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 as0 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.
David Austin completed his Ph.D. at the University of Utah in 1989, supervised by Ronald Stern and working in gauge theory and its applications to low-dimensional topology. He then completed a two-year visiting position at the Institute for Advanced Study on a National Science Foundation Postdoctoral Fellowship. David next joined the mathematics faculty at the University of British Columbia in 1991 and was promoted to Associate Professor in 1995. During that time, David was a Visiting Fellow for one term at St. Catherine's College, Oxford. In 1999, David moved to Grand Valley State University where he is currently a Professor of Mathematics. Over time, his mathematical interests have grown to include more applied topics including numerical linear algebra. In 2021, David was awarded the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America.
| Preface Chapter 1. Foundations of Limits Formal (ε–Δ) Definition One-Sided Limits Infinite Limits and Limits at Infinity Continuity of Functions Intermediate Value Theorem (IVT) Chapter 2. The Derivative Definition Quotient Geometric Interpretation: Tangent Line Product and Quotient Rules Chain Rule Implicit Differentiation Chapter 3. Applications of Derivatives Critical Points First Derivative Test Concavity and Inflection Points Curve Sketching Optimization Theory Chapter 4. Integrals: Introduction and Techniques Antiderivatives The Definite Integral Terminology and Notation Fundamental Theorems of Calculus Trigonometric Integrals Partial Fraction Decomposition Chapter 5. Applications of Integration Area Between Curves Volume by Disks and Washers Volume by Cylindrical Shells Arc Length Surface Area of Revolution |
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