explore the concept of invariant points under isometries, examining the scenarios where certain points remain fixed despite the transformative operations. We classify isometries based on their behavior towards invariant points, distinguishing between those that possess invariant points and those that do not. Furthermore, we analyze isometries as compositions of symmetries, illuminating the underlying symmetrical structures embedded within these transformative processes.

Delving into the fundamental concepts of geometry and symmetry, our lecture explores the properties and applications of isometries, including rotations and geometric symmetry. From understanding the definition and properties of isometries to exploring their compositions and invariant points, this lecture provides a thorough examination of these transformative operations. Whether you’re seeking insights into math rotations, mathematical symmetry, or geometry rotation, our lesson offers valuable resources to deepen your understanding.

Course Outline:
1 Definition and properties of an isometry
1.1 Definition of an isometry of the plane
1.2 Isometries and scalar product
1.3 Inverse of an isometry
1.4 Characterization of an isometry
2 Composition of isometries
2.1 General properties of isometries composition
2.2 Composition of two orthogonal symmetries
3 Isometries and invariant points
3.1 Isometries having invariant points
3.2 Isometries having no invariant point
3.3 Isometries as compositions of symmetries

21 Displacements and antidisplacements

PDF of Math lesson on « Displacements and antidisplacements » for final High school year.

In this lecture on « Displacements and Antidisplacements, » we explore the fundamental concepts of transformations in geometry, focusing on the preservation of distances and orientations.

Definition and Properties: In this section, we lay the groundwork by defining what constitutes a displacement or an antidisplacement and examining their key properties. We delve into the classification of isometries, providing a framework for understanding the different types of transformations that preserve distances. Furthermore, we explore the characterization of displacements and antidisplacements, elucidating the distinguishing features that set them apart from each other.
Displacements: Building upon the foundation established in the previous section, we delve deeper into the realm of displacements. Here, we investigate the angle of a displacement, shedding light on how these transformations affect the orientation of geometric objects in the plane. Additionally, we explore the composition of displacements, uncovering the rules governing the combination of multiple displacement operations. Furthermore, we examine various methods for representing displacements, particularly focusing on their representation in complex numbers.
Antidisplacements: Concluding our exploration, we turn our attention to antidisplacements, which are transformations that reverse the orientation of geometric objects while preserving distances. We delve into the unique properties of antidisplacements, contrasting them with displacements. Through detailed analysis, we aim to provide a comprehensive understanding of these transformative operations and their significance in geometric transformations.

Through detailed analysis, we examine the properties and classifications of displacements and antidisplacements, shedding light on their role in preserving geometric structures. We delve into the intricate dynamics of composition of transformations, exploring how complex numbers provide a powerful framework for representing these operations. Furthermore, we investigate the angle of displacement, uncovering its significance in understanding the orientation-preserving properties of certain transformations.

Course Outline:
1 Definition and properties
1.1 Classification of isometries
1.2 Characterization of a displacement or an antidisplacement
2 Displacements
2.1 Angle of a displacement
2.2 Composition of displacements
2.3 Representation of a displacement in C
3 Antidisplacements

22 Similarities

PDF of Math lesson on « Similarities » for final High school year.

In this lecture on « Similarities, » we explore fundamental concepts in geometry related to transformations that preserve shape but not necessarily size or orientation. We’ll delve into the fascinating world of similarities, examining both direct and indirect transformations and their characteristic elements.

Introduction to Similarities: In this section, we lay the groundwork by introducing the basic concepts of similarities. We begin by exploring the fundamental transformations of homotheties and translations, which serve as building blocks for understanding similarities. Next, we define similarities as transformations that preserve shape while possibly altering size or orientation. Additionally, we distinguish between direct and indirect similarities, offering insights into the different types of transformations that fall under the umbrella of similarities.
Direct Similarities: Building upon the foundation established in the previous section, we delve deeper into direct similarities. Here, we examine the characteristic elements that define a direct similarity, including the preservation of shape and orientation. We explore the reduced form of direct similarities and their representation in the complex plane (C), providing students with a comprehensive understanding of how these transformations operate geometrically and algebraically.
Indirect Similarities: Concluding our exploration, we turn our attention to indirect similarities, which involve transformations that preserve shape but reverse orientation. We delve into the unique properties of indirect similarities and contrast them with direct similarities, highlighting their distinct characteristics and applications in geometry. Through detailed analysis and illustrative examples, students will gain a thorough understanding of both direct and indirect similarities, laying a solid foundation for further exploration in geometry and related fields.

Through detailed analysis, we examine how similarities preserve shape while potentially altering size or orientation, highlighting the importance of shape preservation, size preservation, and orientation preservation in geometric transformations. From exploring characteristic elements of direct similarities to understanding their representation in the complex plane (C), we provide a thorough examination of these transformative operations.

Course Outline:
1 Introduction to Similarities
1.1 Homotheties and translations
1.2 Definition of similarities
1.3 Direct or indirect similarities
2 Direct similarities
2.1 Characteristic elements of a direct similarity
2.2 Reduced form and representation in C
3 Indirect similarities