This course comprehensively treats limits, continuity and differentiability of real-valued single-variable functions. You will learn how to: - Define and classify real-valued functions and their properties, such as domain, range, and graphs - Identify and compare different kinds of real-valued functions, such as composites, piecewise functions, polynomials, rational functions, algebraic functions, transcendental functions, odd and even functions - Find the limit of a function as the independent variable approaches a certain value, and use the formal (epsilon-delta) and informal definitions of limits - Find the limit of a function using various methods and techniques, such as direct substitution, theorems, graphing, factorization, conjugates, L’Hôpital’s rule, use of known special limits, the squeeze theorem, two-sided limits, etc. - Find the limit of a function at infinity or an infinite limit, and use the formal and informal definitions of limits at infinity and infinite limits - Find the continuity of a function at a point or on an interval, and use the graphical, formal, and informal definitions of continuity - Identify and understand different types of discontinuities, such as removable, jump, and infinite discontinuities - Apply the max-min theorem and the intermediate-value theorem to continuous functions and their graphs - Find the differentiability of a function at a point or on an interval, and use the graphical and formal definitions of differentiability - Find the derivative of a function as the slope of the tangent line or the rate of change of the function, and use the first-principle method and the rules of differentiation to calculate derivatives - Apply the mean-value theorem and the Rolle's theorem to differentiable functions and their graphs - Find the higher-order derivatives of a function by applying the differentiation rules repeatedly, and use the notation and terminology for higher derivatives - Apply the Leibniz's formula to find the derivatives of products of functions - Find the Taylor and Maclaurin series expansions of differentiable functions and use them to approximate functions and their values 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 the course, you will have a firm grasp of advanced single-variable differential calculus concepts and their applications. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve analyzing the behavior, trends, and optimization of functions and their graphs. 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.

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.