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Since January 2024
Instructor since January 2024
This course is designed for students who want to learn complex numbers from scratch in a clear, structured way with practical applications.
We will begin with the historical motivation for the topic, understanding why complex numbers emerged and how mathematicians such as Cardano, Bombelli, Euler, and Gauss contributed to their development.
Throughout the course you will learn:
What is the imaginary unit?
How to write complex numbers in form
Basic operations: addition, subtraction, multiplication, and division
Conjugate and modulus of a complex number
Representation on the complex plane
Trigonometric form and polar form
Euler's formula
Powers and roots of complex numbers
In addition, practical applications will be presented in areas such as physics, engineering, and telecommunications, showing the real importance of these concepts.
The course is geared towards step-by-step teaching, with solved examples and guided exercises, ideal for secondary school students, university students or anyone interested in strengthening their mathematical foundations.
We will begin with the historical motivation for the topic, understanding why complex numbers emerged and how mathematicians such as Cardano, Bombelli, Euler, and Gauss contributed to their development.
Throughout the course you will learn:
What is the imaginary unit?
How to write complex numbers in form
Basic operations: addition, subtraction, multiplication, and division
Conjugate and modulus of a complex number
Representation on the complex plane
Trigonometric form and polar form
Euler's formula
Powers and roots of complex numbers
In addition, practical applications will be presented in areas such as physics, engineering, and telecommunications, showing the real importance of these concepts.
The course is geared towards step-by-step teaching, with solved examples and guided exercises, ideal for secondary school students, university students or anyone interested in strengthening their mathematical foundations.
My name is Miguel Angel, and I offer tutoring and academic support in Mathematics and Physics. I'd be happy to help you with any courses you need in these subjects. I have over 25 years of experience preparing students of all levels.
Start of the Integration Course from scratch for students and teachers.
Course: Integration From Scratch
General Objective:
Guide the student from the basics to a solid mastery of indefinite and definite integrals, techniques, and applications.
Module 1: Previous Foundations and Motivation
Goals:
Understand the concept of area under the curve as a motivation for the integral.
Establish a basis with functions and derivatives.
Contents:
1. What is integration? (visual intuition)
2. Review of functions (linear, quadratic, exponential)
3. Derivative and its relationship with the integral
4. Introduction to the integration symbol
Exercises:
Identify areas under simple curves (visually).
Relate known derivatives with antiderivatives.
Module 2: Indefinite Integrals
Goals:
Learn to calculate antiderivatives of common functions.
Contents:
1. Definition of indefinite integral
2. Integration rules:
Constant
Powers
Addition/subtraction
3. Integration by substitution
4. Integration by parts (simple introduction)
Exercises:
15 graded exercises with feedback.
Module 3: Definite Integrals and Fundamental Theorem
Goals:
Calculate exact areas under curves using definite integrals.
Understand and apply the fundamental theorem of calculus.
Contents:
1. Definition of definite integral
2. Geometric interpretation
3. Properties (linearity, additivity)
4. Fundamental Theorem of Calculus
Exercises:
10 visual exercises + 10 algebraic exercises.
Module 4: Integration Techniques
Goals:
Use advanced techniques to integrate more complex functions.
Contents:
1. Trigonometric substitution
2. Partial fractions
3. Integration by parts (formal)
4. Numerical methods: trapezoidal rule
Exercises:
20 problems per technique, with guided solutions and free practice.
Module 5: Applications
Goals:
Apply integration to real-world problems.
Contents:
1. Calculation of areas between curves
2. Volume of solids of revolution
3. Work and energy
4. Physical and economic problems
Exercises:
10 practical problems with real-life context.
Module 6: Evaluation and Final Challenges
Goals:
Measure learning and reinforce skills.
Contents:
1. Final multiple-choice exam and development
2. Level-based challenges
3. Corrections in video or PDF
Additional resources:
Formula notebook
Bank of integrals
Explanatory videos
Downloadable PDFs with theory and exercises
Course: Integration From Scratch
General Objective:
Guide the student from the basics to a solid mastery of indefinite and definite integrals, techniques, and applications.
Module 1: Previous Foundations and Motivation
Goals:
Understand the concept of area under the curve as a motivation for the integral.
Establish a basis with functions and derivatives.
Contents:
1. What is integration? (visual intuition)
2. Review of functions (linear, quadratic, exponential)
3. Derivative and its relationship with the integral
4. Introduction to the integration symbol
Exercises:
Identify areas under simple curves (visually).
Relate known derivatives with antiderivatives.
Module 2: Indefinite Integrals
Goals:
Learn to calculate antiderivatives of common functions.
Contents:
1. Definition of indefinite integral
2. Integration rules:
Constant
Powers
Addition/subtraction
3. Integration by substitution
4. Integration by parts (simple introduction)
Exercises:
15 graded exercises with feedback.
Module 3: Definite Integrals and Fundamental Theorem
Goals:
Calculate exact areas under curves using definite integrals.
Understand and apply the fundamental theorem of calculus.
Contents:
1. Definition of definite integral
2. Geometric interpretation
3. Properties (linearity, additivity)
4. Fundamental Theorem of Calculus
Exercises:
10 visual exercises + 10 algebraic exercises.
Module 4: Integration Techniques
Goals:
Use advanced techniques to integrate more complex functions.
Contents:
1. Trigonometric substitution
2. Partial fractions
3. Integration by parts (formal)
4. Numerical methods: trapezoidal rule
Exercises:
20 problems per technique, with guided solutions and free practice.
Module 5: Applications
Goals:
Apply integration to real-world problems.
Contents:
1. Calculation of areas between curves
2. Volume of solids of revolution
3. Work and energy
4. Physical and economic problems
Exercises:
10 practical problems with real-life context.
Module 6: Evaluation and Final Challenges
Goals:
Measure learning and reinforce skills.
Contents:
1. Final multiple-choice exam and development
2. Level-based challenges
3. Corrections in video or PDF
Additional resources:
Formula notebook
Bank of integrals
Explanatory videos
Downloadable PDFs with theory and exercises
I'm Miguel Ángel, a Master of Science in Mathematics with extensive teaching experience. I provide personalized tutoring and tutoring classes in Mathematics and Physics, both online and in-person, tailored to any academic level.
I specialize in helping students understand everything from basic to advanced topics with clear explanations, ongoing support, and strategies that actually work.
What do I offer?
100% personalized advice
Support with homework, exams, and projects
Preparation for midterms and finals
Practical and theoretical classes
I work with all academic levels: secondary, pre-university, university, and graduate.
Subjects I cover
Basic Mathematics and Geometry
Differential and Integral Calculus
Linear and Nonlinear Algebra
Series and Differential Equations
Complex Variable
Numerical Mathematics
Probabilities and Statistics
Laplace, Fourier and Z transforms
Physics I, II and III
I specialize in helping students understand everything from basic to advanced topics with clear explanations, ongoing support, and strategies that actually work.
What do I offer?
100% personalized advice
Support with homework, exams, and projects
Preparation for midterms and finals
Practical and theoretical classes
I work with all academic levels: secondary, pre-university, university, and graduate.
Subjects I cover
Basic Mathematics and Geometry
Differential and Integral Calculus
Linear and Nonlinear Algebra
Series and Differential Equations
Complex Variable
Numerical Mathematics
Probabilities and Statistics
Laplace, Fourier and Z transforms
Physics I, II and III
Do you want to learn Differential Equations from scratch?
This course is for you!
What will you learn?
Basic concepts and what is a differential equation
Simple methods for solving ordinary differential equations
Practical applications in real life (physics, biology, economics)
Step-by-step exercises to help you understand everything easily
Who is it addressed to?
Students beginning advanced mathematics
Professionals who want to strengthen their foundations
Anyone interested in understanding these key tools
Why choose this course?
Clear and uncomplicated explanations
Personalized support during the course
Study material and practical exercises included
Don't wait any longer and take the first step toward mastering differential equations!
Reserve your spot now!
This course is for you!
What will you learn?
Basic concepts and what is a differential equation
Simple methods for solving ordinary differential equations
Practical applications in real life (physics, biology, economics)
Step-by-step exercises to help you understand everything easily
Who is it addressed to?
Students beginning advanced mathematics
Professionals who want to strengthen their foundations
Anyone interested in understanding these key tools
Why choose this course?
Clear and uncomplicated explanations
Personalized support during the course
Study material and practical exercises included
Don't wait any longer and take the first step toward mastering differential equations!
Reserve your spot now!
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