Calculus is the branch of mathematics which is concerned with the study of rates of change. It is further divided in two branches, namely, differential calculus and integral calculus. Differential calculus studies the rate of changes of functions with respect to their variables through the use of derivatives and differentials. Slopes and curves of graphs are a part of differential calculus. Integral calculus focuses on accumulation of quantities and studies the area under and between the curves. Calculus finds its applications across various domains such as robotics, video games, artificial intelligence, planetary motion, aerospace, bridge engineering, business and economics. This book unfolds the innovative aspects of calculus which will be crucial for the holistic understanding of the subject matter. Some of the diverse topics covered herein address the varied branches that fall under this category. Those in search of information to further their knowledge will be greatly assisted by this book. Calculus is an extremely powerful tool for solving a host of practical problems in fields as diverse as physics, biology, and economics, to mention just a few. In this rigorous but accessible text, a noted mathematician introduces undergraduate-level students to the problem-solving techniques that make a working knowledge of calculus indispensable for any mathematician. The author first applies the necessary mathematical background, including sets, inequalities, absolute value, mathematical induction, and other "precalculus" material. The text begins with the actual study of differential calculus with a discussion of the key concept of function, and a thorough treatment of derivatives and limits. Differentiation is used as a tool; among the topics covered here are velocity, continuous and differentiable functions, the indefinite integral, local extrema, and concrete optimization problems. The book treats integral calculus, employing the standard definition of the Riemann integral, and deals with the mean value theorem for integrals, the main techniques of integration, and improper integrals. It offers a brief introduction to differential equations and their applications, including problems of growth, decay, and motion. Concise and well written, this text is ideal as a primary text or as a refresher for anyone wishing to review the fundamentals of this crucial discipline. This book offers a concise approach to calculus that focuses on major concepts, and supports those concepts with precise definitions, patient explanations, and carefully graded problems.
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