Differential Calculus by B. C. Das and B. N. Mukherjee is a classic textbook on differential calculus that covers various topics such as limits, continuity, differentiation, maxima and minima, mean value theorems, curve tracing, and applications of derivatives. The book was first published in 1975 by U.N. Dhur and has been revised and updated several times since then[^1^]. The book is suitable for undergraduate students of mathematics, physics, engineering, and other related fields.
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The book has 448 pages and is divided into 18 chapters. Each chapter begins with an introduction that outlines the main concepts and results to be discussed. The chapters also contain numerous solved examples that illustrate the methods and techniques of differential calculus. The book also provides exercises at the end of each chapter for practice and self-assessment. The book also includes a key to the exercises and an index for easy reference.
One of the strengths of the book is its clear and concise presentation of the theory and applications of differential calculus. The book covers both basic and advanced topics in a logical and systematic manner. The book also provides a historical perspective on the development of differential calculus and its contributions to various fields of science and technology. The book also uses simple and elegant language that makes it accessible to a wide range of readers.
One of the limitations of the book is that it does not cover some topics that are relevant to modern differential calculus, such as differential forms, vector analysis, complex analysis, and differential geometry. The book also does not provide any graphical or visual aids to complement the analytical approach. The book also assumes some prior knowledge of algebra, trigonometry, and coordinate geometry that may not be available to all readers.
Overall, Differential Calculus by Das and Mukherjee is a comprehensive and rigorous textbook on differential calculus that can serve as a valuable resource for students and teachers alike. The book provides a solid foundation for further studies in mathematics and its applications.
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In this section, we will review some of the main topics and results of differential calculus that are covered in the book by Das and Mukherjee. We will also provide some examples and applications of these topics and results.
The concept of limit is one of the fundamental notions of differential calculus. It is used to describe the behavior of a function as its argument approaches a certain value. For example, the limit of the function f(x) = x^2 as x approaches 2 is 4, which means that the values of f(x) get closer and closer to 4 as x gets closer and closer to 2. The limit of a function may or may not exist, depending on the function and the value that the argument approaches.
The concept of continuity is closely related to the concept of limit. A function is said to be continuous at a point if its value at that point is equal to its limit at that point. For example, the function f(x) = x^2 is continuous at x = 2, because f(2) = 4 and lim x->2 f(x) = 4. A function is said to be continuous on an interval if it is continuous at every point in that interval. For example, the function f(x) = x^2 is continuous on the interval [0, 3], because it is continuous at every point in that interval.
Limits and continuity are important concepts in differential calculus because they allow us to study the properties and behavior of functions without having to know their exact values. They also enable us to define other concepts such as derivatives and integrals.
The concept of differentiation is another fundamental notion of differential calculus. It is used to measure the rate of change of a function with respect to its argument. For example, the derivative of the function f(x) = x^2 with respect to x is f'(x) = 2x, which means that the rate of change of f(x) with respect to x is 2x. The derivative of a function may or may not exist, depending on the function and the point at which it is evaluated.
The concept of differentiation has many applications in various fields of science and technology. For example, differentiation can be used to find the slope of a curve, the velocity and acceleration of a moving object, the rate of growth or decay of a population, the marginal cost or revenue of a business, and so on.
Differentiation also has many rules and techniques that simplify the process of finding derivatives. For example, some of these rules are: the sum rule (the derivative of a sum is the sum of the derivatives), the product rule (the derivative of a product is the product of the first function and the derivative of the second function plus the product of the second function and the derivative of the first function), the chain rule (the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function), and so on.
One of the applications of differentiation is finding maxima and minima of functions. A maximum (or minimum) of a function is a point where the function attains its highest (or lowest) value in a given interval or domain. For example, the maximum value of 8cf37b1e13