This course provides a rigorous foundation in the concept of functions, the building block of calculus. We begin with the real number system - rationals, irrationals, intervals, and infinity - before defining real-valued functions, their domains, and ranges. You will systematically study all key function families: polynomial, rational, algebraic, piecewise-defined, and transcendental, including exponential, logarithmic, trigonometric, and hyperbolic types. A command of functions is essential for all technical and quantitative disciplines. These principles are directly applied in physics to model motion, in economics for supply-demand curves, in engineering for signal processing, and in computer science to analyse algorithm complexity. This course delivers the core mathematical tools required for effective modelling and problem-solving in these fields and beyond. By the end of this course, you will be able to: calculate the domain and range of any given function; classify functions and identify their fundamental properties; analyse functions for key characteristics like symmetry (odd and even); and accurately construct graphs for a wide range of standard and piecewise-defined functions directly from their equations. This course is designed for first-year undergraduates in science, technology, engineering, mathematics (STEM), and economics. It is also critical for any student preparing for advanced calculus or for professionals who require a solid mathematical base for roles in finance, data science, or technical research.

This course provides a rigorous introduction to the concept of the limit, the theoretical bedrock upon which all of calculus is built. We will move from an intuitive understanding of what it means for a function to 'approach' a value to the algebraic techniques required for precise evaluation. The course covers the limit laws, methods for handling indeterminate forms, and the behavior of functions at infinity. Understanding limits is non-negotiable for any serious study of calculus. The principles developed here are not merely abstract; they are the tools used to formally define the core concepts of calculus that are used to model the mechanics of change that govern engineering, physics, economics, and computer science. By the end of this course, you will be able to: evaluate limits graphically, numerically, and algebraically; apply the Squeeze Theorem and L’Hôpital’s Rule; analyse the end-behavior of functions; and identify vertical and horizontal asymptotes. This course is designed for first-year undergraduates in STEM fields who are beginning their calculus sequence. It is an essential prerequisite for subsequent courses on continuity and differentiability and is also invaluable for any student or professional seeking to rebuild their mathematical foundation from first principles.

This course provides a focused and rigorous examination of continuity, a fundamental property of functions that underpins the whole of differential and integral calculus. We move from an intuitive understanding of "unbroken" graphs to the formal, limit-based definition of continuity. The course systematically explores the different types of discontinuities and the powerful theorems that apply only to continuous functions. A command of continuity is essential for understanding why calculus works. This concept ensures that functions are predictable and well-behaved, a necessary condition for modeling real-world phenomena and for the validity of the major theorems of calculus. It is the bridge between the concept of a limit and the concept of a derivative, explaining why a function must be continuous to be differentiable. By the end of this course, you will be able to use the three-part definition to test for continuity at a point, identify and classify removable, jump, and infinite discontinuities, and apply the Intermediate Value Theorem and the Extreme Value Theorem to analyse function behavior on a closed interval. This course is designed for first-year undergraduates in science, technology, engineering, and mathematics who have completed a course on limits. It is a critical prerequisite for the study of differentiability and is essential for any student seeking a deep understanding of calculus theory.

This course provides a rigorous, theoretical introduction to differential calculus. We will move beyond computation to understand the fundamental nature of the derivative, beginning with its formal definition as the limit of a difference quotient. The curriculum is designed to build a deep conceptual understanding of what it means for a function to be differentiable and the profound consequences of this property. The primary focus is on formal proofs and theoretical results. We will systematically derive the rules of differentiation, including the product, quotient, and chain rules, directly from first principles. This course emphasizes the "why" behind the mechanics of calculus, establishing the logical framework upon which all applications are built, including the cornerstone theorems of Rolle, the Mean Value Theorem, and Taylor's Theorem. By the end of this course, you will be able to: explain the derivative from its limit definition; prove the relationship between differentiability and continuity; formally derive all major differentiation rules; and understand the theoretical significance of the Mean Value Theorem and Taylor's Theorem. This course is designed for first-year undergraduates in mathematics, physics, and engineering who require a deep theoretical foundation for their studies. It is the ideal precursor to subsequent courses on differentiation techniques and applications, providing the essential logical underpinnings for those more practical subjects.

Do you want to learn how to work with infinite sums of numbers and functions and their properties, operations, and applications? Do you want to understand the concepts of progressions, sequences, series, and power series, and how they relate to the approximation, convergence, and divergence of functions? Do you want to master the skills of finding and applying various tests and criteria of convergence and divergence to different types of series, such as arithmetic, geometric, harmonic, p-series, and alternating series? If you answered yes to any of these questions, then this course is for you! This course reviews the fundamental concepts of finite progressions and provides a thorough treatise of the convergence of infinite real sequences and series. You will learn how to: - Define and classify progressions and their properties, such as common difference, common ratio, and sum to infinity - Define and classify sequences and their properties, such as terms, general term, and boundedness - Find the convergence or divergence of a sequence using the formal definition and various theorems and examples - Define and classify series and their properties, such as partial sums, absolute and conditional convergence, and uniform convergence - Find the convergence or divergence of a series of real numbers using various methods and techniques, such as direct comparison, integral test, p-series test, ratio test, Raabe's test, and alternating series test - Define and classify sequences and series of functions and their properties, such as pointwise and uniform convergence, and term-by-term differentiation and integration - Find the convergence or divergence of a sequence or series of functions using various methods and techniques, such as Weierstrass M-test, Abel's test, and Dirichlet's test - Define and classify power series and their properties, such as radius and interval of convergence, and term-by-term differentiation and integration - Find the convergence or divergence of a power series using various methods and techniques, such as ratio test, root test, and Cauchy-Hadamard theorem This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of this course, you will have a solid foundation of the theory and practice of calculus and its operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve infinite sums of numbers and functions and their approximation, convergence, and divergence. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

This course introduces multi-variable real-valued functions and thoroughly addresses their limits, continuity, partial and total derivatives, applications and related concepts. You will learn how to: - Define and classify real-valued functions of several variables and their properties, such as domain, range, graphs, level curves and surfaces - Find the limit of a function of several variables as the independent variables approach certain values, and use the formal and informal definitions of limits - Find the continuity of a function of several variables at a point or on a set, and use the graphical, formal, and informal definitions of continuity - Find the partial derivative of a function of several variables with respect to one of the independent variables, and use the formal definition and various rules of partial differentiation - Find the higher-order partial derivatives of a function of several variables by applying the partial differentiation rules repeatedly, and use the notation and terminology for higher partial derivatives - Find the partial derivative of a function of several variables that is composed of other functions, and use the chain rule to differentiate composite functions - Find the partial derivative of a function of several variables that is homogeneous, and use the Euler's theorem to simplify the calculation - Find the total differential and derivative of a function of several variables, and use them to approximate the change in the function value - Find the partial derivative of a function of several variables that is defined implicitly by an equation - Find the Jacobian determinant of a function of several variables, and use it to obtain derivatives of implicit functions - Find the Taylor series expansion of a function of several variables, and use it to approximate the function value and its derivatives - Find the equation of the tangent plane and the normal line to a surface defined by a function of two variables, and use them to analyze the local behaviour of the surface - Find the extreme values of a function of several variables, and use the partial derivatives and the second derivative test to determine the nature of the extrema - Find the extreme values of a function of several variables subject to some constraint, and use the partial derivatives and the method of Lagrange multipliers to solve the constrained optimization problem This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of this course, you will have a solid understanding of derivatives of functions of several variables. You will also be able to apply the knowledge and skills you gain to real-world problems and challenges that involve functions of several variables and their derivatives. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

Numerical solution of non-linear equations, integration.

A comprehensive treatise of ordinary differential equations covering solutions of first-order and second-order ordinary differential equations, and applications of first-order ordinary differential equations.